Continuous Semigroups of Holomorphic Self-maps of the Unit Disc (Springer Monographs in Mathematics) 3030367819, 9783030367817

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Springer Monographs in Mathematics

Filippo Bracci Manuel D. Contreras Santiago Díaz-Madrigal

Continuous Semigroups of Holomorphic Self-maps of the Unit Disc

Springer Monographs in Mathematics Editors-in-Chief Isabelle Gallagher, Paris, France Minhyong Kim, Oxford, UK Series Editors Sheldon Axler, San Francisco, USA Mark Braverman, Princeton, USA Maria Chudnovsky, Princeton, USA Tadahisa Funaki, Tokyo, Japan Sinan C. Güntürk, New York, USA Claude Le Bris, Marne la Vallée, France Pascal Massart, Orsay, France Alberto A. Pinto, Porto, Portugal Gabriella Pinzari, Padova, Italy Ken Ribet, Berkeley, USA René Schilling, Dresden, Germany Panagiotis Souganidis, Chicago, USA Endre Süli, Oxford, UK Shmuel Weinberger, Chicago, USA Boris Zilber, Oxford, UK

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Filippo Bracci Manuel D. Contreras Santiago Díaz-Madrigal •

•

Continuous Semigroups of Holomorphic Self-maps of the Unit Disc

123

Filippo Bracci Dipartimento di Matematica Università di Roma “Tor Vergata” Roma, Italy

Manuel D. Contreras Departamento de Matemática Aplicada II and IMUS Universidad de Sevilla Sevilla, Spain

Santiago Díaz-Madrigal Departamento de Matemática Aplicada II and IMUS Universidad de Sevilla Sevilla, Spain

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-030-36781-7 ISBN 978-3-030-36782-4 (eBook) https://doi.org/10.1007/978-3-030-36782-4 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

A Niccolò, che, da grande, prima vuole guidare una ruspa e poi fare matematica. Filippo A mis hijos, Carlos y Elena, por todo y por tanto. Manolo A Flora, Javi y Sara. Santi

Acknowledgements

It was a sunny and hot day in Nahariya some years ago when we started discussing the idea of writing a book about semigroups of holomorphic self-maps of the unit disc. Since the wonderful books on the subject by Marco Abate, Mark Elin, Simeon Reich, and David Shoikhet, there had been no sources in book form containing the various advances of the intervening years, and many colleagues seemed interested in having an updated complete reference source. We subsequently worked on the raw material which finally became the present book. During the years needed to see the “light at the end of the tunnel”, we profited from and very much enjoyed discussions with colleagues and friends. It is our pleasant duty to thank all of those who helped us. Special thanks are due to our friend Pavel “Pasha” Gumenyuk who gave us priceless comments, ideas, and constructive criticisms, besides offering much philosophical advice about the book. Certainly this book—and life—would have been very different without his help. It was our great privilege to have the opportunity of knowing and profiting from the experience, encouragement, and help of our friend Christian Pommerenke, from whom we learned a lot. We cannot forget our beloved friend Sasha Vasil’ev, whose constant encouragement was essential to us. Wherever you are now, we can imagine you are taking a look at the book with a cerveza in your hand! We wish to thank Hervé Gaussier, from whom we learned a lot about Gromov’s hyperbolicity theory. The hours and hours spent at the coffee bar in Seville discussing hyperbolic geometry led us to simplify many proofs in the book, as well as other aspects. Thanks are also owed to Andy Zimmer, whose enthusiasm and skill allowed us to understand much better part of the theory. We would also like to thank Marco Abate: besides promising to read the book, his work has always been a great inspiration for all of us. Leandro Arosio helped us with many interesting comments and clever ideas. Thanks!

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Acknowledgements

We further wish to thank the following great friends, collaborators, and masters, who, in one way or another, contributed a lot to help us: Dimitri Betsakos, Mark Elin, Pietro Poggi-Corradini, Simeon Reich, David Shoikhet, and Aristos Siskakis. We also thank Graziano Gentili for his constant support and friendship. Filippo wants to thank his parents, Renzo and Anna, for always being there, and his wife Ele for her patience (at least, sometimes), support, and love. Manolo wants to express his deepest gratitude to Mara José, his wife, for her constant support and encouragement, especially during the time dedicated to writing this book. Life would not be so beautiful without her. Thanks! Santi is thankful and deeply indebted to his wife Flora for her endless patience and love. Last but not least, we want to express our gratitude to the institutions that have supported us during these years of work: Departamento de Matemática Aplicada II and the Instituto de Matemáticas IMUS, Universidad de Sevilla; Dipartimento di Matematica, Università di Roma “Tor Vergata” (and the related MIUR Excellence Department Project MATH@TOV), and the ERC grant “HEVO”. Rome, Italy Seville, Spain October 2019

Filippo Bracci Manuel D. Contreras Santiago Díaz-Madrigal

Introduction

Continuous one-parameter semigroups of holomorphic self-maps of the unit disc D in the complex plane C have been a subject of study since early 1900s, both for their intrinsic interest in complex analysis and for applications. In recent years, there has been a lively development of the theory, and deep achievements about boundary behavior and dynamical aspects have been obtained. Various key results are contained in different research papers and the aim of this book is to give a systematic and unified account of the subject, from the very first definitions up to the latest results. The theory of local groups (or flows) is intrinsically related to the theory of differential equations. Given a vector field X defined and smooth on some open set U of Rn (or more generally on a manifold), and a point x0 2 U, the Cauchy problem
@xðtÞ @t

¼ XðxðtÞÞ; xð0Þ ¼ x0

has a unique smooth solution uðx0 ; Þ defined in some interval Ix0 of R containing 0 such that uðx0 ; 0Þ ¼ x0 . The map ðx; tÞ 7! uðx; tÞ is well-defined and smooth on V  J, where V is a neighborhood of x0 and J is a small interval containing 0. By the uniqueness of solutions of the Cauchy problem, uðx; t þ sÞ ¼ uðuðx; tÞ; sÞ for all s; t 2 J such that s þ t 2 J and uðx; tÞ 2 V. The family ðuð; tÞÞ is a one-parameter local group, the flow of the vector field X. If Ix0 ¼ R for every x0 , then the maps x 7! uðx; tÞ are diffeomorphisms of U for all t 2 R, and ðuð; tÞÞ is a one-parameter group. In other words, X defines a continuous action of the group R on U. On the other hand, if Ix0 contains ½0; þ 1Þ for every x0 2 U, then ðuð; tÞÞ is a one-parameter semigroup and ut :¼ uð; tÞ is an injective smooth map of U into U for each t  0. The semigroup equation, ut þ s ¼ ut  us , t; s  0 implies that un ¼ un 1 ¼ u1  . . .  u1 (n times composition of u1 with itself), and the behavior of the orbits of fun 1 g is strictly related to the analytic properties of the vector field X, which can be easier to understand. Nevertheless, it is often the case that a dynamical system is described by a single self-map f : U ! U, and one is interested in

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understanding the asymptotic behavior of the orbits ff n g. If f can be embedded into a continuous group (or a semigroup) of functions, the orbits of f can then be studied using vector fields. However, the embedding problem is usually very hard and not always solvable, not even locally. In the complex setting, Koenigs [91] was one of the first mathematicians to solve the local embedding problem, by finding solutions to the so-called Schröder equation. Namely, he proved that, given a holomorphic self-map f of the unit disc D fixing the origin and whose derivative f 0 ð0Þ ¼ ek (with k 2 C, Rek [ 0—and that is the only possibility unless f is linear, by the Schwarz Lemma), one can find a holomorphic function u : D ! C, locally invertible at 0 such that ðu  f ÞðzÞ ¼ ek uðzÞ for all z 2 D. Since z 7! ekt z, t 2 R, is a group of automorphisms of C, Koenigs’ result implies that f can be locally embedded into a continuous group. At the beginning of the twentieth century, Tricomi [124] dealt with problems which, translated into modern language, were related to the asymptotic behavior of continuous one-parameter semigroups of holomorphic self-maps of the unit disc. In 1923, Loewner [95] introduced what is nowadays called “Loewner theory” to tackle extremal problems in complex analysis. Such a theory, as developed in particular by Pommerenke [102], contains the germ of elliptic semigroups theory and relates semigroups to certain ordinary differential equations. Later on, in 1943, Kufarev [94] introduced an ordinary differential equation whose solutions are pretty much related to continuous non-elliptic semigroups. In 1939, Wolff [128] studied continuous iteration in the half-plane and proved a type of continuous Denjoy-Wolff theorem, which describes the asymptotic behavior of the trajectories of continuous one-parameter semigroups of holomorphic self-maps of the unit disc. In 1968, Karlin and McGregor [85, 86] studied the (global) embedding problem of the probability generating function of a simple discrete time Markov process into a continuous one-parameter semigroup of holomorphic self-maps of the unit disc. Such a process, introduced in 1875 by Galton and Watson [126] in connection with the probability of extinction of family names, can be formulated in mathematical terms by means of a discrete dynamical system given by a holomorphic self-map of the unit disc. The interest in the Galton-Watson branching process flourished when it was proposed as a basic model for many other physical processes, and this created a need to understand the theory of continuous one-parameter semigroups of holomorphic self-maps of the unit disc. In 1978, Berkson and Porta [11] studied continuous one-parameter semigroups of holomorphic self-maps of the unit disc in connection with composition operators. They proved the existence of the infinitesimal generator and what is now referred to as the Berkson-Porta Formula. In 1981, Heins [82] investigated continuous one-parameter semigroups of holomorphic self-maps of Riemann surfaces and proved the existence of the Koenigs function for semigroups of the unit disc. Since then, many papers and books focusing on different aspects of this theory have been written. We cite here the book by Abate [1], which mainly focuses on the asymptotic behavior of orbits of discrete dynamical systems and contains a chapter

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about semigroups, the book by Shoikhet [118] which is devoted to the so-called generation theory and relations between semigroups and geometric function theory, the book by Reich and Shoikhet [111], which contains extension of the theory to Banach spaces, and the book by Elin and Shoikhet [65] where the emphasis is on linearization models of semigroups. In recent decades, applications of the theory of continuous one-parameter semigroups of holomorphic self-maps of the unit disc appeared in mathematics and other sciences. For instance, apart from Galton-Watson evolution type models which we already mentioned, in the paper [96] the growth of leaves of the Arabidopsis plant is modeled using semigroups of linear fractional maps of the unit disc. A general Loewner theory, which is, in fact, the non-autonomous version of the theory of continuous one-parameter semigroups, has been introduced in [26, 49, 28]. Semigroups of holomorphic self-maps are also strictly related to operator theory, via the so-called composition operators. Indeed, every one-parameter semigroup of holomorphic self-maps of the unit disc gives rise to a semigroup of composition operators on Hardy spaces of holomorphic functions over the disc, or on more general functional spaces. This connection between composition operators and semigroups allows one to translate functional analytical questions (such as spectral properties, operator ideal properties, compactness, cyclicity, and so on) into corresponding dynamical questions for semigroups (see, e.g., [120, 92, 121, 4, 74]). The theory of continuous one-parameter semigroups has also been studied for holomorphic self-maps of complex spaces different from the unit disc. However, in the category of Riemann surfaces, the theory essentially makes sense only for the unit disc. In fact, Heins [82, 1] proved that Riemann surfaces with non-Abelian fundamental group admit no non-trivial continuous one-parameter semigroups of holomorphic self-maps, while for non-hyperbolic Riemann surfaces every continuous one-parameter semigroup is a group of simple form. Finally, in the case of an annulus, every semigroup is a group of rotations, and, in the case of the punctured disc, every semigroup is just the restriction of a semigroup of the unit disc which fixes the origin. In higher dimensional complex manifolds, and in infinitely dimensional complex Banach spaces, there are contributions to the theory from many authors, although presently there is not a well-outlined and complete theory as in the unit disc. We refer the reader to [23, 27, 1, 111] and references therein. The beauty of the theory lies, in fact, in the interplay among dynamical properties of semigroups, analytical properties of infinitesimal generators, and geometrical properties of Koenigs functions. The main objective of this book is to study these relations. The book includes precise descriptions of the behavior of trajectories, backward orbits, petals, and boundary behavior in general, aiming to give a rather complete picture of all interesting phenomena that occur. In order to fulfill this task, we choose to introduce a new point of view, which is mainly based on the intrinsic dynamical aspects of semigroups in relation with the Gromov hyperbolicity theory, harmonic measure theory, and Carathéodory prime ends topology.

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Content of the book. The book is divided into two parts. Part I contains all preliminary results, stated with detailed proofs and suitably adapted to our needs, and introduces the core of the book, which is Part II. We briefly describe the content of Part II. A continuous one-parameter semigroup of holomorphic self-maps of the unit disc, or, briefly from now on, a semigroup in D, is a family ð/t Þ depending on a parameter t 2 ½0; þ 1Þ such that /t : D ! D is holomorphic for all t  0, /s þ t ¼ /s  /t for s; t 2 ½0; þ 1Þ, /0 ¼ idD and ½0; þ 1Þ 3 t 7! /t is continuous with respect to the Euclidean topology of ½0; þ 1Þ and the topology of uniform convergence on compacta in the space of holomorphic maps of D. The first main result is the continuous Denjoy-Wolff theorem, which says that if ð/t Þ is a semigroup in D which is not a group of hyperbolic rotations, then there exists a unique point s 2 D such that limt! þ 1 /t ðzÞ ¼ s for all z 2 D. The point s is called the Denjoy-Wolff point of ð/t Þ. If s 2 D, then /t ðsÞ ¼ s and /0t ðsÞ ¼ ekt for all t  0, where k 2 C is a number such that Re k [ 0. If s 2 @D, then /t has non-tangential limit s at s for all t  0, so that s is, in a sense, a fixed point of the semigroup, and /0t has non-tangential limit at s given by ekt , for some k  0. Thus we have a first important classification of semigroups in D: elliptic if s 2 D, hyperbolic if s 2 @D and k [ 0, and parabolic if s 2 @D and k ¼ 0. The number k is intrinsically related to the dynamics of the semigroup. Indeed, in the non-elliptic case, it measures the rate of divergence of the trajectories of the semigroup in terms of the hyperbolic distance of D. Next, we exploit an abstract construction, known as basin of attraction or space of orbits, which allows us to define a universal holomorphic model, i.e., roughly speaking, a holomorphic conjugation of ð/t Þ to a group ðwt Þ of D or C, in such a way that any holomorphic conjugation factorizes through this model. This idea is somehow classical in dynamical systems and was used by Cowen [56] in order to construct linear models for discrete iteration in the unit disc, and pushed forward in [8] and by Arosio in [5, 6]. Since this model is defined using properties of the hyperbolic metric, it allows us to move easily any intrinsic dynamical information of ð/t Þ to ðwt Þ and conversely. More concretely, the model’s construction gives an essentially unique univalent function h : D ! C, which we call the Koenigs function of the semigroup ð/t Þ, in such a way that either ðh  /t ÞðzÞ ¼ ekt hðzÞ in the elliptic case, or ðh  /t ÞðzÞ ¼ hðzÞ þ it for all z 2 D and t  0 in the non-elliptic case. In the elliptic case, hðDÞ is a k-spirallike domain with respect to 0. In the hyperbolic case, hðDÞ is a domain starlike at infinity which is contained in a strip of width p=k and cannot be contained in any smaller strip. In the parabolic case, hðDÞ is a domain starlike at infinity and either is contained in a half-plane with boundary parallel to the imaginary axis (in the so-called positive hyperbolic step case) or cannot be contained in any such half-plane (in the so-called zero hyperbolic step case). Moreover, due to the universality of the model, any other holomorphic map g : D ! C (or more generally any other complex manifold) which conjugates ð/t Þ to a group of automorphisms can be factorized through h.

Introduction

xiii

The importance of the models relies on the possibility of translating all dynamical questions on ð/t Þ to questions on geometrical properties of hðDÞ. Koenigs functions can also be obtained directly from the differential equation defining the infinitesimal generator of a semigroup, as was done by Siskakis [120], or from iterative processes like in the discrete case: see Valiron [125], Pommerenke [104], and Baker-Pommerenke [9]. However, these methods do not provide immediately the universality of the model and the essential uniqueness of the Koenigs function. A first basic use of the model is the construction of the so-called infinitesimal generators, relating semigroups with differential equations. This can be done easily from models. For instance, in the elliptic case, ð/t Þ is conjugated via h to the group f 7! ekt f. Fixing f 2 C, the curve y : ½0; þ 1Þ 3 t 7! ekt f is the positive-time solution of the Cauchy problem @yðtÞ @t ¼ HðyðtÞÞ, yð0Þ ¼ f, where HðwÞ :¼ kw is a holomorphic vector field1 in C. Since hðDÞ is invariant under the map f 7! ekt f for t  0, the curve xðtÞ :¼ h1 ðyðtÞÞ ¼ /t ðh1 ðfÞÞ, t 2 ½0; þ 1Þ is contained in D for all t  0 and it is the positive-time solution of the Cauchy problem @xðtÞ hðxðtÞÞ ¼ GðxðtÞÞ :¼ dðh1 ÞhðxðtÞÞ ðHÞ ¼ k 0 ; @t h ðxðtÞÞ

xð0Þ ¼ h1 ðfÞ:

In this way, to any semigroup in D one associates a unique holomorphic vector field G on D, called the infinitesimal generator of ð/t Þ such that for all t  0 and z 2 D, @/t ðzÞ ¼ Gð/t ðzÞÞ: @t Holomorphic vector fields G on D which define semigroups (that is, for which the associated Cauchy problem admits a solution defined for every positive time regardless of the initial data) can be characterized by means of various formulas and inequalities. The basic one is the so-called Berkson-Porta Formula, which establishes that GðzÞ ¼ ðz  sÞðsz  1ÞpðzÞ; where p is a holomorphic function with non-negative real part and s 2 D. Such a formula can be obtained directly from the model. For example, in the elliptic case, assuming that hð0Þ ¼ 0, GðzÞ ¼ k hhðzÞ 0 ðzÞ. Since hðDÞ is k-spirallike with respect to   hðzÞ 0, this shows that Re k zh0 ðzÞ \0 and GðzÞ ¼ zpðzÞ for some holomorphic function p with positive real part. 1 @ Formally HðwÞ ¼ kw @w . However, since the tangent bundle of C is globally trivial and can be @ naturally identified with C, in this book we omit writing the global base @w .

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Introduction

We also present several other representation formulas of infinitesimal generators, including Abate’s Formula, Aharonov-Elin-Reich-Shoikhet’s Formula, and other characterizations via hyperbolic distance and Euclidean inequalities. To push the theory further and obtain information on the boundary behavior of semigroups, we exploit Carathéodory’s prime ends topology, which, roughly speaking, allows one to extend every univalent function f : D ! C as a homeomorphism from D to the closure of f ðDÞ in a suitable topology. The natural idea of using Carathéodory’s prime end theory in this context first appeared in [47] and was then strongly developed by Gumenyuk [81]. Using this tool, we study impression and principal parts of prime ends of Koenigs functions, proving that Koenigs functions as well as every iterate of a semigroup have non-tangential limit at every point of @D. This allows us to extend both the semigroup equation and the model functional equation up to the boundary (in the sense of non-tangential limit). It also allows us to define boundary fixed points for semigroups. Surprisingly enough, boundary fixed points of semigroups in general do not correspond to zeros of the associated infinitesimal generators. However, there are certain “regular” fixed points which do. At every regular fixed point, other than the Denjoy-Wolff point, /0t has a non-tangential limit equal to elt for some l\0, t  0. These points are the repelling fixed points of a semigroup and the number l is the repelling spectral value of the semigroup at the corresponding fixed point. We characterize repelling fixed points in terms of infinitesimal generators and obtain several formulas which generalize the Berkson-Porta formula. Moreover, taking a point of view inspired by Goryaĭnov [78], we give constraints for the repelling spectral values. From a geometrical point of view, repelling fixed points are in one-to-one correspondence with (hyperbolic) petals, that is, simply connected domains contained in D which are completely invariant for the semigroup. The restriction of a semigroup to a petal is a hyperbolic group, whose divergence rate is exactly the opposite of the repelling spectral value, l. Exploiting a construction of Poggi-Corradini [98] for discrete dynamics, we show that every petal is the image of a univalent map from the unit disc (which, except in a very special case, is always a homeomorphism up to the boundary) which conjugates a hyperbolic group of D with divergence rate l to the restriction of the semigroup to the petal—what we call a pre-model. This allows us to use the model to geometrically characterize petals in terms of the image of the Koenigs function h. For instance, in the non-elliptic case, a petal with repelling spectral value l is a maximal strip in hðDÞ of width p=l and, conversely, every maximal strip contained in hðDÞ gives rise to a petal. Since every iterate of a semigroup extends non-tangentially to the boundary, we can consider the non-tangential limit of a semigroup at each boundary point. We show that the continuous Denjoy-Wolff theorem holds (in the non-tangential limits sense) at every boundary point which is not fixed. Next, one can consider contact arcs, namely, open arcs on @D whose non-tangential image under the semigroup is still on @D, at least for short time. It turns out that such contact arcs can be

Introduction

xv

characterized in terms of holomorphic extension of the associated infinitesimal generator, and in terms of geometrical properties of the associated Koenigs function. In particular, both the semigroup and its Koenigs function extend holomorphically through contact arcs. Moreover, every super-repelling fixed point (that is, a fixed point which is not regular) which starts a contact arc gives rise to a boundary critical point of the infinitesimal generator. Afterward, we turn our attention to other special boundary points: those which give rise to “regular” poles for infinitesimal generators. These points can be characterized in terms of analytical behavior of semigroups and geometrical properties of Koenigs functions. The understanding of the dynamical behavior of trajectories of semigroups to the Denjoy-Wolff point in the non-elliptic cases is another deep and important issue addressed in this book. In order to achieve this aim we exploit harmonic measure theory and Gromov’s hyperbolicity theory. More concretely, we define the total speed at t  0 of a (non-elliptic) semigroup ð/t Þ as the hyperbolic distance between 0 and /t ð0Þ. Such a speed is essentially the sum of the orthogonal speed and the tangential speed of the semigroup. The tangential speed is the hyperbolic distance between /t ð0Þ and the segment L joining 0 to the Denjoy-Wolff point of ð/t Þ (which is, in fact, a hyperbolic geodesic in D). This speed measures how far a trajectory is from converging non-tangentially at the Denjoy-Wolff point. If xt is the point in L which realizes the distance between L and /t ð0Þ, the orthogonal speed is just the hyperbolic distance between 0 and xt . The understanding of the asymptotic behavior of these three speeds is the so-called “rate of convergence problem”. It turns out that the orthogonal speed can be investigated quite well using harmonic measure theory, as was firstly shown by Betsakos [12, 13], while Gromov’s hyperbolicity theory (in particular, the use of quasi-geodesics and the corresponding Shadowing Lemma) works well to understand the tangential speed [34, 35]. Using these two techniques, one can obtain lower and upper estimates for the orthogonal speed of semigroups, as described in the book. A strictly related problem is the “slope problem”, namely, the understanding of the slope of landing of the trajectories of a semigroup at the Denjoy-Wolff point. In the book we mainly concentrate on solving this problem by looking at the geometry of the image of the Koenigs function of a semigroup. It turns out that the convergence of the trajectories of a (non-elliptic) semigroup to the Denjoy-Wolff point is non-tangential if and only if the image X of the Koenigs map is “almost symmetric” with respect to any vertical line which has non-empty intersection with X. This allows us also to construct examples of semigroups whose trajectories land oscillating as much as wanted at the Denjoy-Wolff point. According to the type (hyperbolic, parabolic of positive hyperbolic step, parabolic of zero hyperbolic step) more precise results are also presented. Finally, we consider topological invariants of semigroups, namely, those objects (divergence rate, fixed points, contact arcs, and so on) which are invariant under topological conjugation.

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How to read this book. This book is intended both as a reference source for researchers and as an introductory book for beginners with a (undergraduate) background in real and complex analysis. For this purpose, the book is self-contained and all non-standard (and, mostly, all standard) results are proved in detail. The core of the book is Part II, starting at Chap. 8. The reader, especially the expert, could start reading from this chapter on and go back to Part I when needed. However, we advise the reader to be familiar with the material of Chaps. 1–3 before moving to Part II. For instance, a first course on the theory of continuous one-parameter semigroups of holomorphic maps in the unit disc could be done using Chaps. 1, 2, 3, 8, 9, and 10. Chapters 11 to 15 and Chap. 18 are also based on Chaps. 4 and 5, while Chaps. 16 and 17 strongly require the material contained in Chaps. 6 and 7. The literature on the subject is quite wide. Nonetheless, we have tried to be as precise as possible in attributing correct credits to the authors of the various results presented in the book. Any missing reference is to be attributed solely to our lack of knowledge.

Contents

Part I 1

2

3

Preliminaries

Hyperbolic Geometry and Iteration Theory . . . . . . . . . . . . . 1.1 Riemann Surfaces, Riemann Sphere and the Group of Möbius Transformations . . . . . . . . . . . . . . . . . . . . . . 1.2 The Schwarz Lemma and the Automorphism Group of the Unit Disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Hyperbolic Metric and Hyperbolic Distance . . . . . . . . . . 1.4 Horocycles and Julia’s Lemma . . . . . . . . . . . . . . . . . . . 1.5 Non-Tangential Limits and Lindelöf’s Theorem . . . . . . . 1.6 Poisson Integral and Fatou’s Theorem . . . . . . . . . . . . . . 1.7 Angular Derivatives and Julia-Wolff-Carathéodory’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Iteration in the Unit Disc and the Denjoy-Wolff Theorem 1.9 Boundary Regular Contact Points . . . . . . . . . . . . . . . . . 1.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holomorphic Functions with Non-Negative Real Part . . 2.1 The Herglotz Representation Formula . . . . . . . . . . 2.2 Growth Estimates for Functions with Non-Negative Real Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Finite Contact Points of Holomorphic Functions with Non-Negative Real Part . . . . . . . . . . . . . . . . . 2.4 Boundary Behavior . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Univalent Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Univalent Functions and Simply Connected Domains 3.2 No Koebe Arcs Theorem . . . . . . . . . . . . . . . . . . . . . 3.3 Boundary Behavior . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Distortion Theorems . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

3.5 3.6

Convergence of Univalent Mappings . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

Carathéodory’s Prime Ends Theory . . . 4.1 Prime Ends . . . . . . . . . . . . . . . . . . 4.2 The Carathéodory Topology . . . . . 4.3 Carathéodory Extension Theorems . 4.4 Cluster Sets at Boundary Points . . . 4.5 Notes . . . . . . . . . . . . . . . . . . . . . .

5

Hyperbolic Geometry in Simply Connected Domains . . . . . 5.1 Hyperbolic Metric and Geodesics in Simply Connected Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Estimates for the Hyperbolic Metric . . . . . . . . . . . . . . . 5.3 Estimates for the Hyperbolic Distance . . . . . . . . . . . . . 5.4 Hyperbolic Geometry in the Half-Plane . . . . . . . . . . . . 5.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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91 91 100 104 108 115

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117 127 128 129 132

Quasi-Geodesics and Localization . . . . . . . . . . . . . . . . . . . . . . . 6.1 Symmetric Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Hyperbolic Sectors and Non-Tangential Convergence . . . . . 6.3 Quasi-Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Orthogonal Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Hyperbolic Projections, Tangential and Orthogonal Speeds of Curves in the Disc . . . . . . . . . . . . . . . . . . . . . . . 6.6 Localization of Hyperbolic Metric and Hyperbolic Distance 6.7 Hyperbolic Geometry in the Strip . . . . . . . . . . . . . . . . . . . 6.8 Some Localization Results . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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133 133 134 137 147

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155 160 162 164 169

Harmonic Measures and Bloch Functions . . . . . . . . . . . 7.1 Harmonic Measures in the Unit Disc . . . . . . . . . . . 7.2 Harmonic Measures in Simply Connected Domains 7.3 Bloch Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Diameter Distorsion for Univalent Functions . . . . . 7.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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171 171 177 187 198 201

Semigroups of Holomorphic Functions . . . . . . . . . . . . . . . . 8.1 Semigroups in the Unit Disc . . . . . . . . . . . . . . . . . . . . 8.2 Groups in the Unit Disc . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Continuous Version of the Denjoy-Wolff Theorem . 8.4 Semigroups in Riemann Surfaces . . . . . . . . . . . . . . . . .

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205 205 214 218 222

Part II 8

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Semigroups

Contents

8.5 8.6

xix

Semigroups of Linear Fractional Maps . . . . . . . . . . . . . . . . . . . 225 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 . . . . . . . . . .

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229 230 235 244 251 260 264 267 268 271

10 Infinitesimal Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Infinitesimal Generators and the Berkson-Porta Formula 10.2 Characterizations of Infinitesimal Generators . . . . . . . . 10.3 Infinitesimal Generators of Groups . . . . . . . . . . . . . . . . 10.4 Infinitesimal Generators of Semigroups of Linear Fractional Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Convergence of Infinitesimal Generators . . . . . . . . . . . 10.6 The Product Formula . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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273 273 282 289

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292 297 302 305

11 Extension to the Boundary . . . . . . . . . . . . . . . . . . . . 11.1 Prime Ends and Koenigs Functions . . . . . . . . . . 11.2 Boundary Extensions of Semigroups . . . . . . . . . 11.3 Continuous Boundary Extensions of Semigroups 11.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

Models and Koenigs Functions . . . . . . . . . . . . . . . . . . . . 9.1 The Divergence Rate and Hyperbolic Steps . . . . . . . 9.2 Holomorphic Models . . . . . . . . . . . . . . . . . . . . . . . 9.3 Canonical Models and Koenigs Functions . . . . . . . . 9.4 Basic Properties of Koenigs Functions . . . . . . . . . . . 9.5 Semigroups of Linear Fractional Maps . . . . . . . . . . . 9.6 Non-Canonical Holomorphic Semi-Models . . . . . . . . 9.7 Holomorphic Conjugations and Holomorphic Models 9.8 Topological Models and Topological Conjugations . . 9.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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307 307 316 318 325

12 Boundary Fixed Points and Infinitesimal Generators . . . 12.1 Inner and Boundary Fixed Points . . . . . . . . . . . . . . . 12.2 Boundary Fixed Points and Infinitesimal Generators . 12.3 Synchronization Formulas . . . . . . . . . . . . . . . . . . . . 12.4 Non-Regular Critical Points Versus Super-Repelling Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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13 Fixed 13.1 13.2 13.3 13.4 13.5 13.6

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Points, Backward Invariant Sets and Petals . . . . . . Backward Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pre-Models at Repelling Fixed Points . . . . . . . . . . . . Maximal Invariant Curves . . . . . . . . . . . . . . . . . . . . . Petals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Petals and the Geometry of Koenigs Functions . . . . . . Analytic Properties of Koenigs Functions at Boundary Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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13.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 13.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 14 Contact Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 The Boundary Denjoy-Wolff Theorem . . . . . . . . . . . . . . 14.2 Maximal Contact Arcs . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Infinitesimal Generators and Maximal Contact Arcs . . . . 14.4 Super-Repelling Fixed Points and Maximal Contact Arcs 14.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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407 407 409 417 421 425 427

15 Poles 15.1 15.2 15.3 15.4 15.5 15.6

of the Infinitesimal Generators . . . . . . . . . . . . . . . . Regular Poles and b-Points . . . . . . . . . . . . . . . . . . . Tips of Isolated Radial and Spiral Slits . . . . . . . . . . Measure-Theoretic Characterization of Regular Poles Dual Infinitesimal Generators . . . . . . . . . . . . . . . . . Radial Multi-Slits Semigroups . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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16 Rate 16.1 16.2 16.3 16.4 16.5

of Convergence at the Denjoy-Wolff Point . . . . . . . . . . . . Speeds of Convergence of Orbits . . . . . . . . . . . . . . . . . . . . Total Speed of Convergence . . . . . . . . . . . . . . . . . . . . . . . Orthogonal Speed of Convergence of Parabolic Semigroups Trajectories on the Boundary . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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17 Slopes of Orbits at the Denjoy-Wolff Point . . . . . . . . . . . . . . 17.1 Euclidean Geometry of Domains Starlike at Infinity . . . . 17.2 Quasi-Geodesics in Starlike Domains at Infinity . . . . . . . 17.3 Convergence to the Denjoy-Wolff Point for Non-Elliptic Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 The Slope of Hyperbolic Semigroups . . . . . . . . . . . . . . . 17.5 The Slope of Parabolic Semigroups . . . . . . . . . . . . . . . . 17.6 Parabolic Semigroups of Zero Hyperbolic Step with Prescribed Slopes . . . . . . . . . . . . . . . . . . . . . . . . . 17.7 The Shift of Non-Elliptic Semigroups . . . . . . . . . . . . . . 17.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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18 Topological Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Extension of Topological Conjugation for Non-Elliptic Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Topological Invariants for Non-Elliptic Semigroups . . . . . 18.3 Exceptional Maximal Contact Arcs and the Denjoy-Wolff Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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18.4 Elliptic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 18.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

Notation

All non-standard (and often standard) notations are introduced in the book along with the development of the theory. There are however certain standard notations that we use throughout the book without introducing them explicitly along the way, which we briefly introduce here. The set2 N :¼ f1; 2; 3; . . .g. When we need to consider the set N together f0g, we use the notation N0 :¼ f0g [ N. If we say that n is a natural number, the reader can harmlessly choose either n 2 N or n 2 N0 according to her/his own taste. The set of integer numbers is denoted by Z :¼ f0; 1; 2; . . .g, the set of rational numbers is denoted by Q, the set of real numbers is R and C is the set of complex numbers. For a complex number z ¼ x þ iy, x; y 2 R, we denote by Re z :¼ x the real part of z, and by Im z :¼ y the imaginary part. As usual, the modulus of z is pffiffiffiffiffiffiffiffiffiffiffiffiffiffi jzj :¼ x2 þ y2 . We denote by z :¼ x  iy the complex conjugate of z. For z 2 Cnf0g one can write z ¼ qeih with q [ 0 and h 2 ½p; pÞ. The (principal) argument of z is ArgðzÞ :¼ h: In order to simplify the notation, given a sequence of complex numbers whose elements are labeled by z0 ; z1 ; z2 ; . . ., we denote such a sequence by fzn gn2N0 , or simply by fzn g when there is no risk of confusion. Unless otherwise specified, the topology on C is the Euclidean topology. All the previous sets are endowed with the topology given by the set theoretical inclusions N  N0  Z  Q  R  C. For a sequence of natural numbers fmn g we simply write mn ! 1 to mean limn!1 mn ¼ 1. Such a notation is used sometimes also for sequences of real numbers, for instance, if ftn g is a sequence of real numbers, we write tn ! þ 1 if limn!1 tn ¼ þ 1 in the Euclidean topology. 2

Whether 0 has to be considered a natural number or not is a large source of debate, even among the authors of the present book. We decided not to enter into this question, but just name the sets we are using.

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Notation

Given z0 2 C and r 2 R, r [ 0, the disc of center z0 and radius r is ff 2 C : jf  z0 j\rg. The unit disc, i.e., the disc of center 0 and radius 1 is denoted by D. The circle of center z0 and radius r [ 0 is just the boundary of the disc with the same center and same radius, that is, ff 2 C : jf  z0 j ¼ rg. An open arc of a circle C of center z0 2 C and radius r [ 0 with end points p; q 2 C, (possibly p ¼ q), is a connected component of Cnfp; qg. The closure of an open arc is a closed arc. When we do not need to specify open or closed arc, we simply say arcs. By definition, a domain in C is a connected open set. A simply connected domain is a domain whose fundamental group is trivial.

Symbols

þz Cr ðzÞ :¼ rrz , Cayley transform 2

Eðr; RÞ :¼ fz 2 D : jrzj \Rg, horocycle in the unit disc 1jzj2 E H ð1; RÞ :¼ fw 2 H : Re w [ Rg, horocycle at infinity in the right half-plane EzX0 ðy; RÞ, horocycle in a simply connected domain 1

L½z :¼ fw 2 C : Re w ¼ Re zg Sðr; RÞ :¼ fz 2 D : jrzj 1jzj \Rg, Stolz region SX ðc; RÞ :¼ fw 2 X : kX ðw; cðða; þ1ÞÞÞ\Rg, hyperbolic sector around a geodesic c TðrÞ, the life-time of r 2 @D under the action of a semigroup az Ta ðzÞ :¼ 1az , canonical automorphism of D for jaj\1

U open set in the Carathéodory topology Vða; r0 Þ :¼ fqeih : q [ r0 ; jhj\ag, horizontal sector VðaÞ :¼ Vða; 0Þ ¼ fqeih : q [ 0; jhj\ag Wða; bÞ :¼ fqeih : q [ 0; a\h\bg ArgðzÞ, principal argument of a complex number Argk ðwÞ :¼ h; the k-spirallike argument of w ¼ ekt þ ih C1 , the Riemann sphere D :¼ fz 2 C : jzj\1g, unit disc in C Cðc; bÞ :¼ fp 2 C1 : 9ftn g  ½a; bÞ : limn!1 tn ¼ b; limn!1 cðtn Þ ¼ pg, cluster set of a curve c GenðDÞ, set of infinitesimal generators H :¼ fw 2 C : Re w [ 0g, the right half-plane H :¼ fz 2 C : Re z\0g, the left half-plane HolðS1 ; S2 Þ, the set of all holomorphic functions from S1 to S2 HolðD; CÞ, space of holomorphic functions in D N :¼ f1; 2; 3; . . .g N0 :¼ f0g [ N Xa;b;R :¼ C n fz 2 C : Re z 2 fa; a þ Rg; Im z  bg

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Symbols

IðxÞ; impression of the prime end x PðxÞ, the principal part of the prime end x PðGÞ, the set of regular poles of the infinitesimal generator G PC ðGÞ, the set of regular poles of mass C of the infinitesimal generator G k  k1 , L1 -norm of integrable functions k  kB , semi-norm of a Bloch functions a/ ðrÞ :¼ lim inf z!r 1j/ðzÞj 1jzj for / : D ! D, boundary dilation coefficient vA , the characteristic function of a set A þ ~ dX;p ðtÞ :¼ inffjz  ðp þ itÞj : Re z  Re p; z 2 C n Xg, right distance in X  ~ d ðtÞ :¼ inffjz  ðp þ itÞj : Re z  Re p; z 2 C n Xg, left distance in X X;p

~ d X;p ðtÞ :¼ minfdX;p ðtÞ; tg, averaged sides distances in X dX ðzÞ :¼ inf w2CnX jz  wj, the Euclidean distance from the boundary Rt ‘X ðc; ½s; t Þ :¼ s jX ðcðrÞ; c0 ðrÞÞdr, hyperbolic length of a curve in a simply connected domain diam S ðKÞ :¼ supz;w2K dS ðz; wÞ, spherical diameter of the set K diam E ðHÞ :¼ supfjz  wj : z; w 2 Hg, Euclidean diameter of the set H jvj jD ðz; vÞ :¼ 1jzj 2 , the hyperbolic norm of v at z in the unit disc jX ðz; ðv; wÞÞ, the hyperbolic metric of a simply connected domain X jX ðz; vÞ, the hyperbolic norm of v at z in a simply connected domain X C, the set of complex numbers Q, the set of rational numbers R, the set of real numbers Z, the set of integer numbers Kp :¼ C n ff 2 C : Re f ¼ Re p; Im f  Im pg, the Koebe domain AreaðXÞ, area of the set X Slope½c; r :¼ CðArgð1  rcðtÞÞ; bÞ, the slope of the curve c at b Spir½l; 2a; h0 :¼ fetl þ ih : t 2 R; jh  h0 j\ag, a l-spirallike sector spirk ½c :¼ feks c : s 2 Rg [ f0g [ f1g, spiral passing through the point c lðz; A; XÞ, harmonic measure of A at z 2 X relative to X þ jTw ðzÞj , the hyperbolic distance in the unit disc Xðz; wÞ :¼ 12 log 11jT w ðzÞj 1

X , the closure of a set X in C1 @C X, the Carathéodory boundary of a simply connected domain X @1 X, boundary of a domain in C1 /n , n-th iterate of / pc ðzÞ, hyperbolic projection of a point z onto a geodesic c Sq :¼ ff 2 C : 0\Re f\qg, the strip of width q [ 0 S :¼ S1 ¼ ff 2 C : 0\Re f\1g SM R :¼ ff 2 SR : Im f [ Mg, the semi-strip of width R [ 0 and height M 2 R xr , the prime end associated with r 2 @D b :¼ X [ @C X X

Symbols

xxvii

fVn g basis of open neighborhoods of a prime end given by interior parts of null chains cX ð/t Þ :¼ lims!þ1 kX ð/ssðzÞ;zÞ, divergence rate of the semigroup ð/t Þ dS ðz; wÞ, the spherical distance in C1 kS ðz; wÞ, hyperbolic distance on a Riemann surface S p 4 q, orientation on a contact arc su ð/t ; zÞ :¼ limr!þ1 kX ð/r ðzÞ; /r þ u ðzÞÞ, the u-th hyperbolic step 2

1jzj ur ðzÞ :¼  jrzj 2 , Poisson kernel

vðtÞ :¼ Xð0; /t ð0ÞÞ, the total speed of a non-elliptic semigroup vT ðtÞ :¼ Xð/t ð0Þ; pc ð/t ð0ÞÞÞ, the tangential speed of a non-elliptic semigroup vo ðtÞ :¼ Xð0; pc ð/t ð0ÞÞÞ, the orthogonal speed of a non-elliptic semigroup vTX;z0 ðg; tÞ, tangential speed of a curve g in a simply connected domain voX;z0 ðg; tÞ, orthogonal speed of a curve g in a simply connected domain Bb ðf Þ, the set of all b-points of a holomorphic map f AutðDÞ :¼ {T : D ! D such that T is a biholomorphism} LFMðDÞ :¼ fF : C1 ! C1 : F is a Möbius transformation, FðDÞ  Dg idS , identity map in S

Part I

Preliminaries

Chapter 1

Hyperbolic Geometry and Iteration Theory

In this chapter we introduce some basic tools necessary for our study. We start recalling the concept of Riemann surfaces, focus mainly on the geometry of the unit disc, the complex plane and the Riemann sphere. Next, from Schwarz’s Lemma, we define the hyperbolic metric and hyperbolic distance of the unit disc, and extend, à la Kobayashi, the concept of hyperbolic distance to Riemann surfaces. We turn then our attention to the analytical and dynamical properties of holomorphic self-maps of the unit disc. We introduce the notion of horocycles and Stolz’s regions, and we prove the Lindelöf Theorem, which allows one to infer the existence of non-tangential limits provided the limit along some curve exists. Then we prove Julia’s Lemma and the Julia-Wolff-Carathéodory Theorem, which can be seen as boundary version of the Schwarz Lemma. With those tools at hand, we consider iteration of holomorphic self-maps of the unit disc, and prove the Denjoy-Wolff Theorem, which says that, except trivial cases, the orbits of a holomorphic self-map of the unit disc converge to a same point on the closed unit disc. Finally, we discuss boundary fixed points (and, more generally, boundary contact points) of holomorphic self-maps of the unit disc when no continuous extension to the boundary is assumed.

1.1 Riemann Surfaces, Riemann Sphere and the Group of Möbius Transformations Although we are mainly interested in studying maps from the unit disc into itself, in some of our constructions we need to deal with Riemann surfaces and with the geometry of the Riemann sphere. Therefore we briefly recall the definition and basic properties of these objects in this section.

© Springer Nature Switzerland AG 2020 F. Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4_1

3

4

1 Hyperbolic Geometry and Iteration Theory

Definition 1.1.1 A connected, Hausdorff, second countable topological space S is called a Riemann surface if there exist an open covering {Uα } of S, continuous maps ψα : Uα → C such that ψα (Uα ) is an open subset of C and ψα : Uα → ψα (Uα ) is a homeomorphism and such that for every α, β with Uα ∩ Uβ = ∅ the map ψα ◦ ψβ−1 : ψβ (Uα ∩ Uβ ) → ψα (Uα ∩ Uβ ) is holomorphic. The family {Uα , ψα } is called a holomorphic atlas for S and ψα is a holomorphic chart of S on Uα . A continuous map f : S1 → S2 between two Riemann surfaces S1 and S2 is holomorphic if for every p ∈ S1 there exist a holomorphic chart (U, ψ) of S1 with p ∈ U and a holomorphic chart (V, η) of S2 with f ( p) ∈ V such that the function η ◦ f ◦ ψ −1 : ψ(U ∩ f −1 (V )) → η(V ) is holomorphic. If S1 , S2 are Riemann surfaces, we denote by Hol(S1 , S2 ) the set of all holomorphic maps from S1 to S2 . A holomorphic map f : S1 → S2 is a biholomorphism if it admits a holomorphic inverse. This is equivalent to say that f is holomorphic, injective and surjective. A holomorphic and injective map is also called univalent. Every open subset U ⊆ C is a Riemann surface, with an atlas composed by only one chart: (U, idU ), where idU denotes the identity map in U . In particular, the unit disc D := {z ∈ C : |z| < 1} and C are Riemann surfaces. Another important example of a Riemann surface is provided by the Riemann sphere C∞ . Its construction is done as follows. On C2 \ {(0, 0)} define the following equivalence relation: (w1 , w2 ) ∼ (w1 , w2 ) if w1 w2 − w2 w1 = 0. Let C∞ := (C2 \ {(0, 0)})/ ∼ be the set of equivalence classes, endowed with the natural quotient topology. Let [w1 : w2 ] denote the equivalence classes of (w1 , w2 ) ∈ C2 \ {(0, 0)}. Let π : C2 \ {(0, 0)} → C∞ be the surjective map defined by π((w1 , w2 )) := [w1 : w2 ]. Then U ⊂ C∞ is open if and only if π −1 (U ) is open in C2 \ {(0, 0)}. It is easy to see that C∞ is a connected, simply connected, Hausdorff, second countable topological space, and it is compact. In fact, using the stereographic projection, one can prove that C∞ is homeomorphic to the Euclidean sphere {(x1 , x2 , x3 ) ∈ R3 : |x1 |2 + |x2 |2 + |x3 |2 = 1}. We endow C∞ with a Riemann surface structure in the following way. Let U1 := {[w1 : w2 ] ∈ C∞ : w1 = 0} and U2 := {[w1 : w2 ] ∈ C∞ : w2 = 0}. The two sets U1 , U2 are open and C∞ = U1 ∪ {[0 : 1]} = {[1 : 0]} ∪ U2 = U1 ∪ U2 . Therefore, {U1 , U2 } is an open covering of C∞ . Let ψ1 : U1 → C be given by ψ1 ([w1 : w2 ]) := ww21 and ψ2 : U2 → C be given by ψ2 ([w1 : w2 ]) := ww21 . It is easy to check that ψ1 (respectively ψ2 ) is well defined and it is a homeomorphism from U1 to C (resp. from U2 to C). Moreover, ψ1 (U1 ∩ U2 ) = ψ2 (U1 ∩ U2 ) = C \ {0} and for all ζ ∈ C \ {0} it holds ψ2 (ψ1−1 (ζ )) = ψ2 ([1 : ζ ]) =

1 , ζ

1.1 Riemann Surfaces, Riemann Sphere and the Group of Möbius Transformations

5

which implies that ψ2 ◦ ψ1−1 : ψ1 (U1 ∩ U2 ) → ψ2 (U1 ∩ U2 ) is holomorphic. A similar argument shows that also ψ1 ◦ ψ2−1 : ψ2 (U1 ∩ U2 ) → ψ1 (U1 ∩ U2 ) is holomorphic. Hence, {(U j , ψ j ) j=1,2 } is a holomorphic atlas for C∞ . Definition 1.1.2 The Riemann surface C∞ is called the Riemann sphere. Using the holomorphic chart (U2 , ψ2 ) it is easy to check that the map ι : C → C∞ given by ι(z) := [z : 1] is univalent and, in fact, ι = ψ2−1 , proving that ι is a homeomorphism on its image. This allows to consider C∞ as the holomorphic one-point compactification of C by adding a point at infinity. To make this sentence rigorous, let ∞ := [1 : 0]. Then, C∞ = ι(C) ∪ {∞}. As customary, with a slight abuse of notation, we will forget to write ι and simply write C∞ = C ∪ {∞}. This means that z ∈ C can be thought of as the point [z : 1] ∈ C∞ and ∞ = [1 : 0]. In particular, if R > 0 and VR := {z ∈ C : |z| > R} ∪ {∞}, the family {VR } is a basis of open neighborhoods of ∞, and thus a sequence {z n } ⊂ C converges to ∞ in C∞ if and only if |z n | → ∞. One can define a distance on C∞ which is equivalent to the Euclidean distance on bounded sets. Definition 1.1.3 Let z, w ∈ C∞ . If z, w = ∞, let d S (z, w) := 

2|z − w| (1 + |z|2 )(1 + |w|2 )

.

If w = ∞ and z = ∞, let d S (z, ∞) = d S (∞, z) = 

2 1 + |z|2

,

and d S (∞, ∞) = 0. We call d S (z, w) the spherical distance between z and w. The function d S is a distance, that is, it satisfies d S (z, w) = d S (w, z) for all z, w ∈ C∞ , d S (z, w) = 0 if and only if z = w and d S (z, w) ≤ d S (z, u) + d S (u, w) for all z, w, u ∈ C∞ . It is, in fact, induced by the spherical distance on the sphere {(x1 , x2 , x3 ) ∈ R3 : |x1 |2 + |x2 |2 + |x3 |3 = 1} via the stereographical projection (see e.g. [3, Sect. 2.4]). Let K ⊂ C∞ . The spherical diameter diam S (K ) is defined by diam S (K ) := sup d S (z, w). z,w∈K

If K is a bounded subset of C, then there exists C K > 0 such that 1 d S (z, w) ≤ |z − w| ≤ C K d S (z, w) 2

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1 Hyperbolic Geometry and Iteration Theory

for all z, w ∈ K . Namely, the spherical distance is equivalent to the Euclidean distance on bounded subsets of C. In particular, it is easy to see that the spherical distance d S induces on C∞ the topology defined above. By Liouville’s Theorem, the unit disc D can not be biholomorphic to the complex plane C. Moreover, since C∞ is compact, it can not be biholomorphic to D or C. Thus, D, C and C∞ are examples of non-biholomorphic simply connected Riemann surfaces. The Uniformization Theorem assures that these are the only ones: Theorem 1.1.4 (Uniformization Theorem) Every simply connected Riemann surface is biholomorphic either to the unit disc D, or to the complex plane C, or to the Riemann sphere C∞ . A proof can be found in, e.g., [71, 72]. Now we turn our attention to the group of automorphisms of C∞ . Let a, b, c, d ∈ C be such that ad − bc = 0. The map F : C∞ → C∞ defined by F : [w1 : w2 ] → [aw1 + bw2 : cw1 + dw2 ] is well defined and invertible, and, using the holomorphic charts previously introduced, one can easily see that it is holomorphic. Therefore, it is an automorphism of C∞ . Using the decomposition C∞ = C ∪ {∞} one can write . In particular, taking into account the definition of C ∪ {∞}, such a map as z → az+b cz+d = ac , and BA = ∞ if A ∈ C \ {0} and B = 0. we have F(∞) := lim z→∞ az+b cz+d Definition 1.1.5 A map F : C∞ → C∞ of the form F(z) =

az + b , cz + d

(1.1.1)

for some a, b, c, d ∈ C with ad − cb = 0, is called a Möbius transformation or a linear fractional map. Lemma 1.1.6 Let f : C → C be univalent. Then there exist a, b ∈ C, a = 0, such that f (z) = az + b. Proof Let g(z) := f (1/z). Then g : C \ {0} → C is univalent. Assume 0 is an essential singularity for g. Let w0 := g(1). On the one hand, by Weierstrass’ Theorem (see, e.g., [3, Theorem 9, p. 129]), there exists a sequence {z n } converging to 0 such that limn→∞ g(z n ) = w0 . On the other hand, since g is holomorphic, it is open. Therefore, letting D := {z ∈ C : |z − 1| < 1/2}, it follows that g(D) is an open neighborhood of w0 . Hence, the sequence {g(z n )} is eventually contained in g(D). Since g is injective, it follows that {z n } = {g −1 (g(z n ))} is eventually contained in D, a contradiction. Hence 0 is not  an essential singularity for g. j of f at 0, a j ∈ C for j ≥ 0. Then Let f (z) = ∞ j=0 a j z be the Taylor expansion ∞ the Laurent expansion of g at 0 is g(z) = j=0 a j z − j . Since 0 is not an essential singularity, it follows that f is a polynomial. If the degree of f were greater or equal than two, then f would not be injective by the Fundamental Theorem of Algebra. Therefore, f is a polynomial of degree one.  A straightforward corollary of the previous lemma is the following result.

1.1 Riemann Surfaces, Riemann Sphere and the Group of Möbius Transformations

7

Corollary 1.1.7 Every automorphism of C is a Möbius transformation which fixes ∞. As we saw before, all Möbius transformations are automorphisms of C∞ . The converse is true as well. Proposition 1.1.8 Let F be an automorphism of C∞ . Then F is a Möbius transformation. Proof Let G be a Möbius transformation such that G(F(∞)) = ∞. Hence, G ◦ F is an automorphism of C∞ which fixes ∞. Therefore, its restriction to C is an automorphism of C and, by Corollary 1.1.7, it is a Möbius transformation H . Hence,  F = G −1 ◦ H is a Möbius transformation. Remark 1.1.9 Every Möbius transformation maps circles and lines in C onto either circles or lines in C (circles can be mapped both onto circles and onto lines and similarly for lines). Proposition 1.1.10 Let F be a Möbius transformation different from the identity. Then one and only one of the following cases is possible. (1) F has two distinct fixed points in C∞ . In this case there exist a Möbius transformation G and λ ∈ C \ {0} such that G ◦ F ◦ G −1 (z) = λz. (2) F has only one fixed point in C∞ . In this case there exists a Möbius transformation G such that G ◦ F ◦ G −1 (z) = z + i. Proof If F is a Möbius transformation given by (1.1.1), the equation for fixed points is az + b = z(cz + d), which has exactly two solutions (counting multiplicity) in C∞ if c = 0 and one solution if c = 0. If F has two distinct fixed points in C∞ , say σ1 , σ2 ∈ C∞ , and σ1 , σ2 = ∞, let z−σ1 . In case one of the fixed points is ∞, say σ2 = ∞, let G(z) = z − σ1 . G(z) = z−σ 2 Then G(σ1 ) = 0 and G(σ2 ) = ∞. Therefore the Möbius transformation G ◦ F ◦ G −1 fixes 0 and ∞, hence it is of the form z → λz for some λ = 0. If F has only one fixed point (regardless multiplicity), say σ ∈ C∞ , let H (z) = 1 if σ = ∞, or let H (z) = z if σ = ∞. Hence H (σ ) = ∞. Therefore the Möbius z−σ transformation H ◦ F ◦ H −1 fixes ∞ and it is of the form z → az + b. If a = 1 then H ◦ F ◦ H −1 would have two fixed points in C∞ , which would force F to have two fixed points in C∞ . Hence a = 1 and necessarily b = 0. Finally, let S(z) = z . Hence (S ◦ H ◦ F ◦ H −1 ◦ S −1 )(z) = z + i and taking G = S ◦ H we have the −ib result.  Remark 1.1.11 If f is an automorphism of C, not the identity, by Corollary 1.1.7, f is a Möbius transformation of C∞ such that f (∞) = ∞. Hence, f has either one (simple) fixed point in C or it does not fix any point in C. In the first case f (z) = az + b with a = 1, while, in the second case f (z) = z + b with b = 0. The previous proposition has a direct consequence about the derivatives at fixed points of Möbius transformations. We do not introduce the concept of derivative of

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1 Hyperbolic Geometry and Iteration Theory

a holomorphic function on a Riemann surface, but we content ourselves to define the derivative of a holomorphic function on the Riemann sphere at ∞: if f : C∞ → C∞ is holomorphic, using the decomposition C∞ = C ∪ {∞}, we define f  (∞) := z lim z→∞ z( f (z) − f (∞)) in case f (∞) ∈ C, and f  (∞) := lim z→∞ f (z) in case 1  . f (∞) = ∞. Moreover, if ζ ∈ C and f (ζ ) = ∞, we set f (ζ ) = lim z→ζ f (z)(z−ζ ) A direct computation shows that the chain rule holds for these derivates. Hence, as a direct consequence of Proposition 1.1.10 we have Corollary 1.1.12 Let F be a Möbius transformation. If F has only one fixed point ζ ∈ C∞ then F  (ζ ) = 1. If F has two distinct fixed points ζ0 , ζ1 ∈ C∞ then F  (ζ0 ) · F  (ζ1 ) = 1. Using Möbius transformations one can simplify much computations. The typical example is the so called Cayley transform, which often allows to transform non-linear expressions into affine ones. We define it here for future reference. Let H := {w ∈ C : Re w > 0} be the right half-plane. The Cayley transform Cσ : D → H with respect to σ ∈ ∂D, is defined by Cσ (z) :=

σ +z . σ −z

(1.1.2)

The map Cσ is a Möbius transformation which is a biholomorphism between D and H whose inverse is given by Cσ−1 (w) = σ

w−1 . w+1

Moreover, Cσ (σ ) = ∞ in C∞ . Remark 1.1.13 Let σ ∈ ∂D. Then T : D → D is an automorphism of D if and only if Cσ ◦ T ◦ Cσ−1 is an automorphism of H .

1.2 The Schwarz Lemma and the Automorphism Group of the Unit Disc Theorem 1.2.1 (Schwarz’s Lemma) Let φ : D → D be holomorphic. Assume that φ(0) = 0. Then (1) |φ(z)| ≤ |z| for all z ∈ D. Moreover, equality holds at some—and hence any— z = 0 if and only if there exists θ ∈ R such that φ(z) = eiθ z. (2) |φ  (0)| ≤ 1. Moreover, |φ  (0)| = 1 if and only if there exists θ ∈ R such that φ(z) = eiθ z. for z = 0 and f (0) = φ  (0). The map f : D → C is holoProof Let f (z) := φ(z) z morphic. Let 0 < r < 1 and |z| < r . By the Maximum Principle it holds

1.2 The Schwarz Lemma and the Automorphism Group of the Unit Disc

| f (z)| ≤ max | f (z)| = max | f (z)| = max |z|≤r

|z|=r

|z|=r

9

1 |φ(z)| ≤ . r r

Letting r → 1 we obtain | f (z)| ≤ 1, namely, |φ(z)| ≤ |z| and |φ  (0)| = | f (0)| ≤ 1. Again by the Maximum Principle, if |φ(z)| = |z| for some z ∈ D \ {0}, then | f | attains its maximum 1 in D, hence it is constant and (1) holds. Similarly, if |φ  (0)| = 1 then | f (0)| = 1 and again the Maximum Principle implies that f is constant, and hence φ is a rotation.  Now we examine the automorphism group of the unit disc Aut(D), that is, Aut(D) := {T : D → D such that T is a biholomorphism}. Note that Aut(D) is a group under composition of functions. For a ∈ D, let Ta : D → C be the holomorphic map defined by Ta (z) :=

a−z . 1 − az

(1.2.1)

It is easy to see that Ta (D) = D and Ta−1 (z) = Ta (z), that is, Ta is an automorphism of D. Note also that Ta (a) = 0, Ta (0) = a. Proposition 1.2.2 Let T ∈ Aut(D). Then there exists θ ∈ R and a ∈ D such that T (z) = eiθ Ta (z).

(1.2.2)

In particular, every automorphism of D extends as a homeomorphism from D to D. Moreover, Aut(D) acts transitively on D—that is, for every ζ, ξ ∈ D there exists T ∈ Aut(D) such that T (ζ ) = ξ —and double transitively on ∂D—that is, for every σ1 , σ2 ∈ ∂D and σ1 , σ2 ∈ ∂D with σ1 = σ2 and σ1 = σ2 , there exists T ∈ Aut(D) such that T (σ1 ) = σ1 and T (σ2 ) = σ2 . Proof Let T ∈ Aut(D). Let a = T −1 (0). Then γ := T ◦ Ta ∈ Aut(D) and γ (0) = 0. By Theorem 1.2.1, |γ (z)| ≤ |z| for all z ∈ D. On the other hand, γ −1 ∈ Aut(D) and γ −1 (0) = 0, therefore, also |γ −1 (z)| ≤ |z| for all z ∈ D. This implies that |γ (z)| = |z| for all z ∈ D and again by Theorem 1.2.1, there exists θ ∈ R such that γ (z) = eiθ z and (1.2.2) is proved. From this formula, all assertions follow easily, except the double transitivity of Aut(D) on ∂D. In order to see this, it is enough to prove that for every couple σ1 , σ2 ∈ ∂D, σ1 = σ2 , there exists T ∈ Aut(D) such that T (1) = σ1 and T (−1) = σ2 . Moreover, up to composing with a rotation, we can assume σ1 = 1. Thus, we look for an automorphism T of D such that T (1) = 1 and T (−1) = σ2 . Using the Cayley transform C1 : D → H with respect to 1, defined in (1.1.2), see Remark 1.1.13, the statement amounts to prove that there exists an automorphism S of the right halfplane H such that S(∞) = ∞ and S(0) = C1 (σ2 ) =: iβ, for some β ∈ R. Define S(w) := w + iβ, then clearly S is an automorphism of H which satisfies the needed  condition. The automorphism T in D is then given by C1−1 ◦ S ◦ C1 .

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Using automorphisms, one can prove a general version of the Schwarz Lemma: Theorem 1.2.3 (Schwarz-Pick’s Lemma) Let φ : D → D be holomorphic. Then, for all z 1 , z 2 ∈ D,      φ(z 1 ) − φ(z 2 )   z 1 − z 2    ≤ (1.2.3)  1 − φ(z )φ(z )   1 − z z  1

and, for all z ∈ D,

2

1 2

|φ  (z)| 1 ≤ . 1 − |φ(z)|2 1 − |z|2

(1.2.4)

Moreover, equality in (1.2.3) holds for some—and hence any—z 1 , z 2 ∈ D, z 1 = z 2 or equality in (1.2.4) holds for some—and hence any—z ∈ D if and only if φ is an automorphism of the unit disc. Proof Let w ∈ D and f := Tφ(w) ◦ φ ◦ Tw , where Ta is the automorphism defined in (1.2.1). Then f : D → D is holomorphic and f (0) = 0. The statements follow then by applying Schwarz’s Lemma (Theorem 1.2.1) to f .  Corollary 1.2.4 Let φ : D → D be holomorphic, φ = idD . Then there is at most one point ζ ∈ D such that φ(ζ ) = ζ . Proof Assume φ(ζ ) = ζ and φ(ξ ) = ξ for some ζ, ξ ∈ D, ζ = ξ . By (1.2.3), φ is an automorphism of D fixing two points in D. Hence, Tζ ◦ φ ◦ Tζ is an automorphism of D fixing 0, thus it is a rotation. But since it fixes as well the point Tζ (ξ ), it is the identity. Hence, φ is the identity, a contradiction. 

1.3 Hyperbolic Metric and Hyperbolic Distance Definition 1.3.1 The hyperbolic metric (or Poincaré metric) is defined for v, w ∈ C and z ∈ D by vw . κD2 (z; (v, w)) := (1 − |z|2 )2 From a differential geometric point of view, κD2 can be thought of as a Kähler metric κD2 : T D ⊗ T D → C of constant Gaussian curvature −4. However, we are not going to use this point of view in the rest of the book. We only point out that the function κD2 (z; (v, w)) is continuous with respect to z, v, w and, when z is fixed, it gives a Hermitian product on C. The hyperbolic norm of a vector v ∈ C at z ∈ D is defined by κD (z; v) :=



κD2 (z; (v, v)) =

|v| . 1 − |z|2

We use the hyperbolic norm to define a length for C 1 -smooth curves:

1.3 Hyperbolic Metric and Hyperbolic Distance

11

Definition 1.3.2 Let γ : [a, b] → D be a piecewise C 1 -smooth curve. We define the hyperbolic length of γ to be:  D (γ ) :=

b





κD (γ (t); γ (t))dt =

a

a

b

|γ  (t)| dt. 1 − |γ (t)|2

Using hyperbolic length of curves we can define the hyperbolic distance as follows: Definition 1.3.3 Let z, w ∈ D. The hyperbolic distance (or Poincaré distance) from z to w is ω(z, w) := inf D (γ ), γ

where γ runs over all piecewise C 1 -smooth curves in D which join z and w. From the definition it is clear that ω is a non-negative symmetric function and satisfies the triangle inequality. Also, there is a simple expression for the hyperbolic distance in D. In order to find it, we need a preliminary lemma: Lemma 1.3.4 Every automorphism of D is an isometry for the hyperbolic metric of D. That is, for all v ∈ C, z ∈ D and T ∈ Aut(D), κD (T (z); T  (z)v) = κD (z; v). In particular, for every z, w ∈ D and for every T ∈ Aut(D), ω(T (z), T (w)) = ω(z, w). Proof If T ∈ Aut(D), according to Proposition 1.2.2, there exist a ∈ D and θ ∈ R such that T (z) = eiθ Ta (z), where Ta (z) is given by (1.2.2). A straightforward computation shows that 1 − |T (z)|2 =

(1 − |a|2 )(1 − |z|2 ) , |1 − az|2

and |T  (z)| =

1 − |a|2 . |1 − az|2

Hence, |1 − az|2 |T  (z)v| (1 − |a|2 )|v| = 2 2 1 − |T (z)| |1 − az| (1 − |a|2 )(1 − |z|2 ) |v| = = κD (z; v), 1 − |z|2

κD (T (z); T  (z)v) =

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1 Hyperbolic Geometry and Iteration Theory

and thus T is an isometry for κD . According to the definition of ω, it follows that T is also an isometry for ω.  Using this lemma we can find an explicit expression for ω. Theorem 1.3.5 Let z, w ∈ D, and let Tw be given by (1.2.1). Then ω(z, w) =

1 1 + |Tw (z)| log . 2 1 − |Tw (z)|

(1.3.1)

Proof Since automorphisms are isometries for ω by Lemma 1.3.4, it follows that ω(z, w) = ω(Tw (z), Tw (w)) = ω(Tw (z), 0). Therefore, we only need to find the expression of ω(0, z) for z ∈ D. Since rotations around the origin are automorphisms of D, hence isometries for the hyperbolic distance, we can also assume that z = |z| > 0. Therefore, we are left to find the expression of ω(0, r ) for r ∈ (0, 1). Let γ : [a, b] → D be a piecewise C 1 -smooth curve such that γ (a) = 0 and γ (b) = r . Write γ (t) := γ1 (t) + iγ2 (t), with γ j : [a, b] → R, j = 1, 2. Note that γ1 : [a, b] → D is a piecewise C 1 -smooth curve which joins 0 to r . Then |γ  (t)| ≥ |γ1 (t)| and 1 − |γ (t)|2 ≤ 1 − |γ1 (t)|2 . Hence,  D (γ ) = a

b

|γ  (t)| dt ≥ 1 − |γ (t)|2

 a

b

|γ1 (t)| dt = D (γ1 ). 1 − |γ1 (t)|2

(1.3.2)

Let [α, β] = γ ([a, b]). Note that α ≤ 0 and β ≥ r . By the usual rule of change of variables in integrals (see, e.g., [122, (6.95) p. 325]),  D (γ1 ) =

b

a

 =

α

β

 b |γ1 (t)| γ1 (t) dt ≥ dt 2 1 − |γ1 (t)|2 a 1 − γ1 (t)  r 1+r dt dt 1 . ≥ = log 2 2 1−t 2 1−r 0 1−t

Let γ˜ (t) := t, t ∈ [0, r ]. A direct computation shows that D (γ˜ ) = fore, (1.3.2) and (1.3.3) show that ω(0, r ) = inf D (γ ) = D (γ˜ ) = γ

and (1.3.1) follows.

1 2

(1.3.3)

log 1+r . There1−r

1+r 1 log , 2 1−r 

Remark 1.3.6 Theorem 1.3.5 shows that ω : D × D → [0, +∞) is a smooth function, and it is a complete distance. Namely, ω is a symmetric function which satisfies the triangle inequality and ω(z, w) = 0 if and only if z = w, and, the adjective “complete” refers to the fact that for every ζ ∈ D and R > 0 the hyperbolic disc

1.3 Hyperbolic Metric and Hyperbolic Distance

13

D(ζ, R) := {z ∈ D : ω(z, ζ ) < R} is relatively compact in D. Moreover, the topology induced by ω on D is equivalent to the Euclidean topology on D. Putting together Theorem 1.2.3 and Theorem 1.3.5 we have: Theorem 1.3.7 Let φ : D → D be holomorphic. Then (1) For every z, w ∈ D, it holds ω(φ(z), φ(w)) ≤ ω(z, w), with equality for some z = w if and only if φ ∈ Aut(D). (2) For every z ∈ D and v ∈ C it holds κD (φ(z); φ  (z)v) ≤ κD (z; v), with equality for some v = 0 if and only if φ ∈ Aut(D). The hyperbolic distance in D is a convex function: Lemma 1.3.8 Let z 1 , z 2 , w1 , w2 ∈ D and λ ∈ [0, 1]. Then ω(λz 1 + (1 − λ)z 2 , λw1 + (1 − λ)w2 ) ≤ max{ω(z 1 , w1 ), ω(z 2 , w2 )}. Proof Suppose that max{ω(z 1 , w1 ), ω(z 2 , w2 )} = ω(z 1 , w1 ). If z 1 = w1 , the result follows immediately. Therefore, we can assume z 1 = w1 . For a ∈ D, let Ta be the automorphism given by (1.2.1). Then ω(0, |Tz2 (w2 )|) = ω(0, Tz2 (w2 )) = ω(Tz2 (0), w2 ) = ω(z 2 , w2 ) ≤ ω(z 1 , w1 ) = ω(Tz1 (0), w1 ) = ω(0, Tz1 (w1 )) = ω(0, |Tz1 (w1 )|). Thus |Tz2 (w2 )| ≤ |Tz1 (w1 )|. This inequality implies that the function  φ(z) = Tz2

 Tz2 (w2 ) Tz1 (z) , z ∈ D, Tz1 (w1 )

is a holomorphic self-map of the unit disc and satisfies φ(z 1 ) = z 2 and φ(w1 ) = w2 . If λ ∈ [0, 1], then the function ϕ(z) = λz + (1 − λ)φ(z), z ∈ D, is also a holomorphic self-map of the unit disc. Hence, by Theorem 1.3.7, ω(λz 1 + (1 − λ)z 2 , λw1 + (1 − λ)w2 ) = ω(ϕ(z 1 ), ϕ(w1 )) ≤ ω(z 1 , w1 ), and the proof is completed.



For the aims of this book, we need to extend the definition of hyperbolic distance to other Riemann surfaces. Let S be a Riemann surface and let z, w ∈ S. A finite family C of holomorphic functions f j : D → S, j = 1, . . . , n, for some n ∈ N is

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1 Hyperbolic Geometry and Iteration Theory

called a chain of holomorphic discs in S joining z and w if there exist t j ∈ (0, 1), j = 1, . . . , n, such that f 1 (0) = z, f 1 (t1 ) = f 2 (0), . . . , f n−1 (tn−1 ) = f n (0), f n (tn ) = w. The hyperbolic length of the chain C is defined by (C ) :=

n

ω(0, t j ).

j=1

We denote by Γ S (z, w) the set of all chains of holomorphic discs in S joining z and w. Definition 1.3.9 Let S be a Riemann surface. Let z, w ∈ S. The hyperbolic distance between z and w is defined as k S (z, w) =

inf

C ∈Γ S (z,w)

(C ).

Since S is connected by definition and hence each two points in S can be joined by a compact curve in S which can be covered by a finite number of holomorphic charts, it turns out that k S (z, w) < +∞ for all z, w ∈ S. Moreover, k S (z, z) = 0 for all z ∈ S. Also, as a direct consequence of the definition we have that k S : S × S → [0, +∞) is a symmetric function which satisfies the triangle inequality. The hyperbolic distance is contracted by holomorphic functions, and preserved by biholomorphisms: Proposition 1.3.10 Let S, S˜ be Riemann surfaces. If φ : S → S˜ is holomorphic, then for all z, w ∈ S k S˜ (φ(z), φ(w)) ≤ k S (z, w). Moreover, if φ is a biholomorphism then equality holds for all z, w ∈ S. Proof If C := { f j } is a chain of holomorphic discs in S joining z and w, then φ ◦ C := {φ ◦ f j } is a chain of holomorphic discs in S˜ joining φ(z) and φ(w), namely, φ ◦ Γ S (z, w) ⊂ Γ S˜ (φ(z), φ(w)). Moreover, if C ∈ Γ S (z, w) then (C ) = (φ ◦ C ). Therefore k S˜ (φ(z), φ(w)) = =

inf

C ∈Γ S˜ (φ(z),φ(w))

inf

C ∈Γ S (z,w)

(C ) ≤

inf

C ∈φ◦Γ S (z,w)

(C )

(C ) = k S (z, w).

If φ is a biholomorphism, then clearly φ ◦ Γ S (z, w) = Γ S˜ (φ(z), φ(w)), and the equal ity holds between k S (z, w) and k S˜ (φ(z), φ(w)). Remark 1.3.11 From the definition of hyperbolic distance for Riemann surfaces it follows immediately that if S, S˜ are two Riemann surfaces and S ⊂ S˜ then k S˜ (z, w) ≤ k S (z, w) for all z, w ∈ S. We study now the hyperbolic distance for simply connected Riemann surfaces. By Theorem 1.1.4, every simply connected Riemann surface is biholomorphic either

1.3 Hyperbolic Metric and Hyperbolic Distance

15

to D, or to C or to C∞ . Since the hyperbolic distance is invariant under biholomorphisms, we just need to consider these three cases. Proposition 1.3.12 The following hold: (1) kD (z, w) = ω(z, w) for all z, w ∈ D, (2) kC (z, w) = 0 for all z, w ∈ C, (3) kC∞ (z, w) = 0 for all z, w ∈ C∞ . Proof (1) Let z, w ∈ D, with z = w. Let Tz be the automorphism of D given by (1.2.1) and let λ := |TTzz (w) . Then T (ζ ) := λTz (ζ ), ζ ∈ D, is an automorphism of D (w)| such that T (z) = 0 and r := T (w) > 0. Therefore, by Lemma 1.3.4 and Proposition 1.3.10, ω(z, w) = ω(0, r ) and kD (z, w) = kD (0, r ). Thus, it is enough to prove that ω(0, r ) = kD (0, r ) for all 0 < r < 1. Since the identity map can be seen as a chain of holomorphic discs in D joining 0 and r , it follows that kD (0, r ) ≤ ω(0, r ). In order to prove the converse inequality, let C = { f j } j=1,...,n be a chain of holomorphic discs in D joining 0 and r . By definition, there exist t1 , . . . , tn ∈ (0, 1) such that f 1 (0) = 0, f 1 (t1 ) = f 2 (0), . . . , f n−1 (tn−1 ) = f n (0) and f n (tn ) = r . Hence by Theorem 1.3.7, ω(0, t j ) ≥ ω( f j (0), f j (t j )) for all j = 1, . . . , n, and a repeated use of the triangle inequality gives (C ) = ω(0, t1 ) + ω(0, t2 ) + . . . + ω(0, tn ) ≥ ω( f 1 (0), f 1 (t1 )) + ω( f 2 (0), f 2 (t2 )) + . . . + ω( f n (0), f n (tn )) ≥ ω( f 1 (0), f n (tn )) = ω(0, r ). By the arbitrariness of C , it follows that kD (0, r ) ≥ ω(0, r ), and we are done. (2) Let z, w ∈ C. For n ∈ N, let f n : D → C be given by f n (ζ ) := n(ζ z − (ζ −

1 )w). n

Then f n : D → C is holomorphic, f n (0) = w and f n ( n1 ) = z. Hence, 1 kC (z, w) ≤ ω(0, ) → 0 for n → ∞, n therefore kC ≡ 0. (3) Let z, w ∈ C∞ , z = w. There exists a Möbius transformation T such that T (z), T (w) ∈ C. Therefore, by Proposition 1.3.10 and Remark 1.3.11 kC∞ (z, w) = kC∞ (T (z), T (w)) ≤ kC (T (z), T (w)) = 0, and we are done.



Proposition 1.3.10 and Proposition 1.3.12 can be used to find an explicit expression for the hyperbolic distance of simply connected domains (in some cases).

16

1 Hyperbolic Geometry and Iteration Theory

Example 1.3.13 Let σ ∈ ∂D and let Cσ : D → H be the Cayley transform with respect to σ given by (1.1.2). Since Cσ is a biholomorphism, it follows that for all w1 , w2 ∈ H,    w1 −w2  1 +  w1 +w2  1 .  (1.3.4) kH (w1 , w2 ) = ω(Cσ−1 (w1 ), Cσ−1 (w2 )) = log  2 2 1 −  ww11 −w  +w2 In the following we will need two properties of the hyperbolic distance. The first one is about continuity: Proposition 1.3.14 Let S be a Riemann surface. Then k S : S × S → [0, +∞) is continuous. Proof For all z 0 , w0 , z, w ∈ S, by the triangle inequality, |k S (z 0 , w0 ) − k S (z, w)| ≤ k S (z 0 , z) + k S (w0 , w). Hence it is sufficient to prove that for each z 0 ∈ S the function S  z → k S (z, z 0 ) is continuous at z 0 . Let (U, ψ) be a holomorphic chart of S such that z 0 ∈ U . Since ψ(U ) is an open set of C containing ψ(z 0 ), there exists an Euclidean disc D ⊂ ψ(U ) such that ψ(z 0 ) ∈ D. Let ε > 0. Since D is biholomorphic to the unit disc D—and hence, by Proposition 1.3.10 and Proposition 1.3.12, k D is continuous—there exists a neighborhood V of ψ(z 0 ) in D such that k D (ψ(z 0 ), ζ ) < ε for all ζ ∈ V . Hence, by Remark 1.3.11 and Proposition 1.3.10, for all z ∈ ψ −1 (V ), it holds k S (z 0 , z) ≤ kψ −1 (D) (z 0 , z) = k D (ψ(z 0 ), ψ(z)) < ε, and thus k S (z 0 , ·) is continuous at z 0 .



Another property of the hyperbolic distance we will need is the following: Proposition 1.3.15 Let S be a Riemann surface. Suppose that {Sm }, m ∈ N, is a family of open subsets of S such that Sm ⊂ Sm+1 for all m ∈ N and that S = ∪m∈N Sm . Then for every z, w ∈ S there exists m 0 ∈ N such that z, w ∈ Sm for all m ≥ m 0 and k S (z, w) = lim k Sm (z, w). m→∞

Proof Let z, w ∈ S. Then there exists m 0 ∈ N such that z, w ∈ Sm for all m ≥ m 0 . By Remark 1.3.11, the sequence {k Sm (z, w)}m≥m 0 is decreasing, and hence it has a limit, say ρ ≥ 0, and k S (z, w) ≤ ρ. In order to show that ρ = k S (z, w), we will prove that for any ε > 0 there exists m ≥ m 0 such that k Sm (z, w) ≤ k S (z, w) + ε, from which it immediately follows that ρ ≤ k S (z, w).

1.3 Hyperbolic Metric and Hyperbolic Distance

17

Let thus ε > 0 be fixed, and let C = { f j }nj=1 be a chain of holomorphic discs in S joining z and w such that (C ) ≤ k S (z, w) + 2ε . By definition there exist t j ∈ (0, 1), j = 1, . . . , n, such that f 1 (0) = z, f 1 (t1 ) = f 2 (0), . . . , f n−1 (tn−1 ) = f n (0) and f n (tn ) = w. Let 0 < r < 1, close to 1, be such that t j < r and ω(0,

tj ε ) ≤ ω(0, t j ) + r 2n

for all j = 1, . . . , n. Let Dr := {z ∈ C : |z| < r }. Since the set ∪nj=1 f j (Dr ) is compact in S, it is contained in Sm r for some m r ≥ m 0 . Let f jr : D → Sm r be defined by f jr (ζ ) = f j (r ζ ), j = 1, . . . , n. Then C r := { f jr } is a chain of holomorphic discs in r ( tn−1 ) = f nr (0) and f nr ( trn ) = w. Sm r such that f 1r (0) = z, f 1r ( tr1 ) = f 2r (0), . . . , f n−1 r Then k Smr (z, w) ≤ (C r ) =

n j=1

tj ε ε )≤ ω(0, t j ) + = (C ) + ≤ k S (z, w) + ε, r 2 2 n

ω(0,

j=1



and we are done.

The story about Riemann surfaces and hyperbolic distance does not end here, and we will come back to this in Chap. 5 where we define and study hyperbolic metric and distance in simply connected domains. However, for the time being, the previous simple facts are just those we need. Using the uniformization theorem, one can show that every Riemann surface is holomorphically covered by either D, or C or C∞ . The hyperbolic distance is a genuine distance on a Riemann surface S—that is, k S (z, w) = 0 if and only if z = w—which is complete and induces the natural topology of S if and only if S is holomorphically covered by D (in this case S is also called hyperbolic). In the other cases, k S ≡ 0. The interested reader can check, e.g., [71] or [1].

1.4 Horocycles and Julia’s Lemma Definition 1.4.1 Let σ ∈ ∂D and R > 0. The horocycle E(σ, R) of center σ and (hyperbolic) radius R > 0 is E(σ, R) := {z ∈ D :

|σ − z|2 < R}. 1 − |z|2

σ of radius R/(R + 1) It is easy to see that E(σ, R) is an Euclidean disc of center 1+R contained in D and tangent to ∂D at σ . Horocycles can be expressed in terms of hyperbolic distance:

Proposition 1.4.2 Let σ ∈ ∂D. Then for every z ∈ D

18

1 Hyperbolic Geometry and Iteration Theory

  |σ − z|2 1 . lim [ω(z, w) − ω(0, w)] = log w→σ 2 1 − |z|2

(1.4.1)

In particular, for every R > 0 E(σ, R) = {z ∈ D : lim [ω(z, w) − ω(0, w)] < w→σ

1 log R}. 2

(1.4.2)

Proof Let z ∈ D and let Tz be the canonical automorphism of D such that Tz (z) = 0 [see (1.2.1)]. Then   1 + |Tz (w)| 1 − |w| 1 · . ω(z, w) − ω(0, w) = ω(0, Tz (w)) − ω(0, w) = log 2 1 − |Tz (w)| 1 + |w| Therefore, lim [ω(z, w) − ω(0, w)] =

w→σ

    1 − |w| 1 − |w|2 1 1 lim log = lim log . 2 w→σ 1 − |Tz (w)| 2 w→σ 1 − |Tz (w)|2

A direct computation shows that (1 − |w|2 ) ·

|1 − zw|2 1 |1 − zw|2 2 = = (1 − |w| ) · , 1 − |Tz (w)|2 (1 − |w|2 )(1 − |z|2 ) 1 − |z|2

hence (1.4.1) holds. Finally, (1.4.2) follows at once from (1.4.1).



There is another useful way to express horocycles. Let σ ∈ ∂D and let u σ : D → R− be the Poisson kernel, defined as u σ (z) := −

1 − |z|2 = Re |σ − z|2



z+σ z−σ

 .

(1.4.3)

Note that u σ is a negative harmonic function in D, which extends continuously to ∂D \ {σ } as u σ (z) = 0 for z ∈ ∂D \ {σ }. By definition, for all R > 0 E(σ, R) = {z ∈ D : u σ (z) < −1/R}.

(1.4.4)

In particular, horocycles are the sublevel sets of the Poisson kernel. Definition 1.4.3 Let φ : D → D be holomorphic and let σ ∈ ∂D. The boundary dilation coefficient of φ at σ is given by αφ (σ ) := lim inf z→σ

1 − |φ(z)| . 1 − |z|

1.4 Horocycles and Julia’s Lemma

19

Example 1.4.4 Let T be an automorphism of D and σ ∈ ∂D. By Proposition 1.2.2, there exist a ∈ D and θ ∈ R such that T (z) = eiθ Ta (z) for all z ∈ D. A direct computation shows 1 − |Ta (z)| 1 − |Ta (z)| = lim z→σ 1 − |z| 1 − |z| 2 1 − |a|2 1 − |Ta (z)| 1 + |z| = = lim = −u σ (a). 2 z→σ 1 − |z| 1 + |Ta (z)| |1 − aσ |2

αT (σ ) = lim inf z→σ

Lemma 1.4.5 Let φ : D → D be holomorphic and let σ ∈ ∂D. Then 0 < αφ (σ ) ≤ +∞. Moreover, 1 log αφ (σ ) = lim inf [ω(0, w) − ω(0, φ(w))]. w→σ 2

(1.4.5)

Proof Since holomorphic maps contract the hyperbolic distance, and by the triangle inequality, for every w ∈ D, we have ω(0, w) − ω(0, φ(w)) ≥ ω(φ(0), φ(w)) − ω(0, φ(w)) ≥ −ω(0, φ(0)). In particular, lim inf [ω(0, w) − ω(0, φ(w))] > −∞. w→σ

(1.4.6)

Now,     1 − |φ(w)| 1 1 + |w| 1 + log . (1.4.7) ω(0, w) − ω(0, φ(w)) = log 2 1 − |w| 2 1 + |φ(w)| Since

1 2

≤

1+|w| 1+|φ(w)|

≤ 2 for every w ∈ D,

  1 − |φ(w)| 1 1 1 log + log( ) ≤ ω(0, w) − ω(0, φ(w)) 2 1 − |w| 2 2   1 − |φ(w)| 1 1 ≤ log + log 2. 2 1 − |w| 2 Hence, αφ (σ ) = +∞ if and only if lim inf w→σ [ω(0, w) − ω(0, φ(w))] = +∞. Moreover, from (1.4.6), it follows that αφ (σ ) > 0. On the other hand, if {z n } ⊂ D is a sequence converging to σ such that either n )| exists finitely or limn→∞ [ω(0, z n ) − ω(0, φ(z n ))] exists finitely, limn→∞ 1−|φ(z 1−|z n | then |φ(z n )| → 1 as n → ∞ and, by (1.4.7),   1 1 − |φ(z n )| . lim [ω(0, z n ) − ω(0, φ(z n ))] = lim log n→∞ n→∞ 2 1 − |z n |

20

1 Hyperbolic Geometry and Iteration Theory



From this (1.4.5) follows at once.

Remark 1.4.6 Let φ : D → D be holomorphic and let σ ∈ ∂D. If {z n } ⊂ D is a sequence converging to σ , then, by the triangle inequality and (1.4.5), 1 | log αφ (σ )| ≤ lim inf ω(z n , φ(z n )). n→∞ 2 Theorem 1.4.7 (Julia’s Lemma) Let φ : D → D be holomorphic and let σ ∈ ∂D. Assume αφ (σ ) < +∞. Then there exists a unique η ∈ ∂D such that for all R > 0 φ(E(σ, R)) ⊆ E(η, αφ (σ )R),

(1.4.8)

or, equivalently, for all z ∈ D, |η − φ(z)|2 |σ − z|2 ≤ α (σ ) . φ 1 − |φ(z)|2 1 − |z|2

(1.4.9)

Moreover, there exists z ∈ ∂ E(σ, R) ∩ D such that φ(z) ∈ ∂ E(η, αφ (σ )R) if and only if φ is an automorphism of D if and only if φ(E(σ, R)) = E(η, αφ (σ )R) for some—and hence every—R > 0. Finally, if {z n } ⊂ D is a sequence converging to σ such that lim sup n→∞

1 − |φ(z n )| < +∞, 1 − |z n |

(1.4.10)

then limn→∞ φ(z n ) = η. Proof If αφ (σ ) < ∞ then there exists a sequence {wk } ⊂ D converging to σ such that 1 − |φ(wk )| = αφ (σ ). lim k→∞ 1 − |wk | By (1.4.7), this is equivalent to saying that lim [ω(0, wk ) − ω(0, φ(wk )] =

k→∞

1 log αφ (σ ). 2

(1.4.11)

Note that limk→∞ |φ(wk )| = 1, thus, up to extracting converging subsequences, we can assume that limk→∞ φ(wk ) = η for some η ∈ ∂D. Let z ∈ E(σ, R). We want to show that φ(z) ∈ E(η, αφ (σ )R). By (1.4.2), this is equivalent to showing that lim [ω(φ(z), φ(wk )) − ω(0, φ(wk ))]
0. In order to show that η is the unique point with such a property, assume η˜ ∈ ∂D \ {η} satisfies φ(E(σ, R)) ⊂ E(η, ˜ αφ (σ )R) for all R > 0. In particular, for all R > 0, φ(E(σ, R)) ⊂ E(η, ˜ αφ (σ )R) ∩ E(η, αφ (σ )R). However, if η = η, ˜ there exists R > 0 such that E(η, ˜ αφ (σ )R) ∩ E(η, αφ (σ )R) = ∅, a contradiction. Therefore η = η˜ and the uniqueness is proved. Now, assume that φ(E(σ, R)) = E(η, αφ (σ )R) for some R > 0. By the Open Mapping Theorem, for every p ∈ ∂ E(η, αφ (σ )R) \ {η} there exists q ∈ ∂ E(σ, R) \ {σ } such that φ(q) = p. Assume that there exists z 0 ∈ ∂ E(σ, R) \ {σ } such that φ(z 0 ) ∈ ∂ E(η, αφ (σ )R). Let u σ , u η be the Poisson kernels defined in (1.4.3). Let v : D → R− be the harmonic function given by 1 u σ (z). v(z) := u η (φ(z)) − αφ (σ ) Since (1.4.8) is equivalent to (1.4.9) for all z ∈ D, by (1.4.3) it follows that (1.4.8) is equivalent to v(z) ≤ 0 (1.4.13) for all z ∈ D. Since z 0 ∈ ∂ E(σ, R) \ {σ } is such that φ(z 0 ) ∈ ∂ E(η, αφ (σ )R), it follows that v(z 0 ) = 0 and by the Maximum Principle for harmonic functions, v ≡ 0. By (1.4.3), this is equivalent to  Re

1 z+σ φ(z) + η − φ(z) − η αφ (σ ) z − σ

 = 0.

Since the function D  z → φ(z)+η − αφ1(σ ) z+σ is holomorphic, the previous equaφ(z)−η z−σ tion implies that there exists b ∈ R such that for all z ∈ D, φ(z) + η 1 z+σ = + ib. φ(z) − η αφ (σ ) z − σ

(1.4.14)

22

1 Hyperbolic Geometry and Iteration Theory

Let Φ : H → H be the automorphism given by Φ(w) = can be rewritten as φ = Cη−1 ◦ Φ ◦ Cσ ,

1 w αφ (σ )

− ib. Then (1.4.14)

where Cσ and Cη are the Cayley transforms introduced in (1.1.2), proving that φ is an automorphism of D. Finally, let {z n } ⊂ D be a sequence converging to σ which satisfies (1.4.10). Up to extracting subsequences, we can assume limn→∞ φ(z n ) = η for some η ∈ D. Clearly, (1.4.10) implies η ∈ ∂D. Then, repeating the previous argument substituting {wk } with {z k }, one can prove that φ(E(σ, R)) ⊆ E(η , L R) where L = n )| . But then, for all R > 0 it holds φ(E(σ, R)) ⊆ E(η , L R) ∩ lim supn→∞ 1−|φ(z 1−|z n | E(η, αφ (σ )R). Since for R small enough E(η , L R) ∩ E(η, αφ (σ )R) = ∅ if η = η ,  this implies that η = η . Remark 1.4.8 If φ is an automorphism of D, the point η ∈ ∂D in Theorem 1.4.7 is just η = φ(σ ). In the following we will need to consider horocycles at infinity in the right halfplane H. Let σ ∈ ∂D and let Cσ : D → H be the Cayley transform with respect to σ given by (1.1.2). Then Cσ (σ ) = ∞ and Cσ maps the horocycles of center σ onto unbounded open subsets of H which we call horocycles of H at ∞. An explicit computation shows that for R > 0 E H (∞, R) := Cσ (E(σ,

1 )) = {w ∈ H : Re w > R}. R

(1.4.15)

1.5 Non-Tangential Limits and Lindelöf’s Theorem In studying boundary behavior of holomorphic maps, it is useful to define different approaching regions to boundary points. The most common approaching ways to the boundary are the radial limits, the non-tangential limits and the unrestricted limits, which we are going to define. Recall that d S denotes the spherical distance in C∞ . Also, we say that a sequence {z n } ⊂ D converges non-tangentially to σ ∈ ∂D if limn→∞ z n = σ and there exists C > 1 such that |σ − z n | < C(1 − |z n |) for all n ∈ N. Definition 1.5.1 Let f : D → C be a function and let σ ∈ ∂D. We say that L ∈ C∞ is the radial limit of f at σ , and we write lim

(0,1)r →1

f (r σ ) = L

if lim(0,1)r →1 d S ( f (r σ ), L) = 0. We say that L ∈ C∞ is the non-tangential limit of f at σ , and we write ∠ lim f (z) = L , z→σ

1.5 Non-Tangential Limits and Lindelöf’s Theorem

23

Fig. 1.1 Stolz region

if limn→∞ d S ( f (z n ), L) = 0 for every sequence {z n } ⊂ D converging nontangentially to σ . Finally, we say that f has unrestricted limit (or just limit) L ∈ C∞ at σ , and we write lim f (z) = L z→σ

if lim z→σ d S ( f (z), L) = 0. As customary, if σ ∈ ∂D and f : D → C we say that a (radial, non-tangential or unrestricted) limit L is finite if L ∈ C. We say that the limit L is infinite if L = ∞. A Stolz region S(σ, R) of vertex σ ∈ ∂D and amplitude R > 1 is given by (see Fig. 1.1) |σ − z| < R}. S(σ, R) := {z ∈ D : 1 − |z| Hence a function f : D → C has non-tangential limit L at σ ∈ ∂D if and only if the unrestricted limit (in C∞ ) of f | S(σ,R) at σ is L for all R > 1. Lemma 1.5.2 Let σ ∈ ∂D. (1) S(σ, R) ⊂ E(σ, R 2 ) for every R > 1, (2) for every R > 0 and for every M > 1 there exists δ > 0 such that S(σ, M) ∩ {z ∈ D : |z − σ | < δ} ⊂ E(σ, R). Proof Since |σ − z|2 = 1 − |z|2 and

1−|z| 1+|z|



|σ − z| 1 − |z|

2

≤ 1, the two statements follow at once.

·

1 − |z| , 1 + |z| 

When dealing with boundary behavior of holomorphic functions, it is often convenient to use biholomorphisms to move D to other simply connected domains in

24

1 Hyperbolic Geometry and Iteration Theory

such a way that a point σ ∈ ∂D is mapped to ∞, and it is then important to understand the shape of non-tangential approach regions in other contexts. For instance: Remark 1.5.3 Let Cσ : D → H be the Cayley transform with respect to σ ∈ ∂D defined in (1.1.2). Then Cσ (σ ) = ∞ and, since Cσ is a Möbius transformation, it follows that a sequence {z n } ⊂ D converges to σ non-tangentially if and only if {Cσ (z n )} converges to ∞ and there exists M > 0 such that |Im Cσ (z n )| ≤ MRe Cσ (z n ) for all n ∈ N. Remark 1.5.4 A direct computation using (1.4.1) shows that for every M > 1 and σ ∈ ∂D, S(σ, M) = {z ∈ D : lim [ω(z, w) − ω(0, w)] + ω(0, z) < log M}. w→σ

Stolz regions can be also describe in terms of geodesics for the hyperbolic metric (see Sect. 6.2). Stolz regions are well behaving under holomorphic self-maps of the unit disc at boundary points where the boundary dilation coefficient is finite: Proposition 1.5.5 Let φ : D → D be holomorphic. Let σ ∈ ∂D and assume that αφ (σ ) < +∞. Let η ∈ ∂D be given by Theorem 1.4.7 and set

K := 4 Then for every M > 1,

αφ (σ ) . 1 − |φ(0)|2

φ(S(σ, M)) ⊂ S(η, K M).

Proof Let first assume that φ(0) = 0. Then by (1.4.9) |σ − z|2 (1 − |z|)2 |η − φ(z)|2 (1 − |φ(z)|)2 · ≤ αφ (σ ) · , 2 2 (1 − |φ(z)|) 1 − |φ(z)| (1 − |z|)2 1 − |z|2 hence, by Schwarz’ lemma (see Theorem 1.2.1), 1 − |z| |η − φ(z)|2 |σ − z|2 1 + |φ(z)| |σ − z|2 · ≤ α ≤ α (σ ) · (σ ) . φ φ (1 − |φ(z)|)2 (1 − |z|)2 1 + |z| 1 − |φ(z)| (1 − |z|)2  Taking the square root, we obtain φ(S(σ, M)) ⊂ S(η, αφ (σ )M). Assume now that φ(0) = a = 0. Let Ta be the automorphism of D defined in (1.2.1). Let f := Ta ◦ φ : D → D. Then f (0) = 0. Moreover, let {z n } ⊂ D be a n )| . By Theorem 1.4.7, sequence converging to σ such that αφ (σ ) = limn→∞ 1−|φ(z 1−|z n | {φ(z n )} is converging to η. By Example 1.4.4, L := αTa (η) =

1−|a|2 . |1−aη|2

Hence,

1.5 Non-Tangential Limits and Lindelöf’s Theorem

25

1 − | f (z)| 1 − |Ta (φ(z n ))| 1 − |φ(z n )| ≤ lim · z→σ n→∞ 1 − |z| 1 − |φ(z n )| 1 − |z n | 1 − |a|2 = · αφ (σ ) = Lαφ (σ ) < +∞. |1 − aη|2

α f (σ ) = lim inf

By Theorem 1.4.7, there exists p ∈ ∂D such that f (E(σ, R)) ⊂  E( p, α f (σ )R) for all R > 0 and, for what we already proved, f (S(σ, M)) ⊂ S( p, α f (σ )M) for all M > 1. Therefore, recalling that Ta = Ta−1 , for every M > 1 it holds   φ(S(σ, M)) ⊂ Ta (S( p, α f (σ )M)) ⊆ Ta (S( p, αφ (σ )L M)).

(1.5.1)

Moreover, |Ta ( p) − Ta (z)| |Ta ( p) − Ta (z)| |1 − az| |z − p| = (1 + |Ta (z)|) ≤ 2 1 − |Ta (z)| 1 − |Ta (z)|2 1 − |z| (1 + |z|)|1 − a p| | p − z| 4 . ≤ |1 − a p| 1 − |z| 4 Therefore, Ta (S( p, N )) ⊂ S(Ta ( p), |1−a N ) for all N > 1. p| We claim that Ta ( p) = η. By Theorem 1.4.7 and Remark 1.4.8, for all R > 0,

f (E(σ, R)) = Ta (φ(E(σ, R))) ⊂ Ta (E(η, αφ (σ )R)) = E(Ta (η), αTa (η)αφ (σ )R). Thus, by the uniqueness statement in Theorem 1.4.7, it follows that p = Ta (η), and hence Ta ( p) = η. Finally, let

√  αφ (σ ) 4 . K := αφ (σ ) L =4 |1 − a p| 1 − |φ(0)|2 Hence, by (1.5.1), it holds φ(S(σ, M)) ⊂ S(η, K M) for all M > 1, and we are done.  One of the basic results about the boundary behavior of holomorphic functions is the Lindelöf Theorem. We start with a lemma: Lemma 1.5.6 Let a > 0 and let a := {z ∈ C : |Re z| < a}. Let f : a → C be a bounded holomorphic function. Let γ : [0, 1) → a be a continuous curve such that limt→1 Im γ (t) = +∞. Suppose that limt→1 f (γ (t)) = L ∈ C exists. Then for every 0 < δ < a it holds limRy→+∞ f (x + i y) = L uniformly in |x| < a − δ. Proof Without loss of generality, up to composition with affine transformations, we can assume a = 1, L = 0 and | f (z)| ≤ 1 for all z ∈ a . Set  := 1 . Fix δ ∈ (0, 1) and let 0 < ε < 1. Then there exists t0 ∈ (0, 1) such that | f (γ (t))| < ε for all t ∈ (t0 , 1). Let y0 := maxt∈[0,t0 ] Im γ (t). Hence, if Im γ (t) > y0 , necessarily t > t0 , and then | f (γ (t))| < ε. We claim that for all y > y0 and all |x| ≤ 1 − δ it holds

26

1 Hyperbolic Geometry and Iteration Theory δ

| f (x + i y)| ≤ ε 4 .

(1.5.2)

The statement of the lemma follows clearly from (1.5.2). In order to prove (1.5.2), fix y1 > y0 . We are going to construct an auxiliary function, which depends on y1 such that, using the Maximum Principle, allows to obtain the needed estimate. Up to a vertical translation we can assume y1 = 0. Let t1 ∈ [t0 , 1) be such that Im γ (t1 ) = 0 and Im γ (t) > 0 for t ∈ (t1 , 1) and let E := {γ (t) : t ∈ [t1 , 1)}. Let E be the reflection of E through the real axis. Assume first that x ∈ (Re γ (t1 ), 1 − δ]. Let μ > 0 and define gμ :  → C by 1+z

f (z) f (z)ε 2 . 1 + μ(1 + z)

gμ (z) :=

(1.5.3)

The map gμ is holomorphic.Moreover,  | f (z)| < 1 for all z ∈ , hence | f (z)| < 1 1+z   1 for all z ∈ , |ε 2 | < 1 and  1+μ(1+z)  < 1 for all z ∈ . Hence, |gμ (z)| < 1 for all z ∈ . Moreover, | f (z)| < ε for z ∈ E and | f (z)| < ε for z ∈ E. Hence |gμ (z)| < ε on E ∪ E. Also, lim sup |gμ (z)| ≤ lim sup |ε z∈,Re z→1

z∈,|Im z|→∞

| = lim sup eRe (

z∈,Re z→1

Finally, lim sup

1+z 2

|gμ (z)| ≤

1+z 2 ) log ε

= .

z∈,Re z→1

1 = 0. z∈,|Im z|→∞ |1 + μ(1 + z)| lim sup

Hence, if  is the connected component of  \ (E ∪ E) containing x, since  cannot intersect {z ∈ C : Re z = −1}, it follows that gμ | :  → D is bounded by ε close to ∂ and lim z∈,|Im z|→∞ |gμ (z)| = 0. Thus, by the Maximum Principle, |gμ (z)| ≤ ε for all z ∈ . In particular, |gμ (x)| ≤ ε, and from (1.5.3) we obtain | f (x)|2 ≤ ε

1 + μ(1 + x) ε

1+x 2

.

Taking the limit for μ → 0 and bearing in mind that |x| ≤ 1 − δ we get | f (x)|2 ≤ ε

1−x 2

δ

≤ ε2,

and (1.5.2) is proved for the case x ∈ (Re γ (t1 ), 1 − δ]. In case x ∈ [−1 + δ, Re γ (t1 )), one can argue exactly as before using the function 1−z

g˜ μ (z) := instead of gμ .

f (z) f (z)ε 2 1 + μ(1 − z) 

1.5 Non-Tangential Limits and Lindelöf’s Theorem

27

Theorem 1.5.7 (Lindelöf’s Theorem) Let f : D → C be a holomorphic function. Assume that f (D) is contained in a half-plane. Let γ : [0, 1) → D be a continuous curve such that limt→1 γ (t) = σ ∈ ∂D. Suppose that limt→1 f (γ (t)) = L ∈ C∞ exists. Then ∠ lim z→σ f (z) = L. Proof Let H ⊂ C be a half-plane such that f (D) ⊆ H . Then there exists a Möbius transformation C such that C(D) = H . The map g := C −1 ◦ f : D → D is holomorphic and limt→1 g(γ (t)) = C −1 (L) ∈ D exists. If we prove that g has non-tangential limit C −1 (L) at σ , it follows at once that f has non-tangential limit L at σ . Therefore, we can assume that f is bounded. Let log : H → C denote the principal value of the logarithm on the right halfplane H. Note that log : H → {z ∈ C : |Im z| < π/2} is a biholomorphism. Let Cσ be the Cayley transform with respect to σ and let h : D → C be the univalent map defined by σ +z 2i 2i log Cσ (z) = log . (1.5.4) h(z) := π π σ −z Then  := h(D) = {z ∈ C : |Re z| < 1}. Let γ˜ (t) := h(γ (t)). The continuous curve γ˜ : [0, 1) →  satisfies limt→1 Im γ˜ (t) = +∞. Let f˜ := f ◦ h −1 :  → C. The map f˜ is bounded, holomorphic and f˜(γ˜ (t)) → L as t → 1. Using Remark 1.5.3, it is not difficult to see that a sequence {z n } ⊂ D converges non-tangentially to σ if and only if there exists δ ∈ (0, 1) such that, setting xn := Re h(z n ) and yn := Im h(z n ), it holds yn → +∞ and |xn | ≤ 1 − δ for all n ∈ N. The statement of the theorem is then equivalent to proving that for every δ ∈ (0, 1) it holds lim y→∞ f˜(x + i y) = L, in |x| ≤ 1 − δ, and this follows from Lemma 1.5.6.  The previous proof can be easily adapted to obtain a similar result for maps which are not bounded, but are bounded on Stolz regions: Proposition 1.5.8 Let f : D → C be a holomorphic function. Let σ ∈ ∂D. Assume that for every R > 1 there exists M R > 0 such that | f (z)| ≤ M R for all z ∈ S(σ, R). Also, suppose that lim(0,1)r →1 f (r σ ) = L ∈ C exists. Then ∠ lim z→σ f (z) = L. Proof By Lemma 1.5.2, S(σ, M) ⊂ E(σ, M 2 ), and E(σ, M 2 ) is affinely equivalent to D. Thus, up to restrict f to E(σ, M 2 ), we can assume that f extends continuously at every p ∈ ∂D \ {σ }. Fix M > 1. We want to show that lim S(σ,M)z→σ f (z) = L. Let Cσ : D → H be the Cayley transform with respect to σ . For β ∈ (0, π/2) let V (β) := {z ∈ H : |Arg(z)| < β}. Since Cσ is a Möbius transformation, it is easy to see that there exist M  > M, N > 0 and β, β  ∈ (0, π/2), β < β  , such that Cσ (S(σ, M)) ⊂ V (β) and (V (β  ) ∩ {w ∈ H, Re w > N }) ⊂ Cσ (S(σ, M  )). Hence, f ◦ Cσ−1 : H → C is bounded on V (β  ). Let log : H → C denote the principal value of the logarithm on the right half-plane H. There exists a ∈ (0, 1) such that 2iπ log(V (β  )) = a := {z ∈ C : |Re z| < a} and 2iπ log(V (β)) = b for some

28

1 Hyperbolic Geometry and Iteration Theory

0 < b < a. Let h be given by (1.5.4). The function f ◦ h −1 |a : a → C and the curve (0, 1)  r → h(r σ ) satisfy the hypotheses of Lemma 1.5.6, and arguing as in the proof of Theorem 1.5.7 the result follows. 

1.6 Poisson Integral and Fatou’s Theorem In this section, if f is a real valued integrable function on ∂D, we denote its L 1 (∂D)norm by  2π | f (eiθ )|dθ.  f 1 := 0

We use the same notation  f 1 to denote the L 1 ([a, b])-norm of any integrable real valued function f : [a, b] → R, a < b. Moreover, if A ⊂ ∂D is a (Lebesgue) measurable subset, we denote by λ(A) its Lebesgue measure. In other words, if χ A : ∂D → R is the characteristic function of A defined by χ A ( p) = 1 if p ∈ A, χ A ( p) = 0 otherwise,  2π

λ(A) =

χ A (eiθ )dθ.

0

Definition 1.6.1 Let f be a real integrable function on ∂D. The Poisson integral of f is 1 P[ f ](z) := 2π





2π

Re 0

eiθ + z eiθ − z



1 f (e )dθ = − 2π



2π

iθ

u eiθ (z) f (eiθ )dθ,

0

where u eiθ is the negative Poisson kernel with pole at eiθ , see (1.4.3). By the Lebesgue Dominated Convergence Theorem, D  z →

1 2π



2π 0

eiθ + z f (eiθ )dθ eiθ − z

is holomorphic. Its real part is P[ f ](z). Hence, D  z → P[ f ](z) is harmonic. Moreover, fix z ∈ D. Then, ∞ eiθ + z = 1 + 2 e−iθn z n , θ ∈ R. eiθ − z n=1

Since the convergence is uniform in θ ∈ [0, 2π ], 1 2π

 0

2π

 ∞ eiθ + z 1 n 2π −inθ dθ = 1 + 2 z e dθ = 1. eiθ − z 2π 0 n=1

1.6 Poisson Integral and Fatou’s Theorem

Therefore 1 2π



29



2π

Re 0

eiθ + z eiθ − z

 dθ = 1.

(1.6.1)

Theorem 1.6.2 Let f be a real-valued integrable function on ∂D. Let θ0 ∈ R. Suppose there exist f + (eiθ0 ) := lim+ f (ei(θ0 +θ) ) and f − (eiθ0 ) := lim− f (ei(θ0 +θ) ). θ→0

θ→0

Then lim P[ f ](r eiθ0 ) =

r →1

1  + iθ0 f (e ) + f − (eiθ0 ) . 2

(1.6.2)

Moreover, if f + (eiθ0 ) = f − (eiθ0 ), that is, if f is continuous at eiθ0 , then lim P[ f ](z) = f (eiθ0 ).

z→eiθ0

(1.6.3)

In particular, if f is continuous on ∂D, then P[ f ] extends continuously on ∂D and the extension coincides with f on ∂D.  Proof Write A := 21 f + (eiθ0 ) + f − (eiθ0 ) . Let u(z) := P[ f ](z). Let ε > 0. Then there is δ > 0 such that | f (eiθ ) − f + (eiθ0 )| < ε/4, (1.6.4) for all θ0 < θ < θ0 + δ, and | f (eiθ ) − f − (eiθ0 )| < ε/4,

(1.6.5)

for all θ0 − δ < θ < θ0 . Write I1 := (θ0 − δ, θ0 ), I2 := (θ0 , θ0 + δ) and I := (θ0 − δ, θ0 + δ). Let {rn } in (0, 1) be a sequence converging to 1 and let z n := rn eiθ0 . Clearly 

1 − |z n |2 iθ : n ∈ N, e ∈ ∂D \ I < +∞. sup |eiθ − z n |2 Therefore, the Lebesgue Dominated Convergence Theorem shows that 1 lim n→∞ 2π



1 − |z n |2  f (eiθ ) − A dθ = iθ 2 [0,2π]\I |e − z n |  1 − |z n |2  1 f (eiθ ) − A dθ = 0. lim iθ = 2 2π [0,2π]\I n→∞ |e − z n |

In particular, there is N ∈ N such that if n ≥ N , then    1   2π

[0,2π]\I

  1 − |z n |2  iθ  ≤ ε/2. f (e ) − A dθ  |eiθ − z n |2

(1.6.6)

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1 Hyperbolic Geometry and Iteration Theory

Moreover, for all z ∈ D, (1.6.1) and (1.6.4) imply      ε 1  1 1 − |z|2 1 − |z|2 iθ + iθ0   ≤ ( f (e ) − f (e ))dθ dθ ≤ ε/4.  4 2π  2π iθ 2 iθ 2 I2 |e − z| I |e − z| (1.6.7) Similarly, using (1.6.5) instead of (1.6.4), for all z ∈ D,      ε 1  1 1 − |z|2 1 − |z|2 iθ − iθ0 ≤  ( f (e ) − f (e ))dθ dθ ≤ ε/4.  4 2π  2π iθ 2 iθ 2 I1 |e − z| I |e − z| (1.6.8) Now, since z n = rn eiθ0 , and |eiθ − rn |2 = |e−iθ − rn |2 for all θ ∈ R, we have  I1

 δ 1 − rn2 1 − rn2 dθ = dθ i(θ0 +θ) − r eiθ0 |2 iθ 2 n 0 |e 0 |e − r n |  δ  δ 1 − rn2 1 − rn2 = dθ = dθ −iθ − r |2 i(θ0 −θ) − r eiθ0 |2 n n 0 |e 0 |e  1 − |z n |2 dθ. = iθ 2 I2 |e − z n |

1 − |z n |2 dθ = |eiθ − z n |2



δ

Therefore, 1 2π

 I1

1 − |z n |2 1 dθ = iθ 2 |e − z n | 2π

 I2

1 − |z n |2 1 1 dθ = iθ 2 |e − z n | 2 2π

 I

1 − |z n |2 dθ. |eiθ − z n |2

Thus 

1 − |z n |2 ( f (eiθ ) − A)dθ = iθ 2 I |e − z n |  1 − |z n |2 1 ( f (eiθ ) − f − (eiθ0 ))dθ + = iθ − z |2 2π |e n I1   1 − |z n |2 1 − |z n |2 iθ + iθ0 − iθ0 + ( f (e ) − f (e ))dθ + f (e ) dθ + iθ 2 iθ 2 I2 |e − z n | I1 |e − z n |    1 − |z n |2 1 − |z n |2 + f + (eiθ0 ) dθ − A dθ iθ 2 iθ 2 I2 |e − z n | I |e − z n |  1 − |z n |2 1 ( f (eiθ ) − f − (eiθ0 ))dθ + = iθ 2 2π I1 |e − z n |   1 − |z n |2 iθ + iθ0 + ( f (e ) − f (e ))dθ . iθ 2 I2 |e − z n |

1 2π

1.6 Poisson Integral and Fatou’s Theorem

31

Therefore, given n ≥ N and using (1.6.1), (1.6.6), (1.6.7), and (1.6.8)    2π  2π  1  1 − |z n |2 1 − |z n |2 1 iθ  |u(z n ) − A| =  f (e )dθ − Adθ  iθ 2 iθ 2 2π 0 |e − z n | 2π 0 |e − z n |    2π 2   1  1 − |z | n iθ  = f (e ) − A dθ  2π 0 |eiθ − z n |2     1  1 − |z n |2 iθ + ≤  ( f (e ) − A)dθ  iθ 2 2π [0,2π]\I |e − z n |     1  1 − |z n |2 iθ  + ( f (e ) − A)dθ  2π |eiθ − z |2 n

I

≤ ε/2 + /2 = ε (1.6.9) proving that (1.6.2) holds. In case f is continuous at eiθ0 , the proof is similar and easier. Indeed, using the notations previously introduced, A = f (eiθ0 ). This time, we let {z n } be any sequence in D such that limn→∞ z n = eiθ0 . Arguing as before, we obtain (1.6.6), and      ε 1  1 1 − |z|2 1 − |z|2 iθ iθ0   ≤ ( f (e ) − f (e ))dθ dθ ≤ ε/2.  2 2π  2π iθ 2 iθ 2 I |e − z| I |e − z| 

Therefore, we can conclude as in (1.6.9).

Corollary 1.6.3 Let u be a real continuous function on D which is harmonic in D. Then u(z) = P[u](z) for all z ∈ D. Proof The function w(z) := u(z) − P[u](z) is harmonic in D and by Theorem 1.6.2, w is continuous on D and w(eiθ ) = 0 for all θ ∈ R. Hence, by the Maximum Principle for harmonic functions, w(z) = 0 for all z ∈ D, and we are done.  If the function f in the above theorem is not continuous at eiθ0 ∈ ∂D, then the limit (1.6.3) might not exist. Nevertheless, the celebrated result known as Fatou’s Theorem guarantees that non-tangential limits exist at almost every point of the boundary of the unit disc. In order to prove such a result, we need some preliminary lemmas. For each R > 1, the non-tangential maximal function of u : D → C at ζ ∈ ∂D is u ∗R (ζ ) := sup |u(z)|.

(1.6.10)

z∈S(ζ,R)

Clearly, if u has a finite non-tangential limit at ζ , then u ∗R (ζ ) < +∞ for all R > 1. Given a real-valued integrable function f on ∂D and ζ ∈ ∂D, let M f (ζ ) := sup I ζ

1 λ(I )

 | f | dθ I

32

1 Hyperbolic Geometry and Iteration Theory

where the supremum is taken over all open arcs I ⊂ ∂D that contains ζ . The function M f is called the Hardy-Littlewood maximal function of f . Lemma 1.6.4 Let f be a real valued integrable function on ∂D. Then, for every R > 1 and ζ ∈ ∂D, P[ f ]∗R (ζ ) ≤ 2(1 + R)M f (ζ ). (1.6.11) Proof Let u := P[ f ]. Replacing f by | f |, we may assume that f ≥ 0. Also, without loss of generality, we may assume ζ = 1. Fix z = r eiθ0 ∈ S(1, R) with θ0 = Arg(z). To simplify the construction, we assume that θ0 ≥ 0 (the other case is similar). Define  1+r qz (θ ) :=

1−r

,

max

2 1−r 2 , 1−r |eiθ −z|2 |e−iθ −z|2

 |θ | ≤ θ0 , , θ0 < |θ | ≤ π.

The function qz has the following properties: (1) qz is an even function on [−π, π ], (2) qz is a decreasing function on [0, π ], 2 (3) qz (θ ) ≥ |e1−r iθ −z|2 for all θ ∈ [−π, π ]. We still denote by qz the 2π -periodic extension of qz . Take now a sequence {tn } formed by all rational numbers in the interval (θ0 , π ). For each n, take θ1n < θ2n < · · · < θnn such that {t1 , t2 , ..., tn } = {θ1n , θ2n , ..., θnn }. Let θ0n := θ0 . For j = 0, ..., n − 1, let cnj := qz (θ nj ) − qz (θ nj+1 ), cnn := qz (θnn ) − qz (π ). The monotonicity of qz implies that cnj > 0 for all j and k. Consider the step function kn : [−π, π ] → R given by kn (θ ) = qz (π ) +

n

cnj χ(−θ nj ,θ nj ) (θ ).

j=0

The graph of k5 is drawn in Fig. 1.2. As we did with the function, qz , with a little abuse of notation, we still denote by kn the 2π -periodic extension of kn . Note that, kn+1 ≥ kn and the sequence {kn } converges pointwise to qz . Moreover, it is clear that n 2θ nj cnj . kn 1 = 2πqz (π ) + j=0

Therefore, 

π

−π

 f (eiθ )kn (θ ) dθ = qz (π )

π −π

f (eiθ ) dθ +

≤ 2πqz (π )M f (1) +

n j=0

n j=0

 cnj

θ nj −θ nj

f (eiθ ) dθ

cnj 2θ nj M f (1) = kn 1 M f (1).

1.6 Poisson Integral and Fatou’s Theorem

33

Fig. 1.2 Graph of the function k5

By the Monotone Convergence Theorem, 1 2π



π −π

f (eiθ )

1 − r2 1 qz 1 M f (1). dθ ≤ iθ 2 |e − z| 2π

(1.6.12)

Now, we estimate qz 1 . Assume that −π/2 ≤ θ0 ≤ π/2, θ0 = 0. Let β be the angle formed by the real segment joining 0 to z and the real segment joining z to 1. By the law of sines (applied to the triangle with vertices 0, 1 and z), we have sin θ0 = sin1 β . Therefore |1−z| |θ0 | Rπ sin θ0 Rπ Rπ |θ0 | ≤R ≤ = sin β ≤ . 1−r |1 − z| 2 |1 − z| 2 2 The previous inequality trivially holds if θ0 = 0. If π/2 ≤ |θ0 | ≤ π and z ∈ S(1, R), then |1 − z| ≥ 1 and |θ0 | |θ0 | ≤R ≤ Rπ. 1−r |1 − z| Hence, using (1.6.1), 

π

1 − r2 1+r + 4π ≤ 4π(1 + R). dθ ≤ 2|θ0 | iθ 2 1−r |θ0 | |e − z| (1.6.13) Finally, by (3), (1.6.12), and (1.6.13), qz 1 = 2|θ0 |

1+r +2 1−r

 2π  π 1 1 − |z|2 1 − |z|2 1 iθ f (e ) dθ = f (eiθ ) dθ iθ 2 2π 0 |e − z| 2π −π |eiθ − z|2  π 1 qz (θ ) f (eiθ ) dθ ≤ 2(1 + R)M f (1). ≤ 2π −π

u(z) =

34

1 Hyperbolic Geometry and Iteration Theory

By the arbitrariness of z ∈ S(1, R), Eq. (1.6.11) follows.



Lemma 1.6.5 Let μ be a positive Borel measure on ∂D and let {I j } be a finite sequence of open arcs in ∂D. Then {I j } contains a pairwise disjoint subfamily {Jk } such that  μ(Jk ). μ ∪I j ≤ 3 Proof We may assume that no I j is contained in the union of the others. Writing I j = {eiθ : θ ∈ (a j , b j )} we may also assume that 0 ≤ a1 < a2 < · · · < an < 2π. Then b j+1 > b j , because otherwise I j+1 ⊂ I j , and b j−1 < a j+1 , because otherwise I j ⊂ I j−1 ∪ I j+1 . If n > 1, then bn < b1 + 2π and bn−1 < a1 + 2π . Therefore, the family of even-numbered arcs I j is pairwise disjoint. The family of odd-numbered arcs I j is almost pairwise disjoint because only the first and last arcs—that is, I1 and In —can intersect. If 1  μ(I j ) ≥ μ ∪I j , 3 j even taking the even-numbered arcs as subfamily {Jk }, we are done. Otherwise,

μ(I j ) ≥

j odd

In that case, if μ(I1 ) ≤

2  μ ∪I j . 3

1 μ(I j ), 2 j odd

we take for {Jk } the family of odd-numbered arcs, omitting I1 , while if μ(I1 ) > we take {Jk } to be the single arc {I1 }.

1 μ(I j ), 2 j odd 

Proposition 1.6.6 Let f be a real valued integrable function on ∂D. Then, for every α > 0, 3 f 1 . (1.6.14) λ({ζ ∈ ∂D : M f (ζ ) > α}) ≤ α Proof Let K be a compact subset of E α := {ζ ∈ ∂D : M f (ζ ) > α}. For each ζ ∈ E α there is an open arc I such that ζ ∈ I and

1.6 Poisson Integral and Fatou’s Theorem

λ(I )
1 and α > 0 λ({ζ ∈ ∂D : P[ f ]∗R (ζ ) > α}) ≤ 6

1+ R  f 1 . α

(1.6.16)

Proof Inequality (1.6.16) follows from Lemma 1.6.4 and Proposition 1.6.6. Let us show that (1.6.16) implies (1.6.15). If h is a real valued integrable function on ∂D, for each ζ ∈ ∂D such that h(ζ ) ∈ R, we set Wh,R (ζ ) := lim sup |v(z) − h(ζ )|, S(ζ,R)z→ζ

where v := P[h] is the Poisson integral of h. Then Wh,R (ζ ) ≤ v∗R (ζ ) + |h(ζ )|. Let α > 0 and let Aα := {ζ ∈ ∂D : |h(ζ )| ≥ α}. Since   α dθ ≤ |h| dθ ≤ h1 , αλ(Aα ) = Aα

Aα

we get λ({ζ : |h(ζ )| > α}) ≤

h1 . α

Therefore, by (1.6.16) λ({ζ : Wh,R (ζ ) > α}) ≤ λ({ζ : v∗R (ζ ) > α/2}) + λ({ζ : |h(ζ )| > α/2}) 7 + 6R h1 . ≤2 α (1.6.17)

36

1 Hyperbolic Geometry and Iteration Theory

Fix ε > 0. Since real valued continuous functions on ∂D are dense in L 1 (∂D), there exists g ∈ C(∂D) such that  f − g1 ≤ ε2 . By Theorem 1.6.2, Wg,R ≡ 0, and hence W f,R = W f −g,R . Applying (1.6.17) to the function f − g we obtain λ({ζ : W f,R (ζ ) > ε}) ≤ 2

(7 + 6R)ε2 = 2(7 + 6R)ε. ε

Therefore, the set E R ⊂ ∂D of points ζ ∈ ∂D such that lim sup |P[ f ](z) − f (ζ )| > 0 S(ζ,R)z→ζ

has Lebesgue measure zero. Since S(ζ, R) ⊂ S(ζ, R  ) for 1 < R < R  , it follows that ∠ lim z→ζ P[ f ](z) = f (ζ ) for all ζ ∈ ∂D \ (∪n∈N E n+1 ). Since the latter set has full measure, we are done.  Proposition 1.6.8 Let u : D → R be a bounded harmonic function. Then (1) For almost every ζ ∈ ∂D, there exists u ∗ (ζ ) := ∠ lim u(z), z→ζ

(1.6.18)

(2) u ∗ ∈ L ∞ (∂D), (3) u is the Poisson integral of u ∗ , that is, 1 u(z) = 2π



2π 0

1 − |z|2 ∗ iθ u (e )dθ, z ∈ D. |eiθ − z|2

(1.6.19)

Proof Let {rn } be a sequence in [0, 1] converging to 1. Let u n (z) := u(rn z) for all z ∈ D. Hence the u n ’s are continuous on D and harmonic in D. By Corollary 1.6.3, for all z ∈ D (1.6.20) u(rn z) = P[u n ](z). Since the {u n }’s have L ∞ (∂D)-norm uniformly bounded by u∞ , by the BanachAlaoglu Theorem, there exists u ∗ ∈ L ∞ (∂D) such that u ∗ ∞ ≤ u∞ and {u n |∂D } converges, up to extracting subsequences, in the weak-∗ topology to a function u ∗ ∈ L ∞ (∂D). Since the linear functional L ∞ (∂D)  v → P[v](z) is continuous, it follows by (1.6.20) that for all z ∈ D, u(z) = lim u(rn z) = lim P[u n ](z) = P[u ∗ ](z). n→∞

n→∞

Hence, u is the Poisson integral of u ∗ , and (1) follows at once by Theorem 1.6.7.  Proposition 1.6.9 Let f : D → C be holomorphic and assume that f (D) is contained in a half-plane. Then for almost every σ ∈ ∂D there exists f ∗ (σ ) :=

1.6 Poisson Integral and Fatou’s Theorem

37

∠ lim z→σ f (z). Moreover, f ∗ is constant on a set of non-zero measure on ∂D if and only if f is constant. Proof Let H be a half-plane such that f (D) ⊂ H . Let C be a Möbius transformation such that C(H ) = D. Clearly, it is enough to prove the result for C ◦ f . Hence, we can assume that f (D) ⊂ D. Since Re f and Im f are bounded real valued harmonic function, it follows by Proposition 1.6.8 that f has non-tangential limit almost everywhere on ∂D. In order to prove the second part we argue by contradiction. Therefore, we can assume that f ∗ = 0 on a set of positive Lebesgue measure A and f ≡ 0. Up to replacing f by fz(z) n , where n is the order of f at zero, we can also assume that f (0) = 0. For 0 < r < 1, let fr (z) := f (r z), z ∈ D. For what we already proved, fr |∂D converges to f ∗ a.e. in ∂D. By Fatou’s Lemma 1 2π



2π

 2π 1 | log | fr (eiθ )|| dθ r →1 2π 0    2π 1 ≤ lim inf − log | fr (eiθ )| dθ . r →1 2π 0

| log | f ∗ (eiθ )|| dθ ≤ lim inf

0

(1.6.21)

Fix r ∈ (0, 1) such that f has no zeros on |z| = r and let {a1 , a2 , ..., an } be the zeros of z−a f in the open disc r D, listed according to their multiplicities. Write B j (z) = r r 2 −a jj z , z ∈ D. Notice that |B j (0)| = |a j |/r < 1 and |B j (w)| = 1 whenever |w| = r . The  function F(z) := f (z)/( nj=1 B j (z)) is holomorphic in a neighborhood of r D and it has no zeros in r D. Then log |F| is harmonic in r D. Therefore ⎛ log | f (0)| = log ⎝|F(0)|

n 

⎞ |B j (0)|⎠ = log |F(0)| + log

j=1

≤ log |F(0)| =

1 2πr



n 

|B j (0)|

j=1 2π

log |F(r eiθ )| dθ =

0

1 2πr



2π

log | fr (eiθ )| dθ.

0

Taking the liminf as r → 1, the previous inequality and (1.6.21) imply −∞ < log | f (0)| ≤ Hence 1 2π



2π

1 2π



2π

log | fr (eiθ )| dθ.

0

| log | f ∗ (eiθ )|| dθ ≤ − log | f (0)| < +∞.

0

Therefore log | f ∗ | ∈ L 1 (∂D) and in particular it assumes finite values almost everywhere on ∂D. However, by hypothesis f ∗ is zero on the set of positive Lebesgue measure A, a contradiction, and we are done. 

38

1 Hyperbolic Geometry and Iteration Theory

1.7 Angular Derivatives and Julia-Wolff-Carathéodory’s Theorem In this section we consider boundary limits of incremental ratios of holomorphic maps of D. We start with the definition of angular derivative: Definition 1.7.1 Let f : D → C be holomorphic. Let σ ∈ ∂D. We say that f has angular derivative f  (σ ) at σ ∈ ∂D if f (σ ) := ∠ lim z→σ f (z) exists finitely and f (z) − f (σ ) z−σ

f  (σ ) := ∠ lim

z→σ

exists in C∞ . In the definition of angular derivative, we do not exclude the case f  (σ ) = ∞. However, in case f  (σ ) = ∞, the following result shows that f  (σ ) is also the nontangential limit of f  at σ , justifying its name: Theorem 1.7.2 Let f : D → C be holomorphic and let σ ∈ ∂D. Then f has finite angular derivative A at σ if and only if f  has finite non-tangential limit B at σ . If this is the case, A = B. Proof Let us first assume that f  has finite non-tangential limit B at σ . For all z, w ∈ D 

w

f (w) = f (z) +

f  (ζ )dζ = f (z) + (w − z)

z



1

f  (tw + (1 − t)z)dt.

0

Since for all z ∈ D there exists a Stolz region S(σ, Rz ) for some Rz > 1 such that the segment {ζ ∈ D : ζ = tσ + (1 − t)z, 0 ≤ t ≤ 1} is contained in S(σ, Rz ), the previous formula, together with the Lebesgue Dominated Convergence Theorem, proves that f has finite non-tangential limit f (σ ) at σ given by 

1

f (σ ) = f (z) + (σ − z)

f  (tσ + (1 − t)z)dt.

0

Hence, by the Lebesgue Dominated Convergence Theorem, f (z) − f (σ ) = ∠ lim ∠ lim z→σ z→σ z−σ



1

f  (tσ + (1 − t)z)dt = B,

0

showing that f has finite angular derivative B at σ . Conversely, assume that f has finite angular derivative A at σ . By definition, f f (σ ) exists finitely. has finite non-tangential limit f (σ ) at σ and A = ∠ lim z→σ f (z)− z−σ Let h(z) := f (z) − f (σ ) − A(z − σ ).

1.7 Angular Derivatives and Julia-Wolff-Carathéodory’s Theorem

39

Hence, h : D → C is holomorphic and satisfies ∠ lim

z→σ

h(z) = 0. z−σ

(1.7.1)

Fix β ∈ (0, π/2) and let Vβ := {z ∈ D : |Arg(1 − σ z)| < β}. Note that Vβ is a region in D bounded by two lines passing through σ and which form an angle of ±β with the line passing through 0 and containing σ . Let z ∈ Vβ . Let β  ∈ (β, π/2). Let D(z, r (z)) ⊂ Vβ  be the Euclidean disc of center z and radius r (z) > 0 which is tangent to ∂ Vβ  . Denote by Γ (z) its boundary. From Cauchy’s Formula we have    f (ζ ) − f (σ ) h(ζ ) A 1 1 1 dζ = dζ + dζ 2πi Γ (z) (ζ − z)2 2πi Γ (z) (ζ − z)2 2πi Γ (z) ζ − z   A(z − σ ) h(ζ ) 1 1 dζ = dζ + A =: I (z) + A. + 2πi Γ (z) (ζ − z)2 2πi Γ (z) (ζ − z)2

f  (z) =

Hence, lim Vβ z→σ f  (z) = A is equivalent to lim I (z) = lim

Vβ z→σ

Vβ z→σ

1 2πi

 Γ (z)

h(ζ ) dζ = 0. (ζ − z)2

(1.7.2)

In order to prove (1.7.2), let ε > 0. Then, by (1.7.1), there exists δ > 0 such that for every z ∈ Vβ ∩ {w ∈ D : |w − σ | < δ} it holds |h(ζ )| ≤ ε|ζ − σ | for all ζ ∈ Γ (z). Therefore,   ε |ζ − σ | ε maxζ ∈Γ (z) |ζ − σ | |I (z)| ≤ |dζ | ≤ |dζ | 2π Γ (z) |ζ − z|2 2π [r (z)]2 Γ (z) ε |z − σ | maxζ ∈Γ (z) |ζ − σ | ≤ (r (z) + |z − σ |) = ε(1 + ). =ε r (z) r (z) r (z) Now, by simple geometric considerations, |I (z)| ≤ ε(1 +

r (z) |z−σ |

≥ sin(β  − β), hence,

1 ). sin(β  − β)

By the arbitrariness of ε, it follows that lim Vβ z→σ I (z) = 0. Since every sequence {z n } ⊂ D which converges non-tangentially to σ is contained in Vβ for some β ∈ (0, π/2), equation (1.7.2) holds and we are done.  Theorem 1.7.3 (Julia-Wolff-Carathéodory’s Theorem) Let φ : D → D be a holomorphic function and σ ∈ ∂D. Then the following statements are equivalent: (1) αφ (σ ) < +∞; (2) there exists η ∈ ∂D such that ∠ lim z→σ (3) ∠ lim z→σ φ  (z) exists finitely.

η−φ(z) σ −z

exists finitely;

40

1 Hyperbolic Geometry and Iteration Theory

Moreover, if the previous conditions hold, then η is the point given by Theorem 1.4.7, ∠ lim z→σ φ(z) = η and ∠ lim

z→σ

η − φ(z) = ∠ lim φ  (z) = αφ (σ )ησ . z→σ σ −z

Proof Assume (2) holds. This implies that ∠ lim z→σ φ(z) = η, and Theorem 1.7.2 implies (3) and the equality between the angular derivative of φ at σ and the nontangential limit of φ  at σ . Also, since for every r ∈ (0, 1) 1 − |φ(r σ )| |η − φ(r σ )| ≤ , 1 − |r σ | 1−r it follows that αφ (σ ) < +∞. If (3) holds then (2) holds as well by Theorem 1.7.2. Assume that (1) holds. Let η ∈ ∂D be the point given by Theorem 1.4.7. First of all, by Lemma 1.5.2.(2), every sequence {z n } converging non-tangentially to σ is eventually contained in E(σ, R) for all R > 0. By Theorem 1.4.7, it follows that the sequence {φ(z n )} is eventually contained in E(η, αφ (σ )R) for all R > 0, hence it converges to η. Therefore, ∠ lim z→σ φ(z) = η. Now, let f (z) := ηφ(σ z). It is clear that α := α f (1) = αφ (σ ), and ∠ lim z→1 f (z) = 1. Moreover, by Theorem 1.4.7, φ(E(σ, R)) ⊂ E(η, α R) and hence f (E(1, R)) ⊂ E(1, α R), for all R > 0. It is easy to check that if f has angular derivative α at 1, then (2) follows. Therefore, we have to prove that f has angular derivative α at 1. Let r ∈ (0, 1). Then by Theorem 1.4.7, |1 − f (r )|2 (1 − r )2 1−r 1 − | f (r )| ≤ . ≤α =α 2 2 1 + | f (r )| 1 − | f (r )| 1−r 1+r

(1.7.3)

This implies that α ≤ lim inf r →1

Hence,

1 − | f (r )| 1 − | f (r )| 1 + | f (r )| ≤ lim sup ≤ lim sup α = α. 1−r 1 − r 1+r r →1 r →1 1 − | f (r )| = α. (0,1)r →1 1−r lim

Now, by (1.7.3), |1 − f (r )|2 |1 − f (r )|2 1 − | f (r )|2 1 − r 2 = (1 − r )2 1 − | f (r )|2 1 − r 2 (1 − r )2

(1.7.4)

1.7 Angular Derivatives and Julia-Wolff-Carathéodory’s Theorem

41

(1 − r )2 1 − | f (r )| 1 + | f (r )| 1 − r 2 1 − r2 1−r 1 + r (1 − r )2 1 − | f (r )| 1 + | f (r )| . =α 1−r 1+r ≤α

Hence, by (1.7.4), we obtain lim supr →1 α = lim

r →1

|1− f (r )| 1−r

≤ α. But,

1 − | f (r )| |1 − f (r )| |1 − f (r )| ≤ lim inf ≤ lim sup ≤ α, r →1 1−r 1−r 1−r r →1

proving that limr →1

|1− f (r )| 1−r

= α. By (1.7.4), this implies that |1 − f (r )| = 1. r →1 1 − | f (r )|

(1.7.5)

lim

Since Re f (r ) > 0 for r close to 1, |1 − f (r )|2 [Im f (r )]2 2 = 1 + , (1 − | f (r )|)2 | f (r )| + Re f (r ) (1 − | f (r )|)2 taking the limit for r → 1, by (1.7.5), it holds limr →1 1− f (r ) limr →1 1−| f (r )|

lim

= 1, and hence by (1.7.4)

(0,1)r →1

|Im f (r )| 1−| f (r )|

= 0. Therefore,

1 − f (r ) 1 − f (r ) 1 − | f (r )| = lim = α. (0,1)r →1 1 − | f (r )| 1−r 1−r

(1.7.6)

f (z) Now, (1.7.6) implies that the holomorphic function 1−1−z : D → C has radial limit f (z) α at 1. If we prove that 1−1−z is bounded on every Stolz region, by Proposition 1.5.8, f (z) it follows that ∠ lim z→1 1−1−z = α and the proof is completed. 1− f (z) In order to show that 1−z is bounded on each Stolz region, let us fix M > 1 and let z ∈ S(1, M). Set R = M|1 − z|. Then, since |1 − z| < M(1 − |z|)

|1 − z|2 =

R |1 − z| < R(1 − |z|) < R(1 − |z|2 ). M

Therefore, z ∈ E(1, R). By Theorem 1.4.7, f (z) ∈ E(1, α R). Now, the Euclidean 2α R , hence, diameter of E(1, α R) is 1+α R |1 − f (z)| ≤ Hence,

|1− f (z)| |1−z|

2α R ≤ 2α R = 2α M|1 − z|. 1 + αR

≤ 2α M for z ∈ S(1, M), and we are done.



42

1 Hyperbolic Geometry and Iteration Theory

Proposition 1.7.4 Let φ : D → D be holomorphic and σ ∈ ∂D. Suppose there exists η ∈ ∂D such that ∠ lim z→σ φ(z) = η. Then ησ φ  (σ ) = αφ (σ ) = ∠ lim

z→σ

1 − |φ(z)| 1 − Re (ηφ(z)) = ∠ lim , z→σ 1 − Re (σ z) 1 − |z|

where φ  (σ ) := ∠ lim z→σ η−φ(z) ∈ C∞ is the angular derivative of φ at σ . In parσ −z ticular, if ∠ lim z→σ φ(z) exists and belongs to ∂D, the angular derivative of φ at σ always exists (finite or infinite). Proof Assume that αφ (σ ) = +∞. This means that lim

z→σ

Since,

1 − |φ(z)| = +∞. 1 − |z|

1 − |z| 1 − |φ(z)| |η − φ(z)| ≥ , |σ − z| |σ − z| 1 − |z|

and, since if {z n } ⊂ D is a sequence converging non-tangentially to σ there exists 1−|z| ≥ C, it follows immediately that φ  (σ ) = ∞. Similarly, since C > 0 such that |σ −z| 1 − Re (σ z) ≤ |σ − z| and 1 − Re (ηφ(z)) 1 − |z| 1 − |φ(z)| ≥ , 1 − Re (σ z) |σ − z| 1 − |z| (ηφ(z)) we have ∠ lim z→σ 1−Re = ∞. 1−Re (σ z) Assume now that αφ (σ ) < +∞. Then η is the point given by Theorem 1.7.3 and ησ φ  (σ ) = αφ (σ ) by the same theorem. Therefore, we are left to show that from every sequence {z n } in D converging non-tangentially to σ one can extract a subsequence {z n k } such that

1 − |φ(z n k )| 1 − Re (ηφ(z n k )) = ∠ lim . k→∞ 1 − |z n k | k→∞ 1 − Re (σ z n k )

αφ (σ ) = ∠ lim

Now, fix an arbitrary sequence {z n } in D converging non-tangentially to σ and denote xn := Re  (σ z n ) and yn := Im (σ z n ), for all n. Then, there exists C > 0 such  yn  that supn  1−xn  < C. Take an accumulation point p ∈ [−C, C] of that sequence yn k 1−xn k

= p and also

1 − i 1−xn 1 − σ znk 1 − ip k = lim  lim 2 = 1 + p 2 .  k→∞ |1 − σ z n k | k→∞ y 1 + 1−xnkn

(1.7.7)

yn { 1−x }. Therefore, there is a subsequence {z n k } such that limk→∞ n yn k

k

1.7 Angular Derivatives and Julia-Wolff-Carathéodory’s Theorem

43

Since {σ z n } tends to one, the formula 1 − Re (σ z n ) 1 + |z n | = yn 1 − |z n | 1 + xn − yn 1−x n implies limk→∞

1−Re (σ z n k ) 1−|z n k |

lim

k→∞

= 1. Hence, taking the real part in (1.7.7), we have

1 1 − Re (σ z n k ) 1 − |z n k | = lim = . k→∞ |σ − z n k | |σ − z n k | 1 + p2

(1.7.8)

Then, applying twice Theorem 1.7.3, lim

k→∞

1 − ip 1 − ηφ(z n k ) η − φ(z n k ) |σ − z n k | 1 − σ z n k = ησ lim = . k→∞ σ − z n k |η − φ(z n k )| |σ − z n k | |1 − ηφ(z n k )| 1 + p2

Moreover, by Proposition 1.5.5, the sequence {φ(z n k )} converges non-tangentially to η. Therefore, repeating the above argument with this sequence, we find that lim

k→∞

1 1 − Re (ηφ(z n k )) 1 − |φ(z n k )| = lim = . k→∞ |η − φ(z n k )| |η − φ(z n k )| 1 + p2

(1.7.9)

Hence, combining (1.7.8), (1.7.9) and Theorem 1.7.3, lim

k→∞

1 − |φ(z n k )| 1 − |φ(z n k )| |σ − z n k | |η − φ(z n k )| = lim = αφ (σ ), k→∞ |η − φ(z n k )| 1 − |z n k | |σ − z n k | 1 − |z n k |

and

lim

k→∞

1 − Re (ηφ(z n k )) 1 − Re (ηφ(z n k )) |σ − z n k | |η − φ(z n k )| = lim = αφ (σ ), 1 − Re (σ z n k ) |η − φ(z n k )| 1 − Re (σ z n k ) |σ − z n k | k→∞



and we are done.

Proposition 1.7.4 and Lindelöf’s Theorem 1.5.7 assure that if φ : D → D is holomorphic, σ ∈ ∂D and lim(0,1)r →1 φ(r σ ) = η ∈ ∂D, then φ has angular derivative φ  (σ ) (finite or infinite) at σ . Moreover, Theorem 1.7.3 implies that if φ  (σ ) is finite, then also ∠ lim z→σ φ  (z) exists and equals φ  (σ ). On the other hand, if φ  (σ ) = ∞, the non-tangential (or even the radial) limit of φ  at σ might not exist, but, as the following result shows, |φ  (r σ )| cannot be bounded as r → 1. Proposition 1.7.5 Let φ : D → D be holomorphic and let σ ∈ ∂D. Suppose that there exists η ∈ ∂D such that lim(0,1)r →1 φ(r σ ) = η and lim sup |φ  (r σ )| < ∞.

(0,1)r →1

44

1 Hyperbolic Geometry and Iteration Theory

Then αφ (σ ) < +∞. Proof Let C := supr ∈[0,1) |φ  (r σ )|. By hypothesis, C < +∞. Fix r ∈ [0, 1). For all u ∈ (r, 1) we have 

uσ

φ  (z)dz  1 φ  ((ut + (1 − t)r )σ )dt. = φ(r σ ) + (u − r )σ

φ(uσ ) = φ(r σ ) +

rσ

(1.7.10)

0

Hence, for all r < u < 1,    1  φ(uσ ) − φ(r σ )   ≤ |φ  ((ut + (1 − t)r )σ )|dt ≤ C.  (u − r )σ  0 Taking the limit as u → 1, we have    η − φ(r σ )     σ − σ r  ≤ C. Proposition 1.7.4 implies that the angular derivative φ  (σ ) of φ at σ exists (finite or infinite), and by the previous inequality it has to be finite. Therefore, again by  Proposition 1.7.4, αφ (σ ) < +∞. As a corollary of the previous results we also have Corollary 1.7.6 Let φ : D → D be holomorphic. Let σ ∈ ∂D. Suppose that lim

(0,1)r →1

φ  (r σ ) = L ∈ C.

Then there exists p ∈ D such that ∠ lim z→σ φ(z) = p. Moreover, if p ∈ ∂D then αφ (σ ) < +∞ and L = 0. Proof Using (1.7.10) with r = 0, and since [0, 1)  s → φ  (sσ ) extends continuously on [0, 1], taking the limit as u → 1, we obtain  lim

(0,1)u→1

φ(uσ ) = φ(0) + σ

1

φ  (tσ )dt,

0

and hence φ has radial limit at σ . By Theorem 1.5.7, there exists p ∈ D such that ∠ lim z→σ φ(z) = p. If p ∈ ∂D then αφ (σ ) < +∞ by Proposition 1.7.5, and L = 0 by Proposition 1.7.4.  Proposition 1.7.7 Let φ j : D → D be holomorphic, j = 1, 2, and σ1 ∈ ∂D. Assume that σ2 := ∠ lim z→σ1 φ1 (z) ∈ ∂D.

1.7 Angular Derivatives and Julia-Wolff-Carathéodory’s Theorem

45

(1) If αφ1 (σ1 ) = +∞ or αφ2 (σ2 ) = +∞, then αφ2 ◦φ1 (σ1 ) = +∞. (2) If both αφ1 (σ1 ) and αφ2 (σ2 ) are finite, then αφ2 ◦φ1 (σ1 ) = αφ1 (σ1 )αφ2 (σ2 ). Proof Since 1 αφ ◦φ (σ1 ) = lim inf [ω(0, z) − ω(0, φ2 (φ1 (z)))] ≥ lim inf [ω(0, z) − ω(0, φ1 (z))] z→σ1 z→σ1 2 2 1 + lim inf [ω(0, φ1 (z)) − ω(0, φ2 (φ1 (z)))], z→σ1

and the boundary dilation coefficient is always strictly positive by Lemma 1.4.5, (1) follows at once. Now, assume that αφ1 (σ ) and αφ2 (σ2 ) are finite. By Theorem 1.4.7, there exists σ3 := ∠ lim z→σ2 φ2 (z) ∈ ∂D. Take any sequence {z n } in D converging nontangentially to σ1 . Then, for every n, σ3 − φ2 (φ1 (z n )) σ3 − φ2 (φ1 (z n )) σ2 − φ1 (z n ) = . σ1 − z n σ2 − φ1 (z n ) σ1 − z n By Proposition 1.5.5, {φ1 (z n )} converges non-tangentially to σ2 thus the right hand side of the above equality has limit and, indeed, by Theorem 1.7.3, lim

n→∞

σ3 − φ2 (φ1 (z n )) = αφ2 (σ2 )σ3 σ2 αφ1 (σ1 )σ2 σ1 . σ1 − z n

Therefore, again by Theorem 1.7.3, we conclude that αφ2 ◦φ1 (σ1 ) = αφ1 (σ1 ) · αφ2 (σ2 ).  It is often useful to work in the right half-plane H instead of the unit disc. We translate here the Julia-Wolff-Carathéodory Theorem into the right half-plane H “centered at ∞”. As a matter of notation, for θ0 ∈ (0, π/2), let V (θ0 ) := {ρeiθ : ρ > 0, |θ | < θ0 }. We say that a sequence {wn } ⊂ H converges non-tangentially to ∞ if there exists θ0 ∈ (0, π/2) such that {wn } ⊂ V (θ0 ) and |wn | → ∞ as n → ∞. Let Cσ : D → H be the Cayley transform with respect to σ ∈ ∂D. By Remark 1.5.3, {wn } ⊂ H converges non-tangentially to ∞ if and only if {Cσ−1 (wn )} converges non-tangentially to σ . For a function f : H → C we write ∠ limw→∞ f (w) = L ∈ C∞ and we say that f has non-tangential limit L at ∞, provided that for every sequence {wn } ⊂ H which converges non-tangentially to ∞ it holds limn→∞ f (wn ) = L. With this notation at hand, we have: Theorem 1.7.8 Let φ : H → H be holomorphic. The following are equivalent: (1)

1 2

log α := lim inf Hw→∞ [kH (1, w) − kH (1, φ(w))] < +∞,

46

1 Hyperbolic Geometry and Iteration Theory

β := inf w∈H ReReφ(w) > 0, w ˜ β := lim supRr →+∞ Re rφ(r ) > 0, ∠ limw→∞ φ(w) exists and it is different from 0, w ∠ limw→∞ φ  (w) exists and it is different from 0.

(2) (3) (4) (5)

Moreover, if the previous conditions hold, then ∠ lim

w→∞

φ(w) 1 = ∠ lim φ  (w) = β = β˜ = , w→∞ w α

and Re (φ(w) − βw) ≥ 0 for all w ∈ H, with equality at one—and hence any— w ∈ H if and only if φ is an automorphism of H. In particular, for every R > 0, φ(E H (∞, R)) ⊆ E H (∞, β R). Proof Let C1 : D → H be the Cayley transform given by (1.1.2) (with σ = 1), and consider the holomorphic self-map φ˜ : D → D defined by φ˜ := C1−1 ◦ φ ◦ C1 . Since ω(z, w) = kH (C1 (z), C1 (w)) for all z, w ∈ D by Proposition 1.3.10, it follows by (1.4.5) that α = αφ˜ (1). It is clear that β˜ ≥ β, hence (2) implies (3). Straightforward computations show that, setting w = C1 (z), ˜ φ(w) 1 − z 1 + φ(z) = , ˜ w 1+z 1 − φ(z) and φ (w) = φ˜  (z) 



1−z ˜ 1 − φ(z)

(1.7.11)

2 .

(1.7.12)

(1) implies (4). If αφ˜ (1) = α < +∞, then Theorem 1.7.3 and (1.7.11) imply at once (4). (4) implies (1), (2) and (5). Indeed, by (1.7.11) and Remark 1.5.3, if (4) holds, ˜ = 1 and then ∠ lim z→1 φ(z) ∠ lim

w→∞

1 φ(w) 1−z 1 = ∠ lim = , = ˜ z→1 1 − φ(z) w αφ˜ (1) α

where the penultimate equality follows from Theorem 1.7.3. In particular, by (4), α = αφ˜ (1) < +∞ and (1) holds. Hence (5) follows at once by (1.7.12) and Theorem 1.7.3, and (2) follows from Theorem 1.4.7 and (1.4.15). (5) implies (4). Suppose not. We claim that ˜ |1 − φ(z)| = +∞. z→1 |1 − z|

∠ lim

1.7 Angular Derivatives and Julia-Wolff-Carathéodory’s Theorem

47

Otherwise, there exists a sequence {z n } ⊂ D converging non-tangentially to 1 and ˜ n )| φ(z < +∞. Since {z n } converges non-tangentially, it follows such that limn→∞ |1− |1−z n | that lim supn→∞

|1−z n | 1−|z n |

lim

n→∞

< +∞. Hence

˜ n )| ˜ n )| |1 − z n | 1 − |φ(z |1 − φ(z ≤ lim < +∞. n→∞ 1 − |z n | |1 − z n | 1 − |z n |

Therefore, αφ˜ (1) < +∞ and Theorem 1.7.3, together with (1.7.11) and Remark 1.5.3, implies (4). Therefore, the claim holds. By the claim, (1.7.11) and Remark 1.5.3, ∠ lim

w→∞

φ(w) = 0. w

By (5), either limr →+∞ Re φ  (r ) = 0 or limr →+∞ Im φ  (r ) = 0 (or both). We assume limr →+∞ Re φ  (r ) = 0 (in case limr →+∞ Im φ  (r ) = 0 the proof is similar considering u(r ) = Im φ(r ) in the next argument). Set u(r ) = Re φ(r ) for r ∈ (0, +∞). The previous equation implies that for every fixed r0 ∈ (0, +∞) u(r ) − u(r0 ) lim = 0. r →+∞ r − r0 However, by (5), there exists r0 > 0 and c > 0 such that |u  (s)| ≥ c for all s ≥ r0 . Now, let sr ∈ (r0 , r ) be such that u(r ) − u(r0 ) = u  (sr )(r − r0 ). Hence, 0 = lim

r →+∞

|u(r ) − u(r0 )| = lim |u  (sr )| ≥ c, r →+∞ r − r0

contradiction, and (4) holds. (3) implies (1). By (1.7.11) it follows ˜ )| ˜ )| 1 − |φ(r |1 − φ(r ≤ lim inf < +∞. (0,1)r →1 (0,1)r →1 1−r 1−r

α = αφ˜ (1) ≤ lim inf

Now, if (1)—(5) hold, then by (1.7.11), (1.7.12), Theorem 1.7.3 and Remark 1.5.3, we have φ(w) 1 ∠ lim = ∠ lim φ  (w) = β˜ = , w→+∞ w w→+∞ α and, by Theorem 1.4.7 and (1.4.15), ˜ w Re φ(w) ≥ βRe for all w ∈ H, with equality at some—and hence any—w ∈ H if and only if φ is an ˜ Since β ≤ β, ˜ we are done.  automorphism of H. This implies at once that β ≥ β.

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1.8 Iteration in the Unit Disc and the Denjoy-Wolff Theorem Let  be a Riemann surface and let φ :  →  be holomorphic. Let n ∈ N. We denote by φ ◦n :  →  the n-th iterate of φ, that is, φ ◦n := (φ ◦ · · · ◦ φ) .   n -times Also, φ ◦0 := id denotes the identity map. If φ is an automorphism, then we also denote by φ −◦n or φ ◦(−n) the n-th iterate of φ −1 , n ∈ N. The aim of this section is to study the behavior of the sequence {φ ◦n } of iterates when φ is a holomorphic self-map of the unit disc. Lemma 1.8.1 Let T ∈ Aut(D) \ {idD }. Then T has at least one fixed point in D. Moreover, if T has no fixed points in D, then it has two fixed points τ, σ ∈ ∂D, possibly τ = σ , such that T  (σ ) · T  (τ ) = 1. Proof Let T ∈ Aut(D). By (1.2.2), T = λTa for some a ∈ D and λ ∈ C, |λ| = 1. Then T (z) = z if and only if az 2 − (1 + λ)z + λa = 0. If a = 0, since T is not the identity map, then the unique fixed point of T in C is z = 0. If a = 0, the above equation has two solutions z 1 , z 2 ∈ C which satisfy z 1 z 2 = λ aa ∈ ∂D (in particular, |z 1 ||z 2 | = 1) and az 1 + az 2 = 1 + λ. Moreover, T  (z 1 )T  (z 2 ) = λ2

(1 − |a|2 )2 = 1. ((1 − az 1 )(1 − az 2 ))2

From this the statement follows at once.



Definition 1.8.2 Let T ∈ Aut(D) \ {idD }. Then we say that (1) T is elliptic if it has a fixed point in D, (2) T is parabolic if it has a unique fixed point in ∂D, (3) T is hyperbolic if it has two different fixed points in ∂D. We investigate now the dynamics of an automorphism of D according to the previous classification. Assume T is an elliptic automorphism with a fixed point τ ∈ D. The map Tτ ◦ T ◦ Tτ is an automorphism that fixes the origin. So, there exists λ ∈ ∂D such that (Tτ ◦ T ◦ Tτ )(z) = λz, for all z ∈ D. Thus T ◦n (z) = Tτ (λn Tτ (z)), for all z ∈ D. The automorphism T is then holomorphically conjugated to a rotation. Assume that T is hyperbolic. Let τ, σ ∈ ∂D be its fixed points, τ = σ . Since T  (τ )T  (σ ) = 1, we may assume that |T  (τ )| ≤ 1. Using the Cayley transform Cτ :

1.8 Iteration in the Unit Disc and the Denjoy-Wolff Theorem

49

D → H, we can conjugate T to an automorphism of the right half-plane Φ = Cτ ◦ T ◦ Cτ−1 . The function Φ is a Möbius transformation in the Riemann sphere that fixes the point ∞. So Φ(w) = aw + b, for all w ∈ C. Since Φ(H) = H we conclude that Re b = 0, Im a = 0, and a > 0. Also, a = 1, for otherwise Φ would not have fixed points in C, contradicting the fact that T also fixes σ = τ , and hence Φ has to fix Cτ (σ ) ∈ ∂H ∩ C as well. Now, a direct computation shows that a = T 1(τ ) . This implies that T  (τ ) is a positive real number and thus T  (τ ) ∈ (0, 1). Also, n b for all w ∈ H, T  (σ ) = T 1(τ ) ∈ (1, +∞). Moreover, since Φ ◦n (w) = a n w + 1−a 1−a ◦n it follows that {Φ (w)} converges to ∞ for all w ∈ H. Hence, {T ◦n (z)} converges to τ for all z ∈ D. One can note that the convergence of {Φ ◦n } is in fact uniform on compacta of H, hence, {T ◦n } converges uniformly on compacta to the constant map D  z → τ . Finally assume that T is parabolic, with a unique fixed point τ ∈ ∂D. Notice that T  (τ )2 = 1. Arguing as in the hyperbolic case, the function Φ = Cτ ◦ T ◦ Cτ−1 is a Möbius transformation in the Riemann sphere that has only one fixed point in C∞ , that is ∞, and Φ(H) = H. Hence, Φ(w) = aw + b, for all w ∈ H, with a = T 1(τ ) > 0 and Re b = 0. In particular, T  (τ ) = 1. Thus Φ(w) = w + b and Φ ◦n (w) = w + nb, which implies that Φ ◦n (w) converges to ∞ for all w ∈ H. Namely, {T ◦n (z)} converges to τ for all z ∈ D, and, even in this case, {T ◦n } converges uniformly on compacta to the constant map D  z → τ . After having analyzed the behavior of the iterates of automorphisms of D, we turn our attention to general holomorphic self-maps of the unit disc. We start with the case φ has a fixed point in D. By Corollary 1.2.4 a holomorphic self-map of the unit disc has at most one fixed point in D, unless it is the identity. Proposition 1.8.3 Let φ : D → D be holomorphic, not an automorphism. Suppose there exists τ ∈ D such that φ(τ ) = τ . Then {φ ◦n } converges uniformly on compacta to the constant map D  z → τ . Proof First assume that τ = 0, i.e., φ(0) = 0. Since φ is not an automorphism, Theorem 1.2.1 implies that |φ(z)| < |z| for every z ∈ D, z = 0. Fix 0 < r < 1, let M(r ) = max{|φ(z)| : |z| ≤ r } and write δ := M(r )/r . Theorem 1.2.1 guarantees z) , z ∈ D. It is clear that ψ : D → D is holomorphic, that δ < 1. Let ψ(z) := φ(r M(r ) continuous up to D and fixes the origin. Again by Theorem 1.2.1, we deduce that |ψ(z)| ≤ |z| for all z ∈ D. Thus, for z ∈ r D, M(r ) z |z| ≤ δ|z|. |φ(z)| = M(r )|ψ( )| ≤ r r Iterating the last inequality yields: |φ ◦n (z)| ≤ δ|φ ◦(n−1) (z)| ≤ δ 2 |φ ◦(n−2) (z)| ≤ · · · ≤ δ n |z| ≤ δ n for each z ∈ r D. Since δ < 1, it follows that φ ◦n tends to zero uniformly on r D. The arbitrariness of r implies that {φ ◦n } converges to zero uniformly on compact subsets of D.

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1 Hyperbolic Geometry and Iteration Theory

In case φ fixes a point τ = 0, we only need to apply the previous argument to the function ϕ = Tτ ◦ φ ◦ Tτ , where Tτ is given by (1.2.1). The resulting holomorphic function ϕ maps D into itself, fixes the origin, and it is not an automorphism, therefore, its iterates ϕ ◦n tends to zero uniformly on compacta of D. Hence, φ ◦n = Tτ ◦ ϕ ◦n ◦ Tτ converges uniformly on compacta of D to Tτ (0) = τ .  Theorem 1.8.4 (Denjoy-Wolff’s Theorem) Let φ : D → D be holomorphic. Assume φ has no fixed points in D. Then there exists a unique point τ ∈ ∂D such that αφ (τ ) ≤ 1 and, for every R > 0, φ(E(τ, R)) ⊆ E(τ, R),

(1.8.1)

with φ(∂ E(τ, R) \ {τ }) ∩ ∂ E(τ, R) = ∅ if and only if φ is a parabolic automorphism of D. Moreover, the sequence of iterates {φ ◦n } converges uniformly on compacta to the constant map D  z → τ . Finally, ∠ lim z→τ φ(z) = τ and ∠ lim

ζ →τ

φ(ζ ) − τ = ∠ lim φ  (ζ ) = αφ (τ ). ζ →τ ζ −τ

Proof First we claim that for every z ∈ D the sequence {|φ ◦n (z)|} converges to 1. Indeed, if this is not the case, there exist z 0 ∈ D and a subsequence such that limk→∞ φ ◦n k (z 0 ) = p for some p ∈ D. In particular, by the previous discussion on dynamical behavior of automorphisms, we deduce that φ is not an automorphism. By Theorem 1.3.7, N  n → ω(φ ◦n (z 0 ), φ ◦(n+1) (z 0 )) is not increasing. In particular, there exists δ ≥ 0 such that limn→∞ ω(φ ◦n (z 0 ), φ ◦(n+1) (z 0 )) = δ. On the one hand, ω( p, φ( p)) = lim ω(φ ◦n k (z 0 ), φ ◦(n k +1) (z 0 )) = δ. k→∞

On the other hand, ω(φ( p), φ ◦2 ( p)) = lim ω(φ ◦(n k +1) (z 0 ), φ ◦(n k +2) (z 0 )) = δ. k→∞

Thus ω( p, φ( p)) = ω(φ( p), φ ◦2 ( p)). Since φ is not an automorphism, this is possible only if φ( p) = p, a contradiction. Therefore, for every z ∈ D, the sequence {φ ◦n (z)} accumulates only on ∂D. In particular, this holds for z = 0. Let wn := φ ◦n (0). Since {|wn |} converges to 1, and wn+1 = φ(wn ), it is easy to see that there exists a subsequence {wn k } with the property that |φ(wn k )| > |wn k | for all n k ∈ N. Up to extracting subsequences, we can assume that {wn k } converges to a point τ ∈ ∂D. We claim that {φ(wn k )} converges to τ as well. Indeed, if there were a subsequence {φ(wn k  )} converging to some p = τ , then limn k  →∞ ω(φ(wn k  ), wn k  ) = +∞. On the other hand, by Theorem 1.3.7, for all nk ∈ N ω(φ(wn k ), wn k ) = ω(φ ◦n k (0), φ ◦(n k +1) (0)) ≤ ω(0, φ(0)),

1.8 Iteration in the Unit Disc and the Denjoy-Wolff Theorem

51

getting a contradiction. Therefore, both {wn k } and {φ(wn k )} converges to τ . Moreover, by construction, 1 − |φ(wn k )| < 1 − |wn k |, hence, αφ (τ ) ≤ lim sup k→∞

1 − |φ(wn k )| ≤ 1. 1 − |wn k |

By Theorem 1.4.7, for all R > 0 it holds φ(E(τ, R)) ⊆ E(τ, R), and there exists p ∈ ∂ E(τ, R) \ {τ } such that φ( p) ∈ ∂ E(τ, R) if and only if φ is an automorphism, and, in this case αφ (τ ) = 1 and φ is parabolic by our previous discussion on automorphisms of D. Now, we deal with the uniqueness of τ . Assume that there are two points τ, τ  ∈ ∂D such that (1.8.1) holds for both. Since horocycles are discs in D tangent to ∂D, given R > 0 there exists R  > 0 such that E(τ, R) ∩ E(τ  , R  ) = {z 0 }, for some z 0 ∈ D. Therefore, using (1.8.1) for both τ and τ  , we see that φ(z 0 ) ∈ E(τ, R) ∩ E(τ  , R  ) = {z 0 }, that is, φ(z 0 ) = z 0 , a contradiction. The last statement follows directly from Theorem 1.7.3. We are only left to prove that {φ ◦n } converges uniformly on compacta to the constant map z → τ . By Vitali’s theorem, it is enough to prove that for every z 0 ∈ D the sequence {φ ◦n (z 0 )} converges to τ . Let z 0 ∈ D. Then there exists R > 0 such that z 0 ∈ E(τ, R). Hence, by (1.8.1), φ ◦n (z 0 ) ∈ E(τ, R) for all n ∈ N. Since {φ ◦n (z 0 )} accumulates only on ∂D and E(τ, R) ∩ ∂D = {τ }, it follows that limn→∞ φ ◦n (z 0 ) = τ .  The previous results allow us to give the following basic definitions: Definition 1.8.5 Let φ : D → D be holomorphic, not the identity. (1) If φ has a fixed point in D, then its unique fixed point is called the Denjoy-Wolff point of φ. (2) If φ has no fixed points in D, then the unique point τ ∈ ∂D given by Theorem 1.8.4 is called the Denjoy-Wolff point of φ. Moreover, φ is (1) elliptic, if its Denjoy-Wolff point belongs to D, (2) hyperbolic, if its Denjoy-Wolff point τ belongs to ∂D and αφ (τ ) ∈ (0, 1), (3) parabolic, if its Denjoy-Wolff point τ belongs to ∂D and αφ (τ ) = 1. The previous definition for an automorphism of D is coherent with the similar classification given in Definition 1.8.2, according to the description of the dynamics of automorphisms. In the hyperbolic case, the information provided by the Denjoy-Wolff Theorem 1.8.4 about the convergence to the Denjoy-Wolff point can be improved as follows. A preliminary lemma is needed. Lemma 1.8.6 Let {z n } and {wn } be two sequences in D. Let C := lim sup ω(z n , wn ). n→∞

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1 Hyperbolic Geometry and Iteration Theory

(1) If C < +∞ and {z n } converges to σ , then {wn } converges to σ . (2) If C < +∞ and {z n } converges to σ non-tangentially, then {wn } converges to σ non-tangentially. " ! (3) If C = 0 and the limit limn→∞ Arg (1 − σ z n ) = α ∈ − π2 , π2 exists, then also limn→∞ Arg (1 − σ wn ) = α. Proof (1) Take an accumulation point p ∈ D of the sequence {wn } and assume that p = σ . Therefore, there exists a subsequence {wn k } such that limk→∞ wn k = p. By hypothesis, supn ω (z n , wn ) < +∞ thus there exists c ∈ (0, 1) such that |z n k − wn k | ≤ c|1 − z n k wn k |, for all k. Taking limits, we have | p − σ | ≤ c|1 − σ p| = c| p − σ |. A contradiction, because 0 < c < 1 and p = σ .  to σ . For all n, write z n = 1 − rn eiθn σ (2) We already know that {wn } converges   and wn = 1 − sn eiξn σ with θn , ξn ∈ − π2 , π2 , rn , sn > 0 and limn→∞ rn = limn→∞ sn = 0. If {z n } is converging non-tangentially to σ , up to extracting subsequences if needed, we can assume that limn→∞ θn = α ∈ (−π/2, π/2). Now, let β ∈ [− π2 , π2 ] be an accumulation point of the sequence {Arg (1 − σ wn )}. Therefore, there is a subsequence of natural numbers {n k } such that β = limk→∞ ξn k . From the definition of ω, we have that  ω(zn , wn ) is a bounded sequence if and  zn −wn  only if there exists δ ∈ (0, 1) such that  1−z  ≤ δ for all n ∈ N0 . Hence, n wn       z n k − wn k   −rn k eiθnk + sn k eiξnk    = δ≥   1 − z n k wn k   rn k e−iθnk + sn k eiξnk − rn k sn k ei (ξnk −θnk )    rn k sn k   iθn k iξn k − rn +s e + e   rn k +sn k nk k . =  rn s r s  n n n  rn +sk n e−iθnk + rn +sk n eiξnk − rn k+skn ei (ξnk −θnk )  k

k

k

k

k

(1.8.2)

k

  rn k Since the terms of the sequence rn +s are in (0, 1), all of its accumulation points n k k belong to [0, 1]. Let λ ∈ [0, 1] be one of them. Taking the limit in (1.8.2) along such a subsequence, it follows that |λeiα − (1 − λ)eiβ | ≤ δ. |λe−iα + (1 − λ)eiβ | From which, we deduce that, necessarily λ ∈ (0, 1), and |β| < π/2. In particular, no subsequence of {wn } can converge to σ tangentially, i.e., {wn } converges nontangentially to σ .    zn −wn  (3) If C = 0, then limn→∞  1−z  = 0. Hence, the previous argument shows that n wn 1−λ 1−λ λeiα − (1 − λ)eiβ = 0. This implies ei(α−β) = . Since ∈ (0, +∞) and λ λ α − β ∈ [−π, π ], that equation shows that α = β as claimed. 

1.8 Iteration in the Unit Disc and the Denjoy-Wolff Theorem

53

Proposition 1.8.7 Let φ be a holomorphic self-map of D with Denjoy-Wolff point τ ∈ ∂D. Assume that αφ (τ ) < 1. Then for every compact set K ⊂ D and every sequence {z n } in K , the sequence {φ ◦n (z n )} converges non-tangentially to τ . Namely, {φ ◦n } converges uniformly non-tangentially to τ on compacta. In particular, for every z 0 ∈ D, the sequence {φ ◦n (z 0 )} converges non-tangentially to τ . Proof For each natural number n ∈ N, denote z n := φ ◦n (z 0 ) and wn := Cτ (z n ) = z n +τ ∈ H, where Cτ is the Cayley transform defined in (1.1.2). The Denjoy-Wolff τ −z n Theorem 1.8.4 guarantees that the sequence {z n } converges to τ . Hence, we only have to prove that the sequence lies entirely inside a Stolz region with vertex at τ . In fact, our proof will show that it lies in a lens-shaped region with vertexes at τ and −τ . Define qn = n (wn+1 ), where n is the automorphism of the right half-plane wn . Proposition 1.3.10 and Theorem 1.3.7 show that given by n (w) = w−iIm Re wn kH (qn , 1) = kH (n (wn+1 ), n (wn )) = kH (wn+1 , wn ) = ω(z n+1 , z n ) ≤ ω(z 1 , z 0 ), n ∈ N.

(1.8.3)

Theorem 1.4.7 implies that, for every z ∈ D,  Re

φ(z) + τ τ − φ(z)

This means that Re qn =



1 ≥ Re αφ (τ )



z+τ τ −z

 .

Re wn+1 1 > 1, n ∈ N. ≥ Re wn αφ (τ )

(1.8.4)

By (1.8.3) and (1.8.4), the closure of the set {qn − 1 : n ∈ N} is a compact subset of H. Thus, there is M > 0 such that qn ∈ 1 + S, for all n, where S := {w ∈ H : |Im w| ≤ MRe w}. We may assume that w1 ∈ S. Therefore, wn+1 ∈ wn + S for all n. Iterating we obtain that wn ∈ w1 + S ⊂ S for all n. Since S + S ⊂ S, we deduce that wn ∈ S for all n. Thus the points z n belong to the lens Cτ−1 (S). Fix K ⊂ D a compact set. Assume that there is a sequence {z n } in K such that {φ ◦n (z n )} does not converge non-tangentially to τ . Up to taking a subsequence, we may assume that {z n k } converges to z 0 ∈ K ⊂ D. Notice that lim sup ω(φ ◦n k (z n k ), φ ◦n k (z 0 )) ≤ lim sup ω(z n k , z 0 ) = 0. k→∞

k→∞

Since, by the first part of the proof, the sequence {φ ◦k (z 0 )} converges non-tangentially to τ , the sequence {φ ◦k (z n k )} also converges non-tangentially to τ (otherwise, apply Lemma 1.8.6 to suitable subsequences). A contradiction. Therefore, {φ ◦n (z n )} converges non-tangentially to τ .  For parabolic maps the situation is rather different as shows next example.

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Example 1.8.8 Consider the linear fractional maps Φ1 (w) := w + i and Φ2 (w) := w + 1 and φk (z) := C1−1 ◦ Φk ◦ C1 , k = 1, 2, where C1 is the Cayley transform , z ∈ D. Since Φk sends the right half-plane into itself, the function φk is C1 (z) := 1+z 1−z a holomorphic self-map of the unit disc for k = 1, 2. Moreover, limn→∞ Φk◦n (w) = ∞, for all w, and thus limn→∞ φk◦n (z) = 1, for all z ∈ D. That is, 1 is the DenjoyWolff point of both φ1 and φ2 . In fact, φ1 (1) = φ2 (1) = 1, so that both are parabolic. Finally, the sequence {φ1◦n (0)} = {C1−1 (1 + ni)} converges to 1 tangentially and the sequence {φ2◦n (0)} = {C1−1 (1 + n)} converges to 1 orthogonally.

1.9 Boundary Regular Contact Points Holomorphic self-maps of the unit disc need not to be continuous up to the boundary. Thus, strictly speaking, it makes sense to talk about fixed points of holomorphic selfmaps of D only for points in D. However, by Corollary 1.2.4, a holomorphic self-map which is not the identity, can have at most one fixed point in D. In this section we introduce a weaker notion of fixed points which makes sense also for boundary points without assuming any extra hypothesis about continuity up to the boundary. Definition 1.9.1 Let φ : D → D be holomorphic. A point σ ∈ ∂D is called a boundary fixed point of φ if lim− φ(r σ ) = σ. r →1

According to Theorem 1.5.7, σ is a boundary fixed point of φ if and only if ∠ lim z→σ φ(z) = σ . When there is no risk of confusion, we call a boundary fixed point simply a fixed point. Also, when needed, we call a fixed point z 0 ∈ D (the Denjoy-Wolff point of the map) an inner fixed point. One can consider the more general notion of contact points: Definition 1.9.2 Let φ : D → D be holomorphic. A point σ ∈ ∂D is called a contact point of φ if there exists η ∈ ∂D such that lim φ(r σ ) = η.

r →1−

As before, the radial limit can be replaced by non-tangential limit. If z 0 ∈ D is an inner fixed point of φ, we define its multiplier as φ  (z 0 ). The following proposition allows to define the multiplier also at boundary contact points: Proposition 1.9.3 Let φ : D → D be holomorphic and σ ∈ ∂D a contact point of φ. Then, the following angular limit always exists φ  (σ ) := ∠ lim

z→σ

φ(z) − η ∈ C∞ \ {0}. z−σ

1.9 Boundary Regular Contact Points

55

Moreover, if φ  (σ ) ∈ C, then σ ηφ  (σ ) ∈ (0, +∞). Proof Consider the boundary dilation coefficient αφ (σ ) ∈ (0, +∞]. If αφ (σ ) < +∞, the result follows from Theorem 1.7.3. Otherwise, αφ (σ ) = +∞ and lim

z→σ

1 − |φ(z)| = +∞. 1 − |z|

Take {z n } a sequence in D converging non-tangentially to σ . Therefore, there exists C > 0 such that |σ − z n | ≤ C(1 − |z n |), for every n. Hence,    φ(z n ) − η  1 − |φ(z n )| , ≤ C  1 − |z n | zn − σ  

and the result follows.

Definition 1.9.4 Let φ : D → D be holomorphic and σ ∈ ∂D a contact point of φ. We call φ  (σ ) the multiplier of φ at σ . Boundary points with finite multiplier deserve a special name: Definition 1.9.5 A boundary regular fixed point (respectively regular contact point) of a holomorphic self-map φ of D is a boundary fixed point (resp. contact point) of φ with finite multiplier. Remark 1.9.6 Given σ ∈ ∂D, by Theorem 1.7.3, σ is a regular contact point if and only if αφ (σ ) < +∞. Remark 1.9.7 For a boundary regular fixed point σ ∈ ∂D of a holomorphic selfmap φ of D, there is the following dichotomy: either φ  (σ ) = αφ (σ ) ∈ (1, +∞) or φ  (σ ) = αφ (σ ) ∈ (0, 1]. In the latter case, φ has no inner fixed points and σ is its Denjoy-Wolff point. If φ : D → D is non-elliptic, this is a consequence of Theorems 1.8.4 and 1.7.3, and, if it is elliptic, of Proposition 1.8.3 and Theorems 1.4.7 and 1.7.3. For our next results, we need to recall the definition and properties of the cluster set of a curve. Definition 1.9.8 Let −∞ < a < b ≤ +∞. Let γ : [a, b) → C be a continuous curve. The cluster set of γ at b is Γ (γ , b) := { p ∈ C∞ : ∃{tn } ⊂ [a, b) : lim tn = b, lim γ (tn ) = p}. n→∞

n→∞

Lemma 1.9.9 Let −∞ < a < b ≤ +∞. Let γ : [a, b) → C be a continuous curve. Then the cluster set Γ (γ , b) is a compact connected subset of C∞ . Proof For r ∈ [a, b), let K r be the closure in C∞ of the set γ ([r, b)). Let r ∈ [a, b). Since γ ([r, b)) is connected, K r is connected. Moreover, K r ⊆ K s for a ≤ s < r

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< b. Hence, ∩r ∈[a,b) K r is a closed non-empty connected subset of C∞ . Since clearly Γ (γ , b) = ∩r ∈[a,b) K r , 

the result follows.

Proposition 1.9.10 Let φ j : D → D be holomorphic, j = 1, 2. Let φ := φ2 ◦ φ1 . Suppose that σ ∈ ∂D is a (regular) contact point of φ. Then σ is a (regular) contact point of φ1 and the point φ1 (σ ) ∈ ∂D is a (regular) contact point of φ2 . Moreover, φ(σ ) = ∠ lim z→φ1 (σ ) φ2 (z) and αφ (σ ) = αφ2 (φ1 (σ )) · αφ1 (σ ). Proof Let γ (r ) := φ1 (r σ ) for r ∈ [0, 1). Let Γ (γ , 1) be the cluster set of γ . We claim that there exists σˆ ∈ ∂D such that Γ (γ , 1) = {σˆ }. Indeed, if q ∈ Γ (γ , 1) ∩ D, and {rk } ⊂ (0, 1) is a sequence converging to 1 such that γ (rk ) → q, then φ(rk σ ) = φ2 (γ (rk )) → φ2 (q) ∈ D, contradicting the hypothesis that φ has a contact point at σ . Therefore Γ (γ , 1) is a connected compact subset of ∂D, namely, it is a closed arc A, possibly reducing to a point (see Lemma 1.9.9). Suppose that A is not a point. Thus λ(A) > 0, where λ is the Lebesgue measure on ∂D. By Proposition 1.6.9, there exists a subset A ⊆ A such that λ(A) = λ(A ) and φ2 has radial limit / ∂ A (where the boundary is taken in ∂D). Since at all points q ∈ A . Let q ∈ A , q ∈ q ∈ A = Γ (γ , 1) but it is not an extreme of the arc A, then the radial segment Γq := {sq : s ∈ (0, 1)} intersects the curve γ infinitely many times, that is, there exists a sequence {rk } ⊂ (0, 1) converging to 1 and such that γ (rk ) ∈ Γq . Therefore lim

(0,1)r →1

φ2 (rq) = lim φ2 (γ (rk )) = lim φ(rk σ ) = φ(σ ). k→∞

k→∞

Hence, again Proposition 1.6.9, φ2 ≡ φ(σ ), a contradiction. Thus A reduces to a point σ ∈ ∂D. This shows that φ1 has radial limit σˆ ∈ ∂D and Lindelöf’s theorem 1.5.7 implies that ∠ lim φ1 (z) = σˆ =: φ1 (σ ). z→σ

By the same token, φ2 has limit φ(σ ) along the curve (0, 1)  r → φ1 (r σ ) which converges to φ1 (σ ), hence it has non-tangential limit φ(σ ) at φ1 (σ ). We end the proof just applying Proposition 1.7.7.  Remark 1.9.11 The converse of above proposition does not hold even for boundary fixed points. Namely, there exists a univalent function φ : D → D for which σ = 1 is a boundary fixed point of φ but it is not a boundary fixed point of φ ◦2 (see [52, Example 1]). The last proposition, which will be used in the next chapters, shows that holomorphic self-maps of the unit disc are “semi-conformal” at regular contact points: Proposition 1.9.12 Let φ : D → D be holomorphic with a regular contact point σ ∈ ∂D. Define η := ∠ lim z→σ φ(z).

1.9 Boundary Regular Contact Points

57

(1) If {z n } is a sequence in D converging to σ non-tangentially with an angle  limn→∞ Arg (1 − σ z n ) = θ ∈ − π2 , π2 , then # $   αφ (σ ) + e−2iθ  + αφ (σ ) − 1 1  .   lim ω (z n , ησ φ (z n )) = log  αφ (σ ) + e−2iθ  − αφ (σ ) − 1 n→∞ 2 (2) If φ is parabolic and σ is its Denjoy-Wolff point. If {z n } is a sequence in D converging non-tangentially to σ , then lim ω(z n , φ(z n )) = 0.

n→∞

(3) Assume that αφ (σ ) = 1. A sequence {z n } in D converges orthogonally to σ (i.e., limn→∞ Arg (1 − σ z n ) = 0) if and only if {z n } converges non-tangentially to σ and lim ω(z n , ησ φ(z n )) =

n→∞

 1  log(αφ (σ )) . 2

Proof By Theorem 1.7.3, αφ (σ ) < +∞. First of all, consider the holomorphic selfmap of D given by D  z → ϕ(z) := ησ φ(z) and note that σ is a boundary regular fixed point of ϕ and αφ (σ ) = αϕ (σ ). (1) A simple calculation shows that (1 − |z n |2 )(1 − |ϕ(z n )|2 ) |1 − z n ϕ(z n )|2    zn σ − 1 ϕ(z n ) − σ −2 + zn = (1 + |z n |)2  zn − σ zn − σ   2 1 − |z n | 1 − |ϕ(z n )|2 . |z n − σ | 1 − |z n |2

1 − |Tzn (ϕ(z n ))|2 =

 Write z n = 1 − rn eiθn σ where θn = Arg (1 − σ z n ). We know that limn→∞ rn = −1 n| 0 and limn→∞ θn = θ . Then limn→∞ zznnσ−σ = σ e−2iθ and limn→∞ |z1−|z = cos(θ ). n −σ | Thus, applying Theorem 1.7.3 and Proposition 1.7.4, cos2 (θ ) A := lim (1 − |Tzn (ϕ(z n ))|2 ) = 4   αϕ (σ ) ∈ [0, 1]. n→∞ αϕ (σ ) + e−2iθ 2 Therefore

  αϕ (σ ) + e−2iθ 2 − 4 cos2 (θ )αϕ (σ )   lim |Tzn (ϕ(z n ))| = 1 − A = αϕ (σ ) + e−2iθ  n→∞  αϕ (σ )2 + 2αϕ (σ ) cos(2θ ) + 1 − 4 cos2 (θ )αϕ (σ )   = αϕ (σ ) + e−2iθ  |αϕ (σ ) − 1| . = αϕ (σ ) + e−2iθ  √

58

1 Hyperbolic Geometry and Iteration Theory

By Theorem 1.3.5, #   $ √ αϕ (σ ) + e−2iθ  + αϕ (σ ) − 1 1 1 1+ 1− A    . = log  lim ω(z n , ϕ(z n )) = log √ αϕ (σ ) + e−2iθ  − αϕ (σ ) − 1 n→∞ 2 2 1− 1− A

(2) By hypothesis, σ = η and αϕ (σ ) = 1. Hence, the result follows immediately from (1). (3) It follows directly from (1) since # $   αφ (σ ) + e−2iθ  + αφ (σ ) − 1 1    (−π, π )  θ → log  αφ (σ ) + e−2iθ  − αφ (σ ) − 1 2 is equal to

1 2

  log(αφ (σ )) only at θ = 0 for αφ (σ ) = 1.



1.10 Notes The material of this chapter is essentially classical, although some results and some proofs might be new and the exposition has been chosen to suitably fit our needs in the next chapters. For the material of this chapter, we took our inspiration and benefited very much from the books [1, 3, 75, 106]. The hyperbolic distance on Riemann surfaces has been introduced using a “several complex variables” point of view due to Kobayashi (see [90]), which avoids all technicalities with the classical approach via covering maps. Also, the proof of the Lindelöf Theorem and the JuliaWolff-Carathéodory Theorem have been adapted from the book [114].

Chapter 2

Holomorphic Functions with Non-Negative Real Part

In this chapter we introduce the basic properties of holomorphic functions with non-negative real part.

2.1 The Herglotz Representation Formula In this section we state and prove the classical Herglotz Representation Formula. Theorem 2.1.1 (Herglotz’s Representation Theorem) Let p : D → H be holomorphic. Then there exists a unique finite non-negative Borel measure μ on ∂D such that  ζ +z dμ(ζ ) + iIm p(0), z ∈ D. p(z) = ∂D ζ − z Proof Let u = Re p and set u r (z) = u(r z) for 0 < r < 1. Every u r is harmonic in D, continuous on D and non-negative. Thus, by the mean value property 1 Re p(0) = u r (0) = 2π



2π

u r (eiθ ) dθ.

0

Let Λr : C(∂D) → R be the continuous linear functional given by Λr f =

1 2π



2π

f (eiθ )u r (eiθ ) dθ,

f ∈ C(∂D).

0

The set {Λr : 0 < r < 1} is bounded in C(∂D)∗ because the net {u r } is uniformly bounded in L 1 (∂D). Since the unit ball of C(∂D)∗ endowed with the weak-* topology is compact and metrizable (see [127, pp. 29 and 32]), given a sequence {rn } converging © Springer Nature Switzerland AG 2020 F. Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4_2

59

60

2 Holomorphic Functions with Non-Negative Real Part

to 1, there exist a subsequence (which, with a slight abuse of notation, we still denote by {rn }) and a functional Λ ∈ C(∂D)∗ such that 1 Λ( f ) = lim Λrn ( f ) = lim n→∞ n→∞ 2π



2π

f (eiθ )u rn (eiθ ) dθ,

f ∈ C(∂D).

0

By the Riesz-Markov-Kakutani Representation Theorem (see, e.g., [113, Theorem 2.14, p. 40]), there is a measure μ on ∂D such that  Λf =

∂D

f (ζ ) dμ(ζ ),

f ∈ C(∂D).

Since u r is non-negative for all 0 < r < 1, it is clear that μ is a non-negative measure on ∂D. In particular, μ(∂D) = Re p(0). we obtain Then, integrating the (positive) Poisson kernel Re ζζ +z −z  Re ∂D

 2π ζ +z 1 eiθ + z dμ(ζ ) = lim u r (eiθ ) dθ Re iθ n→∞ 2π 0 ζ −z e −z n = lim u rn (z) = u(z) = Re p(z), n→∞

which gives the desired representation formula. To prove the uniqueness of the measure μ, let suppose that μ˜ is another finite non-negative Borel measure on ∂D which represents p. Let η := μ − μ. ˜ Then for all z ∈ D  ζ +z dη(ζ ) = 0, Re ∂D ζ − z which implies that  ∂D

ζ +z dη(ζ ) = iγ , ζ −z

for all z ∈ D,

 n n for some real number γ . Since ζζ +z =1+2 ∞ n=1 ζ z , and η is a real measure, −z  n we conclude that ∂D ζ dη(ζ ) = 0 for all n ∈ Z. It follows that η annihilates every trigonometric polynomial, hence, by the Weierstrass Approximation Theorem, it also annihilates every continuous function on ∂D. Since continuous functions are dense in L 1 (∂D), this implies that η(E) = 0 for any measurable subset E of ∂D. Thus η is the zero measure and μ = μ. ˜  Remark 2.1.2 There is one-to-one correspondence between the set of all Borel measures on an interval [a, b] onto the set of all nondecreasing, real-valued, right continuous functions on [a, b] which map a to 0. In fact, if μ is one such a measure and β : [a, b] → R is its associated function, then μ((x, y]) = β(y) − β(x) for all a ≤ x < y ≤ b (see [83, Theorem 19.48]). Moreover, the Riemann-Stieltjes integrals with respect to those functions β are the only non-negative linear functionals on

2.1 The Herglotz Representation Formula

61

the space of continuous real valued functions on [a, b] (see [83, Theorem 19.50]). Therefore the above proof shows that if p : D → H is holomorphic, then there exists a unique nondecreasing, real-valued, right continuous functions β : [0, 2π ] → R such that β(0) = 0, β(2π ) = 2π Re p(0) and 

1 p(z) = 2π

2π 0

eiθ + z dβ(θ ) + iIm p(0), eiθ − z

z ∈ D.

(2.1.1)

In addition, if β is absolutely continuous, then dβ(θ ) = β  (θ )dθ (see [83, Theorem 19.61]). A first consequence of the Herglotz’s Representation Theorem is a useful tool in geometric function theory and, in particular, will be frequently used in this book. Proposition 2.1.3 Let p : D → H be holomorphic. Let μ be the finite non-negative Borel measure associated with p given by Theorem 2.1.1. Let σ ∈ ∂D. Then, (1) the following limit does exist: λ := ∠ lim (1 − σ z) p(z) = 2μ({σ }) ∈ [0, +∞). z→σ

(2) For all z ∈ D,

λ 1 − |z|2 = Re pσ (z), 2 |σ − z|2

Re p(z) ≥

(2.1.2)

(2.1.3)

+z , for all z ∈ D. where pσ (z) := λ2 σσ −z (3) The function p − pσ has non-negative real part and

∠ lim (1 − σ z)( p(z) − pσ (z)) = 0. z→σ

(2.1.4)

(4) The equality holds in (2.1.3) for some (and hence for all) z ∈ D if and only if p = pσ + iIm p(0). Proof Denote by μσ the non-negative Borel measure on ∂D defined by μσ (A) := μ(A ∩ {σ }) for any Borel subset A ⊂ ∂D. Let μ I be the measure defined by μ I (A) := μ(A) − μσ (A). It is clear that for any arc C centered at σ ,  lim (1 − σ z)

z→σ

∂D\C

ζ +z dμ I (ζ ) = 0. ζ −z

Let (Cn ) be a sequence of arcs in ∂D, centered at σ , with Cn+1 ⊆ Cn for all n, and whose Euclidean lengths go to zero as n tends to ∞. Let us denote by m the Lebesgue measure of ∂D. Notice that μ I ({σ }) = 0. Since μ I is a finite measure, 0 = μ I ({σ }) = μ I (∩n Cn ) = lim μ I (Cn ). n→∞

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2 Holomorphic Functions with Non-Negative Real Part

Therefore, given ε > 0 there exists δ > 0 such that μ I (C) < ε for any arc C of Euclidean length less than or equal to δ and centered at σ . Let M > 1 and let Δ := S(σ, M) be the Stolz region of vertex σ and amplitude M. Then for any arc C of length less than or equal to δ and centered at σ ,      ζ +z   |z − σ |    dμ I (ζ ) ≤ lim sup 2 dμ I (ζ ) ≤ 2Mε. lim sup |1 − σ z|   1 − |z| C Δ z→σ Δ z→σ C ζ −z 

Therefore lim (1 − σ z)

Δ z→σ

∂D

ζ +z dμ I (ζ ) = 0. ζ −z

Hence, from Theorem 2.1.1, 

+z dμ(ζ ) −z ∂D +z dμ I (ζ ) = lim (1 − σ z) Δ z→σ −z ∂D  ζ +z dμσ (ζ ) + lim (1 − σ z) Δ z→σ ∂D ζ − z  ζ +z dμσ (ζ ) = lim (1 − σ z) Δ z→σ ∂D ζ − z σ +z μ({σ }) = 2μ({σ }), = lim (1 − σ z) Δ z→σ σ −z

lim (1 − σ z) p(z) = lim (1 − σ z)

Δ z→σ

Δ z→σ

ζ ζ ζ ζ

and (1) is proved. Sinceμ I is a finite non-negative Borel measure on ∂D, the holomorphic function +z dμ I (ζ ) + iIm p(0) has non-negative real part. Moreover, q(z) := ∂D ζζ −z  ∂D

λ σ +z ζ +z σ +z dμσ (ζ ) = μσ ({σ }) = = pσ (z), z ∈ D. ζ −z σ −z 2 σ −z

Since Re pσ (z) = (2), (3), and (4).

λ 1−|z|2 , 2 |σ −z|2

for all z ∈ D, and p = q + pσ , we immediately obtain 

2.2 Growth Estimates for Functions with Non-Negative Real Part In this section we provide growth estimates for functions with non-negative real part and obtain some useful consequences. Theorem 2.2.1 Let p : D → H be holomorphic. Then, for every z ∈ D,

2.2 Growth Estimates for Functions with Non-Negative Real Part

Re p(0)

1 + |z| 1 − |z| ≤ | p(z) − iIm p(0)| ≤ Re p(0) , 1 + |z| 1 − |z| |Im p(z) − Im p(0)| ≤ Re p(0) | p  (z)| ≤ 2 Re p(0)

2|z| , 1 − |z|2

1 . (1 − |z|)2

63

(2.2.1)

(2.2.2)

(2.2.3)

p(0) Proof Let q(z) := p(z)−iIm , z ∈ D. Set φ := C1−1 ◦ q, where C1 is the Cayley Re p(0) transform with respect to 1 defined in (1.1.2). Since φ is a holomorphic self-map of the unit disc that fixes 0, Schwarz’s Lemma (see Theorem 1.2.1) implies that |φ(z)| ≤ |z| for all z ∈ D. This inequality means that if |z| ≤ r , then |C1−1 (q(z))| ≤ r . 2 2r Since C1 maps the disc {z ∈ C : |z| ≤ r } onto the disc of center 1+r and radius 1−r 2, 1−r 2 we deduce that 1+r 1−r 2r ≤ |q(z)| ≤ . , |Im q(z)| ≤ 1 − r2 1+r 1−r

These inequalities are equivalent to (2.2.1) and (2.2.2). Applying now Schwarz-Pick’s Lemma (see Theorem 1.2.3), we obtain the inequality (1 − |z|2 )|φ  (z)| ≤ 1 − |φ(z)|2 for all z ∈ D. Thus (1 − |z|2 )|q  (z)| = (1 − |z|2 )|C1 (φ(z))||φ  (z)| ≤ (1 − |φ(z)|2 )|C1 (φ(z))|, z ∈ D. Therefore, |q  (z)| ≤

2 1 − |φ(z)|2  1 − |φ(z)|2 |C1 (φ(z))| = . 2 1 − |z| 1 − |z|2 |1 − φ(z)|2

Hence, taking into account that |φ(z)| ≤ |z|, we have 1 − |φ(z)|2 (1 − |z|)2 1 − |φ(z)| ≤2 (1 − |z|) 2 2 1 − |z| |1 − φ(z)| |1 − φ(z)|2   1 − |φ(z)| 2 ≤2 ≤ 2. |1 − φ(z)|

|q  (z)|(1 − |z|)2 ≤ 2



Thus, (2.2.3) holds. A corollary of the above result is the following:

Corollary 2.2.2 Let p : D → C be a holomorphic function. Then Re p(z) ≥ 0 for all z ∈ D if and only if there is a function μ : D → (0, +∞) such that Re (zp  (z) + μ(z) p(z)) ≥ 0, for all z ∈ D.

(2.2.4)

64

2 Holomorphic Functions with Non-Negative Real Part

Proof Suppose that Re p(z) ≥ 0 for all z ∈ D. We may assume that Re p has no zeros in the unit disc. By Theorem 2.2.1, | p  (z)| ≤ 2

Re p(z) , z ∈ D. 1 − |z|2

2 Thus the inequality (2.2.4) holds taking μ(z) = 1−|z| 2. iθ0 Conversely, assume that there is z 0 = r0 e ∈ D such that Re p(z 0 ) < 0. Since Re p(0) ≥ 0, there is r1 ∈ (0, r0 ) such that Re p(r1 eiθ0 ) < 0 and Re p  (r1 eiθ0 )eiθ0 = 1 Re p  (r1 eiθ0 )r1 eiθ0 < 0. But this contradicts (2.2.4).  r1

2.3 Finite Contact Points of Holomorphic Functions with Non-Negative Real Part In this section we introduce (regular) contact points of holomorphic functions from the unit disc with non-negative real part different from infinity and give some measure-theoretic characterizations of them. We start with a definition. Definition 2.3.1 Let p : D → H be holomorphic. A point σ ∈ ∂D is a finite contact point of p if the angular limit p(σ ) := ∠ lim z→σ p(z) exists and belongs to iR. Moreover, σ is a regular finite contact point of p if σ is a finite contact point of p and p has finite angular derivative at σ , that is, p  (σ ) := ∠ lim

z→σ

p(z) − p(σ ) ∈ C. z−σ

(2.3.1)

A (regular) finite contact point σ such that p(σ ) = 0 is also called a (regular) zero of p. Remark 2.3.2 Let p : D → H be holomorphic. Let φ := C1−1 ◦ p = p−1 , where p+1 C1 (w) = (1 + w)/(1 − w) is the Cayley transform with respect to 1 defined in (1.1.2). The function φ is a holomorphic self-map of the unit disc. It is clear that σ ∈ ∂D is a finite contact point of p if and only if σ is a contact point of φ in the sense of Definition 1.9.2 and φ(σ ) = 1. Moreover, if σ ∈ ∂D is a finite contact point of p, then a straightforward computation shows that p(z) − p(σ ) φ(z) − φ(σ ) 2 = , z−σ ( p(z) + 1)( p(σ ) + 1) z−σ hence, σ is a regular finite contact point of p if and only if it is a regular contact point of φ and φ(σ ) = 1.

2.3 Finite Contact Points of Holomorphic Functions with Non-Negative Real Part

65

The choice if C1 is arbitrary, in the sense that, in the previous considerations, one can replace C1 with Cτ for any τ ∈ ∂D and hence it turns out that σ ∈ ∂D is a finite (regular) contact point of p if and only if σ is a (regular) contact point of Cτ−1 ◦ p whose image is different from τ . Remark 2.3.3 If p is a holomorphic function such that p(D) ⊂ H and σ ∈ ∂D is a finite contact point of p, then q(z) := 1/( p(z) − p(σ )) is a well-defined holomorphic function on D with non-negative real part. By Proposition 2.1.3, the nontangential limit p  (σ ) := ∠ lim

z→σ

p(z) − p(σ ) p(z) − p(σ ) 1 = −σ ∠ lim = −σ ∠ lim , z→σ z→σ (1 − σ z)q(z) z−σ 1 − σz

exists and it is either equal to ∞ or equal to −σ C for some C > 0. In particular, if σ ∈ ∂D is a regular finite contact point of a holomorphic function p : D → H, then p(r σ ) − p(σ ) = C, lim− r →1 1−r hence, lim−

r →1

Im p(r σ ) − Im p(σ ) = 0. 1−r

On the other hand, if p ≡ ai for some real number a, then every point σ ∈ ∂D is a regular finite contact point of p and p  (σ ) = 0. Note also that if p is identically equal to some z 0 ∈ H, then p is a holomorphic function with non-negative real part without contact points but p  (σ ) = 0, for all σ ∈ ∂D. Proposition 2.3.4 Let p : D → H be holomorphic and σ ∈ ∂D. Then the following are equivalent: (1) (2) (3) (4)

p(r σ ) lim(0,1) r →1 Re1−r and limr →1 Im p(r σ ) exist finitely, p(r σ ) lim inf (0,1) r →1 Re1−r < +∞, ∠ lim z→σ p(z) ∈ iR and ∠ lim z→σ p  (z) ∈ C, σ is a finite regular contact point of p.

Moreover, if one—and hence any—of the previous holds, then lim(0,1) r →1 −σ p  (σ ).

Re p(r σ ) 1−r

=

Proof If p(z) ≡ i y for some y ∈ R, there is nothing to prove, therefore, we can suppose that p(D) ⊂ H. It is clear that (1) implies (2). Assume that (2) holds. Set φ := C1−1 ◦ p = p−1 , where C1 is the Cayley transform p+1 with respect to 1 defined in (1.1.2). The function φ is a holomorphic self-map of the unit disc and

66

2 Holomorphic Functions with Non-Negative Real Part

1 − |φ(z)| 1 − |φ(r σ )| 1 − |φ(r σ )|2 ≤ lim inf ≤ lim inf z→σ r →1 r →1 1 − |z| 1−r 1−r  2 | p(r σ ) − 1| 1 1 4Re p(r σ ) = lim inf 1− = lim inf r →1 1 − r r →1 1 − r | p(r σ ) + 1|2 | p(r σ ) + 1|2 Re p(r σ ) < +∞. ≤ 4 lim inf r →1 1−r

αφ (σ ) = lim inf

By Julia-Wolff-Carathéodory’s Theorem 1.7.3, there exists η := ∠ lim z→σ φ(z) ∈ ∂D and φ(z) − η φ  (σ ) := ∠ lim = ∠ lim φ  (z) = αφ (σ )ησ > 0. z→σ z − σ z→σ Therefore, ∠ lim p(z) = ∠ lim C1 (φ(z)) = z→σ

where, as usual, we set If η = 1 then

z→σ

1+η 1−η

= ∞ in case η = 1. 

lim Re ( p(r σ )(1 − r )) = lim Re

r →1

1+η ∈ iR ∪ {∞}, 1−η

r →1

 2 1−r (1 + φ(r σ )) =  ∈ (0, +∞). 1 − φ(r σ ) φ (σ )

p(r σ ) In particular this implies that limr →1 Re p(r σ ) = +∞, hence lim inf r →1 Re1−r = +∞, against our hypothesis. Therefore, η ∈ ∂D \ {1} and ∠ lim z→σ p(z) exists and belongs to iR. Moreover,  p  (z) = φ 2(z) ( p(z) + 1)2 and

φ  (z) 1 ( p(z) + 1)2 = αφ (σ )ησ ∠ lim p (z) = ∠ lim z→σ z→σ 2 2 



2 1+η + 1 ∈ C. 1−η

Thus, (3) holds. = If (3) holds, writing ai = ∠ lim z→σ p(z), by Theorem 1.7.2, ∠ lim z→σ p(z)−ai z−σ ∠ lim z→σ p  (z) exists finitely, hence σ is a finite regular contact point of p and (4) holds. Finally, if (4) holds, clearly so does (1). by σ and taking the real part, we deduce Moreover, if (4) holds, multiplying p(z)−ai z−σ Re p(r σ ) that limr →1 r −1 exists finitely and equals −σ p  (σ ).  The next result gives a measure theoretic characterization of regular finite contact points: Theorem 2.3.5 Let p : D → H be holomorphic. Let μ be the finite non-negative Borel measure associated with p given by Theorem 2.1.1 and σ ∈ ∂D. Then σ is a regular finite contact point of p if and only if the function ∂D ξ →

1 ∈ [0, +∞] |ξ − σ |2

2.3 Finite Contact Points of Holomorphic Functions with Non-Negative Real Part

67

is μ-integrable. Moreover, if σ ∈ ∂D is a regular finite contact point of p, then μ({σ }) = 0,  1 dμ(ξ ), p  (σ ) = −2σ |ξ − σ |2 ∂D and

 Im p(σ ) = 2 ∂D

Im (ξ σ ) dμ(ξ ) + Im p(0). |ξ − σ |2

Proof If p is identically equal to ai for some real number a, then μ is the null measure and the result is trivial. Thus, we assume p(D) ⊂ H. By Herglotz’s Representation Formula (see Theorem 2.1.1),  ξ +z dμ(ξ ) + i Im p(0), z ∈ D. (2.3.2) p(z) = ∂D ξ − z Therefore, for every r ∈ (0, 1), we have Re p(r σ ) 2 r = (1 + r ) 1−r

 ∂D

r2 dμ(ξ ). |ξ − r σ |2

(2.3.3)

Now, we consider for r ∈ (0, 1), the family of μ-integrable non-negative functions ∂D ξ → fr (ξ ) :=

r2 ∈ [0, +∞). |ξ − r σ |2

Since fr (ξ ) ≤ f s (ξ ), whenever ξ ∈ ∂D and 0 < r < s < 1, Beppo Levi’s Monotone Convergence Theorem implies that ∂D fr (ξ )dμ(ξ ) tends to the integral (which takes values in [0, +∞]) of the function ξ → f (ξ ) := |ξ − σ |−2 with respect to μ. Hence, by (2.3.3), the first part of the theorem follows directly from Proposition 2.3.4. Assume now σ is a regular finite contact point of p. The formula for p  (σ ) is also a consequence of Proposition 2.3.4, bearing in mind the first part of the theorem. Since σ is the unique preimage of ∞ by f , the integrability of f gives μ({σ }) = 0. For the last formula, using (2.3.2) and taking imaginary parts, we have for any r ∈ (0, 1),  r Im p(r σ ) = ∂D

2r 2 Im (ξ σ ) dμ(ξ ) + r Im p(0). |ξ − r σ |2

Note that for r ∈ (0, 1),    r 2 Im (ξ σ )      ≤ f (ξ ), ξ ∈ ∂D \ {σ }.  (ξ − r σ )2 

68

2 Holomorphic Functions with Non-Negative Real Part

Since f is μ-integrable and μ({σ }) = 0, by using Lebesgue’s Dominated Conver gence Theorem, and taking the limit as r → 1− we obtain the formula. Corollary 2.3.6 Let p : D → H be holomorphic. Let μ be the finite non-negative Borel measure associated with p given by Theorem 2.1.1 and σ ∈ ∂D. Then σ is a regular zero of p if and only if the function ∂D ξ →

1 ∈ [0, +∞] |ξ − σ |2

is μ-integrable and

 Im p(0) = −2 ∂D

Im (ξ σ ) dμ(ξ ). |ξ − σ |2

2.4 Boundary Behavior We examine now the boundary behavior along an arc of holomorphic functions with non-negative real part. We start with the following lemma about the so-called typically real functions: Lemma 2.4.1 (Rogosinski’s Theorem for typically real functions) Let f : D → C be holomorphic. Assume f (0) = 0, f  (0) = 1, f (x) ∈ R for all x ∈ (−1, 1) and Im f (z) · Im z > 0 for z ∈ D \ {(−1, 1)}. Then there exists p : D → H holomorphic such that z p(z), z ∈ D. f (z) = 1 − z2 Proof Let h(z) :=

z 1−z 2

for z ∈ D. We have, 1 = −2i sin θ, θ ∈ R. h(eiθ )

For r ∈ (0, 1), let pr (z) := over,

f (r z) . It is clear that h(z)

lim pr (z) = lim

z→0

z→0

pr is holomorphic on D \ {0}. More-

f (r z) z = r. z h(z)

Therefore, pr extends to a holomorphic function—still denoted by pr — on D. We also have for θ ∈ R Re pr (eiθ ) = 2 sin θ Im f (r eiθ ). Since by hypothesis Im f (r eiθ ) · Im r eiθ > 0 for θ = 0 mod π , we obtain Re pr (eiθ ) ≥ 0 for all θ ∈ R. By the maximum principle for harmonic functions,

2.4 Boundary Behavior

69

f (z) pr (D) ⊆ H. Let p(z) := h(z) , z ∈ D. Hence p is holomorphic, and, since pr → p uniformly on compacta, we also have Re p(z) ≥ 0 for all z ∈ D. 

Proposition 2.4.2 Let p : D → H be holomorphic. Let A  ∂D be a open arc with end points x1 , x2 . Suppose that for all σ ∈ A, lim Re p(z) = 0.

z→σ

Then p extends holomorphically through A. Moreover, L j := lim A x→x j p(x) exists (finite or infinite) and ∠ lim z→x j p(z) = L j , j = 1, 2. Proof We may assume that p is not constant, otherwise the result is trivial. Let C be a Möbius transformation which maps D onto iH := {w ∈ C : Im w > 0} and such that C(A) = (−a, a) for some a > 0, C(x1 ) = −a, C(x2 ) = a. Let p˜ : iH → iH be defined by p(w) ˜ := i p(C −1 (w)). By definition, for all x ∈ (−a, a) ˜ = 0. Let Ω := C \ {x ∈ R : |x| ≥ a}. Note that Ω is it holds limiH w→x Im p(w) simply connected. By the Schwarz’s Reflection Principle (see, e.g., [113, Theorem 11.14]), there is q : Ω → C holomorphic such that q(w) = p(w) ˜ if w ∈ iH and ˜ if w ∈ iH. q(w) = p(w) ∈ R, moreover, taking into account that Note that q  (0) = lim(−a,a) r →0 q(r )−q(0) r Im q(0) = 0 and Im q(ir ) > 0 for r > 0, we have Re q  (0) =

 lim

(0,1) r →0

Re

q(ir ) − q(0) ir

 =

lim

(0,1) r →0

Im q(ir ) ≥ 0. r

Therefore, q  (0) ≥ 0. 2az Let g : D → C be defined by g(z) = 1+z 2 . It is easy to see that g is univalent and g(D) = Ω. Moreover, g((−1, 1)) = (−a, a), g(0) = 0, g  (0) > 0 and Im g(z) · Im z > 0 for all z ∈ D \ (−1, 1). Let assume first that q  (0) > 0. Let f := q ◦ g : D → C. By construction, f ((−1, 1)) ⊂ R and Im f (z) · Im z > 0 for z ∈ D \ {(−1, 1)}. Moreover, f (0) = . By construction f˜ satisfies the hypotheq(0) and f  (0) > 0. Let f˜(z) := f (z)−q(0) f  (0) sis of Lemma 2.4.1. Therefore there exists a holomorphic function p0 : D → H such z that f˜(z) = 1−z 2 p0 (z) for all z ∈ D. Namely, f (z) = f  (0)

z p0 (z) + q(0). 1 − z2

Since q  (0) > 0, p0 ≡ 0 (otherwise q ≡ q(0) and q  (0) = 0). Therefore, Re 0. By Proposition 2.1.3, ∠ lim (1 − z 2 ) z→±1

1 p0 (z)

≥

1 = T ± ∈ [0, +∞). p0 (z)

Therefore, the angular limit B ± of f at ±1 exists (finite or infinite). Taking into account that q(x) = f (g −1 (x)) for x ∈ (−a, a) and lim(−a,a) x→±a g −1 (x) = ±1,

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2 Holomorphic Functions with Non-Negative Real Part

it follows that lim(−a,a) x→±a q(x) = B ± . From this it follows easily that L j := lim A x→x j p(x) exists (finite or infinite), j = 1, 2. Moreover, let γ : [0, 1) → D ∩ {w ∈ C : Im w > 0} be a continuous curve such that limt→1− γ (t) = 1 and limt→1− Arg(1 − γ (t)) = π/4. Let η(t) := g(γ (t)). Note that η(t) ∈ iH and limt→1− η(t) = a. We have lim q(η(t)) = lim− f (g −1 (η(t)) = lim− f (γ (t)) = B + .

t→1−

t→1

t→1

Since q(η(t)) = i p(C −1 (η(t))), it follows that p has limit along the curve [0, 1) t → C −1 (η(t)) ∈ D, which converges to x2 for t → 1− , and such a limit equals L 2 . By Theorem 1.5.7, p has non-tangential limit L 2 at x2 . A similar argument allows to prove that p has non-tangential limit L 1 at x1 . In case q  (0) = 0, since q is not constant, we can find x0 ∈ (−a, a) such that  q (x0 ) = 0. Let r0 ∈ (−1, 1) be such that g(r0 ) = x0 . Then we can find a hyperbolic automorphism T of D such that T (±1) = ±1 and T (0) = r0 . Since T ((−1, 1)) = (−1, 1), we can consider g ◦ T instead of g and repeat the previous argument. 

2.5 Notes Holomorphic functions on the disc with non-negative real part and normalized so that 0 is mapped to 1 form the so-called Carathéodory class, which is a classical subject of study in geometric function theory. Although they do not appear explicitly in this form, the results in Sect. 2.3 are inspired by the book [115]. Lemma 2.4.1 is the “easy part” of the so-called Rogosinski’s Theorem, which asserts, in fact, that also the converse to Lemma 2.4.1 is true. Proposition 2.4.2 has been proved in [37], with the less stringent hypothesis that lim(0,1) r →1 Re p(r σ ) = 0 for all σ ∈ A.

Chapter 3

Univalent Functions

In this chapter we describe some properties of univalent functions from the unit disc whose images are contained in C. The choice of the topics is based on the material we need in this book and not on the intrinsic relevance of the topics themselves inside the theory of univalent functions. We first prove the No Koebe Arcs Theorem, from which we obtain several results about pre-images of slits via univalent maps. Then we present the so-called Koebe Distortion Theorems. Finally, we consider families of univalent functions and prove the Carathéodory Kernel Convergence Theorem.

3.1 Univalent Functions and Simply Connected Domains Let f : D → C be univalent (that is holomorphic and injective). Let Ω = f (D). Since non-constant holomorphic functions are open, it follows that Ω is a domain in C (that is, an open, connected subset of C) and f : D → Ω is a biholomorphism. Hence, Ω is simply connected. The Riemann Mapping Theorem states that the converse is also true. A proof can be found in [113, Theorem 14.8]. Theorem 3.1.1 (Riemann’s Mapping Theorem) Let Ω ⊂ C be a simply connected domain such that Ω = C. Then for every w0 ∈ Ω there exists a unique univalent function f : D → C such that f (D) = Ω, f (0) = w0 and f  (0) > 0. Given a simply connected domain Ω  C, a Riemann map of Ω is any univalent map f : D → C such that f (D) = Ω. Hence, from a certain point of view, simply connected domains in C different from C and univalent functions from D are one and the same thing. The following lemma allows in many cases to restrict the attention to bounded univalent functions. © Springer Nature Switzerland AG 2020 F. Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4_3

71

72

3 Univalent Functions

Lemma 3.1.2 Let f : D → C be an univalent function. Then there exist a never vanishing bounded univalent function h : D → C, w0 , z 0 ∈ C such that  f (z) = w0 +

1 − z0 h(z)

2 for all z ∈ D.

(3.1.1)

Proof Let Ω := f (D). Since Ω = C, there exists w0 ∈ C \ Ω. Since Ω is simply connected and the holomorphic function Ω  w → w − w0 has no zeros in Ω, there exists g : Ω → C holomorphic such that [g(w)]2 = w − w0 for all w ∈ Ω (see, e.g., [113, Theorem 13.11]). The function g is univalent. Indeed, if g(w1 ) = g(w2 ), then w1 − w0 = [g(w1 )]2 = [g(w2 )]2 = w2 − w0 , hence w1 = w2 . The same argument shows that there do not exist w1 , w2 ∈ Ω such that g(w1 ) = −g(w2 ). Let q ∈ Ω, and let z 0 = g(q). Note that z 0 = 0, for otherwise w0 ∈ Ω. Since g is holomorphic, there exists r ∈ (0, |z 0 |) such that the Euclidean disc D(z 0 , r ) of center z 0 and radius r is contained in g(Ω). Hence, D(−z 0 , r ) ∩ g(Ω) = 1 ˜ for w ∈ Ω. Then by construction, h˜ is univalent and bounded. ∅. Let h(w) := g(w)+z 0 In particular, defining ˜ f (z)) = h(z) := h(

1 , z ∈ D, g( f (z)) + z 0

the function h : D → C is univalent and bounded and a simple computation shows that (3.1.1) holds.  We state the following very simple but useful sufficient condition for a holomorphic function to be univalent: Theorem 3.1.3 (Noshiro-Warschawski’s Theorem) Let Δ ⊂ C be a convex domain. Let f : Δ → C be holomorphic. If Re f  (z) > 0 for all z ∈ Δ then f is univalent in Δ. Proof Let z, w ∈ Δ, z = w. Assume that f (z) = f (w). Then 

 1 f  (ζ )dζ = f  (t z + (1 − t)w)(z − w)dt w 0  1 = (z − w) f  (t z + (1 − t)w)dt.

0 = f (z) − f (w) =

z

0

Therefore,

1 0

f  (t z + (1 − t)w)dt = 0, and, in particular, 

0 = Re 0

1

f  (t z + (1 − t)w)dt =



1

Re f  (t z + (1 − t)w)dt,

0

which is impossible since by hypothesis Re f  (t z + (1 − t)w) > 0 for t ∈ [0, 1]. 

3.2 No Koebe Arcs Theorem

73

3.2 No Koebe Arcs Theorem Let H ⊂ C. We denote its Euclidean diameter by diamE (H ) := sup{|z − w| : z, w ∈ H }. A Jordan arc is a continuous injective function γ : [a, b] → C∞ for some a, b ∈ R and a < b. A Jordan curve is a continuous function γ : [a, b] → C∞ such that γ (a) = γ (b) and γ : [a, b) → C∞ is injective. In other words, a Jordan curve is a continuous injective function from ∂D to C∞ . As customary, when no risk of confusion occurs, with a slight abuse of notation we will call Jordan arc (or Jordan curve) also the image Γ := γ ([a, b]) of a Jordan arc (or of a Jordan curve). The Jordan Theorem (see, e.g., [105, p. 31]) states that every Jordan curve divides C∞ into two open connected components: Theorem 3.2.1 (Jordan’s Theorem) Let Γ be a Jordan curve in C∞ . Then C∞ \ Γ is the union of two open connected components in C∞ whose common boundary is Γ . Definition 3.2.2 Let f : D → C be holomorphic. A sequence {Cn } of Jordan arcs, Cn ⊂ D for all n ∈ N, is called a sequence of Koebe arcs for f if there exists K > 0 such that for all n ∈ N diamE (Cn ) ≥ K and if there exists L ∈ C∞ such that for every sequence {z n }, with z n ∈ Cn , lim f (z n ) = L .

n→∞

We say that f has no Koebe arcs if there does not exist any sequence of Koebe arcs for f . One of the basic facts about univalent functions which is going to be used in this book is that univalent functions have no Koebe arcs. In order to prove the result we need the following maximum principle: Lemma 3.2.3 Let f : D → C be holomorphic. Assume there exists R > 0 such that supz∈D | f (z)| ≤ R. Let D be an Euclidean disc such that ∂ D ∩ D intersects orthogonally ∂D in two points. Let G be a simply connected domain in D \ ∂ D such that G is relatively compact in D. Assume that ∂G = A ∪ B, where B is a Jordan arc with end points {a1 , a2 }, A ∩ B = {a1 , a2 }, and A = ∂ D ∩ ∂G is a closed arc (see Fig. 3.1). Suppose that there exists N > 0 such that supz∈B | f (z)| ≤ N . Then max | f (z)| ≤ max{N , z∈G

√

N R}.

74

3 Univalent Functions

Fig. 3.1 Domain G such that G = A ∪ B

Proof Let ∂D ∩ ∂ D = { p, q}. By Proposition 1.2.2 there exists an automorphism T of D such that T (−1) = q and T (1) = p. Since T is a Möbius transformation and ∂ D is orthogonal to ∂D, it follows that ∂ D ∩ D = T ((−1, 1)). Hence, G  := T −1 (G) is contained either in {z ∈ C : Im z > 0} or in {z ∈ C : Im z < 0}. We can assume G  ⊂ D ∩ {Im z > 0}, since this does not affect the proof. Let A := T −1 (A) and B  := T −1 (B). Then A = [a, b] for some −1 < a < b < 1. Let ψ(z) := f (T (z)). By construction, ψ : D → C is holomorphic and |ψ(z)| ≤ N on B  .  ∪ (a, b). Note  be the reflection of G  in the real axis and let E := G  ∪ G Let G that E is a domain. Define F(z) := ψ(z)ψ(z) for z ∈ E. Then F is holomorphic in E. Moreover, by hypothesis, |ψ(z)| ≤ N for z ∈ B  and hence |ψ(z)| ≤ N for  , the reflection of B  in the real axis. Since |ψ(z)| ≤ R and |ψ(z)| ≤ R for all z∈B z ∈ E, it follows by the Maximum Principle that sup |F(z)| ≤ sup |F(z)| = sup |ψ(z)||ψ(z)| ≤ N R. z∈E

 z∈B  ∪ B

z∈∂ E

In particular, for r ∈ [a, b] we have |ψ(r )|2 = |F(r )| ≤ N R. Therefore, |ψ(r )| ≤

√

N R for all r ∈ [a, b]. Hence, by the Maximum Principle

sup |ψ(z)| = max |ψ(z)| ≤ max{N , z∈∂G

z∈G 

Thus, | f (z)| ≤ max{N ,

√

N R} for all z ∈ G.

√

N R}. 

Theorem 3.2.4 (no Koebe Arcs Theorem) Let f : D → C be univalent. Then f has no Koebe arcs.

3.2 No Koebe Arcs Theorem

75

Proof First, assume that f is bounded. Assume by contradiction that {Cn } ⊂ D is a sequence of Koebe arcs for f , with diamE (Cn ) ≥ K for some K > 0 and for all n ∈ N. Then there exists L ∈ C such that for every sequence {z n }, with z n ∈ Cn , it holds limn→∞ f (z n ) = L. Up to replace f with f − L, we can assume L = 0. Moreover, since f is bounded, there exists R > 0 such that | f (z)| ≤ R, for all z ∈ D. Hence, up to replace f with Rf , we can assume R = 1. Now, let rn := inf{|z| : z ∈ Cn }. We claim that rn → 1 as n → ∞. If not, there exists r ∈ (0, 1) such that Cn ∩ {z ∈ D : |z| ≤ r } = ∅ for infinitely many Cn ’s. In particular, we can find a sequence {z n k } such that z n k ∈ Cn k , |z n k | ≤ r , and {z n k } converges to some point z 0 ∈ D. Then f (z 0 ) = limn k →∞ f (z n k ) = 0. On the other }. Since the diameter of each Cn is bigger than K , it hand, let s := min{ K2 , 1−r 2 follows that Cn k ⊂ {z ∈ D : |z − z n k | < s}. Therefore, there exists wn k ∈ Cn k such that |wn k − z n k | = s for all n k . We can assume that {wn k } is converging to some z 1 ∈ D, for all n k . Clearly, z 1 = z 0 . But f (z 1 ) = limk→∞ f (wn k ) = 0, since |wn k | ≤ 1+r 2 contradicting the univalency of f . Hence, lim rn = 1.

n→1

(3.2.1)

Since diamE (Cn ) ≥ K , there exist an , bn ∈ Cn such that |an − bn | = K for all n ∈ N. Let Γn denote the circle containing an and bn and orthogonal to ∂D. By (3.2.1), if n is sufficiently large, then an and bn lie on different arcs of Γn ∩ {z ∈ D : rn ≤ |z| ≤ 1}. Hence we can find a subarc Bn ⊂ Cn such that Bn intersects each arc of Γn ∩ {z ∈ D : rn ≤ |z| ≤ 1} in exactly one point. Thus Γn ∩ Bn = { pn , qn }. Let An be the arc in Γn joining pn and qn and such that An ∩ {z ∈ D : |z| < rn } = ∅. Then An ∪ Bn is a Jordan curve, and we denote by G n its bounded part (see Fig. 3.2).

Fig. 3.2 Domain whose boundary is An ∪ Bn with An ⊂ Γn and Bn ⊂ Cn

76

3 Univalent Functions

Note that G n is relatively compact in D and its boundary satisfies the condition of Lemma 3.2.3. Therefore, taking into account that by our hypothesis | f (z)| ≤ 1 for all z ∈ D, by Lemma 3.2.3 we have sup | f (z)| ≤ z∈G n

 max | f (z)|. z∈Bn

(3.2.2)

Let z n ∈ Bn be such that | f (z n )| = maxz∈Bn | f (z)|. Since z n ∈ Cn for all n ∈ N, limn→∞ | f (z n )| = 0. Then, by (3.2.2), for every sequence {wn } ⊂ D such that wn ∈ G n , it holds (3.2.3) lim f (wn ) = 0. n→∞

Now, let Dn be the disc bounded by Γn . By construction, G n contains Dn ∩ {z ∈ D : |z| < rn }. Since an , bn ∈ Γn , the diameter of Dn ∩ D is at least K , hence, there exists r ∈ (0, 1) such that for every n ∈ N it holds G n ∩ {z ∈ D : |z| ≤ r } = ∅. In particular, arguing as in the proof of (3.2.1), we can find two sequences {wn }, {wn } ⊂ D such that {wn } converges to w0 ∈ D and {wn } converges to w1 ∈ D with w0 = w1 and wn , wn ∈ G n for all n ∈ N. Thus, by (3.2.3), f (w0 ) = f (w0 ) = 0, a contradiction. Therefore, every bounded univalent function in D has no Koebe arcs. Suppose now that f is an unbounded univalent function. Assume by contradiction that {Cn } ⊂ D is a sequence of Koebe arcs for f . Then there exists L ∈ C∞ such that for every sequence {z n }, with z n ∈ Cn , it holds limn→∞ f (z n ) = L. We claim that L ∈ C∞ \ f (D). Indeed, if L ∈ f (D) there exists z 0 ∈ D such that f (z 0 ) = L. Let ε0 > 0 be such that D(z 0 , ε0 ) := {z ∈ C : |z − z 0 | < ε0 } ⊂ D. Hence, by the Open Mapping Theorem, for all ε ∈ (0, ε0 ), the set f (D(z 0 , ε)) is an open neighborhood of L. Since f is univalent and hence f −1 ( f (D(z 0 , ε))) = D(z 0 , ε), it follows easily that every sequence {z n } such that z n ∈ Cn , has to converge to z 0 . But then, Cn is eventually contained in D(z 0 , ε) for every fixed ε ∈ (0, ε0 ), thus the diameter of Cn tends to 0 as n → ∞, contradicting the definition of Koebe arcs. Therefore, L ∈ C∞ \ f (D). If L = ∞, let T be a Möbius transformation such that T (L) = ∞. Since L ∈ / f (D), the function T ◦ f : D → C is univalent. Hence, replacing f with T ◦ f if this is the case, we can assume L = ∞. By Lemma 3.1.2, there exists a bounded univalent function h : D → C which satisfies (3.1.1). Let {z n } be a sequence with z n ∈ Cn . By (3.1.1),  lim

n→∞

1 − z0 h(z n )

2 = lim ( f (z n ) − w0 ) = ∞, n→∞

which implies limn→∞ h(z n ) = 0. Hence, {Cn } is also a sequence of Koebe arcs for h, which is a contradiction because we already saw that bounded univalent functions in D do not have Koebe arcs. 

3.3 Boundary Behavior

77

3.3 Boundary Behavior In this section we study the boundary behavior of univalent functions. Theorem 3.3.1 (Lehto-Virtanen’s Theorem) Let f : D → C be univalent and −∞ < a < b ≤ +∞. Let γ : [a, b) → D be a continuous curve such that limt→b γ (t) = σ ∈ ∂D. Suppose that limt→b f (γ (t)) = L for some L ∈ C∞ . Then ∠ lim f (z) = L . z→σ

Proof Let h, z 0 , w0 be given by Lemma 3.1.2. We claim that there exists A ∈ C such that limt→b h(γ (t)) = A. By (3.1.1), for all t ∈ [a, b),  f (γ (t)) − w0 =

1 − z0 h(γ (t))

2 .

Hence, if L = ∞, then necessarily h(γ (t)) → 0 as t → b, and the claim is proved with A = 0. If L ∈ C, the previous equation shows that if A is in the cluster set of h(γ (t)) at b, then ( A1 − z 0 )2 = L − w0 (and, in particular, A = 0). Thus, the cluster set of h(γ (t)) at t = b contains at most two points. However, the cluster set is connected by Lemma 1.9.9. Hence, the claim is proved. Then limt→b h(γ (t)) = A for some A ∈ C. By Theorem 1.5.7, h has nontangential limit A at σ . Hence, if {z n } ⊂ D is a sequence converging non-tangentially to σ , by (3.1.1) it holds  lim f (z n ) = lim

n→∞

and we are done.

n→∞

 w0 +

1 − z0 h(z n )

2

 = w0 +

1 − z0 A

2 = L, 

The following result is a generalization of the classical theorem of Fatou about almost everywhere existence of radial limits for bounded holomorphic functions: Proposition 3.3.2 Let f : D → C be univalent. Then for almost every σ ∈ ∂D (with respect to the Lebesgue measure of ∂D) the non-tangential limit ∠ lim z→σ f (z) exists. Moreover, there exist no L ∈ C∞ and a subset A ⊂ ∂D of positive measure such that ∠ lim z→σ f (z) = L for all σ ∈ A. Proof By Theorem 3.3.1, it is enough to prove that f has radial limit at almost every point in ∂D. By Lemma 3.1.2, it is enough to show that every bounded univalent function has radial limit at almost every point in ∂D, and this follows immediately from Proposition 1.6.9. The last statement follows again by Lemma 3.1.2 and Proposition 1.6.9 for bounded holomorphic maps. 

78

3 Univalent Functions

After having considered boundary limits of univalent functions, we concentrate now on preimages of Jordan arcs. ∞ Let Ω be a domain in C. We denote by Ω its closure in C∞ (which coincides with the closure Ω in C if Ω is bounded). The boundary of Ω in C∞ is ∂∞ Ω := Ω

∞

∞

∩ (C∞ \ Ω ).

If Ω is bounded, and, as usual, ∂Ω denotes its boundary in C, then ∂∞ Ω = ∂Ω. Proposition 3.3.3 Let f : D → C be univalent, and let Ω := f (D). Let γ : [a, b) → Ω be a continuous curve, −∞ < a < b ≤ +∞. Assume there exists p ∈ ∂∞ Ω such that limt→b γ (t) = p. Then there exists σ ∈ ∂D such that limt→b f −1 (γ (t)) = σ and ∠ lim z→σ f (z) = p. Proof If limt→b f −1 (γ (t)) = σ ∈ ∂D, let γ˜ (t) := f −1 (γ (t)). Then γ˜ : [a, b) → D is a continuous curve converging to σ . Moreover, lim f (γ˜ (t)) = lim f ( f −1 (γ (t))) = lim γ (t) = p

t→b

t→b

t→b

and, by Theorem 3.3.1, f has non-tangential limit p at σ . In order to show that there exists σ ∈ ∂D such that limt→b f −1 (γ (t)) = σ ∈ ∂D, let Γ := Γ ( f −1 ◦ γ , b) denote the cluster set of f −1 (γ (t)) at t = b. We have to prove that Γ is a singleton. For r ∈ (0, 1), let D(r ) := {z ∈ C : |z| < r }. Since f is a proper map, it follows / D(r ) for all that for every r ∈ (0, 1) there exists tr ∈ (a, b) such that f −1 (γ (t)) ∈ t ∈ (tr , b). In particular, Γ ⊂ ∂D. Let σ ∈ Γ . Assume by contradiction that η ∈ Γ \ {σ }. Then ∂D \ {σ, η} = A1 ∪ A2 where A1 , A2 are open non-empty disjoint arcs. Since the cluster set Γ is connected (see Lemma 1.9.9) either A1 or A2 is contained in Γ . Hence, we can assume without loss of generality that A1 ⊂ Γ . We claim that for all points q in A1 \ {σ, η}, but at most one point q0 ∈ A1 \ {σ, η}, q q there exists a sequence {tn } ⊂ [a, b) converging to b such that f −1 (γ (tn )) converges (1) (2) to q non-tangentially. Indeed, if there are two, let us say q0 and q0 , denoting L the segment that joints both points, it holds that f −1 (γ (t)) must cut the segment L infinitely many times when t goes to b. Therefore, there is a sequence {tn } converging to b such that f −1 (γ (tn )) ∈ L, for all n, and the sequence { f −1 (γ (tn ))} converges to a point of Γ . Since Γ ⊂ ∂D, then { f −1 (γ (tn ))} converges to one of the points q0(1) or q0(2) , getting a contradiction. By Proposition 3.3.2, f has non-tangential at almost every point of A1 . Let q ∈ A1 \ {q0 , σ, η} be one of such points where f has non-tangential limit. Then ∠ lim f (z) = lim f ( f −1 (γ (tnq ))) = p. z→q

n→∞

Hence, f has constant limit on a set of positive measure, against Proposition 3.3.2. 

3.3 Boundary Behavior

79

The previous result says that the preimage of a continuous curve in Ω ending at one boundary point is a continuous curve in D ending at one boundary point. In fact, one can say a little bit more: the preimage of two continuous curves ending at two different boundary points of Ω are two continuous curves in D ending at two different points. The formal statement is the following: Corollary 3.3.4 Let f : D → C be univalent, and let Ω := f (D). Let γ j : [a j , b j ) → Ω be a continuous curve, with −∞ < a j < b j ≤ +∞, j = 1, 2. Suppose there exist p1 , p2 ∈ ∂∞ Ω, p1 = p2 such that limt→b j γ j (t) = p j , j = 1, 2. Then there exist σ1 , σ2 ∈ ∂D, σ1 = σ2 such that limt→b j f −1 (γ j (t)) = σ j , j = 1, 2. Proof According to Proposition 3.3.3, there exist two points σ1 , σ2 ∈ ∂D such that limt→b j f −1 (γ j (t)) = σ j and ∠ lim z→σ j f (z) = p j for j = 1, 2. Hence, if σ := σ1 = σ2 , then lim(0,1)r →1 f (r σ ) = p1 = p2 , a contradiction. Therefore,  σ1 = σ2 . On the other hand, it is also sometimes useful to have conditions which imply that the pre-image of two curves have the same ending point: Proposition 3.3.5 Let f : D → C be univalent, and Ω := f (D). Let γ j : [a j , b j ) → Ω be two injective continuous curves, with −∞ < a j < b j ≤ +∞, j = 1, 2. Suppose that (1) γ1 (a1 ) = γ2 (a2 ), (2) there exists p ∈ ∂∞ Ω such that limt→b j γ j (t) = p, j = 1, 2, (3) γ1 ((a1 , b1 )) ∩ γ2 ((a2 , b2 )) = ∅. Let Γ be the Jordan curve in C∞ given by the union of the closure of the images ∞ ∞ of γ1 and γ2 , i.e, Γ = γ1 ((a1 , b1 )) ∪ γ2 ((a2 , b2 )) . If one of the two connected components of C∞ \ Γ is contained in Ω, then there exists σ ∈ ∂D such that limt→b j f −1 (γ j (t)) = σ , j = 1, 2. Proof By Proposition 3.3.3 there exist σ1 , σ2 ∈ ∂D such that limt→b j f −1 (γ j (t)) = σ j and ∠ lim z→σ j f (z) = p, j = 1, 2. Assume by contradiction that σ1 = σ2 . Let D ⊂ h(D) be the connected component of C∞ \ Γ . Suppose first that p ∈ C. Let C( p, n) := {ζ ∈ C : |ζ − p| = n1 }, n ∈ N. Let n 0 ∈ N be such that | p − γ1 (a1 )| > n10 . Then for n ≥ n 0 , D ∩ C( p, n) is the union of countably many open arcs on the circle C( p, n). For every n ≥ n 0 , let An be one of such arcs, with the property that one end point of An , say wn1 , belongs to γ1 ((a1 , b1 )) and the other, say wn2 , belongs to γ2 ((a2 , b2 )). Let K n := f −1 (An ), n ≥ n 0 . Since j { f −1 (wn )} converges to σ j , j = 1, 2 and σ1 = σ2 , it follows that there exists c > 0 such that diamE (K n ) ≥ c for all n ≥ n 0 . On the other hand, by construction, for every sequence {z n } such that z n ∈ K n , we have limn→∞ f (z n ) = p. Namely, {K n } is a sequence of Koebe arcs for f , contradicting Theorem 3.2.4. Hence, σ1 = σ2 . In case p = ∞, we can reduce to the previous case using Lemma 3.1.2. 

80

3 Univalent Functions

3.4 Distortion Theorems In this section we will give useful estimates, which go under the name of “Distortion Theorems”, about the growth of univalent functions and their derivatives. We are going to prove only the estimates that we need for our aims, but we point out that such estimates are also sharp, and the extremal functions which realize equality in the estimates are just rotations of the Koebe function D  z → z/(1 − z)2 . We start with the two following Area Theorems. The first one deals with univalent functions from the complement of the unit disc to the Riemann sphere, fixing ∞. As it is well known, such functions can be written as power series in z and 1z at z = ∞. Theorem 3.4.1 (Exterior Area Theorem) Let g : C∞ \ D → C∞ be univalent and g(∞) = ∞. Moreover, assume that g has the following expansion at ∞ g(z) = z +

∞

bn

n=0

1 . zn



Then

0 ≤ Area(C \ g(C \ D)) = π 1 −

∞

n|bn |2 .

n=1

In particular, |b1 | ≤ 1. Proof Fix r > 1 and let Δr := C \ g(Dr ), where Dr := {z ∈ C : |z| ≥ r }. Note that ∂Δr = g(∂ Dr ) for every r . Also, since ∂Δr is a closed curve, it follows that 

 ∂Δr

xd x =

∂Δr

ydy = 0.

Therefore, writing dz = d x + idy, we have 1 2i

 ∂Δr

zdz =

1 2i

 ∂Δr

1 2

[i(xdy − yd x) + xd x + ydy] =

 ∂Δr

(xdy − yd x).

Hence, by Green’s Theorem, and substituting the expression of g into the integrals— and taking into account that we can integrate term by term because the series converges uniformly on compacta of C∞ \ D—, we have  Area(Δr ) =

Δr

 ∂Δr

(xdy − yd x) =

1 2i

 zdz ∂Δr

 1 2π g(z)g (z)dz = g(r eit )g  (r eit )r eit dt 2 0 ∂ Dr    ∞ ∞



1 2π nbn r −n e−int bn r −n eint dt = r eit − r e−it + 2 0 n=1 n=0

1 = 2i



1 2

d xd y = 

3.4 Distortion Theorems

81

 =π r − 2

∞

2 −2n

n|bn | r

.

n=1

N 2 −2n Since 0 ≤ Area(Δr ) = π(r 2 − ∞ ), we see that n=1 n|bn |2 r −2n ≤ n=1 n|bn | r

N 2 + r for every N ∈ N. Hence, taking the limit for r → 1 we get n=1 n|bn |2 ≤ 1 for

∞ every N . Thus, taking the limit for N → ∞, the series n=1 n|bn |2 is convergent. Since Δr  ⊂ Δr for 1 < r  < r , and hence Area(Δr ) → Area(∪r >1 Δr ),

2 −2n ) taking the limit for r → 1+ in the expression Area(Δr ) = π(r 2 − ∞ n=1 n|bn | r we have the result.

The last statement follows easily from the fact that |b1 | ≤ 1 − ∞ n=2 n|bn | and  n|bn | ≥ 0 for every n ≥ 2. Corollary 3.4.2 Let f : D → C be univalent. Assume that f (0) = 0 and f  (0) = 1 n and let f (z) = z + ∞ n=2 an z be the expansion of f at z = 0. Then |a2 | ≤ 2. Proof Let D  z →

f (z 2 ) . z2

Such a function is holomorphic and it is different from  2) which is a holomorphic 0 for every z ∈ D. Then, there exists a square root f (z z2 function in D which sends 0 to 1 (see, e.g., [113, Theorem 13.11]). Let h(z) =  z

f (z 2 ) . z2

Then 

h(z) = z

  ∞

 f (z 2 ) 1 a2 1 + = z an z 2n−2 = z(1 + a2 z 2 + . . .) = z + z 3 + . . . 2 z 2 2 n=2

Notice that h(z) = 0 only if z = 0. We claim that h is univalent. Indeed, preliminarily note that h(−z) = −h(z) for all z ∈ D. Hence, if h(z 1 ) = h(z 2 ), then h(z 1 )2 = h(z 2 )2 , which means f (z 12 ) = f (z 22 ), and, since f is univalent, z 12 = z 22 , that is, z 1 = ±z 2 . However, if z 1 = −z 2 then h(z 2 ) = h(z 1 ) = h(−z 2 ) = −h(z 2 ), forcing h(z 2 ) = 0. Hence z 2 = z 1 and h is univalent. 1 for |z| > 1. Then g : C∞ \ D → C∞ is univalent, Finally, let g(z) := h(1/z) g(∞) = ∞, and a2 1 + ··· . g(z) = z − 2 z By Theorem 3.4.1, | a22 | ≤ 1, and we are done.



The previous result can be extended easily to provide estimates at points different from the origin: Corollary 3.4.3 Let f : D → C be univalent with f (0) = 0 and f  (0) = 1. Then

82

3 Univalent Functions

    f (z) 4|z| 2|z|2  z  f  (z) − 1 − |z|2  ≤ 1 − |z|2 , z ∈ D.

(3.4.1)

   2|z|2 − 4|z| 2|z|2 + 4|z| f (z) ≤ ≤ Re z , z ∈ D. 1 − |z|2 f  (z) 1 − |z|2

(3.4.2)

In particular,

Proof The function g : D → C g(ζ ) :=

 f

ζ +z 1+zζ



− f (z)

(1 − |z|2 ) f  (z)

, ζ ∈D

(3.4.3)

is also univalent, g(0) = 0 and g  (0) = 1. Indeed,   ζ +z f  1+zζ 1 g  (ζ ) =  f (z) (1 + zζ )2 and g  (ζ ) =

f 



ζ +z 1+zζ

f  (z)



  ζ +z f  1+zζ 1 − |z|2 z −2 . 4  (1 + zζ ) f (z) (1 + zζ )3

In particular, g  (0) =

f  (z) (1 − |z|2 ) − 2z. f  (z)

Since |g  (0)|/2 ≤ 2 (see Corollary 3.4.2), we obtain (3.4.1). In order to get (3.4.2) we notice that, for all z ∈ D, (3.4.1) gives       2|z|2 2|z|2 f (z) 4|z| f (z) 4|z| − + Re z  ≤ and − Re z ≤ , f (z) f  (z) 1 − |z|2 1 − |z|2 1 − |z|2 1 − |z|2



and we are done.

The previous bound on the second coefficient of a normalized univalent function is useful to obtain the first distortion theorem, in its qualitative form: Theorem 3.4.4 (Koebe 1/4-Theorem) Let f : D → C be univalent. Suppose f (0) = 0 and f  (0) = 1. Then 41 D ⊂ f (D). Proof Let ζ ∈ C \ f (D). We want to show that |ζ | ≥ 14 . Let h(z) :=

ζ f (z) . ζ − f (z)

Clearly, h : D → C is univalent, since it is the composition of f with the Möbius w n transformation w → ζζ−w . Moreover, if we write f (z) = z + ∞ n=2 an z , a simple

3.4 Distortion Theorems

83

computation shows

  1 2 z + .... h(z) = z + a2 + ζ

Hence, by Corollary 3.4.2, it follows that     a2 + 1  ≤ 2.  ζ Moreover, since, again by Corollary 3.4.2, |a2 | ≤ 2 we have      1    ≤ a2 + 1  + |a2 | ≤ 4, ζ   ζ from which |ζ | ≥

1 4



and we are done.

Now we deal with the analytical form of the Koebe Distortion Theorem. We need a simple lemma: Lemma 3.4.5 Let p : [0, 1) → [0, +∞) be a continuous function and γ : [0, 1] → D a Jordan arc from 0 to z ∈ D such that γ is absolutely continuous in [0, 1]. Then 

|z|



1

p(t) dt ≤

0

p(|γ (t)|)|γ  (t)| dt.

0

Proof Since γ is absolutely continuous, so is its modulus [0, 1]  t → |γ (t)|. Moreover, γ (t) = 0, if t = 0 thus, for almost every t ∈ [0, 1], it holds dtd (|γ (t)|2 ) = 2Re γ  (t)γ (t) and d (|γ (t)|) = Re dt



γ (t) γ (t) |γ (t)|





≤ |γ  (t)|.

Therefore, 

|z| 0



1

p(t) dt = 0

d p(|γ (t)|) (|γ (t)|) dt ≤ dt



1

p(|γ (t)|)|γ  (t)| dt.

0

 Theorem 3.4.6 (Koebe’s Distortion Theorem) Let f : D → C be univalent. Suppose f (0) = 0 and f  (0) = 1. Let z ∈ D. Then |z| |z| ≤ | f (z)| ≤ , (1 + |z|)2 (1 − |z|)2

(3.4.4)

1 − |z| 1 + |z| ≤ | f  (z)| ≤ . (1 + |z|)3 (1 − |z|)3

(3.4.5)

84

3 Univalent Functions

Proof We begin by checking (3.4.5). The function f  is analytic and nonvanishing on the unit disc, therefore we can choose an analytic branch of its logarithm such that log 1 = 0. Hence, for r > 0 and θ ∈ R     ∂ ∂ ∂  iθ  iθ  iθ log | f (r e )| = Re log f (r e ) = Re log f (r e ) ∂r ∂r ∂r       f  r eiθ f  r eiθ iθ 1 iθ  e   re . = Re = Re r f  r eiθ f  r eiθ From (3.4.2), we have 2r − 4 ∂ 2r + 4 log | f  (r eiθ )| ≤ ≤ . 1 − r2 ∂r 1 − r2 By integrating in r , and taking into account that log f  (0) = log 1 = 0, we have 1+r 1−r ≤ log | f  (r eiθ )| ≤ log , (1 + r )3 (1 − r )3

log

from which (3.4.5) follows at once. Now we prove (3.4.4). We assume that z = 0. If z = r eiθ , then 

r

f (z) =

f  (ρeiθ )eiθ dρ

0

and, by (3.4.5), we have 

r

| f (z)| ≤ 0

| f  (ρeiθ )|dρ ≤



r 0

1+ρ r dρ = , (1 − ρ)3 (1 − r )2

which gives the upper bound. As for the lower bound, note that if z ∈ D then |z| ≤ 1/4 because the map [0, 1]  t → t/(1 + t)2 is non-decreasing. There(1+|z|)2 fore it suffices to establish the inequality under the assumption that | f (z)| < 1/4. Fix z ∈ D with | f (z)| < 1/4. Koebe’s 1/4-Theorem 3.4.4 implies that {z : |z| < 1/4} ⊆ f (D). Thus, the curve γ (t) = f −1 (t f (z)) is well-defined for 0 ≤ t ≤ 1. Since f (γ (t)) = t f (z) we have f  (γ (t))γ  (t) = f (z) for all t. Thus inequalities (3.4.5) and Lemma 3.4.5 imply  | f (z)| = 

    f (γ (t))γ  (t) dt ≥

0

0



1 0

|z|

≥ and we are done.

1

1 − |γ (t)| |γ  (t)|dt (1 + |γ (t)|)3

1−t |z| dt = , (1 + t)3 (1 + |z|)2 

3.4 Distortion Theorems

85

In terms of hyperbolic distance, Koebe’s Distortion Theorem can be rewritten as Theorem 3.4.7 Let f : D → C be univalent and z 1 , z 2 ∈ D. Then 1 | f (z 1 ) − f (z 2 )| tanh(ω(z 1 , z 2 )) . tanh(ω(z 1 , z 2 )) ≤ ≤ 4 (1 − |z 1 |2 )| f  (z 1 )| (1 − tanh(ω(z 1 , z 2 ))2 z 1 −z , z ∈ D, and denote Proof Take the automorphism of the unit disc T (z) := 1−z 1z w := T (z 2 ) ∈ D. Notice tanh(ω(z 1 , z 2 )) = |T (z 2 )| = |w|. Consider the univalent map  f = f ◦ T . By Koebe’s Distortion Theorem 3.4.6,

| f  (0)|

|w| |w| ≤ | f (w) −  f (0)| ≤ |  f  (0)| . (1 + |w|)2 (1 − |w|)2

Bearing in mind that T ◦ T is the identity in D and | f  (0)| = | f  (z 1 )||T  (0)| = | f  (z 1 )|(1 − |z 1 |2 ), 

the result follows at once. As a matter of notation, if Ω ⊂ C is a domain, for z ∈ Ω, we let δΩ (z) := inf |z − w|, w∈C\Ω

the Euclidean distance of z from the boundary ∂Ω. Proposition 3.4.8 Let f : D → C be univalent and z ∈ D. Denote Ω := f (D). Then δΩ ( f (z)) = lim inf | f (u) − f (z)|. |u|→1

Proof Let a ∈ ∂Ω be such that |a − f (z)| = δΩ ( f (z)) and let {u n } ⊂ D be such that limn→∞ |u n | = 1 and limn→∞ f (u n ) = a. Then lim inf | f (u) − f (z)| ≤ lim | f (u n ) − f (z)| = δΩ ( f (z)). |u|→1

n→∞

Being trivial the other inequality, we are done.



We end this section with two results that show how the derivative of a univalent map control the distance to the boundary. Theorem 3.4.9 Let f : D → C be univalent and z ∈ D. Denote Ω := f (D). Then 1 (1 − |z|2 )| f  (z)| ≤ δΩ ( f (z)) ≤ (1 − |z|2 )| f  (z)|. 4

86

3 Univalent Functions

Proof Let z ∈ D. Take the canonical automorphism of the unit disc T such that T (z) = 0 (see (1.2.1)) and consider the univalent map  f = f ◦ T . By Koebe’s Distortion Theorem 3.4.6, |w| 1 | f (w) −  f (0)| |w| ≤ ≤ , w∈D   4 (1 + |w|)2 | f (0)| thus, taking u = T (w), 1 |T (u)|(1 − |z|2 )| f  (z)| ≤ | f (u) − f (z)|, u ∈ D. 4 Since lim inf |u|→1 |T (u)| = 1, taking limits in the above expression and using Proposition 3.4.8, we have 1 (1 − |z|2 )| f  (z)| ≤ δΩ ( f (z)). 4 



f (0) On the other hand, let g(w) := f (w)− , w ∈ D and h(w) := g(w) , if w = 0 and  w f  (0) h(0) := g  (0) = 1. Since h(w) = 0 for all w ∈ D, the function 1/ h(w) is holomorphic in D. Hence, by the Maximum Principle

1 1 ≤ lim sup , |h(0)| |w|→1 |h(w)| or, equivalently,

lim inf |h(w)| ≤ |h(0)| = |g  (0)| = 1. |w|→1

Finally, since lim inf |u|→1 |T (u)| = 1, | f (w) −  f (0)| 1 lim inf 2  (1 − |z| )| f (z)| |w|→1 |w| δΩ ( f (z)) 1 | f (u) − f (z)| lim inf = , = (1 − |z|2 )| f  (z)| |u|→1 |T (u)| (1 − |z|2 )| f  (z)|

lim inf |h(w)| = |w|→1

and we are done.



3.5 Convergence of Univalent Mappings In this section, we see how the convergence on compacta of D of sequences of univalent functions can be characterized in terms of the convergence, in a suitable sense, of their images. This brings to the notion of kernel of domains and the Carathéodory Kernel Convergence theorem.

3.5 Convergence of Univalent Mappings

87

We start with the following simple result: Theorem 3.5.1 Let { f n } be a sequence of univalent functions in the unit disc which converges uniformly on compacta of D to a function f . Then f is either univalent or constant. Proof Suppose there exist two different points z 1 , z 2 ∈ D such that c := f (z 1 ) = f (z 2 ) and that f is not constant. For z ∈ C and r > 0 let D(z, r ) := {w ∈ C : |w − z| < r } denote the Euclidean disc of center z and radius r . Let ε > 0 be such that D(z j , ε) ⊂ D, j = 1, 2 and D(z 1 , ε) ∩ D(z 2 , ε) = ∅. Since f is not constant, we can assume that f (z) = c for all z ∈ ∂ D(z 1 , ε) ∪ ∂ D(z 2 , ε). Let N ( f, z j ) (respectively, N ( f n , z j )) be the number of zeros of f (z) − c (respect., f n (z) − c) in D(z j , ε), j = 1, 2. Hence, taking into account that { f n } converges uniformly to f on ∂ D(z 1 , ε) ∪ ∂ D(z 2 , ε), we have for j = 1, 2,  f  (ζ ) 1 dζ 1 ≤ N ( f, z j ) = 2πi ∂ D(z j ,ε) f (ζ ) − c  1 f n (ζ ) = lim dζ = lim N ( f n , z j ). n→∞ 2πi ∂ D(z ,ε) f n (ζ ) − c n→∞ j Therefore, for n sufficiently big, N ( f n , z j ) ≥ 1, which implies that there exist z 1n ∈ D(z 1 , ε) and z 2n ∈ D(z 2 , ε) such that f n (z 1n ) = c = f n (z 2n ), contradicting the  univalency of f n . Definition 3.5.2 Let w0 ∈ C and let {Ωn } be a sequence of domains in C such that w0 ∈ Ωn for every n. Let G be the (possibly empty) set of all points w ∈ C such that there exists an open connected set Δ ⊂ C with the property that {w0 , w} ⊂ Δ and Δ ⊂ Ωn for all but a finite number of n ∈ N. The kernel of {Ωn } with respect to w0 is the union of G and {w0 }. We say that {Ωn } kernel converges to Ω with respect to w0 if every subsequence of {Ωn } has the same kernel. Notice that the kernel is either a single point or a domain in C. Even if every sequence of domains have always a kernel by the very definition, it might not be kernel convergent, as the following example shows: Example 3.5.3 For n ∈ N, let Ω2n := D and let Ω2n−1 := D \ [1/2, 1). The kernel of {Ωn } with respect to 0 is clearly D \ [1/2, 1). However, the kernel of {Ω2n } is D, hence, {Ωn } is not kernel convergent. In case of increasing or decreasing sequences of domains, the kernel convergence is assured: Example 3.5.4 Let {Ωn } be a sequence of domains in C such that Ωn ⊆ Ωn+1 for all n ∈ N and let w0 ∈ Ω1 . Then {Ωn } kernel converges with respect to w0 to ∪n∈N Ωn .

88

3 Univalent Functions

Proposition 3.5.5 Let {Ωn } be a sequence of domains in C such that Ωn+1 ⊆ Ωn for all n ∈ N. Assume w0 ∈ ∩n∈N Ωn , and let Λ be the interior of ∩n∈N Ωn . Then either w0 ∈ Λ and {Ωn } kernel converges with respect to w0 to the connected component Ω of Λ that contains w0 or w0 ∈ / Λ and {Ωn } kernel converges with respect to w0 to {w0 }. Proof Due to the monotonicity of the sequence {Ωn }, the set Ω remains unchanged if we replace {Ωn } by any subsequence {Ωn k }, n k → ∞. Therefore it is sufficient to / Λ, it is clear that the show that Ω is the kernel of {Ωn } with respect to {w0 }. If w0 ∈ kernel is just the set {w0 }. Assume now that w0 ∈ Λ. The set Ω is a subset of the kernel because we can choose Δ = Ω in Definition 3.5.2. Take w = w0 a point of the kernel, and let Δ be chosen according to the Definition 3.5.2. Then Δ ⊂ ∩n∈N Ωn because {Ωn } is decreasing. Since Δ is a connected and w0 ∈ Δ, it follows that Δ ⊂ Ω and therefore w ∈ Ω.  Example 3.5.6 The kernel convergence depends on the choice of the point w0 . As an example, if Ωn := C \ ((−∞, −1/n] ∪ [1/n, +∞)) , n ∈ N then {Ωn } kernel converges with respect to w0 to {w ∈ C : Im w > 0}, {0}, {w ∈ C : Im w < 0},

if w0 = i; if w0 = 0; if w0 = −i.

Now we are going to describe how convergence on compacta of univalent maps is related to kernel convergence of their images. In order to state and prove the main result of this section, we need a lemma. Lemma 3.5.7 Let w0 ∈ C and let { f n } be a sequence of univalent functions in D such that f n (0) = w0 for every n ∈ N. Assume that { f n } converges uniformly on compacta of D to a function f . Suppose there exists a domain Δ ⊂ ∩n∈N f n (D) which contains w0 . Then: (1) f is univalent, (2) Δ is contained in f (D) and also in the kernel of { f n (D)} with respect to w0 , (3) { f n−1 |Δ } converges uniformly on compacta of Δ to f −1 |Δ . Proof First of all note that, since Δ is a domain in C such that w0 ∈ Δ ⊂ ∩n∈N f n (D), it follows that Δ is contained in the kernel of { f n (D)} with respect to w0 . Denote by gn the restriction of the inverse function f n−1 to the domain Δ. Since gn (Δ) ⊂ D, by Montel’s Theorem [113, Theorem 14.6, p. 282], we can find a subsequence {gn k } that converges locally uniformly in Δ to a holomorphic function g : Δ → D. Since gn (w0 ) = 0 for all n ∈ N, it follows g(w0 ) = 0 and thus g(Δ) ⊂ D. Fix w ∈ Δ. Hence, { f n k } converges uniformly to f in a disc centered at g(w). Since {gn k (w)} converges to g(w) and w = f n k (gn k (w)), we conclude that

3.5 Convergence of Univalent Mappings

89

w = f (g(w)). Hence, g is injective and Δ ⊂ f (D)—so that, in particular, f is not constant. Hence, Theorem 3.5.1 implies that f is univalent. Therefore, f restricted to g(Δ) is the inverse of g. In particular, any convergent subsequence of gn has the same limit. Thus, {gn } converges uniformly on compacta to f −1 |Δ .  Theorem 3.5.8 (Carathéodory Kernel Convergence Theorem) Let w0 ∈ C. Let { f n } be a sequence of univalent functions on D such that f n (0) = w0 and f n (0) > 0. Let Ωn := f n (D), for n ∈ N. Then { f n } converges uniformly on compacta of D to a holomorphic function f if and only if {Ωn } kernel converges with respect to w0 to some set Ω and Ω = C. Furthermore, if this is the case, then f (D) = Ω. Moreover, if Ω = {w0 }, then (1) f is univalent, f (0) = w0 and f  (0) > 0; (2) for every domain D such that w0 ∈ D and D ⊂ Ωn for every but a finite number of n ∈ N0 , the sequence { f n−1 |D } converges uniformly on compacta to f −1 |D . Proof Let us first assume that { f n } converges uniformly on compacta of D to a function f : D → C. Let Ω := f (D). We first prove that Ω is the kernel of {Ωn } with respect to w0 . If f is constant, it is clear that Ω is a subset of the kernel of {Ωn } with respect to w0 . Assume that f is not constant. By Theorem 3.5.1, f is univalent. Take w ∈ Ω, w = w0 and ζ ∈ D such that f (ζ ) = w. Choose r such that |ζ | < r < 1. Then the domain Λ := { f (z) : |z| < r } contains w and w0 . We show that Λ ⊂ Ωn for n large enough. Suppose by contradiction this is not the case. Then there exist a subsequence {n k } ⊂ N converging to ∞ and a sequence {wk } ∈ Λ, which we may / Ωn k for all k. assume converging to some point w∗ ∈ Λ ⊂ f (D), such that wk ∈ Since w∗ ∈ f (D) from Rouché’s Theorem we obtain wk ∈ f n k (D) = Ωn k for all k sufficiently large, a contradiction. Therefore Ω is a subset of the kernel of {Ωn } with respect to w0 . If the kernel is a single point {w0 }, since f (D) is contained in the kernel of {Ωn } for what we already proved, it follows that f is constant and Ω = {w0 }. In case the kernel does not reduce to a single point, let w = w0 be a point in the kernel of {Ωn } with respect to w0 . Then there is a domain Δ and n 0 ∈ N such that w0 , w ∈ Δ ⊂ Ωn for n ≥ n 0 . By Lemma 3.5.7, Δ ⊂ Ω. In particular, w ∈ Ω, and, by the arbitrariness of w, it follows that the kernel of {Ωn } with respect to w0 is contained in Ω. Therefore Ω is the kernel of {Ωn } with respect to w0 . Since the limit function f is either univalent or constant it follows that Ω = C and since every subsequence of { f n } also converges to f it also follows that every subsequence of {Ωn } has the same kernel Ω with respect to w0 so that {Ωn } kernel converges with respect to w0 to Ω. Moreover, Lemma 3.5.7 implies (1) and (2). Assume now that {Ωn } kernel converges with respect to w0 to a set Ω  C. First, we show that the sequence { f n } is normal, i.e., from every subsequence of { f n } we f  (0) can extract a converging subsequence. By Koebe 1/4-Theorem 3.4.4, n4 D + w0 ⊂ f n (D) = Ωn . Therefore, if there exists a subsequence { f n k } such that f nk (0) → +∞, it follows that the kernel of {Ωn k } is C, and, since {Ωn } is kernel converging to Ω, we

90

3 Univalent Functions

get Ω = C, a contradiction. Thus, the sequence { f n (0)} is bounded. By the Koebe Distortion Theorem 3.4.6 we have | f n (z) − w0 | ≤ f n (0)

|z| , z ∈ D. 1 − |z|2

Since { f n (0)} is bounded, we deduce that the sequence { f n } is uniformly bounded on compacta of D, so that it is normal. We claim that { f n } converges uniformly on compacta of D. Since { f n } is uniformly bounded on compacta of D, we can extract converging subsequences by Montel’s Theorem (see, e.g., [113, Theorem 14.6, p. 282]). Let { f n k } be a subsequence of { f n } which converges uniformly on compacta of D to a function f . Since {Ωn } kernel converges with respect to w0 to Ω, by the previous part of the proof, we have Ω = f (D). In particular, if Ω = {w0 }, this shows that { f n } converges uniformly on compacta to the constant function z → w0 . Otherwise, every limit of { f n } is a univalent map from D to Ω which maps 0 to w0 and whose derivative at 0 is strictly positive. From the uniqueness part of the Riemann’s Mapping Theorem 3.1.1, such a map is unique and { f n } is therefore converging uniformly on compacta of D.  Example 3.5.9 Let Ωn := D ∪ {w ∈ C : Re w > 0, |Im w| < n1 } for n ∈ N. Since {Ωn } is a decreasing sequence of simply connected domains, by Proposition 3.5.5, {Ωn } kernel converges with respect to 0 to D. Therefore, if f n : D → Ωn is the Riemann map such that f n (0) = 0 and f n (0) > 0, n ∈ N, it follows that { f n } converges uniformly on compacta of D to the identity map.

3.6 Notes The material in this chapter is classical and most of it can be found in any book about univalent functions. We took inspirations mainly from Pommerenke’s books [105, 106] and Duren’s book [59].

Chapter 4

Carathéodory’s Prime Ends Theory

The aim of this chapter is to introduce prime ends and the Carathéodory topology of simply connected domains and see how impressions of prime ends are related to unrestricted limits and principal parts of prime ends can be used to understand the non-tangential behavior of univalent functions. Finally, we prove Carathéodory’s extension theorems.

4.1 Prime Ends Definition 4.1.1 Let Ω ⊂ C be a simply connected domain. Let γ : [0, 1] → C∞ be a Jordan arc or a Jordan curve. Its image C := γ ([0, 1]) is a cross cut for Ω if γ ((0, 1)) ⊂ Ω and γ (0), γ (1) ∈ ∂∞ Ω. The points γ (0) and γ (1) are called end points of C. Note that, by definition, the end points of a cross cut might be equal. Remark 4.1.2 Let Ω1 , Ω2 be two simply connected domains in C and let f : Ω1 → Ω2 be a biholomorphism. Assume C is a cross cut in Ω1 , given by a Jordan arc or Jordan curve γ : [0, 1] → C. The curve f ◦ γ : (0, 1) → Ω2 is continuous and injective. In general, the limit of f ◦ γ at t = 0 or t = 1 does not exist. However, in case both limits exist, then necessarily f (γ (0)), f (γ (1)) ∈ ∂∞ Ω2 and f ◦ γ : [0, 1] → C is either a Jordan arc or a Jordan curve. In this case, with a slight abuse of notation, we say that f (C) is a cross cut in Ω2 . A cross cut divides a simply connected domain in two connected components: Lemma 4.1.3 Let Ω ⊂ C be a simply connected domain, Ω = C. Let C be a cross cut for Ω. Then Ω \ C consists of two open connected components A, B such that ∂ A ∩ Ω = ∂ B ∩ Ω = C ∩ Ω.

© Springer Nature Switzerland AG 2020 F. Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4_4

91

92

4 Carathéodorys Prime Ends Theory

Proof By Theorem 3.1.1, there exists a univalent map f : D → C such that Ω = −1 for w ∈ Ω. Then h : Ω → C is a homeomorphism. Let f (D). Let h(w) := 1−|f f −1(w) (w)| γ : [0, 1] → C∞ be a Jordan arc or Jordan curve such that γ ((0, 1)) ⊂ Ω and C = γ ([0, 1]). Then h ◦ γ : (0, 1) → h(Ω) = C is a continuous injective curve. Additionally, since limt→0 | f −1 (γ (t))| = limt→1 | f −1 (γ (t))| = 1, by Proposition 3.3.3, lim |h(γ (t))| = lim |h(γ (t))| = ∞.

t→0

t→1

Hence, setting h(γ (0)) = h(γ (1)) = ∞, the curve h ◦ γ : [0, 1] → C∞ is a Jordan curve in C∞ . By Theorem 3.2.1, it divides C∞ into two open connected components ˜ and B = h −1 ( B) ˜ are two open connected components of ˜ B. ˜ Hence, A := h −1 ( A) A, Ω such that A ∪ B = Ω \ C and ∂ A ∩ Ω = ∂ B ∩ Ω = C ∩ Ω.  If C is a cross cut of Ω and Q 1 , Q 2 ⊂ Ω are subsets of Ω, we say that C separates Q 1 and Q 2 if, given the decomposition Ω \ C = A ∪ B from Lemma 4.1.3, it follows that either Q 1 ⊂ A and Q 2 ⊂ B or Q 1 ⊂ B and Q 2 ⊂ A. Definition 4.1.4 Let Ω ⊂ C be a simply connected domain, Ω = C. A null chain (Cn ) for Ω is a sequence of cross cuts Cn for Ω, n ∈ N0 , such that (1) Cn ∩ Cm = ∅, n = m, (2) Cn separates C0 ∩ Ω and Cn+1 ∩ Ω for all n ≥ 1, (3) limn→∞ diam S (Cn ) = 0. The interior part Vn of Cn , n ≥ 1, is the connected component of Ω \ Cn which does not contain C0 ∩ Ω. If the domain Ω is bounded in C, we can replace the condition (3) on the spherical diameter limn→∞ diam S (Cn ) = 0 with the equivalent condition on the Euclidean diameter limn→∞ diamE (Cn ) = 0. Moreover, if Vn is the interior part of Cn , condition (2) is equivalent to Vn+1 ⊂ Vn , n ≥ 1.

(4.1.1)

Remark 4.1.5 Note that, with the above the notation, we have Cm ∩ Ω ⊂ Vn ,

for all m > n,

Cm ∩ Ω ⊂ Ω \ Vn ,

for all m < n.

We stress out again that, according to our definition of cross cuts, if (Cn ) is a null chain for Ω, then Cn might have coincident end points: 1 }. Then (Cn ) Example 4.1.6 Let Ω := D \ [0, 1). Define Cn := {z ∈ C : |z| = n+2 1 , is a null chain and all the cross cuts Cn are Jordan curves whose ends points are n+2 n ∈ N0 .

If (Cn ) is a null chain for Ω, although diam S (Cn ) → 0 as n → ∞, the spherical diameter of Vn might not converge to 0.

4.1 Prime Ends

93

1 Example 4.1.7 For n ∈ N0 , let Γn := {z ∈ C : Re z = n+1 , Im z ∈ (−∞, 0]}. Let Γ := ∪n∈N Γn . Then set Ω := H \ Γ . The domain Ω ⊂ C∞ is simply connected and, by Theorem 3.1.1, it is biholomorphic to D. Therefore, there exists a univalent 1 and function f : D → C such that f (D) = Ω. Let Cn be the segment joining n+1 i . Clearly, Cn is a cross cut for Ω and (Cn ) is a null chain. However, the spherical n+1 diameter of Vn does not tend to 0 as n → ∞, since for every n ∈ N the set Vn contains 2n+3 + it : t ∈ (−∞, 0]}. the half line {z ∈ C : z = 2(n+1)(n+2)

Remark 4.1.8 It is not hard to see that if (Cn ) is a null chain for D, then diam S (Vn ) → 0 as n → ∞. Moreover, Cn is a Jordan arc for n ≥ 1, that is, except at most C0 , the end points of Cn are distinct for every n ≥ 1. Definition 4.1.9 Let Ω be a simply connected domain, Ω = C. Let (Cn ) and (Cn ) be two null chains for Ω. Denote by Vn the interior part of Cn and by Vn the interior part of Cn , n ≥ 1. We say that (Cn ) and (Cn ) are equivalent, and we write (Cn ) ∼ (Cn ), if for every n ≥ 1 there exists m ∈ N such that Vm ⊂ Vn , Vm ⊂ Vn . It is easy to check that the relation ∼ just introduced defines an equivalence relation between null chains. If (Cn ) is a null chain, then for every m ∈ N, (Cn )n≥m is another null chain equivalent to (Cn ). Among all representatives of any equivalence class of null-chains, we can always select a very simple one, as we are going to show. Definition 4.1.10 Let Ω ⊂ C be a simply connected domain, Ω = C. A circular null chain (Cn ) for Ω centered at p ∈ C∞ is a null chain (Cn ) for Ω such that there exists a decreasing sequence of positive numbers {rn } converging to 0 such that: (1) if p ∈ C, Cn ⊂ {z ∈ C : |z − p| = rn } for all n ∈ N0 , (2) if p = ∞, Cn ⊂ {z ∈ C : |z| = r1n } for all n ∈ N0 . Proposition 4.1.11 Let Ω ⊂ C be a simply connected domain, Ω = C. Then every null chain for Ω is equivalent to a circular null chain for Ω. Proof Let (Cn ) be a null chain for Ω. Take a sequence {z n } such that z n ∈ Cn for every n ∈ N. Since C∞ is compact, there exists a converging subsequence, {z n k }, and we call p ∈ C∞ its limit. Let r > 0. If p ∈ C, we let D( p, r ) := {z ∈ C : |z − p| < r }, while, if p = ∞, we let D(∞, r ) := {z ∈ C∞ : |z| > r1 }. Notice that D(∞, r ) is an open neighborhood of ∞. Since diam S (Cn ) → 0 as n → ∞, it follows that for every ε > 0 the sequence (Cn k ) is eventually contained in D( p, ε). This implies in particular that p ∈ ∂∞ Ω because for every k the end points of Cn k belong to ∂∞ Ω. The null chain (Cn k ) is clearly equivalent to (Cn ). To avoid burdening the notation, we can assume that (Cn ) is already converging to p without passing to a subsequence. We denote by Vn the interior part of Cn , n ≥ 1.

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4 Carathéodorys Prime Ends Theory

Note that p ∈ / Cn for all n ∈ N0 , for, if p ∈ Cn for some n ∈ N0 , then p ∈ / Vn+1 . Hence, for every m ∈ N0 and r > 0 there exist rm ∈ (0, r ) and Nm ∈ N0 such that Cm ∩ Ω ⊂ Ω \ D( p, rm ), and Cn ⊂ D( p, rm ) for all n ≥ Nm . Now, let B(r ) := ∂ D( p, r ), for r > 0. We claim that there exists a closed arc Γ (rm ) in B(rm ) forming a cross cut for Ω such that one connected component of Ω \ Γ (rm ) contains Cm and the other connected component—call it Wm —has the property that Vn ⊂ Wm for all n ≥ Nm . Assume the claim for the moment, the proof ends as follows. First, let take r = 1 and m = 0 and let r0 , N0 be the numbers given by the claim. Set G 0 := Γ (r0 ). Then let r = r20 and m = N0 and let N1 := N N0 and r N1 be the numbers given by the claim and define G 1 := Γ (r N1 ). Then, continue inductively the process for n ≥ 1: set r r = N2n and m = Nn and let Nn+1 := N Nn and r Nn+1 be the numbers given by the claim and define G n := Γ (r Nn ) for all n ≥ 2. Note that for every n ≥ 0 the interior part Wn of G n by construction contains Vm for all m sufficiently large, hence, Wn+1 ⊂ Wn for all n ≥ 1. Hence, (G n ) is a circular null chain and, by construction, it is equivalent to (Cn ). We are left to prove the claim. Let z 0 ∈ Cm ∩ Ω and w0 ∈ CNm ∩ Ω. It is enough to prove that the arc Γ (rm ) separates z 0 and w0 . In order to do this, note that B(rm ) intersects Ω in at most countable (open) arcs. The closure of each such an arc is a cross cut for Ω. Since Ω is open and connected in C, we can find a finite number of points, {ζ0 , . . . , ζs } ⊂ Ω \ B(rm ) such that ζ0 = w0 , ζs = z 0 and the segments S j := {tζ j + (1 − t)ζ j+1 : t ∈ [0, 1]} are contained in Ω for all j = 0, . . . , s − 1. Let γ := ∪s−1 j=0 S j . The curve γ is a Jordan arc which joins z 0 with w0 . Since w0 ∈ D( p, r m ) and z0 ∈ / D( p, rm ), the curve γ intersects B(rm ) at least once. Moreover, since B(rm ) is a circle, and γ is the union of segments, the set γ ∩ B(rm ) is finite. By construction, each segment S j might intersect B(rm ) at no points, at one point or at two points. / D( p, rm ) and ζ j+1 ∈ D( p, rm ) Let us say that a segment S j is pointing inside if ζ j ∈ / D( p, rm ). Let S be the set of and it is pointing outside if ζ j ∈ D( p, rm ) and ζ j+1 ∈ all segments which point inside or outside. Since the curve γ starts in D( p, rm ) and ends outside of D( p, rm ), there exists a minimal j0 ∈ {0, . . . , s} such that S j0 points outside. Then, it is clear that for each segment S j with j ∈ { j0 + 1, . . . , s − 1} which points inside there exists a segment which point outside, and hence, the number of elements in S is odd. For any open arc A ⊂ B(rm ) ∩ Ω let us denote by S A the subset of S formed by those segments which intersect A. Since the cardinality of S is odd, it follows that there exists A ⊂ B(rm ) ∩ Ω such that the number of elements of S A is odd. We claim that the cross cut Γ (rm ) := A of Ω separates z 0 and w0 . Indeed, notice / S A then the end points of S j belong to the same connected component of that if S j ∈ Ω \ Γ (rm ), while, if S j ∈ S A then the end points of S j belong to different connected components of Ω \ Γ (rm ). Since S A contains an odd number of elements, it is then  clear that z 0 and w0 belong to different connected components of Ω \ Γ (rm ). Remark 4.1.12 One can adapt the previous proof to show that if p ∈ ∂∞ Ω and there exists a Jordan arc γ in Ω ∪ { p} which ends at p then one can construct a

4.1 Prime Ends

95

circular null chain centered at p (see also Theorem 4.4.14). Otherwise, the previous construction can fail in general. Now we examine the behavior of null chains under univalent maps. Recall our slight abuse of notation explained in Remark 4.1.2. We first show that null chains behave well under preimages: Lemma 4.1.13 Let f : D → C be univalent and Ω := f (D). If (Cn ) is a null chain for Ω then ( f −1 (Cn )) is a null chain for D. Moreover, two null chains (Cn ), (Cn ) for Ω are equivalent if and only if ( f −1 (Cn )) and ( f −1 (Cn )) are equivalent. Proof Let (Cn ) be a null chain for Ω. By Proposition 3.3.3, f −1 (Cn ) is a cross cut of D for every n ∈ N0 and f −1 (Cn ) ∩ f −1 (Cm ) = ∅ for every n = m by Corollary 3.3.4. Hence, the sequence { f −1 (Cn )} satisfies conditions (1) and (2) of Definition 4.1.4. We are left to show that diam S ( f −1 (Cn )) → 0 as n → ∞. Suppose by contradiction that there exists δ > 0 such that diam S ( f −1 (Cn k )) ≥ δ for a subsequence {n k } ⊂ N converging to ∞. Since diam S (Cn ) → 0 as n → ∞, we can assume that there exists L ∈ C∞ such that for every sequence {wn k } ⊂ Ω with wn k ∈ Cn k it holds limn k →∞ wn k = L. Hence, { f −1 (Cn k )} is a sequence of Koebe arcs for f , contradicting Theorem 3.2.4. The last statement follows at once since f −1 is a homeomorphism and hence maps  the interior part of Cn onto the interior part of f −1 (Cn ). The converse of the previous lemma is not true in general. Indeed, if C is a cross cut in D, the closure of f (C ∩ D) might not be a Jordan arc or a Jordan curve. However, for every null chain for D one can always find a suitable equivalent null chain such that its image under f is again a null chain: Lemma 4.1.14 Let f : D → C be univalent, Ω := f (D). For every null chain (Cn ) for D there exists an equivalent circular null chain (G n ) such that for every n ∈ N0 the closure of f (G n ∩ D), which we denote by f (G n ), is a cross cut of Ω and ( f (G n )) is a null chain for Ω. Proof By Proposition 4.1.11, we can assume that (Cn ) is circular. Hence, there exist a sequence of positive numbers {rn } converging to 0 and σ ∈ ∂D such that Cn = ∂ D(σ, rn ) ∩ D for all n ∈ N0 , where, D(σ, rn ) = {z ∈ C : |z − σ | < rn }. It is clear that, if {tn } is a decreasing sequence of positive numbers converging to 0 such that t0 ≤ 1, then, setting G n := ∂ D(σ, tn ) ∩ D, the sequence of cross cuts (G n ) is a null chain for D equivalent to (Cn ). Therefore, we are left to show that we can find the tn ’s in such a way that f (G n ) is a null chain for Ω. By Proposition 3.3.2, we can find t0 ∈ (0, 1) such that f (G 0 ) is a cross cut in Ω. Now, suppose that we already found t0 > t1 > . . . > tn−1 > 0 such that f (G j ) is a cross cut in Ω for j = 1, . . . , n − 1, f (G j ) ∩ f (G k ) = ∅ for j = k, f (G j ) separates f (G 0 ) from R for j = 1, . . . , n − 1 f (G j+1 ) for all j = 1, . . . , n − 2 and diam S ( f (G j )) ≤ j+1 and for some fixed constant R > 0. We show how to construct G n such that the previous properties are satisfied for j = 0, . . . , n, and hence, by induction, we are done.

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4 Carathéodorys Prime Ends Theory

With no loss of generality, up to translations, we can assume that 0 ∈ Ω. The Möbius transformation T : z → 1z is an isometry for the spherical distance, and T (∂∞ Ω) is bounded in C (in case Ω is already bounded in C, we do not need to use the transformation T , but since the proof is essentially the same, we can use it anyway). Starting the procedure from a smaller t0 if necessary, we can assume that | f −1 (0) − σ | > t0 . Let D := {z ∈ D : |σ − z| < t0 }. Therefore, g := T ◦ f : D → C is univalent and there exists M > 0 such that g(D) ⊂ MD. In particular, Area(g(D)) ≤ π M 2 . Let r > 1 and let A(r ) := {z ∈ D :

tn−1 tn−1 }. ≤ |z − σ | ≤ r2 r

For q ∈ [ tn−1 , tn−1 ], denote by L(q) the curve D ∩ {z ∈ C : |z − σ | = q}. Note that r2 r A(r ) = ∪q∈[ tn−1 , tn−1 ] L(q). r2

r

Let us denote by a 1j , a 2j the end points of g(G j ), j = 0, . . . , n − 1. Let us agree , tn−1 ] if g has non-tangential limit at both end to say that g has limits at q ∈ [ tn−1 r2 r points of L(q). Let W (r ) be the set of q in [ tn−1 , tn−1 ] such that g has limits at q r2 r 1 2 1 2 }. Note that g(L(q)) is a and such limits are different from {a0 , a0 , . . . , an−1 , an−1 cross cut for T (Ω) for every q ∈ W (r ). Also, by Proposition 3.3.2, the measure of , tn−1 ] \ W (r ) is zero. [ tn−1 r2 r Denote by L (ρ) the length of the image under g of the curve L(ρ), ρ ∈ [ tn−1 , tn−1 ]. r2 r By Schwarz’s inequality 

|g (z)||dz|

L (ρ)2 =

2



L(ρ)

(σ +ρeiθ )∈D

|g (z)|2 |dz|

|dz| L(ρ)



≤ 2πρ



≤

L(ρ)

|g (σ + ρe )| ρdθ. iθ

2

Hence, 

 inf L (q)

2

q∈W (r )

 ≤

W (r )



≤ 2π

L (ρ)2 tn−1 r



Therefore,

log r = dρ = ρ





tn−1 r2

= 2π



A(r )

(σ +ρeiθ )∈D

 inf L (q)

q∈W (r ) tn−1 r

tn−1 r2

L (ρ)2

2

tn−1 r tn−1 r2

dρ = ρ



 inf L (q)

q∈W (r )

dρ ρ

|g (σ + ρeiθ )|2 ρdθ dρ

|g (z)|2 d xd y = 2π Area(g(A(r )) ≤ 2π 2 M 2 .

2 W (r )

dρ ρ

4.1 Prime Ends

97

√

2π M inf L (q) ≤ √ . log r

(4.1.2)

q∈W (r )

Now, since g(L(q)) is contained in the closed Euclidean disc of radius M, there exists a constant K > 0 such that diam S (g(L q )) ≤ K diamE (g(L q )) ≤ K L (q), for every q ∈ W (r ). Therefore, if r > 1 is chosen large enough so that by (4.1.2), there exists tn ∈ W (r ) such that diam S (g(L tn )) ≤

√ 2π M K √ log r


0 there exists δ > 0 such that for every z, w ∈ X with d S (z, w) < δ there exists a connected set A ⊂ X such that z, w ∈ A and diam S (A) < ε. Lemma 4.1.22 Let X ⊂ C∞ be a compact subset. Then X is locally connected if and only if it is connected im kleinen. Proof Given z ∈ C∞ and ε > 0, we write D(z, ε) := {w ∈ C∞ : d S (z, w) < ε}. Assume first that X is locally connected. Suppose by contradiction that X is not connected im kleinen. Then there exists ε > 0 such that for every n ∈ N there exist z n , wn ∈ X such that d S (z n , wn ) < n1 and every connected set in X joining z n and wn has spherical diameter bigger than ε. Since X is compact, up to extracting convergent subsequences, we can assume that {z n } converges to z 0 and {wn } converges to w0 . Since d S (z n , wn ) < n1 , then d S (z 0 , w0 ) = 0 and z 0 = w0 . Being X locally connected, there exists an open set V ⊂ C∞ such that V ∩ X is connected and z 0 ∈ V ∩ X ⊂ D(z 0 , 4ε ) ∩ X . But, for n large enough, z n , wn ∈ V ∩ X , hence we get a contradiction. Assume now X is connected im kleinen. We will prove that every connected component of every open subset of X is open. From this, the locally connectivity of X follows at once. Let U be an open set in X and let C be a connected component of U . Let z ∈ C. Fix ε > 0 such that D(z, ε) ∩ X ⊂ U . Since X is connected im kleinen, there exists δ > 0, which we can assume less than or equal to ε, such that for all w ∈ X with d S (z, w) < δ there exists a connected set A z,w ⊂ D(z, ε) ∩ X such that z, w ∈ A z,w . Thus, for each w ∈ D(z, δ) ∩ U , A z,w ⊂ C. Therefore, D(z, δ) ∩ X ⊂ C. By the arbitrariness of z, this implies that C is open.  Proposition 4.1.23 Let Ω ⊂ C be a simply connected domain, Ω = C. Assume that ∂∞ Ω is locally connected. Then for every x ∈ ∂C Ω there exists p ∈ ∂∞ Ω such that I (x) = { p}. Proof Again, given z ∈ C∞ and ε > 0, we write D(z, ε) := {w ∈ C∞ : d S (z, w) < ε}. Let x ∈ ∂C Ω. By Proposition 4.1.11 there exists a circular null chain (Cn ) centered at some p ∈ ∂∞ Ω which represents x. Let Vn be the interior part of Cn , n ≥ 1. It is clear that the impression of x is the point p if and only if diam S (Vn ) → 0 as n → ∞. In order to show that diam S (Vn ) → 0, we prove that for every ε > 0 there exists n 0 ∈ N such that Vn ⊂ D( p, ε) := {z ∈ C∞ : d S (z, p) < ε} for all n ≥ n 0 . In fact, since Vm ⊂ Vn 0 for all m ≥ n 0 , it is enough to show that this condition holds for n 0 . Fix ε > 0. We can assume that C0 ∩ (Ω \ D( p, ε)) = ∅. By Lemma 4.1.22, since ∂∞ Ω is locally connected, it is also connected im kleinen. Hence, there exists δ > 0 such that every z, w ∈ ∂∞ Ω with d S (z, w) < δ are joined ε . We may assume that δ < ε/4. by a connected subset A of ∂∞ Ω with diam S (A) < 10 1 “Im

kleinen” is a German expression which means “on a small scale”.

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Since (Cn ) is circular, there exists n 0 ∈ N such that Cn 0 ⊂ D( p, 2δ ). In particular, ε which joins there exists a connected set A ⊂ ∂∞ Ω of spherical diameter less than 10 the two end points of Cn 0 (if the two end points coincide, just set A to be equal to that end point). Hence, A ⊂ D( p, 2ε ). / If Vn 0 is not contained in D( p, ε), there exists a point w0 ∈ Vn 0 such that w0 ∈ D( p, ε). Let w1 ∈ Cn 0 ∩ Ω. Let ρ > 0 be such that D(w1 , ρ) ∩ A = ∅, D(w1 , ρ) ∩ C0 = ∅ and D(w1 , ρ) ⊂ Ω. Let Γ0 be a closed segment in D(w1 , ρ) containing w1 in its interior and orthogonal to Cn 0 at w1 . Let a, b be the end points of Γ0 . By construction one of the end points of Γ0 , say a, belongs to Vn 0 and the other, b, belongs to Ω \ Vn 0 . Let Γ1 be a Jordan arc in (Vn 0 \ Γ0 ) ∪ {a} with end points w0 and a. Next, let Γ2 be a Jordan arc in Ω \ (Vn 0 ∪ Γ0 ) ∪ {b} whose ends points are b and a point w2 ∈ C0 \ D( p, ε). We may assume that Γ1 and Γ2 are the union of a finite number of segments. Let Γ := Γ0 ∪ Γ1 ∪ Γ2 . By construction, Γ is a Jordan arc contained in / D( p, ε), Ω, which is the union of a finite number of segments. Hence, since w0 , w2 ∈ we can construct another Jordan arc Γ in C∞ which joins w0 and w2 and does not intersect Γ ∪ D( p, ε). In particular, since A ⊂ D( p, ε), it holds Γ ∩ A = ∅. Hence Γ ∪ Γ is a Jordan curve in C∞ which does not intersect A and, by Theorem 3.2.1, divides C∞ into two open connected components, say U1 , U2 . We claim that U1 ∩ A and U2 ∩ A are not empty. If this is true, U1 ∩ A and U2 ∩ A are two non empty, disjoint, connected components of A such that A = (U1 ∩ A) ∪ (U2 ∩ A). Hence A is not connected, a contradiction. In order to prove the claim, we show that one of the end points of Cn 0 belongs to U1 and the other to U2 . Since they also belong to A, the claim follows. By construction, there exists ρ ∈ (0, ρ) such that D(w1 , ρ ) ∩ (Γ ∪ Γ ) is a segment L and (Cn 0 ∩ D(w1 , ρ )) \ L is the union of two arcs, one contained in U1 and the other in U2 . Since Cn 0 intersects Γ ∪ Γ only at the point w1 , it follows that the end points of Cn 0 belong to different connected components of C∞ \ (Γ ∪ Γ ). 

4.2 The Carathéodory Topology  which coincides with the Euclidean topoNow we aim to define a topology on Ω logy of Ω when restricted to Ω and allows every univalent function to extend as a homeomorphism up to the Carathéodory’s boundary. If U ⊂ Ω is open, let U ∗ be the union of U with all prime ends x of ∂C Ω such that there exist a null chain (Cn ) representing x and n 0 ∈ N such that Vn ⊂ U for n ≥ n 0 , where Vn is the interior part of Cn . We let T ∗ (Ω) := {U ∗ : U is open in Ω}. Also, we denote by T (Ω) the topology of Ω, namely, U ∈ T (Ω) if and only if U is open in Ω.

4.2 The Carathéodory Topology

101

 is the topology generated by Definition 4.2.1 The Carathéodory topology of Ω T (Ω) ∪ T ∗ (Ω).  if and By construction, a set U ⊂ Ω is open in the Carathéodory topology of Ω only if it is open in the Euclidean topology of Ω. Remark 4.2.2 Let (Cn ) be a null chain for Ω and x = [(Cn )] ∈ ∂C Ω, and let Vn denote the interior part of Cn for n ≥ 1. By the definition of open sets in the Carathéodory topology, {Vn∗ } is a countable basis of open neighborhoods of x. In  is first countable, that is, every point in particular, the Carathéodory topology of Ω  Ω has a countable basis of open neighborhoods. Moreover, a sequence {z m } ⊂ Ω converges to x in the Carathéodory topology if and only if, given any null chain (Cn ) representing x, for every n ∈ N there exists m n ∈ N such that z m ∈ Vn for all m ≥ m n . Also, a sequence {x m } ⊂ ∂C Ω converges to x ∈ ∂C Ω if and only if, for every null chain (Cn ) representing x and for every n 0 ∈ N there exist m 0 ∈ N, u m 0 ∈ N and null chains (Cnm ) representing x m such that Vum ⊂ Vn 0 for all m ≥ m 0 and u ≥ u m 0 , where Vnm denotes the interior part of Cnm . By Lemmas 4.1.13 and 4.1.14, we already know how a univalent map f : D → C, such that Ω = f (D), transfers null chains from D to Ω and vice versa. In fact, this map is a homeomorphism: Theorem 4.2.3 Let f : D → C be univalent, Ω = f (D). Then there exists a home with respect to the Carathéodory topology, such that for omorphism  f : D→Ω  every z ∈ D it holds f (z) = f (z). → Proof It is more natural to first define  f −1 : Ω D. Set  f −1 (w) := f −1 (w) for −1 f (x) ∈ ∂C D to be the every w ∈ Ω. If x ∈ ∂C Ω, by Lemma 4.1.13, we define  equivalence class of ( f −1 (Cn )) for any null chain (Cn ) for Ω representing x. Note that for every null chain (Cn ) for Ω, if Vn denotes the interior part of Cn , n ≥ 1, it follows that  f −1 (Vn∗ ) = ( f −1 (Vn ))∗ . Since f −1 (Vn ) is the interior part of f −1 (Cn ),   by Remark 4.2.2, this implies that for every w ∈ Ω, f −1 maps a basis of open neighborhoods of w onto a basis of open neighborhoods of  f −1 (w), hence, it is open. By the same Lemma 4.1.13 it follows immediately that  f −1 is injective. In order to see that it is surjective, let ζ ∈ ∂C D. By Lemma 4.1.14 there exists a circular null chain (G n ) representing ζ such that ( f (G n )) is a null chain in Ω. Let f −1 (x) = ζ . x := [ f (G n )]. Clearly,   Denote by f the inverse of  f −1 .  Then f is continuous and bijective. It remains only to show that  f is open. However, this can be done easily as before by proving that  f maps a basis of open neighborhoods of each point in  D onto a basis of open neighborhoods of the corre using the fact that, by Lemma 4.1.14, every prime end ζ ∈ ∂C D sponding point in Ω, can be represented by a circular null chain whose image under f is the null chain  representing  f (ζ ).

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In order to make the previous result useful, we need to relate the Carathéodory topology and the Euclidean topology of the closure of the domain. This is the case if the boundary is nice: Theorem 4.2.4 Let Ω ⊂ C be a simply connected domain, Ω = C. Assume that ∂∞ Ω is locally connected. Then the impression of every prime end is a point, and  → Ω ∞ defined by the map Φ : Ω  Φ(z) :=

z if z ∈ Ω, I (z) if z ∈ ∂C Ω,

(4.2.1)

is continuous. Proof By Proposition 4.1.23, for every x ∈ ∂C Ω the impression I (x) is a point in  with the Carathéodory topology is first countable by Remark 4.2.2, it ∂∞ Ω. Since Ω  is a sequence converging to z ∈ Ω  then {Φ(z m )} is enough to prove that if {z m } ⊂ Ω converges to Φ(z) in the spherical distance. If z ∈ Ω then {z m } is eventually contained in Ω, hence Φ(z m ) = z m → z = Φ(z) as m → ∞. Let x ∈ ∂C Ω and let (Cn ) be a null chain representing x. Let Vn be the interior part of Cn , n ≥ 1. Since Φ(x) = I (x) = p for some p ∈ ∂∞ Ω, then for every ε > 0 there exists n 0 ∈ N such that Vn ⊂ D( p, ε) := {w ∈ C∞ : d S ( p, w) < ε}, n ≥ n 0 .

(4.2.2)

Let {z m } ⊂ Ω be a sequence converging to x in the Carathéodory topology. We want to show that for every ε > 0 there exists N ∈ N such that Φ(z m ) = z m ∈ D( p, ε) for all m ≥ N . Fix ε > 0 and let n 0 ∈ N be given by (4.2.2). By Remark 4.2.2, {Vn∗ } is a basis of open neighborhood of x. Hence, there exists N ∈ N such that z m ∈ Vn 0 for all m ≥ N . By (4.2.2), z m ∈ D( p, ε) for all m ≥ N , and we are done. Finally, assume {x m } ⊂ ∂C Ω is a sequence converging to x in the Carathéodory topology. Let pm := I (x m ) = Φ(x m ) ∈ ∂∞ Ω. We want to show that for every ε > 0 there exists M ∈ N such that pm ∈ D( p, 2ε) for all m ≥ M. Fix ε > 0 and let n 0 ∈ N be given by (4.2.2). By Remark 4.2.2, there exist M ∈ N, u m ∈ N and null chains (Cnm ) representing x m such that Vumm ⊂ Vn 0 for all m ≥ M, where Vnm denotes the interior part of Cnm . By (4.2.2), this implies that pm ∈ Vumm ⊂ D( p, ε) ⊂ D( p, 2ε), for all m ≥ M. Hence, Φ is continuous. We end up this section by showing that D is homeomorphic to  D: Proposition 4.2.5 The map Φ :  D → D defined in (4.2.1) is a homeomorphism.



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103

Proof For p ∈ ∂D and r > 0 we let D( p, r ) := {z ∈ C : |z − p| < r }. Moreover, let Cn ( p) := ∂ D( p,

1 ) ∩ D, n ∈ N0 , n+1

1 ) ∩ D. and Vn ( p) := D( p, n+1 We divide the proof in some steps: Step (1): For every x ∈ ∂C D there exists p ∈ ∂D with I (x) = { p} such that (Cn ( p)) is a null chain representing x. Moreover, the interior part of Cn ( p) is Vn ( p). Indeed, by Proposition 4.1.11, there exists a circular null chain (Cn ) centered at some p ∈ ∂D which represents x. Then it is easy to see that (Cn ( p)) is a null chain equivalent to (Cn ), and the proof of Step 1 is concluded. Step (2): For every p ∈ ∂D there exists x p ∈ ∂C D such that I (x p ) = { p}. Indeed, the sequence (Cn ( p)) is a null chain for D and the corresponding prime end has impression { p}. Step (3): Given x, y ∈ ∂C Ω, if I (x) = I (y) then x = y. It follows at once from Step (1). Step (4): Define the map Θ : D →  D as follows:

 Θ(z) :=

z xz

if z ∈ D, if z ∈ ∂D.

The map is well defined by Step (2). Moreover, it is easy to see that Φ ◦ Θ = id and, by Step (3), Θ ◦ Φ = id. Therefore, Φ is bijective. Step (5). The map Φ is a homeomorphism. We already know that Φ is continuous by Theorem 4.2.4. Thus, we have only to show that Θ = Φ −1 is continuous, and it is enough to show that it is sequentially continuous. Recalling Remark 4.2.2, and Step (1), this is now a simple task. We only prove that if { pm } ⊂ ∂D is a sequence converging to p ∈ ∂D, then Θ( pm ) → Θ( p) in the Carathéodory topology, being the other cases similar and simpler. By Step (1), Θ( pm ) is represented by (Cn ( pm )) and Θ( p) is represented by (Cn ( p)). Fix N ∈ N. 1 ) for m ≥ m 0 . Hence, for every Then there exists m 0 ∈ N such that pm ∈ D( p, 10N m ≥ m 0 we can find u m ∈ N such that Vu m ( pm ) = D( pm ,

1 1 ) ∩ D ⊂ D( p, ) ∩ D = VN ( p). um + 1 N +1

By Remark 4.2.2 and by the arbitrariness of N , this means that {Θ( pn )} converges to Θ( p) in the Carathéodory topology.  A straightforward consequence of Theorem 4.2.3 and Proposition 4.2.5 is the following:

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 is a Corollary 4.2.6 Let Ω ⊂ C be a simply connected domain, Ω = C. Then Ω compact, Hausdorff, second countable, arcwise connected and locally connected topological space. Remark 4.2.7 Let Ω ⊂ C be a simply connected domain, Ω = C. Let {wn } ⊂ Ω  Let be a sequence converging in the Carathéodory topology of Ω to some p ∈ Ω. q ∈ ∂C Ω \ { p}. Let (Cm ) be a null chain in Ω representing q and let Vm be the / Vm interior part of Cm , m ≥ 1. Then there exists m 0 ∈ N and n 0 ∈ N0 such that wn ∈ for all n ≥ n 0 and all m ≥ m 0 .  is Hausdorff by Corollary 4.2.6, it follows that there exist two open Indeed, since Ω  sets U, U ⊂ Ω such that q ∈ U , p ∈ U and U ∩ U = ∅. Since by Remark 4.2.2, {Vm∗ } is a basis of open neighborhood of q, there exists m 0 ∈ N such that Vm∗ ⊂ U for all m ≥ m 0 . Moreover, there exists n 0 ∈ N such that wn ∈ U for n ≥ n 0 . Hence, / Vm for all n ≥ n 0 and all m ≥ m 0 . wn ∈

4.3 Carathéodory Extension Theorems In the previous section we defined a topology on the Carathéodory compactification of simply connected domains. We proved that a univalent map f : D → Ω extends as a homeomorphism in the Carathéodory topology, and that, if the boundary of Ω is nice enough, the Carathéodory topology is well related to the Euclidean topology of the closure of Ω. Putting together these results we have the Carathéodory Extension Theorem. Recall that a topological space is locally arcwise connected if every point admits a basis of arcwise connected open neighborhoods. Theorem 4.3.1 (Carathéodory’s Extension Theorem) Let f : D → C be univalent and let Ω = f (D). The following are equivalent: (1) (2) (3) (4)

∂∞ Ω is locally arcwise connected, ∂∞ Ω is locally connected, ∞ there exists a continuous function f˜ : D → Ω such that f˜|D = f , there exists a continuous surjective function g : ∂D → ∂∞ Ω.

Proof Clearly (1) implies (2).  → Ω ∞ the continuous map Φ given by TheoIf (2) holds, denote by ΦΩ : Ω D → D be the continuous map given by Theorem 4.2.4 rem 4.2.4 for Ω. Let ΦD :   be f : D→Ω for D. By Proposition 4.2.5, ΦD is a homeomorphism. Finally, let  the homeomorphism given by Theorem 4.2.3. Then the continuous extension of f is given by f ◦ (ΦD )−1 . f˜ := ΦΩ ◦  Therefore (2) implies (3). If (3) holds, since f is a homeomorphism, then  f maps ∂D onto ∂∞ Ω. Therefore setting g :=  f |∂D statement (4) holds.

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105

Assume (4) holds. We preliminarily observe that if every point of ∂∞ Ω has a basis of arcwise connected neighborhoods, then it is locally arcwise connected, that is, for every point in ∂∞ Ω, one can find a basis of arcwise connected open neighborhoods. ⊆W Indeed, let p ∈ ∂∞ Ω and let W ⊂ ∂∞ Ω be an open neighborhood of p. Let W be the arcwise connected component of W which contains p. We show, and it is  is open. Since we are assuming that p has a basis of arcwise connected enough, that W  there exists an arcwise connected neighborhood neighborhoods, for every q ∈ W , V ⊂ W of q. Hence, every point of V can be joined by an arc to q and since q ∈ W  . Since V is a neighborhood of q, it can be joined by an arc to p. Therefore, V ⊂ W  is a neighborhood of each of its it contains an open neighborhood of q. Hence W points, therefore it is open. Now, arguing by contradiction, suppose that ∂∞ Ω is not locally arcwise connected. Then, by the previous observation, there exists ζ0 ∈ ∂∞ Ω and a neighborhood W ⊂ ∂∞ Ω of ζ0 such that W does not contain any arcwise connected neighborhood of ζ0 .  be the arcwise connected component of W containing ζ0 . Hence there exists a Let W  converging to ζ0 . Since g : ∂D → ∂∞ Ω is surjective there sequence {ζn } ⊂ W \ W exists a sequence {σn } ⊂ ∂D such that g(σn ) = ζn for all n ∈ N. Up to extracting subsequences, we can assume that {σn } converges to a point σ ∈ ∂D. Clearly g(σ ) = ζ0 . Thus, since g is continuous, g −1 (W ) is a neighborhood of σ . Therefore, there exists a connected arc A ⊂ g −1 (W ) containing σ . Now, g(A) is arcwise connected,  . However, for n sufficiently large, σn ∈ A, thus ζn = g(σn ) ∈ g(A), hence g(A) ⊂ W obtaining a contradiction. Therefore, (1) holds.  In case the domain is a Jordan domain, one can say more. Recall that a simply connected domain Ω ⊂ C∞ is a Jordan domain if ∂∞ Ω is a Jordan curve. Note that the boundary of a Jordan domain is locally connected. Also, we give the following definition of simple boundary points: Definition 4.3.2 Let Ω ⊂ C be a domain. We say that a point p ∈ ∂∞ Ω is a simple boundary point if for every sequence {z n } ⊂ Ω converging to p there exists a continuous curve γ : [0, 1) → Ω such that z n ∈ γ ([0, 1)) for all n ∈ N and limt→1 γ (t) = p. Theorem 4.3.3 Let f : D → C be univalent and let f (D) = Ω. The following are equivalent: ∞ (1) there exists a homeomorphism f˜ : D → Ω such that f˜|D = f , (2) Ω is a Jordan domain, (3) every point p ∈ ∂∞ Ω is a simple boundary point.

Proof Clearly, (1) implies (2). Let us prove that (2) implies (1). Thus, assume Ω is a Jordan domain, and let ∞ f˜ : D → Ω be the continuous extension of f given by Theorem 4.3.1. It is clear ∞ that f˜ is surjective. Since Ω is Hausdorff and D is compact, if we show that f˜ is injective, then f˜ is automatically a homeomorphism. Since ∂∞ Ω is a Jordan curve, C∞ \ ∂∞ Ω is the union of two open connected ∞ components. In particular, there exists a open disc D ⊂ C∞ \ Ω . Take η ∈ D and

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1 let H (z) := z−η . Then H is a Möbius transformation such that H (η) = ∞ and H (Ω) is a bounded domain in C. Replacing f with H ◦ f if necessary, we can thus assume that Ω is bounded. Now, assume by contradiction that p, q ∈ ∂D, p = q are such that f˜( p) = f˜(q). Let S be the segment joining p and q. Then S divides D into two connected components. Let T := f˜(S). Hence, T is a Jordan curve and C∞ \ T has two open connected components, one bounded and the other unbounded. Let U be the bounded connected component of C∞ \ T . By construction, T is also a cross cut for Ω with coincident end points w0 := f˜( p) = f˜(q), which divides Ω into two connected components, one of them contained in U . Hence, there exists a connected component of D \ S, call it V , such that f (V ) ⊂ U . We claim that, in fact, f (V ) = U . Indeed, if this were not the case, then ∂Ω ∩ U = ∅. Now let g : ∂D → ∂Ω be the homeomorphism whose existence is guaranteed by being ∂Ω a Jordan curve. Let ζ0 ∈ ∂D be such that g(ζ0 ) = w0 and let ζ1 ∈ ∂D / U . Then there exists an arc A ⊂ ∂D with end points ζ0 , ζ1 be such that g(ζ1 ) ∈ such that g(A) ∩ U = ∅. However, by construction, ∂Ω ∩ ∂U = ∂Ω ∩ T = {w0 }. But then there exists ζ ∈ A \ {ζ0 , ζ1 } such that g(ζ ) = w0 , a contradiction. Hence, U = f (V ). This implies that U ∩ ∂Ω = {w0 }. Now, take any z 0 ∈ ∂D ∩ V and let {z n } ⊂ V be a sequence converging to z 0 . Assume also that { f (z n )} converges to L ∈ C. Hence

L = lim f (z n ) ∈ f (V ) ∩ ∂∞ Ω = U ∩ ∂Ω = {w0 }, n→∞

that is, lim z→z0 f (z) = w0 . In particular, the limit of f is constant on ∂D ∩ V , which is a non-empty arc, against Proposition 3.3.2. Therefore, f˜ is injective and (1) holds. If (1) holds, let p ∈ ∂∞ Ω and let {z n } ⊂ Ω be a sequence converging to p. Then { f −1 (z n )} is a sequence converging to f˜−1 ( p). It is clear that there exists a continuous curve γ : [0, 1) → D such that f −1 (z n ) ∈ γ ([0, 1)) for every n ∈ N (simply take the curve given by linear interpolation of the f −1 (z n )’s) and limt→1 γ (t) = f˜−1 ( p). Hence, z n ∈ f (γ ([0, 1))) for all n ∈ N and limt→1 f (γ (t)) = p. Therefore, (3) holds. Finally, assume that (3) holds. First we prove that f extends to ∂D. To this aim, let σ ∈ ∂D and let {z n }, {wn } ⊂ D be two sequences converging to σ . We claim that there exists p ∈ ∂∞ Ω such that limn→∞ f (z n ) = limn→∞ f (wn ) = p. Assume this is not the case. Up to extracting converging subsequences if necessary, we can assume that limn→∞ f (z n ) = p1 = p2 = limn→∞ f (wn ). Since p1 , p2 ∈ ∂∞ Ω are simple boundary points, there exist two continuous curves, γ1 , γ2 : [0, 1) → Ω such that f (z n ) ∈ γ1 ([0, 1)) and f (wn ) ∈ γ2 ([0, 1)) for all n ∈ N and limt→1 γ j (t) = p j , j = 1, 2. By Proposition 3.3.3, the limit limt→1 f −1 (γ1 (t)) exists and, since z n ∈ f −1 (γ ([0, 1)) for all n ∈ N, it follows limt→1 f −1 (γ1 (t)) = limn→∞ z n = σ . Similarly, limt→1 f −1 (γ2 (t)) = limn→∞ wn = σ . But this contradicts Corollary 3.3.4. By the arbitrariness of the two sequences {z n } and {wn }, this means that lim z→σ f (z)

4.3 Carathéodory Extension Theorems

107

∞ exists. Thus, we can well define a function f˜ : D → Ω as follows: f˜(z) := f (z) if z ∈ D and f˜(σ ) := lim z→σ f (z) if σ ∈ ∂D. Note that f˜(∂D) = ∂∞ Ω. Next, we prove that f˜ is injective on D. By definition, it is injective on D, thus we have only to show that it is injective on ∂D. Assume by contradiction that there j exist σ1 , σ2 ∈ ∂D such that p := f˜(σ1 ) = f˜(σ2 ). Let {z n } ⊂ D be a sequence converging to σ j , j = 1, 2. Then the sequence {wn } ⊂ Ω defined by w2n = f (z n1 ) and w2n+1 = f (z n2 ), n ∈ N, converges to p. Since p is a simple boundary point, there exists a continuous curve γ : [0, 1) → Ω such that wn ∈ γ ([0, 1)) for all n and limt→1 γ (t) = p. But then, by Proposition 3.3.3, f −1 (γ (t)) converges to some point η ∈ ∂D. Since both {z n1 } and {z n2 } are contained in f −1 (γ ([0, 1)), this is impossible. Hence, f˜ is injective. ∞ ∞ Finally, we show that f˜ : D → Ω is continuous. Since f˜ is bijective, Ω is Hausdorff and D is compact, this would imply that f˜ is a homeomorphism and (1) holds. In order to see that f˜ is continuous, we show that if {σn } ⊂ ∂D converges to σ ∈ ∂D and { f˜(σn )} is convergent, then its limit is f˜(σ ). Since ∂∞ Ω is compact, this would give the result. Since lim z→σ f (z) = f˜(σ ), for all m ∈ N there exists δm > 0 such that for all z ∈ D with |z − σ | < δm it holds d S ( f˜(σ ), f (z)) < m1 . 1 δm , 2 }. Since Fix m ∈ N. Then there exists n(m) ∈ N such that |σn(m) − σ | < min{ 2m lim z→σn(m) f (z) = f˜(σn(m) ), we can find a point z k(m) ∈ D such that |z k(m) − σ | < δm and d S ( f˜(σn(m) ), f (z k(m) )) < m1 . Therefore,

2 d S ( f˜(σ ), f˜(σn(m) )) ≤ d S ( f˜(σ ), f (z k(m) )) + d S ( f (z k(m) ), f˜(σn(m) )) < . m Therefore, the subsequence { f˜(σn(m) )} converges to f˜(σ ). Since the sequence { f˜(σn )} is convergent by hypothesis, it follows that its limit is f˜(σ ), and we are done.  Remark 4.3.4 Let Ω ⊂ C be a bounded convex domain. Every sequence in Ω converging to a boundary point of Ω in C can be joined by a piecewise linear curve converging to the same point. Hence every point of ∂Ω is a boundary simple point and it follows by the previous result that every bounded convex domain is a Jordan domain. We end this section with a result which allows to count the number of preimages of boundary points: Proposition 4.3.5 Let f : D → C be univalent and let Ω = f (D). Suppose that ∂∞ Ω is locally connected, so that f extends continuously on ∂D and f (∂D) = ∂∞ Ω. Let p ∈ ∂∞ Ω. Then f −1 ( p) is finite if and only if ∂∞ Ω \ { p} has finitely many connected components. Moreover, if this is the case, the number of points in f −1 ( p) equals the number of connected components of ∂∞ Ω \ { p}. Proof Let P := f −1 ( p) = {σ ∈ ∂D : f (σ ) = p}. Note that, by Proposition 3.3.2, P has zero Lebesgue measure. Moreover, since f is continuous on ∂D, P is closed

108

4 Carathéodorys Prime Ends Theory

 and we can write ∂D \ P = k≥1 Ak , where Ak are non-empty open arcs in ∂D, pairwise disjoint. Since f is continuous, f (Ak ) is connected. We claim that f (A j ) ∩ f (Ak ) = ∅ if j = k. Indeed, assume by contradiction that q ∈ f (A j ) ∩ f (Ak ) and let q j ∈ A j , qk ∈ Ak be such that f (q j ) = f (qk ) = q. Let C be the segment joining q j with qk . Let L be the segment joining the end points of A j . Note that L and C intersect in exactly one point z 0 ∈ D and they are transverse at z 0 . Hence, p ∈ f (L) and f (L) is a Jordan curve contained in Ω ∪ { p}. Thus, / f (L), C∞ \ f (L) is the union of two connected components, say U1 , U2 . Since q ∈ we can assume that q ∈ U1 . By construction, q ∈ f (C) and f (C) is a Jordan curve contained in Ω ∪ {q} which intersects f (L) only at one point w0 = f (z 0 ) = q. Hence, f (C) ⊂ U1 ∪ {w0 }. However, L and C intersect transversally at z 0 and since f is conformal at z 0 , it follows that f (L) and f (C) intersect transversally at w0 , which implies that f (C) ∩ U2 = ∅, a contradiction. Now, we have f (Ak ). ∂∞ Ω \ { p} = f (∂D \ P) = k≥1

Therefore, since the f (Ak )’s are connected and pairwise disjoint, the number of connected components of ∂∞ Ω \ { p} equals the number of Ak ’s. Call m ≥ 1 (possibly ∞) such a number. Finally, note that P is finite if and only if m is finite, and, in this case, m is the cardinality of P. By the previous argument, m equals also the number of connected  components of ∂∞ Ω \ { p}, and we are done.

4.4 Cluster Sets at Boundary Points It is often useful to localize the previous extension result. In order to do so, we need a definition: Definition 4.4.1 Let f : D → C be a function. Let σ ∈ ∂D. The cluster set of f at σ is Γ ( f, σ ) := {L ∈ C∞ : ∃{z n } ⊂ D, lim z n = σ, lim f (z n ) = L}. n→∞

n→∞

The non-tangential cluster set of f at σ is Γ N ( f, σ ) := {L ∈ C∞ : ∃ M > 1, ∃{z n } ⊂ S(σ, M), lim z n = σ, lim f (z n ) = L}. n→∞ n→∞

The radial cluster set of f at σ is Γ R ( f, σ ) := {L ∈ C∞ : ∃{rn } ⊂ (0, 1), lim rn = 1, lim f (rn σ ) = L}. n→∞

n→∞

In other words, the non-tangential cluster set of a f is the union of all the limit values of f at σ along sequences converging non-tangentially to σ , while the radial

4.4 Cluster Sets at Boundary Points

109

cluster set of a f is the union of all the limit values of f at σ along sequences converging radially to σ . Remark 4.4.2 Clearly, Γ ( f, σ ), Γ N ( f, σ ) and Γ R ( f, σ ) are non-empty. Moreover, the unrestricted limit of f at σ exists and it is equal to L ∈ C∞ if and only if Γ ( f, σ ) = {L}. Similarly, the non-tangential limit of f at σ exists and it is equal to L ∈ C∞ if and only if Γ N ( f, σ ) = {L}, while the radial limit of f at σ exists and it is equal to L ∈ C∞ if and only if Γ R ( f, σ ) = {L}. Finally, Γ R ( f, σ ) ⊂ Γ N ( f, σ ) ⊂ Γ ( f, σ ). As one might suspect, the cluster set of a univalent function at a boundary point is related to the impression of the corresponding prime end. By Proposition 4.2.5, at every point on the boundary of the unit disc one can associate a unique prime end whose impression is the point itself, that is Definition 4.4.3 Let σ ∈ ∂D. We denote by x σ ∈ ∂C D the unique prime end such that I (x σ ) = {σ }. Moreover, recall that, if f : D → Ω is a biholomorphism, then f extends to a  by Theorem 4.2.3. With this notations at hand, we can homeomorphism  f : D→Ω prove: Proposition 4.4.4 Let f : D → C be univalent, Ω = f (D). Let σ ∈ ∂D. Then Γ ( f, σ ) = I (  f (x σ )). Proof Let {z m } be a sequence converging to σ and assume { f (z m )} converges to D → D be the homeomorphism given by Proposition 4.2.5. p ∈ ∂∞ Ω. Let ΦD :  D. Hence, {ΦD−1 (z m )} converges to ΦD−1 (σ ) = x σ in the Carathéodory topology of   to  f (x σ ). Therefore, {  f (ΦD−1 (z m ))} converges in the Carathéodory topology of Ω f |D = f it follows that {  f (ΦD−1 (z m ))} = f (z m ) for all m ∈ N. Since ΦD |D = idD and  f (x σ ) in the Carathéodory By Remark 4.2.2, the sequence { f (z m )} converges to  f (x σ ), the topology if and only if for every null chain (Cn ) for Ω representing  sequence { f (z m )} is eventually contained in Vn for every n ∈ N, where Vn is the interior part of Cn , n ≥ 1. Therefore, { f (z m )} is eventually contained in Vn for all f (x σ )). n ≥ 1, hence, p ∈ I (  f (x σ )). Thus, Γ ( f, σ ) ⊆ I (  f (x σ ). If On the other hand, let (Cn ) be a null chain for Ω representing  p ∈ I( f (x σ )), then there exists a sequence {wm } ⊂ Ω converging to p such that {wm } is eventually contained in Vn for every n ≥ 1. By Remark 4.2.2, this means  Therefore, f (x σ ) in the Carathéodory topology of Ω. that {wm } converges to  −1   { f (wm )} converges to x σ in the Carathéodory topology of D. Note that by definition,  f −1 (wm ) = f −1 (wm ) for all m ∈ N. Now, ( f −1 (Cn )) is a null chain representing x σ . Hence, { f −1 (wm )} is eventually contained in the interior part of ( f −1 (Cn )) for every n ≥ 1. Therefore, limm→∞ f −1 (wm ) = I (x σ ) = {σ } by Remark 4.2.2. Setting z m := f −1 (wm ), m ∈ N, we have thus a sequence in D converging to σ such f (x σ )) ⊂ Γ ( f, σ ), and we are that { f (z m )} = {wm } converges to p. Therefore, I (  done. 

110

4 Carathéodorys Prime Ends Theory

Non-tangential cluster sets are also related to prime ends. In order to see how, we need to introduce the notion of principal part of a prime end: Definition 4.4.5 Let Ω ⊂ C be a simply connected domain, Ω = C. Let x ∈ ∂C Ω. The principal part of x, denoted by Π (x), is the set of points p ∈ C∞ with the following property: there exists a null chain (Cn ) representing x such that for every ε > 0 there exists n 0 ∈ N such that Cn ⊂ {z ∈ C∞ : d S (z, p) < ε} for all n ≥ n 0 . Remark 4.4.6 In other words, since the spherical diameter of null chains tends to zero, a point p ∈ ∂∞ Ω belongs to the principal part of a prime end x ∈ ∂C Ω if and only if there exists a null chain (Cn ) representing x and a sequence {wn } ⊂ Ω such that wn ∈ Cn for every n ∈ N and limn→∞ wn = p. Since Cn belongs to the closure of its interior part, it follows at once from this that Π (x) ⊂ I (x). Remark 4.4.7 Let Ω ⊂ C be a simply connected domain, Ω = C and x ∈ ∂C Ω. If (Cn ) is a circular null chain representing x, centered at p ∈ C∞ , then clearly p ∈ Π (x). Example 4.4.8 The prime end defined by the null chain (Cn ) in Example 4.1.7 (see also Example 4.1.20) has non trivial impression, the principal part is however one point, the point 0 ∈ ∂∞ Ω. Theorem 4.4.9 Let f : D → C be univalent, Ω = f (D). Let σ ∈ ∂D. Then, Γ R ( f, σ ) = Γ N ( f, σ ) = Π (  f (x σ )). f (x σ ) such Proof Let p ∈ Π (  f (x σ )). Let (Cn ) be a null chain for Ω representing  that for every sequence {wn } ⊂ Ω such that wn ∈ Cn , the sequence {wn } converges to p. Then ( f −1 (Cn )) is a null chain for D by Lemma 4.1.13 representing x σ , hence I (x σ ) = {σ }. By Lemma 4.1.11, there exists a circular null chain for D centered at σ and equivalent to ( f −1 (Cn )). In particular, the interior part of f −1 (Cn ) contains the intersection of an open disc centered at σ and D for n ≥ 1. Therefore, the radial curve [0, 1)  r → r σ intersects f −1 (Cn ) in at least one point z n for all n ∈ N sufficiently large. Hence, { f (z n )} is a sequence converging to p. This implies that Π(  f (x σ )) ⊂ Γ R ( f, σ ).

(4.4.1)

Now, let p ∈ Γ N ( f, σ ), and let {z m } be a sequence in D converging nonf (x σ )). tangentially to σ such that limm→∞ f (z m ) = p. We want to show that p ∈ Π (  Once we see this, by the arbitrariness of p, we have f (x σ )). Γ N ( f, σ ) ⊂ Π (  And, from (4.4.1) we obtain

4.4 Cluster Sets at Boundary Points

111

Π(  f (x σ )) ⊂ Γ R ( f, σ ) ⊂ Γ N ( f, σ ) ⊂ Π (  f (x σ )), proving the result. In order to show that p ∈ Π (  f (x σ )), we will construct a circular null chain (Cm ) representing x σ such that limm→∞ d S ( f (z m ), f (Cm )) = 0. From this it follows at once that p ∈ Π (  f (x σ )). In order to simplify notations, up to replace f with D  z → f (zσ ) and z m with σ z m , we can assume that σ = 1. Moreover, up to replace f with z → f 1(z) restricted to z ∈ D(1, ε) := {z ∈ D : |z − 1| < ε} as in the proof of Lemma 4.1.14, we can assume that f is bounded (and hence replace the spherical diameter with the Euclidean diameter in the following computations). Now, the sequence {z m } converges non-tangentially to 1, therefore there exist ) such that {z m } is eventually contained in the region α ∈ ( π2 , π ) and β ∈ (π, 3π 2 {1 + r eiθ : r ∈ (0, 1], θ ∈ (α, β)}. Let {rn } be a decreasing sequence of positive numbers converging to 0 such that r0 < 1 and 2rn < rn−1 < 4rn for every n ∈ N. Arguing exactly as in the proof of Lemma 4.1.14, see (4.1.2), we can find tn ∈ [rn , 2rn ], for every n ∈ N, such that G n := D ∩ {|z − 1| = tn } has the following properties: f (G n ) is a cross cut for Ω for every n ∈ N, f (G k ) ∩ f (G j ) = ∅, j = k, and limn→∞ diamE ( f (G n )) = 0. Note that the sequence {tn } strictly decreases to 0. ), set For every θ ∈ ( π2 , 3π 2 L(θ, n) := {1 + r eiθ : r ∈ [tn , tn−1 ]} and denote by L (θ, n) the length of the curve f (L(θ, n)). Now, let α ∈ ( π2 , α). Let T (n) := {1 + r eiθ : r ∈ [tn , tn−1 ], θ ∈ [α , α]}. Note that, for n sufficiently large, T (n) is compact in D (see Fig. 4.1). We claim that there exists a sequence {θn } ⊂ [α , α] such that lim L (θn , n) = 0.

n→∞

In order to prove the claim, we first notice that, since f is bounded, ∞

n=0

hence,

Area( f (T (n))) < +∞,

(4.4.2)

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4 Carathéodorys Prime Ends Theory

Fig. 4.1 The sets T (n) and L(θ, n)

lim Area( f (T (n))) = 0.

(4.4.3)

n→∞

Moreover, by Schwarz’s Inequality, and bearing in mind that tn ≥ rn , tn−1 ≤ 2rn−1 and rn−1 < 4rn , so that tn−1 < 8, we have tn ˆ n) (α − α ) inf L (θˆ , n) = inf L (θ,

θˆ ∈[α ,α]

1 ≤ tn



θˆ ∈[α ,α]

1 | f (z)|d xd y ≤ tn T (n)

 



T (n)

α α

 dθ ≤

| f (z)| d xd y

1 1 1 (Area( f (T (n)))) 2 (Area(T (n))) 2 ≤ tn √ 1 < 8 π (Area( f (T (n)))) 2 . =



2

α α



tn−1

| f (1 + r eiθ )|dr dθ

tn

 21  

 21 d xd y T (n)

√ π tn−1 1 (Area( f (T (n)))) 2 tn

Hence by (4.4.3), lim

ˆ n) ≤ inf L (θ,

n→∞ θ∈[α ,α] ˆ

√ 8 π 1 lim (Area( f (T (n)))) 2 = 0, α − α n→∞

and (4.4.2) follows. ) we can find a sequence ηn ∈ [β, β ] With a similar argument, given β ∈ (β, 3π 2 such that (4.4.4) lim L (ηn , n) = 0. n→∞

Let H (n) := {1 + r eiθ : r ∈ [tn , tn−1 ], θ ∈ [θn , ηn ]},

4.4 Cluster Sets at Boundary Points

113

and let n 0 ∈ N be such that for n ≥ n 0 the set H (n) is compact in D. Note that diamE ( f (H (n)) = diamE ( f (∂ H (n)) for n ≥ n 0 . Moreover, the boundary ∂ H (n) of H (n) is composed by a closed arc in G n , a closed arc in G n−1 and the two segments L(θn , n) and L(ηn , n). Since the diameter of f (G n ) tends to zero and, by (4.4.2), (4.4.4), the diameter of f (∂ H (n)) tends to zero, lim diamE ( f (H (n))) = 0.

n→∞

(4.4.5)

By construction, there exists ε > 0 such that for every z ∈ {1 + r eiθ : r ∈ (0, 1], θ ∈ (α, β)} ∩ {z ∈ D : |z − 1| < ε} there exists n(z) ∈ N such that z ∈ H (n(z)). In particular, for every m ∈ N there exists n m ∈ N such that z m ∈ H (n m ). Moreover, up to passing to a subsequence if necessary, we can assume that for every n ∈ N there exists at most one m ∈ N such that z m ∈ H (n). With this assumption, if we set Cm := G n m , the sequence (Cm ) is a null chain for D representing x 1 . Moreover, f (x 1 ). Also, since Cm by construction, ( f (Cm )) is a null chain for Ω representing  intersects H (n m ) and z m ∈ H (n m ), by (4.4.5), d S ( f (z m ), f (Cm )) ≤

and we are done.

sup

w∈ f (Cm ∩H (n m ))

| f (z m ) − w| ≤ diamE ( f (H (n m ))) → 0, m → ∞,



Since the radial cluster set of a univalent function f : D → C at σ ∈ ∂D is the cluster set at 1 of the curve [0, 1) → f (r σ ), by Theorem 4.4.9 and Lemma 1.9.9 it follows: Corollary 4.4.10 Let Ω ⊂ C be a simply connected domain, Ω = C. Let x ∈ ∂C Ω. Then Π (x) ⊂ ∂∞ Ω is compact and connected. In practice it might be complicated to prove at hand that the principal part of a prime end is trivial, that is, consists of one point. The next theorem gives a characterization of such a property. Definition 4.4.11 Let Ω ⊂ C be a simply connected domain, Ω = C. A point p ∈ ∂∞ Ω is an accessible point via a Jordan arc Γ , if there exists a Jordan arc γ : [0, 1] → C∞ such that Γ = γ ([0, 1]), γ ([0, 1)) ⊂ Ω and γ (1) = p. A prime end x ∈ ∂C Ω is accessible via a Jordan arc Γ if there exists a Jordan arc γ : [0, 1] → C∞ such that Γ = γ ([0, 1]), γ ([0, 1)) ⊂ Ω and limt→1 γ (t) = x in the Carathéodory  topology of Ω. In other words, a prime end x is accessible if, given any null chain (Cn ) for Ω representing x and denoting by Vn the interior part of Cn , n ≥ 1, there exists a Jordan arc Γ in C∞ , contained in Ω except one of its end points, such that Γ ∩ Ω is eventually contained in Vn for every n. Note that this is the case if and only if Cn ∩ Γ = ∅ for all n sufficiently large. Remark 4.4.12 Let Ω ⊂ C be a simply connected domain, Ω = C. Then the set of accessible points of ∂∞ Ω is dense in ∂∞ Ω. Indeed, assume first that p ∈ ∂∞ Ω \

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4 Carathéodorys Prime Ends Theory

{∞} is a non-accessible point, and let ε > 0. Let D( p, ε) denote the Euclidean disc of center p and radius ε. Then, since p ∈ ∂∞ Ω, there exists w ∈ Ω such that w ∈ D( p, 2ε ). Let q ∈ ∂∞ Ω be such that |w − q| = minζ ∈∂∞ Ω |w − ζ |. Note that |w − q| ≤ |w − p| < 2ε . Now, the segment joining w with q is contained in Ω ∪ {q}, and, since it is a Jordan arc, it follows that q is an accessible point. By the triangle inequality, |q − p| ≤ |q − w| + | p − w| < ε. By the arbitrariness of ε, this shows that p is in the closure of the accessible points of ∂∞ Ω. Finally, if ∞ ∈ ∂∞ Ω is a non-accessible point, then for every ε > 0 the open set {z ∈ C : |z| > 1/ε} contains points of ∂∞ Ω, and hence, it contains accessible points for what we already saw. Therefore, the claim is proved. Remark 4.4.13 By Remark 4.1.12, if p ∈ ∂∞ Ω is an accessible point via the Jordan arc Γ , then one can create a circular null chain (Cn ) centered at p such that Γ intersects Cn for all n ≥ 0. Thus, if x ∈ ∂C Ω is the prime end defined by (Cn ), it follows that x is an accessible prime end and p ∈ Π (x). Theorem 4.4.14 Let f : D → C be univalent, Ω = f (D). Let σ ∈ ∂D and let  f (x σ ) ∈ ∂C Ω be the corresponding prime end. Then the following are equivalent: (1) (2) (3) (4)

f has radial limit p1 ∈ ∂∞ Ω at σ , f has non-tangential limit p2 ∈ ∂∞ Ω at σ , Π(  f (x σ )) consists of a single point p3 ∈ ∂∞ Ω, the prime end  f (x σ ) is accessible via a Jordan arc ending at p4 ∈ ∂∞ Ω.

If one—and hence any—of the previous occurs, then p := p1 = p2 = p3 = p4 is an accessible point. Proof We already saw that (1) is equivalent to (2) and p1 = p2 by Theorem 3.3.1. (2) and (3) are equivalent by Theorem 4.4.9. If (1) holds, the curve γ : [0, 1]  r → f (r σ ) is a Jordan arc which lies in Ω except for its end point f (σ ) which belongs to ∂∞ Ω. Let (G n ) be a circular null f (x σ ) (see Lemma chain for x σ such that ( f (G n )) is a null chain for Ω representing  4.1.14). Then the curve (0, 1)  r → r σ is eventually contained in the interior part of G n for every n, so that γ is eventually contained in the interior part of f (G n ) for every n, hence, γ converges to  f (x σ ) in the Carathéodory topology, proving that  f (x σ ) is an accessible prime end, hence (4) holds and p1 = p4 . Finally, assume that (4) holds. Let Γ be the Jordan arc which makes  f (x σ ) accessible. Let p4 ∈ ∂∞ Ω be the end point of Γ outside Ω. Let (Cn ) be a null chain representing  f (x σ ). Then, by definition, Γ ∩ Cn = ∅ for all n sufficiently large. Since diam S (Cn ) → 0, it is easy to see that for every ε > 0 there exists n 0 ∈ N such f (x σ )) = { p4 }, hence (3) holds that Cn ⊂ {z ∈ C∞ : d S (z, p) < ε}. This proves Π (  with p3 = p4 .  Corollary 4.4.15 Let f : D → C be univalent and let Ω = f (D). Then 

p ∈ C∞ : ∃σ ∈ ∂D : ∠ lim f (z) = p z→σ

is dense in ∂∞ Ω.

4.4 Cluster Sets at Boundary Points

115

Proof By Remark 4.4.12, the set of accessible points of ∂∞ Ω is dense in ∂∞ Ω. By Remark 4.4.13, for every accessible point p ∈ ∂∞ Ω there exists an accessible prime end whose principal part contains p. By Theorem 4.4.14, there exists σ ∈ ∂D such  that ∠ lim z→σ f (z) = p. Hence the statement follows.

4.5 Notes The material in this chapter is classical, although we present it here in a possibly new form which is both useful for our aims and self-contained. We took inspirations mainly from Pommerenke’s books [105, 106] and Collingwood and Lohwater’s book [45].

Chapter 5

Hyperbolic Geometry in Simply Connected Domains

In this chapter we introduce hyperbolic metric and hyperbolic distance in simply connected domains and we introduce the basic notion of geodesics as distance minimizing length curves. It turns out that there is a one-to-one correspondence between geodesic rays and prime ends and the cluster set of each geodesic is the principal part of the corresponding prime end. Then we consider useful estimates of hyperbolic metric and distance in simply connected domains depending on the distance from the boundary. The chapter ends with some detailed estimates on the hyperbolic distance in the half-plane.

5.1 Hyperbolic Metric and Geodesics in Simply Connected Domains In this section we extend the notion of hyperbolic metric from the unit disc to simply connected domains of C and we introduce the notion of geodesics. Let Ω  C be a simply connected domain. Recall that, by the Riemann Mapping Theorem 3.1.1, for every z ∈ Ω there exists a unique univalent function f : D → C such that f (D) = Ω, f (0) = z and f  (0) > 0. Using this, we can extend the definition of hyperbolic metric and hyperbolic norm to any simply connected domain. Definition 5.1.1 Let Ω  C be a simply connected domain. Let z ∈ Ω and let f : D → Ω be the unique biholomorphism such that f (0) = z and f  (0) > 0. The hyperbolic metric in Ω at z is defined for all v, w ∈ C as κΩ2 (z; (v, w)) := κD2 (0; (( f −1 ) (z)v, ( f −1 ) (z)v)) =

© Springer Nature Switzerland AG 2020 F. Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4_5

vw . ( f  (0))2

117

118

5 Hyperbolic Geometry in Simply Connected Domains

Similarly, the hyperbolic norm in Ω of v at z is κΩ (z; v) :=



κΩ2 (z; (v, v)) = κD (0; ( f −1 ) (z)v) =

|v| . f  (0)

Remark 5.1.2 As it is clear from the definition, if Ω  C is a simply connected domain, for every z ∈ Ω and v ∈ C, κΩ (z; v) = |v|κΩ (z; 1). The definition makes use of the uniqueness of Riemann maps with prescribed values at 0, but, as we immediately see, the same Riemann map gives the value of the hyperbolic metric and norm at each point, namely: Lemma 5.1.3 Let Ω, Ω˜  C be simply connected domains. If g : Ω → Ω˜ is a biholomorphism, then for every z ∈ Ω and v, w ∈ C, κΩ2 (z; (v, w)) = κΩ2˜ (g(z); (g  (z)v, g  (z)w)), κΩ (z; v) = κΩ˜ (g(z); g  (z)v). In particular, if Ω˜ = D, then κΩ2 (z; (v, w)) =

|g  (z)|2 vw |g  (z)||v| , κ (z; v) = , Ω (1 − |g(z)|2 )2 1 − |g(z)|2

and Ω  z → κΩ (z; 1) is a strictly positive C ∞ function. Proof Let f : D → Ω be the biholomorphism such that f (0) = z and f  (0) > 0 and let f˜ : D → Ω˜ be the biholomorphism such that f˜(0) = g(z) and f˜ (0) > 0. Let φ := f˜−1 ◦ g ◦ f : D → D. Note that φ is an automorphism of D and φ(0) = 0. Moreover, φ  (0) = ( f˜−1 ) ( f˜(0)) · g  (z) · f  (0) =

g  (z)  f (0). f˜ (0)

By Schwarz’s Lemma (see Theorem 1.2.1), |φ  (0)| = 1, that is, |g  (z)| 1 =  .  ˜ f (0) f (0)

(5.1.1)

Hence, by definition of hyperbolic metric, κΩ2 (z; (v, w)) = and we are done.

vw |g  (z)|2 vw = = κΩ2˜ (g(z); (g  (z)v, g  (z)w)), ( f  (0))2 ( f˜ (0))2 

5.1 Hyperbolic Metric and Geodesics in Simply Connected Domains

119

The hyperbolic norm decreases under the action of holomorphic maps: Proposition 5.1.4 Let Ω, Ω˜  C be simply connected domains. If f : Ω → Ω˜ is holomorphic, then for every z ∈ Ω and v ∈ C, κΩ˜ ( f (z); f  (z)v) ≤ κΩ (z; v), with equality for some v = 0 if and only if f is a biholomorphism. In particular, if Ω ⊂ Ω˜ then for every z ∈ Ω and v ∈ C, κΩ˜ (z; v) ≤ κΩ (z; v), with equality for some v = 0 if and only if Ω˜ = Ω. Proof Let g : D → Ω and g˜ : D → Ω˜ be biholomorphisms. Then φ := g˜ −1 ◦ f ◦ g : D → D is holomorphic. By Theorem 1.3.7(2), κD (φ(ζ ); φ  (ζ )w) ≤ κD (ζ ; w) for all ζ ∈ D and w ∈ C, with equality for some w = 0 if and only if φ is an automorphism of D. Let z ∈ Ω and v ∈ C. Since g is a biholomorphism, there exist ζ ∈ D and w ∈ C such that g(ζ ) = z and g  (ζ )w = v. Hence, by Lemma 5.1.3, κΩ˜ ( f (z); f  (z)v) = κΩ˜ ( f (g(ζ )); f  (g(ζ ))g  (ζ )w) = κg˜ −1 (Ω) ˜ −1 ( f (g(ζ ))); (g˜ −1 ) ( f (g(ζ ))) f  (g(ζ ))g  (ζ )w) ˜ (g = κD (φ(ζ ); φ  (ζ )w) ≤ κD (ζ ; w) = κΩ (g(ζ ); g  (ζ )w) = κΩ (z; v), with equality for v = 0 if and only if φ — hence f — is a biholomorphism. ˜ we apply the previous result to the holomorphic map Ω  Finally, if Ω ⊂ Ω, ˜ z → z ∈ Ω.  Definition 5.1.5 Let Ω  C be a simply connected domain. Let −∞ < a < b < +∞ and let γ : [a, b] → Ω be a piecewise C 1 -smooth curve. For a ≤ s ≤ t ≤ b, we define the hyperbolic length of γ in Ω between s and t as  Ω (γ ; [s, t]) :=

t

κΩ (γ (u); γ  (u))du.

s

In case the length is computed in all the interval [a, b] of definition of the curve, we will simply write Ω (γ ) := Ω (γ ; [a, b]). The hyperbolic length of a curve does not depend on its parameterization. Indeed, let γ : [a, b] → Ω be a C 1 -smooth curve defined in some interval [a, b] ⊂ R. Let ˜ → [a, b] be a C 1 -diffeomorphism, −∞ < a˜ < b˜ < +∞. We claim that θ : [a, ˜ b] ˜ There are two possibilities: either θ  (t) > 0 for ˜ b]). Ω (γ ; [a, b]) = Ω (γ ◦ θ ; [a,

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5 Hyperbolic Geometry in Simply Connected Domains

˜ and θ (a) ˜ = b (orientation preserving) or θ  (t) < 0 for all all t ∈ [a, ˜ b] ˜ = a, θ (b) ˜ and θ (a) ˜ = a (orientation reversing). We deal with the second t ∈ [a, ˜ b] ˜ = b, θ (b) case, the first being similar. By Remark 5.1.2, ˜ = ˜ b]) Ω (γ ◦ θ ; [a,

 a˜

b˜

κΩ (γ (θ (u)); γ  (θ (u)))|θ  (u)|du



a

=−  =

κΩ (γ (u); γ  (u))du

(5.1.2)

b b

κΩ (γ (u); γ  (u))du = Ω (γ ; [a, b]).

a

Among all possible parameterizations of a C 1 -smooth curve, there is one which is more natural than the others: Definition 5.1.6 Let Ω  C be a simply connected domain. A C 1 -smooth curve γ : (a, b) → Ω is said to be parameterized by arc length if for every t ∈ (a, b), κΩ (γ (t); γ  (t)) = 1.

(5.1.3)

Note that if a C 1 -smooth curve γ : (a, b) → Ω has an arc length parameterization then γ  (t) = 0 for all t ∈ (a, b). Remark 5.1.7 The arc length parameter is essentially unique. Indeed, let γ : (a, b) → Ω be a C 1 -smooth curve such that κΩ (γ (t); γ  (t)) = 1 for all t ∈ (a, b) ˜ → (a, b) is a diffeomorphism such that κΩ (γ (θ (t)); (γ ◦ and suppose θ : (a, ˜ b) ˜ Then by (5.1.2), for all t ∈ (a, ˜ we have t − a˜ = ˜ b). ˜ b) θ ) (t)) = 1 for all t ∈ (a, |θ (t) − θ (a)|, ˜ from which it follows that θ (t) = θ (a) ˜ ± (t − a). Every regular C 1 -smooth curve admits an arc length parameterization: Lemma 5.1.8 Let Ω  C be a simply connected domain and −∞ ≤ a < b ≤ +∞. Let γ : (a, b) → Ω be a C 1 -smooth curve such that γ  (t) = 0 for all t ∈ (a, b). Then there exist −∞ ≤ A < B ≤ +∞ and a diffeomorphism θ : (A, B) → (a, b) such that θ  (t) > 0 for all t ∈ (A, B) and (A, B)  t → (γ ◦ θ )(t) is parameterized by arc length. Proof Let x0 ∈ (a, b). Note that the function 

x0

(a, x0 ]  t → −Ω (γ ; [t, x0 ]) = −

κΩ (γ (u); γ  (u))du

t

is strictly decreasing because κΩ (γ (u); γ  (u)) > 0 for all u ∈ [t, x0 ] since γ  (u) = 0 for all u ∈ [t, x0 ] and κΩ (z; v) = 0 if and only if v = 0 by Lemma 5.1.3. Let A := limt→a + −Ω (γ ; [t, x0 ]). Similarly, [x0 , b)  t → Ω (γ ; [x0 , b)) is strictly increasing. Let B := limt→b− Ω (γ ; [x0 , b)).

5.1 Hyperbolic Metric and Geodesics in Simply Connected Domains

121

The function F : (a, b) → (A, B) defined by  F(t) :=

t

κΩ (γ (u); γ  (u))du,

x0

has the property that F  (t) = κΩ (γ (t); γ  (t)) > 0 for all t ∈ (a, b). Hence, F is an orientation preserving diffeomorphism between (a, b) and (A, B). Therefore, we can consider the new parameterization of γ given by γ ◦ F −1 : (A, B) → Ω. For every A < s1 < s2 < B, using Remark 5.1.2, we have Ω (γ ◦ F −1 ; [s0 , s1 ]) =



s1

κΩ (γ (F −1 (u)), γ  (F −1 (u)))(F −1 ) (u)du

s

=

 0s1

du = s1 − s0 ,

s0

hence θ := γ ◦ F −1 is an arc length parameterization of γ .



Remark 5.1.9 Let Ω, Ω˜  C be simply connected domains. Let −∞ < a < b < +∞ and let γ : [a, b] → Ω be a piecewise C 1 -smooth curve. If g : Ω → Ω˜ is holomorphic, then as a direct consequence of Proposition 5.1.4, we see that for all a ≤ s ≤ t ≤ b we have Ω (γ ; [s, t]) ≥ Ω˜ (g ◦ γ ; [s, t]). In particular, if g is a biholomorphism, we have Ω (γ ; [s, t]) = Ω˜ (g ◦ γ ; [s, t]). As in the unit disc, the hyperbolic distance between two points of a simply connected domain can be obtain as the infimum of the length of all piecewise smooth curves which join the two points: Proposition 5.1.10 Let Ω  C be a simply connected domain. Let z, w ∈ Ω. Let f : D → Ω be a biholomorphism. Then kD ( f −1 (z), f −1 (w)) = kΩ (z, w) = inf Ω (γ ; [a, b]), γ

where γ runs over all piecewise C 1 -smooth curves γ : [a, b] → Ω such that γ (a) = z, γ (b) = w, for some −∞ < a < b < +∞. Proof By Proposition 1.3.10, kD ( f −1 (z), f −1 (w)) = kΩ (z, w). Let Γ denote the set of all piecewise C 1 -smooth curves γ : [a, b] → Ω such that γ (a) = z and γ (b) = w, for some −∞ < a < b < +∞. Hence, by definition of hyperbolic distance in D, Proposition 1.3.10 and Remark 5.1.9, kΩ (z, w) = kD ( f −1 (z), f −1 (w)) = inf D ( f −1 ◦ γ ; [a, b]) = inf Ω (γ ; [a, b]), γ ∈Γ

and we are done.

γ ∈Γ



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5 Hyperbolic Geometry in Simply Connected Domains

Definition 5.1.11 Let Ω  C be a simply connected domain. A C 1 -smooth curve γ : I → Ω, where I is an interval in R, such that γ  (t) = 0 for all t ∈ I is called a geodesic of Ω if for every s, t ∈ I , s ≤ t, Ω (γ ; [s, t]) = kΩ (γ (s), γ (t)). Moreover, if z, w ∈ Ω and there exist s < t such that γ (s) = z and γ (t) = w, we say that γ |[s,t] is a geodesic which joins z and w. With a slight abuse of notation, we call geodesic also the image of γ in Ω. Remark 5.1.12 Let Ω, Ω˜  C be simply connected domains and let g : Ω → Ω˜ be a biholomorphism. It follows immediately from Lemma 5.1.3 and Proposition 1.3.10 ˜ that if γ : (a, b) → Ω is a geodesic of Ω, then g ◦ γ : (a, b) → Ω˜ is a geodesic of Ω. Now we are going to study some properties of geodesics in simply connected domains. We start with the unit disc. Lemma 5.1.13 Let −∞ ≤ a < b ≤ +∞. (1) If η : (a, b) → D is a geodesic, then η(a) := lim+ η(t), η(b) := lim− η(t) t→a

t→b

exist. Moreover, if η(a), η(b) ∈ D then ω(η(a), η(b)) = lim+ D (η; [a + ε, b − ε]). ε→0

(2) If η : (a, b) → D is a geodesic such that η(a), η(b) ∈ ∂D, then either η(a) = −η(b) and η((a, b)) is the segment in D joining η(a) and η(b), or η(a) = −η(b), η(a) = η(b) and η((a, b)) is the arc in D of the circle which contains η(a) and η(b) and is orthogonal to ∂D at η(a) and η(b). (3) For any z, w ∈ D, z = w, there exists a real analytic geodesic γ : (a, b) → D such that γ (a) = z and γ (b) = w. Moreover, such a geodesic is essentially ˜ → D is another geodesic joining z and w, then unique, namely, if η : (a, ˜ b) ˜ γ ([a, b]) = η([a, ˜ b]). (4) If γ : (a, b) → D is a geodesic such that either γ (a) ∈ D or γ (b) ∈ D (or both), ˜ → D such that η(a), ˜ ∈ ∂D and such then there exists a geodesic η : (a, ˜ b) ˜ η(b) ˜ that γ ([a, b]) ⊂ η([a, ˜ b]). Proof Let γ0 : (−1, 1) → D be defined by γ0 (t) = t. Using Theorem 1.3.5, we have for every −1 < s < t < 1 

t

D (γ0 ; [s, t]) =

κD (γ0 (u); γ  (u))du = ω(γ0 (s), γ0 (t)).

s

Therefore, γ0 is a geodesic in D. Note that, trivially, limt→±1 γ0 (t) exists and belongs to ∂D.

5.1 Hyperbolic Metric and Geodesics in Simply Connected Domains

123

Let η : (a, b) → D be a geodesic in D and suppose that η(a  ) ∈ (−1, 1) and η(b ) ∈ (−1, 1) for some a < a  < b < b. Write η(t) = η1 (t) + iη2 (t), with η1 := Re η and η2 := Im η. By (1.3.2), we have D (η; [a  , b ]) ≥ D (η1 ; [a  , b ]), and equality holds if and only if η1 ≡ η on [a  , b ]. Hence, since D (η; [a  , b ]) = ω(η(a  ), η(b )) = inf D (γ ), γ

where γ runs the set of piecewise C 1 -smooth curves joining η(a  ) and η(b ), it follows that η(t) ∈ (−1, 1) for all t ∈ [a  , b ]. Since, by definition of geodesics, η (t) = 0 for all t ∈ (a, b), it follows that  [a , b ]  t → η(t) ∈ (−1, 1) is either strictly increasing or strictly decreasing. In the first case η([a  , b ]) = [η(a  ), η(b )] and, in the second case, η([a  , b ]) = [η(b ), η(a  )]. Summing up, we have proved: (∗) for every −1 < s < t < 1, γ0 |[s,t] is a geodesic joining s and t and any other geodesic η joining s and t has image [s, t]. Now, take z 0 , w0 ∈ D, z 0 = w0 . Let T (z) = eiθ Tz0 (z), z ∈ D, where Tz0 is the automorphism of D defined in (1.2.1) and θ ∈ R is such that eiθ Tz0 (w0 ) = |Tz0 (w0 )|. Note that T is an automorphism of D. Hence, by Lemma 1.3.4, ω(z 0 , w0 ) = ω(T (z 0 ), T (w0 )) = ω(0, |Tz0 (w0 )|). By Remark 5.1.12, (T −1 ◦ γ0 )|[0,|Tz0 (w0 )|] is a geodesic joining z 0 and w0 and, by (∗), every geodesic joining z 0 and w0 has image equal to T −1 (γ0 ([0, |Tz0 (w0 )|])). In particular, since T is a Möbius transformation, it follows that T −1 (γ0 ([0, |Tz0 (w0 )|])) is either an interval in D contained in a line passing through 0 or it is the arc in D of a circle which intersects orthogonally ∂D in two points. Now, let η : (a, b) → D be a geodesic and assume η(a  ) = 0 and η(b ) > 0 for some a < a  < b < b. We claim that η((a, b)) ⊂ (−1, 1). Indeed, let t ∈ (a, a  ) (a similar argument works for t ∈ (b , b)). Hence, for what we already proved, η([t, b ]) is either contained in a line passing through 0 or it is contained in an arc in D of a circle which intersects orthogonally ∂D in two points. Since η([a  , b ]) ⊂ (−1, 1) by (∗), this implies that η([t, b ]) ⊂ (−1, 1), and hence η(t) ∈ (−1, 1). By the arbitrariness of t, the claim is proved. Since η (t) = 0 for all t ∈ (a, b), and η(b ) > η(a  ), the function (a, b)  t → η(t) is increasing, hence, limt→a + η(t) and limt→b− η(t) exist. Finally, if η : (a, b) → D is any geodesic, arguing as before, we can find an automorphism T such that T ◦ η(a  ) = 0 and T ◦ η(b ) > 0 for some a < a  < b < b. Taking into account that T is a Möbius transformation and T ◦ η is a geodesic by Remark 5.1.12, for what we have proved, it follows that limt→a + η(t) and limt→b− η(t) exist. Moreover, since ω(·, ·) is continuous in D × D, if η(a), η(b) ∈ D, we have

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5 Hyperbolic Geometry in Simply Connected Domains

 lim+ D (η; [a + ε, b − ε]) = lim+

ε→0

ε→0

b−ε

κD (η(u); η (u))du

a+ε

= lim+ ω(η(a + ε), η(b − ε)) = ω(η(a), η(b)). ε→0

This proves (1). Also, parts (2) and (4) follows easily from what we have proved so far. As for (3), we already proved it in case z, w ∈ D. In case z ∈ D and w ∈ ∂D, we can use an automorphism of D to map z to 0 and w to 1 (such an automorphism being given by the composition of Tz with a rotation) to reduce to the case z = 0 and w = 1, for which γ0 works. Finally, in case z, w ∈ ∂D, we can reduce to the case z = 1 and w = −1 thanks to Proposition 1.2.2.  We can easily translate the previous lemma to any simply connected domain using Carathéodory theory: Proposition 5.1.14 Let Ω  C be a simply connected domain. Let −∞ ≤ a < b ≤ +∞. (1) If η : (a, b) → Ω is a geodesic, then η(a) := lim+ η(t), η(b) := lim− η(t) t→a

t→b

exist as limits in the Carathéodory topology of Ω. Moreover, if η(a), η(b) ∈ Ω then kΩ (η(a), η(b)) = lim+ Ω (η; [a + ε, b − ε]). ε→0

(2) If η : (a, b) → Ω is a geodesic such that η(a), η(b) ∈ ∂C Ω, then η(a) = η(b).  z = w, there exists a real analytic geodesic γ : (a, b) → Ω (3) For any z, w ∈ Ω, such that γ (a) = z and γ (b) = w. Moreover, such a geodesic is essentially ˜ → Ω is another geodesic joining z and w, then unique, namely, if η : (a, ˜ b) ˜  γ ([a, b]) = η([a, ˜ b]) in Ω. (4) If γ : (a, b) → Ω is a geodesic such that either γ (a) ∈ Ω or γ (b) ∈ Ω (or ˜ → Ω such that η(a), ˜ ∈ ∂C Ω ˜ b) ˜ η(b) both), then there exists a geodesic η : (a, ˜ in Ω.  and such that γ ([a, b]) ⊂ η([a, ˜ b]) (5) If γ : (a, b) → Ω is a geodesic such that γ (a) ∈ ∂C Ω then the cluster set Γ (γ , a) = Π (γ (a)), the principal part of the prime end γ (a) (and similarly for b in case γ (b) ∈ ∂C Ω). Proof Since every simply connected domain is biholomorphic to the unit disc, using the Carathéodory topology, as a straightforward consequence of Lemma 5.1.13, Proposition 5.1.10 and Theorem 4.2.3, (1)–(4) follow immediately. (5) Let γ : (a, b) → Ω be a geodesic in Ω such that γ (a) ∈ ∂C Ω. Let f : D → Ω be a biholomorphism. Hence, by Remark 5.1.12, f −1 ◦ γ is a geodesic of D. Take a < s1 < s2 < b. Let T (z) = eiθ T f −1 (γ (s1 )) (z), z ∈ D, where T f −1 (γ (s1 )) is the automorphism of D defined in (1.2.1) and θ ∈ R is such that

5.1 Hyperbolic Metric and Geodesics in Simply Connected Domains

125

eiθ T f −1 (γ (s1 )) ( f −1 (γ (s2 ))) = |T f −1 (γ (s1 )) ( f −1 (γ (s2 )))|. Again by Remark 5.1.12, T ◦ f −1 ◦ γ is a geodesic in D hence, by Lemma 5.1.13, it follows that T ( f −1 (γ (a, b))) ⊆ (−1, 1). Therefore, up to replace f with f ◦ T −1 we can assume that f −1 (γ ((a, b))) ⊆ (−1, 1). Moreover, by Theorem 4.2.3, the prime end γ (a) in Ω corresponds to a prime end x in D, in such a way that limt→a + f −1 (γ (t)) = x in the Carathéodory topology of D. By Proposition 4.2.5, if σ ∈ ∂D is the point which corresponds to x, it follows that limt→a + f −1 (γ (t)) = σ (in the Euclidean topology). Since f −1 (γ ((a, b))) ⊆ (−1, 1), this implies that σ = ±1. We can assume σ = 1. Hence, Γ R ( f, 1) = Γ ( f ◦ f −1 ◦ γ , a) = Γ (γ , a), and the result follows by Theorem 4.4.9.



Remark 5.1.15 Let Ω  C be a simply connected domain, z 0 ∈ Ω and f : D → Ω a biholomorphism. If γ : R → Ω is a geodesic parameterized by arc length, by Proposition 5.1.14(1) it follows that limt→±∞ γ (t) exists in the Carathéodory topology of Ω. Moreover, since kD ( f −1 (γ (0)), f −1 (γ (t))) = kΩ (γ (0), γ (t)) = |t|, we have limt→±∞ f −1 (γ (t)) ∈ ∂D, and hence, by Theorem 4.2.3, limt→±∞ γ (t) ∈ ∂C Ω. We end this section by studying convergence of sequences of geodesics with a given fixed point. Proposition 5.1.16 Let Ω  C be a simply connected domain and let z 0 ∈ Ω. Let {Rn } be an increasing sequence of positive real numbers converging to +∞. For every n ∈ N, let γn : [0, Rn ) → Ω be a geodesic parameterized by arc length such that γn (0) = z 0 . Then there exists p ∈ ∂C Ω and a subsequence {γn k } which converges uniformly on compacta of [0, +∞) to a geodesic γ : [0, +∞) → Ω parameterized by arc length such that limt→+∞ γ (t) = p in the Carathéodory topology of Ω. Proof By Lemma 5.1.13 and Theorem 4.2.3, it is enough to prove the result for Ω = D and z 0 = 0. Let then γn : [0, Rn ) → D be geodesics in D such that γn (0) = 0. By Lemma 5.1.13 we can extend continuously γn up to Rn and ω(γn (s), γn (t)) = t − s for all 0 ≤ s < t ≤ Rn . Let θn ∈ [0, 2π ] be such that eiθn γn (Rn ) = |γn (Rn )|. Up to extracting a subsequence, we can assume that lim n→∞ θn = θ exists. Let ηn : [0, Rn ] → D be defined by ηn (t) := eiθn γn (t). Since z → eiθn z is an automorphism of D, it follows that ηn is a geodesic of D for all n ∈ N by Remark 5.1.12. Hence, it is enough to prove that one can extract a sequence from {ηn } which converges uniformly on compacta to a geodesic of D. By Lemma 5.1.13, ηn ([0, Rn ]) ⊂ (−1, 1) and since ηn (t) = 0 for all t ∈ [0, Rn ] and ηn (0) = 0, ηn (Rn ) > 0, it follows that [0, Rn ]  t → ηn (t) ∈ [0, 1) is strictly increasing. Moreover,

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5 Hyperbolic Geometry in Simply Connected Domains

ω(ηn (0), ηn (t)) = D (ηn ; [0, t]) = t, for all t ∈ [0, Rn ], which immediately implies that ηn (Rn ) → 1 as n → ∞. Note that since ηn (t) ∈ D for all t ∈ [0, Rn ], then the sequence {ηn } is equibounded by 1 for all n. Fix R > 0 and let n 0 be such that Rn ≥ R for all n ≥ n 0 . We claim that there exists c ∈ (0, 1) such that for all n ≥ n 0 , and for all s ∈ [0, R], |ηn (s)| ≤ c.

(5.1.4)

Indeed, since 1 + |ηn (s)| 1 log = ω(0, ηn (s)) = ω(ηn (0), ηn (s)) = s < R, 2 1 − |ηn (s)| equation (5.1.4) follows at once. Moreover, by (5.1.3) and since the ηn ’s are non negative and strictly increasing, ηn (s) |ηn (s)| = = κD (ηn (s); ηn (s)) = 1, 1 − ηn (s)2 1 − |ηn (s)|2

(5.1.5)

which, together with (5.1.4), implies that {|ηn |} is equibounded on [0, R] for all n ≥ n 0 . Hence, {ηn } is equicontinuous on [0, R] for all n ≥ n 0 . Therefore, Arzelà-Ascoli’s Theorem (see, e.g. [113, Theorem 11.28 pag. 245]) implies that there exists a subsequence {ηn k } converging uniformly on compacta of [0, +∞) to some continuous function η : [0, +∞) → (−1, 1) such that η(0) = 0. Since ω(·, ·) is continuous, we have for all 0 ≤ s < t < +∞ t − s = lim ω(ηn k (s), ηn k (t)) = ω(η(s), η(t)). k→+∞

(5.1.6)

This proves that η is monotone and, in fact, it is strictly increasing since the ηn ’s are. Moreover, η([0, +∞)) = [0, 1) and limt→+∞ η(t) = 1. Now, (5.1.5) implies that {ηn k (s)} has pointwise limit at every s ∈ [0, +∞) and such a limit is 1 − η(s)2 > 0. Since {ηn k } is equibounded on compacta, by Lebesgue’s Dominated Convergence Theorem, we have for all t ∈ [0, +∞),  η(t) = lim ηn k (t) = lim k→∞

k→∞ 0

t

ηn k (u)du =



t

(1 − η(u)2 )du.

0

This proves that η is C 1 -smooth and η (t) = 1 − η(t)2 > 0 for all t ∈ [0, +∞). Hence, κD (η(t); η (t)) = 1 for all t ∈ [0, +∞). This, together with (5.1.6), implies at once that η is a geodesic parameterized by arc length, and the proof is concluded. 

5.2 Estimates for the Hyperbolic Metric

127

5.2 Estimates for the Hyperbolic Metric In this section we obtain lower and upper estimates for the hyperbolic norm of a simply connected domain in C, and some better lower estimates for convex domains. Theorem 5.2.1 Let Ω  C be a simply connected domain. Then for every z ∈ Ω and v ∈ C, |v| |v| ≤ κΩ (z; v) ≤ . 4δΩ (z) δΩ (z) Proof Let z ∈ Ω and let f : D → Ω be a biholomorphism such that f (0) = z and f  (0) > 0. By definition, κΩ (z; v) = f |v|  (0) . Hence the result follows at once by Theorem 3.4.9.  In case of convex domains, we have a better lower estimate. Recall that a domain Ω ⊂ C is called convex if for every z, w ∈ Ω and for all t ∈ [0, 1] the point t z + (1 − t)w ∈ Ω. Also a direct computation, using the Cayley transform from D to H and Proposition 5.1.4, shows that |v| . (5.2.1) κH (z; v) = 2Re z Theorem 5.2.2 Let Ω  C be a convex simply connected domain. Then for every z ∈ Ω and v ∈ C, |v| |v| ≤ κΩ (z; v) ≤ . 2δΩ (z) δΩ (z) Proof The right hand side estimate follows from Theorem 5.2.1. As for the left hand side estimate, let z ∈ Ω and let p ∈ ∂Ω be such that | p − z| = δΩ (z). z− p (w − p). The map T is a biholoConsider the affine transformation T : w → |z− p| morphism between Ω and T (Ω), and it is easy to see that T (Ω) is still convex. By Lemma 5.1.3, T is an isometry for the hyperbolic metrics of Ω and T (Ω) and, moreover, δΩ (z) = δT (Ω) (T (z)). Therefore, we can assume without loss of generality that p = 0, z = r for some r > 0 and r = δΩ (r ) (since 0 ∈ ∂Ω is the point of ∂Ω which is closest to r ). We claim that Ω ⊂ H. Indeed, suppose there is a point ζ0 ∈ Ω \ H. Since Ω is open, we can assume that Re ζ0 < 0. Consider the line L = {uζ0 : u ∈ R}. It is clear that L ∩ {ζ ∈ C : |ζ − r | < r } = ∅. Let ζ1 ∈ L ∩ {ζ ∈ C : |ζ − r | < r }. By definition of distance, {ζ ∈ C : |ζ − r | < r } ⊂ Ω, hence, ζ1 ∈ Ω. By convexity, the segment between ζ0 and ζ1 is in Ω, hence, 0 ∈ Ω, a contradiction. Therefore, Ω ⊂ H and by Proposition 5.1.4 and (5.2.1), κΩ (r ; v) ≥ κH (r ; v) = and the proof is concluded.

|v| |v| = , 2r 2δΩ (r ) 

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5 Hyperbolic Geometry in Simply Connected Domains

5.3 Estimates for the Hyperbolic Distance In this section, we use the estimates for the hyperbolic metric we discussed before in order to obtain useful estimates for the hyperbolic distance. Theorem 5.3.1 (Distance Lemma) Let Ω  C be a simply connected domain. Then for every w1 , w2 ∈ Ω,    |w1 − w2 | 1 |dw| log 1 + ≤ kΩ (w1 , w2 ) ≤ , 4 min{δΩ (w1 ), δΩ (w2 )} Γ δΩ (w) where Γ is any piecewise C 1 -smooth curve in Ω joining w1 to w2 . Proof Let w1 , w2 ∈ Ω. By Proposition 5.1.14 there exists a real analytic geodesic γ : [a, b] → Ω of Ω such that γ (a) = w1 and γ (b) = w2 . Hence, by Theorem 5.2.1, 



|γ  (t)| dt. a a δΩ (γ (t)) (5.3.1) Now, without loss of generality, we can assume that δΩ (w1 ) ≤ δΩ (w2 ). By the triangle inequality, for all p ∈ C \ Ω and t ∈ (a, b), we have kΩ (w1 , w2 ) = Ω (γ ; [a, b]) =

b

κΩ (γ (t); γ  (t))dt ≥

1 4

b

δΩ (γ (t)) ≤ |γ (t) − p| ≤ |w1 − γ (t)| + |w1 − p|, hence, δΩ (γ (t)) ≤ δΩ (w1 ) + |w1 − γ (t)| for all t ∈ (a, b). Moreover, if E (t) :=



t

|γ  (s)|ds

a

denotes the Euclidean length of γ ([a, t]) for t ∈ [a, b], then |w1 − γ (t)| ≤ E (t). Hence, from (5.3.1),   E 1  (b) |γ  (t)|dt dt 1 b = 4 a δΩ (w1 ) + E (t) 4 0 δΩ (w1 ) + t     1 E (b) 1 |w1 − w2 | = log 1 + ≥ log 1 + , 4 δΩ (w1 ) 4 δΩ (w1 )

kΩ (w1 , w2 ) ≥

and the lower estimate is proved. The upper estimate follows Proposition 5.1.10.

at

once

from

Theorem

5.2.1

and 

Remark 5.3.2 Theorem 5.3.1 implies immediately that   |z − ζ |  1  kΩ (z, w) ≥ sup log |w − ζ |  , z, w ∈ Ω. ζ ∈C\Ω 4

(5.3.2)

5.3 Estimates for the Hyperbolic Distance

129

The same proof using Theorem 5.3.3 instead of Theorem 5.2.1 proves a better estimate for convex domains: Theorem 5.3.3 (Distance Lemma for convex domains) Let Ω  C be a convex simply connected domain. Then for every w1 , w2 ∈ Ω,    1 |dw| |w1 − w2 | log 1 + ≤ kΩ (w1 , w2 ) ≤ , 2 min{δΩ (w1 ), δΩ (w2 )} Γ δΩ (w) where Γ is any piecewise C 1 -smooth curve in Ω joining w1 to w2 .

5.4 Hyperbolic Geometry in the Half-Plane As we already saw in many instances in the previous chapters, it is often useful to move to the right half-plane in order to do actual computations. Therefore, in this section, we discuss hyperbolic geometry in the right half-plane H and prove some useful statements that we collect in the next lemma. Since H is biholomorphic to D via a Cayley transform, we already noticed [(see (1.3.4) and (5.2.1)] that    w1 −w2  1 +  w1 +w2  1  , w1 , w2 ∈ H,  kH (w1 , w2 ) = log  2 2 1 −  ww11 −w +w2  and κH (w; v) =

|v| , w ∈ H, v ∈ C. 2Re w

(5.4.1)

Moreover, since a Cayley transform is a Möbius transformation, it follows from Lemma 5.1.13 that the geodesics in H are either intervals contained in semi-lines in H parallel to the real axis, or arcs in H of circles intersecting orthogonally the imaginary axis. Lemma 5.4.1 Let β ∈ (− π2 , π2 ). (1) Let 0 < ρ0 < ρ1 and let Γ := {ρeiβ : ρ0 ≤ ρ ≤ ρ1 }. Then, H (Γ ) = log

ρ1 ρ1 1 . In particular, kH (ρ0 , ρ1 ) = log . ρ0 2 ρ0

1 2 cos β

1 1 log . 2 cos β (3) Let ρ0 > 0 and α ∈ (− π2 , π2 ). Then, (0, +∞)  ρ → kH (ρeiα , ρ0 eiβ ) has a minimum at ρ = ρ0 , it is increasing for ρ > ρ0 and decreasing for ρ < ρ0 .

(2) Let ρ0 , ρ1 > 0. Then, kH (ρ0 , ρ1 eiβ ) − kH (ρ0 , ρ1 ) ≥

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5 Hyperbolic Geometry in Simply Connected Domains

(4) Let θ0 , θ1 ∈ (− π2 , π2 ) and ρ > 0. Then kH (ρeiθ0 , ρeiθ1 ) = kH (eiθ0 , eiθ1 ). Moreover, kH (1, eiθ ) = kH (1, e−iθ ) for all θ ∈ [0, π/2) and [0, π/2)  θ → kH (1, eiθ ) is strictly increasing. (5) Let β0 , β1 ∈ (− π2 , π2 ) and 0 < ρ0 < ρ1 . Then kH (ρ0 eiβ0 , ρ1 eiβ1 ) ≥ kH (ρ0 , ρ1 ). 1 1 1 (6) For all ρ > 0 we have kH (ρ, ρeiβ ) ≤ log + log 2. 2 cos β 2 Proof (1) Setting γ (ρ) := ρeiβ , we have  H (Γ ) = H (γ ; [ρ0 , ρ1 ]) =

ρ1

ρ0

ρ1 1 1 log . dρ = iβ 2Re ρe 2 cos β ρ0

In particular, since for β = 0, Γ is a geodesic of H, H (Γ ; [ρ0 , ρ1 ]) = kH (ρ0 , ρ1 ). (2) We have,  iβ  2  ρ1 e −ρ0    

 iβ 1 +  ρ1 eiβ +ρ0  ρ1 e + ρ0  + ρ1 eiβ − ρ0  2 1 1 iβ kH (ρ0 , ρ1 e ) = log     iβ 2 = log  ρ1 eiβ + ρ0 2 − ρ1 eiβ − ρ0 2  2 2 0 1 −  ρρ11 eeiβ −ρ  +ρ0  2 2 ρ + ρ + ρ14 + ρ04 − 2ρ02 ρ12 cos(2β) 0 1 1 . = log 2 2ρ0 ρ1 cos β 

Assume ρ0 ≤ ρ1 and set x =

ρ0 ρ1

(in case ρ0 > ρ1 , set x =

1 + x2 + 1 kH (ρ0 , ρ1 e ) − kH (ρ0 , ρ1 ) = log 2 iβ

ρ1 ). ρ0

Hence,

1 + x 4 − 2x 2 cos(2β) . 2 cos β

Since the numerator inside the logarithm is strictly increasing in x and x ∈ (0, 1], the estimate follows. (3) We have  iβ iα   ρ0 e −ρe  1 +  ρ0 eiβ +ρe−iα  1 iα iβ  iβ iα  . kH (ρe , ρ0 e ) = log  −ρe  2 1 −  ρρ00eeiβ +ρe −iα  Since the derivative of [0, 1)  x → prove the statement for the function (0, +∞)  ρ →

1 2

log 1+x is strictly positive, it is enough to 1−x

ρ02 + ρ 2 − 2ρρ0 cos(β − α) |ρ0 eiβ − ρeiα |2 , = |ρ0 eiβ + ρe−iα |2 ρ02 + ρ 2 + 2ρρ0 cos(β + α)

and this follows immediately from a direct computation. (4) From a straightforward computation from the very definition of hyperbolic distance in H, we have

5.4 Hyperbolic Geometry in the Half-Plane

131

1+ 1 kH (ρe , ρe ) = log 2 1− iθ0

iθ1

|eiθ0 −eiθ1 | |eiθ0 +e−iθ1 | |eiθ0 −eiθ1 | |eiθ0 +e−iθ1 |

= kH (eiθ0 , eiθ1 ).

This proves the first part of the statement. Alternatively, this follows from the fact that the multiplication by ρ is a biholomorphism of H. Next, since   iθ e − 1  = 1 − cos θ = tan(θ/2),  [0, π/2)  θ →  iθ e + 1 1 + cos θ is strictly increasing, using the fact that (0, 1)  x → 21 log 1+x is strictly increas1−x ing in x and from the very definition of kH (1, eiθ ) it follows that [0, π/2)  θ → kH (1, eiθ ) is strictly increasing. Moreover, the previous formula also shows that kH (1, eiθ ) = kH (1, e−iθ ) for all θ ∈ [0, π/2). is strictly increasing in x and from (5) Using the fact that (0, 1)  x → 21 log 1+x 1−x the very definition of kH , it is enough to prove that ρ1 − ρ0 |eiβ0 ρ0 − eiβ1 ρ1 | ≥ . |eiβ0 ρ0 + e−iβ1 ρ1 | ρ0 + ρ1 Setting a := ρ02 + ρ12 and b = 2ρ0 ρ1 , and taking the square in the previous inequality, this amounts to show that a−b a − b cos(β0 − β1 ) ≥ . a + b cos(β0 + β1 ) a+b Using that a + b cos(β0 + β1 ), after simple computations, this is equivalent to cos β1 cos β0 +

b sin β1 sin β0 ≤ 1. a

Since b ≤ a, the result follows. (6) Since the curve η : (−π/2, π/2)  θ → ρeiθ is a geodesic in H, assuming β > 0 (the case β < 0 is analogous), we have 

β



β ρ κH (η(θ ); η (θ ))dθ = dθ 2Re ρeiθ 0 0    β 1 1 dθ 1 = log + tan β = 2 0 cos θ 2 cos β 1 1 1 + log 2. ≤ log 2 cos β 2

kH (ρ, ρe ) = iβ





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5 Hyperbolic Geometry in Simply Connected Domains

5.5 Notes The material of this chapter is rather classical. The proof of Theorem 5.3.1 is inspired by Shapiro’s book [117]. One can define the notion of (hyperbolic) rectifiable curves in simply connected domains as follows. Let Ω  C be a simply connected domain and let γ : [a, b] → Ω be a curve (no regularity required so far). For a partition P of [a, b] given by a = t0 < . . . < t N = b of [a, b], one can define Ω (P) :=

N −1 

kΩ (γ (t j ), γ (t j+1 )).

j=0

We say that γ is rectifiable if L := supP Ω (P) < +∞. If this is the case, we define the hyperbolic length of γ to be Ω (γ ; [a, b]) := L. Rectifiable curves can always be parameterized by hyperbolic arc length, in the sense that for every a ≤ s ≤ t ≤ b, we have t − s = Ω (γ ◦ F −1 , [s, t]). Indeed, the function F(t) := Ω (γ ; [a, t]) for t ≥ a is non-decreasing and constant only where γ is constant. Thus one can “invert” F safely removing the interval where it is constant, and consider γ ◦ F −1 : [0, Ω (γ ; [a.b])] → Ω, which turns out to be parameterized by hyperbolic arc length. By the very definition of hyperbolic length, it turns out that if γ is rectifiable, Ω (γ ; [a, b]) ≥ kΩ (γ (a), γ (b)). In case γ is regular enough (for instance Lipschitz continuous), the hyperbolic length of γ equals the integral of κΩ (γ (t); γ  (t)) on [a, b]. In this book, we won’t use this point of view, but, in the next chapter, we will define directly the hyperbolic length of a Lipschitz curve as integral of the “hyperbolic norm” of its tangent.

Chapter 6

Quasi-Geodesics and Localization

In this chapter we show that hyperbolic neighborhoods of geodesic rays correspond to Stolz regions and thus geodesics can be used to understand non-tangential and orthogonal behavior of pre-images under Riemann maps of sequences or curves converging to the boundary in simple connected domains. As it is essentially impossible to detect geodesics in simply connected domains, we introduce the notion of Gromov’s quasi-geodesics, which are usually much simpler to find, and prove the so-called Shadowing Lemma, which states that close to every quasi-geodesic there is a geodesic. We also prove a “Pythagoras’ Theorem” in hyperbolic geometry which says that, given two points z, w in a simply connected domain, and a geodesic γ which contains z, if u ∈ γ is the closest point to w, then the hyperbolic distance from z and w is, up to a universal constant, the sum of the hyperbolic distance between z and u and the hyperbolic distance between u and w. We use such a result to define three “speeds” of convergence of a curve to a boundary point: the total speed, the orthogonal speed and the tangential speed (which measures how far is the curve from converging non-tangentially). Given a simply connected domain, the hyperbolic metric at one point or the hyperbolic distance between two points depends on the whole domain, so that, in general, it is impossible to explicitly compute such a quantity. However, if the domain has some regular shape in a neighborhood of the points, then one can estimate hyperbolic metric and hyperbolic distance of the given domain in terms of the hyperbolic metric and distance of the regular part. This process goes usually under the name of localization, and it is the last content of this chapter.

6.1 Symmetric Domains Given a simply connected domain, it is in general a hard task to find geodesics. In some special cases, this is however possible, and, in this section, we show that the “axis” of symmetry of a domain is indeed (the image of) a geodesic. © Springer Nature Switzerland AG 2020 F. Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4_6

133

134

6 Quasi-Geodesics and Localization

Let L = {ζ ∈ C : ζ = teiθ + a, t ∈ R} be a real line, where θ ∈ R and a ∈ C. If p ∈ L we set p L := p. If p ∈ C \ L, let q ∈ L be such that | p − q| = δC\L ( p), (recall that by definition δC\L ( p) is the Euclidean distance from p to L). We let p L := p + 2(q − p). The map C  p → p L ∈ C is the reflection through L. Example 6.1.1 If L = {ζ ∈ C : Im ζ = 0}, the reflection of a point z ∈ C through L is given by z → z. Definition 6.1.2 Let Ω  C be a simply connected domain. Let L ⊂ C be a real line. We say that Ω is symmetric with respect to L provided p ∈ Ω if and only if p L ∈ Ω. Proposition 6.1.3 Let Ω  C be a simply connected domain and let L be a real line. Suppose that Ω is symmetric with respect to L. Then L ∩ Ω is connected and it is (the image of) a geodesic of Ω. Proof Suppose L = {ζ ∈ C : ζ = teiθ + a, t ∈ R}, where θ ∈ R, a ∈ C. Let T (z) := e−iθ (z − a). Note that T is an automorphism of C, the composition of a translation and a rotation, hence an isometry for the Euclidean metric of C. Moreover, T (L) = R, the real axis. Then it follows that T (Ω) is symmetric with respect to R. Moreover, since T : Ω → T (Ω) is a biholomorphism, by Remark 5.1.12, L ∩ Ω is a geodesic in Ω if and only if T (Ω) ∩ R is a geodesic in T (Ω). Therefore, without loss of generality, we can assume L = R and Ω symmetric with respect to L. By Example 6.1.1, this implies that z ∈ Ω if and only if z ∈ Ω. Fix x0 ∈ Ω ∩ R. Let f : D → Ω be a biholomorphism such that f (0) = x0 , f  (0) > 0. The function g(z) := f (z), z ∈ D, is univalent. Moreover, since Ω is symmetric with respect to R, it follows that g(D) = Ω. Hence, the function φ := f −1 ◦ g : D → D is holomorphic and φ(0) = 0, φ  (0) = 1. Therefore, by Schwarz’s Lemma (Theorem 1.2.1), φ(z) = z for all z ∈ D. This implies that f (z) = g(z) for all z ∈ D and, in particular, f (r ) ∈ R for all r ∈ (−1, 1). Hence, by Remark 5.1.12, f ((−1, 1)) ⊆ Ω ∩ R is the image of a geodesic of Ω. In order to end the proof, we are left to show that f ((−1, 1)) = Ω ∩ R. Let s ∈ Ω ∩ R, and let z ∈ D be such that f (z) = s. Hence, f (z) = f (z) = s = s. Since f (z) = f (z) and f is injective, it follows that z = z. Therefore, z ∈ (−1, 1) and s ∈ f ((−1, 1)). By the arbitrariness of s, we are done. 

6.2 Hyperbolic Sectors and Non-Tangential Convergence The aim of this section is to provide an intrinsic way to define non-tangential limits in simply connected domains. More precisely, the question we consider here is the following: let Ω be a simply connected domain, Ω = C, and f : D → Ω a Riemann map. Let {z n } ⊂ Ω be a sequence such that { f −1 (z n )} converges to σ ∈ ∂D. How is it possible to determine whether { f −1 (z n )} converges non-tangentially to σ looking at the geometry of Ω?

6.2 Hyperbolic Sectors and Non-Tangential Convergence

135

We start with a definition which allows us to extend the notion of non-tangential limit to any simply connected domain: Definition 6.2.1 Let Ω  C be a simply connected domain. Let γ : (a, +∞) → Ω, a ≥ −∞ be a geodesic with the property that limt→+∞ kΩ (γ (t), γ (t0 )) = +∞, for some t0 > a. A hyperbolic sector around γ of amplitude R > 0 is SΩ (γ , R) := {w ∈ Ω : kΩ (w, γ ((a, +∞))) < R}. Note that, if f : D → C is a Riemann map of Ω such that f (0) = γ (a + 1), then by Lemma 5.1.13, σ˜ := limt→+∞ f −1 (γ (t)) ∈ ∂D exists and f −1 (γ ) is a geodesic in D. Up to composing f with a rotation, we can assume σ˜ = 1. Hence, by Lemma 5.1.13, f −1 (γ (a, +∞)) ⊂ (−1, 1). Now, let C : D → H be the Cayley transform such that C(0) = 1, C(1) = ∞. The map h := f ◦ C −1 : H → Ω is a biholomorphism, h(1) = γ (a + 1), h −1 (γ (a, + ∞)) ⊂ (0, +∞) and limt→+∞ h −1 (γ (t)) = ∞. Moreover, since h is an isometry for the hyperbolic distance, SΩ (γ , R) = h(SH (h −1 ◦ γ , R)). Definition 6.2.2 Let β ∈ (0, π ) and r0 ∈ [0, +∞), let V (β, r0 ) := {ρeiθ : ρ > r0 , |θ | < β},

(6.2.1)

be a horizontal sector of angle 2β symmetric with respect to the real axis and with height r0 . Lemma 6.2.3 Let γ : [0, +∞) → H be a geodesic such that γ ([0, +∞)) = [r0 , + ∞) and γ (0) = r0 for some r0 > 0. Then for every R > 0 there exists β ∈ (0, π/2), with kH (1, eiβ ) = R, such that hyp

SH (γ , R) = V (β, r0 ) ∪ DH (r0 , R),

(6.2.2)

hyp

where DH (r0 , R) := {w ∈ H : kH (r0 , w) < R} is the hyperbolic disc in H of center r0 and radius R. Proof Let w ∈ H, w = ρeiθ for some ρ > 0 and θ ∈ (−π/2, π/2). Hence, by Lemma 5.4.1(3), kH (w, (0, +∞)) = kH (ρeiθ , ρ) = kH (eiθ , 1). Let β ∈ (0, π/2) be such that kH (1, eiβ ) = R. Therefore, given ρ > 0, by Lemma 5.4.1(4) and the previous equation, kH (ρeiθ , (0, +∞)) < R if and only if |θ | < β. This implies at once that V (β, r0 ) ⊂ SH (γ , R). hyp Moreover, let w ∈ DH (r0 , R). Hence, M := kH (r0 , w) < R. Let r ∈ (r0 , +∞) be such that kH (r, r0 ) < R − M. Hence, by the triangle inequality, kH (w, r ) ≤ kH (w, r0 ) + kH (r0 , r ) < M + R − M = R,

136

6 Quasi-Geodesics and Localization hyp

proving that w ∈ SH (γ , R). Therefore, V (β, r0 ) ∪ DH (r0 , R) ⊂ SH (γ , R). On the other hand, let w = ρeiθ ∈ SH (γ , R) with ρ > 0 and θ ∈ (−π/2, π/2). If ρ > r0 , by Lemma 5.4.1(3) and (4), it follows immediately that w ∈ V (β, r0 ). If ρ ≤ r0 , the condition w ∈ SH (γ , R) implies that there exists r > r0 such that kH (w, r ) < R. Hence, by Lemma 5.4.1(3), kH (ρeiθ , r0 ) < kH (ρeiθ , r ) < R and w ∈ hyp hyp  DH (r0 , R). This proves that SH (γ , R) ⊂ V (β, r0 ) ∪ DH (r0 , R). Remark 6.2.4 It follows easily from Lemma 5.1.13 and using a Cayley transform from D to H that every geodesic γ : [0, +∞) → H such that limt→+∞ γ (t) = ∞, has the property that there exist x0 ∈ (0, +∞) and y0 ∈ R such that γ ([0, +∞)) = [x0 , +∞) + i y0 . Therefore, by Lemma 6.2.3, it follows immediately for every R > 0 and for every geodesic γ : [0, +∞) → H such that limt→+∞ γ (t) = ∞ there exists t0 ≥ 0 such that γ (t) ∈ SH (γ0 , R) for every t ≥ t0 , where γ0 : [1, +∞) → H is the geodesic defined by γ0 (t) := t. As a consequence, we have the following characterization of non-tangential convergence: Proposition 6.2.5 Let Ω  C be a simply connected domain and let f : D → Ω be a Riemann map. Let {z n } ⊂ Ω be a sequence with no accumulation points in Ω. Then { f −1 (z n )} converges non-tangentially to σ ∈ ∂D if and only if there exist R > 0 and a geodesic γ : [0, +∞) → Ω such that limt→+∞ γ (t) = fˆ(x σ ) in the Carathéodory topology of Ω and {z n } is eventually contained in SΩ (γ , R). Here,  is the homeomorphism induced by f and x σ ∈ ∂C Ω is the prime end fˆ :  D→Ω corresponding to σ . Proof Since the condition that {z n } is eventually contained in SΩ (γ , R) is invariant under isometries for the hyperbolic distance and f is an isometry between ω and kΩ , it is enough to prove the statement for Ω = H and a Cayley transform f : D → H which maps σ to ∞. By Remark 1.5.3, {z n } ⊂ D converges non-tangentially to σ if and only if {C(z n )} is eventually contained in a horizontal sector in H. The result follows then at once from Lemma 6.2.3.  As a corollary we have: Corollary 6.2.6 Let Ω  C be a simply connected domain and let f : D → C be a Riemann map of Ω. Let σ ∈ ∂D and let γ : [0, +∞) → Ω be a geodesic such that limt→+∞ γ (t) = fˆ(x σ ) in the Carathéodory topology of Ω. Let {z n } ⊂ Ω be a sequence such that limn→∞ f −1 (z n ) = σ . Then { f −1 (z n )} converges tangentially to σ if and only if lim kΩ (z n , γ ([0, +∞))) = +∞. n→∞

Proof Since { f −1 (z n )} converges to σ , then {z n } converges to fˆ(x σ ) in the Carathéodory topology of Ω. If { f −1 (z n )} converges tangentially to σ , by Proposition 6.2.5, no subsequences of {z n } can be eventually contained in a hyperbolic sector around γ of a fixed radius, hence kΩ (z n , γ ([0, +∞))) → +∞ as n → ∞.

6.2 Hyperbolic Sectors and Non-Tangential Convergence

137

Conversely, the latter condition implies that no subsequence of {z n } is eventually contained in a hyperbolic sector around γ of a fixed radius, hence, by Proposition 6.2.5, no subsequence of { f −1 (z n )} converges non-tangentially. Therefore, { f −1 (z n )} converges tangentially to σ . 

6.3 Quasi-Geodesics In this section we introduce the concept of quasi-geodesics. These are curves which are closely related to geodesics, but much easier to detect and to use. For future aims, we need to consider curves which are Lipschitz instead of just piecewise C 1 -smooth. Given −∞ ≤ a < b ≤ +∞, recall that a curve γ : (a, b) → C is Lipschitz if there exists C > 0 such that |γ (t) − γ (s)| ≤ C|t − s|, for all s, t ∈ (a, b). In particular, a Lipschitz curve is absolutely continuous, it is differentiable almost everywhere and |γ  (t)| ≤ C for almost every t ∈ (a, b). Definition 6.3.1 Let Ω  C be a simply connected domain, a < b. Let γ : [a, b] → Ω be a Lipschitz continuous curve. We define its hyperbolic length as  Ω (γ ; [a, b]) :=

b

κΩ (γ (t); γ  (t))dt.

a

Notice that, if γ : [a, b] → Ω is Lipschitz, with Lipschitz constant C > 0, then  a

b

κΩ (γ (t); γ  (t))dt ≤ C



b

κΩ (γ (t); 1)dt < +∞,

a

hence, the previous definition is well given. Note also that the previous definition agrees with the one given for piecewise C 1 -smooth curves (see Definition 5.1.5). Moreover, using a Riemann map to move to D and arguing exactly as in the proof of Theorem 1.3.5, (since (1.3.2) and (1.3.3) hold for Lipschitz curves), one can easily see that Ω (γ ; [a, b]) ≥ kΩ (γ (a), γ (b)). Finally, arguing as in (5.1.2) one can see that the hyperbolic length of a Lipschitz curve does not depend on the parameterization (as long as the change of parameterization is chosen to be a homeomorphism). Therefore, in the next sections, not to burden the notation, we can always assume that the Lipschitz curves we deal with are parameterized in [0, +∞). Definition 6.3.2 Let Ω  C be a simply connected domain. A Lipschitz curve γ : [0, +∞) → Ω such that limt→+∞ kΩ (γ (0), γ (t)) = +∞, is a quasi-geodesic if there exist A ≥ 1 and B ≥ 0 such that for every 0 ≤ s ≤ t < +∞,

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Ω (γ ; [s, t]) ≤ AkΩ (γ (s), γ (t)) + B. If we need to specify the constants A, B for which γ is a quasi-geodesic, we will say that γ is a (A, B)-quasi-geodesic. Every geodesic in Ω is a (1, 0)-quasi-geodesic. Moreover, if γ is a (A, B)-quasigeodesic, then it is a (A , B  )-quasi-geodesic for every A ≥ A and B  ≥ B. Also, since automorphisms are isometries for the hyperbolic distance, if γ is a (A, B)quasi-geodesic in Ω then Φ ◦ γ is a (A, B)-quasi-geodesic in Ω for every automorphism Φ of Ω. For future reference we also need the following observation: Remark 6.3.3 If γ : [0, +∞) → Ω is a Lipschitz curve such that there exists A ≥ 1, B ≥ 0 and T > 0 such that γ |[T,+∞) is a (A, B)-quasi-geodesic, then, taking B  := Ω (γ ; [a, T ]), it is easy to see that γ is a (A, B + B  )-quasi-geodesic. In order to better understand the concept of quasi-geodesic, we give some examples: Example 6.3.4 Let β0 ∈ (0, π/2), and let γ : [1, +∞) → H be given by γ (t) = teiβ0 . By Lemma 5.4.1(1), for every 1 ≤ s ≤ t, H (γ ; [s, t]) =

1 t log . 2 cos β0 s

On the other hand, by Lemma 5.4.1(5) kH (seiβ0 , teiβ0 ) ≥ kH (s, t) =

t 1 log . 2 s

Therefore, γ is a ( cos1β0 , 0)-quasi-geodesic. Example 6.3.5 Fix μ > 0 and consider the curve γμ : [0, +∞) → H defined as γμ (t) = μ + it. For 0 ≤ s ≤ t, taking into account (5.4.1), we have 

t

H (γμ ; [s, t]) = s

κH (γμ (r ); γμ (r ))dr =

 s

t

t −s dr = . 2μ 2μ

μ+i(t−s) On the other hand, let θ (s, t) ∈ (0, π/2) be such that eiθ(s,t) = |μ+i(t−s)| . Hence, taking into account that z → z − is is an automorphism of H, and by Lemma 5.4.1(1) and (6),

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139

kH (μ + is, μ + it) = kH (μ, μ + i(t − s)) ≤ kH (μ, |μ + i(t − s)|) + kH (μ + i(t − s), |μ + i(t − s)|) 1 |μ + i(t − s)| 1 1 1 ≤ log + log + log 2. 2 μ 2 cos θ (s, t) 2 Note that |μ + i(t − s)| cos θ (s, t) = μ. Hence, by the previous inequality,     t −s 2 1 . kH (μ + is, μ + it) ≤ log 2 1 + 2 μ If γμ were a quasi-geodesic, then there would exist A ≥ 1, B ≥ 0 such that for every 0 ≤ s ≤ t,     A t −s 2 t −s ≤ log 2 1 + + B, 2μ 2 μ which is clearly impossible. Therefore, there exist no A ≥ 1, B ≥ 0 such that γμ is a (A, B)-quasi-geodesic. The two previous examples are, in a sense, comprehensive of the possible behaviors of quasi-geodesics and non-quasi-geodesics. To better understand such a statement, we need a couple of results which estimate how far can be a quasi-geodesic from being the curve of Example 6.3.4 and how far a quasi-geodesic has to be from the curve of Example 6.3.5. First of all, we estimate how much “time” a quasi-geodesic can spend outside a horosphere at infinity for H: Lemma 6.3.6 Let A ≥ 1, B ≥ 0. Then, for every n, there exists Rn (A, B) > 0 and n 0 (A, B) ∈ N with the following properties: (1) if n ≥ n 0 (A, B),

Rn (A, B) ≤ n − 2

3

(6.3.1)

(2) If γ : [0, +∞) → H is any (A, B)-quasi-geodesic such that there exists [tn , tn+1 ] ⊂ [0, +∞) satisfying • Re γ (t) ≤ n −2 for t ∈ [tn , tn+1 ], • Re γ (tn ) ≥ (n + 1)−2 , • Re γ (tn+1 ) ≥ (n + 1)−2 , then |Im γ (tn+1 ) − Im γ (tn )| ≤ Rn (A, B). Proof We give the proof in case Im γ (tn+1 ) > Im γ (tn ), the other case being similar. Let xn := Re γ (tn ), yn := Im γ (tn ), xn+1 := Re γ (tn+1 ), yn+1 := Im γ (tn+1 ). Then, by hypotheses, (n + 1)−2 ≤ xn ≤ n −2 and (n + 1)−2 ≤ xn+1 ≤ n −2 . Let also

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6 Quasi-Geodesics and Localization

h n := yn+1 − yn > 0. Since δH (γ (t)) ≤

1 n2

for all t ∈ [tn , tn+1 ], by Theorem 5.2.1,

H (γ ; [tn , tn+1 ]) ≥

n2 2



tn+1

|γ  (t)|dt ≥

tn

n2 hn . 2

(6.3.2)

On the other hand, taking into account that z → z − i yn is an automorphism of H, kH (γ (tn ), γ (tn+1 )) = kH (xn , xn+1 + i h n ). Let ρeiθ = xn+1 + i h n . In particular, notice that  1 = cos θ and

2 xn+1 + h 2n

xn+1

 ≤ (n + 1)2 n −4 + h 2n ,

(n + 1)−4 + h 2n ≤ ρ 2 ≤ n −4 + h 2n .

Then, by Lemma 5.4.1(1) and (6), kH (xn , ρeiθ ) ≤ kH (xn , 1) + kH (1, ρ) + kH (ρ, ρeiθ ) 1 1 1 2 1 ≤ log + | log ρ| + log 2 xn 2 2 cos θ

1 1 ≤ 2 log(n + 1) + log max n −4 + h 2n , 4 (n + 1)−4 + h 2n 1 1 + log 2 + log(n −4 + h 2n ). 2 4 Since γ is a (A, B)-quasi-geodesic, the previous inequality together with (6.3.2) implies

A 1 n2 hn ≤ 2 A log(n + 1) + log max n −4 + h 2n , 2 4 (n + 1)−4 + h 2n A A + log 2 + log(n −4 + h 2n ) + B. 2 4

(6.3.3)

Now, fix n ∈ N. Consider (6.3.3) as an inequality in the variable h n . It is easy to see that there exists a value Rn (A, B) > 0 such that (6.3.3) is satisfied only for h n ≤ Rn (A, B), and this proves (2). 3 As for (1), let h n = n − 2 . Consider again (6.3.3) as an inequality in n. Again, it is easy to see that there exists n 0 (A, B) ∈ N such that for n ≥ n 0 (A, B) the inequality 3  is not verified. Hence, Rn (A, B) ≤ n − 2 for n ≥ n 0 (A, B).

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141

The following corollary explains why the curve in Example 6.3.5 is not a quasigeodesic: Corollary 6.3.7 Let A ≥ 1, B ≥ 0 and T > 0. Then there exists C = C(A, B, T ) > 0 such that if γ : [0, +∞) → H is any (A, B)-quasi-geodesic with the properties that γ (0) ∈ (0, T ) and Re γ (t) < T for all t ∈ [0, +∞) then |γ (t)| < C for all t ∈ [0, +∞). Proof Using the automorphism of H given by z → Tz , we can assume that T = 1 (and γ (0) ∈ (0, 1)). For n ∈ N, let Rn := Rn (A, B) > 0 be given by Lemma 6.3.6. Note that, by (6.3.1), ∞  Rn < +∞. R := n=1 2R Let θ ∈ (0, π/2) be such that tan θ = 2R and let C = sin . Note that if z ∈ H has θ the property that Re z < 1 and |z| > C, then |Im z| > 2R. Seeking for a contradiction, we let γ : [0, +∞) → H be a (A, B)-quasi-geodesic such that Re γ (t) < 1 for every t ∈ [0, +∞) and |γ (t)| > C for some t ∈ [0, +∞). In particular, we can find [t0 , t1 ] ⊂ [0, +∞) such that Im γ (t0 ) = 0,

|Im γ (t1 )| > 2R, and 0 < |Im γ (t)| < |Im γ (t1 )| for all t ∈ (t0 , t1 ). Let r1 := max{Re γ (t) : t ∈ [t0 , t1 ]},   let n 1 ∈ N be such that (n 1 + 1)−2 ≤ r1 ≤ n −2 1 and y1 := Im γ (t ) where t ∈ [t0 , t1 ]  is such that Re γ (t ) = r1 . Let

B1 := {z ∈ C : 0 < Re z ≤ n −2 1 , y1 − Rn 1 < Im z < y1 + Rn 1 }. Since Re γ (t) ≤ n −2 1 for all t ∈ [t0 , t1 ], by Lemma 6.3.6 it follows that if Re γ (t) ≥ (n 1 + 1)−2 for some t ∈ [t0 , t1 ], then |y1 − Im γ (t)| ≤ Rn 1 . Namely, γ (t) ∈ B1 . Therefore, Re γ (t) < (n 1 + 1)−2 for all t ∈ [t0 , t1 ] such that γ (t) ∈ / B1 . Now, we proceed by induction in constructing the boxes B j as follows. Suppose j−1 B1 , . . . , B j−1 have been constructed. If γ ([t0 , t1 ]) ⊂ k=1 Bk , the construction ends at this point. Otherwise, let r j := max{Re γ (t) : t ∈ [t0 , t1 ], γ (t) ∈ /

j−1

k=1

Bk }.

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 Then, we let n j ∈ N be such that (n j + 1)−2 ≤ r j ≤ n −2 j and y j := Im γ (t ) where j−1 t  ∈ [t0 , t1 ] is such that Re γ (t  ) = r j and γ (t  ) ∈ / k=1 Bk . Finally, we let

B j := {z ∈ C : 0 < Re z ≤ n −2 j , y j − Rn j < Im z < y j + Rn j }. Note that, by Lemma 6.3.6, if t ∈ [t0 , t1 ] is such that Re γ (t) ≥ (n j + 1)−2 , then j γ (t) ∈ k=1 Bk . In particular, n j > n j−1 and γ ([t0 , t1 ]) ⊂ Nj=1 B j , where either N N ∈ N in case γ ([t0 , t1 ]) ⊂ k=1 Bk or N = ∞ otherwise. Therefore, since the height of B j is 2Rn j , it follows that the height of Nj=1 B j is ∞ at most j=1 (2Rn j ) ≤ 2R. Thus, 2R ≥ Im γ (t1 ) > 2R, 

a contradiction, and we are done. We are now ready to state and prove the main result of this section:

Theorem 6.3.8 (Shadowing Lemma) Let A ≥ 1, B ≥ 0. Then there exists R > 0, which depends only on A, B, such that the following property holds. Let Ω  C be a simply connected domain. Let γ : [0, +∞) → Ω be a (A, B)-quasi-geodesic. Then there exists a prime end x ∈ ∂C Ω such that γ (t) → x in the Carathéodory topology of Ω as t → +∞ and, if η : [0, +∞) → Ω is a geodesic of Ω such that η(0) = γ (0) and limt→+∞ η(t) = x in the Carathéodory topology of Ω, then γ (t) ∈ SΩ (η, R) for every t ∈ [0, +∞). Proof Since the statement is invariant under biholomorphisms, we can assume that Ω = H and γ (0) = 1. By definition of quasi-geodesic, limt→+∞ kΩ (γ (0), γ (t)) = +∞. Hence, the cluster set Γ (γ ; +∞) of γ (t) for t → +∞ is contained in ∂∞ H. Up to composition with a suitable automorphism of H fixing 1, we can assume that ∞ ∈ Γ (γ ; +∞). Namely, lim supt→+∞ |γ (t)| = +∞. In particular, Corollary 6.3.7 implies that lim supt→+∞ Re γ (t) = +∞. Let θ0 ∈ (0, π/2) be such that tan θ0 > 4 A.

(6.3.4)

Write γ (t) = ρ(t)eiθ(t) , for ρ(t) > 0 and θ (t) ∈ (−π/2, π/2).

Step 1. There exists N = N (A, B) > 0 with the following property. Suppose there exist t0 < t1 such that (1) either ρ(t1 ) cos θ (t1 ) ≥ ρ(t0 ) and |θ (t0 )| = θ0 , or ρ(t0 ) cos θ (t0 ) ≥ ρ(t1 ) and |θ (t1 )| = θ0 , (2) |θ (t)| ≥ θ0 for all t ∈ [t0 , t1 ]. Then, kH (ρ(t0 ), ρ(t1 )) ≤ N .

6.3 Quasi-Geodesics

143

We give the proof in case ρ(t1 ) cos θ (t1 ) ≥ ρ(t0 ), |θ (t0 )| = θ0 , the other being similar. Note that ρ(t1 ) > ρ(t0 ) and, since | sin θ (t1 )| ≥ | sin θ (t0 )|, |Im γ (t1 )| ≥ |Im γ (t0 )|. Hence, for every t ∈ [t0 , t1 ], δH (γ (t)) = ρ(t) cos θ (t) ≤

|Im γ (t)| . tan θ0

Therefore, by Theorem 5.2.1,  1 t1 4 t0 tan θ0 ≥ 4 tan θ0 ≥ 4

H (γ ; [t0 , t1 ]) ≥

Since z →

z ρ(t0 )

 |γ  (t)| tan θ0 t1 |Im γ  (t)| dt ≥ dt δH (γ (t)) 4 t0 |Im γ (t)|     log Im γ (t1 )  = tan θ0 log ρ(t1 )| sin θ (t1 )|  Im γ (t0 )  4 ρ(t0 )| sin θ (t0 )| ρ(t1 ) log . ρ(t0 )

(6.3.5)

is an automorphism of H, by the triangle inequality and Lemma 5.4.1, ρ(t1 ) iθ(t1 ) ) e ρ(t0 ) ρ(t1 ) ρ(t1 ) iθ(t1 ) ρ(t1 ) ≤ kH (1, eiθ0 ) + kH (1, ) + kH ( ,e ) (6.3.6) ρ(t0 ) ρ(t0 ) ρ(t0 ) 4 ρ(t1 ) 1 1 . ≤ log + log 2 cos θ0 2 ρ(t0 ) cos θ (t1 )

kH (γ (t0 ), γ (t1 )) = kH (eiθ0 ,

Since γ is a (A, B)-quasi-geodesic, by (6.3.5), (6.3.6), tan θ0 ρ(t1 ) A 4 ρ(t1 ) A log ≤ log + B. + log 4 ρ(t0 ) 2 cos θ0 2 ρ(t0 ) cos θ (t1 ) Bearing in mind that we are assuming 

1 cos θ(t1 )

≤

ρ(t1 ) , ρ(t0 )

the previous equation implies

 tan θ0 ρ(t1 ) A 4 − A log ≤ log + B. 4 ρ(t0 ) 2 cos θ0

By (6.3.4), it follows that there exists N  > 0 (which depends only on A, B) such ρ(t1 ) ≤ N  . From Lemma 5.4.1(1), the statement of Step 1 follows at once. that ρ(t 0)

Step 2. Suppose there exists t1 ∈ (0, +∞) such that |θ (t1 )| > θ0 . Let N > 0 be given by Step 1 and let M := max{N , kH (1, cos θ0 )}. Then there exists [a, b] ⊂ [0, +∞) such that a < t1 < b, |θ (a)| = |θ (b)| = θ0 , |θ (t)| > θ0 for all t ∈ (a, b) and kH (ρ(a), ρ(b)) ≤ M.

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6 Quasi-Geodesics and Localization

Since the curve γ is continuous and γ (0) = 1, there exists a ∈ (0, +∞) such that a < t1 , |θ (a)| = θ0 and |θ (t)| > θ0 for all t ∈ (a, t1 ]. Suppose |θ (t)| > θ0 for all t ≥ t1 . Since lim supt→+∞ Re γ (t) = +∞, there exists a strictly increasing sequence {tn } ⊂ [t1 , +∞) converging to +∞ such that limn→∞ ρ(tn ) cos θ (tn ) = +∞. Hence, there exists n  ∈ N such that ρ(tn ) cos θ (tn ) ≥ ρ(a) for all n ≥ n  . Then, by Step 1, kH (ρ(tn ), ρ(a)) ≤ N for all n ≥ n  , which is a contradiction since limn→∞ ρ(tn ) = +∞. Therefore, there exists b > t1 such that |θ (b)| = θ0 and |θ (t)| > θ0 for all t ∈ (a, b). If either ρ(b) cos θ0 ≥ ρ(a) or ρ(a) cos θ0 ≥ ρ(b), then kH (ρ(a), ρ(b)) ≤ N ≤ M by Step 1. Otherwise, assume ρ(a) ≤ ρ(b) (the case ρ(b) < ρ(a) is similar) and suppose ρ(b) cos θ0 < ρ(a). Since (0, +∞) is a geodesic in H, and ρ(b) cos θ0 < z is an automorphism of H, we ρ(a) ≤ ρ(b), and taking into account that z → ρ(b) have kH (ρ(a), ρ(b)) < kH (ρ(b) cos θ0 , ρ(b)) = kH (cos θ0 , 1) ≤ M, and Step 2 is proved. Step 3. Let M be as in Step 2. Let β ∈ (θ0 , π/2) be such that 1 cos θ0 2 log > A log + AM + B. 2 2 cos β cos θ0

(6.3.7)

Then |Arg γ (t)| < β for all t ∈ [0, +∞). Suppose this is not the case. Then there exists t1 ∈ (0, +∞) such that |θ (t1 )| ≥ β. We assume θ (t1 ) ≥ β (the case θ (t1 ) ≤ −β being similar). By Step 2, we can find 0 < a < t1 < b such that θ (a) = θ (b) = θ0 and θ (t) > θ0 for all t ∈ (a, b). Moreover kH (ρ(a), ρ(b)) ≤ M. Now, by Lemma 5.4.1, kH (γ (a), γ (t1 )) = kH (ρ(a)eiθ0 , ρ(t1 )eiθ(t1 ) ) ≥ kH (eiθ0 , eiθ(t1 ) ) ≥ kH (1, eiθ(t1 ) ) − kH (1, eiθ0 ) 1 1 1 1 1 − log ≥ log − log 2 2 cos θ (t1 ) 2 cos θ0 2 1 cos θ0 1 cos θ0 = log ≥ log . 2 2 cos θ (t1 ) 2 2 cos β

(6.3.8)

Also, again by Lemma 5.4.1, kH (γ (a), γ (b)) = kH (ρ(a)eiθ0 , ρ(b)eiθ0 ) ≤ kH (ρ(a)eiθ0 , ρ(a)) + kH (ρ(a), ρ(b)) + kH (ρ(b), ρ(b)eiθ0 ) 2 ≤ log + M. cos θ0

6.3 Quasi-Geodesics

145

Since γ is a (A, B)-quasi-geodesic, the previous inequality together with (6.3.8) implies 1 cos θ0 log ≤ kH (γ (a), γ (t1 )) ≤ H (γ ; [a, t1 ]) 2 2 cos β ≤ H (γ ; [a, b]) ≤ AkH (γ (a), γ (b)) + B ≤ A log

2 + AM + B, cos θ0

contradicting (6.3.7). Therefore, Step 3 holds. Note that, by construction, β depends only on A, B. Step 3 implies in particular that the only points in Γ (γ ; +∞) can be 0 and ∞. Since Γ (γ ; +∞) is connected (see Lemma 1.9.9) and we are assuming ∞ ∈ Γ (γ ; +∞), it follows that limt→+∞ γ (t) = ∞.

Step 4. there exists 0 < r0 < 1 (which depends only on β, hence only on A, B) such that |γ (t)| > r0 for all t ∈ [0, +∞). Indeed, assume there exists r ∈ (0, 1) such that |γ (s)| = r for some s ∈ (0, +∞). Since γ (t) → +∞ as t → +∞, there exists t > s such that |γ (t)| = 1. By Step 3 and Lemma 5.4.1(6), 1 2 kH (1, γ (t)) ≤ log . 2 cos β On the other hand, by Lemma 5.4.1(5) H (γ ; [0; t]) ≥ H (γ ; [0, s]) ≥ kH (1, γ (s)) ≥

1 1 log . 2 r

The latter two displayed inequalities, taking into account that γ is a (A, B)-quasigeodesic, imply 1 A 2 1 log ≤ log + B, 2 r 2 cos β from which Step 4 follows at once. From Step 3 and Step 4, it follows that γ (t) ∈ V (β; r0 ) for all t ∈ [0, +∞), where, V (β, r0 ) is a horizontal sector (see (6.2.1)). Hence, setting R := kH (1, r0 eiβ ), it follows from Lemma 6.2.3 that γ (t) ∈ SH (η, R) for all t ≥ 0, where η : [0, +∞) → H is the geodesic given by η(s) = s + 1.  As a direct consequence of the previous theorem, Proposition 6.2.5, Corollary 6.2.6 and the triangle inequality, we have the following corollary: Corollary 6.3.9 Let Ω  C be a simply connected domain and let f : D → C be a Riemann map of Ω. Let σ ∈ ∂D. Suppose η : [0, +∞) → Ω is a continuous curve such that limt→+∞ f −1 (η(t)) = σ and let {tn } be a sequence of positive numbers converging to +∞. Let γ : [0, +∞) → Ω be a quasi-geodesic such that limt→+∞ γ (t) = fˆ(x σ ) in the Carathéodory topology of Ω. Then,

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6 Quasi-Geodesics and Localization

(1) { f −1 (η(tn ))} converges non-tangentially to σ if and only if there exists C > 0, such that for all, n ∈ N, kΩ (η(tn ), γ ([0, +∞))) < C. In particular, f −1 (η(t)) → σ non-tangentially as t → +∞ if and only if lim sup kΩ (η(t), γ ([0, +∞))) < +∞. t→+∞

(2) { f −1 (η(tn ))} converges tangentially to σ if and only if lim kΩ (η(tn ), γ ([0, +∞))) = +∞.

n→∞

In particular, f −1 (η(t)) → σ tangentially as t → +∞ if and only if lim kΩ (η(t), γ ([0, +∞))) = +∞.

t→+∞

As an application of the theory developed so far, we prove the following result: Proposition 6.3.10 Let Ω  C be a simply connected domain such that H ⊂ Ω, it ∈ / Ω for t ≤ 0 and it ∈ Ω for t > 0. Let f : D → C be a Riemann map of Ω. Then there exists σ ∈ ∂D such that f −1 (it) converges non-tangentially to σ as t → +∞ if and only if lim supt→+∞ δΩ t(it) < +∞. In particular, the convergence of f −1 (it) as t → +∞ is non-tangential if and only if Ω contains a vertical sector symmetric with respect to the imaginary axis. Proof Let η0 (t) := t, for t ≥ 1. We claim that η0 is a (4, 0)-quasi-geodesic. Indeed, by Theorem 5.3.1, for every 1 ≤ s ≤ t kΩ (η0 (s), η0 (t)) ≥

    t −s 1 t 1 log 1 + = log . 4 s 4 s

On the other hand, by Theorem 5.2.1, 

t

Ω (η0 ; [s, t]) = s

κΩ (η0 (r ); η0 (r ))dr ≤



t s

dr = δΩ (r )

 s

t

  dr t , = log r s

and the claim is proved. Now, by Proposition 3.3.3, there exist σ, σ  ∈ ∂D such that f −1 (it) → σ and −1 f (t) → σ  as t → +∞. If we let γ : [0, +∞) → Ω be the curve defined as γ (t) = 1 − t + ti for 0 ≤ t ≤ 1 and γ (t) = it for t ≥ 1, it follows that γ , η0 satisfy the hypothesis of Proposition 3.3.5. Hence, σ = σ  . Therefore, by Corollary 6.3.9, f −1 (it) → σ non-tangentially if and only if there exists M > 0 such that for every t ≥ 1 there exists st ≥ 1 such that kΩ (it, st ) ≤ M.

6.3 Quasi-Geodesics

147

Now, by Theorem 5.3.1, for every t ≥ 1 and s ≥ 1,     |it − s| 1 t 1 ≥ log . kΩ (it, s) ≥ log 1 + 4 min{s, δΩ (it)} 4 δΩ (it) Therefore, if kΩ (it, st ) ≤ M for all t ≥ 1 then lim supt→+∞ δΩ t(it) < +∞. Assume now that δΩ t(it) ≤ C < +∞ for all t ≥ 1, and fix t ≥ 1. Let η1 (r ) := r + it, r ∈ [0, t]. Let η2 : [0, 1] → H be the geodesic in H such that η2 (0) = t (1 + i) and η2 (1) = t. Hence, again by Theorem 5.3.1 and taking into account Lemma 5.4.1,  t du kΩ (it, t) ≤ Ω (η1 ) + Ω (η2 ) ≤ + H (η2 ) δ (η 0 Ω 1 (u))  t du t + kH (t (1 + i), t) = + kH (1 + i, 1) ≤ C + kH (1 + i, 1). ≤ δΩ (it) 0 δΩ (it)

Thus, we get kΩ (it, t) ≤ C  := C + kH (1 + i, 1), and we are done.



6.4 Orthogonal Convergence The aim of this section is to find conditions for ensuring that the pre-image of a slit under a Riemann map converges orthogonally to the boundary. To this aim, we first prove a preliminary result and then extend to the general setting by extending the notion of horocycles to any simply connected domains. Lemma 6.4.1 Let Ω  C be a simply connected domain and let f : D → C be a Riemann map of Ω. Suppose that H + a ⊂ Ω ⊂ H for some a > 0. Then there exists ξ ∈ ∂D such that f −1 (t) converges orthogonally to ξ as t → +∞. Proof We can assume without loss of generality that a = 1 and let U := H + 1. Note that, for every z, w ∈ U , kU (z, w) = kH (z − 1, w − 1).

(6.4.1)

We divide the proof in several steps: Step 1. Let β ∈ (0, π/4). Then there exists a constant K (β) > 0 such that for every θ0 , θ1 ∈ [−β, β] and for every ρ ≥ 2, kU (ρeiθ0 , ρeiθ1 ) < K (β). Moreover, limβ→0 K (β) = 0. ˜ ˜ Fix β ∈ (0, π4 ) and ρ ≥ 2. Let β˜ = β(ρ) ∈ (0, π2 ) be such that ei β = Hence, 2 sin β sin β ≤ , sin β˜ = iβ |e − 1/ρ| |2eiβ − 1|

˜ which shows that limβ→0 supρ≥2 β(ρ) = 0.

ρeiβ −1 . |ρeiβ −1|

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6 Quasi-Geodesics and Localization

Fig. 6.1 Construction of Q for β = π/5 and ρ = 2

˜ Note that A1 is the arc of the circle of Let A1 := {|eiβ ρ − 1|eiθ + 1 : |θ | ≤ β}. ˜ iβ center 1, radius |e ρ − 1| with end points p0 := eiβ ρ = |eiβ ρ − 1|ei β + 1 and p1 := ˜ e−iβ ρ = |eiβ ρ − 1|e−i β + 1. ˜ Note that A2 is the arc of the circle of center Let A2 := {(ρ − 1)eiθ + 1 : |θ | ≤ β}. ˜ ˜ 1 and radius ρ − 1 with end points q0 := (ρ − 1)ei β + 1 and q1 := (ρ − 1)e−i β + 1. Note that by construction, A1 , A2 are arcs of circles which intersect orthogonally ∂U , hence they are geodesics for the hyperbolic distance kU . ˜ ˜ Let B1 := {r ei β + 1 : ρ − 1 ≤ r ≤ |eiβ ρ − 1|} and let B2 := {r e−i β + 1 : ρ − iβ 1 ≤ r ≤ |e ρ − 1|}. By construction, A1 ∪ B1 ∪ A2 ∪ B2 is a Jordan curve which bounds a simply connected domain Q ⊂ C. Moreover, by simple geometric considerations, the curve Γ := {ρeiθ : |θ | ≤ β} ⊂ Q (see Fig. 6.1 with β = π/5 and ρ = 2). Hence, kU (ρeiθ0 , ρeiθ1 ) ≤ diamU (Q), where diamU (Q) := supz,w∈Q kU (z, w) is the hyperbolic diameter of Q. Clearly, diamU (Q) ≤ U (A1 ) + U (A2 ) + U (B1 ) + U (B2 ). Now, since A1 , A2 are geodesics for U , it follows that U (A1 ) = kU ( p0 , p1 ) and U (A2 ) = kU (q0 , q1 ). Hence, by (6.4.1)

6.4 Orthogonal Convergence

149 ˜

˜

U (A1 ) = kU ( p0 , p1 ) = kH ( p0 − 1, p1 − 1) = kH (|eiβ ρ − 1|ei β , |eiβ ρ − 1|e−i β ), ˜ ˜ and, by Lemma 5.4.1(4), kH (|eiβ ρ − 1|ei β , |eiβ ρ − 1|e−i β ) depends on β˜ and goes to 0 as β goes to 0. Similarly, U (A2 ) goes to 0 as β goes to 0. On the other hand, by (6.4.1) and Lemma 5.4.1(1),

˜

U (B1 ) = H (B1 − 1) = H ({r ei β : ρ − 1 ≤ r ≤ |eiβ ρ − 1|}) =

˜ ≤ Since sin(β)

2 sin(π/4) |2eiπ/4 −1|

 U (B1 ) ≤

=



2√ , 5−2 2

˜ ≥ we have that cos(β)

1 2 cos β˜



√ 3−2√2 5−2 2

log

|eiβ ρ − 1| . ρ−1

and

 √ √ 5−2 2 5−2 2 |eiβ ρ − 1| ≤ √ log √ log |2eiβ − 1|, ρ−1 3−2 2 3−2 2

for every ρ ≥ 2, which shows that U (B1 ) goes to 0 as β goes to 0. A similar argument shows that also U (B2 ) goes to 0 as β goes to 0 and Step 1 follows.

Step 2. Let β ∈ (0, π/2). Let αβ := (1 − cos2 β)−1 . Then for every x1 > x0 ≥ max{2, αβ } the geodesic in Ω joining x0 and x1 is contained in V (β, 0). Fix x0 ≥ 2 and x1 > x0 . Let σ : [0, 1] → Ω be the geodesic for Ω such that σ (0) = x0 and σ (1) = x1 . Assume that σ ([0, 1]) is not contained in V (β, 0). Hence, there exist 0 < t1 ≤ t2 < 0 such that σ (t j ) ∈ ∂ V (β, 0), j = 1, 2 and {σ (t) : t1 ≤ t ≤ t2 } ∩ V (β, 0) = ∅. Since V (β, 0) disconnects H in two connected components, we can assume without loss of generality that σ (t1 ) = y1 eiβ and σ (t2 ) = y2 eiβ for some y1 , y2 > 0 (possibly y1 = y2 ). Let b := kH (1, eiβ ) and let R := {r : r > 0}. Hence, by Lemma 5.4.1(3) and (4), kH (σ (t), R) ≥ b t1 ≤ t ≤ t2 , and kH (σ (t j ), R) = kH (σ (t j ), y j ) = b,

j = 1, 2.

Let γ1 be the segment in R joining x0 and y1 , namely, if y1 ≥ x0 , let γ1 := {r : x0 ≤ r ≤ y1 }, while, if y1 < x0 , let γ1 := {r : y1 ≤ r ≤ x0 }. Let γ2 = {y1 eiθ : θ ∈ [0, β]}. Let γ3 be the segment on ∂ V (β) joining σ (t1 ) = y1 eiβ with σ (t2 ) = y2 eiβ , i.e., if for instance y1 ≤ y2 , γ3 := {r eiβ : y1 ≤ r ≤ y2 }. Then, let γ4 := {y2 eiθ : θ ∈ [0, β]}. Finally, let γ5 be the segment on R joining y2 with x1 . Let Γ = γ1 ∪ γ2 ∪ γ3 ∪ γ4 ∪ γ5 . Hence, Γ is a piecewise smooth curve in H which joins x0 and x1 . Now, since σ is a geodesic in Ω, Ω ⊂ H and by Lemma 5.4.1,

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6 Quasi-Geodesics and Localization

Ω (σ ) = kΩ (x0 , y1 eiβ ) + kΩ (y1 eiβ , y2 eiβ ) + kΩ (y2 eiβ , x1 ) ≥ kH (x0 , y1 eiβ ) + kH (y1 eiβ , y2 eiβ ) + kH (y2 eiβ , x1 ) ≥ kH (x0 , y1 eiβ ) + kH (y1 , y2 ) + kH (y2 eiβ , x1 ) = kH (x0 , y1 ) + kH (y1 , y2 ) + kH (y2 , x1 ) + [kH (x0 , y1 eiβ ) − kH (x0 , y1 )] + [kH (y2 eiβ , x1 ) − kH (y2 , x1 )] ≥ kH (x0 , x1 ) + [kH (x0 , y1 eiβ ) − kH (x0 , y1 )] + [kH (y2 eiβ , x1 ) − kH (y2 , x1 )] 1 , ≥ kH (x0 , x1 ) + log cos β where the penultimate inequality follows from the triangle inequality, and the last inequality follows from Lemma 5.4.1(2). Moreover, a direct computation shows that 1− 1 kH (x0 , x1 ) = kU (x0 , x1 ) − log 2 1−

1 x1 1 x0

.

Therefore, taking into account that U ⊂ Ω, from the previous inequality we have 1− 1 kΩ (x0 , x1 ) = Ω (σ ) ≥ kU (x0 , x1 ) − log 2 1− ≥ kΩ (x0 , x1 ) − which forces

1− 1 log 2 1−

1 x1 1 x0

1− 1 ≤ log log cos2 β 1−

+ log

1 x1 1 x0

1 x1 1 x0

+ log

1 cos β

1 , cos β

.

However, if x0 ≥ αβ , log

1− 1 ≤ log 2 cos β 1−

1 x1 1 x0

≤ log

1−

1 x1

cos2 β

,

getting a contradiction, and Step 2 follows.

Step 3. Let β ∈ (0, π/4) and let 2 ≤ x0 < x1 . Let σ : [0, 1] → Ω be a geodesic for Ω such that σ (0) = x0 and σ (1) = x1 . Let K (β) be the constant defined in Step 1 and let c ∈ (0, x0 e−K (β) ). Suppose σ ([0, 1]) ⊂ V (β, 0). Then |σ (t)| > c for all t ∈ [0, 1]. Assume by contradiction that there exists t1 ∈ (0, 1) such that |σ (t1 )| = c. Then σ (t1 ) = ceiθ1 for some θ1 ∈ (−β, β). Moreover, by continuity of σ , there ˜ ˜ exist t˜1 ∈ [0, t1 ) and t˜2 ∈ (t1 , 1) such that σ (t˜1 ) = x0 ei θ1 , σ (t˜2 ) = x0 ei θ2 for some θ˜1 , θ˜2 ∈ (−β, β) and |σ (t)| ≤ x0 for all t ∈ [t˜1 , t˜2 ].

6.4 Orthogonal Convergence

151

Now, since σ is a geodesic in Ω, and Ω ⊂ H, ˜ ˜ kΩ (x0 ei θ1 , x0 ei θ2 ) = Ω (σ ; [t˜1 , t˜2 ]) = Ω (σ ; [t˜1 , t1 ]) + Ω (σ ; [t1 , t˜2 ]) ˜

˜

˜

˜

= kΩ (x0 ei θ1 , ceiθ1 ) + kΩ (ceiθ1 , x0 ei θ2 ) ≥ kH (x0 ei θ1 , ceiθ1 ) + kH (ceiθ1 , x0 ei θ2 ) ≥ kH (x0 , c) + kH (c, x0 ) = 2kH (x0 , c) = log

x0 , c

where the last inequality follows from Lemma 5.4.1. On the other hand, since U ⊂ Ω, and by Step 1, ˜

˜

˜

˜

kΩ (x0 ei θ1 , x0 ei θ2 ) ≤ kU (x0 ei θ1 , x0 ei θ2 ) ≤ K (β). ≤ K (β), which contradicts the choice of c and Step 3 follows. Step 4. For every δ > 0 there exists μδ ≥ 2 such that for every x1 > x0 ≥ μδ , if σ : [0, 1] → Ω is a geodesic of Ω such that σ (0) = x0 and σ (1) = x1 , then for every x ∈ [x0 , x1 ] there exists tx ∈ [0, 1] such that kΩ (x, σ (tx )) < δ. In order to prove Step 4, we first claim that for every ν ∈ (0, π4 ) there exists μν ≥ 2 such that for every x1 > x0 ≥ μν , σ ([0, 1]) ⊂ V (ν, 0) + 1. If the claim is true, since σ is continuous, for every x ∈ [x0 , x1 ] there exist |θx | < ν and tx ∈ [0, 1] such that σ (tx ) = (x − 1)eiθx + 1. Hence, by (6.4.1) and Lemma 5.4.1(4), and recalling that U ⊂ Ω,

Hence, log

x0 c

kΩ (σ (tx ), x) ≤ kU ((x − 1)eiθx + 1, (x − 1) + 1) = kH ((x − 1)eiθx , x − 1) = kH (eiθx , 1) < kH (eiν , 1). Since kH (eiν , 1) → 0 as ν → 0, Step 4 follows. In order to prove the claim, given ν ∈ (0, π4 ), let β ∈ (0, ν). Hence, there exists α > 1 such that V (β, 0) ∩ {w ∈ U : |w| > α} ⊂ V (ν, 0) + 1. Let αβ be given by Step 2. Let μν > e K (β) max{αβ , α}, where K (β) is given by Step 1. Hence, by Step 2, for every x1 > x0 > μν the geodesic σ for kΩ joining x0 , x1 is contained in V (β, 0). By Step 3, σ ([0, 1]) ⊂ {w : |w| > α}, hence, σ is contained in V (ν, 0) + 1. By Proposition 3.3.3, there exists ξ ∈ ∂D such that limt→+∞ f −1 (t) = ξ . Let γ : [0, 1) → D be defined by γ (t) = tξ . By Lemma 5.1.13, γ is a geodesic in D. Step 5. For every  > 0 there exists t > 0 such that f −1 (t) ∈ SD (γ , ) for all t ≥ t . Fix  > 0. Let δ = 3 and let μδ ≥ 2 be the point defined in Step 4. Let {xn } be an increasing sequence of positive real numbers converging to +∞. Let σn : [0, Rn ] → Ω be the geodesic in Ω parameterized by arc length such that σn (0) = μδ and σn (Rn ) = xn . By Proposition 5.1.16, up to extracting a subsequence, we can assume that {σn } converges uniformly on compacta of [0, +∞) to a geodesic σ : [0, +∞) →

152

6 Quasi-Geodesics and Localization

Ω, parameterized by arc length such that σ (0) = μδ and lims→+∞ σ (s) = y ∈ ∂C Ω in the Carathéodory topology of Ω. In particular, for every fixed T > 0 there exists n T ∈ N such that for every n ≥ n T we have Rn ≥ T and for every s ∈ [0, T ], kΩ (σn (s), σ (s)) < δ.

(6.4.2)

By Step 4, for every t ∈ [μδ , xn ] there exists stn ∈ [0, Rn ] such that kΩ (σn (stn ), t) < δ. We claim that, for every fixed x1 > μδ there exists C x1 > 0 such that for all n ∈ N and all t ∈ [μδ , x1 ], we have stn ≤ C x1 . Indeed, since [μδ , x1 ] is compact in Ω, C0 := maxx∈[μδ ,x1 ] kΩ (x, μδ ) < +∞. Hence, recalling that σn is parameterized by arc length, for all t ∈ [μδ , x1 ], we have stn = kΩ (σn (stn ), σn (0)) = kΩ (σn (stn ), μδ ) ≤ kΩ (σn (stn ), t) + kΩ (t, μδ ) ≤ δ + C0 =: C x1 . Therefore, fix x1 > μδ , and set T := C x1 . By (6.4.2), for all t ∈ [μδ , x1 ] we have kΩ (σ (stn T ), t) ≤ kΩ (σ (stn T ), σn T (stn T )) + kΩ (σn T (stn T ), t) < 2δ. By the arbitrariness of x1 , this proves that t ∈ SΩ (σ, 2δ) for all t ≥ μδ . Since f is an isometry for the hyperbolic distance, f −1 ◦ σ is a geodesic in D parameterized by arc length and f −1 (t) ∈ SD ( f −1 ◦ σ, 2δ) for all t ≥ μδ . In particular, for every t ≥ μδ there exists st ∈ [0, +∞) such that ω( f −1 (t), f −1 (σ (st ))) < 2δ.

(6.4.3)

Note that this implies in particular that st → +∞ as t → +∞. By Lemma 1.8.6, limt→+∞ f −1 (σ (st )) = ξ , hence limt→+∞ f −1 (σ (t)) = ξ by Lemma 5.1.13. Using a Cayley transform from D to H which maps ξ to ∞, it follows directly from Remark 6.2.4 that there exists s1 ≥ 0 such that f −1 (σ (s)) ∈ SD (γ , δ) for all s ≥ s1 . In particular, for every s ≥ s1 , there exists rs ∈ [0, 1) such that ω( f −1 (σ (s)), γ (rs )) < δ.

(6.4.4)

Let t ≥ μδ be such that st ≥ s1 for all t ≥ t . Then by (6.4.3) and (6.4.4), for all t ≥ t , ω(γ (rst ), f −1 (t)) ≤ ω( f −1 (σ (st )), γ (rst )) + ω( f −1 (t), f −1 (σ (st ))) < 3δ = , and Step 5 follows. From Step 5 we see that lim supt→+∞ ω( f −1 (t), [0, 1)ξ ) = 0, and Lemma 1.8.6  implies then that f −1 (t) converges to ξ orthogonally.

6.4 Orthogonal Convergence

153

In order to extend the previous result to a more intrinsic setting, note that by (1.4.15), the condition in Lemma 6.4.1 means that Ω is contained in H and contains a horocycle at infinity of H. In order to replace H with a general simply connected domain, we need to extend the notion of horocycles. Remark 6.4.2 Let Ω  C be a simply connected domain, z 0 ∈ Ω and let f : D → Ω be a Riemann map such that f (0) = z 0 . Let y ∈ ∂C Ω be a prime end of Ω. By Theorem 4.2.3 and Proposition 4.2.5, there exists exactly one σ ∈ ∂D which corresponds to a prime end x σ ∈ ∂C D such that fˆ(x σ ) = y. Moreover, a sequence {wn } ⊂ Ω converges to y in the Carathéodory topology of Ω if and only if { f −1 (wn )} converges to σ (in the Euclidean topology). Therefore, if {z n } and {wn } are two sequences in Ω which converge to y in the Carathéodory topology of Ω, taking into account that f is an isometry for the hyperbolic distance, by (1.4.1), we have for every z ∈ Ω, lim [kΩ (z, wn ) − kΩ (z 0 , wn )] = lim [kD ( f −1 (z), f −1 (wn )) − kD (0, f −1 (wn ))]

n→∞

n→∞

= lim [kD ( f −1 (z), f −1 (z n )) − kD (0, f −1 (z n ))] n→∞

= lim [kΩ (z, z n ) − kΩ (z 0 , z n )]. n→∞

Moreover, by the same equation (1.4.1), lim [kD ( f −1 (z), f −1 (wn )) − kD (0, f −1 (wn ))] ∈ (−∞, +∞),

n→∞

hence limn→∞ [kΩ (z, wn ) − kΩ (z 0 , wn )] ∈ (−∞, +∞). By the previous remark, the following definition is well posed (that is, it is independent of the sequence {wn } chosen): Definition 6.4.3 Let Ω  C be a simply connected domain and z 0 ∈ Ω. Let y ∈ ∂C Ω be a prime end of Ω, and let {wn } ⊂ Ω be a sequence which converges to y in the Carathéodory topology of Ω. Let R > 0. The horocycle E zΩ0 (y, R) of center y, base point z 0 and hyperbolic radius R > 0 is E zΩ0 (y, R) := {z ∈ Ω : lim [kΩ (z, wn ) − kΩ (z 0 , wn )] < n→∞

1 log R}. 2

Note that, by (1.4.2), for every σ ∈ ∂D and R > 0, E(σ, R) = E 0D (x σ , R).

(6.4.5)

The base point z 0 in the definition of horocycles is essentially irrelevant: Lemma 6.4.4 Let Ω  C be a simply connected domain, y ∈ ∂C Ω and z 0 , z 1 ∈ Ω. Let {wn } ⊂ Ω be any sequence converging to y in the Carathéodory topology of Ω. Then the following limit exists

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6 Quasi-Geodesics and Localization

1 log A := lim [kΩ (z 0 , wn ) − kΩ (z 1 , wn )]. n→∞ 2 Moreover, A ∈ (0, +∞) and for every R > 0 E zΩ0 (y, R) = E zΩ1 (y, A R). Proof By Remark 6.4.2, the limit exists and A ∈ (0, +∞). Moreover, by the same remark, for every z ∈ Ω, lim [kΩ (z, wn ) − kΩ (z 1 , wn )] = lim [kΩ (z, wn ) − kΩ (z 0 , wn )]

n→∞

n→∞

+ lim [kΩ (z 0 , wn ) − kΩ (z 1 , wn )], n→∞



from which the result follows at once. We can thus re-interpret Lemma 6.4.1 in a general way:

Theorem 6.4.5 Let Ω, U  C be two simply connected domains. Let y ∈ ∂C U . Suppose there exists z 0 ∈ U and R > 0 such that E zU0 (y, R) ⊂ Ω ⊆ U. Assume f : D → C is a Riemann map of Ω. Then there exists σ ∈ ∂D such that if γ : [0, +∞) → U is a geodesic in U with limt→+∞ γ (t) = y in the Carathéodory topology of U , then γ (t) is eventually contained in Ω and f −1 (γ (t)) converges orthogonally to σ . Proof Using a Riemann map from D onto U composed with a Cayley transform from D onto H, we can assume that U = H, y ∈ ∂C H is the prime end which corresponds to ∞ and γ (t) = t + 1. Hence, by Lemma 6.4.4, (6.4.5) and (1.4.15), we have H + a ⊂ Ω ⊆ H, for some a > 0, and the result follows immediately from Lemma 6.4.1.  We end this section by describing horocycles in some domains. Example 6.4.6 Let H be the right half-plane. We denote by ∞ the prime end in H corresponding to the null chain (Cn ) defined by Cn := {ζ ∈ H : |ζ | = n + 1} for n ∈ N0 . Note that {n} converge to ∞ in the Carathéodory topology of H. Hence for R>0 E 1H (∞, R) = {w ∈ H : lim [kH (w, n) − kH (wn , 1)] < n→∞

1 log R}. 2

The Cayley transform C(z) = 1+z is a biholomorphism from D onto H, which maps 0 1−z to 1. Moreover, C maps the prime end 1 ∈ ∂C D, which corresponds to 1 ∈ ∂D, to ∞. Since C is an isometry between ω and kH , it follows that E 1H (∞, R) = C(E 0D (1, R)) and, by (1.4.15),

6.4 Orthogonal Convergence

155

E 1H (∞, R) = E H (∞,

1 1 ) = {z ∈ H : Re z > }. R R

Example 6.4.7 Let p ∈ C. Consider the Koebe domain defined by K p := C \ {ζ ∈ C : Re ζ = Re p, Im ζ ≤ Im p}. The domain K p is symmetric with respect to (0, +∞)  t → p + it, which, by Proposition 6.1.3 is a geodesic in K p . Such a geodesic converges as t → +∞ in the Carathéodory topology of K p to a prime end ∞ ∈ ∂C K p . We want to describe the K horocycles E p+ip (∞, R), for R > 0. Let f : H → K p be defined as f (z) := i z 2 + p. √ −1 is f (w) = −i(w − p), Clearly, f is a biholomorphism from H to K p . Its inverse √ where the branch of the square root is chosen so that 1 = 1. Note that f (1) = p + i and f maps the prime end ∞ in ∂C H (see Example 6.4.6) to the prime end ∞ in ∂C K p . Since f is an isometry between kH and kK p , by Example 6.4.6, we have  1 K E p+ip (∞, R) = f (E 1H (∞, R)) = {w ∈ K p : Re −i(w − p) > }. R Now, for w ∈ K p , write w = p + iρeiθ , with ρ > 0 and θ ∈ (−π, π ). Hence, w ∈ √ K E p+ip (∞, R) if and only if ρ cos θ2 > R1 , which is equivalent to ρ(1 + cos θ ) > R22 . In other words, since ρ cos θ = Re (−i(w − p)) = Im (w − p), K

E p+ip (∞, R) = {w ∈ K p : |w − p| + Im (w − p) >

2 }. R2

A simple computation shows that, in particular, there exist a > 0, b > Im p (depending on R and converging to +∞ as R → 0+ ) such that K

E p+ip (∞, R) ⊂ C \ {ζ ∈ C : −a ≤ Re (ζ − p) ≤ a, Im ζ ≤ b}.

(6.4.6)

6.5 Hyperbolic Projections, Tangential and Orthogonal Speeds of Curves in the Disc In what follows, for not burdening the notation, we will consider geodesics parameterized by (hyperbolic) arc length, but, as it will be clear, this is not relevant, and any parameterization of geodesics would work as well.

156

6 Quasi-Geodesics and Localization

Definition 6.5.1 Let Ω  C be a simply connected domain. Let γ : R → Ω be a geodesic parameterized by arc length. Let z ∈ Ω. The hyperbolic projection πγ (z) ∈ γ (R) of z onto γ is the closest point (in the hyperbolic distance) of γ to z, namely, kΩ (πγ (z), z) = min kΩ (γ (t), z). t∈R

The next proposition shows that the hyperbolic projection onto a geodesic is uniquely defined and can be determined geometrically: Proposition 6.5.2 Let Ω  C be a simply connected domain. Let γ : R → Ω be a geodesic in Ω parameterized by arc length and let z ∈ Ω. Then πγ (z) is the point of intersection of γ with the geodesic γ˜ containing z and intersecting γ orthogonally (in the Euclidean sense). Proof Let C be a univalent map such that C(Ω) = H and such that C(γ ) = (0, +∞) and C(z) = ρeiβ for some ρ > 0 and β ∈ (−π/2, π/2). By Lemma 5.4.1(3), ρ is the closest point of C(γ ) to C(z). Since ρ is the intersection of C(γ ) with {w = ρeiθ : |θ | < π2 }, which is the only geodesic in H containing C(z) and orthogonal to C(γ ), the result follows by recalling that C is an isometry for the hyperbolic distance, maps geodesics of Ω onto geodesics of H and preserves orthogonality.  Although hyperbolic projections onto geodesics are not holomorphic maps, they do not increase the hyperbolic distance: Proposition 6.5.3 Let Ω  C be a simply connected domain, γ : R → Ω a geodesic parameterized by arc length. Then for every z, w ∈ Ω, we have kΩ (πγ (z), πγ (w)) ≤ kΩ (z, w). Proof Since the statement is invariant under isometries for the hyperbolic distance, using a univalent map, we can assume Ω = H and the image of γ is (0, +∞). We can write z = ρ0 eiβ0 with ρ0 > 0 and β0 ∈ (−π/2, π/2) and w = ρ1 eiβ1 with ρ1 > 0 and β1 ∈ (−π/2, π/2). By Lemma 5.4.1(3), πγ (z) = πγ (ρ0 eiβ0 ) = ρ0 and πγ (w) = πγ (ρ1 eiβ1 ) = ρ1 . Hence the result follows immediately from Lemma 5.4.1(5).  Remark 6.5.4 Let x ∈ (0, 1) and denote by S − (respectively, S + ) the (image of the) geodesic in D through −x (resp., x) and orthogonal to (−1, 1) at −x (resp., at x). Let γ : R → D be a geodesic parameterized by arc length so that γ (R) = (−1, 1). Then, as a direct consequence of Propositions 6.5.2 and 6.5.3, we have for every z − ∈ S − and z + ∈ S + , ω(z − , z + ) ≥ ω(πγ (z − ), πγ (z + )) = ω(−x, x). The next result shows a uniform decomposition of the distance between two points with respect to a given geodesic containing one of the points:

6.5 Hyperbolic Projections, Tangential and Orthogonal Speeds of Curves in the Disc

157

Proposition 6.5.5 Let Ω  C be a simply connected domain, γ : R → Ω a geodesic parameterized by arc length, x0 ∈ γ and z ∈ Ω. Then kΩ (x0 , πγ (z)) + kΩ (z, γ ) −

1 log 2 ≤ kΩ (x0 , z) ≤ kΩ (x0 , πγ (z)) + kΩ (z, γ ), 2

where kΩ (z, γ ) := inf t∈R kΩ (z, γ (t)) = kΩ (z, πγ (z)). Proof Since the statement is invariant under isometries for the hyperbolic distance, using a univalent map, we can transfer our considerations to H, and we can assume that γ (R) = (0, +∞) and x0 = 1. Let z ∈ H, and write z = ρeiβ with ρ > 0 and β ∈ (−π/2, π/2). By Lemma 5.4.1(3), πγ (ρeiβ ) = ρ. Hence, by the triangle inequality, kH (1, ρeiβ ) ≤ kH (1, ρ) + kH (ρ, ρeiβ ) = kH (1, πγ (ρeiβ )) + kH (γ , ρeiβ ). On the other hand, by Lemma 5.4.1(2), kH (1, ρeiβ ) ≥ kH (1, ρ) +

1 1 log . 2 cos β

The previous equation, together with Lemma 5.4.1(6), gives 1 1 1 log ≥ kH (1, ρ) + kH (ρ, ρeiβ ) − log 2 2 cos β 2 1 iβ iβ = kH (1, πγ (ρe )) + kH (γ , ρe ) − log 2, 2

kH (1, ρeiβ ) ≥ kH (1, ρ) +



and we are done.

The previous proposition gives sense to the following definition and the subsequent remarks. Definition 6.5.6 Let Ω  C be a simply connected domain and let z 0 ∈ Ω. Let η : [0, +∞) → Ω be a continuous curve such that η(t) converges in the Carathéodory topology of Ω to a point x ∈ ∂C Ω as t → +∞. Let γ : (−∞, +∞) → Ω be the geodesic of Ω parameterized by arc length such that γ (0) = z 0 and γ (t) → x in the o Carathéodory topology of Ω as t → +∞. The orthogonal speed vΩ,z (η; t) of η is 0 o (η; t) := kΩ (z 0 , πγ (η(t))). vΩ,z 0 T (η; t) of η is The tangential speed vΩ,z 0 T (η; t) := kΩ (γ , η(t)). vΩ,z 0

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6 Quasi-Geodesics and Localization

Remark 6.5.7 Let Ω, z 0 , x, γ and η be as in Definition 6.5.6. (1) The orthogonal speed and the tangential speed of a curve do not depend on the parameterization of the geodesic γ . Therefore, as a consequence of Proposition 5.1.14(3), the definition of orthogonal speed and tangential speed depend only on Ω, z 0 and x. (2) If Ω, Ω   C are simply connected domains, z 0 ∈ Ω, z 0 ∈ Ω  and f : Ω → Ω  o o is a biholomorphism such that f (z 0 ) = z 0 then vΩ,z (η; t) = vΩ  ,z  ( f ◦ η; t) and 0 0 T T vΩ,z0 (η; t) = vΩ  ,z  ( f ◦ η; t) for all t ≥ 0. This follows immediately since f is 0 an isometry for the hyperbolic distances of Ω and Ω  . The actual orthogonal speed and tangential speed of a curve depend on the base point chosen, but, asymptotically they do not: Lemma 6.5.8 Let Ω  C be a simply connected domain and let z 0 , z 1 ∈ Ω. Then for every x ∈ ∂C Ω and for every continuous curve η : [0, +∞) → Ω converging to x in the Carathéodory topology of Ω, we have o (η; t) = +∞, (1) limt→+∞ vΩ,z 0 T T (2) limt→+∞ |vΩ,z0 (η; t) − vΩ,z (η; t)| = 0, 1 o o (η; t)| ≤ kΩ (z 0 , z 1 ). (3) lim supt→+∞ |vΩ,z0 (η; t) − vΩ,z 1

Proof By Remark 6.5.7(2), up to composing with a biholomorphism from H to Ω, we can assume Ω = H, z 0 = 1 and x is the prime end of H which corresponds to “∞”, namely, the prime end defined by the null chain {(n + 1)eiθ : |θ | < π/2}n∈N . Hence, limt→+∞ |η(t)| = +∞. Moreover, the geodesic in H which joins 1 to x is γ0 (r ) := r , r ∈ (0, +∞). While, the geodesic in H which joins z 1 := x + i y to x is γ1 (r ) := r + i y, r ∈ (0, +∞). From Lemma 5.4.1(3), we have πγ0 (η(t)) = |η(t)|. This shows in particular that o (η; t) = kH (1, πγ0 (η(t))) = kH (1, |η(t)|) → +∞, vH,1

as t → +∞, and (1) follows. On the other hand, using the automorphism z → z − i y which maps γ0 onto γ1 and taking into account that it is an isometry for kH , we see that πγ1 (η(t)) = |η(t) − i y| + i y. Therefore, T (η; t) − v T |vH,1 H,x+i y (η; t)| = |kH (η(t), πγ0 (η(t))) − kH (η(t), πγ1 (η(t)))|

≤ kH (πγ0 (η(t)), πγ1 (η(t))) = kH (|η(t)|, |η(t) − i y| + i y).

Taking into account that limt→+∞ |η(t)| = +∞, a direct computation shows that lim kH (|η(t)|, |η(t) − i y| + i y) = 0,

t→+∞

and hence (2) follows.

(6.5.1)

6.5 Hyperbolic Projections, Tangential and Orthogonal Speeds of Curves in the Disc

159

Now, using the triangle inequality, o o |vH,1 (η; t) − vH,x+i y (η; t)| = |kH (1, πγ0 (η(t))) − kH (x + i y, πγ1 (η(t)))|

= |kH (1, πγ0 (η(t))) − kH (x + i y, πγ0 (η(t))) + kH (x + i y, πγ0 (η(t))) − kH (x + i y, πγ1 (η(t)))| ≤ kH (1, x + i y) + kH (πγ0 (η(t)), πγ1 (η(t))) = kH (1, x + i y) + kH (|η(t)|, |η(t) − i y| + i y), and thus (3) follows from (6.5.1).



The reason for the name “tangential speed” follows from the following property: Proposition 6.5.9 Let η : [0, +∞) → D be a continuous curve converging to a point σ ∈ ∂D. Let t0 := inf{s ≥ 0 : Re (σ η(t)) ≥ 0 ∀t ∈ [s, +∞)}. Then t0 ∈ [0, +∞) and for all t ≥ t0 ,     1 1 ω(0, η(t)) − 1 log  ≤ log 2,  2 1 − |η(t)|  2    o  1 1 v (η; t) − 1 log  ≤ log 2,  D,0 2 |σ − η(t)|  2    T  v (η; t) − 1 log |σ − η(t)|  ≤ 3 log 2.  D,0 2 1 − |η(t)|  2 Proof Since η(t) → σ as t → +∞, it follows that t0 < +∞. The first equation follows immediately from the very definition of ω (see (1.3.1)). Indeed, for every t ≥ 0,    1  1  = log(1 + |η(t)|) < 1 log 2. ω(0, η(t)) − 1 log  2 1 − |η(t)|  2 2 In order to prove the other two equations, up to change η with σ η, we can assume without loss of generality that σ = 1. Let C : D → H be the Cayley transform given . For every t ≥ 0, let us write ρt eiθt := C(η(t)), with ρt > 0 and by C(z) = 1+z 1−z θt ∈ (−π/2, π/2). This implies in particular, that ρt ≥ 1 for all t ≥ t0 . Then, for t ≥ t0 we have 1 log ρt 2 |1 + η(t)| 1 1 , = log |C(η(t))◦ | = log 2 2 |1 − η(t)|

o o vD,0 (η; t) = vH,1 (ρt eiθt ; t) = kH (1, ρt ) =

(6.5.2)

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6 Quasi-Geodesics and Localization

where, the first equality follows from Remark 6.5.7(2), the second equality follows from the definition of orthogonal speed and since the hyperbolic projection of ρt eiθt onto the geodesic (0, +∞) is ρt by Lemma 5.4.1(3) and the third equality follows from Lemma 5.4.1(1). Therefore, by (6.5.2), and taking into account that for t ≥ t0 we have |1 + η(t)| ≥ 1 + Re η(t) ≥ 1,    1  o 1  = log |1 + η(t)| ≤ 1 log 2. v (η; t) − 1 log  D,0 2 |1 − η(t)|  2 2 As for the last inequality, from Proposition 6.5.5 we have o T o (η; t) ≤ vD,0 (η; t) ≤ ω(0, η(t)) − vD,0 (η; t) + ω(0, η(t)) − vD,0

1 log 2, 2

o and using the previous two inequalities for the estimates of ω(0, η(t)) and vD,0 (η; t), we get the result. 

Remark 6.5.10 As a consequence of the previous proposition, we have that if η : [0, +∞) → D is a continuous curve such that limt→+∞ η(t) = σ ∈ ∂D, then T (η; t) < +∞. η converges to σ non-tangentially if and only if lim supt→+∞ vD,0

6.6 Localization of Hyperbolic Metric and Hyperbolic Distance In this section we prove a localization result which allows us to get information on the hyperbolic metric and hyperbolic distance of a simply connected domain in a portion of the domain itself. We start with the notion of totally geodesics subsets: Definition 6.6.1 Let Ω  C be a simply connected domain. A domain U ⊂ Ω is said to be totally geodesic in Ω if for every z, w ∈ U the geodesic of Ω joining z and w is contained in U . Let Ω  C be a simply connected domain and let γ : R → Ω be a geodesic parameterized by arc length. By Proposition 5.1.14 and Remark 5.1.15, there exist p + , p − ∈ ∂C Ω, p + = p − , such that limt→±∞ γ (t) = p ± in the Carathéodory topol∞

ogy of Ω. However, this does not mean in general that γ (R) Proposition 5.1.14(5)). Nonetheless, we have the following:

is a Jordan arc (see

Lemma 6.6.2 Let Ω  C be a simply connected domain. Let γ : R → Ω be a geodesic parameterized by arc length. Then Ω \ γ (R) consists of two simply connected components which are totally geodesic in Ω.

6.6 Localization of Hyperbolic Metric and Hyperbolic Distance

161

Proof Let f : D → Ω be a biholomorphism. Then, f −1 ◦ γ : R → D is a geodesic parameterized by arc length. By Lemma 5.1.13, up to pre-composing with an automorphism of D, we can assume that f −1 (γ (R)) = (−1, 1). Consider D+ := {ζ ∈ D : Re ζ > 0}. Since the geodesic in D joining two points z, w ∈ D is the arc of a circle containing z, w and meeting ∂D orthogonally, it is easy to see that D+ is totally geodesic in D. A similar argument shows that D− := {ζ ∈ D : Re ζ < 0} is totally geodesic. Moving back to Ω via f and recalling that f is an isometry for the hyperbolic distance, we have the result.  Now we can state and prove the localization result for the hyperbolic metric and the hyperbolic distance: Theorem 6.6.3 (Localization Lemma) Let Ω  C be a simply connected domain.  which contains p. Assume that U ∗ ∩ Ω Let p ∈ ∂C Ω and let U ∗ be an open set in Ω is simply connected. Let C > 1. Then there exists an open neighborhood V ∗ ⊂ U ∗ of p such that for every z, w ∈ V ∗ ∩ Ω and all v ∈ C, κΩ (z; v) ≤ κU ∗ ∩Ω (z; v) ≤ CκΩ (z; v),

(6.6.1)

kΩ (z, w) ≤ kU ∗ ∩Ω (z, w) ≤ CkΩ (z, w).

(6.6.2)

In particular, if Ω is a Jordan domain then for every σ ∈ ∂∞ Ω, for every U ⊂ C∞ open set such σ ∈ U and U ∩ Ω is simply connected, and every C > 1, there exists an open neighborhood V ⊂ U of σ such that (6.6.1) and (6.6.2) hold (with U ∗ = U and V ∗ = V ). Proof The inequalities on the left in (6.6.1) and (6.6.2) follows immediately from Proposition 5.1.4. In order to deal with the inequalities on the right, we first assume Ω = D. By Proposition 4.2.5, the identity map extends to a homeomorphism Φ between  D and D. Hence, there exists an open set (for the Euclidean topology) W ⊂ C such that σ := Φ( p) ∈ W and Φ(U ∗ ) = W ∩ D. Since by hypothesis U ∗ ∩ D is simply connected, then Φ(U ∗ ∩ D) = W ∩ D is simply connected as well. For every r ∈ (0, 1), let D(0, r ) := {ζ ∈ D : |ζ | < r }. Given r ∈ (0, 1), the map ζ → r ζ is a biholomorphism between D and D(0, r ). Hence, by Lemma 5.1.3, for all v ∈ C, 1 (6.6.3) κ D(0,r ) (0; v) = κD (0; v). r In particular, since automorphisms of D are isometries for κD , if D hyp (z 0 , R) := {z ∈ D : ω(z, z 0 ) < R}, then there exists an automorphism T of the unit disc such that T (D hyp (z 0 , R)) = D hyp (0, R) = {ζ ∈ D : |ζ | < tanh R}. Therefore, by (6.6.3), for all v ∈ C, κ D hyp (z0 ,R) (z 0 ; v) = (tanh R)−1 κD (z 0 ; v).

Claim A: given R > 0 such that (tanh R)−1 < C, there exists an open set X ⊂ W , σ ∈ X , such that for every z ∈ X ∩ D the hyperbolic disc D hyp (z, R) ⊂ W ∩ D.

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6 Quasi-Geodesics and Localization

Once Claim A is proved, it follows from Proposition 5.1.4 that for all z ∈ X ∩ D and v ∈ C, κW ∩D (z; v) ≤ κ D hyp (z,R) (z; v) = (tanh R)−1 κD (z; v) < CκD (z; v).

(6.6.4)

In order to prove Claim A, we argue by contradiction. If Claim A is not true, there / W ∩ D for all exist two sequences {z n }, {wn } ⊂ D such that limn→∞ z n = σ , wn ∈ n ∈ N and ω(z n , wn ) < R. Hence, Lemma 1.8.6 implies that {wn } converges to σ , a contradiction and Claim A follows. Now, let  ∈ (0, π/4) and let γ : (−∞, +∞) → D be a geodesic in D parameterized by arc length such that limt→−∞ γ (t) = ei σ and limt→+∞ γ (t) = e−i σ (see Lemma 5.1.14). Since γ (R) is the arc in D on the circle passing through ei σ and e−i σ and orthogonal to ∂D at ei σ, e−i σ , it follows easily that if  is sufficiently small, then γ (R) ⊂ X . By Lemma 6.6.2, D \ γ (R) is the union of two simply connected components. Since γ (R) does not contain σ , it follows that σ belongs to the closure of one and only one of the connected components of D \ γ (R). Call Y such a component. Note that, by construction, Y ⊂ X ∩ D. By Lemma 6.6.2, Y is totally geodesic in D. Therefore, for every z, w ∈ Y , the geodesic η : [0, 1] → D of D such that η(0) = z, η(1) = w is contained in Y ⊂ X ∩ D. Hence, by (6.6.4), 

1

κW ∩D (η(t); η (t))dt k W ∩D (z, w) ≤ W ∩D (η; [0, 1]) = 0  1 ≤C κD (η(t); η (t))dt = C kD (z, w). 0

By the arbitrariness of z, w, we have proved the result for the unit disc, setting V ∗ := Φ −1 (Y˜ ∩ D), where Y˜ is any open set in C such that Y˜ ∩ D = Y . Now, if Ω  C is any simply connected domain, using a biholomorphism from D to Ω, taking into account Theorem 4.2.3 and that biholomorphisms are isometries for the hyperbolic metric and hyperbolic distance, we easily obtain the result from the corresponding result for the unit disc. Finally, if Ω is a Jordan domain, the result follows since, by Theorem 4.3.3 and ∞  are homeomorphic.  Proposition 4.2.5, Ω and Ω

6.7 Hyperbolic Geometry in the Strip In this section we study hyperbolic geometry in the strip, proving some results which will be used for localizing the hyperbolic metric and distance. Definition 6.7.1 For ρ > 0 we define the strip of width ρ

6.7 Hyperbolic Geometry in the Strip

163

Sρ := {ζ ∈ C : 0 < Re ζ < ρ}. For ρ = 1 we simply write S := S1 . Proposition 6.7.2 Let a ∈ R and R > 0. The curve γ0 : R  t → a + geodesic of S R + a and, for every s < t, kS R +a (a +

R 2

+ it is a

R R π(t − s) + is, a + + it) = . 2 2 2R

log z + R2 + a is a biholomorProof The holomorphic function f : H  z → Ri π phism from H to S R + a. Since S R + a is symmetric with respect to the line {z ∈ C : Re z = a + R2 }, it follows from Proposition 6.1.3 that γ0 is a geodesic. The formula for the hyperbolic distance follows at once by a direct computation using f and the corresponding  expression of kH . The following result gives a link between the boundary behavior of a hyperbolic holomorphic self-map of D and the geometry in the strip: Proposition 6.7.3 Let φ : D → D be hyperbolic with Denjoy-Wolff point τ ∈ ∂D. Suppose αφ (τ ) ∈ (0, 1) (that is, φ is hyperbolic) and let λ > 0 be such that e−λ = αφ (τ ). If {z n } ⊂ D is a sequence converging non-tangentially to τ with angle limn→∞ Arg(1 − σ z n ) = θ ∈ (−π/2, π/2), then lim ω(z n , φ(z n )) = kS π ( λ

n→∞

Proof Let L := that

π , λ

π θ π θ + + i, + ). λ 2λ λ 2λ

and let S := S L − L2 . Fix θ ∈ (−π/2, π/2) and let θ˜ := λθ . Note kS L (

L θ L θ + + i, + ) = k S (θ˜ + i, θ˜ ). λ 2 λ 2

Let g(z) := i

  1+z L log , z ∈ D. π 1−z

Hence g : D → S is a Riemann map and π

g −1 (z) :=

e−i L z − 1 , z ∈ S. π e−i L z + 1

In particular,  ⎞   −1 ˜ 1 + (g ( θ + i)) T  −1 ˜ g (θ) 1 −1 ˜ −1 ˜ ⎝ ˜ ˜  ⎠ . k S (θ , θ + i) = ω(g (θ), g (θ + i)) = log  2 −1 (θ˜ + i)) 1 − Tg−1 (θ) (g ˜ ⎛

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6 Quasi-Geodesics and Localization

Moreover     g −1 (θ˜ ) − g −1 (θ˜ + i)       −1  Tg−1 (θ) ˜ (g (θ˜ + i)) =   1 − g −1 (θ˜ )g −1 (θ˜ + i)    π  2e−i πL θ˜ − 2e−i πL θ˜ e πL  1 − e− L   = =   2ei πL θ˜ + 2e−i πL θ˜ e πL  |e−2 πL θi˜ + e− πL | = Hence,

1 − e−λ . + e−λ |

|e−2θi

    e−λ + e−2iθ  + e−λ − 1 1 ˜ θ˜ + it) = log   .   k S (θ, e−λ + e−2iθ  − e−λ − 1 2

Therefore, the result follows from Proposition 1.9.12.

(6.7.1) 

Remark 6.7.4 From (6.7.1), it follows immediately that if a, b ∈ (−π/2, π/2) and kS π ( λ

π a π π b π a b + + i, + ) = kS π ( + + i, + ), λ λ 2λ λ 2λ λ 2λ λ 2λ

then a = ±b.

6.8 Some Localization Results In this section we prove some localization results which will be used in the study of the slope of convergence of orbits of semigroups. Lemma 6.8.1 Let p ∈ D and R > 0 and let D hyp ( p, R) := {z ∈ D : ω(z, p) < R}. Then, for all c > 1 and M > 0, there exists R > M such that for all p ∈ D, we have k D hyp ( p,R) (z, w) ≤ c ω(z, w) for all z, w ∈ D hyp ( p, M). Proof Using an appropriate automorphism of D, we can assume p = 0. Fix R > 0, 2R and let m(R) = ee2R −1 . Hence, D hyp (0, R) = {z ∈ D : |z| < m(R)}, and f : D → +1 hyp D (0, R) given by f (z) = m(R)z is a biholomorphism. Then for every z, w ∈ D hyp (0, R), k D hyp (0,R) (z, w) = ω(z/m(R), w/m(R)).

6.8 Some Localization Results

165

Moreover, if z, w ∈ D, then z, w ∈ D hyp (0, R) for R sufficiently large, and ω(z, w) = lim k D hyp (0,R) (z, w), R→+∞

where the limit is uniform on compacta. Therefore, given c > 1 and M > 0 there exists R = R(c, M) > 0 such that k D hyp (0,R) (z, w) ≤ c ω(z, w) for all z, w ∈ D hyp (0, M).



Based on the previous lemma, we have the following: Proposition 6.8.2 Let R > 0. For every c > 1, S > 0, T ∈ [0, R) there exists r ∈ (T, R) such that for every z, w ∈ Sr such that T1 ≤ Re z ≤ T , T1 ≤ Re w ≤ T and |Im z − Im w| < S, kSr (z, w) ≤ ckS R (z, w). Proof Let K := {z ∈ C : T1 ≤ Re z ≤ T, |Im z| ≤ S}. Let x0 ∈ R be such that x0 ∈ K . Note that K is compact in S R . Then there exists M > 0 such that hyp

K ⊂ DS R (x0 , M) := {z ∈ S R : kS R (z, x0 ) < M}. Fix c > 1. By Lemma 6.8.1 and since the Riemann mappings are an isometries for the hyperbolic distance, there exists A > M such that, for all z, w ∈ K , k D hyp (x0 ,A) (z, w) ≤ ckS R (z, w).

(6.8.1)

SR

hyp

hyp

Since DS R (x0 , A) ⊂ S R , we can find r ∈ (T, R) such that DS R (x0 , A) ⊂ Sr . Therefore, by (6.8.1) for all z, w ∈ K , kSr (z, w) ≤ k D hyp (x0 ,A) (z, w) ≤ ckS R (z, w). SR

Now, suppose z 0 , w0 ∈ S R are such that T1 ≤ Re z 0 ≤ T , T1 ≤ Re w0 ≤ T and |Im z 0 − Im w0 | < S. The map Sρ  z → z − iIm w0 is an automorphism in Sρ for all ρ > 0. Since z 0 − iIm w0 and Re w0 belong to K , the previous inequality gives kSr (z 0 , w0 ) = kSr (z 0 − iIm w0 , Re w0 ) ≤ ckS R (z 0 − iIm w0 , Re w0 ) = ckS R (z 0 , w0 ), and we are done. Let a, b ∈ R and R > 0. Let Ωa,b,R := C \ {z ∈ C : Re z ∈ {a, a + R}, Im z ≤ b}.



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6 Quasi-Geodesics and Localization

Proposition 6.8.3 For every c > 1 there exists D = D(c) > 0, such that for every R > 0, a, b ∈ R, κΩa,b,R (z; v) ≤ κS R +a (z; v) ≤ cκΩa,b,R (z; v) for every z ∈ (S R + a) such that Im z ≤ b − R D and for every v ∈ C. Moreover, kΩa,b,R (z, w) ≤ kS R +a (z, w) ≤ ckΩa,b,R (z, w) for every z, w ∈ (S R + a) such that Im z, Im w ≤ b − R D. Proof The inequalities on the left hand side follow from Proposition 5.1.4 and Remark 1.3.11, since S R + a ⊂ Ωa,b,R . Assume now R = 1, a = b = 0 and let Ω := Ω0,0,1 . For n ∈ N0 let Cn := {ζ ∈ C : 0 ≤ Re ζ ≤ 1, Im ζ = −n}. Clearly, (Cn ) is a null  defined chain in Ω, which represents a prime end x of Ω. Let S∗ be the open set in Ω by S (see Sect. 4.2). Hence, S∗ is an open neighborhood of x, since, by construction, the interior part of Cn belongs to S for all n ≥ 1. Moreover, S∗ ∩ Ω = S, which is simply connected. Therefore, we can apply Theorem 6.6.3 to x and S∗ and come up  which contains x and such that with an open set V ∗ ⊂ S∗ in Ω κS (z; v) ≤ cκΩ (z; v), kS (z, w) ≤ ckΩ (z, w),

(6.8.2)

for all z, w ∈ V := V ∗ ∩ Ω and v ∈ C. Note that since V ∗ is an open neighborhood of x, by Remark 4.2.2, there exists n 0 ∈ N such that the interior part of Cn is contained in V for all n ≥ n 0 . In particular, (6.8.2) holds for every ζ ∈ S such that Im ζ ≤ −(n 0 + 1). Hence, we proved the result with D := −(n 0 + 1) for R = 1, a = b = 0. Now, assume R > 0 and a, b ∈ R. The map C  z → R1 (z − a − ib) ∈ C restricted to Ωa,b,R is a biholomorphism between Ωa,b,R and Ω, which maps (S R + a) onto S. Hence, by Lemma 5.1.3 and Proposition 1.3.10, for all z, w ∈ Ωa,b,R and v ∈ C we have v 1 κΩa,b,R (R; v) = κΩ ( (z − a − ib); ), R R 1 1 kΩa,b,R (z, w) = kΩ ( (z − a − ib), (w − a − ib)), R R and

v 1 κS R +a (z; v) = κS ( (z − a − ib); ), R R 1 1 kS R +a (z, w) = kS ( (z − a − ib), (w − a − ib)). R R

Since moreover the set {ζ ∈ S : Im ζ ≤ −D} is mapped onto {ζ ∈ S R + a : Im ζ ≤ b − R D}, the previous equations and (6.8.2) imply immediately the result. 

6.8 Some Localization Results

167

The next localization result is similar to the previous one, and localizes the hyperbolic metric in a semi-strip contained in a strip. We start with a definition: Definition 6.8.4 Let M ∈ R, R > 0. The semi-strip of width R and height M is SM R := {ζ ∈ C : 0 < Re ζ < R, Im ζ > M}.

Proposition 6.8.5 For every c > 1 there exists D = D(c) > 0, such that for every R > 0, a, M ∈ R, we have κS R +a (z; v) ≤ κS MR +a (z; v) ≤ cκS R +a (z; v), for every z ∈ (S R + a) such that Im z ≥ R D + M and for all v ∈ C, and kS R +a (z, w) ≤ kS MR +a (z, w) ≤ ckS R +a (z, w), for every z, w ∈ (S R + a) such that Im z, Im w > R D + M. Proof The proof is similar to the proof of Proposition 6.8.3. We first prove the result for a = M = 0 and R = 1 and then, exactly as in the proof of Proposition 6.8.3, we rescale using the transformation z → Rz + a + i M to handle the general case. Suppose a = M = 0 and R = 1. For n ∈ N0 let Cn := {ζ ∈ C : 0 ≤ Re ζ ≤ 1, Im ζ = n}. Clearly, (Cn ) is a null chain in S, which represents a prime end x of S. Let (S01 )∗ be the open set in  S defined by S01 (see Sect. 4.2). Hence, (S01 )∗ is an open neighborhood of x, since, by construction, the interior part of Cn belongs to S01 for all n ≥ 1. Moreover, (S01 )∗ ∩ S = S01 , which is simply connected. Therefore, we can apply Theorem 6.6.3 to x and (S01 )∗ and come up with an open set S which contains x and such that κS MR +a (z; v) ≤ cκS R +a (z; v) and V ∗ ⊂ (S01 )∗ in  kS MR +a (z, w) ≤ ckS R +a (z, w) hold for all z, w ∈ V := V ∗ ∩ S and v ∈ C. Note that since V ∗ is an open neighborhood of x, by Remark 4.2.2, there exists n 0 ∈ N such that the interior part of Cn is contained in V for all n ≥ n 0 . In particular, the estimate hold with D = n 0 + 1.  The next localization result we need is a sort of converse of the previous one: we choose the part we want to localize and come up with a constant for the localization. Proposition 6.8.6 For every E > 0 there exists c = c (E) > 1 such that for every a ∈ R, M ∈ R and R > 0, we have κS R +a (z; v) ≤ κS MR +a (z; v) ≤ c κS R +a (z; v), for every z ∈ (S R + a) such that Im z ≥ R E + M and for all v ∈ C, and kS R +a (z, w) ≤ kS MR +a (z, w) ≤ c kS R +a (z, w),

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6 Quasi-Geodesics and Localization

for every z, w ∈ (S R + a) such that Im z, Im w > R E + M. Proof The left-hand side estimates follow immediately from Proposition 5.1.4 and Remark 1.3.11. In order to prove the right-hand side estimates, arguing as in Proposition 6.8.3, it is enough to prove the result for R = 1, a = 0, M = 0 and then use the affine map z → R1 (z − a − i M) to pass to the general case. Fix E > 0. Let K := {z ∈ C : E ≤ Re z ≤ 1 − E, E ≤ Im z ≤ 1} (possibly K is / K , we have δS01 (z) = δS1 (z), empty if E > 1). For z ∈ S01 such that Im z ≥ E and z ∈ hence, from Theorem 5.2.2, κS01 (z; v) ≤

|v| |v| =2 ≤ 2κS1 (z; v). δS01 (z) 2δS1 (z)

For points in K , in case K is non-empty, notice that K is compact in S01 and in S1 . The hyperbolic metric is continuous in z, hence, they are well defined q := min κS1 (z; 1), z∈K

Q := max κS01 (z; 1). z∈K

Moreover, q > 0 (for otherwise the hyperbolic norm of 1 would be 0 at an interior point). Hence, for z ∈ K and v ∈ C, κS01 (z; v) = |v|κS01 (z; 1) ≤

|v|Q |v|Q Q q≤ κS1 (z; 1) = κS1 (z; v). q q q

Taking c = max{2, Qq } we have the first estimate. In order to prove the second inequality, note that (0, 1) + i E is a geodesic in S1 by Proposition 6.1.3 (since S1 is symmetric with respect to (0, 1) + i E). Then, Lemma 6.6.2 guarantees that S1E is totally geodesic in S1 . Therefore, given z, w ∈ S1E , let γ : [0, 1] → S1 be the geodesic for S1 which joins z and w. Hence, γ ([0, 1]) ⊂ S1E . Therefore, for what we have already proved,  1 κS01 (γ (t); γ  (t))dt kS01 (z, w) ≤ S01 (γ ; [0, 1]) = 0  1 ≤ c κS1 (γ (t); γ  (t))dt = c kS1 (z, w), 0

and we are done.



6.9 Notes

169

6.9 Notes The content of this chapter is essentially taken from [33, 34]. Although we do not use this explicitly in the book, the results proved in this chapter are based on a property of simply connected domains seen as metric spaces endowed with the hyperbolic metric: they are proper geodesic spaces which are Gromov’s hyperbolic. This means that every geodesic triangle is “thin”. Namely, given three points p1 , p2 , p3 , let γ1 be a geodesic which joins p2 , p3 , γ2 a geodesic which joins p1 , p3 and γ3 a geodesic which joins p1 , p2 . Then, there exists a universal constant δ > 0 such that every point of γ1 is at hyperbolic distance at most δ from γ2 ∪ γ3 (and similar for every point of γ2 and γ3 ). Every Gromov’s hyperbolic space enjoys the Shadowing Lemma (Theorem 6.3.8). The proof of the Shadowing Lemma we presented in this book is however rather different from the ones we found in the literature. We refer the reader to the book [76] for more on Gromov’s hyperbolicity. The “Pythagoras Theorem” (Proposition 6.5.5) seems to appear for the first time in this book, and it is valid as well in Gromov’s hyperbolic spaces. Principles of localization are very much used in several complex variables, and we took our point of view from that setting.

Chapter 7

Harmonic Measures and Bloch Functions

In this chapter we introduce the last two tools we need in our study of semigroups throughout the book. The first one comes from potential theory: the harmonic measure of a simply connected domain in C related to a subset of its boundary. The second one is the notion of Bloch function and related maximum principles and distortion theorems.

7.1 Harmonic Measures in the Unit Disc Before introducing the notion of harmonic measure, we present a slightly improvement of the classical Maximum Principle for harmonic functions. Lemma 7.1.1 (Lindelöf’s Maximum Principle) Let Ω be a domain in C such that Ω = C. Let u be a harmonic function on Ω which is bounded from above. Let F be a finite subset of ∂Ω and suppose that lim sup u(z) ≤ 0

(7.1.1)

z→ζ

for all ζ ∈ ∂Ω \ F. Then u(z) ≤ 0 for all z ∈ Ω. Proof We may assume that Ω is bounded. If this is not the case, fix z 0 ∈ / Ω. Then the map Ω  z → f (z) := 1/(z − z 0 ) maps Ω into a bounded domain and we may replace Ω with f (Ω) and u by u ◦ f . If (7.1.1) holds for all ζ ∈ ∂Ω, then the result is nothing but the ordinary Maximum Principle for harmonic functions. Write F = {ζ1 , ζ2 , . . . , ζ N }, take ε > 0 and let u ε (z) = u(z) − ε

N  j=1

 log

 diamE (Ω) , z ∈ Ω. |z − ζ j |

© Springer Nature Switzerland AG 2020 F. Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4_7

171

172

7 Harmonic Measures and Bloch Functions

Then u ε is harmonic on Ω and, since u is bounded from above, lim supz→ζ u ε (z) ≤ 0 for all ζ ∈ ∂Ω. Therefore, u ε ≤ 0 for all ε, u(z) ≤ lim ε ε→0

N 

 log

j=1

diamE (Ω) |z − ζ j |

 = 0, z ∈ Ω, 

and we are done. As an immediate consequence we have

Proposition 7.1.2 Let Ω be a domain in C such that Ω = C. Let u, v be two harmonic and bounded functions on Ω. Let F be a finite subset of ∂Ω. If for all ζ ∈ ∂Ω \ F, lim u(z) = lim v(z) z→ζ

z→ζ

then u = v on Ω. Now we can introduce the notion of harmonic measure and state its first properties. Definition 7.1.3 Let A be a Borel set in ∂D. The harmonic measure1 of A at z ∈ D is defined by μ(z, A, D) := P[χ A ](z) =

1 2π

 0

2π

1 − |z|2 χ A (eiθ )dθ, |eiθ − z|2

where χ A is the characteristic function of A. Proposition 7.1.4 Let A be a Borel set in ∂D. Then: (1) The function D  z → μ(z, A, D) is harmonic and 0 ≤ μ(z, A, D) ≤ 1 for all z ∈ D. (2) If A is open, then lim μ(z, A, D) = 1, ζ ∈ A, z→ζ

lim μ(z, A, D) = 0, ζ ∈ ∂D \ A.

z→ζ

(3) For each z ∈ D, μ(z, ·, D) is a probability Borel measure on ∂D (in fact, it is absolutely continuous with respect to the Lebesgue measure). (4) If T ∈ Aut(D) then μ(T (z), T (A), D) = μ(z, A, D) for all z ∈ D. Proof (1) and (2) follow from Theorem 1.6.2 applied to the function χ A and from the classical maximal principle for harmonic functions. (3) is obvious from the very definition of probability measure. 1 Harmonic measure is usually denoted by ω(z,

A, D). Since in this book we denote by ω(z, w) the hyperbolic distance in D between z, w ∈ D, in order to avoid misunderstandings, we prefer to use a less standard notation for the harmonic measure.

7.1 Harmonic Measures in the Unit Disc

173

(4) Let T ∈ Aut(D) be an automorphism of the unit disc. By Proposition 1.2.2, a−z , z ∈ D. A straightforward there exists λ ∈ ∂D and a ∈ D such that T (z) = λ 1−az computation shows that 1 − |z|2 1 − |T (z)|2 |T (ζ )| = , z ∈ D, ζ ∈ ∂D. |T (ζ ) − T (z)|2 |ζ − z|2 Therefore, changing variables (see, e.g. [3, Sect. 4.1]), we get  2π 1 μ(z, A, D) = 2π 0  2π 1 = 2π 0  2π 1 = 2π 0

1 − |z|2 χ A (eiθ )dθ |eiθ − z|2 1 − |T (z)|2 |T (eiθ )|χ A (eiθ )dθ |T (eiθ ) − T (z)|2 1 − |T (z)|2 χT (A) (eiθ )dθ = μ(T (z), T (A), D), |eiθ − T (z)|2 

and we are done.

Remark 7.1.5 Due to the previous proposition, given A ⊂ ∂D a Borel set and z ∈ D, the harmonic measure μ(z, A, D) can be interpreted as the probability that one can hit A starting from z with a random path. Example 7.1.6 Let A be an arc in ∂D and z ∈ D. Denote by E (A) the Euclidean length of A. Then μ(0, A, D) =

1 2π



2π

χ A (eiθ )dθ =

0

1 E  (A). 2π

(7.1.2)

If θ0 ∈ (0, π ] and A ⊂ ∂D is the arc in the upper half-plane with end points 1 and eiθ0 , then  iθ0  1 |e − 1| 1 E θ0  (A) = = arcsin . μ(0, A, D) = 2π 2π π 2

(7.1.3)

In addition if B is the arc in ∂D with end points e−iθ0 and eiθ0 and containing 1, then μ(0, B, D) = μ(0, A, D) + μ(0, A, D) =

θ0 , π

where A = {ζ : ζ ∈ A}. If T is an automorphism of the unit disc that interchanges the points 0 and z, then μ(z, A, D) = μ(0, T (A), D) =

1 E  (T (A)). 2π

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7 Harmonic Measures and Bloch Functions

Example 7.1.7 Let ∂D− = {ζ ∈ ∂D : Im ζ ≤ 0}. Then μ(z, ∂D− , D) =

1 π



     π 1+z 2 Im z 1 π , z ∈ D, − Arg = − arctan 2 1−z π 2 1 − |z|2

where, as usual throughout the book, Arg(z) denotes the principal argument of z. maps ∂D− onto −i[0, ∞) ∪ Indeed, note that the Cayley transform C(z) = 1+z 1−z    is harmonic in the unit disc, {∞}. Hence, the function v(z) := π1 π2 − Arg 1+z 1−z lim z→ζ v(z) = 1 for all ζ ∈ ∂D− \ {−1, 1} and lim z→ζ v(z) = 0 for all ζ ∈ ∂D \ ∂D+ . Thus, by Propositions 7.1.4(2) and 7.1.2, μ(z, ∂D− , D) = v(z) for all z. Proposition 7.1.8 Let φ : D → D be holomorphic and let A, B ⊂ ∂D be Borel sets. If φ(ζ ) := ∠ lim z→ζ φ(z) ∈ B for almost every ζ ∈ A, then μ(z, A, D) ≤ μ(φ(z), B, D), z ∈ D.

(7.1.4)

Proof Since for all z ∈ D, μ(z, ·, D) is a Borel measure, μ(z, B, D) = inf{μ(z, V, D) : V is open and B ⊂ V }.

(7.1.5)

By hypothesis, there exists a Borel set M of measure zero such that φ(ζ ) exists for every ζ ∈ A \ M and φ(ζ ) ∈ B. Thus, up to removing a Borel set of measure zero, we may assume that φ(ζ ) exists for all ζ ∈ A and φ(A) ⊂ B. Let V be any open subset of ∂D such that B ⊂ V . Then z → μ(z, V, D) is continuous on D ∪ V by Proposition 7.1.4. The function u(z) := μ(φ(z), V, D) is harmonic in D and 0 ≤ u ≤ 1. By Proposition 1.6.8, there exists the radial limit u ∗ (ζ ) := limr →1 u(r ζ ) for almost every ζ ∈ D and, for all z ∈ D, u(z) =

1 2π



2π 0

1 − |z|2 ∗ iθ u (e )dθ, z ∈ D. |eiθ − z|2

If ζ ∈ A, then φ(ζ ) ∈ B ⊂ V so that u has radial limit 1 at ζ by Proposition 7.1.4. In particular, u ∗ (q) ≥ χ A (q) for almost all q ∈ ∂D. Hence,  2π 1 − |z|2 ∗ iθ 1 u (e )dθ μ(φ(z), V, D) = u(z) = 2π 0 |eiθ − z|2  2π 1 − |z|2 1 χ A (eiθ )dθ = μ(z, A, D), ≥ 2π 0 |eiθ − z|2 and the result follows from (7.1.5).



As a direct corollary of the previous proposition and (7.1.2) we have Corollary 7.1.9 (Loewner’s Lemma) Let φ : D → D be holomorphic and φ(0) = 0. Let A, B ⊂ ∂D be Borel sets. If φ(ζ ) := ∠ lim z→ζ φ(z) ∈ B for almost every ζ ∈ A, then E (A) ≤ E (B).

7.1 Harmonic Measures in the Unit Disc

175

Lemma 7.1.10 Let S be a geodesic in D joining two different points ξ1 , ξ2 ∈ ∂D. Let A be one of the two closed arcs in ∂D with end points ξ1 and ξ2 . Let x ∈ S. Then μ(x, A, D) =

1 . 2

Moreover, let D ⊂ D be the bounded component of C \ {A ∪ S}. Then, for every z ∈ D, 1 μ(z, A, D) > , 2 while, for any z ∈ D \ D, μ(z, A, D)
0}. Let b := −Re a + iIm a. Finally, let Ix+ ⊂ ∂D ∩ {Im w > 0} be the closed arc with end points a, b and Ix− := Ix+ (see Fig. 7.1). Then, for all t ∈ (−x, x), μ(t, Ix+ , D) = μ(t, Ix− , D) >

1 . 6

Proof Denote by A x the closed arc in ∂D containing 1 and with end points a and a. Since t is real, μ(t, Ix+ , D) = μ(t, Ix+ , D) = μ(t, Ix− , D).

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7 Harmonic Measures and Bloch Functions

Fig. 7.1 Construction of I x+ and I x− in Theorem 7.1.11

Now suppose 0 ≤ t < x (the argument for −x < t < 0 is similar). Recall that S is a geodesic in D. Hence, by Lemma 7.1.10, μ(t, A x , D)
T (−x), simple geometric considerations show that T (−A x )  −A x . Therefore, by the very definition of harmonic measure and by (7.1.2), μ(t, −A x , D) = μ(0, T (−A x ), D) < μ(0, −A x , D) =

E (A x ) . 2π

(7.1.7)

Since μ(z, ·, D) is a probability measure which is absolutely continuous with respect to the Lebesgue measure, μ(z, Ix+ , D) = 1 − μ(z, Ix− , D) − μ(z, A+ x , D) − − + , D) for all z ∈ D. Taking into account that μ(t, I , D) = μ(t, I , μ(z, A− x x x D), we − , D) − μ(t, A , D)) for all z ∈ D. Thus, have, in fact, μ(t, Ix+ , D) = 21 (1 − μ(t, A+ x x by (7.1.6) and (7.1.7),

7.1 Harmonic Measures in the Unit Disc

177

1 (1 − μ(t, A x , D) − μ(t, −A x , D)) 2     1 1 E (A x ) 1 1 π/3 1 > 1− − ≥ = , − 2 2 2π 2 2 2π 6

μ(t, Ix+ , D) =

where the penultimate inequality follows since x > E (A √3 ) = π/3. 2

√ 3 2

= cos(π/6) and E (A x ) ≤ 

7.2 Harmonic Measures in Simply Connected Domains In this section we extend the notion of harmonic measure to a simply connected domain biholomorphic to D. Let us recall that given a simply connected domain Ω in C, with Ω = C, and f : D → C a univalent function with f (D) = Ω, by Proposition 3.3.2, there exists the non-tangential limit f (ζ ) := ∠ lim z→ζ f (z), for almost every ζ ∈ ∂D. This fact allows us to extend f to ∂D at almost every point of ∂D, and such extension is a measurable function (being the limit almost everywhere of the sequence of continuous functions { f ((1 − 1/n)ζ )}, ζ ∈ ∂D). Therefore, given a Borel set A ⊂ ∂∞ Ω, the set f −1 (A) := {ζ ∈ ∂D : f (ζ ) exists and f (ζ ) ∈ A}

(7.2.1)

is Borel. Remark 7.2.1 Notice that by Lehto-Virtanen’s Theorem 3.3.1, the set f −1 (A) defined in (7.2.1) coincides with {ζ ∈ ∂D : there is a continuous curve γ in D ending at ζ such that

lim f (z) exists and belongs to A}.

γ z→ζ

Definition 7.2.2 Let Ω be a simply connected domain, Ω = C, and let f : D → C be a univalent function such that f (D) = Ω. Let A be a Borel set on ∂∞ Ω. We define the harmonic measure of A at z ∈ Ω relative to Ω by μ(z, A, Ω) := μ( f −1 (z), f −1 (A), D). It follows from Proposition 7.1.4(4) that this definition does not depend on the choice of the univalent function f mapping D onto Ω. Proposition 7.2.3 Let Ω be a simply connected domain, Ω = C, and let A be a Borel set on ∂∞ Ω. Then: (1) The function Ω  z → μ(z, A, Ω) is harmonic and 0 ≤ μ(z, A, Ω) ≤ 1 for z ∈ Ω.

178

7 Harmonic Measures and Bloch Functions

(2) For each z ∈ Ω, μ(z, ·, Ω) is a probability measure on ∂∞ Ω. (3) The harmonic measure is conformally invariant in the following sense: if Ω is a Jordan domain, f : Ω → Ω˜ is a biholomorphism, B is a Borel subset of ∂∞ Ω˜ and f −1 (B) := {ζ ∈ ∂∞ Ω : there is a continuous curve γ in Ω ending at ζ such that

lim f (z) exists and belongs to B},

γ z→ζ

then f −1 (B) is a Borel set and ˜ z ∈ Ω. μ(z, f −1 (B), Ω) = μ( f (z), B, Ω), (4) If A ⊂ ∂∞ Ω is a Borel set and P ⊂ A is a countable set, then μ(z, A, Ω) = μ(z, A \ P, Ω) for all z ∈ Ω. Proof (1) and (2) follow immediately from Proposition 7.1.4 and the very definition of harmonic measure. (3) If Ω is a Jordan domain, by Theorem 4.3.3, the Riemann map g : D → Ω extends as a homeomorphism. Moreover, f ◦ g : D → Ω˜ is a Riemann map. Hence, by definition of harmonic measure, ˜ = μ(g −1 (z), ( f ◦ g)−1 (B), D), μ( f (z), B, Ω) while,

μ(z, f −1 (B), Ω) = μ(g −1 (z), g −1 ( f −1 (B)), D).

Therefore, we have to show that ( f ◦ g)−1 (B) = g −1 ( f −1 (B)). Let σ ∈ ( f ◦ g)−1 (B). By definition, this means that the non-tangential limit p := ∠ lim z→σ f (g(z)) exists and p ∈ B. Consider the continuous curve γ (t) := g(tσ ), t ∈ (0, 1). Since g extends continuously at σ , limt→1 γ (t) = g(σ ) ∈ ∂∞ Ω exists and and limt→1 f (γ (t)) = limt→1 f (g(tσ )) = p. Therefore, g(σ ) ∈ f −1 (B). Hence, σ ∈ g −1 ( f −1 (B)). On the other hand, if σ ∈ g −1 ( f −1 (B)), then, by definition, lim z→σ g(z) = g(σ ) ∈ f −1 (B). Hence, by definition of f −1 (B), there exists a continuous curve γ : (0, 1) → Ω such that limt→1 γ (t) = g(σ ) and p := limt→1 f (γ (t)) exists and p ∈ B. Since g is a homeomorphism, Γ˜ := g −1 ◦ γ : (0, 1) → D is a continuous curve ending at σ . Hence, lim ( f ◦ g)(Γ˜ (t)) = lim f (γ (t)) = p.

t→1

t→1

Therefore, by Theorem 3.3.1, f ◦ g has non-tangential limit at σ and this limit is p ∈ B. In other words, σ ∈ ( f ◦ g)−1 (B). (4) Since μ(z, ·, Ω) is a measure,

7.2 Harmonic Measures in Simply Connected Domains

179

μ(z, A, Ω) = μ(z, A \ P, Ω) + μ(z, P, Ω). It is enough to show that μ(z, P, Ω) ≡ 0. To this aim, let f : D → Ω be a surjective univalent map. We claim that f −1 (P) has zero measure, and hence μ(z, P, Ω) = μ( f −1 (z), f −1 (P), D) ≡ 0, since μ( f −1 (z), ·, D) is absolutely continuous with respect to the Lebesgue measure. Indeed, f −1 (P) is the disjoint union of countably many sets given by f −1 ( p), p ∈ P. Since by Proposition 3.3.2 each of the sets f −1 ( p) has zero measure, it  follows that f −1 (P) has zero measure. Example 7.2.4 Let Ω = {z ∈ D : Im z > 0}. We want to compute μ(z, [−1, 1], Ω). Let C(z) = 1+z be the Cayley transform with respect to 1. It maps the unit disc onto 1−z the right half-plane. A simple computation shows that C(Ω) = {w ∈ C : Re w > 0, Im w > 0} and C([−1, 1]) = [0, +∞) ∪ {∞}. Hence, Ω  z → −iC(z)2 ∈ H is a biholomorphism from Ω onto H which maps [−1, 1] onto −i[0, +∞) ∪ {∞}. Therefore, k(z) := C −1 (−iC 2 (z)) is a univalent function from Ω onto D which maps [−1, 1] onto ∂D− . Thus, by Proposition 7.2.3(3) and Example 7.1.7, for z ∈ Ω, μ(z, [−1, 1], Ω) = μ(k(z), ∂D− , D)     π 1 π 1 π − Arg −iC(z)2 = − Arg C(z)2 + = π 2 π 2 2 1 π π = − 2Arg (C(z)) + π 2   2 2 π 1+z = − Arg . π 2 1−z Example 7.2.5 Fix R > 0 and the let Γ1 := {is : s ≥ R} and Γ2 := {is : |s| ≥ R}. sends conformally H onto D and Γ1 onto the arc A in ∂D The map H  w → w−1 w+1 i R−1 that joins 1 with 1+i R in the upper half-plane. Hence, by Proposition 7.2.3(3) and Example 7.1.6 μ(1, Γ1 , H) = μ(0, A, D) =

  iR −1 1 Arg . 2π 1+iR

Moreover, denoting Γ1 = {is : s ≤ −R}, μ(1, Γ2 , H) = μ(1, Γ1 ∪ Γ1 , H) = μ(1, Γ1 , H) + μ(1, Γ1 , H)   iR −1 1 . = 2μ(1, Γ1 , H) = Arg π 1+iR In particular, if R > 1 then

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7 Harmonic Measures and Bloch Functions

μ(1, Γ1 , H) =

1 arctan 2π



2R R2 − 1

 and μ(1, Γ2 , H) =

1 arctan π



 2R . R2 − 1

Example 7.2.6 Let Ω = {w ∈ C : Im w > 0} and  two real numbers a < b.  w−btake is harmonic and The map v : Ω → R given by v(w) := π1 Im log w−a lim v(w) = 1 if x ∈ (a, b) and

w→x

lim v(w) = 0 if x ∈ R \ [a, b].

w→x

Let g : D → Ω be the biholomorphism given by g(z) := i 1+z . Hence, μ(w, (a, b), 1−z Ω) = μ(C −1 (w), C −1 ((a, b)), D). In particular, by Proposition 7.1.4(2), μ(w, (a, b), Ω) → 1 as w → x ∈ (a, b) and μ(w, (a, b), Ω) → 0 as w → x ∈ R \ [a, b]. Therefore, by Proposition 7.1.2, for all w ∈ Ω,     w−b w−b 1 1 μ(w, (a, b), Ω) = v(w) = Im log = Arg . π w−a π w−a

(7.2.2)

Notice that π μ(w, (a, b), Ω) measures the angle between the segments [w, a] and [w, b] (see Fig. 7.2). Moreover, since μ(w, ·, Ω) is a probability measure, μ(w, R \ (a, b), Ω) = 1 −

  w−b 1 Arg , for all w ∈ Ω. π w−a

Following a similar idea, using Ω  w →

1 Arg(w), π

(7.2.3)

we obtain that, if a ∈ R,

1 Arg(w − a), for all w ∈ Ω, π 1 μ(w, [a, +∞), Ω) = 1 − Arg(w − a), for all w ∈ Ω. π

μ(w, (−∞, a], Ω) =

Fig. 7.2 Harmonic measures and angles

(7.2.4)

7.2 Harmonic Measures in Simply Connected Domains

181

Using the previous example one can characterize the angle of convergence of a sequence to a boundary point in terms of harmonic measures: Proposition 7.2.7 Let σ ∈ ∂D and ξ ∈ ∂D \ {σ }. Let J + be the open arc in ∂D that goes clockwise from ξ to σ and let J − = ∂D \ J + be the open arc in ∂D that goes counterclockwise from ξ to σ . Let {z n } be a sequence in D converging to σ . Let k ∈ (0, 1). Then the following are equivalent: (1) limn→∞ Arg(1 − σ z n ) = π k − π2 , (2) limn→∞ μ(z n , J + , D) = 1 − k, (2) limn→∞ μ(z n , J − , D) = k. is a biholomorphism from D onto Proof The Möbius transformation T (z) := i ξξ +z −z Ω := {w ∈ C : Im w > 0} and T (J + ) = (a, +∞), T (J − ) = (−∞, a), where a := T (σ ). Since T is conformal at σ , we see that limn→∞ Arg(1 − σ z n ) = π k − π2 for some 0 < k < 1 if and only if limn→∞ Arg(T (z n ) − a) = π k. Indeed, if 1 − σ z n = ρn eiθn , with limn→∞ ρn = 0 and limn→∞ θn = θ := π k − π2 , then

 2iρn eiθn 2 − 2Re (ξ σ ) − ρn e−iθn (1 − σ ξ ) . T (z n ) − a = |ξ − σ (1 − ρn eiθn )|2 |ξ − σ |2 Thus

  Arg(T (z n ) − a) = Arg ieiθn 2 − 2Re (ξ σ ) − ρn e−iθn (1 − σ ξ ) and limn→∞ Arg(T (z n ) − a) = Arg(ieiθ ) = π k. Taking into account Proposition 7.2.3(4), the result follows then from (7.2.4) and Proposition 7.2.3(3).  Example 7.2.8 In this example we compute the harmonic measure of a slit halfplane. Namely, we compute μ(w, [1 + ∞), H \ [1, +∞))  when w ∈ H \ [1, +∞). To this end, consider the Joukowski function J (w) = 21 w + w1 . If w = r eiθ , with r > 0 and 0 ≤ θ ≤ π/2, then J (w) =

    1 1 1 1 cos(θ ) r + + sin(θ ) r − i. 2 r 2 r

This implies that the points of ∂D in the first quadrant {w ∈ C : Re w > 0, Im w > 0} are mapped to the segment (0, 1), the points of r ∂D, with r > 1, in the first quadrant are mapped to the arc of the ellipse x2 y2 + =1 1/2 (2 (r + 1/r )) (2 (r − 1/r ))1/2 in the first quadrant and, finally, the points of r ∂D, with 0 < r < 1, in the first quadrant are mapped to the arc of the same ellipse in the fourth quadrant. Thus, J maps con-

182

7 Harmonic Measures and Bloch Functions

formally the first quadrant onto H \ [1, +∞). In addition, J ((0, +∞)) = [1, +∞) and J ((0, +∞)i) = iR. Therefore, composing J with the principal branch of the square root we see that the function H (w) = 21 w1/2 + w11/2 sends conformally the upper-half plane onto H \ [1, +∞), H ((0, +∞)) = (1, +∞) and H ((−∞, 0)) = iR. Thus, using (7.2.4) and the conformal invariance of harmonic measures, we deduce that, for all w ∈ H \ [1, +∞), μ(w, [1, +∞), H \ [1, +∞)) = μ(H −1 (w), [0, +∞), {z ∈ C : Im z > 0})   1 = 1 − Arg H −1 (w) . π (7.2.5) If we take 0 < x < 1, then H −1 (x) = ei2 arccos(x) and   1 Arg H −1 (x) π 2 2 = 1 − arccos(x) = arcsin(x). π π

μ(x, [1, +∞), H \ [1, +∞)) = 1 −

(7.2.6)

Example 7.2.9 Let λ > 0 and consider the strip Sπ/λ and the arc Γ = i[R, +∞) for some R ∈ R. The map w → exp (iλw) sends conformally Sπ/λ onto the upper halfplane and Γ onto the segment (0, e−λR ]. Therefore, from (7.2.2) and since harmonic measures are conformal invariant, μ(w, Γ, Sπ/λ ) = μ(exp (iλw) , [0, e−λR ], {w ∈ C : Im w > 0})   1 = Arg 1 − e−λ(R+iw) , for all w ∈ Sπ/λ . π

(7.2.7)

In particular, if w ∈ (0, π/λ),   e−λR sin(λw) 1 μ(w, Γ, Sπ/λ ) = arctan π 1 − e−λR cos(λw) (7.2.8)   1 sin(λw) sin(λw) 1 ≤ e−λR . ≤ arctan e−λR π 1 − cos(λw) π 1 − cos(λw) Harmonic measures enjoy a monotonicity property with respect to domains: Proposition 7.2.10 Let Ω1 and Ω2 be two simply connected domains in C. If Ω1 ⊂ Ω2 , Ω2 = C, and A ⊂ ∂∞ Ω1 ∩ ∂∞ Ω2 is a Borel set, then μ(z, A, Ω1 ) ≤ μ(z, A, Ω2 ),

z ∈ Ω1 .

Proof Let f j , j = 1, 2, map D conformally onto Ω j . Let φ := f 2−1 ◦ f 1 . Note that φ is a univalent self-map of D and φ ◦ f 1−1 = f 2−1 . Take ζ ∈ f 1−1 (A). By the very definition, a := limr →1 f 1 (r ζ ) ∈ A. By Proposition 3.3.3, there exists ζ˜ := limr →1 f 2−1 ( f 1 (r ζ )) = limr →1 φ(r ζ ) ∈ ∂D. Clearly, limr →1 f 2 (φ(r ζ )) = a. Then,

7.2 Harmonic Measures in Simply Connected Domains

183

using again Proposition 3.3.3, we deduce ∠ lim z→ζ˜ f 2 (z) = a. In particular, ζ˜ ∈ f 2−1 (A). Hence, by Proposition 7.1.8, μ(z, A, Ω1 ) = μ( f 1−1 (z), f 1−1 (A), D) ≤ μ(φ( f −1 (z)), f 2−1 (A), D) = μ( f 2−1 (z), f 2−1 (A), D) = μ(z, A, Ω2 ), 

and we are done.

In the sequel we need the following topological fact, whose proof is based, as the proof of Jordan’s Theorem, on Janiszewski’s Theorem (see, e.g. [105, p. 31]): Theorem 7.2.11 Let Ω ⊂ C be a simply connected domain and let Γ be a Jordan arc such that (1) there exists p ∈ C∞ such that Γ ∩ ∂∞ Ω = { p}; (2) Γ \ { p} ⊂ Ω. Then Ω \ Γ is simply connected. Harmonic measures are conformally invariant under biholomorphisms between slit domains: Lemma 7.2.12 Let Ω, Ω˜  C be two simply connected domains. Let Γ, Γ˜ be Jordan arcs such that (1) there exist p, p˜ ∈ C∞ such that Γ ∩ ∂∞ Ω = { p} and Γ˜ ∩ ∂∞ Ω˜ = { p}; ˜ ˜ (2) Γ \ { p} ⊂ Ω and Γ˜ \ { p} ˜ ⊂ Ω. ˜ Then for Suppose f : Ω → Ω˜ is a biholomorphism such that f (Γ ∩ Ω) = Γ˜ ∩ Ω. every z ∈ Ω \ Γ , μ(z, Γ, Ω \ Γ ) = μ( f (z), Γ˜ , Ω˜ \ Γ˜ ). Proof By Theorem 7.2.11, Ω \ Γ and Ω˜ \ Γ˜ are simply connected. Since Γ \ (Γ ∩ Ω) is one point, it follows from Proposition 7.2.3(4) that for all z ∈ Ω \Γ, μ(z, Γ, Ω \ Γ ) = μ(z, Γ ∩ Ω, Ω \ Γ ), and, similarly, for all z ∈ Ω˜ \ Γ˜ , ˜ Ω˜ \ Γ˜ ). μ(z, Γ˜ , Ω˜ \ Γ˜ ) = μ(z, Γ˜ ∩ Ω, Therefore, it is enough to show that for every z ∈ Ω \ Γ , ˜ Ω˜ \ Γ˜ ). μ(z, Γ ∩ Ω, Ω \ Γ ) = μ( f (z), Γ˜ ∩ Ω, Let g : D → Ω \ Γ be a biholomorphism. By hypothesis, g˜ := f ◦ g : D → Ω˜ \ Γ˜ is a biholomorphism. Thus, by definition of harmonic measure, for all z ∈ Ω \ Γ ,

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7 Harmonic Measures and Bloch Functions

μ(z, Γ ∩ Ω, Ω \ Γ ) = μ(g −1 (z), g −1 (Γ ∩ Ω), D), and

˜ Ω˜ \ Γ˜ ) = μ(g˜ −1 ( f (z)), g˜ −1 (Γ˜ ∩ Ω), ˜ D) μ( f (z), Γ˜ ∩ Ω, ˜ D). = μ(g −1 (z), ( f ◦ g)−1 (Γ˜ ∩ Ω),

˜ Hence, we are left to show that g −1 (Γ ∩ Ω) = ( f ◦ g)−1 (Γ˜ ∩ Ω). Taking into account that f is univalent on Ω, we see that if x ∈ g −1 (Γ ∩ Ω) then ˜ While, if x ∈ ( f ◦ g)−1 (Γ˜ ∩ Ω) ˜ then ∠ lim z→x f (g(z)) ∠ lim z→x f (g(z)) ∈ Γ˜ ∩ Ω. ˜ hence, = ζ ∈ Γ˜ ∩ Ω, ∠ lim g(z) = ∠ lim f −1 ( f (g(z))) = f −1 (ζ ) ∈ Γ ∩ Ω. z→x

z→x

˜ and we are done. Namely, g −1 (Γ ∩ Ω) = ( f ◦ g)−1 (Γ˜ ∩ Ω),



Now, we can state and prove a basic result about harmonic measures which we need in our study: Theorem 7.2.13 Let Γ be a Jordan arc in D connecting a point ζ ∈ D to 1 such that 0 ∈ / Γ . Let Γ be the geodesic in D whose closure joins ζ and 1 and assume that 0∈ / Γ . Also, let π : D → (−1, 1) be the hyperbolic projection on (−1, 1) and let K be the geodesic in D orthogonal to (−1, 1) and containing π(ζ ). Finally, let γ be the open arc in ∂D joining K ∩ ∂D and 1 and not containing −1 (see Fig. 7.3). Then μ(0, Γ, D \ Γ ) >

1 |ζ − 1| 1 μ(0, Γ , D \ Γ ) > μ(0, γ , D) > arcsin . (7.2.9) 2 π 2

Fig. 7.3 Construction of arcs in Theorem 7.2.13

7.2 Harmonic Measures in Simply Connected Domains

185

Proof We assume that Im ζ ≥ 0, the other case being similar. Let us start by showing that μ(0, Γ, D \ Γ ) > 21 μ(0, Γ , D \ Γ ). Let λ := 1−ζ and f (z) := λTζ (z), z ∈ D, ζ −1

ζ −z , z ∈ D. Note that f ∈ where Tζ is the automorphism of the unit disc Tζ (z) := 1−ζ z Aut(D), f (1) = 1 and f (ζ ) = 0. Let Γ1 := f (Γ ). Note that Γ1 is a Jordan arc connecting 0 to 1. Let β1 = f (0). By Lemma 7.2.12,

μ(0, Γ, D \ Γ ) = μ(β1 , Γ1 , D \ Γ1 ). Since D \ Γ1 is simply connected by Theorem 7.2.11 and does not contain 0, one can define a holomorphic branch g of the square root in D \ Γ1 , in such a way that g(∂D \ {1}) = ∂D+ := ∂D ∩ {w ∈ C : Im w > 0}. Therefore, k := g ◦ f : D \ Γ → C is a univalent map such that ∂k(D \ Γ ) is the union of ∂D+ and a Jordan arc Γ2 passing through −1, 0, 1. In particular, k(D \ Γ ) is a Jordan domain and (k −1 )−1 (Γ ) = Γ2 in the sense of Proposition 7.2.3(3). Moreover, by definition, k 2 (z) = f (z) for all z ∈ D\Γ. Let ∂D− := {w ∈ ∂D : Im w ≤ 0} and β2 = k(0). Notice that β22 = f (0). Then, by Proposition 7.2.3(3) (applied to k −1 : k(D \ Γ ) → D \ Γ ), we have μ(0, Γ, D \ Γ ) = μ(β2 , Γ2 , k(D \ Γ )). Moreover, taking into account that μ(β2 , ·, k(D \ Γ )) is a probability measure and Proposition 7.2.10, we have μ(0, Γ, D \ Γ ) = μ(β2 , Γ2 , k(D \ Γ )) = 1 − μ(β2 , ∂D+ , k(D \ Γ )) ≥ 1 − μ(β2 , ∂D+ , D) = μ(β2 , ∂D− , D). Now if Im β2 ≤ 0, by Example 7.1.7, we have μ(β2 , ∂D− , D) =

1 π



  1 π 1 + β2 1 ≥ > μ(0, Γ , D \ Γ ). − Arg 2 1 − β2 2 2

In case Im β2 > 0, we argue as before, setting Γ1 := f (Γ ) = [0, 1] (where the last equality follows since f ∈ Aut(D) and automorphisms map geodesics onto √ geodesics). In √ particular, g(z) = z is defined in D \ [0, 1], where the branch √ is chosen so that −1 = i, and g(D \ Γ1 ) = {w ∈ D : Im w > 0}. Since, k := f (z), in this case we have (k −1 )−1 ([−1, 1]) = Γ and Im k(0) > 0. Since k 2 (0) = f (0), it follows that β2 = k(0). Hence, arguing as before, and setting Ω = D ∩ {w ∈ C : Im w > 0}, we have μ(0, Γ , D \ Γ ) = μ(β2 , [−1, 1], Ω) =

2 π



  π 1 + β2 , − Arg 2 1 − β2

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7 Harmonic Measures and Bloch Functions

where the last equality follows from Example 7.2.4. Therefore, again by Example 7.1.7,    1 1 + β2 1 π = μ(β2 , [−1, 1], Ω) − Arg μ(β2 , ∂D , D) = π 2 1 − β2 2 1 = μ(0, Γ , D \ Γ ). 2 −

Hence, also in this case, μ(0, Γ, D \ Γ ) > 21 μ(0, Γ , D \ Γ ). Let us see that 21 μ(0, Γ , D \ Γ ) > μ(0, γ , D). In this case, we perform the computations in the right half-plane. To this end consider the Cayley transform which carries D onto the right half-plane H. There is a positive real C(z) = 1−z 1+z number r such that C sends K onto the semicircle {w ∈ H : |w| = r }, γ onto the interval γ1 = (−ir, 0) and Γ onto a circular geodesic arc Γ3 connecting 0 to the point C(ζ ) = r eiφ with − π2 < φ ≤ 0. With this notation, by the conformal invariance of harmonic measures (see Proposition 7.2.3(3) and Lemma 7.2.12), we are left to prove that 1 μ(1, Γ3 , H \ Γ3 ) > μ(1, γ1 , H). (7.2.10) 2 By Example 7.2.6, μ(1, γ1 , H) = μ(i, (0, r ), iH) =

1 1 Arg(1 + ir ) = arctan(r ). π π

r 1 + i tan(φ) sends conformally H On the other hand, the map w → T (w) := cos(φ) w onto itself and Γ3 onto the half-line [1, ∞). Hence, by Lemma 7.2.12 and (7.2.5),

μ(1, Γ3 , H \ Γ3 ) = μ(T (1), [1, +∞), H \ [1, +∞))   r + i tan(φ), [1, +∞), H \ [1, +∞) =μ cos(φ)    r 1 −1 + i tan(φ) , = 1 − Arg H π cos(φ) where H (w) =

1 2

 1/2 w +

1 w1/2



(see Example 7.2.8). In order to simplify the expo r + i tan(φ) = ρe2iα . sition, take α ∈ (−π/2, π/2) and ρ > 0 such that H −1 cos(φ) Hence, (7.2.10) is equivalent to π 2

Notice that cot(α) = tan From the equation

  2α 1− > arctan(r ). π

π  1− 2

2α π

(7.2.11)

 , so that we have to prove that cot(α) > r .

7.2 Harmonic Measures in Simply Connected Domains

187

 1  1/2 iα r + i tan(φ) = ρ e + ρ −1/2 e−iα , cos(φ) 2 r = 21 cos(α)(ρ 1/2 + ρ −1/2 ) and tan(φ) = we deduce that cos(φ) Therefore, a simple calculation shows

r2 cos2 (φ) cos2 (α) and

−

1 2

sin(α)(ρ 1/2 − ρ −1/2 ).

tan2 (φ) =1 sin2 (α)

cos2 (α) 2 sin (φ) + cos2 (α) cos2 (φ) sin2 (α)  cos2 (α)  2 sin (φ) + sin2 (α) cos2 (φ) < cot 2 (α), = 2 sin (α)

r2 =

proving that (7.2.11)—and hence (7.2.10)—holds. ˜ Finally, note that by (7.1.3), μ(0, γ , D) = π1 arcsin |ζ −1| , where ζ˜ = K ∩ γ . So 2 ˜ that the last inequality of (7.2.9) is indeed equivalent to |ζ − 1| > |ζ − 1|. ˜ . Then In order to prove this last inequality, consider the Cayley map C(z) = 1+z 1−z ˜ C(K ) is the semicircle {w ∈ H : |w| = R} for some R > 0. Hence, K can be iθ −1 parameterized by β(θ ) = C˜ −1 (Reiθ ) = Re , with θ ∈ (−π/2, π/2). Now, there is Reiθ +1 −1 iθ ˜ ζ˜ ) = Ri, θ ∈ (−π/2, π/2) such that ζ = C˜ (Re ) and, taking into account that C( we have 4 4 = 2 |ζ − 1|2 = iθ 2 |Re + 1| R + 2R cos(θ ) + 1 4 = |ζ˜ − 1|2 , < 2 R +1 

and we are done.

7.3 Bloch Functions We devote this section to the so-called Bloch functions, a large family of holomorphic maps from the unit disc into C satisfying non-trivial maximum and minimum principles. As a byproduct of such principles, Bloch functions satisfy a Lehto-Virtanen type Theorem and nice estimates of the diameters of the curves in the image. Definition 7.3.1 A holomorphic function f : D → C is a Bloch function if  f B := sup{(1 − |z|2 )| f (z)| : z ∈ D} < +∞. Note that  · B defines a seminorm and Bloch functions form a Banach space B where the norm is given by | f (0)| +  f B .

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7 Harmonic Measures and Bloch Functions

Proposition 7.3.2 If f : D → C is a Bloch function and T is an automorphism of the disc, then f ◦ T belongs to the Bloch space and  f ◦ T B =  f B . Proof By Proposition 1.2.2, |T (z)| =

1−|T (z)|2 . 1−|z|2

Therefore

 f ◦ T B = sup{(1 − |z|2 )| f (T (z))||T (z)| : z ∈ D} = sup{(1 − |T (z)|2 )| f (T (z))| : z ∈ D} = sup{(1 − |z|2 )| f (z)| : z ∈ D} =  f B , 

and we are done. The following results give examples of Bloch functions.

Proposition 7.3.3 Let f : D → C be univalent and a ∈ / f (D). Then, D  z → log( f (z) − a) is a Bloch function and  log( f − a)B ≤ 4. Proof By Theorem 3.4.9, for all z ∈ D d δ f (D) ( f (z)) | f (z)| (1 − |z| ) log( f (z) − a) = (1 − |z|2 ) ≤4 ≤ 4. dz | f (z) − a| | f (z) − a| 2

Therefore  log( f − a)B ≤ 4, 

and we are done.

Proposition a ∈ / f (D) and ξ ∈ ∂D. Then, z →  7.3.4 Let f : D → C be univalent,  f (z)−a f (z)−a log z−ξ is a Bloch function and  log z−ξ B ≤ 8. Proof Since ξ ∈ / D, by Proposition 7.3.3 we have  log( f − a)B ≤ 4 and  log(idD − ξ )B ≤ 4. Bearing in mind that  log

f (z) − a z−ξ



for some k ∈ Z, we have

and we are done.

= log( f (z) − a) − log(z − ξ ) + 2kπi, z ∈ D,

 

f −a

log

≤ 8,

idD − ξ B 

Proposition 7.3.5 Let f : D → C be univalent. Then, log f is a Bloch function and  log f B ≤ 6.

7.3 Bloch Functions

189

Proof Using Corollary 3.4.3, we have (1 − |z|2 ) f (z) − 2z ≤ 4. f (z) Therefore

 log f B ≤ 6, 

and we are done.

Now we turn our attention to some maximum/minimum principles for Bloch functions. Theorem 7.3.6 Let C be a circle or a line intersecting ∂D at two different points and let G be a domain such that G ⊂ D and G ⊂ D \ C. Let γ ∈ (0, π ) be the angle formed by C and ∂D in the connected component of D \ C which contains G (see Fig. 7.4). If f : D → C belongs to the Bloch space let c :=

γ  f B . 2 sin(γ )

Then: (1) If there exists a ≥ ec such that | f (z)| ≥ a, for all z ∈ ∂G \ C, then | f (z)| ≥ ae , for all z ∈ G. (2) If there exists 0 < a ≤ c such that | f (z)| ≤ a, for all z ∈ ∂G \ C, then | f (z)| ≤

2c , for all z ∈ G. log(c/a) + 1

(3) dist( f (z), f (∂G \ C)) ≤ ec, for all z ∈ G. Fig. 7.4 Domain determined by the circle C in Theorem 7.3.6

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7 Harmonic Measures and Bloch Functions

Proof We can assume that f is not constant, for otherwise the result is trivially true. We may also assume that C intersects ∂D at the points ±i and G is contained in the connected component of D \ C whose closure contains 1. Indeed, let M be an automorphism of the unit disc which maps the two points of C ∩ ∂D to ±i (see Proposition 1.2.2). Hence, M(C) is a circle or a line intersecting ∂D at ±i and divides D into two connected components, call U the one whose closure contains 1. If M(G) ⊂ U , we replace f with f ◦ M −1 . If M(G) ⊂ D \ U , we replace f (z) with f ((M −1 (−z))), z ∈ D. Since M preserves angles and the Bloch semi-norm by Proposition 7.3.2, there is no loss of generality in making the previous assumption. z . Note that T maps i Now, let T : D → H be the Cayley transform T (z) := 1+i 1−i z to 0 and −i to ∞ and T ((−1, 1)) = {eiθ : θ ∈ (−π/2, π/2)}. Also, given θ ∈ (0, π ), denote by C(θ ) the unique circle or line intersecting ∂D at ±i such that the angle formed by C(θ ) and ∂D in the connected component of D \ C(θ ) whose closure contains 1 is θ . Then, T (C(θ )) = {ρei(π/2−θ) : ρ > 0}. In particular, if we define u : D → (0, π ) as u(z) =

π − Arg(T (z)), 2

it follows that u is harmonic and u(z) = θ for all z ∈ C(θ ) and θ ∈ (0, π ). Moreover, u(x) =

  π π 2x = − 2arctg(x), x ∈ (−1, 1). − arctg 2 2 1−x 2

Note also that for every θ ∈ (0, π ) there exists a unique x(θ ) ∈ (−1, 1) such that C(θ ) ∩ (−1, 1) = {x(θ )}. Since u(x(θ )) = θ , it follows that    1 + i x(θ ) 1 + i x(θ ) π − Arg = cos Arg sin θ = sin 2 1 − i x(θ ) 1 − i x(θ ) 2 1 − x(θ ) 1 + i x(θ ) = = Re . 1 − i x(θ ) 1 + x(θ )2 

(7.3.1)

Finally, note that, given any θ ∈ (0, π ) and z 0 ∈ C(θ ), there exists an automorphism Q z0 ,θ of D such that Q z0 ,θ (±i) = ±i, Q z0 ,θ (z) = x(θ ) and Q z0 ,θ (C(η)) = C(η) for all η ∈ (0, π ). In fact, Q z0 ,θ (w) := T −1 ( |z10 | T (w)). Moreover, consider the strictly increasing function x ϕ(x) := x log , x ∈ [a/e, +∞). a Note that ϕ(a/e) = −a/e and ϕ(a) = 0. Since lim x→+∞ ϕ(x) = +∞ and c > 0, there exists b > a such that ϕ(b) = c. Moreover, since a ≥ ec, there exists b∗ ∈ [a/e, a) such that ϕ(b∗ ) = −c. With all these preliminary considerations at hand, we can now prove (1). By hypothesis, | f (z)| ≥ a for all z ∈ ∂G \ C. Since b∗ ≥ a/e, it is enough to check that | f (z)| ≥ b∗ , for z ∈ G.

7.3 Bloch Functions

191

Assume this is not the case. Hence, there exists z ∈ G such that | f (z)| < b∗ . Since by hypothesis | f (z)| ≥ a > b∗ on ∂G \ C, it follows that the compact set K b∗ := {z ∈ G : | f (z)| = b∗ } is non-empty and K b∗ ∩ G = ∅. Let z 0 ∈ K b∗ be such that γ := u(z 0 ) ≤ u(z) for all z ∈ K b∗ . Since K b∗ ∩ G = ∅, we have 0 < γ < γ . Moreover, up to replace f with f ◦ Q −1 z 0 ,γ (as we remarked at the beginning of the proof, composition with automorphisms does not change angles and the Bloch norm), we can assume that x0 := z 0 ∈ (−1, 1). Note that x0 ∈ G because | f (x)| > b∗ for all x ∈ (−1, 1) ∩ (∂G \ C). Let C := C(γ ). The curve C divides D into two connected components. Call r U the one whose closure contains 1 and let G := G ∩ U r . By definition of x0 , we have that | f (z)| > b∗ for all z ∈ G . Let Z := {z ∈ D : f (z) = 0}. Note that Z ∩ G = ∅. Let v : G → R be defined by v(z) :=

cu(z) + log | f (z)|. b∗ γ

The function v is harmonic in G and continuous in G . By the Maximum Principle for harmonic functions, −v has a maximum on ∂G —hence v has a minimum on ∂G . On the one hand, if z ∈ ∂G \ C ⊂ ∂G \ C, then v(z) ≥ log | f (z)| ≥ log a. On the other hand, if z ∈ ∂G ∩ C , v(z) =

−ϕ(b∗ ) cu(z) + log | f (z)| ≥ + log b∗ = log a. b∗ γ b∗

Therefore, v(z) ≥ log a = v(x0 ) for all z ∈ G . Since x0 ∈ G, there exists δ > 0 such that (x0 , x0 + δ) ⊂ G . Thus, v(x) ≥ log a = v(x0 ) for all x0 < x < x0 + δ and hence v (x0 ) ≥ 0. Then 0 ≤ v (x0 ) = −

1 2c + Re ∗ b γ 1 + x02



f (x0 ) f (x0 )

 ≤−

2c 1  f B + . 2 ∗ b γ 1 + x0 (1 − x02 )b∗

Hence, taking into account (7.3.1), γ 1 + x02 γ  f B γ  f B = c ≤  f B = 2 sin(γ ) 2 2 sin(γ ) 1 − x02 which contradicts γ < γ , since the function x → x/ sin(x) is strictly increasing in (0, π ). The proof of (2) is quite similar. By hypothesis, | f (z)| ≤ a for all z ∈ ∂G \ C. Since c = ϕ(b) = b log (b/a), we have

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7 Harmonic Measures and Bloch Functions

       1 b 1 b 1  c 1 = log ≥ log + log (log(b/a)) + 1 = log +1 . b c a 2c a 2c a Taking into account that a ≤ c, it follows that the right-hand side is positive and, therefore, it is enough to check that | f (z)| ≤ b, for z ∈ G. Assume by contradiction this is not the case. Arguing as before, we find 0 < γ < γ and x0 ∈ G ∩ (−1, 1) ∩ C(γ ) with the following properties. Let C := C(γ ). The curve C divides D into two connected components. Call U r the one whose closure contains 1 and let G := G ∩ U r . Then | f (x0 )| = b and | f (z)| ≤ b for all z ∈ G . Let Z := {z ∈ D : f (z) = 0} and let v∗ (z) := −

cu(z) + log | f (z)|. bγ

Note that v∗ : G \ Z → R is harmonic, continuous on G \ Z and v∗ (z) → −∞ as z → z 0 ∈ Z . So, by the Maximum Principle, it has a maximum on ∂G \ Z . Thus v := −v∗ has a minimum on ∂G \ Z . / Z , then On the one hand, if z ∈ ∂G \ C ⊂ ∂G \ C and z ∈ v(z) ≥ − log | f (z)| ≥ − log a. / Z, On the other hand, if z ∈ ∂G ∩ C and z ∈ v(z) =

c cu(z) − log | f (z)| ≥ − log b = − log a. bγ b

Therefore, v(z) ≥ − log a = v(x0 ) for all z ∈ G \ Z . Since x0 ∈ G and f (x0 ) = 0, there exists δ > 0 such that (x0 , x0 + δ) ⊂ G and f (x) = 0 for all x ∈ [x0 , x0 + δ). Thus, v(x) ≥ − log a = v(x0 ) for all x0 < x < x0 + δ. In particular, v (x0 ) ≥ 0, that is   2c 1 1 2c f (x0 )  f B ≤− . − Re + 0 ≤ v (x0 ) = − 2 2 bγ 1 + x0 f (x0 ) bγ 1 + x0 (1 − x02 )b Hence, taking into account (7.3.1), γ 1 + x02 γ  f B γ  f B = c ≤  f B , = 2 sin(γ ) 2 2 sin(γ ) 1 − x02 which contradicts γ < γ , since the function x/ sin(x) is strictly increasing in (0, π ). Finally, in order to prove (3), we can assume c > 0 (the case c = 0 corresponds to f constant). Let z ∈ G ∩ D and suppose that dist( f (z), f (∂G \ C)) > ec. Then | f (z) − f (ξ )| ≥ ec, for all ξ ∈ ∂G \ C. Hence, by (1), | f (z) − f (ξ )| ≥ c > 0 for  all ξ ∈ G ∩ D, contradicting z ∈ G ∩ D.

7.3 Bloch Functions

193

Corollary 7.3.7 Let C be a circle or a line intersecting ∂D at two different points ξ1 , ξ2 and let G be a domain such that G ⊂ D \ C and ∂G ⊂ D ∪ {ξ1 , ξ2 }. Let γ ∈ (0, π ) be the angle formed by C and ∂D in the connected component of D \ C which contains G. If f : D → C belongs to the Bloch space let c :=

γ  f B . 2 sin(γ )

Then: (1) If there exists a > ec such that | f (z)| ≥ a, for all z ∈ ∂G \ C, then | f (z)| ≥ ae , for all z ∈ G \ {ξ1 , ξ2 }. (2) If there exists 0 < a < c such that | f (z)| ≤ a, for all z ∈ ∂G \ C, then | f (z)| ≤

2c , for all z ∈ G \ {ξ1 , ξ2 }. log(c/a) + 1

(3) dist( f (z), f (∂G \ C)) ≤ ec, for z ∈ G \ {ξ1 , ξ2 }. Proof We prove (1), the others being similar. Up to pre-composing f with an automorphism (as in the proof of Theorem 7.3.6) we may also assume that C intersects ∂D at ±i and G is contained in the connected component of D \ C whose closure contains 1. Let x0 be the point of intersection between (−1, 1) and C and let {xn } ⊂ (x0 , 1) be a strictly decreasing sequence converging to x0 such that every Cn := (xn − x0 ) + C intersects G. For all n, let Unr be the connected component of D \ Cn whose closure contains 1 and let G n := G ∩ Unr . Let γn ∈ (0, π ) be the angle formed by Cn and ∂D γn  f B . Since a > ce, there in Unr . Note that γn → γ as n → ∞. Let cn := 2 sin(γ n) exists n 0 ∈ N such that a ≥ cn e for all n ≥ n 0 . Taking into account that ∂G n \ Cn ⊂ ∂G \ C, we have | f (z)| ≥ a on ∂G n \ Cn and a ≥ cn e for all n 0 . It follows that each connected component of G n for n ≥ n 0 satisfies the hypotheses of Theorem 7.3.6, and hence | f (z)| ≥ a/e for all z ∈ G n ,   n ≥ n 0 . Since n≥n 0 G n = G, we are done. Theorem 7.3.8 Let I be a closed arc of ∂D and let f : D → C be a Bloch function. Then there exists ξ ∈ I such that | f (r ξ ) − f (0)| ≤ 4π e

 f B , for 0 < r < 1. E (I )

Proof We can assume f is not constant, otherwise the result is trivially true. Up to replace f (z) with ( f (z) − f (0))/ f B , z ∈ D, we can assume that f (0) = 0 and  f B = 1. Suppose initially that E (I ) < π . Pre-composing f with a rotation if necessary, and by Proposition 7.3.2, we may additionally assume that I is the closed arc between eiθ and e−iθ passing through 1, for some θ ∈ (0, π/2).

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7 Harmonic Measures and Bloch Functions

Let C be the circle passing through eiθ , e−iθ and 0. Let W be the connected component of D \ C whose closure contains I and let γ be the angle formed by C and ∂D at e±iθ in W . Note that γ = π − θ . Let a := eπ > 0 and let G be the connected component of U := {z ∈ W ∩ D : 2θ | f (z)| < a} which contains (0, ε) for some small ε > 0 (recall that f (0) = 0). Let A := ∂G \ C. We claim that I ∩ ∂G = ∅. Since π −θ π ≥ , θ sin(π − θ ) γ we find a ≥ e 2 sin(γ . If the claim is false, ∂G ⊂ D and | f (z)| ≥ a, for all z ∈ A. ) Therefore, by Theorem 7.3.6(1), | f (r )| ≥ ae , for r ∈ (0, ε). Since f (0) = 0, we have a contradiction. Therefore, there exists a sequence {z n } ⊂ G such that limn→∞ z n =: ξ ∈ I ∩ ∂G. Fix r ∈ (0, 1). Fix n ∈ N. If r z n ∈ G, then

| f (r z n )| ≤ a =

eπ eπ = E . 2θ  (I )

In case r z n ∈ D \ G, since f (0) = 0, there exists s ∈ (0, r ) such that sz n ⊂ G. Since G is connected, there exists a Jordan arc γ : [0, 1] → G such that γ (0) = sz n and γ (1) = z n . Let C1 := {ρz n : ρ ∈ R}. Taking into account that C1 ∩ Γ is compact, we can define t0 := max{t ∈ [0, 1] : γ (t) = uz n , u ∈ (0, r )}, and t1 := min{t ∈ [0, 1] : γ (t) = uz n , u ∈ (r, 1]}. Note that 0 ≤ t0 < t1 ≤ 1. Then P := γ ([t0 , t1 ]) is a Jordan arc which intersects C1 in two points: γ (t0 ) = u 0 z n , with u 0 ∈ (0, r ) and γ (t1 ) = u 1 z n with u 1 ∈ (r, 1]. It turns out that J := P ∪ {ρz n : ρ ∈ [u 0 , u 1 ]} is a Jordan curve. Let G 1 be the bounded connected component of C \ J . By construction, r z n ∈ ∂G 1 , G 1 ⊂ D, G 1 ⊂ D \ C1 and ∂G 1 \ C1 ⊂ G. Note that C1 forms an angle γ1 = π/2 with ∂D in the connected component γ1  f B = π4 (since  f B = 1). Hence, of D \ C1 containing G 1 . Let c1 := 2 sin(γ 1) Theorem 7.3.6(3) implies that dist( f (r z n ), f (∂G 1 \ C1 )) ≤ ec1 = eπ . 4 Therefore, there exists p ∈ P such that, taking into account that 2 > θ , | f (r z n )| ≤ | f ( p)| + dist( f (r z n ), f (∂G 1 \ C1 )) ≤ a + ec1 =

eπ eπ eπ + ≤ . 2θ 4 θ

Taking limits in n, we have | f (r ξ )| ≤ eπ = 2eπ E (I ) . θ E In case  (I ) ≥ π , we take a closed subinterval J ⊂ I whose length is the half of that of I . Then, applying the previous result to J , we obtain that there exists

7.3 Bloch Functions

195

Fig. 7.5 Geometric elements in Theorem 7.3.9

ξ ∈ J ⊂ I such that, for all 0 < r < 1 | f (r ξ )| ≤

4eπ 2eπ = E , E (J )  (I ) 

and we are done.

Theorem 7.3.9 There exists a universal constant K 1 > 0 such that for every x ∈ √ ( 23 , 1), every f : D → C univalent and every t ∈ (−x, x), there exists ξ ∈ Ix such that | f (t) − f (z)| ≤ K 1 δΩ ( f (t)), where Ω := f (D), Ix is any of the two closed arcs in ∂D, with Ix ∩ R = ∅, determined by the closure of the geodesic through −x and x orthogonal to R, and z belongs to the geodesic in D determined by t and ξ (see Fig. 7.5). Proof Fix x. Let J be a closed arc in ∂D and g : D → C a univalent map. Recall that, by Example 7.1.6, 2π μ(0, J, D) = E (J ). By Proposition 7.3.5 and Theorem 7.3.8, there exists ξ ∈ J such that, for every 0 < s < 1     log |g (sξ )| ≤ log g (sξ ) = | log g (sξ ) − log g (0)| |g (0)| g (0) ≤

4π e · 6 12π e = . 2π μ(0, J, D) μ(0, J, D)

This inequality implies that if |g (sξ )| > |g (0)| then  12π e . |g (sξ )| ≤ |g (0)| exp μ(0, J, D)





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7 Harmonic Measures and Bloch Functions

Since the same inequality holds trivially if |g (sξ )| ≤ |g (0)|, by Theorem 3.4.9 we have for all r ∈ (0, 1) |Re g(r ξ ) − Re g(0)| ≤ r max |Re g (sξ )| ≤ max |g (sξ )| s∈[0,r ] s∈[0,r ]     12π e 12π e ≤ |g (0)| exp ≤ 4δg(D) (g(0)) exp . μ(0, J, D) μ(0, J, D) A similar argument holds for |Im g(r ξ ) − Im g(0)|, hence, for all r ∈ (0, 1), √ |g(r ξ ) − g(0)| ≤ 4 2δg(D) (g(0)) exp



 12π e . μ(0, J, D)

(7.3.2)

Now, consider t ∈ (−x, x) and let Ix be the corresponding arc in the upper halfplane (the other case is similar). Moreover, let T be an automorphism of the disc such that T (0) = t. Applying the previous result to g := f ◦ T and J := T −1 (Ix ), we find a point ξt, f := ξ ∈ J such that (7.3.2) holds for all r ∈ (0, 1). Hence, by Proposition 7.1.4 and Theorem 7.1.11,   √ 12π e |( f ◦ T )(r ξt, f ) − ( f ◦ T )(0)| ≤ 4 2δ( f ◦T )(D) (( f ◦ T )(0)) exp μ(0, J, D)   √ 12π e = 4 2δΩ ( f (t)) exp μ(t, Ix , D) √ ≤ δΩ ( f (t))4 2 exp (12π e · 6) . Since {T (r ξt, f ) : r ∈ (0, 1)}√is the geodesic in D which joins t and T (ξt, f ) ∈ Ix , we are done by taking K 1 := 4 2 exp (72π e) and ξ := T (ξt, f ).  Now we are ready to show that every asymptotic value of a Bloch function is taken as non-tangential limit at a suitable boundary point. Theorem 7.3.10 Let Γ : [a, b] → C be a Jordan arc such that Γ ([a, b)) ⊂ D and ξ := limt→b Γ (t) ∈ ∂D. If f : D → C is a Bloch function and L := limt→b f (Γ (t)) ∈ C∞ exists, then ∠ lim z→ξ f (z) = L. Proof We can assume f is not constant, for otherwise the result is trivially true. We first deal with the case L = ∞. Up to pre-composing f with a rotation and by Proposition 7.3.2, we assume ξ = 1. If C is a circle which is the boundary of a Euclidean disc D, and C intersects ∂D at two different points, we let Dom(C) := D ∩ D. For θ ∈ (0, π/2) and γ ∈ (π/2, π ), we denote by C + (θ, γ ) (respectively − C (θ, γ )) the circle passing through 1 and eiθ (resp. 1 and e−iθ ) and forming with ∂D in those two points an angle γ in Dom(C + (θ, γ )) (respect., in Dom(C − (θ, γ ))). Let Dom± (θ, γ ) := Dom(C ± (θ, γ )) and Dom(θ, γ ) := Dom+ (θ, γ ) ∩ Dom− (θ, γ ).

7.3 Bloch Functions

197

For any R > 0, there exists γ ∈ (π/2, π ) with the property that for every θ ∈ (0, π/2) there is δ > 0 such that S(1, R) ∩ {z ∈ D : |z − 1| < δ} ⊂ Dom(θ, γ ). Therefore, in order to prove the result it is enough to show that for every γ ∈ (π/2, π ) and ε > 0 there exists θ ∈ (0, π/2) such that | f (z) − L| < ε for all z ∈ Dom(θ, γ ). γ  f − LB > 0, we can take 0 < Fix γ ∈ (π/2, π ) and ε > 0. Since c := 2 sin(γ ) ε1 < min{c, ε} such that 2c  < ε. 1 + log εc1 Now, consider θ ∈ (0, π/2) close enough to 0 so that there exists a1 ∈ (a, b) with | f (Γ (t)) − L| < ε1 , t ∈ [a1 , b), for all t ∈ (a1 , b), Γ (t) ∈ Ω(θ, γ ) := Dom+ (θ, γ ) ∪ Dom− (θ, γ ), and Γ (a1 ) ∈ ∂Ω(θ, γ ). Let Γ ∗ := Γ ([a1 , b)) and denote by V + (respectively V − ) the connected component of Dom+ (θ, γ ) \ Γ ∗ (resp. Dom− (θ, γ ) \ Γ ∗ ) whose boundary contains the arc A+ := {eis : s ∈ (0, θ } (resp. A− := {eis : s ∈ (−θ, 0)}). Moreover, let G + := Dom+ (θ, γ ) \ (V + ∪ Γ ∗ ), and G − := Dom− (θ, γ ) \ − (V ∪ Γ ∗ ). We claim that Dom(θ, γ ) ⊂ G + ∪ G − ∪ Γ ∗ . Indeed, if there existed a point p in Dom(θ, γ ) belonging to V + ∩ V − , then it would be possible to connect A+ and A− with a continuous curve in Ω(θ, γ ) without intersecting Γ ∗ , which is clearly absurd. Now, G + is either empty or the countable union of connected components (G i+ ) such that G i+ ⊂ D \ C + (θ, γ ) and ∂G i+ ⊂ C + (θ, γ ) ∪ Γ ∗ ∪ {1}. Similarly for G − . Therefore, applying Corollary 7.3.7(2) to the Bloch function f − L with respect to each G i± with a := ε1 , we have that, for every z ∈ G + ∪ G − , | f (z) − L| ≤

2c 1 + log

 < ε.

In particular, this holds for every z ∈ Dom(θ, γ ).

c ε1

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7 Harmonic Measures and Bloch Functions

The case L = ∞ can be handled in a similar way using statement (1) in Corollary 7.3.7 instead of (2).  As a direct consequence of Proposition 7.3.4 and Theorem 7.3.10 we have: Corollary 7.3.11 Let f : D → C be univalent, ξ ∈ ∂D and a ∈ / f (D). If there exist a Jordan arc Γ : [a, b] → C such that Γ ([a, b)) ⊂ D and ξ = limt→b Γ (t), and L ∈ C∞ with f (Γ (t)) − a , L = lim t→b Γ (t) − ξ then L = ∠ lim z→ξ

f (z) − a . z−ξ

7.4 Diameter Distorsion for Univalent Functions In this section we prove some distortion theorems for univalent functions related to diameters of Jordan arcs. Given x ∈ (−1, 1), let us denote by Sx the (only) geodesic containing x and orthogonal to (−1, 1) at x. Theorem√7.4.1 There exists a universal constant K 2 > 0 with the following property. Let 23 < x < 1, and let C be a Jordan arc in D with end points z − ∈ S−x and z + ∈ Sx . If f : D → C is univalent then diamE ( f ([−x, x])) ≤ K 2 diamE ( f (C)). √

Proof Fix x ∈ ( 23 , 1) and let γ : [0, 1] → D be a Jordan arc such that z − = γ (0) ∈ S−x and z + = γ (1) ∈ Sx , C := γ ([0, 1]). Let ρ := diamE ( f (C)) ∈ (0, +∞). By Theorem 3.4.7 1 tanh(ω(z − , z + ))(1 − |z + |2 )| f (z + )| ≤ | f (z − ) − f (z + )| ≤ ρ. 4 A direct computation shows that ω(−x, x) ≥ 1. It follows then by Remark 6.5.4 2 . Thus, the previous that ω(z − , z + ) ≥ 1. Therefore, tanh(ω(z − , z + )) ≥ tanh 1 = ee2 −1 +1 inequality implies 4(e2 + 1) ρ. (1 − |z + |2 )| f (z + )| ≤ 2 e −1 Let p ∈ ∂ f (D) be such that | p − f (z + )| = δ f (D) ( f (z + )). Hence, by Theorem 3.4.9, | p − f (z + )| = δ f (D) ( f (z + )) ≤ (1 − |z + |2 )| f (z + )| ≤

4(e2 + 1) ρ. e2 − 1

(7.4.1)

7.4 Diameter Distorsion for Univalent Functions

199

Let now t ∈ [−x, x] and let g(z) := 1/( f (z) − p). Note that g : D → C is univalent. Let Ix+ (respectively, Ix− ) be the closed arc in ∂D contained in the upper (resp., lower) half-plane and whose end points are Sx ∩ ∂D ∩ {w ∈ C : Im w > 0} and S−x ∩ ∂D ∩ {w ∈ C : Im w > 0} (resp., Sx ∩ ∂D ∩ {w ∈ C : Im w < 0} and S−x ∩ ∂D ∩ {w ∈ C : Im w < 0}). Since 0 ∈ / g(D), we have δg(D) (g(t)) ≤ |g(t)|. Then, according to Theorem 7.3.9, there exists ξt+ ∈ Ix+ (resp., ξt− ∈ Ix− ) such that for every z belonging to the geodesic I1t (resp., I2t ) determined by t and ξt+ (resp., t and ξt− ), we have |g(z)| ≤ |g(t)| + |g(t) − g(z)| ≤ |g(t)| + δg(D) (g(t))K 1 ≤ (1 + K 1 )|g(t)|. That is, for every t ∈ [−x, x] and z ∈ I1t ∪ I2t , | f (t) − p| ≤ (1 + K 1 )| f (z) − p|.

(7.4.2)

Moreover, since I1t ∪ I2t is a cross cut for D which does not intersect S−x ∪ Sx , and C connects S−x with Sx , it follows that there exists z t ∈ C such that z t ∈ I1t ∪ I2t . Therefore, by (7.4.2) and (7.4.1), | f (t) − p| ≤ (1 + K 1 )| f (z t ) − p| ≤ (1 + K 1 )(| f (z t ) − f (z + )| + | f (z + ) − p|)   4(e2 + 1) 4(e2 + 1) ρ) = (1 + K 1 ) 1 + 2 diamE f (C). ≤ (1 + K 1 )(ρ + 2 e −1 e −1

Finally, if t1 , t2 ∈ [−x,x] are such that diamE ( f ([−x, x])) = | f (t1 ) − f (t2 )|, set2 +1) , from the previous inequality we have ting K 2 := 2(1 + K 1 ) 1 + 4(ee2 −1 diamE ( f ([−x, x])) = | f (t1 − f (t2 )| ≤ | f (t1 ) − p| + | f (t2 ) − p| ≤ K 2 diamE f (C),



and we are done.

Theorem 7.4.2 There exists a universal constant K 2 > 0 (in fact, the same as in Theorem 7.4.1) with the following property. If ξ1 , ξ2 ∈ ∂D, ξ1 = ξ2 , γ : (0, 1) → D is a continuous injective curve such that limt→0 γ (t) = ξ1 and limt→1 γ (t) = ξ2 , C = γ ((0, 1)), and f : D → C is univalent, then diamE ( f (S)) ≤ K 2 diamE ( f (C)), where S denotes the (unique) geodesic in D whose closure contains ξ1 and ξ2 . Proof Let T be an automorphism of D such that T (ξ1 ) = −1 and T (ξ2 ) = 1. There := T ◦ C is a cross cut for D with end fore, T (S) = (−1, 1) and the closure of C points −1 and 1. √ Now, let {xn } ⊂ ( 23 , 1) be a sequence converging to 1. Moreover, for n ∈ N, let Sn+ (respectively, Sn− ) be the geodesic through xn (resp., −xn ) orthogonal to (−1, 1).

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7 Harmonic Measures and Bloch Functions

 connects −1 to 1 and Sn+ ∩ Sn− = ∅, for every n ∈ N we can find a Jordan Since C  whose end points belong to Sn+ and Sn− . Since  n ⊂ C f := f ◦ T −1 is univalent, arc C by Theorem 7.4.1, we have n )) ≤ K 2 diamE ( f (C)). f ([−xn , xn ])) ≤ K 2 diamE (  f (C diamE (  Taking supremum in the left-hand side as n → ∞, we have the result.



Lemma 7.4.3 Let ξ = eiθ with 0 < θ ≤ π/2 and let S be the geodesic in D whose closure contains ξ and ξ . Let a := S ∩ (−1, 1). Then a=

Re ξ ∈ [0, 1). 1 + Im ξ

Proof Let T : D → H, T (z) = 1+z be the Cayley transform. Taking into account 1−z that S lies on the circle which intersects ∂D orthogonally at ξ, ξ , we see that T (S) is iIm ξ −iIm ξ and T (ξ ) = 1−Re and orthogcontained in the circle passing through T (ξ ) = 1−Re ξ ξ onal to iR. Hence T (S) = {w ∈ H : |w| = |T (ξ )|}. Since T ((−1, 1)) = (0, +∞) Im ξ , we have and T (S) ∩ (0, +∞) = |T (ξ )| = 1−Re ξ a = S ∩ (−1, 1) = T

−1



Im ξ 1 − Re ξ

 =

Im ξ − 1 + Re ξ Re ξ = , Im ξ + 1 − Re ξ 1 + Im ξ

where the last equality follows taking into account that (Re ξ )2 + (Im ξ )2 = 1.  Theorem 7.4.4 There exists a universal constant K 3 > 0 such that if f : D → C is univalent and γ : (a, b) → D is and injective continuous curve such that η := limt→a γ (t) ∈ ∂D and σ := limt→b γ (t) ∈ ∂D, then, setting C := γ ((a, b)), δ f (D) ( f (0))|σ − η|2 ≤ K 3 diamE ( f (C)). Proof Denote Ω := f (D). Up to pre-composing f with a rotation, we may assume that σ = ξ and η = ξ , for some ξ = eiθ with 0 < θ ≤ π/2. Let S be the geodesic in D whose closure contains ξ and ξ and let a := S ∩ (−1, 1). Let {z n } ⊂ S be a sequence converging to ξ . Then, by Theorems 3.4.7 and 3.4.6, for all n ∈ N, 1 diamE ( f (S)) ≥ | f (z n ) − f (a)| ≥ (1 − a 2 )| f (a)| tanh(ω(z n , a)) 4 1−a 1 ≥ (1 − a 2 ) | f (0)| tanh(ω(z n , a)). (1 + a)3 4 Re ξ By Lemma 7.4.3, a = 1+Im ∈ [0, 1). Hence, limn→∞ tanh(ω(z n , a)) = 1. Thereξ fore, passing to the limit as n → ∞ in the previous inequality and taking into account Theorem 3.4.9, we have,

7.4 Diameter Distorsion for Univalent Functions

201

(1 − a)2 (1 − a)2 | f (0)| ≥ δΩ ( f (0)) 24 24 2 1 (1 − Re ξ + Im ξ ) δΩ ( f (0)) ≥ 4 2 (1 + Im ξ )2 1 (Im ξ )2 δΩ ( f (0)) ≥ 4 2 (1 + Im ξ )2 1 1 ≥ 6 (Im ξ )2 δΩ ( f (0)) = 8 |ξ − ξ |2 δΩ ( f (0)). 2 2

diamE ( f (S)) ≥

Hence, by Theorem 7.4.2, δΩ ( f (0))|ξ − ξ |2 ≤ 28 K 2 diamE ( f (C)). The result follows setting K 3 = 28 K 2 .



7.5 Notes Harmonic measures can be generalized to domains of any connectivity. This more general setting is not based on conformal mappings (see, e.g., the books [75, 108]). In this book, we followed the approach of [106]. Theorem 7.2.13 is due to Gaier and appeared in [73]. The name “Bloch functions” derives from the close connections of such functions with the so-called “Bloch constant problem”. Such a notion was introduced as a technique to tackle the normality of a wide family of holomorphic functions. Its connection with the study of conformal mappings comes from the fact that a holomorphic function in the unit disc f is a Bloch function if and only if there exist a constant α and a univalent function g such that f = α log(g ) (see, e.g., [103]). The main results stated in this chapter about Bloch functions follow the approach of [106].

Part II

Semigroups

Chapter 8

Semigroups of Holomorphic Functions

In this chapter we introduce the primary subject of our study: continuous one-parameter semigroups of holomorphic self-maps of the unit disc. We establish their main basic properties and extend to this context the Denjoy-Wolff Theorem. Then we characterize groups of automorphisms and more generally of linear fractional self-maps of the unit disc. We also briefly consider continuous semigroups of holomorphic self-maps of C and C∞ , proving that they reduce to groups of Möbius transformations, and we explain why a non-trivial theory of continuous semigroups of holomorphic maps only makes sense for self-maps of the unit disc.

8.1 Semigroups in the Unit Disc Definition 8.1.1 An algebraic semigroup (φt ) of holomorphic self-maps in the unit disc is a homomorphism between the additive semigroup of non-negative real numbers and the composition semigroup of all holomorphic self-maps of the unit disc. In other words: (1) φt ∈ Hol(D, D) for all t ≥ 0; (2) φ0 = idD , that is, φ0 is the identity in D; (3) φs+t = φs ◦ φt , for all s, t ≥ 0. Every φt is called an iterate of the semigroup. Moreover, the semigroup (φt ) is said to be continuous if, additionally, the map [0, +∞)  t → φt ∈ Hol(D, D) is continuous when [0, +∞) is endowed with the Euclidean topology and Hol(D, D) with the topology of uniform convergence on compacta.

© Springer Nature Switzerland AG 2020 F. Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4_8

205

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8 Semigroups of Holomorphic Functions

Remark 8.1.2 It is worth pointing out that, using Vitali’s Theorem, the continuity of the map [0, +∞)  t → φt ∈ Hol(D, D) at t0 ≥ 0 is equivalent to the pointwise continuity at t0 of the maps [0, +∞)  t → φt (z) ∈ D, for every z ∈ D. In the rest of the book, the expression “semigroup in D” will always mean a continuous semigroup of holomorphic self-maps of the unit disc.

Remark 8.1.3 If φt := idD , for all t ≥ 0, the family (φt ) defines a semigroup in D which is called the trivial semigroup. Definition 8.1.4 Let (φt ) be a semigroup in D and z ∈ D. The curve [0, +∞)  t → φt (z) is the orbit or trajectory of (φt ) with starting point z. Example 8.1.5 A non-trivial but rather simple example of a semigroup in D is the semigroup of Euclidean rotations with angular velocity ω ∈ R \ {0} defined by φt (z) := e−iωt z, t ≥ 0, z ∈ D. Note that if we fix a point z ∈ D \ {0}, the curve t → φt (z) describes a rotation around the origin; clockwise if ω > 0 and counterclockwise whenever ω < 0. Using automorphisms of the unit disc, it is possible to generalize this type of semigroups. Namely, take a ∈ D and consider Ta the canonical automorphism of the disc which maps a to the origin (see (1.2.1)). Let φt (z) := Ta (e−iωt Ta (z)), t ≥ 0, z ∈ D. Then (φt ) is a semigroup in D called the semigroup of hyperbolic rotations around a with angular velocity ω. The name comes from the fact that, for z = a, the curve t → φt (z) describes an Euclidean circle which is the boundary of a hyperbolic disc centered at a. The main idea behind the above example can be stated in a more general context and provides a fundamental way to generate semigroups in D. Proposition 8.1.6 Let h be a biholomorphism from D onto a domain Ω of the complex plane. Assume that (φt ) (t ≥ 0) is a family of holomorphic self-maps of Ω such that: (1) φ0 = idΩ , (2) φs+t = φs ◦ φt , for all s, t ≥ 0, and (3) the map [0, +∞)  t → φt ∈ Hol(Ω, Ω) is continuous. Then, φth := h −1 ◦ φt ◦ h is a semigroup in D. In particular, if (φt ) is a semigroup in D and T is an automorphism of the unit disc, the family φtT := T −1 ◦ φt ◦ T is a semigroup in D.

8.1 Semigroups in the Unit Disc

207

Using the above proposition, we describe two other examples of semigroups in D with a strong geometrical and dynamical meaning. Example 8.1.7 Consider the family of holomorphic self-maps of the right half-plane H defined by ϕt (w) := w + i2tα, t ≥ 0, w ∈ H, where α is a non-zero real number. Note that for every w ∈ H, the curve t → ϕt (w) describes a translation parallel to the imaginary axis; upwards if α > 0 and downwards whenever α < 0. If Cσ is the Cayley map with respect to σ ∈ ∂D (see (1.1.2)), by Proposition 8.1.6, the family φt := Cσ−1 ◦ ϕt ◦ Cσ , t ≥ 0, is a semigroup in D. Note that, for every t ≥ 0, σ is a fixed point of φt , φt (σ ) = 1 and, σ φt

(σ ) = i2αt. Example 8.1.8 Fix a non-zero real number α and consider the family of holomorphic self-maps of the right half-plane defined by ϕt (w) := eαt w, t ≥ 0, w ∈ H. Note that for every w ∈ H, the curve [0, +∞)  t → ϕt (w) is a line from 0 to ∞ if α > 0 and from ∞ to 0 whenever α < 0. Geometrically speaking, these maps are dilations in the right half-plane of (normalized) factor α. Using again the Cayley map with respect to σ ∈ ∂D and Proposition 8.1.6, we find that the family φt := Cσ−1 ◦ ϕt ◦ Cσ , t ≥ 0, is a semigroup in D. Note that all the φt ’s are Möbius transformations and {σ, −σ } ⊂ ∂D are (the unique) two common fixed points of all the iterates of the semigroup in the closed unit disc. This approach can be extended to two different and arbitrary points σ1 , σ2 ∈ ∂D. Let T be a Möbius transformation such that T (D) = H with T (σ1 ) = 0 and T (σ2 ) = ∞ (see Proposition 1.2.2). Then, by Proposition 8.1.6, the family φtT (z) := T −1 (eαt T (z)), t ≥ 0, z ∈ D is also a semigroup in D. Moreover, {σ1 , σ2 } are the unique common fixed points of all the iterates of (φtT ) in the closed unit disc. Also, for every t ≥ 0, (φtT ) (σ1 ) = eαt and (φtT ) (σ2 ) = e−αt . The next example also uses the technique described in Proposition 8.1.6 but this time the domain Ω is not as simple as the right half-plane.

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Example 8.1.9 Consider the Koebe function k(z) := i

z , z∈D (1 − z)2

which maps injectively the unit disc onto C \ {it : t ∈ (−∞, −1/4]}. Then, (1) e−t k(D) ⊂ k(D), for every t ≥ 0, (2) k(D) + it ⊂ k(D), for every t ≥ 0. Bearing in mind that k is univalent, these two geometric inclusions allows us to define two different semigroups in D. Namely, (1) φt(1) (z) := k −1 (e−t k(z)), t ≥ 0, z ∈ D; (2) φt(2) (z) := k −1 (k(z) + it), t ≥ 0, z ∈ D. We notice that all the iterates of (φt(1) ) fix the origin but none of the iterates of (t > 0) has fixed points in D. As we will see in Chap. 9, these two examples are quite significative and reflect in part the essential structure of all semigroups in D. (φt(2) )

For our next example, and also for some further results in this section, we need to study the Cauchy functional equation f (x + y) = f (x) + f (y), for all x, y ≥ 0,

(8.1.1)

where f is a function from [0, +∞) into R. Trivial solutions are those of the form f (x) = ax, for some real a. As we will see soon, if f is Lebesgue measurable, then f has to be a trivial solution. However, without additional conditions on f , there also exist “exotic” solutions (see Example 8.1.13). Lemma 8.1.10 Let A be a Lebesgue measurable subset of R with positive measure. Then there exists δ > 0 such (−δ, δ) ⊂ A − A := {a1 − a2 : a1 , a2 ∈ A}. Proof By inner regularity, there exists a compact subset K ⊂ A such that λ(K ) > 0 (here λ denotes the Lebesgue measure of R). Moreover, by outer regularity applied to K , there exists an open subset U ⊃ K such that λ(U ) < 2λ(K ). Denote by δ ≥ 0, the distance between K and R \ U . Since K ⊂ U , K is compact and U is open, δ > 0. Therefore, (−δ, δ) + K ⊂ U . We claim that (x + K ) ∩ K = ∅ for every x ∈ (−δ, δ). Otherwise, there would exist |x0 | ≤ δ such that x0 + K and K are disjoint. Then 2λ(K ) = λ(K ) + λ(x0 + K ) = λ(K ∪ (x0 + K )) ≤ λ(U ) which contradicts our election of K and U . Hence, (−δ, δ) ⊂ K − K ⊂ A − A. 

8.1 Semigroups in the Unit Disc

209

Theorem 8.1.11 Let f : [0, +∞) → R be a Lebesgue measurable function verifying f (x + y) = f (x) + f (y), for all x, y ≥ 0. Then, there exists c ∈ R such that f (x) = cx, for every x ≥ 0. Proof The proof will be divided in three steps. Step (1): f is continuous at zero. Firstly, note that f (0) = 0. Fix ε > 0. Then, take I := (−ε/2, ∞ ε/2) and let of the rational numbers. Since R = {qn } be an enumeration n=1 (qn + I ), we  −1 f (q + I ). Moreover, by hypothesis, every f −1 (qn + I ) have [0, +∞) = ∞ n n=1 is Lebesgue measurable thus there exists a natural number N such that W := f −1 (J ) has positive Lebesgue measure, where J := q N + I . Then, by Lemma 8.1.10, there exists δ > 0 such that (−δ, δ) ⊂ W − W . Therefore, any x ∈ (0, δ) belongs to (W − W ) ∩ [0, +∞). Since f solves the Cauchy functional equation (8.1.1), (W − W ) ∩ [0, +∞) ⊂ f −1 (J − J ) = f −1 (I − I ) = f −1 ((−ε, ε)). Thus, | f (x)| < ε. Step (2): f is continuous at every point of [0, +∞). Fix a > 0. Let {xn } be a sequence converging to a on the right, in particular xn ≥ a for all n ∈ N. Then f (xn ) = f (xn − a) + f (a) and, by Step (1), the sequence { f (xn )} converges to f (a). On the other hand, assume {xn } converges to a on the left, in particular, xn ≤ a. Then, f (xn ) = f (a) − f (a − xn ) and, again by Step (1), the sequence { f (xn )} converges to f (a). Step (3): There exists c ∈ R such that f (x) = cx, for every x ≥ 0. By induction, f (n) = cn, where c := f (1) and n ∈ N. If p/q is a positive rational number (thus p, q ∈ N), then cp = f ( p) = f

    p p = qf . q q q

Hence, f (x) = cx, for every rational number x ≥ 0. Since Q is dense in R, and f is continuous by Step (2), we deduce that f (x) = cx, for every x ≥ 0.  Remark 8.1.12 The argument given in the second step of the above proof can be adapted to show that any solution of the Cauchy functional equation (8.1.1) which is continuous at a point t0 ∈ [0, +∞) is in fact continuous in the whole interval. Example 8.1.13 In this example we describe an algebraic non-continuous semigroup of holomorphic self-maps of D by means of non-measurable solutions of the Cauchy functional equation. Consider R as a vector space over Q and choose a basis {xi }i∈I . The dimension of R over the field Q is infinite and, indeed, the cardinality of the index I is uncountable. Now, take i 0 ∈ I and define f : R → R by f (r ) := qi0 xi0 ,

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 where r = i∈I qi xi and qi ∈ Q. Hence f is a well-defined Q-linear map and the restriction of f (which we still denote by f ) to [0, +∞) verifies the Cauchy functional equation (8.1.1). If f were measurable, then, by Theorem 8.1.11, there would exist a real number c such that f (x) = cx, for any x ≥ 0. In particular, taking i ∈ I different from i 0 xi0 = f (xi + xi0 ) = c(xi + xi0 ), which is absurd. Hence, f is not measurable. Now, define φt (z) := ei f (t) z, t ≥ 0, z ∈ D, It is easy to check that (φt ) is an algebraic semigroup in D (note that f (0) = 0). However, (φt ) is not continuous, since f is not continuous. In this section, we will also consider some other functional equations related to the Cauchy functional equation. Proposition 8.1.14 Consider the equation f (x + y) = f (x) f (y), x, y ≥ 0.

(8.1.2)

(1) If f : [0, +∞) → ∂D is Lebesgue measurable and solves the equation, then there exists λ ∈ R such that f (x) = eiλx , for every x ≥ 0. (2) If f : [0, +∞) → C is Lebesgue measurable and solves the equation, then either f (x) = 0 for x > 0 and f (0) ∈ {0, 1}, or there exists λ ∈ C such that f (x) = eλx , for every x ≥ 0. (3) If f : [0, +∞) → R is Lebesgue measurable and solves the equation, then either f (x) = 0 for x > 0 and f (0) ∈ {0, 1}, or there exists λ ∈ R such that f (x) = eλx , for every x ≥ 0. Proof (1) First, note that f (0) = 1. Moreover, since, for 0 ≤ y ≤ x | f (x) − f (y)| = | f (x − y) f (y) − f (y)| = | f (y)|| f (x − y) − 1| = | f (x − y) − 1|,

the continuity of f at zero would imply the continuity of f in the whole interval [0, +∞). We also note that, for 0 ≤ y ≤ x, f (x) = f (x − y) f (y), and thus f (x − y) = f (x) f (y). Let us check that f is continuous at zero. Fix ε > 0 and take θ0 > 0 such that the diameter of the arc A := {eiθ : |θ | < θ0 } is lessthan ε. Likewise, let iqn {qn } be an enumeration the rational numbers. Since ∂D = ∞ A), we have n=1 (e ∞ −1 of iqn [0, +∞) = n=1 f (e A). The measurability hypothesis implies the existence of a natural number N such that W := f −1 (J ) has positive Lebesgue measure, where J := eiq N A. Then, by Lemma 8.1.10, there exists δ > 0 such that (−δ, δ) ⊂ W − W . Let x ∈ (0, δ). Since x belongs to (W − W ) ∩ [0, +∞), there exist w1 , w2 ∈ W such that x = w1 − w2 . Hence f (x) = f (w1 ) f (w2 ). Finally, | f (x) − 1| = | f (w1 ) f (w2 ) − 1| = | f (w1 ) − f (w2 )| ≤ diamE (J ) = diamE (A) < ε.

8.1 Semigroups in the Unit Disc

211

Therefore, f is continuous on [0, +∞). Since the continuous curve [0, +∞)  x → f (x) ∈ ∂D do not pass through the origin, we can choose a continuous selection of the argument, namely, there exists a continuous function ω : [0, +∞) → R such that ω(x) is an argument of f (x), for every x ≥ 0 (see, for instance, [41, Theorem 4.1]). In particular, eiω(x+y) = eiω(x) eiω(y) , for all x, y ≥ 0. Since ω is continuous, this implies that ω(x + y) = ω(x) + ω(y) + 2kπ , for some integer k independent of x and y. Thus [0, +∞)  x → ω(x) + 2kπ ∈ R is a continuous solution of the Cauchy functional equation (8.1.1). By Theorem 8.1.11, there exists λ ∈ R such that ω(x) = λx − 2kπ . Therefore, f (x) = eiλx , for every x ≥ 0. (2) Clearly, f (0) ∈ {0, 1}. Moreover, if f (0) = 0, the functional equation implies f (x) = 0 for all x > 0. Assume there exists x0 > 0 such that f (x0 ) = 0. Equation (8.1.2) clearly implies that f (2−n x0 ) = 0, for every natural number n. Hence, for any x1 > 0, we can find a positive number x2 such that x2 < x1 and f (x2 ) = 0. Then f (x1 ) = f (x1 − x2 ) f (x2 )=0. ∈ Now, we assume f (x) = 0 for every x ≥ 0. Then, consider g1 (x) := | ff (x) (x)| ∂D, x ≥ 0. The measurable function g1 satisfies (8.1.2) as well. Hence, by (1), there exists b ∈ R such that g1 (x) = eibx . On the other hand, since | f (x)| > 0, g2 (x) := log | f (x)| is a well-defined Lebesgue measurable function from [0, +∞) into R. Moreover, g2 solves the Cauchy functional equation (8.1.1) thus, by Theorem 8.1.11, there exists a ∈ R such that g2 (x) = ax. Hence, f (x) = g1 (x)e g2 (x) = e(a+ib)x , for every x ≥ 0. (3) Arguing as before, we see that if f (x0 ) = 0 for some x0 > 0, then f (x) = 0 for all x > 0 and f (0) ∈ {0, 1}. If f (x) = 0 for all x > 0, by (2), there exists λ ∈ C  such that f (x) = eλx , for every x ≥ 0. Since f takes only real values, λ ∈ R. The continuity of an algebraic semigroup (φt ) can be characterized in different ways via the map (t, z) → φt (z). Theorem 8.1.15 Let (φt ) be an algebraic semigroup of holomorphic self-maps in D. Then, the following are equivalent: (1) The semigroup (φt ) is continuous. (2) The map [0, +∞) × D  (t, z) → φt (z) ∈ D is continuous. (3) For every z ∈ D, the map [0, +∞)  t → φt (z) ∈ D is continuous at t = 0. Proof It is trivial that (1) implies (2) and (2) implies (3) thus only (3) implies (1) requires a proof. The proof is divided in four steps. Step (1): For every z ∈ D, the map [0, +∞)  t → φt (z) ∈ D is continuous on the right in [0, +∞). Fix z ∈ D and T ≥ 0 and consider {tn } a sequence in [T, +∞) converging to T . Then, by hypothesis, lim φtn (z) = lim φtn −T (φT (z)) = φT (z).

n→∞

n→∞

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Step (2): For every compact subset K ⊂ D, there exists TK > 0 and R K ∈ (0, 1) such that |φs (z)| ≤ R K , z ∈ K , s ∈ [0, TK ], |φs (z)| ≤ 2, |z| ≤ R K , s ∈ [0, TK ]. Without loss of generality, we can assume K := {z ∈ D : |z| ≤ r }, for some r ∈ (0, 1). Define R K = (1 + r )/2. By hypothesis and Vitali’s Theorem, we have that limt→0+ φt = idD in the topology of uniform convergence on compacta, so that there exists T1 > 0 such that, for every t ∈ [0, T1 ] and every |z| ≤ r , |φt (z) − z| ≤ R K − r (hence, |φt (z)| ≤ R K ). Moreover, appealing to Weierstrass’ Theorem, we also find limt→0+ φt = 1 in the topology of uniform convergence on compacta. So there exists T2 > 0 such that, for every t ∈ [0, T2 ] and every |z| ≤ R K , |φt (z) − 1| ≤ 1 (hence, |φt (z)| ≤ 2). The claim is proved by taking TK := min{T1 , T2 }. Step (3): For every compact subset K ⊂ D there exists TK > 0 such that the map [0, +∞)  t → φt (z) ∈ D is continuous in [0, TK ], for every z ∈ K . First of all, by Step (1), for every z ∈ D, the map [0, +∞)  t → φt (z) ∈ D is a measurable function and, indeed, belongs to L ∞ ([0, +∞), R). Fix the compact subset K in D and take the numbers TK > 0 and R K given in Step (2). Now, fix T ∈ (0, TK ] and consider a sequence {tn } in (0, T ] converging to T . Then, for every natural number n and every z ∈ K , again by Step (2),    1  tn   φ (φ (z)) − φ (φ (z)) du u t −u u T −u n  tn  0   tn

1 ≤ |φv (w)| |φtn −u (z) − φT −u (z))|du sup tn 0 |w|≤R K ; 0≤v≤TK  2 tn ≤ |φu (z) − φu+T −tn (z)|du tn 0  TK 2 χ[0,tn ] (u)|φu (z) − φu+T −tn (z))|du, = tn 0

|φtn (z) − φT (z)| =

where χ[0,tn ] denotes the characteristic function of the interval [0, tn ]. Now, consider the bounded sequence { f n } in L ∞ ([0, TK ]; R) defined, for n ∈ N, as 2 f n (u) := χ[0,tn ] (u)|φu (z) − φu+T −tn (z)|, u ∈ [0, TK ]. tn

8.1 Semigroups in the Unit Disc

213

Thanks to Step (1), the sequence { f n } converges pointwise to zero as n goes to ∞. Hence, Step (3) follows at once from Lebesgue’s Dominated Convergence Theorem. Step (4): For every z ∈ D, the map [0, +∞)  t → φt (z) ∈ D is continuous in [0, +∞). Fix z ∈ D and T > 0 and consider {tn } a sequence in (0, T ] converging to T . By Step (3), and taking the singleton {z} as a compact subset, there exists 0 < Tz < T such that [0, +∞)  t → φt (z) ∈ D is continuous in [0, Tz ]. Then, there exist a sequence of non-negative integers {Nn } and a sequence {sn } in (0, Tz ) such that tn = Nn Tz + sn , for every n ∈ N. Since {tn } converges to T , the sequence {Nn } is eventually constant and we may assume that tn = N Tz + sn for some non-negative integer N and for every n. Hence, {sn } converges to some s ∈ [0, Tz ] thus limn→∞ φsn (z) = φs (z). Finally, lim φtn (z) = lim φ N Tz (φsn (z)) = φ N Tz (φs (z)) = φT (z).

n→∞

n→∞

Hence, by Remark 8.1.2, the semigroup (φt ) is continuous.



Continuity of a semigroup can also be checked using derivatives: Theorem 8.1.16 Let (φt ) be an algebraic semigroup of holomorphic self-maps in D. Then, (φt ) is continuous if and only if there exists a ∈ D such that lim φt (a) = 1.

t→0

Proof Assume that the semigroup is continuous. Then, the implication follows from Theorem 8.1.15 and Weierstrass’ Theorem. Now, let us consider the reverse implication. We initially assume that a = 0. Consider the Taylor expansion of each iterate: φt (z) =

∞

an (t)z n , t ≥ 0, z ∈ D.

n=0

 2 Since every iterate belongs to the Hardy space H 2 (D), it holds ∞ n=0 |an (t)| ≤ 1. Then, for every t ≥ 0 and for every z ∈ D and using Schwarz’s Inequality,

|φt (z) − z| ≤ |a0 (t)| + |a1 (t) − 1| + 2

2

∞

|an (t)|

n=2

1 ≤ (2 − 2Re (a1 (t)))1/2 (1 − |z|2 )1/2 √ 2 ≤ |1 − φt (0)|1/2 . (1 − |z|2 )1/2

2

1/2 ∞ n=0

1/2 |z|

2n

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The result follows from our hypothesis and Theorem 8.1.15. Finally, for a general a ∈ D, consider ϕt := Ta ◦ φt ◦ Ta (t ≥ 0), where Ta is the canonical automorphism sending a to the origin (see (1.2.1)). It is straightforward to check that this new family (ϕt ) is an algebraic semigroup of holomorphic self-maps in D and satisfies the hypothesis for a = 0. Therefore, by our previous argument,  (ϕt ) is continuous, so is (φt ) by Proposition 8.1.6. The last result of this section shows that non-injective self-maps cannot be imbedded into a semigroup: Theorem 8.1.17 Every iterate of a semigroup in D is univalent. Proof Let (φt ) be a semigroup in D and suppose that there exist t0 > 0 and two different points z 1 , z 2 ∈ D such that φt0 (z 1 ) = φt0 (z 2 ). Consider T := inf{t ≥ 0 : φt (z 1 ) = φt (z 2 )}. By the continuity in t, the infimum is a minimum. Moreover, T > 0, since φ0 is the identity. Denote ξ := φT (z 1 ) and take a sequence {tn } in [0, T ) converging to T . Since {φT −tn } converges to idD in the topology of uniform convergence on compacta and using Rouche’s Theorem, there exist an open disc D in D centered at ξ and a natural number N such that φT −tn is injective in the disc D for every n ≥ N . Moreover, we may also assume that φtn (z 1 ), φtn (z 2 ) ∈ D for every n ≥ N . Now, φT −tn (φtn (z 1 )) = φT (z 1 ) = φT (z 2 ) = φT −tn (φtn (z 2 )). Therefore, φtn (z 1 ) = φtn (z 2 ), for every n ≥ N and this contradicts the definition of T . 

8.2 Groups in the Unit Disc Definition 8.2.1 An algebraic (respectively continuous) group (φt ) of holomorphic self-maps in the unit disc is an algebraic (resp. continuous) semigroup in D such that every φt is an automorphism of D. Since each iterate of a group belongs to Aut(D), for every t ≥ 0, we can introduce the notation φt := (φ−t )−1 ∈ Aut(D), for t < 0. Proposition 8.2.2 Let (φt ) be a continuous group in D. Then: (1) φs+t = φs ◦ φt , for all s, t ∈ R. (2) The map R  t → φt ∈ Aut(D) is continuous, where R is endowed with the Euclidean topology and Aut(D) with the topology of uniform convergence on compacta.

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215

Proof (1) By the very definition φs+t = φs ◦ φt , for all s, t ≥ 0. Moreover, if both s, t ≤ 0, then φs+t = (φ−s−t )−1 = (φ−t ◦ φ−s )−1 = (φ−s )−1 ◦ (φ−t )−1 = φs ◦ φt . There are two other possibilities: s ≤ 0 ≤ t and t ≤ 0 ≤ s. We will only consider the first case (the other can be addressed in a similar way). Assume, additionally, |s| ≤ t. Then, φt = φ−s ◦ φt+s . Therefore, φs+t = (φ−s )−1 ◦ φt = φs ◦ φt . If |s| > t, consider the expression φ−s = φt ◦ φ−s−t and repeat the argument. (2) Since (φt ) is a group, we only have to check the continuity in (−∞, 0]. Fix t0 ≤ 0 and consider a sequence {tn } converging to t0 on the left (that is, tn ≤ t0 ). Then, {φt0 −tn } converges uniformly on compacta to idD . By Montel’s Theorem (see, e.g., [113, Theorem 14.6, p. 282]), {φtn : n ∈ N} is a relatively compact subset of Hol(D, C). Take any function g ∈ Hol(D, C), which is the limit of a subsequence {φtnk } in the topology of uniform convergence on compacta. Therefore, for every z ∈ D, φt0 (z) = lim φt0 −tnk (φtnk (z)) = g(z). k→∞

Hence, (φtn ) converges to φt0 . The case when {tn } converges to t0 on the right (tn ≥ t0 ) can be handled in a similar way.  Since all the iterates of a group in D are automorphisms, each of them is well defined and univalent in a disc centered at zero with radius, depending on the iterate, strictly larger than one. Moreover: Proposition 8.2.3 Let (φt ) be a group in D. Then: (1) Given s, t ∈ R, there exists r (s, t) > 1, such that, for all |z| < r (s, t)

(z) = φs (φt (z))φt (z). φs+t (z) = φs (φt (z)), φs+t

(2) For every t ∈ R,

s→t

sup{|φt (z) − φs (z)| : |z| ≤ 1} −→ 0. Proof (1) Note that, for z ∈ D, the expression φs+t (z) = φs (φt (z)) holds by Proposition 8.2.2 and the other one by applying the Chain Rule for derivatives. Since φs , φt and φs+t are univalent beyond the closed unit disc and φt (∂D) = ∂D, the Identity Principle implies that both expressions in (1) still hold in some disc D(0, r ) := {z ∈ C : |z| < r } where r = r (s, t) > 1. (2) Fix t ∈ R and consider any sequence of real numbers {tn } converging to t. Then, by Proposition 1.2.2, φt = λTa for some a ∈ D and λ ∈ ∂D and φt is welldefined and univalent in |z| < r0 , for some r0 > 1. In the same way, φtn = λn Tan with

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an ∈ D and λn ∈ ∂D. By Proposition 8.2.2, limn→∞ an = a and limn λn→∞ = λ, thus there exists r1 ∈ (1, r0 ) such that the whole sequence (φtn ) is well-defined in the open disc D(0, r1 ) := {z ∈ C : |z| < r1 } and converges to φt in the Riemann sphere. Therefore, (φtn ) converges uniformly to φt on compact subsets of D(0, r1 ) and hence  in D. Note that if (φt ) is a group in D, the family (φ−t )t≥0 is also a semigroup (indeed, a group) in D. Theorem 8.2.4 Let (φt ) be a semigroup in D. Then, (φt ) is a group in D if and only if there exists t > 0 such that φt is an automorphism of D. Proof Assume that φt0 ∈ Aut(D) for some t0 > 0. By Theorem 8.1.17, every iterate is univalent thus, in order to show that each φt is an automorphism, it is enough to check that it is surjective. Fix t > 0 different from t0 . On the one hand, if t < t0 , we have D = φt0 (D) = φt (φt0 −t (D)) ⊂ φt (D) and thus φt (D) = D. On the other hand, if 0 < t0 < t, we can write t = kt0 + u for ◦ φu , and, some non-negative integer k and u ∈ (0, t0 ). Hence φt = φkt0 ◦ φu = φt◦k 0 for what we have just shown, we find that φt is the composition of a finite number of elements of Aut(D), hence it is also a member of Aut(D).  Remark 8.2.5 As a consequence of Theorem 8.2.4, if (φt ) is a semigroup in D which is not a group, then φt is not surjective for every t > 0. In particular, φt = idD for every t > 0. Theorem 8.2.6 Let (φt ) be a non-trivial group in D. Then, (φt ) has one of the following three mutually exclusive forms: (1) There exist τ ∈ D and ω ∈ R \ {0} such that φt (z) =

(e−iωt − |τ |2 )z + τ (1 − e−iωt ) , t ≥ 0, z ∈ D. τ (e−iωt − 1)z + 1 − |τ |2 e−iωt

Moreover, if (ωt)/(2π ) ∈ / Z, φt is not the identity and it is the unique elliptic automorphism of D with τ as Denjoy-Wolff point and φt (τ ) = e−iωt . (2) There exist τ, σ ∈ ∂D (τ = σ ) and α > 0 such that φt (z) =

(σ − τ eαt )z + τ σ (eαt − 1) , t ≥ 0, z ∈ D. (1 − eαt )z + σ eαt − τ

Moreover, φt is the unique hyperbolic automorphism of D with τ as the DenjoyWolff point, σ the other fixed point and φt (τ ) = e−αt . (3) There exist τ ∈ ∂D and α ∈ R non-zero such that φt (z) =

(1 − iαt)z + iαtτ , t ≥ 0, z ∈ D. −iατ t z + 1 + iαt

8.2 Groups in the Unit Disc

217

Moreover, φt is the unique parabolic automorphism of D with τ as Denjoy-Wolff point and φt

(τ ) = i2tατ . Proof In this proof, we will repeatedly appeal to Proposition 8.2.2. For t ∈ R, let Ft := {z ∈ D : φt (z) = z} be the set of fixed points of φt in the closed unit disc. If z ∈ Ft , then for every s ∈ R it holds φs (z) = φs (φt (z)) = φt (φs (z)). Therefore, φs (Ft ) ⊂ Ft , for all s, t ∈ R. Since the group is non-trivial, we can find t0 > 0 such that φt0 is not the identity. According to Sect. 1.8, there are three possibilities: φt0 is elliptic, hyperbolic or parabolic. If φt0 is elliptic, then Ft0 = {τ } ⊂ D. Then, τ is also a fixed point of all the iterates of the group. Let Tτ be the canonical automorphism of the unit disc which maps τ to 0 given in (1.2.1). Then, ϕt := Tτ ◦ φt ◦ Tτ is a family of automorphisms of D and every ϕt fixes 0. Therefore, by Schwarz’s Lemma 1.2.1, there exists λt ∈ ∂D such that ϕt (z) = λt z for every z ∈ D and every t ≥ 0. Moreover, [0, +∞)  t → λt ∈ ∂D is a continuous function satisfying λs+t = λs λt , for all s, t ≥ 0. Therefore, by Proposition 8.1.14, there exists ω ∈ R such that ϕt (z) = e−iωt z. Since the group is non-trivial, ω = 0. In order to obtain the formula in (1) it is enough to explicitly compute φt (z) = Tτ (e−iωt Tτ (z)). If φt0 is hyperbolic, then Ft0 = {τ, σ } ⊂ ∂D with τ = σ and φt 0 (τ ) < 1. Let s ∈ R \ {0}. Since φs (Ft0 ) ⊂ Ft0 , it follows that either Fs = Ft0 or φs (τ ) = σ and φs (σ ) = τ . In the latter case, by Lemma 1.8.1, φs has a fixed point ζ ∈ D \ {σ, τ }. This implies that φ2s = φs◦2 fixes τ, σ, ζ . Therefore, again by Lemma 1.8.1, φ2s = idD . Hence, the orbit of 0 under φs , {φs◦n (0)}, does not converge to a boundary point of D, which, by Theorem 1.8.4, implies that φs is elliptic. For what we proved above, it follows that φt0 is elliptic as well, a contradiction. Therefore, Fs = Ft0 . Let C(z) :=

τ +σ τ +z − , z ∈ D. τ −z τ −σ

Then C is a Möbius transformation which maps D onto the right half-plane and sends τ to ∞ and σ to 0. Therefore, ϕt := C ◦ φt ◦ C −1 is a family of automorphisms of the right half-plane and the unique fixed points (in the Riemann sphere) of ϕt (t > 0) are 0 and ∞. Hence, there exists λt ∈ R such that ϕt (w) = λt w for every w ∈ H and every t ≥ 0. Moreover, [0, +∞)  t → λt ∈ R is a continuous function verifying λs+t = λs λt . Therefore, by Proposition 8.1.14, there exists α ∈ R such that ϕt (w) = eαt w. Since ϕt (∞) = φt (τ ) < 1, for all t > 0, we find α > 0. In order to obtain the formula in (2) just compute explicitly φt (z) = C −1 (eαt C(z)). If φt0 is parabolic, then Ft0 = {τ } ⊂ ∂D. Then, τ is also a fixed point of all of the iterates of the group. Let Cτ be the Cayley transform with respect to τ . Then, ϕt := Cτ ◦ φt ◦ Cτ−1 is a family of automorphisms of the right half-plane (hence Möbius transformations) and the unique fixed point (in the Riemann sphere) of every

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ϕt (t > 0) is ∞. Therefore, there exists λt ∈ R such that ϕt (w) = w + iλt for every w ∈ H and every t ≥ 0. Moreover, [0, +∞)  t → λt ∈ R is a continuous function verifying λs+t = λs + λt . Therefore, by Theorem 8.1.11, there exists α ∈ R such that ϕt (w) = w + i2αt. Since the group is non-trivial, α = 0. The formula in (3)  follows immediately computing explicitly φt (z) = Cτ−1 (Cτ (z) + i2αt). By the uniqueness in the statements of the above theorem, it follows at once that every automorphism of the unit disc can be embedded into a semigroup of D: Corollary 8.2.7 Let φ be an element of Aut(D) different from the identity in D. Then, there exists a unique group (φt ) in D such that φ = φ1 . Definition 8.2.8 Let (φt ) be a non-trivial group in D. Then we say that (1) (φt ) is elliptic if all the iterates of the group have a common fixed point in D, (2) (φt ) is parabolic if all the iterates of the group have a unique common fixed point in ∂D, (3) (φt ) is hyperbolic if all the iterates of the group have two different common fixed points in ∂D. The group (φt ) is called non-elliptic if it is hyperbolic or parabolic. Remark 8.2.9 Elliptic groups are the (semi)groups discussed in Example 8.1.5. The semigroups introduced in Example 8.1.8 are the hyperbolic groups and those in Example 8.1.7 are the parabolic groups. A straightforward consequence of Theorem 8.2.6 is the following result about convergence of the orbits of a group. Corollary 8.2.10 Let (φt ) be a non-elliptic group in D. (1) If the group is hyperbolic, then, for every z ∈ D, lim φt (z) = τ,

t→+∞

lim φt (z) = σ.

t→−∞

(2) If the group is parabolic then, for every z ∈ D, lim φt (z) = lim φt (z) = τ.

t→+∞

t→−∞

8.3 The Continuous Version of the Denjoy-Wolff Theorem The concept of Denjoy-Wolff point also plays a prominent role in the theory of semigroups. Theorem 8.3.1 Let (φt ) be a non-trivial semigroup in D. Then, all iterates different from the identity have the same Denjoy-Wolff point τ ∈ D. Moreover:

8.3 The Continuous Version of the Denjoy-Wolff Theorem

219

(1) If τ ∈ D, then there exists λ ∈ C \ {0} with Re λ ≥ 0 such that φt (τ ) = e−λt ,

for every t ≥ 0.

In particular, either |φt (τ )| = 1, for every t > 0, or |φt (τ )| < 1, for every t > 0. (2) If τ ∈ ∂D, then there exists λ ≥ 0 such that αφt (τ ) = ∠ lim φt (z) = e−λt , z→τ

for every t ≥ 0.

In particular, either αφt (τ ) = 1, for every t > 0, or αφt (τ ) ∈ (0, 1), for every t > 0. Proof By Theorem 8.2.6, we may assume that (φt ) is not a group in D. Thus, by Remark 8.2.5, none of the iterates of (φt ) is the identity. For t > 0 let us denote by τt ∈ D the Denjoy-Wolff point of φt . Let s, t > 0. Let us assume first that τs ∈ D. Then φs (φt (τs )) = φt (φs (τs )) = φt (τs ). Therefore, φt (τs ) is a fixed point of φs and since φs is not the identity map, it turns out that φt (τs ) = τs . By the same token, τt = τs . Now, denote by τ ∈ D the common Denjoy-Wolff point of the semigroup and consider a(t) := φt (τ ) (t ≥ 0). Using Weierstrass’ Theorem and the Chain Rule, we find that a is a continuous function from [0, +∞) into C \ {0} and satisfies a(s + t) = a(s)a(t), for every s, t ≥ 0. Then, by Proposition 8.1.14, there exists λ ∈ C such that a(t) = e−λt , for every t ≥ 0. Since |φt (τ )| ≤ 1, for all t ≥ 0, we conclude Re λ ≥ 0. Since the semigroup is non-trivial, λ = 0. Now, we assume that τs ∈ ∂D. Fix z 0 ∈ D and consider the compact subset K 0 := {φu (z 0 ) : 0 ≤ u ≤ s} ⊂ D. By the (discrete) Denjoy-Wolff Theorem 1.8.4, τs = limn→∞ φson (z) = limn→∞ φns (z) uniformly on z ∈ K 0 . Fix t > 0. For every natural number n, there exists a non-negative integer kn and u n ∈ (0, s) such that nt = kn s + u n . Bearing in mind that limn→∞ kn = +∞, we have lim φnt (z 0 ) = lim φkn s (φu n (z 0 )) = τs .

n→∞

n→∞

Since z 0 was arbitrary and applying again Theorem 1.8.4, we find that τs is the Denjoy-Wolff point of φt , so τs = τt . Denote by τ ∈ ∂D the common DenjoyWolff point of the semigroup and consider α(t) := ∠ lim z→τ φt (z) (t ≥ 0). Note that by Theorem 1.7.3, α(t) is just the boundary dilation coefficient of φt at τ . Since α(t) = limn→∞ φt ((1 − 1/n)τ ) and using Weierstrass’ Theorem, we find that α is a measurable function from [0, +∞) into (0, 1]. Moreover, for every s, t ≥ 0 and every r ∈ (0, 1), the Chain Rule gives

(r τ ) = φs (φt (r τ ))φt (r τ ). φs+t

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Since φt (r τ ) tends non-tangentially to τ as r goes to 1 by Proposition 1.5.5, we can take limit r → 1 in the above expression and obtain that α(s + t) = α(s)α(t). Therefore, [0, +∞)  t → log(α(t)) ∈ (−∞, 0] is a measurable solution of the Cauchy functional equation (8.1.1) thus, by Theorem 8.1.11, there exists λ ≥ 0 such that log(α(t)) = −λt, for every 0 ≤ t.  Definition 8.3.2 Let (φt ) be a non-trivial semigroup in D. The common DenjoyWolff point τ ∈ D of all those iterates different from the identity is called the DenjoyWolff point of the semigroup. Moreover, the number λ defined in Theorem 8.3.1 is called the spectral value of the semigroup. If (φt ) is the trivial semigroup, we define its spectral value to be λ = 0. Definition 8.3.3 Let (φt ) be a non-trivial semigroup in D and τ ∈ D its DenjoyWolff point. Then we say that (1) (φt ) is elliptic if τ ∈ D, (2) (φt ) is parabolic if τ ∈ ∂D and the spectral value λ = 0, (3) (φt ) is hyperbolic if τ ∈ ∂D and the spectral value λ > 0. The semigroup (φt ) is called non-elliptic if it is either hyperbolic or parabolic. Remark 8.3.4 Let (φt ) be a non-trivial semigroup in D. By Theorem 8.3.1, if the semigroup (φt ) is elliptic according to Definition 8.3.3, then for every t ≥ 0 the iterate φt is either the identity or elliptic according to Definition 1.8.5. Similarly, if (φt ) is hyperbolic (respectively parabolic) according to Definition 8.3.3, then for every t > 0 the iterate φt is hyperbolic (resp. parabolic) according to Definition 1.8.5. Remark 8.3.5 By Remark 8.2.5 and Theorem 8.3.1, if (φt ) is a non-trivial semigroup in D such that φt0 = idD for some t0 > 0, then (φt ) is an elliptic group with spectral value λ = iω such that ωt0 = 2π k for some k ∈ Z \ {0}. Theorem 8.3.6 Let (φt ) be a non-trivial semigroup different from an elliptic group in D. Let τ ∈ D be its Denjoy-Wolff point. Then (φt ) converges uniformly on compact subsets of D to the constant map D  z → τ as t tends to +∞. Proof Let τ ∈ D be the Denjoy-Wolff point of the semigroup and denote φ := φ1 . Since (φt ) is not an elliptic group, by Theorem 8.3.1, either τ ∈ ∂D or τ ∈ D and φ is not an automorphism of D. Therefore, we can apply the Denjoy-Wolff Theorem (Theorem 1.8.4 and Proposition 1.8.3) to φ. Thus, for every z ∈ D, {φn } converges to τ uniformly on the compact subset {φu (z) : u ∈ [0, 1]}. Now, take any sequence {tn } of positive real numbers converging to +∞. Then, for every natural n, there exist a non-negative integer kn and u n ∈ [0, 1) such that limn→∞ kn = +∞ and tn = kn + u n . Therefore lim φtn (z) = lim φkn (φu n (z)) = τ.

n→∞

n→∞

Appealing to Vitali’s Theorem, the convergence of (φt ) to the constant function τ , when t goes to +∞, holds uniformly on compact subsets of D. 

8.3 The Continuous Version of the Denjoy-Wolff Theorem

221

For hyperbolic semigroups the previous result can be improved: Proposition 8.3.7 Let (φt ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then for every z ∈ D, φt (z) goes to τ non-tangentially as t → +∞. Proof Fix z ∈ D. The continuous version of the Denjoy-Wolff Theorem (Theorem 8.3.6) guarantees that limt→+∞ φt (z) = τ . Let us prove that the converge is nontangential. By Theorem 8.1.15, the set K := {φr (z) : 0 ≤ r ≤ 1} is a compact subset of D. Take any sequence of non-negative real numbers {tn } converging to +∞. For each n, there is m(n) ∈ N and r (n) ∈ [0, 1] such that tn = m(n) + r (n). Proposition 1.8.7 implies that φtn (z) = φ1◦m(n) (φr (n) (z)) converges non-tangentially to τ . Thus,  φt (z) goes to τ non-tangentially. We end up this section by giving a simple description of groups in D according to the type and the spectral value. As a matter of notation, we let H− := {z ∈ C : Re z < 0}. Proposition 8.3.8 Let (φt ) be a non-trivial semigroup in D. Then (1) (φt ) is an elliptic group with spectral value iθ , with θ ∈ R \ {0}, if and only if there exists an automorphism T of D such that (T ◦ φt ◦ T −1 )(z) = e−θit z for all t ≥ 0 and z ∈ D, (2) (φt ) is a hyperbolic group with spectral value λ > 0 if and only if there exists a Möbius transformation T such that T (D) = H and (T ◦ φt ◦ T −1 )(w) = eλt w for all t ≥ 0 and w ∈ H, (3) (φt ) is a parabolic group if and only if there exists a Möbius transformation T such that either T (D) = H or T (D) = H− and (T ◦ φt ◦ T −1 )(w) = w + it for all t ≥ 0 and w ∈ H. Proof (1) It follows immediately from Remark 8.2.9. (2) It follows from Remark 8.2.9 and Example 8.1.8 taking T the Möbius transformation such that T (D) = H and T (τ ) = ∞, T (σ ) = 0, where τ ∈ ∂D is the DenjoyWolff point of (φt ) and σ ∈ ∂D \ {τ } is the other common fixed point of (φt ). (3) From Remark 8.2.9 and Example 8.1.7, it follows that (φt ) is a parabolic group if and only if there exists a Cayley transform Cτ with respect to the Denjoy-Wolff point τ ∈ ∂D of (φt ) such that ψt (w) := (Cτ ◦ φt ◦ Cτ−1 )(w) = w + 2itα for some α ∈ R \ {0} and all t ≥ 0. Let A(w) := 2|α|w. Then A is a Möbius transformation such that A(H) = H. Moreover, (A−1 ◦ ψt ◦ A)(w) = w + it if α > 0 and (A ◦ ψt ◦ A−1 )(w) = w − it if α < 0. In the first case, we let T = A ◦ Cτ . In the second case,  we let T = −A ◦ Cτ .

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8.4 Semigroups in Riemann Surfaces As we have seen in Sect. 8.1 (see Examples 8.1.8 and 8.1.7), some semigroups in D are better understood when they are conjugated to other semigroups defined in other domains of the complex plane. This suggests to introduce and study the notions of algebraic and continuous semigroups in general Riemann surfaces. However, the results of this section clearly show that a really rich theory of semigroups is only possible for the unit disc or, obviously, for those Riemann surfaces biholomorphically equivalent to the unit disc. Definition 8.4.1 Let S be a Riemann surface. An algebraic semigroup (φt ) of holomorphic self-maps in S is a homomorphism between the additive semigroup of nonnegative real numbers and the composition semigroup of all holomorphic self-maps of S. In other words: (1) φt ∈ Hol(S, S) for all t ≥ 0, (2) φ0 = id S , that is, φ0 is the identity in S, (3) φs+t = φs ◦ φt , for all s, t ≥ 0. Every φt is called an iterate of the semigroup. Moreover, the semigroup (φt ) is said to be continuous if, additionally, the map [0, +∞)  t → φt ∈ Hol(S, S) is continuous when [0, +∞) is endowed with the Euclidean topology and Hol(S, S) with the topology of uniform convergence on compacta. Definition 8.4.2 Let S be a Riemann surface. An algebraic (respectively continuous) group (φt ) of holomorphic self-maps in S is an algebraic (resp. continuous) semigroup in S such that every φt is an automorphism of S. The semigroup (in fact, a group) where all the iterates are the identity in S is called the trivial semigroup in S. The proof of Theorem 8.1.17 can be adapted to this general context. Theorem 8.4.3 All the iterates of a continuous semigroup in a Riemann surface are univalent. Also, one can easily generalize Theorem 8.2.4 to Riemann surfaces. Theorem 8.4.4 Let S be a Riemann surface. Let (φt ) be a continuous semigroup in S. Then, (φt ) is a group if and only if some iterate φt with t > 0 is an automorphism of S. Theorem 1.1.4 shows that, up to biholomorphisms, there are exactly three nonequivalent simply connected Riemann surfaces: the unit disc D, the complex plane C and the Riemann sphere C∞ . Here we describe the semigroups in C and C∞ .

8.4 Semigroups in Riemann Surfaces

223

Theorem 8.4.5 Let (φt ) be a non-trivial continuous semigroup in C. Then there exists an affine transformation T in C such that either: (1) T ◦ φt ◦ T −1 (z) = z + it, or (2) T ◦ φt ◦ T −1 (z) = eat z, for some non-zero a ∈ C. In particular, every continuous semigroup in C is a continuous group. Proof By Theorem 8.4.3, every φt is a univalent entire function. Thus, by Lemma 1.1.6, it is a polynomial of degree one. Therefore, there exist two continuous functions a : [0, +∞) → C \ {0}, b : [0, +∞) → C such that φt (z) = a(t)z + b(t), for every z ∈ C. Moreover, for every s, t ≥ 0 a(t + s) = a(t)a(s), b(t + s) = a(t)b(s) + b(t). According to Proposition 8.1.14, there exists a complex number a such that a(t) = eat , for every t ≥ 0. If a = 0, then b(t + s) = b(t) + b(s) thus, by Theorem 8.1.11 there exists μ ∈ C such that b(t) = μt, for every t ≥ 0. Since the semigroup is non-trivial, μ = 0. Therefore, φt (z) = z + μt. Take ξ ∈ ∂D such that ξ μ = i|μ| ξ and consider the affine map T (z) = |μ| z. Then, T ◦ φt ◦ T −1 (z) = z + it, for every t ≥ 0. If a = 0, take t0 > 0. Then, b(t + t0 ) = a(t)b(t0 ) + b(t) = a(t0 )b(t) + b(t0 ), for every t ≥ 0. Therefore b(t0 ) b(t) = at (eat − 1). e 0 −1 at at 0) − 1). Consider the affine map T (z) = Let b := eb(t at0 −1 . Hence, φt (z) = e z + b(e −1 at z + b. Then, T ◦ φt ◦ T (z) = e z, for every t ≥ 0. 

Theorem 8.4.6 Let (φt ) be a non-trivial continuous semigroup of C∞ . Then, there exists a Möbius transformation T such that either (1) T ◦ φt ◦ T −1 (z) = z + it, or (2) T ◦ φt ◦ T −1 (z) = eat z, for some non-zero a ∈ C. In particular, every continuous semigroup in C∞ is a continuous group. Proof By Theorem 8.4.3, every φt is univalent. Since C∞ is compact and by the Open Mapping Theorem φt is open, it follows that φt (C∞ ) is an open and closed subset of C∞ , hence φt (C∞ ) = C∞ for every t ≥ 0. Therefore, φt is surjective for all t ≥ 0, thus φt is an automorphism of C∞ and we conclude that φt is in fact a Möbius transformation by Proposition 1.1.8. Since the semigroup is non-trivial, there exists t0 > 0 such that φt0 is different from the identity. According to Proposition 1.1.10, φt0 admits either two different fixed points or just a unique fixed point. If φt0 has a unique fixed point z 0 ∈ C∞ , applying Proposition 1.1.10, there exists a Möbius transformation T1 that sends z 0 to ∞ such that T1 ◦ φt0 ◦ T1−1 (z) = z + i. Since every iterate φt commutes with φt0 , z 0 is also the unique fixed point of φt except in case φt is the identity. Hence φt is either the identity or a Möbius transformation

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8 Semigroups of Holomorphic Functions

which fixes z 0 and we deduce that, for every t ≥ 0, T1 ◦ φt ◦ T1−1 (z) = z + a(t), for some complex number a(t). Moreover, [0, +∞)  t → a(t) ∈ C is a continuous function satisfying a(s + t) = a(s) + a(t). Therefore, by Theorem 8.1.11 there exists a complex number a such that a(t) = at. Moreover, since a(t0 ) = i, a = i/t0 = 0. Consider the Möbius transformation T2 (z) = −t0 z. Let T := T2 ◦ T1 . Then, T ◦ φt ◦ (T )−1 (z) = z + it, for every t ≥ 0. If φt0 has two different fixed points—say z 1 , z 2 ∈ C∞ —applying again Proposition 1.1.10, there exist a Möbius transformation T and λ ∈ C \ {0} such that T ◦ φt0 ◦ T −1 (z) = λz. Now, every iterate φt commutes with φt0 , hence, either z 1 and z 2 are fixed points of φt or φt (z 1 ) = z 2 and φt (z 2 ) = z 1 . Therefore, the map [0, +∞)  t → φt (z 1 ) ∈ C∞ is continuous with discrete range, thus constant. Hence, T ◦ φt ◦ T −1 fixes 0 and ∞ for every t ≥ 0, and T ◦ φt ◦ T −1 (z) = a(t)z, for some non-zero complex number a(t). Moreover, [0, +∞)  t → a(t) ∈ C \ {0} is a continuous function satisfying a(s + t) = a(s)a(t). By Proposition 8.1.14, there exists a complex number a such that a(t) = eat . Since the semigroup is not trivial, we conclude that a = 0.  The previous results show that the theory of continuous semigroups in C and C∞ is quite simple. In the unit disc, the situation is much richer and more complicated as we will see in the next chapters of this book. One might wonder what the story is in other Riemann surfaces. By the Uniformization Theorem, every Riemann surface is holomorphically covered by D, C or C∞ . Using this and the theory of covering spaces, one can classify all the Riemann surfaces covered by D, C and C∞ . It follows that, apart from those biholomorphic to D, every Riemann surface which is covered by D is either a Riemann surface with non-Abelian fundamental group, or it is biholomorphic to the punctured unit disc or to an annulus. A Riemann surface not biholomorphic to C and covered by C is either biholomorphic to the punctured complex plane or to a torus. Finally, every Riemann surface covered by C∞ is, in fact, biholomorphic to C∞ . The next theorem, whose proof can be found, e.g., in [1, Sect. 1.4.3], shows that the theory of semigroups is really interesting only for those Riemann surfaces biholomorphic to the unit disc. Theorem 8.4.7 Let (φt ) be a continuous semigroup in a Riemann surface S. (1) If S is the punctured unit disc, then either (φt ) is the trivial semigroup or (φt ) is the restriction to S of a non-trivial elliptic semigroup in D having zero as the Denjoy-Wolff point. (2) If S is an annulus, then (φt ) is the restriction to S of a group of rotations in C. (3) If S is a Riemann surface covered by the unit disc with non-abelian fundamental group, then (φt ) is the trivial (semi)group. (4) If S is the punctured complex plane, then (φt ) is the restriction to S of a continuous group in C. (5) if S is a torus, then (φt ) is a continuous group. Moreover, every iterate φt can be lifted to a map φ˜ t : C → C in such a way that (φ˜ t ) is a continuous group in C.

8.5 Semigroups of Linear Fractional Maps

225

8.5 Semigroups of Linear Fractional Maps In this section, we consider the family of Möbius transformations which map D into D: LFM(D) := {F : C∞ → C∞ : F is a M¨obius transformation, F(D) ⊂ D}. By Proposition 1.2.2, Aut(D) ⊂ LFM(D). By Theorem 8.2.4, if (φt ) is a semigroup in D such that φt0 ∈ Aut(D) for some t0 > 0 then (φt ) is a group in D. The aim of this section is to show that the same rigidity result holds in the class LFM(D). To start with, we need a preliminary result about extensions of semigroups: Proposition 8.5.1 Let D and Δ be two Riemann surfaces such that D ⊂ Δ. Let (φt ) be a continuous semigroup in D. Assume there exists an automorphism ψ of Δ with the following properties: (1) ψ(D) ⊂ D, (2) for every compact subset K ⊂ Δ there exists n ∈ N such that ψ ◦n (K ) ⊂ D, (3) for every t ∈ [0, +∞) it holds ψ ◦ φt = φt ◦ ψ.

t ) in Δ such that φ

t | D = φt . Then there exists a continuous semigroup (φ Proof Let z ∈ Δ. Let n ∈ N0 be such that ψ ◦n (z) ∈ D. For t ≥ 0 define

t (z) := (ψ −◦n ◦ φt ◦ ψ ◦n )(z). φ The definition does not depend on the number n ∈ N such that ψ ◦n (z) ∈ D. Indeed, assume m ≥ n, then (ψ −◦m ◦ φt ◦ ψ ◦m )(z) = (ψ −◦m ◦ φt ◦ ψ ◦(m−n) ◦ ψ ◦n )(z) = (ψ −◦m ◦ ψ ◦(m−n) ◦ φt ◦ ψ ◦n )(z) = (ψ −◦n ◦ φt ◦ ψ ◦n )(z).

t : Δ → Δ is well defined. It is also holomorphic. Indeed, for Therefore the map φ every z ∈ Δ there exist a relatively compact open subset U ⊂ Δ and n ∈ N such that

t |U = (ψ −◦n ◦ φt ◦ ψ ◦n )|U , which proves that φ

t z ∈ U and ψ ◦n (U ) ⊂ D, hence φ

is holomorphic at z. By the same token, it is easy to see that (φt ) is a continuous

t | D = φt because if z ∈ D one can take n = 0 semigroup in Δ. It is also clear that φ in the definition.  We want to apply the previous extension result to the case of a semigroup in D such that some of its iterates is a Möbius transformation. In order to do that, we need to describe the dynamics of elements in LFM(D) which are not automorphisms: Proposition 8.5.2 Let F ∈ LFM(D) \ Aut(D). Then there exists a simply connected open set Δ ⊂ C∞ , D ⊂ Δ, such that

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(1) F(Δ) ⊂ Δ and F|Δ is an automorphism of Δ, (2) for every compact subset K ⊂ Δ there exists n ∈ N such that F ◦n (K ) ⊂ D, (3) there exists a Möbius transformation C such that either C(Δ) = C or C(Δ) = D. Moreover, (1) if F is elliptic, then there exists σ ∈ C∞ \ D such that F(σ ) = σ . In this case, Δ = C∞ \ {σ }. (2) if F is hyperbolic with Denjoy-Wolff point τ ∈ ∂D then there exists σ ∈ C∞ \ D such that F(σ ) = σ . In this case, Δ is either the disc containing D and such that σ, τ ∈ ∂Δ, if σ = ∞, or the half-plane containing D such that τ ∈ ∂Δ, whenever σ = ∞. (3) if F is parabolic with Denjoy-Wolff point τ ∈ ∂D then τ is the only fixed point of F in C∞ . In this case, Δ = C∞ \ {τ }. Proof If F is elliptic with Denjoy-Wolff point τ ∈ D, by Theorem 1.2.3, |F (τ )| < 1. By Corollary 1.1.12 there exists another fixed point σ ∈ C∞ \ {τ }, and, by Corollary 1.2.4, such σ does not belong to D. Let Δ := C∞ \ {σ } and let G be the Möbius z−τ if σ ∈ C or by G(z) := z − τ if σ = transformation defined either by G(z) := z−σ ∞. Then G maps τ to 0 and Δ to C. Therefore the Möbius transformation G ◦ F ◦ G −1 fixes ∞ and 0, and has derivative λ := F (τ ) at 0. Thus, (G ◦ F ◦ G −1 )(z) = λz. From this the statement follows at once. Assume now F is non-elliptic with Denjoy-Wolff point τ ∈ ∂D. Let Cτ be the Cayley transform with respect to τ defined in (1.1.2). Let G := Cτ ◦ F ◦ Cτ−1 . Then G is a Möbius transformation such that G(H) ⊂ H and G(∞) = ∞. Therefore, G(w) = aw + b for some a, b ∈ C, a = 0. A direct computation shows that a = 1 = α F1(τ ) ≥ 1, and the condition G(H) ⊂ H implies then that Re b > 0 (note F (τ ) that F is not an automorphism). b ∈ C \ H is a fixed point for G—hence If F is hyperbolic, that is a > 1, 1−a −1 b σ := Cτ ( 1−a ) ∈ C∞ \ D is a fixed point for F. Let Δ be either the disc containing D and such that σ, τ ∈ ∂Δ if σ = ∞, or the half-plane containing D such that τ ∈ ∂Δ, whenever σ = ∞. Then F(Δ) = Δ and thus F is an automorphism of Δ. b } and it is easy to see that for Moreover, Cτ (Δ) = V := {w ∈ C : Re w > Re 1−a every compact subset K ⊂ V there exists n ∈ N such that G ◦n (K ) ⊂ H. Since Cτ is a biholomorphism between Δ and V and conjugates F with G, the result follows. Finally, if F is parabolic, by Corollary 1.1.12 and Proposition 1.1.10, its only fixed point in C∞ is τ . In this case G(z) = z + b, and b = 0. If b were purely imaginary, then G would be an automorphism of H, hence F would be an automorphism of D, against our hypothesis. Therefore, Re b > 0. Thus, for every compact set K ⊂ C there exists n ∈ N such that G ◦n (K ) ⊂ H. Setting Δ := Cτ−1 (C) we have the result.  Now we can state the rigidity theorem for linear fractional maps: Theorem 8.5.3 Let (φt ) be a semigroup in D. Suppose there exists t0 > 0 such that φt0 ∈ LFM(D). Then φt ∈ LFM(D) for all t ≥ 0.

8.5 Semigroups of Linear Fractional Maps

227

Proof If φt0 is an automorphism, then (φt ) is a group by Theorem 8.2.4 and the / Aut(D). theorem follows from Theorem 8.2.6. Therefore we can assume φt0 ∈ Let Δ be the simply connected open set containing D relative to the Möbius transformation φt0 given by Proposition 8.5.2. Then we are in the hypotheses of Proposition 8.5.1 with D = D and ψ = φt0 . Thus, there exists a continuous semi t |D = φt for all t ≥ 0. Since φ

t0 is an automorphism of Δ,

t ) of Δ such that φ group (φ

t is an automorphism of Δ for all t ≥ 0. But Δ is by Theorem 8.4.4, it follows that φ

t ◦ G −1 biholomorphic to either D or C via a Möbius transformation G, hence G ◦ φ is an automorphism of either C or D for all t ≥ 0, hence a Möbius transformation

t , and hence φt , is a Möbius by Corollary 1.1.7 and Proposition 1.2.2. Therefore, φ transformation for all t ≥ 0. 

8.6 Notes It is hard to trace back the birth of the theory of continuous semigroups in the unit disc. At the beginning of the twentieth century, Tricomi [124] dealt with problems which, translated in modern language, were related to the asymptotic behavior of continuous semigroups. In 1923, Loewner [95] introduced the nowadays called “Loewner theory” to tackle extremal problems in complex analysis. Such a theory, as developed in particular by Pommerenke [102], contains the germ of elliptic semigroups theory and relate semigroups to certain ordinary differential equations. Later on, in 1943, Kufarev [94] introduced an ordinary differential equation whose solutions are pretty much related to continuous non-elliptic semigroups. A general theory containing both the classical Loewner-Kufarev theory and semigroups theory has been introduced in [28]. In 1939, Wolff [128], studied continuous iteration in the half-plane and proved a type of continuous Denjoy-Wolff theorem. The first paper systematically dealing with the theory of continuous semigroups in the unit disc is due to Berkson and Porta [11]. They proved Theorem 8.3.6 and related every semigroup in D with a certain holomorphic vector field, called the infinitesimal generator of the semigroup, via an autonomous ordinary differential equation (we will discuss infinitesimal generators in Chap. 10). Another context where continuous semigroups appear naturally is the so-called problem of embedding which is strictly related to the existence of fractional iterates. More precisely, given φ : D → D holomorphic, the problem of embedding is to find a continuous semigroup (φt ) in D such that φ1 = φ. The question about existence of fractional iterates is whether and how it makes sense to define a holomorphic function φ ◦t for every positive real number t in such a way that it satisfies the natural composition properties. By Theorem 8.1.17, a necessary condition to solve the embedding problem is that φ be univalent. A complete answer to this problem can be obtained by using models, as explained in the Notes of the next chapter, but an answer in terms of analytical properties of φ is still unknown. This point of view has been taken in [56, 123].

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8 Semigroups of Holomorphic Functions

The proof of Theorem 8.1.16 was told in a private conversation with the authors by Ch. Pommerenke. Siskakis in his PhD Thesis (see [120, Theorem 1.7]) proved Theorem 8.3.1 in a slightly different form. The material in Sect. 8.4 was first proved in [82]. Theorem 8.5.3 was proved, with different methods, in [24].

Chapter 9

Models and Koenigs Functions

In this chapter we construct models for semigroups in D. Starting from a given semigroup, the basic idea is to define an abstract space (the space of orbits of a semigroup or abstract basin of attraction) which inherits a complex structure of simply connected Riemann surface, in such a way that the semigroup is conjugated to a continuous group of automorphisms of such a Riemann surface. Moreover, our construction is universal, which implies that all possible (semi-)conjugations of the semigroup factorize through the model. Also, the model respects basic properties of the semigroup, in particular, the divergence rate, which is a measure in the hyperbolic distance of the rate of convergence to the Denjoy-Wolff point. After having defined and discussed models (and the more general concept of semi-models), we concentrate in studying the canonical models, where it makes its appearance the second main subject of our study: the Koenigs function of a semigroup, which intertwines the semigroup to a very simply group of automorphisms of either C, or of a half-plane or of a strip, depending on the properties of the starting semigroup. Those Koenigs functions are univalent functions which are either spirallike or starlike at infinity, and we spend some time in understanding properties of those maps. The upshot is to translate dynamical properties of a semigroup into geometrical properties of the image of the associated Koenigs function. We will also study in details models for semigroups of linear fractional maps and non-canonical semi-models with base space D and C and we see how conjugation can be naturally reads through models. We end up the chapter by considering topological models for semigroups, showing that, from a topological point of view, there are only three possible models: the model of hyperbolic rotations, the model of elliptic semigroups (which are not groups) with spectral value 1 and the model of hyperbolic semigroups with spectral value π .

© Springer Nature Switzerland AG 2020 F. Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4_9

229

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9 Models and Koenigs Functions

9.1 The Divergence Rate and Hyperbolic Steps We are going to introduce a quantity, called divergence rate, which, roughly speaking, measures the average hyperbolic speed of escape of an orbit of a semigroup. In order to do this, we need to use the following version of Fekete’s Theorem. Recall that a function v : [0, +∞) → R is subadditive provided v(x + y) ≤ v(x) + v(y) for all x, y ≥ 0. Lemma 9.1.1 Let v : [0, +∞) → [0, +∞) be a continuous subadditive function. Then limt→+∞ v(t) exists finitely and t lim

t→+∞

v(t) v(t) = inf . t>0 t t

Proof Let β := inf t>0 v(t) . Clearly β < +∞. Let b > β. Let t0 > 0 be such that t v(t0 )/t0 < b. For t > 3t0 let m = [t/t0 ] ∈ N be the integer part of t/t0 and set n := m − 2 ∈ N. Let M = max |v(s)| for s ∈ [2t0 , 3t0 ]. Note that t − nt0 ∈ [2t0 , 3t0 ] for all t > 3t0 . Hence, for t > 3t0 , v(t) = v(nt0 + t − nt0 ) ≤ v(nt0 ) + v(t − nt0 ) ≤ nv(t0 ) + M. Thus β≤

M nt0 v(t0 ) v(t) + . ≤ t t t0 t

Passing to the limit as t → +∞ we obtain β ≤ lim inf t→+∞

because limt→+∞

nt0 t

v(t) v(t) ≤ lim sup ≤ b, t t t→+∞

= 1. By the arbitrariness of b > β, the result follows.



Lemma 9.1.2 Let (φt ) be a continuous semigroup on a Riemann surface Ω. Then for all z ∈ Ω the limit kΩ (φs (z), z) (9.1.1) cΩ (φt ) := lim s→+∞ s exists independently of z, cΩ (φt ) ∈ [0, +∞) and moreover cΩ (φt ) = inf

s>0

kΩ (φs (z), z) . s

(9.1.2)

Proof Fix z ∈ Ω. The function [0, +∞)  t → kΩ (z, φt (z)) is continuous (see Proposition 1.3.14). Moreover, for s, t ≥ 0, using the triangle inequality and the contractiveness property of kΩ under holomorphic maps (see Proposition 1.3.10),

9.1 The Divergence Rate and Hyperbolic Steps

231

kΩ (z, φt+s (z)) ≤ kΩ (z, φt (z)) + kΩ (φt (z), φt+s (z)) = kΩ (z, φt (z)) + kΩ (φt (z), φt (φs ((z)))) ≤ kΩ (z, φt (z)) + kΩ (z, φs (z)).

Hence, the function [0, +∞)  t → kΩ (z, φt (z)) is a non-negative continuous subadditive function. By Lemma 9.1.1, the limit kΩ (φt (z), z) kΩ (φt (z), z) = inf t→+∞ t>0 t t

c(z) := lim

exists finitely. It remains to show that c(z) is independent of z ∈ Ω. To this aim, let w ∈ Ω be another point. Then, using again the triangle inequality and the contractiveness of the hyperbolic distance, we have kΩ (z, φt (z)) ≤ kΩ (z, w) + kΩ (w, φt (w)) + kΩ (φt (z), φt (w)) ≤ kΩ (w, φt (w)) + 2kΩ (z, w). Therefore, dividing by t and taking the limit as t → +∞, we see that c(z) ≤ c(w). Changing z with w and repeating the previous argument we get c(z) = c(w).  Definition 9.1.3 Let (φt ) be a continuous semigroup on a Riemann surface Ω. The number cΩ (φt ) defined in (9.1.1) is called the divergence rate of (φt ). Remark 9.1.4 It is clear from the definition that if (φt ) is a continuous semigroup in Ω such that there exists z ∈ Ω with φt (z) = z for all t ≥ 0, then the divergence rate of (φt ) is cΩ (φt ) = 0. In particular, elliptic semigroups in D have zero divergence rate. We introduce now another measure of “hyperbolicity” of a semigroup on a Riemann surface Ω, and relate it to the divergence rate. Let (φt ) be a continuous semigroup on a Riemann surface Ω. Note that for r ≥ r ≥ 0, kΩ (φr (z), φr +u (z)) = kΩ (φr (z), φr (φu (z))) = kΩ (φr −r (φr (z)), φr −r (φr (φu (z)))) ≤ kΩ (φr (z), φr (φu (z))) = kΩ (φr (z), φr +u (z)). Hence, the function r → kΩ (φr (z), φr +u (z)) is decreasing in r and therefore the limit exists. Taking this into account, we can define the hyperbolic step of a semigroup: Definition 9.1.5 Let (φt ) be a continuous semigroup on a Riemann surface Ω. Let u ≥ 0. The hyperbolic step of order u (or u-th hyperbolic step) of (φt ) at z ∈ Ω is defined as su (φt , z) := lim kΩ (φr (z), φr +u (z)). r →+∞

The 1-st hyperbolic step s1 (φt , z) is just called hyperbolic step.

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9 Models and Koenigs Functions

Remark 9.1.6 If (φt ) is a continuous group of automorphisms of a Riemann surface Ω, it holds su (φt , z) = kΩ (z, φu (z)). On the other hand, if (φt ) is an elliptic semigroup, not a group, in D then su (φt , z) ≡ 0 for all z ∈ D and u ≥ 0. Indeed, if τ ∈ D is the Denjoy-Wolff point of (φt ) then φt (τ ) = τ for all t ≥ 0 and hence su (φt , τ ) = 0. If z ∈ D is not the Denjoy-Wolff point of (φt ), then by Theorem 8.3.6, limt→+∞ ω(φt (z), φt+u (z)) = ω(τ, τ ) = 0, hence su (φt , z) = 0. Proposition 9.1.7 Let (φt ) be a continuous semigroup on a Riemann surface Ω. Then for all z ∈ Ω, su (φt , z) . cΩ (φt ) = lim u→+∞ u Proof Clearly kΩ (z, φu (z)) ≥ su (φt , z) for all u ≥ 0 because of the contractiveness of the hyperbolic distance with respect to holomorphic functions. Then, cΩ (φt ) ≥ lim sup u→+∞

su (φt , z) . u

In order to prove the converse, let m ∈ N and u > 0. By the triangle inequality, m−1 kΩ (z, φum (z)) 1  kΩ (φu j (z), φu( j+1) (z)). ≤ um um j=0

Note that kΩ (φu j (z), φu( j+1) (z)) → su (φt , z) as j → ∞, hence, by the Cesàro Means Theorem the right-hand side of the previous inequality converges to su (φut ,z) as m → ∞. Therefore, for any u > 0, lim

t→+∞

su (φt , z) kΩ (z, φt (z)) kΩ (z, φmu (z)) = lim ≤ . Nm→∞ t um u

This proves that cΩ (φt ) ≤ lim inf u→+∞

su (φt , z) , u 

and the result follows.

In case of a non-elliptic semigroup in D the divergence rate is essentially equal to the spectral value. We need a preliminary lemma: Lemma 9.1.8 Let {am } be a sequence of positive real numbers. Then lim inf m→∞

am+1 ≤ lim inf (am )1/m . m→∞ am

9.1 The Divergence Rate and Hyperbolic Steps

233

Proof If lim inf m→∞ aam+1 = 0 the result is trivially true. Otherwise, let 0 < T < m am+1 lim inf m→∞ am . Then, for any m sufficiently large, aam+1 ≥ T , that is am+1 ≥ T am . m By recurrence, am+k ≥ T k am . Therefore am+k ≥ T m+k am /T m . This implies 1

(am+k ) m+k ≥ T

1  a  m+k

m

Tm

.

Hence 1

1

lim inf (ak ) k = lim inf (am+k ) m+k ≥ T lim inf k→∞

k→∞

k→∞

1  a  m+k

m

Tm

= T. 

By the arbitrariness of T , the result follows.

Theorem 9.1.9 Let (φt ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D and spectral value λ ≥ 0. Let cD (φt ) denote the divergence rate of (φt ). Then cD (φt ) =

1 λ. 2

(9.1.3)

In particular, if (φt ) is hyperbolic then cD (φt ) > 0 while if (φt ) is parabolic then cD (φt ) = 0. Proof Set c := cD (φt ). Let z m := φm (0), m ∈ N. By the Denjoy-Wolff Theorem 8.3.6, limm→∞ z m = τ . Moreover, e−λ is the boundary dilatation coefficient of φ1 at its Denjoy-Wolff point τ . Hence lim inf m→∞

1 − |z m+1 | 1 − |φ1 (z m )| = lim inf ≥ e−λ , m→∞ 1 − |z m | 1 − |z m |

and, by Lemma 9.1.8, lim inf (1 − |z m |)1/m ≥ lim inf m→∞

m→∞

1 − |φ1 (z m )| ≥ e−λ . 1 − |z m |

Thus, ω(0, φt (0)) ω(0, φm (0)) ω(0, z m ) = lim = lim m→∞ Nm→∞ t m m 1/m  1 1 + |z m | 1 1 lim log = ≤ − log e−λ = λ. m→∞ 2 1 − |z m | 2 2

c = lim

t→+∞

(9.1.4)

If (φt ) is parabolic, that is λ = 0, (9.1.4) implies that c = 0 as well. Now, assume (φt ) is hyperbolic, and thus λ > 0. Let R ∈ (0, 1) and let E(τ, R) be the horocycle of center τ and radius R. Since the point of E(τ, R) closest to 0 is 1−R τ, 1+R 1 1− R inf ω(0, z) = ω(0, τ ) = − log R. (9.1.5) z∈E(τ,R) 1+ R 2

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9 Models and Koenigs Functions

Now, 0 ∈ ∂ E(τ, 1), and by Theorem 1.4.7, z m = φm (0) = φ1◦m (0) ∈ E(τ, e−mλ ) for all m ∈ N. Hence, by (9.1.5), 1 1 − e−mλ 1 1 1 ω(0, z m ) ≥ ω(0, log e−mλ = λ. τ) = − −mλ m m 1+e 2m 2 Therefore, c = lim

m→∞

1 1 ω(0, z m ) ≥ λ, m 2

which, together with (9.1.4), implies the statement.



Formula (9.1.3) allows easily to see how the spectral value of a semigroup behaves under conjugacy. To set up properly the terminology, we give the following definition: Definition 9.1.10 Let (φt ) be a semigroup in D. Let Ω be a Riemann surface and let (ϕt ) be a semigroup in Ω. We say that (φt ) is semi-conjugated to (ϕt ) if there exists a holomorphic map g : D → Ω, called a semi-conjugation map, such that for all t ≥ 0 it holds g ◦ φt = ϕt ◦ g. In other words, (φt ) is semi-conjugated to (ϕt ) if there exists a holomorphic map g such that the following diagram commutes for all t ≥ 0: D

φt

g

Ω

D g

ϕt

Ω.

Proposition 9.1.11 Let (φt ) be a semigroup in D. Suppose that (φt ) is semiconjugated to a continuous semigroup (ϕt ) of a Riemann surface Ω via the semiconjugation map g. Then (9.1.6) cD (φt ) ≥ cΩ (ϕt ). Proof Since g : D → Ω is holomorphic, it contracts the hyperbolic distance. Hence, kD (0, φt (0)) ≥ kΩ (g(0), g(φt (0))) = kΩ (g(0), ϕt (g(0)). Dividing by t and taking the limit as t → +∞, formula (9.1.6) follows at once.  Corollary 9.1.12 Let (φt ), (ϕt ) be two semigroups in D and suppose that (φt ) is semi-conjugated to (ϕt ). If (ϕt ) is hyperbolic with spectral value η > 0, then (φt ) is hyperbolic and its spectral value λ satisfies λ ≥ η.

9.1 The Divergence Rate and Hyperbolic Steps

235

Proof Let c denote the divergence rate of (φt ) and c˜ that of (ϕt ). By (9.1.3), c˜ = 21 η > 0. Hence, by (9.1.6), c ≥ c˜ > 0, which implies in particular that (φt ) is non-elliptic. Therefore, again by (9.1.3), λ = 2c ≥ 2c˜ = η > 0, and (φt ) is hyperbolic.



9.2 Holomorphic Models The idea underlying the construction of models for semigroups is to find simple groups in some domains in C such that every continuous semigroup in D is conjugated to one of such groups, and in such a way that the information on the dynamical behavior of the original semigroup is encoded both in the model group and in the geometrical properties of the image of the conjugating map. In particular, if one succeeds in such a construction using very simple model groups such as rotations and translations, then the core of the dynamical behavior of the original semigroup is encrypted in the shape of the image of the unit disc via the conjugation map. Despite the simplicity of this idea, one has to be very careful in defining what a model is and how representative it should be. In order to explain this, consider the following example: Example 9.2.1 Let ϕt (z) = z + it, for t ≥ 0. Note that (ϕt ) is a group of automorphisms of C. Let C1 : D → H be the Cayley transform with respect to 1 defined in (1.1.2). Then define φt := C1−1 ◦ ϕt ◦ C1 , t ≥ 0. Since (ϕt ) is a group of automorphisms in H without fixed points, then (φt ) is a non-elliptic group in D. Define φ˜ t (z) := C1−1 (−iϕt (iC1 (z))). Then (φ˜ t ) is a non-elliptic semigroup in D, which is not a group since for t > 0 the map φ˜ t is not surjective. By construction, both (φt ) and (φ˜ t ) are conjugated with the group of automorphisms (ϕt ) of C, the first via the intertwining map C1 and the second via the map z → iC1 (z). Therefore, if we considered the group (ϕt ) of C a “model” for both semigroups, we could not hope to obtain too much information. However, note that ∪t≤0 ϕt (C1 (D)) = H = C while ∪t≤0 ϕt (iC1 (D)) = C. In a sense that will be clear in a while, the group of automorphisms (ϕt ) of C is representative of the semigroup (φ˜ t ), but not of (φt ), because the condition ∪t≤0 ϕt (iC1 (D)) = C tells somewhat that the dynamical behavior in iC1 (D) can be extended to all C. The previous example should make clear the reason for the following definition: Definition 9.2.2 Let (φt ) be a semigroup in D. A holomorphic semi-model for (φt ) is a triple (Ω, h, ψt ) where Ω is a Riemann surface, h : D → Ω is holomorphic,

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9 Models and Koenigs Functions

(ψt ) is a continuous group of automorphisms of Ω such that h ◦ φt = ψt ◦ h, t ≥ 0, and



ψ−t (h(D)) = Ω.

(9.2.1)

(9.2.2)

t≥0

We call the manifold Ω the base space and the mapping h the intertwining mapping. If h : D → Ω is univalent we call the triple (Ω, h, ψt ) a holomorphic model for (φt ). In the definition of semi-model we do require the group (ψt ) to be continuous a priori. However, starting with a continuous semigroup, the continuity of the group in any semi-model is automatic: Lemma 9.2.3 Let (φt ) be a semigroup in D. Let Ω be a Riemann surface, let h : D → Ω be a holomorphic map and let (ψt ) be an algebraic group of automorphisms of Ω. If (9.2.1) and (9.2.2) are satisfied, then (ψt ) is a continuous group of automorphisms of Ω. Proof First, we note that h is not constant. Otherwise, there exists a ∈ Ω such that h(z) = a, for every z ∈ D. Using (9.2.1), it follows that a = h(φt (z)) = ψt (h(z)) = ψt (a) for all t ≥ 0. By (9.2.2), this implies that Ω = {a} a contradiction. Next, since ψt (h(D)) = h(φt (D)) ⊂ h(D) for all t ≥ 0, it follows that ((ψt )|h(D) ) is an algebraic semigroup in h(D). We claim that ((ψt )|h(D) ) is a continuous semigroup in h(D). Assuming the claim for the moment, we show how to obtain the continuity of (ψt ) in all Ω. Let K ⊂ Ω be a compact set. By (9.2.2) there exists u ∈ [0, +∞) such that ψu (K ) ⊂ h(D). The set ψu (K ) is compact, and, by the claim, ψt → ψs uniformly on ψu (K ) as t → s. Since ψt = ψ−u ◦ ψt ◦ ψu , for every t ≥ 0, it follows immediately that ψt → ψs uniformly on K as t → s. Now we are left to prove the claim. Let Dr := h({z ∈ D : |z| < r }) for r ∈ (0, 1). Since h : D → Ω is holomorphic and non-constant, it is open, hence {Dr } is an open covering of h(D). In particular, for every compact set K ⊂ h(D) there exists r (K ) ∈ (0, 1) such that K ⊂ Dr (K ) . Assume by contradiction the claim is not true. Hence, there exist a sequence of non-negative real numbers {tn } converging to s ≥ 0 and a sequence {wn } included in some compact subset K of h(D) such that inf n |ψtn (wn ) − ψs (wn )| ≥ δ > 0. Our previous remark shows that there is r ∈ (0, 1) and z n ∈ D (not necessarily unique) with |z n | ≤ r such that h(z n ) = wn for all n ∈ N. Without loss of generality, we may assume that {z n } converges to some z 0 ∈ D (in fact, |z 0 | ≤ r ). In particular, limn→∞ φtn (z n ) = φs (z 0 ). Then, using (9.2.1), δ ≤ |ψtn (wn ) − ψs (wn )| ≤ |h(φtn (z n )) − h(φs (z 0 ))| + |h(φs (z 0 )) − h(φs (z n ))|.

9.2 Holomorphic Models

237

Since the right-hand side terms tend to 0 as n → ∞ because (φt ) is continuous, we get a contradiction.  Some first simple properties of semi-models are contained in the following proposition: Proposition 9.2.4 Let (φt ) be a semigroup in D and let (Ω, h, ψt ) be a holomorphic semi-model for (φt ). Then (1) For every t ≥ 0, ψt (h(D)) ⊆ h(D). In particular, if (Ω, h, ψt ) is a model, then φt (z) = h −1 (ψt (h(z))), for all z ∈ D.

(9.2.3)

(2) If (φt ) is elliptic then (ψt ) has a common fixed point in Ω. Proof (1) Follows at once from ψt (h(D)) = h(φt (D)) ⊂ h(D), for all t ≥ 0. In particular, if h is univalent then h is invertible on ψ(h(D)) and (9.2.3) follows from the functional equation (9.2.1). (2) If (φt ) is elliptic, then there exists z 0 ∈ D such that φt (z 0 ) = z 0 for all t ≥ 0. From (9.2.1), ψt (h(z 0 )) = h(φt (z 0 )) = h(z 0 ), proving that h(z 0 ) is a fixed point for ψt for all t ≥ 0.



A semigroup in D might have different holomorphic semi-models: Example 9.2.5 Let C1 : D → H be the Cayley transform with respect to 1. Let φt (z) := C1−1 (et C1 (z)), for t ≥ 0. Then (φt ) is a hyperbolic group in D (see Proposition 8.3.8). By construction, it is clear that (H, C1 , z → et z) is a holomorphic model for (φt ). Let log denote the principal branch of the logarithm, and let h : D → Sπ := {z ∈ C : 0 < Re z < π } be the biholomorphism given by h(z) := i log C1 (z) +

π . 2

By construction, for all t ≥ 0, h(φt (z)) = i log(et C1 (z)) +

π π = i log C1 (z) + + it = h(z) + it. 2 2

Therefore, (Sπ , h, z → z + it) is also a holomorphic model for (φt ). One can construct semi-models for (φt ) which are not models as follows. Let ˜ λ ∈ C \ {0}. Then define h(z) := exp(−λi h(z)). Hence ˜ ˜ t ) = exp(−λi(h(z) + it)) = eλt exp(−λi h(z)) = eλt h(z). h(φ ˜ If we take, for instance, λ = 4, then h˜ is not univalent but h(D) = C∗ = C \ {0}. Thus ∗ 4t ˜ (h, C , z → e z) is a holomorphic semi-model for (φt ), and it is not a model. Note

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also that, on the one hand, the divergence rate of (φt ) is 1 (since it is a group of hyperbolic automorphisms of D with spectral value 1). On the other hand, kC∗ ≡ 0—as it can be easily seen from the definition of hyperbolic distance by using holomorphic discs of the form D  z → exp( f (z)) where f : D → C is holomorphic. Hence, 4t z) limt→+∞ kC∗ (z,e = 0. t At a first sight the previous example might induce to think that the notion of (semi-)models is not so useful, since, even for a simple case, there are plenty of different semi-models. However, all (semi-)models are pleasantly related each other. In order to see this, we need to introduce the notion of morphisms of semi-models:   t ) be two holomorphic semi-models for h, ψ Definition 9.2.6 Let (Ω, h, ψt ) and (Ω, a semigroup (φt ) in D. A morphism of holomorphic semi-models ηˆ : (Ω, h, ψt ) →  such that   t ) is given by a holomorphic map η : Ω → Ω (Ω, h, ψ

and

 h = η ◦ h,

(9.2.4)

t ◦ η = η ◦ ψt , t ≥ 0. ψ

(9.2.5)

  t ) be a morphism of holomorphic h, ψ Remark 9.2.7 Let ηˆ : (Ω, h, ψt ) → (Ω, semi-models for a semigroup in D. Then t ◦ η = η ◦ ψt , t ∈ R. ψ

(9.2.6)

t ◦ η = η ◦ ψt . Therefore Indeed, let t > 0. Then by (9.2.5), it holds ψ t−1 ◦ ψ t ◦ η ◦ (ψt )−1 = ψ t−1 ◦ η ◦ ψt ◦ (ψt )−1 = ψ t−1 ◦ η, η ◦ (ψt )−1 = ψ t−1 = ψ −t , formula (9.2.6) holds. and, since (ψt )−1 = ψ−t , ψ Example 9.2.8 Let (Ω, h, ψt ) be a holomorphic semi-model for a semigroup in D. ˆ Ω : (Ω, h, ψt ) → (Ω, h, ψt ) is the morphism of semi-models defined by the Then id identity map idΩ : Ω → Ω, idΩ (z) = z for all z ∈ Ω. j

Let (Ω j , h j , ψt ), j = 1, 2, 3 be three holomorphic semi-models for a given semij j+1 group in D. Assume that ηˆ j : (Ω j , h j , ψt ) → (Ω j+1 , h j+1 , ψt ) for j = 1, 2 are two morphisms of holomorphic semi-models. It is easy to see that 1 3 ηˆ 2 ◦ ηˆ 1 := η 2 ◦ η1 : (Ω1 , h 1 , ψt ) → (Ω3 , h 3 , ψt )

is a morphism of holomorphic semi-models.   t ) be a morphism of holomorphic h, ψ Definition 9.2.9 Let ηˆ : (Ω, h, ψt ) → (Ω, semi-models for a semigroup in D. We say that ηˆ is an isomorphism of holomorphic semi-models if there exists a morphism of holomorphic semi-models ˆ Ω and ηˆ ◦ μˆ = id ˆ Ω .   t ) → (Ω, h, ψt ) such that μˆ ◦ ηˆ = id μˆ : (Ω, h, ψ

9.2 Holomorphic Models

239

Morphisms among semi-models are very rigid, as the following results show.   t ) be Lemma 9.2.10 Let (φt ) be a semigroup in D. Let (Ω, h, ψt ) and (Ω, h, ψ   t ) is a h, ψ two holomorphic semi-models for (φt ). Assume ηˆ : (Ω, h, ψt ) → (Ω,  is surjective. morphism of semi-models. Then η : Ω → Ω Proof From (9.2.2), (9.2.4) and (9.2.6), = Ω



t ( ψ h(D)) =

t≤0

 t≤0

t (η(h(D))) = ψ



η(ψt (h(D))) = η(Ω),

t≤0

and the statement holds.



  t ) be two Lemma 9.2.11 Let (φt ) be a semigroup in D. Let (Ω, h, ψt ) and (Ω, h, ψ    ˆ μˆ : (Ω, h, ψt ) → (Ω, h, ψt ) are two holomorphic semi-models for (φt ). Suppose η, morphisms of semi-models. Then ηˆ = μ. ˆ Proof By (9.2.4), η ◦ h = μ ◦ h. Hence, by (9.2.6), for every t ∈ R, t ◦ η ◦ h = ψ t ◦ μ ◦ h = μ ◦ ψt ◦ h. η ◦ ψt ◦ h = ψ Now, if z ∈ Ω, by (9.2.2) there exist ζ ∈ D and t ≥ 0 such that z = ψ−t (h(ζ )), and the previous equation then shows that η(z) = μ(z). Hence, η = μ and ηˆ = μ. ˆ  As a simple consequence of these lemmas, we can characterize isomorphisms of semi-models as follows:   t ) be Corollary 9.2.12 Let (φt ) be a semigroup in D. Let (Ω, h, ψt ) and (Ω, h, ψ   t ) is a h, ψ two holomorphic semi-models for (φt ). Assume ηˆ : (Ω, h, ψt ) → (Ω, morphism of semi-models. Then the following are equivalent: (1) η is a biholomorphism, (2) ηˆ is an isomorphism of holomorphic semi-models,   t ) → (3) there exists a morphism of holomorphic semi-models μˆ : (Ω, h, ψ (Ω, h, ψt ). Proof Assume (1) holds. Then it is easy to see that η−1 defines a morphism of −1 : (Ω,   t ) → (Ω, h, ψt ) which inverts η. h, ψ ˆ Hence (2) holds. semi-models η If (2) holds then (3) follows trivially. Assume (3) holds. Then μˆ ◦ ηˆ : (Ω, h, ψt ) → (Ω, h, ψt ) is a morphism of semi models. Hence μˆ ◦ ηˆ = id Ω by Lemma 9.2.11. Therefore, μ ◦ η = idΩ . This implies that η is injective. By Lemma 9.2.10, η is also surjective. Hence (1) holds.  The previous result shows that one can give a natural partial order among isomorphism classes of semi-models for a given semigroup (φt ) in D, defined in the   t ) if there exists a morphism of semi-models h, ψ following way: (Ω, h, ψt ) ≥ (Ω,   t ). h, ψ ηˆ : (Ω, h, ψt ) → (Ω,

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9 Models and Koenigs Functions

Remark 9.2.13 The partial order ≥ on the isomorphism classes of semi-models for a given semigroup (φt ) in D is not total. Indeed, for any n ∈ N it is easy to see that (D, z n , z → eiθnt z) is a holomorphic semi-model for the group (φt ) of D defined by φt (z) := eiθt z, for all t ≥ 0 and for some fixed θ ∈ R \ {0}. If n, m ∈ N, n, m ≥ 2 and m, n are relatively prime, then there exist no morphisms of holomorphic semi-models between (D, z m , z → eiθmt z) and (D, z n , z → eiθnt z). Indeed, suppose by contradiction that there exists a morphism of holomorphic semi-models hˆ : (D, z m , z → eiθmt z) → (D, z n , z → eiθnt z). Then there exists a non-constant holomorphic function h : D → D such that for all t ≥ 0 and z ∈ D, h(eiθmt z) = eiθnt h(z).

(9.2.7)

j In particular, h(0) = 0. Writing the expansion of h at 0 as h(z) = ∞ j=1 a j z , and substituting in (9.2.7), it follows that for every j ∈ N such that a j = 0, eiθm jt = eiθnt for every t ≥ 0. This is the case if and only if m j = n. Since m and n are relatively prime, it follows that a j = 0 for all j ∈ N and h ≡ 0, a contradiction. The main basic result is that there exists a unique (up to isomorphisms of semimodels) maximal element for this partial ordering, and such a maximal element is a holomorphic model. In order to see this, we first deal with uniqueness by showing that holomorphic models, if exist, are indeed maximal: Proposition 9.2.14 Let (φt ) be a semigroup in D. Suppose (Ω, h, ψt ) is a holomor  t ) is a holomorphic semi-model for (φt ), then there h, ψ phic model for (φt ). If (Ω,   t ). h, ψ exists a unique morphism of holomorphic semi-models ηˆ : (Ω, h, ψt ) → (Ω, Proof Uniqueness of the morphism follows from Lemma 9.2.11. To prove existence, for t ≥ 0, let Ωt := ψ−t (h(D)). Note that, if s > t ≥ 0, then Ωt ⊂ Ωs . Indeed, let z ∈ Ωt . Then ψt (z) ∈ h(D). By Proposition 9.2.4, ψs (z) = ψs−t (ψt (z)) ∈ h(D), thus, z ∈ Ωs . Hence, by (9.2.2), Ω is the growing union of the Ωt ’s.  by Now, fix t ≥ 0 and define a holomorphic map ηt : Ωt → Ω −t ◦  h ◦ h −1 |h(D) ◦ ψt . ηt := ψ Since ψt (Ωt ) = h(D) and h is univalent, the map ηt is well defined and holomorphic. Let s ≥ t ≥ 0. Then −s ◦  −s ◦  η s | Ωt = ψ h ◦ h −1 |h(D) ◦ ψs |Ωt = ψ h ◦ h −1 |h(D) ◦ ψs−t ◦ ψt |Ωt (9.2.3)

−s ◦  −s ◦ ψ s−t ◦  = ψ h ◦ φs−t ◦ h −1 |h(D) ◦ ψt |Ωt = ψ h ◦ h −1 |h(D) ◦ ψt |Ωt −t ◦  =ψ h ◦ h −1 |h(D) ◦ ψt |Ωt = ηt . Hence, ηs coincides with ηt on Ωt . Thus, we can well define a holomorphic map  by setting η:Ω→Ω

9.2 Holomorphic Models

241

η(z) := ηt (z)

if z ∈ Ωt .

In order to conclude the proof, we have to show that η defines a morphism of holo  t ). h, ψ morphic semi-model ηˆ : (Ω, h, ψt ) → (Ω, To start with, note that for all t ≥ 0 −t η◦h =ψ −t =ψ

−t ◦  ◦ h ◦ h −1 |h(D) ◦ ψt ◦ h = ψ h ◦ h −1 |h(D) ◦ h ◦ φt −t ◦ ψ t ◦  ◦ h ◦ φt = ψ h = h,

hence, (9.2.4) holds. Finally, fix t ≥ 0. Let s > t. Then on Ωs , t ◦ η = ψ t ◦ ψ −s ◦  ψ h ◦ h −1 |h(D) ◦ ψs −(s−t) ◦  =ψ h ◦ h −1 |h(D) ◦ ψs−t ◦ ψt = ηs−t ◦ ψt = η ◦ ψt . By the arbitrariness of s, (9.2.5) holds.



Corollary 9.2.15 Let (φt ) be a semigroup in D. Then a holomorphic model for (φt ), if it exists, is unique up to isomorphisms of holomorphic semi-models. In particular, the base space of the model is unique up to biholomorphisms.   t ) are two holomorphic models for (φt ). Proof Assume that (Ω, h, ψt ) and (Ω, h, ψ By Proposition 9.2.14, there exist a morphism of semi-models ηˆ : (Ω, h, ψt ) →   t ) → (Ω, h, ψt ). Hence, by   t ) and a morphism of semi-model μˆ : (Ω, h, ψ (Ω, h, ψ  is a biholomorphism and the two models Corollary 9.2.12 it follows that η : Ω → Ω are isomorphic.  The previous result allows to characterize groups of automorphisms: Corollary 9.2.16 Let (φt ) be a semigroup in D. Let (Ω, h, ψt ) be a holomorphic model for (φt ). Then (φt ) is a group of automorphisms if and only if h(D) = Ω. Proof If (φt ) is a group of automorphisms, then (D, idD , φt ) is another holomorphic model for (φt ). Hence the result follows from Corollaries 9.2.15 and 9.2.12. On the other hand, if h(D) = Ω, by (9.2.3) it follows at once that (φt ) is an automorphism for all t ≥ 0.  A straightforward consequence of the maximality of holomorphic models proved in Corollary 9.2.15, is the following result, which will be useful later on. Corollary 9.2.17 Let (φt ) be a semigroup in D. Suppose (φt ) admits two models h, ψt ). with the same base and same automorphisms group, say (Ω, h, ψt ) and (Ω,  Then there exists an automorphism ν : Ω → Ω such that ν ◦ ψt = ψt ◦ ν for all t ≥ 0 and h˜ = ν ◦ h. We are now ready to prove the main result of this section.

242

9 Models and Koenigs Functions

Theorem 9.2.18 Let (φt ) be a semigroup in D. Then there exists a holomorphic model (Ω, h, ψt ) for (φt ). The model (Ω, h, ψt ) is unique up to isomorphisms of holomorphic semi-models. Moreover, either Ω is biholomorphic to D or Ω is biholomorphic to C. Proof Uniqueness of the model follows from Corollary 9.2.15. Thus we can concentrate on the existence of the model. Endow [0, +∞) with the discrete topology and consider the relation ∼ on D × [0, +∞) defined as follows: (z, t) ∼ (w, s) if there exists u ∈ [0, +∞), u ≥ max{t, s} such that φu−t (z) = φu−s (w). It is easy to see that ∼ is an equivalence relation on D × [0, +∞). Let Ω := D × [0, +∞)/ ∼ be the quotient space, endowed with the quotient topology. Let π : D × [0, +∞) → Ω be the natural projection which maps (z, t) ∈ D × [0, +∞) to its equivalence class π((z, t)) in Ω. For t ∈ [0, +∞), define h t : D → Ω by h t (z) := π((z, t)). Clearly, h t is continuous and it is also open because [0, +∞) is endowed with the discrete topology. Also, h t is injective. Indeed, if h t (z) = h t (w) for some z, w ∈ D, then there exists u ∈ [0, +∞), u ≥ t such that φu−t (z) = φu−t (w). But then, z = w since φu−t is injective by Theorem 8.1.17. Let Ωt := h t (D). Therefore, h t : D → Ωt ⊂ Ω is a homeomorphism for all t ∈ [0, +∞). Since clearly (φt−s (z), t) ∼ (z, s) for all t ≥ s, h s = h t ◦ φt−s , 0 ≤ s ≤ t.

(9.2.8)

This implies that Ωs ⊂ Ωt ,

for all 0 ≤ s ≤ t.

(9.2.9)

In particular, Ω = ∪t∈N Ωt . From this, it follows easily that Ω is a Hausdorff, second countable and pathwise connected topological space. Moreover, if for all t ≥ 0 we declare h −1 t : Ωt → D to be a chart of Ω, then the )} of Ω is holomorphic by (9.2.8). Hence, Ω has the structure of a atlas {(Ωt , h −1 t Riemann surface. Let h := h 0 : D → Ω0 ⊂ Ω. By construction, the map h is univalent. Now we want to define a continuous group of automorphisms of Ω which satisfies (9.2.1) and (9.2.2). Fix s ≥ 0. Let t ≥ s. Then (9.2.8)

−1 −1 = h t−s ◦ φt−s ◦ h −1 h t−s ◦ h −1 t |Ωs = h t−s ◦ h t ◦ h s ◦ h s s (9.2.8)

−1 −1 = (h t−s ◦ φt−s ◦ h −1 = h 0 ◦ h −1 s = h ◦ hs . 0 ) ◦ h0 ◦ hs

Therefore, for every t ≥ s, the holomorphic maps h t−s ◦ h −1 t : Ωt → Ω coincide on Ωs . Thus, the holomorphic map h ◦ h −1 s : Ωs → Ω extends holomorphically to a map ψs : Ω → Ω, which is defined as

9.2 Holomorphic Models

243

ψs (z) := (h t−s ◦ h −1 t )(z), z ∈ Ωt , for t ≥ s. The map ψs is injective. Indeed, assume z, w ∈ Ω and ψs (z) = ψs (w). Then there exists t ≥ s such that z, w ∈ Ωt . Hence −1 h t−s (h −1 t (z)) = ψs (z) = ψs (w) = h t−s (h t (w)),

which implies z = w since h t−s ◦ h −1 t is univalent on Ωt . The map ψs is also surjective. Indeed, let w ∈ Ω. From the definition of ψs , it holds ψs ◦ h t = h t−s for all t ≥ s. Let t ≥ s be such that w ∈ Ωt−s . Hence, there exists ζ ∈ D such that h t−s (ζ ) = w. Therefore, setting z = h t (ζ ) we have ψs (z) = ψs (h t (ζ )) = h t−s (ζ ) = w, proving that ψs is surjective. Therefore, ψs : Ω → Ω is an automorphism for all s ≥ 0. Fix t ≥ 0. Since h(D) ⊂ Ωt , (9.2.8)

−1 ψt ◦ h = (h ◦ h −1 t ) ◦ h = h ◦ (h t ◦ h) = h ◦ φt ,

hence (9.2.1) holds. Moreover, since ψ−t = h t ◦ h −1 on h(D), then (9.2.9)

∪t≥0 ψ−t (h(D)) = ∪t≥0 h t (h −1 (h(D))) = ∪t≥0 h t (D) = ∪t≥0 Ωt = Ω, and also (9.2.2) holds. Now we show that the family (ψt ) is an algebraic group of automorphisms of Ω. Indeed, let s, t ≥ 0 and let z ∈ Ω. There exists u > s + t such that z ∈ Ωu . Therefore, −1 ψs (z) = h u−s (h −1 u (z)) and ψt (z) = h u−s−t (h u−s (z)). Hence, −1 −1 ψt (ψs (z)) = (h u−s−t ◦ h −1 u−s ◦ h u−s ◦ h u )(z) = (h u−s−t ◦ h u )(z) = ψs+t (z),

proving that (ψt ) is an algebraic group of automorphisms of Ω. The continuity of the group (ψt ) follows from Lemma 9.2.3. It remains to show that Ω is either biholomorphic to D or biholomorphic to C. Since every loop γ in Ω is contained in Ωt for some t large enough, it follows that Ω is a simply connected Riemann surface. Also, Ω is not compact, for otherwise one could extract a finite covering from the open covering {Ωt } of Ω. But then Ω = Ωt for some t ∈ [0, +∞) by (9.2.9), hence Ω would be compact and biholomorphic to the unit disc, a contradiction. By Theorem 1.1.4, Ω is either biholomorphic to D or to C. 

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9 Models and Koenigs Functions

9.3 Canonical Models and Koenigs Functions In the previous section we proved that every semigroup in D admits a holomorphic model with simple base space. Now we are going to refine this result according to the different types of semigroups. We start with the following preliminary result: Lemma 9.3.1 Let (φt ) be a semigroup in D. Let (Ω, h, ψt ) be a holomorphic model for (φt ). Then kΩ (h(z), h(w)) = lim ω(φt (z), φt (w)) z, w ∈ D. t→+∞

(9.3.1)

Moreover, cD (φt ) = cΩ (ψt ). Proof By (9.2.1), for 0 ≤ s ≤ t, we have ψ−s (h(D)) = ψ−t (ψt−s (h(D))) = ψ−t (h(φt−s (D))) ⊂ ψ−t (h(D)), hence the Riemann surface Ω is the growing union of the open sets {ψ−t (h(D))}t≥0 . Therefore, by Proposition 1.3.15 and taking into account that h is an isometry between ω and kh(D) , we have kΩ (h(z), h(w)) = lim kψ−t (h(D)) (h(z), h(w)) = lim kh(D) (ψt (h(z)), ψt (h(w))) t→+∞

t→+∞

= lim kh(D) (h(φt (z)), h(φt (w))) = lim ω(φt (z), φt (w)), t→+∞

t→+∞

which proves (9.3.1). Finally, let z ∈ D. Then for all u ≥ 0, the u-th hyperbolic step of (φt ) satisfies su (φt , z) = lim ω(φv (z), φv+u (z)) = lim ω(φv (z), φv (φu (z))) v→+∞

v→+∞

(9.3.1)

= kΩ (h(z), h(φu (z))) = kΩ (h(z), ψu (h(z))) = kΩ (ψv (h(z)), ψv+u (h(z))) = lim kΩ (ψv (h(z)), ψv+u (h(z))) v→+∞

= su (ψt , h(z)), where we used that ψv is an isometry for kΩ for all v ≥ 0. Hence, cD (φt ) = cΩ (ψt ) by Proposition 9.1.7.  A first consequence of the previous result and the results of the previous section is that holomorphic models detect the type of the semigroups: Proposition 9.3.2 Let (φt ) be a non-trivial semigroup in D and let (Ω, h, ψt ) be a holomorphic model for (φt ). Then (φt ) is elliptic if and only if (ψt ) has one and only one common fixed point in Ω. Moreover if (φt ) is non-elliptic, then (1) (φt ) is hyperbolic if and only if cΩ (ψt ) > 0,

9.3 Canonical Models and Koenigs Functions

245

(2) (φt ) is parabolic if and only if cΩ (ψt ) = 0. Proof The existence of a holomorphic model follows from Theorem 9.2.18. Moreover, by Proposition 9.2.4, if (φt ) is elliptic then (ψt ) has a common fixed point. Conversely, if there exists w0 ∈ Ω such that ψt (w0 ) = w0 for all t ≥ 0, by (9.2.2), it follows that w0 ∈ h(D). Hence by (9.2.3), h −1 (w0 ) is a fixed point for φt for all t ≥ 0. In particular, (φt ) is either trivial or elliptic, and, clearly, it is not trivial if and only if w0 is the only fixed point of (ψt ) in Ω. If (φt ) is non-elliptic, then the result follows directly from Theorem 9.1.9 and Lemma 9.3.1.  A second interesting consequence of Lemma 9.3.1 is: Corollary 9.3.3 Let (φt ) be a non-elliptic semigroup in D. If there exist z ∈ D and u > 0 such that the hyperbolic step of order u is equal to 0, that is su (φt , z) = 0, then for every v ≥ 0 and w ∈ D it holds sv (φt , w) = 0. Proof Let (Ω, h, ψt ) be a holomorphic model for (φt ) given by Theorem 9.2.18, with either Ω = D or Ω = C. Let v ≥ 0, and z ∈ D. Set w := φv (z). By Lemma 9.3.1 kΩ (h(z), h(w)) = lim ω(φt (z), φt (w)) t→+∞

= lim ω(φt (z), φt+v (z)) = sv (φt , z).

(9.3.2)

t→+∞

Assume su (φt , z) = 0 for some u > 0 and z ∈ D, and let w := φu (z). Note that w = z, since (φt ) is not elliptic. Moreover, h is injective, hence h(z) = h(w). Therefore, by (9.3.2), the domain Ω has two different points whose hyperbolic distance is zero. Hence Ω = C and kΩ ≡ 0 by Proposition 1.3.12. Then by (9.3.2) the result holds.  Definition 9.3.4 Let (φt ) be a semigroup in D. We say that (φt ) is of positive hyperbolic step (or it is of automorphic type) if there exist z ∈ D and u ≥ 0 such that su (φt , z) > 0. Otherwise, we say that (φt ) is of zero hyperbolic step (or it is of non-automorphism type). By Remark 9.1.6, groups of automorphisms of D are of positive hyperbolic step, while elliptic semigroups in D which are not groups, are always of zero hyperbolic step. By Remark 9.1.6 and Corollary 9.3.3, (φt ) is of positive hyperbolic step if and only if su (φt , z) > 0 for all u > 0 and z ∈ D, z different from the Denjoy-Wolff point of the semigroup. Let us recall that H := {ζ ∈ C : Re ζ > 0}, H− := {ζ ∈ C : Re ζ < 0} and, given ρ > 0, Sρ := {ζ ∈ C : 0 < Re ζ < ρ}. For ρ = 1 we simply write S := S1 . Theorem 9.3.5 Let (φt ) be a semigroup in D. Then (1) (φt ) is the trivial semigroup if and only if (φt ) has a holomorphic model (D, idD , z → z).

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9 Models and Koenigs Functions

(2) (φt ) is a group of elliptic automorphisms with spectral value iθ , for θ ∈ R \ {0}, if and only if (φt ) has a holomorphic model (D, h, z → e−iθt z). (3) (φt ) is elliptic, not a group, with spectral value λ, for λ ∈ C with Re λ > 0, if and only if (φt ) has a holomorphic model (C, h, z → e−λt z). (4) (φt ) is hyperbolic with spectral value λ > 0 if and only if it has a holomorphic model (S πλ , h, z → z + it). (5) (φt ) is parabolic of positive hyperbolic step if and only if it has a holomorphic model either of the form (H, h, z → z + it) or of the form (H− , h, z → z + it). (6) (φt ) is parabolic of zero hyperbolic step if and only if it has a holomorphic model (C, h, z → z + it). Proof (1) It is obvious. (2) If (φt ) is an elliptic group in D, with common fixed point τ ∈ D and spectral value iθ , the model is given by (D, Tτ , z → e−iθt z). On the other hand, if the model is given by (D, h, z → e−iθt z) it follows that e−itθ h(D) ⊂ h(D) by Proposition 9.2.4, showing that h(D) is a disc centered at 0. Hence, using (9.2.2), it holds D = ∪t≥0 eiθt h(D) = h(D). By Corollary 9.2.16 it follows that (φt ) is a group of automorphisms. Equation (9.2.3) implies that (φt ) is an elliptic group with fixed point h −1 (0). Moreover, differentiating (9.2.3) in z at h −1 (0) it follows immediately that the spectral value of (φt ) is iθ . (3) If (φt ) is a semigroup, not a group, in D with Denjoy-Wolff point τ ∈ D and spectral value λ, by Theorem 9.2.18, there exists a holomorphic model (Ω, h, ψt ), with either Ω = D or Ω = C. By Theorem 8.3.6, for every z, w ∈ D it holds limt→+∞ ω(φt (z), φt (w)) = 0, hence, by (9.3.1), the hyperbolic distance of Ω is identically zero, therefore, Ω = C. By Proposition 9.2.4, the group ψt has a fixed point in C that, up to conjugation with a translation, we can assume to be 0. Therefore, by Theorem 8.4.5, ψt (z) = eμt z for some μ ∈ C. Differentiating (9.2.3) in z at τ it follows immediately that μ = λ. On the other hand, if (φt ) has a holomorphic model of the form (C, h, z → e−λt z), by (9.2.2), the point 0 belongs to h(D), say h(τ ) = 0 for some τ ∈ D. By (9.2.1), φt (τ ) = τ for all t ≥ 0, hence (φt ) is elliptic with Denjoy-Wolff point τ . As before, it follows easily that the spectral value of (φt ) is λ. (4) Suppose (φt ) is hyperbolic, with Denjoy-Wolff point τ ∈ ∂D and spectral value λ > 0. By Theorem 9.2.18, there exists a holomorphic model (Ω, h, ψt ) for (φt ) with Ω = C or Ω = D. By (9.1.3) and Lemma 9.3.1, it holds 0
0, the group (ψt ) can not be elliptic (see Remark 9.1.4). Thus, again by (9.1.3), (ψt ) is a hyperbolic group of automorphisms of D with the same spectral value λ. Using a Möbius transformation C such that C(D) = H (see Proposition 8.3.8), we can find an isomorphic holomorphic model for (φt ) given by (H, C ◦ h, z → eλt z). Let log z denote the principal branch of the logarithm on H, and let f : H → S πλ be the biholomorphism given by

9.3 Canonical Models and Koenigs Functions

f (z) :=

247

π i log z + . λ 2λ

(9.3.3)

Note that f (eλt z) = f (z) + it for all z ∈ H and t ≥ 0. Therefore it is easy to see that (S πλ , f ◦ C ◦ h, z → z + it) is a holomorphic model for (φt ). On the other hand, if (S πλ , h, z → z + it) is a holomorphic model for (φt ), let f be the map defined in (9.3.3). Then (H, f −1 ◦ h, z → eλt z) is a holomorphic model for (φt ). By Proposition 8.3.8, the group (z → eλt z) is conjugated via a Möbius transformation C which maps D onto H to a hyperbolic group (ψt ) of D with spectral value λ. Therefore, (D, C −1 ◦ f −1 ◦ h, ψt ) is a holomorphic model for (φt ). By (9.1.3) and Lemma 9.3.1, it follows that cD (φt ) = λ2 > 0 and hence, since (φt ) can not be elliptic by Remark 9.1.4, (φt ) is a hyperbolic semigroup in D with spectral value λ. (5) Assume (φt ) is parabolic with positive hyperbolic step. By Theorem 9.2.18, there exists a holomorphic model (Ω, h, ψt ), with Ω = D or C. By (9.3.1), kΩ (h(z), ψ1 (h(z))) = kΩ (h(z), h(φ1 (z))) = lim ω(φt (z), φt+1 (z)) = s1 (φt , z) > 0, t→+∞

hence Ω = D. The group (ψt ) has divergence rate cD (ψt ) = cD (φt ) = 0 by Lemma 9.3.1. Hence (ψt ) is either elliptic or parabolic. The first case is excluded by the previous point (1) for otherwise (φt ) would be a group of elliptic automorphisms. Hence, (ψt ) is a group of parabolic automorphisms of D. By Proposition 8.3.8, conjugating with a suitable Möbius transformation, it follows that a holomorphic model for (φt ) is either (H, h, z → z + it) or (H− , h, z → z + it). Conversely, if (φt ) admits a holomorphic model of the forms (H, h, z → z + it) or (H− , h, z → z + it), then by conjugating with a Möbius transformation (see Proposition 8.3.8), it follows that (φt ) has a holomorphic model (D, h, ψt ) with (ψt ) a group of parabolic automorphisms of D. By Lemma 9.3.1, it follows then that (φt ) is parabolic with positive hyperbolic step. (6) The proof follows from an argument similar to (5), just noting that by Lemma 9.3.1, (φt ) is of zero hyperbolic step if and only if the domain Ω given in Theorem 9.2.18 is C. Moreover, (ψt ) can not have a fixed point in C, for otherwise (φt ) would be elliptic for what we already proved (2). Therefore (ψt ) is a group of translations in C by Theorem 8.4.5.  Remark 9.3.6 The two holomorphic models appearing in (5) are not isomorphic. To see this, assume that (H− , h, z → z + it) is a holomorphic model of a semigroup (φt ) in D. Let g : H− → H be the biholomorphism given by g(z) = −z. Then (H, g ◦ ˆ and then it is also a h, z → z − it) is isomorphic to (H− , h, z → z + it) via g, ˜ z → z + it) were a holomorphic model for holomorphic model for (φt ). If (H, h, (φt ), then by Corollary 9.2.15, there would exist an isomorphism of models νˆ : ˜ z → z + it) → (H, h, z → z − it) defined by a biholomorphism ν : H → (H, h, H, which is a linear fractional map. In particular, the oriented line R  r → ir having H on the right would be mapped onto the oriented line R  r → ν(ir ) having H on the right. However, by (9.2.6), it holds ν(ir ) = ν(0) − ir for all r ∈ R, a contradiction.

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9 Models and Koenigs Functions

One striking consequence of Theorem 9.3.5 is that a semigroup of zero hyperbolic step must be elliptic or parabolic: Corollary 9.3.7 Let (φt ) be a hyperbolic semigroup in D. Then (φt ) is of positive hyperbolic step. Proof By Theorem 9.3.5, (φt ) has a holomorphic model of the form (Sρ , h, z → z + it) for some ρ > 0. In particular, by (9.3.1), s1 (φt , 0) = lim ω(φt (0), φt+1 (0)) = kSρ (h(0), h(φ1 (0))) = kSρ (h(0), h(0) + i) > 0, t→+∞

hence (φt ) is of positive hyperbolic step.



The previous theorem shows that the model for a parabolic semigroup of zero hyperbolic step is (C, h, z → z + it). Therefore, taking into account that kC ≡ 0, Lemma 9.3.1 implies: Corollary 9.3.8 Let (φt ) be a parabolic semigroup in D of zero hyperbolic step. Then, for all z, w ∈ D, lim ω(φt (z), φt (w)) = 0. t→+∞

Theorem 9.3.5 provides simple holomorphic models for a semigroup (φt ) in D, which also give information on hyperbolic and dynamical properties of (φt ). Indeed, by (9.2.3), the intertwining map h (and its image) contains all the information about the semigroup (φt ). These models deserve a special name: Definition 9.3.9 Let (φt ) be a semigroup in D. The holomorphic model for (φt ) given by Theorem 9.3.5 is called the canonical model of (φt ) and the intertwining map h is the Koenigs function associated with (φt ). At a first sight, the use of the definite article “the” for denoting a canonical model might not seem to be a good idea. Indeed, although Theorem 9.3.5 assures that every canonical model for a given semigroup of D has the same base space and group of automorphisms, a priori, there could be other “Koenigs functions” intertwining the given semigroup with the same group of automorphisms. This is the case, in fact, but it turns out that all Koenigs functions are essentially unique up to a constant, so that the use of the definite article “the” is well justified: Proposition 9.3.10 Let (φt ) be a non-trivial semigroup in D. Let (Ω, h, ψt ) be a canonical model for (φt ) and let h˜ : D → Ω be holomorphic. Then (1) If (φt ) is an elliptic group then h˜ is a Koenigs function for (φt ) if and only if ˜ there exists θ ∈ R such that h(z) = eiθ h(z) for all z ∈ D. (2) If (φt ) is elliptic, not a group, then h˜ is a Koenigs function for (φt ) if and only if ˜ there exists a ∈ C \ {0} such that h(z) = ah(z) for all z ∈ D. (3) If (φt ) is either hyperbolic or parabolic of positive hyperbolic step, then h˜ is ˜ = a Koenigs function for (φt ) if and only if there exists a ∈ R such that h(z) h(z) + ai for all z ∈ D.

9.3 Canonical Models and Koenigs Functions

249

(4) If (φt ) is parabolic of zero hyperbolic step, then h˜ is a Koenigs function for (φt ) ˜ if and only if there exists a ∈ C such that h(z) = h(z) + a for all z ∈ D. Proof The “if” implications of the statements are clear. Thus, assume h˜ is a Koenigs function for (φt ). By Corollary 9.2.17, there exists an automorphism ν : Ω → Ω such that h˜ = ν ◦ h and ν ◦ ψt = ψt ◦ ν for all t ≥ 0. In case (φt ) is elliptic, being Ω = C (or Ω = D in case (φt ) is a group) and ψt (z) = e−λt z for some λ ∈ C \ {0}, with Re λ ≥ 0, it follows that ν(0) = 0. Hence, ν is a linear map and the statements (1) and (2) hold. Now we examine the case (φt ) is non-elliptic. In this case, for all z ∈ Ω and all t ≥ 0, ν(z + it) = ν(z) + it. (9.3.4) Differentiating (9.3.4) in t and setting t = 0 we obtain ν (z) ≡ 1. Integrating in z, we obtain ν(z) = z + c for some c ∈ C. Moreover, if (φt ) is either hyperbolic or parabolic of positive hyperbolic step, since Ω is a half-plane or a strip, and ν(Ω) = Ω, then c is pure imaginary. From this, (3) and (4) hold.  If (φt ) is an elliptic group with Denjoy-Wolff point τ ∈ D, then the canonical model is given by (D, Tτ , z → e−iθt z) for some θ ∈ R. There are elliptic semigroups, not groups, which have the same Koenigs function. For instance, the semigroup (ϕt ) defined by ϕt (z) := Tτ−1 (e−t Tτ (z)), for z ∈ D. Its canonical model is (C, Tτ , z → e−t z). Thus, Koenigs functions by themselves do not characterize elliptic groups. However, for non-elliptic semigroups, Koenigs functions do characterize groups. In order to see this, we start with a lemma. +z , z ∈ D, Lemma 9.3.11 Let a ∈ C, a = 0, and τ ∈ ∂D. Then the map h(z) = ai ττ −z is the Koenigs function of a non-elliptic semigroup if and only if Re a ≥ 0. In such a case, setting φt (z) := h −1 (h(z) + it), for z ∈ D and t ≥ 0, (φt ) is a parabolic semigroup of linear fractional self-maps of D with Denjoy-Wolff point τ ∈ ∂D and τ φt (τ ) = at for all t ≥ 0.

Proof By Theorem 9.3.5, the function h is the Koenigs function of a non-elliptic semigroup in D if and only if (1)  h(D) + it ⊂ h(D) for all t ≥ 0, and (2) t≥0 (h(D) − it) is equal to either S πλ for some λ > 0, or H, or H− or C.  Since h(D) = ai H, it is clear that condition (2) is always satisfied with t≥0 (h(D) − it) equals to either H, or H− or C, but it can never be equal to S πλ for any λ > 0. In particular, a non-elliptic semigroup in D having h as Koenigs function, is necessarily parabolic. Multiplying by ai , condition (1) is equivalent to H + at ⊂ H for all t ≥ 0, that is, Re a ≥ 0. Therefore, h is the Koenigs function of a parabolic semigroup in D if and only if Re a ≥ 0. , we deduce that, for z ∈ D and Now, suppose Re a ≥ 0. Since h −1 (w) = τ aw−i aw+i t ≥ 0,

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9 Models and Koenigs Functions

φt (z) = h −1 (h(z) + it) = τ

2z + at (τ − z) a(h(z) + it) − i =τ . a(h(z) + it) + i 2τ + at (τ − z)

In particular, (φt ) is a semigroup of linear fractional maps. Thus, φt (z) =

4τ 2 8τ 2 at , φ (z) = , t (2τ + at (τ − z))2 (2τ + at (τ − z))3

z ∈ D, t ≥ 0.

Hence, φt (τ ) = τ , φt (τ ) = 1 and τ φt (τ ) = at.



Proposition 9.3.12 Let (φt ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D and spectral value λ ∈ [0, +∞). Let h be the Koenigs function of (φt ). Then (1) (φt ) is a hyperbolic group with other fixed point σ ∈ ∂D \ {τ } if and only if   i τ +z τ +σ π h(z) = log − + , λ τ −z τ −σ 2λ where log : H → C denotes the principal branch of the logarithm. (2) (φt ) is a parabolic group if and only if there exists a ∈ C \ {0}, with Re a = 0, such that i τ +z h(z) = . aτ −z In this case, a = τ φ1 (τ ). Proof Assume (φt ) is a hyperbolic group with Denjoy-Wolff point τ ∈ ∂D and other − ττ +σ . Then C is a Möbius transformation such fixed point σ ∈ ∂D. Let C(z) := ττ +z −z −σ that C(D) = H and C(τ ) = ∞ and C(σ ) = 0. Therefore, (C ◦ φt ◦ C −1 ) is a group (whose iterates are Möbius transformations) in H which fixes 0 and ∞. It follows that (C ◦ φt ◦ C −1 )(w) = μt w for some μt > 0. By the Chain Rule for derivatives μt+s = μt μs for every s, t ≥ 0 hence by Proposition 8.1.14, μt = eλt for all t ≥ 0. Now, if f is given by (9.3.3), then f (eλt w) = f (w) + it. Hence, h := f ◦ C : D → S πλ is the Koenigs function of (φt ). Conversely, if (φt ) is a hyperbolic semigroup with canonical model (Ω, h, z → z + it) with h as in (1), then h(D) = S πλ . Since this domain is invariant under the map z → z + it for t ∈ R, it follows by (9.2.2) that Ω = h(D). Hence, by Corollary 9.2.16, (φt ) is a group. Now, assume (φt ) is a parabolic group. Let Cτ : D → H be the Cayley transform with respect to τ . Then (Cτ ◦ φt ◦ Cτ−1 ) is a group (whose iterates are Möbius transformations) in H which fixes ∞ and has no other fixed points in H. Therefore, ψt (w) := (Cτ ◦ φt ◦ Cτ−1 )(w) = w + iμt . Hence, μt+s = μt + μs for all s, t ≥ 0, and μt = itα for some α ∈ R \ {0} and all t ≥ 0 by Theorem 8.1.11. Let A(w) := wα . Then A is a Möbius transformation such that A(H) = H if α > 0 and A(H) = H− if α < 0. Moreover, (A ◦ ψt ◦ A−1 )(w) = w + it. Hence, h = A ◦ Cτ is the Koenigs function of (φt ) and setting a = αi we have the claimed form of h in (2).

9.3 Canonical Models and Koenigs Functions

251

Conversely, if (φt ) is a parabolic semigroup with canonical model (Ω, h, z → z + it) with h as in (2), then h(D) = H or h(D) = H− . Since these domains are invariant under the map z → z + it for t ∈ R, it follows by (9.2.2) that Ω = h(D). Hence, by Corollary 9.2.16, (φt ) is a group.  Finally, the equality a = τ φ1 (τ ) follows from Lemma 9.3.11.

9.4 Basic Properties of Koenigs Functions The aim of this section is to give both a geometric and an analytic characterization of those (univalent) functions which are Koenigs functions of a semigroup with a prescribed Denjoy-Wolff point. We first consider the case of an interior Denjoy-Wolff point, and then of a boundary Denjoy-Wolff point. We start with the following simple fact: Proposition 9.4.1 Let (φt ) be an elliptic semigroup in D with Denjoy-Wolff point τ ∈ D and spectral value λ ∈ C \ {0} with Re λ ≥ 0. Let h be the associated Koenigs function. Then h(τ ) = 0. Proof For all t ≥ 0,

h(τ ) = h(φt (τ )) = e−λt h(τ )

which implies h(τ ) = 0.



Definition 9.4.2 Let λ ∈ C \ {0} be such that Re λ ≥ 0. A domain Ω ⊂ C such that 0 ∈ Ω is λ-spirallike if e−λt Ω ⊆ Ω for all t ≥ 0. A λ-spirallike domain with λ ∈ (0, +∞) is also called starlike. A map h : D → C is λ-spirallike with respect to ζ ∈ D if it is univalent, h(ζ ) = 0 and h(D) is a λ-spirallike domain. A λ-spirallike map with λ ∈ (0, +∞) is also called starlike. Theorem 9.4.3 Let h : D → C be the Koenigs function of an elliptic semigroup (φt ) in D with Denjoy-Wolff point τ ∈ D and spectral value λ ∈ C \ {0} with Re λ ≥ 0. Then h is λ-spirallike with respect to τ . Conversely, if h : D → C is λ-spirallike with respect to τ ∈ D, for some λ ∈ C \ {0} with Re λ ≥ 0, let φt (z) := h −1 (e−λt h(z)) for z ∈ D and t ≥ 0. Then (φt ) is an elliptic semigroup in D with Denjoy-Wolff point τ and spectral value λ. Moreover, (1) if Re λ > 0 then (φt ) is a semigroup which is not a group and h is its Koenigs function, (2) if Re λ = 0 then (φt ) is a group, h is bounded, and there exists α ∈ (0, +∞) such that αh is the Koenigs function of (φt ). Proof Let h be the Koenigs function of an elliptic semigroup (φt ) in D with DenjoyWolff point τ ∈ D and spectral value λ. By Theorem 9.3.5 for all t > 0 e−λt h(D) = h(φt (D)) ⊂ h(D),

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9 Models and Koenigs Functions

hence h(D) is λ-spirallike. Moreover, h(τ ) = 0 by Proposition 9.4.1, and h is a λ-spirallike function with respect to τ . Conversely, let us assume h is λ-spirallike with respect to τ ∈ D. It is clear that (φt ) is a (continuous) semigroup in D, with Denjoy-Wolff point τ . Moreover, differentiating in z at z = τ the equation φt (z) = h −1 (e−λt h(z)), we obtain φt (τ ) = e−λt , showing that λ is the spectral value of (φt ). Now, assume Re λ > 0. Then (φt ) is not a group, and in order to see that h is its Koenigs function we have to show that h satisfies (9.2.1)—which is obvious from the construction—and (9.2.2) with Ω = C. But, h(D) is an open set containing 0 and Re λ > 0, hence ∪t≥0 eλt h(D) = C. Finally, in case Re λ = 0, the semigroup (φt ) is a group. Moreover, for every z ∈ h(D) the curve [0, +∞)  t → eλt z is a circle of center 0 and radius |z|. Hence, h(D) is a disc of center 0 and some “radius” a ∈ (0, +∞]. However, since h is univalent, if a = +∞, then h would be a biholomorphism between D and C, impossible. Therefore, a < +∞. Let α = a1 . Then αh(D) = D and αh satisfies (9.2.1) and (9.2.2)  with Ω = D, proving that αh is the Koenigs function of (φt ). Now, we aim to give an analytic characterization of spirallike functions. In order to do this, we need a preliminary lemma. Lemma 9.4.4 Let p : D → C be a holomorphic function such that Re p(z) > 0 for all z ∈ D. Then, for each z ∈ D \ {0}, the initial value problem d x(t) = −x(t) p(x(t)), x(0) = z, dt

(9.4.1)

has a solution x z (t) with strictly decreasing modulus on the interval 0 ≤ t < +∞, tending to zero as t → +∞. Proof It is clear that the initial value problem (9.4.1) has a maximal solution x defined in [0, δz ) for some δz > 0. We prove that δz = +∞. Take u(t) = |x(t)|2 , for all t ∈ [0, δz ). Then d (u(t)) = −2u(t)Re p(x(t)) dt and, integrating,  t  u(t) = |z| exp −2 Re p(x(s)) ds , t ∈ [0, δz ). 0

Thus |x(t)| decreases strictly as t increases. In particular, the set {x(t) : t ∈ [0, δz )} is contained in the compact set D := {w ∈ D : |w| ≤ |z|} and this implies that δz = +∞. Moreover, since Re p(z) > 0 for all z ∈ D and p is continuous in D, there exists ε > 0 such that Re p(w) ≥ ε for all w ∈ D. In particular, Re p(x(t)) ≥ ε for all t ∈ [0, +∞). Hence, |u(t)| ≤ |z|e−2εt → 0, as t → +∞. Therefore, limt→+∞ x(t) = 0. 

9.4 Basic Properties of Koenigs Functions

253

Theorem 9.4.5 Let ζ ∈ D. Let h : D → C be holomorphic, h(ζ ) = 0, h (ζ ) = 0. Let λ ∈ C \ {0} with Re λ ≥ 0. Then h is λ-spirallike with respect to ζ if and only if for every z ∈ D such that h(z) = 0 it holds  Re

1 h (z) (z − ζ )(1 − ζ z) λ h(z)

 ≥ 0.

(9.4.2)

Moreover, equality holds at some—and hence any—z ∈ D, with h(z) = 0, if and only if Re λ = 0 and ζ −z h(z) = aTζ (z) = a (9.4.3) 1 − ζz for some a ∈ C \ {0}. Proof First we prove the statement for ζ = 0. Suppose that h is λ-spirallike with respect to 0. For each t > 0, consider the function φt (z) = h −1 (e−λt h(z)), z ∈ D. (9.4.4) Then φt : D → D is holomorphic and φt (0) = 0. By Schwarz’s Lemma 1.2.1, it holds |φt (z)|≤ |z| for  all z ∈ D. Fix z ∈ D \ {0}. Hence, the real analytic function satisfies g(t) ≤ 0 for all t ≥ 0, and g(0) = 0. Differentiating g(t) := Re φt (z)−z z g(t) in t at t = 0+ , we have

 Re

 1 ∂φt (z)

≤ 0. z ∂t t=0

Since h is univalent, we have that (9.4.4) in t, we obtain h (φt (z))

∂φt (z) ∂t t=0

(9.4.5)

= 0. Differentiating now both sides of

∂φt (z) = −λe−λt h(z), ∂t

z ∈ D.

Thus, by (9.4.5),  Re

1 zh (z) λ h(z)



 = −Re

zh (z)



eλt h (φt (z)) ∂φ∂tt (z) ⎛   zh (z) ⎜ = −Re lim+ = −Re ⎝ ∂φt (z) λt 1 t→0 e h (φt (z)) ∂t

⎞ 1

∂φt (z) z ∂t t=0

and (9.4.2) holds.

⎟ ⎠ ≥ 0,

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9 Models and Koenigs Functions

Now suppose (9.4.2) holds for ζ = 0. Let Z = {z ∈ D : h(z) = 0}. Notice that Z is a discrete subset of D since h ≡ 0. Hence, by (9.4.2) the holomorphic func (z) has range in H. Let C : D → H be a Cayley transform. tion q : D \ Z  z → λ1 zhh(z) −1 Then ψ := C ◦ q : D \ Z → D is holomorphic. By the Riemann Removable Singularities Theorem, ψ extends to a holomorphic function from D to D. There are two cases: either there exists σ ∈ ∂D such that ψ(z) = σ for all z ∈ D, or ψ(D) ⊆ D. In the first case, q(z) = C(ψ(z)) = C(σ ) for all z ∈ D. In particular, C(σ ) = (z) = λ1 for all z ∈ D. Hence, it follows that this is the case if and q(0) = lim z→0 λ1 zhh(z) only if Re λ = 0 and σ = C −1 (1/λ). Moreover, since q(z) = q(0) for all z ∈ D, it follows that h(z) = zh (z) for all z ∈ D and thus h = a idD for some a ∈ C \ {0}. Hence h is λ-spirallike and h satisfies (9.4.3). In the second case, ψ(D) ⊆ D. Hence, Re q(z) = Re C(ψ(z)) > 0 for all z ∈ D. 1 is well defined and holomorphic from D to H. Therefore, the function p(z) := q(z) z For each z ∈ D, let x : [0, +∞) → D be the solution of the Eq. (9.4.1) and let w z (t) := h(x z (t)) for all t ≥ 0. Then w z (0) = h(z) and, for all t ≥ 0, d z d (w (t)) = h (x z (t)) (x z (t)) = −x z (t) p(x z (t))h (x z (t)) dt dt = −λh(x z (t)) = −λw z (t), which implies that w z (t) = e−λt h(z) for all t ≥ 0. This shows that for each z ∈ D, the function h maps the curve [0, +∞)  t → x z (t) onto the arc of the λ-spiral from h(z) to 0. Thus h(D) is a λ-spirallike domain. Now we show that h is univalent. Suppose by contradiction that h(z 1 ) = h(z 2 ) for some points z 1 , z 2 ∈ D. Then w z1 (t) = w z2 (t) for all t ≥ 0. Since h (0) = 0, there is ε > 0 such that h is univalent in the disc εD. By Lemma 9.4.4, limt→+∞ x z1 (t) = limt→+∞ x z2 (t) = 0. Hence, if t is large enough in such a way that |x z1 (t)| < ε and |x z2 (t)| < ε, the univalency of h in εD implies that x z1 (t) = x z2 (t). By the uniqueness of solutions for the problem (9.4.1), it follows that x z1 (t) = x z2 (t) for all t ≥ 0, which, in turn, implies z 1 = z 2 . Hence h is univalent in D. Therefore, h is λ-spirallike with respect to 0. Thus the result holds in case ζ = 0. The general case ζ = 0 can be deduced from the previous one by considering the auxiliary function g : D → D defined by g(w) := h(Tζ (w)), where Tζ is the canonical automorphism of D which maps ζ to 0, see (1.2.1). Note that g(D) = h(D), g(0) = 0 and g (0) = (|ζ |2 − 1)g (ζ ). Hence, g is λ-spirallike with respect to 0 if and only if h is λ-spirallike with respect to ζ . Moreover, write w = Tζ (z). Since Tζ ◦ Tζ = idD , it holds g(Tζ (z)) = h(z). Hence, a direct computation shows

9.4 Basic Properties of Koenigs Functions

 Re

1 wg (w) λ g(w)

255



 = Re

 = Re

1 Tζ (z) g (Tζ (z))Tζ (z) λ Tζ (z) g(Tζ (z))



 1 (z − ζ )(1 − ζ z) h (z) . λ 1 − |ζ |2 h(z)

Therefore the theorem holds for λ-spirallike functions with respect to ζ ∈ D if and only if it holds for λ-spirallike functions with respect to 0, and we already proved the latter case.  Now we consider Koenigs functions associated with non-elliptic semigroups. We start with the following simple result. Proposition 9.4.6 Let (φt ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Let h be the associated Koenigs function. Then lim sup Im h(z) = +∞.

(9.4.6)

z→τ

Proof Let γ : [0, ∞) → D be the continuous curve defined by γ (t) := φt (0). Note that limt→+∞ γ (t) = τ by Theorem 8.3.6. Hence, lim Im h(γ (t)) = lim Im h(φt (0))) = lim Im (h(0) + it) = +∞,

t→+∞

t→+∞

t→+∞



and the result holds.

Remark 9.4.7 Arguing as in the proof of the previous proposition and using the Lehto-Virtanen Theorem 3.3.1, it follows that under the hypotheses of Proposition 9.4.6, ∠ lim h(z) = ∞, z→τ

where the limit has to be understood in C∞ . However, such a condition by itself does not characterize the Denjoy-Wolff point of (φt ), while (9.4.6) does, as we will see later on. For hyperbolic semigroups, one can say something more: Proposition 9.4.8 Let (φt ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D and Koenigs function h. Then ∠ lim Im h(z) = +∞. z→τ

(9.4.7)

Proof By Theorem 9.3.5, |Re h(z)| ≤ π/λ for all z ∈ D, where λ > 0 is the spectral value of (φt ). Hence, by Remark 9.4.7, ∠ lim z→τ Im h(z) is either +∞ or −∞. Since, by Proposition 8.3.7, the curve [0, +∞)  t → φt (0) converges to τ nontangentially as t → +∞, arguing as in the proof of Proposition 9.4.6, we see that ∠ lim supz→τ Im h(z) = +∞. Hence ∠ lim z→τ Im h(z) = +∞ and we are done. 

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Definition 9.4.9 A domain Ω ⊂ C is starlike at infinity if Ω + it ⊆ Ω for all t ≥ 0. A map h : D → C is starlike at infinity with respect to σ ∈ ∂D if it is univalent, lim supz→σ Im h(z) = +∞ and h(D) is starlike at infinity. Maps which are starlike at infinity and Koenigs functions of non-elliptic semigroups in D are one and the same: Theorem 9.4.10 Let h : D → C be the Koenigs function of a non-elliptic semigroup (φt ) in D with Denjoy-Wolff point τ ∈ ∂D. Then h is starlike at infinity with respect to τ . Conversely, if h : D → C is starlike at infinity with respect to τ ∈ ∂D, let φt (z) := h −1 (h(z) + it) for t ≥ 0. Then (φt ) is a non-elliptic semigroup in D with DenjoyWolff point τ . Moreover, let a = inf z∈D Re h(z) and b = supz∈D Re h(z), then (1) if a = −∞, b = +∞, then (φt ) is a parabolic semigroup of zero hyperbolic step and h is its Koenigs function, (2) if a = −∞ and b < +∞, then (φt ) is a parabolic semigroup of positive hyperbolic step, and its canonical model is (H− , h − b, z → z + it), (3) if a > −∞ and b = +∞, then (φt ) is a parabolic semigroup of positive hyperbolic step, and its canonical model is (H, h − a, z → z + it), (4) if a > −∞ and b < +∞ then h is a hyperbolic semigroup, πλ = b − a and h − a is its Koenigs function and the spectral value is λ. Proof Let h be the Koenigs function of a non-elliptic semigroup (φt ) in D with Denjoy-Wolff point τ ∈ ∂D. By Theorem 9.3.5, for all t > 0 h(D) + it = h(φt (D)) ⊂ h(D), hence h(D) is starlike at infinity. Moreover, lim supz→τ Im h(z) = +∞ by Proposition 9.4.6. Hence h is starlike at infinity with respect to τ . Conversely, let us assume h is starlike at infinity with respect to τ . Define φt (z) := h −1 (h(z) + it), for all z ∈ D and t ≥ 0. Now, it is clear that (φt ) is a (continuous) semigroup in D, without fixed points in D. First, note that if z 0 ∈ h(D) then ∪t≥0 (z 0 − it) = {z ∈ C : Re z = z 0 , Im z ∈ (−∞, Im z 0 ]}. Moreover, since h(D) + it ⊂ h(D) for all t ≥ 0, it follows that if z 0 ∈ h(D) then ∪t≥0 (h(D) − ti) contains the line {z ∈ C : Re z = Re z 0 }. Therefore, bearing in mind that h(D) is connected, Ω := ∪t≥0 (h(D) − ti) = {z ∈ C : a < Re z < b}. From this and from Theorem 9.3.5, taking into account that h + α satisfies (9.2.1) for all α ∈ R, implications (1), (2), (3) and (4) follow easily. As an example, let us prove (2). In this case, Ω = {z ∈ C : Re z < b}. Therefore, if we let h˜ := h − b, it ˜ − it) = H− , (9.2.2) is satisfied follows that h˜ satisfies (9.2.1), and since ∪t≥0 (h(D) − as well. Therefore, (H , h − b, z → z + it) is the canonical model of (φt ), and (φt ) is a parabolic semigroup with positive hyperbolic step by the Theorem 9.3.5.

9.4 Basic Properties of Koenigs Functions

257

Finally, we are left to show that τ is the Denjoy-Wolff point of (φt ). Assume this is not the case, and let σ ∈ ∂D be the Denjoy-Wolff point of (φt ). Let h˜ be the Koenigs function of (φt ). By Proposition 9.4.6, there exists a sequence ˜ n ) = +∞. As we already noticed {wn } ⊂ D converging to σ such that limn→∞ Im h(w above, h = h˜ − α for some α ∈ R, therefore, limn→∞ Im h(wn ) = +∞ as well. Since h is starlike at infinity with respect to τ , there exists a sequence {z n } ⊂ D converging to τ such that limn→∞ Im h(z n ) = +∞. The connected domain h(D) being starlike at infinity, there exists a curve Γn ⊂ h(D) joining z n to wn such that min Im ζ = min{Im h(z n ), Im h(wn )}.

ζ ∈Γn

In particular, for every R > 0 there exists n R ∈ N such that Im ζ ≥ R for all n ≥ n R and all ζ ∈ Γn . This implies that if {ζn } is a sequence such that ζn ∈ Γn for all n ∈ N, then (9.4.8) lim ζn = ∞ in C∞ . n→∞

Now, let Cn := h −1 (Γn ). By construction, Cn joins z n to wn . Since z n → τ and wn → σ and τ = σ , there exists K > 0 such that diamE (Cn ) ≥ K for all n ∈ N. Moreover, for any sequence {ξn } such that ξn ∈ Cn , it holds h(ξn ) ∈ Γn . Hence, by (9.4.8), the sequence {h(ξn )} converges to ∞ in C∞ . Therefore, {Cn } is a sequence of Koebe arcs for h, contradicting the No Koebe Arcs Theorem 3.2.4. Hence τ = σ and we are done.  Theorem 9.4.11 Let h : D → C be non-constant and holomorphic and let σ ∈ ∂D. Then h is starlike at infinity with respect to σ if and only if for all z ∈ D, Im [σ (σ − z)2 h (z)] ≥ 0.

(9.4.9)

Moreover, equality holds at some—and hence any—z ∈ D if and only if h(z) = a

σ +z + c, z ∈ D σ −z

(9.4.10)

for some a ∈ R \ {0} and c ∈ C. Proof Let Cσ : D → H be the Cayley transform with respect to σ given by (1.1.2). Given h : D → C holomorphic, we define a new holomorphic map g : H → C by setting g(w) := h(Cσ−1 (w)). Thus, writing z = Cσ−1 (w), g (w) = h (Cσ−1 (w)) (Cσ−1 (w)) = Therefore, (9.4.9) is equivalent to

σ h (z) = (σ − z)2 h (z). Cσ (z) 2

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9 Models and Koenigs Functions

Im g (w) ≥ 0, for all w ∈ H.

(9.4.11)

Moreover, it is easy to see that h is starlike at infinity with respect to σ ∈ ∂D if and only if g is starlike at infinity, namely, g is univalent, for every t ≥ 0 it holds g(H) + it ⊂ g(H) and lim supw→∞ Im g(w) = +∞. Thus, in order to prove the result, we have to prove that g is starlike at infinity if and only if (9.4.11) holds. To start with, assume that (9.4.11) holds. First of all, note that if there exists w0 ∈ H such that Im g (w0 ) = 0, by the Maximum Principle for harmonic functions, Im g (w) ≡ 0, and hence g (w) ≡ a for some a ∈ R. Namely, g(w) = aw + b for some b ∈ C. Since g is not constant, a = 0 and it is easy to see that g is starlike at infinity. Moreover, a direct computation shows that h has the form (9.4.10). Assume now that Im g (w) > 0 for all w ∈ H. By Theorem 3.1.3, the function −ig is univalent, and so is g. Also, let y > 0 and r ∈ R. Since ∂ Re g(y + ir ) = Re (ig (y + ir )) = −Im g (y + ir ) < 0, ∂r it follows that the function R  r → Re g(y + ir ) is strictly decreasing. This implies that the curve R  r → g(y + ir ), that parameterizes ∂g(E H (∞, y)), intersects every vertical line at most in one point—here E H (∞, y) denotes the horocycle of H at ∞ of radius y (see (1.4.15)). Therefore, either g(y + ir ) + it ∈ g(E H (∞, y)) for all t < 0 or g(y + ir ) + it ∈ g(E H (∞, y)) for all t > 0. However, the first possibility is excluded because g preserves the orientation. Therefore, for every t > 0, and for every y > 0, g(∂ E H (∞, y)) + it ⊂ g(E H (∞, y)).

(9.4.12)

By the arbitrariness of y, it follows that g(H) + it ⊂ g(H) for all t ≥ 0. Therefore, in order to prove that g is starlike at infinity, it remains only to show that lim supw→∞ Im g(w) = +∞. Suppose by contradiction that lim supw→∞ Im g(w) = A < +∞. We claim that this implies that g(H) is contained in {w ∈ C : Im w < A}, which is clearly impossible since g(H) is starlike at infinity. Suppose the claim is false. Hence there exists w0 = x0 + i y0 ∈ H such that Im g(w0 ) > A + ε, for some ε > 0. Since Im g (w) > 0 for all w ∈ H, the curve [x0 , +∞)  r → Im g(r + i y0 ) is increasing, hence lim supr →+∞ Im g(r + i y0 ) ≥ A + ε, a contradiction. This proves that g is starlike at infinity, as needed. Assume now that g is starlike at infinity. If Im g = 0, the result is clear. Thus, we will suppose that Im g > 0. We want to show that (9.4.11) holds. First, we claim that (∗) g starlike at infinity implies (9.4.12). Assume the claim (∗) is true. Fix y > 0. Then the function R  r → Re g(y + ir ) is either strictly monotone or constant. Indeed, assume it is not constant, and that, by contradiction, there exist r0 , r1 ∈ R, r0 = r1 , such that Re g(y +

9.4 Basic Properties of Koenigs Functions

259

ir0 ) = Re g(y + ir1 ). Since g is univalent, Im g(y + ir0 ) = Im g(y + ir1 ) and we can assume that Im g(y + ir0 ) > Im g(y + ir1 ). Let t := Im g(y + ir0 ) − Im g(y + ir1 ) > 0. By (9.4.12), g(y + ir0 ) = g(y + ir1 ) + it ∈ g(E H (∞, y)), which is a contradiction since g(y + ir0 ) ∈ g(∂ E H (∞, y)). Therefore, if not constant, the function R  r → Re g(y + ir ) is strictly monotone. Moreover, by (9.4.12), it is clear that g maps E H (∞, y) onto the connected component of C \ g(∂ E H (∞, y)) which contains the curve (0, +∞)  r → g(y + ir ). Since univalent maps preserve orientation, this implies that, if not constant, the function R  r → Re g(y + ir ) is strictly decreasing. In particular, Im g (y + ir ) = − ∂r∂ Re g(y + ir ) ≥ 0, and (9.4.11) holds. Now, we show that claim (∗) holds. Fix y > 0. By Theorem 9.4.10, there exists α ∈ R such that h + α is the Koenigs function of a non-elliptic semigroup (φt ) in D with Denjoy-Wolff point σ . Note that (φt ) is not a group since Im g > 0. Therefore, for all t > 0, taking into account that φt (z) = h −1 (h(z) + it) and Julia’s Theorem 1.4.7,   1 1 h −1 h(∂ E(σ, )) + it ⊆ E(σ, ). y y Hence, by (1.4.15), 1 g(∂ E H (∞, y)) + it = h(Cσ−1 (∂ E H (∞, y))) + it = h(∂ E(σ, )) + it y 1 1 ⊆ h(E(σ, )) = g(Cσ (E(σ, ))) = g(E H (∞, y)), y y and claim (∗) is proved. Finally, it is clear that if h is given by (9.4.10), then Im [σ (σ − z)2 h (z)] = 0 for all z ∈ D.  Summing up Proposition 9.3.12, Theorems 9.4.3, 9.4.5, 9.4.10 and 9.4.11, we have: Proposition 9.4.12 Let (φt ) be a semigroup in D with Denjoy-Wolff point τ ∈ D and spectral value λ. Let h be the Koenigs function of (φt ). Then   1 (z − τ )(1 − τ z) h (z) ≥ 0, (1) either τ ∈ D and Re λ 1 − |τ |2 h(z) (2) or τ ∈ ∂D and Im [τ (τ − z)2 h (z)] ≥ 0. Conversely, (i) if h : D → C is a holomorphic function which satisfies (1), h(τ ) = 0 and h (τ ) = 0, setting φt (z) := h −1 (e−λt h(z)) for t ≥ 0, (φt ) is an elliptic semigroup with Denjoy-Wolff point τ and spectral value λ. Moreover, the inequality in (1) is equality for some z ∈ D if and only if (φt ) is an elliptic group.

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9 Models and Koenigs Functions

(ii) if h : D → C is a holomorphic function which satisfies (2), then, setting φt (z) := h −1 (h(z) + it) for t ≥ 0, (φt ) is a non-elliptic semigroup with Denjoy-Wolff point τ . Moreover, the inequality in (2) is equality for some z ∈ D if and only if (φt ) is a parabolic group.

9.5 Semigroups of Linear Fractional Maps In this section we consider semigroups in D whose iterates are in LFM(D). It is worth recalling that in Theorem 8.5.3 we saw that if (φt ) is a semigroup in D then there exists t0 > 0 such that φt0 ∈ LFM(D) if and only if φt ∈ LFM(D) for all t ≥ 0. The canonical model for semigroups of Möbius transformations are quite easy to be described: Proposition 9.5.1 Let (φt ) be a semigroup, not a group, in D. Suppose that φt ∈ LFM(D) for all t ≥ 0. Then: (1) If (φt ) is elliptic with Denjoy-Wolff point τ ∈ D and spectral value λ ∈ C \ {0}, Re λ > 0, then its canonical model is (C, h, z → e−λt z), with the Koenigs function h given by z−τ h(z) = , (9.5.1) 1 − σz where σ ∈ C∞ \ D is the other common fixed point of (φt ) (and, with the usual notation, we set σz ≡ 0 in case σ = ∞). (2) If (φt ) is hyperbolic, with Denjoy-Wolff point τ ∈ ∂D and spectral value λ > 0 then its canonical model is (S πλ , h, z → z + i), with the Koenigs function h given by   τ +z 1 + τ/σ π i + + , (9.5.2) h(z) = log λ τ −z 1 − τ/σ 2λ where log denotes the principal branch of the logarithm and σ ∈ C∞ \ D is the other common fixed point of (φt ) (and, with the usual notation, we set στ = 0 in case σ = ∞). (3) If (φt ) is parabolic with Denjoy-Wolff point τ ∈ ∂D, then (φt ) is of zero hyperbolic step and its canonical model is (C, h, z → z + i), with the Koenigs function h given by i τ +z , (9.5.3) h(z) = aτ −z where a ∈ C \ {0} is given by a = τ φ1 (τ ). Proof (1) Let us assume (φt ) is elliptic. By Proposition 8.5.2, there exists σ ∈ C∞ \ D such that φ1 (σ ) = σ . Note that φ1 ◦ φt = φt ◦ φ1 in C∞ for all t ≥ 0 since the same relation holds in the open disc D. Since

9.5 Semigroups of Linear Fractional Maps

261

φt (σ ) = φt (φ1 (σ )) = φ1 (φt (σ )), it follows that φt (σ ) is equal to either τ or σ . However, since φt (τ ) = τ and φt is an automorphism of C∞ , it follows that φt (σ ) = σ for all t ≥ 0. Let h be defined in (9.5.1). Then h ◦ φt ◦ h −1 is a Möbius transformation which fixes 0 and ∞ for all t ≥ 0, hence (h ◦ φt ◦ h −1 )(z) = at z for some at ∈ C. Moreover, by the Chain Rule for derivatives in C∞ and Proposition 8.1.14, at = e−λt . Since (φt ) is not a group, and hence Re λ > 0 and 0 ∈ h(D), it follows that ∪t≥0 etλ h(D) = C. Therefore, (C, h, z → e−λt z) is the canonical model of (φt ). (2) Let us assume now that (φt ) is hyperbolic with spectral value λ > 0 and Denjoy-Wolff point τ ∈ ∂D. By Proposition 8.5.2, there exists σ ∈ C∞ \ D such that φ1 (σ ) = σ and, arguing as before, it follows that φt (σ ) = σ for all t ≥ 0. If σ = ∞, let Δ be the disc containing D and such that σ, τ ∈ ∂Δ, while, if σ = ∞, let Δ be the half-plane containing D such that τ ∈ ∂Δ. Then Δ is invariant under φt for all t ≥ 0 and, in fact, φt |Δ is an automorphism of Δ for all t ≥ 0. Moreover, by Proposition 8.5.2, ∪t≥0 (φt |Δ )−1 (D) = Δ. Hence, if we let ι : D → Δ be defined by ι(z) = z, it follows that (Δ, ι, φt |Δ ) is a holomorphic model for (φt ) (seen as a 1+ τ + 1− στ (which is a translation of the Cayley semigroup in D). Now, let H (z) := ττ +z −z σ

transformation with respect to τ ). Then H (Δ) = H and (H ◦ φt ◦ H −1 )(w) = eλt w for every t ≥ 0, since (H ◦ φt ◦ H −1 ) is a Möbius transformation which fixes 0 and ∞ and by the Chain Rule for derivatives in C∞ its derivative at ∞ equals 1/φt (τ ) = eλt . Now, let f be given by (9.3.3). Hence f (H) = S πλ and f (eλt z) = z + it for all t ≥ 0, so that, setting h = f ◦ H we have the statement. (3) Now we assume (φt ) is a parabolic semigroup in D, not a group, with Denjoy. For every Wolff point τ ∈ ∂D. Consider the Cayley transformation Cτ (z) := z+τ z−τ −1 t ≥ 0, the maps ψt := Cτ ◦ φt ◦ Cτ form a semigroup of Möbius transformations with only one fixed point in C∞ , which is ∞. Therefore, ψt (z) = z + β(t) for some continuous function β : [0, +∞) → C such that β(t + s) = β(t) + β(s) for all t, s ≥ 0. Thus β(t) = at for some a ∈ C \ {0}, with Re a ≥ 0, by Theorem 8.1.11. Let i τ +z i . h(z) := Cτ (z) = a aτ −z Hence, (h ◦ φt ◦ h −1 )(z) = z + it for all z ∈ h(D). In order to conclude the proof and show that (C, h, z → z + it) is the canonical model, we only need to verify that  (h(D) − it) = C.

(9.5.4)

t≥0

Now, h(∂D) is a line L in C, and h(D) is contained in one of the two half-planes bounded by L, let A := h(D) be such a half-plane. The half-plane A is invariant under the transformation z → z + it, for all t ≥ 0 since h(φt (z)) = h(z) + it for all t ≥ 0. Hence, (9.5.4) holds if and only if L is not a vertical line. If L were a vertical line, then h(D) = A were invariant also under the transformation z → z − it, t ≥ 0. Therefore,

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φt (z) := h −1 (h(z) + it) would be an automorphism of D, against our assumption. Hence, L is not vertical and (9.5.4) holds. Finally, the equality a = τ φ1 (τ ) follows from Lemma 9.3.11.  There is another interesting question that can be completely solved for linear fractional self-maps of the unit disc, the question of embedding: given a holomorphic self-map φ : D → D, does there exist a semigroup (φt ) in D such that φ1 = φ? Clearly, the map φ has to be univalent in D by Theorem 8.1.17, but such a condition is not sufficient. In this section we solve the following question: Let φ : D → D be linear fractional. Does there exist a semigroup (φt ) in D such that φ1 = φ? By Theorem 8.5.3, if such a semigroup exists, all its iterates are linear fractional. We first examine the non-elliptic case, where the previous question has always a positive answer: Theorem 9.5.2 Let φ : D → D be a non-elliptic linear fractional map. Then there exists a semigroup (φt ) in D such that φ1 = φ. Proof By Corollary 8.2.7 we can suppose that φ is not an automorphism. If φ is hyperbolic, with Denjoy-Wolff point τ and dilation e−λ for some λ > 0, by Propo− ττ +σ sition 8.5.2, there exists σ ∈ C∞ \ D such that φ(σ ) = σ . Let g(z) = ττ +z −z −σ 1 if σ ∈ C and g(z) = z−τ if σ = ∞. Hence g is a Möbius transformation such that g(σ ) = 0 and g(τ ) = ∞. Therefore, g ◦ φ ◦ g −1 is a Möbius transformation which fixes 0 and ∞. Namely, g ◦ φ = βg for some β ∈ C. Therefore, recalling the definition of derivatives in the Riemann sphere, we have 1 1 1 g (τ ) = lim = lim z→τ βg(z)(z − τ ) z→τ g(φ(z))(z − τ ) β φ(z) − τ 1 · = g (τ )φ (τ ). = lim z→τ g(φ(z))(φ(z) − τ ) z−τ Thus, β1 = φ (τ ) = e−λ and eλ g(z) = g(φ(z)) for all z ∈ D. Let h(z) = λi log g(z) + π , z ∈ D. Then it is easy to check that h(φ(z)) = h(z) + i for all z ∈ D, and h(D) 2λ is invariant under the transformation z → z + it for all t ≥ 0. Hence, the setting φt (z) := h −1 (h(z) + it) for t ≥ 0 defines a semigroup in D such that φ1 = φ. 1 . Since the If φ is parabolic with Denjoy-Wolff point τ ∈ ∂D. Let g(z) := z−τ −1 Möbius transformation g ◦ φ ◦ g has only one fixed point in C∞ and fixes ∞ then (g ◦ φ ◦ g −1 )(z) = z + b for some b ∈ C \ {0}. Thus h(z) := bi g(z) satisfies h(φ(z)) = h(z) + i for all z ∈ D. The domain h(D) is a half-plane whose boundary is given by the line L = h(∂D) and it is invariant under the transformation z → z + it for all t ≥ 0. Hence, the setting φt (z) := h −1 (h(z) + it) for t ≥ 0 defines a  semigroup in D such that φ1 = φ. For elliptic linear fractional self-maps of D, the question of embeddability does not always have a positive answer. Every elliptic automorphism of D can be embedded

9.5 Semigroups of Linear Fractional Maps

263

into a group by Corollary 8.2.7. Thus, we concentrate on non-automorphic elliptic maps. τ −z of D is a Möbius transformation, Given τ ∈ D, the automorphism Tτ (z) := 1−τ z that is, an automorphism of C∞ . As such, if τ = 0, we have Tτ (∞) = lim

z→∞

τ −z 1 = , 1 − τz τ

while, if τ = 0, T0 (∞) = ∞. With this convention, we can now state and prove the following embedding result: Theorem 9.5.3 Let φ : D → D be a linear fractional map, not an automorphism of D, with Denjoy-Wolff point τ ∈ D. Let σ ∈ C∞ \ D be the other fixed point of φ and let φ (τ ) = e−λ , for some λ ∈ C such that Re λ > 0. Then there exists a semigroup (φt ) in D such that φ1 = φ if and only if |λ| ≤ Re λ

|σ − τ | , |1 − σ τ |

(9.5.5)

|σ −τ | = +∞ in case τ = 0 and where, with a slight abuse of notation, we let Re λ |1−σ τ| σ = ∞.

Proof If T is an automorphism of D then φ is embeddable into the semigroup (φt ) if and only if T ◦ φ ◦ T −1 is embeddable into the semigroup (T ◦ φt ◦ T −1 ). Let Tτ be the automorphism given by (1.2.1) which maps τ to 0 and Tτ = Tτ−1 . Let ˜ φ := Tτ ◦ φ ◦ Tτ . Then φ˜ is a linear fractional self-map of D which fixes the origin. Moreover, φ˜ (0) = e−λ . The other fixed point of φ˜ is β := Tτ (σ ) = (τ − σ )/(1 − τ σ ) (with the previous convention). Therefore, (9.5.5) is equivalent to |λ| ≤ Re λ|β|.

(9.5.6)

Thus, the statement of the theorem is equivalent to: φ˜ is embeddable into a semigroup in D if and only if (9.5.6) holds. Assume first that φ˜ is embeddable into a semigroup (φ˜ t ) in D. Then, by Theorem 8.5.3, φ˜ t is a linear fractional map with 0 and β as fixed points for all t ≥ 0. Moreover, by (9.5.1), the Koenigs function h of (φ˜ t ) is given by h(z) =

z 1−

z β

.

(9.5.7)

By Theorem 9.4.3, h is λ-spirallike with respect to 0, hence, by (9.4.2), for all z ∈ D it satisfies     1 β 1 zh (z) = −Re , (9.5.8) 0 ≤ Re λ h(z) λz−β

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9 Models and Koenigs Functions

β with the usual convention that z−β = −1 if β = ∞. In particular, if β = ∞, the previous equation reduces to Re λ ≥ 0, which is always the case. Otherwise, in case β = ∞, the previous equation is equivalent to

  |λ||β| = sup Re λβz ≤ |β|2 Re λ, z∈D

hence, since β = 0, (9.5.6) holds. Conversely, assume (9.5.6) holds. Let h be given by (9.5.7). Then, (9.5.6) is equivalent to (9.5.8), hence, by Theorem 9.4.5, the function h is λ-spirallike with respect to 0. In particular, setting φ˜ t (z) := h −1 (e−λt h(z)) for t ≥ 0, it follows by ˜ Indeed, Theorem 9.4.3 that (φ˜ t ) is a semigroup in D. Finally, we notice that φ˜ 1 = φ. −1 ˜ h ◦ φ ◦ h is a linear fractional map fixing 0, ∞, and whose derivative at z = 0 is  e−λ , hence (h ◦ φ˜ ◦ h −1 )(z) = e−λ z = (h ◦ φ˜ 1 ◦ h −1 )(z), and we are done.

9.6 Non-Canonical Holomorphic Semi-Models In Theorem 9.3.5 we have seen that every semigroup in D admits a canonical model, and that canonical models are unique (up to isomorphisms of holomorphic models). One might be interested in understanding the other possible holomorphic semiconjugations. In this section we examine semi-conjugations with prescribed simple base spaces. Let (φt ) be a semigroup in D. Let X be a Riemann surface, (ϕt ) a group of automorphisms of X and let η : D → X be a holomorphic map such that for all t ≥ 0 it holds η ◦ φt = ϕt ◦ η. If the map η is not constant, the image η(D) is an open subset of X and Λ := t≥0 ϕ−t (η(D)) is thus a Riemann surface which is completely invariant under the action of (ϕt ). Therefore, (Λ, η, ϕt |Λ ) is a holomorphic semi-model for (φt ). Hence, in order to study (non-trivial) holomorphic semi-conjugations, it is enough to study holomorphic semi-models and, by Proposition 9.2.14, this amounts to study morphisms of holomorphic semi-models from the canonical models. As Example 9.2.1 shows, in general semi-models can be quite wild, however, if the prescribed base space is as simple as D or C, semi-models are very rigid. In fact, we classify all possible non-trivial holomorphic semi-models with base space D or C. It follows that for non-elliptic semigroups there exist no holomorphic semimodels which are not models (thus, up to isomorphisms of semi-models, the only possible holomorphic semi-model is the canonical one). While, for the elliptic case, holomorphic semi-models are essentially powers of the canonical model: Proposition 9.6.1 Let (φt ) be a non-trivial semigroup in D. Let (Λ, η, ϕt ) be a holomorphic semi-model for (φt ) with either Λ = D or Λ = C and η non-constant. Let (Ω, h, ψt ) be the canonical model for (φt ).

9.6 Non-Canonical Holomorphic Semi-Models

265

(1) If (φt ) is an elliptic semigroup with spectral value λ, for some λ ∈ C \ {0} with Re λ ≥ 0, then there exists m ∈ N such that (Λ, η, ϕt ) is isomorphic to the holomorphic semi-model (Ω, h m , z → e−mλt z). (2) If (φt ) is a non-elliptic semigroup then (Λ, η, ϕt ) is a holomorphic model, isomorphic to (Ω, h, ψt ). Proof Case (I). Assume (φt ) is an elliptic group. In this case, according to Theorem 9.3.5, Ω = D and there exists θ ∈ R with λ = iθ and such that ψt (z) = e−iθt z for all t ≥ 0. First, we want to show that Λ cannot be equal to C. Indeed, assume by contradiction that Λ = C. By Proposition 9.2.4, (ϕt ) is a group of automorphisms of C with a common fixed point z 0 ∈ C. Hence, the automorphism A of C, defined as A(z) := z − z 0 , satisfies (A ◦ ϕt ◦ A−1 )(z) = eμt z for all t ≥ 0, and for some μ ∈ C \ {0}. Therefore, A defines an isomorphism Aˆ of semi-models between (C, η, ϕt ) and (C, A ◦ η, z → eμt z). By Proposition 9.2.14 there exists a morphism of holomorphic semi-models fˆ : (D, h, z → e−iθt z) → (C, A ◦ η, z → eμt z). By the functional equation (9.2.6), we have (9.6.1) f (e−iθt z) = eμt f (z),

for all t ∈ R and z ∈ D. Write the expansion of f at z = 0 as f (z) = j≥ j0 a j z j , for some j0 ∈ N and a j0 = 0. Plugging this expansion in (9.6.1), we obtain the equality a j e−iθt j = a j eμt , which holds for all j ≥ j0 and t ∈ R. Hence, μ = −iθ j0 , and a j = 0 for all j > j0 . This implies that f (z) = a j0 z j0 , and then f (D) = C. In particular, f is not surjective, contradicting Lemma 9.2.10. Therefore, Λ cannot be equal to C. In case Λ = D, by Proposition 9.2.4, (ϕt ) is a group of automorphisms of D with a common fixed point z 0 ∈ D. If A := Tz0 is the canonical automorphism of D given by (1.2.1), then, as before, A defines an isomorphism Aˆ of semi-models between (D, η, ϕt ) and (D, A ◦ η, z → eμt z), for some μ ∈ iR \ {0}. By Proposition 9.2.14 there exists a morphism of holomorphic semi-models fˆ : (D, h, z → e−iθt z) → (D, A ◦ η, z → eμt z). Then, arguing exactly as before, one can show that μ = −iθ j0 , and f (z) = a j0 z j0 for some j0 ∈ N, a j0 = 0. Since by Lemma 9.2.10, f (D) = D, it follows that |a j0 | = 1. Set m := j0 and q := a j0 . Then, by definition of morphism of holomorphic semimodels and for what we proved above, A ◦ η = f ◦ h = qh m and μ = −iθ m. Thus, the holomorphic semi-model (D, η, ϕt ) is isomorphic to the holomorphic semi-model (D, qh m , z → e−iθmt z). Finally, this latter holomorphic semi-model is clearly isomorphic to the holomorphic semi-model (D, h m , z → e−iθmt z) via the isomorphism of semi-model induced by the automorphism of D given by z → qz. Hence, in case (φt ) is an elliptic group, the result is proved. Case (II). If (φt ) is an elliptic semigroup which is not a group, then, according to Theorem 9.3.5, Ω = C and ψt (z) = e−λt z. The argument is very similar to the one

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9 Models and Koenigs Functions

given in Case (I), and we omit it. We only note that in this case Λ cannot be equal to D. Indeed, by Proposition 9.2.14 there exists a morphism of holomorphic semimodels fˆ : (C, h, z → e−λt z) → (Λ, η, ϕt ). Since f : Ω → Λ is a holomorphic and surjective map by Lemma 9.2.10, Liouville’s Theorem implies that, in fact, Λ = C. Case (III). Assume that (φt ) is non-elliptic and Λ = C. According to Theorem 9.3.5, ψt (z) = z + it for all t ≥ 0. Since Λ = C, then by Theorem 8.4.5, there exists an automorphism T of C such that either (T ◦ ϕt ◦ T −1 )(z) = z + it or (T ◦ ϕt ◦ T −1 )(z) = eat z for some a ∈ C \ {0}. In the first case, T defines an isomorphism of holomorphic semi-models Tˆ : (C, η, ϕt ) → (C, T ◦ η, z → z + it) and, by Proposition 9.2.14, there exists a morphism of holomorphic semi-models fˆ : (Ω, h, z → z + it) → (C, T ◦ η, z → z + it). The holomorphic map f : Ω → C is surjective by Lemma 9.2.10 and by (9.2.6) it satisfies f (z + it) = f (z) + it, for all t ∈ R and all z ∈ Ω. Differentiating in t and setting t = 0, we obtain f (z) ≡ 1. Hence, f (z) = z + b for some b ∈ C. In particular, f is injective. It follows that f : Ω → C is an affine biholomorphism. Therefore Ω = C, and Corollary 9.2.12 implies that fˆ is indeed an isomorphism of semi-model. Therefore, in this case, (Λ, η, ϕt ) is isomorphic to (Ω, h, ψt ). In case (T ◦ ϕt ◦ T −1 )(z) = eat z for some a ∈ C \ {0}, Proposition 9.2.14 implies the existence of a morphism of holomorphic semi-models fˆ : (Ω, h, z → z + it) → (C, T ◦ η, z → eat z) such that, by (9.2.6), f (z + it) = eat f (z), for all z ∈ Ω and t ∈ R. Since f : Ω → C is surjective by Lemma 9.2.10, there exists z 0 ∈ Ω such that f (z 0 ) = 0. But then, f (z 0 + it) = eat f (z 0 ) = 0 for all t ∈ R, which implies that f ≡ 0 on Ω, a contradiction. Therefore, the case (T ◦ ϕt ◦ T −1 )(z) = eat z for some a ∈ C \ {0} cannot occur. Case (IV). Assume that (φt ) is non-elliptic and Λ = D. According to Theorem 9.3.5, ψt (z) = z + it for all t ≥ 0. Since (ϕt ) is a group in D, by Theorem 9.3.5, it has a canonical model (Θ, , ϕ˜t ), where, either Θ = D and ϕ˜t (z) = e−iθt z for some θ ∈ R and for all t ≥ 0, or Θ = H, H− , Sρ for some ρ > 0 and ϕ˜t (z) = z + it for all t ≥ 0. Since (ϕt ) is a group, by Corollary 9.2.16, the map  : D → Θ is a biholomorphism, and hence it defines an isomorphism of holomorphic semi-models for (φt ) between (D, η, ϕt ) and (Θ,  ◦ η, ϕ˜t ). Therefore, by Proposition 9.2.14, there exists a morphism of holomorphic semimodels fˆ : (Ω, h, z → z + it) → (Θ,  ◦ η, ϕ˜t ). Now, by (9.2.6), for all z ∈ Ω and t ∈ R, f (z + it) = ϕ˜t ( f (z)).

9.6 Non-Canonical Holomorphic Semi-Models

267

Arguing as in Case (III), we see that the only possibility is that ϕ˜t (z) = z + it for all t ≥ 0 and that f : Ω → Θ be a biholomorphism. Therefore, in this case, (Λ, η, ϕt )  is isomorphic to (Ω, h, ψt ).

9.7 Holomorphic Conjugations and Holomorphic Models The concept of holomorphic models allows also to detect semigroups in D which are holomorphically conjugated. We officially start with a definition: t ) be two semigroups in D. We say that (φt ) and Definition 9.7.1 Let (φt ) and (φ  (φt ) are holomorphically conjugated if there exists an automorphism T of D such t ◦ T for all t ≥ 0. that T ◦ φt = φ There is a very simple and useful criterion for detecting when two semigroups are holomorphically conjugated using holomorphic models. In order to state it, we need a definition: Definition 9.7.2 Let (φt ) and (φ˜ t ) be two semigroups in D. Let (Ω, h, ψt ) be a holo  t ) a holomorphic model for (φ t ). We say that h, ψ morphic model for (φt ) and (Ω,   t ) are holomorphically conjugated if there exists a biholoh, ψ (Ω, h, ψt ) and (Ω, t ◦ g and g(h(D)) =   such that g ◦ ψt = ψ h(D). The map g morphism g : Ω → Ω is called a holomorphic conjugation of models. Conjugated semigroups correspond to conjugated models: t ) be two semigroups in D, with holomorphic models Proposition 9.7.3 Let (φt ), (φ    (Ω, h, ψt ) and (Ω, h, ψt ). The following are equivalent:   t ) are holomorphically conju(1) the holomorphic models (Ω, h, ψt ) and (Ω, h, ψ gated; t ) are holomorphically conjugated. (2) the semigroups (φt ) and (φ Proof (1) implies (2). Let g be a holomorphic conjugation of models between   t ), that is, g : Ω → Ω  is a biholomorphism such that (Ω, h, ψt ) and (Ω, h, ψ   h −1 ◦ g ◦ h. It is easy to see that g ◦ ψt = ψt ◦ g and g(h(D)) = h(D). Define T :=   T is a holomorphic conjugation between (φt ) and (φt ). t ). Let (2) implies (1). Let T be a holomorphic conjugation between (φt ) and (φ −1 −1    Ωt := ψt (h(D)) and Ωt := ψt (h(D)), t ≥ 0. Note that, if t ≥ s then Ωs ⊆ Ωt , since ψt = ψt−s ◦ ψs and h(D) is invariant under ψt for all t ≥ 0. Moreover,  ψt (Ωt ) = h(D) and, by the definition of model, Ω = ∪t≥0 Ωt . Similarly for Ω.  For t ≥ 0, define gt : Ωt → Ωt by t−1 ◦  h ◦ T ◦ h −1 ◦ ψt . gt := ψ t . Also, by definition, ψ t ◦ g0 = Clearly, gt is a biholomorphism from Ωt and Ω g0 ◦ ψt |Ω0 for all t ≥ 0.

268

9 Models and Koenigs Functions

Let 0 ≤ s ≤ t and let w ∈ Ωs . Then t−1 (g0 (ψt−s (ψs (w)))) = ψ t−1 (ψ t−s (g0 (ψs (w)))) = gs (w). gt (w) = ψ  defined by Therefore, for 0 ≤ s ≤ t it holds gt |Ωs = gs . Hence, the map g : Ω → Ω  g|Ωt := gt is well defined and it is a biholomorphism from Ω onto Ω. Finally, by a similar argument as before, one can show that for all t ≥ 0, gt ◦ s ◦ gt for all s ≥ 0, hence g ◦ ψs = ψ s ◦ g for all s ≥ 0, concluding the ψs |Ωt = ψ proof.  In particular, we have the following corollary: t ) be two semigroups in D. Assume (Ω, h, ψt ) is the Corollary 9.7.4 Let (φt ), (φ t ) are holomorphically conjugated if and canonical model for (φt ). Then, (φt ) and (φ only if there exists an automorphism T of D such that (Ω, h ◦ T, ψt ) is the canonical t ). model for (φ t ) are holomorphically conjugated. Then there exists an autoProof Assume (φt ), (φ t = T −1 ◦ φt ◦ T for all t ≥ 0. Therefore, for all t ≥ 0, morphism T of D such that φ t = (h ◦ φt ) ◦ T = ψt ◦ (h ◦ T ). (h ◦ T ) ◦ φ Moreover, since h(T (D)) = h(D) it is easy to see that (Ω, h ◦ T, ψt ) is the canonical t ). model for (φ Conversely, if there exists an automorphism T such that (Ω, h ◦ T, ψt ) is the t ), then the identity function idΩ : Ω → Ω defines a holocanonical model for (φ morphic conjugation between the holomorphic model (Ω, h, ψt ) for (φt ) and the t ). Hence the result follows from Proposiholomorphic model (Ω, h ◦ T, ψt ) for (φ tion 9.7.3. 

9.8 Topological Models and Topological Conjugations The careful reader might have realized that in the constructions of semi-models, the property of being holomorphic does not play any fundamental role. In fact, one can repeat all the constructions in Sect. 9.2 assuming just continuity of the intertwining maps. We do not repeat all those constructions, but concentrate on “topological models”: Definition 9.8.1 Let (φt ) be a semigroup in D. A topological model for (φt ) is a triple (Ω, h, ψt ) such that Ω is a domain in C, ψt is a group of (holomorphic) automorphisms of Ω and h : D → h(D) ⊂ Ω is a homeomorphism onto its image (that is, h is open, continuous and injective), h ◦ φt = ψt ◦ h and ∪t≥0 ψt−1 (h(D)) = Ω.

9.8 Topological Models and Topological Conjugations

269

Also, we can define isomorphisms of topological models:   t ) be two topological models for a Definition 9.8.2 Let (Ω, h, ψt ) and (Ω, h, ψ semigroup (φt ) in D. An isomorphism of topological models ηˆ : (Ω, h, ψt ) →  such that   t ) is given by a homeomorphism η : Ω → Ω (Ω, h, ψ

and

 h = η ◦ h,

(9.8.1)

t ◦ η = η ◦ ψt , t ≥ 0. ψ

(9.8.2)

A first result is the following: Lemma 9.8.3 Let (φt ) be a semigroup in D. Then there exists a topological model for (φt ), unique up to isomorphisms of topological models. Proof Since every holomorphic model is in particular a topological model, the existence follows at once from Theorem 9.2.18. The uniqueness up to isomorphisms of topological models follows arguing exactly as in the proof of Corollary 9.2.15 and Proposition 9.2.14.  Now we classify all possible topological models: Theorem 9.8.4 Let (φt ) be a non-trivial semigroup in D. (1) (φt ) is a group of elliptic automorphisms of D with spectral value iθ , θ ∈ R \ {0}, if and only if (φt ) has a topological model given by (D, h, z → eit|θ| z). (2) (φt ) is elliptic, not a group, if and only if (φt ) has a topological model given by (C, h, z → e−t z). (3) (φt ) is non-elliptic if and only if (φt ) has a topological model given by (S, h, z → z + it). Proof If (φt ) is an elliptic group, it has the canonical model (D, h, z → e−itθ z) by Theorem 9.3.5. If θ > 0, the map z → z conjugates the holomorphic model to the model (D, h, z → eit|θ| z). In case (φt ) is elliptic, not a group, let (C, h, z → e−λt z) be the canonical model for (φt ) given by Theorem 9.3.5, with λ = a + ib, a > 0 and b ∈ R. Define  ϕ(ρeiθ ) := exp

  b 1 +i log ρ eiθ , ρ = 0, a a

and ϕ(0) = 0. It is easy to see that ϕ : C → C is a homeomorphism and that ϕ , z → e−t z) is the topoϕ(e−λt w) = e−t ϕ(w) for all t ∈ R and w ∈ C. Then (C,  ϕ = ϕ ◦ h. logical model of (φt ), where  Assume that (φt ) is non-elliptic. Let (Ω, h, z → z + it) be the canonical model for (φt ) given by Theorem 9.3.5. If Ω = C, define ϕ : C → S by ϕ(x + i y) = θ (x) + i y, where θ : R → (0, 1) is any homeomorphism. Then clearly ϕ(z + it) = ϕ(z) +

270

9 Models and Koenigs Functions

it for all t ∈ R. Now, let  ϕ := ϕ ◦ h. It is then easy to see that (S,  ϕ , z → z + it) is a topological model for (φt ). In case Ω = H, it is enough to replace θ with any homeomorphism θ : (0, +∞) → (0, 1). While, if Ω = H− one can replace θ with any homeomorphism θ : (−∞, 0) → (0, 1). If Ω = Sρ , just define ϕ(x + i y) = x/ρ + i y. To prove the converse implications, by Lemma 9.8.3, it is enough to show that the various topological models are not topologically isomorphic each other. In fact, ˜ z → let (φt ) be a semigroup in D, and assume that (S, h, z → z + it) and (Ω, h, eμt z) are two topological models for (φt ), with Ω = D or Ω = C and μ ∈ C \ {0}. By Lemma 9.8.3 there exists a homeomorphism η : S → Ω such that η(z + it) = eμt η(z) for all t ≥ 0 (see (9.8.2)). Since η is surjective, there exists ζ ∈ C such that h(ζ ) = 0. But then, for all t ≥ 0 it holds η(ζ + it) = eμt h(ζ ) = 0, which is impossible because η is injective. A similar argument holds in case the two topological models are (C, h, z → e−t z) ˜ z → eiθt z) for some θ ∈ R \ {0}. and (D, h, Finally, we are left to prove that if two models (D, h 1 , z → eiθ1 t z) and (D, h 2 , z → iθ2 t e z) with θ1 , θ2 ∈ R \ {0} are topological isomorphic then θ2 = ±θ1 . Indeed, if the two models are isomorphic, there exists a homeomorphism T : D → D such that T (eiθ1 t z) = eiθ2 t T (z) for all t > 0 and all z ∈ D, by (9.8.2). Let t = 2π/θ1 . Then T (z) = e2πθ2 i/θ1 T (z) for all z ∈ D. Hence θ2 /θ1 =: m ∈ Z. Taking t = 2π/(mθ1 )  we deduce that T (e2πi/m z) = T (z) for all z ∈ D. Therefore m = ±1. As in the holomorphic case, topological models can be used to study topological conjugations. Just to formally state what we mean, we give the following definition: t ) be two semigroups in D. We say that (φt ) and Definition 9.8.5 Let (φt ) and (φ t ) are topologically conjugated if there exists a homeomorphism T : D → D such (φ t ◦ T for all t ≥ 0. that T ◦ φt = φ Also, Definition 9.8.6 Let (φt ) and (φ˜ t ) be two semigroups in D. Let (Ω, h, ψt ) be a   t ) a topological model for (φ t ). We say that h, ψ topological model for (φt ) and (Ω,    (Ω, h, ψt ) and (Ω, h, ψt ) are topologically conjugated if there exists a homeomort ◦ τ and τ (h(D)) =   such that τ ◦ ψt = ψ h(D). The map τ is phism τ : Ω → Ω called a topological conjugation of models. As in the holomorphic case, conjugated semigroups correspond to conjugated models: t ) be two semigroups in D, with topological models Proposition 9.8.7 Let (φt ), (φ   t ). The following are equivalent: h, ψ (Ω, h, ψt ) and (Ω,   t ) are topologically conjugated; (1) the topological models (Ω, h, ψt ) and (Ω, h, ψ

9.8 Topological Models and Topological Conjugations

271

t ) are topologically conjugated. (2) the semigroups (φt ) and (φ 

Proof The proof is the same as of Proposition 9.7.3. As a consequence we have the following result: Corollary 9.8.8 The following holds:

(1) Every group of elliptic automorphisms in D with spectral value θi, where θ ∈ R \ {0} is topologically conjugated to the elliptic group (z → e|θ|it z). (2) Every elliptic semigroup, not a group, in D is topologically conjugated to an elliptic semigroup in D with spectral value 1. (3) Every non-elliptic semigroup in D is topologically conjugated to a hyperbolic semigroup in D with spectral value π . Proof We only give the proof of (3), the others being similar. Let (φt ) be a non-elliptic semigroup in D. By Theorem 9.8.4, it admits a topological model (S, h, z → z + it). Since h : D → h(D) is a homeomorphism, then h(D) is a simply connected domain and by Theorem 3.1.1 there exists a biholomorphism h˜ : D → h(D). Since h(φt (z)) = h(z) + it for all z ∈ D and t ≥ 0, it holds h(D) + it ⊂ h(D). Therefore, ˜ ˜ h(D) + it = h(D) + it ⊂ h(D) = h(D),

for all t ≥ 0.

˜ + it) is a well defined (holomorphic) semigroup in D. Hence, φ˜ t (z) := h˜ −1 (h(z) Note that   ˜ (h(D) − it) = (h(D) − it) = S. t≥0

t≥0

˜ z → z + it). By Theorem 9.3.5, it folHence, the canonical model of (φ˜ t ) is (S, h, ˜ lows that (φt ) is a hyperbolic semigroup with spectral value π . Finally, by Proposition 9.8.7, in order to show that (φt ) and (φ˜ t ) are topologically ˜ z → z + conjugated, we have to show that their models (S, h, z → z + it) and (S, h, ˜ it) are topologically conjugated. Since h(D) = h(D), the identity map idS : S → S provides the needed topological conjugation. 

9.9 Notes In 1884, Koenigs [91] proved that given a holomorphic self-map φ of D such that φ(0) = 0 and 0 < |φ (0)| < 1 there exists a unique holomorphic map h : D → C satisfying the Schröder equation [116] h ◦ φ = φ (0)h and such that h(0) = 0, h (0) = 1. Clearly ∪m∈N φ (0)−m h(D) = C. If the function φ is hyperbolic with Denjoy-Wolff point τ ∈ ∂D, in 1931, Valiron [125] proved that there exists a holomorphic map h : D → H which satisfies the Valiron equation h ◦ f = αφ (τ )h. The solution h also satisfies ∪m∈N αφ (τ )−m h(D) = H

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9 Models and Koenigs Functions

and is essentially unique in the sense that any other holomorphic solution is a positive multiple of h by [38, Proposition 2.4]. In 1979, Pommerenke and Baker [9, 104] dealt with the parabolic case, proving that in such a case the Abel equation h ◦ φ = h + i admits a holomorphic solution h : D → C. For any solution h of the Abel equation the domain ∪m∈N (h(D) − mi) is either the whole C or is biholomorphic to D, depending on whether φ is of zero hyperbolic step or of nonzero hyperbolic step. The uniqueness of the solutions of the Abel functional equations has been investigated in [53, 101]. These three functional equations are examples of intertwining models, and since φ is intertwined with linear fractional maps, they are called linear fractional models. There is a geometric way of approaching the problem of intertwining models which was proposed in 1981 by Cowen [56] exploiting a categorial construction (the tail space, also known as the abstract basin of attraction). His approach allows to treat the three previous functional equations in a unique framework. Cowen’s approach has been refined in the categorial sense presented in this book in [8]. In 1996, Bourdon and Shapiro [19] considered the case of holomorphic self-maps of the unit disc with no interior fixed points having some regularity at the DenjoyWolff point. Assuming regularity at the Denjoy-Wolff point allows one to obtain better knowledge of the intertwining map and of the shape of the corresponding image domain. Such results, aside an intrinsic interest, have been applied in many ways, for instance to study dynamics, commuting maps and properties of the associated composition operators, see, e.g., [18, 40, 57, 58, 65, 107, 117]. In 1939, Wolff [128], studying continuous iteration, proved the existence of the Koenigs function of a semigroup in the elliptic case. Heins [82] in 1981 and Siskakis [120] in 1985 proved, using infinitesimal generators (see Chap. 10), the existence of Koenigs functions for continuous semigroups of D. The existence of linear fractional models for univalent functions allows to solve the embedding problem in a geometric way. Indeed, a univalent self-map of D is embeddable into a semigroup of D if and only if the intertwining map of its linear fractional model is the Koenigs function of a semigroup in D (for studies on the embedding problem following this point of view, see [65, pp. 96–99], [56, 88, 123]). The material in Sects. 9.1, 9.2 and 9.3 has been elaborated from [8] (see also [31]). The proof of Lemma 9.1.1, which holds more generally for measurable subadditive functions, is taken from [93, Theorem 16.2.9]. The proof of Theorem 9.4.5 was taken from [59]. Theorem 9.4.11 was first proved under slightly different hypotheses in [80] for holomorphic functions from the upper half-plane and in [43] for the unit disc case. The material in Sect. 9.5 is taken from [24]. Sections 9.7 and 9.8 are taken from [30]. Corollary 9.8.8(1) follows also from Naishul’s theorem (see, e.g., [21, Theorem 2.29]), which states that two germs of elliptic holomorphic maps in C fixing 0 which are topologically conjugated by an orientation preserving map, must have the same derivative at 0. The proof of such a result is however much more complicated than the corresponding results for groups.

Chapter 10

Infinitesimal Generators

After having defined the Koenigs function of a semigroup in the previous chapter, now we turn our attention to the second characteristic feature of a semigroup: the infinitesimal generator. We see how to relate semigroups to Cauchy problems, showing that every semigroup is completely determined by a holomorphic vector field in the unit disc, its infinitesimal generator. Once shown the existence of such a vector field, we will focus on different descriptions and characterizations of infinitesimal generators and discuss several of their properties and examples.

10.1 Infinitesimal Generators and the Berkson-Porta Formula Definition 10.1.1 Let G : D → C be a holomorphic function and I an open interval = G(x(t)) that contains the point 0. A function x : I → D of class C 1 such that d x(t) dt for all t ∈ I is called a solution of w = G(w). Given a holomorphic function G : D → C and z ∈ D, the Cauchy problem 

w = G(w), w(0) = z

(10.1.1)

has a (unique) continuous solution x z defined on a maximal open interval J that contains the point 0. Namely, if y : I → C is another solution of the same Cauchy problem, with 0 ∈ I and y(0) = z, then I ⊆ J and y(t) = x z (t) for all t ∈ I . The solution x z is also called the maximal solution and J the maximal interval of existence of the solution, see, e.g., [45, Chap. 2]. Definition 10.1.2 Let G : D → C be a holomorphic function. We say that G is a semicomplete vector field if the maximal interval of existence of the solution of the © Springer Nature Switzerland AG 2020 F. Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4_10

273

274

10 Infinitesimal Generators

Cauchy problem (10.1.1) contains [0, +∞) for all z ∈ D. We say that G is a complete vector field if the maximal interval of existence of the solution of the Cauchy problem (10.1.1) is R for all z ∈ D. Remark 10.1.3 From a differential geometric point of view, a holomorphic vector field X in D is a holomorphic section of the holomorphic tangent bundle T D, that is, a holomorphic map X : D → T D such that X (z) ∈ Tz D for all z ∈ D. Since T D is holomorphically trivial, namely T D = D × C, the vector field X can be written as X (z) = (z, G(z)) for some holomorphic map G : D → C. This is the reason why holomorphic functions in D can be thought of as holomorphic vector fields on D and vice versa. Theorem 10.1.4 Let (φt ) be a semigroup in D. Then the function [0, +∞) × D  (t, z) −→ φt (z) ∈ D is real analytic and there exists a unique holomorphic semicomplete vector field G : D → C such that ∂φt (z) = G(φt (z)), ∂t

z ∈ D, t ∈ [0, +∞).

(10.1.2)

Conversely, given a holomorphic semicomplete vector field G, there exists a unique semigroup (φt ) in D such that (10.1.2) is satisfied. Moreover, if h is the Koenigs function of (φt ), then (1) in case (φt ) is the trivial semigroup, G ≡ 0, (2) in case (φt ) is an elliptic semigroup with spectral value λ, for all z ∈ D, G(z) = −λ

h(z) , h  (z)

(3) while, in case (φt ) is a non-elliptic semigroup, for all z ∈ D, G(z) =

i h  (z)

.

Proof Assume that (φt ) is a semigroup in D. According to Theorem 9.3.5, the semigroup (φt ) has a canonical model. If (φt ) is either trivial or elliptic, then its canonical model is (Ω, h, z → e−λt z), where λ is the spectral value of (φt ) and either Ω = D or Ω = C. Hence φt (z) = h −1 (e−λt h(z)). Clearly, the map (t, z) → φt (z) is real  analytic in [0, ∞) × D. Let G(z) := ∂φ∂tt (z)  = −λ hh(z)  (z) , z ∈ D. This function is t=0 holomorphic and e−λt h(z) h(φt (z)) ∂φt (z) = −λ  = −λ  = G(φt (z)), z ∈ D, t ≥ 0. ∂t h (φt (z)) h (φt (z))

10.1 Infinitesimal Generators and the Berkson-Porta Formula

275

That is, (φt ) solves (10.1.2). If (φt ) is non-elliptic, by Theorem 9.3.5, its canonical model is (Ω, h, z → z + it), with either Ω = Sρ for some ρ > 0 or Ω = H or Ω = H− or Ω = C. In particular, this means that φt (z) = h −1 (h(z) + it). Again, this implies  that the ∂φt (z)  map (t, z) → φt (z) is real analytic in [0, ∞) × D. Let G(z) := ∂t  = h  i(z) , t=0 z ∈ D. This function is holomorphic and, as before, we deduce again that (φt ) solves (10.1.2). Assume now that G is a semicomplete holomorphic vector field in D. This implies that for all z ∈ D the Cauchy problem 

= G(x(t)), x(0) = z d x(t) dt

has a solution x z : [0, +∞) → D. Define φt (z) := x z (t). Fix t, s > 0. By uniqueness z of solutions of Cauchy problems, x z (t + s) = x x (s) (t). Therefore φt+s = φt ◦ φs . Finally, for every t ≥ 0 the function φt : D → C is holomorphic thanks to the analytic dependence with respect to the initial values of the above Cauchy problem (see, e.g., [44, Theorem 8.2, p. 35]).  Definition 10.1.5 Let (φt ) be a semigroup in D. The unique holomorphic semicomplete vector field G given by Theorem 10.1.4 is called the infinitesimal generator of (φt ). We denote by Gen(D) the set of all infinitesimal generators in D. Remark 10.1.6 From (10.1.2) it follows immediately that if (φt ) is a semigroup in D and G is the associated infinitesimal generator, then z 0 ∈ D is a fixed point of φt for all t ≥ 0 if and only if G(z 0 ) = 0. In particular, if (φt ) is not the trivial semigroup, then G has at most one zero in D. A first fallout of the existence of infinitesimal generators and canonical models is the next characterization of the continuity of algebraic semigroups in the unit disc. Proposition 10.1.7 Let (φt ) be an algebraic semigroup of holomorphic self-maps in D. Then, the following are equivalent: (1) The semigroup (φt ) is continuous. (2) For every T ∈ (0, +∞) and every ε > 0 there exists δ > 0 such that supz∈D |φt (z) − φs (z)| < ε for every s, t ∈ [0, T ] with |t − s| < δ. (3) limt→0+ supz∈D |φt (z) − z| = 0. (4) The map [0, +∞)  t → φt ∈ Hol(D, D) is Lipschitz-continuous uniformly on compact subsets of D. That is, for every T > 0 and 0 < r < 1, there exists C = C(T, r ) > 0 such that |φs (z) − φt (z)| ≤ C(t − s), |z| ≤ r, 0 ≤ s ≤ t ≤ T.

276

10 Infinitesimal Generators

Proof It is clear that (2) implies (3) taking s = 0. Moreover, by Theorem 8.1.15, (3) implies (1). Let us assume (1) holds. We want to show that (2) holds as well. Assume this is not the case. Then there exist T ∈ [0, +∞), ε > 0, two sequences {sn }, {tn } ⊂ [0, T ] and a sequence {z n } ⊂ D such that limn→∞ |sn − tn | = 0 and for all n ∈ N, (10.1.3) |φtn (z n ) − φsn (z n )| ≥ ε. Let (Ω, h, ψt ) be the canonical model for (φt ) given by Theorem 9.3.5. Then, either ψt (z) = e−λt z for some λ ∈ C with Re λ ≥ 0, or ψt (z) = z + it for all t ≥ 0. We give the proof for the case ψt (z) = z + it, being the other similar. Equations (9.2.3) and (10.1.3) imply that for all n ∈ N |h −1 (h(z n ) + itn ) − h −1 (h(z n ) + isn )| ≥ ε.

(10.1.4)

Up to extracting subsequences, we can assume that the sequence {h(z n )} converges to some L ∈ C∞ and that {sn }—and hence {tn }—converges to some point t0 ∈ [0, T ]. n be the segment joining h(z n ) + isn with h(z n ) + itn . Since h(D) For n ∈ N, let C n ⊂ h(D) for all n ∈ N. Let is starlike at infinity by Theorem 9.4.10, it follows that C −1  Cn := h (Cn ), n ∈ N. Since limn→∞ h(z n ) = L and limn→∞ sn = limn→∞ tn = t0 , it follows that for every sequence {ζn } such that ζn ∈ Cn , n ∈ N, limn→∞ h(ζn ) = L + it0 (where, if L = ∞, we use the usual convention that ∞ + it0 = ∞). However, (10.1.4) implies that diamE (Cn ) ≥ ε for all n ∈ N. Thus, {Cn } is a sequence of Koebe arcs for h, contradicting Theorem 3.2.4. Hence, (2) holds. Next, by the very definition of continuous semigroups, (4) implies (1). Assume (1) holds and fix r < 1 and T < +∞. By Theorem 8.1.15 there exists R = R(T, r ) < 1 such that |φt (z)| ≤ R whenever |z| ≤ r and t ≤ T . Let G be the infinitesimal generator of (φt ). Take C = C(T, r ) such that |G(z)| ≤ C for all |z| ≤ R. Then, if |z| ≤ r and s ≤ t ≤ T we have  t   t   |G(φu (z))| du ≤ C(t − s). |φs (z) − φt (z)| =  G(φu (z)) du  ≤ s

s



Thus, (4) holds.

Theorem 10.1.4 shows that the trajectories of every semigroup are solutions of a Cauchy problem. It is also possible to show that they are the solutions of a partial differential equation, known as Kolmogorov’s Backward Equation. Moreover, this equation leads to a useful expression of the derivatives of the iterates of a semigroup in terms of the derivative of the infinitesimal generator. Proposition 10.1.8 Let (φt ) be a semigroup in D with associated infinitesimal generator G. (1) For all z ∈ D and t ≥ 0,

∂φt (z) = G(z)φt (z). ∂t

(10.1.5)

10.1 Infinitesimal Generators and the Berkson-Porta Formula

277

(2) For all z ∈ D and t ≥ 0, φt (z)



t

= exp





G (φs (z))ds .

(10.1.6)

0

Proof Consider the equation φs+t (z) = φt (φs (z)). Differentiating with respect to s, we obtain ∂φs+t (z) ∂φs (z) = φt (φs (z)) . ∂s ∂s Taking s = 0, we deduce (10.1.5). By Theorem 10.1.4, the function (t, z) → φt (z) is real analytic in [0, +∞) × D and, in particular, ∂ ∂φt (z) ∂ ∂φt (z) ∂φt (z) = = . ∂z ∂t ∂t ∂z ∂t Hence, differentiating (10.1.2) with respect to z, we obtain ∂φt (z) = G  (φt (z))φt (z), z ∈ D, t ≥ 0. ∂t Fix z ∈ D. Since φt is univalent for all t ≥ 0, the curve [0, +∞)  t → φt (z) admits a smooth selection of the argument θ : [0, +∞) → R. Moreover, we may assume θ (0) = 0 because φ0 (z) = 1. Using real logarithms, we define  [0, +∞)  t → f (t) := log |φt (z)| + iθ (t). Notice that φt (z) = e f (t) and f (0) = 0. Then, for all t ≥ 0 f  (t) =

∂  φ (z) ∂t t φt (z)

= G  (φt (z)).

Integrating and taking exponentials in this expression, we deduce (10.1.6).



Our next aim is to give a useful description of infinitesimal generators. First of all, we get rid of constant maps: Lemma 10.1.9 Let G : D → C be constant. Then G is an infinitesimal generator if and only if G ≡ 0. In this case, the associated semigroup is the trivial semigroup. Proof Clearly G ≡ 0 is the infinitesimal generator associated with the trivial semigroup. Let assume that G is an infinitesimal generator and let G(z) = c for all z ∈ D. From (10.1.2) it follows that φt (z) = z + ct. Since φt (D) ⊂ D then c = 0.  Next we give a first fundamental description of the non-constant functions that belong to Gen(D).

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10 Infinitesimal Generators

Theorem 10.1.10 (Berkson-Porta’s Formula) Let G : D → C be a non-constant holomorphic function. Then G ∈ Gen(D) if and only if there exist a point τ ∈ D and a non vanishing holomorphic function p : D → H such that G(z) = (z − τ )(τ z − 1) p(z).

(10.1.7)

The point τ and the function p are univocally determined by G and the decomposition formula (10.1.7) is unique. Moreover, if G ∈ Gen(D) and (φt ) is the associated semigroup, then the point τ in (10.1.7) is the Denjoy-Wolff point of (φt ). Proof Assume firstly that G is the infinitesimal generator of a semigroup (φt ). Let τ be its Denjoy-Wolff point. We split the proof in two cases. Case I: (φt ) is elliptic. Let h be the Koenigs function of (φt ). By Theorem 10.1.4, G(z) = −λ hh(z)  (z) where λ is the spectral value of (φt ). Let τ ∈ D be the Denjoy-Wolff point of (φt ). Since by Proposition 9.4.1, h(τ ) = 0, it follows that G(τ ) = 0. Hence the function p : D \ {τ } → C defined by p(z) :=

G(z) λh(z) λh(z) =− = (z − τ )(τ z − 1) (z − τ )(τ z − 1)h  (z) (z − τ )(1 − τ z)h  (z)

p(z) 1 extends holomorphic to τ and p(z) = 0 for all z ∈ D. Note that Re p(z) = Re for | p(z)|2 all z ∈ D. Since h is λ-spirallike with respect to τ by Theorem 9.4.3, it follows from 1 ≥ 0 for all z ∈ D, and hence Re p(z) ≥ 0 for all z ∈ D. Theorem 9.4.5 that Re p(z) Thus (10.1.7) is proved for this case. Case II: (φt ) is non-elliptic. Let h be the Koenigs function of (φt ). By Theorem 10.1.4, G(z) = h  i(z) for all z ∈ D. Let τ ∈ ∂D be the Denjoy-Wolff point of (φt ). Define p : D → C by i , z ∈ D. p(z) :=  h (z)(z − τ )(τ z − 1)

By Theorem 9.4.10 the function h is starlike at infinity with respect to τ , hence, by Theorem 9.4.11, Re p(z) =

Im (h  (z)(z − τ )(τ z − 1)) ≥ 0, |h  (z)(z − τ )(τ z − 1)|2

z ∈ D.

Summing up, we have proved that if G ∈ Gen(D), then it has a decomposition of the form (10.1.7). Suppose now that G is a non-constant holomorphic map given by (10.1.7). First, assume that τ = 0. Let λ := p(0) and consider the holomorphic function in the unit disc  z    λ 1 h(z) = z exp − 1 dξ , z ∈ D. p(ξ ) 0 ξ

10.1 Infinitesimal Generators and the Berkson-Porta Formula

Then

279

1 zh  (z) 1 = , z ∈ D. λ h(z) p(z) 

(z) ≥ 0, for all z ∈ D. Since Re p(z) ≥ 0 for all z ∈ D, this clearly implies that Re λ1 zhh(z) Hence, by Theorem 9.4.5, h is λ-spirallike with respect to 0. By Theorem 9.4.3, up to a factor, the function h is the Koenigs function of the semigroup with Denjoy-Wolff point 0 defined by φt (z) = h −1 (e−λt h(z)), t ≥ 0, z ∈ D. A direct computation shows that G and (φt ) satisfy (10.1.2), and hence G is an infinitesimal generator. If τ ∈ D \ {0}, let q := (1 − |τ |2 ) p ◦ Tτ , where Tτ is the canonical automorphism of D given by (1.2.1). Since Re q(z) ≥ 0 for all z ∈ D, for what we already proved, the holomorphic function F(z) = −zq(z) is the infinitesimal generator of a semigroup (ϕt ) with Denjoy-Wolff point 0. Consider the family of functions φt = Tτ ◦ ϕt ◦ Tτ for all t ≥ 0. Clearly, (φt ) is a semigroup with Denjoy-Wolff point τ . Moreover

∂φt (z) ∂ϕt (Tτ (z)) = Tτ (ϕt (Tτ (z))) , ∂t ∂t

z ∈ D, t ≥ 0.

Taking t = 0, for all z ∈ D we have  ∂φt (z)  = Tτ (Tτ (z))(−Tτ (z))q(Tτ (z)) ∂t t=0 Tτ (z) = −  (1 − |τ |2 ) p(z) = (τ z − 1)(z − τ ) p(z). Tτ (z) Thus, the function G(z) = (τ z − 1)(z − τ ) p(z) is an infinitesimal generator. z i dξ , for all z ∈ D. Since G(z) = 0 Finally, assume τ ∈ ∂D. Define h(z) = 0 G(ξ ) for all z ∈ D, the function h is holomorphic. Then Im τ (z − τ )2 h  (z) = Im

1 i = Re ≥ 0, p(z) p(z)

z ∈ D.

Thus, by Theorem 9.4.11, h is starlike at infinity with respect to τ ∈ ∂D. By Theorem 9.4.10, up to a translation, h is the Koenigs function of the semigroup (φt ) with Denjoy-Wolff point τ where φt (z) = h −1 (h(z) + it) for all z ∈ D and t > 0. A direct computation shows that G and (φt ) satisfy (10.1.2), and hence G is an infinitesimal generator. Finally, by the uniqueness of the Denjoy-Wolff point of a non-trivial semigroup, it follows easily that τ , p and the decomposition (10.1.7) are unique.  When the Denjoy-Wolff point of the semigroup belongs to the boundary of the unit disc, Berkson-Porta’s Formula can be interpreted in terms of the Poisson kernel introduced in (1.4.3):

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10 Infinitesimal Generators

Corollary 10.1.11 Let G : D → C be a non-constant holomorphic function and τ ∈ ∂D. Then G is the infinitesimal generator of a semigroup with Denjoy-Wolff point τ if and only if (du τ )z · G(z) ≤ 0 for all z ∈ D.  Proof Write p(z) := (z−τG(z) for all , z ∈ D. Recalling that u τ (z) = Re z+τ )(τ z−1) z−τ z ∈ D, a direct computation shows that  (du τ )z · G(z) = −2 Re

G(z) (z − τ )(τ z − 1)

 = −2 Re p(z), z ∈ D.

Thus, the result follows from Theorem 10.1.10.



Corollary 10.1.12 Let (φt ) be a non-trivial semigroup with associated infinitesimal generator G and Denjoy-Wolff point τ . Let λ be its spectral value. (1) If τ ∈ D then the only zero of G in D is τ and λ = −G  (τ ). In particular, Re G  (τ ) ≤ 0. (2) If τ ∈ ∂D, then G has no zeros in D, ∠ lim G(z) = 0, and − λ = ∠ lim z→τ

z→τ

G(z) = ∠ lim G  (z) ∈ (−∞, 0]. z→τ z−τ

Proof Let h : D → C be the Koenigs function of (φt ). If τ ∈ D, by Theorem 10.1.4, G(z) = −λ hh(z)  (z) and hence   h(z)h  (z) . G (z) = −λ 1 − h  (z)2 

Since h(τ ) = 0 by Proposition 9.4.1, we conclude that G(τ ) = 0 and G  (τ ) = −λ. Suppose now that τ ∈ ∂D. By Theorem 10.1.10, there exists a non vanishing holomorphic function p : D → H such that G satisfies (10.1.7). Since p has no zeros in D, we deduce that G(z) = 0 for all z ∈ D. By Proposition 2.1.3, there exists the limit β := ∠ lim (τ z − 1) p(z) ∈ (−∞, 0]. z→τ

Thus, ∠ lim z→τ

G(z) = ∠ lim (τ z − 1) p(z) = β. z→τ z−τ

Then, clearly, ∠ lim z→τ G(z) = 0 and, by Theorem 1.7.2, β = ∠ lim z→τ G  (z). By (10.1.6), log(φt (r τ ))



t

=

G  (φs (r τ )) ds, 0 < r < 1.

0

Now, αφs (τ ) ≤ 1 for all 0 ≤ s ≤ t and {φs (0)}s∈[0,t] is compact. Therefore, by Proposition 1.5.5, there exists a Stolz region S(τ, L) for some L > 1 such that the curves

10.1 Infinitesimal Generators and the Berkson-Porta Formula

281

r → φs (r τ ) are contained in S(τ, L) for all r ∈ [0, 1) and all 0 ≤ s ≤ t. Hence, the Lebesgue Dominated Convergence Theorem implies −λt = lim− r →1

log(φt (r τ ))

 =

t





lim− G (φs (r τ )) ds =

0 r →1

t

β ds = βt.

0

Thus β = −λ.



The above corollary allows an interesting characterization of hyperbolic semigroups in terms of Koenigs functions: Corollary 10.1.13 Let (φt ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D, canonical model (Ω, h, z → z + it) and spectral value λ ≥ 0. Then the following are equivalent: (1) The semigroup is hyperbolic; (2) there exists ρ > 0 such that

lim

(0,1)r →1

(3) lim inf |h  (r τ )|(1 − r ) < +∞.

Im h(r τ ) = ρ; − log(1 − r )

(0,1)r →1

Moreover, if the previous conditions are satisfied then ρ −1 = λ. Proof Let G be the infinitesimal generator of the semigroup. By Corollary 10.1.12, −λ = ∠ lim z→τ

We know that G(z) =

i h  (z)

G(z) ∈ (−∞, 0]. z−τ

for all z ∈ D. Therefore, ∠ lim z→τ

i h  (z)(z

− τ)

= −λ.

Thus, the semigroup is hyperbolic if and only if (3) holds. Consider the function g(r ) := Im h(r τ ), r ∈ (0, 1). By L’Hôpital’s Rule [122, p. 180], it follows that Im h(r τ ) g(r ) = lim = lim g  (r )(1 − r ) r →1 − log(1 − r ) r →1 − log(1 − r ) r →1 rτ − τ  , = lim Im τ h (r τ )(1 − r ) = − lim Re r →1 r →1 G(r τ ) lim

from which we get the equivalence between (1) and (2), and also that ρ −1 = λ.  Next corollary provides a sufficient condition for the membership in Gen(D). Corollary 10.1.14 Let G : D → C be a holomorphic function such that G(0) = 0 and Re G  (z) ≤ 0 for all z ∈ D. Then G ∈ Gen(D).

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10 Infinitesimal Generators

Proof The function p(z) := − G(z) is holomorphic in the unit disc. According to z Theorem 10.1.10, G is an infinitesimal generator of a semigroup in D that fixes 0 if Re p(z) ≥ 0 for all z ∈ D. But Re (zp  (z) + p(z)) = −Re G  (z) ≥ 0 for all z ∈ D. Thus Corollary 2.2.2 shows that Re p(z) ≥ 0 for all z ∈ D.



10.2 Characterizations of Infinitesimal Generators The Berkson-Porta Formula gives a useful description of infinitesimal generators. In this section we provide other characterizations of the elements of Gen(D). For a ∈ D, let Ta denote the canonical automorphism of D defined in (1.2.1). Definition 10.2.1 The hyperbolic pseudo-distance  ω : D × D → [0, 1) is defined by  ω(z, w) := |Tw (z)|, z, w ∈ D. ω(z,w) . Clearly, the function  ω : D × D → [0, 1) is conNote that 2ω(z, w) = log 1+ 1− ω(z,w) tinuous on D × D and of class C ∞ on D × D \ {(z, z) : z ∈ D}. In order to deduce the different characterizations of infinitesimal generators we are aiming to, we need a couple of preliminary lemmas.

Lemma 10.2.2 Let z, w ∈ D, z = w. Let γ j : [0, 1] → D, j = 1, 2 be two smooth curves with γ1 (0) = z and γ2 (0) = w. The function g : [0, 1] → [0, 1) defined by g(t) :=  ω(γ1 (t), γ2 (t)) is differentiable at t = 0 and 

γ1 (0) − γ2 (0) γ1 (0)w + zγ2 (0) + z−w 1 − zw  γ1 (0)z γ1 (0)w + zγ2 (0) γ2 (0)w (1 − |z|2 )(1 − |w|2 ) Re + − = . |z − w||1 − zw| 1 − |z|2 1 − |w|2 1 − zw

|z − w| Re g (0) = |1 − zw| 

Proof Let h(t) := Tγ2 (t) (γ1 (t)) =

γ2 (t) − γ1 (t) 1 − γ2 (t)γ1 (t)

.

Thus g(t)2 = h(t)h(t), hence, differentiating in t, we deduce that g(t)g  (t) = Re h  (t)h(t) for all t. Since

10.2 Characterizations of Infinitesimal Generators

h  (t) =

γ2 (t) − γ1 (t) 1 − γ2 (t)γ1 (t)

+

283

γ2 (t) − γ1 (t) (1 − γ2 (t)γ1 (t))2

 γ2 (t)γ1 (t) + γ2 (t)γ1 (t) ,

we have  w − z w−z  γ2 (0) − γ1 (0)  + . γ (0)z + wγ1 (0) h (0)h(0) = 1 − wz (1 − wz)2 2 1 − wz 



Dividing by g(0) we get the first expression of g  (0). For the second one, consider the identity: (1 − |γ1 (t)|2 )(1 − |γ2 (t)|2 ) . g(t)2 = 1 − |1 − γ2 (t)γ1 (t)|2 Differentiating again at t = 0, and taking into account that γ1 (0) = z and γ2 (0) = w, we obtain    γ1 (0)z(1 − |w|2 ) + γ2 (0)w(1 − |z|2 ) 2g  (0)g(0) = 2Re |1 − zw|2  (1 − |z|2 )(1 − |w|2 )(γ1 (0)w + γ2 (0)z)(1 − zw) − 2Re , |1 − zw|4 and the second expression of g  (0) follows.



Lemma 10.2.3 Let T > 0 and let g : [0, T ] → R be a function such that (1) for all a, b ∈ [0, T ] and λ ∈ [0, 1] we have g(λa + (1 − λ)b) ≤ max{g(a), g(b)}; (2) there exists ε > 0 such that the restriction g|[0,ε] is C 1 -smooth and g  (0) > 0. Then g is non decreasing. Proof By (2), there is 0 < δ < T such that g is strictly increasing in [0, δ]. Assume that there are δ ≤ t1 < t2 ≤ T such that g(t1 ) > g(t2 ). Take t3 < δ so that g(t3 ) < g(δ). Since t3 < δ ≤ t1 < t2 , (1) implies that g(δ) ≤ max{g(t3 ), g(t1 )} = g(t1 ),

g(t1 ) ≤ max{g(δ), g(t2 )} = g(δ)

where, in the last inequality, we have used that g(t1 ) > g(t2 ). Thus g(t1 ) = g(δ). Applying (1) again, we have that g(δ) ≤ max{g(t3 ), g(t2 )}. But this is impossible because g(δ) > g(t3 ) and g(δ) = g(t1 ) > g(t2 ).



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10 Infinitesimal Generators

Theorem 10.2.4 Let G : D → C be a holomorphic function. The following are equivalent: (1) G ∈ Gen(D); (2) there exist ε > 0 and a family { f t }t∈[0,ε] of holomorphic self-maps of the unit disc D such that f 0 (z) = z for all z ∈ D, and G(z) = lim+ t→0

f t (z) − z , for all z ∈ D; t

(3) (d ω)(z,w) · (G(z), G(w)) ≤ 0 for all z, w ∈ D, z = w; (4) (dω)(z,w) · (G(z), G(w)) ≤ 0 for all z, w ∈ D, z = w; (5) the function G is ω-monotone, i.e., for all z, w ∈ D and for all r > 0 such that z − r G(z), w − r G(w) ∈ D it holds ω(z − r G(z), w − r G(w)) ≥ ω(z, w); (6) for all z, w ∈ D, z = w,  Re

G(z) − G(w) z−w



 + Re

G(z)w + zG(w) 1 − zw

 ≤ 0.

(10.2.1)

Proof Clearly (1) implies (2) with f t = φt , t ∈ [0, +∞) where (φt ) is the semigroup generated by G. Assume that (2) holds. Fix z, w ∈ D, z = w, and let g(t) :=  ω( f t (z), f t (w)) for t ∈ [0, ε]. By Schwarz-Pick Lemma (see Theorem 1.2.3), g(t) ≤ g(0). Since g is differentiable at t = 0, it follows (d ω)(z,w) · (G(z), G(w)) = g  (0) = lim+ t→0

g(t) − g(0) ≤ 0, t

and (3) holds. Next, since ω(z, w) = f ◦  ω(z, w) where f (x) = 21 log 1+x for all x ∈ [0, 1) and 1−x f is an increasing function, (3) is clearly equivalent to (4). Now we want to show that (4) implies (5). Since it is clear that (5) holds automatically for z = w, fix z, w ∈ D, z = w, and r > 0 such that z − r G(z) ∈ D and w − r G(w) ∈ D. Since D is convex, z − t G(z) ∈ D and z − t G(w) ∈ D for every t ∈ [0, r ]. Hence, the curve g(t) := ω(z − t G(z), w − t G(w)) is a well-defined continuous curve, C 1 close to 0. Note that if a, b ∈ [0, r ] and λ ∈ [0, 1], applying Lemma 1.3.8, we have g(λa + (1 − λ)b) = ω(z − (λa + (1 − λ)b)G(z), w − (λa + (1 − λ)b)G(w)) = ω(λ(z − aG(z)) + (1 − λ)(z − bG(z)), λ(w − aG(w)) + (1 − λ)(w − bG(w))) ≤ max{ω(z − aG(z), w − aG(w)), ω(z − bG(z), w − bG(w))} = max{g(a), g(b)}.

Therefore, g satisfies hypothesis (1) of Lemma 10.2.3. Moreover, by (4), g  (0) = −(dω)(z,w) · (G(z), G(w)) ≥ 0. Hence, if (dω)(z,w) · (G(z), G(w)) < 0, g is nondecreasing (by Lemma 10.2.3) and (5) follows. In case (dω)(z,w) · (G(z), G(w)) = 0,

10.2 Characterizations of Infinitesimal Generators

285

let F : D → C be an infinitesimal generator in D such that (dω)|(z,w) · (F(z), F(w)) < 0, for instance, F(z) = − 2z . Now fix ε > 0 small enough such that the infinitesimal generator H := G + εF satisfies z − r H (z), w − r H (w) ∈ D. By construction, (dω)|(z,w) · (H (z), H (w)) < 0 and, for what we already proved, ω(z − r H (z), w − r H (w)) ≥ ω(z, w) for all r > 0 such that z − r H (z), w − r H (w) ∈ D. Now, letting ε tend to 0 we obtain (5). If (5) holds, let z, w ∈ D, z = w and let r > 0 be such that z − r G(z) ∈ D and w − r G(w) ∈ D. Using again the convexity of D, for every t ∈ [0, r ] it holds z − t G(z) ∈ D and z − t G(w) ∈ D. Thus ω(z + t G(z), w + t G(w)) − ω(z, w) t→0 t ω(z − t G(z), w − t G(w)) − ω(z, w) ≤ 0. = lim+ t→0 −t

(dω)(z,w) · (G(z), G(w)) = lim

Hence (4) holds. Assume (3) holds. Let z ∈ D. The maximal solution x z of the Cauchy-Problem  d x(t)

= G(x(t)), dt x(0) = z.

(10.2.2)

is defined in some interval (az , δz ) for some az < 0 < δz . We have to prove that δz = +∞ for all z ∈ D. To this aim, let z, w ∈ D with z = w and let δ = min{δz , δw } > 0. Let g : [0, δ)  t →  ω(x z (t), x w (t)). By the uniqueness of solutions of the above Cauchy problems, z x (t) = x w (t) for all t ∈ [0, δ). Differentiating with respect to t we obtain g  (t) = (d ω)(x z (t),x w (t)) · (G(x z (t)), G(x w (t))) ≤ 0. Therefore g is non-increasing in t. Hence, ω(x z (0), x w (0)) =  ω(z, w) < 1.  ω(x z (t), x w (t)) ≤ 

(10.2.3)

This implies that δz = δw because, if for instance δz < δw , then as t → δz it would ˜ z (t), x w (t)) → 1 follow that |x z (t)| → 1 while x w (t) → x w (δz ) ∈ D, and then ω(x contradicting (10.2.3). By the arbitrariness of z, w ∈ D, this means that for all z ∈ D we have δz = δ. Thus z x z (t + s) = x x (s) (t) for all s, t ≥ 0 such that s + t < δ and z ∈ D. This implies that δ = +∞. Indeed, if δ < +∞, let 2δ > t0 ≥ δ and let s ≥ 0 be such that t0 − s < δ, z s < δ. Define γ (t) := x z (t) for 0 ≤ t ≤ s and γ (t) = x x (s) (t − s) for s ≤ t ≤ t0 . Then γ is well defined and solves the Cauchy problem with initial data z in the interval [0, t0 ], against the definition of δ. Thus we have proved that (1), (2), (3), (4) and (5) are equivalent. The equivalence between (3) and (6) is clear bearing in mind that given two different points z, w in the unit disc and considering the curves γ1 (t) = z + t G(z)

286

10 Infinitesimal Generators

and γ2 (t) = w + t G(w), defined for t ∈ [0, r ] for some small r > 0 such that z + r G(z), w + r G(w) ∈ D, by Lemma 10.2.2, we have |z − w| Re (d ω)(z,w) · (G(z), G(w)) = |1 − zw|



G(z) − G(w) G(z)w + zG(w) + , z−w 1 − zw 

and we are done.

Remark 10.2.5 The role of the hyperbolic distance ω in the statement (4) of above theorem can be played by other important functions in Complex Analysis. Assume that f : (0, +∞) → R is a non-decreasing function of class C 1 , and define γ (z, w) := f ( ω(z, w)) for all z, w ∈ D, z = w. The above proof shows that, given a holomorphic function G : D → C, then G ∈ Gen(D) if and only if (dγ )(z,w) · (G(z), G(w)) ≤ 0 for all z, w ∈ D, z = w. In particular, since the Green ω(z, w), we deduce a characterfunction in the unit disc is given by GD (z, w) = log  ization of Gen(D) in terms of GD . Now we characterize infinitesimal generators in terms of the hyperbolic norm κD . Since κD : D × C → [0, +∞) is a continuous function, smooth outside D × {0}, if G : D → C is a holomorphic map, one can consider the function D  z → κD (z; G(z)) = (κD ◦ G)(z) ∈ [0, +∞). Note that such a function is continuous and smooth outside the points z ∈ D such that G(z) = 0. Theorem 10.2.6 (Abate’s Formula) Let G : D → C be a holomorphic function. The following are equivalent: (1) G ∈ Gen(D); (2) d(κD ◦ G)z · G(z) ≤ 0 for all z ∈ D such that G(z) = 0; (3) for all z ∈ D,   Re 2zG(z) + (1 − |z|2 )G  (z) ≤ 0.

(10.2.4)

Proof Assume that (1) holds. Letting w converge to z in (10.2.1), we obtain (3). Now, assume (3) holds. Fix z ∈ D such that G(z) = 0 and let γ : [0, ε) → D be a curve such that γ  (t) = G(γ (t)) and γ (0) = z. In other words, γ is a solution of the Cauchy Problem (10.2.2). Hence, by (10.2.4), for all t ∈ [0, ε),    dγ (t) 2

  dt  d 2|G(γ (t))|2 2 )G  (γ (t)) ≤ 0. = Re 2γ (t)G(γ (t)) + (1 − |γ (t)| dt (1 − |γ (t)|2 )2 (1 − |γ (t)|2 )3 

Thus the function t → κD (γ (t); γ (t)) = d(κD ◦ G)z · G(z) =

   dγ (t)   dt  1−|γ (t)|2

is not increasing. In particular,

d κD (γ (t); γ  (t))|t=0 ≤ 0. dt

10.2 Characterizations of Infinitesimal Generators

287

Therefore, (2) holds. In order to complete the proof we have to check that (2) implies (1). We will show that (2) implies that the vector field G is semicomplete, and hence, by Theorem 10.1.4, G ∈ Gen(D). Fix z ∈ D, and let x : (−δ  , δ) → D be a maximal solution of the Cauchy Problem (10.2.2), δ  , δ > 0. If G(z) = 0, then x(t) ≡ z and hence δ = +∞, thus we can assume that G(z) = 0. By the uniqueness of solutions of the Cauchy Problem, x  (t) = G(x(t)) = 0 for all t ∈ [0, δ). Hence, the function [0, δ)  t → κD (x(t); x  (t)) is differentiable in t and by (2) d κD (x(t); x  (t)) = d(κD ◦ G)x(t) · G  (x(t)) ≤ 0. dt Therefore, [0, δ)  t → κD (x(t); x  (t)) is non increasing. Now, given s < δ, let γ be the C 1 -smooth curve [0, s]  t → x(t). By definition of hyperbolic distance,  s ω(z, x(s)) ≤ D (γ ) = κD (γ (t); γ  (t))dt 0  s ≤ κD (γ (0); γ  (0))dt = sκD (z; G(z)). 0

Therefore, if δ < +∞, it would follow that x([0, δ)) is contained in a compact subset of D, contradicting the maximality of the solution x (see, for example, [44]). Hence, δ = +∞ and G is semicomplete.  As customary, let Hol(D, C) denote the set of all holomorphic functions in D endowed with the topology of uniform convergence on compacta. A first consequence of the above theorem is: Corollary 10.2.7 Gen(D) is a closed convex cone in Hol(D, C) with vertex in 0. Proof Let {G n } ⊂ Gen(D) be a sequence of infinitesimal generators which converges uniformly on compacta  G : D → C, then by  to a (holomorphic) function Abate’s formula (10.2.4), Re 2zG n (z) + (1 − |z|2 )G n (z) ≤ 0 for all n ∈ N. Taking the limit for n → ∞, it follows that Re 2zG(z) + (1 − |z|2 )G  (z) ≤ 0 and hence G ∈ Gen(D) by Theorem 10.2.6. Moreover, if G, H ∈ Gen(D) and α, β ≥ 0, then by (10.2.4)   Re 2z(αG(z) + β H (z)) + (1 − |z|2 )(αG  (z) + β H  (z))     = αRe 2zG(z) + (1 − |z|2 )G  (z) + βRe 2z H (z) + (1 − |z|2 )H  (z) ≤ 0, hence αG + β H ∈ Gen(D).



Lemma 10.2.8 Given a ∈ C \ {0}, the function G(z) = a − az 2 belongs to Gen(D). Proof Write a = ρeiθ . Then

288

10 Infinitesimal Generators

G(z) = a − az 2 = (z − eiθ )(e−iθ z − 1)ρ

eiθ + z eiθ − z

, it follows that p : D → H and Theorem 10.1.10 guarThus, setting p(z) := ρ eeiθ +z −z antees that G ∈ Gen(D).  iθ

In Sect. 10.3 we will see that, in fact, the function G in above lemma is the infinitesimal generator of a hyperbolic group, but this additional information is not relevant right now. Next corollary gives another useful explicit description of Gen(D). In fact, it shows that any generator is the sum of the infinitesimal generator of an hyperbolic group and the infinitesimal generator of a semigroup that fixes the origin: Corollary 10.2.9 (Ahoronov-Elin-Reich-Shoikhet’s Formula) Let G : D → C be a holomorphic function. Then G ∈ Gen(D) if and only if there exist a ∈ C and a holomorphic function p : D → H such that G(z) = a − az 2 − zp(z), z ∈ D.

(10.2.5)

Proof Assume that G ∈ Gen(D). Clearly, if G ≡ 0 the statement is true by setting a = 0 and p ≡ 0. Therefore, we can assume that G ≡ 0, and let a := G(0). Lemma 10.2.8 implies that the function F(z) := −a + az 2 belongs to Gen(D). Thus, by Corollary 10.2.7, H := G + F is an infinitesimal generator and H (0) = 0 by construction. By Theorem 10.1.10, there exists p : D → H holomorphic such that H (z) = −zp(z) for all z ∈ D. Thus, G(z) = a − az 2 − zp(z) for z ∈ D. Conversely, assume that (10.2.5) is satisfied. Since F1 (z) = a − az 2 and F2 (z) = −zp(z) belongs to Gen(D) (using again Lemma 10.2.8 and Theorem 10.1.10, or Lemma 10.1.9 if p ≡ 0), by Corollary 10.2.7, the function G = F1 + F2 belongs to Gen(D).  Theorem 10.2.10 Let G : D → C be a holomorphic function. The following are equivalent: (1) G ∈ Gen(D); (2) G satisfies lim sup Re (zG(z)) ≤ 0, for all σ ∈ ∂D;

(10.2.6)

z→σ

(3) Re G(z)z ≤ Re G(0)z(1 − |z|2 ) for all z ∈ D; (4) for all z, w ∈ D,    zG(w) + wG(z) G(w)w G(z)z ≤ Re + Re . 1 − |z|2 1 − |w|2 1 − zw 

Proof First, we show that (1) implies (4). Take two points z, w ∈ D, z = w, and consider the curves γ1 (t) = z − t G(z) and γ2 (t) = w − t G(w), defined for t ∈ [0, r ]

10.2 Characterizations of Infinitesimal Generators

289

for some small r > 0 such that z − r G(z), w − r G(w) ∈ D. By Lemma 10.2.2, the derivative of the function g(t) =  ω(γ1 (t), γ2 (t)) at t = 0 is (1 − |z|2 )(1 − |w|2 ) Re g (0) = |z − w||1 − zw| 



γ1 (0)z γ1 (0)w + zγ2 (0) γ2 (0)w + − . 1 − |z|2 1 − |w|2 1 − zw

By Theorem 10.2.4, (1) implies that g(t) ≥ g(0) for t ∈ [0, r ]. In particular g  (0) ≥ 0 and, since γ1 (0) = −G(z) and γ2 (0) = −G(w), we obtain (4). If (4) holds, setting w = 0, (3) holds at once. Moreover, it is clear that (3) implies (2). Assume that (2) holds. Write a = G(0) and define p : D → C by p(z) = −az 2 +a−G(z) . Clearly, the function p is holomorphic in the unit disc. Moreover, z zG(z) = −|z|2 p(z) − (az|z|2 − az), z ∈ D. Therefore, given σ ∈ ∂D, by (10.2.6), we have 0 ≥ lim sup Re (zG(z)) = lim sup Re (− p(z)). z→σ

z→σ

By the Maximum Principle for harmonic functions, Re p(z) ≥ 0, for all z ∈ D. Thus, by Corollary 10.2.9, G ∈ Gen(D). Hence, (1) holds.  A direct consequence of (10.2.6) is the following interesting fact: Corollary 10.2.11 Let φ : D → D be an holomorphic function. Then G := φ − idD belongs to Gen(D). This corollary, together with Corollary 10.2.7, implies that for every holomorphic function φ : D → D and for every α ≥ 0 the holomorphic function α(φ − idD ) belongs to Gen(D). Combining this fact with statement (2) in Theorem 10.2.4, it follows Gen(D) = {α(φ − idD ) : φ ∈ Hol(D, D), α ≥ 0}, where the closure is taken in the topology of uniform convergence on compact subsets of D.

10.3 Infinitesimal Generators of Groups We start with the following characterization of infinitesimal generators of groups: Proposition 10.3.1 Let G : D → C be a holomorphic function. The following are equivalent: (1) G is the infinitesimal generator of a group;

290

(2) (3) (4) (5) (6)

10 Infinitesimal Generators

  Re 2zG(z) + (1 − |z|2 )G  (z) = 0, for all z ∈ D; lim z→σ Re (zG(z)) = 0 for all σ ∈ ∂D; Re G(z)z = Re G(0)z(1 − |z|2 ), for all z ∈ D; (dω)(z,w) · (G(z), G(w)) = 0 for all z, w ∈ D, z = w; there exist a ∈ C and b ∈ R such that G(z) = az 2 + ibz − a, for all z ∈ D.

Proof As we saw in Sect. 8.2, if (φt ) is a group in the unit disc, then (ϕt ) := (φ−t ) is also a group. A simple computations shows that if G is the infinitesimal generator (φt ), then −G is the infinitesimal generator of (ϕt ). Therefore, G ∈ Gen(D) is the infinitesimal generator of a group if and only if −G also belongs to Gen(D). As an immediate consequence of this and the main results in Sect. 10.2, it follows that (1), (2), (3), (4) and (5) are equivalent. Now, if there exist a ∈ C and b ∈ R such that G(z) = az 2 + ibz − a, for all z ∈ D, then lim Re (zG(z)) = lim Re (az|z|2 + ib|z|2 − az) = Re (aσ |σ |2 + ib|σ |2 − aσ ) = 0,

z→σ

z→σ

for all σ ∈ ∂D. Hence (6) implies (3). Finally, if G is the infinitesimal generator of a group, by Corollary 10.2.9 there are a ∈ C and p : D → C holomorphic with non-negative real part, such that G(z) = az 2 − a − zp(z), for all z ∈ D, and by (4), we deduce that Re (az|z|2 − az) − |z|2 Re p(z) = Re (zG(z)) = Re (−az(1 − |z|2 )), z ∈ D. Therefore, Re p(z) = 0 for all z ∈ D, and thus there exists b ∈ R such that Re p(z) = ib for all z and we are done.  A neat formulation of the Berkson-Porta Formula for groups however requires some work. The reason is that the Denjoy-Wolff point of the group generated by G might differ from that of the group generated by −G, as in the case of a hyperbolic group. In the next corollary we explicitly describe infinitesimal generators of elliptic, hyperbolic and parabolic groups: Corollary 10.3.2 Let G : D → C be a non-constant holomorphic function. (1) G is the infinitesimal generator of an elliptic group with Denjoy-Wolff point τ ∈ D and spectral value ωi for some ω ∈ R \ {0} if and only if G(z) = (z − τ )(τ z − 1)

iω . 1 − |τ |2

(10.3.1)

(2) G is the infinitesimal generator of a hyperbolic group with Denjoy-Wolff point τ ∈ ∂D, other fixed point σ ∈ ∂D and spectral value λ > 0 if and only if G(z) =

λ (z − τ )(z − σ ). σ −τ

(10.3.2)

10.3 Infinitesimal Generators of Groups

291

(3) G is the infinitesimal generator of a parabolic group with Denjoy-Wolff point τ ∈ ∂D if and only if there exists a ∈ C \ {0}, with Re a = 0, such that G(z) =

a τ (z − τ )2 . 2

(10.3.3)

In this case, a = τ φ1 (τ ). Proof Let G be an infinitesimal generator of a non-trivial group (φt ). Therefore, by Theorem 10.1.10, there exist τ ∈ D and a never vanishing holomorphic function p : D → H such that the Berkson-Porta Formula (10.1.7) holds. If p ≡ iα for some α ∈ R, α = 0, then G is given either by (10.3.1)—if τ ∈ D, or by (10.3.3)—if τ ∈ ∂D. In the first case, the group is elliptic with Denjoy-Wolff point τ and, by Corollary 10.1.12, its spectral value is −G  (τ ) = iω. In the second case, the Denjoy-Wolff point is τ ∈ ∂D and, again by Corollary 10.1.12, its spectral value is ∠ lim z→τ G  (z) = G  (τ ) = 0, and the group is parabolic. Assume now Re p(z) > 0 for all z ∈ D. If τ ∈ D, by Proposition 10.3.1(2), it follows Re p(τ ) = 0. Since this is not possible, τ ∈ ∂D. Write g(r ) = Re p(r τ ), −1 < r < 1. Taking z = r τ in Proposition 10.3.1(2), we deduce that −2Re p(r τ ) + (1 − r 2 )Re τ p  (r τ ) = 0,

−1 < r < 1.

That is, −2g(r ) + (1 − r 2 )g  (r ) = 0 for −1 < r < 1. Integrating this ordinary diffor −1 < r < 1 and for some ferential equation we find Re p(r τ ) = g(r ) = c 1+r 1−r + ib for some b ∈ R. In particular, since c ∈ R. It follows that p(z) = c 1+zτ 1−zτ Re p > 0, we deduce that c > 0, and then G is of the form  1 + zτ + ib = (z − τ ) [−c(1 + zτ ) + ib(τ z − 1)] G(z) = (z − τ )(τ z − 1) c 1 − zτ   2c c + ib = τ (ib − c)(z − τ ) z − τ = (z − τ )(z − σ ), ib − c σ −τ 

c+ib where we set σ := τ ib−c ∈ ∂D. By Corollary 10.1.12, the spectral value of the semigroup generated by G is λ = −∠ lim z→τ G  (z) = −G  (τ ) = 2c > 0. In particular, the group (φt ) is hyperbolic, with spectral value 2c. The other fixed point of (φt ) is σ , because G extends holomorphically through D and G(σ ) = 0, implying that φt (σ ) = σ for all t ≥ 0, hence, (10.3.2) holds. Finally, recalling that G(z) = h  i(z) where h is the Koenigs function of the semigroup in the non-elliptic case and applying Proposition 9.3.12, we deduce that a = τ φ1 (τ ) in (3). Conversely, by Proposition 10.3.1, it is clear that every G of the form (10.3.1), (10.3.2) or (10.3.3) is the infinitesimal generator of a group in D, and Corollary 10.1.12 shows the type of the group G generates. 

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The above result also shows that G is the infinitesimal generator of a hyperbolic group if and only if there are τ ∈ ∂D and two real numbers a, b, with a > 0, such that z ∈ D. G(z) = aτ (z 2 − τ 2 ) + ibτ (z − τ )2 , . This means In fact, the other fixed point of the semigroup is given by σ = τ ib−a a+ib that the infinitesimal generator of an arbitrary hyperbolic group is the sum of the infinitesimal generator of a hyperbolic group with fixed points τ and −τ and the infinitesimal generator of a parabolic group with Denjoy-Wolff point τ . Remark 10.3.3 Let G ∈ Gen(D). By Corollary 10.2.9, G has a decomposition of the form G(z) = a − az 2 − zp(z), for some constant a and a holomorphic function with non-negative real part p. Notice that G  (0) = − p(0). Thus G  (0) = 0 if and only if there exist a positive real number α and a point τ ∈ ∂D, chosen so that a = ατ , such that G(z) = ατ (τ 2 − z 2 ), that is, by (10.3.2), G is the infinitesimal generator of a group of hyperbolic automorphisms with fixed points τ and −τ .

10.4 Infinitesimal Generators of Semigroups of Linear Fractional Maps In this section we consider semigroups of linear fractional maps and provide a complete characterization of their infinitesimal generators. For further reference we state here the following simple fact. Lemma 10.4.1 Let p(z) = mz + n be a complex polynomial. Then Re p(z) ≥ 0 for every z ∈ D if and only if Re (n) ≥ |m|. Our first result shows that the infinitesimal generators of semigroups of linear fractional maps are polynomials of degree at most two. Theorem 10.4.2 Let G : D → C be a holomorphic function. The following are equivalent: (1) The map G is the infinitesimal generator of a semigroup of linear fractional maps. (2) The map G is the infinitesimal generator of a semigroup in D and it is a polynomial of degree at most two. (3) The map G is a polynomial of the form G(z) = a − az 2 − z(mz + n), z ∈ D with a, m, n ∈ C and Re (n) ≥ |m|. (4) The map G is a polynomial of the form G(z) = αz 2 + βz + γ with α, β, γ ∈ C and Re (β) + |α + γ | ≤ 0.

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293

(5) The map G is a polynomial of degree at most two and satisfies the following condition Re (G(σ )σ ) ≤ 0, for all σ ∈ ∂D. Moreover, Re (n) = m = 0 in statement (3) if and only if Re (β) = |α + γ | = 0 in statement (4) if and only if equality holds for all σ ∈ ∂D in statement (5) if and only if G is the infinitesimal generator of a group. Proof Assume that G is the infinitesimal generator of a semigroup of linear fractional maps (φt ) with Denjoy-Wolff point τ ∈ D. If (φt ) is a group, then G is a polynomial of degree two by Corollary 10.3.2. If (φt ) is either elliptic or parabolic, not a group, by Proposition 9.5.1, its Koenigs function is given by a linear fractional map h(z) = az+b λ , with ad − cb = 0. In the first case, G(z) = λ hh(z)  (z) = ad−cb (az + b)(cz + d), cz+d . In both cases, G is a for some λ, and in the second case G(z) = h  i(z) = − λi (cz+d) ad−cb polynomial of degree at most two. In case (φt ) is hyperbolic, not a group, by (9.5.2) its Koenigs function is given by 2

  τ +z τ +σ π i − + , h(z) = log λ τ −z τ −σ 2λ for some λ > 0, where log denotes the principal branch of the logarithm. Since G(z) = h  i(z) , it follows also in this case that G is a polynomial of degree at most two. Finally, if (φt ) is the trivial semigroup then G ≡ 0. Thus, (1) implies (2). Let us prove that (2) implies (1). Let (φt ) be the semigroup generated by G. If G is constant, by Lemma 10.1.9, G ≡ 0 and (φt ) is the trivial semigroup, thus (1) holds in this case. If G is not constant, let τ ∈ D be the Denjoy-Wolff point of (φt ). Assume that G ∈ Gen(D) is a polynomial of degree one. Being G(τ ) = 0, G(z) = −λ(z − τ ) for some λ ∈ C \ {0} and, by Corollary 10.1.12, λ is the spectral value of (φt ). If τ ∈ D, by Theorem 10.1.4, h  (z)λ(z − τ ) = λh(z), where h is the Koenigs function of (φt ). Hence, h(z) = c(z − τ ), for some c ∈ C \ {0}, and φt (z) = h −1 (e−λt h(z)) = e−λt z + τ (1 − e−λt ) for all z ∈ D. If τ ∈ ∂D, by Theorem 10.1.4, the Koenigs function h of the semigroup (φt ) satisfies −λ(z − τ ) =

i h  (z)

,

z ∈ D.

Then h(z) = − λi log(1 − τ z), where log denotes the principal branch of the logarithm. Then a direct computation, keeping in mind that h −1 (w) = τ (1 − exp(iλw)), shows that φt (z) = h −1 (h(z) + it) = τ (1 − (1 − τ z)e−λt ) = e−λt z + τ (1 − e−λt ) for t ≥ 0 and z ∈ D. Hence, (1) holds in case G is a polynomial of degree one. Assume now that G ∈ Gen(D) is a polynomial of degree two. Since G(τ ) = 0, it follows that G(z) = μ(z − τ )(z − β) for some μ ∈ C \ {0} and β ∈ C. We claim that β ∈ / D. Indeed, if β ∈ D by Remark 10.1.6, β would be a common fixed point of (φt )—implying in particular that (φt ) is either trivial or elliptic. But, if (φt ) is trivial, then G ≡ 0, hence, (φt ) has to be elliptic and τ = β. However, if this is the

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case, then G  (τ ) = 0 and by Corollary 10.1.12 the spectral value of (φt ) would be 0, forcing (φt ) to be trivial, again, a contradiction. Therefore, β ∈ C \ D. If τ ∈ D, since G  (τ ) = μ(τ − β), by Corollary 10.1.12 the spectral value of (φt ) is −μ(τ − β), and then the Koenigs function h of the semigroup satisfies h  (z)μ(z − τ )(z − β) = μ(τ − β)h(z) by Theorem 10.1.4. Therefore h(z) = z−τ . Since h is a linear fractional map, so is each iterate of the semi(β − τ ) β−z group. If τ ∈ ∂D and β = τ , then μ(z − τ )2 = h  i(z) by Theorem 10.1.4. Thus 1 h(z) = −i + c for some constant c. Again h is a linear fractional map and hence μ z−τ the iterates of the semigroup are linear fractional maps. Finally, if τ ∈ ∂D and β = τ , then μ(z − τ )(z − β) = h  i(z) and h(z) =

  1 − τz i log , μ(τ − β) 1 − β −1 z

1−γ where log denotes the principal branch of the logarithm. Since h −1 (w) = τ −β −1 γ , −1 where γ = exp(μ(β − τ )iw), an easy computation shows that φt (z) = h (h(z) + it) is a linear fractional map for all t ≥ 0. Therefore, (2) implies (1). Assume that G ∈ Gen(D) is a polynomial of degree at most two. By Corollary 10.2.9, there is a holomorphic function p with non-negative real part such that G(z) = a − az 2 − zp(z), z ∈ D. Since G is a polynomial of degree two, p(z) = mz + n for some m, n ∈ C. By Lemma 10.4.1, Re (n) ≥ |m|. Therefore, (2) implies (3). Clearly, again by Corollary 10.2.9, the converse implication holds. Identifying the coefficients in statements (3) and (4), we have that α = −a − m and β = −n. Hence, (3) is equivalent to (4). If G is a polynomial of the form G(z) = αz 2 + βz + γ and |σ | = 1, then

Re G(σ )σ = Re (ασ + γ σ ) + Re β = Re ((α + γ )σ ) + Re β. This equality, which holds for every σ ∈ ∂D, implies at once that (4) and (5) are equivalent.  The previous theorem shows that not every complex polynomial of degree two is an infinitesimal generator of a semigroup in D. In the next two propositions we analyze in details the complex polynomials of degree one and two which are infinitesimal generators. We start with degree one polynomials: Proposition 10.4.3 Let G(z) = −λ(z − τ ), λ ∈ C \ {0}, be a complex polynomial of degree one. Then G is the infinitesimal generator of a semigroup in D if and only if |λτ | ≤ Re λ. In this case, the associated semigroup (φt ) has Denjoy-Wolff point τ ∈ D and spectral value λ and φt (z) = e−λt z + τ (1 − e−λt ), Moreover,

z ∈ D.

(10.4.1)

10.4 Infinitesimal Generators of Semigroups of Linear Fractional Maps

295

(1) if Re λ = 0 then τ = 0 and (φt ) is a group of rotations, (2) if Re λ > 0 and τ ∈ D, then (φt ) is an elliptic semigroup which is not a group, (3) if τ ∈ ∂D then λ ∈ (0, +∞) and (φt ) is a hyperbolic semigroup which is not a group. Proof Theorem 10.4.2 implies that G(z) = −λ(z − τ ) is an infinitesimal generator of a semigroup in D (which is necessarily a semigroup of linear fractional maps) if and only if −Re λ + |λτ | ≤ 0. Note that the latter inequality implies that τ ∈ D and Re λ ≥ 0. Moreover, if Re λ = 0 then τ = 0. Now assume that G is the infinitesimal generator of the semigroup (φt ) in D. Since the infinitesimal generators of groups are complex polynomials of degree two except in case the Denjoy-Wolff point is 0 (see Corollary 10.3.2), then (φt ) can not be a group except in case τ = 0. Moreover, the unique solution of G(z) = 0 is z = τ , hence, Corollary 10.1.12 implies that τ is the Denjoy-Wolff point of (φt ) and its spectral value is λ. In particular, if τ ∈ ∂D then λ ∈ (0, +∞) (because we already pointed out that Re λ > 0) and hence (φt ) is hyperbolic. Finally, it is clear that if (φt ) is given by (10.4.1), then it solves the equation ∂φ∂tt (z) = G(φt (z)) for all z ∈ D and all t ≥ 0. Hence, by Theorem 10.1.4, the semigroup  generated by G is (φt ) given by (10.4.1). The previous proposition shows that there is a one to one correspondence between infinitesimal generators which are complex polynomials of degree one and affine semigroups in D. Now, we examine infinitesimal generators which are complex polynomials of degree two. Proposition 10.4.4 Let G(z) = μ(z − c1 )(z − c2 ) be a complex polynomial of degree two with μ = |μ|eiθ , μ = 0, and |c1 | ≤ |c2 |. Then, G is the infinitesimal generator of a semigroup in D if and only if Re (eiθ c1 + eiθ c2 ) ≥ 0 and (|c1 |2 − 1)(1 − |c2 |2 ) ≥ [Im (eiθ c1 − eiθ c2 )]2 . (10.4.2) In this case, if (φt ) is the semigroup in D generated by G with Denjoy-Wolff point τ ∈ D, then the following are the only possible cases: (1) if c1 ∈ D, then τ = c1 , (φt ) is an elliptic semigroup and c2 ∈ C \ D. Moreover: (a) if c2 ∈ ∂D, then the semigroup is not a group, it has two fixed points in D and eiθ (c2 − c1 ) ∈ (0, +∞); (b) if c2 ∈ C \ D, and c2 c1 = 1 then the semigroup is not a group, it has only one fixed point in D and Re (eiθ (c1 + c2 )) ∈ (0, +∞), Im (eiθ (c1 − c2 )) ∈ [−β, β],  where β := (|c1 |2 − 1)(1 − |c2 |2 ) > 0; (c) if c2 ∈ C \ D and c2 c1 = 1, then the semigroup is a group of automorphisms and Re (eiθ c1 ) = 0.

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(2) If c1 = c2 ∈ ∂D, then τ = c1 , the semigroup (φt ) is parabolic, and Re (eiθ c1 ) ≥ 0. Moreover in this case, (φt ) is a parabolic group if and only if Re (eiθ c1 ) = 0. (3) If c1 ∈ ∂D and c2 ∈ C \ (D ∪ {c1 }) then (φt ) is a hyperbolic semigroup. Moreover, c1 = τ if and only if eiθ (c2 − c1 ) ∈ (0, +∞). In this case, (φt ) is a hyperbolic group if and only if c2 ∈ ∂D. Proof Since Gen(D) is a cone, G(z) = μ(z − c1 )(z − c2 ) belongs to Gen(D) if and only if eiθ (z − c1 )(z − c2 ) does. By Theorem 10.4.2, this happens if and only if  |e−iθ + eiθ c1 c2 | ≤ Re eiθ (c1 + c2 ) . This inequality is equivalent to Re (eiθ c1 + eiθ c2 ) ≥ 0 Since

and

  2 |e−iθ + eiθ c1 c2 |2 ≤ Re eiθ (c1 + c2 ) . (10.4.3)

|e−iθ + eiθ c1 c2 |2 = 1 + 2Re (e2iθ c1 c2 ) + |c1 |2 |c2 |2

and 2   Re eiθ (c1 + c2 ) = |c1 |2 + |c2 |2 − [Im (eiθ c1 − eiθ c2 )]2 + 2Re (e2iθ c1 c2 ), inequality (10.4.2) holds if and only if (10.4.3) does and (10.4.2) is therefore equivalent to G being an infinitesimal generator. Now assume that the complex polynomial of degree two G is the infinitesimal generator of a semigroup (φt ) in D, necessarily formed by linear fractional maps by Theorem 10.4.2. Let τ ∈ D be its Denjoy-Wolff point. First, we prove (1). Bearing in mind that any non constant infinitesimal generator has at most one zero in D by Remark 10.1.6, if c1 ∈ D, then τ = c1 and |c2 | ≥ 1. By Corollary 10.3.2, if (φt ) is an elliptic group, then c1 c2 = 1. If |c2 | = 1, by (10.4.2), eiθ (c1 − c2 ) is a positive real number, and hence c1 c2 = 1, which implies that (φt ) cannot be a group, and a) follows. Similarly, we deduce b) and c). If c1 = c2 , then necessarily |c1 | = 1, for otherwise G(c1 ) = G  (c1 ) and Corollary 10.1.12 implies that the Denjoy-Wolff point of (φt ) is c1 and the spectral value is 0, that is, (φt ) is the trivial semigroup and hence G ≡ 0. Therefore, c1 = c2 ∈ ∂D again by Corollary 10.1.12, (φt ) is a parabolic semigroup. By Corollary 10.3.2, it is a group if and only if Re (eiθ c1 ) = 0. Therefore (2) holds. Finally, if c1 ∈ ∂D and c2 ∈ C \ (D ∪ {c1 }) then by Corollary 10.1.12, (φt ) is a hyperbolic semigroup. Since G  (c1 ) = |λ|eiθ (c1 − c2 ) and G  (c2 ) = |λ|eiθ (c2 − c1 ), it follows that the Denjoy-Wolff point is c1 if and only if eiθ (c1 − c2 ) < 0. Also, by  Corollary 10.3.2, the semigroup is a group if and only if c2 ∈ D.

10.5 Convergence of Infinitesimal Generators

297

10.5 Convergence of Infinitesimal Generators In this section, we analyze the convergence of infinitesimal generators, where we endow Gen(D) with the topology of uniform convergence on compacta. In what follows, if G ∈ Gen(D), G not identically zero, using the Berkson-Porta Formula (10.1.7), we write G(z) = (z − τG )(τG z − 1) pG (z), z ∈ D, where τG ∈ D and pG : D → H is holomorphic. Finally, let us denote by 0 the zero function, that is, 0(z) = 0 for all z ∈ D. Proposition 10.5.1 The following two maps are continuous B Pτ : Gen(D) \ {0} → D, Gen(D) \ {0}  G → B Pτ (G) := τG ; B Pp : Gen(D) \ {0} → Hol(D, C), Gen \ {0}  G → B Pp (G) := pG . Proof By the uniqueness of the Berkson-Porta representation, the maps B Pτ and B Pp are well-defined. Let {G n } be a sequence in Gen(D) \ {0} converging to G ∈ Gen(D) \ {0}. Let τn := B Pτ (G n ), τ := B Pτ (G), pn := B Pp (G n ), and p := B Pp (G). We have to show that τn → τ and pn → p. To this aim, it is enough to show that any converging subsequences of {τn } and { pn } converges to τ and p, respectively. Since the right half-plane is biholomorphic to D via a Cayley transform, the family { pn : n ∈ N} is a normal family in Hol(D, C) by Montel’s Theorem (see, e.g., [113, Theorem 14.6, p. 282]). Therefore there exist a strictly increasing sequence of natural numbers {n k } and a point α ∈ D such that τn k → α and pn k converges uniformly on compacta either to a holomorphic function q : D → C or to ∞. Since {G n } converges to a function, we deduce that this last option cannot be possible and hence pn k → q with Re q ≥ 0. Therefore, for all z ∈ D, G(z) = lim G n k (z) = (z − α)(αz − 1)q(z). k→∞

On the other hand, G(z) = (z − τG )(τG z − 1) pG (z). By the uniqueness of the Berkson-Porta representation, we conclude that α = τ and q = pG as wanted.  In the next propositions we give two simple conditions for a sequence of infinitesimal generators to have a converging subsequence. Proposition 10.5.2 Let {G n } be a sequence in Gen(D) such that there are two different points z 0 , z 1 ∈ D and two sequences {u n } and {vn } in D with limn u n = z 0 and limn vn = z 1 such that sup |G n (u n )| < +∞ and n

sup |G n (vn )| < +∞. n

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Then there exists a subsequence {G n k } converging uniformly on compacta to an infinitesimal generator G ∈ Gen(D). Proof We may assume that G n is not identically zero for all n. By Berkson-Porta’s Theorem (Theorem 10.1.10), there are points τn ∈ D and holomorphic maps pn : D → C, with Re pn ≥ 0, such that G n (z) = (z − τn )(τn z − 1) pn (z) for all z ∈ D. Since the sequence {τn } is bounded and { pn : n ∈ N} is a normal family by Montel’s Theorem (see, e.g., [113, Theorem 14.6, p. 282]), there exist a strictly increasing sequence of natural numbers {n k } and a point τ ∈ D such that τn k → τ and pn k converges uniformly on compacta either to a holomorphic function p : D → C or to ∞. Suppose that { pn k } compactly diverges to ∞. Since z 0 and z 1 are different, we may assume that τ = z 0 . Then we have that +∞ > sup |G n (u n )| ≥ lim |G n k (u n k )| = lim |(u n k − τn k )(τn k u n k − 1) pn k (u n k )| k→∞

n

k→∞

= |(z 0 − τ )(τ z 0 − 1)| lim | pn k (u n k )| = +∞. k→∞

A contradiction. So ( pn k ) converges uniformly on compacta to a holomorphic function p : D → C with Re p ≥ 0. Letting G(z) = (z − τ )(τ z − 1) p(z), it follows then that (G n k ) converges uniformly on compacta to G and, again by Berkson-Porta’s Theorem, G is an infinitesimal generator.  Proposition 10.5.3 Let {G n } be a sequence in Gen(D). Suppose that there exist z 0 ∈ D and a sequence {z n } in D, with limn→∞ z n = z 0 , such that sup |G n (z n )| < +∞ and n

sup |G n (z n )| < +∞. n

Then there exists a subsequence {G n k } converging uniformly on compacta to an infinitesimal generator G ∈ Gen(D). Proof Assume firstly that z n = 0 for all n. By Corollary 10.2.9, there are holomorphic functions pn : D → C with Re pn (z) ≥ 0 for all z ∈ D such that G n (z) = G n (0) − G n (0)z 2 − zpn (z), (z ∈ D). Notice that G n (0) = − pn (0). So by Theorem 2.2.1 and Montel’s Theorem (see, e.g., [113, Theorem 14.6, p. 282]), there is a subsequence {G n k } that converges uniformly on compacta to a holomorphic function G : D → C such that G(z) = a − az 2 − zp(z) with Re p(z) ≥ 0 for all z ∈ D (a is the limit of G n k (0) and p is the limit of { pn k } in Hol(D, C)). Using again Corollary 10.2.9, we deduce that G ∈ Gen(D). z n −z , for all z ∈ D. If (φt(n) ) is the Let us pass to the general case. Let Tzn (z) = 1−z nz semigroup with associated infinitesimal generator G n , write ϕt(n) = Tzn ◦ φt(n) ◦ Tzn .

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299

Then (ϕt(n) ) is the semigroup with associated infinitesimal generator 2 n (z) := Tz (Tzn (z))G n (Tzn (z)) = (1 − z n z) G n (Tzn (z)), z ∈ D. G n |z n |2 − 1

n (0) = 12 G n (z n ) and G n (0) = −2z2 n G n (z n ) + G n (z n ), the previous Since G |z n | −1 |z n | −1 n k } converging to an infinitesimal argument shows that there is a subsequence {G  ∈ Gen(D) uniformly on compacta. Now, a direct argument shows that generator G the sequence {G n k } converges uniform on compacta to a holomorphic function H . By Corollary 10.2.7, H ∈ Gen(D).  Remark 10.5.4 It is worth noticing that Propositions 10.5.2 and 10.5.3 are sharp, in the sense that the given conditions for convergence are minimal. The first one would not be true if we assumed that there is only one point z 0 ∈ D and one sequence (u n ) in D with limn u n = z 0 such that sup |G n (u n )| < +∞. n

For example, consider the sequence of infinitesimal generators given by G n (z) := −nz, for all z ∈ D (G n is the infinitesimal generator of the semigroup ϕt (z) = e−nt z). This example also shows that the conclusion of Proposition 10.5.3 is not true if we only assume that |G n (0)| is a bounded sequence. Moreover, considering the sequence of infinitesimal generators of hyperbolic groups {G n (z)} = {n − nz 2 }, the sequence {|G n (0)|} is bounded but the sequence of infinitesimal generators has no converging subsequences. The last result of this section shows that the pointwise convergence of a sequence of infinitesimal generators not only implies that the limit is an infinitesimal generator but also the convergence of the associated semigroups. The main two ingredients in the proof are Proposition 10.5.2 and the following Gronwall’s Lemma: Lemma 10.5.5 (Gronwall’s Lemma) Let θ, k, f : [a, b] → R be continuous functions such that k is non negative and f is increasing. If 

t

θ (t) ≤ f (t) +

k(s)θ (s) ds, for all t ∈ [a, b],

(10.5.1)

a



then θ (t) ≤ f (t) exp

t

 k(s) ds ,

for all t ∈ [a, b].

(10.5.2)

a

t Proof Consider the auxiliary function g(t) = exp(− a k(s) ds). Note that g  (t) = −g(t)k(t) for all t ∈ [a, b] and g(a) = 1. Then, for all t ∈ [a, b],

300

d dt

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 g(t)

t

 k(s)θ (s) ds

= g  (t)



t

k(s)θ (s) ds + g(t)k(t)θ (t)  t k(s)θ (s) ds + g(t)k(t)θ (t) = −g(t)k(t) a   t  = g(t)k(t) − k(s)θ (s) ds + θ (t) ≤ g(t)k(t) f (t),

a

a

a

where the last inequality follows from (10.5.1). Integrating between a and t, we have  g(t) a

t

 k(s)θ (s) ds ≤

t

g(s)k(s) f (s) ds.

a

This inequality, (10.5.1) and the monotonicity of f imply 

 t 1 θ (t) ≤ f (t) + k(s)θ (s) ds ≤ f (t) + g(s)k(s) f (s) ds g(t) a  a     t  t 1 1 f (t)  ≤ f (t) 1 + g(s)k(s) ds = f (t) 1 − g (s) ds = , g(t) a g(t) a g(t) t



and (10.5.2) holds.

Theorem 10.5.6 Let {G n } be a sequence in Gen(D) that converges pointwise to a function G : D → C. Then: (1) The sequence {G n } converges to G uniformly on compacta of the unit disc, G is holomorphic and belongs to Gen(D). (2) If (φt(n) ) denotes the semigroup associated with G n , for n ∈ N, and (φt ) denotes the semigroup associated with G, then φt (z) = lim φt(n) (z), n→∞

and the convergence is uniform on compacta of {(t, z) ∈ [0, ∞) × D}. Proof (1) Take z 0 = 0 and z 1 = 1/2 and consider the constant sequences {u n } = {z 0 } and {vn } = {z 1 }. By Proposition 10.5.2, every subsequence of {G n } has a subsequence that converges uniformly on compact subsets of the unit disc to an infinitesimal generator. Necessarily such infinitesimal generator has to be the function G and the pointwise convergence of {G n } is then uniformly on compacta. (2) First of all, we claim the following: given R ∈ (0, 1) there exists M(R) > 0 such that |G n (z) − G n (z  )| ≤ M(R)|z − z  |, for all |z| ≤ R, |z  | ≤ R, n ∈ N. ∈ (0, 1). Since (G n ) converges to G uniIndeed, fix R ∈ (0, 1), and let R ∗ := 1+R 2 formly on compact subsets of D and G is bounded on the closure of the disc

10.5 Convergence of Infinitesimal Generators

301

D(0, R ∗ ) := {z ∈ C : |z| < R ∗ }, there exists M R > 0 such that sup{|G n (z)| : |z| ≤ R ∗ , n ∈ N} ≤ M R . Let |z|, |z  | ≤ R. Since R < R ∗ , for all n, the Cauchy Integral Formula gives    G n (ξ ) G n (ξ ) 1 dξ − G n (z) − G n (z ) = 2πi ∂ D(0,R ∗ ) ξ − z ξ − z  z − z G n (ξ ) dξ. = 2πi ∂ D(0,R ∗ ) (ξ − z)(ξ − z  ) 

Therefore |G n (z) − G n (z  )| ≤ |z − z  |

MR 2π R ∗ 2(1 + R)M R = |z − z  | ∗ 2 2π (R − R) (1 − R)2

R setting M(R) := 2(1+R)M , the claim follows. (1−R)2 Now, we pass to the proof of (2). In what follows, for n ∈ N, t ∈ [0, +∞) and z ∈ D, let 

t

αn (t, z) :=

|G(φs (z)) − G n (φs (z))| ds.

0

Fix T > 0, r ∈ (0, 1) and ε > 0. Consider the compact subset of D given by K := {|φt (z)| : t ∈ [0, T ], |z| ≤ r } and let 0 < R K < 1 be such that K is contained in {z ∈ D : |z| ≤ R K }. Let R ∗K := 1+R K ∈ (R K , 1). 2 According to the previous claim, there exists M(R ∗K ) > 0 such that   G n (z) − G n (z  ) ≤ M(R ∗ )|z − z  |, for all |z|, |z  | ≤ R ∗ , n ∈ N. K K Now choose ε1 > 0 such that ∗

ε1 e M(R K )T
0 and, by continuity, max{|φt(n) n |z| ≤ r } = R ∗K . We claim that tn = T for all n ≥ N . Suppose on the contrary that there exists n ≥ N with tn < T . Then, for |z| ≤ r and t ∈ [0, tn ], |φt (z) −

φt(n) (z)|

 t     (n)  G(φs (z)) − G n (φs (z)) ds  = 0  t     (n)  ≤ αn (t, z) +  G n (φs (z)) − G n (φs (z)) ds  0  t   φs (z) − φ (n) (z) ds ≤ αn (T, z) + M(R ∗K ) s 0  t   φs (z) − φ (n) (z) ds. ≤ ε1 + M(R ∗K ) s 0

By Lemma 10.5.5 |φt (z) − φt(n) (z)| ≤ ε1 e M(R K )t ≤ ε1 e M(R K )T < ∗

∗

R ∗K − R K . 2

(10.5.3)

Therefore, taking t = tn , (z)| < R ∗K , |φt(n) n

for |z| ≤ r,

contradicting the properties of tn . Hence, by the very definition of tn and since tn = T for all n ≥ N , it follows that |φt(n) (z)| ≤ R ∗K whenever t ∈ [0, T ], |z| ≤ r and n ≥ N . Thus, by (10.5.3), |φt (z) − φt(n) (z)| ≤ ε1 e M(R K )t ≤ ε1 e M(R K )T ≤ ε, ∗

and, by the arbitrariness of ε, we are done.

∗



10.6 The Product Formula One of the remarkable consequences of Theorem 10.2.4, is that if { f t }t∈[0,δ) , δ > 0, is a family of holomorphic self-maps of D such that f 0 (z) = z for all z ∈ D and { f t } is pointwise differentiable in t at t = 0 for every z ∈ D, and its derivative G is holomorphic in D, then G is in fact an infinitesimal generator. In the next theorem we relate the semigroup (φt ) generated by G with the family { f t }.

10.6 The Product Formula

303

Theorem 10.6.1 (Product Formula) Let { f t }t∈[0,δ) , δ > 0, be a family of holomorphic self-maps of the unit disc D such that for all z ∈ D the following limit exists G(z) := lim+ t→0

f t (z) − z . t

Then G is holomorphic and, indeed, G ∈ Gen(D). Moreover the semigroup (φt ) generated by G has the property that for every sequence of positive real numbers {tn } converging to t ∈ [0, δ), (10.6.1) φt = lim f t◦n n /n n→∞

where the limit is uniform on compact subsets of the unit disc. Proof By Corollaries 10.2.11 and 10.2.7, for each t > 0, the function 1t ( f t − idD ) = G(z) uniformly on belongs to Gen(D). Thus, by Theorem 10.5.6, limt→0+ ft (z)−z t compact subsets of D, G is holomorphic and G ∈ Gen(D). Let (φt ) be the semigroup generated by G. Fix t0 ∈ [0, δ) and r ∈ (0, 1). We have to show that, if {tn } is a sequence of positive real numbers converging to t0 then (z) uniformly in |z| ≤ r . φt0 (z) = limn→∞ f t◦n n /n ◦l Let T > t0 . Taking into account that φt/m = φlt/m and since [0, T ] × {z ∈ D : |z| ≤ r } is compact in [0, +∞) × D, by Theorem 8.1.15 there exists R > 0 such that for all m ∈ N, l ∈ N0 , 0 ≤ l ≤ m, 0 ≤ t ≤ T and z ∈ D such that |z| ≤ r , we have ◦l (z), 0) ≤ R. (10.6.2) ω(φt/m Now, we claim that for each ε > 0 there exists η ∈ (0, δ) such that for all z ∈ D with ω(z, 0) ≤ R, 0 ≤ s ≤ η. (10.6.3) ω(φs (z), f s (z)) ≤ sε, In order to prove the claim, since {z ∈ D : ω(z, 0) ≤ R} is compact, it is enough to f s (z)) = 0 uniformly on compacta. Now, show that lims→0+ ω(φs (z), s    φs (z)− fs (z)  1 +   1 1−φ (z) f (z)  s s . ω(φs (z), f s (z)) = log  φs (z)− fs (z)  2 1 −  1−φ  (z) f (z) s

s

Hence, using Taylor expansion, it is enough to show that lim+

s→0

  1  φs (z) − f s (z)  = 0, s  1 − φs (z) f s (z) 

uniformly on compacta. In order to prove (10.6.4), we note that

(10.6.4)

304

10 Infinitesimal Generators

G(z) = lim+ h→0

φh (z) − z f h (z) − z = lim+ h→0 h h

and both limits are uniform on compact subsets of D. Hence, for any compact K ⊂ D there exists s0 > 0 such that, for all z ∈ K , φs (z) = z + sG(z) + q1 (s, z) q (s,z) and f s (z) = z + sG(z) + q2 (s, z), with lims→0+ j s = 0 uniformly in z ∈ K , j = 1, 2. Therefore,   1  φs (z) − f s (z)  |q1 (s, z) − q2 (s, z)| = lim+ lim = 0, s→0+ s  1 − φs (z) f s (z)  s→0 s|1 − φs (z) f s (z)| uniformly on K , and (10.6.4)—and hence the claim (10.6.3)—is proved. Fix ε > 0. Let 0 < t < min{δ, T } and take N > 0 so large that Nt ≤ η. Then for all |z| ≤ r and m > N , since, by Theorem 1.3.7, f x contracts the hyperbolic distance for all x ∈ [0, δ), we have ◦m ◦m ◦m ω( f t/m (z), φt (z)) = ω( f t/m (z), φt/m (z))

≤

m 

◦(k−1) ◦(m−k) ◦(k−1) ◦(m−k) ω( f t/m ( f t/m (φt/m (z))), f t/m (φt/m (φt/m (z))))

k=1

≤

m 

ω( f t/m (z k ), φt/m (z k )) ≤ tε,

k=1 ◦(m−k) where we set z k := φt/m (z) and the last inequality follows from (10.6.3) because ω(z k , 0) ≤ R for every k = 1, . . . , m by (10.6.2). Now let {tm } be a sequence converging to t0 such that tm < δ for all m ∈ N. Since t0 < T , we can assume that tm < T for all m ∈ N. Then, for m ≥ N ,

(z), φt0 (z)) ≤ ω( f t◦m (z), φtm (z)) + ω(φtm (z), φt0 (z)) ω( f t◦m m /m m /m ≤ tm ε + ω(φtm (z), φt0 (z)). Since {φtm } converges to φt0 uniformly on compacta and ε is arbitrary, we are done.  Corollary 10.6.2 Let G, H ∈ Gen(D). Denote by (φt ) the semigroup generated by G and by (ϕt ) the semigroup generated by H . Then the holomorphic vector field G + H is the infinitesimal generator of the semigroup (ψt ) given by ψt = lim (φt/m ◦ ϕt/m )◦m , m→∞

where the limit is uniform on compact subsets of the unit disc. Proof Let f t := φt ◦ ϕt for t ∈ [0, +∞). Then { f t } is a family of holomorphic selfmaps of D which converges uniformly on compacta to idD as t → 0+ . Moreover, by (10.1.2), for all t ≥ 0 and z ∈ D,

10.6 The Product Formula

305

∂ f t (z) = G(φt (ϕt (z))) + φt (ϕt (z))H (ϕt (z)). ∂t Since φt (z) → z and ϕt (z) → z as t → 0+ uniformly on compacta of D, it follows that f t (z) − z ∂ f t (z) lim = |t=0 = G(z) + H (z), t→0+ t ∂t uniformly on compacta of D. Therefore, the result follows at once from Theorem 10.6.1. 

10.7 Notes The interaction via a non-autonomous ordinary differential equation between families of holomorphic self-maps of the unit disc satisfying certain semigroup-like properties and holomorphic vector fields has been successfully introduced first by Loewner [95], who proved, using his method, the bound 3 in the Bieberbach conjecture for the third coefficient in the expansion of any normalized univalent map of the unit disc. In 1939, Wolff [128] studied “continuous iteration” using autonomous differential equations. The theory of semigroups of holomorphic self-maps in D got new life from the paper of Berkson and Porta [11]. Using the point of view of differential equations, they proved Theorems 10.1.4 and 10.1.10 and the algebraic structure given in Corollary 10.2.7. Corollary 10.1.14 is a private communication to the authors from D. Shoikhet. Theorem 10.2.4 is proved in [27, 109]. Abate’s formula (10.2.1) is taken from [1, Theorem 1.4.14]. The key decomposition formula given in Corollary 10.2.9 was obtained in [2]. The results of Sect. 10.4 are taken from [24]. Propositions 10.5.1 and 10.5.2 were proved in [28] and Proposition 10.5.3 appears in [36]. The proof of Theorem 10.6.1 is inspired from [7], where it is proved for Kobayashi hyperbolic converges uniformly complex manifolds under the stronger hypothesis that ft (z)−z t on compacta to G. Such a formula already appeared in the context of semigroups of holomorphic self-maps in [110]. Using Möbius transformations of D one can assume that the Denjoy-Wolff point of a given semigroup is either τ = 0 or τ = 1. In fact, we can further reduce, up to some extent, the case of interior Denjoy-Wolff point (τ = 0) to the case of boundary Denjoy-Wolff point (τ = 1). This is the meaning of the following non-elementary proposition. Proposition 10.7.1 Let (φt ) be a non-trivial semigroup with the Denjoy-Wolff point τ = 0. Let h be its Koenigs function and G its infinitesimal generator. Then there exists a unique semigroup (φ˜ t ) in the right half-plane H with the Denjoy-Wolff point τ˜ = ∞ such that for all t ≥ 0 and all z˜ ∈ H we have   exp − φ˜ t (˜z ) = φt exp(−˜z ) .

306

10 Infinitesimal Generators

Moreover, the Koenigs function h˜ 0 and the infinitesimal generator G˜ of the oneparameter semigroup (φ˜ t ) are given by  ˜ G exp(−˜z ) ˜ z) = − ˜h 0 (˜z ) = − h(˜z ) , G(˜ G  (0) exp(−˜z )

for all z˜ ∈ H,

 where h˜ : H → C is a holomorphic lifting of H  z˜ → h exp(−˜z ) ∈ C∗ with respect to the covering map C  w˜ → exp(−w) ˜ ∈ C∗ . By means of the above proposition one can often reduce the study of boundary behavior of elliptic semigroups to that of non-elliptic ones. We are not using this argument in the book, but we refer the reader to the papers [37, 42, 81].

Chapter 11

Extension to the Boundary

In this chapter we study the boundary extension of the iterates of a semigroup and of the associated Koenigs function. After studying the impression and the principal part of prime ends of domains defined by Koenigs functions, we prove that every Koenigs function and every iterate of a semigroup have non-tangential limit at every boundary point. Moreover, the semigroup functional equation and the functional equation defined by the canonical model extend in the non-tangential limits sense up to the boundary. In the last part of the chapter we analyze conditions for a tout court continuous extension of iterates of a semigroup up to the boundary.

11.1 Prime Ends and Koenigs Functions The aim of this section is to describe the impressions and principal parts of every prime end of the range of a Koenigs function. We start with a topological lemma which we will use several times: Lemma 11.1.1 Let Ω ⊂ C be a simply connected domain. Let C be a cross cut for Ω. Let Ω \ C = A ∪ B, where A, B are the open nonempty and connected sets given by Lemma 4.1.3, such that A ∩ B = ∅. Assume there exists a Jordan curve Γ ⊂ C∞ such that C ⊂ Γ and Γ \ C ⊂ C∞ \ Ω. Let U + , U − denote the two connected components of C∞ \ Γ given by Jordan’s Theorem 3.2.1. Then, either A = U + ∩ Ω and B = U − ∩ Ω or A = U − ∩ Ω and B = U + ∩ Ω. Proof Since Ω \ C = Ω \ Γ , Jordan’s Theorem 3.2.1 implies A ∪ B = Ω \ C = Ω \ Γ = (Ω ∩ C∞ ) \ Γ = Ω ∩ (C∞ \ Γ ) = Ω ∩ (U + ∪ U − ) = (Ω ∩ U + ) ∪ (Ω ∩ U − ). Now, we claim that Ω ∩ U + = ∅ and Ω ∩ U − = ∅. In order to prove the claim, let us assume by contradiction that Ω ∩ U + = ∅. Hence, since by Jordan’s Theorem, © Springer Nature Switzerland AG 2020 F. Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4_11

307

308

11 Extension to the Boundary

∂∞ U + = Γ , it follows that, for every w0 ∈ C ∩ Ω, there exists a sequence {wn } ⊂ U + ∩ C such that wn → w0 . Since Ω ∩ U + = ∅, then {wn } ⊂ C \ Ω, and this is a contradiction because w0 ∈ Ω. Since (Ω ∩ U + ) ∩ (Ω ∩ U − ) = Ω ∩ (U + ∩ U − ) = ∅, the previous displayed equation shows that Ω ∩ U + and Ω ∩ U − are connected, for otherwise A ∪ B would be the union of more than two (non-empty) connected components. Hence, the result follows since Ω \ C is then the union of the two disjoint, non-empty, connected  components Ω ∩ U + and Ω ∩ U − . Let h : D → C be a univalent map and let Ω = h(D). Recall that by Theorem 4.2.3  which extends h. Moreover, for every there exists a homeomorphism hˆ :  D→Ω prime end ζ ∈ ∂C D there exists a unique point σ ∈ ∂D such that ΦD (ζ ) = σ by Proposition 4.2.5. Therefore, h defines a bijective correspondence between prime h −1 (x)). In order to avoid ends of Ω and points of ∂D given by ∂C Ω x → ΦD ( burdening the notation, we say that a prime end x ∈ ∂C Ω corresponds to σ ∈ ∂D h −1 (x)) = σ . under h if ΦD ( Given λ ∈ C with Re λ > 0 and c ∈ C, c = 0, we define the spiral   spir λ [c] = e−λs c : s ∈ R ∪ {0} ∪ {∞} . Notice that the function R s → |e−λs c| is strictly decreasing. Theorem 11.1.2 Let (φt ) be an elliptic semigroup with Denjoy-Wolff point τ ∈ D, Koenigs function h and spectral value λ ∈ C with Re λ > 0. Let x be a prime end of Ω := h(D). Then there exists p ∈ ∂∞ Ω such that Π (x) = { p}. If p = ∞, then I (x) = {∞} while, if p ∈ C, then there is 0 < | p| ≤ R ≤ +∞ such that ∞

I (x) = {w ∈ spir λ [ p] : | p| ≤ |w| < R} . Proof Let x be a prime end of Ω. By Proposition 4.1.11, there exists a circular null chain (Cn ) centered at some p ∈ C∞ which represents x. Let Vn be the interior / Vn for all part of Cn . By Lemma 4.1.18, since 0 ∈ Ω, we might assume that 0 ∈ n ≥ 1. Since Cn is circular, there exist 0 ≤ an < bn ≤ 2π and a sequence of positive numbers {rn } converging to 0 such that, if p ∈ C, Cn = { p + rn eiθ : θ ∈ [an , bn ]} and q1n := p + rn eian , q2n := p + rn eibn are the end points of Cn , while, if p = ∞, then Cn = { r1n eiθ : θ ∈ [an , bn ]} and q1n := r1n eian , q2n := r1n eibn are the end points of Cn . / spir λ [q1n ]. Let L nj := spir λ [q nj ] ∩ {w ∈ C∞ : |w| ≥ |q nj |}, Assume firstly that q2n ∈ / spir λ [q1n ], then L n1 ∩ j = 1, 2. Observe that, by definition, ∞ ∈ L n1 ∩ L n2 . Since q2n ∈ n n n L 2 = {∞} and the curve L 1 ∪ L 2 ∪ Cn is a Jordan curve in C∞ . Hence, it divides C∞ in two connected regions, let us call Un the one which does not contain 0. If q2n ∈ spir λ [q1n ], we may assume, up to switch q1n with q2n , that |q1n | ≤ |q2n |. In this case, let L n := spir λ [q1n ] ∩ {w ∈ C : |q1n | ≤ |w| ≤ |q2n |}. Hence L n ∪ Cn is a Jordan curve which divides C in two connected components, and we let Un be the one which does not contain 0.

11.1 Prime Ends and Koenigs Functions

309

We claim that Vn ⊆ Un . Indeed, since Ω is λ-spirallike by Theorem 9.4.3, / spir λ [q1n ], then L nj ⊂ C∞ \ Ω, j = 1, 2, and hence by it follows that, if q2n ∈ ∞ Lemma 11.1.1 it follows that either Vn = Ω ∩ Un or Vn = Ω ∩ (C∞ \ Un ). Since / Un , we have Vn = Un ∩ Ω and in particular, Vn ⊂ Un . Similarly, we 0∈ / Vn and 0 ∈ can argue in case q2n ∈ spir λ [q1n ]. In particular, this implies immediately that if p = ∞, I (x) = Π (x) = {∞}. Assume that p ∈ C. Since Cn+1 ∩ Ω ⊂ Vn ⊂ Un , the end points q1n+1 , q2n+1 of Cn+1 belong to Un . Being Ω a λ-spirallike domain and Cn circular, it can be seen that Un+1 is contained in Un . Simple considerations show that Un is bounded if and only if q2n ∈ spir λ [q1n ]. is boundedas well for all Therefore, if there exists n such that Un is bounded, then Um ∞ ∞ m ≥ n and it is easy to see that q1m , q2m ∈ spir λ [q1n ]. Hence Vn ⊆ Un = { p} and I (x) = Π (x) = { p}. Otherwise, Un is not bounded for all n and 

Vn

∞

⊆



Un

∞

∞

= spir λ [ p] ∩ {w ∈ C : |w| ≥ | p|} .

Since I (x) is connected and compact, this proves the assertion about the impression. By construction, p ∈ Π (x). If q is another point of Π (x), then q ∈ C, for otherwise I (x) = Π (x) = {∞}. By Remark 4.4.7, there exists a circular null chain centered at q. Repeating the previous argument, we deduce that I (x) = ∞ {w ∈ spir λ [q] : |q| ≤ |w| ≤ R  } for some R  ∈ [|q|, +∞]. Hence p = q, and we are done.  Now we turn our attention to the case of the Koenigs function of a non-elliptic semigroup. As a matter of notation, given z ∈ C, we denote by L[z] := {w ∈ C : Re w = Re z} ∪ {∞}. We need a lemma on the prime end corresponding to the Denjoy-Wolff point: Lemma 11.1.3 Let (φt ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D and Koenigs function h. Let Ω := h(D). Let (Cn ) be a null chain for Ω. Then the following are equivalent: (1) the null chain (Cn ) represents the prime end of Ω corresponding to τ under h; (2) limn→∞ supw∈Cn ∩C Im w = +∞; (3) for every w0 ∈ Ω there exists n 0 ∈ N such that L[w0 ] ∩ {w ∈ C : Im w ≥ Im w0 } ∩ Cn = ∅, n ≥ n 0 .

(11.1.1)

Proof Let us prove that (2) implies (3). Note that, if w0 ∈ Ω, since Ω is starlike at infinity by Theorem 9.4.10, the curve Γ := L[w0 ] ∩ {w ∈ C : Im w ≥ Im w0 } ⊂ Ω. Assume by contradiction that (11.1.1) does not hold for some w0 ∈ Ω. Hence there exists a subsequence {n m } ⊂ N such that Cn m ∩ Γ = ∅. Since (Cn m ) is equivalent to

310

11 Extension to the Boundary

(Cn ), we can assume that Cn ∩ Γ = ∅ for all n ≥ 0. Let q1n , q2n be the end points of Cn , which we may assume to be in C. If q1n = q2n , then Cn is a Jordan curve and divides C into two connected components, Un+ , Un− , such that Un− is bounded in C. Hence, by Lemma 11.1.1, either Vn = Un+ ∩ Ω or Vn = Un− ∩ Ω. If Vn ⊂ Un− , then Cm ∩ Ω ⊂ Vn ⊂ Un− for all m > n, and hence sup{Im w : w ∈ Cn , n ≥ m} < +∞, a contradiction. Therefore, Vn = Un+ ∩ Ω. Note also that Γ ⊂ Un+ because Γ is unbounded and by hypothesis it does not intersect Cn . If q1n = q2n and Re q1n = Re q2n , we can assume Im q1n < Im q2n . Since Ω is starlike at infinity, it follows that L n := L[q1n ] ∩ {w ∈ C : Im q1n ≤ Im w ≤ Im q2n } ⊂ C \ Ω. The curve L n ∪ Cn is a Jordan curve and divides C into two connected components, Un+ , Un− , where Un− is the bounded one. Arguing as before, we see that Vn = Un+ ∩ Ω and Γ ⊂ Un+ . If Re q1n = Re q2n , since Ω is starlike at infinity, the line L nj := (L[q nj ] ∩ {w ∈ C : Im w ≤ Im q nj }) ∪ {∞} satisfies L nj ⊂ C∞ \ Ω for j = 1, 2. The curve L n1 ∪ L n2 ∪ Cn is thus a Jordan curve in C∞ dividing C∞ into two connected components, let us say Un+ and Un− , and we let Un− be the one such that supw∈Un− Im w < +∞. By Lemma 11.1.1, it follows that either Vn = Un− ∩ Ω or Vn = Un+ ∩ Ω. If it were Vn ⊂ Un− then it would follow that Cm ∩ C ⊂ Vn ⊂ Un− and hence supw∈Cm Im w ≤ supw∈Un− Im w < +∞, for all m > n, a contradiction. Therefore, Vn = Un+ ∩ Ω. Moreover, since supw∈Γ Im w = +∞, Γ ⊂ Un+ . Summing up, we have proved that, if (11.1.1) does not hold, then Vn = Un+ ∩ Ω and Γ ⊂ Un+ . Therefore, Γ ⊂ Vn for all n. In particular, w0 ∈ Vn for all n, against Lemma 4.1.18. Hence, (11.1.1) holds. (3) implies (1). Let y ∈ ∂C Ω be the prime end represented by (Cn ). Let σ ∈ ∂D be the point corresponding to y under h. Let w0 ∈ Ω. Since w0 + it ∈ Ω for all t ≥ 0, by (11.1.1), there exist a sequence {n m } of natural numbers converging to ∞, and a sequence {tm } of positive real numbers converging to +∞ such that w0 + itm ∈ Cn m for all m ∈ N. The null chain (Cn m ) is equivalent to (Cn ). The null chain (h −1 (Cn m )) for D represents then the prime end hˆ −1 (y) by Lemma 4.1.13 and according to the definition of the map hˆ in Theorem 4.2.3. In particular, σ = I (hˆ −1 (y)) and every sequence {z m } ⊂ D such that z m ∈ h −1 (Cn m ) converges to σ . Therefore, limm→∞ h −1 (w0 + itm ) = σ . However, since h(φt (z)) = h(z) + it for all z ∈ D and t ≥ 0, it follows that σ = lim h −1 (w0 + itm ) = lim φtm (h −1 (w0 )) = τ. m→∞

m→∞

(1) implies (2). By hypothesis, ΦD (hˆ −1 (x)) = τ . Thus, (h −1 (Cn ∩ Ω)) is a null chain in D with the property that for every wn ∈ h −1 (Cn ∩ Ω), limn→∞ wn = τ . By the Denjoy-Wolff Theorem 8.3.1, there exists a sequence {tn } of positive real numbers converging to +∞ such that φtn (0) ∩ h −1 (Cn ∩ Ω) = ∅ for n large enough. Since h(0) + itn = h(φtn (0)) ∈ Cn , we clearly obtain that  limn→∞ supw∈Cn ∩C Im w = +∞.

11.1 Prime Ends and Koenigs Functions

311

Theorem 11.1.4 Let (φt ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D and Koenigs function h. Let x be a prime end of Ω := h(D). Then there exists p ∈ C∞ such that Π (x) = { p}. Moreover, (1) if p ∈ C then there exists R ∈ [−∞, Im p] such that ∞

I (x) = {w ∈ L[ p] : R < Im w ≤ Im p} ; (2) if p = ∞ and x does not correspond to τ under h then I (x) = Π (x) = {∞}; (3) if x corresponds to τ under h then p = ∞ and: (i) if (φt ) is hyperbolic with spectral value λ ∈ (0, +∞) then π I (x) ⊂ L[0] ∪ L[ ]; λ (ii) if (φt ) is parabolic of positive hyperbolic step, then I (x) ⊂ L[0]; (iii) if (φt ) is parabolic of zero hyperbolic step, then I (x) = {∞}. Proof (1) Let us assume there exists p ∈ Π (x) ∩ C. By Remark 4.4.7, there exists a circular null chain (Cn ) centered at p which represents x. Since Cn is circular, there exist 0 ≤ an < bn ≤ 2π and a sequence of positive numbers {rn } converging to 0 such that Cn = { p + rn eiθ : θ ∈ [an , bn ]} and q1n := p + rn eian , q2n := p + rn eibn are the end points of Cn . Note that by Lemma 11.1.3, since (Cn ) is bounded in C, the prime end x does not correspond to τ under h. Let Vn be the interior part of Cn , n ≥ 1. By Lemma 4.1.18, since h(0) ∈ Ω, we might assume that h(0) ∈ / Vn for all n ≥ 1. / L[q1n ]. Let L nj := (L[q nj ] ∩ {w ∈ C : Im w ≤ Im q nj }) ∪ Assume firstly that q2n ∈ / L[q1n ], then {∞}, j = 1, 2. Observe that, by definition, ∞ ∈ L n1 ∩ L n2 . Since q2n ∈ n n n n L 1 ∩ L 2 = {∞} and the curve L 1 ∪ L 2 ∪ Cn is a Jordan curve in C∞ . Hence, it divides C∞ in two connected components, let us call Un the one which does not contain h(0). If q2n ∈ L[q1n ], we may assume, up to switch q1n with q2n , that Im q1n ≤ Im q2n . In this case, let L n := L[q1n ] ∩ {w ∈ C : Im q1n ≤ Im w ≤ Im q2n }. Hence L n ∪ Cn is a Jordan curve which divides C in two connected components, and we let Un be the one which does not contain h(0). / L[q1n ], since Ω is starlike at infinWe claim that Vn = Un ∩ Ω. Indeed, if q2n ∈ n ity by Theorem 9.4.10, it follows that L j ⊂ C∞ \ Ω, j = 1, 2, and hence, by

312

11 Extension to the Boundary ∞

Lemma 11.1.1, either Vn = Un ∩ Ω or Vn = (C∞ \ Un ) ∩ Ω. Since h(0) ∈ / Vn and h(0) ∈ / Un , it follows that indeed Vn = Un ∩ Ω. Similarly, we can argue for the case q2n ∈ L[q1n ]. Now, we claim that Un+1 ⊂ Un . Indeed, since Cn+1 ∩ Ω ⊂ Vn ⊂ Un , the end points q1n+1 , q2n+1 of Cn+1 belong to Un . Since Ω is starlike at infinity and Cn is circular, it is easy to see that Un+1 is contained in Un . If there exists n such that q2n ∈ L[q1n ], then Un is bounded, q1m , q2m ∈ L[q1n ], Um  ∞  ∞ is bounded and contained in Un for all m ≥ n. Hence Vn ⊆ Un = { p} and I (x) = Π (x) = { p}. Otherwise, Un is not bounded for all n and 

Vn

∞

⊆



Un

∞

∞

= L[ p] ∩ {w ∈ C : Im w ≤ Im p} .

Since I (x) is connected and compact, this proves the assertion about the impression. By construction, p ∈ Π (x). If q is another point of Π (x) ∩ C, repeating the previous argument with a circular null chain centered at q, we deduce that I (x) = ∞ L[q] ∩ {w ∈ C : R  < Im w ≤ Im q} for some R  ∈ [−∞, Im q]. Hence p = q. / Π (x), Finally, since Π (x) is connected by Corollary 4.4.10, we deduce that ∞ ∈ for otherwise Π (x) = { p, ∞}. (2) Assume now that Π (x) = {∞} and that x does not correspond to τ under h. By Remark 4.4.7, there exists a circular null chain (Cn ) centered at ∞ which represents x. Hence, there exists a sequence of positive real numbers {rn } converging to 0 such that Cn ⊆ {z ∈ C : |z| = r1n } for all n ≥ 0. Let q1n , q2n be the end points of Cn , n ≥ 0. By Lemma 11.1.3, (11.1.2) K := sup sup Im w < +∞. n

w∈Cn

We claim that there exists no increasing subsequence {n m } ⊂ N converging to ∞ such that Re q1n m = Re q2n m . Indeed, assume this is the case and, without loss of generality, assume Im q1n ≤ Im q2n for all n ∈ N. Let L m := (L[q1n m ] ∩ {w ∈ C : Im q1n m ≤ Im w ≤ Im q2n m }). Let Γm := L m ∪ Cn m . Then Γm is a Jordan curve which divides C into two connected components, denote by Um− the bounded connected component of C \ Γm and by Um+ the unbounded one. By Lemma 11.1.1, either Vn m = Um− ∩ Ω n +j n +j or Vn m = Um+ ∩ Ω. Since q1 m ∈ Vn m for all j ≥ 0 and |q1 m | = 1/rn m + j → ∞ + as j → ∞, it follows that Vn m = Um ∩ Ω. Let w0 ∈ Ω be such that Im w0 > K . By (11.1.2), w0 ∈ Um+ for all m ∈ N, hence, ∩m Vn m = Ω ∩ (∩m Um+ ) = ∅, getting a contradiction. Therefore, there exists n 0 such that Re q1n = Re q2n for all n ≥ n 0 and, up to start the null chain from n 0 , we can assume n 0 = 0. Up to relabelling, we can assume Re q1n < Re q2n . Let L nj := (L[q nj ] ∩ {w ∈ C : Im w ≤ Im q nj }) ∪ {∞}, j = 1, 2. Then the curve n L 1 ∪ L n2 ∪ Cn is a Jordan curve in C∞ , and divides C∞ into two open connected components, one of them contained in {w ∈ C : Im w < K + 1}. Let Un− be such a connected component and let Un+ be the other connected component. Since Ω is

11.1 Prime Ends and Koenigs Functions

313

starlike at infinity by Theorem 9.4.10, it follows that L nj ⊂ C∞ \ Ω, j = 1, 2. Hence, by Lemma 11.1.1, either Vn = Un− ∩ Ω or Vn = Un+ ∩ Ω. We claim that there exists a subsequence {n m } ∈ N such that Vn m ⊂ Un−m for every m ∈ N. Indeed, if this is not the case, it means that there exists n 0 ∈ N such that Vn = Un+ ∩ Ω for all n ≥ n 0 . The null chain (Cn )n≥n 0 is equivalent to (Cn ), hence we can assume n 0 = 0. But then, every point w ∈ Ω such that Im w > K + 1 would be contained in Vn for all n, a contradiction to Lemma 4.1.18. Therefore the claim is proved. Since the null chain (Cn m ) is equivalent to (Cn ), we can assume that Vn = Un− ∩ Ω for all n ≥ 1. This implies that Cn+1 ∩ Ω ⊂ Un− for all n ≥ 1 and − ⊂ Un− . Note that since Im q1n , Im q2n ≤ K , and −∞ < Re q10 ≤ Re q1n < hence Un+1 n Re q2 ≤ Re q20 < +∞ for all n ∈ N, it follows that Im q nj → −∞ for j = 1, 2. Hence, since diam S (Cn ) → 0, for every R ∈ R there exists n R such that Un− ⊂ {w ∈ C : Im w < R} for all n ≥ n R . Therefore, I (x) =



Vn

∞

⊂



Un−

∞

= {∞}.

Since ∞ ∈ Π (x) ⊆ I (x), we have I (x) = {∞}. (3) Finally, assume that x corresponds to τ under h. If Π (x) contains a finite point, say p ∈ C, then by Remark 4.4.7, there exists a circular null chain (Cn ) centered at p which represents x. In particular, for every ε > 0, Cn is contained in the Euclidean disc of center p and radius ε for all n large enough. But then, if w0 ∈ Ω and Im w0 > Im p + 2ε, it follows that L[w0 ] ∩ {w ∈ C : Im w ≥ Im w0 } ∩ Cn = ∅ for n large, contradicting Lemma 11.1.3. Thus Π (x) = {∞}. As in (2), there exists a circular null chain (Cn ) centered at ∞, with Cn ⊆ {z ∈ C : |z| = r1n } for all n ∈ N, which represents x. By Lemma 11.1.3, lim sup Im w = +∞.

n→∞ w∈C

n

By Theorem 9.3.5, Ω ⊂ {z ∈ C : a < Re z < b} with either a = −∞ and b = 0 or b = +∞, or a = 0 and b = πλ or b = +∞ (see also Theorem 9.4.10). We claim that, if p ∈ ∂Ω and a < Re p < b then p ∈ / I (x). Indeed, assume by contradiction that p ∈ ∂Ω, a < Re p < b and p ∈ I (x). Let s := sup{Im w : w ∈ L[ p] ∩ (C \ Ω)}. Since Ω is starlike at infinity, and, hence, if w0 ∈ L[ p] ∩ (C \ Ω) then L[ p] ∩ {w ∈ C : Im w ≤ Im w0 } ⊂ (C \ Ω), it follows that s < +∞, for otherwise the line L[ p] would disconnect Ω. Let n 1 ∈ N be such that r1n > | p| + 1 for all n ≥ n 1 . The circle {z ∈ C : |z| = r1n } intersects L[ p] ∩ {w ∈ C : Im w ≤ s} at one point ζn with Re ζn = Re p and Im ζn < Im p. By (11.1.1), there exists n 0 ∈ N such that the curve Γ := L[ p] ∩ {w ∈ C : Im w ≥ s + 1} intersects Cn for all n ≥ n 0 . Since Cn is circular, then either Γ is tangent to Cn , or Γ intersects Cn transversally at one point, or Γ intersects Cn in two points.

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11 Extension to the Boundary

However, for n ≥ max{n 0 , n 1 } the curve Γ intersects Cn transversally at one point ηn . Let q1n and q2n be the end points of Cn . Arguing as in the proof of Lemma 11.1.3, we construct two open connected and disjoint sets Un+ , Un− such that Vn = Ω ∩ Un+ and supw∈Un− Im w < +∞. If q1n = q2n , then, by construction, ζn = q1n = q2n . In this case, Un+ is the exterior of the disc centered at 0 and with radius 1/rn , while Un− is the open disc of center 0 and radius 1/rn . Hence, Vn ⊂ Un+ , while, by construction, p ∈ Un− . In particular, / I (x). p∈ / Vn and hence p ∈ Assume q1n = q2n . Note that Cn is a closed arc of a circle, not containing ζn but containing ηn in its interior, and Re ζn = Re ηn . Then necessarily Re q1n = Re q2n . We can assume Re q1n < Re q2n . In this case the boundary of Un+ is given by Cn ∪ L n1 ∪ L n2 where L nj := L[q nj ] ∩ {w ∈ C : Im w ≤ Im q nj } ∪ {∞}, j = 1, 2. Also, simple geometric considerations show that Re q1n ≤ Re ηn = Re ζn ≤ Re q2n (and Re q nj = Re ηn if and only if q nj = ζn , j = 1, 2). Therefore, p ∈ Un− . Since Vn = Un+ ∩ Ω, it follows that p ∈ / Vn and hence p ∈ / I (x). The claim is proved. From this, and taking into account the various cases of a, b and Theorem 9.3.5, the proposition is finally proved.  Remark 11.1.5 Let (φt ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D and Koenigs function h. Let x be a prime end of h(D) which is represented by σ ∈ ∂D under h. Suppose τ = σ and assume that Π (x) = {∞}. The proof of (2) in Theorem 11.1.4 shows that there exists a (circular) null chain (Cn ) representing x such that, denoting by Vn the interior part of Cn for n ≥ 1, then for every R < 0 there exists n R such that Vn ⊂ {w ∈ C : Im w < R} for all n ≥ n R . Remark 11.1.6 Let (φt ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D and Koenigs function h. Let x be a prime end of Ω := h(D). Let σ ∈ ∂D be the point corresponding to x under h. Suppose τ = σ and assume that Π (x) = {∞}. Then lim Im h(z) = −∞, −∞ < lim inf Re h(z) ≤ lim sup Re h(z) < +∞.

z→σ

z→σ

z→σ

Indeed, in the proof of case (2) of Theorem 11.1.4 we saw that Vn ⊂ U1− for all n ≥ 1. The set U1− is contained in a strip {w ∈ C : a < Re w < b} for some −∞ < D → D be the a < b < +∞. Let {z m } ⊂ D be a sequence converging to σ . Let ΦD :  ˆ −1 (z m ))} homeomorphism given by Proposition 4.2.5. Then, by Theorem 4.2.3, {h(Φ D  converges to x in the Carathéodory topology of Ω. Therefore, by Remark 4.2.2, {h(z m )} eventually belongs to Vn and hence to U1− for every n ∈ N. Therefore, a ≤ lim inf n→∞ Re h(z n ) ≤ lim supn→∞ Re h(z n ) ≤ b. Moreover, since I (x) = {∞}, by Proposition 4.4.4, it follows that for every sequence {z n } ⊂ D converging to σ , limn→∞ |h(z n )| = ∞. Since a ≤ Re h(z n ) ≤ b and Im h(z n ) is bounded from above because h(z m ) ∈ U1− , it follows that necessarily lim z→σ Im h(z) = −∞. By Theorems 11.1.2 and 11.1.4, the principal parts of every prime end of the image of a Koenigs function is a singleton. Therefore, by Theorem 4.4.9 we have:

11.1 Prime Ends and Koenigs Functions

315

Corollary 11.1.7 Let (φt ) be a semigroup in D with Koenigs function h. Then, for every σ ∈ ∂D, there exists ∠ lim z→σ h(z) ∈ C∞ . In the non-elliptic case, one can detect the Denjoy-Wolff point by using h: Proposition 11.1.8 Let (φt ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D and Koenigs function h. Let σ ∈ ∂D. Then σ = τ if and only if lim sup Im h(z) = +∞. z→σ

Proof If σ = τ then lim supz→σ Im h(z) = +∞ by Proposition 9.4.6. If σ = τ , let x be the prime end of h(D) corresponding to σ under h. By Theorem 11.1.4 there are two cases to be considered. If Π (x) = ∞, then by Remark 11.1.6, lim z→σ Im h(z) = −∞. If Π (x) = { p} for some p ∈ C, then I (x) ⊂ {w ∈ C : Im w ≤ Im p} ∪ {∞}. Since by Proposition 4.4.4, the cluster set of h at σ is given by I (x) it follows that  lim supz→σ Im h(z) = Im p < +∞. Another straightforward consequence of Theorem 11.1.4, Remark 11.1.6, Proposition 4.4.4 and Theorem 4.4.9 is the following: Proposition 11.1.9 Let (φt ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D and Koenigs function h. Let σ ∈ ∂D \ {τ }. (1) If ∠ lim z→σ h(z) = p ∈ C, then lim z→σ Re h(z) = Re p and lim sup Im h(z) ≤ Im p. z→σ

(2) If ∠ lim z→σ h(z) = ∞, then lim z→σ h(z) = ∞, lim z→σ Im h(z) = −∞, and −∞ < lim inf Re h(z) ≤ lim sup Re h(z) < +∞. z→σ

z→σ

The next example shows that it is possible to have two lines as impression of a prime end corresponding to the Denjoy-Wolff point of a hyperbolic semigroup. Similar constructions can be performed in case of parabolic semigroups of positive hyperbolic steps. Example 11.1.10 Consider the domain Ω := {w ∈ C : 0 < Re w < 1} \

∞ 

(L 1n ∩ L 2n ),

n=2

where L 1n := {w ∈ C : Re w = 1 − 1/n, Im w ≤ n} and L 2n := {w ∈ C : Re w = 1/n, Im w ≤ n}. Let h : D → C be a Riemann map of Ω. The domain Ω is starlike at infinity, hence, the map h is starlike at infinity with respect to some τ ∈ ∂D. Therefore, by Theorem 9.4.10, h is the Koenigs function of a hyperbolic semigroup (φt ) of D with Denjoy-Wolff point τ and dilation λ = π . For each n, take the cross

316

11 Extension to the Boundary

cut given by the segment [1/n + in, (1 − 1/n) + in]. Then (Cn ) is a null chain such that supn supw∈Cn Im w = +∞. Hence, by Lemma 11.1.3, the prime end represented by (Cn ) corresponds under h to the Denjoy-Wolff point τ of (φt ). A simple argument shows that I ([(Cn )]) = L[0] ∪ L[1].

11.2 Boundary Extensions of Semigroups This section is devoted to a first analysis of the boundary behavior of the iterates of a semigroup. In particular, we prove that every iterate can be extended to the boundary in the non-tangential sense. Theorem 11.2.1 Let (φt ) be a semigroup. Then for any t ≥ 0 and any σ ∈ ∂D there exists the non-tangential limit φt (σ ) := ∠ lim z→σ φt (z). Moreover, for each σ ∈ ∂D and each Stolz region S of vertex σ the convergence φt (z) → φt (σ ) as S z → σ is locally uniform in t ∈ [0, +∞), i.e., for every ε > 0 and T > 0 there exists δ > 0 such that |φt (z) − φt (σ )| < ε, for all t ∈ [0, T ] and z ∈ S such that |z − σ | < δ. Proof Let h be the Koenigs function of (φt ). Fix σ ∈ ∂D. By Corollary 11.1.7 the non-tangential limit ∠ lim z→σ h(z) = L ∈ C∞ exists. We divide the proof in two steps. Step (1): Let σ ∈ ∂D and t ≥ 0. Then φt (σ ) := ∠ lim z→σ φt (z) exists. Indeed, let (D, h, z → ψt (z)) be the canonical model of (φt ) given by Theorem 9.3.5, where D = D, C, H, H− or a strip and ψt (z) = e−λt z for some λ ∈ C \ {0} or ψt (z) = z + it. Since h has non-tangential limit at σ , there exists L ∈ C∞ such that lim(0,1) r →1 h(r σ ) = L. On the one hand, φt (z) = h −1 (ψt (h(z))) for all z ∈ D. On the other hand, lim(0,1) r →1 ψt (h(r σ )) = ψt (L) (where we set ψt (L) = ∞ if L = ∞). We claim that lim(0,1) r →1 h −1 (ψt (h(r σ ))) exists. This is clear if ψt (L) ∈ h(D), and it follows from Proposition 3.3.3 if ψt (L) ∈ ∂h(D). In any case, there exists lim

(0,1) r →1

φt (r σ ) =

lim

(0,1) r →1

h −1 (ψt (h(r σ ))).

Step (1) follows then by Theorem 1.5.7. Step (2): Fix T < +∞ and let S be a Stolz region of vertex σ . Let {tn } ⊂ [0, T ] be a sequence converging to some t0 and let {z n } ⊂ S be a sequence converging to σ . Then {φtn (z n )} converges to φt (σ ). We argue by contradiction. Then, there exist ε > 0, a sequence {z n } ⊂ S converging to σ and a sequence of non-negative real numbers {tn } converging to some t0 ∈ [0, T ] such that |φtn (z n ) − φt0 (σ )| ≥ ε. Since limn→∞ φt0 (z n ) = φt0 (σ ) by Step (1), we may assume that |φtn (z n ) − φt0 (z n )| ≥ ε/2 for all n. Let us denote by In the

11.2 Boundary Extensions of Semigroups

317

interval in R with extreme points tn and t0 . Let Cn be the Jordan arc in D parameterized by γn : In → D, where γn (t) = φt (z n ), n ∈ N. By construction, Cn has end points φtn (z n ) and φt0 (z n ) and, in particular, diamE (Cn ) ≥ ε/2. Moreover, given u n ∈ In we have lim h(γn (u n )) = lim ψu n (h(z n )) = ψt0 (h(σ )) ∈ C∞ . n→∞

n→∞

Therefore (Cn ) is a sequence of Koebe arcs for h, contradicting Theorem 3.2.4.  Using the above theorem, we can extend the iterates of a semigroup (φt ) up to ∂D in a non-tangential sense. In order to avoid burdening notations, we still denote this extension by φt . More precisely, if σ ∈ ∂D, we let φt (σ ) := ∠ lim φt (z). z→σ

As we will see in the Sect. 11.3, the φt ’s do not need to be continuous on ∂D. However, The curve t → φt (z) is continuous in t for any z ∈ D. In fact, this is a straightforward byproduct of above theorem and Proposition 10.1.7. Proposition 11.2.2 Let (φt ) be a semigroup. Then (1) For every T ∈ (0, +∞) and every ε > 0 there exists δ > 0 such that supz∈D |φt (z) − φs (z)| < ε for every s, t ∈ [0, T ] with |t − s| < δ. (2) limt→0+ supz∈D |φt (z) − z| = 0. In particular, the curve [0, +∞) t → φt (z) is continuous for all z ∈ D. The previous results allow us to extend the semigroups functional equations to the boundary. Recall that, as a matter of notation, if f : D → C is a function and σ ∈ ∂D is a point such that the non-tangential limit of f at σ exists in C∞ , we set f (σ ) := ∠ lim f (z). z→σ

Theorem 11.2.3 Let (φt ) be a semigroup in D. Let (Ω, h, z → ψt (z)) be the canonical model of (φt ) given by Theorem 9.3.5. Let σ ∈ ∂D, then (1) φt (φs (σ )) = φt+s (σ ) for all t, s ≥ 0. (2) If h(σ ) ∈ C then h(φt (σ )) = ψt (h(σ )) for all t ≥ 0. (3) If h(σ ) = ∞ then φt (σ ) ∈ ∂D and h(φt (σ )) = ∞ for all t ≥ 0. Proof (1) By Theorem 11.2.1, for every t ≥ 0, the non-tangential limit φt (σ ) := ∠z→σ φt (z) exists. Hence, for all t ≥ 0, φt+s (σ ) = lim− φt+s (r σ ) = lim− φt (φs (r σ )). r →1

r →1

Since limr →1− φs (r σ ) = φs (σ ), by Theorem 1.5.7 and the previous equation it follows limr →1− φt (φs (r σ )) = φt (φs (σ )).

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11 Extension to the Boundary

(2–3) We may assume that the semigroup is not an elliptic group. By Corollary 11.1.7, there exists h(σ ) = ∠ lim z→σ h(z) ∈ ∂∞ h(D). Taking into account that for all t ≥ 0, lim z→∞ ψt (z) = ∞ ∈ C∞ , we have, for all t ≥ 0, lim h(φt (r σ )) = lim− ψt (h(r σ )) = ψt (h(σ )),

r →1−

r →1

where, we set ψt (h(σ )) = ∞ if h(σ ) = ∞. It follows then by Theorem 1.5.7, h(φt (σ )) = ψt (h(σ )) t ≥ 0. Finally, it is clear that φt (σ ) ∈ ∂D for all t ≥ 0 if h(σ ) = ∞ because h(D) ⊂ C. 

11.3 Continuous Boundary Extensions of Semigroups In spite of the remarkable results of the previous sections, the extension of φt to the boundary using non-tangential limits is not necessarily continuous on ∂D. In other words, the unrestricted limits of iterates do not need to exist in general on ∂D. In this section, we analyze when the non-tangential extension to the boundary can be replaced by a tout court extension and when the iterates of a semigroup belong to the disc algebra, that is, are continuous up to the boundary. The next result shows that the continuity of the Koenigs function at a boundary point implies as well the continuity of z → φt (z), locally uniformly in t. The proof is essentially the same of Theorem 11.2.1, so we only sketch it. Proposition 11.3.1 Assume that the Koenigs function h of a semigroup (φt ) has unrestricted limit, finite or infinite, at a point σ ∈ ∂D. Then φt also has unrestricted limit at σ for every t ≥ 0. Moreover, the convergence φt (z) → φt (σ ) as D z → σ is locally uniform with respect to t ≥ 0, namely, for every ε > 0 and every T > 0 there exists δ > 0 such that |φt (z) − φt (σ )| < ε for all t ∈ [0, T ] and all z ∈ D such that |z − σ | < δ. Proof In order to show that the function φt has unrestricted limit at σ , using an elementary argument of reductio ad absurdum, we reduce ourselves to prove that given any continuous curve γ : [0, 1) → D converging at σ , the function φt has a limit φt (σ ) at σ along γ . This can be done as in Step (1) of Theorem 11.2.1 replacing r σ with γ (r ). The proof of the locally uniform convergence in t is exactly the same of Step (2) in Theorem 11.2.1, simply replace {z n } ⊂ S by any sequence {z n } ⊂ D converging to σ .  Except at the Denjoy-Wolff point, the converse to the above proposition is also true: Proposition 11.3.2 Let (φt ) be a semigroup with Koenigs function h and DenjoyWolff point τ . Let σ ∈ ∂D, σ = τ . Then the following are equivalent:

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319

(1) the unrestricted limit lim z→σ φt (z) exists for every t ≥ 0, (2) there exists t0 > 0 such that the unrestricted limit lim z→σ φt0 (z) exists, (3) the unrestricted limit lim z→σ h(z) exists. Proof Clearly (1) implies (2), and (3) implies (1) by Proposition 11.3.1. Therefore, we are left to show that (2) implies (3). Suppose (2) holds. We first show that for every n ∈ N0 , the unrestricted limit lim z→σ φt0 /2n (z) exists. We prove it by induction. It is true for n = 0. Assume it is true for n ∈ N. Hence, Q := lim z→σ φt0 /2n (z) exists. By contradiction, we assume that lim z→σ φt0 /2n+1 (z) j does not exist. Hence, there exist two sequences {ζm } ⊂ D, j = 1, 2 such that j j limm→∞ ζm = σ and limm→∞ φt0 /2n+1 (ζm ) = p j ∈ D, j = 1, 2 with p1 = p2 . Let G˜ m be the segment joining ζm1 to ζm2 . Note that for every sequence {˜z m } such that z˜ m ∈ G˜ m it follows limm→∞ z˜ m = σ , hence, limm→∞ φt0 /2n (˜z m ) = Q. Let G m := φt0 /2n+1 (G˜ m ). Since φt0 /2n+1 is injective, it follows that {G m } is a sequence of Jordan arcs in D. Moreover, since p1 = p2 , there exists K > 0 such that diamE (G m ) > K for all m ∈ N. Now, let am ∈ G m for all m ∈ N. By construction, for every m ∈ N there exists a˜ m ∈ G˜ m such that φt0 /2n+1 (a˜ m ) = am . Therefore, lim φt0 /2n+1 (am ) = lim φt0 /2n+1 (φt0 /2n+1 (a˜ m )) = lim φt0 /2n (a˜ m ) = Q.

m→∞

m→∞

m→∞

This implies that {G m } is a sequence of Koebe arcs for φt0 /2n+1 , contradicting Theorem 3.2.4. Therefore, for every n ∈ N0 , the unrestricted limit lim z→σ φt0 /2n (z) exists. Now, we suppose (2) holds but the limit lim z→σ h(z) does not exist. Suppose first that (φt ) is non-elliptic. Let x be the prime end in h(D) such that hˆ −1 (x) is the prime end in D corresponding to σ . Let (Cm ) be a circular null chain representing x. By Proposition 4.4.4, I (x) is not a single point and, therefore, since σ = τ , by Theorem 11.1.4, there exist a, b ∈ R and R ∈ [−∞, b) such that ∞

I (x) = {w ∈ C : Re w = a, R < Im w ≤ b} . Moreover, Π (x) = {a + ib}. Let {wn } ⊂ h(D) be such that wn ∈ Cn for all n. Since Π (x) = {a + ib}, it follows from the definition of principal part of a prime end that limn→∞ wn = a + ib. Moreover, by Remark 4.2.2, {wn } converges to x in the Carathéodory topology of h(D). Therefore, {h −1 (wn )} converges in the Carathéodory topology of D to hˆ −1 (x), and, by Proposition 4.2.5, {h −1 (wn )} converges to σ (in the Euclidean topology). Let c ∈ (R, b) and let m ∈ N be such that t := t0 /2m has the property that t < b − c. Taking into account that φt (z) = h −1 (h(z) + it) for all z ∈ D, we have φt (σ ) = lim φt (h −1 (wn )) = lim h −1 (wn + it). n→∞

n→∞

(11.3.1)

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By definition of impression, there exists a sequence {z n } ⊂ D converging to σ such that {h(z n )} converges to a + ic. Note that {h(z n )} converges to x in the Carathéodory topology of h(D). Since φt is continuous at σ for what we already proved, limn→∞ φt (z n ) = φt (σ ), and, by (11.3.1), taking into account that φt (z) = h −1 (h(z) + it), we have lim h −1 (wn + it) = lim h −1 (h(z n ) + it).

n→∞

n→∞

By Proposition 4.2.5, this implies that {h −1 (wn + it)} and {h −1 (h(z n ) + it)} converge to a same point y in the Carathéodory topology of D. Hence, by Theorem 4.2.3, ˆ in the Carathéodory topology of h(D). {wn + it} and {h(z n ) + it} converge to h(y) Since a + i(t + b) ∈ / I (x), it follows from the definition of impression of a prime end and Remark 4.2.2 that {wn + it} does not converge to x in the Carathéodory ˆ = x. topology, that is, h(y) On the other hand, we claim that {h(z n ) + it} converges to x in the Carathéodory topology, giving a contradiction. In fact, by Remark 4.4.7, since Π (x) = {a + ib}, there exists a decreasing sequence {rm } of positive real numbers converging to 0 such that Cm ⊂ {ζ ∈ C : |ζ − (a + ib)| = rm }. Let Vm be the interior part of Cm , m ≥ 1. Since {h(z n )} converges to x, by Remark 4.2.2, it follows that for every m ∈ N there exists n m ∈ N such that h(z n ) ∈ Vm for all n ≥ n m . ˆ = x, by Remark 4.2.7 On the other hand, since {h(z n ) + it} is converging to h(y) / Vm for every m ≥ m 0 and there exist m 0 ∈ N and n 0 ∈ N0 such that h(z n ) + it ∈ n ≥ n0. Let Nm := max{n m , n 0 }, m ≥ m 0 . Since h(D) is starlike at infinity, the segment {h(z n ) + is : 0 ≤ s ≤ t} ⊂ h(D) for all n ∈ N. Therefore, since h(z Nm ) ∈ Vm and / Vm for all m ≥ m 0 , there exists sm ∈ (0, t] such that h(z Nm ) + sm i ∈ h(z Nm ) + it ∈ Cm for all m ≥ m 0 . Namely, |h(z Nm ) + sm i − (a + ib)| = rm for all m ≥ m 0 . Up to extracting subsequences, we can assume that limm→∞ sm = s0 ∈ [0, t]. Hence, 0 = lim rm = lim |h(z Nm ) + sm i − (a + ib)| = |c + s0 − b| > (b − c) − t > 0, m→∞

m→∞

a contradiction, and the claim holds. If the semigroup is elliptic, one can easily adapt the previous argument replacing Theorem 11.1.4 by Theorem 11.1.2.  The request that σ = τ in Proposition 11.3.2 is necessary for proving that (2) implies (3), as the following example shows: Example 11.3.3 Let us consider again Example 11.1.10. With the notation introduced there, consider the semigroup (φt ) where φt (z) := h −1 (h(z) + it), for all t ≥ 0 and z ∈ D. Using the symmetry of the domain, we may assume that Re h(r ) = 1/2 for all r ∈ (−1, 1) and Im h(r ) increases with r . Hence φt (r ) ∈ R for all r ∈ (−1, 1). Thus, it is easy to deduce that the Denjoy-Wolff point of (φt ) is τ = 1. In

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Example 11.1.10, we saw that h does not extend continuously to 1. Let Vn be the interior part of Cn . Fix t > 0. If {z n } is a sequence in the unit disc that converges to 1, for each m ∈ N, there is n 0 ∈ N such that if n ≥ n 0 , then h(z n ) ∈ Vm . Thus h(z n ) + it ∈ Vm for all n ≥ n 0 . Therefore limn→∞ φt (z n ) = limn→∞ h −1 (h(z n ) + it) = 1. That is, all the iterates of the semigroup have a continuous extension to D ∪ {1}. Indeed, it is not difficult to show that the functions φt belong to the disc algebra for all t. Example 11.3.4 We construct a semigroup (φt ) in the unit disc such that every iterate φt (t > 0) has no unrestricted limit at an uncountable number of points of ∂D. Let C be the classical ternary Cantor set in [0, 1]. It is known that [0, 1] \ C is the union of an infinite countable number of disjoint open subintervals (In ) of [0, 1]. Moreover, if E denotes the collection of the extremes of every In , then E ⊂ C and, for every point p ∈ C \ (E ∪ {0, 1}) there exists a strictly increasing sequence {an ( p)} ⊂ E converging to p and a strictly decreasing sequence {bn ( p)} ⊂ E also converging to p. Consider the following simply connected domain of the complex plane Ω := {x + yi ∈ C : 0 < x < 1, y > 0} ∪

∞ 

{x + yi ∈ C : x ∈ In }

n=1

and let h be a Riemann map from D onto Ω. Let φt (z) := h −1 (h(z) + it), for z ∈ D and t ≥ 0. By Theorem 9.3.5, we see that (φt ) is a hyperbolic semigroup in D with spectral value π and h is its Koenigs function. Note that C \ {0, 1} ⊂ ∂Ω and for every p ∈ C \ {0, 1}, p + it ∈ Ω for all t > 0. Hence, by Proposition 3.3.3, for each p ∈ C \ {0, 1} the limit σ p := limt→0+ h −1 ( p + it) ∈ ∂D exists. For each p ∈ C \ {0, 1}, by Theorem 11.1.4, there exists the limit ∠ lim z→σ p h(z). Taking into account that limt→0+ h(h −1 ( p + it)) = p, we deduce, from Corollary 3.3.4, that ∠ lim z→σ p h(z) = p and σ p = σq for p = q. Moreover, for every point p ∈ C \ (E ∪ {0, 1}), consider the cross cuts given by 



i i i ∪ an ( p) + , bn ( p) + ∪ Cn = an ( p), an ( p) + n+1 n+1 n+1

 i , bn ( p) . ∪ bn ( p) + n+1 Clearly, (Cn ) is a null chain for Ω which represents a prime end p ∈ ∂C Ω. Moreover, Π ( p) = { p} and I ( p) = p − i[0, +∞). Thus, by Proposition 4.4.4, the unrestricted limit of h does not exist at σ p . Therefore, using Proposition 11.3.2, we conclude every iterate φt (t > 0) has no unrestricted limit at every σ p . Since, as we already notice, C \ {0, 1} p → σ p is bijective, it turns out that every φt , t > 0 cannot be extended continuously at an uncountable number of points of ∂D.

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The remarkable fact we are going to prove now is that every iterate of a semigroup is continuous at its Denjoy-Wolff point, regardless the continuity of the associated Koenigs function. We start with the following: Proposition 11.3.5 Let (φt ) be a non-elliptic semigroup with Koenigs function h and Denjoy-Wolff point τ ∈ ∂D. Let Ω = h(D). Then the following are equivalent: (1) Ω does not contain any half-plane bounded by a line parallel to iR, (2) limn→∞ φtn (z n ) = τ , for every sequence {z n } in D converging to τ and every sequence {tn } in [0, +∞), (3) limn→∞ φtn (z n ) = τ , for every sequence {z n } in D converging to τ and every sequence {tn } converging to +∞. In particular, if one—and hence any—of the previous conditions hold, φt (z) converges to τ as z → τ uniformly in t, namely, for every ε > 0 there exists δ > 0 such that |φt (z) − τ | < ε for every t ≥ 0 and z ∈ D such that |z − τ | < δ. Proof Throughout the proof, given two points w1 , w2 ∈ C, we denote by [w1 , w2 ] the segment that joins w1 to w2 . Assume (1) does not hold, that is, there exists a half-plane H ⊂ Ω whose boundary is parallel to iR. There is a ∈ R such that either a + H ⊂ H or a + H− ⊂ H . Let us consider the first case (the second follows from a similar argument). By Theorem 8.3.6, lims→+∞ h −1 (a + is) = lims→+∞ φs (h −1 (a)) = τ . On the other hand, by Proposition 3.3.3, there exists σ := lims→−∞ h −1 (a + is). Let γ : [0, 1) → H be the continuous curve such that γ is affine on each interval [1 − 1/n, 1 − 1/(n + 1)] and γ (1 − 1/2n) = a + n, γ (1 − 1/(2n − 1)) = a + (−1)n ni, for n ≥ 1. It is clear that limt→1− γ (t) = ∞ and γ ([0, 1)) ⊂ h(D). Therefore, by Proposition 3.3.3, the limit limr →1 h −1 (γ (r )) exists. Thus, since, by construction, a subsequence of h −1 (γ (r )) converges to τ , we have σ = τ . Finally, writing z n := h −1 (a − in) for all n, we have that the sequence {z n } converges to τ and φn (z n ) = h −1 (a) for all n. So limn→∞ φn (z n ) = τ . This implies that (3) implies (1). Being clear that (2) implies (3), it remains to prove that (1) implies (2). We may n the connected component assume that w0 := h(0) ∈ R. For each n ∈ N, denote by C of the set {w ∈ Ω : Im w = n} that contains the point in + w0 . The following four cases exhaust all possibilities: n is bounded. Case (A): for each n ∈ N the set C Case (B): there exists n 0 ∈ N such that for all n > n 0 , n } = +∞. n } ∈ R, sup{Re w : w ∈ C inf{Re w : w ∈ C

Case (C): there exists n 0 ∈ N such that for all n > n 0 , n } ∈ R. n } = −∞, sup{Re w : w ∈ C inf{Re w : w ∈ C

Case (D): there exists n 0 ∈ N such that for all n > n 0 ,

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n } = −∞, sup{Re w : w ∈ C n } = +∞. inf{Re w : w ∈ C n are of the form C n = (xn + in, xn + in), where −∞ < In Case (A), the sets C   < xn < +∞, and we set Cn := [xn + in, xn + in] for all n ∈ N. Since Ω + it ⊂   , xn+1 ) for every n ∈ N. Ω, (xn , xn ) ⊂ (xn+1 In Case (B), condition (1) implies that there exists a strictly increasing unbounded sequence (xn ) ⊂ (0, +∞) such that xn + in ∈ Ω for all n and L[xn ] is not a subset n }, yn := inf{Im w : w ∈ of Ω for every n ≥ n 0 . Now, set xn := inf{Re w : w ∈ C  L[xn ] ∩ Ω} and let xn

Cn := [xn + in, xn + in] ∪ [xn + in, xn + i yn ].   , xn+1 ) for every n ≥ n 0 . Note that again we have (xn , xn ) ⊂ (xn+1 The construction in Case (C) is similar to Case (B) and we omit it. In Case (D), there exist a strictly increasing unbounded sequence (xn ) ⊂ (0, +∞) and a strictly decreasing unbounded sequence (xn ) ⊂ (−∞, 0) such that xn + in, xn + in ∈ Ω for all n and L[xn ] and L[xn ] are not included in Ω for n ≥ n 0 . Now, set yn := inf{Im w : w ∈ L[xn ] ∩ Ω}, yn := inf{Im w : w ∈ L[xn ] ∩ Ω} and let Cn := [xn + i yn , xn + in] ∪ [xn + in, xn + in] ∪ [xn + in, xn + i yn ].

Clearly, in all the cases, (Cn )n≥n 0 is a null chain for Ω. By Lemma 11.1.3, (Cn ) represents the prime end in Ω corresponding to τ under h. For each n ≥ n 0 , let G n := {w ∈ C : Im w < n, xn < Re w < xn }.   , xn−1 )⊂ By construction Cn+1 ∩ G n = ∅, while Cn−1 ∩ Ω ⊂ G n because (xn−1   (xn , xn ). Hence, for each n ≥ n 0 + 1, the interior of Cn is given by Vn = Ω \ G n (see Lemma 11.1.1). Since G n ⊂ G n + it for any t ≥ 0, we see that Vn + it ⊂ Vn for t ≥ 0. Hence, φt (Un ) ⊂ Un for all t ≥ 0 and all n ≥ n 0 , where Un := h −1 (Vn ).  be the D→Ω Now, let x τ ∈ ∂C D be the prime end defined by τ , and let hˆ :  homeomorphism defined by h. By Remark 4.2.2, {Vn∗ } is a basis of open neighborˆ τ ), hence, {Un∗ } is a basis of open neighborhoods of x τ . In particular, if hoods of h(x {z m } ⊂ D is a sequence converging to τ , then for any n ∈ N, there exists m n ∈ N0 such that z m ∈ Un for all m ≥ m n . The previous considerations imply that, given any sequence {tm } ⊂ [0, +∞) converging to +∞, φtm (z m ) ∈ Un for all m ≥ m n . Thus {φtm (z m )} converges to τ . Therefore (2) holds and we are done. 

In case h(D) contains a half-plane with boundary parallel to the imaginary axis, the continuity of the iterates of a semigroup at the Denjoy-Wolff point is still granted, but, in general, not (globally) uniform in t:

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Proposition 11.3.6 Let (φt ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then φt (z) converges to τ as z → τ locally uniformly in t, namely, for every T > 0 and ε > 0 there exists δ > 0 such that |φt (z) − τ | < ε for all t ∈ [0, T ] and all z ∈ D such that |z − τ | < δ. Proof Denote by h the Koenigs function of the semigroup and write Ω = h(D). If the semigroup is parabolic of zero hyperbolic step, by Theorem 11.1.4, the function h has a unrestricted limit at τ . Therefore, the result follows by Proposition 11.3.1. Proposition 11.3.5 guarantees that the conclusion of the theorem holds if Ω does not contain any half-plane bounded by a line parallel to iR. In particular, the result holds if the semigroup is hyperbolic. Assume now that the semigroup is parabolic of positive hyperbolic step and there exist a half-plane H with ∂ H parallel to iR such that H ⊂ Ω. Hence, there is a real number x0 such that either H := {w ∈ C : Re w > x0 } or H := {w ∈ C : Re w < x0 }. As the two cases are similar, we just consider the first. Note that, in this case, h(D) ⊂ H by Theorem 9.3.5. m be the connected component of {w = x + im ∈ Ω : x ≤ For each m ∈ Z, let C m } ≥ 0. By the transx0 } that contains the point x0 + im. Let xm := inf{Re w : w ∈ C lational invariance of Ω, we have xm ≤ xk whenever m > k. For n ∈ N, let  − in, x0 − in]. Cn := [xn + in, x0 + in] ∪ {w : Re w ≥ x0 , |w − x0 | = n} ∪ [x−n

Clearly, (Cn ) is a null chain for Ω that, by Lemma 11.1.3, represents the prime end in Ω corresponding to τ under h. It follows that for any n ≥ 2, Cn+1 ∩ Ω ⊂ Ω \ G n and Cn−1 ⊂ G n , where G n := {w : Re w ≥ x0 , |w − x0 | < n} ∪ {w :|Im w| < n, xn ≤ Re w < x0 }  }, ∪{w : Im w ≤ −n, xn < Re w < x−n  with the convention that the last set is empty if xn = x−n . The interior of each Cn is thus given by Vn = Ω \ G n (see Lemma 11.1.1). We also notice that if k > n, then G n ⊂ G k + it for all t ∈ [0, k − n]. It follows that

Vk + it = (Ω \ G k ) + it = (Ω + it) \ (G k + it) ⊂ Ω \ (G k + it) ⊂ Ω \ G n = Vn

for all t ∈ [0, k − n]. Fix N ∈ N. According to above chain of inclusions, φt (Uk(n) ) ⊂ Un for all t ∈ [0, N ] and all n ∈ N, where Un := h −1 (Vn ) and k(n) := n + N + 1. Arguing now as in the proof of Proposition 11.3.5, we conclude the proof.  As an immediate consequence, we obtain Corollary 11.3.7 Let (φt ) be a semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then, for each t > 0, there exists lim z→τ φt (z) = τ . That is, the function φt has a continuous extension to D ∪ {τ }.

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The previous corollary, Propositions 11.3.1 and 11.3.2 imply Theorem 11.3.8 Let (φt ) be a semigroup in D with Koenigs function h and DenjoyWolff point τ ∈ D. The following are equivalent: (1) The map h has unrestricted limit, finite or infinite, at a every point of ∂D \ {τ }. (2) Every φt belongs to the disc algebra, that is, every φt has a continuous extension to D. (3) There exists t > 0 such that φt belongs to the disc algebra. By Theorem 4.3.1, we have Corollary 11.3.9 Let (φt ) be an elliptic semigroup in D with Koenigs function h. The following assertions are equivalent: (1) The map h has unrestricted limit, finite or infinite, at a every point of ∂D. (2) ∂∞ Ω is locally connected. (3) Every φt belongs to the disc algebra, that is, every φt has a continuous extension to D. (4) There exists t > 0 such that φt belongs to the disc algebra. We end this chapter with an example of semigroup whose iterates do not belong to the disc algebra. Example 11.3.10 Take any sequence {σn } in ∂D converging to σ0 := 1 and σn = 1 for all n ∈ N. Let Ω := D \ ∪n∈N0 [1/2, 1)σn . Note that Ω is a simply connected domain, 0 ∈ Ω and the boundary of Ω is not locally connected. Let h : D → Ω be a Riemann map such that h(0) = 0. By Theorem 4.3.1, the map h does not extend continuously to ∂D. Since Ω is starlike with respect to 0, h is the Koenigs function of an elliptic semigroup (φt ) in D with Denjoy-Wolff point 0 by Theorem 9.4.3. Theorem 11.3.8 implies that none of the iterates of (φt ) belongs to the disc algebra.

11.4 Notes The results in this chapter are mainly based on the recent papers [47] and [81]. The impressions of a prime end for spirallike functions was described in [47] and for starlike functions at infinity in [81]. In those papers the proofs are based on representation formulas while in this book we rely only on Carathéodory’s theory. Based on classical ideas of starlike functions, Kim and Sugawa obtained Corollary 11.1.7 for the elliptic case in [89]. The non-elliptic case is due to Gumenyuk [81]. With the exception of Theorem 11.3.8 and Corollary 11.3.9 (that appeared firstly in [47]), the results of Sects. 11.2 and 11.3 were first proved by Gumenyuk [81].

Chapter 12

Boundary Fixed Points and Infinitesimal Generators

In this chapter, we continue the analysis of the boundary behavior of semigroups, and we concentrate on boundary fixed points. We show that if a positive iterate of a semigroup has a boundary fixed point (in the sense of non-tangential limit), then such a point is indeed fixed for all the iterates of the semigroup. Moreover, the boundary dilation coefficients of the semigroup at a boundary fixed point are either identically infinity or they are of the form e−λt for some λ ∈ R. The latter is the case of boundary regular fixed points. We prove that there is a one-to-one correspondence between boundary regular fixed points and boundary regular critical points of infinitesimal generators. We also prove several synchronization formulas, that is, decomposition formulas for infinitesimal generators in terms of repelling fixed points and their spectral values. The chapter ends with two examples which show that in general there is no relation between non-regular boundary fixed points of a semigroup and non-regular boundary critical points of the associated infinitesimal generator.

12.1 Inner and Boundary Fixed Points Recall that, by Theorem 11.2.1, every iterate of a semigroup in D has non-tangential limit at each point of ∂D. Moreover, as a matter of notation that we will use freely through all the chapter, if (φt ) is a semigroup in D and σ ∈ ∂D, we set φt (σ ) := ∠ lim φt (z). z→σ

Definition 12.1.1 Let (φt ) be a semigroup in D. A point σ ∈ D is called a fixed point of the semigroup if φt (σ ) = σ for all t ≥ 0. In fact, if σ ∈ D, we say that σ is an inner fixed point of the semigroup while, if σ ∈ ∂D, we say σ is a boundary fixed point of the semigroup. © Springer Nature Switzerland AG 2020 F. Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4_12

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Remark 12.1.2 Every point of D is an inner fixed point and every point in ∂D is a boundary fixed point of the trivial semigroup. Moreover, by Theorem 8.3.6, Corollary 1.2.4 and Theorem 8.2.4, a non-trivial semigroup in D is either elliptic and the only inner fixed point is its Denjoy-Wolff point or is non-elliptic and has no inner fixed points. Remark 12.1.3 The analysis of boundary fixed points is only interesting for semigroups in D different from elliptic groups. In fact, Proposition 8.3.8 shows that if (φt ) is an elliptic group, then either (φt ) is an irrational rotation—and hence no iterate has boundary fixed points—or it is a rational rotation—and hence every iterate different from the identity has no boundary fixed points. Theorem 12.1.4 Let (φt ) be a semigroup in D which is not an elliptic group. If σ ∈ ∂D is a boundary fixed point of φt0 for some t0 > 0, then σ is a boundary fixed point of the semigroup. Proof Let (Ω, h, ψt ) be the canonical model given by Theorem 9.3.5. By Theorem 11.2.3, we have (12.1.1) h(φt (σ )) = ψt (h(σ )), t ≥ 0, where, if p ∈ ∂D then h( p) denotes the non-tangential limit of h at p and, if h(σ ) = ∞, we set ψt (h(σ )) = ∞ for all t ≥ 0. Assume that σ is a boundary fixed point for φt0 for some t0 > 0. Since h(σ ) ∈ ∂∞ h(D) and ψt (w) = w for all t > 0 and w ∈ ∂h(D) (because (φt ) is not an elliptic group), it follows at once from (12.1.1) that h(σ ) = ∞. Hence, by Theorem 11.2.3, φt (σ ) ∈ ∂D and h(φt (σ )) = ∞ for all t ≥ 0. Assume by contradiction that φt1 (σ ) = σ for some t1 > 0. By Proposition 11.2.2, the curve [0, +∞)  t → φt (σ ) ∈ ∂D is continuous and it is not constant because φt0 (σ ) = σ = φt1 (σ ). Hence, there exists a non-empty arc A ⊂ {φt (σ ) : t ∈ [0, +∞)}. But then h has constant non-tangential limit ∞ at every point of A, contradicting Proposition 3.3.2.  Remark 12.1.5 The proof of the above theorem shows that if σ ∈ ∂D is a fixed point of φt for some t > 0 then ∠ lim z→σ h(z) = ∞. In Chap. 8, we introduced the spectral value of a semigroup as a concept intimately related to the Denjoy-Wolff point of the semigroup and the associated multipliers of the iterates. Now, we are going to extend this notion to an arbitrary boundary fixed point. Recall that, by Proposition 1.9.3 and Theorem 1.7.3, if φ : D → D is a holomorphic self-map, σ ∈ ∂D is such that ∠ lim z→σ φ(z) = σ , then φ (σ ) = αφ (σ ), where φ (σ ) is the multiplier of φ at σ and αφ (σ ) is the boundary dilation coefficient of φ at σ . Proposition 12.1.6 Let (φt ) be a semigroup in D which is not an elliptic group and let σ ∈ ∂D be a boundary fixed point of the semigroup different from the Denjoy-Wolff point. Then,

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329

(1) either αφt (σ ) = φt (σ ) = ∞, for every t > 0, (2) or, there exists λ ∈ (−∞, 0) such that αφt (σ ) = φt (σ ) = e−λt , for every t ≥ 0. Proof Suppose there exists t > 0 such that φt (σ ) = ∞. By Theorem 1.7.3, αφt (σ ) = +∞. This implies, using repeatedly Proposition 1.7.7, that αφt/2n (σ ) = +∞, for every natural n. Fix s > 0 and take n such that T := 2tn < s. Since φs = φT ◦ φs−T , by Proposition 1.7.7, we have αφs (σ ) = +∞ and, again by Theorem 1.7.3, φs (σ ) = ∞. If φt (σ ) = ∞ for some t > 0, the previous argument also shows that φs (σ ) = ∞, for all s > 0. Since σ is not the Denjoy-Wolff point of the semigroup, by Remark 1.9.7, M(s) := φs (σ ) ∈ (1, +∞), for all s > 0. Moreover, M(s) = limn→∞ φs ((1 − 1/n)σ ) by Theorem 1.7.3, and, using Weierstrass’ Theorem, we find that M is a measurable function from [0, +∞) into (1, +∞). For every s, t ≥ 0 and every r ∈ (0, 1), the Chain Rule gives

(r σ ) = φs (φt (r σ ))φt (r σ ). φs+t

Since φt (r σ ) tends non-tangentially to σ as r goes to 1 by Proposition 1.5.5, we can take limit r → 1 in the above expression and obtain that M(s + t) = M(s)M(t). Therefore, [0, +∞)  t → log(M(t)) ∈ (0, +∞) is a measurable solution of the Cauchy functional equation (8.1.1) thus, by Theorem 8.1.11, there exists λ < 0 such that log(M(t)) = −λt, for every t ≥ 0.  Definition 12.1.7 Let (φt ) be a semigroup in D which is not an elliptic group, and let σ ∈ ∂D be a boundary fixed point of the semigroup different from the Denjoy-Wolff point. If φt (σ ) is finite for some t > 0, we say that σ is a repelling fixed point of the semigroup and call the number λ defined in Proposition 12.1.6 the repelling spectral value of the semigroup at σ . If φt (σ ) = ∞ for some t > 0, we say that σ is a super-repelling fixed point and we say that its spectral value at σ is −∞. Definition 12.1.8 A boundary fixed point σ ∈ ∂D of a semigroup in D is called regular if σ is a boundary regular fixed point of every iterate. Hence, a boundary regular fixed point is either a repelling fixed point or the Denjoy-Wolff point of a non-elliptic semigroup.

12.2 Boundary Fixed Points and Infinitesimal Generators In the theory of differential equations it is a basic fact that stationary trajectories (fixed points) are in one-to-one correspondence with zeros (critical points) of the corresponding vector field. In Chap. 10, we have seen that the Denjoy-Wolff point of a non-elliptic semigroup in D is a non-tangential zero of the associated infinitesimal

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12 Boundary Fixed Points and Infinitesimal Generators

generator (see Corollary 10.1.12). In this section we see to which extent this result holds for boundary fixed points other than the Denjoy-Wolff point. Definition 12.2.1 Let G be the infinitesimal generator of a semigroup in D. A point σ ∈ ∂D is said to be a boundary critical point of G (or boundary critical point of the semigroup) if the radial limit limr →1− G(r σ ) = 0. Proposition 12.2.2 Let G be the infinitesimal generator of a semigroup in D and σ ∈ ∂D a boundary critical point of G. Then ∠ lim z→σ G(z) = 0. Proof Assume the semigroup is non-trivial so that G is not identically vanishing and let τ ∈ D be the Denjoy-Wolff point of the associated semigroup. If σ = τ , the result follows from Corollary 10.1.12. Assume now that σ = τ . By Berkson-Porta’s Formula (Theorem 10.1.10), there exists a non vanishing holomorphic function p : D → H such that G(z) = (z − τ )(τ z − 1) p(z), z ∈ D. Since σ is a boundary critical point of G, we deduce that limr →1− (r σ − τ )(r τ σ − 1) p(r σ ) = 0 and, since limr →1− (r σ − τ )(r τ σ − 1) = −σ |σ − τ |2 = 0, we deduce that limr →1− p(r σ ) = 0. By Lindelöf’s Theorem (see Theorem 1.5.7), we have that  ∠ lim z→σ p(z) = 0. Therefore ∠ lim z→σ G(z) = 0. Assume that G ∈ Gen(D) is not identically vanishing, with a Berkson-Porta’s Decomposition Formula (Theorem 10.1.10) given by G(z) = (z − τ )(τ z − 1) p(z), z ∈ D, where τ ∈ D is the Denjoy-Wolff point of the associated semigroup and p : D → H is a non vanishing holomorphic function. Fix σ ∈ ∂D, σ = τ . The holomorphic function 1/ p has non-negative real part and, by Proposition 2.1.3, there exists λ ∈ 1 . Note that (−∞, 0] such that λ = ∠ lim z→σ (σ z − 1) p(z) G(z) p(z) = σ (z − τ )(τ z − 1) . z−σ σz − 1 Thus if λ < 0, then ∠ lim z→σ G(z) = −|σ − τ |2 λ1 ∈ (0, +∞) and, if λ = 0, then z−σ = ∞. In any case, the following limit always exists ∠ lim z→σ G(z) z−σ G (σ ) := ∠ lim

z→σ

G(z) ∈ (0, +∞) ∪ {∞}. z−σ

(12.2.1)

Definition 12.2.3 Let G be the infinitesimal generator of a semigroup in D and σ ∈ ∂D a boundary critical point of G. We say that σ is a boundary regular critical point of G if G (σ ) is finite.

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331

The next result provides different characterizations of boundary regular critical points. Proposition 12.2.4 Let G be the infinitesimal generator of a semigroup in D and σ ∈ ∂D. Then the following are equivalent: (1) σ is a boundary regular critical point of the infinitesimal generator, that is, G (σ ) = ∠ lim z→σ G(z) ∈ C; z−σ (2) ∠ lim z→σ G(z) = 0 and ∠ lim z→σ G (z) ∈ C; σ )| < +∞; (3) lim inf r →1− |G(r 1−r (4) limr →1− G(r σ ) = 0 and lim supr →1− |G (r σ )| < +∞. If the above statements hold, then G (σ ) = ∠ lim z→σ G (z) and G (σ ) ∈ R. Proof We assume that the semigroup is non-trivial so that G is not identically vanishing. If σ is the Denjoy-Wolff point of the semigroup, Corollary 10.1.12 shows that statements (1)–(4) hold with G (σ ) = ∠ lim z→σ G (z) ∈ R. Therefore, we assume that σ is not the Denjoy-Wolff point of the semigroup. By Theorem 1.7.2, statements (1) and (2) are equivalent and, in fact, G (σ ) = ∠ lim z→σ G (z) ∈ R by (12.2.1). It is clear that (1) implies (3) and that (2) implies (4). Now, by (12.2.1), lim inf − r →1

      G(r σ )   |G(r σ )|  = ∠ lim G(z)  .  = lim inf    z→σ z − σ  r →1− 1−r σ − rσ

Hence, (3) implies (1). Finally, if (4) holds and M := supr ∈(0,1) |G (r σ )|, then for 0 < r < s < 1 we have   |G(r σ ) − G(sσ )| = 

r

s

  G (xσ ) d x  ≤ (s − r )M.

Taking limit when s goes to 1, we get |G(r σ )| ≤ (1 − r )M. Thus, (3) is satisfied, and we are done.  For boundary regular fixed points of a semigroup, the relationship with critical points of the infinitesimal generator is quite natural and we have: Theorem 12.2.5 Let (φt ) be a semigroup in D with associated infinitesimal generator G and σ ∈ ∂D. Then the following are equivalent: (1) σ is a boundary regular fixed point of the semigroup; (2) σ is a boundary regular critical point of the infinitesimal generator; (3) there exists μ ∈ R such that  Re

σ G(z) (σ − z)2

 ≥

μ 1 − |z|2 for all z ∈ D. 2 |σ − z|2

(12.2.2)

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12 Boundary Fixed Points and Infinitesimal Generators

If one—and hence any—of the previous holds, and φt (σ ) = e−λt for all t ≥ 0, then and −λ = ∠ lim z→σ G(z) z−σ    σ G(z) 2|σ − z|2 λ = max μ ∈ R : μ ≤ Re , 1 − |z|2 (σ − z)2

 for all z ∈ D . (12.2.3)

Proof We assume that the semigroup is non-trivial so that G is not identically vanishing.  1−|z|2 for all z ∈ D, we have = Re σσ +z Let us see that (3) implies (2). Since |σ −z|2 −z that   σ G(z) μσ +z ≥0 Re − (σ − z)2 2 σ −z and thus p(z) := (σσ G(z) − μ2 σσ +z is a holomorphic function in the unit disc with non−z)2 −z negative real part. In particular, by Proposition 2.1.3, there is β ∈ [0, +∞) such that β = ∠ lim z→σ (1 − σ z) p(z). Therefore ∠ lim

z→σ

 G(z) μ = −∠ lim (1 − σ z) p(z) + σ (σ + z) = −(β + μ) ∈ R. z→σ z−σ 2 (12.2.4)

Thus σ is a boundary regular critical point of G. Let us now check that (1) implies (3). Thus, assume that σ ∈ ∂D is a boundary regular fixed point for (φt ). By Theorem 8.3.1 —in case σ is the Denjoy-Wolff point of the semigroup— and by Proposition 12.1.6 —if σ is a repelling fixed point–, there is λ ∈ R such that φt (σ ) = e−λt for every t ≥ 0. By Julia’s Lemma (see Theorem 1.4.7), we have φt (E(σ, R)) ⊂ E(σ, e−λt R), That is,

2 |σ − φt (z)|2 −λt |σ − z| ≤ e 1 − |φt (z)|2 1 − |z|2

for all t, R > 0.

for all t > 0 and z ∈ D.

−φt (z)| This means that the function [0, +∞)  t → g(t) := eλt |σ satisfies g(t) ≤ 1−|φt (z)|2

g(0) for all t ≥ 0. Hence, g (0) ≤ 0. A direct computation shows that 2

12.2 Boundary Fixed Points and Infinitesimal Generators

333

|σ − z|2 −2(1 − |z|2 )Re (G(z)(σ − z)) + 2|σ − z|2 Re (G(z)z) + 1 − |z|2 (1 − |z|2 )2     2 2 G(z) zG(z) |σ − z| |σ − z| −Re + Re =λ +2 1 − |z|2 1 − |z|2 σ −z 1 − |z|2   ◦  2 z |σ − z| 1 = − λ + 2Re G(z) 1 − |z|2 1 − |z|2 σ −z   zσ − 1 |σ − z|2 λ + 2Re G(z) = 1 − |z|2 (1 − |z|2 )(σ − z)   2 2 |σ − z| σ G(z) |σ − z| λ − 2 . Re = 1 − |z|2 1 − |z|2 (σ − z)2

g (0) = λ

Thus λ−2 

and Re

|σ − z|2 Re 1 − |z|2

σ G(z) (σ − z)2

 ≥



σ G(z) (σ − z)2

 ≤0

λ 1 − |z|2 , for all z ∈ D, 2 |σ − z|2

(12.2.5)

and (12.2.2) holds. In order to end the proof, we see that (2) implies (1). We may assume that the semigroup is not an elliptic group. Suppose that σ is a boundary regular critical point. By Proposition 12.2.4, β = ∠ lim z→σ G (z) ∈ R. Then the function G is bounded on the radial segment [0, 1)σ , that is, there is c > 0 such that |G (xσ )| ≤ c for all 0 ≤ x < 1. Since G(σ ) := ∠ lim z→σ G(z) = 0,   |G(xσ )| = |G(xσ ) − G(σ )| = 

σ

xσ

  G (s) ds  ≤ c|xσ − σ | = c(1 − x)

for all 0 ≤ x < 1. By Kolmogorov’s Backward Equation (10.1.5) and Schwarz-Pick’s Lemma (Theorem 1.2.3), we see that    ∂φt (xσ )  



         ∂t  = |G(xσ )| φt (xσ ) ≤ c(1 − x) φt (xσ ) 1 − |φt (xσ )|2 ≤ c(1 − x) ≤ 2c (1 − |φt (xσ )|) . 1 − x2 Therefore,

   ∂φt (xσ )      ∂  ∂t  log (1 − σ φt (xσ )) =  ∂t  |1 − σ φ (xσ )| ≤ 2c, t

where log denotes the principal branch of the logarithm. Since φ0 = idD , we obtain

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12 Boundary Fixed Points and Infinitesimal Generators

|log (1 − σ φt (xσ )) − log (1 − x)| ≤ 2ct

for all t ≥ 0.

(12.2.6)

t (xσ ) Hence | log σ −φ | ≤ 2ct for all t ≥ 0, which implies that φt (σ ) = σ , for all t ≥ 0. σ −xσ Moreover, since 1 − σ φt (xσ ) ∈ H for all t ≥ 0 and x ∈ [0, 1), we have that Im log(1 − σ φt (xσ )) = Arg(1 − σ φt (xσ )). Hence, again by (12.2.6),

|Arg(1 − σ φt (xσ ))| ≤

π , 4

for all 0 < x < 1, 0 ≤ t ≤

π . 8c

Thus, for every sequence {tn } such that 0 ≤ tn ≤ π/8c, and every sequence {xn }, xn ∈ (0, 1), converging to 1, the sequence {φtn (xn σ )} converges to σ non-tangentially. We claim that there exists C > 0 such that |G (φt (xσ ))| ≤ C for all t ∈ [0, π/8c] and x ∈ (0, 1). Indeed, if this were not the case, there would exist a sequence {tn } such that 0 ≤ tn ≤ π/8c, and a sequence {xn }, xn ∈ (0, 1), converging to 1 such that limn→∞ |G (φtn (xn σ ))| = ∞. However, since we already noticed that {φtn (xn σ )} converges to σ non-tangentially, we would have G (σ ) = limn→∞ G (φtn (xn σ )) = ∞, against the hypothesis that σ is a regular critical point. Thus, by the Lebesgue Dominated Convergence Theorem,  lim

x→1 0

t





t

G (φs (xσ )) ds =

G (σ ) ds = G (σ )t, t ≤ π/8c.

0

Therefore, by Proposition 10.1.8, and for t ≤ π/8c,

e G (σ )t = lim e x→1

t 0

G (φs (xσ )) ds

= lim φt (xσ ). x→1

That is, σ is a boundary regular fixed point of φt for t small enough. By Theorem 12.1.4 and Proposition 12.1.6, σ is a boundary regular fixed point of the semigroup. Moreover, −G (σ ) is either the spectral value of the semigroup—in case σ is the Denjoy-Wolff point—or the repelling spectral value at σ —in case σ is a repelling fixed point. In the first part of the proof [see (12.2.4)], we got that −λ = G (σ ) ≤ −μ for any μ satisfying (12.2.2). But equality (12.2.5) shows that λ belongs to the set   2|σ −z|2 σ G(z)  μ ∈ R : μ ≤ 1−|z|2 Re (σ −z)2 , for all z ∈ D . Thus (12.2.3) holds. Due to the previous theorem, with a slight abuse of notation, if σ ∈ ∂D is a regular critical point of an infinitesimal generator G so that the associated semigroup (φt ) has a repelling fixed point at σ with repelling spectral value λ ∈ (−∞, 0), we sometimes say that λ is the repelling spectral value of G at σ . Corollary 12.2.6 Let (φt ) be a semigroup in D with associated infinitesimal generator G and σ ∈ ∂D. Then the following are equivalent: (1) σ is a boundary regular fixed point of the semigroup; (2) there exist λ ∈ R and a unique p : D → H holomorphic, with ∠ lim z→σ (z − σ ) p(z) = 0, such that

12.2 Boundary Fixed Points and Infinitesimal Generators

335

λσ +z , for all z ∈ D. G(z) = (σ z − 1)(z − σ ) p(z) + 2σ −z

(12.2.7)

If one—and hence any—of the previous holds, then φt (σ ) = e−λt , for every t ≥ 0. Proof Assume that σ is a boundary regular fixed point of the semigroup with spectral λ. By Theorem 12.2.5, −λ = ∠ lim z→σ G(z) and z−σ  Re

σ G(z) (σ − z)2

 ≥

λ 1 − |z|2 , for all z ∈ D. 2 |σ − z|2

 1−|z|2 for all z ∈ D, we have that p(z) := (σσ G(z) = Re σσ +z − Since |σ −z|2 −z −z)2 holomorphic function in the unit disc with non-negative real part and ∠ lim (z − σ ) p(z) = ∠ lim z→σ

z→σ

λ σ +z 2 σ −z

is a

σ G(z) λ + (σ + z) = −λσ + λσ = 0. z−σ 2

Assume that there exist another real number μ ∈ R and a holomorphic function q : D → H, with ∠ lim z→σ (z − σ )q(z) = 0, such that

μσ +z G(z) = (σ z − 1)(z − σ ) q(z) + , for all z ∈ D. 2 σ −z

(12.2.8)

Then, using again Theorem 12.2.5 and this decomposition formula, −λ = ∠ lim

z→σ

 G(z) μ = ∠ lim (σ z − 1)q(z) − σ (σ + z) = −μ. z→σ z−σ 2

Hence, μ is the spectral value and it follows immediately that p = q. Conversely, if (2) is satisfied, then  Re

σ G(z) (σ − z)2

 −

μ 1 − |z|2 = Re ( p(z)) ≥ 0, for all z ∈ D, 2 |σ − z|2

and (1) holds by Theorem 12.2.5.



The next two results provide two characterizations of boundary regular critical points which complement Proposition 12.2.4. Proposition 12.2.7 Let G be the infinitesimal generator of a semigroup in D and σ ∈ ∂D. Then the following are equivalent: (1) σ is a boundaryregularcritical point of the infinitesimal generator; σ) (2) lim inf r →1− Re σ G(r < +∞ and limr →1− Im (σ G(r σ )) = 0. r −1 Proof We assume that G is not identically vanishing. If (1) holds, by Proposition σ )| < +∞ and limr →1 G(r σ ) = 0. Thus (2) is clearly satisfied. 12.2.4, lim inf r →1 |G(r 1−r

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12 Boundary Fixed Points and Infinitesimal Generators

Assume that (2) holds. Let τ ∈ D be the Denjoy-Wolff point of the semigroup. By Berkson-Porta’s Theorem 10.1.10, there exists a holomorphic function p : D → H such that G(z) = (z − τ )(τ z − 1) p(z), z ∈ D. If σ = τ , then σ is a boundary regular fixed point of the semigroup and then a boundary regular critical point of the semigroup. Therefore, let us assume that σ = τ . Notice that σ G(r σ ) = (r − σ τ )(τr σ − 1) p(r σ ) and lim (r − σ τ )(τr σ − 1) = −|σ − τ |2 = 0.

r →1

To simplify the notation, write Ar := Re (r − σ τ )(τr σ − 1) = (r 2 + 1)Re (τ σ ) − Br = 2r and Br := Im (r − σ τ )(τr σ − 1) = (1 − r 2 )Im (τ σ ). Since [0, 1)  r → 1−r (1 + r )Im (τ σ ) is bounded, −Ar

Br Re (σ G(r σ )) Re ( p(r σ )) + Im (σ G(r σ )) = (Ar2 + Br2 ) , r −1 1−r 1−r

and limr →1 Ar = −|σ − τ |2 < 0, we have that lim inf r →1 p(r σ ) only if lim inf r →1 Re1−r < +∞. Moreover,

Re (σ G(r σ )) r −1

< +∞ if and

−Br Re (σ G(r σ )) + Ar Im (σ G(r σ )) = (Ar2 + Br2 )Im ( p(r σ )). By Theorem 2.2.1, the map [0, 1)  r → (1 − r ) p(r σ ) is bounded. Clearly this implies that the map [r, 1)  r → Br Re (σ G(r σ ))) = (1 − r 2 )Im (τ σ )Re (σ G (r σ ))) is also bounded and we conclude that lim supr →1 Im p(r σ ) < +∞. Thus, Proposition 2.3.4 shows that there is a ∈ R and L ∈ C such that ai = ∠ lim z→σ p(z) and L = ∠ lim z→σ p (z). Therefore, ∠ lim G(z) = −σ |σ − τ |2 ai, ∠ lim G (z) = ai(2τ σ − |τ |2 − 1) − σ |σ − τ |2 L ∈ C. z→σ

z→σ

Finally, since limr →1− Im (σ G(r σ )) = 0, we get that a = 0 and thus ∠ lim G(z) = 0, ∠ lim G (z) ∈ C. z→σ

z→σ

Hence, by Proposition 12.2.4, σ is a boundary regular critical point of the infinitesimal generator.  Remark 12.2.8 The proof of above proposition shows that the condition  lim inf Re − r →1

σ G(r σ ) r −1

 < +∞,

12.2 Boundary Fixed Points and Infinitesimal Generators

337

by itself does not guarantee that σ is a boundary regular critical point. For example, consider the function with non-negative real part p(z) := 1 − z + i, z ∈ D, and the semigroup in the unit disc with infinitesimal generator G(z) = −zp(z). Clearly, ) = 1 < +∞ and 1 is not a boundary regular critical point of G limr →1− Re G(r r −1 because G(1) = −i = 0. Corollary 12.2.9 Let G be the infinitesimal generator of a semigroup in D and σ ∈ ∂D a boundary critical point of G. Then the following are equivalent: (1) (2) (3) (4)

σ is a boundaryregularcritical point of the infinitesimal generator; σ) lim inf r →1− Re σ G(r < +∞; 1−r lim supr →1− Re G (r σ ) < +∞; lim inf r →1− (1 − r )(du σ )r σ · G(r σ ) < +∞, where u σ is the Poisson kernel introduced in (1.4.3).

Proof The equivalence between (1) and (2) is nothing but Proposition 12.2.7. (1) implies (3) follows from Proposition 12.2.4. Let us see that (3) implies (2). By hypothesis, c := supr ∈(0,1) Re G (r σ ) < +∞. Fix 0 < r < s < 1. Then  s    ∂  |Re (σ G(sσ )) − Re (σ G(r σ ))| =  G(xσ ))) d x (σ (Re  r ∂x  s    =  Re (G (xσ )) d x  ≤ c(s − r ). r

Since σ is a critical point, taking limit as s goes to 1, we get |Re (σ G(r σ ))| ≤ c(1 − r ). That is, (2) holds.  for all z ∈ D, a direct computation shows that Recalling that u σ (z) = Re z+σ z−σ  (du σ )z · G(z) = −2 Re

 G(z) , z ∈ D. (z − σ )(σ z − 1)

In particular, taking z = r σ ,  (1 − r )(du σ )r σ · G(r σ ) = 2 Re Thus, (2) and (4) are clearly equivalent.

 σ G(r σ ) , z ∈ D. r −1 

Next we show a characterization of repelling fixed points in terms of the Poisson kernel introduced in (1.4.3). An analogous characterization for the Denjoy-Wolff point was obtained in Corollary 10.1.11.

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12 Boundary Fixed Points and Infinitesimal Generators

Corollary 12.2.10 Let (φt ) be a semigroup in D with associated infinitesimal generator G and σ ∈ ∂D. Then, σ is a repelling fixed point of the semigroup if and only if there exists β < 0 such that d(u σ )z · G(z) − βu σ (z) ≤ 0,

for all z ∈ D.

Moreover, if σ is a repelling fixed point of (φt ) and λ is the repelling spectral value of (φt ) at σ , then, λ = max{β : d(u σ )z · G(z) − βu σ (z) ≤ 0,

for all z ∈ D}.

 for all z ∈ D. Thus, if h Proof Let us recall that, given σ ∈ ∂D, u σ (z) = Re z+σ z−σ is holomorphic in the unit disc, a direct computation shows that  (du σ )z · h(z) = −2 Re

 h(z) , z ∈ D. (z − σ )(σ z − 1)

Therefore, given z ∈ D,     z+σ G(z) − βRe (du σ )z · G(z) − βu σ (z) = −2 Re (z − σ )(σ z − 1) z−σ     β z+σ σ G(z) G(z) β 1 − |z|2 − = −2 Re = −2 Re − (z − σ )(σ z − 1) 2σ −z (z − σ )2 2 |σ − z|2 

and the result follows from Theorem 12.2.5.

12.3 Synchronization Formulas In this section we are going to prove a representation formula for infinitesimal generators, involving several boundary regular fixed points. Definition 12.3.1 Let p : D → C be a non-zero holomorphic function with nonnegative real part and σ ∈ ∂D. We say that σ is a regular zero of p if ∠ lim z→σ p(z) = p(z) ∈ C. 0 and p (σ ) = ∠ lim z→σ z−σ By Proposition 2.1.3 (applied to the function 1/ p) we have ∠ lim z→σ [0, +∞). Thus σ is a regular zero of p if and only if σ p (σ ) = −∠ lim

z→σ

p(z) 1−σ z

∈

p(z) ∈ (−∞, 0). 1 − σz

Proposition 12.3.2 Let σ1 , σ2 , . . . , σn ∈ ∂D such that σk = σl , if k = l. Let p : D → C be a non-zero holomorphic function with non-negative real part and β ∈ (0, +∞). Then the following are equivalent:

12.3 Synchronization Formulas

339

 (1) σk is a regular zero of p for each . . . , n and nk=1 σk p (σk ) = −β, kn = 1, 1 (2) there are λ1 , . . . , λn > 0 with j=1 λ j = 2β and there exists a holomorphic function q : D → H, with ∠ lim z→σk (z − σk )q(z) = 0, for k = 1, . . . , n, such that −1  n  σk + z + q(z) p(z) = λk , for all z ∈ D. (12.3.1) σk − z k=1 In such a case, λk = − 2σk p1 (σk ) . Proof Assume (2) holds. Then, for every j = 1, . . . , n, ∠ lim

z→σ j

p(z) 1 1

 = = ∠ lim > 0. n z→σ j σk +z 1 − σjz 2λ j (1 − σ j z) λ + q(z) k k=1 σk −z

Therefore each σ j is a regular zero of the function p and −σ j p (σ j ) = 2λ1 j , for each  j = 1, . . . , n. Since nk=1 λ1k = 2β, we conclude that (1) holds. Assume (1) holds. Suppose firstly that n = 1 so that σ1 p (σ1 ) = −β. Therefore, 1 σ1 +z 1z = β1 . Write p(z) ˜ := 2β , z ∈ D. By Proposition 2.1.3, the func∠ lim z→σ1 1−σ p(z) σ1 −z 1 tion q := p − p˜ has non-negative real part and ∠ lim z→σ1 (1 − σ 1 z)q(z) = 0. Thus (2) holds for n = 1. σk +z , z ∈ D. Assume now that n > 1. Fix k ∈ {1, . . . , n}. Write pk (z) := 2σk −1 p (σk ) σk −z Applying the previous case to σk , we get that the holomorphic function qk := 1p − pk has non-negative real part and ∠ lim z→σ1 (1 − σ k z)qk (z) = 0. Notice that n n n 1 1 1 1 1 = = qk (z) + pk (z), z ∈ D. p(z) n k=1 p(z) n k=1 n k=1

 Write q˜ := n1 nk=1 qk . Clearly q˜ is a holomorphic function in the unit disc with non-negative real part and, by Proposition 2.1.3, there exists μ j := ∠ lim z→σ j (1 − σ j z)q(z) ˜ ≥ 0, for j = 1, . . . , n. Since every σk is a regular zero of p, −σk p (σk ) = ∠ lim

z→σk

=

p(z) 1

= ∠ lim z→σk 1 n 1 − σk z (1 − σ k z) q(z) ˜ − 2n j=1 1

μk −

=

1 μk −

1 1 n σk p (σk )



.

− 1 σk p1 (σk ) for k = 1, . . . , n. Applying n times Proposition 2.1.3,  μ σ +z we see that the function q(z) := q(z) ˜ − nj=1 2j σ jj −z , z ∈ D, has non-negative real part and ∠ lim z→σ j (1 − σ j z)q(z) = 0, for j = 1, . . . , n. Finally, for z ∈ D,

Hence, μk =

1

1 1 2 2n σk p (σk )

σ j +z 1 σ j p (σ j ) σ j −z

n

340

12 Boundary Fixed Points and Infinitesimal Generators n σj + z 1  1 1 = q(z) ˜ −

p(z) 2n j=1 σ j p (σ j ) σ j − z n n  σj + z μj σj + z 1  1 −

(σ ) σ − z 2 σ − z 2n σ p j j j j=1 j=1 j   n n σj + z σj + z 1 1  1 1 1 −1 − = q(z) + 2 j=1 n σk p (σk ) σ j − z 2n j=1 σ j p (σ j ) σ j − z

= q(z) +

σj + z 1 1 , 2 j=1 σ j p (σ j ) σ j − z n

= q(z) −



what provides the desired equality (12.3.1).

Theorem 12.3.3 Let G : D → C be a holomorphic function, τ ∈ D, σ1 , σ2 , . . . , σn ∈ ∂D such that σk = τ , for all k, and σk = σl , if k = l, and λ1 , . . . , λn ∈ (−∞, 0). Then the following are equivalent: (1) G is the infinitesimal generator of a semigroup in D with Denjoy-Wolff point τ and with boundary regular critical points at σ1 , . . . , σn with repelling spectral values λ1 , . . . , λn , respectively, (2) there exists a holomorphic function p : D → H, with ∠ lim z→σk (z − σk ) p(z) = 0, for k = 1, . . . , n, such that for all z ∈ D 

1  |τ − σk |2 σk + z G(z) = (z − τ )(τ z − 1) p(z) + 2 k=1 |λk | σk − z n

−1 .

(12.3.2)

Proof Let us see that (1) implies (2). By Theorem 10.1.10, there is a non-zero holomorphic function P : D → H such that G(z) = (z − τ )(τ z − 1)P(z) for all z ∈ D. Since σk is a repelling fixed point, σk = τ for every k. By Proposition 2.1.3 1 ∈ (applied to the function 1/P), for each l, there exists μl := ∠ lim z→σl (1 − σ l z) P(z) [0, +∞). Since each σl is a boundary regular critical point of G, using Theorem 12.2.5, we have |λl | = ∠ lim

G(z)

z→σl z − σl

= ∠ lim

z→σl

(z − τ )(τ z − 1)P(z) (σl − τ )(τ σl − 1) |τ − σl |2 = = . z − σl −σl μl μl

l| Thus, μl = |τ −σ = 0, for l = 1, . . . , n. In particular, each σl is a regular zero of |λl | P. Applying Proposition 12.3.2, we obtain the decomposition (12.3.2). Conversely, assume that (2) holds. Then, for each z ∈ D, 2

 Re

1  |τ − σk |2 σk + z p(z) + 2 k=1 |λk | σk − z n



1  |τ − σk |2 1 − |z|2 ≥ 0. 2 k=1 |λk | |σk − z|2 n

= Re p(z) +

12.3 Synchronization Formulas

341

−1

 2 k | σk +z Thus, Re p(z) + 21 nk=1 |τ −σ ≥ 0, for all z ∈ D. Hence, by Theorem |λk | σk −z 10.1.10, G is the infinitesimal generator of a semigroup in D with Denjoy-Wolff point τ . Moreover, ∠ lim

z→σl

G(z) (z − τ )(τ z − 1) 

= ∠ lim  2 z→σ l z − σl k | σk +z (z − σl ) p(z) + 21 nk=1 |τ −σ |λk | σk −z =

(σl − τ )(τ σl − 1) 0−

1 |τ −σl |2 2σl 2 |λl |

= |λl |.

Therefore, Theorem 12.2.5 shows that each σl is a boundary regular critical point of G with repelling spectral value λl , and (1) holds. Conversely, assume that (2) holds. Then, for each z ∈ D,  Re

1  |τ − σk |2 σk + z p(z) + 2 k=1 |λk | σk − z n



1  |τ − σk |2 1 − |z|2 ≥ 0. 2 k=1 |λk | |σk − z|2 n

= Re p(z) +

−1

 2 k | σk +z Thus, Re p(z) + 21 nk=1 |τ −σ ≥ 0, for all z ∈ D. Hence, by |λk | σk −z Theorem 10.1.10, G is the infinitesimal generator of a semigroup in D with DenjoyWolff point τ . Moreover, ∠ lim

z→σl

G(z) (z − τ )(τ z − 1) 

= ∠ lim  2 z→σ l z − σl k | σk +z (z − σl ) p(z) + 21 nk=1 |τ −σ |λk | σk −z =

(σl − τ )(τ σl − 1) 0−

1 |τ −σl |2 2σl 2 |λl |

= |λl |.

Therefore, Theorem 12.2.5 shows that each σl is a boundary regular critical point of  G with repelling spectral value λl , and (1) holds. Corollary 12.3.4 Let (φt ) be a non-trivial elliptic semigroup in D with DenjoyWolff point τ ∈ D, τ = 0, spectral value λ ∈ C, and different repelling fixed points σ1 , . . . , σn ∈ ∂D with repelling spectral values λ1 , . . . , λn ∈ (−∞, 0). Let G be the infinitesimal generator of (φt ). Then    ⎡ ⎤    n n 2 2     1 − |τ | |τ | τ Im (σ τ ) |τ − σ | 2τ k k ≤ Im ⎣Re ⎦. − − −  λ G(0) |λk |  1 − |τ |2 G(0) |λk |  k=1 k=1

(12.3.3) In particular,

n  |τ − σk |2 τ ≤ 2 Re . |λk | G(0) k=1

(12.3.4)

342

12 Boundary Fixed Points and Infinitesimal Generators

This inequality is sharp and equality holds if and only if the infinitesimal generator of the semigroup is 

 |τ − σk |2 σk + z 2τ + G(z) = 2(z − τ )(τ z − 1) iIm G(0) k=1 |λk | σk − z n

−1 ,

z ∈ D.

Proof By Theorems 10.1.10 and 12.3.3, there are two holomorphic functions p, q : D → H, with ∠ lim z→σk (z − σk )q(z) = 0, for k = 1, ..., n, p non-zero, such that, for all z ∈ D, ⎡

⎤−1 n 2 σ +z  |τ − σ | 1 k k ⎦ . G(z) = (z − τ )(τ z − 1) p(z) = (z − τ )(τ z − 1) ⎣q(z) + 2 |λk | σk − z k=1

Therefore

1  |τ − σk |2 σk + z 1 = q(z) + , p(z) 2 k=1 |λk | σk − z n

z ∈ D.

By Theorem 2.2.1, |Im q(τ ) − Im q(0)| ≤ Re q(0) On the one hand,

2|τ | , 1 − |τ |2

(12.3.5)

1  |τ − σk |2 1 = q(0) + . p(0) 2 k=1 |λk | n

 2 k| Since 21 nk=1 |τ −σ is real, we have that Re |λk | 1 Im p(0) = Im q(0). On the other hand, 1 1 Im = Im q(τ ) + Im p(τ ) 2 = Im q(τ ) +

1 p(0)

= Re q(0) +

n k=1

|τ −σk |2 |λk |

and

 n   |τ − σk |2 σk + τ |λk | σk − τ k=1

n  Im (σk τ ) k=1

1 2

|λk |

.

Therefore, (12.3.5) can be rewritten as     n n   1 2|τ | 1  1  |τ − σk |2 Im (σk τ ) 1  − − Im − Re . Im ≤  p(τ ) k=1 |λk | p(0)  1 − |τ |2 p(0) 2 k=1 |λk |

12.3 Synchronization Formulas

343

Since G(0) = τ p(0) and −λ = G (τ ) = (|τ |2 − 1) p(τ ), this inequality is nothing but (12.3.3). Equality holds in (12.3.4) if and only if Re q(0) = 0 if and only if there is β ∈ R such that n 1 1  |τ − σk |2 σk + z = iβ + , z ∈ D. p(z) 2 k=1 |λk | σk − z τ = β. This ends the proof. Clearly, Im G(0)



Corollary 12.3.5 Let (φt ) be a non-trivial elliptic semigroup in D with Denjoy-Wolff point τ = 0 and spectral value λ ∈ C \{0} with Re λ < 0. Let σ1 , . . . , σn ∈ ∂D be different repelling fixed points with repelling spectral values λ1 , . . . , λn ∈ (−∞, 0). Let G be the infinitesimal generator of the semigroup. Then    n n  G

(0)  σk  1  1  ≤ 2 Re − . − Im   2λ2 |λk |  λ k=1 |λk | k=1 In particular,

n  1 1 ≤ 2 Re . |λ | λ k k=1

(12.3.6)

(12.3.7)

This inequality is sharp and equality holds if and only if the infinitesimal generator of the semigroup is 

2  1 σk + z G(z) = −2z −iIm + λ k=1 |λk | σk − z n

−1 ,

z ∈ D.

Proof By Theorem 10.1.10, there is a non-zero holomorphic function p : D → H, such that, G(z) = −zp(z), for all z ∈ D. Take a sequence {r j } in the interval (0, 1) such that lim j→∞ r j = 0 and, for each j, consider the holomorphic function G j (z) = (z − r j )(r j z − 1) p(z), for all z ∈ D. By Theorem 10.1.10, G j is the infinitesimal generator of a semigroup in the unit disc with Denjoy-Wolff point r j . In fact, G j (r j ) = (r 2j − 1) p(r j ). Notice that lim j→∞ G (r j ) = G (0) = −λ. Fix k ∈ {1, 2, ..., n}. Since σk is a repelling fixed point of (φt ) with spectral value λk , by Theorem 12.2.5, we have ∠ lim

z→σk

G j (z) 1 G(z) = ∠ lim (z − r j )(r j z − 1) z→σk z − σk (−z) z − σk = (σk − r j )(r j σk − 1)(−σ k )(−λk ) = −|σk − r j |2 λk .

That is, σk is a repelling fixed point of the semigroup generated by G j with repelling spectral value |σk − r j |2 λk .

344

12 Boundary Fixed Points and Infinitesimal Generators

Therefore, by Corollary 12.3.4, applied to G j , we have      n  1 − r 2j rj Im (σ k r j )   − − Im   r j p(0) |σk − r j |2 |λk |  (1 − r 2j ) p(r j ) k=1   (12.3.8) n  rj |r j − σk |2 rj ≤ − 2 Re . r j p(0) k=1 |σk − r j |2 |λk | 1 − r 2j Simplifying and dividing by r j , we obtain      n   1 1 Im (σ ) k  Im − −   2 |λ |  r j p(r j ) r j p(0) |σ − r | k j k k=1   n  1 1 1 ≤ − 2 Re . p(0) k=1 |λk | 1 − r 2j Since 1 r j→∞ j lim



1 1 − p(r j ) p(0)



p (0) 1 p(0) − p(r j ) 1 G

(0) 1 G

(0) =− = = , 2 G (0)2 2 λ2 j→∞ r j p(r j ) p(0) p(0)2

= lim

taking limits in above inequality we deduce (12.3.6) and thus (12.3.7). Assume that equality holds in (12.3.7). Then, the right term in the inequality (12.3.8) is zero and by Corollary 12.3.4, we have that, for each j and each z ∈ D, 

 |r j − σk |2 σk + z 2r j + G j (z) = 2(z − r j )(r j z − 1) i Im r j p(0) k=1 |r j − σk |2 |λk | σk − z n

−1 .

Taking limits when j goes to ∞ we have 

 1 σk + z 2 + G(z) = −2z iIm

G (0) k=1 |λk | σk − z n

−1 ,

z ∈ D.

Finally, some elementary calculations show that if the infinitesimal generator of the  semigroup is given by the previous expression then nk=1 |λ1k | = 2 Re λ1 . Corollary 12.3.6 Let (φt ) be a non-trivial semigroup in D with Denjoy-Wolff point τ ∈ ∂D and different repelling fixed points σ1 , . . . , σn ∈ ∂D with repelling spectral values λ1 , . . . , λn ∈ (−∞, 0). Let G be its infinitesimal generator. Then there exists 3 ∈ [0, +∞) and β := ∠ lim z→τ τ(τ2−z) G(z) β+

n  τ |τ − σk |2 ≤ 2 Re . |λ | G(0) k k=1

(12.3.9)

12.3 Synchronization Formulas

345

This inequality is sharp. In fact, generator of the semigroup is

n k=1

|τ −σk |2 |λk |

= Re



2τ G(0)

if and only if the infinitesimal

 |τ − σk |2 σk + z 2τ G(z) = 2(z − τ )(τ z − 1) iIm + G(0) k=1 |λk | σk − z n

−1 ,

z ∈ D.

Proof By Theorem 10.1.10, there is a non-zero holomorphic function p : D → H, such that, G(z) = (z − τ )(τ z − 1) p(z), for all z ∈ D. By Proposition 2.1.3, applied to the function 1/ p, β := ∠ lim z→τ

1 − τz (τ − z)3 = ∠ lim 2 ∈ [0, +∞). z→τ τ G(z) p(z)

Take a sequence {r j } in the interval (0, 1) such that lim j→∞ r j = 1 and, for each j, consider the holomorphic function G j (z) = (z − r j τ )(r j τ z − 1) p(z), for all z ∈ D. By Theorem 10.1.10, G j is the infinitesimal generator of a semigroup in the unit disc with Denjoy-Wolff point r j τ . Since σk is a repelling fixed point of (φt ) with spectral value λk , by Theorem 12.2.5, we have ∠ lim

z→σk

G j (z) (z − r j τ )(r j τ z − 1) G(z) = ∠ lim z→σk z − σk (z − τ )(τ z − 1) z − σk |σk − r j τ |2 = −λk . |σk − τ |2

That is, σk is a repelling fixed point of the semigroup generated by G j with spectral value λk

|σk −r j τ |2 . |σk −τ |2

By Corollary 12.3.4, we get that

n  rjτ rjτ τ |τ − σk |2 ≤ 2 Re = 2 Re = 2 Re . |λk | G j (0) r j G(0) G(0) k=1

This proves (12.3.9) if β = 0. If β > 0, then ∠ lim z→τ

(1 − r j )2 G j (z) p(z) = ∠ lim (z − r j τ )(r j τ z − 1) = ∈ (0, +∞). z→τ z−τ z−τ β

These means that τ is another repelling fixed point of the semigroup generated by (1−r )2 G j with spectral value − β j . Therefore, Corollary 12.3.4 implies that β+

n n   |r j τ − τ |2 rjτ |τ − σk |2 |τ − σk |2 τ = + ≤ 2 Re = 2 Re , 2 |λk | (1 − r j ) /β k=1 |λk | G j (0) G(0) k=1

and (12.3.9) holds if β > 0.

346

12 Boundary Fixed Points and Infinitesimal Generators

 2 2τ k| The equality nk=1 |τ −σ = Re G(0) holds if and only if the equality holds in |λk | (12.3.4) for each j and if and only if, for each j, 

 |τ − σk |2 σk + z 2τ + G j (z) = 2(z − r j τ )(r j τ z − 1) iIm G(0) k=1 |λk | σk − z This holds if and only if proof.

1 p(z)

= iIm

n

2τ G(0)

+

n k=1

|τ −σk |2 σk +z , |λk | σk −z

−1 ,

z ∈ D.

z ∈ D, finishing the 

Remark 12.3.7 Following the proof of Corollary 12.3.6, it can be seen that, if β = 0, equality holds in (12.3.9) if and only if the infinitesimal generator of the semigroup is ⎡

⎤−1 n 2 σ +z  1 τ + z |τ − σ | 2τ k k ⎦ , + + G(z) = 2(z − τ )(τ z − 1) ⎣iIm G(0) |λk | σk − z β τ −z

z ∈ D,

k=1

and, in fact, the semigroup is parabolic. Indeed ∠ lim z→τ

G(z) τ 2 G(z) (z − τ )2 1 = −∠ lim = − 0 = 0. z→τ (τ − z)3 z−τ τ2 β

Remark 12.3.8 The right hand side terms of the inequalities (12.3.4), (12.3.7), and (12.3.9) are zero if and only if the Berkson-Porta’s Decomposition Formula of the infinitesimal generator is given by G(z) = (z − τ )(τ z − 1)ai, z ∈ D, for some a ∈ R \ {0}. In particular, the above corollaries show that the associated semigroup has no repelling fixed point. In fact, by Corollary 10.3.2, if τ ∈ D, then G is the infinitesimal generator of an elliptic group while, by Proposition 10.4.4, if τ ∈ ∂D, then G is the infinitesimal generator of a parabolic semigroup of linear fractional maps. An interesting direct consequence of Corollary 12.3.5 [see (12.3.7)] and Corollary 12.3.6 [see (12.3.9)] is the following: Proposition 12.3.9 Let (φt ) be a semigroup in D which is not an elliptic group. Let σ ∈ ∂D be a repelling fixed point of (φt ), with repelling spectral value λ ∈ (−∞, 0). Then there exists an open neighborhood U of σ such that, if p ∈ ∂D ∩ U \ {σ } is a repelling fixed point of (φt ) with repelling spectral value μ ∈ (−∞, 0), then μ ≤ λ. Now we consider separately the elliptic and non-elliptic cases. Corollary 12.3.10 Let (φt ) be a non-trivial elliptic semigroup in D with DenjoyWolff point τ ∈ D and spectral value λ ∈ C \{0} with Re λ < 0. Let σ1 , . . . , σn ∈

12.3 Synchronization Formulas

347

∂D be different repelling fixed points with repelling spectral values λ1 , . . . , λn ∈ (−∞, 0). Then n  1 1 ≤ 2 Re . |λk | λ k=1 This inequality is sharp and equality holds if and only if the infinitesimal generator of the semigroup is 

n  |τ − σk |2 σk + z G(z) = 2(z − τ )(τ z − 1) ai + |λk | σk − z k=1

−1 ,

z ∈ D,

for some a ∈ R. Proof Let G be the infinitesimal generator of the semigroup. We know that −λ = . Moreover, Re λ ≥ 0 and λ = 0. By Theorem 12.3.3, there is a holo∠ lim z→τ G(z) z−τ morphic function q : D → H, with ∠ lim z→σk (z − σk )q(z) = 0, for k = 1, ..., n, such that, for all z ∈ D, 

1  |τ − σk |2 σk + z G(z) = (z − τ )(τ z − 1) q(z) + 2 k=1 |λk | σk − z Therefore

n

−1 , z ∈ D.

G(z) τz − 1 = n |τ −σk |2 σk +z , z ∈ D. 1 z−τ q(z) + 2 k=1 |λk | σk −z

Taking limits in this equality we deduce −λ = Thus

q(τ ) +

|τ |2 − 1 = |τ −σk |2 σk +τ 1 n 2

k=1

|λk |

σk −τ

q(τ ) +

|τ |2 − 1 .  n 1−|τ |2 +2iIm (σ k τ ) 1 2

k=1

|λk |

 1  1 2 2 ≥ . Re = Re q(τ ) + λ 1 − |τ |2 |λk | |λk | k=1 k=1 n

n

Notice that, in particular, the equality holds if and only if Re q(τ ) = 0 if and only if there is constant c ∈ R such that q(z) = ci for all z ∈ D.  Corollary 12.3.11 Let (φt ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D, spectral value λ > 0, and different repelling fixed points σ1 , ..., σn ∈ ∂D with repelling spectral values λ1 , ...., λn ∈ (−∞, 0). Then n  1 1 ≤ . |λ | λ k k=1

348

12 Boundary Fixed Points and Infinitesimal Generators

This inequality is sharp and equality holds if and only if the infinitesimal generator of the semigroup is  G(z) = 2(z − τ )(τ z − 1) 2i

n  Im (τ σk )

|λk |

k=1

n  |τ − σk |2 σk + z + |λk | σk − z k=1

−1 ,

z ∈ D.

Proof Let G be the infinitesimal generator of the semigroup. By Theorem 12.3.3, there is a holomorphic function q : D → H, with ∠ lim z→σk (z − σk )q(z) = 0, for k = 1, ..., n, such that, for all z ∈ D, 

1  |τ − σk |2 σk + z G(z) = (z − τ )(τ z − 1) q(z) + 2 k=1 |λk | σk − z Therefore

n

, z ∈ D.

G(z) τz − 1 , z ∈ D. = 2 1 n k | σk +z z−τ q(z) + 2 k=1 |τ −σ |λk | σk −z

We know that −λ = ∠ lim z→τ 

G(z) , z−τ

where λ ∈ (0, +∞). Thus

1  |τ − σk |2 σk + z 0 = ∠ lim q(z) + z→τ 2 k=1 |λk | σk − z n



 Im (τ σk ) 1  |τ − σk |2 σk + τ = ∠ lim q(z) − i. z→τ 2 k=1 |λk | σk − τ |λk | k=1 n

= ∠ lim q(z) + z→τ

That is, q(τ ) := ∠ lim z→τ q(z) = n Im (τ σk ) k=1 |λk | . Moreover 1 τz − 1

−1

n

n k=1

Im (τ σk ) i |λk |

∈ iR. To simplify, we write β :=

 n 1  |τ − σk |2 σk + z + βi 2 k=1 |λk | σk − z  n  n 1 1  |τ − σk |2 σk + z 1  |τ − σk |2 σk + τ = − τ z − 1 2 k=1 |λk | σk − z 2 k=1 |λk | σk − τ  n   1  |τ − σk |2 σk + z σk + τ 1 − = τ z − 1 2 k=1 |λk | σk − z σk − τ  n  n  1  (σk − τ )(σ k − τ ) 2σk (z − τ ) τ 1 . = =− z − τ 2 k=1 |λk | (σk − z)(σk − τ ) |λk | k=1



12.3 Synchronization Formulas

Since

349

q(z) − iβ 1 z−τ = + G(z) τz − 1 τz − 1



 n 1  |τ − σk |2 σk + z + βi , 2 k=1 |λk | σk − z

we deduce that there exists A := ∠ lim z→τ q(z)−iβ and taking limits in the above τ z−1 equality we deduce n  1 1 − = A− . λ |λ k| k=1  τ) ≤ 0. Thus, λ1 ≥ nk=1 |λ1k | and the equality In fact, A = Re A = limr →1 Rer q(r −1 holds if and only if A = 0. On the one hand, if q − iβ is the null function then A = 0. On the other hand, if q − iβ is not the null function, we can apply Proposi1 to deduce that tion 2.1.3 to the holomorphic function D  z → q(z)−iβ 1 τz − 1 = ∠ lim ∈ [0, +∞) z→τ q(z) − iβ A and thus A = 0. Therefore, A = 0 if and only if q(z) = iβ for all z ∈ D.



As a last application of the decomposition formula (12.2.7) we state a rigidity theorem at boundary regular fixed points. Theorem 12.3.12 Let G be an infinitesimal generator of a semigroup (φt ) in D and σ ∈ ∂D a boundary regular critical point of the semigroup with φt (σ ) = e−λt , for all t ≥ 0, for some real number λ. Assume that there exists a sequence {z n } converging to σ non-tangentially such that lim

n→∞

G(z n ) + λ(z n − σ ) + 21 λσ (z n − σ )2 = 0. (z n − σ )3

(12.3.10)

λ σ (σ 2 − z 2 ), 2

(12.3.11)

Then G(z) =

z ∈ D.

In particular, if λ = 0, G is the infinitesimal generator of an hyperbolic group with fixed points σ and −σ . Proof By Corollary 12.2.6, there exists p : D → H such that

λσ +z , G(z) = (z − σ )(σ z − 1) p(z) + 2σ −z

z ∈ D.

350

12 Boundary Fixed Points and Infinitesimal Generators

Thus

1 σ 1 λσ +z G(z) + λ(z − σ ) + λσ (z − σ )2 = σ (z − σ )2 p(z) + +λ + λ 2 2σ −z z−σ 2 = σ (z − σ )2 p(z).

That is,

G(z) + λ(z − σ ) + 21 λσ (z − σ )2 p(z) , z ∈ D. =σ (z − σ )3 z−σ

Assume that p is not identically zero. By Proposition 2.1.3, applied to the function with non-negative real part 1/ p, we have ∠ lim

z→σ

1 − σz (z − σ )3 ∈ [0, +∞). = −σ 2 ∠ lim z→σ G(z) + λ(z − σ ) + 1 λσ (z − σ )2 p(z) 2

This fact clearly contradicts (12.3.10) and, thus, p must be zero. Hence

λ λσ +z = σ (σ 2 − z 2 ), G(z) = (z − σ )(σ z − 1) 0 + 2σ −z 2

z ∈ D.

Finally, Corollary 10.3.2 shows that if λ = 0, then G is the infinitesimal generator of a hyperbolic group with fixed points σ and −σ (notice that σ is the Denjoy-Wolff point if and only if λ < 0, otherwise the Denjoy-Wolff point is −σ ).  Remark 12.3.13 The hypotheses of above theorem are satisfied if G has second and third derivatives at σ with values −λσ and 0, respectively. As a byproduct we can get the following Burns-Krantz’s Rigidity Theorem for holomorphic self-maps of the unit disc. Corollary 12.3.14 (Burns-Krantz’s Rigidity Theorem) Let φ : D → D be a holomorphic function and σ ∈ ∂D. Assume that there exists a sequence {z n } that converges non-tangentially to σ and such that φ(z n ) − z n = 0, n→∞ (z n − σ )3 lim

then φ = idD . Proof By Corollary 10.2.11, the function G := φ − idD belongs to Gen(D). Our hypothesis implies that φ has a boundary fixed point at σ with angular derivative 1. Therefore, G has a boundary regular fixed point at σ with spectral value λ = 0. The result follows at once using Theorem 12.3.12. 

12.4 Non-Regular Critical Points Versus Super-Repelling Fixed Points

351

12.4 Non-Regular Critical Points Versus Super-Repelling Fixed Points As shown in Theorem 12.2.5, there is a one-to-one correspondence between regular critical points of an infinitesimal generator and boundary regular fixed points of the associated semigroup. In this section, we present two examples which show that, in general, there is no such a connection between (non-regular) critical points and superrepelling fixed points. However, as we will see in Corollary 14.4.3, super-repelling fixed points which have no backward orbits landing on them, are also (non-regular) critical points. Example 12.4.1 There exists a semigroup with no boundary fixed point and such that its infinitesimal generator has a boundary (non-regular) critical point. Indeed, √ √ 1 − z, z ∈ D, where w = consider the self-map of the unit disc φ(z) = 1 − 1 exp 2 log(w) for all w ∈ H and log is the principal branch of the logarithm. By √ Corollary 10.2.11, the function G(z) = 1 − z − 1 − z belongs to Gen(D). For t ≥ 0, define φt : D → D by

2 √ φt (z) = 1 − 1 − e−t/2 + e−t/2 1 − z . It is clear that φt is holomorphic for all t ≥ 0 and that φ0 (z) = z for all z ∈ D. Moreover, ∂φt (z) = G(φt (z)). ∂t √ √ On the other hand, 1 − φs (z) = 1 − e−s/2 + e−s/2 1 − z and

2  φt (φs (z)) = 1 − 1 − e−t/2 + e−t/2 1 − φs (z)

2 √ = 1 − 1 − e−t/2 + e−t/2 (1 − e−s/2 + e−s/2 1 − z)

2 √ = 1 − 1 − e−(t+s)s/2 + e−(t+s)/2 1 − z) = φt+s (z), z ∈ D, t, s ≥ 0.

Thus (φt ) is a semigroup in D with infinitesimal generator G. Its Denjoy-Wolff point is 0 and it has no boundary fixed points. However, the infinitesimal generator satisfies = ∞. ∠ lim z→1 G(z) = 0 and ∠ lim z→1 G(z) z−1 To build the next example we have to introduce some notation and a preliminary lemma. Given n ∈ Z and α ∈ R, denote L[n, α] := {w ∈ C : Re w ≤ n, Im w = α}. Let I denote the set of sequences α = {αn } in [0, 1) of one the following three types: (i) αn = 0 for all n; (ii) there exists n such that α1 > α2 > ... > αn > 0 and αm = 0 for all m > n; (iii) αn > αn+1 for all n and limn→∞ αn = 0.

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12 Boundary Fixed Points and Infinitesimal Generators

Likewise, we denote by D the family of domains associated to I as follows (i) Ωα = {w ∈ C : |Im w| < 1} if α = {0}; (ii) if there exists n such that α1 > α2 > ... > αn > αm = 0 for all m > n, then Ωα = {w ∈ C : |Im w| < 1} \

n  

L[− j, α j ] ∪ L[− j, −α j ] ;

j=1

(iii) if αn > αn+1 for all n, then Ωα = {w ∈ C : |Im w| < 1} \

∞  

L[− j, α j ] ∪ L[− j, −α j ] .

j=1

Finally, if α ∈ I , let h α stand for the Riemann mapping of D onto Ωα such that h α (0) = 0 and h α (0) > 0. Using the symmetry of the domain Ωα , we see that the function g(z) := h α (z) is another Riemann map of Ωα with g(0) = 0 and g (0) > 0. Thus g = h α and h α ((−1, 1)) = R. Moreover, since h α (0) > 0, we have −1 −1

limRu→+∞ h −1 α (u) = 1, lim Ru→−∞ h α (u) = −1 and (h α ) (u) > 0 for all u ∈ R. In the following lemma and example, we will make use of this notation and properties without making any explicit mention to them. Lemma 12.4.2 Fix n ∈ N and 0 < αn < αn−1 < ... < α1 < 1. Then there is δ = δ(α1 , ..., αn ) ∈ (0, αn ) such that if 0 < αn+1 < δ and β = {βm } ∈ I is such that βm = αm for all m ≤ n + 1 and βm = 0 for all m > n + 1, then there exists a point

u n ∈ (−(n + 1), −n) such that (h −1 β ) (u n ) > n. Proof Assume the result is not true. Then there are αm such that αm > αm+1 for all m ≥ n, with limm→∞ αm = 0, and a sequence {βm } in I with the n + 1 first terms  

  ≤n given by α1 , ..., αn , αm and βm = 0 for all m > n + 1, such that  h −1 (u) βm  for all u ∈ (−(n + 1), −n). Denote by Ω0 the limit of {Ωβm } with respect to 0 in the sense of kernel convergence. Then ⎡ Ω0 = {w ∈ C : |Im w| < 1} \ ⎣

n  



⎤

L[− j, α j ] ∪ L[− j, −α j ] ∪ L[−(n + 1), 0]⎦ .

j=1

Let us denote by h the univalent map from D onto Ω0 such that h(0) = 0 and h (0) > 0. Arguing as above, we can prove that h((−1, 1)) = (−(n + 1), +∞). Moreover, since h (0) > 0, we have that (h −1 ) (u) > 0 for all u > −(n + 1), limRu→+∞ h −1 (u) = 1 and limRu→−(n+1)+ h −1 (u) = −1. We claim that lim

u→−(n+1)+

(h −1 ) (u) = +∞.

(12.4.1)

12.4 Non-Regular Critical Points Versus Super-Repelling Fixed Points

353

−1 Assuming the claim for the moment, by Theorem 3.5.8, {h −1 βm } converges to h on compact subsets of Λ := {w ∈ C : |Im w| < αn , Re w > −(n + 1)}. Therefore,

to (h −1 ) on compact subsets of Λ. But then (12.4.1) contradicts {(h −1 βm ) } converges   

  (u) ≤ n for all u ∈ (−(n + 1), −n) and m > n. that  h −1 βm Proof of the claim (12.4.1). Consider the domain

Ω = {w : |Im w| < αn } \ L[−(n + 1), 0]. is univalent and maps D onto C \ L[−1/4, 0], 1/2  z 5 − 1 − n is a Riemann the map g : D → C given by g(z) = αn π2 log (1−z) 2 + 4 map of Ω that sends the segment (−1, 1) onto the half-line (−(n + 1), +∞) and Since the Koebe map D  z →

αn g (x) = π



z (1−z)2

x 5 + (1 − x)2 4

−1

x +1 > 0, x ∈ (−1, 1). (1 − x)3

Note that limr →−1+ g(rr)+n+1 = g (−1) = 0. +1 −1 The function φ = h ◦ g sends the unit disc into itself, φ((−1, 1)) = (−1, 1) and limr →−1+ φ(r ) = −1. By Lemma 1.4.5, αφ (−1) > 0. On the one hand, if αφ (−1) < +∞ then, by Julia-Wolff-Carathéodory’s Theorem (Theorem 1.7.3), we have lim x→−1 φ(x)+1 = αφ (−1). On the other hand, by the very definition, if x+1 αφ (−1) = +∞, then lim x→−1 1−|φ(x)| = lim x→−1 1+φ(x) = ∞. Therefore, in any 1−|x| 1+x case, h(x) + n + 1 h(φ(x)) + n + 1 = lim h (−1) = lim x→−1 x→−1 x +1 φ(x) + 1 1 1 + x g(x) + n + 1 = g (−1) = 0. = lim x→−1 φ(x) + 1 x +1 αφ (−1) Finally, since (h −1 ) (h(x))h (x) = 1 for all x ∈ (−1, 1), we have (12.4.1).



Example 12.4.3 There exists a semigroup in D with a boundary (super-repelling) fixed point which is not a boundary critical point for its infinitesimal generator. In order to construct such an example, take α1 := 1/2 and αn := δ(α1 , ..., αn−1 )/2 where δ is the function defined in the statement of Lemma 12.4.2. Set Ω = {w : |Im w| < 1} \

∞ 

(L[−n, αn ] ∪ L[−n, −αn ])

n=1

and let h be the conformal mapping of D onto Ω such that h(0) = 0 and h (0) > 0.

By Lemma 12.4.2, there exist points u n ∈ (−(n + 1), −n) such that h −1 (u n ) > n, for all n. Using the symmetry of the domain, we have that the function g(z) := h(z) is another Riemann map of Ω with g(0) = 0 and g (0) > 0. Thus g = h and h((−1, 1)) = R. Moreover, since h (0) > 0, we have that limRu→+∞ h −1 (u) = 1

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12 Boundary Fixed Points and Infinitesimal Generators

and limRu→−∞ h −1 (u) = −1. Consider the function φt (z) := h −1 (h(z) + t), for z ∈ D and t ≥ 0. It is clear that (φt ) is a semigroup in D which satisfies limr →1 φt (−r ) = limr →1 h −1 (h(−r ) + t) = −1 and limt→+∞ φt (0) = limt→+∞ h −1 (t) = 1. Thus τ = 1 is the Denjoy-Wolff point of the semigroup and σ = −1 is a boundary fixed point. Its infinitesimal generator G is given by G(z) = h 1(z) . Writing vn := h −1 (u n ) we have limn→∞ vn = −1 and G(vn ) = h (v1 n ) = (h −1 ) (u n ). Thus limn→∞ G(vn ) = ∞ and ∠ lim z→−1 G(z) = 0. That is, −1 is not a critical point of the infinitesimal generator G.

12.5 Notes The main result in Section 12.1 states that a boundary fixed point of an iterate of a semigroup which is not an elliptic group is a boundary fixed point of the semigroup (Theorem 12.1.4). This result appeared, with a different proof, for the first time in [56] and [51]. The connection between boundary fixed points of a semigroup and boundary critical points of its infinitesimal generator has been a very active research topic in recent decades. There is an important number of papers addressing this relationship. For listing some of them we mentioned [25, 37, 52, 68, 77–79, 119]. Some proofs of Sects. 12.2 and 12.3 are inspired from [48, 52, 78, 79, 119] and Example 12.4.3 is taken from [37]. Theorem 12.3.12 and Corollary 12.3.14 are examples of the so-called BurnsKrantz’ rigidity theorems (see [10]). Theorem 12.3.12 was proved in [39].

Chapter 13

Fixed Points, Backward Invariant Sets and Petals

In this chapter, we study the behavior at the boundary of semigroups from a dynamical point of view. Given a semigroup (φt ) in D and a point z ∈ D, one can follow the “backward” trajectory up to a boundary point. The union of the backward trajectory and the forward trajectory of z is a maximal invariant curve for the semigroup. In case the backward trajectory is defined for all negative times, it is called a backward orbit. In the first section we study such backward orbits and prove that every repelling fixed point admits a special regular backward orbit landing at such a point. Using this special backward orbits, we construct pre-models at repelling fixed points. Next, we examine the backward invariant set of (φt ), namely, the set of points which admit a maximal invariant curve defined for all negative times. We prove that this set is formed by petals, which are the images of pre-models, and “isolated” backward orbits. We show that the boundary of a petal is locally connected (and, in fact, except a very special case, a Jordan curve), always contains the Denjoy-Wolff point and can contain at most another fixed point of the semigroup which has to be repelling. Next, using these tools, we characterize petals and repelling fixed points via the geometry of the image of the Koenigs function. This allows also to completely describe the analytic behavior of the Koenigs function at boundary fixed points. Explicit examples are provided in the last section of the chapter when, the theory being previously sufficiently developed, we will be able to get precise information on the dynamics near fixed points just looking at the picture of a domain which is either spirallike or starlike at infinity.

13.1 Backward Orbits Definition 13.1.1 Let (φt ) be a semigroup in D. A continuous curve γ : [0, +∞) → D is called a backward orbit if for every t ∈ (0, +∞) and for every 0 ≤ s ≤ t, φs (γ (t)) = γ (t − s). © Springer Nature Switzerland AG 2020 F. Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4_13

355

356

13 Fixed Points, Backward Invariant Sets and Petals

A backward orbit γ is said to be a regular backward orbit if V (γ ) := lim sup ω(γ (t), γ (t + 1)) < +∞. t→+∞

We call V (γ ) the hyperbolic step of γ . Remark 13.1.2 Let (φt ) be a semigroup in D and let γ : [0, +∞) → D be a backward orbit for (φt ). Let (Ω, h, ψt ) be the canonical model of (φt ) given by Theorem 9.3.5. For all t ≥ 0, h(γ (t)) ∈ h(D), hence, ψt (h(γ (t))) = h(φt (γ (t))) = h(γ (0)) and h(γ (t)) = ψ−t (h(γ (0))), for all t ≥ 0. In particular, if (φt ) is elliptic, not a group, and λ ∈ C, Re λ > 0 is its spectral value, then h(γ ([0, +∞))) = spir λ [h(γ (0))] ∩ {w ∈ C : |w| ≥ |h(γ (0))|}. While, if (φt ) is non-elliptic, h(γ ([0, +∞))) = L[h(γ (0))] ∩ {w ∈ C : Im w ≤ Im h(γ (0))}. We start considering the case of groups: Lemma 13.1.3 Let (φt ) be a non-trivial group in D and let τ ∈ D be the DenjoyWolff point of (φt ). Let γ : [0, +∞) → D be a backward orbit. Then γ is regular. Moreover, (1) if (φt ) is elliptic, then either γ (t) ≡ τ or the image of γ is the boundary of a hyperbolic disc centered at τ . (2) If (φt ) is hyperbolic and σ ∈ ∂D \ {τ } is the other fixed point of (φt ), then limt→+∞ γ (t) = σ and there exists α ∈ (−π/2, π/2) such that limt→+∞ Arg(1 − σ γ (t)) = α. (3) If (φt ) is parabolic, then limt→+∞ γ (t) = τ and the image of γ is contained in ∂ E(τ, R) for some R > 0. In particular, γ (t) converges to τ tangentially. Proof Let z 0 := γ (0). Then, φt (γ (t)) = γ (0) = z 0 , hence γ (t) = φ−t (z 0 ) for all t ≥ 0. In particular, ω(γ (t), γ (t + 1)) = ω(φ−t (z 0 ), φ−t (φ−1 (z 0 ))) = ω(z 0 , φ−1 (z 0 )) < +∞. Therefore the backward orbit γ is regular. In order to prove (1)—(3), we make the following simple observation. Let T : D → C be the Möbius transformation given by Proposition 8.3.8. Let ψt := T ◦ φt ◦ T −1 . Then T (γ ) is a backward orbit for (ψt ), i.e., ψs (T (γ (t))) = T (γ (t − s)) for 0 ≤ s ≤ t. If (φt ) is an elliptic group, then ψt (z) = e−iθt z, for some θ ∈ R \ {0}. Since T is an isometry for the hyperbolic distance, (1) follows at once.

13.1 Backward Orbits

357

If (φt ) is hyperbolic and σ ∈ ∂D \ {τ } is the other fixed point of (φt ), then ψt (z) = eλt z for some λ > 0, and T (τ ) = ∞, T (σ ) = 0. Hence γ (t) = φ−t (z 0 ) = T −1 (ψ−t (T (z 0 ))) = T −1 (e−λt T (z 0 )), from which it follows at once that limt→+∞ Arg(1 − σ γ (t)) = α for some α ∈ (−π/2, π/2). Finally, if (φt ) is a parabolic group then, ψt (z) = z + it, and the statement follows at once.  Remark 13.1.4 Let (φt ) be a hyperbolic group with Denjoy-Wolff point τ ∈ ∂D and other fixed point σ ∈ ∂D \ {τ }. Then for every α ∈ (−π/2, π/2) there exists z 0 ∈ D such that [0, +∞) t → φ−t (z 0 ) is a regular backward orbit which converges to σ and such that limt→+∞ Arg(1 − σ φ−t (z 0 )) = α. Indeed, let C : D → H be the Cayley transform such that C(τ ) = ∞ and C(σ ) = 0. Hence C(φt (C −1 (w))) = eλt w, where λ > 0 is the spectral value of (φt ). It follows that if w0 ∈ H is such that Arg(w0 ) = α, the curve [0, +∞) t → e−λt w0 converges to 0 forming an angle of α with the real axis. Let z 0 := C −1 (w0 ). Hence, φ−t (z 0 ) = C −1 (e−λt w0 ) forms an angle α with the segment C −1 ([0, +∞)) at σ . Since C −1 ([0, +∞)) is tangent to the segment [0, σ ], the result follows. Now, we examine the case of semigroups which are not groups. We start with two lemmas: Lemma 13.1.5 Let (φt ) be a semigroup in D which is not a group. Let τ ∈ D be its Denjoy-Wolff point. Let γ : [0, +∞) → D be a backward orbit for (φt ). Then, • either τ ∈ D and γ (t) ≡ τ for all t ≥ 0, • or there exists σ ∈ ∂D (possibly σ = τ ) such that σ is a fixed point of (φt ) and limt→+∞ γ (t) = σ . Proof Let (Ω, h, ψt ) be the canonical model for (φt ), where Ω is either C, H, H− or the strip Sρ for some ρ > 0 and either ψt (w) = w + it or ψt (w) = e−λt w for some λ ∈ C with Re λ > 0. Let w0 := h(γ (0)). Let t ≥ 0. By Remark 13.1.2, h(γ (t)) = ψ−t (w0 ) for all t ≥ 0. In particular, by the form of ψt , it follows that either ψ−t (w0 ) ≡ w0 = 0 (and hence γ (t) ≡ τ ∈ D) or limt→+∞ ψ−t (w0 ) = ∞ in C∞ . Therefore, by Proposition 3.3.3, there exists σ ∈ ∂D such that lim γ (t) = lim h −1 (h(γ (t)) = lim h −1 (ψ−t (w0 )) = σ.

t→+∞

t→+∞

t→+∞

Now, let s > 0. Then lim φs (γ (t)) = lim γ (t − s) = σ.

t→+∞

t→+∞

Therefore, by Theorem 1.5.7, ∠ lim z→σ φs (z) = σ for all s ≥ 0, hence σ is a bound ary fixed point of (φt ).

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13 Fixed Points, Backward Invariant Sets and Petals

Lemma 13.1.6 Let (φt ) be a semigroup in D which is not a group. Let τ ∈ D be its Denjoy-Wolff point. Let γ : [0, +∞) → D be a regular backward orbit for (φt ) converging to a point σ ∈ ∂D \ {τ }. If σ is a regular fixed point of (φt ), then γ converges to σ non-tangentially. Proof We claim that there exists a Stolz region S(σ, M ), M > 1, such that γ ([0, +∞)) ⊂ S(σ, M ). That is, γ (t) converges non-tangentially to σ . Indeed, let z n := γ (n), n ∈ N0 . Claim: {z n } converges to σ non-tangentially. Assume the claim is true for the moment. In particular, there exists M > 1 such that {z n } ⊂ S(σ, M) for all n ∈ N0 . By Proposition 1.5.5, there exists M > 1 such that for all s ∈ [0, 1] it holds φs (S(σ, M)) ⊂ S(σ, M ).

(13.1.1)

Now, let t ≥ 0. We can write t = n − s, with n ∈ N and s ∈ (0, 1]. Hence, γ (t) = γ (n − s) = φs (γ (n)) = φs (z n ). Therefore γ (t) ∈ S(σ, M ) by (13.1.1). We are left to prove the Claim. In order to simplify notations, set φ = φ1 . Since σ is a repelling fixed point of φ and {z n } converges to σ , lim inf n→∞

1 − |z n | 1 − |φ(z n+1 )| = lim inf ≥ φ (σ ) > 1. n→∞ 1 − |z n+1 | 1 − |z n+1 |

Therefore, we can find δ1 > 0 and N ∈ N such that, for all n ≥ N , 1 − |z n | ≥ (1 + δ1 )(1 − |z n+1 |). This inequality also implies that |z n+1 | > |z n |, for all n ≥ N ; in particular z n = z n+1 . Since supn∈N ω(z n , z n+1 ) < +∞, there exists δ2 ∈ (0, 1) such that for every n ∈ N,    z n − z n+1    (13.1.2)  1 − z z  ≤ δ2 n n+1 This means that z n+1 belongs to the closed Euclidean disc Tz−1 n (D(0, δ2 )) = z n −z Tzn (D(0, δ2 )), where Tzn is the automorphism given by Tzn (z) = 1−z , z ∈ D, and nz D(0, δ2 ) denotes the Euclidean disc centered at 0 and radius δ2 . Given θ ∈ R, |z n |2 + δ22 − 2δ2 Re (z n e−iθ ) 1 + |z n |2 δ22 − 2δ2 Re (z n e−iθ )   |z n | + δ2 2 |z n |2 + δ22 + 2δ2 |z n | = , ≤ 1 + δ2 |z n | 1 + |z n |2 δ22 + 2δ2 |z n |

|Tzn (δ2 eiθ )|2 =

13.1 Backward Orbits

359

where the inequality follows from the fact that the function x → decreasing in the interval [−|z n |, |z n |]. Therefore |z n+1 | ≤ Hence 1 − |z n+1 | ≥ (1 − δ2 )

|z n |2 +δ22 −2δ2 x 1+|z n |2 δ22 −2δ2 x

is

|z n | + δ2 . 1 + δ2 |z n |

1 − |z n | 1 − δ2 (1 − |z n |). ≥ 1 + δ2 |z n | 1 + δ2

Moreover, by (13.1.2),    z n − z n+1 2 (1 − |z n |2 )(1 − |z n+1 |2 )  ≥ 1 − δ2 .  =1− 2 |1 − z n z n+1 |2 1 − z n z n+1  Then, by the previous inequalities and (13.1.2), for every n ≥ N , we have  (|z n+1 | − |z n |)2 = (1 − |z n+1 |)2

2 1 − |z n | −1 1 − |z n+1 |

1 − δ2 (1 − |z n |)(1 − |z n+1 |) 1 + δ2 δ 2 1 − δ2 (1 − |z n |2 )(1 − |z n+1 |2 ) |1 − z n z n+1 |2 ≥ 1 |z n − z n+1 |2 4 1 + δ2 |1 − z n z n+1 |2 |z n − z n+1 |2 δ 2 1 − δ2 1 − δ22 ≥ 1 |z n − z n+1 |2 . 4 1 + δ2 δ22 ≥ δ12

Therefore, denoting δ :=

δ1 (1−δ2 ) 2δ2

|z m | − |z n | =

> 0, we find that, for every m > n ≥ N , m−1 

(|z k+1 | − |z k |)

k=n

≥δ

m−1 

|z k+1 − z k | ≥ δ|z m − z n |.

k=n

Taking limits when m goes to ∞, this implies that the whole sequence {z n } is included in a Stolz region S(σ, R) for some R ≥ 1δ thus {z n } converges nontangentially to σ .  Now we are ready to study regular backward orbits: Proposition 13.1.7 Let (φt ) be a semigroup in D which is not a group. Let τ ∈ D be its Denjoy-Wolff point. Let γ : [0, +∞) → D be a regular backward orbit for (φt ). Then,

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13 Fixed Points, Backward Invariant Sets and Petals

(1) either τ ∈ D and γ (t) = τ for all t ∈ [0, +∞), (2) or, there exists a unique point σ ∈ ∂D \ {τ } such that γ (t) converges to σ nontangentially. Moreover, σ is a repelling fixed point of (φt ) with repelling spectral value λ ≥ −2V (γ ), (3) or, τ ∈ ∂D and limt→+∞ γ (t) = τ . Moreover, in this case, the convergence of γ to τ is tangential and (φt ) is parabolic. Proof Suppose γ ≡ τ . Hence, by Lemma 13.1.5 there exists a boundary fixed point σ ∈ ∂D of (φt ) such that limt→+∞ γ (t) = σ . By Lemma 1.4.5 and since γ is regular, 1 log αφ1 (σ ) ≤ lim sup[ω(0, γ (t)) − ω(0, φ1 (γ (t)))] 2 t→+∞ ≤ lim sup ω(γ (t), γ (t − 1)) = V (γ ) < +∞.

(13.1.3)

t→+∞

Hence, σ is a boundary regular fixed point for φ1 . By Proposition 12.1.6, it follows that σ is a boundary regular fixed point for (φt ). If σ is a repelling fixed point and λ ∈ (−∞, 0) is the repelling spectral value of (φt ) at σ , taking into account that αφ1 (σ ) = φ1 (σ ) = e−λ by Theorem 1.7.3, it follows that λ ≥ −2V (γ ) by (13.1.3). If σ = τ , γ (t) converges non-tangentially to σ by Lemma 13.1.6. γ := If σ = τ , then let Cτ : D → H be a Cayley transform which maps τ to ∞, let  Cτ ◦ γ and ψt := Cτ ◦ φt ◦ Cτ−1 . Recall that Cτ (E(τ, R1 )) = E H (∞, R) = {w ∈ H : γ (t) = +∞ and, by Re w > R} for every R > 0 [see (1.4.15)]. Hence, limt→+∞  Theorem 1.8.4, for every t ≥ 0 and every R > 0, ψt (E H (∞, R)) ⊂ E H (∞, R), or, which is the same, (13.1.4) Re ψt (w) > Re w, for all w ∈ H. Moreover, for 0 ≤ s ≤ t, ψs (γ˜ (t)) = Cτ (φs (Cτ−1 (Cτ (γ (t))))) = Cτ (φs (γ (t))) = Cτ (γ (t − s)) = γ˜ (t − s). Hence, (13.1.4) implies γ (t)) > Re  γ (t), Re  γ (t − s) = Re ψs ( that is, t → Re  γ (t) is a decreasing function. Let a ≥ 0 be its limit for t → γ (t) = a. Apply+∞. Hence,  γ ([0, +∞)) ⊂ {w ∈ H : Re w > a} and limt→∞ Re  ing Cτ−1 , it follows that if a > 0 then γ ([0, +∞)) ⊂ E(τ, a1 ) and γ converges to τ tangentially to ∂ E(τ, a1 ) —hence tangentially to ∂D. In case a = 0, γ converges to τ tangentially to ∂D. Finally, assume by contradiction that (φt ) is hyperbolic. Let (Sρ , h, z → z + it) be the canonical model of (φt ), for some ρ > 0. Let w0 := h(γ (0)). Hence, for every t ≥0 h(γ (t)) + it = h(φt (γ (t))) = h(γ (0)) = w0 .

13.1 Backward Orbits

361

Therefore, h(γ (t)) = w0 − it for all t ≥ 0. Let η(t) := w0 + it, t ∈ R. Since h is starlike at infinity with respect to τ it follows that η(−∞, +∞) ⊂ h(D). Let C0 be the closure of the connected component of h(D) ∩ {w : Im w = 0} which intersects η((−∞, +∞)). Then for n ∈ N let Cn+ be the closure of the connected component of h(D) ∩ {w : Im w = n} which intersects η((−∞, +∞)) and let Cn− be the closure of the connected component of h(D) ∩ {w : Im w = −n} which intersects η((−∞, +∞)). It is clear that (Cn+ ) and (Cn− ) defines two non equivalent null chains for h(D). Moreover, η(t) converges in the Carathéodory topology for t → +∞ to the prime end defined by (Cn+ ), while η(t) converges in the Carathéodory topology for t → −∞ to the prime end defined by (Cn+ ). The map h extends to a homeomorˆ → h(D) ˆ  in the Carathéodory topology by Theorem 4.2.3. Moreover, D phism hˆ : D is homeomorphic to D by Proposition 4.2.5. Therefore, taking into account that lim h −1 (η(t)) = lim h −1 (h(γ (−t))) = lim γ (−t) = τ,

t→−∞

t→−∞

t→−∞

we have limt→+∞ h −1 (η(t)) = p ∈ ∂D \ {τ }. Hence, lim sup Im h(z) ≥ lim Im h(h −1 (η(t))) = +∞. z→ p

t→+∞

But, by Proposition 11.1.8, this forces τ = p, a contradiction. Therefore, (φt ) is parabolic, and the proof is concluded.  The previous proposition allows us to give the following definition: Definition 13.1.8 Let (φt ) be a non-trivial semigroup in D with Denjoy-Wolff point τ ∈ D. Let γ : [0, +∞) → D be a non-constant regular backward orbit for (φt ). We say that γ is exceptional if limt→+∞ γ (t) = τ ∈ ∂D and we say that γ is nonexceptional if limt→+∞ γ (t) = τ . We saw in Proposition 13.1.7 that every non-exceptional regular backward orbit lands at a repelling fixed point, and that the hyperbolic step of the orbit controls the repelling spectral value of the semigroup at the fixed point. Now we prove the converse: Proposition 13.1.9 Let (φt ) be a semigroup, not a group, in D. Let σ ∈ ∂D. Assume σ is a repelling fixed point of (φt ) with repelling spectral value λ ∈ (−∞, 0). Then there exists a non-exceptional regular backward orbit γ : [0, +∞) → D for (φt ) such that limt→+∞ γ (t) = σ and with hyperbolic step V (γ ) = − 21 λ. Proof By Proposition 12.3.9, there exists  ∈ (0, 21 ) such that the Euclidean disc D of center σ and radius  has the property that A := D ∩ ∂D does not contain repelling fixed points for (φt ) with repelling spectral value greater than or equal to λ. Without loss of generality, we can also assume that the Denjoy-Wolff point τ ∈ D of (φt ) is not contained in D. Let B := ∂ D ∩ D.

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13 Fixed Points, Backward Invariant Sets and Petals

Let α < 0 be such that E(σ, eλn+α ) ⊂ D for all n ∈ N0 . Let rn ∈ (0, 1) be such / D, that rn σ ∈ ∂ E(σ, eλn+α ). Since limt→+∞ φt (rn σ ) = τ ∈ sn := sup{s ∈ [0, +∞) : φs (rn σ ) ∈ D} < +∞, and φsn (rn σ ) ∈ B. We claim that sn > n. Indeed, by Theorem 1.4.7, taking into account that (φt ) is not a group, for 0 ≤ t ≤ n, φt (rn σ ) ∈ φt (E(σ, eλn+α ) \ {σ }) ⊂ E(σ, eλ(n−t)+α ) ⊂ E(σ, eα ) ⊂ D. We let z n := φsn (rn σ ) ∈ B. We claim that {z n } is relatively compact in D. Otherwise, we can extract a subsequence {z n k } which converges to a point p ∈ B ∩ ∂D. Now, since φ1 (z n k ) = φ1 (φsnk (rn k σ )) = φsnk (φ1 (rn k σ )), we have lim sup ω(z n k , φ1 (z n k )) = lim sup ω(φsnk (rn k σ ), φsnk (φ1 (rn k σ ))) k→∞

k→∞

λ ≤ lim sup ω(rn k σ, φ1 (rn k σ )) = − , 2 k→∞

(13.1.5)

where the last equality follows from Proposition 1.9.12 and the fact that αφ1 (σ ) = e−λ . In particular this implies that limk→∞ φ1 (z n k ) = p by Lemma 1.8.6. Moreover, since by Lemma 1.4.5 1 log αφ1 ( p) ≤ lim sup[ω(0, z n k ) − ω(0, φ1 (z n k ))] ≤ lim sup ω(z n k , φ1 (z n k )), 2 k→∞ k→∞ Theorem 1.4.7 and (13.1.5) imply that p is a boundary regular fixed point for φ1 with αφ1 ( p) ≤ e−λ . Hence, by Theorem 12.1.4 and Proposition 12.1.6, the point p ∈ ∂D is a boundary regular fixed point for (φt ) with repelling spectral value greater than or equal to λ. This contradicts the definition of D. Hence {z n } is relatively compact in D. Therefore, there exists an infinite subset I0 ⊂ N0 such that {z n }n∈I0 converges to a point w0 ∈ D. Since sn > n for all n ∈ N0 , given j ∈ N, φsn − j (rn σ ) is well defined for all n ≥ j. We claim that for every j ∈ N there exists an infinite subset I j of N0 such that I j ⊂ I j−1 and {φsn − j (rn σ )}n∈I j converges to a point w j ∈ D. Indeed, fix j ∈ N and suppose we constructed I j . Arguing as in (13.1.5), we see that lim ω(φsn −( j+1) (rn σ ), φsn − j (rn σ )) ≤

I j n→∞

λ lim ω(rn σ, φ1 (rn σ )) = − . 2

I j n→∞

Hence, by the same argument as for {z n k }, it follows that {φsn −( j+1) (rn σ )}n∈I j is relatively compact in D and thus we can extract an infinite subset I j+1 from I j such that {φsn −( j+1) (rn σ )}n∈I j+1 converges to a point w j+1 ∈ D.

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363

We have φ1 (w j+1 ) =

lim

I j+1 n→∞

φ1 (φsn −( j+1) (rn σ )) =

lim

I j+1 n→∞

φsn − j (rn σ ) = w j

(13.1.6)

for every j ∈ N. Also, by the definition of sn , we have φsn −( j+1) (rn σ ) ∈ D for all j ∈ I j , and for all j ∈ N. Hence, w j ∈ D for all j ∈ N0 . For every t > 0, let jt ∈ N0 be such that jt < t ≤ jt + 1. Now, let γ (0) = w0 and define, for every t > 0 γ (t) = φ jt +1−t (w jt +1 ). If 0 ≤ s < t then jt−s ≤ jt and φs (γ (t)) = φs+ jt +1−t (w jt +1 ) = φ jt−s +1−(t−s) (φ jt − jt−s (w jt +1 )) ◦( jt − jt−s )

= φ jt−s +1−(t−s) (φ1

(w jt +1 )) = φ jt−s +1−(t−s) (w jt−s +1 ) = γ (t − s).

Hence, γ is a backward orbit for (φt ). Moreover, by (13.1.6), wj =

lim φsn − j (rn σ ) =

I j n→∞

lim

I j+1 n→∞

φsn − j (rn σ ).

Therefore, as before, ω(w j , w j+1 ) = ≤

lim

ω(φsn − j (rn σ ), φsn −( j+1) (rn σ ))

lim

λ ω(rn σ, φ1 (rn σ )) = − . 2

I j+1 n→∞

I j+1 n→∞

Hence, λ ω(γ (t), γ (t + 1)) = ω(φ jt +1−t (w jt +1 ), φ jt +2−(t+1) (w jt +2 )) ≤ ω(w jt +1 , w jt +2 ) ≤ − , 2

which proves that V (γ ) ≤ − λ2 . Since τ ∈ / D and γ ( j) = w j ∈ D for all j ∈ N, by Proposition 13.1.7, γ is nonexceptional and converges to a repelling fixed point p ∈ D ∩ ∂D with repelling spectral value μ ≥ −2V (γ ) ≥ λ. But, by definition of D, the only repelling fixed point of (φt ) in D ∩ ∂D with repelling spectral value greater than or equal to λ is σ itself. Hence, p = σ and μ = λ. This also proves that λ = −2V (γ ).  The previous results show that that for every repelling fixed point of a semigroup, there exists a regular backward orbit converging to such a point. Moreover, every backward orbit lands at a boundary fixed point of the semigroup. One might wonder whether, to close the circle, for every super-repelling fixed point there exists a backward orbit converging to it. In fact, this is not the case, as we will see later in the next chapter.

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13 Fixed Points, Backward Invariant Sets and Petals

13.2 Pre-Models at Repelling Fixed Points In order to properly introduce the concept of pre-model, we need first to introduce the notion of semi-conformality. Definition 13.2.1 Let g : D → D be holomorphic. Let σ ∈ ∂D and assume that ∠ lim z→σ g(z) = η ∈ ∂D. We say that g is semi-conformal at σ if ∠ lim Arg z→σ

1 − ηg(z) = 0. 1 − σz

Remark 13.2.2 Let g : D → D be holomorphic. Let σ ∈ ∂D. If αg (σ ) < +∞, Theorem 1.7.3 implies at once that g is semi-conformal at σ . Lemma 13.2.3 Let σ, η ∈ ∂D. If g : D → D is univalent, ∠ lim z→σ g(z) = η and g is semi-conformal at σ , then for every  > 0, M > 1 and M > M there exists δ > 0 such that S(η, M) ∩ {ζ ∈ D : |ζ − η| < δ} ⊂ g(S(σ, M ) ∩ {ζ ∈ D : |ζ − σ | < }). Proof Suppose by contradiction that there exist  > 0 and M > M > 1 and a sequence {z n } ⊂ S(η, M) converging to η such that / g(S(σ, M ) ∩ {ζ ∈ D : |ζ − σ | < }) zn ∈ for all n ∈ N. Let 0 < α0 < π/2 be such that S(σ, M) ⊂ {σ (1 − ρeiθ ) : ρ > 0, |θ | < α0 }. Moreover, let ρ0 ∈ (0, 1) and β ∈ (α0 , π/2) be such that the set U := {σ (1 − ρeiθ ) : ρ ∈ (0, ρ0 ), |θ | < β} satisfies U ⊂ S(σ, M ) ∩ {ζ ∈ D : |ζ − σ | < } ∪ {σ }. Up to extracting subsequences, we can assume that α := limn→∞ Arg(1 − σ z n ) exists. Since {z n } ⊂ S(σ, M), it follows that |α| ≤ α0 . Note that U is a Jordan domain whose boundary at σ forms an angle of amplitude 2β symmetric with respect to the segment joining 0 to η. Since g is semi-conformal at σ , it follows that g(U ) is a Jordan domain, whose boundary at σ is formed by two curves which form an angle of amplitude 2β symmetric with respect to the segment joining 0 to σ . Hence, {z n } is eventually contained in g(U ) ⊂ g(S(σ, M ) ∩ {ζ ∈ D : |ζ − σ | < }), a contradiction.  An useful method to check whether a function is semi-conformal is the following: Lemma 13.2.4 Let g : D → D be holomorphic. Let σ ∈ ∂D and assume that ∠ lim z→σ g(z) = σ ∈ ∂D. Then g is semi-conformal at σ if and only if for all z ∈ D,

13.2 Pre-Models at Repelling Fixed Points

365

 lim Arg(1 − σ g(ψt (z))) = Arg

t→∞

1−z 1+z

 ,

(13.2.1)

where (ψt ) is a hyperbolic group in D with Denjoy-Wolff point σ and other fixed point −σ . Proof Up to replace g(z) with σ g(σ z), we can assume σ = 1. By Theorem 8.2.6.(2) (with Denjoy-Wolff point 1 and other fixed point −1), there exists λ > 0 (the spectral value of (ψt )) such that 1 − ψt (z) =

eλt

1−z 2 . λt +1 e − 1 eλt −1 + z

Hence, limt→+∞ Arg(1 − ψt (z)) = Arg 1−z , for all z ∈ D. In particular, if g is semi1+z conformal at 1 then (13.2.1) holds. . Then C is a Cayley transformation mapping D Conversely, let C : z → 1−z 1+z onto H and C(−1) = ∞, C(1) = 0. Moreover, by Proposition 8.3.8, (C ◦ ψt ◦ C −1 )(w) = e−λt w, t ∈ R, w ∈ H, where λ > 0 is the spectral value of (ψt ). Let α ∈ (−π/2, π/2) and let {z n } ⊂ D be a sequence which converges to 1 and such that limn→∞ Arg(1 − z n ) = α. Write C(z n ) = ρn eiθn , where ρn > 0 and θn ∈ (−π/2, π/2). Hence, limn→∞ ρn = 0, and limn→∞ θn = α. Let tn ∈ R be such that ρn = e−λtn , n ∈ N. Therefore, C(ψtn (C −1 (eiα ))) = e−λtn eiα . Since ρn → 0 as n → ∞, we have tn → +∞ as n → ∞. Moreover, by Lemma 5.4.1(4), ω(z n , ψtn (C −1 (eiα ))) = kH (C(z n ), C(ψtn (C −1 (eiα )))) = kH (ρn eiθn , e−λtn eiα ) = kH (ρn eiθn , ρn eiα ) = kH (eiθn , eiα ). Therefore, limn→∞ ω(z n , ψtn (C −1 (eiα ))) = 0. Since, lim ω(g(z n ), g(ψtn (C −1 (eiα )))) ≤ lim ω(z n , ψtn (C −1 (eiα ))) = 0,

n→∞

n→∞

it follows by Lemma 1.8.6 that lim Arg(1 − g(z n )) = lim Arg(1 − g(ψtn (C −1 (eiα )))) = Arg

n→∞

n→∞

and g is semi-conformal.

1 − C −1 (eiα ) = α, 1 + C −1 (eiα ) 

Now we can define the very important notion of pre-model: Definition 13.2.5 Let (φt ) be a semigroup in D which is not a group. Let σ ∈ ∂D be a repelling fixed point for (φt ) with repelling spectral value λ ∈ (−∞, 0). We say that the triple (D, g, ηt ) is a pre-model for (φt ) at σ if

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13 Fixed Points, Backward Invariant Sets and Petals

(1) (ηt ) is the unique hyperbolic group with Denjoy-Wolff point −σ , other fixed point σ and spectral value −λ, (2) g : D → D is univalent, ∠ lim z→σ g(z) = σ , and g is semi-conformal at σ , i.e., ∠ lim Arg z→σ

1 − σ g(z) = 0, 1 − σz

(3) g ◦ ηt = φt ◦ g for all t ≥ 0. Remark 13.2.6 If (D, g, ηt ) is a pre-model for a semigroup (φt ) at a repelling fixed point σ ∈ ∂D with repelling spectral value λ ∈ (−∞, 0), it follows that the repelling spectral value of (ηt ) at σ is λ. Indeed, from a direct computation from Theorem 8.2.6.(2), we have ηt (σ ) = e−λt for all t ≥ 0. The next result shows that pre-models always exist at repelling fixed points. They are also (essentially) unique, as we will see in Corollary 13.4.13. Theorem 13.2.7 Let (φt ) be a semigroup in D which is not a group. Let σ ∈ ∂D be a repelling fixed point for (φt ) with repelling spectral value λ ∈ (−∞, 0). Then there exists a pre-model (D, g, ηt ) for (φt ) at σ . Proof In order to simplify notations, up to conjugation with the rotation z → σ z, we can assume σ = 1. Let λ ∈ (−∞, 0) be the repelling spectral value of (φt ) at 1. By Proposition 13.1.9, there exists a regular backward orbit γ : [0, +∞) → D for (φt ) which converges to 1, with hyperbolic step V (γ ) = − λ2 . For t ≥ 0, let z − γ (t) . Tt (z) := 1 − γ (t)z Note that {Tt } is a family of automorphisms of the unit disc, Tt (γ (t)) = 0 and Tt−1 (z) =

z + γ (t) 1 + γ (t)z

.

For t ≥ 0, we also define ηt (z) :=

(1 + e−λt )z + (1 − e−λt ) . (1 − e−λt )z + (1 + e−λt )

By Theorem 8.2.6, (ηt ) is a hyperbolic group in D with Denjoy-Wolff point −1 and other fixed point 1. The spectral value of the group is −λ, and the repelling spectral value of (ηt ) at 1 is λ. We first prove that lim

t→+∞

1 − γ (t) 1 − γ (t)

= 1 and

lim

t→+∞

1 − γ (t) = 1. |1 − γ (t)|

(13.2.2)

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367

Indeed, let {tn } ⊂ [0, +∞) be an increasing sequence converging to +∞ and let wn := γ (tn ), n ∈ N0 . By Proposition 13.1.7, {wn } converges to 1 non-tangentially. Moreover, V (γ ) = − λ2 and Lemma 1.4.5 imply −

λ 1 1 = log e−λ = log αφ1 (1) = lim inf [ω(0, z) − ω(0, φ1 (z))] z→1 2 2 2 ≤ lim inf ω(z, φ1 (z)) ≤ lim sup ω(wn+1 , φ1 (wn+1 )) z→1

n→∞

λ = lim sup ω(γ (tn+1 ), γ (tn+1 − 1)) ≤ lim sup ω(γ (t + 1), γ (t)) = − . 2 n→∞ t→+∞ Therefore,

1 log αφ1 (1). 2

lim ω(φ1 (wn+1 ), wn+1 ) =

n→∞

Hence, by Proposition 1.9.12, limn→∞ Arg(1 − γ (tn )) = 0. Therefore, if we write γ (tn ) = 1 − ρn eiθn with ρn > 0 and limn→1 ρn = 0 and θn ∈ (−π/2, π/2), we have limn→0 θn = 0. Hence, 1 − γ (tn )

lim

n→∞

Also, lim

n→∞

1 − γ (tn )

eiθn = 1. n→∞ e−iθn

= lim

1 − γ (tn ) = lim eiθn = 1. |1 − γ (tn )| n→∞

By the arbitrariness of {tn }, Eq. (13.2.2) holds. Claim A. For every s ≥ 0, the family {Tt+s ◦ Tt−1 } converges uniformly on compacta −1 } converges uniformly on compacta to ηs as t → +∞ and, hence, the family {Tt ◦ Tt+s to η−s as t → +∞. In order to prove Claim A, we first show that for every s ≥ 0, lim

t→+∞

Indeed,

1 − γ (t) = e−λs . 1 − γ (t + s)

(13.2.3)

1 − φs (γ (t + s)) 1 − γ (t) = , 1 − γ (t + s) 1 − γ (t + s)

and, taking into account that γ (t) converges to 1 non-tangentially, (13.2.3) follows at once from Theorem 1.7.3. A direct computation shows that  Tt+s (Tt−1 (z))

=

1−γ (t+s)γ (t) 1−γ (t+s) γ (t)−γ (t+s) 1−γ (t+s)





z+

z+

γ (t)−γ (t+s) 1−γ (t+s)

1−γ (t+s)γ (t) 1−γ (t+s)

.

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13 Fixed Points, Backward Invariant Sets and Petals

Now, by (13.2.2) and (13.2.3) 1 − γ (t) 1 − γ (t) 1 − γ (t + s)γ (t) = 1 + γ (t + s) → 1 + e−λs , 1 − γ (t + s) 1 − γ (t) 1 − γ (t + s) γ (t) − γ (t + s) 1 − γ (t) =1− → 1 − e−λs , 1 − γ (t + s) 1 − γ (t + s) γ (t) − γ (t + s) 1 − γ (t + s) 1 − γ (t) 1 − γ (t) = − → 1 − e−λs , 1 − γ (t + s) 1 − γ (t + s) 1 − γ (t) 1 − γ (t + s) 1 − γ (t + s)γ (t) 1 − γ (t) 1 − γ (t + s) = + γ (t) → e−λs + 1, 1 − γ (t + s) 1 − γ (t + s) 1 − γ (t + s) and hence {Tt+s ◦ Tt−1 } converges uniformly on compacta to ηs . Finally, since Tt+s ◦ −1 = idD for all t ≥ 0 and limt→∞ Tt+s ◦ Tt−1 = ηs , it follows that {Tt ◦ Tt−1 ◦ Tt ◦ Tt+s −1 Tt+s } converges to η−s . Claim B. For every s ≥ 0, the family {Tt+s ◦ ηs−1 ◦ Tt−1 } converges uniformly on compacta to idD as t → +∞. Indeed, a direct computation shows (Tt ◦ η−s ◦ Tt−1 )(z) =

[(γ (t) − γ (t))(1 − eλs ) + (1 − |γ (t)|2 )(1 + eλs )]z + (1 − γ (t)2 )(1 − eλs ) 2

(1 − γ (t) )(1 − eλs )z + [(γ (t) − γ (t))(1 − eλs ) + (1 − |γ (t)|2 )(1 + eλs )

.

Divide both the numerator and the denominator by 1 − γ (t) and take the limit as t → +∞. Note that, taking into account (13.2.2), we have 1 − γ (t) γ (t) − γ (t) =1− → 0, 1 − γ (t) 1 − γ (t) 1 − |γ (t)|2 1 − γ (t) = 1 + γ (t) → 2, 1 − γ (t) 1 − γ (t) therefore Tt ◦ η−s ◦ Tt−1 → η−s . This, together with Claim A, implies Tt+s ◦ η−s ◦ Tt−1 = (Tt+s ◦ Tt−1 ) ◦ (Tt ◦ η−s ◦ Tt−1 ) → ηs ◦ η−s = idD , and Claim B holds. Now, let {tk } ⊂ [0, +∞) be a sequence which converges to +∞ such that {φtk ◦ } converges, and let g be the limit. Tt−1 k

Claim C. For every s ≥ 0, the sequence {φtk +s ◦ Tt−1 } converges uniformly on comk +s pacta to g as t → +∞.

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369

First of all, notice that for all z ∈ D, ω(γ (t), Tt−1 (z)) = ω(Tt−1 (0), Tt−1 (z)) = ω(0, z) < +∞. Therefore, since γ (t) → 1 non-tangentially, Tt−1 (z) → 1 non-tangentially as t → +∞ by Lemma 1.8.6. Now, ω(φtk +s (Tt−1 (z)), φtk (Tt−1 (z)) ≤ ω(φs (Tt−1 (z)), Tt−1 (z))) k +s k k +s k ≤ ω(φs (η−s (Tt−1 (z))), Tt−1 (z)) + ω(φs (η−s (Tt−1 (z))), φs (Tt−1 (z))) (13.2.4) k k k k +s (z))), Tt−1 (z)) + ω(η−s (Tt−1 (z)), Tt−1 (z)). ≤ ω(φs (η−s (Tt−1 k k k k +s Note that ∠ lim z→1 φs ◦ η−s (z) = 1 and αφs ◦η−s (1) = e−λs eλs = 1 by (z) → 1 non-tangentially, Proposition 1.7.7. Therefore, taking into account that Tt−1 k −1 (z))), T (z)) = 0 by Proposition 1.9.12. Moreover, we have limk→∞ ω(φs (η−s (Tt−1 tk k by Claim B, (z)), Tt−1 (z)) = lim ω(Ttk +s (η−s (Tt−1 (z))), z) = 0. lim ω(η−s (Tt−1 k k +s k

k→∞

k→∞

Hence limk→∞ ω(φtk +s (Tt−1 (z)), φtk (Tt−1 (z))) = 0 by (13.2.4), and Claim C is k +s k proved. Now, from Claim C, ◦ ηs ) = lim (φs ◦ φtk ◦ Tt−1 ◦ Ttk ◦ Tt−1 ◦ ηs ). g ◦ ηs = lim (φtk +s ◦ Tt−1 k +s k k +s k→∞

k→∞

Since φtk ◦ Tt−1 → g and Ttk ◦ Tt−1 → η−s by Claim A, we have k +s k g ◦ ηs = φs ◦ g. We are left to prove that g is non constant, ∠ lim z→1 g(z) = 1 and that g is semiconformal at 1. We have ω(φt (Tt−1 (η−s (0))), γ (s)) = ω(φt (Tt−1 (η−s (0))), φt (γ (t + s))) −1 (0)) ≤ ω(Tt−1 (η−s (0)), γ (t + s)) = ω(Tt−1 (η−s (0)), Tt+s −1 = ω(η−s (0), Tt (Tt+s (0))) → ω(η−s (0), η−s (0)) = 0,

by Claim A. Therefore, for every s ≥ 0, (η−s (0))) = γ (s). g(η−s (0)) = lim φtk (Tt−1 k k→∞

(13.2.5)

From here it follows that g is not constant. Hence, as a limit of univalent functions, g is univalent. Since lims→+∞ η−s (0) = 1 and lims→+∞ γ (s) = 1, Theorem 1.5.7 implies ∠ lim z→1 g(z) = 1.

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13 Fixed Points, Backward Invariant Sets and Petals

Finally, in order to prove semi-conformality, for every t ≥ 0, consider the univalent function f t := Tt ◦ g ◦ η−t : D → D. Note that f t (0) = Tt (γ (t)) = 0 by (13.2.5). On the other hand, for t ≥ 1 and by (13.2.5), −1 (0)). f t (η1 (0)) = Tt (g(η−t (η1 (0)))) = Tt (g(η−t+1 (0))) = Tt (γ (t − 1)) = Tt (Tt−1 −1 By Claim A, Tt (Tt−1 (0)) → η1 (0) for t → +∞. Therefore, every limit of { f t } for t → +∞ fixes both 0 and η1 (0), hence by Schwarz’s Lemma (see Theorem 1.2.1) { f t } converges to the identity for t → +∞. That is, Tt ◦ g ◦ η−t → idD uniformly on compacta for t → +∞. This implies that for t → +∞,

ω(g(η−t (z)), Tt−1 (z)) = ω(Tt (g(η−t (z))), z) → 0. Lemma 1.8.6 implies then lim [Arg(1 − g(η−t (z))) − Arg(1 − Tt−1 (z))] = 0.

(13.2.6)

t→+∞

Now, Arg(1 −

Tt−1 (z))

= Arg

1 − γ (t) − (1 − γ (t))z 1 + γ (t)z

= Arg

1−γ (t) |1−γ (t)|

−

1−γ (t) z |1−γ (t)|

1 + γ (t)z

,

and, taking into account (13.2.2), passing to the limit for t → +∞, we obtain lim Arg(1 −

t→+∞

Tt−1 (z))



1−z = Arg 1+z

 .

Lemma 13.2.4 and (13.2.6) implies that g is semi-conformal at 1.



Given a pre-model (D, g, ηt ) for (φt ) at σ , the map g is in general not regular (or conformal) at σ (see Example 13.7.11). A consequence of Theorem 13.2.7 is the following: Theorem 13.2.8 Let (φt ) be a semigroup, not a group, in D. Let σ ∈ ∂D be a repelling fixed point of (φt ) with repelling spectral value λ ∈ (−∞, 0). Let (D, g, ηt ) be a pre-model for (φt ) at σ . Then, for every α ∈ (−π/2, π/2) there exists a nonexceptional backward orbit γ : [0, +∞) → D for (φt ), with γ ([0, +∞)) ⊂ g(D), converging to σ such that lim Arg(1 − σ γ (t)) = α.

t→+∞

(13.2.7)

Conversely, if γ : [0, +∞) → D is a backward orbit converging to σ , then γ is regular if and only if it converges non-tangentially to σ . Moreover, in this case, γ ([0, +∞)) ⊂ g(D) and there exists α ∈ (−π/2, π/2) such that (13.2.7) holds.

13.2 Pre-Models at Repelling Fixed Points

371

Proof By Remark 13.1.4 there exists z 0 ∈ D such that [0, +∞) t → η−t (z 0 ) converges to σ and limt→+∞ Arg(1 − σ η−t (z 0 )) = α. Let γ (t) := g(η−t (z 0 )). Note that for 0 ≤ s ≤ t, φs (γ (t)) = φs (g(η−t (z 0 ))) = g(ηs (η−t (z 0 ))) = g(ηs−t (z 0 )) = γ (t − s). Hence γ is a backward orbit. Since η−t (z 0 ) → σ non-tangentially, then γ (t) = g(η−t (z 0 )) → σ . Moreover, g is semi-conformal at σ , hence limt→+∞ Arg(1 − σ γ (t)) = α. Finally, since γ ([0, +∞)) ⊂  := g(D), ω(γ (t), γ (t + 1)) = ω(g(η−t (z 0 )), g(η−(t+1) (z 0 ))) ≤ ω(η−t (z 0 ), η−(t+1) (z 0 )) = ω(z 0 , η−1 (z 0 )), hence γ is a regular backward orbit satisfying the statement of the corollary. Conversely, if γ is a regular backward orbit converging to σ , then it converges non-tangentially to σ by Proposition 13.1.7. On the other hand, assume now that γ : [0, +∞) → D is a backward orbit which converges to σ non-tangentially. In particular, there exists M > 1 and t0 ≥ 0 such that γ (t) ∈ S(σ, M) for all t ≥ t0 . By Lemma 13.2.3, it follows that there exists t1 ≥ t0 such that γ (t) ∈ g(D) for all t ≥ t1 . Now, let s ∈ [0, t1 ). Then γ (s) = φt1 −s (γ (t1 )) = g(ηt1 −s (g −1 (γ (t1 )))) ∈ g(D). Hence, γ ([0, +∞)) ⊂ g(D). Note that for all t ≥ 0, φt (g(D)) = g(ηt (D)) = g(D), hence, ηt ◦ g −1 = g −1 ◦ φt |g(D) for all t ≥ 0. This easily implies that g −1 ◦ γ is a backward orbit for (ηt ). Hence, by Lemma 13.1.3, it is regular. Therefore, lim sup ω(γ (t), γ (t + 1)) ≤ lim sup k g(D) (γ (t), γ (t + 1)) t→+∞

t→+∞

= lim sup ω(g −1 (γ (t)), g −1 (γ (t + 1))) < +∞, t→+∞

and γ is regular. Finally, since g −1 ◦ γ is a backward orbit for (ηt ), by Lemma 13.1.3, it converges to σ with a certain angle. But g is semi-conformal at σ , hence γ converges to σ with the same angle, that is, (13.2.7) holds. 

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13 Fixed Points, Backward Invariant Sets and Petals

13.3 Maximal Invariant Curves The aim of this section is to study curves which are invariant and maximal, and see how these curves read the boundary dynamics of a semigroup. We start with a definition: Definition 13.3.1 Let (φt ) be a semigroup, not a group, in D, with Denjoy-Wolff point τ ∈ D. Let a ∈ [−∞, 0). A continuous curve γ : (a, +∞) → D is called a maximal invariant curve for (φt ) if φs (γ (t)) = γ (t + s) for all s ≥ 0 and t ∈ (a, +∞), limt→+∞ γ (t) = τ and there exists p ∈ ∂D such that limt→a + γ (t) = p. We call p the starting point of γ . Remark 13.3.2 Let a ∈ [−∞, 0). Let γ : (a, +∞) → D be a maximal invariant curve for a semigroup (φt ) which is not a group and let lim t→a + γ (t) = p ∈ ∂D. Then γ is injective. Indeed, if not, say γ (t) = γ (s) for some a < s < t. Then φt−s (γ (s)) = γ (t) = γ (s), implying that γ (s) = τ ∈ D, the Denjoy-Wolff point of (φt ). But then, for all a < t0 < s, φs−t0 (γ (t0 )) = γ (s) = τ . Since φs−t0 is injective in D, this forces γ (t0 ) = τ for all a < t0 < s, a contradiction since limt→a + γ (t) ∈ ∂D. Let h : (0, 1) → (a, +∞) be any orientation preserving homeomorphism. Then, setting (0) = p, (t) := γ (h(t)) and (1) = τ , it follows that  : [0, 1] → D is a Jordan arc (or a Jordan curve if p = τ ). Remark 13.3.3 Every non-constant backward orbit gives rise to a maximal invariant curve with starting point a boundary fixed point. Indeed, if (φt ) is a semigroup in D, not a group, and γ : [0, +∞) → D is a non-constant backward orbit for φt , then define the continuous curve μ : (−∞, +∞) → D as follows: μ(t) = γ (−t) for t ≤ 0 and μ(t) = φt (γ (0)) for t ≥ 0. By Lemma 13.1.5, limt→−∞ μ(t) = p ∈ ∂D, where p is a boundary fixed point for (φt ) and by Theorem 8.3.1, limt→+∞ μ(t) = τ . Moreover, for all s ≥ 0, t ≥ 0, φs (μ(t)) = φs (φt (γ (0))) = φt+s (γ (0)) = μ(t + s). Also, for t < 0 and 0 ≤ s ≤ −t, φs (μ(t)) = φs (γ (−t)) = γ (−t − s) = μ(t + s). Finally, if t < 0 and s > −t, φs (μ(t)) = φs+t (φ−t (γ (−t))) = φs+t (γ (0)) = μ(t + s). Hence, μ is a maximal invariant curve for (φt ). Definition 13.3.4 Let (φt ) be a semigroup, not a group, in D. Let W := ∩t≥0 φt (D). We call W the backward invariant set of (φt ).

13.3 Maximal Invariant Curves

◦

373

If W is the backward invariant set of a semigroup, as customary, we denote by

W the interior of W . Let (φt ) be a semigroup. Recall that φt has non-tangential limit at for every p ∈ ∂D for all t ≥ 0 by Theorem 11.2.1, and such a limit is denoted by φt ( p). The following proposition relates maximal invariant curves with backward invariant sets. Proposition 13.3.5 Let (φt ) be a semigroup, not a group, in D with Denjoy-Wolff point τ ∈ D and backward invariant set W . Then for every z 0 ∈ D \ {τ } there exists a unique maximal invariant curve γ : (a, +∞) → D for (φt ) such that γ (0) = z 0 . Moreover, z 0 ∈ W if and only if a = −∞. Also, denote by p ∈ ∂D the starting point of γ . Then (1) if z 0 ∈ / W then γ (t + a) = φt ( p) for all t ∈ (0, +∞), in particular p is not a / W for all t > a, fixed point for (φt ), and γ (t) ∈ (2) if z 0 ∈ W then [0, +∞) t → γ (−t) is a backward orbit and p is a boundary ◦

◦

fixed point of (φt ). Also, z 0 ∈W if and only if γ (t) ∈W for all t ∈ R. ◦

◦

◦

◦

In particular, φt (W ) =W , φt (W \ W ) = W \ W and φt (D \ W ) ⊂ D \ W for all t ≥ 0. Proof First we show that if there exists a maximal invariant curve γ for (φt ) such ˜ +∞) → D is another maximal that γ (0) = z 0 , then it is unique. Assume γ˜ : (a, invariant curve such that γ˜ (0) = z 0 , for some a˜ < 0. Then for t ≥ 0, γ (t) = φt (γ (0)) = φt (z 0 ) = φt (γ˜ (0)) = γ˜ (t). For max{a, a} ˜ < t < 0, φ−t (γ (t)) = γ (0) = z 0 = γ˜ (0) = φ−t (γ˜ (t)). Since φ−t is injective, we have γ (t) = γ˜ (t). If a˜ > a, then lim γ˜ (z) = lim+ γ (t) ∈ D,

t→a˜ +

t→a˜

a contradiction. Similarly, if a < a, ˜ we get a contradiction. Then, if exists, such a maximal invariant curve is unique. Now we construct a maximal invariant curve. Let (Ω, h, ψt ) be the canonical model for (φt ), where, Ω is either C, H, H− or the strip Sρ for some ρ > 0 and either ψt (w) = w + it or ψt (w) = e−λt w for some λ ∈ C with Re λ > 0. Let w0 := h(z 0 ) and a := inf{t < 0 : (ψ−t )−1 (w0 ) ∈ h(D)}. By the geometry of Ω (spirallike or starlike at infinity) and the form of ψt , it follows that ψt (w0 ) ∈ h(D) for every t > a. Hence, we can well define

374

13 Fixed Points, Backward Invariant Sets and Petals

γ (t) := h −1 (ψt (w0 )), t ∈ (a, +∞). Since h −1 ◦ ψs = φs ◦ h −1 for all s ≥ 0 on h(D), it is easy to see that φs (γ (t)) = γ (s + t) for all s ≥ 0 and t ∈ (a, +∞). This implies at once that limt→+∞ γ (t) = τ by Theorem 8.3.1. Assume that a > −∞. Then q := ψa (w0 ) ∈ ∂h(D) ∩ C and w0 = ψ−a (q). By construction, the curve (a, +∞) t → ψt−a (q) is contained in h(D) and limt→a + ψt−a (q) = q. By Corollary 3.3.4, there exists p ∈ ∂D such that limt→a + h −1 (ψt−a (q)) = p. Hence, lim γ (t) = lim h −1 (ψt (w0 )) = lim h −1 (ψt (ψ−a (q))) = lim h −1 (ψt−a (q)) = p.

t→a +

t→a +

t→a +

t→a +

Moreover, for all s ≥ 0 and t > a, φs (γ (t)) = φs (h −1 (ψt (w0 ))) = φs (h −1 (ψt−a (q))) = h −1 (ψt+s−a (q)). Hence, limt→a + φs (γ (t)) = h −1 (ψs (q)) ∈ D. Therefore, by Theorem 1.5.7, for all s > 0, φs ( p) = h −1 (ψs (q)) = h −1 (ψs+a (w0 )) = γ (s + a). / W for all s > 0, hence, γ (t) ∈ / W for all t > a. It is clear that φs ( p) ∈ Assume now a = −∞. Clearly, [0, +∞) t → γ (−t) is a backward orbit for (φt ), hence either it is constantly equal to τ ∈ D, which implies at once that z 0 = τ , against our hypothesis, or it converges to a point p ∈ ∂D which is fixed for (φt ) by Proposition 13.1.7. Now, let t ∈ R and s ≥ 0. Then γ (t) = φs (γ (t − s)), which implies that γ (t) ∈ W for all t ∈ R, in particular, z 0 ∈ W . The previous argument shows that for all t ≥ 0 φt (W ) = (φt )−1 (W ) = W .

(13.3.1)

◦

Now, suppose z 0 ∈W . Then there exists an open set U ⊂ W such that z 0 ∈ U . Let z 1 = γ (t1 ) for some t1 ∈ (a, +∞). If t1 > 0, then φt1 (U ) ⊂ W is an open neigh◦

borhood of φt1 (z 0 ) = γ (t1 ) = z 1 , hence, z 1 ∈W . If t1 < 0, taking into account that U ⊂ φt1 (D), it follows that V := (φt1 )−1 (U ) is an open neighborhood of z 1 . By ◦

(13.3.1), V ⊂ W , hence z 1 ∈W .



Remark 13.3.6 The previous proof shows that if (φt ) is a non-elliptic semigroup in D and h : D → C is its Koenigs function, then every maximal invariant curve of (φt ) is of the form {z ∈ D : Re h(z) = b}, for some b ∈ R. In fact, if γ is such a maximal invariant curve, then h(γ ) is a line of a half-line. Remark 13.3.7 Let (φt ) be a semigroup, not a group, in D, with infinitesimal generator G. Let γ : (a, +∞) → D be a maximal invariant curve. Then γ is the maximal solution to the Cauchy problem dtd x(t) = G(x(t)), x(0) = γ (0). It fol-

13.3 Maximal Invariant Curves

375

lows at once differentiating in s the expression φs (γ (t)) = γ (t + s). Conversely, by the uniqueness property of the solution to the Cauchy problem, given z 0 ∈ D and γ : (−, +∞) → D,  ∈ (0, +∞] the maximal solution to the Cauchy problem d x(t) = G(x(t)), x(0) = z 0 , then γ is a maximal invariant curve of (φt ). dt Remark 13.3.8 Let (φt ) be a semigroup, not a group, in D. Let z 0 ∈ D \ {τ } and let γ : (a, +∞) → D be the unique maximal invariant curve for (φt ) such that γ (0) = z 0 , a < 0. Let t0 ∈ (a, +∞). It is easy to check that γ˜ : (a + t0 , +∞) → D (where, if a = −∞, we set a + t0 = −∞) is a maximal invariant curve for (φt ) such that γ˜ (0) = γ (t0 ). In other words, for every z ∈ γ ((a, +∞)), the image of the maximal invariant curve for (φt ) which values z at time 0 is γ ((a, +∞)). Remark 13.3.9 The uniqueness of a maximal invariant curve holds also at the starting point in case this is not a fixed point. More precisely, let (φt ) be a semigroup, not a group. Let γ : (a, +∞) → D be a maximal invariant curve for (φt ) with a < 0 and starting point σ ∈ ∂D. Assume a > −∞. If γ˜ : (a, ˜ +∞) → D is a maximal invariant curve for (φt ), a˜ < 0, such that limt→a˜ + γ˜ (t) = σ then a˜ > −∞ and γ ((a, +∞)) = γ˜ ((a, ˜ +∞)). Indeed, by Proposition 13.3.5, a > −∞ implies σ is not a fixed point of (φt ) which in turn, by the same token, implies a˜ > −∞. Then, by Proposition 13.3.5(1), γ (t + ˜ for all t ≥ 0, from which the previous statement follows at a) = φt (σ ) = γ˜ (t + a) once.

13.4 Petals In this section we deeply examine the interior part of the backward invariant set of a semigroup. Definition 13.4.1 Let (φt ) be a semigroup, not a group, in D. A non-empty connected ◦

component of W , the interior of the backward invariant set W of (φt ), is called a petal of (φt ). Proposition 13.4.2 Let (φt ) be a semigroup, not a group, in D with Denjoy-Wolff point τ ∈ D. Let  be a petal of (φt ). Then (1)  is a simply connected domain, (2) φt () =  for all t ≥ 0 and (φt | ) is a continuous group of automorphisms of , (3) τ ∈ ∂, (4) there exists σ ∈ ∂D ∩ ∂ (possibly σ = τ ) such for every z ∈  the curve [0, +∞) t → (φt | )−1 (z) is a regular backward orbit for (φt ) which converges to σ . Moreover, σ is a boundary regular fixed point and, if σ = τ then (φt ) is parabolic.

376

13 Fixed Points, Backward Invariant Sets and Petals

Proof (1) Seeking for a contradiction, we assume  is not simply connected. Hence, there exists a Jordan curve  ⊂  such that the bounded connected component of C \ / W , the backward invariant set of (φt ). By Proposition 13.3.5,  contains a point z 0 ∈ there exists a maximal invariant curve γ : (a, +∞) → D such that lim z→a − γ (t) = / W for all t > a. But γ ((a, +∞)) has to p ∈ ∂D and γ (0) = z 0 . Moreover, γ (t) ∈ intersect , a contradiction. Therefore,  is simply connected. ◦

(2) By Proposition 13.3.5, φt () ⊂W for all t ≥ 0. Hence, since  is connected, if z 0 ∈  then [0, +∞) t → φt (z 0 ) is contained in . That is, φt () ⊆  for all t ≥ 0. On the other hand, if z 0 ∈  \ {τ }, let γ : (−∞, +∞) →  be the maximal invariant curve such that γ (0) = z 0 , given by Proposition 13.3.5. Hence, since  ◦

is connected and γ ((−∞, +∞)) ⊂W , it follows that γ (t) ∈  for all t ∈ R. Thus, φt (γ (t)) = γ (0) = z 0 for every t ≥ 0. Namely, φt () =  for all t ≥ 0. Therefore, (φt | ) is a continuous semigroup whose iterates are automorphisms of . By Theorem 8.4.4, it extends to a continuous group (φt | ) of . (3) If z ∈ , then φt (z) ∈  for all t ≥ 0 and φt (z) → τ . Hence, τ ∈ . In particular, if (φt ) is not elliptic, then τ ∈ ∂. Assume (φt ) is elliptic and τ ∈ . If τ = 0, let T : D → D be an automorphism such that T (0) = τ . Then (T −1 ◦ φt ◦ T ) is a semigroup in D with Denjoy-Wolff point 0 and T −1 () is a petal of (T −1 ◦ φt ◦ T ). Clearly, 0 ∈ T −1 () if and only if τ ∈ . Therefore, we can assume τ = 0. Since  is simply connected, there exists a univalent map f : D → C such that f (D) =  and f (0) = 0. Hence, by (2), ( f −1 ◦ φt ◦ f ) is a group of D with Denjoy-Wolff point 0. In particular, by Proposition 8.3.8, 1 = |( f −1 ◦ φt ◦ f ) (0)| = |φt (0)|. By Theorem 1.2.1, it follows that (φt ) is a group, against our assumption. (4) Let z ∈ . Taking into account that φt | is an automorphism of  for all t ≥ 0, we have ω((φt | )−1 (z), (φt+1 | )−1 (z)) ≤ k ((φt | )−1 (z), (φt+1 | )−1 (z)) = k (z, (φ1 | )−1 (z)) < +∞. Moreover, φs ((φt | )−1 (z)) = (φt−s | )−1 (z) for all 0 ≤ s ≤ t. Therefore, [0, +∞)

t → (φt | )−1 (z) is a regular backward orbit for (φt ). Hence, by Proposition 13.1.7, there exists σ ∈ ∂D ∩ ∂ such that limt→+∞ (φt | )−1 (z) = σ —and, if σ = τ then (φt ) is parabolic. Finally, if w ∈ , for all t ≥ 0, ω((φt | )−1 (z), (φt | )−1 (w)) ≤ k ((φt | )−1 (z), (φt | )−1 (w)) = k (z, w) < +∞, hence limt→+∞ (φt | )−1 (w) = σ by Lemma 1.8.6.



Remark 13.4.3 By Proposition 13.4.2(2), if (φt ) is a semigroup in D which is not a group and  is a petal of (φt ), then  = D.

13.4 Petals

377

The previous result allows to give the following definition: Definition 13.4.4 Let (φt ) be a semigroup, not a group, in D with Denjoy-Wolff point τ ∈ D. Let  be a petal for (φt ). We say that  is a hyperbolic petal if ∂ contains a repelling fixed point of (φt ). Otherwise, we call  a parabolic petal. Remark 13.4.5 By Proposition 13.4.2, a petal contains no (inner) fixed points of the semigroup. Moreover, only parabolic semigroup can have parabolic petals. Our aim now is to describe the boundary of petals. We start with the following result which, taking into account Remark 13.3.3, shows that exceptional backward orbits have to be necessarily contained in the closure of a parabolic petal: Proposition 13.4.6 Let (φt ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Let γ : (−∞, +∞) → D be a maximal invariant curve for (φt ) such that limt→−∞ γ (t) = τ . Let J be the Jordan curve defined by γ , i.e., J := γ (R). Let V be the bounded connected component of C \ J . Then there exists a parabolic petal  of (φt ) such that V ⊆ . In particular, J is contained in the closure of a parabolic petal of (φt ) and the semigroup is parabolic. Proof By Remark 13.3.2, J is a Jordan curve. Fix ζ0 ∈ V . Let η : (a, +∞) → D be the maximal invariant curve for (φt ) such that η(0) = ζ0 . Note that, since J is the closure of the image of a maximal invariant curve which does not contain ζ0 , we have η((a, +∞)) ∩ J = ∅ by Remark 13.3.8. Hence, η((a, +∞)) ⊂ V . By Proposition 13.3.5, the initial point w0 of η belongs to ∂D, hence, the only possibility is w0 = τ . In this case, again by Proposition 13.3.5, we have ζ0 ∈ W , the backward invariant ◦

set of (φt ). By the arbitrariness of ζ0 , this implies that V ⊂W . Therefore, there exists a petal  of (φt ) such that V ⊆ . Finally, let z ∈ V . The curve [0, +∞) t → (φt | )−1 (z) is a backward orbit of (φt ) by Proposition 13.4.2. By Remark 13.3.3, it extends to a maximal invariant curve of (φt ). Hence, for what we already proved, such a curve converges to τ . It follows that  is parabolic, again by Proposition 13.4.2(4).  Now, we focus on the boundary of petals. To this aim, and for the subsequent results, we need a lemma: Lemma 13.4.7 Let (φt ) be a semigroup, not a group, in D with Denjoy-Wolff point τ ∈ D. Suppose  ⊂ D is a petal and z 0 ∈ ∂ ∩ (D \ {τ }). Let γ : (a, +∞) → D, a < 0, be the maximal invariant curve of (φt ) such that γ (0) = z 0 . Then γ ((a, +∞)) ⊂ ∂. Proof Let W be the backward invariant set of (φt ). Since z 0 ∈ ∂, it follows that ◦

◦

z0 ∈ / W for all t ∈ (a, +∞). In particular, γ (t) ∈ / / W . By Proposition 13.3.5, γ (t) ∈ for all t ∈ (a, +∞). Let t ≥ 0 and fix δ > 0. Since z 0 ∈ ∂, there exists w ∈  such that ω(z 0 , w) < δ. Then ω(φt (w), γ (t)) = ω(φt (w), φt (z 0 )) ≤ ω(w, z 0 ) < δ.

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13 Fixed Points, Backward Invariant Sets and Petals

Since φt (w) ∈  for all t ≥ 0 by Proposition 13.4.2, by the arbitrariness of δ, we have γ (t) ∈ ∂ for all t ≥ 0. Now, let t ∈ (a, 0). Assume by contradiction that γ (t) ∈ / ∂. Then there exists an open neighborhood U of γ (t) such that U ∩  = ∅. Since φ−t (γ (t)) = γ (0) = z 0 , and φ−t (U ) is open, there exists w ∈ φ−t (U ) ∩ . Since φ−t | is an automorphism of , it follows that (φ−t | )−1 (w) ∈ . But φ−t is injective, hence (φ−t | )−1 (w) ∈ U , a contradiction. Therefore, γ (t) ∈ ∂ for all t ∈ (a, +∞).  We first consider the case in which the boundary of a petal contains a maximal invariant curve starting at the Denjoy-Wolff point: Proposition 13.4.8 Let (φt ) be a parabolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Let  ⊂ D be a petal of (φt ). Suppose there exists a maximal invariant curve γ : (−∞, +∞) → D of (φt ) such that γ (0) ∈ ∂ and limt→−∞ γ (t) = τ . Let J := γ (R) (a Jordan curve) and let V be the bounded connected component of C \ J . Then  is parabolic,  = V and J = ∂. In particular, the boundary of a hyperbolic petal cannot contain maximal invariant curves with starting point τ . Proof By Lemma 13.4.7, J ⊂ ∂. Since  is connected and J ∩  = ∅, either  ⊆ V or  ⊆ D \ V . But, by Proposition 13.4.6, V is contained in a parabolic petal. Therefore, if  ⊂ V , then, in fact, V =  and the statement holds. Thus, assume by contradiction that  ⊆ U := D \ V . Note that by Proposition 13.4.6, V is contained in a petal of (φt ). Since by Lemma 13.4.7, J = ∂ V ⊂ ∂ and  ∩ V = ∅, it follows that V is a petal of (φt ). Let z 0 ∈ J ∩ D. We claim that there exists an open neighborhood A of z 0 such that A ∩ U ⊂ . Assume the claim is true. Let W be the backward invariant set of (φt ). Note ◦

that J ∩ D ⊂ W \ W . But, we have: A ∩ U ⊂  ⊂ W (by the claim), A ∩ V ⊂ W (because V is a petal) and J ∩ D ⊂ W (by Proposition 13.3.5). Hence, A ⊂ W , ◦

which implies z 0 ∈W , a contradiction. We are left to prove the claim. If  = U , then set A = D. Otherwise, there exists ζ0 ∈ ∂ ∩ U . Let η0 : (a, +∞) → D, a < 0, be the maximal invariant curve such that η0 (0) = ζ0 . Let C0 be its closure (it is a Jordan curve by Remark 13.3.2). By Lemma 13.4.7, C0 ⊂ ∂. The Jordan curve C0 divides D in / J , we two connected components, one of them, call it B0 , contains . Since ζ0 ∈ have C0 ∩ J ∩ D = ∅ by Remark 13.3.8, and hence, V ∩ D ⊂ B0 . Let p0 be the starting point of η0 . If p0 = τ , then B0 is the bounded connected component of D \ C0 , hence, it is contained in a parabolic petal by Proposition 13.4.6. ◦

This implies J ∩ D ⊂W , a contradiction. Therefore, p0 = τ . Also, B0 = , for otherwise V ⊆ B0 =  against the hypothesis  ⊂ D \ V . Hence, there exists ζ1 ∈ ∂ ∩ B0 , and, arguing as before, we obtain another Jordan curve C1 ⊂ ∂, which does not intersect C0 ∩ D and J ∩ D. Let B1 be the connected component of D \ C1 which contains . By construction, such connected component contains V ∩ D and C0 ∩ D.

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379

Let p1 be the starting point of the maximal invariant curve whose closure defines C1 . As before, p1 = τ . By construction,  ⊂ G := B0 ∩ B1 ∩ U . Note that G is a simply connected domain whose boundary is given by J ∪ C0 ∪ C1 ∪ T , where T is a closed arc in ∂D with end points p0 and p1 (we set T = { p0 } in case p0 = p1 ). We claim that G = , and hence the claim is proved by setting A = G ∪ (J ∩ D) ∪ V . Indeed, if there were a point ζ2 ∈ G ∩ ∂, the Jordan curve C2 , defined as before by ζ2 , would divide G into two connected components and  would belong to one of the two connected components. But this is impossible because J, C0 , C1 ⊂ ∂.  Now we are in good shape to describe the boundary of petals: Proposition 13.4.9 Let (φt ) be a semigroup, not a group, in D with Denjoy-Wolff point τ ∈ D. Let  be a petal for (φt ). Then one and only one of the following cases can happen: (1) τ ∈ D and there exists a Jordan arc J with end points τ and a point p ∈ ∂D such that ∂ = J ∪ ∂D. In particular, in this case,  = D \ J is the only petal of (φt ); (2) τ ∈ ∂D and there exists a Jordan arc J with end points τ and a point p ∈ ∂D \ {τ } such that ∂ = J ∪ A where, A is an arc in ∂D with end points τ and p. In this case,  is the bounded connected component of C \ (J ∪ A); (3) there exist two Jordan arcs J1 , J2 with end points p ∈ ∂D \ {τ } and τ such that J1 ∩ J2 = {τ, p} and ∂ = J1 ∪ J2 . In this case,  is the bounded connected component of C \ (J1 ∪ J2 ); (4) there exist two Jordan arcs J1 , J2 such that J j has end points τ and p j ∈ ∂D \ {τ }, j = 1, 2, p1 = p2 , with J1 ∩ J2 = {τ } and ∂ = J1 ∪ J2 ∪ A, where A ⊂ ∂D is a closed arc with end points p1 and p2 . In this case,  is the bounded connected component of C \ (J1 ∪ J2 ∪ A); (5) τ ∈ ∂D and there exists a Jordan curve J such that J ∩ ∂D = {τ }, ∂ = J and  is the bounded connected component of C \ J . In this case,  is a parabolic petal. In particular, ∂ is locally connected and, in cases (2), (3), (4) and (5),  is a Jordan domain. Proof Since (φt ) is not a group by hypothesis, it follows that  = D by Proposition 13.4.2. Hence, ∂ ∩ D = ∅. Since  is simply connected, ∂ ∩ D cannot reduce to one point. Let z 0 ∈ ∂ ∩ D \ {τ }. Let γ : (a, +∞) → D be the maximal invariant curve such that γ (0) = z 0 , a < 0, given by Proposition 13.3.5. Let J be the closure of γ ((a, +∞)). By Remark 13.3.2, J is a Jordan arc with end points τ and p ∈ ∂D (if p = τ then J is a Jordan curve). By Lemma 13.4.7, J ⊂ ∂. By Proposition 13.4.8, if p = τ then we are in case (5). If ∂ \ J ⊂ ∂D and p = τ , then we are in case (1) or (2). In this case, if τ ∈ D then ∂ is necessarily the union of J with ∂D and, moreover, D \  = J , which, since J has no interior, implies that  is the only petal of (φt ). On the other hand, if τ ∈ ∂D, and p = τ , since  is connected, and J disconnects D in two connected

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13 Fixed Points, Backward Invariant Sets and Petals

components by Lemma 4.1.3, it follows that ∂ is the union of J with an arc in ∂D with end points τ and p. Now, assume there exists z 0 ∈ (∂ \ J ) ∩ D. We repeat the above argument, in order to obtain another Jordan arc (or Jordan curve) J2 ⊂ ∂ which contains z 0 and J2 is contained in D except, at most, the two ends points. By Remark 13.3.8, since / J , it follows J ∩ D and J2 ∩ D are (images of) maximal invariant curves, and z 0 ∈ that J ∩ J2 ⊂ ∂D ∪ {τ } (that is, J and J2 can have in common only the end points). Let p2 ∈ ∂D be the initial point of J2 . By Proposition 13.4.8, p2 = τ . We claim that (∂ \ (J ∪ J2 )) ⊂ ∂D. Indeed, if this is not the case, one can find a point w0 ∈ (∂ \ (J ∪ J2 )) ∩ D. Repeating the above argument, we end up with another Jordan curve J3 such that J ∩ J3 ∩ D = J2 ∩ J3 ∩ D = ∅. Recalling that J, J2 , J3 have a common end point τ and another end point on ∂D \ {τ }, it is easy to see that we reach a contradiction. For instance, in case τ ∈ D, the Jordan curve J ∪ J2 divides D into two connected components, and  has to stay in one of them, call it U . Then J3 ∩ D is contained in U , and divides U into two connected components, one whose boundary is J, J3 and (possibly) an arc on ∂D, and the other whose boundary is J, J2 and (possibly) an arc on ∂D. Since  is connected, it has to stay in one of the two components, say the first, but then J2 can not be contained in ∂, a contradiction. The other cases are similar. If J1 and J2 have the same end points, then we are in case (3). If J1 has a different end point than J (they both have τ as common end point), then we are in case (4).  Every case given by Proposition 13.4.9 actually happens, as we will see in the examples of the last section. Proposition 13.4.10 Let (φt ) be a semigroup, not a group, in D with Denjoy-Wolff point τ ∈ D. Let  be a petal for (φt ). (1) If  is a hyperbolic petal, then there exists a repelling fixed point σ ∈ ∂ of (φt ) such that ∂ \ {τ, σ } does not contain any (repelling or super-repelling) boundary fixed point of (φt ). (2) If  is a parabolic petal, then ∂ \ {τ } does not contain any (repelling or superrepelling) boundary fixed point of (φt ). Proof Since  is simply connected by Proposition 13.4.2, there exists a univalent function g : D → C such that g(D) = . Let ψt := g −1 ◦ φt ◦ g, t ≥ 0. Since (φt | ) is a continuous group of automorphisms of  by Proposition 13.4.2, it follows that (ψt ) is a group in D. Moreover, limt→+∞ φt (g(0)) = τ ∈ ∂, and hence ψt (0) = g −1 (φt (g(0))) can accumulate only on ∂D. Therefore, (ψt ) is a non-elliptic group in D. By Proposition 13.4.9, ∂ is locally connected, hence, by Theorem 4.3.1, g extends to a continuous and surjective function, which we still denote by g, from D to . In particular, for all p ∈ ∂D, g(ψt ( p)) = lim− g(ψt (r p)) = lim− φt (g(r p)). r →1

r →1

13.4 Petals

381

By Theorem 1.5.7, it follows that the non-tangential limit of φt at g( p) is g(ψt ( p)), that is, (13.4.1) g(ψt ( p)) = φt (g( p)) for all p ∈ ∂D, t ≥ 0. Let p ∈ ∂D be such that g( p) is a fixed point of (φt ). Hence, by (13.4.1), g(ψt ( p)) = g( p) for all t ≥ 0. We claim that this implies that ψt ( p) = p for all t ≥ 0. Otherwise, the image [0, +∞) t → ψt ( p) would be an arc in ∂D where g is constant, contradicting Proposition 3.3.2. Since (ψt ) is a non-elliptic group in D, it has at most two fixed points on ∂D, hence, there exist at most two fixed points for (φt ) on ∂. From this, (1) follows at once. Now, assume that  is parabolic. By Remark 13.4.5, (φt ) is necessarily parabolic, and τ ∈ ∂D. Hence, by Proposition 13.4.9,  is a Jordan domain and Theorem 4.3.3 implies that, in fact, g : D →  is a homeomorphism. Now, by Proposition 13.4.2(4), there exists a regular backward orbit γ : [0, +∞) →  such that limt→+∞ γ (t) = τ . It is easy to see that g −1 ◦ γ is a backward orbit for (ψt ) which converges to g −1 (τ ). Since ψt (0) = g −1 (φt (g(0))) → g −1 (τ ), it follows that (ψt ) has a backward orbit which converges to its Denjoy-Wolff point. Hence, (ψt ) is a parabolic group by Lemma 13.1.3. Therefore, (ψt ) has only one fixed point on ∂D, and so (φt ) has a unique fixed point on ∂ which is τ .  As a direct consequence of Proposition 13.4.10, we have Corollary 13.4.11 Let (φt ) be a semigroup, not a group, in D with Denjoy-Wolff point τ ∈ D. Suppose  is a petal of (φt ). Let J ⊂ ∂ ∩ (D \ {τ }) be a connected component. Then the closure of J is a Jordan arc (or Jordan curve) with end points τ and p0 ∈ ∂D. Moreover, (1) if  is hyperbolic and σ ∈ ∂ is the unique (repelling) fixed point of (φt ) contained in , then, either p0 = σ or p0 is not a fixed point of (φt ). (2) if  is parabolic, then either p0 = τ or p0 is not a fixed point of (φt ). Proof By Proposition 13.4.9, the closure of J is a Jordan arc or Jordan curve. (1) If  is hyperbolic, let σ ∈ ∂D be the unique repelling fixed point of (φt ) contained in . If p0 is a fixed point of (φt ), then either p0 = σ or p0 = τ by Proposition 13.4.10. However, by Lemma 13.4.7, J is the image of a maximal invariant curve. Hence, Proposition 13.4.8 excludes the case p0 = τ . (2) It follows immediately from Proposition 13.4.9.  The previous results show that the closure of every hyperbolic petal contains exactly one repelling fixed point, now we show the converse: Proposition 13.4.12 Let (φt ) be a semigroup, not a group, in D with Denjoy-Wolff point τ ∈ D. Suppose σ ∈ ∂D is a repelling fixed point for (φt ). Then there exists a unique hyperbolic petal  such that σ ∈ ∂. Moreover, for all M > 1 there exists  > 0 such that S(σ, M) ∩ {ζ ∈ C : |ζ − σ | < } ⊂ .

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13 Fixed Points, Backward Invariant Sets and Petals

Also, if (D, g, ηt ) is a pre-model for (φt ) at σ then g(D) = . Proof By Theorem 13.2.7, there exists a pre-model (D, g, ηt ) for (φt ) at σ . For all t ≥ 0, (13.4.2) φt (g(D)) = g(ηt (D)) = g(D). This implies that g(D) ⊂ W , the backward invariant set of (φt ). Since g(D) is open and simply connected, there exists a petal  such that g(D) ⊂ . Moreover, η−t (0) converges non-tangentially to σ as t → +∞ by Lemma 13.1.3, hence, since ∠ lim z→σ g(z) = σ , we have lim g(η−t (0)) = σ,

t→+∞

that is, σ ∈ g(D). Since  contains no fixed points of (φt ) by Remark 13.4.5, it follows that σ ∈ ∂, proving the first part of the statement. Now we show that g(D) = . Let z 0 ∈ . By Proposition 13.4.2, the curve [0, +∞) t → (φt | )−1 (z 0 ) is a regular backward orbit for (φt ) which converges to σ , and, by Proposition 13.1.7, the convergence to σ is non-tangential. Therefore, by Theorem 13.2.8, (φt | )−1 (z 0 ) ∈ g(D) for all t ≥ 0. In particular, z 0 ∈ g(D) and hence  = g(D) by the arbitrariness of z 0 . Moreover, by Lemma 13.2.3, for all M > 1 there exists  > 0 such that S(σ, M) ∩ {ζ ∈ C : |ζ − σ | < } ⊂ g(D) = . Finally, we are left to show that  is the unique petal which contains σ on its boundary. Assume by contradiction this is not the case and let  be a petal different from  such that σ ∈ ∂ . Note that  ∩  = ∅ (since they are different open connected components of the interior of the backward invariant set of (φt )). We claim that  has to be a Jordan domain. Indeed, looking at Proposition 13.4.9, we see if  is not a Jordan domain, then  is the only petal of (φt ), forcing  =  . Let f : D → C be univalent such that f (D) =  . Hence, by Theorem 4.3.3, f extends as a homeomorphism—which we still denote by f —from D to  . By Proposition 13.4.2, τ ∈ ∂ and (φt | ) is a continuous group of automorphisms of  . Hence, arguing as in the proof of Proposition 13.4.10, it is easy to see that ( f −1 ◦ φt ◦ f ) is a group in D, with fixed points f −1 (τ ) and f −1 (σ ). Therefore, ( f −1 ◦ φt ◦ f ) is a hyperbolic group and by Remark 13.1.4, it has a regular backward orbit γ converging to f −1 (σ ). It is easy to see that f ◦ γ is a backward orbit for (φt ) converging to σ and it is regular because for all t ≥ 0, ω( f (γ (t)), f (γ (t + 1))) ≤ ω(γ (t), γ (t + 1)). By Proposition 13.1.7, f ◦ γ converges to σ non-tangentially. Theorem 13.2.8 implies then that f (γ ([0, +∞))) ⊂ g(D) = . Hence,  ∩  = ∅, a contradiction. 

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383

A first consequence of the previous proposition is the (essential) uniqueness of a pre-model: Corollary 13.4.13 Let (φt ) be a semigroup, not a group, in D with Denjoy-Wolff point τ ∈ D. Suppose σ ∈ ∂D is a repelling fixed point for (φt ). Let (D, g, ηt ) and (D, g, ˜ ηt ) be two pre-models for (φt ). Then there exists a hyperbolic automorphism ϕ of D with fixed points σ, −σ such that g˜ = g ◦ ϕ. Proof By Proposition 13.4.12,  := g(D) = g(D). ˜ Hence, ϕ := g −1 ◦ g˜ : D → D is an automorphism of D. Moreover, by definition of pre-model, for all t ≥ 0, g˜ ◦ ηt ◦ g˜ −1 | = g ◦ ηt ◦ g −1 | . Hence, ϕ ◦ ηt = ηt ◦ ϕ for all t ≥ 0. In particular, lim ϕ(ηt (0)) = lim ηt (ϕ(0)) = −σ.

t→+∞

t→+∞

Since ηt (0) → −σ as t → +∞, this proves that ϕ(−σ ) = −σ . A similar argument for t → −∞ implies that ϕ(σ ) = σ , and we are done.  With the results obtained so far we can studied the rate of convergence when approaching a repelling fixed point. Recall that, given a petal  for a semigroup (φt ), (φt | ) is a continuous group of automorphisms of  by Proposition 13.4.2. Therefore, given z ∈ , it is well-defined φt | (z) for all t ∈ R. With a slight abuse of notation, we write φt (z) to denote φt | (z) for all t ∈ R when z ∈ . Proposition 13.4.14 Let (φt ) be a semigroup in D with a repelling fixed point σ ∈ ∂D, repelling spectral value λ ∈ (−∞, 0) at σ , and associated hyperbolic petal  (see Proposition 13.4.12). Then, for all z ∈ , lim

t→−∞

1 log (1 − σ φt (z)) = −λ. t

(13.4.3)

Proof Let G be the infinitesimal generator of the semigroup. By Theorem 12.2.5 and Proposition 12.2.4, −λ = ∠ lim

z→σ

G(z) ∈ (0, ∞). z−σ

Moreover, by Lemma 13.1.6, the regular backward orbit [0, +∞) t → φ−t (z) converges to σ non-tangentially, so that G(φt (z)) , z ∈ . t→−∞ φt (z) − σ

−λ = lim

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13 Fixed Points, Backward Invariant Sets and Petals

Given z ∈  and t ∈ R,

t

0

s=t −σ G(φs (z)) ds = log(1 − σ φs (z)) s=0 1 − σ φs (z) = log(1 − σ φt (z)) − log(1 − σ z).

Then, using L’Hôpital’s Rule and the non-tangential convergence, we obtain 1 t→−∞ t



t

lim

0

−σ G(φs (z 0 )) G(φt (z 0 )) ds = lim = −λ. t→+∞ φt (z 0 ) − σ 1 − σ φs (z 0 )

Hence, limt→−∞ 1t log (1 − σ φt (z)) = −λ.



As a last result of this section we prove that a super-repelling fixed point can be the limit of at most one backward orbit: Proposition 13.4.15 Let (φt ) be a semigroup, not a group, in D with Denjoy-Wolff point τ ∈ D. Suppose σ ∈ ∂D is a super-repelling fixed point of (φt ). Assume γ j : [0, +∞) → D, j = 1, 2, are backward orbits of (φt ) converging to σ . Then, either γ1 ([0, +∞)) ⊆ γ2 ([0, +∞)) or γ2 ([0, +∞)) ⊆ γ1 ([0, +∞)). In particular, up to re-parameterization, there is at most one maximal invariant curve for (φt ) with starting point σ . Proof Suppose by contradiction that the statement is not true. Let define η j (t) = γ j (−t) for t ≥ 0 and η j (t) = φt (γ j (0)) for t > 0, j = 1, 2. By Remark 13.3.3, η1 , η2 are maximal invariant curves for (φt ). Hence, by Remark 13.3.8, either η1 ((−∞, +∞)) = η1 ((−∞, +∞)) or they are disjoint, and by our hypothesis, the latter case holds. Since limt→+∞ η j (t) = τ and limt→−∞ η j (t) = σ , j = 1, 2, by Remark 13.3.2 it follows that the closure of η1 ((−∞, +∞)) ∪ η2 ((−∞, +∞)), call it J , is a Jordan curve such that J ∩ ∂D = {τ, σ }. Let D be the bounded connected component of C \ J . We claim that D ⊂ W , the backward invariant set of (φt ). Assuming the claim, it follows at once that D is contained in a petal . But σ ∈ ∂, hence, by Proposition 13.4.10,  is hyperbolic and σ is repelling, contradiction. In order to prove the claim, let z 0 ∈ D and let η : (a, +∞) → D be the maximal invariant curve such that η(0) = z 0 , with a ∈ [−∞, 0). Let p ∈ ∂D be the starting point of η. Since D ∩ ∂D = {σ, τ }, it follows that p ∈ {τ, σ }, hence, Proposition 13.3.5 implies that z 0 belongs to the backward invariant set of (φt ). The last statement follows at once from what we already seen and Remark 13.3.3. 

13.5 Petals and the Geometry of Koenigs Functions In this section we see how the geometry of the image of the Koenigs function of a semigroup detects petals. Recall that by Proposition 13.4.12 for every repelling fixed point of a semigroup there exists a unique hyperbolic petal whose boundary

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385

contains that fixed point. Moreover, the restriction of the semigroup to any petal is a continuous group of automorphisms of such a petal (Proposition 13.4.2). The basic observation is contained in the following lemma: Lemma 13.5.1 Let (φt ) be a semigroup, not a group, in D. Let (Ω, h, ψt ) be the canonical model of (φt ) (where Ω and ψt are given by Theorem 9.3.5). Suppose σ ∈ ∂D is a repelling fixed point for (φt ) with repelling spectral value λ ∈ (−∞, 0). Let  be the hyperbolic petal of (φt ) such that σ ∈ ∂. Let A := h(). Then ψt (A) = A for all t ≥ 0 and (ψt | A ) is a continuous group of automorphisms of A. Moreover, the divergence rate satisfies λ c (φt | ) = c A (ψt | A ) = − . 2 Proof Let (D, g, ηt ) be a pre-model for (φt ) at σ . By Proposition 13.4.12, g(D) = . Hence, (, g, φt | ) is a model for (ηt ). By Lemma 9.3.1, cD (ηt ) = c (φt | ). By Theorem 9.1.9, it follows that cD (ηt ) = − λ2 , since, by definition of pre-model, (ηt ) is a hyperbolic group with spectral value −λ. Hence, λ c (φt | ) = − . 2

(13.5.1)

Now, let A = h(). From A = h() = h(φt ()) = ψt (h()) = ψt (A), t ≥ 0, it follows that (ψt | A ) is a continuous group of automorphisms of A and that (A, h ◦ g, (ψt | A )) is a model for (ηt ). Hence, the last statement on the rates of divergence follows by Lemma 9.3.1 and (13.5.1).  In order to properly set the results, we need to introduce spirallike sectors. See Section 11.1 for the definition of the spiral spir μ [eiθ ]. Definition 13.5.2 Let μ ∈ C, Re μ > 0, α ∈ (0, π ] and θ0 ∈ [−π, π ). A μspirallike sector of amplitude 2α and center eiθ0 is Spir[μ, 2α, θ0 ] := {etμ+iθ : t ∈ R, θ ∈ (−α + θ0 , α + θ0 )} spir μ [eiθ ] ∩ (C \ {0}) = θ∈(θ0 −α,θ0 +α)

(see Fig. 13.1). Remark 13.5.3 Notice that a μ-spirallike sector is not a μ-spirallike domain in the sense of Definition 9.4.2 since the point 0 is in the boundary of the domain, but not inside.

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13 Fixed Points, Backward Invariant Sets and Petals

Spir Fig. 13.1 μ-spirallike sector

Lemma 13.5.4 Let D := Spir[μ, 2α, θ0 ] be a μ-spirallike sector for some μ ∈ C, Re μ > 0 and α ∈ (0, π ]. Let ψt (z) = e−μt z, for z ∈ C. Then (ψt | D ) is a continuous group of automorphisms of D and c D (ψt | D ) =

|μ|2 π . 4αRe μ

Proof The map z → e−iθ0 z is a biholomorphism between D and Spir[μ, 2α, 0] and conjugates (ψt | D ) to (ψt |Spir[μ,2α,0] ). Since a biholomorphism is an isometry with respect to the hyperbolic distance, the divergence rate of (ψt | D ) and (ψt |Spir[μ,2α,0] ) is the same. Hence, we can assume θ0 = 0. It is clear that ψt (D) = D for all t ≥ 0, hence, (ψt | D ) is a continuous group of automorphisms of D. Since D is simply connected and 0 ∈ / D, it is well defined a Im μ 1−i Re μ holomorphic branch of f : D w → w ∈ C. A straightforward computation shows that D := f (D) = {ρeiθ : ρ > 0, θ ∈ (−α, α)}. Moreover, ( f ◦ ψt ◦ f −1 ) is a continuous group of automorphisms of D . A direct computation shows −μt ψ˜ t (z) := ( f ◦ ψt ◦ f −1 )(z) = e



Im μ 1−i Re μ



|μ|2

z = e−t Re μ z,

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387 π

for all t ≥ 0. Now, consider the function g : D z → z − 2α ∈ C. Then g(D ) = H and (g ◦ ψ˜ t ◦ g −1 ) is a continuous group of automorphisms of H. A direct computaπ|μ|2 tion shows that ηt (z) := (g ◦ ψ˜ t ◦ g −1 )(z) = et 2αRe μ z for all t ≥ 0. By Proposition 8.3.8, (ηt ) is a group in H which is conjugated via the Cayley transform C : D → H, given by C(z) = (1 + z)/(1 − z), to a hyperbolic group (η˜ t ) in D with spectral value π|μ|2 . Hence, by Theorem 9.1.3, 2αRe μ cD (η˜ t ) =

π |μ|2 . 4αRe μ

Since by construction, (D, f −1 ◦ g −1 ◦ C, (ψt | D )) is a holomorphic model for (η˜ t ),  it follows c(η˜ t ) = c D (ψt | D ) by Lemma 9.3.1, and the result is proved. We are now ready to relate petals of a semigroup with the shape of the image of the corresponding Koenigs function. As a matter of notation, if D ⊂ C is a μ-starlike domain with respect to 0 for some μ ∈ C, Re μ > 0, we say that a μ-spirallike sector Spir[μ, 2α, θ0 ] ⊂ D (for some α ∈ [0, π ) and θ0 ∈ [−π, π )) is maximal in D if there exist no θ1 ∈ [−π, π ), β ∈ (0, 2π ] such that Spir[μ, 2α, θ0 ] ⊂ Spir[μ, β, θ1 ] ⊂ D and Spir[μ, 2α, θ0 ] = Spir[μ, β, θ1 ]. Similarly, if D ⊂ C is starlike at infinity, z 0 ∈ C, ρ > 0, the strip (Sρ + z 0 ) ⊂ D is maximal in D if there exist no r > 0 and z 1 ∈ C such that (Sρ + z 0 ) ⊂ (Sr + z 1 ) ⊂ D and (Sρ + z 0 ) = (Sr + z 1 ). Theorem 13.5.5 Let (φt ) be a semigroup, not a group, in D. Let τ ∈ D be the Denjoy-Wolff point of (φt ) and μ its spectral value. Let h be the Koenigs function of (φt ). Suppose  is a hyperbolic petal for (φt ), let σ ∈ ∂ ∩ ∂D be the unique repelling fixed point for (φt ) in , and let λ ∈ (−∞, 0) be the repelling spectral value of (φt ) at σ . (1) If τ ∈ D, then there exists θ0 ∈ [−π, π ) such that h() is a maximal μ-spirallike |μ|2 π sector of h(D) of center eiθ0 and amplitude − λRe , i.e., μ h() = Spir[μ, −

|μ|2 π , θ0 ]. λRe μ

(2) If τ ∈ ∂D, then there exists z 0 ∈ C such that h() is a maximal strip z 0 + S− πλ in h(D), i.e., h() = z 0 + S− πλ . Proof We first consider the elliptic case. The canonical model is (C, h, z → e−μt z). Since φt () =  for all t ≥ 0, it follows that e−μt h() = h() and z → e−μt z is an automorphism of h() for all t ≥ 0. In particular, e−μt h() = h() for all t ∈ R. It follows easily that h() = Spir[μ, 2α, θ0 ] for some θ0 ∈ [−π, π ) and α ∈ (0, π ]. |μ|2 π . By Lemma 13.5.1 and Lemma 13.5.4 it follows at once that 2α = − λRe μ

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13 Fixed Points, Backward Invariant Sets and Petals

|μ| π We are left to prove that Spir[μ, − λRe , θ0 ] is maximal. Suppose this is not the μ case. Therefore, there exist θ1 ∈ [−π, π ), β ∈ (0, 2π ] such that D := Spir[μ, β, θ1 ] |μ|2 π , θ0 ] is properly contained in D. Therefore, (e−μt z| D ) is a ⊂ h(D) and Spir[μ, − λRe μ continuous group of automorphisms of D. Since D ⊂ h(D), it follows φt (h −1 (z)) = h −1 (e−μt z) for every z ∈ h(D), hence φt (h −1 (D)) = h −1 (D) for all t ≥ 0. This implies that h −1 (D) is an open connected component in the backward invariant set of (φt ) which properly contains , a contradiction. The proof in case (φt ) is non-elliptic is similar and we just sketch it. In this case the model is (Ω, h, z → z + it), where Ω = C, H, H− or Sμ . Hence, either h() = z 0 + Sρ for some ρ > 0 or h() is a half-plane. In this latter case however, by Proposition 9.3.2, the divergence rate of (z + it) on h() is 0 against Lemma 13.5.1. Hence, h() = z 0 + Sρ for some ρ > 0. By Lemma 13.5.1, (z → z + it) is a continuous group of automorphisms of z 0 + Sρ whose divergence rate is −λ/2. Up to a translation of z 0 , by Theorem 9.3.5, this is the canonical model of a hyperbolic group of D with divergence rate −λ/2, that is, with spectral value −λ by Theorem 9.1.9. Hence, by Theorem 9.3.5, ρ = −π/λ. Arguing as in the elliptic case, one can easily see that such a strip is maximal in h(D).  2

The converse of the previous theorem is also true: Theorem 13.5.6 Let (φt ) be a semigroup, not a group, in D. Let τ ∈ D be the Denjoy-Wolff point of (φt ) and μ its spectral value. Let h be the Koenigs function of (φt ). (1) If τ ∈ D and there exist θ0 ∈ [−π, π ) and β ∈ (0, 2π ] such that Spir[μ, β, θ0 ] ⊂ h(D) is a maximal μ-spirallike sector of h(D), then h −1 (Spir[μ, β, θ0 ]) is a hyperbolic petal for (φt ). Moreover, if σ ∈ ∂D ∩ ∂ is the unique repelling fixed point of (φt ) contained in , then the repelling spectral value of (φt ) at σ |μ|2 π is λ = − βRe . μ (2) If τ ∈ ∂D and there exist z 0 ∈ C and ρ > 0 such that z 0 + Sρ ⊂ h(D) is a maximal strip of h(D), then h −1 (z 0 + Sρ ) is a hyperbolic petal for (φt ). Moreover, if σ ∈ ∂D ∩ ∂ is the unique repelling fixed point of (φt ) contained in , then the repelling spectral value of (φt ) at σ is λ = − πρ . Proof (1) The canonical model is (C, h, z → e−μt z). Let D := Spir[μ, β, θ0 ]. Since φt (h −1 (z)) = h −1 (e−μt z) for all z ∈ h(D) and t ≥ 0, it follows at once that, for all t ≥ 0, φt (h −1 (D)) = h −1 (D). Hence, there exists a petal  such that h −1 (D) ⊆ . However, if  = h −1 (D), then by Theorem 13.5.5, h() is a spirallike sector properly containing D, against the maximality of D. Therefore,  = h −1 (D) and, the result follows from Theorem 13.5.5. (2) The argument is similar and we omit it.  Finally, we turn our attention to parabolic petals. Recall that only parabolic semigroups can have parabolic petals (Remark 13.4.5). As a matter of notation, if W ⊂ C is a domain starlike at infinity and a ∈ R, we say that a half-plane {w ∈ C : Re w > a} ⊂ W (respectively {w ∈ C : Re w < a} ⊂ W )

13.5 Petals and the Geometry of Koenigs Functions

389

is maximal if {w ∈ C : Re w > b} ⊂ W for every b < a (resp. {w ∈ C : Re w < b} ⊂ W for every b > a). Theorem 13.5.7 Let (φt ) be a parabolic semigroup, not a group, in D, with DenjoyWolff point τ ∈ ∂D. Let h be the Koenigs function of (φt ). If  is a parabolic petal for (φt ) then h() is a maximal half-plane in h(D). Conversely, if H ⊂ h(D) is a maximal half-plane in h(D) then h −1 (H ) is a parabolic petal for (φt ). Moreover, (φt ) can have at most two parabolic petals and, if this is the case, (φt ) has zero hyperbolic step. Proof Let (Ω, h, z → z + it) be the canonical model of (φt ), where Ω = C, H or H− . Let  be a parabolic petal for (φt ). Since φt () =  for all t ≥ 0, and h() = h(φt ()) = h() + it for all t ≥ 0, it follows that h() + it = h() for all t ≥ 0, and, hence, h() + it = h() for all t ∈ R. Therefore, either h() is a strip Sρ + z 0 for some ρ > 0 and z 0 ∈ h(D) or h() is a half-plane. Arguing as in the proof of Theorem 13.5.5 it is easy to see that h() is maximal in h(D). Therefore, if h() is a maximal strip, the petal  is hyperbolic by Theorem 13.5.6, contradicting our hypothesis. Hence, h() is a maximal half-plane in h(D). Conversely, if H ⊂ h(D) is a maximal half-plane in h(D), then arguing as in the proof of Theorem 13.5.6, it follows that h −1 (H ) is a petal. Moreover, by Theorem 13.5.5, h −1 (H ) cannot be hyperbolic, hence, it is parabolic. It is clear that a domain starlike at infinity (different from C) can contain at most two maximal half-planes, one given by {w ∈ C : Re w < a} and the other given by {w ∈ C : Re w > b}, for some −∞ < a ≤ b < +∞. Hence, (φt ) can have at most two parabolic petals. Finally, assume that (φt ) has two parabolic petals 1 , 2 . Hence, there exist −∞ < a ≤ b < +∞ such that h(1 ) = {w ∈ C : Re w < a} and h(2 ) = {w ∈ C : Re w > b}. This implies that h(D) is not contained in H or H− , and, by Theorem  9.3.5, it follows that Ω = C and (φt ) has zero hyperbolic step.

13.6 Analytic Properties of Koenigs Functions at Boundary Fixed Points In this section we study the analytic behavior of Koenigs functions at boundary fixed points. We start with the following characterization: Proposition 13.6.1 Let (φt ) be a semigroup, not a group, in D with Denjoy-Wolff point τ ∈ D and Koenigs function h. Then σ ∈ ∂D is a fixed point of (φt ) if and only if ∠ lim z→σ h(z) = ∞ ∈ C∞ . Moreover, if σ = τ , then the unrestricted limit lim z→σ h(z) = ∞. Proof By Remark 12.1.5, if σ ∈ ∂D is a fixed point of (φt ) then ∠ lim z→σ h(z) = ∞. The converse is essentially contained in the proof of Theorem 12.1.4: suppose ∠ lim z→σ h(z) = ∞. Then by Theorem 11.2.3, φt (σ ) ∈ ∂D and h(φt (σ )) = ∞ for

390

13 Fixed Points, Backward Invariant Sets and Petals

all t ≥ 0. By Proposition 11.2.2, the curve [0, +∞) t → φt (σ ) ∈ ∂D is continuous, and hence, if σ is not a fixed point, its image contains an open arc A ⊂ ∂D. But then h would have constant non-tangential at each point of A, contradicting Proposition 3.3.2. Finally, if σ = τ , let x σ ∈ ∂C D be the prime end corresponding to σ . Let ˆ → h(D)  be the homeomorphism in the Carathéodory topology defined by ˆh : D ˆ σ )) = {∞}. Hence, by Theorem h. Then, by Theorem 4.4.9, the principal part (h(x ˆ σ )) = {∞}. By Proposition 4.4.4, 11.1.2 and Theorem 11.1.4, the impression I (h(x  lim z→σ h(z) = ∞. Proposition 13.6.1 asserts that, if (φt ) is elliptic, then a point σ ∈ ∂D is a fixed point of (φt ) if and only if the associated Koenigs function has unrestricted limit ∞ at σ . In the non-elliptic case, the latter condition implies that σ is a boundary fixed point, but does not assure that σ = τ . One can characterize boundary fixed points other than the Denjoy-Wolff point as follows: Proposition 13.6.2 Let (φt ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D and Koenigs function h. Let σ ∈ ∂D. Then the following are equivalent: (1) σ is a boundary fixed point of (φt ) and σ = τ , (2) limr →1 Im h(r σ ) = −∞, (3) lim z→σ Im h(z) = −∞. Proof Assume that σ is a boundary fixed point of the semigroup other than τ . By Proposition 13.6.1, lim z→σ h(z) = ∞, and (3) follows from Proposition 11.1.9. Being clear that (3) implies (2), it remains to show that (2) implies (1). Taking into account that h has non-tangential limit at σ by Corollary 11.1.7, condition (2) clearly implies ∠ lim z→∞ h(z) = ∞, hence, by Proposition 13.6.1, σ is a boundary fixed point of (φt ). If σ = τ , let (0, 1) r → g(r ) := Im h(r σ ). Then Theorem 9.4.11 implies g (r ) = Im σ h (r σ ) =

 1 Im σ (r σ − σ )2 h (r σ ) ≥ 0, 0 < r < 1. 2 (1 − r )

and limr →1 Im h(r σ ) cannot be −∞. Thus σ = τ .



Now we characterize repelling and super-repelling fixed points via Koenigs functions. In order to properly deal with the elliptic case we need to introduce some terminology. Let λ ∈ C, Re λ > 0. Every point w ∈ C \ {0} can be written in a unique way in λ-spirallike coordinates as w = e−λt+iθ , where t ∈ R and θ ∈ [−π, π ). We define Argλ (w) := θ, and we call it the λ-spirallike argument of w .

13.6 Analytic Properties of Koenigs Functions at Boundary Fixed Points

391

Theorem 13.6.3 Let (φt ) be an elliptic semigroup in D, not a group, with DenjoyWolff point τ ∈ D and spectral value λ ∈ C with Re λ > 0. Let h be the associated Koenigs function and σ ∈ ∂D. The following are equivalent: (1) σ is a repelling fixed point of (φt ), (2) lim z→σ |h(z)| = ∞ and ∠ lim inf Argλ (h(z)) = ∠ lim sup Argλ (h(z)), z→σ

z→σ

(3) lim z→σ |h(z)| = ∞ and lim inf Argλ (h(z)) = lim sup Argλ (h(z)). z→σ

z→σ

Moreover, if σ is a repelling fixed point for (φt ) with repelling spectral value ν ∈ (−∞, 0), then there exists θ0 ∈ [−π, π ) such that if {z n } ⊂ D is a sequence converging to σ and limn→∞ Arg(1 − σ z n ) = β ∈ (−π/2, π/2), then lim Argλ (h(z n )) = θ0 +

n→∞

β|λ|2 νRe λ

mod [−π, π ).

Proof (1) implies (2). Suppose σ is a repelling fixed point for (φt ) with spectral value ν ∈ (−∞, 0). By Proposition 13.6.1, lim z→σ |h(z)| = ∞. By Theorem 13.5.5 there exists a hyperbolic petal  such that σ ∈ ∂ and h() = Spir[λ, 2α, θ0 ] is a maximal λ-spirallike sector of amplitude 2α and center eiθ0 in h(D), where θ0 ∈ [0, 2π ) and 2α = −π |λ|2 /νRe λ. Let (D, g, ηt ) be a pre-model for (φt ) at σ . By Im λ Proposition 13.4.12, g(D) = . Now, take a holomorphic branch of f : z → z 1−i Re λ such that the image of Spir[λ, 2α, θ0 ] is V = {ρeiθ : ρ > 0, θ ∈ (θ0 − α, θ0 + α)}. Let then k : V → H be defined by k(w) = w−π/2α . By construction, C := k ◦ f ◦ h ◦ g : D → H is a biholomorphism. Hence, C is a Möbius transformation, and, looking at the definition and taking into account that lim z→σ |h(z)| = ∞, we see that C(σ ) = 0 and C(−σ ) = ∞. Therefore, C(z) =

σ −z σ +z

Now, let {z n } ⊂ D be a sequence converging to σ such that limn→∞ Arg(1 − σ z n ) = β ∈ (−π/2, π/2). By Proposition 13.4.12, {z n } is eventually contained in g(D) and, without loss of generality, we can assume {z n } ⊂ g(D). Let wn := g −1 (z n ). Since ∠ lim z→σ g(z) = σ and g is semi-conformal at σ , it follows at once that limn→∞ Arg(1 − σ wn ) = β. Hence, limn→∞ Arg(C(wn )) = β. Therefore, taking into account that Argλ f −1 (w) = Arg(w) for all w ∈ V , we have lim Argλ (h(z n )) = lim Argλ ( f −1 ◦ k −1 ◦ C ◦ g −1 )(z n ) = lim Arg(k −1 ◦ C)(wn ). n→∞ n→∞

n→∞

392

13 Fixed Points, Backward Invariant Sets and Petals

Taking into account that k −1 (z) = eiθ0 z −2α/π , the previous equation gives immediately 2αβ lim Argλ (h(z n )) = θ0 − mod [−π, π ). n→∞ π This proves (2) and, since 2α = −π |λ|2 /νRe λ, the final part of the statement. Clearly, (2) implies (3). (3) implies (1). Since lim z→σ |h(z)| = ∞ implies lim z→σ h(z) = ∞ in C∞ , Proposition 13.6.1 immediately implies that σ is a boundary fixed point of (φt ). We have to show that σ is repelling. Let x σ ∈ ∂C D be the prime end representing  be the homeomorphism in the σ , given by Proposition 4.2.5, and let hˆ : D → h(D) Carathéodory topology defined by h. Since lim z→σ h(z) = ∞, by Proposition 4.4.4, ˆ σ )) = {∞}. Hence, by Proposition 4.1.11, we can find a circular null we have I (h(x ˆ σ ) such that there exists an increasing sequence of poschain (Cn ) representing h(x itive real numbers {Rn } converging to +∞ such that Cn ⊂ {z ∈ C : |z| = Rn } for 1 2 every n ∈ N0 . For every n ∈ N0 , let e−λtn +iθn and e−λtn +iθn be the end points of Cn , j −tn Re λ = Rn and θn ∈ [−π, π ) j = 1, 2 with θn1 ≤ θn2 . where tn ∈ R is such that e 1 If θn1 = θn2 for all n ∈ N, then Cn ∩ h(D) = {z ∈ C : |z| = Rn } \ {e−λtn +iθn } for all n ∈ N. Since h(D) is λ-spirallike, it follows that for all n ∈ N, h(D) ∩ (spir λ [e−λtn +iθn ] ∩ {w ∈ C : |w| ≥ Rn }) = ∅. 1

Hence, the only possibility is that θn1 is constant for all n, say, θn1 =: θ0 . That is, h(D) = C \ (spir λ [eiθ0 ] ∩ {w ∈ C : |w| ≥ R}), for some R ∈ (0, R0 ]. It is then clear that C \ spir λ [eiθ0 ] is a maximal λ-spirallike sector in h(D) of amplitude 2π . By Theorem 13.5.6, there exists a hyperbolic petal  ⊂ D such that h() = C \ spir λ [eiθ0 ]. Moreover, if J := (spir λ [eiθ0 ] ∩ {w ∈ C : |w| < R}), then  = D \ h −1 (J ). Note that, by Proposition 3.3.3, the closure of h −1 (J ) is a Jordan arc with end points τ ∈ D and a point p ∈ ∂D. Hence, ∂ = h −1 (J ) ∪ ∂D. By Proposition 13.4.10, ∂ contains only one boundary fixed point of (φt ), which is repelling. Since σ ∈ ∂ is a fixed point, it follows that σ is repelling. Now, we assume that there exists n 0 ∈ N such that θn10 < θn20 . Up to considering the equivalent null chain (Cn )n≥n 0 , we can assume n 0 = 0. Since h(D) is λ-spirallike, j h(D) ∩ (spir λ [eiθ0 ] ∩ {w ∈ C : |w| ≥ R0 }) = ∅, j = 1, 2. Hence, it is easy to see that 1 θn1 < θn2 for all n ≥ 0. Let Vn be the interior part of Cn , n ≥ 1. Since spir λ [eiθ0 ] ∪ 2 spir λ [eiθ0 ], forms a Jordan curve J in C∞ containing 0 and ∞, taking into account that h(D) is λ-spirallike, it follows that Vn is contained in one of the connected component of C \ J . Thus, taking also into account that for every w ∈ Cn , we have (spir λ [w] ∩ {w ∈ C : |w| < Rn }) ⊂ h(D), we have two possibilities. Either θn2 + θn1 ] ∩ {w ∈ C : |w| > Rn }), 2 θ 2 + θn1 ] ∩ {w ∈ C : |w| < Rn }) ⊂ h(D) (Spir[λ, θn2 − θn1 , n 2 Vn ⊆ (Spir[λ, θn2 − θn1 ,

(13.6.1)

13.6 Analytic Properties of Koenigs Functions at Boundary Fixed Points

or, setting ξn :=

θn2 +θn1 +2π 2

393

mod 2π ,

Vn ⊆ (Spir[λ, θn1 − θn2 + 2π, ξn ] ∩ {w ∈ C : |w| > Rn }), (Spir[λ, θn1 − θn2 + 2π, ξn ] ∩ {w ∈ C : |w| < Rn }) ⊂ h(D).

(13.6.2)

If (13.6.1) holds for some n = n 0 , since Vn ⊂ Vn 0 for all n ≥ n 0 , (13.6.1) implies that θn1 < θn2 and (13.6.1) holds for all n ≥ n 0 . Assume we are in this case—the proof for the case (13.6.2) holds for every n is similar and we omit it. As before, we can assume n 0 = 0. By hypothesis, there exist two sequences {z m }, {wm } ⊂ D converging to σ such that α := limm→∞ Argλ (h(z m )) and β := limm→∞ Argλ (h(wm )) exist and α < β. Let us write h(z m ) = e−λrm +iαm and h(wm ) = e−λsm +iβm , where rm , sm ∈ R and αm , βm ∈ [−π, π ) such that limm→∞ αm = α and limm→∞ βm = β. Since {z m } and {wm } are converging to σ in the Euclidean topology, it follows by Proposition 4.2.5 that they also converge to x σ in the Carathéodory ˆ σ ) in the Carathéodory topology of D. Hence, {h(z m )} and {h(wm )} converge to h(x topology of h(D). By Remark 4.2.2, it follows that for all n ∈ N there exists m n ∈ N such that h(z m ), h(wm ) ∈ Vn for all m ≥ m n . By (13.6.1), we have θn1 < αm , βm < θn2 for all n and m ≥ m n , and, taking the limit in m, we obtain θn1 ≤ α < β ≤ θn2 , n ∈ N.

(13.6.3)

Let θ0 := (β + α)/2 and a ∈ (0, β − α). Equation in (13.6.3) implies at once that Spir[λ, a, θ0 ] ∩ {w ∈ C : |w| > Rn } ⊂ Vn for all n ∈ N and Spir[λ, a, θ0 ] ⊂ h(D). By Theorem 13.5.6 there exists a hyperbolic petal  ⊂ D such that Spir[λ, a, θ0 ] ⊆ h(). Moreover, by Proposition 13.4.10, ∂ contains only one boundary fixed point of (φt ) which is repelling. Thus, if we prove that σ ∈ ∂, it follows that σ is repelling. To this aim, consider the curve γ : (−∞, 0) t → e−tλ+iθ0 . Since for all n ∈ N there exists tn ∈ (−∞, 0) such that γ (t) ∈ Spir[λ, a, θ0 ] ∩ {w ∈ C : |w| > Rn } ⊂ Vn for all t ≤ tn , it follows by Remark 4.2.2 that γ (t) converges in the Carathéodory ˆ σ ) as t → −∞. Hence, by Proposition 4.2.5, h −1 (γ (t)) → σ topology of h(D) to h(x as t → −∞, proving that σ ∈ ∂.  As an immediate corollary of Proposition 13.6.1 and Theorem 13.6.3 we have: Corollary 13.6.4 Let (φt ) be an elliptic semigroup in D, not a group. Let h be the associated Koenigs function and σ ∈ ∂D. The following are equivalent: (1) σ is a super-repelling fixed point of (φt ), (2) lim z→σ |h(z)| = ∞ and lim z→σ Argλ (h(z)) = θ0 for some θ0 ∈ [−π, π ]. Next, we characterize repelling fixed points of elliptic semigroups in terms of the derivative of the associated Koenigs function: Proposition 13.6.5 Let (φt ) be an elliptic semigroup in D, not a group, with DenjoyWolff point τ ∈ D, Koenigs function h and infinitesimal generator G. Let σ ∈ ∂D. Then the following are equivalent:

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13 Fixed Points, Backward Invariant Sets and Petals

(1) the point σ is a repelling fixed point of (φt ); log |h(r σ )| = ρ; (2) there exists ρ > 0 such that lim (0,1) r →1 Re G (τ ) log(1 − r ) |h (r σ )|(1 − r ) (3) lim sup > 0. |h(r σ )| (0,1) r →1 Moreover, if the previous conditions are satisfied, then |G(r σ )| > 0, r →1 1 − r

 := lim ρ −1 =  and ∠ lim

z→σ

G (τ ) h (z)(z − σ ) = . h(z) 

Proof By Proposition 12.2.4 and Theorem 12.2.5, σ is a boundary regular fixed point of the semigroup if and only if G (σ ) = ∠ lim

z→σ

G(z) ∈ R. z−σ

Moreover, since σ is not the Denjoy-Wolff point of the semigroup, G (σ ) ∈ (0, +∞). By Theorem 10.1.4, G(z) = −λ hh(z) (z) for all z ∈ D, where λ is the spectral value of (φt ). Therefore, σ is a boundary regular fixed point of the semigroup if and only if ∠ lim (−λ) z→σ

h(z) ∈ (0, +∞). h (z)(z − σ )

Hence, (1) clearly implies (3). Conversely, if (3) holds, then |G(r σ )| |h(r σ )| = lim inf < +∞ (0,1) r →1 |λ|(1 − r ) (0,1) r →1 |h (r σ )|(1 − r ) lim inf

and Proposition 12.2.4 guarantees that σ is a boundary regular fixed point of the semigroup. h(z) Since the function D z → z−τ has no zeros, the function   h(r σ ) g(r ) := Re log r σ −τ , r ∈ (0, 1), is well-defined. By L’Hôpital’s Rule [122, p. 180], it follows that lim

r →1

log |h(r σ )| g(r ) = lim = lim g (r )(1 − r ) r →1 − log(1 − r ) − log(1 − r ) r →1   σ (1 − r ) σ (1 − r ) σ (1 − r ) − = − lim Re λ , = lim Re −λ r →1 r →1 G(r σ ) rσ − τ G(r σ )

from which we get the equivalence between (1) implies (2), and also that ρ −1 =  ) = G (τ ) .  and ∠ lim z→σ h (z)(z−σ h(z)

13.6 Analytic Properties of Koenigs Functions at Boundary Fixed Points

395

Now we turn our attention to non-elliptic semigroups. The proofs of the next results are similar in spirit to those for the elliptical case. In fact, roughly speaking, in the non-elliptic case, the role of the modulus is played by the imaginary part and that of the λ-argument by the real part. Theorem 13.6.6 Let (φt ) be a non-elliptic semigroup in D, not a group, with DenjoyWolff point τ ∈ ∂D and spectral value λ ≥ 0. Let h be the associated Koenigs function and σ ∈ ∂D. The following are equivalent: (1) σ is a repelling fixed point of (φt ), (2) lim z→σ Im h(z) = −∞ and −∞ < ∠ lim inf Re h(z) = ∠ lim sup Re h(z) < +∞, z→σ

z→σ

(3) lim z→σ Im h(z) = −∞ and −∞ < lim inf Re h(z) = lim sup Re h(z) < +∞. z→σ

z→σ

Moreover, if σ is a repelling fixed point for (φt ) with repelling spectral value ν ∈ (−∞, 0), then there exists a ∈ R such that if {z n } ⊂ D is a sequence converging to σ and limn→∞ Arg(1 − σ z n ) = β ∈ (−π/2, π/2), then lim Re h(z n ) = a +

n→∞

β . ν

Proof (1) implies (2). Assume σ is a repelling fixed point for (φt ) with repelling spectral value ν ∈ (−∞, 0). By Proposition 13.6.2, lim z→σ Im h(z) = −∞. Moreover, by Proposition 11.1.9, lim inf z→σ Re h(z) > −∞ and lim supz→σ Re h(z) < +∞. We are left to prove that ∠ lim inf z→σ Re h(z) = ∠ lim supz→σ Re h(z). By Theorem 13.5.5 there exists a hyperbolic petal  such that σ ∈ ∂ and h() = Sρ + z 0 is a maximal strip in h(D) with ρ = − πν . Let (D, g, ηt ) be a pre-model π log z − 2ν + z0 . for (φt ) at σ . By Proposition 13.4.12, g(D) = . Let f (z) := −i ν −1 Then f : H → Sρ + z 0 is a biholomorphism. Thus, C := f ◦ h ◦ g : D → H is a biholomorphism, hence, it is a Möbius transformation. By construction (taking into account that lim z→σ Im h(z) = −∞) it follows that C(σ ) = 0 and C(−σ ) = ∞, i.e., . C(z) = σσ −z +z Now, let {z n } ⊂ D be a sequence converging to σ such that limn→∞ Arg(1 − σ z n ) = β ∈ (−π/2, π/2). By Proposition 13.4.12, {z n } is eventually contained in g(D) and, without loss of generality, we can assume {z n } ⊂ g(D). Let wn := g −1 (z n ). By Lemma 13.2.3, the sequence {wn } converges non-tangentially to σ . Since ∠ lim z→σ g(z) = σ and g is semi-conformal at σ , it follows at once that limn→∞ Arg(1 − σ wn ) = β. Hence, limn→∞ Arg C(wn ) = β. Therefore, we have lim Re h(z n ) = lim Re ( f ◦ C ◦ g −1 )(z n ) = lim Re ( f ◦ C)(wn ) =

n→∞

n→∞

Setting a := Re z 0 −

n→∞

π 2ν

β π + Re z 0 − . ν 2ν

∈ R we have (2) and the final part of the statement.

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13 Fixed Points, Backward Invariant Sets and Petals

Clearly (2) implies (3). In order to prove that (3) implies (1), we see that lim z→σ Im h(z) = −∞ implies that σ is a boundary fixed point of (φt ) by Proposition 13.6.2. We are left to show that σ is repelling. Let x σ ∈ ∂C D be the prime end representing  be the homeomorphism in the σ , given by Proposition 4.2.5, and let hˆ : D → h(D) Carathéodory topology defined by h. Since lim z→σ h(z) = ∞, by Proposition 4.4.4, ˆ σ )) = {∞}. Hence, by Remark 11.1.5, we can find a circular null we have I (h(x ˆ σ ) such that for every M < 0 there exists n R such that chain (Cn ) representing h(x Cn ⊂ {w ∈ C : Im w < M} for all n ≥ n R . Let {Rn } be an increasing sequence of positive real numbers converging to +∞ such that Cn ⊂ {z ∈ C : |z| = Rn } for every n ∈ N0 . For every n ∈ N0 , let an1 + ibn1 and an2 + ibn2 be the end points of Cn , where j j an , bn ∈ R, j = 1, 2, and an1 ≤ an2 . Without loss of generality, up to replace (Cn ) by the equivalent null chain (Cn )n≥n 0 , we can assume Cn ⊂ {w ∈ C : Im w < 0} for all n ∈ N. Hence, Cn ⊂ {z ∈ C : |z| = Rn , Im z < 0}, which implies at once that an1 < an2 and Im ζ ≤ max{bn1 , bn2 } for all ζ ∈ Cn , n ∈ N0 . Let Vn be the interior part of Cn , n ≥ 1. Since h(D) is starlike at infinity, the previous considerations show Vn ⊂ {w ∈ C : an1 < Re w < an2 , Im w < max{bn1 , bn2 }},

(13.6.4)

{w ∈ C : an1 < Re w < an2 , Im w > max{bn1 , bn2 }} ⊂ h(D).

(13.6.5)

and By hypothesis, there exist two sequences {z m }, {wm } ⊂ D converging to σ such that α := limm→∞ Re h(z m ) and β := limm→∞ Re h(wm ) exist and α < β. Let write h(z m ) = αm + ism and h(wm ) = βm + irm , where αm , βm , rm , sm ∈ R, limm→∞ αm = α, limm→∞ βm = β and limm→∞ sm = limm→∞ rm = −∞. Since {z m } and {wm } are converging to σ in the Euclidean topology, it follows by Proposition 4.2.5 that they also converge to x σ in the Carathéodory topology of D. Hence, {h(z m )} and ˆ σ ) in the Carathéodory topology of h(D). By Remark 4.2.2, {h(wm )} converge to h(x it follows that for all n ∈ N there exists m n ∈ N such that h(z m ), h(wm ) ∈ Vn for all m ≥ m n . By (13.6.4), we have an1 < αm , βm < an2 for all n and m ≥ m n , and, taking the limit in m, we obtain an1 ≤ α < β ≤ an2 , n ∈ N.

(13.6.6)

By (13.6.5), taking into account that limn→∞ max{bn1 , bn2 } = −∞, we see that the strip Sβ−α + (α + β)/2 is contained in h(D). Moreover, since h(D) is starlike j j at infinity and {z = a0 + it, t ∈ (−∞, b0 )} ⊂ C \ h(D), j = 1, 2 it follows that Sβ−α + (α + β)/2 cannot be contained in a maximal half-plane of h(D). Hence, there exists a maximal strip S of h(D) which contains Sβ−α + (α + β)/2. By Theorem 13.5.6 there exists a hyperbolic petal  ⊂ D such that (Sβ−α + α) ⊆ h(). Moreover, by Proposition 13.4.10, ∂ contains only one boundary fixed point of (φt ) which is repelling. Thus, if we prove that σ ∈ ∂, it follows that σ is repelling. To this aim, consider the curve γ : (0, +∞) t → (α + β)/2 − it. Since for all

13.6 Analytic Properties of Koenigs Functions at Boundary Fixed Points

397

n ∈ N there exists tn ∈ (0, +∞) such that γ (t) ∈ (Sβ−α + α) ∩ {w ∈ C : Im w < min{bn1 , bn2 }} ⊂ Vn for all t ≥ tn , it follows by Remark 4.2.2 that γ (t) converges in ˆ σ ) as t → +∞. Hence, by Proposition the Carathéodory topology of h(D) to h(x −1  4.2.5, h (γ (t)) → σ as t → +∞, proving that σ ∈ ∂. As an immediate corollary of Proposition 13.6.1, Proposition 11.1.9 and Theorem 13.6.6 we have: Corollary 13.6.7 Let (φt ) be a non-elliptic semigroup in D, not a group. Let h be the associated Koenigs function and σ ∈ ∂D. The following are equivalent: (1) σ is a super-repelling fixed point of (φt ), (2) lim z→σ Im h(z) = −∞ and lim z→σ Re h(z) = a for some a ∈ R. Finally, we characterize repelling fixed points of a non-elliptic semigroup in terms of the derivative of the associated Koenigs function: Proposition 13.6.8 Let (φt ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D, Koenigs function h and infinitesimal generator G. Let σ ∈ ∂D \ {τ }. Then the following are equivalent: (1) the point σ is a boundary regular fixed point of (φt ); Im h(r σ ) (2) there exists ρ > 0 such that lim = ρ; (0,1) r →1 log(1 − r ) (3) lim sup |h (r σ )|(1 − r ) > 0. (0,1) r →1

Moreover, if the previous conditions are satisfied, then  := lim

r →1

ρ −1 =  and ∠ lim h (z)(z − σ ) = z→σ

|G(r σ )| > 0, 1−r

1 . 

Proof By Proposition 12.2.4 and Theorem 12.2.5, σ is a boundary regular fixed point of the semigroup if and only if G (σ ) = ∠ lim

z→σ

G(z) ∈ R. z−σ

Moreover, since σ is not the Denjoy-Wolff point of the semigroup, G (σ ) ∈ (0, +∞). By Theorem 10.1.4, G(z) = h i(z) for all z ∈ D. Therefore, σ is a boundary regular fixed point of the semigroup if and only if ∠ lim

z→σ

i h (z)(z

− σ)

∈ (0, +∞).

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13 Fixed Points, Backward Invariant Sets and Petals

Hence, (1) clearly implies (3). Conversely, if (3) holds, then lim inf

(0,1) r →1

|G(r σ )| 1 = lim inf < +∞ (0,1) r →1 |h (r σ )|(1 − r ) 1−r

and Proposition 12.2.4 guarantees that σ is a boundary regular fixed point of the semigroup. Consider the function g(r ) := Im h(r σ ), r ∈ (0, 1). By L’Hôpital’s Rule [122, page 180], it follows that lim

r →1

Im h(r σ ) g(r ) = lim = lim g (r )(1 − r ) r →1 − log(1 − r ) − log(1 − r ) r →1 rσ − σ , = lim Im σ h (r σ )(1 − r ) = − lim Re r →1 r →1 G(r σ )

from which we get the equivalence between (1) and (2), and also that ρ −1 =  and ) = G (τ ) .  ∠ lim z→σ h (z)(z−σ h(z) We end this section with a characterization of points which are not boundary regular fixed points. Corollary 13.6.9 Let (φt ) be a semigroup, not an elliptic group, in D with DenjoyWolff point τ ∈ D and let h be its Koenigs function. Let σ ∈ ∂D. (1) If τ ∈ D and λ is the spectral value of (φt ), then lim z→σ Argλ h(z) exists if and only if σ is not a boundary regular fixed point. (2) If τ ∈ ∂D, then lim z→σ Re h(z) exists finitely if and only if σ is not a boundary regular fixed point. Proof (1) Theorem 13.6.3 implies that if lim z→σ Argλ h(z) exists then σ is not a boundary regular fixed point. Conversely, if σ is super-repelling, by Corollary 13.6.4, lim z→σ Argλ h(z) exists. If σ is not a boundary fixed point, by Proposition 13.6.1 and ˆ σ )) is contained in a λ-spiral so Theorem 11.1.2, ∠ lim z→σ h(z) = p ∈ C and I (h(x that lim z→σ Argλ h(z) exists by Proposition 4.4.4. (2) Assume that lim z→σ Re h(z) = a ∈ R. By Theorem 13.6.6, σ is not a repelling fixed point. If σ = τ and z ∈ D, then a = lim Re h(φt (z)) = lim Re (h(z) + it) = lim Re h(z) = Re h(z), t→+∞

t→+∞

t→+∞

so that h is constant. A contradiction. Therefore, σ is not a boundary regular fixed point. Conversely, assume that σ is not a boundary regular fixed point. Then, either σ is a super-repelling fixed point or σ is not a boundary fixed point. In the first case, by Corollary 13.6.7, lim z→σ Re h(z) = a for some a ∈ R, while in the second case, Proposition 13.6.1 and Theorem 11.1.4 show that ∠ lim z→σ h(z) = p ∈ C. Thus  Proposition 11.1.9 implies lim z→σ Re h(z) = Re p.

13.7 Examples

399

13.7 Examples With all the theory developed in the previous sections, we are finally able to provide examples. The main point here is to use the geometry of the image of a Koenigs function to reconstruct properties of the associated semigroup. Let (φt ) be a semigroup, not a group, in D and let h be its Koenigs function. Recall that, by Theorem 13.5.5 and Theorem 13.5.6 there is a one-to-one correspondence between hyperbolic petals of (φt ) and maximal strips in the non-elliptic case (or maximal spirallike sectors in the elliptic case) in h(D). Moreover, the repelling spectral value can be read by the width of the strip (or the angle of the spirallike sector). Also, by Theorem 13.5.7, there is a one-to-one correspondence between parabolic petals and maximal half-planes. The previous developed theory allows also to read information on the boundary of a petal using directly the image of h. We summarize and translate here the results in a suitable handable way. We start with the elliptic case: Proposition 13.7.1 Let (φt ) be an elliptic semigroup, not a group, with Denjoy-Wolff point τ ∈ D and spectral value μ ∈ C, Re μ > 0 and let h be its Koenigs function. Let  be a hyperbolic petal which corresponds to the maximal μ-spirallike sector Spir[μ, 2α, θ0 ], for some α ∈ [0, π ) and θ0 ∈ [−π, π ). Let σ ∈ ∂D ∩ ∂ be the only repelling fixed point of (φt ) contained in . Let S := spir μ [ei(θ0 +α )] ∩ (C \ {0}) or S := spir μ [ei(θ0 −α )] ∩ (C \ {0}). Then one and only one of the following happens: (1) There exists a > 0 such that S ∩ {w ∈ C : |w| < a} ⊂ h(D) and S ∩ {w ∈ C : |w| ≥ a} ∩ h(D) = ∅. This is the case if and only if h −1 (S ∩ {w ∈ C : |w| < a}) is a connected component of ∂ ∩ (D \ {τ }) whose closure is a Jordan arc with end points τ and a non-fixed point p ∈ ∂D such that ∠ lim z→ p h(z) = S ∩ {w ∈ C : |w| = a}. (2) S ⊂ h(D). This is the case if and only if h −1 (S) is a connected component of ∂ ∩ (D \ {τ }) whose closure is a Jordan arc with end points τ and σ . Proof Assume S := spir μ [ei(θ0 +α )] ∩ (C \ {0}) (the other case is similar). (1) Clearly, h −1 (S ∩ {w ∈ C : |w| < a}) is a connected component of ∂ ∩ (D \ {τ }). By Corollary 13.4.11, the closure of h −1 (S ∩ {w ∈ C : |w| < a}) is a Jordan arc joining τ with a point p ∈ ∂D which can be either σ or a nonfixed point. Let γ : (−∞, t0 ) t → eμt+i(θ0 +α) be a parameterization of S, with t0 ∈ R such that eμt0 +i(θ0 +α) = S ∩ {w ∈ C : |w| = a}. Then limt→t0 h −1 (γ (t)) = p. Since limt→t0 h(h −1 (γ (t))) = S ∩ {w ∈ C : |w| = a}, Theorem 1.5.7 implies that ∠ lim z→ p h(z) = S ∩ {w ∈ C : |w| = a}. In particular, ∠ lim z→ p |h(z)| < +∞, and hence p is not a fixed point by Proposition 13.6.1. (2) The argument is similar and we omit the proof.  A similar argument allows to handle the non-elliptic case: Proposition 13.7.2 Let (φt ) be a non-elliptic semigroup, not a group, with DenjoyWolff point τ ∈ ∂D and let h be its Koenigs function. Let  be a hyperbolic petal which

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13 Fixed Points, Backward Invariant Sets and Petals

corresponds to the maximal strip S = {w ∈ C : a1 < Re w < a2 } for some a1 , a2 ∈ R, a1 < a2 . Let σ ∈ ∂D ∩ ∂ be the only repelling fixed point of (φt ) contained in . Fix j ∈ {1, 2}. Then one and only one of the following happens for j = 1, 2: (1) {w ∈ C : Re w = a j } ∩ h(D) = ∅. (2) There exists r ∈ R such that {w ∈ C : Re w = a j , Im w > r } ⊂ h(D) and {w ∈ C : Re w = a j , Im w ≤ r } ∩ h(D) = ∅. This is the case if and only if h −1 ({w ∈ C : Re w = a j , Im w > r }) is a connected component of ∂ ∩ D whose closure is a Jordan arc with end points τ and a non-fixed point p ∈ ∂D such that ∠ lim z→ p h(z) = a j + ir . (3) {w ∈ C : Re w = a j } ⊂ h(D). This is the case if and only if h −1 ({w ∈ C : Re w = a j }) is a connected component of ∂ ∩ D whose closure is a Jordan arc with end points τ and σ . Finally, we deal with parabolic petals: Proposition 13.7.3 Let (φt ) be a parabolic semigroup, not a group, with DenjoyWolff point τ ∈ ∂D and let h be its Koenigs function. Let  be a parabolic petal which corresponds to the maximal half-plane H = {w ∈ C : Re w > a} for some a ∈ R. Then one and only one of the following happens: (1) There exists r ∈ R such that {w ∈ C : Re w = a, Im w > r } ⊂ h(D) and {w ∈ C : Re w = a, Im w ≤ r } ∩ h(D) = ∅. This is the case if and only if h −1 ({w ∈ C : Re w = a, Im w > r }) is a connected component of ∂ ∩ D whose closure is a Jordan arc with end points τ and a non-fixed point p ∈ ∂D such that ∠ lim z→ p h(z) = a + ir . (2) {w ∈ C : Re w = a} ⊂ h(D). This is the case if and only if h −1 ({w ∈ C : Re w = a}) is a connected component of ∂ ∩ D whose closure J is a Jordan curve with J ∩ ∂D = {τ }. Proof The proof is similar to that of Proposition 13.7.1, so we leave it to the reader. The only issue here is to show that the case {w ∈ C : Re w = a} ∩ h(D) = ∅ cannot happen. Indeed, if this is the case, then h(D) = H and (φt ) is a parabolic group.  A similar result holds in case the maximal half-plane associated with the parabolic petal is H = {w ∈ C : Re w < a} for some a ∈ R. Now we are ready to give examples. z Example 13.7.4 Consider the Koebe function h(z) = (1−z) 2 , z ∈ D. Clearly h is holomorphic in the unit disc and, by Theorem 9.4.5, it is starlike with respect to 0 because   1+z h (z) = Re > 0, z ∈ D. Re z h(z) 1−z

In particular, it is univalent and it is easy to see that h(D) = C \ (−∞, −1/4]. Consider the semigroup whose model is (C, h, z → e−t z), that is, φt (z) := h −1 (e−t h(z)), for all z ∈ D and t ≥ 0. Since ∩t≥0 e−t h(D) \ {0} = C \ (−∞, 0] = Spir[1, 2π, −π ] and it is a maximal spirallike sector of h(D), Theorem 13.5.6 shows that

13.7 Examples

401

h −1 (Spir[1, 2π, −π ]) = D \ (−1, 0] is a hyperbolic petal for (φt ). Clearly, it is the unique petal of the semigroup. Therefore, (φt ) has a unique boundary fixed point σ ∈ ∂D, which is repelling with repelling spectral value −1/2. Since lim(0,1) r →1 h(r ) = ∞, we get σ = 1 by Proposition 13.4.10. This petal is an example of the type described in Proposition 13.4.9(1). Example 13.7.5 Consider the domain Ω = S ∪ (1 + iR) ∪ {w ∈ S + 1 : Im w(1 − Re w) > 1} ⊂ S2 . Let h : D → C be univalent such that h(D) = Ω. By construction, Ω + it ⊂ Ω for all t ≥ 0 and ∪t≥0 (Ω − it) = S2 . Consider the semigroup whose model is (S2 , h, z → z + it), that is, φt (z) := h −1 (h(z) + it), for all z ∈ D and t ≥ 0. Let us call τ ∈ ∂D its Denjoy-Wolff point. Since ∩t≥0 (Ω + it) = ∪x∈(0,1] (x + iR) = S ∪ (1 + iR), h(D) contains a unique maximal strip, S, whose boundary is iR ∪ (1 + iR). Let  = h −1 (S). Therefore,  is the unique hyperbolic petal of the semigroup (φt ). Let us denote by σ the repelling fixed point associated with  given by Proposition 13.4.10. By Proposition 13.7.2, the maximal invariant curve R t → h −1 (1 + it) is a connected component of ∂ ∩ D whose closure is a Jordan arc with end points τ and σ . It divides the unit disc in two connected components, one of them is the petal, and the other one is B = h −1 ({w ∈ S + 1 : Im w(1 − Re w) > 1}). Clearly,  ∩ ∂D and B ∩ ∂D are the two Jordan arcs in ∂D that joins σ and τ . Let us denote by J the one which is included in ∂. Then ∂ = {τ, σ } ∪ h −1 (1 + iR) ∪ J . Thus  is an example of a petal of the type described in Proposition 13.4.9(2). Example 13.7.6 Consider the domain Ω = {w ∈ S : Im w > 0} ∪ (1 + i(0, +∞)) ∪ (H + 1) ⊂ H. ∞

Let h : D → C be univalent such that h(D) = Ω. Since Ω is a Jordan domain, ∞ ˜ D = h, by Theorem 4.3.3, there exists a homeomorphism h˜ : D → Ω such that h| Let (φt ) be the parabolic semigroup whose model is (H, h, z → z + it), that is, φt (z) := h −1 (h(z) + it), for all z ∈ D and t ≥ 0. Let τ ∈ ∂D be its Denjoy-Wolff point. Note that ∩t≥0 (Ω + it) = ∪x∈(1,+∞) (x + iR) = H + 1. Let  = h −1 (H + 1). Therefore,  is the unique (parabolic) petal of the semigroup (φt ). Write J = h −1 (1 + i(0, +∞)) ⊂ D, σ = h˜ −1 (1) and A = h˜ −1 (1 + i(−∞, 0)) ⊂ ∂D. Then ∂ = {τ } ∪ J ∪ {σ } ∪ A. Thus  is an example of a petal described in Proposition 13.4.9(2). The difference with Example 13.7.5 is that σ in this case is not a boundary fixed point. Example 13.7.7 Consider the domain Ω ={w ∈ S : Im w(Re w − 1) < Re w} ∪ (1 + iR) ∪ (S + 1)∪ ∪ (2 + iR) ∪ {w ∈ S + 2 : Im w(Re w − 2) > Re w − 3} ⊂ S3 .

402

13 Fixed Points, Backward Invariant Sets and Petals

Let h : D → C be univalent such that h(D) = Ω. By construction, Ω + it ⊂ Ω for all t ≥ 0 and ∪t≥0 (Ω − it) = S3 . Consider the semigroup whose model is (S3 , h, z → z + it), that is, φt (z) := h −1 (h(z) + it), for all z ∈ D and t ≥ 0. Let τ ∈ ∂D be its Denjoy-Wolff point. Since ∩t≥0 (Ω + it) = ∪x∈[1,2] (x + iR), its interior has a unique maximal strip whose boundary is (1 + iR) ∪ (2 + iR). Let  = h −1 (∪x∈(1,2) (x + iR)) = h −1 (S + 1). Therefore,  is the unique (hyperbolic) petal of the semigroup (φt ). Let us denote by σ the repelling fixed point associated with  given by Proposition 13.4.10. By Proposition 13.7.2(3), h −1 (1 + iR) and h −1 (2 + iR) are connected components of ∂ ∩ D whose closures are Jordan arcs with end points τ and σ . According to Proposition 13.4.9, ∂ = {τ, σ } ∪ h −1 (1 + iR) ∪ h −1 (2 + iR). Thus  is an example of a petal described in Proposition 13.4.9(3). Example 13.7.8 Consider the domain Ω ={w ∈ S : Im w > 0} ∪ (1 + i(0, +∞)) ∪ (S + 1)∪ ∪ (2 + i(0, +∞)) ∪ {w ∈ S + 2 : Im w > 0} ⊂ S3 . Let h : D → C be univalent such that h(D) = Ω. Consider the semigroup whose model is (S3 , h, z → z + it), that is, φt (z) := h −1 (h(z) + it), for all z ∈ D and t ≥ 0. Denote by τ ∈ ∂D its Denjoy-Wolff point. Note that ∩t≥0 (Ω + it) = ∪x∈(1,2) (x + iR) = S + 1. Let  = h −1 (S + 1). Therefore,  is the unique hyperbolic petal of the semigroup (φt ). Let us denote by σ the repelling fixed point associated with  given by Proposition 13.4.10. Write J j = h −1 ( j + i(0, +∞)) ⊂ D, j = 1, 2. Proposition 13.7.2(2) guarantees that J1 and J2 are connected components of ∂ ∩ D with end points τ and a non-fixed point p j ∈ ∂D such that ∠ lim z→ p j h(z) = j. Taking into account Proposition 13.4.9, we see that  is an example of a petal described in Proposition 13.4.9(4). Therefore, if A ⊂ ∂D is the closed arc with end points p1 and p2 which does not pass through τ , we have that ∂ = {τ } ∪ J1 ∪ J2 ∪ A. Example 13.7.9 Consider the domain Ω = {w ∈ S : Im w(Re w − 1) < Re w} ∪ (H + 1) ⊂ H. Let h : D → C be univalent such that h(D) = Ω. Consider the parabolic semigroup whose model is (H, h, z → z + it), that is, φt (z) := h −1 (h(z) + it), for all z ∈ D and t ≥ 0. Let τ ∈ ∂D be its Denjoy-Wolff point. Note that ∩t≥0 (Ω + it) = ∪x∈[1,+∞) (x + iR) = H + 1 and write  = h −1 (H + 1). Therefore,  is the unique parabolic petal of the semigroup (φt ). Let A = h −1 (1 + iR) ⊂ D. Then Proposition 13.7.3(2) shows that A is a connected component of ∂ ∩ D and ∂ = {τ } ∪ A. Thus  is an example of a petal described in Proposition 13.4.9(5). Given a semigroup in D which is not a group with a repelling fixed point σ , Theorem 13.2.7 provides a pre-model (D, g, ηt ) for (φt ) at σ . By the very definition, g is semi-conformal at σ . We end this section with an example where g is not regular

13.7 Examples

403

(or conformal) at σ , so that the regularity of g given in Theorem 13.2.7 cannot be improved. We need a preliminary lemma. In the next lemma and example, we will use without mentioning explicitly Remark 1.3.11 which says that given two domains Ω1 and Ω2 such that Ω1 ⊂ Ω2 then kΩ2 (z, w) ≤ kΩ1 (z, w) for all z, w ∈ Ω1 . Lemma 13.7.10 There exist two strictly decreasing sequences {yk } and {αk } of real numbers, both converging to −∞, such that, for each k ∈ N,             1 1 1 1 1 kΩ αk + αk + i, αk − i ≤ 1+ k Ωk i, αk − i , 2 2 2k 2 2 where, for k ∈ N, L k+ := {w ∈ C : Re w = 1 + 1/k, Im w ≤ yk }, L k− := {w ∈ C : Re w = −(1 + 1/k), Im w ≤ yk }, Ωk := C \ (L k+ ∪ L k− ), and Ω := ∩k∈N Ωk . Proof Given p ∈ Ωk and R > 0, we write Bk ( p, R) := {w ∈ Ωk : kΩk ( p, w) < R}. Let Sk = {w ∈ C : |Re z| < 1 + 1/k}. Clearly Sk ⊂ Ωk for all k. Hence, by Proposition 6.7.2, for any choice of the sequence {αk }, kΩk ((αk + 1/2)i, αk i) ≤ k Sk ((αk + 1/2)i, αk i) =

π 1 < π. 2 4(1 + 1/k)

(13.7.1)

Set y1 = 0 and α1 = −1/2. By Lemma 6.8.1 and the invariance of hyperbolic distance with respect to biholomorphisms, there exists R1 > π such that for all z, w ∈ B1 (−i/2, π ),   1 kΩ1 (z, w). k B1 (−i/2,R1 ) (z, w) ≤ 1 + 2 In particular, by (13.7.1),   1 kΩ1 (0, −i). k B1 (−i/2,R1 ) (0, −i) ≤ 1 + 2 Since the closure of B1 (−i/2, R1 ) is compact in Ω1 , we have β1 := inf{Im w : w ∈ B1 (−i/2, R1 )} > −∞. From now on, we choose y j < β1 for all j ≥ 2. Hence, B1 (−i/2, R1 ) ⊂ Ω, for any admissible choice of the sequence {yk }. Therefore,

404

13 Fixed Points, Backward Invariant Sets and Petals

  1 kΩ1 (0, −i), kΩ (0, −i) ≤ k B1 (−i/2,R1 ) (0, −i) ≤ 1 + 2 and the statement holds for k = 1. Now we choose y2 and α2 . Let y2 < β1 and let f 2 be a Riemann map of the domain Ω2 . By Proposition 3.3.3, there exists p := limt→−∞ f 2−1 (it). Let 1+ := f 2−1 (L 1+ ) and 1− := f 2−1 (L 1− ). For each n ∈ N0 , let Jn be the segment in Ω2 with extreme points ±(1 + 1/2) + (y2 − n)i and Cn = f 2−1 (Jn ). Then (Cn ) is a null chain and its impression is { p}. Clearly, 1+ ∪ 1− does not intersect the interior part of Cn for all n and then p ∈ / + − 1 ∪ 1 . / Let R2 be given by Lemma 6.8.1 with M = π and c = 1 + 1/4. Since p ∈ 1+ ∪ 1− , there is α2 < y2 such that D hyp ( f 2−1 (α2 i), R2 ) ∩ 1+ ∪ 1− = ∅. That is, B2 (α2 i, R2 ) ⊂ Ω1 ∩ Ω2 . As before, this implies that kΩ ((α2 − 1/2)i, (α2 + 1/2)i) ≤ (1 + 1/4)kΩ2 ((α2 − 1/2)i, (α2 + 1/2)i). Let β2 = inf{Im w : w ∈ B2 (α2 i, R2 )} > −∞. Since α2 < y2 < β1 , we get that  β2 < β1 and the lemma is proved repeating this argument by induction. Example 13.7.11 Let {yk } and {αk } be the sequences given by Lemma 13.7.10. Let h : D → C be univalent such that h(D) = Ω, h(0) = 0 and −i h (0) > 0. Since Ω is symmetric with respect to iR, then h((−1, 1)) = iR by Proposition 6.1.3 and limr →1 Im h(r ) = +∞ and limr →1 Im h(−r ) = −∞. Consider the semigroup given by φt (z) := h −1 (h(z) + it), for all z ∈ D and t ≥ 0. Then τ = 1 is its Denjoy-Wolff point and σ = −1 is a repelling fixed point associated with the maximal strip S := {w ∈ C : −1 < Re w < 1} = S2 − 1. Let (D, g, ηt ) be a pre-model for (φt ) at σ given by Theorem 13.2.7. Notice that g(0) = 0 and h(g(D)) = S. We claim that g is not regular at σ . Assume on the contrary that 1 − |g(z)| < +∞ αg (σ ) = lim inf z→σ 1 − |z| (see Theorem 1.7.3). Then, by Proposition 1.7.4, αg (σ ) = ∠ lim

z→σ

1 − |g(z)| . 1 − |z|

In particular, since η−t (0) goes non-tangentially to σ , as t tends to +∞,

13.7 Examples

405

1 − |g(η−t (0))| = αg (σ ). t→+∞ 1 − |η−t (0)| lim

In other words, by (1.4.5), lim

t→+∞

 1 ω(0, η−t (0)) − ω(0, g(η−t (0))) = log αg (σ ). 2

Notice that, by the very definition, h(g(ηt (0))) = h(φt (g(0))) = h(g(0)) + it = it for all t ≥ 0. Therefore, h(g(ηt (0))) = it for all t ∈ R. Since ω(0, η−t (0)) − ω(0, g(η−t (0))) = k g(D) (g(0), g(η−t (0))) − ω(0, g(η−t (0))) = kh(g(D)) (h(g(0)), h(g(η−t (0)))) − kh(D) (h(0), h(g(η−t (0)))) = k S (0, −it) − kΩ (0, −it), we conclude that lim [k S (0, −it) − kΩ (0, −it)] =

t→+∞

1 log αg (σ ) < +∞. 2

(13.7.2)

On the other hand, since R t → −it is a geodesic for the hyperbolic distance of S, we have that for any u 1 < u 2 < u 3 k S (−iu 1 , −iu 3 ) = k S (−iu 1 , −iu 2 ) + k S (−iu 2 , −iu 3 ). Therefore, by Remark 1.3.11, Lemma 13.7.10 and Proposition 6.7.2, we have k S (0, i(αk + 1/2)) − kΩ (0, i(αk + 1/2)) ≥ ≥

k−1 



k S (i(α j − 1/2), i(α j + 1/2)) + k S (i(α j − 1/2), i(α j+1 + 1/2))

j=1

−kΩ (i(α j − 1/2), i(α j + 1/2)) − kΩ (i(α j − 1/2), i(α j+1 + 1/2)) ≥

k−1 



k S (i(α j − 1/2), i(α j + 1/2)) − kΩ (i(α j − 1/2), i(α j + 1/2))





j=1

≥

  1 kΩ j (i(α j − 1/2), i(α j + 1/2)) ] k S (i(α j − 1/2), i(α j + 1/2)) − 1 + 2j

k−1  j=1

  1 k S j (i(α j − 1/2), i(α j + 1/2)) ] k S (i(α j − 1/2), i(α j + 1/2)) − 1 + 2j j=1 ⎡ ⎤  k−1 k−1  π  1 π π  1 ⎣ − 1+ ⎦ =  . = 4 2j 4 1 + 1 8 1+ j

≥

k−1 

j=1

j

j=1

406

13 Fixed Points, Backward Invariant Sets and Petals

Thus lim [k S (0, i(αk + 1/2)) − kΩ (0, (αk + 1/2)t)] = +∞,

k→∞

and, bearing in ming that limk→∞ αk = −∞, we get a contradiction to (13.7.2). Hence g is not regular at σ . By Corollary 13.4.13, every pre-model for (φt ) is not regular at σ .

13.8 Notes In [98] Poggi-Corradini constructed a pre-model for discrete iteration. The proofs of the basic Proposition 13.1.9 and Theorem 13.2.7 were adapted from that paper. The study of abstract backward sequences was performed in [20], with the aim of proving a conjecture of Cowen [57] about common boundary fixed points of commuting holomorphic maps. The proof of Proposition 13.1.7 follows the line of [20] (for a systematic treatment of backward iteration sequences in discrete iteration see [99, 100]). In the discrete (elliptic, starlike type) case, Theorem 13.5.5 was proven in [97]. Theorems 13.5.5 and 13.5.6 were first proved with a direct argument in [46], see also [61, 69, 70]. The main results of this chapter come from [32].

Chapter 14

Contact Points

In the previous chapters we studied boundary points which are either fixed or the initial points of maximal invariant curves for a semigroup. In this chapter we examine the other points, which turn out to be contact points, and we show that super-repelling fixed points can be divided into two separated sets: those which are the landing point of a backward orbit and those which are the initial point of a maximal contact arc (in the latter case they are also critical points for the infinitesimal generators). We also discuss the behavior of the Koenigs function and the infinitesimal generator at the end points of maximal contact arcs. The chapter ends with some examples and, in particular, with the construction of a semigroup with an uncountable set of super-repelling fixed points.

14.1 The Boundary Denjoy-Wolff Theorem To start with, we extend the continuous Denjoy-Wolff Theorem 8.3.1 up to boundary points which are not fixed. Recall that, by Theorem 11.2.1, every iterate of a semigroup (φt ) in D has non-tangential limit at every σ ∈ ∂D, and such a limit is denoted by φt (σ ). Theorem 14.1.1 Let (φt ) be a semigroup in D which is not an elliptic group. Let τ ∈ D be its Denjoy-Wolff point. Let σ ∈ ∂D. If σ is not a boundary fixed point of (φt ), then the curve [0, +∞)  t → φt (σ ) is continuous, injective and lim φt (σ ) = τ.

t→+∞

Proof By Proposition 11.2.2, the curve [0, +∞)  t → φt (σ ) is continuous. Let (Ω, h, ψt ) be the canonical model given by Theorem 9.3.5. Since σ is not a fixed point for (φt ), Proposition 13.6.1 implies that h(σ ) := ∠ lim z→σ h(z) ∈ C (such a limit exists by Corollary 11.1.7). By Theorem 11.2.3, we have © Springer Nature Switzerland AG 2020 F. Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4_14

407

408

14 Contact Points

h(φt (σ )) = ψt (h(σ )), t ≥ 0,

(14.1.1)

and, since ψt (h(σ )) = ∞ for all t ≥ 0, it follows again by Proposition 13.6.1 that for all t ≥ 0, φt (σ ) is not a boundary fixed point of (φs ). We claim that [0, +∞)  t → φt (σ ) in injective. Indeed, if this is not the case, then φt (σ ) = φs (σ ) for some 0 ≤ s < t. Hence, by Theorem 11.2.3, φt−s (φs (σ )) = φt (σ ) = φs (σ ), which proves that φs (σ ) is a fixed point of (φt ). For what we saw before, the only possibility is τ = φs (σ ) ∈ D and (φt ) is elliptic. But, h(σ ) ∈ ∂h(D), hence h(σ ) = 0 and, if λ ∈ C, Re λ > 0, is the spectral value of (φt ), e−λs h(σ ) = ψs (h(σ )) = h(φs (σ )) = h(τ ) = 0, a contradiction. Therefore, the curve is injective. Now, if φt (σ ) ∈ D for some t > 0, then φs+t (σ ) = φs (φt (σ )) ∈ D by Theorem 11.2.3 for all s ≥ 0. Hence, Theorem 8.3.1 implies limt→+∞ φt (σ ) = τ . Assume that φt (σ ) ∈ ∂D for all t ≥ 0. Note that, if τ ∈ D, that is, if (φt ) is elliptic, then ψt (h(σ )) → 0 as t → +∞. Hence, (14.1.1) implies that there exists t > 0 such that h(φt (σ )) ∈ h(D), that is, φt (σ ) ∈ D. Therefore, τ ∈ ∂D and (φt ) is non-elliptic. Hence, ψt (z) = z + it and h(D) is starlike at infinity. By (14.1.1), h(φt (σ )) = h(σ ) + it for all t ≥ 0, which implies that {a + i(t + Im h(σ )), t ≥ 0} ⊂ ∂h(D), where a = Re h(σ ). Since h(D) is starlike at infinity, this also implies that either h(D) ⊂ {w ∈ C : Re w > a} or h(D) ⊂ {w ∈ C : Re w < a}, where a = Re h(σ ). Now {a + i(t + Im h(σ )), t ≥ 0} ⊂ ∂h(D), and Ω = I × R with either I = (−∞, 0), or I = (0, +∞), or I = R, or I = (0, ρ) for some ρ > 0. Hence, it follows that the semigroup (φt ) is either hyperbolic, with a = 0 or a = ρ, or it is parabolic of positive step with a = 0. We may suppose h(D) ⊂ H and a = 0 (the other cases are similar). Let x t ∈ ∂C D denote the prime end corresponding to φt (σ ) in the homeomorphism  be the homeomorD given by Proposition 4.2.5. Let hˆ :  D → h(D) between D and  phism in the Carathéodory topology defined by h (see Theorem 4.2.3). Let (Cnt ) be a ˆ t ) for all t ≥ 0. By Theorem 4.4.9, Π (h(x ˆ t )) = circular null chain representing h(x {h(φt (σ ))}. Hence, by Remark 4.4.7, for each t ≥ 0 the center of the null chain (Cnt ) is h(φt (σ )). In other words, for every t ≥ 0 there exists a decreasing sequence {rnt }, with rnt ∈ (0, 1) converging to 0 such that Cnt ⊂ {z ∈ C : |z − h(φt (σ ))| = rnt } for all n ≥ 0 and t ≥ 0. Since h(φt (σ )) = h(σ ) + it, this implies in particular that for every M > 0 there exists t M > 0 such that inf{Im w : w ∈ Cnt } ≥ M for all t ≥ t M and n ≥ n t0 . Since ∂D is compact and [0, +∞)  t → φt (σ ) is continuous and injective, it follows easily that there exists p ∈ ∂D \ {φt (σ )}t∈(0,+∞) such that limt→+∞ φt (σ ) = p. Let x p be the prime end representing p. Then, {x t } converges in the Carathéodory

14.1 The Boundary Denjoy-Wolff Theorem

409

ˆ t ) converges to h(x ˆ p ) in the Carathéodory topology of  D to x p which implies that h(x  topology of Ω as well. ˆ p ). Let Vm be the interior Let (G m ) be a null chain in h(D) which represents h(x part of G m , m ≥ 1. By Remark 4.2.2, for every m ≥ 1 there exist Tm and n tm such that Cnt ∩ h(D) ⊂ Vm for all t ≥ Tm and n ≥ n tm . The previous considerations imply that we can find a sequence {wn } ⊂ h(D) such that Im wn ≥ n for all n ∈ N and such that for all m ∈ N there exists n m ∈ N such that wn ∈ Vm for all n ≥ n m . Indeed, it is enough to choose wm ∈ Cnt ∩ h(D) for t ≥ max{tm , Tm } and n ≥ max{n t0 , n tm } for every m ∈ N. Therefore, the sequence {wm } ˆ p ) in the Carathéodory topology by Remark 4.2.2. Hence, {h −1 (wm )} converges to h(x converges to x p in the Carathéodory topology, and, taking into account Proposition 4.2.5, this implies that {h −1 (wm )} converges to p. Since limm→∞ Im h(h −1 (wm )) = +∞, Proposition 11.1.8 implies that p = τ .  Remark 14.1.2 The previous proof shows that if (φt ) is an elliptic semigroup (not a group) or a parabolic of zero hyperbolic step semigroup in D, then for every σ ∈ ∂D which is not a fixed point of (φt ) there exists t > 0 such that φt (σ ) ∈ D. As a consequence of the boundary Denjoy-Wolff Theorem we have Corollary 14.1.3 Let (φt ) be a semigroup in D with Denjoy-Wolff point τ ∈ D. If σ ∈ D is not a fixed point of (φt ), then φs (σ ) is not a fixed point of (φt ) for all s ≥ 0. In particular, if σ ∈ D \ {τ } then φs (σ ) = τ for all s ≥ 0. Proof Assume σ is not a fixed point of (φt ). If σ ∈ D, since φs is univalent for all s ≥ 0, it follows easily that φs (σ ) ∈ D is not a fixed point of (φt ) for all s ≥ 0. On the other hand, if σ ∈ ∂D and φt0 (σ ) = p for some t0 > 0, where p is a fixed point of (φt ), then the curve [0, +∞)  t → φt (σ ) would not be injective since, by Theorem 11.2.3, φt+t0 (σ ) = φt (φt0 (σ )) = φt ( p) = p for all t ≥ 0, contradicting Theorem 14.1.1. In particular, if σ ∈ D \ {τ } is not a fixed point of (φt ), then φs (σ ) = τ for all s ≥ 0. While, if σ ∈ ∂D \ {τ } is a boundary fixed point of (φt ), then, clearly, φs (σ ) = σ = τ for all s ≥ 0, and we are done. 

14.2 Maximal Contact Arcs Theorem 14.1.1 justifies the following definition: Definition 14.2.1 Let (φt ) be a semigroup in D and σ ∈ ∂D. The life-time T (σ ) of σ under the action of (φt ) is T (σ ) := sup{t ∈ [0, +∞) : φt (σ ) ∈ ∂D}.

410

14 Contact Points

Clearly, if σ ∈ ∂D is a fixed point of a semigroup (φt ) in D, then T (σ ) = +∞. If (φt ) is a group, then for every σ ∈ ∂D, it follows that T (σ ) = +∞. The converse is also true, as it follows at once from the following proposition where we prove basic facts about the life-time: Proposition 14.2.2 Let (φt ) be a semigroup, not a group, in D with Denjoy-Wolff point τ ∈ D. (1) If σ ∈ ∂D, then T (σ ) = 0 if and only if σ is the starting point of a maximal invariant curve of (φt ). (2) There exists at least one point σ ∈ ∂D such that T (σ ) = 0. (3) If there exists a point σ ∈ ∂D such that σ is not a fixed point of (φt ) and T (σ ) = +∞, then either (φt ) is hyperbolic or it is parabolic of positive hyperbolic step. (4) If σ ∈ ∂D \ {τ } and 0 < T (σ ) ≤ +∞ then for every t ∈ (0, T (σ )) the unrestricted limit lim z→φt (σ ) φs (z) = φs+t (σ ) exists for all s ≥ 0. (5) If T (σ ) ∈ (0, +∞) then for all t ∈ [0, T (σ )] we have T (φt (σ )) = T (σ ) − t. Proof (1) It follows at once by Proposition 13.3.5. (2) Since (φt ) is not a group, there exists a point z 0 ∈ D which does not belong to the backward invariant set of (φt ). Then, the starting point σ of the maximal invariant curve for (φt ) containing z 0 satisfies T (σ ) = 0. (3) It follows at once by Theorem 14.1.1 and Remark 14.1.2. (4) If σ is a fixed point of (φt ), by Proposition 13.6.1, lim z→σ h(z) exists, hence by Proposition 11.3.2, lim z→σ φt (z) exists for all t ≥ 0. Assume σ is not fixed. Fix t0 ∈ (0, T (σ )) and let p := φt0 (σ ). By Proposition 11.3.2, it is enough to prove that there exists s > 0 such that lim z→ p φs (z) exists. Let ε ∈ (0, T (σ ) − t0 ) and s ∈ (0, ε). Let x p ∈ ∂C D be the prime end defined by p. By Lemma 4.1.14, there exists a circular null chain (Cn ) centered at p representing x p such that (φs (Cn )) is a null chain in φs (D) representing φˆs (x p ) ∈ ∂C φs (D). Hence, there exists a decreasing sequence {rn }, rn ∈ (0, 1), converging to 0 such that Cn = {|z − p| = rn , z ∈ D}, n ∈ N. Let an , bn be the end points of Cn . Since φs has non-tangential limit at every point of ∂D and Cn is circular, it follows that limCn z→an φs (z) = φs (an ), and limCn z→bn φs (z) = φs (bn ). Namely, φs (Cn ) has end points φs (an ) and φs (bn ). Without loss of generality, we can assume that r0 is such that r0 < min{| p − σ |, | p − φT (σ ) |} (the last number is strictly greater than 0 because the curve [0, +∞)  t → φt (σ ) is injective by Theorem 14.1.1). With this choice, an , bn ∈ {φt (σ ) : t ∈ (0, T (σ ))}, say, an = φαn (σ ) and bn = φβn (σ ) for αn ∈ (0, t0 ) and βn ∈ (t0 , T (σ )) and αn → t0 , βn → t0 . Note that φs (bn ) = φs+βn (σ ) and φs (an ) = φs+αn (σ ). Let n 0 ∈ N0 be such that βn < T (σ ) − s for all n ≥ n 0 (it is possible because T (σ ) − s > t0 and βn → t0 ). Hence, φs (an ), φs (bn ) ∈ ∂D for all n ≥ n 0 . Let An = {φt (an ) : t ∈ [s, s + βn ]}. Since [0, +∞)  t → φt (σ ) is injective and continuous, An is a closed arc on ∂D which joins φs (an ) and φs+βn (σ ) = φs (bn ). Then Yn := φs (Cn ) ∪ An is a Jordan curve, call Vn the bounded connected component of C \ Yn . Since by construction φs (an+1 ), φs (bn+1 ) belong to the interior of An and φs (Cn+1 ) ∩ D is a curve in D \ φs (Cn ) whose closure joins φs (an+1 ) and

14.2 Maximal Contact Arcs

411

φs (bn+1 ), it follows that φs (Cn+1 ) ∩ D ⊂ Vn . Hence, Vn is the interior part of φs (Cn ), n ≥ 1. Now, ∠ lim z→ p φs (z) = φs ( p), therefore, by Theorem 4.4.9, Π (φˆs (x p )) = {φs ( p)} and hence, by definition of principal part of a prime end, for every δ > 0 there exists n δ such that φs (Cn ) ⊂ {z ∈ C : |z − φs ( p)| < δ} for all n ≥ n δ . This implies at once that ∩n∈N Vn = {φs ( p)}. Namely, I (φˆs (x p )) = {φs ( p)}. By Proposition 4.4.4, lim z→ p φs (z) = φs ( p), and the result follows. (5) Since φT (σ ) (σ ) = φT (σ )−t (φt (σ )) for all t ∈ [0, T (σ )] by Theorem 11.2.3, the result follows at once.  Definition 14.2.3 Let (φt ) be a semigroup in D. A point σ ∈ ∂D is a contact point of (φt ) if 0 < T (σ ) ≤ +∞. Note that, by definition, σ ∈ ∂D is a contact point of a semigroup (φt ) if and only if there is t0 > 0 such that σ is a contact point of φt0 according to Definition 1.9.2 (and, in this case, T (σ ) ≥ t0 ). Every fixed point of a semigroup is a contact point, and every boundary point is a contact point for every group of automorphisms of D. We concentrate on contact points which are not fixed in case of semigroups which are not groups. By Proposition 14.2.2 and Theorem 14.1.1, it follows that if σ ∈ ∂D is a contact point which is not a fixed point, then for all t ∈ (0, T (σ )), φt (σ ) is a contact point which is not fixed and the image of (0, T (σ ))  t → φt (σ ) is an open arc in ∂D. This suggests the following definition: Definition 14.2.4 Let (φt ) be a semigroup in D. A non-empty open arc A ⊂ ∂D is called a contact arc of (φt ) if every point p ∈ A is a contact point of (φt ) which is not fixed. A contact arc A of (φt ) is called maximal if there exist no contact arcs B such that A  B. We stress that by definition contact arcs are open and contains no fixed points. Clearly, every contact arc is contained in a maximal contact arc. By Theorem 14.1.1, every non-elliptic group (φt ) defines a natural orientation on every contact arc A. That is, if p, q ∈ A, we say that p  q if there exists s ≥ 0 such that φs ( p) = q. This natural orientation allows to select initial and final points of a contact arc. More precisely, if A  ∂D, is a contact arc for (φt ), with the natural orientation induced by (φt ), let x0 (A) and x1 (A) be the end points of A. We say that x0 (A) is the starting point of A if limn→∞ pn = x0 (A) for one—and hence any—sequence { pn } ⊂ A such that pn+1  pn and { pn } has no accumulation points in A. Similarly, one can define the final point x1 (A) of a contact arc. If (φt ) is an elliptic group, then ∂D is a maximal contact arc. If (φt ) is a hyperbolic group with Denjoy-Wolff point τ ∈ ∂D and other fixed point σ ∈ ∂D, then there are two maximal contact arcs given by the connected components of ∂D \ {τ, σ }, the starting point of any such maximal arc is σ and the final point is τ.

412

14 Contact Points

If (φt ) is a parabolic group with Denjoy-Wolff point τ ∈ ∂D then there is a maximal contact arc given by ∂D \ {τ }. In this case, τ is both the starting and the final point of the maximal contact arc. Remark 14.2.5 By the previous considerations and by Proposition 14.2.2, if (φt ) is a semigroup in D for which ∂D is a maximal arc then (φt ) is an elliptic group. In general, we have the following possibilities, which follow immediately from Theorem 14.1.1: Proposition 14.2.6 Let (φt ) be a semigroup, not a group, in D. Let τ ∈ D be the Denjoy-Wolff point of (φt ). Let A ⊂ ∂D be a maximal contact arc. Then: (1) The starting point x0 (A) is either a fixed point of (φt ) (possibly x0 (A) = τ ) or a contact point which is not fixed. (2) The final point x1 (A) is either a non-fixed point of (φt ) which is the starting point of a maximal invariant curve, or x1 (A) = τ (in this latter case, (φt ) is necessarily non-elliptic). Semigroups having (maximal) contact arcs with the same starting and final points are very special: Proposition 14.2.7 Let (φt ) be a semigroup in D. Then (φt ) has a contact arc A with x0 (A) = x1 (A) if and only if (φt ) is a parabolic group with Denjoy-Wolff point x0 (A). Proof One direction is clear. So we assume that A is a contact arc for a semigroup (φt ) in D with q := x0 (A) = x1 (A). Suppose (φt ) is not a group. By Proposition 14.2.6, q = x0 (A) is either a fixed point of (φt ) or a contact point which is not fixed. However, by the same proposition, q = x1 (A) is either the Denjoy-Wolff point of (φt ) or a non-fixed point which is the starting point of a maximal invariant curve. In the latter case, in particular, q is not a contact point. Therefore the only possibility is that q is the Denjoy-Wolff point of (φt ), which is thus non-elliptic. Now, let z ∈ D and let γ : (a, +∞) → D, a ∈ [−∞, 0), be the maximal invariant curve of (φt ) such that γ (0) = z. Let p := limt→a + γ (t) ∈ ∂D be the starting point of γ . By Proposition 13.3.5, p can not be a contact point of (φt ), hence the only possibility is p = x0 (A) = x1 (A). Again by Proposition 13.3.5, it follows that z belongs to the backward invariant set W of (φt ). By the arbitrariness of z, it follows W = D and hence D is a petal for (φt ), against Remark 13.4.3. Thus (φt ) is a group which clearly is parabolic.  Definition 14.2.8 Let (φt ) be a semigroup in D with Denjoy-Wolff point τ ∈ D. A maximal contact arc A  ∂D is called exceptional if x1 (A) = τ . Remark 14.2.9 Notice that if T (σ ) ∈ (0, +∞) (respectively T (σ ) = +∞ and σ is not a fixed point), then σ belongs to a non-exceptional (resp. exceptional) maximal arc.

14.2 Maximal Contact Arcs

413

Contact arcs can be characterized in terms of geometry of the image of Koenigs functions and analytic behavior of infinitesimal generators. As a matter of notation, if h is the Koenigs function of a semigroup and A ⊂ ∂D, taking into account that h has non-tangential limit at every point of ∂D (see Corollary 11.1.7), we set h(A) := {w ∈ C∞ : w = ∠ lim h(z), σ ∈ A}. z→σ

Theorem 14.2.10 Let (φt ) be a semigroup in D which is not an elliptic group, with Denjoy-Wolff point τ ∈ D, Koenigs function h and infinitesimal generator G. Let A  ∂D be an open non-empty arc. Then the following are equivalent: (1) A is a contact arc of (φt ), (2) In case τ ∈ D, h(A) = spir λ [w0 ] ∩ {w ∈ C : a < |w| < b}, where λ ∈ C, Re λ > 0 is the spectral value of (φt ), w0 ∈ C \ {0} and 0 < a < b ≤ +∞. In case τ ∈ ∂D, h(A) = L[w0 ] ∩ {w ∈ C : a < Im w < b}, where w0 ∈ C and −∞ ≤ a < b ≤ +∞, (3) lim z→σ Re (σ G(z)) = 0 and lim sup(0,1)r →1 |G(r σ )| = 0 for every σ ∈ A. Moreover, if one—and hence any—of the previous holds, then h and G extend holomorphically through A. Proof Let (Ω, h, ψt ) be the canonical model given by Theorem 9.3.5. (1) implies (2). If A is a contact arc, by Proposition 14.2.2, φt extends continuously at σ , for all σ ∈ A. Hence, by Proposition 11.3.2, h has unrestricted limit at every point of A. Moreover, since A does not contain fixed points of (φt ) by definition, Proposition 13.6.1 implies that h(σ ) ∈ C for all σ ∈ A. Fix σ ∈ A. Then, by Theorem 11.2.3, we have h(φt (σ )) = ψt (h(σ )) for all t ∈ [0, T (σ )). From this (2) follows at once. Note in particular that h extends continuously to A and the image is a real analytic arc, therefore, Schwarz’s Reflection Principle implies that h extends holomorphically through A and h (σ ) = 0 for all σ ∈ A (because h −1 extends as well through h(A)). (2) implies (1). Let σ ∈ A. By hypothesis, for t > 0 sufficiently small, ψt (h(σ )) ∈ ∂h(D). Since by Theorem 11.2.3 we have h(φt (σ )) = ψt (h(σ )) for all t ≥ 0, it follows that φt (σ ) ∈ ∂D. By Proposition 13.6.1, σ is not a fixed point of (φt ) because h(σ ) = ∞ by hypothesis. Hence every point of A is a contact point of (φt ) which is not fixed, and (1) holds. (1) implies (3). We already saw that (1) implies that h extends holomorphically through A and h (σ ) = 0 for all σ ∈ A. By Theorem 10.1.4, G extends holomorphically through A as well and G(σ ) = 0 for all σ ∈ A. We are left to show that Re (σ G(σ )) = 0 for all σ ∈ A, from which (3) follows.

414

14 Contact Points

Let assume τ ∈ D. In this case, G(z) = −λ hh(z)

(z) and h(A) = spir λ [w0 ] ∩ {w ∈ C : a < |w| < b}, for some w0 ∈ C \ {0} and 0 < a < b ≤ +∞. Fix σ ∈ A. Let θ0 ∈ R be such that eiθ0 = σ , and let ε > 0 be such that (θ0 − ε, θ0 + ε)  θ → eiθ ∈ A. Then Argλ (h(eiθ )) = const. That is to say, h(eiθ ) = e−λr (θ)+iα with r (θ ) ∈ R depending on θ and α ∈ [−π, π ]. Since e−Re λr (θ) = |h(eiθ )|, we have λ

e− 2Re λ log |h(e

iθ

)|2

h(eiθ ) = eiα .

Differentiating with respect to θ and setting θ = θ0 , we obtain λh(σ )Im [h (σ )h(σ )σ ] + i h (σ )σ = 0. Re λ|h(σ )|2  Which is equivalent to Re



h (σ )σ λh(σ ) i . In h (z)

= 0, and hence Re (σ G(σ )) = 0.

this case, Re h(eiθ ) = const. Differentiating in If τ ∈ ∂D, then G(z) = θ and setting θ = θ0 , we obtain immediately Re (σ G(σ )) = 0. (3) implies (2). Let σ ∈ A. By the Berkson-Porta formula (see Theorem 10.1.10), G(z) = (z − τ )(τ z − 1) p(z), where p : D → H is holomorphic. Note that lim z→σ (z − τ )(τ z − 1)σ = −|1 − σ τ |2 = 0. Hence, lim Re p(z) =

z→σ

1 lim Re (G(z)σ ) = 0. −|1 − σ τ |2 z→σ

By Proposition 2.4.2, p extends holomorphically through A. Hence, G extends holomorphically through A and, by hypothesis, G(σ ) = 0 for all σ ∈ A. By Theorem 10.1.4, it follows that also h extends holomorphically through A. We give details for the case τ ∈ D, the other case being similar and easier. We have 1 h (z) =− , G(z) λh(z)

(z) which implies that hh(z) extends holomorphically through A and it is different from 0. Fix a point σ ∈ A and let z 0 ∈ D and ε > 0 be such that σ ∈ D(z 0 , ε) := {ζ ∈

(z) is holomorphic and non-zero in D(z 0 , ε). Then, for all C : |ζ − z 0 | < ε} and hh(z) z ∈ D(z 0 , ε),  z h (ζ ) log h(z) − log h(z 0 ) = dζ, z 0 h(ζ )

which implies that log h(z) extends holomorphically through σ , and so does h. Finally, using again Theorem 10.1.4 and arguing as in the proof of “(1) implies (3)”, we see that the condition Re (σ G(σ )) = 0 implies that h(A) is either a segment on a spiral (in case τ ∈ D) or a segment on a half-line (in case τ ∈ ∂D), and (2) holds. 

14.2 Maximal Contact Arcs

415

Corollary 14.2.11 Let (φt ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Let (I × R, h, z → z + it) be the canonical model of (φt ), where I = R, (0, +∞), (−∞, 0) or (0, ρ) for some ρ > 0. (1) If (φt ) is hyperbolic, and hence I = (0, ρ), and A is an exceptional maximal contact arc for (φt ), then h(A) ⊂ L[0] ∪ L[ρ]. (2) If (φt ) is parabolic of positive hyperbolic step, and hence either I = (0, +∞) or I = (−∞, 0), and A is an exceptional maximal contact arc, then h(A) ⊂ L[0]. (3) If (φt ) is parabolic of zero hyperbolic step, and hence I = R, then (φt ) has no exceptional maximal contact arc. Proof Let A be an exceptional maximal contact arc for (φt ). By Theorem  14.2.10, h(A) ⊂ L[r ] for some r ∈ I . Since h(D) is simply connected and t≥0 (h(D) − it) = I × R, it follows at once that if r belongs to the interior of I , then there exists a ∈ R such that max Im h(x) ≤ a. x∈A

Taking into account that h extends holomorphically through A and h(φt (x)) = h(x) + it for all x ∈ A and t ≥ 0 by Theorem 11.2.3, it follows that h(x1 (A)) = r + ia. Hence, by Proposition 11.1.8, x1 (A) = τ , a contradiction. Therefore, h(A) ⊆ L(r ) with r ∈ ∂ I , from which the statement follows at once.  Remark 14.2.12 Let (φt ) be a semigroup in D which is not an elliptic group, with infinitesimal generator G. Let A ⊂ ∂D be a contact arc for (φt ). By Theorem 14.2.10, G extends holomorphically through A and Re (G(σ )σ ) = 0 for all σ ∈ A. This means that the vector field G(σ ) is tangent to ∂D at σ . For σ ∈ ∂D, let xσ : (a, b) → D, −∞ ≤ a < 0 < b ≤ +∞ be the maximal solution to the Cauchy problem d x(t) = G(x(t)), x(0) = σ. dt Since G is holomorphic at σ , and hence the solution to the Cauchy problem depends holomorphically on the initial data, it follows that φt (σ ) = xσ (t) for all t ≥ 0. In other words, the semigroup differential equation ∂φ∂tt (z) = G(φt (z)) holds also for all z ∈ A. We end this section by studying the behavior of the Koenigs function at starting and final points of maximal contact arcs: Proposition 14.2.13 Let (φt ) be a semigroup in D which is not an elliptic group, with Denjoy-Wolff point τ ∈ D and Koenigs function h. Let A  ∂D be a maximal contact arc for (φt ). Then for j = 0, 1,

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14 Contact Points

lim

Ax→x j (A)

h(x) = ∠ lim h(z), z→x j (A)

(14.2.1)

In particular, the starting point x0 (A) of A is a fixed point of (φt ) if and only if • in case τ ∈ D,

h(A) = spir λ [w0 ] ∩ {w ∈ C : |w| > a},

where λ ∈ C, Re λ > 0 is the spectral value of (φt ), w0 ∈ C \ {0} and a > 0; • in case τ ∈ ∂D, h(A) = L[w0 ] ∩ {w ∈ C : Im w < b}, where w0 ∈ C and b ∈ (−∞, +∞]. Proof We can assume that (φt ) is not a parabolic group, as in this case the result is clear. According to Theorem 14.2.10, h extends holomorphically through A and, moreover, by Corollary 11.1.7, ∠ lim z→x j (A) h(z) exists, finite or infinite, for j = 0, 1. Fix j ∈ {0, 1}. Since h(A) is a connected subset of either a line or a spiral, and h(φt ( p)) = ψt (h( p)) for all t ≥ 0 and p ∈ A, where either ψt (z) = e−λt z (in the elliptic case, λ being the spectral value of (φt )) or ψt (z) = z + it (in the non-elliptic case) by Theorem 11.2.3, it follows that the limit Q := lim Ax→x j (A) h(x) exists in C∞ and h( p) = Q for all p ∈ A. Fix p ∈ A. Let A j be the open subarc in A that joins p and x j (A). Let U := {ζ ∈ D : ζ = r x, x ∈ A j , r ∈ (0, 1)}. Note that U is a Jordan domain, hence, by Theorem 4.3.3, there exists a homeomorphism g : D → U such that g|D : D → U is a biholomorphism. The map h ◦ g : ∞ D → h(U ) is a biholomorphism. Moreover, let Γ j := {h(r x j (A)) : r ∈ [0, 1)} and ∞ Γ := {h(r p) : r ∈ [0, 1)} . Note that, since h admits non-tangential limit at every point in ∂D, Γ j and Γ are Jordan arcs. Therefore (h(A j ) ∪ Γ j ∪ Γ ∪ {Q}) is closed and (h(A j ) ∪ Γ j ∪ Γ ∪ {Q}) ⊆ ∂∞ h(U ) = ∂∞ (h(g(D)). Applying Corollary 4.4.15 to h ◦ g, we get that h(A j ) ∪ Γ j ∪ Γ is dense in ∂∞ h(U ), hence, h(A) ∪ Γ j ∪ Γ ∪ {Q} = h(A) ∪ Γ j ∪ Γ

∞

= ∂∞ h(U ).

(14.2.2)

We claim that Q ∈ Γ ∪ Γ j . Indeed, assume by contradiction that this is not the case. Hence, there exists an open set V ⊂ C∞ such that Q ∈ V and V ∩ (Γ ∪ Γ j ) = ∅. Therefore, by (14.2.2), ∂∞ h(U ) ∩ V = (h(A) ∪ {Q}) ∩ V . Now, if Q ∈ C and r > 0, let D(Q, r ) := {ζ ∈ C : |ζ − Q| < r }. If Q = ∞ and r > 0, let D(Q, r ) := {ζ ∈ C : |ζ | > 1/r }. We can find r > 0 such that D(Q, r ) ⊂ V . Let {rn } be a strictly decreasing sequence of positive real numbers converging to 0 such that rn < r for all n ∈ N. Let Cn := ∂ D(Q, rn ) \ h(A). Since h(A)

14.2 Maximal Contact Arcs

417

is a connected subset of either a line or a spiral and Cn ⊂ V , it follows that Cn = ∂ D(Q, rn ) \ {h(ηn )} for some ηn ∈ A. Hence, (Cn ) is a circular null-chain for h(g(D)) centered at Q and the impression of the prime end represented by (Cn ) is {Q}. In particular, there exists σ ∈ ∂D such that I (h ◦ g(x σ )) = {Q}, where h ◦g is the homeomorphism defined by h ◦ g in the Carathéodory topology and x σ ∈ ∂C D is the prime end representing σ . Since by construction h is continuous on ∂U except at most x j (A) and h(η) = Q for all η ∈ ∂U \ {x j (A)} (because Q ∈ / Γ ∪ Γ j by hypothesis and Q ∈ / h(A)) it follows that g(σ ) = x j (A). Hence, by Proposition 4.4.4 and taking into account that g is a homeomorphism from D to U , lim

U z→x j (A)

h(z) = lim h(g(z)) = Q. z→σ

Therefore, Q = h(x j (A)) ∈ Γ j , a contradiction. Hence, Q ∈ Γ ∪ Γ j . Moreover, since Q ∈ ∂∞ h(D), the only possibilities are Q = h(x j (A)) or Q = h( p). Since the second case cannot occur (because h( p) = Q for all p ∈ A), we have Q = h(x j (A)), and Eq. (14.2.1) follows. Finally, the last part of the statement follows immediately from (14.2.1) and Proposition 13.6.1. 

14.3 Infinitesimal Generators and Maximal Contact Arcs Now we turn our attention to the behavior of infinitesimal generators at the end points of a maximal contact arc. Theorem 14.3.1 Let (φt ) be a semigroup, not a group, in D with Denjoy-Wolff point τ ∈ D. Let G be the infinitesimal generator of (φt ). Suppose A  ∂D is a maximal contact arc with starting point x0 (A) and final point x1 (A). Then (1) (2) (3) (4)

L 0 := ∠ lim z→x0 (A) G(z) exists finite, if x0 (A) is a boundary fixed point of (φt ) then L 0 = 0, L 1 := ∠ lim z→x1 (A) G(z) exists, finite or infinite, L 1 = 0 if and only if x1 (A) = τ (in particular, (φt ) is non-elliptic).

Proof By Theorem 14.2.10, G extends holomorphically through A and for every σ ∈ A we have Re (σ G(σ )) = 0 and G(σ ) = 0. Moreover, by the Berkson-Porta formula (see Theorem 10.1.10), G(z) = (z − τ )(τ z − 1) p(z), where p : D → H is holomorphic. Note that lim z→σ (z − τ )(τ z − 1)σ = −|1 − σ τ |2 = 0. Hence, lim Re p(z) =

z→σ

1 lim Re (G(z)σ ) = 0. −|1 − σ τ |2 z→σ

Therefore, Proposition 2.4.2 implies that ∠ lim z→x j (A) p(z) = lim Ax→x j (A) p(z) exists, finite or infinite, j = 0, 1. Hence,

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14 Contact Points

L j = ∠ lim G(z) = z→x j (A)

lim

Ax→x j (A)

G(z)

(14.3.1)

exists, finite or infinite, j = 0, 1, and, in particular, (3) holds. Now, let h be the Koenigs function of (φt ). By Theorem 10.1.4, either G(z) = λh(z) , where λ ∈ C, Re λ > 0, is the spectral value of (φt ) in case (φt ) is elliptic, or h (z) G(z) = i/ h (z) in case (φt ) is non-elliptic. Hence, Theorem 14.2.10 guarantees that for all x ∈ A, i λh(x) or G(x) = . G(x) = h (x) h (x) Therefore, by (14.3.1), L0 =

λh(x) Ax→x0 (A) h (x)

or

lim

L0 =

lim

i

Ax→x0 (A) h (x)

.

(14.3.2)

By (14.2.1) and Proposition 13.6.1, lim

Ax→x0 (A)

|h(x)| = ∞

if and only if x0 (A) is a fixed point of (φt ). Assume first that x0 (A) is a fixed point for (φt ) and let eiθ0 = x0 (A) for some θ0 ∈ [0, 2π ) and suppose without loss of generality that A = {eiθ : θ ∈ (θ0 , θ0 + δ)} for some δ > 0. Hence, lim+ |h(eiθ )| = ∞. θ→θ0

In case τ ∈ D, this implies that limθ→θ0+ log |h(eiθ )| = +∞. Differentiating in θ , we conclude that lim sup Ax→x0 (A) |Re h (x)/ h(x)| = +∞, and, by (14.3.2), L 0 = 0. In case τ ∈ ∂D, by Theorem 14.2.10, we have Re h(eiθ ) = α for all θ ∈ (θ0 , θ0 + δ) and for some fixed α ∈ R. Hence, lim supθ→θ0+ |Im h(eiθ )| = +∞. Let θ1 ∈ (θ0 , θ0 + δ). Since for all θ ∈ (θ0 , θ1 ) we have  Im h(eiθ1 ) − Im h(eiθ ) =

θ

θ1

Im (ieiu h (eiu ))du,

it follows that lim supθ→θ0+ |Im h (eiθ )| = +∞ and, in turn, lim sup Ax→x0 (A) |h (x)| = +∞. Thus, by (14.3.2), L 0 = 0. This proves (2). In order to prove (1), we need to show that |L 0 | < +∞ and, thanks to (2), we can assume that x0 := x0 (A) is not a fixed point of (φt ). Hence, Proposition 13.6.1 implies that h(x0 ) ∈ C (and h(x0 ) = 0 in case (φt ) is elliptic). Thus, (14.3.1) and Theorem 10.1.4 imply that ∠ lim z→x0 h (z) exists, finite or infinite, and |L 0 | < +∞ if and only if ∠ lim h (z) = 0. z→x0

14.3 Infinitesimal Generators and Maximal Contact Arcs

419

Suppose by contradiction that ∠ lim z→x0 h (z) = 0. For α > 0, in case (φt ) is elliptic, let Rα := spir λ [h(x0 )] ∩ {w ∈ C : |w| ≥ |h(x0 )| − α}, while, in case (φt ) is non-elliptic, let Rα := L[h(x0 )] ∩ {w ∈ C : Im w ≤ Im h(x0 ) + α}. Taking into account that either h(D) is λ-spirallike (in case (φt ) is elliptic) or starlike at infinity (in case (φt ) is non-elliptic), Theorem 14.2.10 and Proposition 14.2.13 imply that there exists α > 0 such that Rα ∩ h(D) = ∅. Fix such an α, and let g : D → C \ Rα be a Riemann map. Hence, ϕ := g −1 ◦ h is a well defined univalent self-map of D. Moreover, let γ : [0, 1) → h(D) be defined by γ (r ) = h(r x0 ). Note that limr →1 γ (r ) = h(x0 ) and there exists σ ∈ ∂D such that g extends holomorphically in a neighborhood of σ , g (σ ) = lim z→σ g (z) exists and it is different from 0, and limr →1 g −1 (γ (r )) = σ (this follows at once by looking at g(D) = C \ Rα and using Schwarz’s Reflection Principle). Also, note that lim ϕ(r x0 ) = lim− g −1 (γ (r )) = σ.

r →1−

r →1

Therefore, x0 is a contact point of ϕ (maybe non-regular), and, by Theorem 1.7.2 and Proposition 1.9.3, we have 0= =

lim

(0,1)r →1

lim

(0,1)r →1

h(x0 ) − h(r x0 ) (0,1)r →1 1−r g(σ ) − g(ϕ(r x0 )) σ − ϕ(r x0 ) = g (σ )ϕ (x0 ) = 0, σ − ϕ(r x0 ) 1−r

h (r x0 ) =

lim

a contradiction, and (1) holds. It remains to prove (4) Let x1 := x1 (A). If x1 = τ , then L 1 = 0 by Corollary 10.1.12. We assume x1 = τ . Hence, by Proposition 14.2.6, x1 is a non-fixed point which is the starting point of a maximal invariant curve of (φt ). In particular, h(x1 ) ∈ C and, arguing as before, we see that ∠ lim z→x1 h (z) exists, finite or infinite. Moreover, L 1 = 0 if and only if ∠ lim h (z) = ∞. z→x1

Now, in case (φt ) is elliptic, let θ1 := Argλ (h(x1 )). For α > 0 let Spir[λ, 2α, θ1 − α] = {etλ+iθ : t ∈ R, θ ∈ (−2α + θ1 , θ1 )} be the λ-spirallike sector of amplitude 2α and center ei(θ1 −α) . Similarly, define the λ-spirallike sector Spir[λ, 2α, θ1 + α].

420

14 Contact Points

For a > 0 let + := Spir[λ, 2α, θ1 + α] ∩ {w ∈ C : |w| < a} Sα,a − Sα,a := Spir[λ, 2α, θ1 − α] ∩ {w ∈ C : |w| < a}.

By Theorem 14.2.10 and taking into account that h(D) is λ-spirallike, it follows that + − ⊂ h(D) or Sα,a ⊂ h(D). there exist α ∈ (0, π ] and a > |h(x1 )| such that either Sα,a + ⊂ h(D). We can assume, without loss of generality, that Sα,a In case (φt ) is non-elliptic, again by Theorem 14.2.10 and taking into account that h(D) is starlike at infinity, there exist a < Im h(x1 ) and α > 0 such that, setting + Sα,a : = {w ∈ C : Im w > a, Re h(x1 ) < Re w < Re h(x1 ) + α} − Sα,a : = {w ∈ C : Im w > a, Re h(x1 ) − α < Re w < Re h(x1 )}, + − we have that either Sα,a ⊂ h(D) or Sα,a ⊂ h(D). We can assume, without loss of + generality, that Sα,a ⊂ h(D). + . Note that S is simply In both the elliptic and non-elliptic case, let S := Sα,a connected, and let g : D → S be a Riemann mapping. Since h(x1 ) belongs to a real analytic curve on ∂ S, it follows by the Schwarz Reflection Principle that there exists a point σ ∈ ∂D such that g extends holomorphically through σ , g (σ ) = 0 and g(σ ) = h(x1 ). Now, let ϕ := h −1 ◦ g : D → D. Since S ⊂ h(D), the map ϕ is well defined and univalent and it is easy to see that ∠ lim z→σ ϕ(z) = x1 . Therefore, σ is a contact point of ϕ, and hence, by Proposition 1.9.3 and since h ◦ ϕ = g, we have

g(σ ) − g(r σ ) 1−r h(ϕ(σ )) − h(ϕ(r σ )) x1 − ϕ(r σ ) = lim (0,1)r →1 x1 − ϕ(r σ ) 1−r h(x ) − h(ϕ(r σ )) 1 . = ϕ (σ ) lim (0,1)r →1 x1 − ϕ(r σ )

g (σ ) =

lim

(0,1)r →1

In particular, since ϕ (σ ) = 0 and g (σ ) = ∞, it follows that L :=

lim

(0,1)r →1

h(x1 ) − h(ϕ(r σ )) x1 − ϕ(r σ )

exists, finite. By Corollary 7.3.11, ∠ lim

z→x1

h(x1 ) − h(z) = L. x1 − z

14.3 Infinitesimal Generators and Maximal Contact Arcs

421

Hence, by Theorem 1.7.2, ∠ lim h (z) = L = ∞, z→x1



and we are done.

14.4 Super-Repelling Fixed Points and Maximal Contact Arcs Theorem 14.4.1 Let (φt ) be a semigroup, not a group, in D. Let σ ∈ ∂D be a superrepelling fixed point. Then, one and only one of the following cases holds: (1) there exists a unique (up to re-parameterization) maximal invariant curve for (φt ) with starting point σ , (2) there exists a unique maximal contact arc with starting point σ . Proof We divide the proof in four steps. Step 1. The point σ cannot be at the same time both the starting point of a maximal contact arc and the starting point of a maximal invariant curve of (φt ). Assume by contradiction this is the case. Let γ : (−∞, +∞) → D be a maximal invariant curve such that limt→−∞ γ (t) = σ and let A ⊂ ∂D be a maximal contact arc such that x0 (A) = σ . By Proposition 13.4.15 the curve γ is, up to re-parameterization, the unique maximal invariant curve of (φt ) with starting point σ . Let p := x1 (A) be the final point of A. By Proposition 14.2.6, either p = τ , the Denjoy-Wolff point of (φt ), or p = τ is the starting point of a maximal invariant curve η : (a, +∞) → D, a ∈ (−∞, 0). In case p = τ , let J be the union of the closures of A and γ ((−∞, +∞)). In case p = τ , by Remark 13.3.9, up to re-parameterization, η is the unique maximal invariant curve of (φt ) with starting point p. In this case, we let J = A ∪ γ ((−∞, +∞)) ∪ η((a, +∞)). By Remark 13.3.2, J is a Jordan curve containing τ . Let V ⊂ D be the bounded connected component of C \ J . Let z 0 ∈ V and let γ˜ : (a, ˜ +∞) → D, a˜ < 0, be the unique maximal invariant curve for (φt ) / J for all t ∈ (a, ˜ +∞). Therefore, such that γ˜ (0) = z 0 . By Remark 13.3.8, γ˜ (t) ∈ limt→a˜ + γ˜ (t) = q ∈ J ∩ ∂D. Since φt (ζ ) ∈ ∂D for all ζ ∈ A and t > 0 sufficiently small, it follows that q ∈ / A. By the uniqueness up to re-parameterization of γ , it follows that q = σ . Moreover, if p = τ , by the uniqueness up to re-parameterization of η, we have q = p. Therefore, the only possibility is q = τ . Proposition 13.3.5 implies that a˜ = −∞ and z 0 ∈ W , the backward invariant set of (φt ). By the arbitrariness ◦

of z 0 , we have V ⊂ W . In particular, since V is connected, there exists a petal Δ of (φt ) such that V ⊆ Δ. But σ ∈ ∂Δ is a super-repelling fixed point, contradicting Proposition 13.4.10. Step 2. The point σ can be the starting point of, at most, one maximal contact arc. The argument is similar to the one in Step 1, and we omit the details.

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14 Contact Points

Step 3. Suppose that τ ∈ ∂D and σ is not the starting point of a maximal invariant curve for (φt ). Then σ is the starting point of a maximal contact arc. Since by Proposition 1.2.2 the group Aut(D) acts double transitively on ∂D, there exists T ∈ Aut(D) such that T (τ ) = 1 and T (σ ) = −1. Taking into account that T is a Möbius transformation, it is easy to see that −1 is a super-repelling fixed point of the semigroup (T ◦ φt ◦ T −1 ) and 1 is its Denjoy-Wolff point. Moreover, −1 is the starting point of a maximal contact arc (respectively, of a maximal invariant curve) for (T ◦ φt ◦ T −1 ) if and only if σ is the starting point of a maximal contact arc (resp., of a maximal invariant curve) for (φt ). In other words, we can assume without loss of generality that τ = 1 and σ = −1. Let (Ω, h, z → z + it) be the canonical model of (φt ) given by Theorem 9.3.5, where Ω is either C, H, H− or a strip Sρ for some ρ > 0. By Proposition 13.6.7, there is α ∈ R such that lim z→−1 Re h(z) = α and lim z→−1 Im h(z) = −∞. Fix r ∈ (0, 1). If Re h(−r ) = α, let Sr := {h(−r )}. If Re h(−r ) < α, let Sr := {ζ ∈ C : Re h(−r ) ≤ Re ζ < α, Im ζ = Im h(−r )}, while, if Re h(−r ) > α, let Sr := {ζ ∈ C : α < Re ζ ≤ Re h(−r ), Im ζ = Im h(−r )}. Claim (): Sr ⊂ h(D) for all r ∈ (0, 1). The claim is obviously true if Sr := {h(−r )}, so assume this is not the case. Note that, by Proposition 9.4.12, Im [(1 + r )2 h (−r )] ≥ 0. That is, (0, 1)  r → Im h(−r ) is decreasing in r .

(14.4.1)

Now, fix x0 ∈ R such that x0 + iIm h(−r ) ∈ Sr . Since (0, 1)  s → Re h(−s) is a continuous curve which converges to α as s → 1− , there exists s ∈ [r, 1) such that Re h(−s) = x0 . Taking into account that h(D) is starlike at infinity, it follows that x0 + it ∈ h(D) for all t ≥ Im h(−s). Since Im h(−s) ≤ Im h(−r ), we have that x0 + iIm h(−r ) ∈ h(D), and the claim is proved. Now we claim that there exists y0 ∈ R such that {α + it : t ≤ y0 } ∩ h(D) = ∅. Assume by contradiction this is not the case. Taking into account that h(D) is starlike at infinity, this implies that Pα := {α + it : t ∈ R} ⊂ h(D). Note that R  t → h −1 (α + it) is a maximal invariant curve for (φt ), because φs (h −1 (α + it)) = h −1 (α + i(t + s)). Since by hypothesis there exist no maximal invariant curves for (φt ) starting at −1, it follows that limt→−∞ h −1 (α + it) = p ∈ ∂D \ {−1}. In par/ h −1 (Pα )—hence h(−r ) ∈ / Pα —for all ticular, there exists r0 ∈ (0, 1) such that −r ∈ r ∈ [r0 , 1). Note that, since the curve [r0 , 1) → Re h(−r ) is continuous, this implies that either Re h(−r ) > α or Re h(−r ) < α for all r ∈ [r0 , 1). By Claim () and (14.4.1), Sr0 ⊂ h(D) and Im h(−s) ≤ Im h(−r0 ) for all s ∈ [r0 , 1). Taking into account that h(D) is starlike at infinity, we can then find a Jordan arc Γ in h(D) which joins h(−r0 ) with α + iIm h(−r0 ) and whose interior does not intersect Pα and h((−1, −r0 ]). Now let γ : (0, +∞) → h(D) be a continuous injective curve such that γ (0) = α + iIm h(−r0 ), γ ([0, r0 ]) = Γ and γ (t) = h(−t) for t ≥ r0 . Let ∞

∞

J = γ ([0, +∞)) ∪ Pα ∩ {w ∈ C : Im w < Im h(−r0 )} .

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423

By construction, J is a Jordan curve in C∞ containing ∞. By Claim (), and since either Re h(−r ) > α or Re h(−r ) < α for all r ∈ [r0 , 1), it follows that one of the two connected components of C∞ \ J is contained in h(D). By Proposition 3.3.5, −1 = limt→+∞ h −1 (γ (t)) = limt→+∞ h −1 (α − it) = p, a contradiction. Therefore, there exists y0 ∈ R such that {α + it : t ≤ y0 } ∩ h(D) = ∅. In fact, by Claim (), {α + it : t ≤ y0 } ⊂ ∂h(D). Let r1 ∈ (0, 1) be such that Im h(−r1 ) ≤ y0 . As before, either Re h(−r ) < α or Re h(−r ) > α for all r ∈ [r1 , 1). Let U :=



Sr

r ∈(r1 ,1)

Note that U is open, simply connected and U ⊂ h(D). Moreover, by construction, {α + it : t ≤ Im h(−r1 )} ⊂ ∂U and ∂U \ {α + it : t ≤ Im h(−r1 )} ⊂ h(D). Schwarz’s Reflection Principle implies that the function h −1 |U extends holomorphically through {α + it : t < Im h(−r1 )} and hence there exists an open arc A ⊂ ∂D such that h(A ) = {α + it : t < Im h(−r1 )}. By Theorem 14.2.10, A is a contact arc for (φt ). Let A ⊂ ∂D be the maximal contact arc for (φt ) which contains A . Let {rn } ⊂ (−1, −r1 ) be a decreasing sequence converging to −1 and let ζn := α + iIm h(−rn ). By (14.4.1), ζn ∈ {α + it : t < Im h(−r1 )} for all n ∈ N. Hence, h −1 (ζn ) ∈ A for all n ∈ N. Note that {Srn } is a sequence of Jordan arcs whose spherical diameters tend to 0. Therefore, by the no Koebe Arcs Theorem 3.2.4, the Euclidean diameters of the Jordan arcs {h −1 (Srn )} tend to zero as well. Since h −1 (−rn ) ∈ Srn and h −1 (−rn ) → −1 as n → ∞, it follows that h −1 (ζn ) → −1 as n → ∞. In other words, −1 ∈ A. By the definition of contact arcs and Proposition 14.2.6, it follows that x0 (A) = −1, and we are done. Step 4. Suppose that τ ∈ D and σ is not the starting point of a maximal invariant curve for (φt ). Then σ is the starting point of a maximal contact arc. One can argue as in Step 3, considering spirals instead of lines parallel to the imaginary axis, but here we present a different argument which allows to reduce to Step 3. Let σ = eiθ0 for some θ0 ∈ [−π, π ). Let ε ∈ (0, π/2). Consider the arc Z := iθ {e : θ ∈ (θ0 , θ0 + ε)}. By Proposition 3.3.2, there exists θ1 ∈ (θ0 , θ0 + ε) such that eiθ1 is not a fixed point of (φt ). If the life-time T (eiθ1 ) = 0, then set p1 := eiθ1 . If T (eiθ1 ) > 0, then there exists a maximal contact arc A  ∂D whose closure contains eiθ1 . Let q be the starting point of A. If q = σ we are done. Otherwise, q = eiθ2 for some θ2 ∈ (θ0 , θ1 ]. Let Z := {eiθ : θ ∈ (θ0 , θ2 )}. As before, we can find θ3 ∈ (θ0 , θ2 ) such that T (eiθ3 ) < +∞. If T (eiθ3 ) = 0, set p1 = eiθ3 , otherwise, there exists a maximal contact arc A  ∂D whose closure contains eiθ3 ; let p1 be its end point. Since A ∩ A = ∅, it follows that p1 = eiθ4 for some θ4 ∈ (θ0 , θ2 ). In any case, if σ is not the starting point of a maximal contact arc of (φt ), there exists p1 = eiβ1 for some β1 ∈ (θ0 , θ0 + ε) which is the starting point of a maximal invariant curve of (φt ).

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14 Contact Points

A similar argument shows that there exists p2 = eiβ2 for some β2 ∈ (θ0 − ε, θ0 ) which is the starting point of a maximal invariant curve of (φt ). Note that p1 = p2 . Consider the Jordan curve J given by the union of the maximal invariant curve starting at p1 , the one starting at p2 and the arc in ∂D with end points p1 and p2 and containing σ . Note that τ ∈ J . Let V be the bounded connected component of C \ J . By Remark 13.3.8, for all z 0 ∈ V , φt (z 0 ) ∈ V for all t ≥ 0. Namely, φt (V ) ⊆ V . Let g : D → V be a Riemann map. Since V is a Jordan domain, g extends as a homeomorphism—still denoted by g—from D onto V by Theorem 4.3.3. Let φ˜ t := g −1 ◦ φt ◦ g, t ≥ 0. Then (φ˜ t ) is a semigroup in D and, since lim φ˜ t (0) = lim g −1 (φt (g(0))) = g −1 (τ ) ∈ ∂D,

t→+∞

t→+∞

it follows that (φ˜ t ) is non-elliptic. Since σ is a super-repelling fixed point of (φt ), it follows from Proposition 13.6.1 and Proposition 11.3.1 that lim z→σ φt (z) = σ for all t ≥ 0. Thus, lim z→g−1 (σ ) φ˜ t (z) = g −1 (σ ) for all t ≥ 0. Namely, σ˜ := g −1 (σ ) is a boundary fixed point of (φ˜ t ). If there exists a maximal invariant curve γ : (−∞, +∞) → D for (φ˜ t ) such that limt→−∞ γ (t) = σ˜ , it is easy to see that g ◦ γ is a maximal invariant curve for (φt ) which starts at σ , against our hypothesis. Therefore, σ˜ is a super-repelling fixed point of (φ˜ t ) which is not the starting point of any maximal invariant curve of (φ˜ t ). For what we already proved, there exists a maximal contact arc A˜  ∂D for (φ˜ t ) ˜ ∩ ∂D. which starts at σ˜ . Let A be the interior part of the (non-empty) arc g −1 ( A) Note that, by construction, A is a non-empty open arc in ∂D and σ is one of its end points. If we prove that A is a contact arc, by Proposition 14.2.6, σ is its starting point, and we are done. Let ζ ∈ A and let ζ˜ ∈ A˜ be such that g(ζ˜ ) = ζ . Let t > 0 be small enough such that φ˜ t (ζ˜ ) ∈ g −1 (A). Then, taking into account Proposition 14.2.2(4), lim φt (r ζ ) = lim− g(φ˜ t (g −1 (rg(ζ˜ )))) = g(φ˜ t (ζ˜ )) ∈ A.

r →1−

r →1

By the arbitrariness of ζ , it follows that A is a contact arc.



Proposition 14.2.6 and Theorem 14.4.1 allow to give the following definition: Definition 14.4.2 Let (φt ) be a semigroup in D. Suppose σ ∈ ∂D is a super-repelling fixed point of (φt ), then, • we say that σ is a super-repelling fixed point of the first type if it is the starting point of a maximal invariant curve of (φt ), • we say that σ is a super-repelling fixed point of the second type if it is the starting point of a maximal contact arc of (φt ) which is not exceptional, • we say that σ is a super-repelling fixed point of the third type if it is the starting point of an exceptional maximal contact arc of (φt ). We saw in Theorem 12.2.5 that if σ ∈ ∂D is a repelling fixed point of a semigroup (φt ) then σ is a (regular) critical point of the infinitesimal generator G of (φt ). We

14.4 Super-Repelling Fixed Points and Maximal Contact Arcs

425

also saw that in general boundary critical points do not correspond to boundary fixed points. However, for super-repelling fixed points of the second and third types, the result continues to hold. Indeed, as an immediate consequence of Theorems 14.3.1 and 14.4.1, we have: Corollary 14.4.3 Let (φt ) be a semigroup, not a group, in D with associated infinitesimal generator G. If σ ∈ ∂D is a super-repelling fixed point of (φt ) of second or third type, then σ is a boundary critical point of G. In particular, if σ ∈ ∂D is a super-repelling fixed point of (φt ) and there are no backward orbits of (φt ) converging to σ then σ is a boundary critical point of G.

14.5 Examples In this section we construct some examples in order to illustrate the objects defined and studied in this chapter. Example 14.5.1 Let (φt ) be the semigroup constructed in Example 13.7.5. Namely, let h be a Riemann map of the domain Ω = S ∪ (1 + iR) ∪ {w ∈ S + 1 : Im w(1 − Re w) > 1} ⊂ S2 , and let φt (z) := h −1 (h(z) + it), for all z ∈ D and t ≥ 0. Let us denote by τ ∈ ∂D its Denjoy-Wolff point. The arc J constructed in Example 13.7.5 is an exceptional maximal contact arc with starting point σ and final point τ . Example 14.5.2 Example 13.7.6 provides an exceptional maximal contact arc with starting point a contact point which is not fixed, namely with the notation introduced there, the arc is B = h −1 (i[0, +∞)) ⊂ ∂D. As a consequence of Proposition 12.3.9, the set of repelling fixed points of a semigroup is always countable. In neat contrast, a semigroup can have an uncountable set of boundary fixed points as the next example shows. Example 14.5.3 Consider the set of rational numbers of the interval (0, 1), say {α(n) : n ∈ N} . For each n, let Γn := α(n) + i(−∞, −n] and define the domain Ω := S \ (∪n∈N Γn ) . Take a Riemann map h of the domain Ω. Consider the semigroup (φt ) whose canonical model is (S, h, z → z + it), that is, φt (z) := h −1 (h(z) + it), for all z ∈ D and t ≥ 0. Let τ be the Denjoy-Wolff point of (φt ). For each irrational number x ∈ (0, 1) the curve R  t → h −1 (x − it) is a maximal invariant curve for (φt ). By Lemma 13.1.5, there is a boundary fixed point σx such that limt→+∞ h −1 (x − it) = σx . Since the semigroup is hyperbolic, by Proposition 13.4.6, σx = τ .

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14 Contact Points

By construction and Theorem 13.5.5, σx is not repelling, thus, it has to be superrepelling. Finally, Proposition 13.4.15 guarantees that the map (0, 1) \ Q  x → σx is one-to-one. Thus, the semigroup (φt ) has an uncountable set of boundary fixed points. In addition, all of them are super-repelling fixed point of the first type. Take x = [(Cn )] a prime end of Ω. Assume that x does not correspond to τ under h. By Theorem 11.1.4, there is p ∈ C∞ such that Π (x) = { p} and, in fact, if p = ∞ then I (x) = {∞}. Now, if p ∈ C, the geometry of Ω implies that there is ε > 0 such that ∂Ω ∩ D ⊂ L[ p], where D = {w ∈ C : |w − p| < ε}. Moreover, we may assume that Cn is included in D for all n. Therefore the two endpoints of Cn belong to L[ p] for all n, and this implies that the interior of Cn is included in the disc D. Thus I (x) ⊂ D. The arbitrariness of ε allows to conclude that I (x) = { p}. Summing up, the map h has unrestricted limit at every point of ∂D \ {τ } (also in τ , but we do not need this fact), and by Theorem 11.3.8, every φt has a continuous extension to D. In the next two examples we construct semigroups with super-repelling fixed point of the second and third type. Example 14.5.4 Consider the domain Ω := {w ∈ H : Im w Re w > −1} , and take a Riemann map h from D onto Ω. Let (φt ) be the semigroup in D defined by φt (z) := h −1 (h(z) + it), z ∈ D, t ≥ 0. Let τ be its Denjoy-Wolff point. Since there is no vertical line in Ω, the backward invariant set of (φt ) is empty. Let (0, 1]  t → γ (t) := h −1 (t + i(−1/t + 1)) and let σ := limt→0 γ (t) ∈ ∂D (see Proposition 3.3.3). Then limt→0 Im h(γ (t)) = −∞. By Theorem 11.1.4 and Proposition 11.1.9, lim z→σ Im h(z) = −∞. Thus Proposition 13.6.2 implies that σ is a boundary fixed point of the semigroup (φt ). Since the semigroup has no petals, σ is a super-repelling fixed point. Let J be the Jordan arc in ∂D joining σ with τ such that h(J ) = iR. The final point of J is τ and σ is its starting point. Thus J is an exceptional maximal contact arc with final point τ , hence σ is a super-repelling fixed point of the third type. The construction of a semigroup with a super-repelling fixed point of the second type is a slight modification of the previous example. Example 14.5.5 Consider the domain Ω := {w ∈ H : 0 < Re w ≤ 1, Im w > 0} ∪ ({w ∈ H : Im w > log(Re w)} + 1), and take a Riemann map h from D onto Ω. Now, let (φt ) be the semigroup defined by φt (z) := h −1 (h(z) + it), z ∈ D, t ≥ 0.

14.5 Examples

427

Let us denote by τ its Denjoy-Wolff point. Since there is no vertical line in Ω, the backward invariant set of (φt ) is empty. Take (0, 1]  t → γ (t) := h −1 (t + 1 + i(log(t) + 1)) and let σ := limt→0 γ (t) ∈ ∂D (see Proposition 3.3.3). Then limt→0 Im h(γ (t)) = −∞. By Theorem 11.1.4 and Proposition 11.1.9, lim z→σ Im h(z) = −∞. Thus Proposition 13.6.2 implies that σ is a boundary fixed point of the semigroup. Since the semigroup has no petal, it turns out that σ is a super-repelling fixed point. Take σ˜ the point in ∂D such that h(σ˜ ) = 1 and J the Jordan arc in ∂D joining σ with σ˜ such that h(J ) = 1 + i(−∞, 0]. Hence, σ is a super-repelling fixed point of the second type.

14.6 Notes Theorem 14.1.1 was first proved in [81]. The rest of the chapter is taken basically from [30, 37]. Condition (3) in Theorem 14.2.10 can be weaken assuming that for every σ ∈ A lim(0,1)r →1 Re (σ G(r σ )) = 0 and lim sup(0,1)r →1 |Im G(r σ )| = 0 (see [37, Proposition 3.6]).

Chapter 15

Poles of the Infinitesimal Generators

In this chapter, we introduce the notion of regular (boundary) poles for infinitesimal generators of semigroups. We characterize such regular poles in terms of β-points (i.e. pre-images of values with a positive (Carleson-Makarov) β-numbers) of the associated semigroup and of the associated Koenigs function. We also define a natural duality operation in the cone of infinitesimal generators and show that the regular poles of an infinitesimal generator correspond to the regular critical points of the dual generator. Finally we apply such a construction to study radial multi-slits and give an example of a non-isolated radial slit semigroup whose tip has not a positive (Carleson-Makarov) β-number. In order to study radial multi-slits semigroup, we exploit a representation formula for starlike functions which we are going to prove in this chapter.

15.1 Regular Poles and β-Points Definition 15.1.1 Let G ∈ Gen(D). A point σ ∈ ∂D is a regular pole of G of mass C > 0 if ∠ lim inf |G(z)(σ − z)| = C. z→σ

We denote by PC (G) the set of regular poles of G of mass C. Moreover, we let P(G) := ∪C>0 PC (G) be the set of regular poles of G. Berkson-Porta’s Formula (see Theorem 10.1.10) and Proposition 2.1.3 imply:

© Springer Nature Switzerland AG 2020 F. Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4_15

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15 Poles of the Infinitesimal Generators

Lemma 15.1.2 If G ∈ Gen(D), then for all σ ∈ ∂D the non-tangential limit ∠ lim G(z)(σ − z) = a z→σ

exists, with |a| ∈ [0, +∞). Remark 15.1.3 By Lemma 15.1.2, if G ∈ Gen(D) and C > 0 then PC (G) = {σ ∈ ∂D : ∠ lim |G(z)(σ − z)| = C}. z→σ

Moreover, note that, if σ ∈ ∂D is a regular pole then ∠ lim |G(z)| = ∞. z→σ

In addition, again by Berkson-Porta’s Formula (see Theorem 10.1.10) and Proposition 2.1.3, for every α > 1 ∠ lim G(z)(z − σ )α = 0. z→σ

Hence, G has no pole of order strictly greater than 1. Proposition 15.1.4 Let G ∈ Gen(D) be given by the Berkson-Porta’s Formula G(z) = (τ − z)(1 − τ z) p(z) for some τ ∈ D and p : D → C holomorphic with Re p(z) ≥ 0 for all z ∈ D. Let A j > 0, j = 1, . . . , m. Let σ j ∈ P A j (G) for j = 1, . . . , m. Then m  Aj ≤ Re p(0). (15.1.1) 2 2|σ j − τ| j=1 Proof Note that p is not constant, for otherwise ∠ lim z→σ j |G(z)| < +∞ and then ∠ lim z→σ j |G(z)(σ j − z)| = 0, for j = 1, . . . , m. For j = 1, . . . , m, let L j :=

lim

(0,1)r →1

1−r p(r σ j ). 2

By Proposition 2.1.3 such a limit exists finite, it is non-negative, and the function f (z) := p(z) −

m  j=1

Lj

σj + z σj − z

is holomorphic in D and Re f (z) ≥ 0 for all z ∈ D. Taking into account that Aj =

lim

(0,1)r →1

|G(r σ j )|(1 − r ) = 2|σ j − τ |2 L j ,

15.1 Regular Poles and β-Points

431

inequality (15.1.1) follows at once because Re f (0) = Re p(0) −

m j=1

L j ≥ 0. 

A direct consequence of the previous proposition is the following:   Corollary 15.1.5 Let G ∈ Gen(D). Then for all C > 0 the set ∪ D≥C P D (G) is finite. In particular, P(G) is, at most, countable. Definition 15.1.6 Let f : D → C be holomorphic. A point σ ∈ ∂D is a CarlesonMakarov β-point (or, simply, a β-point) if ∠ lim sup z→σ

| f (z)| = L < +∞. |σ − z|

We call L the mass of f at σ and we denote by Bβ ( f ) the set of all β-points of f . We are going to prove that regular poles of infinitesimal generators correspond to β-points of the associated semigroups. Recall that, φt (σ ) := ∠ lim z→σ φt (z) exists for all t ≥ 0 by Theorem 11.2.1. Theorem 15.1.7 Let (φt ) be a semigroup in D with infinitesimal generator G ∈ Gen(D). Let σ ∈ ∂D. The following are equivalent: (1) lim sup |G(r σ )|(1 − r ) > 0, (0,1)r →1

(2) σ ∈ P(G), (3) σ ∈ Bβ (φt ) for some—and hence any—t > 0, (4) ∠ lim z→σ φt (z) = 0, ∠ lim z→σ φt (z) = L ∈ C for some—and hence any—t > 0, (5) lim(0,1)r →1 |φt (r σ )| = 0, lim sup(0,1)r →1 |φt (r σ )| < +∞ for some—and hence any—t > 0, |φt 0 (r σ )| < +∞. (6) there exists t0 > 0 such that lim sup (0,1)r →1 1 − r Moreover, if the previous conditions hold, then, for all t > 0, (a) φt (σ ) ∈ D, ∠ lim z→σ φt (z) = 0, (b) if σ ∈ P|A| (G) with A = ∠ lim z→σ G(z)(z − σ ) then ∠ lim φt (z) = ∠ lim z→σ

z→σ

G(φt (σ )) φt (z) = . z−σ A

In particular, if σ ∈ P(G) then σ is the starting point of a maximal invariant curve of (φt ). Proof By Lemma 15.1.2, (1) is equivalent to (2). Moreover, clearly, (3) implies (6). Assume (6) holds. In particular, lim(0,1)r →1 φt 0 (r σ ) = 0. Hence, φt0 (σ ) ∈ D by Corollary 1.7.6. By Proposition 10.1.8, for each r ∈ (0, 1), we obtain

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15 Poles of the Infinitesimal Generators

φt 0 (r σ ) G(r σ )(1 − r ) = G(φt0 (r σ )). 1−r

(15.1.2)

Now, limr →1 G(φt0 (r σ )) = G(φt0 (σ )) ∈ C. Since φt0 (σ ) is not the Denjoy-Wolff point of (φt ) by Corollary 14.1.3, it follows that G(φt0 (σ )) = 0 and (1) follows immediately from (15.1.2). Now, assume that (2) holds. Let A := ∠ lim G(z)(σ − z) = 0. z→σ

For t ≥ 0, let

M(t) := lim sup |φt (r σ )|. (0,1)r →1

We claim that M(t) = 0, for all t > 0.

(15.1.3)

Assume for the moment that (15.1.3) holds. Fix t > 0. Then lim(0,1)r →1 φt (r σ ) = 0 and hence Corollary 1.7.6 implies that φt (σ ) ∈ D. By Proposition 10.1.8, ∠ lim

z→σ

G(φt (σ )) φt (z) G(φt (z)) = ∠ lim = z→σ G(z)(σ − z) σ −z A

proving that σ ∈ Bβ (φt ) for all t > 0, that is, (3) holds. Moreover, Theorem 1.7.2 (applied to φt ) implies that (4) and (b) hold as well. In order to prove that (15.1.3) holds, we argue by contradiction. Suppose there exists t0 > 0 such that M(t0 ) > 0. Let {rn } ⊂ (0, 1) be a sequence converging to 1 such that lim |φt 0 (rn σ )| = M(t0 ). n→∞

Using again Proposition 10.1.8, we have for n ∈ N |φt 0 (rn σ )| =

|G(φt0 (rn σ ))||1 − rn | . |G(rn σ )||1 − rn |

(15.1.4)

Now, limn→∞ |G(rn σ )||1 − rn | = C > 0 by hypothesis. Therefore, if φt0 (σ ) ∈ D then limn→∞ |G(φt0 (rn σ ))||1 − rn | = 0, which implies M(t0 ) = 0, against our hypothesis on M(t0 ). Hence φt0 (σ ) ∈ ∂D. Let G(z) = (τ − z)(1 − τ z) p(z) be the Berkson-Porta decomposition of G. Let g(z) := −zp(φt0 (z)). By Theorem 10.1.10, g ∈ Gen(D). Hence, by Lemma 15.1.2, lim |g(rn σ )||1 − rn | = b < +∞.

n→∞

Therefore there exists M > 0 such that

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433

|G(φt0 (rn σ ))||1 − rn | = |g(rn σ )||1 − rn |

|τ − φt0 (rn σ )||1 − τ φt0 (rn σ )| ≤M rn

for all n ∈ N. Thus (15.1.4) shows that M(t0 ) < +∞. By Proposition 1.7.5, σ is a regular contact point of φt0 . If σ is a boundary regular fixed point for φt0 then it is a boundary regular fixed point for the semigroup by Theorem 12.1.4 and Proposition 12.1.6. Therefore, by Theorem 12.2.5, ∠ lim z→σ |G(z)| = 0, against σ ∈ P(G). Hence, σ is a regular non-fixed contact point for φt0 . This implies that T (σ ) > 0 so that σ is a contact point of (φt ). Moreover, since the curve [0, +∞)  t → φt (σ ) is injective by Theorem 14.1.1, it follows that φt (σ ) = φs (σ ) for all t = s. Let t ∈ (0, t0 ). Since φt0 = φt ◦ φt0 −t , by Proposition 1.7.7, αφt (σ ) < +∞ for all t ∈ (0, t0 ). Namely, σ is a regular non-fixed contact point of φt for all t ∈ (0, t0 ]. By Proposition 1.5.5, the curve (0, 1)  r → φt (r σ ) converges non-tangentially to φt (σ ) as r → 1. Therefore, taking into account Lemma 15.1.2 and Theorem 1.7.3, we have by Proposition 10.1.8 ∠ lim |G(z)||φt (σ ) − z| = z→φt (σ )

=

lim

|G(φt (r σ ))||φt (σ ) − φt (r σ )|

lim

|G(r σ )|(1 − r )|φt (r σ )|

(0,1)r →1

(0,1)r →1

|φt (σ ) − φt (r σ )| 1−r

= |A|(αφt (σ ))2 . Taking into account that αφt0 (σ ) > 1 by Remark 1.9.7, it follows that the φt (σ )’s are distinct regular poles of G of mass strictly greater than |A| for t ∈ (0, t0 ), contradicting Corollary 15.1.5. Therefore, (15.1.3) holds. So far, we have seen that (1), (2), (3), and (6) are equivalent, that (2) implies (4) and (b). Note also, that, by Corollary 1.7.6, (6) implies (a). If (4) holds for some t0 > 0, then, clearly, (5) holds for t = t0 . Finally, if (5) holds for some t0 > 0, arguing exactly as in the proof of Proposition 1.7.5 one can see that for all 0 < r < u < 1, |φt 0 (uσ ) − φt 0 (r σ )| ≤ C, u −r where C := sups∈[0,1) |φt 0 (sσ )| < +∞. Hence, taking the limit for u → 1, we have |φt 0 (r σ )| ≤ C, 1−r for all r ∈ (0, 1), and (6) holds.



Now we are going to relate the regular poles of an infinitesimal generator to the β-points of the associated Koenigs function:

434

15 Poles of the Infinitesimal Generators

Theorem 15.1.8 Let G be the infinitesimal generator of a semigroup in D with holomorphic model (Ω, h, ψt ) and Denjoy-Wolff point τ ∈ D. Let σ ∈ ∂D. The following are equivalent: (1) σ ∈ P(G), (2) σ ∈ Bβ (h), |h (r σ )| < +∞. (3) lim inf (0,1)r →1 1 − r (4) ∠ lim z→σ h(z) ∈ C, ∠ lim z→σ h (z) = 0 and ∠ lim z→σ h (z) = ∈ C, (5) lim(0,1)r →1 |h (r σ )| = 0 and lim sup(0,1)r →1 |h (r σ )| < +∞. If the previous conditions hold and σ ∈ P|A| (G) with A = ∠ lim z→σ G(z)(z − σ ), then (a) If τ ∈ D then ∠ lim h (z) = ∠ lim z→σ

z→σ

h(σ )G (τ ) h (z) = = 0, z−σ A

where h(σ ) = ∠ lim z→σ h(z). (b) If τ ∈ ∂D then ∠ lim h (z) = ∠ lim z→σ

z→σ

h (z) 1 = . z−σ A

Proof Clearly, (2) implies (3). In the case where τ ∈ ∂D, then G(z) = i/ h (z), hence i h (z) = . z−σ G(z)(z − σ )

(15.1.5)

From this equation and Lemma 15.1.2 it follows at once that (1) is equivalent to (2) and (3). In the case τ ∈ D, G(z) = G (τ )h(z)/ h (z) and h(τ ) = 0. Therefore G (τ ) h (z) 1 = . z − σ h(z) G(z)(z − σ )

(15.1.6)

Assume (1) holds and let (φt ) be the semigroup generated by G. Since h(φt (z)) = e G (τ )t h(z), from Theorem 15.1.7(a), it follows that φt (σ ) ∈ D and φt (σ ) = τ , for all t > 0, hence ∠ lim h(z) = e−G (τ )t h(φt (σ )) ∈ C. z→σ

Therefore (2) follows from (15.1.6). Conversely, if (3) holds, since ∠ lim inf z→σ |h(z)| > 0 (because h is univalent and h(τ ) = 0), (15.1.6) and Lemma 15.1.2 imply (1).

15.1 Regular Poles and β-Points

435

Now, if (1)–(3) hold, then (a) and (b) follow from (15.1.6), (15.1.5) and Theorem 1.7.2. Hence, (4) and (5) hold as well. Clearly, (4) implies (5). Finally, if (5) holds, arguing as in the last part of the proof of Theorem 15.1.7, we get (3). 

15.2 Tips of Isolated Radial and Spiral Slits In this section we provide a simple sufficient geometric condition on the shape of the image of Koenigs functions in order to detect a regular pole. We start with a definition. Definition 15.2.1 Let Ω ⊂ C be a simply connected domain. Let λ ∈ C, Re λ > 0. Let −∞ < a < +∞, z 0 ∈ C \ {0}, w0 ∈ C and Γ := {z ∈ C : z = eλs z 0 + w0 , s ∈ [a, +∞)}.

(15.2.1)

We say that Γ defines an isolated λ-spiral slit of Ω if there exists ε > 0 such that {ζ ∈ C : |ζ − (eλa z 0 + w0 )| < ε} ∩ ∂Ω = {ζ ∈ C : |ζ − (eλa z 0 + w0 )| < ε} ∩ Γ. The point eλa z 0 + w0 is the tip of Γ . In case λ ∈ R, we say that Γ defines an isolated radial slit of Ω Remark 15.2.2 Let (φt ) be a semigroup in D, with Koenigs function h. If (φt ) is either elliptic with real spectral value, or non-elliptic, then h is either starlike with respect to 0 or starlike at infinity. Hence, if h(D) has an isolated spiral slit, this is necessarily a radial slit. On the other hand, if (φt ) is elliptic with spectral value λ ∈ C, Re λ > 0, Im λ = 0, then h is λ-spirallike with respect to 0. Hence, if h(D) has an isolated spiral slit, this is necessarily a λ-spiral slit defined by Γ := {z ∈ C : z = eλs z 0 , s ∈ [a, +∞)} for some z 0 ∈ C \ {0}. We first describe the behavior of Riemann maps close to the tip of an isolated spiral slit. Lemma 15.2.3 Let h : D → C be univalent and λ ∈ C, Re λ > 0. Suppose Γ is given by (15.2.1) and defines an isolated λ-spiral slit of Ω with tip T . Then there exist δ > 0, a Jordan domain U ⊂ D, such that ∂U is the disjoint union of A ∪ E, where A is a non-empty open arc A ⊂ ∂D with end points q1 , q2 , E is a Jordan arc in D such that E ∩ ∂D = {q1 , q2 }, and a unique point σ ∈ A with the following property: h extends continuously on A, h(A) = Γ ∩ {ζ ∈ C : |ζ − T | < δ}, h(U ) = {ζ ∈ C : |ζ − T | < δ} \ Γ, and h(σ ) = T .

436

15 Poles of the Infinitesimal Generators

Proof Let Ω := h(D). Up to rotation and translation and dilation, we can assume that Γ := {z = eλs ∈ C : s ≥ 0}. Hence, T = 1. By definition, there exists ε > 0 such that, if D(1, ε) := {ζ ∈ C : |ζ − 1| < ε}, then ∂Ω ∩ D(1, ε) = {z = eλs : s ∈ [0, a)}, for some a > 0 such that |eλa − 1| = ε. Let δ ∈ (0, ε/2) be such that C := {ζ ∈ C : |ζ − 1| = δ} has the property that C intersects Γ at one point eλs0 for some s0 ∈ (0, a). Hence, C := C \ {eλs0 } ⊂ Ω. By Proposition 3.3.3, the closure of h −1 (C ) is a Jordan arc (or a Jordan curve) in D which intersects ∂D in either one or two points. We claim that the first possibility cannot occur, that is, h −1 (C ) ∩ ∂D = {q1 , q2 } with q1 = q2 . Indeed, let G := {ζ ∈ C : |ζ − eλs0 | = 4δ }. By construction, G intersects Γ in two points, say c− , c+ , and G \ {c− , c+ } is formed by two connected components, G + , G − such that G + , G − ⊂ Ω. Moreover, since G + intersects transversally C at exactly one point in Ω, it follows by Corollary 3.3.4, that h −1 (G + ) is a Jordan arc in D which intersects ∂D in two different points and h −1 (C ) intersects transversally h −1 (G + ) at exactly one point in D. Since h −1 (G + ) divides D in two connected components, and h −1 (C ) intersects both of them, it follows easily from the previous considerations that h −1 (C ) intersects ∂D in two points. Now, by construction, C divides Ω in two connected components, one of them being D(1, δ) \ {z = eλs : s ∈ [0, a)}. Hence, h −1 (C ) divides D into two connected components, one of them, say U , being such that h(U ) = D(1, δ) \ {z = eλs : s ∈ [0, a)}. Note that ∂U is formed by E := h −1 (C ) and an open arc A ⊂ ∂D with end points q1 , q2 . In particular, U is a Jordan domain. Let g : D → U be a Riemann map. By Theorem 4.3.3, g extends as a homeomorphism from D onto U and h ◦ g : D → D(1, δ) \ {z = eλs : s ∈ [0, a)} is a Riemann map, which extends continuously on ∂D by Theorem 4.3.1. Therefore, h is continuous on A. Moreover, by Proposition 4.3.5, there exists only one point η ∈ ∂D such that h(g(η)) = 1. Since, clearly, h(A) = {z = eλs : s ∈ [0, a)}, it follows that σ := g(η) ∈ A is the unique point in A such that h(σ ) = 1, and we are done.  In the following example, we study the tip of a very special domain. Example 15.2.4 Let p ∈ C and Γ := {ζ ∈ C : Re ζ = Re p, Im ζ ≤ Im p}. Consider the Koebe domain defined by K p := C \ Γ.

15.2 Tips of Isolated Radial and Spiral Slits

437

Note that Γ is an isolated radial slit of K p and its tip is p. A Riemann map of this domain is given by the Koebe function k p (z) := i

z i + + p, z ∈ D. (1 − z)2 4

Note that k p (−1) = p. Since ∠ lim sup z→−1

|k p (z)| |1 + z|

= ∠ lim sup z→−1

1 1 = < +∞, |1 − z|3 8

it turns out that σ := −1 is a β-point of k p . In other words, the point in ∂D which corresponds to the tip of the isolated radial slit Γ under the Riemann map k p , is a β-point of k p . In order to extend the previous example we need the following “localization” result for β-points. Proposition 15.2.5 Let h 1 : D → C be univalent. Let A ⊂ ∂D be an arc. Suppose h 1 extends continuously on A. Let h 2 : D → C be another univalent map such that h 2 (D) ⊂ h 1 (D), and assume there exists an open arc A ⊆ A such that h 2 extends continuously on A and h 2 (A ) ⊂ h 1 (A). Let σ be a point of A such that h 1 (σ ) = h 2 (σ ). Then σ ∈ Bβ (h 1 ) if and only if σ ∈ Bβ (h 2 ). Proof The function φ := h −1 1 ◦ h 2 : D → D is univalent. We claim that φ extends continuously on A and φ(A ) ⊂ ∂D. Indeed, let η be a point in A . Since h 2 is continuous at η, the curve [0, 1)  r → h 2 (r η) is an injective continuous curve in h 2 (D) converging to some point p ∈ ∂∞ h 2 (D). Hence, by Proposition 3.3.3, there exists η ∈ ∂D such that lim φ(r η) = lim h −1 1 (h 2 (r η)) = η .

r →1

r →1

By Theorem 1.5.7, ∠ lim z→η φ(z) = η . In order to prove that φ extends continuously at η, let {z n } ⊂ D be a sequence converging to η. We need to show that limn→∞ φ(z n ) = η . We already know that the result is true for sequences converging non-tangentially to η, therefore we can reduce to consider the case that either Im (ηz n ) > 0 for all n ∈ N or Im (ηz n ) < 0 for all n ∈ N, and Re (ηz n ) is strictly increasing. We deal with the case Im (ηz n ) > 0 for all n ∈ N, the other being similar. We set z 0 = 0. Let γ : [0, +∞) → D be the curve given by γ (t) = (1 + n − t)z n + (t − n)z n+1 for t ∈ [n, n + 1), n ≥ 0. By contruction, γ is an injective continuous curve, γ (0) = 0, limt→+∞ γ (t) = η and γ ((0, +∞)) ∩ (0, 1)η = ∅. Hence, Γ := γ ([0, +∞)) ∪ [0, 1]η is a Jordan curve. Let V be the bounded connected component of C∞ \ Γ . Note that V ⊂ D. Moreover, since h 2 is continuous at η, h 2 (V ) is a simply connected domain contained in h 2 (D) and ∂∞ (h 2 (V )) = h 2 (Γ ). Moreover, limt→+∞ h 2 (γ (t)) = p. Therefore, by Proposition 3.3.5, it follows that

438

15 Poles of the Infinitesimal Generators lim φ(γ (t)) = lim h −1 1 (h 2 (γ (t))) = η ,

t→+∞

t→+∞

from which limn→∞ φ(z n ) = η . Hence, φ extends continuously on A and φ(A ) ⊂ ∂D. Then, by Schwarz’ Reflection Principle, φ extends holomorphically through A . In particular, φ(σ ) = σ and φ (σ ) exists finite and, by Julia’s lemma (see Theorem 1.4.7), φ (σ ) = αφ (σ ) > 0. Now, for all z ∈ D, we have |h 1 (φ(z))| |σ − φ(z)| |h (z)| |φ (z)| = 2 . |σ − φ(z)| |σ − z| |σ − z|

(15.2.2)

Since by Proposition 1.5.5, φ maps Stolz regions of vertex σ into Stolz regions of vertex σ , it follows at once from (15.2.2) and Theorem 1.7.3, that, if σ is a β-point of h 1 , then σ is a β-point of h 2 . Conversely, assume σ is a β-point of h 2 . Then, by (15.2.2) and Theorem 1.7.3, ∠ lim sup z→σ

|h 1 (φ(z))| =: L < +∞. |σ − φ(z)|

Let {z n } ⊂ D be a sequence converging to σ and contained in a Stolz region S(σ, M) for some M > 1. Since σ is a regular fixed point of φ, by Remark 13.2.2, φ is semi-conformal at σ . Hence, by Lemma 13.2.3, there exist N ∈ N and a sequence {wn } ⊂ S(σ, M ) converging non-tangentially to σ and such that φ(wn ) = z n for all n ≥ N . Therefore, lim sup n→∞

|h 1 (z n )| |h (φ(wn ))| = lim sup 1 ≤ L, |σ − z n | n→∞ |σ − φ(wn )|

and the arbitrariness of {z n } implies that σ is a β-point of h 1 .



Now we are in good shape to prove the main result of this section: Theorem 15.2.6 Let (φt ) be a semigroup in D with infinitesimal generator G and Koenigs function h. Suppose h(D) has an isolated radial or spiral slit with tip T . Then there exists a point σ ∈ ∂D such that h extends continuously on ∂D in a neighborhood of σ , h(σ ) = T and σ is a regular pole of G. Proof Let Γ , given by (15.2.1), be the set which defines an isolated radial or spiral slit in h(D) with tip T . By Remark 15.2.2, Γ is either a radial slit (in case (φt ) is either elliptic with real spectral value or non-elliptic) or Γ is a λ-spiral slit (with w0 = 0), where λ ∈ C \ R, Re λ > 0 is the spectral value of (φt ) (and (φt ) is elliptic). Case 1: Γ is a radial slit. Let δ > 0, U ⊂ D, A ⊂ ∂D and σ ∈ A be given by Lemma 15.2.3. Let V := {ζ ∈ C : |ζ − T | < δ} \ Γ and let g : D → V be a Riemann map. Note that g(D) ⊂ h(D).

15.2 Tips of Isolated Radial and Spiral Slits

439

By Theorem 4.3.1, g extends continuously on ∂D and, by Proposition 4.3.5, g −1 (T ) contains only one point, which, up to pre-composing g with a rotation, we can assume to be σ . Therefore, by construction, we can find a non-empty open arc A ⊂ ∂D such that σ ∈ A and g(A ) ⊂ h(A). Hence, by Proposition 15.2.5, σ is a β-point of h if and only if it is a β-point of g. We claim that σ is a β-point of g. Indeed, let W := C \ Γ . There exists an affine transformation M of C such that M(W ) = K0 , the Koebe domain. Let k0 : D → M(W ) be the Riemann map defined in Example 15.2.4. Up to composing k0 with a rotation, we can assume that k0 (σ ) = 0 = M(T ). Hence, as shown in Example 15.2.4, σ is a β-point of k0 , and, therefore, σ is a β-point of M −1 ◦ k0 . Since g(D) ⊂ W and there exists an open arc A ⊂ ∂D containing σ such that M −1 (k0 (A )) ⊂ g(A ), another application of Proposition 15.2.5 shows that σ is a β-point of g. Therefore, σ is a β-point of h, and the result follows at once from Theorem 15.1.8. Case 2: Γ is a λ-spiral slit, λ ∈ C \ R, Re λ > 0. Arguing exactly as before, using Lemma 15.2.3 and Proposition 15.2.5, we can reduce the problem to the following one. Let Ω := C \ {eλs : s ≥ 0} and let f : D → Ω be a Riemann map. Note that {eλs : s ≥ 0} is an isolated λ-spiral slit for Ω with tip 1. By Theorem 4.3.1, f extends continuously on ∂D and, by Proposition 4.3.5, f −1 (1) = {σ } for some σ ∈ ∂D. We have to show that σ is a β-point of f . Let S := Spir[λ, π, 0] = {etλ+iθ : t ∈ R, θ ∈ (−π/2, π/2)} be a λ-spirallike sector of amplitude π and center 1. Let W := S \ {eλs : s ≥ 0} and let g : D → W be a Riemann map. As above, g extends continuously on ∂D, we can assume that g(σ ) = 1 and, since W ⊂ Ω, σ is a β-point for f if and only if it is a β-point for g. Im λ Since 0 ∈ / S, the map g1 : S → H defined by g1 (w) = w1−i Re λ , chosen so that g1 (1) = 1, is a biholomorphism, which maps {eλs : s ≥ 0} onto [1, +∞). Let g2 : D → H \ [1, +∞) be a Riemann map such that g2 (0) = g1 (g(0)). As before, g2 extends continuously on ∂D. We can assume, up to pre-composing with a rotation, that g2 (σ ) = 1. Hence, by construction, g2−1 ◦ g1 ◦ g is an automorphism of D which fixes 0 and σ , therefore, g = g1−1 ◦ g2 . Hence, |g (z)| 1 |g (z)| = 2 . |z − σ | |z − σ | |g1 (g1 (g2 (z)))| Since lim z→σ |g1 (g1 (g2 (z)))| = |g1 (g1 (1))| = |g1 (1)| = 0, it follows from the previous equality that σ is a β-point of g if and only if it is a β-point of g2 . By Case 1,  σ is a β-point of g2 , and we are done.

440

15 Poles of the Infinitesimal Generators

15.3 Measure-Theoretic Characterization of Regular Poles In this section we give a measure-theoretic characterization of regular poles coming from the Herglotz representation formula of holomorphic functions with positive real part. Let G(z) = (z − τ )(τ z − 1) p(z) be the Berkson-Porta decomposition of an infinitesimal generator G. Then p : D → C is holomorphic and Re p(z) ≥ 0 for all z ∈ D. If G has regular poles, in fact, Re p(z) > 0 for all z ∈ D. Hence, Re (1/ p(z)) > 0 for all z ∈ D. Therefore, by Theorem 2.1.1, there exists a positive finite Borel measure μ on ∂D such that 1 1 = p(z) 2π



π −π

1 eiθ + z dμ(θ ) + i Im . eiθ − z p(0)

(15.3.1)

Proposition 15.3.1 Let G be an infinitesimal generator in D, given by G(z) = (z − τ )(τ z − 1) p(z), where τ ∈ D, p : D → C is holomorphic with Re p(z) > 0 for all z ∈ D. Let σ := eiθ ∈ ∂D \ {τ } for some θ ∈ R. Let μ be the positive finite Borel measure on ∂D defined by (15.3.1). Then σ ∈ P(G) if and only if the function ξ ∈ ∂D →

1 ∈ [0, +∞] |ξ − σ |2

is μ-integrable and

 Im p(0) = −2 ∂D

Im (ξ σ ) dμ(ξ ). |ξ − σ |2

Proof Let q(z) := 1/ p(z). The map q : D → H is holomorphic. Moreover, since σ ∈ P(G) if and only if limr →1 (1 − r )| p(r σ )| > 0, it follows that σ ∈ P(G) if and only if σ is a regular zero of q. The result follows then by Corollary 2.3.6.  In order to concretely use the previous result to construct other examples of regular poles, we need the following representation formula for starlike functions. Proposition 15.3.2 Let h : D → C be holomorphic such that h(0) = 0 and h (0) > 0. If h is a starlike function with respect to zero, then there exists a unique selection arg(h(r eiθ )) of the argument of h(r eiθ ) which is continuous for (r, θ ) ∈ (0, 1) × [0, 2π ], limr →0+ arg(h(r )) = 0 and β(θ ) := lim arg(h(r eiθ )) r →1

(15.3.2)

exists for all θ ∈ [0, 2π ]. Moreover, β : [0, 2π ] → R is an increasing function, β(2π ) − β(0) = 2π , and      1 2π −iθ h(z) = h (0)z exp − log 1 − e z dβ(θ ) , z ∈ D, π 0

(15.3.3)

15.3 Measure-Theoretic Characterization of Regular Poles

441

where log denotes the principal branch of the logarithm and 

1 zh (z) = h(z) 2π

2π

0

eiθ + z dβ(θ ), z ∈ D, eiθ − z

(15.3.4)

where the integrals must be understood in the Riemann-Stieltjes sense. ˜ ˜ Conversely, if β˜ : [0, 2π ] → R is increasing and β(2π ) − β(0) = 2π then      1 2π −iθ ˜ h(z) := Az exp − log 1 − e z d β(θ ) , z ∈ D, π 0

(15.3.5)

represents a starlike function with respect to zero, for every A > 0 and lim

˜ )+π − arg(h(r e )) = β(θ iθ

(0,1)r →1

1 2π



2π

˜ β(s)ds.

(15.3.6)

0

Proof Let h be a starlike function with respect to zero such that h(0) = 0 and h (0) > (z) satisfies Re q(z) ≥ 0 0. By Theorem 9.4.5, the holomorphic function q(z) := zhh(z) for all z ∈ D and q(0) = 1, thus Re q(z) > 0 for all z ∈ D. Since the holomorphic has no zeros and can be extended holomorphically at 0 as function D \ {0}  z → h(z) z such that Im (log h (0)) = h (0) > 0, considering the branch of the logarithm of h(z) z 0, we have    h(z) q(z) − 1 log = , z ∈ D. z z Hence



h(z) log z



− log h (0) =

 0

z

1 (q(ξ ) − 1)dξ, ξ

(15.3.7)

for all z ∈ D and, h(z) = z exp h (0)

 0

z

 q(ξ ) − 1 dξ , z ∈ D. ξ

(15.3.8)

By Remark 2.1.2, since q(0) ∈ R, there exists an increasing function V : [0, 2π ] → R with V (2π ) = 2π and V (0) = 0, such that q(z) =

1 2π



2π

0

eiθ + z d V (θ ), eiθ − z

z ∈ D,

(15.3.9)

where the above integral must be understood in the Riemann-Stieltjes sense. Being V integrable in [0, 2π ] in the Riemann sense,  C := 0

2π

(V (θ ) − θ ) dθ ∈ R.

442

15 Poles of the Infinitesimal Generators

C , θ ∈ [0, 2π ]. Note that U is increasing, U (2π ) − U (0) = Let U (θ ) := V (θ ) − 2π 2π ,  2π iθ 1 e +z dU (θ ), z ∈ D, (15.3.10) q(z) = 2π 0 eiθ − z and



2π

(U (θ ) − θ ) dθ = 0.

(15.3.11)

0

Since 1 =

U (2π)−U (0) 2π

1 1 q(z) − 1 = z z 2π

=



2π 0

1 2π



2π 0

dU (θ ), by (15.3.10),

  eiθ + z 1 2π 1 − 1 dU (θ ) = dU (θ ), z ∈ D. eiθ − z π 0 eiθ − z

Therefore, given z ∈ D, Fubini’s Theorem implies 

z

0

   2π 1 z q(ξ ) − 1 1 dξ = dU (θ ) dξ ξ π 0 eiθ − z 0  z   1 1 2π dξ dU (θ ) = iθ π 0 0 e −z  1 2π =− log(1 − e−iθ z)dU (θ ), π 0

(15.3.12)

where log denotes the principal branch of the logarithm. Since 

2π

log(1 − e−iθ z)dθ = 0,

0

it follows from (15.3.7) and (15.3.12) that, for all z ∈ D,  log

h(z) z



− log h (0) = −

1 π



2π

log(1 − e−iθ z)d (U (θ ) − θ ) .

0

Bearing in mind that U (2π ) − 2π = U (0) − 0, integration by parts for the RiemannStieltjes integral shows 

2π 0



2π

d log(1 − e−iθ z)dθ dθ 0  2π e−iθ z = −i dθ. (U (θ ) − θ ) 1 − e−iθ z 0

log(1 − e−iθ z)d (U (θ ) − θ ) = −

Hence, by (15.3.11), we have, for all z ∈ D

(U (θ ) − θ )

15.3 Measure-Theoretic Characterization of Regular Poles

 log

h(z) z



443

 2π  2π i e−iθ z i (U (θ) − θ)dθ + (U (θ) − θ)dθ π 0 1 − e−iθ z 2π 0  2π iθ e +z i (U (θ) − θ)dθ. = 2π 0 eiθ − z

− log h (0) =

Therefore, for r ∈ (0, 1) and θ ∈ [0, 2π ], 

h(r eiθ ) Im log r eiθ

 = P[ f ](r eiθ ),

where f (eiθ ) := U (θ ) − θ , θ ∈ [0, 2π ]. Note that f is a well defined real-valued integrable function on ∂D since U (2π ) − 2π = U (0) − 0. We let arg(h(r eiθ )) := θ + P[ f ](r eiθ ). Then, arg(h(r eiθ )) is a selection of the argument of h(r eiθ ) which is continuous for (r, θ ) ∈ (0, 1) × [0, 2π ] and limr →0+ arg(h(r )) = 0. Taking into account that U is increasing, the following limits exist: U + (θ0 ) := lim U (θ) ∈ R, θ0 ∈ [0, 2π ); U − (θ0 ) := lim U (θ) ∈ R, θ0 ∈ (0, 2π ], θ→θ0+

θ→θ0−

and, hence, f + (θ0 ) := limθ→θ0+ f (θ ), θ0 ∈ [0, 2π ) and f − (θ0 ) := limθ→θ0− f (θ ), θ0 ∈ (0, 2π ] exist as well, and 1 + 1 ( f (θ ) + f − (θ )) = (U + (θ ) + U − (θ )) − θ, θ ∈ (0, 2π ); 2 2 1 + 1 1 − ( f (2π ) + f (2π )) = ( f + (0) + f − (0)) = (U + (0) + U − (2π ) − 2π ). 2 2 2 Therefore, by Theorem 1.6.2 and for all θ ∈ [0, 2π ], there exists the limit β(θ ) := θ + lim P[ f ](r eiθ ) = θ + r →1

f + (θ ) + f − (θ ) . 2

Hence, setting U − (0) := U − (2π ) − 2π and U + (2π ) := U + (0) + 2π , we have 1 + (U (θ ) + U − (θ )), θ ∈ (0, 2π ), 2 1 1 β(2π ) = 2π + (U + (0) + U − (2π ) − 2π ) = (U + (2π ) + U − (2π )), 2 2 1 1 β(0) = (U + (0) + U − (2π ) − 2π ) = (U + (0) + U − (0)). 2 2 β(θ ) =

444

15 Poles of the Infinitesimal Generators

Note that β(2π ) − β(0) = 2π and β(θ ) = 21 (U + (θ ) + U − (θ )). Since U is increasing and U ([0, 2π ]) ⊂ [0, 2π ], both U + and U − are increasing in [0, 2π ], so β is increasing as well. Taking into account (see, e.g., [83, Remark 8.20(a)]) that for every continuous function g : [0, 2π ] → R, 

2π



2π

g(θ )dU (θ ) =

0

0

+



2π

g(θ )dU (θ ) =

g(θ )dU − (θ )

0

equations (15.3.3) and (15.3.4) follow immediately from the previous considerations. Conversely, let A > 0 and let β˜ : [0, 2π ] → R be an increasing function with ˜ ˜ β(2π ) − β(0) = 2π and      1 2π −iθ ˜ h(z) := Az exp − log 1 − e z d β(θ ) , z ∈ D. π 0 Clearly, h defines a holomorphic function in the disc such that h(0) = 0 and h (0) = A. Let g(z) := −

1 π



2π

  ˜ ), z ∈ D. log 1 − e−iθ z d β(θ

0

By hypothesis, g : D → C is a holomorphic branch of the logarithm of the natural extension to 0). Moreover, for z ∈ D

h(z) Az

(with

zh (z) = h(z) + zg (z)h(z). Therefore, in order to see that h is a starlike function with respect to zero, we only have to check that Re (1 + zg (z)) ≥ 0 for all z ∈ D (see Theorem 9.4.5). ˜ ˜ Now, since β(2π ) − β(0) = 2π , for each z ∈ D,    2π  1 1 2π −e−iθ ˜ ˜ d β(θ ) 1 + zg (z) = d β(θ ) + z − 2π 0 π 0 1 − e−iθ z   2π   2π iθ 2e−iθ z 1 e +z 1 ˜ )= ˜ ). 1+ d β(θ d β(θ = −iθ 2π 0 1−e z 2π 0 eiθ − z

Since the Poisson kernel is non-negative, we obtain Re (1 + zg (z)) ≥ 0 as needed. Finally, (15.3.6) follows from (15.3.5), looking at the way we defined the argument in the previous part.  The function β in (15.3.2) might not be continuous. However, since it is increasing, it is continuous except on a countable set where the function must have jump discontinuities. If θ0 ∈ (0, 2π ) is such a discontinuity, the above proof shows that

15.3 Measure-Theoretic Characterization of Regular Poles

β(θ0 ) =

1 2



445

 lim+ β(θ ) + lim− β(θ ) .

θ→θ0

θ→θ0

As a by product of Propositions 15.3.1 and 15.3.2, we have Corollary 15.3.3 Let (φt ) be an elliptic semigroup with Denjoy-Wolff point 0 and real spectral value. Let G be the infinitesimal generator of (φt ) and h the Koenigs function, with h(0) = 0 and h (0) > 0. Let β(θ ) := lim(0,1)r →1 arg h(r eiθ ), θ ∈ [0, 2π ] (where arg denotes the argument in the sense of Proposition 15.3.2). Let σ := eiα ∈ ∂D. Then σ ∈ P(G) if and only if 

2π

0

dβ(θ ) < +∞ and iθ |e − eiα |2

 0

2π

sin(α − θ ) dβ(θ ) = 0. |eiθ − eiα |2

(15.3.13)

Example 15.3.4 Let α > 0. Let β(θ ) := −π 1−α (−θ + π )α , for 0 ≤ θ ≤ π , and β(θ ) := π 1−α (θ − π )α , for π ≤ θ ≤ 2π . Note that β is increasing, absolutely continuous and β (θ ) = απ 1−α |θ − π |α−1 > 0 for θ ∈ [0, 2π ] \ {π }. Therefore (15.3.5) defines a starlike function h, with h (0) = 1. Consider the elliptic semigroup (φt ) with Denjoy-Wolff point 0, spectral value 1 and Koenigs function h, defined as φt (z) := h −1 (e−it h(z)), t ≥ 0. Let G(z) = −zp(z) denote the infinitesimal generator of (φt ). Then, by (15.3.6) and (15.3.4), 1 2π



π

1 zh (z) eiθ + z dβ(θ ) = = . eiθ − z p(z) h(z)

−π

Take σ = eiπ = −1. On the one hand, 

2π 0



β (θ )dθ = |eiθ + 1|2

π

0

β (θ )dθ = απ 1−α 1 + cos(θ )



π

0

(π − θ )α−1 dθ. 1 + cos(θ )

Taking into account that 1 + cos(θ ) ≈ 21 (π − θ )2 for θ close to π , we have that 2π β (θ)dθ 0 |eiθ +1|2 < +∞ if and only if α > 2. On the other hand,  0

2π

sin(π − θ )β (θ )dθ = 0, |eiθ + 1|2

for all α. Thus, by Corollary 15.3.3, σ ∈ P(G) if and only if α > 2. In Fig. 15.1, using (15.3.5) and numerical integration, we draw the boundary of h(D) for α = 0.5 (left), α = 1.5 (center), and α = 2.5 (right).

446

15 Poles of the Infinitesimal Generators

Fig. 15.1 The boundary of h(D) in Example 15.3.4

15.4 Dual Infinitesimal Generators In this section we introduce the concept of “dual infinitesimal generator”, and show how regular critical points of an infinitesimal generator are related to regular poles of its dual. Definition 15.4.1 Let G ∈ Gen(D), G ≡ 0. Let G(z) = (τ − z)(1 − τ z) p(z) be the Berkson-Porta decomposition of G, where τ ∈ D is the Denjoy-Wolff point of the associated semigroup and p : D → C holomorphic with Re p(z) ≥ 0 for all z ∈ D. The dual infinitesimal generator is 1 ˆ , z ∈ D. G(z) := (τ − z)(1 − τ z) p(z) Remark 15.4.2 Let G ∈ Gen(D), G ≡ 0 and let τ ∈ D be the Denjoy-Wolff point of the associated semigroup. Then Gˆ ∈ Gen(D) and the associated semigroup has Denjoy-Wolff point τ . Moreover, Gˆˆ = G. In fact, this follows at once from the Berkson-Porta’s Formula as soon as one realizes that 1/ p : D → C is holomorphic (because p(z) = 0 for some z ∈ D if and only if p ≡ 0 if and only if G ≡ 0) and Re (1/ p(z)) ≥ 0 for all z ∈ D. As a matter of notation, if G ≡ 0 is an infinitesimal generator in D, we denote by ˆ (φˆ t ) the semigroup and by hˆ the Koenigs function associated with G. Notice that ˆ G(z)G(z) = (τ − z)2 (1 − τ z)2 . From this formula we immediately have: Proposition 15.4.3 Let G ∈ Gen(D), G ≡ 0 with Denjoy-Wolff τ ∈ D. Let σ ∈ ∂D \ {τ }. Then the following are equivalent: (1) σ is a boundary regular critical point for G with repelling spectral value := limr →1 |G(r σ )|/(1 − r ) > 0, (2) σ is a regular pole for Gˆ of mass |τ − σ |4 / .

15.5 Radial Multi-Slits Semigroups

447

15.5 Radial Multi-Slits Semigroups Let Γ be a radial slit, namely Γ = {z ∈ C : z = r T, r ∈ [s, +∞)}, where T ∈ C \ {0} and s > 0. A radial m-slit domain Ω is given by C \ ∪mj=1 Γ j where the Γ j ’s are (pairwise disjoint) radial slits. Since Ω is starlike with respect to 0, there exists a unique Riemann map h : D → Ω such that h(0) = 0 and h (0) > 0. Up to dilation, we can assume that h (0) = 1. Moreover, ∂Ω is locally arcwise connected, hence h extends continuously up to ∂D (see Theorem 4.3.1). We call T1 one of the tips of the m radial slits with minimal real part, and label the other tips T j , j = 2, . . . , m, in such a way that T j follows T j−1 clockwise. Write M = max{|T j | : j = 1, . . . , m}. We let σ1 , . . . , σm ∈ ∂D be such that h(σ j ) = T j for each j. Such a point σ j is unique by Proposition 4.3.5. Fix j = 1, . . . , m. For j n ∈ N, we denote by Cn the arc of the circle |z| = n + M between the slits Γ j and j Γ j+1 (with j counted mod m). The prime end [(Cn )] has impression ∞, thus there is ξ j ∈ ∂D such that h(ξ j ) = ∞. In fact, by Proposition 4.3.5, the points {ξ1 , . . . , ξm } are the unique pre-images of ∞ under h. It is clear that in the arc [σ j , σ j+1 ] (with j counted mod m) there is only one of such points, and we label them in such a way that ξ j ∈ [σ j , σ j+1 ] for j = 1, . . . , m − 1 and ξm ∈ [σm , σ1 ]. Also, let 2π α j , with α j > 0, j = 1, . . . , m − 1 be the amplitude of the angle formed by the vectors T j and T j+1 for j = 1, . . . , m − 1 and let 2π αm be the amplitude of the angle formed by the vectors Tm and T1 . By definition, mj=1 α j = 1. Consider the semigroup in D defined by φt := h −1 (e−t h(z)). We call it a radial m-slits semigroup. Clearly, 0 is the Denjoy-Wolff point of (φt ) and the spectral value of the semigroup is 1. Using the notations just introduced we have: Proposition 15.5.1 Let (φt ) be a radial m-slits semigroup, and let G(z) = −zp(z) be its infinitesimal generator. Then G has m boundary regular critical points ξ1 , . . . , ξm ∈ ∂D with repelling spectral values, respectively, 2α1 , . . . , 2αm . Moreover, G has m regular poles σ1 , . . . , σm with mass, respectively, 2μ1 , . . . , 2μm > 0 such that mj=1 μ j = 1. In addition, p(z) =

m  j=1

μj

σj + z , σj − z

 ξj + z 1 = . αj p(z) ξj − z j=1 m

(15.5.1)

 σ +z Conversely, if p(z) is given by p(z) = mj=1 μ j σ jj −z with σ j ∈ ∂D, σ j = σk , for m k = j, μ j > 0 with j=1 μ j = 1, then the infinitesimal generator G(z) = −zp(z) generates a radial m-slits semigroup. Proof First of all, notice that p(0) = 1. By Proposition 15.3.2, there exists a selection of the argument of h(r eiθ ) which is continuous for (r, θ ) ∈ (0, 1) × [0, 2π ] and limr →0+ arg(h(r )) = 0 such that β(θ ) := lim arg(h(r eiθ )) r →1

(15.5.2)

448

15 Poles of the Infinitesimal Generators

exists for all θ ∈ [0, 2π ], and β : [0, 2π ] → R is an increasing function. Clearly, on the arc (ξ j , ξ j+1 ) ( j = 1, . . . , m mod m). Hence, it follows eiβ(θ) is constant  easily that dβ = mj=1 α j δξ j where δξ j is the Dirac measure concentrated in ξ j of 2π eit +z 1 1 mass 1. Since p(z) = 2π 0 eit −z dβ(t), (15.5.1) holds for 1/ p(z). Now we want to show that p(z) has the claimed form. By Theorem 15.2.6, the points σ1 , . . . , σm are regular poles of G, with some mass μ1 , . . . , μm > 0. Since all z ∈ C, it follows that these are the only poles of p. Hence 1/ p(z) = −1/ p(1/z) for the map q(z) := p(z) − mj=1 μ j (σ j + z)/(σ j − z) is a bounded rational map, thus it is constant. Now, by Proposition 2.1.3, Re q(z) ≥ 0 for all z ∈ D. Hence, q(z) ≡ r with Re r ≥ 0. Now, for z = ξ1 (one of the zeros of p) we have r = p(ξ1 ) −

m 

μ j (σ j + ξ1 )/(σ j − ξ1 ) = −

j=1

m 

μ j (σ j + ξ1 )/(σ j − ξ1 ) ∈ iR,

j=1

from which it follows that r = 0, and p has the desired  form. Moreover, since p(0) = 1, we have mj=1 μ j = mj=1 α j = 1. To end up the proof, assume that p satisfies (15.5.1). Then let q(z) = 1/ p(z). By (15.3.1), it follows that the measure μ associated with q has atoms at σ1 , . . . , σm with mass μ1 , . . . , μm respectively. Let h : D → C be the Koenigs map associated with the infinitesimal generator −zq(z) such that h(0) = 0, h (0) = 1. Hence, by (15.3.4) and Proposition 15.3.2, the function β(θ ) := limr →1 arg h(r eiθ ) is constant on the arc (σ j , σ j+1 ) ( j = 1, . . . , m mod m), therefore the associated domain is a radial multi-slits domain. Applying what we already proved, we obtain that q(z) is of the form given in (15.5.1), hence, 1/ p(z) = q(z) is of the same type and, repeating the above argument, we obtain that −zp(z) generates a radial multi-slit domain.  In the following example we show that the tip of a non-isolated radial slit might not correspond to a regular pole: Example 15.5.2 Let {θ j } ⊂ (0, 1/2) be a sequence monotonically decreasing to 0. Let T∞ := 1 and let T j := e2πiθ j for j ∈ N. Let Γ j := {sT j : s ≥ 1} for j ∈ N ∪ {∞} and set

m  Ωm := C \ Γk ∪ Γk ∪ Γ∞ , m ∈ N ∪ {∞}. k=1

For a fixed m ∈ N, the domain Ωm is a radial (2m + 1)-slit domain symmetric with respect to the real axis, and, as m → ∞, the sequence {Ωm } converges in the kernel sense to the simply connected domain Ω∞ , which has infinitely many isolated radial slits collapsing to a non-isolated one, Γ∞ . Fix m ∈ N ∪ {∞}. Let h m : D → Ωm be the unique Riemann mapping normalized so that h m (0) = 0, h m (0) > 0. Since Ωm is symmetric with respect to the real axis, it follows that h m (z) = h m (z) for all z ∈ D. From this, it is not difficult to see that lim(0,1)r →1 h m (r ) = T∞ for all m ∈ N ∪ {∞}. By Carathéodory Extension Theorem 4.3.1, h m is continuous up to ∂D for m ∈ N while h ∞ is continuous on

15.5 Radial Multi-Slits Semigroups

449

D \ {1} and has non-tangential limit T∞ at 1. Again by symmetry, if σ j,m ∈ ∂D with j = 1, . . . , m are such that h m (σ j,m ) = T j , then Im σ j,m > 0 and h m (σ j,m ) = T j . Moreover, Re σ j,m > Re σk,m for all k < j ∈ {1, . . . , m}. In order to see this, let L j,1 be the segment between 1 and T j . Then h −1 m (L j,1 ) is a crosscut in D ending in 1 and σ j,m which divides D in two regions A0 and A1 , say, with 0 ∈ A0 . Since h m (0) = 0, h m (A1 ) is the region in Ωm contained in the open convex hull between Γ∞ and Γ j . As this region does not contain Tk , it follows that σk,m ∈ A0 , that is Re σk,m < Re σ j,m . Also, by Theorem 7.4.4, |1 − σ j,m |2 ≤ K |1 − T j |,

j = 1, . . . , m,

(15.5.3)

for a certain constant K > 0 (which does not depend on m). Note that the angle between T j and T j+1 (clockwise orientation) is 2π(θ j − θ j+1 ) for j ∈ N, while the angle between T1 and T1 is 2π(1 − 2θ1 ). Let denote by 2π θ∞,m the angle between Tm and Tm . −t m For each fixed m ∈ N ∪ {∞}, let φtm (z) := h −1 m (e h m (z)). Then (φt ) is an elliptic semigroup of D and we denote by G m (z) := −zpm (z) its infinitesimal generator. It h m (z) follows that pm (z) = zh (z) for all z ∈ D \ {0} and thus pm (z) = pm (z) for all z ∈ D. m If m = ∞, by Proposition 15.5.1, the infinitesimal generator G m has 2m + 1 boundary regular critical points. By symmetry, it is easy to see that −1 is one of such points. Moreover, if we label by ξ j,m the boundary regular critical point of G m contained in the arc between σ j,m and σ j+1,m with j = 1, . . . , m − 1 and by ξm,m the boundary regular critical point contained in the arc between σm,m and 1, then ξ j,m for j = 1, . . . , m are the other boundary regular critical points of G m . Hence, using the notations previously introduced, Proposition 15.5.1 implies that 1 (1 − 2θ1 ) 1 − z = pm (z) 2 1+z m−1  1 − z2 1 − z2 + + θ∞,m . (θ j − θ j+1 ) (z − ξ j,m )(z − ξ j,m ) (z − ξm,m )(z − ξm,m ) j=1 (15.5.4) As the sequence of domains {Ωm } kernel converges to Ω∞ , the Carathéodory Kernel Convergence Theorem 3.5.8 implies that {h m } converges uniformly on compacta of D to h ∞ . As a consequence, { pm } converges uniformly on compacta to p∞ , which, from (15.5.4), it is not difficult to be seen having the form ∞

(1 − 2θ1 ) 1 − z  1 1 − z2 = + (θ j − θ j+1 ) , p∞ (z) 2 1+z (z − ξ j )(z − ξ j ) j=1

(15.5.5)

where ξ j = limm→∞ ξ j,m . By Theorem 13.5.6, the semigroup (φt∞ ) has a sequence of boundary regular fixed points with dilation θ j − θ j+1 , j ∈ N. From this and from the fact that the function p∞ extends meromorphic on C \ {1} and its zeros are the

450

15 Poles of the Infinitesimal Generators

ξ j ’s, it follows that the ξ j ’s are actually boundary regular fixed points and also ξ j belongs to the arc with extremes σ j,∞ and σ j+1,∞ for each j ∈ N. Now we want to show that, for a suitable choice of {θ j }, the point 1, which corresponds to the (non isolated) tip h ∞ (1) = T∞ is not a regular pole of G ∞ . This is the case if and only if lim

(0,1)r →1

1 = ∞. p∞ (r )(1 − r )

Now, by (15.5.5) and by Fatou’s Lemma, this condition holds if ∞  θ j − θ j+1 = ∞. |1 − ξ j |2 j=1

However, by (15.5.3) ∞ ∞ ∞ ∞    θ j − θ j+1  θ j − θ j+1 θ j − θ j+1 θ j − θ j+1  ≥ ≥ , 2 2 |1 − ξ j | |1 − σ j,∞ | K |1 − T j | θj j=1 j=1 j=1 j=1

and the last series diverges if for instance θ j = 1/j.

15.6 Notes The results included in this chapter are based on [29]. In [37] it was obtained a complete characterization of regular poles in terms of the geometry of the Koenigs function. In order to state properly the result, we have to introduce some notation. Let h be a univalent map, and σ ∈ ∂D a point such that ξ := ∠ lim z→σ h(z) exists finitely. Fix γ > 1 and, for k ∈ N, denote with αk the opening of the smallest angle with vertex at ξ containing the set {w ∈ / h(D) : γ −k−1 ≤ |w| ≤ γ −k }. With this notation in mind, we can state [37, Proposition 6.11]: Theorem 15.6.1 Let (φt ) be a semigroup in D with associated infinitesimal generator G and Koenigs function h. Then a point σ ∈ ∂D is a regular pole of G if and only if ∞  αn < +∞. n=1

15.6 Notes

451

The proof of this result is based on a geometric characterization of the angular derivative given by Rodin and Warschawski [112] and a study of β-points due to Bertilson [17]. In [37], it was also introduced and studied a more general notion of singularities defined as follows: a point σ ∈ ∂D is called a regular (fractional) singularG(r σ ) exists finitely and nonzero. ity of order α ∈ R \ {0} for G provided limr →1− (1−r )α An infinitesimal generator can have regular fractional singularities only of order α ∈ [−1, 1] \ {0}, where α = −1 corresponds to a regular pole and α = 1 to a regular critical point. In [37], the authors provide different characterizations of regular singularities of order α, for α ∈ (−1, 1) \ {0} and give sufficient and necessary geometric characterizations of them.

Chapter 16

Rate of Convergence at the Denjoy-Wolff Point

The aim of this chapter is to study the rate, or speed, of convergence of orbits of nonelliptic semigroups to the Denjoy-Wolff point, considering both “orthogonal speed” and “total speed” as introduced in Definition 6.5.6. As we see, in the hyperbolic case the speed of convergence follows strict rules, while, in the parabolic case the situation is more complicated.

16.1 Speeds of Convergence of Orbits To start with, we recall that in Proposition 6.5.9 we showed that the speed of convergence can be studied in terms of the euclidean distance. Namely, given a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D, a point z ∈ D and t0 such that Re (τ φt (z)) ≥ 0 for all t ≥ t0 (such a point t0 exists due to the Denjoy-Wolff Theorem 8.3.6), then    1  1  ≤ log 2, ω(0, φt (z)) − 1 log  2 1 − |φt (z)|  2    o  1 1 v (η; t) − 1 log  ≤ log 2, (16.1.1)  D,0 2 |τ − φt (z)|  2    T  v (η; t) − 1 log |τ − φt (z)|  ≤ 3 log 2,  D,0 2 1 − |φt (z)|  2 where η : [0, +∞) → D is the continuous curve defined by η(t) := φt (z). Remark 16.1.1 When z = 0, we can choose t0 = 0 in above argument. Indeed, let R > 0 be such that 0 ∈ ∂ E(τ, R). Notice that Re (τ z) > 0 for every z ∈ E(τ, R) (because E(τ, R) is an euclidean disc in D whose boundary contains τ and 0). Since τ is the Denjoy-Wolff point of (φt ), it follows from the Denjoy-Wolff Theorem 1.8.4 that φt (E(τ, R)) ⊆ E(τ, R) for all t ≥ 0. Hence, Re (τ φt (0)) ≥ 0 for all t ≥ 0. © Springer Nature Switzerland AG 2020 F. Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4_16

453

454

16 Rate of Convergence at the Denjoy-Wolff Point

Next, we show that the orthogonal speed and the tangential speed of an orbit of a semigroup do not depend on the starting point: Lemma 16.1.2 Let (φt ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Let z 1 , z 2 ∈ D and let η j : [0, +∞) → D be the continuous curve defined by η j (t) := φt (z j ), j = 1, 2. Then, for every t ≥ 0, o o (η1 ; t) − vD,0 (η2 ; t)| ≤ ω(z 1 , z 2 ), |vD,0 T T |vD,0 (η1 ; t) − vD,0 (η2 ; t)| ≤ 2ω(z 1 , z 2 ).

Proof Let γ : (−1, 1) → D be the geodesic of D defined by γ (r ) = r τ . For z ∈ D let πγ (z) be the hyperbolic projection of z onto γ . Then, by the very definition of orthogonal speed of curves and Proposition 6.5.3, we have o o |vD,0 (η1 ; t) − vD,0 (η2 ; t)| = |ω(0, πγ (η1 (t))) − ω(0, πγ (η2 (t)))|

≤ ω(πγ (η1 (t)), πγ (η2 (t))) ≤ ω(η1 (t), η2 (t)) = ω(φt (z 1 ), φt (z 2 )) ≤ ω(z 1 , z 2 ). A similar argument proves the second inequality. Namely, T (η ; t) = ω(φ (z ), π (φ (z ))) vD,0 t 1 γ t 1 1

≤ ω(φt (z 1 ), φt (z 2 )) + ω(φt (z 2 ), πγ (φt (z 2 ))) + ω(πγ (φt (z 2 )), πγ (φt (z 1 ))) T (η ; t) + ω(π (φ (z )), π (φ (z ))) = ω(φt (z 1 ), φt (z 2 )) + vD,0 γ t 2 γ t 1 2 T (η ; t). ≤ 2ω(z 1 , z 2 ) + vD,0 2 T T That is, vD,0 (η1 ; t) − vD,0 (η2 ; t) ≤ 2ω(z 1 , z 2 ). Changing the role of z 1 and z 2 , we obtain the second inequality of the statement. 

Lemmas 16.1.2 and 6.5.8 show that, in order to study the asymptotic behavior of the speed of convergence of semigroups’ orbits to the Denjoy-Wolff point, it is enough to study the orbit starting at 0 and considering the speed with respect to 0. In other words, the following definition makes sense: Definition 16.1.3 Let (φt ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. For t ≥ 0, we let v(t) := ω(0, φt (0)), and call v(t) the total speed of (φt ). Also, let γ : (−1, 1) → D be the geodesic of D defined by γ (r ) := r τ and let πγ : D → γ ((−1, 1)) be the hyperbolic projection. For t ≥ 0, we let o (φs (0); t) := ω(0, πγ (φt (0))), vo (t) := vD,0

16.1 Speeds of Convergence of Orbits

455

and call vo (t) the orthogonal speed of (φt ). Finally, we let T (φs (0); t) := ω(φt (0), πγ (φt (0))), v T (t) := vD,0

and call v T (t) the tangential speed of (φt ). Remark 16.1.4 It follows immediately from Remark 6.5.10 that the orbit [0, +∞)  t → φt (z) converges non-tangentially to τ for some—and hence any—z ∈ D if and only if lim supt→+∞ v T (t) < +∞. By Proposition 6.5.5, if (φt ) is a non-elliptic semigroup we have vo (t) + v T (t) −

1 log 2 ≤ v(t) ≤ vo (t) + v T (t). 2

(16.1.2)

A second less immediate relation between the orthogonal speed and the tangential speed is contained in the following proposition: Proposition 16.1.5 If (φt ) is a non-elliptic semigroup in D then, for every t ≥ 0, v T (t) ≤ vo (t) + 4 log 2.

(16.1.3)

Proof Let τ ∈ ∂D be the Denjoy-Wolff point of (φt ) and let λ ≥ 0 be its spectral value. We first note that, by Proposition 1.5.5, for every t ≥ 0  |τ − φt (0)| e−λt ≤4 , 1 − |φt (0)| 1 − |φt (0)|2 which is equivalent to eλt Applying the function x → t ≥ 0,

16 1 + |φt (0)| ≤ . 1 − |φt (0)| |τ − φt (0)|2 1 2

log x to the previous inequality, we obtain for every

1 λt 1 1 1 1 log ≤ + log + log(1 + |φt (0)|) 2 1 − |φt (0)| 2 2 1 − |φt (0)| 2 1 1 . ≤ log 16 + log 2 |τ − φt (0)| Therefore, by (16.1.1) and Remark 16.1.1, we have for all t ≥ 0,

456

16 Rate of Convergence at the Denjoy-Wolff Point

1 1 1 log + log 2 2 1 − |φt (0)| 2 1 1 1 ≤ log 16 + log + log 2 2 |τ − φt (0)| 2 3 7 1 ≤ log 16 + log 2 + 2vo (t) = 2vo (t) + log 2. 2 2 2

v(t) ≤

Hence, by (16.1.2), we have for all t ≥ 0, vo (t) + v T (t) ≤ v(t) +

1 1 7 log 2 ≤ 2vo (t) + log 2 + log 2. 2 2 2

Finally, the previous equation implies that v T (t) ≤ vo (t) + 4 log 2 for all t ≥ 0, and we are done.  Proposition 16.1.6 Let (φt ) and (ψt ) be two non-elliptic semigroups in D. Suppose there exists M ∈ Aut(D) such that φt = M −1 ◦ ψt ◦ M for all t ≥ 0. Denote by v(t), vo (t), v T (t) (respectively, v˜ (t), v˜ o (t), v˜ T (t)) the total speed, orthogonal speed and tangential speed of (φt ) (resp. of (ψt )). Then there exists C > 0 such that for all t ≥0 |v(t) − v˜ (t)| < C, |vo (t) − v˜ o (t)| < C, |v T (t) − v˜ T (t)| < C. Proof Let τ ∈ ∂D be the Denjoy-Wolff point of (φt ) and τ˜ ∈ ∂D that of (ψt ). Let γ : (0, +∞) → D (respectively, γ˜ : (0, +∞) → D) be the geodesic in D parameterized by arc length such that γ (0) = 0 (resp., γ˜ (0) = 0) and limt→+∞ γ (t) = τ (resp., limt→+∞ γ (t) = τ˜ ). Since M is an isometry for the hyperbolic distance, for all t ≥ 0, v(t) = ω(0, φt (0)) = ω(0, (M −1 ◦ ψt ◦ M)(0)) = ω(M(0), ψt (M(0)). Hence, for all t ≥ 0, |v(t) − v˜ (t)| = |ω(M(0), ψt (M(0)) − ω(0, ψt (0))| ≤ |ω(M(0), ψt (M(0)) − ω(0, ψt (M(0))| + |ω(0, ψt (M(0)) − ω(0, ψt (0))| ≤ ω(M(0), 0) + ω(ψt (M(0)), ψt (0)) ≤ 2ω(M(0), 0) =: C0 . Moreover, since M is an isometry for the hyperbolic distance, the curve γ1 : (0, +∞) → D defined by γ1 := M ◦ γ is a geodesic in D parameterized by arc length. Hence, for all t ≥ 0, v T (t) = ω(φt (0), γ ) = ω(M(φt (0)), γ1 ) = ω(ψt (M(0)), γ1 ).

16.1 Speeds of Convergence of Orbits

457

By Lemma 6.5.8, limt→+∞ |˜v T (t) − ω(ψt (M(0)), γ1 )| = 0, thus there exists C1 > 0 such that |v T (t) − v˜ T (t)| < C1 for all t ≥ 0. Finally, by (16.1.2) we have for all t ≥ 0, vo (t) − v˜ o (t) ≤ v(t) − v T (t) +

1 1 log 2 − v˜ (t) + v˜ T (t) ≤ C0 + C1 + log 2. 2 2

The same argument proves that v˜ o (t) − vo (t) ≤ C0 + C1 + 21 log 2, and we are done.  Remark 16.1.7 Let (φt ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D and z ∈ D. We started this section by showing the relationship between the total and orthogonal speed in terms of the functions t → 1 − |φt (z)| and t → |1 − τ φt (z)|, respectively (see (16.1.1)). We will show that the tangential speed is related to the function t → Arg(τ φt (z)). In fact, take t0 such that Re (τ φt (z)) > 0 for all t > t0 , then   1 2 |φt (z)||Arg(τ φt (z))| ≤ |φt (z) − τ | √ 1 − |φt (z)| + π (16.1.4) 2 ≤ 1 − |φt (z)| + |Arg(τ φt (z))|. It is worthwhile comparing these estimates with (16.1.2). To get these inequalities, it is enough to see that given w ∈ D with Re w > 0 (in our case w = τ φt (z)), then |w − 1| ≤ 1 − |w| + |eiArg(w) − 1| ≤ 1 − |w| + |Arg(w)| and 4 |w − 1|2 = (1 − |w|)2 + 4|w| sin2 (Arg(w)/2) ≥ (1 − |w|)2 + 2 |w|Arg(w)2 π √  2 2 |w| 1 1 − |w| + |Arg(w)| . ≥ 2 π Bearing in mind (16.1.1), (16.1.4) and Remark 16.1.4, it follows that the orbit [0, +∞)  t → φt (z) converges non-tangentially to τ if and only if lim supt→+∞ φt (z))| v T (t) < +∞ if and only if lim supt→+∞ |Arg(τ < +∞. In addition, if there is a 1−|φt (z)| sequence {tn } such that either limn→∞ v T (tn ) = +∞ or limn→∞ then    T |Arg(τ φtn (0))|  1  < +∞. sup v (tn ) − log 2 1 − |φtn (0)|  n

|Arg(τ φtn (0))| 1−|φtn (0)|

= +∞,

458

16 Rate of Convergence at the Denjoy-Wolff Point

16.2 Total Speed of Convergence In this section we consider the total speed of convergence of orbits of hyperbolic and parabolic semigroups to the Denjoy-Wolff point. Proposition 16.2.1 Let (φt ) be a non-elliptic semigroup in D, with Denjoy-Wolff point τ ∈ ∂D and spectral value λ ≥ 0. Then v(t) vo (t) 1 λ 1 = lim = − lim log (1 − τ φt (z)) = , t→+∞ t t→+∞ t 2 t→+∞ t 2 lim

and

(16.2.1)

v T (t) = 0. t→+∞ t lim

Proof By Lemma 9.1.2, c(φt ) = lim

t→+∞

ω(0, φt (0)) v(t) = lim . t→+∞ t t

Since c(φt ) = λ2 (see Theorem 9.1.9), we conclude that limt→+∞ v(t) = λ2 . t In case λ = 0, that is, when (φt ) is parabolic, it follows immediately from (16.1.2) that vo (t) v T (t) lim = lim = 0. t→+∞ t→+∞ t t In case λ > 0, that is, when (φt ) is hyperbolic, by Proposition 8.3.7 the curve t → φt (0) converges to τ non-tangentially and then lim supt→+∞ v T (t) < +∞. Thus o T and limt→+∞ v t(t) = 0. from (16.1.2) we have limt→+∞ v t(t) = limt→+∞ v(t) t Finally, the second equation of (16.1.1) implies that 1 vo (t) log |1 − τ φt (z)| = − lim t→+∞ t→+∞ t 2 t lim

and, since the function t → Arg(1 − τ φt (z)) is bounded, we have − and we are done.

1 λ 1 lim log (1 − τ φt (z)) = , t→+∞ 2 t 2 

According to the type of the semigroup, we have also a simple lower bound on the total speed: Proposition 16.2.2 Let (φt ) be a non-elliptic semigroup in D, with Denjoy-Wolff point τ ∈ ∂D.

16.2 Total Speed of Convergence

459

• If (φt ) is hyperbolic with spectral value λ > 0, then lim inf [v(t) − t→+∞

λ t] > −∞, 2

• if (φt ) is parabolic of positive hyperbolic step, then lim inf [v(t) − log t] > −∞, t→+∞

• if (φt ) is parabolic of zero hyperbolic step, then lim inf [v(t) − t→+∞

1 log t] > −∞. 4

Proof Let (φt ) be hyperbolic with spectral value λ > 0. The canonical model of (φt ) is (S πλ , h, z → z + it). Hence, for every t ≥ 0, v(t) = ω(0, φt (0)) = kh(D) (h(0), h(φt (0))) = kh(D) (h(0), h(0) + it) ≥ kSπ/λ (h(0), h(0) + it) π π π π + it) − kSπ/λ ( , h(0)) − kSπ/λ (h(0) + it, + it) ≥ kSπ/λ ( , 2λ 2λ 2λ 2λ λ π = t − 2kSπ/λ ( , h(0)), 2 2λ where the last equality follows from Proposition 6.7.2 and taking into account that π π kSπ/λ (h(0) + it, 2λ + it) = kSπ/λ (h(0), 2λ ) for all t ∈ R since z → z + it is an autoπ morphism of S λ . From this, the result for hyperbolic semigroups follows at once. Now, assume that (φt ) is parabolic of positive hyperbolic step. We can assume that its canonical model is (H, h, z → z + it) (in case the canonical model is (H− , h, z → z + it) the argument is similar). Arguing as in the hyperbolic case, we see that v(t) ≥ kH (1, 1 + it) + C, for some constant C ∈ R and every t ≥ 0. Now, write 1√ + it = ρt eiθt for ρt > 0 and 1 . θt ∈ [0, π/2). A simple computation shows that ρt = 1 + t 2 and cos θt = √1+t 2 Therefore, by Lemma 5.4.1(1) and (2), we have    1 kH (1, 1 + it) ≥ kH (1, 1 + t 2 ) + log 1 + t 2 = log 1 + t 2 ≥ log t, 2 and the result follows in this case as well. Finally, in case (φt ) is parabolic of zero hyperbolic step, the canonical model is (C, h, z → z + it). Since h(D) is starlike at infinity and is different from C, there exists p ∈ C such that p − it ∈ / h(D) for all t ≥ 0 and p + it ∈ h(D) for all t > 0. Hence, h(D) ⊆ K p , where K p is the Koebe domain C \ {ζ ∈ C : Re ζ =

460

16 Rate of Convergence at the Denjoy-Wolff Point

Re p, Im ζ ≤ Im p}. Therefore, arguing as in the previous cases, we find C ∈ R such that for every t ≥ 0, v(t) ≥ kK p ( p + i, p + it) + C = kK 0 (i, it) + C. √ Taking into account that the map K0  z → √ −i z ∈ H is a biholomorphism, where the branch of the square root is chosen so that 1 = 1, we have by Lemma 5.4.1(1) that, for all t ≥ 1, √ 1 kK 0 (i, it) = kH (1, t) = log t, 4 

and we are done.

Remark 16.2.3 The bounds given by Proposition 16.2.2 are sharp. Indeed, as it is clear from the proof, if (φt ) is a hyperbolic group in D with spectral value λ > 0 then there exists C > 0 such that |v(t) − λ2 t| < C for every t ≥ 0, while, if (φt ) is a parabolic group then there exists C > 0 such that |v(t) − log t| < C for every t ≥ 0—so that, in this sense, non-elliptic groups in D have the lowest total speed. Moreover, the semigroup (φt ) in D defined as φt (z) := h −1 (h(z) + it), z ∈ D, where h : D → K0 is a Riemann map for the Koebe domain K0 , has the property that there exists C > 0 such that |v(t) − 14 log t| < C for all t ≥ 0. As it is clear from the proof of the previous theorem, one can get lower or upper estimates on the total speed of convergence according to the geometry of the image of the Koenigs function using the domain monotonicity of the hyperbolic distance. We provide here an example of such situation. In order to state the result, we need the following notation for non-symmetric sectors: for α, β ∈ [0, π ], with α + β > 0, we denote   W (α, β) := r eiθ : r > 0, −α < θ < β . Proposition 16.2.4 Let (φt ) be a parabolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D and Koenigs function h. • Assume that h(D) is contained in a sector p + i W (α, β), for some p ∈ C and α, β ∈ [0, π ], with α + β > 0. (1) If α and β are both positive then lim inf [v(t) − t→+∞

π log t] > −∞. 2(α + β)

(2) If either α or β is zero then lim inf [v(t) − t→+∞

π +α+β log t] > −∞. 2(α + β)

16.2 Total Speed of Convergence

461

• Assume that h(D) contains a sector p + i W (α, β), for some p ∈ C and α, β ∈ [0, π ], with α + β > 0. (1) If α and β are both positive then lim sup[v(t) − t→+∞

π log t] < +∞. 2(α + β)

(2) If α or β is zero then lim sup[v(t) − t→+∞

π +α+β log t] < +∞. 2(α + β)

Proof Let V := p + i W (α, β). Hence, if h(D) ⊂ V we have for t ≥ 0 v(t) = ω(0, φt (0)) = kh(D) (h(0), h(0) + it) ≥ k V (h(0), h(0) + it), while, if V ⊂ h(D) and h(z 0 ) ∈ V for some z 0 ∈ D, we have for t ≥ 0 by Lemma 16.1.2, v(t) = ω(0, φt (0)) ≤ ω(z 0 , φt (z 0 )) + ω(0, z 0 ) = kh(D) (h(z 0 ), h(z 0 ) + it) + ω(0, z 0 ) ≤ k V (h(z 0 ), h(z 0 ) + it) + ω(0, z 0 ). The results then follow by computing k V (h(z 0 ), h(z 0 ) + it) for z 0 ∈ D such that h(z 0 ) ∈ V . In fact, it is enough to compute k V (w, w + it) for any w ∈ V . Indeed, taking into account that z → z + it is a holomorphic self-map of V , we have |k V (h(z 0 ), h(z 0 ) + it) − k V (w, w + it)| ≤ |k V (h(z 0 ), h(z 0 ) + it) − k V (h(z 0 ), w + it)| + |k V (h(z 0 ), w + it) − k V (w, w + it)| ≤ k V (h(z 0 ) + it, w + it) + k V (h(z 0 ), w) ≤ 2k V (h(z 0 ), w).

(16.2.2)

Moreover, without loss of generality, up to a translation, we can assume that p = 0. In case α, β > 0, we can compute k V (i, i + it). Note that V = R(W ), where R(z) = iei(β−α)/2 z and W := {ρeiθ : ρ > 0, |θ | < (α + β)/2}. Hence, k V (i, i + it) = k W (ei(α−β)/2 , ei(α−β)/2 (1 + t)). The map f : W → H given by f (w) := wπ/(α+β) is a biholomorphism. Therefore, , we have if we set θ0 := π(α−β) 2(α+β)

462

16 Rate of Convergence at the Denjoy-Wolff Point

k V (i, i + it) = kH (eiθ0 , eiθ0 (1 + t)π/(α+β) ). Now, by Lemma 5.4.1, |kH (eiθ0 , eiθ0 (1 + t)π/(α+β) ) − kH (1, (1 + t)π/(α+β) )| ≤ |kH (eiθ0 , eiθ0 (1 + t)π/(α+β) ) − kH (1, eiθ0 (1 + t)π/(α+β) )| + |kH (1, eiθ0 (1 + t)π/(α+β) ) − kH (1, (1 + t)π/(α+β) )| ≤ kH (1, eiθ0 ) + kH (eiθ0 (1 + t)π/(α+β) , (1 + t)π/(α+β) ) ≤ 2 kH (1, eiθ0 ). Since kH (1, (1 + t)π/(α+β) ) = 21 log(1 + t)π/(α+β) , the previous computations and (16.2.2) show that there exists C > 0 such that |k V (h(z 0 ), h(z 0 ) + it) −

π log t| < C 2(α + β)

for all t ≥ 1, and we are done in case α, β > 0. Now we assume that β = 0 (the case α = 0 being similar). In this case, we compute k V (ei(π−α)/2 , ei(π−α)/2 + it) (note that (0, +∞)  t → tei(π−α)/2 is the symmetry axis of V ). Arguing as before, one can see that k V (ei(π−α)/2 , ei(π−α)/2 + it) = k W (1, 1 + teiα/2 ). We write 1 + teiα/2 = ρt eiθt . Since f : W → H defined as f (w) = wπ/α is a biholomorphism, we have π/α i(θt π)/α

k W (1, 1 + teiα/2 ) = kH (1, ρt

e

).

By Proposition 6.5.5, π/α i(θt π)/α

|kH (1, ρt

e

π/α

) − kH (1, ρt

π/α

Thus, we are left to compute kH (1, ρt we have π/α

kH (1, ρt

)=

π/α

) − kH (ρt

π/α

) + kH (ρt

π/α i(θt π)/α

, ρt

e

π/α i(θt π)/α

, ρt

e

)| ≤

1 log 2. 2

). By Lemma 5.4.1,

π π/α π/α log ρt , kH (ρt , ρt ei(θt π)/α ) = kH (1, ei(θt π)/α ), 2α

and |kH (1, ei(θt π)/α ) −

1 1 1 log | < log 2. 2 2 cos( θαt π )

Therefore, by (16.2.2), there exists C > 0 such that

16.2 Total Speed of Convergence

463

|k V (h(z 0 ), h(z 0 ) + it) − Now, ρt =

1 π 1 log ρt − log | < C. 2α 2 cos( θαt π )

 1 + cos(α/2)t t 2 + 2 cos(α/2)t + L 2 , cos θt = . ρt

π π log ρt goes like 2α log t as t → Clearly, limt→+∞ ρtt = 1, which implies that 2α 1 1 +∞. Let us analyze the asymptotic behavior of the term 2 log cos( θt π ) . Notice that α

limt→+∞ cos θt = cos(α/2) and limt→+∞ (ρt − t) = cos(α/2). Applying the Mean Value Theorem to the function g(x) = arccos(x), we deduce that for each x ∈ [0, 1] there is a point ξ in the interval of extremes points x and cos(α/2) such that g(x) −

α = g (ξ )(x − cos(α/2)). 2

Taking x = cos(θt ) we deduce that there is ξt in the interval of extremes points cos θt and cos(α/2) such that θt −

1 α = − (cos(θt ) − cos(α/2)). 2 1 − ξt2

Clearly, we have that limt→+∞ ξt = cos(α/2). Thus, cos(θt ) − cos(α/2) = − lim 1 − ξt2 = − sin(α/2). t→+∞ t→+∞ θt − α2 lim

Therefore lim t cos(

t→+∞

Thus,

1 2

θπ



π θt π α  cos( αt ) π α )= lim t θt − lim = − t θ − t α α t→+∞ 2 θt πα − π2 α t→+∞ 2 π lim t (cos θt − cos(α/2)) = α sin(α/2) t→+∞ t π lim = (1 + cos(α/2)(t − ρt )) α sin(α/2) t→+∞ ρt

π π = 1 − cos2 (α/2) = sin(α/2) ∈ (0, +∞). α sin(α/2) α

log cos(1θt π ) goes like α

1 2

log t as t → +∞ and the result follows.



As a direct consequence of (16.1.1) and Proposition 16.2.4 we have Corollary 16.2.5 Let (φt ) be a parabolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D and Koenigs function h.

464

16 Rate of Convergence at the Denjoy-Wolff Point

• Assume that h(D) is contained in a sector p + i W (α, β), for some p ∈ C and α, β ∈ [0, π ], with α + β > 0. (1) If α and β are both positive then for every z ∈ D, there exist C = C(α, β, z) > 0 such that, for all t ≥ 0, C 1 − |φt (z)| ≤ π/(α+β) . t (2) If α or β is zero then for every z ∈ D, there exist C = C(α, β, z) > 0 such that, for all t ≥ 0, 1 C 1 − |φt (z)| ≤ . t t π/(α+β) • Assume that h(D) contains a sector p + i W (α, β), for some p ∈ C and α, β ∈ [0, π ], with α + β > 0. (1) If α and β are both positive then for every z ∈ D, there exist C = C(α, β, z) > 0 such that, for all t ≥ 1, C 1 − |φt (z)| ≥ π/(α+β) . t (2) If α or β is zero then for every z ∈ D, there exist C = C(α, β, z) > 0 such that, for all t ≥ 1, 1 C . 1 − |φt (z)| ≥ t t π/(α+β) Another consequence of Proposition 16.2.4 is the following: Corollary 16.2.6 Let (φt ) be a non-elliptic semigroup in D with Koenigs function h. Suppose h(D) = p + i W (α, β) for some α, β ∈ [0, π ] with α + β > 0. (1) If α > 0, β > 0 then there exists C > 0 such that v T (t) ≤ C and |vo (t) −

π π log t| ≤ C, |v(t) − log t| ≤ C, 2(α + β) 2(α + β)

for all t ≥ 1. (2) If either α = 0 or β = 0 then there exists C > 0 such that for all t ≥ 1 π +α+β log t| ≤ C, 2(α + β) π log t| ≤ C, |vo (t) − 2(α + β) 1 |v T (t) − log t| ≤ C. 2 |v(t) −

Proof Up to a translation, we can assume that p = 0. Let τ ∈ ∂D be the Denjoy-Wolff point of (φt ) and let T : D → H, T (z) = (τ + z)/(τ − z) be the Cayley transform which maps τ to ∞.

16.2 Total Speed of Convergence

465

Let W := {ρeiθ : ρ > 0, |θ | < (α + β)/2}. Note that W (α, β) = R(W ), where R(z) = iei(β−α)/2 z. Moreover, the map f : W → H given by f (w) := wπ/(α+β) is a biholomorphism. Therefore, g := R ◦ f −1 ◦ T is a biholomorphism from D onto W (α, β). Note that the geodesic (−1, 1)τ in D is mapped by g onto the geodesic Γ := iei(β−α)/2 (0, +∞) in W (α, β) (which is, in fact, the symmetry axis of W (α, β)). Therefore, by the invariance under isometries of the tangential and orthogonal speeds and by Lemma 6.5.8, for every w ∈ W (α, β) there exists K ≥ 0 such that for all t ≥ 0, |vo (t) − k W (α,β) (g(0), πΓ (w + it))| ≤ K , |v T (t) − k W (α,β) (w + it, πΓ (w + it))| ≤ K , where πΓ : V → Γ is the hyperbolic projection on Γ . . Then f ◦ R −1 (i + it) = In case α > 0, β > 0, we pick w = i. Let θ0 := π(α−β) 2(α+β) iθ0 π/(α+β) . Moreover, Γ is mapped onto the geodesic (0, +∞) of H. Since e (1 + t) for every w ∈ H the hyperbolic projection of w onto (0, +∞) is given by |w| (see Lemma 5.4.1), we have for all t ≥ 0, k W (α,β) (i + it, πΓ (i + it)) = kH (eiθ0 (1 + t)π/(α+β) , |eiθ0 (1 + t)π/(α+β) |) = kH (eiθ0 , 1). Hence, (1) follows immediately from Proposition 16.2.4 and (16.1.2). In case β = 0 (the case α = 0 is similar), we pick w = ie−α/2 , and we let ρt eiθt := f ◦ R −1 (ie−α/2 + it). Hence, taking into account that Γ is mapped onto the geodesic (0, +∞) of H by f ◦ R −1 and Lemma 5.4.1, we see that k W (α,β) (g(0), πΓ (ie−α/2 + it)) = kH (1, ρt ), and k W (α,β) (ie−α/2 + it, πΓ (ie−α/2 + it)) = kH (ρt eiθt , ρt ) = kH (eiθt , 1). The asymptotic behavior of those quantities has been computed in the last part of the proof of Proposition 16.2.4, and (2) follows.  Another direct consequence of Propositions 16.2.1 and 16.2.2 is the following: Corollary 16.2.7 Let (φt ) be a non-elliptic semigroup in D. Then lim inf t→+∞

v(t) v(t) > 0, lim sup < +∞. log t t t→+∞

466

16 Rate of Convergence at the Denjoy-Wolff Point

In Proposition 16.2.1 we showed that if (φt ) is a parabolic semigroup in D, then v(t)/t → 0 as t → +∞. This is essentially the only possible upper bound, as the following proposition shows: Proposition 16.2.8 Let g : [0, +∞) → [0, +∞) be a function such that limt→+∞ g(t) = +∞ and limt→+∞ g(t) = 0. Then there exists a parabolic semigroup (φt ) in t D of zero hyperbolic step such that lim sup t→+∞

v(t) = +∞. g(t)

Proof Let {a j } be a strictly increasing sequence of positive real numbers, a1 > 0, lim j→∞ a j = +∞. Let {b j } be a strictly increasing sequence of positive real numbers to be chosen later on. Let ⎛ ⎞ ∞  Ω := C \ ⎝ {z ∈ C : Re z = ±a j , Im z ≤ b j }⎠ . j=1

Note that Ω is simply connected and starlike at infinity. Let h : D → Ω be a Riemann −1 map such that h(0) = 0, and let φ t (z) := h (h(z) + it) for z ∈ D and t ≥ 0. Then (φt ) is a semigroup in D and, since t≥0 (Ω − it) = C, it follows that (φt ) is parabolic of zero hyperbolic step. In order to estimate the total speed v(t) of (φt ), note that Ω is symmetric with respect to the imaginary axis iR, hence, by Proposition 6.1.3, the orbit [0, +∞)  t → it is a geodesic in Ω, and so is [0, +∞)  t → φt (0) in D. In particular, if we set γ (t) = it, we have  v(t) = ω(0, φt (0)) = kΩ (0, it) =  1 t dr ≥ , 4 0 δΩ (ir )

t

κΩ (γ (r ); γ (r ))dr

0

(16.2.3)

where the last inequality follows from Theorem 5.2.1. Now, we claim that we can choose the b j ’s in such a way that for every j ≥ 1 there exists x j ∈ (b j , b j+1 ) such that δΩ (it) = a j+1 for every t ∈ [x j , b j+1 ] and such that b j+1 − x j ≥ ja j+1 g(b j+1 ).

(16.2.4)

Indeed, set b1 = 1. Let x1 > 1 be such that |i x1 − (a1 + ib1 )| = a2 . Notice that for any later choice of b j , j ≥ 3, simple geometric consideration shows that, if we take b2 > x1 then δΩ (it) = a2 for every t ∈ [x1 , b2 ]. Moreover, since g(t)/t → 0 as t → +∞, we can find b2 > x1 such that a2 g(b2 ) + x1 < 1. b2

16.2 Total Speed of Convergence

467

Therefore, there exist x1 , b2 such that (16.2.4) is satisfied for j = 1. Now, we can argue by induction is a similar way. Suppose we constructed b1 , . . . , b j and x1 , . . . , x j−1 for j > 1. Then we select x j in such a way that |i x j − (a j + ib j )| = a j+1 and, again since g(t)/t → 0 as t → +∞, we choose b j+1 > x j such that ja j+1 g(b j+1 )+x j < 1. b j+1 Thus, by (16.2.3) and (16.2.4), we have 1 v(b j+1 ) ≥ 4



b j+1 0

1 dr ≥ δΩ (ir ) 4

Therefore,

hence lim supt→+∞



b j+1 xj

b j+1 − x j jg(b j+1 ) dr . = ≥ a j+1 4a j+1 4

v(b j+1 ) j ≥ , g(b j+1 ) 4 v(t) g(t)

= +∞, and we are done.



For some hyperbolic semigroups it is possible to get a tighter rate of convergence to the Denjoy-Wolff point than the one in (16.2.1). Such a rate of convergence can be characterized in terms of the conformality of the function exp(iλh) at the DenjoyWolff point of the semigroup, where λ is the spectral value of the semigroup and h is its Koenigs map. Theorem 16.2.9 Let (φt ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D and canonical model (Sπ/λ , h, z → z + it). Then, for each z ∈ D, the following limit exists, uniformly on compacta of D, K (z) := lim eλt (1 − τ φt (z)). t→+∞

(16.2.5)

Moreover, the function K is either identically zero or a univalent function with Re K > 0. In addition the following are equivalent: lim supt→+∞ (1 − |φt (0)|)eλt > 0; lim inf t→+∞ [ω(0, φt (z)) − kSπ/λ (h(0), h(z) + it)] < +∞ for all z ∈ D; There exists C > 0 such that |v(t) − λ2 t| < C for all t ≥ 0; The function g := −ieiλh is conformal at τ , that is, there exists g (τ ) := g(z) ∈ C \ {0}; ∠ lim z→τ z−τ (5) for each z ∈ D, the limit (1) (2) (3) (4)

K (z) := lim eλt (1 − τ φt (z)) t→+∞

(16.2.6)

exists and K is a univalent function with Re K > 0; (6) there exist z ∈ D and {tn } converging to +∞ such that the limit lim eλtn (1 − τ φtn (z)) ∈ C \ {0};

n→∞

(16.2.7)

468

16 Rate of Convergence at the Denjoy-Wolff Point

(7) for each z ∈ D, the limit

k(z) := lim eλt φt (z)

(16.2.8)

t→+∞

exists and belongs to C \ {0}; (8) there exist z ∈ D and {tn } converging to +∞ such that the limit lim eλtn φt n (z) ∈ C \ {0}.

(16.2.9)

n→∞

In such a case, K (z) =

g(z) g (τ )

and k(z) = −τ K (z), for all z ∈ D. −iλw

−1 Proof The function T (w) = ie , where w ∈ Sπ/λ , maps conformally the strip ie−iλw +1 Sπ/λ onto the unit disc so that f := T ◦ h is a holomorphic self-map of the unit disc and limr →1 f (r τ ) = 1, because limr →1 Im h(r τ ) = +∞ by Proposition 9.4.8. As usual, we denote by α f (τ ) the boundary dilation coefficient of f at τ . For all t ≥ 0 and z ∈ D, we have   [ω(0, φt (z)) − ω(0, f (φt (z)))] − [ω(0, φt (z)) − kS (h(0), h(z) + it)] π/λ

= |ω(0, f (φt (z))) − kSπ/λ (h(0), h(z) + it)| = |kSπ/λ (T −1 (0), h(z) + it) − kSπ/λ (h(0), h(z) + it)| ≤ kSπ/λ (T −1 (0), h(0)).

(16.2.10) Therefore, (2) is equivalent to α f (τ ) < +∞. Moreover, if α f (τ ) < +∞, taking into account that φt (z) converges to τ non-tangentially as t → +∞ by Proposition 8.3.7, it follows by Proposition 1.9.12 that for every z ∈ D there exists C ∈ R (depending on z) such that |ω(0, φt (z)) − ω(0, f (φt (z)))| ≤ ω(φt (z), f (φt (z))) ≤ C .

(16.2.11)

Next, by Theorem 1.3.5 and Proposition 6.7.2, for all t ≥ 0, v(t) −

π π λ t = ω(0, φt (0)) − kSπ/λ ( , + it) 2  2λ 2λ   (16.2.12) 1 + |φt (0)| λt 1 1 1 + |φt (0)| − = log e−λt . = log 2 1 − |φt (0)| 2 2 1 − |φt (0)|

Hence, taking into account that z → z + it is an automorphism of Sπ/λ , the triangular inequality and Proposition 1.3.10 show that

16.2 Total Speed of Convergence

469

    [ω(0, φt (z)) − kS (h(0), h(z) + it)] − [v(t) − λ t] π/λ  2  π π π ≤ |ω(0, φt (z)) − ω(0, φt (0))| + |kSπ/λ ( , + it) − kSπ/λ (h(0), + it)| 2λ 2λ 2λ π + |kSπ/λ (h(0), + it) − kSπ/λ (h(0), h(z) + it)| 2λ π π ≤ ω(0, z) + kSπ/λ ( , h(0)) + kSπ/λ ( , h(z)). 2λ 2λ (16.2.13) Therefore, (3) implies (2). And, conversely, if (2) holds, then α f (τ ) < +∞ and (16.2.11), (16.2.10) and (16.2.13) imply (3). Moreover, by (16.2.12), (3) implies (1) while, (1) implies lim inf t→+∞ [v(t) − λ t] < +∞, and hence (16.2.13) implies (2). 2 Therefore, (1), (2) and (3) are equivalent. Now, by Proposition 1.7.4, τ α f (τ ) = ∠ lim z→τ

2 1 − f (z) 1 = ∠ lim . z→τ τ − z ie−iλh(z) + 1 τ −z

(16.2.14)

Hence, if (2) holds (and hence α f (τ ) < +∞) ∠ lim z→τ

i eiλh(z) = − τ α f (τ ) ∈ C \ {0}. z−τ 2

Namely, (4) holds. Notice that, in fact, the above argument shows that the conformality of g is equivalent to α f (τ ) < +∞. Hence, (4) is equivalent to (2). Now, assume (4) holds. Using again the non-tangential convergence of the orbits to the Denjoy-Wolff point in the hyperbolic case (Proposition 8.3.7), we obtain immediately that eiλh(φt (z)) eiλh(z)−λt i = lim . − τ α f (τ ) = lim t→+∞ φt (z) − τ t→+∞ φt (z) − τ 2 Therefore, lim eλt (1 − τ φt (z)) =

t→+∞

−2ieiλh(z) . α f (τ )

Since Re eλt (1 − τ φt (z)) > 0 for all z and t, and the limit is not constant, we deduce that the function K (z) = −2ieiλh(z) /α f (τ ) has non-negative real part, it is not constant and then univalent, hence (5) holds. Clearly (5) implies (6). Assume now that (6) holds and let B = limn→∞ eλtn (1 − τ φtn (z)). Since 1 − f (φtn (z)) 1 2 = λt , τ − φtn (z) e n (1 − τ φtn (z)) ie−iλh(z) + e−λtn

470

16 Rate of Convergence at the Denjoy-Wolff Point

2 iλh(z) it follows immediately from (16.2.14) and Proposition 8.3.7 that τ α f (τ ) = Bi e ∈ C. In particular, α f (τ ) < +∞. Hence, (2) holds. Thus, the first six statements are equivalent. Since (6) implies (7) and (8) is a particular case of (7), we are left to show that (8) implies (6). Take z ∈ D and {tn } converging to +∞ such that the limit (16.2.9) is not zero. Denote by G the infinitesimal generator of the semigroup. Then, by Proposition 10.1.8,

K tn (z) = eλtn (1 − τ φtn (z)) = τ eλtn G(φtn (z)) = τ eλtn φt n (z)G(z)

τ − φtn (z) G(φtn (z))

τ − φtn (z) . G(φtn (z))

Therefore, in view of the non-tangential convergence of {φt (z)} to τ as t → +∞, we conclude that limn→∞ K tn (z) = λτ G(z) limn→∞ eλtn φt n (z) ∈ C \ {0}. Summing up, we have proved that statements (1) to (8) are equivalent. Suppose now that the equivalent statements (1) to (8) do not hold. Hence α f (τ ) = +∞ and (16.2.14) implies at once that K ≡ 0.  Example 16.2.10 We construct a hyperbolic semigroup in D with Denjoy-Wolff point 1 such that, for all z ∈ D, lim eλt (1 − φt (z)) = 0.

t→+∞

(16.2.15)

Consider the strip S2 = {w ∈ C : 0 < Re w < 2}. Given 0 < x < 1, we let Sx = (k+1)2 {w ∈ C : x < Re w < 2 − x}. Take ck := (k+1) 2 −1 and x k := 1/(k + 1) for all k ∈ N. Fix k and assume we have chosen y1 , y2 , . . . , yk−1 . Write Ωk,b = C \ {z ∈ C : Re z ∈ {xk , 2 − xk }, Im z ≤ b}. By Proposition 6.8.3, there is a constant Dk = D(ck ) > 0, independent of b, such that (16.2.16) k Sxk (z, w) ≤ ck kΩk,b (z, w) for every z, w ∈ Sxk with Im z, Im w ≤ b − Dk 2(1 − xk ). Choose yk > k + 1 + Dk 2(1 − xk ). Let

where



Ω = S2 \ ∪k∈N L k− ∪ L k+ , L k+ := {w ∈ C : Re w = 2 − xk , Im w ≤ yk }, L k− := {w ∈ C : Re w = xk , Im w ≤ yk }.

16.2 Total Speed of Convergence

471

Clearly, Ω ⊂ Ωk,yk , so that kΩk,yk (z, w) ≤ kΩ (z, w) for all z, w ∈ Ω. Note that Ω is symmetric with respect to 1 + iR. Hence, 1 + iR is the image of a geodesic in Ω by Proposition 6.1.3. Therefore, we can find a Riemann map h of Ω such that h(0) = 1, h((−1, 1)) = 1 + iR and limr →1 Im h(r ) = +∞. Consider the semigroup defined by φt (z) := h −1 (h(z) + it), for all z ∈ D and t ≥ 0. By construction, τ = 1 is the Denjoy-Wolff point of (φt ). Moreover, S2 = ∪t≤0 (h(D) − it). In other words, (S2 , h, z → z + it) is the canonical model of (φt ), which is then a hyperbolic semigroup in D with spectral value λ = π/2. Hence, in order to show (16.2.15), by Theorem 16.2.9, it is enough to prove that lim [ω(0, φt (0)) − kS2 (1, 1 + it)] = +∞.

t→+∞

(16.2.17)

Since the curve iR is a geodesic in Ω and kS2 (z, w) ≤ kΩ (z, w) for all z, w ∈ Ω, Proposition 6.7.2 and (16.2.16) imply that, for all N ∈ N, ω(0, φ N (0)) − kS2 (1, 1 + i N ) = kΩ (1, 1 + i N ) − kS2 (1, 1 + i N ) =

N −1 

[kΩ (1 + ki, 1 + (k + 1)i)

k=0

 − kS2 (1 + ki, 1 + (k + 1)i) N −1   π kΩ (1 + ki, 1 + (k + 1)i) − = 4 k=0 N −1   π kΩk,yk (1 + ki, 1 + (k + 1)i) − 4 k=0   N −1  1 π ≥ k S (1 + ki, 1 + (k + 1)i) − ck xk 4 k=0   N −1 N −1   1 π π π = − = . ck 4(1 − xk ) 4 4(k + 1) k=0 k=0

≥

Passing to the limit, (16.2.17) holds.

16.3 Orthogonal Speed of Convergence of Parabolic Semigroups In this section we give estimates on the orthogonal speed of convergence of semigroups. Since the orbits of hyperbolic semigroups converge non-tangentially to the Denjoy-Wolff point (see Proposition 8.3.7), it follows from (16.1.2) that the total and

472

16 Rate of Convergence at the Denjoy-Wolff Point

the orthogonal speeds of hyperbolic semigroups have the same asymptotic behavior. Therefore, we concentrate on parabolic semigroups. Following the notation introduced in Chap. 5 and in order to simplify the notation, for any α ∈ (0, π ], we write V (α) := V (α, 0) = {w = ρeiθ : ρ > 0, |θ | < α}. Theorem 16.3.1 Let (φt ) be a parabolic semigroup, not a group, in D with DenjoyWolff point τ ∈ ∂D and Koenigs function h. Suppose that h(D) is contained in a sector p + i V (α), p ∈ C, α ∈ (0, π ]. Then for every z ∈ D, there exists a constant C = C( p, α, z) > 0 such that |φt (z) − τ | ≤

C t π/(2α)

In particular, lim inf [vo (t) − t→+∞

, t ∈ (0, +∞).

π log t] > −∞. 4α

(16.3.1)

(16.3.2)

Proof Up to a translation, and with a simple geometric argument, we see that we can assume that the domain h(D) is contained in i V (α) and there is t0 > 0 such that it0 ∈ h(D). Assume for the moment that τ = 1 and h(0) = it0 . Using (16.1.1) and Lemma 16.1.2, it is enough to show that (16.3.1) holds for z = 0. For each t > 0, consider the Jordan arc Γt := {φs (0) : s ≥ t}. Notice that h(Γt ) = i[t + t0 , +∞). By Remark 16.1.1, Γt ⊂ D ∩ H for all t > 0. By Theorem 7.2.13, |φt (0) − 1| ≤ 2 arcsin

|φt (0) − 1| ≤ 2π μ(0, Γt , D \ Γt ). 2

Let k be a Riemann map of the domain D \ Γt . Notice that h ◦ k is a Riemann map of the domain h(D) \ h(Γt ) and, since h is univalent, we deduce that (h ◦ k)−1 (h(Γt )) = k −1 (Γt ). Thus, by the very definition of harmonic measure, μ(0, Γt , D \ Γt ) = μ(k −1 (0), k −1 (Γt ), D) = μ(h ◦ k −1 (h(0)), (h ◦ k)−1 (h(Γt )), D) = μ(h(0), h(Γt ), h(D) \ h(Γt )).

16.3 Orthogonal Speed of Convergence of Parabolic Semigroups

473

Therefore, using the domain monotonicity of the harmonic measure (Proposition 7.2.10), we obtain |φt (0) − 1| ≤ 2π μ(h(0), h(Γt ), h(D) \ h(Γt )) = 2π μ(h(0), i[t + t0 , +∞), h(D) \ i[t + t0 , +∞)) ≤ 2π μ(it0 , i[t + t0 , +∞), i V (α) \ i[t + t0 , +∞)) = 2π μ(t0 , [t + t0 , +∞), V (α) \ [t + t0 , +∞)). The map w → (w/t0 )π/(2α) sends V (α) to the right half-plane, the point t0 to 1 and the half-line [t + t0 , +∞) to [(1 + t/t0 )π/(2α) , +∞). Thus, bearing in mind (7.2.6), we obtain |φt (0) − 1| ≤ 2π μ(1, [(1 + t/t0 )π/(2α) , +∞), H \ [(1 + t/t0 )π/(2α) , +∞)) = 2π μ((1 + t/t0 )−π/(2α) , [1, +∞), H \ [1, +∞)) = 4 arcsin((1 + t/t0 )−π/(2α) ). Since arcsin(x) ≤

π 2

x for 0 ≤ x ≤ 1, |φt (0) − 1| ≤ 2π(1 + t/t0 )−π/(2α) .

Finally, (16.3.2) follows at once from (16.1.1) and (16.3.1). This conclude the proof for the particular case we have considered. In order to deal with the general case, let z 0 ∈ D be such that h(z 0 ) = it0 and take T ∈ Aut(D) such that T (0) = z 0 and T (1) = τ . Note that T (z) = Tz0 (λz), where Tz0 is given by (1.2.1) and λ = Tz0 (τ ). By Proposition 16.1.6 the orthogonal speed of the semigroup (φ˜ t ) := (T −1 ◦ φt ◦ T ) is comparable to that of (φt ), hence (16.3.2) holds in the general case as well. Finally, (16.3.1) follows from (16.1.1) and (16.3.2).  Remark 16.3.2 The previous bounds are sharp, as shown by Corollary 16.2.6. In general, we have the following bounds: Theorem 16.3.3 Let (φt ) be a parabolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. (1) For each z ∈ D, there exists a constant C = C(z) > 0 such that, for every t ≥ 0, t 1/2 |1 − τ φt (z)| ≤ C.

(16.3.3)

In particular, lim inf [vo (t) − t→+∞

1 log t] > −∞. 4

(16.3.4)

(2) If, in addition, the semigroup is of positive hyperbolic step, then for each z ∈ D, there exists a constant C = C(z) > 0 such that, for every t ≥ 0,

474

16 Rate of Convergence at the Denjoy-Wolff Point

t|1 − τ φt (z)| ≤ C.

(16.3.5)

In particular, lim inf [vo (t) − t→+∞

1 log t] > −∞. 2

(16.3.6)

Proof Let (Ω, h, z → z + it) be the canonical model of the semigroup. (1) Take a point p ∈ C \ h(D). Since h is starlike at infinity (Theorem 9.4.10), h(D) ⊂ p + i V (π ) and the result follows immediately from Theorem 16.3.1. (2) By (16.1.2) and (16.1.3) we have v(t) ≤ 2vo (t) + 4 log 2. Hence, by Proposition 16.2.2, lim inf [vo (t) − t→+∞

1 1 log t] ≥ lim inf [v(t) − log t − 2 log 2] > −∞. 2 2 t→+∞

Thus, (16.3.6) holds. Finally, (16.3.5) follows at once by Lemma 16.1.2 and (16.1.1).  Remark 16.3.4 The bounds given by Theorem 16.3.3 are sharp (see Corollary 16.2.6). Remark 16.3.5 Proposition 16.2.2, (16.3.4) and (16.1.2) imply at once that if (φt ) is a non-elliptic semigroup in D and there exists a constant C > 0 such that for all t ≥0 1 |vo (t) − log t| < C, 4 then lim supt→+∞ v T (t) < +∞ and hence [0, +∞)  t → φt (z) converges non-tangentially to the Denjoy-Wolff point for every z ∈ D. A sufficient condition of different nature to estimate the rate of convergence is given in Corollary 16.3.8. To this aim, we need some preliminary results. Lemma 16.3.6 Let p : D → H be holomorphic and σ ∈ ∂D. Assume that for some k ∈ R, the angular limit (16.3.7) ∠ lim (1 − σ z)k p(z) = a z→σ

exists finitely and is different from zero. Then k ∈ [−1, 1] and |Arg(a)| ≤

π (1 2

− |k|).

Proof Assume firstly that p is constant. Then k = 0 and the result is clear. Thus, let us assume that p is not constant. By Theorem 2.2.1, taking z = r σ , we have | p(r σ ) − iIm p(0)| ≤ Re p(0)

1+r , r ∈ [0, 1). 1−r

16.3 Orthogonal Speed of Convergence of Parabolic Semigroups

475

If k > 1, then |a| = lim |(1 − r )k p(r σ )| = lim (1 − r )k | p(r σ ) − iIm p(0)| r →1

r →1

≤ lim Re p(0)(1 + r )(1 − r )k−1 = 0. r →1

A contradiction because a = 0. Therefore, k ≤ 1. Now, ∠ lim (1 − σ z)−k z→σ

1 1 = p(z) a

exists finitely and is different from zero. Thus, the previous argument shows that −k ≤ 1. Hence k ∈ [−1, 1]. Define the holomorphic function q : H → H by q = p ◦ Cσ−1 , where Cσ is the for w ∈ H. Since Cayley transform given in (1.1.2). Recall that Cσ−1 (w) = σ w−1 w+1

(1 − σ Cσ−1 (w))k p(Cσ−1 (w)) =

2k q(w), (1+w)k

Eq. (16.3.7) can be rewritten as

a 1 1 = ∠ lim q(w) = ∠ lim k q(w). w→∞ (1 + w)k w→∞ w 2k

(16.3.8)

In particular, 2ak = limRr →+∞ r1k q(r ) ∈ H and thus a ∈ H \ {0}. Therefore, |Arg(a)|

≤ π/2. Now fix β ∈ − π2 , π2 . Using again (16.3.8), we deduce that 1 a = lim q(r eiβ )e−iβk . k Rr →+∞ r k 2 Hence |Arg(a) + βk| = |Arg(aeiβk )| ≤ π/2.

Since β ∈ − π2 , π2 is arbitrary, we conclude that |Arg(a)| ≤

π (1 2

− |k|).



Proposition 16.3.7 Let (φt ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D, canonical model (Ω, h, z → z + it) and infinitesimal generator G. Fix α ∈ R, α = 0. Then the following are equivalent: τ G(z) ∈ C \ {0}, (1 − τ z)1+α α (2) b := ∠ lim h(z)(1 − τ z) ∈ C \ {0}. (1) a := ∠ lim z→τ

z→τ

If (1) or (2)—and hence both—holds, α ∈ (0, 2], the semigroup (φt ) is parabolic, abα = i, and |Arg(a)| ≤ π2 min{α, 2 − α}. Proof Assume (1) holds. By Berkson-Porta’s Formula (Theorem 10.1.10) there is a non-vanishing holomorphic function p : D → H such that G(z) = (z − τ ) (τ z − 1) p(z), z ∈ D. Thus,

476

16 Rate of Convergence at the Denjoy-Wolff Point

∠ lim (1 − τ z)1−α p(z) = ∠ lim z→τ

z→τ

τ G(z) = a ∈ C \ {0}. (1 − τ z)1+α

By Lemma 16.3.6, we have that 1 − α ∈ [−1, 1] and |Arg(a)| ≤ π2 (1 − |1 − α|). Since α = 0, it follows that α ∈ (0, 2] and |Arg(a)| ≤ π2 min{α, 2 − α}. Finally, taking into account that G(z) = h i(z) , z ∈ D, ∠ lim z→τ h(z) = ∠ lim z→τ (1 − τ z)−α = ∞ and L’Hôpital’s Rule we conclude that ∠ lim z→τ

1 (1 − τ z)−α ατ (1 − τ z)−1−α = ∠ lim = −iaα. = ∠ lim z→τ z→τ h(z)(1 − τ z)α h(z) h (z)

Thus (2) holds and αab = i. Assume now (2) holds. Let f (z) := h(z)(1 − τ z)α+1 , z ∈ D. By (2), we have that f (z) = −τ b. Thus, by Theorem 1.7.2, ∠ lim z→τ f (z) = −τ b. Since ∠ lim z→τ z−τ f (z) = h (z)(1 − τ z)α+1 − τ (α + 1)h(z)(1 − τ z)α , z ∈ D, we have, ∠ lim h (z)(1 − τ z)α+1 = ∠ lim f (z) + ∠ lim τ (α + 1)h(z)(1 − τ z)α = τ αb z→τ

z→τ

and ∠ lim z→τ

z→τ

τ G(z) τi i . = ∠ lim = z→τ h (z)(1 − τ z)1+α (1 − τ z)1+α αb

Therefore, (1) holds. Finally, since α > 0, we have ∠ lim z→τ

τ G(z) τ G(z) = ∠ lim (1 − τ z)α = 0 z→τ 1 − τz (1 − τ z)1+α

and we conclude that the semigroup is parabolic (see Corollary 10.1.12).



Notice that the above proposition does not hold if α = 0. Indeed, by Theorem τ G(z) 11.1.4, ∠ lim z→τ h(z) = ∞ while ∠ lim z→τ (1−τ is always a finite real number. z) Corollary 16.3.8 Let (φt ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D and canonical model (Ω, h, z → z + it). Assume there exists α ∈ R such that (16.3.9) b := ∠ lim h(z)(1 − τ z)α ∈ C \ {0}. z→τ

Then, the semigroup (φt ) is parabolic, α ∈ (0, 2] and π π |1 − α| ≤ Arg(b) ≤ π − |1 − α|. 2 2

16.3 Orthogonal Speed of Convergence of Parabolic Semigroups

477

Moreover, (1) If there exists z 0 such that φt (z 0 ) converges to τ non-tangentially at t → +∞, then limt→+∞ t (1 − τ φt (z 0 ))α = −ib. (2) If b = lim h(z)(1 − τ z)α ∈ C \ {0}, z→τ

then, for every z ∈ D, limt→+∞ t (1 − τ φt (z))α = −ib. In particular, there exists C > 0 such that for all t ≥ 0, |vo (t) −

1 log t| < C. 2α

Proof Theorem 11.1.4 shows that ∠ lim z→τ h(z) = ∞. Thus, α = 0. By Proposition 16.3.7, the semigroup (φt ) is parabolic, α ∈ (0, 2] and |Arg(i/b)| ≤ π2 min{α, 2 − α}. Therefore, π π |1 − α| ≤ Arg(b) ≤ π − |1 − α|. 2 2 Fix z ∈ D. Then lim

t→+∞

h(φt (z)) h(z) + it = lim = i. t→+∞ t t

Therefore, bearing in mind that t (1 − τ φt (z))α =

t h(φt (z))

h(φt (z))(1 − τ φt (z))α

we deduce immediately (1) and the first statement of (2). Finally, from the first statement of (2) and (16.1.1) we obtain the final statement in (2).  As a corollary we have the following result (which could also be proved directly by a straightforward computation). Corollary 16.3.9 Let (φt ) be a parabolic group in D with Denjoy-Wolff point τ ∈ ∂D. Then, for every z ∈ D, lim t (1 − τ φt (z)) =

t→+∞

2 . τ φ1

(τ )

Moreover, there exists C > 0 such that for all t ≥ 0 |vo (t) −

1 log t| ≤ C, 2

|v T (t) −

1 log t| ≤ C. 2

and

478

16 Rate of Convergence at the Denjoy-Wolff Point

Proof Let a = τ φ1

(τ ). By Proposition 9.3.12, the Koenigs function of the group is +z , z ∈ D. Then given by h(z) = ai ττ −z lim h(z)(1 − τ z) =

z→τ

2i ∈ C \ {0}. a

Therefore, the limit and the first equation in the statement follow at once from Corollary 16.3.8. Finally, since by Remark 16.2.3, there exists C > 0 such that |v(t) − log t| ≤ C for all t ≥ 0, the last equation in the statement follows from the previous ones and (16.1.2).  Example α 16.3.10 Fix 0 < α ≤ 2 and consider the holomorphic function h(z) = i 1+z , z ∈ D. Since 1−z Im [(1 − z)2 h (z)] = 2αRe



1+z 1−z

α−1

≥ 0, z ∈ D,

by Theorem 9.4.11, the function h is starlike at infinity with respect to τ = 1. Consider the semigroup (φt ) given by φt (z) = h −1 (h(z) + it). Its Denjoy-Wolff point is 1 and ∠ lim h(z)(1 − z)α = i∠ lim (1 + z)α = i2α ∈ C \ {0}. z→1

z→1

Moreover, h(D) = i V ( απ ) (where, as usual, V (q) := {z ∈ C : |z| > 0, |Arg(z)| < 2 q}, q ∈ (0, π ]). Hence, it is symmetric with respect to iR and by Proposition 6.1.3, it follows that [0, +∞)  t → i + it is a geodesic in h(D). Therefore, [0, +∞)  t → φt (h −1 (i)) is a geodesic in D and converges non-tangentially to 1. In particular, by Remark 16.1.4, [0, +∞)  t → φt (z) converges to 1 non-tangentially for all z ∈ D. Hence, Corollary 16.3.8 implies that lim t→+∞ t (1 − φt (z))α = 2α . Moreover, 1 log t| < C (cfr. Corollary 16.2.6 taking into there exists C > 0 such that |vo (t) − 2α απ account that h(D) = i V ( 2 )).

16.4 Trajectories on the Boundary Let (φt ) be a non-elliptic semigroup in D. By Theorem 11.2.1, for every t ≥ 0 and every σ ∈ ∂D, the angular limit φt (σ ) := ∠ lim φt (z) z→σ

exists. Moreover by Theorem 14.1.1, for every z ∈ D, the function [0, +∞)  t → φt (z) ∈ D

16.4 Trajectories on the Boundary

479

is continuous. So it makes sense to consider the trajectory γz : [0, +∞) → D, γz (t) = φt (z) for every z ∈ D. If z ∈ ∂D and the trajectory γz enters the unit disc (namely, for some t > 0, γz (t) ∈ D), then the study of the rate of convergence to the Denjoy-Wolff point is reduced to the same problem for trajectories starting from an interior point. In case the trajectory stays on ∂D for all t ≥ 0 (i.e., if (φt ) has an exceptional maximal contact arc), one cannot estimate the speeds of convergence as they are defined in terms of hyperbolic distance, but can still estimate the Euclidean counterparts. Recall that only hyperbolic semigroups and parabolic semigroups of positive hyperbolic step can have exceptional maximal contact arcs (see Corollary 14.2.11). Theorem 16.4.1 Let (φt ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Suppose (φt ) has an exceptional maximal contact arc A ⊂ ∂D and σ ∈ A. Then one of the following two cases occurs: (1) The semigroup (φt ) is hyperbolic, and there exists a constant C1 = C1 (σ ) > 0 such that (16.4.1) |φt (σ ) − τ | ≤ C1 e−πt/λ , t ∈ (0, +∞), where λ is the spectral value of the semigroup. (2) The semigroup (φt ) is parabolic of positive hyperbolic step and there exists a constant C2 = C2 (σ ) > 0 such that |φt (σ ) − τ | ≤

C2 , t ∈ (0, +∞). t

(16.4.2)

Proof Let (Ω, h, z → z + it) be the canonical model of the semigroup. Up to a translation, we may assume that h(0) ∈ R. For t > 0, let E (Γt ) be the length of the circular arc Γt ⊂ A on the unit circle with end points τ and φt (σ ). By Example 7.1.6, |φt (σ ) − τ | ≤ E (Γt ) = 2π μ(0, Γt , D) = 2π μ(h(0), h(Γt ), h(D)).

(16.4.3)

By Corollary 14.2.11, h(Γt ) is a vertical half-line starting from the point h(φt (σ )) ∈ C and lies on the boundary of h(D). Therefore, the whole vertical line passing from the point h(φt (σ )) is contained in the complement of h(D). Assume that the semigroup is hyperbolic with spectral value λ so that Ω = Sπ/λ . We may assume that Re h(σ ) = 0, and h(Γt ) = {y : y ≥ Im h(σ ) + t}. By the domain monotonicity of the harmonic measure and Example 7.2.9, μ(h(0), h(Γt ), h(D)) ≤ μ(h(0), h(Γt ), Sπ/λ ) ≤ C1 e−πt/λ . Hence the result follows from (16.4.3). In case the semigroup is parabolic of positive hyperbolic step, we may assume that Ω = H, hence Re h(σ ) = 0, h(0) > 0 (since we assumed h(0) ∈ R) and

480

16 Rate of Convergence at the Denjoy-Wolff Point

h(Γt ) = {y : y ≥ Im h(σ ) + t}. By the domain monotonicity of the harmonic measures, taking into account that z → h(0)z is an automorphism of H, the conformal invariance of harmonic measures and Example 7.2.5 imply μ(h(0), h(Γt ), h(D)) ≤ μ(h(0), h(Γt ), H) = μ(1, and the result follows again from (16.4.3).

1 C2 h(Γt ), H) ≤ , h(0) t 

16.5 Notes The point of view of considering hyperbolic “speeds” is taken from [22]. Theorem 16.3.3 is due to Betsakos [12]. Betsakos (see [12–14]) was the first author who used harmonic measures to tackle different open problems concerning semigroups of holomorphic self-maps of the unit disc. The rate of convergence in the Euclidean sense, under certain hypotheses on the infinitesimal generator of the semigroup, has been also investigated in [63]. Theorem 16.3.1 is taken from [16] where a more general results for non-symmetric sectors is proven. Also, Theorem 16.4.1 is in [16]. Proposition 16.2.4 is a generalization of a result in [16]. A related result to Corollary 16.3.8 appeared in [61] (see also [60, 67]).

Chapter 17

Slopes of Orbits at the Denjoy-Wolff Point

In this chapter we consider the slope of orbits of non-elliptic semigroups when converging to the Denjoy-Wolff point. In other words, we study the possible angles of approach of the trajectories of a semigroup toward its Denjoy-Wolff point. We show that the angle of approach of the orbits of a hyperbolic semigroup is a harmonic function whose level sets are exactly the maximal invariant curves of the semigroup and whose range is (−π/2, π/2). While, the orbits of a parabolic semigroup of positive hyperbolic step always converge tangentially to the Denjoy-Wolff point. The situation becomes more complicated for parabolic semigroups of zero hyperbolic step. In such a case all orbits of the same semigroup have the same slopes, but an orbit can behave in all possible ways: it can converge tangentially, non-tangentially, have some subsequence which converges tangentially and some non-tangentially. Moreover, using harmonic measure theory, for any closed interval I of [−π/2, π/2] we construct an example of a parabolic semigroup of zero hyperbolic step whose slope is I . We will show however how to detect the type of convergence from the image of the Koenigs function of a semigroup: for instance, the convergence is nontangential if and only if the image of the Koenigs function is “quasi-symmetric” with respect to vertical axes. These types of results are based on the construction of suitable quasi-geodesics in starlike domains at infinity and on suitable estimates of the hyperbolic distance. Finally, we study the shift of a semigroup. A non-elliptic semigroup is of finite shift if one, and hence any, orbit converges to the Denjoy-Wolff point outside a horocycle. All non-elliptic semigroups, but the parabolic ones of positive hyperbolic step, are of infinite shift. Parabolic semigroups of positive hyperbolic step might or might not be of finite shift. We provide simple geometric conditions on the image of the associated Koenigs function in order to detect when a parabolic semigroup of positive hyperbolic step is or not of finite shift.

© Springer Nature Switzerland AG 2020 F. Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4_17

481

482

17 Slopes of Orbits at the Denjoy-Wolff Point

17.1 Euclidean Geometry of Domains Starlike at Infinity In this section we define the “right” and “left” distance for points of a domain starlike at infinity. Definition 17.1.1 Let Ω  C be a domain starlike at infinity and p ∈ C. For t ≥ 0, let + δ˜Ω, p (t) := inf{|z − ( p + it)| : Re z ≥ Re p, z ∈ C \ Ω}, − δ˜Ω, p (t) := inf{|z − ( p + it)| : Re z ≤ Re p, z ∈ C \ Ω}. + ˜− Note that, if p + it ∈ C \ Ω then δ˜Ω, p (t) = δΩ, p (t) = 0. While, for p ∈ Ω and t ≥ 0, + ˜− δΩ ( p + it) = min{δ˜Ω, p (t), δΩ, p (t)}. + The function δ˜Ω, p (t) can be equal to +∞, in case {z ∈ C : Re z ≥ Re p} ⊂ Ω. For instance, if Ω = C \ {z ∈ C : Re z = 0, Im z ≤ 0}, for every point p = x + i y with + ˜− x > 0, δ˜Ω, p (t) = +∞ for all t > 0. Similarly, the function δΩ, p (t) can be equal to + − +∞. However, if Ω = C, either δ˜Ω, p (t) or δ˜Ω, p (t) has to be finite.

Remark 17.1.2 Let p ∈ C. The distance function [0, +∞)  t → δΩ ( p + it) ∈ [0, +∞) is continuous and, since Ω is starlike at infinity, non-decreasing. ± The functions δ˜Ω, p (t) are continuous and non-decreasing as well. Indeed, it is easy to see that, if we let Ω − p := Ω ∪ {z ∈ C : Re z > Re p}, then − − δ˜Ω, p (t) = δΩ p ( p + it), + and similarly for δ˜Ω, p (t).

In order to avoid to deal with the value +∞, we introduce the following: Definition 17.1.3 Let Ω  C be a domain starlike at infinity. For p ∈ C and t ≥ 0 we let + − ˜+ ˜− δΩ, p (t) := min{δΩ, p (t), t}, δΩ, p (t) := min{δΩ, p (t), t}. ± Note that, by Remark 17.1.2, (0, +∞)  t → δΩ, p (t) is non-decreasing.

Lemma 17.1.4 Let Ω be a domain starlike at infinity. For all p, q ∈ Ω there exist 0 < c < C such that for all t ≥ 0 ± ± ± cδΩ, p (t) ≤ δΩ,q (t) ≤ CδΩ, p (t).

(17.1.1)

± ± Proof Notice that since p, q ∈ Ω, there is t0 > 0 such that δΩ,q (t) = δΩ, p (t) = t ± ± for t ≤ t0 . Since δΩ,q (t) and δΩ, p (t) are positive for t ≥ t0 , we can prove (17.1.1) for t large enough.

17.1 Euclidean Geometry of Domains Starlike at Infinity

483

+ + Assume Re p ≤ Re q. Hence, δΩ, p (t) ≤ δΩ,q (t) + | p − q|. Therefore, for t ≥ t0 + δΩ, p (t) + δΩ,q (t)

≤1+

| p − q| | p − q| ≤1+ + , + δΩ,q (t) δΩ,q (t0 )

which proves one inequality. Now we show the converse inequality. Since Ω is starlike at infinity, there exists B ≥ Im p such that W := {z ∈ C : Re p ≤ Re z ≤ Re q, Im z ≥ B} ⊂ Ω. If there + + (t) = δΩ, exist no z ∈ C \ Ω such that Re z ≥ Re p, then δΩ,q p (t) = t. Let C := min{| p − w| : Im w = B, Re p ≤ Re w ≤ Re q} = B − Im p ≥ 0. Assume there exists z ∈ C \ Ω such that Re z ≥ Re p. If Re z ≥ Re q, then + + (t) − | p − q| ≥ δΩ,q (t) − | p − q|. | p + it − z| ≥ |q + it − z| − | p − q| ≥ δ˜Ω,q

If Re p ≤ Re z < Re q, let t > 0 be such that p + it ∈ W . Then | p + it − z| ≥ Thus

min

Im w=B,Re p≤Re w≤Re q

+ | p + it − w| ≥ t − C ≥ δΩ,q (t) − C.

+ + δ˜Ω, p (t) ≥ δΩ,q (t) − C.

+ + + Since t ≥ δΩ,q (t), we also have that δΩ, p (t) ≥ δΩ,q (t) − C, and the converse inequality follows. − , and we are done.

A similar argument holds for δΩ

17.2 Quasi-Geodesics in Starlike Domains at Infinity The aim of this section is to construct a quasi-geodesic in a domain Ω  C starlike at infinity which converges in the Carathéodory topology to “+∞” and to get useful estimates on the hyperbolic distance from this curve to a vertical axis. In all this section, we assume that Ω ⊂ C is a domain starlike at infinity such that 0∈ / Ω and it ∈ Ω for all t > 0. We define σ : [1, +∞) → Ω by σ (t) :=

+ − δΩ,0 (t) − δΩ,0 (t)

2

+ it.

(17.2.1)

The aim of this section is to show that σ is a quasi-geodesic in Ω. The proof is quite long and requires several lemmas.

484

17 Slopes of Orbits at the Denjoy-Wolff Point

Lemma 17.2.1 The curve σ is 2-Lipschitz. In particular, |σ  (t)| ≤ 2 for almost every t ≥ 1. ± Proof For all s, t ≥ 1, using the triangle inequality we have δΩ,0 (t) ≤ |t − s| + ± ± ± δΩ,0 (s) and δΩ,0 (t) ≥ −|t − s| + δΩ,0 (s). Therefore, ± ± (t) − δΩ,0 (s)| ≤ |t − s|. |δΩ,0

From this it follows immediately that σ is 2-Lipschitz. Let



+ − ω(t) := δΩ,0 (t) + δΩ,0 (t).

Lemma 17.2.2 For t ≥ 1

1 δΩ (σ (t)) ≥ √ ω(t). 2 2

+ − (t) ≥ δΩ,0 (t), which implies that Proof Fix t ≥ 1. First consider the case δΩ,0 Re σ (t) ≥ 0. If z ∈ ∂Ω and Re (z) > 0, then + (t) − |z − σ (t)| ≥ |z − it| − |it − σ (t)| ≥ δΩ,0

Now, for z ∈ C define

+ − (t) − δΩ,0 (t) δΩ,0 ω(t) = . 2 2

z1 = |Re z| + |Im z|.

Then |z| ≤ z1 ≤

√

2|z|.

If z ∈ ∂Ω and Re z ≤ 0, then 1 |z − σ (t)| ≥ √ z − σ (t)1 . 2 Further, since Re z ≤ 0 ≤ Re σ (t) we have z − σ (t)1 = |Re z − Re σ (t)| + |Im z − Im σ (t)| = Re σ (t) − Re z + |Im z − t| − (t) = Re σ (t) + z − it1 ≥ Re σ (t) + |z − it| ≥ Re σ (t) + δΩ,0

=

Hence

+ − (t) − δΩ,0 (t) δΩ,0

2

− + δΩ,0 (t) =

1 ω(t). 2

1 |z − σ (t)| ≥ √ ω(t). 2 2

17.2 Quasi-Geodesics in Starlike Domains at Infinity

485

+ − The case when δΩ,0 (t) ≤ δΩ,0 (t) is similar.



As a direct consequence of the previous lemma, Lemma 17.2.1 and Theorem 5.3.1, we have: Lemma 17.2.3 If 1 ≤ a < b < ∞, then √  Ω (σ ; [a, b]) ≤ 4 2

b

a

1 dt. ω(t)

We can now prove that σ is a quasi-geodesic in Ω in a simple case: Proposition 17.2.4 Suppose that there exist α, T0 > 0 such that ω(t) ≥ αt for all t ≥ T0 . Then σ is a quasi-geodesic in Ω. ± Proof We have δΩ,0 (t) ≤ t for all t ≥ 1, hence, + − (t) − δΩ,0 (t)| |δΩ,0

t Therefore, for all t ≥ 1,

≤

+ |δΩ,0 (t)|

t

+

− |δΩ,0 (t)|

t

≤ 2.

t ≤ |σ (t)| ≤ 2t.

So, by (5.3.2) and taking into account that 0 ∈ C \ Ω, for all 1 ≤ a ≤ b,   |σ (b)|  1 b 1 1  ≥ log − log 2. kΩ (σ (a), σ (b)) ≥ log  4 |σ (a)| 4 a 4

(17.2.2)

On the other hand, if T0 ≤ a ≤ b, then by Lemma 17.2.3, √  Ω (σ ; [a, b]) ≤ 4 2 a

b

√  √ 4 2 b dt 4 2 b dt ≤ = log . ω(t) α a t α a

(17.2.3)

From this last inequality, (17.2.2) and Remark 6.3.3, it follows at once that σ is a quasi-geodesic in Ω.

Remark 17.2.5 For future reference, we make the following observations. If there exist α, T0 > 0 such that ω(t) ≥ αt for all t ≥ T0 , then

486

17 Slopes of Orbits at the Denjoy-Wolff Point

(1) by the same token we obtained (17.2.2), we have max{kΩ (ia, σ (b)), kΩ (σ (a), ib)} ≥

b 1 1 log − log 2. 4 a 4

Hence, by (17.2.3), there exist constants A, B > 0 such that for every T0 ≤ a ≤ b we have kΩ (σ (a), σ (b)) ≤ Ω (σ ; [a, b]) ≤ A min{kΩ (ia, σ (b)), kΩ (σ (a), ib)} + B.

(17.2.4)

(2) Also, again arguing as in (17.2.2), we have  a

b

4 dt ≤ kΩ (ia, ib). ω(t) α

(17.2.5)

Now we make the following assumption: Assumption There does not exist α, T0 > 0 such that ω(t) ≥ αt for all t ≥ T0 . Assuming this condition, there exists T0 > 0 such that ω(T0 ) < T0 . In particular, + − (T0 ), δΩ,0 (T0 )} < T0 . Hence, for every t ≥ T0 we have max{δΩ,0 ± ± (t) ≤ t − T0 + δΩ,0 (T0 ) < t − T0 + T0 = t. δΩ,0

Therefore, for every t ≥ T0 ,

± (t) < t. δΩ,0

(17.2.6)

Step 1: constructing sequences. Fix a, b ∈ [T0 , ∞) with a < b. We define a sequence of positive numbers {tn } a = t0 < t1 < t2 < · · · and complex numbers {z n± } ⊂ C \ Ω such that for all n ≥ 0 (0) (1) (2) (3) (4) (5)

Im z n+ = Im z n− , Re z n− < 0 < Re z n+ , ± (tn ), |Re z n± | ≤ δΩ,0 yn ≤ tn , where yn := Im z n + = Im z n− , max{|σ (tn ) − z n+ |, |σ (tn ) − z n− |} ≤ 2ω(tn ), and for all n ≥ 1, + − |, |σ (tn ) − z n−1 |} = 6ω(tn ). min{|σ (tn ) − z n−1

We first explain the construction of these sequences and then verify that they have the desired properties (see Fig. 17.1). We define tn , z n+ , and z n− sequentially as follows. If n = 0, then define t0 := a. Otherwise, define

17.2 Quasi-Geodesics in Starlike Domains at Infinity

487

Fig. 17.1 Step 1

 1 + − tn := max t ≥ tn−1 : ω(s) ≥ min{|σ (s) − z n−1 |, |σ (s) − z n−1 |} 6  for all s ∈ [tn−1 , t] .

(17.2.7)

Next pick an , bn ∈ C \ Ω such that Re (an ) ≤ 0 ≤ Re (bn ), − |an − itn | = δΩ,0 (tn ), + |bn − itn | = δΩ,0 (tn ).

Since tn ≥ T0 , by (17.2.6) we have Re (an ) < 0 < Re (bn ). Then let yn := min{Im (an ), Im (bn )}. Since Ω is starlike at infinity, max{Im (an ), Im (bn )} ≤ tn , hence yn ≤ tn . Then define z n+ := Re (bn ) + i yn , z n− := Re (an ) + i yn .

488

17 Slopes of Orbits at the Denjoy-Wolff Point

We now verify that the resulting sequences have the desired properties. Claim 1: a = t0 < t1 < t2 < · · · . First, note that Property (4) implies that the set in (17.2.7) is non-empty. Hence each tn exists. We next show that tn < +∞. If n = 0, then tn = a < +∞. If n > 0, then the definition of σ implies that + − |, |σ (t) − z n−1 |} ≥ t − tn−1 min{|σ (t) − z n−1

for all t ≥ tn−1 . Then, since we assume that there does not exist α, T0 > 0 such that ω(t) ≥ αt for all t ≥ T0 , we see that tn < +∞. Finally, we show that if n > 0, then tn−1 < tn . By Property (4)   + − |, |σ (tn−1 ) − z n−1 | ≤ 2ω(tn−1 ). min |σ (tn−1 ) − z n−1 So by the continuity of ω we see that tn > tn−1 . Now Properties (0)–(3) and (5) hold by the construction. So we only have to verify Property (4). Claim 2: max{|σ (tn ) − z n+ |, |σ (tn ) − z n− |} ≤ 2ω(tn ) for all n ≥ 0. We first argue that max{|itn − z n+ |, |itn − z n− |} ≤ ω(tn ).

(17.2.8)

Indeed, assume that yn = Im bn (a similar argument works in case yn = Im an ). Then, + (tn ), while |itn − z n+ | = δΩ,0 − |itn − z n− | ≤ |itn − an | + |an − (Re an + i yn )| = δΩ,0 (tn ) + (Im an − yn ) − − − + ≤ δΩ,0 (tn ) + (tn − yn ) ≤ δΩ,0 (tn ) + |itn − bn | = δΩ,0 (tn ) + δΩ,0 (tn ) = ω(tn ).

Also, clearly |σ (tn ) − itn | ≤ ω(tn ). This last inequality, together with (17.2.8), implies |σ (tn ) − z n± | ≤ |σ (tn ) − itn | + |itn − z n± | ≤ 2ω(tn ). This completes the construction of the sequences. Step 2: key estimates. We now establish key estimates on the sequences constructed in the previous step. Lemma 17.2.6 For n ≥ 1 we have 3ω(tn ) ≤ yn − tn−1 ≤ min{tn − tn−1 , yn − yn−1 }. In particular, t0 < y1 ≤ t1 < y2 ≤ t2 < · · · and limn→∞ yn = ∞.

17.2 Quasi-Geodesics in Starlike Domains at Infinity

489

Proof Fix n ≥ 1. By Property (5) in Step 1, + − |, |σ (tn ) − z n−1 |} = 6ω(tn ). min{|σ (tn ) − z n−1 + | = 6ω(tn ). Then, by (17.2.8) and taking into First assume that |σ (tn ) − z n−1 account that ω(tn ) ≥ ω(tn−1 ), we have

yn − tn−1 = |i yn − itn−1 | + + ≥ |σ (tn ) − z n−1 | − |σ (tn ) − i yn | − |itn−1 − z n−1 | ≥ 6ω(tn ) − 2ω(tn ) − ω(tn−1 ) ≥ (6 − 3) ω(tn ) = 3ω(tn ).

By Property (3) in Step 1, yn − tn−1 ≤ min{tn − tn−1 , yn − yn−1 }. The case when − | = 6ω(tn ) is essentially the same. |σ (tn ) − z n−1 Finally, the previous estimates show that {yn } is an increasing sequence and 0 < 3ω(t0 ) ≤ 3 lim ω(tn ) ≤ lim (yn − yn−1 ). n→∞

n→∞

Hence limn→∞ yn = ∞.



As straightforward consequence of the previous lemma and taking into account that ω(tn ) ≥ ω(tn−1 ), we see that  log

yn − yn−1 ω(tn−1 )

≥ log 3 > 1

(17.2.9)

for every n ≥ 1. Lemma 17.2.7 If n ≥ 1 and t ∈ [yn , tn ], then ω(t) ≤ ω(tn ) ≤ 2ω(t). Proof The first inequality follows from the fact that Ω is starlike at infinity. Since tn−1 < yn ≤ tn it follows from (17.2.7) and the fact that σ is 2-Lipschitz (see Lemma 17.2.1) that 1 + − |, |σ (t) − z n−1 |} min{|σ (t) − z n−1 6 1 1 + − ≥ min{|σ (tn ) − z n−1 |, |σ (tn ) − z n−1 |} − |σ (tn ) − σ (t)| 6 6  1 1 1 1 ω(tn ) ≥ ω(tn ), ≥ ω(tn ) − 2 (tn − t) ≥ ω(tn ) − |itn − i yn | ≥ 1 − 6 3 3 2

ω(t) ≥

and the proof is completed.

Step 3: A lower bound on distance. Define



490

17 Slopes of Orbits at the Denjoy-Wolff Point

δn := Re (z n+ ) − Re (z n− ). By Property (2) in Step 1, + − (tn ) + δΩ,0 (tn ) = ω(tn ). δn ≤ δΩ,0

Fix N ≥ 0 such that y N ≤ b < y N +1 (recall that we fixed a, b ∈ [T0 , ∞) with a < b). Lemma 17.2.8 Suppose u ∈ {ia, σ (a)} and v ∈ {ib, σ (b)}. If N = 0, then    1 1 b − y0 kΩ (u, v) ≥ − log (2) + log max 1, . 4 4 ω(a) If N ≥ 1, then

   N −1 1 yk+1 − yk y1 − y0 + kΩ (u, v) ≥ log − log 2 + log 4 ω(a) δk k=1    b − yN . + log max 1, δN Proof First suppose that N = 0. If b − y0 ≤ ω(a) there is nothing to prove. So suppose that b − y0 ≥ 1. ω(a) By (17.2.8) and Property (4) in Step 1,   max |u − z 0+ |, |u − z 0− | ≤ 2ω(a).

(17.2.10)

Next, since |v − z 0± | ≥ |Im v − Im z 0± | = b − y0 , we have   min |v − z 0+ |, |v − z 0− | ≥ b − y0 .

(17.2.11)

Putting together (5.3.2) with (17.2.10) and (17.2.11), we have kΩ (u, v) ≥

    v − z 0+  1  ≥ − 1 log (2) + 1 log b − y0 . log  4 4 4 ω(a) u − z 0+ 

Next suppose that N > 0. Let γ : [0, T ] → Ω be a geodesic parameterized by hyperbolic arc-length such that γ (0) = u and γ (T ) = v. For k = 1, . . . , N define τk := min{t ≥ 0 : Im(γ (t)) = yk }. Note that a < τ1 < τ2 < · · · < τ N < b.

17.2 Quasi-Geodesics in Starlike Domains at Infinity

491

Then, since Ω is starlike at infinity, Re (z k− ) < Re (γ (τk )) < Re (z k+ ).

(17.2.12)

Also, since |γ (τk+1 ) − z k± | ≥ |Im γ (τk+1 ) − Im z k± | = yk+1 − yk , we have   min |γ (τk+1 ) − z k+ |, |γ (τk+1 ) − z k− | ≥ yk+1 − yk .

(17.2.13)

Moreover, by (17.2.12) we have |γ (τk ) − z k± | = |Re γ (τk ) − Re z k± | ≤ δk , hence   max |γ (τk ) − z k+ |, |γ (τk ) − z k− | ≤ δk .

(17.2.14)

Now, by (5.3.2), (17.2.13) and (17.2.10) we have     γ (τ1 ) − z 0+  1 1 y1 − y0 1   ≥ − log (2) + log . (17.2.15) kΩ (u, γ (τ1 )) ≥ log  4 4 4 ω(a) u − z 0+  For k ≥ 1, (5.3.2), (17.2.13) and (17.2.14) imply that     γ (τk+1 ) − z k+  1 1 yk+1 − yk   kΩ (γ (τk+1 ), γ (τk )) ≥ log  ≥ log . 4 δk γ (τk ) − z k+  4

(17.2.16)

Finally, (5.3.2), (17.2.14) imply that     v − z+  1 b − yN 1 N   kΩ (γ (τ N ), v) ≥ log  ,  ≥ 4 log 4 δN γ (τ N ) − z + N    b − yN 1 . log max 1, 4 δN

and hence kΩ (γ (τ N ), v) ≥

(17.2.17)

Since γ is a geodesic, we have kΩ (u, v) = kΩ (u, γ (τ1 )) +

N −1 

kΩ (γ (τk ), γ (τk+1 )) + kΩ (γ (τ N ), v),

k=1

The statement then follows from (17.2.15), (17.2.16), (17.2.17).

Step 4. An upper bound on length Lemma 17.2.9 If T ∈ [a, y1 ], then  a

T

   T − y0 dt ≤ 1 + 6 log max 1, . ω(t) ω(a)



492

17 Slopes of Orbits at the Denjoy-Wolff Point

Proof Notice that by (17.2.8), (a − y0 ) = (t0 − y0 ) ≤ |it0 − z 0± | ≤ ω(t0 ), hence, y0 ≤ a ≤ y0 + ω(a) and if a ≤ t, then ω(a) ≤ ω(t). So 

y0 +ω(a)

a

dt ≤ ω(t)



y0 +ω(a)

a

dt ≤ 1. ω(a)

Now if t ∈ [a, y1 ], then by (17.2.7), 1 1 min{|σ (t) − z 0+ |, |σ (t) − z 0− |} ≥ (t − y0 ). 6 6

ω(t) ≥

So if T ≥ y0 + ω(a), then 

T

dt ≤6 ω(t)

y0 +ω(a)



T y0 +ω(a)

 dt T − y0 , = 6 log t − y0 ω(a)



and we are done. Lemma 17.2.10 For k ≥ 1, 

yk+1 yk

dt ≤ 8 log ω(t)



yk+1 − yk ω(tk )

.

Proof By Lemma 17.2.6, yk + ω(tk ) ≤ yk + 3ω(tk ) ≤ yk+1 . Further, by Lemma 17.2.7, if t ∈ [yk , tk ], then ω(t) ≥ ω(tk )/2 and, since Ω is starlike at infinity, if t ≥ tk , then ω(t) ≥ ω(tk ). Therefore, ω(t) ≥ ω(tk )/2 when t ≥ yk . Thus 

yk +ω(tk ) yk

1 dt ≤ ω(t)



yk +ω(tk ) yk

2 dt = 2. ω(tk )

(17.2.18)

Next consider t ∈ [yk + ω(tk ), yk+1 ]. By (17.2.8), we have tk − yk = |itk − i yk | ≤ |itk − z k± | ≤ ω(tk ). Then yk + ω(tk ) ≥ tk . So t ∈ [tk , yk+1 ] and yk+1 ≤ tk+1 . Hence, by (17.2.7),

17.2 Quasi-Geodesics in Starlike Domains at Infinity

493

1 1 min{|σ (t) − z k+ |, |σ (t) − z k− |} ≥ (t − yk ). 6 6

ω(t) ≥ Therefore, 

yk+1 yk +ω(tk )

dt dt ≤ 6 ω(t)



yk+1 yk +ω(tk )

dt = 6 log t − yk



yk+1 − yk ω(tk )

.

(17.2.19)

Thus by (17.2.18), (17.2.19) and (17.2.9), 

yk+1 yk

dt dt ≤ 2 + 6 log ω(t)



yk+1 − yk ω(tk )



 ≤ 8 log

yk+1 − yk ω(tk )

,



and we are done. Repeating the proof of the previous lemma one can prove: Lemma 17.2.11 If N ≥ 1, then 

b yN

   b − yN dt ≤ 2 + 6 log max 1, . ω(t) ω(t N )

We are now ready to prove the main result of this section: Theorem 17.2.12 The curve [1, +∞)  t → σ (t) is a quasi-geodesic in Ω. Proof By Lemma 17.2.3 we have √  Ω (σ ; [a, b]) ≤ 4 2 a

b

1 dt. ω(t)

b Hence, in order to prove Theorem 17.2.12, it is enough to show that a ω(t)−1 dt is comparable to the lower bounds in Lemma 17.2.8. This can be done using the estimates in the previous three lemmas. Indeed, recall that a, b ∈ [T0 , ∞) with a < b and N ≥ 0 is a natural number such that y N ≤ b < y N +1 . If N = 0, then Lemma 17.2.9 implies    √ √ b − y0 , Ω (σ ; [a, b]) ≤ 4 2 + 24 2 log max 1, ω(a) while if N > 0, then Lemmas 17.2.10 and 17.2.11 imply

(17.2.20)

494

17 Slopes of Orbits at the Denjoy-Wolff Point

y1 − y0 ω(a)  N −1 √  yk+1 − yk log + 32 2 ω(tk ) k=1    √ b − yN . + 24 2 log max 1, ω(t N )

√ √ Ω (σ ; [a, b]) ≤ 12 2 + 24 2 log



(17.2.21)

Finally, Lemma 17.2.8 and the fact that δk ≤ ω(tk ) imply that there exist A > 1 and B > 0 such that for every T0 ≤ a ≤ b, Ω (σ ; [a, b]) ≤ AkΩ (σ (a), σ (b)) + B.



Now, by Remark 6.3.3, we are done.

Remark 17.2.13 We also notice that by (17.2.20), (17.2.21) and Lemma 17.2.8, there exist constants A, B > 0 such that for every 1 ≤ a ≤ b, kΩ (σ (a), σ (b)) ≤ Ω (σ ; [a, b]) ≤ A min{kΩ (σ (a), ib), kΩ (σ (b), ia)} + B.

(17.2.22)

As a consequence of the previous results, we have the following: Proposition 17.2.14 Assume there exist c, C > 0 such that for all t ≥ 1 − + − cδΩ,0 (t) ≤ δΩ,0 (t) ≤ CδΩ,0 (t)

Then βi : [0, +∞)  t → i + it is a quasi-geodesic in Ω. − + − + (t), δΩ,0 (t)} and δΩ,0 (t) is comparable to δΩ,0 (t), Proof Since δΩ (it) = min{δΩ,0  there exists C > 1 such that for every t ≥ 1,

ω(t) ≤ C  δΩ (it). In particular, by Theorem 5.3.1, we have for every 0 ≤ a ≤ b,  Ω (βi ; [a, b]) ≤ a

b

1 dt ≤  δΩ (it) C

 a

b

dt . ω(t)

Therefore, in case there exist α, T0 > 0 such that ω(t) ≥ αt for all t ≥ T0 , Eq. (17.2.5) implies that βi is a quasi-geodesic. On the other hand, if there exist no α, T0 > 0 such that ω(t) ≥ αt for all t ≥ T0 , Lemmas 17.2.8, 17.2.9, 17.2.10 and 17.2.11 imply again that βi is a quasi-geodesic.

We end this section with a useful estimate. For t ≥ 1, let st ∈ [1, +∞) be such that

17.2 Quasi-Geodesics in Starlike Domains at Infinity

kΩ (σ (st ), it) =

min kΩ (σ (r ), it).

r ∈[1,+∞)

495

(17.2.23)

Proposition 17.2.15 There exist α > 1, β > 0 such that for every t ≥ 1, kΩ (σ (t), it) ≤ αkΩ (σ (st ), it) + β. Proof Either by (17.2.4), or by (17.2.22), for t ≥ 1, we have kΩ (σ (t), σ (st )) ≤ AkΩ (it, σ (st )) + B. Therefore kΩ (it, σ (t)) ≤ kΩ (it, σ (st )) + kΩ (σ (st ), σ (t)) ≤ (A + 1)kΩ (it, σ (st )) + B,



and we are done.

17.3 Convergence to the Denjoy-Wolff Point for Non-Elliptic Semigroups In this section we establish results which relate the type of convergence of orbits of a non-elliptic semigroup with the shape of the image of its Koenigs function. Theorem 17.3.1 Let (φt ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D and Koenigs function h and let Ω := h(D). Suppose that {tn } is a sequence converging to +∞. Then (1) the sequence {φtn (z)} converges non-tangentially to τ as n → ∞ for some—and hence any—z ∈ D if and only if for some—and hence any— p ∈ Ω there exist 0 < c < C such that for all n ∈ N + − + cδΩ, p (tn ) ≤ δΩ, p (tn ) ≤ CδΩ, p (tn ).

(2) limn→∞ Arg(1 − τ φtn (z)) = π2 (in particular, {φtn (z)} converges tangentially to τ as n → ∞) for some—and hence any—z ∈ D if and only if for some—and hence any— p ∈ Ω, + δΩ, p (tn ) = 0, lim − n→∞ δ Ω, p (tn ) while, limn→∞ Arg(1 − τ φtn (z)) = − π2 (in particular, {φtn (z)} converges tangentially to τ as n → ∞) for some—and hence any—z ∈ D if and only if for some—and hence any— p ∈ Ω,

496

17 Slopes of Orbits at the Denjoy-Wolff Point

lim

n→∞

+ δΩ, p (tn ) − δΩ, p (tn )

= +∞.

± Proof From Lemma 17.1.4 it follows that the conditions on δΩ, p (t) do not depend on p ∈ Ω. We can suppose that (φt ) is not a group of automorphisms of D, for otherwise the result is clear. In this case, there exists p ∈ C such that p ∈ / Ω and p + it ∈ Ω for all t > 0. Up to ± ± (t) = δΩ,0 (t) a translation, we can assume p = 0. In particular, this implies that δ˜Ω,0 for every t > 0. Step 1. The sequence {φtn (h −1 (i))} converges to τ as n → +∞ non-tangentially (respectively, tangentially) if and only if for every z ∈ D the sequence {φtn (z)} converges to τ as n → +∞ non-tangentially (resp., tangentially). Since kD (φtn (h −1 (i)), φtn (z))) ≤ kD (h −1 (i), z) < +∞ for every n ∈ N, it follows that φtn (z) is contained in a fixed hyperbolic neighborhood of {φtm (h −1 (i)) : m ∈ N} for all n ∈ N. Therefore the result follows at once from the triangle inequality and from Proposition 6.2.5 (respectively, Corollary 6.2.6) with f = id and γ : [0, 1) → D the geodesic γ (r ) := r τ . Step 2. Let now σ be the curve defined in (17.2.1). We claim that

lim h −1 (σ (t)) = τ.

t→+∞

(17.3.1)

Indeed, since limt→+∞ |σ (t)| = ∞, the limit x := limt→+∞ h −1 (σ (t)) exists by Proposition 3.3.3. Suppose for a contradiction that x = τ . For n ∈ N, consider the segments C˜ n (s) := in + s

+ − δΩ,0 (n) − δΩ,0 (n) , 0 ≤ s ≤ 1. 2

Note that C˜ n ⊂ Ω for all n ∈ N and C˜ n (1) = σ (n). In particular, h −1 (C˜ n (1)) → x as n → ∞. On the other hand, since h(φn (h −1 (i))) = i + in, we have for n ≥ 1, h −1 (C˜ n (0)) = h −1 (in) = h −1 (h(φn−1 (h −1 (i)))) = φn−1 (h −1 (i)). Therefore, h −1 (C˜ n (0)) → τ as n → ∞. Let Cn := h −1 (C˜ n ), n ∈ N. Since x = τ , by the previous considerations, the Euclidean diameter of (Cn ) is bounded from below by a constant K > 0. Moreover, for every sequence {z n } such that z n ∈ Cn , limn→∞ |h(z n )| = ∞. Therefore, (Cn ) is a sequence of Koebe’s arcs for h, contradicting the no Koebe Arcs Theorem 3.2.4. The claim is proved. Step 3. The sequence {φtn (z 0 )} converges non-tangentially to τ as n → ∞ for all z 0 ∈ D if and only if there exists C > 0 such that for every n ∈ N kΩ (itn , σ ([1, +∞))) ≤ C.

17.3 Convergence to the Denjoy-Wolff Point for Non-Elliptic Semigroups

497

Moreover, the sequence {φtn (z 0 )} converges tangentially to τ as n → +∞ for all z 0 ∈ D if and only if for every M > 0 there exists n M ≥ 1 such that for all n ≥ n M , kΩ (itn , σ ([1, +∞))) > M. This follows immediately from Corollary 6.3.9 and since σ is a quasi-geodesic in Ω by Theorem 17.2.12 such that limt→+∞ h −1 (σ (t)) = τ by Step 2. Now, for t ≥ 1 let st be defined as in (17.2.23), that is, kΩ (it, σ ([1, +∞))) = kΩ (it, σ (st ))). Then, by Proposition 17.2.15 and the Distance Lemma (see Theorem 5.3.1), we have for all t ≥ 1, 1 β kΩ (it, σ (t)) − α α

+ − |δΩ,0 (t) − δΩ,0 (t)| 1 β log ≥ − . 4α 2 min{δΩ (it), δΩ (σ (t))} α

kΩ (it, σ ([1, +∞))) ≥

In other words, there exist A, B > 0 such that for every t ≥ 1,

+ − |δΩ,0 (t) − δΩ,0 (t)| kΩ (it, σ ([1, +∞)) ≥ A log 2δΩ (it)

− B.

(17.3.2)

Now, for t ≥ 1 let ηt : [0, 1] → Ω be defined as ηt (r ) := it + r

+ − δΩ,0 (t) − δΩ,0 (t) . 2

For all t ≥ 1 we have kΩ (σ (st ), it) ≤ kΩ (σ (t), it) ≤ Ω (ηt ; [0, 1]).

(17.3.3)

We want to estimate Ω (ηt ; [0, 1]). In order to do so, we claim that for every t ≥ 1 and for every r ∈ [0, 1] we have δΩ (ηt (r )) ≥ δΩ (it).

(17.3.4)

+ − + − (t) ≥ δΩ,0 (t) (the case δΩ,0 (t) ≤ δΩ,0 (t) is Indeed, fix t ≥ 1 and assume that δΩ,0 similar and we omit it). Fix r ∈ [0, 1]. Notice that Re ηt (r ) ≥ 0. Therefore, if z ∈ C \ Ω and Re z ≤ 0, then − (t) = δΩ (it). |ηt (r ) − z| ≥ |it − z| ≥ δΩ,0

498

17 Slopes of Orbits at the Denjoy-Wolff Point

+ On the other hand, if z ∈ C \ Ω and Re z > 0, then |it − z| ≥ δΩ,0 (t). Therefore,

|ηt (r ) − z| ≥

inf

+ |w−it|=δΩ,0 (t),Re w>0

+ |ηt (r ) − w| = δΩ,0 (t) − Re ηt (r )

+ (t) − Re σ (t) = ≥ δΩ,0

 1 + − δ (t) + δΩ,0 (t) ≥ δΩ (it), 2 Ω,0

and we are done. By (17.3.4) and the Distance Lemma (Theorem 5.3.1), for every t ≥ 1,  Ω (η; [0, 1]) = ≤

1

κΩ (η(r ); η (r ))dr ≤

0 + |δΩ,0 (t)

− δΩ,0 (t)|

− 2δΩ (it)

+ − |δΩ,0 (t) − δΩ,0 (t)|



2

0

1

dr δΩ (η(r ))

.

This latter inequality together with (17.3.3) and (17.3.2) implies that for every t ≥ 1,

+ − |δΩ,0 (t) − δΩ,0 (t)| A log 2δΩ (it)

− B ≤ kΩ (it, σ ([1, +∞))) + − (t) − δΩ,0 (t)| |δΩ,0 . ≤ 2δΩ (it)

(17.3.5)

Part (1) of Theorem 17.3.1 follows now directly from Step 3 and (17.3.5). δ + (tn ) Also, by the same token, we see that φtn (z) → τ tangentially if and only if δΩ,0 − Ω,0 (tn ) converges either to 0 or +∞ as n → ∞. We are left to show that + δΩ,0 (tn ) lim − = +∞ (17.3.6) n→∞ δ Ω,0 (tn ) if and only if

π lim Arg(1 − τ φtn (z)) = − . 2

n→∞

(17.3.7)

To this aim, we extend σ to all of (0, ∞) in the obvious way: σ (t) =

+ − δΩ,0 (t) − δΩ,0 (t) + it. 2

Since 0 ∈ / Ω and it ∈ Ω for all t > 0, limt→0+ σ (t) = 0. Then σ ((0, ∞)) divides Ω into the connected domains U + = {x + i y ∈ Ω : x > Re σ (y)}

17.3 Convergence to the Denjoy-Wolff Point for Non-Elliptic Semigroups

and

499

U − = {x + i y ∈ Ω : x < Re σ (y)}.

Hence, Γ := h −1 (σ (0, +∞)) divides D into two connected components D + := h (U + ) and D − := h −1 (U − ). Also, there exists τ˜ ∈ ∂D, τ˜ = τ such that limt→0− h −1 (σ (t)) = τ˜ by Corollary 3.3.4. Since σ : [1, +∞) → Ω is a quasi-geodesic in Ω, by Theorem 6.3.8 and Proposition 6.2.5 it follows that h −1 (σ (t)) converges to τ non-tangentially as t → +∞. This implies that Γ is contained in the set −1

{z ∈ D : |Arg(1 − τ z)| ≤ θ } ∪ {τ, τ˜ } for some θ ∈ (0, π/2). Notice that this set is an angular sector of amplitude 2θ with vertex τ symmetric with respect to segment joining −τ with τ . Since h preserves orientation, it follows that D + contains all the sequences converging tangentially to τ with slope π/2 while D − contains all the sequences converging tangentially to τ with slope −π/2. Therefore, if (17.3.6) holds, then itn ∈ U − for n sufficiently big, hence, φtn (z) ∈ − D eventually and (17.3.7) holds. Conversely, if (17.3.7) holds then φtn (z) ∈ D − eventually, hence, itn ∈ U − eventually and (17.3.6) holds. This concludes the proof of the theorem.

The previous result allows to study the slope of non-elliptic semigroups according to their types (hyperbolic, parabolic of positive hyperbolic step, parabolic of zero hyperbolic step). This will be the aim of the next sections. In this section we content ourselves to state a couple of consequences. The first one follows immediately from part (2) of Theorem 17.3.1: Corollary 17.3.2 Let (φt ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D and Koenigs function h and let Ω := h(D). Then the following are equivalent: (1) limt→+∞ Arg(1 − τ φt (z)) = π/2 (respectively = −π/2) for some—and hence any—z ∈ D, and, in particular, [0, +∞)  t → φt (z) converges tangentially to τ as t → +∞, δ + p (t) δ + p (t) = 0 (resp. limt→+∞ δΩ, = +∞). (2) limt→+∞ δΩ, − − (t) (t) Ω, p

Ω, p

The second consequence states that, in case of non-tangential convergence, the orbits are quasi-geodesics in D: Theorem 17.3.3 Let (φt ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D and Koenigs function h and let Ω := h(D). Then the following are equivalent: (1) for some—and hence any—z ∈ D, the orbit [0, +∞)  t → φt (z) converges non-tangentially to τ as t → +∞,

500

17 Slopes of Orbits at the Denjoy-Wolff Point

(2) for some—and hence any—z ∈ D, the curve [0, +∞)  t → φt (z) is a quasigeodesic in D, (3) for some—and hence any— p ∈ Ω there exist 0 < c < C such that for all t ≥ 0, + − + cδΩ, p (t) ≤ δΩ, p (t) ≤ CδΩ, p (t).

Proof (1) and (3) are equivalent as a consequence of part (1) in Theorem 17.3.1. Moreover, by Theorem 6.3.8 and Proposition 6.2.5, (2) implies (1). In order to end the proof, we show that (3) implies (2). We need to prove that the orbit [0, +∞)  t → φt (z) is a quasi-geodesic in D for every z ∈ D. Since h is an isometry between ω and kΩ , the latter statement is equivalent to proving that, setting p = h(z), the curve β p : [0, +∞)  t → p + it is a quasi-geodesic in Ω. We can assume that (φt ) is not a group, the result being clear otherwise. Hence, up to a translation, we can assume 0 ∈ / Ω and it ∈ Ω for all t > 0. We need to prove that for every p ∈ Ω, there exists A p > 1 and B p > 0 such that for all 0 ≤ s ≤ t, Ω (β p ; [s, t]) ≤ A p kΩ (β p (s), β p (t)) + B p . Fix p ∈ Ω. By Proposition 17.2.14, there exists A ≥ 1 and B ≥ 0 such that Ω (βi ; [s, t]) ≤ AkΩ (βi (s), βi (t)) + B,

(17.3.8)

for all 0 ≤ s ≤ t. Now, for 0 ≤ s ≤ t, kΩ (i + is, i + it) ≤ kΩ ( p + is, i + is) + kΩ ( p + is, p + it) + kΩ (i + it, p + it) ≤ kΩ ( p + is, p + it) + 2kΩ ( p, i), where the last inequality follows from the fact that Ω  z → z + it is a holomorphic self-map of Ω. Therefore, there exists B1 > 0 such that for all s, t ≥ 0, kΩ (i + is, i + it) ≤ kΩ ( p + is, p + it) + B1 .

(17.3.9)

By Lemma 17.1.4 there exists c > 0 such that δΩ (i + it) ≤ cδΩ ( p + it) for all t ≥ 0. Hence, by the Distance Lemma (Theorem 5.3.1), for 0 ≤ s ≤ t, 

t

Ω (β p ; [s, t]) = s



≤c s

κΩ (β p (r ); β p (r ))dr t

dr ≤ 4c δΩ (i + ir )

 ≤ s



t s

t

dr δΩ ( p + ir )

κΩ (βi (r ); βi (r ))dr = 4cΩ (βi ; [s, t]).

17.3 Convergence to the Denjoy-Wolff Point for Non-Elliptic Semigroups

501

Therefore, by (17.3.8) and (17.3.9) Ω (β p ; [s, t]) ≤ 4cΩ (βi ; [s, t]) ≤ 4c AkΩ (i + is, i + it) + 4cB ≤ 4c AkΩ ( p + is, p + it) + 4c AB1 + 4cB, for all 0 ≤ s ≤ t.



17.4 The Slope of Hyperbolic Semigroups In this and the next section, we study the slope of the curve [0, +∞)  t → φt (z) as t → +∞ for non-elliptic semigroups (φt ) in D. We start with a definition: Definition 17.4.1 Let γ : [a, b) → D be a continuous curve, −∞ < a < b ≤ +∞ and assume there exists σ ∈ ∂D such that limt→b γ (t) = σ . The slope of γ at σ , is the cluster set of the real curve [a, b)  t → Arg(1 − σ γ (t)) at t = b. Namely, Slope[γ , σ ] := Γ (Arg(1 − σ γ (t)), b).   In other words, θ ∈ − π2 , π2 belongs to Slope[γ , σ ] if there exists a sequence {tn } ⊂ [a, b) converging to b such that limn→∞ Arg(1 − σ γ (tn )) = θ .   By Lemma 1.9.9, Slope[γ , σ ] is either a point or a closed interval of − π2 , π2 . Lemma 17.4.2 Given σ ∈ ∂D and a sequence {z n } in D that converges to σ, the following assertions are equivalent: (1) there exists α = limn→∞ Arg (1 − σ z n ); 1−σ z n ; (2) there exists m = limn→∞ |1−σ zn |

(3) there exists μ = limn→∞

Im (σ z n ) . 1−Re (σ z n )

Moreover, if one of the above holds, then eiα = m and μ = − tan (α) . Proof Since, tan(Arg(1 − σ z n )) = − The result follows at once.

Im (σ z n ) and 1 − Re (σ z n )

eiArg(1−σ zn ) =

1 − σ zn , |1 − σ z n |



In this section, we focus on the slope of orbits of hyperbolic semigroups. Taking into account Theorem 9.3.5, from Theorem 17.3.3 it follows at once that every orbit of a hyperbolic semigroup converges non-tangentially to the Denjoy-Wolff point (this follows also from Proposition 8.3.7). Moreover, again by Theorem 17.3.3, the orbits are quasi-geodesic in D. However, we can better describe the slopes of orbits of hyperbolic semigroups. We start with a lemma:

502

17 Slopes of Orbits at the Denjoy-Wolff Point

Lemma 17.4.3 Let (φt ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then for every z ∈ D there exists θ (z) ∈ (−π/2, π/2) such that Slope[t → φt (z), τ ] = lim Arg (1 − τ φt (z)) = θ (z). t→+∞

Proof By Lemma 17.4.2, this is equivalent to prove that the function γ : [0, +∞) → Im (τ φt (z)) R given by γ (t) := 1−Re has a limit when t goes to +∞. Take a sequence (τ φt (z)) {tn } in [0, +∞) going to +∞ such that there exists μ := limn→∞ γ (tn ). Write u n := Re (τ φtn (z)) and vn := Im (τ φtn (z)). Then, lim

n→∞

vn = μ, 1 − un

lim u n = 1,

n→∞

lim vn = 0.

n→∞

Let an := Re (τ φtn +1 (z)) and bn := Im (τ φtn +1 (z)). Since φt (z) converges to τ nontangentially, φ1 (φt (z)) − τ 1 − τ φ1 (φt (z)) 1 − (an + ibn ) = lim = lim . t→+∞ t→+∞ n→∞ 1 − (u n + ivn ) φt (z) − τ 1 − τ φt (z)

φ1 (τ ) = lim Therefore

1 − an n→∞ 1 − u n lim

Hence

 1−i

bn 1 − an

1 − an = φ1 (τ ), n→∞ 1 − u n lim



= (1 − iμ)φ1 (τ ).

bn = μφ1 (τ ). n→∞ 1 − u n lim

Take s > t. By Schwarz-Pick’s Lemma (Theorem 1.2.3), applied to the function φs−t , we have      φs+1 (z) − φs (z)   φt+1 (z) − φt (z)  ≤    1 − φ (z)φ (z)   1 − φ (z)φ (z)  . s+1 s t+1 t This implies that the function f : [0, +∞) → [0, 1) given by    φt+1 (z) − φt (z)   f (t) :=  1 − φt+1 (z)φt (z)  is non-increasing and there exists l := limt→+∞ f (t) ∈ [0, 1). Since |an + ibn − (u n + ivn )| , f (tn ) =    1 − (an + ibn )(u n + ivn )

17.4 The Slope of Hyperbolic Semigroups

503

we conclude that     1 − 1 − an + i bn − i vn   1 − un 1 − un 1 − un    1 − an vn bn vn = f (tn ) 1 + u n − ian + iu n − bn 1−u 1−u 1−u 1−u

  . 

n

n

n

n

Taking limits, we have     1 − φ  (τ ) + iμφ  (τ ) − iμ = l 1 + φ  (τ ) − iμ + iμφ  (τ ) . 1

Thus

1

1

1

     (φ (τ ) − 1)(1 − iμ) = l 1 + φ  (τ ) + iμ(φ  (τ ) − 1) . 1

1

1

Therefore   (φ1 (τ ) − 1)2 (1 + μ2 ) = l 2 (1 + φ1 (τ ))2 + μ2 (φ1 (τ ) − 1)2 and μ = 2

1 1−l 2

  2  2 1+φ1 (τ ) l 1−φ  (τ ) − 1 . This shows that the cluster set of the curve γ 1

has at most two points. Since the cluster set is connected (see Lemma 1.9.9), it has to be a singleton.

Theorem 17.4.4 Let (φt ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Let θ (z) := Slope[t → φt (z), τ ] ∈ (−π/2, π/2). Then θ : D → (−π/2, π/2) is a surjective harmonic function whose level sets are the maximal invariant curves of (φt ). Namely, θ (z 1 ) = θ (z 2 ) if and only if either z 2 = φt (z 1 ) or z 1 = φt (z 2 ) for some t ≥ 0. Furthermore, if {z n } ⊂ D is such that limn→∞ Arg(1 − τ z n ) = β ∈ (−π/2, π/2) then lim θ (z n ) = β. n→∞

Moreover, if h is the Koenigs function of (φt ) and λ > 0 is the spectral value of (φt ), then for all z ∈ D, π θ (z) + . Re h(z) = λ 2λ Proof Let (S πλ , h, z → z + it) be the canonical model of (φt ) and let θ (z) be the function given by Lemma 17.4.3 such that θ (z) = lim Arg(1 − τ φt (z)). t→+∞

Step 1. Let β ∈ (−π/2, π/2). Then there exists z 0 ∈ D such that θ (z 0 ) = β. In particular, θ : D → (−π/2, π/2) is surjective.

504

17 Slopes of Orbits at the Denjoy-Wolff Point

Let {z n } ⊂ D be such that limt→+∞ Arg(1 − τ z n ) = β. Since h(D) ⊂ Sπ/λ , up to extracting subsequences, we can assume that limn→∞ Re h(z n ) = x ∈ [0, π/λ]. We claim that, in fact, x ∈ (0, πλ ). Assume this is not the case. By Proposition 1.9.12, there exists M > 0 such that for all n ∈ N, ω(φ1 (z n ), z n ) < M. Therefore, by the Distance Lemma (see Theorem 5.3.1), for all n, M ≥ ω(z n , φ1 (z n )) = kh(D) (h(z n ), h(z n ) + i)

|h(z n ) − h(z n ) − i| 1   ≥ log 1 + 4 min δh(D) (h(z n )) , δh(D) (h(z n ) + i)   1 1 1 1 = log 1 + ≥ log 1 + . 4 δh(D) (h(z n )) 4 δSπ/λ (h(z n )) But, since we are assuming that Re h(z n ) converges either to 0 or to π/λ, it follows that δSπ/λ (h(z n )) = δSπ/λ (Re h(z n )) → 0 as n → +∞, contradicting the previous inequality. Therefore, x ∈ (0, π/λ). Let yn := Im h(z n ). By Proposition 9.4.8, yn → +∞, and we can assume, passing to a subsequence if necessary, that {yn } is strictly increasing. Since by definition of canonical model, for every compact set K ⊂ Sπ/λ there exists t0 such that K + it0 ⊂ h(D), we can find 0 < r1 < r2 < π/λ, n 0 ∈ N, M ∈ R such that TM,r1 ,r2 := {z ∈ C : r1 < Re z < r2 , Im z > M} ⊂ h(D), x + i yn 0 ∈ TM,r1 ,r2 and h(z n ) ∈ TM,r1 ,r2 for all n ≥ n 0 . Fix c > 1. Let S r1 ,r2 := {z ∈ C : r1 < Re z < r2 }. By Proposition 6.8.5, there exists n 1 ≥ n 0 such that for all n ≥ n1, kh(D) (h(z n ), x + i yn ) ≤ k TM,r1 ,r2 (h(z n ), x + i yn ) ≤ ck Sr1 ,r2 (h(z n ), x + i yn ). Since z → z − i yn is an automorphism of S r1 ,r2 , it follows that lim kh(D) (h(z n ), x + i yn ) ≤ c lim k Sr1 ,r2 (h(z n ), x + i yn )

n→∞

n→∞

= c lim k Sr1 ,r2 (Re h(z n ), x) = 0. n→∞

Let {tn } be a sequence of non-negative real numbers converging to +∞ such that yn 0 + tn = yn for n ≥ n 0 . Let w0 := x + i yn 0 and z 0 = h −1 (w0 ). Hence, by the previous equation, ω(z n , φtn (z 0 )) = kh(D) (h(z n ), h(z 0 ) + itn ) = kh(D) (h(z n ), x + i yn ) → 0, n → ∞.

17.4 The Slope of Hyperbolic Semigroups

505

Step 1 follows then from Lemmas 17.4.3 and 1.8.6. Step 2 For all t ≥ 0, lim ω(φt+s (z), φs (z)) = kS π (Re h(z) + it, Re h(z)). λ

s→+∞

(17.4.1)

Fix t ≥ 0 and c > 1. Note that Re h(φt+s (z)) = Re h(z) = Re h(φs (z)) for all s ≥ 0, and |Im h(φt+s (z)) − Im h(φs (z))| = |Im (h(z) + i(t + s)) − Im (h(z) + is)| = t. Therefore, Proposition 6.8.2 implies that there exists r ∈ (0, π/λ) such that for all s ≥ 0, kSr (h(φt+s (z)), h(φs (z))) ≤ ckS π (h(φt+s (z)), h(φs (z))). λ

On the other hand, arguing as in Step 1, (and possibly taking r larger, but still in (0, π/λ)) there exists s0 ≥ 0 such that for all s ≥ s0 kh(D) (h(φt+s (z)), h(φs (z))) ≤ ckSr (h(φt+s (z)), h(φs (z))). The last two inequalities, and since h(D) ⊂ S πλ , imply kS π (h(φt+s (z)), h(φs (z))) ≤ kh(D) (h(φt+s (z)), h(φs (z))) λ

≤ c2 kS π (h(φt+s (z)), h(φs (z))).

(17.4.2)

λ

Since w → w + i(s − Im h(z)) is an automorphism of S πλ , kS π (h(φt+s (z)), h(φs (z))) = kS π (h(z) + i(t + s), h(z) + is) λ

λ

= kS π (Re h(z) + it, Re h(z)). λ

Hence, (17.4.2) and the arbitrariness of c > 1 imply lim ω(φt+s (z), φs (z)) = lim kh(D) (h(φt+s (z)), h(φs (z)))

s→+∞

s→+∞

= kS π (Re h(z) + it, Re h(z)), λ

and Step 2 follows. Step 3 kS π (Re h(z) + i, Re h(z)) = kS π ( λ

λ

π θ (z) π θ (z) + + i, + ). λ 2λ λ 2λ

(17.4.3)

By Theorem 9.3.5, αφ1 (τ ) = e−λ . Hence, the result follows at once from (17.4.1), Lemma 17.4.3 and Proposition 6.7.3.

506

17 Slopes of Orbits at the Denjoy-Wolff Point

Step 4. θ (z 1 ) = θ (z 2 ) if and only if z 1 , z 2 belong to the same maximal invariant curve of (φt ) and, for all z ∈ D, π θ (z) + . (17.4.4) Re h(z) = λ 2λ π By (17.4.3) and Remark 6.7.4 it follows that either Re h(z) = θ(z) + 2λ or Re h(z) = λ θ(z) π − λ + 2λ . Let a ∈ [−∞, 0) and let γ : (a, +∞) → D be a maximal invariant curve for (φt ). Note that, by Proposition 13.4.8 and Remark 13.4.5 the starting point of γ is in ∂D \ {τ }. Moreover, by Remark 13.3.6 there exists b ∈ (0, π/λ) such that

h(γ (a, +∞)) = {w ∈ h(D) : Re w = b}. Hence, since γ is continuous, the previous consideration shows that θ (γ (t)) is constant for every t ∈ (a, +∞). Let γ0 : (a0 , +∞) → D be the maximal invariant curve such that θ (γ0 (t)) = 0 for π for all t ∈ (a0 , +∞). some—hence any—t ∈ (a0 , +∞). Note that Re h(γ0 (t)) = 2λ Moreover, γ0 ((a, +∞)) divides D into two connected components, say D + , D − in such a way that if {z n } ⊂ D converges to τ and limn→∞ Arg(1 − τ z n ) = β ∈ (−π/2, π/2), then {z n } is eventually contained in D + if β > 0 and eventually contained in D − if β < 0. π ) and let γb : (ab , +∞) → D be the maximal invariant curve of (φt ) Let b ∈ (0, 2λ such that Re h(γb (t)) = b for all t ∈ (ab , +∞). In particular, this implies that θb := θ (γb (t)) = 0. Since h preserves orientation, it follows at once that γb ((ab , +∞)) ⊂ D − and θb < 0. That is, for all t ∈ (ab , +∞), Re h(γb (t)) =

π θ (γb (t)) + . λ 2λ

π π A similar argument works for b ∈ ( 2λ , λ ). Hence, since every z ∈ D belongs to a maximal invariant curve of (φt ), Step 4 follows. Step 1 and Step 4 imply that θ : D → (−π/2, π/2) is surjective and its level sets are maximal invariant curves for (φt ). Moreover, (17.4.4) implies that θ is harmonic. In order to conclude the proof we are left to show that if {z n } ⊂ D is such that limn→∞ Arg(1 − τ z n ) = β ∈ (−π/2, π/2) then

lim θ (z n ) = β.

n→∞

Up to extracting subsequences, we can assume that limn→∞ θ (z n ) exists. Again up to extracting subsequences, we can also assume that there exists x ∈ [0, π/λ] such that Re h(z n ) → x. Repeating the argument in Step 1, we see that x ∈ (0, π/λ). Let yn := Im h(z n ), and recall that yn → +∞ as n → ∞. Thus, there exists n 0 such that wn := Re h(z n ) + i yn 0 ∈ h(D) for all n ≥ n 0 and w0 := x + i yn 0 ∈ h(D). Let z 0 := h −1 (w0 ) and ζn := h −1 (wn ). In Step 1 we showed that θ (z 0 ) = β. Since

17.4 The Slope of Hyperbolic Semigroups

507

ζn → z 0 as n → ∞, and θ is continuous, limn→∞ θ (ζn ) = β. Now, (17.4.4) implies that for all n ≥ n 0 , π π π = λRe wn − = λRe h(z n ) − = θ (z n ), 2 2 2

θ (ζn ) = λRe h(ζn ) −

hence, limn→∞ θ (z n ) = β, and we are done.



As a direct corollary of Theorem 17.4.4, we have the following result which says that the Koening function of a hyperbolic semigroup is semi-conformal at the Denjoy-Wolff point (cfr. Theorem 13.6.6): Proposition 17.4.5 Let (φt ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D, spectral value λ > 0 and Koenigs function h. If {z n } ⊂ D is a sequence converging to τ with an angle β = limn→∞ Arg(1 − τ z n ) ∈ (−π/2, π/2), then lim Re h(z n ) =

n→∞

π β + . λ 2λ

Remark 17.4.6 By Proposition 6.7.3 and a direct computation of the hyperbolic distance in S πλ , it follows that if (φt ) is a hyperbolic semigroup of D with spectral value λ > 0 then for every z ∈ D  sin(2θ (z)) = 4 sinh(λ/2)

2

1 + e2s(z) 1 − e2s(z)

2 + 2 sinh(λ)

where s(z) := s1 (φt , z) > 0. Theorem 17.4.4 has an interesting consequence in terms of characterization of hyperbolic semigroups via the slope of orbits: Corollary 17.4.7 Let (φt ) and (φ˜ t ) be two hyperbolic semigroups in D with DenjoyWolff point τ ∈ ∂D and let λ > 0 (respectively, λ˜ > 0) be the spectral value of (φt ) (resp. (φ˜ t )). Let α := λ˜ /λ. Then the following are equivalent: (1) for every z ∈ D lim Arg(1 − τ φt (z)) = lim Arg(1 − τ φ˜ t (z)).

t→+∞

t→+∞

(2) φ˜ t (z) = φαt (z) for all z ∈ D and t ≥ 0. In particular, if (1) holds and λ = λ˜ then (φ˜ t ) = (φt ). Proof Let h be the Koenigs function and let G be the infinitesimal generator of (φt ) ˜ G˜ be the corresponding functions for (φ˜ t ). and let h, By Theorem 17.4.4, (1) is equivalent to 1 ˜ = 0, Re ( h(z) − h(z)) α

508

17 Slopes of Orbits at the Denjoy-Wolff Point

for all z ∈ D. Hence, by Theorem 10.1.4, (1) is equivalent to ˜ G(z) = αG(z), for all z ∈ D. Since φ0 (z) = z for all z ∈ D and ∂φαt (z) ˜ αt (z)), = αG(φαt (z)) = G(φ ∂t it follows by the uniqueness of solutions to ordinary differential equations that (1) is equivalent to (2).

Remark 17.4.8 Let (φt ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D and canonical model (Sπ/λ , h, z → z + it). The function g(z) = −ieiλh(z) , z ∈ D, has non-negative real part and ∠ lim z→τ g(z) = 0. By Proposition 2.1.3, there exists ∠ lim z→τ

1 − τz ∈ [0, +∞), g(z)

g(z) so that ∠ lim z→τ z−τ ∈ C∞ \ {0}. Thus, Proposition 2.3.4 (notice that limr →1 Im g(r τ ) = 0) and Theorem 1.7.2 show that the following are equivalent

(1) (2) (3) (4)

g is conformal at τ , that is, there exists g  (τ ) := ∠ lim z→τ ∠ lim z→τ g  (z) ∈ C; g(r τ ) ∈ R; limr →1 Re1−r g(r τ ) lim inf r →1 Re1−r < +∞.

g(z) z−τ

∈ C \ {0};

Conformality of g at τ can also be characterized in terms of the Koenigs function: Proposition 17.4.9 Let (φt ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D and canonical model (Sπ/λ , h, z → z + it). Let g(z) = −ieiλh(z) , z ∈ D. The following are equivalent (i) g is conformal at τ , that is, there exists g  (τ ) := ∠ lim z→τ (ii) A1 := ∠ lim (iλh(z) − log(1 − τ z)) ∈ C;

g(z) z−τ

∈ C \ {0};

z→τ

(iii) A2 := lim (iλh(r τ ) − log(1 − r )) ∈ C; r →1

(iv) A3 := lim (λIm h(r τ ) + log |1 − r |) ∈ R. r →1

π and A3 = −Re A1 . In such a case, A1 = A2 = log(−g  (τ )τ ) + i 2λ

Proof Given z ∈ D,

g(z) eiλh(z)−iπ/2 = log(1−τ z) = e f (z) , 1 − τz e

17.4 The Slope of Hyperbolic Semigroups

509

where f (z) := iλh(z) − iπ/2 − log(1 − τ z). Theorem 17.4.4 implies   π ∠ lim Im f (z) = ∠ lim λRe h(z) − − Arg(1 − τ z) = 0. z→τ z→τ 2 g(z) Assume (i) holds. Then β := ∠ lim z→τ 1−τ = ∠ lim z→τ e f (z) ∈ (0, +∞). Therez fore, β = ∠ lim z→τ |e f (z) | = ∠ lim z→τ eRe f (z) . Hence,

∠ lim f (z) = log β + i0 z→τ

and

π π = log β + i 2 2 π  = log(−g (τ )τ ) + i . 2

∠ lim (iλh(z) − log(1 − τ z)) = ∠ lim f (z) + i z→τ

z→τ

That is, (i) implies (ii). Clearly (ii) implies (iii). Since by Theorem 17.4.4 we have that limr →1 Re h(r τ ) = π/(2λ) ∈ R, it follows that (iii) is equivalent to (iv). Assume that (iii) holds. Then lim

r →1

g(r τ ) = lim e f (r τ ) ∈ C \ {0}. r →1 1−r

Thus, Remark 17.4.8 shows that g is conformal at τ .



Remark 17.4.10 Let (φt ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D and canonical model (Sπ/λ , h, z → z + it). As before, let g(z) = −ieiλh(z) , z ∈ D. Let G(z) = (z − τ )(τ z − 1) p(z), with Re p ≥ 0, be the infinitesimal generator of (φt ). Then 

g(z) log 1 − τz



1 1 1 g  (z) + = −λ + g(z) τ −z G(z) τ − z   τ τ p(z) − λ . =− (τ z − 1) p(z) τ −z

=

Notice that ∠ lim z→τ (τ z − 1) p(z) = ∠ lim z→τ G(z) = −λ = 0. Therefore, the conz−τ formality of g at τ is equivalent to the existence of the finite angular limit 

z

∠ lim z→τ

0

  τ τ p(z) − λ dζ. (τ z − 1) p(z) τ −z

510

17 Slopes of Orbits at the Denjoy-Wolff Point

17.5 The Slope of Parabolic Semigroups In case of parabolic semigroups, the slope does not depend on the initial point of the orbit: Theorem 17.5.1 Let (φt ) be a parabolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then, for every z 1 , z 2 ∈ D, Slope[t → φt (z 1 ), τ ] = Slope[t → φt (z 2 ), τ ]. If, in addition, the semigroup is of positive hyperbolic step then • Slope[t → φt (z), τ ] = {−π/2} for every z ∈ D if and only if the canonical model is (H, h, z → z + it), • Slope[t → φt (z), τ ] = {π/2} for every z ∈ D if and only if the canonical model is (H− , h, z → z + it). In particular, the orbits of every parabolic semigroup of positive hyperbolic step converge tangentially to τ . Proof If (φt ) is parabolic of positive hyperbolic step, its canonical model is either (H, h, z → z + it) or (H− , h, z → z + it). In the first case, for all p ∈ h(D) there − + exists C p > 0 such that δh(D), p (t) ≤ C p for all t < 0, while δh(D), p (t) → +∞ as t → +∞. Therefore, according to Corollary 17.3.2, lim Arg(1 − τ φt (z)) = −

t→+∞

π 2

for all z ∈ D. Similarly for the second case. If (φt ) is parabolic of zero hyperbolic step, let z 1 , z 2 ∈ D and suppose {tn } is a sequence of positive real numbers converging to +∞ such that limn→∞ Arg(1 − τ φtn (z 1 )) = θ ∈ [−π/2, π/2]. Since (φt ) is of zero hyperbolic step, by Corollary 9.3.8, lim ω(φtn (z 1 ), φtn (z 2 )) = 0. n→∞

Therefore, by Lemma 1.8.6, limn→∞ Arg(1 − τ φtn (z 2 )) = θ .



In the case of a parabolic semigroups of zero hyperbolic step, the slope (which according to the previous theorem does not depend on z ∈ D) can be either a single point or a closed interval in [−π/2, π/2], and the convergence of the orbits to the Denjoy-Wolff point can be both non-tangential and tangential, as the following examples show. Example 17.5.2 Let Ω1 := {ζ ∈ C : Im (ζ ) > (Re (ζ ))2 } (see Fig. 17.2). Note that Ω1 is starlike at infinity. Let h : D → C be a Riemann map of Ω1 , and φt (z) := h −1 (h(z) + it), z ∈ D. Since

17.5 The Slope of Parabolic Semigroups

511

Fig. 17.2 The set Ω1

 (Ω1 + it) = C, t≤0

it follows that (C, h, z → z + it) is the canonical model of (φt ). Namely, (φt ) is a parabolic semigroup of zero hyperbolic step. Note that Ω1 is symmetric with respect to the imaginary axis, hence, by Proposition 6.1.3 γ : [0, +∞)  t → (1 + t)i is a geodesic in Ω1 . It follows then from Theorem 6.4.5 (with U = Ω = h(D)) that h −1 (γ (t)) = φt (h −1 (i)) converges orthogonally to τ . Hence, by Theorem 17.5.1, Slope[t → φt (z), τ ] = {0} for all z ∈ D. Example 17.5.3 Let Ω2 (see Fig. 17.3) be defined by Ω2 := {ζ ∈ C : Re (ζ ) > 0} ∪ {ζ ∈ C : Im (ζ ) > (Re (ζ ))2 }. Note that Ω2 is starlike at infinity. Let h : D → C be a Riemann map of Ω2 , and φt (z) := h −1 (h(z) + it), z ∈ D. As in the previous example, (φt ) is a parabolic semigroup of zero hyperbolic step.  1 + − For every t > 4, δΩ2 ,0 (t) = t and δΩ2 ,0 (t) = t − . Hence 4 lim

t→∞

+ δΩ (t) 2 ,0 − δΩ (t) 2 ,0

= +∞.

512

17 Slopes of Orbits at the Denjoy-Wolff Point

Fig. 17.3 The set Ω2

It follows from Theorem 17.3.1 that Slope[t → φt (z), τ ] = {−π/2} for all z ∈ D. In particular, the orbits of the semigroup φt (z) converge tangentially to the DenjoyWolff point τ ∈ ∂D. Example 17.5.4 Let Ω3 (see Fig. 17.4) be defined by Ω3 := Ω2 ∪n≥1 Sn , where for every n ≥ 1, Sn is the vertical strip Sn := {ζ ∈ C : an < Re (ζ ) < bn < 0}, and the sequences (an ) and (bn ) are constructed inductively asfollows. − First consider t1 > 4. Then δΩ (t ) = |it1 − ζ1 | = t1 − 14 with ζ1 = 2 ,0 1  − t1 − 21 + i(t1 − 21 ).  Let b1 = −1 − t1 − 21 and η1 := b1 + ib12 . Select the unique s1 > t1 such that

|is1 − η1 | = 21 s1 . We can now choose a1 < b1 such that for every ζ ∈ {z ∈ C : Re (z) ≤ a1 , Im (z) = (Re (z))2 }, we have |is1 − ζ | > 21 s1 . In particular we will have 1 1 + − (s1 ) = |is1 − η1 | = s1 = δΩ (s1 ). δΩ 3 ,0 2 2 3 ,0 − Now, we may choose t2 > s1 and ζ2 ∈ ∂Ω2 , with Re (ζ2 ) < a1 , such that δΩ (t ) 2 ,0 2   1 1 2 = |it2 − ζ2 | = t2 − 4 . Then let b2 = −1 − t2 − 2 and η2 := b2 + ib2 . From b2

17.5 The Slope of Parabolic Semigroups

513

Fig. 17.4 The set Ω3

and ζ2 , we construct a2 < b2 and s2 , exactly as we constructed a1 and b1 from b1 and ζ1 . In particular we will have − (s2 ) = |is2 − η2 | = δΩ 3 ,0

1 1 + s 2 = δΩ (s2 ). 2 2 3 ,0

The construction of sequences (an ) and (bn ) is completed by induction. By construction, limn→∞ an = limn→∞ bn = −∞ and limn→∞ tn = limn→∞ sn = +∞. Let h : D → C be a Riemann map of Ω3 , and φt (z) := h −1 (h(z) + it), z ∈ D. Then, (φt ) is a parabolic semigroup of zero hyperbolic step.

514

17 Slopes of Orbits at the Denjoy-Wolff Point

+ − Note that, for every t ≥ 0, δΩ,0 (t) ≥ δΩ,0 (t), which, according to Theorem 17.3.1, means that there are no subsequences of any orbit of (φt ) converging to τ with slope π/2. On the other hand, for every n ≥ 1, we have: + (t ) δΩ 3 ,0 n

− δΩ (t ) 3 ,0 n

=

tn tn −

1 4

→ +∞

as n → ∞,

which, again by Theorem 17.3.1, implies that φtn (z) → τ with slope −π/2. Finally, since − (sn ) δΩ sn /2 1 3 ,0 = = , + s 2 δΩ (s ) n n 3 ,0 for every z ∈ D the sequence {φsn (z)} converges non-tangentially to τ . In particular, the slope of (φt ) is Slope[t → φt (z), τ ] = [−π/2, α] for some −π/2 < α < π/2. The previous example can be easily adapted to construct parabolic semigroups of zero hyperbolic step whose slope is [a, π/2] for a ∈ (−π/2, π/2) and [−π/2, π/2]. With an argument based on harmonic measure theory, in the next section we show that for every −π/2 ≤ a < b ≤ π/2 one can construct examples of parabolic semigroups whose slope is [a, b]. Now we prove that some regularity of the infinitesimal generator of a parabolic semigroup at the Denjoy-Wolff point detects the slope of convergence of the orbits. Let G be the infinitesimal generator of a parabolic semigroup (φt ) with DenjoyWolff point τ ∈ ∂D. As customary, we say that G extends C 2 -smooth at τ if there exists a C 2 -smooth function G˜ defined in a neighborhood U of τ such that G(z) = ˜ G(z) for all z ∈ U ∩ D. In particular, by Corollary 10.1.12, it follows that if G extends C 2 -smooth at τ , then G(τ ) = G  (τ ) = 0. Proposition 17.5.5 Let (φt ) be a parabolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D and infinitesimal generator G. Suppose G extends C 2 -smooth at τ and G  (τ ) = 0. Then Arg(τ G  (τ )) ∈ [−π/2, π/2] and Slope[t → φt (z), τ ] = {−Arg(τ G  (τ ))}. In particular, the orbits of (φt ) converge tangentially to τ if and only if τ G  (τ ) is pure imaginary. Proof Let μ = G  (τ ). Let h be the Koenigs function of (φt ). By Proposition 16.3.7, lim h(z)(1 − τ z) =

z→τ

τi . μ

Hence, by Corollary 16.3.8, Arg(τ μ) ∈ [−π/2, π/2] and lim t (1 − τ φt (z)) =

t→+∞

1 , τμ

17.5 The Slope of Parabolic Semigroups

515

from which it follows immediately that limt→+∞ Arg(1 − τ φt (z)) = Arg(1/(τ μ)), and we are done.

We end this section with some sufficient conditions in terms of the image of the Koenigs function forcing the orbits to converge orthogonally to the Denjoy-Wolff point. For β ∈ (0, π ), we let V (β) := V (β, 0) := {ρeiθ : ρ > 0, |θ | < β}, the horizontal sector of angle 2β. Proposition 17.5.6 Let (φt ) be a parabolic semigroup in D of zero hyperbolic step, with Koenigs function h and Denjoy-Wolff point τ ∈ ∂D. If (1) either there exists a > 0 such that (iH + ia) ⊂ h(D) ⊂ iH, (2) or, there exist −∞ < a < b < +∞ and c ∈ R such that ∂(h(D)) is contained in the semistrip {ζ ∈ C : a < Re ζ < b, Im ζ < c}, then Slope[t → φt (z), τ ] = {0} (i.e., the orbits of (φt ) converge orthogonally to τ ) for every z ∈ D. Proof (1) follows directly from Lemma 6.4.1. (2) since h(D) is starlike at infinity and different from C, there exists p ∈ ∂(h(D)) such that h(D) ⊂ K p , where K p = C \ {ζ ∈ C : Re ζ = Re p, Im ζ ≤ Im p} is a Koebe domain. Moreover, by (6.4.6), it follows that there exists R > 0 such that K

E p+ip (∞, R) ⊂ h(D). Since (0, +∞)  t → p + it is a geodesic in K p which converges to ∞ in the Carathéodory topology of K p (see Example 6.4.7), (2) follows at once from Theorem 6.4.5.



17.6 Parabolic Semigroups of Zero Hyperbolic Step with Prescribed Slopes The aim of this section is to show that for every −π/2 ≤ a < b ≤ π/2 there exists a parabolic semigroup of zero hyperbolic step whose slope is [a, b]. The construction of such examples depends on a characterization of the slope in terms of the values of a certain harmonic measure evaluated along the trajectory and the existence of suitable starlike domains with prescribed asymptotic behavior of such harmonic measure.

516

17 Slopes of Orbits at the Denjoy-Wolff Point

We start by characterizing the slope in terms of the values of a certain harmonic measure evaluated along the trajectory. Proposition 17.6.1 Let (φt ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D. Let ξ = τ and let J ⊂ ∂D be the open arc which goes counterclockwise from ξ to τ . Let a1 = lim inf t→+∞ μ(φt (z), J, D) and a2 = lim supt→+∞ μ(φt (z), J, D). Then for every z ∈ D, 1 1 Slope[t → φt (z), τ ] = [π(a1 − ), π(a2 − )]. 2 2

(17.6.1)

Proof This result is a straightforward consequence of the equivalence between (1) and (3) in Proposition 7.2.7 (with σ = τ ) and the very definition of Slope[t → φt (z), τ ]. Indeed, if {tn } converges to +∞ and limn→∞ Arg(1 − σ φtn (z)) = θ , by Proposition 7.2.7, limn→∞ μ(φtn (z), J, D) = 21 + πθ . Thus a1 ≤

θ 1 + ≤ a2 . 2 π

Therefore, Slope[t → φt (z), τ ] ⊂ [π(a1 − 1/2), π(a2 − 1/2)]. Conversely, if {tn } converges to +∞ and a2 = limn→∞ μ(φtn (z), J, D), then limn→∞ Arg(1 − σ φtn (z)) = π(a2 − 1/2) (again by Proposition 7.2.7). Therefore, π(a2 − 1/2) ∈ Slope[t → φt (z), τ ]. Similarly, π(a1 − 1/2) ∈ Slope[t → φt (z), τ ]. Since Slope[t → φt (z), τ ] is closed and arcwise connected, it follows that [π(a1 − 1/2), π(a2 − 1/2)] ⊂

Slope[t → φt (z), τ ]. The above result can be re-written in terms of the Koenigs function. Given a domain Ω and w ∈ C, we let ∂w+ Ω := ∂Ω ∩ {z ∈ C : Re z > Re w}. Recall that, if h : D → C is the Koenigs function of a non-elliptic semigroup then for every σ ∈ ∂D, the non-tangential limit h(σ ) := ∠ lim z→σ h(z) exists by Corollary 11.1.7. Moreover, given z ∈ D, + + h(D)) := {σ ∈ ∂D : h(σ ) ∈ ∂h(z) h(D)}. h −1 (∂h(z) + + In particular, h −1 (∂h(z) h(D)) is a Borel set, since ∂h(z) h(D) is open in ∂∞ h(D).

Lemma 17.6.2 Let h : D → C be the Koenigs function of a non-elliptic semigroup + + (φt ) in D. Fix z ∈ D and assume ∂h(z) h(D) = ∅ and ∂h(z) h(D) = ∂h(D). Then there exist an open arc J ⊂ ∂D going counterclockwise from a point ξ ∈ ∂D \ {τ } to τ + h(D)) = J \ Z . and a set of zero Lebesgue measure Z ⊂ J such that h −1 (∂h(z) Proof Let L := {w ∈ h(D) : Re w = Re h(z)}. Note that, since h(D) is starlike at infinity, there exists s ∈ [−∞, Im h(z)) such that

17.6 Parabolic Semigroups of Zero Hyperbolic Step with Prescribed Slopes

L := {w ∈ C : Re w = Re h(z), s < Im w}.

517

(17.6.2)

By Remark 13.3.6, h −1 (L) is (the image of) a maximal invariant curve for (φt ). Let η ∈ ∂D be its starting point. If η = τ , by Proposition 13.3.5, there exists a maximal invariant curve γ : (−∞, +∞) → D such that γ ((−∞, +∞)) = h −1 (L). Since limt→−∞ γ (t) = η = τ , it follows from Proposition 13.4.6 that (φt ) has a parabolic petal Δ such that h −1 (L) ⊂ Δ. By Theorem 13.5.7, H := h(Δ) is a maximal half-plane. Since L ⊂ H , it follows that either H := {w ∈ C : Re w > a} ⊂ h(D) for some a ≤ Re h(z) or H := {w ∈ C : Re w < a} ⊂ h(D) for some a ≥ Re h(z). In the first case, + + h(D) = ∅, while, in the second case, ∂h(z) h(D) = ∂h(D). Hence, both cases ∂h(z) lead to a contradiction, and therefore ξ = τ . Therefore, h −1 (L) is (the image of) a maximal invariant curve starting from η = τ and ending at τ . The closure h −1 (L) is a cross cut in D and by Lemma 4.1.3 divides D into two connected components, say U + , U − , such that ∂U + = h −1 (L) ∪ J + and ∂U − = h −1 (L) ∪ J − , where J + ⊂ ∂D is the open arc which goes clockwise from η to τ and J − ⊂ ∂D is the open arc which goes counterclockwise from η to τ . Since h preserves orientation, h(U − ) = {w ∈ h(D) : Re w > Re h(z)}, h(U + ) = {w ∈ h(D) : Re w < Re h(z)}.

(17.6.3)

Now, by (17.6.3), if σ ∈ J + , then either h(σ ) = ∞ or h(σ ) ∈ C and Re h(σ ) ≤ + h(D). In other words, Re h(z). In both cases, h(σ ) ∈ / ∂h(z) + h(D)) ⊆ J − . h −1 (∂h(z)

(17.6.4)

Let s ∈ [−∞, Im h(z)) be given by (17.6.2). Hence, lim h(h −1 (h(z) + it)) = Re h(z) + is,

t→s +

where, if s = −∞, with a slight abuse of notations, we let Re h(z) + is = ∞. Since (s, +∞)  t → h −1 (h(z) + it) is a maximal invariant curve starting at η, it follows by Lehto-Virtanen’s Theorem 3.3.1 that h(η) = Re h(z) + is. There are three cases: Case 1: s = −∞. In this case, {w ∈ C : Re w = Re h(z)} ⊂ h(D). Let Z := {σ ∈ J − : h(σ ) = ∞}. By Proposition 3.3.2, Z has zero Lebesgue measure. If σ ∈ J − \Z , by (17.6.3), Re h(σ ) ≥ Re h(z), but, since every point of type Re h(z) + it, t ∈ (−∞, +∞) is contained in h(D), it follows that, in fact, Re h(σ ) > Re h(z). + + h(D). Since h(Z ) = ∞ ∈ / ∂h(z) h(D), and bearing in mind Namely, h(J − \ Z ) = ∂h(z) − (17.6.4), the statement holds in this case taking J = J and ξ = η. Case 2: s > −∞ and η is not contained in the closure of a maximal contact arc. Let σ ∈ J − and assume h(σ ) ∈ C. We claim that Re h(σ ) > Re h(z). Indeed, assume

518

17 Slopes of Orbits at the Denjoy-Wolff Point

by contradiction that Re h(σ ) = Re h(z). Note that since h(D) is starlike at infinity, s < Im h(σ ). Let t0 ∈ (s, Im h(z)) be such that |h −1 (Re h(z) + t0 i) − σ | = min{|ζ − σ |, ζ ∈ h −1 (L)}. Let Γ be union of h −1 (Re h(z) + i(s, t0 ]) and the closed segment joining h −1 (Re h(z) + t0 i) with σ . By construction, Γ is a cross cut for D with end points η, σ and Γ ⊂ U − . Hence, Γ divides D into two connected components, one of them, say A, has the property that A ⊂ U − and ∂ A ∩ ∂D is the closed arc I ⊂ J − which goes counterclockwise from η to σ . Let T := {w ∈ C : Re w = Re h(z), Im w = t, t ∈ [t0 , Im h(σ )]} ∪ h(Γ ). By construction, T is a Jordan curve. Let V be the bounded connected component of C \ V . Since h preserves orientation and by (17.6.3) it follows that h(A) ⊆ V ⊂ {w ∈ C : Re w > Re z}. Let p ∈ I \ {η, σ }. Hence, h( p) ∈ V . If h( p) ∈ V ∪ h(Γ ∩ D)—which implies Re h( p) > Re h(z)—taking into account that h(D) is starlike at infinity, it follows that Re h( p) + it ∈ / h(D) for all t ≤ Im h( p). But then there exists t ≤ Im h( p) such that Re h( p) + it ∈ ∂ V = T , and, since Re h( p) > Re h(z), the only possibility is that Re h( p) + it ∈ h(Γ ∩ D) ⊂ h(D), a contradiction. Therefore, h( p) ∈ {w ∈ C : Re w = Re h(z), Im w = t, t ∈ [Im h(σ ), t0 ]}. Thus, by Schwarz’s Reflection Principle, h extends holomorphically through I \ {η, σ } and h(I \ {η, σ }) is a segment in {w ∈ C : Re w = Re h(z)}. By Theorem 14.2.10, I \ {η, σ } is contained in a maximal contact arc. Therefore, η is contained in the closure of a maximal contact arc, contradicting our hypothesis. The claim is proved. Now, as before, let Z 1 := {σ ∈ J − : h(σ ) = ∞}, a set of zero measure. By the + h(D). Hence, bearing in mind (17.6.4), the claim we just proved, h(J − \ Z 1 ) ⊆ ∂h(z) statement follows by setting Z := Z 1 ∪ {η} ∪ {τ }, ξ = η and J = J − . Case 3: s > −∞ and η is contained in the closure of a maximal contact arc. Since η is the starting point of a maximal invariant curve, the life-time T (η) = 0 by Proposition 14.2.2, hence it cannot be contained in a maximal contact arc, and, by Proposition 14.2.6, it cannot be either the starting point of a maximal contact arc. Therefore, η is the final point of a maximal contact arc M ⊂ J − . By Corollary 14.2.11 and + h(D), it follows that M cannot be exceptional, that is, the our hypotheses on ∂h(z) initial point of M is a point ξ ∈ J − . Let J ⊂ J − be the open arc which goes counterclockwise from ξ to τ . Note that J = J − \ M. Let Z 1 := {σ ∈ J : h(σ ) = ∞} and let Z := Z 1 ∪ {ξ, τ }. Again, Z has zero Lebesgue measure. We claim that for every σ ∈ J \ Z 1 , we have Re h(σ ) > Re h(z). From this, taking into account (17.6.4), the statement follows in this case as well. The argument to prove the claim is essentially the same used in Case 2. Arguing by contradiction and repeating the construction of Case 2, if σ ∈ J \ Z 1 and Re h(σ ) =

17.6 Parabolic Semigroups of Zero Hyperbolic Step with Prescribed Slopes

519

Re h(z), we come up with a closed arc I ⊂ J − going counterclockwise from η to σ so that for every p ∈ I , Re h( p) = Re h(z). Hence, I is contained in a maximal contact arc. However, ξ ∈ I and ξ cannot be contained in a maximal contact arc, and we have a contradiction.

Corollary 17.6.3 Let (φt ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ + h(D) = ∂D and holomorphic model (Ω, h, z → z + it). Fix z ∈ D and assume ∂h(z) + + ∅ and ∂h(z) h(D) = ∂h(D). Let a1 = lim inf t→+∞ μ(h(z) + it, ∂h(z) h(D), h(D)) and + h(D), h(D)). Then a2 = lim supt→+∞ μ(h(z) + it, ∂h(z) 1 1 Slope[t → φt (z), τ ] = [π(a1 − ), π(a2 − )]. 2 2

(17.6.5)

Proof By definition of harmonic measure, and since h −1 (h(z) + it) = φt (z), + + h(D), h(D)) = μ(φt (z), h −1 (∂h(z) h(D)), D). μ(h(z) + it, ∂h(z)

By Lemma 17.6.2, there exist an open arc J ⊂ ∂D going counterclockwise from a point ξ ∈ ∂D \ {τ } to τ and a set of zero Lebesgue measure Z ⊂ J such that + h(D)) = J \ Z . Since the harmonic measure in D is absolutely continuous h −1 (∂h(z) with respect to the Lebesgue measure, + h(D)), D) = μ(φt (z), J, D). μ(φt (z), h −1 (∂h(z)



Hence the result follows from Proposition 17.6.1.

Thanks to Corollary 17.6.3 we can translate the problem of constructing parabolic semigroups of zero hyperbolic step with prescribed slope to the problem of constructing starlike domains at infinity with prescribed harmonic measures of the “right boundary” along the orbit t → z + it. In order to solve this problem, we need to introduce some notations and a localization lemma for harmonic measures. Given wl , wr > 0, consider the strip S(wl , wr ) := {w ∈ C : −wl < Re (w) < wr }. A simple computation from the definition of harmonic measure and the biholomorr (log 1+z + i π2 ) − wl i, allows to phism f : D → S(wl , wr ) given by f (z) = wl +w π 1−z see that μ(w, iR + wr , S(wl , wr )) =

Re (w) + wl , wr + wl

w ∈ S(wl , wr ).

Take now u ∈ R and v > 0 and define the rectangle R(u, v, wl , wr ) := {w ∈ S(wl , wr ) : u < Im w < u + v}.

(17.6.6)

520

17 Slopes of Orbits at the Denjoy-Wolff Point

We denote by R the family of all such rectangles and call v, wl and wr , respectively, the height, the left width and the right width of R(u, v, wl , wr ). Moreover, we define v c(R(u, v, wl , wr )) := i(u + ) 2

(17.6.7)

and call it the pseudocenter of the rectangle R(u, v, wl , wr ) with respect to wl and wr . Finally, we let ∂V R(u, v, wl , wr ) := {z ∈ C : u ≤ Im z ≤ u + v, Re z ∈ {−wl , wr }}, the “vertical” boundary of the rectangle. Lemma 17.6.4 Let wl , wr > 0. Then for any ε > 0 there exists v0 = v0 (wl , wr , ε) > 0 with the following property. Fix v > v0 , and u ∈ R. Then for every domain Ω ⊂ C starlike at infinity such that (1) R(u, v, wl , wr ) ⊂ Ω, and (2) ∂V R(u, v, wl , wr ) ⊂ ∂Ω,    wl  μ(i(u + v ), ∂ + Ω, Ω) − < ε,  2 wr + wl 

it holds

(17.6.8)

where ∂ + Ω := ∂Ω ∩ H. Proof Up to a translation, using the conformal invariance of the harmonic measure (see Proposition 7.2.3), we can assume that u = 0. Also, for s ≥ 0, we let As = [−wl , wr ] + is and Bs = wr + i[0, s]. Let α(v) =

v μ(y + i , A0 ∪ Av , R(0, v, wl , wr )). 2 y∈(−wl ,wr ) sup

Let f : D → R := R(0, v, wl , wr ) be a biholomorphism. Since R is a Jordan domain, by Theorem 4.3.3, f extends to a homeomorphism, still denoted by f , from D onto R. Let L := A0 ∪ Av ∪ Bv . Let L  := L \ {−wl , −wl + iv}. Note that L  is an open connected subset of ∂ R, hence, f −1 (L  ) is an open arc in ∂D. By Proposition 7.2.3(4), μ(z, L , R) = μ(z, L  , R). Hence, for every x ∈ L  by Proposition 7.1.4(2), lim μ(z, L , R) =

z→x

lim

w→ f −1 (x)

μ(w, f −1 (L  ), D) = 1.

Similarly, lim z→x μ(z, L , R) = 0 for all x ∈ ∂ R \ L. With a similar argument we see that μ(z, A0 ∪ Av , R) → 1 as Im z → 0 or Im z → v, Re z → x ∈ (−wl , wr ) and μ(z, A0 ∪ Av , R) → 0 as z → x ∈ ∂ R \ (A0 ∪ Av ). Also, μ(z, wr + iR, S(wl , wr )) → 1 as Re z → wr and μ(z, wr + iR, S(wl , wr )) → 0 as Re z → −wl . Therefore, the harmonic function

17.6 Parabolic Semigroups of Zero Hyperbolic Step with Prescribed Slopes

521

R  z → θ (z) := μ(z, A0 ∪ Av ∪ Bv , R) − μ(z, iR + wr , S(wl , wr )) − μ(z, A0 ∪ Av , R), has the property that lim z→x θ (z) = −μ(x, iR + wr , S(wl , wr )) < 0 for all x ∈ (A0 ∪ Av ) \ {−wl , wr , wr + iv, −wl + iv}, lim z→x θ (z) = 0 for all x = −wl + is or x = wr + is, s ∈ (0, v). In other words, lim z→x θ (z) ≤ 0 for all x ∈ ∂ R \ {−wl , wr , wr + iv, −wl + iv}. By Lindelöf’s Maximum Principle (see Lemma 7.1.1), θ (z) ≤ 0 for all z ∈ R. Hence, v v 0 ≤ μ(y + i , A0 ∪ Av ∪ Bv , R(0, v, wl , wr )) − μ(y + i , iR + wr , S(wl , wr )) 2 2 v ≤ μ(y + i , A0 ∪ Av , R(0, v, wl , wr )) ≤ α(v). 2 In other words, v v − μ(y + i , iR + wr , S(wl , wr )) ≤ −μ(y + i , A0 ∪ Av ∪ Bv , R(0, v, wl , wr )) + α(v). 2 2

(17.6.9)

Let Σ := {w ∈ S(wl , wr ) : Im w > 0}. A similar argument as before, using Lindelöf’s Maximum Principle, shows v v 0 ≤ μ(y + i , iR + wr , S(wl , wr )) − μ(y + i , Bϕ , Σ) 2 2 v ≤ μ(y + i , A0 ∪ Ah , R(0, v, wl , wr )) ≤ α(v). 2 Therefore, v v μ(y + i , Bv , Σ) − μ(y + i , iR + wr , S(wl , wr )) ≥ −α(v). 2 2

(17.6.10)

Also, since Ω is starlike at infinity, by hypothesis (1), Σ ⊂ Ω and, by hypothesis (2), Bv ⊂ ∂Σ ∩ ∂Ω. Hence, by Proposition 7.2.10, for all z ∈ Σ, μ(z, Bv , Σ) ≤ μ(z, Bv , Ω). Taking into account that Bv ⊂ ∂ + Ω, we have then v v μ(y + i , Bv , Σ) ≤ μ(y + i , ∂ + Ω, Ω). 2 2

(17.6.11)

Putting together (17.6.10) and (17.6.11) we obtain v v μ(y + i , ∂ + Ω, Ω) − μ(y + i , iR + wr , S(wl , wr )) ≥ −α(v). 2 2

(17.6.12)

Now, we claim that for all x = wr + is with s ∈ (0, v), lim μ(z, ∂ + Ω, Ω) = 1.

Rz→x

(17.6.13)

522

17 Slopes of Orbits at the Denjoy-Wolff Point

In order to see this, by hypothesis (1), R ⊂ Ω and, by hypothesis (2), Bv ⊂ ∂ R ∩ ∂Ω. Hence, by Proposition 7.2.10, for all z ∈ R, μ(z, Bv , R) ≤ μ(z, Bv , Ω). Taking into account that Bv ⊂ ∂ + Ω (and hence μ(z, Bv , Ω) ≤ μ(z, ∂ + Ω, Ω) for all z ∈ Ω), we have then 1 = lim μ(z, Bv , R) ≤ lim μ(z, Bv , Ω) ≤ lim μ(z, ∂ + Ω, Ω), z→x

z→x

z→x

and, since μ(z, ∂ + Ω, Ω) ≤ 1, the claim follows. Next, we claim that for all x = −wl + is with s ∈ (0, v), lim μ(z, ∂ + Ω, Ω) = 0.

Ωz→x

(17.6.14)

To prove the claim, let t0 := inf{s ∈ R : is ∈ Ω}. By hypothesis (1), and taking into account that u = 0, t0 ∈ [−∞, 0]. Let H := (Ω ∩ H) ∪ (H− \ {w ∈ C : Re w = −wl , Im w ≤ v}) ∪ i(t0 , +∞). Note that H is a starlike at infinity domain and Ω ⊂ H (since Ω is starlike at infinity as well). Since ∂ + Ω ⊂ ∂ H ∩ ∂Ω, by Proposition 7.2.10, for all z ∈ Ω, 0 ≤ μ(z, ∂ + Ω, Ω) ≤ μ(z, ∂ + Ω, H ). Let K := {w ∈ C : Re w = −wl , Im w ≤ v}. By the previous inequality, it is enough to prove that for every x ∈ K , lim μ(z, ∂ + Ω, H ) = 0.

Ωz→x

(17.6.15)

To this aim, let g : D → H be a biholomorphism. Fix 0 < η < wl . By Proposition 3.3.3, ξ − := limt→−∞ g −1 (it − η) ∈ ∂D and ξ + := limt→+∞ g −1 (it − η) ∈ ∂D exist and ∠ lim z→ξ ± g(z) = ∞. Let Γ := {w ∈ C : Re w = −η} ⊂ H . It follows by the previous consideration that g −1 (Γ ) is a cross-cut for D with end points ξ − and ξ + . The cross-cut g −1 (Γ ) divides D into two connected components, say, A− , A+ with g(A− ) ⊂ H− − η and g(A+ ) ⊂ H+ − η. Let J − := ∂ A− ∩ ∂D and J + := ∂ A+ ∩ ∂D. Note that, J + and J − are closed arcs and J + ∩ J − = {ξ − , ξ + }. In particular, g −1 (∂ + Ω) ⊂ J + and g −1 (K ) ⊂ J − . By Proposition 3.3.2, ξ − = ξ + . Since g(ξ + ) = g(ξ − ) = ∞, we have g −1 (K ) ⊂ ∂D \ J + . Therefore, by Proposition 7.1.4(2), for every σ ∈ g −1 (K ), lim μ(w, J + \ {ξ + , ξ − }, D) = 0.

w→σ

Since g −1 (∂ + Ω) ⊂ J + , for every z ∈ H ,

(17.6.16)

17.6 Parabolic Semigroups of Zero Hyperbolic Step with Prescribed Slopes

μ(z, ∂ + Ω, H ) = μ(g −1 (z), g −1 (∂ + Ω), D) ≤ μ(g −1 (z), J + , D) = μ(g −1 (z), J + \ {ξ + , ξ − }, D).

523

(17.6.17)

Now, let x ∈ K and let {z n } ⊂ H be such that lim n→∞ z n = x and limn→∞ g −1 (z n ) = σ . Since eventually {z n } ⊂ H− − η, it follows that σ ∈ J − . Up to extracting subsequences, we can assume that Re z n < −η for all n and either Re z n > −wl or Re z n = −wl or Re z n < −wl for all n ∈ N. We consider the first case, the others being similar. Let γ : [1, +∞) → H be such that γ (n) = z n and γ (t) = γ (n)(1 − t + n) + γ (n + 1)(t − n) for t ∈ [n, n + 1), n ∈ N. Note that −wl < Re γ (t) < −η for all t. Hence, γ is a continuous curve and γ ([1, +∞)) ⊂ H . Moreover, limt→+∞ γ (t) = x. Hence, by Proposition 3.3.3 and Lehto-Virtanen’s Theorem 3.3.1, limt→+∞ g −1 (γ (t)) exists—and it has to be equal to σ since g −1 (γ (n)) → σ —and g(σ ) := ∠ lim z→σ g(z) = x. Therefore, σ ∈ g −1 (K ) and, by (17.6.16) and (17.6.17), lim μ(z n , ∂ + Ω, H ) = 0. n→∞

By the arbitrariness of x and {z n }, (17.6.15)—and hence (17.6.14)—follows. Arguing as before with the aim of the Lindelöf’s Maximum Principle, we have v v μ(y + i , ∂ + Ω, Ω) ≤ μ(y + i , A0 ∪ Av ∪ Bv , R(0, v, wl , wr )). 2 2

(17.6.18)

From (17.6.18) and (17.6.9) we have v v μ(y + i , ∂ + Ω, Ω) − μ(y + i , iR + wr , S(wl , wr )) ≤ α(v). 2 2

(17.6.19)

Thus, by (17.6.12) and (17.6.19), and taking into account (17.6.6), we have    wl  μ(i v , ∂ + Ω, Ω) − =  2 wr + wl    v v   = μ(i , ∂ + Ω, Ω) − μ(i , iR + wr , S(wl , wr )) ≤ α(v). 2 2 In order to end the proof, we are left to show that limv→+∞ α(v) = 0. To this aim, let ωv (z) := μ(z, A0 ∪ Av , R(0, v, wl , wr )), for all z ∈ R(0, v, wl , wr ). Given k > v, and arguing as before using the Lindelöf Maximum Principle, we have ωk (z) ≤ ωv (z) for all z ∈ R(0, v, wl , wr ). Therefore, {ωn (z)}Nn>v is a pointwise decreasing sequence of bounded harmonic functions defined on R(0, v, wl , wr ). Hence, by Harnack’s Theorem (see [55, p. 261]), the sequence converges uniformly on compacta on R(0, v, wl , wr ) to a harmonic function ω : R(0, v, wl , wr ) → [0, 1]. Since ω ≤ ωv , it follows that lim z→x ω(z) = 0 for all x ∈ {−wl + i(0, v)} ∪ {wr + i(0, v)}. Repeating this argument for all v > 0, we conclude that the limit function ω does not depend on v and it is well defined on the strip S(wl , wr ). Moreover, 0 ≤ ω ≤ 1

524

17 Slopes of Orbits at the Denjoy-Wolff Point

and lim z→x ω(z) = 0 for all x ∈ ∂ S(wl , wr ). By the Lindelöf Maximum Principle, ω ≡ 0. Hence, for all z ∈ R(0, v, wl , wr ), lim ωk (z) = 0.

k→∞

(17.6.20)

Now, for all v > 0, let f v : [−wl , wr ] → [0, 1] given by f v (y) := ωv (y + i v2 ), if y ∈ (−wl , wr ), and f v (−wl ) = f v (wr ) := 0. Note that f v is continuous and limv→+∞ f v (y) = 0 for all y ∈ [−wl , wr ] by (17.6.20). Thus, by Dini’s Theorem [122, p. 143], { f v } converges uniformly to 0 on [−wl , wr ]. Hence, limv→+∞ α(v) = 0, and we are done.

Lemma 17.6.5 For every 0 ≤ a1 ≤ a2 ≤ 1 there exist two sequences {αn }, {βn } of positive real numbers converging to +∞ such that lim

n→∞

αn = a1 , αn + βn+1

lim

n→∞

αn = a2 . αn + βn

Proof We consider four cases: (A) 0 < a1 < a2 < 1, (B) 0 = a1 < a2 < 1, (C) 0 < a1 < a2 = 1, (D) 0 = a1 < a2 = 1. a2 Case (A): Set c := 1−a ∈ (0, +∞). Since 0 < a1 < a2 < 1, we have 2 γ :=

1 − a1 a2 ∈ (1, +∞). a1 1 − a2

Then, we define αn := cγ n , βn := γ n , n ∈ N. Since γ > 1, the sequences {αn }, {βn } are increasing and converge to +∞. Moreover, for all n c αn c αn = a1 , = a2 . = = αn + βn+1 c+γ αn + βn c+1

Case (B): Set c :=

a2 1−a2

∈ (0, +∞) and let αn := cn!, βn := n!, n ∈ N.

Hence, {αn }, {βn } are two increasing sequences of positive real numbers converging to +∞. Moreover, c αn c αn n→∞ = a2 , for all n. = −→ 0 and = αn + βn+1 c+n+1 αn + βn c+1 1 Case (C): Let c := 1−a ∈ (0, +∞) and take p > 0 such that c(1 + p) ≥ 2. Morea1 over, let γn := c(n + p) for all n and also cn := n + 1 + p for all n, and

αn := cn γ1 · · · γn , βn := γ1 · · · γn , n ∈ N.

17.6 Parabolic Semigroups of Zero Hyperbolic Step with Prescribed Slopes

525

Hence, {αn }, {βn } are two increasing sequences of positive real numbers converging to +∞. Moreover, 1 cn n+1+ p αn = = a1 , for all n, = = αn + βn+1 cn + γn+1 n + 1 + p + c(n + 1 + p) 1+c and

cn αn n + 1 + p n→∞ = −→ 1. = αn + βn cn + 1 n+1+ p+1

Case (D): Consider the sequences αn := n(n!)2 , βn := (n!)2 , n ∈ N. Clearly, {αn }, {βn } are two increasing sequences of positive real numbers converging to +∞. Moreover n αn n n→∞ αn n→∞ = −→ 0 and = −→ 1, αn + βn+1 n + (n + 1)2 αn + βn n+1



and we are done. Given ξ ∈ C, we denote the lower vertical half-line starting at ξ by L l [ξ ] := {ξ + it : t ≤ 0}.

Moreover, given two complex sequences {ξn+ }, {ξn− } such that {Re ξn+ } is a strictly increasing sequence of positive real numbers converging to +∞ and {Re ξn− } is a strictly decreasing sequence of negative real numbers converging to −∞, we define Ω[{ξn+ }, {ξn− }] := C \

∞  

 L l [ξn+ ] ∪ L l [ξn− ] .

n=1

Observe that Ω[{ξn+ }, {ξn− }] is a domain starlike at infinite containing the imaginary axis and ∞  ∂ + Ω[{ξn+ }, {ξn− }] = L l [ξn+ ]. n=1

Theorem 17.6.6 Given 0 ≤ a1 < a2 ≤ 1, there exists two complex sequences {ξn+ }, {ξn− } such that {Re ξn+ } is a strictly increasing sequence of positive numbers converging to +∞ and {Re ξn− } is a strictly decreasing sequence of negative real numbers converging to −∞ and lim inf μ(it, ∂ + Ω[{ξn+ }, {ξn− }], Ω[{ξn+ }, {ξn− }]) = a1 , t→+∞

526

17 Slopes of Orbits at the Denjoy-Wolff Point

lim sup μ(it, ∂ + Ω[{ξn+ }, {ξn− }], Ω[{ξn+ }, {ξn− }]) = a2 . t→+∞

Proof Fix 0 ≤ a1 ≤ a2 ≤ 1 and let {αn } and {βn } be the sequences given by Lemma 17.6.5. By Lemma 17.6.4, for every k ∈ N, there exists vk > 0 such that: (C1 ) for every domain Ω ⊂ C starlike at infinite with ∂ + Ω = ∅ and every rectangle R ∈ R (see page 520) with height v ≥ vk , left width αk and right width βk such that R ⊂ Ω and ∂V R ⊂ ∂Ω,     μ(c(R), ∂ + Ω, Ω) − αk  ≤ 1 ,  αk + βk  2k where c(R) is the pseudocenter of R as defined in (17.6.7). (C2 ) for every domain Ω ⊂ C starlike at infinite with ∂ + Ω = ∅ and every rectangle R ∈ R with height v ≥ vk , left width αk and right width βk+1 such that R ⊂ Ω and ∂V R ⊂ ∂Ω,     αk ≤ 1 . μ(c(R), ∂ + Ω, Ω) −  αk + βk+1  2k Moreover, again by Lemma 17.6.4, we may assume that {vk } is strictly increasing and converges to +∞.  Let ρ0 = 0 and, for n ∈ N, let ρn := nj=1 v j and Ω0 := Ω[{−αn + iρ2n }, {βn + iρ2n−1 )}]. Note that +

∂ Ω0 =

∞ 

L l [βn + iρ2n−1 ].

n=1

Let O L be the cluster set at +∞ of the curve [0, +∞)  t → μ(it, ∂ + Ω0 , Ω0 ). We are left to show that O L = [a1 , a2 ]. For every n ∈ N, let  Rn :=

R(ρ2k−2 , v2k−1 , αk , βk ), n = 2k − 1 with k ∈ N, R(ρ2k−1 , v2k , αk , βk+1 ), n = 2k with k ∈ N.

Note that, for every n ∈ N, Rn ∈ R, Rn ⊂ Ω0 and ∂V Rn ⊂ Ω0 . From (C2 ), for k ∈ N (taking into account that v2k > vk ), we have     αk ≤ 1 . μ(c(R2k ), ∂ + Ω0 , Ω0 ) −  αk + βk+1  2k

17.6 Parabolic Semigroups of Zero Hyperbolic Step with Prescribed Slopes

527

On the other hand, from (C1 ), and taking into account that v2k−1 ≥ vk , for k ∈ N, we have     μ(c(R2k−1 ), ∂ + Ω0 , Ω0 ) − αk  ≤ 1 .  αk + βk  2k In particular, lim μ(c(R2k ), ∂ + Ω0 , Ω0 ) = a1 ,

k→∞

lim μ(c(R2k−1 ), ∂ + Ω0 , Ω0 ) = a2 .

k→∞

Since limk→∞ vk = +∞, we have limk→∞ Im (c(Rk )) = +∞. Being O L a closed subinterval of [0, 1], we deduce [a1 , a2 ] ⊂ O L . Now, we show that O L ⊂ [a1 , a2 ]. Fix x ∈ O L . Hence, there exists a strictly increasing sequence {tn } of positive real numbers converging to +∞ such that x = lim μ(itn , ∂ + Ω0 , Ω0 ). n→∞

Up to passing to a subsequence of {tn }, we can assume that either there exists an increasing sequence of odd natural numbers { jn } converging to +∞ such that tn ∈ [Im (c(R jn )), Im (c(R jn +1 )]), for all n, or there exists an increasing sequence of even natural numbers { jn } converging to +∞ such that tn ∈ [Im (c(R jn )), Im (c(R jn +1 ))], for all n. We deal with the first situation (the second being similar). Fix k ∈ N and take an arbitrary t ∈ [Im (c(R2k−1 )), Im (c(R2k ))]. As we remarked, R2k−1 ⊂ Ω0 and ∂V R2k−1 ⊂ Ω0 . Now, consider the domain starlike at infinite ⎛ Ω0(1)

:= C \ ⎝

∞  n=1,n=k

L l [βn + iρ2n−1 ] ∪ L l [βk + iρ2k ] ∪

∞ 

⎞ L l [−αn + iρ2n ]⎠ .

n=1

Note that Ω0(1) = Ω0 \ {w ∈ C : Re w = βk , Im w ∈ [ρ2k−1 , ρ2k ]}, hence, Ω0(1) ⊂ Ω0 . Moreover, let  1 (1) R2k−1 := R t − v2k−1 , v2k−1 , αk , βk ∈ R. 2 (1) (1) Note that the pseudocenter of R2k−1 is it. By construction, R2k−1 ⊂ Ω0(1) and (1) ⊂ ∂Ω0(1) . Therefore, by (C1 ), ∂V R2k−1

    ω(it, ∂ + Ω (1) , Ω (1) ) − αk  ≤ 1 . 0 0  αk + βk  2k Now, ∂Ω0 \ ∂ + Ω0 = ∂Ω0(1) \ ∂ + Ω0(1) . Hence, by the domain monotonicity of harmonic measures (see Proposition 7.2.10),

528

17 Slopes of Orbits at the Denjoy-Wolff Point

μ(it, ∂ + Ω0 , Ω0 ) = 1 − μ(it, ∂Ω0 \ ∂ + Ω0 , Ω0 ) (1)

(1)

(1)

(1)

≤ 1 − μ(it, ∂Ω0 \ ∂ + Ω0 , Ω0 ) = 1 − μ(it, ∂Ω0 \ ∂ + Ω0 , Ω0 ) αk 1 (1) (1) + . = μ(it, ∂ + Ω0 , Ω0 ) ≤ αk + βk 2k

Therefore, x = lim μ(itn , ∂ + Ω0 , Ω0 ) ≤ lim n→∞



n→∞

α jn 1 + α jn + β jn 2 jn

= a2 .

Now, we consider the domain starlike at infinite ⎛ ⎞ ∞ ∞   Ω0(2) := C \ ⎝ L l [βn + iρ2n−1 ] ∪ L l [−αn + iρ2n ]⎠ . n=1,n=k

n=1

Clearly, Ω0(2) ⊃ Ω0 . Moreover, let  1 (2) R2k−1 := R t − v2k−1 , v2k−1 , αk , βk+1 ∈ R. 2 (2) (2) Note that it is the pseudocenter of R2k−1 . By construction, R2k−1 ⊂ Ω0(2) and (2) ⊂ ∂Ω0(2) . Therefore, by (C2 ), ∂V R2k−1

    αk μ(it, ∂ + Ω (2) , Ω (2) ) − ≤ 1 . 0 0  αk + βk+1  2k Note ∂Ω0 \ ∂ + Ω0 = ∂Ω0(2) \ ∂ + Ω0(2) . Hence, using again the domain monotonicity of harmonic measures, μ(it, ∂ + Ω0 , Ω0 ) = 1 − μ(it, ∂Ω0 \ ∂ + Ω0 , Ω0 ) (2)

(2)

(2)

(2)

≥ 1 − μ(it, ∂Ω0 \ ∂ + Ω0 , Ω0 ) = 1 − μ(it, ∂Ω0 \ ∂ + Ω0 , Ω0 ) αk 1 (2) (2) − . = μ(it, ∂ + Ω0 , Ω0 ) ≥ αk + βk+1 2k

Therefore, 

+

x = lim μ(itn , ∂ Ω0 , Ω0 ) ≥ lim n→∞

n→∞

α jn 1 − α jn + β jn +1 2 jn

= a1 .

Summing up, x ∈ [a1 , a2 ] thus O L ⊂ [a1 , a2 ], and we are done. As a direct consequence of Corollary 17.6.3 and Theorem 17.6.6 we have:



17.6 Parabolic Semigroups of Zero Hyperbolic Step with Prescribed Slopes

529

Corollary 17.6.7 Given −π/2 ≤ c1 < c2 ≤ π/2, there exists a parabolic semigroup (φt ) in D of zero hyperbolic step such that Slope[t → φt (z), τ ] = [c1 , c2 ].

17.7 The Shift of Non-Elliptic Semigroups Definition 17.7.1 Let (φt ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. We say that (φt ) is of finite shift if there exist R > 0 and z ∈ D such that / E(τ, R) for all t ≥ 0. If (φt ) is not of finite shift, we say that (φt ) is of infinite φt (z) ∈ shift. Remark 17.7.2 The rationale for the name comes from the half-plane translation of the definition. Indeed, if (φt ) is a continuous semigroup of holomorphic self-maps of H with Denjoy-Wolff point at ∞, taking into account that an horocycle at ∞ in H is given by E H (∞, R) = {w ∈ H : Re w > R} for some R > 0, it follows that (φt ) is of finite shift if there exist R > 0 and w ∈ H such that Re φt (w) < R for all t ≥ 0. Since Stolz regions are eventually contained in any horocycle, it follows that if the orbits of a semigroup converge non-tangentially to the Denjoy-Wolff point, then the semigroup is of infinite shift. Therefore, hyperbolic semigroups in D are of infinite shift. On the other hand, parabolic groups in D are of finite shift. The next result shows that parabolic semigroups of zero hyperbolic step are also of infinite shift, regardless the type of convergence of their orbits to the Denjoy-Wolff point: Proposition 17.7.3 Let (φt ) be a parabolic semigroup in D of zero hyperbolic step with Denjoy-Wolff point τ ∈ ∂D. Then, for each z ∈ D and R > 0, there is t0 > 0 such that φt (z) ∈ E(τ, R) for all t ≥ t0 . In particular, (φt ) is of infinite shift. Proof Considering the Cayley transform Cτ that maps D onto H and τ to ∞ given by (1.1.2) we can define the semigroup Φt := Cτ ◦ φt ◦ Cτ−1 in H having ∞ as DenjoyWolff point. The statement is then equivalent to the following: for each w ∈ H and R > 0, there is t0 > 0 such that Re Φt (w) > R for all t ≥ t0 . Since the semigroup is parabolic, for all t ≥ 0, lim inf [kH (1, w) − kH (1, Φt (w))] = lim inf [ω(0, Cτ−1 (w)) − ω(0, φt (Cτ−1 (w)))] w→+∞

w→+∞

= lim inf [ω(0, z) − ω(0, φt (z))] = 0. z→τ

(17.7.1) Assume the result does not hold. Then there is a point w0 ∈ H and R > 0 such that Re Φt (w0 ) ≤ R for all t > 0. Let xt := Re Φt (w0 ) and yt := Im Φt (w0 ) for all t ≥ 0. By (17.7.1), Theorem 1.7.8 guarantees that, given t > s ≥ 0, xt = Re Φt (w0 ) = Re Φt−s (Φs (w0 )) > Re Φs (w0 ) = xs .

530

17 Slopes of Orbits at the Denjoy-Wolff Point

Therefore, the function t → xt is strictly non-decreasing and there exists L := limt→+∞ xt ∈ (x0 , R]. Fix w ∈ H. Since the semigroup (φt ) is of zero hyperbolic step and Cτ is an isometry for the hyperbolic distance, by Corollary 9.3.8, lim kH (Φt (w), Φt (w0 )) = 0.

t→+∞

By (1.3.4), this implies       Φt (w) − Φt (w0 )    = lim  ((Φt (w) − i yt )/xt ) − 1  . 0 = lim    t→+∞ Φ (w) + Φ (w ) t→+∞ ((Φt (w) − i yt )/x t ) + 1  t t 0 yt Hence, limt→+∞ Φt (w)−i = 1. Thus limt→+∞ Re Φt (w) = L. xt Now, let w := R + 1. Since t → Re Φt (R + 1) is non-decreasing (again by Theorem 1.7.8), we conclude that

R + 1 ≤ lim Re Φt (R + 1) = L ≤ R. t→+∞



A contradiction, and we are done.

The previous discussions imply that the notion of finite shift really makes sense only for parabolic semigroups of positive hyperbolic step. For such semigroups, the orbits converge tangentially to the Denjoy-Wolff point (Theorem 17.5.1). This is the reason why, sometimes, semigroups of finite shift are also called “strongly tangential”. We see in a moment that one can characterize finite shift in terms of regularity of the Koenigs function. Before this, we prove that the finiteness of the shift does not depend on the chosen point. Lemma 17.7.4 Let (φt ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Suppose (φt ) is of finite shift. Then, for every z ∈ D there exists R = R(z) > / E(τ, R) for all t ≥ 0. 0 such that φt (z) ∈ Proof Suppose z 0 ∈ D, R > 0 are such that φt (z 0 ) ∈ / E(τ, R) for all t ≥ 0. By (1.4.2), it follows that for all t ≥ 0 lim [ω(φt (z 0 ), w) − ω(0, w)] ≥

w→τ

Let z ∈ D, z = z 0 , and let β > 0 be such that w ∈ D,

1 2

1 log R. 2

log β = −ω(z, z 0 ). Since for every

ω(φt (z), w) − ω(φt (z 0 ), w) ≥ −ω(φt (z), φt (z 0 )) ≥ −ω(z, z 0 ) = we have for all t ≥ 0,

1 log β, 2

17.7 The Shift of Non-Elliptic Semigroups

531

[ω(φt (z), w) − ω(0, w)] ≥ [ω(φt (z 0 ), w) − ω(0, w)] +

1 log β. 2

Taking the limit for w → τ , the previous inequality implies that φt (z) ∈ / E(τ, β R) for all t ≥ 0.

As a matter of notation, in what follows, if the canonical model of (φt ) is (H, h, z → z + it) then C denotes a Cayley transform from D onto H which maps τ to ∞, while, if the canonical model of (φt ) is (H− , h, z → z + it) then C denotes a Cayley transform from D onto H− which maps τ to ∞. Theorem 17.7.5 Let (φt ) be a parabolic semigroup in D of positive hyperbolic step, with Denjoy-Wolff point τ ∈ ∂D and Koenigs function h. Then the following are equivalent: (1) the holomorphic map C −1 ◦ h : D → D has a boundary regular fixed point at τ , (2) (φt ) is of finite shift. Proof We assume that the canonical model of (φt ) is (H, h, z → z + it), the other case being similar. (1) implies (2). Let h˜ := h ◦ C −1 : H → H. Note that C(0) = 1. Since C is an isometry between ω and kH , we have for all z ∈ D ˜ ω(0, z) − ω(0, C −1 (h(z))) = kH (1, C(z)) − kH (1, h(C(z))).

(17.7.2)

Hence, C −1 ◦ h has a boundary regular fixed point at τ if and only if 1 ˜ log α := lim inf [kH (1, w) − kH (1, h(w))] < +∞. Hw→∞ 2 Let R > 0. By Theorem 1.7.8 and (1.4.15), this latter condition implies ˜ ˜ H (∞, h(E(τ, R)) = h(C(E(τ, R))) = h(E ⊆ E H (∞,

1 )) R

1 1 ) = {w ∈ H : Re w > }. αR αR

 Since t≥0 (h(D) + it) = H, it follows that there exists p ∈ h(D) such that Re p < 1 . Hence, by the previous equation, φt (h −1 ( p)) = h −1 ( p + it) ∈ / E(τ, R) for all αR t ≥ 0, and (2) holds. (2) implies (1). Let h˜ : H → H be the holomorphic map introduced before. By ˜ < +∞ (17.7.2), it is enough to show that lim inf Hw→∞ [kH (1, w) − kH (1, h(w))] ˜ Re h(w) and, by Theorem 1.7.8, the latter condition is equivalent to inf w∈H Re w > 0. Let ψt := C ◦ φt ◦ C −1 . Then (ψt ) is a continuous semigroup of holomorphic ˜ z → z + it) is a model for (ψt ). Moreover, taking into self-maps of H and (H, h, account (1.4.15), by Theorems 8.3.1 and 1.8.4 we have

532

17 Slopes of Orbits at the Denjoy-Wolff Point

Re ψt (w) ≥ Re w,

(17.7.3)

for all t ≥ 0 and w ∈ H. In particular, since Re ψt+s (w) = Re ψt (ψs (w)) for all 0 ≤ s, t and w ∈ H, the previous equation implies that [0, +∞)  t → Re ψt (w) is non-decreasing for all w ∈ H. Thus, x∞ (w) := limt→+∞ Re ψt (w) exists and belongs to (0, +∞]. However, since (φt ) is of finite shift (see Remark 17.7.2), for every w ∈ H lim Re ψt (w) = x∞ (w) ∈ (0, +∞). t→+∞

Moreover, from (9.3.1), we have for all w0 , w1 ∈ H lim kH (ψt (w0 ), ψt (w1 )) = lim ω(φt (C −1 (w0 )), φt (C −1 (w1 )))

t→+∞

t→+∞

= kh(D) (h(C

−1

(w0 )), h(C

−1

˜ 0 ), h(w ˜ 1 )). (w1 )) = kh(H) (h(w ˜

(17.7.4)

Now, for w ∈ H, let xt (w) := Re ψt (w) and yt (w) := Im ψt (w). Since w → w − i yt (1) is an automorphism of H for all t ≥ 0, we have kH (ψt (1), ψt (w)) = kH (xt (1), xt (w) + i(yt (w) − yt (1))).

(17.7.5)

Hence, (17.7.4) implies that yt (w) − yt (1) is bounded. We claim that, in fact, there exists c(w) ∈ R such that lim [yt (w) − yt (1)] = c(w).

t→+∞

Indeed, let c ∈ R be a point in the cluster set of yt (w) − yt (1) at +∞. By (17.7.4) and (17.7.5), we have then ˜ ˜ (h(1), h(w)) =: R(w) kH (x∞ (1), x∞ (w) + ic) = kh(H) ˜ which forces x∞ (w) + ic to stay on the boundary of the hyperbolic disc in H of center x∞ (1) and radius R(w). Since the boundaries of hyperbolic discs in H are circles, this implies that the cluster set of yt (w) − yt (1) at +∞ contains at most two points. But, the cluster set of a continuous curve is connected (see Lemma 1.9.9), hence yt (w) − yt (1) has limit for t → +∞. For w ∈ H, let g(w) := x∞ (w) + ic(w). Note that g(w) = lim [ψt (w) − i yt (1)]. t→+∞

(17.7.6)

In particular, g(H) ⊂ H and g is the limit of univalent self-maps of H. Since H is biholomorphic to D via C, it follows from Montel’s Theorem that g is holomorphic and, being the limit of univalent functions, it is either constant or univalent. However, by (17.7.5) and (17.7.4), we have for all w ∈ H,

17.7 The Shift of Non-Elliptic Semigroups

533

˜ ˜ kh(H) (h(1), h(w)) = kH (x∞ (1), g(w)), ˜ hence g is not constant and it is, in fact, univalent. For s ≥ 0, let βs := c(ψs (1)). By (17.7.6), for all w ∈ H and s ≥ 0, g(ψs (w)) = lim [ψt (ψs (w)) − i yt (1)] t→+∞

= lim [ψt+s (w) − i ys+t (1) + i(yt (ψs (1)) − yt (1))] t→+∞

(17.7.7)

= g(w) + iβs . Note that, in particular, iβs = g(ψs (1)) − g(1). Hence, [0, +∞)  s → βs is continuous. Moreover, for s, t ≥ 0 g(1) + iβs+t = g(ψs+t (1)) = g(ψs (ψt (1))) = g(ψt (1)) + iβs = g(1) + i(βt + βs ). Therefore, βs+t = βt + βs for all s, t ≥ 0. Thus, by Theorem 8.1.11, there exists a ∈ R such that βs = as for all s ≥ 0. Since g is univalent, it follows from (17.7.7) that a = 0. Thus, by (17.7.7), g(ψs (w)) = g(w) + ias for all w ∈ H. If a > 0, this implies that g(H) is starlike at infinity. While, if a < 0, this implies that for every w ∈ g(H), w − it ∈ g(H) for all t ≥ 0, or, in other words, −g(H) is starlike at infinity. The function x∞ |(0,+∞) = Re g|(0,+∞) : (0, +∞) → (0, +∞) is continuous, so that its image is an interval in [0, +∞). Moreover, given r ∈ (0, +∞), we have that x∞ (r ) = limt→+∞ xt (r ) ≥ r by (17.7.3). This implies that there exists r0 ∈ [0, +∞) such that (r0 , +∞) ⊆ x∞ ((0, +∞)). From this, if a > 0 and since g(H) is starlike at infinity, it is easy to see that there exists b ≥ 0 such that H + b = t≥0 (g(H) − iat). In case a 0, while t≥0 (g(H) and t ≥ 0, (g˜ ◦ C ◦ φt )(z) = (g˜ ◦ ψt ◦ C)(z) = (g˜ ◦ C)(z) + it. Since we are assuming that the canonical model of (φt ) is (H, h, z → z + it), by Remark 9.3.6 we have a > 0. Moreover, by Proposition 9.3.10, there exists q ∈ R such that for every w ∈ H ˜ ˜ + iq. h(w) = h(C −1 (w)) = g(w)

534

Therefore, inf w∈H

17 Slopes of Orbits at the Denjoy-Wolff Point ˜ Re h(w) Re w

> 0 if and only if Re g(w) ˜ > 0. w∈H Re w inf

Now, by (17.7.3), we have for every w ∈ H, Re ψt (w) Re g(w) = lim ≥ 1. t→+∞ Re w Re w Hence, by Theorem 1.7.8, ∠ limw→∞ ∠ lim

w→∞

g(w) w

exists and

g(w) Re g(w) = inf ≥ 1. w∈H Re w w

Since ∠ lim

w→∞

again Theorem 1.7.8 implies that

g(w) ˜ g(w) 1 = ∠ lim , w a w→∞ w Re g(w) ˜ Re w

≥ a for all w ∈ H, and we are done.



In order to construct examples of parabolic semigroups with positive hyperbolic step and finite or infinite shift, we provide two simple conditions on the image of the associated Koenigs function. Let β ∈ (0, π ). A right vertical semi-sector Wβ+ of amplitude β is given by Wβ+ := i V (β) ∩ H, where V (β) = {ρeiθ ∈ C : ρ > 0, |θ | < β} is a horizontal sector of amplitude 2β. A left vertical semi-sector Wβ− of amplitude β is given by Wβ− := i V (β) ∩ H− . We say that a domain Ω ⊂ C contains a vertical semi-sector if there exist p ∈ Ω and β ∈ (0, π ) such that either (Wβ+ + p) ⊂ Ω or (Wβ− + p) ⊂ Ω. We also say that Ω contains a vertical half-plane if there exists a ∈ R such that either {w ∈ C : Re w > a} ⊂ Ω or {w ∈ C : Re w < a} ⊂ Ω. Theorem 17.7.6 Let (φt ) be a parabolic semigroup in D of positive hyperbolic step, with Denjoy-Wolff point τ ∈ ∂D and Koenigs function h. (1) If h(D) does not contain a vertical semi-sector then (φt ) is of infinite shift. (2) If h(D) contains a vertical half-plane then (φt ) is of finite shift. Proof (1) We assume that the canonical model of (φt ) is (H, h, z → z + it) (the case when the canonical model is (H− , h, z → z + it) is similar).

17.7 The Shift of Non-Elliptic Semigroups

535

By hypothesis, h(D) does not contain a vertical semi-sector. By Theorem 17.7.5, (φt ) is of finite shift if and only if C −1 ◦ h has a boundary regular fixed point at τ . By Lemma 1.4.5, this latter condition is equivalent to +∞ > lim inf [ω(0, w) − ω(0, C −1 (h(w)))] = lim inf [ω(0, w) − kH (C(0), h(w))] w→τ

w→τ

= lim inf [kh(D) (h(0), h(w)) − kH (C(0), h(w))], w→τ

and, by Proposition 1.7.4 and Eq. (1.4.7), it is, in fact, equivalent to the existence of L ∈ R such that ∠ lim [kh(D) (h(0), h(w)) − kH (C(0), h(w))] = L . w→τ

(17.7.8)

Now, arguing by contradiction, we assume that (φt ) is of finite shift, that is, (17.7.8) holds. Since h(D)  H, there exists p ∈ H \ h(D) such that p + it ∈ h(D) for all t > 0. Let a := Re p > 0 and let Ω := h(D) − p. Then Ωis starlike at infinity, 0 ∈ / Ω and it ∈ Ω for all t > 0. Moreover, Ω ⊂ (H − a) and t≥0 (Ω − it) = H − a. Since h(D) does not contain a vertical semi-sector, then Ω does not contain a − (t) ≤ a for all t > 0, the latter condition implies vertical semi-sector. Since δΩ,0 lim inf t→+∞

+ δΩ,0 (t)

t

= 0.

(17.7.9)

+ − The curve σ : [1, +∞) → Ω defined by σ (t) = (δΩ,0 (t) − δΩ,0 (t))/2 + it is a quasi-geodesic in Ω by Theorem 17.2.12. Hence, [1, +∞)  t → σ (t) + p is a quasi-geodesic in h(D) and, by (17.3.1), limt→+∞ h −1 (σ (t) + p) = τ . Moreover, by Theorem 6.3.8, h −1 (σ (t) + p) converges to τ non-tangentially as t → +∞. Therefore, by (17.7.8),

L = lim [kh(D) (h(0), h(h −1 (σ (t) + p))) − kH (C(0), h(h −1 (σ (t) + p)))] t→+∞

= lim [kΩ (h(0) − p, σ (t)) − kH (C(0), σ (t) + p)]. t→+∞

By the triangle inequality, the previous equality implies lim sup[kΩ (i, σ (t)) − kH (1, σ (t) + p)] < +∞.

(17.7.10)

t→+∞

Now, write σ (t) + p = ρt eiθt , with ρt > 0 and θt ∈ (−π/2, π/2). By Lemma 5.4.1(1) and (6), kH (1, σ (t) + p) ≤ kH (ρt , ρt eiθt ) + kH (1, ρt ) ≤

2ρt 1 log . 2 cos θt

(17.7.11)

536

17 Slopes of Orbits at the Denjoy-Wolff Point

+ − Note that Re σ (t) + a → +∞ as t → +∞, since δΩ,0 (t) → +∞ and δΩ,0 (t) ≤ a. Therefore, there exists t0 ≥ 1 such that, for t ≥ t0 ,

1 ρt ≤ ρt . = cos θt Re σ (t) + a ± Moreover, taking into account that δΩ,0 (t) ≤ t, a straightforward computation shows that for t ≥ a, √ ρt ≤ 2 2 t.

Therefore, from (17.7.11) we obtain that for t ≥ max{t0 , a}, kH (1, σ (t) + p) ≤ log t + log 4.

(17.7.12)

On the other hand, since σ is a quasi-geodesic in Ω, there exist A ≥ 1, B > 0 such that for all t ≥ 1, 1 B kΩ (i, σ (t)) ≥ Ω (σ ; [1, t]) − . A A By Theorem 5.2.1, and since |σ  (t)| ≥ 1 for almost all t ≥ 1,  Ω (σ ; [1, t]) =

t

κΩ (σ (τ ); σ  (τ ))dτ ≥

1

1 4

 1

t

dτ . δΩ (σ (τ ))

Taking into account that Ω ⊂ (H − a), we have + (t) + a. δΩ (σ (t)) ≤ δH−a (σ (t)) = Re σ (t) + a ≤ δΩ,0

The three previous inequalities imply 1 kΩ (i, σ (t)) ≥ 4A



t

1

dτ B − . + A δΩ,0 (τ ) + a

(17.7.13)

Now, let n 0 ∈ N be such that for all n ≥ n 0 , e1/n < 1 +

1 . 8A

(17.7.14)

We claim that there exists a sequence {tn }n≥n 0 converging to +∞ such that tn 0 ≥ 1, tn+1 ≥ e1/n tn for all n ≥ n 0 , and such that for every t ∈ [tn , e1/n tn ], + (t) + a ≤ δΩ,0

Indeed, fix n ≥ n 0 and T ≥ 1. Let also

t . 8A

17.7 The Shift of Non-Elliptic Semigroups

Cn :=

537

1 . 1 − 8A(e1/n − 1)

Note that Cn > 1 by (17.7.14). tn + (tn ) + a ≤ 8AC . Hence, for every By (17.7.9), there exists tn ≥ T such that δΩ,0 n 1/n stn , with s ∈ [1, e ], by the triangle inequality and the definition of Cn , we have tn + (s − 1)tn 8ACn stn tn ≤ , − 1)tn = 8A 8A

+ + (stn ) + a ≤ δΩ,0 (tn ) + (s − 1)tn + a ≤ δΩ,0

≤

tn + (e1/n 8ACn

and the claim follows at once. Now, by the claim, for every n ≥ n 0 we have 1 4A

 1

n

n  1/j n  e1/j t j  dτ dτ dτ 1  e tj ≥ ≥ 2 + + 4 A j=n t j τ δΩ,0 (τ ) + a δΩ,0 (τ ) + a j=n t j 0

=2

n  j=n 0

log ei/j

0

n  1 ≥ 2 log n + D, =2 j j=n 0

for some D < 0 which depends only on n 0 . Therefore, by (17.7.13), kΩ (i, σ (n)) ≥ 2 log n −

B + D, A

and hence, by (17.7.12), lim sup[kΩ (i, σ (n)) − kH (1, σ (n) + p)] n→+∞

≥ lim sup[2 log n − log n − log 4 − n→+∞

B + D] = +∞, A

contradicting (17.7.10). Therefore, (φt ) is of infinite shift. (2) As before, we might assume that the canonical model of (φt ) is (H, h, z → z + it). By hypothesis, there exists a > 0 such that (H + a) ⊂ h(D). We have to show that (φt ) is of finite shift. By Theorem 17.7.5, and arguing as in (1), it is enough to show that lim inf [kh(D) (h(0), h(w)) − kH (C(0), h(w))] < +∞. w→τ

Since H + a ⊂ h(D), the curve γ : [a + 1, +∞) → C defined by γ (t) = t is contained in h(D). We claim that limt→+∞ h −1 (γ (t)) = τ . Indeed, let η(t) := a + 1 + it. Note that γ (a + 1) = η(0), and

538

17 Slopes of Orbits at the Denjoy-Wolff Point

lim γ (t) = lim η(t) = ∞ ∈ C∞ .

t→+∞

t→+∞

Moreover, γ ((a + 1, +∞)) ∩ η((0, +∞)) = ∅ and the simply connected region in ∞ ∞ C∞ bounded by γ ([a + 1, +∞)) ∪ η([0, +∞)) which contains a + 2 + i is contained in H + a, hence, in h(D). Therefore, by Proposition 3.3.5, lim h −1 (γ (t)) = lim h −1 (η(t)) = lim φt (h −1 (a + 1)) = τ,

t→+∞

t→+∞

t→+∞

and the claim follows. Now, since H + a ⊂ h(D), by Proposition 5.1.4, and since z → z − a is a biholomorphism between H + a and H, we have for t ≥ a + 1, kh(D) (a + 1, h −1 (h(t))) ≤ kH+a (a + 1, t) = kH (1, t − a). Therefore, using the triangle inequality and Lemma 5.4.1(1), lim inf [kh(D) (h(0), h(w)) − kH (C(0), h(w))] w→τ

≤ lim inf [kh(D) (a + 1, h(h −1 (γ (t)))) + kh(D) (h(0), a + 1) − kH (C(0), h(h −1 (γ (t))))] t→+∞

≤ lim inf [kH (1, t − a) − kH (C(0), t)] + kh(D) (h(0), a + 1) t→+∞

≤ lim inf [kH (1, t − a) − kH (1, t)] + kh(D) (h(0), a + 1) + kH (C(0), 1) t→+∞

1 1 = lim inf [ log(t − a) − log t] + kh(D) (h(0), a + 1) + kH (C(0), 1) < +∞, t→+∞ 2 2

and we are done.



Remark 17.7.7 By Theorem 13.5.7, h(D) contains a vertical half-plane if and only if (φt ) has a parabolic petal. Therefore (2) in Theorem 17.7.6 can be rephrased as follows. If (φt ) is a parabolic semigroup in D of positive hyperbolic step which has a parabolic petal, then (φt ) is of finite shift.

17.8 Notes The results in Sect. 17.2, Theorem 17.3.1 and its consequences, are taken from [35]. The asymptotic behavior of orbits of hyperbolic semigroups and parabolic semigroups of positive hyperbolic step was first studied in [46, 54, 62, 66]. Corollary 17.4.7, with a different proof, is in [64, Theorem 7.4]. The first examples of parabolic semigroups such that the slope of a trajectory is not a singleton appear in the papers [14, 50]. Indeed, in those examples the slope is the interval [−π/2, π/2]. An example of a parabolic semigroup with slope [c1 , c2 ], for some −π/2 < c1 < c2 < π/2, was constructed in [34], using methods of localization

17.8 Notes

539

of hyperbolic metrics. Corollary 17.6.7 and the construction of semigroups with arbitrary given slopes appeared in [87]. Proposition 17.5.6 is taken from [33], although (2) was already proved directly in [14], using methods of harmonic measure theory. Proposition 17.5.5 (and several generalizations of it, with different degrees of regularity at the Denjoy-Wolff point) can be found in [64]. Proposition 17.7.3 and Theorem 17.7.5 are the continuous versions of the analogous results for discrete iteration proved in [54]. In [15], Betsakos gives some characterizations of finite shift, in particular, he proves that a parabolic semigroup of positive hyperbolic step has finite shift if and only if the image of a horocycle via the Koenigs map is contained in a vertical half-plane whose closure does not contain the imaginary axis. In [84], Karamanlis, using methods of extremal length, gives a characterization of finiteness of shift in case the image of the Koenigs function of a parabolic semigroup of positive hyperbolic step contains certain angles around (0, +∞).

Chapter 18

Topological Invariants

In the previous chapters we introduced and studied various properties of semigroups in the unit disc such as the dynamical type (elliptic, hyperbolic, parabolic), the hyperbolic step, the Denjoy-Wolff point, boundary (regular or super-repelling) fixed points, regular poles, maximal contact arcs. All these objects are holomorphic invariants of a semigroup of holomorphic selfmaps. Namely, if two semigroups of holomorphic self-maps are holomorphically conjugated through an automorphism of the unit disc, there is a one-to-one correspondence among the objects mentioned above and the way they are displaced along the boundary of the disc. This follows easily from the fact that automorphisms of the unit disc are linear fractional maps. One might expect that lowering the regularity of the conjugation map, the number of invariants decreases. In this chapter we are interested in studying topological invariants of semigroups in the unit disc. Namely, we consider properties of holomorphic semigroups which are invariant under conjugation via homeomorphisms of the unit disc, without any assumption on the regularity of the conjugacy map on the boundary of the unit disc. One might expect that all holomorphic invariants related to the boundary behavior are destroyed. However, and quite surprisingly, most of them survive and are topological invariants. Roughly speaking, the holomorphic invariants which are related to the isometric (with respect to the hyperbolic distance) nature of holomorphic conjugacies are destroyed under topological conjugation, but, those invariants which are related to the dynamics survive, with the exception of exceptional maximal contact arcs.

18.1 Extension of Topological Conjugation for Non-Elliptic Semigroups In what follows we will use the following characterization of points belonging to exceptional maximal contact arcs in terms of Carathéodory’s prime end theory. © Springer Nature Switzerland AG 2020 F. Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4_18

541

542

18 Topological Invariants

Let (φt ) be a non-elliptic semigroup, let us denote by M(φt ) ⊂ ∂D the set of points which belong to an exceptional maximal contact arc for (φt ) (note that M(φt ) is open and possibly empty). Let τ ∈ ∂D be the Denjoy-Wolff point of (φt ) and let E(φt ) := M(φt ) ∪ {τ }. Proposition 18.1.1 Let (φt ) be a non-elliptic semigroup in D. Let (I × R, h, z → z + it) be the holomorphic model of (φt ), where I = R, (0, +∞), (−∞, 0) or (0, ρ) for some ρ > 0. Let σ ∈ ∂D be different from the Denjoy-Wolff point of (φt ). Then σ ∈ / E(φt ) if and only if the following condition holds: (L) there exist a null chain (Cn ) representing the prime end of h(D) corresponding to σ under h, a compact subinterval I  ⊂ I and N ∈ N such that Vn ⊂ I  × R for all n ≥ N , where Vn denotes the interior part of Cn . Proof Assume M is a non-empty exceptional maximal contact arc for (φt ). By Corollary 14.2.11, I = R and we can assume without loss of generality that I = (0, ρ), with ρ ∈ (0, +∞] and that h(M) = {it : t > d} for some d ∈ [−∞, +∞). If σ ∈ M, since, by Theorem 14.2.10, h extends holomorphically through M and h(M) ⊂ {it : t > d}, every null chain (Cn ) representing the prime end of h(D) corresponding to σ under h converges to {h(σ )}, hence it is not possible to find a closed subinterval I  ⊂ I and N ∈ N such that Vn ⊂ I  × R for all n ≥ N . If σ is either the starting or the final point of M, consider a null chain (Cn ) representing the prime end of h(D) corresponding to σ under h. Then, for each n, one of the end points of h −1 (Cn ), let us say σn , belongs to M. Since Re h(σn ) = 0, it turns out that it is not possible to find a closed subinterval I  ⊂ I and N ∈ N such that Vn ⊂ I  × R for all n ≥ N . Now assume σ ∈ / M. / M, σ is not Suppose first σ is a boundary regular fixed point of (φt ). Since σ ∈ the Denjoy-Wolff point of (φt ). Therefore, by Theorem 13.5.5, there exist a maximal invariant vertical strip S = {z ∈ C : a < Re z < b} ⊂ h(D), with 0 ≤ a < b ≤ ρ, such that (a, b) ⊂ I , and for every c ∈ (a, b), limt→−∞ h −1 (c + it) = σ . If a = 0, then σ would be the initial point of the exceptional maximal contact arc defined by {z ∈ ∂D : z = h −1 (a + it), t ∈ R}, therefore a > 0. Similarly one can see that b < ρ. Now, let Cn be the connected component of {Im z = −n} which intersects S. Then (Cn ) is a null chain representing the prime end of h(D) corresponding to σ under h. By construction, (Cn ) satisfies condition (L). In case σ is not a boundary regular fixed point of (φt ), by Corollary 13.6.9, there exists  ∈ [0, ρ] such that lim z→σ Re h(z) = . If  ∈ (0, ρ), take 0 < ε < min{, ρ − }. We claim that for every null chain (Cn ) which represents σ under h, there exists N ∈ N such that, for n ≥ N , Cn ⊂ {w ∈ C :  − ε ≤ Re w ≤  + ε}, from which (L) holds. Indeed, if this were not the case, there would exist a sequence {ζn } such that ζn ∈ Cn for all n and Re ζn <  − ε or Re ζn >  + ε for all n. But, h −1 (ζn ) → σ as n → ∞, hence Re ζn = Re h(h −1 (ζn )) → , leading to a contradiction.

18.1 Extension of Topological Conjugation for Non-Elliptic Semigroups

543

We are left to prove that  = 0 and  = ρ. Arguing by contradiction, we assume  = 0 (the case  = ρ is similar). If σ is a super-repelling fixed point of (φt ), then lim z→σ Im h(z) = −∞ by Corollary 13.6.7, while, if σ is not a fixed point of (φt ), there exists t0 ∈ R such that ∠ lim z→σ h(z) = it0 by Proposition 13.6.2. Let α := −∞ in case σ is a super-repelling fixed point of (φt ) and α = t0 in case σ is not a fixed point. Let Γ = h([0, 1)σ ). Since Γ + it ⊂ h(D) for all t ≥ 0, it follows that {z ∈ C : Re z = 0, Im z > α} ⊂ ∂h(D) and, by Schwarz’ Reflection Principle, there exists an open arc A ⊂ ∂D such that h extends holomorphically through A and h(A) = {z ∈ C : Re z = 0, Im z > α}. By Theorem 14.2.10, A is a contact arc of (φt ). Let M be the maximal contact arc of (φt ) which contains A. By Proposition 14.2.6, the final point x1 (M) of M is either a non-fixed point of (φt ) or the Denjoy-Wolff point. Since ∠ lim z→x1 (M) h(z) = limt→+∞ h(h −1 (it)) = +∞ by Proposition 14.2.13, it follows that x1 (M) is the Denjoy-Wolff point of (φt ) by Proposition 13.6.1. Hence, M is an exceptional maximal arc and h(M) = {z ∈ C : Re z = 0, Im z > β} for some β ∈ [−∞, α]. Let x0 (M) be the initial point of M. By Proposition 14.2.13, iβ = ∠ lim z→x0 (M) h(z). For every t ∈ (β, +∞), let rt := sup{x > 0 : (0, x) + it ⊂ h(D)}. Since h(D) is starlike at infinity, rt ≥ rs for all t ≥ s and hence rt > 0 for all t > β. Let D := ∪t>β ((0, rt ) + it). Clearly, D is a domain starlike at infinity. If β < α (which means in particular that α > −∞), then Γ is eventually contained in D, from which it follows immediately that σ ∈ M. Hence, β = α. Now, let T := h([0, 1]x0 (M)). If x0 (M) = σ then T ∩ Γ = {h(0), iα}. In particular, T ∪ Γ is a Jordan curve, which intersects iR at most in iα (if α > −∞). Let U be the connected component of C∞ \ (T ∪ Γ ) which does not contain it for t = α. We claim that U ⊂ h(D). Indeed, if there is w0 ∈ C \ h(D) such that w0 ∈ U , / h(D) for all t ≥ 0 since h(D) is starlike at infinity. If α > −∞, U then w0 − it ∈ is relatively compact in C, hence, there exists s < Im w0 such that Re w0 + is ∈ (Γ ∪ T ) ∩ (C \ h(D)). Since Re w0 > 0, and by construction (Γ ∪ T ) ∩ {z ∈ C : Re z > 0} ⊂ h(D), we have a contradiction. In case α = −∞, since Re w0 > 0, it follows that either lim inf (0,1) r →1 Re h(r σ ) ≥ Re w0 or lim inf (0,1) r →1 Re h(r x0 (M)) ≥ Re w0 . Since lim(0,1) r →1 Re h(r σ ) = 0, we have then lim inf (0,1) r →1 Re h(r x0 (M)) ≥ Re w0 (which means that T is eventually contained in {z ∈ C : Re z > Re w0 }). Let ζ ∈ M and let V ⊂ D be the connected component of D \ ([0, 1)x0 (M) ∪ [0, 1)ζ ) such that the intersection of its boundary with ∂D is contained in M. Set B = V ∩ ∂D. Since h extends holomorphic through B ∪ {ζ } and by Proposition 14.2.13, it follows that ∞ R := T ∪ h([0, 1]ζ ) ∪ h(B) ⊆ ∂∞ h(V ) and R is a Jordan curve in C∞ . Therefore it divides C∞ into two connected components, one of them, say W , contains h(V ). By construction, w0 ∈ W \ h(V ). Moreover, there exists s ≥ Im w0 such that, setting w1 := Re w0 + is, we have {z ∈ C : Re z = Re w0 , Im z > Im w1 } ⊂ h(D) / h(D). Clearly, w1 ∈ W . By Proposition 3.3.3, there exists ζ0 ∈ ∂D such and w1 ∈ that ∠ lim z→ζ0 h(z) = w1 . However, this is impossible because, by construction,

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∠ lim z→ p h(z) ∈ / W for all p ∈ ∂D \ B, and ∠ lim z→ p h(z) ∈ R for all p ∈ B. We reach then a contradiction. Thus, U ⊂ h(D). But then, by Proposition 3.3.5, σ = x0 (M). Summing up, we proved that if  = 0 then σ is contained in the closure of an  exceptional maximal contact arc of (φt ), hence  = 0, and we are done. Lemma 18.1.2 Let (φt ) and (φ˜ t ) be two non-elliptic semigroups in D. Let (Ω1 , h 1 , z → z + it) be the holomorphic model of (φt ) and let (Ω2 , h 2 , z → z + it) be the holomorphic model of (φ˜ t ). Write Ω j = I j × R, j = 1, 2, where I j is R, (−∞, 0), (0, +∞), or (0, ρ), with ρ > 0. Then (φt ) and (φ˜ t ) are topologically conjugated if and only if there exist a homeomorphism u : I1 → I2 and a continuous function v : I1 → R such that g(z) := u(Re z) + i(Im z + v(Re z)), z ∈ D,

(18.1.1)

satisfies g(h 1 (D)) = h 2 (D). Proof Let Q 1 := h 1 (D) and Q 2 := h 2 (D). By Proposition 9.8.7, (φt ) and (φ˜ t ) are topologically conjugated if and only if there exists a homeomorphism g : Ω1 → Ω2 such that g(z + it) = g(z) + it for all t ∈ R, z ∈ Ω1 and g(Q 1 ) = Q 2 . Assume that (φt ) and (φ˜ t ) are topologically conjugated. Write z = x + i y with x ∈ I1 and y ∈ R. Then g(x + i y) = g(x) + i y. Define u(x) := Re (g(x)) and v(x) = Im (g(x)), for all x ∈ I1 . Then u : I1 → I2 and v : I1 → R are continuous. Since also g −1 : Ω2 → Ω1 satisfies g −1 (z + it) = g −1 (z) + it for all t ∈ R, it follows that u is one-to-one. Indeed, if u(x1 ) = u(x2 ), then x1 = g −1 (g(x1 )) = g −1 (u(x1 ) + iv(x1 )) = g −1 (u(x2 ) + iv(x2 )) + i(v(x1 ) − v(x2 )) = x2 + i(v(x1 ) − v(x2 )). Taking real part, we get that x1 = x2 . In addition, if x ∈ I2 , then x = g(g −1 (x)) = g(Re g −1 (x) + iIm g −1 (x)) = g(Re g −1 (x)) + iIm g −1 (x) = u(Re g −1 (x)) + i(Im g −1 (x) + v(Re g −1 (x))), and x = u(Re g −1 (x)). Thus, u is a homeomorphism. Conversely, given an homeomorphism u : I1 → I2 and a continuous function v : I1 → R such that g(z) := u(Re z) + i(Im z + v(Re z)) satisfying g(h 1 (D)) = h 2 (D), it is clear that g and the map z → z + it commute and  g : Ω1 → Ω2 is a homeomorphism. Recall that, if (φt ) is a semigroup in D then for every σ ∈ ∂D the non-tangential limit φt (σ ) := ∠ lim z→σ φt (z) exists for all t ≥ 0.

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Theorem 18.1.3 Let (φt ) and (φ˜ t ) be two non-elliptic semigroups of holomorphic self-maps of D. Suppose (φt ) and (φ˜ t ) are topologically conjugated via the homeomorphism f : D → D. Then f extends to a homeomorphism f : D \ E(φt ) → D \ E(φ˜ t ). Moreover, for all σ ∈ D \ E(φt ), the life-time T (σ ) = T ( f (σ )) and f (φt (σ )) = φ˜ t ( f (σ )) for all t ≥ 0. Proof Let (Ω1 = I1 × R, h 1 , z → z + it) be the holomorphic model of (φt ) and let (Ω2 = I2 × R, h 2 , z → z + it) be the holomorphic model of (φ˜ t ), where I1 , I2 are intervals of the form (−∞, 0), (0, +∞), (0, ρ), with ρ > 0, or R. As usual, let hˆ 1 denote the homeomorphism from ∂D to the Carathéodory prime-ends boundary of h 1 (D) defined by h 1 . Let σ ∈ ∂D \ E(φt ). Let (Cn ) be a null chain in h 1 (D) representing the prime end corresponding to σ under h 1 and denote by Vn the interior part of Cn . By Lemma 4.1.13, (h −1 (Cn )) is a null chain for D such that I ([(h −1 (Cn ))]) = {σ }. Since σ is not the Denjoy-Wolff point of (φt ), by Proposition 11.1.8, lim supz→σ Im h 1 (z) < K for some constant K < +∞. By Theorem 11.1.4, it follows that sup{Im z : z ∈ I ([(Cn )])} < +∞.

(18.1.2)

By Proposition 18.1.1, we can assume that Vn ⊂ I1 × R for some compact subinterval I1 ⊂ I1 and for all n ∈ N. By Proposition 9.7.3, f = h −1 2 ◦ g ◦ h 1 , where, by Lemma 18.1.2, g is given by (18.1.1). In particular, setting g(∞) := ∞, it follows ∞ that g is uniformly continuous on I1 × R . From this it follows easily that g(Cn ) is a null chain in h 2 (D). Moreover, if Wn is the interior part g(Cn ), then Wn = g(Vn ) and Wn is contained in u(I1 ) × R for all n ∈ N, with u(I1 ) being a compact subinterval of I2 . Let σ˜ ∈ ∂D be such that (g(Cn )) represents the prime end corresponding to h 2 (σ˜ ) under h 2 . Then g(I ([(Cn )])) = I ([(g(Cn ))]), and by (18.1.2), sup{Im z : z ∈ I ([(g(Cn ))])} < +∞, which, in turns, implies that lim supz→σ˜ Im h 2 (z) < +∞. Since the non-tangential limit of Im h 2 at the Denjoy-Wolff point of (φ˜ t ) is +∞, this implies that σ˜ is not the Denjoy-Wolff point of (φ˜ t ). Hence, by Proposition 18.1.1, σ˜ does not belong to the closure of an exceptional maximal compact arc for (φ˜ t ). Now we show that, in fact, lim z→σ f (z) = σ˜ . Let {z n } be a sequence converging to σ . Then {h 1 (z n )} is eventually contained in Vm for all m ∈ N. Thus {g(h 1 (z n ))} is eventually contained in g(Vm ) = Wm for all m ∈ N, which implies that { f (z n )} = ˜ , that is, lim z→σ f (z) = σ˜ . {h −1 2 (g(h 1 (z n )))} converges to σ Next, if σ1 , σ2 ∈ ∂D \ E(φt ) are two different points, the null chains (Cn1 ), (Cn2 ) in h 1 (D) are not equivalent. Therefore, if we choose those null chains satisfying condition (L) in Proposition 18.1.1 for all n ∈ N, it follows easily that (g(Cn1 )) and

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(g(Cn2 )) are not equivalent. Thus, if σ˜ j ∈ ∂D is such that (g(Cn )) represents hˆ 2 (σ˜ j ), j = 1, 2, it follows that σ˜ 1 = σ˜ 2 . Hence, the extension of f to ∂D \ E(φt ) is injective. Finally, we prove that f : D \ E(φt ) → D \ E(φ˜ t ) is continuous. In order to prove continuity of f , it is enough to show that if {σm } ⊂ ∂D \ E(φt ) converges to σ ∈ ∂D \ E(φt ) then limm→∞ f (σm ) = f (σ ). Let σ˜ m := f (σm ), σ˜ := f (σ ). For each m ∈ N, let (Cnm ) be a null chain in h 1 (D) representing hˆ 1 (σm ), chosen so that condition (L) of Proposition 18.1.1 is satisfied for all m, and similarly let (Cn ) be a null chain which satisfies condition (L) and represents hˆ 1 (σ ). As shown before, (g(Cnm )) is a null chain in h 2 (D) representing hˆ 2 (σ˜ m ) and (g(Cn )) is a null chain in h 2 (D) representing hˆ 2 (σ˜ ). Fix an open subset U ⊂ h 2 (D) such that g(Cn ) ⊂ U for all large enough n. Then, g −1 (U ) eventually contains (Cn ). Since σm → σ , this implies that for every big enough m, (Cnm ) is eventually contained in g −1 (U ). Hence (g(Cnm )) is eventually contained in U for big enough m. Therefore, σ˜ m → σ and f is continuous. Since the same applies to f −1 , it follows that f : D \ E(φt ) → D \ E(ϕt ) is a homeomorphism. In order to prove the last equations, fix σ ∈ D \ E(φt ) and fix t ≥ 0. Let r ∈ (0, 1). Then φ˜ t ( f (r p)) = f (φt (r σ )) → f (φt (σ )) as r → 1. Therefore, the limit of φ˜ t along the continuous curve r → f (r σ ) (which converges to f (σ )) is f (φt (σ )). By Theorem 3.3.1, φ˜ t ( f (σ )) = ∠ lim z→ f (σ ) φ˜ t (z) = f (φt (σ )), and thus the functional equation holds at σ . From this, it follows at once that T (σ ) = T ( f (σ )).  j

18.2 Topological Invariants for Non-Elliptic Semigroups In this section we describe invariants of semigroups in D under a topological conjugation. Proposition 18.2.1 Let (φt ) and (φ˜ t ) be two non-elliptic semigroups in D. Suppose (φt ) and (φ˜ t ) are topologically conjugated via the homeomorphism f : D → D. Then Δ ⊂ D is a petal of (φt ) if and only if f (Δ) is a petal of (φ˜ t ). Proof Let W be the backward invariant set of (φt ) and W˜ the backward invariant set of (φ˜ t ). Since f is a homeomorphism, f (D) = D and f ◦ φt ◦ f −1 = φ˜ t , t ≥ 0, we have W˜ = ∩t≥0 φ˜t (D) = ∩t≥0 f (φt (D)) = f (∩t≥0 φt (D)) = f (W ). Moreover, since f is open, it maps connected components of the interior part of W onto connected components of W˜ , and the statement follows at once.  Proposition 18.2.2 Let (φt ) and (φ˜ t ) be two non-elliptic semigroups in D. Suppose (φt ) and (φ˜ t ) are topologically conjugated via the homeomorphism f : D → D. If M is a maximal contact arc for (φt ) which is not exceptional, then f extends to a

18.2 Topological Invariants for Non-Elliptic Semigroups

547

homeomorphism from M onto f (M) and f (M) is a maximal contact arc for (φ˜ t ) which is not exceptional. Proof Since M is not exceptional, then M ⊂ ∂D \ E(φt ). Thus, by Theorem 18.1.3, f extends as a homeomorphism on M and f (M) ⊂ ∂D \ E(φ˜ t ). Clearly, f (M) is an arc in ∂D. Let σ ∈ M. By Theorem 18.1.3, T (σ ) = T ( f (σ )) > 0, hence φ˜ s ( f (σ )) = f (φs (σ )), s ∈ (0, T (σ )). Therefore, f (M) is a contact arc. Let M  be a maximal contact arc for (φ˜ t ) such that f (M) ⊆ M  . Since f (M) ⊂ ∂D \ E(φ˜ t ), then M  is not exceptional. Repeating the argument with f −1 and M  , we have that f −1 (M  ) is contained in a maximal contact arc M  for (φt ). Taking into account that M ⊆ f −1 (M  ), by the maximality of M we have M  = M, and then f (M) = M  . Thus f (M) is a maximal contact arc and we are done.  Now we consider boundary fixed points and their type. Proposition 18.2.3 Let (φt ) and (φ˜ t ) be two non-elliptic semigroups in D. Suppose (φt ) and (φ˜ t ) are topologically conjugated via the homeomorphism f : D → D. / E(φt ). Then the Let σ ∈ ∂D be a boundary fixed point for (φt ). Suppose that σ ∈ unrestricted limit f (σ ) := lim f (z) ∈ ∂D z→σ

exists and f (σ ) ∈ ∂D \ E(φ˜ t ) is a boundary fixed point for (φ˜ t ). Moreover, (1) if σ is a repelling fixed point for (φt ), then f (σ ) is a repelling fixed point for (φ˜ t ); (2) if σ is a boundary super-repelling fixed point of first type (respectively of second type) for (φt ), then f (σ ) is a boundary super-repelling fixed point of first type (respectively of second type) for (φ˜ t ). Proof First assume σ is a boundary fixed point which does not belong to the closure of an exceptional maximal contact arc. By Theorem 18.1.3, f extends continuously at σ , f (σ ) is not in the closure of any exceptional maximal contact arc and T (σ ) = T ( f (σ )). Therefore, f (σ ) is a boundary fixed point for (φ˜ t ) different from the Denjoy-Wolff point of (φ˜ t ). If σ is a repelling fixed point of (φt ) then by Proposition 13.4.12, there exists a unique hyperbolic petal Δ for (φt ) such that σ ∈ ∂Δ. Moreover, by Proposition 13.4.10, ∂Δ does not contain any boundary fixed point of (φt ) but σ and the Denjoy-Wolff point of (φt ). By Proposition 18.2.1, f (Δ) is a petal for (φ˜ t ). Since f (σ ) ∈ ∂ f (Δ) and f (σ ) is a fixed point of (φ˜ t ) different from the Denjoy-Wolff point of (φ˜ t ), it follows again by Proposition 13.4.10 that f (σ ) is a repelling fixed point. If σ is a super-repelling fixed point of first type for (φt ), then it is the initial point of a maximal invariant curve Γ , hence it is easy to see that f (σ ) is a super-repelling fixed point for (φ˜ t ) which starts the maximal invariant curve f (Γ ), therefore it is of first type.

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The same argument implies that f −1 maps repelling fixed points onto repelling fixed points and super-repelling fixed points of the first type onto super-repelling fixed points of the first type. Therefore, f maps super-repelling fixed points of second type  for (φt ) onto super-repelling fixed points of second type for (φ˜ t ). Another immediate consequence of Theorem 18.1.3 is that the continuity of a semigroup at a boundary point is a topological invariant: Proposition 18.2.4 Let (φt ) and (φ˜ t ) be two non-elliptic semigroups in D. Suppose (φt ) and (φ˜ t ) are topologically conjugated via the homeomorphism f : D → D. Let σ ∈ ∂D \ E(φt ) be such that the unrestricted limit lim z→σ φt (z) exists for all t ≥ 0. Then the unrestricted limit f (σ ) = lim z→σ f (z) ∈ ∂D \ E(φ˜ t ) and the unrestricted limit lim z→ f (σ ) φ˜ t (z) exist for all t ≥ 0. Proof By Theorem 18.1.3, f has unrestricted limit at σ , f (σ ) ∈ / E(φ˜ t ) and f −1 −1 extends continuously at f (σ ), f ( f (σ )) = σ . Since φ˜ t (z) = f (φt ( f −1 (z))), z ∈ D, the result follows. 

18.3 Exceptional Maximal Contact Arcs and the Denjoy-Wolff Point In this section we examine exceptional maximal contact arcs and boundary DenjoyWolff points under topological conjugation. Roughly speaking, we will see that the behavior of the topological intertwining map on an exceptional maximal contact arc can be quite wild. We start with the following result: Proposition 18.3.1 Let (φt ) and (φ˜ t ) be two non-elliptic semigroups in D. Suppose (φt ) and (φ˜ t ) are topologically conjugated via the homeomorphism f : D → D. (1) If σ ∈ ∂D is a repelling fixed point for (φt ) which starts an exceptional maximal contact arc, then the non-tangential limit f (σ ) = ∠ lim z→σ f (z) exists and f (σ ) is a boundary regular fixed point for (φ˜ t ) (possibly the Denjoy-Wolff point of (φ˜ t )). (2) If σ ∈ ∂D is the Denjoy-Wolff point of (φt ) then the non-tangential limit f (σ ) = ∠ lim z→σ f (z) exists and f (σ ) ∈ ∂D is the Denjoy-Wolff point of (φ˜ t ). Proof (1) Let (Ω1 = I1 × R, h 1 , z → z + it) be the holomorphic model of (φt ) and let (Ω2 = I2 × R, h 2 , z → z + it) be the holomorphic model of (φ˜ t ), where I1 , I2 are (possibly unbounded) open intervals in R. Assume M is the exceptional maximal contact arc for (φt ) such that σ is its initial point. By Corollary 14.2.11, I = R and we can assume without loss of generality that I1 = (0, ρ), with 0 < ρ ≤ +∞ and h(M) = {it : t ∈ R}.

18.3 Exceptional Maximal Contact Arcs and the Denjoy-Wolff Point

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By Theorem 13.5.6, there are a petal Δ associated with σ and 0 < b ≤ ρ, b < +∞, such that h(Δ) = {ζ ∈ C : 0 < Re ζ < b}. Note that b = ρ if and only if h(D) = Ω1 , that is, (φt ) is a (hyperbolic) group of automorphisms. By Proposition 9.7.3 and Lemma 18.1.2, f = h −1 2 ◦ g ◦ h 1 , with g(z) = u(Re z) + i(Im z + v(Re z)), where u : I1 → I2 is a homeomorphism. With no loss of generality we can assume that u is orientation preserving, and we let a˜ = lim x→0+ u(x) and b˜ = lim x→b− u(x). By Proposition 18.2.1, f (Δ) is a petal for (φ˜ t ) and D˜ := g(h 1 (Δ)) = {ζ ∈ C : ˜ Let σ ( D) ˜ denote the only boundary regular fixed point for (φ˜ t ) a˜ < Re ζ < b}. defined by D˜ (see Theorem 13.5.6). Let γ : [0, 1) → D be a continuous curve which converges non-tangentially to σ . By Theorem 13.6.6, limt→1 Im h 1 (γ (t)) = −∞ and there exist a < a  < b < b and t0 < 1 such that a  ≤ Re h 1 (γ (t)) ≤ b for all t ∈ (t0 , 1). Therefore, there exist a˜ < a˜  < b˜  < b˜ such that a˜  ≤ Re g(Re (h 1 (γ (t)))) ≤ b˜  for all t ∈ (t0 , 1). Thus, the continuous curve g ◦ h 1 ◦ γ is eventually contained in the strip {ζ ∈ C : a˜  < Re ζ < b˜  } and its imaginary part converges to −∞. It follows then from Proposition 3.3.3 applied to h 2 that ˜ lim f (γ (t)) = lim h −1 2 (g(h 1 (γ (t))) = σ ( D).

t→1

t→1

˜ The arbitrariness of the curve γ implies that ∠ lim z→σ f (z) = σ ( D). (2) In case σ is the Denjoy-Wolff point of (φt ), the argument is similar and we omit the proof.  Remark 18.3.2 In the hypothesis of Proposition 18.3.1, if σ ∈ ∂D is a repelling fixed point for (φt ) which starts an exceptional maximal contact arc and f (σ ) is the Denjoy-Wolff point of (φ˜ t ), then (φ˜ t ) is parabolic. Indeed, the previous proof shows that if Δ is the hyperbolic petal of (φt ) whose closure contains σ then f (σ ) is a regular fixed point of (φ˜ t ) contained in the closure ˜ Moreover, f (σ ) = limt→+∞ h −1 of the petal f (Δ) and h 2 ( f (Δ)) = D. 2 (w0 − it) for ˜ ˜ with |a|, ˜ < ˜ any w0 ∈ D. If (φt ) is hyperbolic, then D˜ = {ζ ∈ C : a˜ < Re ζ < b}, ˜ |b| ˜ +∞. Therefore, by Theorem 13.5.6, f (Δ) is a hyperbolic petal of (φt ) and f (σ ) is the only repelling fixed point contained in its closure. In the following examples we show that the unrestricted limit of the homeomorphism f might no exist at the Denjoy-Wolff point or at boundary regular fixed points which start an exceptional maximal contact arc. Example 18.3.3 (Non existence of unrestricted limit at Denjoy-Wolff point) Let Ω = {z ∈ S : Im z Re z > 1} and let h : D → Ω be a Riemann map of Ω. Consider the semigroup (φt ) defined by φt (z) := h −1 (h(z) + it), t ≥ 0. The point τ = limt→+∞ h −1 ( 21 + it) is the Denjoy-Wolff point of (φt ). Let v : (0, 1) → R be defined by v(x) = −1/x for all x ∈ (0, 1), and define g(z) := Re z + i(Im z + v(Re z)). Write Ω˜ = g(Ω) = {z ∈ S : Im z > 0} and let h˜ : D → Ω˜ be a Rie˜ ˜ The semigroup (φ˜ t ) defined by φ˜ t (z) := h˜ −1 (h(z) + it), z ∈ D mann map of Ω.

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and t ≥ 0, is topologically conjugated to (φt ) via f := h˜ −1 ◦ g ◦ h. We claim that the unrestricted limit of f at τ does not exist. By Proposition 18.3.1, the point τ˜ := ∠ lim z→τ f (z) is the Denjoy-Wolff point of (φ˜ t ). Notice that h and h˜ can be ˜ respectively by Theorem 4.3.3. extended to homeomorphisms of D onto Ω and Ω,   ˜ We still denote by h and h those extensions. Take z n = h −1 n1 + (1 + n)i ∈ D. By the properties of h, we deduce that the sequence {z n } converges to τ˜ . Moreover, f (z n ) = h˜ −1 (g(h(z n ))) = h˜ −1



1 +i n



→ h˜ −1 (i) = τ˜ .

Example 18.3.4 (Non existence of unrestricted limit at boundary regular fixed points starting an exceptional maximal contact arc) Let h : D → S be the Rie. Let (φt ) be the (semi)group of hypermann map given by h(z) = 21 + πi log 1+z 1−z bolic automorphisms of D defined by φt (z) := h −1 (h(z) + it), t ≥ 0. Then 1 = limIm z→+∞ h −1 (z) is the Denjoy-Wolff point of (φt ), while −1 = limIm z→−∞ h −1 (z) is a boundary regular fixed point of (φt ). Let g : S → S be the homeomorphism defined by g(z) = Re z + i(Im z + Re1 z ). Notice that g(w + it) = g(w) + it for all w ∈ S and t ∈ R. Then f := h −1 ◦ g ◦ h is a homeomorphism of D and f ◦ φt (z) = h −1 ◦ g ◦ h ◦ h −1 (h(z) + it) = h −1 ◦ g(h(z) + it) = h −1 (g ◦ h(z) + it) = h −1 (h(h −1 ◦ g ◦ h(z)) + it) = h −1 (h( f (z)) + it) = φt ◦ f (z), z ∈ D, t ≥ 0. By Proposition 18.3.1, f has non-tangential limit at −1 and it is easy to see that ∠ lim z→−1 f (z) = −1. Consider now the curve γ : (0, 1) → S defined by γ (s) := (1 − s) + i(s − 1)−1 . Then h −1 (γ (s)) is a curve in D which converges (tangentially) to −1 as s → 1. However, lim f (h −1 (γ (s))) = lim h −1 (g(γ (s))) = lim h −1 (1 − s) = h −1 (0) = −1.

s→1

s→1

s→1

Therefore, f is not continuous at −1. As we already proved in Theorem 9.8.4, every non-elliptic semigroup of holomorphic self-maps can be topologically conjugated to a hyperbolic one (with S as model domain). It follows that every non-elliptic semigroup can be topologically conjugated to one whose model domain is C (that is, a parabolic semigroup of zero hyperbolic step). In this case, by Corollary 14.2.11, there exist no exceptional maximal contact arcs, hence, exceptional maximal contact arcs are not topological invariants. However, if one stays in the class of hyperbolic semigroups, the question whether an exceptional maximal contact arc is a topological invariant is less trivial, as the following example shows:

18.3 Exceptional Maximal Contact Arcs and the Denjoy-Wolff Point

551

Example 18.3.5 Let Ω = {z ∈ S : Im z Re z > −1} and let h : D → Ω be a Riemann map of Ω. For t ≥ 0, let φ˜ t (z) := h −1 (h(z) + it). Clearly, (φ˜ t ) is a semigroup in D. The point σ = limt→0 h −1 (t − i/2t) is a boundary super-repelling fixed point of third type for this semigroup (see Example 14.5.4). Let v : (0, 1) → R be a continuous function and define g(z) := Re z + i(Im z + v(Re z)). Write Ωv := g(Ω) = {z ∈ S : Im z > − Re1 z + v(Re z)}. Let h v : D → Ωv be a Riemann map of Ωv . Let φt (z) := h −1 v (h v (z) + it), z ∈ D and t ≥ 0. The semigroup (φt ) is clearly topologically conjugated to (φ˜ t ). On the one hand, taking v(x) := 2/x, we obtain a semigroup whose unique fixed point is its Denjoy-Wolff point and the map f = h −1 v ◦g◦h sends the exceptional maximal contact arc starting at σ to the Denjoy-Wolff point of (φt ). In particular, the boundary super-repelling fixed point of third type σ is sent to the Denjoy-Wolff point. On the other hand, taking v(x) = 1/x we obtain a semigroup whose unique fixed point is the Denjoy-Wolff point and the map f = h −1 v ◦g◦h sends the exceptional maximal contact arc starting at σ onto an exceptional maximal contact arc for (φt ) having a non-fixed point as initial point. In particular, the boundary super-repelling fixed point of third type σ is mapped to a contact, not fixed, point. Proposition 18.3.6 Let (φt ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Let M be an exceptional maximal contact arc for (φt ) whose initial point is σ ∈ ∂D. (1) If σ is a repelling fixed point for (φt ) and (φ˜ t ) is a hyperbolic semigroup of holomorphic self-maps of D, topologically conjugated to (φt ) via the homeomorphism f : D → D, then (φ˜ t ) has an exceptional maximal contact arc whose initial point ∠ lim z→σ f (z) is a repelling fixed point. (2) If σ is not a boundary regular fixed point for (φt ), then there exists a homeomorphism f : D → D such that (φ˜ t ) := ( f −1 ◦ φt ◦ f ) is a hyperbolic semigroup of holomorphic self-maps of D and lim z→ p f (z) = τ for all p ∈ M. Proof (1) By Remark 18.3.2, f (σ ) := ∠ lim z→σ f (z) is a repelling fixed point of (φ˜ t ). If f (σ ) does not belong to the closure of an exceptional maximal contact arc for (φ˜ t ), by Theorem 18.1.3, f −1 has unrestricted limit at f (σ ) and σ =

lim

(0,1) r →1

f −1 ( f (r σ )) = f −1 ( f (σ )) ∈ / E(φt ),

a contradiction. Therefore f (σ ) belongs to the closure of an exceptional maximal contact arc M for (φ˜ t ). Being f (σ ) a fixed point, it cannot sit in M, and, by Proposition 14.2.6, is in fact the initial point of M. (2) Let (Ω = (0, ρ) × R, h, z → z + it), 0 < ρ < +∞, be the holomorphic model of (φt ) and let Q := h(D). Let M be an exceptional maximal contact arc with initial point σ . By Theorem 14.2.10, we can assume without loss of generality that Re h(z) = 0 for all z ∈ M. Assume σ is not a boundary regular fixed point for (φt ).

552

18 Topological Invariants

If σ is a super-repelling boundary fixed point for (φt ), let / Q, s ∈ (0, 1/n]}. λn = sup{y : s + i y ∈ Since σ is super-repelling, by Corollary 13.6.7, we have that limr →1 Re h(r σ ) = 0 and limr →1 Im h(r σ ) = −∞. Thus λn tends to −∞. For each n ∈ N, take 0 < sn ≤ / Q and yn ≥ λn − 1/n. Take a monotone subsequence 1/n and yn such that sn + i yn ∈ (sn k ) of (sn ) and a continuous function v : (0, ρ) → R such that v(sn k ) = −2λn k . Define g : Ω → Ω by g(x + i y) := x + i(y + v(x)). Note that g is a homeomorphism. Let h 2 : D → g(Q) be a Riemann map of g(Q). Let (φ˜ t ) be the semigroup ˜ in D defined by φ˜ t (z) := h −1 2 (h 2 (z) + it). Clearly, (φt ) is topologically conjugated / g(Q) and to (φt ). Since wn = g(sn + i yn ) = g(sn ) + i yn ∈ lim Im (wn k ) = lim(yn k + v(sn k )) ≥ lim(−λn k − 1/n k ) = +∞, k

k

k

it follows that f maps the exceptional maximal contact arc M to the Denjoy-Wolff point of (φ˜ t ). Next, suppose that σ is not a boundary fixed point. Then, by Theorem 11.1.4, there exists limr →1 h(r σ ) = i y0 with y0 ∈ R. Therefore there exist points xn + i yn ∈ Ω \ Q such xn goes to 0 and yn goes to y0 . Take a continuous function v : (0, ρ) → R such that lim x→0 v(x) = +∞. As before, define g : Ω → Ω by g(x + i y) := x + i(y + v(x)) and let h 2 : D → g(Q) be any Riemann map. By construction, g(xn + i yn ) ∈ Ω \ h 2 (D) and Im g(xn + i yn ) goes to +∞. Therefore the semigroup (φ˜ t ) defined by φ˜ t (z) := h −1 2 (h 2 (z) + it) for t ≥ 0 is topologically conjugated to (φt ) and f maps  the exceptional maximal contact arc M to the Denjoy-Wolff point of (φ˜ t ). The cluster set of f at an exceptional maximal contact arc is described by the following proposition: Proposition 18.3.7 Let (φt ) and (φ˜ t ) be two non-elliptic semigroups in D with holomorphic models (Ω1 = I1 + iR, h 1 , z → z + it) and (Ω2 = I2 + iR, h 2 , z → z + it), respectively. Suppose (φt ) and (φ˜ t ) are topologically conjugated via the homeomorphism f : D → D. Let τ and τ˜ be the Denjoy-Wolff points of (φt ) and (φ˜ t ), respectively. Assume that M is an exceptional maximal contact arc for (φt ). Denote by S = {z ∈ D \ {0} : z/|z| ∈ M} and let σ be the initial point of M. Then the set   E(M) = w ∈ D : ∃{z n } ⊂ S, z n → z ∈ M, f (z n ) → w is a compact connected arc in ∂D containing τ˜ which is either equal to {τ˜ } or it is contained in the closure of an exceptional maximal contact arc for (φ˜ t ). Proof Let Q 1 := h 1 (D) and let Q 2 := h 2 (D). By Lemma 18.1.2, (φt ) and (φ˜ t ) are topologically conjugated if and only if there exists a homeomorphism g : Ω1 → Ω2 given by (18.1.1) and g(Q 1 ) = Q 2 . By Proposition 9.7.3, f = h −1 2 ◦ g ◦ h1.

18.3 Exceptional Maximal Contact Arcs and the Denjoy-Wolff Point

553

Since ∠ lim z→τ f (z) = τ˜ by Proposition 18.3.1, it follows that τ˜ ∈ E(M). Moreover, it is easy to see that E(M) is a compact subset of ∂D. Therefore we are left to check that E(M) is connected. Indeed, assume this is not the case. Then there exist two compact sets A and B such that E(M) = A1 ∪ A2 and A1 ∩ A2 = ∅. Denote by k > 0 the euclidean distance between A and B. For j = 1, 2, take w j ∈ A j , z n, j ∈ D, z for all n, with |zn,n, jj | ∈ M, {z n, j } → z j ∈ M and { f (z n, j )} → w j . We may assume that | f (z n, j ) − w j | < k/4 for all j and n. Let Cn be the arc in M that joins z n,2 and |z n,2 | Γn = [z n,1 , rn z n,1 ] ∪ rn Cn ∪ [z n,2 , rn z n,2 ]

z n,1 |z n,1 |

with

where rn = max{|z n,1 |, |z n,2 |}. Notice that Γn is connected. Consider the continuous function l : Γn → R being l(z) the distance between f (z) and A1 . Since l(z n,1 ) < k/4 and w2 ∈ A2 , we have l(z n,2 ) ≥ k − |z n,2 − w2 | ≥ k − k/4 = 3k/4. Thus there is αn ∈ Γn such that l(αn ) = k/2. Since |αn | ≥ min{|z n,1 |, |z n,2 |}, we can take a subsequence such that αn k → z ∈ M and f (αn k ) → w. Clearly, w ∈ E(M) and l(w) = k/2. A contradiction. Hence E(M) is connected. Finally, if E(M) = {τ˜ }, E(M) is contained in the closure of an exceptional maximal contact arc for (φ˜ t ), for otherwise f −1 would map points of ∂D \ E(φ˜ t ) into  E(φt ). Example 18.3.8 Let Ω1 = {z ∈ S : Im z > 0} and h 1 : D → Ω1 a Riemann map of Ω1 . Consider the semigroup (φt ) defined by φt (z) := h −1 1 (h 1 (z) + it), t ≥ 0. The arc M = h −1 1 ([0, ∞)i) is an exceptional maximal contact arc for (φt ). Define g : S → S as g(x + i y) := x + i(y − 1/x), Ω2 := g(Ω1 ) = {z ∈ S : Im z Re z > −1} and let h 2 : D → Ω2 be a Riemann map of Ω2 . The semigroup (φ˜ t ) defined by φ˜ t (z) := h −1 2 (h 2 (z) + it), t ≥ 0, is topological conjugated to (φt ) via the homeomorphism ˜ ˜ f = h −1 2 ◦ g ◦ h 1 . The semigroup (φt ) has an exceptional maximal contact arc M = −1 h 2 (Ri) with initial point a fixed point σ . Notice that for all w ∈ M it holds σ = lim z→w f (z), namely, the map f sends the arc M to the point σ . This does not contradict the previous proposition, since M˜ is the cluster set of f at the DenjoyWolff point of (φt ).

18.4 Elliptic Case In this final section we show how to recover the results of Sects. 18.1 and 18.2 in case of elliptic semigroups which are not groups. The key point is to replace Lemma 18.1.2 by the following lemma (whose proof is similar to that of Lemma 18.1.2): Lemma 18.4.1 Let (φt ) and (φ˜ t ) be two elliptic semigroups in D, which are not groups. Let (C, h 1 , z → eλ1 t z) be a holomorphic model of (φt ) and let (C, h 2 , z → eλ2 t z) be a holomorphic model of (φ˜ t ). Then, (φt ) and (φ˜ t ) are topologically conjugated if and only if there exist a homeomorphism u of the unit circle ∂D ◦ g0 ◦ θλ1 satisfies and a continuous map v : ∂D → (0, +∞) such that g := θλ−1 2

554

18 Topological Invariants

g(h 1 (D)) = h 2 (D), where  g0 (z) :=

0, z=0 , z ∈ C, |z|u(z/|z|)v(z/|z|), z = 0

and, given λ = a + ib with a < 0,   b θλ (z) := z|z|−(1+1/a) exp −i Log|z| , z ∈ C, z = 0, and θλ (0) := 0. a As already remarked, if the semigroup (φt ) is elliptic, the set E(φt ) = ∅. Using the previous lemma and mimicking the proof of Theorem 18.1.3, one can prove the following extension result: Theorem 18.4.2 Let (φt ) and (φ˜ t ) be two elliptic semigroups in D, which are not groups. Suppose (φt ) and (φ˜ t ) are topologically conjugated via the homeomorphism f : D → D. Then f extends to a homeomorphism f : D → D. Moreover, for all σ ∈ ∂D the life-time T (σ ) = T ( f (σ )) and f (φt (σ )) = φ˜ t ( f (σ )) for all t ≥ 0. And also, Proposition 18.4.3 Let (φt ) and (φ˜ t ) be two elliptic semigroups in D, which are not groups. Suppose (φt ) and (φ˜ t ) are topologically conjugated via the homeomorphism f : D → D. (1) If M is a maximal contact arc for (φt ), then f extends to a homeomorphism from M onto f (M) and f (M) is a maximal contact arc for (φ˜ t ). (2) Let σ ∈ ∂D be a boundary fixed point for (φt ). Then the unrestricted limit f (σ ) := lim f (z) ∈ ∂D z→σ

exists and f (σ ) ∈ ∂D is a boundary fixed point for (φ˜ t ). Moreover, (i) if σ is a boundary regular fixed point for (φt ), then f (σ ) is a boundary regular fixed point for (φ˜ t ); (ii) if σ is a boundary super-repelling fixed point of first type (respectively of second type) for (φt ), then f (σ ) is a boundary super-repelling fixed point of first type (respectively of second type) for (φ˜ t ). We end this section showing that Theorem 18.4.2 is no longer true for groups of elliptic automorphisms.

18.4 Elliptic Case

555

Example 18.4.4 Consider the group of automorphisms (φt ) where φt (z) = eit z for all t ∈ R and z ∈ D and the continuous function f : D → D given by f (z) = z exp(i ln(1 − |z|)) for all z ∈ D. It is clear that f is a homeomorphism of the unit disc with inverse function f −1 (z) = z exp(−i ln(1 − |z|)) and f (φt (z)) = φt ( f (z)), t ≥ 0, z ∈ D. However, f has no continuous extension at any point of ∂D.

18.5 Notes The chapter is based on [30].

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Index

A Accessible point via a Jordan arc , 113 Accessible prime end, 113 Algebraic group, 222 Algebraic semigroup, 214, 222 Angular derivative, 38 Arc length parameter of a curve, 120 Area Theorem, 80 Argument of a complex number, xxiii

B Backward invariant set of a semigroup, 372 Backward orbit, 355 Base space of a holomorphic semi-model, 236 Berkson-Porta Formula, 278 β-point, 431 Bloch function, 187 Bloch space B, 187 Boundary critical point, 330 Boundary dilation coefficient, 18 Boundary fixed point of a self-map of the unit disc, 54 Boundary fixed point of a semigroup, 327 Boundary regular critical point, 330 Boundary regular fixed point of a self-map, 55 Boundary regular fixed point of a semigroup, 329

Carathéodory Kernel Convergence Theorem, 89 Carathéodory topology, 101 Cauchy’s Functional Equation, 208 Cayley transform, 8 Circle, xxiv Circular null chain, 93 Cluster set of a curve, 55 Cluster set of a function, 108 Complete vector field, 274 Conjugated semigroups, 267 Connected im kleinen, 99 Contact arc of a semigroup, 411 Contact point of a self-map of the unit disc, 54 Contact point of a semigroup, 411 Continuous group, 222 Continuous semigroup, 205, 222 Convex domains, 127 Cross cut, 91

D Denjoy-Wolff point, 51 Denjoy-Wolff point of the semigroup, 220 Disc, xxiv Divergence rate, 231 Domain, xxiv Dual infinitesimal generator, 446

E Elliptic function, 51 C Elliptic groups, 218 Canonical model, 248 Elliptic semigroups, 220 Carathéodory boundary of a simply connected domain, 97 Embedding, 262 © Springer Nature Switzerland AG 2020 F. Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4

563

564 End points of a cross cut, 91 End points of an open arc, xxiv Euclidean diameter diamE , 73 Exceptional regular backward orbit, 361

F Final point of a contact arc, 411 Finite contact point of a holomorphic function with non-negative real part, 64 Fixed point of a semigroup, 327 Fixed points of a holomorphic self-map, 54

G Geodesics in simply connected domains, 122 Green function, 286 Group in a Riemann surface, 222 Group in D, 214

H Hardy-Littlewood maximal function, 32 Harmonic measure in D, 172 Harmonic measure in simply connected domains, 177 Herglotz representation, 59 Holomorphically conjugated semigroups, 267 Holomorphic conjugation of holomorphic models, 267 Holomorphic model, 236 Holomorphic semi-model, 235 Horizontal sector, 135 Horocycle, 17 Horocycle in a simply connected domain, 153 Hyperbolic distance of points in D, 11 Hyperbolic distance on Riemann surfaces, 14 Hyperbolic function, 51 Hyperbolic groups, 218 Hyperbolic length of a curve, 11 Hyperbolic length of a curve in a simply connected domain, 119 Hyperbolic length of Lipschitz curves, 137 Hyperbolic metric, 10 Hyperbolic metric in a simply connected domain, 117 Hyperbolic norm of a vector in a simply connected domain, 118 Hyperbolic petal of a semigroup, 377

Index Hyperbolic projection of a point onto a geodesic, 156 Hyperbolic pseudo-distance, 282 Hyperbolic sector around a geodesic, 135 Hyperbolic semigroups, 220 Hyperbolic step of a regular backward orbit, 356 Hyperbolic steps of order u, 231

I Infinitesimal generator, 275 Abate’s Formula, 286 Ahoronov-Elin-Reich-Shoikhet’s Formula, 288 characterizations, 275, 280, 284, 288 of a group, 289 of a semigroup of linear fractional maps, 292 Inner fixed point of a semigroup, 327 Interior part of a null chain, 92 Intertwining mapping, 236 Isolated radial slit, 435 Isolated spiral slit, 435 Isomorphism of topological models, 269 Iterate of a semigroup in a Riemann surface, 222 Iterate of a semigroup in D, 205

J Jordan arc, 73 Jordan curve, 73 Jordan domain, 105

K Kernel convergence of a sequence of domains, 87 Kernel of a sequence of domains, 87 Koebe arcs, 73 Koebe domain, 155 Koebe’s Distortion Theorem, 83 Koebe 1/4-Theorem, 82 Koenigs function, 248 uniqueness, 248 Kolmogorov’s Backward Equation, 276

L Life-time of a boundary point, 409 Linear fractional map, 6

Index Locally arcwise connected space, 104 Locally connected space, 99

M Maximal half-plane, 389 Maximal invariant curve, 372 Maximal spirallike sector, 387 Maximal strip, 387 Möbius transformation, 6 Multiplier of an inner fixed point, 54 Multiplier of a self-map at a boundary point, 55

N Non-elliptic groups, 218 Non-elliptic semigroups, 220 Non-exceptional regular backward orbit, 361 Non-tangential cluster set of a function, 108 Non-tangential converges to ∞ in H, 45 Non-tangential limit of a function f : D → C, 22 Non-tangential limit of a function f : H → C at ∞, 45 Non-tangential limit of a sequence, 22 Non-tangential maximal function, 31 Null chain, 92

O Open arc, xxiv Orbit of a semigroup, 206 Orthogonal speed of a curve in the disc, 157 Orthogonal speed of a non-elliptic semigroup, 455

P Parabolic function, 51 Parabolic groups, 218 Parabolic petal of a semigroup, 377 Parabolic semigroups, 220 Petal of a semigroup, 375 Poisson integral, 28 Poisson kernel, 18, 280 Pole of a infinitesimal generator, 429 Positive hyperbolic step, 245 Prime end, 97 Prime end impression, 97 Principal part of a prime end, 110 Product Formula, 303

565 Q Quasi-geodesics, 137 Question of embedding, 262 R Radial cluster set, 108 Radial cluster set of a function, 108 Radial limit of a function f : D → C, 22 Reflection through a line, 134 Regular backward orbit, 356 Regular contact point of a self-map, 55 Regular finite contact point of a holomorphic function with non-negative real part, 64 Regular pole of a infinitesimal generator, 429 Regular zero, 338 Regular zeros of a holomorphic function with non-negative real part, 64 Repelling fixed point of the semigroup, 329 Repelling spectral value of a semigroup at a boundary fixed point, 329 Riemann sphere, 5 Riemann surface, 4 S Semicomplete vector field, 273 complete, 274 Semi-conformality at a boundary point, 364 Semi-conjugation map, 234 Semigroup, 206 algebraic in the unit disc, 205 characterizations, 211, 213, 275 elliptic, 220 hyperbolic, 220 in D, 206 iterate, 205 non-elliptic, 220 of hyperbolic rotations, 206 of positive hyperbolic type, 245 of zero hyperbolic type, 245 parabolic, 220 semi-conjugated, 234 trivial, 206 Semigroup of automorphic type, 245 Semigroup of non-automorphism type, 245 Semigroups of linear fractional maps, 226 Semi-strip of width R and height M, 167 Shift of a semigroup, 529 Simple boundary point, 105 Slope of a curve, 501 Spectral value of the semigroup, 220 Spherical diameter, 5

566 Spherical diameter d S , 5 Spherical distance, 5 Spirallike, 251, 253 Spirallike argument, 390 Spirallike sector, 385 Starlike, 251 Starlike at infinity, 256, 257 Starting point of a contact arc, 411 Starting point of a maximal invariant curve, 372 Stolz region, 23 Strip, 162 Subadditive, 230 Super-repelling fixed point of the first type, 424 Super-repelling fixed point of the second type, 424 Super-repelling fixed point of the semigroup, 329 Super-repelling fixed point of the third type, 424 Symmetry of a simply connected domain with respect to a line, 134

Index T Tangential speed of a curve in a simply connected domain, 157 Tangential speed of a non-elliptic semigroup, 455 Tip of an isolated spiral or radial slit, 435 Topological conjugation of topological models, 270 Topologically conjugated semigroups, 270 Total speed of a non-elliptic semigroup, 454 Trajectory of a semigroup, 206

U Univalent, 4 Unrestricted limit of a function f : D → C, 23

Z Zero hyperbolic step, 245