Meromorphic Dynamics: Volume 2 : Elliptic Functions with an Introduction to the Dynamics of Meromorphic Functions 1009215981, 9781009215985

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Meromorphic Dynamics Volume II This text, the second of two volumes, builds on the foundational material on ergodic theory and geometric measure theory provided in Volume I, and applies all the techniques discussed to describe the beautiful and rich dynamics of elliptic functions. The text begins with an introduction to topological dynamics of transcendental meromorphic functions before progressing to elliptic functions, discussing at length their classical properties, measurable dynamics, and fractal geometry. The authors then look in depth at compactly nonrecurrent elliptic functions. Much of this material is appearing for the first time in book or paper form. Both senior and junior researchers working in ergodic theory and dynamical systems will appreciate what is sure to be an indispensable reference. J a n i n a Ko t u s is Professor of Mathematics at the Warsaw University of Technology, Poland. Her research focuses on dynamical systems, in particular holomorphic dynamical systems. Together with I.N. Baker and Y. L¨u, she laid the foundations for iteration of meromorphic functions. M a r i u s z U r b a n´ s k i is Professor of Mathematics at the University of North Texas, USA. His research interests include dynamical systems, ergodic theory, fractal geometry, iteration of rational and meromorphic functions, random dynamical systems, open dynamical systems, iterated function systems, Kleinian groups, Diophantine approximations, topology, and thermodynamic formalism. He is the author of 8 books, 7 monographs, and more than 200 papers.

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Jean Bertoin, B´ela Bollob´as, William Fulton, Bryna Kra, Ieke Moerdijk, Cheryl Praeger, Peter Sarnak, Barry Simon, Burt Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit www.cambridge.org/mathematics.

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S. Berhanu, P. D. Cordaro and J. Hounie An Introduction to Involutive Structures A. Shlapentokh Hilbert’s Tenth Problem G. Michler Theory of Finite Simple Groups I A. Baker and G. W¨ustholz Logarithmic Forms and Diophantine Geometry P. Kronheimer and T. Mrowka Monopoles and Three-Manifolds B. Bekka, P. de la Harpe and A. Valette Kazhdan’s Property (T) J. Neisendorfer Algebraic Methods in Unstable Homotopy Theory M. Grandis Directed Algebraic Topology G. Michler Theory of Finite Simple Groups II R. Schertz Complex Multiplication S. Bloch Lectures on Algebraic Cycles (2nd Edition) B. Conrad, O. Gabber and G. Prasad Pseudo-reductive Groups T. Downarowicz Entropy in Dynamical Systems C. Simpson Homotopy Theory of Higher Categories E. Fricain and J. Mashreghi The Theory of H(b) Spaces I E. Fricain and J. Mashreghi The Theory of H(b) Spaces II J. Goubault-Larrecq Non-Hausdorff Topology and Domain Theory ´ J. Sniatycki Differential Geometry of Singular Spaces and Reduction of Symmetry E. Riehl Categorical Homotopy Theory B. A. Munson and I. Voli´c Cubical Homotopy Theory B. Conrad, O. Gabber and G. Prasad Pseudo-reductive Groups (2nd Edition) J. Heinonen, P. Koskela, N. Shanmugalingam and J. T. Tyson Sobolev Spaces on Metric Measure Spaces Y.-G. Oh Symplectic Topology and Floer Homology I Y.-G. Oh Symplectic Topology and Floer Homology II A. Bobrowski Convergence of One-Parameter Operator Semigroups K. Costello and O. Gwilliam Factorization Algebras in Quantum Field Theory I J.-H. Evertse and K. Gy¨ory Discriminant Equations in Diophantine Number Theory G. Friedman Singular Intersection Homology S. Schwede Global Homotopy Theory M. Dickmann, N. Schwartz and M. Tressl Spectral Spaces A. Baernstein II Symmetrization in Analysis A. Defant, D. Garc´ıa, M. Maestre and P. Sevilla-Peris Dirichlet Series and Holomorphic Functions in High Dimensions N. Th. Varopoulos Potential Theory and Geometry on Lie Groups D. Arnal and B. Currey Representations of Solvable Lie Groups M. A. Hill, M. J. Hopkins and D. C. Ravenel Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem K. Costello and O. Gwilliam Factorization Algebras in Quantum Field Theory II S. Kumar Conformal Blocks, Generalized Theta Functions and the Verlinde Formula P. F.X. M¨uller ardy Martingales T. Kaletha and G. Prasad Bruhat–Tits Theory J. Schwermer Reduction Theory and Arithmetic Groups J. Kotus and M. Urba´nski Meromorphic Dynamics I J. Kotus and M. Urba´nski Meromorphic Dynamics II

Meromorphic Dynamics Elliptic Functions with an Introduction to the Dynamics of Meromorphic Functions volume ii

JA N I NA KOT U S Warsaw University of Technology ´ SKI M A R I U S Z U R BA N University of North Texas

Shaftesbury Road, Cambridge CB2 8EA, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 103 Penang Road, #05–06/07, Visioncrest Commercial, Singapore 238467 Cambridge University Press is part of Cambridge University Press & Assessment, a department of the University of Cambridge. We share the University’s mission to contribute to society through the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781009215978 DOI: 10.1017/9781009215985 © Janina Kotus and Mariusz Urba´nski 2023 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press & Assessment. First published 2023 Printed in the United Kingdom by TJ Books Limited, Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication data Names: Kotus, Janina, author. | Urba´nski, Mariusz. author. Title: Meromorphic dynamics / Janina Kotus, Warsaw University of Technology, Mariusz Urba´nski, University of North Texas. Description: First edition. | Cambridge, United Kingdom ; New York, NY, USA: Cambridge University Press, 2023. | Series: Nmm new mathematical monographs | Includes bibliographical references and index. | Contents: Volume 1. abstract ergodic theory, geometry, graph directed Markov systems, and conformal measures – Volume 2. Elliptic functions with an introduction to the dynamics of meromorphic functions. Identifiers: LCCN 2022031269 (print) | LCCN 2022031270 (ebook) | ISBN 9781009216050 (set ; hardback) | ISBN 9781009215916 (volume 1 ; hardback) | ISBN 9781009215978 (volume 2 ; hardback) | ISBN 9781009215930 (volume 1 ; epub) | ISBN 9781009215985 (volume 2 ; epub) Subjects: LCSH: Functions, Meromorphic. Classification: LCC QA331 .K747 2023 (print) | LCC QA331 (ebook) | DDC 515/.982–dc23/eng20221102 LC record available at https://lccn.loc.gov/2022031269 LC ebook record available at https://lccn.loc.gov/2022031270 ISBN – 2 Volume Set 978-1-009-21605-0 Hardback ISBN – Volume I 978-1-009-21591-6 Hardback ISBN – Volume II 978-1-009-21597-8 Hardback Cambridge University Press & Assessment has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Janina Kotus dedicates this book to the memory of her sister Barbara. ´ Mariusz Urbanski dedicates the book to his family.

Contents

Volume II Preface Acknowledgments Introduction

page xvii xxii xxiii

PART III TOPOLOGICAL DYNAMICS OF MEROMORPHIC FUNCTIONS 13

14

15

Fundamental Properties of Meromorphic Dynamical Systems 13.1 Basic Iteration of Meromorphic Functions 13.2 Classification of Periodic Fatou Components 13.3 The Singular Sets Sing(f −n ), Asymptotic Values, and Analytic Inverse Branches

3 3 13 34

Finer Properties of Fatou Components 14.1 Properties of Periodic Fatou Components 14.2 Simple Connectedness of Fatou Components 14.3 Baker Domains 14.4 Fatou Components of Class B and S of Meromorphic Functions

67 67 72 75

Rationally Indifferent Periodic Points 15.1 Local and Asymptotic Behavior of Analytic Functions Locally Defined Around Rationally Indifferent Fixed Points 15.2 Leau–Fatou Flower Petals

85

vii

77

85 104

viii

Contents of Volume II

15.3 15.4

Fatou Flower Theorem and Fundamental Domains Around Rationally Indifferent Periodic Points Quantitative Behavior of Analytic Functions Locally Defined Around Rationally Indifferent Periodic Points: Conformal Measures Outlook

110

116

PART IV ELLIPTIC FUNCTIONS: CLASSICS, GEOMETRY, AND DYNAMICS 16

Classics of Elliptic Functions: Selected Properties 16.1 Periods, Lattices, and Fundamental Regions 16.2 General Properties of Elliptic Functions 16.3 Weierstrass ℘-Functions I 16.4 The Field of Elliptic Functions 16.5 The Discriminant of a Cubic Polynomial 16.6 Weierstrass ℘-Functions II

125 125 133 140 149 156 166

17

Geometry and Dynamics of (All) Elliptic Functions 17.1 Forward and Inverse Images of Open Sets and Fatou Components 17.2 Fundamental Structure Results 17.3 Hausdorff Dimension of Julia Sets of (General) Elliptic Functions 17.4 Elliptic Function as a Member of A(X) for Forward Invariant Compact Sets X ⊆ C 17.5 Radial Subsets of J (f ) and Various Dynamical Dimensions for Elliptic Functions f : C →  C 17.6 Sullivan Conformal Measures for Elliptic Functions 17.7 Hausdorff Dimension of Escaping Sets of Elliptic Functions 17.8 Conformal Measures of Escaping Sets of Elliptic Functions

173 175 182 187 189 191 193 208 214

PART V COMPACTLY NONRECURRENT ELLIPTIC FUNCTIONS: FIRST OUTLOOK 18

Dynamics of Compactly Nonrecurrent Elliptic Functions 18.1 Fundamental Properties of Nonrecurrent Elliptic Functions: Ma˜ne´ ’s Theorem

221 222

Contents of Volume II

18.2

18.3 18.4 19

Compactly Nonrecurrent Elliptic Functions: Definition, Partial Order in Critc (J (f )), and Stratification of Closed Forward-Invariant Subsets of J (f ) Holomorphic Inverse Branches Dynamically Distinguished Classes of Elliptic Functions

Various Examples of Compactly Nonrecurrent Elliptic Functions 19.1 The Dynamics of Weierstrass Elliptic Functions: Some Selected General Facts 19.2 The Dynamics of Square Weierstrass Elliptic Functions: Some Selected Facts 19.3 The Dynamics of Triangular Weierstrass Elliptic Functions: Some Selected Facts 19.4 Simple Examples of Dynamically Different Elliptic Functions 19.5 Expanding (Thus Compactly Nonrecurrent) Triangular Weierstrass Elliptic Functions with Nowhere Dense Connected Julia Sets 19.6 Triangular Weierstrass Elliptic Functions Whose Critical Values Are Preperiodic, Thus Being Subexpanding 19.7 Weierstrass Elliptic Functions Whose Critical Values Are Poles or Prepoles, Thus Being Subexpanding, Thus Compactly Nonrecurrent 19.8 Compactly Nonrecurrent Elliptic Functions with Critical Orbits Clustering at Infinity 19.9 Further Examples of Compactly Nonrecurrent Elliptic Functions

ix

242 248 254 263 263 265 269 277

284 288

291 295 301

PART VI COMPACTLY NONRECURRENT ELLIPTIC FUNCTIONS: FRACTAL GEOMETRY, STOCHASTIC PROPERTIES, AND RIGIDITY 20

Sullivan h-Conformal Measures for Compactly Nonrecurrent Elliptic Functions 20.1 Existence of Conformal Measures for Compactly Nonrecurrent Elliptic Functions 20.2 Conformal Measures for Compactly Nonrecurrent Elliptic Functions and Holomorphic Inverse Branches

307 308 309

x

Contents of Volume II

20.3

21

22

23

Conformal Measures for Compactly Nonrecurrent Regular Elliptic Functions: Atomlessness, Uniqueness, Ergodicity, and Conservativity

Hausdorff and Packing Measures of Compactly Nonrecurrent Regular Elliptic Functions 21.1 Hausdorff Measures 21.2 Packing Measure I 21.3 Packing Measure II Conformal Invariant Measures for Compactly Nonrecurrent Regular Elliptic Functions 22.1 Conformal Invariant Measures for Compactly Nonrecurrent Regular Elliptic Functions: The Existence, Uniqueness, Ergodicity/Conservativity, and Points of Finite Condensation dμh 22.2 Real Analyticity of the Radon–Nikodym Derivative dm h 22.3 Finite and Infinite Condensation of Parabolic Periodic Points with Respect to the Invariant Conformal Measure μh 22.4 Closed Invariant Subsets, K(V ) Sets, and Summability Properties 22.5 Normal Subexpanding Elliptic Functions of Finite Character: Stochastic Properties and Metric Entropy, Young Towers, and Nice Sets Techniques 22.6 Parabolic Elliptic Maps: Nice Sets, Graph Directed Markov Systems, Conformal and Invariant Measures, Metric Entropy 22.7 Parabolic Elliptic Maps with Finite Invariant Conformal Measures: Statistical Laws, Young Towers, and Nice Sets Techniques 22.8 Infinite Conformal Invariant Measures: Darling–Kac Theorem for Parabolic Elliptic Functions Dynamical Rigidity of Compactly Nonrecurrent Regular Elliptic Functions 23.1 No Compactly Nonrecurrent Regular Function is Esentially Linear 23.2 Proof of the Rigidity Theorem

316 335 336 338 340 348

349 360

367 375

393

411

428 445 458 459 472

Contents of Volume II

xi

Appendix A A Quick Review of Some Selected Facts from Complex Analysis of a One-Complex Variable

489

Appendix B Proof of the Sullivan Nonwandering Theorem for Speiser Class S

494

References Index of Symbols Subject Index

503 510 513

xii

Contents of Volume I

Volume I Preface Acknowledgments Introduction

xv xx xxi

PART I ERGODIC THEORY AND GEOMETRIC MEASURES 1

2

3

Geometric Measure Theory 1.1 Measures, Integrals, and Measure Spaces 1.2 Measures on Metric Spaces: (Metric) Outer Measures and Weak∗ Convergence 1.3 Covering Theorems: 4r, Besicovitch, and Vitali Type; Lebesgue Density Theorem 1.4 Conditional Expectations and Martingale Theorems 1.5 Hausdorff and Packing Measures: Hausdorff and Packing Dimensions 1.6 Hausdorff and Packing Measures: Frostman Converse-Type Theorems 1.7 Hausdorff and Packing Dimensions of Measures 1.8 Box-Counting Dimensions Invariant Measures: Finite and Infinite 2.1 Quasi-invariant Transformations: Ergodicity and Conservativity 2.2 Invariant Measures: First Return Map (Inducing); Poincar´e Recurrence Theorem 2.3 Ergodic Theorems: Birkhoff, von Neumann, and Hopf 2.4 Absolutely Continuous σ -Finite Invariant Measures: Marco Martens’s Approach Probability (Finite) Invariant Measures: Basic Properties and Existence 3.1 Basic Properties of Probability Invariant Measures 3.2 Existence of Borel Probability Invariant Measures: Bogolyubov–Krylov Theorem 3.3 Examples of Invariant and Ergodic Measures

3 3 7 13 23 30 36 44 50 55 55 64 78 92 102 102 104 107

Contents of Volume I

4

xiii

Probability (Finite) Invariant Measures: Finer Properties 4.1 The Law of the Iterated Logarithm 4.2 Decay of Correlations and the Central Limit Theorems: Lai-Sang Young Towers 4.3 Rokhlin Natural Extension

122 122

5

Infinite Invariant Measures: Finer Properties 5.1 Counterexamples to Ergodic Theorems 5.2 Weak Ergodic Theorems 5.3 Darling–Kac Theorem: Abstract Version 5.4 Points of Infinite Condensation: Abstract Setting

134 134 141 146 157

6

Measure-Theoretic Entropy 6.1 Partitions 6.2 Information and Conditional Information Functions 6.3 Entropy and Conditional Entropy for Partitions 6.4 Entropy of a (Probability) Measure-Preserving Endomorphism 6.5 Shannon–McMillan–Breiman Theorem 6.6 Abramov’s Formula and Krengel’s Entropy (Infinite Measures Allowed)

160 160 161 166

Thermodynamic Formalism 7.1 Topological Pressure 7.2 Bowen’s Definition of Topological Pressure 7.3 Basic Properties of Topological Pressure 7.4 Examples 7.5 The Variational Principle and Equilibrium States

192 192 214 224 225 228

7

125 129

168 184 191

PART II COMPLEX ANALYSIS, CONFORMAL MEASURES, AND GRAPH DIRECTED MARKOV SYSTEMS 8

Selected Topics from Complex Analysis 8.1 Riemann Surfaces, Normal Families, and Montel’s Theorem 8.2 Extremal Lengths and Moduli of Topological Annuli 8.3 Koebe Distortion Theorems 8.4 Local Properties of Critical Points of Holomorphic Functions

249 250 266 282 291

xiv

Contents of Volume I

8.5 8.6 9

10

11

12

Proper Analytic Maps and Their Degree Riemann–Hurwitz Formula

Invariant Measures for Holomorphic Maps f in A(X) or in Aw (X) 9.1 Preliminaries 9.2 Three Auxiliary Partitions 9.3 Pesin’s Theory 9.4 Ruelle’s Inequality 9.5 Volume Lemmas and Hausdorff Dimensions of Invariant Measures Sullivan Conformal Measures for Holomorphic Maps f in A(X) and in Aw (X) 10.1 General Concept of Conformal Measures 10.2 Sullivan Conformal Measures for Holomorphic Maps f in A(X) and in Aw (X) 10.3 Radial Subsets of X, Dynamical Dimensions, and Sullivan Conformal Measures for Holomorphic Maps f in A(X) and in Aw (X) 10.4 Conformal Pairs of Measures

293 296 316 317 321 327 333 336 342 343 364

368 371

Graph Directed Markov Systems 11.1 Subshifts of Finite Type over Infinite Alphabets: Topological Pressure 11.2 Graph Directed Markov Systems 11.3 Conformal Graph Directed Markov Systems 11.4 Topological Pressure, θ-Number, and Bowen’s Parameter 11.5 Hausdorff Dimensions and Bowen’s Formula for GDMSs 11.6 Conformal and Invariant Measures for CGDMSs 11.7 Finer Geometrical Properties of CGDMSs 11.8 The Strong Open Set Condition 11.9 Conformal Maximal Graph Directed Systems 11.10 Conjugacies of Conformal Graph Directed Systems

378

Nice Sets for Analytic Maps 12.1 Pre-Nice and Nice Sets and the Resulting CGMDS: Preliminaries 12.2 The Existence of Pre-Nice Sets 12.3 The Existence of Nice Sets

433

378 383 385 390 393 400 416 419 424 431

434 438 455

Contents of Volume I

12.4 12.5

The Maximal GDMSs Induced by Nice Sets An Auxiliary Technical Result

References Index of Symbols Subject Index

xv

458 467 469 478 480

Preface

The ultimate goal of our book is to present a unified approach to the dynamics, ergodic theory, and geometry of elliptic functions mapping the complex plane C onto the Riemann sphere  C. We consider elliptic functions as the most regular class of transcendental meromorphic functions. Poles, infinitely many of them, form an essential feature of such functions, but the set of critical values is finite and an elliptic function is “the same” in all of its fundamental regions. In a sense, this is the class of transcendental meromorphic functions whose resemblance to rational functions on the Riemann sphere is the largest. This similarity is important since the class of rational functions has been, from the dynamical point of view, extensively investigated since the pioneering works of Pierre Fatou [Fat1] and Gaston Julia [Ju]; also see the excellent historical accounts in [Al] and [AIR] on the early days of holomorphic dynamics. This similarity can be, and frequently was, a good source of motivation and guidance for us when we were dealing with elliptic functions. On the other hand, the differences are striking in many respects, including topological dynamics, measurable dynamics, fractal geometry of Julia sets, and more. We will touch on them in the course of this Preface. We would just like to stress here that elliptic functions belong to the class of transcendental meromorphic functions and posseses many dynamical and geometric features that are characteristic for this class. Indeed, the study of iteration of transcendental meromorphic functions, more precisely of transcendental entire functions, began with the pioneering works of Pierre Fatou ([Fat2] and [Fat3]). Then for about two decades, beginning with paper [Ba1], I.N. Baker was actually the sole mathematician dealing with the dynamics of transcendental entire functions. It was Janina Kotus’s idea to study the iteration of meromorphic functions despite the existence of poles that, in general, cause the second iterate to not be defined

xvii

xviii

Preface

everywhere; more precisely, because it has finite essential singularities, thus, it is not a meromorphic function defined on the complex plane. This is the phenomenon that had been deterring mathematicians from dealing with the iteration of general meromorphic functions. Janina was not afraid and as a result, to our knowledge, the first, and quite systematic, account of the dynamics of general transcendental meromorphic functions was set up in a series of works by I.N. Baker, J. Kotus, and Y. L¨u ([BKL1]–[BKL4]). Since then this subfield of dynamical systems has been flourishing. Of great importance for the development of this subject was the excellent expository article by Walter Bergweiler [Ber1]. We would also like to mention an early paper by Alexander Eremenko and Misha Lyubich [EL2], who introduced and studied class B of transcendental entire functions. The definition of class B literally extends to the class of general meromorphic functions and plays an important role in this field too. The area of transcendental meromorphic dynamical systems is also a beautiful and vast field for investigations of the measurable dynamics they generate and the fractal geometry of their Julia sets and their significant subsets. Here, the early papers by Misha Lyubich [Ly] and Mary Rees [Re] on measurable dynamics come to the fore. The study of the fractal geometry of Julia sets of meromorphic functions began with two papers by Gwyneth Stallard ([Sta1] and [Sta2]) and has been continued ever since by her, Krzysztof Bara´nski, Walter Bergweiler, Bogusia Karpi´nska, Volker Mayer, Phil Rippon, Lasse Rempe-Gillen, Anna Zdunik, the authors of this book, and many more mathematicians, obtaining interesting and sophisticated results; we are not able to list all of them here. We would like, however, to mention the early paper by Anna Zdunik and Mariusz Urba´nski [UZ1], where the concept of conformal measures was adapted for and used in transcendental dynamics, and also Hausdorff and packing measures of (radial) Julia sets were studied in detail. We would also like to single out a paper by Krzysztof Bara´nski [Ba], who initiated the use of thermodynamic formalism in transcendental meromorphic dynamics. More papers using and developing thermodynamic formalism then followed, among them those by Janina Kotus [KU1], Anna Zdunik [UZ2], and Volker Mayer ([MyU4] and [MyU5]), all written jointly with Mariusz Urba´nski. As a source of detailed information on many aspects of the use of this method in meromorphic dynamics, we recommend the expository article by Volker Mayer and Mariusz Urba´nski [MyU6]. As already stated at the beginning of this Preface, we ultimately focus in this book on the dynamics, ergodic theory, and geometry of elliptic functions. To

Preface

xix

our knowledge, the first dynamical result specific to elliptic functions appeared in Janina Kotus’s 1995 paper [Ko3]. It gave a good lower bound for the Hausdorff dimensions of the Julia setd of elliptic functions. Its was refined, using the theory of countable alphabet conformal iterated function systems, in [KU3]. A later paper [MyU1] presents a form of thermodynamic formalism for elliptic functions. A quite long series of papers by Jane Hawkins and her collaborators, studying the dynamics and geometry of Weierstrass ℘-functions, began in 2002 with [HK1]. Our book stems from, has been motivated by, and largely develops our 2004 paper [KU4] We devote the first chapter of the second volume to a rather short and compressed, albeit with proofs, introduction to the topological dynamics of transcendental meromorphic functions. We then move on to elliptic functions, giving first some short, but with proofs, expositions of classical properties of such functions, and then we deal with their measurable dynamics and fractal geometry. We single out several dynamically significant subclasses of elliptic functions, primarily nonrecurrent, compactly nonrecurrent, subexpanding, and parabolic. We devote one long chapter to describing examples of elliptic functions with various properties of their dynamics, Fatou Connected Components, and the geometry of Julia sets. Our approach to measurable dynamics and the fractal geometry of elliptic functions is founded on the concept of the Sullivan conformal measures. We prove their existence for all (nonconstant) elliptic functions and provide several characterizations, in dynamical terms, of the minimal exponent for which these measures exist. By using the method of conformal iterated function systems with a countable alphabet and essentially reproducing the proof from [KU3], we provide a simple lower bound, expressed in terms of orders of poles, of the Hausdorff dimensions of Julia sets of all elliptic functions. It follows from this estimate that such Hausdorff dimensions are always strictly larger than 1. We also provide a closed exact formula for the Hausdorff dimension of the set of points escaping to ∞. We then deal with the class of nonrecurrent elliptic functions f : C →  C and their subclasses such as compactly nonrecurrent, subexpanding, and parabolic ones. This is the ultimate object of our interests in the book, especially in Volume II. We would like to add that we do not give separate attention to hyperbolic/expanding elliptic functions; none of them allow critical points or rationally indifferent periodic points in their Julia sets. Indeed, doing this would take up a lot of pages and preparations, and a good account of the thermodynamic formalism of quite general classes of meromorphic functions is given in [MyU4] and [MyU5], as well as [MyU3] and [MyU6]. In this book, we focus on elliptic functions that may have critical

xx

Preface

points and rationally indifferent periodic points in the Julia sets; we do allow them, and our main objective is to deal with and study the various phenomena that they cause. Our presentation of the theory of nonrecurrent elliptic functions is based on an appropriate version, which we prove, of Ma˜ne´ ’s Theorem, which roughly speadking asserts that the connected components of all inverse images of all orders of all sufficiently small sets remain small. Having a Sullivan conformal measure m with a minimal exponent, we prove its uniqueness and atomlessness for compactly nonrecurrent regular elliptic functions. Next, we prove the existence and uniqueness (up to a nonzero multiplicative factor) of a σ -finite invariant measure μ that is absolutely continuous with respect to the conformal measure m. We prove its ergodicity and conservativity. Restricting our attention to the classes of subexpanding and parabolic functions, in fact to some natural subclasses of them, we prove much more refined stochastic properties of the dynamical system (f ,μ). Our approach here stems from and largely develops the methods developed in papers [ADU], [DU4], [U3], and [U4]. It is, however, significantly enlarged and enriched, via the powerful tool of nice sets, by the methods of countable alphabet conformal iterated function systems and by graph directed Markov systems as developed and presented in [MU1] and [MU2]. These in turn are substantially based on the theory of countable alphabet thermodynamic formalism developed in [MU5] and [MU2]. When dealing with subexpanding functions, especially with the exponential shrinking property, the paper by Przytycki and RiveraLetelier [PR] was also very useful. The finer stochastic properties mentioned above are primarily the exponential decay of correlations, the Central Limit Theorem, and the Law of the Iterated Logarithm in the case of subexpanding elliptic functions. In the case of parabolic elliptic functions for which the invariant measure μ is finite, we prove the Central Limit Theorem. All of these are achieved with the help of the Lai-Sang Young tower techniques from [LSY3]. In the case of parabolic elliptic functions for which the invariant measure μ is infinite, we prove an appropriate version of the Darling–Kac Theorem, establishing the strong convergence of weighted Birkhoff averages to Mittag–Leffler distributions. Last, we would like to mention finer fractal geometry. For both subexpanding and parabolic elliptic functions, we give a complete description and characterization of conformal measures and Hausdorff and packing measures of Julia sets. Because the Hausdorff dimension of the Julia set of an elliptic function is strictly larger than 1, this picture is even simpler than for the subexpanding, parabolic, and nonrecurrent rational functions given in [DU4], [DU5], and [U3].

Preface

xxi

In order to comprehensively cover the dynamics and geometry of elliptic functions described above, we extensive large preparations. This is primarily done in the first volume of the book, which consists of two parts: Part I, “Ergodic Theory and Geometric Measures” and Part II, “Complex Analysis, Conformal Measures, and Graph Directed Markov Systems.” We intend our book to be as self-contained as possible and we use essentially all the major results from Volume I in Volume II when dealing with dynamics, ergodic theory, and the geometry of elliptic functions. Our book can thus be treated not only as a fairly comprehensive account of dynamics, ergodic theory, and the fractal geometry of elliptic functions but also as a reference book (with proofs) for many results of geometric measure theory, finite and infinite abstract ergodic theory, Young towers, measuretheoretic Kolmogorov–Sinai entropy, thermodynamic formalism, geometric function theory (in particular the Koebe Distortion Theorems and Riemann– Hurwitz Formulas), various kinds of conformal measures, conformal graph directed Markov systems and iterated function systems, the classical theory of elliptic functions, and the topological dynamics of transcendental meromorphic functions. The material contained in Volume I of this book, after being substantially processed, collects, with virtually all proofs, the results that are essentially known and have been published. However, Chapter 5 contains material on infinite ergodic theory that, to the best or our knowledge, has not been included, with full proofs, in any prior book. Also, Chapter 12, which treats nice sets, is strongly processed and goes, in many respects, far beyond the existing knowledge and use of nice sets and nice families in conformal dynamics. The need for such far-reaching extensions of this method comes from the need for its applications to parabolic elliptic functions. Most of the material at the end of the second volume of this book is actually new, although we borrow, use, and apply much from the previous results, methods, and techniques. Indeed, Chapter 17, except its last two sections, is entirely new. Also, to the best of our knowledge, Section 19.4 is purely original, providing large classes of a variety of simple examples of various kinds of dynamically elliptic functions. Part VI is, indeed, entirely original.

Acknowledgments

We are very indebted to Jane Hawkins, who gave for us the four images of Julia sets of various Weierstrass elliptic ℘-functions that are included in Chapter 19 of Volume II of this book. We also thank her very much for fruitful discussions about the dynamics and Julia sets of Weierstrass elliptic ℘-functions. Mariusz Urba´nski would like to thank William Cherry for helpful discussions about normal families and Montel’s Theorems. Last, but not least, we want to express our gratitude to the referees of this book, whose numerous valuable remarks, comments, and suggestions prompted us to improve the content of the book and the quality of its exposition. The research of Janina Kotus was supported by the National Science Center, Poland, decision no. DEC-2019/33/B/ST1/00275. The reseach of Mariusz Urba´nski was supported in part by the NSF grant DMS 1361677.

xxii

Introduction

In this introduction, we describe in detail the content of the second volume of the book, simultaneously highlighting its sources and the interconnections between various fragments of the book. Most of this volume is devoted to a direct exploration of the dynamics and geometry of elliptic functions. Indeed, all but one, the first, parts of the volume, i.e., Parts IV–VI, do this. The first part of this volume, i.e., Part III, “Topological Dynamics of Meromorphic Functions,” is devoted to the iteration of arbitrary meromorphic functions. Indeed, it provides a relatively short and condensed account of the topological dynamics of almost all meromorphic functions with an emphasis on Fatou domains, including a detailed account of Baker domains that are exclusive for transcendental functions and do not occur for rational functions. We actually do this for all meromorphic functions, occasionally restricting our attention to the class of transcendental meromorphic functions all of whose prepoles (that include poles) form an infinite set. Essentially, all results of this part are supplied with full proofs. In particular, we provide a complete proof of Fatou’s classification of Fatou Periodic Components. We do a thorough analysis of the singular set of the inverse of a meromorphic function and all its iterates; in particular, we study at length asymptotic values and their relations to transcendental tracts. We analyze the structure of these components and the structure of their boundaries in greater detail. In particular, we provide a very detailed qualitative and quantitative description of the local behavior of locally and globally defined analytic functions around rationally indifferent periodic points and of the structure of corresponding Leau–Fatou flower petals, including the Fatou Flower Petal Theorem. Such an analysis will turn out to be an indispensable tool in the last three sections of Chapter 22 in Part VI, where we deal with the ergodic theory of parabolic elliptic functions. We also distinguish Speiser class S and Eremenko–Lyubich class B of meromorphic functions, which play a seminal role in the recent development xxiii

xxiv

Introduction

of the theory of iteration of transcendental meromorphic functions, proving their fundamental properties, which include some structural theorems about their Fatou components such as no existence of Baker domains and wandering domains (the Sullivan Nonwandering Theorem) for class S. The proof of the latter theorem, because of its length and high technicality, is however relegated to Appendix B. To the best of our knowledge, there is no systematic book account of the topological dynamics of transcendental meromorphic functions. Some results, with and without proofs, can be found in [BKL1]–[BKL4] and in [Ber1]. As we have already said, essentially all results in Part III of our book are supplied with proofs. In Part IV, we move on to elliptic functions and stay with them until the end of the book. The first chapter of this part, i.e., Chapter 16, which is interesting on its own, is devoted to presenting an account of the classical theory of elliptic functions. Almost no dynamics is involved here. We will actually not need this chapter anywhere else in the book except in Chapter 19, where we provide many examples of elliptic functions, including mainly but, we want to emphasize this, not only Weierstrass ℘ functions. Here, we primarily follow the classical books [Du] and [JS]. We would also like to draw the reader’s attention to the books [AE] and [La]. Throughout the whole of Chapter 17, we deal with general nonconstant elliptic functions, i.e., we impose no constraints on a given nonconstant elliptic function. We first systematically deal with forward and, more importantly, backward images of open connected sets, especially those with connected components of the latter. We mean to consider such images under all iterates f n , n ≥ 1, of a given elliptic function f . We do a thorough analysis of the singular set of the inverse of a meromorphic function and all its iterates; in particular, we study at length asymptotic values and their relations to transcendental tracts. We also provide sufficient conditions for the restrictions of iterates f n to such components to be proper or covering maps. Both of these methods, the latter allowing the use of the machinery of Section 8.6 from Volume I, are our primary tools to study the structure of connected component backward images of open connected sets. In particular, they prove the existence of holomorphic inverse branches if “there are no critical points.” Holomorphic inverse branches will be one of the most common tools used throughout the rest of the book. We then apply these results to study images and backward images of connected components of the Fatou set. Section 17.2 continues this theme, providing some structural theorems about Fatou and Julia sets of elliptic functions. Some of these are the immediate consequences of the results obtained in Part III, “Topological Dynamics of

Introduction

xxv

Meromorphic Functions,” once we observed that each elliptic function belongs to Speiser class S, while others are more technically complicated. The rest of Chapter 17 is actually devoted to analyzing in greater detail the fractal properties of any nonconstant elliptic function. Following the paper [KU3], by associating with a given elliptic function an infinite alphabet conformal iterated function system, and heavily utilizing its θ number, we provide a strong, somewhat surprising, lower bound for the Hausdorff dimension of the Julia sets of all nonconstant elliptic functions. In particular, this estimate shows that the Hausdorff dimension of the Julia sets of any nonconstant elliptic function is strictly larger than 1. We also provide a simple closed formula for the Hausdorff dimension of the set of points escaping to infinity under iteration of an elliptic function. In the last section of this chapter, we prove that no conformal measure of an elliptic function charges the set of escaping points. However, the central focus of this chapter is Section 17.6, where we prove the existence of the Sullivan conformal measures with a minimal exponent for all elliptic functions and we characterize the value of this exponent in several dynamically significant ways. Section 17.6 depends on the preparatory work in Sections 17.4 and 17.5, which are also interesting on their own. It also heavily depends on Chapter 10 in the first volume. In Part V, “Compactly Nonrecurrent Elliptic Functions: First Outlook,” we define the class of nonrecurrent and, more notably, the class of compactly nonrecurrent elliptic functions. This is the class of elliptic functions that will be dealt with by us from the moment compactly nonrecurrent elliptic functions are defined until the end of the book. Its history goes back to the papers [U3], [U4], and [KU4]. One should also mention the paper [CJY]. Similarly to all the papers that our treatment of nonrecurrent elliptic functions is based on, the fact that this is possible at all is due to an appropriate version of the breakthrough Ma˜ne´ ’s Theorem that was proven in [M1] in the context of rational functions. Without Ma˜ne´ ’s Theorem, such treatment would not be possible. In our setting of elliptic functions, this is Theorem 18.1.6. The first section of Chapter 18 is entirely devoted to proving this theorem, its first most fundamental consequences, and some other results surrounding it. The next two sections of this chapter, also relying on Ma˜ne´ ’s Theorem, provide us with further refined technical tools to study the structure of Julia sets and holomorphic inverse branches. The last section of this chapter, i.e., Section 18.4, has a somewhat different character. It systematically defines and describes various subclasses of the, mainly compactly nonrecurrent, elliptic functions that we will be dealing with in Part VI of the book. Mostly, but not exclusively, these classes of elliptic functions are defined in terms of how strongly expanding these functions

xxvi

Introduction

are. We would like to add that while in the theory of rational functions such classes pop up in a natural and fairly obvious way, e.g., metric and topological definitions of expanding rational functions describe the same class of functions, in the theory of iteration of transcendental meromorphic functions such a classification is by no means obvious as the topological and metric analogs of rational function concepts do not usually coincide and the definitions of expanding, hyperbolic, topologically hyperbolic, subhyperbolic, etc. functions vary from author to author. Our definitions seem to us to be quite natural and fit well with our purpose of the detailed investigation of the dynamical and geometric properties of the elliptic functions that they define. In this section, we also define the class of regular elliptic functions. The condition defining them is quite simple but, although very frequently holding, it does not look natural. It is, in fact, tailor-made for the proof of the possibly (in a sense) richest properties of the Sullivan conformal measures obtained in Section 20.3 to go through. The purpose of Chapter 19 is to provide examples of elliptic functions with prescribed properties of the orbits of critical points (and values). We primarily focus on constructing examples of the various classes of compactly nonrecurrent elliptic functions discerned in Section 18.4. All these examples are either Weierstrass ℘ elliptic functions or their modifications. The dynamics of such functions depends heavily on the lattice  and varies drastically from  to . The first three sections of this chapter have a preparatory character and, respectively, describe the basic dynamical and geometric properties of all Weierstrass ℘ elliptic functions generated by square and triangular lattices . In Section 19.4, we provide simple constructions of many classes of elliptic functions discerned in Section 18.4. We essentially cover all of them. All these examples stem from Weierstrass ℘ functions. We then, starting with Section 19.5, also provide some different, interesting on their own, and historically first examples of various kinds of Weierstrass ℘ elliptic functions and their modifications. These come from the series of papers [HK1], [HK2], [HK3], [HKK], and [HL] by Hawkins and her collaborators. Part VI, “Compactly Nonrecurrent Elliptic Functions: Fractal Geometry, Stochastic Properties, and Rigidity,” is entirely devoted to getting the dynamical, geometric/fractal, and stochastic properties of dynamical systems generated by compactly nonrecurrent elliptic functions, primarily subexpanding and parabolic ones. In Chapter 20, we use the fruits of the existence of the Sullivan conformal measures with a minimal exponent proven in Section 17.6 and its dynamical characterizations obtained therein. Having compact nonrecurrence, we are able to prove in the first section of this chapter that this minimal exponent is equal

Introduction

xxvii

to the Hausdorff dimension HD(J (f )) of the Julia set J (f ), which we always denote by h. We also obtain in this section strong restrictions on the possible locations of atoms of such conformal measures. Section 20.3, the last section in Chapter 20, is a culmination of our work on the Sullivan conformal measures for elliptic functions treated on their own. There, and from then onward, we assume that our compactly nonrecurrent elliptic function is regular, which is the concept introduced in Section 18.4. For this class of elliptic functions, we prove the uniqueness and atomlessness of hconformal measures along with their first fundamental stochastic properties such as ergodicity and conservativity. The results of Chapter 20 are not, however, the last word on the Sullivan conformal measures. Left alone, these measures would be a kind of curiosity that is perhaps only worthy of shrugging shoulders and raised eyebrows. Their true power, meaning, and importance come from their geometric characterizations, and, more accurately, from their usefulness – one could even say indispensability – for understanding geometric measures on Julia sets, i.e., their Hausdorff and packing h-dimensional measures, where, we recall, h = HD(J (f )). This is fully achieved in Chapter 21 for compactly nonrecurrent regular elliptic functions. Having said this, Chapter 21 can be viewed from two perspectives. The first is that we provide therein a geometrical characterization of the h-conformal measure mh , which, with the absence of parabolic points, turns out to be a normalized packing measure; the second is that we give a complete description of geometric, Hausdorff, and packing measures of the Julia sets J (f ). All of this is contained in Theorem 21.0.1, which gives a simple clear picture. Because of the fact that the Hausdorff dimension of the Julia set of an elliptic function is strictly larger than 1, this picture is even simpler than for nonrecurrent rational functions of [U3]; see also [DU5]. Throughout the whole of Chapter 22, f : C →  C is assumed to be a compactly nonrecurrent regular elliptic function. This chapter is, in a sense, the core of our book. Taking the fruits of what has been done in all previous chapters, we prove in Chapter 22 the existence and uniqueness, up to a multiplicative constant, of a σ -finite f -invariant measure μh equivalent to the h- conformal measure mh . Furthermore, still heavily relying on what has been done in all previous chapters, particularly on conformal graph directed Markov systems, nice sets, first return map techniques, and Young towers, we provide here a systematic account of the ergodic and refined stochastic properties of the dynamical system (f ,μh ) generated by such subclasses of compactly nonrecurrent regular elliptic functions as normal subexanding elliptic functions of finite character and parabolic elliptic functions. By stochastic properties, we

xxviii

Introduction

mean here the exponential decay of correlations, the Central Limit Theorem, the Law of the Iterated Logarithm for subexpanding functions, the Central Limit Theorem for those parabolic elliptic functions for which the invariant measure μh is finite (probabilistic after normalization), and an appropriate version of the Darling–Kac Theorem that establishes the strong convergence of weighted Birkhoff averages to Mittag–Leffler distributions for those parabolic elliptic functions for which the invariant measure μh is infinite. In Chapter 23, the last actual chapter of the book, we deal with the problem of dynamical rigidity of compactly nonrecurrent regular elliptic functions. The issue at stake is whether a weak conjugacy such as a Lipschitz one on Julia sets can be promoted to a much better one such as an affine conjugacy on the whole complex plane C. This topic has a long history and goes back at least to the seminal paper [Su4] by Sullivan, who treated, among others, the dynamical rigidity of conformal expanding repellers. Its systematical account can be found in [PU2]. A large variety, in many contexts, both smooth and conformal, of dynamical rigidity theorems have been proved. The literature abounds. Our approach in this chapter stems from the original article by Sullivan [Su4]. It is also influenced by [PU1], where the case of tame rational functions has actually been done, and [SU], where the equivalence of statements (1) and (4) of Theorem 23.0.1 was established for all tame transcendental meromorphic functions. Being tame means that the closure of the postsingular set does not contain the whole Julia set; in particular, each nonrecurrent elliptic function is tame. We would, however, like to emphasize that, unlike [SU], we chose in our book the approach that does not make use of the dynamical rigidity results for conformal iterated function systems proven in [MPU]. In Appendix A, “A Quick Review of Some Selected Facts from Complex Analysis of a One Complex Variable,” we collect for the convenience of the reader many basic and fundamental theorems of complex analysis. We provide no proofs, but we give detailed references (quite arbitrarily chosen) where the proofs can be found. We use these theorems throughout the book, frequently without directly referring to them. The content of Appendix B is clear from its title. It stems from the Sullivan breakthrough paper [Su1] and follows closely the proof presented in [BKL4].

P A R T III Topological Dynamics of Meromorphic Functions

13 Fundamental Properties of Meromorphic Dynamical Systems

In this chapter, we provide a relatively short and condensed account of the topological dynamics of all meromorphic functions, with an emphasis on Julia sets and Fatou domains, including Baker domains that are exclusive for transcendental functions and do not occur for rational functions. We do this for all meromorphic functions. In particular, we provide a complete proof of Fatou’s classification of Fatou Periodic Components. We analyze the structure of these components and the structure of their boundaries in greater detail. We also do a thorough analysis of the singular set of the inverse of a meromorphic function and all its iterates; in particular, we study at length asymptotic values and their relations to transcendental tracts. The results of this analysis will be very frequently used to study the topological structure of connected components (and their boundaries) of Fatou sets in this part of the book and a countless number of times when we move on to dealing with elliptic functions. To the best of our knowledge, there is no systematic book account of the topological dynamics of transcendental meromorphic functions. Some results, with and without proofs, can be found in [BKL1]–[BKL4] and in [Ber1]. For the iteration of rational functions, the reader may consult [CaGg], [Bea], [Ste], and [Mil1]. Essentially, all results in this chapter are supplied with proofs.

13.1 Basic Iteration of Meromorphic Functions In this section, we define Fatou and Julia sets of meromorphic functions. We also classify all periodic points of such functions. We prove some basic, rather elementary facts about of all of them.

3

4

Part III Topological Dynamics of Meromorphic Functions Given a meromorphic function f : C −→  C and a set A ⊆  C, the set f −1 (A) := {z ∈ C : f (z) ∈ A}

has the standard meaning. We then define f −0 (A) := A and the sets f −n (A), n ≥ 2, by induction as follows: f −n (A) := f −(n−1) (f −1 (A)).

(13.1)

Then, for every n ≥ 0, there is a well-defined nth-folded composition function C) −→  C f n : f −n ( and f −n ( C) = C\

n−1  k=1

f −k (∞) =  C\

n−1 

f −k (∞).

(13.2)

k=0

C). Since In particular, whenever we write f n (z), we assume that z ∈ f −n (  the function f : C −→ C is continuous, by using (13.1) and (13.2), along with the fact that complements of countable subsets in  C (and C) are connected, we directly obtain the following. Observation 13.1.1 Let f : C −→  C be a meromorphic function and n ≥ 1 be an integer. Then C) is an open connected subset of C. (1) f −n (  −k (∞) is countable and closed in C. (2) The set n−1 k=1 f C) −→  C is meromorphic, in particular conti(3) The function f n : f −n ( nuous. Consequently, C too). (4) If A ⊆  C is an open set, then the set f −n (A) is open in C (and  (5) If A ⊆ C is a closed set, then the set f −n (A) is also closed (in C). Definition 13.1.2 Let f : C −→  C be a meromorphic function. Given a natural number n ≥ 0, we call the elements of the set f −n (∞) the prepoles of f of order n. Note that a pole is just an order 1 prepole and ∞ is the sole prepole of order 0. We shall prove the following slight strengthening of Observation 13.1.1(4). Lemma 13.1.3 For every integer n ≥ 0, the set of accumulation points of f −n (∞) in  C is contained in

13 Fundamental Properties of Meromorphic Dynamical Systems

{∞} ∪

n−1 

5

f −k (∞).

k=0

Proof Fix n ∈ N. Let z ∈ C be an accumulation of f −n (∞). Then there −n (∞) exists a sequence (zk )∞ k=1 of mutually distinct elements in the set f such that z = lim zk . k→∞

Seeking contradiction, suppose that z∈ /

n−1 

f −k (∞).

k=0

Then there exists a neighborhood U of z such that f n restricted to U is a meromorphic function. But this is a contradiction since f n has poles at all points zk , k ≥ 1, and infinitely many of them belong to U . The proof is complete.  We recall that a meromorphic function f : C −→  C is transcendental if and only if ∞ is its essential singularity. The most fundamental definitions in this volume of the book are these. Definition 13.1.4 The Fatou set F (f ) of a meromorphic function f : C −→  C is defined in exactly the same manner as for rational functions: F (f ) is the set of all z ∈ C for which all the iterates f n of f are defined, i.e.,  points ∞ −n ( C) , and form a normal family on some neighborhood of z. z ∈ Int n=0 f Definition 13.1.5 The Julia set J (f ) of a meromorphic function f : C −→  C   is defined to be the complement of F (f ) in C, i.e., J (f ) = C\F (f ). Thus, the Fatou set F (f ) is open while the Julia set J (f ) is closed. We adopt the following definition. Definition 13.1.6 A meromorphic function f : C −→  C is said to be (1) entire if and only if f (C) ⊆ C; (2) nearly entire if and only if it is either entire or f −1 (∞) is a singleton whose (only) element is an omitted value; (3) nonnearly entire if and only if it is transcendental and not nearly entire. The class of nonnearly entire functions will be denoted by NNE. In what follows, we will try to impose on the meromorphic function f : C −→  C as weak assumptions as possible. Eventually, we will assume that

6

Part III Topological Dynamics of Meromorphic Functions

f ∈ NNE. Although we regret it, we do this for two interrelating reasons. First, since at some point the development and particular proofs for nonnearly entire functions diverge from those that are either entire or rational; second, since ultimately we will deal in this book with elliptic functions and these all are nonnearly entire. In addition, the literature on entire and rational functions is quite rich (see the introduction to this chapter), so we do not feel too guilty. For all z ∈  C, we define O + (z) := {f n (z) : n ≥ 0}, i.e., the forward orbit of z with the convention, taken here, that f (∞) = ∞, and O − (z) :=

∞ 

f −n (z),

n=0

i.e., the backward orbit of z. We provide here the following characterization of the class of nonnearly entire functions which is a straightforward consequence of Picard’s Great Theorem. Theorem 13.1.7 If f :  C −→  C is a transcendental meromorphic function, then the following statements are equivalent. (1) (2) (3) (4)

f ∈ NNE. f −2 (∞) = ∅. The set f −2 (∞) is infinite. The set O − (∞) is infinite.

As an immediate consequence of this theorem and Montel’s Theorem II, i.e., Theorem 8.1.16, we get the following. Theorem 13.1.8 If f :  C −→  C is a nonnearly entire meromorphic function, then J (f ) =

∞ 

f −n (∞)

(13.3)

n=0

and

 F (f ) = Int

∞ n=0

f

−n

( C) .

(13.4)

13 Fundamental Properties of Meromorphic Dynamical Systems

7

We now recall the basic properties of the Fatou set and the Julia set. It can be directly seen from the definitions that the Fatou set F (f ) is completely invariant while f −1 (J (f )) ⊂ J (f ) and f (J (f )\{∞}) = J (f ).

(13.5)

We shall prove the following. Theorem 13.1.9 If f : C −→  C is a meromorphic function, then either  J (f ) = C or J (f ) has an empty interior (is nowhere dense). Proof Suppose that J (f ) has a nonempty interior. Denote this interior by  n  W . Let W∞ = ∞ n=0 f (W ). If C\W∞ contains three distinct points, then Montel’s Theorem II, i.e., Theorem 8.1.16, yields that the family of iterates {f n |W } is normal; therefore, W ⊆ F (f ). This is a contradiction; thus,  C. The proof C\W∞ contains at most two points. Hence, J (f ) ⊇ W∞ =  is complete.  We say that a point z ∈ C is exceptional if and only if the set O − (z) is finite. Picard’s Great Theorem tells us that a transcendental meromorphic function can have at most two exceptional values. Again, Montel’s Theorem II, i.e., Theorem 8.1.16, along with (13.5) imply that if z is not exceptional and z ∈ J (f ), then J (f ) = O − (z).

(13.6)

We recall that a subset of a topological space is called perfect if and only if it contains no isolated points. We shall prove the following. Theorem 13.1.10 If f : C −→  C is a nonnearly entire meromorphic function, then the Julia set J (f ) is perfect. Proof Fix z ∈ J (f ) arbitrary. Let U be an open neighborhood of z. As, by Theorem 13.1.7, O − (∞) is infinite, we can find three mutually distinct points: n } is not normal, z ∈ z1,z2,z3 ∈ O − (∞)\O + (z). Since the sequence {f|U j + − O (U ) for at least one j ∈ {1,2,3}. Hence, O (zj ) ∩ (U \{z}) = ∅. As O − (zj ) ⊆ J , this entails that J ∩ (U \{z}) = ∅. Hence, z is not isolated in J . Thus, J (f ) is perfect and the proof is complete.  Now we provide the classical classification of periodic points. A point ξ ∈ C is called periodic if f p (ξ ) = ξ

8

Part III Topological Dynamics of Meromorphic Functions

for some p ≥ 1. In this case, the number p is called a period of ξ , and the smallest p with this property is called the minimal (or prime) period of ξ . If p = 1, then ξ is also called (naturally) a fixed point of f . We denote by Per(f ), Perp (f ), and Per∗p (f ), respectively, the set of all periodic points of f , all periodic points of f of period p, and all periodic points of f of period prime p. If ξ is a periodic point of f of prime period p, then the complex number (f p ) (ξ ) is called the multiplier of ξ . We classify periodic points of f as follows. Definition 13.1.11 Let ξ be a periodic point of a meromorphic function f:C→ C with minimal period p ≥ 1. The periodic point ξ is called (1) (2) (3) (4)

attracting, super-attracting (of course, being super-attracting yields attracting), indifferent (or neutral), or repelling,

respectively, as the modulus of its multiplier is less than 1, equal to 0, equal to 1, or greater than 1. Definition 13.1.12 Writing the multiplier of an indifferent periodic point ξ in the form e2π iα where 0 ≤ α < 1, we say that (a) ξ is rationally indifferent (parabolic) if α is rational, and (b) ξ is irrationally indifferent if α is irrational. (c) If ξ is a rationally indifferent fixed point of f and f  (ξ ) = 1, then ξ is called a simple rationally indifferent (parabolic) fixed point of f . If ξ is an attracting periodic point of f with minimal period p ≥ 1, then, for all sufficiently small R > 0, we have that  1 + |(f p ) (ξ )| R ⊂ B(ξ,R). f (B(ξ,R)) ⊂ B ξ, 2 p

Thus, it follows from Montel’s Theorem II, i.e., Theorem 8.1.16, that B(ξ,R) ⊂ F (f ). So, if we define 

A(ξ ) := z ∈ C : lim f pn (z) = ξ , n→∞

13 Fundamental Properties of Meromorphic Dynamical Systems

9

then B(ξ,R) ⊂ A(ξ ) ⊂ F (f ); furthermore, if we denote by A∗ (ξ ) the connected component of A(ξ ) that contains ξ , then B(ξ,R) ⊂ A∗ (ξ ) and A(ξ ) =

∞ 

f −n (B(ξ,R)) =

n=0

∞ 

f −n (A∗ (ξ )).

n=0

Since all limit points of iterates of f on A(ξ ) are constant functions (with values in {ξ,f (ξ ), . . . ,f p−1 (ξ )}), we conclude that no point on the boundary of A∗ (ξ ) may belong to the Fatou set F (f ). Thus, the set A∗ (ξ ) is a connected component of the Fatou set F (f ). We collect these observations in the following theorem. Theorem 13.1.13 If ξ ∈ C is an attracting periodic point of a meromorphic function f : C −→  C, then A∗ (ξ ) and A(ξ ) are open sets, ∗

ξ ∈ A (ξ ) ⊂ A(ξ ) =

∞ 

f −n (A∗ (ξ )) ⊂ F (f ),

n=0

and

A∗ (ξ )

is a connected component of the Fatou set F (f ).

In particular, the attracting periodic point of a meromorphic function f belongs to its Fatou set. For repelling periodic points, just the opposite is true. Theorem 13.1.14 Each repelling periodic point of a meromorphic function f:C→ C belongs to the Julia set J (f ) of f . Proof Seeking contradiction, suppose that a repelling periodic point ξ of f belongs to the Fatou set F (f ). Denote the minimal period  n of∞ ξ by p. Let U be such an open neighborhood of ξ that the family f |U n=0 is normal.  Then ∞exists a meromorphic function g : U → C to which some sequence  pk there n f |U n=0 converges uniformly on compact subsets of U . Then g(ξ ) = ξ ; therefore, g is holomorphic and, in particular, g  (ξ ) ∈ C. But, on the other hand,    |g  (ξ )| = lim  f pkn (ξ ) = lim |(f p ) (ξ )|kn = +∞ k→∞

k→∞

as |(f p ) (ξ )| > 1. This contradiction finishes the proof.



We also have the following. Theorem 13.1.15 Each rationally indifferent periodic point of a meromorphic function f : C −→  C belongs to the Julia set J (f ) of f . Proof Changing coordinates by a translation, we may assume without loss of generality that this periodic point is equal to 0. Passing to a sufficiently high

10

Part III Topological Dynamics of Meromorphic Functions

iterate, we may further assume that 0 is a simple parabolic fixed point of f . Then the Taylor series expansion of f about 0 takes on the form f (z) = z + azp+1 + higher terms of z, where a = 0 and p ≥ 1 is an integer. We shall show by induction that f n (z) = z + nazp+1 + higher terms of z. Indeed, this is, of course, true for n = 1. Assuming that it is true for some n ≥ 1, and denoting higher than k ≥ 0 terms of w (a power series of w starting with w k+1 ) by H Tk (w), we get that f n+1 (z) = f (f n (z)) = f n (z) + a(f n (z))p+1 + H Tp+1 (f n (z)) = f n (z) + a(f n (z))p+1 + H Tp+1 (z) = z + nazp+1 + H Tp+1 (z) p+1  + H Tp+1 (z) + a z + nazp+1 + H Tp+1 (z) = z + (n + 1)azp+1 + H Tp+1 (z). The inductive proof is complete. Consequently, (f n )(p+1) (0) = a(p + 1)! n. Therefore, limn→∞ |(f n )(p+1) (0)| = +∞; as f (0) = 0, the proof can now be concluded in the same way as the proof of Theorem 13.1.14.  Definition 13.1.16 Let ξ be a periodic point of a meromorphic function f : C −→  C with minimal period p ≥ 1. The map f p is called linearizable near the periodic point ξ if and only if f p is topologically conjugate to its differential z −→ g(z) := ξ + (f p ) (ξ )(z − ξ ) in some (sufficiently small) neighborhood of ξ . Theorem 13.1.17 An irrationally neutral periodic point ξ of a meromorphic function f : C −→  C with minimal period p ≥ 1 belongs to the Fatou set F (f ) if and only if f p is linearizable near ξ . If this holds, the point ξ is called a Siegel periodic point of f . The corresponding topological conjugacy then also yields a holomorphic one. Proof Replacing f by f p we may assume without loss of generality that p = 1, i.e., that ξ is a fixed point of f . Furthermore, changing coordinates by a translation, we may assume without loss of generality that ξ = 0. Write f  (0) := γ , |γ | = 1.

13 Fundamental Properties of Meromorphic Dynamical Systems

11

First, we assume that f is linearizable near 0. This means that H ◦ f ◦ H −1 (z) = γ z, z ∈ D, where D is a sufficiently small disk centered at 0 and H : D → D is a homeomorphism. Iterating this equation, we get, for every n ≥ 0, that H ◦ f n ◦ H −1 (z) = γ n z, z ∈ D. Equivalently, f n (z) = H −1 (γ n H (z)), z ∈ D. In f n (D) ⊆ D for every n ≥ 0. Therefore, the family of iterates ∞  nparticular, f |D n=0 is normal, so 0 ∈ F (f ). We now assume that 0 ∈ F (f ). Then there is a neighborhood of 0 on which the sequence (f n )∞ n=0 is equicontinuous; from this, we see that there exists some ball B(0,r) (0 < r < 1) of 0 such that, for all n ≥ 0 and all z ∈ U , we have that |f n (z)| = |f n (z) − f n (0)| < 1.

(13.7)

Now, for every n ≥ 1, define a function Tn : B(0,r) → C by the formula   Tn (z) := n−1 z + γ −1 f (z) + γ −2 f 2 (z) · · · + γ −(n−1) f n−1 (z) . Note that, as |γ | = 1, we have that Tn (B(0,r)) ⊆ B(0,1)

(13.8)

for every n ≥ 1. A direct verification shows that the functions Tn satisfy the following: (n/γ )Tn (f (z)) + z = (n + 1)Tn+1 (z) = nTn (z) + γ −n f n (z). Hence, Tn (f (z)) − γ Tn (z) = n−1 (γ 1−n f n (z) − γ z). Since |γ | = 1 and invoking (13.7), we, thus, conclude that Tn (f (z)) − γ Tn (z) → 0

(13.9)

uniformly on B(0,r) as n → ∞. Next, (13.8) implies that the sequence  ∞ Tn |B(0,r) n=1 is normal on U . It, thus, follows that there exists (kn )∞ n=1 , an increasing sequence of positive integers, such that Tkn : B(0,r) −→ B(0,1) converges locally uniformly on B(0,r) to some holomorphic function H : B(0,r) −→ B(0,1). By (13.9), it satisfies H ◦ f (z) = γ H (z)

12

Part III Topological Dynamics of Meromorphic Functions

for all z ∈ U . Since Tn (0) = 1, we also have that H  (0) = 1, so H is a homeomorphism on a sufficiently small neighborhood of 0. This completes the proof.  Definition 13.1.18 An irrationally neutral periodic point ξ of a meromorphic function f : C −→  C with minimal period p ≥ 1 belonging to J (f ) is called a Cremer periodic point of f . If p ≥ 1 denotes the prime period of ξ , then near ξ the map f p is not topologically conjugate to its differential (see Theorem 13.1.17). From all the above, we have the following. Theorem 13.1.19 All attracting and Siegel periodic points of a meromorphic function f : C −→  C are in the Fatou set of f , while repelling, rationally indifferent, and Cremer periodic points are in the Julia set of f . Also, a point ξ ∈  C is called preperiodic if f n (ξ ) is periodic for some n ≥ 0. If f is a nonnearly entire meromorphic function and n ≥ 2, then f has infinitely many periodic points of mininimal period n. In fact, f has infinitely many repelling periodic points of minimal period n and the Julia set of f is the closure of the set of all repelling periodic points of f . We shall now prove these results. Moreover, the Julia set of f will turn out to be perfect. Theorem 13.1.20 If f : C −→  C is a nonnearly entire meromorphic function, then J (f ), the Julia sets of f , is the closure of the set of all repelling periodic points of f . In order to prove Theorem 13.1.20 and the preceding statement about repelling periodic points, we need the following well-known “Five-Island Theorem ” of Ahlfors (see [Ah]) from complex analysis. Theorem 13.1.21 (Ahlfors Five-Island Theorem) If f : C →  C is a transcendental meromorphic function and D1,D2, . . . ,D5 are any five simply connected domains in C with mutually disjoint closures, then there exists at least one j ∈ {1,2, . . . ,5} such that, for every R > 0, there exists a simply connected domain G ⊆ {z ∈ C : |z| > R} for which the map f |G : G → Dj is a conformal homeomorphism. If f has only finitely many poles, then the number “5” may be replaced by “3.” The next lemma follows from Theorem 13.1.21. Lemma 13.1.22 Suppose that f : C →  C is a transcendental meromorphic function and that some five points z1,z2, . . . ,z5 ∈ O −1 (∞)\{∞} are mutually

13 Fundamental Properties of Meromorphic Dynamical Systems

13

distinct. Define nj ≥ 1 uniquely by the property that f nj (zj ) = ∞. Then there exists j ∈ {1, . . . ,5} such that zj is a limit point of repelling periodic points of f with minimal period equal to nj + 1. If f has only finitely many poles then “5” may be replaced by “3.” Proof In order to deduce Lemma 13.1.22 from Theorem 13.1.21, we choose the Dj to be disks around zj , where the radii are chosen so small that the Dj do not contain critical points of f nj nor other poles of f nj apart from zj , and that their closures are pairwise disjoint. Since each zj is a pole of f nj , there exists R > 0 such that nj −1   k nj 1 f (Dj ) ⊆ B(0,R/4) and f Dj ⊃ {z : |z| > R/2} ∪ {∞} 2 k=0

(13.10) for all j = 1,2, . . . ,5. We pick j and G according to the Ahlfors Five Island Theorem, i.e., Theorem 13.1.21. Then we can find an open connected set H ⊆ (1/2)Dj such that f nj (H ) = G. So, the map f nj +1 |H : H −→ Dj is a holomorphic surjection without critical points. Since the open set Dj is simply connected, the Monodromy Theorem implies that the inverse map 1 Dj 2 is well defined and, of course, holomorphic. By the Brouwer Fixed Point −(n +1) Theorem (an alternative argument would be to use contraction of f∗ j in a hyperbolic metric on Dj and to apply the Banach Contraction Principle), −(nj +1)

f∗

: Dj −→ H ⊆

−(n +1)

in (1/2)Dj , and by Schwarz’s Lemma there exists a fixed point ξ of f∗ j this point is attracting. Hence, ξ is a repelling fixed point of f nj +1 . In other words, ξ is a repelling periodic point of f of period nj + 1, and by (13.10) its minimal period is equal to nj + 1. Because the disks Dj can be chosen arbitrarily small, we, therefore, now conclude that zj is the limit of a sequence of repelling periodic points with minimal period equal to nj +1. This completes the proof of the lemma.  Theorem 13.1.20 now directly follows from Lemma 13.1.22, (13.3), and Theorem 13.1.14.

13.2 Classification of Periodic Fatou Components In this section, given a meromorphic function f : C −→  C, we want to describe and analyze in detail the structure of periodic connected components

14

Part III Topological Dynamics of Meromorphic Functions

of the Fatou set F (f ). First, we prove some short introductory results about backward and forward iterates of such components and, more generally, about iterates of arbitrary open connected sets. Then we study at length periodic connected components of Fatou sets. Theorem 13.2.1 Let f : C −→  C be a meromorphic function and V ⊆  C be an open connected set. If n ≥ 1 is an integer and U is a connected component of f −n (V ) such that, for each 0 ≤ k ≤ n − 1, the set f k (U ) is bounded (this precisely means that ∞ ∈ / f k (U )), then f n (∂U ) ⊆ ∂V; equivalently, the map f n |U : U → V is proper, in particular surjective. Proof Since, by our assumptions, the function f n |U → V is continuous, we have that f n (U ) ⊆ f n (U ) ⊆ V . Therefore, f n (∂U ) ⊆ f n (U ) ⊆ V . Now seeking contradiction, suppose that f n (∂U ) ⊆ ∂V . So, f n (∂U )∩V = ∅. Hence, there exists a point ξ ∈ V ∩ f n (∂U ). Thus, there exists a point z ∈ ∂U

(13.11)

such that f n (z) ∈ V . Since all points z,f (z),f 2 (z), . . . ,f n−1 (z) are in C, it follows from (iterated) continuity of f that there exists r > 0 such that f n (Be (z,r)) ⊆ V . But then as a union of two intersecting connected sets, the set U ∪ B  e (z,r) is connected  and also f n U ∪ Be (z,r) ⊆ V . It, therefore, follows from the definition of U that Be (z,r) ⊆ U . In particular, z ∈ U , contrary to (13.11) since U is open. The proof of the first assertion of our theorem is complete. The second one follows immediately from Theorem 8.6.2.  Theorem 13.2.2 Let f : C −→  C be a transcendental meromorphic function and V ⊆  C be a connected component of the Fatou set F (f ). Then f n (V ) is contained in a unique connected component Vn (f ) of F (f ), each connected component U of f −n (V ) is a connected component of the Fatou set F (f ), and ∂U ⊆ J (f ). Proof The first assertion of our theorem is obvious. To prove the third one, take a point z ∈ ∂U . If f n (z) = ∞, then z ∈ J (f ) as ∞ ∈ J (f ). If

13 Fundamental Properties of Meromorphic Dynamical Systems

15

f n (z) = ∞, then all points z,f (z),f 2 (z), . . . ,f n−1 (z) are in C, whence it follows from (iterated) continuity of f that f n (z) ∈ V .

(13.12)

Now seeking contradiction, suppose that f n (z) ∈ V . Then, by an already established continuity of f n at z and the openness of the set V , there would exist δ > 0 such that f n (Be (z,δ)) ⊆ V . But then U ∪ Be (z,δ) would be a connected set with f n U ∪ Be (z,δ) ⊆ V . Thus, by the definition of U , we would have that U ∪ Be (z,δ) ⊆ U . Consequently, z ∈ U , contrary to the facts / V ; it, thus, follows from (13.12) that that U is open and z ∈ ∂U . So, f n (z) ∈ n f (z) ∈ ∂V . But ∂V ⊆ J (f ), whence z ∈ J (f ), and the proof of the third assertion of our theorem is complete. To prove the second assertion of our theorem, note that there exists a unique connected component Uˆ of F (f ) such that U ⊆ Uˆ . Seeking contradiction, suppose that U ⊆ Uˆ . Since U is open in Uˆ and since Uˆ is connected, this implies that ∂Uˆ U = ∅. But since ∂Uˆ U ⊆ Uˆ ∩ ∂C U , we get that Uˆ ∩ ∂C U = ∅. But this is a contradiction since Uˆ ⊆ F (f ) and, by the already proven third  assertion, ∂C U ⊆ J (f ). The proof of Theorem 13.2.2 is complete. Remark 13.2.3 It follows from Theorems 13.2.1 and 13.2.2 that, with the notation of the latter, if n = 1 and U is bounded, then U is a connected component of the Fatou set F (f ) and f (U ) = V . This is generally no longer true if U is not bounded. Indeed, if the map f is of the form z → λez with λ ∈ (0,1/e), then F (f ) has exactly one connected component V , this component is the basin of immediate attraction to an attracting fixed point, and 0 ∈ V . Then f (U ) ⊆ V , but f (U ) = V since obviously 0 ∈ / f (U ). Here, the missing point 0 is an omitted value of f , but, in general, such a point need not be an omitted value. Examples of such maps and an extended discussion of images of connected components of Fatou sets can be found in [Ber1], [Ber2], [Bol1], and [Bol2]. We would only like to add here that Herring proved in [Her] that if f ∈ NNE, then the set V \f (U ) contains at most two points. As Bolsch pointed out, this also follows from a result of Heins in [Heins]. We will show, see Theorem 17.1.10, that, for elliptic functions that belong to class NNE (see Theorem 17.0.1) and that are for us the primary object of interest in this volume of the book, f (U ) = V . Definition 13.2.4 Let f : C −→  C be a meromorphic function. A connected component U of F (f ) is called preperiodic if and only if there exist n > m ≥ 0 such that Un (f ) = Um (f ).

16

Part III Topological Dynamics of Meromorphic Functions

In particular, if m = 0, then U is called periodic with period n. The finite set {U0 (f ),U1 (f ), . . . ,Un−1 (f )} is then called a (periodic) cycle of components. The smallest n ≥ 1 with this property is called the minimal period of U . Observe that then f n (U ) ⊆ U . In the case when n = 1, i.e., if U1 (f ) = U or, equivalently, f (U ) ⊂ U, the connected component U is called f -invariant or just invariant. A connected component of F (f ) that is not preperiodic is called a wandering component. Now we formulate and will prove the fundamental classification theorem of all periodic components of the Fatou set F (f ). This theorem is essentially due to Cremer [Cre] and Fatou [Fat2]. Fatou proved in [Fat2, §56, p. 249] that if f a rational function, V ⊆  C, is an open connected set, and the sequence  is ∞ f n |V n=0 has only constant limit functions, then V is the immediate basin of attraction either to an attracting periodic point or to a rationally indifferent periodic point, i.e., a Fatou–Leau domain. His proof shows that the only other possibility in the case of transcendental functions is that of a Baker domain. ∞ Cremer proved in [Cre, p. 317] that if the sequence f n |V n=0 has nonconstant limit functions, then V is either a Siegel disk or a Herman ring. Neither Fatou nor Cremer stated the full classification theorem, but T¨opfer’s remarks in [T¨o, p. 211] come quite close to it. Theorem 13.2.5 (Fatou Periodic Components) Let U be a periodic connected component of the Fatou set F (f ), of some period p ≥ 1, of a nonaffine (different from z → az + b) meromorphic function f : C −→  C. Then we have one of the following possibilities. (1) U contains an attracting periodic point ξ of period p. Then lim f np (z) = ξ

n→∞

for all z ∈ U . We recall then that U = A∗ (ξ ) is called the immediate basin of attraction to ξ . (2) ∂U contains a periodic point ξ of period p and lim f np (z) = ξ

n→∞

13 Fundamental Properties of Meromorphic Dynamical Systems

17

for all z ∈ U . Then (f p ) (ξ ) is a root of unity, meaning that ξ is a rationally indifferent periodic point of f . In this case, the component U is commonly called a Leau–Fatou domain of ξ . (3) There exists an analytic homeomorphism H : B(0,1) −→ U such that the following diagram commutes: B(0,1)

Rα

H

B(0,1) H

U

fp

U

i.e., H −1 (f p (H (z))) = e2π iα z, z ∈ B(0,1), for some α ∈ R\Q, where Rα : B(0,1) −→ B(0,1) is the rotation Rα (z) = e2π iα z. In this case, U is called a Siegel disk of f . (4) There exists an analytic homeomorphism H : A(0;1,r) −→ U , r > 1, such that the following diagram commutes: A(0;1,r)

Rα

A(0;1,r)

H

H fp

U

U

i.e., H −1 (f p (H (z))) = e2π iα z, z ∈ A(0;1,r), for some α ∈ R\Q. In this case, U is called a Herman ring of f . (5) There exists ξ ∈ ∂U such that lim f np (z) = ξ

n→∞

for all z ∈ U , but there is no continuous extension of f p from U to ξ . In this case, U is called a Baker domain of f . We then have that ∞ ∈ ∂U ∪ ∂(f (U )) ∪ ∂(f 2 (U )) ∪ · · · ∪ ∂(f p−1 (U )). This classification theorem follows from Theorem 13.2.8 below. The reader may also consult the books [CaGg], [Bea], [Ste], and [Mil1] for the case of rational functions. We start the proof with the following. Lemma 13.2.6 Let f : C −→  C be a nonaffine (different from any map of the form z → az + b) meromorphic function. Suppose that D is a forward f -invariant connected component of the Fatou set F (f ) of f .

18

Part III Topological Dynamics of Meromorphic Functions

 ∞ If there exists some nonconstant limit function of the sequence f n |D n=0 , and the then the function f |D : D → D is a conformal homeomorphism  n ∞ identity map IdD on D is a limit point of the sequence f |D n=0 . Proof Let g : D −→  C be a nonconstant (analytic) limit function of the ∞ n sequence f |D n=0 ; by our hypotheses, at least one such function exists. Therefore, there exists (nj )∞ j =1 , an unbounded increasing sequence of positive integers such that f nj |D −→ g uniformly on compact subsets of D as j → ∞. Since f nj (D) ⊆ D for all j ≥ 1 and since the limit function g is nonconstant, it follows from Theorem A.0.21 in Appendix A that g(D) ⊆ D.

(13.13)

By passing to a subsequence and relabeling it, we may assume that lim (nj − nj −1 ) = +∞.

j →∞

For every j ≥ 1, put mj := nj − nj −1 .  ∞ Since the family f mj |D j =1 is normal, by the same token as for the  n ∞ sequence f |D n=0 , there are some holomorphic h : D → D and   functions m j some infinite set N ⊆ N such that the sequence f |D j ∈N converges to h uniformly on compact subsets of D. Now take any z ∈ D. Since limj →+∞ f nj (z) = g(z) with g(z) ∈ D by (13.13), there exists s > 0 such that 

B(g(z),2s) ⊆ D and f nj (z) ∈ B(g(z),s) for all j ≥ 1 large enough. Therefore, lim

Nj →+∞

f mj = h

uniformly on B(g(z),s). In conclusion, h(g(z)) =

lim

Nj →+∞

f mj (f nj −1 (z)) =

lim

Nj →+∞

f nj (z) = g(z).

(13.14)

13 Fundamental Properties of Meromorphic Dynamical Systems

19

As g is not constant, h must be the identity map IdD on D, and this proves the second claim of our lemma. The fact that the function f |D : D → D is a conformal homeomorphism follows now quite easily. Injectivity first. Assume that a,b ∈ D and f (a) = f (b). Then f mj (a) = f mj −1 (f (a)) = f mj −1 (f (b)) = f mj (b). So, letting j → ∞ in N, we obtain, by the already proved part of our lemma, that a = IdD (a) = IdD (b) = b. So, f |D : D → D is injective. In order to prove surjectivity of f |D : D → D, seeking contradiction, suppose that f (D) = D. Since f (D) is an open set and D = h(D) is not a singleton, it would then follow from Theorem A.0.21 that D = h(D) ⊆ f (D) ⊆ D. This contradiction finishes the proof of surjectivity of f |D . Thus, f |D : D → D is a conformal homeomorphism and the proof of Lemma 13.2.6 is complete.  Lemma 13.2.7 For every r ∈ (0,1), the only conformal homeomorphisms of the annulus Ar := {z ∈ C : r < |z| < 1} that are of infinite order are Euclidean rotations by angles which are irrational multiples of π. Proof Let κ := log(1/r). We use the notation of Proposition 8.2.9. In particular, the map g : H −→ H is given by the formula g(z) := kz, where



2π 2 k := exp κ



and κ : H −→ Ar

20

Part III Topological Dynamics of Meromorphic Functions

is the analytic covering map of Ar given by the formula κ  i log z . κ (z) := exp π

(13.15)

Let G : Ar −→ Ar be an infinite-order conformal homeomorphism of Ar . Let ˜ : H −→ H G be a lift of G, i.e., such a continuous map that ˜ G ◦ κ = κ ◦ G, i.e., the following diagram commutes: H

˜ G

κ

C∗

H κ

G

C∗

Then, for any two points z,w ∈ H with κ (z) = κ (w), we have that ˜ ˜ κ (G(z)) = G ◦ π(z) = G ◦ π(w) = κ (G(w)). Therefore, because of Proposition 8.2.9, ˜ ˜ G(z) = g m (G(w)) with some integer m. It, in turn, follows from this that, for each z ∈ H, there is an integer m(z) such that ˜ ˜ G(kz) = k m(z) G(z). So, the function  −1 ˜ ˜ G  z −→ k m(z) = G(kz) G(z) ∈C is continuous. But, in fact, this function takes values in (0,+∞) and 

˜  m(z)  1 1 G(kz) m(z) = = log k log . ˜ log k log k G(z)

(13.16)

13 Fundamental Properties of Meromorphic Dynamical Systems

21

So, we conclude that the function H  z −→ m(z) ∈ Z is continuous. Therefore, since the space H is connected, the function H  z −→ m(z) ∈ Z is constant. Denote its only value by m. Formula (13.16) then takes on the form ˜ ˜ G(kz) = k m G(z);

(13.17)

˜ ˜ n z) = k mn G(z) G(k

(13.18)

So, by immediate induction:

˜ is a M¨obius map and suppose for a for all z ∈ H and all n ∈ Z. Recall that G moment that m = 0. Letting n → +∞, it would then follow from (13.18) that ˜ is a constant function whose only value is G(∞), ˜ ˜ G contrary to the fact that G is an invertible M¨obius map. Thus, m = 0. Recalling again that H˜ is a M¨obius map and letting n → +∞, we conclude ˜ from (13.18) that G(∞) is either ∞ (if m > 0) or 0 (if m < 0). By letting ˜ ˜ fixes n → −∞, we conclude that G(0) is either 0 or ∞. Therefore, either G ˜ both 0 and ∞ or interchanges them. This means that G is one of the form  C  z −→ az ∈  C or  C  z −→ b/z ∈  C, where a > 0 and b < 0. Using (13.15), we get in the former case that κ  ˜ = κ (az) = exp i log(az) G(κ (z)) = κ ◦ G(z) π  κ κ i log(a) exp i log(z) = exp π  π κ i log(a) κ (z) = exp π for all z ∈ H. Likewise, in the latter case, G(κ (z)) =

exp

κ

π i log(b)



κ (z)

for all z ∈ H. Hence, G(w) = eiα w and G(w) = eiβ /w

22

Part III Topological Dynamics of Meromorphic Functions

with α := πκ log(a) ∈ R and β := πκ log(b) ∈ R for all w ∈ Ar , respectively, in C is of order 2, the former and the latter case. But the map  C  w → eiβ /w ∈  whence, as H is of infinite order, it follows that G(w) = eiα w 

for all w ∈ Ar . The proof is complete. Now we shall prove the following, already announced, theorem.

Theorem 13.2.8 Let f : C −→  C be a nonaffine (different from z → az + b) meromorphic function and V ⊆  C be a periodic connected component of the Fatou set F (f ) of f .  ∞ (a) If all the limit functions of the sequence f n |V n=0 are constant and, in the case when ∞ is an essential singularity of f , each of them is different from ∞, then V is either the immediate basin of attraction of an attracting periodic point of f or V is a Leau–Fatou ∞ of f .  ndomain (b) If the limit functions of the sequence f |V n=0 contain a nonconstant function, then V is either a Siegel disk or a Herman ring. (c) If ∞ is an essential singularity of f ,i.e., if f is transcendental, all the ∞ limit functions of the sequence f n |V n=0 are constant, and at least one of them is equal to ∞, then V is a Baker domain. In particular, ∞ ∈ ∂V . Proof

We first assume that V is f -invariant, i.e., f (V ) ⊆ V .

Item (a). If ∞ is a limit function, then, by our hypothesis, f is a rational function. In what follows, we actually work entirely on the Riemann sphere  C. Since V is a connected component of the Fatou set F (f ), the family ∞

n C n=0 (13.19) f |V : V −→  is normal. Hence, the set L of all limit functions of this family, which are all by our hypotheses constant, is nonempty. We shall show that each element of L is a fixed point of f . Indeed, let ξ ∈ L. Then there exists (nk )∞ k=1 , an unbounded increasing sequence, such that lim f nk (z) = ξ

k→∞

for all z ∈ V . So, take any w ∈ V . Then f (w) ∈ f (V ) ⊆ V and   f (ξ ) = f lim f nk (w) = lim f nk (f (w)) = ξ . k→∞

k→∞

13 Fundamental Properties of Meromorphic Dynamical Systems

23

Now again take any point w ∈ V . Since V is arcwise connected, there exists a continuous map (even a homeomorphic embedding) p : [0,1] −→ V such that p(0) = w and p(1) = f (w). Next, extend this function to a function C p∞ : [0,+∞) −→  by the formula p∞ (t) := f [t] (p(t − [t])). It is immediate from this definition that the function p∞ is continuous. Since the set of all accumulation points of p∞ (as t → +∞) is a closed (so compact) connected subset of  C, either it is a singleton or it contains uncountably many points. Note that the set of accumulation ∞coincides with the set of  points of p∞ accumulation points of the sequence f n (p([0,1])) n=0 , which is equal to L as p([0,1]) is a compact subset of V . Thus, either L is a singleton or it contains uncountably many points. Seeking contradiction, suppose that the set L is uncountable. Consider a meromorphic function h : C →  C defined as h(z) := f (z) − z. Then h(ξ ) = 0 for all ξ ∈ L. So, h has an uncountable set of zeros and, therefore, is identically equal to zero. Thus, f (z) = z for all z ∈ C, contrary to our hypotheses. Therefore, the set L is a singleton. In accordance with our considerations up to now, denote its only element by ξ . Of course, ξ ∈ V. Suppose first that ξ ∈ V. Fix a number R > 0 such that B(ξ,R) ⊆ V . By the definition of ξ , there exists an integer m ≥ 1 such that f m (B(ξ,R)) ⊆ B(ξ,R/2).

24

Part III Topological Dynamics of Meromorphic Functions

So, by Schwarz’s Lemma, |f  (ξ )|m = |(f m ) (ξ )| < 1. Hence, |f  (ξ )| < 1 and ξ ∈ V is an attracting fixed point of f . As we already know, the full sequence

n ∞ f |V : V −→  C n=0 converges to ξ on V uniformly on compact subsets of V . Thus, V is the basin of immediate attraction of the attracting fixed point ξ ∈ V . Suppose, in turn, that ξ ∈ ∂V . Since the full sequence

n ∞ f |V : V −→  C n=0

converges to ξ on V uniformly on compact subsets of V and f (ξ ) = ξ , it remains to prove that f  (ξ ) = 1. Obviously, |f  (ξ )| ≥ 1 since, otherwise, ξ would lie in F (f ). Seeking contradiction, suppose that |f  (ξ )| > 1. Then since, also fixing any z ∈ V , we know that limn→∞ f n (z) = ξ , we conclude that f n (z) = ξ for some integer n ≥ 0. But as ξ ∈ J (f ), so also z ∈ J (f ). This contradicts the fact that z ∈ V ⊆ F (f ) and we get that |f  (ξ )| = 1. Conjugating (changing coordinates) by an affine function (z → az+b, a = 0), we may assume without loss of generality that ξ = 0.

13 Fundamental Properties of Meromorphic Dynamical Systems

25

Let λ := f  (0). Then there exists R > 0 such that f (z) = λz + a2 z2 + a3 z3 + · · · = λz + z2 H (z) for all z ∈ B(0,R), where H : B(0,R) −→ C is some bounded holomorphic function. Since f  (0) = 0, we may require R > 0 to be so small that the map f |B(0,R) : B(0,R) −→ C is one-to-one. Since f |nV −→ 0 uniformly on compact subsets of V as n → +∞ and 0 ∈ V , there exist w ∈ V \{0} and r > 0 such that B(w,2r) ∩ B(0,2r) = ∅,

(13.20)

B(w,4r) ⊆ B(0,R), and, moreover, f n (B(w,4r)) ⊆ B(0,R) for every integer n ≥ 0 large enough, say n ≥ N. In particular, the function f |nB(w,4r) : B(w,2r) −→ B(0,R) is holomorphic for every integer n ≥ N and 0∈ / f n (B(w,4r)) for every integer n ≥ N. Therefore, we can, for every integer n ≥ N , define a function gn : B(w,4r) −→ C by the formula gn (z) :=

f n (z) . f n (w)

Of course, all functions gn are holomorphic and injective and gn (w) = 1, 0,∞ ∈ / gn (B(w,4r)), and g n (1) = {w}.

26

Part III Topological Dynamics of Meromorphic Functions

It, thus, follows from the 14 -Koebe Distortion Theorem (Theorem 8.3.3) that 1  |g (w)| ≤ 1. 4 n So, it follows from the Koebe Distortion Theorem (Theorem 8.3.8) that   gn (B(w,2r)) ⊆ B 1,K|gn (w)| ⊆ B(1,4K) ⊆ B(0,1 + 4K). Thus, the sequence of holomorphic functions  ∞ gn |B(w,2r) n=0 is normal. Let (nk )∞ k=1 be any increasing sequence of positive integers such that the sequence (gnk )∞ k=1 converges uniformly on compact subsets of B(w,2r). Let g := lim gnk . k→∞

Of course, g : B(w,2r) → C is a holomorphic function. For every n ≥ 0 and every z ∈ B(w,r), we have that f (f n (z)) λf n (z) + (f n (z))2 H (f n (z)) = n f (f (w)) λf n (w) + (f n (w))2 H (f n (w))   f n (z) λ + f n (z)H (f n (z))   = n f (w) λ + f n (w)H (f n (w)) λ + f n (z)H (f n (z)) = gn (z) . λ + f n (w)H (f n (w))

gn+1 (z) =

n Since the function H is bounded, ∞0, and limn→∞ f (z) = 0, we, therefore,  λ = conclude that the sequence gnk +1 k=1 also converges uniformly on compact subsets of B(w,2r) and

lim gnk +1 = g.

(13.21)

k→∞

Also, f n (f (z)) f n (w) f n (f (z)) = f n (w)

gn+1 (z) =

= gn (f (z)) ·

· ·

f n (w) f n+1 (w) f n (w) f n (w)(λ + f n (w)H (f n (w))) 1

λ + f n (w)H (f n (w))

.

13 Fundamental Properties of Meromorphic Dynamical Systems

27

Again, since H is bounded, λ = 0, and limn→∞ f n (w) = 0, using (13.21), we conclude that 1 g(z) = lim gnk +1 (z) = g(f (z)) , k→∞ λ i.e., g ◦ f = λg

(13.22)

on B(w,2r). Iterating this equality, we get that g ◦ f n = λn g

(13.23)

on B(w,2r). Seeking contradiction, suppose that g is injective. Then (13.23) is equivalent to f n (z) = g −1 (λn g(z)) for every z ∈ B(w,2r). In particular, f n (w) = g −1 (λn g(w)) = g −1 (λn ) ∈ g −1 (∂D) ⊆ B(w,2r). Invoking also (13.20), we, thus, conclude that 

n f (w) : n ≥ 0 ⊆ C\B(0,2r), contrary to the fact that limn→∞ f n (w) = 0. So, the function g is not injective, and, as g(w) = 1, it follows from Hurwitz’s Theorem that g = 11 on B(w,2r). Therefore, the sequence ∞  gn |B(w,2r) n=0 converges to the function 11 uniformly on compact subsets of B(w,2r). It, thus, follows from (13.22) that λ = 1. We are, thus, done in this case, namely (a). Item (b). Since J (f ) is infinite, the unit disk D is the universal covering space of V . Denote the corresponding cover group (the group of deck transformations) by . Let π : D −→ V be the corresponding covering projection map. Changing coordinates, we may assume without loss of generality that 0 ∈ V and that π(0) = 0. For every integer n ≥ 0, let f˜n : D −→ D be a lift of f n |V : V −→ V via π , i.e., such a continuous, in fact holomorphic, map from D to D that π ◦ f˜n = f n ◦ π,

(13.24)

28

Part III Topological Dynamics of Meromorphic Functions

i.e., the following diagram commutes: D

f˜n

D

π

V

π fn

V

Let R > 0 be so small that the restriction πR := π|B(0,2R) : B(0,2R) −→ V is injective. By our assumptions and Lemma 13.2.6, the set of limit functions of the sequence  n ∞ f |V n=0 contains the identity map IdV on V . This means that there exists an unbounded increasing sequence (nk )∞ k=1 of positive integers such that IdV = lim f nk |V k→∞

(13.25)

uniformly on compact subsets of V . So, disregarding finitely many terms, and also since π(B(0,R)) is an open subset of V containing 0, we may assume without loss of generality that f nk (π(B(0,R))) ⊆ π(B(0,2R)) for all integers k ≥ 1. So,   π f˜nk (B(0,R)) = f nk (π(B(0,R))) ⊆ π(B(0,2R)). Therefore, there exists a deck transformation gk : D −→ D of V such that gk ◦ (B(0,R)) ⊆ B(0,2R). Replacing now f˜nk by gk ◦ f˜nk , we will have (13.24) still satisfied and f˜nk (B(0,R)) ⊆ B(0,2R). It follows from (13.25) and (13.24) that lim π ◦ f˜nk = π

k→∞

(13.26)

13 Fundamental Properties of Meromorphic Dynamical Systems

29

with the convergence being uniform on all compact subsets of D. Along with (13.26) and injectivity of πR , we get that lim f˜nk |B(0,R) = IdB(0,R) .

(13.27)

lim f˜nk (0) = 0.

(13.28)

k→∞

In particular, k→∞

 ∞ Since the sequence f˜nk k=1 is normal, it follows from (13.27) that lim f˜nk = IdD

k→∞

(13.29)

with the convergence being uniform on all compact subsets of D. There are two cases to consider. Namely: Case 1. f˜nk (0) = 0 for some k ≥ 1 and Case 2. f˜nk (0) = 0 for all k ≥ 1. First consider Case 1. With k ≥ 1 as therein, we have that   f nk (0) = f nk (π(0)) = π f˜nk (0) = π(0) = 0, meaning that 0 is a fixed point of f nk . So, since 0 is in the Fatou set of f nk , it is either attracting or irrationally indifferent.  nBut∞it cannot be attracting since then all the limit functions of the sequence f |V n=1 would be constant and equal to 0, f (0), . . . ,f nk −2 (0), or f nk (0). Thus, 0 is an irrationally indifferent fixed point of f nk . It then follows from Theorem 13.1.17 that V is a Siegel disk for f nk . In particular, f nk |V is conjugate to some irrational rotation; hence, it can have only one fixed point, namely 0. Consequently, 0 must be a fixed point of f and, moreover, an irrationally indifferent one. So, applying Theorem 13.1.17 again finishes the proof in this case. It remains to consider Case 2. It follows from (13.29) and Hurwitz’s Theorem that the holomorphic maps f˜nk : D −→ D are conformal homeomorphisms for all k ≥ 1 sufficiently large. In fact, it follows from Lemma 13.2.6 that f : V → V is a conformal homeomorphism and so is each of its iterates f n : V → V , n ≥ 0. It is well known (and easy to prove) in algebraic topology that a lift of any homeomorphism is a homeomorphism. So, disregarding finitely many terms (if using only Hurwitz’s Theorem), we may

30

Part III Topological Dynamics of Meromorphic Functions

assume without loss of generality that all holomorphic maps f˜nk : D −→ D, k ≥ 1, are conformal homeomorphisms. Then the inverse maps f˜−nk : D −→ D, k ≥ 1, are well defined and holomorphic and form a normal family in the sense of Montel. Let H : D −→ D be any limit function of this sequence, say −n H = lim f˜ kj j →∞

(13.30)

for some unbounded increasing sequence (kj )∞ j =1 . Fix a compact set S ⊆ D and r > 0 so small that B(H (S),2r) ⊆ D. It follows from (13.30) that f

−nkj

(S) ⊆ B(H (S),r) ⊆ D

for all j ≥ 1. Having this and noting that the set B(H (S),r) ⊆ D is compact, it follows from (13.30) and (13.29) that  −n n  −n H |S = lim IdD ◦ f˜ kj |S = lim f˜ kj  ◦ f kj |S = lim IdS . j →∞

j →∞

B(H (S),r)

j →∞

Thus, H = IdD and it follows from (13.30) that lim f˜−nk = IdD

k→∞

(13.31)

with the convergence being uniform on all compact subsets of D. Now, for every integer k ≥ 1, both sets f˜nk ◦  and  ◦˜ f nk constitute the collection of all lifts of f nk to D. In particular, f˜nk ◦  =  ◦ f˜nk . Therefore, for any element γ ∈ , f˜−nk ◦ γ ◦ f˜nk ∈ . In addition, we conclude from (13.29) and (13.31) that lim f˜−nk ◦ γ ◦ f˜nk = γ

k→∞

(13.32)

with the convergence being uniform on all compact subsets of D. Since the -orbit of any point in D cannot accumulate in ∂BD (is discrete therein), we deduce that, for all sufficiently large k ≥ 1, say for k ≥ k(γ ), we have that f˜−nk ◦ γ ◦ f˜nk = γ , meaning that f˜nk and γ commute.

(13.33)

13 Fundamental Properties of Meromorphic Dynamical Systems

31

It is convenient now to replace D (as the universal covering space of V ) by the upper half-plane H. The “new” cover groups (acting on H and depending on the choice of a covering map from H onto V ) of V are equal to h ◦  ◦ h−1, where h ranges over arbitrary M¨obius transformations from  C onto  C such that h(D) = H. However, for the simplicity of exposition, we continue to use the notation  and f˜nk , where now f˜nk , k ≥ 1, are conformal homeomorphisms of H (and f˜nk is still a lift of f nk ). By choosing the map h suitably, we may assume that the group  contains one of the maps H  z −→ z + 1 ∈ H or H  z −→ kz ∈ H with some real number k > 1. Assume first that  contains the former, i.e., the map γ : H → H given by the formula γ (z) = z + 1 belongs to . Let ρ ∈  be arbitrary. Recall (see (13.33)) that, for large k ≥ max{k(γ ),k(ρ)}, the map f˜nk commutes with both γ and ρ. A straightforward computational reasoning shows that the only M¨obius maps which preserve H and which commute with a map z → z + t, t ∈ R, are all the maps of the form z → z + s, s ∈ R. Thus, f˜nk (z) = z + t, z ∈ H with some t ∈ R; hence, ρ(z) = z + s, z ∈ H with some s ∈ R. This means that  is a discrete group of real translations and so, up to conjugacy, is generated by the map H  z → z + 1 ∈ H. In this case, the quotient space H/  is, up to conformal conjugacy, the punctured unit disk D0 := {z ∈ C : 0 < |z| < 1} and the corresponding quotient map is given by the formula H  z −→ exp(2π iz) ∈ D0 . It follows that there is a conformal homeomorphism Q : D0 −→ V from D0 onto V . Since the point 0 ∈ D is neither pole nor an essential singularity of Q, it must be a removable singularity. Therefore, Q extends to a meromorphic map from D onto V ∪ {Q(0)} and Q(0) is an isolated point of

32

Part III Topological Dynamics of Meromorphic Functions

∂V . Hence, Q(0) is an isolated point of J (f ), contrary to Theorem 13.1.10. Thus,  must contain an element γ of the form H  z −→ γ (z) = kz ∈ H with real numbers k > 1. The argument follows now essentially the same lines as above. The only M¨obius maps which commute with γ and preserve H are of the form H  z −→ tz,

(13.34)

where t > 0, and the maps of the form H  z −→ u/z,

(13.35)

where u < 0. Now maps of the latter type are of order 2, so if  contains such an element and k ≥ 1 is large enough, then, by (13.32), f˜2nk = IdH . Thus, f 2nk |V = IdV . Since both functions f 2nk : C →  C and IdC : C →  C are meromorphic, this implies that f 2nk = IdC, contrary to the fact that the degree of f is larger than 1. It follows that each map f˜nk : H → H with k ≥ 1 large enough is of the form from (13.34). Now, for every k ≥ 1 large enough, any element ρ ∈  commutes with f˜nk , so ρ is one of the maps of (13.34) or (13.35). If ρ ∈ \{IdH }, then ρ has no fixed points in H. So, ρ must be of the form from (13.34), in addition with t = 1. Hence,  is a discrete subgroup of the maps of (13.34). Thus, it is a cyclic group generated by some map H  z −→ kz ∈ H, k ∈ (0,+∞)\{1}. Write 2π 2 κ with some unique κ > 0. It, therefore, follows from Proposition 8.2.9 that the quotient space H/  is conformally equivalent to the annulus A(0;e−κ ,1). Since it is also conformally equivalent to V , item (b) is now proved by a direct application of Lemma 13.2.7. k=

13 Fundamental Properties of Meromorphic Dynamical Systems

33

Proving item (c), note first that if the family of (13.19) has a limit function which is constant, then the reasoning from the beginning of proving item (a) shows that the collection of all limit functions of this family is a singleton. But this is a contradiction since ∞ is such a function. Thus, lim f n (z) = ∞

n→∞

for all z ∈ V uniformly on all compact subsets of V . The proof of item (c) is complete; simultaneously, the proof of the whole theorem is also complete in the case of the forward invariant Fatou components V . If V is periodic with some period p ≥ 1, then we pass to the pth iterate f p and apply the already proven case of invariant domains (i.e., of period 1). One should only notice that the fact that f p need no be longer meromorphic (prepoles of f of orders ≤ p − 1 become essential singularities for f p ) causes no problems. The proof of Theorem 13.2.8 is complete.  We end this section with the following result, which is interesting on its own and which forms an ingredient in the proof, given in Appendix B, of Theorem 14.4.4, which asserts that no functions in Speiser class S have wandering domains. Proposition 13.2.9 If f : C −→  C is a meromorphic function, is a con n V∞ nected component of the Fatou set F (f ), and the sequence f |V n=0 has a nonconstant limit function, then the component V is preperiodic. Proof It follows from the hypotheses of our proposition that there exists some strictly increasing sequence (nj )∞ j =1 of positive integers such that the sequence  n ∞ j f j =1 converges to some nonconstant analytic function g : V −→ C uniformly on each compact subset of V . Then there exists a point ξ ∈ V such that g  (ξ ) = 0. Thus, there exists r > 0 such that B(ξ,2r) ⊆ V and g(z) = g(ζ ) for all z ∈ B(ξ,r)\{ζ }; in particular, for all z ∈ ∂B(ξ,r). Since the latter set is compact, we, thus, have that

 inf |g(w) − g(ξ )| : w ∈ ∂B(ξ,r) > 0. ∞  So, since the sequence f nj |∂B(ξ,r) j =1 converges uniformly to g|∂B(ξ,r) , we will have, for all integers j ≥ 1 large enough and all z ∈ ∂B(ξ,r), that

34

Part III Topological Dynamics of Meromorphic Functions

       nj  f (z) − g(ξ ) − g(z) − g(ξ )  =  f nj (z) − g(z) 

 < inf |g(w) − g(ξ )| : w ∈ ∂B(ξ,r) ≤ |g(z) − g(ξ )|. Hence, by Rouch´e’s Theorem, the two functions B(ξ,r)  z −→ g(z) − g(ξ ) ∈ C and B(ξ,r)  z −→ f nj (z) − g(ξ ) ∈ C have the same number of zeros for all j ≥ k. Since the first of these two functions vanishes at ξ , the second one must also have at least one zero in B(w,r). Denote it by ζj . So, f nj (ζ ) = g(ξ ). But then also f nj +1 (ζj +1 ) = g(ξ ). Thus, in particular, f nj (V ) ∩ f nj +1 (V ) = ∅. Hence, Vnj +1 (f ) = Vnj (f ) and the proof is complete.



13.3 The Singular Sets Sing(f −n ), Asymptotic Values, and Analytic Inverse Branches Let f : C −→  C be a meromorphic function. We assume throughout this whole section that meromorphic entails being nonconstant. This section is crucial for all subsequent considerations in this book. It lays down the foundations for obtaning the existence (the uniqueness is fairly automatic) of holomorphic branches of f −n , n ≥ 1, the structure of connected components U of inverse images of open connected sets V ⊆  C, especially those that are also simply connected, and the structure of forward images of these connected components. Our exposition here is, therefore, very careful and detailed. It is related to Iversen’s Theory [Iv]. The crucial concepts involve inverse connected chains, both algebraic and transcendental, transcendental tracts, asymptotic points and values, the singular set Sing(f −1 ), and, more generally, the singular sets Sing(f −n ), n ≥ 1. As the ultimate outcome of this section, we give sufficient, and actually also necessary, conditions for the maps f n |U : U −→ V to be covering or (conformal) homeomorphisms. We do this in terms of critical values and asymptotic points and values. We

13 Fundamental Properties of Meromorphic Dynamical Systems

35

also characterize the singular sets Sing(f −n ) in these terms. These theorems will be used very frequently to study the topological structure of connected components (and their boundaries) of the Fatou set of f and a countless number of times when we move on to dealing with elliptic functions.

13.3.1 Asymptotic Points and Asymptotic Values Theorem 13.3.1 If f : C −→  C is a meromorphic function, V ⊆  C is an open connected set, and U is a connected component of f −1 (V ), then f (U ) is a dense subset of V ; more precisely, f (U ) = U . Proof Of course, f (U ) ⊆ U . Seeking contradiction, suppose that f (U )  V . Then also f (U )  V . So, V ∩ f (U )  V . Hence, V ∩ f (U )  V .

(13.36)

But, since V ∩ f (U ) = f (U )V , and the set f (U ) is connected, we conclude that the set V ∩ f (U ) is connected. It then follows from (13.36) that ∂V (V ∩ f (U )) = ∅. Thus, there exists a point ξ ∈ ∂V (V ∩ f (U )) = V ∩ (V ∩ f (U )) ∩ (V ∩ (V \(V ∩ f (U )))) ⊆ V ∩ f (U ) ∩ (V \f (U )). Hence, there exists r > 0 such that Bs (ξ,r) ⊆ V ,Bs (ξ,r) ∩ f (U ) = ∅,

and

Bs (ξ,r) ∩ (V \f (U )) = ∅. (13.37)

So, there exists a point w ∈ U such that f (w) ∈ Bs (ξ,r)\{∞} and f  (w) = ∅. Since the set V \f (U ) is open, it follows from the last assertion of (13.37) that if

 Z := θ ∈ [0,2π ) : (f (w) + Reiθ ) ∩ Bs (ξ,r) ∩ (V \f (U )) = ∅ , then Leb(Z) > 0. Thus, by the Gross Star Theorem (see Theorem A.0.29 in Appendix A), there exists θ ∈ Z such that the holomorphic germ fw−1 has an analytic continuation along the whole ray f (w) + Reiθ . Then fixing an arbitrary point y ∈ (f (w) + Reiθ ) ∩ Bs (ξ,r) ∩ (V \f (U )),

(13.38)

36

Part III Topological Dynamics of Meromorphic Functions

we have that [f (w),y] ⊆ Bs (ξ,r) ⊆ V and there exist s > 0 such that Be ([f (w),y]) ⊆ Bs (ξ,r) ⊆ V

(13.39)

and g : Be ([f (w),y]) −→ C, a holomorphic branch of f −1 , such that g and fw−1 coincide on some open neighborhood of f (w). Then g([f (w),y]) is a connected set contained in f −1 (V ) and containing w (which belongs to U ). Thus, g([f (w),y]) ⊆ U . In particular, y = f (g(y)) ∈ f (U ), contrary to (13.38). We are done.  By a fairly straightforward induction, we shall prove the following important generalization of this theorem. Theorem 13.3.2 If f : C −→  C is a meromorphic function, n ≥ 1 is an integer, V ⊆ C is an open connected set, and U is a connected component of f −n (V ), then f n (U ) is a dense subset of V ; more precisely, f n (U ) = V . Proof We will proceed by induction on n. For n = 1, this is exactly Theorem 13.3.1. So, suppose that it holds for some n ≥ 1. Then let V ⊆  C be an open −(n+1) connected set and U be a connected component of f (U ). Then f n (U ) n is a connected set such that f (f (U )) ⊆ V . Therefore, there exists a unique connected component Un of f −1 (V ) such that f n (U ) ⊆ Un . By Theorem 13.3.1, we have that f (Un ) = V .

(13.40)

Also, let U0 be the unique connected component of f −n (Un ) containing U . Then f n+1 (U0 ) = f (f n (U0 )) ⊆ f (f n (f −n (Un ))) ⊆ f (Un ) ⊆ f (f −1 (V )) ⊆ V . Hence, U0 ⊆ U . Thus, U0 = U . So, by the inductive hypothesis set, f n (U ) C) −→  C is also continuous is dense in Un . Since the function f n : f −n ( −n  (and Un ⊆ f (C)), we, therefore, obtain that f (f n (U )) = f (Un ). Thus, by applying (13.40), we finally get that f n+1 (U ) = f (f n (U )) = f (Un ) = V . The proof of Theorem 13.40 is complete.



For future use in this section, we adopt the following definition. Definition 13.3.3 If f : C −→  C is a meromorphic function, n ≥ 1 is an integer, V ⊆  C is a connected set, and U is a connected component of f −n (V ), then, for every integer 0 ≤ k ≤ n, we denote by fˆk ([U ;V ]) the only connected component of f −(n−k) (V ) containing f k (U ).

13 Fundamental Properties of Meromorphic Dynamical Systems

37

Of course, fˆ0 ([U ;V ]) = U and fˆn ([U ;V ]) = V . We shall easily prove the following. Lemma 13.3.4 If f : C −→  C is a meromorphic function, n ≥ 1 is an  integer, and V ⊆ C is an open connected set, then, for every integer 0 ≤ k ≤ n, every connected component U of f −n (V ) is a connected component of f −k (fˆk ([U ;V ])). Proof Let  be the connected component of f −k (fˆk ([U ;V ])) containing U . Then f n () = f n−k (f k ()) ⊆ f n−k (fˆk ([U ;V ])) ⊆ V . So, since  is connected, we conclude that  ⊆ U . Thus,  = U and we are done.  Lemma 13.3.5 Let f : C −→  C be a meromorphic function. Let H ⊆  C be an open connected set. If n ≥ 0 and G is a connected component of f −n (H ), C) ∩ ∂G) ⊆ ∂H . then f n (f −n ( Proof By items (1) and (3) of Observation 13.1.1, we have that   f n (f −n ( C) ∩ ∂G) ⊆ f n (f −n ( C) ∩ G) = f n clf −n ( (G) ⊆ f n (G) ⊆ H . C) C) ∩ ∂G. Seeking contradiction, suppose that f n (w) ∈ H . Now let w ∈ f −n ( Thus, again by items (1) and (3) of Observation 13.1.1, there exists r > 0 such C) and f n (Be (w,r)) ⊆ H . But then Be (w,r) ∪ G is a that Be (w,r) ⊆ f −n ( connected set such that f n (Be (w,r)∪G) ⊆ H . Hence, w ∈ Be (w,r)∪G ⊆ G and this contradiction shows that C) ∩ ∂G) ⊆ H \H = ∂G. f n (f −n ( 

The proof of Lemma 13.3.5 is complete.

Definition 13.3.6 Let f : C −→  C be a meromorphic function. Let H ⊆  C be an open connected set. Fix an integer n ≥ 0 and a connected component G of f −n (H ). If w ∈ ∂ C G, ξ ∈ H , and there exists a continuous function γ : [0,+∞) −→ G such that lim γ (t) = w

t→+∞

and

lim f n (γ (t)) = ξ,

t→+∞

(13.41)

then the point w is called an asymptotic point for the map f n |G : G −→ H while ξ is an asymptotic value for the map f n |G : G −→ H . Let us record several immediate consequences of Lemma 13.41.

38

Part III Topological Dynamics of Meromorphic Functions

Observation 13.3.7 Let f : C −→  C be a meromorphic function. Let H ⊆  C be an open connected set. If n ≥ 0, G is a connected component of f −n (H ), and w is an asymptotic point for the map f n |G : G −→ H , then  −k (∞). w ∈ n−1 k=0 f For n = 0, this observation gives the following. Corollary 13.3.8 Let f : C −→  C be a meromorphic function. Let H ⊆  C be −1 an open connected set. If G is a connected component of f (H ) and w is an asymptotic point for the map f |G : G −→ H , then w = ∞. Observation 13.3.7 also gives the following. Observation 13.3.9 Let f : C −→  C be a meromorphic function. Fix an integer n ≥ 0. Let G ⊆ H ⊆  C be two open connected sets. Let D be a connected component of f −n (G) and E be the unique connected component of f −n (H ) containing D. If w is an asymptotic point for the map f n |D : D −→ G, then w ∈ ∂H and w is an asymptotic point for the map f n |E : E −→ H . C) and w and ξ are, respectively, an If G is a connected component of f −n ( : f −n ( C) −→ asymptotic point and an asymptotic value for the map f n f −n ( C)  C, then w and ξ are, respectively, called just an asymptotic point and an asymptotic value of f n . As an immediate consequence of Observation 13.3.9, we get the following. Corollary 13.3.10 Let f : C −→  C be a meromorphic function. Let H ⊆  C be an open connected set. If n ≥ 0 is an integer, G is a connected component of f −1 (H ), and w is an asymptotic point for the map f n |G : G −→ H , then w is an asymptotic point of f n . Similarly for asymptotic values. Finally, Observation 13.3.7 gives the following. Observation 13.3.11 If f : C −→  C is an entire function (i.e., f (C) ⊆ C), then ∞ is the only possible asymptotic point for the maps f n , n ≥ 1. We can do more than claimed in Observation 13.3.7. We shall prove the following. Lemma 13.3.12 Let f : C −→  C be a meromorphic function. Let H ⊆  C be an open connected set. If n ≥ 0, G is a connected component of f −n (H ), w ∈ ∂G is an asymptotic point for the map f n |G : G −→ H , and ξ ∈ H is a corresponding asymptotic value, then there exist a unique 0 ≤ k ≤ n − 1 and a continuous function γ : [0,+∞) −→ G such that w ∈ f −k (∞),

13 Fundamental Properties of Meromorphic Dynamical Systems lim γ (t) = w,

t→+∞

lim f k (γ (t)) = ∞,

t→+∞

and

39

lim f n (γ (t)) = ξ .

t→+∞

(13.42) In particular, ∞ is an asymptotic point for the map f n−k |Gk : Gk −→ H , where, we recall, Gk is the unique connected component of f −(n−k) (H ) containing f k (G) and ξ is the corresponding asymptotic value. Proof By Observation 13.3.7, there exists a unique 0 ≤ k ≤ n − 1 such that w ∈ f −k (∞). By Definition 13.3.6, there exists a continuous function γ : [0,+∞) −→ G such that the first and the third formulas of (13.42) hold. C), it then follows from items (1) and (3) of Observation Since w ∈ f −k ( 13.1.1 along with the first formula of (13.42) that the second formula of (13.42) holds. The proof of Lemma 13.3.12 is complete.  Now we shall easily show that the properness entails the absence of asymptotic values (and asymptotic points). Proposition 13.3.13 Let f : C −→  C be a meromorphic function. Let H ⊆  C be an open connected set. If n ≥ 0, G is a connected component of f −n (H ), and the map f n |G : G −→ H is proper, then this map has no asymptotic value and no asymptotic points. Proof Proving by contrapositive, suppose that the map f n |G : G −→ H has an asymptotic value. Then, by Definition 13.3.6, there exists w ∈ ∂G, ξ ∈ H , and the continuous function γ : [0,+∞) −→ G such that both formulas of (13.41) hold. But then the set f n (γ ([0,+∞))) = f n (γ ([0,+∞))) ∪ {ξ } ⊆ H is compact, while n f n |−1 / G, G (f (γ ([0,+∞))))C ⊇ γ ([0,+∞)) ∪ {w}  w ∈ n whence the set f n |−1 G (f (γ ([0,+∞)))), contained in G, cannot be compact. We are done. 

Now we shall investigate in greater detail the nature of asymptotic points, asymptotic values, and the other singularities, i.e., the critical points and critical values, of meromorphic functions and their iterates. Definition 13.3.14 Let f : C −→  C be a meromorphic function. Let H ⊆  C be an open connected set, n ≥ 1 be an integer, and G be a connected component of f −n (H ). Fix ξ ∈ H . We say that a collection

 U := Ur : r ∈ (0,dists (ξ,  C\H ))

40

Part III Topological Dynamics of Meromorphic Functions

(if H =  C then dists (ξ,  C\ C) = +∞) of subsets of G is an inverse connected chain at ξ (of order n) for the map f n |G : G −→ H if and only if C\H )), the set Ur is a connected component of (1) for every r ∈ (0,dists (ξ,  f −n (Bs (ξ,r)) contained in G, and C\H ), then Ur ⊆ Us . (2) if 0 < r ≤ s < dists (ξ,  Lemma 13.3.15 Let f : C −→  C be a meromorphic function. Let H ⊆  C be an open connected set, n ≥ 1 be an integer, and G be a connected component of f −n (H ). Fix ξ ∈ H and s ∈ (0,dists (ξ,  C\H )). If U is a connected component of f −1 (Bs (ξ,r)) contained in G, then there exists an inverse connected C) −→  C such that Us = U chain U = {Ur }r>0 at ξ for the map f n : f −n ( and

  2l(0,dists (ξ,  C\H )) := Ur : r ∈ (0,dists (ξ, C\H )) is an inverse connected chain at ξ for the map f n |G : G −→ H . Proof For any t ≥ s, define Ut to be the only connected component of f −n (Bs (ξ,t)) containing U . Note then that Ut1 ⊆ Ut2 , whenever 0 < r ≤ t1 ≤ t2 . Defining Ur for r < s is a somewhat subtler task. We first define inductively the sets U ks , k ≥ 1, as follows. Suppose 13.3.2, the  thatU ks has been defined. Then,  by Theorem  s s n set f (U ks ) ∩ Bs ξ, k+1 is nonempty. So, U ks ∩ Bs ξ, k+1 = ∅. Define then

s s to be the only connected component of f −n (Bs (ξ, s U k+1 k+1 )) intersecting U k . But then s ⊆ Us . U k+1 k

Now, given any r ∈ (0,s), let k ≥ 1 be an integer, e.g., the least one, such that s/k ≤ r. Define then Ur to be the connected component of f −n (B(s,r)) containing U ks . Note then that Ur1 ⊆ Ur2 whenever 0 < r1 ≤ r2 ≤ s. It follows from this construction that Ur ∩ U = ∅ for every r > 0. So, Ur ∩ G = ∅ for every r > 0. Since also Ur ⊆ f −n (B(s,r)) ⊆ f −n (H )

13 Fundamental Properties of Meromorphic Dynamical Systems

41

for every r ∈ (0,dists (ξ,  C\H )), we conclude that Ur ⊆ G for every r ∈ C\H )). The proof of Lemma 13.3.15 is complete.  (0,dists (ξ,  Since the sets Ur appearing in this proof are uniquely defined for all r ≥ s, as an immediate consequence of this lemma, we get the following. Lemma 13.3.16 Let f : C −→  C be a meromorphic function. Let H ⊆  C be an open connected set, n ≥ 1 be an integer, and G be a connected component of f −n (H ). Fix ξ ∈ H . If U = {Ur }r>0 is an inverse connected C) −→ C such that Us ∩ G = ∅ for some chain at ξ for the map f n : f −n ( C\H )), then the connected chain s ∈ (0,dists (ξ, 

  U|(0,dists (ξ,  C\H )) := Ur : r ∈ (0,dists (ξ, C\H )) n : G −→ H . is an inverse connected chain at ξ for the map f|G

 Conversely, for every inverse connected chain Ur : r ∈ (0,dists (ξ,  C\H )) at ξ for the map f n |G : G −→ H , there exists a unique inverse connected C) −→ C such that Ur = Ur for all chain {Ur }r>0 at ξ for the map f n : f −n (  r ∈ (0,dists (ξ, C\H )). We call  this chain the maximal extension of the chain C\H )) . We then have that Ur : r ∈ (0,dists (ξ, 

  Ur : r ∈ (0,dists (ξ,  C\H )) = Ur .

(13.43)

r>0

Observation 13.3.17 Let f : C −→  C be a meromorphic function. Let H ⊆ C be an open connected set, n ≥ 1 be an integer, and G be a connected component of f −n (H ). Fix ξ ∈ H .  C\H )) is an inverse connected chain at ξ (of If U = Ur : r ∈ (0,dists (ξ,  n order n) for the map f |G : G −→ H , then G ∩ U r ⊆ Us whenever 0 < r < s < (0,dists (ξ,  C\H )). Proof Since the function f n : f −n (C) −→  C is continuous and G ⊆ f −n ( C), we have that f n (G ∩ U r ) = f n (clG (U )) ⊆ f n (Ur ) ⊆ Bs (ξ,r) ⊆ Bs (ξ,s). Thus, G ∩ U r ⊆ f −1 (Bs (ξ,r)). Since also Ur ⊆ G ∩ U r ⊆ U r and the set Ur is connected, we conclude that the set G ∩ U r is connected and Ur ⊆ Us . The proof of Observation 13.3.17 is complete.  As an immediate consequence of this observation, we get the following.

42

Part III Topological Dynamics of Meromorphic Functions

Observation 13.3.18 Let Let f : C −→  C be a meromorphic function. Let H ⊆ C be an open connected set, n ≥ 1 be an integer, and G be a connected C\H ))} is an component of f −n (H ). Fix ξ ∈ H . If U = {Ur : r ∈ (0,dists (ξ,  inverse connected chain at ξ (of order n) for the map f n |G : G −→ H , then 

 Ur : r ∈ (0,dists (ξ,  C\H )) = G ∩ U r : r ∈ (0,dists (ξ,  C\H ))

 = G ∩ U r : r ∈ (0,dists (ξ,  C\H )) 

C\H )) . = clG (Ur ) : r ∈ (0,dists (ξ,  Definition 13.3.19 Let f : C −→  C be a meromorphic function. Let H ⊆  C be an open connected set, n ≥ 1 be an integer, and G be a connected component of f −n (H ). Fix ξ ∈ H . An inverse connected chain U at ξ for the map f n |G : G −→ H is said to be algebraic if and only if U = ∅. U ∈U

Otherwise, i.e., if and only if

U = ∅,

U ∈U

it is called transcendental. We shall now investigate in detail the nature of algebraic and, especially, transcendental inverse connected chains. We start with the following general fact. Proposition 13.3.20 Let f : C −→  C be a meromorphic function. Let n ≥ 1  be an integer and ξ ∈ C. If U = {Ur }r>0 is an inverse connected chain at ξ for C) −→  C, then the map f n : f −n ( lim sup diams (Ur ) = 0 r−→0

and the intersection

 cl C (Ur ) ⊆ C

(13.44)

r>0

is a singleton whose only element we will denote by ξU . Proof Since all the sets Ur , r > 0, are nonempty and connected, their closures cl C (Ur ), r > 0, are continua, i.e., nonempty compact connected sets. Therefore, by invoking item (2) of Definition 13.3.14, we conclude that the

13 Fundamental Properties of Meromorphic Dynamical Systems

intersection limit, and



r>0 cl C (Ur )

43

is not empty, the upper limit in (13.44) is in fact a 

lim diams (Ur ) = diams

r−→0



cl C (Ur ) .

r>0

It, therefore, suffices to prove the second (last) assertion of our proposition.  Proceeding by contradiction, suppose that the set r>0 cl C (Ur ) contains at C) −→ C is continuous, we get, least two points. Since the function f n : f −n ( for every s > 0, that



  n −n  n −n  cl (Ur ) = f f (C) ∩ cl (Ur ) f f (C) ∩ C

C

r>0

 =f

n

r>0



  n clf −n ( (U ) ⊆ f (U ) cl −n  r s C) f (C)

r>0

⊆

f n (U

s)

⊆ B(ξ,s). (13.45)

Therefore, f

 n

f

−n

( C) ∩



cl C (Ur ) ⊆

r>0



B(ξ,s) = {ξ }.

(13.46)

s>0

 But since the set r>0 cl C (Ur ) is a nondegenerate, i.e., nonsingleton, conC) is tinuum, it is uncountable, in fact of cardinality c, and the set C\f −n ( countable (possibly finite or even empty), we see that the set C) ∩ cl f −n ( C (Ur ) r>0

is uncountable, and we conclude from (13.46) that the function C) −→ C is constant. But then also the function f : C −→  C is f n : f −n ( constant and this contradiction finishes the proof.  Having this proposition, we can provide a simple and illuminating characterization of algebraic and transcendental inverse connected chains. Theorem 13.3.21 Let f : C −→  C be a meromorphic function. Let H ⊆  C be an open connected set, n ≥ 1 be an integer, and G be a connected component of f −n (H ). Fix ξ ∈ H . If U is an inverse connected chain U at ξ for the map f n |G : G −→ H , then (1) U is algebraic if and only if ξU ∈ G; equivalently, (2) U is transcendental if and only if ξU ∈ ∂ C G.

44

Part III Topological Dynamics of Meromorphic Functions

In addition, (3) If U is algebraic, then f n (ξU ) = ξ .  Proof If U is algebraic, then G ⊇ U ∈U (U ) = ∅, whence it follows from  Proposition 13.3.20 that it is the only element of U ∈U U , and, thus, belongs to G. On the other hand, if ξU ∈ G, then it follows from Proposition 13.3.20 that cl C (V ) ⊆ G for some V ∈ U. But then cl C (U ) ⊆ G for all U  U ⊆ (U ) = cl (U ) for all such U . It, therefore, follows from V . Hence, G ∩ cl  C C Observation 13.3.18 and Proposition 13.3.20 that



U=

U ∈U

U=

V ⊃U ∈U

U ∈U

cl C (U ) = {ξU }.

 In particular, U ∈U U = ∅, meaning that the inverse connected chain U is algebraic. Item (3) is a direct consequence of the definition of algebraic chains. The proof of Theorem 13.3.21 is complete.  Now we are going to relate transcendental inverse connected chains with asymptotic points and asymptotic values. We start with the following. Theorem 13.3.22 Let f : C −→  C be a meromorphic function. Let H ⊆  C be an open connected set, n ≥ 1 be an integer, and G be a connected component of

 f −n (H ). Fix ξ ∈ H . If U = Ur : r ∈ (0,dists (ξ,  C\H )) is a transcendental inverse connected chain at ξ for the map f n |G : G −→ H , then there exists a (unique) 0 ≤ k ≤ n − 1 such that ξU ∈ f −k (∞); moreover,    k  cl C f (Ur ) : r ∈ (0,dists (ξ, C\H )) = {∞}. Proof Seeking contradiction, suppose that there is no 0 ≤ k ≤ n − 1 such that ξU ∈ f −k (∞). Then ξU ∈ f −k ( C). So, by invoking the last assertion of Proposition 13.3.20 and repeating the calculation of (13.45), we get, for every s > 0, that  f (ξU ) ∈ f n

n

f

−n

( C) ∩







cl C (Ur ) = f

n

r>0



f

−n

( C) ∩ cl C (Ur )

r>0

 =f

n



clf −n ( C) (Ur )

r>0

  ⊆ f n clf (−n) ( C) (Us ) ⊆ f n (Us ) ⊆ B(ξ,s).

13 Fundamental Properties of Meromorphic Dynamical Systems

Therefore, f (ξU ) ∈ f n

 n

f

−n

( C) ∩



cl C (Ur ) ⊆

r>0



45

B(ξ,s) = {ξ },

s>0

whence f n (ξU ) = ξ . But then since ξU ∈ f −n ( C), f −n ( C) is an open set, ξ ∈ H and H is open, −n −n C) −→  C is continuous, we see that there exists δ > 0 and the map f : f ( C) and f n (Be (ξU,δ)) ⊆ H . It follows from this that such that B(ξU,δ) ⊆ f −n ( C is connected (as a union of two connected sets having the set G∪Be (ξU,δ) ⊆  a common point ξ ) and f n (G ∪ Be (ξU,δ)) ⊆ H . Thus, ξU ∈ G ∪ Be (ξU,δ) = G, which contradicts item (2) of Theorem 13.3.21. Thus, the first assertion of our theorem is proved. We shall now prove the second (last) assertion of our theorem. Since C) and f −k ( C) is an open subset of  C, it follows from ξU ∈ f −k (∞) ∈ f −k ( (U ) ⊆ f −k ( C). Then Proposition 13.3.20 that there exists s > 0 such that cl s C −k  cl (C) C (Ur ) ⊆ cl C (Us ) ⊆ f

(13.47)

for all r ∈ (0,s]. Hence, for all such rs, we have that clf −k ( C) (Ur ) = cl C (Ur ) and it follows from continuity of the map f k : f −k ( C) −→  C and compactk ness of the sets clf −k ( C) (Ur ) that the sets f (clf −k ( C) (Ur )) are compact and, thus, closed; therefore,   k k (f (U )) = f (U ) . cl cl −k  r r C f (C) It then follows from (both parts of) Proposition 13.3.20, (13.47), and continuity C) −→  C at the point ξU that of the map f k : f −k (  k k  cl cl C (f (Ur )) : r ∈ (0,dists (ξ, C\H )) = C (f (Ur )) r∈(0,s]

=



  f k clf −k ( C) (Ur )

r∈(0,s]

= {f k (ξU )} = {∞}. The proof of Theorem 13.3.22 is complete.



Theorem 13.3.23 Let f : C −→  C be a meromorphic function. Let H ⊆  C be an open connected set, n ≥ 1 be an integer, and G be a connected component

46

Part III Topological Dynamics of Meromorphic Functions

of f −n (H ). Fix ξ ∈ H . Then an inverse connected chain U = {Ur ,r ∈ C\H ))} at ξ for the map f n |G : G −→ H is transcendental if and (0,dists (ξ,  only if there exists 0 ≤ k ≤ n − 1 and a continuous map γ : [0,+∞) −→ G (a curve) such that lim f k (γ (t)) = ∞,

t−→+∞

lim f n (γ (t)) = ξ

t−→+∞

(13.48)

and, for every r ∈ (0,dists (ξ,  C\H )), there exists tr ≥ 0 such that γ ([tr ,+∞)) ⊆ Ur .

(13.49)

Proof Suppose, first, that such a curve γ exists and, proceeding by way of construction, assume that U is algebraic. Then, by Theorem 13.3.21, ξU ∈ G. So, since G is open, it follows from Proposition 13.3.20 that there exists r ∈ C\H )) such that cl (0,dists (ξ,  C (Ur ) ⊆ G. Then, by (13.49), we have that γ ([tr ,+∞)) ⊆ cl C (Ur ) ⊆ G. So, since G ⊆ f −n ( C) ⊆ f −k ( C) and 0 ≤ k ≤ n − 1, we get from the C) →  C, and first formula of (13.49), continuity of the function f k : f −k ( compactness of the set cl C (Ur ) that ∞ ∈ f k (γ ([tr ,+∞))) ⊆ f k (γ ([tr ,+∞))) k = f k (cl C (Ur )) = f (cl C (Ur ))

⊆ f k (G) ⊆ C. This contradiction establishes the first implication. Now suppose that U is transcendental. We shall define a required function γ inductively by assigning its values on the intervals of the form [j,j + 1], j ≥ 0. Fix any number s ∈ (0,dists (ξ,  C\H )). In order to define γ on [0,1], pick arbitrarily a point x0 ∈ Us and ξ1 ∈ Us/2 . Since Us/2 ⊆ Us and the set U1 is arcwise connected, there exists a one-to-one continuous function (we do not really care about it being one-to-one, mere continuity suffices) γ0 : [0,1] −→ Us such that γ0 (0) = x0 and γ0 (1) = x1 . For an inductive step, suppose that j ≥ 0 and a continuous function γj : [j,j + 1] −→ Us2−j has been defined such that γj (j + 1) ∈ Us2−(j +1) .

13 Fundamental Properties of Meromorphic Dynamical Systems

47

Fix an arbitrary point xj +2 ∈ Us2−(j +2) . Since Us2−(j +2) ⊆ Us2−(j +1) and the set Us2−(j +1) ⊆ C is arcwise connected, there exists a continuous function γj +1 : [j + 1,j + 2] −→ Us2−(j +1) such that γj +1 (j + 1) = γj (j + 1) and γj +1 (j + 2) = xj +2 ∈ Us2−(j +1) . Define then a function γ : [0,+∞) −→ G by declaring that γ |[j,j +1) := γj |[j,j +1) . By our construction, the function γ |[0,j ] : [0,j ] −→ G is continuous for every j ≥ 0; therefore, the function γ : [0,+∞) −→ G is continuous. Also, by the construction, we have that γ ([0,+∞)) ⊆ Us2−j . Thus,

(13.50)

  f n γ ([0,+∞)) ⊆ f n (Us2−j ) ⊆ Bs (ξ,s2−j ).

Hence, lim f n (γ (t)) = ξ .

t−→+∞

(13.51)

Now if r > 0 is arbitrary, choose any integer j ≥ 0, e.g., the least one, such that s2−j ≤ r. Set tr := j . Then, by virtue of (13.50), we have that γ ([tr ,+∞)) = γ ([j,+∞)) ⊆ Us2−j ⊆ Ur . We are left to show that the first formula of (13.48) holds. Let 0 ≤ k ≤ n − 1 be produced in Theorem 13.3.23. By applying this theorem, we get that  (f k (γ [t,+∞))) = f k (γ [tr ,+∞)) : r ∈ (0,dists (ξ,  C\H )) t≥0

⊆



 f k (γ (Ur )) : r ∈ (0,dists (ξ,  C\H ))

= {∞}. This means that the first formula of (13.48) holds and the proof of Theorem 13.3.23 is complete.  Now we want to deal for a moment with the forward and backward iterates of inverse connected chains.

48

Part III Topological Dynamics of Meromorphic Functions

Definition 13.3.24 Let f : C −→  C be a meromorphic function. Let H ⊆  C be a connected set, n ≥ 1 be an integer, and G be a connected component C\H ))} be an inverse of f −n (H ). Fix ξ ∈ H . Let U = {Ur ,r ∈ (0,dists (ξ,  connected chain at ξ for the map f n |G : G −→ H . Then, for every 0 ≤ k ≤ n − 1, we denote by fˆk [U] the inverse connected chain at ξ for the map f n−k : fˆn−k ([G;H ]) −→ H consisting of all sets fˆn−k ([Ur ;Bs (ξ,r)]), C\H )). We then call fˆk [U] the kth image of U and U the kth r ∈ (0,dists (ξ,  inverse image of fˆk [U]. As a direct consequence of this definition and Definition 13.3.19, we get the following. Theorem 13.3.25 Let f : C −→  C be a meromorphic function. Let H ⊆  C be a connected set, n ≥ 1 be an integer, and G be a connected component of f −n (H ). Fix ξ ∈ H . If an inverse connected chain U at ξ for the map f n |G : G −→ H is algebraic, then, for every 0 ≤ k ≤ n − 1, the chain fˆk [U] is algebraic. By contrapositive, if, for some 0 ≤ k ≤ n − 1, the chain fˆk [U] is transcendental, then U is transcendental. On the other hand, as an immediate consequence of Theorem 13.3.23, we get the following. Theorem 13.3.26 Let f : C −→  C be a meromorphic function. Let H ⊆ C be a connected set, n ≥ 1 be an integer, and G be a connected component of f −n (H ). Fix ξ ∈ H . If an inverse connected chain U at ξ for the map f n |G : G −→ H is transcendental, then, for every j ≥ 0 which is less than or equal to the largest k of Theorem 13.3.23, the chain fˆj [U] is transcendental. Remark 13.3.27 It directly follows from (13.49) and Proposition 13.3.20 that, for the curve γ : [0,+∞) −→ G, constructed in the proof of Theorem 13.3.23, we have that lim γ (t) = ξU .

t−→+∞

(13.52)

The first main result is the following. Theorem 13.3.28 Let f : C −→  C be a meromorphic function. Let H ⊆  C be an open connected set, n ≥ 1 be an integer, and G be a connected component of f −n (H ). Then a point ξ ∈ H is an asymptotic value for the map f n |G : g −→ H if and only if there exists a transcendental inverse connected chain at ξ for the map f n |G : g −→ H . Proof The implication (⇐) is a direct consequence of Theorem 13.49, (13.52) of Remark 13.3.27, and item (2) of Theorem 13.3.21.

13 Fundamental Properties of Meromorphic Dynamical Systems

49

For the implication (⇒), let γ : [0,+∞) −→ C be the curve following on from Definition 13.3.6 and w ∈ ∂ C G be the corresponding asymptotic point. It then follows from the second formula of (13.41) that, for every r > 0, there exists ur ≥ 0 such that f n (γ ([ur ,+∞))) ⊆ Bs (ξ,r). Equivalently, γ ([ur ,+∞)) ⊆ f −n (B(ξ,r)). So, since the set γ ([ur ,+∞)) is connected, there exists a unique connected component Ur of f −n (B(ξ,r)) such that γ ([ur ,+∞)) ⊆ Ur .

(13.53)

Now if 0 ≤ r ≤ s, then ∅ = γ ([max{ur ,us }, + ∞)) ⊆ Ur ∩ Us . So, since Ur is a connected set contained in f −n (Bs (ξ,s)) and Us is a connected component of f −n (Bs (ξ,s)), we conclude that Ur ⊆ Us . Thus, U = {Ur }r>0 is an inverse connected chain at ξ . Furthermore, if C\H )), then Bs (ξ,r) ⊆ H and, by virtue of (13.53) and r ∈ (0,dists (ξ,  the fact that γ ([0,+∞)) ⊆ G, Ur ∩ G = ∅, we conclude that U ∪ G is a connected set such that f n (Ur ∪ G) ⊆ H . Thus, Ur ⊆ G, whence V := U|(0,dists (ξ,  C\H )) is an inverse connected chain at ξ for the map f n |G : G −→ H . It is left for us to show that the chain V is transcendental. Indeed, it follows from the first formula of (13.41) and (13.53) that w ∈ cl C (γ [ur ,∞)) ⊆ cl C (Ur )  for all r > 0. Therefore, w ∈ r>0 cl C (Ur ). So, it follows from Proposition 13.3.20 that ξV = w ∈ ∂G. Thus, the chain V is transcendental by virtue of item (2) of Theorem 13.3.21. The proof of Theorem 13.3.28 is complete.  Now we shall explicitly rewrite the above results for the case when n = 1. They look somewhat simpler and, perhaps, easier to grasp. First, rewrite verbatim Corollary 13.3.8. Corollary 13.3.29 Let f : C −→  C be a meromorphic function. Let H ⊆  C be an open connected set. If G is a connected component of f −1 (H ) and w is an asymptotic point for the map f |G : G −→ H , then w = ∞. Having this, Definition 13.3.8 takes on, for n = 1, the following form. Definition 13.3.30 Let f : C −→  C be a meromorphic function. Let H ⊆  C −1 be an open connected set. If G is a connected component of f (H ), then a

50

Part III Topological Dynamics of Meromorphic Functions

point ξ ∈ H is an asymptotic value for the map f |G : G −→ H if and only if there exists a continuous function γ : [0,+∞) −→ G such that lim γ (t) = ∞

r−→0

and

lim f (γ (t)) = ξ .

r−→0

Remember that if H =  C, then G = C (necessarily) and we simply refer then to ξ as an asymptotic value of f . Proposition 13.3.13 takes on the following form. Proposition 13.3.31 Let f : C −→  C be a meromorphic function. Let H ⊆  C be an open connected set. If G is a connected component of f −1 (H ) and the map f |G : G −→ H is proper, then this map has no asymptotic values. Definition 13.3.19 takes on the following form. Definition 13.3.32 Let f : C −→  C be a meromorphic function, H ⊆  C be an open connected set, and G be a connected component of f −1 (H ). Fix a point ξ ∈ H . An inverse connected chain at U for the map f |G : G −→ H is said to  be algebraic if and only if U ∈U U = ∅ and it is said to be transcendental if  and only if U ∈U U = ∅. Remember that if H =  C, then G = C. Proposition 13.3.20 takes on the following form. Proposition 13.3.33 Let f : C −→  C be a meromorphic function. Let ξ ∈  C. If U = {Ur }r>0 is an inverse connected chain at ξ for the map f , then lim sup diams (Ur ) = 0 r−→0

and the intersection



 cl C (U r) ⊆ C

r>0

is a singleton, whose only element is denoted by ξU . Remember that if H =  C, then G = C. Theorems 13.3.21 and 13.3.23 give the following. Theorem 13.3.34 Let f : C −→  C be a meromorphic function. Let H ⊆  C be an open connected set and G be a connected component of f −1 (H ). Fix ξ ∈ H . If U is an inverse connected chain U at ξ for the map f |G : G −→ H , then (1) U is algebraic if and only if ξU ∈ G; equivalently, (2) U is transcendental if and only if ξU = ∞.

13 Fundamental Properties of Meromorphic Dynamical Systems

51

In addition, (3) If U is algebraic, then f (ξU ) = ξ . Remember that if H =  C, then G = C Theorem 13.3.23 takes on the following form. Theorem 13.3.35 Let f : C −→  C be a meromorphic function. Let H ⊆  C be an open connected set and G be a connected component of f −1 (H ). Fix C\H ))} at ξ ∈ H . Then an inverse connected chain U = {Ur ,r ∈ (0,dists (ξ,  ξ for the map f |G : G −→ H is transcendental if and only if there exists a continuous map γ : [0,+∞) −→ G (a curve) such that lim γ (t) = ∞,

t−→+∞

lim f (γ (t)) = ξ

t−→+∞

and for every r ∈ (0,dists (ξ,  C\H )) there exists tr ≥ 0 such that γ ([tr ,+∞)) ⊆ Ur . Remember that if H =  C, then G = C Theorem 13.3.28 takes on the following form. Theorem 13.3.36 Let f : C −→  C be a meromorphic function. Let H ⊆  C −1 be an open connected set and G be a connected component of f (H ). Then a point ξ ∈ H is an asymptotic value for the map f |G : G −→ H if and only if there exists a transcendental inverse connected chain at ξ for the map f |G : G −→ H . Remember that if H =  C, then G = C. Picard’s Little Theorem says that if f : C −→  C is a meromorphic function,  then the set C\f (C) contains at most two points. These points are commonly referred to as omitted values, Picard’s points, or omitted Picard’s points of f . As a direct consequence of Theorem 13.3.34(3) and Theorem 13.3.36 we get the following. Theorem 13.3.37 Each omitted value of a meromorphic function f : C −→  C is an asymptotic value of f . Moreover, each inverse connected chain at each omitted value is transcendental (at least one always exists since the difference  C\f (C) contains at most two points and because of Lemma 13.3.15). However, the converse is not true: an asymptotic value need not be omitted. Indeed, 0 is an asymptotic value of the entire function C  z −→ zez but it is not omitted. Being almost at the end of this subsection, we want to prove one more characterization of algebraic and transcendental inverse connected chains.

52

Part III Topological Dynamics of Meromorphic Functions

Lemma 13.3.38 Let f : C −→  C be a meromorphic function. Fix ξ ∈  C. If U = {Ur }r>0 is an inverse connected chain at ξ , then the following statements are equivalent. (1) (2) (3) (4)

U is algebraic. There exists s > 0 such that diame (Us ) < +∞. limr−→0 diame (Ur ) = 0. limr−→0 Dists (∞,Ur ) > 0.

Proof The equivalence (1) ⇔ (4) is a direct consequence of Proposition 13.3.20 and Theorem 13.3.34. By the same token, (1) ⇔ (3). Of course, (3) ⇒ (2). If (2) holds, then ∞ ∈ / cl C (Us ), whence Dists (∞,Ur ) ≥ dists (∞,Ur ) ≥ dists (∞,Us ) > 0 for all r ∈ (0,s]. Thus, lim Dists (∞,Ur ) ≥ dists (∞,Us ) > 0,

r−→0

meaning that (4) holds. The proof of Lemma 13.3.38 is complete.



As the contrapositive of this lemma, we get the following. Lemma 13.3.39 Let f : C −→  C be a meromorphic function. Fix ξ ∈  C. If U = {U }r>o is an inverse connected chain at ξ , then the following statements are equivalent. (1) (2) (3) (4)

U is transcendental. For every s > 0, diame (Ur ) = +∞. limr−→0 diame (Ur ) > 0. limr−→0 Dists (∞,Ur ) = 0.

We close this subsection with the following remark. Remark 13.3.40 The elements of a transcendental inverse connected chain U for a meromorphic function f : C −→  C are frequently called (asymptotic) tracts. But this is slightly ambiguous since sometimes the chain U is called a tract.

13.3.2 Analytic Inverse Branches, the Singular Sets Sing(f −n ), and Asymptotic Values Let f : C −→  C be a meromorphic function. If H ⊆  C is an open set and n ≥ 0 is an integer, then any analytic function f∗−n : H −→ C such that the map f n is well defined on f∗−n (H ) and

13 Fundamental Properties of Meromorphic Dynamical Systems

53

f n ◦ f∗−n = IdH is called an analytic (or holomorphic, or meromorphic) branch of f −n . We shall prove the following. Lemma 13.3.41 If f : C −→  C is a meromorphic function, H ⊆  C is an open connected set, n ≥ 0 is an integer, and f∗−n : H −→ C is an analytic inverse branch of f −n , then (1) The map f∗−n : H −→ f∗−n (H ) is a conformal homeomorphism. (2) The map f n : f∗−n (H ) −→ H is a conformal homeomorphism.  −1 . (3) f∗−n = f n |f∗−n (H )

(4) f∗−n (H ) is a connected component of f −n (H ). (5) If g : H −→ C is an analytic inverse branch of f −n such that g(w) = f∗−n (w) for some point w ∈ H , then g = h. (6) Furthermore, if g : H −→ C is an analytic inverse branch of f −n such that g(H ) ∩ f∗−n (H ) = ∅, then g = f∗−n . Proof Items (1), (2), and (3) are straightforward. We shall prove item (4). Let G be a connected component of f −n (H ) containing f∗−n (H ). Seeking contradiction, suppose that G\f∗−n (H ) = ∅. But then since G is connected, f∗−n (H ) is connected, and f∗−n (H ) is open, we must have that ∂G f∗−n (H ) = ∅. So, since ∂G f∗−n (H ) ⊆ G ∩ ∂f∗−n (H ), there exists a point ξ ∈ G ∩ ∂f∗−n (H ).

(13.54)

/ f∗−n (H ). But since f n (ξ ) ∈ Hence, since the set f∗−n (H ) is open, ξ ∈ n −n n f (G) ⊆ H , the point f∗ (f (ξ )) is well defined; furthermore, f∗−n (f n (ξ )) ∈ f∗−n (H ) and f∗−n (f n (ξ )) = ξ . So, there exists a radius r > 0 so small that   ξ∈ / Bs f∗−n (f n (ξ )),2r ⊆ f∗−n (H ). (13.55)   Therefore, since f n Be (f∗−n (f n (ξ )),r) is an open neighborhood of f n (ξ ), there exists a radius s > 0 so small that   Be (ξ,s) ∩ Be f∗−n (f n (ξ )),r = ∅ (13.56) and   f n (Be (ξ,s)) ⊆ f n Be (f∗−n (f n (ξ )),r) . But then     f n f∗−n (H ) ∩ Be (ξ,s) ⊆ f n Be (f∗−n (f n (ξ )),r)

54

Part III Topological Dynamics of Meromorphic Functions

and, as the set f∗−n (H ) ∩ Be (ξ,s) is, by (13.54), nonempty,     f n f∗−n (H ) ∩ Be (ξ,s) ∩ f n Be (f∗−n (f n (ξ )),r) = ∅. This, however, along with (13.56) and (13.55), contradicts the injectivity of f n on f∗−n (H ), which is a part of item (2). Item (4) is, thus, proved. Proving item (5), first notice that items (1) and (2) imply that the derivatives of f n on on f∗−n (H ) and f∗−n on H nowhere vanish; therefore, the identity part of the Inverse Function Theorem implies that g and f∗−n (H ) coincide on some open neighborhood of w contained in H . So, since H is connected, the Identity Theorem for analytic functions implies that g = f∗−n , meaning that item (5) is proved. To Prove item (6), take an arbitrary point z ∈ g(H ) ∩ f∗−n (H ). Then take two points a,b ∈ H such that g(a) = z = f∗−n (b). By applying f n , we, thus, get that a = f n (g(a)) = f n (z) = f n (f∗−n (b)) = b. Hence, g(a) = z = f∗−n (b) = f∗−n (a) and, so, item (6) follows from item (5). The proof of Lemma 13.3.41 is complete.  We record the following immediate observation. Observation 13.3.42 Let f : C −→  C be a meromorphic function. Let G ⊆ H ⊆ C be two open connected sets and n ≥ 0 be an integer. If f∗−n : H −→ C is an analytic inverse branch of f −n , then f∗−n |G : G −→ C is also an analytic inverse branch of f −n . The rest of this subsection, and, in fact, of the whole section, can actually be viewed as addressing the issue of the existence of analytic inverse branches. We will need the following auxiliary fact, which has, strictly speaking, already been proved in this section and which is quite interesting on its own. Lemma 13.3.43 Let f : C −→  C be a meromorphic function and n ≥ 1 be an integer. If α : [0,+∞) −→ f −n ( C) is a continuous function such that the limit b :=

lim f n ◦ α(t)

t−→+∞

exists, then the limit a := also exists.

lim α(t) ∈  C

t−→+∞

13 Fundamental Properties of Meromorphic Dynamical Systems

Proof The set  :=



55

   cl C α([t,+∞)) ⊆ C

t≥0

is a continuum, i.e., a nonempty compact connected set that is an intersection of a descending family of continua. Seeking contradiction, suppose that  is not a singleton. Then  is uncountable, even of cardinality c. Then the set  ∩ f −n ( C) is also uncountable since C\f −n ( C) is a countable set. But     −n f n  ∩ f −n (C) ⊆ f n cl (C) C (α([t,+∞))) ∩ f t≥0

=



t≥0

⊆



  f n clf −n ( C) (α([t,+∞)))  n  cl C f (α([t,+∞))) = {b}.

t≥0

C) −→  C is constant identically equal to b. Therefore, the function f n : f −n (  Hence, the function f : C −→ C is also constant, and this contradiction finishes the proof.  Denote by f2−1 (∞) the set of poles of f whose orders are greater than 1, i.e., greater than or equal to 2. Note that Crit(f ) ∪ f2−1 (∞) is precisely the set of branching points of f , i.e., all points in C that admit no neighborhood on which the map f is one-to-one. Our next theorem of this section is the following. Theorem 13.3.44 Let f : C −→  C be a meromorphic function. Let H ⊆  C be a connected, simply connected open set. Fix an integer n ≥ 1. If G ⊆ C is a connected component of f −n (H ), then the following statements are equivalent. (1) (2) (3) (4)

The map f n |G : G −→ H is a conformal homeomorphism. The map f n |G : G −→ H is a homeomorphism. The map f n |G : G −→ H is bijective.   G ∩ Crit(f n ) ∪ f −(n−1) (f2−1 (∞)) = ∅ and the map f n |G : −→ H has no asymptotic values (equivalently, no asymptotic points).

Proof Of course, (1) ⇒ (2) ⇒ (3). (3) ⇒ (1) because the map f n |G : G −→ H is open. Of course, (1) implies that   G ∩ Crit(f n ) ∪ f −(n−1) (f2−1 (∞)) = ∅. Item (1) also implies that the map f n |G : G −→ H is proper. Therefore, this map has no asymptotic values by virtue of Proposition 13.3.13. We have, thus,

56

Part III Topological Dynamics of Meromorphic Functions

proved that (1) ⇒ (4). We are left to show that (4) ⇒ (1). Because of Lemma 13.3.41, it suffices to show that there exists an analytic branch f∗n : H −→ C of f −n such that f∗−n (H ) ∩ G = ∅. We shall do this by applying the Monodromy Theorem. We pick up the arbitrary point w ∈ G. Then f n (w) ∈ H and, because of the first part of item (4) and the Inverse Function Theorem, there exist R > 0 and an analytic branch fw−n : Bs (f n (w),R) −→ C of f −n such that f −n (f n (w)) = w. Then, by the definition of G, we have that w ∈ fw−n (Bs (f n (w),R)) ⊆ C. Let β : [0,+∞) −→ H be any continuous function such that β(0) = f n (w). We shall show that the germ of fw−n has an analytic continuation along the whole curve β. Seeking contradiction, suppose that this is not the case. This means that, denoting by u ≥ 0 the supremum of such s ≥ 0 that fw−n has an analytic continuation along β|[0,s) , we have that u < +∞. Of course, u > 0. Furthermore, the germ of f∗−n has an analytic continuation along the whole curve β|[0,u) , and this continuation (uniquely) induces a continuous function g : [0,u) −→ G. Indeed, if gt , t ∈ [0,u), is an element of this continuation, then g(t) = gt (t). In particular, g(0) = w and f n ◦ g ◦ β|[0,u) = β|[0,u) . Therefore, lim f n (g ◦ β(t)) = lim β(t) = β(u),

t−→u−

t−→u−

whence the limit x = lim g ◦ β(t) t−→u−

exists by virtue of Lemma 13.3.43. We know that x ∈ cl C (G) and, since the n map f |G : G −→ H has no asymptotic values, we must have that x ∈ G. But then f n (x) = β(u), So again by the Inverse Function Theorem, there exist

13 Fundamental Properties of Meromorphic Dynamical Systems

57

r > 0 and an analytic branch fx−n : Bs (β(u),r) −→ C of f −n of f −n such that fx−n (β(u)) = x. Now, since both sets Bs (β(u),r) and f −n (Bs (β(u),r)) are open neighborhoods, respectively, of β(u) and x, there exists δ ∈ (0,u) such that       β [u − δ,u + δ] ⊆ Bs (β(u),r) and g ◦ β [u − δ,u) ⊆ fx−1 Bs (β(u),r) . So, if t ∈ [u − δ,u) and gt is the corresponding germ of the analytic continuation of fw−n along β|[0,u) , then f n (gt (β(t))) = β(t) and f n (f −n (β(t))) = β(t); also, gt (β(t)) = g(β(t)) ∈ fx−n (Bs (β(u),r)) and fx−n (β(t)) ∈ fx−n (Bs (β(u),r)). Therefore, since the function f n |f −n B x

gt (β(t)) = fx−n (β(t)). Hence, setting

s (β(u),r)

 is one-to-one, we get that

ga := fx−n for all a ∈ [u,u + δ) gives an analytic continuation of the germ of fw−n along β|[0,u+δ) , contrary to the definition of u. Thus, the germ of fw−n is an analytic continuation along the whole curve β. So, since the open set H ⊆ C is simply connected, it follows from the Monodromy Theorem that there exists an analytic function ϕ : H −→ G such that ϕ|Bs (f n (w),R) = fw−n .

(13.57)

Since f −n ◦ fw−n = IdBs (f n (w),R) , we conclude that f n ◦ ϕ = IdH . This means that the map ϕ : H −→ G is an analytic branch of f −n . The proof of Theorem 13.3.44 is now conducted by applying Lemma 13.3.41.  The analytic branch (f n |G )−1 of f −n resulting from Theorem 13.3.44 will be frequently denoted by fG−n . Generalizing the concept of regular and singular points dealt with in Volume I, we introduce the following definition. Definition 13.3.45 Let f : C −→  C be a meromorphic function. Fix an integer n ≥ 1. We say that a point ξ ∈  C is a regular point of f −n if and only if there exists r > 0 such that, for every connected component C of f −n (Bs (ξ,r)), the restriction f n |C : C −→ Bs (ξ,r)

58

Part III Topological Dynamics of Meromorphic Functions

is a (conformal) homeomorphism from C onto Bs (ξ,r). Otherwise, we say that y is a singular point of f −n and we denote by Sing(f −n ) the set of all such singular points. We also set ∞  f n (Sing(f −1 )) (13.58) PS(f ) := n=0

and call it the post-singular set of f . Let us record the following immediate observation. Observation 13.3.46 Let f : C −→  C be a meromorphic function. Fix an integer n ≥ 1. Let ξ ∈  C be a regular point of f −n and r > 0 be the corresponding radius coming from Definition 13.3.45. Then, for every open connected set V ⊆ Bs (ξ,r) and every connected component U of f −n (V ), the map f n |U : U → V is a (conformal) homeomorphism. One of its immediate consequences is this. Observation 13.3.47 If f : C −→  C is a meromorphic function, then, for every integer n ≥ 1, the singular set Sing(f −n ) is closed in  C. We will now systematically explore the singular sets Sing(f −n ) and their complements. First, we shall prove the following two fairly easy but remarkable theorems. Theorem 13.3.48 Let f : C −→  C be a meromorphic function. If n ≥ 1 is −n  an integer, H ⊆ C\Sing(f ) is an open connected set, and G is a connected component of f −n (H ), then the map f n |G : G −→ H is covering. Proof Because of the definition of Sing(f −n ), all we need to check is that / the map f n |G : G −→ H is surjective. Indeed, fix a point ξ ∈ H . Since ξ ∈ Sing(f −n ) and since the set H is open, it follows from Observation 13.3.46 that there exists r > 0 witnessing Definition 13.3.45 for the point ξ such that Bs (ξ,r) ⊆ H . It follows from Theorem 13.3.2 that f n (G) ∩ Bs (ξ,r) = ∅. So, if D is a connected component of f −n (Bs (ξ,r)) contained in G, then the map f n |D : D −→ Bs (ξ,r) is a homeomorphism. In particular, ξ ∈ f n (D) ⊆ f n (G), implying that the map f n |G : G −→ H is surjective indeed. We are done.  As an immediate consequence of this theorem and Theorem 8.2.21 (see also Proposition 1.32 in [Ha]), we get the following.

13 Fundamental Properties of Meromorphic Dynamical Systems

59

Theorem 13.3.49 Let f : C −→  C be a meromorphic function. If n ≥ 0 is an integer, H ⊆  C\Sing(f −n ) is an open connected, simply connected set, and G is a connected component of f −n (H ), then the map f n |G : G −→ H is a homeomorphism. In particular, fG−n := (f n |G )−1 : H −→ G is an analytic inverse branch of f −n from H to G and the open connected set G is simply connected. Equivalently, if ξ ∈ H and w ∈ f −n (H ), then there exists fw−n : H −→ C, a unique holomorphic branch of f −n defined on H , such that fw−n (ξ ) = w. Thus, also (we invoke here Lemma 13.3.41) fw−n (H ) is the connected component of f −n (H ) containing w. We will need one theorem more of similar flavor. In order to prove it, we will need the following straightforward theorem to prove the well-known result from topology. Theorem 13.3.50 Let V ⊆  C be an open connected set and A ⊆  C be a countable set disjoint from V . (1) If V is simply connected and V ∪ A is an open connected set, then either A = ∅ or A is a singleton and V ∪ A =  C. (2) If V is doubly connected, i.e., V is a topological annulus, and V ∪ A is an open connected set, then either (a) A = ∅, or (b) A is a singleton and the set V ∪ A is simply connected, or (c) A is a doubleton and V ∪ A =  C. Now we can prove the following. Theorem 13.3.51 Let f : C −→  C be a transcendental meromorphic function. If n ≥ 1 is an integer and H ⊆  C is an open connected, simply connected set containing at most one element of the singular set Sing(f −n ) and missing at least two points in  C, then each connected component of f −n (H ) is simply connected. Proof If H ∩ Sing(f −n ) = ∅, then we are immediately done because of Theorem 13.3.49. So, suppose that H ∩ Sing(f −n ) is a singleton and denote

60

Part III Topological Dynamics of Meromorphic Functions

the only point of this intersection by ξ . Then H \{ξ } is a topological annulus with modulus Mod(H \{ξ }) = +∞.

(13.59)

Let G be any connected component of f −n (H ). Then Gξ := G\f −n (ξ ) is the only connected component of f −n (H \{ξ }) contained in G. If Gξ is simply connected, then G is too by virtue of Theorem 13.3.50(1). So, assume that Gξ is not simply connected, meaning that its fundamental group π1 (Gξ ) = 0. But, by Theorem 13.3.48, the map f n |Gξ : Gξ −→ H \{ξ } is covering, thus, by Theorem 8.2.19, inducing a monomorphism (one-to-one homeomorphism)  n  f |Gξ ∗ : π1 (Gξ ) −→ π1 (H \{ξ }) of the corresponding fundamental groups. Therefore, π1 (Gξ ) is isomorphic to a nonzero subgroup π1 (H \{ξ }), which is isomorphic to Z. Hence, π1 (Gξ ) is isomorphic to Z. It, therefore, follows from Theorem 8.2.2 that Gξ is a topological annulus. So, if G\Gξ = ∅, then Case (a) of Theorem 13.3.50 is excluded, and, since G ⊆ C, Case (c) is excluded too. Then the bounded connected component of Gξ must be a singleton belonging to f −n (ξ ), and the set G is simply connected. We are, therefore, left to deal with the case G = Gξ , equivalently G ∩ f −n (ξ ) = ∅.

(13.60)

Then the mapf n |G : G −→ H \{ξ } is covering; invoking (13.59), we conclude from Theorems 8.2.8 and 8.2.23 that there exists an open connected, simply ˆ ⊆  ˆ connected set G C such that G\G is a singleton. We denote its element n C\H has at by w. Since f (G) ⊆ H \{ξ } and, by our hypothesis, the set  least three elements, it follows from Picard’s Great Theorem that f n has C extends to a either a removable singularity pole at w, i.e., f n |G : G −→  n n  ˆ ˆ ˆ meromorphic function f : G −→ C. But then either f (w) ∈ H \{ξ } or fˆn (w) = ξ ∈ H . But, in the case that w = ∞, we have that fˆn = f n , and this formula would give that w ∈ G, contrary to the definition of w. If, on the other hand, w = ∞, then, using this formula, we would conclude that ∞ is not an essential singularity of f n , thus not of f either, meaning that the function f : C −→  C is not transcendental. This contradiction finishes the proof of Theorem 13.3.49  Theorem 13.3.52 Let f : C −→  C be a transcendental meromorphic function. Let H ⊆ C be a connected, simply connected set. Fix an integer n ≥ 0. If G ⊆ C is a connected component of f −n (H ),   G ∩ Crit(f n ) ∪ f −(n−1) (f2−1 (∞)) = ∅,

13 Fundamental Properties of Meromorphic Dynamical Systems

61

and the map f n |G : G −→ H has no asymptotic values (equivalently, no asymptotic points), then the map f n |G : G −→ H is covering. Proof Because of Theorem 13.3.44, we need to check that the map f n |G : G −→ H is surjective. To do this, fix a point ξ ∈ H . Then take r > 0 so small that Bs (ξ,r) ⊆ H . It follows from Theorem 13.3.2 that f n (G) ∩ Bs (ξ,r) = ∅. So, if D is a connected component of f −n (Bs (ξ,r)) contained in G, then the map f n |D : D −→ Bs (ξ,r) is, by virtue of Theorem 13.3.44, a homeomorphism. In particular, ξ ∈ f n (D) ⊆ f n (G), implying that the map f n |G : G −→ H is surjective indeed. We are done.  The first structural result about the singular set Sing(f −n ) is the following. Theorem 13.3.53 Let f : C −→  C be a meromorphic function. Fix an integer n ≥ 1. Then Sing(f −n ) =

n−1 

f k (Sing(f −1 )).

(13.61)

k=0

Proof The inclusion  C\

n−1 

f k (Sing(f −1 )) ⊆  C\Sing(f −1 )

k=0

follows by a straightforward induction from Theorem 13.3.49 and Lemma 13.54. By contrapositive, Sing(f −n ) ⊆

n−1 

f k (Sing(f −1 )).

k=0

Proving the opposite inclusion, assume by way of contradiction that n−1 

f k (Sing(f −1 ))  Sing(f −1 ).

k=0

Then there exists a point ξ∈

n−1  k=0

f k (Sing(f −1 ))\Sing(f −1 ).

(13.62)

62

Part III Topological Dynamics of Meromorphic Functions

Hence, there exist 0 ≤ k ≤ n − 1 and y ∈ Sing(f −1 ) such that f k (y) = ξ .

(13.63)

Also, since the set Sing(f −1 ) is closed, there exists s > 0 such that C\Sing(f −1 ). Bs (ξ,s) ⊆ 

(13.64)

Because of (13.63) there exists a unique connected component Vy of f −k (Bs (ξ,s)) containing y. Since Vy is also open and y ∈ Sing(f −1 ), there exist r > 0 and a connected component Ur of f −1 (Bs (y,r)) such that Bs (y,r) ⊆ Vy and the map f |Ur : Ur −→ Bs (y,r) is not a homeomorphism. Let  be the unique connected component of f −1 (Vy ) containing Ur . Now let C be an arbitrary connected component of f −(n−1−k) (). Then f n (C) = f k+1 (f (n−1−k) (C)) ⊆ f k+1 () = f k (f ()) ⊆ f k (Vy ) ⊆ Bs (ξ,s). Therefore, there exists a unique connected component C ∗ of f −n (Bs (ξ,s)) containing C. Because of (13.64), the map f n |C∗ : C ∗ −→ Bs (ξ,s) is C and a homeomorphism. Therefore, both maps f (n−1−k) |C ∗ : C ∗ −→  C are injective. Hence, both maps f n−k : C ∗ −→  C and f (n−1−k) ◦ (f n |C ∗ )−1 : Bs (ξ,s) −→  (n−k) n ∗ −1 f ◦ (f |C ) : Bs (ξ,s) −→  C are holomorphic and injective. Since also f k+1 ◦ f (n−1−k) ◦ (f n |C ∗ )−1 = IdBs (ξ,s) and f k ◦ f (n−k) ◦ (f n |C ∗ )−1 = IdBs (ξ,s), both these compositions are holomorphic branches, respectively, of f −(k+1) and f −k defined on Bs (ξ,s). But f (n−k) ◦ (f n |C ∗ )−1 (Bs (ξ,s)) = f n−k (C ∗ ) ⊇ f n−k (C) and f n−k (C) = f (f n−1−k (C) ⊆ f ()) ⊆ Vy . Hence, f n−k ◦ (f n |C ∗ )−1 (Bs (ξ,s)) ∩ Vy = ∅. So, it follows from the definition of Vy and Lemma 13.54 that f n−k ◦ (f n |C ∗ )−1 (Bs (ξ,s)) = Vy . Similarly, f n−1−k ◦ (f n |C ∗ )−1 (Bs (ξ,s)) = f n−1−k (C ∗ ) ⊇ f n1 −k (C) and f n−1−k (C) ⊆ .

13 Fundamental Properties of Meromorphic Dynamical Systems

63

Thus, f n−1−k ◦ (f n |C ∗ )−1 (Bs (ξ,s)) ∩  = ∅.

(13.65)

But 

  −1 : Vy −→  C f n−1−k ◦ (f n |C ∗ )−1 ◦ f n−1−k ◦ (f n |C ∗ )−1

is a holomorphic injective map and    −1 f ◦ f n−1−k ◦ (f n |C ∗ )−1 ◦ f n−1−k ◦ (f n |C ∗ )−1    −1 = f n−k ◦ (f n |C ∗ )−1 ◦ f n−1−k ◦ (f n |C ∗ )−1 = IdV , whence 

  −1 f n−k ◦ (f n |C ∗ )−1 ◦ f n−1−k ◦ (f n |C ∗ )−1

is a holomorphic branch of f −1 defined on Vy . So it follows from the definition of , (13.65), and Lemma 13.54 that f n−1−k ◦ (f n |C ∗ )−1 (Bs (ξ,s)) = . This, however, contradicts the defining property of the radius r and shows that n−1 

f k (Sing(f −1 )) ⊆ Sing(f −n ).

k=0

Along with (13.62), this gives n−1 

f k (Sing(f −1 )) ⊆ Sing(f −1 ),

k=0

ending the proof of Theorem 13.3.53.



As an immediate consequence of this theorem, we get the following. Corollary 13.3.54 If f : C −→  C is a meromorphic function, then, for every integer n ≥ 1, Sing(f −n ) ⊆ Sing(f −(n+1) ). Now we will characterize the singular sets Sing(f −n ) in terms of critical C) −→ C. We denote the and asymptotic values of the function f n : f −n ( −1 n latter set by AsymptV(f ). Note that the set f (f2 (∞)) is either the singleton

64

Part III Topological Dynamics of Meromorphic Functions

∞ or it is empty according to whether f2−1 (∞) is not empty or empty. Note also that n       f n Crit f n : f −n ( C) −→ C = f j Crit(f ) ∩ f −j ( C) . (13.66) j =1

The next main result of this section is the following. Theorem 13.3.55 Let f : C −→  C be a meromorphic function. Fix an integer n ≥ 1. Then Sing(f

−n

)=

AsymptV(f n ) ∪

n 

  f j Crit(f ) ∩ f −j ( C) ∪ f (f2−1 (∞)).

j =1

(13.67) Proof

The inclusion

Sing(f −n ) ⊆ AsymptV(f n ) ∪

n 

  f j Crit(f ) ∩ f −j ( C) ∪ f (f2−1 (∞))

j =1

(13.68) follows immediately from Theorem 13.3.44 and (13.66). In order to prove the opposite inclusion, take any point y ∈  C\Sing(f −n ) and let r > 0 be the corresponding number following from Definition 13.3.45. Of course, ⎛ ⎞ n    (13.69) f j Crit(f ) ∩ f −j ( C) ∪ f (f2−1 (∞))⎠ = ∅. Bs (y,r) ∩ ⎝ j =1

Seeking contradiction, suppose that Bs (y,r) ∩ AsymptV(f −n ) = ∅. Fix an arbitrary ξ ∈ Bs (y,r) ∩ AsymptV(f −n ). By virtue of Theorem 13.3.28, there exists a transcendental inverse chain 

C))) U = Us : s ∈ (0,dists (ξ,∂f −n ( C) −→ C at ξ . Take t ∈ (0,dists (ξ,∂f −n ( C)) so small for the map f n : f −n ( that Bs (ξ,t) ⊆ Bs (y,r).

(13.70)

13 Fundamental Properties of Meromorphic Dynamical Systems

65

Let  be the unique connected component of f −n (B(y,r)) containing Ut . Then, on the one hand, by (13.70) and the definition of r, the map f n |Ut : Ut −→ Bs (ξ,t) is homeomorphic and, thus, proper; on the other hand, the collection {Uu : u ∈ (0,t)} is a transcendental inverse chain at ξ for the map f n |Ut : Ut −→ Bs (ξ,t). Hence, again by virtue of Theorem 13.3.28, ξ is an asymptotic value for the map f n |Ut : Ut −→ Bs (ξ,t). So, by Proposition 13.3.13, this map is not proper. This contradiction shows that Bs (y,r) ∩ AsymptV (f −n ) = ∅. Along with (13.69), this implies that ⎛ Bs (y,r) ∩ ⎝AsymptV (f −n ) ∪

n 









⎞

f j Crit(f ) ∩ f −j ( C) ∪ f (f2−1 (∞))⎠ = ∅.

j =1

Thus, ⎛ y∈ / ⎝AsymptV (f −n ) ∪

n 

⎞

f j Crit(f ) ∩ f −j ( C) ∪ f (f2−1 (∞))⎠

j =1

and the inclusion ⎛ ⎝AsymptV (f −n ) ∪

n 





⎞

f j Crit(f ) ∩ f −j ( C) ∪ f (f2−1 (∞))⎠ ⊆ Sing(f −n )

j =1

is proved. Along with (13.3.55), this shows that Sing(f −n ) = AsymptV (f −n ) ∪

n 

  f j Crit(f ) ∩ f −j ( C) ∪ f (f2−1 (∞)).

j =1

The proof of Theorem 13.3.55 is complete.



As a direct consequence of this theorem and Theorem 13.3.53, we get the following. Theorem 13.3.56 Let f : C −→  C be a meromorphic function. Fix an integer n ≥ 1. Then Sing(f

−n

)=

n−1  k=0

  f k AsymptV(f ) ∪ f (Crit(f )) ∪ f (f2−1 (∞)) . (13.71)

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Part III Topological Dynamics of Meromorphic Functions

Finally, as a direct consequence of Theorem 13.3.49, we get the following. Theorem 13.3.57 Let f : C −→  C be a meromorphic function. If V ⊆  C is an open connected, simply connected set disjoint from the post-singular set PS(f ), then, for every integer n ≥ 1 and every connected component U of f −n (V ), the map f n |U : U −→ V is homeomorphic; in particular, the open connected set U is simply connected. Equivalently, for every point ξ ∈ V , every integer n ≥ 1, and every point w ∈ f −n (ξ ), there exists a unique holomorphic branch fw−n : V −→ C of f −n defined on V such that fw−n (ξ ) = w.

14 Finer Properties of Fatou Components

In this chapter, we analyze the structure of Fatou components and the structure of their boundaries in greater detail. In particular, we study simple connectedness of such components. We also bring up the definitions of Speiser class S and Eremenko–Lyubich class B and we prove some structural theorems about their Fatou components. In particular, we prove no existence of Baker domains and wandering domains (the Sullivan Nonwandering Theorem) for class S; the proof of the latter is in Appendix B.

14.1 Properties of Periodic Fatou Components The importance of periodic components of Fatou sets is multi-fold. The present section is evidence of this. But there are many more reasons. For example, later, in Section 14.4, we will see that any meromorphic function in Speiser class S (to be defined in Section 14.4) has no wandering Fatou components; in other words, that every connected component of its Fatou set is preperiodic. The periodic domains of the Fatou set F (f ) of f are closely related to the set Sing(f −1 ) of singular values of f defined and dealt with in Section 13.3. The following two theorems show the significance of the set Sing(f −1 ) when studying attracting components of the Fatou set. Its importance (see all subsequent chapters) goes far beyond these. Theorem 14.1.1 Let f : C −→  C be a meromorphic function. If ξ ∈ C is an attracting periodic point of f , then A∗p (ξ ), the basin of immediate attraction to the forward orbit of ξ , contains at least one singular point of f −1 whose forward orbit under f either does not contain ξ or coincides with the forward orbit of ξ . In particular, in the former case this orbit is infinite, while in the latter case the orbit of ξ is super-attracting. 67

68

Part III Topological Dynamics of Meromorphic Functions

Proof Assume, first, that ξ is an attracting fixed point of f . If ξ ∈ Sing(f −1 ), then we are done. So, suppose that ξ ∈ / Sing(f −1 ). Then, in particular, f  (ξ ) = 0. Hence, there exists U0 , an open connected, simply connected, neighborhood of ξ contained in A∗p (ξ ) such that f (U0 ) ⊆ U0 and the map f |U0 : U0 → U0 is one-to-one. Observe then that U0 ∩

∞ 

f −n (ξ ) = {ξ }.

n=0

Now, by way of contradiction, suppose that all singular points of f −1 lying in A∗ (ξ ) are eventually mapped onto ξ under some iterates of f , i.e., that they  −n (ξ ). We shall prove by induction that there exists belong to the set ∞ n=1 f an ascending sequence (Un )∞ n=0 of connected, simply connected, open subsets of A∗ (ξ ) such that: (an ) ξ ∈ Un for all n ≥ 0. (bn ) For every n ≥ 1, there exists fn−1 : Un−1 −→ Un , a unique surjective holomorphic inverse branch of f such that fn−1 (ξ ) = ξ . (cn ) Un ∩

∞ 

f −n (ξ ) = {ξ }

n=0

for all integers n ≥ 0. Indeed, the set U0 has already been defined satisfying (a0 ), (c0 ), and, vacuously, (b0 ). For the inductive step, suppose that n ≥ 0 and that connected, simply connected subsets U0 ⊂ U1 ⊂ U2 ⊂ · · · ⊂ Un ⊂ A∗ (ξ ), satisfying conditions (a), (b), and (c), with appropriate subscripts, have been defined. Since the set A∗ (ξ ) is open and connected and it is a connected component of f −1 (A∗ (ξ )), by using (cn ), we conclude that there exists a connected component Un+1 of f −1 (Un ) containing ξ . So, Un+1 ⊆ A∗ (ξ ) and since, in addition, the open, connected, and simply connected set Un contains

14 Finer Properties of Fatou Components

69

no singular values of f |A∗ (ξ ) , we conclude from Theorem 13.3.44 that there exists a holomorphic branch −1 fn+1 : Un −→ Un+1 ⊆ A∗ (ξ ) −1 of f −1 uniquely determined by the requirement that fn+1 (ξ ) = ξ . We immediately see from this definition that Un+1 is an open connected, simply connected subset of A∗ (ξ ) and that −1 −1 ξ = fn+1 (ξ ) ∈ fn+1 (Un ) = Un+1 .

In conclusion, item (bn+1 ) holds. Now we shall show that Un ⊆ Un+1 . Indeed, −1 −1 Un = fn−1 (Un−1 ) = fn+1 (Un−1 ) ⊆ fn+1 (Un ) = Un+1 .

So, we are only left to show that (cn+1 ) holds. Indeed, suppose that ∞ 

z ∈ Un+1 ∩

f −n (ξ ).

n=0

Then, as f (ξ ) = ξ and f (Un+1 ) = Un , we have that f (z) ∈ Un ∩

∞ 

f −n (ξ ).

n=0

Hence, by (cn ), f (z) = ξ . Thus, −1 z = fn+1 (ξ ) = ξ

and item (cn+1 ) is established The inductive construction of the sequence (Un )∞ n=0 is, thus, complete. It follows from all the above properties of the sequence (Un,fn+1 )∞ n=0 that, for all n ≥ 1, the composition −1 fξ−n := fn−1 ◦ fn−1 ◦ · · · ◦ f2−1 ◦ f1−1 : U0 → Un ⊂ A∗ (ξ )

is well defined and holomorphic. Since A∗ (ξ ) ⊂ F (f ) and since the Julia set J (f ) contains at least three points (because it is perfect, by Theorem 13.1.10), it follows from Montel’s Theorem II (Theorem 8.1.16) that the family of maps (fξ−n : U0 → A∗ (ξ ))∞ n=0 is normal. This, however, produces a contradiction since lim (fξ−n ) (ξ ) = lim ((f n ) (ξ ))−1 = ∞.

n→∞

n→∞

70

Part III Topological Dynamics of Meromorphic Functions

We are, thus, done in the case when ξ is a fixed point of f . In general, if ξ is a periodic point of f , say of minimal period p ≥ 1, then, as we have just actually proved, the map f p : A∗ (ξ ) −→ A∗ (ξ ) (the function  −n ( C) and the f p need not be meromorphic but A∗ (ξ ) ⊆ F (F ) ⊆ ∞ n=0 f −p same proof goes through) contains a singular point of f . Moreover, as A∗p (ξ ) =

p−1 

A∗ (f j (ξ )) =

p−1 

j =0

f j (A∗ (ξ ))

j =0

and f p = f ◦ f ◦ · · · ◦ f (n-fold composition), we, thus, conclude that A∗p (ξ ) contains a singular value of f and the proof is complete.  Later on, in Section 15.2, after appropriate preparations, we shall prove Theorem 15.2.5 as an analogous result for basins of immediate attraction to rationally indifferent periodic points. Now we shall prove a somewhat analogous theorem for Siegel disks and Herman rings. Theorem 14.1.2 Let f : C −→  C be a meromorphic function. If {U0,U1, . . . ,Up−1 } is a cycle of Siegel disks or Herman rings, then ∂Uj ⊂ PS(f ) for all j ∈ {1, . . . ,p − 1}, where the set PS(f ) was defined in (13.58). Proof

We aim to apply Lemma 8.1.19. We set R := C,

S := C\f −1 (∞),

and keep f the same, i.e., to be really formally correct, when applying Lemma 8.1.19, we consider f |R . We take Q as an arbitrary finite subset of Per(f )(⊆ C) with at least three elements. Seeking contradiction, suppose that ∂Uj  PS(f ) for some j ∈ {0,1, . . . ,p − 1}. Denote D := ∂Uj . Let  := B(0,1) if D is a Siegel disk and  := A(0;1,r)

(14.1)

14 Finer Properties of Fatou Components

71

with r > 1 coming from Theorem 13.2.5 if D is a Herman ring. Let H :  −→ D be the analytic homeomorphism resulting from Theorem 13.2.5(3) and (4), respectively, in the Siegel or Herman case. Because of (14.1), there exists a point ξ ∈ ∂D\PS(f ).

(14.2)

Using the fact that D is either simply connected or doubly connected and r ∈ (1,+∞), we deduce that ξ is not an isolated point of ∂D; in fact, ∂D has no isolated points. Therefore, since O + (Q) ∪ f −1 (∞) is a countable set, we may assume, in addition, that ξ∈ / O + (Q) ∪ f −1 (∞), so that Lemma 8.1.19 applies. Fix s > 0 so small that Be (ξ,3s) ∩ PS(f ) = ∅. Pick a point w ∈ B(ξ,s) ∩ D.

(14.3)

B(w,2s) ∩ PS(f ) = ∅.

(14.4)

  F := H {z ∈  : |z| = |H −1 (w)|} .

(14.5)

Then

Let

Note that F is a compact set (homeomorphic to a circle) and F ⊆ D. Since w ∈ D, for every n ≥ 1, the intersection D ∩ f −n (w) is a singleton. Denoting its only element by wn , we will have that wn ∈ F .

(14.6)

By (14.4) and Theorem 13.3.57, for every n ≥ 1, there exists a unique holomorphic branch fn−n : B(w,2s) −→ C

(14.7)

of f −n sending w to wn . By (14.2), ξ ∈ J (f ), whence fn−n (ξ ) ∈ J (f ). Thus, / D; therefore, using also (14.6) and (14.7), we conclude that fn−n (ξ ) ∈   diame fn−n (B(w,2s)) ≥ diste (wn,fn−n (ξ )) ≥ diste (F,C\D) > 0.

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Part III Topological Dynamics of Meromorphic Functions

Therefore,

  lim inf diame fn−n (B(w,r)) ≥ diste (F,C\D) > 0. n→∞

It, thus, follows from the Koebe Distortion Theorem, i.e., Theorem 8.3.8, that lim inf |(f n ) (w)| < +∞, n→∞

contrary to Lemma 8.1.19. The proof of Theorem 14.1.2 is, thus, finished. 

14.2 Simple Connectedness of Fatou Components In this section, we provide several sufficient conditions for a Fatou component (especially periodic) of a meromorphic function to be simply connected. We start with the following. ¯ be a meromorphic function. If W is a Theorem 14.2.1 Let f : C −→ C periodic connected component of the Fatou set F (f ) of f which is neither a Herman ring nor a Baker domain and if W contains at most one singular point of f −1 , then W is simply connected. Proof Passing to an iterate of f (even though such an iterate may fail to be meromorphic), we may assume without loss of generality that W is a forward invariant connected component of F (f ), i.e., f (F (f )) ⊆ F (f ). The assertion of our theorem is immediate if W is a Siegel disk because of Definition 13.1.16 and Theorems 13.1.17 and 13.2.5(3). It is true for all Siegel disks. Assume now that W is the basin of immediate attraction to an attracting fixed point. Denote it by ξ . We proceed somewhat similarly to the proof of Theorem 14.1.1. Similarly to therein, fix U0 , an open connected, simply connected neighborhood of ξ contained in W , such that f (U0 ) ⊆ U0 .

(14.8)

We shall construct inductively a sequence (Un )∞ n=0 of open subsets of C with the following properties. (an ) (bn ) (cn ) (dn )

Un+1 ⊃ Un . Un ⊆ A∗ (ξ ) = W . Un+1 is a connected component of f −1 (Un ). Un is simply connected.

Indeed, suppose that the open sets U0,U1, . . . ,Un have been defined such that all conditions (ak )–(dk ) hold for all 0 ≤ k ≤ n − 1 along with (bn ) and (dn ). Look at f −1 (Un ). By (an ) and (cn−1 ),

14 Finer Properties of Fatou Components

73

f −1 (Un ) ⊃ f −1 (Un−1 ) ⊃ Un . Let Un+1 be the connected component of f −1 (Un ) containing Un . Then, immediately, conditions (an ) and (cn ) hold. Conditions (bn ) and (dn ) hold because of our inductive assumption. By (an ) and (bn ), we have that Un+1 ∩ A∗ (ξ ) = ∅. Since Un+1 ⊆ F (f ) and A∗ (ξ ) is a connected component of F (f ), this implies that Un+1 ⊆ A∗ (ξ ), meaning that bn+1 holds. Item dn+1 follows directly from Theorems 13.3.49 and 13.3.51. Let U∞ :=

∞ 

Un .

n=0

Then U∞ is open and also connected and simply connected as being the union of an ascending sequence of open connected, simply connected sets. Thus, in order to finish the proof in the current case, it suffices to show that U∞ = W . By conditions (bn ), we have that U∞ ⊆ W . Seeking contradiction, suppose that U∞  W . Since both sets, W and U∞ , are connected, this gives that ∂ W U∞ = ∅. So, there exists a point z ∈ ∂ W U∞ .

(14.9)

By (cn ), we have that f (Un+1 ) = Un , whence f (U∞ ) = U∞ . Therefore, f (U ∞ ) ⊆ f (U∞ ) = U ∞ . Hence, for every integer k ≥ 0, f k (∂ W U∞ ) ⊆ U ∞ ∩ W .

(14.10)

Since z ∈ W = A∗ (ξ ), there exists an integer l ≥ 0 such that f l (z) ∈ U0 . In particular, f l (z) ∈ U∞ . Let n ≥ 0 be the least integer with this property. It follows from (14.9) that n ≥ 1; so, with the help of (14.10), f n−1 (z) ∈ ∂ W U∞ . So, replacing z by f n−1 (z), we may assume that z ∈ ∂ W U∞

and

f (z) ∈ U∞ .

(14.11)

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Part III Topological Dynamics of Meromorphic Functions

Hence, there exists n ≥ 0 such that f (z) ∈ Un .

(14.12)

Since the map f is continuous, there, thus, exists an ε > 0 such that B(z,ε) ⊆ f −1 (Un ) ⊆ F (f ).

(14.13)

Since the sequence (Un )∞ n=0 is ascending, it follows from the first part of (14.11) that B(z,ε) ∩ Us+1 = ∅

(14.14)

for some s ≥ n. Using again the fact that the sequence (Un )∞ n=0 is ascending, it follows from (14.13) that B(z,ε) ⊆ f −1 (Us ) . Along with (cs ), this yields B(z,ε) ∪ Us+1 ⊆ f −1 (Us ). Since both sets, B(z,ε) and Us+1 , are connected, it follows from (14.14) that the set B(z,ε) ∪ Us+1 is connected. Combining this, (14.13), and (cs ) again, we conclude that B(z,ε) ⊆ Us+1 . Hence, z ∈ U∞ contrary to the first part of (14.11) and openness of U∞ in W . We are done in this case. Because of Theorem 13.2.5, we are left only with the case of ξ being a rationally indifferent fixed point of f . Passing to yet a larger iterate, we may assume without loss of generality that ξ is simple, i.e., f  (ξ ) = 1. We now use some notation and results from Chapter 15 that are logically independent of the current section. First, W = A∗j (ξ ) j

with some j ∈ {1,2, . . . ,p(ξ )}. Recall that Sa (ξ,α), α ∈ (0,π), is the sector defined by (15.36). Taking j

U0 := Sa (ξ,α) and making use of Lemma 15.2.1 and Theorem 15.2.3, we may proceed in exactly the same way as in the case of an attracting fixed point, to conclude that the set W = A∗j (ξ ) is simply connected. The proof of Theorem 14.2.1 is complete.  As a fairly immediate consequence of Theorem 14.2.1, we get the following.

14 Finer Properties of Fatou Components

75

Corollary 14.2.2 Let f : C −→  C be a meromorphic function. If W is a periodic connected component of the Fatou set F (f ) of f which is neither a Herman ring nor a Baker domain and each connected component of the set W∞ :=

∞ 

f −n (W )

n=0

contains at most one singular value of f, then each of these components is simply connected. Proof For each connected component  of W∞ , let N ≥ 0 be the least integer such that f N () ⊆ W . We proceed by induction with respect to N . If N = 0, then  = W , and  is simply connected because of Theorem 14.2.1. So, fix n ≥ 0 and suppose that the assertion of Corollary 14.2.2 holds for all components G of W∞ for which NG ≤ n. Let  be a connected component of W∞ for which N = n + 1. Then f () ⊆ W∞ and, with the notation of Definition 13.3.3, Nfˆ([;W ]) = n. So, fˆ([;W ]) is simply connected. Since it contains at most one singular value of f and since  is a connected component of f −1 (fˆ([;W ])), it follows directly from Theorems 13.3.49 and 13.3.51 that  is simply connected, and the proof of Corollary 14.2.2 is complete.  We would like to end this section by pointing out that interesting and far-reaching results about the simple connectedness of all Fatou components of a meromorphic function, and, equivalently, about the connectivity of Julia sets, were obtained in [BFJK1]. In particular, it follows from these results that the Julia set of the Newton’s method of any entire function is connected. We will also deal with the simple connectedness of Fatou components later, in Section 14.4.

14.3 Baker Domains In this section, we deal with Baker domains. Indeed, except for Baker domains, all the periodic components of Fatou sets described in Theorem 13.2.5 have already appeared in the realm of the dynamics of rational functions. The first example of an entire transcendental function with a Baker domain was given previously by Fatou [Fat3], who considered the entire function f : C −→ C given by the formula f (z) = z + 1 + e−z

76

Part III Topological Dynamics of Meromorphic Functions

and proved that lim f n (z) = +∞

n→∞

for every z ∈ C with Re(z) > 0. This implies that the right half-plane is contained in an invariant Baker domain. For ergodic and fractal properties of the function f , see [KU5]. An example of a Baker domain of higher period was given in [BKL3], where it was shown that the function f (z) := z−1 − ez has a cycle U0,U1 of Baker domains such that 2n 2n f|U −−−−→ ∞ and f|U −−−−→ 0. 0 1 n→∞

n→∞

Now we explore some general properties of Baker domains. Let f : C −→  C be a meromorphic function. Assume that {U0,U1, . . . ,Up−1 } is a periodic cycle of Baker domains, and denote by ξj the limit point corresponding to Uj , i.e., lim f np (z) = ξj

n→∞

for all z ∈ Uj . Clearly, f (ξj ) = ξj +1 if zj = ∞, where ξp := ξ0 and there exists at least one j ∈ {0,1, . . . ,p − 1} such that ξj = ∞ and, for all j ∈ {0,1, . . . ,p − 1}, there exists l = l(j ) ∈ {0,1, . . . ,p − 1} such that f l (ξj ) = ∞. We claim that each set Uj contains a (topological) curve γj converging to ξj such that f p (γj ) ⊆ γj and f p (z) → ξj as z → ξj in γj . To see this, we fix w ∈ U0 and then a closed (compact) topological arc σ ⊆ U0 that joins w0 and f p (w). We define γ0 :=

∞ 

f np (σ ) and γj := f j (γ0 )

n=0

for j ∈ {1,2, . . . ,p − 1}. Then the curves γj have the desired properties. Moreover, lim

γj z→ξj

f (z) = ξj +1 .

14 Finer Properties of Fatou Components

77

We deduce that if ξj = ∞, then ξj +1 is an asymptotic value of f , the asymptotic path being contained in Uj . We have, thus, proved the following theorem. Theorem 14.3.1 Assume that f : C −→  C is a meromorphic function with {U0,U1, . . . ,Up−1 }, a periodic cycle of Baker domains. Denote by ξj , j = 0,1, . . . ,p − 1, the limit corresponding to Uj and define ξp := ξ0 . Then ξj ∈

p−1 

f −n (∞)

n=0

for all j ∈ {0,1, . . . ,p − 1} and ξj = ∞ for at least one j ∈ {0,1, . . . ,p − 1}. If ξj = ∞, then ξj +1 ∈ AsymptV(f ), i.e., it is an asymptotic value of f . The two following consequences of this theorem are immediate. Corollary 14.3.2 If a meromorphic function f : C −→  C has a periodic cycle {U0,U1, . . . ,Up−1 } of Baker domains such that n −→ ∞ as n → +∞, f|U 0

then ∞ is an asymptotic value of f . In particular, this is the case if f has a Baker domain which is f -invariant . Corollary 14.3.3 If a meromorphic function f : C −→  C has a cycle {U0,U1, . . . ,Up−1 } of Baker domains such that n −→ ∞ as n → +∞, f|U 0

then f has a finite asymptotic value.

14.4 Fatou Components of Class B and S of Meromorphic Functions It is fairly common and more and more frequent to consider in transcendental dynamics the following two classes of transcendental meromorphic functions. First,

S := f : C −→  C : f is transcendental meromorphic and  C ∩ Sing(f −1 ) is finite . This collection of functions is usually referred to as Speiser class S.

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Part III Topological Dynamics of Meromorphic Functions

Speiser class S was investigated systematically by Nevanlinna, Teichm¨uller, and others. It plays an important role in the value distribution theory [Nev] and [Wi]. It was introduced to dynamics in [EL1] and [GK]. Second,

B := f : C −→  C : f is transcendental meromorphic and  C ∩ Sing(f −1 ) is bounded . This collection of functions is usually called Eremenko–Lyubich class B. It was introduced in [EL2] and has been widely used since then. Of course, S ⊆ B. As an immediate consequence of Theorem 13.3.48, we get the following simple but useful results. Proposition 14.4.1 If f : C −→  C is a meromorphic function belongingto −1 Speiser class S, then, for every  open connected set H ⊆ C\Sing(f −1 ) in −1 particular H = C\Sing(f ) and every connected component G of f (H ), the map f : G → H is covering. Proposition 14.4.2 If f : C −→  C is a meromorphic function belonging to class B, then, for every open connected set H ⊆ C\Be 0,Diste (0,Sing(f −1 )    in particular H = C\Be 0,Diste (0,Sing(f −1 )) and every connected component G of f −1 (H ), the map f : G → H is covering. As an immediate consequence of Theorems 14.1.1 and 15.2.5, we get the following. Theorem 14.4.3 Any meromorphic function in Speiser class S has only finitely many attracting and rationally indifferent periodic points. The fundamental theorem about Speiser class S is the following. Theorem 14.4.4 No functions in Speiser class S have wandering domains. This theorem has been conjectured by Fatou in [Fat1] and [Fat2] for rational functions and was proved in this form by Sullivan in [Su1]. In fact, it holds for all meromorphic functions in Speiser class S and was proved by Baker, Kotus, and L¨u in [BKL4]; see this paper and references therein for more historical and bibliographical information. Examples of wandering components for analytic self-maps of C∗ were provided by Kotus in [Ko1] and by Baker, Kotus, and L¨u in [BKL2]. We provide in Appendix B a proof of Theorem 14.4.4 stemming from [BKL4].

14 Finer Properties of Fatou Components

79

For every integer n ≥ 1, we introduce the following auxiliary class of functions.

C : f is meromorphic transcendental with Bn := f : C −→   (14.15) C ∩ Sing(f −n ) bounded . Of course, B1 = B is the class B introduced above. Because of Corollary 13.3.54, Bn ⊆ Bn+1 for every n ≥ 1 and also S⊆

∞

Bn .

n=1

Similarly as for class B, we have the following. Proposition 14.4.5 If f : C −→  C is a meromorphic function belonging to class Bn with some n ≥ 1, then, for every open connected set H ⊆ C\  Be 0,Diste (0,Sing(f −n )) in particular, H = C\Be 0,Diste (0,Sing(f −n )) and every connected component G of f −n (H ), the map f n : G → H is covering. For every R > 0, we define ∗ C : |z| > R} and B∞ (R) := {z ∈ C : |z| > R}. B∞ (R) := {z ∈ 

Aiming to prove Theorem 14.4.10 about nonexistence of Baker domains for functions from Speiser class S, we first need several auxiliary results which concern class B and its subclasses Bn . Our exposition here stems from Rippon and Stallard’s paper [RS]. For every R > 0, we define HR := {w ∈ C : Re(w) > log R}. We start dynamical results with the following. ˆ be a transcendental meromorphic Proposition 14.4.6 Let f : C −→ C function. If n ∈ N, R > 0, and C ∩ Sing(f −n ) ⊆ B(0,R), then, for every ∗ (R)), there exists a holomorphic function connected component V of f −n (B∞ h : HR −→ V such that h(HR ) = V

(14.16)

f n ◦ h = exp ,

(14.17)

and

80

Part III Topological Dynamics of Meromorphic Functions

i.e., the following diagram commutes: h

HR exp

V fn

(14.18)

∗ (R) B∞

Furthermore, the connected component Vˆ of f −n (B∞ (R)) containing V is simply connected. ∗ (R)∩Sing(f −n ) ⊆ {∞}, the last assertion of this proposition Proof Since B∞ ∗ (R) directly follows from Theorem 13.3.51. Since the map f n |V : V −→ B∞ is, by Theorem 13.3.48, covering, the existence of a continuous map h making the diagram (14.18) commute follows from the fact that the space HR is simply connected. Finally, the map h : HR −→ V is holomorphic because both maps ∗ (R) and f n | : V −→ B ∗ (R) are holomorphic and locally exp : HR −→ B∞ V ∞ invertible. The proof of Proposition 14.4.6 is complete. 

As an immediate consequence of this proposition, we get the following interesting corollary that is useful on its own. ˆ is a meromorphic function in class B, Corollary 14.4.7 If f : C −→ C then, for every R > Diste (0,Sing(f −1 )), each connected component of f −1 (B∞ (R)) in C is simply connected. Lemma 14.4.8 If f : C −→  C is a transcendental meromorphic function, then there exists Rf such that if R > Rf , n ≥ 1, Sing(f −n ) ⊆ B(0,R), and |z|,|f n (z)| > R 2, then |(f n ) (z)| ≥

|f n (z)| log |f n (z)| . 16π|z|

Proof First fix ξ ∈ J (f ), a periodic point of f , and take Rf ≥ 0 so large that |f n (ξ )| < Rf for each n ∈ N. Now take any R > Rf and any z ∈ C. Suppose that Sing(f −n ) ⊆ B(0,R) and |z|,|f n (z)| > R 2 . Let V be a connected component of f −n (B∞ (R)). Then ξ ∈ / V and so, with the notation of Proposition 14.4.6, the holomorphic function h−ξ : HR −→ V

14 Finer Properties of Fatou Components

81

nowhere vanishes. Since the open connected set HR is also simply connected, there, thus, exists (in fact, there exist countably infinitely many) G := log(h − ξ ) : HR −→ C, a holomorphic branch of logarithm of h − ξ . Fix z ∈ V . By Proposition 14.4.6, there exists w ∈ HR such that z = h(w). We then have, also by this lemma, f n (z) = f n (h(w)) = ew .

(14.19)

Hence, (f n ) (z)h (w) = (f n ) (h(w))h (w) = ew = f n (z). But, by the definition of G, G (w) =

h (w) h (w) = . h(w) − ξ z−ξ

So, (f n ) (z)G (w)(z − ξ ) = f n (z). Hence, |f n (z)| = |(f n ) (z)| · |G (w)| · |z − ξ |.

(14.20)

Now since G(HR ) does not contain any disk of radius greater than π , it follows from Theorem A.0.12 (Bloch’s Theorem) and (A.1) that |G (w)| ≤

4π 4π = . Re(w) − log R log |f n (z)| − log R

Inserting this into (14.20), we get that |f n (z)| ≤

4π|(f n ) (z)| · |z − ξ | . log |f n (z)| − log R

Equivalently, |(f n ) (z)| ≥

|f n (z)|(log |f n (z)| − log R) . 4π |z − ξ |

(14.21)

But |z − ξ | ≤ |z| + |ξ | ≤ |z| + Rf ≤ |z| + |ξ | ≤ |z| + R ≤ 2|z| and log |f n (z)| − log R ≥ 12 log |f n (z)| since log |f n (z)| > 2 log R. Since also, by (A.1), β > 1/4, we, thus, get that |(f n ) (z)| ≥ The proof is complete.

|f n (z)| log |f n (z)| . 16π|z| 

82

Part III Topological Dynamics of Meromorphic Functions

Rearranging terms in (14.21), we get that     f n (z) 4π    (z − ξ )(f n ) (z)  ≤ log |f n (z)| − log R .

(14.22)

Lemma 14.4.9 If n ≥ 1 and f ∈ Bn , then there is no connected component U of the Fatou set F (f ) such that f mn |U → ∞ uniformly on compact subsets of U as m → ∞. Proof Since f ∈ Bn , there exists R > max(e64,Rf ) with Sing(f −n ) ⊆ B(0,R), where Rf > 0 comes from Lemma 14.4.8. Seeking contradiction, suppose that the Fatou set F (f ) of f has a connected component U such that f mn |U → ∞ uniformly on compact subsets of U as m → ∞. Hence, there exist v ∈ F (f ) and r > 0 such that   ∗ Um := f mn B(v,r) ⊆ B∞ (R 2 )

(14.23)

n for all integers m ≥ 0. Fix an integer m ≥ 0.  Since f (Um ) = Um+1 ⊆ ∗ 2 ∗ −n ∗ B∞ (R) . Then let Vm be the B∞ (R ) ⊆ B∞ (R), we have that Um ⊆ f  ∗  −n connected component of f B∞ (R) containing Um and Vˆm be the con−n B∞ (R) containing Vm . Let ξ be the same periodic nected component of f point as in the proof of Lemma 14.4.8. Denote by Tξ : C → C the translation / Tξ (Vˆm ) and since the open given by the formula Tξ (z) = z − ξ . Since 0 ∈ ˆ connected set Vm is, by Proposition 14.4.6, simply connected, there exists

logm (Tξ ) : Vm → C, a holomorphic branch of the logarithm of Tξ . Now put Gm := logm (Tξ )(Um ). We have that     f n exp(Gm ) + ξ = f n Tξ (Um ) + ξ = f n (Um ) = Um+1 ⊆ Vˆm+1 . Therefore, the composition Fm+1 := logm+1 (Tξ ) ◦ f n ◦ (exp +ξ ) : Gm −→ C is well defined and Fm+1 (Gm ) = logm+1 (Tξ )(Um+1 ) = Gm+1 . It, therefore, follows from (14.22) and (14.23) that if w ∈ Gm , then, for z = ξ + ew ∈ Um , we get that

14 Finer Properties of Fatou Components

83

 n        (f ) (ξ + ew )ew   (f n ) (z)(z − c)   (f n ) (z)(z − c)   = ≥  (w)| =  n |Fm+1  f (ξ + ew ) − ξ   f n (z) − ξ   2f n (z) log R 1 (log |f n (z)| − log R) ≥ ≥ 8π 8π ≥ 2. Hence, using the Chain Rule, we obtain that |(Fm ◦ · · · ◦ F1 ) (w)| ≥ 2m for all w ∈ U0 and all integers m ≥ 1. Thus, by Bloch’s Theorem, each set Gm = Fm ◦ · · · ◦ F1 (G0 ) contains a disk of radius const · 2m diverging to infinity as m → ∞. This is, however, a contradiction since the set Gm = logm (Tξ )(Um ) is contained in a horizontal strip of height 2π; thus, contains no disk of radius greater than π . We are done.  ∞ Since S ⊆ n=1 Bn , as an immediate consequence of Lemma 14.4.9 and Theorems 13.2.5 and 13.2.8(c), we get the following remarkable result. Theorem 14.4.10 No functions in Speiser class S have Baker domains. As an immediate consequence of Corollary 14.2.2 and Theorem 14.4.10, we get the following. ˆ be a meromorphic function in Speiser Corollary 14.4.11 Let f : C −→ C class S. If W is a periodic connected component of the Fatou set F (f ) of f which is not a Herman ring and each connected component of W∞ :=

∞ 

f −n (W )

n=0

contains at most one singular value of f, then each of these components is simply connected. Since the Julia set is, by definition, the complement of the Fatou set, as an immediate consequence of this corollary along with Theorems 14.4.4 and 13.2.5, we get the following. ˆ be a meromorphic function in Speiser Theorem 14.4.12 Let f : C −→ C class S. If f has no Herman rings and each connected component of the Fatou set F (f ) of f contains at most one critical value of f , then each connected component of F (f ) is simply connected, and (therefore) the Julia set J (f ) of f is connected. In subsequent sections, we will be dealing with ω-limit sets and α-limit sets of points. For a given z ∈ C, the ω-limit set ω(z) is the set of

84

Part III Topological Dynamics of Meromorphic Functions

accumulation points of O + (z) in  C. Analogously, the α-limit set α(z) is the C. We end this section with a concept set of accumulation points of O − (z) in  which will play an important role in the part of the book devoted to elliptic functions, particularly to the Hausdorff dimension of their Julia sets, ⎧ ⎫ ⎨ ⎬  I∞ (f ) := z ∈ C : z ∈ f −n (∞) or lim f n (z) = ∞ . (14.24) n→∞ ⎩ ⎭ n≥0

The set I∞ (f ) is called the set of points escaping to ∞ under iterates of f , or just the escaping set of f . As an immediate consequence of Theorems 14.4.4, 14.4.10, and 13.2.5, we get the following. Theorem 14.4.13 If f ∈ S, then set I∞ (f ) ⊆ J (f ). Although, in this book, we are primarily interested in transcendental meromorphic functions that do have poles, even infinitely many of them, we would, however, like to bring up here the following theorem, which is very close to the spirit of the current section and whose proof uses Proposition 14.4.6. Theorem 14.4.14 If f : C → C is a transcendental entire function in class B, then the Julia set J (f ) of f is connected. Equivalently, all connected components of the Fatou set F (f ) are simply connected. This theorem has been formulated in [EL2] as Proposition 3 and derived as a consequence of Baker’s results from [Ba4] and [Ba3].

15 Rationally Indifferent Periodic Points

In this chapter, we provide a very detailed qualitative and quantitative description of the local behavior of iterates of locally and globally defined analytic functions around their rationally indifferent periodic points. We also examine the structure of the corresponding Leau–Fatou flower petals, including the Fatou Flower Petal Theorem. These will be frequently used in further chapters of the book that are devoted to the study of compactly nonrecurrent parabolic elliptic functions.

15.1 Local and Asymptotic Behavior of Analytic Functions Locally Defined Around Rationally Indifferent Fixed Points 15.1.1 Quantitative Local Behavior In this section, we want to bring up some basic results about the local behavior of meromorphic functions, about their parabolic (rationally indifferent) fixed points, or, somewhat more generally, about parabolic periodic points. As a matter of fact, our analysis is so local that all that we will be assuming throughout this section is that our given analytic map ϕ is defined on some neighborhood of its parabolic fixed point. In accordance with Definition 13.1.12, we call a holomorphic map ϕ, defined around a point ω ∈ C, a locally holomorphic map parabolic at ω (or rationally indifferent at ω) if and only if ω is a periodic point of ϕ, meaning that ϕ p (ω) = ω for some p ≥ 1, (ϕ p ) (ω) is a root of unity, and no iterate of ϕ is equal to the identity map. We then also call ω parabolic or rationally indifferent. We call the locally holomorphic parabolic map ϕ, or the point ω, simple parabolic if and only if 85

86

Part III Topological Dynamics of Meromorphic Functions

(a) ϕ(ω) = ω, (b) ϕ  (ω) = 1, and (c) ϕ is not the identity map. Note that some sufficiently high iterate of any locally holomorphic parabolic map is simple. Therefore, in order to analyze the behavior of locally holomorphic parabolic maps, it essentially suffices to do this for simple parabolic maps only. Thus, throughout this section, the map ϕ (and its fixed point ω) is always assumed to be locally holomorphic simple parabolic. Then, on a sufficiently small neighborhood of ω, the map φ has the following Taylor series expansion: ϕ(z) = z + a(z − ω)p+1 + b(z − ω)p+2 + · · ·

(15.1)

with some integer p = p(ω) ≥ 1 and a ∈ C\{0}. Being in the circle of ideas related to the Fatou Flower Theorem (see [Al] for extended historical information), we now want to analyze qualitatively and especially quantitatively the behavior of ϕ in a sufficiently small neighborhood of the parabolic point ω. Let us recall that the rays coming out from ω and forming the set {z ∈ C : a(z − ω)p < 0} are called attracting directions and the rays forming the set {z ∈ C : a(z − ω)p > 0} are called repelling directions. Fix an attracting direction, say  p A := ω + −a −1 (0,∞), √ where p · is a holomorphic branch of the pth radical defined on C\a −1 (0,∞). In order to simplify our analysis, let us change the system of coordinates with the help of the affine map  p ρA (z) = −(ap)−1 z + ω. In other words, we define ϕA,0 := ρA−1 ◦ ϕ ◦ ρA . We then get that ϕA,0 (z) = ρA−1 ◦ ϕ ◦ ρA (z) = z −

∞

1 p+1  + an zp+n z p n=2

(15.2)

15 Rationally Indifferent Periodic Points

and

87

  ϕA,0 (0) = 0, ϕA,0 (0) = 1,

i.e., 0 is a simple parabolic fixed point of ϕA,0 . In addition, ρ −1 (A) = (0,∞) is an attracting direction for ϕA,0 . We first want to analyze the behavior of ϕA,0 on sufficiently small neighborhoods of 0. In order to do this, similarly to the previous section, we conjugate ϕA,0 on C\(−∞,0] to a map defined “near” infinity. More precisely, √ we consider p ·, to be the holomorphic branch of the pth radical defined on C\(−∞,0] and leaving the point 1 fixed. We will also frequently denote this branch by z1/p . Then we define the map 1 −1 H (z) := √ = z p. p z This map has a meromorphic inverse H −1 :  C −→  C, which is given by the formula H −1 (z) = z−p . Consider the conjugate map ϕ˜A := H −1 ◦ ϕA,0 ◦ H = (ρA ◦ H )−1 ◦ ϕ ◦ (ρA ◦ H ),

(15.3)

defined on Vϕ \(−∞,0], where Vϕ ⊆ C is a sufficiently small neighborhood of 0 on which ϕ0 is defined. We shall prove the following technical but very useful result. Lemma 15.1.1 If ϕ is a locally holomorphic simple parabolic map and A is an attracting direction of ϕ, then, shrinking the neighborhood Vϕ if necessary, function B : Vϕ −→ C such that, for  there exists  a holomorphic −1 all z ∈ C\(−∞,0] ∩ H (Vϕ ), we have that ϕ˜A (z) = z + 1 + B(H (z)) and  ϕ˜A (z) = 1 + (B ◦ H ) (z)

with − p1

|B(H (z))| ≤ M|H (z)| = M|z| and

|(B ◦ H ) (z)| = |B  (H (z))| · |H  (z)| ≤ Mp −1 |z| with some appropriate constant M ∈ (0,+∞).

− p+1 p

88

Part III Topological Dynamics of Meromorphic Functions

Proof We have that H −1 (z) = z−p . For all z ∈ C\(−∞,0] ∩ H −1 (Vϕ ), we then get that 

∞    1 an H (z)p+n ϕ˜A (z) = H −1 ϕA,0 (H (z)) = H −1 H (z) − H (z)p+1 + p n=2 

∞  p+1 p+n 1 1 − − − an z p = H −1 z p − z p + p n=2  

∞ 1 −1  − p1 − p+n−1 −1 p an z =H z 1− z + p n=2 

−p ∞ 1 −1  − p+n−1 p an z . (15.4) =z 1− z + p n=2

Set w := H (z) = z

− p1

and put ∞

G(w) = 1 −

1 p  an w p+n−1, w ∈ Vϕ . w + p

(15.5)

n=2

Now G(0) = 1,

d k G(w)   = 0 for all k = 1,2, . . . ,p − 1, and dw k 0 d p G(w)   = −(p − 1)! . dw p 0

Therefore, G−p (0) = 1,

d k (G−p )(w)   = 0 for all k = 1,2, . . . ,p − 1, and 0 dw k d p (G−p )(w)   = p! 0 dw p

and G−p (w) = 1 + wp +

∞ 

bn w p+n

n=1

with some appropriate coefficients bn , n ≥ 1, where the series ∞  n=1

bn w p+n

(15.6)

15 Rationally Indifferent Periodic Points

89

converges uniformly absolutely on some sufficiently small neighborhood of 0; shrinking Vϕ , if needed, we may indentify this neighborhood with Vϕ . Then the series ∞  B(w) := bn w n n=1

also converges uniformly absolutely on V and represents a holomorphic function. Shrinking V even more if needed, we will have a constant M ∈ (0,+∞) such that |B(w)| ≤ M|w| and |B  (w)| ≤ M

(15.7)

Vϕ . Going back to the variable C\(−∞,0] ∩ H −1 (Vϕ )

for all w ∈ we get from (15.4)–(15.6) that −p

ϕ˜A (z) = zG



(H (z)) = z 1 + H (z) + p



∞

1 1 bn H (z)n =z 1+ + z z



∞ 

 z = w −p ,

bn (z)H (z)

p+n

n=1

(15.8)

n=1

= z + 1 + B(H (z)). It also follows from (15.7) that − p1

|B(H (z))| ≤ M|H (z)| = M|z|

(15.9)

and |(B ◦ H ) (z)| = |B  (H (z))| · |H  (z)| ≤ Mp−1 |z|

− p+1 p

.

(15.10) 

The proof is complete. Given now a point x ∈ (0,∞) and an angle α ∈ (0,π), let S(x,α) := {z ∈ C \ {x} : − α < arg(z − x) < α}.

By Lemma 15.1.1, for all κ ∈ (0,1) and all α ∈ (0,π), there exists x(α,κ) ∈ (0,+∞) such that, for every x ≥ x(α,κ), we have that ϕ˜A (S(x,α)) ⊆ S(x + 1 − κ,α) ⊆ S(x,α)

(15.11)

Re(ϕ˜A (z)) ≥ Re(z) + 1 − κ

(15.12)

and

for all z ∈ S(x,α). Then, by an immediate induction, n (z)) ≥ Re(z) + (1 − κ)n Re(ϕ˜A

(15.13)

90

Part III Topological Dynamics of Meromorphic Functions

for all z ∈ S(x,α) and all integers n ≥ 0. Summarizing, we have the following two statements. Proposition 15.1.2 Let ϕ be a locally holomorphic simple parabolic map and A be an attracting direction of ϕ. Fix α ∈ (0,π), κ ∈ (0,1), and x ≥ x(α,κ). n : S(x,α) −→ C, n ≥ 0, are well defined and Then all the iterates ϕ˜A n ϕ˜A (S(x,α)) ⊆ S(x + (1 − κ)n,α) ⊆ S(x,α). n (z) −−−−→ ∞ uniformly on S(x,α). In particular, ϕ˜A n→∞

Lemma 15.1.3 If ϕ is a locally holomorphic simple parabolic map and A is an attracting direction of ϕ, then, for all α ∈ (0,π) and κ > 0, we have that n n (z)| ≥ Re(ϕ˜A (z)) ≥ Rez + (1 − κ)n |ϕ˜A

for all n ≥ 0 and all z ∈ S(x(α,κ),α). We shall prove the following. Lemma 15.1.4 If ϕ is a locally holomorphic simple parabolic map and A is an attracting direction of ϕ, then, for all α ∈ (0,π), κ ∈ (0,1), n ≥ 1, and z ∈ S(x(α,κ),α), we have that   1− 1 n ϕ˜A (z) = z + n + ORe(z) max{n p , log(n + 1)} , where Ot is a big O symbol depending decreasingly on t ∈ R and converging to 0 when t −→ +∞. Proof Fix z ∈ S(x(α,κ),α). By Lemmas 15.1.1 and 15.1.3, we have, for every n ≥ 1, that  1 − n+1 n (z) = ϕ˜A (z) + 1 + ORe(z) n p . ϕ˜A Therefore, by induction, n (z) ϕ˜ A

= z + n + ORe(z)

n−1 

k

− p1

k=1

= z + n + ORe(z) (max{n

1− p1

, log(n + 1)}).

The proof is complete.



As an immediate consequence of this lemma, we get the following. Lemma 15.1.5 If ϕ is a locally holomorphic simple parabolic map and A is an attracting direction of ϕ, then, for all α ∈ (0,π), κ ∈ (0,1), and

15 Rationally Indifferent Periodic Points

91

t ∈ R, there exists a constant C = C(α,κ,t) such that, for all n ≥ 1 and all z ∈ S(x(α,κ),α) with Re(z) ≥ t, we have that n C −1 n ≤ |ϕ˜ A (z)| ≤ Cn.

For every R > 0, let S(x,α,R) := S(x,α) ∩ B(0,R). We shall prove the following. Lemma 15.1.6 Let ϕ be a locally holomorphic simple parabolic map and let A be an attracting direction of ϕ. Fix α ∈ (0,π), κ ∈ (0,1), and R > 0. Then

n   0 < inf |(ϕ˜A ) (z)| : z ∈ S(x(α,κ),α,R), n ≥ 1 n  ) (z)| : z ∈ S(x(α,κ),α,R), n ≥ 1} < +∞. ≤ sup{|(ϕ˜A

Furthermore, for every γ > 1,

n   ) (z) − 1| : z ∈ S(x,α,γ x), n ≥ 1 = 0 lim sup |(ϕ˜ A x→+∞

and

n   ) (z) − 1| : z ∈ S(x,π/2), n ≥ 1 = 0. lim sup |(ϕ˜ A

x→+∞

Proof For every z ∈ S(x(α,κ),α), let  g(z) := ϕ˜A (z) − 1.

By the Chain Rule, we have, for every n ≥ 1, that n  (ϕ˜A ) (z) =

n−1 

 ϕ˜A (ϕ˜A (z)) = j

j =0

n−1 

j

(1 + g(ϕ˜ A (z))).

j =0

But by Lemmas 15.1.1 and 15.1.3, we have that j

j

|g(ϕ˜ A (z))|  |ϕ˜A (z)|

− p+1 p

.

For every x > 0, let r = r(x,α) be the radius of the maximal ball centered at 0 which is disjoint from S(x,α). Of course, r(x,α) ≤ ux with some constant u ∈ (0,+∞) depending only on α. Let kR ≥ 0 be the least integer such that −R + (1 − κ)(kR + 1) ≥ R.

92

Part III Topological Dynamics of Meromorphic Functions

Then −R + (1 − κ)kR < R; therefore, 2R . 1−κ

kR
0.  (z) ∈ C\{0} is holo(3) Therefore, the function S(x(α,κ),α)  z −→ ϕ˜A,∞ morphic. (4) lim

z∈S(x(α,κ),α) Rez→+∞

 ϕ˜A,∞ (z) = 1.

(5) So, if, in addition, α ∈ (0,π/2), then lim

S(x(α,κ),α)z→∞

 ϕ˜A,∞ (z) = 1.

Proof Fix an arbitrary ε > 0. By Lemma 15.1.6, there exists x ≥ 0 such that   n  (ϕ˜ ) (ξ ) − 1 < ε (15.14) A for all ξ ∈ S(x,π/2) and all integers n ≥ 0. By Lemma 15.1.3, there exists k ≥ 0 so large that φ˜ n (z) ≥ x, i.e., φ˜ n (z) ∈ S(x,π/2), for all z ∈ S(x(α,κ),α) ∩ {w ∈ C : Re(w) ≥ t} and all n ≥ k. Then, for every i ≥ k and every j ≥ 0, we have that  i+j   (ϕ˜ ) (z)     A  j − 1 = (ϕ˜A ) (φ˜ i (z)) − 1 < ε.  i   (ϕ˜A ) (z) 

94

Part III Topological Dynamics of Meromorphic Functions

 n  ∞ Therefore, the sequence (ϕ˜A ) (z) n=0 is uniformly quotient Cauchy on S(x(α,κ),α) ∩ {w ∈ C : Re(w) ≥ t} and the first assertion of our lemma is proved. Assertions (2) and (3) then follow immediately, while (4) is a direct consequence of (15.14) and (5) is a direct consequence of (4).  It immediately follows from this lemma that   ϕ˜A,∞ (z) = ϕ˜A,∞ (ϕ(z)) ˜ φ˜  (z)

(15.15)

for all z ∈ S(x(α,κ),α). In particular, the holomorphic function S(x(α,κ),α)   (z) and its modulus are not constant. z −→ ϕ˜∞ For every x ∈ (0,∞), α ∈ (0,π), and R > 0, let

and

S0 (x,α) := H (S(x,α)),

(15.16)

SϕA (x,α) := ρA ◦ H (S(x,α)) = ρA (S0 (x,α)),

(15.17)

  S0 (x,α,R) := S0 (x,α) ∩ B c (0,R) = H S(x,α,R −p ) ,

(15.18)

while

  SϕA (x,α,R) := SϕA (x,α) ∩ B c (ω,R) = ρA ◦ H S(x,α,|a|−1 R −p ) . (15.19)

The regions S0 (x,α) and SϕA (x,α) look like flower petals that are, respectively,  symmetric about the rays (0,∞) and A = ω + p −a −1 (0,∞), that contain initial segments of these rays, and that form with them two “angles” of measure α/p at the points 0 and ω, respectively. The following proposition is an immediate consequence of Proposition 15.1.2. Proposition 15.1.8 Let ϕ be a locally holomorphic simple parabolic map at ω ∈ C and A be an attracting direction of ϕ. Fix α ∈ (0,π) and κ ∈ (0,1). Then, for every x ≥ x(α,κ), all iterates ϕ n : SϕA (x,α) −→ C, n ≥ 0, are well defined, ϕ n (SϕA (x,α)) ⊆ SϕA (x,α),

∞  and the sequence ϕ n (z) n=0 converges to ω uniformly on SϕA (x,α). A direct elementary calculation based on (15.1) yields this. Lemma 15.1.9 If ϕ is a locally holomorphic simple parabolic map at ω ∈ C π and A is an attracting direction of ϕ, then, for every 0 ≤ α < 2p(ω) , there exists Rα (ω) > 0 such that

15 Rationally Indifferent Periodic Points

95

|ϕ(z) − ω| < |z − ω| and |ϕ  (z)| < 1   for all z ∈ SφA (x,α) ∩ B(ω,Rα (ω)) . We shall now prove the following. Proposition 15.1.10 Let ϕ be a locally holomorphic simple parabolic map at ω ∈ C and A be an attracting direction of ϕ. Fix α ∈ (0,π) and κ ∈ (0,1). Recall that p = p(ω). For all z ∈ SϕA (x(α,κ),α), we have that  p+1  −1 lim n p |ϕ n+1 (z) − ϕ n (z)| = |a| p

n→∞

and  1  −1 lim n p |ϕ n (z) − ω| = |a| p .

n→∞

In addition, in the two limits above, the convergence is uniform on the set SϕA (x(α,κ),α,R) for every R > 0. Proof We have by Lemma 15.1.4, for all ξ ∈ S0 (x(α,κ),α,R), that n |ϕA,0 (ξ )| = |H ◦ ϕ˜ n ◦ H −1 (ξ )| − 1   p  = ϕ˜ n (H −1 (ξ ))

  − p1  1− 1 = H −1 (ξ ) + n + O−R −p max{n p , log(n + 1)}  . Therefore, as |H −1 (ξ )| ≤ R −p , we get that  1  n lim n p |ϕA,0 (ξ )| = lim n−1 H −1 (ξ ) + 1

n→∞

n→∞

 − p1 −1 =1 + O−R −p max{n p ,n−1 log(n + 1)} 

and the convergence is uniform on ξ ∈ S0 (x(α,κ),α,R). Since, for all points z ∈ SϕA (x(α,κ),α,R), we have that n ϕ n (z) − ω = ρ(ϕA,0 (ρ −1 (z))) − ω =

 p

n −a −1 ϕA,0 (ρ −1 (z)),

the second formula of our proposition follows along with the appropriate uniform convergence. Turning our attention to the first formula, we have by (15.1), by the first formula of the proposition, and by the last assertion of Proposition 15.1.10 that

96

Part III Topological Dynamics of Meromorphic Functions

lim n

n→∞

p+1 p

|ϕ n+1 (z) − ϕ n (z)|

= lim n



= |a| lim

n→∞

= |a| lim



n→∞

= |a||a|

  a(ϕ n (z) − ω)p+1 + O(|ϕ n (z) − ω|p+2 )

p+1 p

n→∞

1

n p |ϕ n (z) − ω| 1 p

n |ϕ (z) − ω|

− p+1 p

n

= |a|

− p1

p+1 p+1

 1  + lim O (n p |ϕ n (z)−ω|)p+1 |ϕ n (z)−ω| n→∞

,

where the big O symbol represents an absolute constant depending only on the Taylor series expansion (15.1) of φ; moreover, the convergence is uniform on ξ ∈ S0 (x(α,κ),α,R). The proof is complete.  For every z ∈ SϕA (x(α,κ),α), put   (z) := ϕ˜A,∞ (ρ ◦ H −1 (z)). ϕa,∞

(15.20)

Our last result about the iterates of ϕ is this. Proposition 15.1.11 Let ϕ be a locally holomorphic simple parabolic map at ω ∈ C and A be an attracting direction of ϕ. Fix α ∈ (0,π) and κ ∈ (0,1). Recall that p = p(ω). Then, for every z ∈ SϕA (x(α,κ),α), lim n

p+1 p

n→+∞

|(ϕ n ) (z)| = |a|

− p+1 p

 |z − ω|−(p+1) |ϕA,∞ (z)|

(15.21)

and the convergence is uniform on SϕA (x(α,κ),α,R) for every R > 0. In addition, if α ∈ (0,π/2), then lim

SϕA (x(α,κ),α)z→ω

Proof

 ϕA,∞ (z) = 1.

(15.22)

By virtue of Lemma 15.1.4, we get, for all z ∈ S0 (x(α,κ),α), that |z|p+1 n

p+1 p

n |(ϕA,0 ) (z)|

p+1 p

n n  |H  (ϕ˜A (H −1 (z)))(ϕ˜A ) (H −1 (z))(H −1 ) (z)| p+1 1 − p+1 n n  = |z|p+1 n p |ϕ˜ A (H −1 (z))| p |(ϕ˜A ) (H −1 (z))|p|z|−(p+1) p

=n

=n

p+1 p

n |ϕ˜A (H −1 (z))|

− p+1 p

n  |(ϕ˜ A ) (H −1 (z))|

15 Rationally Indifferent Periodic Points

=n

p+1 p

97

  n  |(ϕ˜ A ) (H −1 (z))|H −1 (z) + n

 − p+1 p 1− 1 + ORe(H −1 (z)) max{n p , log(n + 1)}   −1  H (z) n  −1 = |(ϕ˜A ) (H (z))|  +1 n  p+1  − p − p1 −1 . +ORe(z−p ) max{n ,n log(n + 1)} 

(15.23)

So, if z ∈ SϕA (x(α,κ),α,R), then (15.23) takes on the form  −1 p+1  H (z) p+1 p n  n  −1 n |(ϕA,0 ) (z)| = |(ϕ˜A ) (H (z))|  |z| n 

− p1

+1 + O−R −p max{n

−1

,n

 p+1  − p log(n + 1)}  .

Therefore, n

p+1 p

|(ϕ n ) (z)| = n

p+1 p

n |(ϕA,0 ) (ρ −1 (z))|   = |ρ −1 (z)|−(p+1) (ϕ˜ n ) (ρ ◦ H )−1 (z) A

 p+1   (ρ ◦ H )−1 (z)   − p − p1 −1  + 1 + O−R −p max{n ,n log(n + 1)}  × n  n   − p+1 = |a| p |z − ω|−p+1 (ϕ˜A ) (ρ ◦ H )−1 (z)  p+1   (ρ ◦ H )−1 (z)   − p − p1 −1  . + 1 + O−R −p max{n ,n log(n + 1)}  × n (15.24) Since also

    (ρ ◦ H )−1 SϕA (x,α,R) = S x,α,(|a|R)−1/p ,

the proof is, thus, concluded by applying Lemma 15.1.7. We are done.



15.1.2 Fatou Coordinates The next theorem establishes the existence of functions called Fatou coordinates around simple parabolic fixed points. It is primarily used in this book in the description of basins of attractions to parabolic points. This theorem is, however, of local character and we formulate and prove it here. For every α ∈ (0,π), put

98

Part III Topological Dynamics of Meromorphic Functions Qα := S(x(α,1/2),α) and Qα,t := Qα ∩ {z ∈ C : Re(z) ≥ t}.

Theorem 15.1.12 Let ϕ be a locally holomorphic simple parabolic map at ω ∈ C and A be an attracting direction of ϕ. Then, for every α ∈ (0,π), there exists a unique, up to an additive constant, holomorphic function !α : Qα −→ C such that F !α + 1 !α ◦ ! ϕA = F F

(15.25)

!α (z) F =1 Qα,t z→+∞ z

(15.26)

and lim

for every t ∈ R. Furthermore, replacing x(α,1/2) by a sufficiently large real number and keeping the same symbol x(α,1/2) for that number, the map !α : Qα −→ C F is univalent. !α , α ∈ (0,π), are called the Fatou coordinates of the The functions F parabolic map ! ϕA . Proof First of all, notice that, because of Proposition 15.1.2 and (15.12), we have that ! ϕA (Qα ) ⊆ Qα and ! ϕA (Qα,t ) ⊆ Qα,t

(15.27)

n , n ≥ 0, of ! ϕA are well defined on Qα for all t ∈ R. In particular, all iterates ! ϕA !α : Qα → C and Qα,t . For uniqueness, suppose that a holomorphic function F satisfies (15.25) and (15.26). Fix an arbitrary ε > 0. In view of the definition of Qα and (15.26), there exist β ∈ (0,α) small enough and s > max{2,x(α,1/2)} large enough such that if z ∈ Qβ,s , then   !α (z) F  ε  B(z,|z|/2) ⊆ Qα and  − 1 < . z 3

The Cauchy Integral Formula then yields   1 ! |Fα (z) − 1| =  2π i ∂B(z,|z|/2)   1 =  2π i

 !α (ξ )  F dξ − 1 (ξ − z)2  !α (ξ ) − ξ  F  dξ  2 ∂B(z,|z|/2) (ξ − z) !α (ξ ) − ξ | |F 1 |dξ | ≤ 2π ∂B(z,|z|/2) |ξ − z|2

15 Rationally Indifferent Periodic Points

99

2 !α (ξ ) − ξ | |dξ | |F π|z|2 ∂B(z,|z|/2) 2 ε ≤ |ξ | |dξ | 2 π |z| ∂B(z,|z|/2) 3 2 ε 3|z| ≤ π |z| = ε. 2 3 2 π |z|

=

(15.28)

Now also fix w ∈ Qβ,s . Using (15.25), we obtain that n n !α (w) = F !α (! !α (! !α (z) − F ϕA (z)) − F ϕA (w)) F

= n

!α (ξ )dξ = F

n

n n =! ϕA (z) − ! ϕA (w) +

!α (ξ )))dξ (1 + (1 − F

n

!α (ξ ))dξ, (1 − F

n (w) to ! n (z). So, applying (15.28), where n is the oriented segment from ! ϕA ϕA it follows that       n n  F   !α (w) − (! ! !α (z) − F ϕA (z) − ! ϕA (w)) ≤  (1 − Fα (ξ ))dξ  n

   F !α (ξ ) − 1 |dξ |

≤ n

≤ ε |n |   n ϕ n (w) . = ε ! ϕ (z) − ! A

Thus,

A

  ! !  F  α (z) − Fα (w) − 1 ≤ ε. n n  ! ϕA (z) − ! ϕA (w)

n (z) − ! n (w))∞ exists and ϕA Hence, the limit of the sequence (! ϕA n=0 n n !α (w) = lim (! !α (z) − F ϕA (z) − ! ϕA (w)). F n→∞

The uniqueness part is, thus, established. This argument also gives us a hint as to how to prove the existence part. Keep t ∈ R fixed and also fix w ∈ Qβ,s as above. Consider the functions n n ϕA (z) − ! ϕA (w), n ≥ 0. Qα  z −→ ψn (z) := !

(15.29)

We then have that n n n ϕA (! ϕA (z)) −(! ϕA (z) + 1)) −(! ϕA (! ϕ n (w)) −(! ϕA (w) +1)). ψn+1 (z) −ψn (z) = (!

100

Part III Topological Dynamics of Meromorphic Functions

So, making use of the Mean Value Inequality, by virtue of Lemma 15.1.1, (15.27), and (15.13), we obtain that |ψn+1 (z) − ψn (z)|   n n n n ϕA (z)) − (! ϕA (z) + 1)) − (! ϕA (! ϕA (w)) − (! ϕA (w) + 1) = ! ϕA (!   n n (z))) − B(H (! ϕA (w))) = B(H (! ϕA   n n (z)) − B ◦ H (! ϕA (w)) = B ◦ H (! ϕA  n  − p+1 n ≤ Mp−1 sup{|ξ | p : ξ ∈ n } ! ϕA (z) − ! ϕA (w)  − p+1  n ϕ (z) − ! ϕ n (w) ≤ Ct n p ! A

A

− p+1 p

= Ct |ψn (z)| n

with some constant Ct ∈ (0,+∞) independent of z and n but depending (in general) on t. Equivalently,     ψn+1 (z) − p+1   p . (15.30)  ψ (z) − 1 ≤ Ct n n " − p+1 p converges, the estimate (15.30) implies that the Since the series ∞ n=1 n ∞ sequence (ψn )n=1 converges uniformly on Qα,t . So, the limit !α := lim ψn F n→∞

(15.31)

is a holomorphic function on Qα,t . Letting t → −∞, we conclude that (15.31) !α from Qα to C. Moreover, owing to (15.31) defines a holomorphic function F and Lemma 15.1.1 combined with (15.13), for all z ∈ Qα , we obtain the following equality:  n+1  n !α (! ϕ (z) − ! ϕA (w) ϕA (z)) = lim ψn (! ϕA (z)) = lim ! F n→∞ n→∞ A     n+1 n+1 n+1 n (z) − ! ϕA (w) + ! ϕA (w) − ! ϕA (w) = lim ! ϕA n→∞

!α (z) + 1. = lim ψn+1 (z) + 1 = F n→∞

This proves (15.25). In order to obtain (15.26), we must do a little bit more work. Keep again t ∈ R and z ∈ Qα,t fixed. Based on the first and third assertions of Lemma 15.1.1, we deduce by induction that  n  ! ϕA (z) − (z + n) ≤ Lt,n |z|−1/p,

(15.32)

15 Rationally Indifferent Periodic Points

101

where Ln,t > 0 is some constant independent of z ∈ Qα,t but that may depend on t and n. Let ε > 0. By (15.31) and (15.30), there exists k ∈ N so large that   ! F   α (z)/z − 1 < ε ∀ z ∈ Qα,t . (15.33)  ψ (z)/z  2 k But, by (15.32) and (15.29), we infer that      ! k k (ζ )   ψk (z) ϕA   ϕA (z) − !   − 1 = − 1 lim  lim   Qα,t z→∞ Qα,t z→∞  z z    ! k   ϕ (z) − 1 = lim  A  Qα,t z→∞  z  k  ! ϕA (z) − z = lim Qα,t z→∞ |z| ≤

k + Lt,k |z|−1/p Qα,t z→∞ |z| lim

= 0. So, limQα,t z→∞ (ψk (z)/z) = 1 and, hence, (15.33) yields   ! F   α (z) − 1 < ε  z  for all z ∈ Qα,t with sufficiently large moduli. Since ε > 0 is arbitrary, (15.26) follows. !α : Qα → C is univalent. As the main It only remains to show that F ingredient in the proof of this statement, we shall prove the following. Claim. The map ! ϕA |Qα : Qα −→ C is univalent. −1

Proof Assume that x(α,1/2) ≥ 1 is so large that M|z| p < 1/2 for every z ∈ Qα , where M > 0 is the constant in Lemma 15.1.1. In particular, observe that Qα is convex. Note also that if w ∈ Qα , then it follows from Lemma 15.1.1 that − p+1 p

 |! ϕA (w) − 1| ≤ M|w|

− p1

≤ M|w|


1/2. In particular, Re(! ϕA ϕA (ξ ) for some w,ξ ∈ Qα , then it follows from Lemma Now, if ! ϕA (w) = ! 15.1.1 that

|w − ξ | ≤ 2Mt

− p1

< 1.

(15.34)

102

Part III Topological Dynamics of Meromorphic Functions

So, it suffices to prove injectivity of ! ϕA on the convex set B(ξ,1) ∩ Qα for an arbitrary ξ ∈ Qα . Let w1,w2 ∈ B(ξ,1) ∩ Qα and denote by  the line segment from w1 to w2 . Then        !   ϕA (w2 ) − ! ϕA (w1 ) =  ! ϕA (w) dw      1     ! ϕA (w1 + t (w2 − w1 ))dt  = |w2 − w1 | ·   0  1

≥ |w2 − w1 | · Re 0 1

= |w2 − w1 | · 0

 ! ϕA (w1 + t (w2 − w1 ))dt

   Re ! ϕA (w1 + t (w2 − w1 )) dt

1 ≥ |w2 − w1 | · . 2 Therefore, the function ! ϕA is injective on B(ξ,1) ∩ Qα . This completes the proof of the claim.  !α is not In view of this claim, of (15.25), which implies that the function F ∞ constant, and of the uniform convergence of the sequence (ψn )n=1 on Qα,t for every t ∈ R, an application of Hurwitz’s Theorem (Theorem A.0.23) gives !α |Qα,t is also univalent for all t ∈ R. Hence, the function that the function F ! Fα : Qα −→ C is univalent too. This completes the proof of our theorem.  !α are called As was indicated in the formulation of this theorem, functions F the Fatou coordinates of the parabolic map ! ϕA . We shall now establish one !α : Qα −→ C. important property of the functions F Lemma 15.1.13 Let ϕ be a locally holomorphic simple parabolic map at ω ∈ C and A be an attracting direction of ϕ. Then, for every α ∈ [π/2,π) and every t ∈ R, there exists s := s(t) > 0 such that !α (Qα,t ) ⊇ {z ∈ C : Re(z) > s}. F Proof that

By virtue of (15.26), there exists s ≥ max{0,4(x(α,1/2) + t)} so large   ! F   α (z) − 1 ≤ 1  z  8

whenever Re(z) > s/4. Take now an arbitrary number w ∈ C with (w) > s. Observe that B(w,2|w|/3) ⊆ Qα,t .

(15.35)

15 Rationally Indifferent Periodic Points

103

If z ∈ ∂B(w,|w|/2), then

  !α (z)     F  (F      !α (z) − w) − (z − w) = F !α (z) − z = |z|  z − 1 |w| + |w|/2 3 |z| ≤ ≤ |w| ≤ 8 8 16 1 < |w|. 4

Hence,  |w|  ! > Fα (z) − w − (z − w). 2 !α (z) − w has the Thus, Rouch´e’s Theorem asserts that the function z → F same number of zeros in B(w,|w|/2) ⊆ Qα,t as the function z → z − w. But the latter function has exactly one zero; namely, z = w. Thus, there exists !α (z) = w. This means that z ∈ B(w,|w|/2) ⊆ Qα,t such that F |w − z| =

!α (Qα,t ) ⊇ {z ∈ C : Re(z) > s}. F 

The proof is complete. Fix α ∈ (0,π), κ = 1/2, and denote by p(ω)

Sa1 (ω,α), . . . ,Sa

(ω,α)

(15.36)

the corresponding attracting sectors for ϕ defined in (15.17) with x := x(α,1/2). Denote also by p(ω)

Sr1 (ω,α), . . . ,Sr

(ω,α)

(15.37)

the analogous attracting sectors for ϕ −1 . We will also frequently call them the repelling sectors for ϕ (at the parabolic point ω). As an immediate consequence of their definition, we get the following. Lemma 15.1.14 If ϕ is a locally holomorphic simple parabolic map at ω ∈ C, then, for every α ∈ (0,π), both collections $ # $ # p(ω) p(ω) Sa1 (ω,α), . . . ,Sa (ω,α) and Sr1 (ω,α), . . . ,Sr (ω,α) consist of mutually disjoint sets. Putting !α ◦ ρ ◦ H −1 Fα := F

(15.38)

and passing to the original parabolic function ϕ, as an immediate consequence of Proposition 15.1.2, Theorem 15.1.12, and Lemma 15.1.13, we obtain the following result.

104

Part III Topological Dynamics of Meromorphic Functions

Theorem 15.1.15 If ϕ is a locally holomorphic simple parabolic map at ω ∈ C, then, for every α ∈ (0,π), we have that j

j

ϕ(Sa (ω,α)) ⊂ Sa (ω,α)

(15.39)

for all j = 1, . . . ,p(ω) and Fα ◦ ϕ = Fα + 1

(15.40)

j

on Sa (ω,α). Also, (1) With appropriate normalizations (fixing the values of Fα at one point), if 0 < β,γ < π , then Fγ |S j (ω,β)∩S j (ω,γ ) = Fβ |S j (ω,β)∩S j (ω,α) . a

a

a

a

(2) Increasing the point x(α,1/2) ∈ R appropriately, we have that the map Fα |S j (ω,α) is univalent. a (3) If α ∈ [π/2,π), then there exists s > 0 such that j

Fα (Sa (ω,α)) ⊃ {z ∈ C : Re(z) > s}.

15.2 Leau–Fatou Flower Petals In this section, we continue our study of parabolic fixed and periodic points undertaken in the previous section. However, up to this point, our considerations have had entirely local character in the sense that the map f (denoted so far by ϕ) was defined only on some small neighborhood of its simple rationally indifferent fixed point ω. Throughout this section, f is instead global, i.e., f : C −→  C is a meromorphic function and has a simple rationally indifferent fixed point ω. We revisit, in this section, the corresponding Leau–Fatou Flower Petals and study their dynamical and topological structure in greater detail. It follows from Proposition 15.1.8 that, for every j ∈ {1, . . . ,p(ω)}, j Sa (ω,α) is a subset of the Fatou set F (f ) of f . We then define the set A∗j (ω) to be the connected component of the Fatou set F (f ) that contains j

Sa (ω,α) for some (equivalently, for every) α ∈ (0,π). In view of Proposition 15.1.8 and Theorem 13.2.5, A∗j (ω) is a Leau–Fatou domain of ω. Let δf > 0 be so small that there exists fω−1 : B(ω,δf ) −→ C,

15 Rationally Indifferent Periodic Points

105

a unique holomorphic inverse branch of f that sends ω to ω. We require, in addition, that the map f |B(ω,2δf ) j

is injective. Recall that the sets Sr (ω,α), j ∈ {0, . . . ,p(ω) − 1}, are the attracting sectors for the map fω−1 : B(ω,δf ) −→ C. Invoking Theorem 13.2.5 and Proposition 15.1.8 again, we obtain the following. Lemma 15.2.1 If f : C −→  C is a meromorphic function and ω is its simple rationally indifferent fixed point, then f (A∗j (ω)) ⊆ A∗j (ω) for every j ∈ {1, . . . ,p(ω)} and f n −−−−→ ω n→∞

uniformly on compact subsets of

p(ω)−1 j =0

A∗j (ω).

We further define Aj (ω) :=

∞ 

f −n (A∗j (ω)).

n=0

The open set Aj (ω) is called the basin of attraction to ω in the direction determined by j , and A∗j (ω) is called the immediate basin of attraction to ω in this direction. We shall now prove the following. Lemma 15.2.2 If f : C −→  C is a meromorphic function and ω is a simple parabolic fixed point of f , then, for every α ∈ (0,π), we have that ⎞ ⎛ p(ω) p(ω) ∞    j A∗ (ω) := A∗j (ω) ⊆ f −n ⎝ Sa (ω,α)⎠ . j =1

j =1

n=0

Proof Suppose, by way of contradiction, that this inclusion does not hold. This means that there exist some i ∈ {1, . . . ,p(ω)} and some ⎞ ⎛ p(ω) ∞   j f −n ⎝ Sa (ω,α)⎠ . (15.41) z ∈ A∗i (ω)\ n=0

j =1

Since f (A∗j (ω)) = A∗j (ω) and since, by Lemma 15.2.1, limk→∞ f k (z) = ω, passing to a sufficiently large iterate of z, we may assume without loss of generality that f k (z) ∈ B(ω,δf )

(15.42)

106

Part III Topological Dynamics of Meromorphic Functions

for all integers k ≥ 0. Again, since limk→∞ f k (z) = ω, we conclude from (15.41) that f n (z) ∈

p(ω) 

j

Sr (ω,π − α)

(15.43)

j =1

for all n ≥ 0 large enough, say n ≥ q ≥ 1. But, in light of Proposition 15.1.8, for all k ≥ 0 large enough, we have that ⎞ ⎛ p(ω)  j (15.44) Sr (ω,π − α)⎠ ⊂ B(ω,|z − ω|/2). fω−k ⎝ j =1

Since, by virtue of (15.42), we have that z = fω−k (f k (z)) for all k ≥ 0, we, thus, obtain from (15.43) and (15.44) that z ∈ B(ω,|z − ω|/2). This contradiction finishes the proof.  We call A∗ (ω) the immediate basin of attraction to the parabolic fixed point ω. Theorem 15.2.3 If f : C −→  C is a meromorphic  p(ω) function and ω is a simple parabolic fixed point of f , then the sets A∗j (ω) j =1 , called Leau–Fatou petals or Leau domains, are mutually disjoint; moreover, for every j ∈ {1, . . . ,p(ω)} and every α ∈ (0,π), we have that A∗j (ω) ⊆

∞ 

 j  f −n Sa (ω,α) .

n=0

 ∞

 −n

p(ω) j Sa (ω,α) j =1 is a finite collection of mutually p(ω)  disjoint open sets, and since all the sets A∗j (ω) j =1 are connected, making use of Lemma 15.2.2, it follows that each set A∗i (ω) is contained in exactly one set      ∞ −n S j (ω,α) . Since A∗ (ω) ∩ −n S i (ω,α) = ∅, of the form ∞ a a n=0 f n=0 f i we, thus, conclude that Proof

Since

n=0 f

A∗i (ω) ⊆

∞ 

  f −n Sai (ω,α) .

n=0

 p−1 This also implies that all the sets A∗j (ω) j =0 are mutually disjoint and the proof is complete.  Coming back to Theorem 15.1.15, if the map ϕ is global, i.e., if ϕ : C →  C is a meromorphic function, we can now say more. It turns out, as we shall

15 Rationally Indifferent Periodic Points

107

prove momentarily, that all the maps Fα extend holomorphically to the entire petals A∗j (ω). Theorem 15.2.4 If f : C −→  C is a meromorphic function and ω ∈ C is a simple rationally indifferent fixed point of f , then, for every j ∈ {1, . . . ,p(ω)}, there exists a holomorphic function F : A∗j (ω) −→ C such that F ◦f =F +1

(15.45)

and, for all α ∈ (0,π), the function F |S j (ω,α) is univalent; in addition, if α ∈ a [π/2,π), then there exists s > 0 such that j

F (Sa (ω,α)) ⊇ {z ∈ C : Re(z) > s}.

(15.46)

Proof Fix z ∈ A∗j (w). By virtue of Theorem 15.2.3, there exists k ≥ 0 such j

that f k (z) ∈ Sa (ω,α). We then define F (z) := Fα (f k (z)) − k.

(15.47)

To see that this definition is independent of k, notice that Theorem 15.1.15 implies that, for every j ≥ 0, F (z) = Fα (f k+j (z)) − (k + j ). So, F (z) := lim Fα (f n (z)) − n

(15.48)

n→∞

is independent of k and (15.45) holds. Formula (15.46) is an immediate consequence of Theorem 15.1.15(3). Since (15.47) directly implies that the map F is holomorphic, the proof is, thus, complete.  Being more general, let ω ∈ C be a rationally indifferent periodic point of f . Then there exists the least integer  ≥ 1 such that ω is a simple parabolic fixed point of f  . We define A∗ (ω) to be the immediate basin of attraction to ω under the iterates of f  . We also define A∗j (ω), j ∈ {1,1, . . . ,p(ω)}, to be the Leau–Fatou petals of ω considered as a simple parabolic fixed point of f  . We further define A(ω) :=

∞ 

f −j (A∗ (ω)), A∗p (ω) :=

j =0

−1  j =0

and Ap (ω) :=

−1  j =0

A(f j (ω)).

A∗ (f j (ω)),

108

Part III Topological Dynamics of Meromorphic Functions

Now we are in a position to prove the following theorem. Theorem 15.2.5 If f : C −→  C is a meromorphic function of degree at least equal to 2 and ω ∈ C is a rationally indifferent periodic point of f , then each Leau–Fatou domain of ω contains at least one singular point of f −1 . Proof The first part of the proof parallels the corresponding part of the proof of Theorem 14.1.1, which concerns attracting periodic orbits. The argument generating a contradiction is, however, more difficult. So, assume first that ω is a simple rationally indifferent fixed point of f . Fix j ∈ {1,2, . . . ,p(ω)}. Assume, for contradiction, that A∗j (ω) contains no singular point of f −1 . We shall prove by induction that there exists an ascending sequence (Un )∞ n=0 of open connected, simply connected subsets of A∗j (ω) such that (a) ω ∈ U0 . (b) For every n ≥ 1, there exists fn−1 : Un−1 −→ Un , a surjective holomorphic inverse branch of f . −1 |Un−1 = fn−1 . (c) For every n ≥ 1, fn+1 Indeed, set U0 := Sa (ω,π/2) ⊆ A∗j (ω). j

By virtue of (15.39) in Theorem 15.1.15, we have that U0 ⊆ fω−1 (U0 ) ⊆ A∗j (ω), where, we recall, fω−1 is the unique local holomorphic inverse branch of f that sends ω to ω. Therefore, setting f1−1 := fω−1 |U0 , as U0 is simply connected, we have that U1 := f1−1 (U0 ) ⊇ U0 is also simply connected and the base case of the induction is complete. For the inductive step, suppose that n ≥ 1 and U0 ⊆ U1 ⊆ · · · ⊆ Un, an ascending sequence of open connected, simply connected sets have been constructed such that properties (a) and (b) are satisfied for all 1 ≤ j ≤ n, whereas (c) is satisfied for all 1 ≤ j ≤ n − 1. Since A∗j (ω) ∩ f −1 (Un ) contains no singular points of f −1 , since f : A∗j (ω) −→ A∗j (ω) is a holomorphic surjective map, and since Un ⊃ Un−1 is open connected, simply connected, there

15 Rationally Indifferent Periodic Points

109

−1 exists, by virtue of Theorem 13.3.49, fn+1 : Un −→ A∗j (ω), a holomorphic inverse branch of f such that −1 fn+1 |Un−1 = fn−1 .

(15.49)

−1 Un+1 := fn+1 (Un−1 ),

(15.50)

So, setting

we only need to check that Un+1 ⊇ Un . Invoking (15.49), (15.50), and property (b) above, we infer that −1 Un+1 = fn+1 (Un−1 ) ⊇ fn−1 (Un−1 ) = Un .

The inductive construction is, thus, finished. Since, by Lemma 15.2.2, ∞ ∗ n=0 Un = Aj (ω), conditions (a)–(c) along with ascendance of the sequence (Un )∞ n=0 yield the existence of a holomorphic bijective function f∗−1 : A∗j (ω) −→ A∗j (ω) such that f∗−1 |Un−1 = fn−1 for every n ≥ 1. Since also f ◦ f∗−1 = IdA∗j (ω) , it follows that the function f : A∗j (ω) −→ A∗j (ω) is also injective. Since we already know that it is surjective, we finally conclude that the holomorphic function f : A∗j (ω) −→ A∗j (ω) is bijective. Iterating (15.45) of Theorem 15.2.4 yields F ◦ fn = F + n

(15.51)

for all n ≥ 0. Taking now any two distinct points w,z ∈ A∗j (ω), we find n ≥ 0 j

so large that both points f n (w) and f n (z) belong to Sa (ω,π/2). It then follows from bijectivity of the map f n : A∗j (ω) → A∗j (ω), (15.51), and the middle placed assertion of Theorem 15.2.4 that F (w) = F (z). In consequence, the holomorphic function F : A∗j (ω) −→ C (produced in Theorem 15.2.4) is univalent. Formula (15.51) also yields F ◦ f∗−n = F − n.

110

Part III Topological Dynamics of Meromorphic Functions

It, therefore, follows from (15.46) of Theorem 15.2.4 that F (A∗j (ω)) ⊇ F (Un ) = F (f∗−n (U0 )) = F (U0 ) − n ⊇ {z ∈ C : Re(z) > s} − n = {z ∈ C : Re(z) > s − n}. Taking the union over all n ≥ 1, we, therefore, obtain that F (A∗j (ω)) = C. Since we already know that the map F : A∗j (ω) → C is univalent, we, thus, have that the inverse F −1 : C → A∗j (w) is holomorphic. Since A∗j (ω) ∩ J (f ) = ∅ and J (f ), being perfect (see Theorem 13.1.10), contains at least three distinct points, we, therefore, conclude from Picard’s Little Theorem that the function F −1 : C → A∗j (ω) is constant. This contradiction finishes the proof in the case when ω is a simple parabolic fixed point of f . The general case follows from the preceding one (simple parabolic fixed point) by applying it to f l , where l ≥ 1 is the least integer making ω a simple  parabolic fixed point of f l . The proof is complete.

15.3 Fatou Flower Theorem and Fundamental Domains Around Rationally Indifferent Periodic Points In this section, f : C −→  C is an entirely arbitrary (global) meromorphic function; in particular, any elliptic one is allowed. The results we formulate here (with proofs) are of fairly classical nature (except those concerning semiconformal measures) and are scattered widely in the literature; you may find them, for example, in [ADU], [DU5], and [DU6]. Let (f ) denote the set of rationally indifferent periodic points of f . Throughout this section, unless otherwise stated, ω is assumed to be a simple parabolic fixed point of f , i.e., we recall that f (ω) = ω and f  (ω) = 1. We denote the set of all simple parabolic fixed points of f by 0 (f ). On a sufficiently small neighborhood V of ω, a unique holomorphic inverse branch fω−1 : V −→ C, which sends ω to ω, is well defined. Thus, all the results of Section 15.1 apply with both ϕ := fω−1 and ϕ := f . Fix α ∈ (0,π) (κ = 1/2) and recall that p(ω)

Sa1 (ω,α), . . . ,Sa

(ω,α)

15 Rationally Indifferent Periodic Points

111

are the corresponding attracting sectors for f defined in (15.36) and p(ω)

Sr1 (ω,α), . . . ,Sr

(ω,α)

are the corresponding repelling sectors defined in (15.37). Equivalently, these are the attractive sectors for fω−1 . Since the family of iterates of f is not normal on any neighborhood of any point in the Julia set, as a fairly immediate consequence of Proposition 15.1.8, we get the following celebrated classical result. Theorem 15.3.1 (Fatou Flower Theorem) Let f : C −→  C be a meromorphic function. If ω ∈ (f ), i.e., if ω is a rationally indifferent periodic point of f , then, for every α ∈ (0,π), there exists θα (ω) ∈ (0,θf ) such that p(ω)

J (f ) ∩ B(ω,θα (ω)) ⊆ Sr1 (ω,α) ∪ · · · ∪ Sr

(ω,α).

Proof Passing to a sufficiently high iterate of f , we may assume without loss of generality that ω is a simple parabolic fixed point of f . Seeking contradiction, suppose that there exists a sequence (zn )∞ n=1 of points in p(ω) j J (f )\ ∪j =1 Sr (ω,α) such that lim |zn − ω| = 0.

n→+∞

Then, fixing β ∈ (0,α), for every n ≥ 1 large enough, there would exist i ∈ {1,2, . . . ,p(ω)} such that zn ∈ Sai (ω,π − β). Fix one such n and i. Then, by Proposition 15.1.8, Sai (ω,π − β) is a subset of the Fatou set of f . Thus, / J (f ). This contradiction finishes the proof.  zn ∈ Since the Julia set J (f ) is fully invariant (f −1 (J (f )) ⊆ J (f ) and f (J (f )\ {∞}) = J (f )), we infer from the Fatou Flower Theorem (Theorem 15.3.1), Proposition 15.1.8, and (15.2) (actually the derivative of φ0 ) that, for every  ω ∈ 0 (f ), there exists θ (ω) ∈ 0,θπ/4 (ω) such that, for every 0 < R ≤ θ (ω), we have that fω−1 (J (f ) ∩ B(ω,R)) ⊆ J (f ) ∩ B(ω,R).

(15.52)

fω−n : J (f ) ∩ B(ω,R) −→ J (f ) ∩ B(ω,R),

(15.53)

Thus, all iterates

n = 0,1,2, . . . , are well defined. From Theorem 15.3.1 and (15.52), we obtain, for all α > 0 and all j ∈ {1,2, . . . ,p(ω)}, that   j j (15.54) fω−1 J (f ) ∩ Sr (ω,α) ⊆ J (f ) ∩ Sr (ω,α).

112

Part III Topological Dynamics of Meromorphic Functions

By Proposition 15.1.8,  j  j fω−1 Sr (ω,α) ⊆ Sr (ω,α);

(15.55)

fω−n : Sr (ω,α) −→ Sr (ω,α),

(15.56)

therefore, all iterates j

j

n ≥ 0, are well defined. Furthermore, it follows from Proposition 15.1.8 that lim f −n (z) n→∞ ω

=ω

(15.57)

j

uniformly on z ∈ Sr (ω,α), and from Lemma 15.1.9 that |fω−1 (z) − ω| < |z − ω| and |(fω−1 ) (z)| < 1  j  π for every z ∈ Sr (ω,α) ∩ B(ω,Rα (ω)) \{ω} if α < 2p(ω) . Put 

θ = θ (f ) := min min{θ (ω),Rπ/4 (ω)} : ω ∈ (f ) .

(15.58)

(15.59)

As an immediate consequence of the Fatou Flower Theorem (Theorem 15.3.1) and (15.53), we also get the following. Lemma 15.3.2 Let f : C −→  C be a meromorphic function. If τ > 0 is sufficiently small (with θ > 0 decreased if necessary), then, for every ω ∈ 0 (f ) and every z ∈ J (f ) ∩ B(ω,θ), there exists j ∈ {1,2, . . . ,p(ω)} such that j

Be (z,2τ |z − ω|) ⊆ B(ω,θ) ∩ Sr (ω,π/2). In addition, all the local holomorphic inverse branches fω−n : B(z,2τ |z − ω|) −→ C are well defined for all integers n ≥ 0. As an immediate consequence of Proposition 15.1.10 and the Fatou Flower Theorem (Theorem 15.3.1), we get the following. Lemma 15.3.3 If f : C −→  C is a meromorphic function, then, for every j ω ∈ 0 (f ), every j ∈ {1,2, . . . ,p(ω)}, and every z ∈ Sr (ω,α), in particular for every z ∈ J (f ) ∩ B(ω,θ), with p := p(ω), we have that lim (n

p+1 p

n→∞

|fω−(n+1) (z) − fω−n (z)|) = |a|

and 1

− p1

lim (n p |fω−n (z) − ω|) = |a|

n→∞

.

− p1

15 Rationally Indifferent Periodic Points

113 j

In addition, in these two limits, the convergence is uniform on the set Sr (ω,α)∩ B c (ω,R), in particular on the set J (f ) ∩ B(ω,θ) ∩ B c (ω,R), for every R > 0. As an immediate consequence of Proposition 15.1.11 and the Fatou Flower Theorem (Theorem 15.3.1), we get the following. Proposition 15.3.4 If f : C −→  C is a meromorphic function, then, for every ω ∈ 0 (f ), there exists a holomorphic function 

p(ω)   ∗ Aj (ω) −→ C\{0} fω−∞ : i=1

with the following properties. Fix α ∈ (0,π) and j ∈ {1,2, . . . ,p(ω)}. Then, j for every z ∈ Sr (ω,α), in particular for every z ∈ J (f ) ∩ B(ω,θ), with p := p(ω), we have that lim n

n→+∞

p+1 p

|(fω−n ) (z)| = |a|

− p+1 p

  |z − ω|−(p+1) fω−∞ (z)

(15.60)

j

and the convergence is uniform on Sr (ω,α) ∩ B c (ω,R) for every R > 0. In addition, if α ∈ (0,π/2), then  −∞  fω (z) = 1. (15.61) lim j

Sr (ω,α)z→ω

As the last result of the local behavior of f around ω, we will prove the following. Proposition 15.3.5 (local expansiveness at rationally indifferent parabolic points) Let f : C −→  C be a meromorphic function. If ω ∈ 0 (f ), z ∈ J (f ), and f n (z) ∈ B(ω,θ) for all integers n ≥ 0, then z = ω. Proof Since z ∈ J (f ), we have that f n (z) ∈ J (f ) for every n ≥ 0. It, therefore, follows from Theorem 15.3.1 (Fatou Flower Theorem) and (15.57) (the uniform convergence does matter) that ω = lim fω−n (f n (z)) = lim z = z. n→∞

n→∞



We end this section with the following two lemmas, which are interesting on their own and which will be used substantially in the “parabolic” sections of Chapter 22. Let Sr (ω,α) :=

p(ω)  j =1

j

Sr (ω,α).

114

Part III Topological Dynamics of Meromorphic Functions

Lemma 15.3.6 If f : C −→  C is a meromorphic function,  ω ∈ 0 (f ), and α ∈ (0,π), then the set J (f ) ∩ Sr (ω,α)\fω−1 (Sr (ω,α)) is a fundamental domain for the map fω−1 acting on J (f ) near ω. More precisely, (a) If (i,k),(j,l) ∈ {1,2, . . . ,p(ω)} × N0 and (i,k) = (j,l), then   j   fω−k Sri (ω,α)\fω−1 (Sr (ω,α)) ∩ fω−l Sr (ω,α)\fω−1 (Sr (ω,α)) = ∅ and (b)





ω ∈ IntJ (f ) J (f ) ∩ {ω} ∪

∞ 

fω−n





Sr (ω,α)\fω−1 (Sr (ω,α))



.

n=0

Proof We start by proving (a). Assume, first, that i = j but k = l; furthermore, without loss of generality that k < l. Using (15.3) and (15.17), we get that     fω−k Sri (ω,α)\fω−1 (Sr (ω,α)) ∩ fω−l Sri (ω,α)\fω−1 (Sr (ω,α))      = fω−k ρ ◦ H S(x(α,κ),α) \fω−1 ρ ◦ H (S(x(α,κ),α))    ∩ fω−l ρ ◦ H (S(x(α,κ),α))\fω−1 ρ ◦ H (S(x(α,κ),α))      = fω−k ρ ◦ H S(x(α,κ),α) \ρ ◦ H f˜ω−1 (S(x(α,κ),α))      ∩ fω−l ρ ◦ H S(x(α,κ),α) \ρ ◦ H f˜ω−l (S(x(α,κ),α))      = fω−k ρ ◦ H S(x(α,κ),α) \f˜ω−1 S(x(α,κ),α)      ∩ fω−l ρ ◦ H S(x(α,κ),α) \f˜ω−l S(x(α,κ),α)      = ρ ◦ H f˜ω−k S(x(α,κ),α) \f˜ω−1 S(x(α,κ),α)      ∩ ρ ◦ H f˜ω−l S(x(α,κ),α) \f˜ω−1 S(x(α,κ),α)      = ρ ◦ H f˜ω−k S(x(α,κ),α) \f˜ω−1 S(x(α,κ),α)    ∩ f˜ω−l S(x(α,κ),α)\f˜ω−1 S(x(α,κ),α)    = ρ ◦ H ◦ f˜ω−k S(x(α,κ),α)\f˜ω−1 S(x(α,κ),α)    ∩ f˜ω−(l−k) S(x(α,κ),α)\f˜ω−1 S(x(α,κ),α)   ⊆ ρ ◦ H ◦ f˜ω−k S(x(α,κ),α)\f˜ω−1 (S(x(α,κ),α)) ∩ f˜ω−(l−k) (S(x(α,κ),α)).

15 Rationally Indifferent Periodic Points

115

Now, since l − k ≥ 1 and employing (15.11), we continue as follows:     fω−k Sri (ω,α)\fω−1 (Sr (ω,α)) ∩ fω−l Sri (ω,α)\fω−1 (Sr (ω,α))     ⊆ ρ ◦ H ◦ f˜ω−k S(x(α,κ),α)\f˜ω−1 (S(x(α,κ),α)) ∩ f˜ω−1 S(x(α,κ),α) = ρ ◦ H ◦ f˜ω−k (∅) = ∅. We are, thus, done with (a) in this case. Now assume that i = j . Then we are immediately done by virtue of Lemma 15.1.14 and (15.39) of Theorem 15.1.15. Part (b) is an immediate consequence of 15.55 and the Fatou Flower Theorem, i.e., Theorem 15.3.1. The proof is complete.  Lemma 15.3.7 If f : C −→  C is a meromorphic function, ω ∈ 0 (f ), and α ∈ (0,π), then there exists u > 0 such that denoting, for every j = 1, 2, . . . ,p(ω),     j j j (ω,α) := j (ω,α,u) = Sr (ω,α)\fω−1 (Sr (ω,α)) ∩ C\B(ω,u) and (ω) := (ω,α) :=

p(ω) 

j (ω,α),

j =1

we have that the set (ω,α) is a fundamental domain for the map fω−1 acting on J (f ) near ω. More precisely, (a) If (i,k),(j,l) ∈ {1,2, . . . ,p(ω)} × N0 and (i,k) = (j,l), then fω−k (j (ω,α)) ∩ fω−l (j (ω,α)) = ∅ and     −n (b) ω ∈ IntJ (f ) J (f ) ∩ {ω} ∪ ∞ n=0 fω ((ω,α)) . Proof Item (a) is an immediate consequence of Lemma 15.3.6. We shall now prove (b). Since fω−1 is a C 1 -diffeomorphism, there is u ∈ (0,θα (ω)) such that B(ω,u) ⊆ fω−1 (B(ω,θα (ω))). It then follows from the Fatou Flower Theorem, i.e., Theorem 15.3.1, that J (f ) ∩ B(ω,u) ⊆

p(ω)  j =1

fω−1 (Sr (ω,α)) = f −1 (Sr (ω,α)). j

(15.62)

116

Part III Topological Dynamics of Meromorphic Functions

Because of Lemma 15.3.6(b), it is enough to show that

∞ ∞     −n −1 fω Sr (ω,α)\fω (Sr (ω,α)) ⊆ fω−n ((ω,α)). J (f ) ∩ n=0

n=0

So, let z belong to the left-hand side of this inclusion. Then there exists n ≥ 0 such that   z ∈ J (f ) ∩ fω−n Sr (ω,α)\fω−1 (Sr (ω,α)) . But then, because of (15.62), we get that   f n (z) ∈ J (f ) ∩ Sr (ω,α)\fω−1 (Sr (ω,α))     ⊆ J (f ) ∩ Sr (ω,α)\fω−1 (Sr (ω,α)) ∩ C\B(ω,u) ⊆ (ω,α). Hence, z ∈ f −n ((ω,α)) and the proof is complete.



15.4 Quantitative Behavior of Analytic Functions Locally Defined Around Rationally Indifferent Periodic Points: Conformal Measures Outlook We now, in this section, move on to dealing with the local behavior of conformal measures and its generalizations around parabolic points. Recall that, in Definition 10.4.1, we introduced the concept of semi-conformal and conformal measures. To deal with them, we shall prove the following. Lemma 15.4.1 Let m be a semi-t-conformal measure for a meromorphic map f : C −→  C defined on some neighborhood of a simple parabolic fixed point ω of f . Then, for every R > 0, there exists a constant C = C(t,ω,R) ≥ 1 such that, for every 0 < r ≤ R, m(B(ω,r)\{ω}) ≤ C, r αt (ω) where αt (ω) := t + p(ω)(t − 1). If m is t-conformal, then, in addition, m(B(ω,r)\{ω}) ≥ C −1 . r αt (ω) Proof

Take R > 0 so small, e.g., R = θ/2, that Be (ω,R) ⊆ B(ω,θ) and let $ # (15.63) P := J (f ) ∩ z ∈ C : R(2fω )−1 ≤ |z − ω| ≤ R ,

15 Rationally Indifferent Periodic Points

where

117

# $ fω ∞ := sup |f  (z)| : z ∈ B(ω,θ) ∩ J (f ) .

Fix τ > 0 so small, as spcified in Lemma 15.3.2. Let δ = τ inf{|z − ω| : z ∈ P } > 0. Since P is compact, there are finitely many points z1, . . . ,zq in P such that   P ⊆ J (f ) ∩ B(z1,δ) ∪ · · · ∪ B(zq ,δ) , and we may assume that δ is so small that fω−n (B(zi ,δ)) ∩ B(zi ,δ) = ∅ for i = 1, . . . ,q and n = 1,2, . . .. Denote p := p(ω). For every n ≥ 1, define % & 1 1 −1 Pn := z ∈ B(ω,θ) ∩ J (f ) : |a| p n−1/p ≤ |fω−n (z) − ω| ≤ 2|a| p n−1/p . 2 By the local behavior of f around a parabolic point, we conclude that, for every z ∈ (B(ω,R)\{ω}) ∩ J (f ), there exists an integer l ≥ 0 such that R(2fω ∞ )−1 < |f l (z) − ω| < R, i.e., f l (z) ∈ P . Therefore, the set 

J (f ) ∩ z : R(2fω ∞ )−1 < |z − ω| < R is nonempty (Be (ω,R) ∩ (J (f )\{ω}) is nonempty since J (f ) is perfect). Moreover, since it is open in J (f ), we deduce that, for some 1 ≤ j ≤ q, the set Be (zj ,δ) ∩ P has a nonempty interior in J (f ). Hence, M := me (Be (zj ,δ) ∩ P ) > 0. By Lemma 15.3.3, there is an integer n0 ≥ 1 such that fω−n (z) ∈ Pn for every n ≥ n0 and and every z ∈ P . In other words, this means that Pn ⊃ fω−n (P ) for n ≥ n0 . Thus, q ∞ ∞ ∞       −1 B ω,2|a| p n−1/p ⊃ Pk ⊃ fω−k (P ) ⊃ fω−n (B(zi ,δ) ∩ P ). k=n

k=n

k=n i=1

(15.64)

118

Part III Topological Dynamics of Meromorphic Functions

On the other hand, because of Proposition 15.3.5, for every z ∈ J (f ) ∩ (B(ω,R)\{ω}) there exists an integer l ≥ 0 such that f l (z) ∈ P . Let l(z) ≥ 0 be such a least integer. Take n1 ≥ n0 so large that if z ∈ B(ω,2|a| then l(z) ≥ n0 . Consider now any − p1 −1/p

z ∈ J (f ) ∩ B(ω,2|a|

n

− p1 −1/p n1 ),

)\{ω} −l(z)

with n ≥ n1 . Since l(z) ≥ n0 and f l(z) (z) ∈ P , we see that z = fω − p1

(f l(z) (z)) ∈ Pl(z) . Therefore, 12 |a|p l(z)−1/p ≤ 2|a| n−1/p ; consequently, l(z) ≥ 4−p |a|2 . Hence,  −1 J (f ) ∩ B(ω,2|a| p n−1/p ) ⊆ {ω} ∪ fω−l (P ) l≥4−p |a|2 n q  

= {ω} ∪

(15.65) fω−l (B(zi ,δ)).

i=1 l≥4−p |a|2 n

Since, for every 1 ≤ j ≤ q, the sets {fω−n (J (f ) ∩ B(zj ,δ))}∞ n=0 are mutually disjoint, it follows from Theorem 8.3.8, Proposition 15.3.4, and semiconformality of the measure m that ⎛ ⎞ q   1   − f −l (B(zi ,δ))⎠ m B(ω,2|a| p n−1/p )\{ω} ≤ m ⎝ ω,i

i=1 l≥4−p |a|2 n



≤ qK t C t

l

− p+1 p t

l≥4−p |a|2 n

≤ C  (n−1/p )αt (ω), where C > 0 and C  are some constants. If, in addition, m is t-conformal, we have that ∞    −1 me (fω−k (Be (zj ,δ) ∩ P )) m B(ω,2|a| p n−1/p )\{ω} ≥ k=n

≥

∞ 

K −t C −t (k

− p+1 t p

)M

k=n

≥ MK −t C −t

2n  − p+1 (k p )t k=n

≥ MK

−t

− p+1 p t

=2

−t

− p+1 p t

C n((2n)

)

MK −h C −t (n−1/p )αt (ω) .

15 Rationally Indifferent Periodic Points

The proof is now concluded by observing that limn→∞

(n+1)−1/p n−1/p

119

= 1.



As a sub-proof of this proof, we obtained the following. Lemma 15.4.2 Let m be a semi-t-conformal measure for a meromorphic map f : C −→  C defined on some neighborhood of (f ). Then, for all R > 0 sufficiently small, there exists a constant C(R) > 0 such that, for all ω ∈ (f ) and all k ≥ 0, we have that 1− p(ω)+1 p(ω) t

m(fω−q(ω)k (B(ω,R) ∩ J (f )\{ω})  C(R)(k + 1)

,

where q(ω) is the smallest integer ≥ 1 turning ω into a simple parabolic point and p(ω) is the number of its petals. As an immediate consequence of (15.64) and (15.65), we get the following. Lemma 15.4.3 If f : C −→  C is a meromorphic map defined on some neighborhood of a simple parabolic fixed point ω of f , then, for every integer n ≥ 0, we have that

  ∞  −k IntJ (f ) J (f ) ∩ {ω} ∪ fω (P ) = ∅, k=n

where the set P is defined by (15.63). Lemma 15.4.4 If m is a semi-t-conformal measure for a meromorphic map f : C −→  C, defined (i.e., m) on some neighborhood of (f ), then ∀ β > 0 ∃ Cβ ≥ 1 ∀ ω ∈ (f ) ∀ z ∈ J (f ), m(B(z,β|z − ω|)) ≥ Cβ |z − ω|αt (ω) . Proof Since the set (f ) is finite, it suffices to show the lemma for some fixed ω ∈  and every 0 < β < 1. Furthermore, passing to a sufficiently high iterate of f , we may assume that ω is simple. Denote p := p(ω). For any ξ ∈ C, let

 A(ξ ) := x ∈ C : (1 − β)|ξ − ω| ≤ |x − ω| ≤ (1 + β)|ξ − ω| . Observe that there exists 0 < α ∈ (0,π) such that if j ∈ {1,2, . . . ,p(ω)} and j ξ ∈ Sr (ω,α), then j

A(ξ ) ∩ Sr (ω,α) ⊆ B(ξ,2β|ξ − ω|).

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Part III Topological Dynamics of Meromorphic Functions

Now apply the construction of the proof of Lemma 15.4.1. We use the same notation and, owing to Theorem 15.3.1, may assume in addition that P ⊆ B(ω,θα (ω)) and that the radius δ of balls B(z1,δ), . . . ,B(zq ,δ) is so small that each of j these balls B(zi ,δ) is contained in exactly one sector Sr (ω,α), j ∈ {1, . . . ,p}. Choose ε > 0 so small that  p p  e−ε e+ε 1, s := (1 + β)(e − ε) (1 − β)(e + ε) where e = |a|−1/p(ω) . With these definitions, by virtue of Lemma 15.3.3, there exists n0 ≥ 1 so large that, for every n ≥ sn0 and every y ∈ P , [un] − ([sn] + 1) ≥

u−s n, 2

where [t] denotes the integer part of t, and (e − ε)n−1/p ≤ |fω−n (y) − ω| ≤ (e + ε)n−1/p, where p = p(ω). For every n ≥ 1, let 

An := x ∈ C : (1 − β)(e + ε)n−1/p ≤ |x − ω| ≤ (1 + β)(e − ε)n−1/p . By Theorem 15.3.1, for every ξ ∈ J (f ) ∩ B(ω,θα (ω))\{ω}, there exists j (z) exactly  j (ξ ) such that 1 ≤ j (ξ ) ≤ p such that ξ ∈ Sr (ω,α). Moreover,  −n one j fω (ξ ) = j (ξ ) for every ξ ∈ J (f ) ∩ B(ω,θα (ω))\{ω} and every integer n ≥ 0. Hence, for every y ∈ P and every n ≥ n0 , we have that   j (y) An ∩ Sr (ω,θα (ω)) ⊆ B fω−n (y),2β|zn − ω| . Now consider an arbitrary z∈



fω−n (P ).

n≥n0

Then z ∈ fω−n (P ) for some n ≥ n0 . Let x := f n (z) and choose 1 ≤ i ≤ q such that x ∈ B(zi ,δ). Having the radius δ > 0 small enough, it follows from Theorem 15.3.1 that, for every l ≥ 0, we have that fω−l (J (f ) ∩ B(zi ,δ)) ⊆ Sr

j (x)

(ω,α).

If y ∈ P and sn0 ≤ sn ≤ l ≤ un, then, by the choice of ε and n0 , we have that |fω−l (y) − ω| ≤ (e + ε)l −1/p ≤ (e + ε)s −1/p n−1/p = (1 + β)(e − ε)n−1/p

15 Rationally Indifferent Periodic Points

121

and |fω−l (y) − ω| ≥ (e − ε)l −1/p ≥ (e − ε)u−1/p n−1/p = (1 − β)(e + ε)n−1/p . Thus, fω−l (P ) ⊆ An , whence  j (x) fω−l (J (f ) ∩ B(zi ,δ)) ⊆ An ∩ Sr (ω,α) sn≤l≤un

   ⊆ B fω−n (x),2β fω−n (x) − ω = B(z,2β|z − ω|).

Since the sets fω−l (J (f ) ∩ B(zi ,δ)), l = [sn] + 1, . . . ,[un] + 1, are mutually disjoint, it follows from Proposition 15.3.4 and the Koebe Distortion Theorem (Theorem 8.3.8) that, with some constant C4 > 0, ⎛ ⎞ [un]   − p+1 h m⎝ fω−l (J (f ) ∩ B(zi ,δ))⎠ ≥ C4 l p ; sn≤l≤un

l=[sn]

therefore, u − s − p+1 h − p+1 h nu p n p 2 u − s − p+1 h = C4 u p (n−1/p )h+p(h−1) 2  −n  u−s − p+1 h f (x) − ωα(ω) ≥ C4 u p ω α(ω) 2(e + ε) u−s − p+1 h = C4 u p |z − ω|α(ω) . 2(e + ε)α(ω)

m(B(z,2β|z − ω|)) ≥ C4

This proves the lemma for all points in the set  fω−n (P ). G := J (f ) ∩ n≥n0

Since, for ω itself, there is nothing to prove, the statement of the lemma holds on the set J (f ) ∩ ({ω} ∪ G), which is, by Lemma 15.4.3, a neighborhood of ω in J (f ). Since m is positive on nonempty open sets, for z ∈ / {ω} ∪ G we have that m(B(z,2β|z − ω|)) ≥ const > 0, finishing the proof of our lemma.



Lemma 15.4.5 Suppose m is a semi-t-conformal measure for the meromorphic map f : C −→  C defined on some neighborhood of (f ). Then, for every ω ∈ , every R > 0, and every 0 < σ ≤ 1, there exists L = L(ω,R,σ ) > 0

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Part III Topological Dynamics of Meromorphic Functions

such that, for every 0 < r ≤ R, the point ω is (r,σ,L)-αt (ω)-s.l.e. with respect to measure m.   Proof Let z ∈ Be (ω,r). If σ r ≥ 2|z − ω|, then Be (z,σ r) ⊇ Be ω, σ2 r ; therefore, by Lemma 15.4.1,  σ αt (ω)  σ αt (ω) νe (Be (z,σ r)) ≥ C(R/2) = C(R/2)r αt (ω) . (15.66) r 2 2 In order to deal with the opposite case, first notice that always Be (z,σ r) ⊇ Be (z,σ |z − ω|); therefore, by Lemma 15.4.4, we have that νe (Be (z,σ r)) ≥ C −1 (σ )|z − ω|αt (ω) . As σ r < 2|z − ω|, this implies that

 α (ω) νe (Be (z,σ r)) ≥ C −1 (σ ) σ/2 t r αt (ω) .

So, putting L(ω,R,σ ) = (σ/2)αt (ω) min{C(R/2),C −1 (σ )} finishes the proof.



P AR T IV Elliptic Functions: Classics, Geometry, and Dynamics

16 Classics of Elliptic Functions: Selected Properties

In this chapter, using a variety of contemporary sources, we present some parts of a well-known classical theory of elliptic functions, with especial emphasis on Weierstrass elliptic functions. For dynamical, topological, and fractal properties, this chapter will be primarily used in Chapter 19, which is devoted to providing many (classes of) examples of compactly nonrecurrent elliptic functions that exhibit a variety of topological and dynamical properties. We primarily follow here the classical books [Du] and [JS]. We would also like to draw the reader’s attention to the books [AE] and [La].

16.1 Periods, Lattices, and Fundamental Regions Let f be a function defined on the complex plane C. Then a complex number w is called a period of f if f (z + w) = f (z) for all z ∈ C. The function f is called periodic if it has a period w = 0. For example, the maps C  z −→ sin(z) ∈ C and C  z −→ cos(z) ∈ C have period 2π , the map C  z −→ ez ∈ C has period 2π i, while, for any complex number w = 0, the map C  z −→ sin(2π z/w) has period w. The set f of all periods of a function f has two important properties: one algebraic, valid for all f , and one topological, valid for 125

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Part IV Classics, Geometry, and Dynamics

nonconstant meromorphic functions. These properties are given in Theorems 16.1.1 and 16.1.2. Theorem 16.1.1 For every function f : C −→  C, the set f is a subgroup of the additive group C. Proof

Let α,β ∈ f . Then f (z + (α + β)) = f ((z + α) + β) = f (z + α) = f (z)

and so α + β ∈ f . Moreover, f (z − α) = f ((z − α) + α) = f (z), and so −α ∈ f . Finally, f (z + 0) = f (z), and so 0 ∈ f . Thus, f is a subgroup of C.  A subset  of a topological space is called discrete if every point x ∈  has a neighborhood U such that U ∩  = {x}. For example, (a) the integers Z form a discrete subset of R; (b) any finite subset of Rn is discrete; (c) {1/n; n ∈ Z, n = 0} is a discrete subset of R. However, the set {1/n; n ∈ Z} ∪ {0} is not discrete, since every neighborhood of 0 contains a real number of the form 1/n, n ∈ Z. Let f : C →  C be a meromorphic function. In particular, f is continuous; therefore, the set f is closed. Note that, for every ξ ∈ C ∩ f (C) and every z ∈ f −1 (ξ ), z + f ⊆ f −1 (ξ ). Since the set z + f is discrete if and only if f is, and since zeros of any nonconstant meromorphic function form a discrete set, we get the following. Theorem 16.1.2 For every nonconstant meromorphic function f : C −→  C, f is a closed discrete subset of C. To summarize, the periods of a nonconstant meromorphic function form a discrete subgroup of C. We will now show that there are exactly three types of discrete additive subgroups of C: isomorphic to {0}, Z, and Z × Z. Given two complex number w1,w2 ∈ C, we denote [w1,w2 ] := {mw1 + nw2 : m,n ∈ Z}. Of course, [w1,w2 ] is an additive subgroup of C, but we will have more to say about that.

16 Classics of Elliptic Functions: Selected Properties

127

Theorem 16.1.3 Let  be a discrete subgroup of C. Then one of the following holds: (1)  = {0}. (2) There exists w ∈ C such that  = [w,w] = {nw : n ∈ Z}; in particular,  is isomorphic to Z. (3)  = [w1,w2 ] for some w1,w2 ∈ C that are linearly independent over R, / R; in particular, the additive group  is i.e., w1 = 0 = w2 and w1 /w2 ∈ isomorphic to Z × Z. Proof Suppose that  = {0}. Since  is a discrete and closed subset of C, there exists w1 ∈ \{0} with a least value of |w1 |. Of course, w1 is not unique: −w1 , for example, will do equally well. Now let L = {tw1 : t ∈ R} be the line through 0 and w1 in C. Then {nw1 : n ∈ Z} ⊆  ∩ L. First, suppose that  ⊆ L. We then claim that  = {nw1 : n ∈ }, so that  is of type (2). For if  contains some w = nw1 , then, since  ⊆ L, we have that w = tw1 for some t ∈ R\Z. So, n < t < n + 1 for some n ∈ Z. Since  is a group containing w and nw1 , it contains w − nw1 = (t − n)w1 . However, 0 = |(t − n)w1 | < |w1 |, which contradicts the minimality of |w1 |. Hence,  = {nw1 ; n ∈ Z}, and we are in Case (2). Now suppose that   L. The set \L is a discrete subset of C because it is a subset of the discrete set . Thus, the set \L is closed in C. Hence, \L has an element w2 with a least modulus. Then  ⊇  := {mw1 + nw2 : m,n ∈ Z}. Since w2 ∈ / L, the vectors w1 and w2 are linearly independent over R. So  consists of the vertices of a tessellation of C by congruent parallelograms. We will now show that  = .

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Part IV Classics, Geometry, and Dynamics

If this is not the case, then there exists w ∈ \. Let w = λw1 + μw2 with λ,μ ∈ R. Then, by adding suitable integral multiples of w1 and w2 , we may assume that 1 1 |λ| ≤ and |μ| ≤ . 2 2 If μ = 0, then w = λw1 ∈ L, with |w| = |λw1 | < |w1 |. By minimality of |w1 |, we then have that w = 0, and hence w ∈ , contrary to our assumption. If λ = 0, then w = μw2 , and again w = 0, this time by minimality of |w2 |. Hence, λw1 and μw2 are nonzero and, therefore, linearly independent over R, giving 1 1 1 1 |w1 | + |w2 | ≤ |w2 | + |w2 | = |w2 |. 2 2 2 2 Now w ∈ \L since μ = 0. So, by minimality of |w2 |, we have that w = 0, contradicting the fact that w ∈ / . Thus,  =  and we are in Case (3).  |w| < |λw1 | + |λw2 | ≤

Definition 16.1.4 Let f : C −→  C be a meromorphic function. C is called simply • If f has its set f of periods of type (2), then f : C −→  periodic. • If f is of type (3), then f is called doubly periodic. • Any group  of type (3) is called a lattice, and any pair {w1,w2 } of complex numbers such that  = [w1,w2 ] is called a basis, or a generator, for the lattice . The above results show that a nonconstant meromorphic periodic function is either simply periodic or doubly periodic. We now turn to the study of doubly periodic meromorphic functions, starting with a closer look at some of the algebraic and geometric ideas involved. Let  be a group of type (3). Write  = [w1,w2 ], where {w1,w2 } is some generator of . Of course, there are other bases for  besides {w1,w2 }. For instance, {w1,w1 + w2 } is also a basis because if λ ∈ [w1,w2 ], then w = mw1 + nw2 = (m − n)λ1 + n(w1 + w2 ) with m − n ∈ Z. In general, if w1 ,w2 ∈ [w1,w2 ], then w1 = aw1 + bw2 where a,b,c,d are integers.

and

w2 = cw1 + w2,

(16.1)

16 Classics of Elliptic Functions: Selected Properties

129

Theorem 16.1.5 Equations (16.1) define a basis {w1 ,w2 } for [w1,w2 ] if and only if ad − bc = ±1. Proof It is convenient to write (16.1) using matrix notation    w1 w1 =A , w2 w2 where



a c

b d

(16.2)

.

If ad − bc = ±1, then A−1 has integer coefficients and we have that       w1 d −b w1 w1 = A−1 = ± . −c a w2 w2 w2 Thus, w1,w2 ∈ [w1 ,w2 ]; hence, [w1,w2 ] ⊆ [w1 ,w2 ]. The reverse inclusion is obvious, so [w1,w2 ] = [w1 ,w2 ]; hence, {w1 ,w2 } is a basis. Conversely, suppose that (16.2) defines a basis {w1 ,w2 } for [w1,w2 ]. Expressing the elements w1,w2 in terms of the basis, we have that    w1 w1 =B w2 w2 for some fixed matrix B with integer coefficients. So,   w1 w1 = BA . w2 w2

(16.3)

Since w1 and w2 form a basis of C considered as a two-dimensional vector space over R, we, thus, have that BA = Id. Therefore, det(B) · det(A) = 1. Since both A and B have integer coefficients, their determinants are integers; therefore, det(A) = ±1, i.e., ad − bc = ±1.  If  is a lattice and a ∈ C, then we define a := {aw : w ∈ }. This is a lattice if and only if a = 0.

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Part IV Classics, Geometry, and Dynamics

Definition 16.1.6 Two lattices  and  are called similar if and only if  = a for some a = 0. Keep {w1,w2 } a generator of a lattice  and {w1 ,w2 } a generator of a lattice  . Linear independence of w1 and w2 over R means that Im(w1 /w2 ) = 0. Interchanging w1 and w2 , if necessary, we may assume that Im (w1 /w2 ) > 0. We define the modulus of the basis {w1,w2 } to be τ := w1 /w2, where the numbering is such that Im(τ ) > 0. Each lattice  determines its set of moduli, the moduli of all its bases. Since μw1 /μw2 = w1 /w2 , similar lattices have the same set of moduli. Putting τ  := w1 /w2 , the modulus of the basis {w1 ,w2 } for  , we see that  and  are similar if and only if τ =

aτ + b , cτ + d

(16.4)

where a,b,c,d ∈ Z and ad − bc = ±1. Now both τ and τ  , being moduli, lie in the upper half-plane H := {z ∈ C : Imz > 0}, and if ad − bc = −1, then the M¨obius transformation az + b   ∈C C  z −→ T (z) := cz + d maps H onto the lower half-plane, so we must, therefore, have that ad − bc = 1. Conversely, since a,b,c,d ∈ Z and ad − bc = 1, we have that w1 = μ(aw2 + bw1 ), w2 = μ(cw2 + dw1 ) gives the basis [w1 ,w2 ] for a lattice  similar to . The M¨obius transformations az + b   C  z −→ T (z) := ∈ C, a,b,c,d ∈ Z, ad − bc = 1, cz + d form a discrete group, called the modular group PSL(2,Z).

(16.5)

16 Classics of Elliptic Functions: Selected Properties

131

Theorem 16.1.7 If  = [w1,w2 ] and  = [w1 ,w2 ] are lattices in C, with respective moduli τ = w2 /w1 and τ  = w2 /w1 , both in H, then the lattices  and  are similar if and only if τ  = T (τ ) for some T ∈ PSL(2,Z). Given a lattice , we define z1,z2 ∈ C to be congruentmod , written as z1 ∼ z2, if and only if z1 − z2 ∈ . The congruence mod  is obviously an equivalence relation on C, and the equivalence classes are co-sets z +  of  in the additive group C. Alternatively, we may regard  as acting on C as a transformation group. Each w ∈  induces the transformation C  z −→ Tw (z) = z + w ∈ C of C onto itself. Since Tw1 +w2 = Tw1 ◦ Tw2 , we have a group isomorphism   w −→ (Tw : C −→ C). Then two points z1,z2 ∈ C are congruent mod  if and only if they lie in the same orbit under this action of . Definition 16.1.8 A closed connected subset R of C is called a fundamental region for a lattice  ⊆ C if and only if (1) for each z ∈ C, R contains at least one point in the same -orbit as z, i.e., every point z ∈ C is congruent to some point in R, (2) no two points in the interior of R are in the same coset of , i.e., no two distinct points in the interior of R are congruent mod . If, as is usually the case in applications, R is also a Euclidean polygon, with a finite numbers of sides, then we call R a fundamental parallelogram for . For example, the parallelogram R with vertices 0,w1,w2,w1 + w2 is a fundamental parallelogram for the lattice [w1,w2 ]. Conditions (1) and (2) of the above definition ensure that if R is a fundamental region for a lattice , then R and its image under action of  (i.e., its translates R + w, w ∈ )

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Part IV Classics, Geometry, and Dynamics

cover the plane C completely, overlapping only at their boundaries. This type of covering is known as tessellation of C. By using (16.1), we can obtain fundamental parallelograms of different shapes, and hence obtain different tessellations of C. If R is any fundamental region for , then, for a fixed t ∈ C, the set R + t = {z + t : z ∈ R} is also a fundamental region. This is useful when we need to find a fundamental region containing or avoiding certain specified points, e.g., we can always find a fundamental parallelogram for  with 0 in its interior. We have seen that a fundamental region for a lattice is not unique. However, the next theorem shows that its area is unique, and may, therefore, be regarded as a function of the lattice alone. For notational convenience, we will write Tw (X) for X + w; since the translation Tw (z) = z + w is an isometry of C, we have that S(Tw (X)) = S(X), where S denotes the planar Lebesgue measure on C. Theorem 16.1.9 If R1 and R2 are two fundamental regions for a lattice  whose boundaries are of planar Lebesgue measure zero, e.g., if R1 and R2 are polygons, then S(R1 ) = S(R2 ). Proof

Because of our hypotheses, we have that S(R1 ) = S(IntR1 ) and S(R2 ) = S(IntR2 ).

We have that R1 ⊇ R1 ∩



Tw (IntR2 ) =

w∈



R1 ∩ Tw (Q2 ).

w∈

As IntR2 is the interior of a fundamental region, the sets {R1 ∩ Tw (IntR2 ) : w ∈ } are pairwise disjoint. Hence,       S(R1 ) ≥ S R1 ∩ Tw (IntR2 ) = S T−w (R1 ) ∩ IntR2 w∈

w∈

= 

   S Tw (R1 ) ∩ IntR2 .

w∈

Now as R1 is a fundamental region, w∈ Tw (R1 ) = C; hence,  Tw (R1 ) ∩ IntR2 = IntR2 . w∈

16 Classics of Elliptic Functions: Selected Properties

Therefore,



133

  S(Tw (R1 ) ∩ IntR2 ) ≥ S IntR2 = S(R2 ),

w∈

giving S(R1 ) ≥ S(R2 ). Interchanging R1 and R2 , we have that S(R2 ) ≥ S(R1 ). Thus, S(R1 ) = S(R2 ) 

and the theorem is proved.

The proof of the next, well-known, theorem can be found, for instance, in [JS]; see Theorem 5.8.4 therein. Theorem 16.1.10 The set % & 1 F := z ∈ H : |z| ≥ 1 and |Rez| ≤ 2 is a fundamental region for the modular group PSL(2,Z), having analogous meaning as in Definition 16.1.8 with  replaced by PSL(2,Z).

16.2 General Properties of Elliptic Functions Definition 16.2.1 A nonconstant meromorphic function f : C −→  C is called elliptic if and only if it is doubly periodic. This means that f , the set of periods of f , is a lattice in C. Thus, if f : C −→  C is an elliptic function, then f (z + w) = f (z) for all z ∈ C and all w ∈ f . It is actually a substantial problem to construct (nonconstant) elliptic functions. We will construct them, but, before doing this, we will examine some of the elementary properties that elliptic functions must possess. Recall from Volume I of this book that, given a lattice  ⊆ C, we denote T := C/, the quotient space generated by the lattice . So, T is a parabolic compact (connected) Riemann surface, called a complex torus, and it is topologically the two-dimensional torus. Let  =  : C −→ T be the corresponding quotient/projection covering map.

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Part IV Classics, Geometry, and Dynamics

Now keep f : C −→  C, an elliptic function. We define Tf := Tf = C/f and f := f : C −→ Tf .

(16.6)

Denote ˆ f := Tf \f (f −1 (∞)). T Of course, the function f : C −→  C induces a unique continuous map fˆ : Tˆ f −→ Tf such that the following diagram commutes: f

C\f −1 (∞)

C

f

f fˆ

Tˆ f

(16.7)

Tf

ˆ f −→ Tf is a holomorphic map. Actually, even more, we Evidently, fˆ : T have a holomorphic map C f˜ : Tf −→ 

(16.8)

such that the following diagram commutes: f˜

ˆf T fˆ

C f

(16.9)

Tf

This means that fˆ = f ◦ f˜.

(16.10)

Fix c ∈  C. Then the solutions of the equation f (z) = c are isolated, and each solution has finite multiplicity, with congruent solutions having the same multiplicity. Since the solutions are isolated, any fundamental polygon R for f contains only finitely many solutions (since R is compact). Thus, replacing R by R + t with some appropriate t ∈ C if necessary, we may and we do assume that there are no solutions on the boundary ∂R of R. Let {z1,z2, . . . ,zr } := R ∩ f −1 (c).

16 Classics of Elliptic Functions: Selected Properties

135

Let k1,k2, . . . ,kr denote respective multiplicities of solutions z1,z2, . . . ,zr . Let N (c) := k1 + k2 + · · · + kr . Then we say that there are N (c) solutions (counted with multiplicities) to the equation f (z) = c. Of course, N (c) is also the number of solutions (counted with multiplicities) to the equation fˆ(z) = f (c) ˆ and on the torus T, fˆ−1 (f (c)) = f (f −1 (c)) = f ({z1,z2, . . . ,zr }). In particular, the number N (c) is independent of the fundamental region R. Definition 16.2.2 The order ord(f ) of an elliptic function f : C →  C is the number N (∞) of solutions (counted with multiplicities) to the equation f (z) = ∞, i.e., the sum of multiplicities of the congruence classes of poles of f . This is analogous to the degree deg(f ) of a rational function f , which is equal to the number of solutions of f (z) = ∞, counting multiplicities. For the rest of this section, we assume that f is a (nonconstant) elliptic function. We denote by R a fundamental parallelogram for f with vertices t, t + w1, t + w2, t + w1 + w2, where [w1,w2 ] is a basis for f and t is chosen so that ∂R contains no zeros or poles of f . Theorem 16.2.3 If f : C →  C is an elliptic function, then ord(f ) ≥ 1. In particular, there is no entire elliptic function. Proof Seeking contradiction, suppose that ord(f ) = 0. Then f has no poles, whence f is holomorphic on C. So, since R is compact and f is continuous, f (R) is a compact subset of C and is, therefore, bounded. Since f (C) = f (R), it follows that f is bounded on C. Therefore, Liouville’s Theorem implies that f must be constant. This contradiction finishes the proof.  Theorem 16.2.4 The sum of the residues of any elliptic function f : C −→  C within its fundamental polygon R is equal to zero.

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Part IV Classics, Geometry, and Dynamics

Proof Since f is meromorphic and is holomorphic on some neighborhood of ∂R, we have that 1 2π i

∂R

f (z)dz

is equal to the sum of residues within R. Now let 1 , 2 , 3 , and 4 be the sides of R from t to t + w1 , t + w1 to t + w1 + w2 , t + w1 + w2 to t + w2 , and t + w2 to t, respectively, so that

∂R

f (z)dz =

4 

f (z)dz,

j =1 i

where the direction along j is consistent with the positive (i.e., anticlockwise) orientation on ∂R. Now, since w2 is a period of f and since 3 = −1 + w2 , where −1 is 1 with the reverse orientation, we can write f (z)dz = 3

=

−1 +w2 −1

f (z)dz =

−1

f (z)dz = −

f (z + w2 )dz

f (z)dz. 1

' ' By the same token, 4 f (z)dz = − 2 f (z)dz. Hence, the sum of the residues is equal to zero.  Corollary 16.2.5 There are no elliptic functions f of order ord(f ) = 1. Proof If f : C →  C were elliptic of order 1, it would have a single pole of order 1 in R, say, at a point a ∈ C. Then f (z) =

∞ 

aj (z − a)j

j =−1

for near z sufficiently close to a, with a−1 = 0. Thus, the sum of the residues of f within R would be equal to a−1 , which is nonzero, contradicting Theorem 16.2.4.  Theorem 16.2.6 If f : C →  C is an elliptic function, c ∈  C and  −1 then, for −1 each fundamental region R of f such that ∂R ∩ f (c) ∪ f (∞) = ∅, the number of solutions (counted with multiplicities) to the equation f (z) = c in R is equal to ord(f ). Equivalently, the number of solutions (counted with multiplicities) to the equation fˆ(z) = f (c) is equal to ord(f ). Proof This is the definition of ord(f ) if c = ∞, so we may assume that c ∈ C. Replacing f by f − c (which has the same order as f ), we may assume

16 Classics of Elliptic Functions: Selected Properties

137

that c = 0. Now f  /f is meromorphic and, since ∂R contains no poles or zeros of f , the function f  /f is analytic on ∂R . We may, therefore, integrate f  /f . Hence, applying the argument used in the proof of Theorem 16.2.4, we get that

∂R

f  (z) dz = 0. f (z)

This means that the sum of the residue f  /f within R is equal to zero. Now f  /f has poles at zeros and poles of f , and nowhere else. Suppose that f has zero of some multiplicity k at a point a ∈ R. Then f (z) = (z − a)k g(z) for all z sufficiently close to a, where g is an analytic function defined on some neighborhood of a and g(a) = 0. Then f  (z) = k(z − a)k−1 g(z) + (z − a)k g  (z) for all z sufficiently close to a, and so k g  (z) f  (z) = + . f (z) z−a g(z) Thus, f  /f has residue k at a. A similar argument, using local representation f (z) = (z − a)−l h(z), shows that f  /f has residue −l at each pole of f of multiplicity l. Since the sum of the residue of f  /f is zero, the number of zeros of f must be equal to the number of poles (both counting with multiplicities). Thus, the equation f (z) = 0 has ord(f ) solutions, as required. The proof is complete.  Theorem 16.2.7 Let f and g be elliptic functions with respect to some lattice . If f −1 (∞) = g −1 (∞) and both f and g have the same principal parts at each pole, then f − g = c for some constant c ∈ C. Proof The function f − g is doubly periodic with no poles. Thus, f − g is constant by Theorem 16.2.3.  Theorem 16.2.8 Let f and g be elliptic functions with the following properties. • f = g . −1 −1 −1 −1 • f (0) = g (0) and f (∞) = g (∞). • f and g have equal multiplicities at their zeros and their poles. Then f (z) = cg(z) for some constant c = 0. Proof Replace f − g by f/g in the proof of Theorem 16.2.7.



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A rational function f :  C −→  C, which is not identically zero, must have finitely many zeros (say at a1, . . . ,ar with multiplicities k1, . . . ,kr ) and finitely many poles (say at b1, . . . ,bs with multiplicities l1, . . . ,ls ). Conversely, C and multiplicities given any choice of points a1, . . . ,ar ,b1, . . . ,bs ∈  k1, . . . ,kr ,l1, . . . ,ls ≥ 1, there exists a rational function f with these zeros and poles, with these multiplicities, provided that (1) k1 + · · · + kr = l1 + · · · + ls (both must be equal to the degree of f ), and (2) the sets {a1, . . . ,ar } and {b1, . . . ,bs } are disjoint (zeros and poles cannot overlap). We just take f (z) :=



(z − aj )kj

(

j

(z − bj )lj ,

j

where these products range over all j such that aj ,bj ∈ C, but exclude factors where aj = ∞ or bj = ∞. If an elliptic function f : C −→  C is to have its zeros and poles at the congruent classes [a1 ], . . . ,[ar ] and [b1 ], . . . ,[bs ] with multiplicities k1, . . . ,kr and l1, . . . ,ls , then condition (1) is necessary by Theorem 16.2.6; corresponding to (2), we have the following necessary condition: (2 ) the sets [a1 ] ∪ · · · ∪ [ar ] and [b1 ] ∪ · · · ∪ [bs ] are disjoint. The next result shows that, in contrast to the situation for rational functions, these conditions are not sufficient for the existence of f . Theorem 16.2.9 Let the congruence classes of zeros and poles of an elliptic function f , respectively, be [a1 ], . . . ,[ar ] and [b1 ], . . . , ∪ [bs ] with respective multiplicities k1, . . . ,kr and l1, . . . ,ls . Then r 

kj aj ∼f

j =1

s 

lj bj .

j =1

Proof As usual, let R be a fundamental parallelogram for f , chosen so that f has no zeros or poles on ∂R. The effect of replacing any aj or −bj by a congruent point is to add an element of f to r  j =1

kj aj −

s 

lj bj .

j =1

This does not affect the condition of the theorem, so we may assume that aj , bj ∈ R for all j . First, we prove that

16 Classics of Elliptic Functions: Selected Properties r 

kj aj −

j =1

s 

lj bj =

j =1

1 2π i

139

zf  (z) dz. ∂ R f (z)

Indeed, the poles of zf  (z)/f (z) are the zeros and poles of f . If f has a zero of multiplicity k at z = a, then f (z) = (z − a)k g(z) near z = a, with g analytic and g(a) = 0. Then z zf  (z) = (k(z − a)k−1 g(z) + (z − a)k g  (z)) f (z) (z − a)k g(z) kz zg  (z) = + (z − a) g(z) near z = a, with zg  /g analytic at a, so zf  /f has residue ka at z = a. Similarly, if f has a pole of multiplicity l at z = b, then zf  /f has residue −lb at z = b. Now the zeros and poles of f within R are a1, . . . ,ar and b1, . . . ,bs with multiplicities k1, . . . ,kr and b1, . . . ,bs , so 1 2π i

zf  (z) dz, ∂ R f (z)

which is equal to the sum of the residue of zf  /f and takes the value r  j =1

kj aj −

s 

lj bj .

j =1

We label the sides of R as in the proof of Theorem 16.2.4. Since −4 , the path 4 with reverse orientation, is just the path 2 − w1 , we get that

2

zf  (z) dz = f (z)

(z − w1 )f  (z) (w1 )f  (z) dz + dz f (z) f (z) 2 2 (z − w1 )f  (z) = dz + w1 [log f (z)]2 . f (z) −4 +w1

Now we will calculate both summands separately. First, since w1 is a period of both f and f  , we get that

−4 +w1

(z − w1 )f  (z) dz = f (z)

zf  (z + w1 ) dz −4 f (z + w1 ) zf  (z + w1 ) =− dz = − 4 f (z + w1 )

Second, w1 [log f (z)]2 = 2π n1 iw1

4

zf  (z) dz. f (z)

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Part IV Classics, Geometry, and Dynamics

for some n1 ∈ Z since log f (z) changes its value by an integer multiple of 2π i as z travels along 2 from t + w1 to t + w1 + w2 (as f (t + w1 ) = f (t + w1 + w2 )). Thus, 2

zf  (z) dz = − f (z)

4

zf  (z) dz + 2π n1 iw1 . f (z)

3

zf  (z) dz + 2π n2 iw2 f (z)

Similarly,

1

zf  (z) dz = − f (z)

for some n2 ∈ Z. Hence, r  j =1

kj aj −

s 

lj bj =

j =1

1 2π i

4 zf  (z) 1  dz = 2π i ∂ R f (z) j =1

2

zf  (z) dz f (z)

1 (2π n1 iw1 + 2π n2 iw2 ) 2π i = n1 iw1 + n2 iw2,

=

which is an element of f as required. The proof is complete.



When we come to construct elliptic functions, we shall see that the conditions (1) and (2 ), together with the condition "r "s (3) j =1 kj aj ∼ j =1 lj bj , are sufficient for the existence of an elliptic function f with prescribed zeros and poles.

16.3 Weierstrass ℘-Functions I Let  = [w1,w2 ] be a lattice with a basis {w1,w2 } and R be a fundamental parallelogram for  with no elements of  on ∂R. Our goal in this section is to construct nonconstant functions f which are elliptic with respect to . By Theorem 16.2.3, we know that such a function cannot be analytic and so it must have poles in R. By Corollary 16.2.5, we know that f cannot just have one simple pole in R. So, the possibly simplest elliptic functions must have order 2, with either two simple poles or else a single pole of order 2. In this section, we shall introduce the Weierstrass ℘ (z)-function, which is elliptic of order 2 with respect to  and has a single pole of order 2 in R. This will be our basic elliptic function in the sense that every function which

16 Classics of Elliptic Functions: Selected Properties

141

is elliptic with respect to  is a rational function of ℘ and its derivative ℘  (z) (see Theorem 16.4.1). The sets

 r := aw1 + bw2 : a,b ∈ R and max(|a|,|b|) = r , for integers r ≥ 1, are similar parallelograms centered on 0. Defining r :=  ∩ r , we have that r = {mw1 + nw2 : m,n ∈ Z and max(|m|,|n|) = r}. Now  is a disjoint union  = {0} ∪ 1 ∪ 2 ∪ 3 ∪ · · · , and for each r ≥ 1 we have that #r = 8r.

(16.11)

We can order the elements of  by starting at 0 and then listing the elements of 1,2, . . . in turn, rotating around each r in the order rw1,rw1 + w2, . . . ,rw1 − w2 so that the sequence spirals outwards from 0. If we denote this ordering by w (0),w (1),w (2), . . ., then w (0) = 0, w(1) = w1, w (2) = w1 + w2, w (3) = w2, . . . , w (8) = w1 − w2, w (9) = 2w1, and w (10) = 2w1 + w2, . . . . Clearly, limk→+∞ |w (k) | = +∞. By 





and

w∈

,

w∈

we shall, naturally, mean the sum over all (respectively, all nonzero) lattice points w taken in the above order. Thus, 

h(w) =

w∈

∞ 

h(w (k) )

k=0

for any function h; similarly, 



w∈

h(w) =

∞  k=1

h(w (k) ).

142

By

Part IV Classics, Geometry, and Dynamics

 w∈



and



,

w∈

we shall mean the product over all (respectively all nonzero) lattice points, again in the above order. For convenience, we will abbreviate the notation " " to , , etc., the particular lattice  being understood. In practice, the particular ordering of  will not often be important, as the sums and products which concern us are usually absolutely convergent and, hence, invariant under rearrangements. The convergence properties of the Weierstrass functions depend on the following result, which is a two-dimensional analogue of the fact that the series "∞ −s defining the Riemann zeta-function converges if and only if s > 1. n=1 n Its proof is straightforward and left for the reader as an exercise. " Theorem 16.3.1 If  ⊆ C is a lattice and s ∈ R, then the series w∈ |w|−s converges if and only if s > 2. It is now easy to construct an elliptic function of any order ≥ 3. Theorem 16.3.2 For each integer N ≥ 3, the function  C  z −→ FN (z) := (z − w)−N ∈  C w∈

is elliptic with respect to the lattice  and its order ord(f ) is equal to N. Proof If K is any compact subset of C\, then the terms (z − w)−N are analytic and, therefore, bounded on K. Since K is bounded, there exists a set  ⊆  such that \ is finite and |w| ≥ 2|z| for all w ∈  and all z ∈ K. It follows that |z − w| ≥ |w| − |z| ≥

1 |w| 2

for all z ∈ K and all w ∈ . Hence, |(z − w)−N | ≤ 2N |w|−N for all w ∈  and all z ∈ K. So, if N ≥ 3, then Theorem 16.3.1 and the comparison test imply that the series  (z − w)−N w∈

16 Classics of Elliptic Functions: Selected Properties

143

converges absolutely uniformly on K. Since each term (z − w)−N is analytic on K, this implies that FN (z) is analytic on C\. Fix an element ξ ∈ . Since  is closed and discrete, there exists r > 0 so small that B(ξ,r) ∩ B(\{ξ },r) = ∅. Then  (z − w)−N . FN (z) = (z − ξ )N + w∈\{ξ }

The first term of this sum is meromorphic on B(ξ,r), while the second one can be shown as above to be absolutely uniformly convergent on B(ξ,r). Thus, the function FN restricted to B(ξ,r) is meromorphic as a sum of meromorphic and analytic function. Therefore, the function FN is meromorphic on C. Since, as we have shown, the series defining FN converges absolutely at each point of C\, we can rearrange its terms arbitrarily and get, for every ξ ∈ , that   ((z + ξ ) − w)−N = (z + w )−N = FN (z) FN (z + ξ ) = w ∈

w∈

as w  = w\ξ ranges over  as w does (though in a different order). The proof is complete.  Clearly, this method fails to produce an elliptic function F2 (z) of order 2, since Theorem 16.3.1 cannot be used to prove convergence of  (z − w)−2 . w∈

In order to guarantee convergence, we make the terms of this series smaller, replacing (z − w)2 by (z − w)2 + w −2 for each w = 0. Thus, the resulting series is  1 1 1  − 2 . ℘ = ℘ := 2 + (16.12) z (z − w)2 w w∈

Theorem 16.3.3 Given a lattice  ⊆ C, the series ℘ in (16.12) defines a meromorphic function from C to  C. Moreover, this function • is periodic with respect to the lattice ; in fact, the set of periods of ℘ coincides with , i.e., ℘ = , has poles of order 2 at all elements of , and • is holomorphic in C\. • This function is called the Weierstrass elliptic function induced by the lattice . If we want to stress the importance of the lattice, we denote it by ℘ rather than merely by ℘.

144

Part IV Classics, Geometry, and Dynamics For every w ∈ , we have that      1   z(z − 2w)  1  − = .  (z − w)2 w 2   (z − w)2 w 2 

Proof

So, if |w| ≥ 2|z|, then      1 1   z(z − 2w)  3|w| −3   (z − w)2 − w 2  =  (z − w)2 w 2  ≤ |z| |w|4 = 3|z| · |w| . It, therefore, follows from Theorem 16.3.1 that, for each point z ∈ C, there exists r > 0 so small that the series of (16.12) can be represented on B(z,r) as a sum of its finitely many terms, which form a meromorphic function (even on C) and an absolutely uniformly convergent series of analytic functions. Thus, (16.12) defines a meromorphic function in C. Checking periodicity of ℘, we rearrange terms to get, for all z ∈ C\ and all ξ ∈ , that ℘ (z + ξ )

 1 1 1  + − (z + ξ )2 ((z + ξ ) − w)2 w2 w∈   1 1 1 1 1  + − + 2− = 2 2 2 (z + ξ ) (z − (w − ξ )) (w − ξ ) z (−ξ )2 w∈\{ξ }   1 1 1 1  = 2+ + − z (z + ξ )2  (z − w )2 (w  )2 =

w ∈

1 1 1 − + − (z + ξ )2 (−ξ )2 (−ξ )2   1 1 1  = 2+ − z (z − w )2 (w  )2  w ∈

= ℘ (z). This means that  ⊆ ℘ . The statement about poles and analyticity in C\ is obvious, and this has further consequences. Namely, since 0 is a pole and 0 ∈ ℘ , we see that each element of ℘ is a pole of ℘, i.e., ℘ ⊆ ℘ −1 (∞). But we already know that ℘ −1 (∞) ⊆ . So, ℘ ⊆  and the equality ℘ =  is established. The proof is complete.  Theorem 16.3.4 Any Weierstrass ℘-function has order 2 and its derivative ℘  has order 3.

16 Classics of Elliptic Functions: Selected Properties

145

Proof ℘ has a single congruence class of poles, namely the lattice . And since each element of  is of multiplicity 2, the function ℘ has order 2.  = −2F has a single class of poles, all of multiplicity 3. Thus, Similarly, ℘ 3   ℘ has order 3.  (z) We shall now derive an important equation connecting ℘ (z) and ℘ obtained from the Laurent series for ℘ (z) near 0. We start by finding the Laurent series for the function  1  1 z 1 ζ (z) := + + + 2 . (16.13) z z−w w w w∈

Let

 m = min |w| : w ∈ \{0} > 0. Since z2 1 z 1 , + + 2 = 2 z−w w w w (z − w) " we see, by comparison with  |w|−3 , that the series 



w∈

1 1 z + + 2 z−w w w

is absolutely convergent for each z ∈ C\. Moreover, for each w ∈ \{0}, the binomial series 1 z2 1 z = − − 2 − 3 − ··· z−w w w w is absolutely convergent for z ∈ B(0,m), so we may substitute this in (16.13) and reverse the order of summation to obtain that  1  z3 1 z2 ζ (z) = + − 3 − 4 − · · · = − G3 z2 − G4 z3 − · · · z z w w w∈

for all z ∈ B(0,m), where Gk = Gk () :=





w −k .

w∈

These series Gk , k ≥ 3, are called the Eisenstein series for ; these are absolutely convergent by Theorem 16.3.1. For odd integers k ≥ 3, the terms

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Part IV Classics, Geometry, and Dynamics

w −k and (−w)−k cancel each other out, giving Gk = 0. So, the Laurent series of ζ near 0 is ∞

ζ (z) =

1  G2n z2n−1 . − z

(16.14)

n=2

Hence, ℘ (z) = −ζ  (z) =

∞

 1 + (2n − 1)G2n z2n−2 . z2

(16.15)

n=2

This is, thus, the Laurent series for ℘ (z), valid for all z ∈ B(0,m). Now straightforward calculations give  (z) = − ℘

2 + 6G4 z + 20G6 z3 + · · · z3

and 4 24G4 − 2 − 80G6 + z2 φ1 (z), 6 z z 4 36G4 4℘ (z)3 = 6 + 2 + 60G6 z + z2 φ2 (z), z z 60G4 + z2 φ3 (z), 60G4 ℘ (z) = z2  ℘ (z)2 =

where φ1 (z),φ2 (z) are some power series convergent in B(0,m). These last three equations give ℘  (z)2 − 4℘ (z)3 + 60G4 ℘ (z) + 140G6 = z2 φ(z), where φ(z) = φ1 (z) − φ2 (z) + φ3 (z) is some power series convergent in  are periodic with respect to , the function B(0,m). As ℘ and ℘  f (z) = ℘ (z)2 − 4℘ (z)3 + 60G4 ℘ (z) + 140G6

is also periodic with respect to . Since f (z) = z2 φ(z) in D, with φ being analytic, the function f vanishes at 0 and, hence, at all w ∈ . However, by its construction, f can have poles only where ℘ or ℘  have poles, i.e., at the lattice points . Therefore, f has no poles and so, by Theorem 16.2.3, f (z) is constant, which must be zero since f (z) = 0. Thus, we have proved the following.

16 Classics of Elliptic Functions: Selected Properties

147

 (z)2 = 4℘ (z)3 − 60G ℘ (z) − 140G . Theorem 16.3.5 ℘  4  6

This is the differential equation for ℘ (z). It is customary to define   −4 g2 = g2 () := 60G4 = 60 w

(16.16)

w∈

and g3 = g3 () := 140G6 = 140





w −6,

(16.17)

w∈

so that  (z)2 = 4℘ (z)3 − g2 ℘ (z) − g3 . ℘

(16.18)

If we put z = ℘ (t), then this means that  2 dz = 4z3 − g2 z − g3 . dt So, locally inverse branches of ℘-functions take on the form ℘ −1 (z) = t =

dz , √ p(z)

where p(z) is the cubic polynomial 4z3 − g2 z − g3 . This shows how the local inverses of elliptic functions appear as indefinite integrals. Theorem 16.3.6 Let  = [w1,w2 ] and w3 = w1 + w2 . If R is a fundamental parallelogram for  with & % 1 1 1 0, w1, w2, w3 ⊆ Int(R), 2 2 2 then 1 1 1 w1 , w2 , w3 2 2 2  in R. are the only critical points of ℘ in R, i.e., the only zeros of ℘  (z) has order 3; hence, it Proof By Theorem 16.3.4, the elliptic function ℘ has exactly three (counting with multiplicities) zeros in R. If w ∈ R, then, because 12 w ∼ − 12 w, we have that   1 1   w = ℘ − w . ℘ 2 2

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Part IV Classics, Geometry, and Dynamics

 (z) is an odd function, we have that ℘  (− 1 w) = Since, in addition, ℘  2  (− 1 w); hence, ℘  (− 1 w) = 0. Since also ℘ −1 (∞) =  and {w /2, −℘ 1   2 2 w2 /2,w3 /2} ∩  = ∅, we must, therefore, have that  1  ℘ − wi = 0 2

for all j = 1,2,3. The proof is complete.



We define ej := ℘ (wj /2), j = 1,2,3;

(16.19)

as an immediate consequence of Theorem 16.3.6, we get the following. Theorem 16.3.7 If  = [w1,w2 ] and w3 = w1 + w2 , then  

 ℘ Crit(℘ ) = {e1,e2,e3 } = ℘ (w1 /2),℘ (w2 /2),℘ (w3 /2) . Corollary 16.3.8 For each c ∈  C\{e1,e2,e3 }, the equation ℘ (z) = c has two simple solutions, while for c = e1,e2,e3 or ∞ this equation has one double solution. Proof Since ℘ is elliptic of order 2, it takes each value c ∈  C twice by Theorem 16.2.6, giving either two simple solutions (some z and −z since ℘ is even) or one double solution. If c ∈ C, then ℘ (z) = c has a double solution  (z) = 0, giving z ∼ 1 w , j = 1,2,3. So, c = e . The pole if and only if ℘  2 j j of order 2 at 0 shows that ℘ (z) = ∞ has a double solution.  Theorem 16.3.9 The complex numbers e1,e2 , and e3 are mutually distinct. Proof

Let fj (z) = ℘ (z) − ej

for j = 1,2,3. Like ℘ , the functions fj are of order 2. As fj (wj /2) = fj (wj /2) = 0, the functionfj has double zeros on the  equivalence class [w/2], and hence has no other zeros. In particular, fj (wk /2) = 0 for j = k. Since fj (wk /2) =  ℘ (wk /2) − ej = ek − ej , it follows that ej = ek for j = k. We want to end this section by noting that, by (16.18), the polynomial p(z) = 4z3 − g2 z − g3  (t) = 0, so p(z) has three distinct zeros has zero at z = ℘ (t), where ℘ z = e1,e2 , and e3 .

16 Classics of Elliptic Functions: Selected Properties

149

16.4 The Field of Elliptic Functions In this section, we consider a fixed lattice . An elliptic function will mean a function which is elliptic with respect to . If f and g are elliptic, then so are f + g, f − g, and f g; if g is not identically zero, then 1/g is elliptic. Thus, the set of all elliptic functions is a field, which we shall denote by E(). This field contains a subfield E1 () consisting of even elliptic functions. The constant functions form a subfield of E1 () isomorphic to C, so we may regard E() and E1 () as an extension field of C. Since E1 () contains ℘ (z) := ℘ (z), it contains all rational functions of ℘ with complex coefficients; these rational functions form a field C(℘), the smallest field containing ℘ and the constant functions C. Similarly, E() contains ℘ and ℘  and, hence, contains the field C(℘,℘  ) of rational functions of ℘ and ℘  . This is the smallest field containing ℘,℘  , and C. We start with the main result of this section. Theorem 16.4.1 For any lattice  ⊂ C the following are true. (1) If f is an even elliptic function with respect to , then f = R1 ◦ ℘ for some rational function R1 ; thus, E1 () = C(℘ ). (2) If f is any elliptic function with respect to , then f = R1 ◦ ℘ + ℘  R2 ◦ ℘,  ). where R1 and R2 are rational functions; thus, E() = C(℘,℘

Proof (1) If k ∈ C, then the equation f (z) = k has multiple roots only at points z for which f  (z) = 0, and this occurs at only finitely many congruence classes of points z. Thus, f (z) = k has its roots simple for all but finitely many values of k. We can, therefore, choose two distinct complex numbers c and d so that the roots of f (z) = c and of f (z) = d are all simple, and so that none of these roots are congruent to 0 or wj /2, j = 1,2,3. Since f is even, a complete set of roots of f (z) = c will have the form {a1,−a1, . . . ,an,−an }, these being simple and mutually noncongruent, and similarly for the roots {b1,−b1, . . . ,bn,−bn } of f (z) = d. Hence, the elliptic function g(z) :=

f (z) − c f (z) − d

has simple zeros at a1,−a1, . . . ,an, − an and simple poles b1,−b1, . . . , bn,−bn . Now, Corollary 16.3.8 implies that the equations ℘ (z) = ℘ (ai ) and ℘ (z) = ℘ (bi ) have simple roots, respectively, at z = ±ai and z = ±bi for every 1 ≤ i ≤ n. So, the elliptic function

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Part IV Classics, Geometry, and Dynamics

h(z) :=

(℘ (z) − ℘ (a1 ))(℘ (z) − ℘ (a2 )), . . . ,(℘ (z) − ℘ (an )) (℘ (z) − ℘ (b1 ))(℘ (z) − ℘ (b2 )), . . . ,(℘ (z) − ℘ (bn ))

has the same zeros and poles as g, with the same multiplicities (all simple). Hence, Theorem 16.2.8 implies that g = μh for some constant μ = 0. Solving f (z) − c (℘ (z) − ℘ (a1 ))(℘ (z) − ℘ (a2 )), . . . ,(℘ (z) − ℘ (an )) =μ f (z) − d (℘ (z) − ℘ (b1 ))(℘ (z) − ℘ (b2 )), . . . ,(℘ (z) − ℘ (bn )) for f (z), we see that f (z) = R1 (℘ (z)) with some rational function R1 . (2) If f is odd, then f/℘  is even, so by (1) we have that f = ℘  R2 ◦ ℘ for some rational function R2 . In general, if f is any elliptic function, then f (z) =

1 1 (f (z) + f (−z)) + (f (z) − f (−z)), 2 2

where 12 (f (z) + f (−z)) is even and elliptic while 12 (f (z) − f (−z)) is odd and elliptic. So, by the above arguments, we have that f = R1 ◦ ℘ + ℘  R2 ◦ ℘,  where R1 and R2 are some rational functions. The proof is complete. Using the differential equation (℘  )2 = 4℘ 3 − g2 ℘ − g3 , we can reduce any rational function of ℘ and ℘  to the form R1 (℘) + ℘  R2 (℘) by eliminating powers of ℘  , e.g., (4℘ 4 − g2 ℘ 2 − g3 ℘) − ℘  ℘ ℘℘  ℘℘  (℘  − 1) = . =  2 ℘+1 (℘ ) − 1 4℘ 3 − g2 ℘ − g3 ℘ − 1 We can view Theorem 16.3.5 as an algebraic equation between the functions ℘ and ℘  . We now show that any two functions in E() are connected by an algebraic equation. Theorem 16.4.2 If f ,g ∈ E(), then there exists a nonzero irreducible polynomial φ(x,y) with complex coefficients, such that φ(f ,g) is identically zero. Proof

If we choose any polynomial in two variables x,y, say F (x,y) =

n m  

αkl x k y l ,

αkl ∈ C,

k=1 l=1

then the function h(z) = F (f (z),g(z)) is an elliptic function with poles only at the poles of f or g. If f and g have M and N poles, respectively, then h has at most mM + nN poles counting with multiplicities in each case. Therefore, unless h is identically zero, it has at most mM + nN zeros by Theorem 16.2.6. We now show that if m and n are

16 Classics of Elliptic Functions: Selected Properties

151

large enough, then we can choose the coefficients αkl , so that h has more than mM + nN zeros and hence h(z) ≡ 0. To do this, we let z1, . . . ,zmn−1 be mn − 1 noncongruent points distinct from the poles of f and g. Now we regard h(zj ) =

n m  

αkl f (zj )k g(zj )l = 0, j = 1, . . . ,mn − 1,

(16.20)

k=1 l=1

as a set of mn − 1 homogeneous linear equations in the mn unknowns αkl . As there are more unknowns than equations, this set of equations has a nontrivial solution, i.e., there exists αkl , not all zero, satisfying (16.20). Thus, f (x,y) is not identically zero, but F (f (z),g(z)) = h(z) = 0 at the points z = z1, . . . ,zmn−1 . Now for m,n large enough, mn − 1 > mM + nN , and so by choosing the coefficients αkl as above we must have h(z) identically equal to 0, i.e., F (f ,g) = 0. We can factorize F (x,y) within the polynomial ring C[x,y] as product F (x,y) = F1 (x,y)F2 (x,y), . . . ,Fr (x,y) of irreducible polynomials. Thus, F1 (f ,g)F2 (f ,g), . . . ,Fr (f ,g) = 0 within  the field E(), so some Fi (f ,g) = 0, and we can take φ to be Fi . The Weierstrass σ -function is defined as   σ (z) : = z g(w,z), where w∈

 z 1  z 2 z exp + g(w,z) := 1 − . w w 2 w 

(16.21)

The factor (1 − (z/w)) is introduced in g(w,z) to give g(z), a simple zero at each lattice point w ∈ , while the exponential factor is included to guarantee convergence of the infinite product. In order to see this convergence, consider ∈  with large moduli and denote by log0 the holomorphic branch of the logarithm defined on a neighborhood of 1 and sending 1 to 0. Define then log∗ , a holomorphic branch of log g, as follows:  1  z 2 z z log∗ g(w,z) := log0 1 − . + + w w 2 w   Expanding now log0 1 − wz into its Taylor series near zero, we obtain that 1  z 3 log∗ g(w,z) = + higher terms. 3 ω

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Part IV Classics, Geometry, and Dynamics

This guarantees the convergence of the product defining σ (z). We have that  σ  (z) d 1  1 z 1 = log(σ (z)) = + + + 2 = ζ (z). σ (z) dz z z−w w w w∈

After differentiating (term by term), we recover (16.15) from the previous section: ζ  (z) = −℘ (z). The functions ζ and σ which were just introduced are not elliptic, because, as we shall show, they are not invariant under the translations C  z → z + w ∈ C, w ∈ . However, an examination of their behavior under translations will enable us to construct elliptic functions with prescribed properties. Since ζ  (z) = −℘ (z), we have that ζ  (z + wj ) = ζ  (z) for j = 1,2, so the integration with respect to z gives ζ (z + wj ) = ζ (z) + ηj , j = 1,2, where η1,η2 are constants independent of z. If w ∈ , then w = mw1 + nw2 , where m,n ∈ Z; hence, ζ (z + w) = ζ (z) + η,

(16.22)

η = mη1 + nη2 .

(16.23)

where

Let R be a fundamental parallelogram for , containing 0 in its interior, with vertices ξ , ξ + w1 , ξ + w1 + w2, ξ + w2 , ξ ∈ C. Denote its edges by 1 , C is 2 , 3 , 4 , with anti-clockwise orientation. Since the function ζ : C −→  meromorphic and has a single pole in R at 0 with residue equal to 1, we have that 2π i =

∂R

ζ (z)dz =

4 

ζ (z)dz.

j =1 j

Now ζ (z)dz = − 3



(ζ (z + w2 ))dz = − 1

 ζ (z) + η2 dz

1

So, ζ (z)dz + 1

ζ (z)dz = 3

η2 dz = −η2 w1 . 1

16 Classics of Elliptic Functions: Selected Properties

153

Similarly, ζ (z)dz + 2

ζ (z)dz = η1 w2 ; 4

hence, η1 w2 − η2 w1 = 2π i.

(16.24)

This equation is usually referred to as Legendre’s relation. It implies that at least one of η1,η2 is nonzero, so ζ (z) is not elliptic. To see how σ (z) behaves under translations, we use the formula σ  (z) = ζ (z). σ (z) From this and (16.22), we get that σ  (z) σ  (z + w) = + η, σ (z + w) σ (z) where η = ηw := mη1 + nη2 for w = mw1 + nw2 . Integrating this, we obtain that log σ (z + w) = log σ (z) + ηz + c, where logs are any fixed branches of the logarithm and where c is a constant depending only on w. Hence, σ (z + w) = σ (z) exp(ηz + c). We will now evaluate c. First suppose that w/2 ∈ / , so that σ (w/2) = 0. Putting z = −(w/2) and using the fact that σ is an odd function, we obtain that    1 1 1 w = −σ w exp − ηw + c . σ 2 2 2 So, canceling σ (w/2), we get that

1 ηw . exp(c) = − exp 2

Thus,



 1 σ (z + w) = −σ (z) exp η(z + w , 2

(16.25)

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Part IV Classics, Geometry, and Dynamics

provided that w/2 ∈ / . Repeating this, we get that     1 3 σ (z + 2w) = σ (z) exp η z + w exp η z + w 2 2 = σ (z) exp(2η(z + w)). Turning to the notation η = ηw , a straightforward induction then yields:   (16.26) σ (z + 2kw) = σ (z) exp 2kηw (z + kw) for every integer k ≥ 1 whenever w/2 ∈ / . Assume now that w/2 ∈ \{0}. Then there exists a unique integer l ≥ 1 such that / . 2−l w ∈  but 2−(l+1) w ∈ Applying then (16.26) with w replaced by 2−l w and k = 2l−1 , we get that     σ (z + w) = σ z + 2 · 2l−1 (w/2l ) = σ (z) exp 2l ηw/2l (z + 2l−1 (w/2l ))   w  . = σ (z) exp ηw z + 2 Combining this along with (16.25) and including the trivial case of w = 0, we get that   1 σ (z + w) = εσ (z) exp ηw z + w , (16.27) 2 where

% ε :=

+1 if 12 w ∈  −1 otherwise.

Of course, w/2 ∈  if and only if both m and n are even in the representation w = mw1 + nw2 . So, ε = (−1)mn+m+n . We now turn to the problem, posed in Section 16.2, of finding an elliptic function f ∈ E() with given zeros and poles. We will show that the conditions (1), (2 ), and (3), given in Section 16.2, are not only necessary but also sufficient for the existence of such an f . This is in contrast to the situation for rational functions on the sphere, where conditions (1) and (2) of Section 16.2 are already necessary and sufficient. Theorem 16.4.3 Let [a1 ], . . . ,[ar ] and [b1 ], . . . ,[bs ] be elements of C/ for a lattice  and k1, . . . ,kr ,l1, . . . ,ls be positive integers. If conditions (1), (2 ), and (3) of Section 16.2 hold, then there exists an elliptic function f ∈ E() with zeros of multiplicity kj at each [aj ], poles of multiplicity lj at each bj , and no other zeros or poles.

16 Classics of Elliptic Functions: Selected Properties

155

Proof Let u1, . . . ,un be the elements a1, . . . ,ar , each aj being listed kj " times, so that n = rj =1 kj . Similarly, let v1, . . . ,un be the elements b1, . . . ,bs counting with multiplicities lj . Then (3) takes on the form n 

uj −

j =1

n 

vj = w ∈ ;

j =1

there is no loss in replacing u1 by v1 + w, so that (3) now reads n 

uj =

j =1

n 

vj .

(16.28)

j =1

Now consider the function f (z) :=

σ (z − u1 ), . . . ,σ (z − un ) . σ (z − v1 ), . . . ,σ (z − vn )

Since σ (z) is an analytic function, f (z) is meromorphic. By (16.27), we have that σ (z − uj + wi )

  1 = −σ (z − uj ) exp ηi z − uj + wi , 2

j = 1, . . . ,n,

i = 1,2,

and similarly for σ (z − vj + wi ). Hence, with the use of (16.28), for i = 1,2 we have that "  n 1 (−1)n exp η (z − u + w ) i i i j =1 2 "  · f (z) f (z + wi ) = n 1 n (−1) exp η (z − v + w ) i i i j =1 2 ⎛ ⎞ n  = exp ⎝ηi (vj − uj )⎠ f (z) j =1

= exp (ηi · 0) f (z) = f (z). Thus, the function f is doubly periodic with respect to ; hence, f is elliptic. Applying Theorem A.0.26 to infinite product (16.21) for σ (z), we see that σ (z) has simple zeros at the lattice points z ∈  and that σ (z) = 0 for z ∈ / . Hence, zeros and poles of f (z) are at [a1 ], . . . ,[ar ] and [b1 ], . . . ,[bs ] with  multiplicities k1, . . . ,kr and l1, . . . ,ls , respectively. If g is any other elliptic function with the same zeros and poles as f , then, by Theorem 16.2.8, g(z) = cf (z) for some constant c = 0.

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Part IV Classics, Geometry, and Dynamics

16.5 The Discriminant of a Cubic Polynomial Let  ⊆ C be a lattice. It follows from Theorem 16.3.5 that the Weierstrass  = √p, where p is a cubic elliptic function ℘ satisfies the equation ℘ polynomial of the form p(z) = 4z3 − c2 z − c3,

c2,c3 ∈ C,

z = ℘ .

(16.29)

Any polynomial of the form (16.29) is said to be in Weierstrass normal form. By means of the substitution θ (z) := az + b, a,b ∈ C, a = 0, any cubic polynomial may be brought into this form. Now, the map θ : C → C is a bijection, preserving the multiplicities of the roots of polynomials. So, without loss of generality, we can restrict our attention to cubic polynomial p in Weierstrass normal form. If e1,e2,e3 are the roots of the polynomial p in (16.29), then we define the discriminant of p to be p = 16(e1 − e2 )2 (e2 − e3 )2 (e3 − e1 )2 .

(16.30)

Clearly, these roots are distinct if and only if p = 0. We shall prove the following. Theorem 16.5.1 If p(z) = 4z3 − c2 z − c3 , then p = c23 − 27c32 . Proof

Since p(z) = 4(z − e1 )(z − e2 )(z − e3 ),

(16.31)

by equating coefficients between this and (16.29), we have that e1 + e2 + e3 = 0, e1 e2 + e2 e3 + e3 e1 = − e1 e2 e3 =

c2 , 4

(16.32)

c3 . 4

The remaining symmetric functions of the roots may be obtained from (16.32), e.g., e12 + e22 + e32 = (e1 + e2 + e3 )2 − 2(e1 e2 + e2 e3 + e3 e1 ) =

c2 2

and e12 e22 + e22 e32 + e32 e12 = (e1 e2 + e2 e3 + e3 e1 )2 − 2e1 e2 e3 (e1 + e2 + e3 ) =

c22 . 16

16 Classics of Elliptic Functions: Selected Properties

157

Now, differentiating (16.29) and (16.31) at z = e1 , we have that 4(e1 − e2 )(e1 − e3 ) = p  (e1 ) = 12e12 − c2 with similar expression for p  (e2 ) and p  (e3 ). Hence, 1 p = − p (e1 )p (e2 )p (e3 ) 4 3 1 =− (12ei2 − c2 ) 4 i=1

1 = − 1728(e1 e2 e3 )2 − 144c2 (e12 e22 + e22 e32 + e32 e12 ) 4  + 12c22 (e12 + e22 + e32 ) − c23 1 = − (108c32 − 9c23 + 6c23 − c23 ) 4 = c23 − 27c32 .



Corollary 16.5.2 A polynomial p(z) = 4z3 − c2 z − c3 has distinct roots if and only if c23 − 27c32 = 0. One can give a direct proof of Corollary 16.5.2 without introducing p , by eliminating z between the equation p(z) = 0 and p  (z) = 0, thus giving a necessary and sufficient condition for p and p to have a common root. We have chosen the above proof since the discriminant is an interesting function in its own right. By Theorem 16.3.5, (16.16), and (16.17),  the Weierstrass function ℘  = p(℘ ), where p is a cubic associated with a lattice  satisfies ℘  polynomial in Weierstrass normal form p(z) = 4z3 − g2 z − g3 with g2 = g2 () = 60





(16.33)

w −4

w∈

and g3 = g3 () = 140





w −6 .

w∈

If we write () for the discriminant p of p, then, by Theorem 16.5.1, we have that () = g2 ()3 − 27g3 ()2 .

(16.34)

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Part IV Classics, Geometry, and Dynamics

Theorem 16.3.9 implies that p has distinct roots, so () = 0 by Corollary 16.5.2; hence, we may define a function J :  −→ J () by J () :=

g2 ()3 g2 ()3 . = () ()2 − 27g3 ()2

(16.35)

This function J :  −→ J () is called the modular function associated with the lattice . For a similar lattice μ, μ = 0, we have that   g2 (μ) = 60 (μw)−4 = μ−4 g2 () (16.36) w∈

and g3 (μ) = 140





(μw)−6 = μ−6 g3 (),

(16.37)

w∈

so that (μ) = μ−12 (). Thus, J (μ) = J ()

(16.38)

for all μ ∈ C\{0}, so that similar lattices determine the same values of J . We, therefore, may, and now will, regard g1,g2 , , and J as functions of the modulus τ ∈ H by evaluating them on the lattice  = [1,τ ], which has τ as one of its moduli. Thus, g2 (τ ) = 60





(m + nτ )−4,

m,n

g3 (τ ) = 140





(m + nτ )−6,

(16.39)

m,n

where

" m,n

denotes summation over all (m,n) ∈ Z×Z except for (0,0). Then (τ ) = g2 (τ )3 − 27g3 (τ )2

(16.40)

and J (τ ) =

g23 (τ ) . (τ )

(16.41)

16 Classics of Elliptic Functions: Selected Properties

159

In [JS], it is proved (see formula (6.4.9)) that expansion of J (τ ) has the form  1 1 J (τ ) = + 744 + 196884q + · · · . (16.42) 1728 q By Theorem 16.1.7, for every T ∈ P SL(3,Z), the lattices  = [1,τ ] and [1,T (τ )] are similar; hence, J (T (τ )) = J (τ ) by (16.38). So, we have the following. Theorem 16.5.3 J (T (τ )) = J (τ ) for all τ ∈ P SL(2,Z). This precisely means that J (τ ) is invariant under the action of the modular group P SL(2,Z). We will now show that the functions g2 (τ ),g3 (τ ), and (τ ) come close to sharing this property. If aτ + b  ∈C T: C  τ −→ cτ + d is an element of P SL(3,Z), then   aτ + b −4  m+n g2 (τ ) = 60 cτ + d m,n   = 60(cτ + d)−4 (m(cτ + d) + n(aτ + b))−4 m,n

= 60(cτ + d)−4





((md + nb) + (mc + na)τ )−4 .

m,n

Since ad − bc = 1, the transformation (m,n) −→ (md + nb,mc + na) merely permutes the elements of the indexing set Z × Z\{(0,0)}. Hence, by Theorem 16.1.5 and by absolute convergence (implying unconditional convergence), we have that   (m + nτ + b)−4 = (cτ + d)−4 g2 (τ ). g2 (T (τ )) = 60(cτ + d)−4 m,n

(16.43) Similarly, g3 (T (τ )) = (cτ + d)−6 g3 (τ );

(16.44)

(T (τ )) = (cτ + d)−12 (τ ),

(16.45)

hence,

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Part IV Classics, Geometry, and Dynamics

from which we immediately obtain an alternative proof of Theorem 16.5.3. In the special case where a = b = d = 1 and c = 0, we have that T (τ ) = τ + 1, giving the following theorem. Theorem 16.5.4 The functions g2 (τ ), g3 (τ ), and J (τ ) are periodic with respect to Z. It is also useful to determine the effect on these functions of the orientationreserving transformation of H of the form T (τ ) =

aτ + b , cτ + d

a,b,c,d ∈ Z, ad − bc = −1.

(16.46)

Calculations similar to those above give that g2 (T (τ )) = (cτ + d)−4 g2 (τ ), g3 (T (τ )) = (cτ + d)−6 g3 (τ ),

(16.47)

(T (τ )) = (cτ + d)−13 (τ ), J (T (τ )) = J (τ ). Now we shall prove the following.

Theorem 16.5.5 The functions g2 , g3 , , and J : H −→ C are analytic on H. Proof

Given any τ0 ∈ H, let δ :=

1 Im(τ0 ), 2

so that δ > 0, and let D(τ0,δ) := {τ ∈ H : |τ − τ0 | ≤ δ}. Now the functions H  τ −→ (m + nτ )−4 ∈ C and H  τ −→ (m + nτ )−6 ∈ C are holomorphic for all (m,n) ∈ Z×Z\{(0,0)}, so if we can show that the series (16.39) defining the functions g2 and g3 are absolutely uniformly convergent on each set D(τ0,δ), τ0 ∈ H, then these two functions are analytic on H. For all m,n ∈ Z with n = 0, we have that −m/n ∈ R; hence, m     + τ0  ≥ Im(τ0 ) = 2δ. n Therefore, for all m,n ∈ Z (including n = 0) and τ ∈ D(τ0,δ), we have that |(m + nτ ) − (m + nτ0 )| = |n||τ − τ0 | ≤ |n|δ ≤

1 |m + nτ0 |; 2

16 Classics of Elliptic Functions: Selected Properties

161

so the triangle inequality gives |m + nτ | ≥ |m + nτ0 | − |(m + nτ ) − (m + nτ0 )| ≥

1 |m + nτ0 |. 2

Thus, for any r > 0, we have that |m + nτ |−2r ≤ 22r |m + nτ0 |−2r for all τ ∈ D(τ0,δ) and (m,n) = (0,0). By Theorem 16.3.1, the series   |m + nτ0 |−2r m,n

converges for each r > 1, so, by the Weierstrass test, the series   |m + nτ0 |−2r m,n

converges absolutely uniformly on D(τ0,δ). Putting r = 2 and r = 3, we, thus, see that the functions g2 and g3 are analytic on H. It immediately follows from (16.40) that (τ ) is analytic on H; since (τ ) = 0 on H by Theorem 16.3.9 and Corollary 16.5.2, it follows from (16.41) that the function J is analytic on H.  Lemma 16.5.6 We have the following two properties. (1) If 2Re(τ ) ∈ Z, then g2 , g3 , , and J are all real. (2) If |τ | = 1, then g2 (τ ) = τ 4 g2 (τ ), g3 (τ ) = τ 6 g3 (τ ), (τ ) = τ 12 (τ ), and J (τ ) = J (τ ). Proof (1) If 2Re(τ ) = n ∈ Z, then τ is fixed by reflection T (τ ) = n − τ , which is of type (16.46) with a = −1, b = n, and d = 1, so the result follows from (16.47). (2) If |τ | = 1, then τ is fixed by the inversion T (τ ) = 1/τ in the unit circle; putting a = d = 0 and b = c = 1 in (16.47), we have that g2 (τ ) = g2 (1/τ ) = τ −4 g2 (τ ) = τ 4 g2 (τ ), and similarly for the other three functions.



Theorem 16.1.10 asserts that the modular group P SL(2,Z) has a fundamental region % & 1 F := z ∈ H : |z| ≥ 1 and |Rez| ≤ . 2

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Part IV Classics, Geometry, and Dynamics

We, thus, immediately have the following corollary. Corollary 16.5.7 J (τ ) is real whenever τ is on the imaginary axis or on the boundary ∂R of R. Corollary 16.5.8 g2 () = g3 (i) = J () = 0 and J (i) = 1, where  = e2π i/3 . Proof Part (1) of Lemma 16.5.6 shows that g2 and g3 both take real values at i and , while part (2) shows that g2 (ρ) = ρg2 () and g3 (i) = −g3 (i). Thus,  g2 (ρ) = g3 (i) = 0, so (16.41) gives J () = 0 and J (i) = 1. Let L := L1 ∪ L2 ∪ L3, where

% & 1 L1 := τ ∈ H : |τ | ≥ 1 and Re(τ ) = − , 2 % & 1 L2 := τ ∈ H : |τ | = 1 and − ≤ Re(τ ) ≤ 0 , 2 L3 := {τ ∈ H : |τ | = 1

and

Re(τ ) = 0}.

By Corollary 16.5.7, we have that J (L) ⊆ R, but, in fact, we can prove equality. Theorem 16.5.9 J (L) = R. Proof

If τ ∈ L3 , then we have that τ = iy with some y ≥ 1. Then q =: e2π iτ = e−2πy −→ 0

through positive real numbers as y → +∞. So (16.42) gives J (τ ) = (q −1 + 744 + · · · )/1728 −−−−−→ +∞. y→+∞

(16.48)

Similarly, on L1 , we have that τ = − 12 + iy with some y ≥ 0 and q := −e2πy −−−−−→ 0 y→+∞

and the convergence is through negative real numbers. Hence, lim J (τ ) = −∞.

y→+∞

(16.49)

Since J (L) ⊆ R, J |L : T −→ R is continuous as being analytic on H by Theorem 16.5.5, and since L is connected, it follows from (16.48) and (16.49) that J (L) = R. 

16 Classics of Elliptic Functions: Selected Properties

163

By Theorem 16.5.3, J is constant on each orbit of P SL(2,Z) in H. We shall prove the following theorem. Theorem 16.5.10 For each c ∈ C, there is exactly one orbit of P SL(2,Z) in H on which J takes the value c. Proof Each orbit of  meets the fundamental region F either at a unique point in the interior of F or else at one or two equivalent points on ∂F. First, suppose that c ∈ C\R. Since J (∂F) ⊂ R by Corollary 16.5.7, it is sufficient to show that there is a unique solution to the equation J (τ ) = c in IntF. By Theorem 16.5.5 and Corollary 16.5.8, J is analytic and not identically equal to c, so the function g(τ ) =

J  (τ ) J (τ ) − c

is meromorphic on H. We can use (16.42) to express g(τ ) as a function of q = e2π iτ , which is meromorphic at q = 0 since J (τ ) is. Hence, g(τ ) is analytic for sufficiently small nonzero |q|, i.e., provided that Imτ is sufficiently large, say Imτ ≥ K for some K > 1. Thus, the poles of g(τ ) in F all lie in the interior of G := {τ ∈ F : Imτ ≤ K}. So, the sum of the residues of g(τ ) in F (and, hence, the number of solutions, counting multiplicities, of J (τ ) = c in F) is equal to 1 2π i

g(τ )dτ,

(16.50)

∂G

where the boundary ∂G is given the positive orientation. Now the sides Reτ = −1/2 and Reτ = 1/2 of G are equivalent under the transformation τ −→ τ + 1 of , so J (τ ) and, hence, g(τ ) take the same values at equivalent points on these sides. Hence, the integrals of g(τ ) along these sides cancel in (16.50), and similarly the integral along the unit circle from ρ to i cancels with the integral from i to ρ + i, using the transformation τ → 1/τ . Hence, g(τ )dτ = ∂G

g(τ )dτ, γ

where γ is the side Im(τ ) = K of G oriented from 12 + iK to − 12 + iK. Away from the poles of g(τ ), each branch of the logarithm function of J satisfies

164

Part IV Classics, Geometry, and Dynamics J  (τ ) d (log J (τ ) − c) = = g(τ ). dτ J (τ ) − c

So, G(τ )dτ = [log(J (τ ) − c)]γ , γ

which is the change in the value of log(J (τ ) − c) arising from analytic continuation along γ . As τ follows γ , q winds once (in the negative direction) around the circle δ given by |q| = e−2π K , starting and finishing at −e−2π K . By (16.42), q(J (τ ) − c) is analytic and nonzero for 0 ≤ |q| ≤ e−2π K . Since this set is simply connected, the Monodromy Theorem A.0.28 implies that [log q(j (τ ) − c)]γ = 0. So, [log q(j (τ ) − c)]γ = [log q(j (τ ) − c) − log q] = [log q]γ = 2π i. Hence, (16.50) shows that the number of solutions of J (τ ) = c in F is equal to (1/2π i) = 1, as required. Finally, suppose that c ∈ R. By Theorem 16.5.9, there is at least one orbit of  on which J takes the value c. If there were more than one such orbit, there would be two inequivalent solutions τ1,τ2 of J (τ ) = c, so, by choosing c ∈ C\R sufficiently close to c, we would have two inequivalent solutions τ1 and τ2 of J (τ ) = c , close to τ1 and τ2 , respectively. We have already shown that this is impossible, so the orbit is unique.  We record the following two consequences of this theorem. Corollary 16.5.11 If c2,c3 ∈ C satisfy c23 − 27c32 = 0, then there is a lattice  ⊆ C with gk () = ck for k = 2,3. Proof First suppose that c2 = 0, so that c3 = 0. By Corollary 16.5.8, g2 (ρ) = 0 and, hence, g3 (ρ) = 0 since g2 (τ )3 − 27c32 (τ ) does not vanish on H. We can, therefore, choose μ ∈ C\{0} such that μ−6 g3 (ρ) = c3 , so putting  := μ[1,τ ] = [μ,μτ ] we have that g2 () = μ−4 g2 (ρ) = 0 = c2 ; hence, g3 () = μ−6 g3 (ρ) = c3 , as required. Similarly, if c3 = 0, then c2 = 0. We have that g3 (i) = 0 = g2 (i), so we can choose μ ∈ C\{0} satisfying μ−4 g2 (i) = c2 , and then  = [μ,μi] satisfies g2 () = μ−4 g2 (i) = c2 and g3 () = 0 = c3 .

16 Classics of Elliptic Functions: Selected Properties

165

Finally, we consider the general case, where c2 = 0 = c3 . By Theorem 16.5.10, there exists τ ∈ H such that c23

J (τ ) =

c23 − 27c32

.

(16.51)

For any μ ∈ C\{0}, the lattice  = [μ,μτ ] satisfies g2 () = μ−4 g2 (τ ) and g3 () = μ−6 g3 (τ ), which are both nonzero since J (τ ) = J () =

c23 c23

− 27c32

, (16.52)

g23 () g23 () − 27g32 ()

does not take the value 0 or 1 by (16.51) and the fact that c2 = 0 = c3 . We can, therefore, choose μ = 0 so that μ2 =

c2 g3 (τ ) ; c3 g2 (τ )

hence, c2 g2 (τ ) μ−4 g2 (τ ) = . = −6 g3 (τ ) c3 μ g3 (τ ) Thus, gk () = λck (k = 2,3) for some λ = 0, so substituting in (16.52) and using (16.51) we have that J () =

c23 c23 − 27c32

and J () =

λ3 c23 λ3 c23 − 27λ2 c32

=

c23 c23 − 27λ−1 c32

.

Hence, λ = 1 and so gk () = ck , k = 2,3, as required.



Corollary 16.5.12 The numbers g2 and g3 form a set of full invariants for lattices in C. More precisely, if , are two lattices in C, then ) *  =  ⇐⇒ g2 ( ) = g2 () and g3 ( ) = g3 () . In addition, given ξ ∈ C\{0}, ) *  = ξ  ⇐⇒ g2 ( ) = ξ 4 g2 () and g3 ( ) = ξ 6 g2 () .

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16.6 Weierstrass ℘-Functions II In this section, we collect and summarize, for the convenience of the reader, some properties of the Weierstrass ℘-functions which have been proved in the present chapter. We actually bring up only those properties that will be used in Chapter 19 in our constructions of dynamically significant examples. Recall that, given any lattice  in C, the Weierstrass elliptic function ℘ is defined by the following formula:   1 1 1 − ∈ C. C  z −→ ℘ (z) = 2 + z (z − w)2 w2 w∈\{0}

Replacing z by −z in the definition, we see that ℘ is an even function which is analytic in C\ and has poles of order 2 at each w ∈  (see Theorems 16.3.3 and 16.3.4). The derivative of the Weierstrass elliptic function ℘ is also an elliptic function, periodic with respect to , and is expressible by the series  1  (z) = −2 . (16.53) ℘ (z − w)3 w∈

The Weierstrass elliptic function ℘ and its derivative are related by the differential equation  (z)2 = 4℘ (z)3 − g2 ℘ (z) − g3, ℘

(16.54)

where (see (16.16), (16.17), and (16.18)), we recall that   w −4 and g3 () = 140 w −6 . g2 () = 60 w∈\{0}

w∈\{0}

 are called the critical points of ℘ and the set of Recall that the zeros of ℘  all of them is denoted by Crit(℘ ). As an immediate consequence of Theorem 16.3.6, we get the following.

Theorem 16.6.1 If  = [w1,w2 ] ⊆ C is a lattice, then, with w3 = w1 + w2 , we have that w  w  w  1 2 3 + ∪ + ∪ + Crit(℘ ) = 2 2 2 and   ℘ Crit(℘ ) = {e1,e2,e3 }, where e1 := ℘

w  1

2

, e2 := ℘

w  2

2

, and e3 := ℘

w  3

2

.

16 Classics of Elliptic Functions: Selected Properties

167

For any lattice , the Weierstrass elliptic function and its derivative satisfy the following properties: for every z ∈ C and every s ∈ C\{0}, we have that 1 ℘ (z) − homogeneity of ℘, (16.55) s2 1    . (sz) = 3 ℘ (z) − homogeneity of ℘ (16.56) ℘s s Verification of these properties can be done by substituting into the corresponding series definitions. We discern below some distinguished types of lattices. ℘s (sz) =

Definition 16.6.2 We define the following five classes of lattices. (0) A lattice  is called real if and only if  = . (1) A lattice  is called real rectangular if and only if there exist λ1,λ2 ∈ C such that  = [λ1,λ2 ] and λ1 ∈ R and λ2 ∈ iR. Then (obviously)  is real. Any lattice similar to a real rectangular lattice is called rectangular. (2) A lattice  is called real rhombic if and only if  = [λ,λ] with some λ ∈ C\{0}. Then (obviously)  is real. Any lattice similar to a real rhombic lattice is called rhombic. (3) A lattice  is called a square lattice if and only if i = . Equivalently,  is a square lattice if and only if it is similar to the lattice [1,i]. In addition, a square lattice is real if and only if it is of the form [λ,λi], λ ∈ (0,+∞). (4) A lattice  is called triangular if and only if  = e2π i/3 . In each of the Cases (1)–(3), the numbers λ1 and λ2 can be chosen so that the fundamental parallelogram with vertices 0,λ1,λ2 , and λ3 := λ1 + λ2 is a rectangle, rhombus, or square, respectively. In Case (4), the fundamental parallelogram comprises two equilateral triangles.

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Definition 16.6.3 A meromorphic function f : C −→  C is called real if and only if f (¯z) = f (z) for all z ∈ C. The proof of the following proposition can be found in [Du]. Proposition 16.6.4 For a lattice  ⊆ C, the following conditions are equivalent. (1) (2) (3) (4)

 is real.  is real rectangular or real rhombic. g2 and g3 are real. The Weierstrass ℘ -function is real.

As an immediate consequence of the definition of real matrices, we get the following. Observation 16.6.5 If a lattice  ⊆ C is real, then so is the lattice i. Proposition 16.6.6 If  ⊆ C is a lattice, then the following three conditions are equivalent. (1)  is triangular. (2)

+ , + πi , 2π i πi  = λ,λe 3 = λe 3 ,λe− 3 ,

with some λ ∈ C\{0}. (3) g2 () = 0. In addition, (4) For every g ∈ C\{0}, there exists a unique triangular lattice  ∈ C such that g3 () = g. (5) The (three) critical values of the Weierstrass ℘ -function coincide with the cubic roots of g3 ()/4. If g3 () ∈ R, then e3 () are the real cubic roots of g3 ()/4. (6) A triangular lattice  ∈ C is real if and only if , + πi , + 2π i πi  = λ,λe 3 = λe 3 ,λe− 3 , with some λ ∈ (0,+∞) ∪ i(0,+∞).

16 Classics of Elliptic Functions: Selected Properties

169

Then we have the following: (6a) g3 (λ[1,]) > 0 ⇔ λ ∈ (0,+∞) and g3 (λ[1,]) < 0 ⇔ λ ∈ i(0,+∞). (6b)   λ  ∩ R = λZ, Crit ℘ ∩ R = + λZ, and ℘  (z) > 0 2 for all z ∈ R\λZ, and (6c) For every k ∈ Z, the real-valued function ℘ |[λk,λ(k+1)] (6c1) (6c2) (6c3) (6c4)

is continuous, * ) is strictly decreasing on ) λk,λk + λ2 , * is strictly increasing on λk + λ2 ,λ(k + 1) , has a unique absolute minimum at the point λk + value e3 (). In addition,

(6c5)

℘

λk,λk +

λ 2

λ 2

with the

. = [e3 (),+∞) . λ = ℘ λk + ,λ(k + 1) . 2

Proof The implication (2)⇒(1) is obvious. The implication (1)⇒(3) directly follows from (16.36) applied with μ = . Formula (16.37) yields g3 ([μ,μ]) = μ−6 g3 ([1,]).

(16.57)

So, knowing that the lattices [μ,μ], μ = 0, are all triangular, by varying μ over all C\{0}, we conclude, from Corollaries 16.5.11 and 16.5.12 and the already proven implication (1)⇒(3), that item (4) holds, that (1) implies (2), and that (3) implies (2). Item (5) directly follows from (16.54) and item (3) of the current proposition. The proof is complete. Proving the equivalence (first) part of item (6) and also item (6a), the fact that the lattices [λ,λ] = [λ,λ], λ ∈ (R ∪ iR)\{0}, are all real is obvious. Since g3 () ∈ R\{0} for every real lattice , by varying μ over all (0,+∞) ∪ i(0,+∞) in (16.57), we obtain the converse implication too and item (6a) as well.

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Part IV Classics, Geometry, and Dynamics

The first two formulas of item (6b) are immediate with the use of Theorem 16.6.1 and the realness and triangularity of . The last formula of item (6b),  (z) > 0, directly follows from (16.53) after differentiating it. Items i.e., ℘ (6c1)–(6c4) are immediate consequences of item (6b).  Proposition 16.6.7 If  ⊆ C is a lattice, then the following three conditions are equivalent. (1)  is a square lattice. (2)  = [λ,λi] with some λ ∈ C\{0}. (3) g3 () = 0. In addition, (4) For every g ∈ C\{0}, there exists a unique square lattice  ∈ C such that g2 () = g. (5) If  is a square lattice, then the (three) critical values of the Weierstrass ℘ -function are: e1 =

1 1 g2 () 2 , e2 = −e1, and e3 = 0. 2

In particular, e3 is a pole of ℘ . If, in addition, λ ∈ (0,+∞), then g2 () > 0 and e1 is a real positive number. (6) A square lattice  ∈ C is real if and only if  = [λ,λi] πi

with some λ ∈ (0,+∞) ∪ e 4 (0,+∞). Then we have the following: (6a) g2 (λ[1,i]) > 0 ⇔ λ ∈ (0,+∞) and g2 (λ[1,i]) < 0 ⇔ λ ∈ i(0,+∞). (6b)   λ  ∩ R = λZ, Crit ℘ ∩ R = + λZ, and ℘  (z) > 0 2 for all z ∈ R\λZ, and (6c) For every k ∈ Z, the real-valued function ℘ |[λk,λ(k+1)]

16 Classics of Elliptic Functions: Selected Properties

(6c1) (6c2) (6c3) (6c4)

is continuous, * ) is strictly decreasing on ) λk,λk + λ2 , * is strictly increasing on λk + λ2 ,λ(k + 1) , has a unique absolute minimum at the point λk + value e1 (). In addition,

(6c5)

℘

λk,λk +

λ 2

171

λ 2

with the

. = [e1 (),+∞) . λ = ℘ λk + ,λ(k + 1) . 2

Proof Implication (2)⇒(1) is obvious. Implication (1)⇒(3) directly follows from (16.36) applied with μ = i. Formula (16.37) yields g2 ([μ,μi]) = μ−4 g2 ([1,i]).

(16.58)

So, knowing that the lattices [μ,iμ], μ = 0, are all square, by varying μ over all C\{0}, we conclude, from Corollaries 16.5.11 and 16.5.12 and the already proven implication (1)⇒(3), that item (4) holds, that (1) implies (2), and that (3) implies (2). Now we shall prove item (5). Since, by item (3) of the current proposition g3 () = 0, the equation (16.54) takes on the form 4ei2 − g2 = 0, i = 1,2,3. So, it is only left to show that if λ ∈ (0,+∞), then g2 () > 0 and e1 > 0. But these follow by a direct calculation. πi Proving item (6), the fact that the lattices [λ,λi], λ ∈ (R ∪ e 4 R)\{0}, are all real is obvious. Since g2 () ∈ R\{0]} for every real lattice , by varying μ πi over all (0,+∞)∪e 4 (0,∞) in (16.58), we obtain the converse implication too. The first two formulas of item (6b) are immediate with the use of Theorem 16.6.1 and the realness and squareness of . The last formula of item (6b), i.e.,  (z) > 0, directly follows from (16.53) after differentiating it. Items (6c1)– ℘ (6c4) are immediate consequences of item (6b). The proof is complete.  We call a lattice  ⊆ C real rhombic square if and only if it is real rhombic and square. The following proposition is an immediate consequence of the previous one. Proposition 16.6.8 If  ⊆ C is a lattice, then the following three conditions are equivalent.

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Part IV Classics, Geometry, and Dynamics

(1)  ⊆ C is a real rhombic square lattice. π (2)  = ae 4 i [i,1] = b[1 + i,1 − i] for some a,b ∈ (0,+∞). (3) g3 () = 0 and g2 () < 0, and these two values determine  uniquely. In addition, (4) If  ⊆ C is a real rhombic square lattice, then the (three) critical values of the Weierstrass ℘ -function are: 1 1 g2 () 2 , e2 () = −e1 (), and e3 () = 0. 2 In particular, both e1 () and e2 () are pure imaginary numbers, e2 () = −e1 (), e1 ()i ∈ (0,+∞), and e3 is a pole of ℘ .

e1 () =

17 Geometry and Dynamics of (All) Elliptic Functions

Throughout the whole of Chapter 17, we deal with general elliptic functions, i.e., we impose no constraints on a given elliptic function. We first honestly deal with forward and, more importantly, backward images of open sets, especially with connected components of the latter. We mean to consider such images under all iterates f n , n ≥ 1, of a given elliptic function f . We provide sufficient conditions for the restrictions of iterates f n to such components to be proper or covering maps. The former property enables us to employ the whole machinery of Section 8.6 from Volume I to study the structure of such components and prove the existence of holomorphic inverse branches if “there are no critical points.” Holomorphic inverse branches will be one of our most common tools used throughout the rest of the book. We then apply these results to study images and backward images of connected components of the Fatou set. Section 17.2 continues this theme by providing some structural theorems about Fatou and Julia sets of elliptic functions. Some of these are immediate consequences of the results obtained in Part III, once we observe that each elliptic function belongs to Speiser class S, and some are more technical. The rest of this chapter is actually devoted to analyzing in greater detail the fractal properties of any elliptic function. Following the paper [KU3], by associating a given elliptic function with infinite alphabet conformal iterated function systems, and heavily utilizing its θ number, we provide a strong, somewhat surprising, lower bound for the Hausdorff dimension of the Julia sets of all elliptic functions. In particular, this estimate shows that the Hausdorff dimension of the Julia sets of any elliptic function is strictly larger than 1. We also provide a simple closed formula for the Hausdorff dimension of the set of points escaping to infinity of an elliptic function. In the last section of this chapter, we prove that no conformal measure of an elliptic function charges

173

174

Part IV Classics, Geometry, and Dynamics

the set of escaping points. However, the main theme of this chapter is Section 17.6, where we prove the existence of the Sullivan conformal measures with a minimal exponent for all elliptic functions and we characterize the value of this exponent in several dynamically significant ways. This section depends on the preparatory work done in Sections 17.4 and 17.5, which are also interesting on their own. It also heavily depends on Chapter 10 in the first volume. As indicated in the title, throughout this chapter, f : C −→  C is a (nonconstant) elliptic function, i.e., a nonconstant doubly periodic and meromorphic function. Let f ⊆ C be the set of all periods of f . We know from the first two sections of Chapter 16 that there then exist two unique vectors λ1, λ2 , Im(λ1 /λ2 ) = 0, such that f = [λ1,λ2 ] = {mλ1 + nλ2 : m,n ∈ Z }. In particular, f (z) = f (z + mλ1 + nλ2 ) for all z ∈ C and all n,m ∈ Z. Recall that the set f is called the lattice of the elliptic function f . We recall from Chapter 16 that z ∼f w if and only if w − z ∈ f and that ∼f is an equivalence relation. We will frequently use the abbreviated notation ∼f := ∼f .

(17.1)

Keeping {λ1,λ2 } as the basis of the lattice f , let Rf := {t1 λ1 + t2 λ2 : 0 ≤ t1,t2 ≤ 1} be the corresponding fundamental parallelogram of f . It follows from the periodicity of f that f (C) = f (Rf ). Therefore, the set f (C) is simultaneously compact, whence closed, and open in  C. Since  C is connected, the set f (C) is equal to  C. This means that each elliptic function is surjective, and also that C. f (C) = f (Rf ) = 

(17.2)

In particular, the set f −1 (∞) of all poles of the elliptic function f is not empty and, since it is f -invariant, it is infinite. So, invoking Definition 13.1.6 and Theorem 13.1.8, we immediately get the following.

17 Geometry and Dynamics of (All) Elliptic Functions

175

Theorem 17.0.1 Each elliptic function f : C −→  C is nonnearly entire and J (f ) =

∞ 

f −n (∞).

n=0

Recall that Crit(f ) is the set of critical points of f , i.e., Crit(f ) = {z ∈ C : f  (z) = 0}. Its image, f (Crit(f )), has been called the set of critical values of f

17.1 Forward and Inverse Images of Open Sets and Fatou Components In this section, which deals with arbitrary elliptic functions, we prove the fundamental results about backward and forward iterates of arbitrary open connected sets in  C. These results are important and interesting on their own and will have numerous applications in subsequent sections and chapters. The first applications will be given in the current section to study forward and backward images of connected components of Fatou sets. First, we shall deal with connected components of inverse images of open connected sets with sufficiently small diameters. These results are interesting on their own and will be frequently applied throughout the whole book. Their first consequences will be in studying, in the present section, connected components of inverse images of all open connected sets, i.e., with no further restrictions. We start with the following. For every subset A of a topological space X, we denote by Comp(A) the collection of all connected components of A. We shall easily prove the following. Proposition 17.1.1 If f : C −→  C is an elliptic function, V ⊆  C is an open connected set, n ≥ 1 is an integer, and U ∈ Comp(f −n (V )), then λ + U ∈ Comp(f −n (V )) for every λ ∈ f . Proof Since f n (λ + U ) = f n (U ) ⊆ V and since λ + U is a connected subset of  C, there exists a (unique) connected component Uλ of f −n (V ) containing λ + U . But then −λ + Uλ ⊇ −λ + (λ + U ) = U ;

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Part IV Classics, Geometry, and Dynamics

by the same token as above, −λ + Uλ is contained in a (unique) connected component Uλ of f −n (V ). So, U ⊆ −λ + Uλ ⊆ Uλ . Therefore, from the very definition of connected components, U = Uλ . Hence, also U = −λ + Uλ , whence Uλ = λ + U and the proof of Proposition 17.1.1 is complete.  Overusing a little bit of notation and having an elliptic function f : C −→  C, we now naturally extend the relation ∼f to subsets of C by declaring that A ∼f B if and only if there exists λ ∈ f such that B = λ + A.

(17.3)

Obviously, ∼f is an equivalence relation on all subsets of C. Our next results shed further light on the structure of connected components of the sets f −n (V ). Proposition 17.1.2 Let f : C −→  C be an elliptic function, V ⊆  C be an −n open connected set, and n ≥ 1 be an integer. If U1,U2 ∈ Comp(f (V )), then the following statements are equivalent. (1) There exists λ ∈ f such that U2 = λ + U1 . (2) For every x ∈ U1 , there exists λ ∈ f such that λ + x ∈ U2 . (3) There exist x ∈ U1 and λ ∈ f such that λ + x ∈ U2 . Proof Of course, (1) ⇒ (2) ⇒ (3). In order to complete the proof, suppose that (3) holds. Then (λ + U1 ) ∩ U2 = ∅. But U2 is a connected component of f −1 (V ) and, by Proposition 17.1.1, λ + U1 is too. So, λ + U1 = U2 and we are done.  The key technical result of this section is the following. Proposition 17.1.3 If f : C −→  C is an elliptic function, then, for every ε > 0 and every n ≥ 0, there exists δ ∈ (0,ε) such that (1) ∀z ∈  C ∀(0 ≤ k ≤ n) Every connected component U of f −k (Bs (z,δ)) has spherical diameter smaller than ε. (2) ∀z ∈ C ∀(0 ≤ k ≤ n) Every connected component U of f −k (Be (z,δ)) has Euclidean (thus, spherical too) diameter smaller than ε. (3) ∀z ∈  C ∀(1 ≤ k ≤ n) Every connected component U of f −k (Bs (z,δ)) has Euclidean (thus, spherical too, implying (1)) diameter smaller than ε.

17 Geometry and Dynamics of (All) Elliptic Functions

177

(4) In Case (1) or (3) the map f k : U −→ Bs (z,δ) is proper and f k (U ) = Bs (z,δ), while in Case (2) the map f k : U −→ Be (z,δ) is proper and f k (U ) = Be (z,δ). Proof We can, and do, assume that ε > 0 is so small that, with f = [λ1,λ2 ], the four balls B(0,2e), B(λ1,2ε), B(λ2,2ε), and B(λ1 + λ2,2ε) are mutually disjoint. Then also all the balls B(λ,2ε), λ ∈ f , are mutually disjoint. Consequently, we get the following. Claim 1◦ For every w ∈ C, all the balls B(w + λ,2ε), λ ∈ f , are mutually disjoint. Now define the set   ε Uz := Be w, ⊆ C. 2ord(f ) −1 w∈f

(z)

We shall prove the following. Claim 2◦ If  is a connected component of Uz , then diame () < ε. Proof Seeking contradiction, suppose that diame () ≥ ε

(17.4)

for some connected component  of Uz . Fix arbitrarily one ξ ∈ f −1 (z) such that  ε ⊆ . Be ξ, 2ord(f ) Let fξ−1 (z) be the set of all points w in f −1 (z) that are not congruent with ξ modulo f and also the point ξ . Let Uz (ξ ) be the connected component of   ε Be w, ⊆ C. 2ord(f ) −1 w∈fξ (z)

Then Uz (ξ ) ⊆ Uz ; since the collection



% Be

ε w, 2ord(f )

& w∈fξ−1 (z)

contains no more that ord(f ) elements, we conclude that   diame Uz (ξ ) < ε.

(17.5)

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Part IV Classics, Geometry, and Dynamics

It then follows from (17.4) that there exists a point w ∈ f −1 (z)\{ξ } congruent modulo f with ξ and such that  ε = ∅. Uz (ξ ) ∩ Be w, 2ord(f ) But then, using (17.5), we conclude that Be (ξ,2ε) ∩ Be (w,2ε) = ∅, contrary to Claim 1◦ . The proof of Claim 2◦ is complete.



We will now prove the following. Claim 3◦ For every ε > 0, there exists δ ∈ (0,ε) such that, for every z ∈  C, every connected component of f −1 (Bs (z,δ)) has Euclidean diameter smaller than ε. Proof

Since f + Uz = Uz , we also have that f + (C\Uz ) = (C\Uz ) and f (Uz ) ∩ f (C\Uz ) = ∅.

But since Uz is an open set and f (Uz ) is an open set containing z, we, thus, conclude, with R being the closure of a fundamental domain of f , that   f (C\Uz ) = f (C\Uz ) ∩ R is a compact subset of C not containing z. Hence, there exists δ ∈ (0,ε) such that Bs (z,δ) ∩ f (C\Uz ) = ∅. Hence, f −1 (Bs (z,δ)) ⊆ Uz . Therefore, for each connected component D of f −1 (Bs (z,δ)), there exists a ˆ So, it follows from (unique) connected component Dˆ of Uz such that D ⊆ D. ◦ Claim 2 that ˆ < ε. diame (D) ≤ diame (D) The proof of Claim 3◦ is complete.



Now, since spherical diameters do not exceed Euclidean ones, item (3) follows from Claim 3◦ by immediate induction while items (1) and (2) follow from item (3). Item (4) follows now directly from Theorem 13.2.1. The proof of Proposition 17.1.3 is complete. 

17 Geometry and Dynamics of (All) Elliptic Functions

179

As an immediate consequence of this proposition and Theorem 13.2.1, we get the following. Proposition 17.1.4 If f : C −→  C is an elliptic function, then, for every ε > 0 and every n ≥ 0, there exists δ ∈ (0,ε) such that if V is an open connected subset of  C with diams (V ) ≤ δ, then, for every integer 1 ≤ k ≤ n, the following statements hold. (1) Every connected component U of f −k (V ) has spherical diameter smaller than ε. (2) Every connected component U of f −k (V ) has Euclidean (thus, spherical too) diameter smaller than ε. (3) Every connected component U of f −k (V )) has Euclidean (thus, spherical too, implying (1)) diameter smaller than ε. (4) In each case the map f k : U −→ V is proper and f k (U ) = V . Remark 17.1.5 We would like to alert the reader to the fact that, in general, a connected component of an inverse image of a bounded open set under the action of an elliptic function f : C −→  C can be unbounded. Indeed, consider a straight line  parallel to the sides of a fundamental parallelogram of f   avoiding poles of f . Then r := diste f ,f −1 (∞) > 0; therefore, the set   V := f Be (,r/2) is open and bounded. However, one of the connected components of f −1 (V ) contains the set Be (,r/2), which is unbounded. Also, as an immediate consequence of Proposition 17.1.3 along with Theorem 13.3.28, Lemma 13.3.39, and Theorem 13.3.56, we get the following fundamental result. Theorem 17.1.6 If f : C −→  C is an elliptic function, then f −1 (in fact, f −n for every n ≥ 1) has no asymptotic values and Sing(f −1 ) = f (Crit(f )) ∪ f (f2−1 (∞)). Moreover, Sing(f −n ) =

n−1 

  f k f (Crit(f )) ∪ f (f2−1 (∞)) .

(17.6)

k=0

We would like to say that this result, whose essence depends on there being no existence of asymptotic values, could have been proved directly without the use of Section 13.3. We would also like to record, which we have already done

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in Section 13.3, that the set f (f2−1 (∞)) is either equal to {∞} or it is empty depending on whether or not the map f has poles of order larger than 1. Now we shall prove a fundamental theorem, which may look a little bit inconspicuous but which is interesting on its own and will have quite profound consequences and applications throughout the whole book. Theorem 17.1.7 Let f : C →  C be an elliptic function. If V ⊆  C is an open connected set, n ≥ 1 is an integer, and U is a connected component of f −n (V ), then (1) f n (U ) = V . (2) If, in addition,   U ∩ Crit(f n ) ∪ f −(n−1) (f2−1 (∞)) = ∅, or just U ∩ Crit(f n ) = ∅ if V ⊆ C, e.g., if ⎛

n−1 

V ∩⎝

⎞ f j (Crit(f )) ∪ f −1 (f2−(n−1) (∞))⎠ = ∅,

j =1

or just V ∩

n−1 

f j (Crit(f )) = ∅

j =1

if V ⊆ C, then the map U : U −→ V is covering. (3) Furthermore, if, additionally, V is simply connected, then the map f n |U : U −→ V is a conformal homeomorphism and, so, U is simply connected too. f n|

Proof Items (2) and (3) of this theorem are immediate consequences of Theorems 17.1.6, 13.3.44, and 13.3.52. In order to prove item (1), fix a point z ∈ V . Then let δ > 0 be so small, as required by Proposition 17.1.3, with this point z and ε := 1 and also so small that Bs (z,δ) ⊆ V . So, it follows from Theorem 13.3.2 that f n (U ) ∩ Bs (z,δ) = ∅. So, if D is a connected component of f −n (Bs (z,δ)) contained in U , then the map f n |D : D −→ Bs (z,δ) is, by virtue of Proposition 17.1.3(4), surjective. Thus,

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z ∈ f n (D) ⊆ f n (U ). Hence, V ⊆ f n (U ) and item (1) of our theorem is proved. The proof of Theorem 17.1.7 is complete.  As an immediate consequence of Theorem 17.1.7, we get the following. Theorem 17.1.8 Let f : C −→  C be an elliptic function and V ⊆ C be an open connected, simply connected set. If n ≥ 0 is an integer, ξ ∈ f −n (V ), and the (unique) connected component Uξ of f −n (V ) containing ξ contains  j no critical points of f n (e.g., if V ∩ n−1 j =1 f (Crit(f )) = ∅), then (1) the map f n |Uξ : Uξ → V is a conformal homeomorphism, (2) there exists a unique holomorphic branch fξ−n : V → Uξ (equal to f |−n Uξ ) −n n of f sending f (ξ ) to ξ , and (3) the map fξ−n : V → Uξ is a conformal homeomorphism; (4) in particular, fξ−n (V ) = Uξ . Having Proposition 17.1.4, we can also easily prove the following. Proposition 17.1.9 If f : C −→  C is an elliptic function, then there exists η0 (f ) > 0 so small that the following hold. (1) 

min |w − z| : w,z ∈ Crit(f ) ∪ f −1 (∞) : w = z > 2η0 (f ). (2) If V is an open connected subset of  C with diams (V ) ≤ η0 (f  ), then the Euclidean diameter of each connected component U of f −1 V ) is finite, the map f : U → V is proper, and f (U ) = V . (3) If, in addition, the set V is simply connected, then each connected component U of f −1 (V ) is simply connected. Proof Since the set Crit(f ) ∪ f −1 (∞) is discrete and  f -invariant, the number min |w − z| : w,z ∈ Crit(f ) ∪ f −1 (∞) : w = z is well defined and positive. The current proposition then immediately follows from Proposition 17.1.4 and Corollaries 8.6.18 and 8.6.19.  Now we shall apply Theorem 17.1.7 to study forward and inverse images of connected components of Fatou sets of elliptic functions. Theorem 17.1.10 Let f : C −→  C be an elliptic function. If V is a connected component of the Fatou set F (f ) of f , then, for every integer n ≥ 0, the following two statements hold. (1) The set f n (V ) is a connected component of the Fatou set F (f ) of f .

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(2) Each connected component U of f −n (V ) is a connected component of the Fatou set F (f ) and f n (U ) = V . Proof We already know, and this is, in fact, obvious, that f n (V ) is contained in a unique connected component Vn of the Fatou set F (f ) of f . Let Wn be the connected component of f −n (Vn ) containing V . Then Wn is also a connected component of F (f ), whence Wn = V . It, therefore, follows from item (1) of Theorem 17.1.7 that f n (V ) = f n (Wn ) = V , whence item (1) is proved. Item (2) now follows immediately from item (1) and the fact that U is a connected component of F (f ) following from Theorem 13.2.2. The proof is complete 

17.2 Fundamental Structure Results In this section, we provide fundamental results about the structure of elliptic functions which show how this class is situated among all meromorphic functions. Along with the results of Part III this description will capture profound features of the nature of the Fatou set of f . Indeed, in Theorem 17.1.6 of the previous section, we proved that if f : C −→  C is an elliptic function, −1 ) = f (Crit(f ))∪f (f2−1 (∞)), and, then f has no asymptotic values, Sing(f   n−1 k f (Crit(f )) ∪ f (f −1 (∞)) . We repeat moreover, Sing(f −n ) = k=0 f 2 that, since each elliptic function is meromorphic, all considerations of Chapter 13 apply to them. Since the set Rf ∩ Crit(f ) is finite and since f (Crit(f )) = f (Rf ∩ Crit(f )), the set of critical values f (Crit(f )), is, therefore, also finite, we obtain from Theorem 17.1.6 the following. Theorem 17.2.1 Each elliptic function belongs to Speiser class S. As an immediate consequence of this theorem and Theorem 14.4.3, we get the following. Theorem 17.2.2 Any elliptic function has only finitely many attracting and rationally indifferent periodic points. Because of this same theorem, i.e., Theorem 17.2.1, all considerations of Section 14.4 apply to them. Therefore, as an immediate consequence of Theorems 14.4.4 and 14.4.10, we get the following. Theorem 17.2.3 No elliptic functions have Baker or wandering domains.

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Now we shall prove the following. Theorem 17.2.4 No even elliptic function has a cycle of Herman rings. Proof Suppose that f has a cycle of Herman rings {U0,U1, . . . ,Up−1 } of some period p ≥ 1. Then, for each i = 0,1, . . . ,p − 1, the iterate f p : Ui → Ui is conjugate to an irrational rotation of an annulus A(0;1,R) with some R > 1; thus, it is bijective and ∞ ∈ / U0 ∪ U1 ∪ · · · ∪ Up−1 . The preimages under these conjugacies from A(0;1,R) to Ui , i = 0,1, . . . ,p−1, of the circles {z ∈ C : |z| = r}, 1 < r < R, foliate the rings Ui with f p forward invariant leaves on which f is bijective. Let γ be an f p forward invariant leaf of U0 and Bγ denote the bounded component of the complement of γ ; remember / γ . If Bγ contained no prepole of f p , then the standard that ∞ ∈ / U0 , so ∞ ∈ Maximal Modulus Theorem argument would yield f pk (Bγ ) ⊆ Bγ for all k ≥ 0. Hence, Bγ would be contained in the Fatou set of f , contrary to the definition of a Herman ring. Thus, Bγ contains a prepole of f p . Therefore, there is a smallest integer n ≥ 0 such that Bf n (γ ) ∩ f = ∅. Then fix a point ω ∈ Bf n (γ ) ∩ f . Since f n (U0 ) is (exactly) one of the rings Ui , i = 0,1, . . . , p − 1, there exists exactly one j = 0,1, . . . ,p − 1 such that f n (U0 ) = Uj . Since the function f is even, −Uj is also a connected component of the Fatou set F (f ). Since f is -invariant, so is the set −Uj + 2ω. In addition, ω ∈ −Bf n (γ ) + 2ω. If ω belongs to one of the sets Uj or −Uj + 2ω, then it also belongs to the other, and so Uj = −Uj + 2ω

(17.7)

as both these sets are components of the Fatou set F (f ). Otherwise, ω∈ / Uj ∪ (−Uj + 2ω). Hence, ω belongs to the bounded connected components both of the complement of Uj and of −Uj + 2ω. Therefore, supposing that Uj and −Uj + 2ω are disjoint, we conclude that one of the sets Uj or −Uj + 2ω is contained in the bounded component of the complements of the other. This, however, is a contradiction since sup{|z − ω| : z ∈ Uj } = sup{|z − ω| : z ∈ −Uj + 2ω}. Since any two components of the Fatou set of f are either disjoint or equal, this yields (17.7) in this case too. But then, for every z ∈ Uj , −z +2ω is also in Uj ,

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the points z and −z+2ω (if only z = ω) are different, and f (z) = f (−z+2ω).  This contradicts the bijectivity of f on Uj and finishes the proof. We would, however, like to remark that (not even) elliptic functions can have Herman rings. In fact, Rocha proves through a quasiconformal surgery in [Ro] that there are elliptic functions f of order ord(f ) ≥ 3 that admit Herman rings. She also proves that the number of such Herman rings cannot exceed ord(f ) − 2. In particular, no elliptic function of order 2 has a Herman ring. Remark 17.2.5 For elliptic functions f : C →  C, we frequently adopt a slightly modified definition of the Julia set J (f ) of f , i.e., we let J (f ) be the complement of the Fatou set F (f ) in C. Thus, ∞ ∈ / J (f ). This is a particularly convenient convention for our considerations concerning conformal measures and geometric measures supported on Julia sets. A remarkable fact closely related to (17.2) is the following. Proposition 17.2.6 Each elliptic function f : C −→  C is topologically exact in the sense that if U ⊆ C is an open set intersecting the Julia set J (f ), then there exists an integer l ≥ 1 such that f l (U ) =  C. Proof It follows from Theorem 17.0.1 that, for some integer l ≥ 2, the image f l−1 (U ) contains an open neighborhood of ∞ in  C. Thus, it contains at least one (in fact, infinitely many) congruent copy of the fundamental parallelogram Rf of f . Consequently, by (17.2), we get that f l (U ) ⊇ f (Rf ) =  C. 

The proof is complete.

Now we analyze in greater detail the behavior of elliptic functions near poles. It follows from the periodicity of f that    f −1 (∞) = Rf ∩ f −1 (∞) + mλ1 + nλ2 . m,n∈Z

For every pole b of f , let qb denote its multiplicity. We define q = qmax (f ) := sup{qb : b ∈ f −1 (∞)} = max{qb : b ∈ f −1 (∞) ∩ Rf }. For every R > 0, we have defined in Section 14.4 the following two sets: ∗ B∞ (R) = {z ∈  C : |z| > R} and B∞ (R) = {z ∈ C : |z| > R}.

Given b ∈ f −1 (∞), let Bb (R) be the connected component of f −1 (B∞ (R)) containing b

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and Bb∗ (R) := Bb (R)\{b}. More generally, given k ≥ 1 and b ∈ f −k (∞), let Bbk (R) be the connected component of f −k (B∞ (R)) containing b and Bb∗k (R) := Bbk (R)\{b}. We shall prove the following useful auxiliary result. Lemma 17.2.7 If f : C −→  C is an elliptic function, then there exists T (f ) ∈ (0,η0 (f )/2) so large that, for every b ∈ f −1 (∞) and every R ≥ T (f ), (1) diame (Bb (R)) < η0 (f )/2 < +∞, (2) the open set Bb (R) is simply connected, and (3) there exists a bounded holomorphic function gb : Bb (R) −→ C\{0} such that f (z) = (z − b)qb gb (z) for all z ∈ Bb (R), (4) all the sets Ba (R), a ∈ f −1 (∞), are mutually disjoint and (5) /

/

sup max ga ∞,ga ∞,

1 1 ,  inf |ga (z)| : z ∈ Ba (R) inf |gb (z)| : z ∈ Ba (R)

00 < +∞,

where the supremum is taken over all a ∈ f −1 (∞). Proof Items (1) and (4) are immediate consequences of Proposition 17.1.3(3) and the definition of the number η0 (f ). Item (2) immediately follows from Theorem 13.3.51 by taking T (f ) > 0 so large that f (Crit(f )) ∩ B∞ (R) = ∅. Items (3) and (5) then follow from the definition of a pole and the fact that the poles f −1 (∞) of f form an isolated set and the function f is doubly periodic.  Since the set f (Crit(f )) is finite, we can further assume that T (f ) is so large that   (17.8) B∞ T (f )/4 ∩ f (Crit(f )) = ∅,   i.e., B∞ T (f )/4 contains no critical values of f . It follows from Lemma 17.2.7 that the map f |B∞ (T (f )) is qb -to-one

(17.9)

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Part IV Classics, Geometry, and Dynamics

and there exists A1 = A1 (f ) ∈ [1,+∞) such that −qb ≤ |f (z)| ≤ A1 |z − b|−qb A−1 1 |z − b|

(17.10)

and −(qb +1) A−1 ≤ |f  (z)| ≤ A1 |z − b|−(qb +1) 1 |z − b|

(17.11)

for all b ∈ f −1 (∞) and all z ∈ Bb (T (f )). Consequently,   − 2+ q1

A1

b

|f (z)|

qb +1 qb

2+ q1

≤ |f  (z)| ≤ A1

b

|f (z)|

qb +1 qb

(17.12)

for all b ∈ f −1 (∞) and all z ∈ Bb (T (f )). From (17.8)–(17.12) and Theorem 17.1.8, we directly get the following. ∗ (T (f )) is Lemma 17.2.8 If f : C −→  C is an elliptic function and U ⊆ B∞ an open connected, simply connected set, then, for every pole b ∈ f −1 (∞), all −1 −1 , . . . ,fb,U,q of f are well defined on the holomorphic inverse branches fb,U,1 b U . In addition, there exists A2 = A2 (f ) ≥ 1 such that, for every 1 ≤ j ≤ qb and all z ∈ U , we have that

A−1 2 |z|

−

qb +1 qb

−1 ≤ |(fb,U,j ) (z)| ≤ A2 |z|

−

qb +1 qb

.

(17.13)

As its immediate consequence, we get that qb −1

(2A2 )

−1

qb −1

qb −1

|z| qb |z| qb |z| qb −1 ∗ ≤ |(f ) (z)| ≤ 2A ≤ 2A 2 2 b,U,j 1 + |b|2 1 + |b|2 |b|2

(17.14)

for every 1 ≤ j ≤ qb and all z ∈ U . Set A(f ) := max{A1 (f ), A2 (f )}.

(17.15)

Also based on the congruency of poles, a slightly more general and straightforward observation from the local behavior around poles is that, for every k ≥ 1, there exist constants Lk ≥ 1 and Rk ≥ T (f ), both monotone increasing with k, such that, for all b ∈ f −k (∞) and all R ≥ Rk , we have that ≤diame (Bbk (R)) ≤ Lk R

− q1

(1 + |b|2 )−1 ≤diams (Bbk (R)) ≤ Lk R

− q1 b

L−1 k R L−1 k R

− q1 b

− q1

b

b

, (1 + |b|2 )−1 .

(17.16)

Frequently, we will write L for L1 . Towards the end of this section, we record the following technical observation, which directly follows from the double periodicity of an elliptic function and Theorem 17.1.6.

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Observation 17.2.9 Let f : C →  C be an elliptic function. If  z∈ C\ f (Crit(f )) ∪ f (f2−1 (∞)) , then there exist rz > 0 such that, for every w ∈ f −1 (z), the map f |Be (w,rz ) is injective. Finally, we would like to bring up the holomorphic torus map fˆ : Tˆ f −→ Tf introduced and dealt with in Section 16.2. It will be our frequent tool from now on throughout the book, but now primarily from the dynamical point of view. In line with Volume I, we denote by J (fˆ) the Julia set of the map fˆ : Tˆ f −→ Tf . We record the following straightforward observation. Observation 17.2.10 If f : C →  C is an elliptic function, then PC(fˆ) = f (PC(f )) and J (fˆ) = f (J (f )).

17.3 Hausdorff Dimension of Julia Sets of (General) Elliptic Functions In this section, we apply the results of Section 11.5 (Bowen’s Formula (Theorem 11.5.3) and Theorem 11.5.4) to provide a strong, somewhat surprising, lower bound for the Hausdorff dimension of the Julia sets of all elliptic functions. The idea is to associate with each elliptic function an iterated function system and to apply the above-mentioned theorems. This result is taken from [KU3] and its proof closely follows the one from that paper. It strengthens the earlier result from [Ko3], where the first inequality of Theorem 17.3.1 was not sharp. We also provide, in Section 17.7, a closed formula (obtained for the first time in [GK1]) for the Hausdorff dimension of I∞ (f ), the set of points escaping to infinity under iteration of f . These two results, in particular, show that I∞ (f ) is a very thin subset of the Julia set J (f ). Theorem 17.3.1 If f : C −→  C is an elliptic function, then HD(J (f )) >

2qmax (f ) ≥ 1. qmax (f ) + 1

Proof Take R1 ≥ R0 := T (f ) so large that − q1

LR1 for all b ∈ f −1 (∞).

b

< R0

(17.17)

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Part IV Classics, Geometry, and Dynamics

Given two poles b1,b2 ∈ B∞ (2R1 ), denote by fb−1 : B(b1,R0 ) −→ C, 2,b1,j −1 1 ≤ j ≤ qb , all the holomorphic branches of f resulting from Lemma 17.2.8. It then follows from (17.16) and (17.17) that   fb−1 B(b1,R0 ) ⊆ Bb2 (2R1 − R0 ) ⊆ Bb2 (R1 ) ⊆ B(b2,R0 ). (17.18) 2,b1,j Fix a pole a ∈ B∞ (2R1 ) with qa = q = qmax (f ). For every pole b ∈ B∞ (2R1 ) ∩ f −1 (∞) with qb = q, fix the inverse branches −1 −1 fb,a,1 : B(a,R0 ) −→ C and fa,b,1 : B(b,R0 ) −→ C

of f . In view of (17.18),   −1  −1  fb,a,1 B(a,R0 ) ⊆ B(b,R0 ) and fa,b,1 B(b,R0 ) ⊆ B(a,R0 ). The family

−1  −1 S = fa,b,1 ◦ fb,a,1 : B(a,R0 ) −→ B(a,R0 ) b∈B

2R1 ∩f

−1 (∞)

thus forms a conformal infinite iterated function system. Given t ≥ 0, we consider the function  Z1 (t) = φb t∞ b∈B∞ (2R1 )∩f −1 (∞)

and the number θS = inf{t ≥ 0 : Z1 (t) < ∞}. 2q Our proof is based on demonstrating that θS = q+1 and Z1 (θS ) = +∞. In view of (17.13), we can write  − q+1 t − q+1 t |a| q |b| q Z1 (t)  b∈B∞ (2R1 )∩f −1 (∞)





|b|

− q+1 q t

.

b∈B∞ (2R1 )∩f −1 (∞)

" − q+1 t But the series b∈B∞ (2R2 )∩f −1 (∞) |b| q converges if and only if t > therefore, the formulas θS =

2q ≥ 1 and Z1 (θS ) = +∞ q +1

2q q+1 ;

(17.19)

are proved. Now applying the second assertion of Theorem 11.5.4, the desired result follows.  As an immediate consequence of this theorem, we get the following.

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Corollary 17.3.2 If  is a lattice in C and K is the field of all elliptic functions with respect to L, then sup{HD(J (f )) : f ∈ K } = 2.

17.4 Elliptic Function as a Member of A(X) for Forward Invariant Compact Sets X ⊆ C We start with the following main technical result of this section, the one due to which all other results of this section can be (easily) proved and whose proof is also easy and short. Lemma 17.4.1 If f : C −→  C is an elliptic function  and X ⊆ J (f ) ∩ C is a compact f -forward invariant set such that X∗ f |X ∩ Crit(f ) = ∅, then f |X ∈ A(X). Proof Only item (e) of Definition 9.0.1 requires a proof. But it follows immediately from Proposition 17.1.9 if we set Y := C, U (f ) := C\f −1 (∞),  and ιf := η0 (f )/4. Remark 17.4.2 In the context of Lemma 17.4.1, the set X contains no poles of f and we will always take the open set C\f −1 (∞) as U (f ) and the C as a holomorphic extension of f |X function f |C\f −1 (∞) : C\f −1 (∞) −→  to U (f )(= C\f −1 (∞)). Also, Y := C. Of course, if X has no isolated points (i.e., it is perfect), then such an extension is unique and must be equal to f |C\f −1 (∞) . As an immediate consequence of Lemma 17.4.1, we get the following. Lemma 17.4.3 If f : C −→  C is an elliptic function and X ⊆ J (f ) ∩ C is a compact f -forward invariant set such that X ∩ Crit(f ) = ∅, then f |X ∈ A0 (X). Now keeping f : C −→  C, an elliptic function, let V ⊆  C be an open neighborhood of ∞ in  C such that Crit(f ) ∩ ∂V = ∅.

(17.20)

Define KJ (V ) := J (f ) ∩



f −n ( C\V ) = {z ∈ J (f ) : ∀n ≥ 0 f n (z) ∈ / V }.

n≥0

(17.21)

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Part IV Classics, Geometry, and Dynamics

These sets will play an important role throughout the book and they will be more systematically treated in Section 22.4, where some historical outlook will also be given. Obviously, f (KJ (V )) ⊆ KJ (V ).

(17.22)

Since f : C −→  C is continuous, since V is open, and since ∞ ∈ V , we readily see that KJ (V ) is a compact subset of C. Since the elliptic map f : C −→  C is open, we get that KJ (V )∗ (f ) ⊆ ∂V .

(17.23)

We, thus, using Lemma 17.4.1, obtain the following. Lemma 17.4.4 Let f : C −→  C be an elliptic function. If V ⊆  C is an  open neighborhood of ∞ in C such that Crit(f ) ∩ ∂V = ∅, then f | KJ (V ) ∈   A f |KJ (V ) . Lemma 17.4.5 If f : C −→  C is an elliptic function and X ⊆ J (f ) ∩ C is a compact f -forward invariant set, then there exists R > 0 such  that ∂B∞ (R) ∩ Crit(f ) = ∅, X ⊆ KJ (B∞ (R)), and f |KJ (V ) ∈ A f |KJ (V ) . Proof This lemma is an immediate consequence of Lemma 17.4.4 once one notices that the set Crit(f ) is countable.  Since the topological support of an invariant measure is forward invariant, as an immediate consequence of Lemma 17.4.5, we get the following. Lemma 17.4.6 If f : C −→  C is an elliptic function and μ is a Borel probability f -invariant measure whose topological support supp(μ) is a compact subset of C, then there exists R > 0 such that ∂B∞ (R) ∩ Crit(f ) = ∅, supp(μ) ⊆ KJ (B∞ (R)), and f |KJ (V ) ∈ A f |KJ (V ) . In addition, if supp(μ) ∩ Crit(f ) = ∅, then f ∈ A0 (supp(μ)). Since the topological support of an invariant measure is forward invariant, as an immediate consequence of Lemma 17.4.3 and, respectively, of Corollary 9.1.3 and Theorems 9.4.1 and 9.5.1 we get the following. Theorem 17.4.7 If f : C −→  C is an elliptic function and μ is a Borel probability f -invariant ergodic measure whose topological support supp(μ) is a compact subset of C, then χμ (f ) ≥ 0.

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Theorem 17.4.8 (Ruelle’s Inequality) If f : C −→  C is an elliptic function and μ is a Borel probability f -invariant ergodic measure whose topological support supp(μ) is a compact subset of C, then hμ (f ) ≤ 2 max{0,χμ (f )}. If, in addition, f ∈ A0 (X), then hμ (f ) ≤ 2χμ (f ). Theorem 17.4.9 (Volume Lemma) Let f : C −→  C be an elliptic function and μ be a Borel probability f -invariant ergodic measure whose topological support supp(μ) is a compact subset of C. If χμ (f ) > 0, then hμ (f ) log μ(B(x,r)) = r"0 log r χμ (f ) lim

for μ-a.e. x ∈ X. In particular, the measure μ is dimensional exact and, by Proposition 1.7.8, HD(μ) = PD(μ) =

hμ (f ) . χμ (f )

17.5 Radial Subsets of J (f ) and Various Dynamical Dimensions for Elliptic Functions f : C →  C This section is modeled on Section 10.3 and adapted for elliptic functions. Like Section 10.3 but even more so, it will play an important role in our study of the Sullivan conformal measures for elliptic functions. Definition 17.5.1 If f : C −→  C is an elliptic function, then we say that a point z ∈ J (f ) is radial (or conical) if and only if there exists η > 0 such that, for infinitely f n restricted to the connected component  −n manyn n ≥ 1,the map Compz f (B(f (z),η)) of f −n (B(f n (z),η)) containing z is one-to-one. Denote the set of such ns by Nz (f ). We call a radial point z ∈ J (f ) expanding if and only if there exists a number λ > 1 such that |(f n ) (z)| ≥ λn for infinitely many ns in Nz (f ).

(17.24)

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Part IV Classics, Geometry, and Dynamics

We, respectively, denote by Jr (f ) and Jer (f ) the sets of radial and expanding radial points in X. Given η > 0, we define Jr (f )(η) to be the set of all radial points of f witnessing, in this definition, the number 2η. Then also Jer (f )(η) := Jr (f )(η) ∩ Jer (f ). The radial Julia sets for rational functions were indirectly introduced in [Lj] and, independently, openly in [U2] by analogy with radial/conical sets in the theory of Kleinian groups (see also [DMNU] and [McM2]). In the realm of transcendental meromorphic functions, these were, for the first time, introduced and dealt with in [UZ1] and then in many papers, especially those by Mayer, Rempe-Gilen, Zdunik, and the second named author of this book. In this book, radial Julia sets, just defined, primarily serve as a good concept and tool to characterize the minimal exponent f for which the Sullivan conformal measures exist for elliptic functions. Among many other purposes, they have served the same purpose for rational functions. Analogously to the definition given by (10.35), given an elliptic function f : C −→  C we call the number DDh (J (f )) := sup{HD(μ)}

(17.25)

the first dynamical dimension of J (f ), where the supremum is taken over all Borel probability f -invariant ergodic measures μ on J (f ) whose topological support supp(μ) is compact and contained in C, and for which hμ (f ) > 0. Similarly, we call the number DDχ (J (f )) := sup{HD(μ)}

(17.26)

the second dynamical dimension of J (f ), where the supremum is taken over all Borel probability f -invariant ergodic measures μ on J (f ) whose topological support supp(μ) is compact and contained in C, and for which χμ (f ) > 0. Because of Theorem 17.4.8 (Ruelle’s Inequality), we have the following immediate inequality: DDχ (J (f )) ≤ DDh (J (f )).

(17.27)

As an immediate consequence of Proposition 10.3.3 and Lemma 17.4.5, we obtain the following. Proposition 17.5.2 If f : C −→  C is an elliptic function, then     DDh (J (f )) ≤ DDχ (J (f )) ≤ HD Jer (f ) ≤ HD Jr (f ) .

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17.6 Sullivan Conformal Measures for Elliptic Functions In this chapter, we deal systematically with one of the primary concepts of the book; namely, that of the Sullivan conformal measures for (all) elliptic functions f : C →  C. We will prove their existence; more precisely, we will prove the existence of a Sullivan conformal measure m with a minimal exponent which will be denoted by sf . We will also provide several dynamical/geometric characterizations of this exponent and will rule out some points in  C as possible atoms of m. We have already met the concept of conformal measures in Chapter 10 of Volume I. We treated these measures in a very general setting in the first section of this chapter, while in its second section we treated them in a setting and spirit quite close to the one we will be dealing with in the current section. In particular, in the current section, we will make significant use of the results in Chapter 10. We gave in Chapter 10 (Volume I) quite an extended historical account of the concept of conformal measures, particularly the Sullivan ones. We repeat a part of it here for the sake of completeness and convenience for the reader. Conformal measures were first defined and introduced by Patterson in his seminal paper [Pat1] (see also [Pat2]) in the context of Fuchsian groups. Sullivan extended this concept to all Kleinian groups in [Su2] and [Su4]. He then, in papers [Su5] and [Su7], defined conformal measures for all rational functions of the Riemann sphere  C; he also proved their existence therein. Both Patterson and Sullivan came up with conformal measures in order to get an understanding of geometric measures, i.e., Hausdorff and packing measures. Although Sullivan had already noticed that there are conformal measures for Kleinian groups that are not equal, nor even equivalent to any Hausdorff or packing (generalized) measure, the main purpose of dealing with them is still to understand Hausdorff and packing measures. The second purpose is actually purely dynamical. In Part VI, we will see that, for regular compactly nonrecurrent elliptic functions, conformal measures admit equivalent invariant measures and these generate very interesting and rich measure-preserving dynamical systems exhibiting stochastic behavior and satisfying such stochastic laws as the Central limit Theorem, the Law of the Iterated Logarithm, or the exponential decay of correlations. Chapter 11 in Volume I and the whole Part VI provide good evidence and examples of how these two goals can be achieved. Conformal measures, in the sense of Sullivan, have been studied in the context of rational functions in greater detail in [DU3], where, in particular, the structure of the set of their exponents was examined.

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Part IV Classics, Geometry, and Dynamics

Since then, conformal measures in the context of rational functions have been studied in numerous research works. We list here only a very few of them that appeared in the early stages of the development of their theory: [DU1], [DU5], [DU6]. Subsequently, the concept of conformal measures, in the sense of Sullivan, has been extended to countable alphabet iterated function systems in [MU1] and to conformal graph directed Markov systems in [MU2]. These were treated at length in Chapter 11. This was, furthermore, extended to some transcendental meromorphic dynamics in [KU2], [UZ1], and [MyU3]; see also [UZ2], [MyU4], and [BKZ1]. Last, the concept of conformal measures also found its place in random dynamics; we cite only [MSU]. In this section, we prove the existence of a Sullivan conformal measure with a minimal exponent for all elliptic functions and we provide several dynamically transparent characterizations of this exponent. Let H : U1 → U2 be an analytic map of open subsets U1 , U2 of the complex plane C. We recall from Definition 10.4.1 that, given t ≥ 0, a Borel measure νe , finite on bounded sets of C, is a Euclidean semi-t-conformal measure if and only if |H  |t dνe

νe (H (A)) ≥ A

for every Borel subset A of U1 such that H |A is one-to-one and is called t-conformal if the “≥” sign can be replaced by an “=” sign. As long as we are far from poles and infinity, it does not really matter whether our pair of measures is spherical (semi)-conformal or Euclidean (semi)-conformal. This is explained by the following. Observation 17.6.1 If (mG , mH ) is a Euclidean (semi)-t-conformal pair of measures for a meromorphic map f : G −→ H , then (m∗G , m∗H ) is a spherical t-conformal pair of measures for f , where the measures m∗G and m∗H are, respectively, defined by dm∗G (z) = (1 + |z|2 )−t dmG

(17.28)

dm∗H (z) = (1 + |z|2 )−t dmH

(17.29)

and

and they satisfy m∗H (f (A)) = for each special set A ⊆ G.

A

|f  |ts dm∗G

17 Geometry and Dynamics of (All) Elliptic Functions

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A similar conversion holds if we start from a spherical t-conformal pair of measures. In what follows, we respect the convention that spherical conformal measures (or their weaker versions) are labeled with the subscript “s” whereas Euclidean conformal measures (and their weaker versions) are labeled with the subscript “e.” If no subscript is used, the conformal measure under consideration can be spherical equally well as Euclidean. Obviously, me is equivalent to ms and infinite but σ -finite. In fact, by double periodicity of elliptic functions, it is finite on all bounded subsets of C. Assume now that f : C −→  C is an elliptic function and that m is some t-conformal measure for f . Let  be the corresponding lattice. Then, for every w ∈ , we get that |f  |t dme = me (f (A)) = me (f (A + w)) = A

|f  |t dme . A+w

Since the derivative f  is periodic with respect to the lattice , we, thus, get the following. Proposition 17.6.2 If f : C −→  C is an elliptic function and m is some t-conformal measure for f , then the Euclidean t-conformal measure me is Tw -invariant for every w ∈ , where Tw : C −→ C is the translation about the vector w given by the formula Tw (z) = z + w. As an immediate consequence of this proposition and Theorem 1.2.12, we get the following. Corollary 17.6.3 With the hypotheses of the previous proposition, for every r > 0, we have that M(me,r) := inf{me (Be (z,r)) : z ∈ J (f )} > 0. In order to provide all our characterizations of the least exponent of all Sullivan conformal measures for general elliptic functions, we need the following lemma involving conformal measures and radial Julia sets Jr (f ). Lemma 17.6.4 Let f : C −→  C be an elliptic function. If t ≥ 0 and ν is a t-conformal measure, either Euclidean or spherical, then t ≥ HD(Jr (f )) and Ht |Jr (f ) is absolutely continuous with respect to ν, i.e., Ht |Jr (f ) ≺ ν. Proof Since the measures νe and νs are equivalent, as are Hte and Hts , it suffices to prove the lemma for Euclidean measures νe and Hte . Now fix z ∈ Jr (f ). Then there exists η > 0 such that z ∈ Jr (f )(η), which, along with the infinite subset Nz (f ) of natural numbers, comes from Definition 10.3.1. Fix

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Part IV Classics, Geometry, and Dynamics

n ∈ Nz (f ). It follows then from the Koebe Distortion Theorem I (Euclidean version) that     fz−n Be (f n (z),η) ⊆ Be z,Kη|(f n ) (z)|−1 . Applying the Koebe Distortion Theorem I (Euclidean version) again as well as Corollary 17.6.3 and conformality of the measure m, we, thus, get that      νe Be z,Kη|(f n ) (z)|−1 ≥ K −t |(f n ) (z)|−t νe Be (f n (z),η) ≥ K −t |(f n ) (z)|−t M(νe,η) = M(ν,η)K −2t η−t (Kη|(f n ) (z)|−1 )t . Thus, letting n ∈ Nz (f ) tend to +∞, we get that lim sup r→0

νe (Be (z,r)) ≥ M(νe,η)K −2t η−t . rt

Ht |

Therefore, Jr (f )(η) is absolutely continuous with respect to ν, i.e., Ht |Jr (f )(η) ≺ ν because of Theorem 1.6.3(1). Since Jr (f ) =

∞ 

Jr (f )(1/k),

k=1



the proof of Lemma 17.6.4 is complete.

In order to prove Theorem 17.6.7, we will now single out in a separate definition the property (17.39) enjoyed by the measures mV appearing in its proof below and also in further investigations of geometrical properties of the conformal measure constructed in the proof of Theorem 17.6.7. Definition 17.6.5 Let f : C −→  C be an elliptic function, Y ⊆ J (f ) be a finite set, and Vˆ ⊆  C be an open neighborhood of Y such that the closure of Vˆ is disjoint from at least one fundamental parallelogram of f . If me is a semi-tconformal measure for which there exists an open neighborhood Vme ⊆ Vˆ of Y such that |f  |t dme

me (f (A)) = A

for every Borel set A ⊆ J (f ) such that both f |A is one-to-one and A ∩ V me = ∅, then the semi-t-conformal measure me is said to be almost t-conformal with respect to the triple (Y , Vˆ , Vme ). Then, for every Borel set A such that f |A is one-to-one and A ∩ V me = ∅ and for every w ∈ f , we have that |f  |t dme = me (f (A)) = me (f (A + w)) ≥ A

|f  |t dme, A+w

(17.30)

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and the last inequality sign becomes an equality either if, in addition, (A + w) ∩ V me = ∅ or if me is a t-conformal measure, and we assume only that f |A is one-to-one. Since f  is periodic with respect to a lattice  of f , all of the above statements and assumptions lead to the following. Lemma 17.6.6 Let f : C −→  C be an elliptic function, Y ⊆ J (f ) be a finite set, Vˆ ⊆  C be an open neighborhood of Y such that the closure of Vˆ is disjoint from at least one fundamental parallelogram of f , and Vme ⊆ Vˆ be an open neighborhood of Y . Then, for almost every t-conformal measure m with respect to the triple (Y , Vˆ , Vme ), every w ∈ f , and every Borel set A ⊆ C such that A ∩ V me = ∅, we have that me (A + w) ≤ me (A).

(17.31)

If, in addition, either (A + w) ∩ V me = ∅ or m is h-conformal and we assume only that f |A is one-to-one, then this inequality becomes an equality. Furthermore, for every r > 0, there exists M(r) ∈ (0,∞) independent of almost any t-conformal measure m such that me (F ) ≤ M(r)

(17.32)

for every Borel set F ⊆ C with diameter ≤ r. If, in addition, m is h-conformal, then, for every R > 0, there exist constants Q(R) and Qh (R) such that me (Be (x,r)) ≥ Q(R)r 2 ≥ Qh (R)r h

(17.33)

for all x ∈ J (f ) and all r ≥ R. Proof Inequality (17.31) as well as its equality counterpart are an immediate application of (17.30). Formula (17.32) follows directly from (17.31) and the fact that V is disjoint from at least one fundamental parallelogram. The second part of (17.33) is clear. In order to prove the first one, fix a fundamental parallelogram R of f and notice that T (R) := inf{me (B(z,R)) : z ∈ J (f ) ∩ R} > 0. Hence, if R ≤ r ≤ 4diame (R), then, for any x ∈ J (f ), me (Be (x,r)) ≥ me (Be (x,R)) ≥ T (R) =

T (R) 2 1 T (R) 2 r , r ≥ 2 16 diam2e (R) r

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Part IV Classics, Geometry, and Dynamics

and we are done in this case. So, suppose that r ≥ 4diam(R). Then each ball Be (x,r) contains at least

2  √ 2r r2 = 2diame (R) 2diam2e (R) nonoverlapping f -congruent copies of R. Therefore, me (Be (x,r)) ≥

r2 2diam2e (R)

me (R) =

me (R) 2 r . 2diame (R) 

We are done.

Let f2−1 (∞) be the set of all those poles b of f such that the order qb of b is ≥2. Note that the set Crit(f ) in the sense of Definition 10.1.4 is equal to Crit(f ) ∪ f2−1 (∞), where the latter Crit(f ) is equal to {c ∈ C : f  (c) = 0}. Now we shall prove the main result of this section. Theorem 17.6.7 If f : C −→  C is an elliptic function, then     DDh (J (f )) = DDχ (J (f )) = HD Jer (f ) = HD Jr (f ) and, denoting this common value by sf , there exists an sf -conformal measure m on J (f ) for f such that m((f ) ∪ {∞}) = 0 (remember that its spherical version, as for all spherical conformal measures considered in this book, is finite). In addition, sf is the least exponent t ≥ 0 for which there exists a tconformal measure for f supported on J (f ). Proof With the help of Section 10.2, particularly Lemma 10.2.5, and by utilizing the method of K(V ) sets developed in [DU3] (see also [KU6]), we shall first construct an s-conformal measure m with s ≤ DDh (J (f )).

(17.34)

In order to begin, we call Y ⊆ {∞} ∪ (f ) ∪

∞ 

f n (Crit(J (f )))

(17.35)

n=1

a crossing set if Y is finite and the following four conditions are satisfied. (y1) ∞ ∈ Y . (y2) Y ∩ {f n (x) : n ≥ 1} is a singleton for all x ∈ Crit(J (f )).

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(y3) Y ∩ Crit(f ) = ∅. (y4) (f ) ⊆ Y . Since the set f (Crit(f )) is finite, crossing sets do exist. Let V ⊆  C be an open neighborhood of Y such that Crit(f ) ∩ ∂V = ∅.

(17.36)

Having Lemmas 17.4.4 and 10.2.4, and applying Lemma 10.2.5 with X = KJ (V ) and U (f ) = C, we directly obtain a number   0 ≤ s(V ) ≤ DDh KJ (V ) ≤ DDh (J (f ))

(17.37)

and a Borel probability measure mV supported on KJ (V ) such that mV (f (A)) ≥ A

) |f  |s(V dmV , s

for every special set A ⊆ KJ (V ), and mV (f (A)) = A

) |f  |s(V dmV , s

for every special set A ⊆ KJ (V )\∂V . Treating the measure mV as supported on J (f ), with a direct calculation, we, thus, get, for every special set A ⊆ J (f ), that   mV (f (A)) ≥ mV f (A ∩ KJ (V )) ≥ A∩KJ (V )

) |f  |s(V dmV = s

A

) |f  |s(V dmV s

(17.38)

and    mV (f (A)) = mV f A ∩ f −1 (KJ (V )) = A

) |f  |s(V dmV s

(17.39)

for every special set A ⊆ J (f )\V , using the fact that then A∩f −1 (KJ (V )) ⊆ KJ (V )\∂V . From now on throughout the entire proof, we fix a crossing set Y and we consider an open neighborhood Vˆ ⊆  C of Y so small that the closure of Vˆ is disjoint from at least one fundamental parallelogram of f and that the limit set JS of the iterated function system S defined in the proof of Theorem 17.3.1 is contained in KJ (Vˆ ). We are free to require the point ξ , used in the construction of conformal measures in Section 10.2, particularly in

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Part IV Classics, Geometry, and Dynamics

Lemma 10.2.5, to belong to the set JS . Then, with the notation of Chapter 11, Section 10.2, and the proof of Theorem 17.3.1, we get that cξ (t) ≥ lim sup n→∞

 1 1 φω t = PS (t). log 2n 2 |ω|=n

Thus, by virtue of Theorem 11.5.3 and of the first part of (17.19), we obtain that s(V ) ≥ h∗ (f ) := HD(JS ) > θS ≥ 1. Along with (17.37), this gives 1 < h∗ (f ) ≤ s(V ) ≤ DDh (J (f )).

(17.40)

All neighborhoods of Y considered in this proof will always be assumed to be contained in this set Vˆ . We also fix from now on (Vn )∞ n=1 , a descending sequence of open neighborhoods of Y contained in Vˆ , satisfying (17.36), and such that diams (Vn ) ≤ 1/n

(17.41)

for every integer n ≥ 1. In view of (17.37), by passing to a subsequence, we may assume without loss of generality that the sequence (s(Vn ))∞ n=1 converges. Denote its limit by s(Y ). Because of (17.40), we then have that 1 < h∗ (f ) ≤ s(Y ) ≤ DDh (J (f )).

(17.42)

Passing to yet another subsequence, we may assume that the sequence (mVn )∞ n=1 , treated as consisting of probability measures on the compact space  C. We shall now C, converges weakly to a Borel probability measure mY on  prove the following claim. Claim 1◦ The limit measure mY enjoys the following properties. ' s(Y ) (a) mY (f (A)) ≥ A |f  |s dmY for every special set A ⊆ J (f ). ' ) (b) mY (f (A)) = A |f  |s(Y dmY for every special set A ⊆ J (f )\Y . s (c) mY (∞) = 0. (d) mY ((f )) = 0. Proof Let Sing(f ) be the singular set of f as defined in Definition 10.1.4. s(V ) Note that the sequence of continuous functions gn := |f  |s n , n ≥ 1, defined on J (f ), converges uniformly to the continuous function ) g := |f  |s(Y : J (f ) −→ [0,+∞). s

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Fix any integer k ≥ 1. It then follows from (17.41) and, respectively, from (17.38) and (17.39) that, for every n ≥ k, we have that mVn (f (A)) ≥

A

n ) dm , |f  |s(V Vn s

(17.43)

|f  |ss(Vn ) dmVn ,

(17.44)

for every special set A ⊆ J (f ), and mVn (f (A)) =

A

for every special set A ⊆ J (f )\Vk . Therefore, since also J (f )∗ (f ) = ∅ and g|Crit(f )∪f −1 (∞) = gn |Crit(f )∪f −1 (∞) = 0, 2

2

it follows from Lemma 10.1.9 that mY (f (A)) ≥ A

) |f  |s(Y dmY , s

(17.45)

) |f  |s(Y dmY , s

(17.46)

for every special set A ⊆ J (f ), and mY (f (A)) = A

for every special set A ⊆ J (f )\Vk . Item (a) of our claim is proved since it asserts exactly the same as (17.45). Aiming to prove item (b) of our claim, let A be a special subset of J (f ) such that A ∩ Y = ∅.

(17.47)

Then, by virtue of (17.41), there exists k ≥ 1 such that A ⊆ J (f )\Vk . So, item (b) holds by virtue of (17.46). Now, completely generally, let A be a special subset of J (f )\Y . Then A=

∞ 

A\B(Y,1/j ),

j =1

 ∞ the sequence A\B(Y,1/j ) j =1 is ascending, J (f )\B(Y,1/j ) ⊆ J (f )\Y . Therefore,   mY (f (A)) = lim mY f (A\B(Y,1/j ))

and

A\B(Y,1/j ) ⊆

j →∞

= lim

j →∞ A\B(Y,1/j )

and item (b) is proved.

) |f  |s(Y dmY = s

A

) |f  |s(Y dmY , s

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Part IV Classics, Geometry, and Dynamics

We shall now prove that (c) holds. It is immediate if f2−1 (∞) = ∅ since, for every point z ∈ f2−1 (∞), we have that |f ∗ (z)| = 0, and then, by item (b) (note that (17.35) along with (y1), (y2), and (y3) imply that Y ∩ f −1 (∞) = ∅), mY (∞) = |f ∗ (z)|s(Y ) mY ({z}) = 0. In the fully general case, the proof is much more involved. We start it now. For every n ≥ 1, set msn := mVn , men := (mVn )e, and sn := s(Vn ). For every k ≥ 0, consider Sk , the square centered at the origin whose edges are parallel to the coordinates’ axes and are of length 2k . Since, by (17.43) and (17.44), each measure msn is almost sn -conformal with respect to the triple (Y , Vˆ , Vˆ ) and since each “annulus” Ak = Sk+1 \Sk is a union of 3 × 4k squares whose edges are parallel to the coordinates’ axes and are of length 1, it follows from (17.32) of Lemma 17.6.6 that, for all k ≥ 1 and all n ≥ 1, √ men (Ak ) ≤ 3M( 2)4k . Consequently, msn (Ak )

√ √ √ √ 3M( 2)4k 3M( 2)4k 3M( 2)4k ≤ ≤ ≤ = 3M( 2)4(1−h∗ (f ))k . k s ks kh (f ) n n ∗ (1 + 4 ) 4 4

Since h∗ (f ) > 1 (see, again, (17.40)), we, thus, get, for all j ≥ 1 and all n ≥ 1, that ⎛ ⎞ ∞ ∞   msn ( C\Sj ) = msn ⎝ Ak ⎠ = msn (Ak ) k=j

≤

∞ 

k=j

√ 3M( 2)4(1−h∗ (f ))k

k=j

√ = 3M( 2)(1 − 41−h∗ (f ) )−1 4(1−h∗ (f ))j . Since limj →∞ 4(1−h∗ (f ))j = 0, this implies that mY (∞) = 0. We, thus, are done with item (c). Proving item (d), keep the same measures mn as in the proof of item (c). Since the measures mn are semi-sn -conformal (see (17.38)), it follows from Lemma 15.4.1 that there exists a constant C > 0 such that, for every ω ∈ (f ) and every r ∈ (0,1], we have that msnn (B(ω,r)) ≤ Cr sn +p(ω)(sn −1) ≤ Cr h∗ (f )+p(ω)(h∗ (f )−1),

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203

where the second inequality was written as a result of (17.40). Hence, passing to the limit, msY (B(ω,r)) ≤ Cr h∗ (f )+p(ω)(h∗ (f )−1) . Since, also by (17.40), h∗ (f ) > 1, by letting r " 0, this implies that mY (ω) = 0. The proof of item (d) and, simultaneously, of the whole Claim 1◦ is complete.  We continue the proof with the construction of a conformal measure m. Under some additional hypothesis (see Claim 2◦ below), which will be satisfied for compactly nonrecurrent elliptic functions (most important for us and dealt with in the last chapters of the book), it will be just mY . In the fully general case, we will need one more limiting procedure. Suppose that y ∈ Y ∩ {f n (c) : n ≥ 1} for some c ∈ Crit(f ). Then there exists n ≥ 1 such that y = f n (c) and Y ∩ {c,f (c),f 2 (c), . . . ,f n−1 (c)} = ∅. It then follows from item (b) of Claim 1◦ that mY ({y}) = |f ∗ (c)|s(Y ) mY (c) = 0 · mY (c) = 0. Along with items (c) and (d) of Claim 1◦ , this implies that mY (Y ) = 0.

(17.48)

Therefore, since, in addition, f ((f )) = (f ), in order to prove s(Y )conformality of the measure m, it would suffice to show that mY (f (Y \(f ))) = 0.

(17.49)

We consider now an, already announced, special case. Since the forward orbit of any point in the Julia set contains only finitely many critical points, we may find a crossing set Y such that Crit(f ) ∩

∞ 

f n (Y ) = ∅.

(17.50)

lim sup |(f n ) (y)| = +∞

(17.51)

n=0

We shall prove the following. Claim 2◦ If n→∞

for every y ∈ Y \((f ) ∪ {∞}), then the measure mY is s(Y )-conformal. Proof If mY (f (y)) > 0 for some y ∈ Y \((f ) ∪ {∞}), then we would have that lim sup mY,e ((f n )(y)) = +∞. n→∞

(17.52)

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Part IV Classics, Geometry, and Dynamics

This would, however, contradict (17.32) of Lemma 17.6.6 since the measure mY is almost s(Y )-conformal. Thus, (17.49) holds and the measure mY is s(Y )conformal.  We now move on to the fully general case. We will not appeal to, and we will not use, what we have done under hypothesis (17.51). We decompose the set Y \(f ) into three disjoint sets: Y∞ := {y ∈ Y : ∞ ∈ ω(y)}, Yc := {y ∈ Y : ω(y) ∩ Crit(f ) = ∅}\Y∞, and

 Y0 := Y \(Y∞ ∪ Yc ) ⊆ y ∈ Y : ω(y) ∩ Crit(f ) = ∅ . ∞ Now, for every y ∈ Y , fix a strictly ∞ sequence (nk (y))k=1 of positive  nincreasing (y)(y) k converges and integers such that the sequence f k=1

lim f nk (y) = ∞

k→∞

if y ∈ Y∞ , while lim f nk (y) (y) ∈ Crit(f )

k→∞

if y ∈ Yc . Observe that, for every k ≥ 1, 

Yk := (f ) ∪ f nk (y) (y) : y ∈ Y∞ ∪ Yc ∪ Yn is a crossing set. Let (kl )∞ increasing sequence of integers l=1 be a strictly  ∞ such that the sequence of numbers s Ykl l=1 converges and the sequence  ∞ of measures mYkl l=1 converges in the weak* topology of Borel probability measures on  C. Denote the former limit by s and the latter by m. Because of (17.42), we have that 1 < h∗ (f ) ≤ s ≤ DDh (J (f )).

(17.53)

We shall show that m is a desired conformal measure. First, we shall check its conformality. For every y ∈ (f ), let yˆ := y, and, for every y ∈ Y∞ ∪ Yc ∪ Yn , let yˆ := lim f nk (y) (y). k→∞

Let



 Yˆ := yˆ : y ∈ (f ) ∪ Y∞ ∪ Yc ∪ Y0 = (f ) ∪ {∞} ∪ yˆ : y ∈ ∪Yc ∪ Y0 .

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Now let U be an open neighborhood of Yˆ . Then Yk ⊆ U for all k ≥ 1 large enough, say k ≥ q ≥ 1. Noting that      s  s Y kl  |f |s Crit(f )∪f −1 (∞) = |f |s = 0, Crit(f )∪f −1 (∞) 2

2

1◦

with Y being the sets Ykl , l ≥ q, and the measures and using then Claim mkl , l ≥ q, it follows from Lemma 10.1.9, applied with  := U , that the limit measure m enjoys the following properties: m(f (A)) ≥ A

|f  |ss dm,

(17.54)

|f  |ss dm,

(17.55)

for every special set A ⊆ J (f ), and m(f (A)) = A

for every special set A ⊆ J (f )\U . Now let A be a special subset of J (f )\Yˆ . Then A=

∞ 

ˆ A\B(Y,1/j ),

j =1

 ∞ ˆ sequence A\B(Y,1/j ) j =1 is ascending, ˆ J (f )\B(Y,1/j ) ⊆ J (f )\Y . Therefore,   ˆ )) m(f (A)) = lim mY f (A\B(Y,1/j

the

and

ˆ A\B(Y,1/j ) ⊆

j →∞

= lim

j →∞ A\B(Y,1/j ˆ )

|f  |ss dm =

A

|f  |ss dm.

(17.56)

So, in order to establish s-conformality of the measure m, all that is left is for us to show that ˆ s m({y}) ˆ m({f (y)}) ˆ = |f ∗ (y)|

(17.57)

for every y ∈ (f ) ∪ Yc ∪ Yn . For every l ≥ 1, put

  ml := mYkl and sl := s Ykl .

Since the measures ml are semi-sl -conformal, it follows from Lemma 15.4.1 that there exists a constant C > 0 such that, for every ω ∈ (f ) and every r ∈ (0,1], we have that ml (B(ω,r)) ≤ Cr sl +p(ω)(sl −1) ≤ Cr h∗ (f )+p(ω)(h∗ (f )−1),

206

Part IV Classics, Geometry, and Dynamics

where the second inequality was written as a result of (17.42). Hence, passing to the limit, m(B(ω,r)) ≤ Cr h∗ (f )+p(ω)(h∗ (f )−1) . Since, also by (17.42) for example, h∗ (f ) > 1, by letting r " 0, this implies that m({ω}) = 0. So, we have that m((f )) = 0

(17.58)

and, using also the fact that f ((f )) = (f ), (17.57) holds for all y ∈ (f ). Now let y ∈ Yc . Since the set Yˆ is finite and the congruence class of yˆ modulo f is infinite, there exists z ∈ Crit(f )\U congruent with yˆ modulo ˆ if U is a sufficiently small open neighborhood of f , i.e., such that z ∼f y, ˆ the finite set Y . Then, by using (17.55) (and also the fact that z ∈ J (f )), we get that   ˆ ss m({y}). ˆ m {f (y)} ˆ = m({f (z)}) = |f  (z)|ss m({z}) = 0 = |f  (y)| Thus, (17.57) holds also for all y ∈ Yc . Finally, let y ∈ Y0 . We conclude from the definition of Y0 that ω(Y0 ) is a forward invariant (f (ω(Y0 )) ⊆ ω(Y0 )) compact subset of the complex plane C such that ω(Y0 ) ∩ Crit(f ) = ∅. It then directly follows from Lemma 17.4.3 that f |ω(Y0 ) ∈ A0 (ω(Y0 )). Therefore, in view of Corollary 9.1.4, we could have chosen such a point yˆ that this point is either periodic or not preperiodic, ˆ s ≥ 1. lim sup |(f n ) (y)| n→∞

Therefore,   ˆ s > 0 lim sup  f n ) (f k (y) n→∞

for all integers k ≥ 0. We will need this property only for k = 0 and 1. Suppose first that yˆ is not a preperiodic point of f . Let w be either yˆ or f (y). ˆ Then there n ∗ would exist η > 0 and an infinite set Nw ⊆ N such that |(f ) (w)| ≥ η for every n ∈ Nw . So, if seeking contradiction, we assume that m({w}) > 0, then, with the use of (17.54), we would obtain that       m {f n (w)} ≥ ηs m({w}) = +∞. 1 ≤ m {f n (w) : n ∈ Nw } = n∈Nw

n∈Nw

This contradiction shows that m({w}) = 0. Therefore,   m {f (y)} ˆ = 0 = |f  (y)| ˆ ss m({y}). ˆ

17 Geometry and Dynamics of (All) Elliptic Functions

207

Now suppose that yˆ is not a periodic point of f . Denote its period by p. If ˆ > 1, then, again with the use of (17.54), we would obtain that |(f p )∗ (y)| ∗    1 ≥ lim sup m {f n (y)} ˆ ≥ lim sup | f np (y)| ˆ s m({y}) ˆ n→∞ n→∞  ∗ = lim sup | f p (y)| ˆ sn m({y}). ˆ n→∞

Therefore,  ∗ m({y}) ˆ ≤ lim inf | f p (y)| ˆ −sn = 0. n→∞

Likewise, since = f (y) ˆ and |(f p )∗ (f (y))| ˆ = |(f p )∗ (y)| ˆ > 1, we have that m({f (y)}) ˆ = 0. Therefore,   m {f (y)} ˆ = 0 = |f ∗ (y)| ˆ s m({y}). ˆ f p (f (y)) ˆ

So, finally, assume that |(f p )∗ (y)| ˆ = 1. Then, again by virtue of (17.54), we get that   m {f (y)} ˆ ≥ |f ∗ (y)| ˆ s m({y}) ˆ and s   ∗    m({y}) ˆ = m f p−1 (f (y)) ˆ ≥  f p−1 (f (y)) ˆ  m {f (y)} ˆ   ˆ −s m {f (y)} ˆ . = |f ∗ (y)| Therefore,   m {f (y)} ˆ = |f ∗ (y)| ˆ s m({y}) ˆ and the proof of s-conformality of the measure m is complete. We shall now prove that m({∞}) = 0.

(17.59)

The proof is almost entirely the same as the proof of item (c) of Claim 1◦ . It is immediate if f2−1 (∞) = ∅ since, for every point z ∈ f2−1 (∞), we have that |f ∗ (z)| = 0, and then, by the already proven s-conformality of m, m(∞) = |f ∗ (z)|s(Y ) mY ({z}) = 0. In the fully general case, the proof is much more involved. We start it now. For every k ≥ 0, consider Sk , the square centered at the origin whose edges are parallel to the coordinates’ axes and are of length 2k . Since, by Claim 1◦ and the definition of the set Yˆ , each measure msl is almost sl -conformal with respect to the triple (Yˆ , U , U ) if U is a sufficiently small neighborhood of Yˆ , and since each “annulus” Ak = Sk+1 \Sk is a union of 3 × 4k squares whose

208

Part IV Classics, Geometry, and Dynamics

edges are parallel to the coordinates axes and are of length 1, it follows from (17.32) of Lemma 17.6.6 that, for all k ≥ 1 and all l ≥ 1, √ mel (Ak ) ≤ 3M( 2)4k . Consequently, msn (Ak )

√ √ √ √ 3M( 2)4k 3M( 2)4k 3M( 2)4k ≤ ≤ ≤ = 3M( 2)4(1−h∗ (f ))k . k s ks kh (f ) l l ∗ (1 + 4 ) 4 4

Since h∗ (f ) > 1 (see, again, (17.40)), we, thus, get, for all j ≥ 1 and all n ≥ 1, that ⎛ ⎞ ∞ ∞   C\Sj ) = msl ⎝ Ak ⎠ = msl (Ak ) msl ( k=j

≤

∞ 

k=j

√ 3M( 2)4(1−h∗ (f ))k

k=j

√ = 3M( 2)(1 − 41−h∗ (f ) )−1 4(1−h∗ (f ))j . Since limj →∞ 4(1−h∗ (f ))j = 0, this implies that m(∞) = 0, i.e., (17.59) holds. Formulas (17.59) and (17.58) imply that m((f ) ∪ {∞}) = 0. Now a direct application of Proposition 17.5.2, Lemma 17.6.4, and (17.53) completes the proof of Theorem 17.6.7. 

17.7 Hausdorff Dimension of Escaping Sets of Elliptic Functions We prove, in this section, that the Hausdorff dimension of the set of points escaping to infinity under the action of an elliptic function f is precisely equal to 2qmax (f ) . qmax (f ) + 1 This fact was first obtained in [GK1]; see also [GK2] for related results. Our method of proof is somewhat different and considerably simpler. We first provide an upper bound for this dimension and then the lower bound. Along with the estimate of the previous section, this shows that I∞ (f ), the set of points escaping to infinity (unlike the case of exponential functions,

17 Geometry and Dynamics of (All) Elliptic Functions

209

for example (see [McM1])), is a very thin subset of the Julia set. We now recall, from (14.24) of Section 14.4, the formal definition of I∞ (f ), ⎧ ⎫ ⎨ ⎬  f −n (∞) or lim f n (z) = ∞ . I∞ (f ) = z ∈ C : z ∈ n→∞ ⎩ ⎭ n≥0

As an immediate consequence of Theorems 14.4.13 and 17.2.1, we get the following. Theorem 17.7.1 If f : C −→  C is a nonconstant elliptic function, then I∞ (f ) ⊆ J (f ). As announced, we start with the upper bound. Lemma 17.7.2 If f : C −→  C is a nonconstant elliptic function, then HD(I∞ (f )) ≤

2qmax (f ) . qmax (f ) + 1

Proof As in the previous section, put q := qmax (f ). Keep R1 ≥ R0 the same as in the proof of Theorem 17.3.1, i.e., satisfying : B(b1,R0 ) −→ C, 1 ≤ (17.17). Likewise, the holomorphic branches fb−1 2,b1,j j ≤ qb , of f −1 . For every real number R > 0, set IR (f ) := {z ∈ C : ∀n≥0 |f n (z)| > R}. Since the series 2q q+1 ,

"

b∈f −1 (∞)\{0} |b|

−s

converges for all s >

2q q+1 ,

given t >

there exists R∗ ≥ R1 such that q(4A2 )t



|b|

− q+1 q t

≤ 1.

(17.60)

b∈B∞ (2R∗ )∩f −1 (∞)

Fix R ≥ 4R∗ . Put E := f −1 (∞) ∩ B∞ (R/2). It follows from (17.16)–(17.18) that, for every l ≥ 1, the family % fb−1 ,b

l l−1 ,jl

&   ◦ fb−1 ,b ,j · · · ◦ fb−1,b ,j Bb0 (R/2) : bi ∈ ER : 1 ≤ ji ≤ qbi , i = 0,1, . . . ,l , l−1 l−2 l−1 1 0 1

denoted by Wl , is well defined and covers IR (f ). Applying (17.14), the second part of (17.16), and noting that Be (b,R0 ) ⊆ Be (0,2|b|) for every b ∈ E, we may now estimate as follows:

210

Part IV Classics, Geometry, and Dynamics

"l (R) q

:=

q

bl  

···

bl ∈E jl =1



 diamts fb−1 ,b

  ◦ fb−1 ,b ,j · · · ◦ fb−1,b ,j Bb0 (R/2) l l−1 ,jl l−1 l−2 l−1 1 0 1

b1 ∈E j1 =1 b0 ∈E 1 qb  1  1   1

qb

 l

b1   

−1 −1 1 f −1 1 bl ,bl−1 ,jl ◦ fbl−1 ,bl−2 ,jl−1 · · · ◦ fb1 ,b0 ,j1 bl ∈E jl =1 b1 ∈E j1 =1 b0 ∈E 1   × diamts Bb0 (R/2) ⎛ qb −1 ⎞t l qb qb q l 1  ⎜     (2|bl−1 |) bl ⎟ ⎜ ⎟ ≤ ··· ⎜2A2 ⎟ ⎝ ⎠ |bl |2

≤

···

bl ∈E jl =1

b1 ∈E j1 =1 b0 ∈E

⎛

qb

l−1

−1

⎜ q ⎜ (2|bl−2 )| bl−1 ·⎜ ⎜2A2 2 |b ⎝ l−1 |  ≤ Lt  ≤ Lt

2 R 2 R

t

q

⎞t

⎛ qb −1 1 ⎟ qb ⎜ ⎟ 1 (2|b |) ⎜ 0 ⎟ · · · ⎜2A2 ⎟ ⎝ |b1 |2 ⎠

q

(4A2 )lt

t

q

1t 1 1 1 1 BR 1 0 ∞

#    

(4A2 )lt

bl   bl ∈E jl =1 qb l  

⎞t  − t ⎟ qb R 1 ⎟ 0 ⎟ × Lt ⎠ 2 |b0 |2t 

q

···

b1   

|bl |−2t

q+1 q+1 t − q t − · · · |b0 | q



|bl−1 |

b1 ∈I j1 =1 b0 ∈E q

···

b1   



q+1 q+1 q+1 − q t − − t t |bl−1 | q · · · |b0 | q



|bl |

bl ∈E jl =1 b1 ∈E j1 =1 b0 ∈E ⎛ ⎞l t   − q+1 t 2 q ≤ Lt (4A2 )lt ⎝ |b| q ⎠ q l R b∈E

 ≤ Lt

2 R

t

⎛

q ⎜

t ⎝q(4A2 )



⎞l q+1 − q+ t ⎟ |b| ⎠ .

b∈B∞ (2R∗ )∩f −1 (∞)

Applying (17.60), we, therefore, get that "l ≤ Lt (2/R)t/q . Since the diameters (in the spherical metric) of the sets of the covers Wl converge uniformly to 0 as l " ∞, we, therefore, infer that Hts (IR (f )) ≤ Lt (2/R)t/q , where the subscript s indicates that the Hausdorff measure is considered with respect to the spherical metric. Consequently, HD(IR (f )) ≤ t; if we put # $  IR,e (f ) := z ∈ C : lim inf |f n (z)| > R = f −k (IR (f )), n→∞

k≥1

then also HD(I∞ (f )) ≤ HD(IR,e (f )) = HD(IR (f )) ≤ t. Now letting t " 2q  q+1 finishes the proof. As an immediate consequence of this theorem, we obtain the following. Corollary 17.7.3 If f : C →  C is an elliptic function, h := HD(J (f )), then Hh (I∞ (f )) = 0; consequently, S(I∞ (f )) = 0, S being a planar Lebesgue measure on C.

17 Geometry and Dynamics of (All) Elliptic Functions

211

This corollary and the previous lemma show that the escaping set I∞ (f ) is a fairly small subset of the Julia set J (f ). Now we shall prove the opposite inequality. Lemma 17.7.4 If f : C −→  C is a nonconstant elliptic function, then HD(I∞ (f )) ≥

2qmax (f ) . qmax (f ) + 1

Proof As in the previous proof, keep q := qmax (f ). Let fq−1 (∞) be the set of all poles of f of order q. Fix R0 so large, as required in (17.8), (17.10), (17.14), and (17.13). Then fix R ≥ 4 max{R0,diam(RF )}.

(17.61)

Our goal is to apply Theorem 1.7.3. We perform an inductive construction, required by this theorem, as follows. Fix ξ ∈ fq−1 (∞). Let ˆ E0 := C. As an inductive assumption, suppose that, for some n ≥ 1, a collection E0, . . . ,En−1 of mutually disjoint compact connected subsets of E0 has been defined with the following properties. (a) For every F ∈ En , there exists a unique set F− ∈ En−1 such that F ⊆ F− . The set F− will be referred to as the parent of F and F as a child of F− . (b) If F ∈ En , then there exists a unique pole in (ξ +f )∩(B¯ ∞ (2n+1 A1 R q )\ ¯ F ,R −1 ) and B∞ (2n+2 A1 R q )), denoted by bF , such that f n (F ) = B(b there exists a compact connected set Fˆ containing F such that ¯ F ,2R −1 ) and (c) f n (Fˆ ) = B(b n ˆ ¯ F ,2R −1 ) is one-to-one (and, consequently, a homeo(d) f | ˆ : F −→ B(b F

morphism). As the inductive step, for every F ∈ En , define En+1 (F ) to be the collection of all the sets of the form −1 ¯ Fb := (f n |F )−1 (fb−1 (B(b,R ))) ⊆ F F ,B(b,1),1

(17.62)

and −1 ¯ ˆ (B(b,2R ))) ⊆ F, Fˆb := (f n |Fˆ )−1 (fb−1 F ,B(b,1),1

(17.63)

where b ranges over all elements of the set (ξ + f ) ∩ (B¯ ∞ (2n+2 A1 R q )\B∞ (2n+3 A1 R q )).

(17.64)

212

Part IV Classics, Geometry, and Dynamics

Obviously, all elements of En+1 (F ) are compact and mutually disjoint. Let ¯ F ,R −1 )).  := f (B(b −1 )) for all a ∈ (ξ +  ) and, by (17.10), that ¯ Note that  = f (B(a,R f   −1 ¯ )  ⊃ B∞ A1 (R −1 )−q = B∞ (A1 R q ) ⊃ B(b,2R

for every b ∈ (ξ + f ) ∩ B¯ ∞ (2A1 R q ) ⊇ (ξ + f ) ∩ (B¯ ∞ (2n+2 A1 R q )\B∞ (2n+3 A1 R q )). This, along with (a), implies that, for such bs, we have that f n+1 (Fb ) = −1 ), f n+1 (Fˆ ) = B(b,2R −1 ), and, obviously, the map f n+1 | ¯ ¯ B(b,R b ˆ is oneFb

to-one. Thus, defining  Fb : b ∈ (ξ + f ) ∩ (B¯ ∞ (2n+2 A1 R q )\B∞ (2n+3 A1 R q )) En+1 := F ∈En

finishes our inductive construction so that (a), (b), (c), and (d) hold. Fix n ≥ 1 and F ∈ En . The conditions (a)–(d) imply that the Koebe Distortion Theorem (Theorem 8.3.8) applies to the map f n−1 |Fˆ− and f n |Fˆ . We, therefore, get from (17.13) and (17.62) that   ¯ F ,R −1 ))) diame (f n−1 |F− )−1 (fb−1 (B(b diame (F ) F− ,B(bF ,1),1     = diame (F− ) diame (f n−1 |F− )−1 B¯ bF− ,R −1   ¯ F ,R −1 )) diame fb−1 ( B(b ,B(b ,1),1 F− F    ≤K diame B¯ bF− ,R −1 ) (17.65) ≤ KA2 (|bF | − R −1 ) ≤ KA2 (2n A1 R q ) − q+1 q n

= γ1 2

− q+1 q

− q+1 q

R −1

R −1

,

where − q+1 q

γ1 := KA2 A1

R −(q+2) .

So, an immediate induction yields diame (F ) ≤ γ1n

n  k=1

− q+1 q k

2

.

(17.66)

17 Geometry and Dynamics of (All) Elliptic Functions

213

Hence, with the notattion of Theorem 1.7.3, we get that dn = max{diame (F ) : F ∈ En } ≤ γ1n

n 

− q+1 q k

2

.

(17.67)

k=1

So, in conclusion, {En }∞ n=1 is a McMullen sequence, and Theorem 1.7.3 applies indeed. Still keeping F ∈ En , by the same arguments as those generating (17.65), we get that    ¯ F ,R −1 )) S (f n−1 |F− )−1 fb−1 (B(b 0 S(F ) F− ,B(bF ,1),1   = S(F− ) S (f n−1 |F− )−1 (B(bF− ,R −1 ))   ¯ F ,R −1 )) S fb−1 ( B(b ,B(b ,1),1 F− F   ≥ K −2 S B(bF− ,R −1 ) (17.68) −1 ≥ K −2 A−2 2 (|bF | + R ) n+3 ≥ K −2 A−2 A1 R q ) 2 (2 − q+1 q n

= γ2−2 4

−2 q+1 q

−2 q+1 q

,

2 q+1

q+1

where γ2 := KA2 A1 q R 2(q+1) 64 q and S, as always, denotes a planar Lebesgue measure on C. Since, with some γ3 > 0, there are at least γ3 4n (see (17.61) and (17.64)) children of each element in En , we, thus, get from (17.68) that, for each G ∈ En , we have that   S H ∈En+1(G) H − q+1 n −n n (G) := ≥ γ3 γ2−2 4n 4 q = γ3 γ2−2 4 q . S(G) Therefore, − qn

n := min{n (G) : G ∈ En } ≥ γ3 γ2−2 4

.

Combining this with (17.67), we, thus, get that " "n−1 (n − 1) log(γ3 γ2−2 ) − q1 log 4 n−1 k=1 k k=1 log k lim ≤ lim " q+1 n−1 n→∞ n→∞ log dn n log γ1 − q log 2 k=1 k =

1 q log 4 q+1 q log 2

=

2 . q +1

Thus, "n−1 2 − lim

n→∞

log n 2 2q ≥2− = . log dn q +1 q +1

k=1

(17.69)

214

Part IV Classics, Geometry, and Dynamics

As in Theorem 1.7.3, denote by E∞ the set ∞ 

F.

n=1 F ∈En

It follows from Theorem 1.7.3, applied with a planar Lebesgue measure S on 2q C, and (17.69) that HD(E∞ ) ≥ q+1 . Since also, by (b) (|bF | ≥ 2n+1 A1 R q for all F ∈ En ) and by (c), we have that E∞ ⊆ I∞ (f ), we, thus, conclude that HD(I∞ (f )) ≥

2q . q +1 

The proof is complete.

Along with Lemma 17.7.2, this gives the following main result of the current section. Theorem 17.7.5 If f : C −→  C is a nonconstant elliptic function, then HD(I∞ (f )) =

2qmax (f ) ∈ [1,2). qmax (f ) + 1

We would like to remark that this theorem has recently found a beautiful, at least in our eyes, generalization in [MyU7] for the whole class of functions considered in [BK]. Elliptic functions form its most regular and most transparent subclass. Indeed, in [MyU7], a closed formula for the Hausdoff dimension of points was obtained by expresing it in terms of poles. For elliptic functions, it gives just Theorem 17.7.5.

17.8 Conformal Measures of Escaping Sets of Elliptic Functions In Chapter 20, we will start presenting a systematic account of the theory of conformal measures for nonrecurrent elliptic functions. In this section, we still deal with arbitrary elliptic functions and we will “only” prove that any conformal measure for any elliptic function vanishes on its set of escaping points. This fact is interesting on its own and will be needed in the proof of ergodicity and conservativity of conformal measures (see Theorem 20.3.11). Its proof is similar to the proof of Theorem 17.7.2. Let  f −n (∞). (17.70) I− (f ) := n≥1

Recall that conformal measures were introduced in Definition 10.4.1. The result, which we stated would be proved in this section, is the following.

17 Geometry and Dynamics of (All) Elliptic Functions

215

Lemma 17.8.1 If m is a t-conformal measure for an elliptic function f : C −→  C, then m(I∞ (f )\I− (f )) = 0. Furthermore, there exists R > 0 such that m({z ∈ C : lim inf |f n (z)| > R}) = 0. n→∞

Proof As in the previous sections, we denote q = qmax (f ). It suffices to prove the lemma for the spherical measure ms . Let b be a pole of f : C → C. We shall obtain first an upper estimate on ms (Bb (R)) similar to the second inequality in (17.16). And, indeed, fix any R ≥ T (f ) and consider the following two open connected, simply connected sets: ∗ ∗ (R) : Imz > 0} and BR− = {z ∈ B∞ (R)\ : Imz < 1}. BR+ = {z ∈ B∞ −1 ± For every j = 1,2, . . . ,qb , let fb,B ± : BR −→ C be the holomorphic ,j R

branches of f −1 resulting from Lemma 17.2.8. So, for every j = 1,2, . . . ,qb , we obtain that   −1   ± −1 (f ± ) t dms . 1 ≥ ms fb,B ±,j BR = (17.71) b,B ,j s BR±

R

R

Now also using (17.14), we obtain that 1≥

BR±



 −1  (f ± ) t dms ≥ b,B ,j s R

=

(2A2 )−t (1 + |b|2 )t

and BR±

 −1  (f ± ) t dms ≤ b,B ,j s R

BR±

(2A2 )−1 qbq−1 |z| b 1 + |b|2

 BR±

BR±

2A2 |b|−2 |z|

≤ (2A2 ) |b| t

−2t BR+

|z|

dms (z)

qb −1 qb t

qb −1 qb

|z|

t

dms (z)

(17.72)

t

q−1 q t

dms (z) (17.73) dms (z).

Taking any pole b of maximal multiplicity, i.e., with qb = q, we see from (17.72) that BR±

|z|

q−1 q t

dms (z) ≤ (2A2 )t (1 + |b|2 )t < +∞.

216

Part IV Classics, Geometry, and Dynamics

Put / "R := max

0 BR+

|z|

q−1 q t

dms (z),

BR−

|z|

q−1 q t

dms (z) ∈ [0,+∞).

Since BR∗ = BR+ ∪ BR− , we obtain that qb 

Bb∗ (R) =

j =1

qb  +   − −1 −1 B ∪ BR . fb,B fb,B ± ± R ,j ,j R

j =1

R

Hence, using also (17.71) and (17.73), we obtain, for any pole b of f , that ms (Bb∗ (R)) ≤

qb  + j =1 BR

qb   −1  (f + ) t dms + b,B ,j s

− j =1 BR

R

 −1  (f − ) t dms b,B ,j s R

≤ 2(2A2 )t q"R |b|−2t .

(17.74)

Now the argument goes essentially in the same way as in the proof of Lemma 17.7.2. We present it here for the sake of completeness. Keep R1 ≥ R0 the same as in the proof of Theorem 17.3.1, i.e., satisfying (17.17). Recall that, for every real number R > 0, we put in the proof of Lemma 17.7.2 IR (f ) = {z ∈ C : ∀n≥0 |f n (z)| > R}. Since the series 2q q+1 ,

"

b∈f −1 (∞)\{0} |b|

−s

converges for all s >

2q q+1 ,

given t >

there exists R∗ ≥ R1 such that q(4A2 )t



|b|

− q+1 q t

≤ 1/2.

(17.75)

b∈B∞ (2R∗ )∩f −1 (∞)

Fix R ≥ 4R∗ . Put E := f −1 (∞) ∩ B∞ (R/2). It follows from (17.16)–(17.18) that, for every l ≥ 1, the family % fb−1 ,b

l l−1 ,jl

&   ◦ fb−1 ,b ,j · · · ◦ fb−1,b ,j Bb0 (R/2) : bi ∈ ER : 1 ≤ ji ≤ qbi , i = 0,1, . . . ,l , l−1 l−2 l−1 1 0 1

denoted by Wl , is well defined and covers IR (f ). Applying (17.14) and (17.74) and noting that Be (b,R0 ) ⊆ Be (0,2|b|) for every b ∈ E, we may now estimate as follows:

17 Geometry and Dynamics of (All) Elliptic Functions

217

ms (IR (f )) q

≤

bl   bl ∈E jl =1

q

···

b1   

b1 ∈E j1 =1 b0 ∈E

   ms fb−1 ◦ fb−1 ,b ,j ◦ · · · ◦ fb−1,b ,j ◦ fb−1,b ,j Bb0 (R/2) l ,bl−1 ,jl l−1 l−2 l−1 2 1 2 1 0 1

1 1t qb 1  ∗  1  1   1 −1 1  −1 −1 −1 1 1  ≤ ··· 1 fbl ,bl−1 ,jl ◦ fbl−1 ,bl−2 ,jl−1 ◦ · · · ◦ fb2 ,b1 ,j2 ◦ fb1 ,b0 ,j1  1 1 1 B (R/2) bl ∈E jl =1 b1 ∈E j1 =1 b0 ∈E b0 ∞   × ms Bb0 (R/2) ⎞ ⎛ ⎛ ⎛ qb −1 t qb −1 ⎞t qb −1 ⎞t l−1 l 1 qb qb ⎜ q q 1  ⎜ |b | bl ⎟ ⎜ |b | bl−1 ⎟ ⎜ |b | qb1 ⎟  l   ⎟ ⎜ l−1 ⎟ ⎜ l−2 ⎟ ⎜ 0 lt ⎟ ≤ ··· (4A2 ) ⎜ ⎟ ·⎜ ⎟ ⎟ ···⎜ ⎝ ⎠ ⎝ ⎠ ⎝ |b1 |2 |bl |2 |bl−1 |2 ⎠ qb l

 

bl ∈E jl =1

b1 ∈E j1 =1 b0 ∈E

× 2q(2A2 )t "R |b0 |−2t q

= 2q(2A2 )t "R (4A2 )lt

q

bl  

···

bl ∈I jl =1 qb

≤ 2q(2A2 )t "R (4A2 )lt

 l

···

b1   

q+1 q+1   − − t t |bl |−2t |bl−1 | q · · · |b0 | q

b1 ∈I j1 =1 b0 ∈E qb 1    

q+1 q+1 q+1  − q t − − t t |bl−1 | q · · · |b0 | q

|bl |

bl ∈I jl =1

b1 ∈I j1 =1 b0 ∈E ⎞l  − q+1 t |b| q ⎠ q l ≤ 2q(2A2 )t "R (4A2 )lt ⎝ b∈E

⎛

⎛

⎞l q+1 − q t⎟ |b| ⎠ .



⎜ ≤ 2q(2A2 )t "R ⎝q(4A2 )t

b∈B∞ (2R1 )∩f −1 (∞)

Applying (17.75), we, therefore, get that ms (IR (f )) ≤ 2q(2A2 )t "R 2−l . Letting l → ∞, we, thus, get that ms (IR (f )) = 0. Since ms ◦ f −1 is absolutely continuous with respect to ms and since ∞  

n f −j (IR (f )), z ∈ C : lim inf |f (z)| > R = n→∞

we conclude that

j =0

  ms {z : lim inf |f n (z)| > R} = 0. n→∞

The proof is complete.



P AR T V Compactly Nonrecurrent Elliptic Functions: First Outlook

18 Dynamics of Compactly Nonrecurrent Elliptic Functions

In this chapter, we define the class of nonrecurrent and, more notably, the class of compactly nonrecurrent elliptic functions. This is the class of elliptic functions that we will deal with in great detail from now until the end of the book. Its history goes back to the papers [U3] and [U4], by the second named author, and [KU4]. One should also mention the paper [CJY]. Similarly to all these papers, our treatment of nonrecurrent elliptic functions is based on, in fact, is possible at all, an appropriate version of the breakthrough Ma˜ne´ ’s Theorem proven in [M1] in the context of rational functions. In our setting of elliptic functions, this is Theorem 18.1.6. The first section of the current chapter is entirely devoted to proving this theorem, its first most fundamental consequences, and some other results surrounding it. The next two sections of this chapter, also relying on Ma˜ne´ ’s Theorem, provide us with further refined technical tools to study the structure of Julia sets and holomorphic inverse branches. Our proof of Theorem 18.1.6 stems from the one contained in [KU4] and is closely modeled on the original proof of Ma˜ne´ . A very good general version of Ma˜ne´ ’s Theorem is now known; see [RvS] for all meromorphic functions. Some other papers on the subject include [SL], [GKS], [Ki], and [Ok]. In this book, we will only need the version of this theorem for elliptic functions and so we decided to provide its proof in such a, restricted, generality only. The last section of this chapter has a somewhat different character. It systematically defines and describes the various subclasses of, mainly compactly nonrecurrent, elliptic functions that we dealt with in Part IV of the book. Mostly, these classes of elliptic functions will be defined in terms of how strongly expanding these functions are. We would like to add that while, in the theory of rational functions, such classes pop up in a natural and fairly obvious way, e.g., metric and topological definitions of expanding rational functions describe the same class of functions, in the theory of 221

222

Part V First Outlook

iteration of transcendental meromorphic functions such a classification is by no means obvious, topological and metric analogs of rational function realm concepts do not usually coincide, and the definitions of expanding, hyperbolic, topologically hyperbolic, subhyperbolic, etc. functions vary from author to author. Our definitions seem to us to be quite natural and fit well our purpose of the detailed investigation of the dynamical and geometric properties of the elliptic functions that they define.

18.1 Fundamental Properties of Nonrecurrent Elliptic ˜ e’s Theorem Functions: Man´ In this section, we prove the results forming the fundamental dynamical and topological properties of elliptic functions. They stem from the breakthrough Ma˜ne´ ’s Theorem proven in [M1] in the context of rational functions. Our results in this section are based on an appropriate version of Ma˜ne´ ’s Theorem for elliptic functions, i.e., Theorem 18.1.6. This theorem, roughly speaking, asserts that the connected components of sufficiently small balls centered at points of Julia sets, different from rationally indifferent periodic points and not belonging to the omega limit set of recurrent critical points, have small diameters. It has enormously powerful consequences for the structure of such components, gives some sort of hyperbolicity, and is, along with its technical consequences, an indispensable tool for us throughout the rest of the book. In particular, see Theorem 18.1.15. This is not the most significant consequence of Ma˜ne´ ’s Theorem; rather, this theorem rules out the existence of Siegel disks, Herman rings, and Cremer points for all elliptic functions. Denote by Critr (f ) the set of all recurrent critical points of an elliptic function f : C −→  C, i.e., the set of such critical points c of f that c ∈ ω(c). Note that, except for periodic attracting points, all nonrecurrent critical points of f are contained in the Julia set J (f ) of f . Essentially, from now on throughout the remainder of the book, we will deal with the class of nonrecurrent elliptic functions. Here is their definition. Definition 18.1.1 We say that a (nonconstant) elliptic function f : C −→  C is nonrecurrent (NR) if and only if (NR) Critr (f ) ∩ J (f ) = ∅. Directly from Theorems 17.2.3 and 13.2.5, we obtain the following.

18 Dynamics of Compactly Nonrecurrent Elliptic Functions

223

Observation 18.1.2 If f : C −→  C is a nonrecurrent elliptic function and if c ∈ Crit(f ) ∩ F (f ), then there exists either an attracting periodic point ω of f or a rationally indifferent periodic point ω of f , in either case such that ω(c) ⊆ {f n (ω) : n ≥ 0}. The class of nonrecurrent elliptic functions has several important subclasses. These are particularly transparent in the realm of rational functions on the Riemann sphere  C. In the context of meromorphic transcendental functions, in particular, elliptic functions, their analogs, are not uniquely obvious and definitions vary from author to author and from paper to paper. We will primarily deal with the subclass of compactly nonrecurrent elliptic functions and some of its subclasses such as various subclasses of subexpanding and parabolic elliptic functions, which will be defined in the next section. As has already been stated, similarly to the paper [KU4], the basic indispensable technical tool in the present book for dealing with the geometry and dynamics of nonrecurrent elliptic functions is an appropriate version of Ma˜ne´ ’s Theorem, originally proved by him in [M1] for the class of rational functions. We start with several preparatory results, then we prove the elliptic version of Ma˜ne´ ’s Theorem, and then we derive many of its consequences for recurrent elliptic functions. Recall that, for any set A ⊆ C, we denote  f n (A). O+ (A) = n≥0

Now we shall prove an appropriate version of Przytycki’s Lemma, stated and proved in the realm of rational functions in [P3]. This is an important ingredient of the proof of Theorem 18.1.6. Note that there are places in the several proofs we provide below where one has to proceed in a more subtle way than in the case of rational functions. Let Attr(f ) denote the set of all attracting periodic points of an elliptic function f . Lemma 18.1.3 If f : C −→  C is an elliptic function, then ∀N ∈ N ∀λ ∈ (0,1) ∀ε > 0 ∀κ > 0 ∃ δ0 > 0 ∀δ ∈ (0,δ0 ] if   x ∈ C\Be Attr(f ) ∪ (f ),κ , then, for every integer n ≥ 0 and every connected component W of f −n (Be (x,δ)) such that f n |W has at most N critical points,

224

Part V First Outlook

each connected component W  of f −n (Be (x,λδ)) contained in W satisfies diame (W  ) ≤ ε and lim diame (W  ) = 0

n→∞

uniformly with respect to x and W  . Proof Because of Proposition 17.1.3, it suffices to prove the second, i.e., the last, assertion of this lemma. Proving it, suppose, for a contrary, that there exist N ∈ N, λ ∈ (0,1), and also   (1) a sequence (xn )∞ n=1 of points belonging to C\Be Attr(f ) ∪ (f ),κ , (2) a decreasing to 0 sequence (δn )∞ n=1 , of positive integers diverging to infinity, (3) a sequence {kn }∞ n=1 (4) a sequence of connected components Wn , n ∈ N, of f −kn (Be (xn,δn )) such that the number of critical points (counted with multiplicities) of each map f kn restricted to Wn is bounded above by N, and −n (B (x ,λδ )) con(5) a sequence (Wn )∞ e n n n=1 of connected components of f tained in Wn such that the following limit exists and lim inf diame (Wn ) > 0. n→∞

(18.1)

By item (4), there exist two open Euclidean balls   Be xn,λδn ) ⊆ Bn(1) ⊆ Bn(2) ⊆ Be xn,δn ) such that Mod(A(n)) =

  − log λ and Crit f kn |A(n) = ∅, N (2)

(18.2)

(1)

where A(n) is the geometric annulus Bn \Bn . Because of Proposition 17.1.1, we may assume without loss of generality that all the components Wn intersect (1) (2) the fundamental region Rf of f . Let Wn and Wn be the connected components, respectively, of     f −kn Bn(1) and f −kn Bn(2) containing Wn . Put An := Wn(2) \Wn(1) and, for all 0 ≤ m ≤ kn and i = 1,2, let (i) := f kn −m (Wn(i) ), Wn,m

(2) (1) An,m := f kn −m (An ) = Wn,m \Wn,m ,

(18.3)

18 Dynamics of Compactly Nonrecurrent Elliptic Functions

225

where the last equality sign “=” is validated by item (1) of Theorem 17.1.7. For each integer n ≥ 0, let 0 ≤ m = m(n) ≤ kn be the least integer such that (1) diame (Wn,m ) ≥ η0 (f ),

where η0 (f ) is the number introduced in Proposition 17.1.9. If no such integer exists, put m(n) := kn . Therefore, by virtue of Proposition 17.1.9(1) and (2), for every integer 1 ≤ t ≤ (i) (i) m(n) − 1 the maps f |W (i) : Wn,t −→ Wn,t−1 , i = 1,2, are proper and have n,t at most one critical point. Thus, by Proposition 17.1.9(3) and straightforward (1) (2) induction, both sets Wn,t and Wn,t are simply connected. Hence, for every integer 1 ≤ t ≤ m(n) − 1, the set An,t is a topological annulus. Therefore, invoking (18.3), the fact that An,kn = An , (18.2), Theorem 17.1.7(2), Theorem 8.2.23, and item (4) above, we conclude that  − log λ Mod An,m(n)−1 ) ≥ . N2

(18.4)

(1)

Since diame (Wn,m(n)−1 ) < δ0 (f ), by virtue of item (2) above, it follows from Proposition 17.1.3 that lim (m(n) − 1) = +∞.

n→∞

(18.5)

Since all the connected components Wn , n ≥ 1, intersect the fundamental region Rf , we can pick, for every n ≥ 1, a point zn ∈ Wn ∩ Rf .

(18.6)

It then follows from the compactness of Rf that there exists an increasing  ∞ unbounded sequence of integers (ns )∞ s=1 such that the sequence zns s=1 converges. Denote its limit by z. Because of (18.1), (18.4), and (18.6),  it follows from Theorem 8.2.15 that there exists η > 0 such that Be zns ,2η ⊆ Wn(2) . Hence, for all s ≥ 1 large enough, s ,m(ns )−1 (2)

Be (z,η) ⊆ Wns ,m(ns )−1 . Hence,   f m(ns )−1 (Be (z,η)) ⊂ Be xns ,δns .

(18.7)

Thus, the family of functions {f m(ns )−1 : Be (z,η) −→ C}∞ s=1 is normal; consequently, looking also at (18.8), we conclude that Be (z,η) cannot intersect the Julia set J (f ). If Be (z,η) were contained in a preimage of a Siegel

226

Part V First Outlook

disk or a Herman ring, the limit of diameters of iterates f m(ns )−1 (Be (z,η)) would be positive contrary to item (2) and (18.7). Thus, because of Theorems 17.2.3 and 13.2.5, D is contained in the basin of attraction to an attracting periodic orbit or a parabolic periodic orbit. In either case, as s → ∞, the sets f m(ns )−1 (Be (z,η)) would have some limit point being either an attracting periodic point or a rationally indifferent periodic point. Because of (18.7) and item (2), the sequence (xn )∞ n=1 would also have some limit point being either an attracting periodic point or a rationally indifferent periodic point. This would, however, contradict item (1). The proof is complete.  Our second preparatory result is the following obvious observation. Observation 18.1.4 Let f : C −→  C be an elliptic function. If x ∈ J (f )\ω(Critr (f )), then there exists η1 (f ,x) ∈ (0,η0 (f )] such that (1) there is no critical point c of f for which there exist two integers 0 < n1 ≤ n2 satisfying |f n1 (c) − c| ≤ η1 (f ,x)

and

|f n2 (c) − x| ≤ η1 (f ,x).

If Critr (f ) = ∅, i.e., if the elliptic function f is nonrecurrent, then we set #   η1 (f ) := min η0 (f ),diste J (f ),Attr(f ) , $

min diste (c,O+ (f (c)) : c ∈ J (f ) ∩ Crit(f ) (1 )

and item (1) takes on the following stronger form: |f n (c) − c| > η1 (f ) for all c ∈ J (f ) ∩ Crit(f ) and all integers n ≥ 1.

Now let Nf be the number of equivalence classes of the relation ∼f between critical points of an elliptic function f : C →  C. In other words, Nf = #(Crit(f ) ∩ R) for any fundamental domain R of f whose boundary contains no critical points of f . Of course, Nf is a finite number and Nf ≤ ord(f  ). Our last preparatory result, interesting on its own, is this. Lemma 18.1.5 Let f : C −→  C be an elliptic function. Let x ∈ J (f )\ ω(Critr (f )). If V ⊂ C is an open connected, simply connected neighborhood

18 Dynamics of Compactly Nonrecurrent Elliptic Functions

227

of x, n ≥ 0 is an integer, and U is a connected component of f −n (V ) for which diame (f i (U )) ≤ η1 (f ,x)

(18.8)

for all 0 ≤ i ≤ n, where η1 (f ,x) comes from Observation 18.1.4, then (1) Each set f i (U ), 0 ≤ i ≤ n, is simply connected and contains at most one critical point of f . (2) The equivalence class of the equivalence relation ∼f of each critical point of f intersects at most one of the sets f i (U ), 0 ≤ i ≤ n. (3) If f i (U ) ∩ Crit(f ) = ∅ for some 0 ≤ i ≤ n, then the map f |f i (U ) : f i (U ) −→ C is one-to-one. (4) Consequently,    pc , deg f n |U ≤ Nf∗ := c

where the product is taken over any (fixed) selector of the relation ∼f between critical points of f , e.g., over the fundamental region Rf . (5) Hence,   # U ∩ Crit(f n ) ≤ Nf∗ − 1. Proof The second assertion of item (1) is immediate from the definition of η0 (f ) and since η1 (f ,x) ≤ η0 (f ). Then the first one follows immediately from Corollaries 8.6.18 and 8.6.19 because of (18.8), Theorem 17.1.7(1), and Theorem 13.2.1. In order to prove item (2), suppose for a contradiction that c1 ∈ Crit(f ) ∩ f k1 (U ), c2 ∈ Crit(f ) ∩ f k2 (U ), and c1 ∼f c2 , where 0 ≤ k1 < k2 ≤ n. But then also f k2 −k1 (c2 ) = f k2 −k1 (c1 ) ∈ f k2 (U ). So, |f k2 −k1 (c2 ) − c2 | ≤ η1 (f ).

(18.9)

  f n−k1 (c2 ) = f n−k1 (c1 ) ∈ f n−k1 f k1 (U ) = U .

(18.10)

On the other hand,

So, |f n−k1 (c2 ) − x| ≤ diam(U ) ≤ η1 (f ). Since also k2 − k1 > 0 and n − k1 ≥ k2 − k1 , both (18.9) and (18.10) contradict Observation 18.1.4 and the proof of item (2) is finished. Item (3) now follows immediately from the first part

228

Part V First Outlook

of item (1) and Corollary 8.6.18 because of (18.8), Theorem 17.1.7(1), and Theorem 13.2.1. Item (4) is an immediate consequence of items (1)–(3) along with Corollaries 8.6.18 and 8.6.19 because of (18.8), Theorem 17.1.7(1), Theorem 13.2.1, and multiplicity of degree. Item (5) is now an immediate consequence of (8.70) from Corollary 8.6.17.  Now we are ready to prove the theorem that is fundamental and absolutely indispensable for the remainder of the book. Theorem 18.1.6 (Ma˜ne´ ’s Theorem for Elliptic Functions) Let f : C −→  C be an elliptic function and, as always, let (f ) denote the set of all rationally indifferent periodic points of f . If x ∈ J (f )\((f ) ∪ ω(Critr (f )), then ∀ ε > 0 ∃ δ > 0 such that (a) For all integers n ≥ 0, every connected component of f −n (Be (x,δ)) has Euclidean diameter ≤ ε.  (b) deg f n |V ≤ Nf∗ for all integers n ≥ 0 and for every connected component V of f −n (Be (x,δ)). Proof The core of this theorem is item (a). Then item (b) follows immediately from item (a) and Lemma 18.1.5. Given an open set U ⊂  C, denote, in this proof, by (1) c(U,n) the collection of all connected components of f −n (U ). Of course, if V ∈ c(U,n), then f j (V ) ∈ c(U,n − j ) for all 0 ≤ j ≤ n. Recall that, given any real number α > 0 and an open ball B = B(z,r), we denote by αB the ball B(z,αr). If B is an open ball with radius r, then denote by L(B) the family of all open balls contained in 32 B\B with radii equal to r/4. Denote further by L∗ (B) the family of all squares 32 D with D ∈ L(B). Take η2 (f ,x) ∈ (0,η1 (f ,x)] so small that (2) diste (x,(f ) ∪ Attr(f )) > 10η2 (f ,x).

18 Dynamics of Compactly Nonrecurrent Elliptic Functions

229

Because of Proposition 17.1.3, applied with n = 1, for every ε > 0, there exists ε1 > 0 satisfying the following two conditions. # $ ε η2 (f ,x) , 10 . (3) 0 < ε1 < min 10 (4) If U is an open connected set with diame (U ) ≤ 2ε1 , then diame (W ) ≤ η2 (f ,x) for all W ∈ c(U,1). Let δ > 0 be given by # $ ,x) (5) δ := min η2 (f 10 ,δ0 , where δ0 > 0 is the number produced in Lemma 18.1.3 for N = Nf∗ , λ = 2/3, ε = ε1 /(20Nf∗ ), and κ = η2 (f ,x). Let B0 be the square with center x and radius δ. Suppose that Theorem 18.1.6(a) fails. Then there exists an integer n ≥ 0 and a set V ∈ c(B0,n) with diame (V ) ≥ ε ≥ 10ε1. Hence, there exists an integer n0 ≥ 0 such that there exists V0 ∈ c 32 B0,n0 satisfying (6) diame (f −(n0 −i) (B0 ) ∩ f i (V0 )) ≤ ε1 for all 1 ≤ i ≤ n0 , and (7) diame (f −n0 (B0 ) ∩ V0 ) > ε1 . Since, by (6), diame (B0 ) = 2δ < ε1 , it follows that n0 ≥ 1. Now starting  ∞ with B0 , we shall construct inductively a sequence of squares Bj j =0 and  ∞ a monotone decreasing sequence of positive integers nj j =0 satisfying the following. (8) Bj +1 ∈ L∗ (Bj ) and

(9) there exists Vj ∈ c

3 2 Bj ,nj

 such that

diame (f −(nj −i) (Bj ) ∩ f i (Vj )) ≤ ε1 for all 1 ≤ i ≤ nj and diame (f −nj (Bj ) ∩ Vj ) > ε1 . If we construct such a sequence of squares and then Theorem  integers, ∞ 18.1.6(a) will be proved as follows. The sequence nj j =0 must stabilize, i.e., nj = ni for all j ≥ i and some i ≥ 0. But the definition of the operation L∗ implies that the radius of Bj is equal to (3/8)j δ. In particular, lim diame (Bj ) = 0.

(18.11)

j →∞

On the other hand, by (9),



3 Vj ∈ c Bj ,nj 2





3 Bj ,ni =c 2



230

Part V First Outlook

and diame (f −ni (Bj ) ∩ Vj ) = diame (f −nj (Bj ) ∩ Vj ) > ε1 . It, therefore, follows that   3 Bj ,ni ≥ ε1 > 0. lim diame c 2 j →∞ This, however, contradicts Proposition 17.1.3.  ∞ (18.11)and∞ The sequences Bj j =0 and nj j =0 will be constructed by induction starting with B0 . It follows from (6) and (7) that B0 satisfies (9). For the inductive step, fix an integer j ≥ 0 and suppose that Bi and ni are constructed for all 0 ≤ i ≤ j satisfying (8) and (10). In order to find Bj +1 and nj +1 , we begin by observing that, by (8) and from (a) and the definition of the operation L∗ , we have, for all z ∈ B ∈ L∗ (Bj ), that |z − x| ≤

j +1  i=0

diame (Bi ) =

j +1  i  3

8

i=0

=2

diame (B0 )

j +1  i  3 i=0

8

δ ≤ 4δ < η2 (f ,x).

This means that B ⊆ B(x,η2 (f ,x)). Hence, by invoking (2) of Observation 18.1.4, we get that   (10) diste B,(f ) ∪ Attr(f ) > 9η2 (f ,x). For the induction step (i.e., the construction of Bj +1 and nj +1 ), we shall prove the following. Claim 10 . There exists a ball B ∈ L(Bj ) for which there exist an integer 1 ≤ n ≤ nj and a set V ∈ c(B,n) such that diame (V ) ≥

ε1 . 10Nf∗

Proof Seeking contradition, suppose that the claim is false. Then, invoking also (9), we see that, for all 1 ≤ i ≤ nj , we get that   diame (f i (Vj )) ≤ diame f −(nj −i) (Bj ) ∩ f i (Vj ) 

+ sup diame (W ) : B ∈ L∗ (Bj ) and W ∈ c(B,nj − i) ε1 ≤ ε1 + ≤ 2ε1 . 10Nf∗

18 Dynamics of Compactly Nonrecurrent Elliptic Functions

231

From this inequality, applied to i = 1, and from property (4), we get that diame (Vj ) ≤ η2 (f ,x). In addition, since 2ε1 ≤ η2 (f ,x) (by (3)), we have that diame (f i (Vj )) ≤ η2 (f ,x) for all 1 ≤ i ≤ nj ; hence, in consequence, for all 0 ≤ i ≤ nj . By Lemma 18.1.5, this proves that   deg f nj |Vj ≤ Nf∗ . (18.12) Hence,   #{W ∈ c(Bj ,nj ) : W ⊂ Vj } ≤ deg f nj |Vj ≤ Nf∗ . (18.13)   Also, since Vj ∈ c 23 Bj ,nj , it follows from (10), (18.12), Lemma 18.1.5(5), and Lemma 18.1.3 that ε1 [W ∈ c(Bj ,nj ) and W ⊂ Vj ] ⇒ diame (W ) ≤ (18.14) 20Nf∗ and [S ∈ L(Bj ) and G ∈ c(B,nj )] ⇒ diame (G) ≤

ε . 20Nf∗

(18.15)

Now observe that Vj is the union of all sets W ∈ c(Bj ,nj ), where W ⊂ Vj , along with the sets G ∈ c(B,nj ), where G ⊂ Vj and B ∈ L(Bj ). Denote the former family of these sets by F1 and the latter by F2 . Then, for any two sets W  , W  in F1 ∪ F2 , there exist an integer k ≥ 0 and mutually distinct sets W  = W0,W1, . . . ,Wk = W  alternately belonging to either F1 or F2 and such that W l ∩ W l+1 = ∅ or all 0 ≤ l < k. Then k ≤ 2Nf∗ by (18.13) and, by (18.14) and (18.15), we get that diame (Vj ) ≤ (2Nf∗ + 1)

ε1 3 ≤ ε1 . ∗ 10Nf 10

This contradicts the last inequality in condition (9) and the proof of Claim 1◦ is finished. 

232

Part V First Outlook

  Now take V produced in Claim 1◦ . Denote by V˜ the only set in c 32 B,n   containing V . If deg f n |V˜ ≤ Nf∗ , then, by (8.70) of Corollary 8.6.17, by Lemma 18.1.3, and by condition (5), we get that ε1 diame (V ) ≤ 20Nf∗     since V ∈ c 23 32 S ,n and it is contained in V˜ . This, however, contradicts Claim 1◦ and proves that   deg f n |V˜ ≥ Nf∗ + 1. It then follows from Lemma 18.1.5 that diame (f l (V˜ )) > η2 (f ,x) for some 0 ≤ l ≤ n. Now we define 3 B ∈ L∗ (Bj ). 2 Then f i (V˜ ) ∈ c(Bj +1,n − l) and diame (f l (V˜ )) > δ0 ≥ 10ε1 . Moreover, diame (Bj +1 ) ≤ 2δ < ε1 . Hence, there exists 0 ≤ nj +1 ≤ n − l ≤ nj − l and Vj +1 ∈ 32 c(Bj +1,nj +1 ) such that Bj +1 :=

diame (f −nj +1 (Bj +1 ) ∩ Vj +1 ) > ε1 and diame (f −nj +1 +i (Bj +1 ) ∩ f i (Vj +1 )) ≤ ε1   for all 1 ≤ i ≤ nj +1 . Observe that nj +1 > 0 since diame Bj+1 < ε1 . This ∞ completes the construction of the sequences {Bj }∞ j =0 and {nj }j =0 and, simultaneously, the proof of the entire Theorem 18.1.6.  As a kind of maximal consequence of this theorem, we shall prove the following. Theorem 18.1.7 Let f : C −→  C be an elliptic function. If   X ⊆ J (f )\ (f ) ∪ ω(Critr (f )) is compact (remember that now ∞ ∈ J (f )), then, for every ε > 0, there exists δ > 0 such that

   sup diame Comps (z,f n,δ) : n ≥ 1, z ∈ f −n (X) ≤ ε, where the subscript s above indicates that this is a connected component of f −n (Bs (z,δ)).

18 Dynamics of Compactly Nonrecurrent Elliptic Functions

233

Proof Assume first that Case 1◦ .   X ⊆ J (f )\ (f ) ∪ ω(Critr (f )) ∪ {∞} . Then, by virtue of Theorem 18.1.6, for every x ∈ X, there exists δx > 0 such that, for every integer n ≥ 0, all the connected components of f −n (Bs (x,δx )) have Euclidean diameters ≤ ε. Let 2δ > 0 be a Lebesgue number of the cover {Be (x,δx )}x∈X . Then, for every x ∈ X, there exists y ∈ X such that Bs (x,δ) ⊆ Be (y,δy ). So, for every integer n ≥ 0, each connected component of f −n (Bs (x,δ)) is contained in a (unique) connected component of f −n (Be (y,δy )), whence its Euclidean diameter is ≤ ε. For the general case, let    := diste (f ) ∪ ω(Critr (f )), f −1 (∞) > 0. In view of (17.13) and (17.16), there exists R > 0 so large that if |f (z)| ≥ R/2, then, for some b ∈ f −1 (∞), z ∈ Bb (R/2), |f  (z)| ≥ 2 and diame (Bb (R/2)) ≤ /2.

(18.16)

Consider now the compact set      Y := X ∪ J (f )\ Be (f ) ∪ ω(Critr (f )), /2 ∪ B∞ (R) . Let 0 < δ1 ≤ min{ε,R/2} be the number δ ascribed to the set Y and the number min{ε,R/2,δ1 } according to Case 1◦ . Let δ2 ∈ (0,δ1 ] be the number δ > 0 ascribed to min{ε,R/2,δ1 } and the integer n = 1 according to Proposition 17.1.3. Finally, let 0 < δ ≤ δ2 be ascribed to the set Y and the number δ2 according to Case 1◦ . Because of Case 1◦ , in order to complete the proof of our theorem, it suffices to prove the following. Claim 1◦ . If x ∈ B∞ (R), then, for each integer n ≥ 1, the Euclidean diameter of each connected component V of f −n (Bs (x,δ)) does not exceed ε. Proof Fix w ∈ f −n (x) ∩ V . Let 0 ≤ k ≤ n be the least integer such / B∞ (R), provided that it exists. Otherwise, set k = n. Since that f n−k (w) ∈ n f (w) = x ∈ B∞ (R), we have that k ≥ 1. We shall show by induction that   diame f n−j (V ) ≤ δ1 (18.17) for every 1 ≤ j ≤ k.

234

Part V First Outlook

For j = 1, this formula is true since δ ≤ δ2 and because of the second part of Proposition 17.1.3. For the inductive step, suppose that (18.17) holds for some 1 ≤ j ≤ k − 1. Since f n−j (w) ∈ B∞ (R) and since diame (f n−j (V )) ≤ δ1 ≤ R/2, we conclude that f n−j (V ) ⊆ B∞ (R/2).

(18.18)

It, therefore, follows from the first part of (18.16) that    1  1 diame f n−(j +1) (V ) ≤ diame f n−j (V ) ≤ δ1 ≤ δ1 . 2 2 This proves (18.17). In the case when k = n, Claim 1◦ follows from (18.17) since δ1 ≤ ε. Otherwise, i.e., if 1 ≤ k ≤ n − 1, note first that (18.18) also holds for j = 0, and then that it follows from (18.18), applied with j = k − 1, and from the second part of (18.16) that   f n−k (w) ∈ C\Be (f ) ∪ ω(Critr (f )), /2 . Since we also know that f n−k (w) ∈ / B∞ (R), we conclude that f n−k (w) ∈ Y . It, thus, follows from (18.17) and the, already proven, Case 1◦ that diame (V ) ≤ min{ε,R/2,δ1 } ≤ ε. 

We are done. As a fairly immediate consequence of this theorem, we get the following. Theorem 18.1.8 Let f : C −→  C be a nonrecurrent elliptic function. If X ⊆ J (f )\(f )

is compact (remember that now ∞ ∈ J (f )), then, for every ε > 0, there exists δ > 0 such that (1)

   sup diame Comps (z,f n,δ) : n ≥ 1,z ∈ f −n (X) ≤ ε and (2) 

  lim sup diame Comps (z,f n,δ) : z ∈ f −n (X) = 0,

n→∞

where the subscript s above   indicates that this is a connected component of the set f −n Bs (z,f n,δ) .

18 Dynamics of Compactly Nonrecurrent Elliptic Functions

235

Proof Item (1) is indeed an immediate consequence of Theorem 18.1.7. For item (2), note that the number η1 (f ) > 0 of Observation 18.1.4 is now well defined and we may assume without loss of generality that δ ≤ ε < η1 (f ). Now since (f ) is f -invariant, we have that f −1 (X) ⊆ J (f )\(f ) and, as f −1 (X) is closed and (f ) is finite, we have that diste (f −1 (X),(f )) > 0. Therefore, both Lemmas 18.1.5 and 18.1.3, the latter used for the set of points x ∈ f −1 (X), apply to yield item (2).  Let us now take, first, the basic fruits of Theorem 18.1.8. From now on throughout this chapter, and in fact throughout the entire book, f : C →  C is assumed to be a nonrecurrent elliptic function. As an immediate consequence of Theorems 18.1.8 and 8.3.8, we get the following. Lemma 18.1.9 Let f : C −→  C be a nonrecurrent elliptic function. If X ⊆ J (f )\PC(f ) is compact (remember that now ∞ ∈ J (f ), so, in particular, both X and PC(f ) may contain infinity), then, for every δ ∈ (0,diste (X,PC(f )), we have that (1)    

sup diame fz−n Be (f n (z),δ) : n ≥ 1,z ∈ f −n (X) < +∞, (2)  

  lim sup diame fz−n Be (f n (z),δ) : z ∈ f −n (X) = 0,

n→∞

(3) 

inf |(f n ) (z)| : n ≥ 1,z ∈ f −n (X) > 0, and (4)

 lim inf |(f n ) (z)| : z ∈ f −n (X) = +∞.

n→∞

Remark 18.1.10 We could have alternatively easily deduced this lemma from Lemmas 8.1.19 and 12.5.1, using the latter after projecting on the torus.

236

Part V First Outlook

In (15.59), the number θ (f ) was defined. We now want to utilize it. However, we need a (possibly) smaller number, so we redefine θ (f ) to be & % 1 θ = θ (f ) := min min{θ (ω) : ω ∈ (f )}, diste ((f ),Crit(f )) > 0. 2 (18.19) We denote A = A(f ) := max{A(f ,c), A(f ,b) : c ∈ Crit(f ), b ∈ f −1 (∞) }, (18.20) where A(f ,c) was defined by (8.34) and (8.35) in Section 8.4, while A(f ,b) was defined in (17.15). Recall from Chapter 16 that two points, z,w ∈ C, are equivalent modulo the lattice f associated with the elliptic function f if and only if w − z ∈ f . We then write z ∼f w. Obviously, z ∼f w implies that O+ (f (z)) = O+ (f (w)) and ω(z) = ω(w). Since the number Nf of equivalence classes of critical points with respect to the relation ∼f is finite, each of the following four numbers below is positive: η1 (f ),θ (f )/2, min{(A(f ,c)R(f ,c))1/pc : c ∈ Crit(f )}, min{|c − c | : c,c ∈ Crit(f ) and c = c }, where pc = p(f ,c) is the order of the critical point c of f . Both pc and R(f ,c) were defined in Section 8.4. Fix a positive constant β = βf

(18.21)

smaller than all four of these numbers. It immediately follows from Theorem 18.1.8 that there exists 0 < γ = γf < 1/4 such that, if n ≥ 0 is an integer, / Be ((f ),θ ), then z ∈ J (f ), and f n (z) ∈   (18.22) diame Comp(z,f n,2γf ) < βf . From now on, fix also 0 < τf < θ −1 (f ) min{βf ,γf }

(18.23)

18 Dynamics of Compactly Nonrecurrent Elliptic Functions

237

so small as required in Lemma 15.3.2 for every ω ∈ (f ) and so small that, for every z ∈ J (f ),   diame Comp(z,f ,θ (f )τf ) < min{βf ,γf }. (18.24) As an immediate consequence of Lemma 18.1.5 and item (1) of Theorem 17.1.7 (to get the last assertion of item (1)), we get the following. Lemma 18.1.11 Let f : C −→  C be a nonrecurrent elliptic function. If n ≥ 0 is an integer, η > 0, z ∈ J (f ), and, for every k ∈ {0,1, . . . ,n},   diame Comp(f k (z),f n−k ,η) ≤ βf , then (1) Each connected component Comp(f k (z),f n−k ,η), k ∈ {0,1, . . . ,n}, is simply connected, contains at most one critical point of f , and   f n−k Comp(f k (z),f n−k ,η) = B(f n (z),η). (2) The equivalence class of the equivalence relation ∼f of each critical point of f intersects at most one of the sets Comp(f k (z),f n−k ,η), k ∈ {0,1, . . . ,n}. (3) If Comp(f k (z),f n−k ,η) ∩ Crit(f ) = ∅, for some k ∈ {0,1, . . . ,n}, then the map f restricted to Comp(f k (z),f n−k ,η) is one-to-one. (4) Consequently,   deg f n |Comp(z,f n,η) ≤ Nf∗ − 1. (5) Hence,   # Comp(z,f n,η) ∩ Crit(f n ) ≤ Nf∗ − 1. The argument provided in the proof below is very similar to the one used in the proof of Lemma 18.1.3. Lemma 18.1.12 Let f : C −→  C be a nonrecurrent elliptic function. If z ∈ J (f ), n ≥ 0 is an integer, and f n (z) ∈ / Be ((f ),θ (f )), then, for every r ∈ (0,γf ], (1) theset Comp(z,f n,2r)  is simply connected, (2) f n Comp(z,f n,2r) = Be (f n (z),2r), (3) Comp(z,f n,2r)\Comp(z,f n,r) is a topological annulus, and (4)   log 2 Mod Comp(z,f n,2r)\Comp(z,f n,r) ≥ . (Nf∗ )2

238

Part V First Outlook

Proof The first two assertions follow immediately from Lemma 18.1.11(1) and the definition of γ , i.e., (18.22). The third assertion is an immediate consequence of the first one. By Lemma 18.1.11(5), there exists a geometric annulus R ⊆ Be (f n (z),2r)\Be (f n (z),r) centered at f n (z) of modulus log 2/Nf∗ such that Rn ∩ Crit(f n ) = ∅, where Rn := f −n (R)∩Comp(z,f n,2r) is, also because of Lemma 18.1.11(1), a topological annulus and, because of Theorem 17.1.7, the map f n : Rn −→ R is covering. Hence, applying Corollary 8.2.4 (monotonicity of modulus) and Theorem 8.2.23 along with Lemma 18.1.11(4), we conclude that   log 2 log 2 Mod Comp(z,f n,2r)\Comp(z,f n,r) ≥ Mod(Rn ) ≥ ∗ (Nf∗ )−1 = , Nf (Nf∗ )2 (18.25) meaning that the last item of our lemma and, simultaneously, the whole lemma are proved.  As an immediate consequence of this lemma and Theorem 8.3.15, we get the following. Lemma 18.1.13 Let f : C −→  C be a nonrecurrent elliptic function. Suppose that z ∈ J (f ), n ≥ 0 is an integer, and f n (z) ∈ / Be ((f ),θ ). If 0 ≤ k ≤ n and f k : Comp(z,f n,2γ ) −→ Comp(f k (z),f n−k ,2γ ) is univalent, then

⎛ |(f k ) (y)| |(f k ) (x)|

⎞

⎜ log 2 ⎟ ≤ w ⎝  2 ⎠ Nf∗

for all x,y ∈ Comp(z,f n,γ ), where w : (0,+∞) −→ [1,+∞) is the function produced in Theorem 8.3.15. Lemma 18.1.14 Let f : C −→  C be a nonrecurrent elliptic function. Suppose that z ∈ J (f ), n ≥ 0 is an integer, and f n (z) ∈ / Be ((f ),θ ). Suppose also (2) that Q(1) ⊆ Q(2) ⊆ B(f n (z),γ ) are connected sets. If Qn is a connected (1) component of f −n (Q(2) ) contained in Comp(z,f n,γ ) and Qn is a connected (2) component of f −n (Q(1) ) contained in Qn , then

18 Dynamics of Compactly Nonrecurrent Elliptic Functions

239

    diame Q(1) diame Q(1) n  .  (2)  $ diame Q(2) diame Qn Proof Let 1 ≤ n1 ≤ · · · ≤ nu ≤ n be all the integers k between 1 and n such that Crit(f ) ∩ Comp(f n−k (z),f k ,2γ ) = ∅. Fix 1 ≤ i ≤ u. If j ∈ [ni ,ni+1 − 1] (we set nu+1 = n − 1), then, by Lemma 18.1.13, there exists a universal constant T > 0 such that (1)

diame (Qj ) (2)

diame (Qj )

≥T

diame (Q(1) ni ) (2)

.

(18.26)

diame (Qni )

Since, in view of Lemma 18.1.11, with the use of (18.22), we have that u ≤ #(Crit(f )∩R), in order to conclude the proof it is enough to show the existence of a universal constant E > 0 such that, for every 1 ≤ i ≤ u, (1)

diame (Qni ) (2)

diame (Qni )

≥E

diame (Q(1) ni −1 ) (2)

.

diame (Qni −1 )

Indeed, let c be the critical point in Comp(f n−ni (z),f ni ,2γ ) and pc ≥ 2 be, (1) (2) as always, its order. Since both sets Qni and Qni are connected, we get, for j = 1,2, that diame (Qni −1 )  diame (Qni ) sup{|f  (x)| : x ∈ Qni } (j )

(j )

(j )

p −1

c  diame (Q(i) ni )Diste

(c,Q(i) ni ).

Hence, diame (Q(2) ni )

p −1

(1)

(1)

diame (Qni )



diame (Qni −1 ) p −1

Diste c

(c,Q(1) ni )

·

Diste c

(1)

(2)

(c,Qni )

diame (Q(2) ni −1 )

≥

diame (Qni −1 ) diame (Q(2) ni −1 )

. 

The proof is finished.

An important consequence of Theorem 18.1.8 is the following result, which, along with Theorem 17.2.3, sheds a lot of light on the structure of Fatou and Julia sets of nonrecurrent elliptic functions. Theorem 18.1.15 If f : C −→  C is a nonrecurrent elliptic function, then f has no Siegel disks, Herman rings, or Cremer periodic points. Proof Seeking contradiction, suppose that the function f has a Siegel disk or a Herman ring. Denote it by D. Let  := B(0,1)

240

Part V First Outlook

if D is a Siegel disk and  := A(0;1,r) with r > 1 coming from Theorem 13.2.5 if D is a Herman ring. Let H :  −→ D be the analytic homeomorphism resulting from Theorem 13.2.5(3) and (4), respectively, in the Siegel or Herman case. Using the fact that either D is simply connected or doubly connected and r ∈ (1,+∞), we deduce that ξ is not an isolated point of ∂D; in fact, ∂D has no isolated points. Therefore, there exists a point ξ ∈ ∂D\(f ).

(18.27)

Fix δ > 0 so small, as required in Theorem 18.1.8(2) for the set X := {ξ }, and fix a point w ∈ D ∩ Bs (ξ,δ). Let   (18.28) F := H {z ∈  : |z| = |H −1 (w)|} . Note that F is a compact set (homeomorphic to a circle) and F ⊆ D. Since w ∈ D, for every n ≥ 1, the intersection D ∩ f −n (w) is a singleton. Denoting its only element by wn , we will have that wn ∈ F .

(18.29)

Of course, wn ∈ Comps (wn,f n,δ).   But since, by Theorem 17.1.7, f n Comps (wn,f n,δ) = Bs (w,δ) and since ξ ∈ Bs (w,δ) ∩ J (f ), we conclude that J (f ) ∩ Comps (wn,f n,δ) = ∅. Therefore,   diame Comps (wn,f n,δ) ≥ diste (wn,J (f )) ≥ dist(F,C\D) > 0. Hence,   lim inf diame Comps (wn,f n,δ) ≥ diste (F,C\D) > 0, n→∞

contrary to Theorem 18.1.8(2). Siegel disks and Herman rings are, thus, ruled out.

18 Dynamics of Compactly Nonrecurrent Elliptic Functions

241

Now proceeding again by way of contradiction, suppose that f has a periodic Cremer point. Call it ξ and denote by q ≥ 1 one of its periods. So, we have that ξ ∈ J (f ), f q (ξ ) = ξ, |(f q ) (ξ )| = 1,

(18.30)

and (f q ) (ξ ) is not a root of unity. In particular, ξ∈ / (f ). Take ε > 0 so small that the map f q |Be (ξ,4ε) : Be (ξ,4ε) −→ C is one-to-one. Then let δ > 0 be ascribed to this ε > 0 and the set X = {ξ } according to Theorem 18.1.8. Because of this theorem, we have that   diame Compe (ξ,f qn,δ) ≤ ε for every integer n ≥ 1. In particular,   Compe ξ,f qn,δ ⊆ Be (ξ,ε). It then follows by immediate induction that the restriction   f qn |Comp ξ,f qn,δ  : Compe ξ,f qn,δ −→ Be (ξ,δ) e

is a one-to-one map. So, its inverse −qn

fξ

  : B(ξ,δ) −→ Compe ξ,f qn,δ ⊆ C

is a well-defined holomorphic injective map sending ξ to ξ . Therefore, it follows from the 14 -Koebe Distortion Theorem, i.e., Theorem 8.3.3, that  1 −qn −qn  fξ (Be (ξ,δ)) ⊃ Be ξ, δ|(fξ ) (ξ )| = Be (ξ,δ/4). 4 Hence,

   f qn Be ξ,δ/4 ⊆ Be (ξ,δ)

for every integer n ≥ 1. Thus, the family of maps

qn ∞ f |Be (ξ,δ/4) n=1 is normal. Hence, ξ ∈ F (f q ) = F (f ), contrary to the first item of (18.30). The proof of Theorem 18.1.15 is finished.  As an immediate consequence of this theorem and Theorems 17.2.2, 17.2.3, and 13.2.5, we get the following.

242

Part V First Outlook

Theorem 18.1.16 Any nonrecurrent elliptic function has only finitely many Fatou periodic components and all of them are basins of attraction to either attracting or rationally indifferent periodic points.

18.2 Compactly Nonrecurrent Elliptic Functions: Definition, Partial Order in Critc (J (f )), and Stratification of Closed Forward-Invariant Subsets of J (f ) From now on throughout the remainder of the book, we will need a stronger property than mere nonrecurrence. Indeed, from now on, all chapters deal almost exclusively with compactly nonrecurrent elliptic functions and some of their subclasses such as regular, subexpanding, and parabolic; see Section 18.4 for their definitions and some basic properties. In this section, we lay down the foundations of such functions, which will be used throughout the reminder of the book. Compact nonrecurrence is given by the following definition. Definition 18.2.1 We say that an elliptic function f : C −→  C is compactly nonrecurrent (CNR) if and only if whenever c ∈ Crit(f ) ∩ J (f ), then either (1) ω-limit set ω(c) is a compact subset of C (i.e., ∞ ∈ / ω(c)) and c ∈ / ω(c), or  (2) c ∈ n≥1 f −n (∞), or (3) limn→ ∞ f n (c) = ∞. Of Course, we have the following. Observation 18.2.2 Every compactly nonrecurrent elliptic function is nonrecurrent. Definition 18.2.3 Given an arbitrary elliptic function f : C −→  C, the set of critical points captured by items (1), (2), or (3) in Definition 18.2.1 will be, respectively, referred to as Critc (f ), Critp (f ), or Crit∞ (f ). Let us record the following immediate observation. Observation 18.2.4 If f : C −→  C is an arbitrary elliptic function, then Critp (f ) ∪ Crit∞ (f ) ⊆ J (f ). Proof Critp (f ) ⊆ J (f ) because all poles of f are contained in J (f ) and f −1 (J (f )) ⊆ J (f ), while the inclusion Crit∞ (f ) ⊆ J (f ) immediately follows from Theorems 17.2.3 and 13.2.5. 

18 Dynamics of Compactly Nonrecurrent Elliptic Functions

243

Keeping f : C −→  C, an arbitrary elliptic function, for every c ∈ Critp (f ), we, in fact, have that  c∈ f −n (∞). n≥2

Let n(c) ≥ 2 be the only integer such that f j (c) is well defined for all 0 ≤ j ≤ n(c) and f n(c) (c) = ∞. Set  {f j (c) : j ≥ 1}, PCc (f ) := c∈Critc (f )

:= Critc (f ) ∪ PCc (f ),  PCp (f ) := {f j (c) : 1 ≤ j ≤ n(c) − 1}, PC0c (f )

c∈Critp (f )

:= Critp (f ) ∪ PCp (f ),  {f j (c) : j ≥ 1}, PC∞ (f ) := PC0p (f )

(18.31)

c∈Crit∞ (f )

:= Crit∞ (f ) ∪ PC∞ (f ),  PC(f ) := {f j (c) : j ≥ 1},

PC0∞ (f )

c∈Crit(f )

PC (f ) := Crit(f ) ∪ PC(f ). 0

Throughout this section, we assume that an elliptic function f : C −→  C is compactly nonrecurrent. Unless otherwise explicitly stated, all the distances and all closures are understood with respect to a Euclidean metric and topology on the Euclidean complex plane C. In particular, if limn→∞ f n (z) = ∞ or  −n (∞), then ω(z) = ∅. Also, dist(A,∅) = 0. z∈ ∞ n=1 f We record the following immediate observation. Observation 18.2.5 If f : C −→  C is a compactly nonrecurrent elliptic function, then PC(f ) := PCc (f ) ∪ PCp (f ) ∪ PC∞ (f ), PC0 (f ) := PC0c (f ) ∪ PC0p (f ) ∪ PC0∞ (f ). In this section, we introduce an order in the set of critical points and a stratification of the Julia set. Both of these are crucial for inductive proofs in forthcoming sections and chapters. The results and proofs provided in the present section closely follow those from Section 2.4 of [KU4]. Set Critc (J (f )) := Critc (f ) ∩ J (f ).

244

Part V First Outlook

Lemma 18.2.6 If f : C −→  C is a compactly nonrecurrent elliptic function, then the set ω(Critc (J (f ))) is nowhere dense in J (f ). Proof Suppose that the interior (relative to J (f )) of ω(Critc (J (f ))) is nonempty. Then there exists c ∈ Critc (J (f )) such that ω(c) has a nonempty interior. But then, because of Proposition 17.2.6, there would exist n ≥ 0 such that f n (ω(c)) = J (f ); consequently, ω(c) = J (f ). This, however, is a contradiction, as c ∈ / ω(c).    Since O+ J (f ) ∩ Crit(f ) is the union of ω(Critc (J (f ))) and the forward orbit of J (f ) ∩ Crit(f ), which is a countable set, is an immediate consequence of this lemma and the Baire Category Theorem, we get the following. Proposition 18.2.7 If f : C −→  C is a compactly nonrecurrent elliptic function, then the set   O+ J (f ) ∩ Crit(f ) is nowhere dense in J (f ). Now we introduce in Critc (J (f )) a relation c3 > · · · . But this contradicts Lemma 18.2.9, proving (d). Now, if Cr1 = ∅, then it would follow from (18.33) by a straightforward induction that Cri = ∅ for every i ≥ 0. Then Critc (J (f )) = ∅ by (d). But this would contradict our hypothesis that Critc (J (f )) = ∅, whence proving (e).  As an immediate consequence of the definition of the sequence p {Cri (f )}i=0 , we get the following simple lemma.

246

Part V First Outlook

Lemma 18.2.12 Let f : C −→  C be a compactly nonrecurrent elliptic function. If c,c ∈ Cri (f ), then it is not the case that c < c . For each point z ∈ J (f ), define the set Critc (z) := {c ∈ Critc (J (f )) : c ∈ ω(z)}. Lemma 18.2.13 Let f : C −→  C be a compactly nonrecurrent elliptic function. If z ∈ J (f )\I∞ (f ), then either  (1) z ∈ n≥0 f −n ((f )) or (2) ω(z)\{∞} is not contained in O+ (f (Critc (z))) ∪ (f ).  Proof Suppose that z ∈ / n≥0 f −n ((f )) ∪ I∞ (f ). Then ω(z)\{∞} = ∅ and, by Proposition 15.3.5, the set ω(z)\{∞} is not contained in (f ). So, if we assume that ω(z)\{∞} ⊆ O+ (f (Critc (z))) ∪ (f ),

(18.34)

then, as ω(z)\{∞} = ∅, we conclude that Crit(z) = ∅. Let c1 ∈ Critc (z). / (f ), it follows from (18.34) that there This means that c1 ∈ ω(z); as c1 ∈ exists c2 ∈ Critc (z) such that either c1 ∈ ω(c2 ) or c1 = f n1 (c2 ) for some n1 ≥ 1. Iterating this procedure, we obtain an infinite sequence {cj }∞ j =1 such n j that, for every j ≥ 1, either cj ∈ ω(cj +1 ) or cj = f (cj +1 ) for some nj ≥ 1. Consider an arbitrary block ck ,ck+1, . . . ,cl such that cj = f nj (cj +1 ) for every k ≤ j ≤ l − 1 and suppose that l − (k − 1) ≥ Nf . Then there are two indexes k ≤ a < b ≤ l such that ca ∼f cb . Then f na +na+1 +···+nb−1 (ca ) = f na +na+1 +···+nb−1 (cb ) = ca ; consequently, as na + na+1 + · · · + nb−1 ≥ b − a ≥ 1, ca is a superattracting periodic point of f . Since ca ∈ J (f ), this is a contradiction; in consequence, the length of the block ck ,ck+1, . . . ,cl is bounded above by Nf . Hence, there exists an infinite subsequence {nk }k≥1 such that cnk ∈ ω(cnk +1 ) for every k ≥ 1 or, equivalently, cnk < cnk+1 for every k ≥ 1. This, however, contradicts Lemma 18.2.9 and we are done.  Recall that the integer p was defined in Lemma 18.2.11. Define, for every i = 0,1, . . . ,p, Si (f ) = Cr0 (f ) ∪ · · · ∪ Cri (f )

(18.35)

18 Dynamics of Compactly Nonrecurrent Elliptic Functions

247

and, for every i = 0,1, . . . ,p − 1, consider  c ∈ ω(c) ∩ Critc (J (f )). c∈Cri+1 (f )

Then there exists c ∈ Cri+1 (f ) such that c ∈ ω(c), which means that c < c. Thus, by (18.33), we get c ∈ Si (f ). So  ω(c) ∩ (Critc (J (f ))\Si (f )) = ∅. (18.36) c∈Cri+1 (f )

 Therefore, since the set c∈Cri+1 (f ) ω(c) ⊆ C is compact and Critc (J (f ))\Si (f ) has no accumulation points in C, ⎞ ⎛  δi := diste ⎝ ω(c),Critc (J (f ))\Si (f )⎠ > 0. (18.37) c∈Cri+1 (f )

Set ρ :=

#  $ 1 min min{δi : i = 0,1, . . . ,p − 1},diste O+ (Critc (f )),Critp (f ) ∪ Crit∞ (f ) . 2 (18.38)

Fix a closed forward-invariant subset E of J (f ) and, for every i = 0,1, . . . ,p, define 

  Ei (f ) := z ∈ E : diste O+ (z),Critc (J (f ))\Si (f ) ≥ ρ . Let us now prove the following two lemmas concerning the sets Ei (f ), i = 0, . . . ,p. Lemma 18.2.14 If f : C −→  C is a compactly nonrecurrent elliptic function, then E0 (f ) ⊆ E1 (f ) ⊆ · · · ⊆ Ep (f ) = E(f ). Proof Since Si+1 (f ) ⊇ Si (f ), the inclusions Ei (f ) ⊆ Ei+1 (f ) are obvious.  Since Sp (f ) = Critc (J (f )), it holds that Ep = E. We are done. Lemma 18.2.15 If f : C −→  C is a compactly nonrecurrent elliptic function, then there exists l = l(f ) ≥ 1 such that, for every i = 0,1, . . . ,p − 1, we have that  ω(c) ⊆ O+ (f l (Cri+1 (f ))) ⊆ PCc (f )i ⊆ PC0c (f )i . c∈Cri+1 (f )

Proof The left-hand inclusion is obvious regardless of whatever l(f ) ≥ 1 is. In order to prove the right-hand inclusion, fix i ∈ {0,1, . . . ,p − 1}. By the definition of ω-limit sets, there exists li ≥ 1 such that, for every c ∈ Cri+1 (f ), we have that

248

Part V First Outlook ⎛



diste ⎝O+ (f li (c)),

⎞ ω(c)⎠ < δi /2.

c∈Cri+1 (f )

Thus, by (18.37),   diste O+ (f li (c)),Critc (J (f ))\Si (f ) > δi /2. Since ρ ≤ δi /2 and since, for every z ∈ O+ (f li (c)), also O+ (z) ⊆ O+ (f li (c)), we, therefore, get that O+ (f l (Cri+1 (f ))) ⊆ PC0c (f )i . So, by putting l(f ) = max{li : i = 0,1, . . . ,p − 1}, the proof is completed. 

18.3 Holomorphic Inverse Branches This section has a technical character. The main (technical) result, Proposition 18.3.3, of this section concerns compactly nonrecurrent elliptic functions and provides us with an abundance of holomorphic inverse branches of iterates of elliptic functions. It will be used many times in what follows. However, at the beginning of this section, our considerations are fairly general. So, let f : C −→  C be an elliptic function. Set    − f −n (f ) ∪ Crit(J (f )) ∪ f −1 (∞) Sing (f ) := n≥0

and recall that I− (f ) =



f −n (∞).

n≥1

Consider any T ≥ T (f ), where T (f ) > 0 comes from Lemma 17.2.7 and (17.8). For every b ∈ f −1 (∞) and every w ∈ Bb∗ (2T ), let −1 fb,w : Be (f (w),T ) −→ Bb∗ (T )

be the holomorphic inverse branch of f sending f (w) to w whose existence is guaranteed by Lemma 17.2.8. It follows from (17.10), (17.13), and (17.16) that there exists a constant L ≥ 1 so large that, for every b ∈ f −1 (∞), B(b,L−1 ) ⊆ Bb (R0 ) and the following properties are satisfied: diame (Bb (T )) ≤ LT −1/qb ,

Bb (2T ) ⊃ Be (b,L−1 T −1/qb ),

(18.39)

18 Dynamics of Compactly Nonrecurrent Elliptic Functions

249

and, for every R ∈ (0,T ] sufficiently small,   q +1 q +1 − bq − bq −1 −1 Be w,L R|f (w)| b ⊂ fb,w (Be (f (w),R)) ⊂ Be w,LR|f (w)| b ⊂ Be (w,R), (18.40) qb +1

where the last inclusion was written assuming that |f (w)| ≥ L qb . Since there are only finitely many equivalence classes of the relation ∼f on the poles of f , there exists R1 > 0 so small that if w ∈ Be (f −1 (∞),R1 ), then |f (w)| ≥ L. Using now (18.40) and the right-hand side of (18.39) with T replaced by 2T , a straightforward induction gives the following. Lemma 18.3.1 If f : C −→  C is a compactly nonrecurrent elliptic function, then there exists  % & 1 −1 −1 R2 ∈ 0, min T ,(2LT ) ,R1, diste (f (∞),Crit(f )) 2 so small that if n ≥ 1, z ∈ f −n (C), and {f j (z) : 0 ≤ j ≤ n − 1} ⊆ Be (f −1 (∞),R2 ), then there exists a unique holomorphic inverse branch fz−n : Be (f n (z),R2 ) −→ Be (z,R2 ) of f n sending f n (z) to z. Now we shall prove the following. Lemma 18.3.2 If f : C −→  C is an elliptic function, then, for every ε > 0, there exists a = a(ε) ≥ 1 such that if z ∈ C\Be (f −1 (∞),ε), then z∈ /

∞ 

Be (f j (Crit∞ (f )),5).

j =a+1

Proof Suppose, on the contrary, that there exists ε > 0 and, for every a ≥ 1,  j −1 (∞),ε). Since the sets there exists za ∈ ∞ j =a+1 Be (f (Crit∞ (f )),5)\Be (f f j (Crit∞ (f )) converge to ∞ when j → ∞, it follows that lima→∞ za = ∞. But then za ∈ Be (f −1 (∞),ε) for all a ≥ 1 large enough. This contradiction finishes the proof.  We now move on to dealing with compactly nonrecurrent elliptic functions. Since, for all such functions, the sets PCc (f ) and PCp (f ) are bounded, the number

250

Part V First Outlook

1 Diste (PCc (f ) ∪ PCp (f ),0) 2 is finite. The main result of this section is the following. D :=

Proposition 18.3.3 Let f : C −→  C be a compactly nonrecurrent elliptic function. If z ∈ J (f )\Sing− (f ), then there exist: (a) η(z) > 0. of positive integers. (b) (nj = nj (z))∞ j =1 , an increasing sequence   ∞ (c) A sequence (xj (z))j =1 ⊆ J (f )\ (f ) ∪ ω(Critc (z)) with the following properties: (1) (2) (3) (4) (5)

Comp(z,f nj ,9η(z)) ∩ Crit(f nj ) = ∅. |f nj (z)−xj (z)| < η(z) for all j ≥ 1 and limj →∞ |f nj (z)−xj (z)| = 0. If |xj (z)| ≥ 2D for all j ≥ 1, then η(z) ≥ min{2,R2 }. If the sequence {xj (z)}∞ j =0 is bounded, then it is constant. / I∞ (f ), then xj (z) ∼f xk (z) for all If limj →∞ xj (z) = ∞ and z ∈ j,k ≥ 1.

Proof If z ∈ I∞ (f ) and, equivalently, if limn→∞ diste (f n (z),f −1 (∞)) = 0, then, in view of Lemma 18.3.1, we are done by setting nj = j +u and xj (z) = f j +u (z) with some u ≥ 0 large enough, so that diste (f j +u (z),f −1 (∞)) < R2 . So, suppose that there exists an ε > 0 such that the set 

S := k ≥ 0 : diste (f k (z),f −1 (∞)) > ε is infinite. Now suppose that AdS , the set of limit points of AS = {f k (z) : k ∈ S}, is unbounded. Then there exists w ∈ AdS and a(ε) ≥ 1 (a(ε) comes from Lemma 18.3.2) such that ⎛ ⎞ a(ε)  Be (w,5) ∩ ⎝ f j (Crit∞ (f )) ∪ PCc (f ) ∪ PCp (f )⎠ = ∅ (18.41) j =0

and |w| ≥ 4D. There also exists an increasing sequence {nj }∞ j =1 ⊆ S such that lim f nj (z) = w.

j →∞

Disregarding finitely many elements of this sequence, we may assume without loss of generality that f nj (z) ∈ Be (w, min{1,D}) for all j ≥ 1. In view of Lemma 18.3.2 and (18.41), we conclude from Theorem 17.1.8 that, for every −n j ≥ 1, there exists a holomorphic inverse branch fz j : Be (f nj (z),4) −→ C n n / ω(Crit(f )) and we are of f j sending f j (z) to z. On the same premise, w ∈ done in this case by setting xj (z) := w for all j ≥ 1 and η(z) := 36.

18 Dynamics of Compactly Nonrecurrent Elliptic Functions

251

So, suppose that the set AdS is bounded. Assume, first, that lim inf diste (f n (z),f −1 (∞)) = 0.

n→∞ ∞ Then there exists {kj }j =1 , an increasing sequence of positive integers, such that

diste (f kj +1 (z),f −1 (∞)) > ε

(18.42)

(i.e., kj + 1 ∈ S, j ≥ 1) and we require f kj (z) to be so close to f −1 (∞) (assuming ε > 0 to be sufficiently small) that |f kj +1 (z)| ≥ 4D + 2

(18.43)

for all j ≥ 1. Passing to a subsequence, we may assume without loss of generality that the sequence {f (f kj +1 (z))}∞ j =1 on the torus Tf converges to some point y ∈ Tf , where, we recall, f : C −→ Tf = C/ ∼f is the canonical projection from C onto the torus Tf . Clearly, there exists a sequence kj +1 (z) − x (z)| = 0 and  (x (z)) = y for {xj (z)}∞ j f j j =1 such that limj →∞ |f all j ≥ 1. Assume, first, that the sequence {f kj +1 (z)}∞ j =1 is unbounded. Passing to a subsequence, we may assume without loss of generality that limj →∞ f kj +1 (z) = ∞. Then, applying (18.42), Lemma 18.3.2, (18.43), and the definition of D, we are done with nj = kj + 1 and η(z) = 2. So, assume that the sequence {f kj +1 (z)}∞ j =1 is bounded. We already know that ⎛ ⎞ ∞  Be (f kj +1 (z),2) ∩ ⎝PCc (f ) ∪ PCp (f ) ∪ f j (Crit∞ (f ))⎠ = 0. j =a(ε)+1

Since the second term of this intersection is forward invariant, we conclude that no accumulation point of the sequence {f kj −a(ε) (z)}∞ j =1 belongs to PCc (f ) ∪ PCp (f ) ∪ PC∞ (f ) = PC(f ). If the sequence {f kj −a(ε) (z)}∞ j =1 is unbounded, we may complete the argument in exactly the same way as above with kj + 1 replaced by kj − a(ε). If the sequence {f kj −a(ε) (z)}∞ j =1 is bounded, we are immediately done by passing to a converging subsequence. So assume that lim inf diste (f n (z),f −1 (∞)) > 0. n→∞

Then lim infn→∞ |f n (z)| < ∞ and the ω-limit set is compact in the plane C. In view of Lemma 18.2.13, there exists x ∈ ω(z)\((f ) ∪ O+ (f (Critc (z))) ∪ {∞}). The number

252

Part V First Outlook

η(z) ˆ :=

  1 diste x,(f ) ∪ O+ (f (Critc (z))) 2

is positive since ω(Critc (z)) is a compact subset of C and (f ) is finite. Then there exists an infinite increasing sequence {mj }j ≥1 such that lim f mj (z) = x

j →∞

and ˆ ∩ Be (f mj (z), η(z))



f n (Critc (z)) = ∅.

(18.44)

(18.45)

n≥1

Now we claim that there exists η(z) such that, for every j ≥ 1 large enough, Comp(z,f mj ,9η(z)) ∩ Crit(f mj ) = ∅.

(18.46)

Otherwise, we would find an increasing to infinity subsequence {mji }∞ i=1 of ∞ such and a decreasing to zero sequence of positive numbers {η } {mj }∞ i j =1 i=1 that ηi < η and Comp(z,f mji ,ηi ) ∩ Crit(f mji ) = ∅. For every i ≥ 1 let c˜i ∈ Comp(z,f mji ,ηi ) ∩ Crit(f mji ). Then there exists ci ∈ Crit(f ) such that f pi (c˜i ) = ci for some 0 ≤ pi ≤ mji − 1. Since the set f −1 (x) is at a positive distance from (f ) and since ηi → 0, it follows from Theorem 18.1.8 that lim c˜i = z.

i→∞

 Since z ∈ / n≥0 f −n (Crit(f )), this implies that limi→∞ pi = ∞. But then, using Theorem 18.1.8 again and the formula f pi (c˜i ) = ci , we conclude that the set of all accumulation points of the sequence (ci )∞ i=1 is contained in ω(z). Hence, passing to a subsequence, we may assume that the limit c = limi→∞ ci exists. But since c ∈ ω(z), since ω(z) is a compact subset of C, and since ∞ is the only accumulation point of Crit(f ), we conclude that the sequence (ci )∞ i=1 is eventually constant. Thus, dropping some finite number of initial terms, we may assume that this sequence is constant. This means that ci = c for all i = 1,2, . . .. Since c = f pi (c˜i ), we get that |f mji (z) − f mji −pi (c)| = |f mji (z) − f mji (c˜i )| < ηi . Since limi→∞ ηi = 0 and since ω(z) is a compact subset of C, we conclude that lim |f mji (z) − f mji −pi (c)| = 0.

i→∞

18 Dynamics of Compactly Nonrecurrent Elliptic Functions

253

Since c ∈ Crit(z), in view of (18.45), this implies that mji − pi ≤ 0 for all i large enough. So, we get a contradiction as 0 ≤ pi ≤ mji − 1 and (18.46) is proved.  As an immediate consequence of this proposition and Theorem 17.1.8, we get the following. Proposition 18.3.4 With the hypotheses and notation of Proposition 18.3.3, for every integer j ≥ 1, there exists a unique holomorphic inverse branch −nj

fz

: Be (xj (z),η(z)) −→ C

of f nj sending f nj (z) to z. For every integer j ≥ 1, let Tj : C → C be a translation given by the formula Tj (w) = w + xj (z). We shall prove the following. Lemma 18.3.5 With the hypotheses and notation of Proposition 18.3.4, the sequence  −nj ∞ fz ◦ Tj : Be (0,8η(z)) −→ C j =1 converges uniformly to the constant function identically equal to z. Proof Decreasing η(z) > 0 if necessary, we can always find a periodic orbit of f of period ≥ 3 disjoint from all the balls Be (xj (z),9η(z)). Then this orbit is also disjoint from all the sets −nj

fz

−nj

◦ Tj (Be (0,9η(z))) = fz

(Be (xj (z),9η(z))).

Hence, by Montel’s Theorem II (Theorem 8.1.16), the family 

−n  Fz := fz j ◦ Tj : Be (0,9η(z)) −→ C  j ≥ 1 is normal. Since also f nj (z) − xj (z) ∈ Be (0,η(z)) (by item (2) of Proposition 18.3.3) and   −n fz j ◦ Tj f nj (z) − xj (z) = z, we conclude that if the proposition failed, then there would exist a radius r > 0 and an increasing subsequence {njk }∞ k=1 such that −njk

fz

◦ Tjk (Be (0,9η(z))) ⊇ Be (z,r).

254

Part V First Outlook

Equivalently, Tj−1 ◦ f njk (Be (z,r)) ⊂ Be (0,9η(z)) k or f njk (Be (z,r)) ⊆ Tjk (Be (0,9η(z))). Passing to yet another subsequence, we may assume without loss of generality that the set  C\

∞ 

Tjk (Be (0,9η(z)))

k=1

has a nonempty interior and, consequently, contains at least three points. Thus, the family {f njk : Be (z,r) −→ C}∞ k=1 would be normal, contrary to the fact that z ∈ J (f ). We are done.



As an immediate consequence of this lemma, we get the following. Corollary 18.3.6 Let f : C −→  C be a compactly nonrecurrent elliptic function. If z ∈ J (f )\Sing− (f ) and the sequence (nj (z))∞ j =1 is taken from Proposition 18.3.4, then lim |(f nj (z) ) (z)| = +∞.

j →∞

18.4 Dynamically Distinguished Classes of Elliptic Functions In this section, we recall already defined, and define new classes of, dynamically significant elliptic functions. These will form essentially all major classes of elliptic functions dealt with in this book. Mostly, they will be defined in terms of how strongly expanding these functions are. We would like to add that while, in the theory of rational functions, such classes pop up in a natural and fairly obvious way, e.g., metric and topological definitions of expanding rational functions describe the same class of functions, in the theory of iteration of transcendental meromorphic functions such a classification is by no means obvious, topological and metric analogs of rational function realm concepts do not usually coincide, and the definitions of expanding, hyperbolic, topologically hyperbolic, subhyperbolic, etc., functions vary from author to author. Our definitions seem to us to be quite natural and fit

18 Dynamics of Compactly Nonrecurrent Elliptic Functions

255

well our purpose of the detailed investigation of the dynamical and geometric properties of the elliptic functions that they define. We start with the following obvious observation. Observation 18.4.1 If f : C −→  C is an elliptic function, then, for every  −n point z ∈ C\ n≥1 f (∞), we have that lim f n (z) = ∞ if and only if lim sup diste (f n (z),f −1 (∞)) = 0.

n→∞

n→∞

Let

 δ(f −1 (∞)) := min |b − a| : a,b ∈ f −1 (∞) and a = b > 0.

(18.47)

Using Observation 18.4.1, we see that, for every c ∈ Crit∞ (f ) and all integers n ≥ 1 large enough, there exists a unique pole bn of f such that |f n (c) − bn | < δ(f −1 (∞))/2.

(18.48)

qc = lim sup qbn ,

(18.49)

Let n→∞

where, we recall, qbn is the multiplicity of the pole bn . The integer pc ≥ 2, i.e., the multiplicity of the critical point c, was defined at the beginning of Section 8.4. Let l∞ := max{pc qc : c ∈ Crit∞ (f )}

(18.50)

C (if Crit∞ (f ) = ∅, then l∞ = 0). We say that the elliptic function f : C −→  is regular if and only if h := HD(J (f )) >

2l∞ . l∞ + 1

(18.51)

Regularity of elliptic functions will be our standing hypothesis from now on, essentially through to the end of the book. We will now provide the definitions of all those classes of elliptic functions f dealt with in this section, for which (f ) = ∅, where, we recall, (f ) is the set of all rationally indifferent periodic points of f . Later in this section, we will deal with classes of elliptic functions for which (f ) = ∅. Definition 18.4.2 We say that an elliptic function f : C −→  C is normal if and only if 

∞   f −1 (∞) ∪ Crit(f ) ∩ f n (Crit(f )) = ∅. n=1

(18.52)

256

Part V First Outlook

Definition 18.4.3 We say that an elliptic function f : C −→  C is of finite character if and only if Crit∞ (f ) = ∅. Definition 18.4.4 We say that an elliptic function f : C −→  C is weakly semi-expanding if and only if it is nonrecurrent and (f ) = ∅. Definition 18.4.5 We say that an elliptic function f : C −→  C is semiexpanding if and only if it is compactly nonrecurrent and (f ) = ∅. Definition 18.4.6 We say that an elliptic function f : C −→  C is subexpanding if and only if it is semi-expanding and   ω f (Crit(f )) ∩ J (f ) ∩ Crit(f ) = ∅. (18.53) Definition 18.4.7 We say that an elliptic function f : C −→  C is expanding if and only if J (f ) ∩ PC(f ) = ∅;

(18.54)

equivalently, J (f ) ∩ PC0 (f ) = ∅, (18.55) ∞  n where, we recall, PC(f ) = n=1 f n (Crit(f )) and PC0 (f ) = ∞ n=0 f (Crit(f )). We shall now bring up some inclusion relations between these classes of elliptic functions. All of them except one are obvious. We start with the following. Observation 18.4.8 Every elliptic function of finite character is regular. Proposition 18.4.9 Each expanding elliptic function is normal subexpanding of finite character. Proof Let f : C −→  C be an expanding elliptic function. It is compactly nonrecurrent because Crit(f )∩J (f ) = ∅. It follows from Theorem 15.2.5 and (18.54) that (f ) = ∅. Hence, f is semi-expanding. It is subexpanding since J (f ) ∩ Crit(f ) = ∅. By the same token,f is normal and of finite character.  So, we get the following. Proposition 18.4.10 Referring to elliptic functions, the following inclusions hold: expanding ⊆ subexpanding ⊆ semi-expanding ⊆ compactly nonrecurrent ⊆ nonrecurrent

18 Dynamics of Compactly Nonrecurrent Elliptic Functions

257

and weakly semi-expanding ⊆ semi-expanding. We shall now provide the same useful characterizations of some of these classes of elliptic functions. We will first prove the following fact, which is interesting by itself and which will be frequently used in this and the next section. In the theory of iteration of rational functions of the Riemann sphere, this fact is commonly referred to as the exponential shrinking property. We follow this tradition. Theorem 18.4.11 (Exponential Shrinking on C) If f : C −→  C is a weakly semi-expanding elliptic function, then there exist 0 < γˆ ≤ γf and Mf > 0 such that diame (Comp(z,f n, γˆ )) ≤ 4eγf e−Mf n for all integers n ≥ 0 and all z ∈ f −n (J (f )). Proof Let γ˜ := min{δ0 /2,γf }, where δ0 corresponds to ε = 1 and λ = 1/2 appearing in Lemma 18.1.3. Then, in view of the last assertion of this lemma, there exists an integer l ≥ 1 such that diame (Comp(z,f j , γ˜ )) < γ˜ /2 for all j ≥ l and all z ∈

f −k (J (f )).

(18.56)

For every k ≥ 1, let

Bk (z) := Comp(z,f kl , γ˜ ) and

Ak (z) := Bk (z)\B k+1 (z).

(18.57)

Since f kl (Bk+1 (z)) is a connected subset of C containing the point f kl (z) and since, by Lemma 18.1.12(2), f l (f kl (Bk+1 (z))) = B(f (k+1)l (z), γ˜ ), we get that f kl (Bk+1 (z)) ⊆ Comp(f kl (z),f l , γ˜ ) ⊆ Be (f kl (z), γ˜ /2) for every z ∈ f −(k+1)l (J (f )). Hence, f kl (B k+1 (z)) ⊆ B e (f kl (z), γ˜ /2) ⊆ Be (f kl (z), γ˜ ).

(18.58)

Since B k+1 (z) is a connected subset containing z, we, thus, conclude from the second inclusion of (18.58) that B k+1 (z) ⊆ Bk (z).

(18.59)

Since also, by Lemma 18.1.12, Ak (z) and Comp(z,f kl ,γ )\Comp(z,f kl ,γ /2) are topological annuli and, by the first inclusion of(18.58), Ak (z) ⊃ Comp(z,f kl , γ˜ )\Comp(z,f kl , γ˜ /2),

258

Part V First Outlook

with the inclusion being essential, we conclude from Corollary 8.2.4 and Lemma 18.1.12 that   Mod(Ak (z)) ≥ Mod Comp(z,f kl , γ˜ )\Comp(z,f kl , γ˜ /2) ≥ Mf∗ , (18.60) where Mf∗ := log 2/(Nf∗ )2 . By virtue of (18.56) and (18.57), we get that Ak (z) ⊆ Be (z, γ˜ /2)\B k+1 (z)

(18.61)

with the inclusion being essential. Fixing now an integer q ≥ 1 and z ∈ f −ql (J (f )), we conclude from q−1 (18.56), (18.57), and (18.59) that (Ak (z))k=0 is a sequence of mutually disjoint annuli separating z from ∂B(z, γ˜ /2). It also follows from (18.59) and (18.61) that Ak (z) ⊆ Be (z, γ˜ /2)\B q (z) for all k = 0,1, . . . ,q − 1 and the inclusion is essential. Hence, by applying Theorem 8.2.3 and (18.60), we get that    Mod Be (z, γ˜ /2)\B q (z) ≥ Mod(Ak (z)) ≥ Mf∗ q. q−1 k=0

So, by invoking Theorem 8.2.14, we get that ∗

∗

diame (B q (z)) ≤ 2Diste (z,B q (z)) ≤ 4γ˜ e−Mf q ≤ 4γf e−Mf q .

(18.62)

Now, by Proposition 17.1.3, there exists γˆ ∈ (0, γ˜ ] such that if 0 ≤ i ≤ l is an integer, then, for every w ∈ f −i (J (f )),   diame Comp(w,f i , γˆ ) < γ˜ . (18.63) Take now any integer n ≥ 1 and z ∈ f −n (J (f )). By the division algorithm, there then exist integers 0 ≤ qn ≤ n and 0 ≤ rn ≤ l − 1 such that n = qn l + rn .

(18.64)

It immediately follows from (18.63) that

  Comp(z,f n, γˆ ) ⊆ Comp z,f qn l , γˆ = Bqn (z).

We conclude from this, (18.62), and (18.64) that

   rn  ∗ n diame (Comp(z,f , γˆ )) ≤ 4γf e = 4γf exp −Mf −  l l 

∗ Mf n . ≤ 4γf e exp − l n

−Mf∗ qn

Hence, setting Mf := Mf∗ / l finishes the proof.



18 Dynamics of Compactly Nonrecurrent Elliptic Functions

259

As an immediate consequence of this theorem, we get the following. Corollary 18.4.12 (Exponential Shrinking on Tf ) If f : C −→  C is a weakly semi-hyperbolic elliptic function, then diam(Comp(z, fˆn, γˆ )) ≤ 4eγf e−Mf n for all n ≥ 0 and all z ∈ fˆ−n (J (fˆ)), where γˆ ∈ (0,γf ] and Mf > 0 come from Theorem 18.4.11. Now we shall prove the following characterization of expandingness, which fully justifies the name. Theorem 18.4.13 An elliptic function f : C →  C is expanding if and only if −q ∃q ≥ 1 ∀ξ ∈ J (f ) ∀z ∈ f (ξ ) |(f q ) (z)| ≥ 2.

(18.65)

Proof Assume first that f : C −→  C is expanding. It then follows from Definition 18.4.2 that (f ) = ∅ and J (f )∩Crit(f ) = ∅. It, therefore, follows from Theorems 13.2.5 and 17.2.3 and the finiteness of the set f (Crit(f )) that ω(Crit(f )) ⊆ A ⊆ F (f ),

(18.66)

where A is the finite set of all attracting points of f . Along with the definition expandingness, this implies that R=

  1 diste PC(f ),J (f ) > 0. 2

(18.67)

Hence, it follows from Theorem 17.1.7 that, for all n ≥ 1, all ξ ∈ J (f ), and all z ∈ f −n (ξ ), there exists fz−n : Be (ξ,2R) −→ C, a unique holomorphic branch of f −n sending ξ to z. The proof (18.65) is now concluded by applying Theorem 18.4.11 along with the Koebe Distortion Theorem (Theorem 8.3.8). For the opposite direction, suppose that (18.65) holds. Then J (f ) ∩ Crit(f ) = ∅ and J (f ) ∩ (f ) = ∅. So, exactly as in the first part of the proof, we conclude that (18.66) holds and, furthermore, (18.67) holds. In particular, O + (Crit(f )) ∩ J (f ) = ∅, meaning that f : C →  C is expanding. The proof of Theorem 18.4.13 is complete.  Now we shall prove a similar characterization of subexpanding elliptic functions.

260

Part V First Outlook

Theorem 18.4.14 An elliptic function f : −→  C is subexpanding if and only if there exists q ≥ 1 such that, for all ξ ∈ J (f ) ∩ ω(Crit(f )) and all z ∈ f −q (ξ ) ∩ J (f ) ∩ ω(Crit(f )), we have that |(f q ) (z)| ≥ 2

(18.68)

and, for every point c ∈ J (f ) ∩ Crit(f ), either ω(c) is a compact subset of  −n (∞), or the sequence (f n (c))∞ converges to ∞ (meaning C, c ∈ ∞ n=1 f n=1 that c ∈ Crit∞ (f )). Proof Assume first that f : C −→  C is expanding. So, f is compactly nonrecurrent, and we only need to show that (18.68) holds. Since the set Crit(f ) has only finitely many congruent classes modulo f , since diste (Crit(f ),f −1 (∞)) > 0, and since f is compactly nonrecurrent, it follows from (18.53) that   1 R = diste ω(Crit(f )) ∩ J (f ),Crit(f ) > 0. (18.69) 2 Since, also, (f ) = ∅, it follows from Theorem 18.1.8, applied with   X := ω(Crit(f )) ∩ J (f ) ∪ {∞} and ε := 12 R, that there exists δ > 0 such that, for all n ≥ 1, also 0 ≤ k ≤ n, all ξ ∈ J (f ) ∩ ω(Crit(f )), and all z ∈ f −n (ξ ), we have that   diame f k (Comp(z,f n,2δ)) < R. So, if, in addition, z ∈ J (f ) ∩ ω(Crit(f )), then     f k Comp(z,f n,2δ) ⊆ Be J (f ) ∩ ω(Crit(f )),R for all 0 ≤ k ≤ n. Along with (18.69), this gives   f k Comp(z,f n,2δ) ∩ Crit(f ) = ∅ for all 0 ≤ k ≤ n. Hence, by virtue of Theorem 17.1.8, there exists fz−n : Be (ξ,2δ) −→ C, a unique holomorphic branch of f −n sending ξ to z. The proof of (18.68) is now concluded by applying Theorem 18.4.11 along with the Koebe Distortion Theorem (Theorem 8.3.8). For the opposite implication, suppose that its hypotheses hold. We then only need to show that (18.53) holds and (f ) = ∅, but both of them follow immediately from (18.68) and the fact that (f ) ⊆ ω(Crit(f )) ∩ J (f ) (which, in turn, follows from Theorem 15.2.5). The proof of Theorem 18.4.14 is complete. 

18 Dynamics of Compactly Nonrecurrent Elliptic Functions

261

We shall now prove the following Theorem 18.4.15 If f : C −→  C is a normal elliptic function of finite character, then f is subexpanding if and only if PC(f ) ∩ J (f ) is a compact subset of C, (f ) = ∅, and   (18.70) diste f −1 (∞) ∪ Crit(f ),PC(f ) ∩ J (f ) > 0. Proof Suppose first that f : C −→  C is subexpanding. Then (f ) = ∅. Since f is normal and of finite character, we have that Critp (f ) = Crit∞ (f ) = ∅. Hence, since f is compactly nonrecurrent, it follows that ω(Crit(f ))∩J (f ) = ωc (Crit(f ) ∩ J (f )) is a compact subset of C and f −1 (∞) ∩ ω(Crit(f )) ∩ J (f ) = ∅.

(18.71)

Thus, PC(f ) ∩ J (f ) is a bounded subset of C, whence PC(f ) ∩ J (f ) is a compact subset of C. Since ω(f (Crit(f )) ∩ J (f )) = ω(f (Crit(f )) ∩ J (f )) is a compact subset of C and the set Crit(f ) ∪ f −1 (∞) ⊆ C is closed, it follows from (18.53) and (18.71) that   diste f −1 (∞) ∪ Crit(f ),ω(f (Crit(f )) ∩ J (f )) > 0. Having this and invoking normality of f , we deduce that (18.70) holds. Suppose, in turn, that PC(f ) ∩ J (f ) is a compact subset of C, (f ) = ∅, and (18.70) holds. Then, immediately, (18.53) holds and f is compactly nonrecurrent, and, moreover, semi-expanding. This means that f is subexpanding. The proof of Theorem 18.4.15 is complete.  As an immediate consequence of this theorem and Observation 17.2.10, along with Theorems 17.2.3 and 13.2.5, we get the following. Theorem 18.4.16 If a normal elliptic subexpanding function f : C −→  C is of finite character, then PC(fˆ) ∩ J (fˆ) is a compact subset of Tˆ f and     dist f f −1 (∞) ∪ Crit(f ) ,PC(fˆ) > 0. The last class of elliptic functions that we will be dealing with in this book, more precisely in Sections 22.6–22.8, are parabolic elliptic functions, i.e., all elliptic functions f : C −→  C for which Crit(f ) ∩ J (f ) = ∅ and (f ) = ∅.

262

Part V First Outlook

As an immediate consequence of the first half of this definition, we get the following observation. Observation 18.4.17 Every parabolic elliptic function is regular normal compactly nonrecurrent of finite character. We furthermore discern in Sections 22.6–22.8, mentioned above, two disjoint subclasses: of finite class and of infinite class, respectively, depending on whether the invariant measure μh of Theorem 22.6.7 is finite or infinite.

19 Various Examples of Compactly Nonrecurrent Elliptic Functions

The purpose of this chapter is to provide examples of elliptic functions with prescribed properties of the orbits of critical points (and values). We are primarily focused on constructing examples of various classes of compactly nonrecurrent elliptic functions discerned in Section 18.4. All these examples are either Weierstrass ℘ -elliptic functions or their modifications. The dynamics of such functions depends heavily on the lattice  and varies drastically from  to . The first three sections of this chapter have a preparatory character and, respectively, describe the basic dynamical and geometric properties of all Weierstrass ℘ -elliptic functions, i.e., those generated by square lattices and triangular lattices. In Section 19.4, we provide simple constructions of many classes of elliptic functions discerned in Section 18.4. We essentially cover all of them. All these examples stem from Weierstrass ℘-functions. We then, starting with Section 19.5, also provide some different, interesting on their own, and historically first examples of various kinds of Weierstrass ℘-elliptic functions and their modifications. These come from the series of papers [HK1], [HK2], [HK3], [HKK], and [HL] by Hawkins and her collaborators. Throughout the whole of this chapter, we use the notation and terminology introduced in Chapter 16, particularly in Sections 16.3–16.6.

19.1 The Dynamics of Weierstrass Elliptic Functions: Some Selected General Facts As an immediate consequence of Theorem 17.2.3, we get the following. Theorem 19.1.1 If ℘ : C −→  C is a nonconstant Weierstrass elliptic function, then f has no Baker or wandering domains. 263

264

Part V First Outlook

As an immediate consequence of this theorem and Theorems 13.2.5 (Fatou Periodic Components), 14.1.1, 15.2.5, 16.6.1, and 17.1.6, we get the following. Theorem 19.1.2 If ℘ : C −→  C is a nonconstant Weierstrass elliptic function that has no Siegel disks or Herman rings, then either (1) the Julia set J (℘) is the whole sphere  C or else all Fatou connected components are basins of attractions to (super)-attracting periodic points or rationally indifferent periodic points, or (2) there are at most three cycles of periodic components of the Fatou set F (℘) and each of them contains a critical value of ℘ which is not preperiodic. Invoking, in addition, Theorem 14.1.2, we get the following. Theorem 19.1.3 If ℘ : C −→  C is a nonconstant Weierstrass elliptic function that has three periodic orbits, each of which is either (super)-attracting or rationally indifferent, then the collection of Fatou periodic connected components of ℘ consists of immediate basins of attraction to these three periodic orbits and the Fatou set of ℘ is the union of basins of attraction to these three periodic orbits. As an immediate consequence of Theorems 19.1.2 and 18.1.15, we get the following. Theorem 19.1.4 If ℘ : C −→  C is a nonrecurrent Weierstrass elliptic function, then (1) ℘ has no Baker or wandering domain and no Siegel disks, Herman rings, or Cremer periodic points. (2) Either the Julia set J (℘) is the whole sphere  C or else all Fatou connected components are basins of attractions to (super)-attracting periodic points or rationally indifferent periodic points. (3) There are at most three periodic components of the Fatou set F (℘) and each of them contains a critical value of ℘ which is not preperiodic. Since each Weierstarss ℘-function is even, as an immediate consequence of Theorem 17.2.4, we get the following. Theorem 19.1.5 For any lattice , the Weierstarss ℘ -function has no cycle of Herman rings. Since each Weierstrass elliptic function is even, as an immediate consequence of Theorem 14.4.12, we get the following.

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 265 Theorem 19.1.6 If ℘ : C −→  C is a Weierstrass elliptic function such that each connected component of the Fatou set F (℘ ) contains at most one critical value of ℘, then the Julia set J (℘) is connected.

19.2 The Dynamics of Square Weierstrass Elliptic Functions: Some Selected Facts Square Weierstrass elliptic functions will play an important role in creating many of our examples. We will need several of their properties. The first one is the following. Proposition 19.2.1 If  is a square lattice, then the Fatou set F (℘ ) and the Julia set J (℘ ) of the Weierstrass ℘ -function are invariant either under multiplication by i or, in geometric terms, under rotation about the angle π/2, i.e., iJ (℘ ) = J (℘ ) and iF (℘ ) = F (℘ ). In addition, n n (iz) = i℘ (z) ℘

(19.1)

n (z) is well defined. for all integers n ≥ 0 and all z ∈ C whenever ℘

Proof Formula (19.1) directly follows, by a straightforward induction, from the homogeneity property (16.55). z ∈ F (℘ ) and an open neighborhood  n Take ∞ |U n=1 is normal. By (19.1), we have that U of z such that the family ℘ n n (iU ) = i℘ (U ) ℘  n ∞ |iU n=1 is normal. So, iz ∈ F (℘ ). Hence, for all n ≥ 1. Thus, the family ℘ iF (℘ ) ⊆ F (℘ ). The proof of the converse inclusion is analogous. Thus, the second assertion of our proposition is established. The first one then follows  immediately since J (℘ ) = C\F (℘ ). The proof is complete.

We shall now prove the following. Theorem 19.2.2 Let ℘ : C −→  C be a square Weierstrass elliptic function. If ξ ∈ C is either a superattracting, attracting, or rationally indifferent periodic point, then (1) iξ is also, respectively, a superattracting, attracting, or rationally indifferent periodic point and both ξ and iξ lie on the same periodic orbit.

266

Part V First Outlook

(2) ℘  (ξ ) = ℘  (iξ ). (3) If ξ is a superattracting periodic point of ℘, then (a) there is exactly one superattracting periodic orbit of ℘; namely, ξ ; (b) there is exactly one periodic connected component of the Fatou set F (℘); namely, the basin of immediate attraction of ℘ to ξ ; (c) all connected components of the Fatou set F (℘) are basins of attraction of ℘ to ξ . (4) If ξ is an attracting (but not superattracting) periodic point of ℘, then (a) there are exactly three attracting (not superattracting) periodic orbits of ℘; namely, ξ ; (b) there is exactly one periodic connected component of the Fatou set F (℘); namely, the basin of immediate attraction of ℘ to ξ ; (c) all connected components of the Fatou set F (℘) are basins of attraction of ℘ to ξ . (5) If ξ is a rationally indifferent periodic point of ℘, then (a) there is exactly one rationally indifferent periodic cycle of ℘; namely, ξ ; (b) there is exactly one periodic connected component of the Fatou set F (℘); namely, the basin of immediate attraction of ℘ to ξ ; (c) all connected components of the Fatou set F (℘) are basins of attraction of ℘ to ξ . Proof

Since ℘ is a square function, there is a square lattice  ⊆ C such that ℘ = ℘ .

We first will show that if f has one (super)-attracting or rationally indifferent periodic point ξ ∈ C with some period p ≥ 1, then iξ are also, respectively, (super)-attracting or rationally indifferent periodic points and ℘  (ξ ) = ℘  (iξ ). Furthermore, the periodic orbits of ξ and iξ coincide. So, let ξ ∈ C be such a periodic point with period p ≥ 1. Because of (19.1), we get that p

p

℘ (iξ ) = i℘ (ξ ),

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 267

meaning that iξ are also periodic points of ℘ with period p. Differentiating now both equations in (19.1) and using the Chain Rule, we get that  p   p   p  i ℘ (ξ ) = ℘ ◦ i (ξ ) = i ℘ (iξ ). Hence, 

 p  p  ℘ (iξ ) = ℘ (ξ ).

Now, since, by Proposition 16.6.7(6), e1 () =  and since the elliptic  −e2 () function ℘ is even, we have that ℘ e1 () = ℘ e2 () . Since, also by Proposition 16.6.7(6), the third (and the last) critical value e3 () = 0 of ℘ is a pole of ℘ , we directly deduce all the remining assertions of our theorem from Theorems 13.2.5 (Fatou Periodic Components), 14.1.1, 15.2.5, 16.6.1, and 17.1.6. The proof is complete.  As an immediate consequence of this theorem and Theorems 19.1.4, and 18.1.15, we get the following. Theorem 19.2.3 If ℘ : C −→  C is a square Weierstrass nonrecurrent elliptic function, then exactly one of the following holds. (1) J (℘) = C. (2) ℘ has exactly one (super)-attracting periodic orbit. Denote one of its points by ξ . Then (a) there is exactly one periodic connected component of the Fatou set F (℘); namely, the basin of immediate attraction of ℘ to ξ ; (b) all connected components of the Fatou set F (℘) are basins of attraction of ℘ to ξ . (3) ℘ has exactly one rationally indifferent periodic orbit. Denote one of its points by ξ . Then (a) there is exactly one rationally indifferent periodic cycle of ℘; namely, ξ ; (b) there is exactly one periodic connected component of the Fatou set F (℘); namely, the basin of immediate attraction of ℘ to ξ ; (c) all connected components of the Fatou set F (℘) are basins of attraction of ℘ to ξ . We recall from Definition 16.6.2(0) that a lattice  ⊆ C is called real if and only if  = .

268

Part V First Outlook

We need the following properties of square lattices and properties of Weierstrass ℘ -functions generated by such lattices. Proposition 19.2.4 Let  ⊆ C be a square lattice. (1) Then the critical values of the Weierstrass ℘ function are 1 1 g2 () 2 , e2 = −e1, and e3 = 0. 2 In particular, e3 is a pole of ℘ . (2) If the lattice  ⊆ C is real, then

e1 =

PC(℘ ) ⊆ {−e1 } ∪ {0} ∪ [e1,+∞] ⊆ R ∪ {+∞}. Proof Item (1) is a reformulation of item (6) of Proposition 16.6.7. Proving item (2), it follows from Proposition 16.6.7(5) and (6c5) that ℘ ([e1,+∞]) ⊆ [e1,+∞] ⊆ R ∪ {+∞}.

(19.2)

Therefore, n O+ (e1 ) = {℘ (e1 ) : n ≥ 0} ⊆ [e1,+∞].

Also, because of Proposition 16.6.7(5) and the evenness of ℘ , e3 = 0, e2 = − e1 , and ℘L (e2 ) = ℘L (e2 ), item (2) follows. The proof is complete.  Using this proposition, we get the following. Theorem 19.2.5 If ℘ : C −→  C is a real square Weierstrass elliptic function, then exactly one of the following holds. (1) J (℘) = C. (2) ℘ has exactly one (super)-attracting periodic orbit. Denote one of its points by ξ . Then (a) there is exactly one periodic connected component of the Fatou set F (℘); namely, the basin of immediate attraction of ℘ to ξ ; (b) all connected components of the Fatou set F (℘) are basins of attraction of ℘ to ξ . (3) ℘ has exactly one rationally indifferent periodic orbit. Denote one of its points by ξ . Then (a) there is exactly one rationally indifferent periodic cycle of ℘; namely, ξ ; (b) there is exactly one periodic connected component of the Fatou set F (℘); namely, the basin of immediate attraction of ℘ to ξ ; (c) all connected components of the Fatou set F (℘) are basins of attraction of ℘ to ξ .

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 269

Proof The same arguments as for Theorem 19.2.3 work, so all we need to show is that ℘ has no Siegel disks or Herman rings. We will do this now. So, seeking contradiction, suppose that U is either a Siegel disk or a Herman ring of the function ℘ . By Proposition 19.2.4 and Theorem 14.1.2, ∂U ⊆ {−e1 } ∪ {0} ∪ [e1,+∞] ⊆ R ∪ {+∞},

(19.3)

where e1 ∈ (0,+∞). Since ∂U ⊆ J (℘) and J (℘) has no isolated points, we further conclude that ∂U ⊆ [e1,+∞] ⊆ R ∪ {+∞}.

(19.4)

Since the point 0 is a pole of ℘ , it belongs to the Julia set J (℘ ), whence it does not belong to U . But, by (19.4) and since e1 ∈ (0,+∞), the origin 0 does not belong to the boundary ∂U either. Therefore, 0 ∈ C\(U ∪ ∂U ).

(19.5)

On the other hand, since U is not empty and open in C, it is not contained in [e1,+∞]. So, there exists a point ξ ∈ U \[e1,+∞].

(19.6)

Let [0,ξ ] be the closed line segment joining 0 and ξ . Then, on the one hand, [0,ξ ] ⊆ C\[e1,+∞] ⊆ C\∂U .

(19.7)

On the other hand, by (19.5) and (19.6), we have that [0,ξ ] ∩ U = ∅ and [0,ξ ] ∩ (C\U ) = ∅. Along with (19.7), this produces a contradiction since the segment [0,ξ ] is connected. We are done. 

19.3 The Dynamics of Triangular Weierstrass Elliptic Functions: Some Selected Facts We recall from Definition 16.6.2(4) that a lattice  ⊆ C is called triangular if and only if  = , where, we recall, =e

2π i 3

.

(19.8)

270

Part V First Outlook

Triangular Weierstrass elliptic functions will play an important role in creating many of our examples. We will need several of their properties. The first one is the following. Proposition 19.3.1 If  is a triangular lattice, then the Fatou set F (℘ ) and the Julia set J (℘ ) of the Weierstrass ℘ -function are invariant either under 2π i multiplication by  = e 3 or, in geometric terms, under rotation about the angle 2π/3, i.e.,  2 J (℘ ) = J (℘ ) = J (℘ ) and  2 F (℘ ) = F (℘ ) = F (℘ ). In addition, n n ℘ (z) = ℘ (z) and

n 2 n ℘ ( z) =  2 ℘ (z)

(19.9)

n (z) is well defined. for all integers n ≥ 0 and all z ∈ C whenever ℘

Proof Formula (19.9) directly follows, by a straightforward induction, from the homogeneity property (16.55). z ∈ F (℘ ) and an open neighborhood  n Take ∞ |U n=1 is normal. By (19.9), we have that U of z such that the family ℘ n n n (V ) = ℘ (U ) = ℘ (U ) ℘  n ∞ for all n ≥ 1. Thus, the family ℘ |U n=1 is normal. So, z ∈ F (℘ ). Hence, F (℘ ) ⊆ F (℘ ). The proof of the converse inclusion is analogous. Thus, the second assertion of our proposition is established. The first one then follows immediately since J (℘ ) = C\F (℘ ). This immediately implies that

 2 F (℘ ) = F (℘ ) and  2 J (℘ ) = J (℘ ) = J (℘ ). 

The proof is complete. We shall now prove the following.

Theorem 19.3.2 Let ℘ : C −→  C be a triangular Weierstrass elliptic function. If ξ ∈ C is either a superattracting, attracting, or rationally indifferent periodic point, then (1) ξ and  2 ξ are also, respectively, superattracting, attracting, or rationally indifferent periodic points. (2) ℘  (ξ ) = ℘  (ξ ) = ℘  ( 2 ξ ). (3) Either some two of the three periodic orbits of ξ , ξ , or  2 ξ intersect or all three of them are mutually disjoint.

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 271

Assume the last case holds. Then (A) If ξ is a superattracting periodic point of ℘, then (a) there are exactly three superattracting periodic orbits of ℘ and these are equal to the periodic orbits of ξ , ξ , and  2 ξ , (b) the periodic connected components of the Fatou set F (℘) are, respectively, the three basins of immediate attraction of ℘ to the periodic orbits of ξ , ξ , and  2 ξ , (c) all connected components of the Fatou set F (℘) are, respectively, the three basins of attraction of ℘ to the periodic orbits of ξ , ξ , and  2 ξ . (B) If ξ is an attracting (but not superattracting) periodic point of ℘, then (a) there are exactly three attracting (none of which is superattracting) periodic orbits of ℘ and these are equal to the periodic orbits of ξ , ξ , and  2 ξ , (b) the periodic connected components of the Fatou set F (℘) are, respectively, the three basins of immediate attraction of ℘ to the periodic orbits of ξ , ξ , and  2 ξ , (c) all connected components of the Fatou set F (℘) are, respectively, the three basins of attraction of ℘ to the periodic orbits of ξ , ξ , and  2 ξ , (d) the sets of derivatives of ℘ for each of these three attracting periodic orbits of ξ , ξ , and  2 ξ are equal. (C) If ξ is a rationally indifferent periodic point of ℘, then (a) there are exactly three rationally indifferent periodic cycles of ℘, (b) the periodic connected components of the Fatou set F (℘) are, respectively, the three basins of immediate attraction of ℘ to the periodic orbits of ξ , ξ , and  2 ξ , (c) all connected components of the Fatou set F (℘) are, respectively, the three basins of attraction of ℘ to the periodic orbits of ξ , ξ , and  2 ξ , (d) the sets of derivatives of ℘ for each of these three attracting periodic orbits of ξ , ξ , and  2 ξ are equal. Proof Since ℘ is triangular, there is a triangular lattice  ⊆ C such that ℘ = ℘ . We will first show that if f has one (super)-attracting or rationally indifferent periodic point ξ ∈ C with some period p ≥ 1, then ξ and  2 ξ are also, respectively, (super)-attracting or rationally indifferent periodic points, the derivative of ℘ at each of these three periodic points is the same, and if any

272

Part V First Outlook

two of the three periodic orbits of ξ , ξ or  2 ξ intersect, then all these three periodic orbits coincide. So, let ξ ∈ C be such a periodic point with period p ≥ 1. Because of (19.9), we get that p

p

p

p

℘ (ξ ) = ℘ (ξ ) = ξ and ℘ ( 2 ξ ) =  2 ℘ (ξ ) = ξ, meaning that ξ and  2 ξ are also periodic points of ℘ with period p. Differentiating now both equations in (19.9) and using the Chain Rule, we, respectively, get that  p   p   p   ℘ (ξ ) = ℘ ◦  (ξ ) =  ℘ (ξ ). Hence,

Likewise,

 

 p  p  ℘ (ξ ) = ℘ (ξ ).

 p  p  ℘ ( 2 ξ ) = ℘ (ξ ).

Now suppose that some two of the three periodic orbits of ξ , ξ , or  2 ξ intersect. We may assume without loss of generality that these are orbits of ξ and ξ . But any two periodic orbits that intersect, coincide. Thus, in particular, the orbits of ξ and ξ are equal. But then there exists an integer 1 ≤ p such that ℘ k (ξ ) = ξ . Having this and using (19.9), we get that ℘ k (ξ ) = ℘ k (ξ ) =  2 ξ . Therefore, all the orbits of ξ , ξ , or  2 ξ coincide, and we have proved what we claimed. Having this and assuming that the orbits of ξ , ξ , and  2 ξ do not intersect, we are immediately done by virtue of Theorem 19.1.3. The proof is complete.  The case in the above theorem when the three periodic orbits are mutually distinct is typical, while the case when they coincide is exceptional. It is not hard to calculate that such an exceptional phenomenon occurs, for example, for the triangular lattice with g3 = 5.5. This lattice is not real. It is illustrated on Figure 19.1. The only attracting, in fact superattracting, periodic orbit with period 3 is marked in black. This lattice is a rotation of the real triangular lattice with g3 = 5.5, having instead three superattracting fixed points. As an immediate consequence of the previous theorem, we get the following.

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 273

Figure 19.1 J (℘ ) for the triangular lattice  with g3 () = 5.5. The only attracting, in fact superattracting, periodic orbit with period 3 is marked in black.

Theorem 19.3.3 Let ℘ : C −→  C be a triangular nonrecurrent Weierstrass elliptic function. (1) If ℘ has a (super)-attracting fixed point ξ ∈ C, then (a) ℘ has exactly three (super)-attracting fixed points ξ , ξ , and  2 ξ , (b) the periodic connected components of the Fatou set F (℘) are the three basins of immediate attraction of ℘ to these fixed points ξ , ξ , and  2 ξ ; (c) all connected components of the Fatou set F (℘) are, respectively, the three basins of attraction of ℘ to these fixed points ξ , ξ , and  2 ξ . (d) in consequence, the elliptic function ℘ is expanding, so is normal compactly nonrecurrent of finite character. (2) If ℘ has a rationally indifferent fixed point, then (a) ℘ has exactly three rationally indifferent fixed points ξ , ξ , and  2 ξ , (b) the periodic connected components of the Fatou set F (℘) are the three basins of immediate attraction of ℘ to these fixed points ξ , ξ , and  2 ξ , (c) all connected components of the Fatou set F (℘) are, respectively, the three basins of attraction of ℘ to these fixed points ξ , ξ , and  2 ξ ; (d) in consequence, the elliptic function ℘ is parabolic.

274

Part V First Outlook

We recall from Definition 16.6.2(0) that a lattice  ⊆ C is called real if and only if  = . We need the following properties of triangular lattices and properties of Weierstrass ℘ -functions generated by such lattices. Proposition 19.3.4 Let  ⊆ C be a triangular lattice. (1) Then the critical values of the Weierstrass ℘ -function are the cubic roots of g3 ()/4. So, e3 ∈ C\{0}, e1 = e4π i/3 e3, and e2 = e2π i/3 e3 . In particular, if g3 () = 4 (then  is real by Proposition 16.6.6), then the critical values of the Weierstrass ℘ -function are the cubic roots of unity, i.e., e1 =  2, e2 = , and e3 = 1, 2π i

where, we recall,  := e 3 . (2) If the lattice  ⊆ C is real, then the postcritical set PC(℘ ) of the Weierstrass ℘ -function is contained in the union of the following three ℘ -forward invariant rays: [e3,+∞] ⊆ R ∪ {∞}, [e3,+∞], and  2 [e3,+∞]. Proof Item (1) is a reformulation of item (6) of Proposition 16.6.6. Proving item (2), it follows from Proposition 16.6.6(5) and (6c5) that ℘ ([e3,+∞]) ⊆ [e3,+∞] ⊆ R.

(19.10)

Therefore, n (e3 ) : n ≥ 0} ⊆ [e3,+∞]. O+ (e3 ) = {℘

Since  = , the homogeneity property (16.55) implies that 1 ℘ (z) = ℘ (z) 2 for all z ∈ C. Similarly (or consequently), ℘ (z) =

℘ ( 2 u) =  2 ℘ (u) for all z ∈ C. These two above-displayed formulas along with (19.10) yield ℘ ([e3,+∞]) ⊆ [e3,+∞] and ℘ ( 2 [e3,+∞]) ⊆  2 [e3,+∞]. (19.11)

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 275

Thus, ℘ (e1 ) = ℘ ( 2 ) =  2 ℘ (1) ∈  2 [e3,+∞] and ℘ (e2 ) = ℘ () = ℘ (1) ∈ [e3,+∞]. Along with (19.10) and (19.11), these imply that PC(℘ ) = O+ (e1 ) ∪ O+ (e2 ) ∪ O+ (e3 ) ⊆ [e3,+∞] ∪ [e3,+∞] ∪  2 [e3,+∞]. 

The proof is complete.

Now we shall prove one more theorem in the form of Theorems 19.3.2 and 19.3.3. Theorem 19.3.5 If  is a (real) triangular lattice with g3 () > 0, then the Weierstrass elliptic ℘ -function has no Siegel disk or Herman ring. Moreover, if ξ ∈ C is either a superattracting, attracting, or rationally indifferent periodic point, then (1) ξ and  2 ξ are also, respectively, superattracting, attracting, or rationally indifferent periodic points, (2)    (ξ ) = ℘ (ξ ) = ℘ ( 2 ξ ), ℘

(3) either some two of the three periodic orbits of ξ , ξ , or  2 ξ intersect or all three of them are mutually disjoint. Assume the last case holds. Then (A) If ξ is a superattracting periodic point of ℘ , then (a) there are exactly three superattracting periodic orbits of ℘ and these are equal to the periodic orbits of ξ , ξ , and  2 ξ ,   (b) the periodic connected components of the Fatou set F ℘ are, respectively, the three basins of immediate attraction of ℘ to the periodic orbits of ξ , ξ , and  2 ξ ,   (c) all connected components of the Fatou set F ℘ are, respectively, the three basins of attraction of ℘ to the periodic orbits of ξ , ξ , and  2 ξ , (B) If ξ is an attracting (but not superattracting) periodic point of ℘ , then

276

Part V First Outlook

(a) there are exactly three attracting (none of which is superattracting) periodic orbits of ℘ and these are equal to the periodic orbits of ξ , ξ , and  2 ξ ,  (b) the periodic connected components of the Fatou set F (℘ are, respectively, the three basins of immediate attraction of ℘ to the periodic orbits of ξ , ξ , and  2 ξ ,   (c) all connected components of the Fatou set F ℘ are, respectively, the three basins of attraction of ℘ to the periodic orbits of ξ , ξ , and  2 ξ , (d) the sets of derivatives of ℘ for each of these three attracting periodic orbits of ξ , ξ , and  2 ξ are equal. (C) If ξ is a rationally indifferent periodic point of ℘ , then (a) there are exactly three rationally indifferent periodic cyclesof ℘ , (b) the periodic connected components of the Fatou set F ℘ are, respectively, the three basins of immediate attraction of ℘ to the periodic orbits of ξ , ξ , and  2 ξ ,   (c) all connected components of the Fatou set F ℘ are, respectively, the three basins of attraction of ℘ to the periodic orbits of ξ , ξ , and  2 ξ , (d) the sets of derivatives of ℘ for each of these three attracting periodic orbits of ξ , ξ , and  2 ξ are equal. Proof The same arguments as for Theorems 19.1.4 and 19.3.3 work, so all we need to show is that ℘ has no Siegel disks or Herman rings. We will do this now. So, seeking contradiction, suppose that U is either a Siegel disk or a Herman ring of the function ℘ . By Proposition 19.3.4 and Theorem 14.1.2, ∂U ⊆ [e3,+∞] ∪ [e3,+∞] ∪  2 [e3,+∞],

(19.12)

where e3 ∈ (0,+∞). Since the point 0 is a pole of ℘ , it belongs to the Julia set J (℘ ), whence it does not belong to U . But, by (19.12) and since e3 ∈ (0,+∞), the origin 0 does not belong to the boundary ∂U either. Therefore, 0 ∈ C\(U ∪ ∂U ).

(19.13)

On the other hand, since U is not empty and open in C, it is not contained in [e3,+∞] ∪ [e3,+∞] ∪  2 [e3,+∞]. So, there exists a point   ξ ∈ U \ [e3,+∞] ∪ [e3,+∞] ∪  2 [e3,+∞] . (19.14) Let [0,ξ ] be the closed line segment joining 0 and ξ . Then, on the one hand,   [0,ξ ] ⊆ C\ [e3,+∞] ∪ [e3,+∞] ∪  2 [e3,+∞] ⊆ C\∂U . (19.15)

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 277

On the other hand, by (19.13) and (19.14), we have that [0,ξ ] ∩ U = ∅ and [0,ξ ] ∩ (C\U ) = ∅. Along with (19.15), this produces contradiction since the segment [0,ξ ] is connected. We are done.  As an immediate consequence of this theorem, we get the following. Theorem 19.3.6 Let  be a (real) triangular lattice with g3 () > 0. Then the following hold. (1) If ℘ has a (super)-attracting fixed point ξ ∈ C, then (a) ℘ has exactly three (super)-attracting fixed points ξ , ξ, and  2 ξ , (b) the periodic connected components of the Fatou set F ℘ are the three basins of immediate attraction of ℘ to these fixed points ξ , ξ , and  2 ξ , (c) all connected components of the Fatou set F (℘) are, respectively, the three basins of attraction of ℘ to these fixed points ξ , ξ , and  2 ξ ; (d) in consequence, the elliptic function ℘ is expanding, so is normal compactly nonrecurrent of finite character. (2) If ℘ has a rationally indifferent fixed point, then (a) ℘ has exactly three rationally indifferent fixed points ξ ,ξ , and  2 ξ , (b) the periodic connected components of the Fatou set F ℘ are the three basins of immediate attraction of ℘ to these fixed points ξ , ξ , and  2 ξ ,   (c) all connected components of the Fatou set F ℘ are, respectively, the three basins of attraction of ℘ to these fixed points ξ , ξ , and  2 ξ ; (d) in consequence, the elliptic function ℘ is parabolic.

19.4 Simple Examples of Dynamically Different Elliptic Functions Let  ⊆ C be a lattice. Let γ ∈ C\{0}. Finally, let s ∈ C\{0}. Then, by the homogeneity property (16.55), for any z ∈ C\{0}, we have that ℘ (sz) = sz

⇐⇒

℘ (z) = s 3, z

278

Part V First Outlook

while, by the homogeneity property (16.55), we have that  ℘ (sz) = γ

 γ −1 ℘ (z) = s 3 .

⇐⇒

Therefore, we get the following. Lemma 19.4.1 If  ⊆ C is a lattice and γ ∈ C\{0}, then there exists s ∈ −1 C\{0} and ξ ∈ C\( ∪ ℘ (0)) such that ℘s (sξ ) = sξ

and

 ℘s (zξ ) = γ

if and only if the equation  (z) z℘ =γ ℘λ (z) −1 (0)). has its solution in C\( ∪ ℘

ˆ be the meromorphic function defined by the formula Let F : C −→ C F (z) :=

z℘λ (z) . ℘ (z)

Our first result is this. Lemma 19.4.2 If  ⊆ C is a lattice and t ∈ (0,+∞), then there exists Gt , a nonempty open (in the relative topology) subset of {z ∈ C : |z| = t} such that, for every w ∈ Gt , the equation  (z) z℘ =w ℘ (z)

(19.16)

−1 (0)). has a solution in C\( ∪ ℘

ˆ is Proof Since the set C is open and connected and the function F : C → C meromorphic, we get the following. Claim 1◦ The set F (C) is open and connected. Obviously, Claim 2◦ F (Crit(℘ )) = {0} and F (\{0}) = {∞}. ˆ it follows from Since the set {z ∈ C : |z| = t} separates 0 and ∞ ∈ C, ◦ ◦ Claim 1 and Claim 2 that {z ∈ C : |z| = t} ∩ F (C) is a nonempty open set in the relative topology on {z ∈ C : |z| = t}. Now there are at least two arguments for the set

 −1 (0))) z ∈ C : |z| = t ∩ F (C\( ∪ ℘ to be nonempty. The first one is to observe that the set

 −1 (0))) z ∈ C : |z| = t ∩ F (C\( ∪ ℘

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 279

is a countable Fσ subset of

 z ∈ C : |z| = t ∩ F (C) and to apply the Baire Category Theorem. The second one is to note that the set {z ∈ C : |z| = t} ∩ F (C) is uncountable. We are done.  As an immediate consequence of this lemma and Lemma 19.4.1, we get the following. Lemma 19.4.3 If  ⊆ C is a lattice, t ∈ (0,+∞), and w ∈ Gt , the set produced in Lemma 19.4.2, then there exists s ∈ C\{0} and ξ ∈ C\( ∪ −1 (0)) such that ℘ ℘ (sξ ) = sξ

and

℘ξ  (sξ ) = w.

Let us take several remarkable fruits of this lemma and our theorems about triangular lattices. Indeed, as an immediate consequence of this lemma and Theorem 19.3.2, we get the following four theorems. Theorem 19.4.4 For every t ∈ (0,1), there exists a triangular lattice  ⊆ C such that (1) There exist three attracting fixed points ξ1,ξ2,ξ3 ∈ C of ℘ such that  ℘λ (ξ1 ) = ℘λ (ξ2 ) = ℘λ (ξ3 ) and |℘ (ξ1 )| = t.

(2) The periodic Fatou components of ℘ consist of the three basins of immediate attraction to ξ1,ξ2,ξ3 . (3) The Fatou set F (℘ ) of ℘ is the union of basins of attraction to ξ1 , ξ2 , and ξ3 . (4) The Weierstrass elliptic function ℘ is expanding, thus is normal compactly nonrecurrent of finite character. Theorem 19.4.5 There exists a triangular lattice  ⊆ C such that (1) There exist three rationally indifferent fixed points ξ1,ξ2,ξ3 ∈ C of ℘ such that    ℘ (ξ1 ) = ℘ (ξ2 ) = ℘ (ξ3 ).

(2) The periodic Fatou components of ℘ consist of the three basins of immediate attraction to ξ1,ξ2 , and ξ3 . (3) The Fatou set F (℘ ) of ℘ is the union of basins of attraction to ξ1,ξ2 , and ξ3 . (4) The Weierstrass elliptic function ℘ is parabolic.

280

Part V First Outlook

Theorem 19.4.6 For each lattice  ⊆ C, there exists a lattice  ⊆ C, similar to , such that the Weierstrass elliptic function ℘ has at least one Siegel disk whose center is a fixed point of ℘ . Theorem 19.4.7 For each lattice  ⊆ C, there exists a lattice  ⊆ C, similar to , such that the Weierstrass elliptic function ℘ has at least one Cremer fixed point. We would like to have a little bit more than Theorem 19.4.5; namely, we would like to have a parabolic triangular Weierstrass elliptic function with a specific, explicitly known value of the derivative of its rationally indifferent fixed points. We shall prove the following. Theorem 19.4.8 For every (real) triangular lattice  with g3 () > 0, there exists a real triangular lattice , similar to , such that the Weierstrass elliptic function ℘ is parabolic and has three rationally indifferent fixed points ξ1,ξ2 , and ξ3 such that ℘ (ξ1 ) = ℘ (ξ2 ) = ℘ (ξ3 ) = 1. Proof

Because of Proposition 16.6.6(6),  = λ[, −1 ]

with some λ ∈ (0,+∞). Fix arbitrary k ∈ N. By Proposition 16.6.6(8), we have that  λ F = 0. (19.17) 2 Since ℘ has a pole of order 2 at λ, we have that ℘ (z) = A(z − λ)−2 + O(|z − λ|−1 )

(19.18)

and  ℘ (z) = −2A(z − λ)−3 + O(|z − λ|−2 )

with some A ∈ R. Hence,  (x) ℘ −2A(x − λ)−3 + O(|x − λ|−2 ) = λ lim Rx%λ ℘ (x) Rx%λ A(x − λ)−2 + O(|z − λ|−1 )

lim F (x) = λ lim

Rx%λ

−2A(x − λ)−1 + O(1) Rx%λ A

= λ lim

= O(1) − 2λ lim (x − λ)−1 = +∞. Rx%λ

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 281

It follows from this, (19.17), continuity of the function F |[λ/2,λ) : [λ/2,λ) −→ R, and the Intermediate Value Theorem that F ((λ/2,λ)) ⊃ (0,+∞). Hence, there exists x ∈ (λ/2,λ) such that F (x) = 1. Therefore, a direct application of Lemma 19.4.1 completes the proof.  Let  = λ[1,i] be a real square lattice with λ > 0. For every α ∈ C, let hα := ℘ + α. We shall prove the following. Theorem 19.4.9 If  = 2λ[1,i] is a real square lattice with some λ ∈ (0,+∞), then, for every ε ∈ (0,1), there exists Mε > 0 such that if (mk )∞ k=1 is any Mε sequence of positive integers such that mk ≥ 2λ , then there exists β ∈ (−,) such that hkβ (λ) ∈ [2λmk ,2λ(mk + 1)] for every integer k ≥ 2. ˆ by the Proof For every integer k ≥ 1, define the function Gk : C −→ C formula Gk := hkα (λ + λi) with the convention that hα (∞) = ∞. We then have that Gk+1 (α) = ℘ ◦ Gk (α) + α and G1 (α) = ℘ (λ + λi) + α = α,

G2 (α) = ℘ (α) + α,

and

G2 (0) = ℘1 (0) = ∞. So, G2 is meromorphic (and takes on the value ∞) on some open ball BC (0,2δ) with δ ∈ (0,ξ ). In addition, since ℘ is real, G2 ((−2δ,2δ)) ⊆ R and G2 ((−δ,δ)) ⊃ [Mε,+∞) ⊆ R

(19.19)

with some Mε > 0. We shall now inductively define a descending sequence {[ak ,bk ]}∞ k=1 of compact intervals in R such that, for every k ≥ 2,

282

Part V First Outlook

(1) [ak ,bk ] ⊆ [−δ,δ] ⊆ (−ε,ε) and (2) Gk (ak ) = 2λmk , Gk (bk ) = 2λ(mk + 1), and Gk ([ak ,bk ]) = [2λmk ,2λ(mk + 1)] ⊆ R. By our hypothesis, [2λm2,2λ(m2 + 1)] ⊆ [M ,+∞). So, it follows from (19.19) and the continuity of G2 , that there exists a closed interval [a2,b2 ] ⊆ [−δ,δ] such that G2 (a2 ) = 2λm2, G2 (b2 ) = 2λ(m2 + 1), and G2 ([a2,b2 ]) = [2λm2,2λ(m2 + 1)]. We are, thus, done with the base of induction. For the inductive step, suppose that (1) and (2) hold for some integer k ≥ 2. Then, by (2), there exists ck ∈ [ak ,bk ] such that Gk (ck ) = 2λmk + λ. Hence, Gk+1 (ck ) = ℘ ◦ Gk (ck ) + ck = ℘ (2λmk + λ) + ck = ℘ (λ) + ck = e1 () + ck ≤ e1 () + ε ≤ e1 () + 1. Also, by (2), Gk+1 (ak ) = ℘ ◦ Gk (ak ) + ak = ℘ (2λmk ) + ak = ∞. Thus, also using Proposition 16.6.7(6c), we get that Gk+1 ([ak ,bk ]) ⊃ [e1 () + 1,+∞). Therefore, there exists an interval [ak+1,bk+1 ] ⊆ [ak ,bk ] such that Gk+1 (ak+1 ) = 2λmk+1, Gk+1 (bk+1 ) = 2λ(mk+1 + 1), and Gk+1 ([ak+1,bk+1 ]) = [2λmk+1,2λ(mk+1 + 1)].

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 283 So, our inductive construction is complete. Since {[ak ,bk ]}∞ k=1 is a descending sequence of compact intervals contained in [−δ,δ] ⊆ (−ε,ε), we get that ∅ =  :=

∞

[ak ,bk ] ⊆ [−δ,δ] ⊆ (−ε,ε).

k=1

Therefore, for any β ∈ , we have, for all k ≥ 2, that hkβ (λ) = Gk (β) ∈ Gk ([ak ,bk ]) = [2λmk ,2λ(mk + 1)]. 

The proof is complete. We can now prove one of the main results of this chapter.

Theorem 19.4.10 If  = 2λ[1,i] is a real square lattice with some λ ∈ (0, + ∞) which has an attracting periodic point, then, for every ε ∈ (0,1) sufficiently small, there exists β ∈ (−ε,ε) such that the elliptic function ˆ hβ = ℘ + β : C → C has the following properties. (1) hβ has exactly one attracting periodic orbit. (2) β is a critical value of hβ and its orbit {hβ (β) : n ≥ 0} is bounded and infinite. (3) The Julia set J (hβ ) is a proper nowhere dense subset of C. (4) hβ is a normal subexpanding (in particular, compactly nonrecurrent) elliptic function of finite character, and is nonexpanding. Proof By Theorem 19.2.5 and item (6) of Proposition 16.6.7, ℘ has exactly one periodic orbit and both critical values e1 () and e2 () belong to its basin of attraction. Since attracting periodic orbits are stable under locally smart  perturbation, if ε ∈ (0,1) is small enough, then each function hα , α ∈ (−ε, , has an attracting periodic orbit and both its critical values e1 () + α and e2 () + α belong to its basin of attraction. So, (3) and a part of (1) are proved. Now we take (mk )∞ k=1 , a bounded sequence of positive integers which is not ε eventually periodic with m2 ≥ M 2λ . Then let β ∈ (−ε,ε) be the corresponding number produced in Theorem 19.4.9. Then (2) is automatically satisfied. Item (1) then follows since hβ has exactly three critical values. Item (4) follows as well. The proof is complete.  In order to complete the picture, we will now describe a large, uncountable, class of real square lattices  satisfying the hypothesis of Theorem 19.4.10.

284

Part V First Outlook

Theorem 19.4.11 Let  = 2λ[1,i] be a real square lattice with some λ ∈  e1  1 3 , the real cubic root of e1 . (0,+∞). Let m be any odd integer and k := mλ mλ If  := k (a real square lattice), then the corresponding Weierstrass ˆ has a superattracting periodic point mkλ. elliptic function ℘ : C −→ C In consequence, by Theorem 19.4.9, there exists β ∈ R\{0} such that the ˆ has all the properties (1)–(4) of Theorem elliptic function ℘ + β : C −→ C 19.4.9. Proof

By the homogeneity equation (16.56), we have that ℘ (mkλ) = ℘ (kλ) = k −3 ℘ (λ) = k −3 · 0 = 0.

By the homogeneity equation (16.55), we have that ℘ (mkλ) = ℘ (kλ) = k −2 ℘ (λ) = k −2 e1 = k −2 k 3 mλ = mkλ. So, mkλ is indeed a superattracting periodic point of ℘ and the proof is complete.  This theorem/example stemmed from Lemma 7.2 in [HK2]. Theorems 9.3 (with n = 0) and 9.4 in this paper are other sources of examples in Theorem 19.4.9.

19.5 Expanding (Thus Compactly Nonrecurrent) Triangular Weierstrass Elliptic Functions with Nowhere Dense Connected Julia Sets In this section, we shall provide several examples of compactly nonrecurrent elliptic functions with various behaviors of critical points (and critical values), all of them having connected Julia sets. All these examples are motivated by the work of Hawkins and her collaborators, primarily by [HK1]. The first following example was published therein. Theorem 19.5.1 Let  = [w1,w2 ], where w1 = w2 with  = e2π i/3 , be the triangular lattice associated with the invariants g2 = 0 and g3 = 4; see Proposition 16.6.6 for its existence and uniqueness. In particular, by Proposition 16.6.6(6),  is real. Let m be a negative odd integer. If 

1 2w12  2 3 w2 , γ2 := γ1 , and  := [γ1,γ2 ], γ1 := m w1 then

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 285 (1) The corresponding Weierstrass function ℘ : C −→  C has exactly three superattracting fixed points. (2) The periodic connected components of the Fatou set F (℘) are the three basins of immediate attraction of ℘ to these fixed points, and all connected components of the Fatou set F (℘) are the three basins of attraction of ℘ to these fixed points. C is expanding, thus is normal (3) The Weierstrass function ℘ : C −→  compactly nonrecurrent of finite character. (4) The Julia set J (℘ ) is a proper nowhere dense subset of C. (5) The Julia set J (℘ ) is connected. Proof First, we will verify that the lattice  is triangular. Indeed, since w1 w1 = w2 and γ1 = γ2 w , we have that γ1 = γ2 . Thus, 2  = ,

(19.20)

meaning that  is triangular. It then follows from Theorem 16.3.7, (16.19), and Proposition 19.3.4(1) that ℘ (w1 /2) = e1 =  2 . Then the homogeneity equation (16.55) gives  γ  w  1 γ 1 w1 1 1 γ · =℘ 1 =  2 ℘ ℘ w1 2 w1 2 2 γ1 w1

=

1 γ1 w1

2 2  = 

w12  2 2w12  2 m

2 = 3

mγ1 . 2

γ1 1 Since m is an odd integer, we have that ℘ ( mγ 2 ) = ℘ ( 2 ). Thus,  mγ  mγ 1 1 ℘ = , 2 2

meaning that

mγ1 2

is a fixed point of ℘ . Also,  mγ  γ  1 1   ℘ = ℘ = 0, 2 2

(19.21)

(19.22)

(19.23)

1 whence mγ 2 is a superattracting fixed point of ℘ . Therefore, items (1)–(3) directly follow from Theorem 19.3.6. Item (5) follows now directly from Theorem 19.1.6. Item (4) follows from Theorem 13.1.9 since J (℘) = C. 

Figure 19.2 shows the Julia set J (℘ ) of the triangular Weierstrass ℘ function defined in Theorem 19.5.1 for the case when m = −1. The corresponding Weierstrass ℘ -function is expanding and has three superattracting

286

Part V First Outlook

Figure 19.2 J (℘ ), where  = [λ1,λ2 ] with λ1 ≈ 1.1382 + 1.9741i, λ2 ≈ 1.1382 − 1.9714i. ℘ is triangular, expanding, and has three superattracting fixed points.

fixed points at 1.1382, −1.1382, and 1.1382 2 . A fundamental region is highlighted in white (so we can see that it is made up of two equilateral triangles), the three distinct fixed points are indicated with black dots (so we see their symmetry around the origin), and each attracting basin is a different color. The origin is marked, but it is tiny in this format. It is the left vertex of the region. The following theorem has a proof analogous to the proof of Theorem 19.5.1, so its proof will be omitted. 2π i

Theorem 19.5.2 ([HK2], Theorem 8.3) Let  = [λ,λe 3 ], λ > 0, be the triangular lattice associated with the invariants g2 = 0, g3 = 4. For any m,n ∈ Z, if   2π i −1/3 , k = (λ/2) + mλ + nλe 3 then, for  := k, the invariant

  2π i 2 g3 () = 4 λ/2 + mλ + nλe 3 .

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 287

Furthermore, (1) The corresponding Weierstrass function ℘ : C −→  C has exactly three superattracting fixed points. (2) The periodic connected components of the Fatou set F (℘) are the three basins of immediate attraction of ℘ to these fixed points, and all connected components of the Fatou set F (℘) are the three basins of attraction of ℘ to these fixed points. C is expanding, thus is normal (3) The Weierstrass function ℘ : C −→  compactly nonrecurrent of finite character. (4) The Julia set J (℘ ) is a proper nowhere dense subset of C. (5) The Julia set J (℘ ) is connected. Figure 19.3 illustrates the Julia set of the triangular expanding Weierstrass ℘ -function associated with the triangular lattice  with the invariant g3 () ≈ 5.67 + 2.08i. In this case, there are three attracting cycles of period 3. One of these cycles occurs at approximately

 ξ = 1.139 + 0.134i,0.989 + 0.11i,1.131 − 0.068i , 2π i

4π i

while the second cycle is located at e 3 p and the third one at e 3 p. Both parts of Figure 19.3 are the same, with the fundamental period outlined. Each attracting basin is colored differently and the period 3 orbits are marked with points.

Figure 19.3 J (℘ ), where g3 () ≈ 5.67 + 2.08i. ℘ is triangular, expanding, and has three attracting cycles of period 3.

288

Part V First Outlook

19.6 Triangular Weierstrass Elliptic Functions Whose Critical Values Are Preperiodic, Thus Being Subexpanding The main result of this section is the following theorem proved in [HK1] as Theorem 8.6. Theorem 19.6.1 There exists a triangular lattice  such that the critical values of the Weierstrass ℘ -function are preperiodic. More precisely, ℘ (e1 ), ℘ (e2 ), and ℘ (e3 ) are repelling fixed points of ℘ . C is normal subexpanding In consequence, the elliptic function ℘ : C −→  of finite character (in particular, compactly nonrecurrent) and nonexpanding. C. Also, J (℘ ) =  Proof Denote again the real triangular lattice with g3 = 4 (see Proposition 16.6.6 for its existence and uniqueness) by . By Proposition 16.6.6,  = [w1,w2 ], where w1 = te

πi 3

πi

and w2 = te− 3

with some t > 0. Then w3 = w1 + w2 is a real period of the Weierstrass ℘ -function. Using the tables in [MT], we have that w3 ≈ 2.42 . . ., and so 2 < w3 < 3. Given k ∈ C\{0}, let  := k = [kw1,kw2 ] = [λ1,λ2 ]. As usual, denote λ3 := λ1 + λ2 . Then, by Theorem 16.6.1, λ3 /2 is a critical point of ℘ , i.e.,  (λ3 /2) = 0. ℘

  2 = 1 (see The homogeneity equation (16.55) and the property ℘ w1 +w 2 Proposition 19.3.4(1)) imply that   1 1 w 1 + w2 λ3 e3, = ℘ = 2 ℘ = 2. 2 2 k k We are looking for a value of k ∈ (0,+∞) such that ℘ (e3, ) = ℘ (k −2 )

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 289

is a fixed point of the function ℘ . First, we will show that if there exists k ∈ R\{0} such that  1 1 (19.24) ℘ − 2 + 2λ3 = − 2 + 2λ3, k k then

   1 1 = ℘ . ℘ ℘ 2 k k2

(19.25)

Indeed, using -periodicity and symmetry (i.e., evenness) of the function ℘ , we get from (19.24) that       1 1 1 ℘ ℘ − 2 + 2λ3 = ℘ ℘ − 2 = ℘ ℘ k k k2 (19.26) and

   1 1 1 ℘ − 2 + 2λ3 = ℘ − 2 = ℘ . k k k2

(19.27)

So, (19.25) follows. Applying the homogeneity equation (16.55), we get that     1 1 1 1 = 2 ℘ − 3 , ℘ − 2 + 2λ3 = ℘k k − 3 + 2w3 k k k k so we can rewrite (19.24) as  1 1 1 1 ℘ − = − 2 + 2λ3 = − 2 + 2kw3  k2 k3 k k or, equivalently, as

 1 ℘ − 3 = −1 + 2k 3 w3 . k

(19.28)

In order to find k ∈ (0,+∞) satisfying (19.28), we consider two auxiliary real-valued functions 4 1  1 5 2 3 1 3 , −→ R, f ,g : [r,s] := w3 w3 defined as g(k) := ℘ (−1/k 3 ) and f (k) := −1 + 2k 3 w3 . Since 2 < w3 < 3, we have that r > 1/2 and s < 1. Note that g(r) = +∞, g  (s) = 0, and g is monotone decreasing on the interval [r,s]. Hence, there exists δ > 0 such that

290

Part V First Outlook r + δ < s < 1 and g(r + δ) > 10.

Thus, f (r + δ) < −1 + 2w3 < −1 + 6 = 5 and g(r + δ) − f (r + δ) > 0. By Proposition 19.3.4(1), we get that g(s) = ℘ (−w3 /2) = ℘ (w3 /2). Hence, g(s) − f (s) < 0. Since both g and f are continuous functions on the interval [r + δ,s], the Intermediate Value Theorem implies that there exists a number k ∈ (r + δ,s) such that g(k) = f (k). Hence, the condition (19.28) is satisfied; in consequence, (19.24) holds and so does (19.25) too. Thus, we have a triangular lattice  such that e3, = k −2 is preperiodic. Therefore, the homogeneity properties (19.9) and (16.56) imply that n n ℘ (z) = ℘ (z) and

n 2 n ℘ ( z) =  2 ℘ (z),

for all integers n ≥ 0, and   (z) = ℘ (z) and ℘

  ℘ ( 2 z) =  2 ℘ (z).

Therefore, using Proposition 19.3.4(1), we conclude that the critical values e1 () and e2 () are also preperiodic. Since λ3 /2 ∈ (−∞,0) and k −2 ∈ (0, + ∞), we get that λ3 /2 = e3, .

(19.29)

Seeking contradiction, suppose that   ℘ e3, = λ3 /2. Then (19.25) would imply that e3, = ℘ (λ3 /2) = λ3 /2, contrary to (19.29). Thus, the critical point λ3 /2 is strictly preperiodic. Then, by virtue of Theorems 19.3.3, 14.1.1, and we deduce   15.2.5,   that the periodic point ℘ e3, , as well the points ℘ e1, and ℘ e2, , are all repelling.

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 291

This means that the first assertion of our theorem is proved. The second assertion is then an immediate consequence of the definitions of elliptic functions being expanding, subexpanding, normal, compactly nonrecurrent, and of finite character. The third assertion follows from it and Theorem 19.1.1. 

19.7 Weierstrass Elliptic Functions Whose Critical Values Are Poles or Prepoles, Thus Being Subexpanding, Thus Compactly Nonrecurrent We shall provide, in this section, examples of elliptic functions whose critical values are prepoles of arbitrarily high orders. These come from [HKK]. Theorem 19.7.1 There exists a triangular lattice  ⊆ C such that the critical values of the Weierstrass function ℘ : C −→  C are the lattice points (i.e., belong to ), thus poles. C is subexIn consequence, the Weierstrass elliptic function ℘ : C −→  panding (thus, compactly nonrecurrent) of finite character, nonexpanding, and C. not normal. In addition, J (℘ ) =  Proof Let  = [λ,λ], λ > 0, be the real triangular lattice associated with the invariants g2 = 0 and g3 = 4; see Proposition 16.6.6 for its existence and uniqueness. Consider the real triangular lattice  := k := [kλ,kλ], where k := (mλ + nλ)−1/3 and m,m are some fixed nonzero integers. Denote γ := kλ. The homogeneity equation (16.55) and Proposition 19.3.4(1) give   γ  1 1 kλ λ ℘ = ℘k = 2 ℘ = 2 = (mγ + nγ ), 2 2 2 k k where the last equality in (19.30) follows from  2   − 1 1 = mλ + nλ 3 = mλ + nλ mλ + nλ 3 2 k   = k mλ + nλ = mγ + nγ .

(19.30)

292

Part V First Outlook

Thus, e1, = ℘ (γ /2) = mγ + nγ  is a lattice point and so a pole of . Since  is triangular, by applying (19.9), we get that  γ γ    ℘  = ℘ =  mγ + nγ  2 2 and

 γ γ  =  2 ℘ =  2 (mγ + nγ ). ℘  2 2 2     But  mγ + nγ  and  2 mγ + nγ  are also lattice points of  and so poles of ℘ . Hence, γ γ 2γ , , ∈ ℘−2 (∞). 2 2 2

Therefore, all three critical values of the Weierstrass function ℘ are prepoles. The second assertion of our theorem immediately follows from this. The fact C now directly follows from the second assertion and Theorem that J (℘ ) =  19.1.1.  We now pass to such examples based on square and, later, rhombic lattices. We recall that a lattice  ⊆ C is called a square lattice if and only if i = . Now we will give some examples of elliptic functions whose critical values are poles. Theorem 19.7.2 ([HK1], Theorem 8.2) There exists a real square lattice  ⊆ C such that all the critical values of the Weierstrass function ℘ : C −→  C are the lattice points (i.e., belong to ), thus poles. In consequence, the elliptic C is subexpanding (thus, compactly nonrecurrent) of function ℘ : C −→  C. finite character, nonexpanding, and not normal. In addition, J (℘ ) =  Proof Let  = [δ,δi], δ ∈ (0,+∞), be the real square lattice associated with the invariants g2 = 4 (and g3 = 0); see Proposition 16.6.7 for its existence and uniqueness. Consider the lattice  := [γ ,γ i], 1  where γ = δ 2 /m 3 and m ∈ N. The homogeneity equation (16.55) and Proposition 16.6.7(6) give  γ  1 δ ℘ = ℘ = e1 () = 1.   2 2 2 (mδ) 3

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 293

Hence, ℘

γ 

2

= (mδ) 3 = mγ .

2 But mγ is a lattice point of ℘ and, thus, a pole of ℘ . The homogeneity property (16.55) implies that ℘ (iz) =

1 ℘ (z) = −℘ (z). i2

(19.31)

Therefore, ℘ (iγ /2) = −℘ (γ /2) = −mγ , which is also a lattice point and so a pole of ℘ . In addition, it follows from Proposition 16.6.7(6) that e3 = 0, also a pole of ℘ . Hence, {e1,e2,e3 } ⊆ ℘−2 (∞). The second assertion of our theorem immediately follows from this. The fact that J (℘ ) =  C now directly follows from the second assertion and Theorem 19.1.1.  We recall, see Section 16.6, that any lattice  of the form [w,w], w ∈ C\{0}, is called real rhombic and any lattice similar to it is called rhombic. We also recall from this section that any lattice that is real rhombic and square is called real rhombic square. The proofs of the next two results are analogous to the proof of Theorem 19.7.2 (see Proposition 4.1 in [HKK] and Theorem 8.9 in [HK2]). Theorem 19.7.3 There exists a real rhombic square lattice  ⊆ C such that C are the lattice all the critical values of the Weierstrass function ℘ : C −→  points (i.e., belong to ), thus poles. C is subexIn consequence, the Weierstrass elliptic function ℘ : C −→  panding (thus, compactly nonrecurrent) of finite character, nonexpanding, and C. not normal. In addition, J (℘ ) =  Proof Let  := [2b + 2bi,2b − 2bi] = 2b[1 + i,1 − i], b > 0, be the real rhombic square lattice associated with the invariants g2 = −4 and g3 = 0; see Proposition 16.6.8 for its existence and uniqueness. It follows from Proposition 16.6.8 that, for critical points b + bi, b − bi, and 2b of ℘ , we have that ℘ (b + bi) = −i, ℘ (b − bi) = i, and ℘ (2b) = 0.

(19.32)

294

Part V First Outlook

Note that, for any square lattice , p ∈ C is a pole of ℘ if and only if ±pi is a pole. Thus, all the points 2bj , j ∈ Z, are the poles of ℘ . Fix j ∈ N. Let 1

k := (2bj )− 3 .

(19.33)

Let  := k. We will show that, for critical points k(b + bi), k(b − bi), and 2kb of ℘ , we have that   ℘2 {k(b + bi),k(b − bi),2kb} = {∞}.

(19.34)

The homogeneity property (16.55) along with (19.32) yield ℘ (k(b + bi)) = ℘k (k(b + bi)) =

i 1 ℘ (b + bi) = − 2 = −2j kbi k2 k (19.35)

and ℘ (k(b − bi)) = ℘k (k(b − bi)) =

i = 2j kbi. k2

(19.36)

Since ±2j bi ∈ , we have that ±2j kbi ∈ , whence ℘ (±2j kbi) = ∞. Hence,   ℘2 {k(b + bi),k(b − bi)} = ∞.

(19.37)

Analogously, we get that ℘ (2bk) = ℘k (2kb) =

1 ℘ (2b) = 0. k2

(19.38)

Thus, ℘2 (2kb) = ∞.

(19.39)

Formulas (19.37) and (19.39) entail (19.34). Therefore, all three critical values of the Weierstrass function ℘ are prepoles. The second assertion of our C now directly theorem immediately follows from it. The fact that J (℘ ) =  follows from the second assertion and Theorem 19.1.1. 

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 295

19.8 Compactly Nonrecurrent Elliptic Functions with Critical Orbits Clustering at Infinity Theorem 19.8.1, the first one we prove below, apart from being interesting in itself, is also the main ingredient in the construct of the two classes of examples which will follow it. We recall, see Definition 13.1.2, that the prepoles of order n ≥ 0 of a meromorphic function f : C −→  C were defined as

 −n n f (∞) = z ∈ C : f (z) is well defined and f n (z) = ∞ . In particular, the poles coincide with order 1 prepoles. For a Weierstrass function ℘ : C −→  C, we have the following immediate fact which was used several times above: f −1 (∞) = . Fix a lattice  = [γ1,γ2 ] ⊆ C. For any α ∈ C\{0}, set gα := α℘ .

(19.40)

Of course, the poles of gα are the same as for ℘ , i.e., equal to . The critical points of gα are also obviously the same as for ℘ , i.e.,   Crit ℘ = (c1 + ) ∪ (c2 + ) ∪ (c3 + ), but the critical values of gα are, in general, different from critical values of ℘ since      gα Crit(gα ) = α℘ Crit ℘ , where γ1 γ2 γ 1 + γ2 , c2 = , c3 = . 2 2 2 Thus, we can denote all the critical values of gα , respectively, as c1 =

e1 (α) = gα (c1 ) = αe1, e2 (α) = gα (c2 ) = αe2, e3 (α) = gα (c3 ) = αe3 . Recall that B∞ (R) = {z ∈  C : |z| > R}. The first main technical result of this section, which is needed in the next ones, is the following.

296

Part V First Outlook

Theorem 19.8.1 Let  ⊆ C be a lattice and ℘ : C −→  C be the corresponding Weierstrass function. −q If q ≥ 2 is an integer and ξ ∈ ℘ (∞), then, for every ε ∈ (0,1), there ∞ exist a sequence (βk )∞ k=q ⊆ B(1,ε) ⊆ C and a sequence (εk )k= of real positive numbers such that (0) βq = 1 and εq = ε, (1) βk ∈ B(βk−1,εk−1 ) ⊆ B(1,ε) for every k ≥ q + 1, (2) (∞) ξ ∈ gβ−k k for every k ≥ q + 1, (3)     B βk+1,εk+1 ⊆ B βk ,εk ⊆ B(1,ε) for every k ≥ q + 1, and (4) 

  lim inf inf gαk (ξ ) : α ∈ B(βk ,εk ) = +∞. k→∞

Proof We shall prove Theorem 19.8.1 by induction on k ≥ q + 1, i.e., we define inductively the sequences ∞ (βk )∞ k=q ⊆ B(1,ε) and (εk )k=q

so that conditions (1)–(4) are satisfied. First, βq := 1 and εq := ε. Now the base of induction, i.e., k = q + 1 (yes, k = q + 1 rather than k = q). We define the meromorphic function C Gq : B(1,ε) −→  by Gq (α) := gαq (ξ ).

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 297

Therefore, q

q

Gq (1) = g1 (ξ ) = ℘ (ξ ) = ∞.

  It is also immediate from the definition of Gq that Gq B(1,ε)\{1} ⊆ C if ε ∈ (0,1) is small enough. Therefore, for such εs, the function Gq has a a unique pole at α = 1. Consequently, there exists an Rq > 0 such that BRq (∞) ⊂ Gq (B(1,ε)). Thus, fixing a point γq ∈  ∩ BRq (∞), there exists a parameter βq+1 ∈ B(1,ε) such that q

gβq+1 (ξ ) = Gq (βq+1 ) = γq . Therefore, q+1

gβq+1 (ξ ) = ∞, meaning that −(q+1)

ξ ∈ gβq+1

(∞).

So, we have items (1) and (2) established for k = q + 1. For the inductive step, assume that, for some k ≥ q + 1, the numbers βq+1,βq+2, . . . ,βk ∈ C and εq+1, . . . ,εk−1 ∈ (0,1) have been defined such that the following hold: • item (1) has been established for all integers q + 1 ≤ j ≤ k, meaning that βj ∈ B(βj −1,εj −1 ) ⊆ B(1,ε) for all integers q + 1 ≤ j ≤ k, • item (2) has been established for all integers q + 1 ≤ j ≤ k, • item (3) has been established for all integers q + 1 ≤ j ≤ k − 2,  • item (4 ): 

  inf gαj (c1 ) : α ∈ B(βj ,εj ) ≥ j for all integers q + 1 ≤ j ≤ k − 1, • and also nonconstant meromorphic functions  Gj : B(1,ε)\∪3≤i 0 such that   Gk B(βk ,εk ) ⊃ BRk (∞). Hence, fixing a point γk ∈  ∩ BRk (∞), there exists a parameter βk+1 ∈ B(βk ,εk )

(19.43)

such that gβk k+1 (ξ ) = Gk (βk+1 ) = γk . Therefore, gβk+1 (ξ ) = ∞, k+1 meaning that ξ ∈ gβ−(k+1) (∞). k+1 This along with (19.43) means that conditions (1) and (2) are established for k + 1. The inductive step of our construction is complete. Since item (4) follows directly from item (4 ), the proof of Theorem 19.8.1 is finished.  We now move directly on to the announced examples. We first briefly recall the examples of elliptic functions considered in the proof of Theorem 19.7.3. Let  = [2b + 2bi,2b − 2bi], b > 0,

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 299 be the real rhombic square lattice introduced in its proof. We choose j = 4 as the positive even integer appearing in the proof of Theorem 19.7.3. As in this proof,  = k, where, as in (19.33) with j = 4, 1

k := (8b)− 3 . We denote c1 := k(b + bi), c2 := k(b − bi), c3 := 2kb and e1 = ℘ (c1 ), e2 = ℘ (c2 ), e3 = ℘ (c3 ). Then, recalling (19.35) and (19.37), we have that 2

℘ (c1 ) = −(8b) 3 i = e1 ∈  and ℘2 (c1 ) = ℘ (e1 ) = ∞.

(19.44)

Theorem 19.8.2 ([HKK],Theorem 4.9) Let  and c1 , c2 , and c3 be as above. If gα = a℘ , α ∈ C\{0}, is the family of elliptic functions defined in (19.40), then, for every ε ∈ (0,1), there exists a parameter β ∈ B(1,ε) ⊆ C such that     lim gβn (c1 ) = lim gβn βe1 = ∞ = lim gβn (c2 ) = lim gβn βe2 (19.45) n→∞

n→∞

n→∞

n→∞

and e3 = 0,

(19.46)

i.e., the forward iterates of the two critical values βe1 and βe2 of gβ diverge to ∞ under the action of gβ while the third one, i.e., e3 = 0, is its pole. C is nonexpanding, In consequence, the elliptic function gβ : C −→  C. subexpanding, thus compactly nonrecurrent. In addition, J (gβ ) =  Proof Fix an arbitrary ε ∈ (0,1) and, looking at (19.44), apply Theorem 19.8.1 with ξ := c1 and q = 2 to get the sequences (βk )∞ k=2 ⊆ B(1,ε) and ⊆ (0,1). Because of condition (3) of this theorem, we a sequence (εk )∞ k=2 have that ∞

B(βk ,εk ) = ∅.

k=3

Fix any β in this intersection. Then β ∈ B(1,ε) and it follows from item (4) of Theorem 19.8.1 that the first two equality signs of (19.45) hold.

300

Part V First Outlook

Since c2 = c1 i, we can show, analogously to (19.9), that gβn (c2 ) = gβn (c1 ) for all integers n ≥ 2. Thus, the remaining two equality signs of (19.45) also follow. Formula (19.46) follows directly from (19.8) and (19.38). Hence, the C is nonexpanding, subexpanding, thus compactly elliptic function gβ : C −→  nonrecurrent. Furthermore, because of Theorems 19.1.1, 18.1.15, and 13.2.5 (Fatou Periodic Components), and because Crit(gβ ) = {βe1,βe2,e3 }, we have C. The proof is complete.  that J (gβ ) =  We finish this section with a class of examples of nonrecurrent elliptic functions whose critical values all diverge to infinity; compare with Theorem 4.10 in [HKK]. Theorem 19.8.3 If  ⊆ C is a triangular lattice such that the critical value e3 (see Proposition 19.3.4(1)) of the Weierstrass function ℘ : C −→  C belongs −q to ℘ (∞) with some integer q ≥ 2 (see Theorem 19.7.1 for such examples), then, for every ε ∈ (0,1), there exists β ∈ B(1,ε) such that, for the elliptic C, we have that function gβ := β℘ : C −→    lim gβn βei = ∞ n→∞

for all i = 1,2,3, i.e., the forward iterates of all critical values of gβ diverge to ∞. C is nonexpanding, subexIn consequence, the elliptic function gβ : C −→  C. panding, thus compactly nonrecurrent, and J (gβ ) =  Proof Let c3 ∈ C be such a critical point of ℘ that ℘ (c3 ) = e3 . By our hypotheses, c1 is a prepole of ℘ , say of order q ≥ 1. So, we may apply Theorem 19.8.1 in the same way as in the proof of the previous theorem (Theorem 19.8.2), with ξ := c3 and q being q, to conclude that there exists β ∈ B(1,ε) such that   lim gβn (c3 ) = lim gβn βe3 = ∞. n→∞

n→∞

By Proposition 19.3.4 and (19.9), we get that gβn (e1 ) =  2 gβn (e3 ) and gβn (e2 ) = gβn (e3 ). Therefore, lim g n (c1 ) n→∞ β

= ∞ and

lim g n (c2 ) n→∞ β

= ∞.

In conclusion, lim g n n→∞ β



 βe1 = ∞ and

lim g n n→∞ β



 βe2 = ∞.

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 301

The proof can now be completed in exactly the same way as that of the previous theorem. 

19.9 Further Examples of Compactly Nonrecurrent Elliptic Functions Finally, we will provide some examples of nonrecurrent elliptic functions with a nonempty Fatou set, all of which are nonhyperbolic maps. C In Theorem 19.7.3, we considered the Weierstrass function ℘ : C −→  induced by a real rhombic square lattice  = 2kb[1 + i,1 − i], where 1

b ∈ (0,+∞), k = (2bj )− 3 , and j ∈ N. We know that c1 = k(b + bi), c2 = k(b − bi), and c3 = 2kb are critical points of ℘ and every other critical point of ℘ is congruent mod  with one of them. Also, ei = ℘ (ci ), i = 1,2,3, are all critical values of ℘ and they are all poles of ℘ . In addition, e3 = 0, which is also a pole of ℘ . 1 We fix j = 1, so that k = (2b)− 3 . We define C. h := ℘ + 2kbi : C −→ 

(19.47)

The function h is elliptic with poles of order 2 at each point of . We shall prove the following. Theorem 19.9.1 If h : C −→  C is the elliptic function defined by (19.47), then (1) 0, 4kbi, and 2kbi are all its critical values, (2) 0 and 4kbi are also poles of h, (3) h(2kb) = 2kbi and 2kbi is a superattracting fixed point of h. In consequence, the elliptic function h : C −→  C is nonexpanding, subexpanding, thus compactly nonrecurrent, of finite character, and the Julia set J (h) is a proper, nowhere dense subset of  C.

302

Part V First Outlook

Proof

By (19.35) and (19.36), we have that

h (k(b + bi)) = ℘ (k(b + bi)) + 2kbi = −2kbi + 2kbi = 0

(19.48)

h (k(b − bi)) = ℘ (k(b − bi)) + 2kbi = k2bi + 2kbi = 4kbi.

(19.49)

and

By (19.38), we have that h(2kb) = ℘ (2kb) + 2kbi = 0 + 2kbi = 2kbi.

(19.50)

Thus, item (1) follows. Item (2) is obvious since 4kbi = 2kb(1 + i) − 2kb(1 − i) ∈ . By (19.38) and using also the fact that ℘ (iz) = −i℘ (z) (holding because  is a real rhombic lattice), we get that h(2kbi) = ℘ (2kbi) + 2kbi = −i℘ (2kb) + 2kbi = 0 + 2kbi = 2kbi. (19.51) Thus, 2kbi is a fixed point of h. Since  is a square lattice, by the homogeneity equation (16.56), we get that  (2kbi) = i −3 ℘ (2kb) = i℘ (c3 ) = 0. h (2kbi) = ℘ (2kbi) = ℘i

Therefore, 2kbi is a superattracting fixed point of h. All other assertions of the theorem now follow immediately and the proof is complete.  Figure 19.4 illustrates the Julia set of ℘ with g2 () ≈ 26.5626 and g3 () ≈ −26.2672 ( is real rectangular). For these values, ℘ has an attracting fixed point ξ ≈ 1.5566. A fundamental region is (precisely) a 2 × 1 horizontal real rectangle The outline of one fundamental region is shown. Theorem 19.9.2 Let h : C −→  C be the elliptic function defined by (19.47) and, for every α ∈ C\{0}, gα := αh. Then, for every ε ∈ (0,1), there exists β ∈ B(1,ε) such that (1) the critical value β2kbi of gβ is attracted to some attracting fixed point of gβ , (2) the critical value 0 of gβ is its pole, and (3) the critical value β4kbi of gβ diverges to infinity. In consequence, the elliptic function gβ : C −→  C is compactly nonrecurrent and nonexpanding and the Julia set J (gβ ) is a proper, nowhere dense subset of  C. Proof Since attracting fixed points are stable under perturbations that are small near them and since, by Theorem 19.9.1(3), 2kbi is a superattracting

19 Various Examples of Compactly Nonrecurrent Elliptic Functions 303

Figure 19.4 J (℘ ) with g2 () ≈ 26.5626.

fixed point of h, there exists ε ∈ (0,1) so small that, for every α ∈ B(1,ε), the elliptic function gα has an attracting fixed point ξα such that lim ξα = 2kbi

α→1

and α2kbi, a critical value of gα , belongs to the basin of immediate attraction to ξα . So, (1) holds for every α ∈ B(1,ε). By Theorem 19.9.1(1) and the definition of gα , 0 is a critical value of gα . By Theorem 19.9.1(2), it is a pole of gα , meaning that item (2) of our theorem holds. Applying now Theorem 19.8.1 in the same way as it was applied in the proof of Theorem 19.8.2, we produce β ∈ B(1,ε) such that lim g n (β4kbi) n→∞ β The proof is complete.

= ∞. 

P AR T VI Compactly Nonrecurrent Elliptic Functions: Fractal Geometry, Stochastic Properties, and Rigidity

20 Sullivan h-Conformal Measures for Compactly Nonrecurrent Elliptic Functions

In this chapter, we deal systematically with one of the primary concepts of the book, namely that of (Sullivan) h-conformal (as always, h = HD(J (f ))) measures for compactly nonrecurrent elliptic functions. We will prove their existence for this class of elliptic functions. In Section 20.3, we will introduce an important class of regular compactly nonrecurrent elliptic functions. For this class of elliptic functions, we will prove the uniqueness and atomlessness of h-conformal measures along with their first basic stochastic properties such as ergodicity and conservativity. We will then assume an elliptic function to be compactly nonrecurrent regular throughout the entire book, unless explicitly stated otherwise. We have already met the concept of conformal measures in Sections 10.1 and 10.2, where we treated them in a very general setting in the former of these two sections and in the latter in a setting and spirit quite close to the one we will be dealing with in the current chapter. In particular, we will frequently use the results of these two sections in the current chapter. We gave, in Section 10.1, quite an extended historical account of the concept of conformal measures, particularly the Sullivan ones. We repeat a part of it here for the sake of completeness and for the convenience of the reader. Conformal measures were first defined and introduced by Patterson in his seminal paper [Pat1] (see also [Pat2]) in the context of Fuchsian groups. Sullivan extended this concept to all Kleinian groups in [Su2] and [Su4]. He then, in papers [Su5] and [Su7], defined conformal measures for all rational functions of the Riemann sphere  C; he also proved their existence therein. Both Patterson and Sullivan came up with conformal measures in order to get an understanding of geometric measures, i.e., Hausdorff and packing measures. Although Sullivan had already noticed that there are conformal measures for Kleinian groups that are not equal, nor even equivalent to any Hausdorff or packing (generalized) measure, the main purpose of dealing with them 307

308

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

is to understand Hausdorff and packing measures. Chapter 11 in Volume I, Section 17.6, and, especially, the current Part VI of our book provide good evidence. Conformal measures, in the sense of Sullivan, have been studied in the context of rational functions in greater detail in [DU3], where, in particular, the structure of the set of their exponents was examined. Since then, conformal measures in the context of rational functions have been studied in numerous research works. We list here only a very few of them that appeared in the early stages of the development of their theory: [DU1], [DU5], [DU6]. Subsequently, the concept of conformal measures, in the sense of Sullivan, has been extended to countable alphabet iterated function systems in [MU1] and to conformal graph directed Markov systems in [MU2]. These were treated at length in Chapter 11. This was, furthermore, extended to some transcendental meromorphic dynamics in [KU2], [UZ1], and [MyU3]; see also [UZ2], [MyU4], and [BKZ1]. Our current construction fits well with this line of development. Last, the concept of conformal measures also found its place in random dynamics; we cite only [MSU].

20.1 Existence of Conformal Measures for Compactly Nonrecurrent Elliptic Functions In this section, we prove the existence of h-conformal measures for compactly nonrecurrent elliptic functions. We also locate their potential atoms. As a fairly straightforward application of Theorem 17.6.7, we shall prove the following main result of this section. Theorem 20.1.1 If f : C −→  C is a compactly nonrecurrent elliptic function, then     DDh (J (f )) = DDχ (J (f )) = HD Jer (f ) = HD Jr (f ) = h = HD(J (f )) (20.1) and there exists an h-conformal measure mh for f (remember that its spherical version is, as are all spherical conformal measures considered in this book, finite; as a matter of fact, probabilistic) all of whose atoms are contained in the set Zf :=

∞  n=0

∞    f −n Crit(J (f )) ∪ f −n (∞). n=1

20 Sullivan h-Conformal Measures

309

Furthermore, mh is the measure produced in Claim 2◦ stated in the proof of Theorem 17.6.7. In addition, if mh (Crit(J (f )) = 0, then all atoms of mh are contained in the set I− (f ) =

∞ 

f −n (∞).

n=1

Proof Let mh be the measure m produced in Theorem 17.6.7. In fact, because of Corollary 18.3.6, we can, and we do, take the measure mh as produced in Claim 2◦ stated in the proof of Theorem 17.6.7. In view of Proposition 18.3.4, we have that J (f )\Jr (f ) ⊆ Sing− (f ). Since the set Sing− (f ) is countable, it follows that h = HD(J (f )) = HD(Jr (f )) = sf . Thus, by virtue of Theorem 17.6.7, (20.1) holds and the measure mh is h-conformal. Now by applying h-conformality of the measure mh , it follows from (17.32) of Lemma 17.6.6 and Corollary 18.3.6 that if z ∈ J (f )\Sing− (f ), then mh ({z}) = 0. Since, by Theorem 17.6.7, mh ((f )) = 0, we conclude that if z ∈ f −n ((f )) with some integer n ≥ 0, and m({z}) = 0, then z∈

∞ 

  f −n Crit(J (f )) .

n=0

Thus (note also that, by Theorem 17.6.7, mh (∞) = 0), all atoms of mh are contained in the set Zf =

∞  n=0

∞    f −n Crit(J (f )) ∪ f −n (∞). n=1

In order to prove the last assertion of our theorem, assume that mh (Crit(J (f )) = 0, z ∈ J (f ), and f n (z) ∈ Crit(J (f )) for some integer n ≥ 0. Then let 0 ≤ k ≤ n be the least integer such that f k (z) ∈ Crit(J (f )). Since mh ({f k (z)}) = 0, we then conclude, by h-conformality of mh , that  mh ({z}) = 0. The proof of Theorem 20.1.1 is, thus, complete.

20.2 Conformal Measures for Compactly Nonrecurrent Elliptic Functions and Holomorphic Inverse Branches In this section, we keep f : C −→  C, a compactly nonrecurrent elliptic function. Let m be an almost t-conformal measure and me be its Euclidean version.

310

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

The upper estimability and strongly lower estimability will be considered in this section with respect to the measure me . When we speak about lower estimability, we will make a stronger assumption, namely that the measure m is t-conformal. Since the number of parabolic points is finite, passing to an appropriate iteration, we assume without loss of generality, in this and the next section, that all parabolic periodic points of f are simple. Consider a closed forward f -invariant subset E of C such that f  E := sup{|f  (z)| : z ∈ E} < +∞. Such sets will be called f -pseudo-compact. Obviously, each f -invariant compact subset E of C is f -pseudo-compact. Recall that θ = θ (f ) > 0 was defined in (18.19), β = βf > 0 was defined in (18.21), αt (ω) in Lemma 15.4.1, and that τ > 0 is so small as required in Lemma 15.3.2. The proofs of Propositions 4.15 and 4.16 from [KU4] translate verbatim to our current case. We present them now. Proposition 20.2.1 Let f : C →  C be a compactly nonrecurrent elliptic function. Fix an f -pseudo-compact subset E of J (f ). Let z ∈ E, λ > 0, −1 be a real number. Suppose that at least and 0 < r ≤ τ θ min{1,f  −1 E }λ one of the following two conditions is satisfied:  f −n (Crit(J (f )) z ∈ E\ n≥0

or z∈E

and



−1 inf{|(f n ) (z)|−1 : n = 1,2, . . . . r > τ θ min 1,f  −1 E }λ

Then there exists an integer u = u(λ,r,z) ≥ 0 such that r|(f j ) (z)| ≤ λ−1 θ τ for all 0 ≤ j ≤ u and the following four conditions are satisfied:   diame Comp(f j (z),f u−j ,λr|(f u ) (z)|) ≤ β = βf

(20.2)

for every j = 0,1, . . . ,u. Let m be an almost t-conformal measure. Then, for every η > 0, there exists a continuous function [0,∞)  t −→ Bt = Bt (λ,η) > 0 (independent of z, n, and r) such that if f u (z) ∈ Be (ω,θ ) for some ω ∈ (f ), then f u (z) is (ηr|(f u ) (z)|,Bt ) − αt (ω)-u.e.

(20.3)

with respect to the almost t-conformal measure m, and there exists a function Wt = Wt (λ,η) : (0,1] −→ (0,1] (independent of z, n, and r) such that if f u (z) ∈ Be (ω,θ ) for some ω ∈ (f ), then, for every σ ∈ (0,1],

20 Sullivan h-Conformal Measures f u (z) is (ηr|(f u ) (z)|,σ,Wt (σ )) − αt (ω)-s.l.e.

311

(20.4)

with respect to the almost t-conformal measure m. If f u (z) ∈ / Be ((f ),θ ), then (20.3) and (20.4) are also true with αt (ω) replaced by t.

(20.5)

sup{λr|(f j ) (z)| : j ≥ 0} > θτ min{1,f  −1 E }.

(20.6)

Proof Suppose, first, that

Let n = n(λ,z,r) ≥ 0 be the least integer for which λr|(f n ) (z)| > θτ min{1,f  −1 E }.

(20.7)

Then n ≥ 1 (owing to the assumption imposed on r), λr|(f j ) (z)| ≤ θ τ min{1,f  −1 E }

(20.8)

for all 0 ≤ j ≤ n − 1, and also λr|(f n ) (z)| ≤ θ τ .

(20.9)

If f n (z) ∈ / Be ((f ),θ ), set u = u(λ,r,z) := n. Then items (20.3)–(20.5) are obvious in view of (20.7) and (20.9), while (20.2) follows from (18.22) and (18.23) along with (20.9). Thus, we are done in this subcase. So, suppose that f n (z) ∈ Be ((f ),θ ),

(20.10)

say f n (z) ∈ Be (ω,θ ), ω ∈ (f ). Let 0 ≤ k = k(λ,z,r) ≤ n be the least integer such that f j (z) ∈ Be ((f ),θ ) for every j = k,k + 1, . . . ,n. Consider all the numbers ri := |f i (z) − ω||(f i ) (z)|−1, where i = k,k + 1, . . . ,n. Put  f  + E := max{1,f E }.

By (20.7), we have that −1 −1 −1 rn = |f n (z) − ω||(f n ) (z)|−1 ≤ θ f  + λr = f  + λr; Eθ τ Eτ

therefore, there exists a minimal k ≤ u = u(λ,r,z) ≤ n such that ru ≤ −1 λr. In other words, f  + Eτ −1 |f u (z) − ω| ≤ f  + λr|(f u ) (z)|. Eτ

(20.11)

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Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Now suppose that sup{λr|(f j ) (z)| : j ≥ 0} ≤ θ τ min{1,f  −1 E }.

(20.12)

Then it follows from Corollary 18.3.6 and our hypotheses that z∈

∞ 

f −j ((f )).

j =0

Define then the three numbers u(λ,z,r), k(λ,z,r), and n(λ,z,r) to all be equal to the least integer j ≥ 0 such that f j (z) ∈ (f ). Denote ω = f u (z). Notice that, in this case, (20.9) and (20.11) are also satisfied. Our further considerations are valid in both cases (20.6) with (20.10), and (20.12). First note that, by (20.11), we have that −1 −1 η λ)ηr|(f u ) (z)|). Be (f u (z),ηr|(f u ) (z)|) ⊆ Be (ω,(1 + f  + Eτ (20.13)

In view of Lemma 15.4.1 along with (20.8) and (20.9), we get that   −1 −1 αt (ω) η λ) (ηr|(f u ) (z)|)αt (ω) . me Be (f u(z),ηr|(f u )(z)|) ≤ C(1 +f  + Eτ So, item (20.3) is proved. Also applying (20.11), Lemmas 15.4.5 and 10.4.4, and (20.9), we see that the point f u (z) is   + −1 −1 αt (ω) f E τ λr|(f u ) (z)|,σ τ f  −1 L(ω,2f  + E ηλ ,2 E θ,   + −1 −1 σ τ (2f E ) ηλ ) − αt (ω)-s.l.e. −1 λ ≥ η, then, by Lemma 10.4.5, f u (z) is So, if f  + Eτ  −1 −1 αt (ω) λη ) L(ω,2f  + ηr|(f u ) (z)|,σ,(2f  + Eτ E θ,   −1 −1 σ τ (2f E ) η)λ − αt (ω)-s.l.e.

If, instead, f  E τ −1 λ ≤ η, then, again, it follows from (20.11), Lemmas 15.4.5 and 10.4.4, and (20.9) that the point f u (z) is   ηr|(f u ) (z)|,σ,2αt (ω) L(ω,2θ τ λ−1 η,σ/2) − αt (ω)-s.l.e. So, part (20.4) is also proved. In order to prove (20.2), suppose, first, that u = k. In particular, this is  the case if z ∈ j ≥0 f −j ((f )). If k = 0, we are done since λr ≤ τ θ by our hypotheses, while τ θ ≤ βf by (18.23). So, suppose that k ≥ 1. Since 0 ≤ u ≤ n, it then follows from (20.8) and (20.9) that Comp(f k−1 (z),f ,r|(f u ) (z)|) ⊆ Comp(f k−1 (z),f ,θ τ ),

20 Sullivan h-Conformal Measures

313

and, by the choice of k and (15.58), we have that f k−1 (z) ∈ / Be ((f ),θ ). Therefore, (20.2) follows from (18.22)) and (18.23). If u > k (so, we are in the case of (20.6) and (20.10)), then ru−1 > −1 λr ≥ f   τ −1 λr. Also using (15.58), we get that f  + E Eτ ru =

|f u (z) − ω| −1 |f  (f u−1 (z))|−1 ru−1 ≥ f  −1 λr. E ru−1 ≥ τ |f u−1 (z) − ω|

Hence, λr|(f u ) (z)| ≤ τ |f u (z) − ω| and, applying Lemma 15.3.2 and (15.58) u − k times, we conclude that, for every k ≤ j ≤ u,   diame Comp(f j (z),f u−j ,λr|(f u ) (z)|) ≤ θ τ < βf . And now, for all j = k − 1,k − 2, . . . ,1,0, the same argument as in the case of u = k finishes the proof.  Proposition 20.2.2 Let f : C −→  C be a compactly nonrecurrent elliptic function. Fix an f -pseudo-compact subset E of J (f ). Let both ε and λ be positive numbers such that ε < λ min{1,τ −1,θ −1 τ −1 γ }. If 0 < r < −1 and z ∈ E\Crit(J (f )), then there exists an integer τ θ min{1,f  −1 E }λ s = s(λ,ε,r,z) ≥ 1 with the following three properties: |(f s ) (z)| = 0.

(20.14)

If u = u(λ,r,z) of Proposition 20.2.1 is well defined, then s ≤ u(λ,r,z). If either u is not defined or s < u, then there exists a critical point c ∈ Crit(f ) such that |f s (z) − c| ≤ εr|(f s ) (z)|.

(20.15)

In any case,   Comp z,f s ,(KA2 )−1 2−Nf εr|(f s ) (z)| ∩ Crit(f s ) = ∅,

(20.16)

where A was defined in (18.20). Proof Since z ∈ / Crit(J (f )) and in view of Proposition 20.2.1, there exists a minimal number s = s(λ,ε,r,z) for which at least one of the following two conditions is satisfied: |f s (z) − c| ≤ εr|(f s ) (z)|

(20.17)

for some c ∈ Crit(J (f )) or u(λ,r,z) is well defined

and s(λ,ε,r,z) = u(λ,r,z).

Since |(f s ) (z)| = 0, the parts (20.14) and (20.15) are proved.

(20.18)

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Part VI Fractal Geometry, Stochastic Properties, and Rigidity

In order to prove (20.16), notice, first, that no matter which of the two numbers s is, in view of Proposition 20.2.1, we always have that εr|(f s ) (z)| ≤ ελ−1 θ τ . Let us now argue that, for every 0 ≤ j ≤ s,   diame Comp(f s−j (z),f j ,εr|(f s ) (z)|) ≤ βf .

(20.19)

(20.20)

Indeed, if s = u, it follows immediately from Proposition 20.2.1 and (20.2) since ε ≤ λ. Otherwise, |f s (z) − c| ≤ εr|(f s ) (z)| ≤ ελ−1 θ τ < θ ; therefore, / Be ((f ),θ ). Thus, (20.20) follows from (18.22). by (18.19), f s (z) ∈ Now, by (20.20) and Lemma 18.1.11, there exist 0 ≤ p ≤ Nf , an increasing sequence of integers 1 ≤ k1 < k2 < · · · < kp ≤ s, and mutually distinct critical points c1,c2, . . . ,cp of f such that   {cl } = Comp f s−kl (z),f kl ,εr|(f s ) (z)| ∩ Crit(f ), (20.21) for every l = 1,2, . . . ,p, and if j ∈ / {k1,k2, . . . ,kp }, then  s−j  j Comp f (z),f ,εr|(f s ) (z)| ∩ Crit(f ) = ∅.

(20.22)

Setting k0 = 0, we shall show by induction that, for every 0 ≤ l ≤ p,   Comp f s−kl (z),f kl ,(KA2 )−1 2−l εr|(f s ) (z)| ∩ Crit(f kl ) = ∅. (20.23) Indeed, for l = 0, there is nothing to prove. So, suppose that (20.23) is true for some 0 ≤ l ≤ p − 1. Then, by (20.22),   Comp f s−(kl+1 −1) (z),f kl+1 −1,(KA2 )−1 2−l εr|(f s ) (z)| ∩Crit(f kl+1 −1 ) = ∅. So, if cl+1 ∈ Comp(f s−kl+1 (z),f kl+1 ,(KA2 )−1 2−(l+1) εr|(f s ) (z)|), then, by Lemma 8.4.3 applied for holomorphic maps H = f , Q = f kl+1 −1 , and the radius R = (KA2 )−1 2−(l+1) εr|(f s ) (z)| < γ , we get that |f s−kl+1 (z) − cl+1 | ≤ KA2 |(f kl+1 ) (f s−kl+1 (z))|−1 (KA2 )−1 2−(l+1) εr|(f s ) (z)| = 2−(l+1) εr|(f s−kl+1 (z)) | ≤ εr|(f s−kl+1 ) (z)|, which contradicts the definition of s and proves (20.23) for l + 1. In particular, it follows from (20.23) that   Comp z,f s ,(KA2 )−1 2−Nf εr|(f s ) (z)| ∩ Crit(f s ) = ∅. The proof is finished.



20 Sullivan h-Conformal Measures

315

We will also need the following similar result. Lemma 20.2.3 Let f : C −→  C be a compactly nonrecurrent elliptic function. Assume that (f ) = ∅. Then there exist two constants a,ξ > 0 such that the following holds. Suppose that z ∈ J (f )\

∞ 

f −n ({∞} ∪ Crit(f )).

n=0

Suppose also that r ∈ (0,γ (aξ )−1 ), where γ > 0 was defined in (18.22). Then there exists an integer s ≥ 0 with the following properties: (a) raξ |(f s ) (z)| ≥ γ , or (b) raξ |(f s ) (z)| < γ , and (c) there exists a critical point c ∈ Crit(J (f )) such that |(f s )(z) − c| < rξ |(f s ) (z)|, or (d) there exists a pole b ∈ f −1 (∞) such that |(f s )(z) − b| < rξ |(f s ) (z)|. In either case,   Comp z,f s ,2ξ r|(f s ) (z)| ∩ Crit(f s ) = ∅. Proof Put a = 2KA2 2Nf , where A was defined in (18.20). Fix ρ ∈ (0,1/2) so small that, for every w ∈ C\(Crit(f ) ∪ f −1 (∞)), the map f restricted to the set    Be w,2ρ diste w,Crit(f ) ∪ f −1 (∞) is one-to-one. Set ξ = 2−4 ρ. Take λ > 0 in Proposition 20.2.2 such that ε > 0 appearing there can be taken to be equal to aξ . In view of Corollary 18.3.6, there exists a least integer n ≥ 0 such that raξ |(f n ) (z)| ≥ γ . Since r < γ (aξ )−1 , we see that n ≥ 1. If there exists an integer 0 ≤ j ≤ n − 1 satisfying (c) or (d), take s to be the least one. Otherwise, take s = n. By the definition of n, it follows from (18.22) that diame (Comp(z,f k ,2ξ r|(f k ) (z)|)) < βf for all k = 0, . . . ,n − 1. Thus, we see that (20.20) is satisfied if s ≤ n − 1 and the proof of the last formula in our lemma is complete by verbatim repetition of the fragment of the proof of Lemma 20.2.2 from “Now, by (20.20)” to its end. If s = n, the same argument shows that   (20.24) Comp z,f n−1,2ξ r|(f n−1 ) (z)| ∩ Crit(f n−1 ) = ∅.

316

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

By the choice of ξ and the definition of n, we also know that the map f n−1 restricted to the ball Be (f n−1 (z),16ξ r|(f n−1 ) (z)|) is injective. Thus, by the 1 4 -Koebe Theorem (Theorem 8.3.3),   f Be (f n−1 (z),16ξ r|(f n−1 ) (z)|) ⊃ Be (f n (z),4ξ r|(f n ) (z)|); therefore,   Comp f n−1 (z),f ,2ξ r|(f n−1 ) (z)| ⊆ Be (f n−1 (z),16ξ r|(f n−1 ) (z)|). Combining this with (20.24) and injectivity of f restricted to Be (f n−1 (z),16ξ r|(f n−1 ) (z)|), we conclude that

  Comp z,f n,2ξ r|(f n ) (z)| ∩ Crit(f n ) = ∅.

We are done.



20.3 Conformal Measures for Compactly Nonrecurrent Regular Elliptic Functions: Atomlessness, Uniqueness, Ergodicity, and Conservativity In this section, we continue dealing with conformal measures. We already have their existence, and our goal now is to prove their uniqueness, atomlessness, ergodicity, and conservativity. This will require stronger hypotheses than mere compact nonrecurrence. In fact, it will require one more hypothesis. This hypothesis is the regularity of a compactly nonrecurrent elliptic function introduced at the beginning of Section 18.4; see, especially, (18.51). First, we need it to be able to show that the h-conformal measure constructed in Theorem 20.1.1 is atomless. This, in turn, is a prerequisite for, essentially all, our considerations concerning geometric measures (Hausdorff and packing) and measurable dynamics with respect to the measure class generated by the conformal measure mh . In this book, we need regularity from the proof of Lemma 20.3.9 onward. Let us record the following immediate observation. Observation 20.3.1 Every compactly nonrecurrent elliptic function f : C −→  C with Crit∞ (f ) = ∅ is regular. This simple observation starkly indicates that the class of all regular nonrecurrent elliptic functions is large indeed; see also the entire Section 19 devoted to examples of nonrecurrent elliptic functions. As an immediate consequence of Observation 20.3.1, we have the following corollary.

20 Sullivan h-Conformal Measures

317

Corollary 20.3.2 Every expanding and parabolic elliptic function is regular. Another sufficient condition, immediately following from Theorem 17.3.1 for a nonrecurrent elliptic function to be regular, is this. Proposition 20.3.3 If f : C −→  C is a compactly nonrecurrent elliptic function and 2qmax (f ) 2l∞ > , qmax (f ) + 1 l∞ + 1 then f is regular. Now we derive from (18.51) a technical condition, (20.27), which will be directly needed in our considerations involving the continuity of conformal measures. It immediately follows from (18.51) that, for every c ∈ Crit∞ (f ), c qc . Hence, h > p2p c qc +1 pc − 1 h < (qc + 1)h − 2qc . pc So there exists h− ∈ (1,h) such that pc − 1 h− < (qc + 1)h− − 2qc ; pc

(20.25)

therefore, there exists κc > 0 such that pc − 1 h− < κc < (qc + 1)h− − 2qc . pc The right-hand side of this formula is equivalent to the following:   qc + 1 h − − κc > 1. 2 − κc qc

(20.26)

(20.27)

We now pass to more general considerations. Let ms be a Borel probability measure on C and me be its Euclidean version, i.e., dme (z) := (1 + |z|2 )t . dms We shall prove the following. Lemma 20.3.4 If z ∈ C, rn "0, and there are two constants M ≤ M such that M ≤ lim inf n→∞

me (Be (z,rn )) me (Be (z,rn )) ≤ lim sup ≤ M, t rn rnt n→∞

318

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

then

  ms Bs (z,(2(1 + |z|2 ))−1 rn ) ≤ 2t M lim sup ((2(1 + |z|2 ))−1 rn )t n→∞

and

  ms Bs (z,2(1 + |z|2 )−1 rn ) ≥ 2−t M. lim inf n→∞ (2(1 + |z|2 )−1 rn )t

Proof

Since, for every r > 0 sufficiently small, Be (z,2−1 (1 + |z|2 )−1 r) ⊆ Bs (z,r) ⊆ Be (z,2(1 + |z|2 )r)

and since lim

r"0

me (Be (z,r)) = (1 + |z|2 )t , ms (Be (z,r))

we get that

  ms Bs (z,(2(1 + |z|2 ))−1 rn ) ms (Be (z,rn )) lim sup ≤ lim −t = 2t M 2 −1 t n→∞ 2 (1 + |z|2 )−t r t ((2(1 + |z| )) rn ) n→∞ n

and

  m Bs (z,2(1 + |z|2 )−1 rn ) ms (Be (z,rn )) ≥ lim t = 2−t M. lim inf 2 −1 t n→∞ n→∞ (2(1 + |z| ) rn ) 2 (1 + |z|2 )−t rnt 

We are done.

Assuming that the compactly nonrecurrent elliptic function f : C →  C is regular, our first goal is to show that the h-conformal measure m proven to exist in Theorem 20.1.1 is atomless and that Hhs (J (f )) = 0 whenever h < 2. The regularity assumption will be needed only from Lemma 20.3.9 onward. We will now consider for f almost t-conformal measures ν with t ≥ 1. The notion of upper estimability introduced in Definition 10.4.2 is considered with respect to the Euclidean almost t-conformal measure νe . Recall that l = l(f ) ≥ 1 is the integer produced in Lemma 18.2.15 and put # $ Rl (f ) := inf R(f j ,c) : c ∈ Crit(f ) and 1 ≤ j ≤ l(f ) $ # = min R(f j ,c) : c ∈ Crit(f ) ∩ R and 1 ≤ j ≤ l(f ) < ∞ (20.28) and

# $ Al (f ) := sup A(f j ,c) : c ∈ Crit(f ) and 1 ≤ j ≤ l(f ) $ (20.29) # = max A(f j ,c) : c ∈ Crit(f ) ∩ R and 1 ≤ j ≤ l(f ) ,

20 Sullivan h-Conformal Measures

319

where the numbers R(f j ,c) and A(f j ,c) are defined in Section 8.4. Since O+ (f (Critc (J (f )))) is a compact f -invariant subset of C (so disjoint from f −1 (∞)) and since PC0c (f ) = O+ (Critc (J (f ))) = Critc (J (f )) ∪ O+ (f (Critc (J (f )))), we have the following straightforward but useful fact. Lemma 20.3.5 If f : C −→  C is a compactly nonrecurrent elliptic function, 0 then the set PCc (f ) is f -pseudo-compact. Recall, for the purpose of proving the next two lemma, that the sequence p p {Cri (f )}i=1 was defined inductively by (18.33) and the sequence {Si (f )}i=1 was defined by (18.35), while the number p, here and in what follow in this section, comes from Lemma 18.2.11(c). Since the number Nf of equivalence classes of the relation ∼f between critical points of an elliptic function f : C −→  C is finite, looking at Lemmas 18.2.15 and 17.6.6, the following lemma follows immediately from Lemma 10.4.10. Lemma 20.3.6 Let f : C −→  C be a compactly nonrecurrent elliptic function. (u) Fix an integer 1 ≤ i ≤ p − 1. If Ri > 0 is a positive constant and t −→ (u) Ct,i ∈ (0,∞), t ∈ [1,∞), is a continuous function such that all points z ∈ (u)

PC0c (f )i are (r,Ct,i )-t-u.e. with respect to some Euclidean almost t-conformal (u)

measure νe (with t ≥ 1) for all 0 < r ≤ Ri , then there exists a continuous (u) function t −→ C˜ t,i > 0, t ∈ [1,∞), such that all critical points c ∈ Cri+1 (f ) (u) (u) are (r, C˜ )-t-u.e. with respect to the measure νe for all 0 < r ≤ A−1 R . t,i

l

i

In the above lemma, the superscript u stands for “upper.” In the lemma below, it has the same connotation. The number u is also used to denote the value of the function u(λ,r,z) defined in Proposition 20.2.1. This should not cause any confusion. Lemma 20.3.7 Let f : C −→  C be a compactly nonrecurrent elliptic (u) function. Fix an integer 1 ≤ i ≤ p. If Ri,1 > 0 is a positive constant and (u)

[1, + ∞) −→ Ct,i,1 ∈ (0,∞), is a continuous function such that all critical (u)

points c ∈ Si (f ) are (r,Ct,i,1 )-t-u.e. with respect to some Euclidean almost (u) t-conformal measure νe (with t ≥ 1) for all 0 < r ≤ Ri,1 , then there exist a (u) (u) > 0, and R˜ > 0 such that all continuous function [1, + ∞) −→ C˜ t,i,1

i,1

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Part VI Fractal Geometry, Stochastic Properties, and Rigidity

points z ∈ PC0c (f )i are (r, C˜ t,i,1 )-t-u.e. with respect to the measure νe (with t ≥ 1) for all 0 < r ≤ R˜ (u) . (u)

i,1

Proof

Put ε := 2K(KA2 )2Nf ,

where A ≥ 1 was defined in (18.20). Then fix λ > 0 so large that

 (u) ε < λ min 1,τ −1,θ −1 τ −1 min{ρ,Ri,1 /2} ,

(20.30)

where ρ was defined in (18.38). We shall show that one can take % % & & (u) (u) −1  −1 ˜ Ri,1 := min τ θλ min 1,f  0 ,Ri,1 ,1 PCc (f )i

and (u) (u) C˜ t,i,1 := max{K 2 2t Ct,i,1,K 2t Bt },

where Bt = Bt (λ,η) > 0 comes from Proposition 20.2.1 with η = 2K. (u) and z ∈ PC0c (f )i . If z ∈ Crit(J (f )), then z ∈ Consider 0 < r ≤ R˜ i,1 Critc (J (f )) and z ∈ Si (f ), and we are, therefore, done. Thus, we may assume that z ∈ / Crit(J (f )). Let s = s(λ,ε,r,z). By the definition of ε, 2Kr|(f s ) (z)| = (KA2 )−1 2−Nf εr|(f s ) (z)|.

(20.31)

Suppose first that u(λ,r,z) is well defined and s = u(λ,r,z). Then, by item (20.3) in Proposition 20.2.1 or by item (20.5) in Proposition 20.2.1, we see that the point f s (z) is (2Kr|(f s ) (z)|,Bt )-t-u.e. Using (20.31), we obtain, from item (20.16) in Proposition 20.2.2 and Lemma 10.4.7, that the point z is (r,K 2h Bt )-t-u.e.. If either u is not defined or s < u(λ,r,z), then, in view of item (20.16) in Proposition 20.2.2, there exists a critical point c ∈ Critc (J (f )) such that |f s (z) − c| ≤ εr|(f s ) (z)|. Since s ≤ u, by Proposition 20.2.1 and (20.30), we get that 2Kr|(f s ) (z)| ≤ εr|(f s ) (z)| < min{ρ,Ri,1 /2}. (u)

(20.32)

Since z ∈ PC0c (f )i , this implies that c ∈ Si (f ). Therefore, using (20.32), the assumptions of Lemma 20.3.7, and (20.31) and then applying item (20.16) in Proposition 20.2.2 (remember that, by Lemma 20.3.5, the set PC0c (f ) is (u) f -pseudo-compact) and Lemma 10.4.7, we conclude that z is (r,K 2 2t Ct,i,1 )t-u.e. The proof is complete. 

20 Sullivan h-Conformal Measures

321

Given an arbitrary integer k ≥ 1, recall that, for any pole b of f k , the number qb denotes its multiplicity and Bbk (R) is the connected component ∗ (R)) containing b. We have proved Lemma 4.21 in [KU4] with of f −k (B∞ no constraints imposed on the elliptic function f . In fact, the following more general lemma is true (with the same proof), where f −1 is replaced by f −k . Lemma 20.3.8 Let f : C −→  C be an elliptic function. Fix an integer k ≥ 1 and a point b ∈ f −k (∞). b such that If νe is a Euclidean almost t-conformal measure with t > q2q b +1 νe (b) = 0, and if m is the h-conformal measure proven to exist in Theorem 20.1.1, then νe (Bbk (R))  R

2−

qb +1 qb t

and me (B(b,r)) $ r (qb +1)h−2qb for all sufficiently small radii 0 < r ≤ 1. Proof It follows from Lemma 17.6.6 that me ({z ∈ C : R ≤ |z| < 2R})  R 2 and νe ({z ∈ C : R ≤ |z| < 2R})  R 2 for all R > 0 large enough. It, therefore, follows from (17.13) that q +1   − b h me Bbk (R)\Bbk (2R)  R 2 R qb

(20.33)

q +1   − b t νe Bbk (R)\Bbk (2R)  R 2 R qb .

(20.34)

and

Now fix r > 0 so small that R = (r/Lk )−qb is large enough for (20.33) and (20.34) to hold. Using (17.16) and (20.34), we get that ⎞ ⎛  νe (Bbk (R)) = νe ⎝ (Bbk (2j R)\B k (2j +1 R))⎠ b

j ≥0

=

∞ 

νe (Bbk (2j R)\Bbk (2j +1 R))

j =0 ∞ q +1  − b t  (2j R)2 (2j R) qb j =0

=R

2−

qb +1 qb t

∞  j =0

  q +1 j 2− bq t

2

b

322

Part VI Fractal Geometry, Stochastic Properties, and Rigidity   q +1 qb 2− bq t

= Lk

b

r

(qb +1)t−2qb

∞ 

  q +1 j 2− bq t

2

b

j =0

r

(qb +1)t−2qb

,

(20.35)

t > 2. We are done with the where the last comparability sign holds since qbq+1 b first part of our lemma. b because of Now replace νe by me and t by h (which is greater than q2q b +1 Theorem 17.3.1) in the above formula. In this case, the “” sign in (20.35) can, by virtue of (20.33), be replaced by the comparability sign “.” Since the first equality sign in (20.35) becomes “≥” (we have not ruled out the possibility that me (b) > 0 yet) and me (B(b,r)) ≥ me (Bb (R)), we are also done in this case.  From now onward, in all our considerations in this chapter, we assume that f : C −→  C to be a compactly nonrecurrent regular elliptic function. We shall now prove the following. Lemma 20.3.9 If f : C −→  C is a compactly nonrecurrent regular elliptic function, then the h-conformal measure mh , for f : J (f ) −→ J (f ) ∪ {∞}, proven to exist in Theorem 20.1.1, is atomless. Proof By induction on i = 0,1, . . . ,p (remember that p comes from Lemma 18.2.11(c)), it follows from Lemma 20.3.7 (this lemma provides the base of induction as S0 (f ) = ∅ and, simultaneously, contributes to the inductive step), Lemma 20.3.6, and Lemma 18.2.14 that there exists a continuous function t −→ Ct ∈ (0,∞), t ∈ [1,∞), such that if ν is an arbitrary almost t-conformal measure on J (f ), then νe (B(x,r)) ≤ Ct r t

(20.36)

for all x ∈ PC0c (f ) and all r ≤ r0 for some r0 > 0 sufficiently small. Consider now the almost sj -conformal measures msj := mVj , j ≥ 1, and their Euclidean versions   mej := mVj e, both introduced at the beginning of the proof of Theorem 17.6.7, where the numbers sj = s(Vj ) also come from the proof of Theorem 17.6.7. Letting j → ∞ and recalling that, according to Theorem 20.1.1, mh,s is a weak limit, coming from Claim 2◦ stated in the proof of Theorem 17.6.7, of measures msj , j ≥ 1, we see from (20.36) that

20 Sullivan h-Conformal Measures

323

mh,e (B(x,r)) ≤ Ch r h

(20.37)

for all x ∈ PC0c (f ) and all r ≤ r0 . It now follows from Lemma 20.3.4 that lim sup r"0

ms (B(x,r)) ≤ 2h Ch rh

(20.38)

for all x ∈ PC0c (f ). In particular, mh,s (Critc (f )) = 0. (20.39)   b Now fix k ≥ 1, b ∈ f −k (∞), and u ∈ q2q ,h . Consider all integers j ≥ 1 b +1 e e −k so large that sj ≥ u. Since mj (f (∞)) ≤ mj (f −k (Vj )) = 0, it follows from Lemma 20.3.8 that mej (Bbk (R))  R

2−

qb +1 qb sj

≤R

2−

qb +1 qb u

.

Hence, mh,e (b) = 0. Since mh,s and mh,e are equivalent on C, this gives mh,s (b) = 0. Consequently, ⎞ ⎛  (20.40) f −n (∞)⎠ = 0. mh,s ⎝ n≥1

In particular, mh,s (Critp (f )) = 0.

(20.41)

We now move on to dealing with the set Crit∞ (f ). Since sj % h and since h− < h (h− was defined in (20.25)), disregarding finitely many j s, we may assume without loss of generality that sj > h−

(20.42)

for all j ≥ 1. Fix c ∈ Crit∞ (f ). Fix also j ≥ 1 and put t := sj . Since limn→∞ f n (c) = ∞, there exists an integer k ≥ 1 such that qbn ≤ qc (where bn ∈ f −1 (∞), defined in (18.48), is near f n (c), and qc was defined in (18.49)) and

 |f n (c)| > max 1,2Diste (0,f (Crit(f ))) (20.43) for all n ≥ k. We may need in the course of the proof the integer k ≥ 1 to be appropriately bigger. Put a := f k (c). We recall that κc was defined in (20.26). We shall prove the following.

324

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Claim 1◦ . There exists a constant c1 ≥ 1, independent of j , such that mej (Be (a,r)) ≤ c1 r κc for all r > 0 small enough independently of j . Proof Put q = qc . In view of (20.43) and Theorem 17.1.8, for every n ≥ 1, there exists a unique holomorphic inverse branch  1 n −1 n fn : Be f (a), |f (a)| −→ C 2 of f sending f n (a) to f n−1 (a). Then, by Lemma 8.3.13 and (18.48), we have, for every n ≥ k, that    1 K ⊂ Be f n−1 (a), |f n (a)| · |f  (f n−1 (a))|−1 fn−1 Be f n (a), |f n (a)| 4 4  − q+1 ⊆ Be f n−1 (a),C|f n (a)| · |f n (a)| q   −1 = Be f n−1 (a),C|f n (a)| q  1 ⊆ Be f n−1 (a), |f n−1 (a)| 2 with some constant C > 0, where the last inclusion was written assuming that − q1

|f n−1 (a)| ≥ 2C|f n (a)| So, the composition fa−n

=

f1−1

◦ f2−1

, which holds if the integer k is taken large enough.

◦ · · · ◦ fn−1 :

 1 n n Be f (a), |f (a)| −→ C, 4

sending f n (a) to a, is well defined and forms a holomorphic branch of f −n . 1 |a| and let n + 1 ≥ 1 be the least integer such that Take 0 < r < 16 r|(f n+1 ) (a)| ≥

1 n+1 (a)|. |f 16 qb +1

Such an integer exists since |f  (z)|  |f (z)| qb if z is near a pole b. By 1 |a|, we have that definition n ≥ 0 and since r < 16 r|(f n ) (a)|
h− (see (20.42)) and qn ≤ qc , it follows from (20.27) that   qn + 1 t − κc > 1. 2 − κc qn Hence, t−κc qn +1 qn

|f n (a)| < |f n (a)| 2−κc

t−κc

 |f  (f n−1 (a))| 2−κc

t−κc

t−2

 |(f n ) (a)| 2−κc = |(f n ) (a)||(f n ) (a)| 2−κc . Combining this and the Case 1 assumption, we get that r
0 was defined in (18.20) and qmin is the minimal order of all critical points and poles. Put α := 32A

qmin +1 qmin

. Then

Be (f n (a),4r|(f n ) (a)|) ⊆ Be (bn,(4 + c)r|(f n ) (a)|) ⊆ Be (bn,(4 + α)δ(f −1 (∞))) and it follows from Lemma 20.3.8 that mej (Be (f n (a),4r|(f n ) (a)|))  (4r|(f n ) (a)|)(qn +1)t−2qn .

326

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Thus, mej (Be (a,r)) ≤ K t |(f n ) (a)|−t (4r|(f n ) (a)|)(qn +1)t−2qn  r (qn +1)t−2qn |(f n ) (a)|(t−2)qn ≤ r (qn +1)t−2qn . But, as qn ≤ qc and t > h− , it follows from (20.26) that (qn + 1)t − 2qn ≥ (qn + 1)t − 2qc > κc ; therefore, mej (B(a,r)) ≤ r κc . It remains for us to consider the following. Case 3. r|(f n ) (a)| < But then

1 − 32 A

qmin +1 qmin

|f n (a) − bn |.

r|(f n+1 ) (a)| = r|(f n ) (a)||f  (f n (a))| +1 qn +1 1 − qmin < A qmin |f n (a) − bn |(A|f n+1 (a)|) qn 32 +1 1 1 − qmin +1 ≤ A qmin A qn |f n+1 (a)| 32 1 n+1 (a)| |f ≤ 32 1 n+1 (a)| |f ≤ 16 contrary to the definition of n. So, Claim 1◦ is proved.



The last step of our proof is to demonstrate the following. Claim 2◦ . There exist c2 ≥ and R > 0, both independent of j , such that mej (Be (c,r)) ≤ c2 r pc κc +h(1−pc ) for all j ≥ 1 and for all r ≤ R, where pc is the order of critical point c of the map f k . Proof

Let p := pc ≥ 2. There exists R > 0 so small that f k (Be (c),R) ⊆ Be (f k (c),2−4 |f k (c)|)

and that there exists M ≥ 1 such that M −1 |z − c|p ≤ |f k (z) − f k (c)| ≤ M|z − c|p

20 Sullivan h-Conformal Measures

327

and M −1 |z − c|p−1 ≤ |(f k ) (z)| ≤ M|z − c|p−1 for all z ∈ Be (c,R). Thus, for all k ≥ 0 and all r ≤ R,   f k (A(c;2−(l+1) r,2−l r)) ⊆ A f k (c);M −1 r p 2−p(l+1),Mr p 2−pl . k Since the map f|B is p-to-one, using almost conformality of the measure e (c,R) e mj and the right-hand side of (20.26), we get that    mej A f k (c); M −1 r p 2−p(l+1),Mr p 2−pl   1 ≥ M −h (2−(l+1) r)t (p−1) mej A(c;2−(l+1) r,2−l r) p   ≥ p−1 M −h (2−(l+1) r)h(p−1) mej A(c;2−(l+1) r,2−l r) .

Applying Claim 1◦ , we, therefore, get that mej (Be ( c, r)) =

∞ 

mej (A(c,2−(l+1) r,2−l r))

l=0

≤ pM h r h(1−p)

∞ 

   2h(p−1)(l+1) mej A f k (c);M −1 r p 2−p(l+1),Mr p 2−pl

l=0

≤ pM h c1 2h(p−1) r h(1−p)

∞ 

2h(p−1)l (Mr p 2−pl )κc

p=0

= p2h(p−1) c1 M h+κc r h(1−p)+pκc = p2

h(p−1)

c1 M

h+κc

∞ 

2(h(p−1)−pκc )l

l=0 h(p−1)−pκc −1 pκc +h(1−p)

(1 − 2

)

r

,

where writing the last equality sign we used the fact that pκc + h(1 − p) > 0  equivalent to the left-hand side of (20.26). Claim 2◦ is, thus, proved. Repeating again that pκc + h(1 − p) > 0, Claim 2◦ implies that mh (c) = 0. So, mh,s (Crit∞ (f )) = 0.

(20.47)

Along with (20.39)–(20.41) and Theorem 20.1.1, this shows that the measure mh is atomless and the proof of Lemma 20.3.9 is complete.  The argument from the beginning of the proof of this lemma, based on Lemmas 20.3.7 and 20.3.6, gives the following,

328

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Lemma 20.3.10 If f : C −→  C is a compactly nonrecurrent regular elliptic 0 function, then the set PCc (f ) is uniformly h-upper estimable with respect to the measure mh constructed in Theorem 20.1.1. Denote by Tr(f ) ⊆ J (f ) the set of all transitive points of f , i.e., the set of points in J (f ) such that ω(z) = J (f ). The main and last result of this section is the following. Theorem 20.3.11 If f : C −→  C is a compactly nonrecurrent regular elliptic function, then (1)     DDh (J (f )) = DDχ (J (f )) = HD Jer (f ) = HD Jr (f ) = h = HD(J (f )) (20.48) and there exists a unique spherical h-conformal probability measure mh for f : J (f ) −→ J (f ) ∪ {∞}. This measure is atomless. (2) The spherical h-conformal measure mh is weakly metrically exact, in particular ergodic and conservative. (3) All other conformal measures are purely atomic, supported on Sing− (f ) with exponents larger than h. (4) mh (Tr(f )) = 1. In what follows, the h-conformal measure m, either spherical ms or its Euclidean version me , will be denoted by mh . Following the convention of this book, the spherical and Euclidean versions of mh will be, respectively, denoted by mh,s and mh,e . Proof Formula (20.48) is a part of Theorem 20.1.1. In view of Lemma 20.3.9, there exists an atomless h-conformal measure mh for f : J (f ) −→ J (f ) ∪ {∞}. So, the existence part of (1) is done. Continuing the proof, let R > 0 be so large that the ball Be (0,R) contains a fundamental domain of F . For every w ∈ C, fix w  ∈ B(0,R) such that w ∼f w  . Suppose that νe is an arbitrary Euclidean t-conformal measure for f and some t ≥ 0. By Lemma 17.6.4, t ≥ h. For each z ∈ J (f )\Sing− (f ), let (xk (z))∞ k=1 be the sequence produced in Proposition 18.3.3. Define, for every l ≥ 1,

 Zl := z ∈ J (f )\Sing− (f ) : η(z) ≥ 1/ l .

20 Sullivan h-Conformal Measures

329

Fix l ≥ 1 and assume that z ∈ Zl . Disregarding finitely many terms if needed, assume without loss of generality that   n f k (z) − xk (z) < 1 (20.49) 32Kl for all ≥ 1. Then, for each k ≥ 1,   1 1 nk B f (z), ⊆ B xk (z), 2l l and the holomorphic inverse branch  1 fz−nk : Be f nk (z), −→ C 2l produced in Proposition 18.3.4, sending f nk (z) to z, is well defined. Using conformality of the measure ν along with the 14 -Koebe Theorem (Theorem 8.3.3), the Koebe Distortion Theorem I, Euclidean version (Theorem 8.3.8), and Proposition 17.6.2, we get the following:      1 1 νe Be z, |(f nk ) (z)|−1 ≤ νe fz−nk Be f nk (z), 16l 4l   1 ≤ K t |(f nk ) (z)|−t νe Be f nk (z), 4l   1 ≤ K t |(f nk ) (z)|−t νe Be xk (z), 2l   1 = K t |(f nk ) (z)|−t νe Be xk (z), 2l ≤ K t νe (Be (0,R + 1))|(f nk ) (z)|−t . (20.50) Likewise, using Lemma 8.3.13, the Koebe Distortion Theorem I, Euclidean version (Theorem 8.3.8), and Corollary 17.6.3, we get the following:      1 1 nk  −1 −nk nk νe Be z, |(f ) (z)| ≥ νe fz Be f (z), 16l 16Kl   1 −t nk  −t nk ≥ K |(f ) (z)| νe Be f (z), 16Kl   1 −t nk  −t ≥ K |(f ) (z)| νe Be xk (z), 32Kl   1 −t nk  −t  = K |(f ) (z)| νe Be xk (z), 32Kl  1 ≥ K t M t, |(f nk ) (z)|−t , 32Kl (20.51)

330

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

  1 where the constant M t, 32Kl comes from Corollary 17.6.3. Summarizing (20.50) and (20.51), we obtain that   1 |(f nk ) (z)|−1 B(νe,l)−1 |(f nk ) (z)|−t ≤ νe Be z, 16l ≤ B(νe,l)|(f nk ) (z)|−t ,

(20.52)

where B(νe,l) ≥ 1 is some constant depending only on R, νe , and l. Fix now E, an arbitrary bounded Borel set contained in Zl . Since mh,e is outer regular, for every x ∈ E, there exists a radius r(x) > 0 of the form from (20.52) such that

  Be (x,r(x))\E < ε. (20.53) mh,e x∈E

Now, by the Besicovitch Covering Theorem, i.e., Theorem 1.3.5, we can choose a countable subcover {Be (xi ,r(xi ))}∞ i=1, r(xi ) ≤ ε, from the cover {Be (x,r(x))}x∈E of E, of multiplicity bounded by some constant C ≥ 1, independent of the cover. Therefore, by (20.52) and (20.53), we obtain that νe (E) ≤

∞ 

νe (Be (xi ,r(xi ))) ≤ B(νh,e,l)

i=1

∞ 

r(xi )t

i=1

≤ B(νe,l)B(mh,e,l)

∞ 

r(xi )t−h me,h (Be (xi ,r(xi )))

i=1

≤ B(νe,l)B(mh,e,l)Cεt−h mh,e

∞ 



(20.54)

Be (xi ,r(xi ))

i=1

≤ CB(νe,l)B(mh,e,l)εt−h (ε + mh,e (E)). In the case when t > h, letting ε " 0, we obtain that νe (Zl ) = 0. Since J (f )\Sing− (f ) =

∞ 

Zl ,

l=1

we, therefore, get that νe (J (f )\ ∪ Sing− (f )) = 0, which means that νe (Sing− (f )) = 1. Thus, item (3) of our theorem is proved.

20 Sullivan h-Conformal Measures

331

Suppose now that t = h. Then, letting ε " 0, (20.54) takes on the form νe (E) ≤ CB(νe,l)B(mh,e,l)mh,e (E).

(20.55)

Since this holds for every integer l ≥ 1, we, thus, conclude that νe |J (f )\Sing− (f ) ≺ mh,e |J (f )\Sing− (f )  mh,s |J (f )\Sing− (f ) . Reversing the roles of mh,e and νe , we infer that νe |J (f )\Sing− (f )  mh,s |J (f )\Sing− (f ) .

(20.56)

Suppose that νe (Sing− (f )) > 0. Then there exists y ∈ Crit(J (f )) ∪ (f ) ∪ f −1 (∞) such that νs (y) > 0. But then 

|(f n(ξ ) ) (ξ )|−h s < +∞,

ξ ∈y −

 where y − = n≥0 f −n (y) and, for every ξ ∈ y − , n(ξ ) is the least integer n ≥ 0 such that f n (ξ ) = y. Hence, " n(ξ ) ) (ξ )|−h δ ξ s ξ ∈y − |(f νy := " −h n(ξ )  ) (ξ )|s ξ ∈y − |(f is a spherical h-conformal measure supported on y − ⊆ Sing− (f ). This contradicts the, already proven (see (20.56)), fact that the measures νy |J (f )\Sing− (f ) and mh,s |J (f )\Sing− (f ) are equivalent and mh,s (J (f )\Sing− (f )) = 1. Thus, νe and mh,s are equivalent. Let us now prove that any probability spherical h-conformal measure νs is ergodic. Indeed, suppose, to the contrary, that f −1 (G) = G for some Borel set G ⊆ J (f ) with 0 < νs (G) < 1. But then the two conditional measures νG and νJ (f )\G νG (B) :=

νs (B ∩ G) νs (B ∩ (J (f )\G)) and νJ (f )\G (B) := νs (G) νs (J (f )\G)

would be h-conformal and mutually singular; a contradiction. If now νs is again an arbitrary probability spherical h-conformal measure, then, by a simple computation based on the definition of conformal measures, we see that the Radon–Nikodym derivative φ := dνs /dmh,s is constant on grand orbits of f . Therefore, by ergodicity of mh,s , we conclude that φ is constant mh,s -a.e. As both mh,s and νs are probability measures, this implies that φ = 1 a.e.; hence, νs = mh,s . Thus, item (1) of our theorem is established.

332

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Let us now show that the probability spherical h-conformal measure mh,s is conservative. We shall prove first that E, any forward invariant (f (E) ⊆ E) Borel subset of J (f ), is of measure either 0 or 1. Indeed, suppose to the contrary that 0 < mh,s (E) < 1. Let Eˆ := f + E = {w + y : w ∈ f , y ∈ E}. Then the set Eˆ is f -translation invariant, i.e., w + Eˆ = Eˆ

(20.57)

for all w ∈ f . Furthermore, ˆ mh,s (E) ˆ > 0, E ⊆ E, and ˆ = f (E) ⊆ E ⊆ E. ˆ f (E) Since mh,s (E) < 1 and since f maps the sets of measure mh,s equal to zero into sets of measure mh,s equal to zero, it follows from this that ˆ < 1. mh,s (E) Since mh,s (Sing− (f )) = 0, in order to get a contradiction, it suffices to show that − ˆ mh,s (E\Sing (f )) = 0.

Fix an arbitrary point x ∈ J (f ) and an arbitrary radius R > 0. Seeking contradiction, suppose that ˆ = 0. mh,e (Be (x,R)\E) Then also ˆ = 0. mh,s (Be (x,R)\E) By conformality of mh,s , we have that mh,s (f (Y )) = 0 for all Borel sets Y ⊆ C such that mh,s (Y ) = 0. Hence, also using the fact that ˆ ⊇ f n (Be (x,R))\f n (E), ˆ f n (Be (x,R)\E)

(20.58)

20 Sullivan h-Conformal Measures

333

we get that     ˆ ≥ mh,s f n (Be (x,R))\f n (E) ˆ 0 = mh,s f n (Be (x,R)\E)     ˆ ≥ mh,s f n (Be (x,R))\Eˆ ≥ mh,s f n (Be (x,R)) − mh,s (E)

(20.59)

for all n ≥ 0. By virtue of Proposition 17.2.6, there exists an integer l ≥ 1 such that f l (Be (x,R)) =  C. In particular,   mh,s f l (Be (x,R)) = 1. ˆ which is a contradiction. ConseThen (20.59) implies that 0 ≥ 1 − mh,s (E), quently, ˆ > 0. mh,e (Be (x,R)\E)

(20.60)

Denote by Z the Borel set of all points z ∈ E\(I∞ (f ) ∪ Sing− (f )) such that   ˆ ∞ (f ) ∪ Sing− (f ))) mh,e B(z,r) ∩ (E\(I = 1. (20.61) lim r→0 mh,e (B(z,r)) In view of the Lebesgue Density Theorem, i.e., of Theorem 1.3.7, we have that ˆ Since mh,s (E) > 0, there exists at least one point z ∈ Z. mh,s (Z) = mh,s (E). Since z ∈ J (f )\(I∞ (f ) ∪ Sing− (f )), Proposition 18.3.3 applies. Let (xj (z))∞ j =1 , η(z) > 0, and an increasing be given by this proposition. Put sequence (nj )∞ j =1 δ = η(z)/8. It then follows from (20.60) and Proposition 18.3.3 that, for every j ≥ 1 large enough, we have that   (20.62) mh,e Be (xj (z),δ)\Eˆ > 0. Therefore, as f −1 (J (f )\E) ⊆ J (f )\E, the standard application of Theorem 8.3.8 and Lemma 10.4.7 shows that lim sup r→0

ˆ mh,e (B(z,r)\E) > 0, mh,e (B(z,r))

(20.63)

which contradicts (20.61). Thus, either mh,s (E) = 0 or mh,s (E) = 1.

(20.64)

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Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Now conservativity is straightforward. One needs to prove that, for every Borel set B ⊆ J (f ) with mh,s (B) > 0, one has mh,s (G) = 0, where ⎫ ⎧ ⎬ ⎨  11B (f n (x)) < +∞ . G := x ∈ J (f ) : ⎭ ⎩ n≥0

Indeed, suppose that m(G) > 0. For all n ≥ 0, let ⎧ ⎫ ⎨ ⎬  11B (f n (x)) = 0 Gn := x ∈ J (f ) : ⎩ ⎭ k≥n

 = x ∈ J (f ) : f k (x) ∈ / B for all k ≥ n . Since G=



Gn,

n≥0

there exists k ≥ 0 such that mh,s (Gk ) > 0. Since all the sets Gn are forward invariant, we get from (20.64) that mh (Gk ) = 1. But, on the other hand, all the sets f −n (B), n ≥ k, are of positive measure and are disjoint from Gk . This contradiction finishes the proof of conservativity of mh,s . Item (2) is established. Because of (2) and since supp(mh,s ) = J (f ), we have that mh,s (Tr(f )) = 1, i.e., item (4). The proof of Theorem 20.3.11 is complete. 

21 Hausdorff and Packing Measures of Compactly Nonrecurrent Regular Elliptic Functions

From now on, throughout this chapter, and, in fact, throughout the entire book, Hte stands for the t-dimensional Hausdorff measure on C with respect to the Euclidean metric, whereas Hts refers to its spherical counterpart. The same convention is applied to the packing measures te and ts . Note that the measures Hte and Hts as well as te and ts are equivalent to the Radon– Nikodym derivative bounded away from zero and ∞ on compact subsets of C. In particular, the Hausdorff dimension of any subset A of C has the same value no matter whether calculated with respect to the Euclidean or spherical metric; it will be denoted in what follows simply by HD(A). If Ht or t is endowed with neither the subscript “e” nor the subscript “s,” then it refers simultaneously to both the Euclidean as well as the spherical measures. As in the previous chapters, we keep h = HD(J (f )). The goal of this chapter can be viewed as two-fold. The first is to provide a geometrical characterization of the h-conformal measure mh , which, with the absence of parabolic points, turns out to be a normalized packing measure, and the second is to give a complete description of geometric, Hausdorff, and packing measures of the Julia sets J (f ). All of this is contained in the following theorem. Theorem 21.0.1 Let f : C −→  C be a compactly nonrecurrent regular elliptic function. If h = HD(J (f )) = 2, then J (f ) = C. If h < 2, then (a) Hhs (J (f )) = 0. (b) hs (J (f )) > 0. (c) hs (J (f )) < +∞ if and only if (f ) = ∅.

335

336

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

In either Case (c) or if HD(J (f )) = 2, the unique spherical h-conformal measure mh coincides with the normalized packing measure hs /hs (J (f )) restricted to the Julia set J (f ). This theorem has an interesting story: for expanding rational functions f , we always have, essentially because of [Bow2], that 0 < Hh (J (f )),h (J (f )) < +∞, and these two measures coincided up to a multiplicative constant. Their probability version is then the unique h-conformal measure. If f is still a rational function but parabolic or, more generally, nonrecurrent, then (see [DU5] and [U3], respectively): (a) Hh (J (f )) < +∞ and h (J (f )) > 0. (b) Hh (J (f )) = 0 if and only if h < 1 and (f ) = ∅. (c) h (J (f )) = +∞ if and only if h > 1 and (T ) = ∅. So, the descriptions of Hausdorff and packing measures in the cases of both nonrecurrent rational functions and compactly nonrecurrent regular elliptic functions coincide except that, in the latter case, Hh (J (f )) ≤ 1 never holds. For other transcendental meromorphic and entire functions, even hyperbolic (expanding), the situation is generally less clear and varies from case to case; see, for example, [UZ1] and [MyU2]. As an immediate consequence of Theorem 21.0.1, we get the following. Corollary 21.0.2 Let f : C −→  C be a compactly nonrecurrent regular elliptic function. If (f ) = ∅, then the Euclidean h-dimensional packing measure he is finite on each bounded subset of J (f ).

21.1 Hausdorff Measures We start with the proof of the first part of Theorem 21.0.1. Our first preparatory result is the following. Lemma 21.1.1 If f : C −→  C is a compactly nonrecurrent elliptic function, then ∞  f −j (∞)\O+ (Crit(f )) = ∅. j =1

Proof

Seeking contradiction, suppose that f −1 (∞) ⊆ O+ (Crit(f )).

21 Hausdorff and Packing Measures

337

So, for each b ∈ f −1 (∞), there exists cb ∈ Crit(f ) ∩ J (f ) such that b ∈ O+ (cb ). It then follows from Definition 18.2.1 that its items (1) and (3) are ruled out for cb , whence item (2) must hold. We then conclude that b ∈ O+ (f (cb )).

(21.1)

Since then O+ (f (cb )) is a finite set and since f (Crit(f )) is also a finite set, we conclude that  O+ (f (cb )) b∈f −1 (∞)

is a finite set. But, (21.1) implies that 

f −1 (∞) ⊆

O+ (f (cb )).

b∈f −1 (∞)

Since f −1 (∞) is infinite, we arrived at a contradiction, and we are, thus, done.  Proof of part (a) of Theorem 21.0.1 b ∈ f −1 (∞)\O+ (Crit(f )), say

By

Lemma

21.1.1,

there

exists

b ∈ f −1 (∞)\O+ (Crit(f )). Hence, there exists κ > 0 such that Be (b,3κ) ∩ O+ (Crit(f )) = ∅.

(21.2)

Consider an arbitrary point z ∈ Tr(f ). Then there exists an infinite increasing sequence {nj }∞ j =0 such that lim f nj (z) = b and |f nj (z) − b| < κ/2

j →∞

(21.3)

for every j ≥ 1. It follows from this, (21.2), and Theorem 17.1.8 that, for every j ≥ 1, there exists a holomorphic inverse branch −nj

fz

: Be (f nj (z),2κ) −→ C

of f nj sending f nj (z) to z. Let mh be the unique h-conformal atomless measure proven to exist in Theorem 20.3.11. Using now Theorem 8.3.8 and Lemmas 8.3.13, 10.4.7, and 20.3.8, we conclude that

338

Part VI Fractal Geometry, Stochastic Properties, and Rigidity    mh,e Be z,2K|(f nj ) (z)|−1 |f nj (z) − b|  −n    ≥ mh,e fz j Be f nj (z),2|f nj (z) − b|    ≥ K −h mh,e Be f nj (z),2|f nj (z) − b| |(f nj ) (z)|−h   ≥ K −h mh,e Be (b,|f nj (z) − b|) |(f nj ) (z)|−h $ |f nj (z) − b|(qb +1)h−2qb |(f nj ) (z)|−h  h = 2K|(f nj ) (z)|−1 |f nj (z) − b| (2K)−h |f nj (z) − b|qb (h−2) .

Since h < 2, using (21.3), this implies that    mh,e Be z,2K|(f nj ) (z)|−1 |f nj (z) − b| mh,e (Be (z,r)) lim ≥ lim = +∞.  h j →∞ r→0 rh 2K|(f nj ) (z)|−1 |f nj (z) − b| Hence, Hhe (Tr(f )) = 0 in view of Theorem 1.6.3(1). Since, by Theorem 20.3.11, mh,e (J (f )\ Tr(f )) = 0, it follows from Lemma 17.6.4 that Hhe (J (f )\Tr(f )) = 0. In conclusion, Hhe (J (f )) = 0, 

which completes the proof.

21.2 Packing Measure I In this section, we shall prove Proposition 21.2.1, stated below. Its item (3) is just item (b) of Theorem 21.0.1, while item (1) contributes toward the last assertion of this theorem. We shall also prove Lemma 21.2.2, which establishes one side of item (c). Proposition 21.2.1 If f : C −→  C is a compactly nonrecurrent regular elliptic function, then (1) The h-conformal measure mh is absolutely continuous with respect to the packing measure h . Moreover, (2) The Radon–Nikodym derivative dms /dhs is uniformly bounded away from infinity. In particular, (3) h (J (f )) > 0.

21 Hausdorff and Packing Measures

339

Proof Since   J (f ) ∩ ω Crit(f )\Crit(J (f )) = (f ), we conclude from Lemma 18.2.6 that there exists y ∈ J (f ) at a positive distance, and denote it by 8η, from O+ (Crit(f )). Fix z ∈ Tr(f ). Then there exists an infinite sequence (nj )∞ j =1 of increasing positive integers such that f nj (z) ∈ Be (y,η) for every j ≥ 1. Hence, Be (f nj (z),4η) ∩ O+ (Crit(f )) = ∅. Consequently,   Comp z,f nj ,4η ∩ Crit(f nj ) = ∅. Hence, it follows from Lemmas 8.3.13 and 10.4.7 that lim inf r→0

mh,e (Be (z,r)) ≤B rh

for some constant B ∈ (0,∞) and all z ∈ Tr(f ). Applying Lemma 20.3.4, we, therefore, get that lim inf r→0

mh,s (Bs (z,r)) ≤ 2h B. rh

Therefore, by Theorem 1.6.4(1), the measure mh,s |Tr(f ) is absolutely continuous with respect to hs |Tr(f ) . Since, by Theorem 20.3.11, mh,s (J (f )\ Tr(f )) = 0, we are done.  Lemma 21.2.2 If f : C −→  C is a compactly nonrecurrent regular elliptic function and (f ) = ∅, then hs (J (f )) = +∞. Proof Fix ξ ∈ (f ). Since the set 

f −n (ξ )

n≥0

is dense in J (f ) and since, by Lemma 18.2.6, ω(Crit(f )) is nowhere dense in J (f ), there exist an integer s ≥ 0, a real number η > 0, and a point ⎞ ⎛  y ∈ f −s (ξ )\Be ⎝ f n (Crit(f )),η⎠ . n≥0

340

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Since, by Theorem 17.3.1, h > 1, it follows from Lemmas 15.4.1 and 10.4.10 (y may happen to be a critical point of f s !) that me (Be (y,r)) = 0. (21.4) r→0 rh Consider now a transitive point z ∈ J (f ), i.e., z ∈ Tr(f ). Then there exists an infinite increasing sequence nj = nj (z) ≥ 1, j ≥ 1, of positive integers such that lim inf

lim |f nj (z) − y| = 0

j →∞

and

rj = |f nj (z) − y| < η/7

for every j ≥ 1. By the choice of y and Theorem 17.1.8, for all j ≥ 1, there exist holomorphic inverse branches −nj

fz

: Be (f nj (z),6rj ) −→ C

of f nj sending f nj (z) to z. So, applying Lemmas 8.3.13 and 10.4.7 with R = 3rj , we conclude from (21.4) that mh,e (Be (z,r)) = 0. r→0 rh Applying Lemma 20.3.4, we conclude that the same formulas remain true with mh,e replaced by mh,s and Be (z,r) by Bs (z,r). Therefore, it follows from Theorems 20.3.11 (mh,s (Tr(f )) = 1) and 1.6.4(1) that hs (J (f )) = +∞. We are done.  lim inf

21.3 Packing Measure II As before, from now on throughout this section, mh denotes the unique atomless h-conformal measure proven to exist in Theorem 20.3.11. Our aim in this section is to show that, in the absence of parabolic periodic points, the h-dimensional Euclidean packing measure is finite on bounded subsets of J (f ) and that hs (J (f )) < +∞. This will complete item (c) of Theorem 21.0.1. Recall that the numbers Rl (f ) and Al (f ) have been defined by (20.28) and (20.29), respectively. p Recall, for the needs of this section, that the sequence {Cri (f )}i=1 was p defined inductively by (18.33) and the sequence {Si (f )}i=1 was defined by (18.35), while the number p, here and in what follows in this section, comes from Lemma 18.2.11(c). Since the number Nf of equivalence classes of the relation ∼f between critical points of an elliptic function f : C −→  C is finite, looking at Lemmas 18.2.15 and 17.6.6, the following lemma follows immediately from Lemma 10.4.11.

21 Hausdorff and Packing Measures

341

Lemma 21.3.1 Let f : C −→  C be a compactly nonrecurrent regular elliptic (l) (l) function. Fix 0 ≤ i ≤ p − 1. If Ci > 0, 0 < Ri ≤ Rl (f )/3, and 0 < σ ≤ 1 (l)

are three real numbers such that all points z ∈ PC0c (f )i are (r,σ,Ci )-h(l) s.l.e. with respect to the measure mh,e , then there exists C˜ i > 0 such that all (l) ˜ C˜ i )-h-s.l.e. with respect to the measure critical points c ∈ Cri+1 (f ) are (r, σ, (l) mh,e for all 0 < r ≤ Al (f )−1 Ri , where σ˜ was defined in Lemma 10.4.11. Let us prove the following. Lemma 21.3.2 Let f : C −→  C be a compactly nonrecurrent regular elliptic (l) function. Suppose that (f ) = ∅. Fix 0 ≤ i ≤ p. Assume that Ci,1 > 0, (l) Ri,1 > 0, and 0 < σ ≤ 1 are three real numbers such that all critical points (l) c ∈ Si (f ) are (r,σ,Ci,1 )-h-s.l.e. with respect to the measure mh,e for all 0 < (l) (l) (l) r ≤ Ri,1 . Then there exist C˜ i,1 > 0, R˜ i,1 > 0 such that all points z ∈ PC0c (f )i (l) are (r,8K 3 A2 2Nf σ, C˜ i,1 )-h-s.l.e. with respect to the measure mh,e for all (l) 0 < r ≤ R˜ i,1 , where A > 0 was defined in (18.20). Proof Recall that, by Lemma 20.3.5, the set PC0c (f ) is f -pseudo-compact. We shall show that this time one can take

 (l) −1 (l) R˜ i,1 := min τ θ min{1,f  −1 and i }λ ,Ri,1,1 h (l)  (l) 2 N C˜ i,1 := 8(KA )2 f Ci,1, where f  i := f   λ > 0 so large that

PC0c (f )i

. Indeed, take ε := 4K(KA2 )2Nf and then choose

$ # (l) /2} . ε < λ min 1,τ −1,θ −1 τ −1 min{ρ,Ri,1

(21.5)

Consider 0 < r ≤ R˜ i,1 and z ∈ PC0c (f )i . If z ∈ Critc (J (f )), then z ∈ Si (f ) / and we are done. Thus, we may assume that z ∈ / Critc (J (f )), then z ∈ Crit(J (f )). Let s = s(λ,ε,r,z). By the definition of ε, (l)

4Kr|(f s ) (z)| = (KA2 )−1 2−Nf εr|(f s ) (z)|.

(21.6)

Suppose first that u(λ,r,z) is well defined and s = u(λ,r,z). Then, by item (20.4) in Proposition 20.2.1, applied with η = K, we see that the point f s (z) is (Kr|(f s ) (z)|,σ/K 2,Wh (σ/K 2 ))-h-s.l.e. Using (21.6), it follows from item (20.16) in Proposition 20.2.2 and Lemma 10.4.8 that the point z is (r,σ,Wh (σ/K 2 ))-h-s.l.e. If either u is not defined or

342

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

s ≤ u(λ,r,z), then, in view of item (20.15) in Proposition 20.2.2, there exists a critical point c ∈ Crit(f ) such that |f s (z) − c| ≤ εr|(f s ) (z)|. Since s ≤ u, by Proposition 20.2.2 and (21.5), we get that (l) |f s (z) − c| ≤ εr|(f s ) (z)| < min{ρ,Ri,1 /2}.

(21.7)

Since z ∈ PC0c (f )i , this implies that c ∈ Si (f ). Therefore, by the assumptions (l) of Lemma 21.3.2 and by (21.7), we conclude that c is (2εr|(f s ) (z)|,σ,Ci,1 )s h-s.l.e. Consequently, in view of Lemma 10.4.4, the point f (z) is (l) (εr|(f s ) (z)|,2σ,2h Ci,1 )-h-s.l.e. So, by Lemma 10.4.5, this point is (l) (Kr|(f s ) (z)|,2σ ε/K,(2εK −1 )h Ci,1 ) − h-s.l.e.

Using now (21.6) and item (20.16) in Proposition 20.2.2 along with the fact that Kε−1 < 1, we have from Lemma 10.4.8 that the point z is (l)  (r,2Kεσ,(2εK −1 )h Ci,1 )-h-s.l.e. The proof is complete. As a fairly straightforward consequence of these two lemmas, we get the following. Lemma 21.3.3 If f : C −→  C is a compactly nonrecurrent regular elliptic function, then, with some R > 0 and some G > 0, each point of PC0c (f ) (in particular, each point of Critc (f )) is (r,1/2,G)-h-s.l.e. with respect to the measure mh,e for every r ∈ [0,R]. Proof Since S0 (f ) = ∅, starting with σ > 0 as small as we wish, it immediately follows from Lemmas 21.3.2, 21.3.1, and 18.2.14 by induction on i = 0,1, . . . ,p that all the points of Si (f ) and PC0c (f )i are (r,1/2,G)-h-s.l.e. with the same G,R > 0 and all r ∈ [0,R]. We are done.  This lemma and Lemma 20.3.8, taken together, yield the following. Lemma 21.3.4 If f : C −→  C is a compactly nonrecurrent regular elliptic function, then every point of the set Crit(J (f )) ∪ f −1 (∞) is h-s.l.e. with respect to the measure mh,e with σ ∈ (0,1) arbitrary. Fix c ∈ Crit∞ (f ). Since limn→∞ f n (c) = ∞, there exists an integer k ≥ 1 such that qbn ≤ qc (where bn ∈ f −1 (∞), defined in (18.48), is near f n (c), and qc was defined in (18.49)) and |f n (c)| > max{1,2Diste (0,f (Crit(f )))}

(21.8)

21 Hausdorff and Packing Measures

343

for all n ≥ k. Put a := f k (c)

(21.9)

(we may need, in the course of the proof, k ≥ 1 to be bigger). We shall prove the following. Lemma 21.3.5 If f : C →  C is a compactly nonrecurrent regular elliptic function and c ∈ Crit∞ (f ), then there exists a constant C1 ≥ 1 such that mh,e (Be (a,r)) ≥ C1−1 r h for all radii r > 0 small enough, where a is defined by (21.9). Proof Put q := qc . In view of (21.8) and Theorem 17.1.8, for every n ≥ 1, there is a well-defined holomorphic inverse branch  1 n −1 n fn : Be f (a), |f (a)| −→ C 2 of f sending f n (a) to f n−1 (a). Let bn ∈ f −1 (∞) be the unique pole (assuming that k = 1 is large enough) such that |f n (a) − bn | ≤ δ(f −1 (∞))  1, where δ(f −1 (∞)) comes from (18.47). Then, by Theorem 8.3.8,    1 K fn−1 Be f n (a), |f n (a)| ⊂ Be f n−1 (a), |f n (a)||f  (f n−1 (a))|−1 4 4  − q+1 ⊆ Be f n−1 (a),C|f n (a)||f n (a)| q   −1 = Be f n−1 (a),C|f n (a)| q  1 n−1 n−1 ⊆ Be f (a), |f (a)| , 4 where C ∈ (0,+∞)0 is a constant and the last inclusion was written assuming that − q1

|f n−1 (a)| ≥ 4c|f n (a)|

,

which we can assume to hold for all n ≥ k if k is large enough. So, the composition  1 fa−n := f1−1 ◦ f2−1 ◦ · · · ◦ fn−1 : Be f n (a), |f n (a)| −→ C, 4

344

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

sending f n (a) to a, is well defined and forms a holomorphic branch of f −n . Take 0 < r < 8K/|a| and let n + 1 ≥ 1 be the least integer such that r|(f n+1 ) (a)| ≥

K n+1 (a)|. |f 8 qb +1

Such an integer exists since |f  (z)|  |f (z)| qb if z is near a pole b. By definition n ≥ 0 and since r < 8K/|a|, we have that r|(f n ) (a)|
1. We are done.

Proposition 21.3.7 Let f : C −→  C be a compactly nonrecurrent regular elliptic function. If HD(J (f )) = 2, then J (f ) = C. Proof Since 2e and S, the two-dimensional Lebesgue measure on C, coincide up to a multiplicative constant, it follows from (already proved) Theorem 21.0.1(b) that if h = 2, then S(J (f )) > 0. So, in order to prove our proposition, it suffices to show that if J (f )  C, then S(J (f )) = 0. So, suppose that J (f ) = C. We want to show that S(J (f )\Sing− (f ))) = 0. For any integer l ≥ 1, let the set Zl have exactly the same meaning as in the proof of Theorem 20.3.11. Since J (f ) is a f -invariant nowhere densesubset  of C, there exists ε > 0 such that, for every y ∈ C, there exists yε ∈ Be y, 2l1 such that  1 \J (f ). (21.13) Be (yε,ε) ⊆ Be y, 2l

Keep the notation from the proof of Theorem 20.3.11. Fix an arbitrary point z ∈ Zl . By Theorem 8.3.8, the 14 -Koebe Theorem (Theorem 8.3.3), and (21.13), we have that     fz−nk (Be (f nk (z) (z),ε)) ⊆ fz−nk Be f nk (z) (z),(2l)−1 \J (f )   ⊆ Be z,K|(f nk ) (z)|−1 (2l)−1 \J (f )

21 Hausdorff and Packing Measures

and fz−nk (Be (f nk (z)ε,ε))

 ⊃ Be

1 fz−nk (f nk (z)ε ), ε|(f nk (z) ) (z)|−1 4

347

.

Therefore, we see that      S Be z,K|(f nk ) (z)|−1 (2l)−1 \J (f ) εl 2    > 0. ≥ 2K S Be z,K|(f nk ) (z)|−1 (2l)−1 So, z is not a Lebesgue density point for the set Zl ; therefore, S(Zl ) = 0. Hence, ∞

∞     − Zl = S(Zl ) = 0. S(J (f )) = S J (f )\Sing (f ) = S l=1

The proof of Theorem 21.3.6 is complete.

l=1



Theorem 21.0.1 is now a logical consequence of Section 21.1, Proposition 21.2.1, Lemma 21.2.2, and Theorem 21.3.6.

22 Conformal Invariant Measures for Compactly Nonrecurrent Regular Elliptic Functions

Throughout this whole chapter, f : C −→  C is assumed to be a compactly nonrecurrent regular elliptic function. This chapter is, in a sense, a dynamical core of our book. Using the fruits of what has been done in all previous chapters, we prove the existence and uniqueness, up to a multiplicative constant, of a σ -finite f -invariant measure μh equivalent to the h-conformal measure mh , which, in turn, was proven to exist in Theorem 20.3.11. We also provide a simple, easy to check, necessary, and sufficient condition for this invariant measure μh to be finite. Furthermore, still significantly based on what has been done in all previous chapters, particularly nice sets, first return map techniques, and Young towers, we provide here a systematic account of ergodic and refined stochastic properties of the dynamical system (f ,μh ) generated by such subclasses of compactly nonrecurrent regular elliptic functions as normal subexpanding elliptic functions of finite character and parabolic elliptic functions. By stochastic properties, we mean here the exponential decay of correlations, the Central Limit Theorem, and the Law of the Iterated Logarithm for subexpanding functions, the Central Limit Theorem for those parabolic elliptic functions for which the invariant measure μh is finite (probabilistic after normalization), and an appropriate version of the Darling–Kac Theorem establishing the strong convergence of weighted Birkhoff averages to Mittag– Leffler distributions for those parabolic elliptic functions for which the invariant measure μh is infinite.

348

22 Conformal Invariant Measures for CNRR Functions

349

22.1 Conformal Invariant Measures for Compactly Nonrecurrent Regular Elliptic Functions: The Existence, Uniqueness, Ergodicity/Conservativity, and Points of Finite Condensation In this section, we deal with σ -finite invariant measures equivalent to the conformal measure mh proven to exist in Theorem 20.3.11. We prove their existence, uniqueness up to a multiplicative constant, and their metric exactness, implying ergodicity and conservativity. We also study at length the points of their finite and infinite condensation, giving the first outlook on the location of such points. In the context of rational functions, the results of such kind were obtained, for example, in [ADU], [DU1], [DU2], and [U4]. In the context of quite general hyperbolic/expanding transcendental meromorphic and entire functions, see, for example, [UZ2], [MyU3], and [MyU4]. The first result of this chapter is the following. Theorem 22.1.1 If f : C −→  C is a compactly nonrecurrent regular elliptic function, then there exists a unique, up to a multiplicative constant, σ -finite f -invariant measure μh that is absolutely continuous with respect to the h-conformal measure mh . In addition, (1) (2) (3) (4)

μh is equivalent to mh , μh is metrically exact, hence ergodic and conservative, and to Theorem 2.4.4, by (2.37)–(2.39), μh is given, according  μh J (f )\Tr(f ) = 0

Proof Let ξ ∈ C be a periodic point of f with some period p ≥ 3. We put   P3 (f ) := O+ f (J (f ) ∩ Crit(f )) ∪ {ξ,f (ξ ), . . . ,f p−1 (ξ )}.   Since, by Proposition 18.2.7, O+ f (J (f ) ∩ Crit(f )) is a forward-invariant nowhere dense subset of J (f ) and since the h-conformal measure mh,s is positive on nonempty open subsets of J (f ), it follows from ergodicity and conservativity of mh,s (see Theorem 20.3.11) that mh,s (PC(f )) = 0. Since mh,s has no atoms (see Theorem 20.3.11), we, therefore, obtain that mh,s (P3 (f )) = 0.

(22.1)

Aiming to apply Theorem 2.4.4, we shall now construct a Marco Martens cover of Definition 2.4.2 for the map f : J (f ) −→ J (f ) ∪ {∞}. For further purposes, our construction will be more involved and more specific than the

350

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

one which would be needed just for the sake of having a Marco Martens cover. Fix an integer q ≥ 1. For every integer k ≥ 1, set   1 ∩ J (f ). (22.2) Zk = Zk (q) := B e (0,qk)\B P3 (f ), qk Obviously, each set Zk is compact. We shall inductively construct the sets (Xˆ n )n≥0 as follows. First, by compactness of Z1 , we can cover it by finitely many sets Xˆ 0, Xˆ 1, . . . , Xˆ n1 , each of which is an open ball centered at a point Z1 with radius equal to 1/4. For the inductive step, fix j ≥ 1 and suppose that the open balls Xˆ0, . . . , Xˆ n1 , Xˆ n1 +1, . . . , Xˆ n1 +n2 , . . . , Xˆ n1 +···+nj have been constructed with the following properties. (a) ∀0≤i≤j Zi ⊆ Xˆ ni−1 +1 ∪ Xˆ ni−1+2 ∪ · · · ∪ Xˆ ni . (b) ∀0≤i≤j the sets Xˆ ni−1 +1, . . . , Xˆ ni are open balls centered at points Zi with radii all equal to 4−i . (c) ∀0≤i 0, then there exists n ≥ 1 such that f n (Be (z,r)) =  C.

22 Conformal Invariant Measures for CNRR Functions

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Let us check condition (4). Indeed, by our construction, we have that 2Xˆ n ⊆ C\P3 (f ) ⊆ C\PC(f ). Hence, it follows from Theorems 17.1.8 and 17.1.6 that all holomorphic inverse branches of f −n for all n ≥ 1 are well defined on 2Xn . In view of Theorem 8.3.12, there exists a constant Kˆ ≥ 1 such that if i ≥ 0 and f∗−n : Xi −→ C is a holomorphic branch of f −n , then, for all x,y ∈ Xi , we have that |(f∗−n )∗ (y)| ˆ ≤ K. |(f∗−n )∗ (x)|

(22.4)

We, therefore, obtain, for all Borel sets A,B ⊆ Xi with mh,s (B) > 0, and all n ≥ 0, that ' |(f −n )∗ |h dmh,s mh,s (f∗−n (A)) ' = A ∗−n ∗ h −n ms (f∗ (B)) A |(f∗ ) | dmh,s ≤

supAk {|(f∗−n )∗ |h }mh,s (A) infAk {|(f∗−n )∗ |h }mh,s (B)

≤ Kˆ h

mh,s (A) . mh,s (B)

Hence, mh,s (f −n (A)) =

 ∗

mh,s (A) mh,s (f∗−n (A)) ≤ Kˆ h mh,s (A)

ˆ h mh,s (A)

=K

mh,s (B)

mh,s (f

−n

 ∗

mh,s (f∗−n (B))

(B)).

So, condition (4) of Definition 2.4.2 is satisfied, and we have shown that (Xi )∞ i=0 is a Marco Martens cover. Since item (2) of Theorem 20.3.11, the map f : (J (f ),mh,s ) −→ (J (f ),mh,s ), is ergodic and conservative, we see that Theorem 2.4.4 applies, and its application yields items (1) and (3). Item (2) of our current theorem also follows from item (2) of Theorem 20.3.11, this time from weak metrical exactness of mh,s and from Propositions 2.2.4 and 2.2.3. Item (4) follows now from item (4) of Theorem 20.3.11. The proof is complete.  We want to record one particular property of the cover (Xn )∞ n=0 , which follows immediately from our construction. Lemma 22.1.2 If f : C −→  C is a compactly nonrecurrent regular elliptic function, then, for every compact set F ⊆ C\P3 (f ), the set {n ≥ 0 : Xn ∩ F = ∅} is finite.

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Part VI Fractal Geometry, Stochastic Properties, and Rigidity

The following lemma was established in Theorem 2.4.4. Lemma 22.1.3 If f : C −→  C is a compactly nonrecurrent regular elliptic function, then, for every n ≥ 0, we have that 0 < μh (Xn ) < ∞. In particular, μh ∈ M∞ f , meaning that it is a σ -finite f -invariant measure. The following lemma immediately follows from the results of Section 5.4. Lemma 22.1.4 If f : C −→  C is a compactly nonrecurrent regular elliptic function, then Jμh (∞) := Jμ (f )(∞), the set of points of infinite condensation of the measure μh , is (1) closed, (2) f (Jμh (∞)) ⊆ Jμh (∞), and (3) μ(Jμh (∞)) = 0. Lemma 22.1.5 If f : C −→  C is a compactly nonrecurrent regular elliptic function, then Jμh (∞) ⊆ O+ (Crit(f )) ∪ {∞}. Proof Since all the sets Xn , n ≥ 0, are open, it follows from Lemma 22.1.3 that Jμh (∞) ⊆ P3 (f ) ∪ {∞}. Replacing ξ, . . . ,f p−1 (ξ ) by another periodic orbit with period ≥ 3 and using uniqueness of the measure μh (Theorem 22.1.1), we, thus, conclude that Jμh (∞) ⊆ O+ (Crit(f )) ∪ {∞}. The proof is finished.  Now we further investigate in greater detail the structure of the set Jμh (∞) of points of infinite condensation of the measure μh . Fix w ∈ J (f )\Be ((f ),θ (f )), where θ (f ) > 0 comes from (18.19), and an open Jordan domain Q ⊆ Be (w,2γf ), where γf is defined by (18.22). A sequence {Qn }∞ n=0 of connected components of inverse images of f −n (Q), n ≥ 1, is called w-nested if and only if f (Qn+1 ) = Qn for all integers n ≥ 0. The lemma below follows immediately from Lemma 18.1.5, the definition of βf , and (18.22).

22 Conformal Invariant Measures for CNRR Functions

353

Lemma 22.1.6 Let f : C −→  C be a nonrecurrent elliptic function. Let w ∈ J (f )\Be ((f ),θ (f )) and Q ⊆ Be (w,γf ) be a Jordan domain. If −n (Q), {Qn }∞ n=0 is a w-nested sequence of connected components of the sets f then all the sets Qn,n ≥ 0, are open Jordan domains. Let Crith (f ) be the set of all critical points of f that are h-upper estimable with respect to the h-conformal measure mh . Because of Lemma 20.3.10, we have that Critc (f ) ⊆ Crith (f ).

(22.5)

The key fact for what will follow in this section is this. Lemma 22.1.7 Let f : C −→  C be a compactly nonrecurrent regular elliptic function. Let w ∈ J (f )\Be ((f ),θ ), Q ⊆ Be (w,γ ) be a Jordan domain, and −n (Q). {Qn }∞ n=0 be a w-nested sequence of connected components of the sets f −n For every n ≥ 0, let Wn be the connected component of f (Be (w,2γ )) containing Qn . If (Crit(f )\Crith (f )) ∩

∞ 

Wn = ∅,

n=1

then mh,e (Qn ) 

mh,e (Q) diamhe (Q)

diamhe (Qn )

for all n ≥ 0. Proof For n = 0, the required inequality is trivial. Fix k ≥ 0 and n ≥ 0. Suppose that Wk+n contains no critical points of f n . Then, by virtue of Theorem 17.1.7, the map f n |Wk+n : Wk+n −→ Wk is a conformal homeomorphism. Hence, it follows from Theorem 8.3.15 and (18.25) (note that also Q ⊆ Be (w,γ ) that 

mh,e (Qk+n ) ≤ sup |(f n ) (z)|−h : z ∈ Qk+n mh,e (Qk ) 

≤ K∗h inf |(f n ) (z)|−h : z ∈ Qk+n mh,e (Qk ) ≤ K∗h

diamhe (Qk+n )

mh,e (Qk ) diamhe (Qk ) mh,e (Qk ) = K∗h diamhe (Qk+n ), diamhe (Qk )

(22.6)

354

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

with an appropriate universal constant K∗h ≥ 1. Now suppose that Wk+1 contains a critical point c of f . By (18.22) and Lemma 18.1.11, c is the only critical point of f in Wk+1 . Suppose, first, that diste (f (c),Qk ) ≥ 4diame (Qk ). Fix z ∈ Qk . Then Qk ⊆ Be (z,diame (Qk )), f (Crit(f )) ∩ Be (z,2diam(Qk )) = ∅ (assuming that γ ,η > 0 sufficiently small), which makes other (finitely many) critical values lying sufficiently far apart from f (c). Hence, invoking Theorem 17.1.7 and denoting by f∗−1 : Be (z,2γ diame (Q)) −→ C the resulting holomorphic inverse branch of f whose range covers Qk , by using Theorem 8.3.15, we estimate similarly as above 

mh,e (Qk+1 ) ≤ sup |(f∗−1 ) (x)|h : x ∈ Qk+1 mh,e (Qk ) 

≤ K h inf |(f∗−1 ) (x)|h : x ∈ Qk+1 mh,e (Qk ) ≤ Kh

diamhe (Qk+1 )

mh,e (Qk ) diamhe (Qk ) mh,e (Qk ) ≤ Kh diamhe (Qk+1 ). diamhe (Qk )

(22.7)

Now assume that diste (f (c),Qk ) ≤ 4diame (Qk ).

(22.8)

We, thus, get that Qk ⊆ Be (f (c),5diame (Qk )); therefore,   Qk+1 ⊆ Be c,A(5diame (Qk ))1/pc . Hence, making use of h-upper estimability of the point c, we get that h  me (Qk+1 ) ≤ L A(5diame (Qk ))1/pc . It follows from (22.8) that Diste (c,Qk+1 ) ≤ A(Diste (f (c),Qk ))1/pc ≤ A(diste (f (c),Qk ) + diame (Qk ))1/pc ≤ A(5diame (Qk ))1/pc .

(22.9)

22 Conformal Invariant Measures for CNRR Functions

355

Therefore, diame (Qk ) ≤ diame (Qk+1 )A(Diste (c,Qk+1 ))pc −1 pc −1

≤ A2 51/pc diame (Qk+1 )diame pc (Qk ). Thus, 1/pc

diame

(Qk ) ≤ A2 51/pc diame (Qk+1 ).

Inserting this into (22.9), we get that me (Qk+1 ) ≤ L(25)h/pc A3h/pc diamhe (Qk+1 ). Applying this, (22.6), and (22.7) and making use of Lemma 18.1.11 along with (18.22), a straightforward inductive argument yields that, for every j ≥ 1, N

me (Qj ) ≤ max (L(25A3 )1/pc ,K,K∗ )h f diamhe (Qj ). 

The proof is complete.

As an immediate consequence of this lemma, Lemma 18.1.14, and Theorem 8.3.15, we obtain the following. Lemma 22.1.8 Let f : C −→  C be a compactly nonrecurrent regular elliptic function. Having θ ∈ (0,θ (f )) and γ ∈ (0,γf ), let w ∈ J (f )\Be ((f ),θ ), V ⊆ Be (w,γ ) be a Jordan domain, and U be a Jordan domain contained in V . Let {Vn }∞ n=0 be a w-nested sequence of connected components of f −n (V ) and {Un }∞ n=0 , with Un ⊆ Vn , be a w-nested sequence of connected components of f −n (U ). For every n ≥ 0, let Wn be the connected component of f −n (Be (w,2γ )) containing Vn . Suppose that 

∞   Wn = ∅ Crit(f )\Crith (f ) ∩ n=1

and that there exists a Jordan domain U˜ such that U ⊆ U˜ ⊆ V and U˜ ∩ PC(f ) = ∅. Then mh,e (Vn ) 

mh,e (V ) mh,e (Un ), mh,h (U )

and the same inequality remains true (perhaps with a smaller constant on the right-hand side) with mh,e replaced by mh,s since the diameters of all the sets Vn are bounded above by βf . Now we can take the first fruit of this lemma.

356

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Proposition 22.1.9 All the points of the set PC0c (f )\(f ) are of finite condensation with respect to the invariant measure μh . Proof Keep the Marco Martens cover (Xn )∞ n=0 as defined by (22.3). We keep working within the framework of Theorem 2.4.4 and Definition 2.4.2 set up in the proof of Theorem 22.1.1. Take an arbitrary point w ∈ PC0c (f )\(f ). Assuming θ > 0 to be small enough, we will have that w ∈ / Be ((f ),θ ). Fix V ⊆ Be (w,γ ), an open set containing w and disjoint from PCp (f ) ∪ PC∞ (f ). Since, by Proposition 18.2.7, PC0c (f ) is a nowhere dense subset of J (f ), by taking sufficiently large q ≥ 1 in (22.2), we may assume without loss of generality that 2X0 ⊆ V . Invoking (22.5), it immediately follows from Lemma 22.1.8 (note that 2X0 ∩ PC(f ) = ∅) that mh,s (f −n (X0 ) ∩ Vn ) $ mh,s (Vn ) for every integer n ≥ 0, where Vn is a connected component of f −n (V ). Therefore, summing up over all connected components Vn of f −n (V ), we obtain that mh,s (f −n (X0 )) = mh,s (f −n (A0 ) ∩ f −n (V )) $ mh,s (f −n (V )). Consequently, "k mh,s,n (V ) = "kn=0

mh,s (f −n (V ))

n=0 mh,s (f

−n (X )) 0

 1,

where mh,s,n are given by (2.37), and it follows from (2.39) of Theorem 2.4.4  that μh (V ) < +∞, which finishes the proof. Lemma 22.1.10 If f : C →  C is compactly nonrecurrent regular elliptic function, then all the points of the set PC0p (f ) ∪ PC0∞ (f ) are of finite condensation with respect to the f -invariant measure μh . Proof We still keep the Marco Martens cover (Xn )∞ n=0 as defined by (22.3) and are working within the framework of Theorem 2.4.4 and Definition 2.4.2 set up in the proof of Theorem 22.1.1. Fix a point w ∈ PC0p (f ) ∪ PC0∞ (f ). There exists an integer j ≥ 0 so large that   f −j (w) ∩ O+ (Crit(f )) ∪ (f ) = ∅. Therefore, taking into account Corollary 8.6.19, we see that, with θ > 0 and γ > 0 small enough, there exists an open disk V centered at w with the following properties.

22 Conformal Invariant Measures for CNRR Functions

357

(a) For every z ∈ f −j (w), diste (z,(f )) > θ . (b) For every z ∈ f −j (w), if Vz is the connected component of f −j (V ) containing z, then Vz is a Jordan domain and Vz ⊆ Be (z,γ ).  (c) z∈f −j (w) Be (z,2γ ) ∩ PC(f ) = ∅. We may assume without loss of generality that 2X0 ⊆ V . It follows from condition (c) that 

Crit(f ) ∩

∞ 

f −n (Be (z,2γ )) = ∅.

z∈f −j (w) n=0

So, we may apply Lemma 22.1.8 to the pairs (Uz,Vz ), z ∈ f −j (w), where Uz are the connected components of f −j (X0 ) contained in Vz , to get, similarly to the proof of Proposition 22.1.9, that, for every z ∈ f −j (w) and every n ≥ 0, n 

mh,s (f −i (Vz )) 

i=0

n 

mh,s (f −i (Uz )).

i=0

Summing over all z ∈ f −j (w), we, thus, get that j +n i=j

mh,s (f

−i

(V )) =

n 

mh,s (f

−i

(f

−j

(V ))) 

i=0

"j −1

j +n

mh,s (f −i (X0 )).

i=j

"j −1

Since, in addition, both i=0 mh,s (f −i (X0 )) and i=0 mh,s (f −i (V )) are finite, we, thus, get that mh,s,n (V )  1, where mh,s,n are given by (2.37), for all n ≥ 0. It, therefore, follows from (2.39) quoted in Theorem 2.4.4 that  μh (V ) < +∞. The proof is complete. As an immediate consequence of this lemma, Proposition 22.1.9, and Lemma 22.1.5, we get the following. Theorem 22.1.11 If f : C −→  C is a compactly nonrecurrent regular elliptic function, then Jμh (∞) ⊆ (f ) ∪ {∞}. Now we shall deal with the point ∞. Recall that the set Crith (f ) was defined just above (22.5) We shall prove the following. Proposition 22.1.12 If f : C −→  C is a compactly nonrecurrent regular elliptic function and Crit∞ (f ) ⊆ Crith (f ), then ∞ is a point of finite condensation of the f -invariant measure μh . Proof As above, we keep the Marco Martens cover (Xn )∞ n=0 as defined by (22.3) and are working within the framework of Theorem 2.4.4 and

358

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Definition 2.4.2 set up in the proof of Theorem 22.1.1. Recall that γf > 0 γf is determined by (18.22), while θ (f ) > 0 was defined in (18.19). Fix γ ∈ (0,γf ) and θ ∈ (0,θ (f )) so small that      Be (f ),θ (f ) ∪ PC0c (f ) ∩ B(b,2γ ) = ∅ (22.10) b∈f −1 (∞)

and ⎛ ⎝



⎞

⎛

⎞ 

⎜ O+ (c)⎠ ∩ ⎝ b∈f −1 (∞)\

c∈Critp (f )



⎟ B(b,2γ )⎠ = ∅. (22.11)

c∈Critp (f ) O+ (c)

Let R > 0 be so large that Bb (R) ⊆ B(b,γ )

(22.12)

for every b ∈ f −1 (b), where, we recall, Bb (R) is the connected component of f −1 (B∞ (R)) containing b. By taking sufficiently large q ≥ 1 in (22.2), we may assume without loss of generality that 2X0 ⊆ B∞ (R). Of course, ξ := inf{diame (Bb (R)) : b ∈ f −1 (∞)} > 0. For every b ∈ f −1 (∞), let X0b be a connected component of f −1 (X0 ) contained in Bb (R) and 2X0b be the connected component of f −1 (2X0 ) containing X0b . Since the set  Pc := f −1 (∞) ∩ O+ (c) c∈Critp (f )

is finite, proceeding as in the second part of the proof of Lemma 22.1.10, with the pair (X0,V ) replaced by (X0b,Bb (R)), we see that there exists j ≥ 0 (with meaning analogous to that in the proof of Lemma 22.1.10) such that, for all n ≥ 0, +n  j

mh,s (f −i (Bb (R))) ≤

b∈Pc i=j

n 

mh,s (f −i (X0b )).

(22.13)

b∈Pc i=j

Since Crit∞ (f ) ⊆ Crith (f ), invoking (22.2)–(22.12), it follows directly from Lemma 22.1.8 that, for all j ≥ 0, mh,s (V ) 

mh,s (Bb (R)) mh,s (X0b )

mh,s (f −i (X0b ) ∩ V )

22 Conformal Invariant Measures for CNRR Functions

359

for all b ∈ P2 := f −1 (∞)\P1 and all connected components V of f −i (Bb (R)). But mh,e (X0b )  dist(0,X0 )

−

qb +1 qb h

mh,e (X0 )  1,

so, applying Lemma 20.3.8 and Theorem 17.3.1, we get that q +1 mh,e (Bb (R)) mh,s (Bb (R)) 2− bq h b  1.   R mh,s (X0b ) mh,e (X0b )

Therefore, mh,s (V )  mh,s (f −i (X0b ) ∩ V ). Summing this inequality over all connected components V of f −i (Bb (R)), we, thus, get that mh,s (f −i (Vb ))  mh,s (f −i (X0b )). Hence, for all n ≥ 0, n 

mh,s (f −i (Bb (R))) 

b∈P2 i=0

n 

mh,s (f −i (X0b )).

b∈P2 i=0

Adding this inequality and (22.13) side by side, we get that n 

mh,s (f −i (BR )) − F 

i=0

n 

mh,s (f −i (X0 )) − G

i=0

with some positive numbers F and G independent of n, resulting from the fact that we sum in (22.13) from i = j and not from i = 0. Thus, mh,s,n (B∞ (R))  1, where, for all n ≥ 1, the measures mh,s,n are given by (2.37). It, therefore, follows from (2.39) of Theorem 2.4.4 that μh,s (B∞ (R)) < +∞. We are done.  As an immediate consequence of Theorem 22.1.11 and Proposition 22.1.12, we get the following. Corollary 22.1.13 If f : C −→  C is a compactly nonrecurrent regular elliptic function and Crit∞ (f ) ∪ (f ) = ∅, then the invariant measure μh , equivalent to the conformal measure mh,s (which, in this case, coincides, up to a multiplicative constant, with the packing measure h ), is finite. We will then always assume that μh is a probability measure. We also get the following. Corollary 22.1.14 If f : C −→  C is a compactly nonrecurrent elliptic function whose Julia set is equal to the entire complex plane C, then there

360

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

exists a unique Borel probability f -invariant measure μ2 equivalent to the planar Lebesgue measure on C. Proof The function f is regular since h = 2. Hence, since (f ) = ∅ and Crit2 (f ) = Crit(f ), the existence of μ2 follows immediately from Theorem 22.1.11 and Proposition 22.1.12. Uniqueness is guaranteed by Theorem 22.1.1.  Remark 22.1.15 In fact, we have shown in the proof of Proposition 22.1.12 that, for all R > 0 large enough, mh,s,n (B∞ (R))  1, where the measures mh,s,n are given by (2.37).

22.2 Real Analyticity of the Radon–Nikodym Derivative

dμh dmh

Throughout this section, we keep f : C −→  C, a compactly nonrecurrent regular elliptic function, and notation from the previous section; in particular, mh is the unique h-conformal measure for f and μh is the σ -finite f -invariant measure produced in Theorem 22.1.1. The goal of this section is to show dμh has a real-analytic extension to that the Radon–Nikodym derivative dm h,e a neighborhood of J (f )\PC(f ) in C. In the context of conformal iterated systems and conformal graph directed Markov systems such a result has appeared in [MPU]; see also [MU2]. The proof we provide in this chapter stems from the one given in [MPU]. The first step toward this end is to work with the analytic map fˆ : Tˆ f −→ Tf defined at the beginning of Section 16.2 so that the diagram (16.7) commutes. We repeat it here. f

C\f −1 (∞) f

Tˆ f

C f

fˆ

(22.14)

Tf

Recall from Observation 17.2.10 that PC(fˆ) = f (PC(f )) and J (fˆ) = f (J (f )) ⊆ Tf . Define the Borel probability measure m ˆ h on J (fˆ) by the formula m ˆ h (A) := mh,e (−1 f (A) ∩ R),

(22.15)

22 Conformal Invariant Measures for CNRR Functions

361

where R is a fundamental domain of f . This definition is, in fact, independent of the choice of the fundamental region R, and we clearly have the following. Proposition 22.2.1 If f : C −→  C is a compactly nonrecurrent regular elliptic function, then the Borel probability measure m ˆ h is h-conformal with ˆ ˆ respect to the map f : Tf −→ Tf . Because the diagram (22.14) commutes, as an immediate consequence of Theorem 22.1.1, we get the following. Proposition 22.2.2 If f : C −→  C is a compactly nonrecurrent regular elliptic function, then for the Borel σ -finite measure μˆ h := μh ◦ −1 f

(22.16)

we have that μˆ h is fˆ-invariant, μˆ h (J (fˆ)) = 1, ˆ h, μˆ h is equivalent to m  the dynamical system fˆ : J (fˆ) → J (fˆ), μˆ h is metrically exact, hence ergodic,     ˆ h J (fˆ)\Tr(fˆ) = 0. (5) μˆ h J (fˆ)\Tr(fˆ) = m

(1) (2) (3) (4)

We shall prove the following. Lemma 22.2.3 If f : C −→  C is a compactly nonrecurrent regular elliptic μˆ h function, then the Radon–Nikodym derivative ρˆ h := ddm has a real-analytic ˆh   extension to a neighborhood of J (fˆ)\PC(fˆ) in Tf . Proof Since the measure mh,e is ergodic and conservative (Theorem 20.3.11), so is the measure m ˆ h . Since, by Theorem 8.3.9 (Koebe Distortion Theorem I for Parabolic Surfaces), we have bounded distortion on the complement of PC(fˆ), we see that the assumptions of Theorem 2.4.4 are satisfied for the dynamical system fˆ : J (fˆ) −→ J (fˆ) and the conformal measure m ˆ h. Therefore, ρˆh (z) = lim an−1 n→∞

n 



|(fˆk ) (ξ )|−h

k=0 ξ ∈fˆ−k (z)

for every z ∈ J (fˆ)\PC(fˆ), where an :=

n  k=0

m ˆ h (fˆ−k (A0 ))

(22.17)

362

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

with some set A0 ⊆ J (fˆ)\PC(fˆ) as required in Theorem 2.4.4. Fix such an arbitrary point z ∈ J (fˆ)\PC(fˆ) and take r = r(z) > 0 so small that B(z,2r) ∩ PC(fˆ) = ∅. We can assume without loss of generality that A0 ⊆ B(z,r). It follows from Theorem 17.1.8 that, for every integer k ≥ 0 and every ξ ∈ fˆ−k (z), there exists fˆξ−k : B(z,2r) −→ Tf , a unique holomorphic inverse branch of fˆk defined on B(z,2r) and determined by the requirement that fˆξ−k (z) = ξ . Now embed C into C2 by the formula: C  x + iy −→ (x,y) ∈ C2 . For every ξ ∈ fˆ−k (z), consider the map gξ : B(z,2r) → C defined as follows: gξ (w) :=

(fˆξ−k ) (w) (fˆξ−k ) (z)

.

Since the ball B(z,2r) is simply connected, since the Jacobian gξ nowhere vanishes on B(z,2r), and since gξ (z) = 1, there exists log gξ : B(z,2r) −→ C, a unique holomorphic branch of logarithm gξ , such that log gξ (z) = 0. By Theorem 8.3.8 and the Koebe Distortion Theorem 8.3.6 (see (8.28)), there exists a constant Kˆ such that | log gξ | ≤ Kˆ throughout B(z,r). Expand log gξ into its Taylor series: log gξ =

∞ 

un (w − z)n .

n=0

By Cauchy’s estimates ˆ n |un | ≤ K/r

(22.18)

22 Conformal Invariant Measures for CNRR Functions

363

for all integers n ≥ 0. For every point x + iy ∈ B(z,2r), we can write ∞

 n un ((x − Rez) + i(y − Imz)) Re log gξ (x + iy) = Re n=0

=

∞ 





Re up+q

p,q=0

=



p+q q i (x − Rez)p (y − Imz)q q

cp,q (x − Rez)p (y − Imz)q .

In view of (22.18), we have that ˆ −(p+q) 2p+q . |cp,q | ≤ Kr Hence, Re log gξ extends, by the same power series expansion,  DC2 (z,r/3)  (x,y) −→ cp,q (x − Rez)p (y − Imz)q ∈ C ˆ Denote to the polydisk DC2 (z,r/3) and its modulus is bounded above by 4K. 6 this extension by Re log gξ . Now, for every n ≥ 0, consider the function bn : B(z,2r) → C given by the formula bn (w) := an−1

n 



|(fˆξ−k ) (w)|h .

k=0 ξ ∈fˆ−k (z)

Each function bn extends to a holomorphic function Bn : DC2 (z,2r) −→ C as follows: Bn := an−1

n 



6 log gξ ). |(fˆk ) (ξ )|−h exp(hRe

k=0 ξ ∈fˆ−k (z)

Since A0 ⊆ B(z,2r), it follows from (22.17) and the Koebe Distortion Theorem I (Euclidean Version), i.e., Theorem 8.3.8, that L := sup{bn (z)} < +∞. n≥0

Therefore, for every w ∈ DC2 (z,r/3), we get that |Bn (z)| ≤ an−1

n 



6 log gξ (w))| |(fˆk ) (ξ )|−h | exp(hRe

k=0 ξ ∈fˆ−k (z)

≤ an−1

n 



k=0 ξ ∈fˆ−k (z)

6 log gξ (w)|) |(fˆk ) (ξ )|−h exp(h|Re

364

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

ˆ n−1 ≤ exp(4hK)a

n 



|(fˆk ) (ξ )|−h

k=0 ξ ∈fˆ−k (z)

ˆ n (z) = exp(4hK)b ˆ ≤ L exp(4hK). Hence, applying the Cauchy Integral Formula (in DC2 (z,r/2)), we see that the family {Bn }∞ n=0 is equicontinuous on DC2 (z,r/4). Thus, we can choose from ∞ {Bn }n=0 a subsequence uniformly convergent on DC2 (z,r/5). Its limit function Gz : DC2 (z,rz /5) −→ C is analytic and Gz |Jˆ∩D

C2 (z,rz /5)

= ρˆ|Jˆ∩D

C2 (z,rz /5)

So, Gz |B(z,rz /5) is a real-analytic extension of ρˆ|Jˆ∩D

C2

.

(z,rz /5) .

Now if

B(z,rz /10) ∩ B(z,rz /10) = ∅,   z,z ∈ J (fˆ)\f PC(f ) , then choose a point v ∈ B(z,rz /10) ∩ B(z,rz /10). We may assume without loss of generality that rz ≤ rz . Then  rz  rz + ⊆ B(z,rz /5). z ∈B z, 10 10 So, z ∈ B(z,rz /5) ∩ B(z,rz /5); in particular, Jˆ ∩ B(z,rz /5) ∩ B(z,rz /5) = ∅. Since this intersection is not contained in any real-analytic curve (its Hausdorff dimension is larger than 1), we, thus, conclude that Gz |B(z,rz /5)∩B(z,rz /5) = Gz |B(z,rz /5)∩B(z,rz /5) . In particular, Gz |B(z,rz /10)∩B(z,rz /10) = Gz |B(z,rz /10)∩B(z,rz /10) . So, the formula G(w) := Gz (w),

22 Conformal Invariant Measures for CNRR Functions

365

if z ∈ Jˆ\PC(fˆ) and w ∈ B(z,rz /10), provides a well-defined real-analytic function on  B(z,rz /10) z∈Jˆ\(PC(f))

which coincides with ρˆ on Jˆ\PC(fˆ). The proof is complete.



ˆ Definition 22.2.4 Let f : C −→  C be an elliptic function. For every z ∈ C, let f0−1 (z) be a maximal subset of points from f −1 (z) that are mutually incongruent with respect to the equivalence relation ∼f , i.e., modulo the lattice f of f . We also fix R > 0 so large that f (B(0,R)) =  C and we require, in addition, that f0−1 (z) ⊆ B(0,R) for all z ∈  C. Now we can prove the following main result of this section. Theorem 22.2.5 If f : C −→  C is a compactly nonrecurrent regular elliptic dμh function, then the Radon–Nikodym derivative ρh := dm has a real-analytic h,e extension to a neighborhood of J (f )\PC(f ) in  C. In particular, for every r > 0, the function

 ρh J (f )∪{∞}\B(PC(f ),r)

is uniformly continuous with respect to the spherical metric on  C. Proof Fix a point z ∈ J (f )\PC(f ) and put Rz := 12 diste (z,PC(f )) > 0. Then, for every ξ ∈ f −1 (z) and every Borel set A ⊆ Be (z,Rz ), we have that       −1 −1 m ˆ h f (fξ−1 (A)) = mh,e −1 f f (fξ (A)) ∩ Rf = mh,e (fξ (A)), (22.19) where, we recall, Rf is a fundamental domain of f . Also, ⎞ ⎛   ⎟ ⎜ μh (A)| = μh (f −1 (A)) = μh ⎝ w + fξ−1 (A)⎠ ⎛

⎛

⎛

ξ ∈f0−1 (z) w∈

⎜ ⎜ ⎜  = μh ⎝−1  ⎝ ⎝ f f ξ ∈f0−1 (z)

⎞⎞⎞

⎟⎟⎟ fξ−1 (A)⎠⎠⎠

366

Part VI Fractal Geometry, Stochastic Properties, and Rigidity ⎛

⎞⎞

⎛

⎜ ⎜  = μˆ h ⎝f ⎝ 

=

⎟⎟ fξ−1 (A)⎠⎠

ξ ∈f0−1 (z)

μˆ h (f (fξ−1 (A))).

ξ ∈f0−1 (z)

Therefore, using this and (22.19), we get that μh (A) = mh,e (A) =

μˆ h (f (fξ−1 (A)))

 ξ ∈f0−1 (z)

μˆ h (f (fξ−1 (A)))

 ξ ∈f0−1 (z)

=

mh,e (A)

m ˆ h (f (fξ−1 (A))) μˆ h (f (fξ−1 (A)))

 ξ ∈f0−1 (z)

m ˆ h (f (fξ−1 (A)))

·

·

m ˆ h (f (fξ−1 (A))) mh,e (A) mh,e (fξ−1 (A)) mh,e (A)

.

Hence, passing to Radon–Nikodym derivatives, we get that dμh (w) = dmh,e

 ξ ∈f0−1 (z)

d μˆ h (f (fξ−1 (w)))|(fξ−1 ) (w)|h d mˆh,e

(22.20)

for all w ∈ B(z,Rz ) ∩ J (f ). By Lemma 22.2.3, the function B(z, Rˆz )  w −→

d μˆ (f (fξ−1 (w)))|(fξ−1 ) (w)|h ∈ R dm ˆ

is real analytic on some sufficiently small ball B(z, Rˆz ), 0 < Rˆ z ≤ Rz . Hence, since the number of terms in the series (22.20) is finite, bounded above by the (finite) number of elements of the lattice f, this series represents a realanalytic function from B(z, Rˆ z ) to R. We, thus, conclude, exactly as in the last part of the proof of Lemma 22.2.3, that (22.20) gives a real-analytic extension dμh to the open set of the Radon–Nikodym derivative dm h,e 

B(z, Rˆ z ).

z∈J (f )\PC(f )

We are done.



22 Conformal Invariant Measures for CNRR Functions

367

22.3 Finite and Infinite Condensation of Parabolic Periodic Points with Respect to the Invariant Conformal Measure μh Throughout this section, we keep f : C −→  C, a compactly nonrecurrent regular elliptic function, and notation from the previous section; in particular, mh is the unique h-conformal measure for f and μh is the σ -finite f -invariant measure produced in Theorem 22.1.1. We continue the detailed study of the points of their finite and infinite condensation originated at the end of Section 22.1. More precisely, we provide, under some additional mild assumptions, precise characterizations of the locations of such points. In the context of rational functions, the results of such points were obtained, for example, in [ADU], [DU1], [DU2], and [U4]. We again utilize the Marco Martens cover (Xn )∞ n=0 defined by (22.3) and keep working throughout the whole section within the notation and concepts of Theorem 2.4.4 and Definition 2.4.2 as these were used in the proof of Theorem 22.1.1. Let ω ∈ C be a simple parabolic fixed point of f , i.e., we recall that f (ω) = ω and f  (ω) = 1. Let p = p(ω) ≥ 1 be the corresponding number of petals. Fix any α ∈ (0,π). Let (ω,α) :=

p 

j (ω,α)

j =1

be the corresponding fundamental domain, defined in Lemma 15.3.7. Let Sr (ω) := {ω} ∪

p 

j

Sr (ω,α)

j =1

be the union of corresponding sectors, defined by (15.37). In particular, Proposition 15.1.8 holds, i.e., fω−1 (Sr (ω)) ⊆ Sr (ω). For every y ∈ f −1 (ω), let fy−1 (B(ω,θf )) be the connected component of f −1 (B(ω,θf )) containing y. For every set A ⊆ B(ω,θf ), let fy−1 (A) := f −1 (A) ∩ fy−1 (B(ω,θf )). Keep mh to be the h-conformal measure for f . For every Borel set A ⊆ Sr (ω) and every integer n ≥ 0, we can write

368

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Sn mh,s (A) : =

n 

  mh,s f −j (A)

j =0

= mh,s (fω−n (A)) +



(22.21)

n−1 

Sn−(k+1) mh,s (fy−1 (fω−k (A))).

y∈f −1 (ω)\{ω} k=0

We shall first prove the following. Proposition 22.3.1 Let f : C −→  C be a compactly nonrecurrent regular elliptic function. If ω ∈ (f ) \ O+ ( f (Crit(f ) ∩ J (f ) is a simple parabolic fixed point of f , then 

μh fω−l (F )



1− p(ω)+1 p(ω) h

 (l + 1)

 dμh  mh (F ) and l+1 dmh fω−l ((ω)) (22.22)

for every integer l ≥ 0 and every Borel set F ⊆ (ω,α). Proof

Let A := fω−l (F ),

where F is a Borel set contained in j (ω,α) for some 1 ≤ j ≤ p. Fix an integer n ≥ 1. It then follows from Proposition 15.3.4, conformality of the measure mh,s , and the Koebe Distortion Theorem I, Spherical Version (Theorem 8.3.12) that, for all y ∈ f −1 (ω)\{ω} and all integers 0 ≤ k ≤ n − 1, we have that   − p+1 h Sn−(k+l) mh,s fy−1 (fω−k (fω−l (F )))  (k + l) p Sn−(k+1) mh,s (fy−1 (F )). So, substituting this into (22.21), we obtain that mh,s,n (fω−l (F )) =

Sn mh,s (fω−l (F )) Sn mh,s (X0 )

mh,s (fω−n (fω−l (F ))) +  Sn mh,s (X0 ) ×



n−1  − p+1 h (k + l) p

y∈f −1 (ω)\{ω} k=0

Sn−(k+1) mh,s (fy−1 (F )) Sn−(k+1) mh,s (X0 ) Sn−(k+1) mh,s (X0 )

Sn mh,s (X0 )

22 Conformal Invariant Measures for CNRR Functions



mh,s (fω−n (fω−l (F ))) + Sn mh,s (X0 ) ×





n−1 

(k + l)

369

− p+1 p h

y∈f −n (ω)\{ω} k=0

Sn−(k+1) mh,s (fy−1 (X0 )) mh,s (F ) Sn−(k+1) mh,s (X0 ) Sn−(k+1) mh,s (X0 ) mh,s (X0 ) Sn mh,s (X0 )

mh,s (fω−n (fω−l (F ))) Sn mh,s (X0 ) " n−1 −1  y∈f −n (ω)\{ω} Sn−(k+1) mh,s (fy (F )) − p+1 h (k + l) p + Sn−(k+1) mh,s (X0 ) k=0

× 

mh,s (F ) Sn−(k+1) mh,s (X0 ) mh,s (X0 ) Sn mh,s (X0 )

mh,s (fω−n (fω−l (F ))) Sn mh,s (X0 )

  −1 (X ) n−1 f −1  0 p+1 Sn−(k+1) mh,s y∈f (ω)\{ω} y − h + (k + l) p Sn−(k+1) mh,s (X0 ) k=0

× Now Sn−(k+1) mh,s

Sn−(k+1) mh,s (X0 ) mh,s (F ) . Sn mh,s (X0 ) mh,s (X0 ) 

y∈f −1 (ω)\{ω}

fy−1 (X0 )

(22.23)



Sn−(k+1) mh,s (X0 ) Sn−k mh,s (X0 ) mh,s (f −(n−k) (X0 )) = +1 Sn−(k+1) mh,s (X0 ) Sn−(k+1) mh,s (X0 ) 1 ≤ + 1. mh,s (X0 )

≤

Also, since there exists an integer j ≥ 0 such that ⎞⎞ ⎛ ⎛  fy−1 (X0 )⎠⎠ > 0, mh,s ⎝X0 ∩ f −j ⎝ y∈f −1 (ω)\{ω}

by making use of item (4) of Definition 2.4.2, we conclude that   −1 (X ) f Sn−(k+1) mh,s −1 0 y∈f (ω)\{ω} y $ 1. Sn−(k+1) mh,s (X0 )

(22.24)

370

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Taking this and (22.24) together, we obtain that   −1 (X ) f Sn−(k+1) mh,s −1 0 y∈f (ω)\{ω} y Sn−(k+1) mh,s (X0 )

 1.

Inserting this into (22.23), we get that mh,s,n (fω−l (F )) =

mh,s (fω−n (fω−l (F ))) Sn mh,s (X0 ) +

n−1  − p+1 h Sn−(k+1) mh,s (X0 ) (k + l) p + mh,s (F ) Sn mh,s (X0 ) k=0

 o(1) +

n−1 

(k + l)

Sn−(k+1) mh,s (X0 ) − p+1 p h Sn mh,s (X0 )

k=0

mh,s (F ). (22.25)

Therefore, on the one hand, mh,s,n (fω−l (F ))  o(1) +

n−1 

(k + l)

− p+1 p h

mh,s (F )

k=0 1− p+1 p h

≤ o(1) + (l + 1)

mh,s (F ),

where the o(1) symbol is written with respect to n → ∞. So, it follows from (2.39) that 1− p+1 p h

μh (fω−l (F ))  (l + 1)

mh,s (F ).

(22.26)

Hence, μh (fω−l (F ))

mh,s (fω−l (F ))



1− p+1 p h

(l + 1)

− p+1 p h

(l + 1)

mh,s (F )

= (l + 1).

mh,s (F )

Therefore,  dμh   l + 1. dmh,s fω−l ((ω)) On the other hand, there exists q ≥ 1 so large that q  k=0

(k + l)

− p+1 p h

1 ≥ 2



p+1 1− p+1 h h − 1 (l + 1) p . p

(22.27)

22 Conformal Invariant Measures for CNRR Functions

371

Inserting this into (22.25), we get, for every n ≥ q + 1, that mh,s,n (fω−l (F )) $ o(1) +

q  − p+1 h Sn−(k+1) mh,s (X0 ) (k + l) p mh,s (F ) Sn mh,s (X0 ) k=0

Sn−(q+1) mh,s (X0 )  − p+1 h (k + l) p mh,s (F ) Sn mh,s (X0 ) q

$ o(1) +

k=0

$ o(1) + l

Sn−(q+1) mh,s (X0 ) 1− p+1 p h

mh,s (F ) Sn mh,s (X0 ) Sn mh,s (X0 ) − q 1− p+1 p h $ o(1) + l mh,s (F ) Sn mh,s (X0 ) 1− p+1 p h

$ o(1) + (l + 1)

(22.28)

mh,s (F ).

Since fω−l (F ) is contained in a finite union of the sets Xj , j ≥ 0, it follows from (2.40) and (22.28) that μh (fω−l (F )) $ l

1− p+1 p h

mh,s (F ).

Hence, μn (fω−l (F ))

mh,s (fω−l (F )) Therefore,

$

1− p+1 p h

(l + 1)

− p+1 p h

(l + 1)

mh,s (F )

= l + 1.

mh,s (F )

 dμh  $ l + 1. dmh,s fω−l ((ω))

Combining these two formulas with (22.26) and (22.27), we get that 1− p+1 p h

μh (fω−l (F ))  (l + 1) and

mh,s (F )

 dμn   l + 1. dmh fω−l (δ(ω))

The proof of Proposition 22.3.1 is complete.



As an immediate consequence of this proposition, we get the following. Corollary 22.3.2 Let f : C −→  C be a compactly nonrecurrent regular elliptic function. If ω ∈ (f ) \ O+ ( f (Crit(f ) ∩ J (f ) is a simple parabolic fixed point of f , then, for every α ∈ (0,π) and every integer k ≥ 0, we have that

372

Part VI Fractal Geometry, Stochastic Properties, and Rigidity ⎛ μh ⎝

∞ 

⎞ 2− fω−j ((ω))⎠  (k + 1)

p(ω)+1 p(ω) h

j =k

if h >

2p(ω) p(ω)+1

and ⎛ μh ⎝

∞ 

⎞ fω−j ((ω,α))⎠ = +∞

j =k

if h ≤

2p(ω) p(ω)+1 .

Passing to an appropriate iterate if ω is a parabolic periodic point, as an immediate consequence of this proposition, we get the following main result of this section.  Theorem 22.3.3 Let f : C −→  C be a compactly nonrecurrent regular elliptic function. If ω ∈ (f ) \ O+ ( f (Crit(f ) ∩ J (f ) , then ω ∈ Jμh (∞) if and only if h ≤

2p(ω) . p(ω) + 1

Observe that if f is just compactly nonrecurrent, then we can always find a point y ∈ f −1 (ω)\{ω} such that PC(f ) ∩ fy−1 (B(ω,2R)) = ∅ for all R > 0 sufficiently small. We then get the $ part of Proposition 22.3.1. In conclusion, we obtain the following. Theorem 22.3.4 Let f : C −→  C be a compactly nonrecurrent regular elliptic 2p(ω) function. If ω ∈ (f ) and h ≤ p(ω)+1 , then ω ∈ Jμh (∞). We end this section with a proposition showing that critical points eventually landing at parabolic points make the latter more likely to be of infinite condensation. We need the following. Theorem 22.3.5 Let f : C −→  C be a compactly nonrecurrent regular elliptic function. If ω ∈ (f ) and c ∈ J (f ) is a critical point of f s such that f l (c) = ω for some integer l ≥ 1, then mh,s (B(c,r))  r h+qc p(ω)(h−1) . Proof The proof of this theorem follows a simple “integration” method originated in [DU5, Lemma 4.8] and since then frequently used in similar contexts. Recall that, given x ∈ C and 0 < r1 < r2 , we denoted A(x;r1,r2 ) = {z ∈ C : r1 < |z − x| ≤ r2 }.

22 Conformal Invariant Measures for CNRR Functions

373

Set q := qc . Then using Lemma 15.4.1, conformality of the measure mh,s , and its atomlesness, we can write, for every r > 0 sufficiently small, the following:   (r q )h+p(h−1)  mh,s A(f l (c),r q ,(2r)q )      mh,s f l (A(c,r,2r))  mh,s A(c,r,2r) r (q−1)h ;   therefore, mh,s A(c,r,2r)  r h+qp(h−1) . Thus, mh,s (B(c,r)) =

∞ 

  mh,s A(c,2−n r,2−(n−1) r)

n=1



∞ 

r h+qp(h−1) 2−n(h+qp(h−1))

n=1

= r h+qp(h−1)

∞ 

2−n(h+qp(h−1)) .

n=1

Since mh,s (B(c,r)) is finite, it first follows from this formula that qp , and then that h + qp(h − 1) > 0, or, equivalently, that h > 1+qp mh,s (B(c,r))  r h+qp(h−1) . 

We are done.

Proposition 22.3.6 Let f : C −→  C be a compactly nonrecurrent elliptic function with Crit∞ (f ) = ∅. If ω ∈ (f ), l ≥ 1 is an integer, c ∈ f −l (ω) c p(ω) , then ω ∈ Jμh (∞). \PC(f ) is a critical point of f l , and h ≤ q2q c p(ω)+1 Proof There exists R0 > 0 such that PC(f ) ∩ B(ω,2R0 ) = ∅. Fix 0 < R ≤ R0 . We may assume without loss of generality that X0 ⊆ B(c,R) is the set defined by (22.3) with n = 0. As in the previous proof, set q := qc . Let fc−l (B(ω,2R0 )) be the connected component of fc−l (B(ω,2R0 )) containing c. For every set G ⊆ B(ω,2R0 ), let fc−l (G) = fc−l (B(ω,2R0 )) ∩ f −l (G). By virtue of Lemma 22.3.5, assuming that R0 > 0 is small enough, we get, for every k ≥ 0, that h+pq(h−1)     − 1 1−h pq+1 pq mh,s fc−l fω−kl (J (f ) ∩ B(ω,R))  (kl) pq = (kl) k

1−h pq+1 pq

.

374

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Applying Theorem 8.3.12, we, therefore, conclude that, for all integers k,n ≥ 1,    mh,s fc−l fω−kl (J (f ) ∩ B(ω,R)) −l −kl mh,n (fc (fω (J (f ) ∩ B(ω,R))))  ms (X0 )  −l  −kl   mh,s fc fω (J (f ) ∩ B(ω,R)) k

1−h pq+1 pq

(22.29)

,

where mh,n := (mh,s )n are the measures defined by (2.37). Since the sequence   ∞ ms,n fc−l (fω−kl (J (f )) ∩ B(ω,R)) n=0

is bounded and since, by virtue of Lemma 22.1.2, fc−l (J (f ) ∩ B(ω,R)) ∩

∞ 

Yj = ∅

j =s

for all s ≥ 0 large enough, (2.40) and (22.29) yield   μh fc−l (fω−kl (J (f ) ∩ B(ω,R)))  ∞  1−h pq+1 pq . = lB mh,n (fc−l (fω−kl (J (f ) ∩ B(ω,R)))) n=0 $ k But, since the measure μ is f l -invariant, we, therefore, get, for every integer k ≥ 0, that   μh fω−kl (J (f ) ∩ B(ω,R))   = μh f −l (fω−kl (J (f ) ∩ B(ω,R)))   = μh fω−(k+1) (J (f ) ∩ B(ω,R))    + μh fy−l (fω−k (J (f ) ∩ B(ω,R))) y∈f −l (ω)\{ω}

    ≥ μh f −(k+1)l (J (f ) ∩ B(ω,R)) + μh fc−l (fω−k (J (f ) ∩ B(ω,R)))   1−h pq+1 pq , ≥ μh f −(k+1)l (J (f ) ∩ B(ω,R)) + B(k + 1) with some constant B ∈ (0,+∞) independent of k ≥ 0. Therefore, by induction, μh (B(ω,R)) ≥ B

∞ ∞   1−h pq+1 1−h pq+1 pq = B pq = +∞, (k + 1) k k=0

as, by our assumptions, enough, we are done.

1 − h pq+1 pq

k=1

≤ −1. Since this holds for all R > 0 small 

22 Conformal Invariant Measures for CNRR Functions

375

22.4 Closed Invariant Subsets, K(V ) Sets, and Summability Properties In this section, we deal with compact forward-invariant subsets of nonrecurrent elliptic functions. For the sake of future applications, we will be mostly occupied with projected maps on the tori. We will define and characterize expanding and pseudo-expanding invariant sets. We will get good upper estimates of their Hausdorff and box-counting dimensions. Next, similarly to the proof of Theorem 20.1.1, but even more thoroughly and actually working on the projected torus Tf , we will highlight in the current section the method of K(V ) sets developed in [DU3]; see also [KU6]. Its ultimate goal will be Lemma 22.4.21, a summability result which will constitute the main ingredient in Sections 22.5.1, 22.5.2, and 22.7 for proving strong regularity of conformal graph directed Markov systems resulting from nice sets. This, in turn, will be instrumental, in the same sections, for proving the statistical properties of dynamical systems generated by elliptic functions and the finiteness of the corresponding Kolmogorov–Sinai metric entropy. ˆ f −→ Tf is the projection of an elliptic function Recall that fˆ : T f : C −→  C to the torus Tf defined by the diagram (22.14). Denote B∞ (fˆ) := f (f −1 (∞)).

(22.30)

Note that B∞ (fˆ) is a finite set. Given integers l ≥ 1, we will make, in this section, frequent use of holomorphic maps fˆl : T(l) f := Tf \

l−1 

fˆ−j (B∞ (fˆ)) −→ Tf .

(22.31)

j =0

Recall that l−1  

  (l) fˆ−j Crit(fˆ) . Crit(fˆl ) = x ∈ Tf : (fˆl ) (x) = 0 =

(22.32)

j =0

We also record for future use the following fact, which directly results from Theorem 8.2.22. Proposition 22.4.1 If  is a lattice in C and V is an open connected, simply connected subset of T , then, for each connected component U of the inverse image −1  (V ) under the covering projection map  : C −→ T , the map  |U : U −→ V is one-to-one and, thus, a conformal homeomorphism; in consequence, the set U is open, connected, and simply connected.

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Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Now we shall prove for the maps fˆ of complex tori the direct analogs of Theorems 17.1.6–17.1.8. Theorem 22.4.2 If f : C −→  C is an elliptic function, then   Sing(fˆ−1 ) = fˆ(Crit(fˆ)) = f f (Crit(f )) . Moreover, for every integer l ≥ 1, we have that Sing(fˆ−l ) =

l−1  k=0

l−1       fˆk fˆ(Crit(fˆ)) = f f k f (Crit(f )) .

(22.33)

k=0

Proof The second equality sign in (22.33) is obvious. So, by invoking (17.6) of Theorem 17.1.6, we see that, in order to conclude the proof, it suffices to show that   Sing(fˆ−l ) = f C ∩ Sing(f −l ) . So, fix a point ξ ∈ Tf \Sing(fˆ−l ). By the definition of the singular Sing(fˆ−l ), there exists a radius r > 0 so small that, for each connected component  of fˆ−l (B(ξ,r)), the map fˆl | :  −→ B(ξ,r) is a conformal homeomorphism. We can further assume r > 0 to be so small that the projection map f restricted to each connected component of −1 f (B(ξ,r)) is an isometry.

Now fix any point z ∈ −1 f (ξ ). Then Be (z,r) is the connected component of −1  (B(ξ,r)) containing z. Let C be a connected component of fˆ−l (B(ξ,r)). f

Then, using Theorem 17.1.6, we get that fˆl (f (C)) = f (f l (C)) = f (Be (ξ,r)) = B(ξ,r). So, since the set f (C) is connected, there exists a unique connected component Cˆ of fˆ−l (B(ξ,r)) containing f (C). Let C˜ be the connected ˆ ˆ component of −1 f (C) containing C. Since C is simply connected, it follows from Proposition 22.4.1 that the map f |C˜ is one-to-one. Hence, the map fˆl ◦ f |C˜ is one-to-one too. But fˆl ◦ f = f ◦ f l and C˜ ⊇ C, whence the map f ◦ f l |C is one-to-one. But we already know that f l (C) = B(z,r) and the map f |Be (z,r) is one-to-one. Hence, the map f l |C : −→ Be (z,r) is one-to-one. Since, by item (1) of Theorem 17.1.7, it is also surjective, z∈ / Sing(f −l ). In conclusion,   −1 Tf \Sing(fˆ−l ) ⊆ C\Sing(f −l ). f But

  ˆ−l ) = C\−1 (Sing(fˆ−l )), −1 \Sing( f T f f f

22 Conformal Invariant Measures for CNRR Functions

377

ˆ−l whence C ∩ Sing(f −l ) ⊆ −1 f (Sing(f )). Therefore,   Sing(fˆ−l ) ⊇ f C ∩ Sing(f −l ) .

(22.34)

Proving the opposite inclusion, suppose that   ξ ∈ Tf \f C ∩ Sing(f −l ) .   Since Sing(f −l ) is a finite set, so is f C ∩ Sing(f −l ) , and there exists a radius r > 0 so small that   (22.35) B(ξ,r) ∩ f C ∩ Sing(f −l ) = ∅ and the map f restricted to each connected component of f −1 (B(ξ,r)) is an isometry. Let  be a connected component of fˆ−l (B(ξ,r)) and ˜ be an arbitrary connected component of −l f (). Then ˜ = fˆl ◦ f () ˜ ⊆ fˆl () ⊆ B(ξ,r). f ◦ f l () ˜ is connected, there exists a unique connected compoSo, since the set f l () (B(ξ,r)) containing f l () and, by our choice of r, it must be of nent of −1 f the form Be (z,r) with some z ∈ −1 f (B(ξ,r)). It follows from (22.35) that Be (z,r) ∩ Sing(f −l ) = ∅.

(22.36)

˜ Since Then let ∗ be the connected component of f −l (Be (z,r)) containing . l the map f ◦ f |∗ is, by virtue of (22.36) and Theorem 17.1.7(3), one-to-one, and since f ◦ f l = fˆl ◦ f , we get that the map fˆ l ◦ f |∗ is one-to-one. ˜ = , we conclude But then fˆ l |f ∗ is one-to-one and, as f (∗ ) ⊇ f () l ˆ that the map f | :  −→ B(ξ,r) is one-to-one. Also, by item (1) of Theorem 17.1.7, fˆl (f (∗ )) = f (f l (∗ )) = f (B(e (z,r))) = B(ξ,r). Therefore, as f (∗ ) ⊇ , it follows from the definition of  and the connectedness of the set f (∗ ) that  = f (∗ ). Hence, fˆ l () = B(ξ,r). In conclusion, the map fˆl | :  −→ B(ξ,r) is a conformal homeomorphism. Therefore, ξ ∈ / Sing(fˆ−l ). Hence,   Tf \f C ∩ Sing(f −l ) ⊆ Tf \Sing(fˆ−l ). Equivalently,   Sing(fˆ−l ) ⊆ f C ∩ Sing(f −l ) .

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Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Along with (22.34), this gives   Sing(fˆ−l ) = f C ∩ Sing(f −l ) , completing the proof of Theorem 22.4.2.



Having this theorem proved, we could prove the next theorem below analogously to the way that Theorem 17.1.7 was derived from Theorem 17.1.6 with the use of the Monodromy Theorem. However, for the sake of completeness and the convenience of the reader, we will directly deduce it from Theorem 17.1.7. Theorem 22.4.3 Let f : C →  C be an elliptic function. If V ⊆ Tf is an open connected set, l ≥ 1 is an integer, and U is a connected component of fˆ−l (V ), then (1) fˆl (U ) = V . (2) If, in addition, U ∩ (Crit(fˆl )) = ∅, e.g., if V ∩

l−1 

fˆj (Crit(fˆ)) = ∅,

j =1

then the map fˆl |U : U −→ V is covering. (3) Furthermore, if, additionally, V is simply connected, then the map fˆl |U : U −→ V is a conformal homeomorphism and, so, U is simply connected too. Proof

Proving item (1), let U˜ be a connected component of −1 f (U ). Then f ◦ f l (U˜ ) = fˆ l ◦ f (U˜ ) = fˆ l (U ) ⊆ V .

Since the set fˆ l (U˜ ) is connected, there, thus, exists a unique connected l ˜ component H of −1 f (V ) containing f (U ). Then let G be the connected component of f −l (H ) containing U˜ . Then, by Theorem 17.1.7(1), fˆ l ◦ f (G) = f ◦ f l (G) = f (H ) = V .

(22.37)

Since also f (G) ⊇ f (U˜ ) = U and since the set f (G) is connected, we conclude that f (G) = U . So, fˆ l (U ) = V by invoking (22.37). Item (1) of our theorem is proved.

22 Conformal Invariant Measures for CNRR Functions

379

Proving item (3), let U˜ be an arbitrary connected component of −1 f (U ). Then, using the already proven item (1), we get that f ◦ f l (U˜ ) = fˆ l ◦ f (U˜ ) = fˆ l (U ) = V .

(22.38)

So, since the set f l (U˜ ) is connected, it is contained in a unique connected component of −1 f (V ). We denote it by H . But since V is simply connected, it follows from Theorem 8.2.22 that H is simply connected and the map f |H : H −→ V is a (conformal) homeomorphism. Therefore, since also fˆ l (U˜ ) ⊆ H , it follows from (22.38) that H = f l (Uˆ ). Since, by our hypothesis stated in item (2) and by 22.32, Crit(f l ) ∩ f l (Uˆ ) = ∅, it follows from item (3) of Theorem 17.1.7 that the map f l : U˜ ∗ −→ f l (Uˆ ) is a conformal homeomorphism, where U˜ ∗ is the unique connected component of f −l (f l (Uˆ )) containing U˜ . We, thus, immediately conclude that U˜ ∗ = U˜ . We consider the map −1 f ◦ f l |−1 ˜ ◦ f | l

f (U˜ )

U

and we have that  l −1 fˆl ◦ f ◦ f l |−1 ˜ ◦ f ◦ f | l U

f (U˜ )



: V −→ U

(22.39)

−1 = f ◦ f l ◦ f l |−1 ˜ ◦ f | l

=

U f ◦ f |−1l ˜ = f (U )

f (U˜ )

IdV .

This means that this map of (22.39) is a holomorphic branch of fˆ−1 from V to U . Therefore, the proof of item (3) is concluded by applying Lemma 13.54, which holds, with the same proof, for holomorphic maps of complex tori. Item (3) is now an immediate consequence of items (2) and (1). The proof of Theorem 22.4.3 is complete.  As a direct consequence (reformulation) of this theorem, we get the following. Theorem 22.4.4 Let f : C −→  C be an elliptic function and V ⊆ Tf be an open connected, simply connected set. If l ≥ 0 is an integer, ξ ∈ fˆ−l (V ), and the (unique) connected component Uξ of fˆ−l (V ) containing ξ contains  ˆj ˆ no critical points of fˆl (e.g., if V ∩ l−1 j =1 f (Crit(f )) = ∅), then (1) the map fˆl |Uξ : Uξ −→ V is a conformal homeomorphism, (2) there exists a unique holomorphic branch fˆξ−l : V −→ Uξ (equal to fˆ|−l Uξ ) −l l of fˆ sending fˆ (ξ ) to ξ , and (3) the map fˆ−l : V −→ Uξ is a conformal homeomorphism; ξ

(4) in particular, fˆξ−l (V ) = Uξ .

380

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Set (fˆ) := f ((f )). Obviously, (fˆ) is the set of all rationally indifferent periodic points of fˆ : Tˆ f −→ Tf . From now on throughout the remainder of this section, we let f : C −→  C be a nonrecurrent elliptic function. Definition 22.4.5 Fix an integer l ≥ 1. We say that a closed (equivalently (l) (l) compact) set X ⊆ Tf ∩ J (fˆ) in Tf is expanding for fˆl : Tf −→ Tf if and only if (a) fˆl (X) ⊆ X, i.e., X is forward invariant under fˆl and   (b) X ∩ Crit(fˆl ) ∪ (fˆ) = ∅. If, instead of (b), we merely have that   (c) (X\(fˆ)) ∩ Crit(fˆl ) ∪ (fˆ) = ∅, then X is called quasi-expanding for fˆl . Of course, each expanding set is quasi-expanding. We shall now prove some basic properties of expanding and quasi-expanding sets. In particular, we will prove some characterizations of such sets that will fully justify the names expanding and quasi-expanding. We start with the following. Because of (a), X∩

l−1 

fˆ−j (B∞ (fˆ)) = ∅.

(22.40)

j =0

We, therefore, immediately get the following. ˆ be a nonrecurrent elliptic function. Fix Observation 22.4.6 Let f : C −→ C ˆ) is a quasi-expanding set for fˆl , then an integer l ≥ 1. If X ⊆ T(l) ∩ J ( f f X ∩ Crit(fˆl ) = ∅. In addition, looking up at the right-hand side of (22.32), we get the following. ˆ be a nonrecurrent elliptic function. Fix Observation 22.4.7 Let f : C −→ C an integer l ≥ 1. Then (1) If X ⊆ Tf ∩ J (fˆ) is an expanding set for fˆl , then ⎛ ⎞ l−1  ⎝ fˆj (X)⎠ ∩ Crit(fˆ) = ∅. (l)

j =0

22 Conformal Invariant Measures for CNRR Functions

381

ˆ ˆl (2) If X ⊆ T(l) f ∩ J (f ) is a quasi-expanding set for f , then ⎛ ⎝

l−1 

⎞ fˆj (X)⎠ ∩ Crit(fˆ) = ∅.

j =0

We also record the following immediate observation. ˆ be a nonrecurrent elliptic function. Fix Observation 22.4.8 Let f : C −→ C (l) ˆ an integer l ≥ 1. If X ⊆ Tf ∩ J (f ) is a closed set in Tf invariant under fˆl , then the following conditions are equivalent: (1) X is quasi-expanding for fˆl . (2) X ∩ Crit(fˆl ) = ∅ and X\(fˆl ) ⊆ Tf is a closed set. We shall prove the following. ˆ be a nonrecurrent elliptic function. Fix an Lemma 22.4.9 Let f : C −→ C (l) integer l ≥ 1. If X ⊆ Tf ∩ J (fˆ) is a quasi-expanding set for fˆl , then there exists η > 0 such that if x ∈ X, n ≥ 0 is an integer, and fˆln (x) ∈ / (fˆ), then Crit(fˆln ) ∩ Comp(x, fˆln,2η) = ∅.

(22.41)

In particular, the holomorphic branch fˆx−ln : B(fˆ−ln (x),2η) −→ Tf of fˆln sending fˆln (x) back to x is, by Theorem 22.4.4, well defined. Proof Since the set X\(fˆ) is compact, so is the set ˆ X˜ := −1 f (X\(f )) ∪ {∞} ⊆ J (f )\(f ). Also, because of Observation 22.4.6,   ε := dist Crit(fˆl ),X > 0. It, thus, follows Theorem 18.1.8 that there exists η > 0 such that   diame fˆlj (Comp(x, fˆln,2η)) < ε / (fˆ). for every n ≥ 0, every 0 ≤ j ≤ n, and every x ∈ X such that fˆln (x) ∈ Therefore,   Crit(fˆl ) ∩ fˆlj Comp(x, fˆln,2η) = ∅ for every 0 ≤ j ≤ n. Formula (22.41), thus, holds and Lemma 22.4.9 is proved. 

382

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

ˆ be a nonrecurrent elliptic function. Fix an Lemma 22.4.10 Let f : C −→ C (l) integer l ≥ 1. If X ⊆ Tf ∩ J (fˆ) is a quasi-expanding set for fˆl , then there exist constants C ∈ [1,+∞) and β > 0 such that  ln   (fˆ ) (z) ≥ Ceβn for every integer n ≥ 0 and every z ∈ X such that fˆln (z) ∈ / (fˆ). Proof Applying Theorem 18.1.8 in the same way as in the proof of Lemma 22.4.9 but this time also with the help of this lemma itself and the Koebe Distortion Theorem, we conclude that there exists an integer N ≥ 1 such that  lk   (fˆ ) (x) ≥ e (22.42) for every integer k ≥ N and every x ∈ X such that fˆlk (x) ∈ / (fˆ). If n ≥ 0 / (fˆ), then is an integer, z ∈ X, and fˆln (z) ∈ fˆlj (z) ∈ X\(f ) for every integer 0 ≤ j ≤ n. Write uniquely that ln := qlN + r, where q,r ≥ 0 are integers with r ≤ lN − 1. Applying then (22.42) qN times, we get that  ln    qlN    r  qlN  (fˆ ) (z) = (fˆ ) (z) · (fˆ ) (fˆ (z)) ≥ eq |(fˆr ) (fˆqlN (z))| ≥ e−r M lN e N n, 1

 where M := min 1, inf{|(fˆ )(x)| : x ∈ X} > 0 since, because of Observation 22.4.7, ⎛ ⎞ l−1  dist ⎝ f j (X),Crit(fˆ)⎠ > 0. j =0



The proof is complete. As a fairly immediate consequence of this lemma, we get the following.

ˆ be a nonrecurrent elliptic function. Fix Corollary 22.4.11 Let f : C −→ C l ˆ an integer l ≥ 1. A closed f -invariant set X ⊆ J (f ) ∩ T(l) f is expanding for l ˆ f if and only if there exist constants C ∈ [1,+∞) and β > 0 such that  ln   (fˆ ) (z) ≥ Ceβn for every integer n ≥ 0 and every z ∈ X.

(22.43)

22 Conformal Invariant Measures for CNRR Functions

383

Proof The implication (⇒) is a direct consequence of Lemma 22.4.10. Proving (⇐), (22.43) implies that both X∩

l−1 

fˆ−j (Crit(fˆ)) = ∅ and X ∩ (fˆ) = ∅.

j =0

Looking up at the right-hand side of (22.32), the proof is, thus, complete.



We also have the following. ˆ be a nonrecurrent elliptic function. Fix Corollary 22.4.12 Let f : C −→ C l ˆ an integer l ≥ 1. A closed f -invariant set X ⊆ J (f ) ∩ T(l) f is quasi-expandng for fˆl if and only if there exist constants C ∈ [1,+∞) and β > 0 such that  ln   (fˆ ) (z) ≥ Ceβn (22.44) / (fˆ). for every integer n ≥ 0 and every z ∈ X such that fˆln (z) ∈ Proof The implication (⇒) is a direct consequence of Lemma 22.4.10. Proving (⇐), (22.44), as in the proof of the previous corollary, implies that X∩

l−1 

fˆ−j (Crit(fˆ)) = ∅.

j =0

By virtue of the right-hand side of (22.32), this means that X ∩ Crit(fˆl ) = ∅. Seeking contradiction, suppose that (X\(fˆ)) ∩ (fˆ) = ∅.

(22.45)

Passing to an appropriate integral multiple of l, we may assume that each element of (fˆ) is a simple parabolic fixed point of fˆl . Fix an integer n ≥ 1. Then there exists ε > 0 such that    ln   fˆ (x) ≤ 2 (22.46) for all x ∈ B((fˆ),ε). By virtue of (22.45) and the continuity properties of ˆl ˆln fˆl , there exists xn ∈ (X\(fˆl )) / (fˆl ). It  f ),ε) such that f (x) ∈  ∩ B(( then follows from (22.46) that | fˆln (xn )| ≤ 2. Hence,     lim sup  fˆln (xn ) ≤ 2. n→∞

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Part VI Fractal Geometry, Stochastic Properties, and Rigidity

On the other hand, it follows from (22.44) that     lim inf  fˆln (xn ) = +∞. n→∞



This contradiction finishes the proof of Corollary 22.4.12. We will also need the following straightforward observation.

ˆ be a nonrecurrent elliptic function. Fix Observation 22.4.13 Let f : C −→ C (l) ˆ an integer l ≥ 1. If X ⊆ Tf ∩J (f ) is quasi-expanding for fˆl , then fˆl ∈ A(X) in the sense of Chapter 9. We shall now provide a natural way of constructing expanding and quasiexpanding sets for elliptic functions. Let f : C −→  C be an elliptic function. Recall from (17.21) that if V is an open subset of  C, then KJ (V ) = J (f ) ∩ f −n ( C\V ). n≥0

Given an open set Vˆ ⊆ Tf , the set K(Vˆ ) has an analogous meaning to the set K(V ) introduced above: (22.47) fˆ−n (Tf \Vˆ ). K(Vˆ ) := n≥0

More generally, given an integer l ≥ 1 and an open set Vˆ ⊆ Tf , let Kl (Vˆ ) := f −ln (Tf \Vˆ ). n≥0

Of course, Kl (Vˆ ) = K(Vˆ ). We collect the most basic properties of the sets Kl (Vˆ ) in the following. Observation 22.4.14 Let f : C −→  C be a nonrecurrent elliptic function. Fix an integer l ≥ 1 and an open set Vˆ ⊆ Tf . Then (a) The map fˆl is well defined on Kl (Vˆ ) and   fˆl Kl (Vˆ ) ⊆ Kl (Vˆ ). (b) Kl (Vˆ ) ∩

∞ 

fˆ−n (B∞ (fˆ)) = ∅.

n=0 (1) (c) Kl (Vˆ ) is a closed subset of Tf = Tf \B∞ (fˆ). (1) (d) If B∞ (fˆ) ⊆ Vˆ , then Kl (Vˆ ) is a closed, thus compact, subset of T . f

22 Conformal Invariant Measures for CNRR Functions

385

(e) If B∞ (fˆ)∪Crit(fˆl ) ⊆ Vˆ and (Vˆ ∪(fˆ))∩J (fˆ) is an open subset relative to J (fˆ), then the closed f l -invariant set Kl (Vˆ ) is quasi-expanding for fˆl . (f) If, instead of the second hypothesis of (e), we assume more, namely that Vˆ ⊇ (fˆ), then Kl (Vˆ ) is expanding for fˆl . Sometimes, in order to be more precise, we will write Kf (V ) or Kfˆ (Vˆ ) to indicate whether we mean the subsets of  C or Tf . We shall prove the following. Lemma 22.4.15 Let f : C −→  C be a nonrecurrent elliptic function. If ˆ X ⊆ J (f ) is a closed (equivalently compact) fˆ-forward-invariant expanding set, then BD(X) < HD(J (f )). Proof For every j ≥ 1, put   Wˆ j := B B∞ (fˆ) ∪ Crit(fˆ) ∪ (fˆ),1/j ⊆ Tf . Because of Observations 22.4.13 and 22.4.14 the map fˆ|K(Wˆ j ) : K(Wˆ j ) −→ ˆ j ) belongs to A(K(Wˆ j )) in the sense of Chapter 9 (with the ambient K(W Riemann surface Y equal to Tf ). By Lemma 10.2.4, we have that   sj := s fˆ|K(Wˆ j ) ≤ HD(K(Wˆ j )) ≤ HD(J (f )) = h.

(22.48)

Seeking contradiction, assume that sj = h. Let mj be the Borel probability measure produced in Lemma 10.2.5 with X = K(Wˆ j ) for the map fˆ. Since, ˆ j ) is expanding for fˆ, it follows from by Observation 22.4.14, the set K(W Lemma 22.4.9 that, for every j ≥ 1, there exists ηj > 0 such that if z ∈ K(Wj ) and n ≥ 0, then there exists a unique holomorphic inverse branch fˆz−n : B(fˆn (z),2ηj ) −→ Tf of f n defined on B(fˆn (z),2ηj ) that sends fˆn (z) to z. So, applying in a standard way the Koebe Distortion Theorem (Theorem 8.3.8) along with Lemmas 10.2.5 and 1.3.8, we deduce that there exists a constant C ∈ (0,+∞) such that ˆ h (A) mj (A) ≤ C m for every Borel set A ⊆ K(Wˆ j ). Thus, m ˆ h (K(Wˆ j )) ≥ C −1 mj (K(Wˆ j )) = 1/C > 0,

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Part VI Fractal Geometry, Stochastic Properties, and Rigidity

contrary to the last item of Proposition 22.2.2 as K(Wˆ j ) is a nowhere dense subset of J (fˆ). So, (22.49)

sj < h.

ˆ ˆ be a weak limit of the sequence  ≥ 1 so large that X ⊆ K(Wu ). Let m Fix u mj j ≥u treated as Borel probability measures on Tf , i.e., m ˆ := lim mkj j →∞

(22.50)

for some strictly increasing sequence (kj )∞ j =u of positive integers. Substituting the set B(fˆ) ∪ (f ) for Y , the same considerations as those leading to Claim

1◦ in the proof of Theorem 17.6.7, yield, with s := s(Y ), the formula |fˆ |s d m ˆ

m( ˆ fˆ(A)) =

(22.51)

A

  for every special set A ⊆ J (fˆ)\ B(fˆ) ∪ (fˆ) , i.e., formula (b) of Claim 1◦ from the proof of Theorem 17.6.7 will hold; (a), (c), and (d) will hold too. Now, since the map f : C −→  C is nonrecurrent, it cannot have arbitrarily long chains of inverse images consisting of critical points of fˆ. We, therefore, deduce from (22.51) that supp(m) ˆ ⊇

∞ 

ˆ fˆ−n (supp(m)).

n=0

So, since supp(m) ˆ = ∅ and since we conclude that

∞

ˆ−n (z) = J (fˆ) for every z ∈ J (fˆ),

n=0 f

supp(m) ˆ = J (fˆ).

(22.52)

Since X is compact, we can cover it by some finitely many open balls {B(z,ηu /2)}z∈E , where E is some finite subset of X. Now fix x ∈ X arbitrary and a radius r ∈ (0,ηu ]. Since X ⊆ K(Wˆ u ) is compact, fˆ-forward invariant, and disjoint from (fˆ), with the help of (16.8)–(16.10), we deduce from Theorem 18.1.8 (and our choice of ηu ) that there exists a least integer n ≥ 0 such that   (22.53) Comp x, fˆn,ηu ⊆ B(x,r).   Then n ≥ 1 and Comp x, fˆn−1,ηu ⊆ B(x,r). Therefore, using the Koebe Distortion Theorem (Theorem 8.3.8), we get that   r ≤ diam Comp(x, fˆn−1,ηu ) ≤ Kηu |(fˆn−1 ) (x)|−1 ≤ Kηu D|(fˆn ) (x)|−1, (22.54)

22 Conformal Invariant Measures for CNRR Functions

387

where the number D := fˆ |X ∞ is finite since X is compact and disjoint from B(fˆ). Because of (22.50) and (22.52), and the definition of εu , there exists i ≥ u (in fact, all but finitely many) such that 

M := min mki (B(y,ηu /2)) : y ∈ E > 0 and   fˆk Comp(x, fˆn,2ηu ) ∩ K(Wki )0 = ∅ for all k = 0,1, . . . ,n, where the set K(Wki )0 is understood in the sense of Section 10.1. It then follows from Lemma 10.2.5 that   mki Comp(x, fˆn,ηu ) =

B(fˆn (x),ηu )

|(fˆx−n ) |si dmki .

Hence, using (22.53) and the Koebe Distortion Theorem (Theorem 8.3.8), we further obtain that   −s   mki (B(x,r)) ≥ mki Comp(x, fˆn,ηu ) ≥ K −ski  fˆn (x) kim(B(fˆn (x),ηu )). (22.55) Since f n (x) ∈ X, there exists y ∈ E such that fˆn (x) ∈ B(y,ηu /2). Therefore, mki (B(fˆn (x),ηu )) ≥ mki (B(y,ηu /2)) ≥ M. Hence, (22.54) and (22.55) yield   −s mki (B(x,r)) ≥ MK −ski  fˆn (x) ki ≥ M(K 2 ηu D)−si r si .

(22.56)

At this moment, we can invoke Proposition 1.8.7 to conclude that BD(X) ≤ ski . Along with (22.49), applied for j = ki , this finally yields BD(X) < h = HD(J (f )). The proof is complete.  As an immediate consequence of Lemma 22.4.15 and Observation 22.4.14, we get the following. Lemma 22.4.16 Let f : C −→  C be a nonrecurrent elliptic function. If Vˆ ⊆ Tf is an open neighborhood of B∞ (fˆ) ∪ Crit(fˆ) ∪ (fˆ), then   BD J (fˆ) ∩ K(Vˆ ) < HD(J (f )). For the sake of future dealing with parabolic elliptic functions, we will need a slightly stronger result and its slightly more general consequence. We shall prove the following.

388

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Lemma 22.4.17 Let f : C −→  C be a nonrecurrent elliptic function. If Vˆ ⊆ Tf is an open neighborhood of B∞ (fˆ) ∪ Crit(fˆl ) ∪ (fˆ), then   BD J (fˆ) ∩ Kl (Vˆ ) < HD(J (f )) for every integer l ≥ 1. Proof

Because of Observation 22.4.14, the set X := J (fˆ) ∩

l−1 

fˆlj (Kl (Vˆ )) ⊆ J (fˆ)

j =0

is compact and f -forward invariant. Since Kl (Vˆ ) ∩ B∞ (fˆ) = ∅ and Kl (Vˆ ) is f l -forward invariant, we have that X ∩ B∞ (fˆ) = ∅. Since Kl (Vˆ ) ∩

l−1 

fˆ−j (Crit(fˆ)) = ∅,

j =0

we have that X ∩ Crit(fˆ) = ∅. Finally, since fˆ((fˆ)) = (fˆ), we conclude that X ∩ (fˆ) = ∅. Therefore, by Observation 22.4.14, the set X is expanding for fˆ and it also satisfies all the requirements of Lemma 22.4.15. Noting that HD(X) = HD J (fˆ) ∩ Kl (Vˆ ) and applying this lemma completes the proof.  As a fairly immediate consequence of this lemma, we get the following. Lemma 22.4.18 Let f : C −→  C be a nonrecurrent elliptic function. Fix an ˆ ⊆ Tf is an open set containing B∞ (fˆ) ∪ Crit(fˆl ) such that integer l ≥ 1. If V   Vˆ ∪ (fˆ) ∩ J (fˆ) is an open neighborhood of (fˆ) relative to J (fˆ), then   HD J (fˆ) ∩ Kl (Vˆ ) < HD(J (f )). Proof

By our hypotheses, there exists an open set Vˆ ∗ ⊆ T such that   Vˆ ∗ ∩ J (fˆ) = Vˆ ∪ (fˆ) ∩ J (fˆ).

Then Vˆ ∗ satisfies all the hypotheses of Lemma 22.4.17 and ∞    fˆ−n ((fˆ)). J (fˆ) ∩ Kl (Vˆ ) ⊆ J (fˆ) ∩ Kl (Vˆ ∗ ) ∪ n=0

22 Conformal Invariant Measures for CNRR Functions

389

∞ ˆ−n ((fˆ)) is countable and Therefore, remembering that the set n=0 f applying Lemma 22.4.17, we get that / ∞

0      ∗ −n HD J (fˆ) ∩ Kl (Vˆ ) ≤ max HD J (fˆ) ∩ Kl (Vˆ ) ,HD fˆ ((fˆ)) n=0

    = HD J (fˆ) ∩ Kl (Vˆ ∗ ) ≤ BD J (fˆ) ∩ Kl (Vˆ ∗ ) < HD(J (f )). 

The proof is complete. Sticking to Vˆ , an open subset of Tf , and an integer l ≥ 1, let Dl (Vˆ )

denote the family of all connected components of Tf \Kl (Vˆ ). If ˆ is a connected component of Vˆ , denote by ˆ + the only element of Dl (Vˆ ) ˆ Let Vˆ+ be the union of all such components ˆ + . Note that containing . Kl (Vˆ+ ) = Kl (Vˆ )

(22.57)

Dl (Vˆ+ ) = Dl (Vˆ ).

(22.58)

and

We call the open set Vˆ dynamically maximal if Vˆ+ = Vˆ . For every W ∈ Dl (Vˆ ), let kVˆ (W ) ≥ 0 be the least integer n ≥ 0 such that fˆln (W ) ∩ Vˆ = ∅. Equivalently, fˆln (W ) ∩ Vˆ+ = ∅, and then fˆlkVˆ (W ) (W ) is a connected component of Vˆ+ . Given a component Hˆ of Vˆ+ , denote 

   Dl Vˆ , Hˆ := W ∈ Dl Vˆ ) : fˆlkVˆ (W ) (W ) = Hˆ . ˆ ⊆ Tf horizontal of type 1 if there exist a We call an open connected set G positive radius RGˆ > 0 and a point zGˆ ∈ Vˆ such that     Vˆ ⊆ B zGˆ ,RGˆ and B zGˆ ,2RGˆ ∩ PC(fˆ) = ∅.

390

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

We call an open connected set Vˆ ⊆ Tf horizontal of type 2 if each connected component of fˆ−1 (Vˆ ) whose closure is disjoint from (fˆ) is horizontal of type 1. Such sets exist, for example, if   ω∈ / O+ J (fˆ) ∩ Crit(fˆ) . ˆ ⊆ Tf horizontal if it is horizontal of either We call an open connected set G type 1 or type 2. Finally, we call an open set contained in Tf horizontal if all its connected components are horizontal. Keeping Vˆ ⊆ Tf an open set, if b ∈ Vˆ \Kl (Vˆ ), then we denote by Vˆ (b) the connected component of Vˆ containing b. Then Vˆ+ (b) := Vˆ (b)+ is the connected component of Tf \Kl (Vˆ ) containing Vˆb or, equivalently, containing just b. The following observation is immediate from the definition of nice sets defined and extensively dealt with in Chapter 12. Observation 22.4.19 Let f : C −→  C be an elliptic function. If l ≥ 1 is an integer and Vˆ ⊆ Tf is a pre-nice set (in particular, if it is a nice set) for the ˆ map fˆ l : Tˆ (l) f −→ Tf , then V is a dynamically maximal open set, meaning that Vˆ+ = Vˆ . Conversely, if a dynamically maximal open set Vˆ ⊆ Tf satisfies all the conditions (b) and (c) of Definition 12.1.4, then it is a pre-nice set for the ˆ (l) −→ Tf . map fˆ l : T f We record the following immediate consequence of Observation 22.4.14. Observation 22.4.20 Let f : C −→  C be a nonrecurrent elliptic function. Fix an integer l ≥ 1 and an open set Vˆ ⊆ Tf . Then (a) The map fˆl is well defined on Kl (Vˆ ) and   fˆl Kl (Vˆ ) ⊆ Kl (Vˆ ). (b) Kl (Vˆ ) ∩

∞ 

fˆ−n (B∞ (fˆ)) = ∅.

n=0 (1) (c) Kl (Vˆ ) is a closed subset of Tf = Tf \B∞ (fˆ). (1) (d) If B∞ (fˆ) ⊆ Vˆ + , then Kl (Vˆ ) is a closed, thus compact, subset of T . f

22 Conformal Invariant Measures for CNRR Functions

391

(e) If B∞ (fˆ) ∪ Crit(fˆl ) ⊆ Vˆ + and (Vˆ + ∪ (fˆ)) ∩ J (fˆ) is an open subset relative to J (fˆ), then the closed f l -invariant set Kl (Vˆ ) is quasi-expanding for fˆl . (f) If, instead of the second hypothesis of (e), we assume more, namely that Vˆ + ⊇ (fˆ), then Kl (Vˆ ) is expanding for fˆl . Although technical, the ultimate result of this section is the one formulated below. We want to add that the proof we present below is entirely different from the Przytycki and Rivera–Latelier’s proof of an analogous result in [PR]. Lemma 22.4.21 Let f : C −→  C be a nonrecurrent elliptic function. Fix an ˆ ⊆ Tf be an open set containing B∞ (fˆ) ∪ Crit(fˆl ) such integer l ≥ 1. Let V   that Vˆ ∪ (fˆ) ∩ J (fˆ) is an open neighborhood of (fˆ) relative to J (fˆ). If ˆ is a connected component of Vˆ such that ˆ + is horizontal and J (fˆ) ∩    ˆ + = ∅, then there exists a number t ∈ HD J (fˆ) ∩ Kl (Vˆ ) ,HD(J (f )) such that  diamt (W ) < +∞. W ∈Dl (Vˆ , ˆ + )

Proof Since J (fˆ)∩ ˆ + = ∅, there exists ξ ∈ J (fˆ)∩ ˆ + , which is a repelling periodic point of fˆl . Furthermore, there exists a nonempty open set G ⊆ Vˆ such that G ∩ {fˆlj (ξ ) : j ≥ 0} = ∅ and (G ∪ (fˆ)) ∩ J (fˆ) is an open neighborhood of (fˆ) in J (fˆ). Then, Kl (Vˆ ) ⊆ Kl (G), ξ ∈ Kl (G), and, by Observation 22.4.20, Kl (Vˆ ) ∩ J (fˆ) is a closed expanding fˆl -invariant subset of J (fˆ). For every integer n ≥ 0, define −n  (ξ ) ⊆ Kl (G)\(fˆ). En := fˆl |Kl (G) By Observation 22.4.13, fˆl ∈ A(Kl (G)) in the sense of Chapter 9. For every t ≥ 0, let cξ (t) be the corresponding transition parameter as defined at the beginning of Section 10.2. We shall prove the following. Claim 1◦ . The function [0,+∞)  t −→ cξ (t) is strictly decreasing. Proof For every α ≥ 0 and γ ∈ R, let 

(α,γ ) :=

∞    ln  −α −γ n (fˆ ) (x) e . n=0 x∈En

392

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Now fix 0 ≤ s < t and u > cξ (s) + s − t. It then follows from Lemma 22.4.10 that, for every integer n ≥ 0 and every point x ∈ En , we have that   ln  −t  ln    (fˆ ) (x) = (fˆ ) (x)|−s |(fˆln ) (x)s−t ≤ C s−t eβ(s−t)n (fˆln ) (x)−s . Hence, 

(t,u) =

∞  

|(fˆln ) (x)|−t e−un ≤ C s−t

n=0 x∈En

=C

s−t



∞  

 −s e−β(t−s+u)n (fˆln ) 

n=0 x∈En

(s,u + t − s) < +∞,

the last inequality holding because u + t − s > cξ (s). Therefore, cξ (t) ≤ u, whence cξ (t) ≤ cξ (s) + s − t < cξ (s). The proof of Claim 1◦ is complete.  Now it follows from Lemmas 22.4.18 and 10.2.4 and (10.38) that sl ≤ HD(J (fˆ) ∩ Kl (G)) < HD(J (fˆ))

(22.59)

cξ (sl ) = 0,

(22.60)

and

where sl := s(fˆl |Kl (G) ) is defined by (10.34). Fix an arbitrary   t ∈ HD(J (f ) ∩ Kl (G)),HD(J (f )) . It follows from Claim 1◦ and (22.59) and (22.60) that cξ (t) < 0. So, we conclude directly from the definition of c(t) that ∞    ln  −t (fˆ ) (x) < +∞.

(22.61)

n=0 x∈En

Now if n ≥ 1 and W ∈ Dl (Vˆ , ˆ + ) is such that kVˆ (W ) = n, then, since ˆ + is horizontal, there exists a unique holomorphic branch −lk ˆ (W ) : B(z,2R) −→ Tf fˆW V

of fˆlkVˆ (W ) such that f −lkVˆ (W ) (Vˆ + ) = ˆ +, where z = zˆ + and RGˆ + are the parameters witnessing horizontality of ˆ + . Let −lkVˆ (W )

ξw := fˆW

(z) = fˆW−ln (z).

22 Conformal Invariant Measures for CNRR Functions

393

Note that ξW ∈ Kl (G). So, denoting

 (n) Dl (Vˆ , ˆ + ) := W ∈ Dl (Vˆ , ˆ + ) : kVˆ (W ) = n , we have produced a one-to-one map Dl (Vˆ , ˆ + )  W −→ ξW ∈ En . (n)

It, therefore, follows from (22.61) and the Koebe Distortion Theorem that 

diamt (W ) =

W ∈Dl (Vˆ , ˆ + )

∞ 



diamt (W )

n=0 W ∈D (n) (Vˆ , ˆ + ) l

≤ Kt

∞ 



|(fˆln ) (ξW )|−t

n=0 W ∈D (n) (Vˆ , ˆ + ) l

≤ Kt

∞ 



|(fˆln ) (x)|−t

n=0 x∈En

< +∞. The proof is complete.



22.5 Normal Subexpanding Elliptic Functions of Finite Character: Stochastic Properties and Metric Entropy, Young Towers, and Nice Sets Techniques Throughout this section, unless stated otherwise, we assume that f : C −→  C is a normal subexpanding elliptic function of finite character; see Definitions 18.4.2, 18.4.3, and 18.4.6 and Theorem 18.4.15. By virtue of Observation 18.4.8, we have the following. Observation 22.5.1 Every normal subexpanding elliptic function of finite character is regular. Since the function f is compactly nonrecurrent, Theorems 20.3.11 and 22.1.1 apply. Employing these theorems, Observation 22.5.1, Theorem 21.0.1, and Corollary 22.1.13, we get the following. Theorem 22.5.2 If f : C −→  C is a normal subexpanding elliptic function of finite character, then (1) There exists a unique t ≥ 0 for which there exists a t-conformal atomless probability measure for f : J (f ) −→ J (f ) ∪ {∞}; namely, t = h.

394

(2) (3)

(4) (5) (6)

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Furthermore, there exists a unique spherical h-conformal atomless probability measure mt,s for f : J (f ) −→ J (f ) ∪ {∞}. The spherical h-conformal measure mh,s coincides with a (constant) multiple of the (finite and positive) packing measure hs on J (f ). There exists a unique probability f -invariant measure μh absolutely continuous with respect to mh . In addition, μh is atomless and equivalent to mh . The f -invariant Borel probability measure μh is metrically exact; in particular, ergodic and conservative. All other conformal measures are purely atomic, supported on Sing− (f ) with exponents larger than h. μh (Tr(f )) = 1.

This section, based on previous sections, and frequently using the concepts and results about nice sets in Chapter 12, is devoted to exploring the refined   stochastic properties of the dynamical system f |J (f ),μh . Employing the existence of nice sets and their giving rise to conformal maximal graph directed Markov systems in Chapter 11 (all done in Chapter 12), we will show that the Young tower abstract approach, described in Chapter 4, applies, and we will derive many dynamical and stochastic consequences from it, such as the Central Limit Theorem, the Law of the Iterated Logarithm, and the exponential decay of correlations. In the case when J (f ) =  C, item (3) of this theorem was proved in [KS] for all meromorphic Misiurewicz maps.

22.5.1 Geometric and Dynamical Preparations for Young Towers of Finite Character Elliptic Functions This subsection is motivated by the seminal paper [PR]. The absolutely crucial and necessary concept for us in this section is the one of nice sets from Chapter 12. We will work with the torus Tf phase space, and eventually we will transfer the fruits of this work back to C again. In particular, we will prove strong regularity of iterated function systems induced by nice sets. As the last preparation for applications of the Young towers results and techniques (see Chapter 4), we will prove that the measures of points returning in time n to a nice set decay exponentially fast with n. At the very end of this section, we will also prove that the Kolmogorov–Sinai metric entropy hμh is finite. Given an elliptic function f : C −→  C, let B(f ) := Crit(f ) ∪ f −1 (∞)

(22.62)

22 Conformal Invariant Measures for CNRR Functions

and

395

  B(fˆ) := f (B(f )) = f f −1 (∞) ∩ Crit(f ) = B∞ (fˆ) ∪ Crit(fˆ), (22.63)

where B∞ (fˆ) = f (f −1 (∞)) has been defined in (22.30). From now on throughout the remainder of this section, we assume that f : C −→  C is a normal subexpanding elliptic function of finite character. According to Theorem 18.4.16,   1 Rf := distT B(fˆ),PC(fˆ) > 0. (22.64) 2 As an immediate consequence of Lemma 22.4.21, we get the following. Lemma 22.5.3 Let f : C −→  C be a normal subexpanding elliptic function ˆ of finite character. If V ⊆ Tf , a neighborhood of B(fˆ),  is a nice set for the  map fˆ : Tˆ f −→ Tf , then there exists a number t ∈ BD J (fˆ) ∩ K(Vˆ ) , HD(J (f )) such that  diamt (W ) < +∞, W ∈D (Vˆ )

where, we recall, D(Vˆ ) denotes the family of all connected components of Tf \K(Vˆ ). Recall that the concept of nice sets introduced in Chapter 12 concerned all analytic maps of Riemann surfaces whose range is either a parabolic or elliptic Riemann surface, the former including all complex tori and the complex plane while the latter being the Riemann sphere  C. We have proved in this chapter the existence (see especially Corollary 12.3.2) of nice sets for all such maps satisfying some further hypotheses. Now, with Y := Tf and X := Tˆ f , we want to apply these results to the map fˆ : Tˆ f −→ Tf and the set F := B(fˆ). We shall prove the following. Theorem 22.5.4 If f : C −→  C is a normal subexpanding elliptic function of finite character, then, for F = B(fˆ) and every radius r > 0 sufficiently small fulfilling condition (1) of Corollary 12.3.2, there exists a corresponding nice set

396

Part VI Fractal Geometry, Stochastic Properties, and Rigidity Vˆ := Ur

ˆ f −→ Tf . for the quotient map fˆ : T Proof We just wish to check that condition (12.40) (with a fixed λ and κ larger than 1) of Corollary 12.3.2 holds for all 

* r ∈ 0, min γˆ ,Rf /2,uTf sufficiently small, particularly satisfying condition (12.39) of Corollary 12.3.2, where γˆ > 0 comes from Corollary 18.4.12 (Exponential Shrinking on Tf ). Indeed, for every integer n ≥ 1, every ξ ∈ fˆ−n (B(fˆ)) such that   Comp ξ, fˆn,2r ∩ B(fˆ) = ∅, (22.65)   and every z ∈ Comp ξ, fˆn,2r , we get, by virtue of (22.64), Theorem 22.4.3, Lemma 8.3.13, Theorem 8.3.9, and Corollary 18.4.12, that   1 1 (2Rf )(fˆn ) (z)|−1 ≤ K (2Rf )(fˆn ) (ξ )|−1 4 4    ≤ Kdiame Comp ξ, fˆn,Rf ≤ 4eKγf e−Mf n . Therefore,  n  (fˆ ) (z)| ≥ (8eKγf )−1 Rf eMf n . But, since the function f is normal, it follows from condition (22.65) that we can have r ∈ (0, min{γˆ ,Rf /2,uTf }] so small that the number n ≥ 1 is so large that % & 2κ −1 Mf n (8eKγf ) Rf e ≥ max ,λ, . κ −1 This means that condition (12.40) of Corollary 12.3.2 holds and the proof of Theorem 22.5.4 is complete.  Having κ,λ > 1 fixed, we say that the, resulting from Theorem 22.5.4, via Corollary 12.3.2, nice set Vˆ := Ur ⊆ Tf for the analytic map fˆ : Tˆ f −→ Tf is small (so, in particular, r > 0 satisfies formula (1) from the hypotheses of Corollary 12.3.2) if and only if $ 1 # 

(22.66) max diam(Vˆb ) : b ∈ B(fˆ) < min γf ,βf ,Rf ,uTf , 6

22 Conformal Invariant Measures for CNRR Functions

397

where β = βf > 0 was defined in (18.21), γf is determined by (18.22), while uTf > 0 results from Observation 8.1.4. The sets Vˆb denote here the respective connected components of Vˆ containing points b. Let JVˆ := JSVˆ be the limit set of the maximal conformal graph directed Markov system SVˆ , produced by Theorem 12.1.8. Clearly, SVˆ is finitely irreducible and B(fˆ) is ˆ the set of its vertices. We denote the set of all edges of the system SVˆ by E. ˆ For every e ∈ E, we denote by (22.67)

N (e) the unique integer greater than or equal to 1 such that φˆ e : Vˆt (e) −→ Vˆi(e),

the element of SVˆ corresponding to the edge e, is a local holomorphic branch of fˆ−N (e) . With the terminology of Proposition 12.1.7(a), N (e) = n(Vˆi(e) ). We further define N (ω) :=

|ω| 

N (ωj ).

(22.68)

j =1

In other words, N (ω) is the unique integer greater than or equal to 1 such that φω : Vˆt (ω) −→ Vˆi(ω) is a local holomorphic branch of fˆ−N (ω) . First, we shall prove the following. Lemma 22.5.5 Let f : C −→  C be a normal subexpanding elliptic function of finite character. If Vˆ ⊆ T is a small nice set, produced in Theorem 22.5.4, ˆ f −→ Tf , then for the map fˆ : T m ˆ h (JVˆ ) > 0

and

HD(JVˆ ) = HD(J (f )),

where, we recall, the measure m ˆ h has been defined in (22.15). Furthermore, the maximal graph directed Markov system SVˆ is regular (in the sense of ˆ h (JVˆ ) is the h-conformal measure (in the sense of Definition 11.4.4) and m ˆ h /m Theorem 11.9.2 with t = h) for the system SVˆ . Proof By the construction of the system SVˆ , provided in Theorem 12.2.1, and by the definition of the limit set JVˆ , the latter contains the set of all transitive points of the map fˆ : Jˆ(f ) ∩ Tˆ f → Jˆ(f ), denoted by Tr(fˆ), that belong to Vˆ . Since m ˆ h is of full topological support and since f (Tr(f )) ⊇ Tr(fˆ), it, therefore, follows from Theorem 20.3.11 that m ˆ h (JVˆ ) ≥ m ˆ h (Vˆ ∩ Tr(fˆ)) = m ˆ h (Vˆ ) > 0.

398

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Since the measure m ˆ h is h-conformal for the map fˆ : Tˆ f → Tf , it also satisfies condition (f) of Theorem 11.9.2 with t = h and γ = 1. It, therefore, ˆ h (JVˆ ) is the h-conformal follows from this theorem that P(h) = 0 and m ˆ h /m measure for the conformal graph directed system JVˆ . Consequently, the system SVˆ is regular and (the same theorem) h = hSVˆ . But, by Theorem 11.5.3,  hSVˆ = HD(JVˆ ). The proof is, thus, complete. Recall that the concept of strongly regular conformal graph directed systems was defined in Section 11.4. The crucial, technically involved, intermediate fact that opens up the doors for the stochastic properties of the dynamical system f |J (f ),μh is this following. Lemma 22.5.6 Let f : C −→  C be a normal subexpanding elliptic function of finite character. If Vˆ ⊆ Tf , a neighborhood of B(fˆ), is a small nice set, produced in Theorem 22.5.4, for the analytic map fˆ : Tˆ f −→ Tf , e.g., the one proved in Theorem 22.5.4, then SVˆ = {φe }e∈Eˆ , the corresponding maximal conformal graph directed Markov system, is strongly regular in the sense of Definition 11.4.4. Proof As always, let q ≥ 1 be  the maximum of the orders of all poles of f . Fix an arbitrary number t ∈ BD(J (fˆ) ∩ K(Vˆ )),HD(J (f )) produced in Lemma 22.5.3. Since any number in (t,HD(J (f ))) is also good for this lemma, we may assume, because of Theorem 17.3.1, that &  %   2q ,HD(J (f )) . (22.69) t ∈ max BD J (fˆ) ∩ K(Vˆ ) , q +1 For every a ∈ B(fˆ), let Eˆ a := {e ∈ Eˆ : i(e) = a} = {e ∈ Eˆ : φˆ e (Vˆt (e) ) ⊆ Vˆa }. ˆ are mutually disjoint, and their union is equal Of course, all the sets Eˆ a , a ∈ E, −1 ˆ If b ∈ f (f (∞)) and e ∈ Eˆ b , then to E. fˆ(φˆ e (Vˆt (e) )) ∈ D(Vˆ ) and diam(φˆ e (Vˆt (e) )) ≤ diam(fˆ(φˆ e (Vˆt (e) ))), provided that the diameters of the elements of the nice set Vˆ are sufficiently small, so that the sets φˆ e (Vˆt (e) ), e ∈ Eˆ b , lie sufficiently close to the pole b. For every b ∈ f (f −1 (∞)), the map

22 Conformal Invariant Measures for CNRR Functions

399

  Eˆ b  e −→ fˆ φˆ e (Vˆt (e) ) ∈ D(Vˆ ) is at most q-to-one. So,   diamt (φˆ e (Vˆt (e) )) ≤ b∈f (f −1 (∞)) e∈Eˆ b





diamt (fˆ(φˆ e (Vˆt (e) )))

b∈(f −1 (∞)) e∈Eˆ b

≤q



diamt (W ),

(22.70)

W ∈D (Vˆ )

the last estimate being very crude but sufficient for us. By Corollary 18.4.12 and (22.64), there exists 0 < δR ≤ γˆ , the latter coming from this corollary, such that   diam Comp(z, fˆn,δR ) < Rf /2 (22.71) for all n ≥ 0 and all z ∈ fˆ−n (Jˆ(f )), and the function fˆ is injective on every ball B(z,δR ) if z ∈ PC(fˆ). Now, by Lemma 22.5.3 for example, there exists an integer l ≥ 1 so large that diam(W ) < δR /4 for all W ∈ D(Vˆ ) with n(W ) ≥ l. Since Vˆ ∩ PC(fˆ) = ∅ and since the set PC(fˆ) is forward invariant, meaning that f (PC(fˆ)) ⊆ PC(fˆ), we have that  PC(fˆ) ∩ {W : W ∈ D(Vˆ )} = ∅. Since also the family {W ∈ D(Vˆ ) : n(W ) = l} is finite, we conclude that    α := dist PC(fˆ), {W : W ∈ D(Vˆ ) and n(W ) = l} is positive. Let 

1 min βf , γˆ ,α,Rf ,δR ,uTf . 4 Now fix c ∈ Crit(fˆ) and e ∈ Eˆ c with N (e) ≥ l + 1. Since α ≥ κ and since the set PC(fˆ) is forward invariant, there exists a least integer 1 ≤ ke ≤ N (e) − l such that   dist fˆke (c), fˆke (φˆ e (Vˆt (e) )) ≥ κ. (22.72) κ :=

Fix an arbitrary point ξe ∈ φˆ e (Vˆt (e) ) and put we := fˆke (ξe ).      Consider the ballB(we,δR ). Since fˆke φˆ e (Vˆt (e)) ∈ D(Vˆ ) and n fˆke φˆ e (Vˆt (e)) =   N (e) − ke ≥ l, by the choice of l, we have that fˆke φˆ e (Vˆt (e) ) ⊆ B(we,δR /4). Consequently,   (22.73) φˆ e (Vt (e) ) ⊆ Comp ξe,we, fˆke ,δR /4 .

400

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

It follows from the definition of Rf , (22.71), and the choice of ke that, for all 1 ≤ k ≤ ke − 1, we have that dist(Crit(fˆ), fˆk (Comp(ξe, fˆke ,δR ))) ≥ dist(Crit(fˆ), fˆk (c)) − dist(fˆk (c), fˆk (Comp(ξe,we, fˆke ,δR ))) ≥ 2Rf − dist(fˆn (c), fˆk (φˆ e (Vˆt (e) ))) − diam(fˆk (Comp(ξe,we, fˆke ,δR ))) 5 > 2Rf − κ − Rf /2 ≥ Rf > 0. 4 In particular, Crit(fˆ) ∩ fˆk (Comp(ξe, fˆke ,δR )) = ∅ for all 1 ≤ k ≤ ke − 1. Thus, by virtue of Theorem 22.4.3, there exists a unique holomorphic inverse branch fˆe−(ke −1) : B(we,δR ) −→ Tf of fˆke −1 sending we to fˆ(ξe ). It then directly follows from Lemma 8.3.13, −(k −1) applied to the map fˆe e : B(we,δR ) −→ Tf , (22.73), and Corollary 18.4.12 (remember that δR < γˆ ), that     diam fˆke (φˆ e (Vˆt (e) )) · |(fˆke −1 ) (fˆ(ξe ))|−1 C diam fˆ(φˆ e (Vˆt (e) )) ≤ 4eγf e−Mf (ke −1),

(22.74)

with some comparability constant C ≥ 1 independent of e ∈ Eˆ c . Since κ < δR /4, we have that B(we,κ) ⊆ B(we,δR /4), and it, therefore, follows from Lemma 8.3.13 that  1 fˆe−(ke −1) (B(we,κ)) ⊃ B fˆ(ξe ), |(fˆke −1 ) (fˆ(ξe ))|−1 . 4 / B(we,κ), we conclude that Since also, by (22.72), fˆke (c) ∈  1 ˆke −1  ˆ −1 ˆ ˆ f (c) ∈ / B f (ξe ), |(f ) (f (ξe ))| . 4 This means that |fˆ(c) − (fˆ(ξe ))| ≥ 14 |(fˆek−1 ) (fˆ(ξe ))|−1 . Since ξe was an arbitrary point in φe (Vt (e) ), this implies that 1 dist(fˆ(c), fˆ(φˆ e (Vˆt (e) ))) ≥ |(fˆke −1 ) (ξe )|−1 . 4 So, |fˆ (z)| ≥ A−1



1 ke −1  ˆ ) (f (ξe ))| |(f 4

qc −1 qc

= A−1 4

1−qc qc

|(f ke −1 ) (fˆ(ξe ))|

1−qc qc

22 Conformal Invariant Measures for CNRR Functions

401

for all z ∈ φˆ e (Vˆt (e) ). Combining this and (22.74), we get that diam(φˆ e (Vˆt (e) )) ≤ diam(fˆ(φˆ e (Vˆt (e) )))A4 ≤ A4

qc −1 qc

qc −1 qc

|(f ke −1 ) (fˆ(ξe ))|

qc −1 qc

−1+ diam(fˆke −1 (φe (Vt (e) )))|(f ke −1 ) (fˆ(ξe ))|

qc −1 qc

1

− ≤ 4AC|(f ke −1 ) (fˆ(ξe ))| qc diam(fˆke −1 (φˆ e (Vˆt (e) )))

≤ C1 e

−

Mf qc

ke

diam(fˆke −1 (φˆ e (Vˆt (e) )))

(22.75)

ˆ Let with some constant C1 ≥ 1 independent of e ∈ E. Eˆ cl := {e ∈ Eˆ c : N (e) ≥ l + 1}. For all e ∈ Eˆ cl , put

   We := fˆke −1 φˆ e Vˆt (e) .

We shall prove the following. Claim 1◦ . For every c ∈ Crit(fˆ), the function Eˆ cl  e −→ (ke,We ) is at most qc -to-one. Proof Fix c ∈ Crit(fˆ) and suppose, for a contrary, that there are two elements ˆ a,b  that  a = b and  k := ka = kb , Wa = Wb . Suppose also that  ∈ Ee such fˆ φˆ a (Vˆt (a) ) = fˆ φˆ b (Vˆt (b) ) . Since Wa = Wb , there, thus, exists a least 2 ≤ j ≤ k such that    fˆj φˆ a (Vˆt (a) ) = fˆj φˆ b (Vˆt (b) ) . (22.76) Then

  fˆj −1 φˆ a (Vˆt (a) ) ∩ fˆj −1 (φˆ b (Vˆt (b) )) = ∅.

(22.77)

By the definition of ka and kb , and since n(fˆj −1 (φa (Vˆt (a) ))),n(fˆj −1 (φb (Vˆt (b) ))) ≥ l, we have that fˆj −1 (φˆ a (Vˆt (a) )) ∪ fˆj −1 (φˆ b (Vˆt (b) ))  #    $   ⊆ B fˆj −1 (c),κ + max diam fˆj −1 φˆ a (Vˆt (a) ) ,diam fˆj −1 φˆ b (Vˆt (b) )   ⊆ B fˆj −1 (c),κ + δR /4     (22.78) ⊆ B fˆj −1 (c),δR /2 ⊆ B fˆj −1 (c),δR . But, by the choice of δR , the function fˆ restricted to the ball B(fˆj −1 (c),δR ) is one-to-one. This, however, contradicts (22.76)–(22.78) taken together.   Along with the observation that the function Eˆ c  e −→ fˆ φˆ e (Vt (e) ) is  at most qc -to-one, this completes the proof of Claim 1◦ .

402

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Let q := max{qc : c ∈ Crit(fˆ)}. Applying Claim 1◦ and (22.75), we have, for any c ∈ Crit(fˆ), that 

   − Mf tk diamt φˆ e (Vˆt (e) ) ≤ C1t e q e diamt (We )

e∈Eˆ cl

e∈Eˆ cl



≤ qc C1t

diamt (W )

W ∈D (Vˆ )

≤

e

−

Mf q

tk

(22.79)

k=1



C1t qQt

∞ 

diamt (W ).

W ∈D (Vˆ )

Since the set



∈ Eˆ c : N (e) ≤ l} is finite, the sum     := diamt φˆ e (Vˆt (e) )

c∈Crit(f ) {e

 (t) l

c∈Crit(fˆ) e∈Eˆ c : N (e)≤l

is also finite. Combining this along with (22.70) and (22.79), and finally with Lemma 22.5.3, we get the following:        diamt φˆ e (Vˆt (e) ) = diamt φˆ e (Vˆt (e) ) e∈Eˆ

b∈f (f −1 (∞)) e∈Eˆ b

+ ≤q

 (t) l







+

c∈Crit(fˆ) e∈Eˆ c

diamt (W ) +

W ∈D (Vˆ )

+

  diamt φˆ e (Vˆt (e) )





C1t qQt

c∈Crit(fˆ)

= q(1 + C1t Qt )

 (t) l

diamt (W )

W ∈D (Vˆ )



diamt (W ) +

 (t) l

< +∞.

W ∈D (Vˆ )

Since t < HD(J (f )) = HD(Jr ), this implies that our system is strongly regular. The proof of Lemma 22.5.6 is complete.  Now we want to form an appropriate strongly regular conformal maximal graph directed Markov system on C and to obtain from it stochastic laws such as the Law of the Iterated Logarithm and, by the Young tower construction, the

22 Conformal Invariant Measures for CNRR Functions

403

exponential decay of correlations and the Central Limit Theorem. It will also allow us to show that the metric entropy hμh (f ) is finite. Keep Vˆ ⊆ Tf , a small nice set, produced in Theorem 22.5.4, for the map ˆ f : Tˆ f −→ Tf , satisfying (22.66). As the set of vertices take V, any (finite) selector of the partition of −1 ˆ (∞) ∪ (Crit(f ) ∩ J (f )) −1 f (B(f )) = f

into equivalence classes of the equivalence relation ∼f , i.e., choose exactly one point from each such equivalence class intersected with f −1 (∞) ∪ (Crit(f ) ∩ ˆ J (f )). For every v ∈ V, let Vv be the connected component of −1 f (Vv ) containing v. Because of (22.66), for each v ∈ V, the holomorphic map  v := f V : Vv −→ Vˆv v

is one-to-one. As the set of edges, set   

E := e ∈ Eˆ : ∃v ∈ V s.t. f N (e)−1 f˜(Vˆt (e) ) = Vv , C is given by (16.8). For each e ∈ E, let where, we recall, the map f˜ : Tf −→  −1 t (e) be the only element of V ∩f (t (e)), where the latter t (e) is understood to be associated with the graph directed Markov systems SVˆ . Likewise, i(e) is the only element of V ∩ −1 f (i(e)), where the latter i(e) is, as before, understood to be associated with the graph directed Markov systems SVˆ . For each e ∈ E, define ˆ φe := −1 i(e) ◦ φe ◦ t (e) : Vt (e) −→ Vi(e) . Obviously, SV := {φe : Vt (e) −→ Vi(e) }e∈E is a conformal maximal graph directed Markov system with edges E and vertices V. We shall easily prove the following. Theorem 22.5.7 Let f : C −→  C be a normal subexpanding elliptic function of finite character. If Vˆ ⊆ T, a neighborhood of B(fˆ), is a small nice set, ˆ f −→ Tf , then produced in Theorem 22.5.4, for the map fˆ : T mh (JV ) > 0

and

HD(JV ) = HD(J (f )),

where, we recall, mh is the h-conformal measure for the dynamical system f:C→ C. Furthermore, the maximal graph directed Markov system SV is strongly regular (in the sense of Definition 11.4.4) and mh /mh (JV )

404

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

is the h-conformal measure (in the sense of Theorem 11.9.2 with t = h) for the system SV . Proof All but strong regularity follows in the same way as in the proof of Theorem 22.5.5 once we realize that, by the construction of SV ,JV ⊇ Tr(f ) ∩ V . Strong regularity of the system SV is an immediate consequence of the facts that the system SVˆ is strongly regular (see Lemma 22.5.6), PSV (h) = PSVˆ (h) = 0, PSV (t) ≤ PSVˆ (t) for all t ≥ 0, the latter holding because E ⊆ Eˆ   and all the maps v , v ∈ V , are isometries, and both maps θSVˆ ,+∞  t −→  PSV (t),PSVˆ (t) are strictly decreasing. The first consequence of Theorem 22.5.7 is the following. Lemma 22.5.8 Let f : C −→  C be a normal subexpanding elliptic function of finite character. If Vˆ ⊆ Tf , a neighborhood of B(fˆ), is a small nice set, ˆ f −→ Tf , then there exist C > 0 and produced in Theorem 22.5.4, for fˆ : T α > 0 such that ⎛



mh ⎝





⎞

φe Vt (e) ⎠ ≤ Ce−αn

N (e)≥n

for all n ≥ 1, where, we recall, mh is the unique h-conformal measure for the map f : J (f ) → J (f ). Proof that

As HD(Jr ) = h, Theorem 22.5.7 produces a number t ∈ (0,h) such

Z1 (t) =



φe t∞ < +∞.

e∈E

So, employing Theorem 18.4.11 and the Koebe Distortion Theorem, i.e., Theorem 8.3.8, we get, for every n ≥ 1, that ⎛ mh ⎝







⎞

φe Vt (e) ⎠ =

N (e)≥n



mh (φe (Ve )) 

N (e)=n

=



 N (e)=n

φe t∞ φe h−t ∞

N (e)=n





N (e)=n

φe t∞ e−Mf (h−t)n

φe h∞

22 Conformal Invariant Measures for CNRR Functions 

= e−Mf (h−t)n 

405

φe t∞

N (e)=n

≤ Z1 (t)e

−Mf (h−t)n

. 

We are done.

22.5.2 Young Towers for Subexpanding Elliptic Functions of Finite Character In this subsection, we use the results obtained in the previous subsection and in Section 4.2 to establish the exponential decay of correlations, the Central Limit Theorem, and the Law of the Iterated Logarithm for a normal subexpanding elliptic function of finite character, with respect to the invariant measure μh equivalent to the h-conformal measure mh . With the setting of the previous section, we define the map F : JV → JV by the formula FV (φe (z)) = z if e ∈ E and z ∈ JV .   Our goal is to show that the system F,νh := mh |JV can be embedded as the base into an appropriately defined Young tower by fitting it into the abstract framework of Section 4.2. Then to check that the hypotheses of Theorems 4.2.2–4.2.4 are satisfied for this tower. Finally, to show that the original map f : J (f ) → J (f ) is a measure-theoretic factor of the tower dynamical system. The stochastic laws will then follow almost automatically. For every b ∈ B(fˆ), set Jb := Vb ∩ JV . Our Young tower Yf is constructed as follows. (1) The space 0 is now JV , the limit set of the iterated function system SV . (2) The partition P0 consists of the sets e := φe (Jt (e) ), e ∈ E. (3) The measure m is mh , the h-conformal measure mh , as a matter of fact, restricted JV . (4) The map T0 : 0 → 0 is, in our setting, just the map FV . (5) The function R, the return time, is, naturally, defined as R|e := N (e). Fix e ∈ E arbitrary and then two arbitrary points x,y ∈ e = φe (JV ). This means that x = φe (x  ) and y = φε (y  ) with some x ,y  ∈ JV . Since

406

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

t h − 1 = h(t − 1) + O(|t − 1|2 ), and because of Theorem 8.3.7, there exist respective constants C1,C2 > 0 such that we have that          |φ  (x  )|h  J acmh FV (y)    |FV (y)|h   e   = − 1 − 1 − 1 =   J ac F (x)    |F  (x)|h    h  (φe (y )| mh V V       |φ (x )| (22.80) ≤ C1  e  − 1 |φe (y )| ≤ C1 C2 |y  − x  |. ∞ . Put Now write x  = πV (α) and y  = πV (γ ) with appropriate α,γ ∈ EA  k  ω := α ∧ γ and k := |ω|. Write also x := πV (σ (α)) and y := πV (σ k (γ )). Then we get that

|y  − x  | = |φω (y  ) − φω (x  )| ≤ |φω (x  )||y  − x  | |ω|

≤ diame (Vt (ω) )φω ∞ ≤ diame (Vt (ω) )βV s(x ,y  )

= diame (Vt (ω) )βV

(22.81)

s(F (x),F (y))

= diame (Vt (ω) )βV

s(F (x),F (y))

≤ V βV

,

where, we recall, βV ∈ (0,1) is the contracting factor of the system SV and V  := sup{diame (Vv ) : v ∈ B(fˆ)}. Since s(FV (x),FV (y)) = s(x,y) − 1 (as we know that s(x,y) ≥ 1), along with (22.80), then (22.81) gives       J acmh FV (y) s FV (x),FV (y) s(x,y)   = βV−1 V βV .  J ac F (x) − 1 ≤ V βV mh V So, (4.8) is established in our context. The fact that the partition P0 is generating follows either from the contracting property of graph directed Markov systems, or, more directly, from Theorem 18.4.11. The Big Images Property holds because the alphabet of the GDMS SV is finite. The last assumption in Theorem 4.2.2 is that the map T :  → , denoted now by TV , is topologically mixing. We will prove it now, introducing on the way some concepts which will also be needed further in the main proof. Lemma 22.5.9 The dynamical system TV :  →  is topologically mixing in the sense of Section 4.2. Proof

∗ , set For every τ ∈ EA

N (τ ) :=

|τ |  j =1

N (τj ).

22 Conformal Invariant Measures for CNRR Functions

407

By our construction of the tower Yf , we have, for every e ∈ E, that   R( )  TV e e × {0} = TV (e × {R(e )} = FV (e ) × {0} = Jt (e) × {0}. ∗ , that So, by an immediate induction, we get, for every τ ∈ EA  N (τ )  TV φτ (Jt (τ ) ) × {0} = Jt (τ ) × {0}.

(22.82)

Now fix two arbitrary elements a,b ∈ E. Then there exists s ≥ 0 such that   f u φa (Vt (a) ) ⊇ φb (Vt (b) ) for all u ≥ s. Hence, for all such u, there exists a holomorphic branch of f u mapping φb (Vt (b) ) into φa (Vt (a) ). By our construction of the conformal graph directed Markov system SV , this holomorphic branch is an admissible composition of elements of SV . This means that it is equal to φτ (u) : Vt (τ (u)) −→ Vi(τ (u)) ∗ EA

for some τ (u) ∈ with t (τ (u)) = i(b) and i(τ (u)) = i(a). Then, applying (22.82), we get, for every 0 ≤ k ≤ R(a ) − 1, every 0 ≤ l ≤ R(b ) − 1, and every n ≥ s + R(b ) − 1, that       TVn a × {k} = TVn+k a × {0} = TVl+n+k−l a × {0}    ⊇ TVl TVn+k−l φτ (n+k−l) (Jt (τ (n+k−l)) ) × {0}     = TVl Jt (τ (n+k−l)) × {0} ⊇ TVl φb (Jb ) × {0} = φb (Jb ) × {l} = b × {l}. The proof of Lemma 22.5.9 is complete.



Having all the above and also invoking Lemma 22.5.8, which yields mh (R −1 ([n,∞)) ≤ Ce−αn , we conclude from Observation 4.2.1 and Theorems 4.2.2–4.2.4 that •

m ˜ h () < +∞,

(22.83)

where m ˜ h is derived out of mh restricted to JV according to (4.9). • The map TV :  −→  admits a probability T -invariant measure νh , which is absolutely continuous with respect to m ˜ h. Given γ ∈ (0,1), for the dynamical system TV : (,νh ) −→ (,νh ) the • following hold: – The exponential decay of correlations in the form of (4.13) holds. – The Central Limit Theorem is true for all functions g ∈ Cγ () that are not cohomologous to a constant in L2 (νh ).

408

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

– The Law of the Iterated Logarithm is true for all functions g ∈ Cγ () that are not cohomologous to a constant in L2 (νh ). Now consider H :  −→ C, the natural projection from the abstract tower  to the complex plane C given by the formula H (z,n) = f n (z). Then H ◦ T = f ◦ H,

(22.84)

m ˜ h |0 ◦ H −1 = mh |JV , and m ˜ h |e ×{n} ◦ H −1 = mh |e ×{0} ◦ f −n = mh |e ◦ f −n for all e ∈ E and all 0 ≤ n ≤ N (e) − 1. So, m ˜ h |e ×{n} ◦ H −1 is absolutely continuous with respect to mh , with the Radon–Nikodym derivative equal to     −n −n −h   f n |e (z) =  f n (z)  Je,n (z) := J acmh f n |e for all z ∈ f n (e ) and zero elsewhere in J (f ). Therefore, using (22.83), we get that (e)−1  N J (f ) e∈E

Je,n dmh =

(e)−1  N e∈E

n=0

n=0 −1

=m ˜h ◦H

Je,n dmh (J (f )) = m ˜ h () < +∞.

Thus, the function (e)−1  N e∈E

Je,n

n=0

is integrable with respect to the measure mh . This implies immediately that the measure m ˜ h ◦ H −1 is absolutely continuous with respect to the measure mh with the Radon–Nikodym derivative equal to (e)−1  N e∈E

Je,n .

n=0

Hence, the measure νh ◦ H −1 is also absolutely continuous with respect to mh . Since ν is FV -invariant and H ◦ TV = f ◦ H , the measure νh ◦ H −1 is f -invariant. But the measure μh is f -invariant, ergodic, and equivalent to the

22 Conformal Invariant Measures for CNRR Functions

409

conformal measure mh . Hence, νh ◦ H −1 is absolutely continuous with respect to the ergodic measure μh . Therefore, we obtained the following. Lemma 22.5.10 If f : C −→  C is a normal subexpanding elliptic function of finite character, then νh ◦ H −1 = μh . We are now in a position to prove the following. Theorem 22.5.11 If f : C −→  C is a normal subexpanding elliptic function of finite character and if μh is the corresponding probability f -invariant measure equivalent to the h-conformal measure mh , then the dynamical system   f |J (f ),μh satisfies the following. If g : J (f ) → R is a bounded function, H¨older continuous with respect to the Euclidean metric on J (f ), then (1) For every bounded measurable function ψ : J (f ) → R, we have that      ψ ◦ f n · gdμh − gdμh ψdμh  = O(θ n )   for some 0 < θ < 1 depending on α, a H¨older exponent of g. (2) The Central Limit Theorem holds for every H¨older continuous bounded function g : J (f ) → R that is not cohomologous to a constant in L2 (μh ), i.e., for which there is no square integrable function η for which g = const + η ◦ f − η. More precisely, there exists σ > 0 such that ' "n−1 j j =0 g ◦ f − n gdμh −→ N (0,σ ) √ n in distribution, where, as usual, N (0,σ ) denotes the Gauss (normal) distribution centered at 0 with covariance σ . (3) The Law of the Iterated Logarithm holds for every H¨older continuous bounded function g : J (f ) → R that is not cohomologous to a constant in L2 (μh ). This means that there exists a real positive constant Ag such that μh -a.e. ' Sn g − n gdμh = Ag . lim sup √ n log log n n→∞ Proof Let g : J (f ) −→ R and ψ : J (f ) −→ R be as in the hypotheses of our theorem. Define the functions g˜ := g ◦ H :  −→ R and ψ˜ := ψ ◦ H :  −→ R. We shall prove the following.

410

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Claim 1. The function g˜ :  −→ R belongs to the space Cβ for an appropriate exponent β ∈ (0,1). Indeed, consider two arbitrary points (x,k),(y,l) ∈ . We treat the two cases separately depending on whether s((x,k),(y,l)) = 0 or not. If s((x,k),(y,l)) = 0, then we get that |g(y,l) ˜ − g(x,k)| ˜ = |g(H (y,l)) − g(H (x,k))| = |g(f l (y)) − g(f k (x))| ≤ |g(f l (y))| + |g(f k (x))| ≤ 2g∞ = 2g∞ β s((x,k),(y,l))

(22.85)

regardless of what the value of β ∈ (0,1) is, which will be specified in the next case. Indeed, if s((x,k),(y,l)) > 0, then k = l, k < R(x) = R(y), and |g(y,l) ˜ − g(x,k)| ˜ = |g(f k (y)) − g(f k (x))| ≤ Hg |f k (y) − f k (x)|γ , (22.86) where Hg ≥ 0 and γ > 0 are, respectively, the H¨older constant and the H¨older s((x,k),(y,l)) exponent of g. Moreover, x,y ∈ φτ (Vt (τ ) ), with some τ ∈E , and A    k k k k then f (x),f (y) ∈ f φτ (Vt (τ ) ) and the set f φτ (Vt (τ ) ) is contained in a connected component of    f −(R(x)−k) F −(s((x,k),(y,l))) Vt (τ ) ) whose diameter is, by Theorem 18.4.11, generously bounded above by      exp −Mf s((x,k),(y,l)) + R(x) − k ≤ exp −Mf (s(x,k),(y,l)) . In conjunction with (22.86) (and (22.85)), this finishes the proof of Claim 1 by taking β = exp(−γ Mf ). Claim 2. The function g˜ is not cohomologous to a constant in L2 (ν). Indeed, assume without loss of generality that μh (g) = 0. By virtue of Lemma 2.3.7, the fact that g : J (f ) −→ R is not a coboundary in L2 (μh ) equivalently ∞ means that the sequence Sn (g) n=1 is not uniformly bounded in L2 (μh ). But, ˜ L2 (ν) = Sn (g)L2 (μh ) . So, the sequence because of Lemma 22.5.10, Sn (g) ∞  ˜ n=0 is not uniformly bounded in L2 (ν). Thus, by Lemma 2.3.7 again, Sn (g) it is not a coboundary in L2 (ν). Having these two claims, all items, (1), (2), and (3), now follow immediately from Theorem 4.2.3 with the use of Lemma 22.5.10 and (22.84). The proof is finished. 

22 Conformal Invariant Measures for CNRR Functions

411

22.5.3 Metric Entropy In this miniature subsection, by taking the fruits of what has been done so far,  we will easily prove that the metric entropy of the dynamical system f |J (f ),μh is finite. Theorem 22.5.12 If f : C −→  C is a normal subexpanding elliptic function of finite character and if μh is the corresponding Borel probability f -invariant measure equivalent to the h-conformal measure mh , then hμh (f ) < +∞. Proof It is a direct consequence of Theorems 22.5.7 and 11.7.3 to have finite entropy of the induced system FV : JV −→ JV with respect to the probability FV -invariant measure μV = (μh (JV ))−1 μh |JV . Then, as FV : JV −→ JV is the first return map of f from JV to JV , Abramov’s Formula (Theorem 6.6.1)  yields hμh (f ) < +∞. For the case when the Julia set is the whole complex plane C, we get from this the following. Corollary 22.5.13 If f : C −→  C is a normal subexpanding elliptic function of finite character with J (f ) = C, then hμ2 (f ) < +∞, where μ2 is the (unique) Borel probability f -invariant measure on C equivalent to the planar Lebesgue measure on C .

22.6 Parabolic Elliptic Maps: Nice Sets, Graph Directed Markov Systems, Conformal and Invariant Measures, Metric Entropy In this section, we deal with parabolic elliptic functions f : C −→  C, i.e., with all such elliptic functions f : C −→  C for which Crit(f ) ∩ J (f ) = ∅ and (f ) = ∅. As an immediate consequence of the first half of this definition, we get the following observation, already included in Section 18.4 as Observation 18.4.17. Observation 22.6.1 Every parabolic elliptic function is regular normal compactly nonrecurrent of finite character. In consequence, all that we have proved so far for such functions applies to parabolic elliptic functions. Our aim in this section is to construct the appropriate pre-nice and nice sets dealt with at length in Chapter 12, to construct, as has already been done in

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Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Chapter 12, and to study the corresponding conformal maximal graph directed Markov systems in the sense of Chapter 11, showing particularly their strong regularity. Having all these tools, we also prove that the Kolmogorov–Sinai entropy of the measure μh is always finite even if the measure μh itself is infinite. Last, at the end of the section, we briefly deal with Radon–Nikodym derivatives of invariant measures μh with respect to the conformal ones mh . We start by defining appropriate nice families for a parabolic elliptic map f : C −→  C, actually for some of its iterates. Indeed, let l = lf ≥ 1 be an integer such that each element of (f ) is a simple parabolic point of f l . Our first goal is to construct a nice set for the holomorphic map fˆl : T(l) f = Tf \

l−1 

fˆ−j (B∞ (fˆ)) −→ Tf ,

j =0

introduced in (22.31), where, we recall, B∞ (fˆ) = f (f −1 (∞)). We want to construct such a nice set with properties suitable for estimating from above the first entrance time to it sufficiently well so that the machinery of Young towers set up in Section 4.2, and applied in the preceding Section 22.5.2, will be now applicable for those parabolic elliptic maps for which the measure μh is finite. For our construction of nice sets, we will apply Theorem 12.4.4. Sticking to its notation, we set F0 := (fˆ) = f ((f )) = (fˆl )

and

F1 := B∞ (fˆ).

(22.87)

Of course, the finite set (fˆl ) consists of all rationally indifferent fixed points of fˆl . For all b ∈ F1 , we set U0 (b) := B(b,r) with r > 0 so small that

  B(b,6r) ∩ PC(f ) ∪ (fˆ) = ∅,

(22.88)

and as required further in the course of our construction. For every ω ∈ F0 , a fixed α ∈ (π/2,π), a fixed κ ∈ (0,1), and every j ∈ {1, . . . ,p(ω)}, set  j  U0 (ω,j ) := f Sr (ω,α) (22.89) j

where the petals Sr (ω,α) are defined by (15.37). With X :=

(l) Tf

:= Tf \

l−1  j =0

fˆ−j (B∞ (fˆ)), Y = Tf , and f being fˆl ,

22 Conformal Invariant Measures for CNRR Functions

413

all the hypotheses of Theorem 12.4.4 up to (2d) (included) are evidently satisfied for all b ∈ F and the corresponding radii rb > 0, small enough. Hypothesis (2e) follows directly from the definition (22.89). Hypotheses (2f) and (2g) follow immediately from Proposition 15.1.8. Hypotheses (2h), (2i), (6), and (7) are also immediate from our definitions. In order to verify hypothesis (2j), fix any s > 0 and then any x ≥ x(α,κ) so large that p(ω) 



 j j f (Sr (ω,α)) ∪ f (Sa (ω,α)) ⊆ B(ω,s).

j =1

Since α ∈ (π/2,π), we have that ⎞ ⎛ p(ω)   j j f (Sr (ω,α)) ∪ f (Sa (ω,α)) ⎠ = ∅. Int ⎝{ω} ∪ j =1

Therefore, there exists sω− > 0 such that B(ω,sω− ) ⊆ {ω} ∪

p(ω) 



 j j f (Sr (ω,α)) ∪ f (Sa (ω,α)) .

j =1

So, if z ∈ B(ω,sω− )\

p(ω) 

U0 (ω,j ),

j =1

then z∈

p(ω) 

j

f (Sa (ω,α)).

j =1

Therefore, f n (z) ∈

p(ω) 

j

f (Sa (ω,α)) ⊆ B(ω,s)

j =1

for all integers n ≥ 0. Thus, hypothesis (2j) is satisfied too. Hypothesis (3) clearly holds for all rξ > 0, ξ ∈ F0 , small enough, and (4) holds for all rξ > 0, ξ ∈ F1 , small enough, because of (22.88). Hypothesis (5a) holds for all rξ > 0, ξ ∈ F , small enough since PC(fˆ) ∩ J (fˆ) = (fˆ).

(22.90)

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Part VI Fractal Geometry, Stochastic Properties, and Rigidity

We are left to show that (5b) can be satisfied with an appropriate choice of rξ ,ξ ∈ F , small enough. Fixing a and b in F , w ∈ U0 (b), assuming that for them the hypotheses of (5b) hold; already knowing that (5a) holds, we denote −ln fˆa,b := fˆw−ln : B(b,6rb ) −→ Tf .

By (12.55), being a hypothesis of (5b), we have that # $  −ln inf fˆl(n−1) ◦ fˆa,b (b) − b : b ∈ (fˆ),a ∈ (fˆ)\{b},n ≥ 1 > 0. We look for radii rξ , ξ ∈ F , being all equal, i.e., r := ra = rb for all a,b ∈ F . Since fˆ−1 ((fˆ)) ∩ Crit(fˆ) = ∅, since the points of       ˆ−1 (fˆ) \(fˆ) is a fˆ−1 (fˆ) cluster toward B∞ (fˆ), and since −1 f f compact subset of J (f ) disjoint from PC(f ), it follows from Lemma 18.1.9(3) that # $

 (22.91) M := min 1, inf |(fˆ l ) (z)| : z ∈ fˆ−l ((fˆ)) > 0. (l) (l) Let Q ⊆ Tf be a periodic orbit of fˆl : Tf −→ Tf disjoint from (fˆ); it is, of course, disjoint from B∞ (fˆ). So, our analytic map fˆl : T(l) f −→ Tf enjoys the Standard Property. Since the set B∞ (fˆ) is finite and disjoint from

Q ∪ PC(fˆ) (see (22.90)), it follows from Lemma 18.1.9(4) that there exists an integer N1 ≥ 1 such that 8κ |(fˆn ) (ξ )| ≥ M −1 K 2 κ −1

(22.92)

for every n ≥ N1 and every ξ ∈ fˆ−n (B∞ (fˆ)), where K ≥ 1 is, as usual, the Koebe constant corresponding to the scale 1/2. Since (fˆ) is a finite set consisting of fixed points of fˆ only and is disjoint from the finite set B∞ (fˆ), there exist R1 > 0 such that 

N1 l      k ˆ ˆ ˆ ˆ B B∞ (f ),4R1 ∩ PC(f ) ∪ (22.93) f B∞ ((f ),4R1 ) = ∅. k=0

Then, looking up at (22.92) and using Theorem 22.4.4, which gives the existence of a unique holomorphic branch   fξ−n : B fˆn (ξ ),4R1 −→ Tf

22 Conformal Invariant Measures for CNRR Functions

415

  of fˆ−n defined on Be fˆn (ξ ),4R1 , and mapping f n (ξ ) to ξ , along with Theorem 8.3.9 (Koebe Distortion Theorem I for Parabolic Surfaces), we conclude that 8κ (22.94) |(fˆn ) (z)| ≥ M −1 K κ −1   for every n ≥ N1 and every z ∈ fˆ−n B(B∞ (fˆ),2R1 ) . Since the set   Z := fˆ−l ((fˆ))\(fˆ) \B(B∞ (fˆ),R1 ) (22.95) is finite and disjoint from Q ∪ PC(fˆ), it again follows from Lemma 18.1.9(4) that there exists an integer N2 ≥ N1 such that 8κ |(fˆn ) (ξ )| ≥ M −1 K κ −1

(22.96)

for every n ≥ N2 and every ξ ∈ fˆ−n (Z). As before, since (fˆ) is a finite set consisting of fixed points of fˆ only and it is disjoint from Z, there exists R2 ∈ (0,R1 ] such that B(Z,4R2 ) ∩

N2 l 

  fˆk B((fˆ),4R2 ) = ∅.

(22.97)

k=0

Take any r ∈ (0,R2 ].

(22.98)

Dealing still with hypothesis (5b), consider several cases. Case 1. a ∈ F0,b ∈ F1 . Then n ≥ N1 by (22.93). Hence, (12.56) holds because of (22.98), (22.91) (M ≤ 1), (22.94), and Theorem 8.3.9 (Koebe Distortion Theorem I for Parabolic Surfaces). Case 2. a,b ∈ F0 . Then   −ln (b) ∈ fˆ−l ((fˆ))\(fˆ). fˆl(n−1) fˆa,b Suppose first that

  −ln fˆl(n−1) fˆa,b (b) ∈ Z.

−ln Then fa,b (b) ∈ fˆ−l(n−1) (Z). So, it follows from (22.97) that n − 1 ≥ N2 . So, using (22.96) and (22.91), we get that −ln −ln −ln |(fˆln ) (fˆa,b (b))| = |(fˆl(n−1) ) (fˆa,b (b))| · |(fˆl ) (fˆl(n−1) (fˆa,b (b)))|

≥ M −1 K

8κ 8κ M=K . κ −1 κ −1

416

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Invoking now Theorem 8.3.9 (Koebe Distortion Theorem I for Parabolic Surfaces) (and equality r = ra = rb ), (12.56) thus follows in this case. So, suppose that −ln fˆl(n−1) (fˆa,b (b)) ∈ B(B∞ (fˆ),R1 ).

It then follows from (22.93) that n − 1 ≥ N1 . So, using (22.94) and (22.91), we get that −ln −ln −ln (b))| = |(fˆl(n−1) ) (fˆa,b (b))| · |(fˆl ) (fˆl(n−1) (fˆa,b (b)))| |(fˆln ) (fˆa,b

8κ 8κ M=K . κ −1 κ −1 Invoking, as before, Theorem 8.3.9 (Koebe Distortion Theorem I for Parabolic Surfaces) and equality r = ra = rb , (12.56) thus follows in this case too. Case 3. a ∈ F1 , b ∈ F . Assume first that ≥ M −1 K

n ≥ N2 + 1.

(22.99)

Consider three subcases: Case 3A. b ∈ F1 . Then (12.56) holds because of (22.94). Case 3B. b ∈ F0 . Then −ln fˆl(n−1) (fˆa,b (b)) ∈ fˆ−l ((fˆ))\(fˆ)

and assume that −ln (b)) ∈ Z. fˆl(n−1) (fˆa,b −ln Then fˆa,b (b) ∈ fˆ−l(n−1) (Z). Since, by (22.99), n−1 ≥ N2 , and using (22.96) and (22.91), we get that −ln −ln −ln (b))| = |(fˆl(n−1) ) (fˆa,b (b))| · |(fˆl ) (fˆl(n−1) (fˆa,b (b)))| |(fˆln ) (fˆa,b

8κ 8κ M=K . κ −1 κ −1 Invoking now Theorem 8.3.9 (Koebe Distortion Theorem I for Parabolic Surfaces), (12.56) thus follows in this case. ≥ M −1 K

−ln Case 3C. b ∈ F0 and fˆl(n−1) (fˆa,b (b)) ∈ B(B∞ (fˆ),R1 ). Then, using (22.94) and (22.91), we get that −ln −ln −ln (b))| = |(fˆl(n−1) ) (fˆa,b (b))| · |(fˆl ) (fˆl(n−1) (fˆa,b (b)))| |(fˆln ) (fˆa,b

8κ 8κ M=K . κ −1 κ −1 Invoking, as before, Theorem 8.3.9 (Koebe Distortion Theorem I for Parabolic Surfaces), (12.56) thus follows in this case too. ≥ M −1 K

22 Conformal Invariant Measures for CNRR Functions

417

So, finally, assume that n ≤ N2 . ˜ Since −1 f (B∞ (f )) is a compact subset of J (f ) disjoint from PC(f ), the same argument as the one giving (22.91) gives us that $ #  D := inf (fˆlk ) (z) : 0 ≤ k ≤ N2, z ∈ fˆ−lk (F ) > 0. −ln Also, by taking r > 0 small enough so that fˆa,b (b) (with all a, b, and n as in ˆ the considered case) is sufficiently close to B(f ), we will have that  l  −ln  (fˆ ) (fˆ (b)) ≥ D −1 K 8κ . a,b κ −1 Hence,       ln  −ln  (fˆ ) (fˆ (b)) = (fˆl ) (fˆ−ln (b)) ·  fˆl(n−1)  (fˆl (fˆ−ln (b))) a,b a,b a,b

≥ D −1 K

8κ 8κ D=K . κ −1 κ −1

Thus, a direct application of Theorem 8.3.9 (Koebe Distortion Theorem I for Parabolic Surfaces) completes the proof of (12.56). Hypothesis (5b) of Theorem 12.4.4 is therefore verified. Its hypotheses (8) and (9) also obviously hold for all r ∈ (0,R2 ) small enough. So, Theorems 12.4.4 and 12.2.1 apply as well and we have the following. Theorem 22.6.2 If f : C −→  C is a parabolic elliptic map, then the construction started with (22.87) leads, by applying Theorem 12.2.1, to a pre-nice U for the map fˆl : T(l) f −→ Tf , which, in turn, leads, via Theorem 12.4.4, to a (l) maximal conformal graph directed Markov system SˆU (acting on T ) in the f

sense of Definitions 11.3.1 and 11.9.1. Denote the edges of this system by Eˆ and the corresponding contractions by φˆ e : Xt (e) −→ Xi(e) . We need a few more properties concerning nice sets produced in the theorem above. Lemma 22.6.3 Let f : C −→  C be a parabolic elliptic map. If U is the prenice set coming from Theorem 22.6.2, then, for every ω ∈ (fˆ) and every j ∈ {1,2, . . . ,pω }, we have that ⎛ ⎞ pω  ∞  fˆ−ln (X(ω,j ))⎠ J (fˆ) ∩ ⎝{ω} ∪ j =1 n=0

418

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

is a neighborhood of ω in J (fˆ), where the sets X(ω,j ) are the members of the maximal conformal graph directed Markov system SˆU from Theorem 22.6.2; see Theorems 12.4.4 and 12.2.1 and (12.46) for their definition (with the parameter u being skipped).   pω Proof Since, by Lemma 15.3.6, J (fˆ) ∩ {ω} ∪ j =1 U0 (ω,j ) is an open neighborhood of ω ∈ J (fˆ), so is the set ⎛ ⎞ pω  Hω := J (fˆ) ∩ ⎝{ω} ∪ U (ω,j )⎠ . j =1

Hence, there exists u > 0 so small that ⎞ ⎛ pω pω   Vj (ω,u) ∪ f l (Vj (ω,u))⎠ ⊆ Hω . J (fˆ) ∩ ⎝ j =1

(22.100)

j =1

By virtue of Proposition 15.3.5, for every z ∈ Hω \{ω}, there exists k ≥ 1 such that / Hω . f kl (z) ∈

(22.101)

Let k be the minimal integer with this property. Then f (k−1)l (z) ∈ U (ω,j )\ fω−l (U (ω,j )) with some j ∈ {1, . . . ,pω }. It then follows from (22.100) and pω −l / i=1 f (Vj (b,u)). Hence, (22.101) that f (k−1)l (z) ∈   f (k−1)l (z) ∈ U (ω,j )\fω−l (U (ω,j )) \Vj (ω,u) ⊆ X(ω,j ;u). So,

⎛ Hω ⊆ J (fˆ) ∩ ⎝{ω} ∪

pω  ∞ 

⎞ fω−ln (X(ω,j ;u))⎠

j =1 n=0

and, with X(ω,j ) := X(ω,j ;u), the proof is complete.



As an immediate consequence of Lemma 22.4.21, we get the following. ˆ be a parabolic elliptic function and Vˆ be an Lemma 22.6.4 Let f : C −→ C open subset of Tf such that (1) (2) (3) (4)

ˆ ˆ B∞ (f ) ⊆ V , Vˆ ∪ (fˆ) ∩ J (fˆ) is an open neighborhood of (fˆ) in J (fˆ), the open set Vˆ has finitely many connected components, the open set Vˆ + is horizontal.

22 Conformal Invariant Measures for CNRR Functions

419

  Then, for every k ≥ 1, there exists t ∈ BD(J (fˆ) ∩ Kk (Vˆ )),HD(J (f )) such that, with the notation of Section 22.4, we have that  diamt (Q) < +∞. Q∈Dk (Vˆ )

As an immediate consequence of this lemma along with Lemma 22.6.3 and Observation 22.4.19, we get the following. ˆ is a parabolic elliptic function and U is a Lemma 22.6.5 If f : C −→ C pre-nice set of Theorem 22.6.2 for the map fˆl : T(l) f −→ Tf , then there exists   t ∈ BD(Kl (U )),HD(J (f )) such that, with the notation of Section 22.4, we have that  diamt (Q) < +∞. Q∈Dl (U )

We are now able to prove the following. ˆ is a parabolic elliptic function and U is a Lemma 22.6.6 If f : C −→ C pre-nice set produced in Theorem 22.6.2 for the map fˆl : T(l) f −→ Tf , then ˆ the corresponding maximal graph directed Markov system SU , given also by (l) Theorem 22.6.2, for the map fˆl : T −→ Tf is strongly regular. f

Proof Let q ≥ 1 be the maximal multiplicity of a pole of f . Fix the number t ∈ (BD(Kk (V )),HD(J (f ))) produced in Lemma 22.6.5. Since any number in (t,HD(J (f ))) is also good for this lemma, we may assume, because of Theorem 17.3.1, that &  % 2q ˆ ,HD(J (f )) . (22.102) t ∈ max BD(J (f ) ∩ Kl (U )), q +1 For all a,b ∈ F , let

 Ea,b := e ∈ E : p1 (i(e)) = a and p1 (t (e)) = b .

Since, for every e ∈ E(0,1), we have that   fˆl φˆ e (Ut (e) ) ∈ Dl (U ) and the map   E(0,1)  e −→ fˆl φˆ e (Ut (e) ) ∈ Dl (U ) is one-to-one (assuming that the radius of the pre-nice set U is sufficiently l is one-to-one for all b ∈ F0 = (fˆ)), we get from small so that fˆ|B(b,6r) Lemma 22.6.5 that

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Part VI Fractal Geometry, Stochastic Properties, and Rigidity 

  diamt fˆl (φˆ e (Ut (e) )) < +∞.

(22.103)

e∈E(0,1)

Since, for parabolic maps, we have that 

M := inf |fˆ (z)| : z ∈ J (fˆ) > 0,

(22.104)

with the help of Theorem 8.3.9 (Koebe Distortion Theorem I for Parabolic Surfaces), we conclude that, for all e ∈ E(0,1), we have that   (22.105) diamt fˆl (φˆ e (Ut (e) )) ≥ M lt ϕˆe t∞ with some constant  ∈ (0,+∞) independent of e ∈ E(0,1). Inserting this into (22.103), we get that  ϕˆe t∞ < +∞. (22.106) e∈E(0,1)

Moving on to dealing with the set E(1,1), note that, similarly to the previous case,

 (22.107) D1,1 := fˆl (φˆ e (Ut (e) )) : e ∈ E(1,1) ⊆ Dl (U ). For every D ∈ D1,1 , let 

E(D) := e ∈ E(1,1) : fˆl (φˆ e (Ut (e) )) = D . Since t >

2q q+1 ,

we have that 

⎛ |(fˆl ) (ξ )|−t  ⎝



⎞l − q+1 q t

|x|

⎠ < +∞,

ξ ∈f

ξ ∈fˆ−l (z)

for every z ∈ J (f ) with the comparability constant independent of z ∈ J (f ). Hence, using also the Koebe Distortion Theorem, we conclude that ⎛ ⎞l   q+1 − t ϕˆe t∞  ⎝ |x| q ⎠ diamt (D)  diamt (D) (22.108) e∈E(D)

ξ ∈f

holds for all D ∈ D1,1 . Because of this, (22.107), and Lemma 22.6.5, we conclude that     ϕˆe t∞ = ϕˆe t∞  diamt (D) < +∞. e∈E(1,1)

D∈D1,1 e∈E(D)

D∈D1,1

(22.109)

22 Conformal Invariant Measures for CNRR Functions

421

Let us now deal with the set E(0,0). For every e ∈ E(0,0), let k(e) ≥ 0 be the least integer such that fˆlk(e) (ϕˆe (Ut (e) )) ⊆ Ut (e) for every integer k(e) ≤ s ≤ NU (e). For every n ≥ 0, let En (0,0) := {e ∈ E(0,0) : NU (e) − k(e) = n}. Then fˆlk(e) (ϕˆe (Ut (e) )) = fˆp−ln (Ut (e) ). 1 (t (e))

(22.110)

Let ψe : B(p1 (t (e)),6r) −→ Tf be a unique holomorphic branch of fˆ−lk(e) such that . ϕˆe = ψe ◦ fˆp−ln 1 (t (e))

(22.111)

fˆl ◦ ψe (Ut (e) ) ∈ Dl (U ).

(22.112)

Then

Invoking the assumption of the previous case that the radius of the pre-nice set l is one-to-one for all b ∈ F0 = (f ), U is sufficiently small so that fˆ|B(b,6r) we conclude that the map En (0,0)  e −→ fˆl ◦ ψe (Ut (e) ) ∈ Dl (U ) is one-to-one. Therefore, using Theorem 8.3.9 (Koebe Distortion Theorem I for Parabolic Surfaces), (22.110), (22.111), Proposition 15.3.4, and Lemma 22.6.5, we get that  "n (0,0) := diamt (fˆl (φˆ e (Xt (e) ))) e∈En (0,0)





diam (fˆl ◦ ψe (Ut (e) )) t

e∈En (0,0)





(Xt (e) )) diamt (fˆp−ln 1 (t (e)) diamt (Ut (e) )

− diamt (fˆl ◦ ψe (Ut (e) ))(n + 1)

e∈En (0,0) −

= (n + 1)



p(p1 (t (e)))+1 p(p1 (t (e))) t

e∈En (0,0) p(p1 (t (e)))+1 − p(p t 1 (t (e)))

 (n + 1)

.

p(p1 (t (e)))+1 p(p1 (t (e))) t

diamt (fˆl ◦ ψe (Ut (e) ))

(22.113)

422

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Therefore, 

diam (fˆl ◦ ϕˆ e (Xt (e) )) = t

e∈E(0,0)



∞  n=0 ∞ 

"n (0,0) −

(n + 1)

p(p1 (t (e)))+1 p(p1 (t (e))) t

< +∞, (22.114)

n=0

where the last inequality sign was written since, by (22.102), t > 1. Since, in our present case, (22.105) is also true, we conclude from it and (22.114) that  ϕˆe t∞ < +∞. (22.115) e∈E(0,0)

We now deal with the last case: the set E(1,0). This is the case where the difficulties of both cases E(1,1) and E(0,0) come together. This happens, in a sense, by combining these two cases together. For every e ∈ E(1,0), we define 0 ≤ k(e) ≤ NU (e) in exactly the same way as in the case of E(0,0). Similarly to this case, let

 En (1,0) := e ∈ E(1,0) : NU (e) − k(e) = n . Again, as in the E(0,0) case, (22.110) holds, the holomorphic map ψe : B(p1 (t (e)),6r) −→ Tf is defined in the same way, (22.108) and (22.112) hold, and the map En (1,0)  e −→ fˆ ◦ ψe (Ut (e) ) ∈ Dl (U ) is one-to-one. As in the case of E(1,1), we define D1,0 (n) by the following formula corresponding to (22.107). For every n ≥ 0,

 D1,0 (n) := fˆl (ψˆ e (Ut (e) )) : e ∈ En (1,0) ⊆ Dl (U ). Furthermore, analogously, for every D ∈ D1,0 (n), we let

 En (D) := e ∈ En (1,0) : fˆl (ψe (Ut (e) )) = D . Then, by the same reasoning as the one leading to (22.108), we get that  diamt (ψe (Ut (e) ))  diamt (D) e∈En (D)

22 Conformal Invariant Measures for CNRR Functions

423

for all n ≥ 0 and all D ∈ D1,0 (n). Now, similarly to (22.113), we get that  "n (1,0) := diamt (φˆ e (Xt (e) )) e∈En (1,0)

=





diamt (φˆ e (Xt (e) ))

D∈D1,0 (n) e∈En (D)







diamt (ψe (Ut (e) ))

(Xt (e) )) diamt (fˆp−ln 1 (t (e)) diamt (Ut (e) )

D∈D1,0 (n) e∈En (D)







−

diamt (ψe (Ut (e) ))(n + 1)

D∈D1,0 (n) e∈En (D) −

 (n + 1)



p(p1 (t (e)))+1 p(p1 (t (e))) t



p(p1 (t (e)))+1 p(p1 (t (e))) t

diamt (ψe (Ut (e) ))

D∈D1,0 (n) e∈En (D)



p(p1 (t (e)))+1 − p(p t 1 (t (e)))

 (n + 1)

diamt (ψe (Ut (e) ))

D∈D1,0 (n) p(p1 (t (e)))+1 − p(p t 1 (t (e)))

 (n + 1)

.

Now, similarly to (22.114), we get the following:  e∈E(1,0)

diamt (ϕˆe (Xt (e) )) =

∞ 

"n (1,0) 

n=0

∞ 

−

(n + 1)

p(p1 (t (e)))+1 p(p1 (t (e))) t

< +∞.

n=0

The proof of Lemma 22.6.6 is complete.



Now we want to form an appropriate strongly regular conformal maximal graph directed Markov system on C and, by applying the Young tower construction, to obtain from it the stochastic laws such as the exponential decay of correlations and the Central Limit Theorem. It will also allow us to show that the metric entropy hμh (f ) is finite. Keep U , the pre-nice set produced in Theorem 22.6.2, for the map (l) fˆl : Tˆ f −→ Tf and

 SˆU = φˆ e e∈Eˆ , the corresponding conformal maximal graph directed Markov system also produced in Theorem 22.6.2. Set F0∗ := (f ) = (f l ) ˆ and define F1∗ to be a (finite) selector of the partition of −1 f (B∞ (f )) = f −1 (∞) into equivalence classes of the equivalence relation ∼f , i.e., choose

424

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

exactly one point from each such equivalence class intersected with f −1 (∞). Define the set of vertices to be 

V := (ξ,j ) : ξ ∈ F0∗ ∪ F1∗, j = {1,2, . . . ,pf (ξ ) } . If the radius r > 0 appearing in Theorem 22.6.2 is smaller than uTf /8, then all the maps ξ := f |Be (ξ,6r) : Be (ξ,6r) −→ Tf , ξ ∈ F0∗ ∪ F1∗, are isometries, in particular one-to-one. Keeping v = (ξ,j ) ∈ V, put v := ξ and set     ∗ −1 Wv∗ := −1 v W (ξ,j ) and Xv := v X(ξ,j ) . If ξ ∈ F1∗ , we will frequently simply write Xξ∗ and Wξ∗ for Xv∗ and Wv∗ , respectively. As the set of edges, set   

E := e ∈ Eˆ : ∃v ∈ V s.t. f NU (e)−1 f˜(W (t (e))) = Wv∗ , where, we recall, the map f˜ : Tf −→  C is given by (16.8). For each edge e ∈ E, write t (e) = (b,j ), b ∈ F0 ∪ F1 , j ∈ {1,2, . . . ,pb }, and let t (e) := (ξ,j ), where ξ is the only element of (F0∗ ∪ F1∗ ) ∩ −1 f (b). Likewise, for i(e). For each e ∈ E, define ∗ ∗ ˆ φe := −1 i(e) ◦ φe ◦ t (e) : Wt (e) −→ Wi(e) .

Obviously, 

∗ SU := φe : Wt∗(e) −→ Wi(e) e∈E

(22.116)

is a conformal maximal graph directed Markov system with edges E, vertices V, and the sets Xv∗ , v ∈ V. Denote by JU its limit set of SU . Let  X∗ := Xv∗ . (22.117) v∈V

For every x ∈ U , let U (x) be the connected component of U containing x. Because of Observation 22.6.1 and since Crit∞ (f ) = ∅ for all parabolic elliptic functions, as an immediate consequence of Theorems 20.3.11, 22.1.1, and 21.0.1 and Proposition 22.1.12, we get the following.

22 Conformal Invariant Measures for CNRR Functions

425

Theorem 22.6.7 If f : C −→  C is a parabolic elliptic function, then (1) Hhs (J (f )) = 0 and hs (J (f )) = +∞. (2) There exist a unique t ≥ 0 and a unique spherical t-conformal atomless probability measure mt,s for f : J (f ) −→ J (f ) ∪ {∞}. Then t = h. (3) All other conformal measures are purely atomic, supported on Sing− (f ) with exponents larger than h. (4) There exists a unique, up to a multiplicative factor, σ -finite f -invariant measure μh absolutely continuous with respect to mh . C. (5) The measure μh is atomless on  (6) The measure μh is equivalent to mh . (7) The measure μh is metrically exact, in particular ergodic and conservative. (8) μh (Tr(f )) = 1. (9) The measure μh is given, according to Theorem 2.4.4, by (2.37)–(2.39). We first of all need the following. Theorem 22.6.8 Let f : C −→  C be a parabolic elliptic function. If U is the pre-nice set produced in Theorem 22.6.2 for the map fˆl : Tˆ (l) f −→ Tf and SU is the conformal maximal graph directed Markov system, defined by (22.116), then mh (JU ) > 0

and

HD(JU ) = HD(J (f )),

where, we recall, mh is the h-conformal measure for f on J (f ). Furthermore, the maximal conformal graph directed Markov system SU is strongly regular (in the sense of Definition 11.4.4) and mh /mh (JU ) is the h-conformal measure (in the sense of Theorem 11.9.2 with t = h) for the system SU . Proof By the construction of the system SU , and by the definition of the limit set JU , the set JU contains the set of all transitive points of the map f l : J (f ) → J (f ), denoted, as usual, by Tr(f l ), that belong to X∗ . Since mh is of full topological support, it, therefore, follows from Theorem 20.3.11 that ˆ h (X∗ ) > 0. mh (JU ) ≥ mh (X∗ ∩ Tr(f l )) = m C, it also satisfies Since the measure mh is h-conformal for the map f : C →  condition (f) of Theorem 11.9.2 with t = h and γ = 1. It, therefore, follows from this theorem that P(h) = 0 and mh /mh (JU ) is the h-conformal measure for the conformal maximal graph directed Markov system JU . Consequently, the system SU is regular and (the same theorem) h = hSU . But, by Theorem 11.5.3, hSU = HD(JU ).

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Part VI Fractal Geometry, Stochastic Properties, and Rigidity

So, we only need to show that the system SU is strongly regular. Exactly as the proof of strong regularity in Theorem 22.5.7, the proof of strong regularity of the system SV is an immediate consequence of the facts that the system SˆU is strongly regular (see Lemma 22.6.6), PSU (h) = PSˆ (h) = 0, PSU (t) ≤ U PSˆ (t) for all t ≥ 0, the latter holding because E ⊆ Eˆ and all the maps v , U   v ∈ V , are isometries, and both maps θSVˆ ,+∞  t −→ PSU (t),PSˆ (t) are U strictly decreasing. The proof is, thus, complete.  We define the map FU : JU −→ JU by the formula: FU (φe (z)) = z if e ∈ E and z ∈ JU . As an immediate consequence of f -invariance of measure μh , item (4) of Theorem 20.3.11, item (1) of Theorem 22.1.1, the fact that every transitive point of f enters (infinitely often) JU , and of Theorem 2.2.6, we get the following. Lemma 22.6.9 If f : C −→  C is a parabolic elliptic function and U is the (l) pre-nice set produced in Theorem 22.6.2 for the map fˆl : Tˆ f −→ Tf , then the induced system FU : JU −→ JU preserves the Borel probability conditional measure μU := (μh (JU ))−1 μh |JU on JU . Now, as a fairly immediate consequence of Theorem 22.6.8, especially its strong regularity part, we shall prove the following. Theorem 22.6.10 If f : C −→  C is a parabolic elliptic function and if μh is the corresponding Borel σ -finite f -invariant measure equivalent to the h-conformal measure mh , then hμh (f ) < +∞. Proof It is a direct consequence of Theorems 22.5.7 and 11.7.3 to have finite entropy of the induced system FU : JU −→ JU with respect to the probability FU -invariant measure μU = (μh (JU ))−1 μh |JU . Then, as FU : JU −→ JU is the first return map of f from JU to JV , Abramov’s Formula (Theorem 6.6.1)  yields hμh (f ) < +∞. We will need the following two technical facts. The first one is this. Lemma 22.6.11 Let f : C −→  C be a parabolic elliptic function. If U is the pre-nice set of Theorem 22.6.2, then there exists a constant C ∈ (0,+∞) such that 1  1 1 k 1 1 f ◦ φτ 1 ≤ Ce−βU |τ | ∞

22 Conformal Invariant Measures for CNRR Functions

427

∗ and all integers 0 ≤ k ≤ |τ |, where β ∈ (0,1) is the for all τ ∈ EA 1 U contracting factor of the graph directed Markov system SU .

Proof Denoting the infimum of Lemma 18.1.9(3) by C1 , we get, for every x ∈ Xt (τ ) , that     k       f ◦ φτ  (x) =  f k ◦ φτ  φσ (τ ) (x)  ·  φσ (τ )  (x) 1 ≤ C1 e−βU |σ (τ )| = C1 eβU e−βU |τ | . 

So, taking C := C1 eβU finishes the proof. Passing to the second technical fact, let ρs :=

dμh dmh,s

be the Radon–Nikodym derivative of the measure μh with respect to mh,s and let dμh ρe := dmh,e be the Radon–Nikodym derivative of the measure μh with respect to mh,e . We need to know the behavior of these derivatives near infinity. We shall prove the following. Lemma 22.6.12 If f : C −→  C is a parabolic elliptic function, then there exist constants R0 > 0 and M ≥ 1 such that M −1 |z|

− q+1 q h

− q+1 q h

≤ ρe (z) ≤ M|z|

∗ (R ). for all z ∈ J (f ) ∩ B∞ 0

Proof Because of (17.10), there exist R1 > 0 and A ≥ 1 such that A−1 |f (z)|

q+1 q

≤ |f  (z)| ≤ A|f (z)|

q+1 q

(22.118)

for all b ∈ f −1 (∞) with qb = q and all z ∈ Bb (R1 ). Because of Lemma 22.2.3, the function ρˆh : J (f )\PC(fˆ) −→ (0,+∞) is continuous. Hence, it follows from (22.20) that there exists R0 ≥ R1 such ∗ (R ), we have that that, for all z ∈ J (f ) ∩ B∞ 0   1 ρˆh (f (b))|f  (w)|−h 2 −1 −1 b∈f0 (∞) w∈f

(z)∩Bb (R1 )

≤ ρe (z) ≤ 2





ρˆh (f (b))|f  (w)|−h,

b∈f0−1 (∞) w∈f −1 (z)∩Bb (R1 )

428

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

where f0−1 (∞) is given by Definition 22.2.4. Putting

 M− := min ρˆh (f (b)) : b ∈ f −1 (∞) and

 M+ := max ρˆh (f (b)) : b ∈ f −1 (∞) , we further get that M− 2





|f  (w)|−h

b∈f0−1 (∞) w∈f −1 (z)∩Bb (R1 )



≤ ρe (z) ≤ 2M+



|f  (w)|−h .

b∈f0−1 (∞) w∈f −1 (z)∩Bb (R1 )

Finally, invoking (22.118), we get that M− A−h − q+1 h − q+1 h #(f0−1 (∞))|z| q ≤ ρe (z) ≤ 2M+ #(f0−1 (∞))|z| q , 2 whenever |z| ≥ R0 . We are done.



22.7 Parabolic Elliptic Maps with Finite Invariant Conformal Measures: Statistical Laws, Young Towers, and Nice Sets Techniques In this section, we continue dealing with parabolic elliptic functions f : C −→  C. As an immediate consequence of Theorem 22.1.11, Proposition 22.1.12, and Theorem 22.3.3, we get the following simple characterization of parabolic elliptic functions for which the invariant measure μh of Theorem 22.6.7 is finite (and, therefore, also for which it is infinite). Theorem 22.7.1 If f : C −→  C is a parabolic elliptic function, then the f -invariant measure μh of Theorem 22.1.11 is finite if and only if h = HD(J (f )) >

2pmax (f ) , pmax (f ) + 1

where pmax (f ) := max{p(ω) : ω ∈ (f )}. We assume throughout this section that the measure μh of Theorem 22.6.7 is finite. We denote such elliptic functions to be of finite class. We then always normalize the measure μh so that μh (J (f )) = 1.

22 Conformal Invariant Measures for CNRR Functions

429

Similarly to Section 22.5.2, we now want to apply the Young tower machinery of Section 4.2 too. Again, the primary issue is to estimate well enough the measure mh of the set of points in JU whose return time to JU is greater than or equal to n. However, now our approach is different than in Section 22.5.2; it is closer to that of [StU1] and [StU2]. In addition, it is much more convenient with this method, we could actually say that it is indispensable, to work with the invariant measure μh rather than the conformal one, mh . For the sake of this method, we raise in the upcoming section the concept of the first entrance time to the sets JU ; it has, in fact, already been introduced in Section 4.2 in the context of Young towers. We keep the setting and notation of the previous section, i.e., Section 22.6.

22.7.1 The First Entrance and the First Return Times The first entrance time to the set JU is the function Ef l ,JU : J (f ) −→ [0,∞] defined as 

Ef l ,JU (z) := min n ∈ {0,1,2, . . . ,+∞} : f ln (z) ∈ JU . On a set of full measure mh , equivalently μh , this function coincides with the function Ef l ,X : J (f ) → [0,∞], given by the formula: 

Ef l ,X∗ (z) := min n ∈ {0,1,2, . . . ,+∞} : f ln (z) ∈ X∗ . We recall that pmax = max{pf l (ω) : ω ∈ (f )}. The main technical result of this section is the following. Lemma 22.7.2 Let f : C −→  C be a parabolic elliptic function of finite class. ∗ If X is the set defined by (22.117), then there exists a constant C ∈ (0,∞) such that     μh {z ∈ J (f ) : Ef l ,JU (z) ≥ n} = μh {z ∈ J (f ) : Ef l ,X∗ (z) ≥ n} +1 2− pmax pmax h

≤ Cn for every integer n ≥ 1.

  Proof Fix b ∈ F1∗ . Since f l Int(Xb∗ ) ⊆  C is an open set containing ∞  C if l ≥ 2), there exists a fundamental domain R ⊆ C for (f l Int(Xb∗ ) =  f and an open ball B ⊃ R such that   f l Int(Xb∗ ) ⊃ B and 2B ∩ PC(f ) = ∅.

430

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Then, by Theorem 17.1.7, there exists a holomorphic inverse branch fb−l : B −→ C of f such that fb−l (B) ⊆ Int(Xb∗ )

(22.119)

f (B) =  C.

(22.120)

and Since the map f : C −→  C is parabolic, we can partition the set ⎛ ⎞ ∞  p(ω)     ∗ ⎠ J (f )\ ⎝(f ) ∪ fω−ln X(ω,j ) ω∈(f ) j =1 n=0

into countably many Borel sets {Bk }∞ k=1 , such that diam(Bk ) ≤ 1

(22.121)

∞    B Bk ,diam(Bk ) ∩ PC(f ) = ∅,

(22.122)

for all k ≥ 1,

k=1

and lim dist(0,Bk ) = +∞.

k→∞

(22.123)

In particular, by Theorem 17.1.7, all holomorphic branches of f −ln , n ≥ 0, are  well defined on all sets B Bk ,diam(Bk ) , k ≥ 1, and so the Koebe Distortion Theorems apply to all these holomorphic branches restricted to the sets Bk , k ≥ 1. Now fix an integer n ≥ 0 and k ≥ 1. Put k := Bk ∩ Ef−1l ,X∗ ([n,∞]).

  Because of (22.120), there exists a holomorphic branch fk−l : B Bk ,diam(Bk ) −→ C such that    f j fk−l B(Bk ,diam(Bk )) ⊆ B (22.124) for every integer j = 0,1,2, . . . ,l − 1. Hence, looking up at this formula with j = 0, we see that the composition   fb−l ◦ fk−l : B Bk ,diam(Bk ) −→ C is well defined and holomorphic, and it follows from (22.119) and (22.124) that    fb−l ◦ fk−l B Bk ,diam(Bk ) ⊆ Int(Xb∗ ). (22.125)

22 Conformal Invariant Measures for CNRR Functions

431

Fix a point ξk ∈ k . Because of the Koebe Distortion Theorem I, Euclidean Version (Theorem 8.3.8), we have that     −l  ρe fb−l ◦ fk−l (z)  −l −l (f ◦ f −l ) (z)h dμh (z) μh fb ◦ fk (k ) = b k ρ (z) e k  −l  h 1 (22.126) dμh (z), ≥ C1 C2 (fk ) (ξk ) ρ k e (z) where C1 := inf{ρe (x) : x ∈ Int(Xb∗ )} > 0 and C2 := inf{|(fb−l ) (x)| : x ∈ B} > 0. It follows from (22.123) and (22.121) that the sets f l−1 ◦ fk−l (Bk ) accumulate toward poles in B. We can, in fact, choose the holomorphic branches fk−l such that   lim Dist a,f l−1 ◦ fk−l (Bk ) = 0 k→∞

and ∞ 

  f l−1 ◦ fk−l (Bk ) ∩ f −1 (∞)\{a} = ∅,

k=1

where a ∈ f −1 (∞) ∩ B is such a pole that qa = qmax . Then, it follows from the behavior of meromorphic functions around poles and also using (22.124), we conclude that there exists a constant C3 ∈ (0,∞) such that  −l  − q+1  f (ξk )| ≥ C3 |ξk | q (22.127) k for all integers k ≥ 1, where, as usual, we denote qmax by q. On the other hand, it follows from Lemma 22.6.3 that the set V(f ) := (f ) ∪

∞    p(ω)

 ∗  fω−ln X(ω,j )

(22.128)

ω∈ j =1 n=0

is a neighborhood of (f ). Since all sets Bu , u ≥ 1, are disjoint from this neighborhood, we deduce from Theorem 22.2.5 and Lemma 22.6.12 that there exists a constant C4 ∈ (0,∞) such that ρe (z) ≤ C4 |ξu |

− q+1 q h

(22.129)

for all integers u ≥ 1 and all z ∈ Bu . Coming back to our fixed integer k ≥ 1, inserting (22.129) into (22.126), and using (22.127), we get that q+1   − q+1 h h μh fb−l ◦ fk−l (k ) ≥ C1 C2 C3h |ξk | q C4−1 |ξk | q μh (k ) = C1 C2 C3h C4−1 μh (k ).

432

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Since, because of (22.125), Ef l ,X∗ (fb−l ◦ fk−l (k )) ⊆ {0}, we, thus, have that       μh f −2l(k ) ∩Ef−1l ,X∗ ([n + 2,+∞]) ≤ μh f −2l (k ) − μh fb−l ◦ fk−l (k ) ≤ μh (k ) − C1 C2 C3h C4−1 μh (k ) = (1 − C1 C2 C3h Ch−1 )μh (k ). So, setting α := 1 − C1 C2 C3h Ch−1 ∈ [0,1], we have that   μh f −2l (k ) ∩ Ef−1l ,X∗ ([n + 2,+∞]) ≤ αμh (k ). Recalling that {Bk }∞ k=1 is a partition of J (f )\V(f ) , by summing up the above inequality over all k ≥ 1, we obtain that   μh f −2l (J (f )\V(f ) ) ∩ Ef−1l ,X∗ ([n + 2,+∞])   (22.130) ≤ αμh (J (f )\V(f ) ) ∩ Ef−1l ,X∗ ([n,+∞]) . Now suppose that z ∈ V(f ) \(f ). This means that there exists ω ∈ (f ), j ∈ {1,2, . . . ,p(ω)}, and i ≥ 1 such that  ∗  z ∈ fω−li X(ω,j ) . ∗ ∗ Then f li (z) ∈ X(ω,j ) ⊆ X . Therefore,

Ef l ,X∗ (z) ≤ i. Thus, 

  ∗   fω−li X(ω,j V(f ) \(f ) ∩ Ef−1l ,X∗ ([n,∞]) ⊆ ) . i≥n

So, using Proposition 15.3.4, applied to the function f l , and Proposition 22.3.1, we conclude that there exist constants C5,C6 ∈ (0,∞) such that      μh V(f ) ∩ Ef−1l ,X∗ ([n,+∞]) = μh V(f ) \(f ) ∩ Ef−1l ,X∗ ([n,∞]) ≤

  p(ω) ω∈(f ) j =1 i≥n

  ∗  μh fω−li X(ω,j )

22 Conformal Invariant Measures for CNRR Functions

≤ C5

 p(ω) 

i

433

1− p(ω)+1 p(ω) h

ω∈ j =1 i≥n

≤ C5 #p



i

1− p+1 p h

,

i≥n

where p := pmax . Since, by our hypotheses, 1 − obtain, with some constant C6 ∈ (0,∞), that

p+1 p h

< −1, we further

  2− p+1 h μh V(f ) ∩ Ef−1l ,X∗ ([n,∞]) ≤ C6 n p . Since the measure μh is f -invariant, we, therefore, get that     −2    μh f −2 V(f ) ∩ Ef−1 V(f ) ∩ Ef−1 ,U ([n + 2,∞]) ≤ μh f ,U ([n,∞])   = μh V(f ) ∩ Ef−1 ,U ([n,∞]) 2− p+1 p h

≤ C7 n

.

Combining this with (22.130), we get that     2− p+1 p h. μh Ef−1l ,X∗ ([n + 2,∞]) ≤ αμh Ef−1 ,U ([n,∞]) + C6 n So, there exists a constant C7 ∈ [C6,∞) such that   −1   2− p+1 p h. μh Ef−1 ,U ([n + 2,∞]) ≤ αμh Ef l ,X∗ ([n,∞]) + C7 (n + 2) (22.131) Fix an integer s ≥ 2 so large that  β := α

s+2 s

p+1 h−2 p

< 1.

(22.132)

Put

#

p+1 h   $ C := max C7 (1 − β)−1, max k p μh Ef−1l ,X∗ ([k,∞]) : 1 ≤ k ≤ s .

We shall prove by induction that   2− p+1 h μh Ef−1l ,X∗ ([n,∞]) ≤ Cn p

(22.133)

for every integer n ≥ 1. Indeed, for 1 ≤ n ≤ s, this is immediate from the definition of C. So, suppose for the inductive step that n ≥ s and (22.133) holds for all 1 ≤ k ≤ n. Then, also using (22.131) and (22.132), we get that

434

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

    2− p+1 h μh Ef−1l ,X∗ ([n + 1,∞]) ≤ αμh Ef−1l ,X∗ ([n − 1,∞]) + C7 (n + 1) p 2− p+1 p h

+ C7 (n + 1)

2− p+1 p h

+ (1 − β)C(n + 1)

≤ αC(n − 1)

≤ βC(n + 1)

2− p+1 p h

= C(n + 1)

2− p+1 p h 2− p+1 p h

.

The inductive proof of (22.132) is, thus, complete and we are done.



Let NU : JU −→ [1,∞] be the first return time to JU , i.e., ∀z ∈ JU , NU (z) := min{n ≥ 1 : f n (z) ∈ JU }. As an almost immediate consequence of Lemma 22.7.2, we get the following result, which we will not truly need but which is interesting on its own and we will use it in a “negative” way. Lemma 22.7.3 Let f : C −→  C be a parabolic elliptic function of finite class. If U is the pre-nice set of Theorem 22.6.2, then there exists a constant C ∈ (0,+∞) such that   2− pmax +1 h μh z ∈ JU : NU (z) ≥ n ≤ Cn pmax for every integer n ≥ 1. Proof Since

  z ∈ U ∩ J (f ) : NU (z) ≥ n ⊆ z ∈ J (f ) : Ef l ,JU (f l (z)) ≥ n − 1 = f −l (Ef−1l ,J ([n − 1,+∞])), U

f l -invariant,

and since the measure μh is we get from Lemma 22.7.2, for all n ≥ 2, that   2− pmax +1 h 2− pmax +1 h μh {z ∈ JU : NU (z) ≥ n} ≤ C(n − 1) pmax ≤ C  n pmax , with some constant C  ∈ (0,∞), and the lemma follows.



Since the measures μh and mh restricted to our nice set U are equivalent to the log-bounded Radon–Nikodyn derivatives, as an immediate consequence of Lemma 22.7.3, we get the following. Lemma 22.7.4 Let f : C −→  C be a parabolic elliptic function of finite class. If U is the pre-nice set of Theorem 22.6.2, then there exists a constant C ∈ (0,+∞) such that   2− pmax +1 h mh z ∈ JU : NU (z) ≥ n ≤ Cn pmax for every integer n ≥ 1.

22 Conformal Invariant Measures for CNRR Functions

435

22.7.2 Young Towers for Parabolic Elliptic Functions with Finite Invariant Measure μh : Statistical Laws In this subsection, we use the fruits of the results obtained in the previous subsection and in Section 4.2 to establish a polynomial decay of correlations and the Central Limit Theorem for all parabolic elliptic functions f : C −→  C of finite class (i.e., with finite invariant measure μh ) and all H¨older continuous bounded observables. This subsection is very similar to subsection 22.5.2. We, however, present it in full for the sake of completeness, for the convenience of the reader, and since there is one subtlety in the proofs which needs to be taken care of differently than in subsection 22.5.2. We keep, in this section, all hypotheses and notation from the previous one. In particular, f : C −→  C is a parabolic elliptic function of finite class. Furthermore, U is the pre-nice set produced in the previous section, i.e., in Theorem 22.6.2. Let SU be the corresponding conformal graph directed Markov system also produced in Theorem 22.6.2. Keep E to denote the set of its edges and V to denote the set of vertices of SU . We recall that the map FU : JU −→ JU has been defined by the formula: FU (φe (z)) = z if e ∈ E and z ∈ JU . Our goal is to show that the system (FU ,mh ) (in fact, the measure mh is treated here as restricted to JU ) fits into the framework of Section 4.2 and to check that the hypotheses of Theorems 4.2.2–4.2.4 are satisfied for this (induced) system (FU ,mh ). The stochastic laws will then automatically follow. (1) The space 0 is now JU , the limit set of the iterated function system SU . (2) The partition P0 consists of the sets e := φe (JU ), e ∈ E. (3) The measure m is the measure mh restricted to JU ; it is positive because of Theorem 22.6.8. (4) The map T0 : 0 −→ 0 is, in our setting, just the map FU . (5) The function R, the return time, is, naturally, defined as R|e := NU |e . We also write NU (e) := NU |e . Fix e ∈ E arbitrary and then two arbitrary points x,y ∈ e = φe (JU ). This means that x = φe (x  ) and y = φe (y  ) with some x ,y  ∈ JU . Since

436

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

t h − 1 = h(t − 1) + O(|t − 1|2 ), and because of Theorem 8.3.7, there exist respective constants C1,C2 > 0 such that we have that          |(φ ) (x  )|h  J acmh FU (y)    |FU (y)|h   e     J ac F (x) − 1 =  |F  (x)|h − 1 =  |(φ ) (y  )|h − 1 mh U e U   (22.134)   |(φe ) (x  )|  − 1 ≤ C1     |(φe ) (y )| ≤ C1 C2 |y  − x  |. Now write x  = πU (α) and y  = πU (γ ) with appropriate α,γ ∈ E. Put ω := α ∧ γ and k := |ω|. Also write x  := πU (σ k (α)) and y  := πU (σ k (γ )). Then we get that |y  − x  | = |φω (y  ) − φω (x  )| ≤ |φω (x  )||y  − x  |     |ω| ≤ diame Xt∗(ω) φω ∞ ≤ diame Xt∗(ω) βU   s(x ,y  ) = diame Xt∗(ω) βU   s(F (x),FU (y)) = diame Xt (ω) βU U s(FU (x),FU (y))

≤ DU βU

(22.135)

,

where, we recall, βU ∈ (0,1) is the contracting factor of the graph directed Markov system SU and

 DU := max diame (Uv ) : v ∈ F < +∞. Since s(FU (x),FU (y)) = s(x,y) − 1 (as we know that s(x,y) ≥ 1), along with (22.134), (22.135) gives that    J acmh FU (y)    ≤ DU β s(FU (x),FU (y)) = β −1 DU β s(x,y) . − 1 U U U  J ac F (x)  mh U So, (4.8) is established in our context. The fact that the partition P0 is generating follows from the contracting property of the graph directed Markov system SU . The Big Images Property holds because the alphabet of the GDMS SU is finite. The last assumption in Theorem 4.2.2 is that the map T :  −→ , which we now denote by TU , is topologically mixing. The proof of this fact is virtually the same as the proof of Lemma 22.5.9 for subexpanding elliptic maps. We provide it here for the sake of completeness and convenience of the reader. Lemma 22.7.5 The dynamical system TU :  −→  is topologically mixing in the sense of Section 4.2.

22 Conformal Invariant Measures for CNRR Functions

437

∗ , set Proof For every τ ∈ EA

NU (τ ) :=

|τ | 

|τ |    NU (τj ) = R τj .

j =1

j =1

By our construction of the tower Yf , we have, for every e ∈ E, that     TUR(e ) e × {0} = TU e × {R(e )} = FU (e ) × {0} = Jt (e) × {0}. ∗ , that So, by an immediate induction, we get, for every τ ∈ EA   TUNU (τ ) φτ (Jt (τ ) ) × {0} = Jt (τ ) × {0}.

(22.136)

Now fix two arbitrary elements a,b ∈ E. Then there exists s ≥ 0 such that   f lu φa (Xt∗(a) ) ⊇ φb (Xt∗(b) ) for all u ≥ s. Hence, u, there exists a holomorphic branch of f lu  for all such ∗ ∗ mapping φb Xt (b) into φa Xt (a) . By our construction of the conformal graph directed Markov system SU , this holomorphic branch is an admissible composition of elements of SU . This means that it is equal to ∗ φτ (u) : Xt∗(τ (u)) −→ Xi(τ (u)) ∗ with t (τ (u)) = i(b) and i(τ (u)) = i(a). Then, applying for some τ (u) ∈ EA (22.136), we get, for every 0 ≤ p ≤ R(a ) − 1, every 0 ≤ q ≤ R(b ) − 1, and every n ≥ s + R(b ) − 1, that     n+p  q+n+p−q  TUn a × {p} = TU a × {0} = TU a × {0}  q  n+p−q  φτ (n+p−q) (Jt (τ (n+p−q)) ) × {0} ⊇ TU TU   q q = TU Jt (τ (n+p−q)) × {0} ⊇ TU φb (Jb ) × {0}

= φb (Jb ) × {q} = b × {q}. 

The proof of Lemma 22.5.9 is complete.

Since the measures μh and mh , restricted to our nice set U , are equivalent with log-bounded Radon–Nikodyn derivatives, as an immediate consequence of Theorem 20.3.11 (particularly its item (2)), the finiteness of the measure μh , and Theorem 2.2.10(a) (Kac’s Lemma), we get that R dmh  0

R dμh < +∞.

(22.137)

0

Therefore, as an immediate consequence of Observation 4.2.1 and Theorem 4.2.2, we get the following.

438

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Theorem 22.7.6 If f : C −→  C is a parabolic elliptic function of finite class, then ˜ h is derived out of mh restricted to JU according (a) m ˜ h () < +∞, where m to (4.9). (b) There exists a unique probability TU -invariant measure νh , absolutely continuous with respect to m ˜ h. ˜ h is bounded from below by a (c) The Radon–Niokodym derivative dνh /d m positive constant.   (d) The dynamical system TU ,νh is metrically exact, thus ergodic. Now consider H :  −→ C, the natural projection from the abstract tower , generated by the above, to the complex plane C given by the formula H (z,n) = f ln (z).

(22.138)

H ◦ TU = f l ◦ H,

(22.139)

Then

m ˜ h |0 ◦ H −1 = mh |JU , and m ˜ h |e ×{n} ◦ H −1 = mh |e ×{0} ◦ f −ln = mh |e ◦ f −ln for all e ∈ E˜ and all 0 ≤ n ≤ NU (e). Now m ˜ h |e ×{n} ◦ H −1 is absolutely continuous with respect to mh , with the Radon–Nikodym derivative equal to  −1 Je,n := J acmh f ln |e in f n (e ) and zero elsewhere in J (f ). Therefore, using (22.137), we get that (e)−1  NU J (f ) e∈E

Je,n dmh =

(e)−1  NU τ ∈E

n=0

n=0 −1

=m ˜h ◦H

Je,n dmh J (f )

(J (f )) = m ˜ h () < +∞.

Thus, the function (e)−1  NU e∈E

Je,n

n=0

is integrable with respect to the measure mh . This implies immediately that the measure m ˜ h ◦ H −1 is absolutely continuous with respect to the measure mh with the Radon–Nikodym derivative equal to

22 Conformal Invariant Measures for CNRR Functions (e)−1  NU e∈E

439

Je,n .

n=0

Hence, the measure νh ◦ H −1 is also absolutely continuous with respect to mh . Since νh is FU -invariant and H ◦ TU = f l ◦ H , the measure νh ◦ H −1 is f l -invariant. But the measure μh is f -invariant, ergodic, and equivalent to the conformal measure mh . Hence, νh ◦ H −1 is absolutely continuous with respect to the ergodic measure μh . In conclusion, we get the following. Lemma 22.7.7 If f : C −→  C is a parabolic elliptic function of finite class, then νh ◦ H −1 = μh . Recall that the function ETU is given by (4.10). Also using (22.139) and (22.138), we get, for every point (z,k) ∈ , that ETU (z,l) = Ef l ,JU (f lk (z)) = Ef l ,JU (H (z,k)). Therefore, also using Lemma 22.7.7, Theorem 22.7.6, and, at the end, Lemma 22.7.2, we get that        m ˜ h ET−1 (n,+∞]  νh ET−1 (n,+∞] = νh H −1 Ef−1l ,J (n,+∞]) U U U   = μh Ef−1l ,J (n,+∞] U

n

+1 2− pmax pmax h

.

(22.140)

Therefore, we conclude, from Theorem 4.2.3, the following. Theorem 22.7.8 If f : C −→  C is a parabolic elliptic function of finite class, then, for the dynamical system TU : (,νh ) −→ (,νh ), the following hold. (a) The Polynomial Decay of Correlations in the form of (4.12) with the +1 parameter α := pmax pmax h − 2.

max (b) If h > p3p , then, for all γ ∈ (0,1), the Central Limit Theorem is true max +1 for all functions g ∈ Cγ () that are not cohomologous to a constant in L2 (νh ).

Remark 22.7.9 Note that the hypothesis of item (b) has a chance to be satisfied only if pmax = 1. Then this hypothesis says that h > 3/2. In order to verify that such an inequality holds for particular examples, one may, for example, use Theorem 17.3.1.

440

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

We did not formulate the Law of the Iterated Logarithm as, in order to establish it via Theorem 4.2.4 and Lemma 22.7.4, we would have to know that h > 2, which, of course, is never satisfied. We are now in a position to prove the following. Theorem 22.7.10 If f : C −→  C is a parabolic  elliptic function of finite class, then the dynamical system f l |J (f ),μh satisfies the following. If g : J (f ) −→ R is a bounded function, H¨older continuous with respect to the Euclidean metric on J (f ), then (1) (The Polynomial Decay of Correlations) For every bounded measurable function ψ : J (f ) −→ R, we have that   pmax +1    ψ ◦ f ln · gdμh − gdμh ψdμh   n2− pmax h .   (2) (The Central Limit Theorem) If, in addition, g : J (f ) → R is not cohomologous to a constant in L2 (μh ) with respect to f l , i.e., if there is no square integrable function η for which g = const + η ◦ f l − η, then the Central Limit Theorem holds for g. More precisely, there exists σ > 0 such that ' "n−1 lj j =0 g ◦ f − n gdμh −−−−→ N (0,σ ) √ n→∞ n in distribution, where, as usual, N (0,σ ) denotes the Gauss (normal) distribution centered at 0 with covariance σ . Proof

Keep U,

the pre-nice set produced in Theorem 22.6.2, and SU , the corresponding conformal graph directed Markov system also produced in Theorem 22.6.2. Let g : J (f ) −→ R and ψ : J (f ) −→ R be as in the hypotheses of our theorem. Define the functions g˜ := g ◦ H :  −→ R and ψ˜ := ψ ◦ H :  −→ R. We shall prove the following. Claim 1. The function g˜ :  −→ R belongs to the space Cβ for an appropriate exponent β ∈ (0,1). Indeed, consider two arbitrary points (x,k),(y,u) ∈ . We treat two cases separately depending on whether s((x,k),(y,u)) = 0 or not. If s((x,k),(y,u)) = 0, then we get that

22 Conformal Invariant Measures for CNRR Functions

441

|g(y,u) ˜ − g(x,k)| ˜ = |g(H (y,u)) − g(H (x,k))| = |g(f lu (y)) − g(f lk (x))| ≤ |g(f lu (y))| + |g(f lk (x))| ≤ 2g∞ = 2g∞ β s((x,k),(y,u))

(22.141)

regardless of what the value of β is, which will be specified in the next case. Indeed, if s((x,k),(y,u)) > 0, then k = u, 0 ≤ k < R(x) = R(y), and |g(y,u) ˜ − g(x,k)| ˜ = |g(f lk (y)) − g(f lk (x))| ≤ Hg |f lk (y) − f lk (x)|γ , (22.142) where Hg and γ are, respectively, the H¨older constant and H¨older exponent   s((x,k),(y,u)) of the function g. Moreover, x,y ∈ φτ Jt (τ ) , with some τ ∈ EA , and then    f lk (x),f lk (y) ∈ f lk φτ Jt (τ ) ,    0 ≤ k ≤ |τ1 |, and f lk φτ Jt (τ ) is contained in a connected component of   f lk ◦ φτ Wt∗(τ ) whose diameter is, by Lemma 22.6.11, bounded above by   C exp −βU s((x,k),(y,u)) . In conjunction with (22.142) and (22.141), this finishes the proof of Claim 1 by taking β := e−γβU . Claim 2. The function g˜ is not cohomologous to a constant in L2 (ν). Indeed, assume without loss of generality that μh (g) = 0. By virtue of Lemma 2.3.7, the fact that g : J (f ) → R is not a coboundary with  (l) respect ∞ to the l 2 map f in L (μh ) equivalently means that the sequence Sn (g) n=1 is not (l)

uniformly bounded in L2 (μh ), where Sn refers to the nth Birkhoff sum with respect to the map f l . But, because of Lemma 22.7.7, Sn(l) (g) ˜ L2 (νh ) = Sn(l) (g)L2 (μh ) .  (l) ∞ So, the sequence Sn (g) ˜ n=0 is not uniformly bounded in L2 (νh ). Thus, by Lemma 2.3.7 again, it is not a coboundary in L2 (ν). The proof of Claim 2 is finished. Having these two claims, all items, (1), (2), and (3), now follow immediately from Theorem 4.2.3 with the use of Lemma 22.7.7 and (22.139). The proof of Theorem 22.7.10 is finished. 

442

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

As a fairly immediate consequence of Theorem 22.7.10, we get the following. Theorem 22.7.11 If f : C −→  C is a parabolic elliptic function of finite class,  then the dynamical system f |J (f ),μh satisfies the following. If g : J (f ) −→ R is a bounded function, H¨older continuous with respect to the Euclidean metric on J (f ), then (1) (The Polynomial Decay of Correlations) For every bounded measurable function ψ : J (f ) −→ R, we have that   pmax +1    ψ ◦ f n · gdμh − gdμh ψdμh   n2− pmax h .   (2) (The Central Limit Theorem) If, in addition, g : J (f ) −→ R is not cohomologous to a constant in L2 (μh ), i.e., if there is no square integrable function η for which g = const+η ◦f −η, then the Central Limit Theorem holds for g. More precisely, there exists σ > 0 such that ' "n−1 j j =0 g ◦ f − n gdμh −−−−→ N (0,σ ) √ n→∞ n in distribution, where, as usual, N (0,σ ) denotes the Gauss (normal) distribution centered at 0 with covariance σ . Proof Of course, each function ψ ◦ f r : J (f ) −→ R, r > 0, is bounded and measurable and ψ ◦ f r ∞ ≤ ψ∞ . Let Cr ∈ (0,+∞), r ≥ 0, be the constant witnessing formula (1) in Theorem 22.7.10 for the functions ψ ◦ f r and g. Let C := max{Cr : 0 ≤ r ≤ l − 1}. Given n ≥ l, write uniquely that n = lk + r, where r and k are nonnegative integers and r ∈ {0,1, . . . ,l − 1}. Then, by virtue of Theorem 22.7.10, we get that      ψ ◦ f n gdμh − gdμh ψdμh        r lk r  =  (ψ ◦ f ) ◦ f gdμn − gdμh ψ ◦ f dμh  ≤ Cr k

+1 2− pmax pmax h

+1 2− pmax pmax h

≤ Cn

≤ Cr

 n 2− pmax +1 h pmax

2l

22 Conformal Invariant Measures for CNRR Functions

443

pmax +1 −2

for all n ≥ 0 large enough, where C  = C(2l) pmax . Item (1) is, thus, proved. Moving on to item (2), assume without loss of generality that μh (g) = 0. Now, given the integer n ≥ 0, write uniquely that n = lk + r, k = kn ≥ 0, 0 ≤ r ≤ l − 1. Then n−1 

Sn g = Sk(l) (Sl g) +

g ◦ f j = Sk(l) (Sl g) + Sr (g ◦ f n−r ).

(22.143)

j =n−r

Hence, (l)

S (Sl g) Sn g √ = k√ n k

7

k Sr (g ◦ f n−r ) , + √ n n

so       7  (l) (l) S g 1 Sk (Sl g)   Sk (Sl g)   k 1   Sr (g ◦ f n−r )   n − − + ≤ √ √ √ √       . (22.144)  n l l  n k   k  n Likewise, Sn g √ n

7

(l)

S (Sl g) Sr (g ◦ f n−r ) n = k√ + . √ k k k

Therefore,     7 7   S g   Sr (g ◦ f n−r )  n Sk(l) (Sl g)   Sn g   n  n    − l +  √ ≤ √ √ l − √  k.  n n k   n k

(22.145)

Fix now a nondegenerate open interval (a,b) and ε > 0. Since the function g is uniformly bounded, there exists N ≥ 1 such that  7 7 7    k  n   Sr (g ◦ f n−r )  n 1       √  k ≤ ε,  k − l  ≤ ε, and  n − l  ≤ ε (22.146)  n for every n ≥ N. So, if z ∈ J (f ) and Sn g(z) ∈ (a,b), √ n then it follows from (22.145) and (22.146) that (l)

Skn (Sl g(z))   ∈ la − (max{|a|,|b|} + 1)ε,lb + (max{|a|,|b|} + 1)ε . √ kn

444

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Therefore, applying Theorem 22.7.10(2), which is applicable because of Corollary 2.3.8, we get that & % Sn g(z) lim μh ∈ (a,b) z ∈ J (f ) : n→∞ n / (l) S (Sl g(z))  ≤ lim μh z ∈ J (f ) : k √ ∈ la − (max{|a|,|b|} + 1)ε,lb k n→∞ 0

 + (max{|a|,|b|} + 1)ε   = N (0,σl ) la − (max{|a|,|b|} + 1)ε,lb + (max{|a|,|b|} + 1)ε , where σl is the covariance of Theorem 22.7.10(2) ascribed to the dynamical system (f l ,μh ) and observable Sl g. Since the normal distribution has no atoms, by letting ε → 0, it follows from this formula that # $   Sn g(z) lim μh z ∈ J (f ) : ∈ (a,b) ≤ N (0,σl ) (la,lb) n→∞ n   = N (0,σl / l) (a,b) . (22.147) Likewise, if z ∈ J (f ) and (l)

 1 Skn (Sl g(z))  ∈ a + (max{|a|,|b|} + 1)ε,b − (max{|a|,|b|} + 1)ε , √ l kn then it follows from (22.144) and (22.146) that Sn g(z) ∈ (a,b). √ n Therefore, by applying Theorem 22.7.10(2), we get that # $ Sn g(z) lim μh z ∈ J (f ) : ∈ (a,b) n n→∞ / (l) S (Sl g)(z)  ≥ lim μh z ∈ J (f ) : k √ ∈ la + l(max{|a|,|b|} + 1)ε,lb k n→∞ 0

 −l (max{|a|,|b|} + 1)ε   = N (0,σl ) la + l(max{|a|,|b|} + 1)ε,lb − l(max{|a|,|b|} + 1)ε .

22 Conformal Invariant Measures for CNRR Functions

445

Hence, letting ε → 0, we get that # $   Sn g(z) ∈ (a,b) ≥ N (0,σl ) (la,lb) lim μh z ∈ J (f ) : n n→∞   = N (0,σl / l) (a,b) . Along with (22.147), this gives # $   Sn g(z) ∈ (a,b) = N (0,σl / l) (a,b) , lim μh z ∈ J (f ) : n→∞ n and the proof of Theorem 22.7.11(2) is complete. Thus, the proof of the entire Theorem 22.7.11 is complete. 

22.8 Infinite Conformal Invariant Measures: Darling–Kac Theorem for Parabolic Elliptic Functions In this section, similarly to the two previous sections, we deal with parabolic elliptic functions, i.e., with all such elliptic functions f : C −→  C for which Crit(f ) ∩ J (f ) = ∅ and (f ) = ∅. We keep the notation of these two previous sections. Similarly to the previous section, i.e., Section 22.7, our main focus is the invariant measure μh . However, now we no longer assume that the measure μh is finite; indeed, all results in this section concerning measure μh are nontrivial only if this measure is infinite. Our ultimate goal in this section is to prove the Darling–Kac Theorem of Section 5.3 (Theorems 5.3.2 and 5.3.4) for the class of parabolic elliptic functions f : C −→  C with infinite invariant measure μh ; we then say that such functions are of infinite class. We want to apply Theorem 5.3.4; therefore, taking the iterate l ≥ 1 so large that all rationally indifferent periodic points of f become simple fixed points of f l , all we need to do is to find a Borel set Y satisfying the hypotheses of Theorem 5.3.4. Up to the very last theorem of this section, i.e., the Darling–Kac Theorem for Parabolic Elliptic Functions of Infinite Class (Theorem 22.8.11), our considerations are valid regardless of whether the f -invariant measure μh is infinite or finite. To begin with, let

 ∞   −ln fω ((ω)) , (22.148) Y := J (f )\ {} ∪ ω∈ n=1

446

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

where, we recall, given α ∈ (0,π), the sets j (ω,α), ω ∈ , 1 ≤ j ≤ p(ω), are defined in Lemma 15.3.7, p(ω) 

(ω) = (ω,α) =

j (ω,α),

j =1

and =



(ω).

ω∈(f )

It follows from Theorem 22.1.11 that 0 < μh (Y ) < +∞. As in previous sections, let mh,e be the h-conformal Euclidean measure and mh,s be the h-conformal spherical measure. We know that mh,s is finite and it follows from Observation 17.6.1 that 1 dmh,s (z) = . dmh,e (1 + |z|2 )h Recall that ρs :=

dμh dmh,s

is the Radon–Nikodym derivative of the measure μh

dμh with respect to mh,s and that ρe := dm is the Radon–Nikodym derivative of h,e the measure μh with respect to mh,e . Let

Ls : Cb (J (f )) −→ Cb (J (f )) and Le : L1 (mh,e ) −→ L1 (mh,e ) be the transfer (Perron–Frobenius) operators ascribed, respectively, to the quasi-invariant measures mh,s and mh,e of (2.8). They are given, respectively, by the formulas  Ls (g)(z) = |f  (w)|−h s g(w) w∈f −1 (z)

and Le (g)(z) =

 w∈f −1 (z)

|f  (w)|−h g(w).

22 Conformal Invariant Measures for CNRR Functions

447

The Perron–Frobenius operator L0 : Cb (J (f )) −→ Cb (J (f )) ascribed to the measure μh is equal to  g(w)ρs (w)|f  (w)|−h L0 (g)(z) = ρs−1 (z)Ls (gρs )(z) = ρs−1 (z) s w∈f −1 (z)

= ρe−1 (z)Le (gρe )(z)  = ρe−1 (z) g(w)ρe (w)|f  (w)|−h . w∈f −1 (z)

For every point ω ∈  and ξ ∈ f −l (ω)\, let fξ−l : B(ω,θ) −→ C be the unique holomorphic inverse branch of f −l , defined on B(ω,θ) with some θ ∈ (0,θ(f )] sufficiently small, and sending ω to ξ . Now fix z ∈ (ω) ∩ J (f ) and n0 ≥ 1 so large that fω−ln (z) ∈ B(ω,θ) for all n ≥ n0 . According to Proposition 15.3.4 and Theorem 22.2.5, we have that  −l h ρs (fξ−l ◦ fω−ln (z)) −ln   (fξ ◦ fω ) (z)s lim (n + 1) = n→+∞ ρs (z) p(ω)+1  h  −h (1 + |z|2 )h ρs (ξ ) h = lim n p(ω) (fω−ln ) (z) (f l ) (ξ )s n→+∞ (1 + |ω|2 )h ρs (z) p(ω)+1   −h − h = |aω | p(ω) (f l ) (ξ )s (1 + |ω|2 )−h ρs (ξ )  −1   −1 2 h −h(p(ω)+1)  f  (22.149) ω ∞ (z) ρs (z)(1 + |z| ) |z − ω| p(ω)+1 p(ω) h

   uniformly on (ω)∩J (f ), where  fω−1 ∞ (z) comes from Proposition 15.3.4 and (15.20). Denote the product of the first four factors in the last line of (22.149) by Qω (ξ ) and the product of the last four factors by G(z). With the sets Yn (f l ), defined in (5.11) with T = f l , (22.149) then implies that lim n

n→+∞

p(ω)+1 p(ω) h

  Lln 0 11Yn (f l ) (z) = G(z)

 ξ ∈f −l (ω)\

uniformly on (ω) ∩ J (f ). Now let p := min{p(ω) : ω ∈ }

Qω (ξ )

(22.150)

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Part VI Fractal Geometry, Stochastic Properties, and Rigidity

and

⎧ if z ∈ J (f )\ ⎨ 0 " G(z) ξ ∈f −l (ω)\ Qω (ξ ) if z ∈ (ω) ∩ J (f ) and p(ω) = p D(z) := ⎩ 0 if z ∈ (ω) ∩ J (f ) and p(ω) < p.   Since Lln 0 11Yn (f l ) (z) = 0 for all n ≥ 2 and all z ∈ J (f )\{}, it follows from (22.150) that p+1   lim n p Lln (22.151) 0 11Yn (f l ) (z) = D(z) n→∞

uniformly on J (f ). Integrating this over J (f ), we, thus, get that lim n

p(ω)+1 p(ω) h

n→∞

p+1 1 h lim n p μh (Yf l ,n ) n→∞ μh (Y ) p+1 1 h Lln lim n p = 0 (11Yf l ,n )(z)dμh (z) μh (Y ) n→∞ J (f ) p+1 1 h = n p Lln lim 0 (11Yf l ,n )(z)dμh μh (Y ) n→∞ J (f ) 1 = D(z)dμh (z). (22.152) μh (Y ) J (f )

μY (11Yn (f l ) ) =

Along with (22.151), this yields that lim

n→∞

1 μh (Y )D(z) Lln (11 l )(z) = ' μh (Yn (f l )) 0 Yn (f ) J (f ) D(z)dμh (z)

(22.153)

uniformly on J (f ). So, Lemma 5.3.5 applies with the function Hˆ := '

μh (Y )D . J (f ) D(x)dμh (x)

(22.154)

We then have that H := μh (Y )−1 Hˆ = '

D . J (f ) D(z)dμh (z)

(22.155)

Obviously, both functions Hˆ and H are supported on Y . In order to show that the function H : J (f ) −→ [0,+∞) is uniformly sweeping on Y , we will need a more refined knowledge about the behavior of the values of the Perron–Frobenius operator acting on characteristic functions of sufficiently small neighborhoods of poles with maximal order. Recall that q = qf := max{qb : b ∈ f −1 (∞)}.

22 Conformal Invariant Measures for CNRR Functions

449

We start with the following. Lemma 22.8.1 If f : C −→  C is a parabolic elliptic function, then, increasing perhaps R0 > 0 of Lemma 22.6.12, there exists η > 0 such that L0 (11Bb (R) )(z) ≥ η ∗ (R ). whenever b ∈ f −1 (∞) with qb = q, R ≥ R0 , and z ∈ J (f ) ∩ B∞ 0

Proof Keep R0 > 0 so large as required in Lemma 22.6.12 and also, invoking Theorem 22.2.5, so that ρe (w) 1 ≤ ≤2 2 ρe (b) ∗ (R ). Take any R ≥ R . whenever b ∈ J (f ) ∩ f −1 (∞) and w ∈ J (f ) ∩ B∞ 0 0 By virtue of this inequality and Lemma 22.6.12, we, thus, have, for all ∗ (R ), that z ∈ J (f ) ∩ B∞ 0  L0 (11Bb (R) )(z) = ρe−1 (z) ρe (w)|f  (w)|−h w∈f −1 (z)∩Bb (R)

≥ M −1 |z|

q+1 q h

A−h



|z|

− q+1 q h

ρe (w)

w∈f −1 (z)∩Bb (R)



= (MAh )−1

w∈f −1 (z)∩B h −1

≥ (2MA )

ρe (w) b (R)

ρe (b).

So, since, invoking Theorem 22.2.5 again, we see that η := (2MAh )−1 min{ρe (b) : b ∈ f −1 (∞)} is positive, the proof is complete.



We shall prove the following. Lemma 22.8.2 If f : C −→  C is a parabolic elliptic function, then there exists R1 ≥ R0 so large that if R ≥ R1 and b ∈ f −1 (∞) ∩ B∞ (2R) with qb = q, then Ln0 (11Bb (R) )(z) ≥ ηn−1 L0 (11Bb (R) )(z) for all integers n ≥ 1 and all z ∈ J (f ), where η > 0 comes from Lemma 22.8.1. Proof Starting induction on n ≥ 1, the lemma is obviously true for n = 1. The next step is to prove it for n = 2. Toward this end, select R1 ≥ R0 so

450

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

large that if R ≥ R1 and a ∈ f −1 (∞) ∩ B∞ (2R), then |w| > R whenever w ∈ Ba (R). Applying Lemma 22.8.1, we can then estimate as follows with b ∈ f −1 (∞) ∩ B∞ (2R):  L20 (11Bb (R) )(z) = ρe−1 (z)|f  (w)|−h ρe (w)L0 (11Bb (R) )(w) w∈f −1 (z)



≥

w∈f −1 (z)∩B



≥η

ρe−1 (z)|f  (w)|−h ρe (w)L0 (11Bb (R) )(w) b (R)

ρe−1 (z)|f  (w)|−h ρe (w)

w∈f −1 (z)∩Bb (R)

= ηL0 (11Bb (R) )(z). So, our lemma is also proved for n = 2. Proceeding further by induction, suppose that the lemma is true for some n ≥ 2 (or 1, it does not matter). Using monotonicity of the operator L0 , we then get that n n−1 L0 (11Bb (R) )) Ln+1 0 (11Bb (R) ) = L0 (L0 (11Bb (R) )) ≥ L0 (η

= ηn−1 L20 (11Bb (R) ) ≥ ηn L0 (11Bb (R) ). The inductive proof is complete.



Now we record the following. Lemma 22.8.3 If f : C −→  C is a parabolic elliptic function, then, for every n ≥ 1, we have that Ln0 (11Bb (R) )(z) ≥ ηn for all R ≥ R1 , all b ∈ f −1 (∞) ∩ B∞ (2R) with qb = q, and all z ∈ J (f ) ∩ ∗ (R ), where  > 0 comes from Lemma 22.8.1. B∞ 0 Proof

It follows from Lemmas 22.8.1 and 22.8.2 that, for all n ≥ 1, Ln0 (11Bb (R) )(z) ≥ ηn−1 L0 (11Bb (R) )(z) ≥ ηn .

This completes the proof.

(22.156) 

The next lemma is this. Lemma 22.8.4 If f : C −→  C is a parabolic elliptic function, then, for all r > 0, all R ≥ R1 , and all b ∈ f −1 (∞) ∩ B∞ (2R) with qb = q, we have that 

(r) := inf L20 (11Bb (R) )(z) : z ∈ J (f )\B((f ),r) > 0.

22 Conformal Invariant Measures for CNRR Functions

451

Proof Because of Lemma 22.8.3, it suffices to show that 

inf L20 (11Bb (R) )(z) : z ∈ (J (f ) ∩ B(0,R0 ))\B((f ),r) > 0. In order to do this, fix a Euclidean ball B ⊆ f (Bb (R)), centered at a point in J (f ), such that f (B) =  C. So, if    G := Bb (R) ∩ f −1 B ∩ f −1 (J (f ) ∩ B(0,R0 ))\B((f ),r) , then   f 2 (G) = J (f ) ∩ B(0,R0 )\B((f ),r) and (G ∪ f (G)) ∩ ((f ) ∪ f −1 (∞)) = ∅. Therefore,  

 κ := sup (f 2 ) (w) : w ∈ G < +∞; also, invoking Theorem 22.2.5, there exists M ≥ 1 such that ρe (z) ≤ M   for all z ∈ J (f ) ∩ B(0,R0 )\B((f ),r) and ρe (w) ≥ 1/M   for all w ∈ G. Hence, for every z ∈ J (f ) ∩ B(0,R0 )\B((f ),r) , there exists at least one ξz ∈ G such that f 2 (ξz ) = z and  ρe−1 (z)|(f 2 ) (w)|−h ρe (w) L20 (11Bb (R) )(z) = w∈f −2 (z)

≥ The proof is complete.

ρe (ξz ) ≥ M −2 κ −h > 0. ρe (z)|(f 2 ) (ξz )|h 

As a fairly straightforward consequence of this lemma, we can prove the following.

452

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Lemma 22.8.5 If f : C −→  C is a parabolic elliptic function, then, for all r > 0, all R ≥ R1 , all b ∈ f −1 (∞) ∩ B∞ (2R) with qb = q, and all integers n ≥ 1, we have that

 n (r) := inf L2n 0 (11Bb (R) )(z) : z ∈ J (f )\B((f ),r) > 0. Proof We proceed by induction on n ≥ 1. For n = 1, this is just Lemma 22.8.4. Suppose that Lemma 22.8.5 is true for some n ≥ 1. Fix r > 0. Then there exists r  > 0 such that   f −2 J (f )\B((f ),r) ⊆ J (f )\B((f ),r  ). Using Lemma 22.8.4 and our inductive hypothesis, we then have, for all z ∈ J (f )\B((f ),r), that  2(n+1) L0 (11Bb (R) )(z) = ρe−1 (z)|(f 2 ) (w)|−h ρe (w)L2n 0 (11Bb (R) )(w) w∈f −2 (z)

≥ (r  )



ρe−1 (z)|(f 2 ) (w)|−h ρe (w)

w∈f −2 (z)

= (r  )L2n 0 (11Bb (R) )(z) ≥ (r  )n (r). 

The proof is complete.

As a fairly straightforward consequence of this lemma, we can prove the following. Lemma 22.8.6 Let f : C −→  C be a parabolic elliptic function. If U ⊆ J (f ) is a nonempty bounded subset of J (f ) such that U ∩ (f ) = ∅, then there exists an integer s = sU ≥ 2 such that

 inf Ln0 (11U )(z) : z ∈ Y > 0, for all integers n ≥ s, where, we recall, the set Y ⊆ J (f ) is defined by (22.148). Proof Fix b ∈ f −1 (∞) and R ≥ R1 as required in Lemma 22.8.2. Since  −n (∞), there exists an even integer k ≥ 2 such that J (f ) = ∞ n=1 f f k (U ) = J (f ).

(22.157)

Since U is bounded and its closure is disjoint from (f ), it follows from Theorem 22.2.5 that there exists a constant Q ≥ 0 such that ρe (ξ ) ≥ Q

(22.158)

22 Conformal Invariant Measures for CNRR Functions

453

for all ξ ∈ U . Fix now an arbitrary point a ∈ J (f ) ∩ U such that f k (a) = b;  k in particular, {a,f (a), . . . ,f (a)} ∪ {∞} = ∅). Then there exists an open ball B(a,r) ⊆ C such that J (f ) ∩ B(a,r) ⊆ U and {∞} ∪

k 

f l (B(a,r)) = ∅.

l=0

Therefore, there exists a constant T ∈ (0,+∞) such that |(f k ) (ξ )| ≤ T

(22.159)

for all ξ ∈ B(a,r). Since f k ((B(a,r))) is an open neighborhood of b, there exists R ≥ R1 as above and so large that f k (B(a,r)) ⊇ Bb (R). Hence, for every w ∈ Bb (R), there exists at least one ξw ∈ B(a,r) such that f k (ξw ) = w. Using Lemma 22.6.12, (22.158), (22.159), and Lemma 22.8.5, we can write, for all z ∈ Y and all even integers j ≥ 2, that  k+j L0 (11U )(z) = ρe−1 (z)|(f j ) (w)|−h ρe (w)(Lk0 11U )(w) w∈f −j (z)



≥

ρe−1 (z)|(f j ) (w)|−h ρe (w)Lk0 (11U )(w)

w∈f −j (z)∩Bb (R)



≥

ρe−1 (z)|(f j ) (w)|−h ρe (w)

w∈f −j (z)∩Bb (R)

≥ M −1 |w|

q+1 q h

 

QT −h

ρe−1 (w)|(f k ) (ξw )|−h ρe (ξw )



ρe−1 (z)|(f j ) (w)|−h ρe (w)

w∈f −j (z)∩Bb (R)

≥ QR ≥ QR

q+1 q h

(MT h )−1 L0 (11Bb (R) )(z)

q+1 q h

(MT h )−1 j/2 (r),

j

where r > 0 is such that Y ⊆ J (f )\B((f ),r). So, we have proved our lemma for all even integers n ≥ 2 large enough. If we take an odd integer k ≥ 1 satisfying (22.157), then exactly the same reasoning will prove the lemma for all odd integers n ≥ 3 large enough. We are done.  As a main consequence of this lemma, we are now able to easily prove the following.

454

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Proposition 22.8.7 If f : C −→  C is a parabolic elliptic function and the set Y ⊆ J (f ) is defined by (22.148), then the function H : J (f ) −→ [0,+∞), defined by (22.155), is uniformly sweeping on Y for the map f l : J (f ) −→ J (f ) with respect to the invariant measure μh . Proof Aiming to apply Lemma 22.8.6, take as U ⊆ J (f ) any open bounded subset of J (f ) such that U ∩ (f ) = ∅ and U ⊆ V . Fix an integer n ≥ 1 so large  that ln ≥ sU , the number produced in Lemma 22.8.6. Since I := inf H |V > 0 and since L0 is a monotone operator, we get from Lemma 22.8.6, for all z ∈ Y , that ln ln Lln 0 (H )(z) ≥ L0 (I 11U )(z) = I L0 (11U )(z)

≥ I inf{Lln 0 (11U )(w) : w ∈ Y } > 0. 

We are done. Our goal now is to determine the wandering rates wn (Y ) := wn (f l ,Y ),

which, we recall, are defined by (5.10) and the set Y ⊆ J (f ) is defined by (22.148). In order to do this, fix α > 0. The standard elementary integral test gives 

n  1 k −α = if α < 1 for every N ≥ 1, (22.160) lim nα−1 n→∞ 1−α k=N

 lim

n→∞

α−1

n

∞ 

k

−α

k=n

=

1 α−1

if α > 1,

(22.161)

and n ∞ 1  −1 1  −1 k = lim k = 1. n→∞ log n n→∞ log n

lim

k=N

(22.162)

k=n

Let κ :=

p+1 h > 1, p

the inequality holding because, obviously, p+1 p > 1, while h > 1 because of Theorem 17.3.1. We will calculate the asymptotic behavior of the wandering rates (wn (Y ))∞ n−1 in the following two lemmas.

22 Conformal Invariant Measures for CNRR Functions

455

Lemma 22.8.8 If f : C −→  C is a parabolic elliptic function and the set Y ⊆ J (f ) is defined by (22.148), then there exists a constant A1 ∈ (0,+∞) such that lim nκ−2 nμY (τY ≥ n) = A1,

n→∞

where, we recall, τY : Y −→ N = {1,2, . . .} is the first return time to the set Y under the action of the map f l . Proof Put C := μh (Y )−1

D dμh .

(22.163)

Y

Fix an arbitrary ε > 0. By virtue of (22.151), there exists Nε ≥ 1 such that (1 + ε)−1 C ≤ k κ μY (τY = k) ≤ (1 + ε)C

(22.164)

for all k ≥ Nε . Hence, for every n ≥ Nε , (1 + ε)−1 C

∞  k=n

k −κ ≤

∞ 

μY (τY = k) + μY (τY ≥ n) ≤ (1 + ε)C

k=n

∞ 

k −κ .

k=n

It, therefore, follows from (22.161) that C lim inf nκ−1 μY (τY ≥ n) ≤ lim sup nκ−1 μY (τY ≥ n) κ − 1 n→∞ n→∞ C ≤ (1 + ε) . κ −1 Letting ε " 0, this gives (1 + ε)−1

lim sup nκ−2 nμY (τY ≥ n) = n→∞

C κ −1 

and the proof is complete.

Lemma 22.8.9 If f : C −→  C is a parabolic elliptic function and the set Y ⊆ J (f ) is defined by (22.148), then there exists a constant A2 ∈ (0,+∞) such that lim nκ−2

n→∞

n−1 

kμY (τY = k) = A2

k=1

if κ < 2 and 1  kμY (τY = k) = A2 n→∞ log n n−1

lim

k=1

if κ = 2.

456

Part VI Fractal Geometry, Stochastic Properties, and Rigidity

Proof Fix ε > 0. With the constant C defined in (22.163) and Nε as introduced in the proof of Lemma 22.8.8, by virtue of (22.164), for every k ≥ Nε , we have that C(1 + ε)−1 k 1−κ ≤ kμY (τY = k) ≤ C(1 + ε)k 1−κ . Put Sε :=

n−1 

kμY (τY = k).

k=1

Then we have, for all n ≥ Nε + 1, that C(1 + ε)−1

n−1 

k 1−κ ≤

k=Nε

n−1 

kμY (τY = k) ≤ Sε + C(1 + ε)

n−1 

k 1−κ .

k=Nε

k=1

It, therefore, follows from (22.160) and (22.162) that  C kμY (τY = k) ≤ lim sup kμY (τY = k) (1 + ε)−1 lim inf nκ−2 n→∞ 2−κ n→∞ n−1 k=1

≤

C (1 + ε) 2−κ

if κ < 2 and (1 + ε)−1 lim inf n→∞

1  1  kμY (τY = k) ≤ lim sup kμY (τY = k) log n n→∞ log n n−1

n−1

k=1

k=1

≤ C(1 + ε) if κ = 2. Letting ε " 0, the derived result follows.



Since, by Lemma 5.3.1, wn (f ,Y ) = μ(Y ) q

n−1 

kμY (τY = k) + nμY (τY ≥ n) ,

k=1

as a direct consequence of Lemmas 22.8.8 and 22.8.9 , we obtain the following. Theorem 22.8.10 If f : C −→  C is a parabolic elliptic function, the set Y ⊆ J (f ) is defined by (22.148), and κ = p+1 p h ≤ 2 (which precisely means that the f -invariant measure μh is infinite), then wn (f l ,Y ) ∈ (0,∞) n→∞ n2−β lim

if

κ