Rational Iteration: Complex Analytic Dynamical Systems 9783110889314, 9783110137651

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de Gruyter Studies in Mathematics 16 Editors: Heinz Bauer · Jerry L. Kazdan · Eduard Zehnder

de Gruyter Studies in Mathematics 1 Riemannian Geometry, Wilhelm Klingenberg 2 Semimartingales, Michel Metivier 3 Holomorphic Functions of Several Variables, Ludger Kaup and Burchard Kaup 4 Spaces of Measures, Corneliu Constantinescu 5 Knots, Gerhard Burde and Heiner Zieschang 6 Ergodic Theorems, Ulrich Krengel 7 Mathematical Theory of Statistics, Helmut Strasser 8 Transformation Groups, Tammo torn Dieck 9 Gibbs Measures and Phase Transitions, Hans-Otto Georgii 10 Analyticity in Infinite Dimensional Spaces, Michel Herve 11 Elementary Geometry in Hyperbolic Space, Werner Fenchel 12 Transcendental Numbers, Andrei B. Shidlovskii 13 Ordinary Differential Equations, Herbert Amann 14 Dirichlet Forms and Analysis on Wiener Space, Nicolas Bouleau and Francis Hirsch 15 Nevanlinna Theory and Complex Differential Equations, lipo Laine

Norbert Steinmetz

Rational Iteration Complex Analytic Dynamical Systems

W Walter de Gruyter DE G Berlin · New York 1993

Author Norbert Steinmetz Institut für Mathematik Universität Dortmund Postfach 50 05 00 4600 Dortmund 50

Series Editors Heinz Bauer Mathematisches Institut der Universität Bismarckstrasse 1 Vi D-8520 Erlangen, F R G

Jerry L. Kazdan Department of Mathematics University of Pennsylvania 209 South 33rd Street Philadelphia, PA 19104-6395, USA

Eduard Zehnder ETH-Zentrum/Mathematik Rämistrasse 101 CH-8092 Zürich Switzerland

1991 Mathematics Subject Classification: 30-02; 30Cxx, 30D05

©

Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.

Library of Congress Cataloging-in-Publication

Data

Steinmetz, Norbert, 1 9 4 9 Rational iteration : complex analytic dynamical systems / Norbert Steinmetz. p. cm. — (De Gruyter studies in mathematics ; 16) Includes bibliographical references and index. ISBN 3-11-013765-8 1. Iterative methods (Mathematics) 2. Mappings (Mathematics) I. Title. II. Series. QA297.8.S74 1993 51Γ.4—dc20 93-16400 CIP

Die Deutsche Bibliothek — Cataloging-in-Publication

Data

Steinmetz, Norbert: Rational iteration : complex analytic dynamical systems / Norbert Steinmetz. — Berlin ; New York : de Gruyter, 1993 (De Gruyter studies in mathematics ; 16) ISBN 3-11-013765-8 NE: GT

© Copyright 1993 by Walter de Gruyter & Co., D-1000 Berlin 30. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Printing: Gerike GmbH, Berlin. Binding: Lüderitz & Bauer, Berlin. Cover design: Rudolf Hübler, Berlin.

To Christel, Pamela, Niels and Laura

Preface

The theory of rational iteration has its origin in long memoires by FATOU and JULIA, based on the K O E B E - P O I N C A R E Uniformization Theorem, M O N T E L ' S Normality Criterion and earlier work on functional equations due to B Ö T T C H E R , 1 KCENIGS, LEAU, POINCARE and SCHRÖDER at the turn of the century ·. FATOU and JULIA independently discovered the dichotomy of the RIEMANN sphere into the sets now bearing their names, by considering the sequence of iterates of an arbitrary non-linear rational function. More than sixty years after this fundamental work the field attracted new interest in the early eighties, when SULLIVAN announced the solution of the most important problem which had remained open. His no wandering domains theorem, on combination with the classification of periodic domains due to FATOU and C R E M E R , theorems of SIEGEL and A R N O L ' D concerning the existence of rotation domains and SHISHIKURA'S precise bound for the number of periodic cycles, yields a rather complete description of the dynamics of a given iteration sequence ( / n ) , that is, of the complex analytic dynamical system (/, C). The empirical discoveries due to MANDELBROT and the beautiful dynamical colour-plates, for example, in the splendidly illustrated volume by P E I T G E N and R I C H T E R , have probably also stimulated new interest in rational iteration. This book is intended to give a self-contained exposition of the theory of FATOU and JULIA and of more recent developments. Apart from some results of general interest being discussed in the first chapter, and some particular topics referred to in places, the only prerequisites are a good knowledge of analytic function theory as may be found in AHLFORS' Complex Analysis. good reference is

KUCMA'S

book [56].

vi

Preface

Keeping the student in mind as well as the mathematician who wants to become familiar with the basic theory, I have not made great attempts to present those parts of the theory which are based on quasiconformal mappings. These include part of the work of D O U A D Y and H U B B A R D on polynomial-like mappings and SHISHIKURA'S method of quasiconformal surgery *. Because of its extraordinary importance, however, I have included a proof of SULLIVAN'S Theorem, but am conscious that the proof remains unsatisfactory, lack of space having precluded a rigorous presentation of its foundations. Nevertheless, the book may serve as a textbook for a course following a one-year introduction to analytic function theory, and should prove useful for the advanced student as well as the mathematician who wants to become acquainted with this field. However, I do hope that the research worker will also find some aspects new. Many of the results appear for the first time in book form, and indeed some seem to have never been published before. This applies also, as far as I am aware, to some of the proofs of known results. The book is divided into six chapters which are subdivided into sections. Each section is provided with a list of exercises. Most of them are purely mathematical exercises ("Prove t h a t . . . " , and the reader is urgently requested to do this), but also exercises which should stimulate the reader to do experimental mathematics. The figures created by Turbo Pascal programs serve to illustrate various theorems and phenomena (some of the figures have been rotated; the values of parameters are rounded). While the contents of Chapters 2, 3 and 4 are more or less canonical, some of the selected material in Chapters 5 and 6 reflects my own interests. I wish to express my appreciation to several referees for valuable suggestions and criticism. Klaus M E N K E was always ready with TgXnical advice. Wilhelm SCHWICK patiently listened to many attempts at various proofs and provided a lot of helpful comments and remarks. Jim LANGLEY kindly helped with the final English version. Nikolai B U S S E and Christian MATTLER verified some of the proofs and solved the exercises, moving several of them to the right place. Mrs. Helene TREIBER made some typographical corrections. To all of them I want to express my gratitude.

Dortmund, October 1992

Norbert Steinmetz

t i t would, of course, be very interesting to prove some of the quasiconformal results dynamically.

Contents

Preface

ν

List of Special Symbols

ix

C h a p t e r 1 Preliminaries

1

1

Basic Notation

2

2

Proper Mappings

4

3

The Riemann-Hurwitz Formula

7

4

The Poincare Metric

10

5

Capacity and Green's Function

14

C h a p t e r 2 T h e Dynamical Dichotomy

19

1

Two Examples

20

2

Notation

24

3

The Julia Set

28

4

Montel's Criterion

30

5

Repelling Cycles

35

6

Stable Domains

39

7

The Denjoy-Wolff Theorem

42

C h a p t e r 3 T h e F a t o u Set

46

1

Sullivan's Theorem

47

2

The Fatou-Cremer Classification

54

3

Böttcher Domains

60

4

Schröder Domains

66

5

Leau Domains and Leau Flowers

72

6

Indifferent Cycles

78

7

The Centre Problem

82

8

Rotation Domains

85

Chapter 4 The Existence of Rotation Domains

89

1

Siegel's Theorem

90

2

The Bryuno-Riissmann Theorem

96

3

Arnol'd's Theorem

Chapter 5 The Geometry of the Julia Set

102 114

1

Critical Points

115

2

Symbolic Dynamics

120

3

Smooth Julia Sets

125

4

Completely Invariant Stable Domains

132

5

Boundaries of Stable Domains

137

Chapter 6 Miscellanea

146

1

Polynomials

147

2

The Mandelbrot Set

157

3

Lyubich's Invariant Measure

163

4

Stable Julia Sets

173

5

Permutable Rational Functions

177

Bibliography

183

Index

188

List of Special Symbols

(D C

ID Ar Aut(D) II || || χ( , ) [ , }d g{z, zo, D)

C(E) dimn deg fn 0+ 0~ a Λ Μ*

C C+

complex plane R I E M A N N sphere unit disc annulus group of conformal automorphisms EuCLlDean distance norm chordal metric P O I N C A R E metric G R E E N ' S function logarithmic capacity H A U S D O R F F dimension degree (of a rational function) n t h iterate orbit backward orbit cycle multiplier LYUBICH'S invariant measure set of critical points critical limit set

Τ

FATOU s e t

J

JULIA s e t

JC

filled-in

JULIA

set

Μ

MANDELBROT set

E(D) Σ H{X) Rt

exceptional set symbol space space of compact subsets external ray

V

'r J " " .·»

1

Preliminaries

In this introductory chapter we present some material from analytic function theory not usually covered in a one-year course^ on this subject. We confine ourselves to those concepts which are absolutely necessary for understanding. Besides MONTEL's Normality Criterion, which plays the most important role, these are proper mappings, the RIEMANN-HURWITZ Formula and the POINCARE metric. Those results which are not easily, or even not in this form, available in textbooks are provided with complete and rigorous proofs.

tEquivalently, in AHLFORS' book Complex Analysis. or reference may be found there.

As a rule, anything stated without proof

2

1 Basic Notation

1 Basic Notation 1. The field of complex numbers will be denoted by C , and as usual the points of the complex plane are identified with the corresponding complex numbers. The one-point compactification of the plane, the RIEMANN sphere (D = (D U {OO} , will be endowed with the chordal metric χ . By stereographic projection € is metrically equivalent to the two-sphere S2 . A domain is an open and connected subset of (D . If the complement of the domain D consists of η components, then D is called n-connected. The unit disc ID and the upper half-plane IH are simply connected, while the annulus Ar = {z : r < \z\ < 1} is doubly connected. An analytic function is always defined in some domain and holomorphic up to isolated poles.

HM

#··>.,

ifr

FIGURE 1 Some domains of normality of the polynomial sequence ( q n ) :

qo(z) — ζ , qn+i{z) = (qn{z))2-+

ζ.

2. A family of analytic functions having a common domain of definition D is called normal if every sequence in this family contains a locally uniformly convergent subsequence (locally uniform convergence will always be abbreviated by '—>'; there should be no confusion with the convergence of a sequence of numbers or

Chapter 1

Preliminaries

3

with pointwise convergence). The limit function is allowed to be the constant oo, although the constant oo will not be regarded as an analytic function. Equivalent to normality is local equicontinuity of the family with respect to the chordal metric. In particular, normality is a local property. For families of holomorphic functions local boundedness is sufficient for normality, and indeed necessary, if the family is bounded at some point of the domain of definition. Because of its extraordinary importance in rational iteration we shall explicitly formulate The family of all analytic maps of an arbitrary domain D into the three-punctured sphere € \ {a, b, c} is normal.

MONTEL'S NORMALITY CRITERION

A proof can be found in

CARATHEODORY'S

monograph [20], Volume II, or else in

GOLUSIN [43].

We will also have several occasions to use the following theorem on the boundary correspondence of confer mal mappings. Suppose that D is a simply connected domain, conformally equivalent to the unit disc, and that φ : ID —> D is a conformal map. Then α Ε dD is accessible if and only if α equals some radial limit lim φ{νε%α), ι—>1 — and D is bounded by a curve [a J O R D A N curve] if and only if some, and hence each, R I E M A N N map φ extends continuously [homeomorphically] to the closed unit disc. CARATHEODORY'S THEOREM

Here a boundary point α of a domain D is said to be accessible if there exists an arc 7 : [0,1) —> D such t h a t l i m 7 ( t ) = a . For a proof of C A R A T H E O D O R Y ' S Theorem the reader is referred to

P O M M E R E N K E [71] .

EXERCISES 1. Suppose that the sequence of polynomials (q n ) is recursively defined by qo(z) = 1 ,

qi(z) = ζ and qn+i(z)

— (qn(z))2 — qn-i(z).

(Hint: Show that \qn+i(z)\

2. The same for qn+i(z)

Prove that it is normal in \z\ > 2.

> \qn(z)\ for \z\ > 2 . )

= (qn{z))2 + ζ , q0(z) = z.

3. The same for the iteration sequence ( p n ) , defined by p°(z) = ζ and pn+1(z) n

(p (z))

2

=

+ c, where c, subject to |c| < 2, is a constant.

4. (continued) For c = —2 prove normality in the complement of [—2, 2], and in any case prove convergence to 00 in \z\ > 2 . (Hint: VlTALl's Theorem.)

2 Proper Mappings 5. (continued) Show t h a t for c — — 1 the sequence (pn) is normal in \z\ < r and in \z + 1| < r 2 , where r is determined by r 4 + 2r 2 = r (approximately, r = 0.45). (Hint: Prove t h a t |p 2 (z)| < r for \z\ < r.)

2 Proper Mappings 3. It is well-known that if f(z) = wq + a,k(z — zo)k + · · · , α^ Φ 0, is analytic in some neighbourhood of ZQ , then there exist simply connected neighbourhoods V of WQ and U of ZQ , such that every point in V has exactly k preimages in U, which, of course, have to be counted with regard to multiplicity. This will be expressed by writing f :U

^ V .

If this property holds not only locally but globally, then / is called a proper map. To be more precise, an analytic map / of a domain D into some domain G is called proper, if there exists a positive integer k such that / :D - H G . The number k is sometimes called the topological degree of / . For example, the

map ζ h-> - ( ζ + - ) is a proper map of degree 2 2 V ζJ 1 u2 ν2 1 of the annulus {ζ \ - < \z\ < 2\ onto the interior of the ellipse 1 = — , 1 1 1 J F 2 25 9 16 while ζ ι—• z2 is not a proper map of (D \ (—oo, 0] onto C \ {0} . A proper map is called unramißed if it has no critical points, and ramißed otherwise. Here a critical point is a zero of the derivative or a multiple pole. If / assumes the value wo at zq with multiplicity k (in the case k > 1, wo is called a critical value), then ZQ will be counted as a (k — l)-fold critical point. Unramified proper maps are locally univalent, and conformal mapping is synonymous with proper mapping of degree one. k'l A proper map / : D G always maps boundary points to boundary points in the following sense: given any sequence (zn) in D tending to some boundary point of D, the image sequence ( f ( z n ) ) has all its limit points on dG. This will be abbreviated by writing f ( z ) —> dG as ζ —> 3D or f(dD) = dG (which does JOUKOWSKY

Chapter 1

Preliminaries

5

not mean that / is continuous or even defined on the boundary). An equivalent statement is that preimages of compact sets are again compact (see the exercises). More important is that the converse is also true. THEOREM 1 Any analytic function f : D —> G mapping 3D onto dG is a proper

map. Remark: In order that / be a proper map it is thus sufficient that / be continuous on D and map QD onto dG. The topological degree is then the cardinality of f~1({w}) for arbitrary w G G. The reader is referred to the exercises. PROOF of Theorem 1: Since / _ 1 ( { w } ) is compact for each w G G, / assumes ev-

ery value only finitely often. Denote by n(w) the number of preimages of w G G, counted by multiplicities, and let Zi,..., zr be the distinct preimages of Wo • Then k :1 there exist neighbourhoods V of wq and Uj of Zj such that / : Uj > V and n(wo) = k\ -\ 1-kr , and so liminf n(w) > n(wo), with this inequality remaining w—mio true, of course, if the value wq is not taken in D. If (wk) is some sequence converging to wo such that n(wk) > n(wo), then each Wk has an additional preimage r

z'k G D \ (J Uj . Since / maps boundary onto boundary, the sequence (z'k) will 3= 1

r

then have a subsequence converging to some zq Ε D \ (J Uj , which, of course, j=ι is another u;o-point of / . This contradiction shows that the map w ι—» n(w) is continuous, and hence is constant since it is integer-valued. 4. If / is an analytic map of the domain D onto the domain G and if 7 is a curve in G, then it is, in general, not true that there is some curve Γ such that / ο Γ = 7 . If Γ exists it is called a lift of 7 . For unramified proper mappings any curve in G may be lifted. LEMMA 1 If f is an unramißed proper map of D onto G, if 7 is a curve in G starting at wq and if zq is any preimage of wq under f , then there exists a unique lift Γ of η which starts at zq .

PROOF^: Suppose that 7 is parametrized over [0,1] and denote by 7,5 that part of 7 corresponding to the interval [0, • G* is a proper map of degree k* < k, and the boundary of D* consists of one or more boundary components B,B*,... of D and a subset of D . Thus we may conclude that / maps boundary components onto boundary components (see the exercises). If / has only finitely many critical values, we may choose Γ in such a way that G* contains no critical value, and the RIEMANN-HURWITZ Formula (see the next section) then shows that D* is also a ring domain. Thus, given any boundary component Β of D, there exist a ring domain D* C D such that dD* = Β U 7, where 7 is a closed JORDAN curve in D , and a ring domain G* in G with boundary CUT, such that / : D* —> G* is an unramified proper map.

EXERCISES k· 1

1. Let / : D —> G be a proper map, let G* be any subdomain of G, and let D* _ι

k* Ί

be any component of f~ (G*). Prove that / : D* —^ G* (for some k* < k) is a proper map. 2. Prove that a proper mapping indeed maps boundary onto boundary. 3. (continued) Prove that if D and G are finitely connected, then / maps boundary components onto boundary components. 4. (continued) Prove that if D and G are bounded by finitely many analytic JORDAN curves and isolated points, then / is analytic in some domain containing D . 5. Prove that any proper map of degree k of the annulus AR onto AR has the form f(z) = etazk or f(z) = eia(r/z)k . In both cases R = rk is necessary for / to exist. (Hint: Consider one of the harmonic functions log|/(z)| ± Λ log 12:| , where λ = log R / log r .)

Chapter 1

Preliminaries

7

6. Prove t h a t every proper self-map of ID is a BLASCHKE product

k l

e ° Π ί ^ τ , \ajI < 1 for 1 < j < k . J- -L 1 — CLjZ 3= 1

7. Let D be an arbitrary domain and let / be analytic and bounded in D having exactly one, and hence simple, zero. Prove t h a t / maps D conformally onto ID

provided \f(z)\ —> 1 as 2 —> 3D. 8. Prove the Fundamental Theorem of Algebra by looking at a given polynomial as a proper m a p of (D onto itself.

3 The Riemann-Hurwitz Formula 6. Prom the considerations at the end of the last section it may be seen that k-l if / : D —G is a proper map and if D and G have connectivity numbers m and η, respectively, then η < m < kn. In particular, if f is a conformal mapping (k = 1), then m = n, i.e., the connectivity number is a conformal invariant. The R I E M A N N - H U R W I T Z Formula may be regarded as a generalization of this fact. Suppose that f is a proper map of degree k of some m-connected domain D onto some n-connected domain G , / having exactly r critical points in D , counted by multiplicities. Then RIEMANN-HURWITZ FORMULA

m — 2 — k(n — 2) + r .

(1)

Before starting with the proof we will consider some particular cases. For k = 1 and r — 0 the R I E M A N N - H U R W I T Z Formula expresses nothing other than the conformal invariance of the connectivity number. A particular case is πι = η = 2 , where r has to be zero, i.e., there is no ramified proper map between ring domains. If m = η = 1 we must have r = k — 1, which, by the R I E M A N N Mapping Theorem, is equivalent to the fact that a B L A S C H K E product of degree k has exactly k — 1 critical points in the unit disc (the reader is requested to give an elementary proof). 7. For the proof we use the fact that a cross cut either diminishes the connectivity number of a domain or else divides the domain. A cross cut in a domain D is a

3 The Riemann-Hurwitz

8

Formula

arc lying completely in D except for its end points, which belong to the boundary of D . The most important property of a cross cut 7 is that it either divides a given m-connected domain into two domains, whose connectivity numbers add up to m + 1 , or else it does not divide D , and D \ 7 is (m — 1)connected. The interested reader is referred to NEWMAN'S book^. JORDAN

In preparation for the proof of the R I E M A N N - H U R W I T Z Formula we need Suppose that D is an m-connected domain, which is divided by k (in D) mutually disjoint cross cuts into I domains having connectivity numbers mi,..., mi . Then ι - 2) = m - 2 - k . (2) 3=1 LEMMA 1

We proceed by induction. For k = 1 (so that I = 1 or I = 2 ) the assertion is equivalent to the fundamental property of cross cuts. When passing from k — 1 to k we assume first that one of the cross cuts divides D. Then we have two subdomains, which are each divided by less than k cross cuts. Adding up the corresponding equations (2) then gives the assertion. If, however, the domain is not divided, then for any cross cut, 7 say, we may regard the domain D \ 7 as being divided by k — 1 cross cuts. The left hand side of (2) does not change for the new configuration, and this is also true for the right hand side, since both m and k are diminished by 1 . This proves the assertion in the case of k cross cuts. PROOF:

Now we will give a P R O O F * of the RIEMANN-HURWITZ Formula. To achieve a simpler configuration we map D and G conformally, by applying the RIEMANN Mapping Theorem m and η times, onto domains D* and G* , whose boundary components are analytic J O R D A N curves or isolated points. This induces a proper map f* of degree k of D* onto G* , 8.

D

/

-

φ

Φ D*

G

-»• G* /*

which is analytic^ in D* . We may thus assume that / is already analytic in D. In addition, we first assume that / is unramified, and will reduce the case of a ramified / to this first case. TM. Η. A. N E W M A N , Elements of the topology of plane sets of points, Cambridge University Press 1964. ΊFor a completely different proof see O. F O R S T E R , Lectures on R I E M A N N Surfaces, Springer 1981. § Isolated boundary points are removable singularities or poles, and by the S C H W A R Z Reflection Principle / may be continued analytically across analytic boundary curves.

Chapter

1

9

Preliminaries

Again we proceed by induction be continued analytically along the Monodromy Theorem; thus k = πι = 1 and so (1); note that

on η . For η = 1 every local inverse of / may any path in G, and hence is analytic in G by / is a conformal map of D onto G. This proves r — 0.

Now assume η > 1 . By means of a cross cut 7 we diminish the connectivity number of G, the exercises below showing how to do this. Then G* = G \ 7 is (n — l)-connected, and the k preimages (lifts) of 7 are (in D) mutually disjoint cross cuts, the fact that / is unramified and analytic in D being used here, and these cross cuts divide the domain D into domains D i , . . . , D i . Here D j is rajconnected and / is a proper map of degree k j of D j onto G* . In particular, the RIEMANN-HURWITZ Formula holds for ( f , D j , G * ) , namely m

j

-

2

=

k

j

( ( n - l ) - 2 ) ,

and, adding up, equation (2) and the relation k\ + · · · -I- ki = k give ι m

-

2

ι

=

-

2)

+

k

3=1

=

" Y ^ k j { n - S ) +

k

=

k(n

-

2 ) .

3=1

Now if / has critical points, wc consider the (η + s)-connected domain G* , which is G punctured at the distinct critical values w\,..., ws . Then / is an unramified proper map of degree k of D* = f ~ ( G * ) onto G * . To compute the connectivity number m* of D* we remark that each Wj has k preimages, pj of which are 1

s

distinct, and the corresponding multiplicities are q{ . Since r — Ε

Pj

"

j = l i = l

s

implies Σ pj = ks — r , the connectivity number of D* is πι* — m + ks — r . The i=i unramified

RIEMANN-HURWITZ Formula for ( f , D * , G * ) , namely m

+

ks

—

r —2 =

k(n

+

s —

2),

then gives the assertion in the ramified case. EXERCISES 1. Prove that the accessible boundary points of a domain D form a dense subset of dD.

(Hint: Every interior point is the centre of some circle with smallest radius which contains boundary points.) 2. Let A and Β be boundary components of the finitely connected domain D. that there is some cross cut in D joining A and Β .

Prove

3. Prove the RIEMANN-HURWITZ Formula for D = G = C (τη = Η = 0). 4. Prove that a proper map of an n-connected domain, η > 2, onto a ring domain has critical points.

10

4 The Poincare

Metric

5. Give an example of a proper map of degree k mapping some ( k + l)-connected domain onto the punctured disc ID \ { 0 } .

4 The Poincare Metric 9. Since the connectivity number is a conformal invariant, there exists no conformal map of the unit disc onto a multiply connected domain. The conformal map has to be replaced by a so-called universal covering map. An analytic function / : D —> G is called an unramified covering map, if every local inverse of / may be continued analytically along any path within G . We state without proof the following existence theorem for a universal covering map, which is a special case of the famous K O E B E - P O I N C A R E Uniformization Theorem. Proofs may be found in the books of G O L U S I N [ 4 3 ] and T S U J I [ 8 5 ] or else in a recent paper+. Given any hyperbolic domain D , there exists an unramified covering map / of the unit disc onto D. Each such map is called a universal covering map. It is uniquely determined by the normalization /(0) = ZQ and /'(0) >0.

UNIFORMIZATION T H E O R E M

Remark:

Any domain having at least three boundary points is called

hyperbolic.

In the simply connected case the Uniformization Theorem is nothing other than the R I E M A N N Mapping Theorem. For the punctured disc AQ — ID \ { 0 } a universal covering map is given by ζ ^ exp

> while, for D = (D \ { 0 , 1 } , one

solution of the uniformization problem is given by the elliptic modular function, whose existence is needed in most proofs of the Uniformization Theorem. 10. Let Aut(D) denote the group of conformal self-maps of a given domain D . If D is simply connected, the group Aut(D) depends on three real parameters, while the corresponding group for doubly connected domains depends on one parameter only. For m-connected domains, 2 < m < oo, the group Aut(D) is in fact finite. FISHER, J . H. HUBBARD, B . S. WITTNER, A proof of the Uniformization Theorem for Proc. A. M. S. 1 0 4 (1988); see, however, also CARATHEODORY [20] ,

arbitrary plane domains, Vol. II.

Chapter 1

Preliminaries

11

The following characterization of the domains of connectivity one or two may be found in T S U J I ' S book [85]. It will play an important role in the proof of the Classification Theorem in Chapter 3. The group Aut(D) is called non-discrete, if the identity map, and hence each map in Aut(D), may be locally approximated with arbitrary accuracy by other elements of the group, i.e., if the topology of the group, generated by the concept of locally uniform convergence, is non-discrete. THEOREM

conformally

1 Any hyperbolic domain with non-discrete automorphism group is equivalent either to the unit disc ID or else to an annulus Ar .

11. Every continuous and positive function ρ, defined in some domain D , gives rise to a metric 7

sometimes the line element ds — ρ(ζ) \dz\ is called a R i E M A N N i a n metric. Here the infimum has to be taken over all rectifiable (or smooth or analytic) curves joining α and b within D. Locally this metric is comparable to the EuCLiDean or chordal metric, since obviously (for finite a) [α + M ] = e(a)|/i|(l + o(l)) as h —> 0 . We are mainly interested in conformally invariant metrics, defined by the property that conformal self-maps of D are isometries, that is, [f(a)J(b)]

= [a,b].

For the unit disc ID the PoiNCARE metric

7

with the density

is conformally invariant. This is an easy consequence of the well-known

l-\f(z)\2

2 11l -— | zI\z\ |I 2 ''

while for any other analytic function f : ID —> ID strict inequality holds.


, which reflects the properties of the domain D , it is easy to prove that any analytic map of some hyperbolic domain D into some hyperbolic domain G is either a contraction, that is,

[h(a), h{b)]G < [a, b]D

for α φ b,

(3)

or else is an unramified covering map of D onto G. For further applications, however, we will need a slightly more general result. Suppose that h is a function element which may be continued analytically along any path within the hyperbolic domain D, having its values in the hyperbolic domain G. Then given any simply connected subdomain Ω of D , inequality (3) holds for a, b Ε Ω ; if Ω is compactly contained in D, that is, if Ω C D, then there exists a constant q — qfö) < 1 such that I N V A R I A N T L E M M A OF S C H W A R Z

[h(a), h(b)]c < q{a, b]n

fora,beQ,

Chapter 1

Preliminaries

13

except in the case when fp is a universal covering map of ID onto D and (the analytic continuation of) ho fD is a universal covering map of ID onto G . PROOF: By means of universal covering maps fo and fc the function element h will be lifted to the unit disc, i.e., g = /Q 1 oho fD may be continued analytically along any path in ID, having its values in ID, and thus is, by the Monodromy Theorem, an analytic function g : ID — • ID. By the SCHWARZ-PICK Lemma we have [g(a),g(ß)}D

< [α,β]Ό

,

and equality for α φ β is only possible if g is a conformal self-map of ID . The same inequality for h then holds in D (see the exercises), with equality if and only if g is in Aut(ID), which implies that h ο f^ — /q ο g is a universal covering map onto G. If Ω is simply connected and compactly contained in D, and if h ο fD universal covering map, then the continuous function [h(a),h(b)h Q(a,b)

=

[a,b]D ßG(h(a))\h'(a)\

is not a

for α Φ b for a = b

QD{O)

attains its maximum q = Q(a*,b*)

in Ω χ Ω, and hence q < 1.

EXERCISES 1. Determine some universal covering map of ID onto the annulus Ar • (Hint: For r = 0 replace ID by a half-plane and for 0 < r < 1 use a parallel strip.) 2. Let / : ID — > D be analytic and let F be a universal covering map of ID onto D with F(0) = / ( 0 ) . Prove that there exists an analytic function ω : ID — > ID such that \ω(ζ)I < \z\ and f — F ο ω (subordination principle).

f 1 +r 3. Prove that / λ ( ζ ) \dz\ > log

for any smooth path 7 joining the points 0 and

J' ΎΊ

r > 0 in ID, except when 7 is the straight line from 0 to r. > ^ R e 7 ( i ) and | 7 (t)| > | R e 7 ( i ) l ) · (Hint: I | 7 ( i ) 4. (continued) Verify equation (1) and determine the geodesies, i.e., the arcs of shortest [ , ]ID -length joining given points. (Hint: By a MÖBIUS transformation map ID onto itself such that a and b are mapped onto 0 and r — r(a, b) > 0, respectively.) 5. Determine the PoiNCARE density for the upper half-plane and for an annulus AT . 6. Prove that the PoiNCARE density q — Qd satisfies the partial differential equation Δ log ρ = ρ2 . (Hint: Start with D = ID .)

14

5 Capacity and GREEN 'S Function

7. Prove that [α, Ö]D = inf [α,/3]ε> , where the infimum has to be taken over all pairs

(α,ß) e / _ 1 ( Μ ) * R ' M ) or, if α e / ^ ( { a } ) is fixed, over all Β € / " ' ( W ) > where / is a universal covering map of ID onto D . Show also that the infimum is attained. 8. Prove that G C D implies go < QG in G and in particular (choose some suitable subdomain G) QD{Z) < 2 / d i s t ( z , d D ) . 9. (continued) If D is simply connected prove that 2QD{Z) > l/dist(2, 3D). (Hint: If φ : D —> D is a conformal map such that φ(0) = zo , then, by KoEBE's

1/4-Theorem, dist(z 0 ,dD) > \φ'{ϋ)\/Α.)

5 Capacity and Green's Function 14. Let D be a domain and let ZQ be a fixed interior point. A function g harmonic in D \ { Z 0 } is called GREEN'S function for D with pole at ZQ, if it has the following properties: g is continuous in D \ {ZQ} with boundary values 0, and Ζ = ZQ is a removable singularity of g(z) + log \z — zq\ • In the case zo = oo one has to replace ζ — zo by 1 jz. There is at most one GREEN'S function g(z,zo,D), as is easily seen by applying the Maximum Principle. GREEN'S function need not exist, this depending only on D, and not on zo ; if it exists D is called regular. For example, any domain bounded by finitely many JORDAN curves is regular. A sequence (Dn) of regular domains is called a regular exhaustion of D if Dn C Dn+χ,

Dn c D

and

|J Dn = n> 0

D.

It is easily seen (see the exercises) that every domain has a regular exhaustion by domains each bounded by finitely many JORDAN curves. For the proof of the next result we need HARNACK'S Principle, which states that any increasing sequence of harmonic functions in some domain D converges either to + o o or else to a harmonic function. In both cases the convergence is locally uniform. This immediately gives THEOREM 1 Suppose that (Dn) is a regular exhaustion GREEN 'S function of Dn with pole at ZQ . Then g(z) =

lim gn(z) n—foο

of D,

and gn is the

(1)

Chapter 1

Preliminaries

15

exists, locally uniformly in D\{zo} · The function g is independent of the particular exhaustion, and is either Ξ +OO or else harmonic in D \ {ZQ} . If the G R E E N ' S function of D with pole at ZQ exists, it agrees with g. Remark: If the limit g is finite, it is also called the G R E E N ' S function of D, although it may not necessarily have boundary values 0 everywhere. If the limit ( 1 ) is = + o o , we will say that there is no G R E E N ' S function for D .

FIGURE 2

A compact set (black) and some equipotential lines of GREEN'S function for the outer domain.

of Theorem 1: The Maximum Principle, applied to the difference of functions gn — gn+i IN the domain Dn, immediately shows that the sequence (gn) is increasing in every subdomain Ω compactly contained in D, for η > η(Ω). If ( D * ) is another regular exhaustion, then, by the H E I N E - B O R E L Theorem, we have D^ c Dn for η > n(m). This yields g^ < gn < g in D^ by the Maximum Principle and so g* < g. The converse is also true by symmetry, and hence g — g* . PROOF

GREEN'S

If G R E E N ' S function exists then obviously g(z,zo,D) > gn(z) in Dn , and conversely, given ε > 0, the set Ke — {z : g(z,zo,D) > ε} is compactly contained in D, and thus contained in Dn for η > η(ε). Again, the Maximum Principle then shows that gn(z) > g(z, ZQ, D) — ε in Dn , η > η(ε), and hence g(z) — g(z, ZQ, D).

16

5

Capacity

GREEN'S

and

Function

15. For the following considerations the reader is referred to AHLFORS [1] or HILLE [48], Vol. I I .

Suppose that Ε is an infinite and compact subset of (D and let D be its outer domain, i.e., the unique component of C \ Ε containing the point oo. Denote by 1 V(z1,

...,z

n

)

1

zn

z\

=

Π

=

η-1

zn2 ~l

1

Z1

t h e V A N D E R M O N D E determinant

—

Δη

f o r z i , . . . , zn

= m Na x \V(zi,... E

Vn

1

+ o o . Thus there exists a polynomial in Vn having minimum norm. In the exercises it will be shown that there is only one, which is called the nth TCHEBYCHEV polynomial of Ε and is denoted by Tn . As an example we consider the unit circle Ε = 1 .

3=0

The minimum norm 1 is attained by P(z) — zn only. The sequence (||7n||1/'n) converges to its infimum T o o ( E ) = the limit is called the TCHEBYCHEV

of Ε .

'S constant

lim \/\\Tn\\F,α n—>oo V

and

1 6 . If g(z, oo, D) denotes the GREEN'S function of the outer domain of Ε, then g(z,

oo, D) = log

The number 7 is called the ROBIN

\z\

+ 7 + o(l)

'S constant C(E)

=

as ζ —» oo .

of Ε and e~ 7

Chapter 1

Preliminaries

is the logarithmic and C(E) = 0 .

17

capacity of Ε. If D has no G R E E N ' S function we put 7 = +00

Suppose that μ is any probability measure on Ε, i.e., a measure defined on the ring of BoREL-measurable subsets of Ε with total mass β{Ε) = 1 . If the energy integral 1 log άμ(α) dß(b) m = α —b Ε is finite, then the logarithmic

potential f 1 dß{a) / log ζ —α JE

is harmonic in D and there exists a unique measure β such that g(z,oo,D)-i

=

-ρμ(ζ).

FIGURE 3

A compact set and its equilibrium. T h e unique probability measure β is called the equilibrium of Ε . The equilibrium also minimizes the energy integral. If no energy integral is finite, then GREEN'S function does not exist, and conversely, if the equilibrium exists, so does GREEN'S function and the following is true: doc(E) =

Toc{E)

= C{E)

and 7 = Ι\μ\.

5 Capacity

18

and GREEN'S Function

EXERCISES 1. Prove that every domain has a regular exhaustion. (Hint: Cover the plane by a grid of closed squares of width 2~ n , η — 1 , 2 , . . . .) 2. Let Ρ least η (Hint: degree 3.

Ε VN be a polynomial having minimum norm. Prove that there exist at + 1 points in Ε at which |P| attains its maximum ||P|| B . If there are less than Η + 1 points Zj , then there exists a polynomial Q of less than Η interpolating Ρ at Zj . Consider Ρ — EQ for ε > 0.)

(continued) Prove that there exists at most one T C H E B Y C H E V polynomial T N of degree η . (Hint: If there are two, Tn and 7'r* say, consider (Tn + T*)/2 at the extremal points of \Tn\ .)

4. Let E* = dD be the outer boundary of Ε and set Ε = (Γ \ D. TOO(E*) = TOO(E) = TOO(E) .

Prove that

5. Suppose that f(z) — αζ + oo + αι/ζ + · · · ( a ^ O , \z\ large) maps the outer domain D of Ε conformally onto D' with complement E'. Prove that C(E') = |a| C(E). 6. (continued) Compute the logarithmic capacity of Ε = [—c, c], c > 0 . 7. Let P(z) — zm + · · · be a normalized polynomial. Prove that the lemniscate set ER = {Z: \P{Z)\ < R} , R > 0 , has capacity RL/M

.

2

The Dynamical Dichotomy

About 1918 - 20, P. FATOU and G. JULIA independently developed the theory of rational iteration^, their main tool being MONTEL'S Normality Criterion. They discovered the dichotomy of the RIEMANN sphere into the sets now bearing their names. In this chapter we will present the basic results of FATOU and JULIA concerning global properties of FATOU sets and JULIA sets.

^In modern language the theory of complex analytic dynamical systems (/,

€).

1 Two

20

Examples

1 Two Examples 17. A simple application of NEWTON'S method to the equation z 2 — 1 = 0 leads to the sequence (zn), which, for a given ZQ , is defined recursively by _

(

1

" 2 Γ"

1

+

^

However, it is more than a recursive sequence, it is an iteration sequence, generated by iterating the JOUKOWSKY function

w - s K ) · The NEWTON sequence converges to 1 or —1 according to whether zo is sufficiently close to 1 or — 1, respectively. To obtain a quantitative result we put wn

=

Zn - 1 Zn + 1 '

and a short computation leads to a much simpler iteration sequence (w n ), given by wn+1 = wl . It is obvious that the sequence (w n ) converges to 0 if |wo| < 1 and to oo if |uio| > 1, while |IÜO| = 1 leads again to \wn\ — 1 (and in this case the sequence does not ζ - I in general converge). Since the MÖBIUS transformation w = —-J—J- maps the right half-plane onto the interior of the unit circle and the left half-plane onto its exterior, we conclude that Re ZQ > 0 leads to convergence to 1, and Re ZQ < 0 leads to convergence to — 1, while if ZQ is purely imaginary, the sequence (zn) does not in general converge. The change of variables ζ w is called conjugation. 18. It is now quite natural to consider the same problem for the sequence Zn+1 =

i

(2Zn

+j t j ,

which arises from NEWTON'S method applied to ζ3 — 1 = 0 . The question: when does (zn) converge to 1? has the obvious but false answer: if ZQ is chosen in the sector I arg21 < π/3. We will come back to this problem later on and for now just note that it has several times stimulated mathematicians to study rational iteration. That the obvious answer is false easily follows from the fact that the function f(z) — - (2ζ Η—χ ] does not map any sector onto itself, which would be 3 V z2J necessary, however, if the obvious answer were true, since the set of initial values ZQ such that zn —> 1 is clearly invariant under / and f~1.

Chapter 2 The Dynamical

21

Dichotomy

FIGURE 1

method for z 3 + 2z - 2i = 0 . The method fails when starting in one of the white regions. NEWTON'S

19. A similar problem arises from iteration of the polynomial p(z) =Pc{z) = z2 + c, where c is some complex parameter. Setting p°(z) = z, p1(z) = z2 + c, p2(z) = (z2 + c)2 + c>

...

we have, given 2 = zq , to consider the iteration sequence Zn = Pn(z) · The most simple example p(z) = z 2 has already occurred. It is apparent that (z n ) converges to oo, if \zo\ is sufficiently large, and indeed also if \zm\ is large for some m, which may occur even if |2o| is small. As an example we consider the polynomial p(z) — z2 — 2 . A simple discussion of its real graph then shows that the interval [—2,2] is mapped by ρ onto itself and also is its own preimage under ρ , so that —2 < pn(x) < 2 if — 2 < χ < 2. Any other number 2, real or complex, may be written as ζ = w + , where w is uniquely determined subject to > 1, and it follows by induction that ryTX,

zn = w

ejYl

+ w

—»00

as η —> oo .

1 Two Examples

22

It is not possible to deal with any other polynomial pc analytically, and graphicalnumerical experiments show that the initial values ZQ such that zn /> oo form extremely complicated sets. The reader is urged to refer to the exercises.

FIGURE 2

Three filled-in J U L I A sets (white) for z 2 + C; c = - 1 + 0.25 i, c = - 0 . 1 + 0.7 i and c = - 1 . 5 8 .

20. To discuss these examples in more detail we set A(oo) = {z : pn(z) —» oo as η —»· oo} and call J = dA{oo) the J U L I A set of ρ. Though the set J and the ßlled-in J U L I A set JC = C \ A(oo) look very complicated, the following theorem is easy to prove. THEOREM

1

^4(oo) is a completely invariant domain, p(A(oc)) =

and given R >

A(oc)=p-1(A(oo)),

+ Λ/1 + 4 |c| ^ / 2 , it may be written as A( oo) = (J{z:\pm(z)\>R}. m> 0

(1)

Chapter 2 The Dynamical

Dichotomy

23

PROOF: T h e disc \z\ < R contains the zeros of p , and on its boundary we have \p(z)/z2\

> 1 - |c| /R2

> 1 /R \ hence, by the Minimum Principle, \p(z)\ > R —

holds in \z\ > R, and by induction

\pn(z) I > R

(2)

R

follows. T h e inclusion C in (1) is quite obvious. To prove the converse we use (2), which gives \pn+m(z)\

> R

pm(z) R

oo

as η —» oo

for I p m ( z ) \ > R. In particular, A(oo) is open as a union of open sets, and indeed is a domain, for otherwise it would have a bounded component C . Then we would have |p n (z)| < R (n £ IN) on dC, and hence also in C , using the Maximum Principle, and this is absurd. T h e complete invariance of j4(oo) is obvious.

EXERCISES 1. Produce black and white computer graphics of the filled-in J u l i a sets of pc , |c| < 2 , as follows: to each pixel Ρ there corresponds some complex number ζ = χ + iy ( - 2 < χ < 2, - 1 . 5 < y < 1.5). Ρ is coloured black if |p n (z)| < 2 for all η < Ν (20 < Ν < 100, say), and white otherwise. 2. In the same way generate an approximation of the set of those complex numbers ζ such that the sequence (q n {z)) remains bounded. Here qo{z) — 1 , qi{z) — ζ and qn+i — (qn(z))2 — qn-i(z); note that (q n ) is not an iteration sequence. 3. (continued) The same for qo(z) — ζ , qn+i(z) = (q n {z)) 2 + ζ . In both cases we have qn(z) —> oo if and only if |q m (z)| > 2 for some m (proof!). 4. Let (z n ) be the sequence obtained from Newton's method applied to ζ 3 — 1 = 0 (z4 — 1 = 0 , z 5 — 1 = 0 , etc). As in Exercise 1 generate computer graphics of those initial points such that (z n ) converges to 1, in — 1 < χ < 3, —1.5 < y < 1.5, say (here zn —» 1 is numerically equivalent to \zn — 1| < 0.1 for some η < Ν, where 20 < Ν < 50).

2

24

Notation

2 Notation 21. In this section we will give the most important definitions associated with rational iteration. We always denote by

a fixed rational function of degree deg f — d — max (deg P, deg Q) > 1. This holds throughout the whole text unless stated otherwise. Ρ and Q are relatively prime polynomials, and / will be regarded as a ramified proper map

By / ° = i d , f1 = f , f2 = f o f , ...

, f

+ 1

= / o f

= f o / ,

...

we denote the sequence of iterates of / . It is convenient to write fn instead of, say, fn. This should remind one of the fact that the composition ο is understood as the multiplication in the semi-group { / n : η e INo} . There should be no confusion with the ordinary power, which will be explicitly written as ( f ( z ) ) n . Since the degree of a rational function is actually the number of preimages of any w € (D (of course, counted by multiplicities), it is obvious that deg/ n = dn. 22. A point ζ is called normal if the sequence ( f n ) is normal in some neighbourhood of Ζ . The set of normal points is called the FATOU set of / and is denoted by Τ = F f . It is the largest open set of normality. The components of J7 are called stable domains: they are maximal domains of normality. The complement of Τ is denoted by j = jf = c\rf and is called the JULIA sefct of / . It is compact and, by definition, ( / n ) is nonnormal in any domain intersecting J . We will see soon that the definition of the JULIA set of a quadratic polynomial, given in Section 1, agrees with the present definition. α b ϊϊ+1 Φ 0, is any MÖBIUS transformation, then we say that c d cz + d / and g = Τ ο f ο Γ - 1 are conjugate, and will write / ~ g . Obviously, ~ is an equivalence relation. In most cases conjugacy is used to achieve a simpler configuration. Thus to send a given point α to the origin, we consider f(z + a) — a or else, if a = oo , the function l//(l/,z) instead of / . It follows immediately from Τ ο fn = gn ο Τ that Tg = T{Tf) and Jg = T(Jf). For example, any quadratic κ

τ { ζ ) w

=

tThe notation in the literature is not uniform. We adopt that of LYUBICH [62].

Chapter 2 The Dynamical

Dichotomy

25

polynomial is conjugate via some similarity transformation 2 1—>• az + b to a unique polynomial ζ — · > ζ2 + c (see the exercises). 23. A solution zq of the equation f ( z ) = ζ is called a fixpoint of / and Λ = f'(zo) is called its multiplier. This has to be modified if zq — 00: we define Λ to be the multiplier of the fixpoint 0 of the conjugate 2 1—> l / / ( l / z ) .

FIGURE 3

Two conjugate

JULIA

sets.

The orbit of a point zq is the set (sometimes also understood as a sequence) 0+(zo) =

{fn(zo):neJN}·

it is the set of all successors of zq . In the same way the orbit 0+{E) an arbitrary non-empty set Ε .

is defined for

Normality is also equivalent to stability: 2 belongs to the F A T O U set if and only if, given ε > 0, there is some δ > 0 such that χ(ζ, ζ) < δ implies x(fn{z), / n ( C ) ) < ε for η 6 IN, i.e., if the orbit 0 + ( ζ ) stays close to 0+(z) when ζ is close to ζ . The preimages of z0 under / , f2, / 3 , . . . are called predecessors of zq . Analogously to 0+(zo) one may consider the backward orbit (ZQ) ; one has, however, to keep in mind that O~(zo) is a tree rather than a sequence. If zq is some fixpoint of fp , but not a fixpoint of fn for any η, 0 < η < ρ, then a = {zo,f(zo),

...,

ίρ-\ζ0)}

26

2

Notation

is called a cycle of length p. Its multiplier Λ = A(ct) is just defined to be the multiplier of the fixpoint ZQ of fp . Since, by the Chain Rule,

j=0 at least if Ρ(ζο) φ oo for 0 < j < ρ, the value of Λ depends only on α and not on the particular fixpoint zo • It is easily seen that this is also true if oo belongs to α . If / and g are conjugate, Τ ο / = g ο Τ, and if α is some cycle of / of length ρ and multiplier λ , then obviously Τ (a) is a cycle of g with the same multiplier and length. The elements of a cycle are also called periodic points.

^TV

I ,

φ FIGURE 4

The set |J /"({0}) for f(z) - z2 - 08 .2 + 0.16i . 2000

n = -

22

We have to distinguish several cases. The cycle a is called superattracting attracting indifferent repelling

"j I [ J

,ff

( λ = 0 I 0 < |Λ| < 1 ] |Λ| = 1 I |A| > 1.

We also talk about superattracting, attracting ,... fixpoints and periodic points. The indifferent cycles will be subdivided again: a is called a LEAU cycle SIEGEL cycle

λ >

CREMER cycle J

iff

(

A m = 1 for some m € IN


1 , and at the same time a superattracting fixpoint. To see this one has only t o consider the conjugate ζd

1 αο + α^ζ

1

Η

h α,β,ζ

d

a0zd - z ad

Η d

-

had azd

+

in some neighbourhood of ζ = 0 .

EXERCISES 1. Prove that every polynomial of degree d is conjugate to exactly one polynomial of the form cq + C\Z Λ h cd-2zd~2 + zd . 2. Any polynomial \z + z'2 is conjugate to exactly one polynomial z2 + c. Compute A(c) and c(A). 3. Prove that if / ~ g, more explicitly, if T o / = g oT, Tg = T{Ts).

then Jg

= T{Jf)

and

4. Prove that any stable domain (other than A(oo)) of a polynomial is simply connected, .A(oo) being defined as in the quadratic case p(z) = z2 + c. (Hint: Maximum Principle.) dz 5. Let ρ be a polynomial of degree d > 2 . Compute IR = —— I °F for 2m J' \z\ = R - - P(Z-)~Z large R by means of the Residue Theorem, show that IR = 0 if R > Ro , and deduce that ρ has at most d — 1 finite attracting fixpoints. 6. The set of quadratic rational functions

divides into equivalence b0 + biz + b2z2 classes with respect to conjugacy. Determine suitable (and simple) representatives. (Hint: By conjugation one may assume that / has fixpoints 0 and oo.)

7. Prove that each disc Dr = {z : \z — r\ < r } , 0 < r < | , is invariant under p(z) = ζ — ζ2 , that is, p(Dr) C Dr. 8. There are also non-analytic conjugacies such as complex conjugation, namely f*(z) = 7 ( f ) . Prove that J* = {ζ : ζ € J } . 9. Prove that every rational function of degree d has exactly 2d — 2 critical points, counted by multiplicities. (Hint: It suffices to consider / = P/Q , where Q has only simple zeros, and deg Ρ = deg Q = d.)

3 The Julia Set

28

3 The Julia Set In this section we will deduce some simple properties of the JULIA set (associated with /), proceeding as far as elementary arguments will take us. In fact, this will not lead us very far. The reader should keep in mind that / is always some fixed non-linear rational function, defined on and taking values on the compact RIEMANN sphere, and hence is uniformly (and indeed LIPSCHITZ) continuous. 25.

THEOREM 1

The

JULIA

set is non-empty.

Otherwise any subsequence of (/ n ) would contain some uniformly convergent subsequence, whose limit would be either Ξ Ο Ο Ο Γ else rational. Choosing ρ in such a way that fp has distinct fixpoints ZQ and z\ , then fpnk —> g implies that g is non-constant since g(zj) = Zj , and so deg g = s > 1. Passing to another subsequence, if necessary, we may assume that fPnk~s —> h, and since ho fs = g this gives a contradiction, namely that deg h ο fs > ds > s = deg g = deg ho fs . PROOF:

We will see later on that the FATOU set may indeed be empty, although this case may be regarded as pathological. property common to JULIA sets and FATOU sets is complete invariance. We say that some non-empty set Ε C C is invariant, if f(E) C Ε, and backward invariant, if / _ 1 ( £ ) C Ε. If both hold, that is, if f(E) = E = / _ 1 ( £ ) , then Ε is called completely invariant. A

THEOREM 2

The

JULIA

set and the

FATOU

set are completely

invariant.

P R O O F : We will show that Τ and J are backward invariant, and hence completely invariant since they divide the RLEMANN sphere.

Let D C 0} and Fj = { f j ο fnP\D : η > 0}. Then obviously PROOF:

F = FQ U . . . U F P _ ! ,

and since /·? is uniformly continuous, F is normal if and only if FQ is normal. The proof expresses nothing other than the trivial fact that a finite union of families of analytic functions is normal if and only if each of these families is normal. 27. We have not yet been able to decide explicitly whether a given point belongs to the FATOU set or to the JULIA set. If we consider fixpoints or periodic points only, this will be easy to do in the case of non-indifferent fixpoints or periodic points and will turn out to be much more difficult in the case of indifferent fixpoints or cycles. In the first case we have (Super)attracting cycles belong to the cycles belong to the JULIA set.

THEOREM 4

FATOU

set, while repelling

PROOF: In view of Theorem 3 it suffices to consider fixpoints, and by using a suitable conjugation, even the fixpoint ZQ = 0. If |/'(0)| < 1, then there exists some arbitrarily small neighbourhood U of 0 such that f(U) C U, and hence fn(U) C U for η £ IN. This proves normality of ( f n ) in U, and U contains the origin. Incidentally, a simple variant of this proof shows that ( / n ) tends to 0 in U as η —> oo . On the other hand, if |/'(0)| > 1, then (/")'(0) = (/'(0)) n oo as η oo, and nk no subsequence f may converge to an analytic limit φ (note that 0(0) = 0), since {fnk)'(0) would then have to tend to the finite value 0'(O). This is more or less all that can be derived by elementary methods. A more detailed and deeper study is only possible with non-elementary tools. The main role in the subsequent sections will be played by M O N T E L ' S Normality Criterion, which yields a great number of deep and surprising results^.

EXERCISES 1. Prove that if |l — %/l — 4c | < 1, then the filled-in JULIA set of pc(z) = z2 + c has interior points. 2. Let f ( z ) = λζ + 0.2Z2 + · · · , |λ| < 1, be analytic in some neighbourhood of 0. Prove that the sequence of iterates of / is locally defined and that ( / " ) converges to 0 , locally uniformly. 3. Determine all cubic polynomials 2 — (ζ2 — 1 )(az + b) having (super)attracting fixpoints 2 = ± 1 . tOn the other hand, MARTY'S criterion (see AHLFORS [2]) is of no significance, even though (because?) it is necessary and sufficient.

30

4 Montel's Criterion 4. Prove that if the sequence (g n ) of rational functions converges to g , uniformly in € , then g is' rational and dcg gn = deg g for η sufficiently large. 5. Produce some computer graphics of the FATOU sets of the polynomials p(z) = \z + ζ2 , |λ| < 1, as follows: set zn = pn(z) for ζ = χ + iy (—2 < χ < 2, — 1.5 '< y < 1.5). Then the corresponding pixel will be coloured black or white according to whether \zn\ > 2 or \zn\ < 1 — |Λ|, respectively, for some η < Ν (Ν between 25 and 100). The common boundary of these domains is then an approximation of the corresponding JULIA set.

4 Montel's Criterion Normality Criterion turns out to be the appropriate tool in rational iteration. In terms of an iteration sequence of a rational function it will be applied in one of the following forms: 28.

MONTEL'S

Any invariant hyperbolic domain is part of the F A T O U set. Any backward invariant and compact set containing at least three points in fact contains the J U L I A set. For any domain D intersecting J , the set E(D) = z~d, and their conjugates. 2 9 . Before proceeding further with our study of J U L I A sets, we consider a simple polynomial whose J U L I A set is totally disconnected^. The example is p(z) = z2 — 6.

tThis means that the components of J are singletons.

Chapter 2

The Dynamical

Dichotomy

31

2

Since \p(z)\ >\z\ — 6 > 3 for \z\ > 3 , the exterior of the circle \z\ = 3 is invariant, and hence belongs to Τ . By considering the real graph of ρ it is easily seen that Ρ ^ ( [ - 3 , 3])

c

3 , - v / l l U \VS,3

and hence we in fact have J C [ - 3 , - λ / 3 ] U [>/3,3] . Setting J+ = J Π [χ/3,3] and noting the symmetry of the JULIA set, we see that p(J+) = J , and finally, from the fact that p'{x) = 2x > 2\/3 for χ > y/3, it follows that mes J — / p'{x) dx > 2\/3 mes J+ = \/3 mes J . Jj+ Hence mes J = 0, where mes denotes one-dimensional LEBESGUE measure, and J has to be totally disconnected. It is easily seen that the JULIA set is given by J=[-3,3]\

|Jp"n((-V/3,V/3)), n>0

where p~n denotes the preimage under pn . This reminds one strongly of the definition of the classical CANTOR middle-third set. Incidentally, VITALI'S Theorem yields pn —• 00 in T . 30. We will now return to our main concern, the application of MONTEL'S Normality Criterion to iteration sequences of rational functions. THEOREM 1 The JULIA set is either nowhwere dense or else is the whole sphere. PROOF: Suppose that J° Φ 0 and D is any domain contained in J . Then since J is invariant and closed, it follows that J 2 (J fn{D)

= C\E{D)

= C .

n>0

Theorem 1 raises the question as to whether the second case, which seems to be unlikely, may actually occur. The first example of a rational function with empty FATOU set seems to be due to LATTES* and is based on the theory of elliptic functions. It is well-known that any even elliptic function may be represented as a rational function of an appropriate WEIERSTRASS p-function§. Applying this to p(2u) we obtain a rational function / of degree 4 (the reader will be invited as an exercise to determine this function explicitly) such that p(2u) =

f(p(u))

iS. LATTES, Sur i'iteration des substitutions rationelles et les fonctions de Poincare, C. R. Acad. Sei. Paris 16 (1918). W . BERGWEILER has pointed out to me, however, that BÖTTCHER [13] already had a similar example. § Later on it will be easier, but not more elementary, to construct such examples.

4 Montel's

32 holds, and hence p(2nu)

= fn(p(u))

Criterion

by iteration.

Now, if D is an arbitrary domain and Δ denotes any component of ρ~λ(Ό), then for η sufficiently large, 2ηΔ contains some period parallelogram Ρ of ρ, and since ρ assumes in Ρ each value on the sphere, we have fn(D)

= ρ(2ηΔ)

= C ,

and thus fn(D) intersects J . The backward invariance of the J U L I A set then implies that Ό also intersects the J U L I A set, and since J is closed and D was arbitrary, J = C follows. 31. The next theorem is concerned with the structure of the exceptional set E(D). It shows that E(D) may be non-empty only in particular, albeit important, cases. If D Π J Φ 0 , and if the exceptional set E(D) contains exactly one point, then f is conjugate to some polynomial. If E(D) contains two points, then f is conjugate to one of the functions ζ ι—> z±d. THEOREM 2

PROOF: In the first case we may assume, by conjugacy, that E(D) = {oo} , and we have to show that / is a polynomial. This follows almost trivially, for otherwise / would have a finite pole z0 not belonging to U fn(D) = 0 The second case is dealt with in the same manner. Assuming without loss of generality that E(D) = {0, oo} , we find as in the previous case that f(z) = cz±d , which is conjugate to the map ζ t—> z±d. 32. We note that the dependence of E(D) on D is immaterial. Obviously E(D) belongs to the F A T O U set, since the points of E(D) either are superattracting fixpoints or else form a superattracting cycle. We also note an easy consequence of Theorem 2: if D η J φ 0 and E(D) = 0 , then ( / n ( D ) ) n > 0 is an open covering of € , and hence, by the H E I N E - B O R E L Theorem, ) U . . . U

fm(D)

holds for m sufficiently large. THEOREM 3

The

JULIA

set is

perfect.

PROOF: Let Α E J be arbitrary, let D be any neighbourhood of Α and let b satisfy fp(b) = a . Since b lies in J and thus does not belong to E(D), it has some predecessor c Ε D, fn(c) — b for some n. If we are able to show that c φ α, then it follows - note that D was arbitrary - that α is a limit point of J . Conversely, each limit point of J belongs to J since J is closed.

Chapter 2

The Dynamical

Dichotomy

33

We proceed as follows: if fn(a) φ a for all η, we choose any b satisfying f(b) = a. Then obviously fn(a) φ b for every η and so c φ a. If, however, fp(a) = a for some p, then there exists some b φ a such that fp(b) = a. Again, fn(a) = b is impossible, for otherwise we would have b = fn(fp(a))

= fp(fn(a))

= F(b)

= a.

THEOREM 4 For arbitrary α e J the backward orbit 0~ (a) is dense in the JULIA set, that is, we have J = [J /-n({a})

·

n>0

f1 t 1 »v,..1 jr "

„^

Sa» r';

'·«
2 , and is totally disconnected for c > 2 + \p2. (this is not best possible).

5 Repelling Cycles 34. We now come to one of the most important results, but one which cannot be proved completely at the present stage. The missing link will be proved only in Chapter 3, Section 6; we will, nevertheless, repeatedly use the result. The reader is strongly urged to verify that we never use it to prove the missing part, which states that there are only finitely many non-repelling cycles. THEOREM 1

The repelling cycles form a dense subset of the

JULIA

set, that is,

J — {α \ |λ(α)| > 1} . Remark: This theorem corresponds to J U L I A ' S definition of the J U L I A set (of course, he did not use his own name), while F A T O U , as in the present text, started with the domains of normality. As already mentioned we will at present only prove the weaker assertion that J is contained in the closure of the set of periodic points of/. We assume that there exists some α Ε J which is not a limit point of periodic points. Since J is perfect we may assume that ο is not a fixpoint and not a critical value. Then in some disc D centred at α, / has at least two different analytic inverse functions φι and φ^ . As a third^ auxiliary function we consider 3 = i d . If D is sufficiently small then the assumption immediately implies that 4>I(D) Π 0 f c (D) = 0 for j φ k and fn(z) φ φό{ζ) for ζ E ß . The functions PROOF:

if is usually replaced by functions.

f2 . T h e

only reason is to have at our disposal

three

analytic inverse

5 Repelling

36

Cycles

a — c b— c : : denotes the cross ratio of a,b,c and d, omit a —d b— d the values 1, 0 and oo in D , and hence form a normal sequence. This is of course impossible, since ( / n | o ) is n o t normal. This contradiction shows that our assumption was false and the (restricted) assertion is true.

where ( a , b , c , d ) =

35. The next theorem expresses a property of JULIA sets which may be called self-similarity*. Thinking of the sequence ( / n ) as a microscope with levels of magnification f°, f l , f2, . . . , then in each domain D intersecting J it is possible to see the whole JULIA set again.

FIGURE 6

Magnification of the FATOU set of ζ 2 — 1 near 2

THEOREM 2 Suppose that D is some domain intersecting f

n

( D n j ) =

the JULIA set. Then

j

for every sufficiently large η . The PROOF is based on Theorem 1. Diminishing D if necessary we may assume that D is a disc centred at some repelling fixpoint ZQ of f p , and that fp(D) D D t W e are not referring to the definition usually used in the literature.

Chapter 2

The Dynamical

Dichotomy

37

holds. T h e domains fmp(D), m = 1 , 2 , . . . , are increasing and form an open covering of J , and indeed of each compact set not intersecting E(D). T h e HEINEBOREL Theorem then gives the assertion for indices η = p m . For indices η = pm + j the assertion follows from

J = fj(J) =

fmp+i(Dnj).

36. A s a consequence of Theorem 2 we have THEOREM 3

J either is connected or else consists of uncountably many compo-

nents. PROOF: If J is disconnected then there exist non-empty, disjoint and closed sets A and Β such that

J =

AuB.

Of course, relatively with respect to J , A and Β are also open, and replacing / by some appropriate iterate f p we may assume that

f(A)

= f(B)

=

J.

Then A , for example, consists of the subsets

AA = Anf~1(A)

and

AB = ΑΠ

f~1(B),

which are also non-empty, disjoint and closed, and these again consist of subsets

AAA = AA η Γι(Α),

AAB = AA η

f~l{B)

ABB = AB Π

f~l{B),

and

ABA = ABnf~1(A),

respectively. T h e same is true for Β . Proceeding in this way we find that, for every η, J is divided into 2 n non-empty, disjoint and closed subsets, each of which corresponds to exactly one word in {A, B}n . Each component of J is contained in exactly one of these sets, and conversely, each of these sets contains at least one component of J . Given (C1C2 . . . ) G { A , , the corresponding sequence of closed sets (C1C2 •. - C n i s non-increasing, and hence its intersection C1C2 • • • is non-empty and contains at least one component of J . Thus we have constructed a surjective map of the set of components of J onto the uncountable set { A , i?} 1 N , and so J must have uncountably many components.

37. T h e backward orbit of any ζ G J is dense in the JULIA set. This cannot be true, of course, for every forward orbit, but it is a generic property of J , i.e., the orbit 0+{z) is dense in J for each ζ in a countable intersection of relatively open and dense subsets of J .

5 Repelling Cycles

38

THEOREM 4

Each ζ in a dense subset of J has a dense orbit: 0+(z)

= J.

For η — 1 , 2 , . . . we cover J by finitely many chordal discs of radius 1 /η , and thus obtain countably many discs Dj . Then Oj = J Π 0~(Dj) is a relatively open and dense subset of J . By BAIRE'S Category Theorem^ Β = Oj is a dense PROOF:

3

subset of J. Let ε > 0 and b G Β be arbitrary. Then each ζ € J is contained in some disc Dj of radius less than e, and, since fn(b) e Dj for some n , it follows that x{fn(b), ζ) < 2ε; hence the orbit 0+(b) is dense in J. 38. We close this section by proving If D is any domain intersecting the no subsequence which is normal in D . THEOREM 5

JULIA

set, then ( f n ) contains

If D is some domain such that D Η J φ 0, then D contains a repelling fixpoint zq of some iterate f p . Without loss of generality we may assume that zo = 0 and fP(z) = λζ + • • • . Then since fpn(0) = 0 and (/ p n )'(0) = -> oo, nfe there exists no subsequence of the form (/P ) converging locally uniformly in D , and hence there is no convergent subsequence of ( / n ) , for otherwise there would exist a convergent subsequence of the form ( p ο f p r i k ) : here we have used the fact that f i is uniformly continuous. PROOF:

Note that our definition of the J U L I A set may be interpreted as follows: if D is any domain intersecting J , then there exists at least one non-normal subsequence (/ n f e |o) • I n f a c t) n o subsequence is normal.

EXERCISES 1. Prove that the space {0, l} 1 ^ is uncountable. 2. Prove that the J U L I A set of a real rational function is symmetric with respect to the real axis and that the J ULI A set of any even or odd rational function is rotationally symmetric with respect to the origin. 3. Prove that the J U L I A set of z 2 + c for c > 1/4 is disconnected. (Hint: Prove that 1R C Τ and use Exercise 2.) 4. Illustrate Theorem 3 for z2 + 0.26 as follows: let A and Β denote the intersection of J with the upper and lower half-plane, respectively. Each point ζ (pixel P) will be coloured black if Im f10(z) > 0, and white otherwise. 5. Prove that the J U L I A set of a quadratic polynomial is always rotationally symmetric with respect to some point (which one?). §See, e.g.,

R.UDIN [ 7 2 ] .

Chapter 2

The Dynamical Dichotomy

39

6 Stable Domains Normality Criterion was the appropriate tool in the preceding sections dealing with the JULIA set. In this section a comparable role is played by the RLEMANN-HURWITZ Formula and the concept of proper mappings. A first example is 39.

MONTEL'S

THEOREM 1

A rational function maps its stable domains properly onto each

other. If V is any stable domain, then F(V) is part of the FATOU set, and hence is contained in some stable domain. Since f(dV) C J, boundary is mapped onto boundary. This proves the theorem. PROOF:

FIGURE 7

S o m e s t a b l e d o m a i n s of t h e p o l y n o m i a l p(z) = z2 + 0.28 + 0.53 ζ .

If Vo is a non-empty and backward invariant union of stable domains, then J = 8VQ . THEOREM 2

PROOF: Suppose that D is any domain not intersecting VQ (if there is none, then we already have V0 — Τ and there is nothing to prove). By hypothesis / maps D into € \ Vo , and hence this is also true for / n . Thus (/ n |r>) is normal and D is contained in Τ. Since D was arbitrary, this gives J C Vo and indeed J C OVq . The converse inclusion is obvious.

6 Stable Domains

40

40. For each quadratic polynomial z 2 + c, the hypotheses of Theorem 2 are satisfied with Vq = A(OO). Thus in this case, the definition of the JULIA set given in Section 2, namely J = cM(oo), coincides with the general definition. NEWTON'S method applied to a non-linear polynomial ρ may be interpreted as iteration of the rational function

ρ'(ζ) The zeros of ρ are fixpoints of / ; more precisely, an m-fold zero zo has multiplier 1 — 1/m, and hence is superattracting if πι = 1 and attracting otherwise. The set of initial points whose orbits converge to some fixed zero zo of ρ satisfies the hypotheses of Theorem 2. Hence in each neighbourhood of any given point of J there are points whose orbits converge to a prescribed zero. If ρ has k different zeros, then each ζ Ε J is (at least) a fc-corner^. Note that this confirms our earlier remark on NEWTON'S method with reference to the equation z 3 — 1 = 0 . If Vo is simply a stable domain, then much more can be said. THEOREM 3 Suppose that V is a completely invariant stable domain. J = dV , and any other stable domain is simply connected.

Then

PROOF: The first assertion follows from Theorem 2. Now let W Φ V be any other stable domain. If it is not simply connected then there is some JORDAN curve 7 in W separating at least two boundary components of W, i.e., these boundary components belong to different components of € \ 7 . It is easy to construct 7 as a polygon, and hence it is not necessary to use the JORDAN Curve Theorem to show that 7 divides the sphere into two domains D\ and £>2 with common boundary 7 . By construction, both domains intersect the JULIA set and V is contained in D2 , say. Since V is completely invariant we have fn : D\ —> C \ V , and hence ( f n ) is normal in D\. This is impossible, however, and so our assumption must be false and the assertion must be true. As an example we mention any quadratic or, more generally, any non-linear polynomial: by Theorem 3, any (in 1, then since Vj is completely invariant under f p , each other domain Vk is simply connected, that is, all the V& are simply connected. By the p R I E M A N N - H U R W I T Z Formula, each V3 then contains exactly r = d — 1 critical p p points of f . However, since there are only 2(d — 1) critical points, m = 2 follows.

FIGURE 8

Two completely invariant stable domains. The JULIA set is a JORDAN curve.

From the lemma we deduce immediately Ii Τ is non-empty, then there exist one, two or else inßnitely stable domains. THEOREM 4

many

There exist at most two completely invariant stable domains. If there are two, then they are simply connected and each contains d — 1 critical points. THEOREM 5

EXERCISES 1. Construct some polynomial p(z) = z3 + az+b such that the corresponding NEWTON function f ( z ) = ζ — p(z)/p'(z) has a superattracting cycle of length 2.

7 The Denjoy-Wolff

42

Theorem

2. Prove t h a t p(z) = z2 — z3 is a proper m a p of degree 3 of the stable domain containing t h e origin, onto itself. (Hint: Consider ( p n ) in [0,1] .) 3. Consider f ( z ) = z + l / z + c , where c is some complex p a r a m e t e r in t h e right half-plane. Prove t h a t / has a completely invariant stable domain containing the right half-plane. 4. Let D C (9 + 2z + 9 z 2 ) / 2 0 in ID ; u - 1 .

Remark: The second assertion says that the interior of each horocycle containing u is invariant. It is clear that only this has to be proved, since the first assertion is then quite obvious. PROOF^: We consider some increasing real sequence (P^) converging to 1 and put Λ 0 ) =

f{pkz).

§For the notation in (a) and (b) see the next chapter. ^R. B. BURCKEL, Iterating analytic self-maps of discs, Amer. Math. Monthly 88 (1981).

44

7 The Denjoy-Wolff

Theorem

Then fk is analytic in ID and has a bound less than 1 on 0 and 2a+ b — 1). The circle \z\ — 1, its image etc., form a sequence of nested curves shrinking to the point 1. 2. Let / be a conformal self-map of ID having exactly one fixpoint zq £ zo without using the DENJOY-WOLFF Theorem.

Chapter 2

The Dynamical Dichotomy

45

3. (continued) The same if / has two fixpoints z± , zi € 9 D : the sequence ( f converges to the unique fixpoint which is (EuCLiDean) closer to / ( 0 ) . (Hint: In both cases it is convenient to pass to the upper half-plane Η . )

n

)

4. Let / be a conformal self-map of ID , this time having a fixpoint ZQ in ID . Prove that the full orbit (fn(z*) : η an integer }, ζ* ^ z0 fixed, lies on the non-EuCLIDean circle [Z,ZQ]TD = [Z*,ZQ\T> ( P O I N C A R E metric). 5. Let / : ID — > ID be analytic having some fixpoint zo € ID and assume that / is not a conformal self-map of ID. Prove that ( / " ) converges to zg , locally uniformly in ID , and show that \ f'(zo)\ < 1 . d

6. Let f(z)

=

eia

- — ,

K l < 1, be a BLASCHKE product. Prove that / has a

ajZ j=i fixpoint u such that either t i e D and \ f'(u)\

< 1 or else u G 9ID and 0 < f'(u)

< 1.

3

The Fatou Set

The orbit of an arbitrary point in the FATOU set is attracted either by a FATOU cycle or else by a cycle of closed analytic JORDAN curves lying in rotation domains. This is the quintessence of SULLIVAN'S Theorem and the Classification Theorem due to FATOU and CREMER - a com-

bined result. Both theorems will be discussed in the first part of this chapter, the major part of which is devoted to a detailed study of the dynamics (/ n |jr) and, in particular, to a description of the most important components of the FATOU s e t : BÖTTCHER domains, SCHRÖDER

domains, LEAU domains, SIEGEL discs and ARNOL'D-HERMAN rings, each of which is intimately connected with a certain functional equation.

Chapter 3

The Fatou Set

47

1 Sullivan's Theorem 45. Any two stable domains of some rational function / are either disjoint or else equal. In particular, the sequence ( f n ( V ) ) of the successors of a given stable domain either consists of mutually disjoint stable components or else there exist some m > 0 and some ρ > 1 such that fP+Tn(V) = / m ( V ) . A stable domain V is called wandering, if f for n / m , and periodic,

if fp(V)

n

(V)^f

m

(V)

= V for some ρ G EST.

The non-periodic predecessors of a periodic domain are called preperiodic. If, for example, α is a (super)attracting fixpoint of / , then the stable domain containing α is a fixdomain (p = 1). An analogous statement is true if we start from a cycle instead of a fixpoint: those stable domains containing a given (super)attracting cycle form a cycle of periodic domains. 46. One of the most important problems which remained open after the work of FATOU and JULIA was whether there might be wandering domains. This problem was solved in the early eighties by SULLIVAN. We will first prove in preparation for the proof a simple criterion for a given stable domain to be non-wandering. L E M M A 1 Suppose that V is a stable domain and fnk

V . Then (V) intersects the FATOU set there also exists some stable domain W which is intersected by / nfc ( V ) for k > I, that is, fnk (V) = W and so = f m ( y ) , where m = ni and ρ = ηι + 1 — η ι . In 1982 SULLIVAN''' announced his famous theorem that there are no wandering domains, and in 1985 he published a detailed proof The rest of this section is devoted to the proof of SULLIVAN'S T H E O R E M

There are no wandering domains.

47. It was SULLIVAN'S fundamental idea to introduce quasiconformal deformations of a given rational function. We do not make any attempt to give an outline of the theory of quasiconformal mappings, but will only give the basic definitions TO. SULLIVAN, Iteration

des fonctions analytiques complexes, C. R. Acad. Sei. Paris 294

(1982).

ID. SULLIVAN, Quasiconformal homeomorphisms and dynamics I. Solution of the FATOU-JULIA problem on wandering domains, Ann. of Math. 122 (1985); see also L. BERS, On SULLIVAN'S proof of the ßniteness theorem and the eventual periodicity theorem, Amer. J. Math. 109 (1987).

48

1 Sullivan's

Theorem

and state those properties of quasiconformal mappings which are of importance to us. The interested reader should consult the book of LEHTO and VIRTANEN [59]. Let D be any planar domain; then every measurable complex-valued function μ defined in D with norm ||μ|| = esssup D \μ(ζ)\ < 1 is called a BELTRAMI coefficient. Any complex homeomorphism having locally integrable distributional derivatives in D is called quasiconformal, if it satisfies a BELTRAMI equation wj = μ(ζ)ιυζ

a.e.,

where μ is a BELTRAMI coefficient, while wz = (wx

— ivjy)/2

and

wj = (wx + iwy)/2

.

Conversely, given any BELTRAMI coefficient μ, there always exists a quasiconformal solution of the corresponding BELTRAMI equation^, and if w\ is any other quasiconformal solution, then w\ ο w~l is a conformal map of the domain w(D) onto u>i(D). In particular, every quasiconformal mapping with BELTRAMI coefficient 0 is conformal. The composition h = fog of two quasiconformal mappings is again quasiconformal, whenever it is defined, and its BELTRAMI coefficient is given by jjifl

_

e~2iQμf ο g + μ9 ^• , ν 1 + e~ 2u * (μ/ O g) μ9

£1.6* ,

where α = ot(z) — a r g g z ( z ) . In particular, if g (respectively, /) is conformal, then ¥ μΗ = — μ/ O g (respectively, μΗ = μ9). Q We now restrict ourselves to the case of plane mappings. Then every BELTRAMI equation has a unique normalized quasiconformal solution which fixes 0, 1 and oo (w(z) —> oo as 2 —» oo). If this solution is denoted by χυμ, then the map μ ι—» ιυμ is continuous: wßn —> υυμ , locally uniformly, as ||μη — μ|| —• 0 , provided ||μ|| < 1. If μ = μ(ζ, t) depends on some parameter t Ε IR7™ , say, and if, for fixed ζ , the map t ι—» μ(ζ,ί) is continuously differentiate (respectively, real analytic), then the normalized solution of the corresponding BELTRAMI equation also depends in a continuously differentiable (respectively, real analytic) fashion on t. 48. SULLIVAN'S original PROOF was rather long and involved, since he had to consider different cases according to whether a given domain, assumed to be wandering, is simply connected, doubly connected, of finite connectivity greater than two, or else infinitely connected. Using an idea of BAKER^, one can show that it suffices to consider the simply connected case, which simplifies matters considerably. Thus we first prove §This deep existence theorem is sometimes called the Measurable RIEMANN Mapping Theorem. ^See also I. N. B A K E R , J . KOTUS and Y. LÜ, Iterates of meromorphic functions IV: Critical finite functions, Preprint (1991), in a more general setting.

Chapter 3

The Fatou Set

49

If f has a wandering domain, then f also has a simply wandering domain V, and fn , η G IN, is univalent in V.

LEMMA 2

connected

The wandering domain is denoted by V and we set Vn = fn(V). Since / has only finitely many critical points, all but finitely many of the domains Vn are free of critical points, and hence we may assume that all are. We will show that V = Vo is simply connected, so that, by the R I E M A N N - H U E W I T Z Formula, each Vn is simply connected and / is a conformal map of onto V n , or, equivalently, fn is a conformal map of V onto Vn . PROOF:

We may assume that \z\ > 1 is contained in Vo, so that each Vn , η > 0, is contained in ID and / : Vn-\ —> Vn is an unramified proper map. Assuming Vo to be multiply connected, we will derive a contradiction as follows. By hypothesis, Vo contains a closed curve 70 : [0,1] —> Vo which is not homotopic to the point α = 7o(0), i.e., there is no continuous function Η - a homotopy mapping [0,1] x [0,1] into V0, such that 7 0 (i) = H(0,t) and # 0 , 0 ) = H(s, 1) = H(l,t) = α for 0 < s, t < 1. Then ηη = fn ο 70 is not homotopic to the point / η ( α ) , since any homotopy may be lifted in the same way as any curve may be lifted under f n , see Chapter 1, Section 2. By Lemma 1 we have diam η η —> 0 as η —>· oo, where diam denotes the chordal diameter. 49. Denote by Δη the unbounded complementary component of yn and set Dn — Vn U ((D \ Δη). We will prove that f(Dn)

C Dn+1

(1)

if η > m is sufficiently large, depending only on diam η η . This provides a contradiction, since then, for each k, |/ f e (£)| < 1 holds in Dm , contradicting the fact that Dm intersects the JULIA set. Instead of proving (1) it suffices to prove that f(Cn)

C Dn+1,

(2)

where Cn denotes any bounded complementary component of ηη . We first remark that / is L I P S C H I T Z continuous, i.e., there is some positive constant L such that x(f(a),f(b))• φ. This finishes the proof in the first case. ^N. STEINMETZ, On SULLIVAN'S classißcation of periodic stable domains, Complex Variables (1990). When writing this paper I was unaware of CREMER'S work and attributed it to SULLIVAN, who announced the Classification Theorem in his paper Iteration des fonctions analytiques complexes, C. R. Acad. Sei. Paris 294 (1982). 14

Chapter 3

CASE

2

The Fatou Set

57

Every limit function in V is a constant.

Then fnk —>· α implies fnk+1

n k

=f ° f

^ f ( a )

and RFC+1 =

/

N

W - , A ,

locally uniformly in V ; thus each convergent subsequence tends to a fixpoint of / in V . We will first show that the whole sequence ( / n ) converges to a. Let δ > 0 be chosen such that U = {z-.X(z,a)
a to be false, we obtain some subsequence (mk) of the positive integers such that mk A, but mk — 1 £ A and fmk b φ a, contradicting m 1 the fact that f "(zo) = / ( Z ^ " ^ ο ) ) / ( a ) = a. 6 0 . There remain two possibilities left, the first being a e V . By Theorem the sequence ( ( / n ) ' ) then converges to 0, and from

WEIERSTRASS'

(Γ)'(α) = (/'(α)Γ it then follows that |/'(α)| < 1, corresponding to a (super)attracting fixpoint, i.e., to the case of a B Ö T T C H E R domain or a SCHRÖDER domain. The second possibility is that f n ^ a e d V , locally uniformly in V . Then obviously |/'(a)| > 1, because V is maximal, and |/'(o)| < 1, because the sequence of iterates cannot converge to a repelling fixpoint (see the exercises). 6 1 . This case | / ' ( A ) | = 1 will be discussed in the following lemma due to FATOU, which also plays an important role in the study of the dynamics in a LEAU domain - see also the section on external rays of polynomials in Chapter 6.

1 Suppose that f(z) = Xz + 02ζ 1 + · · · , |A| = 1, is a holomorphic selfmap of the hyperbolic domain V, such that fn —> 0 G dV, locally uniformly in V . Then, for every ZQ € V , LEMMA

/"(ζ) locally uniformly in V , and Λ = 1.

,

58

2

The Fatou-Cremer

Classißcation

Remark: For the proof of Lemma 1 it would suffice to know that V Π f(V) is nonempty and that the sequence (/") is defined in V, instead of assuming f(V) C V . PROOF: Since |Λ| = 1 , / is univalent in some disc \z\ < r. Let Δ0 be any domain lying compactly in V and containing ZQ . Then / maps ΔΟ onto the domain Δ\ , and it is easy to construct a domain D which is compactly contained in V and contains ΔΟ U Δ\: take any arc 7 joining ΔΟ a n d ΔΙ in V ; then D consists of Δ0 , Δ ι and some connected neighbourhood of 7 (some corridor). By construction, Dm=

U

fn(D)

is a domain such that f(Dm)

= Dm+i

C

DmCV,

and ( f n ) converges to 0 , uniformly in D0. In particular, Dm is contained in {z : \z\ < r } , at least if m is sufficiently large, and / is univalent in Dm . 62. For simplicity we first assume that DQ is contained in \z\ < R. Then the functions

are univalent in DQ and never assumes the values 0 (which is contained in the boundary of V) and 00, while the value 1 is taken only at ζ = ZQ . Thus (gn) is normal in DQ \ {ZO} by MONTEL'S Normality Criterion. To prove normality at z$ , let \z — ZQ\ < ρ be a disc lying entirely in DQ . Then

M O

_ gn(z0

=

+ ρζ) - 1 P9nizθ)

belongs to the class S and omits the value — (pg'n(zo)) known KOEBE-BIEBERBACH Theorem ", we have IMC)I 00 and•n fn —> 0 give

and z) = }

USEE, E.G.,

DUREN'S

BOOK [34],

N I M fn(zo)

/ n (^o) λ / ^ ο ) + ···

MM) A

Chapter 3

59

The Fatou Set

as η = rik —» oo . 63. We have thus proved that gof

= Xg,

(1)

and by HURWITZ'S Theorem there are two possibilities left: (a)

g is univalent.

Iterating equation (1) gives fn(zo) — g"1(Xn), which contradicts the fact that fn{zο) —^ 0. This contradiction is obvious if λ ρ = 1, since the sequence ( f n ( z o ) ) is not periodic. However, if λ is not a root of unity, then the sequence (A n ) is dense on the unit circle, and hence g(Do) contains l,

(1)

then the sequence ( f n ) is defined and normal in some neighbourhood of the origin and converges to 0 uniformly. Long before FATOU and J U L I A started their investigations of rational functions, B Ö T T C H E R ^ considered the functional equation Φ ° f = α{φγ

(2)

and solved it by iteration. T H E O R E M 1 B Ö T T C H E R 's functional

uniquely determined

equation (2) has a local solution, which is normalization

by the

φ(ζ) = z + a2z2 + •••.

(3)

PROOF: The proof of uniqueness simultaneously gives an idea of how to prove existence constructively. If φ is some solution in \z\ < r, where we may assume that |/(2)| < \z\ / 2 , then iteration of equation (2) gives

ma = ^?Γ5Ζ5ΗΓ =„»?/ *(/"(*)) V a1+"'+fcn-1 V zfcna1+""+fcn_1 ' where the second root is uniquely determined by its value at ζ — 0 : kVl = 1 · Since Φ(/η(ζ))//η(ζ) —> φ'(0) = 1 as η —» oo, uniqueness follows from φ(ζ) = lim

n—>oo

\l

1 y CL '

= 2 lim

tu

\

η—>00

γ Ζ

1+...+kn-i 0 so small that |ft(*)|

=

m

-

1 1 < -

holds in D = {z : \z\ < r } , where

Mz)=

k

2

< 2 / 3 . Then 0 < \}{z)\ < r in D\ {0} and

V f ^ )= z

k

\ j ^

= z + ···

^L. E. BÖTTCHER, The principal laws of convergence of iterates analysis (Russian), Izv. Kazan. Fiz.-Mat. Obshch. 14 (1904).

and their application

to

Chapter 3 The Fatou Set

61

is well-defined and holomorphic in D , the elementary inequality I A/1 + u — 11 < — 1 1 m

for

liil < - , 2

which follows from |l + u|-1~t~1^m < 2, then giving Ψη+ΐ(ζ) Φη(ζ)

- 1

fc"Vl

- 1 < k

+ h(fHz))

-71—1

Thus the infinite product

Π

Φη+ΐ{ζ)

n=0 Φη(ζ)

lim

φη(ζ)

converges, uniformly in D, and its limit φ provides a solution of (2). Obviously φ has the correct normalization at 2 = 0, and the fact that φ satisfies the functional equation (2) for α = 1 follows from = φη ο f . Sometimes φ is called BÖTTCHER's function . 66. We now return to rational iteration and assume that / has a superattracting fixpoint, which we may place at the origin. Sometimes, however, it is more convenient to consider the superattracting fixpoint ζ — oo; this is easily achieved using the conjugation ζ ι—> 1 j z . Then 2 = 0 is contained in some BÖTTCHER domain, and BÖTTCHER'S function defines a local conjugation. Some neighbourhood D of the origin is mapped by φ conformally onto some disc U , and / acts in D like the power ζ t—> α does in U : f —

D Φ

D Φ

U

U C^aC*

THEOREM 2 Suppose that f(z) = azk + • · - , α φ 0 , k > 2, is rational and A ( 0 ) is the stable domain containing the origin (BÖTTCHER domain). Then f . Λ(ο) where m is a non-negative inßnite connectivity.

integer, and A(0)

A(0), is either simply connected or else of

Remark: Even in the simply connected case the local conjugation (2) does not have to hold throughout the whole BÖTTCHER domain. A simple example showing this is the polynomial z2 — z3 ; see also Figure 2.

62

3 Böttcher

Domains

PROOF of Theorem 2: We have only to prove the statement concerning the connectivity number η of A(0). Since / has at least k — 1 critical points in .4(0), ζ = 0 being a (k — l)-fold critical point, the R I E M A N N - H U R W I T Z Formula gives (n - 2) > (k + m)(n - 2) + (k - 1); hence (n - 2)(k + m - 1 ) < -(k - 1 ) < 0 follows and so η = 1, and there are k + m — 1 critical points in A(0).

FIGURE 2 T h e BÖTTCHER d o m a i n A ( 0 ) of ζ ^

1 . 8 8 ζ2 + (1 + i)z3

.

Remark: If the B Ö T T C H E R domain A(0) is simply connected and if φ is a conformal map of .A(O) onto the unit disc ID, normalized by f/>(0) = 0 , then h = i> o / o t A -

is a

BLASCHKE

product

1

: ^ ^

1

©

rn

h(z) =

eioczk

J] γ ^ ; j=1

X

satisfying φ οf — hοφ;

(Jj4 Ζ j

(4)

Chapter 3 The Fatou Set

63

hence ψ/ψ'(0) is B Ö T T C H E R ' S function if and only if m = 0 , i.e., if and only if / has no zero in ^4(0) except at ζ = 0. For the BÖTTCHER domain Λ ( ο ο ) of a polynomial - and fixpoint oo instead of 0 - this is always true. 67. In many cases it is possible to determine GREEN'S function for a BÖTTCHER domain immediately from the dynamics of the sequence of iterates. In general, it is difficult to determine GREEN'S function, even if the domain is defined by simple analytic inequalities, whereas, in the case of a BÖTTCHER domain, GREEN'S function is obtained by a simple limiting process, even though the domain may have extremely complicated boundary. T o construct GREEN'S function we first prove LEMMA 1 Suppose that V is some BÖTTCHER domain and D0 = {z : \z\ < R} is an invariant neighbourhood of the fixpoint ζ = 0 , that is, / ( D o ) c Dq . Then if Dn denotes the component of f~n(Do) containing Do , (Dn) is a regular exhaustion of V , and V is simply connected if and only if each Dn is. PROOF: Set D*n = f~n{D0) Π V; then the fact that fn 0, locally uniformly in V, gives |J £>* = V . If Ε is some compact and connected subset of V intersecting η D0 , then, by the HEINE-BOREL Theorem, there is some η e IN such that Ε C £>* ; hence in fact Ε C Dn. This is also true for an arbitrary compact subset Ε of V, because such a set may always be embedded in some subset as described above (see the exercises). Hence (Dn) is an exhaustion of V , and its regularity follows from the fact that Dn is bounded by JORDAN curves. Moreover, since (Dn) is an increasing sequence, V = |J Dn is simply connected if the Dn are. Conversely, if n> 0

V is simply connected, but Dn is not, then Dn has some complementary component C contained in V. On the boundary of C we have \fn(z)\ = r ; thus \ fn{z)\ < r in the interior of C by the Maximum Principle - where we have tacitly assumed that oo ^V .In the domain Dn U C we therefore have \fn(z)\ < r, and indeed \fn{z)\ < r, contradicting the maximality of Dn . This proves Lemma 1. The next theorem shows how to obtain GREEN'S function for a BÖTTCHER domain dynamically, the additional sufficient condition obviously also being necessary. THEOREM 3 In addition to the hypotheses of Theorem 2 suppose that f has no zero other than at ζ = 0 inside the B Ö T T C H E R domain > 1 ( 0 ) . Then g(z) =

1 lim k~n log oo \fn{z)\

is GREEN'S function for /1(0) with pole at ζ — 0 . PROOF: With the notation of Lemma 1, (Dn) and, since / is zero-free in >1(0) \ { 0 } , 9n{z) = AT" log

is a regular exhaustion of A ( 0 ) ,

r

\fn(z)\

3

64

is GREEN'S function for Dn

, i.e., g =

Böttcher

Domains

lim gn is GREEN'S function for .4(0). n—>00

It is, however, not always true that GREEN'S function has boundary values zero everywhere, if it is defined via some regular exhaustion. To prove that in our case g actually has zero boundary values*, we denote by m = m(z) the first index such that \fm{z)\ < r . Then m(z) —> 00 as ζ —> Ö A ( 0 ) , since otherwise there would be some point α e 9^4(0) and some integer m e IN such that |/ m (o)| < r , contradicting the invariance of J . Prom g(fn(z)) = kng{z) then follows 0 < g(z) < m+1 Μ k~ —> 0 as TO = m(z) —> 00, i.e., as ζ —> cM(0); here Μ is the supremum of g in D i \ D0 . For 00 ^ v4(0) the argument is much simpler: there exist constants 0 < α < b < 00 such that Α < \fn{z)\ < b on CM.(0); hence (gn) is a CAUCHY sequence on 0 A ( 0 ) and so in A ( 0 ) by the Maximum Principle, since the singularity ζ = 0 cancels out for the difference gn — gm . Remark:

FIGURE 3

Equipotential lines of GREEN'S function for the BÖTTCHER domain of — — ^ — X — ^z . 10 — (8 — i)z

68. Any BÖTTCHER domain is either simply connected or else of infinite connectivity, and both possibilities may occur. The following theorem gives a sufficient condition for a given BÖTTCHER domain to be simply connected. It is, however, by no means necessary, since any bounded BÖTTCHER domain of a polynomial is simply connected. iThus, BÖTTCHER domains of this kind are regular in the sense of potential theory.

Chapter 3 The Fatou Set

65

Suppose that f has no critical point in A ( 0 ) other than the superattracting ßxpoint ζ = 0 . Then B Ö T T C H E R ' S function maps ^4(0) conformally onto

THEOREM 4

the disc {w : Η < |α|

, α = / k, r = k — 1 critical points) gives inductively kn~k \ hence Dn is simply connected and so is .4(0). Then by equation (4) the conformal map ψ : 4(0) —• D , PROOF:

normalized by ?/>(0) = 0 and ψ'{0) > 0, satisfies (note that m = 0)

hence φ = ψ/ψ'(0) is the B Ö T T C H E R ' S function associated with Λ ( 0 ) . Finally from φ ο f = ('(0)) fc_ V a , and thus the assertion holds. Remark: The simply connected B Ö T T C H E R domain A(0) of the polynomial z2 — z3 contains two critical points, ζ = 0 and ζ = 2/3 ; hence the hypothesis in Theorem 4 is not necessary for A(0) to be simply connected. Note, however, that in this case B Ö T T C H E R ' S function is not a RIEMANN map. Also Theorem 4 does not remain valid if, in the hypothesis, critical point is replaced by zero. 69. The situation is extremely simple in the case of a polynomial p(z) = adzd + ad-iz'1"1-\

b a0 ,

αΛφ 0.

The fixpoint ζ = oo is superattracting, the corresponding B Ö T T C H E R domain A(OO) is completely invariant, its boundary coincides with the J U L I A set, and g(z) = lim

n—yoo a "

\pn(z)\ = log \z\ + — l o g |ad| + o(l) a — I

(as ζ OO) is G R E E N ' S function for .A(oo). We will not go into further details, since there is an extra section in Chapter 6 devoted to the study of polynomials. Summarizing, we have For any non-linear polynomial p(z) = üdZd Η ments are equivalent: (a) A(oo) is simply connected; (h) J is connected; THEOREM 5

(c) B Ö T T C H E R ' S function maps A(oo) conformally onto {w : (d) p' is zero-free in A{oo).

the following

state-

>

;

|TU|

LA^L1^}

66

4 Schröder

Domains

EXERCISES 1. Determine an interval (α, b), α < Ο < b, and a disc \ζ\ < R inside the BÖTTCHER domain Λ ( 0 ) of p ( z ) = z 2 - z 3 . (Hint: Determine α and b by solving the equations p(a) = b and p(b) = a numerically. Similarly, determine R by the condition \p(z)\ < \z\ for \z\ < R.) 2. Prove that the JULIA sets of z2 + c for —2 < c < 1/4 and c = i are connected. 3. GREEN'S function g(z)

= g(z, oo, -A(OO)) for the BÖTTCHER domain A ( o o ) of

z2+c,

|c| < 2 say, may be illustrated as follows: on a large circle \z\ = R (large means R < 1 0 0 0 ) , g(z) is approximately equal to log/ϊ. Backward iteration of this circle yields approximate level curves. The most interesting window is near the 10
1 . Then

the

equation = \φ

(2)

= z + C2Z2 + •••.

(3)

φοί has a unique

0
n/2 and Re hn(zo) > n j 2 for η > no, and so φη+1(ζ)

=

φη(ζ)

+

[r(h

n

(z)) - r(h

n

(z

0

))]

(6) =

φ

η

( ζ ) [ ΐ + 0 ( η ~ ^ ) ]

.

This follows from the fact that r ' ( £ ) = 0 ( η ~ ι ~ Ί ) in the half-plane Re ζ > n/2, which itself is an easy consequence of CAUCHY'S Integral Formula for the derivative: as path of integration we take the circle centred at ζ with radius nj4. Since oo the product Π (1 + n~1~7) converges, this yields φη —• φ, locally uniformly in Ρ , 71=1 which, however, is not sufficient to deduce the asymptotics of φ from the asymptotics of φη . To do this we put E n = sup \ φη{ζ)/ζ\ , where ζ varies in Ρ , Re 2 > c. ι Ί Then equation (6) yields E n + 1 < E n ( 1 + 0\η~ ~ ), and hence E n is bounded by oo Ε — C2 Π ( 1 + c m " 1 " - ) , where c\ and C2 are positive constants. This and (6) 71=1 imply the estimate Φη+ΐ(ζ)

-

Φη{ζ)


ζ + 1 does in Η :

Ρ

/ —

Ρ

Φ

Φ

Η

Η C^C + i

Remark: ffower.

The domains L& together with their centre

0

form the so-called L E A U

If / ( 0 ) = 0 but / ' ( 0 ) = Λ is some m-th root of unity, then one has to consider / m instead of / . The Flower Theorem then shows that the origin is the centre of some L E A U flower consisting of s = pm L E A U domains, which are permuted cyclically by / , but may form sub-cycles. Each L E A U domain contains at least one critical point of / m , and hence at least one of these domains contains some critical point of / . A similar argument works if α = { ζ χ , . . . , z p } is a rationally indifferent cycle with multiplier Λ , A m = 1. Instead of / we consider j q , where q is the least common multiple of ρ and m . Then each Zj is the centre of a L E A U flower. We have thus proved

Chapter 3 The Fatou Set

77

Suppose that f is rational and α = {ζι,..., zp] is an indifferent cycle. Then to each Zj € α corresponds a L E A U flower centred at Zj and consisting of L E A U domains. Every cycle contains at least one critical point of f .

THEOREM 2

FIGURE 7

The SCHRÖDER domain A(zo), zo = — 0 . 4 1 + 0 . 5 4 i, of f(z) = ζ — ζ2 + z3/zo , and the LEAU domain with fixpoint 0. The reader should be able to recognize the LEAU domain by its geometry.

EXERCISES 1. Discuss Theorem 2 for f(z) = —z + z2 and f(z) = iz + z2 . 2. Determine (up to conjugation) all polynomials of degree 3 having exactly one domain and exactly one SCHRÖDER domain; see Figure 7.

LEAU

3. Let f(z) = ζ + l / z + w , Re w > 0 and w φ 0. Prove that the LEAU flower centred at ζ = oo consists of exactly one completely invariant LEAU domain L . It contains the invariant half-planes Re 2 > 0 and Re wz > 1, and / is a proper map of degree 2 of L onto itself. 4. (continued) Prove that, for |l — w2\ < 1, L is simply connected and that there exists one more completely invariant stable domain.

78

6 Indifferent

Cycles

5. (continued) Prove that, for w = iv , ν > 0 , L is of infinite connectivity and J contained in i IR U {oo} . (Hint: Consider the sequence (/"(—1)).)

is

6. Illustrate the dynamics in the LEAU flowers of f(z) = ζ — z2 , f(z) = ζ — z4 and f(z) — i(z + ζ2), respectively, by computer graphics. Use — 0.11 < 0.1 in the first case (Re yfz > 0) and jz2 — 0.011 < 0.01 in the second case (for some η < 100, say) to decide whether zn = fn(z) —> 0 or not. Let L be a simply connected LEAU domain. Prove that / is locally conjugate to a product which has a fixpoint on V satisfies f(z ο) = ζ a and |/'(zn)| = 1, then / is an unramified covering map of V onto itself. (Hint: Invariant Lemma of SCHWARZ.) 4. (continued) T h e simple connectivity of a SIEGEL disc may be proved as follows: in addition to the solution of (2) already constructed, there exists a solution which is the local inverse of some universal covering map ψ : ID —> V fixing the origin. Prove that ψ is in fact a conformal map.

7 The Centre Problem 81. Actually, Theorem 2 in the previous section is merely a reformulation of the problem. In this form it is known as the Centre Problem (Zentrumproblem). Although some known examples seemed to indicate that the Centre Problem is never solvable, SIEGEL was able to prove in 1942 that it is almost always, in a precise sense, solvable. We will now discuss the class of counter-examples due to CREMER^, and will first make some remarks about continued fractions. The interested reader is referred to the book of NIVEN*.

Let αϊ, a,2, • • • be positive integers; then we define recursively [αϊ] = — αϊ

and

tH. CREMER, Zum Zentrumproblem,

zentren, Math. Ann. 115 (1938).

[αχ,..., α η ] =

—r^ τ, αχ + [ α 2 , . . . , αη\

Math. Ann. 9 8 (1927); Über die Häußgkeit der Nicht-

•tl. NIVEN, Irrational numbers, Carus Monographs 1956.

Chapter 3 The Fatou Set

83

in which the symbol [ ] should not be confused with the greatest integer less than or equal to a given real number. We thus have [01,02] =

[αϊ,«2,03] =

γ-, αϊ Η

γ αχ Η

°2

η α 2 -JΗ

Given any sequence of positive integers (αη), t=

the irrational

1

03 limit

lim [ α ι , . . . , α η ] = [ α ι , α 2 , α 3 , . . . ] η—>·οο

(1)

always exists and, conversely, each irrational number t € (0,1) has a unique continued fraction expansion (1). The continued fraction of any rational number is finite, and periodic continued fractions always represent quadratic algebraic numbers, i.e., solutions of quadratic equations with integer coefficients. As an example we compute the continued

y/E

fraction t — [1,1,1,...] = —

— 1 £

, which plays a prominent role, from t =

(l+i)-1

and the additional information that 0 < t < 1 . 82. If [ α ϊ , . . . , α η ] = pn/qn , denotes the η

pn and qn relatively prime,

convergent of the continued fraction (1), then

2 ( α η + 1 + 1 )ql


weia and the inversion w >—>· r/w, either the stated functional equation holds (which formally looks like SCHRODER'S), rX or else, if / permutes the components of dV, φο f = — is true. Iterating this equa-

Φ

tion, however, leads to φ ο f2 = φ, which is impossible, since f2 is not the identity map in V . The proof of the second part is identical to the proof of Theorem 1. Remark: As in a SIEGEL disc, / acts in an ARNOL'D-HERMAN ring like a rotation, but without centre. It is, however, not known whether φ may be constructed as the 1 χ"

limit of the sequence of arithmetic means φη = — ) η

fc=l

/k

—r . At any rate this sequence XK

is normal for arbitrary Λ = e%a, and each limit function φ satisfies ψ ο f — Χ φ , and hence either coincides with φ or else vanishes identically. Obviously, φ has to vanish if Λ is not the correct rotation number.

EXERCISES 1. By a r a n d o m process determine λ = e 2 7 r l i , 0 < t < 1, a n d consider p(z) = \z + z2 . If the set {zn : 100 < η < 2000} , where zn — pn(—λ/2), looks like a closed curve on t h e screen, t h e n 0 is almost surely a SIEGEL point of ρ . T h u s prove empirically t h a t φ ο ρ = Χφ is solvable for almost every Λ. 2. Prove t h a t if φ ο f = Χφ, |λ| = 1, is solvable for every rational f ( z ) = Xz + · · · , then φ ο g = Χφ is solvable for every rational g(z) = Xz + • • · . 3. Let f(z) = Xz(z — l ) 2 , where ζ — 0 is t h e centre of a SIEGEL disc. Prove t h a t the FATOU set of / consists of t h e BÖTTCHER domain Λ ( ο ο ) , the SIEGEL disc centred at 0 , and the preimages of the latter.

4

4

The Existence of Rotation Domains

The existence of rotation domains is one of the most difficult problems in iteration theory. The first breakthrough was made in 1942 by SIEGEL, who solved SCHRODER'S functional equation for every multiplier satisfying a certain diophantine condition, thus proving the existence of SIEGEL discs. HERMAN was the first to recognize how a local theorem due to ARNOL'D can be transformed into a global result, yielding examples of rational functions having invariant ring domains, which he called ARNOL'D rings and which are now called ARNOL'DHERMAN rings. In this chapter we will prove the theorems of SIEGEL and ARNOL'D under fairly weak hypotheses.

1 Siegel 's Theorem

90

1 Siegel's Theorem 89. The first positive result concerning the solvability of the Centre Problem is due to SIEGEL^. In this section we will give a proof of a more general result due to RÜSSMANN, which may be looked upon as a preparation for the proof of ARNOL'D'S Theorem concerning the existence of invariant ring domains. It indicates that the solvability of SCHRODER'S functional equation is closely connected with the quality of rational approximation of t — (argA)/27r. This phenomenon was already apparent in C R E M E R ' S examples. If t can be well approximated, then the corresponding fixpoint is not a centre, while in the opposite direction, if t is in some sense badly approximated by the convergents of its continued fraction, then the fixpoint is a centre. SIEGEL'S T H E O R E M

Suppose there are positive numbers δ and m such that δ

t-P-

W

—

holds for arbitrary p, q £ IN ; then SCHRODER'S functional equation is solvable for any function f(z) = Az + a2z2 + ••• , A = ε2πίί.

Remark: The condition (1) is sometimes called a diophantine condition of SIEGEL type and is satisfied by almost every t (see the exercises), and in particular, by any non-rational algebraic number. 90. More general than THEOREM

1

SIEGEL'S

— λζ + α2ζ2 + · · · , A = β2πιί,

Suppose that f(z) ~

near ζ — 0 and set A(x) =

result is the following is holomorphic

e-kx

— k= 1 '

— . I f , for some β > 0, the integral '

rß / loglog A(x)dx converges, then

SCHRODER'S

(2)

functional equation has a solution.

We adopt RÜSSMANN'S PROOF*. It is based on the construction of approximate solutions for the inverse of SCHRODER'S functional equation, f οψ =

ψο\,

tC.L. SIEGEL, Iteration of analytic functions, Ann. of Math. 4 3 (1942). TH. RÜSSMANN, Kleine Nenner II: Bemerkungen zur Newtonschen Methode, Wiss. Gött. 1972, Nr. 1 .

Nachr. Akad.

Chapter

4 The Existence

of Rotation

Domains

91

where we have written Λ(Ζ) = Xz, by means of a certain modified N E W T O N ' S method, which we will now describe. If φη is some approximation and φ = φη+Δη is the solution, then 0 = / Ο φη - φη Ο Χ + ( / ' Ο 1 is fixed (later on we will set q = 3/2). Then if ε is sufficiently small, we assert that oo

Σ> 1/2, the right hand side, Gn say, is holomorphic in |ζ| < rn, and thus may be represented by its power series oo G»(0 = I > c fc=2

f c

·

Then obviously b»k>

= Σ

k=2

j r r j

c

94

1 Siegel's

is the unique solution^ of (7) satisfying En(0) — E'n(0) imply the estimate ΐση(οι 1 — χ; the last one is equivalent to e " i n < 1. 95. By definition we have Λ(δη)

— ε " 1 , and so |£η(ΟΙ 0 , the integral (Hint: Show that Λ(2χ)

= Ο (μ(χ)/χ)

2. (continued) Prove that if t -

Ρ > e

fß Γ log log μ(χ)άχ

I

converges.

Jo

as χ -> 0.) q

for all p, q G IN , where 0 < 7 < 1, then

SCHRODER'S functional equation is solvable. 3. Show that if t £ (0,1) does not satisfy SIEGEL'S condition, then given M G IN, there exist some q £ IN and some integer ρ such that ( * )

^ 1 -m-2 < m —9

Prove that, for fixed q, the measure of the set of t Ε (0,1) satisfying ( * ) is less than — < Ί m~1 . m q~ 4. (continued) Prove that almost every (in the sense of LEBESGUE measure) t € (0,1) satisfies some diophantine condition of SIEGEL type.

2

96

5. Suppose λ = e2nit A{x) = 0(x~2~m)

The Bryuno-Rüssmann

Theorem

satisfies some diophantine condition of SIEGEL type. Prove that as χ —> 0 + .

2 The Bryuno-Rüssmann Theorem 9 7 . SIEGEL'S original proof of the solvability of the inverse of SCHRODER'S functional equation, namely

f οψ = ψο\,

(1)

consists of two steps. If the power series oo

f(z)

^ =

= \z + J2akzk> k=2

converges in \z\ < p, say, then in the first step a formal

solution

oo φ{ζ)

= z +

J2ckzk k=2

is determined by equating coefficients, and the second and much more difficult step consists of proving convergence of the formal series in some disc \z\ < r , and so justifying the preceding formal operations. The first step is very simple: equation (1) is equivalent to oo ^ ( λ * - λ )ckzk fc=2

oo

/

= Σ α Ι /=2

oo

\Z + Σ \ j=2

C3zj

and if α is non-rational, the coefficients c^ are uniquely determined by (Afe - X)ck = ak+

Pk(a2,

• ·. , a A _ i , c 2 , • - ·, c f c _ι) ,

(2)

where Pk is a polynomial in the variables 0 , 2 , . . . , ak-i, C2, · • ·, ck-\ with nonnegative integer coefficients, independent of the particular problem. This is easily shown by induction or else by observing that the coefficient of zk in the expansion 00 >. ι

(

3=2

CjZ^ J , 2 < I < k , is a polynomial in C 2 , . . . , ck~i+1 having non-negative '

integer coefficients.

Chapter 4 The Existence of Rotation

Domains

97

98. BRYUNO^ has now adapted and considerably simplified SIEGEL'S proof of convergence, thereby finding the most general sufficiency criterion, which is also necessary* for (1) to be solvable for every f . A completely different proof has been given by RÜSSMANN§.

Let α, 0 < α < 1 , be irrational and let qn of the n-th convergent of the continued fraction of α . Then

BRYUNO-RÜSSMANN

be the denominator

THEOREM

oo ,

i=i η

and so to

n-1 4ω„-ι \cn\ < Σ \cj\ \cn-j\ 3=1

{n > 2 ,

c x = 1),

which reminds one of the equivalences stated without proof in the preceding chapter with regard to the general solvability of SCHRODER'S functional equation and its solvability in the case of f(z) — Xz + z2 . 99. We recursively define two sequences (σ η ) and ( ο·ηιση' Πι +rij + l=n

and δη = ωη1

max 8ni6n> ni+n1 + l=n

Then a simple induction proof gives Cn+1 < 6ηση. that, by definition, oo h(z) = ^2σηζη n=0

To estimate ση we remark

is a solution of the equation h(z)-l

= z(h(z))2,

h( 0) = 1,

and hence h(z) =

—

,

VI =

1,

(4)

Chapter

4

The

Existence

of Rotation

99

Domains

which implies that h has radius of convergence^1 \ . To complete the proof of the theorem we need an estimate of type Sn < r~n oo implying that the power series ψ(ζ) = ζ + Σ CkZk has radius of convergence at k=2

least r/4. 100. This is the most involved part of the proof. We need some more details from the theory of continued fractions, which can be found in NIVEN'S monograph". If qk denotes the denominator of the fc-th convergent [αχ, α2,..., α^] of the continued fraction α = [αϊ, θ2, α^ . . . ] , then (*)

ωη

> uqk

-

for

Qk

η


2