Quasiconformal Surgery in Holomorphic Dynamics

C A M B R I D G E S T U D I E S I N A DVA N C E D M AT H E M AT I C S 1 4 1 Editorial Board ´ S , W. F U LTO N , A . K ...

Author: Bodil Branner | Núria Fagella | 2014

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C A M B R I D G E S T U D I E S I N A DVA N C E D M AT H E M AT I C S 1 4 1 Editorial Board ´ S , W. F U LTO N , A . K ATO K , F. K I RWA N , B. BOLLOBA P. S A R NA K , B . S I M O N , B . TOTA RO

QUASICONFORMAL SURGERY IN HOLOMORPHIC DYNAMICS Since its introduction in the early 1980s, quasiconformal surgery has become a major tool in the development of the theory of holomorphic dynamics, and it is essential background knowledge for any researcher in the field. In this comprehensive introduction the authors begin with the foundations and a general description of surgery techniques before turning their attention to a wide variety of applications. They demonstrate the different types of surgeries that lie behind many important results in holomorphic dynamics, dealing in particular with Julia sets and the Mandelbrot set. Two of these surgeries go beyond the classical realm of quasiconformal surgery and use trans-quasiconformal surgery. Another deals with holomorphic correspondences, a natural generalization of holomorphic maps. The book is ideal for graduate students and researchers requiring a self-contained text including a variety of applications. It particularly emphasizes the geometrical ideas behind the proofs, with many helpful illustrations seldom found in the literature. Bodil Branner is Professor Emerita at the Technical University of Denmark, Lyngby. Her research interests include holomorphic dynamics and complex analysis. She has published in several renowned international journals and given numerous invited talks at conferences, workshops and symposia. Branner has served as Vice-President of the European Mathematical Society, as President of Dansk Matematisk Forening (DMF), and she was one of the founders of European Women in Mathematics. She is an honorary member of DMF, and a Fellow of the AMS. ´ Nuria Fagella is currently Associate Professor at Universitat de Barcelona. Her research is in the area of holomorphic dynamics with an emphasis on the iteration of transcendental functions. She publishes in renowned international journals and with a diverse range of collaborators worldwide. Fagella has been invited to deliver talks and short courses at numerous international conferences and workshops, and has been an organiser of several such events.

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS Editorial Board: B. Bollobás, W. Fulton, A. Katok, F. Kirwan, P. Sarnak, B. Simon, B. Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit: www.cambridge.org/mathematics. Already published 104 A. Ambrosetti & A. Malchiodi Nonlinear analysis and semilinear elliptic problems 105 T. Tao & V. H. Vu Additive combinatorics 106 E. B. Davies Linear operators and their spectra 107 K. Kodaira Complex analysis 108 T. Ceccherini-Silberstein, F. Scarabotti & F. Tolli Harmonic analysis on finite groups 109 H. Geiges An introduction to contact topology 110 J. Faraut Analysis on Lie groups: An introduction 111 E. Park Complex topological K-theory 112 D. W. Stroock Partial differential equations for probabilists 113 A. Kirillov, Jr An introduction to Lie groups and Lie algebras 114 F. Gesztesy et al. Soliton equations and their algebro-geometric solutions, II 115 E. de Faria & W. de Melo Mathematical tools for one-dimensional dynamics 116 D. Applebaum Lévy processes and stochastic calculus (2nd Edition) 117 T. Szamuely Galois groups and fundamental groups 118 G. W. Anderson, A. Guionnet & O. Zeitouni An introduction to random matrices 119 C. Perez-Garcia & W. H. Schikhof Locally convex spaces over non-Archimedean valued fields 120 P. K. Friz & N. B. Victoir Multidimensional stochastic processes as rough paths 121 T. Ceccherini-Silberstein, F. Scarabotti & F. Tolli Representation theory of the symmetric groups 122 S. Kalikow & R. McCutcheon An outline of ergodic theory 123 G. F. Lawler & V. Limic Random walk: A modern introduction 124 K. Lux & H. Pahlings Representations of groups 125 K. S. Kedlaya p-adic differential equations 126 R. Beals & R. Wong Special functions 127 E. de Faria & W. de Melo Mathematical aspects of quantum field theory 128 A. Terras Zeta functions of graphs 129 D. Goldfeld & J. Hundley Automorphic representations and L-functions for the general linear group, I 130 D. Goldfeld & J. Hundley Automorphic representations and L-functions for the general linear group, II 131 D. A. Craven The theory of fusion systems 132 J. Väa¨ nänen Models and games 133 G. Malle & D. Testerman Linear algebraic groups and finite groups of Lie type 134 P. Li Geometric analysis 135 F. Maggi Sets of finite perimeter and geometric variational problems 136 M. Brodmann & R. Y. Sharp Local cohomology (2nd Edition) 137 C. Muscalu & W. Schlag Classical and multilinear harmonic analysis, I 138 C. Muscalu & W. Schlag Classical and multilinear harmonic analysis, II 139 B. Helffer Spectral theory and its applications 140 R. Pemantle & M. C. Wilson Analytic combinatorics in several variables 141 B. Branner & N. Fagella Quasiconformal surgery in holomorphic dynamics 142 R. M. Dudley Uniform central limit theorems (2nd Edition) 143 T. Leinster Basic category theory

Quasiconformal Surgery in Holomorphic Dynamics BODIL BRANNER Technical University of Denmark, Lyngby

´ R I A FA G E L L A NU Universitat de Barcelona

With contributions by Xavier Buff, Shaun Bullett, Adam L. Epstein, Peter Ha¨ıssinsky, Christian Henriksen, Carsten L. Petersen, Kevin M. Pilgrim, Tan Lei and Michael Yampolsky

University Printing House, Cambridge CB2 8BS, United Kingdom Published in the United States of America by Cambridge University Press, New York Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107042919 © Bodil Branner and Núria Fagella 2014 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 Printed and bound in Spain by Grafos SA, Arte Sobre papel A catalogue record for this publication is available from the British Library Library of Congress Cataloging-in-Publication Data Branner, Bodil, author. Quasiconformal surgery in holomorphic dynamics / Bodil Branner and Núria Fagella ; with contributions by Xavier Buff & Christian Henriksen, Shaun Bullett, Adam L. Epstein & Michael Yampolsky, Peter Ha¨ıssinsky, Carsten L. Petersen and Kevin M. Pilgrim & Tan Lei. pages cm ISBN 978-1-107-04291-9 (hardback) 1. Holomorphic mappings. 2. Differentiable dynamical systems. 3. Kleinian groups. I. Fagella, Núria, author. II. Title. QA614.8.B725 2014 5151 .98–dc23 2013017278 ISBN 978-1-107-04291-9 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Dedicated to the memory of Adrien Douady

Contents

List of contributors Preface Acknowledgements List of symbols

page ix xi xiii xv

Introduction 1

Quasiconformal geometry 1.1 The linear case: Beltrami coefficients and ellipses 1.2 Almost complex structures and pullbacks 1.3 Quasiconformal mappings 1.4 The Integrability Theorem 1.5 An elementary example 1.6 Quasiregular mappings 1.7 Application to holomorphic dynamics

2

Boundary behaviour of quasiconformal maps: extensions and interpolations 2.1 Preliminaries: quasisymmetric maps and quasicircles 2.2 Extensions of mappings from their domains to their boundaries 2.3 Extensions of boundary maps

3

Preliminaries on dynamical systems and actions of Kleinian groups 3.1 Conjugacies and equivalences 3.2 Circle homeomorphisms and rotation numbers 3.3 Holomorphic dynamics: the phase space 3.4 Families of holomorphic dynamics: parameter spaces 3.5 Actions of Kleinian groups and the Sullivan dictionary

1 7 7 14 20 39 49 55 60 64 65 69 77 92 94 97 105 126 133 vii

viii

Contents

4

Introduction to surgery and first occurrences 4.1 Changing the multiplier of an attracting cycle 4.2 Changing superattracting cycles to attracting ones 4.3 No wandering domains for rational maps

147 151 162 169

5

General principles of surgery 5.1 Shishikura principles 5.2 Sullivan’s Straightening Theorem 5.3 Non-rational maps

179 180 184 186

6

Soft surgeries 6.1 Deformation of rotation rings

188

Xavier Buff and Christian Henriksen

189 207

6.2 7

Branner–Hubbard motion

Cut and paste surgeries 7.1 Polynomial-like mappings and the Straightening Theorem 7.2 Gluing Siegel discs along invariant curves 7.3 Turning Siegel discs into Herman rings 7.4 Simultaneous uniformization of Blaschke products 7.5 Gluing along continua in the Julia set 7.6 Disc-annulus surgery on rational maps Kevin M. Pilgrim and Tan Lei

7.7 7.8

Perturbation and counting of non-repelling cycles Mating a group with a polynomial Shaun Bullett

8

Cut and paste surgeries with sectors 8.1 Preliminaries: sectors and opening modulus 8.2 Creating new critical points 8.3 Embedding limbs of M into other limbs 8.4 Intertwining surgery Adam Epstein and Michael Yampolsky

9

Trans-quasiconformal surgery 9.1 David maps and David–Beltrami differentials 9.2 Siegel discs via trans-quasiconformal surgery Carsten Lunde Petersen

9.3

218 219 224 235 244 248 267 282 291 307 308 320 337 343 364 365 370

Turning hyperbolics into parabolics Peter Ha¨ıssinsky

References Index

385 400 408

Contributors

Bodil Branner Technical University of Denmark Xavier Buff Université Paul Sabatier, Toulouse Shaun Bullett Queen Mary, University of London Adam Epstein University of Warwick ´ Nuria Fagella Universitat de Barcelona Peter Ha¨ıssinsky Université Paul Sabatier, Toulouse Christian Henriksen Technical University of Denmark Carsten Lunde Petersen Roskilde Universitet Kevin M. Pilgrim Indiana University

ix

x Tan Lei Université d’Angers Michael Yampolsky University of Toronto

Contributors

Preface

The firm intention of writing this book was born in the fall of 2003 during the Ecole Thématique du CNRS Chirurgie holomorphe. This workshop was part of a trimester organized by Adrien Douady at the Institut Henri Poincaré in Paris. Douady was one of the fathers of the theory of holomorphic dynamics and of many of the surgery constructions that the workshop addressed. He used surgery as a tool in a number of ways, but especially to obtain a better understanding of different structures in parameter space. As he said: ‘plough in dynamical spaces and harvest in parameter space’. Douady’s creative and geometric point of view inspired many to explore holomorphic dynamics. Furthermore, he encouraged generous collaboration and believed strongly in the value of sharing ideas. He gathered a large mathematical family around him, and the success of the workshop is a tribute to his influence. Many of those who had originally developed holomorphic surgery presented lectures. It became clear that the content of these wonderful sessions ought to be the core of a book about surgery. We ourselves felt strongly that the book should be more than a collection of papers, though: our goal became to enlist the help of the speakers in creating a comprehensive study of quasiconformal surgery. We are delighted that our wishes have come true in the form of this book, which puts together the foundations of surgery and many of its applications, with, as we had hoped, contributions by a number of the workshop participants who themselves had played such an important part in developing the field. We are grateful for their support. They have our sincere and enthusiastic thanks. Writing and collecting the material and then unifying it into a book was not an easy task for us. Douady’s excitement about the project and his constant encouragement were very important. They kept us going until this very moment. It is to him that we dedicate every word written here, wishing he could have seen the final result. xi

Acknowledgements

This book has taken a long time to write, and we have received a lot of assistance along the way. It is our pleasure to thank many friends and colleagues for their generous help that made it possible. First and foremost we are extremely grateful to Xavier Buff, Shaun Bullett, Adam Epstein, Peter Ha¨ıssinsky, Christian Henriksen, Carsten Lunde Petersen, Kevin Pilgrim, Tan Lei and Michael Yampolsky for agreeing to participate in the project, making the book so much better and special with their contributions. Among them we owe special gratitude to Buff, Ha¨ıssinsky and Henriksen whose valuable suggestions helped to improve other parts of the book. There are many others who should also be mentioned. In particular, the participants in the Quasiconformal Surgery course at the Complex Dynamics Seminar at Universitat de Barcelona, Antonio Garijo, Xavier Jarque, Helena Mihaljević-Brandt, Jörn Peter and Jordi Taixés inspired us with questions and suggestions and read parts of early drafts of the book. There are also those who made valuable comments to different parts of later drafts. These include Anna Miriam Benini, Jordi Canela, Mat´ıas Carrasco, Jonguk Yang and, very specially, Jonathan Brezin, Albert Clop, Ernest Fontich, Linda Keen, Curtis McMullen and Caroline Series. We heartfully thank all of them, and others who helped in various ways, for their effort. Of course, we take full responsibility for any errors that remain. Furthermore, we wish to thank our editor, Roger Astley, for his kindness, enthusiasm and dedication during the whole process. On the technical side we are grateful to Christian Mannes for creating It, the computer program with which many of the illustrations were made. Our thanks also go to the Institut for Matematik og Computer Science at Danmarks Tekniske Universitet, the Departament de Matemàtica Aplicada i Anàlisi at Universitat de Barcelona, the IMUB (Institut de Matemàtica de la UB) and the CRM (Centre de Recerca Matemàtica) for their support. xiii

xiv

Acknowledgements

And finally, but most especially, our sincere gratitude goes to our families, for their encouragement, support and patience throughout this long, but for us so very exciting, project. This book would not have been possible without financial support from different sources. We first and most wish to thank the Marie Curie project CODY (MRTN-CT-2006-035651), which included this book as one of its mathematical training goals. Most of the travel expenses over the last five years were covered by CODY, together with the grant 272-07-0321 from the Danish Research Council for Nature and Universe and by the grants MTM2008-01486 and MTM2006-05849 from the spanish Ministry of Science. Most recently, we were also partially supported by the grant MTM2011-26995-C02-02 from the same source and the catalan grant 2009SGR-792. Bodil Branner and Núria Fagella Kongens Lyngby and Barcelona

Symbols

„ aff ine

„ hyb

affine conjugate hybrid equivalent

„

quasiconformally conjugate

„

topologically conjugate

qc

top

» 1A Af pαq A˝f pαq Af p8q Ar Ar,R B C C˚ p C Cf C r pU q, r C 8 pU q Ccr pU q Critpf q Bz , Bz B, B

conformal equivalence The characteristic function takes the value 1 on A and 0 on CzA The basin of attraction of an attracting p-cycle α “ tα0 , . . . , αp´1 u of f The immediate basin of attraction of a cycle α as above The basin of attraction of infinity of a polynomial f Round open annulus tr ă |z| ă 1u Round open annulus tr ă |z| ă Ru The Bryuno numbers or a Bers’ slice The complex plane The punctured complex plane Czt0u The extended complex plane C Y t8u Set of critical points of f ě 1 The space of r times differentiable maps on U , whose nth derivatives are continuous for all 1 ď n ď r The spaces of functions which belong to C r pU q for all r ě 1 The space of functions in C r pU q with compact support, for 1 ď rď8 Set of finite critical points of f , i.e. in C Ordinary partial derivatives with respect to z and z Partial derivatives in the sense of distributions with respect to z and z

xv

xvi D D˚ Dr Dr pz0 q D ` pU, V q

D0` pU, V q dCp pz, wq Ent Ent˚ extpγ q Ff Fc fn GLp2, Cq H H Hr H Im z intpγ q intpXq Jf Jc Kf Kc Lp{q M Mer

Symbols A Siegel disc or a linearizing domain around an attracting periodic point. The open unit disc t|z| ă 1u in C The punctured unit disc Dzt0u The open disc t|z| ă ru The open disc t|z ´ z0 | ă ru The set of orientation preserving continuous functions f : U Ñ V which are differentiable almost everywhere and whose differential Du f is non-singular almost everywhere and depends measurably on u P U The set of functions in D ` pU, V q which are absolutely continuous with resepct to the Lebesgue measure The spherical distance between two points in the Riemann sphere The set of entire transcendental maps The set of holomorphic transcendental self-maps of C˚ The domain to the right of γ , an oriented Jordan curve The Fatou set of f The Fatou set of Qc f ˝n :“ f ˝ ¨ ¨ ¨ ˝ f , the map f ˇcomposed by itself n times * "ˆ ˙ a b ˇˇ a, b, c, d P C General linear group cd ˇ The upper half plane tIm z ą 0u The left half plane tRe z ă 0u The right half plane tRe z ą 0u The Herman numbers or, in Section Section 8.4, a hyperbolic component The imaginary part of z The domain to the left of γ , an oriented Jordan curve Interior of the set X The Julia set of f The Julia set of Qc The filled Julia set of a polynomial or a polynomial-like map f The filled Julia set of Qc The p{q-limb of the Mandelbrot set The Mandelbrot set The set of transcendental meromorphic maps with at least one pole which is not omitted

Symbols Mer8 mod N OpXq Pf Pol Pold Q Qc R R R˚ p R Rat Ratd Re z Rθ σ pz, wq S1 S1r Singpf ´1 q T Vf Z

xvii

The set of transcendental maps which are meromorphic outside a compact countable set of singularities modulus The natural numbers t1, 2, . . .u The orbit of X, where X is a point or a set The postsingular set or the postcritical set The set of polynomials of degree at least two The set of polynomials of degree d ě 2 The rational numbers The quadratic polynomial Qc pzq “ z2 ` c A Riemann map The real line The punctured real line Rzt0u The extended real line R Y 8 The set of rational maps of degree at least two The set of rational maps of degree d ě 2 The real part of z The rigid rotation by θ P R, represented either by z ÞÑ e2π i θ z, where z P S1 or x ÞÑ x ` θ pmod 1q where x P R The chordal distance between two points in the Riemann sphere The unit circle t|z| “ 1u The circle t|z| “ ru Set of singularities of an inverse map The quotient space R{Z Set of critical values of f The integers t. . . , ´2, ´1, 0, 1, 2, . . .u

Introduction

The theory of one-dimensional complex dynamical systems, understood as the global study of iteration of holomorphic mappings, has its roots in the early twentieth century with the work of Pierre Fatou and Gaston Julia. Local studies had been successfully attempted earlier, but it was Fatou and Julia’s seminal work, inspired by Paul Montel’s notion of normal families of mappings (then relatively new), that set the basis of what is known today as holomorphic dynamics. Both Fatou and Julia studied extensively the basic partition of the dynamical space into the two disjoint, completely invariant subsets, the Fatou set, which is the open set where tame dynamics occur – the set where Montel’s normality appears – and its complement, the Julia set, which is the set of initial values whose orbits are chaotic. Their greatest achievement, arguably, is their detailed description of the geometry and the dynamics of the connected components of the Fatou set, called the ‘Fatou components’. Their Classification Theorem asserts that every periodic Fatou component of a holomorphic map of the Riemann sphere (a rational map) is either (i) a component of an immediate basin of attraction of some attracting or parabolic cycle or (ii) a rotation domain conformally equivalent to a disc or an annulus. Their work also left many interesting open questions, such as Fatou’s No Wandering Domains Conjecture, which states that all Fatou components are eventually periodic, and waited some 60 years for its resolution. Some of their other questions remain open today, but despite their compelling interest, the problems were largely ignored until the late 1970s. The other side of our story, the theory of quasiconformal functions of the plane, began at roughly the same time with the work of Herbert Grötzsch. As the name suggests, quasiconformality is a weakening of the notion of conformality. It is best understood as a geometric condition: conformal maps 1

2

Introduction

preserve angles, and quasiconformal maps distort angles, but only in a bounded fashion. Another important difference is that they do not need to be differentiable everywhere, but only almost everywhere. Even so, many important theorems about conformal mappings, minimally recast, remain true for quasiconformal maps. Grötzsch’s work was soon taken in two different directions. Oswald Teichmüller showed that there is a deep connection between quasiconformality and the function theory of Riemann surfaces, and Charles B. Morrey investigated its relation to the solution of PDEs. Beginning in the 1950s, the geometric side of the theory was developed by Albert Pfluger and Lars Ahlfors. This led to a unification of all of the earlier work into a single, general theory. At the same time, Morrey and Ahlfors, together with Lipman Bers and Bogdan V. Bojarski formulated the celebrated Measurable Riemann Mapping Theorem as it is known today – in this book, it is referred to somewhat more concisely as the Integrability Theorem. As we shall see, it is the essential tool for quasiconformal surgery. Around 1980, two remarkable developments added to this account. On the one hand, computer graphics made it possible to draw pictures of the beautiful phenomena and fractal structures exhibited by the holomorphic dynamics of even very simple maps. This clearly awoke an interest in the subject. On the other hand, and at a much deeper level, in 1981 Dennis Sullivan realized there was a strong connection between holomorphic iteration and the actions of Kleinian groups, introducing what has become known as Sullivan’s dictionary between these two subjects. Inspired by Henri Poincaré’s original (1883) perturbations of Fuchsian groups into quasi-Fuchsian groups, he injected the modern theory of quasiconformal mappings into complex iteration to solve Fatou’s No Wandering Domains Conjecture. With the same arguments, he also gave a new proof of Ahlfors’ Finiteness Theorem, a cornerstone in the theory of Kleinian groups. Sullivan’s technique is now referred to as soft quasiconformal surgery. It deals with quasiconformal deformations of a given map. However, what we call cut and paste quasiconformal surgery is more reminiscent of topological surgery, which is used to produce one manifold from another in some ‘controlled way’. In holomorphic iteration, quasiconformal surgery is used to obtain holomorphic maps with prescribed dynamics that arise from certain model maps which are locally quasiconformal. Often, these model maps are constructed by cutting and pasting different spaces and maps together, explaining the reference to topological surgery. It is worth noticing that, whereas the uniqueness of analytic continuation implies that holomorphic maps are rather

Introduction

3

rigid, quasiconformal maps can be pasted together very flexibly without losing quasiconformality. Sullivan’s first crucial work was merely the beginning of the uses to which quasiconformal surgery could be put in holomorphic dynamics. Quasiconformal surgery has been at the heart of many important advances and is now part of the essential background knowledge of any researcher in the field. All of this brings us to the raison d’etre for this book. While the theory of quasiconformal maps is a mature subject for which an excellent literature exists, a comprehensive treatment of its application to dynamical systems has, until now, only been possible by going back to the original papers. Books about quasiconformal maps, like [Ah5, LV, Le] and [AIM], naturally focus on the function theory, so the surgery techniques are of mainly anecdotal interest. Similarly, general texts on holomorphic dynamics, like [Mi1, Bea, CG, St] and [MNTU], are not the right place to discuss quasiconformal surgery, because the prerequisite theory of quasiconformality would sit very heavily in such a general work. Our goal is to present a text where one can learn from the start about this beautiful, highly geometric and powerful technique and how to apply it to holomorphic dynamics. This book is therefore neither a book about holomorphic dynamics nor about quasiconformal mappings, although it contains introductions to both subjects, together with a brief introduction to Kleinian groups, in order to understand the parallels that often come up. These introductions, however, do not present proofs that can easily be found in other standard texts. Both authors have a stronger background in dynamics than in analysis and this surely introduces a bias in the way these introductions are presented. While no previous knowledge about quasiconformal maps or Kleinian groups is assumed, some familiarity with dynamical systems and especially holomorphic dynamics in one complex variable is useful, in particular when understanding the purpose of the different surgery constructions. While the foundations of the theory and the general description of surgery techniques occupy a good half of the book, the other half is completely dedicated to a variety of applications. These applications are more or less independent of each other, and so each can be read on its own. Our interest is precisely to present the different types of surgery techniques that are required for attacking the most important results in holomorphic dynamics. The applications are grouped by similarities between the constructions they require. Two of them go beyond the classical realm of quasiconformal surgery and use trans-quasiconformal surgery. Even further, another one deals with holomorphic correspondences, a natural generalization of holomorphic maps.

4

Introduction

Some of the sections on applications are written by the authors of the original papers, whom we once again thank. We should also comment on what this book does not cover. Although quasiconformal maps and surgery itself are intimately related to Teichmüller Theory, this deep area of mathematics is not treated in this book. To learn about it, we strongly recommend the recent volume by John H. Hubbard [Hu2]. In connection to the above, there is also no reference to quadratic differentials, although they play an important role in the theory of holomorphic dynamics. Actions of Kleinian groups only appear in the book in a sparse way. We give a brief introduction to the subject so that references to the parallelism between deformations of Kleinian groups and surgery in holomorphic dynamics, expressed by Sullivan’s dictionary, can be understood.

About the book structure Chapter 1, Quasiconformal geometry, contains the basic definitions and results about quasiconformal mappings used in surgery, and those necessary to make a coherent description. We give a geometrical and an analytical approach, give precise references to all important results, and prove those (even if simple), which we could not find proven in the literature. Nevertheless they may be useful for the reader of this text. The same comments apply to Chapter 2, Boundary behaviour of quasiconformal maps: extensions and interpolations. We study how to extend maps to the boundary of their domains, depending on the regularity of the elements involved. Conversely, we see how to extend boundary maps to their neighbouring domains, depending also on the regularity of the different ingredients. These extensions and interpolations are technical results, which are used repeatedly in each and every surgery. Some of the results are celebrated theorems while others are simple exercises; but all are part of the community folklore. We have made it a point to collect them in this chapter, and to prove or indicate proofs of those that can be shown with elementary methods. Chapter 3, Preliminaries on dynamical systems and actions of Kleinian groups, contains the background in dynamics necessary for understanding the sections that follow and especially their motivation. We include general basic background for discrete dynamical systems, circle maps, holomorphic dynamics, and families of such, as well as dynamics of Kleinian groups. As in the previous chapters, there are basically no proofs, but all results are well referenced.

Introduction

5

In Chapter 4, Introduction to surgery and first occurrences, we present the earliest applications of surgery to holomorphic dynamics, although not necessarily the simplest. Inspired by quasiconformal deformations of Kleinian groups, the surgeries lead to conformal parametrizations of hyperbolic components of the Mandelbrot set by D, due to Sullivan, Douady and Hubbard, and to the No Wandering Domains Theorem for rational maps, due to Sullivan. We have chosen to keep a historical flavour, so the surgeries appear in a form fairly close to their original presentation. When applying quasiconformal surgery, there are several paths one may take to accomplish the same result. More precisely, Chapter 5, General principles of surgery, is dedicated to establishing three criteria (due to Mitsuhiro Shishikura and Sullivan) giving conditions under which a surgery can be completed to result in a holomorphic map. The First and Second Shishikura Principles are actually corollaries of the third principle, called Sullivan’s Straightening Theorem. We present all three since their proofs provide different insights. Sullivan’s Straightening Theorem gives a necessary and sufficient condition to decide whether a surgery can be completed. However, these criteria are rarely used in the applications in the book, since certain informations can only be extracted from the specific details of the proof ‘from scratch’. Needless to say, the criteria may be useful in many other instances. Chapters 6 to 9 are entirely dedicated to applications. Sections are pairwise independent, although they all use the common terminology and background of the first part of the book. Some of the applications were presented during the surgery workshop at the Institute of Henri Poincaré in Paris in the fall of 2003 as part of the trimester organized by Adrien Douady. Other applications have been added for completeness. We have tried to show as many different constructions as possible. However, the list is by no means exhaustive. Chapter 6, Soft surgeries, consists of applications where only the complex structure is deformed, so that it is invariant under the given map. The resulting maps are therefore quasiconformally conjugate to the original one. The chapter contains a contribution by Xavier Buff and Christian Henriksen. Chapter 7, Cut and paste surgery, is the longest chapter and contains many groups of applications. The common feature in all of them is that maps and complex structures are both changed. It contains contributions by Kevin M. Pilgrim and Tan Lei, and Shaun Bullett, who present a special surgery application of mating a Kleinian group with a quadratic polynomial. Chapter 8, Cut and paste surgeries with sectors, contains applications where the deformation of the maps and the complex structures are concentrated in sectors. The chapter contains a contribution by Adam Epstein and Michael Yampolsky.

6

Introduction

Chapter 9, Trans-quasiconformal surgery, is dedicated to the contributions by Carsten L. Petersen and Peter Ha¨ıssinsky. The surgeries require maps, called David homeomorphisms, that are not quite quasiconformal, but are ‘close enough’ in a sense made precise there. Many new applications have appeared using this technique. The two we have included are among the earliest and a good introduction to the more recent work.

On how to read this book Readers familiar with quasiconformal geometry and analysis may start in Chapter 3, Preliminaries on dynamical systems and actions of Kleinian groups, and continue from there. Those familiar with dynamics, holomorphic dynamics in particular, may start in Chapter 1, Quasiconformal geometry, read Chapter 2, Extensions and interpolations, only diagonally to see what type of results it contains, and then jump directly to Chapter 4, Introduction to surgery and first occurrences. Chapter 2 can be used as a reference chapter throughout the book. In both cases, if the first applications in Chapter 4 are too difficult to start with, we advise to leave them for later and start with those in Chapter 7, Cut and paste surgery, maybe after having read the introduction to surgery given in Chapter 4. Chapter 5, General principles of surgery, can be read at any point, since the principles are not used often in the constructions, and they may be too abstract for someone not well acquainted with a number of examples. We hope that experts in both quasiconformal mappings and holomorphic dynamics, and even in quasiconformal surgery, may still find some of the applications interesting and in any case a practical collection to have at hand as a reference. It is clear that such readers may skip Chapters 1 to 5. The book can be used, and has been used, several times already, for a graduate course in quasiconformal surgery. If so, it is not recommended to cover the book in a sequential way but to follow the recommendations given above. The lecturer may selectively choose some sections in Chapters 6 to 9 and keep the remaining ones as projects for the students who may be asked to read the original papers, fill in details and do the suggested exercises.

1 Quasiconformal geometry

Since its introduction in the early 1980s quasiconformal surgery has become a major tool in the development of the theory of holomorphic dynamics. The goal of this chapter is to collect the basic definitions and results about quasiconformal mappings used in surgery, and those necessary to make a coherent description. We give a geometrical and analytical approach and give precise references to important results in the vast literature about quasiconformal mappings. In general we only prove those we could not find proven elsewhere, even if simple. However, since the Integrability Theorems (Theorems 1.27 and 1.28), also called the Measurable Riemann Mapping Theorems, are the cornerstone behind every surgery construction we sketch a proof of those. Holomorphic maps are very rigid, due to the property of analytic continuation. For this reason it is not possible to paste different holomorphic maps together along a curve to form a new holomorphic map. However, quasiconformal mappings do have this kind of flexibility and can be pasted together to form new quasiconformal mappings. It is this flexibility that produces the basis for surgery constructions where we change mappings and sometimes also the underlying spaces. When the construction is successful the final goal is to end with a holomorphic map, obtained via the Integrability Theorem.

1.1 The linear case: Beltrami coefficients and ellipses Let CR denote the complex plane, viewed as the two-dimensional oriented Euclidean R-vector space with the orthonormal positively oriented standard basis t1, i u. In CR we shall use as coordinates either px, yq or pz, zq where z “ x ` iy and z “ x ´ i y. 7

8

Quasiconformal geometry

Any R-linear map L : CR Ñ CR can be written, using the coordinates pz, zq, in the form Lpzq “ az ` bz,

with a, b, z P C.

The unit square, spanned by 1 and i , is mapped onto the parallelogram spanned by a ` b and ai ´ bi (see Figure 1.1). pa ´ b qi

i

L a`b

1 Figure 1.1 The unit square is mapped to the parallelogram spanned by ta ` b, pa ´ bqi u.

The absolute value of the determinant of L is the area of the parallelogram, i.e. detpLq “ |a|2 ´ |b|2 . We shall restrict to R-linear maps that are invertible and orientation preserving, i.e. with |a| ą |b|. We define the Beltrami coefficient of L to be μpLq “ ab , and – for reasons which will become clear below – we let θ P R{pπ Zq denote half the argument of μpLq, i.e. ˇ ˇ ˇbˇ μpLq :“ ˇˇ ˇˇ ei 2θ . a Note that μpLq P D when L is orientation preserving, and that L is holomorphic if and only if b “ 0, which occurs if and only if μpLq “ 0. Let EpLq denote the inverse image by L of the unit circle. Then EpLq is an ellipse, and in particular a circle if μpLq “ 0. In order to determine the ellipse EpLq, we set a “ |a| ei α , where α P R{p2π Zq, μ “ μpLq and rewrite L as ` ˘ Lpzq “ ei α |a| z ` |μ| ei 2θ z . ˘ ` Hence, L is the R-linear map Spzq “ |a| z ` |μ| ei 2θ z post-composed with the rotation Rpzq “ ei α z. We have split L into the composition of a selfadjoint linear transformation followed by an orthogonal transformation. (It is easy to check that the 2 ˆ 2 matrix of S in the basis t1, i u is symmetric, and S therefore is self-adjoint.) It follows that S has two real eigenvalues and, if b ‰ 0, that their corresponding eigenvector directions are orthogonal (see Figure 1.2).

1.1 The linear case: Beltrami coefficients and ellipses

9

i pθ` π 2q e |a|p1´|μ|q ei θ |a|p1`|μ|q

S

R

θ

L Figure 1.2 The ellipse EpSq “ EpLq.

It is easy to check that ei θ and ei pθ `π {2q are eigenvectors of S corresponding to the eigenvalues |a|p1 ` |μ|q and |a|p1 ´ |μ|q respectively. It follows that 1 along the direction ei pθ `π {2q EpSq is the ellipse with half major axis |a |p1´| μ|q 1 and half minor axis |a |p1`| along the orthogonal direction ei θ . The ellipse μ|q EpLq equals EpSq, since the unit circle is preserved by the rotation Rpzq “ ei α z. We define the dilation KpLq of L as the ratio of the major axis to the minor axis:

KpLq :“

|a| ` |b| 1 ` |μ| “ , 1 ´ |μ| |a| ´ |b|

and the complex dilatation of L as the Beltrami coefficient μpLq. The dilatation KpLq determines the shape of the ellipse up to scaling, but not the position of its axes. The Beltrami coefficient determines the position and the shape up to scaling. Conversely, if we happen to start with an ellipse E, the Beltrami ´m i 2θ , where M and m are the half coefficient is determined by μpEq “ M M `m e major and half minor axes of E respectively, and θ is the argument of the direction of the minor axis of E chosen in r0, π q. The dilatation KpEq is defined in the natural way (see Figure 1.3).

m

θ

M

Figure 1.3 Given an ellipse E, we define μpEq and KpEq in terms of M, m, and θ. Observe that we choose the argument of the minor axis, θ, to belong to r0, π q.

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We shall denote by σ0 the standard conformal structure of CR , that is to consider CR as a C-vector space with the standard complex scalar multiplication. Any invertible R-linear map L can be used to define a new conformal structure σ pLq on the domain of L, that is a new operation making CR into a C-vector space, extending the R-vector space structure. This is done in the following way: we need to define what it means to ‘multiply’ elements of CR by complex scalars, which reduces (after imposing all the properties that must be satisfied) to define ‘multiplication’ by i . That is, we need to choose an R-linear map J , and define c ˚ z “ Re c z ` Im c J pzq for any c, z P C. It follows from imposing i 2 ˚ z “ i ˚ i ˚ z, that J pJ pzqq “ ´z. The structure induced by L is defined by choosing J “ L´1 ˝ I ˝ L, where I pzq “ i z in the standard way. We will end this linear discussion by considering how Beltrami coefficients and dilatations change under inversion and composition of linear maps. We start with inversion. Given a map L as above, it is easy to check that L´1 pwq “

|a|2

1 paw ´ bwq. ´ |b|2

It follows that μpL´1 q “ ´μpLqei p2 arg a q ,

(1.1)

and hence |μpL´1 q| “ |μpLq|, which implies that KpL´1 q “ KpLq.

(1.2)

Now suppose j P t1, 2u and we have two R-linear maps Lj pzq “ aj z ` bj z with dilatation Kj and Beltrami coefficient μj . The ellipse defined by the composition L1 ˝ L2 is the preimage under L2 of the ellipse defined by the map L1 . Observe that from linear algebra, we can assure that KpL1 ˝ L2 q ď KpL1 qKpL2 q, since the maximal possible stretch is the product of the two maximal stretches of each of the maps, while the corresponding holds for the minimal stretches. If we want to know the Beltrami coefficient for this new ellipse we compute the composition pL1 ˝ L2 qpzq “ pa1 a2 ` b1 b2 qz ` pa1 b2 ` b1 a2 qz,


11

hence μpL1 ˝ L2 q “

b2 ` μ1 a2 a2 ` μ1 b2

(1.3)

or, equivalently μpL1 ˝ L2 q “

μ2 ` μ1 e´2i arg a2 . 1 ` μ1 μ2 e´2i arg b2

(1.4)

We will use these expressions later on when we discuss more general maps. 1.1.1 Geometric interpretation We shall give a geometric interpretation of the conformal structure σ defined by an ellipse, or rather an infinite family of similar concentric ellipses. First suppose we have an R-linear map L as above and the ellipse EpLq. Given a diameter in the unit circle or in the ellipse EpLq, the conjugate diameter is defined as the one that is made up of the midpoints of the cords in the circle or in the ellipse parallel to the given diameter. On the circle, this corresponds to the orthogonal diameter. On a genuine ellipse the only conjugate diameters that are orthogonal to each other are the axes (see Figure 1.4). Alternatively, the conjugate diameter is the one parallel to the line tangent to the ellipse at the point of intersection with the given diameter. The map L sends conjugate diameters of the ellipse EpLq to conjugate diameters of the circle, since parallel lines are mapped to parallel lines and midpoints to midpoints. z

EpLq

Lpzq

L

J pz q

“ L´1 pi Lpzqq i Lpzq J J pzq “ ´z

Figure 1.4 E Ă CR so that LpEq is a circle in CR . A pair of conjugate diameters in E are mapped by L onto the corresponding pair of conjugate diameters in the circle LpEq.

If z P EpLq, then define J pzq to be the vector in EpLq so that z and J pzq are on conjugate diameters, turning from z in the positive direction (less than π ) to J pzq. Clearly J “ L´1 ˝ I ˝ L, where I pzq “ i z. Suppose an ellipse in CR is only defined up to scaling, meaning that we have an infinite family of similar concentric ellipses (see Figure 1.5). Such a

12


Figure 1.5 Some similar concentric ellipses. They are equal up to scaling by a real constant.

family determines a conformal structure σ in CR . For any z ‰ 0, choose the ellipse that contains z. Let J pzq be the point in the positive direction on that ellipse and on the diameter conjugate to the one containing z, positioned as above. The map J clearly satisfies J pJ zqq “ ´z and defines how to multiply a vector by i .

Exercises Section 1.1 1.1.1 Given a, b P C, consider the R-linear map Lpzq “ az ` bz. (i) Show that the matrix of L in the standard basis tp1, 0q, p0, 1qu in R2 , equivalent to t1, i u, is „ j Repa ` bq ´ Impa ´ bq L“ . Impa ` bq Repa ´ bq (ii) Justify that the determinant of the matrix is |a|2 ´ |b|2 . ` ˘ (iii) Consider in particular Spzq “ |a| z ` |μ| ei 2θ z , and confirm that the matrix of S in the standard basis is symmetric. 1.1.2 Given a, b P C with |a| ą |b|, consider the R-linear map Lpzq “ az ` bz. Recall that the norm }L} of L is defined by }L} :“ max |Lpzq|. zPS1

Note that LpS1 q “ EpL´1 q. Show that }L}2 . (1.5) det L We apply the content of this exercise in Exercise 1.3.3 in Section 1.3. 1.1.3 Let L : R2 Ñ R2 be an orientation preserving and invertible R-linear map, which in the standard basis tp1, 0q, p0, 1qu in R2 is given by the matrix „ j αβ L“ , γ δ KpLq “

and let E be the ellipse satisfying LpEq “ S1 .


13

(i) Show that points px, yq P E satisfy the equation pαx ` βyq2 ` pγ x ` δyq2 “ 1. Recall, that the symmetric matrix of the quadratic form in the lefthand side of this equation in the standard basis is equal to j „ 2 α ` γ 2 αβ ` γ δ T . Q“L L“ αβ ` γ δ β 2 ` δ 2 Let λj , j “ 1, 2, denote the eigenvalues of Q. Recall that there exists an orthonormal change of coordinates, so that the equation of E in the new coordinates px1 , y1 q takes the form λ1 x12 ` λ2 y12 “ 1. (ii) Show that both eigenvalues λj , j “ 1, 2, are positive, and label them so that λ1 ě λ2 ą 0. Express the half major and half minor axes M and m respectively in terms of λj . Then conclude that the distortion of L is given as d λ1 M “ KpLq “ . m λ2 1.1.4 (Complex dilatation of an orientation reversing linear map) Given a, b P C with |b| ą |a|, consider the R-linear orientation reversing map Apzq “ az ` bz. Let E be the ellipse satisfying ApEq “ S1 . Then μpEq and KpEq are well defined and we may define (compare with Figure 1.3) μpAq :“ μpEq and KpAq :“ KpEq. Show that μpAq “

á ¯ b

and

KpAq “

|b| ` |a| . |b| ´ |a|

(1.6)

Observe that if A is antiholomorphic (i.e. a “ 0) the ellipse is a circle. Hint: Consider the orientation preserving map Lpzq :“ Apzq and observe that ApEq “ S1 “ LpEq. 1.1.5 (Composition of two orientation reversing linear maps) For j “ 1, 2, let Aj pzq “ aj z ` bj z. Set μ1 “ μpA1 q, the Beltrami coefficient of the 1 1 ellipse E1 “ A´ 1 pS q, as defined in the preceding exercise. Observe that A1 ˝ A2 is orientation preserving and show that μpA1 ˝ A2 q “

μ1 b2 ` a2 μ1 a2 ` b2

.

(1.7)

This formula is the analogue of (1.3) for composition of two orientation preserving linear maps. Note that the Beltrami coefficient of

14

Quasiconformal geometry the complex conjugate ellipse E1 equals μ1 . The formula states the 1 Beltrami coefficient μpA1 ˝ A2 q of the ellipse E “ A´ 2 pE1 q “ pA1 ˝ ´1

1 A2 q´1 pS1 q and shows that A´ 2 pE1 q “ A2 pE1 q.

1.2 Almost complex structures and pullbacks Ť Let U Ă C, and let T U “ uPU Tu U be the tangent bundle over U , i.e. the collection of the tangent spaces over points u P U , each one viewed as a copy of CR . An almost complex structure on U is a measurable field of infinitesimal ellipses E Ă T U . By this we mean an ellipse Eu Ă Tu U defined up to scaling, for almost every point u P U , such that the map u ÞÑ μpuq from U to D is measurable (with respect to the Lebesgue measure), where μpuq denotes the Beltrami coefficient of Eu . Each infinitesimal ellipse defines a conformal structure σ puq on Tu U , as defined and explained in Section 1.1, making the tangent space Tu U into a C-linear vector space. We denote the almost complex structure by σ , and define the dilatation of σ as Kpσ q :“ ess sup Kpuq,

where

Kpuq :“

uPU

1 ` |μpuq| 1 ´ |μpuq|

denotes the dilatation of Epuq. Observe that Kpσ q P r1, 8s. Notice also that any measurable function μ : U Ñ D defines an almost complex structure in the above sense. Now let us see how one can obtain almost complex structures from maps satisfying certain conditions. Consider U, V Ă C and the class D ` pU, V q of continuous orientation preserving functions f from U onto V which are Rdifferentiable almost everywhere, and with a non-singular differential Du f : Tu U Ñ Tf puq V almost everywhere, depending measurably on u. Since we are working in tangent spaces we use the infinitesimal coordinates dz and dz. Then the differential can be written as Du f “ Bz f puqdz ` Bz f puqdz, where Bz f “

1 2

ˆ

Bf Bf í Bx By

˙ ,

Bz f “

1 2

ˆ

Bf Bf ì Bx By

˙ .

Applying the discussion of Section 1.1 to this map, we see that Du f defines an infinitesimal ellipse in Tu U with Beltrami coefficient equal to μf puq “

Bz f puq Bz f puq

(1.8)

1.2 Almost complex structures and pullbacks

15

or, equivalently, a new conformal structure on this tangent space. The dilatation can be written as Kf puq :“ KpDu f q “

1 ` |μf puq| . 1 ´ |μf puq|

Observe that the Cauchy–Riemann equation Bz f puq “ 0 is satisfied at u if and only if μf puq “ 0, or equivalently, if and only if the ellipse is a circle. If we do the same for all points u P U for which f is differentiable, we obtain a measurable field of infinitesimal ellipses, one in almost every tangent space or, as defined above, an almost complex structure σf on U , with Beltrami coefficient μf . We say that σf is the pullback of σ0 (the field of infinitesimal circles, known as the standard complex structure) by f , or equivalently that μf is the pullback of μ0 ” 0 by f (see Figure 1.6). We write for almost every uPU μf puq “ f ˚ μ0 puq

or

σf puq “ f ˚ σ0 puq.

The dilatation of this almost complex structure is then the essential supremum over U of Kf puq, which we denote by Kf . f

pU, σf q

pV , σ0 q

f puq u

Du f

Figure 1.6 The almost complex structure σf in U is the pullback of σ0 in V , under f . The fields of infinitesimal ellipses live in the tangent bundle. The infinitesimal ellipse at Tu U is mapped under Du f (if it exists) to the infinitesimal circle at Tf puq V .

Notice that since the differential does not need to vary continuously with respect to u, neither do the field of infinitesimal ellipses nor the Beltrami coefficient. We can only say that μf is a measurable function (it is a quotient of measurable functions) with respect to u, essentially bounded. The concept of pullback can be slightly generalized, since we may consider the pullback of any almost complex structure σ , not necessarily σ0 , under a map f satisfying an extra condition. We need to require the map f to be absolutely continuous with respect to the Lebesgue measure, that is, to require the preimage by f of any measure zero set to be of measure zero. We denote by D0` pU, V q the subclass of D ` pU, V q consisting of functions with this property. Let f P D0` pU, V q and let μ be the Beltrami coefficient in

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T V corresponding to an almost complex structure σ in V . Let Ev denote the infinitesimal ellipse defined in Tv V for almost every v P V . By the pullback by f of the measurable field of infinitesimal ellipses E, we mean the measurable field of infinitesimal ellipses E 1 with Eu1 “ pDu f q´1 pEf puq q, well defined for almost every u P U (see Figure 1.7). Indeed, Eu1 is defined for all u such that Ef puq is defined and Du f exists and is non-singular. The first property is satisfied on a set of full measure because f is absolutely continuous and so is the second property by hypothesis. f

pU, f ˚ μq

pV , μq

f puq u

Du f

Figure 1.7 The pullback of the Beltrami coefficient μ (in V ) under f is denoted by f ˚ μ (in U ).

When we write f

pU, μ1 q ÝÑ pV , μ2 q we mean that f : U Ñ V and f ˚ μ2 “ μ1 , in the sense explained above. If it happens that μ is given by a certain map g : V Ñ W in the class ` D pV , W q (i.e. μ “ μg ), then we will be looking at f ˚ μg “ f ˚ pg ˚ μ0 q “ pg ˝ f q˚ μ0 “ μg ˝f . From the linear discussion in Section 1.1 it follows that Kg ˝f ď Kf ¨ Kg , and we can write down formulas for Beltrami coefficients (compare with equations (1.3) and (1.4)): f ˚ μpuq “

Bz f puq ` μpf puqqBz f puq Bz f puq ` μpf puqqBz f puq

(1.9)

,

and also μg ˝f puq “

μf puq ` μg pf puqqeí 2 argpBz f puqq 1 ` μf puqμg pf puqqeí 2 argpBz f puqq

.

(1.10)


17

Furthermore, note that if f is holomorphic then (1.9) reduces to f ˚ μpuq “ μpf puqq

Bz f puq . Bz f puq

(1.11)

This formula is important for two reasons: first, we shall often extend an almost complex structure on V with Beltrami coefficient μ by pulling back with a holomorphic map f : U Ñ V to f ˚ μ on U . Note that if |μ| ă k on V then |f ˚ μ| ă k on U . Second, when we extend the previous notions to Riemann surfaces as in Section 1.3.7, everything is expressed in charts and we need to know how an expression is transferred from one chart to another, having a holomorphic overlap with the first. A special and very important case occurs when the pullback of an almost complex structure σ is again σ , as we will see later. Definition 1.1 (f-invariant almost complex structure) Let U be an open subset of C and f : U Ñ U a map in D0` pU, U q. Let σ be an almost complex structure in U with Beltrami coefficient μ. We say that μ (or σ ) is f -invariant if f ˚ μpuq “ μpuq for almost every u P U . We also write f ˚ σ “ σ . Equivalently, we say that f is holomorphic (in fact conformal) with respect to μ or σ . Almost complex structures that are f -invariant will be of crucial importance when we perform surgery. Consider the particular situation with mappings F P D0` pV , V q, G P D0` pU, U q, and f P D0` pU, V q, and suppose the mappings make the following diagram commutative: G

U ÝÝÝÝÑ § § fđ

U § §f đ

F

V ÝÝÝÝÑ V Note that for any F -invariant almost complex structure σ on V with Beltrami coefficient μ, the pullback structure f ˚ σ on U is G-invariant since G˚ pf ˚ μq “ pf ˝ Gq˚ μ “ pF ˝ f q˚ μ “ f ˚ pF ˚ μq “ f ˚ μ. We observe that if the inverse map f ´1 is also absolutely continuous then it makes sense to push forward almost complex structures, which means to pull them back under f ´1 from the tangent bundle T U to T V . We write f˚ “ pf ´1 q˚ . We also have Kf ´1 “ Kf .

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As a final remark, note that we even may allow f to be non-invertible as long as it locally belongs to the class D0` , except around a discrete set of ‘critical points’. Around critical points the pushforward operation is not well defined (because of multiple preimages under f ´1 ) but the pullback by f is. Critical points are not important since almost complex structures are allowed to be defined almost everywhere.

1.2.1 Almost complex structures and symmetries Let μ be the Beltrami coefficient of an almost complex structure defined in C. We would like to define what it means for μ to be symmetric with respect to the real axis or the unit circle. Maps that provide these symmetries are precisely cpzq “ z for R and τ pzq “ 1z for S1 . It might seem reasonable to say that a Beltrami coefficient which is symmetric with respect to, say the unit circle, must be τ -invariant. But the reflections c and τ are antiholomorphic and therefore orientation reversing, and so far we have not defined what it means to pull back almost complex structures under such maps. As a general setup, consider U, V Ă C and an orientation reversing map f P D0´ pU, V q, defined analogously to D0` pU, V q with the differential Du f : Tu U Ñ Tf puq . Let μ denote a Beltrami coefficient in V . Definition 1.2 (Pullback under orientation reversing maps) In the setup above, we define the pullback of μ under f as ˚

f f μ :“ f μ. In particular f f μ0 :“ μf

and

Kf :“ Kf .

Observe that f is orientation preserving and therefore the definitions make formal sense. We refer to Exercises 1.1.4 and 1.1.5 in Section 1.1 for a geometric motivation and to Exercises 1.2.1 to 1.2.4 for the deduction of these and other formulas. Returning to the initial problem, we obtain the following. Definition 1.3 (Symmetries) A Beltrami coefficient μ is symmetric with respect to R if and only if μpzq “ μpzq, or equivalently if cf μ “ μ, where cpzq “ z.


19

We say that μ is symmetric with respect to S1 if and only if μpzq “ μp1{zq

z4 “ μp1{zq e4i arg z . |z|4

or equivalently if τ f μ “ μ, where τ pzq “ 1z .

Exercises Section 1.2 1.2.1 (Complex dilatation of an orientation reversing map) Consider U, V Ă C and an orientation reversing map f P D ´ pU, V q. By applying Exercise 1.1.4 in Section 1.1 observe that Du f defines an ellipse Eu :“ pDu f q´1 pS1 q in Tu U for almost all u P U with Beltrami coefficient ˙ ˆ Bz f puq . μpEu q “ Bz f puq Recall that Bz f “ Bz f and Bz f “ Bz f , in other words, Du f “ Du f . Show that μpEu q “ μf puq as defined in equation (1.8) and KpEu q “ Kf puq. As a consequence, it makes sense to define the pullback of the standard form by an orientation reversing map as f f μ0 :“ μf

Kf :“ Kf .

and

1.2.2 (Pullback under orientation reversing maps) Let f, U and V be as in the exercise above and let μpvq be a Beltrami coefficient corresponding to a field of infinitesimal ellipses pEv qv PV . In analogy with formula (1.7) in Exercise 1.1.5 in Section 1.1, we define the pullback by f as ˚

f f μ :“ f μ. Deduce from (1.9) that f f μpuq “

Bz f puq ` μpf puqqBz f puq Bz f puq ` μpf puqqBz f puq

,

and if f is antiholomorphic, then f f μpuq “ μpf puqq

Bz f puq Bz f puq

.

If follows from Exercise 1.1.5 that pDu f q´1 pEf puq q “ pDu f q´1 pEf puq q. Hence, the pullback definition is justified by this geometrical interpretation.

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1.2.3 (Reflection with respect to R) Let cpzq “ z and μ a Beltrami coefficient defined in the upper half plane. Extend μ to the whole plane by defining $ ’ if Impzq ą 0, ’ &μpzq r pzq :“ cf μpzq if Impzq ă 0, μ ’ ’ %0 if Impzq “ 0. r pzq “ μpcpzqq for z in the lower half plane. Check explicitly that μ r pzq “ μ r pzq for all z P C. Check moreover that cf μ 1.2.4 (Reflection with respect to S1 ) Let τ pzq “ 1z and μ be a Beltrami coefficient defined in the unit disc. Extend μ to the whole plane by defining $ ’ if |z| ă 1, ’ &μpzq r pzq :“ τ f μpzq if |z| ą 1, μ ’ ’ %0 if |z| “ 1. 4

r pzq “ μpτ pzqq |zz|4 for all z P CzD. Check moreover Check explicitly μ r pzq “ μ r pzq for all z P C. that τ f μ 1.2.5 (Pullback of symmetric coefficients under symmetric maps) Let s denote either the reflection with respect to R (spzq “ z) or with respect to S1 (spzq “ 1{z). Consider two open sets U, V Ď C satisfying spU q “ U and spV q “ V . Let μ be a Beltrami coefficient defined on U symmetric with respect to s, i.e. satisfying s f μ “ μ, and suppose f P D0` pV , U q is a map which also satisfies f “ s ˝ f ˝ s. Show that r“μ r. r :“ f ˚ μ defined on V also satisfies s f μ the Beltrami coefficient μ 1.2.6 (Pullback under symmetric maps) Let s and U be as in the exercise above. Consider f P D0` pU, U q and let μ :“ μf “ f ˚ μ0 . Show that if we define fr :“ s ˝ f ˝ s, then μfr “ s f μ.

1.3 Quasiconformal mappings We are now ready to define quasiconformal mappings. There are several equivalent definitions of K-quasiconformality for an orientation preserving homeomorphism φ : U Ñ V between domains of C, where 1 ď K ă 8. As we shall see, a 1-quasiconformal map is conformal. The value of K can be considered as a measure for how near φ is to being conformal. Although a quasiconformal mapping is always R-differentiable almost everywhere and

1.3 Quasiconformal mappings

21

with bounded dilatation, the converse is not true, as illustrated below in the classical counter example in Section 1.3.3. We shall not try to give a self-contained treatment of the theory of quasiconformal mappings, but instead give an overview which should be sufficient for the applications of quasiconformal mappings in holomorphic dynamics, and for quasiconformal surgery in particular. We give precise references to the literature, mainly to the classical textbooks by Ahlfors [Ah5], and by Lehto and Virtanen [LV]. However, we also recommend the reader to consult the modern treatments by Hubbard [Hu2], Lyubich [Ly] and Astala et al. [AIM]. The original idea of a quasiconformal mapping goes back to Grötzsch and is formulated geometrically in terms of moduli of quadrilaterals Q in U and φpQq in V , or alternatively, of moduli of annuli A in U and φpAq in V . The analytic definitions are formulated in terms of distributional derivatives of φ, or alternatively, of ordinary partial derivatives of φ almost everywhere for a mapping which is absolutely continuous on lines, abbreviated ACL. It is not at all obvious why the different definitions are equivalent, but we shall state how the equivalences are proven.

1.3.1 First analytic definition of quasiconformal mappings We start with the most modern (not necessarily the most intuitive) approach to defining quasiconformal mappings, using distributional derivatives. We give a brief introduction to this concept (see e.g. [Co, Sect. 18.4]). A test function is a C 8 function with compact support in U , and the space of test functions is denoted by Cc8 pU q. With the appropriate topology, this is a topological vector space, and distributions are precisely the continuous linear functionals of this space, i.e. L : Cc8 pU q Ñ C. As an example, if φ P L1loc pU q (i.e. is locally integrable like, for example, any continuous function), then φ defines a distribution Lφ phq :“

ż φ ¨ h dm, U

where m denotes the Lebesgue measure. These are actually the distributions we will be interested in although not all distributions are of this form. Often the same symbol is used for the function in L1loc pU q and the distribution it defines. Observe that if h is a test function then Bz h and Bz h are also test functions. It follows that if L is a distribution then h ÞÑ LpBz hq and h ÞÑ LpBz hq are also distributions.

22


Definition 1.4 (Distributional derivatives) If L is a distribution we define its distributional derivatives as the distributions pBLqphq :“ ´LpBz hq and pBLqphq :“ ´LpBz hq. The minus signs in this definition are necessary so that, if φ is a C 1 function then BLφ “ LBz φ and

BLφ “ LBz φ .

But be aware that if φ P L1loc pU q is not C 1 then the distributional derivatives do not need to be defined by functions, although they are still distributions. However, if they happen to be, that is, if there exist φ1 , φ2 P L1loc pU q such that BLφ “ Lφ1 and

BLφ “ Lφ2 ,

then φ1 and φ2 are denoted Bφ and Bφ and called B and B derivatives of φ in the sense of distributions, or the distributional derivatives of φ in L1loc pU q. Precisely, φ1 and φ2 must satisfy ż ż ż ż φ1 ¨ h dm “ ´ φ ¨ Bz h dm and φ2 ¨ h dm “ ´ φ ¨ Bz h dm (1.12) p

for any test function h P Cc8 . More generally, for p ě 1, if φ1 , φ2 P Lloc pU q p exist we say that φ has distributional derivatives in Lloc pU q. The space of funcp p tions in Lloc pU q (or distributions) having distributional derivatives in Lloc pU q 1,p is called the Sobolev space Wloc pU q. p Observe that Lloc pU q Ă L1loc pU q for p ą 1 (see Exercise 1.3.1). 1 If φ P C , its distributional derivatives are precisely its derivatives in the ordinary sense. We are ready for the first definition of a quasiconformal mapping. Definition 1.5 (First analytic definition of K-quasiconformal mapping) Let U and V be domains in C, and let K ě 1 be given. Set k :“ pK ´ 1q{ pK ` 1q. Then φ : U Ñ V is K-quasiconformal if and only if: (i) φ is a homeomorphism; (ii) the partial derivatives Bφ and Bφ exist in the sense of distributions and belong to L2loc (i.e. are locally square integrable); (iii) and satisfy |Bφ| ď k|Bφ| in L2loc .


23

Remarks 1.6 (a) Observe, that if φ is a C 1 homeomorphism then φ is quasiconformal if it satisfies |Bz φ| ď k|Bz φ| for some k ă 1. (b) If φ is a diffeomorphism between compact sets then φ is quasiconformal. To obtain the bound K on the dilatation we simply use the continuity of the derivatives and the fact that a continuous function attains its maximal value on a compact set.

1.3.2 Second analytic definition of quasiconformal mappings A quasiconformal mapping can instead of partial derivatives in the sense of distribution be characterized in terms of absolute continuity on lines. This is a condition that insures the existence of partial derivatives in the ordinary sense almost everywhere. We shall therefore turn to this notion, since it connects to our previous chapter. Definition 1.7 (Absolute continuity on an interval) A continuous complex valued function f defined on an interval I Ă R is said to be absolutely continuous on I if it satisfies the following: for every ą 0 there exists a ř δ ą 0 such that j |f pbj q ´ f paj q| ă for every finite sequence of nonintersecting intervals paj , bj q whose closure are contained in I and have a ř total length i |bj ´ aj | ă δ. It follows from f being absolutely continuous on I that f is of local bounded variation, hence that f has a derivative f 1 pxq in the ordinary sense for almost all x P I . See [LV, Sect. 2.7, Chapt. III]. The above notion of absolute continuity on an interval generalizes to a complex valued function on a domain U as follows. Definition 1.8 (ACL, absolute continuity on lines) A continuous function f : U Ñ C is said to be absolutely continuous on lines if for any family of parallel lines in any disc D compactly contained in U (that is, D Ă U ), f is absolutely continuous on almost all of them. From the interval-discussion above it follows that if f is absolutely continuous on lines, then f has partial derivatives in the ordinary sense almost everywhere in U . The existence of partial derivatives at a point does not necessarily imply differentiability of f at the point. However, the following result by Gehring and Lehto is true (see e.g. [Ah5, pp. 17–18] and in [LV, p. 128]).

24


Theorem 1.9 (Differentiability almost everywhere) Let f : U Ñ V be a continuous open mapping. If f has partial derivatives fx and fy in the ordinary sense almost everywhere then f is R-differentiable almost everywhere. Note that for the partial derivatives fx and fy to exist almost everywhere it is sufficient for f to be absolutely continuous along almost every horizontal and almost every vertical line in every rectangle compactly contained in U . Therefore Bz f and Bz f are defined almost everywhere. Since the integral of the Jacobian over a set is bounded above by the area of the image of this set (see [LV, Lem. III.3.3]), it follows that the Jacobian Jac f “ |Bz f |2 ´ |Bz f |2 is in L1loc . But to really relate the ACL property to the definition of quasiconformality we need to understand the relation of the ordinary derivatives and the distributional ones. It turns out that these two coincide when they are locally integrable. More precisely (see [Ah5, p. 19] and [AIM, Lem. A.5.2]): Lemma 1.10 (ACL characterization of Sobolev spaces) A continuous func1,1 (i.e. has distributional derivatives in L1loc ) if and tion f : U Ñ C is in Wloc only if f is ACL and its ordinary partial derivatives are in L1loc . When this is the case, the distributional and the ordinary derivatives coincide. We are ready to formulate the second analytic definition. Definition 1.11 (Second analytic definition of K-quasiconformal mapping) A mapping φ : U Ñ V is K-quasiconformal if and only if: (1) φ is a homeomorphism; (2) φ is ACL; (3) |Bz φ| ď k|Bz φ| almost everywhere where k :“

K ´1 K `1 .

To conclude that the two definitions are equivalent we must add something to the statements above. Namely, that since φ is ACL, the Jacobian is in L1loc and (3) is fulfilled, it follows that |Bφ|2 ď |Bφ|2 ď

Jac φ , 1 ´ k2

and therefore the partial derivatives are locally square integrable. With this definition we can now relate to our previous section on almost complex structures and pullbacks by a quasiconformal map φ. Since we have that φ is R-differentiable almost everywhere, and as we shall see in Theorem 1.15 with a non-singular differential defined almost everywhere, condition (3) in the second analytic definition requires that the Beltrami coefficient μφ “ Bz φ Bz φ has modulus bounded away from 1 almost everywhere, i.e. }μφ }8 ă 1.


25

Equivalently, in the language of pullbacks, we are asking that the almost complex structure φ ˚ σ0 , induced by φ as the pullback of the standard one, have bounded dilatation, i.e. Kφ “ K ă 8. The larger K is, the further the ellipses are from being circles, that is the further the function φ is from being conformal. However, be aware that this condition alone is not enough. Even if φ ˚ σ has bounded dilatation a.e., ACL is needed to ensure quasiconformality. We will see an example of this fact once we have a geometric definition of quasiconformal mappings. Nevertheless, we will introduce the example at this point to see a mapping that is not ACL.

1.3.3 A devil’s staircase example The following classical example defines a homeomorphism F : CR Ñ CR , which is R-differentiable almost everywhere and with a non-singular Rdifferential almost everywhere. However, it is not absolutely continuous on horizontal lines. Let Ă r0, 1s denote the middle third Cantor set. The Cantor function f : r0, 1s Ñ r0, 1s, also called a devil staircase function, is defined below in two alternative ways. Its graph is shown in Figure 1.8. 1 7 8 3 4 5 8 1 2 3 8 1 4 1 8 1 9

2 9

1 3

2 3

7 9

8 9

1

Figure 1.8 The graph of the Cantor function, a devil’s staircase.

Recall that the Cantor set consists of the points x P r0, 1s which in base 3 have a representation of the form x “ .x1 x2 ¨ ¨ ¨ xn ¨ ¨ ¨ “

x2 x1 xn ` 2 ` ¨¨¨ ` n ` ¨¨¨ 3 3 3

with xn P t0, 2u.

26


Define f as the unique non-decreasing function, which on x P as above takes the value with the following representation in base 2 f pxq “ .x 1 x 2 ¨ ¨ ¨ x n ¨ ¨ ¨ “

x1 x2 xn ` 2 ` ¨¨¨ ` n ` ¨¨¨ 2 2 2

where x n equals 0 if xn “ 0 and 1 if xn “ 2. The Cantor set has measure zero and f pq “ r0, 1s. Alternatively, f is defined as the limiting function f “ lim fn of the uniformly convergent sequence of continuous functions fn : r0, 1s Ñ r0, 1s, defined inductively and starting with f0 pxq “ x, and for n ě 0:

fn`1 pxq :“

$ 1 ’ ’ & 2 fn p3xq 1

2 ’ ’ % 1 ` 1 f p3x ´ 2q 2 2 n

if 0 ď x ď 13 , if if

1 3 2 3

ď x ď 23 , ď x ď 1.

The function f is equal to the constant 21n on any gap of the Cantor set of the form r 3jn , j3`n1 s with j appropriate. Hence the derivative is f 1 pxq “ 0 on r0, 1sz, i.e. almost everywhere. However, f is not absolutely continuous on r0, 1s, since the Cantor set can be covered by a finite set of disjoint open intervals of arbitrarily small total length, but so that the corresponding sum ř 1 j |f pbj q ´ f paj q| is bigger than, say 2 . Now extend the f function to f : R Ñ R by f pxq “ 0 for x ď 0 and f pxq “ 1 for x ě 1. Let F : CR Ñ CR be defined as F px ` i yq “ x ` i py ` f pxqq for x ` i y P CR (see Figure 1.9). Then F is an orientation preserving homeomorphism, with Bz F px ` i yq “ 1 and Bz F px ` i yq “ 0 for x ` i y P CR zp ` i Rq, i.e. almost everywhere. It follows from the one-dimensional example that F is not absolutely continuous on horizontal lines, and therefore F is not ACL.

1.3.4 Geometric definitions of quasiconformal mappings Classically quasiconformal mappings are defined geometrically, without reference to distributions. A quadrilateral Q “ Qpz1 , z2 , z3 , z4 q is a Jordan domain in C with an ordered sequence of boundary points pz1 , z2 , z3 , z4 q called the vertices of Q, with their order agreeing with the positive orientation of Q. Any quadrilateral Q can be mapped conformally onto a rectangle in such a way that vertices map to vertices. This rectangle is unique up to similarity (see for instance


27

F

Figure 1.9 The image of the unit square under F , a two-dimensional devil’s staircase function.

[LV, p. 15]). If we denote by ϕ such a conformal map, then the conformal modulus of Q is defined as mod Qpz1 , z2 , z3 , z4 q :“

|ϕpz1 q ´ ϕpz2 q| . |ϕpz2 q ´ ϕpz3 q|

See Figure 1.10. Two quadrilaterals Q and Q1 are conformally equivalent if and only if they have the same modulus. By rearranging the vertices we obtain mod Qpz2 , z3 , z4 , z1 q “

1 . mod Qpz1 , z2 , z3 , z4 q

(1.13)

Let U, V be domains in C and φ : U Ñ V be an orientation preserving homeomorphism. Then for any quadrilateral Q, compactly contained in U , its mod φ pQq image φpQq is a quadrilateral, compactly contained in V . The ratio mod Q is called the dilatation of Q under φ. The maximal dilation of φ is defined as mod φpQq . QĂU mod Q

Kφ :“ sup

z1

ϕ pz4 q

z4

ϕ pz3 q

ϕ b z2 z3

ϕ pz1 q

a

ϕ pz2 q

Figure 1.10 The modulus of the quadrilateral Qpz1 , z2 , z3 , z4 q is a{b.

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Using equation (1.13) we see that for every dilatation, we also have its inverse, just by considering the same quadrilateral with the vertices rearranged. Thus we conclude that Kφ ě 1. Definition 1.12 (First geometric definition of K-quasiconformal mapping) Let U and V be domains in C, and let K ě 1 be given. Then φ : U Ñ V is Kquasiconformal if and only if φ is an orientation preserving homeomorphism satisfying 1 mod Q ď mod φpQq ď K mod Q, K for all quadrilaterals Q that are compactly contained in U ; or equivalently satisfying Kφ ď K. A proof of the equivalence of the first geometric definition (Definition 1.12) of K-quasiconformality and the second analytic definition (Definition 1.11) can be found in [Ah5, pp. 20–21]. It is a fact proven by Grötzsch ([Ah5, p. 8]) that the affine map from a rectangle Q1 to a rectangle Q2 has the least maximal dilatation (see also Exercise 1.3.5). The second geometric definition of K-quasiconformality refers to annuli instead of quadrilaterals. Let A denote an open annulus in C, i.e. a doubly p Then A can be mapped conformally onto a standard connected domain in C. annulus Ar,R :“ tz P C | 0 ď r ă |z| ă R ď 8u, which is unique up to multiplication by a real constant. If we denote by ϕ such a conformal map, then the conformal modulus of A is defined as the modulus of Ar,R , i.e. # 1 log Rr if r ą 0 and R ă 8, mod A :“ mod Ar,R :“ 2π 8 if r “ 0 or R “ 8. Observe that the two concepts are related in the following way. For a quadrilateral of finite modulus M choose the conformally equivalent rectangle so that the ordered vertices are p0, 2π M, 2π pM ` i q, 2π i q. The rectangle is mapped by the exponential z ÞÑ ez onto the standard annulus A1,R , where log R “ 2π M, identifying points on the horizontal edges of the rectangle. See Figure 1.11, which also shows the straight cylinder obtained from this identification. Note that the modulus of a straight cylinder is the height of the cylinder divided by the circumference of the circle. Definition 1.13 (Second geometric definition of K-quasiconformal mapping) Let U and V be domains in C, and let K ě 1 be given. Then φ : U Ñ V


29

2π i

z ÞÑ ez

»

2π

1

R

log R 0

log R

Figure 1.11 The exponential maps a rectangle of the given form onto a standard annulus, and their moduli are the same.

is K-quasiconformal if and only if φ is an orientation preserving homeomorphism satisfying 1 mod A ď mod φpAq ď K mod A K

(1.14)

for all annuli A that are compactly contained in U .

1.3.5 Revisiting the devil’s staircase example Consider again the devil’s staircase homeomorphism F : CR Ñ CR . We have seen that F is not ACL although it is differentiable and with Bz F “ 0 almost everywhere! This means that the pullback of the standard complex structure by F is the standard complex structure almost everywhere. One would be tempted to say that this is a condition for conformality but, using the second geometric definition of quasiconformality, we show why F is not even quasiconformal. For each n ě 0 we form the two double-infinite cylinders Cn “ pr0, 1{3n s ` i Rq {pi y „ 1{3n ` i yq, Cn1 “ pr0, 1{3n s ` i Rq {pi y „ 1{3n ` i py ` 1{2n qq, and let Fn : Cn Ñ Cn1 denote the homeomorphism induced by F . Let An Ă Cn denote the annulus ´ ¯n An “ pr0, 1{3n s ` i r0, 1{2n sq {pi y „ 1{3n ` i yq with mod An “ 32 . Note that the annulus Fn pAn q is contained in the round annulus Bn Ă Cn1 (see Figure 1.12) bounded by the two line segments of slope p 32 qn through the points Ln p2{3n`1 ` i {2n`1 q and Un p1{3n`1 ` i 3{2n`1 q respectively.

30


1 2n´1

Un

3 2n`1

1 3¨2n`1

Bn

Fn pAn q

1 2n

1 2n`1

Ln

´1 3¨2n`1

2 3n`1

1 3n`1

1 3n

Figure 1.12 The parallelogram inducing the round cylinder Bn in Cn1 .

If F were K-quasiconformal, so would Fn be for any n ě 0. This would imply 1 mod An ď mod Fn pAn q ď mod Bn , K where the left inequality corresponds to the left inequality of (1.14) and the right inequality follows from the Grötzsch inequality (see e.g. [Mi1, Cor. B.6]), using that Fn pAn q is embedded in Bn and of the same homotopy type. Hence K would satisfy Kě

mod An mod Bn

for any n ě 0.

The modulus of the round cylinder Bn is the height divided by the circumference of the circle, hence mod Bn “

p 12 q2n ´ p 13 q2n`1 p 12 q2n ` p 13 q2n

“

1 ´ 13 p 23 q2n 1 ` p 23 q2n

.


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Since mod Bn tends to 1 and mod An tends to 8 when n tends to 8, there is no such K. 1.3.6 Properties of quasiconformal mappings Let φ : U Ñ V be an orientation preserving homeomorphism. We have given four definitions of quasiconformality. However, in practice it is not easy to check any of them. In this section we state some useful properties of quasiconformal mappings. In many instances they help deciding on the quasiconformality of a mapping. In general we refer to [Ah5] for the proofs. We start with some properties which follow easily from the geometric definitions: P1 If φ is K-quasiconformal, then φ ´1 is K-quasiconformal. P2 If φ is K-quasiconformal, then any composition on the left or right with a conformal mapping is K-quasiconformal. P3 The composition of a K1 -quasiconformal mapping and a K2 quasiconformal mapping is K1 K2 -quasiconformal. The next statement comes directly from the first analytic definition, although it is less obvious when we think about it in geometric terms: P4 The homeomorphism φ is K-quasiconformal if and only if φ is locally K-quasiconformal. The next property follows also from the analytic definitions: P5 If φ is K-quasiconformal and of class C 1 then the dilatation of the infinitesimal ellipse Ez in Tz U is bounded by K for all z P U . Indeed, if the R-differential changes continuously, then so does the dilatation of Ez . If the dilatation would be larger than K at some point, this would be true in a neighbourhood of the point, i.e. on a set of positive measure. P6 If φ is K-quasiconformal, then φ satisfies a uniform Hölder condition |φpz1 q ´ φpz2 q| ď M|z1 ´ z2 |1{K on every compact subset of U . P7 If φ is bilipschitz, i.e. if there exist L ą 0 such that L´1 |z1 ´ z2 | ď |φpz1 q ´ φpz2 q| ď L|z1 ´ z2 |, then φ is quasiconformal. The converse is not true. (See also [Roh].)

32


The following is a very useful result in surgery applications (see [Ah5, p. 16]). Theorem 1.14 (Weyl’s Lemma) If φ is 1-quasiconformal, then φ is conformal. In other words, if φ is quasiconformal and Bz φ “ 0 almost everywhere, then φ is conformal. The first statement is proved by using the geometric definition of quasiconformality. The second statement, which is generally known as Weyl’s Lemma, follows from the equivalence of the definitions. In fact the condition of quasiconformality could be substituted by the mapping being an ACL homeomorphism. Notice that the devil’s staircase example illustrates that the quasiconformality (or at least ACL) is a necessary condition in this statement. Indeed, the mapping F in the example satisfies Bz F “ 0 almost everywhere, but F is not conformal. Another important property is that quasiconformal mappings are absolutely continuous with respect to Lebesgue measure, see [Ah5, p. 22]. More precisely, Theorem 1.15 (Quasiconformal maps and measure) If φ is quasiconformal, then it maps sets of measure zero to sets of measure zero. Moreover, for every measurable set E ż Jac φ dm. mpφpEqq “ E

Corollary 1.16 If φ is quasiconformal, then Bz φ ‰ 0 almost everywhere. Moreover, Jac φpzq ą 0 almost everywhere. If φ : U Ñ V is K-quasiconformal then Jac φ ě p1 ´ k 2 q|Bz φ|2 , where k “ pK ´ 1q{pK ` 1q. It follows that Jac φ ą 0 almost everywhere and that φ is orientation preserving. In the analytic definitions of quasiconformality we did not require the homeomorphism φ to be orientation preserving. It is a consequence. In the geometrical definition we needed to add orientation preserving as a requirement. So only now we have the equivalence of the analytic and geometric definitions of quasiconformality. Remark 1.17 Given a Beltrami coefficient μ defined a.e. in a set U Ă C, the pullback φ ˚ pμq under a quasiconformal map φ : U 1 Ñ U is well defined a.e. in U 1 . This is due to Theorem 1.15 and its Corollary. Indeed, if μ is defined on U zX with mpXq “ 0, then φ ˚ pμq is well defined in U 1 zpφ ´1 pXq Y Y q, where Y is the set where φ is not differentiable, hence a set of measure zero. Since φ ´1 pXq also has measure zero, it follows that φ ˚ μ is defined a.e.


33

Finally, the following area distortion estimate due to Astala will be useful later on, see [As]. Theorem 1.18 (Quasiconformal maps and area) Let φ : D Ñ D be a Kquasiconformal mapping such that φp0q “ 0 and E any Borel measurable set. Then, AreapφpEqq ď MpAreapEqq1{K , where M “ MpKq “ 1 ´ OpK ´ 1q. Some facts about quasiconformal mappings are useful, for instance as mentioned above that a C 1 -diffeomorphism between compact sets is quasiconformal (see Remark 1.6). If f : U Ñ V is a C 1 -diffeomorphism between subsets of C that are not assumed to be compact, there are criterions for quasiconformality formulated in Exercises 1.3.3 and 1.3.4. Another useful, although not so simple, fact is that certain sets are removable in terms of quasiconformality. More precisely, a closed set X is quasiconformally removable if any homeomorphism which is defined on a neighbourhood U of X and is quasiconformal on U zX, is quasiconformal on U . It is a fact that quasiconformally removable sets have measure zero, and hence the dilatation of f in U is the same as in U zX. Notice that sets of measure zero do not affect the dilatation condition in the definitions of quasiconformality. But being ACL in U zX does not imply necessarily that the map is ACL in U , even if X has zero measure. This is the main difficulty when showing that a certain set is quasiconformally removable. We shall need the following removability result, see [Ly] or [Hu2, Prop. 4.2.7 and 4.9.9]. Theorem 1.19 (Quasiconformal removability of quasiarcs) If is a quasiarc (the image of a straight line under a quasiconformal mapping) and φ : U Ñ V a homeomorphism that is K-quasiconformal on U z, then φ is K-quasiconformal on U , and hence is quasiconformally removable. In particular, points, lines and smooth arcs are quasiconformally removable. As a consequence, we may construct global quasiconformal mappings by gluing together different quasiconformal mappings along a finite number of boundary pieces, as long as they are sufficiently nice. This is a common technique in holomorphic dynamics as we shall see repeatedly. Another result of this type is Rickman’s Lemma, also known as Bers’s Sewing Lemma, see [DH1].

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Lemma 1.20 (Rickman’s Lemma) Let U Ă C be open, C Ă U compact, φ and two mappings U Ñ C which are homeomorphisms onto their images. Suppose that φ is quasiconformal, that is quasiconformal on U zC, and that φ “ on C. Then is quasiconformal and Bzφ “ Bz almost everywhere on C. The following property deals with families of quasiconformal mappings, see [Hu2, p. 131]. Theorem 1.21 (Compactness) The set of K-quasiconformal homeomorphisms φ : D Ñ D with φp0q “ 0 is compact in the topology of uniform convergence on compact subsets.

1.3.7 Mappings and Beltrami forms on Riemann surfaces Up to now we have defined quasiconformal mappings, Beltrami coefficients and pullbacks of Beltrami coefficients on subsets of C. However, we shall p Therefore, we briefly discuss need the notions also on the Riemann sphere C. how the different terms are extended to Riemann surfaces. They are defined locally in charts, where they agree with the previous definitions. The extra work involved consists in checking that the definitions are independent of the charts in which they are expressed. The checking is easy, we just have to rephrase facts we already know. First some notation: recall that a Riemann surface S is a one-dimensional complex manifold, i.e. a two-dimensional real manifold endowed with a complex analytic structure. Such a structure is given by a collection of charts ϕ : US Ñ U which forms an atlas. For each such chart, ϕ is a homeomorphism from an open subset US of S onto an open subset U of C. The union of all the domains US equals S, and for any pair of overlapping charts the transition map r is a chart such that US X U rS ‰ H then rS Ñ U is conformal, i.e. if ϕr : U rS q Ñ ϕrpUS X U rS q ϕr ˝ ϕ ´1 : ϕpUS X U is conformal. We remark that a topological surface S can be endowed with a quasiconformal (resp. smooth) structure, which means with a collection of charts ϕ : US Ñ U Ă C which forms an atlas with quasiconformal (resp. smooth) transition maps. When dealing with maps between Riemann surfaces, its regularity is always defined in terms of the regularity of the maps between the open sets of C induced by the charts. In this sense, a conformal isomorphism between


35

Riemann surfaces is an isomorphism which is locally conformal (between open sets of C) when expressed in charts. A remarkable theorem by Koebe and Poincaré states that there are only three possible simply connected Riemann surfaces, up to conformal isomorphisms. For a proof, see for instance [Ah3, Chapt. 10], [FK, IV.4.] or [Hu2, Chapt. 1]. Theorem 1.22 (Uniformization Theorem) Let S be a simply connected p Riemann surface. Then S is conformally equivalent to D, C or C. We now turn to analyze quasiconformal mappings between Riemann surfaces. Definition 1.23 (K-quasiconformal mapping between Riemann surfaces) Let S and S 1 be two Riemann surfaces, and let φ : S Ñ S 1 be a homeomorphism. Then φ is quasiconformal if and only if there exists a K ě 1 so that φ is locally K-quasiconformal when expressed in charts. For an arbitrary point s P S choose an arbitrary chart ϕ : US Ñ U defined on a neighbourhood US of s and choose an arbitrary chart ϕ 1 : US 1 Ñ U 1 defined on a neighbourhood US 1 of φpsq. We may assume that φpUS q “ US 1 . By definition φ|US : US Ñ US 1 is K-quasiconformal if ϕ 1 ˝ φ ˝ ϕ ´1 : U Ñ U 1 is Kquasiconformal. The notion is well defined since it is independent of the charts in which we locally check K-quasiconformality. This follows from property P2 in Section 1.3.6: the composition of a K-quasiconformal mapping with conformal mappings left and right is again K-quasiconformal. Changing charts in the domain and the range of φ corresponds exactly to composing ϕ 1 ˝ φ ˝ ϕ ´1 left and right with conformal mappings, i.e. ϕ ˝ ϕ ´1 q´1 “ ϕr1 ˝ φ ˝ ϕr´1 , pr ϕ 1 ˝ ϕ 1´1 q ˝ pϕ 1 ˝ φ ˝ ϕ ´1 q ˝ pr where ϕr and ϕr1 are charts around s and φpsq respectively (see Figure 1.13). Next we see how Beltrami coefficients on subsets of C are generalized to Beltrami forms on Riemann surfaces. Definition 1.24 (Beltrami form on a Riemann surface) A Beltrami form μ, also called a Beltrami differential, on a Riemann surface S is a p´1, 1qdifferential on S. Locally a p´1, 1q-differential is expressed as μpzqdz{dz “ μpzqdz´1 dz. Formally, this means that if ϕ and ϕr are two overlapping charts defined on

36


Figure 1.13 Sketch of a K-quasiconformal mapping φ between Riemann surfaces. The composition ϕ 1 ˝ φ ˝ ϕ ´1 : U Ñ U 1 is K-quasiconformal, while the transition ´1 between the coloured areas are conformal. mappings ϕ ˝ ϕr´1 and ϕ 1 ˝ ϕr1

S with the conformal transition map h “ ϕr ˝ ϕ ´1 then the Beltrami form satisfies the following transformation rule: h1 pzq μϕ pzq “ μϕrpr zq 1 , h pzq

(1.15)

where z “ ϕpsq and r z “ ϕrpsq. In other words, μϕ pzq and μϕrpr zq are the Beltrami coefficients of the Beltrami form when expressed in the z-coordinate in r , in the two overlapping charts. Note that U , respectively the r z-coordinate in U 1 since h is holomorphic, h pzq “ Bz hpzq, and discover the similarity between (1.15) and the pullback formula (1.11) of a Beltrami coefficient by a holomorzq| “ |μϕ pzq| so that }μ}8 is well defined. phic map. Moreover, note that |μϕrpr To justify the transformation rule we consider the geometric information given by a Beltrami form on S with }μ}8 ă 1. The Beltrami form defines in this case an almost complex structure σ on S, i.e. a measurable field of infinitesimal ellipses on the tangent bundle T S. The Beltrami coefficient μϕ pzq 1`|μ pzq|

defines a.e. an infinitesimal ellipse at Tz U with dilatation equal to 1´|μϕ pzq| . ϕ It follows from (1.15) that the infinitesimal ellipse is mapped by the transition r of the charts on the tangent bundles to the infinitesimal map Dz h : Tz U Ñ Trz U r . More precisely, let E zq at Trz U ellipse defined by the Beltrami coefficient μϕrpr be an ellipse representing the infinitesimal ellipse in Tz U , which is well defined r in Trz U r , which is up to scaling. Then E is mapped by Dz h onto the ellipse E 1 1 E scaled by |h pzq| and rotated by the argument of h pzq, see Figure 1.14. The two ellipses have the same dilatation:


37

Figure 1.14 The ellipse E in Tz U is mapped by Dz h : w ÞÑ h1 pzq ¨ w onto r r in T U the ellipse E r z . Hence the ellipses are equal up to multiplication by a complex number.

r KpEq “ KpEq,

(1.16)

r is exactly and twice the argument from the real axis to the minor axis of E zq: equal to the argument of μϕrpr zq. arg μϕ pzq ` 2 arg h1 pzq “ arg μϕrpr

(1.17)

r represents the infinitesimal ellipse defined by the Therefore, the ellipse E zq. It follows that the measurable field of infinitesimal Beltrami coefficient μϕrpr ellipses in T S is well defined and, hence, that the Beltrami form with bounded dilatation determines an almost complex structure on S. The last thing we define is the pullback of a Beltrami form by a quasiconformal mapping. Definition 1.25 (Pullback of Beltrami forms) Let S and S 1 be two Riemann surfaces, and let φ : S Ñ S 1 be a quasiconformal mapping. If μ1 is a Beltrami form on S 1 then φ ˚ μ1 is defined as the Beltrami form on S, which when expressed in charts fits with the previous pullback definition. To make this more precise, let ϕ : US Ñ U and ϕ 1 : US 1 Ñ U 1 be arbitrary charts around s P S and φpsq P S 1 , and set z “ ϕpsq and z1 “ ϕ 1 pφpsqq. Let μ1ϕ 1 pz1 q be the Beltrami coefficient of μ1 in the chart ϕ 1 . Then φ ˚ μ1 is the Beltrami form on S, which in the chart ϕ has the following Beltrami coefficient:

38


pφ ˚ μ1 qϕ pzq “

Bz f pzq ` μ1ϕ 1 pf pzqqBz f pzq Bz f pzq ` μ1ϕ 1 pf pzqqBz f pzq

,

where f pzq “ ϕ 1 ˝ φ ˝ ϕ ´1 pzq. Compare with (1.9). Also this definition is independent of charts (see Figure 1.15). S1

S

Us

Us1

φ s

φ ps q

ϕ ϕ1 U

f “ ϕ 1 ˝ φ ˝ ϕ ´1

U1 z1

z

Dz f

Figure 1.15 The ellipse in Tz U is mapped by Dz f onto the ellipse in Tz1 U 1 . These ellipses are in general different since Dz f is just a linear map of CR .

Recall that the universal covering space of the Riemann sphere minus three points is the unit disc. From this fact, the analogue to Theorem 1.21 follows. Theorem 1.26 (Compactness) The set of K-quasiconformal homeomorp ÑC p fixing three points is compact in the topology of uniform phisms φ : C convergence on compact subsets.

Exercises Section 1.3 p

1.3.1 Prove that Lloc pU q Ă L1loc pU q. Hint: Apply for q satisfying 1{p ` 1{q “ 1 the Hölder inequality ˆż

ż |f ¨ g| dm ď U p

˙1{p ˆż ˙1{q |f |p dm |g|q dm ,

U

U

q

where f P Lloc pU q and g P Lloc pU q.

1.3.2 Let a “ t0 ă t1 ă ¨ ¨ ¨ ă tn “ b be a finite partition of the interval ra, bs. Prove that if f : ra, bs Ñ R is absolutely continuous in every interval rtj , tj `1 s for 0 ă j ă n ´ 1, then f is absolutely continuous in ra, bs. 1.3.3 Let f : U Ñ V be an orientation preserving C 1 -diffeomorphism between subsets of C, which are not assumed to be compact. For each

1.4 The Integrability Theorem

39

u P U consider the linear map Dz f puq : Tu U Ñ Tf puq V . In the following we formulate a criterion, which guaranties quasiconformality of f . We shall use the result from Exercise 1.1.2 in Section 1.1. Assume that the norm of Dz f puq is uniformly bounded and that the determinant of Dz f puq is uniformly bounded away from zero and infinity. Then show that f is quasiconformal by showing that the dilatation KpDz f puqq is uniformly bounded. 1.3.4 Let f : U Ñ V be given as in Exercise 1.3.3 above. The inequality below gives another criterion that guaranties quasiconformality of f . We shall use notation similar to that introduced in Exercise 1.1.3 in Section 1.1. For each u P U let the linear map Lu “ Dz f puq : Tu U Ñ Tf puq V be represented by the matrix, which in the standard basis in Tu U and in Tf puq V is of the form „ Du f “

j αpuq βpuq . γ puq δpuq

Let λj puq, j “ 1, 2, denote the eigenvalues of the symmetric matrix Qu “ pLu qT pLu q, labelled so that λ1 pzq ě λ2 pzq ą 0. Show that if the eigenvalues λj puq, j “ 1, 2, are uniformly bounded away from zero and infinity then f is quasiconformal. 1.3.5 Let Q1 and Q2 be two rectangles of vertical sides of length vj and horizontal sides of length hj , for j “ 1, 2. Let A be the affine map taking Q1 to Q2 . Use the discussion in Section 1.1 to show that KpAq “

modpQ2 q . modpQ1 q

1.4 The Integrability Theorem Let U Ă C be an open set which is either the whole plane or conformally equivalent to the unit disc. In previous sections we have seen how a quasiconformal homeomorphism φ : U Ñ C gives rise to an almost complex structure σφ on U of bounded dilatation, or equivalently, to a Beltrami coefficient μφ “ Bz φ{Bz φ defined a.e. with }μφ }8 “ k ă 1. The following natural question arises. Given an almost complex structure σ in U , under which conditions can one find a quasiconformal homeomorphism φ such that it induces σ almost everywhere? In the pullback language that means φ ˚ σ0 “ σ a.e. Or equivalently, given a Beltrami coefficient μ on U under which conditions can one find φ : U Ñ C quasiconformal (hence in D0` ) such that Bz φpzq “ μpzqBz φpzq

(1.18)

40


for almost every z P U ? Such function φ is said to integrate μ and is therefore called an integrating map. This equation is called the Beltrami equation and the answer to the question is given by the remarkable Integrability Theorem (also known as the Measurable Riemann Mapping Theorem or simply the Mapping Theorem) due to Morrey [Mo], Bojarski [Bo1] and Ahlfors and Bers [AB]. The local version reads as follows. Theorem 1.27 (Integrability Theorem – local version) Let U Ă C be an open set such that U » D (respectively U “ C). Let σ be an almost complex structure on U corresponding to the Beltrami coefficient μ. Suppose the dilatation of σ is uniformly bounded, that is, Kpσ q ă 8 or, equivalently, the essential supremum of |μ| on U is }μ}8 “ k ă 1. Then μ is integrable, i.e. there exists a quasiconformal homeomorphism φ : U Ñ D (respectively onto C) which solves the Beltrami equation, i.e. such that μpzq “

Bz φpzq Bz φpzq

for a.e. z P U . Moreover, φ is unique up to post-composition with automorphisms of D (respectively C). In the pullback language we would say that there exists a quasiconformal homeomorphism φ such that μ “ φ ˚ μ0 a.e. The proof of this theorem is involved and can be found in many classical texts as well as in some more recent ones. See for example [LV, Ah5] or [AIM, Bo2, DB, Hu2, Ly], and Section 1.4.2 for a sketch of the proof. As stated above, the Integrability Theorem applies to Beltrami coefficients in subsets of the complex plane. However, we often deal with rational mappings on the Riemann sphere and are interested in a global change to an almost comp Moreover, we build abstract Riemann surfaces isomorphic plex structure on C. either to the disc, to the complex plane or to the Riemann sphere. Therefore, we are also interested in a global version of the Integrability Theorem. Theorem 1.28 (Integrability Theorem – global version) Let S be a simply connected Riemann surface and σ be an almost complex structure on S with measurable Beltrami form μ. Suppose the dilatation of σ is uniformly bounded, i.e. Kpσ q ă 8 or, equivalently, the essential supremum of |μ| on S is }μ}8 “ k ă 1.


41

Then μ is integrable, i.e. there exists a quasiconformal homeomorphism φ : p which satisfies S Ñ D (respectively onto C or C) φ ˚ μ0 “ μ. p then φ : S Ñ D (respectively If S is isomorphic to D (respectively to C or C) p onto C or C) is unique up to post-composition with automorphisms of D p (respectively of C or C). In this case we also say that φ solves the Beltrami equation and write μ “ Bz φ{Bz φ a.e. (where the equation should be expressed in charts if necessary). The global theorem easily follows from the local one as explained below. Proof The case of the disc or the complex plane follows almost immediately from the local theorem. Indeed, let X be either D or C and let : S Ñ X be a conformal isomorphism. Then for any chart ϕ : US Ñ U in a given atlas for S the map ˝ ϕ ´1 : U Ñ X is holomorphic. Using these conformal maps, we can push forward the Beltrami form μ on S to a Beltrami form μ1 on X. Since the Beltrami coefficients in overlapping charts on S transform as explained in (1.15), the Beltrami form μ1 is well defined on X. It satisfies }μ1 }8 “ }μ}8 :“ k ă 1. The Local Integrability Theorem provides a quasiconformal homeomorphism φ : X Ñ X such that φ ˚ μ0 “ μ1 . Then φ ˝ : S Ñ X is the required integrating map. p and μ is the Beltrami form on S, Now suppose S is isomorphic to C j satisfying }μ}8 :“ k ă 1. Choose a finite atlas tϕj : US Ñ U j u for S so that U j Ă C is conformally isomorphic to D. By definition, μ induces Beltrami coefficients μj on the sets U j , compatible under the transition maps between overlapping charts. We apply the Local Integrability Theorem to each of these Beltrami coefficients and obtain Kj -quasiconformal homeomorphisms φj : U j Ñ D, such that φj˚ μ0 “ μj (see Figure 1.16). It follows from Weyl’s Lemma that, wherever well defined, the maps φj ˝ ϕj ˝ ϕi´1 ˝ φi´1 are conformal, since they are quasiconformal and preserve the j standard complex structure. Thus the maps tφj ˝ ϕj : US Ñ Du form a new atlas for S with conformal transition maps. By the Uniformization Theorem there exists a conformal isomorphism : p meaning that it is conformal when expressed in charts of the new atlas S Ñ C, p Such an expression is j :“ ψj ˝ ˝ ϕ ´1 ˝ φ ´1 : for S and charts for C. j j j p D Ñ Vj , where ψj : pU q Ñ Vj is a chart on C. S

Since the integrating map φj is Kj -quasiconformal, it follows that the maps φj ˝ j : Uj Ñ Vj for any j are K-quasiconformal, where K “ maxj Kj .

42


Figure 1.16 The Integrability Theorem on Riemann surfaces follows from the local version on open sets of the complex plane. In the drawing, tpϕ, i, j q :“ ϕj ˝ ϕi´1 stands for the transition map with respect to the charts tϕu. Likewise, tpφ ˝ ϕ, i, j q is with respect to the new charts tφ ˝ ϕu. These transition maps are only defined on the shaded regions. On the other hand, the map j is defined on the whole disc, and the integrating maps j ˝ φj are defined on the whole Uj .

Hence, with respect to the original atlas on S, the map is K-quasiconformal, p is the required integrating map. and ˚ μ0 “ μ, so : S Ñ C Remark 1.29 Note that we can use the Integrability Theorem to endow a surface S with a complex analytic structure, making it into a Riemann surface, provided it admits a quasiconformal structure. Indeed, suppose there exists an atlas tϕj u with quasiconformal transition maps and a Beltrami form μ which is compatible with the atlas and with }μ}8 ă 1. Let φj be the integrating maps. Then tφj ˝ ϕj u is a new atlas with holomorphic transition maps, and therefore it defines a complex analytic structure on S. 1.4.1 Dependence on parameters In surgery constructions, we are often interested in applying the Integrability Theorem to a family of Beltrami forms depending on parameters. In these cases, it is necessary to know how the solution of the Beltrami equation varies in terms of parameters. Note that an integrating map φ is uniquely determined if it is normalized. If s1 , s2 and s3 are distinct points in S then, in the case of


43

D, a normalization may be chosen by requiring φps1 q “ 0 and φps2 q P R` ; in p the case of C by requiring that φps1 q “ 0 and φps2 q “ 1; and in the case of C by additionally requiring that φps3 q “ 8. Other kinds of normalizations will occur. For the following theorem to hold it is important to refer to some chosen normalization. Theorem 1.30 (Integrability Theorem – dependence on parameters) Let be an open subset of CN (or RN ) for some N ě 1. p and let pμλ qλP (a) Let S be a Riemann surface isomorphic to C (or C), be a family of measurable Beltrami forms on S. Suppose λ ÞÑ μλ psq is holomorphic (respectively continuous, differentiable, real analytic) in λ for each fixed s P S (whenever defined). Moreover, assume there exists a k ă 1 such that }μλ }8 ď k for all λ P . Choose an appropriate norp be the unique quasiconformal malization and let φλ : S Ñ C (or onto C) homeomorphism which integrates μλ . Then for any fixed s P S the map λ ÞÑ φλ psq is holomorphic (respectively continuous, differentiable, real analytic). p and let pμλ qλP be a family of measurable (b) Let U be a Jordan domain in C, Beltrami forms on U . Suppose λ ÞÑ μλ psq is continuous (respectively differentiable, real analytic) in λ for each fixed s P S (whenever defined). Moreover, assume there exists a k ă 1 such that }μλ }8 ď k for all λ P . Choose an appropriate normalization and let φλ : S Ñ D be the unique quasiconformal homeomorphism which integrates μλ . Then for any fixed s P S the map λ ÞÑ φλ psq is continuous (respectively differentiable, real analytic). Remark 1.31 (On D one cannot expect holomorphic dependence) Observe that in the case of the unit disc, we cannot expect holomorphic dependence. Indeed, if s P BU then necessarily φλ psq P S1 . The mapping λ ÞÑ φλ psq cannot be holomorphic, since it is not an open mapping. Remark 1.32 (Dependence of the inverse maps on parameters) It is important to observe that even if the integrating maps depend holomorphically on a parameter λ, their inverses do not necessarily have the same property. Indeed, implicit differentiation shows that we need differentiability of φλ pzq with respect to z in order to ensure differentiability of φλ´1 pwq with respect to λ, and this is something which is not true in general. See the exercises below and at the end of Section 1.7. See also [Hu2, Example 4.8.18].

44

Quasiconformal geometry 1.4.2 On the proof of the Integrability Theorem

The Integrability Theorem is the cornerstone behind every surgery construction. There are several methods to prove the existence of solutions to the Beltrami equation (1.18), depending on the domain and the regularity of μ. We always assume that }μ}8 “ k ă 1. We shall give a short account on different approaches with proper references to the huge available literature. The earliest occurrence of the Beltrami equation, though expressed in real variables, appeared in the work of Gauss (see [Ga]). In 1822 he proved the existence of isothermal coordinates on any surface with a real analytic Riemannian metric. For an orientable surface the existence of isothermal coordinates ensures that the surface can be equipped with the structure of a Riemann surface. Half a century later, Beltrami in his studies of non-Euclidean geometry also considered the equation that now bears his name (see [Bel]). In order to obtain isothermal coordinates one has to solve a system of firstorder partial differential equations (PDEs), which in complex notation is equivalent to solving the Beltrami equation. To ensure the existence of isothermal coordinates the condition on the Riemannian metric was later relaxed by Korn and Lichtenstein. They showed it was enough to require μ to be only Hölder continuous (see [Ko, Li, C]). Lavrentiev [Lavr] showed the same result for continuous μ using approximation by simple functions. See [LV, V.1.3] for a similar approach. Morrey was the first to prove the existence of a homeomorphic solution to the Beltrami equation for μ measurable (see [Mo]). His methods were based on PDEs. The important results accomplished in the 1950s to prove the existence of solutions were all obtained by using singular integral operators: the Cauchy transform denoted by P and the Beurling transform denoted by T (to be defined below). Bojarski was the first to interpret the Beurling transform as an operator on Lp pCq by using the Calderón–Zygmund Inequality (see [Bo1] and [Ah5, Chapt. V]). In the paper by Ahlfors and Bers [AB] they proved the parameter dependence of normalized homeomorphic solutions of the Beltrami equation in terms of the dependence of μ on the parameter. More recently Douady gave a proof of the Integrability Theorem for every measurable μ, using only classical techniques for L2 and Fourier transforms (see [DB]). We now proceed to sketch a proof of the Integrability Theorem. Proof of uniqueness The uniqueness part of Theorems 1.27 and 1.28 is quite simple. Suppose φ1 and φ2 are two solutions of the Beltrami equation (1.18).


45

Then φ1˚ μ0 “ φ2˚ μ0 a.e. and therefore the quasiconformal homeomorphism ϕ :“ φ1 ˝ φ2´1 satisfies ϕ ˚ μ0 “ μ0 a.e. It follows by Weyl’s Lemma (Theorem 1.14) that ϕ is conformal and hence φ1 “ ϕ ˝ φ2 as desired. Proof of existence We now sketch a proof of the existence part of the Integrability Theorem (Theorem 1.27) following mainly the survey in [GRSY] and the proofs in [Ah5] and [LV]. In this proof we shall write φz :“ Bz φ and φz :“ Bz φ, for any function φ for which the derivatives make sense. As we shall see, a solution will be obtained as the fixed point of a strictly contracting operator acting on an appropriate complete metric space, a common tecnique in dynamical systems. Before starting the argument, we introduce the two operators P and T which are used later on (see [LV, III 7.2] for details). The first one is the inverse of the operator Bz , and is given by the Cauchy transform P , defined for h P Lp with p ą 2 as ż hpuq 1 dmpuq, P hpzq “ ´ π C u´z where dm denotes the Lebesgue measure. This is an absolutely convergent integral. The operator P satisfies pP hqz “ h, and if hz , hz P Lp then P hz “ h.

(1.19)

The second linear operator is the derivative of P defined by ż hpuq 1 T hpzq “ pP hqz pzq “ ´ dmpuq π C pu ´ zq2 as a principal value. The Calderón–Zygmund inequality [Ah5, Sect. V.D] ensures that T is well defined in Lp for all p ą 1 and satisfies }T h}p ď Cp }h}p for some constant Cp that fulfills Cp Ñ 1 as p Ñ 2. Case 1: μ has compact support Let μ be a measurable Beltrami coefficient defined on a open set U » D or U “ C, and assume that μ has compact support. We extend μ to C by setting μ “ 0 outside U . The extended μ is then measurable, with compact support and has the same L8 norm k ă 1.

46

Quasiconformal geometry Suppose φ is a homeomorphic solution to the Beltrami equation φ z “ μ φz .

(1.20)

Then φ is conformal in CzU (since μ “ 0 on this set) and 8 is a removable p ÑC p setting singularity. Therefore it can be extended to a function φ : C φp8q “ 8. Observe that any post-composition of φ with an affine map is also a solution of (1.20). Hence, without loss of generality, we may assume that φ 1 p8q “ 1 or equivalently φpzq “ z ` gpzq, where g is conformal in a neighbourhood of infinity. Since φ solves (1.20) then φz “ gz “ μ p1 ` gz q. Note that, because of the behaviour of gz in a neighbourhood of infinity (the Laurent series of gz has lowest term 1{z2 ), the derivative gz belongs to Lp pCq, for every p ą 1, and so does gz . Hence, we may apply the operator P to gz and obtain from (1.19) g “ P gz “ P μ p1 ` gz q.

(1.21)

Differentiating now with respect to z, we have gz “ T μ p1 ` gz q. Therefore, the function h :“ gz satisfies the equation h “ T μ p1 ` hq, or equivalently h is a fixed point of the operator Tr h “ T μ ` T μ h. We now reverse the process and start by showing that the operator Tr does indeed have a fixed point, i.e. there exists h P Lp satisfying Tr h “ h. This follows from the Banach Fixed Point Theorem [Kre, Thm. 5.1-2], since Tr is a strict contraction in the Lp norm for p sufficiently close to 2, and Lp is a complete metric space. More precisely }Tr h1 ´ Tr h2 }p “ }T μph1 ´ h2 q}p ď }T }p }μ}8 }h1 ´ h2 }p ď Cp k}h1 ´ h2 }p , where k “ }μ}8 ă 1 and Cp Ñ 1 as p Ñ 2. Choose p ą 2 so that kCp ă 1, and observe that this is possible because we require k ă 1. Then Tr is a


47

strict contraction in Lp . By the Fixed Point Theorem there exists a unique function h P Lp which is a fixed point of Tr . It can be obtained by successive applications of the operator with an arbitrary initial condition. To obtain g and φ from h “ gz we go back to equation (1.21). Hence φpzq “ z ` gpzq “ z ` P μp1 ` hq is a solution of the Beltrami equation (1.20). We now address the issue of injectivity of φ. First assume that μ P Cc1 . Then its partial derivatives are continuous and in Lp for p ą 2. Under these conditions it can be shown that φ is a C 1 homeomorphism (see [Ah5, Lem. V.B.3]). Indeed, injectivity follows from showing that Jac φ is continuous p This implies that φ is locally injective and hence a global and positive in C. p homeomorphism in C. If μ is only measurable, we approximate μ by a sequence of functions μn P Cc1 such that μn Ñ μ a.e., }μn }8 ď k and μn “ 0 outside a fixed disc. Then the K-quasiconformal homeomorphisms φn , obtained as above and normalized so that they fix 0, 1 and 8, form a normal family and therefore must have a convergent subsequence which converges uniformly to some limit φ. Then, φ is K-quasiconformal since it is the uniform limit of K-quasiconformal maps (see Theorem 1.21). By theorem [LV, IV.5.2] or lemma [AIM, 5.3.5] the complex dilatation of φ coincides with μ almost everywhere, so φ is the desired solution to the Beltrami equation. Case 2: μ “ 0 in a neighbourhood of the origin To get rid of the hypothesis of compact support consider first the case where μ “ 0 in a neighbourhood of 0. Equalities are from now on, if necessary, understood to hold a.e. Set I pzq :“ 1{z, and consider ˆ ˙ 2 1 z r pzq :“ I ˚ μ “ μ . μ z z2 r has compact support in C The formula can be computed using (1.11). Then μ r normalized so that it fixes so it can be integrated by a quasiconformal map φ, 0, 1 and 8. Then, φpzq :“ pI ˝ φr ˝ I ´1 qpzq “

1 r φp1{zq

is a quasiconformal map which fixes 0, 1 and 8 and integrates the given Beltrami form μ. Indeed, r “ I˚ μ r “ μ, φ ˚ μ0 “ pI ´1 q˚ φr˚ I ˚ μ0 “ pI ´1 q˚ φr˚ μ0 “ pI ´1 q˚ μ

48


where we used that I is injective and the pushforward is well defined. This is summarized in the following commutative diagram: φr

r q ÝÝÝÝÑ pC, μ0 q pC, μ § § § § Iđ đI φ

pC, μq ÝÝÝÝÑ pC, μ0 q Case 3: the general case Finally, in the general case, where μ is a measurable Beltrami form on C, we can obtain a solution to the Beltrami equation by combining solutions that correspond to cases 1 and 2 above. Set μ “ μ1 ` μ2 where μ1 “ μ ¨ 1D and μ2 “ μ ¨ 1CzD , with 1A being the characteristic function over the set A Ă C. Since μ2 “ 0 on D there exists, as explained in Case 2, a quasiconformal homeomoprhism φ2 normalized to fix 0, 1 and 8, such that φ2˚ μ0 “ μ2 . Now define r :“ pφ2 q˚ μ, μ r has compact which is well defined since φ2 is injective, and observe that μ support. Indeed, on CzD we have μ “ μ2 and, therefore, on this set r “ pφ2 q˚ μ2 “ μ0 ” 0. μ Hence there exists a unique integrating map φr fixing 0, 1 and 8 so that φr˚ μ0 “ r . The map defined as μ φ :“ φr ˝ φ2 is a quasiconformal homeomorphism that integrates μ. Indeed, r “ φ2˚ pφ2 q˚ μ “ μ. φ ˚ μ0 “ φ2˚ φr˚ μ0 “ φ2˚ μ We can summarize the construction in the following commutative diagram: φ2

pC, μq

/ pC, μ rq HH HH HH φr HH H$ φ pC, μ0 q

This concludes the proof of Theorem 1.27.

1.5 An elementary example

49

Dependence on parameters The chosen approach for the existence proof, using the singular integral operators, has the advantage that it allows to introduce parameters into the Beltrami equation. If μ “ μλ depends with a certain regularity on a parameter λ it can be shown that the normalized solution φλ of the Beltrami equation φz “ μλ φz depends with the same regularity on the parameter λ, proving Theorem 1.30. For the proof we refer to [Ah5, V.C], [Hu2, Sect. 4.7] or [AB].

Exercises Section 1.4 1.4.1 (Integration of symmetric coefficients) Suppose μ is a Beltrami coefficient defined in C satisfying the hypothesis of the Integrability Theorem 1.27 and additionally being symmetric with respect to the real line (resp. unit circle) (see Section 1.2.1). Prove that the integrating map φ normalized to fix 1 and 0 is also symmetric with respect to the real line (resp. unit circle). Hint: Use the unicity of φ and Exercises in Section 1.2, especially Exercise 1.2.6. 1.4.2 For any R ą 1 set R “ t|λ| ą Ru. Consider the family of maps w “ Lλ pzq “ λz ` z with λ P R . Show that this is a family of quasiconformal maps Lλ : C Ñ C which depend holomorphically on λ while the 1 family of inverses z “ L´ λ pwq do not. 1.4.3 Consider a family of φλ : C Ñ C of homeomorphisms, which depend holomorphically on λ P . Show that if w “ φλ pzq also depends holomorphically on z, then the inverses z “ φλ´1 pwq depend holomorphically on both z and λ.

1.5 An elementary example In the example explained in this section we use a few dynamical concepts. They are surveyed in greater generality in Chapter 3. Later on we introduce what is called soft surgery. The elementary example is a special case of soft surgery. The example is elementary in the sense that everything can be computed explicitly. We start with a simple linear contracting map M0 : D Ñ D defined by M0 pzq “ λ0 z for some 0 ă |λ0 | ă 1. We shall define new almost complex structures in D, induced by certain quasiconformal mappings that are all M0 invariant. The left half plane, H , is the universal covering space of D˚ “ Dzt0u under the exponential map. Notice that the exponential map semi-conjugates M0 on

50

Quasiconformal geometry H

D˚

2π i

exp ζ ÞÑ ζ ` ν0

λ0 z ÞÑ λ0 z

ν0 0

Figure 1.17 The exponential map semi-conjugates translation by ν0 on the left half plane H to multiplication by λ0 “ eν0 on D˚ .

D˚ to a translation by ν0 “ log λ0 on H . We choose ν0 so that 0 ď Impν0 q ă 2π (see Figure 1.17). We have the following commutative diagram: zÞÑλ z

D˚ ÝÝÝÝ0Ñ D˚ İ İ §exp § exp§ § ζ ÞÑζ `ν

0 H ÝÝÝÝÝÑ H

Choose any parameter ν in the left half plane H . Consider the R-linear map Lν : H Ñ H which maps the basis tν0 , i u onto the basis tν, i u, and therefore the parallelogram P pν0 q “ ttν0 ` i s | t P p0, 1q, s P p0, 2π qu onto the parallelogram P pνq “ ttν ` i s | t P p0, 1q, s P p0, 2π qu (see Figure 1.18). If we write Lν pζ q “ aζ ` bζ¯ it follows that aν0 ` bν0 “ ν

and

a ´ b “ 1,

hence a“

ν ` ν0 , ν0 ` ν0

b“

ν ´ ν0 , ν0 ` ν0

and the diagram ζ ÞÑζ `ν

0 H ÝÝÝÝÝÑ § § Lν đ

ζ ÞÑζ `ν

H § §L đ ν

H ÝÝÝÝÝÑ H


51

H

H D˚

2π i

Pν0

log

2π i

Lν

Pν

ν0

0

0 ν

ξν Figure 1.18 The standard complex structure in H is first pulled back by Lν and then by the logarithm, to an almost complex structure on D˚ with Beltrami coefficient μν .

commutes. Note that Dζ Lν “ Lν for any ζ , so that the Beltrami coefficient of Lν , is constantly equal to ν ´ ν0 BLν M BLν b “ μpLν q “ “ . Bζ a ν ` ν0 Bζ Set ξν “ Lν ˝ log, where log may be chosen as the branch of the logarithm that maps D˚ onto the half strip ttν ` i s | t P p0, 8q, s P r0, 2π qu. Then we can easily calculate the Beltrami coefficient μν of the pullback ξν˚ μ0 “ log˚ μpLν q at an arbitrary point z P D˚ . From ξν pzq “ a log z ` blog z we obtain Bz ξν “ a

1 , z

Bz ξν “ b

1 , z

and therefore μν pzq “

ν ´ ν0 z . ν ` ν0 z

(1.22)

The modulus of the complex ´ dilatation ¯ μν is constant, but the argument ν ´ν0 varies. At z ‰ 0 it is 2 arg z ` arg ν `ν0 . This means that the argument of the ´ ¯ ν0 minor axes of the infinitesimal ellipse in Tz D is equal to arg z ` 12 arg νν ´ `ν0 .

Hence, along any circle |z| “ r in D˚ the ellipses will make one full turn, and along any radial line´the argument of the minor axis to the radial line is ¯ ν0 . constantly equal to 12 arg νν ´ `ν0 For a given ν P H we have constructed an almost complex structure σν on D˚ corresponding to the Beltrami coefficient μν defined above; we set μν p0q “ 0. The map φν : D Ñ D defined by

52


φν pzq :“

# pexp ˝Lν ˝ logqpzq 0

if z P D˚ if z “ 0

is an integrating map, which by construction makes the following diagram commute: pD˚ , μν q O

M0 :zÞÑλ0 z

exp

exp ζ ÞÑζ `ν0

pH , μpLν qq φν

Lν

pH , μ0 q

/ pH , μpLν qq Lν

ζ ÞÑζ `Êν

exp

pD˚ , μ0 q

/ pD˚ , μν q O

/ pH , μ0 q

φν

exp

zÞÑeν z

/ pD˚ , μ0 q

In other words, M0˚ μν “ μν , and the map Mν “ φν ˝ M0 ˝ φν´1 , obtained as the conjugate to M0 by the integrating map φν , is holomorphic. The commutative diagram also shows that this composition φν ˝ M0 ˝ φν´1 on D˚ is equal to Mν : z ÞÑ eν z, hence it holds on D (see Figure 1.19). Note that whenever ν equals any one of the values of log λ0 , the composition φν ˝ M0 ˝ φν´1 is just M0 again. Hence, it may happen that although we have constructed an M0 -invariant almost complex structure very different from the standard complex structure, we may get the map M0 back when conjugating with the integrating map. The integrating map φν maps the annulus A0 :“ t|λ0 | ď |z| ă 1u onto the annulus Aν :“ t|eν | ď |z| ă 1u. Note that the moduli of the two annuli are mod A0 “

1 1 log 2π |λ0 |

and

mod Aν “

1 1 log ν . 2π |e |

See also Figure 1.20.

1.5.1 Spreading μν by the dynamics In this section we comment on the extension of the almost complex structure given by the M0 -invariant Beltrami coefficient μν in D to an M0 -invariant


53

H

H 2π i

2π i

Lν

ν0

0

0 ν

ζ ÞÑ ζ ` ν

ζ ÞÑ ζ ` ν0

exp

Log

D˚

D˚ φν Me ν

M0

Figure 1.19 The composition exp ˝Lν ˝ Log is precisely the integrating map φν on D˚ .

Aν

A0 | λ0 |

φν

|e ν |

Figure 1.20 The dynamically defined annuli A0 and Aν in the unit disc. For visual reasons, the drawing is out of proportion, compared to the values of ν and ν0 “ Log λ0 in the other drawings.

Beltrami coefficient in C, which we shall also denote by μν . It is geometrically obvious that the almost complex structure given by μν pzq “

ν ´ ν0 z ν ` ν0 z

if z P C˚

is M0 -invariant. (It is also an easy exercise, using (1.11), to show that M0˚ μν “ μν on all of C.) However, we shall use this example to show how this extension can be defined recursively by pullbacks of μν on the annulus A0 by M0n for increasing n P N. We say that the Beltrami coefficient is spread by the dynamics. This way

54


of extending a Beltrami coefficient is very common, and therefore worth illustrating in this example. Set An :“ M0ń pA0 q “

"

1 1 ď |z| ă n ´ 1 |λ0 |n |λ0 |

* ,

and r ν pzq :“ μ

# μν pzq

if z P D,

pM0n q˚ μν pzq

if z P An .

r ν is M0 -invariant by construction and with unchanged bound, i.e. Then μ μν q, since all pullbacks are done by holomorphic maps (see kpμν q “ kpr Figure 1.21).

A2 A1

Figure 1.21 We spread the M0 -invariant Beltrami coefficient μν on D to an M0 -invariant Beltrami coefficient on all of C.

In this example where everything can be calculated explicitly we can also extend the integrating map φν : D Ñ D conjugating M0 to Mν , to an integrating map φrν : C Ñ C, still conjugating M0 to Mν . We set # φν pzq φrν pzq :“ ` ń ˘ Mν ˝ φν ˝ M0n pzq

if z P D, if z P An .

Note that Kφν “ Kφrν , since composing by holomorphic maps does not change the dilatation. Observe that φrν fixes 0 and 1. Hence φν is the uniquely determined integrating map, normalized in the standard way. The global Beltrami r ν which we obtained in this way is the pullback of the standard coefficient μ Beltrami coefficient μ0 under the map φrν . In other words, the following diagram commutes:

1.6 Quasiregular mappings

55

zÞÑλ z

r ν q ÝÝÝÝ0Ñ pC, μ rν q pC, μ § § §r § φrν đ đφν zÞÑeν z

pC, μ0 q ÝÝÝÝÑ pC, μ0 q 1.5.2 Dependence with respect to parameters We end this section by observing how the construction varies with respect to parameters ν P H . Note that for any z P C˚ the map r ν pzq “ ν ÞÑ μ

ν ´ ν0 z ν ` ν0 z

is holomorphic in ν. Hence the integrating map φrν , normalized by fixing 0 and 1, also varies holomorphically with the parameter ν. For z “ 0: ν ÞÑ φrν p0q “ 0. For z P D˚ : ˜ ν ÞÑ φrν pzq “ exp

ν0 log z ´ ν0 log z ν0 ` ν0

¸

ˆ ˙ 2 log |z| exp ν . ν0 ` ν0

For z P An : ν ÞÑ exppńνqφrν pλn0 zq.

1.6 Quasiregular mappings Pullback is a local concept. Global injectivity is not really necessary in order to be able to pull back almost complex structures. This leads to the concept of quasiregular mappings. In this section all quasiregular mappings are denoted by g. Definition 1.33 (First definition of K-quasiregular mapping) Let U Ď C be an open set and K ă 8. A mapping g : U Ñ C is K-quasiregular if and only if g can be expressed as g “ f ˝ φ, where φ : U Ñ φpU q is K-quasiconformal and f : φpU q Ñ gpU q is holomorphic.

56


It follows immediately that g is locally K-quasiconformal except at a discrete set of points φ ´1 pCritpf qq, where Critpf q is the set of points with vanishing derivative, i.e. the set of critical points of f . Around these points the map is not injective. This fact leads to the following alternative definition. Definition 1.34 (Second definition of K-quasiregular mapping) Let U Ď C be an open set and K ă 8. A continuous mapping g : U Ñ C is Kquasiregular if and only if g is locally K-quasiconformal, except at a discrete set of points in U . Proof The first definition implies the second immediately. To see the converse, let be the discrete set of points for which g is not K-quasiconformal in any neighbourhood of such a point. Cover the open set U z by a countable collection of open sets on which g is K-quasiconformal. Such countable collections exists, since any open cover of a subset of R2 has a countable subcover (Lindelöf’s Theorem [Ke, p. 49]). It follows that Bz g and Bz g are well defined almost everywhere in each of the open sets in the subcover and locally belong to L2loc . Since a countable union of sets of measure zero is a set of measure zero, Bz g and Bz g are well defined almost everywhere in U . Moreover, the Beltrami coefficient μpzq “ Bz gpzq{Bz gpzq ´1 satisfies }μ}8 ď k “ K K `1 ă 1 in U . If U is simply connected we can apply the Local Integrability Theorem immediately to obtain a K-quasiconformal homeomorphism φ : U Ñ D (or onto C) integrating the Beltrami coefficient p in C by defining μ. If U is not simply connected we first extend μ to μ p pzq “ 0 if z R U , then apply the Global Integrability Theorem to obtain a μ K-quasiconformal homeomorphism φp : C Ñ C integrating the Beltrami coefp q. Then the composition f :“ g ˝ φ ´1 p U : U Ñ φpU p . Set φ :“ φ| ficient μ is locally K-quasiconformal except at the discrete set of points φ ´1 pq. Moreover f ˚ pμ0 q “ pφ ´1 q˚ g ˚ pμ0 q “ pφ ´1 q˚ μ “ μ0 , as shown by the following diagram: g

pU, μq φ

/ pgpU q, μ0 q q8 qqq q q q qqq f

p q, μ0 q pφpU


57

Hence, by Weyl’s Lemma, f is locally conformal except at a discrete set of points where it is continuous. This implies that f is holomorphic, and it follows that g “ f ˝ φ as we wanted to show. Quasiregular mappings can also be characterized locally by the following equivalent definition. Definition 1.35 (Third definition of K-quasiregular mapping) Let U Ď C be an open set and K ă 8. A mapping g : U Ñ C is K-quasiregular if and only if for every z P U , there exist neighbourhoods Nz and Ng pzq of z and gpzq respectively, a K-quasiconformal mapping ψ : Nz Ñ D and a conformal mapping ϕ : Nf pzq Ñ D, such that pϕ ˝ g ˝ ψ ´1 qpzq “ zd , for some d ě 1. We prove that this definition is indeed equivalent to the two previous ones. Proof Suppose g : U Ñ C satisfies the third definition of quasiregularity. Then g is locally K-quasiconformal at those points where d “ 1. If d ą 1 on Nz , then z is the only point in this neighbourhood where the mapping is not locally invertible. Hence, such points are isolated, and g satisfies the second definition of quasiregularity. Suppose g : U Ñ C satisfies the first definition of quasiregularity and therefore can be written as g “ f ˝ φ on U . We shall prove that g satisfies the third definition of quasiregularity. We distinguish between two cases. Case 1. If φpzq is not in Critpf q, then there exist neighbourhoods Nφ pzq and Ng pzq of φpzq and gpzq respectively such that f : Nφ pzq Ñ Ng pzq is invertible with a conformal inverse. Let Nz “ φ ´1 pNφ pzq q. Then we have the following commutative diagram: Nz § § φđ

g

ÝÝÝÝÑ Ng pzq § § ´1 đf zÞÑz1

Nφ pzq ÝÝÝÝÑ Nφ pzq and we are done. Case 2. If φpzq is in Critpf q then there exist neighbourhoods Nφ pzq and Ng pzq of φpzq and gpzq respectively such that f : Nφ pzq Ñ Ng pzq is a branched covering of degree d ą 1, ramified only at φpzq. All such coverings are equivalent to z ÞÑ zd , in the sense that if ϕ : Ng pzq Ñ D is a Riemann map sending gpzq to 0, then ϕ can be lifted to a conformal map ϕr : Nφ pzq Ñ D, and we have the following commutative diagram:

58


Nz § § φđ

g

ÝÝÝÝÑ Ng pzq § § đId f

Nφ pzq ÝÝÝÝÑ Ng pzq § § §ϕ § ϕrđ đ D

zÞÑzd

ÝÝÝÝÑ

D

Since ϕr ˝ φ is K-quasiconformal and ϕ is conformal, we are done.

Finally for completeness we mention a fourth definition, which is common in functional analysis. Compare to the first analytic definition (Definition 1.5) of quasiconformality. Definition 1.36 (Fourth definition of K-quasiregular mapping) Let U Ď C be an open set and K ă 8. Set k :“ pK ´ 1q{pK ` 1q. A mapping g : U Ñ C is K-quasiregular if and only if the partial derivatives ˇ Bg ˇ andˇ Bgˇ exist in the ˇ ˇ ˇ ˇ 2 sense of distributions and belong to Lloc and satisfy ˇBg ˇ ď k ˇBg ˇ in L2loc . The continuity of the map g is a consequence of the definition, since any 1,2 factorizes as in Definition 1.33. See solution of the Beltrami equation in Wloc [AIM, Sect. 5.5] for details. In what follows, we state some straightforward properties of quasiregular mappings. We have actually used some of them already, but we state them here for future reference. Proposition 1.37 (Properties of quasiregular mappings) Let U, U 1 be open subsets of C. (i) If g1 : U Ñ U 1 and g2 : U 1 Ñ C are K1 - and K2 -quasiregular respectively, then g2 ˝ g1 is K1 K2 -quasiregular. (ii) A mapping g : U Ñ C is holomorphic if and only if g is 1-quasiregular. (iii) If f : U Ñ U 1 is holomorphic and φ : U 1 Ñ C is K-quasiconformal, then g :“ φ ˝ f is K-quasiregular. (iv) If in the third definition, we allow ϕ to be K 1 -quasiconformal (instead of conformal), then g is KK 1 -quasiregular. (v) If g : U Ñ C is quasiconformally conjugate to f : U Ñ C, and f is holomorphic, then g is quasiregular. (vi) (Variant of Weyl’s Lemma) If g : U Ñ C is quasiregular and g ˚ μ0 “ μ0 almost everywhere in U , then g is holomorphic.


59

Proof (i) Follows from the second definition. (ii) If g is holomorphic it is also 1-quasiregular. If g is 1-quasiregular, then g “ f ˝ φ, where f is holomorphic and φ is 1-quasiconformal. By Weyl’s Lemma φ is conformal, and hence g is holomorphic. (iii) Follows from (i). (iv) Locally g can be written as ϕ ´1 ˝ pz ÞÑ zd q ˝ ψ. By the second definition this implies that g is KK 1 -quasiregular. (v) Let φ be the K-quasiconformal conjugacy, so that g “ φ ˝ f ˝ φ ´1 . Since f ˝ φ ´1 is K-quasiregular and φ is K-quasiconformal, it follows that g is K 2 -quasiregular. (vi) We can write g “ f ˝ φ, where f is holomorphic and φ is quasiconformal. Then μ0 “ g ˚ μ0 “ φ ˚ f ˚ μ0 “ φ ˚ μ0 , almost everywhere in U . It follows from Weyl’s Lemma that φ is conformal. Hence g is holomorphic. For mappings between Riemann surfaces, these definitions generalize in the natural way: a mapping is quasiregular if it is locally quasiregular when expressed in charts. We end this section with an observation that is crucial for surgery. Lemma 1.38 (Pullbacks by quasiregular mappings) Quasiregular mappings and their inverse branches send sets of measure zero to sets of measure zero. Consequently, the pullback of a Beltrami form defined a.e., by a quasiregular map, is well defined a.e. Proof A quasiregular map g factors as g “ f ˝ φ with f holomorphic and φ quasiconformal. Since quasiconformal maps (and holomorphic maps in particular) are absolutely continuous with respect to the Lebesgue measure (see Theorem 1.15), it follows that quasiregular maps also have that property. The same arguments apply to inverse branches wherever defined. Arguing as in Remark 1.17, we conclude that pullbacks of Beltrami forms defined a.e., are well defined a.e. In general, when we write f

pS1 , μ1 q ÝÑ pS2 , μ2 q,

60


we mean that f is a quasiregular map between the Riemannn surfaces S1 and S2 and that f ˚ μ2 “ μ1 a.e., where μj for j “ 1, 2 is a Beltrami form defined on Sj .

1.7 Application to holomorphic dynamics Before giving an introduction in Chapter 4 to what quasiconformal surgery is about, we end the current chapter by getting a flavour of how to apply what we have learned so far to dynamical systems. The application is at the heart of surgery constructions. p We are interested in Suppose S is a Riemann surface isomorphic to C or C. dynamical systems generated by the iterates of a mapping f : S Ñ S, which we assume for the moment to be at least quasiregular. In other words, given an initial point s P S we are interested in the asymptotic behaviour of its orbit Opsq “ ts, f psq, f 2 psq, . . . , f n psq, . . .u. In the qualitative study of such dynamical systems, conjugacies play a crucial role, see Section 3.1. We are interested in finding a ‘holomorphic dynamical copy’, i.e. a holomorphic mapping which is quasiconformally conjugate to the given one. The following lemma states conditions under which this is possible. Lemma 1.39 (Key Lemma for surgery) p and let g : S Ñ S (a) Let S be a Riemann surface isomorphic to C or C, be quasiregular. Let μ be a g-invariant Beltrami form on S such that }μ}8 :“ k ă 1. Then there exists a holomorphic mapping f : X Ñ X p such that g and f are quasiconformally conjugate. where X P tC, Cu, (b) Let S 1 Ă S » D be an open set, and let g : S 1 Ñ S be quasiregular. Let μ be a g-invariant Beltrami form on S such that }μ}8 :“ k ă 1. Then there exists a holomorphic mapping f : D 1 Ñ D where D 1 Ă D is open and such that g and f are quasiconformally conjugate. Proof (a) Let φ : S Ñ X be a quasiconformal mapping that integrates μ, given by the Integrability Theorem. Then, if we define f :“ φ ˝ g ˝ φ ´1 , the following diagram commutes: g

pS, μq ÝÝÝÝÑ pS, μq § § §φ § φđ đ f

pX, μ0 q ÝÝÝÝÑ pX, μ0 q

1.7 Application to holomorphic dynamics

61

The mapping f is quasiregular and satisfies f ˚ μ0 “ μ0 almost everywhere. Hence by Weyl’s Lemma f is holomorphic and is quasiconformally conjugate to g via φ (see Figure 1.22). pC, μq

pC, μq g

φ pC, μ0 q

φ pC, μ0 q f

Figure 1.22 Sketch of the Key Lemma for surgery where S “ C.

(b) The proof is analogous to (a) taking D 1 :“ φpS 1 q.

As we shall see in later surgery constructions, such procedures may be performed with dependence on a parameter, say λ P Ă C. The mappings gλ : S Ñ S and the Beltrami forms μλ , which are gλ -invariant, may both depend holomorphically on λ. The Integrability Theorem depending on parameters then provides a family of integrating maps φλ , uniquely determined when normalized appropriately, and varying holomorphically with λ. Applying the Key Lemma to each of these mappings, we obtain a family of holomorphic mappings fλ : X Ñ X satisfying fλ ˝ φλ ” φλ ˝ gλ on S. It is natural to ask whether fλ depends holomorphically on λ. Often, one is able to deduce the form of the resulting family, for example a family of polynomials with coefficients in C, so that the question is equivalent to asking if the coefficients are holomorphic functions of λ. However, the answer is not always simple. As mentioned in Remark 1.32, the inverses of the integrating maps might not depend holomorphically on the parameter. However, we present a particular case where it is possible to prove the holomorphic dependence (cf. Section 4.1.2 and Claim 4.8). We refer to [BC, Prop. 13] for another computation of this type.

62


Lemma 1.40 (Dependence on parameters) Let g be given as in Lemma 1.39(a), and let the Beltrami forms μλ and therefore the integrating mappings φλ , appropriately normalized, depend holomorphically on λ. Then the holomorphic mappings fλ pzq :“ φλ ˝ g ˝ φλ´1 pzq depend holomorphically on λ. Proof Consider the expression fλ ˝ φλ ” φλ ˝ g. We want to show that ¯ λ pzqq “ 0 for all z. The mapping at the left-hand side can be Bfλ {B λpφ written as ¯ φλ pzq, φλ pzqq, ¯ z, zq ÞÑ fλ pλ, λ, pλ, λ, and the mapping at the right-hand side as ¯ z, zq ÞÑ φλ pλ, λ, ¯ gpzq, gpzqq. pλ, λ, Applying the chain rule when computing the derivatives in the sense of distributions we obtain that ˇ ˇ ˇ ˇ ˇ ˇ Bfλ ˇˇ Bφλ ˇˇ Bfλ ˇˇ Bφλ ˇˇ Bφλ ˇˇ Bfλ ˇˇ ` ¨ ` ¨ “ . ˇ Bz ˇφ pzq B λ¯ ˇz Bz ˇφ pzq B λ¯ ˇ B λ¯ ˇφ pzq B λ¯ ˇg pzq λ

λ

λ

z

Since BBφλ¯λ ” 0 and BBfzλ ” 0 we have ˇ Bfλ ˇˇ “ 0. B λ¯ ˇφλ pzq It follows that fλ is varying holomorphically with λ.

The justification for the chain rule in the sense of distributions follows from the fact that quasiconformal mappings can be approximated by C 8 functions (cf. [AIM, Cor. 5.5.8]). We emphasize that in the lemma above, the quasiregular map g is independent of the parameter. However, if it depends holomorphically on the parameter, then the resulting family of maps fλ need not do so. See Exercise 1.7.1 below. In Chapter 8 we shall nevertheless encounter several examples of surgery constructions with a given family of quasiregular mappings that depend holomorphically on a parameter, and so the resulting family of mappings also depend holomorphically on the parameter.

1.7 Application to holomorphic dynamics

63

Exercises Section 1.7 1.7.1 In the setup of Lemma 1.40, assume furthermore that the quasiregular map g depends holomorphically on λ. Prove that in such a case ˇ ˇ ˇ Bφλ ˇˇ Bgλ ˇˇ Bfλ ˇˇ “ ¨ . B z¯ ˇg pzq B λ¯ ˇz B λ¯ ˇφ pzq λ

λ

2 Boundary behaviour of quasiconformal maps: extensions and interpolations

In surgery constructions we often need to interpolate (quasiconformally) between two different maps defined on distinct regions of the plane. This involves finding quasiconformal extensions of given boundary maps, whenever possible, but also knowing whether a map defined on a certain domain extends continuously to its boundary. We address these two types of problems in this chapter. We recall some elementary definitions about curves and their regularity (see [Pom] for details and proofs). A curve γ in the complex plane is the image set of a parametric representation or parametrization. In the following we use γ both as a name for a continuous function γ : ra, bs Ñ C, parametrizing the curve, and for its image γ pra, bsq. The curve γ is closed if γ paq “ γ pbq, in which case it can be reparametrized as γ : T Ñ C; it is a Jordan arc if it has an injective parametrization, and a Jordan curve if in addition it is closed. The Jordan Curve Theorem states that the complement of any Jordan curve γ in the complex plane has exactly two connected components. A bounded domain whose boundary is a Jordan curve is called a Jordan domain. A curve has different degrees of regularity depending on the regularity of a parametrization. For n ě 1 a curve γ is C n if there exists a parametrization of γ which is n times continuously differentiable and satisfies γ 1 ptq ‰ 0 for all t. Curves which are C 1 are often referred to as smooth curves. The highest degree of regularity is achieved when a parametrization can be chosen so that it extends to a holomorphic univalent map in some horizontal band containing ra, bs or, in the closed case, in some annular neighbourhood of the unit circle. The set T “ R{Z is identified with the unit circle S1 :“ tz P C | |z| “ 1u by the parametrization t ÞÑ e2π i t . We often use complex or angular notation indistinctively, if it does not lead to confusion.

64

2.1 Preliminaries: quasisymmetric maps and quasicircles

65

In the problem of extending a given map defined on a domain to a map defined on its closure we use in general the following notation. Let G denote a Jordan domain bounded by a Jordan curve γ , and let f : D Ñ G be a homeomorphism. In general, f does not extend continuously to the boundary of D (see Exercise 2.2.1), but under certain assumptions on the regularity of f and γ it does, and we denote its extension by fp : D Ñ G. In such cases, we shall discuss the regularity of fp |S1 in terms of the regularity of f and γ . Note that whenever a continuous extension exists it is unique. In the problem of extending a given boundary map to the domain we use in general the following notation. Let γ be a Jordan curve bounding the Jordan domain G, and let f : S1 Ñ γ be a homeomorphism. Then f can always be extended continuously to a homeomorphism fp : D Ñ G in different ways. We shall discuss the maximal regularity of the extension in terms of the regularity of f and γ . Finally, we consider the same type of problems for annuli and quadrilaterals, as well as (briefly) covering maps of degree higher than one. The goal of this chapter is to collect all the results about extensions that will be used throughout the surgeries.

2.1 Preliminaries: quasisymmetric maps and quasicircles When dealing with quasiconformal mappings, we are led to the concept of quasisymmetry. Definition 2.1 (First definition of quasisymmetry) A map h : S1 Ñ C is quasisymmetric if h is injective and, for z1 , z2 , z3 P S1 , 0 ‰ |z1 ´ z2 | “ |z2 ´ z3 |

ñ

|hpz1 q ´ hpz2 q| 1 ď ďM M |hpz2 q ´ hpz3 q|

(2.1)

for some M ě 1. Sometimes h is called M-quasisymmetric or quasisymmetric of modulus M. In angular notation, i.e. writing H ptq :“ hpe2π i t q, the condition takes the form |H px ` tq ´ H pxq| 1 ď ď M, M |H pxq ´ H px ´ tq|

(2.2)

for all x, x ` t, x ´ t P T, with t ą 0 (see Figure 2.1). Note that if one of the inequalities is satisfied, the other follows by interchanging the roles of z1 and z3 , or the sign of t. The following is a formally stronger but actually equivalent definition of quasisymmetry, which is some times easier to apply. See [Pom, Chapt. 5].

66

Extensions and interpolations hpz3 q

z2 z1

Þ e2π i x 1 xÑ

0 x´t

x

z3 2π t 2π t

B

h hpz2 q A

x`t

hpz1 q

H Figure 2.1 The definition of a quasisymmetric map of the circle. The quotient of the length of the two segments A and B must be bounded for all triples of equidistant points in S1 , or equivalently for all x P T and all t ą 0.

Definition 2.2 (Second definition of quasisymmetry) A map h : S1 Ñ C is quasisymmetric if h is injective and if there exists a strictly increasing continuous function λ : r0, `8q Ñ r0, `8q such that ˆ ˙ |z1 ´ z2 | |hpz1 q ´ hpz2 q| 1 ´ ¯ď ďλ for z1 , z2 , z3 P S1 . |z ´z | |hpz2 q ´ hpz3 q| |z2 ´ z3 | λ 2 3 |z1 ´z2 |

(2.3) Note that if one of the inequalities in (2.3) is satisfied, the other follows by interchanging the roles of z1 and z3 . Note that if hpS1 q “ S1 and h is quasisymmetric then h´1 is also quasisymmetric. As in the case of quasiconformal maps, the existence of a continuous nonzero derivative is enough to guarantee quasisymmetry. Lemma 2.3 If h : S1 Ñ hpS1 q is a C 1 -diffeomorphism, then it is quasisymmetric. Proof Let H ptq “ hpe2π i t q. We shall prove that H satisfies a bilipschitz condition 1{K |x ´ y| ď |H pxq ´ H pyq| ď K |x ´ y|

(2.4)

for some K ą 0 and all x, y P T. From this it follows that H is K 2 quasisymmetric. Note that, independently of x, t P T we have ˇż t ˇ ˇ ˇ 1 1 1 ˇ |H px ` tq ´ H pxq ´ tH pxq| “ ˇ rH px ` τ q ´ H pxqsdτ ˇˇ ď k|t| 0

for some k ą 0 since

H1

is uniformly continuous on T.

2.1 Preliminaries: quasisymmetric maps and quasicircles Therefore G : T ˆ T Ñ R defined by # |H px q´H py q| Gpx, yq :“

|

|x ý | H 1 pxq |

67

if x ‰ y, if x “ y,

is continuous. Since the torus T ˆ T is compact, G attains both its maximal value and minimum value. The minimal value must be positive since H is injective and H 1 does not vanish on T. Hence, the inequality (2.4) is satisfied for some K ą 0. As we shall see, quasisymmetric mappings on S1 turn out to be precisely the boundary values (i.e. continuous extensions) of quasiconformal homeomorphisms on the unit disc. And, conversely, any quasisymmetric mapping on S1 can be extended to a quasiconformal mapping on D. It is in this sense that quasisymmetric mappings can be thought of as the one-dimensional analogue of quasiconformal mappings. But this analogy is not quite accurate, since quasisymmetric mappings are not as well behaved as quasiconformal mappings. For example, quasisymmetric mappings cannot in general be pasted together to form a quasisymmetric mapping. Consider, for instance, the homeomorphism H : T Ñ S1 defined as # 2 if 0 ď t ď 1{2, e4π i t H ptq :“ 2π i t e if 1{2 ď t ă 1. It is C 1 on pRz 12 Zq{Z but not quasisymmetric. Indeed, for x “ 0 and 0 ă t ď |H px `t q´H px q|

1{2 the fraction |H px q´H px ´t q| “ (see Figure 2.2).

sinp2π t 2 q sinpπ t q

is not bounded below when t Ñ 0

aptq 1 2

2t 2

H ptq

1 2

´ 12

´ 12

t

4π t 2 ´2π t

H p0q

H p´tq

Figure 2.2 Two quasisymmetric maps pasted together do not always produce a 1 argpH ptqq. quasisymmetric map. The left graph shows the map aptq “ 2π

68

Extensions and interpolations

Proposition 2.4 (Compositions of quasisymmetric maps) Let hj : S1 Ñ C, for j “ 1, 2, be quasisymmetric. (a) If h1 pS1 q “ S1 , then h2 ˝ h1 : S1 Ñ C is quasisymmetric. 1 1 1 (b) If γ “ h1 pS1 q “ h2 pS1 q, then h´ 2 ˝ h1 : S Ñ S is quasisymmetric. Proof For j “ 1, 2, let λj : r0, `8q Ñ r1, `8q denote a strictly increasing homeomorphism so that the inequality (2.3) is satisfied for hj . In case (a), set h “ h2 ˝ h1 . Then λ “ λ2 ˝ λ1 is a strictly increasing homeomorphism so that (2.3) is satisfied for h. Hence ´ h is quasisymmetric. ¯ ´1 1 In case (b), set h “ h2 ˝ h1 . Then λ “ 1{ λ´ 2 p1{λ1 q is a strictly increasing homeomorphism. From 1 ´

|hpz q´hpz q|

λ2 |hpz2 q´hpz3 q| 1 2

¯ď

` |z1 ´ z2 | ˘ |h1 pz1 q ´ h1 pz2 q| ď λ1 |h1 pz2 q ´ h1 pz3 q| |z2 ´ z3 |

it follows that |hpz1 q ´ hpz2 q| ď |hpz2 q ´ hpz3 q|

1

¸,

˜ 1 λ´ 2

λ1

´ 1

|z1 ´z2 | |z2 ´z3 |

¯

´ ¯ 1 so that (2.3) is satisfied for h with λ “ 1{ λ´ 2 p1{λ1 q . Hence h is quasisymmetric. Definition 2.5 (Quasiarc, quasicircle and quasidisc) A Jordan arc (resp. Jordan curve) γ in C is a quasiarc (resp. quasicircle) if for some C ą 0 diam γ pz1 , z2 q ď C|z1 ´ z2 | for z1 , z2 P γ , where γ pz1 , z2 q is the arc of smaller diameter of γ joining z1 and z2 . A Jordan domain bounded by a quasicircle is a quasidisc. A piecewise smooth Jordan arc is a quasiarc if and only if it has no cusps (pointing inwards or outwards). See Exercises 2.1.1, 2.1.2 and Figure 2.3. The geometric property associated with quasisymmetric parametrizations is the following, which in fact follows from Theorem 2.9 in the next section. Proposition 2.6 (Quasiarcs and quasicircles are curves with quasisymmetric parametrizations) If h : ra, bs Ñ C (resp. h : S1 Ñ C) is quasisymmetric, then its image γ is a quasiarc (resp. a quasicircle). Conversely, if γ is a quasiarc or a quasicircle, then it is the image of an interval or the unit circle under some quasisymmetric map.

2.2 Extensions of mappings from their domains to their boundaries

69

Figure 2.3 Left: The Koch snowflake is a well-known quasicircle, even if it is nowhere differentiable. Right: A cardioid is not a quasicircle due to its cusp.

Exercises Section 2.1 2.1.1 Let γ “ tpt, |t|α q| ´ 1 ď t ď 1u with α ą 0. Show that γ is a quasiarc if and only if α ě 1, by using the definition. 2.1.2 Let γ “ tpt, |t|α sinp2π {tqq| t P r´1, 0q Y p0, 1su Y tp0, 0qu with α ą 0. Show that γ is not a quasiarc if 0 ă α ă 2. 1 and bn “ 2n1`1 and apply the Hint: Take points of the form an “ 2n definition of quasiarc.

2.2 Extensions of mappings from their domains to their boundaries 2.2.1 Extensions of conformal maps We recall the following theorem (cf. [Pom, Chapt. 2]), which is an essential tool in holomorphic dynamics. We first recall the definition of a locally connected set. Definition 2.7 (Locally connected) A set X Ă C is locally connected if for every point in x P X and any arbitrarily small ą 0, the intersection D pxq X X is connected. Theorem 2.8 (Carathéodory Theorem) Let f : D Ñ G be a conformal isomorphism and G a bounded domain in C. Then f has a continuous extension fp : D Ñ G if and only if BG is locally connected. Moreover, f has a continuous and injective extension to D if and only if BG is a Jordan curve.

70


Maps on the unit disc We start by assuming that f : D Ñ G is a conformal isomorphism, in other words a Riemann mapping, and G Ă C is a Jordan domain with boundary γ :“ BG. It follows from Carathéodory’s Theorem that the map f has a continuous and injective extension to fp : D Ñ G. The extended map fp |S1 : S1 Ñ γ is called a conformal parametrization of γ . Given a Jordan curve γ , its conformal parametrization is uniquely determined up to pre-composition with a Möbius transformation preserving the unit disc. If we assume more regularity for γ we obtain more regularity for the extension, as explained in the following theorem. Theorem 2.9 (Extension of conformal maps on D) Let G Ă C be a Jordan domain with γ :“ BG, and let f : D Ñ G be a conformal isomorphism. Then: (a) f has a quasisymmetric extension to S1 if and only if γ is a quasicircle; (b) if γ is C n`1 for some n ě 0 then the maps f, f 1 , . . . , f pnq have continuous extensions to D; (c) f can be extended conformally to some disc of radius r ą 1 if and only if γ is analytic.

D

f G

Figure 2.4 If a homeomorphism f from D to a Jordan domain G extends continuously to the boundary, the regularity of the extension depends on the regularity of f and the one of γ :“ BG.

Remark 2.10 For part (b) to hold, it is enough that γ is of class C n,α for some 0 ă α ď 1; this means that γ is C n and γ pnq satisfies a Hölder condition of exponent α. The proofs of the different parts of the theorem can all be found in [Pom, Chapt. 3 and 5], more precisely for (a): Proposition 5.10 and Theorem 5.11; for (b) and the remark above: Theorem 3.6; and for (c): Proposition 3.1, a simple application of the Schwarz Reflection Principle. A consequence of this theorem is that if there exists a quasisymmetric (resp. differentiable or analytic) parametrization of γ , then for any Riemann mapping f : D Ñ G, the corresponding conformal parametrization is


71

quasisymmetric (resp. differentiable or analytic). The conformal parametrization is as good as it gets. Remark 2.11 (Extensions of coverings of degree d) Part (c) of Theorem 2.9 is an application of Schwarz Reflection Principle. Lifting the map and using exactly the same arguments, it can be shown that if f is a holomorphic covering of degree d ą 1, it also extends to a larger disc as a holomorphic covering of the same degree. In particular, the restriction to the boundary fp : S1 Ñ γ is an analytic covering map. Remark 2.12 (Piecewise regular curves) If the Jordan curve BG in Theorem 2.9 consists of finitely many arcs, which are either of class C n`1 or analytic, then a conformal map f : D Ñ G extends to the boundary as a piecewise smooth map of class C n or a piecewise analytic map respectively. To see this, observe that the Schwarz Reflection Principle applies to open sets that intersect the unit circle, hence the extension is local. We are also interested in doubly connected domains (annuli) bounded by Jordan curves and simply connected domains such as half strips that are not Jordan domains. We start with the doubly connected case. Jordan annuli A bounded annulus will be called a Jordan annulus (or a quasiannulus) if its outer and inner boundary curves are both Jordan curves (or both quasicircles resp.). For any 0 ă r ă 1 ă R set Ar,R :“ tr ă |z| ă Ru, Ar :“ Ar,1 “ tr ă |z| ă 1u and S1r :“ t|z| “ ru. Recall that any bounded open annulus A of finite modulus is conformally equivalent to a unique standard annulus Ar under a conformal map f : Ar Ñ A, which is uniquely determined up to pre-composition by a rotation. The following is a straightforward corollary of Theorem 2.9. Corollary 2.13 (Extension of conformal maps on annuli) Let A be a Jordan annulus and let f : Ar Ñ A be a conformal equivalence. Let γ i , γ o denote the inner and outer boundary of A. Then: (a) f has a quasisymmetric extension to S1 (S1r resp.) if and only if γ o (γ i resp.) is a quasicircle; (b) If γ o and γ i are C n`1 for some n ě 0 then the maps f, f 1 , . . . , f pnq have continuous extensions to Ar ; (c) f can be extended conformally to some annulus Ar,R where 1 ă R (resp. to Ar 1 where 0 ă r 1 ă r) if and only if γ o (resp. γ i ) is analytic.

72


Proof Let Go denote the Jordan domain of γ o , and let R : D Ñ Go be a p : D Ñ Go . conformal map which extends continuously to the boundary as R 1 o p The boundary map R : S Ñ γ has the properties described in Theorem 2.9. The composition R´1 ˝ f : Ar Ñ R´1 pAq is conformal. By the Schwarz Reflection Principle this map extends to an analytic map from S1 to itself. The boundary map ψ o : S1 Ñ γ o is obtained as the composition of this analytic p The properties for p Therefore ψ o has the same properties as R. map and R. the extended map to the inner boundaries is obtained similarly by using the p exterior domain of γ i in C. Half strips We continue by considering special horizontal half strips. The result is applied in Chapter 8. It suffices to consider the following case. Let γ be a C 2 -Jordan arc, parametrized by y P r0, 1s so that Im γ pyq is strictly, increasing between Im γ p0q “ 0 and Im γ p1q “ 1. A standard half strip of height 1 and bounded by γ is defined by ă :“ tz “ x ` i y | 0 ď y ď 1, x ď Re γ pyqu. 1,γ

See Figure 2.6. Considered as a subset on the Riemann sphere the half strip has three vertices. We denote the vertex at the left end by ´8. Any Riemann mapping, between the interiors of two such half strips, bounded by γ1 and γ2 respectively, extends continuously to the boundaries as piecewise analytic on the horizontal curves and C 1 from γ1 to γ2 (see Remark 2.12). In order to state the property of the horizontal boundary maps we need the following definition. The notion of near translation and its use was introduced by Bielefeld in [Bi1]. Definition 2.14 (Near translation) For j “ 1, 2, let j denote C 1 -Jordan arcs, which are backward invariant under translation by σ where Re σ ą 0, i.e. j ´ σ Ă j as sets. Let h : 1 Ñ 2 be a C 1 -diffeomorphism. Then h is said to be a near translation or C 1 -bounded if there exists a constant C ą 1 such that ˇ ˇ ˇ d hpzptqq ˇ 1 ˇ dt ˇ (2.5) ăˇ d |hpzq ´ z| ă C for any z P 1 and ˇ ă C, ˇ ˇ C dt zptq where t ÞÑ zptq is a C 1 -parametrization of 1 (see Figure 2.5). d d hpzptqq{ dt zptq is independent of the chosen parametrization of Note that dt 1 . Moreover, the composition of two near translations and the inverse of a


1

73

h

σ 2 C z

Figure 2.5 Two backwards σ -periodic Jordan arcs j for j “ 1, 2, and a C 1 -diffeomorphism h : 1 Ñ 2 . If h is a near translation then for any point z P 1 the image hpzq P 2 belongs to the open disc of radius C around z.

near translation are near translations, and any σ -periodic map, i.e. hpz ` σ q “ hpzq ` σ , is a near translation (see Exercise 2.2.2). Proposition 2.15 (Extension of conformal maps on standard half strips ă Ñ ă be the Riemann mapping between of equal height) Let f : 1,γ 1,γ2 1 the interior of the half strips whose continuous extension to the boundaries satisfies fpp´8q “ ´8, fppγ1 p0qq “ γ2 p0q and fppγ1 p1qq “ γ2 p1q. Set fr1 pxq :“ fppxq for x ď γ1 p0q and fr2 pxq :“ fppx ` i q ´ i for x ď Re γ1 p1q. Then frj , j “ 1, 2, are near translations. Proof Let 1 :“ tz “ x ` i y | 0 ď y ď 1u denote the full horizontal strip of height 1. The exponential map : z ÞÑ w “ u ` i v :“ eπ z maps 1 onto ă onto D X Hzt0u, the closure of the upper half plane minus the origin, 1,γ γ1 1 ă onto D X Hzt0u, where D may be chosen as the open Hzt0u, and 1,γ γ2 γj 2 topological disc, which is symmetric with respect to R and bounded by j “ eπ γj in H. Then f is conjugate by to a map F : Dγ1 X Hzt0u Ñ Dγ2 X Hzt0u, which extends to a conformal isomorphism F : Dγ1 Ñ Dγ2 , first by reflection with respect to R (the Schwarz Reflection Principle) and then, using that fpp´8q “ ´8, by continuity, setting F p0q “ 0 (see Figure 2.6). The derivative F 1 p0q is some positive number, since F is mapping an interval of R around 0 diffeomorphically onto an interval of R around 0 with the orientation preserved. By using the expansion of F around 0: F pwq “ F 1 p0qw ` Opw 2 q and by differentiating the equation

F ppzqq “ pf pzqq,

74

Extensions and interpolations xì

γ1 p1q

f px ` i q “ fr2 px q ` i

f

ă 1,γ 1 γ1 p0q

x

γ2 p1q

ă 1,γ 2 γ2 p0q

f px q “ fr1 px q

: z ÞÑ eπ z

: z ÞÑ eπ z 2

1

Dγ2

Dγ1 0

u “ px q

F

F púq

0

F puq

ú “ px ` i q

Figure 2.6 Sketch of the mappings and sets used in the proof of Proposition 2.15. ă we obtain for z P 1,γ 1

f 1 pzq “

F 1 pwq F 1 pwqw “ 1 Ñ 1 as z Ñ ´8 so that w Ñ 0. F pwq F p0q ` Opwq

Similarly, we obtain for x ď Re γ1 p0q, and x ď Re γ1 p1q respectively, that eπ rf1 px q´x s “

e π f px q F puq Ñ F 1 p0q “ π x e u

eπ rf2 px q´x s “

éπ f px ì q ´F púq Ñ F 1 p0q “ π x e u

r

r

p

and

p

as x Ñ ´8 so that u “ eπ x Ñ 0. Hence, for any choice of a constant C ą 1 the inequalities Ć ă frj pxq ´ x ă C

and

1 ă frj1 pxq ă C C

are satisfied for x sufficiently close to ´8, say for x ă x0 . On the remaining compact intervals rx0 , γ1 p0qs and rx0 , Re γ1 p1qs the continuous functions frj1 pxq and frj pxq ´ x attain their maximal and minimal values. Hence, it is easy to adjust the constant C ą 1 such that the inequalities are satisfied. Remark 2.16 (Half strips of different height) Note that if the half strips do not have the same height (here chosen to be 1) then fr1 and fr2 would not be near translations, since |frj pxq ´ x| would be unbounded as x tends to ´8 (see Exercise 2.2.4).


75

We shall see in Lemma 2.26, to be applied in Section 8.1, why it is essential to assume certain boundary mappings to be near translations.

2.2.2 Extensions of quasiconformal maps Next we change the assumption on f , requiring it to be quasiconformal. In this case, we only deal with discs and annuli. First observe that a continuous extension to the boundary may not exist a priori. Indeed Carathéodory’s Theorem does not apply in this more general setting. However, the extension does exist, and can be obtained as a simple and beautiful application of the Integrability Theorem in Section 1.4. The proof can be found for example in [Hu2, Sect. 4.9], but we also include it here. Theorem 2.17 (Homeomorphic extension of a quasiconformal self-map of D) Every K-quasiconformal homeomorphism f : D Ñ D (resp. f : H Ñ p to R). p H) extends continuously as a homeomorphism from S1 to S1 (resp. R Proof Suppose f : H Ñ H is K-quasiconformal, and let μf denote the Beltrami coefficient in H induced by f . Extend μf to C by reflection as follows:

μpzq :“

$ ’ ’ &μf pzq

if Impzq ą 0,

’ ’ %μ pzq f

if Impzq ă 0.

Then μ is symmetric with respect to R and has dilatation bounded by K. Let φ : pC, μq Ñ pC, μ0 q, fixing 0, 1, be the quasiconformal homeomorphism, which integrates μ. Observe that it must preserve the real line because of symmetry (compare with Exercise 1.4.1). Now consider the map g : pH, μ0 q Ñ pH, μ0 q defined as gpzq :“ f ˝ φ ´1 . By Weyl’s Lemma (Theorem 1.14), g is conformal. Applying the Schwarz Reflection Principle, g can be extended to a conformal map gp : C Ñ C, hence a homeomorphism when restricted to R. We therefore define the extension of f to R by setting p Ñ R, p fp :“ gp ˝ φ : R which is a homeomorphism as we wanted to prove. The same result holds for the unit disc, as seen by conjugating with the i Möbius transformation T pzq “ zz´ ì which maps H to D. Using a Riemann map, we obtain the following corollary.

76


Corollary 2.18 (Homeomorphic extension of a quasiconformal map on D) If G is a Jordan domain and f : D Ñ G is quasiconformal, then f extends continuously to a homeomorphism from D to G. Proof Consider a Riemann map R : D Ñ G which extends to the boundary because BG is locally connected. Apply the theorem above to the map R´1 ˝ f and then compose the extensions. In the proof of Theorem 2.17, we could have deduced right away that g is a Möbius transformation and hence globally defined. We used instead the Schwarz Reflection Principle to emphasize that it is not necessary to have the original map defined on the whole upper half plane. Indeed, the following is a corollary of the proof. Corollary 2.19 (Homeomorphic extension of a quasiconformal map on an annulus) Suppose U and V are one-sided neighbourhoods of S1 and f : U Ñ V is a quasiconformal homeomorphism so that |f pzq| Ñ 1 when |z| Ñ 1. Then f extends continuously to a homeomorphism of S1 to itself. Consequently, any quasiconformal map from a standard annulus Ar,R to an arbitrary annulus A extends to a homeomorphism from Ar,R to A. A more delicate problem is to study the regularity of the homeomorphic extension. The following proposition is a well-known result whose proof can be found for example in [Ah5, p. 40] (see also Exercise 2.2.5). Proposition 2.20 (Quasisymmetric extension of a quasiconformal self-map of D) Let f : D Ñ D be quasiconformal. Then f extends continuously to a quasisymmetric map fp : S1 Ñ S1 . Combining all results and ideas above we have the following statement. Corollary 2.21 (a) Let G Ă C be a quasidisc and f : D Ñ G a quasiconformal map. Then f extends continuously as a quasisymmetric map fp : S1 Ñ BG. (b) Let A be a quasiannulus with boundaries γ o and γ i , and let f : Ar Ñ A be a quasiconformal homeomorphism. Then f extends continuously to the boundaries as quasisymmetric maps.

Exercises Section 2.2 2.2.1 Prove that there exists a homeomorphism from D to itself which does not have a continuous extension to the unit circle. Hint: Consider for instance f prei θ q “ rei pθ `s pr qq , where s : r0, 1q Ñ r0, `8q denotes some increasing homeomorphism.

2.3 Extensions of boundary maps

77

2.2.2 Let j , j “ 1, 2, 3, be three Jordan arcs which are backwards invariant under translation by σ for some σ with Repσ q ą 0. Show that the composition h2 ˝ h1 of two near translations: hj : j Ñ j `1 for j “ 1, 2 is a near translation, and that the inverse of a near translation is a near translation. Show that any σ -periodic map h : 1 Ñ 2 is a near translation. 2.2.3 Let 1 “ 2 “ R´ . Give two examples of diffeomorphisms h : 1 Ñ 2 which are not near translations: one where |hpxq ´ x| is unbounded as x tends to ´8 but the derivative condition is satisfied, and another one where |hpxq ´ x| is bounded but the derivative condition is violated. 2.2.4 Go through the proof of Proposition 2.15 in the case where the two strips do not necessarily have the same height. In other words, substitute the ă by ă , where the height m ą 0 is arbitrary. As before strip 1,γ m,γ2 2 let f denote the unique Riemann mapping between the interior of the half strips whose continuous extension to the boundaries maps corners to corners. Observe that we then need two different exponential maps π 1 pzq :“ eπ z and 2 pzq :“ e m z in order to define the map F . Set F ˝ 1 “ 2 ˝ f . Prove that only for m “ 1, the mappings frj , j “ 1, 2, are near translations. More precisely, prove that: (a) f 1 pzq Ñ m as z Ñ ´8, r (b) eπ rf1 px q´x s Ñ F 1 p0q ¨ lim1 px qÑ0` rF p1 pxqqsm´1 as x Ñ ´8, and r as (c) eπ rf2 px q´x s Ñ F 1 p0q ¨ lim1 px qÑ0` r´F p´1 pxqqsm´1 x Ñ ´8. Deduce that |frj pxq ´ x| is unbounded as x Ñ ´8 if and only if m ‰ 1. 2.2.5 Theorem 1 in [Ah5, p. 40] states that every K-quasiconformal self–map of H, extends continuously to R as a quasisymmetric homeomorphism. Deduce Proposition 2.20 from this result. Hint: Lift to the upper half plane under an exponential map and compare the (chordal) distance between two points of the circle, with the length of the arc of S1 that joins them. Relate the latter to the Euclidian distance between two points (angles) on the real line.

2.3 Extensions of boundary maps We now proceed to consider the boundary value problem, first for homeomorphisms of the unit circle and later for general curves. Given an orientation preserving homeomorphism f : S1 Ñ S1 we want to find a continuous extension

78


fp : D Ñ D such that fp “ f on S1 . Clearly an extension, which is a homeomorphism, always exists: just extend radially fppre2π i t q :“ rf pe2π i t q). However, this extension has very bad properties at z “ 0, since the whole variation of fp is concentrated at this point. There are many other possible ways to extend f to a homeomorphism of the disc to itself. The most sophisticated extensions by Beurling–Ahlfors or by Douady–Earle have very good properties. Before addressing the general boundary value problem, we consider the easier cases where the boundaries are circles and the boundary maps are differentiable. Then the extensions can be constructed by interpolation without referring to more sophisticated tools.

2.3.1 Extensions of C 1 -boundary maps Strips and annuli We start by considering the annulus case. Lemma 2.22 (C 1 -interpolation on standard annuli) (a) Suppose 0 ă r1 , r2 ă 1 and f1 : S1 Ñ S1 and f2 : S1r1 Ñ S1r2 are two orientation preserving C 1 -diffeomorphisms. Then there exists a C 1 - diffeomorphism f : Ar1 Ñ Ar2 , which equals f1 on S1 and f2 on S1r1 . In particular, f is quasiconformal. (b) Moreover, suppose r2 pλq is a continuous function of a complex parameter λ P and that the C 1 -diffeomorphisms f2λ : S1r1 Ñ S1r pλq are such that 2

for any fixed z P S1r1 the map λ ÞÑ f2λ pzq is continuous. Then for any fixed z P Ar1 the map λ ÞÑ fλ pzq is continuous, where fλ is the quasiconformal extension above. Proof The extension f can be constructed by linear interpolation when lifted to the universal covering spaces. Consider the exponential map pzq :“ e2π i z r1 which maps the horizontal strip r 1 :“ t0 ď Im z ď r11 “ ´ log 2π u onto the 1

r2 annulus Ar1 , and likewise, the strip r 1 onto Ar2 with r21 “ ´ log 2π . Then 2 f1 and f2 can be lifted to maps F1 and F2 defined on the two respective boundaries of the horizontal strip, satisfying Fj pz ` 1q “ Fj pzq ` 1, for j “ 1, 2. Let us write Fr1 pxq :“ F1 pxq and Fr2 pxq :“ F2 px ` i r11 q ´ i r21 , where Frj pxq, j “ 1, 2, are strictly increasing C 1 -diffeomorphisms of R. We define the extension map F on the strip as the one sending every vertical segment with real part x linearly to the segment joining the points F1 pxq and

2.3 Extensions of boundary maps Ar1

79

f1

Ar2

f r1

r2

1

1

f2

F2 i r11

x ` i r11

F

x ` iy

x

0

i r21

F2 px ` i r11 q

r21 y i r11

F px ` i y q

0

F1 p x q

F1 Figure 2.7 Sketch of the proof of Lemma 2.22: definition of the interpolating map F : r 1 Ñ r 1 on the strips which projects to the interpolating map f : Ar1 Ñ Ar2 1 2 on the annuli with boundary values f1 and f2 .

F2 px ` i r11 q, i.e. scaling the imaginary part accordingly. More precisely (see Figure 2.7), ˆ ˙ r1 y y y F px ` i yq :“ 1 ´ 1 Fr1 pxq ` 1 Fr2 pxq ` i 21 . (2.6) r1 r1 r1 It is easy to check that F is C 1 with the Jacobian ˆˆ ˙ ˙ r21 y r1 y r1 pJac F qpx ` i yq “ 1 1 ´ 1 F1 pxq ` 1 F2 pxq ą 0 r1 r1 r1 for all x ` i y P r 1 . 1

From the fact that Fr1 and Fr2 are strictly increasing we get that F is injective. To see that F is surjective observe that F preserves horizontal lines. Since the image of a horizontal line in r 1 must be a connected set, the image is the 1 entire corresponding horizontal line in r 1 . 2 It follows that F : r 1 Ñ r 1 is a C 1 -diffeomorphism. Clearly F pz ` 1

2

1q “ F pzq ` 1, and hence F projects to a C 1 -diffeomorphism f : Ar1 Ñ Ar2

80


satisfying ˝ F “ f ˝ and with boundary values f1 and f2 . Since f is C 1 on a compact set, f is quasiconformal. Now suppose we have one fixed boundary map f1 : S1 Ñ S1 and other maps f2 “ f2λ : S1r1 Ñ S1r pλq depending continuously on a parameter λ P . Then, 2 equation (2.6) becomes ˙ ˆ r 1 pλqy y y Fλ px ` i yq :“ 1 ´ 1 Fr1 pxq ` 1 Fr2λ pxq ` i 2 1 . r1 r1 r1 The mappings Fλ : r 1 Ñ r 1 pλq depend continuously on λ, and so do the 1

2

projections fλ : Ar1 Ñ Ar2 pλq satisfying ˝ Fλ “ f ˝ .

Remark 2.23 (Differentiability of higher order) Observe that if the original boundary maps in Lemma 2.22 are C n , n ě 1, then the resulting map on the annuli is also. Rectangles and quadrilaterals Using the idea in the proof of the Interpolation Lemma 2.22 above we can extend piecewise C 1 -boundary maps of rectangles to the inside as explained in part (a) of the lemma below; in part (b) the extension is generalized to quadrilaterals. Lemma 2.24 (C 1 -interpolation on rectangles and quadrilaterals) (a) Let R1 , R2 be two rectangles and let f : BR1 Ñ BR2 be an orientation preserving piecewise C 1 -diffeomorphism, mapping corners to corners. Then f extends continuously to a C 1 -diffeomorphism fp : R 1 Ñ R 2 . In particular, the extension is quasiconformal. (b) Let Qj :“ Qj pzj,1 , zj,2 , zj,3 , zj,4 q, j “ 1, 2, be two quadrilaterals with piecewise C 2 boundary curves and let f : BQ1 Ñ BQ2 be an orientation preserving piecewise C 1 -diffeomorphism, mapping z1,k to z2,k . Then f extends continuously to a C 1 -diffeomorphism fp : Q1 Ñ Q2 . In particular, the extension is quasiconformal. Proof (a) Compare with Figure 2.8. Assume without loss of generality that R j “ r0, aj s ˆ r0, bj s for j “ 1, 2. The extension of f is constructed so that every vertical segment with real part x is mapped to the segment joining f pxq and f px ` i b1 q, and every horizontal segment with imaginary part y is mapped to the segment joining f pi yq and f pa1 ` i yq. It is easy to check that the extension is invertible and that both fp and its inverse are C 1 .


i b1

f px ` i b1 q

i b2

x ` i b1

81

f pi yq x ` iy

iy

fppx ` i yq

a1 ` i y

f pa1 ` i yq 0

x

a1

0 f pxq

a2

Figure 2.8 Illustrating how the map fp : R 1 Ñ R 2 is constructed by mapping line segments to line segments.

Since the closures of the rectangles are compact the map is quasiconformal. (see Exercise 2.3.4). (b) Choose rectangles Rj of the same modulus as Qj and similar to those above, and let Rj : Rj Ñ Qj denote the unique Riemann mapping which, when extended continuously to the boundary, maps vertices to vertices. p j : BRj Ñ BQj are pieceFrom Remark 2.12 it follows that the maps R 1 wise C -diffeomorphisms. By applying part (a) to the boundary map p 1 we obtain a C 1 -extension gp : R 1 Ñ R 2 , so that fp “ p ´1 ˝ f ˝ R g :“ R 2 ´ 1 p 2 ˝ gp ˝ R p is a C 1 -extension of f . R 1 The unit disc Using the C 1 -interpolation on annuli (Lemma 2.22), we can solve the extension problem to the disc in the differentiable case. Proposition 2.25 (Piecewise C 1 -extension to D) Let f : S1 Ñ S1 be an orientation preserving C 1 -diffeomorphism. Then there exists an extension fp : D Ñ D, which is a piecewise C 1 -diffeomorphism. In particular, the map fp is quasiconformal on D, and it may be chosen to equal the identity on a disc around z “ 0. Proof Choose any 0 ă r ă 1 and define fppzq :“ z for all z P Dr . Extend fp to the annulus Ar as in Lemma 2.22, with boundary values f1 “ f on S1 and f2 “ Id on S1r . We remark that as r approaches 1, the quasiconformal constant may get larger and larger.

82


Half strips In the proof of Lemma 2.22 (C 1 -interpolation on annuli) we obtain a quasiconformal map from one horizontal strip onto another as an extension of two boundary maps. These boundary maps are lifts of circle maps and hence invariant under translation by 1. We shall need a slight generalization of this lemma where we consider half horizontal strips and horizontal boundary maps which are near translations. Consider two half strips of heights r1 , r2 that are bounded by Jordan arcs γ1 , γ2 respectively, that is răj ,γj “ tz “ x ` i y | 0 ď y ď rj , x ď Re γj pyqu, where the Jordan arcs are parametrized by C 2 -functions with y P r0, rj s and so that Im γj pyq is strictly increasing from Im γj p0q “ 0 to Im γj prj q “ rj . Lemma 2.26 (C 1 -interpolation on half strips) Let fr1 : p´8, γ1 p0qs Ñ p´8, γ2 p0qs and fr2 : p´8, Re γ1 pr1 qs Ñ p´8, Re γ2 pr2 qs be increasing C 2 -diffeomorphisms, which are near translations. The boundary map from Bră1 ,γ1 to Bră2 ,γ2 consisting of the three diffeomorphisms f1 pxq “ fr1 pxq, f2 px ` i r1 q “ fr2 pxq ` i r2 , and fγ pγ1 pyqq “ γ2 p rr21 yq extends to a quasiconformal mapping f : ră1 ,γ1 Ñ ră2 ,γ2 . Proof Choose x0 ď γ1 p0q so that 1ă “ tz “ x ` i y | 0 ď y ď r1 , x ď x0 u is contained in ră1 ,γ1 and 2ă “ tz “ x ` i y | 0 ď y ď r2 , x ď p1 ´ y r y r ă ă ă r1 qf1 px0 q ` r1 f2 px0 qu is contained in r2 ,γ2 . Define f : 1 Ñ 2 by ˙ ˆ y r2 y r f1 pxq ` fr2 pxq ` i y. f px ` i yq “ 1 ´ r1 r1 r1 This is a C 2 -diffeomorphism between the non-compact horizontal half strips extending the boundary map restricted to the boundary of 1ă . Its Jacobi matrix is ff « p1 ´ ry1 qfr11 pxq ` ry1 fr21 pxq r11 pfr2 pxq ´ fr1 pxqq . Dz f “ r2 0 r1 By assumption it follows that there exists a constant C ą 1 so that C1 ă p1 ´ ry1 qfr11 pxq ` ry1 fr21 pxq ă C for all z P 1ă . The Jacobian therefore satisfies r2 1 r2 r1 C ď det Dz f pzq ď r1 C, i.e. it is uniformly bounded away from 0 and 8.


83

Moreover, since all elements in the Jacobi matrix are uniformly bounded it follows that the norm }Dz f pzq} is uniformly bounded. Hence f : 1ă Ñ 2ă is quasiconformal (recall Exercise 1.3.3 in Section 1.3). To obtain the full extension we consider the remaining quadrilaterals, i.e. the closures of răj ,γj zjă . The boundary curves are piecewise C 2 and the boundary map is piecewise C 1 , so we can extend quasiconformally to the interior, as stated in part (b) of Lemma 2.24.

2.3.2 Extensions of quasisymmetric boundary maps Simple approaches as in the preceding section do not work in the general case when the boundary maps are not differentiable. However, it is remarkable that if f is quasisymmetric on the unit circle, an extension can be constructed which is quasiconformal and even real analytic in the interior of the disc, as established in this section.

The unit disc The following important result deals with extensions to the unit disc, of quasisymmetric maps of the unit circle, and is due to Beurling–Ahlfors [BA] and Douady–Earle [DE] (cf. [Pom, Thm. 5.15] or [Hu2, Thm. 4.9.5 and 5.1.2]). Theorem 2.27 (Quasiconformal extensions of quasisymmetric maps) Suppose f : S1 Ñ S1 is an orientation preserving quasisymmetric homeomorphism. Then there is an extension fp : D Ñ D of f which is real analytic in D and has the following properties: (a) if σ, τ P MöbpDq then the extension of σ ˝ f ˝ τ is given by σ ˝ fp ˝ τ (b) fp is quasiconformal in D. The Beurling–Ahlfors extension does not satisfy (a) but it is simple to construct. We shall extract some general facts about Beurling–Ahlfors extensions, which are useful when extending quasisymmetric maps to annuli. The Beurling–Ahlfors extension Given an M-quasisymmetric map f : R Ñ R, which by definition satisfies equation (2.2) for all x, t P R, then (one of) the (possible) Beurling–Ahlfors extension Bf of f to the upper half plane H is given by

84


Bf px ` i yq :“

1 2

ż1 tf px ` tyq ` f px ´ tyqudt 0

ż1 ì

tf px ` tyq ´ f px ´ tyqudt, 0

for all x P R and y ě 0. Clearly its restriction to R is precisely f . Other variants of the extension are obtained by keeping the real part of Bf and changing the imaginary part by multiplying it by any positive constant. Compare with [LV, p. 83], where the constant is 1{2. Our normalization is chosen so that, as we shall see, the horizontal line through i is invariant under Bf when f is a lift of a quasisymmetric map from the circle T “ R{Z onto itself. The Beurling–Ahlfors extension Bf can be rewritten as Bf px ` i yq “

1 2y

ż x `y f psq ds ` i x ý

1 y

ˆż x ` y

żx f psq ds ´

x

˙ f psq ds ,

x ý

which shows that it is continuously R-differentiable, using that f 1 pxq exists almost everywhere since f is quasisymmetric. By direct estimation, using the quasisymmetry of f , one can check that Bf is K-quasiconformal with a K which depends only on M. If f is a lift of a quasisymmetric map from the circle T onto itself, then it satisfies f px ` 1q “ f pxq ` 1. In this case Bf pz ` 1q “ Bf pzq ` 1. Furthermore, one can check that the imaginary part of Bf px ` i q is żx tf pt ` 1q ´ f ptqu dt “ 1, x ´1

so that the horizontal line tIm z “ 1u is invariant. It also follows that f pxq “ x ` θ pxq, where θ pxq is a periodic function of period 1, i.e. θ px ` 1q “ θ pxq. We have that the real part of Bf px ` i q is 1 2

ż x `1 pt ` θ ptqq dt “ x ` α, x ´1

where α is the integral of θ over one period. The map x ` i ÞÑ Bf px ` i q is therefore the translation Tα by α on that horizontal line. Standard annuli A variation of the Beurling–Ahlfors extension can be used to interpolate between quasisymmetric boundary maps of standard annuli, as we show in the following proposition.


85

Proposition 2.28 (Extension of quasisymmetric maps between boundaries of standard annuli) Let 0 ă r1 , r2 ă 1 and suppose f1 : S1 Ñ S1 and f2 : S1r1 Ñ S1r2 are two orientation preserving quasisymmetric homeomorphisms. Then there exists an extension f : Ar1 Ñ Ar2 which is quasiconformal in the interior. Proof We use the notation in the proof of Lemma 2.22. First we lift the annuli to the strip 3 :“ tz | 0 ď Im z ď 3u. We denote by r1 : 3 Ñ Ar1 and r2 : 3 Ñ Ar2 the respective projections, which consist of the exponential composed with a rescaling of the imaginary part. Observe that these projections are not conformal but locally quasiconformal. The maps f1 and f2 lift to maps on the boundaries of the strip denoted by F1 and F2 respectively. Let Fr1 and Fr2 denote the corresponding real mappings defined so that Fr1 pxq :“ F1 pxq and Fr2 pxq :“ F2 px ` 3i q ´ 3i . Further, we may write Frj pxq “ x ` θj pxq for j “ 1, 2, where θj are periodic functions of ş1 period 1. In both cases, set αj “ 0 θj ptq dt. We define the extension map F in three substrips of 3 : one in the strip 1 “ tz | 0 ď Im z ď 1u, another in 1,2 “ tz | 1 ď Im z ď 2u and a third in 2,3 “ tz | 2 ď Im z ď 3u (see Figure 2.9). The idea is as follows: On the bottom substrip 1 we define F :“ B Fr1 , which satisfies F px ` i q “ x ` α1 ` i . On the top substrip 2,3 we define an appropriate modification of the Beurling–Ahlfors extension of Fr2 , namely F pzq :“ c ˝ B Fr2 ˝ c´1 , where c is the map cpzq “ z ` 3i . Hence F px ` 2i q “ x ` α2 ` 2i . Finally, on the middle strip 1,2 we extend F by an affine map, which fits with the boundary values x ` i ÞÑ x ` α1 ` i and x ` 2i ÞÑ x ` α2 ` 2i on its lower and upper boundary respectively. The three pieces then match continuously and are quasiconformal on their domains. Hence the map F is quasiconformal on 3 . Since F satisfies F pz ` 1q “ F pzq ` 1, it projects to 1 f : Ar1 Ñ Ar2 satisfying ´ r2 ˝ F “ f ˝ r1 . We have obtained a quasicon formal map on the annulus with boundary values f1 and f2 as desired. Quasidiscs and quasiannuli Next, we consider similar extension problems from given boundary maps, where the domains are quasidiscs or quasiannuli. First we generalize the definition of a quasisymmetric map. Definition 2.29 (Third definition of quasisymmetry) For j “ 1, 2, let γj be two quasicircles, Gj the corresponding Jordan domains, and Rj : D Ñ Gj two Riemann mappings, giving rise to the conformal parametrizations p j : S1 Ñ γj . An orientation preserving homeomorphism f : γ1 Ñ γ2 is R

86

Extensions and interpolations Ar1

f1

Ar2

f r1

r2

1

1

f2

r2

r1 F2 px ` 3i q “ Fr2 px q ` 3i 3i

3i

B Fr2

2,3

2i

x0 ` 2i

x ` α2

F

1,2 i

2i

x0 ` α2 ` 2i

i x ì B Fr1 0

1

x0 ` α1 ` i x ` α1

0

0 F1 “ Fr1

Figure 2.9 The quasiconformal extension F is defined in three pieces with boundary values F1 , Tα1 , Tα2 and F2 on the horizontal lines of height 0, 1, 2 and 3 respectively. On 1 and 2,3 , the extension F is the Beurling–Ahlfors extension, appropriately defined. On 1,2 we interpolate linearly between Tα1 and Tα2 .

p 1 : S1 Ñ γ2 is quasisymmetric, and equivquasisymmetric if and only if f ˝ R ´ 1 p 1 : S1 Ñ S1 is quasisymmetric. p ˝f ˝R alently if and only if R 2 Observe that the notion is independent of the choices of conformal parametrizations. The proof that the statements are equivalent is left as Exercise 2.3.2. With this definition, the following proposition is straightforward. Proposition 2.30 (Extension of quasisymmetric maps between boundaries of quasidiscs and quasiannuli) (a) Suppose G1 and G2 are quasidiscs bounded by γ1 and γ2 , and let f : γ1 Ñ γ2 be quasisymmetric. Then f extends to a quasiconformal map fp : G1 Ñ G2 . (b) For j “ 1, 2, suppose Aj are open quasiannuli bounded by the quasicircles γji , γjo . Let f i : γ1i Ñ γ2i and f o : γ1o Ñ γ2o be quasisymmetric maps between the inner and outer boundaries respectively. Then there exists a quasiconformal map f : A1 Ñ A2 extending the boundary maps f i and f o .


87

Proof Start by choosing Riemann mappings Rj : D Ñ Gj . Their extensions p j : S1 Ñ γj are quasisymmetric, and so is the composition to the boundaries R ´ 1 p 1 : S1 Ñ S1 . It follows from Theorem 2.27 that g can be p ˝f ˝R g :“ R 2 extended to a quasiconformal map gp : D Ñ D. The map fp “ R2 ˝ gp ˝ R´1 : 1

G1 Ñ G2 is quasiconformal and is an extension of the boundary map f (see also Figure 2.10). f γ2 γ1 p f G2 G1

p1 R

R1

D

R2 gp

p2 R

D

g Figure 2.10 Sketch of the proof of Proposition 2.30(a). Quasisymmetric maps between quasicircles extend to a quasiconformal map between the interior domains.

In the proof of (b) start by choosing ϕj : Arj Ñ Aj to be conformal isomorphisms where rj is uniquely determined by the moduli of the annuli. Let ϕpjo : S1 Ñ γjo and ϕpji : S1rj Ñ γji be the continuous extensions of ϕj to the outer and inner boundaries respectively. It follows from Corollary 2.13(a) that these maps are quasisymmetric, and therefore it follows from Proposition 2.28 that there exists a quasiconformal map g : Ar1 Ñ Ar2 extending the ϕ2o q´1 ˝ f o ˝ ϕp1o and g i :“ pp ϕ2i q´1 ˝ f i ˝ ϕp1i . Hence boundary maps g o :“ pp ´1 f :“ ϕ2 ˝ g ˝ ϕ1 : A1 Ñ A2 is quasiconformal and is a continuous extension of the boundary maps f o and f i (see also Figure 2.11). Remark 2.31 In both cases, if the curves are C 2 and the maps are C 1 , then the extensions can also be (piecewise) C 1 in G1 or A1 respectively. See Theorem 2.9, Corollary 2.13, Proposition 2.25 and Lemma 2.22. We end this section by considering an interpolation depending on parameters, analogous to the case in Lemma 2.22. Proposition 2.32 (Quasiconformal interpolation on quasiannuli) For λ P a complex parameter, let A1 and Aλ2 be bounded annuli of finite modulus,

88

Extensions and interpolations fo A1

γ1i

γ2i

f γ2o

γ1o

A2

fi ϕp1i

ϕ1

ϕp2o

ϕp1o

ϕ2i

ϕ2

go

Ar1 r1

1

g

Ar2 r2

1

gi Figure 2.11 Sketch of the proof of Proposition 2.30(b). Quasisymmetric maps between quasiannuli boundaries extend to quasiconformal maps of the annuli.

bounded by quasicircles γ1o , γ2o , γ1i and γ2i pλq, where the boundary γ2i pλq depends continuously on λ. Let fλi : γ1i Ñ γ2i pλq and f o : γ1o Ñ γ2o be quasisymmetric between the inner and outer boundaries respectively such that, for any fixed z P γ1i , the map λ ÞÑ fλi pzq is continuous. Then there exists a quasiconformal map fλ : A1 Ñ Aλ2 with boundary values fλi and f o , such that for any fixed z P A1 , the map λ ÞÑ fλ pzq is continuous. To prove this proposition we use the following lemma, which we prove at the end of the section. Lemma 2.33 (Continuity of the modulus and uniformizing maps) Let Ă C be open and bounded. Let tAλ uλP be a family of Jordan annuli, with common outer boundary γ o and inner boundary γλi depending continuously on λ. Let 0 ă rpλq ă 1 be so that Aλ – Ar pλq . Then rpλq depends continuously on λ and consequently so does the modulus of Aλ . Moreover, choose an arbitrary point z0 P γ o and normalize the uniformizing maps ϕλ : Ar pλq Ñ Aλ so that their extensions to the boundaries satisfy ϕpλ p1q “ z0 for all λ P . Then λ ÞÑ ϕλ pzq depends continuously on λ for any fixed z whenever it makes sense.


89

Proof of Proposition 2.32 Let r1 and r2 pλq denote the real numbers such that A1 – Ar1 and Aλ2 – Ar2 pλq respectively. Since γ2o is fixed and γ2i pλq depends continuously on λ so does r2 pλq due to the first part of Lemma 2.33. Choose conformal isomorphisms ϕ1 : Ar1 Ñ A1 and ϕ2λ : Ar2 pλq Ñ Aλ2 . These mappings extend to the boundaries. Normalize the mappings ϕ2λ so that the extensions satisfy ϕp2λ p1q “ z0 for some chosen point z0 P γ2o . Then, due to the second part of Lemma 2.33, the map λ ÞÑ ϕ2λ pzq depends continuously on λ for any fixed z whenever it makes sense. As in the proof of Proposition 2.30 define the induced boundary maps from the boundaries of Ar1 to the boundaries of Ar2 pλq . Then apply Lemma 2.22 to obtain the quasiconformal extension gλ : Ar1 Ñ Ar2 pλq , depending continuously on λ. Finally, compose with the uniformizing maps ϕ1´1 and ϕ2λ and the statement follows. We now proceed to prove Lemma 2.33. This lemma can be deduced from [Com, Thm. 3.2] (see also [Ep, Lem. 6]), where the convergence of bounded multiply connected pointed domains in C and the relation to their normalized uniformizing maps is treated in general. However, because of simplicity, we have chosen to give a direct proof in the doubly connected case. 1 Proof of Lemma 2.33 Fix any λ P , and let λn be a sequence of parameters in that converges to λ. We shall suppress λ from the notation and set γni :“ γλin ,

γ i :“ γλi ,

An :“ Aλn ,

A :“ Aλ ,

rn :“ rpλn q,

r :“ rpλq

and ϕn :“ ϕλn : Arn Ñ An ,

ϕ :“ ϕλ : Ar Ñ A, with ϕn p1q “ z0 “ ϕp1q.

It follows from the assumptions that γni converges to γ i in the Hausdorff topology. To prove the lemma it suffices to prove that rn converges to r and that ϕn converges to ϕ uniformly on every compact subset of Ar as n tends to infinity. Choose an arbitrary r 1 P pr, 1q. Then ϕpAr 1 q Ă An for n ě N pr 1 q, where Npr 1 q is some constant large enough. Then the composition ψn :“ ϕn´1 ˝ ϕ : Ar 1 Ñ Arn . is well defined and univalent on Ar 1 . The extended boundary map on the outer boundary is mapping S1 homeomorphically onto itself, fixing 1. It follows that 1 We thank Xavier Buff for providing this proof.

90


mod Arn ě mod Ar 1 and therefore that rn ď r 1 for all n ě N pr 1 q. Since r 1 P pr, 1q is arbitrary, we have lim supn rn ď r. Extracting a subsequence if necessary we may assume that rn tends to r ˚ P r0, rs as n tends to infinity and that ϕn : Arn Ñ An converges uniformly on every compact subset of Ar ˚ to some map ϕ8 : Ar ˚ Ñ A. The latter is obtained by the following argument. On any open subset compactly contained in Ar ˚ , the sequence tϕn u forms a normal family (see Montel’s Theorem 3.23). Hence, it has a convergent subsequence. A diagonal argument shows that there exists a subsequence that converges uniformly on compact subsets of Ar ˚ . In order to prove that r ˚ “ r and ϕ8 “ ϕ we consider again the mappings ψn “ ϕn´1 ˝ ϕ and extend them by reflection in S1 to ψn : Ar 1 ,1{r 1 Ñ Arn ,1{rn . The restriction of ψn to S1 is a homeomorphism, fixing 1. Recall that ψn is defined on Ar 1 ,1{r 1 for all n ě N pr 1 q where r 1 P pr, 1q is arbitrary. By choosing a decreasing sequence rk1 Ñ r as k Ñ 8 and restricting to a subsequence nk ě Nprk1 q with nk Ñ 8 as k Ñ 8, we have ψnk : Ar 1 ,1{r 1 Ñ Arn k

k

k ,1{rnk

.

Extracting once again a subsequence if necessary, we may assume that the sequence ψnk converges uniformly on every compact subset of Ar,1{r to ψ8 : Ar,1{r Ñ Ar ˚ ,1{r ˚ . The restriction of ψ8 to S1 is a homeomorphism, fixing 1. Moreover, it is univalent on Ar,1{r . On Ar the map ϕ8 ˝ ψ8 is the limit of ϕnk ˝ ψnk “ ϕ : Ar 1 Ñ Ank , and therefore ϕ8 ˝ ψ8 “ ϕ : Ar Ñ A. In particular, ϕ8 : k Ar ˚ Ñ A is not constant, thus univalent. It follows that mod Ar ˚ ď mod A and therefore that r ˚ ě r, hence r ˚ “ r. Moreover, ψ8 : Ar,1{r Ñ Ar,1{r is an isomorphism, fixing 1, and therefore equal to the identity. Since ϕ8 ˝ ψ8 “ ϕ on Ar we conclude that ϕ8 “ ϕ : Ar Ñ A as required.

Exercises Section 2.3 2.3.1 (a) In the case where f1 : S1 Ñ S1 and f2 : S1r1 Ñ S1r2 are two orientation preserving C 1 -maps of degree n, construct a C 1 -extension f : Ar1 Ñ Ar2 which is a covering map of degree n. Compare with Lemma 2.22.


91

(b) Let Ar1 and Ar2 be two annuli, compactly contained in C and whose inner and outer boundaries are C 2 curves. Suppose the inner and outer boundary maps f i : BAir1 Ñ BAir2 and f o : BAor1 Ñ BAor2 are given as orientation preserving C 1 -maps of degree n. Then construct a C 1 -extension f : Ar1 Ñ Ar2 , which is a covering map of degree n (see also Exercise 2.3.3). 2.3.2 Using Proposition 2.4, show that Definition 2.29 is independent of the choices of conformal parametrizations, and that the two statements are equivalent. 2.3.3 Let γjo for j “ 1, 2 and γ1i , and γ2i pλq be C 2 curves, with γ2i pλq depending continuously on λ, and being outer and inner boundaries of annuli A1 and Aλ2 in C. Suppose fλi : γ1i Ñ γ2i pλq and f o : γ1o Ñ γ2o are given as orientation preserving C 1 -maps of degree n. Then construct a C 1 extension fλ : A1 Ñ Aλ2 , which is a covering map of degree n, such that for any fixed z P A1 , the map λ ÞÑ fλ pzq is continuous. Compare with Proposition 2.32. 2.3.4 Write an explicit expression for the extension of the boundary map f : BR1 Ñ BR2 in Lemma 2.24 to a map fp : R 1 Ñ R 2 , and check that it is a C 1 -diffeomorphism.

3 Preliminaries on dynamical systems and actions of Kleinian groups

In this book we apply the technique of quasiconformal surgery to holomorphic dynamical systems, or more precisely to the dynamical systems generated by the iteration of a holomorphic map f : S Ñ S, where S is a Riemann surface, i.e. a complex manifold of (complex) dimension 1. For any given initial condition z0 P S we consider the (forward) orbit of z0 under f , that is the sequence Opz0 q “ O` pz0 q “ tz0 , z1 “ f pz0 q, . . . , zn “ f pzn´1 q “: f n pz0 q, . . .u. To understand the dynamical system generated by the iterates of f means to understand the fate of all orbits, in terms of their initial condition, i.e. their asymptotic behaviour when the time n tends to infinity. To fix notation, we mention some examples of long-term behaviour. Under f , the point z0 (or its orbit) can be fixed if f pz0 q “ z0 ; periodic of period p ą 1, or p-periodic, if f p pz0 q “ z0 and p is the smallest number for which this occurs; preperiodic if f k pz0 q is periodic for some k ą 0, but z0 is not; ‘converging’ if f np pz0 q Ñ z˚ as n Ñ 8 for some p ě 1 (then, z˚ is p-periodic if p is minimal with respect to this property). A p-periodic orbit is also called a p-cycle. A point z0 P S is called recurrent if the orbit of z0 intersects any arbitrarily small neighbourhood of z0 . Likewise, U Ă S is a recurrent set if f n pU q intersect U for infinitely many values of n. As a consequence of Montel’s Theorem 3.23, the Uniformization Theorem 1.22, and the study of dynamics on D and on the torus, it can be shown that this theory is interesting and non-trivial in only three cases (up to normalp (rational maps), C (entire transcendental ization) [Mi1, §2,5 and 6], namely C ˚ maps) and C “ Czt0u (transcendental self-maps of the punctured plane). 92

Preliminaries on dynamical systems and actions of Kleinian groups

93

Although polynomials are entire maps we classify them here as particular examples of rational maps since they can be extended to the Riemann sphere. In contrast, entire transcendental maps have an essential singularity at infinity and cannot be extended to the whole sphere. The exponential map and the elementary trigonometric functions are examples. Transcendental self-maps of C˚ have two essential singularities, one at infinity and one at the origin. Examples of such are ez`1{z or complexifications of certain maps of the circle, as we will see later. Meromorphic transcendental maps (for example the tangent map) also present interesting dynamics. The basic definitions take different forms for these maps, since orbits of poles and prepoles are truncated. For this reason, we shall not include them in the general exposition although we might occasionally treat them separately (for example in Chapter 5). p ÑC p is a rational map. Hence f pzq “ P pzq , A holomorphic map f : C Qpzq where P and Q are polynomials with no common factors. The degree of f is defined as d “ maxtdegpP q, degpQqu. p has This is then the topological degree of f in the sense that every point in C exactly d preimages under f when counted with multiplicity. For d “ 1 the map f is a Möbius transformation and it is a conformal isomorphism. Interp require a rational map of degree d ě 2. The topological esting dynamics on C degree of a transcendental map is always infinite. To simplify notation we name the following classes of maps: • • • •

p ÑC p | f is rational of degree at least twou; Rat “ tf : C Ent “ tf : C Ñ C | f is entire transcendentalu; Ent˚ “ tf : C˚ Ñ C˚ | f is holomorphic transcendentalu; p | f is transcendental meromorphic with at least one Mer “ tf : C Ñ C pole, which is not omitted u.

A subset of Rat is the class of polynomials which we denote by Pol. For any degree d ě 2 we shall also consider the classes Ratd and Pold , which denote respectively rational maps and polynomials of degree d. Holomorphic dynamical systems are special cases of discrete dynamical systems f : X Ñ X, where f is only required to be continuous and X is any topological or metric space. Occasionally we shall consider non-holomorphic dynamics like, for example, iteration of functions on the unit circle (see Section 3.2 below). p we consider its standard repreWhen we work on the Riemann sphere C 3 sentation as the unit sphere in R , with the complex plane C as the horizontal

94


equator plane. Points in C are projected to the sphere by stereographic prop we use the spherical metric and the chordal metric, which are jection. On C equivalent to each other. For completeness and later use we give an explicit expression (see [Bea, §2.1]) for the chordal distance σ between two points in p represented by corresponding points z, w P C, namely C σ pz, wq “

2|z ´ w| p1 ` |z|2 q1{2 p1 ` |w|2 q1{2

(3.1)

,

and σ pz, 8q “ lim σ pz, wq “ wÑ8

2 p1 ` |z|2 q1{2

.

(3.2)

A useful relation between σ and the spherical distance dCp is given by 2 d pz, wq ď σ pz, wq ď dCp pz, wq. π Cp

(3.3)

In this chapter we give a brief survey of the basic theory of holomorphic dynamics and analytic circle maps focusing on results that are used in the book. It is not our intention to reproduce the excellent texts which can be found in the literature – instead we refer to them for proofs and for a more detailed exposition than the one given here. The last section contains a brief survey of the actions of Kleinian groups and the Sullivan dictionary, summarizing the striking analogies between holomorphic dynamics and the dynamics of Kleinian groups. Quasiconformal deformations were used in the context of Kleinian groups long before they appeared in holomorphic dynamics. The section should give the reader sufficient background to appreciate the references to the analogies that appear in several places in the book. As general standing references for the basic theory of holomorphic dynamics we recommend [Bea, CG, Mi1, MNTU, St] and also [Bl]; for the background on the iteration of entire transcendental maps (and also meromorphic maps) [Ber, EL1] and [HY]. For more advanced issues we refer to the books and collections [Dev1, FSY, HP, K, Kri, McM2, RS, Sc] and [Ta2]. References for Kleinian groups are mentioned in the dedicated section.

3.1 Conjugacies and equivalences Conjugacies are the main tool used for comparison and classification of dynamical systems. In this section, X and Y are topological or metric spaces and f : X Ñ X and g : Y Ñ Y are continuous maps, unless otherwise

3.1 Conjugacies and equivalences

95

specified. Most of the facts which follow can easily be checked, and those which are not obvious can be found for example in [Ro]. Definition 3.1 (Conjugate dynamical systems) We say that f and g are topologically conjugate if there exists a homeomorphism h : X Ñ Y such that h ˝ f “ g ˝ h, i.e. the following diagram commutes: f

X ÝÝÝÝÑ § § hđ

X § § đh

g

Y ÝÝÝÝÑ Y h

We write f „ g, or f „ g if the topological conjugacy h needs to be specitop

top

fied. If h can be chosen to be C r with 0 ă r ď 8 (resp. linear, affine) we say that f and g are C r -conjugate (resp. linearly conjugate, affine conjugate). If p we shall also use the terms conformally (resp. quasiconformally) X, Y Ă C conjugate with the obvious definition. These definitions also make sense locally. Observe that when X “ Y “ C, conformal conjugacies are affine conjugacies. Conjugate systems are ‘the same’ up to a change of variables. The regularity of the conjugacy makes this identification more or less restrictive. Conjugacies preserve orbits. Indeed, if f „ g then f n „ g n . Therefore, top

top

Of px0 q is mapped bijectively onto Og phpx0 qq by h. In particular, periodic orbits are mapped onto periodic orbits of the same period. Properties which are preserved under conjugacies are important and receive a special name. Definition 3.2 (Conjugacy invariants) A property or a quantity associated to a dynamical system which is preserved under topological (resp. C r , quasiconformal, conformal, etc.) conjugacy is called a topological (resp. C r , quasiconformal, conformal, etc.) invariant. An example of a C 1 -invariant is what is known as the multiplier of a periodic orbit. Lemma 3.3 Suppose X, Y Ă R and f, g P C 1 pRq or X, Y Ă C and f, g are holomorphic. If h is a real (resp. complex) C 1 -conjugacy between f and g and x0 is a p-periodic point then y0 :“ hpx0 q is p-periodic and pf p q1 px0 q “ pg p q1 py0 q.

96


We shall see that many properties of dynamically defined sets are preserved under topological conjugacy. As an example, we mention the following. Definition 3.4 (Invariant sets) A set U Ă X is f -invariant or (forward) invariant under f : X Ñ X if f pU q Ă U . We say that U is backward invariant under f if f ´1 pU q Ă U . Finally, U is totally invariant or completely invariant under f if f pU q “ U “ f ´1 pU q. Lemma 3.5 Topological conjugacies between f and g map f -invariant sets onto g-invariant sets. The same statement is true for backwards and totally invariant sets. Some dynamical systems which are not conjugate may still be semiconjugate. This happens when there exists a continuous function h : X Ñ Y which satisfies h ˝ f “ g ˝ h but is not a homeomorphism. When this is the case, many properties are still transferred from f to g. For example, orbits of f are mapped onto orbits of g, but a p-cycle of f may be mapped onto a p1 -cycle of g, where p1 |p. A weaker concept is that of equivalence between dynamical systems. Definition 3.6 (Equivalent dynamical systems) We say that f and g are topologically equivalent if there exist homeomorphisms h1 and h2 such that the following diagram commutes: f

X ÝÝÝÝÑ § § h1 đ

X § §h đ 2

g

Y ÝÝÝÝÑ Y Equivalences other than topological ones are defined in the obvious way. Equivalences do not preserve orbits and therefore the dynamics of f and g may be very different. However, if f and g are topologically equivalent, there is a bijection between the critical points of f and the critical points of g (a critical point is a point where a map fails to be locally injective), and the same is true for critical values (images of critical points). As we shall see, the number of critical points and values play a very important role in holomorphic dynamics. In this context, being equivalent means, in some sense, to belong to the same ‘family’ of maps.

3.2 Circle homeomorphisms and rotation numbers

97

Exercises Section 3.1 3.1.1 Let f and g be topologically conjugate by a homeomorphism h. Show that critical points of f (points where f is not locally injective) are sent to critical points of g by h. 3.1.2 Let f : X Ñ X and g : Y Ñ Y be topologically conjugate by a homeomorphism h. Suppose U Ă X is dense in X. Show that hpU q is dense in Y . Deduce that if periodic points of f are dense in X then periodic points of g are dense in Y . 3.1.3 Show that a polynomial of degree d ě 2 cd zd ` cd ´1 zd ´1 ` cd ´2 zd ´2 ` ¨ ¨ ¨ ` c1 z ` c0 , with ci P C for i “ 0, . . . , d, is centred (i.e. the sum of the critical points is equal to zero when counted with multiplicity) if and only if cd ´1 “ 0. Show that any polynomial of degree d ě 2 is globally conformally conjugate to a monic (cd “ 1) centred polynomial. Note that for d ą 2 the coefficients are not uniquely determined. In particular, determine the affine conjugacy classes of monic centred cubic polynomials. 3.1.4 Prove that the quadratic family tQc pzq “ z2 ` c | c P Cu has exactly one representative in each affine conjugacy class.

3.2 Circle homeomorphisms and rotation numbers One-dimensional systems generated by the iteration of maps from the unit circle to itself play an important role in general dynamical systems. If a system possesses a simple closed invariant curve, then the dynamics restricted to that curve is conjugate to a circle map. For the content of this section we refer to [Dev2, Sect. 1.14] and [dMvS, Chapt. I]. Consider the unit circle S1 “ tz P C | |z| “ 1u, or equivalently the quotient space T “ R{Z, identified via the exponential map z “ e2π i x . Let π : R Ñ T denote the projection π pxq “ x pmod 1q, and : R Ñ S1 be pxq “ e2π i x . These mappings induce a metric and an orientation on the circle. We study orientation preserving homeomorphisms f : S1 Ñ S1 , or equivalently f : T Ñ T when this is more convenient. The simplest such maps are the rigid rotations Rθ with θ P R, Rθ pzq “ e2π i θ z

or equivalently

Rθ pxq “ x ` θ

pmod 1q.

98


For a rigid rotation, either θ P Q and then all points of S1 are periodic of the same period, or θ P RzQ and no points are periodic. In fact something stronger is true. Theorem 3.7 (Jacobi’s Theorem) If θ P RzQ, all orbits under Rθ are dense in S1 . See Exercise 3.2.2. An orientation preserving homeomorphism of S1 can always be lifted to an orientation preserving homeomorphism of R, via the projection map. Definition 3.8 (Lifts of homeomorphisms) Let f : S1 Ñ S1 be an orientation preserving homeomorphism. A continuous map F : R Ñ R is a lift of f if ˝ F “ f ˝ . Note that this is a semi-conjugacy of infinite degree, since is an infinite degree covering map. As an example, observe that the maps Rk,θ pxq “ x ` θ ` k, for k P Z, denote all the different lifts of the rigid rotation Rθ . This is a general fact: every integer translation of a lift is again a lift, and vice versa. Moreover, if F is a lift then F px ` 1q “ F pxq ` 1 for all x P R and F n is a lift of f n . The rotation number is an important topological invariant associated to a circle map. It measures the average asymptotic rate of rotation of points of the circle. Definition 3.9 (Rotation number) Let f : S1 Ñ S1 be an orientation preserving homeomorphism, and let F : R Ñ R be a lift of f . For x P R we define the rotation number of f as ˙ ˆ F n pxq ´ x pmod 1q. rotpf q “ lim nÑ8 n This is a number in r0, 1q independent of the choices of F and x [Poi]. If f is not a homeomorphism the limit still exists, but different points could result in different rotation numbers. Observe that the rigid rotations have rotpRθ q “ θ

pmod 1q.

The rotation number is very robust. It is in fact invariant both under C 0 perturbations of f (see e.g. [Dev2, Cor. 14.7]) and under topological conjugacies (see Exercise 3.2.3). It is straightforward to check that if f has a fixed point then rotpf q “ 0. In fact, Poincaré showed that much more is true.


99

Proposition 3.10 (Rotation number and periodic points) Let f : S1 Ñ S1 be an orientation preserving homeomorphism, and choose F to be the lift of f so that F p0q P r0, 1q. (1) f has irrational rotation number if and only if f has no periodic points. (2) rotpf q “ p{q with gcdpp, qq “ 1 if and only if f has a p-periodic point z0 so that F q px0 q “ x0 ` p, with px0 q “ z0 . One may ask whether every circle map with an irrational rotation number is actually conjugate to a rigid rotation. This would imply that every circle map with no periodic points has all of its orbits dense in S1 (since homeomorphisms – and therefore conjugacies – map dense sets onto dense sets, see Exercise 3.1.2). The statement turns out to be false. There exist examples, known as Denjoy counterexamples, of circle maps with irrational rotation number, having wandering intervals (intervals which never meet themselves under iteration). This prevents many orbits from being dense in S1 . Hence such maps cannot be conjugate to a rigid irrational rotation. It was Denjoy [Den] who in 1932 specified some additional conditions on the regularity of f to ensure the existence of a conjugacy to a rigid rotation (cf. [dMvS, I.2]). Theorem 3.11 (Denjoy’s Theorem) Let f : S1 Ñ S1 be a diffeomorphism which is C 2 and has an irrational rotation number θ . Then f is conjugate to the rigid rotation Rθ . We are especially interested in analytic circle maps with analytic conjugacies to rigid rotations, because of the applications in holomorphic dynamics. Definition 3.12 (Linearizable circle maps) We say that an analytic circle diffeomorphism without periodic points is (analytically) linearizable if it is analytically conjugate to a rigid rotation. In 1961, Arnol’d [Ar1] found examples of analytic circle diffeomorphisms without periodic points, whose conjugacy to an irrational rotation was not even absolutely continuous. To obtain more regularity for the conjugacies it is necessary to impose further conditions on the rotation number. To state these results we need to introduce some special subsets of irrational numbers. 3.2.1 Continued fractions and subsets of irrational numbers We collect some facts about continued fractions and define certain subsets of irrational numbers which are relevant in the study of circle maps. For more details we refer to [Kin, Lan, Mi1, PM].

100 Preliminaries on dynamical systems and actions of Kleinian groups For θ P pRzQq X p0, 1q, consider the continued fraction expansion 1

θ“

1

a1 ` a2 `

“ ra1 , a2 , a3 , . . .s

1 a3 ` ¨ ¨ ¨

where the coefficients an are uniquely determined positive integers depending on θ . The nth convergent of θ is defined as the irreducible fraction pn :“ ra1 , a2 , . . . , an s. qn Each convergent pn {qn is the best approximation to θ by fractions with denominator at most qn [Mi1, Thm. 11.8]. Definition 3.13 (Diophantine numbers) An irrational number θ is said to be Diophantine of order ď k, or to belong to Dpkq, if there exists ą 0 such that ˇ ˇ ˇ ˇ ˇθ ´ p ˇ ą , for every rational number p . ˇ q ˇ qk q For example, every algebraic number is Diophantine. The set D “ of Diophantine numbers has full measure in p0, 1q. In fact, k ě2 Dpkq Ş Dp2`q “ k ą2 Dpkq has full measure in p0, 1q [Mi1, Lem. 11.7], while the set Dp2q has measure zero [Mi1, Lem. C.8]. The set Dp2q plays an important role in dynamics (not only in holomorphic dynamics and circle maps). It is known as the set of numbers of bounded type (sometimes also referred to a constant type). The motivation for these names comes from the alternative definition of Dpkq in terms of continued fractions (cf. [Mi1, Cor. 11.9]).

Ť

Proposition 3.14 (Alternative definition of Dpkq) An irrational number θ belongs to Dpkq if and only if qn`1 ă Cqnk ´1 for all n ą 0 and for some constant C independent of n. In particular, * " qn`1 ă C ô tan un is bounded. θ P Dp2q ô qn n Irrational numbers which are not Diophantine are often called Liouville numbers. It is a startling fact that the set of Liouville numbers (and many proper subsets thereof) is both generic, i.e. is a countable intersection of open dense sets, and at the same time extremely small – not only a set of measure zero but also of Hausdorff dimension zero (cf. [Mi1, Lem. C.7]).


101

There are other subsets which play an important role in holomorphic dynamics and circle maps. The set B of Bryuno numbers is a superset of D, and is important in the problem of the linearization of fixed points of holomorphic maps (see Section 3.3.2). Definition 3.15 (Bryuno numbers) With tqn un as above, the set of Bryuno numbers is defined as ˇ + # ˇ ÿ logpq n`1 q ˇ ă8 . B “ θ P RzQ ˇ ˇ n qn Finally we mention that Yoccoz [Yo3, Yo4] discovered the so-called set of Herman numbers H when studying the linearization of analytic circle maps. This completed the work started by Herman. The definition is somewhat involved and can be found in [PM, p. 249]. The relative positions of these subsets of irrational numbers are Dp2q Ă D Ă H Ă B Ă pRzQq. 3.2.2 Real analytic circle maps In this section we concentrate on real analytic circle maps f : S1 Ñ S1 , with an irrational rotation number, say θ , hence with no periodic points. Sometimes we assume that f is a diffeomorphism while in other cases we consider critical circle maps, i.e. real analytic circle homeomorphisms with a unique (double) critical point. Due to Denjoy’s Theorem, all analytic diffeomorphisms are conjugate to a rigid rotation Rθ . We are interested in conditions that ensure a certain regularity of the conjugacy such as quasisymmetry (see Definition 2.1), analyticity, etc. The first result is due to Herman [He1] and Yoccoz [Yo3] and gives a precise and optimal condition on the rotation number. Theorem 3.16 (Analytic linearization) Let f : S1 Ñ S1 be a real analytic circle diffeomorphism and θ “ rotpf q. If θ P H then f is analytically linearizable. For an arbitrary θ R H, there exists a real analytic circle diffeomorphism with that rotation number, which is not analytically linearizable. For critical circle maps with irrational rotation number, a theorem of Yoccoz in [Yo2] establishes that the conjugacy to the rigid rotation always exists. It is not difficult to see that it can never be real analytic. This is in fact explained using holomorphic dynamics (see Exercises 3.3.1 and 3.3.2 in Section 3.3).

102 Preliminaries on dynamical systems and actions of Kleinian groups However, in some cases we can obtain quasisymmetric conjugacies, as shown ´ ¸ tek Theorem below (cf. [He3, Sw2] and [Pe2]). in the Herman–Swia Theorem 3.17 (Quasiconformal linearization for critical circle maps) Let f : S1 Ñ S1 be a critical circle map with rotpf q “ θ . Then, θ is of bounded type if and only if f is quasisymmetrically conjugate to the rigid rotation Rθ . We shall see in Section 7.2 that quasisymmetric linearization is all we need in order to perform surgery. The theorem above has been generalized by Petersen [Pe3] for holomorphic critical quasicircle maps, that is holomorphic maps defined on a neighbourhood of a quasicircle having a critical point on the curve (see the section below for the definition of rotation number on a general curve).

3.2.3 Rotation number on invariant curves We mentioned at the beginning of the section that circle maps arise in general dynamics when studying invariant simple closed curves. The rotation number is also defined for such invariant curves. Indeed, suppose γ is an oriented Jordan curve (see the beginning of Chapter 2) on the Riemann sphere and f : γ Ñ γ is a homeomorphism. Let ϕ : S1 Ñ γ denote a conformal parametrization of γ which must be at least a homeomorphism (see Section 2.2.1). Then, fr :“ ϕ ´1 ˝ f ˝ ϕ is a homeomorphism of the unit circle, conjugate to f . We define the rotation number of f on γ , denoted by rotpf, γ q, as the rotation number of fr on the unit circle. Observe that this number is independent of the chosen parametrization ϕ, because the rotation number is a topological invariant. Observe that higher regularity of the curve γ is equivalent to higher regularity of the parametrization ϕ, since conformal parametrizations (boundary values of Riemann maps) are always ‘as good as they can be’ (see Section 2.2.1). In most of our applications, f : γ Ñ γ is the restriction of a holomorphic map defined on a neighbourhood of γ . In this context, the map fr is always real analytic (Exercise 3.2.4). Moreover, if we require some condition on the rotation number then the curve must also be real analytic and f analytically linearizable, as shown by Ghys [Gh] in the following theorem. Theorem 3.18 (Rotation numbers in H and linearizability) Suppose γ is a Jordan curve and F is a univalent map defined on a neighbourhood of γ such that F pγ q “ γ . Let f “ F |γ and θ “ rotpf, γ q P H. Then γ is analytic and F and Rθ are conformally conjugate in a neighbourhood of γ .


103

3.2.4 Families of circle homeomorphisms We end this section by considering families of circle maps in the sense of Arnol’d and Herman. More precisely, let f : S1 Ñ S1 be a circle homeomorphism and define the family fα pzq “ pRα ˝ f qpzq “ e2π i α f pzq, for α P r0, 1q, with the following non-degeneracy condition fαm ı Id |S1 for all α P r0, 1q and for all m ě 0.

(3.4)

This is not a very restrictive condition. It is satisfied, for example, if f is a real analytic homeomorphism of S1 that extends to a non-affine holomorphic map of C˚ (cf. [dMvS, Lem. 4.3(I)]). This is in general the setup in the surgery applications we consider. Given fα as above, define the rotation function ρpαq “ rotpfα q. Since the rotation number varies continuously under C 0 -perturbations, ρpαq is continuous. This function has many other remarkable properties. It is in the same family as the one presented in Section 1.3.3 and is also called a devil’s staircase (cf. [dMvS, Sect. I.4]) (see Figure 3.1). Proposition 3.19 (Properties of the rotation functions) Let f, fα and ρpαq be as above. Then: (1) ρ : T Ñ T is continuous, orientation preserving and onto. (2) If ρpα ˚ q P p0, 1qzQ, then ρ is strictly increasing at α ˚ . Otherwise ρ is locally constant at α ˚ (maybe only on one side). More precisely, the set ρ ´1 pr{sq has non-empty interior for every rational r{s P r0, 1q. (3) The set tα P p0, 1q | ρpαq P p0, 1qzQu is nowhere dense. An immediate consequence of the properties above is the following statement. Theorem 3.20 (Adjusting α to obtain a given rotation number) Let f : S1 Ñ S1 be a circle homeomorphism and fα “ Rα ˝ f the associated family of circle maps satisfying (3.4). Then, for any θ P r0, 1qzQ, there exists a unique α P r0, 1q such that rotpfα q “ θ . To end this section, we state a result related to the linearization problem. This theorem is due to Herman [He2], who constructed exotic examples showing that quasisymmetric conjugacies to rigid rotations cannot always be upgraded.

104 Preliminaries on dynamical systems and actions of Kleinian groups ρ pα q 1.0

Ò

0.8

0.6

0.4

0.2

0.2

0.4

0.6

0.8

1.0Ñ α

Figure 3.1 A devil’s staircase, the graph of the rotation function for 1 sinp2π xq pmod 1q. fα pxq “ α ` x ` 2π

Theorem 3.21 (Quasisymmetric linearization which is not C 2 ) Suppose f : S1 Ñ S1 is a C 8 circle diffeomorphism and fα satisfies (3.2.1). Then there exists α P r0, 1q such that fα is quasisymmetrically conjugate to an irrational rigid rotation, but the conjugacy is not C 2 .

Exercises Section 3.2 3.2.1 Assume θ P p0, 1q is a Bryuno number. Using the continued fraction expansion and the expression for the denominators of the convergents denominators prove that if 1 ´ θ is also a Bryuno number. Hint: For θ P p0, 1{2q and θ “ ra1 , a2 , a3 , . . . , an , . . .s show that 1 ´ θ “ r1, a1 ´ 1, a2 , . . . , an´1 , . . .s. Conclude afterwards that if tpn {qn u are the convergents of θ , and tpn1 {qn1 u are the convergents of 1 ´ θ , then qn1 “ qn´1 for n ě 2. 3.2.2 (Jacobi’s Theorem) Let θ P RzQ. Prove that all orbits under the rigid rotation Rθ pxq “ x ` θ pmod 1q, are dense in T. Hint: First show that every orbit is infinite and therefore there exist two iterates which are arbitrarily close. Rotate to obtain k such that Rθk pxq is arbitrarily close to x, and then iterate to obtain the denseness. 3.2.3 (Rotation numbers are topological invariants) Suppose f, g : S1 Ñ S1 are orientation preserving homeomorphisms and f „ g. Prove that top

rotpf q “ rotpgq.

3.3 Holomorphic dynamics: the phase space

105

Hint: Let F, G and H be lifts of f, g and h respectively, where h is the H conjugacy. Observe that F „ G. In the definition of rotpgq substitute top

Gn by H ˝ F n ˝ H ´1 and add and subtract terms conveniently. Use that H pxq ´ x is bounded for all x and that so is H ´1 pxq ´ x. 3.2.4 (Invariant curves under univalent maps) Let γ be a Jordan curve in p and f a univalent map defined on a neighbourhood of γ such that C f pγ q “ γ . Show that fr : S1 Ñ S1 as defined in Section 3.2.3 is real analytic. Hint: Use the Schwarz Reflection Principle.

3.3 Holomorphic dynamics: the phase space From now on we consider holomorphic mappings f : S Ñ S with the phase p the complex plane space or dynamical space S equal to the Riemann sphere C, ˚ C or the punctured plane C . Most of the basic theory of holomorphic dynamics, especially the local theory, is common for the three settings, although the proofs may vary from one to another. Most texts deal with rational maps, although there are survey references like [Ber, EL1, HY, MNTU], where the differences with transcendental maps are treated. The exposition in this section follows mainly the book by Milnor [Mi1] but see also [Bea] or [Ber].

3.3.1 The dynamical partition A special property of holomorphic dynamical systems is the splitting of the phase space induced by the concept of normal families. p be a Definition 3.22 (Normal family of holomorphic maps) Let U Ă C p domain and F a family of holomorphic maps from U to C. We say that F is a normal family in U if any infinite sequence of elements of F contains a subsequence which converges uniformly on compact sets of U to some limit map. The condition of normality can be phrased in terms of equicontinuity, asking the family to be locally equicontinuous in U with respect to the spherical p (cf. the Arzelà–Ascoli Theorem). metric in C An easy way to check normality is to apply Montel’s Theorem. p be a domain and F a family Theorem 3.23 (Montel’s Theorem) Let U Ă C p p of holomorphic maps from U to C. If there exist three distinct points a, b, c P C p such that f pU q Ă Czta, b, cu for all f P F, then F is a normal family in U .

106 Preliminaries on dynamical systems and actions of Kleinian groups In particular if |f pU q| ď R for all f P F and for some R ą 0, then F is a normal family in U . In holomorphic dynamics the concept of normality is applied to the family of iterates of the given map f . If tf n uně0 is normal in a domain U , orbits of points in U behave in a similar manner. The concept of normality defines the dynamical partition explained in the following definition. Definition 3.24 (Fatou set and Julia set) Given a holomorphic map f : S Ñ S as above, we define the Fatou set of f as Ff “ tz P S | tf n un forms a normal family in a neighbourhood of zu, and the Julia set of f as its complement Jf “ SzFf . For f P Ent Y Ent˚ , the essential singularities are sometimes considered p part of the Julia set, which is then defined in C. The Fatou set is open and the Julia set is closed in S, and both sets are totally invariant (i.e. their orbits do not mix). In fact, J pf q is the smallest closed set which is totally invariant under f . We think of the Fatou set as the set of orbits of f which are in some sense stable or tame, while the Julia set is the set of chaotic orbits. Figures 3.2–3.4 illustrate the splitting of the phase space into the Fatou set and the Julia set of a map f in Pol, Rat and Ent respectively.

Figure 3.2 The dynamical plane of a cubic polynomial. The Fatou set consists of the basin of attraction of infinity (white) and the basin of attraction of an attracting 2-cycle (yellow). The common boundary between these two basins is the Julia set (black).


107

Figure 3.3 The Julia set (in white) of a rational map, which is Newton’s method applied to a cubic polynomial. The Fatou set (in colour) consists of three basins of attraction, one for each root of the cubic polynomial.

Figure 3.4 The Julia set (in black) of z ÞÑ λ exppzq with λ “ 0.64 ` 2.3i consists of a Cantor set of unbounded curves (see [DT]). The Fatou set (white) is the basin of attraction of an attracting 3-cycle.

The Fatou and the Julia sets are preserved under topological conjugacy in the sense that if h is a homeomorphism such that h ˝ f “ g ˝ h, for f, g holomorphic, then J pgq “ hpJ pf qq and Fpgq “ hpFpf qq. As a basic example, consider f pzq “ zd for d ě 2. Then the Fatou set is p 1 , while Jf “ S1 . Indeed, any orbit inside D converges to 0 while equal to CzS p converges to infinity. Orbits of points in S1 stay in S1 and every orbit in CzD follow the dynamics of the map Md : T Ñ T defined by t ÞÑ d ¨ t pmod 1q. Points arbitrarily close to the unit circle have radically different asymptotic

108 Preliminaries on dynamical systems and actions of Kleinian groups behaviour than those on the circle. Although Julia sets In general are fractals, the dynamics on a Julia set share most of the properties of the map Md on the unit circle. In the example above, 0 and 8 are attracting fixed points. In general, if Opz0 q “ tz0 , z1 , . . . , zp´1 u form a p-cycle, we define the multiplier of the cycle as λ “ pf p q1 pzi q “ f 1 pz0 q ¨ f 1 pz1 q ¨ ¨ ¨ f 1 pzp´1 q, for any i P t0, . . . , p ´ 1u, if the orbit lies in C. If the orbit includes the point at infinity, the multiplier is defined as above after a change of variables that moves the orbit into C. The periodic orbits are classified as: • attracting if |λ| ă 1 (superattracting if λ “ 0); • repelling if |λ| ą 1, and • neutral or indifferent if |λ| “ 1, where the last case splits into the following: • parabolic or rationally indifferent if λ “ e2π i p{q with p{q P Q, or • irrationally indifferent if λ “ e2π i θ with θ P RzQ. In Section 3.3.2 we describe how these definitions relate to the local dynamics around the periodic orbits. Observe that the multiplier is zero if and only if the derivative vanishes at one of the points in the cycle. The points z for which f 1 pzq “ 0 are called critical points. The dynamical behaviour of the critical points plays an important role in the study of the dynamics of f . In the basic example the two attracting fixed points at 0 and 8 belong to open sets of orbits whose iterates converge towards the fixed point. Such sets are in general called basins or attraction. Definition 3.25 (Basin of attraction of an attracting cycle) Given an attracting p-cycle Opz0 q of f , we define its basin of attraction A “ Af as the set of points z P S such that f np pzq converges to some zi P Opz0 q as n Ñ 8. The following theorem summarizes some of the basic properties of Julia sets and Fatou sets. All proofs can be found in [Mi1, §4, §12, §14] or [Ber]. In the statement we use the concept of grand orbit. Definition 3.26 (Grand orbit and exceptional set) Given f : S Ñ S and a point z P S, its grand orbit consists of all points in S which are related forwards or backwards with z under iteration of f . More precisely, p | f p pzq “ f q pwq for some p, q P N} . GOpzq “ tw P C


109

We define the exceptional set Epf q as the set of points with a finite grand orbit. Grand orbits are totally invariant. Theorem 3.27 (Properties of Julia sets and Fatou sets) Let the map f : S Ñ S be in Rat, Ent or Ent˚ . (1) (2) (3) (4) (5) (6) (7)

(8) (9) (10) (11) (12) (13)

For any k ą 0, the Julia set Jf k coincides with Jf . Every attracting cycle and its basin of attraction belongs to Ff . If A is the basin of attraction of an attracting cycle, then Jf “ BA. Every repelling and parabolic cycle belongs to Jf . Jf ‰ H. If f P Rat then Jf contains a repelling fixed point or a parabolic fixed point of multiplier 1. The set Epf q has at most two, one or no points if f is in Rat, Ent or Ent˚ respectively. If z0 P Jf and U is a neighbourhood of z0 disjoint from Epf q, then Ť SzEpf q Ă n pf n pU qq. Consequently, if K is a compact set disjoint from Epf q, there exists N ą 0 such that K Ă f n pU q for all n ě N . If f P Rat, then f n pJf X U q “ Jf for all n ě N . Either Jf has no interior point, or Jf “ S. Ť If z0 P SzEpf q, then Jf Ă ną0 f ń pz0 q. Jf has no isolated points. Jf is either connected or has uncountably many components. For z in a generic set of points in Jf , the forward orbit of z is everywhere dense in Jf . Repelling periodic points are dense in Jf .

We end this section defining a special type of conjugacy which is often used in holomorphic dynamics. Definition 3.28 (J -conjugacy) Let f, g P Rat Y Ent X Ent˚ . We say that f and g are J -conjugate if there exists a homeomorphism h : Jf Ñ Jg such that h˝f “g˝h on the respective Julia sets of f and g. We shall discuss a related concept, J -stability in Section 3.4. In some cases the conjugacy h can be extended to a neighbourhood of the Julia set. We shall see examples of this in Section 4.2.2.

110 Preliminaries on dynamical systems and actions of Kleinian groups 3.3.2 Local theory of fixed points In this section we describe the local dynamics of f in a neighbourhood of a p-cycle. After replacing f by f p , we may assume that the orbit is fixed. Furthermore we have Jf p “ Jf . We may also assume that the fixed point is at 0, conjugating by a translation if necessary. Hence, let f be a holomorphic map defined in a neighbourhood of the origin with Taylor expansion f pzq “ λz ` a2 z2 ` a3 z3 ` ¨ ¨ ¨ , where λ “ f 1 p0q is the multiplier of f at the fixed point 0. We are interested in finding normal forms of f . More precisely, we look for conformal changes of variables which conjugate f to a map with a finite number of terms in its power series. The solution to the problem depends on the value of the multiplier λ.

(a) Attracting and repelling fixed points (|λ| ‰ 0,1) In the attracting and repelling cases, f is locally conformally conjugate to its linear part z ÞÑ λz. The precise statement is from 1884, and due to Kœnigs [Kœ] (cf. [Mi1, §8]). Theorem 3.29 (Kœnigs’ Linearization) If the multiplier satisfies |λ| ‰ 0, 1, then there exists a neighbourhood U of 0 and a local conformal conjugacy w “ ϕpzq where ϕ : U Ñ ϕpU q satisfies ϕp0q “ 0 and ϕ ˝ f ˝ ϕ ´1 pwq “ λw in ϕpU X f ´1 pU qq. The conjugacy ϕ is called a linearizing map of f at the fixed point and is unique up to multiplication by a non-zero constant. Moreover, if fα pzq “ λpαqz ` a2 pαqz2 ` ¨ ¨ ¨ depends holomorphically on a complex parameter α, satisfying |λpαq| ‰ 0, 1, then the linearizing map ϕα , appropriately normalized (for instance by ϕα1 p0q “ 1), also depends holomorphically on α. Kœnigs’ Linearization Theorem is clearly a local result. But in the attracting case, the linearizing map can actually be extended to the whole basin of attraction of the fixed point, at the expense of its bijectivity. Theorem 3.30 (Global Linearization) Suppose f is globally defined and has an attracting fixed point z0 with 0 ă |λ| ă 1. Let A be the basin of attraction of z0 . Then there exists a holomorphic map ϕ : A Ñ C with ϕpz0 q “ 0, so that the diagram

3.3 Holomorphic dynamics: the phase space f

A ÝÝÝÝÑ § § ϕđ

111

A § §ϕ đ

wÞÑλw

C ÝÝÝÝÑ C commutes. Moreover, ϕ is biholomorphic in a neighbourhood of z0 (where it determines a local conformal conjugacy). The map ϕ is unique up to multiplication by a non-zero constant. The global linearizing map is constructed as follows. Let ϕ : U Ñ C denote a linearizing map for f in a neighbourhood U of z0 . Then for any z P A define ˆ ˙ ´ ¯ 1 ϕpzq “ ϕ f k pzq , k λ where k is chosen so that f k pzq P U . The value of ϕpzq is independent of the choice of k. Note that the globally defined map is no longer injective. For example, every preimage of z0 under f is mapped to 0 by ϕ. (b) Superattracting fixed points (λ “ 0) The superattracting case corresponds to a holomorphic map f with a fixed critical point at the origin. Then f is locally conformally conjugate to the normal form w ÞÑ wm , where m ´ 1 is the multiplicity of the critical point (its order as a zero of f 1 ). The precise statement is from 1904 and is due to Böttcher [Bö] (cf. [Mi1, §9]). Theorem 3.31 (Böttcher coordinates) Let f pzq “ am zm ` am`1 zm`1 ` ¨ ¨ ¨ with m ě 2 and am ‰ 0. Then there exists a neighbourhood U of 0 and a local conformal conjugacy ϕ : U Ñ C that conjugates f to w ÞÑ wm . The conjugacy ϕ is unique up to multiplication by an pm ´ 1qst root of unity. The map ϕ is called a Böttcher map or coordinate. We can not extend ϕ to the whole basin of attraction as we did in the attracting case where λ ‰ 0 since w ÞÑ wn is not an invertible map. However, it is still possible to extend the real function |ϕ| : U Ñ R` Y t0u to the whole basin. This theorem has important applications to the dynamics of polynomials near infinity, which is a superattracting fixed point for any polynomial of degree ě 2 (see Section 3.3.4). (c) Parabolic fixed points (λ “ e2π i p{q ) The parabolic case corresponds to a holomorphic map f with a fixed point at the origin, where the multiplier λ is

112 Preliminaries on dynamical systems and actions of Kleinian groups a qth root of unity, say e2π i p{q . Taking the qth iterate of the map we reduce to the case where λ “ 1. Hence we consider maps of the form f pzq “ z ` azm`1 ` Opzm`2 q, with m ą 0 and a ‰ 0. The integer m ` 1 is called the multiplicity of the parabolic point with λ “ 1, which is the order of the zero of f ´ Id at the fixed point. To describe the dynamics around the parabolic point, we need the following definition. Definition 3.32 (Attracting and repelling petals) Suppose f is defined and univalent in a neighbourhood U of the origin. An open set P Ă U is called an attracting petal for f at the fixed point if: (a) f pPq Ă P Y t0u; Ş (b) n f n pPq “ t0u. An open set P Ă f pU q is called a repelling petal for f at the fixed point if P is an attracting petal for f ´1 : f pU q Ñ U , where f ´1 denotes the branch of the inverse of f fixing the origin. The following theorem is due to Leau [Lea], Fatou [Fat] and Julia [J] in successive approximations. Theorem 3.33 (Parabolic Flower Theorem) Suppose f has a parabolic fixed point with multiplier λ “ 1 at the origin of multiplicity m ` 1. Then there are 2m petals tPj u2m j “1 , numbered cyclically around the origin and such that Pj is attracting or repelling according to whether j is odd or even. Each petal Pj intersects only its two immediate neighbours Pj ´1 and Pj `1 (indices are taken mod 2m), and are disjoint from the rest. The petals can be chosen so that the union P1 Y ¨ ¨ ¨ Y P2m Y t0u forms an open neighbourhood of the origin (see Figure 3.5). Suppose f is globally defined and has a parabolic fixed point at the origin of multiplier λ “ 1. If the orbit of z ‰ 0 is infinite and converges to 0, it needs to belong to one of the attracting petals P in the Flower Theorem from some iterate and onwards. We say that the orbit converges to 0 through P. Each P has an associated parabolic basin (of attraction).

3.3 Holomorphic dynamics: the phase space P3

P4

P3

113

P2

P2

P5

P1

P6

P4

P1

P10

P7 P8

P5

P9

P6

Figure 3.5 The distribution of the invariant petals around a parabolic point with multiplier λ “ 1 and multiplicity 5 (left) and 3 (right).

Definition 3.34 (Parabolic basin of attraction) If z0 is a parabolic fixed point of f with multiplier λ “ 1 and P is an attracting petal at 0, we define the parabolic basin (of attraction) of z0 associated to P as ď f ń pz0 q | f n pzq ÝÑ z0 through Pu. AP “ tz P Sz nÑ8

ną0

Observe that if the parabolic fixed point has multiplicity m ` 1 then it has exactly m disjoint parabolic basins. Although f is not conjugate to its linear part nor to any normal form in a neighbourhood of the parabolic fixed point, it turns out that some kind of linearization is possible inside each of the petals (cf. [Mi1, Thm. 10.9]). Theorem 3.35 (Parabolic Linearization: Fatou coordinates) For every attracting and for every repelling petal P, there is a conformal embedding ϕ : P Ñ C called the Fatou coordinate in P which conjugates f to the translation w ÞÑ w ` 1 on P X f ´1 pPq. The following diagram commutes: f

P X f ´1 pPq ÝÝÝÝÑ § § ϕđ C

P § §ϕ đ

wÞÑw`1

ÝÝÝÝÝÑ C

If 0 is a parabolic fixed point of f with multiplier λ “ e2π i p{q ‰ 1 and if f q has multiplicity m ` 1 at 0, then the number of attracting and repelling petals in the Flower Theorem is m “ kq for some k P N. The m attracting petals are forward invariant under f q and can be chosen so that they form k forward invariant cycles of petals of period q under f . The petals in a cycle are mapped among themselves with combinatorial rotation number p{q (see Figure 3.6).


P5

P3

P1 P7 P9 Figure 3.6 A parabolic fixed point of multiplier λ “ e2π i 2{5 and multiplicity 5 for f 5 (hence k “ 1). There is one cycle of attracting petals with combinatorial rotation number 2{5.

The k cycles of petals of period q give rise to k disjoint parabolic basins of attraction for the parabolic fixed point. As in the case of basins of attraction, the parabolic ones are subsets of the Fatou set of f . Their boundaries are part of the Julia set (cf. [Mi1, Lem. 10.5]).

(d) Irrationally indifferent fixed points: Cremer points and Siegel discs The irrationally indifferent case corresponds to a holomorphic map f with a fixed point at the origin where the multiplier is λ “ e2π i θ and θ P RzQ. A theorem of N˘aishul’ [N] asserts that the number θ is a topological invariant. It is called the rotation number of the fixed point. The fundamental question is which conditions on θ assure that f is locally conformally conjugate to its linear part z ÞÑ λz. If so, f is (conformally) linearizable around the fixed point. In contrast to the cases treated above, the complete answer is still open in some cases. We first observe that a local holomorphic linearization is possible if and only if the fixed point belongs to the Fatou set (cf. [Mi1, Lem. 11.1]). If this is the case, a neighbourhood of the fixed point is foliated by simple closed invariant curves on which all orbits are dense (since this is the case for the linear map z ÞÑ λz). The maximal neighbourhood of the fixed point where the linearization is defined is called a Siegel disc and the fixed point a Siegel point. Fixed points that are not Siegel are called Cremer points and they always belong to the Julia set. In 1927, Cremer proved that for a generic choice of rotation numbers, linearization is not possible. It was not until 1942 that Siegel [Si] showed that


115

this is not always the case. Indeed he showed that if θ is Diophantine then there is a Siegel disc around the fixed point. This condition was later improved by Bryuno [Bry1] and Rüssman [Rü] in the 1960s, and the improvement was shown to be the best possible by Yoccoz [Yo1] in 1988 (cf. [Mi1, Thm. 11.10, 11.11]). Theorem 3.36 (Existence of Siegel discs) Let f pzq “ e2π i θ z ` Opz2 q be defined in a neighbourhood of the origin with θ P RzQ. If θ P B then f is holomorphically linearizable in a neighbourhood of 0. Conversely, for any θ R B, the quadratic polynomial f pzq “ e2π i θ z ` z2 is not locally holomorphically linearizable. In other words, for the family of quadratic polynomials Pol2 , the Bryuno condition is optimal for linearization. The condition is also optimal for the family of entire maps z ÞÑ λzez [Ge2], but for instance it is an open question whether there exists a cubic polynomial with a Siegel disc whose rotation number is not Bryuno. It is conjectured by Douady that the Bryuno condition is optimal for linearization in the whole class Rat. The local dynamics around Cremer points is complicated and interesting. When linearization is not possible, there is always a compact invariant set with non-trivial topology (known as a hedgehog) around the fixed point, see [PM].

3.3.3 Global theory: periodic cycles, Fatou components and singular values In this section f : S Ñ S is again assumed to be in Rat, Ent or Ent˚ . We shall describe the structure of the Fatou set. We assume that Jf ‰ S, so that the Fatou set is open and non-empty. In general, it has infinitely many connected components, called Fatou components. If f P Rat a Fatou component is mapped onto a Fatou component, since f is an open map and the Fatou and Julia sets are completely invariant. If f P Ent Y Ent˚ , and U and U 1 are Fatou components so that f pU q Ă U 1 , then U 1 zf pU q may consist of at most one point (cf. [Ber, Sect. 4]). Therefore a Fatou component U is: • p-periodic, if f p pU q Ă U for some minimal p ą 0; • (strictly) preperiodic, if f k pU q is periodic for some k ą 0 but U is not; or • wandering, if f k pU q X f m pU q “ H for all k, m ą 0, k ‰ m. A Fatou component which is 1-periodic is said to be fixed or invariant. If U is p-periodic, we denote by OpU q the cycle of Fatou components to which U belongs.

116 Preliminaries on dynamical systems and actions of Kleinian groups The classification of periodic Fatou components is well understood and is essentially due to Cremer [Cr] and Fatou [Fat] (cf. [Mi1, §16] and [Ber]). Theorem 3.37 (Classification of periodic Fatou components) Let f be as above and U be a p-periodic Fatou component of f . Then one and only one of the following possibilities occurs: (1) U contains an attracting p-periodic point z0 and f np pzq Ñ z0 as n Ñ 8 for all z P U . The cycle OpU q is called the immediate basin of attraction of the attracting cycle Opz0 q (denoted by A˝ ). (2) BU contains a periodic point z0 and f np pzq Ñ z0 as n Ñ 8 for all z P U . Then z0 is a parabolic fixed point of multiplier 1 for f p . The cycle OpU q is called the immediate parabolic basin of attraction of the parabolic cycle Opz0 q. (3) There exists a conformal map ϕ : U Ñ D so that pϕ ˝ f p ˝ ϕ ´1 qpzq “ e2π i ω z for some ω P RzQ. The cycle OpU q is called a p-cycle of Siegel discs. (4) There exists 0 ă r ă 1 and a conformal map ϕ : U Ñ Ar :“ tr ă |z| ă 1u so that pϕ ˝ f p ˝ ϕ ´1 qpzq “ e2π i ω z for some ω P RzQ. The cycle OpU q is called a p-cycle of Herman rings. (5) There exists z0 P BU so that f np pzq Ñ z0 as n Ñ 8 for all z P U , but f p pz0 q is not defined. Then z0 is an essential singularity and the cycle OpU q is called a p-cycle of Baker domains. Baker domains do not exist for rational maps since any cycle of Baker domains contains an essential singularity on their boundary. Herman rings do not exist for polynomials due to the Maximum Modulus Principle, nor do they exist for entire transcendental maps, since they require the existence of poles. An example of a rational map with a Herman ring can be seen in Figure 3.8. Figure 7.8 shows the dynamical plane of an entire map with a Siegel disc. It was an open question for a long time whether all Fatou components for a rational map are preperiodic. An affirmative answer was given by Sullivan in 1985 [Su2]. It is known as the Theorem of No Wandering Domains for rational maps. A proof is given in Section 4.1 as one of the earliest examples of quasiconformal surgery. Singularities of the inverse map A global inverse is never well defined for the maps f P Rat Y Ent Y Ent˚ we consider, but local inverse branches often are. A point v P S is called regular if all possible inverse branches of f are well defined in some neighbourhood of v. Otherwise v is called a singular value of f . Its orbit is called a singular orbit.


117

The set of singular values, denoted by Singpf ´1 q, may contain three types of points: (1) critical values, defined as the images of critical points; (2) asymptotic values. A point a P S is an asymptotic value of f if there is an unbounded curve γ ptq ÝÑ 8 such that f pγ ptqq ÝÑ a. Morally, t Ñ8

t Ñ8

asymptotic values have some of their preimages at infinity. A typical example is a “ 0 for the exponential map, with γ ptq being any path whose real part tends to ´8; or (3) limits of the above. If V is a set which does not meet Singpf ´1 q, then f : U Ñ V is a covering for every connected component U of f ´1 pV q. The set of critical points of f is denoted by Cf while Vf denotes its set of critical values. All singular values of maps in Ratd are critical values, and there are only a finite number of them, exactly 2d ´ 2 counted with multiplicity. Transcendental maps may have infinitely many singular values of all three types. For a deeper treatment of singularities of transcendental maps we refer to [BerE]. In holomorphic dynamics, singularities of the inverse map and their orbits play an important role. This is due to the fact that every cycle of Fatou components and every non-repelling cycle is associated to the orbit of some singular value. Definition 3.38 (Postsingular and postcritical set) The postsingular set Pf (also called the postcritical set if f P Rat), is defined as Pf “

ď

ď

f n pvq.

v PSingpf ´1 q ně0

A map f is called Misiurewicz if all singular orbits are preperiodic to repelling cycles. We summarize the relations between Pf and Fatou components in the following theorem (see e.g. [Ber, Sect. 4.3], [Bea, §9.3] or [Mi1, §8-11]). Theorem 3.39 (Cycles and singular values) Let f P Rat Y Ent Y Ent˚ , and let OpU q denote a p-cycle of Fatou components. (a) If A˝ :“ OpU q is the immediate basin of an attracting or parabolic cycle then there exists U 1 P OpU q which contains either a critical point or an asymptotic value.

118 Preliminaries on dynamical systems and actions of Kleinian groups (b) If OpU q is a cycle of Siegel discs or Herman rings, then for every U 1 P OpU q we have BU 1 Ă Pf . (c) If Opz0 q is a Cremer cycle then for every z P Opz0 q we have z P Pf . The following lemma is attributed to Fatou and can be proven by similar arguments as those in part (b) of the theorem above (see [Mi1, Cor. 14.4]). Lemma 3.40 (Small preimages) Let f P Rat Y Ent Y Ent˚ and let U be a simply connected neighbourhood of z0 P Jf which does not intersect the postsingular set Pf . Then, diampf ń pU qq Ñ 0 for any sequence of inverse branches of f defined on U . Indeed, one can show that otherwise, the given sequence forms a normal family and therefore has a subsequence converging to some non-constant holomorphic limit function g : U Ñ V . Hence, for some V 1 Ă V containing gpz0 q and infinitely many n’s, f n pU 1 q Ă U , contradicting that gpz0 q is in the Julia set (see Property (7) in Theorem 3.27). As a consequence of Theorem 3.39, the number of attracting and parabolic periodic points is always bounded above by the number of singular values of the map. (One full singular orbit needs to be contained in each basin and different basins are pairwise disjoint). More is known for rational maps: it is not possible for an orbit of a singular value to accumulate on the boundaries of two different Siegel cycles, Herman rings or Cremer cycles. This is known as the Fatou–Shishikura inequality for rational maps [Sh1], and its proof is a classical example of surgery which we explain in Section 7.7. Theorem 3.41 (Bound on non-repelling cycles and Herman rings) Let f P Ratd . Let natt , np , nsd , nhr and nc denote the number of attracting cycles, parabolic cycles, cycles of Siegel discs, cycles of Herman rings and Cremer cycles respectively. Then natt ` np ` nsd ` 2nhr ` nc ď 2d ´ 2. The generalization of this result for f P Ent with a finite number of singular values is addressed in [Ep2, Ep3]. Analougously to rational maps, maps in this class do not have wandering domains [EL2, GK]. In some cases singular orbits may lie on the boundary of Herman rings or Siegel discs. The proof of this type of results often uses surgery. Some are discussed in Sections 7.2 and 7.3. However, there is a general result which does not involve surgery. It follows from Ghys’ Theorem (Theorem 3.18) about rotation numbers.


119

Theorem 3.42 (Siegel discs and critical points) Let be a Siegel disc with rotation number θ P H, of a holomorphic map f defined at least on a neighbourhood of . If B is a Jordan curve, then it contains a critical point. Indeed, if B did not contain a critical point, then the map f would be injective in a neighbourhood of B, and f would have rotation number θ on the curve B. It then follows from Theorem 3.18 that B is analytic and that f is conformally conjugate to the rigid rotation Rθ in a neighbourhood of B. But this contradicts that the Siegel disc is the maximal domain with that property. Hyperbolicity and related concepts Hyperbolic maps play an important role in dynamics, because of their ‘stability’ under perturbation (the precise concept will be discussed in Section 3.4). The following statement summarizes their properties (see [McM3, Thm. 3.13]). Theorem 3.43 (Definitions of hyperbolicity) A map f P Ratd with d ě 2 is called hyperbolic if any of the following equivalent conditions is satisfied: Pf is disjoint from the Jf ; there are no critical points or parabolic cycles in Jf ; every critical point tends to an attracting cycle under forward iteration; there is a smooth conformal metric ρ defined on a neighbourhood of the Jf such that ||f 1 pzq||ρ ą C ą 1 for all z P Jf ; (e) there is n P N such that f n strictly expands the spherical metric on Jf .

(a) (b) (c) (d)

The last two conditions are to say that f is expanding on its Julia set. The concept of expanding dynamical systems is very general. In this context, it is equivalent to hyperbolicity. For general transcendental maps, those with infinitely many singular values, it is not clear a priori that hyperbolicity makes much sense. In particular, statements (a) and (b) are not equivalent in the presence of Baker or wandering domains. However, for transcendental maps of finite type (those with a finite number of singular values), hyperbolicity is defined analogously. See for example [EL2]. The following two concepts are weaker than hyperbolicity. Definition 3.44 (Subhyperbolic and geometrically finite rational maps) A rational map is subhyperbolic if every critical point is either attracted to an attracting cycle or is preperiodic. It is geometrically finite if all critical points in the Julia set are preperiodic.

120 Preliminaries on dynamical systems and actions of Kleinian groups Observe that geometrically finite rational maps are allowed to have parabolic cycles, while subhyperbolic ones are not. It was proven in [DH2, Chapt. 3] that if a polynomial is subhyperbolic and its Julia set is connected, then it is also locally connected. Tan Lei and Yin [TY] and Pilgrim and Tan Lei [PT2] have proven the much sharper result: if a rational map is geometrically finite then each connected component of its Julia set is locally connected. The concept of geometrically finite rational maps will appear several times in the book (e.g. Section 9.3). Blaschke products A special type of rational maps are called Blaschke products. We shall use them repeatedly throughout the book. Definition 3.45 ((Finite) Blaschke product) A (finite) Blaschke product is a rational map of the form Bpzq “ ei θ

d ź z ´ aj , 1 ´ aj z j “1

for some |aj | ă 1, 1 ď j ď d and some θ P r0, 2π q. Since Bpzq is a product of Möbius transformations which preserve the unit disc, it follows that B is a degree d rational map which preserves D and also p CzD. By Montel’s Theorem, the Julia set must then be contained in the unit circle. It is easy to check that B is symmetric with respect to S1 in the sense that it commutes with the reflection with respect to the unit circle τ pzq “ 1{z. If aj “ 0 for some j (i.e. if z is one of the terms in the product), then 0 and 8 are attracting fixed points and it follows from the Schwarz Lemma that the Julia set is exactly S1 . This is also the case whenever there is a fixed point in p 1 . In this cases, B is expanding on the Julia set. CzS Blaschke products are useful in dynamics in view of the following fact. Proposition 3.46 (Proper maps in simply connected domains) (a) If f : D Ñ D is a proper holomorphic map, then f is a Blaschke product of degree d ě 1. p is a simply connected domain and f : U Ñ U a proper holo(b) If U Ă C morphic map, then f is conformally conjugate on U to a Blaschke product of degree d. Observe that, in the proposition above, f is not required to be rational or even globally defined. In holomorphic dynamics, and also in this book, the concept of Blaschke product is often extended to allow for the points aj to be anywhere in CzS1 .


121

These maps are still symmetric with respect to S1 , but in general they do not preserve the unit disc, which may contain proper preimages of itself and of its complement. In this situation the Julia set is not bound to lie on the unit circle. To distinguish these from the classical Blaschke products we sometimes call them generalized Blaschke products. Examples can be seen in Figures 3.7 and 3.8.

á Figure 3.7 The Julia set of the (generalized) Blaschke product fλ,a pzq “ z3 1z´ az with a “ 2.05133 ` i0.490272. The points z “ 0 and z “ 8 are superattracting fixed points while the other two critical points escape to infinity. The Julia set is symmetric with respect to the S1 and has infinitely many components. The basin of z “ 0 (shaded) is infinitely connected.

The following lemma is related to the proposition above (see [Mi1, Lem. 15.5]). Lemma 3.47 (Rational maps which preserve the unit circle) If a rational map of degree d preserves the unit circle then it is a generalized Blaschke product of degree d.

3.3.4 Polynomial dynamics For the contents of this section we refer to [Mi1, §18]. The general theory explained in the sections above takes a very special form when we deal with polynomial dynamics. For d ě 2, let f P Pold , that is f pzq “ ad zd ` ad ´1 zd ´1 ` ¨ ¨ ¨ ` a1 z ` a0 with leading coefficient ad ‰ 0. Note that f p8q “ 8 “ f ´1 p8q.


`4 Figure 3.8 The Julia set of the (generalized) Blaschke product fλ,a pzq “ λz2 1z` 4z

with λ “ e2π i t , t “ 0.61517 . . .. The Julia set is symmetric with respect to the unit circle. The Fatou set consists of the basins of 0 and 8 together with an invariant Herman ring (shaded) containing the unit circle, and all its preimages.

Then f has d ´ 1 critical points in C when counted with multiplicity and one critical point of multiplicity d ´ 1 at infinity. We let Critpf q denote the set of finite critical points. The point at infinity is always a superattracting fixed point. It turns out that the basin of attraction of infinity p | f n pzq ÝÑ 8u Af p8q :“ tz P C nÑ8

is always connected, since f has no poles, and hence Af p8q does not have any preimage other than itself. The complement p f p8q Kf :“ CzA is called the filled Julia set. It is completely invariant and compact. It is also full p f is connected, or equivalently, that all bounded Fatou which means that CzK components are simply connected. This is a consequence of the Maximum Modulus Principle. The Julia set is the common boundary of these two complementary sets, i.e. Jf “ BAf p8q “ BKf , while the Fatou set Ff consists of the connected component Af p8q and all connected components of the interior of Kf , if any. Since Kf is full, it follows that Kf is connected if and only if Jf is connected. The name ‘filled Julia set’ for Kf refers to the fact that Kf is obtained from the Julia set Jf by filling-in all bounded Fatou components, if any.


123

It turns out that the topological property of connectivity is strongly related to the dynamical behaviour of the critical orbits (cf. [Mi1, Thm. 9.5]). Theorem 3.48 (Connectivity of polynomial Julia sets) Let f P Pold . Then: (a) Kf is connected if and only if Critpf q Ă Kf . In this case, the restriction p p f is conformally conjugate to z ÞÑ zd on CzD; of f to CzK (b) Kf is totally disconnected if Critpf q Ă Af p8q. In this case Jf “ Kf and it is a Cantor set; (c) if at least one critical point of f belongs to Af p8q then both Kf and Jf are disconnected and have uncountably many connected components. Note that items (a) and (c) in the theorem above imply that Af p8q is either simply connected or infinitely connected. Remark 3.49 (The Böttcher coordinate around infinity) Since the point at 8 is a superattracting fixed point for any f P Pold , it follows from Theorem 3.31 that there exists a neighbourhood U of 8 and a local conformal p that conjugates f to z ÞÑ zd . If the polynomial is conjugacy ϕ “ ϕf : U Ñ C monic then we shall refer to the Böttcher coordinate around 8 as the conjugacy that is uniquely determined by the requirement that f pzq{z Ñ 1 as z Ñ 8. Green’s function, equipotentials and external rays Assume that f is monic, p r denote the Böttcher coordinate that conjugates f to and let ϕ : U Ñ CzD d z ÞÑ z in the domain U , which is mapped outside Dr for the maximum r ě 1. If Kf is connected then r “ 1 as stated in Theorem 3.48(a), and if Kf is disconnected then r ą 1 and BU contains a critical point. Set A˚f p8q “ Af p8qzt8u and U ˚ “ U zt8u. The Green’s function for Kf is the real continuous harmonic function g : A˚f p8q Ñ R` that extends log |ϕ| : U ˚ Ñ plog r, `8q as follows: # log |ϕpzq| if z P U ˚ , gpzq “ 1 gpf k pzqq if f k pzq P U ˚ . dk Since gpf pzqq “ d ¨ gpzq for z P U ˚ the function g is well defined on all of A˚f p8q. It may also be extended to the whole plane by setting g ” 0 on Kf . For ρ ą 0, the level set gρ :“ g ´1 pρq “ tz P A˚f p8q | gpzq “ ρu is called the equipotential of potential ρ. If eρ ą r, it is a simple closed curve that surrounds the Julia set.

124 Preliminaries on dynamical systems and actions of Kleinian groups Suppose Kf is connected. Then for t P T “ R{Z, the curve Rf ptq :“ tz P CzKf | argpϕpzqq “ 2π tu is called the (external) ray of (external) argument t. Note that Rf ptq is mapped bijectively under f to Rf pd ¨ tq with arguments in T. In particular, if d p ¨ t ” t pmod 1q then the ray Rf ptq is p-periodic (with minimal p). For example, Rf p0q is always fixed, while Rf pn{dq is always prefixed for any 1 ď n ă d (see Figure 3.9). Rf p 27 q 2 7

Rf p 17 q

Rf p 12 q

Rf p 47 q

1 7

ϕ ´1 ϕ

Rf p0q

-

1 2

0 4 7

Figure 3.9 The filled Julia set of f pzq “ z2 ` c with c „ ´0.12256 ` 0.74486i , also known as the Douady rabbit. This is an example of a locally connected Julia set where all rays land. Some rays and equipotentials are drawn, in particular the fixed ray Rf p0q landing at one fixed point of f , the prefixed ray Rf p 12 q and the 3-cycle of rays landing at the other fixed point of f .

If the limit ´ ¯ γ ptq “ lim ϕ ´1 re2π it r Ñ 1`

exists, we say that Rf ptq lands at the point γ ptq which necessarily belongs to the Julia set. All rays of rational argument land at a repelling or parabolic periodic point (see [Mi1, Thm. 18.10]). If Rf ptq lands at γ ptq, then Rf pd ¨ tq lands at f pγ ptqq. Hence, the landing points of a k-cycle of rays must form a periodic orbit of a period dividing k. Conversely, every repelling or parabolic periodic point z0 is the landing point of at least one periodic ray (see [Mi1, Thm. 8.11] or [Hu1, Thm. I.A]). If k is the period of z0 , only finitely many rays, say q 1 , land at a z0 , and these rays are all periodic of the same period. They are transitively permuted by f k , and this permutation must preserve their circular order since f is a local homeomorphism at z0 ; thus the permutation must send each ray to


125

the one which is p1 further counterclockwise for some p1 ă q 1 . We call p{q the combinatorial rotation number of f at z0 , where p{q “ p1 {q 1 written in lowest terms. There exist polynomials for which not all rays land. But Fatou showed that the set of arguments t P T for which Rf ptq does not land has measure zero. Furthermore, M. and F. Riesz proved that for any given z0 P Jf the set of arguments for which γ ptq “ z0 has measure zero (cf. [CL]). The following theorem summarizes the landing properties of external rays (cf. [Mi1, Thm. 18.3]). Theorem 3.50 (Continuous landing of rays) For any polynomial f with connected Julia set the following conditions are equivalent: (i) every external ray Rt lands at a point γ ptq which depends continuously on the argument t; (ii) the Julia set Jf is locally connected; (iii) the filled Julia set Kf is locally connected; (iv) the inverse of the Böttcher coordinate ϕ ´1 : CzD Ñ CzKf extends continuously to BD and hence induces a continuous parametrization γ : T Ñ Jf of the Julia set, given as γ ptq “ ϕ ´1 pe2π i t q. Remark 3.51 (Carathéodory semi-conjugacy) If the conditions in Theorem 3.50 are satisfied then the map γ : T Ñ Jf is a semi-conjugacy between the map t ÞÑ d ¨ t pmod 1q on T and f on Jf . It is called the Carathéodory semi-conjugacy of Jf (sometimes the Carathéodory loop). This is yet another example of how strongly topological and dynamical properties interplay. Julia sets which are not locally connected are for example those with a Cremer periodic point, or with a Siegel disc whose boundary does not contain a critical point (cf. [Mi1, Cor. 18.6]). Maps with this property are constructed by surgery as explained in Section 7.2.

Hybrid equivalences We end this section by mentioning another important type of conjugacy, although it is called equivalence for historical reasons. Definition 3.52 (Hybrid equivalence) Two polynomials f and g in Pold are hybrid equivalent if there exist neighbourhoods Uf and Ug of Kf and Kg respectively, and a quasiconformal conjugacy φ : Uf Ñ Ug between f and g, satisfying Bφ “ 0 almost everywhere on Kf .

126 Preliminaries on dynamical systems and actions of Kleinian groups Observe that by Weyl’s Lemma (Theorem 1.14) this means that φ is holomorphic in the interior of Kf , if this is non-empty. Hybrid equivalences are the strongest possible type of conjugacy when the Julia sets are connected, as shown by the following theorem [DH3, Prop. 21]. Theorem 3.53 (Hybrid classes are affine classes in the connected case) Let f, g P Pold with connected Julia sets. If f and g are hybrid equivalent then they are affine conjugate.

Exercises Section 3.3 3.3.1 Let f be holomorphic in a neighbourhood U of S1 . Assume S1 is invariant under f and f |S1 is analytically conjugate to the rigid rotation Rθ for some θ P RzQ. Show that f has a Herman ring (or an annular subset of a Herman ring) containing the unit circle. More precisely, let ϕ : S1 Ñ S1 denote an analytic conjugacy and prove that there exists a neighbourhood U 1 Ă U of S1 such that f |U is conformally conjugate to the rigid rotation Rθ on some standard annulus containing S1 and such that the conformal conjugacy ϕr is equal to ϕ on S1 . 3.3.2 Show that a critical circle map can not be analytically conjugate to an irrational rigid rotation. Hint: Use Exercise 3.3.1 above. 3.3.3 Let γ be a Jordan curve that is invariant under a univalent map f defined in a neighbourhood of γ , and assume that rotpf, γ q P H. Prove that f has a Herman ring (or an annular subset of a Herman ring) around γ . Hint: Apply Theorem 3.18

3.4 Families of holomorphic dynamics: parameter spaces We consider holomorphic families of maps in Ratd , where d is fixed, that is p Ñ C, p λ P u Ă Ratd , tfλ : C where the parameter space is a complex manifold of dimension N ě 1 so p ÑC p defined by pλ, zq ÞÑ fλ pzq is holomorphic, i.e. so that the map ˆ C that the coefficients of fλ depend holomorphically on λ. In order to define the notion of J -stability we need to state the general concept of holomorphic motion introduced by Mañe´ , Sad and Sullivan in [MSS]. Definition 3.54 (Holomorphic motion) A holomorphic motion of a set A Ă p over is a mapping H : ˆ A Ñ C p satisfying the following: C

3.4 Families of holomorphic dynamics: parameter spaces

127

(1) for any fixed z P A the mapping λ ÞÑ H pλ, zq is holomorphic; (2) for any fixed λ P the mapping z ÞÑ Hλ pzq :“ H pλ, zq is injective; (3) the mapping Hλ0 “ IdA , for a base point λ0 P . Note that continuity of H is not required in the definition. However, this property follows as a consequence, as shown in the λ-lemma proved in [MSS]. Theorem 3.55 (The λ-lemma) A holomorphic motion of A as above has a unique extension to a holomorphic motion of A. The extended map H : p is continuous, and for each λ P , Hλ : A Ñ C p extends to a ÂÑC p quasiconformal self-map of C. For further results about holomorphic motion and the λ-lemma, see for instance, papers by Bers and Royden [BR] and Słodkowski [Sl], and the survey by Astala and Martin [AM]. Definition 3.56 (J -stability) Consider as above a holomorphic family f : p ÑC p of rational maps in Ratd . For a λ0 P the Julia set Jf is J ˆC λ0 stable if there exists a neighbourhood Uλ0 Ă of λ0 over which the Julia sets p such that Hλ pJλ q “ Jλ and move holomorphically, i.e. H : Uλ0 ˆ Jλ0 Ñ C 0 furthermore Hλ ˝ fλ0 “ fλ ˝ Hλ0 . Hyperbolic rational maps are J -stable. An even stronger statement is proven in Section 4.1 by surgery. The following theorem is also proved in [MSS]. Theorem 3.57 (J -stable set of parameters) The J -stable set of parameters J -stable Ă is an open dense subset of . Our goal is to identify sets of parameter values in for which the corresponding dynamical systems are conjugate or share some given properties. In particular, we shall be interested in conformal and quasiconformal conjugacies, either global or on (neighbourhoods of) the Julia set. 3.4.1 The parameter plane of quadratic polynomials: the Mandelbrot set The best-understood holomorphic family is Pol2 . In the parametrization tQc pzq “ z2 ` c | c P Cu, all quadratic polynomials are uniquely represented up to conformal conjugacy (see Exercise 3.1.4 in Section 3.1). We refer to dedicated chapters in [DH2, Bra1, CG, St] for extensions of the contents of this section.

128 Preliminaries on dynamical systems and actions of Kleinian groups For quadratic polynomials of the form Qc with c P C, we set Kc :“ KQc , Ac :“ AQc p8q and Jc :“ JQc . In the parameter plane C the Mandelbrot set M is defined as M :“ tc P C | Qnc p0q Ñ 8 as n Ñ 8u, i.e. the set of c-values for which the critical orbit tQnc p0qun is bounded. It follows from Theorem 3.48 that if c P M then Kc (and hence also Jc ) is connected, and if c R M then Kc “ Jc is a Cantor set. This means that M “ tc P C | Kc is connectedu. To emphasize this property M is also called the connectedness locus of the quadratic family tQc ucPC (see Figure 3.10). The first basic properties of M were proven by Douady and Hubbard in 1982 (cf. [DH2]). Theorem 3.58 (Basic properties of M) The Mandelbrot set is compact, connected and full. Moreover, it is contained in D2 . Hyperbolic components of M A polynomial Qc can have at most one attracting cycle in C (see Theorem 3.39). It follows that Qc is hyperbolic if and only if Qc has an attracting cycle in C, or if the orbit of the critical point is attracted to the superattracting fixed point at 8 (see Theorem 3.43). In this case the c-value is called a hyperbolic parameter. If Qc0 has an attracting p-cycle in C, then it follows from the Implicit Function Theorem that the p-cycle moves holomorphically for parameter values close to c0 and that the cycle is still attracting. As a consequence, c0 P intpMq. The maximal open set containing c0 , , of parameters for which a p-cycle exists and remains attracting is called a hyperbolic component (of period p) of M. One can check that B Ă BM so that a hyperbolic component is actually a connected component of the interior of M. It is not known if all connected components of intpMq are hyperbolic. It is, however, conjectured to be so. Conjecture 3.59 (Hyperbolicity conjecture) ď intpMq “

hyp.comp.

Equivalently, hyperbolic parameters are dense in C. The conjecture has been proven for parameters on the real line [GS, KSvS].


129

Figure 3.10 The Mandelbrot set can be seen as a catalogue of Julia sets. Filled Julia sets are shaded and attracting cycles are marked. The upper-left picture shows a Siegel disc with some interior orbits drawn.

130 Preliminaries on dynamical systems and actions of Kleinian groups Given a hyperbolic component of period p, the multiplier map is defined as : ÝÑ D, p

c ÞÝÑ pcq “ pQc q1 pzpcqq, where zpcq for each c P denotes a point in the attracting p-cycle so that c ÞÑ zpcq is holomorphic. It follows that is holomorphic. One of the very first surgery constructions was done in order to prove the following theorem as we explain in Section 4.1. Theorem 3.60 (Multiplier map) Let be a hyperbolic component of M of period p. Then the multiplier map is a conformal isomorphism which extends continuously to : Ñ D. It follows that all hyperbolic components are topological discs, each with a centre, the unique parameter value for which the attracting cycle is superattracting, i.e. ´1 p0q. The boundary of has a unique point, called the root of , for which the multiplier of the p-cycle is exactly 1. Attached to every boundary parameter for which the multiplier of the cycle has derivative e2π i r {s we find a new hyperbolic component of period p ¨ s, which explains part of the fractal structure observable near any point of the boundary of the M (see Figure 3.11).

Figure 3.11 If is a hyperbolic component, for every rational number there is a smaller hyperbolic component attached to B.

The boundary of M The boundary of the Mandelbrot set is an intricate and fascinating object which is still the focus of intense research. It has a remarkable fractal structure of Hausdorff dimension equal to 2 [Sh2], the highest possible in the plane. The boundary of the Mandelbrot set can also be characterized in terms of normal families (see Exercise 3.4.1 or [McM2, Thm. 4.6]) as BM “ tc P C | tgn pcq :“ Qnc p0qun is not normal in any neighbourhood of cu.


131

We say that c P M is Misiurewicz if Qc is a Misiurewicz polynomial. This characterization of BM is used to prove the following properties. Theorem 3.61 (Parameters in BM) (a) Misiurewicz points are dense in BM. (b) For each c P BM there exists a sequence tcn un of centres of hyperbolic components such that cn Ñ c as n Ñ 8. (c) If Qc has a parabolic, Siegel or Cremer cycle, then c P BM. The convergence of centres to the boundary points is very fast. This is a property used to compute very accurate approximations to the Mandelbrot set as the ones shown in this section. J -stability can be formulated in many different equivalent ways, see cf. [McM2, Thm. 4.2]. For the quadratic family tQc ucPC , J -stability at c is equivalent to the family of maps gn pcq :“ Qnc p0q being normal at c. Hence CJ -stable “ tc P C | tgn pcqunPN normal in a neighbourhood of cu “ CzBM. The most important conjecture about the Mandelbrot set concerns a topological property. Conjecture 3.62 (MLC) M is locally connected. Conjugacy classes We already mentioned (Exercise 3.1.4 in Section 3.1) that the family tQc ucPC has exactly one representative in each conformal (therefore affine) conjugacy class. Some facts concerning other types of conjugacy are summarized in the following theorem. Theorem 3.63 (Conjugacy classes) (a) Any two polynomials in CzM are hybrid equivalent, but no two different polynomials in M are (see Theorem 3.53). (b) All polynomials in a given hyperbolic component with the exception of its centre are globally quasiconformally conjugate. If we only require the conjugacies to hold in a neighbourhood of the Julia set, then the centre is also included (see Section 4.1 and Figure 3.12). (c) If c P BM and c1 P C and Qc is quasiconformally conjugate to Qc1 , then c “ c1 . In other words, all parameters c P BM are unique representatives of their quasiconformal conjugacy class. It is a central question in holomorphic dynamics whether a topological conjugacy implies a quasiconformal one.


Figure 3.12 Any two polynomials in the same hyperbolic component, different from the centre polynomial, are quasiconformally conjugate. The quasiconformal conjugacy sends one Julia set to the other. In the figure, the basins of attraction are shaded and the periodic cycles are marked.

Conjecture 3.64 (Rigidity conjecture) Two rational maps which are topologically conjugate are quasiconformally conjugate. In the particular case of quadratic polynomials: Rigidity conjecture ùñ MLC ùñ Hyperbolicity conjecture. A weaker type of conjugacy between Julia sets (David conjugacy) is treated in Section 9.3.

3.4.2 Other parameter spaces There are systematic studies of other parameter spaces for particular holomorphic families of polynomials, rational maps or transcendental maps. Here we mention those that are treated in later sections in this book. For a holomorphic family F “ tfλ | λ P u of complex polynomials of a fixed degree d ě 2, the connectedness locus of the family tfλ uλP is CpFq “ tλ P | Kfλ is connectedu.

3.5 Actions of Kleinian groups and the Sullivan dictionary

133

The structure of parameter planes of unicritical polynomials, tz ÞÑ zd ` c | c P C, d ą 2u, is very similar to that of quadratic polynomials. The connectedness loci Md of such families are known as Multibrot sets or generalized Mandelbrot sets. One difference with respect to M is that the multiplier map defined on a hyperbolic component is a degree pd ´ 1q branched covering of the disc, ramified over 0 in D. We refer to Section 4.1.2 for details. In general, any polynomial of degree d ě 2 is affine conjugate to a monic, centred polynomial of the same degree. Hence, the family of such polynomials can be parametrized by pd ´ 1q complex coefficients. The parameter space is therefore Cd ´1 . For cubic polynomials we may choose a parametrization of monic, centred polynomials with marked critical points z ÞÑ z3 ´ 3a 2 z ` b. The critical points are ˘a and the parameter space is C2 . Cubic polynomials are treated in Sections 6.2 and 7.5. We shall also deal with the uniparametric families of polynomials of degree pq ` 1q ě 3 ˙ ˆ z q , with λ P C˚ . Pq,λ pzq “ λz 1 ` q The connectedness loci of tPq,λ uλPC˚ presents many similarities to the Mandelbrot set, some of which are explored and used in Section 8.2.

Exercises Section 3.4 3.4.1 Prove that BM is the set of non-normality of the family G :“ tc ÞÑ gn pcq “ Qnc p0qu. Hint: Using Montel’s Theorem (Theorem 3.23), show that for c R BM there is a neighbourhood of c in which the family is normal. Additionally, show that for c P BM the family can not be normal in any neighbourhood of c.

3.5 Actions of Kleinian groups and the Sullivan dictionary Fatou and Julia were aware of the analogy between iteration of holomorphic mappings and Poincaré’s work on Kleinian groups. But it was Sullivan in 1985 [Su2, MS] who deeply exploited the interplay between these two theories and

134 Preliminaries on dynamical systems and actions of Kleinian groups established a first version of a dictionary between them. The analogy suggested that quasiconformal maps could play a role in holomorphic iteration, as it did in Kleinian groups. This is exemplified by the celebrated No Wandering Domains Theorem for rational maps and a new proof of its analogue for Kleinian groups, Ahlfors’ Finiteness Theorem. In this section we present a brief survey of the theory of Kleinian groups, emphasizing the parallelism with holomorphic iteration. As in the other sections in the current chapter we include mainly statements without proofs. We mostly follow [MNTU, Chapt. 5]. We also refer to [MT] and [Mar] for deep expositions of the subject.

3.5.1 Basic definitions p denote the group of Möbius transformations of the Riemann Let MöbpCq sphere, i.e. ˇ * " ˇ p “ z ÞÑ az ` b ˇ a, b, c, d P C, ad ´ bc “ 1 , MöbpCq cz ` d ˇ which is identified with PSLp2, Cq “SLp2, Cq{t˘I u. The dynamics of Möbius transformations are fairly simple since every elep except the identity, is globally conformally conjugate (i.e. ment in MöbpCq, conjugate by a Möbius transformation) to one of three fundamental types. Proposition 3.65 (Classification of Möbius transformations) Let γ P p γ ‰ Id. Then γ is conformally conjugate to one of the following: MöbpCq, (a) z ÞÑ z ` 1, and we call γ Parabolic; (b) z Ñ Þ λz, |λ| “ 1, λ ‰ 1, and we call γ Elliptic; (c) z Ñ Þ λz, |λ| ą 1, and we call γ Loxodromic or Hyperbolic if λ P R. Parabolic transformations have only one fixed point, while elliptic and loxodromic have two. The only transformations of finite order (γ n “ Id for some n ą 1), different from the identity, are elliptic and conjugate to z ÞÑ e2π i p{q z with p, q P Z. Using that the trace of a matrix is invariant under matrix conjugation it follows that the type of a Möbius transformation γ ‰ Id is also classified by the trace of a representing matrix: • γ is parabolic (resp. elliptic) iff tr = ˘2 (resp. ´2 ă tr ă 2); • γ is loxodromic (in particular hyperbolic) iff | tr |ą 2 (tr P R). We are now ready to define a Kleinian group.


135

p is a Kleinian group Definition 3.66 (Kleinian group) A subgroup of MöbpCq if its identification in PSLp2, Cq is discrete. From the definition it follows that Kleinian groups are countable subsets of PSLp2, Cq and do not contain any elliptic elements of infinite order. It can be shown that a Kleinian group which consists only of elliptic transformations apart from the identity is a finite group. One may also observe that if a Kleinian group contains a loxodromic element γ , then it cannot contain a parabolic one, say η, that shares a fixed point with γ since γ ń ˝ η ˝ γ n Ñ Id as n Ñ 8 (or n Ñ ´8), contradicting the discreteness of the group. 3.5.2 The action of a Kleinian group and the dynamical partition A Kleinian group acts on the Riemann sphere as p ÝÑ C, p ˆC pγ , zq ÞÝÑ γ pzq. We set pzq :“ tγ pzq | γ P u. The dynamics generated by the action of a Kleinian group (also known as conformal dynamics) are different from those generated by iteration of a single holomorphic (usually non-invertible) map. If we apply to a point z we obtain a set of points pzq which can be thought of as the grand orbit of z, but without the order relation that comes with iteration. Invariant sets are still meaningful but forward and backward invariance are the same, since a group contains the inverse of all its elements. p is invariant under Definition 3.67 (Group invariance) We say that U Ă C the Kleinian group if pzq Ă U for all z P U . In this case pU q “ U . Another familiar concept is that of conformal or quasiconformal conjugation. Definition 3.68 (Quasiconformally conjugate groups) We say that the Kleinian groups and 1 are conformally (resp. quasiconformally) conjugate if there exists a Möbius transformation (resp. a quasiconformal map) pÑC p such that 1 “ φφ ´1 :“ tφ ˝ γ ˝ φ ´1 | γ P u. If is given by φ:C a set of generators (called a marking of ), then φ takes the marking of to the marking of 1 . As in iteration theory, normality is the property which induces the dynamical p In the case of families of Möbius transformations, Montel’s partition in C. Theorem has a special formulation (compare with Theorem 3.23).

136 Preliminaries on dynamical systems and actions of Kleinian groups p be a domain and F Theorem 3.69 (Montel’s Theorem adapted) Let U Ă C p such that a family of Möbius transformations. If there exist two points a, b P C p γ pU q Ă Czta, bu for all f P F, then F is a normal family on U . We are now ready to define the analogues to the Fatou and the Julia sets. Definition 3.70 (Ordinary set and limit set) Let be a Kleinian group. We define the ordinary set of , often also called the regular set, as p | is a normal family in some neighbourhood of zu, pq :“ tz P C p and the limit set of as its complement, pq :“ Czpq It follows from the definition that pq is open and consequently that pq is closed. Both sets are invariant under and, in fact, if #ppqq ě 3, then p The definition pq is the smallest non-empty, -invariant closed subset of C. above is chosen to emphasize the analogy between ordinary set and limit set and Fatou set and Julia set respectively. Classically, the ordinary set is defined as the complement of the limit set, which is defined as p | z is a limit pointu. pq “ tz P C p and an infinite Note that z is called a limit point if there exists ζ P C sequence of distinct elements γn P so that z “ limnÑ8 γn pζ q. Compare with Theorem 3.72, Definition 3.71 (Proper discontinuity) We say that a Kleinian group acts p if there exists a neighbourhood U of properly discontinuously at a point z P C z such that the number of elements of satisfying γ pU q X U ‰ H is finite. It can be shown that the ordinary set is precisely the set of points at which acts properly discontinuously. For this reason pq is often called the region of discontinuity. The limit set and the ordinary set are both preserved under conjugation in the sense that if and 1 are Kleinian groups conjugate by a quasiconformal map φ, then p 1 q “ φppqq and p 1 q “ φppqq. As an example consider a loxodromic transformation γ and set “ă γ ą, the group generated by γ . We may assume that γ pzq “ λz with |λ| ą 1, so that γ has fixed points at 0 and 8. Given any open set U Ă Czt0u, the sequence tpγ |U qn uně0 converges to the constant function equal to 8, while the sequence tpγ |U qń uně0 converges to the constant function equal to 0. If, Ť on the other hand, V is a neighbourhood of 0 then γ n pV q “ C and if V is Ť n p It follows that pq “ Czt0u a neighbourhood of 8 then γ pV q “ Czt0u. and pq “ t0, 8u. One can observe the difference with respect to iteration


137

theory where the Julia set of γ consists of the repelling fixed point t0u, while the point at infinity is attracting and hence belongs to the Fatou set. A Kleinian group whose limit set consists of at most two points is called elementary. The group above is an elementary Kleinian group. It can be shown that a non-elementary Kleinian group contains infinitely many loxodromic elements, and that for any loxodromic γ P there exists a loxodromic γ 1 P so that the two transformations have no fixed points in common. In our discussions, Kleinian groups will in general be assumed to be non-elementary. In addition, we shall also assume they contain no elliptic elements, since it can be shown that any non-elementary Kleinian group with elliptic elements always contains a subgroup 1 of finite index in , with no elliptic elements and such that pq “ p 1 q. From the example above it is easy to see that both fixed points of every loxodromic element in a Kleinian group belong to the limit set. Similarly, one can show that the fixed point of every parabolic element also belongs to the limit set. In the following theorem we summarize some of the properties of limit sets and ordinary sets, some of which were already mentioned. Compare with Theorem 3.27. Theorem 3.72 (Properties of limit sets and ordinary sets) Let be a Kleinian group. (1) Fixed points of loxodromic or parabolic elements of belong to pq. Hence pq ‰ H. (2) Fixed points of elliptic elements of belong to pq. (3) If 1 is a component of pq which is invariant under , then pq “ B1 . (4) The number of components of pq is 0, 1, 2 or 8. There are at most two invariant components. p (5) Either pq has no interior point or pq “ C. p If there exists a sequence of distinct elements γn P such (6) Let z, w P C. that γn pwq Ñ z, then z P pq. Assume furthermore that is non-elementary. (7) If U is an open set intersecting pq, then there exist finitely many elements γ1 , . . . , γN P such that pq Ă

N ď n“1

γn pU q.

138 Preliminaries on dynamical systems and actions of Kleinian groups p then there exists a sequence tγn u of such that (8) If z P pq and w P C, γn pwq Ñ z as n Ñ 8. (9) pq has no isolated points. (10) For all z P pq, the set pzq is dense in pq. (11) The set of fixed points of all loxodromic elements are dense in pq. (12) If contains parabolic transformations, then the set of fixed points of all parabolic elements is dense in pq. p so as to see the analogy with Julia sets Special examples where pq ‰ C In what follows we give four examples of Kleinian groups and their limit sets. They will motivate the concepts in the rest of the section and serve as references throughout. In all four cases is a Kleinian group with two generators “ă α, β ą, which are both loxodromic (except in the last example). In each case there exist four circles in the plane Cα , Cα1 , Cβ and Cβ1 with pairwise disjoint interiors such that: • αpCα q “ Cα1 and αpintpCα qq “ extpCα1 q; • βpCβ q “ Cβ1 and βpintpCβ qq “ extpCβ1 q, where int and ext respectively denote the bounded and unbounded components of the complement. Let D :“ intpCα q Y intpCα1 q Y intpCβ q Y intpCβ1 q, p maps to D by α, α ´1 , β and β ´1 . and observe that pq Ă D. Indeed, CzD Thus one can check that every orbit in is completely contained in D or has at most one point in the complement. Consequently one can construct the limit set as the limit of images of the four circles by all possible words in the group formed with α, β and their inverses. For a beautiful exposition about these type of constructions we refer to the book Indra’s Pearls by Mumford, Series and Wright [MSW]. The four examples below follow their recipes. Example 3.73 (pq is a Cantor set) ˙ ˙ ˆ? ˆ? 2? ´2 2 ?i . One can check that Cα and Cα1 and β “ Let α “ í 2 ´ 12 2 can be chosen to be the circles centred at ˘ ?5 with radius ?1 . Likewise, 6

6

3 1 Cβ and Cβ1 can be chosen as the circles centred at ˘i ? with radius ? . 2 2 2 2 The four circles have disjoint closures, and therefore their images form an


Cβ1

Cα1

139

Cβ

Cα

Figure 3.13 Example 3.73. On the left, the four circles Cα , Cα1 , Cβ , Cβ1 and their images under α, α ´1 , β, β ´1 . The small circles at level 1 are coloured depending of which large circle they come from. On the right, several more steps are shown. The limit set is a Cantor set.

uncountable collection of nested circles. The circles are called Schottky circles. Figure 3.13 shows on the left the first step of the sequence, while on the right seven steps are shown. One can show that all ends consist of a single point. Hence the limit set is a Cantor set. Example 3.74 (pq is the unit circle) ˜ 2 1 ¸ ? ˙ ˆ ? ? 2? i 3 3 3 and β “ (see Figure 3.14). In this case Let α “ ?1 ?2 í 3 2 3 3 ? the circles Cα and Cα1 are chosen to be centred at ˘2 with radius 3 while the circles Cβ , Cβ1 are centred at ˘i ?2 with radius ?1 . In this case the unit 3 3 circle is invariant under both α and β. The?example is chosen so that the four circles touch pairwise at the points ˘ 12 ˘ i 23 . These points are parabolic fixed points of the different commutators formed by α, β, α ´1 , β ´1 . The limit set is the unit circle.

Cα

Cβ1

Cα1

Cβ

Figure 3.14 Similar to the previous figure, illustrating the group in Example 3.74. The limit set is the unit circle.

140 Preliminaries on dynamical systems and actions of Kleinian groups Example 3.75 (pq is a quasicircle) ˜? ¸ ˜? Let α “

5 1 2 ?2 5 1 2 2

89 i 16 5 ?5 í 45 589

and

¸

β“ (see Figure 3.15). The circles ? Cα and Cα1 are centred at ˘ 5 with radius 2, and the circles Cβ and Cβ1 ?

are centred at ˘i 489 with radius 54 . The four circles are touching pairwise at parabolic points of the different commutators formed by α, β, α ´1 , β ´1 . The four points are not on a circle. The limit set is a quasicircle.

Cβ1

Cα1

Cα

Cβ

Figure 3.15 Similar to the previous figure, illustrating the group in Example 3.75 The limit set is a quasicircle.

Example 3.76 (pq has infinitely many pinch-points) In this example we have chosen β so that the circles Cβ and Cβ1 meet at the origin at a fixed point (see Figure 3.16). This can be seen as a limit case of Example 3.75, where the repelling and attracting fixed points of β, previously in the interior of the circles, have collapsed at the origin (common boundary point) and thus β has become a parabolic transformation. The transformation α is loxodromic and chosen, as before, so that the four common boundary points (different from the origin) are parabolic fixed points of the commutators. This collapse induces the limit set to ‘pinch’ at infinitely many points. This example is known as ‘Mickey Mouse’.

3.5.3 Fuchsian, quasi-Fuchsian and Schottky groups Example 3.74 above is an example of a special type of Kleinian group called Fuchsian. They are the analogues of Blaschke products in complex iteration.


141

Figure 3.16 Similar to the previous figure, illustrating the group in Example 3.76. The ordinary set has infinitely many bounded components.

Definition 3.77 (Fuchsian group) We say that a Kleinian group is Fuchsian p such that both components of CzS p are invariant under if there is a circle S Ă C (hence so is S). Since pq is minimal, it follows that pq Ă S. We say that is a Fuchsian group of the first kind if pq “ S. Otherwise it is of the second kind. In greater generality, if the limit set of a finitely generated Kleinian group is a Jordan curve, and both complementary components are invariant, then is called a quasi-Fuchsian group (see Example 3.75 above). These are the analogues of maps which are quasiconformally conjugate to Blaschke products (see Definition 3.45). The limit set of a quasi-Fuchsian group is a quasicircle but the corresponding statement is not always true in complex iteration, as shown by the Julia set of Q1{4 pzq “ z2 ` 1{4, where the Fatou set has two completely invariant components, and the Julia set is a Jordan curve which is not a quasicircle since it contains cusps. Another important class of Kleinian groups are those called Schottky groups. They are analogues of hyperbolic rational maps with a connected Fatou set.

142 Preliminaries on dynamical systems and actions of Kleinian groups Definition 3.78 (Schottky group) A Kleinian group is a Schottky group if it is free, finitely generated and all its non-trivial elements are loxodromic. It was shown by Maskit [Mas] that for a Schottky group there always exist 2n disjoint oriented Jordan curves C1 , C11 , . . . , Cn , Cn1 with disjoint interiors, and n Möbius transformations γ1 , . . . , γn such that γj maps the interior of Cj to the exterior of Cj1 , for all 1 ď j ď n, such that “ă γ1 , . . . , γn ą. In this case the limit set pq is a Cantor set contained in the union of the interiors of the 2n Jordan curves. The group in Example 3.73 is a Schottky group for which the Jordan curves can be chosen as circles. These examples exhaust by no means all types of Klenian groups.

3.5.4 Ahlfors’ Finiteness Theorem and no wandering domains Assume is a finitely generated Kleinian group. Definition 3.79 (Fundamental domain) Suppose 1 Ă pq is invariant. We say that F Ă 1 is a fundamental domain of in 1 if F is connected and every orbit pzq Ă 1 intersects F in exactly one point. p Y In Example 3.73, pq has one invariant component. Set F “ pCzDq 1 1 pCα Y Cβ q where D “ intpCα q Y intpCα q Y intpCβ q Y intpCβ q. Then F is a fundamental domain. Consider F and identify Cα1 with Cα , Cβ1 with Cβ under the action of α respectively β. It follows that pq{ is a compact Riemann surface of genus 2. In greater generality, for any Schottky group with g ě 2 generators, the ordinary set pq is invariant and pq{ is a compact Riemann surface of genus g. In fact, every compact Riemann surface of genus g is quasiconformally equivalent to pq{ where is a Schottky group with g generators. (Cf. the Retrosection Theorem in [Be2, Sec. 1.6].) In Example 3.74, pq has two invariant components: 1 “ D and 2 “ p Compare with Figure 3.17. Let Q1 denote the quadrilateral bounded by CzD. the four tangent circles. Then 1 X pQ1 zpCα1 Y Cβ1 qq is a fundamental domain of in 1 . Based on the identifications by α and β on BQ1 it follows that 1 { is isomorphic to a torus with one puncture. Indeed, the four corners correspond to just one point which does not belong to pq, and therefore corresponds to a puncture. By symmetry, 2 { is also isomorphic to a torus with one puncture. Hence the quotient space pq{ is isomorphic to two punctured tori, with their punctures at infinity. The examples above illustrate a general fact known as Ahlfors’ Finiteness Theorem, which was originally proven in [Ah1].


143

Q1

Figure 3.17 The fundamental domain Q1 in Example 3.74 and the Riemann surface which corresponds to 1 pq{.

Theorem 3.80 (Ahlfors’ Finiteness Theorem) For a finitely generated Kleinian group , the set pq{ is a finite union of compact Riemann surfaces each with at most finitely many punctures. In Section 4.3 it is explained how the corollary below follows from Ahlfors’ Finiteness Theorem. Corollary 3.81 (No wandering domains) If is a finitely generated Kleinian group and U is a component of pq, then, there exists a non-trivial element γ P such that γ pU q “ U . In other words, no component of pq is a wandering component. Ahlfors’ Finiteness Theorem is the analogue of Sullivan’s Theorem on the non-existence of wandering domains (Theorem 4.2) for rational maps. In fact Sullivan gave a unified proof of both results [Su2] (cf. [Be3]) using quasiconformal mappings and deformations. We shall explain the ideas and one of the proofs in detail in Section 4.3. 3.5.5 Kleinian groups and quasiconformal mappings: parameter spaces To keep the discussion limited we consider a simple class of groups in order to illustrate the similarities between Kleinian groups and holomorphic dynamics. Let “ă α, β ą be a free group with two loxodromic generators with the condition that αβα ´1 β ´1 is parabolic with trace ´2. Examples 3.74 and 3.75 above consider such groups; their quotient space is a pair of punctured tori. For simplicity, we shall assume that the group is Fuchsian of the first kind and that the limit set is the unit circle as in Example 3.74.

144 Preliminaries on dynamical systems and actions of Kleinian groups p such that αφ :“ φ ˝ α ˝ φ ´1 and βφ :“ Let φ be a quasiconformal map of C ´ 1 φ ˝ β ˝ φ are Möbius transformations (how to choose φ will be explained in Section 4.1.3). Then φ “ă αφ , βφ ą is again Kleinian and satisfies that αφ βφ αφ´1 βφ´1 is parabolic. It follows that pφ q “ φpS1 q is a quasicircle. The quotient space pφ q{φ is the union of two punctured tori φ

T1 “ φpDq{φ

and

φ p T2 “ φpCzDq{ φ.

The group φ is called a quasiconformal deformation of . The space of quasiconformal deformations of is analogous to the space of rational maps with two attracting fixed points, a component of Rat2 which depends on two complex parameters. The generators αφ and βφ each depend on three complex parameters (the coefficients of the matrix) and we can normalize so that three points are fixed. The relation tracepαφ βφ αφ´1 βφ´1 q “ ´2 (corresponding to the commutator being parabolic) is a relation between the remaining three parameters so that tφ u is a two-dimensional space. These two parameters can be thought of as the moduli of the punctured tori. φ p then T2 “ If the quasiconformal map φ is restricted to be conformal in CzD, p p φpCzDq{ φ is conformally equivalent to T2 “ pCzDq. The space of quasiFuchsian groups p p and αφ , βφ P MöbpCqu B :“ tφ “ φφ ´1 | φ is conformal in CzD, is called a Bers slice and is analogous to the main hyperbolic component of the Mandelbrot set. We shall explain these deformations in more detail in Section 4.1. If we consider a sequence φn of quasiconformal maps such that φn P B and so that αφn P φn converges to a parabolic element when n Ñ 8, then φn tends to a point in the boundary of B. Such a boundary group is considered in Example 3.76, which is an example of what is known as a geometrically finite group. To define this concept in the context of Kleinian groups (compare to Definition 3.44 in holomorphic dynamics) one needs to consider the extension of an action of a Kleinian group to the hyperbolic 3-ball, which is an orientation preserving isometry in the hyperbolic metric. The concept of fundamental domains extends to the 3-ball. We then say that a a Kleinian group is geometrically finite if it has a fundamental domain which is a polyhedron in the 3-ball with finitely many sides.


145

Non-geometrically finite groups are called degenerate. Such groups can appear as limits of quasi-Fuchsian groups in Bers’ slices. An analogy in holomorphic dynamics would be Siegel or Cremer polynomials in the cardioid, the boundary of the main hyperbolic component of the Mandelbrot set. The Density Conjecture by Bers, Sullivan and Thurston, states that any finitely generated Kleinian group is a limit of geometrically finite groups. This has recently been proved as the culmination of extensive and remarkable work by many authors, in particular the so-called Tameness Conjecture, see [Mar] for details. The Ahlfors Conjecture, that the limit set of any finitely p or has Lebesgue measure zero, follows generated Kleinian group is either C from previous results of Canary and the Tameness Theorem, see [Mar]. As in the case of rational maps, if the limit set of a geometrically finite Kleinian group is connected, then it is also locally connected [AMa]. (Compare with comments after Definition 3.44.) Using the above machinery, in a long

Table 3.1 Sullivan’s dictionary Complex iteration f P Rat Y Ent f P Rat with degpf q ě 2 Julia set J pf q Fatou set F pf q J pf q is the smallest, closed, nonempty, completely invariant set Repelling periodic points are dense in J pf q #tcomp. of F pf qu “ 0, 1, 2 or 8 F pf q has at most two completely invariant components No Wandering Domains Theorem Blaschke product Hyperbolic rational map with two completely invariant components Hyperbolic rational map with connected F pf q f geometrically finite f hyperbolic rational Main hyperbolic component of the Mandelbrot set

Kleinian groups non-elementary non-elementary and finitely generated Limit set pq Ordinary set or set of discontinuity pq pq is the smallest, closed, non-empty, invariant set. Loxodromic and parabolic fixed points are dense in pq #tcomp. of pqu “ 0, 1, 2 or 8 pq has at most two invariant components Ahlfors’ Finiteness Theorem Fuchsian group Quasi-Fuchsian group Schottky group geometrically finite geometrically finite without parabolic elements Bers’ slice

146 Preliminaries on dynamical systems and actions of Kleinian groups series of papers Mahan Mj has claimed an extension of this result to all finitely generated Kleinian groups. 3.5.6 The Sullivan dictionary We conclude this section with a summary of the terms that appeared in the previous discussion, paired with their analogues in iteration theory (see Table 3.1). The first version of this dictionary appeared in [Su2]. Since then, many empty spaces have been filled in, and several new lines have been added.

4 Introduction to surgery and first occurrences

In this chapter we describe what quasiconformal surgery is about and explain the surgeries which were developed as the first occurrences of this technique: the parametrization of hyperbolic components of the Mandelbrot set and the proof of the No Wandering Domains Theorem. Generalities What is known as quasiconformal surgery in holomorphic dynamics is a technique commonly used to construct holomorphic maps with prescribed dynamics. The ‘prescribed dynamics’ are given by a map f which in general is not holomorphic, although it may be. We shall refer to f as the model map. The word surgery appears because one may need to ‘cut’ and ‘paste’ different spaces and maps together to construct f . This is usually the first step in the construction and is known as topological surgery. We are leaving the holomorphic world in order to have a greater choice for our models, and then checking whether the model map has a ‘holomorphic dynamical copy’, i.e. whether there exists a holomorphic map conjugate to f . The main tool for obtaining ‘holomorphic dynamical copies’ is to apply the Integrability Theorem (Theorem 1.28), which provides a quasiconformal conjugacy to return to the holomorphic setting (see the Key Lemma (Lemma 1.39)). It follows that we should look for models in the space of quasiregular maps (see Proposition 1.37). Let us be more precise. Let S be a Riemann surface, conformally isomorphic p and suppose we have chosen a quasiregular map f : S Ñ S. to D, C or C, We are interested in finding a new map F , which is holomorphic and quasiconformally conjugate to f , hence has qualitatively the same dynamics. In view of the Key Lemma (Lemma 1.39), for this to be possible, we must be able to construct a Beltrami form μ on S, satisfying }μ}8 :“ k ă 1 and being f -invariant (i.e. f ˚ μ “ μ), or equivalently to construct an almost complex 147

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Introduction to surgery and first occurrences

structure σ on S, satisfying Kpσ q ă 8 and being f -invariant. Under these conditions, the existence of F is assured. To define an f -invariant almost complex structure with bounded dilatation is not always possible. It is a natural question to ask under which conditions it can be done. We address this problem in general in Chapter 5. We see later, in Chapter 9, that it is also possible to obtain similar results if the almost complex structure has some areas where the dilatation is not bounded, as long as these regions are sufficiently small. Such cases involve the generalized version of the Integrability Theorem (Theorem 9.7) and the surgery is then called trans-quasiconformal surgery. In this setting, the steps described above are similar but some of them are more involved. Let us recall the elementary example of Section 1.5, and see how it provides – in the simplest case when the model map is holomorphic – an explicit example of what we just explained. We consider the C-linear map f :“ M0 : D Ñ D, where M0 pzq “ λ0 z for some 0 ă |λ0 | ă 1. We define a Beltrami coefficient μν on D, which is the pullback of μ0 , first by an Rlinear map Lν of the plane (hence quasiconformal) and then by a branch of the logarithm (hence holomorphic). The dilatation of μν is the same as the dilatation of the map Lν , and therefore constant. By construction, this Beltrami coefficient μν is M0 -invariant. In this simple example, we are able to write down explicitly an integrating map φν and see that (after composing with a logarithm) it conjugates M0 pzq “ λ0 z to F pzq :“ Mν pzq “ eν z. In general, finding a holomorphic map F that is conjugate to the quasiregular map f is the last step in the surgery construction. The first step, the construction of a model map, often requires some creativity. It is at this stage that the actual ‘surgery’ is involved. There is no general theory on how to do this, since each problem requires its own particular solution. One rough classification of the different surgery procedures comes from the regularity of the model map and its domain. Soft surgery: the model map is holomorphic The simplest type of surgery is a change of complex structure without changing the model map (e.g. the elementary example of Section 1.5). One could argue whether it deserves the name surgery. Start with a Riemann surface S and a holomorphic map f : S Ñ S. Suppose we can find an almost complex structure σ on S (different from σ0 ), which is f -invariant, i.e. f ˚ σ “ σ , and with bounded dilatation. After applying the Integrability Theorem as in the Key Lemma, we obtain a holomorphic map F , which is quasiconformally conjugate to the original map f and is called a quasiconformal deformation of f . One needs to check whether f and F are different maps, since it might happen that we are back


149

to where we started (as seen in the elementary example, when eν “ λ0 ). If we always get the original map f back, no matter which invariant almost complex structure we choose, we say that f is quasiconformally rigid. Soft surgery in holomorphic dynamics is equivalent to its analogue and predecessor in Kleinian groups. Given a Kleinian group , suppose we can find an p which is γ -invariant by all elements γ P , almost complex structure σ on C and with bounded dilatation. If φ is an integrating map given by the Integrability Theorem, the elements φ ˝ γ ˝ φ ´1 are again Möbius transformations and φ :“ tφ ˝ γ ˝ φ ´1 | γ P u is again a Kleinian group, quasiconformally conjugate to . It is called a quasiconformal deformation of . This type of surgery is sometimes used as a building block for other more complicated ones, but it is most interesting when performed so that it depends on one or more parameters. It then allows us, for instance, to parametrize subsets of the parameter space where the maps (or groups) are structurally stable. The hyperbolic components of the Mandelbrot set or the Bers’ slices are examples of this. In this chapter and in Chapter 6 we shall look at several constructions of this kind. The space of quasiconformal deformations of a given map f is of great interest. With some identifications it is known as the Teichmüller space of f , see for instance [MS]. Cut and paste surgery: the model map is quasiregular This is the type of surgery for which most applications have been found. The given model map f : S Ñ S is quasiregular and often obtained by pasting together different holomorphic and quasiconformal maps on the Riemann surface S. Sometimes S itself is constructed by cutting and pasting together different pieces of the complex plane or other Riemann surfaces. We shall look at examples in Chapters 7 and 8. When we obtain results, via surgery, that relate different parameter spaces, the model maps and the Beltrami forms usually depend on a parameter. The Riemann surface may be fixed or also depend on the parameter. The latter case is more involved since it requires the use of the Uniformization Theorem. In Chapter 8 we shall consider examples of this kind, both when the underlying Riemann surface is fixed and when it depends on a parameter. First occurrences Inspired by the quasiconformal deformations of Fuchsian groups (see Section 3.5.5 and the description of soft surgery above), Sullivan was the first to use quasiconformal mappings in holomorphic dynamics, to prove the two results which we explain in this section.

150


The earliest recorded surgery construction in holomorphic dynamics took ´ place in the fall of 1981. Sullivan in a seminar at Institut des Hautes Etudes Scientiques (IHES), and Douady and Hubbard soon after, proved the following theorem (cf. Section 3.4). Theorem 4.1 (The multiplier map) Let be a hyperbolic component of period p of the Mandelbrot set. Then, the multiplier map : Ñ D is a conformal isomorphism. The multiplier map is holomorphic, so what remains is to prove its bijectivity. The proof they gave falls into two parts. Let ˚ “ z´1 p0q consist of the parameters in corresponding to non-superattracting periodic orbits. First we prove (following Sullivan) that : ˚ Ñ D˚ is a holomorphic covering map, and therefore that ´1 p0q is a single point, the centre of . This is an application of the elementary example in Section 1.5 and a soft surgery construction, which can be viewed as changing the modulus of a fundamental domain in the basin of attraction (a torus). The construction is explained in Section 4.1. We finish the argument by showing that the degree of the multiplier map is actually one, using a later result of A. Gleason (see [DH2]). At the end of the section, we sketch the analogous (and earlier) construction for Fuchsian groups, changing the modulus of a representing punctured torus. We also show (Theorem 4.7) how the same construction and similar arguments prove the generalization of Theorem 4.1 to Multibrot sets, i.e. the connectedness loci of unicritical polynomials normalized to be of the form Qd,c pzq “ zd ` c. More precisely, if is a hyperbolic component of period p of a Multibrot set Md , then the multiplier map : Ñ D is a holomorphic branched covering map of degree d ´ 1. In degree two, there is an alternative proof of Theorem 4.1 due to Douady and Hubbard, which uses cut and paste surgery on Blaschke products. We explain this procedure in Section 4.2. Around the same time, Sullivan proved the celebrated No Wandering Domains Theorem [Su2]. Theorem 4.2 (No Wandering Domains Theorem) Let f be a rational map of degree d ě 2. Then, every connected component of the Fatou set of f is preperiodic. The proof involves a soft surgery construction which we explain in Section 4.3. Using the same arguments Sullivan gave a new proof of Ahlfors’ Finiteness Theorem (Theorem 3.80).

4.1 Changing the multiplier of an attracting cycle

151

4.1 Changing the multiplier of an attracting cycle Let be as above, a hyperbolic component of period p of the Mandelbrot set M. The technique we shall use to prove that the multiplier map on a hyperbolic component, : ˚ Ñ D˚ , is a covering map is a soft surgery that can be applied in many other situations. The essential step, for a given quadratic polynomial in ˚ , is to change the multiplier of the attracting cycle. Therefore we shall start by explaining how to change the multiplier in greater generality. p Ñ C, p although To ease notation we choose to work with rational maps f : C the technique also applies to entire transcendental maps of C, or transcendental self-maps of C˚ . p ÑC p denote a rational map of degree d ě 2 with an attracting Let fλ0 : C cycle of period p of multiplier λ0 P D˚ . For any λ P D˚ our first goal is to construct a map fλ quasiconformally conjugate to fλ0 such that the corresponding p-periodic cycle has multiplier λ. For later purposes we restrict to an arbitrary simply connected neighbourhood U0 of λ0 in D˚ so that λ P U0 . First we introduce some notation. Set α “ Opα0 q “ tα0 , α1 , . . . , αp´1 u, an attracting cycle, where αj `1 “ fλ0 pαj q and p is the smallest integer so that αp “ α0 . Let Apαq “ Af pαq denote the basin of attraction of α and A˝j the Ťp´1 connected component of Apαq containing αj , so that A˝ pαq “ j “0 A˝j is the immediate basin of attraction of α. After composing with a Möbius transformation if necessary, we may assume that Apαq does not contain infinity. Let 0 be a neighbourhood of α0 in A˝0 , chosen as the preimage of D under a p linearizing map ψ0 : 0 Ñ D, that conjugates fλ0 to z ÞÑ λ0 z (see Figure 4.1). The linearizing map ψ0 : 0 Ñ D is a conformal isomorphism. A˝1 p 0

fλ

z ÞÑ λ0 z

A˝0 α0

D

ψ0

α1

0

α2

A˝2

Figure 4.1 The immediate basin of the cycle α and the neighbourhood 0 of α0 p mapped onto D under the linearizing map ψ0 of fλ . In the figure p “ 3. 0

As explained in Section 3.3.2, the map ψ0 can be extended to the entire basin p of attraction Af p pα0 q of α0 as a fixed point of fλ0 , so that ψ0 pAf p pα0 qq “ C

152

Introduction to surgery and first occurrences p

p

and so that ψ0 conjugates fλ0 to z ÞÑ λ0 z, i.e. ψ0 pfλ0 pzqq “ λ0 ψ0 pzq for all z P Af p pα0 q. The extended map ψ0 is holomorphic but no longer injective. Most of the preparation for changing the multiplier was done in the elementary example of Section 1.5. We adapt the notation in the example to our situation. Let ν0 be determined so that λ0 “ eν0 and Impν0 q P r0, 2π q, and let L : U0 Ñ C be the branch of the logarithm satisfying Lpλ0 q “ ν0 . For any λ P U0 set ν “ Lpλq. In the example we showed how to change the standard ν0 z complex structure μ0 ” 0 in C to the Beltrami coefficient μλ pzq “ νν ´ `ν0 z¯ ˚ for z P C . (In the example it was denoted μν .) The Beltrami coefficient is invariant under z ÞÑ λ0 z, and the integrating map φλ (in the example called φν ), normalized to fix 0 and 1, is a quasiconformal conjugacy between z ÞÑ λ0 z and z ÞÑ λz. Hence, for all z P C, φλ˚ μ0 “ μλ

and

λ φλ pzq “ φλ pλ0 zq.

We now define a Beltrami form on the basin of attraction Af p pα0 q by pulling back μλ under the holomorphic map ψ0 . By doing so, we obtain a p p λ on Af p pα0 q, as the following commutative fλ0 -invariant Beltrami form μ diagram indicates: p

fλ

0 p λ q ÝÝÝÝ pλq Ñ pAf p pα0 q, μ pAf p pα0 q, μ § § §ψ § ψ0 đ đ 0

zÞÑλ z

pC, μλ q § § φλ đ

ÝÝÝÝ0Ñ

pC, μ0 q

ÝÝÝÝÑ

zÞÑλz

pC, μλ q § §φ đ λ pC, μ0 q

p λ is exactly that of μλ , since we pull back by a Observe that the dilatation of μ holomorphic map. p λ initially p λ can be obtained differently: define μ Remark 4.3 The form μ pλ just on the neighbourhood 0 of the periodic point α0 . Then, spread μ p by the dynamics of fλ0 , to the whole basin of attraction of α0 , similarly to Section 1.5.1. The main difference is that we have multiple preimages of the p annulus 0 zfλ0 p0 q when we pull back. p λ to the whole basin of attraction Apαq by successive If p ą 1 we extend μ p λ as pullbacks by fλ0 . We shall denote the extended Beltrami coefficient by μ p λ is defined as: well. More precisely, μ


153

p λ on Af p pα0 q by fλ0 : Af p pαp´1 q Ñ • on Af p pαp´1 q as the pullback of μ Apα0 q, . . . ; p λ on Af p pα2 q by fλ0 : Af p pα1 q Ñ • on Af p pα1 q as the pullback of μ Af p pα2 q. p

p λ on Af p pα0 q is fλ0 -invariant, it follows that the pullback by fλ0 : Since μ p λ on Af p pα1 q coincides with the μ p λ on Af p pα0 q Af p pα0 q Ñ Af p pα1 q of μ already defined. We set p p λ “ 0 in CzApαq. μ p λ satisfies the hypothesis of the Integrability Theorem, we know Since μ p Ñ C. p Therefore, the mapping fλ “ there exists an integrating map φpλ : C ´1 p p p p φλ ˝ fλ0 ˝ φλ : C Ñ C is holomorphic, hence a rational map of degree d, and quasiconformally conjugate to fλ0 . The following diagram commutes: f

λ0 p μ p μ p λ q ÝÝÝÝ pλq pC, Ñ pC, § § § § φpλ đ đφpλ

fλ p μ0 q ÝÝÝ p μ0 q pC, ÝÑ pC,

Remark 4.4 (1) If in particular fλ0 is a polynomial of degree d to start p λ is equal to μ0 “ 0 in the basin of with, the almost complex structure μ attraction of infinity. Normalizing the integrating map φpλ to fix infinity, it follows that infinity is a superattracting fixed point of fλ , with no other preimages than itself. Hence fλ is also a polynomial of degree d. Moreover, if fλ0 is a monic polynomial and φpλ is furthermore chosen to satisfy φpλ pzq{z Ñ 1 as z Ñ 8 then fλ is also monic. If w “ φpλ pzq then this follows from ˜ ¸d φpλ pfλ0 pφpλ´1 pwqq z fλ pwq φpλ pfλ0 pzqq fλ0 pzq “ “ ÝÑ 1 wd wd fλ0 pzq zd φpλ pzq as w Ñ 8. Furthermore, if fλ0 is a monic centred polynomial and if we normalize φpλ so that the sum of the images by φpλ of the critical points, added with multiplicity, is also zero, then fλ is again a monic centred polynomial. With the above three normalization conditions on φpλ we are ready to apply the Integrability Theorem with holomorphic dependence on the parameter λ. We shall do so in Section 4.1.2. (2) Suppose instead that the given map fλ0 is an entire transcendental map, and normalize again the integrating map to fix infinity. By the topological

154

Introduction to surgery and first occurrences properties of essential singularities, the map fλ has an essential singularity at infinity, and hence fλ is an entire transcendental map. Similarly, if fλ0 : C˚ Ñ C˚ has essential singularities both at the origin and at infinity, and if the integrating map is normalized to fix the origin as well as infinity then fλ : C˚ Ñ C˚ has essential singularities at the origin and infinity.

We need to check that fλ has an attracting p-cycle of multiplier λ. Since fλ is conjugate by φpλ to fλ0 , the fλ -cycle αλ “ tφpλ pα0 u, . . . , φpλ pαp´1 qu is attracting, so the first part is clear. The second part follows from the following fact. Claim 4.5 (Linearizing map) The composition ψλ “ φλ ˝ ψ0 ˝ φpλ´1 : φpλ p0 q ÝÑ D p

is holomorphic and conjugates fλ to z ÞÑ λz. Therefore ψλ is a linearizing p map around the attracting fixed point φpλ pα0 q of fλ . Proof The map ψλ : φpλ p0 q Ñ D preserves, under pullback, the standard complex structure, as shown in the following commutative diagram: p

fλ

pφpλ p0 q, μ0 q O φpλ

p 0

fλ

pλq p0 , μ ψλ

ψ0

zÞÑλ0 z

pD, μλ q φλ

pD, μ0 q

/ pφpλ p0 q, μ0 q O φpλ

/ p0 , μ pλq

ψ0

ψλ

/ pD, μλ q φλ

zÞÑλz

/ pD, μ0 q

Therefore by Weyl’s Lemma, ψλ is holomorphic. Since ψλ is a conformal p map conjugating fλ around φpλ pα0 q to the linear map z ÞÑ λz around the origin, it follows that ψλ is a linearizing map. This concludes our first goal.


155

4.1.1 The monic polynomial case: reading the Böttcher coordinates Suppose fλ0 is a monic polynomial of degree d ě 2. Let φpλ be the integrating map normalized to fix infinity and satisfying φpλ pzq{z Ñ 1 as z Ñ 8. Then by Remark 4.4 above, fλ is a monic polynomial. The basin of attraction Apαq of the cycle α is contained in the filled Julia set Kλ0 :“ Kfλ0 . If the filled Julia set Kλ0 is connected then the Böttcher map ϕλ0 is defined on the entire basin of attraction of infinity of fλ0 , i.e. ϕλ0 : CzKλ0 Ñ CzD. For simplicity we assume that Kλ0 is connected. Claim 4.6 (Böttcher coordinates) Let ϕλ0 : CzKλ0 Ñ CzD be the Böttcher mapping of fλ0 on the basin of attraction of infinity. Then, the map ϕλ “ ϕλ0 ˝ φpλ´1 : Czφpλ pKλ0 q Ñ CzD is the Böttcher mapping of fλ on the basin of attraction of infinity. Proof Recall that a mapping ϕ which conjugates a monic polynomial to z ÞÑ zd in neighbourhoods of infinity is the Böttcher mapping around infinity if and only if it satisfies ϕpzq{z Ñ 1 as z Ñ 8. p λ pzq “ 0 for all points z in CzKλ0 . It follows that Observe that we defined μ the integrating map φpλ is holomorphic on this set, since it transports μ0 “ 0 to itself. Its inverse on this set is holomorphic as well. This means that ϕλ “ ϕλ0 ˝ φpλ´1 is holomorphic and conjugates fλ to z ÞÑ zd on the basin of attraction of infinity of fλ as the following commutative diagram indicates: zÞÑzd

pCzD, μ0 q 9 O ϕλ0

ϕλ0 fλ0

ϕλ

/ pCzD, μ0 q O e

pCzKλ0 , μ0 q

φpλ

pCzφpλ pKλ0 q, μ0 q

fλ

/ pCzKλ , μ0 q 0

ϕλ

φpλ

/ pCzφpλ pKλ q, μ0 q 0

p Ñ 8 since Finally, we have ϕλ pwq{w Ñ 1 as w “ φpzq ϕλ0 pφpλ´1 pwqq ϕλ pzq z “ 0 w z φpλ pzq and both factors tend to 1.

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If the polynomial has a disconnected Julia set, then the Böttcher map is still defined by ϕλ “ ϕ0 ˝ φpλ´1 , but only on a neighbourhood of infinity. 4.1.2 Application to families of unicritical polynomials In this section we prove a generalization of Theorem 4.1 to unicritical polynomials of degree d ě 2. Monic centred unicritical polynomials of degree d are of the form Qc pzq “ Qd,c pzq “ zd ` c. Let Md denote the connectedness locus in the c-plane of Qd,c . Figure 4.2 shows the Multibrot set for d equal to 3 and 4.

Figure 4.2 The Multibrot set for cubic and quartic unicritical polynomials. Notice the central hyperbolic components containing c “ 0 and corresponding to the component in the Mandelbrot set bounded by a cardioid. The bounding curve of the central hyperbolic component of the Multibrot set is always an epicycloid with d ´ 1 cusps.

In the following d is fixed. Let be a hyperbolic component of Md of period p. Then for any c P the polynomial Qc has an attracting p-cycle. Let : Ñ D denote the multiplier map. We shall prove the following theorem. Theorem 4.7 (Multiplier map for unicritical polynomials) Let denote a hyperbolic component of Md . Then the multiplier map “ d : Ñ D is a holomorphic branched covering of degree d ´ 1, ramified over 0 in D.


157

Set ˚ “ ´1 pD˚ q. Proving that : ˚ Ñ D˚ is a covering map is due to Sullivan. The computation of the degree follows from an argument of Gleason The steps in the proof are: (i) construct a local holomorphic inverse to : ˚ Ñ D˚ showing that the map is a covering map; (ii) deduce that ´1 p0q is a single point in ; (iii) determine the degree of the map . Fix a point c0 P ˚ with pc0 q “ λ0 ‰ 0. Choose an arbitrary simply connected neighbourhood U0 of λ0 , compactly contained in D˚ . For any λ P U0 we shall construct a monic centred polynomial Qcpλq pzq “ zd ` cpλq, which is quasiconformally conjugate to Qc0 and satisfies cpλ0 q “ c0 ,

and

pcpλqq “ λ,

λ ÞÑ cpλq is holomorphic.

It follows that cpλq P ˚ , and that we have constructed a holomorphic local inverse of in the neighbourhood U0 of λ0 . Therefore : ˚ Ñ D˚ is a holomorphic covering map (see Figure 4.3). ˚

D˚

cpλq c0 λ0

U0

´1 p0q

0

Figure 4.3 The map λ ÞÑ cpλq is a holomorphic local inverse of the multiplier map .

Notice that the construction of the previous sections, taking fλ0 “ Qc0 , and using the normalization of the integrating map φpλ explained in Remark 4.4 (1) already provides a monic centred polynomial fλ of degree d with one critical point at 0 and with a p-cycle of multiplier λ. These properties imply that fλ pzq “ zd ` cpλq, and moreover cpλq “ φpλ pc0 q, the critical value of fλ . It is clear that, if α0 is one of the points in the p-cycle for Qc0 , then p pcpλqq “ pQcpλq q1 pφpλ pα0 qq “ λ, as we wanted to show.

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The next step in the proof is to draw conclusions from the holomorphic dependence on parameters. p λ satisfy Claim 4.8 There exists k ă 1 such that the Beltrami coefficients μ that ||p μλ ||8 ď k for all λ P U0 . Moreover, p λ pzq is holomorphic in λ for all z P C. λ ÞÑ μ The integrating mappings φpλ with λ P U0 are normalized as to fix 0 and have φpλ pzq z

Ñ 1 as z Ñ 8. It follows that:

(a) λ ÞÑ φpλ pzq is holomorphic in λ for all z P C; (b) in particular the map λ ÞÑ φpλ pc0 q “ cpλq is holomorphic. Proof Recall that L : U0 Ñ C is chosen as the branch of the logarithm satisfying Lpλ0 q “ ν0 where Impν0 q P r0, 2π q. Set ν “ Lpλq for λ P U0 . Then ν depends holomorphically on λ. As it was pointed out in the elementary example of Section 1.5, the Beltrami coefficient μλ in C varies holomorphically with p λ on AQpc pα0 q is obtained respect to ν, hence also with respect to λ. Now μ 0 from μλ on C by pulling back with the extended linearizing map ϕ0 , which is independent of λ. Therefore for z P AQpc pα0 q 0

p λ pzq “ ϕ0˚ μλ pzq λ ÞÑ μ is also holomorphic in λ. If the period p ą 1, this holomorphic dependence is spread to all z P Apαq by successive pullbacks with Qc0 , also independent of p λ pzq “ 0 on CzApαq is trivially holomorphip λ by μ λ. The final extension of μ cally dependent on λ for all z P CzApαq. Set for all λ P U0 ˇ ˇ ˇ Lpλq ´ Lpλ q ˇ 0 ˇ ˇ kpλq “ ˇ ˇ. ˇ Lpλq ´ Lpλ0 q ˇ Then λ ÞÑ kpλq is a continuous real valued function on U0 that takes values in r0, 1q. Moreover, we have ||p μλ ||8 “ ||μλ ||8 “ kpλq. Set k “ sup kpλq. λPU0

Since U0 is compact, the supremum is realized as the maximal value of kpλq μλ ||8 ď k for all λ P U0 . We are now ready to for λ P U0 . Hence k ă 1 and ||p apply the Integrability Theorem with parameters. It follows that λ ÞÑ φpλ pzq is


159

holomorphic in λ for all z P C. Since φpλ is a conjugacy between Qc0 and fλ , it maps the critical value c0 of Qc0 , to the critical value φpλ pc0 q of fλ . In other words, fλ “ Qcpλq

where cpλq “ φpλ pc0 q.

It follows from above, that cpλq depends holomorphically on λ. This finishes the proof of Claim 4.8. (Compare with Lemma 1.40.) Claim 4.9 (Uniqueness of centre) The inverse image by the multiplier map : Ñ D of the centre of D is a single point of , called the centre of . Proof By the Riemann–Hurwitz formula (see e.g. [Bea, Thm. 2.7.1 or 5.4.1.]), the Euler characteristic of ˚ has to be one, which implies that ´1 p0q is a single point. Corollary 4.10 (Quasiconformal conjugacy between polynomials in the same hyperbolic component) Any pair Qc1 , Qc2 of unicritical polynomials of degree d with c1 , c2 P ˚ are quasiconformally conjugate. See Figure 3.12. The last step in the proof of Theorem 4.7 is to prove that the degree of the multiplier map : Ñ D is d ´ 1. Let c˚ denote the centre p of the hyperbolic component, and assume has period p. Then Qc˚ p0q “ 0. The following lemma is due to Gleason. The proof can be found in [BDH, BPer, DH2]. Lemma 4.11 Consider the polynomials Gn pcq “ Qnc p0q for n ě 1. All roots of Gn are simple. Proof

The polynomials Gn are defined recursively by G1 pcq “ c

and

Gn pcq “ pGn´1 pcqqd ` c.

It follows that Gn is monic with integer coefficients. The roots of Gn are therefore algebraic integers over Z, since the ring of algebraic integers A over Z consists of complex numbers which are roots of monic polynomials with integer coefficients. Assume there exist an n ą 1 and a root cr of Gn which is not simple. Then cr must be a root of both Gn and its derivative G1n pcq “ dpGn´1 pcqqd ´1 G1n´1 pcq ` 1.

160


It follows that ´1 . d Since the ring A is closed under addition and multiplication [NZM, Thm. 9.12], it follows from the equation above that ´d1 P A. But this is impossible since Q X A “ Z [NZM, Thm. 9.2]. Hence Gn has no multiple roots. pGn´1 pr cqqd ´1 G1n´1 pr cq “

We are ready to prove that the centre c˚ of is a zero of multiplicity d ´ 1 of the multiplier map . For any c P the points in the attracting p-cycle are solutions of p

F pc, zq “ Qc pzq ´ z “ 0. The map F satisfies the hypothesis of the Implicit Function Theorem at the point pc˚ , 0q, since Fz pc, zq |pc˚ ,0q “ pc˚ q ´ 1 “ ´1 and Fc pc, zq |pc˚ ,0q “ G1p pc˚ q ‰ 0 by Lemma 4.11. In a neighbourhood of c˚ the points in the attracting p-cycle can therefore be expressed as functions of c. Let α0 pcq denote the periodic point in the Fatou component containing the critical point 0, so that α0 pc˚ q “ 0. It follows that α01 pc˚ q ‰ 0, so that α0 pcq has a simple zero at j c “ c˚ . Set αj pcq “ Qc pα0 pcqq for j “ 1, . . . , p ´ 1. The multiplier map can be written as pcq “

p ź j “1

Q1c pαj pcqq “

pź ´1

dpαj pcqqd ´1 .

j “0

The multiplicity of c˚

as a zero of is equal to the sum of the multiplicities of c˚ as a zero of αj pcq, raised to the power d ´ 1. But this is zero for all j ‰ 0 and one for α0 pcq. We conclude that has a zero of multiplicity d ´ 1 at c˚ . This finishes the proof of Theorem 4.7 and in particular of Theorem 4.1. 4.1.3 Changing the modulus of a fundamental domain of a Fuchsian group In this section we sketch the soft surgery used to obtain quasiconformal deformations of a Fuchsian group, making the contents of Section 3.5.5 more precise. We shall use the concepts and notation from Section 3.5. Our goal is to stress the parallelism with the surgery described above, where we changed the multiplier of an attracting cycle. For simplicity, we start with the Fuchsian group “ă α, β ą, where is free, α and β are two loxodromic elements, αβα ´1 β ´1 is parabolic, and pq “ S1 . See for instance Example 3.74 in Section 3.5. Let Q denote a


161

fundamental domain for in D, whose sides are identified in pairs by the generators α and β (see Figure 3.17, where Q “ Q1 ). With these identifications, D{ is conformally equivalent to a punctured torus with marked generators. Hence, there exists a unique ν0 P H so that this torus is conformally equivalent to the torus obtained from the parallelogram T0 in C with vertices at t0, 1, ν0 , 1 ` ν0 u and opposite sides identified linearly. Let ϕ : Q Ñ T0 denote the unique conformal map that induces the conformal equivalence among the tori, the quotient spaces. The number ν0 is in fact the complex modulus of the torus arising from Q, its imaginary part is the (real) modulus of the quadrilateral Q without identifications (see Section 1.3.4). Because the puncture can be at any point, we may identify the punctured torus arising from Q with the unpunctured quotient. If is the group in Example 3.74, then ν0 is purely imaginary due to the symmetry. Choose an arbitrary ν P H, let Tν denote the parallelogram with vertices at t0, 1, ν, 1 ` νu, and let Lν : C Ñ C denote the unique R-linear map which maps T0 to Tν (preserving the order of the vertices). It can be checked that ν ´ ν0 BLν M BLν “ . μpLν q “ Bζ ν0 ´ ν Bζ The Beltrami coefficient μpLν q “ L˚ν μ0 defines a field of infinitesimal ellipses in T0 which can be pulled back to Q under the conformal map ϕ : Q Ñ T0 . Set μν “ ϕ ˚ μpLν q and observe that this defines an almost complex structure in Q of bounded dilatation. We now use the fact that Q is a fundamental domain to spread μν by the action of . More precisely, for every point z P D, there exists a unique γ P such that γ pzq P Q. Indeed if two different elements γ , γ 1 P were to satisfy this condition, mapping z to two different points in Q, then both points would be in the orbit of z, violating the condition that Q is a fundamental domain. On the other hand, if both elements mapped z to the same point, then either γ 1 γ ´1 would be a relation violating that is free or z would be a fixed point of γ 1 γ ´1 and hence lie in the limit set. Thus, for every z P D, it makes sense to define μν pzq “ γ ˚ μν pγ pzqq, where γ is as defined above. This extends μν to the whole unit disc. p and observe that μν , by construction, is invariant under Set μν ” 0 in CzD, the action of . Its dilatation is bounded given that all pullbacks are done by Möbius transformations. We may then apply the Integrability Theorem and obtain a quasiconformal map φν satisfying φν˚ μν “ μ0 . If we normalize φν to fix the two fixed points of α and the attracting fixed point of β, then φν is unique and depends holomorphically on the parameter ν, since μν does.

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Define a new group ν :“ φν φν´1 :“ tφν ˝ γ ˝ φν´1 | γ P u. Because μν is -invariant, it follows that the elements of ν are conformal (and hence Möbius transformations), so that ν is again a Kleinian group. Thus for every ν P H, we obtain a quasiconformal deformation of . Since pν q “ φν ppqq “ φν pS1 q and

pν q “ φν ppqq,

the limit set of the new group is a quasicircle, and therefore ν is a quasip denote the invariFuchsian group. Let ν1 :“ φν pDq and ν2 :“ φν pCzDq ant components of pν q. The quadrilateral Qν :“ φν pQq is a fundamental domain for ν in the component ν1 and ν1 {ν is isomorphic to a punctured torus of complex modulus exactly ν. Indeed, Lν ˝ ϕ ˝ φν´1 is conformal and maps Qν to Tν . Also observe that φν is conformal on CzD, hence no quasiconformal deformation takes place on the invariant component of pν q. This provides a family B of quasi-Fuchsian groups parametrized by the complex parameter in the upper half plane, viewed as the complex modulus of one of the two punctured tori of pq{. This is analogous to the family of quadratic polynomials Qc with c belonging to the main hyperbolic component of the Mandelbrot set. This component is parametrized by a complex parameter in D, viewed as the multiplier of the attracting fixed point αc of Qc , or equivalently, by the complex modulus of the torus pAc zαc q{Qc where Ac is the basin of attraction of αc . As a final remark, if ν 1 “ ν0 ` for some P Z observe that we obtain the same group and the same torus, just with a different set of generators, i.e.with a different marking (recall Definition 3.68). This fact is similar to the parameter periodicity observed when changing the multiplier of an attracting cycle.

4.2 Changing superattracting cycles to attracting ones 4.2.1 Preliminary: Blaschke products In this section we present a simple example of cut and paste surgery. It will serve as a building block for the construction in the next section. Consider the family of Blaschke products Bλ pzq “ z

z`λ 1 ` λz

,

parametrized by λ P “ D.

λ Each Bλ is a quadratic rational map with fixed points at 0, 8 and 11´ P ´λ BD. The fixed points at 0 and 8 are attracting with multipliers λ and λ

4.2 Changing superattracting cycles to attracting ones

163

respectively. (The inversion I : z ÞÑ 1z conjugates Bλ to Bλ .) The third fixed point is repelling. The unit disc D and its inversion CzD are invariant under Bλ and form the basins of attraction of 0 and 8 respectively (by Schwarz Lemma). Hence the unit circle is the Julia set of Bλ . Note that B0 pzq “ z2 . We restrict our attention to the dynamics of Bλ on D. For any 0 ă r ă 1, if we set Dr “ t|z| ă ru, then Dλ1 “ Bλ´1 pDr q compactly contains Dr , i.e. Dr Ă Dλ1 . This is easy to see by observing that if |z| “ r then |Bλ pzq| ă r, since Bλ pzq{z maps D to itself. If λ ‰ 0 and r is small, Dλ1 consists of two connected components. Instead if r ą |λ|, Dλ1 consists of one single connected component and therefore Dλ1 zDr is an annulus. Indeed, when r ą |λ| the second preimage ´λ of zero lies inside Dr , which implies that the map Bλ : Dλ1 Ñ Dr has degree two, the total degree of Bλ . Summarizing, for any parameter λ P D, and for any r ą |λ|, we have Bλ pDr q Ă Dr ,

Bλ pDλ1 q “ Dr ,

Bλ pDzDλ1 q “ DzDr .

See Figure 4.4. We use this family of Blaschke products to illustrate the basic construction of pasting maps together. D Bλ

Dλ1

Bλ p D r q r 0

Figure 4.4 The initial setup for a Blaschke product Bλ , in the case r ą |λ|.

Fix a parameter λ0 P D, and fix |λ0 | ă r ă 1. Then, for all |λ| ă r, we define a quasiregular mapping gλ : D Ñ D, which is the result of pasting together three maps. Denote by Aλ0 the closed annulus Dλ1 0 zDr . We define gλ : D Ñ D as $ ’ ’ &Bλ0 gλ “ Bλ ’ ’ %h λ

on DzDλ1 0 , on Dr , on Aλ0 ,

164


where hλ is a quasiregular map, which coincides with Bλ0 on the outer boundary of Aλ0 and with Bλ on the inner boundary S1r (see Figure 4.5). Observe that these boundary maps are analytic maps of degree two on analytic curves. Moreover, the outer boundary map is fixed and the inner boundary map `λ is varies continuously with λ for each fixed z P S1r , i.e. λ ÞÑ Bλ pzq “ z 1z` λz continuous in λ. Then by Exercise 2.3.3 at the end of Chapter 2 we conclude that hλ can be chosen as a covering map of degree two depending continuously on λ for each z P Aλ0 . D

D Aλ0

Bλ r

1

Aλ r

hλ

1

Bλ0 Figure 4.5 Pasting together different Blaschke products via an interpolating quasiregular map hλ .

We proceed to define a gλ -invariant Beltrami form μλ . We first define it on Dr as μλ “ μ0 . Since gλ “ Bλ is holomorphic on Dr and Bλ pDr q Ă Dr , it follows that μλ is invariant in Dr . Next we define μλ in the region where gλ is not holomorphic, i.e. on the annulus Aλ0 , by setting μλ “ h˚λ μ0 . Finally, we spread μλ to the remaining part DzDλ1 0 , by the dynamics of gλ . To do this it is important to observe that for every point z P DzDr there is a unique n ě 0 such that gλn pzq belongs to the half open annulus Aλ0 zBDr . Moreover, if n ą 0 then gλn pzq “ Bλn0 pzq. Hence μλ is defined recursively on D as $ ’ μ ’ & 0 h˚λ μ0 μλ “ ´ ¯ ’ ’ % Bn ˚ μ λ0

λ

on Dr , on Aλ0 , on Bλ´0n pAλ0 q.

By construction μλ is gλ -invariant. Moreover, for each λ P Dr 1 , the Beltrami coefficient μλ has dilatation bounded away from 1 on the compact annulus Aλ0 . The same is true on all of D, since the dilatation is unchanged when pulling back by the holomorphic map Bλ0 , i.e. }μλ }8 :“ kpλq ă 1. Furthermore, λ Ñ μλ pzq varies continuously with λ for all z P Aλ0 , hence also for all z P D. Therefore kpλq varies continuously with λ. For all λ P Dr 1 the Beltrami coefficient has dilatation bounded by k :“ max|λ|ďr 1 kpλq ă 1.

4.2 Changing superattracting cycles to attracting ones

165

We are ready to apply the Integrability Theorem. Note that for any λ P D we have a quasiconformal integrating map φλ : D Ñ D which fixes 0, and is uniquely determined up to post-composition by a rotation about 0. Any such map extends to the boundary. Let´ us still ¯ denote the extension by λ0 λ . Consequently, the “ 11´ φλ : D Ñ D and choose it such that φλ 11´ ´λ ´λ 0

composition Gλ “ φλ ˝ gλ ˝ φλ´1 is a holomorphic self-map of D of degree two, hence a Blaschke product. The multiplier of the fixed point at 0 is λ, λ is a fixed point of Gλ , it means that since φλ is holomorphic in Dr . Since 11´ ´λ Gλ “ Bλ . The map gλ and the gλ -invariant Beltrami coefficient μλ can be extended to p If this is done we would end by applying the Integrability Theorem all of C. p as suggested in Exercise 4.2.1. depending on parameters for C, In the previous section we described an alternative change of multiplier. The resulting map was always conjugate to the original one. As a consequence, none of the multipliers could be taken to be equal to 0. Note, however, that this construction works for either λ or λ0 equal to 0.

4.2.2 Gluing the Blaschke product into basins of attraction By applying part of the construction from Section 4.2.1 we are able to give an alternative proof of Theorem 4.1, proving in one step that the multiplier map is a covering and that its degree is exactly one. We recall the precise statement in the following proposition. With the technique based on the Blaschke product model we show that, for any c0 P , the multiplier map has a local inverse around λ0 “ pc0 q P D, including the case λ0 “ 0. We start by giving the basic idea in the proof. Let c0 be any point in , and let as before α “ tα0 , . . . , αp´1 u be the attracting p-cycle of Qc0 . Let A˝j be the component of the immediate attracting basin A˝ pαq containing αj , and let the numbering be chosen so that A˝0 contains the critical point 0. Let R0 : A˝0 Ñ D denote the uniquely determined Riemann mapping that maps α0 to 0, and when extended continuously to the boundary of A˝0 , maps the p p λ0 in BD. Then R0 conjugates Qc0 : A˝0 Ñ fixed point of Qc0 in BA˝0 to 11´ ´λ 0

`λ0 , since any A˝0 to the Blaschke product Bλ0 : D Ñ D where Bλ0 pzq “ z 1z` λ0 z holomorphic self-map of D of degree two is a Blaschke product of this form. In order to construct a local inverse to pc0 q “ λ0 in a neighbourhood of λ0 , we paste the two Blaschke products Bλ0 and Bλ together via an interpolating map hλ , for λ in the neighbourhood of λ0 , as explained in Section 4.2.1. We change the complex structure in D accordingly. Then we pull back the construction

166


first by the Riemann mapping R0 to A˝0 , then by Qc0 to all of Apαq, and finally we let the map and the complex structure be left unchanged outside Apαq. We shall give more details only when c0 P ´1 p0q since this is the case that was not covered by the earlier construction. In this case the Blaschke product Bλ0 is equal to B0 pzq “ z2 and the Riemann mapping R0 equals the Böttcher p mapping that conjugates Qc0 : A˝0 Ñ A˝0 to B0 : D Ñ D. Proof of Theorem 4.1 Let c0 belong to ´1 p0q. Choose any 0 ă r 1 ă r ă 1 and |λ| ď r 1 ă r. Then the disc Dr “ t|z| ă ru is compactly contained in Bλ´1 pDr q and Bλ´1 pDr qzDr is an annulus. Using the construction from Section 4.2.1 we obtain the following commutative diagram: fλ

pA˝0 , ηλ q ÝÝÝÝÑ pA˝0 , ηλ q § § § § R0 đ đR0 gλ

pD, μλ q ÝÝÝÝÑ pD, μλ q § § §φ § φλ đ đ λ B

λ pD, μ0 q ÝÝÝÝ Ñ pD, μ0 q

1 ˚ where fλ “ R´ 0 ˝ gλ ˝ R0 and ηλ “ R0 μλ is an fλ -invariant Beltrami form ˝ on A0 . Note that D01 “ B0´1 pDr q “ D?r so that gλ “ B0 on DzD?r , hence fλ “ p

1 ? Qc0 on A˝0 zR´ 0 pD r q. We define Fλ : C Ñ C as follows:

Fλ “

# ˇ´1 pQc0 ˇ qp´1 ˝ fλ

on A˝0 , on CzA˝0 ,

Qc 0

ˇ´ 1 where pQc0 ˇ qp´1 is short for ˇ´1 ˇ´1 ˇ´1 pQc0 ˇ qp´1 “ Qc0 ˇA˝ ˝ ¨ ¨ ¨ ˝ Qc0 ˇA˝ , 1

p´1

and we use that Qc0 : A˝j Ñ A˝j `1 is a bijection for j “ 1, . . . , p ´ 1. The global mapping Fλ is holomorphic everywhere except on the annu1 ? lus R´ 0 pD r zDr q, where it is quasiconformal. In fact Fλ is equal to Qc0 1 ? everywhere except on R´ 0 pD r q. ˝ We spread ηλ on A0 to an Fλ -invariant Beltrami form on C, which we denote again by ηλ . Notice that, for any z P ApαqzA˝0 , there is a unique n ą 0 such that Qnc0 pzq P A˝0 , and define ηλ on C by

4.2 Changing superattracting cycles to attracting ones $ ’ ’ &μ0 η λ “ ηλ ’ ’ %pQn q˚ η λ c0

167

on CzApαq, on A˝0 , ˝ ˝ n on Q´ c0 pA0 qzA0 .

By construction ηλ is an Fλ -invariant Beltrami form with dilatation bounded by the bound for the dilatation of μλ on D?r zDr . We are ready to apply the Integrability Theorem to obtain a quasiconformal integrating mapping φpλ , normalized so that 0 and 8 are fixed and φpλ is tangent to the identity at 8. It follows that the mapping φpλ ˝ Fλ ˝ φpλ´1 is a quadratic polynomial of the form Qcpλq pzq “ z2 ` cpλq with an attracting p-cycle namely φpλ pαq “ tpφλ pα0 q, . . . , φλ pαp´1 qqu of multiplier λ. Finally we add parameters to the problem. We know from Section 4.2.1 that λ ÞÑ gλ pzq is continuous for all z P D. It follows that λ ÞÑ fλ pzq and λ ÞÑ Fλ pzq are continuous for all z in A0 respectively in C. Moreover, μλ pzq depends continuously on λ for each z P D and has a dilatation that is uniformly bounded away from 1. It follows that the same is true for the resulting ηλ pzq for all z P C. Therefore, for each z P C the integrating map φpλ pzq depends continuously on λ. Hence, the attracting p-cycle, φpλ pαq, varies continuously with λ. Note that cpλq “ φpλ ˝ Fλ p0q and that Fλ p0q is independent of λ. Hence cpλq varies continuously with λ and belongs to . We have constructed a local inverse to : λ P Dr 1 ÞÑ cpλq P with pcpλqq “ λ. It follows that ´1 p0q consists of a single point, the centre of .

The dynamics of Qc for c P ˚ is not globally conjugate to the dynamics of the centre polynomial Qc0 . In the first case the critical orbit has infinitely many points, while it has only p points in the second. However, it follows that all polynomials in are quasiconformally conjugate in a neighbourhood of their Julia set. In particular, they are all J -conjugate (see Definition 3.28). Corollary 4.12 (Conjugacy to the centre in a neighbourhood of J ) Any quadratic polynomial Qc with c P is quasiconformally conjugate to the centre polynomial Qc0 of , in some neighbourhoods of their respective Julia sets. Remark 4.13 In Section 7.5 we shall see an alternative construction to the one presented here. There, the gluing is done along the boundary of the immediate basin of attraction, and the resulting map is conjugate to the original only on the Julia set.

168


Exercises Section 4.2 4.2.1 Observe that τ pzq “ 1{z, the reflection with respect to S1 , conjugates `λ to itself. As in Section 4.2.1 choose 0 ă r 1 ă r ă 1 Bλ pzq “ z 1z` λz p ÑC p by defining and λ, λ0 P Dr 1 . Extend gλ : D Ñ D to gλ : C # Bλ0 on BD, gλ “ p τ ˝ gλ ˝ τ on CzD. p depending continuously Define a gλ -invariant Beltrami form μλ on C p on λ for each z P C. Show that this can be done so that with a proper p Ñ C, p the resulting holonormalization of the integrating maps φλ : C ´ 1 p ÑC p is equal to Bλ . morphic map Gλ :“ φλ ˝ gλ ˝ pφλ q : C Remark: The exercises below are special cases of a more general theorem of McMullen (see Section 7.5). 4.2.2 Suppose P is a polynomial with an attracting fixed point α and assume that P has degree d when restricted to the immediate basin of α, say A˝ pαq. Show that there exists a polynomial Q and a quasiconformal homeomorphism φ : C Ñ C conjugating P and Q on a neighbourhood of the respective Julia sets, and in all Fatou components except A˝ pαq, such that φpαq is a superattracting fixed point of Q of multiplicity d, and is the unique critical point in its basin. In particular, the dynamics of Q in the immediate basin of φpαq is conjugate to z ÞÑ zd . Finally observe the analogous result for periodic attracting orbits. Hint: Under a Riemann map, P |A˝ induces a degree d map from D onto itself, which therefore is a Blaschke product. 4.2.3 Suppose P is a polynomial with a superattracting fixed point, say α, whose immediate basin, A˝ pαq, contains no other critical point than itself. Let U be a Fatou component which is mapped to A˝ pαq with degree k ě 3 (i.e. containing at least two critical points). Prove that there exists a polynomial Q and a quasiconformal homeomorphism φ : C Ñ C conjugating P and Q on a neighbourhood of the respective Julia sets, and in all Fatou components except U , such that φpU q contains a unique critical point of multiplicity k ´ 1 which is mapped to φpαq. 4.2.4 Generalize the procedure in Exercise 4.2.2 to other components of the basin of attraction, to show that the same statement would hold if we asked Q to have one single critical grand orbit in the whole basin of attraction φpαq.

4.3 No wandering domains for rational maps

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4.3 No wandering domains for rational maps In this section we prove the celebrated theorem of Sullivan about the non-existence of wandering domains for rational maps (Theorem 4.2). The original proof is in [Su2], using soft surgery. In the same paper he used similar arguments to also prove the non-existence of wandering domains for Kleinian groups as well as Ahlfors’ Finiteness Theorem (Theorem 3.80). We start by giving an idea of the common strategy to prove the absence of wandering components in both settings. Afterwards we give the details in the rational case. Let R denote a rational map and a finitely generated Kleinian group. We recall the definition of a wandering domain: a Fatou component U is wandering for a rational map R if R n pU q ‰ R m pU q for all m ‰ m. Likewise, a component U of pq is a wandering component if γ pU q ‰ U for all γ P different from the identity. The idea is as follows. Start by recalling that Ratd , the space of rational maps of degree d, can be identified with C2d `1 , a complex vector space of finite dimension. Analogously, the space of quasiconformal deformations of a finitely generated Kleinian group is finite dimensional. This is due to the fact that every deformation induces a representation of into PSLp2, Cq, and the space of such representations Homp, PSLp2, Cq is finite dimensional, when is finitely generated (see e.g. [An]). Suppose U is a wandering component of FR (or pq). Then, Sullivan shows that there exists a space of ‘non-equivalent’ Beltrami forms supported on U , of arbitrarily high dimension. One may then use R (or ) to spread these Beltrami forms and obtain an arbitrarily high dimensional space of Beltrami forms supported on FR (or pq). After having applied the Integrability Theorem we obtain an arbitrarily high dimensional space of non-equivalent quasiconformal deformations of R (or ), and hence an arbitrarily high dimensional subspace of Ratd (or Homp, PSLp2, Cq). This is a contradiction. To see the full proof of Ahlfors’ Finiteness Theorem we refer to [Su2, Ah1, Be3] or [MT]. We now proceed to explain in detail the rational case. The proof can be split into three steps: (1) Show that if there were a wandering domain U , all its iterates Un “ f n pU q for large enough n would be simply connected. (2) Construct, for arbitrarily large N , an N -dimensional real analytic family of Beltrami forms μλ , for λ in a subset of RN , invariant by f and with bounded dilatation. After integrating, these Beltrami forms induce a real

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analytic family fλ of quasiconformal deformations of f , all rational maps of the same degree as f . (3) Since the complex dimension of the space Ratd of rational maps of degree d is 2d ` 1, there must be – for N ą 4d ` 2 – a path of Beltrami forms μλpt q , t P r0, 1s, inducing the same rational map g, that is fλpt q “ g. This will contradict the properties of the Beltrami forms we constructed in the second step. There exist several rewritings and simplifications of Sullivan’s proof, although all of them essentially follow the same three steps. We choose to present Baker’s approach to the first step (Proposition 4.14), which is a significant improvement. We present Lemma 4.15 following Steinmetz [St] as part of the second step. The idea is similar to Sullivan’s, but more constructive. For the rest, we choose to follow Sullivan’s original arguments, being consistent with our goal to give some ‘historical flavour’ to this section. Let us start by addressing the first step. Proposition 4.14 (Eventual properties of wandering domains) Let f be a rational map of degree d ě 2. Suppose U “ U0 is a wandering domain and let Un :“ f n pU q for n ě 0. Then, by renaming U0 “ Um if necessary for m large enough, we have that, for all n ě 0, (1) Un contains no critical points; (2) Un is simply connected; (3) f is univalent on Un . Proof Since f has a finite number of critical points, and the Un ’s are pairwise disjoint, we may assume, by renaming U “ f m pU q for some m ą 0 if necessary, that neither U nor its forwards iterates contain critical points. Let us also suppose (conjugating by a Möbius transformation if necessary) that the point at 8 belongs to a component of f ´1 pU q, so that Yně0 Un is bounded in the complex plane. With this set up we shall first prove that any limit function of the normal family tf n |U uně0 is a finite constant; next that this implies that the diameter of the sets Un must tend to 0 as n tends to 8; and finally that all the Un are simply connected and that f |Un : Un Ñ Un`1 is univalent for all n. Claim: A limit function of the normal family tf n |U uně0 is constant. Indeed, suppose this were not the case, and let g be a non-constant limit function of a subsequence, which we denote again by tf n u. Then g is holomorphic in U and g 1 is not identically equal to 0. Choose z0 P U such that


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g 1 pz0 q ‰ 0, and let D Ă U be a compact set around z0 on which g is injective. In particular gpz0 q R gpBDq. Set :“ infzPBD |gpzq ´ gpz0 q| ą 0. Since f n converges uniformly to g in D there exists N so that |f n pzq ´ gpzq| ă for all n ą N and all z P D. Set Fn pzq :“ f n pzq ´ gpzq and Gpzq :“ gpzq ´ gpz0 q. Then for n ą N and for z P BD we have |Fn pzq ´ Gpzq| “ |f n pzq ´ gpzq| ă ď |Gpzq|. By Rouché’s Theorem, it follows that G and Fn have the same number of zeros in D for n ą N . Since G has at least one zero, at z “ z0 , then Fn has at least one zero as well. That is, there exist for n ą N , points z˚ pnq P D such that f n pz˚ pnqq “ gpz0 q. Hence gpz0 q P Un for all n ą N , which contradicts that U is a wandering domain. Claim: The diameter of the sets Un tends to 0 as n tends to 8. Indeed, suppose this were not the case. Let L be a compact subset of U , nj a subsequence tending to 8 such that diampf nj pLqq ě δ for some δ ą 0 and all nj ą N for some N . Take a convergent subsequence of f nj , which we denote again by f nj , converging to the constant limit function c. Then, given any 0 ă ă δ, it follows from uniform convergence that f nj pLq Ă Dc pq for nj sufficiently large. This contradicts that the diameter is bounded from below. Claim: The Un are all simply connected and f |Un : Un Ñ Un`1 are all univalent. Take an arbitrary closed curve γ Ă U . We will show that γ is homotopic in U to a constant curve. Set γn :“ f n pγ q Ă Un . It follows from above that the diameter of the curves γn tends to 0 as n tends to 8. Hence, if Dn p n , then diampDn q tends to denotes a bounded connected component of Czγ 0 as n tends to 8. Therefore, for n large enough, there can be no poles in any of the components Dn . Indeed, f has a finite number of poles, and hence there exists δ ą 0 so that f pDp pδqq Ă f ´1 pU q for any pole p of f , since 8 P f ´1 pU q. p n. Let γnfilled denote the curve γn union all the bounded components of Czγ filled for n It follows that all iterates are holomorphic on a neighbourhood of γn sufficiently large and, since |f n | is bounded on the curves, by the Maximum Modulus Principle it is also bounded on γnfilled . We conclude from Montel’s Theorem that the iterates form a normal family in a neighbourhood of γnfilled and therefore that γnfilled Ă Un . This implies that the curves γn are homotopic in Un to a constant curve for n sufficiently large. Since f n |U : U Ñ Un is a covering map, the homotopy can be lifted to a homotopy in U of γ to a

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constant curve, which shows that U and all its forwards iterates are simply connected. Finally, it follows from the Riemann–Hurwitz formula that all f |Un : Un Ñ Un`1 must be univalent. The fact that f is univalent on all Un and that the Un ’s are pairwise disjoint makes it possible to push forward any almost complex structure that we may define on U . After that, one can pull it back along the grand orbit of U . We proceed to define a real analytic family of almost complex structures on U of real dimension N for some N ą 4d ` 2. These structures are induced by a real analytic family of K-quasiconformal mappings whose properties we state in the following lemma (cf. [St]). The proof of the lemma is given at the end of the section. Lemma 4.15 (Family of diffeomorphisms) Let B “ tλ P RN | ||λ|| ď 1u. Given K ą 1, there exists a family of maps ψ : B ˆ D ÝÑ D, pλ, zq ÞÝÑ ψλ pzq, satisfying: (1) (2) (3) (4) (5)

ψλ is K-quasiconformal on D for all λ P B; λ ÞÑ ψλ pzq is real analytic for all z P D; ψ0 “ Id |D ; for all λ P B and all z “ reiθ P Dzt0u with θ P rπ, 2π s: ψλ pzq “ z; ψλ1 ı ψλ2 if λ1 ‰ λ2 (injectivity property).

Each of these homeomorphisms induces a Beltrami coefficient on D defined by r λ :“ ψλ˚ μ0 “ μ

Bψλ , Bψλ

with dilatation bounded by K. Because of the properties of ψλ , it follows that r λ pzq is real analytic for any z P D. λ ÞÑ μ r λ to the wandering domain U via some Riemann map We now transport μ R : D Ñ U . Let θ1 , θ2 , θ3 P rπ, 2π s be three distinct arguments, for which the radial limit of Rpreiθj q exists as r tends to 1 and are all distinct (existence follows from a theorem by F. and M. Riesz [CL]). Let a1 , a2 , a3 P BU denote the corresponding limiting values. We define the Beltrami coefficient μλ on


173

r λ , the push forward by the Riemann map of μ r λ on D. The U as μλ :“ R˚ μ dilatation of μλ is again bounded by K. p As mentioned before, since f is univalent on Un , Next, we spread μλ to C. we first push μλ on U forward to all Un . Afterwards we pull μλ back to all iterated preimages of Un for any n, in the usual way. For the remaining points p we set μλ ” μ0 . More precisely, we recursively define on C $ n ’ ’ &pf q˚ μλ μλ :“ pf m q˚ μλ ’ ’ %μ 0

on Un , for n ě 0, on any component of f ´m pUn q, for any m, n ě 0, p Ť on Cz f ´m pUn q. n,mě0

Then μλ pzq is a Beltrami form which depends real analytically on λ. Clearly μλ has dilatation bounded by K, since all pullbacks and pushforwards were done by the holomorphic function f . Moreover, μλ is f -invariant by construction. We then apply the Theorem of Integrability with dependence on parameters p ÑC p which integrate the to obtain a family of quasiconformal maps φλ : C ˚ almost complex structures μλ , i.e. such that φλ pμ0 q “ μλ . We define gλ :“ φλ ˝ f ˝ φλ´1 . Since μλ is f -invariant, it follows that μ0 is gλ -invariant, and gλ is holomorp i.e. a rational map of degree d. If we normalize the integrating phic on C, maps so that they fix the three points a1 , a2 , a3 , then gλ can be written as a rational map whose 2d ` 1 coefficients depend real analytically on λ (see Section 1.4). This process defines a map π : B ÝÑ Ratd . Since B has larger real dimension than 2p2d ` 1q, it follows that there must exist a fibre of π containing a non-trivial arc. That is, there exist a rational map of degree d, say g, and a path tλptq, t P r0, 1su Ă B, such that gλpt q “ g for all t P r0, 1s. By defining ϕt :“ φλ´p10q ˝ φλpt q , we obtain a path of K 2 -quasiconformal maps of the sphere fixing a1 , a2 , a3 and such that they commute with f , as shown in the following commutative diagram:

174


p μλp0q q pC, E

f

φλp0q

ϕt

p pC, μ0 q O

φλp0q g

φλptq

p μλpt q q pC,

p μλp0q q / pC, Y p μ0 q / pC, O

ϕt

φλptq f

p μλpt q q / pC,

Note that ϕ0 “ Id. We shall use the following two lemmas which we prove at the end of the section. Lemma 4.16 (Families of topological self-conjugacies) If tϕt ut Pr0,1s is a continuous family of topological conjugacies between a rational map f and itself, such that ϕ0 “ Id, then ϕt fixes all points in the Julia set Jf for all t, and maps each Fatou component onto itself. Lemma 4.17 (Extensions to the unit circle) Suppose U Ă C is an open simply connected set and tϕt : U Ñ U u0ďt ď1 is a continuous family of quasiconformal homeomorphisms which extend to the identity map on the boundary of U . Let R : U Ñ D be a Riemann map and set ϕpt “ R ˝ ϕt ˝ R´1 : D Ñ D. Then, for all t P r0, 1s, the map ϕpt extends to BD as some circle map ϕp independent of t. Hence by Lemma 4.16, the conjugacy ϕt fixes all points in the Julia set and maps U onto itself. In particular, it fixes all points on the boundary of U . Let us now consider the map r λpt q q ÝÑ pD, μ r λp0q q. ϕpt :“ R ˝ ϕt ˝ R´1 : pD, μ By Lemma 4.17 ϕpt extends to a map ϕp on the unit circle, independent of t. Finally set pt :“ ψλp0q ˝ ϕpt : pD, μ r λpt q q ÝÑ pD, μ0 q, ψ which extends to S1 as ψλp0q ˝ ϕp, and compare these maps with the original maps in Lemma 4.15, which are r λpt q q ÝÑ pD, μ0 q. ψλpt q : pD, μ


175

Observe that, for any t, both maps are quasiconformal with the same Beltrami coefficient. Moreover, they both fix three points on the unit circle (in fact they fix the lower half of the unit circle). It follows that they must be identical, since integrating maps on D are unique up to post-compositions with Möbius transformations of the disc. More precisely we have the following commutative diagram: ϕt

pU, μλpt q q R

/ pU, μλpt q q R

ϕpt / pD, μ r λpt q q r λp0q q pD, μ JJ JJ ψpt JJ JJ ψλptq ψλp0q JJ JJ J$ Id / pD, μ0 q pD, μ0 q But this contradicts the injectivity property of the maps ψλpt q . Indeed, since ϕpt pt extends as ϕp to the unit circle, and ψλp0q does not depend on t, it follows that ψ p extends to the boundary as ψλp0q ˝ ϕp for all t. But we just concluded that ψt “ ψλpt q , and the injectivity property assures that the maps ψλpt q cannot coincide for different values of t (and hence different values of λ). This concludes the proof of the No Wandering Domains Theorem, up to the three lemmas we left behind. Proof of Lemma 4.15 For any δ ą 0 we consider the C 8 bump function of R # 2 δ 2 expp x 2δ´δ 2 q for |x| ă δ, ωpxq :“ 0 for |x| ě δ, which satisfies 0 ď ωpxq ď δ 2 {e and |ω1 pxq| ď 8δ{e2 . Choose a partition 0 “ x0 ă x1 ă ¨ ¨ ¨ ă xN ă xN `1 “ π of the interval r0, π s and let δ ă 12 min1ďj ďN `1 pxj ´ xj ´1 q. We define the following N linearly independent functions ωj pxq :“ ωpx ´ xj q,

for j “ 1, . . . , N .

176


Observe that the supports of the functions ωj are pairwise disjoint, namely the compact symmetric intervals around xj of size 2δ. We then define for pλ, θ q P RN ˆ r0, 2π q the real function (see Figure 4.6) pλ, θ q :“

N ÿ

λj ωj pθ q “

j “1

N ÿ

λj ωpθ ´ xj q

where λ “ pλ1 , . . . , λN q.

j “1

0.03

0 0

π 4

π 2

3π 4

π

2π

Figure 4.6 The function pλ, θ q where δ “ 0.3, N “ 3, xj “ j4π for j “ 1, 2, 3 and λ “ p0.5, 0.7, 0.3q.

Observe that this is a linear map in λ and C 8 in θ , which equals 0 for θ P rπ, 2π s. We now use this map to define the family of C 8 diffeomorphisms (for δ small enough) of the punctured disc ψ : B ˆ Dzt0u ÝÑ Dzt0u where

ψλ prei θ q :“ rei pθ `pλ,θ qq ,

for θ P r0, 2π s and 0 ă r ď 1. We extend ψλ to D by setting ψλ p0q “ 0 for all λ P B. It remains to check that ψλ is K-quasiconformal in D. This can be seen either by observing that this map is K-bilipschitz and therefore quasiconformal (see Section 1.3.6) or by computing its Beltrami coefficient directly, obtaining e´2i θ θ pλ, θ q Bz ψλ pzq “ Bz ψλ pzq 1 ` θ pλ, θ q

where z “ rei θ , λ P B and θ “

B . Bθ

It is then clear that the norm of the Beltrami coefficient can be made as small as we want, by taking δ small enough. The rest of the properties are obvious from the construction. Proof of Lemma 4.16 Since ϕt conjugates f to itself it maps the set Pn “ tz P p | f n pzq “ zu onto itself for any n ą 0. Observe that Pn is a discrete set, that C


177

ϕ0 “ Id, and that ϕt varies continuously with respect to t. As a consequence, ϕt must be the identity on the set Pn , for any t P r0, 1s. Since this holds for all n ą 0, and the set of periodic points is dense in the Julia set, by continuity we get that ϕt is the identity on the Julia set for every t P r0, 1s. To prove Lemma 4.17, we first give some preliminary notions on the boundary behaviour of Riemann maps. For details we refer the reader to [CL, Go, Pi] or [Pom]. Let U Ă C be an open, simply connected, bounded set. Let R : D Ñ U be a Riemann map. Carathéodory’s Theorem states that R extends continuously to the boundary of U if and only if BU is locally connected. This means that there is a bijection between points in S1 and points in BU and, even more, the boundary can be continuously parametrized by the unit circle. If the boundary of U is not locally connected this is no longer true. Nevertheless, one can still study how the Riemann maps behaves when we approach the boundary of the unit disc. A theorem by Fatou states that the radial limit at ζ “ ei θ , i.e. limr Ñ1 Rprei θ q, exists for almost every ζ P S1 . The exceptional set E of points for which the radial limit does not exist has measure zero. Even more, a theorem of M. and F. Riesz states that the limit map R : S1 zE Ñ BU cannot be constant on any set of positive measure. A point a P BU is called accessible from U if there exists a curve γ : r0, 1q Ñ U which lands at a, i.e. γ ptq tends to a as t tends to 1. If the boundary of U is locally connected, then all its points are accessible. Following [Go] we say that two such curves γ1 and γ2 are in the same access to a, if for every p of a there exists a curve α : r0, 1s Ñ U X V , such that neighbourhood V Ă C αp0q P γ1 and αp1q P γ2 . Equivalently, an access is a homotopy class within the family of curves γ : r0, 1q Ñ U , such that limt Ñ1 γ ptq “ a. It turns out there is a bijective relation between accesses and points in S1 zE. Clearly every point in S1 for which the radial limit exists, corresponds to an access to an accessible point (the limit point). The following Lindelöf-type theorem says that this correspondence is bijective. Theorem 4.18 (Accesses and radial limits) Let γ : r0, 1q Ñ U be a curve which lands at a point a in the boundary of U . Then the curve R´1 ˝ γ in D lands at some point ζ of BD, and R has the radial limit at ζ equal to a. Moreover, if γ1 , γ2 : r0, 1q Ñ U are curves landing at a common point a P BU , then γ1 and γ2 are in the same access to a if and only if R´1 pγ1 q and R´1 pγ2 q land at the same point in BD. Given an accessible point a P BU we define the fibre of a as the set of accesses corresponding to the point a or, equivalently, the set of points ζ P S1

178


such that the radial limit exists and equals a. Observe that the theorem of F. and M. Riesz, mentioned above, implies that each fibre forms a totally disconnected set in S1 . Indeed, in between two points ζ1 and ζ2 in S1 having both radial limits at a, there must exist a point ζ with a different radial limit. Proof of Lemma 4.17 First observe that the induced maps on the unit disc ϕpt : D Ñ D are quasiconformal and hence they extend to the unit circle. Since ϕt is the identity of the boundary of U it can at most permute the accesses to a given point. This means that the induced maps on the disc ϕpt can at most permute points which belong to the same fibre, which form a totally disconnected set. But at the same time, the maps ϕt depend continuously on t, so the permutation of the accesses cannot vary with t. It then follows that they have to permute these points in the same way for all t. Hence, the maps ϕpt extend to the set S1 zE in the same way for all t. Since this is a dense set, the extension to S1 is also the same map for all t.

5 General principles of surgery

The previous chapters contain examples of quasiregular maps for which an invariant almost complex structure with bounded dilatation exists. It is a natural question to ask under which conditions this can be accomplished, so that a holomorphic dynamical copy is obtained by means of the Integrability Theorem. We first present two statements, both due to Shishikura, describing typical scenarios in surgery constructions. The first one was called the Fundamental Lemma of Quasiconformal Surgery in [Sh1]. It applies to most procedures of cut and paste surgery, where we paste together holomorphic and quasiregular mappings. This is the case, for example, in Section 4.2, where we glue a Blaschke product into a basin of attraction. Shishikura stated his principle for rational maps. We include more general types and slightly modify one of the hypotheses. The second principle may be viewed, in some cases but not all, as a particular case of the first. Finally, we present Sullivan’s Straightening Theorem, also called the Generalized Shishikura Principle, which is the strongest of the three. It gives a necessary and sufficient condition for a quasiregular map f to admit an invariant almost complex structure. Namely, it requires the iterates f n to be uniformly K-quasiregular for some K ă 8. Although the Shishikura principles follow from this theorem, we choose to prove them independently, since the proof illustrates a procedure we shall use in many of the surgeries to come. The principles of this section apply to maps in Rat, Ent and Ent˚ . In fact, appropriately formulated they also apply to maps in Mer, or to holomorphic maps defined on subsets of the Riemann sphere. These cases present no extra difficulties but allowing for them all makes formulation heavier. It is for this reason that we present the statements for rational maps and comment on the differences in the other cases. 179

180

General principles of surgery

To simplify the exposition we use the following terminology. p ÑC p is called Definition 5.1 (Quasirational map) A quasiregular map f : C quasirational if it is quasiconformally conjugate to a rational map. This is equivalent to the existence of an f -invariant almost complex strucp ture of bounded dilatation on C. Likewise, we shall use the terminology quasientire and quasimeromorphic defined in the obvious way.

5.1 Shishikura principles p ÑC p be quasireguProposition 5.2 (First Shishikura Principle) Let f : C lar. Let p ě 1 be given, and suppose there exist: p such that, • U “ U1 Y ¨ ¨ ¨ Y Up consisting of p disjoint open subsets Uj of C for 1 ď j ă p, f pUj q “ Uj `1 ,

f pUp q Ď U1 ;

p and ψ is a quasiconformal homeomorphism r , where U r Ă C, • ψ :U ÑU (the gluing map); rÑU r is quasiregular with H p holomorphic; • H :U satisfying: (i) f |U “ ψ ´1 ˝ H ˝ ψ; and p Ń pU q, for some N ě 0. (ii) Bz f “ 0 a.e. in Czf Then f is quasirational. Comment: In Shihikura’s original formulation p “ 1. When p “ 1 and N “ 0 we can think of this scenario as pasting the holor (see Figure 5.1) onto U . morphic map H via the gluing map ψ : U Ñ U p Allowing N ą 0 adds the possibility of having some regions in CzU where f is not holomorphic, but only quasiregular, as long as orbits of points in these regions after at most N iterates fall into the set U which is invariant under f . Finally, the case p ą 1 allows for some more complicated situations, like maps H that are not necessarily holomorphic, although a certain iterate is (see e.g. [BF2] and Figure 5.2).

5.1 Shishikura principles

f

181 r U

U ψ

H

ψ ´1 H ψ Figure 5.1 The hypothesis in the First Shishikura Principle, for N “ 0 and p “ 1. The gluing map ψ is quasiconformal while H is holomorphic.

ψ r1 U

U1 ψ ´1 H ψ

ψ ´1 H ψ

H

H r2 U

U2 ψ

Figure 5.2 The hypothesis in the First Shishikura Principle, for N “ 0 and p “ 2. In this case H might not be holomorphic but H 2 is.

Proof Suppose ψ is K1 -quasiconformal, H is K2 -quasiregular and f is K3 quasiregular. r almost everywhere (a.e.) We first define an H -invariant Beltrami form μ r r on U . Set Uj :“ ψpUj q for 1 ď j ď p. These sets are disjoint, since ψ is rj `1 and H pU rp q Ď U r1 by rj q “ U a homeomorphism, and they satisfy H pU r construction. Now define a.e. on U , $ rp , ’ μ0 on U ’ ’ ’ &H ˚ pμ q rp´1 , on U 0 r“ μ ’ ¨¨¨ ’ ’ ’ % p ´1 ˚ r1 . pH q pμ0 q on U r is H -invariant since H p is holomorphic, and the dilatation By construction, μ p ´1 r by ψ to a Beltrami form r satisfies Kpr a.e. We pull back μ of μ μq ď K2 ˚ r defined a.e. on U . Given that μ r is H -invariant we have that μ is μ“ψ μ p ´1 f -invariant a.e. on U . Furthermore Kpμq ď K1 K2 . Finally, we spread μ

182


recursively by the dynamics of f as $ ’ on f ´1 pU q, ’f ˚ pμq ’ ’ ’ ’¨ ¨ ¨ ’ & μ “ pf n q˚ pμq on f ń pU q, ’ ’ ’ ’¨ ¨ ¨ ’ ’ ’ %μ ´j pU q . p Ť on Cz 0 j ě0 f p because the All together the Beltrami form μ is well defined a.e. on C, preimages of U form an increasing sequence of sets ď U Ď f ´1 pU q Ď f ´2 pU q Ď ¨ ¨ ¨ Ď f ń pU q Ď ¨ ¨ ¨ Ď f ń pU q, ně0

Ť where μ is f -invariant by construction, and the union ně0 f ń pU q is ´1 forward and backward invariant. We have }μ}8 ă k “ K K `1 , where K “ p ´1

K1 K2 K3N , since f satisfies Bfz “ 0 a.e. outside f Ń pU q, and f is K3 quasiregular inside f Ń pU q. Finally, the Integrability Theorem provides a quasiconformal map p ÑC p is holop ÑC p integrating μ, and therefore F :“ φ ˝ f ˝ φ ´1 : C φ:C morphic and hence a rational map quasiconformally conjugate to f (see the Key Lemma (Lemma 1.39)). Remark 5.3 It follows from the proof that Bφ “ 0 on the set ´j pU q. In particular, F and f are conformally conjugate on p Ť Cz j ě0 f the interior of this set, if any. pÑ Corollary 5.4 (Special case of the First Shishikura Principle) Let f : C p be quasiregular. Let X Ă C p be open and satisfy: C p and f is quasiconformal on X; (a) Bz f “ 0 a.e. on CzX (b) there exists N ě 1 such that maxzPX 7tOpzq X Xu ď N (i.e. orbits of f pass through X at most N times); and (c) there exists M ą 0 such that f n pXq X X “ H for all n ě M. Then f is quasirational. Proof

Set U“

ď

f n pXq,

něM

and observe that U is forward invariant. From the assumption it follows that f is holomorphic on U , since U and X are disjoint. Moreover, it follows that

5.1 Shishikura principles

183

p ´M pU q, since X Ď f ´M pU q. The hypotheses of the first Bz f “ 0 a.e. on Czf principle are now satisfied with p “ 1, ψ “ Id |U and H “ f on U . The second scenario we describe has a similar setting to the Corollary above, but the proof is essentially different. Proposition 5.5 (Second Shishikura Principle) Let f, X and N be as in Corollary 5.4 satisfying paq and pbq but not necessarily pcq. Then, the same conclusion holds. Proof For a point x P X let mx ě 1 be the smallest number of iterates so from then on its orbit does not come back to X, that is f n pxq R X for all n ě mx . This number is always finite by hypothesis, but it could be arbitrarily large, depending on x (this is the difference between this case and Corollary 5.4). p with }μ}8 ă 1. Our goal is to define an f -invariant Beltrami form μ on C We do this by pulling back along grand orbits in such a way that the Beltrami form is well defined a.e. p We may The set of grand orbits induces a partition of the phase space C. think of a grand orbit as an infinite tree where each node is mapped to one single other node, but several nodes (as many as the degree of the map) can be mapped to the same one (see Figure 5.3).

z

x

z0

Figure 5.3 The infinite tree representing the grand orbit of z, where mz “ 7. Points marked with a star are points of X. In this sketch N “ 2, so every biinfinite path in the p tree can contain up to two stars. From z0 forward, all points belong to CzX.

There are two kinds of grand orbits: either GOpzq and X are disjoint sets or the intersection is non-empty but it consists of at most N points along each path of the tree. In the first case we let μ “ μ0 along GOpzq. In the second case for some x P X X GOpzq set z0 “ f mx pxq. Then f n pz0 q R X for all n ě 0. We define μ on GOpxq “ GOpz0 q as follows:

184


μpwq “

# μ0 pf p q˚ pμ0 q

if w “ f n pz0 q for some n ě 0, where f p pwq “ f q pz0 q for some q ě 0.

Note that μ is well defined a.e. By construction μ is f -invariant with dilatation bounded by K N a.e. if f is K-quasiconformal on S. Thus we obtain a Beltrami p with bounded dilatation, and the proposition follows. form on C

5.2 Sullivan’s Straightening Theorem We now state the third and last result, which is a generalization of the two principles above. p ÑC p be Theorem 5.6 (Sullivan’s Straightening Theorem) Let f : C quasiregular. Suppose there exists a constant K such that the iterates f n are K-quasiregular for all n ě 1. Then f is quasirational. Notice that this theorem provides a necessary and sufficient condition. The necessity is clear: if f is conjugate to a holomorphic map F by a K 1 quasiconformal homeomorphism φ, then for every n ě 1, f n “ φ ´1 ˝ F n ˝ φ is pK 1 q2 -quasiregular. For rational maps the proof is outlined in [Su1]. We present P. Tukia’s approach [Tu1], 1 and afterwards adapt it to more general maps. See also [Ge1]. We use the following lemma, which we prove at the end of the section (see also [BrHa, Prop. 2.7]). Recall that the Haudorff metric on the space of non-empty bounded subsets of D is not a metric but a pseudo-metric. Lemma 5.7 Let B Ă D be a non-empty bounded set in the Poincaré metric. Then there exists a unique closed disc of minimal radius which contains B. Its centre cB is called the circumcentre (or centre of Young). The map B ÞÑ cB is continuous with respect to the Hausdorff pseudo-metric. The unit disc D, endowed with the Poincaré metric, comes into play via the Beltrami forms in the following way: let M be the set of Beltrami forms μ on p let Mpzq denote the set of the corresponding p with }μ}8 ă 1; for any z P C, C Beltrami coefficients μpzq at z (expressed in some chosen chart around z). We identify the values μpzq with points in D. p for which either f is not Let Dpf q denote the set of points z P C R-differentiable at z or f is R-differentiable but Dz f is singular. We recall p that if f is quasiregular, then Dpf q has measure zero. For z P CzDpf q set μf pzq “ Bz f pzq{Bz f pzq and λ “ Bz f pzq{Bz f pzq. Notice that |λ| “ 1 and that 1 We are grateful to Peter Ha¨ıssinsky for pointing this out and providing the proof.

5.2 Sullivan’s Straightening Theorem

185

μf pzq is determined once z is given. The formula (1.9) induces a map f ˚ : p M Ñ M, which for every z P CzDpf q restricts to f ˚ |Mpf pzqq : Mpf pzqq Ñ ˚ Mpzq defined by μpf pzqq ÞÑ f μpzq as follows: f ˚ μpzq “

1 λμf pzq ` μpf pzqq ¨ λ 1 ` λμf pzqμpf pzqq

a.e.

The mapping f ˚ |Mpf pzqq : Mpf pzqq Ñ Mpzq is a Möbius transformation of the unit disc onto itself and hence a hyperbolic isometry. We use this fact later in the proof. Ť Proof of Sullivan’s Theorem Set N pf q “ nPZ f n pDpf qq. Then N pf q is a measurable set of measure zero, which is forward invariant, and its complep pf q has full measure and is backwards invariant. Moreover, for ment Y “ CzN any z P Y , the map f n is R-differentiable at z with an invertible differential. In Y we define the sets of Beltrami forms B :“ tpf n q˚ μ0 unPZ Ă M. We note that, by definition and since f is uniformly quasiregular, Bz :“ B X Mpzq is a uniformly bounded set in Mpzq, and f ˚ Bf pzq “ Bz . For all z P Y , let μpzq be the circumcentre of the bounded set Bz in Mpzq. Since f ˚ |Mpf pzqq acts as an isometry, it follows from the uniqueness of the centre and the invariance of B described above that f ˚ μpf pzqq “ μpzq. For z P N pf q, set μpzq “ μ0 pzq. By construction μ is f -invariant and }μ}8 ă 1. It remains to prove that μ is measurable. To do so, let us define for n ě 0, Bpnq “ tpf k q˚ μ0 , |k| ď nu and Bz pnq “ Bpnq X Mpzq, which is a finite set. Let μnz be the circumcentre of Bz pnq. The map z P Y ÞÑ μnz is measurable, since N pf q is measurable and so are the pullbacks pf k q˚ . Also, by continuity of the centre, μnz tends to μpzq, hence the measurability of μ. But N pf q has measure zero, so it follows that f ˚ μ “ μ almost everywhere. Finally, as in the other cases, the Integrability Theorem applied to μ implies the existence of a quasiconformal map φ such that φ ˝ f ˝ φ ´1 is holomorphic. Proof of Lemma 5.7 We may as well assume that B contains at least two points. Let rB ą 0 be the infimum over the radius of any open disc containing B, and let us consider a sequence of open discs Dpxn , rn q such that B Ă Dpxn , rn q, and rn tends to rB . We will prove that txn unPN is a Cauchy sequence. This will suffice to prove the existence and uniqueness of the centre. Let ą 0; one may fix R 1 ă rB and R ą rB such that no geodesic in Dp0, RqzDp0, R 1 q has length larger than {2. This implies that, for every z P D, no geodesic in Dpz, RqzDpz, R 1 q has length larger than {2 since

186


Möbius transformations that preserve D are isometries with respect to the Poincaré metric. We fix m, n large enough so that rn , rm ă R. Let y be the middle point of the geodesic segment γ joining xn to xm . Since R 1 ă rB , there is at least one point z P B such that dpy, zq ą R 1 . By definition of R, one has xn , xm P Dpz, Rq. It follows from the convexity of the distance function that at least half of γ is contained in Dpz, RqzDpz, R 1 q. Therefore, dpxm , xn q ď , and txn unPN is a Cauchy sequence. This proves the existence and uniqueness of the circumcentre. To prove the continuity of the map B ÞÑ cB let us consider a sequence tBn unPN of uniformly bounded sets tending to a bounded set B in the Hausdorff pseudo-metric. This means that given ą 0, for n large enough we have Bn Ă B and B Ă Bn , where the superscript denotes an -neighbourhood of the set. Denote by cn the circumcentre of Bn with its associated radius rn . Define r “ lim inf rn . Extracting a subsequence if necessary, we might as well assume that tcn unPN tends to some c P D. It follows that B Ă Dpc, rq so that rB ď r by minimality. If we had rB ă r, then we could find r 1 P prB , rq such that, for n large enough, rn ą r 1 ; moreover, from the convergence of tBn unPN , it would follow that Bn Ă Dpc, r 1 q for n large enough, contradicting the minimality of rn . Therefore, r “ rB , and by the uniqueness of the circumcentre, c is the circumcentre of B.

5.3 Non-rational maps Both the Shishikura principles and Sullivan’s Straightening Theorem, appropriately formulated, hold for the following cases. Locally defined maps Let f : U 1 Ñ U be quasiregular of degree d ě 2, where U, U 1 » D and U 1 Ă U . The type of maps we have in mind are polynomial-like maps, which will be treated in Section 7.1. Observe that points can be iterated infinitely many times in the set č f ń pU 1 q. ně0

p by the domain of definition. After integratAll proofs go through replacing C ing we obtain a holomorphic map F : D 1 Ñ D, with D 1 Ă D, quasiconformally conjugate to f on the appropriate domains.

5.3 Non-rational maps

187

Transcendental maps with no poles Let f : C Ñ C (or f : C˚ Ñ C˚ ) be quasiregular maps with an essential singularity at infinity (or at infinity and zero). The critical points (understood as points where f fails to be a local homeomorphism) of a map of this type form a discrete set in the plane, so they do not interfere in any pullback operation. However, transcendental maps may have asymptotic values, and even sets of positive measure of asymptotic values. But these do not create any problems either. In the presence of asymptotic values, we pull back along those branches of the inverse, which are well defined, if any. The non-regular branches map the values to infinity, and therefore they do not create new finite points where the pullbacks are not well defined. Additionally, observe that any Beltrami form in C (or C˚ ) can be considered p and integrated as such, where the integrating map is a Beltrami form in C normalized to fix 8 (or 8 and 0). In conclusion, the proofs are transferred to the transcendental case with almost no changes. After integrating we obtain a holomorphic map F P Ent (or F P Ent ˚ ) quasiconformally conjugate to the original one. p be quasiregular, with an Transcendental maps with poles Let f : C Ñ C essential singularity at infinity. On the poles and prepoles, the forward orbit gets truncated after finitely many iterations. On the poles, the differential is not well defined. However, the set of prepoles, say Ppf q, is a countable set and therefore of measure zero. Since the preimage of a set of measure zero has measure zero, repetitive pullbacks still make sense for this type of maps. The proof of the First Shishikura Principle works for transcendental meromorphic maps satisfying the hypothesis of the principle. The f -invariant Beltrami form μ, is defined as μ0 on the set of poles and their iterated preimages, since their orbits cannot eventually fall into the invariant set U . p is a transcendental meromorphic map. The resulting map F : C Ñ C For the Second Principle, observe that poles are points where the infinite tree gets truncated in the forward direction. We define μ “ μ0 on the set p is a of poles, and pull backwards from there. The resulting map F : C Ñ C transcendental meromorphic map. Sullivan’s Straightening Theorem also goes through, after reformulating the hypotheses to require that the nth iterates of f are uniformly quasiregular wherever defined. The set of poles and prepoles is of measure zero and should be included in N pf q.

6 Soft surgeries

In this chapter we treat two different applications of soft surgery. Recall that a soft surgery starts from one given holomorphic map and consists of altering the almost complex structure in an invariant way, in order to obtain a family of invariant Beltrami forms. In the first application, Section 6.1, a contribution by Xavier Buff and Christian Henriksen, we start with one map, which possesses a rotation ring (for example, a Herman ring), and vary the complex structure to produce a one-parameter family of holomorphic maps, parametrized by D, all having a rotation ring. Different parameters result in rings of different modulus and different twist number, a quantity which measures the relative rotation between the two ring boundaries. In the second application, Section 6.2, we start with a polynomial with one or more critical points escaping to infinity and change the complex structure to obtain a one-parameter family of polynomials with the same property but where the Böttcher coordinate of the fastest escaping critical point has been changed. This allows, for example, to parametrize (by D˚ ) a certain one-parameter family of polynomials in the complement of the connectedness locus. In the simplest case, the one-parameter family of quadratic polynomials tQc ucPC , this gives a parametrization of the complement of the Mandelbrot set. Other applications of soft surgery have already been explained in Sections 4.1 and 4.3. We remark that the elementary example in Section 1.5 lies behind all these applications except the one for no wandering domains.

188

6.1 Deformation of rotation rings

189

6.1 Deformation of rotation rings Xavier Buff and Christian Henriksen

In this section, we show how one can deform the complex structure of a rational map having a rotation ring, in order to get a family of rational maps that are quasiconformally conjugate but not conformally conjugate to the initial one. p ÑC p is Recall that the postcritical set Pf of a rational map f : C Pf “

ď ď

f n pvq,

v PVf ně0

where Vf is the set of critical values of f . Definition 6.1 (Rotation ring) A rotation ring for a rational map f is a p f that is an annulus with finite modulus. periodic connected component of CzP Note that if U is a rotation ring of period p, then for any integer k ě 1 and any component Uk of f ´k pU q, the map f k : Uk Ñ U is a covering map. Since U is a component of f ´p pU q, we have that f p : U Ñ U is a covering map. The modulus of the annulus U is preserved, so f p : U Ñ U is an isomorphism. This isomorphism cannot have finite order, since otherwise an p by analytic continuation. iterate of f would be the identity on U , thus on all C p As a consequence, f : U Ñ U is conjugate to an aperiodic rotation and U is contained in a Siegel disc or a Herman ring. The result we are interested in is the following. Theorem 6.2 (Main Theorem) Let f P Ratd have a rotation ring H of modulus m. Then there is an analytic family of rational maps Fτ P Ratd , parametrized by the left half plane, such that: • f is conformally conjugate to Fτ0 with τ0 “ ´2π m; Repτ q • Fτ has a rotation ring of modulus ´ ; 2π • f and Fτ are quasiconformally conjugate, the conjugacy being conformal outside the grand orbit of H . Remark 6.3 The family Fτ is non-trivial since the modulus of the rotation ring is non-constant. There are several cases where this result may be applied. We will illustrate this with two concrete examples: the case of a rational map with a Herman ring and the case of a polynomial with a Siegel disc capturing a critical orbit.

190

Soft surgeries 6.1.1 Examples of rotation rings

Herman rings Consider the family of cubic Blaschke products ft,a pzq :“ e2π i t z

1 ´ az 1 ´ a{z

with t P R

and

a P p0, 1{3q.

When a P p0, 1{3q, the restriction of ft,a to the unit circle S1 is an orientation preserving analytic circle diffeomorphism. When t P Z, there is a fixed point at z “ 1 and the rotation number is 0. According to Theorem 3.20, for each irrational θ P p0, 1q and each a P p0, 1{3q, there is a unique t P p0, 1q such that the rotation number of f “ ft,a is θ . According to Denjoy, in that case, there is a circle homeomorphism ψ : S1 Ñ S1 conjugating Rθ , the rotation of angle θ to f , that is for all z P S1 ,

f ˝ ψpzq “ ψpe2π i θ zq.

(6.1)

If ψ : S1 Ñ S1 is analytic (according to Theorem 3.16 this is true as soon as θ P H), then it has a holomorphic extension to some annulus A :“ A1{R,R “ t1{R ă |z| ă Ru with R sufficiently close to 1. Equation (6.1) holds in A by analytic continuation. In particular, the open set ψpAq is invariant and thus contained in the Fatou set of f . It cannot be contained in a Siegel disc since each p component of CzψpAq contains superattracting fixed points of f (one at 0 and the other at 8). So, ψpAq is contained in a Herman ring Ha (see Figure 3.8). Lemma 6.4 (Herman rings are rotation rings) Assume f “ ft,a has a Herman ring H containing S1 . Then H is a rotation ring for f . Proof Among the four critical points of f , two are fixed (one at zero and the other at infinity). The boundary of H is contained in Pf . In particular, at least one critical point of f is contained in the Julia set. Since f commutes with the map z ÞÑ 1{z, two critical points of f must be contained in the Julia set. So, the Herman ring H does not intersect Pf and H is a rotation ring for f . Siegel captures Let θ P B be a Bryuno number. Consider the family F “ tfa ua PC of cubic polynomials given by fa pzq :“ ρz ` az2 ` z3

with ρ “ e2π i θ .

Such families have been studied by Zakeri [Z1] (see also [BHe]). First, fa p´zq “ ´fá pzq, so that the affine map z ÞÑ ´z conjugates fa to fá . If g is a cubic polynomial with a fixed point with multiplier ρ, there exists a unique b P C such that g is conjugate to f˘?b . So, the a 2 -plane is the moduli space of cubic polynomials having a fixed point of multiplier ρ.


191

Since θ is a Bryuno number, for all a P C, the polynomial fa has a Siegel disc a around the fixed point 0. The orbit of at least one critical point must accumulate on Ba . As a consequence, at most one critical point may have an unbounded orbit. The Julia set is connected if and only if both critical points have bounded orbits. Figure 6.1 shows the connectedness locus of the family F in the a 2 -plane.

Figure 6.1 The connectedness locus of the family F in the a 2 -plane. Points are coloured according to the behaviour of the critical points of fa : white if fa has an unbounded critical orbit; black if fa has a critical orbit that eventually lands on the fixed point at 0; and yellow otherwise.

The following result provides an example of rotation ring (see Figure 6.2).

Figure 6.2 The Julia set of a cubic polynomial fa having a rotation ring (light yellow).

Lemma 6.5 (a contains a rotation ring) If a 2 is sufficiently close, but not equal, to 4ρ, then a zt0u contains a critical value of fa . Its orbit is dense in

192

Soft surgeries

an invariant analytic Jordan curve Ca . The connected component of a zCa which does not contain 0 is a rotation ring for fa . Proof On the one hand, if a 2 “ 4ρ, the cubic polynomial fa has a critical point at á{3 and fa pá{3q “ 0. If a 2 is close but not equal to 4ρ, then fa has a critical value close, but not equal, to 0. On the other hand, according to Bryuno, there is a constant c ą 0 (which only depends on θ ) such that if fa is univalent in Dr , then a contains Dcr . If a 2 is sufficiently close to 4ρ, this disc contains a critical value of fa whose orbit is dense in an invariant analytic Jordan curve Ca . At least one critical point of fa is contained in Jfa since its orbit must accumulate on the boundary of the Siegel disc. It follows that a intersects at most one critical orbit. So, when a 2 is sufficiently close to 4ρ, then a X Pfa coincides with Ca . In that case, the connected component of a zCa which does not contain 0 is an annulus Ha whose boundary components are Ca and Ba . Both boundary components are contained in the postcritical set. And, moreover, Pfa does not intersect Ha . Thus, Ha is a rotation ring for fa . Define the capture component C1 by C1 :“ tb P C | f˘?b maps a critical point into ˘?b u. We shall conclude the section by proving that C1 is a simply connected domain of C by exhibiting a natural conformal isomorphism from D to C1 . 6.1.2 Changing the modulus of the rotation ring Let f P Ratd be a rational map with a rotation ring H . In this paragraph, p so that f remains we show that we can deform the complex structure of C holomorphic with respect to the new structure, but the modulus of the rotation ring H is changed (cf. [FG]). Let m be the modulus of the rotation ring H and set r “ e´2π m . There exists an isomorphism ϕ : H Ñ Ar , where Ar “ tr ă |z| ă 1u. Given α ą 0, we can define an R-differentiable map φα : Ar Ñ Ar α by φα pzq :“ z |z|α ´1 . Note that φα : Ar Ñ Ar α conjugates Rθ : Ar Ñ Ar to Rθ : Ar α Ñ Ar α . In addition, Bz φα “

α ´ 1 α`1 α´1 ´1 z 2 z 2 2

and

Bz φα “

α ` 1 α`1 ´1 α´1 z 2 . z 2 2


193

It follows that Bz φα “ μα Bz φα

with μα “

α´1 z . α`1 z

In particular, ˇ ˇ ˇα ´ 1ˇ ˇă1 }μα }8 “ ˇˇ α ` 1ˇ and φα is quasiconformal. Thus, we can pull back the standard structure σ0 on Ar α to define an almost complex structure σα “ φα˚ σ0 on Ar . Since φα conjugates Rθ : Ar Ñ Ar to Rθ : Ar α Ñ Ar α and σ0 is invariant under Rθ : Ar α Ñ Ar α , we get that σα is invariant under Rθ : Ar Ñ Ar . Pulling back the ellipse field σα by ϕ defines an ellipse field ςα on H . Since σα is invariant under Rθ : Ar Ñ Ar , and ϕ conjugates f p “ H Ñ H to Rθ : Ar Ñ Ar , the ellipse field ςα is invariant under f p : H Ñ H . Pulling back the ellipse field ςα by f along the cycle of H yields an ellipse field which is invariant under f on

H :“

p ď

f k pH q.

k “1

Of course, pulling back p times along the cycle of H , we come back to H (since H is periodic of period p for f ). Moreover, pulling back p times ςα by f , we obtain ςα again since ςα is invariant by f p : H Ñ H . p Next, we extend ςα to an f -invariant ellipse field defined on all of C. ´ 1 ˚ First we extend ςα to f pHq Ą H by letting ςα “ f ςα . This does not redefine ςα on H. Then we spread ςα by the dynamics of f to to the grand orbit of H . On the complement of this grand orbit, we let ςα “ σ0 . Since we pull back by holomorphic maps, the ellipse field ςα has uniformly bounded dilatation. By the Integrability Theorem (Theorem 1.28) there exists a quasiconformal p ÑC p with ςα “ Q˚ σ0 . The ellipse field ςα depends analytically map Qα : C α on α (see Theorem 1.30). Therefore, if Qα is normalized suitably (the choice of normalization depends on the context), it depends analytically on α. The construction is illustrated in Figure 6.3.

194

Soft surgeries

Qα

1 φα ˝ ϕ ˝ Q´ α

ϕ

φα

Figure 6.3 The construction used to change the modulus of the rotation ring. The 1 vertical maps ϕ and φα ˝ ϕ ˝ Q´ α are conformal, whereas the horizontal maps Qα and φα are quasiconformal. The circle/ellipse fields shown at the bottom are drawn by hand, whereas those at the top are computed. 1 Now consider the map Fα :“ Qα ˝ f ˝ Q´ α as shown in the following diagram:

p ςα q pC, f

Qα

p ςα q pC,

p σ0 q / pC,

Qα

Fα

p σ0 q. / pC,

Since the ellipse field ςα is f -invariant and since Qα sends it to the field of circles σ0 , we see that Fα is a quasiregular map that preserves σ0 . According to pÑC p is holomorphic, thus a rational map. We Weyl’s Lemma, the map Fα : C p ÑC p which are quasiconformally constructed a family of rational maps Fα : C


195

p Ñ C, p the conjugacy being conformal outside the grand conjugate to f : C orbit of H . The annulus Hα :“ Qα pH q is a rotation ring for Fα . The map 1 φα ˝ ϕ ˝ Q´ α : Hα Ñ Ar α

is quasiconformal (it is a composition of quasiconformal maps) and preserves the standard complex structure σ0 . Again, according to Weyl’s Lemma, it is holomorphic, thus an isomorphism. This shows that the modulus of the rotation ring Hα is modpHα q “

1 1 log α “ α modpH q. 2π r

In particular, the rational map Fα is not conformally conjugate to f (except when α “ 1) since the modulus of the rotation ring is changed. Figure 6.4 shows an example where the modulus is doubled.

Figure 6.4 We illustrate how one can fatten a rotation ring. Given a cubic polynomial f with a fixed rotation ring, we can construct a quasiconformal map Q2 that conjugates f to a polynomial F2 with a rotation ring with twice the modulus. A circle field on the grand orbit of the rotation ring is illustrated to the right, and the pullback of the circle field by Q2 to the left.

We shall apply this construction to the family of cubic Blaschke products ft,a pzq “ e2π i t z

1 ´ az 1 ´ a{z

with t P R

and

a P p0, 1{3q.

As mentioned above, for each irrational number θ and each a P p0, 1{3q, there is a unique t “ Tθ paq P p0, 1q such that the rotation number of ft,a : S1 Ñ S1 is equal to θ . The following proposition establishes regularity properties of the curve Tθ : p0, 1{3q Ñ p0, 1q.

196

Soft surgeries

Proposition 6.6 (Regularity properties of Tθ ) For each irrational θ , there is an aθ P r0, 1{3s such that: • the curve Tθ is R-analytic on p0, aθ q; and • for pt, aq P p0, 1q ˆ p0, 1{3q, the map ft,a has a Herman ring containing S1 with rotation number θ if and only if a P p0, aθ q and t “ Tθ paq. Proof Fix an irrational number θ . If no map ft,a has a Herman ring containing S1 with rotation number θ , the proposition trivially holds with aθ “ 0. So, let us assume that there is a pair pt0 , a0 q P p0, 1q ˆ p0, 1{3q such that f “ ft0 ,a0 has a Herman ring H containing S1 with rotation number θ . The rotation number of f : S1 Ñ S1 is θ . Thus, t0 “ Tθ pa0 q. It is therefore enough to prove that the curve Tθ is R-analytic in a neighbourhood of a0 and that fTθ pa q,a has a Herman ring containing S1 for all a P p0, a0 s. As mentioned earlier, H is a rotation ring for f . The previous surgery provides a family of ellipse fields ςα depending analytically on α P p0, `8q. p ÑC p be the unique quasiconformal map satisfying ςα “ Q˚ σ0 , Let Qα : C α normalized by the conditions Qα p0q “ 0,

Qα p8q “ 8 and

Qα p1{aq “ 1.

Then, the rational map 1 Fα :“ Qα ˝ f ˝ Q´ α

is a cubic rational map with a superattracting fixed point at 0, a superattracting fixed point at 8 and a Herman ring Hα “ Qα pH q of modulus modpHα q “ α modpH q. Since f has a zero at 1{a, the rational map Fα has a zero at 1. Since f has a pole at a, the rational map Fα has a pole at bα “ Qα paq. It follows that Fα pwq “ ρα w

1´w 1 ´ bα {w

for some complex number ρα P Czt0u depending analytically on α. Lemma 6.7 We have that bα P p0, 1{9q

and

´ á ¯¯ ρα “ exp 2π i Tθ bα .

Proof We shall first prove that bα ą 0. The rational map f commutes with the antiholomorphic involution τ : z ÞÑ 1{z. It follows that the isomorphism ϕ : H Ñ Ar satisfies ϕ ˝ τ ˝ ϕ ´1 “ τr , where τr : Ar Ñ Ar is the antiholomorphic involution of Ar given by τr pzq “ τ pz{rq. Furthermore,


197

φα ˝ τr “ τr α ˝ φα . It follows that the ellipse field σα is invariant by τr and the 1 ellipse field ςα is invariant by τ . Hence the map Qα ˝ τ ˝ Q´ α is antiholomorphic. It exchanges 0 and 8 and also 1 and bα . It follows that 1 Qα ˝ τ ˝ Q´ α pwq “

bα w

and

bα bα

“ 1.

So, bα is real. In addition, ς1 is a field of circles and Q1 is the holomorphic map z ÞÑ w “ az. In particular, b1 “ a 2 ą 0. Since bα ‰ 0 for all α ą 0, we see that bα ą 0 for all α ą 0. The unit circle is the set of fixed points of τ . Its image by Qα is the set ? of fixed points of w ÞÑ bα {w. This is the circle of radius bα . The map f : S1 Ñ S1 is a circle diffeomorphism with rotation number θ . It follows that the restriction of Fα to S1?b is a circle homeomorphism with rotation number θ α and without critical points, thus a circle diffeomorphism. In particular, ? á ¯ a a 1 ´ bα ? “ ρα bα bα “ ρα bα Fα 1 ´ bα { bα ? has modulus bα . This shows that ρα has modulus 1. ? Now, making the change of coordinates z “ w{ bα , the rational map Fα is conjugate to ? 1 ´ bα z ? . z ÞÑ ρα z 1 ´ bα {z The restriction to S1 is a circle diffeomorphism with rotation number θ . So, ´ á ¯¯ a bα P p0, 1{3q and ρα “ exp 2π i Tθ bα

as required. Lemma 6.8 We have that bα tends to 0 as α tends to `8.

Proof The Herman ring Hα of Fα separates the zero at 1 from the pole at bα . As α tends to `8, the modulus modpHα q “ α modpH q tends to `8. It follows that bα tends to 0 as α tends to `8. The map Fα depends analytically on α. In particular, bα , ? Fα p bα q ρα “ ? bα

? bα and

all depend analytically on α. It follows that Tθ is analytic on the image of the ? ? curve α ÞÑ bα . For α “ 1, we have bα “ a0 and as α Ñ `8, we have ? bα Ñ 0. So, Tθ is analytic in a neighbourhood of a0 as required. In addition,

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Soft surgeries

every a P p0, a0 s is equal to bα for some α P p0, `8q and fTθ pa q,a is conjugate ? to Fα via the scaling map z ÞÑ w “ bα z. Since Fα has a Herman ring around ? the circle of radius bα , the map fTθ pa q,a has a Herman ring around S1 . Remark 6.9 The previous argument shows that aθ “ lim

α Ñ0

a

bα . According to

Sections 7.2.3 and 7.3, we have aθ ą 0 if and only if θ P B is a Bryuno number. In addition, when θ P H, we have aθ “ 1{3. According to a deep result by Herman (see Theorem 3.21), there exist θ P BzH such that 0 ă aθ ă 1{3. It is unknown whether this holds for all θ P BzH. 6.1.3 Twisting in the rotation ring We have seen that given a map f with a rotation ring, we can vary its modulus by modifying the complex structure. However, this is not the only way we can deform f quasiconformally. To motivate the construction return to Figure 6.1. Some points in the figure are coloured in red. These are points which, at least 1 log 43 . In numerically, all possess fixed rotation rings with modulus equal to 2π Figure 6.5 the dynamics of four of these polynomials are illustrated. We see very similar pictures; the largest difference from one picture to another seems to be that the two boundary components are rotated with respect to each other. In the following we show how this can be accomplished with a soft surgery. The construction is as follows. As before, let ϕ : H Ñ Ar denote a conformal isomorphism. Passing to logarithmic coordinates, Ar corresponds to the strip SX :“ tx ` i y | X ă x ă 0u

with X “ logprq.

Let Lt : SX Ñ SX be the R-linear map that fixes i and maps X to X ` i t Lt px ` i yq :“ x ` i py ` tx{Xq, or equivalently Lt pzq “ az ` bz

with a “ 1 `

it 2X

and

b“

it . 2X

Then Lt maps vertical lines isometrically to vertical lines, and it restricts to the identity on the imaginary axis. In particular, it descends as a quasiconformal mapping φt : Ar Ñ Ar mapping circles (centred at the origin) to themselves by rigid rotations. On the unit circle, φt coincides with the identity, whereas on the circle of radius r, φt coincides with the rigid rotation Rt .


199

Figure 6.5 The dynamics of four polynomials corresponding to the four marked points in Figure 6.1. The modulus of the rotation rings are all the same (approximately 1 4 2π log 3 ). Inspecting the two boundary components of the rotation rings, the principal difference between the pictures seems to be that the two boundary components are rotated with varying degree with respect to each other. It turns out that this is not far from the truth and that the polynomials can be obtained from each other by quasiconformally twisting the rotation ring.

Proceeding as we did when we modified the modulus of H , we define an almost complex structureσt on Ar by pulling back the standard one by φt . Since φt commutes with Rθ and this preserves σt and σ0 , it follows that ςt “ ϕ ˚ σt is invariant under the action of f p : H Ñ H . We now spread ςt by the p dynamics to H and by the circle field on CzH, so it is defined everywhere and invariant by f . The map Lt is quasiconformal with constant dilatation, hence the dilatation of σt is uniformly bounded, and so is the dilatation of ςt because we pull back by holomorphic mappings. Therefore there exists a quasiconformal homeop ÑC p such that ςt “ Q˚ σ0 . morphism Qt : C t

200

Soft surgeries

As in the previous paragraph, ςt depends analytically on t and so, if suitably normalized, Qt depends analytically on t (see Theorem 1.30). Again, the map 1 p p Ft :“ Qt ˝ f ˝ Q´ :CÑC t

is a rational map. It is quasiconformally conjugate to f , and even conformally outside the grand orbit of H . The construction is illustrated in Figure 6.6 for a cubic polynomial ¸ ˜ ? 5´1 2 3 , f pzq “ ρz ` az ` z with ρ “ exp 2π i 2

Figure 6.6 We illustrate how one can twist a rotation ring. Given a cubic polynomial f with a fixed rotation ring, we can construct a quasiconformal map Qt that conjugates f to a polynomial Ft with a rotation ring with the same modulus. A circle field on the grand orbit of the rotation ring is illustrated to the right, and the pullback of the circle field by Qt to the left.

where a 2 is sufficiently close to 4ρ so that f has a rotation ring. In that case, the grand orbit of H is bounded (it is contained in the filled Julia set of f which is compact in C). It follows that the support of ςα avoids a neighbourhood of infinity, and so we may normalize Qt by the following conditions: Qt p0q “ 0,

Qt pzq “ z ` Op1q as z Ñ 8.

Note that ς0 “ σ0 . As a consequence, Q0 “ Id and F0 “ f . In addition, for all t P R, the cubic rational map Ft fixes 0 and 8 and the only preimage of 8 is 8 itself. So, Ft is a cubic polynomial. It is holomorphically conjugate to f in a neighbourhood of 8 by the map Qt which is tangent to the identity at this point. It follows that Ft is a cubic polynomial. The support of ςt also avoids a


201

neighbourhood of 0, and so the multiplier of 0 as a fixed point of Ft is equal to ρ. This proves that Ft pzq “ ρz ` at z2 ` z3 for some complex number at which depends analytically on t P R. In the next paragraph, we show that the two previous constructions, varying the modulus of the rotation ring and twisting, may be combined in a single operation that depends holomorphically on the parameter. We shall also see that, in some cases, the curve t ÞÑ Ft is periodic of period 2π . 6.1.4 Holomorphic deformation in the rotation ring Fix τ0 in the left half plane H and for τ P H denote by Lτ : C Ñ C the unique R-linear map that fixes i and maps τ0 to τ : Lτ pzq :“

τ ` τ0 τ ´ τ0 z` z. τ0 ` τ 0 τ0 ` τ 0

Observe that this is the same map we considered in the elementary example of Section 1.5. Note that the complex dilatation of Lτ is constant and equal to μτ :“

τ ´ τ0 . τ ` τ0

It is important to note that μτ depends holomorphically on τ . In addition, Lτ maps vertical lines to vertical lines isometrically and, in particular, commutes with the translation z ÞÑ z ` 2π i , and Lτ restricts to the identity on the imaginary axis. For τ P H , denote by Sτ the strip ( Sτ :“ x ` i y | Repτ q ă x ă 0 . Then, Lτ restricts to a quasiconformal homeomorphism Lτ : Sτ0 Ñ Sτ . Since Lτ commutes with the translation by 2π i , it induces a quasiconformal homeomorphism φτ : A|eτ0 | Ñ A|eτ | satisfying exp ˝Lτ “ φτ ˝ exp Lτ

Sτ0

/ Sτ exp

exp

A|eτ0 |

φτ

/ A|eτ | .

Now, the surgery goes as follows. As previously, let ϕ : H Ñ Ar be a conformal isomorphism. Set τ0 “ log r and for τ in the left half plane denote by ςτ the ellipse field on H obtained by pulling back the standard complex structure on A|eτ | by φτ ˝ ϕ

202

Soft surgeries ςτ “ pφτ ˝ φq˚ σ0 .

As previously, ςτ is invariant by f p and may therefore be promoted to a globally f -invariant ellipse field by pulling back ςτ via f on the grand orbit of H and extending by σ0 outside. We may then invoke the Integrability p ÑC p such that Theorem to obtain a quasiconformal homeomorphism Qτ : C ˚ pQτ q σ0 “ ςτ . Since ςτ is invariant by f , the map 1 p p Fτ :“ Qτ ˝ f ˝ Q´ τ :CÑC

is holomorphic – we have built a family of rational maps quasiconformally conjugate to f . As observed previously, μτ depends holomorphically on τ , and so the family of ellipse fields φτ˚ σ0 depends holomorphically on τ . The maps we then use to define ςτ via pullback, i.e. ϕ and f , are holomorphic and do not depend on τ . It follows that ςτ depends holomorphically on τ . So, when suitably normalized, Qτ depends holomorphically on τ , and so does Fτ by Lemma 1.40. Note that Hτ “ Qτ pH q is a rotation ring for Fτ . In addition, the map 1 φτ ˝ ϕ ˝ Q ´ τ : H Ñ A|eτ |

is a conformal isomorphism. In particular, mod pHτ q “

1 Repτ q 1 log τ “ ´ . 2π |e | 2π

Since Lτ0 “ Id, the complex structure φτ˚0 σ0 coincides with σ0 , and so does ςτ0 . As a consequence, Qτ0 is a Möbius transformation and Fτ0 is conformally conjugate to f .

6.1.5 Conformal conjugacies We shall now see that in the two examples we considered (a cubic polynomial with a capture and a cubic rational map with a Herman ring), the rational maps Fτ `2π i and Fτ are conformally conjugate. Note that by assumption, for all k ě 1, the restriction f k : f ´k pH q Ñ H is a covering map since H does not intersect the postcritical set Pf . Lemma 6.10 (Conformal conjugacy) Assume that for all k ě 1 each component of f ´k pH q maps with degree 1 to H by f k . Then, Fτ `2π i is conformally conjugate to Fτ for every τ P H .

6.1 Deformation of rotation rings Proof

203

The quasiconformal homeomorphism g :“ φτ´1 ˝ φτ `2π i : Ar Ñ Ar

commutes with the rigid rotation Rθ . It follows that the quasiconformal homeomorphism h :“ ϕ ´1 ˝ g ˝ ϕ : H Ñ H commutes with f . In addition, h˚ ςτ “ ςτ `2π i . Furthermore, there is a uniform bound on the hyperbolic distance in Ar , between a point and its image under g. So, this also holds for h in H . Comparing hyperbolic and euclidean distances as we approach the boundary, shows that h extends as the identity to BH . According to Rickman’s Lemma 1.20, p Ñ C, p which coincides with h on H and is the homeomorphism h1 : C equal to the identity otherwise, is quasiconformal. Since f : f ´1 pH q Ñ H is a covering map, we may lift h1 : H Ñ H to obtain a quasiconformal automorphism h2 : f ´1 pH q Ñ f ´1 pH q such that the following diagram commutes: f ´1 pH q f

h2

H

/ f ´1 pH q . / H

f

h1

Since each component of f ´1 pH q maps to H with degree 1 by f , we automatically have h2 “ Id on the boundary of f ´1 pH q. Extending h2 by the identity p ÑC p outside f ´1 pH q, we obtain a quasiconformal homeomorphism h2 : C p for which the above diagram commutes on C. Continuing like this, we may p ÑC p such that construct a sequence of quasiconformal automorphisms hk : C ´ k hk “ Id outside f pH q and the following diagram commutes: p C f

p C

hk

hk´1

p / C p / C.

f

204

Soft surgeries

p A point z P C: • either belongs to the grand orbit of H , in which case there is a unique k such that hk `1 pzq ‰ hk pzq; • or belongs to the complement of the grand orbit of H , in which case hk pzq “ z for all k ě 0. p Ñ Cq p converges to some map h8 : C p Ñ C. p It follows that the sequence phk : C p p Since f is holomorphic, the quasiconformal homeomorphisms hk : C Ñ C all have the same bounded dilatation K. By compactness of the set of K-quasiconformal homeomorphisms fixing three points (see Theorem 1.26), p ÑC p is a K-quasiconformal homeomorphism. This limit the limit map h8 : C map coincides with the identity outside the grand orbit of H and conjugates f to itself. p As Now, since h˚ ςτ “ ςτ `2π i on H , we deduce that h˚8 ςτ “ ςτ `2π i on C. a consequence, the quasiconformal homeomorphism p ÑC p ψ :“ Qτ `2π i ˝ h8 ˝ Q´1 : C τ

satisfies ψ ˚ σ0 “ σ0 , thus is a Möbius transformation. In addition, ψ conju gates Fτ `2π i to Fτ , which proves the Lemma. We can always normalize Qτ so that it depends holomorphically on τ and 1 p so that Qτ `2π i ˝ h8 ˝ Q´ τ fixes three points in C. For example, this may be achieved by choosing three repelling periodic points z0 , z1 and z2 of f (those belong to the complement of H ), and then normalizing Qτ by the condition Qτ pz0 q “ 0,

Qτ pz1 q “ 1 and

Qτ pz2 q “ 8.

1 For such a normalization, the Möbius transformation Qτ `2π i ˝ h8 ˝ Q´ τ is the identity, and so Fτ `2π i “ Fτ . Setting

λ :“ eτ

and

gλ :“ Fτ ,

p Ñ Cq p we obtain a well-defined holomorphic family of rational maps pgλ : C parameterized by Dzt0u. On the one hand, the assumptions of Lemma 6.10 are satisfied in the case of the family of cubic Blaschke products presented in Section 6.1.1 1 ´ az , t P p0, 1q and a P p0, 1{3q. ft,a pzq “ e2π i t z 1 ´ a{z Indeed, each point in the unit circle has three preimages, one in the unit circle, one in the unit disc, and one outside the unit disc. It follows that if f “ ft,a has a Herman ring H around the unit circle, then H separates two connected components of f ´1 pH q, one contained in the unit disc and one contained in


205

its complement. Each component maps with degree 1 to H . Normalizing Qτ as in Section 6.1.2, i.e. Qτ p0q “ 0,

Qτ p8q “ 8 and

Qτ p1{aq “ 1,

the rational map gλ “ Fτ takes the form gλ pwq “ ρλ w

1´w 1 ´ bλ {w

for some complex numbers ρλ P Czt0u and bλ P Czt0, 1u which depend holomorphically on λ P Dzt0u (see [BFGH] for further applications of this construction). On the other hand, the assumptions of Lemma 6.10 are not satisfied in the case of the family of cubic polynomials presented in Section 6.1.1 fa pzq “ e2π i θ z ` az2 ` z3 ,

a P C and

θ P B.

Here, the component of fa´1 pH q which surrounds the critical point close to á{2 (the one mapping to the Siegel disc), maps with degree two to H . Even in that case, however, it is still true that the polynomial Fτ `2π i is conformally conjugate to the polynomial Fτ . Indeed, let a be the Siegel disc of fa which contains 0. Extend h1 : H Ñ H to a by the identity. Since h1 fixes the critical value v P a , both h1 ˝ fa : fa´1 pa ztvuq Ñ a ´ tvu and fa : fa´1 pa ztvuq Ñ a ´ tvu are two to one coverings, and we can lift h1 ˝ fa by f to get a homeomorphism h2 : fa´1 pa q Ñ a . Of the two possible lifts, we choose the one that extends as the identity to Bfa´1 pa q. The restriction of f to fań pa qzf ń`1 pa q is a covering map for all n ą 1, and h2 commutes with f . Therefore, we may lift h2 to a homeomorphism hk : fa´k `1 pa q Ñ fa´k `1 pa q which coincides with the identity on Bfa´k `1 pa q, and extend hk by the identity to the complement of fa´k `1 pa q, for k “ 3, 4, . . . . As in the proof of Lemma 6.10, we obtain a sequence of quasiconformal p ÑC p such that the following diagram commutes: homeomorphisms hk : C p C f

p C

hk

hk´1

p / C p / C

f

which is what we used to conclude that Fτ `2π i is conformally conjugate to Fτ . We therefore get a family of cubic polynomials gλ “ Feτ parametrized by the punctured unit disc Dzt0u. We normalize Qτ as in Section 6.1.3 by the condition

206

Soft surgeries Qτ p0q “ 0,

Qτ pzq “ z ` Op1q as z Ñ 8.

Then, gλ takes the form gλ pzq “ ρz ` aλ z2 ` z3

with ρ “ e2π i θ ,

for some complex number aλ depending holomorphically on λ P Dzt0u. The . conformal conjugacy class of gλ is uniquely determined by bλ “ aλ2? Recall that C1 is the set of parameters b P C such that z ÞÑ ρz ˘ bz2 ` z3 has a critical value contained in the Siegel disc ˘?b . We shall conclude this section by proving the following result. Proposition 6.11 (The capture component is simply connected) For each Bryuno number θ , the set C1 is a simply connected domain of C containing 4ρ. Proof As mentioned in Section 6.1.1, 4ρ is contained in the interior of C1 . Let b be a parameter in C1 zt4ρu and let a be a square root of b. Let v be the critical value of fa contained in the Siegel disc a and let H Ă a be the rotation ring of fa that separates the orbit of the critical value v from the boundary of the Siegel disc. Let m be the modulus of the rotation ring. The previous surgery provides a holomorphic family of cubic polynomials gλ pzq “ ρz ` aλ z2 ` z3 , parametrized by the unit disc, with ge´2π m “ fa . The image of the map B : λ ÞÑ aλ2 P C defined for λ P Dzt0u is open and contained in C1 . This shows that C1 is open. The critical value Qτ pvq of gλ and the fixed point 0 are both contained in the bounded component of CzH . The modulus of the rotation ring Qτ pH q 1 log |λ|, which tends to 8 as λ tends to 0. It follows that as λ tends to is ´ 2π 0, Qτ pvq tends to 0. There is a unique parameter b P C such that f˘?b has a critical value at 0: the parameter b “ 4ρ. It follows that aλ2 tends to 4ρ as λ tends to 0. In particular, each connected component of C1 contains 4ρ, and so C1 is connected. At the same time, we see that the map B has a removable singularity at 0 with b0 “ 4ρ. We now prove that B : D Ñ C1 is injective. Suppose bλ “ bλ1 . Clearly, bλ “ 4ρ if and only if λ “ 0, so we can suppose λ and λ1 both differ from 1 0. The quasiconformal map Q´ τ ˝ Qτ 1 commutes with f . Since one critical 1 point is captured by the Siegel disc , and the other is not, Q´ τ ˝ Qτ 1 fixes each critical point, and thus each point in the postcritical set. In particular,

6.2 Branner–Hubbard motion

207

1 Qτ pvq “ Qτ 1 pvq, and Qτ ˝ Q´ τ 1 coincides with the identity on B. Consider the holomorphic map c : A|eτ 1 | Ñ A|eτ | given by the composition 1 ´1 ˝ φτ´1 1 . c :“ φτ ˝ ϕ ˝ Q´ τ ˝ Qτ 1 ˝ ϕ 1 This map extends to the unit circle as the identity, because Q´ τ ˝ Qτ 1 is the ´1 ´ 1 identity on the unit circle, and both ϕ ˝ ϕ and φτ ˝ φτ 1 extend to the unit circle as the identity. Hence c is the identity. Using that Qτ pvq “ Qτ 1 pvq, we get cpφτ ˝ ϕpvqq “ φτ 1 ˝ ϕpvq. Therefore φτ pϕpvqq “ φτ 1 pϕpvqq, and it follows that τ “ τ 1 ` 2ki π for some integer k. In particular, λ “ λ1 . We finally prove that B is surjective. Indeed, if b1 belongs to C1 zt4ρu, the surgery provides an injective map B1 : D Ñ C1 . The map B ´1 ˝ B1 : D Ñ D 1 log |λ| is the modulus of the rotation ring sends 0 to 0. In addition, since ´ 2π corresponding to Bpλq and to B1 pλq, we have that B ´1 ˝ B1 preserves the modulus of λ. According to the Schwarz Lemma, B ´1 ˝ B1 is a rotation. In particular, b1 , which belongs to the image of B1 by construction, also belongs to the image of B.

6.2 Branner–Hubbard motion In this section we construct, by soft surgery, special holomorphic motions, called wring-operations, which are also group actions. They are used to obtain families of holomorphic maps by varying the complex structure in a systematic way in the dynamical space. For polynomials of a fixed degree d ě 2 with at least one critical point escaping to 8, or for mappings with an attracting (not superattracting) cycle, the complex structure is varied in the basin of attraction of 8 and the attracting cycle respectively. In this section we shall treat the first case and restrict to cubic polynomials. The wring-operation was originally introduced in [BH, Chapt. II] in order to deal with polynomials of this kind. Without referring to the wring-operation, a model for changing the multiplier in the attracting linear case is described in Section 1.5, and also applied in Sections 4.1 and 6.1. The wring-operation – also called the Branner–Hubbard motion – has been treated thoroughly in [PeT] in a more general form. We begin the section by the definition of a holomorphic motion respecting a group structure, then we introduce the group underlying the Branner– Hubbard motion. We end the introduction by presenting the model for changing the complex structure in the announced systematic way; in general by wringing, in particular by stretching or turning when restricting to certain subgroups.

208

Soft surgeries 6.2.1 Holomorphic motion respecting a group structure

Recall the definition of holomorphic motion (Definition 3.54). In this section the holomorphic motions will be over a parameter space which is a group Definition 6.12 (Holomorphic motion respecting a group structure) Let X, be two complex analytic manifolds and A a subset of X. Suppose p, ˚q is a group and choose the basepoint λ0 P as the identity element. Let H : ˆ A Ñ X be a holomorphic motion of A over . We say that H is a holomorphic motion respecting the group structure if H satisfies the requirements (1)–(3) for holomorphic motion as well as (4): (1) (2) (3) (4)

For any fixed z P A the mapping λ ÞÑ H pλ, zq is holomorphic. For any fixed λ P the mapping z ÞÑ Hλ pzq :“ H pλ, zq is injective. The mapping Hλ0 “ IdA . H pλ1 , H pλ, zqq “ H pλ1 ˚ λ, zq for all z P A and all λ1 , λ P .

Below we define a group structure on the right half plane Hr as well as the wring-operation, which introduces holomorphic motions on C respecting the group structure on the right half plane Hr . 6.2.2 Group structure on the right half plane Let Hr denote the right half plane with elements ω “ s ` i t, where s ą 0 and t P R. The map „ j s0 ω “ s ` i t P Hr ÞÑ t 1 maps the right half plane bijectively onto the subgroup of the general linear group GL2 pRq of 2 ˆ 2 matrices of the given form under matrix multiplication. The right half plane inherits via the bijection, the group-multiplication: ω ˚ ω1 “ ps ` i tq ˚ ps 1 ` i t 1 q “ ss 1 ` i pts 1 ` t 1 q. We refer to this group as pG, ˚q. The group is non-commutative but has two commutative subgroups that generate pG, ˚q: pS, ˚q :“ pR` , ˚q

and

pT , ˚q :“ p1 ` i R, ˚q.

6.2.3 Wringing, stretching and turning the complex structure Let ω “ s ` i t P Hr and define rω : C Ñ C as the R-linear map rω pzq :“ 12 pω ` 1qz ` 12 pω ´ 1qz .

(6.2)


209

Then rω is the unique R-linear map that maps the ordered triple pi , 0, 1q onto the ordered triple pi , 0, ωq. It equals the identity on the imaginary axis i R and preserves the right half plane Hr as well as the left half plane H . Note that rω pz ` 2π i q “ rω pzq ` 2π i , and that rω1 ˝ rω “ rω1 ˚ω , so that the mapping G ˆ C Ñ C defined by pω, zq ÞÑ rω pzq is a group action of pG, ˚q on the left. It follows from (6.2) that rω is a quasiconformal homeomorphism with constant Beltrami coefficient ω´1 r ω pzq :“ Bz rω {Bz rω pzq “ μ . ω`1 Furthermore, define ω : C Ñ C as ω pzq :“ z ¨ |z|ω´1 ,

(6.3)

so that the following diagram commutes: r

C ÝÝÝωÝÑ § § expđ

C § §exp đ

C˚ ÝÝÝÝÑ C˚ ω

where C˚ “ Czt0u. Then ω is equal to the identity on S1 and preserves the unit disc D as well as the set outside of the unit circle CzD. Moreover, it follows that ω1 ˝ ω “ ω1 ˚ω , hence that G ˆ C Ñ C defined by pω, zq ÞÑ ω pzq is a group action of pG, ˚q on the left. The Beltrami coefficient of ω for z P C˚ is μω pzq :“ Bz ω {Bz ω pzq “

ω´1z , ω`1z

see also (1.22) in Section 1.5. It follows that ω : C˚ Ñ C˚ is a quasiconformal diffeomorphism, and that ω ÞÑ ω pzq

and

ω ÞÑ μω pzq

depend holomorphically on ω P G. (For fixed z ‰ 0 the Beltrami coefficient μω pzq is a Möbius transformation in ω.) Since r1 and 1 equal IdC we conclude that r : G ˆ C Ñ C, pω, zq ÞÑ rω pzq

and

: G ˆ C Ñ C, pω, zq ÞÑ ω pzq are holomorphic motions of C over G, respecting the group structure.

210

Soft surgeries

We say that the complex structure is: • wringed if ω P G “ Hr is arbitrary; • stretched if ω “ s P S “ R` ; • turned if ω “ 1 ` i t P T “ 1 ` i R. Note that ω is mapping the circle |z| “ r onto the circle |z| “ r s , where s “ Re ω, and the radial line z “ rei θ onto the logarithmic spiral z “ r s ei pθ `t q , where r ą 0 and ω “ s ` i t. For a pure stretching any radial line is mapped onto itself. For a pure turning any circle is mapped onto itself. The Beltrami coefficient μω has constant dilatation equal to Kpω pzqq :“ 1`|μω | since |μω | “ |μω pzq| is independent of z. 1´|μω | The ellipse field corresponding to the almost complex structure determined the argument by the Beltrami coefficient μω satisfies that along any radial line ´ ¯ 1 of the minor axis to the radial line is equal to the constant 12 arg ωω´ `1 . This is similar to the ellipse field shown in Figure 1.18 for D˚ (which is easily extended to C˚ ). Since rω : C Ñ C is R-linear it conjugates z ÞÑ kz to itself for any k P R, and hence ω conjugates z ÞÑ zk to itself for any k P R. In particular, we have the following property in any degree d P N. Lemma 6.13 For any ω P G the homeomorphism ω : C Ñ C conjugates z ÞÑ zd to itself. Remark 6.14 The homeomorphisms ω are, up to multiplication by ‘trivial’ maps, the only ones that conjugate z ÞÑ zd to itself. More precisely, suppose : C Ñ C is such a homeomorphism. Then there exists a ω P G and continuous functions ρ ÞÑ αpρq and ρ ÞÑ βpρq with ρ, αpρq, βpρq non-negative and satisfying αpd ¨ ρq “ d ¨ αpρq and βpd ¨ ρq “ d ¨ βpρq so that peρ `2π i θ q “ ω peρ `2π i θ qepα pρ q`2π i β pρ qq . See Exercise 6.2.1. When restricting to the family tω uωPG we have chosen αpρq ” 0 ” βpρq. Lemma 6.15 Let ω P G be given. (1) The R-linear map rω : C Ñ C conjugates the translation z ÞÑ z ` τ with τ P C to the translation z ÞÑ z ` rω pτ q. (2) The homeomorphism ω : C Ñ C conjugates the linear map z ÞÑ λz with λ P C˚ to the linear map z ÞÑ ω pλqz. The proof is straightforward. Note that the model can be used to change the multiplier λ P D˚ of the fixed point at 0 of the linear map z ÞÑ λz to


211

any multiplier ω pλq in D˚ by varying ω P G, similar to how it is done in Section 1.5. 6.2.4 The wring-operation on the family of monic centred marked cubic polynomials In [BH, Sec. 8] the wring-operation is defined on any family of monic centred polynomials of degree d ě 2. We restrict to the case d “ 3. Let P3 denote the family of monic centred cubic polynomials with marked critical points, parametrized by pa, bq P C2 : Ppa,bq pzq “ z3 ´ 3a 2 z ` b, and let Kpa,bq denote the filled Julia set of Ppa,bq . The polynomial has critical points at ˘a. Set Pp0,0q “ P0 . The parameter space C2 splits into the connectedness locus C3 , consisting of parameters for which the corresponding polynomial has a connected filled Julia set, and its complement, the escape locus E3 , consisting of parameters for which at least one critical point escape to infinity, i.e. C3 “ tpa, bq P C2 | ˘ a P Kpa,bq u and E3 “ tpa, bq P C2 | t˘au X pCzKpa,bq q ‰ Hu . Let gpa,bq : C Ñ r0, 8q denote the Green’s function associated to the filled Julia set Kpa,bq of Ppa,bq . It is harmonic on CzKpa,bq and identically zero on Kpa,bq (see Section 3.3.4). Set Gpa, bq “ max gpa,bq p˘aq, i.e. the potential of the fastest escaping critical point, and set rpa,bq “ eGpa,bq . The Böttcher coordinates ϕpa,bq are defined on the neighbourhood Npa,bq of 8 consisting of points of potential larger than Gpa, bq, making the following diagram commutative: Npa,bq § ϕpa,bq § đ

Ppa,bq

ÝÝÝÝÑ

Npa,bq § §ϕpa,bq đ

CzDrpa,bq ÝÝÝÝÑ CzDrpa,bq P0

Let μω,pa,bq denote the Ppa,bq -invariant Beltrami coefficient satisfying μω,pa,bq :“

# ϕp˚a,bq μω μ0

on Npa,bq , on Kpa,bq ,

212

Soft surgeries

and spread to the remaining part of C by the dynamics of Ppa,bq . The Beltrami Ť n p˘aq with bounded dilatation coefficient is well defined on Cz ně0 Ppá,b q Kpμω,pa,bq q “ Kpμω q, since it is constructed through successive pullbacks by holomorphic maps. Note that ω ÞÑ μω,pa,bq pzq depends holomorphically on ω for each z P C. Proposition 6.16 (Integration and normalization) For any pa, bq P C2 and any ω P G there exists a unique integrating map p μω,pa,bq q Ñ pC, p μ0 q, φω,pa,bq : pC, normalized so that: (i) φω,pa,bq p8q “ 8; ´1 P P3 ; (ii) φω,pa,bq ˝ Ppa,bq ˝ φω, pa,bq ´1 is tangent to the identity at infinity. (iii) ω ˝ ϕpa,bq ˝ φω, pa,bq

The map ω ÞÑ φω,pa,bq pzq depends holomorphically on ω for any fixed z P C and pa, bq P C2 . Remark 6.17 (Tangency at infinity) Compare with Remark 4.4 where an integrating map is conformal in a neighbourhood of 8. In that case, we may normalize the integrating map to be tangent to the identity at 8. In the case we consider here the integrating map is quasiconformal in a neighbourhood of 8 so the same requirement would not work. However, for any integrating map the composition in (iii) is conformal in a neighbourhood of 8, hence it makes sense to require this composition to be tangent to the identity at 8. Proof An integrating map is determined up to post-composition by a Möbius transformation. Let φr “ φrω,pa,bq denote an integrating map fixing 8, and φr ˝ Ppa,bq ˝ φr´1 the resulting cubic polynomial. By post-composing φr with an affine map z ÞÑ αz ` β we can obtain a monic centred cubic polynomial. Note that α in such affine maps is determined up to multiplication by a pd ´ 1q-root of unity, hence for d “ 3 multiplication by ˘1, while β is well defined for each choice of α. Abusing notation let φr denote the new integrating map so that φr ˝ Ppa,bq ˝ ´ r φ 1 is a monic centred cubic polynomial. Consider the following commutative diagram:


φr˝Ppa,bq ˝φr´1

r pa,bq q, μ0 q pφpN O φr Ppa,bq

ϕpa,bq

pCzDrpa,bq , μω q ω

s pCzDrpa,bq , μ0 q

/ pNpa,bq , μω,pa,bq q ϕpa,bq

P0

/ pφpN r pa,bq q, μ0 q O φr

pNpa,bq , μω,pa,bq q ϕr

213

ϕr

/ pCzDr , μω q pa,bq ω

P0

~ / pCzDr s , μ0 q pa,bq

The composition ϕr :“ ω ˝ ϕpa,bq ˝ φr´1 conjugates the monic polynomial φr ˝ r pa,bq q. The conjugating map is Ppa,bq ˝ φr´1 to P0 in the neighbourhood φpN therefore tangent to multiplication by δ “ ˘1 at infinity. Set φω,pa,bq pzq :“ r φpδzq. Then φω,pa,bq is the unique integrating map satisfying the requirements in Proposition 6.16. Since ω ÞÑ μω,pa,bq pzq depends holomorphically on ω for any fixed z P C and pa, bq P C2 , it follows from the Integrability Theorem 1.30 with dependence on parameters that ω ÞÑ φω,pa,bq pzq also depends holomorphically on ω for any fixed z P C and pa, bq P C2 . We define the wring-operation W : G ˆ C2 Ñ C2 by Wpω, pa, bqq “ paω , bω q, ´1 where Ppaω ,bω q :“ φω,pa,bq ˝ Ppa,bq ˝ φω, . pa,bq Note that the Böttcher coordinate of Ppaω ,bω q equals ´1 , ϕpaω ,bω q :“ ω ˝ ϕpa,bq ˝ φω, pa,bq s and is mapping φω,pa,bq pNpa,bq q onto CzDrpa,bq , where s “ Re ω. The wring-operation W can be applied to any polynomial Ppa,bq P P3 . For any choice of pa, bq P C2 , the map

pÑC p defined by Hpa,bq pω, zq :“ φω,pa,bq pzq Hpa,bq : G ˆ C p over G, respecting the group structure. is a holomorphic motion of C

214

Soft surgeries

Moreover, it follows from Theorem 6.18, Corollary 6.19 and Remark 6.20 below that H : G ˆ E3 Ñ E3 defined by pω, pa, bqq ÞÑ Wpω, pa, bqq “ paω , bω q is locally injective but in general not globally injective. Hence, the wringoperation on E3 over G is almost a holomorphic motion, respecting the group structure; almost in the sense that we have local but not global injectivity. If we restrict to the subgroup S of real elements of G we do have injectivity. Theorem 6.18 (Properties of the wring) The wring-operation W : G ˆ C2 Ñ C2 has the following properties: (1) the map W is a group action, i.e. Wpω1 , pWpω, pa, bqqq “ Wpω1 ˚ ω, pa, bqq; (2) the map ω ÞÑ paω , bω q is holomorphic in ω for each fixed pa, bq; (3) if pa, bq P C3 then paω , bω q “ pa, bq for all ω P G; (4) if pa, bq P E3 then ω ÞÑ paω , bω q is locally injective. Proof (1) For any pa, bq P C2 and ω P G the quasiregular map Ppa,bq : pC, μω,pa,bq q Ñ pC, μω,pa,bq q with the Ppa,bq -invariant Beltrami coefficient μω,pa,bq is the result of the wring-operation by ω before integrating. The ‘Böttcher coordinate’ of the quasipolynomial Ppa,bq is then s ω ˝ ϕpa,bq : pNpa,bq q, μω,pa,bq q Ñ pCzDrpa,bq , μ0 q.

Composing with the wring-operation by ω1 and applying ω1 ˝ ω “ ω1 ˚ω we obtain ω1 ˝ ω ˝ ϕpa,bq “ ω1 ˚ω ˝ ϕpa,bq : pNpa,bq q, μω1 ˚ω,pa,bq q Ñ pCzDr s 1 s , μ0 q pa,bq

and the result follows. (2) For any pa, bq P C2 and z P C the map ω ÞÑ φω,pa,bq pzq is holomorphic in ω. The critical points and critical values of Ppa,bq are mapped onto the critical points and critical values of Ppaω ,bω q respectively. We find that φω,pa,bq paq “ aω varies holomorphically with ω. If a “ 0, then aω “ 0 and φω,p0,bq pbq “ bω varies holomorphically with ω. If a ‰ á


215

then φω,pa,bq pPpa,bq paqq “ bω ´ 2aω3 varies holomorphically with ω and so does bω . (3) Choose pa, bq P C3 . For any ω P G the quasiconformal mapping φω,pa,bq conjugates Ppa,bq to Ppaω ,bω q and Bφω,pa,bq “ 0 a.e. on Kpa,bq . The polynomials are therefore hybrid equivalent. Since the filled Julia set Kpa,bq is connected they are uniquely determined up to affine conjugation. The polynomial Ppa,bq is affine conjugate to Pp˘a,˘bq . Using that φ1,pa,bq “ IdCp we obtain paω , bω q ” pa, bq for all ω P G. (4) Since the map pω, pa, bqq ÞÑ paω , bω q respects the group action is suffices to prove local injectivity around pa, bq. Note that if paω , bω q “ pa, bq then the critical values must coincide, i.e. bω ˘ 2aω “ b ˘ 2a. We obtain the local injectivity by applying the Böttcher coordinate to the critical value of the fastest escaping critical point. Choose pa, bq P E3 and z P Npa,bq . If ϕpa,bq pzq “ eρ ì θ , then ϕpaω ,bω q pφω,pa,bq pzqq “ ω peρ ì θ q “ esρ ì ptρ `θ q where ω “ s ` i t. It follows that gpaω ,bω q pφω,pa,bq pzqq “ sgpa,bq pzq for all z P C. Applying this to the fastest escaping critical value of P pa, bq, say Ppa,bq paq, with Böttcher coordinate equal to ϕpa,bq pPpa,bq paqq “ rp3a,bq e2π i α , we obtain that Ppaω ,bω q paω q is the fastest escaping critical value of Ppaω .bω q with Böttcher coordinate 3ps ì t q 2π i α

ϕpaω ,bω q pPpaω ,bω q paω qq “ rpa,bq

e

.

If bω ´ 2aω “ b ´ 2a then s ` 2π i 3t P 1 ` 2π i Z. It follows that the wring-operation is locally injective. Corollary 6.19 (Stretching rays and turning curves in E3 ) If pa, bq P E3 then the action of the subgroups S and T on pa, bq defines the stretching ray and the turning curve through pa, bq respectively. The action s ÞÑ pas , bs q is injective on S “ R` . The action of 1 ` i t ÞÑ pa1ì t , b1ì t q may be injective on 1 ` i R or periodic depending on pa, bq. Proof Assume that a is the fastest escaping critical point of Ppa,bq . The function s ÞÑ sgpa,bq paq maps R` bijectively onto R` showing the global

216

Soft surgeries

injectivity of stretching. Examples of turning curves that are globally injective and others that are periodic are given in [Bra2]. Remark 6.20 (Stretching rays and turning curves for quadratic polynomials) If we apply the stretching and turning to c P CzM, the complement of the Mandelbrot set, then the stretching ray through c equals the external ray through c and the turning curve through c equals the equipotential through c. In this case the turning-operation is periodic. For completeness we quote some of the results from [BH] about the topological structure of the parameter space of the cubic escape locus E3 . See [BH, Thm. 11.1]. Theorem 6.21 (Foliation of the escape locus) The mapping G : E3 Ñ R` is a trivial fibration with fibre Sr “ tpa, bq P C2 | Gpa, bq “ log ru for any r ą 1 homeomorphic to S3 . The proof uses that a stretching ray has one point in common with each Sr , and that Sr is homeomorphic to S3 for r sufficiently large. It suffices to understand the structure of Sr for a fixed r in order to understand the topological structure of E3 . The parametrization Ppa,bq is chosen so that the critical points are marked as ˘a. Each Sr splits into the disjoint sets Sr` “ tpa, bq P Sr | gpàq “ log r ą gpáqu, Sr´ “ tpa, bq P Sr | gpáq “ log r ą gpàqu, and their common boundary BSr` “ BSr´ “ tpa, bq P Sr | gpàq “ log r “ gpáqu. The critical value P pàq has three preimages when counted with multiplicity, the critical point à counts for two and its co-critical point ´2a for one. Analogously, the preimages of the critical value P páq are the critical pointá and its co-critical point `2a. Let ψr˘ : Sr˘ Ñ S1 denote the map that determines the argument of the cocritical point of the fastest escaping critical point, i.e. ψr˘ pa, bq :“

ϕpa,bq p¯2aqq |ϕpa,bq p¯2aqq|

.

The next result follows with some modification from [BH, Prop. 11.3 and Sec. 12]. Proposition 6.22 (Fibres are discs) The maps ψr˘ are locally trivial topological fibrations with fibres homeomorphic to D.


217

That the fibrations are locally trivial follows from the fact that a turning curve locally has one point in common with each fibre. It is a deeper result to prove that fibres are homeomorphic to discs.

Exercises Section 6.2 6.2.1 Write z P C˚ as epρ `2π i θ q , where ρ ą 0 is uniquely determined but θ is determined only up to addition by an integer. Suppose peρ `2π i θ q “ eR pρ,θ q`2π i pθ `T pρ,θ qq is a homeomorphism that conjugates z ÞÑ zd to itself. Show that the functions R and T cannot depend on θ so that must be of the form stated in Remark 6.14.

7 Cut and paste surgeries

In this chapter we treat several different applications of cut and paste surgery. The general principle is to paste together dynamics of different maps, under the condition that they agree on the boundaries of their distinct domains of definition. Section 7.1 explains one of the earliest uses of quasiconformal mappings in holomorphic dynamics, due to Douady and Hubbard. It is especially relevant as a building block in many other applications. This celebrated result justifies why polynomial Julia sets appear in the dynamical plane of many non-polynomial families. The parameter version of the Straightening Theorem explains why one can also find copies of the Mandelbrot set in parameter spaces other than that of the quadratic family. Section 7.2 groups several classical results about Siegel discs under one common surgery, due to Ghys. The construction consists in pasting a rigid irrational rotation on a topological disc bounded by an invariant quasicircle: the gluing curve. With this construction one shows, for example, that a Siegel disc of a quadratic polynomial with rotation number of bounded type is a Jordan domain containing the critical point on the boundary. Section 7.3 is dedicated to a construction due to Shishikura, where two rational maps with Siegel discs of a given rotation number are pasted together along an invariant curve, resulting in a map with a Herman ring of the same rotation number. This construction explains, for example, the possible classes of irrational numbers that can be realized as rotation numbers of actual Herman rings. Two cut and paste surgeries due to McMullen are explained in Sections 7.4 and 7.5. The first is inspired by the famous theorem of Bers on Kleinian groups, known as the Simultaneous Uniformization Theorem. In the actual construction two different expanding Blaschke products are pasted together 218

7.1 Polynomial-like mappings and the Straightening Theorem

219

along the unit circle to produce a rational map whose Julia set is a quasicircle. In Section 7.5.2, the dynamics of a rational map is replaced in certain domains by dynamics of other maps, under milder conditions than those in Section 7.2. They are pasted along parts of the given Julia set, even when it is not locally connected. Among the many possible applications, this surgery proves that any given hyperbolic rational map is J -conjugate to a postcritically finite rational map. Section 7.6 is contributed by Kevin M. Pilgrim and Tan Lei. Their surgery replaces the dynamics of a given rational map in certain domains of the Fatou set, increasing the connectivity of some Fatou components of the resulting map. Section 7.7 deals with the celebrated surgery by Shishikura, which provides a bound on the number of non-repelling cycles of any given rational map, in terms of the number of critical points. The surgery consists in constructing an appropriate perturbation of the starting map so that all the non-repelling cycles are turned into attracting ones. The last section in this chapter, Section 7.8, a contribution by Shaun Bullett, goes beyond the realm of holomorphic dynamics and enters the world of holomorphic correspondences, a natural generalization. This surgery construction is special in the sense that we are pasting (two copies of) a limit set of a certain Kleinian group with connected ordinary set together with (two copies of) a Julia set of an appropriate quadratic polynomial with a connected filled Julia set.

7.1 Polynomial-like mappings and the Straightening Theorem The notion of polynomial-like mapping was given in the paper ‘On the dynamics of polynomial-like mappings’ by Douady and Hubbard [DH3], motivated by the observation that rational maps may locally behave qualitatively like a polynomial. Figure 7.1 shows an example of a copy of a polynomial Julia set in the dynamical plane of a certain Newton’s method. How polynomial-like mappings (defined below) are related to actual polynomials is explained by a cut and paste surgery which results in the Straightening Theorem (Theorem 7.4). Definition 7.1 (Polynomial-like mapping) Let U and V be bounded, simply connected subsets of the plane, bounded by analytic curves and such that U Ă V Ă C. The triple pf ; U, V q is called a polynomial-like mapping of degree d if f : U Ñ V is holomorphic and proper of degree d.

220

Cut and paste surgeries

Figure 7.1 Left: Dynamical plane of the rational map NP , which is the Newton’s method of a cubic polynomial P . Initial conditions whose orbits do not converge to any attracting fixed point of NP (roots of P ) are shown in black. There is a clear analogy with the filled Julia set K´1 (the basilica) of the polynomial Q´1 pzq “ z2 ´ 1 (right).

We are only interested in the case d ě 2. Note that f : U Ñ V is a branched covering map with d ´ 1 critical points in U when counted with multiplicity. Remark 7.2 The original definition of polynomial-like mappings did not include assumptions on niceness of the boundaries. However, if U 1 and V 1 are bounded, simply connected subsets of the plane, such that U 1 Ă V 1 Ă C and f : U 1 Ñ V 1 is holomorphic and proper of degree d, then one can consider an open simply connected set V with analytic boundary such that V Ă V 1 and U 1 Ă V . Defining U to be the preimage of V in U 1 , it follows that f : U Ñ V is a polynomial-like mapping of degree d. Note that for any polynomial P of degree d we can choose V “ DR “ t|z| ă Ru for R sufficiently large so that pP |U ; U, V q with U “ P ´1 pV q is polynomial-like of degree d. Definition 7.3 (Filled Julia set and Julia set of a polynomial-like mapping) The filled Julia set of a polynomial-like mapping pf ; U, V q is defined as Kf :“

č

f ń pV q,

ną0

i.e. the set of points z P U for which f n pzq P U for all n ě 0, in other words the points in U that never escape. The Julia set Jf is the boundary of Kf .


221

We are ready to state the Straightening Theorem (see [DH3, Chapt. 1, Thm. 1]). Theorem 7.4 (The Straightening Theorem) (1) Every polynomial-like mapping pf ; U, V q of degree d is hybrid equivalent to a polynomial P of degree d. (2) If Kf is connected then P is unique up to affine conjugation. The proof of the first part of the theorem is by surgery and is presented below. Proof p p (1) Choose any ρ ą 1. Let R : CzV Ñ CzD ρ d be a Riemann map, fixing 8. It follows from Theorem 2.9(c) that R extends continuously to the boundaries as an analytic map, say ψ1 : BV Ñ S1ρ d . Choose a lift of this map ψ2 : BU Ñ S1ρ so that ψ1 pf pzqq “ ψ2 pzqd for all z P BU . Let A0 and Aρ,ρ d denote the annuli A0 “ V zU and Aρ,ρ d “ tρ ă |ζ | ă ρ d u. The boundary maps ψ1 and ψ2 are analytic. By Proposition 2.30(b), they extend continuously to a map ψ : A0 Ñ Aρ,ρ d , which is quasiconformal in intpA0 q. We define a new map F : C Ñ C by gluing the map z ÞÑ zd into V zU , via the gluing map ψ and into CzV via R. More precisely, $ ’ if z P U , ’ &f pzq ` ˘ F pzq :“ R´1 ψpzqd if z P V zU , ’ ’ %R´1 `Rpzqd ˘ if z P CzV . The map F : C Ñ C is quasiregular, since it is the composition of quasiconformal and holomorphic mappings. We define a Beltrami coefficient μ on C as follows. On the annulus A0 we set μ “ ψ ˚ μ0 . On CzV , we set μ “ μ0 “ R˚ pμ0 q. Observe that, up to where it is defined, μ is F -invariant (see Figure 7.2). We now spread μ to the domain U by the dynamics of f . To do so, note that A0 is a fundamental domain of F , since orbits pass through A0 at most once. Indeed, all points in A0 are mapped into CzV by F , and then tend to infinity under iteration by F . As a consequence, the sets An “ tz P U | f n pzq P A0 u are disjoint for different values of n. If Kf is connected these sets are

222

Cut and paste surgeries R V

zd

f

ψ U ρ

ρd

A0

Figure 7.2 Sketch of the proof of the Straightening Theorem.

half-closed annuli; if Kf is disconnected they eventually consist of several connected components. Therefore, it makes sense to define $ ˚ ’ if z P A0 , ’ &ψ μ0 pzq n ˚ μpzq :“ pf q μpzq if z P An , ’ ’ %μ pzq elsewhere. 0

By construction μ is F -invariant. The dilatation of μ is equal to the dilatation of ψ ˚ μ0 on A0 , since all other pullbacks are by holomorphic maps. By the Integrability Theorem there exists a quasiconformal map φ : C Ñ C so that φ ˚ μ0 “ μ. Observe that μ “ μ0 on Kf and therefore Bφ “ 0 on Kf . It follows that the map P :“ φ ˝ F ˝ φ ´1 is a holomorphic map of C of degree d and hence a polynomial. Moreover, it follows that φ is a hybrid equivalence between f and P . (2) Suppose f is hybrid equivalent to two polynomials P1 and P2 with a connected Julia set. Then P1 and P2 are hybrid equivalent to each other and hence, by Theorem 3.53, affine conjugate. Remark 7.5 (Applying the First Shishikura Principle) After defining F we could have skipped the rest of the proof had we applied the First Shishikura p and be a gluing map given Principle (Proposition 5.2). Indeed, let U 1 “ CzU p . Then, F is quasiregular on C p and holomorphic by ψ on V zU and R on CzV


223

p 1 , with U 1 invariant under F . Moreover, F “ ´1 ˝ pz ÞÑ zd q ˝ on on CzU 1 U . From the principle we conclude that F is quasiconformally conjugate to a polynomial. From Remark 5.3 we obtain that the conjugacy is conformal on U and hence a hybrid equivalence. Remark 7.6 (Applying Sullivan’s Straightening Theorem) Likewise, we could have concluded that F is quasiconformally conjugate to a polynomial by directly applying Sullivan’s Straightening Theorem (Theorem 5.6). Upgrading to hybrid equivalence is not so obvious in this case. The Straightening Theorem explains why it is common to find copies of polynomial Julia sets in dynamical spaces of general holomorphic maps (see Figure 7.1). In one-dimensional parameter spaces of holomorphic families of maps we may encounter copies of the Mandelbrot set. In order to explain why it is so we introduce the notion of holomorphic families. Definition 7.7 (Holomorphic family of polynomial-like mappings) Let be a complex analytic manifold and F “ tpfλ ; Uλ , Vλ qu a family of polynomial-like mappings. Set V :“ tpλ, zq | z P Vλ u, U :“ tpλ, zq | z P Uλ u, f pλ, zq :“ pλ, fλ pzqq. F is a holomorphic family of polynomial-like mappings if it satisfies the following properties: (1) U and V are homeomorphic over to ˆ D. (2) The projection from the closure of U in V to is proper. (3) The map f : U Ñ V is holomorphic and proper. If these properties are satisfied, the degree of the polynomial-like mappings is constant and is called the degree of F. By the Straightening Theorem, for each λ the map fλ is hybrid equivalent to a polynomial of the degree of F We define the connectedness locus F as CF :“ tλ P | Kfλ is connected u. Restrict to degree two. Let be a Riemann surface isomorphic to D and F “ tpfλ ; Uλ , Vλ qu be a family of quadratic-like mappings. We then denote the connectedness locus by MF . For each λ P MF , the quadratic-like

224


mapping fλ is hybrid equivalent to a unique polynomial of the form Qc pzq “ z2 ` c. Hence the map χ : MF ÝÑ M, λ ÞÝÑ c “ χ pλq is well defined. The family F is called Mandelbrot-like if it satisfies the requirement in the following theorem, see [DH3, Chapt. IV]. Theorem 7.8 (Homeomorphic copies of the Mandelbrot set) Let K P be a closed set of parameters homeomorphic to a closed disc and containing MF . Let ωλ denote the critical point of fλ , and suppose for each λ P zK, that the critical value fλ pωλ q P Vλ zUλ . Furthermore, assume that the vector fλ pωλ q ´ ωλ turns once around 0 as λ turns once around BK. Then, the map χ is a homeomorphism, and it is holomorphic in the interior of MF .

Figure 7.3 Deformed copy of M in the parameter plane of an entire transcendental family of maps.

Remarks 7.9 (1) The assumption ‘fλ pωλ q P Vλ zUλ if λ P zA’ is equivalent to MF being compact. (2) If the winding number of fλ pωλ q ´ ωλ around 0 is δ ą 1, then MF is a branched covering of degree δ.

7.2 Gluing Siegel discs along invariant curves The surgery we present in this section is due to Ghys [Gh] from 1984. Roughly speaking, starting from an analytic circle diffeomorphism with irrational rotation number, Ghys pasted the corresponding rigid rotation into the unit disc.

7.2 Gluing Siegel discs along invariant curves

225

At the end of the construction he obtained a holomorphic map in a neighbourhood of the origin with a Siegel disc, whose boundary is the image of the unit circle under a quasiconformal map. His motivation was to explain that it is a priori possible to have Siegel discs whose boundary is a Jordan curve not containing any critical point of the map, provided a special class of circle maps exists. Such circle maps were proven to exist by Herman in [He2] (see Theorem 3.21). Therefore the foreseen examples by Ghys are actually realized. We shall explain this in Section 7.2.1. In a Bourbaki seminar in 1987, Douady [Do3] explained that the Ghys surgery also applies to certain real analytic circle homeomorphisms with a critical point, provided they are quasisymmetrically conjugate to an irrational rigid rotation. Douady posed the following question: do there exist irrational rotation numbers, such that these analytic critical circle homeomorphisms are quasisymmetrically conjugate to rigid rotations? A few days after the Bourbaki seminar, Herman gave a positive answer to the question. Using inequalities of ´ ¸ tek [Sw1], Herman [He3] obtained the result known today as the Herman– Swia ´Swia¸tek Theorem (Theorem 3.17), which states that any real analytic circle homeomorphism with rotation number of bounded type is quasisymmetrically conjugate to the corresponding rigid rotation. This led to the celebrated result that Siegel discs of quadratic polynomials with a bounded type rotation number have Jordan curve boundaries containing the critical point (see Section 7.2.2). The same surgery was used later by Shishikura [Sh1] to construct Siegel discs starting from Herman rings (see Section 7.2.3). We begin this section by explaining the Ghys construction in a very general form for invariant quasicircles. Then we will show how to use it to prove the different results mentioned above, as well as others. Let us remark that all constructions in this section may also be viewed as special cases of the McMullen surgery described in Section 7.5. However, as presented here, they form a coherent group with applications of a similar nature.

The Ghys surgery The general set up is as follows: p and f : γ Ñ γ an orientation Let γ denote an oriented Jordan curve in C, preserving homeomorphism. Let θ “ rotpf, γ q denote the rotation number of f on γ (see Section 3.2.3). We refer to the interior (respectively the exterior) of γ , denoted by intpγ q (respectively extpγ q), as the connected component of the complement of γ in p to the left (respectively the right) of γ . Set Gγ :“ intpγ q. C

226


Note that the rotation number of f on γ with the reversed orientation is 1 ´ θ . The Ghys surgery shows how it is possible, under certain conditions, to paste the rigid rotation Rθ of rotation number θ into Gγ . Theorem 7.10 (The Ghys surgery: gluing rigid rotations along quasicirp and f : U Ñ V a holomorcles) Let γ denote an oriented Jordan curve in C, phic map, where U and V are open neighbourhoods of γ , so that f |γ : γ Ñ γ is an orientation preserving homeomorphism. Suppose θ “ rotpf, γ q is irrational, and f |γ is quasisymmetrically conjugate to the rigid rotation Rθ on S1 . Then there exists a quasiconformal map φ : Gγ Y U Ñ D and a holomorphic p such that F has a Siegel disc of rotation number θ containing map F : D Ñ C φpGγ q, and F “ φ ˝ f ˝ φ ´1 on DzφpGγ q . Remarks 7.11 (Classification of the resulting map) (a) If f in Theorem 7.10 is globally defined, then so is F . In fact, if f P Rat, then so is F . If f P Ent, then F is in Ent if 8 R Gγ and in Rat if 8 P Gγ . If f P Ent˚ , then F is either of the same kind or in Ent or in Rat depending on the position of the singularities with respect to γ . (b) By reversing the orientation of γ , we can paste the rigid rotation Rp1´θ q p γ. in CzG Proof By hypothesis, there exists an orientation preserving quasisymmetric map ψ : S1 Ñ γ conjugating Rθ to f |γ . It follows from Ahlfors–Beurling or Douady–Earle and Proposition 2.30 that the quasisymmetric map ψ : S1 Ñ γ extends to a map : D Ñ Gγ , which is quasiconformal in D. We use as the gluing map. Define fr : Gγ Y U Ñ Gγ Y V as follows: # p ˝ Rθ ˝ ´1 qpzq if z P Gγ , frpzq :“ f pzq if z P U zGγ . The map fr is continuous, since ´1 |γ “ ψ ´1 and f |γ “ ψ ˝ Rθ ˝ ψ ´1 |γ . It is quasiregular, because it is the result of pasting together a holomorphic and a quasiconformal map along a quasicircle (see Theorem 1.19). A slight modification of the First Shishikura Principle on locally defined maps applies to this setup. However, we choose an explicit approach so as to deduce the properties of the resulting map. We define a Beltrami form μ on Gγ as the pushforward by of μo ” 0 in D. The dilatation of μ is bounded, because is quasiconformal. Then we spread μ by the dynamics of fr to the rest of U . Observe that this is


227

possible since Gγ is invariant under the map fr. For the recursively defined corresponding Beltrami form μ, we have $ ´1 ˚ ’ ’ &p q μ0 on Gγ , μ :“ pfrn q˚ μ on frń pGγ q, whenever defined ’ Ť ń ’ %μ on U z fr pG q. 0

n

γ

The Beltrami form μ has bounded dilatation in U , since all the pullbacks outside Gγ are by the holomorphic map fr “ f . Moreover, μ is fr-invariant by construction. It follows from the Integrability Theorem (Theorem 1.27) that there exists a quasiconformal map φ : Gγ Y U Ñ D such that φ ˚ μ0 “ μ. We normalize φ so that p0q is mapped to 0. Finally, define F :“ φ ˝ fr ˝ φ ´1 : U 1 Ñ φpGγ Y V q. It is holomorphic by Weyl’s Lemma (see Figure 7.4).

Figure 7.4 The new map F on D, showing the Siegel disc in the case when it is bounded by the curve φpγ q.

Note that F is quasiconformally conjugate to f , via φ, on the complement of φpGγ q. On φpGγ q the map ´1 ˝ φ ´1 preserves μ0 and is therefore conformal. By construction, it conjugates F to the rigid rotation Rθ . Hence φpGγ q is contained in a Siegel disc of F . This concludes the proof. Remarks 7.12 (Properties of the resulting map) (a) Note that F is conformally conjugate to f on any domain that never enters Gγ under iteration of f .

228


(b) Further properties of the map F will depend on the original map f and the curve γ . For example, if f has a critical point on γ , then φpGγ q is maximal and equal to a Siegel disc of F . In other cases, the Siegel disc may extend beyond φpGγ q. This is the case, for instance, if γ is an invariant curve inside a Herman ring. 7.2.1 Application to Siegel discs with rotation numbers θ R H Recall from Chapter 3 that if a holomorphic map has a Siegel disc with rotation number in the Herman class H and if the map is defined in a neighbourhood of the Siegel disc, and if the boundary of the Siegel disc is a Jordan curve, then it contains a critical point (see Theorem 3.42). As explained in the introduction to Section 7.2, the proposed application by Ghys, combined with a theorem of Herman, namely Theorem 3.21, gives the following existence theorem (cf. [Gh, He2]]). Theorem 7.13 (Existence of Jordan boundaries without critical points) There exist holomorphic functions with a Siegel disc of rotation number not in H, whose boundary is a Jordan curve, which does not contain a critical point. In particular, there exist such examples within the family of quadratic polynomials Pθ pzq “ e2π iθ z ` z2 and mappings in the standard family Eθ pzq “ e2π i θ zez , for certain θ R H. Proof For certain values of θ we shall prove the existence of Pθ and Eθ satisfying the statement in the theorem. In both cases we apply the Ghys surgery to certain globally defined maps f and obtain as a result a map F within the respective families. To obtain a quadratic polynomial Pθ , we start with the Blaschke products Ba pzq :“ z2

zá 1 ´ az

with a ą 3.

These cubic rational maps are symmetric with respect to the unit circle, have superattracting fixed points at 0 and 8, and two critical points on the real line, not on the unit circle. They have a pole at z “ 1{a P D and a zero at z “ a P CzD. The unit circle is invariant and, by the Argument Principle, the map has degree one on S1 . In fact, Ba |S1 is a diffeomorphism. By Herman’s Theorem (Theorem 3.21), for each a there exists t P r0, 1q á 1 and θ P r0, 1q such that Ba,t :“ e2π i t Ba “ e2π i t z2 1z´ az restricted to S is quasisymmetrically conjugate to the irrational rigid rotation Rθ , but not C 2 conjugate. Let us fix a ą 3 and t “ tpaq, and define B :“ Ba,t . p ÑC p and γ “ S1 . We paste the We apply the Ghys surgery to f “ B : C p ÑC p and rigid rotation Rθ inside D, and obtain a quasiconformal map φ : C


229

a rational map F of degree two, which has a Siegel disc around the origin containing φpDq. In fact, φpDq is the maximal domain of linearization. If this were not the case, there would exist a neighbourhood U of φpS1 q invariant under F . Since φ is a homeomorphism, and a conjugacy outside φpDq, we have that φ ´1 pU q X pCzDq is invariant under B. But since B is symmetric with respect to S1 , it implies that an annular neighbourhood of S1 is invariant under B. Therefore, B|S1 is analytically conjugate to the rigid rotation, contradicting the choice of tpaq. It follows that the Siegel disc of F is φpDq, that the boundary φpS1 q is a Jordan curve, in fact a quasicircle, and that the unique critical point of F in C is φpcq, where c is the critical point of B in CzD. Since c is at a positive distance from S1 , we have that φpcq is at a positive distance from φpS1 q. Finally note that if we normalize the quasiconformal map φ so that φp0q “ 0, φp8q “ 8 and φpcq “ é2π i θ {2, then the rational map F has a superattracting fixed point at infinity with no finite preimages (hence F is a quadratic polynomial), a fixed point at 0 with derivative e2π i θ , and a critical point at é2π i θ {2. It then follows that F “ Pθ . To obtain a map Eθ P Ent as stated in the theorem, we start with maps in Ent˚ in the Arnol’d family: Ha,t pzq :“ e2π i t zea {2pz´1{zq “: e2π i t Ha

with 0 ă a, t ă 1.

The unit circle is invariant under Ha,t and the map is symmetric with respect to S1 . Actually, Ha,t |S1 is the orientation preserving diffeomorphism x ÞÑ x ` t `

a sinp2π xq 2π

pmod 1q.

For any fixed a, t P p0, 1q, the two critical points of Ha,t are both on the real line, bounded away from the unit circle. By Herman’s Theorem, for each a there exists t “ tpaq and θ P p0, 1q such that Ha,t |S1 is quasisymmetrically conjugate to the irrational rigid rotation Rθ but not C 2 conjugate. Let us fix a and t “ tpaq. We apply the Ghys surgery to f “ Ha,t with γ “ S1 , and obtain a quasiconformal map φ : C Ñ C and a map F P Ent. As before, the boundary of the Siegel disc of F is the Jordan curve φpS1 q at a positive distance from the unique critical point of F . If we set φp0q “ 0 and φpcq “ ´1, where c is the critical point of Ha,t outside the unit disc, one can check that F must be of order one since we have not modified the behaviour of Ha,t near infinity, having a fixed point at 0 with derivative e2π i θ , and a critical point at ´1. The origin is also an asymptotic value and has no other finite preimages than itself,

230


since Ha,t has no zeros. One can check that such a map must be of the form Eθ pzq “ e2π i θ zez . 7.2.2 Application to Siegel discs with rotation number of bounded type As explained in the introduction to Section 7.2, Douady observed that the Ghys surgery applies to real analytic circle homeomorphisms with a critical point. ´ ¸ tek. The following theorem uses results of Douady, Ghys, Herman and Swia Theorem 7.14 (Quadratic bounded type Siegel discs) Let Pθ pzq “ λθ z ` z2 , where λθ “ e2π i θ and θ is an irrational number of bounded type. Then Pθ has a Siegel disc of rotation number θ around z “ 0 whose boundary is a quasicircle containing the critical point. See Figure 7.5.

?

Figure 7.5 The Julia set of Pθ pzq “ e2π i θ z ` z2 for θ “ 52´1 “ r1, 1, 1, . . .s, the golden mean, showing the Siegel disc around the fixed point at 0.

Proof To prove the theorem we start with a special model map in the family of cubic Blaschke products in the previous section, the one with a “ 3. The ´3 rational map B3 pzq “ z2 1z´ 3z has superattracting fixed points at 0 and 8, a double critical point at z “ 1, a zero at z “ 3 and a pole at z “ 1{3. The preimage under B3 of the unit circle consists of three loops meeting at the critical point z “ 1: the unit circle, one loop inside D and one loop outside D. It follows that B3 |S1 : S1 Ñ S1 is a homeomorphism (see Figure 7.6). Consider the Blaschke products Bθ pzq :“ e2π i t pθ q z2

z´3 1 ´ 3z

with 0 ă tpθ q ă 1,


231

1:1 2:1 ´3 obey the following: Figure 7.6 The map B3 and the maps Bθ pzq “ e2π i t pθ q 1z´ 3z each point in the unit disc (the white region to the right), has three preimages in the white regions to the left, one outside the unit disc and two inside the unit disc. Similarly, each point outside the unit disc (the shaded region to the right) has three preimages in the shaded regions to the left, one inside the unit disc and two outside the unit disc.

where tpθ q P p0, 1q is the uniquely determined value so that the rotation number of the circle map Bθ |S1 is precisely θ , the given irrational number of bounded type (see Theorem 3.20). Since θ is of bounded type, it follows ´ ¸ tek Theorem (Theorem 3.17) that Bθ is quasisymmetfrom the Herman–Swia rically conjugate to the rigid rotation Rθ . Hence we apply the Ghys surgery p ÑC p and γ “ S1 . We obtain as explained in the proof of to f “ Bθ : C rθ -invariant Beltrami form μθ rθ , a B Theorem 7.10 a quasipolynomial fr “ B p p normalized to fix 0 and 8 and an integrating quasiconformal map φθ : C Ñ C and to map the critical point z “ 1 of Bθ to z “ ´λθ {2, and a holomorphic rθ ˝ φ ´1 which must be a quadratic rational map. Indeed, map Fθ “ φθ ˝ B θ after gluing the Siegel disc inside S1 , each point in the unit disc has only one preimage in D, and each point outside D has no longer a preimage in D. The point at 8 is a superattracting fixed point of Fθ with no preimage in C. This implies that Fθ is a quadratic polynomial with a fixed point at z “ 0 and the critical point at z “ ´λθ {2. Hence Fθ is of the form Fθ pzq “ apλθ z ` z2 q. Since “ φθ pDq is a Siegel disc of rotation number θ around the fixed point 0, the multiplier of the fixed point must be Fθ1 p0q “ λθ , and we conclude that Fθ “ Pθ . The maximal Siegel disc around 0 is equal to , since its boundary B “ φθ pS1 q contains the critical point ´λθ {2. Moreover, the boundary of is a quasicircle, being the image of S1 under the quasiconformal map φθ . Compare Figure 7.5 to the one to the right in Figure 7.7, and note that Ť rń pS1 qq “ JP . φθ p B ně0

θ

θ

Remark 7.15 (Generalization) The generalization of this result to higher degree polynomials has some extra difficulties. Provided one has the

232


Ť Ť rń pS1 q. The right Figure 7.7 Left: The set ně0 Bθń pS1 q. Right: The set ně0 B θ figure is quasiconformally homeomorphic to Figure 7.5, via the integrating map φθ .

appropriate Blaschke products in hand, the surgery can be performed with no problem. But in degrees higher than 2, there are whole slices in parameter space of polynomials with a Siegel disc of a given rotation number. This means that issues of surjectivity of the surgery procedure need to be addressed. Zakeri [Z1] generalized the result to cubic polynomials. Further generalizations are due to Shishikura, Zakeri [Z2] and Zhang [Zh]. As we saw in the section above, the transcendental analogue of the family of quadratic polynomials is the family of semi-standard maps z ÞÑ λzez , λ P C˚ , for which z “ ´1 is the unique critical point, and z “ 0 is both an asymptotic value and a fixed point of multiplier λ. For any Bryuno number θ the map Sθ pzq “ λθ zez , where λθ “ e2π i θ , has a Siegel disc around the origin. Analogously to the case of quadratic polynomials, Geyer [Ge2] proved the following theorem. Theorem 7.16 (Bounded type Siegel discs for λzez ) Let Sθ pzq “ λθ zez , where λθ “ e2π i θ and θ is an irrational number of bounded type. Then Sθ has a Siegel disc of rotation number θ around z “ 0 whose boundary is a quasicircle containing the critical point. See Figure 7.8 and Exercise 7.2.3.

7.2.3 Turning Herman rings into Siegel discs In the examples so far, we have glued a Siegel disc inside the unit circle. However, in general we may use any oriented quasicircle, as long as its dynamics is quasisymmetrically conjugate to an irrational rotation. The most regular case is given by an invariant curve inside a Herman ring of a holomorphic map. Such a curve is always analytic and the map is actually analytically conjugate to a rigid rotation, so there is no restriction on the rotation number.


233

Figure 7.8 The dynamical plane of Sθ pzq “ e2π i θ zez , showing the Siegel disc around the fixed point at 0.

Versions of the following theorem can be found in [Sh1] for rational maps, and in [DF] for transcendental ones. The converse construction is explained in Section 7.3. Theorem 7.17 (Converting Herman rings into Siegel discs) Suppose f P Ent˚ Y Rat has a Herman ring A of rotation number θ P p0, 1q. Choose γ to be any of the invariant curves in A and let Gγ be one of the two components p . Then there exists a quasiconformal homeomorphism φ : C p ÑC p and of Czγ a map F P Ent Y Rat such that F has a Siegel disc of rotation number θ or 1 ´ θ containing φpGγ q, and F is equal to φ ˝ f ˝ φ ´1 on the complement of φpGγ q. It follows from the construction that if f P Ent˚ then F P Ent, and if f P Rat so is F . The theorem is an immediate consequence of the Ghys surgery in Theorem 7.10. The boundary of the resulting Siegel disc of F is equal to the image under φ of the boundary component of A, which is not in Gγ . We remark that the theorem also holds for meromorphic functions. None of the poles or essential singularities inside Gγ will exist for F . Let us revisit the two cases we discussed above. If we consider the Blaschke family Ba,t pzq “ e2π i t z2

zá , 1 ´ az

where a P p3, 8q, t P p0, 1q,

then Ba,t |S1 is a circle diffeomorphism. We can choose a and t so that the rotation number of the circle map is any predetermined irrational value θ

234


(see Theorem 3.20), so that there is a Herman ring around S1 (for example, for any θ P H). We choose γ to be any of the invariant curves in the Herman ring, with the orientation compatible to the ordinary orientation of S1 , and paste a rigid rotation inside γ . As in the constructions above, it is easy to check that the resulting map is the quadratic polynomial Pθ . If γ were given the opposite orientation, and a rigid rotation were pasted inside γ , then the resulting map would be the quadratic polynomial Pp1´θ q . For any pa, tq for which the circle map Ba,t |S1 is analytically linearizable (i.e. there is a Herman ring), with rotation number θ , it follows that Pθ is analytically linearizable. A similar phenomenon holds for the Arnol’d family Ha,t pzq “ e2π i t zea {2pz´1{zq ,

for a, t P p0, 1q,

in relation to the semi-standard map Sθ pzq “ e2π i θ zez (see Exercise 7.2.4 below).

Exercises Section 7.2 7.2.1 Suppose f P Rat has a Siegel disc centred at 0 of rotation number θ . Check that conjugating by 1{z, we obtain a map fr P Rat with a Siegel disc centred at 8 with the same rotation number. However, conjugating by 1{z, we obtain a rational map with a Siegel disc centred at 8 with rotation number 1 ´ θ . 7.2.2 Suppose f P Rat Y Ent Y Ent˚ has a Siegel disc of rotation number θ . Show that there exists fr of the same type as f having a Siegel disc of rotation number 1 ´ θ . Hint: Consider complex conjugation. 7.2.3 Prove Theorem 7.16, applying the Ghys surgery to the auxiliary model family given by the Arnol’d maps Hθ pzq “ e2π i t pθ q ze1{2pz´1{zq

with 0 ă tpθ q ă 1.

Hint: Proceed analogously to the proof of Theorem 7.14. 7.2.4 Prove that if the Arnol’d circle map is analytically linearizable for some pa, tq and has rotation number θ , then Sθ is analytically linearizable. Deduce also that if θ is of bounded type then each connected component of the boundary of the Herman ring is a quasicircle containing a critical point. 7.2.5 Suppose f P Rat Y Mer. For any given Herman ring A of f with closure A in C, let GA denote the bounded component of CzA. Let A1 and A2 be two nested invariant Herman rings of f with closure in C (i.e. A2 Ă GA1 ). Show that there is a pole in GA2 and another one

7.3 Turning Siegel discs into Herman rings

235

in GA1 zpA2 Y GA2 q. Generalize this fact to a larger number of nested rings. Hint: Glue a Siegel disc in GA2 and apply the Maximum Modulus Principle. 7.2.6 Use surgery to prove that if f is in Rat (or in Mer) with a Herman ring A, then there has to be a critical point in each of the two components of the complement of A. More precisely, let G1 and G2 denote the two complementary components and γ1 and γ2 the two components of BA belonging to G1 and G2 respectively. Then, there exists a critical point in Gj whose orbit accumulates in γj , for j “ 1, 2.

7.3 Turning Siegel discs into Herman rings The cut and paste surgery we explain in this section can be thought as the converse to the one in Theorem 7.17 in the previous section. We start with two holomorphic maps f and fr with invariant Siegel discs of opposite rotation numbers and end up with a map F possessing an invariant Herman ring A. The map F combines the dynamics of f and fr, in the sense that F corresponds to f in one of the connected components of the complement of A, and corresponds to fr in the other connected component. This idea is originally due to Shishikura [Sh1]. He started with rational maps with Siegel discs and constructed rational maps with Herman rings. We explain his construction in the simplest case, i.e. when the Siegel discs are invariant. First we address the case of rational maps and then explain how it adapts to more general maps (see [DF]).

7.3.1 The rational case r ˜ respectively. Suppose both Let f and f be rational maps of degree d and d, r of rotation number θ and 1 ´ θ maps have an invariant Siegel disc, and respectively, for some θ P p0, 1q. Furthermore, assume that both Siegel discs are centred at the origin and that neither of them contains 8. Let ϕ : Ñ D r Ñ D be linearizing coordinates conjugating f and fr to Rθ and and ϕr : R1´θ respectively, i.e. f “ ϕ ´1 ˝ Rθ ˝ ϕ on , and fr “ ϕr´1 ˝ R1´θ ˝ ϕr r on . One might view the surgery in an abstract way as choosing some arbitrary invariant curves γ and γr in the Siegel discs, and obtaining a new Riemann sphere by gluing the two unbounded half-spheres along the chosen curves, via a map that conjugates their dynamics (see Figure 7.9).

236


Figure 7.9 The geometric idea of the main construction.

The special role assigned to 0 and 8 is not essential and could be substituted by any pair of distinct points for f as well as for fr. However, this normalization simplifies the explanation of the surgery. The first step is to obtain a gluing map. Let h : γ Ñ γr be a conjugacy r denote the radii of the between f |γ to fr|γr , defined as follows. Let r and r r r circles Sr “ ϕpγ q and Srr “ ϕrpr γ q. Since the inversion Lpzq “ rz conjugates ´ 1 Rθ to R1´θ , the map h :“ ϕr ˝ L ˝ ϕ is an analytic map conjugating f |γ to fr|γr as shown in the following commutative diagram (see also Figure 7.10): f

/ γ

γ ϕ

h

L

Sr Srr O

Rθ

R1´θ

/ Sr / Srr O

ϕr

ϕ

L

h

ϕr

fr

γr

/ γr

The second step in the construction consists in extending h to a globally p Ñ C, p the gluing map. Let Gγ and Gγr denote defined homeomorphism : C the bounded components of the complement of γ and γr respectively. The map is constructed so that it satisfies: (a) |γ “ h; p γr and pCzG p γ q “ Gγr ; (b) pGγ q “ CzG


fr

f

r

h

Gγ

237

Gγr

ϕr

ϕ

R1´θ

Rθ r

0

1

L

r r

0

1

Figure 7.10 The definition of h : γ Ñ γr.

(c) p0q “ 8 and p8q “ 0; (d) is conformal everywhere outside a closed annular neighbourhood Aγ of γ . To construct the we paste four different maps together. Consider an annular neighbourhood Aγ of γ obtained by choosing two f -invariant curves in . Let γ o and γ i denote the outer and inner boundary of Aγ respectively. Note that Aγ splits into two f -invariant subannuli, denoted by Ai and Ao , bounded by γ and respectively γ i and γ o (see Figure 7.11). Likewise, choose γri rγr “ A ri Y A ro so that A rγr is an fr-invariant annular r and define A and γro in neighbourhood of γr .

Figure 7.11 The global definition of the gluing map , as the result of pasting four maps together.

238


Let p γo Ñ G i Ro : CzG γr

and

p γro Ri : Gγ i Ñ CzG

be chosen as Riemann mappings satisfying Ro p8q “ 0 and Ri p0q “ 8. p o on γ o Since the curves are analytic, Ro and Ri extend to analytic maps R i i p and R on γ respectively (see Theorem 2.9). On the closed annulus Aγ we define ri ho : Ao Ñ A

and

ro hi : Ai Ñ A

p o on γ o and as quasiconformal interpolating maps such that ho agrees with R i i i p with h on γ ; and h agrees with h on γ and R on γ (see Proposition 2.30). The map defined as the combination of the four mappings above: $ p γo, ’ Ro on CzG ’ ’ ’ &ho on Ao , :“ i ’ h on Ai , ’ ’ ’ % i R on Gγ i , has all the desired properties (see Figure 7.11). pÑC p The third step in the construction is to define a quasiregular map g : C with the dynamical properties we have announced for F . We shall define g on the dynamical Riemann sphere of f , by modifying the dynamics in Gγ so it reflects the dynamics of fr outside Gγr . Set # p γ, f on CzG g :“ ´1 ˝ fr ˝ on Gγ . First we observe that g is continuous on γ , because |γ “ h, and h is defined so that h´1 ˝ fr ˝ h “ f on γ . It follows that g is quasiregular. Indeed, g is holomorphic outside Gγ and is quasiregular on Gγ . r is g-invariant, Moreover, note that the topological annulus X ´1 pq and that g is conjugate to the rotation Rθ on this annulus. Indeed, g is equal to f on the ‘outer part’ zGγ , and conjugate to fr on the ‘inner part’ Gγ X r of the annulus. This annulus will become the Herman ring of F after ´1 pq straightening g. Finally, observe that the poles of g are the same as the poles of f , together with the former zeros of fr. Likewise, the zeros of g are the zeros of f together with the former poles of fr.


239

We end the proof by applying Sullivan’s Straightening Theorem (Theorem 5.6). To do that we check that the iterates of g are uniformly K-quasiregular for some K ă 8. Observe that g is holomorphic everywhere except on Ai Y pg ´1 pAi q X Gγ q. On Ai we have that g n “ phi q´1 ˝ frn ˝ phi q and hence pK 1 q2 -quasiconformal, where K 1 is the quasiconformal constant of . On pg ´1 pAi q X Gγ qzAi we have that g “ phi q´1 ˝ fr ˝ Ri and hence it is K 1 -quasiregular. After being mapped to Ai orbits remain there. Hence the iterates of g are uniformly K-quasiregular where K “ pK 1 q3 . We conclude pÑC p conjugating g that there exists a quasiconformal homeomorphism φ : C ´ 1 to the rational F :“ φ ˝ g ˝ φ . Observe that the annulus ¯ ´ r A :“ φ X ´1 pq is F -invariant and topologically conjugate to a rotation of angle θ . The boundary of A belongs to the Julia set of F , since it corresponds to boundaries of the Siegel discs of f and fr respectively. Therefore A is maximal and hence a Herman ring. Alternatively to the use of Sullivan’s Straightening Theorem we could have constructed explicitly a g-invariant Beltrami form (see Exercise 7.3.1). 7.3.2 The transcendental case We extend the construction above, to maps that are not necessarily rational (cf. [DF]). More precisely, we consider f, fr P Rat Y Ent Y Mer, not necessarily in the same subclass. As in the section above we assume that f and fr have invariant Siegel discs of opposite rotation numbers centred at the origin. Since the construction is symmetric, and we already considered the case where both maps are rational, we assume that f P Ent Y Mer and allow fr to be in any of the three classes above. We observe that the surgery construction goes through with only minor modifications. Let us check this step by step. The first and second steps, consisting of the definition of the gluing map , are independent of the maps f and fr, and therefore no modification is needed. The third step, the definition of the quasiregular map g, needs to be slightly modified, since f p8q is not defined, and frp8q might not be defined either. p to be Hence, first we set g : C˚ Ñ C # f on CzGγ , g :“ ´1 ˝ fr ˝ on Gγ zt0u. If fr is defined at 8, i.e. if fr P Rat, we extend the definition of g to C by setting gp0q “ ´1 pfrp8qq. The domain of g is therefore C˚ if fr P Ent Y Mer and

240


C if fr P Rat. Poles of f , if any, are still poles of g, and also any former zero of fr, except for z “ 0 itself, is now a pole of g. Notice that z “ 0 might be an essential singularity, hence all former poles of fr or zeros of f are now preessential points for g, i.e. preimages of the essential singularity at the origin. It follows that g is a quasimeromorphic map. After applying Sullivan’s Straightening Theorem we obtain a new map F with at least one essential singularity at infinity. The class to which F belongs depends on both f and fr. In order to classify the different cases, we need to introduce the class of maps with a small set of essential singularities: p p | B a compact countable set and f meromorphicu. ÑC Mer8 :“ tf : CzB Then one can check that we have the following table: f zfr Pol Rat Ent Mer

Pol Rat Rat Mer Mer8

Rat

Ent

Mer

Rat Mer Ent˚ , Mer8 Mer8 Mer8 Mer8

7.3.3 Applications and extensions The first application in the rational case is in Shishikura’s original paper [Sh1]. His construction is more elaborate and applies to cycles of Siegel discs, which are in turn transformed into cycles of non-nested Herman rings. Using this surgery, Shishikura proves the following theorem. Theorem 7.18 (Existence of periodic Herman rings for rational maps) Suppose there exists a rational map with a p-cycle of Siegel discs of rotation number θ . Then, there exists a rational map with a p-cycle of Herman rings of rotation number θ . In particular, for any Bryuno number θ , and any p ě 1, there exists a rational map with a p-cycle of Herman rings of rotation number θ . The simplest version (the invariant case) is enough for many applications. We show one for the transcendental case and leave others as exercises (cf. [DF]). Proposition 7.19 (Existence of Herman rings for meromorphic maps with exactly one pole) Given any Bryuno number θ , there exists F P Mer with exactly one (non-omitted) pole having a Herman ring of rotation number θ .


241

Moreover, F pzq “ az2

ebz , z`1

for some parameters a, b P C˚ . Proof

We start by f pzq :“ e2π i θ zez

and

frpzq :“ e2π i p1´θ q zp1 ` zq,

where θ is an arbitrary Bryuno number. Observe that f belongs to Ent and has a Siegel disc centred at the origin of rotation number θ . Notice that z “ 0 is also an asymptotic value and has no other preimage than itself. The point z “ ´1 is the only critical point of f . The map fr is a quadratic polynomial with a fixed point at z “ 0 of multiplier 2π e i p1´θ q . Since 1 ´ θ P B (see Exercise 3.2.1), fr has a Siegel disc centred at the origin of rotation number 1 ´ θ . The critical point of fr is at z “ ´1{2, and the other zero is at z “ ´1. Gluing the two maps together as in the construction above, we obtain a new map F with a Herman ring of rotation number θ and with: (a) one essential singularity at 8; (b) one simple pole, since f has no poles and fr has one simple zero, different from 0. The pole is not omitted, because z “ ´1 is not omitted by fr. Thus F P Mer with exactly one pole, which is not omitted To find an expression for F , we choose to normalize the integrating map φ to fix 0, 8 and ´1. Then the point z “ ´1 is the non-omitted pole of F . Observe that the origin is a superattracting fixed point, and also an asymptotic value. Hence, factorizing out the simple pole and the double zero at zero, the remaining map is entire without zeros. Hence F must be of the form F pzq “ az2

ek pzq z`1

for some value a P C˚ and some entire map kpzq. Since we have not modified f around infinity, the order of F is one and hence kpzq “ bz for some b P C˚ . We end this section by considering the special case where we reflect with respect to an invariant curve in the Siegel disc. This is used in [Z2, Zh] to study the properties of boundaries of Siegel discs.

242


Theorem 7.20 (Reflection of f with respect to an invariant curve) Suppose f P Rat (resp. f P Ent) has a Siegel disc, centred at the origin, of rotation number θ . Then there exists F P Rat (resp. F P Ent˚ Y Mer8 ) with a Herman ring of rotation number θ such that F is symmetric with respect to the unit circle, which is therefore invariant under F . Moreover, f and F are quasiconformally conjugate on the complement of the unit disc and conformally conjugate in any open set which does not eventually map to the Herman ring. In spirit this is a special case of the Shishikura construction, gluing f to ‘itself’ (with opposite rotation number). This procedure is used to construct ‘models’ (see Exercises 7.3.4 and 7.3.5). We do the construction in the plane of linearizing coordinates, so that the symmetry is clear. Proof We prove the theorem in the rational case. As before let Sr denote the circle of radius r centred at the origin. Let be the Siegel disc of f and let ϕ : Ñ D3 be a linearizing map, which conjugates f to Rθ . Set γ “ ϕ ´1 pS1 q and γ 1 “ ϕ ´1 pS2 q, let Gγ (resp. Gγ 1 ) denote the bounded component of the complement of γ (resp. γ 1 ), and let A be the annulus bounded by γ and γ 1 . The topological part of the surgery consists in defining a quasiregular symmetric map in the linearizing plane. We first construct the gluing map. Choose a Riemann mapping p 2 fixing infinity, which then extends to an analytic map p γ 1 Ñ CzD R : CzG 1 p R on γ . Let h : A Ñ A1,2 be a quasiconformal interpolation between the p γ 1 and ϕ|γ (see Figure 7.12). Define the gluing map boundary mappings R| p p : C Ñ C by R

γ1 γ

ϕ

Gγ

r A 1

A

2

3

h Figure 7.12 The gluing map equals a Riemann map R outside of γ 1 and a quasiconformal interpolation on the annulus A between the linearizing map ϕ|γ on γ p 1 on γ 1 . and R| γ

7.3 Turning Siegel discs into Herman rings $ ’ ’ &R :“ h ’ ’ %ϕ

243

p γ1, on CzG on A, on Gγ 1 .

Observe that is quasiconformal on the annulus A but conformal everywhere else. Moreover, conjugates f to Rθ on γ . p ÑC p in the unit circle, i.e. τ pzq “ 1{z, we By using the reflection τ : C p ÑC p on the linearizing plane by define a map g : C # p on CzD, ˝ f ˝ ´1 g :“ τ ˝ p ˝ f ˝ ´1 q ˝ τ on D. The map is continuous and quasiregular. It is symmetric with respect to S1 by r :“ A1{2,2 . We construction. It is actually holomorphic outside the annulus A leave to the reader to check that the map # on A1,2 , ϕ ˝ ´1 ϕr :“ ´1 τ ˝ ϕ ˝ ˝ τ on A1{2,1 is a quasiconformal conjugacy between g and Rθ on the given annulus. p recursively, by We define a g-invariant Beltrami form μ on C, $ ˚ ’ r on A, ’ &ϕr μ0 r for n ě 1, μ :“ pg n q˚ μ on g ń pAq ’ Ť ’ %μ r on Xz g ń pAq. 0 ně1

r it follows that μ has bounded Since g is holomorphic everywhere except on A, dilatation. It is g-invariant by construction. Since ϕr and g are symmetric with respect to S1 , so is μ (see Exercise 1.2.5 applied recursively). p ÑC p be the integrating map fixing 0, 1 and 8. Because of the Let φ : C symmetry of μ it follows that φ is also symmetric with respect to S1 , i.e. τ ˝ p Ñ C, p defined as φ ˝ τ “ φ (see Exercise 1.4.1). The holomorphic map F : C ´ 1 1 F “ φ ˝ g ˝ φ, is also symmetric with respect S . It is easy to check that ϕ˜ ˝ r The rotation φ ´1 is conformal and conjugates F to Rθ on the annulus φpAq. r must be symmetric with respect to S1 and be a domain for F containing φpAq Herman ring.

Exercises Section 7.3 7.3.1 In the main construction of Section 7.3.1, define a g-invariant Beltrami form whith bounded dilatation. Also express the linearizing map for

244

7.3.2

7.3.3

7.3.4

7.3.5

7.3.6

7.3.7

Cut and paste surgeries the newly obtained Herman ring in terms of the maps involved in the surgery. Given any set of Bryuno numbers tθ1 , . . . , θN u, with N ą 0, show that there exists f P Mer or f P Rat with exactly N poles and N Herman rings that are not nested. Hint: Construct a polynomial with N invariant Siegel discs of rotation numbers θ1 , . . . , θN . Then apply the construction above N times. á z ´b Let Fα,a,b pzq :“ e2π i α z 1z´ az 1´bz . Prove that for any Bryuno number θ , the parameters α, a and b can be adjusted so that F has two Siegel discs and one Herman ring, all of rotation number θ . Hint: Let fθ pzq “ 1 e2π i θ z zz` ´1 and consider fθ and f1´θ . Let Pθ pzq “ e2π i θ z ` z2 with θ a Bryuno number. Let γ be any invariant curve in the Siegel disc of Pθ . Reflect Pθ with respect to γ as in Theorem 7.20, and show that the resulting map belongs to the Blaschke á family Ba,t pzq “ e2π i t 1z´ az ¯ . 2π i θ z Let fθ pzq “ e ze with θ a Bryuno number. Let γ be any invariant curve in the Siegel disc of fθ . Reflect fθ with respect to γ as in Theorem 7.20, and show that the resulting map belongs to the Arnol’d family Ha,t pzq “ e2π i t zea {2pz´1{zq . Let Ba,t be the Blaschke family of rational maps as in Section 7.2.1. Fix a ą 3 and let t “ tpaq be given by Theorem 3.21, so that Ba,t pa q on the unit circle has rotation number, say θ , and is quasisymmetrically conjugate to Rθ but not C 2 -conjugate. Show that there exist a 1 and t 1 , such that Ba 1 ,t 1 has a Herman ring of rotation number θ , with both boundaries being Jordan curves containing no critical point. Hint: First apply the surgery in Section 7.2.1 and then the one in Section 7.2.3. In the original Shishikura construction in Section 7.3.1, we paste together two rational maps f and fr of degrees d and d˜ respectively. Calculate the degree of the resulting map F .

7.4 Simultaneous uniformization of Blaschke products In this section we explain a cut and paste surgery due to Curtis T. McMullen. The construction is inspired by Bers’ famous Simultaneous Uniformization Theorem, which states that for any two Fuchsian groups 1 and 2 uniformizing two compact Riemann surfaces of the same genus there exists a quasiFuchsian group whose limit set pq is a quasicircle, such that the two Riemann surfaces arising from pq{ are conformally equivalent to the

7.4 Simultaneous uniformization of Blaschke products

245

given ones. The quasi-Fuchsian group is a quasiconformal deformation of 1 and 2 (in the respective invariant components of pq). For an extended statement of Bers’ Simultaneous Uniformization Theorem see [Be1] and in particular the exposition in [Be2]. The notion of Klenian group, limit set and ordinary set is introduced in Section 3.5. It is also mentioned that the analogue in holomorphic dynamics to the action of a Fuchsian group is the iteration of a Blaschke product. McMullen’s construction consists of pasting two expanding Blaschke products along the unit circle respecting the dynamics. The result is a rational map whose Julia set is a quasicircle, separating the two completely invariant Fatou components in which the dynamics is conjugate to the original Blaschke products. More precisely, the theorem is as follows. Theorem 7.21 (Simultaneous uniformization of Blaschke products) Let Bj , j “ 1, 2, be two expanding Blaschke products of degree d ą 1. Then there exists a rational map F of degree d whose Julia set is a quasicircle, and such that F is conjugate to Bj on the uniformizations of the two components of p F respectively. CzJ The manuscript [McM] containing the construction is unpublished; we thank McMullen for providing it to us. The key construction relies on a result about quasicircles, which is the content of Theorem 7.22 below. This is known in the literature as the Sewing Theorem (cf. [LV, p. 92]) or conformal welding (cf. [As, Thm. 5.10.1]). The proof we give here is an application of the Integrability Theorem and is interesting in itself. However, it may also be proved by other means (see e.g. [LV, II.7.5]). The setup is as follows. p and let U, V denote the inteLet γ be an oriented Jordan curve in C, rior (respectively exterior) component of the complement of γ . Choose two p They extend continuRiemann mappings RU : U Ñ D and RV : V Ñ CzD. p V : BV Ñ p U : BU Ñ S1 and R ously to homeomorphisms of the boundaries R pV ˝ R p ´1 : S1 Ñ S1 is an orientation preserving S1 . The composition ψ :“ R U homeomorphism that we call a boundary correspondence associated to γ or a conformal welding (see Figure 7.13). A Riemann mapping is determined up to post-composition with a Möbius transformation that preserves the unit disc. It follows that ψ is determined up to equivalence by Möbius transformations that preserve the unit disc. Indeed, p U and R p 1 “ Mo ˝ R p V , where Mi , Mo : D Ñ D (and therep 1 “ Mi ˝ R let R U V fore they also preserve CzD). Then ψ 1 “ Mo ˝ ψ ˝ Mi´1 is also a boundary correspondence associated to γ .

246

Cut and paste surgeries D

pU R γ

ψ U D

V pV R

Figure 7.13 A boundary correspondance ψ or conformal welding.

Theorem 7.22 (Quasicircles and welding) A function ψ : S1 Ñ S1 is quasisymmetric if and only if there exists an oriented quasicircle γ inducing the boundary correspondence ψ. Proof One direction is immediate, for if γ is an oriented quasicircle, then any associated boundary correspondence is quasisymmetric. p : D Ñ D be Now assume ψ : S1 Ñ S1 is a quasisymmetric map, and let ψ a quasiconformal extension of ψ. Set # p ´1 q˚ μ0 on D, pψ μ :“ p μ0 on CzD, pÑC p satisfying φ ˚ μ0 “ and choose an integrating quasiconformal map φ : C 1 μ a.e. Then γ “ φpS q is an oriented quasicircle. Moreover, the composip p : pD, μ0 q Ñ pφpDq, μ0 q is conformal, and so is φ : pCzD, μ0 q Ñ tion φ ˝ ψ ´ 1 ´ p p p pφpCzDq, μ0 q. Set U :“ φpDq and V :“ φpCzDq, RU “ ψ ˝ φ 1 and pV ˝ R p ´1 “ ψ. Hence ψ is a boundary correRV “ φ ´1 . It follows that R U spondence associated to the quasicircle γ “ φpS1 q. Since the integrating map φ is only determined up to post-composition with a Möbius transformation p Ñ C, p so is γ . M:C It follows from the definition of a boundary correspondence associated to an oriented quasicircle and Theorem 7.22 that we have established a bijection between the quotient spaces QS{ „ ÐÑ

QC ` { «,

7.4 Simultaneous uniformization of Blaschke products

247

where QS denotes the set of quasisymmetric maps ψ : S1 Ñ S1 with the equivalence relation „ defined as ψ1 „ ψ2 if and only if there exist Möbius transformations M 1 , M preserving D such that ψ2 “ M 1 ˝ ψ1 ˝ M; and QC ` denote the set of oriented quasicircles with the equivalence relation « defined as γ1 « γ2 if and only if there exists a Möbius transformation M such that γ2 “ Mpγ1 q. Proof of Theorem 7.21 Given two expanding Blaschke products Bj , with j “ 1, 2, of degree d, choose an orientation preserving homeomorphism ψ : S1 Ñ S1 which conjugates B1 to B2 on S1 . There are d ´ 1 such homeomorphisms, corresponding to the number of fixed points of Bj |S1 and all are quasisymmetric. Then by Theorem 7.22 there exists an oriented quasicircle γ inducing the boundary correspondence ψ, with Riemann maps RU and RV , where U, V denote the interior and exterior components of the complement of γ . We glue the dynamics of B1 , B2 into U, V respectively via the extension of p p V : V Ñ CzD. p U : U Ñ D and R That is, we define the conformal mappings R # pU p ´ 1 ˝ B1 ˝ R on U , R U F :“ ´1 p p R ˝ B2 ˝ R V on V , V

which is well defined on γ (i.e. the two definitions agree) as shown in the following commutative diagram: S1

B1

/ S1 ?

_?? ?

p U? R

??

ψ

RV S1

RU p

γ

ψ

???

p V? R ?

p

B2

? / S1

p which is holomorphic Hence the new map is a continuous self-map of C outside γ , and locally quasiconformal on γ . Since γ has zero measure it follows from Weyl’s Lemma that the map is globally holomorphic, hence a rational map F with Julia set JF “ γ . Example 7.23 We construct an explicit quadratic rational map from two expanding Blaschke products of degree two as in Theorem 7.21. Consider the one-parameter family

248


Bλ pzq :“ z

z`λ 1 ` λz

with λ P D,

corresponding to those Blaschke products of degree two which have attracting fixed points at 0 and 8 (of multiplier λ and λ respectively) and the third fixed point at 1 P S1 . For any pair Bλ1 , Bλ2 we obtain a quadratic rational map whose Julia set is a quasicircle and with the dynamics of Bλ1 , Bλ2 glued into the two Fatou components as explained above. If we normalize the integrating map to fix 0, 1 and 8, then the rational map takes the form Fλ1 ,λ2 pzq :“ z

z ` λ1 1 ` λ2 z

.

Applications An appropriate version of Theorem 7.21 depending on parameters could have interesting applications. As an example, let us fix one of the two Blaschke products to be B0 pzq “ zd . By gluing expanding Blaschke products B of degree d (with a marking, namely the quasisymmetric conjugacy to zd on the unit circle) to B0 , we obtain polynomials of degree d belonging to the main hyperbolic component in the parameter space of complex polynomials of degree d, i.e. the polynomials having one attracting fixed point in C whose immediate basin of attraction contains all critical points in the plane of the polynomial. Defining the appropriate space of ‘marked Blaschke products’ and adding the appropriate parameters and normalizations to the construction, this would lead to a parametrization of the main hyperbolic component of the connectedness locus of Pold .

7.5 Gluing along continua in the Julia set The surgery in this section is a cut and paste construction due to Curtis T. McMullen and is contained in his 1988 paper ‘Automorphisms of rational maps’ [McM1]. It is similar in spirit to some of those in the preceding sections, where we replace the dynamics of a map on a given domain, by the dynamics of a different map, under the condition that the two agree along the gluing curve. In the previous constructions the gluing curve was always a quasicircle. In the surgery in this section the agreement condition of the two maps is milder: the maps only need to agree along the ideal boundary of the domain. The boundary is therefore allowed to be a non-locally connected continuum. In the applications it will be a subset of a Julia set. The original motivation for this surgery was to study the finite group Autpf q of Möbius transformations commuting with a given rational map f . In this

7.5 Gluing along continua in the Julia set

249

section we shall not comment on this, but explain the surgery involved which is interesting in itself. Before making the construction precise, we briefly introduce the concept of ideal boundary and some useful related results. Prime ends and the ideal boundary In this section a disc is an open simply p whose complement contains at least two points. If U connected subset of C is a disc, a Riemann map RU maps U conformally onto D. Carathéodory’s Theorem states that RU extends continuously to BU if and only if BU is locally connected. In that case, BU can be parametrized by the unit circle. If BU is not locally connected there is still a relation with the unit circle, via prime ends of U . (For an introduction to prime ends see [Mi2, Sect. 17], [Co, Sect. 14.3] or [Pom, Sect. 2.4].) Definition 7.24 (Ideal boundary of a disc) Let U be a disc. The ideal boundary of U , denoted by I pU q, is the set of prime ends of U . The ideal boundary I pU q is identified with S1 , via a Riemann map RU . This identification is well defined up to post-composition with a Möbius transformation that preserves D. Induced maps on ideal boundaries Case 1 Any proper holomorphic map between two discs U and V , say f : U Ñ V , induces naturally a real analytic map between their respective ideal boundaries. Indeed, if we consider Riemann maps RU : U Ñ D and 1 RV : V Ñ D, the composition fp :“ RV ˝ f ˝ R´ U : D Ñ D is a proper holomorphic map, which extends to a neighbourhood of S1 via the Schwarz Reflection Principle (see Chapter 2, Remark 2.11). Therefore it induces a real analytic map from the unit circle to itself, i.e. from the ideal boundary I pU q to the ideal boundary I pV q. We shall refer to this map as fp : I pU q Ñ I pV q. Case 2 In fact it is enough to have f defined on a one-sided neighbourhood of BU , which is mapped properly onto a one-sided neighbourhood of BV . We can still uniformize U and V as before, and obtain a map fp from a onesided neighbourhood of S1 to another one-sided neighbourhood of S1 . By the Schwarz Reflection Principle, we again obtain a real analytic map from S1 to itself, hence a real analytic map fp : I pU q Ñ I pV q.

250


Remark 7.25 If the map f in any of the two cases above is quasiconformal, the same construction goes through, and f induces a map fp : I pU q Ñ I pV q, which is quasisymmetric (being the extension to the unit circle of a quasiconformal homeomorphism of D (see Theorem 2.9) or between annular subsets of D (see Corollary 2.13)). In both cases the induced map fp is unique up to equivalence by Möbius transformations that preserve D. The following proposition contains a useful relation between the ideal boundary I pU q and the topological boundary BU . For a proof see [McM1, Prop. 5.2]. Proposition 7.26 (Ideal boundary versus topological boundary) Let U be a disc. If φ : U Ñ U is a quasiconformal homeomorphism, which induces the p identity on I pU q, then φ extended by the identity on CzU is a quasiconformal p homeomorphism on C. In particular, it equals the identity on BU . In greater generality, we can define the ideal boundary of an invariant conp is a non-degenerate continuum if C is a compact, tinuum. Recall, that C Ă C connected set that contains more that one point. p is a Definition 7.27 (Ideal boundary of an invariant continuum) If C Ă C p non-degenerate continuum, then every connected component of CzC is a disc. We define the ideal boundary of C, denoted by I pCq, as the union of the ideal boundary of each of the connected components of its complement. In the applications in this section, C will be a forward invariant connected component of the Julia set of a given rational map f . If so, the connected components of the complement of C might be Fatou components, but in general they are unions of Fatou components and other components of the Julia set. Think for instance of the well-known example of a cubic Blaschke product with an invariant Herman ring (see Figure 3.8). If C is the exterior component of the Julia set, i.e. the boundary of the immediate basin of the superattracting fixed point at infinity, then f pCq “ C and only the unbounded component of the complement is a Fatou component. All the bounded components contain infinitely many Fatou components (annuli) together with their boundaries and accumulations of such. It may also happen that the continuum C is disjoint from the boundary of any Fatou component. This is the case in the McMullen examples of a buried quasicircle, which is an invariant component of the Julia set of a rational map in a special family of rational maps (see Figure 7.14). In this case, C is accumulated by infinitely many Fatou components.


251

Figure 7.14 The Julia set of the maps fλ pzq “ z3 ` λ{z3 consists, for λ ‰ 0 and sufficiently small, of a Cantor set of quasicircles. Uncountably many of them are buried, i.e. they do not intersect the boundary of any Fatou component. There exists a unique buried quasicircle Cλ that is invariant under fλ . These examples were discussed for the first time in [McM1]. The Julia set is drawn for λ “ 0.006 ` 0.002i .

Another possibility is shown in Figure 3.7, where taking C “ S1 we see p that both connected components of CzC are infinitely connected Fatou components, with connected components of the Julia set accumulating on C. If f is a polynomial with connected Julia set, then C can only be the whole Julia set. In this case, every component of the complement is a Fatou components of f . If f is a rational map and C is a forward invariant component of the Julia set, then the preimage f ´1 pCq has finitely many components. The induced map fp : I pCq Ñ I pCq is mapping each component of I pCq onto a component of I pCq. On each of them the map fp is defined as described above. Indeed, let p U be a component of CzC. If U contains no components of f ´1 pCq, then f p maps U properly onto a component V of CzC. In this case fp : I pU q Ñ I pV q is defined as in Case 1 above. If, instead, U does contain at least one component of f ´1 pCq, then let U 1 be the component whose closure contains BU . Then f p In particular, maps U 1 properly onto a subset V 1 of a component V of CzC. f maps a one-sided neighbourhood of BU onto a one-sided neighbourhood of BV as a covering so that fp : I pU q Ñ I pV q is defined as in Case 2 above. In the applications we perform a sequence of surgeries. Therefore, it is convenient to work in the wider class of maps, the quasirational maps, to avoid having to integrate at each step. p ÑC p is quasirational if it is quasiconformally Recall that a map f : C conjugate to a rational map. By the Julia set of f we mean the set which under conjugation is mapped to the Julia set of the rational map. Similarly for the Fatou components of f . Recall also that by Sullivan’s Straightening Theorem

252


(Theorem 5.6), being quasirational is equivalent to the iterates f n for n ě 1 being uniformly K-quasiregular for some K ě 1. Under the same assumptions as above, observe that if the original map f is only assumed to be quasirational, it still makes sense to define the induced map fp : I pCq Ñ I pCq. In this case, however, on each component the map is only quasisymmetrically equivalent to a real analytic map (actually quasisymmetrically conjugate, if the component were mapped to itself). More p ÑC p be an integrating quasiconformal homeomorphism, precisely, let φ : C ´ so that g :“ φ ˝ f ˝ φ 1 is a rational map. Then g induces a real analytic map gp : I pφpU qq Ñ I pφpV qq via Riemann mappings Rφ pU q and Rφ pV q . On the other hand, the maps φ : U Ñ φpU q and φ : V Ñ φpV q induce quasisymmetric maps φpU : I pU q Ñ I pφpU qq and φpV : I pV q Ñ I pφpV qq. The induced map fp : I pU q Ñ I pV q can be written as fp :“ φpV´1 ˝ gp ˝ φpU : I pU q Ñ I pV q. Hence, fp is quasisymmetrically equivalent to the real analytic map gp. Notice that if V “ U then φpV “ φpU and therefore fp : I pU q Ñ I pU q is conjugate to gp. 7.5.1 The surgery The surgery consists in replacing a map f on some disc U by a new map, which is quasiconformally conjugate to a Blaschke product. Setup Let f be quasirational and C a non-degenerate invariant continuum. Its preimage f ´1 pCq consists of C and finitely many other disjoint components. p Let U and V be components of CzC so that, as above, fp : I pU q Ñ I pV q denotes the induced map on the ideal boundaries. Moreover, let B denote a Blaschke product, mapping D onto D. Definition 7.28 (Compatibility of maps on the ideal boundary) In the setup above we say that f and B are compatible on the ideal boundary of U via quasiconformal maps φU : U Ñ D and φV : V Ñ D, if the induced maps φpU : I pU q Ñ S1 and φpV : I pV q Ñ S1 define a quasisymmetric equivalence between fp : I pU q Ñ I pV q and the restriction of B to the unit circle, as shown in the following commutative diagram: fp

I pU q ÝÝÝÝÑ I pV q § § §p § φpU đ đφV S1

B

ÝÝÝÝÑ

S1


253

If V “ U (and hence I pU q is fixed), then we are able to choose φU and φV such that φp :“ φpU “ φpV , and then fp and B are quasisymmetrically conjugate under φp : I pU q Ñ S1 (see Figure 7.15). V

U f

RV

RU I pU q

I pV q

φU

fp

φV

φpV

φpU

S1

S1 B

Figure 7.15 f and B are compatible on the ideal boundary of U via quasiconformal maps φU : U Ñ D and φV : V Ñ D.

We are ready to state and prove the key proposition in the surgery, denoted the Gluing Lemma by McMullen. Proposition 7.29 (Gluing Lemma) In the setup above suppose that: • either I pU q is a fixed component of I pCq under the map fp : I pCq Ñ I pCq; • or I pU q is strictly preperiodic under fp, and f n pV q X U “ H for all n ą 0. Assume f and B are compatible on the ideal boundary of U , via quasiconformal maps φU : U Ñ D and φV : V Ñ D, which in the fixed case satisfy φU “ φV :“ φ. Then the map # f pzq if z R U , gpzq :“ ´1 φV ˝ B ˝ φU pzq if z P U , is quasirational. If I pU q is fixed, then B and g are quasiconformally conjugate on U . Proof Consider f ´1 ˝ g locally near BU , using the branch of f ´1 which fixes the points not in U . This map is the identity outside U , and quasiconformal inside U . It also induces the identity map on the ideal boundary since

254


fp “ φpV´1 ˝ B ˝ φpU “ gp : I pU q Ñ I pV q. Using Proposition 7.26 we deduce that f ´1 ˝ g is locally quasiconformal. By composing with f we obtain that g is locally quasiconformal except on neighbourhoods of the branch points. Hence, g is quasiregular. The dilatation of g is bounded by the maximum of Kpf q and KpφU q ¨ KpφV q. If I pU q is a fixed component of fp, then gpU q “ U and g n pU q “ φ ´1 ˝ n B ˝ φpU q. Hence, we have Kpg n q “ Kpφq2 on U . If z R U , it may have sevp eral iterates in CzU before falling into U . Hence, Kpg n q ď Kpf n q ¨ Kpφq2 ď K ¨ Kpφq2 , where we used that f is quasirational so that Kpf n q ď K for all n and some K ě 1. Hence, g is quasirational (see Figure 7.16). g g“f g “ φV´1 B φU

U

φU

V

φV B

Figure 7.16 The Gluing Lemma.

In the case where fppI pU qq “ I pV q and f n pV q X U “ H for all n ą 0, the dilatation is Kpg n q ď K ¨ KpφU q ¨ KpφV q since no orbit can pass through U more than once. Corollary 7.30 (The Gluing Lemma in the periodic case) In the setup p such that above suppose U0 “ Up , U1 , . . . , Up´1 are components of CzC j p f pI pU0 qq “ I pUj q for 0 ď j ď p ´ 1. For j “ 0, 1, . . . , p ´ 1 let Bj : D Ñ D be Blaschke products so that f |Uj and Bj are compatible on the ideal boundary via the quasiconformal homeomorphisms φj : Uj Ñ D, where φp “ φ0 . Then the map gpzq :“

# f pzq φj´`11

is quasirational.

if z R ˝ Bj ˝ φj pzq

Ťp ´ 1 j “0

Uj ,

if z P Uj for j “ 0, 1, . . . , p ´ 1,


255

Proof Applying the same arguments as in the proof of the Gluing Lemma above we conclude that g is quasiregular. To see that Kpg n q is uniformly bounded, observe that g p is quasiconformally conjugate to a Blaschke řp´1 product of degree d on any of the Uj ’s, where d “ j “0 degpBj q (see Figure 7.17).

g g3

g

g“f g

U0

φ0

g

U1

U2

φ2

φ1

B B1

B0

B2

Figure 7.17 The Gluing Lemma in the periodic case for p “ 3. Notice that g 3 |Uj is conjugate to the Blaschke product Bj `2 ˝ Bj `1 ˝ Bj , where indices are taken mod 3.

Remark 7.31 (1) The degree of g may be lower than the degree of f . (2) Observe that, even if f pU q is not equal to V , then after the surgery gpU q “ V . If I pU q is periodic of period p, we obtain g p pU q “ U so that U is a periodic Fatou component of the quasirational map g.

7.5.2 Application: changing any rational map to one with connected Julia set One application of the surgery above is to simplify the dynamics of a given rational map by pasting in rigid models, defined as follows. Definition 7.32 (Rigid models) We define the three rigid models for proper self-maps of the unit disk D as: (a) the elliptic model z ÞÑ e2π i θ z, for θ P RzQ; (b) the hyperbolic model z ÞÑ zd for some d ą 1; and

256


(c) the parabolic model z ÞÑ Pd pzq for d ą 1, where Pd pzq :“

zd ` a d ´1 , for a “ d 1 ` az d `1

is the unique map of degree d with a single critical point at z “ 0 and a parabolic fixed point of multiplicity three at z “ 1. Remark 7.33 (1) If B is any of the above models, and φ is a quasisymmetric map of S1 , which conjugates B to itself on S1 , then φ must be a rigid rotation z ÞÑ αz with |α| “ 1. For the elliptic model, α can be arbitrary; for the hyperbolic model, α must satisfy α d “ 1, and in the parabolic model, α “ 1. This is the reason for calling the models rigid. Note that different types of rigid models are not quasisymmetrically conjugate on S1 . (2) However, the models are rigid only when we require conjugacy. If we consider equivalence, then the hyperbolic and parabolic models are in à fact conformally equivalent on D. Set ϕ1 “ IdD and ϕ2 pzq “ 1z` az , then d ϕ2 pz q “ Pd pϕ1 pzqq. Analogously, we see that the rigid rotation Rθ is conformally equivalent to IdD . The main application is stated in the following theorem ([McM1, Prop. 6.9]). Theorem 7.34 (Main Theorem) Let f be a rational map and C Ă Jf a forward invariant connected component different from a single point. Then there pÑC p exists a rational map g and a quasiconformal homeomorphism φ : C such that φ conjugates g|Jg to f|C . Moreover, g has the following properties. The Julia set Jg is connected and the periodic Fatou components of g are Siegel discs, basins of superattraction or parabolic basins. On each periodic Fatou component, the first return map is conjugate to one of the three rigid models. Moreover, if c is a critical point in a strictly preperiodic Fatou component, and n is the smallest number for which g n pcq is in a periodic component U , then: (a) if U is a Siegel disc, g n pcq is the indifferent periodic point in U , i.e. the centre of the Siegel disc; (b) if U is a basin of superattraction, g n pcq is the superattracting periodic point; (c) if U is a parabolic basin, g n`k pcq is a critical point, where k is the smallest number for which g k pU q contains a critical point.


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Remark 7.35 (1) The theorem above states that any forward invariant connected component of the Julia set of a rational map is conjugate to the Julia set of a different rational map. (2) If f has a connected Julia set to start with, then we are replacing Fatou components by Fatou components with rigid dynamics. If Jf is disconnected, we are erasing all of the Julia set of f , which is not part of C, and p replacing all components of CzC by Fatou components of g. (3) The replacements could be done with Blaschke products other than the rigid models as long as the induced maps on the ideal boundaries coincide. Corollary 7.36 (Special case of hyperbolic rational maps) If f is a hyperbolic rational map with connected Julia set, then there exists, up to Möbius conjugacies, a unique postcritically finite rational map satisfying the conclusions of the Main Theorem. In fact the same is true if f and C are as in the Main Theorem, and f is expanding on C. To prove the Main Theorem we need to understand what kind of maps f induces on the different components of the ideal boundary of C. The results are summarized in Propositions 7.37 and 7.39 below. We postpone the proofs of the propositions to the end of the section and first finish the proof of the Main Theorem. Dynamics induced by f on the ideal boundary I(C) Suppose f is quasirational and C Ă Jf is a forward invariant connected component different from a single point. Then the preimage of C has finitely many components. The two propositions below are in [McM1, Theorem 6.1]. Proposition 7.37 (Global dynamics of fp on I(C)) p (a) All but finitely many components of CzC map to their images by a quasiconformal homeomorphism. (b) Every component of I pCq is preperiodic and there are only finitely many periodic ones. (c) For any component I pU q of I pCq, there exist quasisymmetrical mappings φp1 and φp2 mapping I pU q and fppI pU qq respectively to S1 providing an equivalence between fp and z ÞÑ zd for some d ě 1. If I pU q is fixed, fp and z ÞÑ zd may not be quasisymmetrically conjugate.

258


Remark 7.38 (1) Note that the equivalence in Proposition 7.37 between fp and z ÞÑ zd for some d ě 1 can be changed to an equivalence between fp and any Blaschke product of degree d on S1 . (2) There is also a lot of freedom in the choices of the quasisymmetric maps φpj for j “ 1, 2 providing the equivalence. If one of them, say φp1 , is changed p1 , then there exists a quasisymmetric to an arbitrary quasisymmetric map ψ p1 , ψ p2 of maps provide the equivalence. p2 so that the pair ψ map ψ Proposition 7.39 (Dynamics of fp on periodic components of I(C)) On each periodic component of I pCq, the first return map is quasisymmetrically conjugate to one of the rigid models. We shall apply Propositions 7.37 and 7.39 in the sequence of surgeries which prove the Main Theorem.

Sequence of surgeries and proof of the Main Theorem The proof of the Main Theorem is divided into a sequence of surgeries. Since we work in the class of quasirational maps we may as well start from a quasirational map f and a forward invariant connected component C Ă Jf different from a single point. After each surgery construction the resulting map g is quasirational, the subset p is unchanged, C is a connected component of Jg , and gp “ fp on I pCq. C of C Hence, after each step we are allowed to denote the resulting map again by f . We start by a survey of the different steps in the proof. Afterwards we make the constructions more precise. We first consider the cycles of components of I pCq under fp, one after the other. It follows from Proposition 7.37(b) that there are only finitely many of them. Suppose I pU q is periodic of period p ě 1. Then from Proposition 7.39 we know that fpp is quasisymmetrically conjugate to exactly one of the rigid models. We then change the dynamics on p the p components of CzC whose ideal boundaries form the p-cycle of I pU q p under f , in such a way that the composition is quasisymmetrically conjugate to the correct rigid model. After applying the Gluing Lemma we obtain a new quasirational map. After a finite number of such surgeries we have dealt with all the cycles of components of I pCq. We are left with the strictly preperiodic components of I pCq under fp. We work backwards along successive preimages of the cycles of I pCq. It follows p from Proposition 7.37(a) that there are only finitely many components of CzC on which the dynamics is not a quasiconformal homeomorphism and has to be


259

replaced. We perform the replacements one after the other, each time obtaining a new quasirational map. The changes are chosen such that the conditions in the Main Theorem are satisfied for the rational map at the end. We now go though the proof with more details. The periodic case Suppose I pU q is periodic of period p ě 1. Following Proposition 7.39 let φp : I pU q Ñ S1 be a quasisymmetric map so that φp ˝ fpp ˝ φp´1 equals exactly one of the rigid models, say BU . Then extend φp to p choose a quasiconformal homeomorphism of D, denote the extension by φ, p a Riemann mapping RU : U Ñ D and set φ :“ φ ˝ RU : U Ñ D. It follows that f p and BU are compatible on the ideal boundary via φ. If p “ 1 we are done: replacing f on U by the map φ ´1 ˝ BU ˝ φ and keeping f outside U , it follows from the Gluing Lemma that we obtain a new quasirational map. If p ą 1 then we have to replace f on the disc components Uj where I pUj q :“ fpj pI pU qq for 0 ď j ď p ´ 1 with maps that are compatible with fp along the ideal boundaries and whose composition agrees with φ ´1 ˝ BU ˝ φ on U . Let dj ě 1 denote the degree of fp : I pUj q Ñ I pUj `1 q for 0 ď j ď řp´1 p ´ 1. Then the sum j “0 dj “ d where d “ 1 in the elliptic case. From Proposition 7.37(c) we know that there exist a pair of quasisymmetric maps mapping I pUj q and I pUj `1 q respectively to S1 and providing an equivalence between fp and Bj pzq “ zdj , for 0 ď j ď p ´ 2. From Remark 7.38(2) we know that we may choose one of the quasisymmetric maps arbitrarily. Let φp0 “ φp : I pU0 q Ñ S1 from above. For j “ 1, . . . , p ´ 1 establish recursively φpj : I pUj q Ñ S1 such that φpj ˝ fp “ Bj ´1 ˝ φj ´1 . Depending on the case we choose Bp´1 to be the Blaschke product defined as $ ’ e2π i θ z in the elliptic case, ’ & d in the hyperbolic case, Bp´1 pzq :“ z p´1 ’ dp´1 ’ z ` a % in the parabolic case, for a “ d ´1 . 1àz

dp´1

d `1

The conditions in Corollary 7.30 are therefore satisfied. Hence, by replacing f on Uj by the map φj´`11 ˝ Bj ˝ φj for j “ 0, 1, . . . , p ´ 1 and keeping f Ťp´1 outside j “0 Uj we obtain a new quasirational map. See Figure 7.17, observing that the choice of U “ U0 is arbitrary. More precisely g p “ φj´1 ˝ B pj q ˝ φj on Uj for any j “ 0, . . . , p ´ 1 where B pj q is the appropriate cyclic composition of B0 , . . . , Bp´1 . The strictly preperiodic case After having dealt with all periodic components of I pCq we work backwards along the preimages of each of them. If I pV q is

260


one of them, and fppI pV qq “ I pW q, then fp is quasisymmetrically equivalent to z ÞÑ zd for some d ě 1. If d “ 1 we keep the quasiconformal homeomorphism f |V : V Ñ W . Only if d ą 1 we shall replace the map. Suppose this has been done up to V . We construct as before gluing maps φ1 and φ2 using Riemann mappings RV and RW and define g : V Ñ W as g :“ φ2´1 ˝ pφ1 qd . The new critical point c equals φ1´1 p0q P V with critical value gpcq equal to φ2´1 p0q. The orbit of c satisfies one of the conditions in the Main Theorem if the orbit of gpcq does. We choose the Riemann mapping RW so that φ2´1 p0q has the correct orbit behaviour. At each step we obtain a new quasirational map. The next surgery starts from the resulting quasirational map. Conclusion of the proof of the Main Theorem After a finite number of steps we have constructed a quasirational map, satisfying the conditions in the Main Theorem. After integration we get the rational map g. Proof of Propositions 7.37 and 7.39 Proof of Proposition 7.37 There are only finitely many components p which intersect f ´1 pCq or contain a critical point. Set V1 , . . . , Vk of CzC, V “ tVj u1ďj ďk . p Proof of (a). If U is a component of CzC not in V, then f maps U p onto another component of CzC as a covering. Since U is a disc, f |U is a quasiconformal homeomorphism onto its image. p Proof of (b). Let U be a component of CzC not in V. Suppose f n pU q is n not in V for any n ě 0. Then f is normal on U and therefore f n pU q must eventually cycle, otherwise U would be a wandering domain and that is not possible for a rational map. Hence, U is eventually mapped onto a Fatou component whose first return map is a quasiconformal homeomorphism, and therefore a Siegel disc. It follows that I pU q is preperiodic. If, on the other hand, f n pU q is eventually mapped onto one of the components in V then I pU q must also be preperiodic. Indeed, if some iterate is mapped onto I pW q for W R V, we can repeat the argument from before, the component W is eventually mapped onto either a Siegel disc or onto one of the components in V. Since there are only finitely many components in V the ideal boundary I pU q will eventually cycle. There is a finite number of periodic components in I pCq, since there is at most finitely many cycles of Siegel discs and finitely many components in V. Proof of (c). If U is not in V then f |U is a quasiconformal homeomorphism and fp : I pU q Ñ f pI pU qq is quasisymmetrically equivalent to the identity. If U is one of the components in V, then we can find an annulus U 1 Ă U such that


261

fp

RV

RU φ1

V

U f

U1

φ2

V1

ϕ1

ϕ2 ϕ2 ˝ f ˝ ϕ1´1

ϕ

ϕ z ÞÑ zd

Figure 7.18 Sketch of the proof of Proposition 7.37(c).

BU Ă BU 1 and f maps U 1 onto f pU 1 q Ă V by a covering. (Since U contains only finitely many critical points and finitely many components of f ´1 pCq then U 1 can be chosen to avoid them all.) If we uniformize U 1 and f pU 1 q separately to standard annuli of the form Ar “ tr ă |z| ă 1u, using maps ϕ1 and ϕ2 respectively, then the induced covering map on the annulus must be conjugate to z ÞÑ zd , with the same extension to S1 as before. Let ϕ denote the conjugacy. 1 Choose Riemann mappings RU and RV . Then φ1 :“ ϕ ˝ ϕ1 ˝ R´ U and ´1 φ2 :“ ϕ ˝ ϕ2 ˝ RV extend to S1 providing an equivalence between fp : I pU q Ñ I pV q and z ÞÑ zd for some d ě 1 (see Figure 7.18). Observe that if I pU q is fixed then the proof provides an equivalence between fp and z ÞÑ zd but not necessarily a conjugacy, since φp1 and φp2 are in general different maps. The remainder of this section is dedicated to proving Proposition 7.39, for which we shall study each type of periodic component of I pCq separately. Let p for which I pU q is periodic under fp. Replacing f by U be a component of CzC an iterate, we may assume that I pU q is fixed. As discussed above, there is an annulus U 1 Ă U with BU Ă BU 1 and such that f is a covering onto its image.

262


Lemma 7.40 Suppose fppI pU qq “ I pU q. If U 1 can be chosen to be a proper subset of f pU 1 q then fp is quasisymmetrically conjugate to z ÞÑ zd for some d ą 1. In the proof of the above lemma we shall apply the following theorem of Shub (cf. [Shu], [dMvS, Chapt. II, Sect. 2]). Theorem 7.41 (Quasisymmetric conjugacy of expanding circle maps) Any expanding C 1`α orientation preserving endomorphism of S1 is quasisymmetrically conjugate to z ÞÑ zd for some d ą 1. Proof of Lemma 7.40 Choose a Riemann mapping RU : U Ñ D. Then RU pU 1 q is a one-sided neighbourhood of S1 , which is mapped properly over 1 itself by the map RU ˝ f ˝ R´ U . By the Schwarz Reflection Principle, we obtain a holomorphic covering F : A1 Ñ A, where A1 and A are annuli surrounding S1 such that A1 is a proper subset of A. It follows that F |S1 is expanding relative to the Poincaré metric on A. Hence by Theorem 7.41 the lemma follows. The next proposition deals with the buried case where BU is not part of the boundary of any Fatou component. Proposition 7.42 (The buried case) Suppose fppI pU qq “ I pU q. If BU is not contained in BV for any Fatou component V Ă U , then fp : I pU q Ñ I pU q is quasisymmetrically conjugate to z ÞÑ zd , for some d ě 1. To prove this proposition we shall apply the following theorem from topology (cf. [Why, p. 35]). Theorem 7.43 (Zoretti’s Theorem) Let C be a component of a compact set J Ă R2 . For all ε ą 0, there exists a simple closed curve γ which encloses C and such that γ X J “ H and distpC, xq ă ε for all x P γ . Proof of Proposition 7.42 Since C is a component of Jf , we can apply Zoretti’s Theorem. Choose a decreasing sequence tn uną0 of positive numbers tending to 0 as n tends to 8. Let γn denote a simple closed curve in the Fatou set, lying in an n -neighbourhood of BU . Since BU is disjoint from the boundary of any Fatou component, these curves lies in infinitely many different Fatou components. For n sufficiently large these components are not periodic, and the open annuli bounded by γn and BU contain no critical values of f . Let V 1 denote such an annulus. Let U 1 denote the unique component of ´ f 1 pV 1 q which lies in U and is bounded by BU and some preimage of γn .


263

Since γn is not contained in a periodic Fatou component, it must be disjoint from its preimages, and therefore either U 1 Ă V 1 or V 1 Ă U 1 . But f maps U 1 properly onto V 1 with some degree. Since they are both annuli, the modulus of V 1 must be larger than or equal to the modulus of U 1 . Hence U 1 must be a proper subset of V 1 . The conclusion then follows from Lemma 7.40. In the remaining cases BU Ă BV , where V Ă U is a Fatou component. By the classification of periodic Fatou components for rational maps V can only be a Siegel disc, a Herman ring, a basin of (super)attraction or a parabolic basin. We deal with these cases separately. Proposition 7.44 (Siegel discs and Herman rings) If V is a fixed Siegel disc or a fixed Herman ring of rotation number θ P RzQ, then fp : I pU q Ñ I pU q is real analytically conjugate to the rigid rotation Rθ pzq “ e2π i θ z. Proof If V is a Siegel disc then U “ V and a Riemann mapping RU , which maps the indifferent fixed point to 0, conjugates f to the rigid rotation on D. Hence fp on S1 is actually equal to Rθ . If V is a Herman ring, then there is a conformal map ϕ : V Ñ Ar “ tr ă |z| ă 1u conjugating f to Rθ . In this case, arguing as we did in Proposition 7.37, we obtain that fp on I pU q is real analytically conjugate to Rθ . Proposition 7.45 (Basins of (super)attraction and certain parabolic basins) If V is an immediate basin of (super)attraction, or a parabolic basin whose parabolic fixed point is not on BU , then fp on I pU q is quasisymmetrically conjugate to z ÞÑ zd for some d ą 1. Proof Let W Ă V be a Jordan domain, for which f pW q is a proper subset of W and W does not intersect BU . In the (super)attracting case W may be chosen as a disc surrounding the (super)attracting fixed point and mapping inside itself. In the parabolic case W may be chosen as an attracting petal in the parabolic basin so that W intersects BV at the parabolic fixed point. Let Wn be the component of f ń pW q containing W , then W Ă W1 Ă W2 ¨ ¨ ¨ Ť

and n Wn “ V . In the parabolic case, BU cannot contain any point in the grand orbit of the parabolic point since f pBU q “ BU , so Wn X BU “ H for all n ą 0. Observe that, if V is not simply connected, Wn will become multiply connected after a certain n0 , since the union of all the Wn ’s must account for all of V .

264


Wn Wn ´ 1 An f

Figure 7.19 Sketch of the annuli An in the superattracting case.

For every n, there is a component of BWn which together with BU bounds an annulus An disjoint from Wn (see Figure 7.19). In general, An may contain infinitely many different components of the Julia set. Each annulus lies in an n -neighbourhood of BU , where n Ñ 0 as n Ñ 8. Otherwise there is a continuum in BV which lies in U and meets BU , contradicting the fact that C is a component of the Julia set. Choose n sufficiently large so that An contains no critical point. Then, An`1 is an annulus properly contained in An and f pAn`1 q “ An . Hence Lemma 7.40 applies and the conclusion follows. The only case left is the general parabolic. Proposition 7.46 (General parabolic basins) If V belongs to a parabolic basin, with a parabolic fixed point on BU , then fp on I pU q is quasisymmetrically conjugate to Pd for some d ą 1. Proof Let RU : U Ñ D be a Riemann map sending the parabolic fixed point to z “ 1 (Riemann maps have radial limits at parabolic fixed points since these are accessible). Then f and V carry over to D where we also denote them by f and V . Observe that f is not defined on the whole disc, and that its domain contains V and a one-sided neighbourhood of S1 . By the Schwarz Reflection Principle, f can be extended to a holomorphic map in a neighbourhood of S1 having a parabolic fixed point at z “ 1. The restriction to the unit circle is precisely fp. We construct by hand a quasisymmetric conjugacy between fp and Pd , where d is the degree of fp on S1 . Consider an attracting petal W , containing z “ 1 on its boundary. For simplicity choose W so that BW is a C 8 curve avoiding the critical orbits. As before, let Wn denote the component of f ń pW q containing W . For n sufficiently large, Wn contains all critical points of V , and hence there is a


265

unique component γn of BWn , lying between BU and Wn , and touching S1 at the n-th preimages of z “ 1. This curve γn is contained in the annulus where f is a covering (see Figure 7.20). P2

f γn`1

An

An

1 γn`1

A1n

A1n

γn1

γn

Wn1

Wn 1

1

An An

A1n

A1n

Figure 7.20 The construction of Wn , γn and An in the parabolic case, drawn for d “ 2.

The curves γn and γn`1 bound a pinched annulus An with several components. This pinched annulus is, in a sense, a fundamental domain, since all forward orbits of points in between S1 and γn`1 must pass once and only once through An Y γn . We may carry out the same discussion on another copy of the unit disc for the map Pd , obtaining analogous sets Wn1 , γn1 , A1n , etc. To construct the conjugacy, we start by fixing n sufficiently large and by choosing a C 1 -diffeomorphism φ from the curve γn onto γn1 , fixing z “ 1, and sending preimages of z “ 1 to their analogues in an appropriate fashion. Using the map f , we define φ on γn`1 so that Pd ˝ φ “ φ ˝ f . We then extend φ quasiconformally to the different components of An sending them to their analogues A1n . Using the dynamics of f and Pd we extend φ recursively to the whole pinched annulus in between γn and S1 , so that it conjugates f and Pd . Such map extends quasiconformally to the whole unit disc, and quasisymmetrically to S1 , conjugating fp and Pd as required.

Exercises Section 7.5 7.5.1 Prove as stated in Remark 7.33(1) that if B is any of the rigid models defined in Definition 7.32 and φ is a quasisymmetric map of S1 conjugating B to itself on S1 , then φ must be a rigid rotation z ÞÑ αz with |α| “ 1. Deduce conditions on α for each type of rigid model.

266


7.5.2 Prove that every Blaschke product of degree d ě 1 mapping D to itself is quasisymmetrically equivalent to z ÞÑ zd on S1 . (This is Remark 7.38(1).) 7.5.3 Suppose f is quasisymmetrically equivalent to z ÞÑ zd on the unit circle. Show that for any quasisymmetric map ψ1 : S1 Ñ S1 , there exists a quasisymmetric map ψ2 : S1 Ñ S 1 such that ψ1 and ψ2 also provide an equivalence between f and z ÞÑ zd . (This is Remark 7.38(2).) 7.5.4 Consider the cubic Blaschke product fλ,a given in the Herman ring example illustrated in Figure 3.8. Let C denote the exterior connected component of the Julia set, i.e. the immediate basin of the superattracting fixed point at infinity. Go through the proof of the Main Theorem (Theorem 7.34) in this case to see that fλ,a only needs to be modified p on one of the disc components of CzC. Observe that the resulting map is quasiconformally conjugate to a quadratic polynomial with a Siegel disc. 7.5.5 Let P be a polynomial of degree d0 ě 2, with an attracting fixed point α. Let A˝ “ A˝ pαq denote the immediate basin of attraction and A “ Apαq the whole basin. Suppose P |A˝ has degree d ě 2. Set C :“ BA˝ ; d b it is a forward invariant continuum in JP . Let Bpzq :“ 1z`` where bzd 1 0 ă b ă dd ´ `1 is arbitrary. (a) Show that B has degree d on D, has an attracting fixed point in p0, 1q whose immediate basin of attraction is D, and has a unique critical point at z “ 0. (b) Prove that P and B are compatible on the ideal boundary of A˝ . (c) Prove that there exists a quasiconformal map φ : C Ñ C such that it conjugates P to a new polynomial Pr for which φpαq is an attracting fixed point, with immediate basin of attraction φpA˝ q, containing a unique critical point.

7.5.6 Let P , A, A˝ , . . . be as in Exercise 7.5.5. Set C :“ BA; it is a forward invariant subset of the Julia set, perhaps disconnected. p are topological discs (a) Observe that all bounded components of CzC and CzA is totally invariant. Therefore, we may define I pCq as the union of the ideal boundaries of all components of A and redefine fp appropriately. (b) Using Exercise 7.5.5 and replacing P on the appropriate preperiodic components, show that there exists a quasiconformal homeomorphism φ : C Ñ C conjugating P to a polynomial Pr, outside A, such that φpAq is a basin of attraction with a single critical grand orbit, having at most one critical point in each component.

7.6 Disc-annulus surgery on rational maps

267

7.6 Disc-annulus surgery on rational maps 1 Kevin M. Pilgrim and Tan Lei

p ÑC p be a rational map. In this section, we introduce a flexible cut Let f : C and paste surgery supported on the Fatou set of f . The resulting new map has dynamics closely related to that of f , although the connectivity of some of the Fatou components may increase. The construction is in many ways similar to a related construction in the context of Kleinian groups. The ‘Klein–Maskit combination of type I using trivial discs’ (cf. [Mar]) takes as input two Kleinian groups 1 , 2 and two r 2 with trivial stabilizer contained in the ordinary sets 1 , 2 . r1 , D round discs D The output is a Kleinian group “ 1 ˚ 2 containing Möbius conjugate copies of 1 and 2 . That is, the dynamics of contains that of 1 and 2 . Moreover, it can easily be shown that the connected components of the limit set of are either translates of connected components of the limit sets j , or points. This operation also has a natural combinatorial analogue in the setting of three-manifolds, namely boundary connect sum. Let Mj “ pH3 Y j q{j , j “ 1, 2, be the quotient three-manifolds. Since j is assumed nonr j descend to discs Dj on empty, Mj has non-empty boundary, and the discs D 3 BMj . The manifold M “ pH Y q{ is obtained from the manifolds Mj by identifying D1 and D2 via an orientation-reversing homeomorphism. In the setting of rational maps, however, the fact that maps with interesting dynamics have degree strictly greater than one makes the formulation of combination theorems taking as input two (or several) rational maps rather subtle. One difficulty is the lack of a natural, simple combinatorial analogue to describe the gluing data, which can be quite complicated to describe. We therefore content ourselves with the more modest program of beginning with a single map f and modifying it (in a more-or-less arbitrary fashion) inside a smoothly bounded disc V Ă Ff to obtain a new map F . Contents In Section 7.6.1 we define precisely the first surgery, which we call disc-annulus surgery. In Section 7.6.2 we give applications and relate the Julia set of F to that of f . In Section 7.6.3 we discuss the dependence of the map F on the various choices made in the construction and prove a uniqueness theorem (Theorem 7.60). In Section 7.6.4 we discuss the inverse operation to disc-annulus surgery. Finally, in Section 7.6.5 we give a related construction in which the modification procedure is slightly more involved. Nevertheless it is useful for the construction of examples such as those found in [PT2]. One such 1 This section is a revised version of the paper [PT1] and is reproduced with the permission of

World Scientific.

268


example is a hyperbolic rational map whose Julia set contains a wandering component which is a Jordan curve but not a quasicircle. 7.6.1 The disc-annulus surgery p We say that V Ă C is a smooth disc if BV is a real-analytic Jordan curve. By a branched covering we mean a proper C 1 -map between smooth, oriented (real) 2-manifolds, possibly with boundary, such that the boundary map is a covering map of (real) 1-manifolds, and such that, on the interior, the map is given in appropriate local (complex) coordinates by z ÞÑ zd for some d. The following two lemmas are the technical ingredients for the definition of the disc-annulus surgery. p be an open annulus bounded by two Lemma 7.47 (Key Lemma) Let A Ă C 1 ˘ C Jordan curves γ , and let W be an open disc bounded by a C 1 Jordan curve η. Give orientations to the curves such that A and W lie to the left of their boundaries. Let f ˘ : γ ˘ Ñ η be two orientation preserving C 1 -coverings of degree d ˘ ě 1. Then there exists a branched covering g : A Ñ W satisfying the following properties: (1) g|γ ˘ “ f ˘ ; (2) gpAq “ W and the degree of g is d ` ` d ´ ; (3) g can be chosen to be C 1 in A and holomorphic and proper in a union of any collection of finitely many smooth discs compactly contained in A with pairwise disjoint closures. See Figure 7.21. We call the map g a covering extension of the boundary maps f ˘ . In practice, we will take g to be holomorphic in a neighbourhood of its critical points. We recall the next lemma which was already mentioned in Remark 2.11 of Chapter 2, as an application of the Schwarz Reflection Principle. p Then Lemma 7.48 (Extension Lemma) Let D, D 1 be two smooth discs in C. 1 a holomorphic proper mapping F : D Ñ D extends to a holomorphic map in a neighbourhood of D. In particular, F : BD Ñ BD 1 is a C 1 -covering. A branched covering F is quasirational if it is quasiconformally conjugate to a rational map. The Julia set of a quasirational map is thus well defined, and has the same qualitative metric and measure theoretic properties as the Julia set of a rational map. The following lemma is standard in the construction of new rational maps via surgeries. It is a restatement of a special case of the Second Shishikura Principle (Proposition 5.5).


269

γ`

A

γ´

f` f´ d` : 1

g

d´ : 1

d` ` d´ : 1

W

η

Figure 7.21 Left: The setup for Lemma 7.47. Right: Example of a branched covering of degree two from an annulus onto a disc, such that each of the boundaries of the annulus is mapped onto the boundary of the disc by degree one. The blue curve is mapped to the blue segment two to one. The outer and inner boundaries of the annulus are mapped to the boundary of the disc as indicated in the figure. The red and green segments are respectively mapped to the red and green slits in the disc two to one. The dotted line is mapped bijectively to the dotted line. Each of the connected domains in the annulus minus the segments are mapped bijectively to the open disc minus the two slits. See the exercises at the end of the section for generalizations.

p ÑC p be a C 1 branched covLemma 7.49 (Shishikura Principle) Let F : C p ering which is holomorphic a.e. in CzB, holomorphic in a neighbourhood of the critical points in B, and for some integer k, F j pBq X B “ H for all j ě k. Then F is quasirational. We now describe disc-annulus surgery. Theorem 7.50 (Disc-annulus surgery) Let pf, V , H, h, gq satisfy the following conditions: • f is a rational map; • V is a smooth disc such that: – BV contains no critical points; – f : V Ñ f pV q is proper; – there exists 1 ď p ď 8 such that f j pV q X V “ H for 0 ă j ă p, and, in case p ă 8, f p pV q Ă V and f p : V Ñ f p pV q is proper;

270


‚ H is a smooth disc with H Ă V zf p pV q; p pV q is a holomorphic proper map, and ‚ h : H Ñ Czf ‚ g : V zH Ñ f pV q is a covering extension of the boundary maps such that g is holomorphic and proper on f p pV q and near the critical points. Then the map

F :“

$ p ’ ’f on CzV &

h on H ’ ’ %g on V zH

is quasirational. Remark 7.51 (V in Ff ) Observe that the hypotheses imply that V is a subset of the Fatou set of f . If p ă 8, by the Schwarz Lemma there is an attracting periodic point in f p pV q, and therefore V is part of a basin of attraction of f (and of F ). If p “ 8 then either (i) V belongs to a strictly preperiodic component of f or (ii) since V is non-recurrent, V is part of a basin of attraction of an attracting or parabolic cycle. In none of the cases V can be part of a periodic rotation domain. Summarizing (see Figure 7.22), f is the original map; V is the disc with conp pV q trolled recurrence on which the modification is supported; h : H Ñ Czf is the new, added dynamics; and g is an interpolating map gluing f outside of V to h on H . We refer to F as a disc-annulus extension of f supported on V . If h is univalent we call the extension F univalent or degree one. Note that p and degpF q “ degpf q ` degphq. F pV q “ C

V

V

F

H

H

h f p pV q

f p pV q f pV q

f pV q

g Figure 7.22 The domains V , f p pV q and H in the case 2 ď p ă 8. The new map F p pV q via h, and V zH to f pV q via g, using Lemma 7.47. On the white sends H to Czf region, F “ f .


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Proof We first extend h to H by Lemma 7.48. If p “ 8, set f p pV q “ H. Then F satisfies the conditions of Lemma 7.49 for B “ V zpH Y f p pV qq and k “ 1. So F is quasirational.

7.6.2 Applications and properties Here is an application of Theorem 7.50 with p “ 8. Corollary 7.52 Let f be a rational map. Let z0 be a point in the Fatou set such that z0 is neither a periodic point nor contained in a rotation domain. Then there is a disc-annulus extension F of f supported on a neighbourhood V of z0 . Proof Choose V a smooth disc containing z0 such that V ztz0 u contains no critical points, f : V Ñ f pV q is proper, and f j pV q X V “ H for all j ą 0. Then any choice of pH, h, gq satisfying the conditions of Theorem 7.50 (for p “ 8) will do. Thus, for any map f and any non-recurrent point z0 in its Fatou set, one can modify f via a disc-annulus surgery supported in a small neighbourhood V of z0 so that on a disc H in this neighbourhood, the new dynamics is more or less completely arbitrary. Theorem 7.50 also applies to recurrent points in the Fatou set. Corollary 7.53 Let f be a rational map. Let z0 be a (super)attracting periodic point of period p. Then there is a disc-annulus extension F of f supported on a neighbourhood V of z0 . Proof Choose V to be a smooth disc containing z0 such that V ztz0 u contains no critical points, f p : V Ñ f p pV q is proper with f p pV q Ă V and f j pV q X V “ H for 0 ă j ă p. Then any choice of pH, h, gq satisfying the conditions of Theorem 7.50 (for p ă 8) will do. We now examine how the dynamics of f and F in the above two Corollaries are related. Given a rational map or branched covering f , recall that the postcritical set of f is defined as ď f n pCf q, Pf “ ną0

where Cf is the set of critical points of f . We say that two compact sets K1 and K2 are conformally homeomorphic if there is a conformal map from a neighbourhood of K1 to a neighbourhood of K2 which sends K1 onto K2 .

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Corollary 7.54 (Properties of disc-annulus extensions) Let f and F be as in Corollaries 7.52 or 7.53. Then: (1) Jf Ă JF , JF is disconnected, and every connected component of Jf is a connected component of JF . (2) If H X Pf “ H and h is univalent, then H X PF “ H. (3) If H X PF “ H, then: • every Julia component of F passing through H infinitely many times is a point, and • every other Julia component of F is conformally homeomorphic to a Julia component of f . Proof (1) The set Jf is equal to the closure of the set of repelling periodic points of f . Since V is contained in the Fatou set of f and F “ f outside V , repelling periodic points of f stay repelling periodic points of F . So Jf Ă p pV q Ą Jf and JF is totally invariant, JF . Since also F pH q “ hpH q “ Czf JF X H ‰ H. Moreover, since f “ F on BV and V is contained in the Fatou set of f , it is easy to show that BV and hence BH are in fact also in the Fatou p q ‰ H, and BH Ă FF . Hence JF is set of F . So JF X H ‰ H, JF X pCzH disconnected. That each component of Jf is also a component of JF follows from the fact that a rational map sends connected components of the Julia set onto such components; see e.g. [Bea] for details. (2) The critical points of F are the union of the critical points of f |Cp zV , g and h. The F -orbit of the g-critical points do not intersect H . So if h is conformal and H X Pf “ H, H X PF “ H. (3) Assume now H X PF “ H. Take a closed disc H 1 Ă H such that pJF X H q Ă H 1 . By a lemma of Fatou, the diameters of the components of F ń pH 1 q tend to zero as n Ñ 8 (see Lemma 3.40). A Julia component J 1 such that F n pJ 1 q Ă H for some n is contained in a component of F ń pH 1 q. Therefore if F n pJ 1 q Ă H for infinitely many n, J 1 is a point. For example, if h is conformal, it has a unique repelling fixed point in H , which is a point component of JF . Now let J 1 be a component of JF such that F pJ 1 q Ă H and F n pJ 1 q X H “ H for n ą . Then F `1 pJ 1 q is a component of Jf , and F `1 is con formal in the component of F ´ pH q containing J 1 . Corollary 7.55 (Connectivity of Fatou components) Denote by Wf , WF the Fatou components of f and F containing BV . If Wf is periodic, then WF is periodic and infinitely connected. If Wf is strictly preperiodic, and h is univalent, then the connectivity of WF is equal to m0 ` m1 , where m0 and m1 are the connectivities of Wf and f pWf q.


273

Proof By assumption V Ă Wf and V zH Ă WF . If Wf is periodic with p ă 8 then WF is also periodic since F p pV zH q “ f p pV q Ă V . If p “ 8 the periodicity is not changed either since F “ f outside V . Since BV is in the Fatou set, but H contains points of the Julia set, it follows that WF is not simply connected. But any invariant component of the Fatou set which is not a Herman ring is either simply connected or infinitely connected ([Bea] Section 7.5). By Remark 7.51, WF is not a Herman ring, hence WF is infinitely connected. Now suppose Wf is strictly preperiodic and h is univalent, and let B “ f pWf qzf pV q, which has m1 ` 1 boundary components. Observe that points in h´1 pBq are in H and belong to WF , since they never come back to Wf under f . In particular, every boundary component of B except Bf pV q, is a boundary component of WF , and H contains no other. All those original boundary components of Wf are also so in WF , and the counting follows. The following corollary has previously been obtained by Baker (cf. [Bea, §11.7]). Corollary 7.56 There exists rational maps with a non-periodic Fatou component of any given number of connectivity. These corollaries provide many examples of maps with disconnected Julia sets. Explicit examples of rational maps with similar properties are given in [Bea, Chapter 11]. The following examples use the freedom of g|f p pV q and h|H to assign interesting dynamics to F . Example 7.57 (Surgery of a polynomial to obtain a higher degree polynomial) Let f be a polynomial of degree greater than 1, so that 8 is a superattracting fixed point. Let V 1 “ t|z| ą Ru, where R is large enough so that V :“ f ´1 pV 1 q Ą V 1 and V zt8u contains no critical point of f . Choose any smooth disc H with H Ă V zf pV q and any holomorphic proper map p pV q. Finally, choose a covering extension g on V zH so that h : H Ñ Czf g is holomorphic and proper from f pV q onto gpf pV qq with 8 as a critical fixed point of local degree degpf q ` degphq; the preimage of the map zdegpf q pz ´ 1qdegphq outside a small neighbourhood of 0 provides a model, showing that such a covering extension exists. This provides a quasirational map F with F ´1 p8q “ 8. So F is quasiconformally conjugate to a polynomial of degree degpf q ` degphq. Example 7.58 (Capturing a critical point) In the setting of Example 7.57, assume that a critical point c of f satisfies f pcq P V zf pV q for some p pV q ą 0. We take H to be a smooth disc containing f pcq and h : H Ñ Czf conformal such that hpf pcqq “ c. Therefore the critical point c, escaping to

274


8 under f , is ‘captured’ back and becomes periodic (hence superattracting) under F . Similarly, one can exploit the same idea to send a critical point c to, for example, a point x P Jf whose orbit is dense in Jf , thereby obtaining a map F whose postcritical set has complicated topology. Note, however, that in such an example the altered critical point lies in the Julia set of F and is non-recurrent. Example 7.59 (Case 1 ă p ă 8) We describe a surgery on the quadratic polynomial f pzq “ z2 ´ 1 to obtain a cubic rational map with disconnected Julia set, and with 0 a double critical point. Let V be a smooth disc containing 0 whose closure is contained in the basin of attraction of 0 so that the second iterate f 2 pV q is relatively compact in V , and f : V Ñ f pV q is a degree two covering ramified at 0. Now take H to be a smooth disc whose closure is contained in V zf 2 pV q. Define h : H Ñ Czf pV q to be conformal, g : V zH Ñ f pV q to be a covering extension which is a holomorphic branched covering of degree three in f 2 pV q, with 0 as a double critical point and ´1 as the critical value. Now pasting together f |Cp zV , h and g gives a quasirational map F . This map has a simple critical point at 8, a double critical point at 0 with orbit 0 ÞÑ ´1 ÞÑ 0, and another simple critical point in V zpH Y f 2 pV qq whose orbit is attracted to the cycle t0, ´1u. The map F is of degree three and the Julia set of F contains a fixed copy J0 of the Julia set Jf , a countable collection of homeomorphic preimages of J0 , and a Cantor set worth of point components.

7.6.3 Uniqueness of disc-annulus extensions The construction of a disc-annulus extension of f depends on the non-canonical choices of V , g, H and h. Clearly, the flexibility of choice in the map g implies that one cannot expect the quasiconformal conjugacy class on the whole sphere to be independent of such choices. With this in mind, we say that two maps F, F 1 are J -conjugate if there is a quasiconformal pÑC p which conjugates F on a neighbourhood of JF homeomorphism φ : C 1 to F on a neighbourhood of JF 1 . Note that this notion of J -conjugacy is stronger than the one used elsewhere in the book. In this section, we establish a prototype uniqueness theorem. Theorem 7.60 (Uniqueness up to J -conjugacy) Let f be a rational map and U a Fatou component of Ff . Then the J -conjugacy class of a univalent disc-annulus extension F supported on V Ă pU zPf q depends only on U and not on V , g, h, H .


275

This fails without the assumption of the univalence of h, or the fact that H X Pf “ H (which is a consequence of the assumption V X Pf “ H). Assume on the contrary that there is a point z P H which is either a point p pV q be another holomorphic of Pf or a critical point of h. Let h1 : H Ñ Czf 1 1 map satisfying h pzq ‰ hpzq and h pzq P Jf . Then V and H are identical but the two resulting extensions F, F 1 are distinct on their Julia sets. Analogy with Kleinian groups The univalent disc-annulus extensions constructed in Theorem 7.60 are analogous to adding a handle to a three-manifold with boundary. Let M1 “ pH3 Y 1 q{1 be the three-manifold with boundary associated to a Kleinian group 1 . Let D, D 1 be disjoint round discs in BM1 , and consider the three-manifold M obtained by gluing D to D 1 via an orientation-reversing homeomorphism (equivalently, join D to D 1 with a solid tube). The resulting manifold admits a hyperbolic structure inherited from the quotient of H3 by a new group “ 1 ˚Z which is an HNN-extension of . The discs D, D 1 yield a compressing disc in M, i.e. its boundary is an essential curve on BM. Indeed, if is a basin of attraction, then V descends to a closed disc on the quotient punctured torus associated to U , which is like the boundary of the quotient three-manifold. The following lemma is the key step in the proof of Theorem 7.60. Lemma 7.61 Let f be a rational map, U a Fatou component of Ff . Suppose (see Figure 7.23): (1) W Ă U is an open subset such that there is 1 ď p ď 8 with (a) f : W Ñ f pW q proper, (b) f j pW q X W “ H for 0 ă j ă p, and in case p ă 8, f p pW q Ă W and f p : W Ñ f p pW q is proper; (2) pV , H q and pV 1 , H 1 q are two pairs of smooth discs in W satisfying the 1 conditions of Theorem 7.50 and the additional condition ´ ¯ that H Y H is contained in a disc or annulus Y Ă W z f p pW q Y Pf . Then any two univalent extensions F “ F pV , g, h, H q, F 1 “ F 1 pV 1 , g 1 , h1 , H 1 q are J -conjugate. Proof of Lemma 7.61 We will construct a combinatorial equivalence (in the sense of McMullen [McM4], Appendix) between the holomorphic coverings F : X1 Ñ X0 , F 1 : X11 Ñ X01 , where X0 , X01 are neighbourhoods of the respective Julia sets and X1 and X11 are respective subsets of X0 and X01 . Such an equivalence consists of a pair of quasiconformal homeomorphisms φj : Xj Ñ Xj1 , j “ 0, 1 such that the diagram

276

Cut and paste surgeries Y W

Pf

H

V

H1

V1

f p pW q

f pW q

Figure 7.23 Setup of Lemma 7.61. Two univalent extensions with the modified domains close enough to each other and avoiding the postcritical set, give rise to two J -conjugate rational maps.

F

X1 ÝÝÝÝÑ § § φ1 đ

X0 § §φ đ 0

F1

X11 ÝÝÝÝÑ X01 commutes and such that φ0 extends to a quasiconformal homeomorphism from X0 to X01 and φ0 is isotopic to φ1 rel. the ideal boundary of X0 . From this, a standard pullback argument ([McM4], Thm. A.1) implies the existence of a conjugacy φ : X0 Ñ X01 , yielding the result. We now proceed to define the combinatorial equivalence. In case p ă 8, let L1 “ f pW q Y ¨ ¨ ¨ Y f p pW q. In case p “ 8, choose L1 to be a union of finitely many closed discs or annuli contained in Ff such that L1 X W “ H, Ť L1 Ą ną0 f n pW q and f pL1 q Ă L1 . Denote by P 1 the union of finitely many disjoint smooth open discs containing Pf such that f pP 1 q Ă P 1 and BP 1 X BL1 “ H. 1 Y P 1 q; note that X Ą J , J 1 , because f pL1 q Ă p Let X0 “ X01 “ CzpL 0 F F 1 n 1 L so f pL q X pH Y H 1 q “ H. Moreover, X0 has finitely many boundary components. Next we lift φ0 to obtain φ1 . We call this the lifting step. Lifting step Define X1 “ F ´1 X0 , X11 “ pF 1 q´1 X0 and φ0 |X0 “ Id. Observe that pL1 Y P 1 q Ă F ´1 pL1 Y P 1 q and therefore X1 Ă X0 . Also p pV q X11 Ă X0 . Extend φ0 to a quasiconformal homeomorphism from Czf p pV 1 q (this is possible since both f pV q and f pV 1 q are compactly to Czf contained in f pW q). Define φ1 “ Id on X0 zW , and on H define φ1 as the lift of φ0 under h, h1 ; this is possible since h, h1 are univalent and indeed we have φ1 |H “ ph1 q´1 ˝ φ0 ˝ h. Observe that X1 Ă pX0 zW q Y H ,


277

X11 Ă pX0 zW q Y H 1 and the following diagram commutes, with the top and the bottom maps being holomorphic coverings: F

X1 ÝÝÝÝÑ § § φ1 đ

X0 § §Id đ X0

F1

X11 ÝÝÝÝÑ X0 Finally we show that φ0 is isotopic to φ1 rel. the ideal boundary of X0 . We call this the isotopy step. Isotopy step There is a C 1 -extension of φ1 on X0 such that φ1 is isotopic to φ0 rel. BX0 . To see this, extend φ1 to W such that φ1 “ Id on W zY and φ1 |Y p Ą BX0 . is isotopic to the identity rel. BY . Then φ1 is isotopic to Id rel. CzY The pair φ0 , φ1 gives the desired combinatorial equivalence, and so we obtain a quasiconformal mapping φ : X0 Ñ X0 such that the following diagram commutes: F

X1 ÝÝÝÝÑ § § φđ

X0 § §φ đ

F1

X11 ÝÝÝÝÑ X0 Extending φ arbitrarily to the whole sphere shows that F and F 1 are J -conjugate. Remark 7.62 In Theorem 7.60, the global quasiconformal conjugacy class depends on the critical orbit relations of F , which are difficult to control since the gluing map g necessarily introduces new critical values in f pV q Ă FF . Also, in the non-univalent case, it may not be possible to carry out the lifting step. Or the lifting step may be possible but the isotopy step impossible. Proof of Theorem 7.60 Let V , V 1 Ă U zPf be any two smooth discs. Set V0 “ V , V1 “ V 1 , and let Vt be a path of smooth discs joining V0 to V1 through U zPf . Any two univalent extensions supported on sufficiently close Vs , Vt are J -conjugate, by Lemma 7.61; the theorem follows by compactness of the interval r0, 1s. 7.6.4 Inverse of disc-annulus extensions: simplification The next result offers an inverse procedure. Theorem 7.63 (Inverse disc-annulus surgery) Let F be a rational map. Let A be a smooth annulus satisfying the following conditions:

278


• F maps A properly onto a disc; • BA contains no critical points; • there is 1 ď p ď 8 such that F j pAq X A “ H for 0 ă j ă p, and, in case p ă 8, F p pAq Ă A and F p : A Ñ F p pAq is proper; j p such that H X Ť • there is a component H of CzA 1ăj ăp F pAq “ H. p Then there is a quasirational map f which coincides with F on CzpA Y Hq and f pA Y H q “ F pAq. Note that degpf q ă degpF q. In the analogy with three-manifolds, one may think of this operation as a special case of cutting along a compressing disc. p Proof Define f to be F on CzpA Y H q and extend f so that f : A Y H Ñ 1 F pAq is a C proper map which is holomorphic in F p pAq. By assumption, f n pBq X B “ H for B “ pA Y H qzF p pAq and n ą 0. So f is quasirational by Lemma 7.49.

7.6.5 Surgery with two gluing regions Here, we describe another, similar surgery construction. In the disc-annulus surgery, the new dynamics h is holomorphic on a disc. Here, we allow for the more flexible setting of adding the new dynamics h which is holomorphic on an annulus. Since the annulus has two boundary components, the gluing map g will be defined on two disjoint pieces. Controlling the recurrence of the additional region where conformal distortion occurs requires our introduction of an additional smooth disc G (see Figure 7.24). h

V H

G

f p pV q

F

V H

G

f p pV q

g f pV q

f pV q

g Figure 7.24 The domains V , f p pV q, H and G in the setup of Theorem 7.64.


279

Theorem 7.64 (Surgery with two gluing regions) Let pf, V , H, G, h, gq satisfy the following conditions: • f is a rational map; • V is a smooth disc contained in a Fatou component U such that: – BV contains no critical points of f ; – there is p ă 8 such that f j pV q X V “ H for 0 ă j ă p, f p pV q Ă V , and f p : V Ñ f p pV q is proper; • H is a smooth annulus such that H Ă V zf p pV q and Bf p pV q is a boundary component of H ; 1 • G is a smooth disc contained iná Fatou component ¯ U such that U and p U 1 have disjoint orbit and mod CzpG Y f pV qq “ d 1 mod H for some integer d 1 ą 0; p Y f pV qq a holomorphic covering of degree d 1 with • h : H Ñ CzpG p hpf pBV qq “ BG and g : V zpH Y f p pV qq Ñ f pV q, g : f p pV q Ñ G a covering extension holomorphic in a neighbourhood of the critical points. Then F :“

$ p ’ ’ &f on CzV

h on H ’ ’ %g on V zH

is quasirational. Moreover, if H X Pf ‰ H and f is hyperbolic, the J -conjugacy class of F depends only on U , U 1 , d 1 and the homotopy class of BH relative to Pf (see Figure 7.24). Proof Set B “ V zH . We have F j pBq X B “ H for j ą p. So by Lemma 7.49 F is quasirational. To prove the unicity, we first establish a local result. Assume pf, V , H, G, h, gq and pf, V 1 , H 1 , G1 , h1 , g 1 q are two sets of choices of the theorem such that: ¯ ´ ¯ ´ p 1 Y f pV 1 q { mod H 1 “ p Y f pV q { mod H “ mod CzG (1) mod CzG d 1 is an integer, and there is a set U 1 which is the union of finitely r and many disjoint closed discs such that G Y G1 Ă intpU 1 q Ă U 1 Ă U 1 1 f pU q Ă U ; (2) there is a smooth annulus Hp containing H Y H 1 such that f ´p pB Hp q X Hp “ H. Then the two maps F and F 1 are J -conjugate.

280


To prove this, set W :“ Hp Y f p pV q. It is a disc containing H Y H 1 Y f p pV q Y f p pV 1 q. By assumption 2, f p pW q is contained in f p pV q X f p pV 1 q. 1 Y f pW q Y ¨ ¨ ¨ Y f p pW q Y P 1 q, where P 1 is the union of p Let X0 “ CzpU finitely many disjoint smooth open discs containing Pf such that f pP 1 q Ă P 1 and BP 1 X BpU 1 Y f pW q Y ¨ ¨ ¨ Y f p pW qq “ H. Note that X0 Ą JF , JF 1 and that X0 has finitely many boundary components.

Lifting step Define X1 “ F ´1 pX0 q, X11 “ F 1´1 pX0 q, and a C 1 diffeomorphism φ0 |X0 isotopic to the identity rel. BX0 , mapping Bf pV q to Bf pV 1 q. p Y f pV qq to Now extend φ0 to a C 1 -diffeomorphism from CzpG 1 Y f pV 1 qq. p CzpG p pV q Y Gq Ñ Define φ1 |X0 zW “ Id and H Ñ H 1 to be a lifting of φ0 : Czpf 1 1 p Czpf pV q Y G q. Extend φ1 to Hp so that it is isotopic to the identity rel. B Hp . Then φ1 is isotopic to Id rel. BX0 . The only problem is that it is not exactly a lift of φ0 (near f ´1 pBf pV qq. The pair φ0 , φ1 gives the desired combinatorial equivalence and hence a quasiconformal-conjugacy of F : X1 Ñ X0 to F 1 : X11 Ñ X0 . Therefore F and F 1 are J -conjugate. From this one can easily obtain the global unicity result. Lemma 7.65 Let pf, V q be as in Theorem 7.64. If there is a Fatou component U whose orbit under f is disjoint from V , then the disc G and the annulus H exist. Proof One can first choose either G or H first. Assume that H and hence modH are given. Choose z0 P U . Since the modulus of the annulus between z0 and Bf pV q is infinite, one can find a smooth disc G containing z0 such p Y f pV qqq “ d 1 mod H for some (probably very large) integer that modpCzpG 1 d ą 0. Of course, we can also start from G. Let G be any smooth disc whose closure is contained in U . There is a minimal integer d 1 such that p Y f pV qqq ă d 1 mod H . Then one can find H in V zf p pV q such modpCzpG p that modpCzpG Y f pV qqq “ d 1 mod H . In practice we may want d 1 to be as small as possible. We will show that in case p “ 1, one can choose V , H, G so that d 1 “ 2 if degpf q ě 3 and d 1 “ 3 if degpf q “ 2. Surgery on the basin of infinity of a polynomial Let f pzq “ z2 ` c and let Kc “ Kf be its filled Julia set, which we assume to have non-empty interior (for example c “ ´1). Fix a smooth disc G with closure in intpKc q. Denote


281

by ϕ the Böttcher coordinate of f in the basin of infinity. Let R be chosen sufficiently large such that ´ ¯ p log R ą 13 mod CzpG Y ϕ ´1 pt|z| ě R 2 uqq ; this is possible since the right-hand side is comparable to 23 log R. Now take V “ ϕ ´1 pt|z| ą Ruq. One can choose H as in Theorem 7.64 with d 1 “ 3. See [PT2] for a computer-generated picture of the Julia set of such a map with c “ ´1 and d 1 “ 3. A similar surgery can be done for any degree d polynomial f with intpKf q ‰ H, or with rational maps with multiple superattracting cycles. Moreover, if d ě 3 one can choose V , G, H so that d 1 “ 2. Finally, note that if d 1 ě 2 then JF may contain preimages of components of Jf under covering maps of positive (indeed, arbitrarily large) degree. We conclude with an easy consequence of Theorem 7.64. Corollary 7.66 (Properties of the extensions) Let F be the quasirational map given by Theorem 7.64. Then JF is not connected, Jf Ă JF , degpF q “ degpf q ` degphq “ degpf q ` d 1 and JF zJf contains uncountably many wandering components which are not points.

Exercises Section 7.6 7.6.1 It is shown in Figure 7.21 how one can construct a branched covering of degree two of an annulus onto a disc such that each of the boundaries of the annulus is mapped onto the boundary of the disc by degree one. Generalize this idea to construct a degree 2d branched covering, mapping each of the boundaries of the annulus onto the boundary of the disc by degree d. Hint: Draw 2d segments in the annulus similar to the two drawn in the figure. Map each connected component of the annulus minus the segments bijectively onto the disc minus the two slits, and map each segment onto one of the two critical slits, alternately. Alternatively, change coordinates so that the annulus separates 0 and 8 and precompose the degree 2 map by zd . 7.6.2 Give a (topological) example of a degree three branched covering map from an annulus onto a disc such that the outer boundary of the annulus is mapped onto the boundary of the disc by degree two while the inner boundary of the annulus is mapped by degree one. Hint: Construct first a branched covering of degree two as in Figure 7.21. Afterwards, make a slit in the annulus from the outer boundary to an interior point. Open up along the slit and glue in a copy of a slitted

282


Figure 7.25 Sketch of the construction in Exercise 7.6.2.

disc, as shown in Figure 7.25. The resulting space is still an annulus. Explain how to map the open annulus 3 : 1 onto the open disc. 7.6.3 For any natural numbers d ` ą 1 and d ´ ą 1, generalize the constructions in the exercises above to give an example of a branched covering of degree d ` ` d ´ from an annulus onto a disc, such that the outer respectively inner boundary of the annulus is mapped onto the boundary of the disc by degree d ` respectively d ´ .

7.7 Perturbation and counting of non-repelling cycles In the paper [Sh1], published in 1987, Shishikura used a cut and paste surgery construction to prove the following celebrated theorem. The result originally appeared in his Masters thesis (in Japanese) in 1985. Theorem 7.67 (Counting non-repelling cycles) Suppose f P Ratd with d ě 2. Let nnr pf q be the number of non-repelling cycles of f . Then nnr pf q ď 2pd ´ 1q. Shishikura proved that every non-repelling cycle of a rational map has a critical point associated to it and that the critical point can not be shared by any other cycle. It is well known that every immediate basin of an attracting or parabolic cycle must contain a critical point in its interior (see Theorem 3.39). This proves that the number of attracting and parabolic cycles of a rational map must be less than or equal to the number of critical points, i.e. 2pd ´ 1q. In fact, Fatou conjectured that one could add Siegel cycles and Cremer cycles to the equation. His idea was to perturb the rational map to convert the irrationally indifferent cycles into attracting ones, without destroying the attracting nature of the others. But he only succeeded in doing so for at least half of them at the same time. Hence, the proof was not complete. Later on, Douady and Hubbard [Do2] proved the statement for polynomials, using polynomial-like mappings

7.7 Perturbation and counting of non-repelling cycles

283

(see Exercise 7.7.1). Although their proof could be used for some rational maps, it did not work in general. Shishikura proved the following theorem, from which Theorem 7.67 follows as a corollary. Theorem 7.68 (Perturbation Theorem) Let f P Ratd with d ě 2. Denote by z0 , . . . , zN all non-repelling periodic points of f . For 0 ă ε ă ε0 , there exist ε in C p such that: fε P Ratd and points z0ε , . . . , zN (a) if ε Ñ 0, then fε Ñ f uniformly and ziε Ñ zi in the spherical metric; (b) if f pzi q “ zj then fε pziε q “ zjε , and each ziε is an attracting periodic point. Let nattr pfε q denote the number of attracting cycles of fε . Since fε P Ratd we have nattr pfε q ď 2pd ´ 1q. By construction we have nnr pf q ď nattr pfε q, and Theorem 7.67 follows. We dedicate this section to the proof of Theorem 7.68. We start by giving the idea of the construction and afterwards fill in the details. We essentially follow the steps presented by Shishikura in [Sh1, Sect. 4], and parts from Beardon in [Ber, Sect. 9.6]. Without loss of generality we may assume that zi ‰ 8 for all i, and therefore the set of non-repelling cycles is contained in a compact set in C. We start by choosing a polynomial h satisfying hpzi q “ 0 and h1 pzi q “ ´1 for all i “ 0, . . . , N .

(7.1)

Let k be the degree of h. For ε P C˚ define a new map gε (the model) as follows: ! ) • gε pzq :“ f pz ` ε hpzqq in the disc Dε “ |z| ď |ε|´1{k ; ! ) • gε pzq :“ f pzq in the set |z| ě 2|ε|´1{k ; map (to be specified) in the annulus Aε :“ • gε is a quasiregular ! ) ´ 1 { k ´ 1 { k A|ε|´1{k ,2|ε|´1{k “ |ε| ď |z| ď 2|ε| , interpolating between the two maps above. By defining gε appropriately we shall prove that, for ε small enough, gε is a quasiregular map with the same topological degree as f . In fact, we set p ÑC p is a quasiconformal homeomorphism, see gε “ f ˝ Hε , where Hε : C Lemma 7.69 (compare with Figure 7.26). Observe that if ε is sufficiently small then Dε contains all the non-repelling cycles of f . Since hpzi q “ 0, it turns out that if zi is p-periodic for f , then it is also p-periodic for gε with multiplier pgεp q1 pzi q “ p1 ´ εqp pf p q1 pzi q.

284

Cut and paste surgeries 8 gε “ f .

Vε A ε gε

z4

gε “ f ˝ Hε . gε pzq “ f pz ` εhpzqq

z5 z3

z0

Eε

z1

z6

Eε

Figure 7.26 Definition of the quasiregular model gε by pasting three different maps together. On the annulus Aε , the map Hε is a quasiconformal auxiliary map so that gε “ f ˝ Hε on Aε interpolates between its two boundary maps. The shaded area Vε is mapped to the invariant set Eε .

If ε is real, positive and small, and the multipliers of the non-repelling cycles of f have moduli less than or equal to one, it follows that z0 , . . . , zN are attracting periodic points of gε . Hence, the model gε satisfies the desired properties, namely having (at least) as many attracting cycles as non-repelling cycles of f , and possessing the same topological degree as f . However, the model is not holomorphic. Therefore, to deduce the existence of one critical point per cycle, we need to find a holomorphic map conjugate to gε . This is where the smoothing of the surgery comes in. Note that gε is holomorphic everywhere except in the annulus Aε , where it is quasiregular. We shall show that, if ε is small enough, orbits of gε pass through Aε at most once (Lemma 7.70). Once this is proven, we may apply the Shishikura Principle, which says that an invariant almost complex structure with bounded dilatation can be defined, and hence a quasiconformal integrating map φε exists, so that fε :“ φε ˝ gε ˝ φε´1 is holomorphic, and hence a rational map of degree d. We finally show that the new map fε preserves the attracting nature of the cycles ziε “ φε pzi q. The proof is then complete. We proceed to make the details of the proof more precise. We start by showing that, for any arbitrary polynomial h of degree k ě 1 we can define a p ÑC p with certain properties, specified family of quasiconformal maps Hε : C in Lemma 7.69 below.


285

Lemma 7.69 (Definition and properties of special perturbations of the identity) Let h be any polynomial of degree k ě 1, and let ρ be a C 8 -function on R satisfying ρpRq “ r0, 1s, ρ “ 1 on p´8, 1s and ρ “ 0 on r2, 8q. Define p ÑC p for ε P C by Hε : C ´ ¯ Hε pzq “ z ` ε hpzq ρ |ε|1{k |z| , for z P C, and Hε p8q :“ 8. Then, for ε small enough, Hε is a quasiconformal homeomorphism. Moreover, Hε Ñ IdCp uniformly w.r.t. the spherical metric and ||μHε ||8 Ñ 0 when ε Ñ 0. For any rational map f set gε :“ f ˝ Hε . It follows that gε is quasiregular and of the same degree as f and, when ε Ñ 0, that gε Ñ f uniformly in the spherical metric. Proof

Note that for ε P C˚ :

! ) • Hε pzq “ z ` ε hpzq in the disc Dε “ |z| ď |ε|´1{k ; ! ) • Hε pzq “ z in the set |z| ě 2|ε|´1{k ; ! ) • Hε is C 8 on the annulus Aε “ |ε|´1{k ď |z| ď 2|ε|´1{k , interpolating between the two maps above. It is therefore enough to prove uniform convergence on the set t|z| ď 2|ε|´1{k u since Hε pzq “ z everywhere else. Choose a constant M ą 1 so that |hpzq| ď M if |z| ď 1 and |hpzq| ď M|z|k and

|h1 pzq| ď M|z|k ´1 if |z| ě 1.

(7.2)

Then, using these estimates and the expression for the chordal distance σ between two points (3.1), and its relation to the spherical distance (3.3) we obtain 2|ε||hpzq|ρ p|ε|1{k |z|q

dCp pHε pzq, zq ď σ pHε pzq, zq “ p1`|z|2 q1{2 p1`|H pzq|2 q1{2 ε 2|ε||hpzq|

ď p1`|z|2 q1{2 ď

2M |ε| maxt1,|z|k u maxt1,|z|u

(7.3)

ď 2M|ε| maxt1, |z|k ´1 u ď 2k M|ε|1{k , which shows that dCp pHε pzq, zq Ñ 0 uniformly. Since any rational map satisfies a Lipschitz condition with respect to the chordal metric (see e.g. [Bea, Thm. 2.3.1]), there exists L ą 1 such that dCp pgε pzq, f pzqq ď σ pf pHε ppzqq, f pzqq ď Lσ pHε pzq, zq ď C|ε|1{k , (7.4)

286


for C “ 2k ML and |z| ď 2|ε|´1{k , which shows that gε Ñ f uniformly in the spherical metric. Observe that if |z| ď 2|ε|´1{k we have |ε||z|k ď 2k and |ε||z|k ´1 ď 2k ´1 |ε|1{k . Putting both observations together, and using (7.2), we obtain that |εhpzq| ď 2k M

and

|εh1 pzq| ď 2k ´1 M|ε|1{k ,

for all z such that 1 ď |z| ď 2|ε|´1{k . In particular, this holds for all points in the annulus Aε if ε is small enough. With these estimates we proceed to show that |μHε | Ñ 0 as ε Ñ 0. It suffices to consider the annulus Aε since Hε is holomorphic in the complement. Differentiating at any point z P C with respect to z and z we obtain Bz Hε “ εhpzq|ε|1{k ρ 1 p|z||ε|1{k q

z 2|z|

and Bz Hε “ 1 ` εh1 pzqρp|z||ε|1{k q ` εhpzqρ 1 p|z||ε|1{k q|ε|1{k

z . 2|z|

Using the estimates above and taking into account that ρ and ρ 1 are bounded, we conclude that for all z such that 1 ď |z| ď 2|ε|´1{k , |Bz Hε | “ Op|ε1{k |q Ñ 0 as ε Ñ 0, and |Bz Hε | “ 1 ` Op|ε1{k |q Ñ 1 as ε Ñ 0. It follows that |μHε | Ñ 0 as ε Ñ 0 as we wanted to show. We now prove that Hε is injective. We know from Sections 1.1 and 1.2 that the Jacobian of Hε is J “ |Bz Hε |2 ´ |Bz Hε |2 . From the estimates above on the partial derivatives, we deduce that the Jacobian is strictly positive for all z for which 1 ď |z| ď 2|ε|´1{k , and hence Hε is a local homeomorphism in this domain. The same is true inside D. Indeed, on this compact set Hε is holomorphic and |h1 pzq| is bounded, so for ε small enough Hε1 pzq “ 1 ` εh1 pzq ‰ 0, and hence Hε has no critical points.


287

On the remaining part of the sphere Hε is the identity. Hence Hε is a C 8 covering map of the sphere. Since the sphere is simply connected, the identity map is a universal covering and therefore any covering map must be of degree one. It then follows that Hε is a global quasiconformal homeomorphism. The previous lemma works for any polynomial, and for any ε P C˚ small enough. From now on we restrict to ε real and positive and we require conditions on the polynomial as in (7.1), namely that hpzi q “ 0 and h1 pzi q “ ´1 for every i “ 0, . . . , N . Recall that such conditions ensure that the non-repelling cycles for f turn into attracting cycles for gε , with the same period. The next step is to show that orbits of gε pass through the annulus Aε at most once. To that end, we construct an invariant set containing the image of p ε. the annulus. Let Vε :“ t|z| ą ε´1{k u “ CzD Lemma 7.70 (Orbits pass through Aε at most once) In the notation above, by further adjusting the polynomial h and the point a :“ f p8q, orbits under gε pass through Aε at most once. More precisely, for ε P p0, ε0 s, there exist open sets Eε uniformly bounded in C, such that Eε is invariant under gε and satisfies: • Eε X Vε “ ∅; • gε pVε q Ă Eε . Compare with Figure 7.26. The proof splits into three cases, depending on the type of non-repelling cycles that f possesses – attracting, parabolic or irrationally indifferent. Case 1: f has an attracting cycle This is the simplest case. Suppose z0 , . . . , zp´1 is an attracting cycle of f . Choose E0 an open disc around z0 such that f p pE 0 q Ă E0 . Let Ei “ f i pE0 q for i “ 1, ¨ ¨ ¨ , p ´ 1, and set Ť E :“ i Ei . We may assume E Ă Dε . Observe that f p pEq Ă E. This is a stable condition under small perturbations. The condition hpzi q “ 0, makes the perturbation arbitrarily small around the periodic cycle, so one can check that for ε small enough, these discs still map compactly inside themselves. Hence define Eε :“ E and add no extra conditions on h (see Figure 7.27) In this case we could require from the start, applying a coordinate transformation if necessary, that f p8q P E and 8 P f ´1 pEqzE. By making ε small enough, we can assure that Vε is mapped compactly inside E by f , and since E does not depend on ε, this is also true for gε . Then the lemma follows. Case 2: f has a parabolic cycle If f has no attracting cycle but there exists a parabolic cycle then we need to work harder to construct Eε . Suppose

288

Cut and paste surgeries gε

f

Eε “ E

z0

Figure 7.27 Definition of Eε :“ E, in the special case of an attracting fixed point. For ε small enough, this open set is mapped inside itself by gε .

z0 , . . . , zp´1 is a parabolic cycle. Let E0 be an attracting petal of f at z0 (see Ť Definition 3.32). Let Ei “ f i pE0 q for i “ 1, . . . , p ´ 1, and set E :“ i Ei . Observe that f p pEq Ă E Y Opz0 q. Away from z0 , this is again a situation that is stable under small perturbations of the map. Since the points zi for i “ 0, . . . , p ´ 1, are attracting periodic points for gε , it follows that there exists a small neighbourhood of z0 , say Nε,0 , which maps strictly inside itself p under gε . Hence we define, Eε,0 :“ Eo Y Nε,0 . p

Observe that, for ε small enough, we have gε pE ε,0 q Ă Eε,0 as required (see Figure 7.28). We define Eε,i :“ gε pEε,i ´1 q for i “ 1, . . . , p ´ 1, and Eε :“ p ´1 Yi “0 Eε,i . We choose Eε,0 so that Eε Ă Dε . f

f E gε Eε z0

Nε z0

gε

Figure 7.28 Definition of Eε , in the case of a parabolic fixed point, as the union of an attracting petal E0 for f and a neighbourhood Nε around the fixed point which maps compactly inside itself under gε . Observe that Eε does not shrink to a point when ε Ñ 0 since it contains the set E0 for all ε.

In this case, we could as before require from the start, that f p8q P E and 8 P f ´1 pEqzE. In this way we can ensure that for ε small enough Eε is uniformly bounded in C, disjoint from Vε , and that gε maps Vε compactly inside Eε , since E does not shrink when ε Ñ 0.


289

Case 3: f has an irrationally indifferent cycle If f has no attracting or parabolic cycles but only irrationally indifferent ones, then we need to work even harder to construct Eε . Suppose z0 , . . . , zp´1 is an irrationally indifferent cycle, and adjust f from the start so that f p8q “ z0 and 8 ‰ zp´1 . By the theory of normal forms (see e.g. [Ar2] or [Bea, Thm. 6.10.5]), there exists a holomorphic local diffeomorphism ψ around 0 such that ψp0q “ z0 and pψ ´1 ˝ f p ˝ ψqpuq “ λu ` Opuk `3 q, where λ is the multiplier of the cycle, and k is the degree of h. A direct calculation of the power series expansion (see [Bea, Lem. 9.6.4]) shows that, close to u “ 0, ´ ¯ pψ ´1 ˝ gεp ˝ ψqpuq “ pψ ´1 ˝ f p ˝ ψqpuq ´ λu ` λup1 ´ εqp ` Opεu2 q. (7.5) Putting together the two expressions we obtain ´ ¯ pψ ´1 ˝ gεp ˝ ψqpuq “ λu p1 ´ εqp ` Opεuq ` Opuk `2 q . Set ε :“ t|u| ă ε1{pk `1q u. For ε sufficiently small, ψ is defined on ε , and p ´1 we may define Eε1 “ ψpε q. Set also Eε :“ Eε1 Y gε pEε1 q Y ¨ ¨ ¨ Y gε pEε1 q. p 1 1 We want to show that if ε is small enough, then gε pE ε q Ă Eε , and hence Eε is invariant under gε . Equivalently, we need to check that pψ ´1 ˝ gεp ˝ ψqpε q Ă ε . 1 p 2 From (7.5), setting t “ k ` 1 and using that p1 ´ εq “ 1 ´ εp ` Opε q, we obtain, for u P ε , ´ ¯ pψ ´1 ˝ gεp ˝ ψqpuq “ λu 1 ´ pε ` Opε1`t q .

It follows that ˇ ˇ ˇ ´1 ˇ ˇpψ ˝ gεp ˝ ψqpuqˇ ă |λu| “ |u| p

and therefore ε is mapped into itself under ψ ´1 ˝ gε ˝ ψ. It follows that p gε pE 1 ε q Ă Eε1 as required. Compared to the other cases, the problems start now, since the set Eε1 shrinks to a point when ε Ñ 0. Even if f p8q “ z0 , we cannot be sure that gε pVε q Ă Eε1 as we want. To guarantee that it happens, we need to estimate the size of the two sets Eε1 and gε pVε q in terms of ε.

290


The size of Eε1 is easily estimated using that ψp0q “ z0 and ψ 1 p0q “ 1. It follows that ψpuq “ z0 ` Opuq for small |u| and, since Dε has radius ε1{pk `1q , we have |z ´ z0 | “ Opε1{pk `1q q for z P Eε1 . Estimating the size of gε pVε q is somewhat more involved. First observe that Vε is a neighbourhood of 8 whose chordal radius is σ p8, ε´1{k q “ ´

2 1 ` ε´2{k

1{k ¯1{2 “ Opε q

(see (3.2)). Since any rational map satisfies a Lipschitz condition with respect to the chordal metric (see e.g. [Bea, Thm. 2.3.1]), we have that f pVε q lies in a disc centred at z0 of chordal size Opε1{k q. Using (7.4) we obtain that gε pVε q lies in a disc around z0 of chordal size Opε1{k q. But in a compact set the chordal metric and the Euclidian one are comparable. Hence, the set gε pVε q lies in a disc with centre z0 and Euclidian radius Opε1{k q. Since ε1{k ă ε1{pk `1q , it follows that for sufficiently small ε, the inclusion gε pVε q Ă Eε1 is satisfied, concluding the proof of Case 3, without adding extra conditions to h. This concludes the proof of Lemma 7.70. We now summarize and conclude the proof of the Perturbation Theorem (Theorem 7.68). Suppose z0 , . . . , zN are all the non-repelling periodic points of f . Choose one specific cycle, say z0 , . . . , zp´1 , and proceed as in one of the described cases, depending on the type of cycle. This defines the polynomial h, fixes the image of infinity, and defines a set Eε , satisfying the properties in Lemma 7.70. At this point everything is arranged so that the annulus Aε , the only region where gε is not holomorphic, is mapped into the invariant set Eε , disjoint from Aε . Choose the standard complex structure on Eε and spread it by the dynamics. Because of the Shishikura Principle this is well defined. The dilatation is bounded, and the structure is gε -invariant. Hence, there exists an integrating map φε , and the map fε :“ φε ˝ gε ˝ φε´1 is a rational map of the same degree as f . We observe the following: • The remaining periodic points zp , . . . , zN are attracting under gε . This means that there exists a collection of neighbourhoods of these points, say Eε1 , disjoint from Eε and Vε , which are also invariant under gε . This ensures that φε is conformal not only on Eε but also on Eε1 . Hence, the new periodic points tziε “ φε pzi quN i “0 are attracting periodic points of fε for every ε. • By the Integrability Theorem depending on parameters (Theorem 1.30), the integrating maps φε appropriately normalized satisfy φε Ñ IdCp and fε Ñ f uniformly when ε Ñ 0.

7.8 Mating a group with a polynomial

291

Exercises Section 7.7 7.7.1 Use polynomial-like mappings to prove that a polynomial P of degree d has at most d ´ 1 non-repelling cycles in C. (A proof can be found in [Do2].) Hint: Suppose N R is the set of non-repelling cycles of P . Consider a perturbation of the form fε “ P ` εQ, choosing ε small enough and an appropriate polynomial Q, so that fε is polynomial-like of degree d and the cycles in N R are attracting cycles of fε . 7.7.2 (Existence of polynomials with certain conditions) Suppose v1 , . . . , vm are distinct points in C. Assume that for some of these points, say the first n, with n ď m, there exist neighbourhoods B1 , . . . , Bn which are closed pairwise disjoint topological discs, and holomorphic functions hj : Bj Ñ C, satisfying hj pvj q “ 0 and h1j pvj q “ ´1. Prove that, for any δ ą 0, there exists a polynomial hpzq such that hpvi q “ 0

h1 pvi q “ ´1

and |h ´ hj | ă δ on Bj ,

for i “ 1, . . . , m and j “ 1, . . . , n. h ´p Hint: Apply Runge’s Theorem to the map j p2 1 , where p1 is a polynomial satisfying p1 pvi q “ 0 and p11 pvi q “ ´1, and let p2 pzq “ śm 2 i “1 pz ´ vi q . Check that h “ p1 ` p2 q satisfies the requirements.

7.8 Mating a group with a polynomial Shaun Bullett

The projective special linear group PSLp2, Cq is the group of 2 ˆ 2 matrices with complex entries and determinant 1, under the equivalence relation ˆ ˙ ˆ ˙ ab á ´b „ . cd ć ´d The elements of PSLp2, Cq are the conformal automorphisms z ÞÑ

az ` b cz ` d

p of the Riemann sphere C. Recall from Section 3.5 that a Kleinian group is a discrete subgroup of p is ordinary if the action of on some neighbourhood PSLp2, Cq; a point z P C of z forms a normal family, otherwise z is a limit point. The ordinary set is denoted pq, while the limit set is denoted pq. Both sets are invariant under the action of . The sets are the analogues of the Fatou set and the Julia set respectively of a rational map

292

Cut and paste surgeries 7.8.1 The modular group, rational maps and holomorphic correspondences

The modular group PSLp2, Zq, which is defined in just the same way as PSLp2, Cq, except that the matrix entries are required to be integers, is an example of a Kleinian group. The modular group is finitely generated: as an abstract group it is the free product of C2 ˚ C3 of a cyclic group C2 of order two, and a cyclic group C3 of order three, generated by ˆ ˙ ˆ ˙ 0 1 ´1 ´1 σ “ and ρ “ ´1 0 1 0 respectively (see Figure 7.29). Equivalently, we may take as generators for PSLp2, Zq the elements τ1 “ σρ ´1 and τ2 “ σρ, that is to say ˆ ˙ ˆ ˙ 11 10 τ1 “ and τ2 “ . 01 11

D P

Q

−2

−1

0

1

2

Figure 7.29 The action of the modular group PSLp2, Zq on the upper half plane. P and Q are the fixed points of σ and ρ respectively. The region D is a fundamental domain. It is also a fundamental domain for the action of PGLp2, Zq “ă σ, ρ, χ ą on p where χ is the involution χ pzq “ 1{z. The region D Y ρpDq Y ρ ´1 pDq is C, outlined in bold.

p (Möbius transformations), τ1 and Regarded as conformal automorphisms of C τ2 are the maps τ1 pzq “ z ` 1 and

τ2 pzq “

z . z`1

To define the notion of a mating between a (finitely generated) Kleinian group and a polynomial map, we shall need to work in a class of holomorphic dynamical systems on the Riemann sphere that is sufficiently general to include finitely generated groups and polynomial maps as special cases.


293

Definition 7.71 (Holomorphic correspondence) A holomorphic corresponp is a multi-valued map z Ñ w defined by a dence on the Riemann sphere C relation of the form P pz, wq “ 0, where P pz, wq is a polynomial in z and w (with coefficients in C) with the property that every non-constant factor involves both the variables z and w. Every rational map z Ñ ppzq{qpzq of degree n ě 1 can be regarded as an n-to-1 correspondence in the obvious way, namely take P pz, wq “ 0 to be the relation wqpzq ´ ppzq “ 0. In particular, a polynomial map z Ñ ppzq becomes a correspondence w ´ ppzq “ 0. Moreover, given any finitely generated Kleinian group , together with a specific choice of generators γj pzq “

aj z ` bj , cj z ` dj

1 ď j ď n,

we may regard as the holomorphic correspondence G : z Ñ w defined by the polynomial relation P pz, wq “ 0, where P pz, wq is the polynomial n ź

pwpcj z ` dj q ´ paj z ` bj qq .

j “1

p are the grand orbits Observe that the orbits of the action of the group on C of the correspondence G, that is to say the orbits under mixed forwards and backwards iteration, z Ñ w and w Ñ z. For example, since the matrices τ1 and τ2 (defined above) generate the p modular group PSLp2, Zq, the orbits of this group on the Riemann sphere C are exactly the same as the grand orbits of the correspondence pw ´ pz ` 1qqpwpz ` 1q ´ zq “ 0. An alternative (but equivalent) definition of the notion of a holomorphic ´1 , correspondence is that it is a multi-valued map z Ñ w of the form π` ˝ π´ where π` and π´ are holomorphic surjections (branched coverings) from a p It is clear that given a polynomial P pz, wq compact Riemann surface S onto C. satisfying the condition in our first definition above, we may satisfy the conditions of this second definition by taking as the Riemann surface S the graph p ˆ C, p and as π´ and π` the coordinate projections tpz, wq | P pz, wq “ 0u Ă C of this graph. The converse is considerably deeper: given a Riemann surface p it can be shown that the graph of the S and projections π` and π´ onto C, ´1 relation π` ˝ π´ is defined by a polynomial equation P pz, wq “ 0 (see [BP2, Thm. 2], and the references cited there).

294


7.8.2 Matings between the modular group and quadratic polynomials Let Kc denote the filled Julia set of a quadratic polynomial Qc pzq “ z2 ` c. Recall that if Kc is connected, that is to say if c lies in the Mandelbrot set, then there is a uniquely defined conformal homeomorphism, the inverse of the Böttcher coordinate p p c, Ñ CzK ψc : CzD which is tangent to the identity at infinity and conjugates the map z Ñ z2 on the complement of the closed unit disc D to the map Qc on the complement of Kc . When Kc is locally connected, ψc extends to a continuous surjection from the unit circle S1 onto the boundary Jc of Kc (by the Carathéodory Theorem (Theorem 2.8), semi-conjugating the argument-doubling map z Ñ z2 on S1 to the action of Qc on Jc . Given a quadratic polynomial Qc which has a connected and locally connected filled Julia set Kc , let K´ _ K` denote the union of two copies K´ and K` of Kc , glued together at the boundary point of external argument 0. Let Hc on K´ _ K` denote the p2 : 2q correspondence defined by sending: • z P K´ to Qc pzq P K´ and to ´z P K` ; 1 • z P K` to the pair of points Q´ c pzq P K` . For example, in the case c “ 0 the filled Julia set K0 is the closed unit disc D, and H0 on D´ _ D` is the p2 : 2q correspondence defined by sending: • z P D´ to z2 P D´ and to ´z P D` ; ? • z P D` to the pair of points ˘ z P D` . The boundary of K´ _ K` is a wedge J´ _ J` of two copies of the Julia set Jc , glued together at the boundary point of external argument 0. The p2 : 2q correspondence Hc on K´ _ K` restricts in the obvious way to J´ _ J` , which in the case c “ 0 is a wedge of two copies of the unit circle. Whenever Kc is connected and locally connected, Jc is a quotient of the unit circle and the action of the correspondence Hc on J´ _ J` is the corresponding quotient of the action of H0 on the wedge of two circles. Given a quadratic polynomial Qc , a mating between Qc and the modup lar group PSLp2, Zq will be defined to be a p2 : 2q correspondence on C obtained by gluing together the p2 : 2q correspondence Hc on pKc q´ _ pKc q` constructed above, and the p2 : 2q correspondence G on the complex upper half-plane defined by the generators τ1 and τ2 of PSLp2, Zq. Before we make this idea precise, we check that the action of the pair of homeomorphisms


295

pτ1 , τ2 q on the boundary Rp “ R Y t8u of the complex upper half plane matches the action of Hc on pJc q´ _ pJc q` . Let h` denote the homeomorphism from r0, 8s to r0, 1s which sends x P r0, 8s, represented by the continued fraction rx0 ; x1 , x2 , . . .s, to t P r0, 1s, the binary number represented by the binary point followed by x0 copies of 0, then by x1 copies of 1, then by x2 copies of 0 and so on. This homeomorphism is sometimes referred to in the literature as the Minkowski question mark map. We shall also need a companion homeomorphism h´ : r´8, 0s Ñ r0, 1s defined by h´ pxq “ 1 ´ h` p´xq. Note that h` and h´ are both orientationreversing. Consider τ1 and τ2 restricted to the completed real axis r´8, 8s. These are defined by: τ1 : x Ñ x ` 1 and

τ2 : x Ñ

x . x`1

Note that on the positive real axis r0, 8s τ1 : rx0 ; x1 , x2 , . . .s ÞÑ rx0 ` 1; x1 , x2 , . . .s, τ2 : rx0 ; x1 , x2 , . . .s ÞÑ r0; 1, x0 , x1 , x2 , . . .s. Let θ1 : r0, 1s Ñ r0, 1{2s and θ2 : r0, 1s Ñ r1{2, 1s denote the halving maps θ1 : t Ñ t{2 and θ2 : t Ñ t{2 ` 1{2 respectively. It is a straightforward exercise to check that the following diagrams commute, and also that the horizontal maps h` are homeomorphisms between the intervals indicated: h`

r0, 8s ÝÝÝÝÑ r0, 1s § § §θ § τ1 đ đ1 h`

r1, 8s ÝÝÝÝÑ r0, 1{2s

h`

r0, 8s ÝÝÝÝÑ r0, 1s § § §θ § τ2 đ đ2 h`

r0, 1s ÝÝÝÝÑ r1{2, 1s

Another straightforward computation shows that the following diagrams also commute. Here the horizontal maps h´ are again homeomorphisms, and the maps θ1´1 and θ2´1 are the doubling maps t ÞÑ 2t and t ÞÑ 2t ´ 1 respectively: h´

r´8, ´1s ÝÝÝÝÑ r1{2, 1s § § § ´1 § τ1 đ đθ2 h´

r´8, 0s ÝÝÝÝÑ r0, 1s

h´

r´1, 0s ÝÝÝÝÑ r0, 1{2s § § § ´1 § τ2 đ đθ1 h´

r´8, 0s ÝÝÝÝÑ r0, 1s

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The remaining restrictions of τ1 and τ2 , namely τ1 : r´1, 0s Ñ r0, 1s and τ2 : r´8, ´1s Ñ r1, 8s, are carried by the pair h´ , h` to the maps t ÞÑ t ` 1{2, from r0, 1{2s to r1{2, 1s, and t ÞÑ t ´ 1{2, from r1{2, 1s to r0, 1{2s, respectively: h´

r´1, 0s ÝÝÝÝÑ r0, 1{2s § § § § τ1 đ Þ t `1{2 đt Ñ h`

r0, 1s ÝÝÝÝÑ r1{2, 1s

h´

r´8, ´1s ÝÝÝÝÑ r1{2, 1s § § § § τ2 đ Þ t ´1{2 đt Ñ r1, 8s

h`

ÝÝÝÝÑ r0, 1{2s

Identifying 0 with 8, the homeomorphism h` induces an orientationreversing homeomorphism h` : r0, 8s{„ Ñ T` , where T` denotes the circle obtained from r0, 1s by identifying its end points. Similarly h´ induces an orientation-reversing homeomrphism h´ : r´8, 0s{„ Ñ T´ , where T´ is a second copy of the circle obtained from r0, 1s by identifying its end points. Together h` and h´ induce a homeomorphism p 0„8 Ñ T` _ T´ , h` _ h´ : R{ p 0„8 to that of the pair of argumentwhich conjugates the action of τ1 , τ2 on R{ halving maps t Ñ t{2, t Ñ pt ` 1q{2 on the circle T` , the argument-doubling map t Ñ 2t on the circle T´ , and the antipodal homeomorphism t Ñ t ` 1{2 from T´ to T` . If we glue together the points 0 and 8 on the boundary of the upper half p 0„8 the action of τ1 and τ2 matches (combiplane, then on this boundary R{ natorially, at least) the boundary action of the p2 : 2q correspondence Hc on the wedge K´ _ K` , constructed earlier in this section. Thus we may glue K´ _ K` to the upper half plane and obtain the action of a p2 : 2q topological correspondence on a topological sphere. The question now is whether this topological mating can be realized by a holomorphic correspondence on the p Riemann sphere C. Definition 7.72 (Mating of a quadratic polynomial and the modular group) Let Qc be a quadratic polynomial with connected filled Julia set Kc , i.e. with c in the Mandelbrot set M. A holomorphic correspondence F : z Ñ w, defined by a polynomial relation P pz, wq “ 0 of bidegree p2 : 2q, is called a mating between Qc and the modular group PSLp2, Zq if: p (a) there exists a completely invariant open simply connected region Ă C and a conformal bijection h from to the upper halfplane conjugating the two branches of F| to the pair of generators τ1 : z ÞÑ z ` 1, τ2 : z ÞÑ z{pz ` 1q of PSLp2, Zq;


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(b) the complement of is the union of two closed sets ´ and ` , which intersect in a single point and are equipped with homeomorphisms h˘ : ˘ Ñ Kc , conformal on interiors, respectively conjugating F, restricted to ´ as domain and codomain, to Qc on Kc , and conjugating F, 1 restricted to ` as domain and codomain, to Q´ c on Kc . Figure 7.30 illustrates an example of a mating of a quadratic polynomial and the modular group.

Figure 7.30 A mating between the modular group and a quadratic polynomial with an attracting cycle of period 2. The plot is of a correspondence in the family (7.6) referred to in the text, with a “ 4.4. The shaded regions ´ and ` , meeting at the origin, are copies of the filled Julia set of the polynomial. Also shown is a tiling of the complementary region by copies of a fundamental domain for PSLp2, Zq.

In this definition we have not specified the identification between the boundaries of the regions and ´ Y ` . This is to allow us maximum flexibility in dealing with the values of c for which the boundaries of the regions ˘ are not locally connected. However, there are two boundary identification conditions which it can sometimes be useful to require: (I) The image of the positive imaginary axis under the conformal bijection p both ends of which h (of condition (a) of the definition) is an arc in C converge to the wedge point ´ X ` . (II) The involution σ : z Ñ ´1{z on and the involution on K´ _ K` which exchanges z P K´ with z P K` , fit together to yield a conformal involup Ñ C. p tion (a Möbius transformation) j : C Notice that if we have a mating which satisfies the boundary condition (I) then by continuity of the action of the correspondence, the homeomorphism

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h from the complex upper half plane to , extends continuously to all rational points of the boundary Rp and maps these to a dense set of points in Bp´ _ ` q. In particular, if Kc is connected and locally connected, then h extends to a continuous surjection Rp Ñ Bp´ _ ` q realizing the boundaryp 0„8 Ñ J´ _ J` constructed earlier in matching surjection h´ _ h` : R{ this section. The convenience of condition (II) is that every p2 : 2q correspondence F : z Ñ w which realizes a mating satisfying this condition has an equation that can be expressed in a canonical form. Condition (II), in conjunction with conditions (a) and (b) in the definition of a mating, guarantees that the compop defines a p3 : 3q equivalence sition j ˝ F together with the identity map on C p relation on C, and hence that j ˝ F has equation Rpwq ´ Rpzq “0 w´z for some degree three rational map R. Conditions (a) and (b) also ensure that R has one double and two single critical points, and hence that up to pre- and post-multiplication by Möbius transformations, R is the map Rpzq “ z3 ´ 3z. Thus up to conjugation the correspondence F has equation of the form w 2 ` wj pzq ` pj pzqq2 “ 3, where j is an involution which has as one of its fixed points the wedge point ´ X ` . Normalizing our coordinate system so that this wedge point becomes the origin, and j becomes the map z ÞÑ ´z, we deduce that our p2 : 2q correspondence F lies in the one (complex) parameter family: ˙ ˙ˆ ˙ ˆ ˙ ˆ ˆ az ` 1 az ` 1 2 aw ´ 1 aw ´ 1 2 ` ` “ 3. (7.6) w´1 w´1 z`1 z`1 Bullett and Penrose formulated the following conjecture [BP1]. Conjecture 7.73 The family (7.6) of p2 : 2q correspondences contains a mating between PSLp2, Zq and every quadratic polynomial having a connected Julia set, that is to say every z Ñ z2 ` c with c P M, the Mandelbrot set. Supporting evidence for Conjecture 7.73 was provided by numerical experiments suggesting a resemblance between the space of matings and the Mandelbrot set [BP2]. To show that all these matings really do exist, it is natural to try to use some form of surgery, modifying the complex structures on the pieces in such a way that they fit together to give an overall complex structure on the topological mating. However, there are technical problems fitting together the complex structures at the ‘wedge point’ ` X ´ and at points on the grand


299

orbit of this point. To circumvent these difficulties we change the question to that of constructing a mating between a generic faithful discrete representation of C2 ˚ C3 and a quadratic polynomial Qc . Using surgery we prove the existence of a mating between every generic discrete faithful representation of C2 ˚ C3 and every Qc with c P M. One can then deduce the existence of a mating between PSLp2, Zq and Qc for a large class of values of c, including all those values for which Qc is hyperbolic, by applying pinching techniques [BHai, BHar] developed by Bullett, Ha¨ıssinsky and Harvey. We shall explain the surgery construction here, and then summarize the results concerning matings between PSLp2, Zq and Qc obtainable by pinching. 7.8.3 Using surgery to construct a mating between z Ñ z2 ` c and a generic faithful discrete representation of C2 ˚ C3 in PSLp2, Cq For simplicity of exposition, we concentrate our attention on representations of C2 ˚ C3 in PSLp2, Rq. Recall our generators σ (of order two) and ρ (of order three) ? of PSLp2, Zq. The fixed points of σ are ˘i and those of ρ are p´1 ˘ i 3q{2. Keeping σ : z Ñ ´1{z unchanged, we can perturb ρ, keeping it of order three, by moving its two fixed points to another pair of conjugate points cos θ ˘ i sin θ on the unit circle. For ´1 ă cos θ ă ´1{2 the group generated by σ and ρ remains discrete and a free product of C2 with C3 . The action of such a perturbed representation is illustrated in Figure 7.31. The difference from the action of PSLp2, Zq (Figure 7.29) is that the parabolic fixed points of PSLp2, Zq (at every point of the rationals Q Ă R) have been opened up to become intervals, with hyperbolic fixed points at either end. The limit set of the perturbed group is a Cantor set on the real axis, with these intervals as the gaps. As in the unperturbed case, the involution χ : z ÞÑ 1{z commutes with

Q

R –1S

P

U T 0

+1

Figure 7.31 The region D Y ρpD q Y ρ ´1 pD q for a perturbed representation of “ă σ, ρ, χ ą.

300


σ and anti-commutes with ρ, and a fundamental domain for the action of the group ă σ, ρ ą on the upper halfplane is a fundamental domain D for the p action of the group “ă σ, ρ, χ ą on C. More generally, one can perturb PSLp2, Zq as a representation of C2 ˚ C3 in PSLp2, Cq and not just in PSLp2, Rq. There is a complex one-dimensional moduli space of such perturbations, parametrized by the cross-ratio of the fixed points of σ with the fixed points of ρ. Provided that the perturbed representation remains faithful and discrete, generically it will have connected ordinary set, and the limit set will be a Cantor set. In fact it will be topologically (indeed quasiconformally) conjugate to the real examples described above. However, at each of the boundary points of the moduli space the limit Cantor set is pinched together and the ordinary set can become disconnected. The simplest such pinching is that at PSLp2, Zq, where the Cantor set joins up to become the whole of the completed real axis Rp and the ordinary set becomes two open discs (the upper and lower half planes). Every discrete faithful representation of C2 ˚ C3 in PSLp2, Cq, not just those in PSLp2, Rq, is equipped with an involution χ which commutes with σ and anti-commutes with ρ. In the Poincaré disc model of hyperbolic 3-space χ is the involution defined by rotation through angle π around the common orthogonal to the axis of ρ and the axis of σ . Recall Definition 3.52 of the term hybrid equivalence and Definition 7.1 of a polynomial-like map, which in degree 2 is called a quadratic-like map. Definition 7.74 (Mating of a quadratic polynomial and certain discrete subgroups of PSLp2, Cq) A p2 : 2q holomorphic correspondence f : z Ñ w is called a mating between a faithful discrete representation r of C2 ˚ C3 in PSLp2, Cq having connected ordinary set pq, and a polynomial Qc having p is the disjoint union of a connected filled Julia set Kc , if the Riemann sphere C connected open set pf q and a closed set pf q made up of two components, ` pf q and ´ pf q, such that each of pf q and pf q is completely invariant under f and: (a) the action of f on pf q is discontinuous and there is a conformal bijection between the grand orbit spaces pf q{f and pq{ (where is the group generated by σ, ρ and χ ); (b) there is a neighbourhood U of ´ pf q such that f restricted to V “ f ´1 pU q X U is a quadratic-like map V Ñ U and there is a quasiconformal homeomorphism from U onto a neighbourhood U 1 of Kc in C which is a hybrid equivalence conjugating the quadratic-like map f to the quadratic-like map Qc ; similarly there is a hybrid equivalence between


301

quadratic-like maps f ´1 on a neighbourhood of ` pf q and Qc on a neighbourhood U 1 of Kc . Notice that the fact that ` and ´ have disjoint neighbourhoods has allowed us to be more demanding in our definition of a mating than in the earlier case of PSLp2, Zq: we ask for hybrid equivalences between f ˘1 on neighbourhoods of ˘ and Qc on neighbourhoods of Kc , whereas earlier we only asked for conjugacies which are conformal on interiors between f ˘1 on ˘ and Qc on Kc . Examples of correspondences satisfying Definition 7.74 are illustrated in Figures 7.32 and 7.33.

Figure 7.32 A mating between a generic faithful discrete representation of C2 ˚ C3 in PSLp2, Rq and a quadratic polynomial with an attracting cycle of period 2. This example belongs to the family (7.7) (referred to later in the text), and is given by the parameter values a “ 3.9, k “ 0.9. Pinching to a point the curve illustrated linking ´ to ` , and pinching its images (‘hairs’ on ` and ´ ) to points, transforms this example into a mating between the modular group and the same quadratic polynomial.

Figure 7.33 Another example of a mating between a generic faithful discrete representation of C2 ˚ C3 , this time in PSLp2, Cq, and a quadratic polynomial. The correspondence again belongs to the family (7.7), and is given by the parameter values a “ 4.6 ` 0.18i, k “ 0.9 ´ 0.05i.

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The following theorem is identical to [BHar, Thm. 1]. Theorem 7.75 For every quadratic map Qc with c P M and for every faithful discrete representation r of C2 ˚ C3 in PSLp2, Cq for which the ordinary set is connected, there exists a p2 : 2q holomorphic correspondence which is a mating between Qc and r. We prove this theorem by constructing the mating using surgery. For simplicity of the description we shall assume the representation of C2 ˚ C3 is in PSLp2, Rq rather than the more general case of PSLp2, Cq, but there is no essential extra complication in the complex case. We first associate an annulus A to Qc . An equipotential for Qc is the image of a circle tRe2π i t | 0 ď t ă 1u under the inverse of the Böttcher coordinate ψc . It is an analytic Jordan curve parameterized by external argument t. The region bounded by such an equipotential is a simply connected domain V , mapped two-to-one by Qc onto a larger domain U Ą V which also has boundary an equipotential parameterized by external argument. Let A denote the annulus U zV , and denote its inner and outer boundaries by B1 A and B2 A respectively. The map Qc sends B1 A analytically, two-to-one, onto B2 A. The map j from B2 A to itself given by t Ñ 1 ´ t on external arguments is a analytic orientation-reversing involution, and there is a conformal involution ι : z Ñ ´z on V sending each z P V to the other point which has the same image in U under Qc . Our next ingredient is an annulus B associated to the group representation r. Recall the fundamental domain D constructed above for the group G “ă σ, ρ, χ ą and illustrated in Figure 7.31. Let B denote the annulus p in C{χ consisting of the three copies D Y ρpD q Y ρ ´1 pD q of D , with the boundary identifications induced by χ (Figure 7.34). The rotations ρ and

0 ∼ ∞

T

U

S ∼ R

−1

Q

P

Figure 7.34 The annulus B, the quotient of D Y ρpD q Y ρ ´1 pD q under the boundary identification χ : z Ñ 1{z. The labelling of points follows that of Figure 7.31.


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ρ ´1 , mapping D Y ρpD q Y ρ ´1 pD q to itself, descend to a p2 : 2q correspondence G on B, mapping each z P B to the pair tρz, ρ ´1 zu (or rather to their equivalence classes under the action of χ ). The set D descends to a ‘fundamental domain’ for the action of G on B. The boundary of B consists of two analytic Jordan curves, an outer boundary B2 B and an inner boundary B1 B. The correspondence G maps each of the three sets, B2 B and two halves of B1 B, to the other two. Thus when its domain is restricted to B1 B, and its range is restricted to B2 B, the correspondence G defines an analytic two-to-one orientation-reversing surjection g. The involution σ descends to an (analytic) involution, which we also denote σ , on the outer boundary B2 B of B, having fixed points Q and S. Any two real analytic orientation-reversing involutions of the circle are real analytically conjugate, so there is an analytic diffeomorphism h : B2 A Ñ B2 B which conjugates the involution j on B2 A to the involution σ on B2 B. By lifting via Qc on one side and σ ˝ g on the other we may extend h to an analytic diffeomorphism BA Ñ BB which conjugates Qc : B1 A Ñ B2 A to σ ˝ g : B1 B Ñ B2 B. Lemma 7.76 The analytic diffeomorphism h : BA Ñ BB extends to a homeomorphism of annuli h : A Ñ B which is quasiconformal on interiors. Proof

This is immediate from Lemma 2.30(b).

We use h to transport our p2 : 2q correspondence G on B to a p2 : 2q correspondence G 1 “ h´1 Gh on A. To construct a mating between Qc and r we first observe that the quotient of U by the boundary identification j is a Riemann surface of genus zero, and thus there is a conformal bijection between this quotient U {j and the Riemann sphere. Taking a double cover of U {j , branched over the fixed points of j , we obtain a Riemann sphere U Y U 1 , equipped with an involution (also denoted j ) exchanging U with a second copy, U 1 , and restricting to the original j on the common boundary. Inside U 1 is a simply connected subdomain V 1 corresponding to V Ă U . Let Q1c “ j ˝ Qc ˝ j : V 1 Ñ U 1 denote the quadratic map corresponding to Qc : V Ñ U and let A1 denote the annulus U 1 zV 1 . To define a p2 : 2q topological correspondence F on U Y U 1 we fit together: • • • •

Qc : V Ñ U , a p2 : 1q correspondence; 1 1 1 pQ1c q´1 “ j ˝ Q´ c ˝ j : U Ñ V , a p1 : 2q correspondence; 1 j ˝ ι : V Ñ V , a p1 : 1q correspondence; and j ˝ G 1 : A Ñ A1 , a p2 : 2q correspondence.

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To construct a complex structure preserved by this topological correspondence we proceed as follows. We define an ellipse field on A by using the quasiconformal homeomorphism h constructed in Lemma 7.76 to transport the standard complex structure (a field of infinitesimal round circles) from the annulus B. Using j we extend this ellipse field to A1 and by pulling back via 1 1´1 we extend it to an ellipse field on the whole of CzpK 1 p Q´ c Y Kc q. c and Qc Finally, we use the standard round circle field on Kc Y Kc1 to extend the ellipse p and observe that this field transforms equivariantly field to the whole of C, under the action of the p2 : 2q correspondence F. By applying the Integrability p respected by F. Theorem we deduce that there exists a complex structure on C This completes the proof of the theorem, since by construction our complex structure is the standard one on Kc and Kc1 , and on their complement pFq it is the structure corresponding to the representation r of C2 ˚ C3 . It is apparent from the proof above that all the correspondences F constructed in this way satisfy a condition we discussed earlier (in Section 7.8.2) in the context of matings between quadratic maps and PSLp2, Zq, namely: (II) The involution σ : z Ñ ´1{z on and the involution on K´ _ K` which exchanges z P K´ with z P K` , fit together to yield a conformal involup Ñ C. p tion (a Möbius transformation) j : C It follows by the same reasoning as in Section 7.8.2 that up to Möbius conjugacy we may write F in the form zÑw

ô

pj wq2 ` pj wqz ` z2 “ 3.

Applying a further conjugacy to transform j to the involution j pzq “ ´z, the equation defining the correspondence F becomes a member of the following family (parameterized by complex a and k): ˙ ˙ˆ ˙ ˆ ˙ ˆ ˆ aw ´ 1 aw ´ 1 2 az ` 1 az ` 1 2 ` ` “ 3k. (7.7) z`1 z`1 w´1 w´1 This is very satisfactory, since we have exactly two complex degrees of freedom in the dynamical systems to be mated – one for the variable c in the quadratic polynomial, and one for the conjugacy class of the representation r of C2 ˚ C3 . The technical difficulty that arises if we try to construct a mating between a quadratic polynomial map and the modular group PSLp2, Zq in an analogous way to the proof of Theorem 7.75, is that the annuli A and B are now pinched, their boundaries contain cusps, and Lemma 7.76 is no longer applicable. Replacing PSLp2, Zq by a generic representation of C2 ˚ C3 allowed us to avoid this problem. But now we can return to the original question and apply


305

pinching techniques to the matings provided by Theorem 7.75 to resolve Conjecture 7.73 in a large class of cases. Two technical restrictions on the quadratic maps Qc are imposed in [BHai] for the pinching techniques to be applicable: (i) if the critical point 0 of Qc is recurrent, the β-fixed point of Qc is not in the ω-limit set of 0; (ii) Qc is weakly hyperbolic, that is, there are constants r ą 0 and δ ă 8 such that, for all z P Jc tpreparabolic pointsu, there is a subsequence of iterates pQnc k qk such that ˙ ˆ n Qc k DpQnc k pzq, rq ď δ, deg Wk pzq ÝÑ nk pDpQnc k pzq, rqq conwhere Wk pzq is the connected component of Q´ c taining z.

The class of weakly hyperbolic quadratic maps is quite large: in addition to all hyperbolic quadratic maps it contains, for example, all quadratic maps which satisfy the Collet–Eckmann condition, and all those which contain parabolic points. We end by quoting the following [BHai, Thm. 1.6]. See details in [BHai]. Theorem 7.77 For every quadratic map Qc satisfying conditions (i) and (ii) above, there exists a mating between Qc and PSLp2, Zq in the family (7.6) of holomorphic correspondences.

Exercises Section 7.8 7.8.1 For the p2 : 2q correspondence F : z Ñ w defined by (7.6) verify that when F ´1 pwq consists of a single point then w is one of the three points 1, 3{pa ` 2q or ´1{pa ´ 2q, and the corresponding point z “ F ´1 pwq is ´1, 0 or ´2{pa ` 1q respectively. When F is a mating between PSLp2, Zq and Qc , the restriction F|´ is a two-to-one map with critical point z “ ´2{pa ` 1q and critical value ´1{pa ´ 2q; under the conjugacy h´ : ´ Ñ Kc these correspond to the critical point 0 and critical value c of Qc . By considering the orbit of ´2{pa ` 1q show that if F is a mating between PSLp2, Zq and: (i) Q0 : z Ñ z2 then a “ 5; (ii) Q´2 : z Ñ z2 ´ 2 then a “ 4; ? (iii) Q´1 : z Ñ z2 ´ 1 then a “ p3 ` 33q{2.

306


As Qc is hyperbolic in all three examples, we know by Theorem 7.77 that these matings occur in the family (7.6), so we may deduce that they occur at these parameter values. 7.8.2 Construct a mating between PSLp2, Zq and Q´2 directly by taking the p by p4 : 4q correspondence G defined on C z Ñ z ` 1, 1{pz ` 1q, z{pz ` 1q, pz ` 1q{z, and pushing it down to a p2 : 2q correspondence on the quotient sphere obtained by identifying z with 1{z. To do this explicitly, write down the polynomial equation P pz, wq “ 0 of G and after dividing through by w2 manipulate it into the form QpZ, W q “ 0 where Z “ z ` 1{z and W “ w ` 1{w. Since G has the same grand orbits as PGLp2, Zq p the p2 : 2q correspondence QpZ, W q “ 0 has limit set the closed on C, line interval from 2 through 8 to ´2 (the image of the real axis under z Ñ Z). It is easy to verify that the action of the correspondence on this limit set and on its complement is that of a mating between Q´2 and PSLp2, Zq. Observe also that a further change of coordinates Z Ñ 1{Z, W Ñ 1{W converts the equation QpZ, W q “ 0 into the form (7.6) with a “ 4.

8 Cut and paste surgeries with sectors

In this chapter we give three different applications of cut and paste surgeries, including their dependence on parameters. We plough in the dynamical planes and harvest in parameter spaces. In all three cases we start by a polynomial in a limb of the connectedness locus of a one-parameter family, or by a polynomiallike map hybrid equivalent to such one. In Section 8.2 we show how it is possible, through surgery, to change a certain given map and its phase space in such a way that the new map on its new phase space has an extra critical point. Adding parameters to the construction, we obtain homeomorphisms between parameter spaces of different families. The first construction of this kind occurred in [BD] and was later generalized in [BF1]. In Section 8.3 we show by a similar construction, depending on parameters, how it is possible to embed a limb of the Mandelbrot set M into other limbs of M. In both sections, the topological surgery part involves a change of phase space for each parameter, either by truncating or extending the given complex plane, hence constructing an abstract Riemann surface for each parameter. The given map is kept in certain regions and replaced by a quasiconformal map in certain sectors outside the filled Julia set. Therefore the complex structure is changed in these sectors, and subsequently spread by the dynamics recursively through iterated pullbacks. In Section 8.4 Adam Epstein and Michael Yampolsky show how to intertwine two quadratic polynomials to obtain a cubic polynomial. Sectors also play an important role in this construction. In the surgeries in Sections 8.3 and 8.4, it is important to control the opening modulus of certain sectors and understand how this guarantees that we can interpolate quasiconformally between certain boundary maps. The opening 307

308

Cut and paste surgeries with sectors

modulus was introduced in [BD]. However, the argument for the quasiconformal interpolation was not correct in that paper. It was corrected by Bielefeld in his PhD thesis [Bi2]. He introduced the notion of near translations and showed how to apply it. In the paper by Epstein and Yampolsky [EY], reviewed in Section 8.4, they show that one can control the opening modulus by choosing a suitable representative of a polynomial in its hybrid class. In Section 8.1 we first collect the tools: sectors outside the filled Julia set of a given slope, opening modulus, near translations and quasiconformal interpolations.

8.1 Preliminaries: sectors and opening modulus Let P be a monic polynomial of degree d ě 2 with connected Julia set J pP q. Let ζ be a repelling k-periodic point. There is a finite number of external rays landing at ζ . They are mapped among themselves by P k with a certain combinatorial rotation number, say p{q (see Section 3.3.4). If the q arguments of one cycle of rays are cyclicly ordered as 0 ď θ1 ă θ2 ă ¨ ¨ ¨ ă θq ă 1, then d ¨ θi mod 1 “ θpi `pq mod q . We may assume ζ to be a fixed point of P . (If not, consider P k , which is a monic polynomial of degree d k .) Then for i “ 1, 2, . . . , q, the arguments above are of the form d qj´1 for certain j P t0, 1, . . . , d q ´ 2u.

8.1.1 Definition of sectors and opening modulus With P , ζ , and p{q as above, we define a bounded sector S with vertex at ζ as a simply connected bounded domain satisfying S Ă P q pSq, bounded by two arcs γ1 and γ2 satisfying γj Ă P q pγj q, for j “ 1, 2, and a third arc connecting the two corners not equal to ζ . The numbering of γ1 and γ2 are chosen so that S is on the left of γ1 when oriented from ζ out. Considering the quotient S{P q , we obtain a fundamental domain conformally equivalent to an annulus AS (see Figure 8.1). We then define the opening modulus of the sector S as the modulus of the annulus AS , i.e. modζ pS, P q q “ modpAS q. The opening modulus of a sector is in general not easy to compute. In what follows we introduce three special types of sectors. In the first case (the one we use most often) the opening modulus is computable.

8.1 Preliminaries: sectors and opening modulus

309

S Pq

γ1

γ2

γ2 {P q

γ1 {P q

AS

ζ Figure 8.1 The definition of a sector with vertex at ζ , invariant under P q , and its opening modulus.

Log-Böttcher sectors Let ψP : CzD Ñ CzKP denote the inverse of the Böttcher coordinates conjugating P0 pzq :“ zd to P on the basin of infinity. Moreover, let Hr denote the right half plane and consider ψP ˝ exp : Hr Ñ CzKP , which conjugates the map Md (multiplication by d), to P . An external ray RP pθ q in CzKP corresponds to a horizontal half line Rpθ q of imaginary part 2π θ and all its vertical translates by 2π . An equipotential of potential ρ corresponds in Hr to the vertical line of real part ρ. In Hr we define the unbounded log-Böttcher sector of slope s centred at Rpθ q as Spθ q “ S s pθ q “ tρ ` 2π i t P Hr | |t ´ θ | ď sρu. In the rest of the discussion we omit the word log-Böttcher. The sector Spθ q is mapped homeomorphically onto the sector Spd ¨ θ q by Md . The unbounded sector SP pθ q in the dynamical plane of P (see Figure 8.2) is defined as SP pθ q “ pψP ˝ expq pSpθ qq. Note that any such sector overlaps itself, no matter how small the slope is. To avoid this and to avoid overlap of different sectors (to be specified below) we consider only bounded sectors. As before let ζ be a repelling fixed point in J pP q. Consider its backwards Ť orbit ně0 P ń pζ q and all rays landing at these points. Their arguments belong to the set "

j d n pd q ´ 1q

*

| j P Z, n P N Y t0u .

310

Cut and paste surgeries with sectors t RP p θ q t “ 2π pθ ` sρ q 2π i θ

Spθ q

SP pθ q

R pθ q

ψP ˝ exp t “ 2π pθ ´ sρ q

0

ρ0

ρ

Figure 8.2 The definition of an unbounded sector in the right half plane and its corresponding unbounded sector in the dynamical plane of P . The shaded region is the piece of sector up to equipotential ρ.

Fix an arbitrary η ą 0. For any n P N Y t0u define the bounded n-sector of slope s centred at Rpθ q as ! η) Sns pθ q :“ Sns pθ, ηq :“ ρ ` 2π i t P Hr | |t ´ θ | ď sρ ; ρ ď n . d It is not difficult to check (see [BF1, Prop. 4.1]) that if the slope s is chosen so that 1 , 2ηpd q ´ 1q ´ ¯ then for any n P N Y t0u the bounded n-sectors Sn d n pdjq ´1q , for j P Z, are pairwise disjoint. Moreover, the bounded sectors ˙ ˙ ˆ ˆ j k s , j P Z, and S , k P Z, k d, n P N, S0s n dq ´ 1 d n pd q ´ 1q s ă smax :“

are all pairwise disjoint, and so are their dynamical counterparts (see Figure 8.3). It is easy to calculate the opening modulus of sectors outside the filled Julia set. Lemma 8.1 (Opening modulus of log-Böttcher sectors of slope s) Let P be a monic polynomial of degree d with connected filled Julia set, ζ a repelling fixed point of combinatorial rotation number p{q, and SPs pθ q a log-Böttcher sector of slope s with vertex at ζ . Then the opening modulus relative to P q is modζ pSPs pθ q, P q q “

2 Arctan p2π sq, q Log d

(8.1)


311

t

ρ

0 η{d

η

Figure 8.3 Disjoint bounded sectors of slope s “ smax in Hr .

hence an increasing function of s. If η tends to 0 and s tends to `8, under the π condition that s η ă 1{2pd q ´ 1q, then modζ pSPs pθ q, P q q tends to q Log d. Proof Since the opening modulus is invariant under holomorphic conjuga2 tion, it suffices to show that mod0 pS s p0q, pMd qq q “ q Log d Arctan p2π sq (see Exercise 8.1.1). Sectors for polynomial-like maps In greater generality, let f : W 1 Ñ W be a polynomial-like map with a repelling fixed point ζ of combinatorial rotation number p{q. Then an invariant sector S Ă W zKf with vertex at ζ is a simply connected domain bounded by a piece of BW and two arcs γ1 and γ2 (named as above) with common endpoint at ζ and satisfying γj Ă f q pγj q for j “ 1, 2. We write S “ zγ1 , γ2 { for the sector between γ1 and γ2 or for simplicity we set BL S :“ γ1 and BR S :“ γ2 , the left and the right boundary curves of S (see Figure 8.4). Sectors in linearizing domains Let denote a neighbourhood of ζ and ϕ : Ñ D a linearizing coordinate, conjugating f : 1 Ñ to the linear map z ÞÑ λz, where 1 is the connected component of f ´1 pq containing ζ and λ “ f 1 pζ q. A sector in is a simply connected domain S Ă with vertex at ζ , which is invariant under f q , i.e. S X “ f q pSq X . In particular, a logBöttcher sector invariant under P q intersects the domain in a sector of this type. However, in general invariant sectors in may contain part of the filled Julia set. If S is a sector for a polynomial-like map as above, S X is a sector in (see Figure 8.4).

312

Cut and paste surgeries with sectors f S

B L S “ γ1

B R S “ γ2

W1

W

ζ

Figure 8.4 A general sector for a polynomial-like map. The neighbourhood denotes a linearizing domain, to be used later.

In this setting we consider the associated quotient torus Tζ “ pztζ uq{f q . The projection of the sector S in Tζ is the annulus AS , whose modulus is modζ pS, f q q, the opening modulus of S (see Figures 8.5 and 8.6). Let S be an invariant sector in under the action of f q . Choose a branch log of the logarithm defined on ϕpS X q. Then the normalized log-linearizing coordinate is the log-linearizing coordinate normalized by dividing by log λq ,

z ÞÑ w “

1 plog ˝ϕqpzq. log λq

It conjugates f q on S to translation T1 by 1 on the horizontal half strip ă , the image of S X under the normalized log-linearizing map. Note that the opening modulus modζ pS, f q q of S is equal to the modulus of the quadrilateral ă zT1´1 p ă q. Let denote the double infinite strip that is invariant under T1 and the extension of the half strip ă . This strip is conformally equivalent to the double infinite straight horizontal strip m with R as its lower boundary and height equal to m :“ modζ pS, f q q and such that the conformal equivalence conjugates T1 on to T1 on m (see Figure 8.5). Observe that if S is the restriction of a log-Böttcher sector in the linearizing domain of a polynomial P , i.e. S “ SPs pθ q X then the two quotient annuli are conformally equivalent and hence, by Lemma 8.1, modζ pS, P q q “

2 Arctanp2π sq. q log d


313

Tζ AS

S

D

ϕ

ζ

0 q

Mλ

fq log

ă

z ÞÑ

z log λq

log λq

z ÞÑ z ` 1

z ÞÑ z ` log λq

m

0

m

Figure 8.5 A sector in the linearization domains and its conformal equivalents.

S

Jc

Rc p0q

Tβc βc

Sp

Figure 8.6 An invariant sector S “ Scs p0q with vertex βc , the landing point of the zero ray Rc p0q.

314

Cut and paste surgeries with sectors 8.1.2 Quasiconformal interpolation on sectors

In the surgeries that follow we need to construct quasiconformal mappings from one invariant sector to another, as an interpolation between given boundary maps. In this subsection we collect the different cases. For j “ 1, 2, let fj be a holomorphic map with a repelling fixed point ζj , and let Sj be a sector that is invariant under fj with vertex at ζj . The boundary of Sj consists of three pieces BL{R Sj , the left and right boundaries of Sj , and Bout Sj , the boundary opposite to the vertex. We may assume that both BL{R S are real analytic and that Bout S is C 2 . Lemma 8.2 (Simple quasiconformal interpolation) Let fj and Sj be as above. Let gL{R : BL{R S1 Ñ BL{R S2 and gout : Bout S1 Ñ Bout S2 be given diffeomorphisms, which are assumed to be analytic on the sides and C 2 on the outer boundary. Furthermore, assume that gL{R ˝ f1 “ f2 ˝ gL{R whenever defined. Then there exists an extension g : S 1 Ñ S 2 , which is quasiconformal in the interior. Proof The map g is defined inductively using the dynamics on the sectors. For n P N let Qj,n denote the closed quadrilateral defined as the clo´pn´1q

pSj qzfjń pSj q. Define g1 : Q1,1 X Q1,2 Ñ Q2,1 X Q2,2 from sure of fj the inner boundary of Q1,1 to the inner boundary of Q2,1 so that gout ˝ f1 “ f2 ˝ g1 is satisfied. Choose a quasiconformal extension g1 : Q1,1 Ñ Q2,1 of the given boundary map which is a piecewise C 2 -diffeomorphism (see Lemma 2.24). Define inductively gn : Q1,n Ñ Q2,n so that f2 ˝ gn´1 “ gn ˝ f1 . Compare with Figure 8.7. The resulting map g : S1 Ñ S2 is

ζ1

ζ2

gn Q1,n

Q2,n

BL S1

BL S2

BR S1 Q1,2

g1

BR S2 Q2,2

Q1,1

Q2,1

Bout S1

Bout S2

Figure 8.7 A simple quasiconformal extension of boundary maps.


315

quasiconformal with distortion equal to that of g1 since the distortion is not changed when pulling back by holomorphic mappings. For j “ 1, 2, let ϕj : j Ñ D be a linearizing coordinate around ζj conjugating fj to z ÞÑ λj z, where λj is the multiplier of fj at the fixed point. Then Sj X j “ fj pSj q X j so that Sj X j is an invariant sector in j . Choose branches of the logarithm defined on Sj X j and consider the corresponding normalized log-linearized coordinates z ÞÑ w “

1 plog ˝ϕj qpzq log λj

for z P Sj X j .

We shall use certain results on extensions, which were explained in Chapter 2. However, we review here the basic concepts we need. Recall that a map between two backwards periodic Jordan arcs (one-sided infinite) is a near translation if, roughly speaking, the map is not too far from the identity map when tending to infinity and with bounded derivative. Periodic maps are always near translations for example. An example of this kind to keep in mind is given by the following proposition. Lemma 8.3 (Example of near translation) Let f : W 1 Ñ W be a quadraticlike map with connected Julia set, ζ a repelling fixed point with combinatorial rotation number p{q and S “ zr, l{ Ă W an invariant sector with vertex ζ r be another quadratic-like map havr1 Ñ W bounded by smooth arcs. Let fr : W ing a repelling fixed point ζr with the same combinatorial rotation number, and Sr “ zr r, r l{ similarly defined. Consider a smooth map ψ : r Ñ r r conjugating the dynamics: ψ ˝ f q “ frq ˝ ψ. In the log-linearizing coordinates the map ψ becomes a near translation. Indeed, the map ψ expressed in log-linearizing coordinates commutes with translation by 1 and is therefore a near translation. More generally, any invariant sector is conformally equivalent to a standard half strip, invariant under translation T1 by 1. Hence, the extension problem is reduced to finding maps between standard half strips. We shall use two results. If two sectors have the same opening modulus – or equivalently if two half strips have the same height – then the unique Riemann map between them, preserving corners when extended to the boundaries (in particular mapping ζ1 to ζ2 ), extends as near translations on the boundaries close to infinity. On the other hand, if two strips have different heights and we are given boundary maps which are near translations, then they extend to the interior as

316


a quasiconformal map (see Chapter 2). In what follows, we apply these results to our present setup. Lemma 8.4 (Near translation boundary maps) Let fj and Sj be as above, and let R : S1 Ñ S2 be the unique Riemann mapping, whose extension to the boundary maps corners to corners, in particular ζ1 to ζ2 . Denote the boundary maps by BL{R R : BL{R S1 Ñ BL{R S2 and Bout R : Bout S1 Ñ Bout S2 . Suppose the opening moduli of the sectors Sj relative to fj at ζj are equal. Then the boundary mappings BL{R R when expressed in normalized log-linearizing coordinates are near translations. Proof We may assume that RpS1 X 1 q Ă S2 X 2 . If not, we shrink 1 and redefine ϕ1 . Consider the restriction of R to S1 X 1 . The standard half strips obtained by the normalized log-linearizing coordinates composed with conformal mappings have the same height. Hence the boundary maps between the standard half strips are near translations (see Proposition 2.15). We now formulate the General Quasiconformal Interpolation Lemma. Notice, that Lemma 8.2 above is a special case of the lemma below and could have been skipped. It is included since it fits well with the common idea of spreading by the dynamics. Lemma 8.5 (General Quasiconformal Interpolation Lemma) Let fj , Sj and ϕj be as above. Let gL{R : BL{R S1 Ñ BL{R S2 be analytic maps and gout : Bout S1 Ñ Bout S2 a C 2 -diffeomorphism. Assume that the gL{R are near translations when expressed in normalized log-linearizing coordinates close to ζj . Then there exists an extension g : S 1 Ñ S 2 , which is quasiconformal in the interior. Proof We may assume that gL{R pBL{R S1 X 1 q Ă BL{R S2 X 2 . Consider the restriction of gL{R to BL{R S1 X 1 . The standard half strips obtained by the normalized log-linearizing coordinates composed with conformal mappings have different heights in general. But the induced boundary maps are near translations by hypothesis. Hence we may extend them to the interior quasiconformally (see Lemma 2.26).

8.1.3 Limbs of the Mandelbrot set In this chapter the surgeries start from quadratic polynomials in a limb of the Mandelbrot set or quadratic-like maps hybrid equivalent to such. We recall


317

the characteristics of quadratic polynomials in limbs. Let 0 denote the main hyperbolic component of the Mandelbrot set, consisting of all c-values for which Qc pzq “ z2 ` c has an attracting fixed point. The boundary B0 is the main cardioid, parametrized by γ0 : t P T ÞÑ pe2π i t {2q ´ pe2π i t {2q2 . For any 0 ă p ă q with gcdpp, qq “ 1 the p{q-limb, Lp{q , of the Mandelbrot set is the connected component of Mz0 whose closure is attached to 0 at γ0 pe2π i p{q q (see Figure 8.8). L1 2 ?

γ0 p 13 q

γ0 p 12 q

L1 3

L1 4

0

γ0 p 23 q

L3 5

Figure 8.8 The limbs of the Mandelbrot set are the connected components of Mz0 .

The polynomials in the p{q-limb Lp{q share the following features. The inner fixed point αc is repelling and the landing point of a cycle of q external rays mapped among themselves by combinatorial rotation number p{q. We denote by Rαc the q rays landing at αc plus the fixed point itself. The only other preimage of αc , apart from itself, is ´αc and it is the landing point of q other external rays which are mapped onto the q-cycle of rays. We denote by R´αc this set of rays plus the pre-fixed point ´αc . The complement of Rαc Y R´αc consists of 2q ´ 1 domains, as shown in Figure 8.9 for p{q “ 1{3. We denote these domains by tVci u0ďi ăq where Vc0 contains the critical point 0, and by Vrci “ ´Vci for 0 ă i ă q, all numbered in cyclic order relative to ˘αc . The quadratic polynomial acts on these domains as 2:1 p Vc0 Ñ Vrc , 1:1 i `ppmod q q for i ‰ q ´ p, Vci , Vrci Ñ Vrc Ťq ´1 q ´p r q ´p 1:1 0 Ñ R´αc Y Vc Y i “1 Vci . Vc , Vc

318

Cut and paste surgeries with sectors Rc p 17 q

1 q Rc p 14

Vrc1 Rc p 27 q

Vc2 αc

Vc0

Rc p 11 14 q

´αc

Vrc2 Rc p 47 q

Vc1 9 q Rc p 14

Figure 8.9 The partition of the plane induced by the set of rays Rαc Y R´αc for p{q “ 1{3.

The set Kc tαc , ´αc u has 2q ´ 1 connected components, one in each set Vci for i “ 0, 1, . . . , q ´ 1, and Vrci for i “ 1, . . . , q ´ 1. We denote the compor i respectively. nents of Kc in Vci and Vrci by Kci and K c 8.1.4 Controlling the opening modulus of sectors In the surgeries to come we need to map certain sectors conformally onto others, in such a way that the boundary maps have nice properties. In particular, we require the opening moduli of these sectors to be equal. In Section 8.3 we shall use polynomials Qc with c P Lp{q zp{q , where p{q is the hyperbolic component with the same root point as the limb Lp{q . In what follows we show how it is possible to control the opening moduli if the equipotential η ą 0 is chosen sufficiently small so that the slope s can be chosen sufficiently large (see Lemma 8.6). Later, in Section 8.4, Epstein and Yampolsky show how one can control the opening moduli by working in the hybrid class of a quadratic polynomial Qc with c P Lp{q . Working with quadratic polynomials directly For any Qc with c P Lp{q , consider the quadratic-like restriction Qc : W 1 Ñ W , where W 1 , W are bounded by equipotentials of potential η{2 and η for an arbitrary choice of η ą 0. Consider the forward invariant set of bounded sectors Scs pθi q of slope s around the q rays Rc pθi q, i “ 1, . . . , q, landing at αc . Let be a linearizing domain q around αc and let Tαc “ ztαc u{Qc denote the quotient torus. Observe that p c : Tα Ñ Tα . The q sectors the polynomial Qc induces a conformal map Q c c p of slope s project to a q-cycle under the map Qc of annuli in Tαc , of common modulus mpsq, given by the formula (8.1) in Lemma 8.1. Observe that mpsq does not depend on c and η. The annuli in between form another q-cycle under


319

p c . Denote the common modulus of these ‘in between’ annuli by the map Q m pc, sq. These moduli are independent of η. Lemma 8.6 The modulus m pc, sq is a continuous function of pc, sq P Lp{q ˆ R` . For fixed c P Lp{q zp{q the modulus m pc, sq tends to 0 as s tends to 8. A proof can be found in [BD, Chap. III, Sect. 13]. Remark 8.7 For any c0 P Lp{q zp{q we can choose a slope s (increasing η if necessary) such that m pc, sq ă mpsq holds for all c in a neighbourhood of q c0 in Lp{q zp{q . We can then split sectors of slope s into three Qc -invariant subsectors SL , SM , SR so that the middle sector SM has opening modulus equal to m pc, sq. Working in the hybrid class of a quadratic polynomial Let f : W 1 Ñ W be a quadratic-like map, which is hybrid equivalent to a quadratic polynomial Qc r i pf q for with c P Lp{q . Let ζf denote the inner fixed point of f , and let K i “ 0, 1, . . . , q ´ 1, denote the q components of Kf ztζf u, numbered cyclicly r 0 pf q. Let Tζ “ pf ztζf uq{f q with the critical point ωf of f contained in K f be the associated quotient torus, and fp : Tζf Ñ Tζf the quotient map induced r i pf q to Tζ . Following Epstein and p i pf q denote the projection of K by f . Let K f Yampolsky (compare with Subsection 8.4.3) we define p i pf q “ inftmod A | K p i pf q Ă Au, modζf K where the infimum is taken over open annuli A in Tζf . The modulus p i pf q is the same for all i and does not depend on the representative modζf K f in the hybrid class. It follows from Lemma 8.6 that the modulus is equal to p i pf q is included in the ‘n between’ annuli zero for all c P Lp{q zp{q , since K for all i. The following lemma is due to Epstein and Yampolsky (cf. [EY]) and is applied in Section 8.4. Lemma 8.8 (Sectors with prescribed opening modulus) Choose c P Lp{q . For each m ą 0 there exists a quadratic-like mapping f : W 1 Ñ W , hybrid equivalent to Qc , such that f admits a q-cycle of disjoint f q -invariant sectors with vertex at ζf , bounded by analytic curves at the sides and with opening modulus relative to f q equal to m. Proof Let as above Qc : V 1 Ñ V be a quadratic-like restriction where V is bounded by an equipotential of potential η ą 0. Then consider the q log-Böttcher sectors Scs pθi q of slope s around the rays Rc pθi q, i “ 1, . . . , q,

320


p1 , . . . , A pq denote the corresponding annuli in the quotient landing at αc . Let A p1 Ñ torus Tαc . Choose ą 0 and a quasiconformal homeomorphism ψ : A ´ 2π p m ` q ă |z| ă 1u is the standard Ae´2π pm`q ,1 , where Ae´2π pm`q ,1 “ te annulus of modulus m ` . Change the complex structure in Scs pθ1 q to the one p1 , and then in the other sectors S s pθi q and all their induced by ψ ˚ pσ0 q in A c preimages by successive pullbacks by Qc . Keep σ0 elsewhere. Through integration we obtain a quadratic-like mapping f : W 1 Ñ W , hybrid equivalent to Qc , with a q-cycle of annuli in the quotient torus Tζf of modulus m ` . Inside this q-cycle we can choose a q-cycle of annuli of modulus m with analytic boundaries at the sides.

Exercises Section 8.1 8.1.1 Consider the linear map z ÞÑ d q z and the unbounded sector S s p0q of slope s centred at Rp0q “ R` . The normalized log-linearizing coordinate map z ÞÑ pLog zq{pLog d q q maps the sector S s p0q onto the standard strip with upper boundary equal to the horizontal line of imaginary part Arg p1 ` i 2π q{ Log d q and lower boundary equal to the horizontal line of imaginary part Arg p1 ´ i 2π q{ Log d q . Show that the height of the strip is equal to the opening modulus and hence mod0 pS s p0q, pMd qq q “ “

˘ 1 1` Logp1 ` i 2π sq ´ Logp1 ´ i 2π sq Log d q i 2 Arctan p2π sq. q Log d

You may recall the formula Arctan z “

1 pLogp1 ´ i zq ´ Logp1 ` i zqq. i

8.2 Creating new critical points The cut and paste surgery in this section involves constructing a truncated complex plane, an abstract Riemann surface. The Integrability Theorem allows p However, when the for Riemann surfaces conformally equivalent to D, C or C. construction depends on a parameter we need to be extra careful. Previously we discussed several examples of surgery, depending on a parameter and inducing well-defined maps between parameter spaces. In all of them we constructed a model map by pasting different maps on one copy of the complex plane or the Riemann sphere.

8.2 Creating new critical points

321

The new feature of the surgery is the creation of an extra critical point, obtained by cutting out part of the original dynamical plane and applying the first return map on the remaining region. This kind of construction was originally done by Branner and Douady (see [BD, Part II]) to explain an observation by Yoccoz, concerning combinatorial similarities between quadratic polynomials in the 1{2-limb L1{2 of the Mandelbrot set and certain cubic polynomials. The construction was later generalized by the authors (see [BF1]), relating quadratic polynomials in any p{q-limb Lp{q of the Mandelbrot set to certain polynomials of degree pq ` 1q. We explain the original surgery and sketch in Section 8.2.4 that the same result can be obtained by pasting different maps on one complex plane, as observed by Shishikura. First we state the content of the main theorem in [BF1]. Then we explain the surgery in the simplest case (q “ 3). Finally we list some properties of the resulting maps between the parameter spaces; we point out the different steps in proving them, but refer to details in the original paper. For any q ě 3 consider the one-parameter family of polynomials of degree pq ` 1q ˙ ˆ z q Pq,λ pzq “ λz 1 ` q

where λ P “ C˚ .

These polynomials of degree pq ` 1q are characterized by having a fixed point at the origin of multiplier λ, a critical point of multiplicity pq ´ 1q at ´q, q mapped onto that fixed point, and a simple free critical point ω “ ´ q ` 1 . Let Cq denote the connectedness locus of this family, Cq “ tλ P C˚ | KPq,λ is connectedu. Notice that for any λ P Dzt0u the two critical points belong to KPq,λ , which is therefore connected. Hence Dzt0u Ă Cq . The unit circle is the analogue of the main cardioid in the Mandelbrot set, and analogously the r{s-limb Lq,r {s of Cq is defined as the connected component of Cq zD attached to the unit circle at e2π i r {s , where either r “ 0 or 0 ă r ă s and gcdpr, sq “ 1. The 0-limb Lq,0 is the focus in the following. Theorem 8.9 (Comparing limbs in the two families) For any q ě 3 and any p ă q such that gcdpp, qq “ 1 there exists a homeomorphism “ p,q : Lp{q ÝÑ Lq,0 , which is holomorphic in the interior of Lp{q .

322


Observe that the image p,q pLp{q q is independent of p. Hence we have the following corollary. Corollary 8.10 (Limbs of M of denominator q are homeomorphic) Given p{q and p1 {q with p, p1 ă q such that gcdpp, qq “ gcdpp1 , qq “ 1, the map r “ r p,p1 ,q “ ´11 ˝ p,q : Lp{q ÝÑ Lp1 {q p ,q is a homeomorphism, which is holomorphic in the interior of Lp{q . One can actually prove this corollary without going through the higher degree families (as observed by Dierk Schleicher at the time), using a cut and paste surgery without leaving the complex plane. This approach is carried out r p,p1 ,q between the limbs in [BF2] in order to prove that the homeomorphisms in the Mandelbrot set preserve their embeddings in the plane. The limbs L1{3 and L3,0 are shown in Figures 8.10 and 8.11 respectively.

Figure 8.10 The 1{3-limb L1{3 of the Mandelbrot set and the filled Julia set of z2 ` c for c “ ´0.127997 ` i 0.871489. The value of c is marked in the limb and the attracting 6-cycle is marked in the filled Julia set.

We organize the section as follows. First we give the characterizations of the polynomials Pq,λ in the 0-limb Lq,0 . Then we explain the surgery for p{q “ 1{3 turning a quadratic polynomial Qc in the 1{3-limb into a quartic polynomial P3,λ in the 0-limb L3,0 . We explain why this gives a well-defined


323

Figure 8.11 The 0-limb L3,0 of C3 , the connectedness locus of the family P3,λ , and the filled Julia set of P3,λ with λ “ 1,3 pcq “ 5.64 ` i 0.54 and c as in Figure 8.10. The value of λ is marked in the limb and the attracting 2-cycle is marked in the filled Julia set.

map “ 1,3 : L1{3 Ñ L3,0 . Finally we sketch how the construction varies with the parameter and what properties the map has. The last part is the most delicate.

8.2.1 The family of polynomials of degree pq ` 1q The 0-limb Lq,0 of the connectedness locus Cq of the family Pq,λ is the connected component of Cq zD whose closure is attached to the unit disc at the point λ “ 1 (see Figure 8.11, where q “ 3). Polynomials Pq,λ in Lq,0 share the following features. The repelling fixed point at z “ 0 is the landing point of a unique invariant ray. As a consequence, the multiple critical point at ´q is the landing point of q rays, which are preimages of the invariant ray. Let ψλ : CzD Ñ CzKPq,λ be the inverse of a choice of Böttcher coordinate, conjugating z ÞÑ zq `1 to Pq,λ and so that the invariant ray has argument 0. We shall denote dynamic rays of argument t by Rq,λ ptq or Rλ ptq if it does not lead to confusion. The preimages of Rq,λ p0q are Rq,λ pj {pq ` 1qq for j “ 1, . . . , q. The q rays landing at ´q partition the plane into q domains, which 0 “ V 0 (the one containing the critical point ω), V 1 “ we denote by Vq,λ q,λ λ q ´1

q ´1

Vλ1 , . . . , Vq,λ “ Vλ (see Figure 8.12, where q “ 3). The dynamics on these domains are as follows: 2:1

Vλ0 ÞÝÑ CzR q,λ p0q, 1:1

Vλi ÞÝÑ CzR q,λ p0q for i “ 1, . . . , q ´ 1.

(8.2)

324

Cut and paste surgeries with sectors Rλ p 34 q

Vλ2 Rλ p0q

0

Vλ0

Rλ p 12 q

´3

Vλ1 Rλ p 14 q

Figure 8.12 Sketch of the partition in the dynamical plane of Pq,λ for q “ 3. The plane is rotated by π to show the resemblance with the partition in the dynamical plane of a quadratic polynomial.

8.2.2 The surgery Starting from a quadratic polynomial Qc with c P L1{3 , we construct a map which has the topological properties of a polynomial Pλ “ P3,λ with λ “ λpcq in L0 “ L3,0 . We show that this determines a well defined map : L1{3 Ñ L0 with pcq “ λpcq. The rays landing at αc and ´αc have external arguments 1 2 4 1 9 11 7 , 7 , 7 and 14 , 14 , 14 respectively. Step 1: Cut and paste to define an abstract Riemann surface CTc For any Qc with c P L1{3 we start by cutting out the domains Vrc1 , Vrc2 and the external ray Rc p 27 q in between, and identifying the rays Rc p 17 q and Rc p 47 q equipotentially. This means that z1 „ z2 if z1 P Rc p 17 q and z2 P Rc p 47 q both have the same potential. Let Rrc denote the set of equivalence classes on the two identified rays, the gluing curve. We obtain the truncated plane CTc “ Vc0 Y Vc1 Y Vc2 {„ . The space CTc has a natural structure as a Riemann surface isomorphic to C (see Exercise 8.2.1). Step 2: Define the first return map on CTc In the truncated complex plane CTc p1q

we let fc denote the first return map, that is, we iterate each point in CTc by Qc until it returns to CTc . On the wedges tVci ui “0,1,2 the map takes the form $ 3 0 ’ ’ &Qc in Vc , p1q fc :“ Q2c in Vc1 , ’ ’ %Q in V 2 . c

c


325

We observe that this map shares many features with the polynomial Pλ . Indeed it is holomorphic in Vc0 Y Vc1 Y Vc2 and maps these sets in the same fashion as (8.2). Roughly speaking it has degree four and the point at ´αc is now topologically a double critical point, since any neighbourhood of ´αc is mapped 3:1 to a neighbourhood of αc . However, the first return map is discontinuous along the three rays landing at ´αc . The discontinuity is of shift type. As 1 q from Vc2 , the an example, if we approach a point of potential ρ on Rc p 14 r images tend to the point of potential 2ρ on the gluing curve Rc . However, if we approach the same point from Vc0 , the images tend to the point of potential 23 ρ on Rrc (see Figure 8.13). As a consequence, the preimage under the first return map of an equipotential curve in CTc of potential ρ will consist of three pieces of equipotential curves of potential 2´3 ρ, 2´2 ρ, 2´1 ρ in Vc0 , Vc1 , Vc2 respectively.

p1q fc

Q3c

1 q Rc p 14

Vc2

Qc

Vc0 Rrc

Rc p0q

´αc

αc

Vc1 Q3c

Q2c

9 q Rc p 14

Figure 8.13 Sketch of the first return map, showing how the preimage of an equipotential curve in CTc consists of three disconnected pieces of equipotential curves. There is a shift discontinuity along the rays landing at ´αc .

p1q

p2q

Step 3: Modify fc to a quasiregular map fc on a restricted region For an arbitrary but fixed choice of potential η ą 0 let Xc denote the restricted region in CTc of points of potential less than or equal to η. On a neighbourhood of each ray of discontinuity (a sector) we modify the first return map to make it continuous. The modifications will be quasiconformal maps, keeping the map unchanged on the truncated KcT :“ Kc X CTc . We use bounded log-Böttcher sectors as neighbourhoods of these rays. As explained in Section 8.1 they can be defined in the right half plane Hr , once and for all, independently of c, with a choice of slope s compatible with the

326

Cut and paste surgeries with sectors θ Rc p 27 q

11 14

Rc p 17 q αc

Rc p 47 q

ψc ˝ exp

9 14 4 7

Sp 47 q

1 q Rc p 14 2 7

Sp 27 q

1 7 1 14

Sp 17 q

´αc 9 q Rc p 14

Rc p 11 14 q

η {4

η {2

η

Figure 8.14 Sectors overlap, but not if we restrict to the inside of a given equipotential curve and an appropriate slope. The right figure shows the sectors in the right half plane. The map ψc ˝ exp brings them to the dynamical plane of Qc .

potential η so none of the sectors and their iterated preimages overlap (see Figure 8.14). Let Sc pθq denote the sectors for the six external arguments corresponding to the rays landing at αc and ´αc . The sectors Sc pθ q, θ “ 17 , 47 , merge into a single one, which we denote by Src . It consists of half of Sc p 17 q and half of Sc p 47 q, joined along the gluing curve Rrc (see Figure 8.15). This sector Src is ‘invariant’ under the first return map, in the sense that points in Src remain in p1q Src until they leave the region Xc . Moreover, fc is holomorphic in Src , where it equals Q3c . Consider the sectors around the rays landing at ´αc . Each of them is mapped, under the first return map, to Src , with a shift discontinuity along the rays. We will modify the first return map inside these sectors. First choose a C 2 -Jordan curve γc , which coincides with the three pieces of equipotentials of potential 2´3 η, 2´2 η, 2´1 η in Xc outside the three sectors landing at ´αc , and connects them with C 2 -Jordan arcs inside these three sectors (see Figure 8.15). Let Xc1 Ă Xc denote the domain bounded by γc . Note that the choice of curve can be made once and for all, independently of c, in Hr , and then transported to the dynamical plane by ψc ˝ exp. Let Sc1 denote the part of the sectors landing at ´αc , which is inside Xc1 , that is ´ ¯ 1 9 q Y Sc p 14 q Y Sc p 11 q X Xc1 . Sc1 :“ Sc p 14 14


Xc

γc “ BXc1

? Src

327

1 q Sc1 p 14

Xc1

rc R ´αc

αc

Sc1 p 11 14 q

9 q Sc1 p 14

Figure 8.15 Sketch of the domain Xc and its preimage Xc1 under the modified map p2q fc . Shadowed regions show the sector Src and its three preimages, disjoint from Src , under the same map.

We now modify the first return map on the sectors Sc1 landing at ´αc . We use the following extension lemma. p2q

Lemma 8.11 For some K ą 1 there exists a K-quasiregular map fc Xc which satisfies the following conditions: p2q

: Xc1 Ñ

p1q

(a) fc pzq “ fc pzq if z P Xc1 zSc1 ; p2q

1 9 11 , 14 , 14 , the map fc (b) for θ “ 14 K-quasiconformal homeomorphism.

: pSc pθ q X Xc1 q Ñ Src

is

a

The proof of this lemma is left as an exercise (see Lemma 8.2 and Exerp2q cise 8.2.2). By working on the right half plane, fc can actually be defined so that it depends holomorphically on the parameter c (see Exercise 8.2.3). p3q

Step 4: Changing the complex structure to obtain a holomorphic map fc We now define a Beltrami form μc on Xc as follows. First define μc “ μ0 on p2q follows that μ0 is invariant. Src . Since fc is holomorphic on´Src , it ¯ p2q ˚

We proceed by defining μc “ fc μ0 on Sc1 . The dilatation is K, from Lemma 8.11. Observe that orbits pass through these three sectors at most once, since they are all mapped to Src which is invariant. Hence we can spread μc p2q recursively by the dynamics of fc , that is

328

Cut and paste surgeries with sectors $ μ ’ ’ ’´ 0 ¯ ˚ ’ ’ & fcp2q μ0 μc :“ ´ p2q ¯˚ ’ ’ pfc qn μc ’ ’ ’ % μ0

on Src , on Sc1 , p2q

on pfc qń pSc1 q, for n ě 1, Ť p2q on Xc1 z n pfc qń pSc1 q. p2q

In particular, μc “ μ0 on KcT . Observe that μc is invariant under fc by construction, and it has bounded dilatation (precisely K), since all pullbacks, except the first one, are done by holomorphic maps. Hence, applying the Integrability Theorem we obtain quasiconformal home˝

omorphisms φc : Xc Ñ D such that φc˚ μ0 “ μc . Observe that, since μc “ μ0 on KcT , it follows that Bφc “ 0 on KcT . ˝

p3q

p3q

Set Dc1 :“ φc pXc1 q and define fc : Dc1 Ñ D as fc Then the following diagram commutes: ˝

f

p2q

p2q

:“ φc ˝ fc

˝ φc´1 .

˝

c ÝÑ pXc , μc q pXc1 , μc q ÝÝÝ § § §φ § φc đ đ c

f

p3q

c pDc1 , μ0 q ÝÝÝ ÝÑ pD, μ0 q

p3q

and the map fc

is holomorphic (see Key Lemma for surgery (Lemma 1.39)). p3q

Step 5: Obtaining a polynomial in L0 Since fc is quasiconformally conp2q p3q jugate to the map fc , the properties of the latter are preserved. Hence, fc is a ramified holomorphic covering of degree four, with two critical points (at z “ 0 and at φc p´αc qq. Moreover, Dc1 is relatively compact in D. It follows p3q

that fc is polynomial-like of degree four. By the Straightening Theorem (Theorem 7.4) there exists a hybrid equivap3q lence χc , conjugating fc to an actual polynomial fc of degree four. Due to the conjugacy, we conclude for fc that: • • • • •

zpcq :“ χc pφc pαc qq is a repelling fixed point, z2 pcq :“ χc pφc p0qq is a simple critical point, z3 pcq :“ χc pφc p´αc qq is a double critical point, z1 pcq :“ fc pz2 pcqq, and z1 pcq is the landing point of a unique fixed ray.


329

Conjugating by the affine map which sends z1 pcq to 0 and z3 pcq to ´3, we obtain a unique polynomial in the 0-limb of C3 , that is fc „ Pλpcq , with λpcq P L0 . affine

See Exercise 8.2.4. Finally we need to check that λpcq is independent of the many choices made during the construction. Proposition 8.12 ( is well defined) The map : L1{3 Ñ L0 defined by c ÞÑ λpcq is well defined. Proof Suppose we made different choices: an equipotential ηr, a slope r s, r 1 , quasiconformal extensions on the sectors a Jordan curve γrc bounding X c r 1 landing at ´αc , resulting in the quasiregular map frc p2q , the induced in X c

p3q

r c and, after integration, the holomorphic map frc , which is Beltrami form μ p2q p2q hybrid conjugate to a unique polynomial Pr . Then fc and frc are hybrid λpcq

equivalent. To see this, observe that the choices only affect the shape of the sectors and the modification of the first return map on those sectors landing at ´αc . Construct recursively a quasiconformal conjugacy between the maps on the sectors (restricting to an outer level corresponding to the smallest of the equipotentials η and ηr, and enlarging to the largest of the slopes s and r s ), so that these conjugating maps equal the identity on the side boundaries of the sectors. p2q p2q Outside the sectors the maps fc and frc coincide with the first return map. Let hc denote the constructed quasiconformal conjugacy on sectors, extended by the identity outside the sectors. Then hc is a hybrid equivalence. p3q p3q It follows that the holomorphic maps fc and frc are also hybrid equiv´1 alent by the conjugating map φrc ˝ hc ˝ φc , and so are the polynomials Pλ

and Prλ obtained by straightening. Since the filled Julia sets, KPλpcq and KPrλpcq , are connected, the polynomials Pλpcq and Prλpcq are affine conjugate (Proposition 3.53). The affine conjugacy must fix 0 and the double critical point ´3 and is therefore the identity. Hence λpcq “ r λpcq. We end this part with a series of remarks, which are consequences of the fact that, on the filled Julia set, the new polynomial acts as the first return map of the quadratic polynomial we started with. Remarks 8.13 (Properties of corresponding polynomials) Let c P L1{3 . (1) If the critical point 0 of Qc is attracted to an attracting cycle, so is the free critical point ω of Pλpcq . The periods of the cycles are not the same, since the first return map skips the first q ´ 1 points in the critical orbit of

330


Qc (and maybe others). However, the multipliers are the same, since all conjugacies are holomorphic in the interior of the filled Julia set. (2) For the same reason, if the critical point of Qc is attracted to a parabolic cycle, so is the free critical point of Pλpcq . Also if Qc has a periodic orbit of Siegel discs, so does Pλpcq . (3) If the critical point of Qc is eventually mapped onto a repelling periodic cycle, so is the free critical point of Pλpcq . 8.2.3 Properties of : L1{3 Ñ L0 – dependence of the parameter We proceed to sketch how to study the map in the parameter plane. All details can be found in [BF1]. We explain how to prove continuity and bijectivity of . Since L0 is compact, the continuity of ´1 follows. Surjectivity To prove that is surjective one needs to perform an inverse surgery. That is, start with a parameter λ P L0 and end up with a quadratic polynomial Qcpλq with cpλq P L1{3 , such that pcpλqq “ λ. This surgery requires to cut the dynamical plane of Pλ along the invariant ray Rλ p0q, open it up and add pieces in order to glue back what we previously removed. p2q To prove injectivity of , we observe that if pcq “ pc1 q, then fc and p2q

fc1 are hybrid equivalent. This means that the first return maps are conformally conjugate in the interior of the truncated filled Julia sets. One can prove that this implies c “ c1 . Continuity The proof of continuity of is the most delicate part, and it makes a difference to be working on an abstract Riemann surface, which varies with the parameter. A na¨ıve approach would be to start by examining the following diagram: ˝

f

p2q

Xc1 § § φc đ

c ÝÝÝ ÝÑ

Dc1 § § χc đ

c ÝÝÝ ÝÑ

f

p3q

˝

Xc § §φ đ c D § §χ đ c

Pλpcq

χc pDc1 q ÝÝÝÝÑ χc pDq If we consider for a moment Xc and Xc1 as subsets of the complex plane, we p2q

can argue that the Beltrami form μc and the map fc

vary holomorphically


331

with the parameter c, since both were constructed once and for all in the complex plane and then transported by the Böttcher coordinates, which vary holomorphically with the parameter. However, this works on the basin of infinity, which itself is moving with c. So we can not make the same statement for all points in the dynamical plane, since a given point z might belong to Kc for a given c, but might be in the basin of infinity for arbitrarily nearby parameters. To make things worse, once the identifications are made, the Riemann surface Xc varies with c. If we ever wanted to apply the Integrability Theorem with parameters, to conclude that φc depends continuously on c, we would need to uniformize Xc to the unit disc. But then we should take into account the dependence of the uniformizing map with respect to c. All these observations indicate that this is not the best approach. In the interior of L1{3 we can prove not only continuity of but also conformality. This is proved by using Remark 8.13 above. Let M denote a hyperbolic component of L1{3 . It is mapped into some hyperbolic component in L0 , say Cq . Let M : D Ñ M and Cq : D Ñ Cq be the respective multiplier maps. They are both conformal isomorphisms. (For Cq this follows by applying the surgery in Section 4.1 to Pλ with λ P Cq , noticing that the free critical point ω is simple.) Therefore 1 |M “ Cq ˝ ´ M

is a conformal isomorphism. It is conjectured that the interior of M is the union of hyperbolic components. As long as the conjecture is not proved, we have to consider the possibility of non-hyperbolic components, also known as queer. In this case a similar argument would show that is continuous and holomorphic in such components. One would use the continuous parametrization obtained when deforming the invariant line field supported on the Julia set (in fact, another soft surgery construction). The remaining case is the continuity on the boundary of L1{3 . It is proved via sequences. Suppose c P BL1{3 , and cn Ñ c, with cn P L1{3 . Set λ “ pcq. Then λ must belong to BL0 , because the inverse map ´1 is mapping the interior of L0 to the interior of L1{3 . If we set λn “ pcn q, we must prove that all convergent subsequences of tλn uně0 have λ as limit. Suppose we have a λ. convergent subsequence (which we denote by λn again), such that λn Ñ r The key is the following rigidity property: if Prλ and Pλ are quasiconformally conjugate, and λ P BL0 , then λ “ r λ. We leave the proof of this rigidity fact as

332


an exercise (see Exercise 8.2.5 or [BF1, Lem. 5.22]). It remains to show that Prλ and Pλ are quasiconformally conjugate. We claim there exists a convergent subsequence Pλnk ÝÑ Pλ˚ , k Ñ8

such that Pλ˚ is quasiconformally conjugate to Pλ . To see the claim, let φn :“ φcn and χn :“ χcn denote the corresponding integrating maps for each of the parameters. Observe that the dilatation of φn is p2q uniformly bounded by K for all n, since fcn was constructed once and for all in the right half plane. To find convergent subsequences of these maps, we first need to fix a common domain for all of them. We use the following lemma. Lemma 8.14 Given c P L1{3 , there exists r “ rpcq ą 0 such that for each t P Dr one can find a quasiconformal homeomorphism Ht : Xc ÝÑ Xc`t , with dilatation bounded by Kt “

r ` |t| r ´ |t|

and H0 “ id|Xc , making p2q

p2q

Ht´1 ˝ fc`t ˝ Ht ÝÑ fc uniformly on compact subsets of Xc as t Ñ 0.

We refer to [BF1, Lem. 5.22] for the proof, which uses holomorphic motions and the λ-lemma. Picking the right tn values so that c ` tn “ cn (with n large enough) we have quasiconformal homeomorphisms Hn :“ Htn : Xc ÝÑ Xcn with dilatation Kn :“ Ktn . Then the maps tφn ˝ Hn : Xc ÝÑ Dun form a K 1 -quasiconformal family, with K 1 ă KKn for all n. We normalize by setting φn Hn pαc q “ 0. Then, the maps tφn Hn un form an equicontinuous family and, by Arzelà– Ascoli’s Theorem, there exists a convergent subsequence φnk ˝ Hnk Ñ φ ˚ . The limit map is a K 1 -quasiconformal homeomorphism. The map p3q

f˚

p2q

:“ φ ˚ ˝ fc

˝ pφ ˚ q´1 ,


333

being the uniform limit of ´ ¯ p3q p2q fnk “ φnk ˝ Hnk ˝ Hn´k 1 ˝ fnk ˝ Hnk ˝ Hn´k 1 ˝ φn´k1 , is holomorphic and p3q φ ˚

f˚ p3q

Hence fc

p2q φc

„ qc fc

p3q

„hb fc .

p3q

„qc f˚ . p3q

Abusing notation, we have a sequence of polynomial-like mappings fn p3q converging to f˚ uniformly on compact subsets of D, and a sequence of hybrid equivalences χn , with uniform bounded dilatation, since the moduli of the exterior annuli are uniformly bounded. We may assume χn is defined in all of D and normalized so that χn p0q “ 0 and χn pϕn pHn p´αc qqq “ ´3. Then, we can find a subsequence χnk converging to a quasiconformal homeomorphism χ ˚ . The polynomials Pλnk tend to the polynomial p3q

P ˚ :“ χ ˚ ˝ f˚

˝ pχ ˚ q´1 .

Therefore χ˚

p3q

P ˚ „ qc f˚

p3q χc

„qc fc

„hb Pλ ,

so it follows that Pλ „qc P ˚ . Finally, note that P ˚ must be of the form Pλ˚ . Hence, Pλn Ñ P ˚ implies λ, and therefore that λn Ñ λ˚ , which proves the claim. It follows that λ˚ “ r we have proved Pλ „qc Prλ . This concludes the proof of continuity at boundary points. 8.2.4 Alternative surgery construction without truncating the plane We may obtain the homeomorphism : Lp{q Ñ Lq,0 by an alternative surgery construction. The idea is due to Shishikura. One may do the construction in the original dynamical plane. As before, starting from any Qc with c P Lp{q the construction should lead to a polynomial-like map of degree pq ` 1q, which is hybrid equivalent to Pλ where λ “ p{q pcq P Lq,0 . In this section we sketch the main steps in this alternative surgery, in the simplest case q “ 3. For c P L1{3 , choose a potential η ą 0 and a slope s so that the bounded sectors Sc pθ q “ Scs pθ q of slope s and argument θ “ 17 , 27 , 47 are pairwise disjoint. Let Yc denote the domain in the dynamical plane of Qc of potential less than η.

334


Spc

αc

1 q Sc1 p 14

´αc

Yc

9 q Sc1 p 14

Sc1 p 11 14 q Figure 8.16 Sketch of the domain Yc and its preimage Yc1 under the modified map Fc , and the sector Spc and its three preimages, disjoint from Spc , under the same map.

Let Yc1 Ă Yc be the simply connected domain inside Yc , which is bounded by the equipotential of potential η{23 everywhere except in Vc1 Y Vc2 Y Sc1 where the bounding curve is chosen similar to the bounding curve of Xc1 (compare Figures 8.16 and 8.15). Set ´ ´ ¯ ´ ¯ ´ ¯¯ Spc “ Sc 17 Y Vrc1 Y Rc 27 Y Vrc2 Y Sc 47 X Yc and Spc1 “ Spc X Yc1 . The sector Spc plays a role similar to the sector Src in the truncated case. The goal is to define a modified map Fc : Yc1 Ñ Yc whose dynamics resembles the dynamics of a polynomial in L3,0 . The map is defined so that it is quasiregular and there exists an Fc -invariant Beltrami form μc of bounded dilatation, and so that after integrating we obtain a polynomial-like map hybrid equivalent to a unique polynomial Pλ , where λ “ 1{3 pcq P L3,0 . We proceed to define the modified map Fc from each of the sectors Spc1 pθ q to Spc . The sector Spc contains part of the filled Julia set Kc and is certainly not invariant under Q3c . However, the sector is locally invariant under Q3c . Let ϕc : c Ñ D be a linearizing coordinate around αc conjugating Qc to z ÞÑ λc z, where λc “ Q1c pαc q is the multiplier of Qc at αc . Note that


335

Spc X c “ Q3c pSpc q X c . We use this fact and construct the map Fc so that it coincides with Q3c in a neighbourhood of αc . For notation compare with Figure 8.17. Let A0L denote the intersection point j j ´1 3 in BYc X BL Spc , and define AL recursively as the preimage of AL under Q´ c j belonging to BL Spc . Define the points AR belonging to BR Spc similarly. Choose N `1 and AL belong to c . Choose γ N as a C 2 -curve inside c N so that AN L{R {R N N `1 denote the preimage of γ N under Q´3 connecting AN c L and AR , and let γ 3 belonging to c . Set Fc :“ Qc on the domain in Spc inside the curve γ N `1 . Let γ 0 and γ 1 be the pieces of equipotentials of potential η and η{23 respectively belonging to the closure of Spc and choose C 2 -curves for j “ 2, . . . , N ´ 1, also belonging to the closure of Spc , pairwise disjoint, and numbered as indicated in Figure 8.17. Denote the closed quadrilateral bounded p j . For j “ 1, . . . , N choose C 2 by γ j , γ j `1 and pieces of BL{R Spc by Q diffeomorphisms from γ j to γ j ´1 . γ0 γ1 γ2 γ3 γ4

p1 Q p3 Q p4 p0 Q p2 Q Q Fc “ Q3c

γ5 αc 5 A A5R L

A4R A3R

A4L

1 2 AL A3L AL

A0L

c

A2R A1R 0 AR

Figure 8.17 Sketch of the subdivision of the sector Spc into pN ` 1q “ 5 p j , j “ 1, . . . , N ` 1, the bounding curves γ j , j “ 0, 1, . . . , N ` 1, quadrilaterals Q j j j connecting the points AL and AR respectively. The points AL{R belong to BL{R Spc j

j ´1

and satisfy Q3c pAL{R q “ AL{R for j “ 1, . . . , N ` 1. The curves γ0 and γ1 are equipotentials. The curves γ2 and γ3 are arbitrary, while γ5 is the preimage of γ4 under Q3c .

pj Ñ Q p j ´1 be a quasiconformal map, which extends the given Let Fc : Q boundary maps, and suppose the map is Kj -quasiconformal for some Kj . This defines Fc : Spc1 Ñ Spc . To complete the definition of the modified map Fc we need to define it on the remaining part of Yc1 . This is done similarly to the original surgery explained above. The map Fc is defined as before on Yc1 zpSpc1 Y Sc1 q, i.e.

336

Cut and paste surgeries with sectors $ 3 ’ ’ &Qc Fc :“ Q2c ’ ’ %Q c

in Vc0 zpSpc1 Y Sc1 q, in Vc1 zSc1 , in Vc2 zSc1 .

1 9 11 Each of the sectors Sc pθ q with θ “ 14 , 14 , 14 are invariant under ´Q3c . Using this and the local invariance of Spc under Q3c it follows that there exist K 1 1 9 11 , 14 , 14 and some K 1 . (Comquasiconformal maps Fc : Sc1 pθ q Ñ Spc for θ “ 14 pare with Lemma 8.2, Lemma 8.5 and Exercise 8.2.2.) ŤN 1 1 pj Set K :“ K 1 ˆ N j “1 Kj , and set Z :“ Sc Y j “1 Q , the subset of Yc on which Fc is quasiconformal; everywhere else Fc is holomorphic. Observe that the orbits of Fc pass through Z at most pN ` 1q times. We can therefore define p 0 by the dynamics to all p c by spreading μ0 on Q an Fc -invariant Beltrami form μ Ť Ť8 ´ n ń p of n“0 Fc pSpc q and leave it as μ0 on the complement Yc1 z 8 n“0 Fc pSc q. The map Fc is K-quasiregular. p c we obtain a polynomialAfter integrating the Fc -invariant Beltrami form μ like map, which is hybrid equivalent to a unique polynomial Pλ with λ P L3,0 . It remains to justify that λ “ 1{3 pcq. It can be checked that Fc is hybrid

p2q

equivalent to fc from the original construction, using the identity map on p2q Xc zSpc and a quasiconformal map from Spc to Src conjugating Fc to fc . By the p2q

same arguments as in Proposition 8.12, the λ-values obtained from fc from Fc are the same.

and

Exercises Section 8.2 8.2.1 Prove that CT is a Riemann surface isomorphic to C. 8.2.2 Prove Lemma 8.11 by constructing quasiconformal extensions as in Lemma 8.2. 8.2.3 Show that the proof of Lemma 8.11 can be done independently of the parameter c. More precisely, define a map f p2q in Hr and conjugate by p2q ψc ˝ exp to obtain fc in Xc zKcT , varying holomorphically with c. 8.2.4 Let q ě 3. Show that any polynomial P of degree pq ` 1q with a fixed point at z “ 0, one simple critical point, and a second critical point at ´q of multiplicity q ´ 1 such that P p´qq “ 0, must be of the form P pzq “ λzp1 ` z{qqq , for some λ P C˚ . 8.2.5 Fix q ě 3 and λ P C˚ . Let Pλ pzq “ λzp1 ` z{qqq . Let Cq denote the connectedness locus of tPλ uλPC˚ . Suppose Pλ and Pλ1 are quasiconformally conjugate and that λ P BCq . Show that λ “ λ1 .

8.3 Embedding limbs of M into other limbs

337

Hint: Let μ denote the Beltrami form induced by the quasiconformal conjugacy in Kλ . Extend it to equal 0 outside Kλ . Consider μt “ tμ for appropriate values of t. Use μt to construct a family Pλpt q of polynomials, all quasiconformally conjugate to Pλ with λptq P Cq . Show that λptq is holomorphic and conclude that it is constant. Compare λp0q and λp1q to deduce, using Theorem 3.53, that λ “ λ1 . 8.2.6 Consider the two one-parameter families of cubic polynomials ´ z ¯2 P2,λ pzq “ λz 1 ` , λ P C˚ . PrA pzq “ Apz3 ´ 3zq, A P C˚ , 2 Note that for PrA the critical points are ˘1 and the critical orbits are symmetric w.r.t. 0. For P2,λ the critical points are ´q “ ´2 and ´q{pq ` 1q “ ´2{3, and the critical point ´2 is mapped to the fixed point 0. Show that PrA and P2,λ are semiconjugate by a quadratic polynomial, which sends both critical points of PrA to the critical point ´2{3 of P2,λ and the preimages of the fixed point 0 of PrA (that are different from 0) to the prefixed critical point ´2 of P2,λ . Conclude that the connectedness locus of the family PrA in the A-plane is the same as the connectedness locus of P2,λ in the λ-plane under the relation λ “ p3Aq2 . Figure 8.23 in Section 8.4 illustrates the connectedness locus of the family PrA , and hence also the connectedness locus of the family P2,λ in the λ-plane.

8.3 Embedding limbs of M into other limbs As in the preceeding section, we present a surgery where the topological part involves abstract Riemann surfaces. The basic construction was done by Branner and Douady (see [BD, part III]) in order to obtain an embedding of the 1/2-limb L1{2 into the 1/3-limb L1{3 of M, so that the image of the interval r´2, ´3{4s Ă L1{2 is a topological arc connecting the landing point of the external ray RM p1{4q to the root point of the limb L1{3 . The interval and the image arc are called the principal veins of L1{2 and L1{3 respectively. Recall from Chapter 3 that the Mandelbrot set is known to be connected and is conjectured to be locally connected. If the MLC Conjecture is proven, it follows immediately that M is arcwise connected, since a closed connected and locally connected set is always arcwise connected. Douady’s main motivation for the surgery was to provide support to the conjecture of arcwise connectivity.

338


The construction was later generalized in [R] showing the exixtence of other veins in M as topological arcs. In the following we concentrate on the original embedding of L1{2 into L1{3 , although it can easily be generalized to an embedding of L1{2 into an arbitrary p{q-limb, and to to an embedding of L1{q into L1{q `1 . The steps in the surgery are similar to those in Section 8.2. However, in this case, one needs to control the opening modulus of certain sectors. The common features for polynomials in Lp{q are given in Section 8.1.3. To distinguish c-values in L1{pq `1q from those in L1{q we shall use c P L1{q and λ P L1{pq `1q . Figure 8.18 shows the domains Vci , Vrci and Vλi , Vrλi and the external rays landing at ˘αc and ˘αλ respectively. Rc p 13 q

Rc p 16 q

Rλ p 17 q

Vrλ1 Vrc1

Vc0

Vc1

Vλ2

Rλ p 27 q

Rλ p 11 14 q

Vλ0 Vrλ2

Rc p 23 q

1 q Rλ p 14

Rc p 56 q

Vλ1 Rλ p 47 q

9 q Rλ p 14

Figure 8.18 Sketch of the partition of dynamical plane for c P L1{2 and λ P L1{3 .

Theorem 8.15 (Embedding of limbs) There exists a map q : L1{q Ñ L1{pq `1q , which is a homeomorphism onto its image and holomorphic in the interior of L1{q . The image N1{pq `1q :“ q pL1{q q consists of parameters λ P L1{pq `1q for which Qnλ p0q R Kλ X Vλ1 for all n P N.

8.3.1 The surgery Recall that the multiplier map : Ñ D from any hyperbolic component Ă M extends continuously to the boundaries. Set q : 1{q Ñ 1 1{pq `1q equal to ´ ˝ 1{q . It remains to define q : L1{q z1{q Ñ 1{pq `1q L1{pq `1q z1{pq `1q . In the following we restrict to q “ 2. Starting from a quadratic polynomial Qc with c P L1{2 z1{2 we construct a map which has the topological properties of a quadratic-like map, hybrid equivalent to a polynomial Qλ with λ “ λpcq P L1{3 .


339

Step 0: Restricting the polynomial to a quadratic-like map We restrict Qc to a quadratic-like map Qc : Wc1 Ñ Wc by a choice of potential η ą 0 so that Wc is bounded by the chosen equipotential. As explained in Section 8.1 we can choose sectors of an arbitrary slope s around the rays Rc p 13 q and Rc p 23 q in Wc , satisfying ηs ă 16 so that the bounded sectors Scs p 13 q and Scs p 23 q and their successive preimages are all disjoint. The sectors are outside the filled Julia set Kc . Applying Lemma 8.6 and Remark 8.7 we may choose η and s so that m pc, sq ă mpsq, where m pc, sq is the opening modulus of cs “ Vrc1 zpScs p 13 q Y Scs p 23 qq and mpsq the opening modulus of the two sectors of slope s, with vertex at αc and relative to Q2c . Note that ´αc is a fixed point of ´Q2c of the same multiplier as αc under Q2c and that the sectors of slope s around Rc p 16 q and Rc p 56 q are ´Q2c -invariant and of opening modulus mpsq. Hence, we can choose a middle sector SM around Rc p 56 q of slope sM ă s so that the opening modulus mod´αc pSM , ´Q2c q “ m pc, sq. We have subdivided the sector around Rc p 56 q into three subsectors SL , SM , SR so that SL Y SM Y SR “ Scs p 56 q X Wc1 (compare with Figure 8.19). Rc p 13 q

Rc p 16 q Scs p 16 q

Scs p 13 q

cs

αc

0

´αc

SL Scs p 23 q

Rc p 23 q

SR

SM

Rc p 56 q

Figure 8.19 The restriction Qc : Wc1 Ñ Wc with sectors and wedge as explained in Step 0.

Step 1: Cut and paste to obtain an abstract Riemann surface CE c We define by extending C as explained below (see Figure 8.20). the space CE c Slit along Rc p 23 q and open the slit up creating two copies of the ray which we denote by Rc p 23 qp1q Ă B Vrc1 and Rc p 23 q Ă BVc0 . Insert a copy of Vrc1 in between these two rays. The original set is denoted by pVrc1 qp1q and the copy by pVrc1 qp2q . The latter is bounded by a copy of each of the original rays, which we denote by Rc p 13 qp2q and Rc p 23 qp2q . We paste by identifying zp2q on Rc p 23 qp2q to the

340

Cut and paste surgeries with sectors Rc p 13 q

Rc p 16 q

pVrc1 qp1q Vc0 pc R

Rc p 23 qp1q „ Rc p 13 qp2q

Vc1

´αc

αc

pVrc1 qp2q Rc p 23 qp2q „ Rc p 23 q

Rc p 56 q

Figure 8.20 The extended plane CE c as explained in Step 1. The arrows show the p1q

discontinuity of the map fc

, as defined in Step 2.

point z on Rc p 23 q, which is on the same equipotential curve as zp2q , and by identifying zp2q on Rc p 13 qp2q to Qc pzqp1q on Rc p 23 qp1q . The identification of Rc p 23 q and Rc p 23 qp2q is still be denoted Rc p 23 q, while the gluing of Rc p 23 qp1q with Rc p 13 qp2q is denoted by Rpc . The sectors around Rc p 13 q and Rc p 23 q are still denoted by Scs p 13 q and Scs p 23 q, while the sector around Rpc is denoted by Spc . The extended space CE c has a natural structure as a Riemann surface isomorphic to C. p1q

Step 2: Define the map fc on the p1q fc by $ ’ Q pzq ’ ’ c ’ &Q pzqp1q p1q c fc pzq :“ ’zp2q ’ ’ ’ % Qc pzq

extended plane We define the map z P Vc1 , z P Vc0 , z “ zp1q P pVrc1 qp1q , z “ zp2q P pVr 1 qp2q . c

It is holomorphic on its domain of definition. By continuous extension, Rc p 16 q is mapped to Rc p 13 q, and Rc p 13 q to Rpc , while Rc p 56 q is a line of discontinuity. p1q Note that fc pzq approaches Rc p 23 q and Rpc respectively, as z approaches Rc p 5 q within V1 pcq and within V 0 respectively. Points in pVr 1 qp2q only have

c c p1q one preimage under fc . p2q p2q Set Kc :“ pKc X pVrc1 qqp2q , and let KcE :“ Kc Y Kc denote p1q filled Julia set. Note that KcE is forward invariant under fc . 6

p1q

p2q

the extended

Step 3: Modify fc to a quasiregular map fc on a restricted domain We start by choosing the domain of definition Xc1 and the range Xc of the modified


341

p2q

map fc (compare with Figure 8.21). The boundary of Xc is chosen as a C 2 -curve, which coincides with the boundary of WcE except in the interior of E pVrc1 qp2q Y Scs p 23 q, where it is chosen to lie in WcE zWc1 . The boundary of Xc1 coincides with the boundary of W 1E except in the interior of pVr 1 qp1q Y Spc , c

p1q

where it is determined by fc p1q

fc

c

mapping Xc1 onto Xc . We shall modify the map

in the sector Scs p 56 q X Wc1 . Rc p 13 q Traces

of

equipot. η{2

Scs p 13 q

pcs qp1q Spc

Rpc

Traces

´αc

αc

SL

pcs qp2q Scs p 23 q

of

equipot. η{2 Traces

SR SM

of

equipot. η

Rc p 56 q

Rc p 23 q

Figure 8.21 The figure illustrates where Xc1 and Xc differ from Wc1E and WcE and p1q

shows the sectors that are important when modifying fc

p2q

to fc

.

Let R : SM Ñ pcs qp2q X Xc denote the unique Riemann mapping, which when extended continuously to the boundaries is mapping corners to corners and so that the vertex ´αc of SM is mapped to the vertex αc of pcs qp2q . Since we arranged the opening moduli to be the same, the boundary maps on the sides are near translations when expressed in log-linearizing coordinates(recall Lemma 8.4). By Lemma 8.5 we can find quasiconformal maps p2q

fc

: SL Ñ Spc X Xc

and

p2q

fc

: SR Ñ Scs p 23 q X Xc p1q

which extend the boundary maps given by BRL{R and fc . This finishes the p2q

construction of fc : Xc1 Ñ Xc , a quasiregular map of degree 2. Note that orbits pass through the side-sectors SL{R at most once, while orbits may pass through SM infinitely often.

342

Cut and paste surgeries with sectors p3q

Step 4: Changing the complex structure to obtain a holomorphic map fc We define a Beltrami form μc on Xc as follows. Set μc :“ μ0 on the three sectors Scs p 13 q, Spc , Scs p 23 q in Xc . The union of these form a forward invariant set ´ ¯ p2q p2q ˚ μ0 on SL{R . Since orbits for fc whenever defined. Then set μc :“ fc pass at most once through SL{R we can spread μc by successive pullbacks Ť p2q p2q by fc to ně1 pfc qń pSL{R q. Set μc :“ μ0 on the remaining part of Xc , Ť p2q i.e. on Xc z ně0 pfc qń pSL{R q. By construction μc is f p2q -invariant. The dilatation of μc is bounded by the dilatation of μc in SL{R , since all pullbacks, except the first one, are done by holomorphic maps. By applying the Integrability Theorem we obtain a quasiconformal home˝

˝

omorphism φc : Xc Ñ D such that φc˚ μ0 “ μc . Set Dc1 :“ φpXc1 q and define p3q

fc

p3q

: Dc1 Ñ D as fc

p2qq

p3q

:“ φc ˝ fc ˝ φc´1 . Then fc Ť p2q In particular, μc “ μ0 on ně0 pfc qń pKcE q.

is holomorphic.

p3q

Step 5: Obtaining a polynomial in L1{3 It follows from above that fc : Dc1 Ñ D is a quadratic-like mapping with a critical point at φc p0q. The critical p3q

orbit belongs to φc pKcE q. Therefore the filled Julia set of fc is connected. By the Straightening Theorem 7.4 there exists a hybrid equivalence χc , conjugatp3q ing fc to a unique quadratic polynomial Qλ with λ P L1{3 . Proposition 8.16 (The map is well defined) The map : L1{2 Ñ L1{3 defined by c ÞÑ λpcq is well defined. Its image satisfies pL1{2 q “ N1{3 :“ tλ P L1{3 | Qnλ p0q R Vλ1 for any n ě 0u. Moreover, if Qc has a periodic orbit, which is attracting, parabolic or Siegel respectively, then so does Qλpcq . Proof (Sketch) Suppose we made other choices throughout the construction (a different equipotential level, slope, extensions, etc.), and obtained a p2q p2q map frc . Then there is a quasiconformal conjugacy hc between fc and p2q frc which is actually a hybrid equivalence. It follows that the holomorp3q p3q phic maps fc and frc are also hybrid equivalent by the conjugating map ´1 φrc ˝ hc ˝ φc , and so are the polynomials Pλ and Pr obtained after straightλ

ening. Since the filled Julia sets, KPλpcq and KPrλpcq , are connected, the polynomials Pλpcq and Prλpcq are affine conjugate (Proposition 3.53). Because of the normalizations the affine conjugacy must be the identity and therefore λpcq “ r λpcq.

8.4 Intertwining surgery

343

8.3.2 Concluding remarks To end the proof of the Embedding Theorem 8.15 for q “ 2 one needs to prove that the map : L1{2 Ñ N1{3 is a bijection and a homeomorphism. To prove surjectivity one performs an inverse surgery, so that starting from λ P N1{3 one ends up with Qcpλq P L1{2 satisfying pcpλqq “ λ. In the inverse construction we either start by removing part of the plane corresponding to what we added before, or we apply a construction similar to the alternative surgery in Section 8.2.4. For details we refer to [BD].

8.4 Intertwining surgery 1 Adam Epstein and Michael Yampolsky

8.4.1 Cubic connectedness locus Our discussion of the parameter space of cubic polynomials follows the detailed presentation by Milnor in [Mi3]. Observe that every cubic polynomial is affine conjugate to a map of the form Fa,b pzq “ z3 ´ 3a 2 z ` b,

(8.3)

with critical points a and á. This normal form is unique up to conjugation by z ÞÑ ´z, which interchanges Fa,b and Fa,´b . The ordered pair of complex numbers A “ a 2 and B “ b2 parametrizes the space of cubic polynomials modulo affine conjugacy. The cubic connectedness locus is the set C Ă C2 of all ordered pairs pA, Bq for which the corresponding polynomial Fa,b has connected Julia set. As in the quadratic case, the connectedness locus is compact and connected with connected complement. These results were obtained by Branner and Hubbard [BH] who showed moreover that this set is cellular, the intersection of a sequence of strictly nested closed discs. On the other hand, Lavaurs [Lav] proved that C is not locally connected. Milnor distinguishes four different types of hyperbolic components, according to the behaviour of the critical points: adjacent, bitransitive, capture and disjoint [Mi3]. We are exclusively interested in the last possibility: a compo˝

nent H Ă C is of disjoint type Dm,n if Fa,b has distinct attracting periodic orbits with periods m and n for every pa 2 , b2 q P H. By definition, the Pern pλqcurve consists of all parameter values for which the cubic polynomial Fa,b has 1 This section is a revised version of the paper [EY] and is reproduced with the permission of the

Annales Scientifiques de l’ENS.

344


a periodic point of period n and multiplier λ. The geography of Per1 p0q was studied in [Mi4] and [Fau]. Notice that any cubic polynomial with real coefficients is affine conjugate to a map of the form (8.3) with A, B P R, that is a, b P R Y i R. Thus we may consider the connectedness locus of real cubic maps, the set of pairs pA, Bq P R2 such that JFa,b is connected. This locus CR is also compact, connected and cellular [Mi3]. We refer the reader to Figure 8.22 which was generated by a computer program of Milnor. The real slices of various hyperbolic components are rendered in different shades of grey. Certain disjoint type components are indicated, as are the curves Per1 p1q X CR and Per2 p1q X CR .

1,1

2,2

Figure 8.22 Connectedness locus CR in the real cubic family.

To avoid ambiguities arising from the choice of normalization, we will actually work in the family of cubics PA,D pwq “ Apw 3 ´ 3wq ` D, A ‰ 0, with marked critical points ´1 and `1. The reparametrization C˚ ˆ C Q pA, Dq ÞÑ pA, AD 2 q “ pA, Bq P C˚ ˆ C is branched over the symmetry locus B “ 0 consisting of normalized cubics which commute with z ÞÑ ´z (see Figure 8.23). In particular, C# “ tpA, Dq Ă C˚ ˆ C| JPA,D is connectedu is a branched double cover of C X pC˚ ˆ Cq. The marking of critical points allows us to label the attracting cycles of maps in disjoint type components


345

Figure 8.23 Symmetry locus in the family PA,D .

H Ă C# , and we denote the corresponding multipliers λ˘ H pA, Dq. It is shown in [Mi5] that the maps H : H Ñ D ˆ D given by ` H pA, Dq “ pλ´ H pA, Dq , λH pA, Dqq

are biholomorphisms. The omitted curve A “ 0, consisting of maps with a single degenerate critical point, is irrelevant to the discussion of disjoint type components. This useful change of variable has the unfortunate side-effect that the values pA, Dq P R˚ ˆ R only account for the first and third quadrants of the real pA, Bq-plane, the second and fourth quadrants being parameterized by R˚ ˆ i R. We are therefore unable to furnish a faithful illustration of the entire locus C#R “ tpA, Dq| pA, AD 2 q P CR u.

8.4.2 Outline of the results In the picture of the real cubic connectedness locus (Figure 8.22) one observes several shapes reminiscent of the Mandelbrot set. We quote Milnor ([Mi3]): these embedded copies tend to be discontinuously distorted at one particular point, namely the period one saddle node point c “ 1{4, also known as the root of the Mandelbrot set. The phenomenon is particularly evident in the lower right quadrant, which exhibits a very fat copy of the Mandelbrot set with the root point stretched out to cover a substantial segment of the saddle-node curve Per2 p1q. . . . As a result of this stretching, the cubic connectedness locus fails to be locally connected along this curve.

346


In this section, we present the following results of [EY] (see also Ha¨ıssinsky [Ha1]), which explain the appearance of these distorted copies of the Mandelbrot set embedded in CR . For p{q P Q{Z we consider the set Cp{q Ă C# consisting of cubic polynomials for which 2q distinct external rays with combinatorial rotation number p{q land at some fixed point ζ . As there can be at most one such point, the various Cp{q are disjoint. Each Cp{q is in turn the disjoint union of subsets Cp{q,m indexed by an odd integer 1 ď m ď 2q ´ 1 specifying how many of these rays are encountered in passing counterclockwise from the critical point ´1 to the critical point `1. In particular, C0 consists of those cubics in C# whose fixed rays RA,D p0q and RA,D p 12 q land at the same fixed point. Recall from Section 8.1.3 that the p{q-limb, Lp{q , of the Mandelbrot set is the connected component of Mz0 whose closure is attached to 0 at the point rootp{q :“ γ0 pe2π i p{q q, its root point, and that p{q denotes the hyperbolic component in Lp{q with root point rootp{q . Theorem 8.17 (Main Theorem on Quadratic Intertwining) Given p{q and m as above, there exists a homeomorphic embedding hp{q,m : Lp{q ˆ Lp{q ÝÑ Cp{q,m which maps the boundary into the boundary. The product of hyperbolic components p{q ˆ p{q is mapped by the embedding onto a component Hp{q,m of type Dq,q . We note that Hp{q,m is the unique Dq,q component contained in Cp{q,m as will follow from Theorem 8.35. The restriction of hp{q,m to p{q ˆ p{q is easily expressed in terms of the multiplier map p{q : p{q Ñ D, which associates the parameter with the multiplier of the attracting cycle 1 hp{q,m pc, crq “ ´ H

p{q,m

p p{q pcq, p{q pr cq q.

Discontinuity of hp{q,m at the corner point p1, 1q is a special case of a phenomenon studied by one of the authors [Ep1]. Theorem 8.18 (Discontinuity at corner point) Each algebraic homeomorphism Hp{q,m : Hp{q,m Ñ D ˆ D


347

extends to a continuous surjection H p{q,m Ñ D ˆ D. The fibre over p1, 1q is the union of two closed discs whose boundaries are real-algebraic curves with a single point in common, and all other fibres are points. The following reasonable conjecture appears to be inaccessible by purely quasiconformal techniques. Conjecture 8.19 Each hp{q,m extends to a continuous embedding Lp{q ˆ Lp{q z tprootp{q , rootp{q qu ÝÑ Cp{q,m . We draw additional conclusions from the natural symmetries of our construction. The central disc in Figure 8.23 is parametrized by the eigenvalue ´3A of the attracting fixed point at 0; this region corresponds to symmetric cubics whose Julia sets are quasicircles. Each value A “ ´ 13 e2π i p{q yields a map with a parabolic fixed point at 0. These parameters are evidently the roots of small embedded copies of M, and our results confirm this observation for odd-denominator rationals. More specifically, it will follow that the latter copies are the images of h0 ˝ δ and hp{q,q ˝ δ for odd q ą 1, where δ

C Q c ÞÑ pc, cq P C2 2 and P 2 is the diagonal embedding. As PA,0 Á,0 are conjugate by z ÞÑ ´z, our construction also accounts for the copies with q ” 2 (mod 4q. Every map in the symmetry locus is semiconjugate, via the quotient determined by the involution, to a cubic polynomial with a fixed critical value. Such maps were studied by Branner and Douady [BD] and are the subject of Section 8.3. Similar considerations applied to the antidiagonal embedding yield results for the real connectedness locus. In view of the fact that real polynomials commute with complex conjugation, C#R X Cp{q “ H unless p{q ” ´p{q (mod 1), and it therefore suffices to consider the real slices of C1{2 and C0 .

Theorem 8.20 There exist homeomorphic embeddings 1{2,1 : L1{2 Ñ C#R X C1{2,1 , 1{2,3 : L1{2 Ñ C#R X C1{2,3 and 0 : Mztrootu Ñ C#R X C0 . It follows from work of Buff [Bu] that these maps are compatible with the standard embeddings in the plane (see the discussion in the 8.4.4). Their

348


projections in CR are indicated in Figure 8.22. Notice that the two images of L1{2 have been identified while the image of Mztrootu has been folded in half.

8.4.3 Intertwining surgery History The intertwining construction was described in the 1990 Conformal Dynamics Problem List [Bi2]: Let P1 be a monic polynomial with connected Julia set having a repelling fixed point x0 which has a ray landing on it with rotation number p{q. Look at a cycle of q rays which are the forward images of the first. Cut along these rays and get q disjoint wedges. Now let P2 be a monic polynomial with a ray of the same rotation number landing on a repelling periodic point of some period dividing q (such as 1 or q). Slit this dynamical plane along the same rays making holes for the wedges. Fill the holes in by the corresponding wedges above making a new sphere. The new map is given by P1 and P2 , except on a neighbourhood of the inverse images of the cut rays where it will have to be adjusted to make it continuous.

A tool: surgery of invariant sectors Let f : W 1 Ñ W be a quadratic-like map with connected Julia set, ζ a repelling fixed point with combinatorial rotation number p{q and associated quotient torus Tζ “ pztζ uq{f q , where is a fixed but otherwise arbitrary linearizing neighbourhood Q ζ . Given p its projection to Tζ ; B Ă W ztζ u with f q pB X q “ B X f q pq, we denote B q 1 p p in particular, Kf , . . . , Kf Ă Tζ are the quotients of the various components of p i1 and A2 Ą K p i2 are isotopic we may speak Kf ztζ u. As any two annuli A1 Ą K f

f

p f if and of a distinguished isotopy class of open annuli A Ă Tζ , namely A „ K i p only if A is isotopic to an annulus containing some Kf . Moreover, it is easy to p i does not separate Tζ then A „ K p f ; it follows then that ζ is on see if A Ă K f

the boundary of an immediate basin of attraction. Consider p i :“ suptmod A| A Ă K pi u modζ K f f and p i :“ inftmod A| A Ą K pi u modζ K f f p f . Notice that these quantities are independent of i. In over open annuli A „ K pf . view of the following we may simply write modζ K p i “ modζ K pi . Lemma 8.21 In this setting modζ K f f

8.4 Intertwining surgery Proof

349

p ď modζ K p for K p“K p i . Conversely, given It is obvious that modζ K f

Rn Œ R8 “ emodζ K there exist conformal embeddings hn : A1,Rn Ñ Tζ such p i , where that hn pA1,Rn q Ą K f p

A1,R “ tz|1 ă |z| ă Ru. It follows from standard estimates in geometric function theory that the hn form a normal family on A1,R8 ; moreover, as all of these embeddings are p isotopic, every limit h8 “ limk Ñ8 hnk is univalent. Clearly, h8 pA1,R8 q Ă K p p and therefore modζ K ď modζ K. p f is defined in terms of the interior of Kf , we observe the As mod K following. p f depends only on the hybrid equivalence Remark 8.22 The quantity modζ K p class; furthermore, modζ Kf ą 0 if and only if ζ lies on the boundary of an immediate basin of attraction. Recall from Section 8.1 that an invariant sector with vertex ζ is a simply connected domain S Ă W bounded by an arc of BW and two additional arcs γ1 and γ2 with γj Ă f q pγj q and a common endpoint at ζ . We write S “ zγ1 , γ2 { for the sector between γ1 and γ2 as listed in counterclockwise order. The quotient Sp Ă Tζ is an open annulus whose modulus is referred to as the opening modulus modζ S “ modζ pS, f q q of the sector S (see Figure 8.6). Let λ “ f 1 pζ q denote the multiplier of the repelling fixed point ζ . Let ϕ : Ñ C be a linearizing map conjugating the action of f in the neighbourhood Q ζ to multiplication by λ in a neighbourhood of the origin. For an invariant sector S Ă select a branch of the mapping

z ÞÑ w “

1 logpϕpzqq q log λ

defined in S. In the log-linear coordinate w the map f q becomes the unit translation τ : w Ñ w ` 1; the image of the sector S is a horizontal strip ă with τ ´1 p ă q Ă ă (compare with Figure 8.5). In what follows we shall need to construct quasiconformal maps between sectors, or equivalently between half strips. In order to do so we will use concepts and results from Chapter 2 and from Section 8.1.2.

350


Construction of a cubic polynomial Fix p{q written in lowest terms and an odd integer m “ 2k ` 1 between 1 and 2q ´ 1. Our aim is to construct a map hp{q,m : Lp{q ˆ Lp{q ÝÑ Cp{q,m . Fixing parameter values c and cr in Lp{q , consider quadratic-like maps f : r1 Ñ W r hybrid equivalent to Qc and Qrc respectively, with W 1 Ñ W and fr : W the choice of the hybrid equivalences to be made below. In what follows we r to obtain a new surface. The reader is invited to follow will identify W and W the construction in the particular case p{q “ 1{2 with m “ 3, as illustrated in Figure 8.24.

r ψ

W N0 zS0

L1

r1 Sr1 zM

r1 R

M0

r W r1 L

r1 M

M1

L0

S1 zM1

r0 L

R1 ψ

r. Figure 8.24 Superimposing the regions W and W

We lose no generality by assuming that 0 is the critical point for both f and fr. Let ζ be the unique repelling fixed point of f with combinatorial rotation number p{q, that is ζ “ βf for p{q “ 0 and ζ “ αf otherwise, and Si ” zri , li {, i “ 0, . . . , q ´ 1 in W zKf a cycle of disjoint invariant sectors with vertex ζ , indexed in counterclockwise order so that the critical point 0 lies in the complementary region between Sq ´1 and S0 . We similarly specify ζr and a cycle of invariant sectors Sri for fr. Let φ:

qď ´1 i “0

li Y r i Ñ

qď ´1 i “0

r ri , li Y r


351

sending li to r ri `q ´k and ri to r li `q ´k ´1 , where k “ pm ´ 1q{2 and indices are understood modulo q, be any smooth conjugacy which therefore satisfies φpf pzqq “ frpφpzqq. The sector Si should now correspond to the component of Kfrztζru containing fri `q ´k p0q. Since Shishikura’s Principle warns against altering the conformal structure on regions of uncontrolled recurrence, we will therer i Ă Sri Ă N ri Ă fore employ invariant sectors Mi Ă Si Ă Ni Ă W zKf and M r zK r to be determined below. For i “ 0, . . . , q ´ 1 denote Li the component W f Ť q ´1 r i the corresponding component of of W z j “0 Nj containing f i p0q, and L Ť q ´1 rz r W j “0 Nj . Let r i `q ´k R i : Mi Ñ L be the Riemann map which extends continuously to the boundary, mapping r i `q ´k X W r q so that ζ maps to ζr, and BpMi X W q to BpL ri : M r i Ñ Li ´ q ` k ` 1 R ri X W r q to BpLi ´q `k `1 X W q so that ζr maps to the Riemann map sending BpM ζ . It remains to fill in the gaps. Proposition 8.23 (Quasiconformal interpolation) For any pair Qc and Qrc as above the hybrid equivalent quadratic-like maps f and fr and the invariant sectors ri Ą Sri Ą M ri Ni Ą Si Ą Mi and N may be chosen so that there exist quasiconformal maps ψ:

r: ψ

qď ´1

qď ´1

i “0

i “0

pSi zMi q Ñ

ri zSri q, pN

qď ´1

qď ´1

i “0

i “0

ri q Ñ pSri zM

pNi zSi q,

with ψ|BMi “ Ri |BMi and ψ|BSi “ φ|BSi , r i | r and ψ| r r “ φ ´1 | r . r r “R ψ| B Mi B Mi B Si B Si

352


Let us complete the construction assuming the truth of Proposition 8.23. We r i as specified above. ri , M r i , and maps ψ, ψ, r Ri and R choose sectors Ni , Mi , N Ťq ´1 Consider the almost complex structure σ on W , given by ψ ˚ σ0 on i “0 Si zMi r r be the almost complex structure on W and by σ0 elsewhere; similarly, let σ Ťq ´1 ˚ r r r σ0 on given by ψ i “0 Si zMi , and σ0 elsewhere. In view of the Integrability Theorem (Theorem 1.28), there exist quasiconformal homeomorphisms r˚ σ0 . Consider the r:W r ÑX r such that σ “ h˚ σ0 and σr “ h h : W Ñ X and h ri , r r Ri , R Riemann surface obtained from W \ W with identifications ψ, ψ, r It has the conformal type of a punctured disc, whose atlas is given by h and h. and we obtain a conformal disc D by replacing the puncture with a point ‹. Setting ¸ ˜ ¸ ˜ qď ´1 qď ´1 1 1 1 r r Si Y W z Si Y t‹u Ă D, D :“ W z i “0

we define a new map F :

i “0

D1

Ñ D by $ 1 Ť q ´1 ’ ’ f pzq for z P W z i “0 Si , & Ť r 1 z q ´1 Sri , F pzq :“ frpzq for z P W i “0 ’ ’ % ‹ for z P t‹, ´ζ, ´ζru.

It is easily verified that F is a three-fold branched covering with critical points r , and holomorphic except on the preimage of 0 P W and 0 P W S :“

qď ´1

r i q. pSi zMi q Y pSri zM

i “0

Recalling that the sectors Si and Sri are invariant and disjoint, we consider the almost complex structure on D defined as # pF n q˚ σ0 on F ń pSq, σp :“ σ0 elsewhere. By construction, the complex dilatation of σp has the same bound as that of F ˚ σ0 , and moreover p. F ˚ σp “ σ It follows from the Integrability Theorem (Theorem 1.28) that there is a quasiconformal homeomorphism ϕ : D Ñ V Ă C with σp “ ϕ ˚ σ0 . Setting U “ ϕpD 1 q, we obtain a cubic-like map G :“ ϕ ˝ F ˝ ϕ ´1 : U Ñ V .


353

In view of the Straightening Theorem (Theorem 7.4), there is a unique hybrid equivalent cubic polynomial PA,D whose critical points ´1 and `1 correspond to the critical point of f and fr respectively. The construction yields extensions of the natural embeddings r : Krc Ñ KPA,D π : Kc Ñ KPA,D and π to neighbourhoods of the filled Julia sets. r are infinitesimally Remark 8.24 By construction, the projections π and π conformal on the respective filled Julia sets: r “ 0 a.e. on Krc . Bπ “ 0 a.e. on Kc and B π We write PA,D « Qc O Qrc for any cubic polynomial so obtained. It is not p{q,m

yet clear that this correspondence is well defined, let alone continuous. These issues will be addressed in Section 8.4.4. Proof of Proposition 8.23 Note first that were it not for the condition of r would quasiconformality, the existence of the interpolating maps ψ and ψ follow without any additional argument. Any smooth interpolations are quasiconformal away from the points of ζ and ζr: the issue is the compatibility of ri . the local behaviour of φ at ζ with that of Ri and R For a proof of the following lemma see Lemma 8.8. Lemma 8.25 Given any c P Lp{q and υ ą 0, there exists a quadratic-like map f which is hybrid equivalent to Qc and admits disjoint invariant sectors Si as above with modζ Si ą υ. Given c, cr P Lp{q , we apply Lemma 8.25 to Qc and Qrc to obtain hybrid equivalent quadratic-like maps f and fr admitting invariant sectors Si and Sri with p rc and modr Sri ą modζ K pc . modζ Si ą modrζ K ζ In view of Remark 8.22 we may then choose disjoint invariant sectors Ni Ą Si ri Ą Sri so that and N r j and modr Sri ą modζ Lj modζ Si ą modrζ L ζ r j as above. Finally, we choose for the complementary invariant sectors Lj ,L r r Mi Ă Si and Mi Ă Si with r j and modr M r i “ modζ Lj . modζ Mi “ modrζ L ζ

354


Note that the boundary maps φ|BSi and φ ´1 |BSri

ri | r . Ri |BMi and R B Mi

are all near translations in log-linearizing coordinates due to Lemma 8.3 and Lemma 8.4 respectively. With this in mind the following lemma easily follows from Lemma 8.5. Lemma 8.26 With this choice of maps and invariant sectors there exist desired quasiconformal interpolations

ψ:

r: ψ

qď ´1

qď ´1

i “0

i “0

pSi zMi q Ñ

ri zSri q, pN

qď ´1

qď ´1

i “0

i “0

ri q Ñ pSri zM

pNi zSi q,

with ψ|BMi “ Ri |BMi and ψ|BSi “ φ|BSi , r i | r and ψ| r r “ φ ´1 | r . r r “R ψ| B Mi B Mi B Si B Si 8.4.4 Renormalization Preliminary facts about external rays The following lemma is proved in [GM]. Lemma 8.27 Let Pt be a continuous family of monic degree d polynomials with continuously chosen repelling periodic points ζt . If the ray of argument θ for Pt0 lands at ζt0 , then for all t close to t0 the ray of argument θ for Pt lands at ζt . The following useful Separation Principle is proven in [GM] for fixed points and in [Kiw] for periodic points, and it directly illustrates why a degree d polynomial admits at most d ´ 1 non-repelling cycles; the latter result was earlier shown by Douady and Hubbard and appropriately generalized to rational maps by Shishikura (see Section 7.7). Theorem 8.28 (The Separation Principle for polynomials) Let P be a polynomial with connected Julia set, N a common multiple of the periods of non-repelling periodic points, the graph formed by the union of all external


355

rays fixed under P N together with their landing points, and let U1 , . . . , Um be Ť ´j pq. Then: the connected components of Cz N j “0 P • each component Ui contains at most one non-repelling periodic point; • given any non-repelling cycle ζ1 , . . . , ζ passing through Ui1 , . . . , Ui , at least one of the components Uik also contains some critical point. We assume henceforth that KP is connected. Let R “ RP pθ q be a periodic external ray landing at the n-periodic point ζ P KP , whose orbit we enumerate ζ “ ζ0 ÞÑ ζ1 ÞÑ ¨ ¨ ¨ ÞÑ ζn “ ζ. Denote by #i Ă Q{Z the set of arguments of the rays in the orbit of R landing at ζi . The iterate P n fixes each point ζi , permuting the various rays landing there while preserving their cyclic order. Equivalently, multiplication by d n carries the set #i onto itself by an order-preserving bijection. For each i we may label the arguments in #i as 0 ď θ 1 ă θ 2 ă ¨ ¨ ¨ ă θ q ă 1; then d n θ i ” θ i `p pmod 1q for some integer p, and we refer to the ratio p{q as the combinatorial rotation number of R. The following theorem of Yoccoz (see [Hu1]) relates the combinatorial rotation number of a ray landing at a period n point ζ to the multiplier λ “ pP n q1 pζ q. Theorem 8.29 (Yoccoz’s Inequality) Let P be a monic polynomial with connected Julia set, and ζ P KP a repelling fixed point with multiplier λ. If ζ is the landing point of m distinct cycles of external rays with combinatorial rotation number p{q then mq Re ρ , ě 2 2 log d |ρ ´ 2π i p{q|

(8.4)

where ρ is the suitable choice of log λ. More geometrically, the inequality asserts that ρ lies in the closed disc of radius log d{pmqq tangent to the imaginary axis at 2π i p{q. Birenormalizable cubics Throughout this section we will work with fixed values of p{q and m as specified above. Here we describe the construction which will provide the inverse to the map hp{q,m . We start with a cubic polynomial P “ PA,D with pA, Dq P Cp{q,m . Let ζ be the landing point of the periodic rays with rotation number p{q. Denote V0 , . . . , V2q ´1 the components of C with the point ζ and the 2q rays landing at ζ removed. We enumerate them in counterclockwise order so that V0 Q ´1.

356


The polynomial P is renormalizable to the left if the forward orbit of the Ťq ´ 1 critical point ´1 is contained in tζ u Y i “0 V2i . In this case the map P has a quadratic-like restriction to a neighbourhood of the critical point ´1, as seen from the thickening construction below. i ĂV Assuming that P is renormalizable to the left, let K´ 2i for i “ 1 0, . . . , q ´ 1 denote the connected component of KP ztζ u. Let RP pθ1i q and i RP pθ2i q be the two external rays landing at ζ which separate K´ 1 from the i i other rays landing at the same point, where the values θ1 and θ2 are chosen so i i i that θ2i ă θ1i and the rays landing at K´ 1 have arguments in rθ2 , θ1 s. Choose a neighbourhood U Q ζ corresponding to a round disc in the local linearizing coordinate. Fix an equipotential E and a small ą 0, and consider the segments of the rays RP pθ1i ` q and RP pθ2i ´ q connecting the boundary of U q ´1 i to E. Let Ą Yi “0 K´ 1 be the region bounded by these two ray segments and the subtended arcs of E and BU , and consider the component 1 of P ´1 pq 1 with 1 Ă . In view of the fact that ζ is repelling, Ă provided that is sufficiently small. Thus, P : 1 Ñ is a quadratic-like map which filled Julia set will be denoted KR . Since q ´1 i tP n p´1qu8 n“0 Ă Yi “0 K´1 , this set is connected, and we refer to the unique hybrid conjugate quadratic polynomial Qc as the left renormalization RpP q. By construction Qc has a fixed point with combinatorial rotation number p{q, that is c P Lp{q . Figure 8.25 illustrates this construction for a cubic polynomial in C0 . Notice that ζ becomes the β-fixed point of the new quadratic polynomial.

Figure 8.25 Construction of the left renormalization for a cubic in C0 . The left renormalization has the hybrid type of a rabbit. The beast on the right is more exotic.


357

The polynomial P is renormalizable to the right if the forward orbit of `1 is Ťq ´1 contained in tζ u Y i “0 V2i `1 , and the set K and the right renormalization pP q are correspondingly defined. It follows from general considerations discussed in [McM2] that the left and right renormalizations do not depend on the choice of thickened domains. A cubic polynomial P is said to be birenormalizable if it is renormalizable on both left and right, in which case ωP p´1q X ωP p`1q Ă KR X K “ tζ u,

(8.5)

and we set q P :“ K

8 ď

P í pKR Y K q.

i “0

The following is an easy consequence of the Goldberg–Milnor–Kiwi Separation Principle (Theorem 8.28) and the standard classification of Fatou components. q P is Lemma 8.30 Let P be a birenormalizable cubic polynomial. Then K q P is repelling. dense in KP , and every periodic orbit in KP zK We denote by Bp{q,m the set of birenormalizable cubics in Cp{q,m , writing R ˆ : Bp{q,m Ñ Lp{q ˆ Lp{q for the map pA, Dq ÞÑ pc, crq, where Qc “ RpPA,D q and Qrc “ pPA,D q. In view of Lemma 8.27 the thickening construction may be performed so that the domains of the left and right quadratic-like restrictions vary continuously for pA, Dq P Bp{q,m . Applying the Straightening Theorem, we obtain the following. Proposition 8.31 R ˆ : Bp{q,m Ñ Lp{q ˆ Lp{q is continuous. We make use of the intertwining surgery in the following result. Proposition 8.32 R ˆ : Bp{q,m Ñ Lp{q ˆ Lp{q is surjective. Proof

Fix c, cr P Lp{q . We saw above that Qc O Qrc « P , p{q,m

for some cubic polynomial P “ PA,D , and we show here that R ˆ pA, Dq “ pc, crq; more precisely, we prove that RpP q “ Qc , the argument for right renormalization being completely parallel.

358

Cut and paste surgeries with sectors π

Let Kc Ă W 1 Ă W ÝÑC be as in Section 8.4.3. By construction, π is a quasiconformal map conjugating Qc |Kc to P |KR and Bπ pzq “ 0 for almost every z P Kc . Let ϕ0 : W Ñ be a quasiconformal homeomorphism with ϕ0 ˝ Qc |BW 1 “ P ˝ ϕ0 |BW 1 , which agrees with π on a small neighbourhood of Kc . As ϕ0 maps the critical value of Qc to the critical value of P |1 there is a unique lift ϕ1 : W 1 Ñ 1 such that ϕ

W 1 ÝÝÝ1ÝÑ § § Qc đ

1 § § đP

ϕ

W ÝÝÝ0ÝÑ commutes and ϕ1 |BW 1 “ ϕ0 |BW 1 . Setting ϕ1 pzq “ ϕ0 pzq for z P W zW 1 , we obtain a quasiconformal homeomorphism ϕ1 : W Ñ with the same dilatation bound as ϕ0 ; moreover, ϕ1 |Kc “ π |Kc . Iteration of this procedure yields a a sequence of quasiconformal homeomorphisms ϕn : W Ñ with uniformly bounded dilatation. The ϕn stabilize pointwise on W , so there is a limiting quasiconformal homeomorphism ϕ : W Ñ . By construction, ϕ ˝ Qc |W 1 “ P ˝ ϕ|W 1 and furthermore ϕ|Kc “ π |Kc ; it follows from Rickman’s Lemma (Lemma 1.20) that ϕ is a hybrid equivalence. Properness Here we deduce the properness of birenormalization from the Separation Theorem (Theorem 8.28). Proposition 8.33 R ˆ : Bp{q,m Ñ Lp{q ˆ Lp{q is proper. In view of the compactness of the connectedness loci, it suffices to prove that if pAk , Dk q P Bp{q,m with pAk , Dk q Ñ pA8 , D8 q P C# and R ˆ pAk , Dk q Ñ pc8 , cr8 q P Lp{q ˆ Lp{q , then pA8 , D8 q P Bp{q,m if and only if c8 ‰ rootp{q ‰ cr8 . Let ζk be the unique repelling fixed point of Pk “ PAk ,Dk where 2q external rays land, its q ´1 multiplier is denoted μk . Denote by tζki ui “0 the periodic orbit of period q q ´1 contained in KR with multiplier λk , and by tζrki ui “0 the similar orbit in K r with multiplier λk . These orbits renormalize to periodic orbits tγki ui Ă Kck and tr γki ui Ă Krck , whose multipliers will be denoted ρk and ρrk . Passing to a


359

subsequence if necessary, we may assume without loss of generality that the ζk converge to a fixed point ζ8 of P8 “ PA8 ,D8 with multiplier μ8 , and ζki i and ζ i with multipliers λ and r r8 and ζrki converge to periodic points ζ8 λ8 . 8 i u and tζ i u r8 Lemma 8.34 In this setting, if c8 ‰ rootp{q ‰ cr8 then ζ8 , tζ8 i i belong to disjoint orbits. Moreover, the fixed point ζ8 is repelling.

Proof It follows from the Implicit Function Theorem that these orbits are i u , tζ i u is parabolic with multiplier 1. Without r8 distinct unless one of tζ8 i i loss of generality, λ8 “ 1, and we may further assume that either |λk | ď 1 for every k or |λk | ą 1 for every k. In the first case, λk “ ρk as straightening preserves the eigenvalues of attracting orbits, and λk Ñ 1 implies c “ rootp{q . In the second case, it similarly follows that |ρk | ą 1 for every k; in view of Yoccoz’s Inequality, the combinatorial rotation numbers of γki are pk {qk Ñ 0, whence ρk Ñ 1 and again c “ rootp{q . i u , ζ and tζ i u lie in distinct orbits, it follows from Theor8 Because tζ8 i 8 i i u is rem 8.28 that at least one of these orbits is repelling. Suppose first that tζ8 i i i ˚ repelling, and let pζk q P KPk be the points which renormalize to ´γk P Kfck . i q˚ , where P ppζ i q˚ q “ P pζ i q, and the rays landing at Then pζki q˚ Ñ pζ8 8 8 8 8 i q˚ separate ζ from the critical point ´1. Similarly, if the orbit the points pζ8 8 i u is repelling then the rays landing at the corresponding points pζ i q˚ r8 tζr8 i separate ζ8 from `1. Applying Theorem 8.28 once again, we conclude that ζ8 is repelling. Continuing with the proof of Proposition 8.33, we observe by Lemma 8.27 that ζ8 is the common landing point of the same two cycles of rays with rotation number p{q. The thickening procedure yields a pair of quadratic-like restrictions r 18 Ñ r 8, P8 : 18 Ñ 8 and P8 : r 18 to be the limits of thickened domains and we may arrange for 18 and 1 1 r for the quadratic-like restrictions of Pk . As P n p´1q P 1 and k and k k k n p´1q P 1 and P n p`1q P r 1 , it follows that P8 r 18 . Thus, P8 Pkn p`1q P 8 8 k is birenormalizable, that is, pA8 , D8 q P Bp{q,m . l Injectivity The time has come to show that the intertwining operations pf, frq ÞÑ f O fr p{q,m

are well defined.

360


Proposition 8.35 R ˆ : Bp{q,m Ñ Lp{q ˆ Lp{q is injective. The relevant pullback argument is formalized as follows. r r Lemma 8.36 Let P “ PA,D and Pr “ PA, rD r where pA, Dq, pA, Dq P Bp{q,m . If r Dq, r R ˆ pA, Dq “ R ˆ pA, then there exists a quasiconformal homeomorphism ϕ : C Ñ C conjugating P qP . ¯ “ 0 almost everywhere on K to Pr with Bϕ Proof We begin by once again restricting P and Pr to domains G Ą KP and r Ą K r bounded by equipotentials. Our first goal is the construction of a G P quasiconformal homeomorphism ϕ0 , which is illustrated in Figure 8.26 for p{q “ 1{2 and m “ 3. Let Rpθ1 q, . . . , Rpθ2q q be the rays landing at ζ , enumerated in counterclockwise order so that the connected component K´1 Q ´1 of KP ztζ u lies between Rpθ1 q and Rpθ2 q; the component K`1 Q `1 then lies between Rpθm`1 q and Rpθm`2 q. We label the remaining components of 1 , . . . , Kq ´1 , so that Ki KP ztζ u as K˘ 1 ˘1 ´1 lies between the rays Rpθ2i `1 q, i Rpθ2i `2 q and similarly for K . The corresponding objects associated to Pr `1

are similarly denoted with an added tilde. It will be convenient to introduce further notation. Let Si Ă G be disjoint invariant sectors centred at Rpθi q, and let Li˘1 be the component of Ť2q i . The thickening procedure yields left and P ´1 pGqzp j “1 Sj q containing K˘ 1 right quadratic-like restrictions P : 1R Ñ R and P : 1 Ñ and r R and Pr : r1 Ñ r . r1 Ñ Pr : R By assumption, there exist hybrid equivalences hR between P |1 and Pr| r1 , R R r and h between P |1 and Pr| 1 . We now replace the domains G and G by r Ť r 1 “ Pr´1 pGq. r We define the map ϕ0 on q ´1 Li Ă G1 , G1 “ P ´1 pGq and G i “0 ˘1 setting it equal to hR pzq for z P Li´1 , and h pzq for z P Li`1 . ri its Let Bi be the quadrilateral of G1 zP ´1 pG1 q contained in Si , and B 1 ´ 1 1 r r ri in r counterpart in G zP pG q. We smoothly extend ϕ0 to map Bi Ñ B i i agreement with the previously specified values of ϕ0 on BpL´1 Y L`1 q and


361

P

G G1

S4

´1

h

`1

0 K` 1

L0´1 S1

K1

Pr

ϕ0

L0`1

S3

Sr4

r1 G

´1

B1

hR

´1

`1

r0 L ´1 h

r G

r0 K `1 r0 L `1

Sr1

r1 K ´1

Sr3

Figure 8.26 Construction of the map ϕ0 in the case p{q “ 1{2 with m “ 3.

ϕ0 ˝ P “ Pr ˝ ϕ0 on the inner boundary of Bi . We now extend ϕ0 to the entire sector Si X X by setting ϕ0 pzq “ Prń ˝ ϕ0 ˝ P n pzq when P n pzq P Bi . r 1 so defined conjugates The quasiconformal homeomorphism ϕ0 : G1 Ñ G ¯ 0 “ 0 almost everywhere on this set, sending KR P to Pr on KR Y K , with Bϕ r r to KR and K to K . We further extend ϕ0 to a quasiconformal homeomorr so that phism from G to G ϕ0 ˝ P |BG1 “ Pr ˝ ϕ0 |BG1 . As ϕ0 is a conjugacy between postcritical sets, there is a unique lift ϕ1 : G1 Ñ r 1 agreeing with ϕ0 on KR Y K such that the following diagram commutes: G ϕ r1 G1 ÝÝÝ1ÝÑ G § § § § r Pđ Pđ ϕ r G ÝÝÝ0ÝÑ G

362


As in the proof of Proposition 8.32, we set ϕ1 pzq “ ϕ0 pzq for z in the annulus GzG1 , and iterate the lifting procedure to obtain a sequence of quasiconformal qP maps ϕn with uniformly bounded dilatation. In view of the density of K r r in KP , the limiting map ϕ : G Ñ G conjugates P to P . As ϕn stabilises q P with ϕ|K “ hR and ϕ|K “ h by construction, it folpointwise on K R ¯ “ 0 almost everywhere lows from Rickman’s Lemma (Lemma 1.20) that Bϕ q on KP . To conclude the proof of Proposition 8.35, we show that the conjugacy just obtained is actually a hybrid equivalence: that any measurable invariant q P has support in a set of Lebesgue measure zero. In line field on KP zK view of Lemma 8.36, it follows from the standard considerations of parameter dependence in the Integrability Theorem (Theorem 1.30) that F “ pR ˆ q´1 pA, Dq is the injective complex-analytic image of a polydisc Dk for some k P t0, 1, 2u; see [MSS] and [MS]. On the other hand, F is compact by Proposition 8.33, whence k “ 0 and F is a single point. l Conclusions Setting hp{q,m pc, crq “ pR ˆ q´1 pc, crq, so that hp{q,m pc, crq “ pA, Dq if and only if PA,D “ Qc O Qrc , we obtain p{q,m

the embeddings hp{q,m : Lp{q ˆ Lp{q ÝÑ Bp{q,m Ă Cp{q,m , whose existence was asserted in Theorem 8.17. As observed in Section 8.4.2, if P “ PA,D is birenormalizable and pA, Dq P C#R then p{q “ 0 or 1{2. It follows by symmetry that R ˆ pA, Dq “ pc, cq ¯ for some c P C; conversely, # if R ˆ pA, Dq “ pc, cq ¯ then pA, Dq P CR . Writing δ¯ for the antidiagonal embedding C Q c ÞÑ pc, cq ¯ P C2 , we define 1{2,1 “ h1{2,1 ˝ δ¯ : L1{2 Ñ C#R X C1{2,1 , 1{2,3 “ h1{2,3 ˝ δ¯ : L1{2 Ñ C#R X C1{2,3 and 0 “ h0 ˝ δ¯ : Mztrootu Ñ C#R X C0 . These are the embeddings whose existence was asserted in Theorem 8.20. Compatibility with the standard planar embeddings is a consequence of the following result of Buff [Bu] (see also [BF2]).


363

Theorem 8.37 Let K1 and K2 be compact, connected, cellular sets in the plane, and ϕ : K1 Ñ K2 a homeomorphism. If ϕ admits a continuous extension to an open neighbourhood of K1 such that points outside K1 map to points outside K2 , then ϕ extends to a homeomorphism between open neighbourhoods of K1 and K2 . Let us sketch the argument for the map 0 . It is easily verified from the explicit expressions in [Mi3, p. 22] that for each ρ P Czt1u there is a unique pair pA, Bq P R2 such that the corresponding polynomial in the normal form (8.3) has a pair of complex conjugate fixed points with multipliers ρ and ρ, ¯ the remaining fixed point having multiplier ν “1´

|ρ ´ 1|2 . 2 Repρ ´ 1q

We may continuously label these multipliers as ρpA, Dq, ρpA, ¯ Dq and νpA, Dq for parameter values pA, Dq P R ˆ i R in a neighbourhood of 0 pMztrootuq; in particular, pA, Dq ÞÑ ρpA, Dq is a homeomorphism on such a neighbourhood. It follows from Yoccoz’s Inequality (8.4) that νpA, Dq ą 1, and therefore ρpA, Dq R r1, 8q, for pA, Dq in 0 pMztrootuq. Similarly, ρp0 pcqq Ñ 1 as c Ñ root, and thus c ÞÑ ρp0 pcqq extends to a embedding ϒ : M Ñ Czp1, 8q, which clearly commutes with complex conjugation. We claim that ϒ ´1 : ϒpMq Ñ M admits a continuous extension meeting the condition of Theorem 8.37. The idea is to allow renormalizations with disconnected Julia sets. We note that the rays RA,D p0q and RA,D p1{2q continue to land at the same fixed point for pA, Dq in a neighbourhood of 0 pMztrootuq. As before, we may construct left and right quadratic-like restrictions with continuously varying domains 1A,D . It is emphasized in Douady and Hubbard’s original presentation [DH3] that straightening, while no longer canonical for maps with disconnected Julia set, may still be continuously defined: it is only necessary to begin with continuously varying quasiconformal homeomorphisms from the fundamental annuli A,D z1A,D to the standard annulus. We thereby obtain a continuous extension to a neighbourhood of ϒpMztrootuq; it is easily arranged that this extension commutes with complex conjugation, so that it is trivial to obtain a further extension to an open set containing the point 1.

9 Trans-quasiconformal surgery

In this chapter we shall refer to the classical Integrability Theorem of Morrey, Bojarski, Ahlfors and Bers (see Section 1.4) as the qc Integrability Theorem in order to distinguish it from its generalization formulated in Theorem 9.7. We shall refer to the latter as the trans-qc Integrability Theorem or the David Integrability Theorem. The qc Integrability Theorem is the key element that makes quasiconformal surgery such a successful technique in holomorphic dynamics. Recall that this remarkable result establishes conditions under which a Beltrami differential μ can be integrated by a quasiconformal homeomorphism φ. This means that φ must satisfy the Beltrami equation Bφ “ μ Bφ, or equivalently, the pullback of μ0 ” 0 under φ, is equal to the starting Beltrami differential μ almost everywhere. The qc Integrability Theorem states that φ exists as long as ||μ||8 :“ kμ ă 1,

(9.1)

or equivalently, when the dilatation Kμ ă 8, where Kμ is the essential supremum of the dilatation over all points in the domain. It is for this reason that quasiregular maps are used as models when doing qc surgery, since those are the ones that may admit invariant almost complex structures satisfying (9.1). One may ask whether it is possible to relax the condition slightly, at the expense of obtaining an integrating map, which is not quasiconformal. This question has a positive answer: integration is possible as long as the dilatation is unbounded in a set of small enough area. Indeed, Guy David in [Da] introduced a new class of Beltrami differentials, the David–Beltrami differentials, which are integrable by David homeomorphisms (he called them 364

9.1 David maps and David–Beltrami differentials

365

μ-homeomorphisms), a wider class of homeomorphisms than the quasiconformal class. As we shall see, David homeomorphisms share some properties with quasiconformal homeomorphisms, obviously not all, but enough to make them useful for surgery. The David Integrability Theorem is a marvellous generalization of the qc Integrability Theorem, which opens a wide range of possibilities for new surgeries to be performed. In its short life, it has already served to prove many theorems in holomorphic dynamics. In this chapter we give two applications. The first one (Section 9.2), is Petersen and Zakeri’s result (Theorem 9.8) extending the class of rotation numbers for which the Siegel disc of a quadratic polynomial is a Jordan domain containing the critical point on the boundary. The second construction (Section 9.3) is Ha¨ıssinsky’s result (Theorem 9.18), which provides David conjugacies between hyperbolic polynomials and parabolic ones on their Julia sets.

9.1 David maps and David–Beltrami differentials The main hypothesis in the qc Integrability Theorem is that the given Beltrami differential μ has essential supremum }μ}8 ă 1 or equivalently bounded dilatation. A wider class of Beltrami differentials is defined by the following condition. Definition 9.1 (David–Beltrami differential) A measurable Beltrami differential μ “ μpzq

dz dz

on a domain U Ă C

with }μ}8 “ 1 is called a David–Beltrami differential if there exist constants M ą 0, α ą 0 and 0 ă 0 ă 1 such that α

Areaptz P U | |μpzq| ą 1 ´ uq ă Me´ for ă 0 .

(9.2)

This condition is often referred to as the area condition. David–Beltrami p to do so one needs differentials can be defined on any subset of the sphere C; on the one hand to take into account that μ is really a p´1, 1q form and on the other hand to replace the Euclidean area by spherical area in the area condition. Recall that a homeomorphism ϕ : U Ñ C belongs to the Sobolev space p1,1q Wloc pU q if its distributional partial derivatives exist and are locally integrable

in U . It follows from Theorem 1.10 that such maps are ACL (absolutely continuous on lines, see Section 1.3.2). Therefore the ordinary partial derivatives

366

Trans-quasiconformal surgery

Bz ϕ, Bz ϕ exist almost everywhere, and even more, ϕ is differentiable almost everywhere. Definition 9.2 (David homeomorphism) An orientation preserving homeop1,1q morphism ϕ : U Ñ C in Wloc pU q is called a David homeomorphism if the induced Beltrami differential Bz ϕ dz μϕ :“ Bz ϕ dz satisfies the inequality (9.2) or, equivalently, if there exist constants M ą 0, α ą 0 and K0 ą 1 such that Areaptz P U | Kϕ pzq ą Kuq ă Me´αK for K ą K0 ,

(9.3)

1`|μ pzq|

where Kϕ pzq “ 1´|μϕ pzq| (see Exercise 9.11). ϕ Remark 9.3 (The Sobolev condition can be replaced by ACL) In the definition of a David homeomorphism one can replace the condition of ϕ being in p1,1q Wloc pU q by the a priori weaker requirement of ϕ being ACL. This is a consequence of the following lemma. Lemma 9.4 Suppose ϕ : U ÝÑ C is an orientation preserving homeomorphism onto its image with partial derivatives a.e. and with a Beltrami coefficient μϕ satisfying (9.2). Then the Jacobian Jac ϕ as well as the partial derivatives Bz ϕ and Bz ϕ belong to L1loc . In particular, if ϕ is ACL and μϕ 1,1 satisfies (9.2), then ϕ P Wloc . Proof By [LV, Lem. 3.3, p. 131] we have for every compact subset KĂU ż Jac ϕ dx dy ď AreapϕpKqq, K

´ |Bz ϕ|2 q P L1loc . Since |Bz ϕ| “ |μϕ ||Bz ϕ| and ϕ is so that Jac ϕ “ p|Bz orientation preserving so that |μϕ | ă 1 a.e. we have ϕ|2

|Bz ϕ|2 “

Jac ϕ Jac ϕ , ď 1 ´ |μϕ | 1 ´ |μϕ |2

and 1

|Bz ϕ| ď pJac ϕq 2 p1 ´ |μϕ |q

´1 2

. 1

By the growth condition (9.2), the function p1 ´ |μϕ |q 2 is locally integrable (see Exercise 9.1.3). It follows that Bz ϕ and Bz ϕ are in L1loc (see Exercise 9.1.4). The same is true for Bz ϕ since |Bz ϕ| ď |Bz ϕ|.


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If ϕ is ACL and μϕ satisfies (9.2), then the partial derivatives Bz ϕ and Bz ϕ are in L1loc and hence coincide with the distributional derivatives of ϕ (see 1,1 Lemma 1.10). Hence ϕ P Wloc . David homeomorphisms differ from quasiconformal maps in many respects. For example, the inverse of a David homeomorphism is not necessarily David. However, they share many convenient properties (such as compactness, see [Tu2]), some of which are important when performing surgery. Proposition 9.5 (David Removability of quasiarcs) 1 Suppose ϕ : U Ñ C is an orientation preserving homeomorphism onto its image. Suppose there exists 1,1 pU zq and a quasiarc Ă U such that the restriction ϕ|pU z q belongs to Wloc 1,1 pU q. its partial derivatives satisfy (9.2) locally in U . Then ϕ belongs to Wloc

Proof Clearly the problem is local. Let z P be arbitrary. After precomposing ϕ with a quasiconformal homeomorphism if necessary, we may assume that z P “ I Ă R and that U “ I 1 ˆ i pá, aq, where a ą 0 and I 1 is an open interval with z P I Ă I 1 . By hypothesis, the ordinary partial derivatives g “ Bz ϕ and h “ Bz ϕ exist a.e. in U , are L1loc functions in U by Lemma 9.4 and are equal to the distributional partial derivatives Bϕ and Bϕ, locally in U zI Ą U zI 1 . Thus we need only to show that g and h are also the distributional partial derivatives of ϕ locally in U . In other words, we need to show that żż żż ψ ¨ g dxdy “ ´ Bz ψ ¨ ϕ dxdy U

U

for any test function ψ P Cc8 (see equation (1.12)). For any 0 ă ă a, set A :“ pá, ´q Y p, aq and Ac

:“ p´, q. Let 1A denote the characteristic function that takes the value 1 in A Ă R and 0 in RzA. Then, for any 0 ă ă a, we split the integral by inserting the characteristic functions 1A and 1Ac (as functions of y) żż żż ψ ¨ g dxdy “ ψ ¨ p1A ` 1Ac q ¨ g dxdy U

U

żż

żż

“ U

ψ ¨ 1A ¨ g dxdy `

U

ψ ¨ 1Ac ¨ g dxdy.

The products ψ ¨ 1A and ψ ¨ 1A are no longer test functions. However, since the characteristic functions can be approximated by C 8 bump functions, the products ψ ¨ 1A and ψ ¨ 1A can be approximated by test functions for which 1 We thank Carsten Lunde Petersen for providing this proposition and its proof.

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the partial derivatives exist. With this in mind, and knowing that g “ Bϕ locally in I 1 ˆ i A , the following equations can be justified: żż U

żż ψ ¨ 1A ¨ g dxdy “ ´

U

Bpψ ¨ 1A q ¨ ϕ dxdy

żż żż “´ Bz ψ ¨ 1A ¨ ϕ dxdy ` ψ ¨ B p1A q ¨ ϕ dxdy. U

U

The distributional derivative of the characteristic functions equal the (shifted) Dirac delta şdistribution at the discontinuity point. The Dirac delta distribution δα satisfies J f ptqδα ptqdt “ f pαq for any continuous function f and any α P J Ă R. Hence we obtain żż U

ψ ¨ B p1A q ¨ ϕ dxdy żż

ψ ¨ 12 i pδ´ ´ δ q ¨ ϕ dxdy

“ U

“ 12 i

ż pψpx, ´q ¨ ϕpx, ´q ´ ψpx, q ¨ ϕpx, qq dx ÝÑ 0. Ñ0

I1

Putting everything together, and using that ψ is C 8 pU q with compact support, ϕ is continuous and g is in L1loc pU q, we conclude żż

żż ψ ¨ g dxdy “ U

U

żż ψ ¨ 1A ¨ g dxdy `

U

ψ ¨ 1Ac ¨ g dxdy ÝÑ Ñ0

żż Bz ψ ¨ ϕ dxdy,

´ U

and hence g is the distributional B derivative of ϕ locally in U . Similarly, it can be shown that h is the distributional B derivative of ϕ.

Proposition 9.6 (Absolute continuity) Every David homeomorphism ϕ and its inverse are absolutely continuous, i.e. for a measurable set E Ă U AreapEq “ 0 if and only if

AreapϕpEqq “ 0.


369

The following is a generalization of the qc Integrability Theorem, which asserts that David–Beltrami differentials can always be integrated by David homeomorphisms [Da]. Theorem 9.7 (David Integrability Theorem) Let μ be a David–Beltrami p Then there exists a David homeomorphism differential on a domain U Ă C. ϕ : U Ñ V , whose complex dilatation μϕ coincides with μ almost everywhere. The integrating map is unique up to post-composition by a conformal map, i.e. if ϕr : U Ñ Vr is another David homeomorphism such that μϕr “ μ almost everywhere, then ϕr ˝ ϕ ´1 : V Ñ Vr is conformal. In the below sections we shall see two examples of trans-quasiconformal surgery where the constructed Beltrami differential μ satisfies }μ}8 “ 1. Both examples follow the same scheme, which we sketch below. Let U Ă C be a domain bounded by a Jordan curve γ which is a quasicircle. (In the first example, γ will actually be the unit circle, while in the second example, γ is the boundary of an immediate basin of attraction for a polynomial.) The model that will be constructed is a continuous map F : C Ñ C, holomorphic on CzU and satisfying F “ H ´1 ˝ R ˝ H

on U ,

where R : H pU q Ñ H pU q is holomorphic and H is a David homeomorphism. Let μ be the Beltrami form on C obtained by defining μ “ H ˚ μ0 on U and then spreading it by the dynamics of F . The first delicate step is to prove that such μ is a David–Beltrami differential, first showing that F has large dilatation only in regions of small area (in the sense of (9.2) and (9.3)), and then proving, as we spread μ by the dynamics, that further preimages of these small sets do not add too much area. This step needs to be done in each special case, since it depends clearly on the maps F and H . Once this is proven, it follows from the David Integrability Theorem that μ can be integrated by a David homeomorphism, say ϕ. The second delicate step is then to show that the composition map ϕ ˝ F ˝ ϕ ´1 is holomorphic. In qc surgery this step is immediate (see Lemma 1.39) from Weyl’s Lemma. In the David case, however, another type of argument is needed, since the hypothesis of quasiconformality is no longer satisfied. In the next sections this problem is handled in Proposition 9.14 and Lemma 9.25.

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Exercises Section 9.1 9.1.1 Check that the conditions on the decay of area expressed in the inequalities (9.2) and (9.3) are equivalent. 9.1.2 By using Theorem 1.18, prove that the composition of a quasiconformal map with a David map is a David map. 9.1.3 Show that if a Beltrami coefficient μ satisfies the area condition (9.2), then the measurable function p1 ´ |μ|q´1 belongs to L1loc , i.e. it is locally integrable. 9.1.4 Apply Hölder’s inequality (stated in Exercise 1.3.1) to show that if f and g are in L1 pU q, then pf ¨ gq1{2 is in L1 pU q.

9.2 Siegel discs via trans-quasiconformal surgery Carsten Lunde Petersen

This section is based on the results in [PZ]. To perform the Douady–Ghys quasiconformal surgery to Blaschke products (see Section 7.2 and Section 7.2.2), the rotation numbers considered must be of bounded type. In this section we extend the construction to a larger class of rotation numbers. In order to do so we need to go beyond the class of quasiconformal maps and enter the realm of David maps, i.e. maps which do not have uniformly bounded dilatation, but whose dilatation is unbounded only in a very controlled manner. This leads however to a new set of complications relative to those encountered in the Douady–Ghys surgery. To be more precise: given θ P r0, 1s, let λθ “ e2π i θ P S1 and Pθ pzq “ λθ z ` z2 . Denote by Aθ p8q the basin of attraction of 8, by Kθ the filled Julia set, that is the compact full complement of Aθ p8q, by Jθ the Julia set, which is the common boundary of the two, and when Pθ is linearizable around 0, let θ be the maximal Siegel disc of Pθ . The ultimate outcome of our labour will be the following theorem, which is the main result in [PZ]. Theorem 9.8 (Main Theorem) There exists an explicit full-measure subset E Ă r0, 1szQ such that for all θ P E, the Julia set Jθ is locally connected, has planar area zero, and the boundary of θ is a Jordan curve passing through the critical point. (The set E is defined in equation (9.6).) The primary goal of this section will be a trans-quasiconformal surgery result, which when combined with other theorems will yield Theorem 9.8. Before we formulate this result it is however convenient to recall the Douady– Ghys cut and paste surgery and introduce some auxiliary notation.

9.2 Siegel discs via trans-quasiconformal surgery

371

9.2.1 The Blaschke model Given an irrational number 0 ă θ ă 1, consider the cubic Blaschke product ˙ ˆ z´3 2π i t pθ q 2 , (9.4) z f :“ fθ : z ÞÑ e 1 ´ 3z which has superattracting fixed points at 0 and 8 and a double critical point at z “ 1. Here 0 ă tpθ q ă 1 is the unique parameter for which the critical circle map f |S1 : S1 Ñ S1 has rotation number θ . By a theorem of Poincaré the restriction is semi-conjugate to the rigid rotation Rθ pzq “ λθ z by a unique continuous map h “ hθ : S1 Ñ S1 fixing 1. And by a theorem of Yoccoz [Yo2], fθ is minimal, i.e. has no wandering intervals, so that h is in fact a conjugacy, i.e. a homeomorphism (see also Section 3.2.2). Let H : D Ñ D be any homeomorphic extension of h and define # F pzq “ Fθ pzq :“ Fθ,H pzq :“

f pzq

if |z| ě 1,

pH ´1 ˝ Rθ ˝ H qpzq

if |z| ă 1.

(9.5)

It is easy to see that F is a degree 2 topological branched covering of the sphere which is holomorphic outside of D and is topologically conjugate to a rigid rotation on D. By imitating the polynomial case, we define AF “ AFθ p8q as the immediate basin of 8, the ‘filled Julia set’ KF “ KFθ,H as the compact, connected and full complement of AF and the “Julia set” JF of F as the common topological boundary of KF and AF : JF “ BKF “ BAF . Note that although the homeomorphism H is by no means canonical, neither JF nor KF nor any of the definitions to follow depend on a particular choice of H . This is simply because the constructions do not involve the values of F on D. The main purpose of introducing F for the following constructions is to forget about the f -preimages of CzD in D. A particular choice of H is only used in the final step of the proofs, where we need H to have some regularity. In the quasiconformal surgery, H should be a quasiconformal homeomorphism and, in our construction here, a David homeomorphism. The following theorem is proved in [Pe1]. Theorem 9.9 (Local connectivity and zero measure) For every irrational θ P r0, 1szQ the ‘Julia set’ J pFθ q is locally connected and of planar Lebesgue measure zero.

372


There are by now several other similar constructions synthesizing other maps by quasiconformal surgery, e.g [Z1] and [YZ] for variants in the case of cubic polynomials and quadratic rational maps, [Ch] for horn maps and [KZ] for sine maps. Recall that the idea in the Douady–Ghys surgery is as follows. (1) Choose H to be quasiconformal whenever it is possible. (2) When this is possible let μ denote the unique F “ Fθ,H invariant Beltrami differential which coincides ˚ with the standard one μ0 “ 0 dz dz on JF Y AF and with H μ0 on D. (3) p ÝÑ C p be the unique integrating quasiconformal homeoLet φ “ φθ,H : C morphism for μ with φp0q “ 0, φp8q “ 8 and φ ˝ F “ φ 2 ` Opzq near 8 (note that μ “ μ0 on a neighbourhood of 8 so that F and φ are holomorphic near 8). (4) Let P “ φ ˝ F ˝ φ ´1 and note that P is holomorphic (by Weyl’s Lemma (Theorem 1.14)) thus a quadratic rational map, which by the choice of normalization equals Pθ . Note also that φpDq “ θ is a quasidisc, whose boundary contains the critical point φp1q “ ´λθ {2. However, H can be chosen to be quasiconformal only when θ is of bounded type, as we shall see later in Section 9.2.3. Consider the continued fraction expansion (see Section 3.2.1) 1

θ “ ra1 , a2 , a3 , . . .s “

1

a1 ` a2 `

1 a3 ` ¨ ¨ ¨

with an “ an pθ q P N. The nth convergent of θ is the irreducible fraction pn {qn :“ ra1 , a2 , . . . , an s. The number θ is said to be of bounded type if tan u is a bounded sequence. The main goal of this section is to extend the surgery above to the class of rotation numbers ? E “ tra1 , a2 , . . .s| log an “ Op nq as n Ñ 8u

(9.6)

at the expense of relaxing the boundedness condition on the F -invariant Beltrami differential. This leads us to the David Integrability Theorem (Theorem 9.7) for unbounded Beltrami differentials with exponential decay of areas supporting large dilatation, which was explained in the previous section. Theorem 9.10 (David conjugation) For every θ “ ra1 , a2 , . . . , an , . . .s P E there exists a David extension H “ Hθ : D ÝÑ D of hθ : S1 ÝÑ S1 and a p ÝÑ C p conjugating Fθ,H to Pθ . David homeomorphism ϕ “ ϕθ : C


373

So why is this result so much better than the one obtained by the Douady– Ghys surgery? The easy answer is that E is a full measure subset of r0, 1s, where as the set of bounded type numbers is a subset of zero measure. Indeed, the following theorem due to Khinchin [Kin] characterizes the asymptotic growth of the sequence tan u for random irrational numbers. Theorem 9.11 (Asymptotic growth of coefficients) Let ψ : N Ñ R be a given positive function. ř 1 (i) If 8 n“1 ψ pnq ă `8, then for almost every irrational 0 ă θ ă 1 there are only finitely many n for which an pθ q ě ψpnq. ř 1 (ii) If 8 n“1 ψ pnq “ `8, then for almost every irrational 0 ă θ ă 1 there are infinitely many n for which an pθ q ě ψpnq. By part (i), taking ψpnq “ eM

?

n,

we obtain the following.

Corollary 9.12 The set E has full measure in r0, 1s. Thus it is a worthwhile extension. But where does the restriction θ P E come into the picture? It turns out we only know how to construct a David extension Hθ of hθ whenever θ P E (see Theorem 9.15 below). We even conjecture that such an extension exists if and only if θ P E. However, we can prove independently, that whenever such an extension H exists, then the unique Fθ,H -invariant Beltrami differential μθ , which coincides with μ0 on the closure of AF and with H ˚ pμ0 q on D is also a David differential. To see this, let f “ fθ and let g0 “ g0,θ : D Ñ U0 denote the unique branch of f ´1 on D with values outside D and image U0 (see Figure 9.1). Define G “ tg : D Ñ CzD | g is a branch of f ń and f n´1 ˝ g “ g0 u,

D

U0 g0

Figure 9.1 The set U0 and the branch g0 of f ´1 .

374


that is, G consists of all the inverse branches of iterates of f factoring through g0 , i.e. g “ gp ˝ g0 , where gp is a branch of f ´pn´1q on U0 . Note that g0 extends univalently to a neighbourhood of Dztf p1qu and that any of the maps gp extends univalently to a neighbourhood of U0 . Define a Borel measure ν supported on D by ÿ νpEq :“ AreapEq ` AreapgpEqq, g PG

for any Borel set E Ă D. That is ν is the sum of the Lebesgue measure on D and of all the pushforwards of Lebesgue measure on each of the components Ť of the interior of KF , i.e. on nPN F ń pU0 q. Then ν is a finite measure since KF is compact and evidently ν is also absolutely continuous with respect to Lebesgue measure. Observe that if E denotes the set where the dilatation of H is larger than 1 ´ ε, then νpEq is the area of the region where μθ has norm larger than 1 ´ ε, since μθ is the spreading of H ˚ μ0 under the dynamics of F , which is holomorphic outside D. However, much more is true. Theorem 9.13 (Power law) The Borel measure ν is dominated by a universal power of Lebesgue measure. That is, there exists a universal constant 0 ă β ă 1 and a constant C (depending on θ ) such that for any Borel set E Ă D, νpEq ď CpAreapEqqβ . Thus it follows that if H is David, then μθ is also David. We shall motivate later why it is not obvious that μθ is a David–Beltrami differential, when Hθ is a David map. Also we shall comment on the principal idea underlying the proof above. For a comprehensive treatment and complete proof of the theorem we refer the reader to the paper [PZ]. If we have obtained an F -invariant David–Beltrami differential μθ , then the David Integrability Theorem provides a unique normalized integrating homeop ÝÑ C, p ϕp0q “ 0, ϕp8q “ 8 and f ˝ ϕ “ ϕ 2 ` Opzq. morphism ϕ “ ϕθ : C It would then be nice if P “ ϕ ˝ F ˝ ϕ ´1 was holomorphic. And indeed it is. Proposition 9.14 (Straightening) In the notation above P “ ϕ ˝ F ˝ ϕ ´1 is holomorphic and thus equals Pθ . In contrast to the classical quasiconformal surgery, where no such consideration occurs, the basic difficulties are that the inverse of a David homeomorphism need not be a David homeomorphism and the absence of a David Gluing Lemma similar to the Quasisymmetric/Quasiconformal Gluing Lemma as Theorem 1.19.


375

Proof P is a degree 2 branched covering of the sphere with P ´1 p8q “ 8, so it will be a quadratic polynomial once we show that it is holomorphic. We will heavily use the uniqueness statement in the David Integrability Theorem. Observe that the hypothesis of the Proposition imply that the two following diagrams commute: R

θ Ñ pD, μ0 q pD, μ0 q ÝÝÝÝ İ İ § § H§ §H

F

pD, μq ÝÝÝÝÑ pD, μq

F

pC, μq ÝÝÝÝÑ pC, μq § § §ϕ § ϕđ đ P

pC, μ0 q ÝÝÝÝÑ pC, μ0 q

Note that the map ϕ ˝ F also integrates the David–Beltrami differential μ. The proof consists in showing that ϕ ˝ F is locally ACL (or that it belongs to 1,1 , see Remark 9.3) in every open set U Ă C where F is injective, and Wloc hence is a David integrating map. Indeed, suppose this is true. Then, both ϕ |U and pϕ ˝ F q |U are David maps integrating the same Beltrami differential μ. Hence, by the uniqueness statement in Theorem 9.7, the composition P “ ϕ ˝ F ˝ ϕ ´1 must be conformal on ϕpU q. It follows that P is locally conformal around every point, except for a discrete number of points, and hence it is holomorphic. This discrete set is the image of the set of critical points of F . 1,1 in Hence we need to show that ϕ ˝ F is locally ACL or belongs to Wloc every open set U Ă C, where F is injective. If U belongs to CzD, then F is 1,1 and a holomorphic and therefore ϕ ˝ F is the composition of a map in Wloc 1,1 holomorphic one, hence in Wloc (see [Ah5, Lem. 3]). If U belongs entirely to D then we have ϕ ˝ F “ ϕ ˝ H ´1 ˝ Rθ ˝ H in U . Again by the uniqueness of David integrating maps, ϕ ˝ H ´1 is conformal since both H and ϕ integrate μ on D. Hence ϕ ˝ F is again the composition of a conformal map with a map in 1,1 1,1 , thus it is also in Wloc . Wloc 1,1 pC˚ zS1 q with Beltrami differential satisfySince ϕ ˝ F belongs to Wloc 1 ing (9.2) and S is a quasiarc, it follows from Proposition 9.5 that ϕ ˝ F P 1,1 pC˚ zt1uq. Wloc Finally let us return to the issue of existence of David extensions Hθ of hθ . Theorem 9.15 (David extension) Let f : S1 Ñ S1 be a critical circle map whose rotation number θ “ ra1 , a2 , a3 , . . .s belongs to the arithmetical class E. Then the normalized linearizing map h : S1 Ñ S1 , which satisfies h ˝ f “ Rθ ˝ h, admits a David extension H : D Ñ D so that

376

Trans-quasiconformal surgery ˇ ˇ + ˇ ˇ BH pzq ˇ α ˇ ˇ ˇ Area z P D ˇ ˇ ˇ ą 1 ´ ε ď M e´ ε for all 0 ă ε ă ε0 . ˇ BH pzq ˇ #

Here M ą 0 is a universal constant, while in general the constant α ą 0 ? depends on lim supnÑ8 plog an q{ n and the constant 0 ă ε0 ă 1 depends on f . Summarizing the proof of Theorem 9.8 Let θ P E be arbitrary. Then by Theorem 9.15 we can choose a David extension H : D Ñ D of the conjugacy hθ : S1 Ñ S1 conjugating fθ |S1 to Rθ |S1 with H p0q “ 0. Let μ denote the unique Fθ “ Fθ,H invariant Beltrami differential, which equals μ0 on AFθ p8q, and which equals H ˚ pμ0 q on D, where μ0 “ 0 dz dz denote the standard Beltrami differential. Then μ is a David–Beltrami differential by Theorem 9.13. Let p ÑC p denote the integrating David homeomorphism for μ, normalized ϕθ : C by ϕθ p8q “ 8, ϕθ p0q “ 0, ϕθ p1q “ éi 2π θ {2. Then from Proposition 9.14, it follows that P “ ϕθ ˝ Fθ ˝ ϕθ´1 “ Pθ and ϕθ pDq “ θ , and by Theorem 9.9, Jθ “ ϕθ pJ pFθ qq is locally connected and of measure zero. We refer the reader to the appendix of the paper [PZ] for a complete proof of this theorem. The proof is based in part on an ingenious geometric construction by Yoccoz in the unpublished manuscript [Yo2]. It is a reminiscence of Carleson squares, but it is specially tailored to the particular geometry of critical circle maps. In the remainder of the section we shall survey this result. 9.2.2 The Yoccoz Conjugacy Theorem: idea of the Construction The idea of the proof is to construct two dynamically defined graphs in the disc linking the backward orbit of the critical point say 1 for f to the backward orbit for 1 under Rθ , in such a way that the conjugacy hθ of f to Rθ extends to a ‘piecewise affine map’ on the graphs. In fact we shall lift the problem to R and the upper half plane by the exponential map. In the upper half plane the extension will be piecewise affine. The graphs will be so that the piecewise affine extension extends quasiconformally to each cell of the associated cell decompositions of the upper half plane. If θ is not of bounded type then the supremum of the dilatations of these extensions necessarily diverge to infinity when we consider cells closer and closer to R. The ingeniousness comes into choosing the right graphs and the right quasiconformal extensions so that we may use as little quasiconformal distortion (dilatation) as possible and also use it in a controlled manner. To be slightly more precise the cells in these decompositions have bounded geometry and are labelled by an integer, called their generation or level.


377

The union of the cells of any generation n is a belt separating the previous generation from R and the following generations. The distance between the nth belt and R tends to zero at least geometrically with n and thus (by bounded geometry) the area of the nth-level belt tends to zero geometrically with n. By making the right constructions the quasiconformal extension of the piecewise affine boundary map on each level n cell can be chosen to have a dilatation bounded by Cp1 ` plog an`1 q2 q for some universal constant C depending only on the derivative f 1 .

9.2.3 Two cell decompositions for the upper half plane Let f : S1 Ñ S1 be a critical circle map with a critical point at 1 and irrational rotation number θ “ ra1 , a2 , a3 , . . .s with convergents pn {qn “ ra1 , a2 , . . . , an s, and let R “ Rθ denote the corresponding rigid rotation. Using the canonical projection e2π i x : R Ñ S1 we identify S1 with the quotient space T “ R{Z and denote the induced maps from T to T by f and R as before. We set xn :“ f ń p0q and x 1 n :“ R ń p0q for all n P Z. The natural partitions n pf q and n pRq of the circle with dividing points txj |0 ď j ă qn ` qn`1 u and tx 1 j |0 ď j ă qn ` qn`1 u respectively are for instance used when studying renormalization of circle maps. However, writing In “ r0, xqn s and I 1 n “ r0, x 1 qn s for n P N, then 1 |I 1 n | 1 ď ď 1 , 1 ` an`1 |I n´1 | an`1 ´ ¸ tek Theorem (see Theorem 3.17 or [Pe2]) there whereas by the Herman–Swia exists some constant C “ Cpf 1 q such that 1 |In | ď ď C. C |In´1 | If the sequence an is unbounded then this shows that h “ hθ can not be quasi-symmetric. This is where the ‘only if’ conclusion of the Herman– ´ ¸ tek Theorem comes from. Swia However, Yoccoz saw that if we instead consider the partitions with dividing points Qn :“ txj | 0 ď j ă qn u

and

Q1 n :“ tx 1 j | 0 ď j ă qn u

378


on T, so that Q0 “ Q1 0 “ t0u, then neighbouring intervals are always of alike lengths for both partitions (see (9.9) below). To see this, first let us note that ¨ ˛ ¨ ˛ ď ď sxj `qn´1 , xj r‚Y ˝ sxj , xj `qn ´qn´1 r‚. T Qn “ ˝ 0ďj ăqn ´qn´1

0ďj ăqn´1

Thus xj and xk , with j ă k, are adjacent in Qn if and only if either k “ j ` qn´1 and 0 ď j ă qn ´ qn´1 , or k “ j ` qn ´ qn´1 and 0 ď j ă qn´1 . It follows that in the first case rxk , xj s X Qn`1 “ txk , xk `qn , xk `2qn , . . . , xk `pan`1 ´1qqn “ xj `qn`1 ´qn , xj u, (9.7) and in the second case rxj , xk s X Qn`1 “ txj , xj `qn , xj `2qn , . . . , xj àn`1 qn “ xk `qn`1 ´qn , xk u. (9.8) As a result, we see that xj and xk , with j ă k, are adjacent in both Qn and Qn`1 if and only if an`1 “ 1, 0 ď j ă qn ´ qn´1 and k “ j ` qn´1 . Since f and R are conjugate, the similar statement holds for Q1 n We lift the sets Qn and Q1 n in T to the two sets in R, which are invariant under the translation z ÞÑ z ` 1: rn :“ Qn ` Z Q

and

Ă1 n :“ Q1 n ` Z. Q

In comparison with the partition n´1 pf q we have removed the intervals which are short for R, when an is large by uniting it with its neighbouring interval across the common end point xj , where qn ď j ă qn ` qn´1 . The rn have lengths gain is that now as before any two adjacent intervals in R Q comparable up to a bound which is asymptotically universal [PZ, Lem. 5.1], " * |I | ˇˇ r max (9.9) ˇ I, J are adjacent in R Qn — 1, |J | Ă1 n (including neighbouring ones) and any two intervals I and J in R Q satisfy 1{2 ă |I |{|J | ă 2. rn , let We now proceed to construct the graphs. For n ě 0 and x P Q Mn pxq :“

1 pxr ´ x q, 2


379

rn immediately to the right and left of x. where xr and x are the points in Q r0 “ Z. Evidently Mn pxq ě Mn`1 pxq. Observe that M0 pxq “ 1 for all x P Q Define zn pxq :“ x ` i Mn pxq

rn . n ě 0, x P Q

Using the sequence tzn u, we shall define an embedded graph in the upper halfplane H as follows: The vertices of are the points tzn pxq | n ě 0 and x P rn u. Note that zn pxq “ zn`1 pxq if and only if Mn pxq “ Mn`1 pxq, in which Q case the corresponding vertex of is doubly labelled. The edges of are the vertical segments rn with Mn pxq ‰ Mn`1 pxqu, trzn pxq, zn`1 pxqs | n ě 0 and x P Q as well as the non-vertical segments rn u. trzn pxq, zn pyqs | n ě 0 and x, y are adjacent in Q By a cell of we mean the closure of any bounded connected component of H . Any cell γ of is uniquely determined by a pair of adjacent points rn with the property that either Mn pxq ‰ Mn`1 pxq or Mn pyq ‰ x ă y in Q Mn`1 pyq. The integer n ě 0 will be called the level of γ , or we say that γ is an n-cell. The top of the n-cell γ is formed by the non-vertical edge rzn pxq, zn pyqs while its bottom is formed by the union of non-vertical edges rzn`1 pt0 q, zn`1 pt1 qs Y rzn`1 pt1 q, zn`1 pt2 qs Y ¨ ¨ ¨ Y rzn`1 ptk ´1 q, zn`1 ptk qs, where the points x “ t0 ă t1 ă ¨ ¨ ¨ ă tk “ y form the intersection rx, ys X rn`1 . The sides of γ are formed by the vertical edge rzn pxq, zn`1 pxqs (which Q collapses to a single point if Mn pxq “ Mn`1 pxq) as well as rzn pyq, zn`1 pyqs (which similarly collapses to a single point if Mn pyq “ Mn`1 pyq). If k “ 1 so rn`1 , then γ is either a triangle or a trapezoid. that x, y are also adjacent in Q Otherwise k ě 2 and by (9.7) or (9.8), γ is a pk ` 3q-gon, where k is either an`1 or an`1 ` 1. The bounds (9.9) imply that the cells of have ‘bounded geometry’ in the sense that there is a constant C ą 1 such that the top, bottom and sides of any n-cell γ of have lengths comparable up to C. Moreover, the slopes of nonvertical edges of γ are bounded by C. The constant C is even asymptotically universal. For every n P N the union of the n-cells, which is invariant under z ÞÑ z ` 1, form a connected set Sn converging geometrically to R in the sense (for proofs

380


see [PZ, Lem. 2.7 and Lem. 5.2]) that there exists C ą 1 and σ1 , σ2 with 0 ă σ1 ă σ2 ă 1 such that * " ˇ σn ˇ 1 n ď y ď Cσ2 . Sn Ă z “ x ` i y ˇ (9.10) C Completely analogously we can construct a graph 1 Ă H for the rigid rotation R with cells generically denoted γ 1 (compare with Figure 9.2). In analogy with the critical circle map case above the cells of 1 have ‘bounded geometry’: there is a universal constant C ą 1 such that the top, bottom, and sides of any cell γ 1 of 1 have lengths comparable up to C. Moreover, the slopes of nonvertical edges of γ 1 are bounded by 1{2. n´1

n n`1

xq1

n´1 ´qn´2

1 x3q

n´1

1 x2q

n´1

xq1

n´1

0 xq1 n

xq1 ´q n n´1

xq1

n´2

Figure 9.2 The embedded graph 1 for the rigid rotation with selected cells and points in Q1n . In this picture an “ 3 and an`1 “ 4. The labels on cells denote their level.

9.2.4 Constructing the extension r r Let h : R Ñ R be the lift to R of the conjugacy h normalized by hp0q “ 0. 1 r r r r Note that h fixes the integer points and hpQn q “ Qn for all n ě 0. We shall r to a homeomorphism Hr between the embedded graphs and 1 extend h by mapping each vertex of to the corresponding vertex of 1 and each edge of affinely to the corresponding edge of 1 . Strictly speaking, for each rn , we define Hr pzn pxqq :“ z1 phpxqq. r Then rz, ws is an edge n ě 0 and x P Q n of if and only if rHr pzq, Hr pwqs is an edge of 1 . Thus we can extend Hr »

further to a homeomorphism ÝÑ 1 by mapping each such edge rz, ws affinely to rHr pzq, Hr pwqs. Note that Hr defined this way is the identity on the horizontal line R ` i so we can define Hr pzq “ z for all z P H with Impzq ě 1. It is easy to check that for each cell γ of , the boundary Bγ is mapped by Hr homeomorphically and edgewise affinely onto the boundary Bγ 1 of a unique cell γ 1 of 1 . Moreover, as the angles of interior corners of cells are (uniformly) bounded from below, the piecewise affine homeomorphism Hr from the boundary of one cell γ of to the boundary of the corresponding cell γ 1 of 1 is quasisymmetric and thus has a quasiconformal extension. The key


381

issue is how much quasiconformal distortion is needed. The following result due to Yoccoz is the key estimate. Theorem 9.16 (Key estimate) There exists a constant C ą 0 with the following property: for any n-cell γ of , the edgewise affine boundary homeomorphism Hr : Bγ Ñ Bγ 1 extends to a quasiconformal homeomorphism Hr : γ Ñ γ 1 whose dilatation is at most Cp1 ` plog an`1 q2 q. The constant C is asymptotically universal. Assuming this result for the moment, let us show how Theorem 9.10 follows. Proof of Theorem 9.10 Consider the extension Hr : H ÝÑ H, obtained by gluing the various extensions to cells given by Theorem 9.16. Clearly Hr is ACL and is chosen so that Hr pz ` 1q “ Hr pzq ` 1 for all z P H. Since ? log an “ Op nq by the assumption, there is a constant C1 ą 0 and an integer N1 ě 1, both depending on θ , such that 1 ` plog an`1 q2 ď C1 n whenever n ą N1 . By Theorem 9.16, there is a universal constant C2 ą 0 and an integer N2 ě 1 depending on f such that the dilatation KHr in the interior of any n-cell of is at most C2 p1 ` plog an`1 q2 q whenever n ą N2 . Finally, by Equation (9.10), there is a universal constant C3 ą 0, a constant 0 ă σ2 ă 1, and an integer N3 ě 1 depending on f such that if n ą N3 , 8 ď

tγ | γ is an m -cell of u Ă tz P H | 0 ă Impzq ď C3 σ2n u.

m“n

Set N :“ maxtN1 , N2 , N3 u and define K0 :“ maxtKHr pzq | z belongs to the interior of an m-cell of with m ď N u. If KHr pzq ą K ą K0 , then either z P (which has Lebesgue measure zero), or else z belongs to the interior of an n-cell of with n ě N , so that K ă KHr pzq ď C1 C2 n, or equivalently n ą K{pC1 C2 q. It follows that K C C2

Areatz P H | 0 ď Repzq ď 1 and KHr pzq ą Ku ď C3 σ2 1

log σ2

“ C3 e C1 C2

K

.

The exponential map z ÞÑ e2π i z does not change the dilatation and has norm of the derivative bounded by 2π when restricted to the upper halfplane H. Therefore, the induced ACL homeomorphism H : D Ñ D satisfies log σ2

Areatz P D | KH pzq ą Ku ď 4π 2 C3 e C1 C2

K

whenever K ą K0 . It follows that H is a David homeomorphism as in (9.3), with M “ 4π 2 C3 , α “ p´ log σ2 q{pC1 C2 q, and K0 defined as above.

382


Moreover, M is universal, α depends on C1 (which in turn depends on ? l lim supnÑ8 plog an q{ n), and K0 depends on f and θ . The remainder of the section is dedicated to sketch the idea and the main steps of the proof of Theorem 9.16. By bounded geometry, if k is small say less than 5, then γ and γ 1 are uniformly quasiconformally equivalent with a constant depending only on the real bounds (9.9).

9.2.5 Near parabolic For critical circle mappings f : S1 ÝÑ S1 Yoccoz proved further the existence of a bound C “ Cpf 1 q (the Yoccoz near parabolic bound) such that for any two rn the intersection neighbouring points x, y P Q rn`1 “ tx “ t0 ă t1 ă ¨ ¨ ¨ ă tk “ yu, rx, ys X Q as above, satisfies, for all 0 ă j ď k, 1 2

Cpmintj, k ` 1 ´ j uq

ď

|tj ´ tj ´1 | C ď |tk ´ t0 | pmintj, k ` 1 ´ j uq2

(9.11)

(see [Yo2] and [dFdM]). This disposition of the orbit, at least in the real analytic case, is the result of the orbit passing the narrow gate of two qn periodic points, which are close to merging into a parabolic point. To see how this a priori bound can be used we consider as a model a rigid rotation Rη pzq “ e2π i η z with pk ` 1qη ă 1 ă pk ` 2qη and k even. We conjugate Rη by a real symmetric Möbius transformation Mpzq “

z`r , 1 ` rz

and obtain T :“ M ˝ Rη ˝ M ´1 with fixed points r and 1r . Let us choose π 1 r “ kk ´ `π . As k tends to infinity the fixed points r and r tend to 1. For k large the conjugate map T is near parabolic. Consider the Rη -orbit through 1, and its segment wj1 “ e2π i pj ´k {2qη corresponding to 0 ď j ď k. (Note that wk1 {2 “ 1.) The orbit is equidistantly separated, and the orbit points w01 “ e´π i kη and wk1 “ eπ i kη are close to ´1 for k large. The points ˘1 are fixed r k points of the Möbius transformation M with derivatives M 1 p´1q “ 11` ´r “ π r π 1 and M 1 p1q “ 11´ `r “ k . Let wj “ Mpwj q, 0 ď j ď k, denote the corresponding segment of the orbit through 1 “ Mp1q, of the conjugate map T (see p. 46–47 in [PZ]). We invite the reader to check that there exists δ ą 0 so that


383

M wk

wk1 w01

´1

0

1 1 “ wk{2

1{r

r

1 1 “ wk{2

w0 Figure 9.3 Changing the fixed points 0, 8 of a rigid rotation to the fixed points r and 1 of a near parabolic rotation. r

´π ` δ ă arg w0 ă ´δ ă 0 ă δ ă arg wk ă π ´ δ, and the arguments of the wj are spaced according to (9.11) (compare with Figure 9.3). Let φ : γ ÝÑ D and φ 1 : γ 1 ÝÑ D be homeomorphically extended Riemann maps of the level n cells γ and γ 1 associated with t0 , . . . , tk and the corresponding set (by Hr ) t01 , . . . , tk1 . Normalize φ by mapping zn`1 pt0 q to í , zn`1 ptk q to i and zn`1 pj0 q to 1, where j0 is the integer part of k{2. And similarly for γ 1 and φ 1 . Then the images wj of zn`1 ptj q will be spaced according to (9.11) for some C 1 depending only on the derivative f 1 . And the images wj1 of zn1 `1 ptj1 q will be approximately equidistant. π obius transformaLet M be as above with r “ kk ´ `π , and let P denote the M¨ tion, which fixes 1, maps í to ´1 and ´1 to i . Note that the inverse Möbius transformation P ´1 satisfies P ´1 pzq “ P pzq. Define ψ : S1 ÝÑ S1 by $ ´1 ’ if 0 ď argpzq ď π {2, ’ &P ˝ M ˝ P pzq ψpzq “ z if π {2 ď argpzq ď 3π {2, ’ ’ %P ´1 ˝ M ˝ P pzq if 3π {2 ď argpzq ď 2π. Note that ψ is a homeomorphism, which is piecewise Möbius with fixed points at 1, ˘i as well as the points on the left halfcircle. Then there exist quasiconformal self-homeomorphisms χ , χ 1 : D ÝÑ D fixing 1 and ˘i and of uniformly bounded (in terms of f 1 ) dilatations such that ´1 ´1 ψpzq “ χ ˝ φ ˝ Hr ´1 ˝ pφ 1 q ˝ pχ 1 q pzq

(compare with Figures 9.4 and 9.5).

384

Trans-quasiconformal surgery H˜ ´1

γ1

t01

χ1

˝ φ1

γ

1 tk{2

tk1

wk1 “ i

−1

t0

tk{2

tk

χ ˝φ

wk “ i

1 1 “ wk{2

1 “ wk{2

w01 “ ´1

w0 “ ´1

Figure 9.4 Uniformizing γ and γ 1 and post-adjusting by quasiconformal homeomorphisms of uniformly bounded distortion.

i P ˝ M ˝ P ´1

1

Id

P ´1 ˝ M ˝ P í Figure 9.5 The map ψ : S1 Ñ S1 , which is piecewise Möbius, to be quasiconformally extended.

To prove the claims above about uniform control of the quasiconformal dilations of χ and χ 1 , it is convenient, because of the ‘squareness’ of the cells γ and γ 1 to map the cells quasiconformally to the upper half plane ‘by hand’ instead of factoring through the Riemann maps, see also [PZ, Sect. 6.3, pp. 46–50]. The problem with making estimates in the above equivalent approach is the non-explicitness of the Riemann maps. To complete the proof

9.3 Turning hyperbolics into parabolics

385

of Theorem 9.16, it suffices to prove that the maps ψ have quasiconformal extensions with dilatations bounded by 2p1 ` plog kq2 q. This is the content of the last Proposition, which is due to K. Strebel. Proposition 9.17 (Extensions to D of piecewise Möbius maps on S1 ) Let ψ : S1 ÝÑ S1 be an orientation preserving homeomorphism, which is piecewise Möbius in the following sense: there exist n ě 2 fixed points x1 “ xn`1 , x2 , . . . , xn ordered counterclockwise and n Möbius transformations ζ 1 , ζ 2 , . . . , ζ n preserving S1 such that ψ|rxj ,xj `1 s “ ζ j for 1 ď j ď n. Let k ą 1 be the largest among the multipliers of the repelling fixed points of the ζ j . Then ψ has a quasiconformal extension : D Ñ D whose dilatation is bounded by 2p1 ` plog kq2 q. Proof Let us first consider a related but easier problem on the horizontal strip S :“ tz P C | 0 ď Impzq ď π {2u with ψpzq “ z on the bottom edge R and ψpzq “ z ` log λ on the top edge R ` i π {2, where λ ą 1. In this case, we can extend ψ to a quasiconformal self-homeomorphism of S by interpolating linearly: pzq “ z `

2 π

Impzq log λ.

It is easy to verify that the dilatation of this is less than 2p1 ` plog λq2 q. (As an exercise, the reader can show that this is the best possible extension.) Back to the original situation, consider the hyperbolic convex set Dj bounded by the interval rxj , xj `1 s Ă S1 and the hyperbolic geodesic ϒj in D with endpoints xj and xj `1 . Each Dj is conformally isomorphic to the strip S above, with rxj , xj `1 s mapping to R ` i π {2 and ϒj mapping to R. The action of ψ on rxj , xj `1 s corresponds to z ÞÑ z ˘ log λj , where λj ą 1 is the multiplier of the repelling fixed point of ζ j . Thus, by the initial construction, ψ can be extended to a quasiconformal homeomorphism : Dj Ñ Dj which interpolates between ψ|rxj ,xj `1 s and the identity on ϒj , with dilatation less Ť that 2p1 ` plog λj q2 q. On D nj“1 Dj , an ideal hyperbolic n-gon, extend as the identity map. Evidently the dilatation of on D is less than 2p1 ` plogpmaxj λj qq2 q “ 2p1 ` plog kq2 q.

9.3 Turning hyperbolics into parabolics Peter Ha¨ıssinsky

Let P be a polynomial of degree d ě 2 and let KP and JP be its filled Julia set and Julia set respectively. We denote by PP its postcritical set, that is the closure of its critical orbits (see Definition 3.38). If α is an attracting or parabolic

386


point, we set Apαq to be its basin of attraction and A˝ pαq its immediate basin. The purpose of this section is to prove the following theorem, Theorem 9.18, and to apply it to the construction of matings (cf. Theorem 9.32). Theorem 9.18 (Main Theorem) Let P be a polynomial of degree d ě 2 with an attracting fixed point α and a repelling fixed point β P BA˝ pαq, β R PP . Suppose also that β is accessible from A˝ pαq. Then there exist a polynomial Q of the same degree and a David map ϕ, such that: (i) ϕpKP q “ KQ ; ϕpβq is a parabolic fixed point with Q1 pϕpβqq “ 1, and ϕpA˝ pαqq “ A˝ pϕpβqq; (ii) for all z R Apαq, ϕ ˝ P pzq “ Q ˝ ϕpzq; in particular, ϕ : JP Ñ JQ is a homeomorphism which conjugates P to Q. Remark 9.19 By a slight and sensible change of the proof, this theorem can be stated replacing the attracting fixed point by several cycles and the repelling point by cycles such that their periods divide those of the attracting points related to them, provided the repelling points that are to become parabolic are not accumulated by the recurrent critical orbits of P . The proof proceeds by a cut and paste surgery where we will only change the map in the immediate basin of the attracting fixed point to create a parabolic behaviour (see Lemma 9.21). The problem that we encounter is that we will have to change drastically the geometry of the Julia set by creating cusps. Hence we will not be able to use the qc Integrability Theorem and, instead, we will rely on David’s generalization, the David Integrability Theorem (Theorem 9.7). There are several technical difficulties to overcome. The first is to check that the assumptions of the theorem are indeed satisfied. This will come from a simple calculation made on a model map (Lemma 9.20), and spread to the whole plane by bounded distortion. The second is that composing non-quasiregular maps together does not imply at once that the resulting function retains enough analytic regularity, e.g. the ACL property. The proof follows [Ha2]. 9.3.1 Regularity of maps between local models Consider the map f : z ÞÑ λz with λ ą 1 (repelling model). For n ą N where N is large, we define the sector * " 1 S :“ z P C | θ ď arg z ď 2π ´ θ and 0 ă |z| ă N , λ


387

f

where 0 ă θ ă π . We write Qn for the quadrilateral bounded by the segments rp1{λn`1 qe˘i θ , p1{λn qe˘i θ s and the circle arcs of radii 1{λn`1 and 1{λn f f contained in S. Note that Qn “ f ń pQ0 q. The map z ÞÑ wpzq “ Log λ{ Log z conjugates f on Dλń zS to g : w ÞÑ

w 1`w

(parabolic model)

on the cusp C “ wpDλń zSq with vertex at the origin (see Figure 9.6). f pzq “ λz

gpwq “ ww`1

χ

S X Dλn g

θ

f

Qn

0

λ´pn`1q ń λ

C X Drn

Qn 0

|w| — n1 1 |w| — n` 1

Figure 9.6 Definition of χn . The total of the shaded region on the left picture corresponds to S X Dλn .

Lemma 9.20 There is a David extension χ of z ÞÑ wpzq to a neighbourhood f of the origin, with Kχ — n on Qn . Proof

We define rn :“ |wpλń e˘i θ q| and observe that rn — n1 . Define ˝

χ : S Ñ DrN zC ρ ¨ ei t ÞÑ

log λ ¨ exp i aρ ptq “ |wpρei θ q| exp i aρ ptq , | log ρ ` i θ |

where aρ ptq is an affine map in t P rθ, 2π ´ θ s, for fixed ρ. The map χ is a homeomorphism, which maps arcs of circles centred at the origin to arcs of circles centred at the origin. g f f Write Qn “ χ pQn q (see Figure 9.6). Observe that in log coordinates, Qn f corresponds to a rectangle Rn of width ln λ and height 2pπ ´ θ q, both indepeng g 1 dent of n. Asymptotically, Qn corresponds to a rectangle Rn of width ln n` n and height 2π . Also note that χ is asymptotically conjugate to an affine map f g from Rn to Rn . It follows that we can estimate Kχ by comparing the modulus f g of Rn with that of Rn . More precisely,

388

Trans-quasiconformal surgery f

Kχ —

mod Rn 1 — n. g :“ Kn — mod Rn lnp1 ` n1 q

Hence, for n0 big enough, ď

Areatz P D | Kχ pzq ě n0 u — Area

něn0

f

Qn “

π ´θ “ c e´2no λ2n0

where c ą 0 is independent of n0 . Therefore χ is David.

ln λ

,

9.3.2 Comparison of hyperbolic and parabolic basins Consider the two Blaschke products Bpar pzq “

zd ` a , 1 ` azd

a “ pd ´ 1q{pd ` 1q

and Bpzq “

zd ` b , 1 ` bzd

0 ă b ă a.

Note that both maps are a composition of z ÞÑ zd with a Möbius transformation of the unit disc onto itself. Hence both leave D invariant as well as CzD and thus S1 . They are branched coverings of D of degree d, with a unique critical point of multiplicity d ´ 1 located at z “ 0. The map B has an attracting fixed point α P p0, 1q, which attracts every point in D (the Schwarz–Pick Lemma), while z “ 1 is a repelling fixed point with b real multiplier λ “ 11´ `b d. On the other hand, Bpar has a parabolic fixed point at z “ 1, which also attracts every point in D. In both cases, observe that the interval r0, 1s is invariant, and the critical orbit marches monotonically along this segment towards z “ α, in the case of B, and towards z “ 1 in the case of Bpar . Observe that, locally, the dynamics of B around z “ 1 is conjugate via linearizing coordinates to the local model f in the section above. Likewise, those of Bpar is conjugate, via Fatou coordinates, to the local parabolic model g. We will use these local models and the map χ constructed in the last section, to build a partial conjugacy between the dynamics of B and those of Bpar in the corresponding basins of attraction. Lemma 9.21 There exist a piecewise C 1 homeomorphism φ : D Ñ D and a sector SB Ă D with vertex at 1, which is a neighbourhood of α, such that:


389

(i) for all z P DzSB , φ ˝ Bpzq “ Bpar ˝ φpzq; (ii) there is a set SB1 which is the intersection of SB with a neighbourhood of 1, Ť Ť such that φ : Dz k B ´k pSB1 q Ñ φpDz k B ´k pSB1 qq is quasiconformal; f 1 (iii) on the quadrilaterals QB n in SB defined as Qn for Lemma 9.20, we have Kφ — n, for all n ě n0 . Remark 9.22 In (ii), the inverse branches are chosen so that no sub-branch fixes 1. Proof Let us first have a look at the dynamics of Bpar . We let : D Ñ C be the Fatou coordinate such that ˝ Bpar pzq “ pzq ` 1 and p0q “ 0 (see Section 3.3.2). Choose R ą 0 large enough so that ´1 t|Im z| “ Ru has two components which bound invariant sepals at the parabolic point 1. Let P “ Cztz P C | Re z ď 1, |Im z| ď Ru, and let Spar be the connected component of ´1 pPq which contains a “ Bpar p0q and 1 in its boundary. Observe that since p0q “ 0, then paq “ 1. We let Epar “ Spar zBpar pSpar q. It is naturally endowed with a quadrilateral structure coming from its image by the Fatou coordinate (pEpar q “ r1, 2q ˆ r´R, Rs) (see Figure 9.7). B

φ

SB EB 0

b

Sp α

1 SB

1

w ÞÑ w ` 1

Bpar Spar Epar 0

a

1 q φ pSB

1

Ri 0

´Ri

1 2

P

pEpar q

Figure 9.7 Sketch of the construction of φ, for the degree d “ 2. The total of the shaded regions correspond to SB , Spar and P respectively. The white region is the 1 , then extended to first preimage of these sets. The map φ is first defined in SB SB and finally extended to D using the dynamics.

Next we look at the dynamics of B. In the quotient torus associated to α, we choose (in a way to be specified later) an annulus which is disjoint from the critical orbit and with pα, 1q{pBq as its equator. We let Sp be the connected component of its lift which contains pα, 1q. It is a simply connected forward invariant set such that the action of B is conjugate to a translation on a strip. Indeed, since the multiplier is real, Sp is conformally mapped to a sector by the

390


linearizing coordinates, which is then sent to an infinite strip by the principal branch of the logarithm. Observe that close to z “ 1, the set Sp belongs to the linearizing domain of this repelling fixed point. We assume that we chose the annulus in such a way that, under the linearizing coordinates around z “ 1, Sp is a straight sector. We then define SB1 Ă Sp and quadrilaterals QB n analogously to what we did in the repelling model in the section above, for values of n large enough. Let be a linearizing domain for α such that b P B. Note that we can p Therefore, BpSB q Ă SB , p We set SB “ Y S. choose the point b in B X B S. and we define EB “ SB zBpSB q. We proceed now to the construction of φ. We first define φ in a neighbourhood of 1 as we did for the model maps (Lemma 9.20). Observe that φ maps 1 the set SB1 and the quadrilaterals QB n in SB to their counterparts in Spar . (Note that the cusp part, is outside Spar .) In particular, it maps the boundary of SB close to 1, to the boundary of Spar also close to 1, conjugating the dynamics of B to the dynamics of Bpar on this boundary. Since linearizing coordinates and Fatou coordinates are conformal, the dilatation of φ is asymptotically n on the 1 quadrilaterals QB n . We will only use φ wherever defined in SB (i.e. on SB ), that is, we forget it on the rest of the neighbourhood of 1 where it was originally defined. We now extend φ to a homeomorphism φ : SB Ñ Spar such that, for all z P BSB , φ ˝ Bpzq “ Bpar ˝ φpzq, and such that φpbq “ a. By the choices above, we can adjust it so that φpEB q “ Epar , still keeping the conjugacy on the boundary. We impose φ to be piecewise C 1 and quasiconformal on the complement of any neighbourhood of 1. We record the following facts: (a) BpSB q Ă SB and Bpar pSpar q Ă Spar ; (b) recall that B : D Ñ D and Bpar : D Ñ D are ramified coverings of the same topological degree, and 0 is the unique critical point for B and Bpar with the same multiplicity, and φpBp0qq “ φpbq “ a “ Bpar p0q; (c) for all z P D, there are minimal iterates nB and npar such that B nB pzq P SB npar and Bpar pzq P Spar . Therefore, we may inductively define φ : D Ñ D so that φ ˝ B “ Bpar ˝ φ on DzSB . There are no topological obstructions for its definition thanks to (a) and (b), and the map is well defined on D and onto thanks to (c). Point (i) is satisfied by construction. Since compositions by holomorphic maps do not change the dilatation, (ii) holds as well. Finally, (iii) follows from the use of Lemma 9.20.


391

9.3.3 Parabolic surgery Let P be as in Theorem 9.18, that is, a polynomial of degree d ě 2 with an attracting fixed point α and a repelling fixed point β on BA˝ pαq, β R PP . It follows from the surgery construction in Section 7.5, due to Curt McMullen, that there is a quasiconformal map ψ, conformal outside Apαq, which conjugates P to another polynomial Pr except on Apαq, and such that Pr has the following properties: there is only one critical grand orbit in ψpApαqq, with at most one critical point in each component of ψpApαqq, and Pr1 pψpαqq ą 0. (See Exercises 7.5.5 and 7.5.6, where the conjugacy is called φ.) Note that in particular, ψpαq is an attracting fixed point of Pr, with immediate basin A˝ pψpαqq “ ψpA˝ pαqq. Also ψpβq is a repelling fixed point on BA˝ pψpαqq, not in the postcritical set of Pr, and ψpβq is accessible from the immediate basin. We also observe that ψpKP q “ KPr and ψpJP q “ JPr , where it conjugates the dynamics of P and Pr. It follows that it is enough to prove Theorem 9.18 for Pr. Therefore, we assume from now on that P “ Pr. Topological surgery Let A˝ be the immediate basin of α. Consider the Riemann map R “ RA˝ from A˝ onto the unit disk D, which conjugates P to a Blaschke product BA˝ . Define the map φ : D Ñ D given by Lemma 9.21, p the conjugate map of which partially conjugates B “ BA˝ to Bpar . Write B p Bpar by φ. Hence, the map B coincides with B except on the sector SB . p ˝ R which coincides with We can then define on A˝ the map F “ R´1 ˝ B P except on the preimage of the sector SB under R. Finally set Gpzq :“

# F pzq

if z P A˝ ,

P pzq

if z R A˝ .

This map G is our topological model: a ramified covering of degree d, piecewise C 1 , and thus ACL. We now need to go back to holomorphic dynamics. Straightening of the almost complex structure and conclusion of the proof of the Main Theorem (Theorem 9.18) Let us emphasize that the full strength of the David Integrability Theorem is needed: the existence of the solution in order to straighten the almost complex structure, and its uniqueness to conclude that the resulting composition map is indeed a complex polynomial (to be seen in Lemma 9.25).

392


We proceed to define a G-invariant almost complex structure in C. Let p We now p “ Bz φ{Bz φ. This Beltrami form is defined in D and invariant by B. μ p . We have the following commutative transport it to A˝ by defining μ “ R˚ μ diagram: Bpar

pD, μ0 q ÝÝÝÝÑ pD, μ0 q İ İ §φ § φ§ § p B

p q ÝÝÝÝÑ pD, μ pq pD, μ İ İ § § R§ §R F

pA˝ , μq ÝÝÝÝÑ pA˝ , μq We now spread it by the dynamics of F (which equals P outside D) to the rest of the dynamical plane. As usual, we recursively define μ :“

# pP n q˚ μ μ0

on P ń pA˝ q, Ť on Cz n P ń pA˝ q.

The map G leaves μ invariant by definition. Lemma 9.23 If β R PP then μ satisfies the hypotheses of the David Integrability Theorem (Theorem 9.7). Proof Let V be a simply connected linearizable neighbourhood of β disjoint from PP . Let U be a connected component of P ´1 pV q compactly contained in V . Let β be the trace of R´1 pSB1 q in U . Set ρ “ |P 1 pβq| and let Kμ be the dilatation ratio of μ; by Koebe’s Distortion Theorem and Lemma 9.21, Areatz P β | Kμ ą nu — p1{ρ 2n q Area β . If y P C satisfies P n pyq “ β, then let y be the connected component of P ń pβ q with vertex y. The map P n : pP ń pV q, yq Ñ pV , βq is conformal, and since P : U Ñ V fixes β, it lifts by P n as a repelling germ of same multiplier in a neighbourhood of y. By bounded distortion, there exists a constant C ą 0 such that Areatz P y , Kμ ą nu ď pC{ρ 2n q Area y , for all preimages y of β.


393

We thus deduce that for n large enough, Areatz P C, | Kμ ą nu “

ÿ

ÿ

Areatz P y , Kμ ą nu

p ě0 P p y “β

ď

ÿ

ÿ

pC{ρ 2n q Area y .

p ě0 P p y “β

Hence, Areatz P C | Kμ ą nu ď pC{ρ 2n q Area X “ C 1 e´2n ln ρ , Ť where X “ pě0, P p py q“β y . But Area X is finite because X is the union of disjoint open sets, all contained in KP , which is a bounded domain. This proves the estimate on Kμ . Remark 9.24 This lemma is also true, with the same proof, if we only assume that β is not accumulated by any recurrent critical point. The David Integrability Theorem (Theorem 9.7) asserts the existence of a map ϕ which integrates μ. However, to obtain a holomorphic function there is still some work to be done. The following lemma is a standard trick (to be compared with Proposition 9.14). Lemma 9.25 (Straightening) If there exists an ACL homeomorphism ϕ : C Ñ C, such that the equation Bz ψ “ μϕ Bz ψ has local ACL solutions, unique up to post-composition by a conformal map, and such that G˚ μϕ “ μϕ a.e., then Q “ ϕ ˝ G ˝ ϕ ´1 is the polynomial of Theorem 9.18. Proof Indeed, on every disk where G is injective, ϕ ˝ G is ACL (see the proof of Proposition 9.14), and has the same Beltrami coefficient as R; hence, there exists a conformal map, say Q, such that ϕ ˝ G “ Q ˝ ϕ (uniqueness of the solution). The map Q is thus holomorphic. Since Q´1 pt8uq “ t8u, it is a polynomial. The dynamical properties of Q are automatically satisfied by construction. To prove the local uniqueness of solutions of μ, it suffices to normalize two solutions on the unit disk and to extend it to the plane by reflection (see Section 1.2.1): the hypotheses of the David Integrability Theorem are still fulfilled, hence these solutions are the same. By Lemma 9.25, the map Q “ ϕ ˝ G ˝ ϕ ´1 is the polynomial we are looking for.

394

Trans-quasiconformal surgery 9.3.4 Application to matings

We will use the techniques above to prove the existence of certain matings of geometrically finite polynomials. Details may be found in [HT]. The different steps involve constructing subhyperbolic and postcritically finite polynomials from the given geometrically finite polynomials. Hence, we start by recalling that a rational map R is said to be geometrically finite if its postcritical set intersects the Julia set JR in a finite set, or equivalently if every critical point is either preperiodic, or attracted to an attracting or parabolic cycle. If R has no parabolic cycles it is called subhyperbolic. If the postcritical set is finite R is said to be postcritically finite. Moreover, recall that if the Julia set of a geometrically finite polynomial is connected then it is always locally connected, see [DH2]. Definition, existence and uniqueness There are many equivalent ways to define matings of polynomials. We have chosen the one presented by Milnor in [Mi2]. Definition 9.26 (The sphere) Consider S2 as the subspace of C ˆ R defined by S2 “ H` Y H´ Y tpz, 0q P C ˆ R | |z| “ 1u, where H` “ tpz, rq P C ˆ R` | |z|2 ` r 2 “ 1u and H´ “ tpz, rq P C ˆ R´ | |z|2 ` r 2 “ 1u. Let τ˘ : C Ñ H˘ be the gnomonic projections defined by: b b τ` pzq “ pz, 1q{ |z|2 ` 1, τ´ pzq “ p¯z, ´1q{ |z|2 ` 1 . The hemispheres H˘ are equipped with conformal structures via τ˘ . p ÑC p be two ramified Definition 9.27 (Topological mating) Let F, G : C coverings of degree d ě 1. Assume that 8 is totally invariant for both F and G, and that Fánd G are¯ holomorphic in a neighbourhood of 8 with local expansion zd 1 ` Op 1z q , i.e. F, G have the same leading term in their local expansion at 8. We define the topological mating F K K G : S2 Ñ S2 to ´1 ´1 2 |H´ . By be the unique extension on S of τ` ˝ F ˝ τ` |H` and τ´ ˝ G ˝ τ´ ´1 abuse of notation we will not distinguish F and τ` ˝ F ˝ τ` |H` (resp. G ´1 |H´ ). An easy calculation shows that the map F K K G is a and τ´ ˝ G ˝ τ´ well defined branched covering of degree d. In particular, if d “ 1 we get a homeomorphism of the sphere. Assume now that f and g are two monic geometrically finite polynomials Kg of degree d ě 2 with connected Julia sets. The sphere S2 on which f K is defined is equipped with the ray equivalence relation, which is defined to be the smallest equivalence relation generated by x „ y if x, y belong to the


395

Kg closure in S2 of an external ray of f or an external ray of g. The map f K preserves this relation. Definition 9.28 (Marked mating and matable polynomials) We say that pf, g, q, Rq is a marked mating, if f, g are monic polynomials of degree d, p is a continuous map such that: R is a degree d rational map, and q : S2 Ñ C (1) (semi-conjugacy) q ˝ pf K K gq “ R ˝ q; (2) (identification) qpxq “ qpyq if and only if x and y are ray-equivalent; (3) (maximal conformality) q is conformal in intpKf q Y intpKg q, p JR , and q ´1 pC p JR q “ intpKf q Y qpintpKf q Y intpKg qq “ C intpKg q. We will say that two monic polynomials f and g are matable if there exist a continuous map q and a rational map R such that pf, g, q, Rq is a marked mating. The basic result of matability is the following, due to Rees, Shishikura and Tan Lei (cf. [Sh3] and [Ta1]). Theorem 9.29 (Matability of postcritically finite polynomials) Let f and g be two monic postcritically finite polynomials of the same degree. Then either fK K g is equivalent to a Lattès example, or f and g are matable if and only if f K K g has no Thurston obstructions. In particular, two postcritically finite quadratic polynomials fc pzq “ z2 ` c and fc1 pzq “ z2 ` c1 are matable if and only if c and c1 do not belong to the same limb of the Mandelbrot set. Uniqueness of R and q Pilgrim has provided examples of non-uniqueness of the conformal conjugacy class of R in a marked mating pf, g, q, Rq in a case where R is a Lattès example. For details, see [Mi2, Appendix B.5]. However, we have the following uniqueness result. Proposition 9.30 (Uniqueness) Let pf, g, q, Rq be a marked mating. If f, g, R are geometrically finite and R is not a Lattès example, then R is unique up to conformal conjugacy, and for a given R, the map q is unique up to postp composition by an automorphism of C. Postcritically finite polynomials associated to geometrically finite polynomials Let f be a monic and centred geometrically finite polynomial of degree d ě 2 with connected Julia set. Recall that there is a unique conformal map (the inverse of Böttcher coordinates) ψf : CzD Ñ CzKf , which is

396


tangent to the identity at infinity and satisfies ψf pzd q “ f ˝ ψf pzq. Since the Julia set is locally connected this map extends continuously to the closure and induces the Carathéodory loop γf : S1 Ñ Jf (see Theorem 3.50 and the remark afterwards). Suppose f has one or more parabolic cycles. We wish to associate to f a canonical monic postcritically finite polynomial T pf q. We will proceed in two steps. The first step follows from [Ha3]. If f is a monic geometrically finite polynomial with connected Julia set, then there exist a monic subhyperbolic polynomial Spf q and an orientation preserving homeomorphism h : Jf Ñ JSpf q such that h ˝ f “ Spf q ˝ h on Jf . The map h is obtained as the continuous extension of the composition ψSpf q ˝ ψf´1 : CzKf Ñ CzKSpf q of the Böttcher coordinates of f and Spf q. So formally h “ γSpf q ˝ γf´1 . Moreover, it follows from the construction in [Ha3] that Spf q satisfies the assumptions for P in Theorem 9.18, giving rise to Q “ f and a David homeomorphism ϕ1 : C Ñ C, which in particular is a homeomorphism on the Julia sets, conjugating Spf q to f . The second step is obtained by a surgery construction that associates a canonical monic postcritically finite polynomial T pf q to Spf q, see for instance Section 4.2 and Exercise 4.2.2, or Section 7.5. From the surgery construction it follows that there is a quasiconformal map ϕ2 : C Ñ C such that ϕ2 ˝ T pf q “ Spf q ˝ ϕ2 is satisfied on some neighbourhoods of their respective Julia sets JT pf q and JSpf q . The subhyperbolic polynomial in the first step in not uniquely determined. However, the different choices give rise to the same T pf q, which is therefore canonically determined. Consider the diagram T pf q

pC, JT pf q q ÝÝÝÝÑ pC, JT pf q q § § §ϕ § ϕ2 đ đ 2 Spf q

pC, JSpf q q ÝÝÝÝÑ pC, JSpf q q § § §ϕ § ϕ1 đ đ 1 pC, Jf q

f

ÝÝÝÝÑ

pC, Jf q

Observe that ϕ :“ ϕ1 ˝ ϕ2 : C Ñ C is a David homeomorphism since it is a composition of a quasiconformal map by a David homeomorphism


397

(see Exercise 9.1.2). Moreover, the diagram commutes on the Julia sets. Therefore we have the following result. Proposition 9.31 (Existence of a David conjugacy on the Julia sets) Let f be a monic geometrically finite polynomial with connected Julia set and let T pf q denote the canonical associated monic postcritically finite polynomial. Then there is a David homeomorphism ϕ : C Ñ C such that ϕ ˝ T pf q “ f ˝ ϕ on JT pf q . We are now ready to state our main application, which will be proved in the next section. Theorem 9.32 (Matability of geometrically finite polynomials) Let f, g be two monic and centred geometrically finite polynomials with connected Julia sets. If T pf q and T pgq are matable, then f and g are also matable. Corollary 9.33 (Precise condition for matability) Two geometrically finite quadratic polynomials fc and fc1 are matable if and only if c¯ and c1 do not belong to the same limb of the Mandelbrot set. Surgery for matings The existence of marked matings of certain postcritically infinite polynomials can now be established through surgery. We state this in the next proposition, which in particular provides us with an extension of Theorem 9.29 to hyperbolic polynomials with infinite critical orbits. Proposition 9.34 (Matability of postcritically infinite polynomials) Let pf, g, q, Rq be a marked mating of polynomials f and g. Let us assume that fp (resp. gp) is a quasiregular map which coincides with f (resp. g) on its basin of infinity, and that μ (resp. ν) is a fp-invariant Beltrami form (resp. gp-invariant) supported on its filled Julia set. If we let ϕ (resp. ψ) be a quasiconformal homeomorphism which integrates μ (resp. ν) normalized at infinity to be tangent to the identity, then the polynomials f1 “ ϕ ˝ fp ˝ ϕ ´1 and g1 “ ψ ˝ gp ˝ ψ ´1 are matable. Proof

Define $ ´1 ’ ’ &τ` ˝ ϕ ˝ τ` K ψ :“ Id ϕK K ψ : S2 Ñ S2 by: ϕK ’ ’ %τ ˝ ψ ˝ τ ´ 1 ´

´

on H` , on the equator, on H´ .

The map ϕ restricted to the basin of 8 of f realizes a conformal conjugacy from f to f1 . The normalization of ϕ guarantees that ϕ maps the external ray of f of argument θ onto the external ray of f1 of the same argument. Therefore

398


the map ϕK K ψ : S2 Ñ S2 is a homeomorphism, conjugating fpK K gp to f1 K K g1 and preserving the external rays of the same argument. p as the pushforward by q as follows: We define the Beltrami form ξ on C $ ’ ’ &q˚ μ on qpintpKf qq, ξ :“ q˚ ν on qpintpKg qq, ’ ’ %μ elsewhere. 0

Since q is conformal on intpKf q Y intpKg q, the Beltrami form ξ is Rinvariant. Therefore, the Integrability Theorem (Theorem 1.28) provides us p a, b, cq Ñ pC, p a, b, cq which with a quasiconformal homeomorphism χ : pC, solves the Beltrami equation associated to ξ . p K ψq´1 : S2 Ñ C. Set R1 “ χ ˝ R ˝ χ ´1 and q1 “ χ ˝ q ˝ pϕK We can then check that pf1 , g1 , q1 , R1 q is a marked mating. Remark 9.35 The proof of Theorem 9.32 will follow the same lines as Proposition 9.34, but instead of using the classical Integrability Theorem (Theorem 1.28), we shall use the David Integrability Theorem (Theorem 9.7) from Section 9.1. Proof of Theorem 9.32 We proceed as in Proposition 9.34. Let us apply Proposition 9.31 to f and g. Let T pf q and Spf q be the associated monic postcritically finite polynomials, and ϕ and ψ the David homeomorphisms that are given by the proposition. By assumption T pf q and T pgq are matable. Let pT pf q, T pgq, q, Rq denote a marked mating. We let μ and ν be the Beltrami forms associated to ϕ and ψ. p and denote the resulting Beltrami form by ξ . We push them forward by q on C Let us define ¯ ´ K pψ ´1 ˝ g ˝ ψq ˝ q ´1 . H :“ q ˝ pϕ ´1 ˝ f ˝ ϕqK This map is well defined and we can check that it is ACL. Moreover, ξ is H -invariant, and we may complete the proof analogously to the proof of Proposition 9.34 using the David Integrability Theorem (Theorem 9.7). The proofs of Lemma 9.23 and Lemma 9.25 apply to this setting as well. Proof of Corollary 9.33 Let fc and fc1 be geometrically finite polynomials. If c and c1 belong to conjugate limbs of the Mandelbrot set their respective α-fixed point have opposite external arguments and therefore S2 { „ray is not a sphere and the marked mating can not exist. See [HT] for further details.


399

On the other hand, if c and c1 are not in conjugate limbs, then it is the same for the parameter values of T pfc q and T pfc1 q, the centre of the hyperbolic component having c, resp. c1 , as its root. Therefore the marked mating of fc and fc1 exists by Theorem 9.29 and Theorem 9.32.

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Index

Page numbers in bold indicate where a concept is defined. (´1, 1)-differential, see Beltrami form absolutely continuous, 15 on an interval, 23 on lines, see ACL access, 177 ACL, 23, 24, 366 affine conjugacy, 95 Ahlfors, L., 21, 40, 44, 134, 364 Conjecture, 145 Finiteness Theorem, 143 almost complex structure, 14, 15 on a Riemann surface, 36 annulus, 28 area condition, 365 area distortion, 32 Arnol’d, V., 99, 103 family, 229, 234 Arzelà–Ascoli Theorem, 105 Astala, K., 21, 127 attracting p-cycle, 108 fixed point, 110 Böttcher coordinates, 111, 123, 155 Baker, I. N., 170, 273 domain, 116 basin of attraction hyperbolic, 388 immediate, 116 of an attracting cycle, 108 parabolic, 113, 388 Beardon, A., 283 Beltrami coefficient, 8, 14 of a Beltrami form, 36

408

symmetric, 18, 20 differential see Beltrami form 364 David, 364 equation, 40 form, 35, 42 Beltrami, E., 44 Bers, L., 40, 44, 127, 145, 244, 364 Sewing Lemma, 34 Beurling–Ahlfors extension, 83 Bielefeld, B., 308 bilipschitz, 31 Blaschke product, 120, 162, 190, 228, 230, 245, 371, 388 generalized, 121 Bojarski, B., 40, 44, 364 boundary connect sum, 267 correspondence, 245 ideal, 249, 250 topological, 250 value problem, 77 bounded type number, 100, 230, 372 branched covering, 268 Bryuno, A.D., 115 number, 101, 190, 240 Buff, X., 189, 347, 362 Bullett, S., 291, 298, 299 Canary, D., 145 Cantor function, 25 capture, 190 component, 192 Carathéodory, C. semi-conjugacy or loop, 125 Theorem, 69

Index Carleson, L., 376 chordal metric, 94 circle homeomorphism, 97 critical, 101 lift of, 98 circle map analytic linearization, 101 critical, 101, 126 expanding, 262 linearizable, 99 real analytic, 101, 190, 230 circumcentre, 186 compactly contained, 23 complex analytic manifold, 34 complex dilatation, 9, 13 complex structure almost, see almost complex structure analytic, 34, 42 invariant, 17 standard, 15 stretching, 208, 215 turning, 208, 215 wringing, 208, 213 conformal conjugacy, 95 conformal welding, 245 conjecture hyperbolicity, 132 MLC, 131, 132 rigidity, 132 conjugacy, 60, 95 J -conjugacy, 167 David, see David conjugacy invariant, 95, 104 quasiconformal, 131 for groups, 135 conjugate diameter, 11 connectedness locus, 128, 132, 191, 211, 321, 343 of polynomial-like mappings, 223 continued fraction, 100, 372 continuum, 250 Cremer, H., 116 fixed point, 114, 115 critical point, 56, 97, 108 free, 321 marked, 133, 344 curve, 64 closed, 64 differentiable, 64 exterior of, 225 interior of, 225 Jordan, see Jordan curve piecewise regular, 71 smooth, 64 cycle, see p-cycle

409

David, G., 364 conjugacy, 372 Beltrami differential, 364, 365 conjugacy, 365, 397 homeomorphism, 364, 366 Integrability Theorem, 364, 369, 386 degenerate group, 145 degree, 93 Denjoy, A., 99, 190 counterexample, 99 Denjoy’s Theorem, 99 Density Conjecture, 145 devil staircase, 25, 29, 32, 104 diameter, 118 dilatation, 9, 14, 15 maximal, 27 of a quadrilateral, 27 Diophantine number, 100, 115 bounded type, see bounded type number constant type, see bounded type number order of, 100 distribution, 21 distributional derivatives, 22 Douady, A., 44, 115, 150, 219, 225, 230, 282, 321, 337, 347, 354 rabbit, 124 Douady–Earle extension, 83 dynamical space, 105 ellipse field, see field of ellipses elliptic Möbius transformation, 134 model, 255 Epstein, A.L., 308, 319, 343 equipotential, 123 equivalence, 96 hybrid, 125, 131, 221 escape locus, 211 foliation, 216 essential singularity, 116 exceptional set, 108 expanding map, 119 extensions Beurling–Ahlfors, 83 David, 375 disc-annulus, 270 Douady–Earle, 83 of boundary maps, 77, 83 of conformal maps, 69, 71 of coverings, 71 of quasiconformal maps, 75 to the unit disc, 81 external argument, 124 ray, 124

410 Fatou component classification of, 116 periodic, 115 preperiodic, 115 wandering, see wandering domain Fatou, P., 112, 118, 282 component, see Fatou component coordinates, 113 set, 107, 109 Fatou–Shishikura inequality, 118 fibre, 177, 216 field of ellipses, 14, 15 fixed point attracting, 110 Cremer, see Cremer fixed point irrationally indifferent, 114 parabolic, see parabolic fixed point repelling, 110 Siegel, 114 superattracting, 111 Flower Theorem, 112 Fuchsian group, 141, 160, 244 fundamental domain, 142 Gauss, C. F., 44 Gehring, F., 23 genus, 244 geometrically finite, 120, 144, 394 Geyer, L., 232 Ghys, E., 102, 224, 228, 230 surgery, 225, 370 Gleason, A., 150, 159 Gluing Lemma, 253, 254 Goldberg, L., 357 Grötzsch, H., 21 grand orbit, 108, 135, 183 Green’s function, 123 Ha¨ıssinsky, P., 184, 346, 365, 385 Harvey, W., 299 hedgehog, 115 Henriksen, C., 189 Herman ring, 116, 122, 126, 189, 190, 232, 233, 235 Herman, M., 101, 103, 198, 225, 228, 230 number, 101, 102, 228 ring, see Herman ring ´ ¸ tek Theorem, 102, 225 Herman–Swia Hölder condition, 31 Hölder inequality, 38 holomorphic correspondence, 293 holomorphic map with respect to μ, 17, 192 holomorphic motion, 126, 208, 332 Hubbard, J.H., 21, 150, 219, 282, 343, 354 hyperbolic component, 128, 131, 151 centre of, 130, 150

Index of disjoint type, 343 root of, 130 hyperbolic model, 255 hyperbolicity, 119 conjecture, 128, 132 ideal boundary, 276 compatibility, 252 indifferent p-cycle, 108 Integrability Theorem, 40 David, 364 dependence on parameters, 42 qc, 364 trans-qc, 364 integrating map, 40 normalization of, 42 interpolation on annuli, 81 on half strips, 82 on quadrilaterals, 80 on standard annuli, 78 quasiconformal, see quasiconformal interpolation invariant set, 96 totally, 107, 109 under a group, 135 J -stability, 127 Jacobi’s Theorem, 98, 104 Jacobian, 24, 32 J -conjugacy, 109 Jordan annulus, 71 arc, 64 curve, 64, 119 Theorem, 64 domain, 64 Julia set, 107 filled, 122 of a polynomial-like mapping, 220 truncated, 325 of a polynomial-like mapping, 220 properties, 109 Julia, G., 107, 112 Key Lemma for surgery, 60 dependence on parameters, 62 Khinchin, A.Y., 373 Kiwi, J., 357 Kleinian group, 135, 245, 267, 291 elementary, 137 Korn, A., 44 Kœnigs’ Theorem, 110 λ-lemma, 127, 332 Lavaurs, P., 343 Lavrentiev, M. A., 44

Index Leau, L., 112 Lehto, O., 21, 23 Lichtenstein, L., 44 lift, 98 limb, 316, 321, 346 embedding, 337, 347 linear conjugacy, 95 linearization, 99, 110 analytic, 101 conformal, 114 parabolic, 113 quasisymmetric, 104 linearizing coordinate, 110, 151, 154 log, 312, 341, 349 Liouville number, 100 local connectivity, see set locally connected loxodromic transformation, 134 Lyubich, M., 21 Möbius transformation, 134 classification of, 134 of finite order, 134 Mañe´ , R., 126 Mandelbrot set, 128, 224 generalized, 133 limb of, 316, 346 Mandelbrot-like family, 224 map R-linear, 8 entire transcendental, 93, 187, 239 expanding, 119 first return, 324 geometrically finite, see geometrically finite hyperbolic, 119 integrating, see integrating map linearizing, see linearizing coordinate meromorphic transcendental, 93, 187, 240 Minkowski question mark, 295 Misiurewiz, see Misiurewicz map multiplier, see multiplier map orientation reversing, 18, 19 proper, 120 quasiconformal, see quasiconformal mapping quasirational, 180, 251 quasiregular, see quasiregular mapping quasisymmetric, see quasisymmetric mapping rational, 93 degree of, 93 subhyperbolic, 119, 394 Martin, G., 127 mating, 292, 294, 296, 394 marked, 395 topological, 394 McMullen, C., 168, 225, 244, 248

411

Gluing Lemma, 253 Measurable Riemann Mapping Theorem, see Integrability Theorem Milnor, J., 105, 343–345, 357, 394 Misiurewicz, M. map, 117 point, 131 polynomial, 131 model map, 147, 369, 386 rigid, 255 modular group, 292, 296 modulus, 30, 192 continuity of, 88 of a quadrilateral, 27 of a torus, 161 of an annulus, 28 opening, 308, 310, 315, 319, 339 Montel’s Theorem, 105, 136 Morrey, C. B., 40, 44, 364 Multibrot set, 133, 156 multiplicity of a parabolic fixed point, 112 multiplier, 108 map, 130, 150, 151, 331 Mumford, D., 138 N˘aishul’, V. I., 114 near translation, 72, 308, 315, 341 neutral p-cycle, 108 Newton’s method, 220 No Wandering Domains Theorem, 169 for Kleinian groups, 143 for rational maps, 116, 150 normal family, 105 normal form, 110 opening modulus, 349 orbit, 60, 92 periodic, see p-cycle singular, 116 p-cycle, 92 classification of, 108 multiplier of, 108 parabolic p-cycle, 108 basin, 388 fixed point, 111 multiplicity of, 112 Flower Theorem, 112 linearization, 113 Möbius transformation, 134 model, 256 surgery, 391 parameter hyperbolic, 128 plane, 127 space, 126

412 parametrization, 64 conformal, 70 quasisymmetric, 68 partial derivatives, 23 Penrose, C., 298 petal, 112 Petersen, C.L., 102, 365, 370 phase space, 105 Pilgrim, K., 120, 267 point accessible, 177 critical, see critical point fixed, see fixed point periodic, see p-cycle pre-essential, 240 preperiodic, 92 recurrent, 92 regular, 116 pole, 240 polynomial, 93 centred, 97 monic, 97 unicritical, 133, 156 polynomial-like mapping, 219, 291 holomorphic family of, 223 postcritically finite, 394 prime end, 249 principal vein, 337 proper discontinuity, 136 pullback, 15, 16, 16, 55 by quasiregular mappings, 59 notation, 16, 60 of Beltrami forms, 37 pushforward, 17 quadrilateral, 26, 80 modulus of, 27 quasi-Fuchsian group, 141, 162, 244 quasiannulus, 71, 85 quasiarc, 33, 68 quasicircle, 68, 245 quasiconformal conjugacy, 60, 95 deformation, 144, 148, 201, 245 interpolation, 87, 314, 316, 351 removability, 33 rigidity, 149 surgery, see surgery quasiconformal mapping analytic definition, 22, 24 between Riemann surfaces, 35 composition, 31 families of, 34, 38 geometric definition, 28 properties, 31 quasidisc, 68, 85 quasientire, 180

Index quasimeromorphic, 180 quasirational, 268 quasiregular mapping, 55–57 properties, 58 pullback by, 59 quasisymmetric mapping, 65, 66, 85, 246 composition, 67 quotient torus, 312, 348 radial limit, 177 recurrent, 92 Rees, M., 395 reflection, 18, 20, 242 renormalization, 354, 356 bi, 355 left, 356 right, 357 repelling p-cycle, 108 fixed point, 110 Rickman’s Lemma, 34 Riemann surface, 34 abstract, 320, 324, 339 atlas, 34 chart, 34 Riesz, F., 172, 177 Riesz, M., 172, 177 rigid rotation, 97 gluing, 226 rigidity, 331 conjecture, 132 rotation function, 103 rotation number, 98, 103, 104, 228 combinatorial, 113, 125, 308, 355 of a fixed point, 114 on invariant curves, 102 rotation ring, 189, 191 twisting, 198 Royden, H. L., 127 Rüssman, H., 115 Sad, P., 126 Schleicher, D., 322 Schottky circles, 139 group, 142 sector, 308 log-Böttcher, 309, 325 opening modulus, 308 slope, 309 semi-conjugacy, 96 Series, C., 138 set cellular, 343 exceptional, 177 full, 122

Index limit, 136, 244, 291 properties, 137 locally connected, 120, 124, 131, 177, 371 ordinary, 136, 291 properties, 137 postcritical, 117 postsingular, 117 recurrent, 92 regular, 136 Sewing Theorem, 245 Shishikura, M., 118, 179, 225, 232, 235, 240, 282, 321, 333, 354, 395 First Principle, 180, 182, 222 for transcendental maps, 186 Perturbation Theorem, 283 Second Principle, 183, 268 Shub, M., 262 Siegel disc, 114, 116, 119, 189, 225, 228, 232, 233, 235, 370 Siegel, C.L., 114 capture, 190 disc, see Siegel disc point, 114 singular orbit, 116 value, 116 singularity, 116 Słodkowski, Z., 127 Sobolev space, 22, 24, 365, 366 spherical metric, 94 standard annulus, 28, 71 half strip, 72 Steinmetz, N., 170 Straightening Theorem, 221, 374, 393 Sullivan, see Sullivan, D. Strebel, K., 385 subhyperbolic, see map subhyperbolic Sullivan, D., 116, 126, 133, 143, 145, 150, 169 dictionary, 146 Straightening Theorem, 184, 223 for transcendental maps, 186 superattracting p-cycle, 108 fixed point, 111 surgery, 147

413

cut and paste, 149, 162, 219, 235, 244, 248, 267, 282, 307, 320, 370, 386 disc-annulus, 267 inverse of, 277 for matings, 397 Ghys, see Ghys, E. intertwining, 343 parabolic, 391 soft, 148, 151, 160, 169, 198, 207 topological, 147 trans-quasiconformal, 148 ´ ¸ tek, G., 230 Swia symmetry, 18, 20 locus, 344 Tameness Conjecture, 145 Tan Lei, 120, 267, 395 Teichmüller space, 149 test function, 21 thickening construction, 356 Thurston, W., 145 topological conjugacy, 95 Tukia, P., 184 twisting, 198 Uniformization Theorem, 35 Simultaneous, 244 value asymptotic, 116 critical, 116 singular, 116 Virtanen, K. I., 21 wandering domain, 115, 170 Weyl Lemma, 32, 61 Wright, D., 138 Yampolsky, M., 308, 319, 343 Yin, Y., 120 Yoccoz, J.-C., 101, 115, 321, 355, 371, 376 inequality, 355 near parabolic bound, 382 Young centre, 186 Zakeri, S., 190, 232, 365 Zhang, G., 232 Zoretti’s Theorem, 262

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