Ergebnisse der Mathematik und ihrer Grenzgebiete

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1 Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge Band 25 ASeries of Modem Surveys in Mathematics Editorial Board E. Bombieri, Princeton S. Fefennan, Stanford M. Gromov, Bures-sur-Yvette H.W. Lenstra, Jr., Berkeley, P.-L. Lions, Paris R. Remmert (Managing Editor), Münster W. Schmid, Cambridge, Mass., J-P. Serre, Paris J.Tits, Paris

2 Welington de Melo Sebastian van Strien One-Dimensional Dynamics With 89 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

3 Welington de Melo Instituto dematemätica Pura e Applicada Estrada Dona Castorina Rio de Janeiro, Brazil Sebastian van Strien Mathematics Department, University of Amsterdam Plantage Muidergracbt TV Amsterdam, Tbe Netberlands MathematIcs Subject Classification (1991): 58Fxx, 34Cxx, 30-xx ISBN-13: e-isbn-13: : / Library of Congress Cataloging-in-Publication Data Melo, Welington deo One-dimensional dynamies Welington de Melo, Sebastian van Strien. p. cm. (Ergebnisse der Mathematik und ihrer Grenzgebiete; 3. Folge, no. 25) Includes bibliographical references and index. ISBN-13: {j Differentiable dynamical systems. I.Strien, Sebastian van, II. Title. III.Series. QA614.8.M l4'.74-dc CIP Tbis work is subjcct to copyright. All rights are reserved, whetherthe whole or part of the material is concerned, spccifically thc rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thercof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its currcnt version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1993 Softcover reprint of the hardcover Ist edition 1993 Tbe use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: From dvi files by the authors I - Printed on acid-free paper

4 To Gilza, Hilary Daniel and David

5 Preface One-dimensional dynamics has developed in the last decades into a subject in its own right. Yet, many recent results are inaccessible and have never been brought together. For this reason, we have tried to give a unified ac count of the subject and complete proofs of many results. To show what results one might expect, the first chapter deals with the theory of circle diffeomorphisms. The remainder of the book is an attempt to develop the analogous theory in the non-invertible case, despite the intrinsic additional difficulties. In this way, we have tried to show that there is a unified theory in one-dimensional dynamics. By reading one or more of the chapters, the reader can quickly reach the frontier of research. Let us quickly summarize the book. The first chapter deals with circle diffeomorphisms and contains a complete proof of the theorem on the smooth linearizability of circle diffeomorphisms due to M. Herman, J.-C. Yoccoz and others. Chapter II treats the kneading theory of Milnor and Thurstonj also included are an exposition on Hofbauer's tower construction and a result on fub multimodal families (this last result solves a question posed by J. Milnor). In the third chapter we analyze structural stability and hyperbolic properties of one-dimensional systems with a simplified proof of a result of R. Marie. This chapter has a section describing the ideas that are used in the theory of rational maps on the Riemann sphere corresponding to those which are used here; it also serves as an introduction to the last chapter. The fourth chapter shows that Denjoy's result for circle diffeomorphisms also holds for non-invertible maps and that the period of the attractors of these maps is bounded. The first part of this result extends work of J. Guckenheimer, J.-C. Yoccoz, A.M. Blokh and M.Yu. Lyubich. The first three sections of Chapter IV give all the distortion tools necessary for the remaining chapters. From Chapter V onwards, we confine ourselves to unimodal maps. The fifth chapter gives an account of the state-of-the-art concerning ergodicity, Cantor attractors and invariant measures. We shall give a unified treatment of results of A.M. Blokh, P. Collet, J.-P. Eckmann, J. Guckenheimer, S.D. Johnson, M.Yu. Lyubich, G. Keller, M. Martens, M. Misiurewicz, T. Nowicki, S. van Strien and partly of the results of Jakohson and S w i ~ talso e k we. give a complete account of the proof of M. Benedicks and 1. Carleson of Jakohson's result on invariant measures in families of maps. In the final chapter, a complete proof is given of D. Sullivan's recent results on the universal structures of one-dimensional systems coming from renormalization. This chapter uses ideas from complex analysis, in particular Teichmüller theory. The necessary background for this book is treated in an appendix. This has resulted in a lengthy volume. However, the reader should not be

6 viii Preface discouraged by this. The book is organized in such a way that each of the chapters is essentially independent of the others. For this reason, we have defined some concepts twice or more. (For example the notion of wandering interval is defined in both Chapter land 11.) We have used Chapter II of this text for an introductory course at first year graduate level on the combinatorial properties of piecewise monotone maps. Viviane Baladi used Chapter land the first part of Chapter II for an 8 week introductory graduate course in dynamical systems. Sections IV.1-IV-3 and V.1-V.6 were used for another course on the metric theory of unimodal maps. A more advanced graduate course on the theory of renormalizations included Sections III.1, IV.1-IV.3 and Chapter VI. Most exercises include substantial hints (some of these hints are even complete proofs). We should emphasize that we have not tried to give a complete account of all developments in one-dimensional dynamics. Most notably we have only given a short introduction to the combinatorial theory of one-dimensional systems; for more references on this see Section 1I.11 and recent monographs by Sarkovskii et al. (1989), Aiseda et al. (1990), Block and Coppel (1992) and by Misiurwicz and Nitecki (1991). We have not touched on thermodynamical theory, singularity spectra, decay of correlations and such matters, though some references to these subjects are given. This book grew from notes written by the first author which were distributed during a course organized by the Brazilian Mathematical Society. The first part of this book retains the same structure as those notes but has been very much expanded. Moreover, the last two chapters are new. In these chapters results are given on ergodic properties of one-dimensional dynamical systems and also a complete account is given of Dennis Sullivan's proof of the renormalization conjectures of Feigenbaum, Coullet and Tresser. Exercises have been added. We wish to thank Klaas Pieter Hart and Lucio Rodrfgues for their help with making 1E;X-macros and Tineke Mulder, Kitty Cijfer and Ruud Schotting for their practical help. We also wish to thank Viviane Baladi, Thierry Barbot, Christi an Bonatti, Karen Brucks, Henk Bruin, Robbert Fokking, Jean-Marc Gambaudo, Ernst Hansen, Toby Hall, Gerhard Keller, Bernd Krauskopf, Ricardo Maiie, Hilary Marland, Marco Martens, Gilza de Melo, Pedro Mendes, Jack Milnor, Tomasz Nowicki, Hiroe Oka, Paulo Sad, Atsuro Sannami, Mitsu Shishikura, Charles Tresser, Masato Tsujii, and Edson Vargas for reading parts of the manuscript. Their comments, in particular those of Viviane Baladi, Henk Bruin, Bernd Krauskopf and Tomasz Nowicki, have helped us to eliminate many mistakes. Duncan Sands' comments on a previous version of the proof of Benedicks and Carleson's result (Section V.6) have been very useful. Conversations with Michael Lyubich were very helpful for us in completing the proof of the Grötzsch Inequality in the proof of Proposition VI.7.1. Frederick Gardiner's comments on the proof of the Coiling Lemma, see Theorem VI.8.2, were also very valuable. We would like to thank Dennis Sullivan for many inspiring discussions and in particular spending many hours with us explaining his renormalization results. The first author would like to thank IHES and the

7 Preface ix Technical University of Delft for the hospitality during several visits that were essential for the preparation of this book. Amsterdam, Rio de Janeiro December 1992 Welington de M elo Sebastian van Strien

8 Table of Contents Chapter O. Introduction Chapter I. Circle Diffeomorphisms The Combinatorial Theory of Poincare The Topological Theory of Denjoy 36 2.a The Denjoy Inequality 47 2.b Ergodicity Smooth Conjugacy Results Families of Circle Diffeomorphisms; Arnol'd tongues Counter-Examples to Smooth Linearizability Frequency of Smooth Linearizability in Families Some Historical Comments and Further Remarks 78 Chapter H. The Combinatorics of One-Dimensional Endomorphisms The Theorem of Sarkovskii Covering Maps of the Circle a.s Dynamical Systems The Kneading Theory and Combinatorial Equivalence a Examples b Hofbauer's Tower Construction Full Families and Realization of Maps Families of Maps and Renormalization Piecewise Monotone Maps can be Modelled by Polynomial Maps The Topological Entropy The Piecewise Linear Model Continuity of the Topological Entropy Monotonicity of the Kneading Invariant for the Quadratic Family Some Historical Comments and Further Remarks 198 Chapter IH. Structural Stability and Hyperbolicity The Dynamics of Rational Mappings Structural Stability and Hyperbolicity Hyperbolicity in Maps with Negative Schwarzian Derivative The Structure of the Non-Wandering Set

9 xii 5. Hyperbolicity in Smooth Maps Misiurewicz Maps are Almost Hyperbolic.. 7. Some Further Remarks and Open Questions Chapter IV. The Structure of Smooth Maps.. 1. The Cross-Ratio: the Minimum and Koebe Principle La Some Facts about the Schwarzian Derivative 2. Distortion of Cross-Ratios... 2.a The Zygmund Conditions Koebe Principles on Iterates Some Simplifications and the Induction Assumption. 5. The Pullback of Space: the Koebe/Contraction Principle 6. Disjointness of Orbits of Intervals Wandering Intervals Accumulate on Turning Points 8. Topological Properties of a Unimodal Pullback. 9. The Non-Existence of Wandering Intervals. 10. Finiteness of Attractors Some Further Remarks and Open Questions Chapter V. Ergodic Properties and Invariant Measures 1. Ergodicity, Attractors and Bowen-Ruelle-Sinai Measures 2. Invariant Measures for Markov Maps Constructing Invariant Measures by Inducing Constructing Invariant Measures by Pulling Back Transitive Maps Without Finite Continuous Measures. 6. Frequency of Maps with Positive Liapounov Exponents in Families and Jakobson's Theorem Some Further Remarks and Open Questions Chapter VI. Renormalization The Renormalization Operator 2. The Real Bounds Bounded Geometry The PullBack Argument 5. The Complex Bounds. 6. Riemann Surface Laminations 7. The Almost Geodesic Principle 8. Renormalization is Contracting 9. Universality of the Attracting Cantor Set. 10. Some Further Remarks and Open Questions Contents

10 Contents Chapter VII. Appendix 1. Some Terminology in Dynamical Systems. 2. Some Background in Topology Some Results from Analysis and Measure Theory 4. Some Results from Ergodic Theory Some Background in Complex Analysis. 6. Some Results from Functional Analysis. Bibliography. Subject Index xiii