2460-1 Advanced School and Workshop in Real and Complex Dynamics 20 - 31 May 2013 Conformal Geometry and Dynamics of Quadratic Polynomials LYUBICH Mikhail State University of New York At Stony Brook NY 11794-3660 Stony Brook USA Conformal Geometry and Dynamics of Quadratic Polynomials Mikhail Lyubich Abstract. Contents Chapter 0. Introduction 5 1. Preface 5 2. Background 8 Part 1. Conformal and quasiconformal geometry 17 Chapter 1. Conformal geometry 19 1. Riemann surfaces 19 2. Holomorphic proper maps and branched coverings 40 3. Extremal length and width 44 4. Principles of the hyperbolic metric 50 5. Uniformization Theorem 59 6. Carath´eodory boundary 62 7. Appendix: Potential theory 67 Chapter 2. Quasiconformal geometry 79 11. Analytic definition and regularity properties 79 12. Geometric definitions 86 13. Further important properties of qc maps 90 14. Measurable Riemann Mapping Theorem 94 15. One-dimensional qs maps, quasicircles and qc welding 101 16. Removability 104 Chapter 3. Elements of Teichm¨uller theory 107 17. Holomorphic motions 107 18. Moduli and Teichm¨uller spaces of punctured spheres 109 19. Bers Embedding 113 Part 2. Complex quadratic family 117 Chapter 4. Dynamical plane 119 20. Glossary of Dynamics 119 21. Holomorphic dynamics: basic objects 123 22. Periodic motions 133 23. Hyperbolic maps 141 24. Parabolic maps 146 25. Misiurewicz maps 146 26. Quasiconformal deformations 146 27. Quadratic-like maps: first glance 150 28. Appendix: Expanding circle maps 153 3 4 CONTENTS Chapter 5. Remarkable functional equations 159 29. Linearizing coordinate in the attracting case 159 30. Existence of Siegel disks 161 31. Global leaf of a repelling point 161 32. Superattractng points and B¨ottcher coordinates 163 33. Parabolic points and Ecale-Voronin´ cylinders 167 Chapter 6. Parameter plane (the Mandelbrot set) 169 28. Definition and first properties 169 29. Connectivity of M 172 30. The Multiplier Theorem 176 31. Structural stability 179 32. Notes 187 Chapter 7. Combinatorics of external rays 189 39. Dynamical ray portraits 189 40. Limbs and wakes of the Mandelbrot set 197 41. Geodesic laminations 197 42. Limbs and wakes of the Mandelbrot set 197 43. Misiurewisz wakes and decorations 202 44. Topological model 202 45. Renormalization 202 Chapter 8. Thurston theory 205 45. Rigidity Theorem 205 46. Rigidity of superattracting polynomials 206 47. Hubbard tree determines f 206 48. Realization of critically periodic maps 207 Part 3. Little Mandelbrot copies 211 Chapter 9. Quadratic-like maps 213 49. Straightening 213 50. External map 220 51. Uniqueness of the straightening 221 52. Weak q-l maps and Epstein class 224 Chapter 10. Quadratic-like families 225 42. Fully equipped families 225 43. QL families over the complex renormalization windows 234 44. Notes 234 Chapter 11. Yoccoz Puzzle 235 46. Combinatorics of the puzzle 235 47. Local connectivity of non-renormalizable Julia sets 240 48. Local connectivity of of M at non-renormalizable points 240 Part 4. Hints and comments to the exersices 241 49. Index 247 CONTENTS 5 Bibliography 251 CHAPTER 0 Introduction 1. Preface In the last quarter of 20th century the complex and real quadratic family 2 fc : z → z + c was recongnized as a very rich and representative model of chaotic dynamics. In the complex plane it exhibits fractal sets of amazing beauty. On the real line, it contains regular and stochastic maps intertwined in an intricate fashion. It also has remarkable universality properties: its small pieces (if to look at the right place) look exactly the same as the whole family. Interplay between real and complex dynamics provide us with deep insights into both. This interplay eventually led to a complete picture of dynamics in the real quadratic family and a nearly compete picture in the complex family. In this book we attempt to present this picture beginning from scratch and supplying all needed background (beyond the basic graduate education). Part 1 of the book contains a necessary background in conformal and quasi- conformal geometry with elements of the Teichm¨uller theory. The main analytical Figure 1. Mandelbrot set. It encodes in one picture all beauty and subtlety of the complex quadratic family. 7 8 0. INTRODUCTION Figure 2. Baby M-set. tools of holomorphic dynamics are collected here in the form suitable for dynami- cal applications: principles of hyperbolic metric and extremal length, the classical Uniformization Theorem and Measurable Riemann Mapping, and various versions of the λ-lemma. Part 2 begins with the classical Fatou-Julia theory (adatpted to the quadratic family): basic properties of Julia sets, classification of periodic motions, important special classes of maps. Then Sullivan’s No Wandering Domains Theorem is proved, which completes description of the dynamcis on the Fatou set. We proceed with a discussion of remarkable functional equations associated with the local dynamics (which were one of the original motivations for the classical theory). In Chapter 2 we pass to the parameter plane, introducing the Mandelbrot set and proving two fundamental theorem about it: Connectivity and the Multiplier Theorem (by Douady and Hubbard). We proceed with the Structural Stability theory (by Man´e-Sad-Sullivan and the author). We conclude this chapter with a proof of the Milnor-Thurston Entropy Monotonicity Conjecture that gives the first illustration of the power of complex methods in real dynamics. The next chapter (3) is dedicated to the combinatorial theory of the quadratic family developed by Douady and Hubbard. It provides us with explicit combinato- rial models for Julia sets and the Mandelbrot set. The problem of local connectivity of Julia sets and the Mandelbrot set (MLC) arises naturally in this context. In the final chapter of this part we introduce a powerful tool of contempo- rary holomorphic dynamics: Yoccoz puzzle, – and prove local connectivity of non- renormalizable quadratic polynomials. 1. PREFACE 9 Figure 3. Real quadratic family as a model of chaos. This picture { n }∞ presents how the limit set of the orbit fc (0) n=0 bifurcates as the parameter c changes from 1/4ontherightto−2 on the left. Two types of regimes are intertwined in an intricate way. The gaps correspond to the regular regimes. The black regions correspond to the stochastic regimes (though of course there are many narrow invisible gaps therein). In the beginning (on the right) you can see the cascade of doubling bifurcations. This picture became symbolic for one-dimensional dynamics. One of the most fascinatining features of the Mandelbrot set, clearly observed on computer pictures, is the presence of the little copies of itself (“baby M-sets”), which look almost identically with the original set (except for possible absence of the main cusp). The complex renormalization theory is designed to explain this phenomenon. In part 3 we develop the the Douady-Hubbard theory of quadratic- like maps and complex renormalization that justifies presence of the baby M-sets, and classify them. (The geometric theory that explains why these babies have a universal shape will be developed later.) This will roughly constitute the 1st volume of the book. In the 2nd volume we plan to prove the Feigenbaum-Coullet-Tresser Renormal- ization Conjecture (by Sullivan, McMullen and the author), density of hyperbolic maps in the real quadratic family, and the Regular and Stochastic Theorem assert- ing that almost any real quadratic map is eitehr regular (i.e., has an attracting cycle 10 0. INTRODUCTION that attracts almost all orbits) or stochastic (i.e., it has an absolutely continuous invariant measure that governs behavior of almost all orbits)– by the author. These results were obtained in 1990’s but recently new insights, particularly by Avila and Kahn, led to much better understanding of the phenomena. We plan to dedicate the 3d volume to recent advances in the MLC Conjecture (based on the work of Kahn and the author). The last volume (if ever written) will be devoted to the measure-theoretic theory of Julia sets and the Mandelbrot set. We will discuss the measure of maximal entropy and conformal measures, Hausdorff dimension and Lebesgue measure of Julia sets and the Mandelbrot set. It would culminate with a construction of examples of Julia sets of positive area, by Buff and Cheritat, and more recently, by Avila and the author (not announced yet). Thisbookcanbeusedinmanyways: • For a graduate class in conformal and quasiconformal geometry illustrated with dynamical examples. This would cover Part I with selected pieces from Part 2. • As the first introduction to the one-dimensional dynamics, complex and real. Then the reader should begin with Part 2 consulting the background material from Part 1 as needed. • As an introduction to advanced themes of one-dimensional dynamics for the reader who knows basics and intends to do research in this field. Such a reader can go through selected chapters of Part 2 proceeding fairly fast to Part 3. • Of course, the book can also be used as a monograph, for reference. 2. Background In this section we collect some standing (usually, standard) notations, defini- tions, and properties. It can be consulted as long as the corresponding objects and properties appear in the text. 2.1. Complex plane and its affiliates. As usually, N = {0, 1, 2,...} stands for the additive semigroup of natural numbers (with the French convention that zero is natural); Z is the group of integers, Z+ and Z− are the sets of positive and negative integers respectively; R stands for the real line; C stands for the complex plane, and Cˆ = C ∪{∞}stands for the Riemann sphere; S2 is a topological sphere, i.e., a topological manifold homeomorphic Cˆ; 2 We let CR ≈ R be the decomplexified C (i.e., C viewed as 2D real vector space). For a ∈ C, r>0, let D(a, r)={z ∈ C : |z − a| <r}; D¯(a, r)={z ∈ C : |z − a|≤r}.