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Complex Dynamics and Renormalization Complex Dynamics and Renormalization by Curtis T. McMullen Contents 1 Introduction 1 1.1 Complex dynamics . 1 1.2 Centralconjectures.................... 3 1.3 Summary of contents . 6 2 Background in conformal geometry 9 2.1 The modulus of an annulus . 10 2.2 Thehyperbolicmetric . 11 2.3 Metric aspects of annuli . 13 2.4 Univalent maps . 15 2.5 Normal families . 17 2.6 Quasiconformal maps . 18 2.7 Measurable sets . 19 2.8 Absolute area zero . 20 2.9 The collar theorem . 22 2.10 The complex shortest interval argument . 27 2.11 Controlling holomorphic contraction . 30 3 Dynamics of rational maps 35 3.1 The Julia and Fatou sets . 36 3.2 Expansion . 38 3.3 Ergodicity . 42 3.4 Hyperbolicity ....................... 44 3.5 Invariant line fields and complex tori . 47 4 Holomorphic motions and the Mandelbrot set 53 4.1 Stability of rational maps . 53 4.2 The Mandelbrot set . 59 i 5 Compactness in holomorphic dynamics 65 5.1 Convergence of Riemann mappings . 66 5.2 Proper maps . 67 5.3 Polynomial-like maps . 71 5.4 Intersecting polynomial-like maps . 74 5.5 Polynomial-like maps inside proper maps . 75 5.6 Univalentlinefields ................... 78 6 Polynomials and external rays 83 6.1 Accessibility........................ 83 6.2 Polynomials . 87 6.3 Eventualsurjectivity . 89 6.4 Laminations . 91 7 Renormalization 97 7.1 Quadratic polynomials . 97 7.2 Small Julia sets meeting at periodic points . 102 7.3 Simple renormalization . 109 7.4 Examples . 113 8 Puzzles and infinite renormalization 117 8.1 Infinite renormalization . 117 8.2 The Yoccoz jigsaw puzzle . 119 8.3 Infinite simple renormalization . 122 8.4 Measure and local connectivity . 123 8.5 Laminations and tableaux . 125 9 Robustness 129 9.1 Simple loops around the postcritical set . 129 9.2 Area of the postcritical set . 132 10 Limits of renormalization 135 10.1 Unbranched renormalization . 137 10.2 Polynomial-like limits of renormalization . 140 10.3 Proper limits of renormalization . 145 10.4 Extracting a univalent line field . 150 ii 11 Real quadratic polynomials 161 11.1 Intervals and gaps . 161 11.2 Real robustness . 165 11.3 Corollaries and generalizations . 168 A Orbifolds 171 A.1 Smooth and complex orbifolds . 171 A.2 Coverings and uniformization . 173 A.3 The orbifold of a rational map . 177 B A closing lemma for rational maps 181 B.1 Quotients of branched coverings . 181 B.2 Criticallyfiniterationalmaps . 184 B.3 Siegel disks, Herman rings and curve systems . 186 B.4 Rationalquotients . .192 B.5 Quotients and renormalization . 194 Bibliography 201 Index 207 iii iv Chapter 1 Introduction 1.1 Complex dynamics This work presents a study of renormalization of quadratic polyno- mials and a rapid introduction to techniques in complex dynamics. Around 1920 Fatou and Julia initiated the theory of iterated ra- tional maps f : C C → on the Riemann sphere. More recently! ! methods of geometric func- tion theory, quasiconformal mappings and hyperbolic geometry have contributed to the depth and scope of research in the field. Thein- tricate structure of the family of quadratic polynomials wasrevealed by work of Douady and Hubbard [DH1], [Dou1]; analogies between rational maps and Kleinian groups surfaced with Sullivan’s proof of the no wandering domains theorem [Sul3] and continue to inform both subjects [Mc2]. It can be a subtle problem to understand a high iterate of a rational map f of degree d>1. There is tension between expanding features of f —suchasthefactthatitsdegreetendstoinfinityunder iteration — and contracting features, such as the presence ofcritical points. The best understood maps are those for which the critical points tend to attracting cycles. For such a map, the tension is resolved by the concentration of expansion in the Julia set or chaotic locus of the map, and the presence of contraction on the rest ofthe sphere. 1 2 Chapter 1. Introduction The central goal of this work is to understand a high iterate of aquadraticpolynomial.Thespecialcaseweconsideristhatof an infinitely renormalizable polynomial f(z)=z2 + c. For such a polynomial, the expanding and contracting properties lie in a delicate balance; for example, the critical point z =0belongs to the Julia set and its forward orbit is recurrent. Moreover high iterates of f can be renormalized or rescaled to yield new dynamical systems of the same general shape as the original map f. This repetition of form at infinitely many scales provides theba- sic framework for our study. Under additional geometric hypotheses, we will show that the renormalized dynamical systems range ina compact family. Compactness is established by combining univer- sal estimates for the hyperbolic geometry of surfaces with distortion theorems for holomorphic maps. With this information in hand, we establish quasiconformal rigid- ity of the original polynomial f.Rigidityoff supports conjectures about the behavior of a generic complex dynamical system, as de- scribed in the next section. The course of the main argument entails many facets of com- plex dynamics. Thus the sequel includes a brief exposition oftopics including: The Poincar´emetric, the modulus of an annulus, and distortion • theorems for univalent maps ( 2); § The collar theorem and related aspects of hyperbolic surfaces • ( 2.9 and 2.10); § § Dynamics of rational maps and hyperbolicity ( 3.1 and 3.4); • § § Ergodic theory of rational maps, and the role of the postcritical • set as a measure-theoretic attractor ( 3); § Invariant line fields, holomorphic motions and stability in fam- • ilies of rational maps ( 3and 4); § § The Mandelbrot set ( 4); • § Polynomial-like maps and proper maps ( 5); • § Riemann mappings and external rays ( 5and 6); • § § 1.2. Central conjectures 3 Renormalization ( 7); • § The Yoccoz puzzle ( 8); • § Real methods and Sullivan’s aprioribounds ( 11); • § Orbifolds (Appendix A); and • Thurston’s characterization of critically finite rational maps • (Appendix B). 1.2 Central conjectures We now summarize the main problems which motivate our work. Let f : C C be a rational map of the Riemann sphere to itself → of degree d>1. The map f is hyperbolic if its critical points tend to attracting periodic! ! cycles under iteration. Within all rational maps, the hyperbolic ones are among the best behaved; for example, when f is hyperbolic there is a finite set A C which attracts all points ⊂ in an open, full-measure subset of the sphere (see 3.4). § One of the central problems in conformal! dynamics is the follow- ing: Conjecture 1.1 (Density of hyperbolicity) The set of hyperbolic rational maps is open and dense in the space Ratd of all rational maps of degree d. Openness of hyperbolic maps is known, but density is not. In some form this conjecture goes back to Fatou (see 4.1). § Much study has been devoted to special families of rational maps, particularly quadratic polynomials. Every quadratic polynomial f is conjugate to one of the form f (z)=z2 + c for a unique c C. c ∈ Even this simple family of rational maps exhibits a full spectrum of dynamical behavior, reflecting many of the difficulties of the general case. Still unresolved is: Conjecture 1.2 The set of c for which z2 + c is hyperbolic forms an open dense subset of the complex plane. 4 Chapter 1. Introduction The Mandelbrot set M is the set of c such that under iteration, n fc (0) does not tend to infinity; here z =0istheuniquecriticalpoint of fc in C.AcomponentU of the interior of M is hyperbolic if fc is hyperbolic for some c in U.Itisknownthatthemapsfc enjoy a type of structural stability as c varies in any component of C ∂M; − in particular, if U is hyperbolic, fc is hyperbolic for every c in U (see 4). It is clear that f is hyperbolic when c is not in M,because § c the critical point tends to the superattracting fixed point atinfinity. Thus an equivalent formulation of Conjecture 1.2 is: Conjecture 1.3 Every component of the interior of the Mandelbrot set is hyperbolic. An approach to these conjectures is developed in [MSS] and [McS], using quasiconformal mappings. This approach has thead- vantage of shifting the focus from a family of maps to the dynamics of a single map, and leads to the following: Conjecture 1.4 (No invariant line fields) Arationalmapf car- ries no invariant line field on its Julia set, except when f is double covered by an integral torus endomorphism. Conjecture 1.4 implies all the preceding conjectures [McS].This conjecture is explained in more detail in 3.5; see also [Mc3]. § The rational maps which are covered by integral torus endomor- phisms form a small set of exceptional cases. For quadratic polyno- mials, Conjecture 1.4 specializes to: Conjecture 1.5 Aquadraticpolynomialcarriesnoinvariantline field on its Julia set. The Julia set J of a polynomial f is the boundary of the set of points which tend to infinity under iteration. A line field on J is the assignment of a real line through the origin in the tangent space to z for each z in a positive measure subset E of J,sothattheslopeis ameasurablefunctionofz.Alinefieldisinvariant if f −1(E)=E, and if f ′ transforms the line at z to the line at f(z). Conjecture 1.5 is equivalent to Conjectures 1.2 and 1.3 (see 4). § Recent progress towards these conjectures includes: 1.2. Central conjectures 5 Theorem 1.6 (Yoccoz) Aquadraticpolynomialwhichcarriesan invariant line field on its Julia set is infinitely renormalizable. See 8. Here a quadratic polynomial is infinitely renormalizable if § there are infinitely many n>1suchthatf n restricts to a quadratic- like map with connected Julia set; see 7. For instance, the much- § studied Feigenbaum example is an infinitely renormalizable polyno- mial (see 7.4). § This work addresses the infinitely renormalizable case.
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