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Conformal Fractals – Ergodic Theory Methods Conformal Fractals – Ergodic Theory Methods Feliks Przytycki Mariusz Urba´nski May 17, 2009 2 Contents Introduction 7 0 Basic examples and definitions 15 1 Measure preserving endomorphisms 25 1.1 Measurespacesandmartingaletheorem . 25 1.2 Measure preserving endomorphisms, ergodicity . .. 28 1.3 Entropyofpartition ........................ 34 1.4 Entropyofendomorphism . 37 1.5 Shannon-Mcmillan-Breimantheorem . 41 1.6 Lebesguespaces............................ 44 1.7 Rohlinnaturalextension . 48 1.8 Generalized entropy, convergence theorems . .. 54 1.9 Countabletoonemaps.. .. .. .. .. .. .. .. .. .. 58 1.10 Mixingproperties . .. .. .. .. .. .. .. .. .. .. 61 1.11 Probabilitylawsand Bernoulliproperty . 63 Exercises .................................. 68 Bibliographicalnotes. 73 2 Compact metric spaces 75 2.1 Invariantmeasures . .. .. .. .. .. .. .. .. .. .. 75 2.2 Topological pressure and topological entropy . ... 83 2.3 Pressureoncompactmetricspaces . 87 2.4 VariationalPrinciple . 89 2.5 Equilibrium states and expansive maps . 94 2.6 Functionalanalysisapproach . 97 Exercises ..................................106 Bibliographicalnotes. 109 3 Distance expanding maps 111 3.1 Distance expanding open maps, basic properties . 112 3.2 Shadowingofpseudoorbits . 114 3.3 Spectral decomposition. Mixing properties . 116 3.4 H¨older continuous functions . 122 3 4 CONTENTS 3.5 Markov partitions and symbolic representation . 127 3.6 Expansive maps are expanding in some metric . 134 Exercises .................................. 136 Bibliographicalnotes. 138 4 Thermodynamical formalism 141 4.1 Gibbsmeasures: introductoryremarks . 141 4.2 Transfer operator and its conjugate. Measures with prescribed Jacobians ............................... 144 4.3 Iteration of the transfer operator. Existence of invariant Gibbs measures................................ 152 4.4 Convergence of n. Mixing properties of Gibbs measures . 155 L 4.5 Moreonalmostperiodicoperators . 161 4.6 Uniquenessofequilibriumstates . 164 4.7 Probability laws and σ2(u, v) ...................168 Exercises .................................. 173 Bibliographicalnotes. 176 5 Expanding repellers in manifolds and Riemann sphere, prelim- inaries 177 5.1 Basicproperties ........................... 177 5.2 Complex dimension one. Bounded distortion and other techniques 183 5.3 Transfer operator for conformal expanding repeller with har- monicpotential............................ 186 5.4 Analytic dependence of transfer operator on potential function . 190 Bibliographicalnotes. 194 6 Cantor repellers in the line, Sullivan’s scaling function, appli- cation in Feigenbaum universality 195 6.1 C1+ε-equivalence .......................... 196 6.2 Scaling function. C1+ε-extensionoftheshiftmap . 202 6.3 Highersmoothness ......................... 206 6.4 Scaling function and smoothness. Cantor set valued scaling function210 6.5 Cantorsetsgeneratingfamilies . 214 6.6 Quadratic-like maps of the interval, an application to Feigen- baum’suniversality. 217 Exercises .................................. 223 Bibliographicalnotes. 224 7 Fractal dimensions 227 7.1 Outermeasures ........................... 227 7.2 Hausdorffmeasures . .. .. .. .. .. .. .. .. .. .. 230 7.3 Packingmeasures .......................... 232 7.4 Dimensions ............................. 234 7.5 Besicovitchcoveringtheorem . 237 7.6 Frostmantypelemmas . 240 CONTENTS 5 Bibliographicalnotes. 245 8 Conformal expanding repellers 247 8.1 Pressurefunctionanddimension . 248 8.2 Multifractal analysis of Gibbs state . 256 8.3 FluctuationsforGibbsmeasures . 266 8.4 BoundarybehaviouroftheRiemannmap . 270 8.5 Harmonic measure; “fractal vs.analytic” dichotomy . .. 274 8.6 Pressure versus integral means of the Riemann map . 283 8.7 Geometric examples. Snowflake and Carleson’s domains . 285 Exercises ..................................289 Bibliographicalnotes. 292 9 Sullivan’s classification of conformal expanding repellers 295 9.1 Equivalentnotionsoflinearity . 295 9.2 RigidityofnonlinearCER’s . 299 Bibliographicalnotes. 305 10 Conformal maps with invariant probability measures of positive Lyapunov exponent 307 10.1 Ruelle’sinequality . 307 10.2Pesin’stheory ............................ 309 10.3 Ma˜n´e’spartition . 312 10.4 Volume lemma and the formula HD(µ) = hµ(f)/χµ(f) ...... 314 10.5 Pressure-like definition of the functional hµ + φ dµ ....... 317 10.6 Katok’s theory—hyperbolic sets, periodic points, and pressure . 320 Exercises ..................................325R Bibliographicalnotes. 325 11 Conformal measures 327 11.1 Generalnotionofconformalmeasures . 327 11.2 Sullivan’s conformal measures and dynamical dimension,I . 333 11.3 Sullivan’s conformal measures and dynamical dimension,II . 335 11.4 Pesin’sformula . .. .. .. .. .. .. .. .. .. .. .. 340 11.5 More about geometric pressure and dimensions . 342 Bibliographicalnotes. 348 Bibliography 349 Index 361 6 CONTENTS Introduction Introduction This book is an introduction to the theory of iteration of expanding and non- uniformly expanding holomorphic maps and topics in geometric measure theory of the underlying invariant fractal sets. Probability measures on these sets yield informations on Hausdorff and other fractal dimensions and properties. The book starts with a comprehensive chapter on abstract ergodic theory followed by chapters on uniform distance expanding maps and thermodynamical formalism. This material is applicable in many branches of dynamical systems and related fields, far beyond the applications in this book. Popular examples of the fractal sets to be investigated are Julia sets for rational functions on the Riemann sphere. The theory which was initiated by Gaston Julia [Julia 1918] and Pierre Fatou [Fatou 1919-1920] has become very popular since appearence of Benoit Mandelbrot’s book [Mandelbrot 1982] with beautiful computer made pictures. Then it has become a field of spectacular achievements by top mathematicians during the last 30 years. Consider for example the map f(z) = z2 for complex numbers z. Then 1 1 1 1 1 the unit circle S = z = 1 is f-invariant, f(S ) = S = f − (S ). For {| |2 } c 0,c = 0 and fc(z)= z + c, there still exists an fc-invariant set J(fc) called ≈ 6 1 1 the Julia set of fc, close to S , homeomorphic to S via a homeomorphism h satisfying equality f h = h f . However J(f ) has a fractal shape. For large ◦ ◦ c c c the curve J(fc) pinches at infinitely many points; it may pinch everywhere to become a dendrite, or even crumble to become a Cantor set. These sets satisfy two main properties, standard attributes of ”conformal fractal sets”: 1. Their fractal dimensions are strictly larger than the topological dimension. 2. They are conformally ”self-similar”, namely arbitrarily small pieces have shapes similar to large pieces via conformal mappings, here via iteration of f. To measure fractal sets invariant under holomorphic mappings one applies probability measures corresponding to equilibria in the thermodynamical for- malism. This is a beautiful example of interlacing of ideas from mathematics and physics. The following prototype lemma [Bowen 1975, Lemma 1.1] stands at the roots of the thermodynamical formalism Lemma 0.0.1. (Finite Variational Principle) For given real numbers φ1,...,φn 7 8 Introduction the quantity n n F (p ,...p )= p log p + p φ 1 n − i i i i i=1 i=1 X X n φi has maximum value P (φ1, ...φn) = log i=1 e as (p1,...,pn) ranges over the n simplex (p1,...,pn) : pi 0, i=1 pi = 1 and the maximum is attained only at { ≥ P } P n φj φi 1 pˆj = e e − i=1 X We can read φi,pi,i = 1,...,n as a function (potential), resp. probability distribution, on the finite space 1,...,n . The proof follows from the strict concavity of the logarithm function.{ } Let us further follow Bowen [Bowen 1975]: The quantity n S = p log p − i i i=1 X is called entropy of the distribution (p1,...,pn). The maximizing distribution (ˆp1, .., pˆn) is called Gibbs or equilibrium state. In statistical mechanics φi = βEi, where β =1/kT , T is a temperature of an external ”heat source” and k − n a physical (Boltzmann) constant. The quantity E = i=1 piEi is the average energy. The Gibbs distribution maximizes thus the expression P 1 S βE = S E − − kT or equivalently minimizes the so-called free energy E kTS. The nature prefers states with low energy and high entropy. It minimizes− free energy. The idea of Gibbs distribution as limit of distributions on finite spaces of con- figurations of states (spins for example) of interacting particles over increasing to infinite, bounded parts of the lattice Zd, introduced in statistical mechan- ics first by Bogolubov and Hacet [Bogolyubov & Hacet 1949] and playing there a fundamental role was applied in dynamical systems to study Anosov flows and hyperbolic diffeomorphisms at the end of sixties by Ja. Sinai, D. Ruelle and R. Bowen. For more historical remarks see [Ruelle 1978] or [Sinai 1982]. This theory met the notion of entropy S borrowed from information theory and introduced by Kolmogorov as an invariant of a measure-theoretic dynamical system. Later the usefulness of these notions to the geometric dimensions has become apparent. It was present already in [Billingsley 1965] but crucial were papers by Bowen [Bowen 1979] and McCluskey & Manning [McCluskey & Manning 1983]. In order to illustrate the idea consider the following example: Let T : I I, i → i = 1,...,n > 1, where I = [0, 1] is the unit interval, Ti(x) = λix + ai, where λi,ai are real numbers chosen in such a way that all the sets Ti(I) are pairwise disjoint and contained in I. Define the limit set Λ as follows
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