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Conformal Fractals – Ergodic Theory Methods
Feliks Przytycki Mariusz Urbanski
May 17, 2009
2
Contents
Introduction
7
0 Basic examples and denitions
15
1 Measure preserving endomorphisms 25
1.1 Measure spaces and martingale theorem . . . . . . . . . . . . . 25
1.2 Measure preserving endomorphisms, ergodicity . . . . . . . . . 28
1.3 Entropy of partition . . . . . . . . . . . . . . . . . . . . . . . . 34
1.4 Entropy of endomorphism . . . . . . . . . . . . . . . . . . . . . 37
1.5 Shannon-Mcmillan-Breiman theorem . . . . . . . . . . . . . . . 41
1.6 Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.7 Rohlin natural extension . . . . . . . . . . . . . . . . . . . . . . 48
1.8 Generalized entropy, convergence theorems . . . . . . . . . . . . 54
1.9 Countable to one maps . . . . . . . . . . . . . . . . . . . . . . . . 58
1.10 Mixing properties . . . . . . . . . . . . . . . . . . . . . . . . . . 61
1.11 Probability laws and Bernoulli property . . . . . . . . . . . . . 63
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2 Compact metric spaces 75
2.1 Invariant measures . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.2 Topological pressure and topological entropy . . . . . . . . . . . 83
2.3 Pressure on compact metric spaces . . . . . . . . . . . . . . . . 87
2.4 Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . 89
2.5 Equilibrium states and expansive maps . . . . . . . . . . . . . . 94
2.6 Functional analysis approach . . . . . . . . . . . . . . . . . . . . 97
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3 Distance expanding maps 111
3.1 Distance expanding open maps, basic properties . . . . . . . . . 112
3.2 Shadowing of pseudoorbits . . . . . . . . . . . . . . . . . . . . . 114
3.3 Spectral decomposition. Mixing properties . . . . . . . . . . . . 116
3.4 Holder 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
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4 Thermodynamical formalism 141
4.1 Gibbs measures: introductory remarks . . . . . . . . . . . . . . 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 L n . Mixing properties of Gibbs measures . . . . 155
4.5 More on almost periodic operators . . . . . . . . . . . . . . . . 161
4.6 Uniqueness of equilibrium states . . . . . . . . . . . . . . . . . . 164
4.7 Probability laws and σ 2 ( u,v ) . . . . . . . . . . . . . . . . . . . 168
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
5 Expanding repellers in manifolds and Riemann sphere, prelim-
inaries 177
5.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.2 Complex dimension one. Bounded distortion and other techniques 183
5.3 Transfer operator for conformal expanding repeller with har-
monic potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.4 Analytic dependence of transfer operator on potential function . 190
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
6 Cantor repellers in the line, Sullivan’s scaling function, appli-
cation in Feigenbaum universality 195
6.1 C 1+ε -equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.2 Scaling function. C 1+ε -extension of the shift map . . . . . . . . 202
6.3 Higher smoothness . . . . . . . . . . . . . . . . . . . . . . . . . 206
6.4 Scaling function and smoothness. Cantor set valued scaling function210
6.5 Cantor sets generating families . . . . . . . . . . . . . . . . . . 214
6.6 Quadratic-like maps of the interval, an application to Feigen-
baum’s universality . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
7 Fractal dimensions 227
7.1 Outer measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
7.2 Hausdor measures . . . . . . . . . . . . . . . . . . . . . . . . . 230
7.3 Packing measures . . . . . . . . . . . . . . . . . . . . . . . . . . 232
7.4 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
7.5 Besicovitch covering theorem . . . . . . . . . . . . . . . . . . . 237
7.6 Frostman type lemmas . . . . . . . . . . . . . . . . . . . . . . . 240
CONTENTS
5
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8 Conformal expanding repellers 247
8.1 Pressure function and dimension . . . . . . . . . . . . . . . . . . 248
8.2 Multifractal analysis of Gibbs state . . . . . . . . . . . . . . . . 256
8.3 Fluctuations for Gibbs measures . . . . . . . . . . . . . . . . . . 266
8.4 Boundary behaviour of the Riemann map . . . . . . . . . . . . . 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. Snowake and Carleson’s domains . . . . . 285
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
9 Sullivan’s classication of conformal expanding repellers 295
9.1 Equivalent notions of linearity . . . . . . . . . . . . . . . . . . . 295
9.2 Rigidity of nonlinear CER’s . . . . . . . . . . . . . . . . . . . . . 299
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
10 Conformal maps with invariant probability measures of positive
Lyapunov exponent 307
10.1 Ruelle’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 307
10.2 Pesin’s theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
10.3 Mane’s partition . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
10.4 Volume lemma and the formula HD( ) = h ( f ) /χ ( f ) . . . . . . 314
10.5 Pressure-like denition of the functional h +
11 Conformal measures 327
11.1 General notion of conformal measures . . . . . . . . . . . . . . . 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’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
11.5 More about geometric pressure and dimensions . . . . . . . . . . 342
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
Bibliography
349
Index
361
φd . . . . . . . 317
10.6 Katok’s theory—hyperbolic sets, periodic points, and pressure . . 320
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

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