Conformal fractals - Ergodic Theory Methods

Title page
Introduction
0 Basic examples and definitions
1 Measure preserving endomorphisms
1.1 Measure spaces and martingale theorem
1.2 Measure preserving endomorphisms, ergodicity
1.3 Entropy of partition
1,4 Entropy of an endomorphism
1.5 Shannon—Mcmillan—Breiman theorem
1.6 Lebesgue spaces
1.7 Rohlin natural extension
1.8 Generalized entropy, convergence theorems
1.9 Countable to one maps
1.10 Mixing properties
1.11 Probability laws and Bernoulli property
Exercises
Bibliographical notes
2 Compact metric spaces
2.1 Invariant measures
2.2 Topological pressure and topological entropy
2.3 Pressure on compact metric spaces
2.4 Variational Principle
2.5 Equilibrium states and expansive maps
2.6 Functional analysis approach
Exercises
Bibliographical notes
3 Distance expanding maps
3.1 Distance expanding open maps, basic properties
3.2 Shadowing of pseudoorbits
3.3 Spectral decomposition. Mixing properties
3.4 Hölder continuous functions
3.5 Markov partitions and symbolic representation
3.6 Expansive maps are expanding in some metric
Exercises
Bibliographical notes
4 Thermodynamical formalism
4,1 Gibbs measures: introductory remarks
4.2 Transfer operator and its conjugate. Measures with prescribed Jacobians
4.3 Iteration of the transfer operator. Existence of invariant Gibbs measures
4.4 Convergence of L^n. Mixing properties of Gibbs measures
4.5 More on almost periodic operators
4.6 Uniqueness of equilibrium states
4.7 Probability laws and σ²(u, v)
Exercises
Bibliographical notes
5 Expanding repellers in manifolds and in the Riemann sphere, preliminaries
5.1 Basic properties
5.2 Complex dimension one. Bounded distortion and other techniques
5.3 Transfer operator for conformal expanding repeller with harmonic potential
5.4 Analytic dependence of transfer operator on potential function
Exercises
Bibliographical notes
6 Cantor repellers in the line, Sullivan’s scaling function, application in Feigenbaum universality
6.1 C^{1+ε}-equivalence
6.2 Scaling function. C^{1+ε}-extension of the shift map
6.3 Higher smoothness
6.4 Scaling function and smoothness. Cantor set valued scaling function 2l2
6.5 Cantor sets generating families
6.6 Quadratic-like maps of the interval, an application to Feigenbaum’s universality
Bibliographical notes
7 Fractal dimensions
7.1 Outer measures
7.2 Hausdorff measures
7.3 Packing measures
7.4 Dimensions
7.5 Besicovitch covering theorem. Vitali theorem and density points
7.6 Frostman-type lemmas
Bibliographical notes
8 Conformal expanding repellers
8.1 Pressure function and dimension
8.2 Multifractal analysis of Gibbs state
8.3 Fluctuations for Gibbs measures
8,4 Boundary behaviour of the Riemann map
8.5 Harmonic measure; “fractal vs. analytic” dichotomy
8.6 Pressure versus integral means of the Riemann map
8.7 Geometric examples. Snowflake and Carleson’s domains
Exercises
Bibliographical notes
9 Sullivan’s classification of conformal expanding repellers
9.1 Equivalent notions of linearity
9.2 Rigidity of nonlinear CER’s
Bibliographical notes
10 Holomorphic maps with invariant probability measures of positive Lyapunov exponent
10.1 Ruelle’s inequality
10.2 Pesin’s theory
10.3 Mañé’s partition
10.4 Volume lemma and the formula HD(μ) = h_μ(f)/χ_μ(f)
10.5 Pressure-like definition of the functional h_μ + int(φ dμ)
10.6 Katok’s theory — hyperbolic sets, periodic points, and pressure
Exercises
Bibliographical notes
11 Conformal measures
11.1 General notion of conformal measures
11.2 Sullivan’s conformal measures and dynamical dimension, I
11.3 Sullivan’s conformal measures and dynamical dimension, II
11.4 Pesin’s formula
11.5 More about geometric pressure and dimensions
Bibliographical notes
Bibliography