Chapter 1. Introduction ......................................... 1
Chapter 2. Balanced functions ................................... 5
2.1. Growth conditions .......................................... 5
2.2. The precise form of α2 ..................................... 6
2.3. Classical families ......................................... 7
2.4. Functions with polynomial Schwarzian derivative ............ 8
2.5. Functions with rational Schwarzian derivative ............. 10
2.6. Uniform balanced growth ................................... 11
Chapter 3. Transfer operator and Nevanlinna Theory ............. 13
3.1. Choice of a Riemannian metric and transfer operator ....... 13
3.2. Nevanlinna Theory and Borel Sums .......................... 14
Chapter 4. Preliminaries, Hyperbolicity and Distortion
Properties .......................................... 17
4.1. Dynamical preliminaries and hyperbolicity ................. 17
4.2. Distortion properties ..................................... 20
4.3. Holder functions and dynamical Holder property ............ 21
Chapter 5. Perron-Probenius Operators and Generalized
Conformal Measures .................................. 23
5.1. Tame potentials ........................................... 23
5.2. Growth condition and cohomological Perron-Probenius
operator .................................................. 24
5.3. Topological pressure and existence of conformal
measures .................................................. 25
5.4. Thermodynamical formalism ................................. 28
5.5. The support and uniqueness of the conformal measure ....... 31
Chapter 6. Finer properties of Gibbs States .................... 35
6.1. The two norm inequality and the spectral gap .............. 35
6.2. Ergodic properties of Gibbs states ........................ 39
6.3. Decay of correlations and Central Limit Theorem ........... 41
6.4. Cohomologies and σ2 = 0 ................................... 45
6.5. Variational principle ..................................... 49
Chapter 7. Regularity of Perron-Frobenius Operators and
Topological Pressure ................................ 55
7.1. Analyticity of Perron-Frobenius operators ................. 55
7.2. Analyticity of pressure ................................... 58
7.3. Derivatives of the pressure function ...................... 60
Chapter 8. Multifractal analysis ............................... 79
8.1. Hausdorff dimension of Gibbs states ....................... 79
8.2. The temperature function .................................. 82
8.3. Multifractal analysis ..................................... 86
Chapter 9. Multifractal Analysis of Analytic Families of
Dynamically Regular Functions ....................... 91
9.1. Extensions of harmonic functions .......................... 91
9.2. Holomorphic families and quasi-conformal conjugacies ...... 93
9.3. Real analyticity of the multifractal function ............. 95
Bibliography .................................................. 103
Index ......................................................... 107
|
