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| Introduction | |
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| Basic examples and definitions | |
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| Measure-preserving endomorphisms | |
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| Measure spaces and the Martingale Theorem | |
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| Measure-preserving endomorphisms; ergodicity | |
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| Entropy of partition | |
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| Entropy of an endomorphism | |
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| Shannon-McMillan-Breiman Theorem | |
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| Lebesgue spaces, measurable partitions and canonical systems of conditional measures | |
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| Rokhlin natural extension | |
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| Generalized entropy; convergence theorems | |
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| Countable-to-one maps, Jacobian and entropy of endomorphisms | |
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| Mixing properties | |
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| Probability laws and Bernoulli property | |
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| Exercises | |
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| Bibliographical notes | |
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| Ergodic theory on compact metric spaces | |
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| Invariant measures for continuous mappings | |
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| Topological pressure and topological entropy | |
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| Pressure on compact metric spaces | |
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| Variational Principle | |
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| Equilibrium states and expansive maps | |
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| Topological pressure as a function on the Banach space of continuous functions; the issue of uniqueness of equilibrium states | |
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| Exercises | |
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| Bibliographical notes | |
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| Distance-expanding maps | |
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| Distance-expanding open maps: basic properties | |
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| Shadowing of pseudo-orbits | |
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| Spectral decomposition; mixing properties | |
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| H�lder continuous functions | |
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| Markov partitions and symbolic representation | |
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| Expansive maps are expanding in some metric | |
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| Exercises | |
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| Bibliographical notes | |
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| Thermodynamical formalism | |
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| Gibbs measures: introductory remarks | |
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| Transfer operator and its conjugate; measures with prescribed Jacobians | |
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| Iteration of the transfer operator; existence of invariant Gibbs measures | |
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| Convergence of L<sup>n</sup>; mixing properties of Gibbs measures | |
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| More on almost periodic operators | |
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| Uniqueness of equilibrium states | |
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| Probability laws and �<sup>2</sup>(u, v) | |
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| Exercises | |
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| Bibliographical notes | |
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| Expanding repellers in manifolds and in the Riemann sphere: preliminaries | |
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| Basic properties | |
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| Complex dimension one; bounded distortion and other techniques | |
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| Transfer operator for conformal expanding repeller with harmonic potential | |
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| Analytic dependence of transfer operator on potential function | |
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| Exercises | |
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| Bibliographical notes | |
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| Cantor repellers in the line; Sullivan's scaling function; application in Feigenbaum universality | |
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| C<sup>1+�</sup>-equivalence | |
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| Scaling function: C<sup>1+�</sup>-extension of the shift map | |
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| Higher smoothness | |
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| Scaling function and smoothness; Cantor set valued scaling function | |
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| Cantor set generating families | |
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| Quadratic-like maps of the interval; an application to Feigenbaum's universality | |
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| Exercises | |
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| Bibliographical notes | |
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| Fractal dimensions | |
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| Outer measures | |
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| Hausdorff measures | |
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| Packing measures | |
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| Dimensions | |
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| Besicovitch Covering Theorem; Vitali Theorem and density points | |
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| Frostman-type lemmas | |
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| Bibliographical notes | |
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| Conformal expanding repellers | |
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| Pressure function and dimension | |
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| Multifractal analysis of Gibbs state | |
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| Fluctuations for Gibbs measures | |
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| Boundary behaviour of the Riemann map | |
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| Harmonic measure; 'fractal vs. analytic' dichotomy | |
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| Pressure versus integral means of the Riemann map | |
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| Geometric examples: snowflake and Carleson's domains | |
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| Exercises | |
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| Bibliographical notes | |
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| Sullivan's classification of conformal expanding repellers | |
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| Equivalent notions of linearity | |
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| Rigidity of non-linear CERs | |
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| Bibliographical notes | |
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| Holomorphic maps with invariant probability measures of positive Lyapunov exponent | |
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| Ruelle's inequality | |
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| Pesin's theory | |
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| Ma��'s partition | |
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| Volume Lemma and the formula HD (�) = h<sup>�</sup>(f)/�<sup>�</sup>(f) | |
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| Pressure-like definition of the functional h<sup>�</sup> + ∫ �d� | |
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| Katok's theory: hyperbolic sets, periodic points, and pressure | |
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| Exercises | |
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| Bibliographical notes | |
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| Conformal measures | |
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| General notion of conformal measures | |
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| Sullivan's conformal measures and dynamical dimension: I | |
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| Sullivan's conformal measures and dynamical dimension: II | |
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| Pesin's formula | |
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| More about geometric pressure and dimensions | |
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| Bibilographical notes | |
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| References | |
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| Index | |