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Conformal Fractals Ergodic Theory Methods

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ISBN-10: 0521438004

ISBN-13: 9780521438001

Edition: 2010

Authors: Feliks Przytycki, Mariusz Urbanski

List price: $94.99
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Description:

This is a one-stop introduction to the methods of ergodic theory applied to holomorphic iteration. The authors begin with introductory chapters presenting the necessary tools from ergodic theory thermodynamical formalism, and then focus on recent developments in the field of 1-dimensional holomorphic iterations and underlying fractal sets, from the point of view of geometric measure theory and rigidity. Detailed proofs are included. Developed from university courses taught by the authors, this book is ideal for graduate students. Researchers will also find it a valuable source of reference to a large and rapidly expanding field. It eases the reader into the subject and provides a vital…    
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Book details

List price: $94.99
Copyright year: 2010
Publisher: Cambridge University Press
Publication date: 5/6/2010
Binding: Paperback
Pages: 366
Size: 6.00" wide x 9.25" long x 0.75" tall
Weight: 1.144

Introduction
Basic examples and definitions
Measure-preserving endomorphisms
Measure spaces and the Martingale Theorem
Measure-preserving endomorphisms; ergodicity
Entropy of partition
Entropy of an endomorphism
Shannon-McMillan-Breiman Theorem
Lebesgue spaces, measurable partitions and canonical systems of conditional measures
Rokhlin natural extension
Generalized entropy; convergence theorems
Countable-to-one maps, Jacobian and entropy of endomorphisms
Mixing properties
Probability laws and Bernoulli property
Exercises
Bibliographical notes
Ergodic theory on compact metric spaces
Invariant measures for continuous mappings
Topological pressure and topological entropy
Pressure on compact metric spaces
Variational Principle
Equilibrium states and expansive maps
Topological pressure as a function on the Banach space of continuous functions; the issue of uniqueness of equilibrium states
Exercises
Bibliographical notes
Distance-expanding maps
Distance-expanding open maps: basic properties
Shadowing of pseudo-orbits
Spectral decomposition; mixing properties
H�lder continuous functions
Markov partitions and symbolic representation
Expansive maps are expanding in some metric
Exercises
Bibliographical notes
Thermodynamical formalism
Gibbs measures: introductory remarks
Transfer operator and its conjugate; measures with prescribed Jacobians
Iteration of the transfer operator; existence of invariant Gibbs measures
Convergence of L<sup>n</sup>; mixing properties of Gibbs measures
More on almost periodic operators
Uniqueness of equilibrium states
Probability laws and �<sup>2</sup>(u, v)
Exercises
Bibliographical notes
Expanding repellers in manifolds and in the Riemann sphere: preliminaries
Basic properties
Complex dimension one; bounded distortion and other techniques
Transfer operator for conformal expanding repeller with harmonic potential
Analytic dependence of transfer operator on potential function
Exercises
Bibliographical notes
Cantor repellers in the line; Sullivan's scaling function; application in Feigenbaum universality
C<sup>1+�</sup>-equivalence
Scaling function: C<sup>1+�</sup>-extension of the shift map
Higher smoothness
Scaling function and smoothness; Cantor set valued scaling function
Cantor set generating families
Quadratic-like maps of the interval; an application to Feigenbaum's universality
Exercises
Bibliographical notes
Fractal dimensions
Outer measures
Hausdorff measures
Packing measures
Dimensions
Besicovitch Covering Theorem; Vitali Theorem and density points
Frostman-type lemmas
Bibliographical notes
Conformal expanding repellers
Pressure function and dimension
Multifractal analysis of Gibbs state
Fluctuations for Gibbs measures
Boundary behaviour of the Riemann map
Harmonic measure; 'fractal vs. analytic' dichotomy
Pressure versus integral means of the Riemann map
Geometric examples: snowflake and Carleson's domains
Exercises
Bibliographical notes
Sullivan's classification of conformal expanding repellers
Equivalent notions of linearity
Rigidity of non-linear CERs
Bibliographical notes
Holomorphic maps with invariant probability measures of positive Lyapunov exponent
Ruelle's inequality
Pesin's theory
Ma��'s partition
Volume Lemma and the formula HD (�) = h<sup>�</sup>(f)/�<sup>�</sup>(f)
Pressure-like definition of the functional h<sup>�</sup> + &#8747; �d�
Katok's theory: hyperbolic sets, periodic points, and pressure
Exercises
Bibliographical notes
Conformal measures
General notion of conformal measures
Sullivan's conformal measures and dynamical dimension: I
Sullivan's conformal measures and dynamical dimension: II
Pesin's formula
More about geometric pressure and dimensions
Bibilographical notes
References
Index