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Dimension Theory in Dynamical Systems : Contemporary Views and Applications - Yakov Pesin

Dimension Theory in Dynamical Systems

Contemporary Views and Applications

Paperback Published: 1st January 1997
ISBN: 9780226662220
Number Of Pages: 314

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The principles of symmetry and self-similarity structure nature's most beautiful creations. For example, they are expressed in fractals, famous for their beautiful but complicated geometric structure, which is the subject of study in dimension theory. And in dynamics the presence of invariant fractals often results in unstable "turbulent-like" motions and is associated with "chaotic" behavior.
In this book, Yakov Pesin introduces a new area of research that has recently appeared in the interface between dimension theory and the theory of dynamical systems. Focusing on invariant fractals and their influence on stochastic properties of systems, Pesin provides a comprehensive and systematic treatment of modern dimension theory in dynamical systems, summarizes the current state of research, and describes the most important accomplishments of this field.
Pesin's synthesis of these subjects of broad current research interest will be appreciated both by advanced mathematicians and by a wide range of scientists who depend upon mathematical modeling of dynamical processes.

Preface
Introduction
Carathéodory Dimension Characteristics
General Carathéodory Construction
Carathéodory Dimension of Sets
Carathéodory Capacity of Sets
Carathéodory Dimension and Capacity of Measures
Coincidence of Carathéodory Dimension and Carathéodory Capacity of Measures
Lower and Upper Bounds for Carathéodory Dimension of Sets; Carathéodory Dimension Spectrum
C-Structures Associated with Metrics: Hausdorff Dimension and Box Dimension
Hausdorff Dimension and Box Dimension of Sets
Hausdorff Dimension and Box Dimension of Measures; Pointwise Dimension; Mass Distribution Principle
C-Structures Associated with Metrics and Measures: Dimension Spectra
q-Dimension and q-Box Dimension of Sets
q-Dimension and q-Box Dimension of Measures
Hausdorff (Box) Dimension and Q-(Box) Dimension of Sets and Measures in General Metric Spaces
C-Structures Associated with Dynamical Systems: Thermodynamic Formalism
A Modification of the General Carathéodory Construction
Dimensional Definition of Topological Pressure; Topological and Measure-Theoretic Entropies
Non-additive Thermodynamic Formalism
Variational Principle for Topological Pressure; Symbolic Dynamical Systems; Bowen's Equation
An Example of Carathéodory Structure Generated by Dynamical Systems
Applications to Dimension Theory and Dynamical Systems
Dimension of Cantor-like Sets and Symbolic Dynamics
Moran-like Geometric Constructions with Stationary (Constant) Ratio Coefficients
Regular Geometric Constructions
Moran-like Geometric Constructions with Non-stationary Ratio Coefficients
Geometric Constructions with Rectangles; Non-coincidence of Box Dimension and Hausdorff Dimension of Sets
Multifractal Formalism
Correlation Dimension
Dimension Spectra: Hentschel-Procaccia, Rényi, and f(alpha)-Spectra; Information Dimension
Multifractal Analysis of Gibbs Measures on Limit Sets of Geometric Constructions
Dimension of Sets and Measures Invariant under Hyperbolic Systems
Hausdorff Dimension and Box Dimension of Conformal Repellers for Smooth Expanding Maps
Multifractal Analysis of Gibbs Measures for Smooth Conformal Expanding Maps
Hausdorff Dimension and Box Dimension of Basic Sets for Axiom A Diffeomrophisms
Hausdorff Dimension of Horseshoes and Solenoids
Multifractal Analysis of Equilibrium Measures on Basic Sets of Axiom A Diffeomorphisms
A General Concept of Multifractal Spectra; Multifractal Rigidity
Relations between Dimension, Entropy, and Lyapunov Exponents
Existence and Non-existence of Pointwise Dimension for Invariant Measures
Dimension of Measures with Non-zero Lyapunov Exponents; The Eckmann-Ruelle Conjecture
Some Useful Facts
Bibliography
Index
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780226662220
ISBN-10: 0226662225
Series: Chicago Lectures in Mathematics
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 314
Published: 1st January 1997
Publisher: The University of Chicago Press
Country of Publication: US
Dimensions (cm): 22.9 x 15.0  x 1.75
Weight (kg): 0.44
Edition Number: 2