Dimension Theory in Dynamical Systems

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ISBN: 9780226662237

Preface
Introduction
Part I: Carathéodory Dimension Characteristics 
Chapter 1. General Carathéodory Construction
1. Carathéodory Dimension of Sets
2. Carathéodory Capacity of Sets
3. Carathéodory Dimension and Capacity of Measures
4. Coincidence of Carathéodory Dimension and Carathéodory Capacity of Measures
5. Lower and Upper Bounds for Carathéodory Dimension of Sets; Carathéodory Dimension Spectrum
Chapter 2. C-Structures Associated with Metrics: Hausdorff Dimension and Box Dimension
6. Hausdorff Dimension and Box Dimension of Sets
7. Hausdorff Dimension and Box Dimension of Measures; Pointwise Dimension; Mass Distribution Principle
Chapter 3. C-Structures Associated with Metrics and Measures: Dimension Spectra
8. q-Dimension and q-Box Dimension of Sets
9. q-Dimension and q-Box Dimension of Measures
Appendix I: Hausdorff (Box) Dimension and Q-(Box) Dimension of Sets and Measures in General Metric Spaces
Chapter 4. C-Structures Associated with Dynamical Systems: Thermodynamic Formalism
10. A Modification of the General Carathéodory Construction
11. Dimensional Definition of Topological Pressure; Topological and Measure-Theoretic Entropies
12. Non-additive Thermodynamic Formalism
Appendix II: Variational Principle for Topological Pressure; Symbolic Dynamical Systems; Bowen's Equation
Appendix III: An Example of Carathéodory Structure Generated by Dynamical Systems
Part II: Applications to Dimension Theory and Dynamical Systems
Chapter 5. Dimension of Cantor-like Sets and Symbolic Dynamics
13. Moran-like Geometric Constructions with Stationary (Constant) Ratio Coefficients
14. Regular Geometric Constructions
15. Moran-like Geometric Constructions with Non-stationary Ratio Coefficients
16. Geometric Constructions with Rectangles; Non-coincidence of Box Dimension and Hausdorff Dimension of Sets
Chapter 6. Multifractal Formalism
17. Correlation Dimension
18. Dimension Spectra: Hentschel-Procaccia, Rényi, and f(alpha)-Spectra; Information Dimension
19. Multifractal Analysis of Gibbs Measures on Limit Sets of Geometric Constructions
Chapter 7. Dimension of Sets and Measures Invariant under Hyperbolic Systems
20. Hausdorff Dimension and Box Dimension of Conformal Repellers for Smooth Expanding Maps
21. Multifractal Analysis of Gibbs Measures for Smooth Conformal Expanding Maps
22. Hausdorff Dimension and Box Dimension of Basic Sets for Axiom A Diffeomrophisms
23. Hausdorff Dimension of Horseshoes and Solenoids
24. Multifractal Analysis of Equilibrium Measures on Basic Sets of Axiom A Diffeomorphisms
Appendix IV: A General Concept of Multifractal Spectra; Multifractal Rigidity
Chapter 8. Relations between Dimension, Entropy, and Lyapunov Exponents
25. Existence and Non-existence of Pointwise Dimension for Invariant Measures
26. Dimension of Measures with Non-zero Lyapunov Exponents; The Eckmann-Ruelle Conjecture
Appendix V: Some Useful Facts
Bibliography
Index