**Cloth**: 978-0-226-66221-3 |

**Paper**: 978-0-226-66222-0 |

**Electronic**: 978-0-226-66223-7

**DOI:**10.7208/chicago/9780226662237.001.0001

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**ABOUT THIS BOOK**

**AUTHOR BIOGRAPHY**

**TABLE OF CONTENTS**

### ABOUT THIS BOOK

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.

### AUTHOR BIOGRAPHY

**Yakov B. Pesin**is professor of mathematics at Pennsylvania State University, University Park. He is the author of

*The General Theory of Smooth Hyperbolic Dynamical Systems*and co-editor of

*Sinai's Moscow Seminar on Dynamical Systems*.

### TABLE OF CONTENTS

**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**

**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 Diffeomorphisms**

**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**