Dimension Theory in Dynamical Systems Contemporary Views and Applications
by Yakov B. Pesin
University of Chicago Press, 1997
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


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.


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.




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