Fractal Dimensions for Poincare Recurrences

This book is devoted to an important branch of the dynamical systems theory : the study of the fine (fractal) structure of Poincare recurrences -instants of time when the system almost repeats its initial state. The authors were able to write an entirely self-contained text including many insights and examples, as well as providing complete details of proofs. The only prerequisites are a basic knowledge of analysis and topology. Thus this book can serve as a graduate text or self-study guide for courses in applied mathematics or nonlinear dynamics (in the natural sciences). Moreover, the book can be used by specialists in applied nonlinear dynamics following the way in the book. The authors applied the mathematical theory developed in the book to two important problems: distribution of Poincare recurrences for nonpurely chaotic Hamiltonian systems and indication of synchronization regimes in coupled chaotic individual systems.

* Portions of the book were published in an article that won the title "month's new hot paper in the field of Mathematics" in May 2004
* Rigorous mathematical theory is combined with important physical applications
* Presents rules for immediate action to study mathematical models of real systems
* Contains standard theorems of dynamical systems theory

1. Introduction

Part 1: Fundamentals

2. Symbolic Systems
3. Geometric Constructions
4. Spectrum of Dimensions for Recurrences

Part II: Zero-Dimensional Invariant Sets

5. Uniformly Hyperbolic Repellers
6. Non-Uniformly Hyperbolic Repellers
7. The Spectrum for a Sticky Set
8. Rhythmical Dynamics

Part III: One-Dimensional Systems

9. Markov Maps of the Interval
10. Suspended Flows

Part IV: Measure Theoretical Results

11. Invariant Measures
12. Dimensional for Measures
13. The Variational Principle

Part V: Physical Interpretation and Applications

14. Intuitive Explanation
15. Hamiltonian Systems
16. Chaos Synchronization

Part VI: Appendices

17. Some Known Facts About Recurrences
18. Birkhoff's Individual Theorem
19. The SMB Theorem
20. Amalgamation and Fragmentation

Index