Entropy in Dynamical Systems

A comprehensive course on entropy in dynamical systems ideal for graduate students learning the subject from scratch.

Tomasz Downarowicz (Author)

9780521888851, Cambridge University Press

Hardback, published 12 May 2011

404 pages, 30 b/w illus. 80 exercises
23.5 x 16 x 2.5 cm, 0.69 kg

"Overall the writing is clear and the author has included motivational and expository material, as well as some examples and exercises. The presentation is nicely unified, and the different parts of the book interact well."
Michael Hochman, Mathematical Reviews

This comprehensive text on entropy covers three major types of dynamics: measure preserving transformations; continuous maps on compact spaces; and operators on function spaces. Part I contains proofs of the Shannon–McMillan–Breiman Theorem, the Ornstein–Weiss Return Time Theorem, the Krieger Generator Theorem and, among the newest developments, the ergodic law of series. In Part II, after an expanded exposition of classical topological entropy, the book addresses symbolic extension entropy. It offers deep insight into the theory of entropy structure and explains the role of zero-dimensional dynamics as a bridge between measurable and topological dynamics. Part III explains how both measure-theoretic and topological entropy can be extended to operators on relevant function spaces. Intuitive explanations, examples, exercises and open problems make this an ideal text for a graduate course on entropy theory. More experienced researchers can also find inspiration for further research.

Introduction
Part I. Entropy in Ergodic Theory: 1. Shannon information and entropy
2. Dynamical entropy of a process
3. Entropy theorems in processes
4. Kolmogorov–Sinai entropy
5. The ergodic law of series
Part II. Entropy in Topological Dynamics: 6. Topological entropy
7. Dynamics in dimension zero
8. The entropy structure
9. Symbolic extensions
10. A touch of smooth dynamics
Part III. Entropy Theory for Operators: 11. Measure theoretic entropy of stochastic operators
12. Topological entropy of a Markov operator
13. Open problems in operator entropy
Appendix A. Toolbox
Appendix B. Conditional S-M-B
List of symbols
References
Index.

Subject Areas: Nonlinear science [PBWR], Differential calculus & equations [PBKJ]