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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.Read more
- Suitable for use as a complete textbook for a graduate course on entropy theory
- Includes a number of open problems at varying levels of difficulty and provides exercises and examples alongside rigorous proofs
- Contains the proof of the Shannon–McMillan–Breiman Theorem in the conditional version, which is unavailable elsewhere
Reviews & endorsements
"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
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- Date Published: June 2011
- format: Hardback
- isbn: 9780521888851
- length: 404 pages
- dimensions: 235 x 160 x 25 mm
- weight: 0.69kg
- contains: 30 b/w illus. 80 exercises
- availability: Available
Table of Contents
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
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