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Entropy in Dynamical Systems

Part of New Mathematical Monographs

  • Date Published: May 2011
  • availability: Available
  • format: Hardback
  • isbn: 9780521888851

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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.

    • 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
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    Product details

    • Date Published: May 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

    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.

  • Resources for

    Entropy in Dynamical Systems

    Tomasz Downarowicz

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  • Author

    Tomasz Downarowicz, Politechnika Wroclawska, Poland
    Tomasz Downarowicz is Full Professor of Mathematics at Wroclaw University of Technology, Poland.

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