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Local entropy theory of a random dynamical system


About this Title

Anthony H. Dooley and Guohua Zhang

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 233, Number 1099
ISBNs: 978-1-4704-1055-1 (print); 978-1-4704-1967-7 (online)
DOI: http://dx.doi.org/10.1090/memo/1099
Published electronically: May 19, 2014
Keywords:Discrete amenable groups, (tiling) Følner sequences, continuous bundle random dynamical systems, random open covers, random continuous functions, local fiber topological pressure, factor excellent and good covers, local variational principles

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Infinite countable discrete amenable groups
  • Chapter 3. Measurable dynamical systems
  • Chapter 4. Continuous bundle random dynamical systems
  • Chapter 5. Local fiber topological pressure
  • Chapter 6. Factor excellent and good covers
  • Chapter 7. A variational principle for local fiber topological pressure
  • Chapter 8. Proof of main result Theorem 7.1
  • Chapter 9. Assumption $(\spadesuit )$ on the family $\mathbf {D}$
  • Chapter 10. The local variational principle for amenable groups admitting a tiling Følner sequence
  • Chapter 11. Another version of the local variational principle
  • Chapter 12. Entropy tuples for a continuous bundle random dynamical system
  • Chapter 13. Applications to topological dynamical systems

Abstract


In this paper we extend the notion of a continuous bundle random dynamical system to the setting where the action of or is replaced by the action of an infinite countable discrete amenable group. Given such a system, and a monotone sub-additive invariant family of random continuous functions, we introduce the concept of local fiber topological pressure and establish an associated variational principle, relating it to measure-theoretic entropy. We also discuss some variants of this variational principle. We introduce both topological and measure-theoretic entropy tuples for continuous bundle random dynamical systems, and apply our variational principles to obtain a relationship between these of entropy tuples. Finally, we give applications of these results to general topological dynamical systems, recovering and extending many recent results in local entropy theory.

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