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Local entropy theory of a random dynamical system
About this Title
Anthony H. Dooley, Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom and Guohua Zhang, School of Mathematical Sciences and LMNS, Fudan University, Shanghai 200433, China
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: https://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,
entropy tuples
MSC: Primary 37A05, 37H99; Secondary 37A15, 37A35
Table of Contents
Chapters
- 1. Introduction
1. Preliminaries
- 2. Infinite countable discrete amenable groups
- 3. Measurable dynamical systems
- 4. Continuous bundle random dynamical systems
2. A Local Variational Principle for Fiber Topological Pressure
- 5. Local fiber topological pressure
- 6. Factor excellent and good covers
- 7. A variational principle for local fiber topological pressure
- 8. Proof of main result Theorem
- 9. Assumption $(\spadesuit )$ on the family $\mathbf {D}$
- 10. The local variational principle for amenable groups admitting a tiling Følner sequence
- 11. Another version of the local variational principle
3. Applications of the Local Variational Principle
- 12. Entropy tuples for a continuous bundle random dynamical system
- 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 $\mathbb {R}$ or $\mathbb {N}$ 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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