# Local Entropy Theory of a Random Dynamical System

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*Anthony H. Dooley; Guohua Zhang*

In this paper the authors 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, they introduce the concept of local
fiber topological pressure and establish an associated variational
principle, relating it to measure-theoretic entropy. They also discuss
some variants of this variational principle.

The authors introduce both topological and measure-theoretic entropy tuples
for continuous bundle random dynamical systems, and apply
variational principles to obtain a relationship between these of
entropy tuples. Finally, they give applications of these results to
general topological dynamical systems, recovering and extending many
recent results in local entropy theory.

#### Table of Contents

# Table of Contents

## Local Entropy Theory of a Random Dynamical System

- Cover Cover11 free
- Title page i2 free
- Chapter 1. Introduction 18 free
- Acknowledgements 411 free

- Part \ 1 . Preliminaries 714
- Part \ 2 . A Local Variational Principle for Fiber Topological Pressure 3542
- Chapter 5. Local fiber topological pressure 3744
- Chapter 6. Factor excellent and good covers 4552
- Chapter 7. A variational principle for local fiber topological pressure 5360
- Chapter 8. Proof of main result Theorem 7.1 5966
- Chapter 9. Assumption (♠) on the family 𝐃 6774
- Chapter 10. The local variational principle for amenable groups admitting a tiling Følner sequence 7178
- Chapter 11. Another version of the local variational principle 7582

- Part \ 3 . Applications of the Local Variational Principle 8188
- Chapter 12. Entropy tuples for a continuous bundle random dynamical system 8390
- Chapter 13. Applications to topological dynamical systems 9198
- 1. Preparations on topological dynamical systems 9198
- 2. Equivalence of a topological dynamical system with a particular continuous bundle random dynamical system 93100
- 3. The equations (7.5) and (7.6) imply main results of [51] 94101
- 4. Local variational principles for a topological dynamical system 97104
- 5. Entropy tuples of a topological dynamical system 100107

- Bibliography 103110

- Back Cover Back Cover1118