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Variational principles of metric mean dimension for random dynamical systems

Part of: Smooth dynamical systems: general theory Topological dynamics Ergodic theory

Published online by Cambridge University Press:  18 September 2025

Yunping Wang ,
Ercai Chen  and
Kexiang Yang
Yunping Wang
Affiliation:
School of Statistics and Data Science, Ningbo University of Technology, Ningbo 315211, Zhejiang, PR China
Ercai Chen
Affiliation:
School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210046, Jiangsu, PR China
Kexiang Yang*
Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai 200433, Shanghai, PR China
*
Corresponding author: Kexiang Yang, email: kxyangs@163.com
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Abstract

In this paper, we establish variational principles for the metric mean dimension of random dynamical systems with infinite topological entropy. This is based on four types of measure-theoretic ϵ-entropies: Kolmogorov-Sinai ϵ-entropy, Shapira’s ϵ-entropy, Katok’s ϵ-entropy and Brin–Katok local ϵ-entropy. The variational principle, as a fundamental theorem, links topological dynamics and ergodic theory.

Keywords

Random dynamical systemmetric mean dimensionmeasure-theoretic ϵ-entropyvariational principle

MSC classification

Primary: 37A35: Entropy and other invariants, isomorphism, classification
Secondary: 37C45: Dimension theory of dynamical systems 37B40: Topological entropy

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 25
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Edinburgh Mathematical Society.

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