Correlation entropy of free semigroup actions

TL;DR

This paper develops the theory of correlation entropy for free semigroup actions on compact metric spaces, generalizing classical results and linking various entropy notions under different conditions.

Contribution

It introduces correlation entropy and local correlation entropy for free semigroup actions and extends classical entropy results to this new setting.

Findings

Established properties of correlation entropy for free semigroup actions

Generalized classical entropy results to free semigroup actions

Linked topological, measure-theoretic, and correlation entropies in this context

Abstract

This paper introduces the concepts of correlation entropy and local correlation entropy for free semigroup actions on compact metric space, and explores their fundamental properties. Thereafter, we generalize some classical results on correlation entropy and local correlation entropy to apply to free semigroup actions. Finally, we establish the relationship between topological entropy, measure-theoretic entropy, correlation entropy, and local correlation entropy for free semigroup actions under various conditions.