Correlation entropy of free semigroup actions

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.