Correlation Entropy of Free Semigroup Actions

This paper introduces the concepts of correlation entropy and local correlation entropy for free semigroup actions on compact metric space, and investigates their underlying properties. Thereafter, we extend certain classical findings on correlation entropy and local correlation entropy to the realm of free semigroup actions. Finally, we establish the interconnections between topological entropy, measure-theoretic entropy, correlation entropy, and local correlation entropy for free semigroup actions under different conditions.