Chain Recurrence Rates and Topological Entropy of Free Semigroup Actions

In this paper, we first introduce the pseudo-entropy of free semigroup actions and show that it is equal to the topological entropy of free semigroup actions defined by Bufetov [9]. Second, for free semigroup actions, the concepts of chain recurrence and chain recurrence time, chain mixing and chain mixing time are introduced, and upper bounds for these recurrence times are calculated. Furthermore, the lower box dimension and the chain mixing time provide a lower bound on topological entropy of free semigroup actions. Third, the structure of chain transitive systems of free semigroup actions is discussed. Our analysis generalizes the results obtained by Misiurewicz [21], Richeson and Wiseman [23], and Bufetov [9] etc.