Topological R-entropy and topological entropy of free semigroup actions

Abstract

We introduce the notion of topological r -entropy for free semigroup actions on a compact metric space and provide some properties of it. By using the skew-product transformation as bridge, we get the following two main results. 1. We extend the result that the topological entropy is the limit of topological r -entropy in [15] to free semigroup actions ( r → 0 ). 2. Let f i , i = 0 , 1 , ⋯ , m − 1 , be homeomorphisms on a compact metric space. We verify that the topological entropy of f 0 , ⋯ , f m − 1 equals the topological entropy of f 0 − 1 , ⋯ , f m − 1 − 1 .