Topological R-entropy and topological entropy of free semigroup actions

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) . (C) 2018 Elsevier Inc. All rights reserved.