Some properties on topological entropy of free semigroup action

The aim of this paper is to examine the topological entropy for a free semigroup action defined by Bufetov using separated and spanning sets. First, this study reveals that such entropy is a topological conjugacy invariant and also can be equivalently defined using open covers. Furthermore, a quantitative analogue of Bowen's theorem for semiconjugacy is provided and we compared the topological entropies presented by Bufetov and Biś. Finally, a formula for the entropy of skew-product transformation with respect to the subshift is obtained.