Topological entropy of free semigroup actions for noncompact sets

Abstract

In this paper we introduce the topological entropy and lower and upper capacity topological entropies of a free semigroup action, which extends the notion of the topological entropy of a free semigroup action defined by Bufetov [ 10 ], by using the Caratheodory-Pesin structure (C-P structure). We provide some properties of these notions and give three main results. The first is the relationship between the upper capacity topological entropy of a skew-product transformation and the upper capacity topological entropy of a free semigroup action with respect to arbitrary subset. The second are a lower and a upper estimations of the topological entropy of a free semigroup action by local entropies. The third is that for any free semigroup action with $m$ generators of Lipschitz maps, topological entropy for any subset is upper bounded by the Hausdorff dimension of the subset multiplied by the maximum logarithm of the Lipschitz constants. The results of this paper generalize results of Bufetov [ 10 ], Ma et al. [ 26 ], and Misiurewicz [ 27 ].