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 Carathéodory-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 skewproduct 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].

Topological Entropy of Free Semigroup Actions Generated by Proper Maps for Noncompact Subsets

“…(2) If X is a compact metric space, the topological entropy, LCTE and UCTE in this paper coincide with those defined by Ju et al [13].…”

“…In this section, by using the C-P structure, we give the notions of the topological entropy, LCTE and UCTE of a free semigroup action generated by proper maps for noncompact subsets. Such works extend the previous notions defined by Ju et al [13], Ma et al [17], Patrão [20] and Tian et al [24]. Moreover, some basic properties of these entropies are provided.…”

“…Related studies include [9,16,22,25,26], etc. By using the C-P structure, Ma et al [19] and Ju et al [13] introduced topological entropy of a free semigroup action in different ways, Xiao et al gave two new notions of topological pressure of a free semigroup action in [27] and [28] respectively, Biś et al [4] studied the topological entropy and upper Carathéodory capacity of a semigroup action.…”

“…

In this paper, we introduce three notions of topological entropy of a free semigroup action generated by proper maps for noncompact subsets, which extends the notions defined by Ju et al [13] and Ma et al [17]. By using the one-point compactification as a bridge, we study the relations of the entropies between two dynamical systems.

…”

Recurrent sets and shadowing for finitely generated semigroup actions on metric spaces

“…Wu et al [26] further studied some chain properties and average shadowing for iterated function systems and proved that an iterated function system with (asymptotic) average shadowing is chain mixing. For more recent results on finitely generated group or semigroup actions on compact metric spaces, we refer the reader to [7,10,12,15,23,30] and references therein.…”