DCDS
Topological pressure and topological entropy of a semigroup of maps
Dongkui Ma Min Wu
Discrete & Continuous Dynamical Systems - A 2011, 31(2): 545-556 doi: 10.3934/dcds.2011.31.545
By using the Carathéodory-Pesin structure(C-P structure), with respect to arbitrary subset, the topological pressure and topological entropy, introduced for a single continuous map, is generalized to the cases of semigroup of continuous maps. Several of their basic properties are provided.
keywords: topological pressure Semigroup of continuous maps C-P structure topological entropy.
DCDS
Topological entropy of free semigroup actions for noncompact sets
Yujun Ju Dongkui Ma Yupan Wang
Discrete & Continuous Dynamical Systems - A 2019, 39(2): 995-1017 doi: 10.3934/dcds.2019041

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 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].

keywords: C-P structure free semigroup actions topological entropy skew-product local entropy Hausdorff dimension

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