On the topological entropy of a semigroup of continuous maps

Wang, Yupan; Ma, Dongkui

doi:10.1016/j.jmaa.2015.02.082

Cited by 23 publications

(10 citation statements)

References 20 publications

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“…Obviously, µ ∈ M (X, g 0 , · · · , g m−1 ). For any ε > 0, w = i 1 · · · i k ∈ F + m and w ′ = i l · · · i k where 1 ≤ l ≤ k, from the equation (4.3) in the proof of Theorem 4.2 by Wang and Ma [31], we have…”

Section: Entropy Of a Free Semigroup Action Generated By Affine Transmentioning

confidence: 99%

On the measure-theoretic entropy and topological pressure of free semigroup actions

Lin¹,

Ma²,

Wang

2016

Ergod. Th. Dynam. Sys.

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Section: Entropy Of a Free Semigroup Action Generated By Affine Transmentioning

confidence: 99%

On the measure-theoretic entropy and topological pressure of free semigroup actions

Lin¹,

Ma²,

Wang

2016

Ergod. Th. Dynam. Sys.

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“…McAndrew [1]. Later, Bowen [7] and Dinaburg [17] defined topological entropy for a uniformly continuous map on metric space and proved that for a compact metric space, they coincide with that defined by Adler et al Since the topological entropy appeared to be a very useful invariant in ergodic theory and dynamical systems, there were several attempts to find its suitable generalizations for other systems such as groups, pseudogroups, graphs, foliations, nonautonomous dynamical systems and so on [3,4,5,6,10,11,12,13,19,20,21,22,23,25,34,35]. Bowen [8] extended the concept of topological entropy for non-compact sets in a way which resembles the Hausdorff dimension.…”

Section: Introduction Topological Entropy Was First Introduced By Admentioning

confidence: 99%

“…Biś [3] and Bufetov [10] gave the definition of the topological entropy of free semigroup actions on a compact metric space, respectively. Related studies include [6,11,12,13,22,23,25,30,34,35], etc.…”

Section: Introduction Topological Entropy Was First Introduced By Admentioning

confidence: 99%

Topological entropy of free semigroup actions for noncompact sets

Ju¹,

Ma²,

Wang³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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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].

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“…Biś [3] and Bufetov [7] introduced the notion of the topological entropy for free semigroup actions on a compact metric space, respectively. 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.…”

Section: Introductionmentioning

confidence: 99%

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

Xie

2022

Taiwanese J. Math.

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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. We then introduce three skew-product transformations, and for a particular subset, the relationship between the upper capacity topological entropy of a free semigroup action generated by proper maps, and the upper capacity topological entropy of a skew-product transformation is given. As applications, we examine the multifractal spectrum of a locally compact separable metric space, and it is shown that the irregular set has full upper capacity topological entropy of a free semigroup action generated by proper maps.

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On the topological entropy of a semigroup of continuous maps

Cited by 23 publications

References 20 publications

On the measure-theoretic entropy and topological pressure of free semigroup actions

On the measure-theoretic entropy and topological pressure of free semigroup actions

Topological entropy of free semigroup actions for noncompact sets

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

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