A variational principle for free semigroup actions

Carvalho, Maria; Rodrigues, Fagner B.; Varandas, Paulo

doi:10.1016/j.aim.2018.06.010

Cited by 29 publications

(34 citation statements)

References 23 publications

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“…For the topological entropy of a pseudogroup introduced in [5], Biś proved a similar result to the Theorem 6.2. In [12], the authors obtained a similar result using the skew product.…”

Section: By Proposition 1(3)mentioning

confidence: 61%

“…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%

“…Acknowledgments. The authors would like to thank Professor P. Varandas for providing the reference [12] and really appreciate the referees' valuable remarks and suggestions that helped a lot.…”

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confidence: 99%

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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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“…For the topological entropy of a pseudogroup introduced in [5], Biś proved a similar result to the Theorem 6.2. In [12], the authors obtained a similar result using the skew product.…”

Section: By Proposition 1(3)mentioning

confidence: 61%

Section: Introduction Topological Entropy Was First Introduced By Admentioning

confidence: 99%

Section: Introduction Topological Entropy Was First Introduced By Admentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Topological entropy of free semigroup actions for noncompact sets

Ju¹,

Ma²,

Wang³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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“…The previous equality extends to the notion of metric mean dimension the formula of Bufetov h top (S, P p ) = h top (T G ) − h top (σ) regarding the topological entropy of a finitely generated free semigroup action with respect to the symmetric Bernoulli random walk P p (cf. [4,7]). In particular, this raises the question of whether…”

Section: Resultsmentioning

confidence: 99%

A variational principle for the metric mean dimension of free semigroup actions

Carvalho¹,

Rodrigues

Varandas³

2021

Ergod. Th. Dynam. Sys.

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We consider continuous free semigroup actions generated by a family $(g_y)_{y \,\in \, Y}$ of continuous endomorphisms of a compact metric space $(X,d)$ , subject to a random walk $\mathbb P_\nu =\nu ^{\mathbb N}$ defined on a shift space $Y^{\mathbb N}$ , where $(Y, d_Y)$ is a compact metric space with finite upper box dimension and $\nu $ is a Borel probability measure on Y. With the aim of elucidating the impact of the random walk on the metric mean dimension, we prove a variational principle which relates the metric mean dimension of the semigroup action with the corresponding notions for the associated skew product and the shift map $\sigma $ on $Y^{\mathbb {N}}$ , and compare them with the upper box dimension of Y. In particular, we obtain exact formulas whenever $\nu $ is homogeneous and has full support. We also discuss several examples to enlighten the roles of the homogeneity, of the support and of the upper box dimension of the measure $\nu $ , and to test the scope of our results.

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Variational principle for BS-dimension of subsets of finitely generated free semigroup actions

Ding,

Liu

2023

Semigroup Forum

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A variational principle for free semigroup actions

Cited by 29 publications

References 23 publications

Topological entropy of free semigroup actions for noncompact sets

Topological entropy of free semigroup actions for noncompact sets

A variational principle for the metric mean dimension of free semigroup actions

Variational principle for BS-dimension of subsets of finitely generated free semigroup actions

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