Generic homeomorphisms have full metric mean dimension

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

doi:10.1017/etds.2020.130

Cited by 12 publications

(24 citation statements)

References 19 publications

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“…The existence of such maps g α i was established in [8]. Take a Bernoulli probability measure P ν on Y N = {1, .…”

Section: 52mentioning

confidence: 99%

“…where the supremum is taken over all Borel probability measures μ on Y . Moreover, if f : [0, 1] → [0, 1] is a continuous map with positive upper metric mean dimension (whose existence is proved in [8,20]),…”

Section: Proof Of Theorem Amentioning

confidence: 99%

“…. The existence of these maps g α was established in [8]. Take Y = {(1/n) : n ∈ N} ∪ {0} and consider the continuous skew product…”

Section: Proof Of Theorem Amentioning

confidence: 99%

“…We remark that, although the metric mean dimension is metric dependent, in this locally constant setting we were able to change the metric and the probability measure in Y 2 (keeping the measure of the interval [0, 1] unchanged), while keeping the upper metric mean dimension of the semigroup action. The existence of such maps g α i was established in [8]. Take a Bernoulli probability measure P ν on Y N = {1, .…”

Section: Proof Of Theorem Amentioning

confidence: 99%

“…d) is not continuous when we endow the space of compact subsets in X with the Hausdorff metric (cf. [8]).…”

Section: Proof Of Corollary IImentioning

confidence: 99%

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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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“…The existence of such maps g α i was established in [8]. Take a Bernoulli probability measure P ν on Y N = {1, .…”

Section: 52mentioning

confidence: 99%

Section: Proof Of Theorem Amentioning

confidence: 99%

“…. The existence of these maps g α was established in [8]. Take Y = {(1/n) : n ∈ N} ∪ {0} and consider the continuous skew product…”

Section: Proof Of Theorem Amentioning

confidence: 99%

Section: Proof Of Theorem Amentioning

confidence: 99%

“…d) is not continuous when we endow the space of compact subsets in X with the Hausdorff metric (cf. [8]).…”

Section: Proof Of Corollary IImentioning

confidence: 99%

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

Carvalho¹,

Rodrigues

Varandas³

2021

Ergod. Th. Dynam. Sys.

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Genericity of Continuous Maps with Positive Metric Mean Dimension

Acevedo

2021

Results Math

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M. Gromov introduced the mean dimension for a continuous map in the late 1990's, which is an invariant under topological conjugacy. On the other hand, the notion of metric mean dimension for a dynamical system was introduced by Lindenstrauss and Weiss in 2000 and this refines the topological entropy for dynamical systems with infinite topological entropy. In this paper we will show if N is an n dimensional compact riemannian manifold then, for any a ∈ [0, n], the set consisting of continuous maps with metric mean dimension equal to a is dense in C 0 (N ) and for a = n this set is residual. Furthermore, we prove some results related to the existence and, density of continuous maps, defined on Cantor sets, with positive metric mean dimension and also continous maps, defined on product spaces, with positive mean dimension.

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