Variational relations for metric mean dimension and rate distortion dimension

Wang, Tao

doi:10.3934/dcds.2021050

Cited by 12 publications

(5 citation statements)

References 14 publications

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“…Inspired by the ideas used in [BS00,WC12], in this paper we introduce the notions of BS metric mean dimension and packing BS metric mean dimension on subsets, which allows us to establish Bowen's equations for Bowen upper mean dimension and packing upper metric mean dimension with potential on subsets. Moreover, two variational principles for BS metric mean dimension and packing BS metric mean dimension on subsets are also obtained analogous to [FH12,WC12,W21]. Finally, we extend Bowen's three important results to the framework of Bowen upper metric mean dimension.…”

Section: Introductionmentioning

confidence: 63%

Bowen’s equations for upper metric mean dimension with potential

Yang

Chen²,

Zhou³

2022

Nonlinearity

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Firstly, we introduce a new notion called induced upper metric mean dimension with potential, which naturally generalises the definition of upper metric mean dimension with potential given by Tsukamoto to more general cases, then we establish variational principles for it in terms of upper and lower rate distortion dimensions and show there exists a Bowen’s equation between induced upper metric mean dimension with potential and upper metric mean dimension with potential. Secondly, we continue to introduce two new notions, called BS metric mean dimension and packing BS metric mean dimension on arbitrary subsets, to establish Bowen’s equations for Bowen upper metric mean dimension and packing upper metric mean dimension with potential on subsets. Besides, we also obtain two variational principles for BS metric mean dimension and packing BS metric mean dimension on subsets. Finally, the special interest about the Bowen upper metric mean dimension of the set of generic points of ergodic measures are also involved.

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Section: Introductionmentioning

confidence: 63%

Bowen’s equations for upper metric mean dimension with potential

Yang

Chen²,

Zhou³

2022

Nonlinearity

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“…Following the ideas of [BS00, WC12], we introduce the notions of BS metric mean dimension and Packing BS metric mean dimension on subsets, which allows us to establish Bowen's equations for upper mean dimension with potential on subsets. Two variational principles for BS metric mean dimension and Packing BS metric mean dimension on subsets are also obtained analogous to [FH12,WC12,W21]. Finally, we extend Bowen's work to the framework of upper metric mean dimension with potential.…”

Section: Introductionmentioning

confidence: 78%

Bowen's equations for upper metric mean dimension with potential

Yang,

Chen,

Zhou

2022

Preprint

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Firstly, we first introduce a new notion called induced upper metric mean dimension with potential, which naturally generalizes the definition of upper metric mean dimension with potential given by Tsukamoto to more general cases. Also, we establish a variational principle for it in terms of upper (and lower) rate distortion dimensions and show induced upper metric mean dimension with potential and upper metric mean dimension with potential are related by a Bowen equation. Secondly, we continue to introduce two new notions, called BS metric mean dimension and Packing BS metric mean dimension on arbitrary subsets, to establish Bowen's equations for upper metric mean dimension with potential on subsets. Besides, two corresponding variational principles for BS metric mean dimension and Packing BS metric mean dimension on subsets are obtained. Finally, the special interest about the upper metric mean dimension of generic points of ergodic measures are also involved.

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“…Shi [Shi22] proved several variational principles for metric mean dimension in terms of Shapira's entropy related to finite overs [Sha07], Katok's entropy and Brin-Katok local entropy [BK83] respectively. Inspired by Feng and Huang's approaches [FH12] in studying Bowen topological entropy of subsets, Wang [Wan21] introduced the Bowen metric mean dimension of subsets. Moreover, some new characterizations for the rate distortion dimension are given and the relation between metric mean dimension of subsets and rate distortion dimension is well investigated in [Wan21].…”

Section: The Constant Dmentioning

confidence: 99%

“…Inspired by Feng and Huang's approaches [FH12] in studying Bowen topological entropy of subsets, Wang [Wan21] introduced the Bowen metric mean dimension of subsets. Moreover, some new characterizations for the rate distortion dimension are given and the relation between metric mean dimension of subsets and rate distortion dimension is well investigated in [Wan21].…”

Section: The Constant Dmentioning

confidence: 99%

A variational principle for weighted topological pressure under -actions

HUO¹,

YUAN²

2022

Ergod. Th. Dynam. Sys.

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Let $k\geq 2$ and $(X_{i}, \mathcal {T}_{i}), i=1,\ldots ,k$ , be $\mathbb {Z}^{d}$ -actions topological dynamical systems with $\mathcal {T}_i:=\{T_i^{\textbf {g}}:X_i{\rightarrow } X_i\}_{\textbf {g}\in \mathbb {Z}^{d}}$ , where $d\in \mathbb {N}$ and $f\in C(X_{1})$ . Assume that for each $1\leq i\leq k-1$ , $(X_{i+1}, \mathcal {T}_{i+1})$ is a factor of $(X_{i}, \mathcal {T}_{i})$ . In this paper, we introduce the weighted topological pressure $P^{\textbf {a}}(\mathcal {T}_{1},f)$ and weighted measure-theoretic entropy $h_{\mu }^{\textbf {a}}(\mathcal {T}_{1})$ for $\mathbb {Z}^{d}$ -actions, and establish a weighted variational principle as $$ \begin{align*} P^{\textbf{a}}(\mathcal{T}_{1},f)=\sup\bigg\{h_{\mu}^{\textbf{a}}(\mathcal{T}_{1})+\int_{X_{1}}f\,d\mu:\mu\in\mathcal{M}(X_{1}, \mathcal{T}_{1})\bigg\}. \end{align*} $$ This result not only generalizes some well-known variational principles about topological pressure for compact or non-compact sets, but also improves the variational principle for weighted topological pressure in [16] from $\mathbb {Z}_{+}$ -action topological dynamical systems to $\mathbb {Z}^{d}$ -actions topological dynamical systems.

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Variational relations for metric mean dimension and rate distortion dimension

Cited by 12 publications

References 14 publications

Bowen’s equations for upper metric mean dimension with potential

Bowen’s equations for upper metric mean dimension with potential

Bowen's equations for upper metric mean dimension with potential

A variational principle for weighted topological pressure under -actions

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