Variational Principle for Topological Pressure on Subsets of Free Semigroup Actions

Zhong, Xing Fu; Chen, Zhi Jing

doi:10.1007/s10114-021-0403-9

Cited by 12 publications

(4 citation statements)

References 25 publications

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“…Following the Brin–Katok formula (see [6, 29]), they defined the measure-theoretic lower entropy and upper entropy and obtained the desired variational principle. Later, [40, 45] extend Feng and Huang’s work to topological pressure. Inspired by [15, 40, 45], to establish the variational principle for weighted topological pressure in the free group setting, one can similarly define a weighted version of measure-theoretic lower entropy and upper entropy by weighted Bowen balls.…”

Section: Final Remarksmentioning

confidence: 99%

“…Later, [40, 45] extend Feng and Huang’s work to topological pressure. Inspired by [15, 40, 45], to establish the variational principle for weighted topological pressure in the free group setting, one can similarly define a weighted version of measure-theoretic lower entropy and upper entropy by weighted Bowen balls. This will avoid the difficulty that the invariant measure under free group actions may fail to exist.…”

Section: Final Remarksmentioning

confidence: 99%

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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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Section: Final Remarksmentioning

confidence: 99%

Section: Final Remarksmentioning

confidence: 99%

A variational principle for weighted topological pressure under -actions

HUO¹,

YUAN²

2022

Ergod. Th. Dynam. Sys.

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“…where h P top (Z, T), h µ (T), and h µ (T) denote respectively the packing topological entropy of Z, measure-theoretical lower and upper entropies of µ. Since then, Feng-Huang's variational principles have been extended to different systems and topological pressures; we refer the reader to [10][11][12][13][14][15][16][17][18] for more details. Tang et al [14] generalized Feng-Huang's variational principle of Bowen topological entropy to Pesin-Pitskel topological pressure: if Z ⊂ X is nonempty and compact then…”

Section: Introductionmentioning

confidence: 99%

Variational principles for topological pressures on subsets

Zhong

Chen

2023

Nonlinearity

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In this paper, we investigate the relations between various types of topological pressures and different versions of measure-theoretical pressures. We extend Feng–Huang’s variational principle for packing entropy to packing pressure and obtain two new variational principles for Pesin–Pitskel and packing pressures respectively. We show that various types of Katok pressures for an ergodic measure with respect to a potential function are equal to the sum of measure-theoretic entropy of this measure and the integral of the potential function. Moreover, we obtain Billingsley type theorem for packing pressure, which indicates that packing pressure can be determined by measure-theoretic upper local pressure of measures, and a variational principle for packing pressure of the set of generic points for any invariant ergodic Borel probability measure.

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“…Inspired by the variational relation between topological entropy and measure-theoretical entropy, Feng and Huang [7] introduced measure-theoretical lower and upper entropies and packing topological entropy, and they obtained two variational principles for Bowen entropy and packing entropy: if Z ⊂ X is nonempty and compact then h B top (Z, T ) = sup{h µ (T ) : µ ∈ M(X ), µ(Z) = 1}, h P top (Z, T ) = sup{h µ (T ) : µ ∈ M(X ), µ(Z) = 1}, where h P top (Z, T ), h µ (T ), and h µ (T ) denote respectively the packing topological entropy of Z, measure-theoretical lower and upper entropies of µ. Since then, Feng-Huang's variational principles have been extended to different systems and topological pressures; we refer the reader to [21,24,5,27,19,25,11,12,8] for more details. Tang, Cheng, and Zhao [19] generalized Feng-Huang's variational principle of Bowen topological entropy to Pesin-Pitskel topological pressure: if Z ⊂ X is nonempty and compact then…”

Section: Introductionmentioning

confidence: 99%

Variational principles for topological pressures on subsets

Zhong¹,

Chen²

2022

Preprint

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In this paper, we investigate the relations between various types of topological pressures and different versions of measure-theoretical pressures. We extend Feng-Huang's variational principle for packing entropy to packing pressure and obtain two new variational principles for Pesin-Pitskel and packing pressures respectively. We show that various types of Katok pressures for an ergodic measure with respect to a potential function are equal to the sum of measure-theoretic entropy of this measure and the integral of the potential function. Moreover, we obtain Billingsley type theorem for packing pressure, which indicates that packing pressure can be determined by measure-theoretic upper local pressure of measures, and a variational principle for packing pressure of the set of generic points for any invariant ergodic Borel probability measure.

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Variational Principle for Topological Pressure on Subsets of Free Semigroup Actions

Cited by 12 publications

References 25 publications

A variational principle for weighted topological pressure under -actions

A variational principle for weighted topological pressure under -actions

Variational principles for topological pressures on subsets

Variational principles for topological pressures on subsets

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