The local variational principle of topological pressure for sub-additive potentials

Chen, Beimei; Ding, Bangfeng; Cao, Yongluo

doi:10.1016/j.na.2010.07.025

Cited by 9 publications

(3 citation statements)

References 18 publications

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“…One of the most important facts known for upper semicontinuous subadditive potentials is the following theorem, which we provide in the most recent formulation from [1] and [2] (see also [3,8,11,14] for the local versions). For non-negative potentials we will provide a new proof based on a new interpretation of the notions involved.…”

Section: Preliminariesmentioning

confidence: 99%

Modeling potential as fiber entropy and pressure as entropy

Downarowicz¹,

Zhang²

2014

Ergod. Th. Dynam. Sys.

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We first prove that topological fiber entropy potential in a relatively symbolic extension (topological joining with a subshift) of a topological dynamical system (which is a non-negative, non-decreasing, upper semicontinuous and subadditive potential $\mathfrak{H}$) yields topological pressure equal to the topological entropy of the extended system. The terms occurring on the other side of the variational principle for pressure are equal to the extension entropies of the invariant measures. Thus the variational principle for pressure reduces to the usual variational principle (for entropy) applied to the extended system. Next we prove our main theorem saying that every non-negative, upper semicontinuous and subadditive potential $\mathfrak{F}$ (we drop the monotonicity assumption) is ‘nearly equal’ to the fiber entropy potential H in some relatively symbolic extension of the system, in the sense that all terms occurring in the variational principle for pressure are the same for both potentials. This gives a new interpretation of all such potentials $\mathfrak{F}$ as a kind of additional information function enhancing the natural information arising from the dynamical system, and provides a new proof of the variational principle for pressure. At the end of the paper we provide examples showing that both assumptions, continuity and additivity, under which so-called lower pressure (defined with the help of spanning sets) equals the pressure, are essential, already in the class of non-negative, upper semicontinuous, subadditive potentials.

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Section: Preliminariesmentioning

confidence: 99%

Modeling potential as fiber entropy and pressure as entropy

Downarowicz¹,

Zhang²

2014

Ergod. Th. Dynam. Sys.

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“…Zhang [50] introduced the notion of measure-theoretic pressure for sub-additive potentials, and studied the relationship between topological pressure and measure-theoretic pressure. Recently, Chen, Ding and Cao [16] studied the local variational principle of topological pressure for sub-additive potential, Liang and Yan [35] introduced the topological pressure for any sub-additive potentials of a countable discrete amenable group action and established a local variational principle for it, and Yan [49] investigated the topological pressure for any sub-additive and asymptotically sub-additive potentials of Z d -actions and established the variational principle for them. Cheng and Li [18] extended the definition of entropy dimension and gave the definition of topological pressure dimension, which is also similar to the fractal measure, and studied the relationships among different types of topological pressure dimension and identifies an inequality relating them.…”

Section: Introductionmentioning

confidence: 99%

Topological pressure dimension for almost additive potentials

2016

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This paper is devoted to the study of the topological pressure dimension for almost additive sequences, which is an extension of topological entropy dimension. We investigate fundamental properties of the topological pressure dimension for almost additive sequences. In particular, we study the relationships among different types of topological pressure dimension and identifies an inequality relating them. Also, we show that the topological pressure dimension is always equal to or greater than 1 for certain special almost additive sequence.

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“…(3) 关于单个连续函数的局部拓扑压的变分原理 (这里的局部是指在定义拓扑压时固定一个开覆盖), 参见文献 [28]. 关于一列次可加势函数的局部拓扑压的变分原理, 参见文献 [29,30]. (4) 当 Φ 是几乎可加的时, 上述变分原理由 Barreira [10] 和 Mummert [11] 各自独立证明.…”

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Topological pressure and dimension estimation

YongLuo¹,

Yun²

2016

Sci. Sin.-Math.

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In this paper, we give a survey of recent results in the dimension theory of dynamical systems, with emphasis on the dimension of repellers and hyperbolic sets and the dimension of invariant measures. In the case of conformal dynamics, the theory is completely well understood. However, it still lacks today a satisfactory general approach for the non-conformal case, although a number of interesting and nontrivial developments have been obtained. Indeed, we only well understand some particular non-conformal repellers, e.g., generalized Sierpiński carpets and average conformal repellers.

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The local variational principle of topological pressure for sub-additive potentials

Cited by 9 publications

References 18 publications

Modeling potential as fiber entropy and pressure as entropy

Modeling potential as fiber entropy and pressure as entropy

Topological pressure dimension for almost additive potentials

Topological pressure and dimension estimation

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