Variational principle for topological pressures on subsets

This paper extends the definition of Bowen topological entropy of subsets to Pesin-Pitskel topological pressure for the continuous action of amenable groups on a compact metric space. We introduce the local measure theoretic pressure of subsets and investigate the relation between local measure theoretic pressure of Borel probability measures and Pesin-Pitskel topological pressure on an arbitrary subset of a compact metric space.

Section: Xiaojun Huang Yuan Lian and Changrong Zhumentioning

confidence: 59%

A Billingsley-type theorem for the pressure of an action of an amenable group

Huang¹,

Lian²,

Zhu³

2019

“…Zhong and Huang [19] introduced a version of invariance pressure in a way resembling Hausdorff dimension, which is called Bowen invariance pressure. On the other hand, Tang, Cheng, and Zhao [16] extended Feng and Huang's result and gave variational principle between Pesin-Pitskel topological pressure (also called Bowen topological pressure) and measure-theoretic lower pressure.…”

Section: (Communicated By Xiangdong Ye)mentioning

confidence: 96%

Variational principles of invariance pressures on partitions

Zhong¹

2020

1. Introduction. Topological feedback entropy was first introduced by Nair et al. [14] by using invariant open covers to characterize the minimal data rate for making a subset of the state space invariant. Later, Colonius and Kawan [5] introduced invariance entropy, which is defined via spanning sets, to describe the exponential growth rate of the minimal number of different control functions sufficient for orbits to stay in a given set when starting in a subset of this set. The fact that these two entropies are equivalent was shown by Colonius, Kawan, and Nair [6]. Recently, Huang and Zhong [10] use the theory of Carathéodory-Pesin structure to obtain a dimension-like characterization for invariance entropy, which is called Bowen invariance entropy. We refer the reader to the monograph written by Kawan [11] for more about invariance entropy.By choosing conditionally invariant measures and quasi-stationary measures, Colonius [2, 3] first introduced four notions of metric invariance entropy in analogy to the topological notion of invariance entropy of deterministic control systems. In [2], Colonius showed that the metric entropy of a given controlled invariant set coincides with the minimal entropy of coder-controllers associated with a quasi-stationary measure rendering that set invariant. Variational principle [17] for topological entropy [1] and measure theoretic entropy [12] in classical dynamical systems states that topological entropy is determined by the supremum of measure theoretic entropies. Feng and Huang gave variational principles between Bowen 2010 Mathematics Subject Classification. Primary: 37A35, 37B40, 37C45; Secondary: 93C55.

“…Later on, Wang and Chen generalized this result to BS-dimension in [25]. Furthermore, still in the setting of Z-actions, Tang, Cheng and Zhao [23] develop the theory for topological pressure on subsets, and proved a corresponding variational principle.…”

Section: Xiaojun Huang Zhiqiang LI and Yunhua Zhoumentioning

confidence: 99%

A variational principle of topological pressure on subsets for amenable group actions

Huang¹,

Li²,

Zhou³

2020

In this paper, we establish a variational principle for topological pressure on compact subsets in the context of amenable group actions. To be precise, for a countable amenable group action on a compact metric space, say G X, for any potential f ∈ C(X), we define and study topological pressure on an arbitrary subset and measure theoretic pressure for any Borel probability measure on X (not necessarily invariant); moreover, we prove a variational principle for this topological pressure on a given nonempty compact subset K ⊆ X.