Variational principles for pointwise preimage entropies

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-08-03 DOI:10.1080/10236198.2023.2263094

Yaling Shi, Kesong Yan, Fanping Zeng

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Abstract

AbstractBased on the preimage structure of the system (X,T), Hurley introduced the notion of pointwise topological preimage entropies hm(T) and hp(T). Furthermore, from the measure-theoretic point of view, Wu and Zhu introduced a notion of pointwise metric preimage entropy hm,μ(T) for a T-invariant measure µ on X, and obtained the variational principle between hm,μ(T) and hm(T) under the condition of uniform separation of preimages. A natural question is whether a variational principle for hm(T) and hm,μ(T) without any additional assumptions. In this paper, we define a new version of topological preimage entropy hm(T|μ) relative to a T-invariant measure µ, and show that the inequality hm,μ(T)⩽hm(T|μ)⩽hp(T) holds for every T-invariant probability measure µ. As a consequence, we obtain that there is a topological dynamical system (X,T) such that the following strict inequality holds: supμ∈M(X,T)hm,μ(T)

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点前像熵的变分原理

摘要基于系统(X,T)的预像结构，Hurley引入了点向拓扑预像熵hm(T)和hp(T)的概念。此外，Wu和Zhu从测度论的角度，对X上的T不变测度引入了点向度量原像熵hm，μ(T)的概念，得到了原像均匀分离条件下hm，μ(T)和hm(T)之间的变分原理。一个自然的问题是，在没有任何额外假设的情况下，hm(T)和hm，μ(T)的变分原理是否。本文定义了相对于T不变测度μ的一个新的拓扑原像熵hm(T|μ)，并证明了不等式hm，μ(T)≤hm(T|μ)≤hp(T)对于每一个T不变概率测度μ都成立。因此，我们得到了存在一个拓扑动力系统(X,T)，使得以下严格不等式成立:supμ∈M(X,T)hm，μ(T)

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.

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