{"title":"Variational principles for pointwise preimage entropies","authors":"Yaling Shi, Kesong Yan, Fanping Zeng","doi":"10.1080/10236198.2023.2263094","DOIUrl":null,"url":null,"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)<hm(T),where M(X,T) denote the set of all T-invariant probability measures.Keywords: Preimage entropyvariational principlenon-invertible map2000 Mathematics Subject Classifications: Primary: 37A25Secondary: 37A3537A05 AcknowledgmentsThe authors would like to thank the anonymous referees for their useful comments and helpful suggestions that improved the manuscript. We also thank Jiehua Mai for the careful reading and helpful suggestions.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThe authors are supported by NNSF of China (12261006) and NSF of Guangxi Province (2018GXNSFFA281008). The second author is also supported by NNSF of China (12171175) and Project of First Class Disciplines of Statistics of Guangxi Province.","PeriodicalId":15616,"journal":{"name":"Journal of Difference Equations and Applications","volume":"70 1","pages":"0"},"PeriodicalIF":1.1000,"publicationDate":"2023-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Difference Equations and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1080/10236198.2023.2263094","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
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)
期刊介绍:
Journal of Difference Equations and Applications presents state-of-the-art papers on difference equations and discrete dynamical systems and the academic, pure and applied problems in which they arise. The Journal is composed of original research, expository and review articles, and papers that present novel concepts in application and techniques.
The scope of the Journal includes all areas in mathematics that contain significant theory or applications in difference equations or discrete dynamical systems.