{"title":"Random expansive measures","authors":"Rafael A Bilbao, Marlon Oliveira, Eduardo Santana","doi":"10.1088/1361-6544/ad673f","DOIUrl":null,"url":null,"abstract":"The notion of expansivity and its generalizations (measure expansive, measure positively expansive, continuum-wise expansive, countably-expansive) are well known for deterministic systems and can be a useful property for studying significant type of behavior, such as chaotic one. This study aims to extend these notions into a random context and prove a relationship between relative positive entropy and random expansive measures and apply it to show that if a random dynamical system has positive relative topological entropy then the <italic toggle=\"yes\">w</italic>-stable classes have zero measure for the conditional measures. We also prove that there exists a probability measure that is both invariant and expansive. Moreover, we obtain a relation between the notions of random expansive measures and random countably-expansive systems.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"49 1","pages":""},"PeriodicalIF":1.6000,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinearity","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1088/1361-6544/ad673f","RegionNum":2,"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
The notion of expansivity and its generalizations (measure expansive, measure positively expansive, continuum-wise expansive, countably-expansive) are well known for deterministic systems and can be a useful property for studying significant type of behavior, such as chaotic one. This study aims to extend these notions into a random context and prove a relationship between relative positive entropy and random expansive measures and apply it to show that if a random dynamical system has positive relative topological entropy then the w-stable classes have zero measure for the conditional measures. We also prove that there exists a probability measure that is both invariant and expansive. Moreover, we obtain a relation between the notions of random expansive measures and random countably-expansive systems.
对于确定性系统而言，扩展性概念及其广义（度量扩展性、度量正扩展性、连续广义扩展性、可数度量扩展性）是众所周知的，对于研究重要的行为类型（如混沌行为）来说是一个有用的属性。本研究旨在将这些概念扩展到随机环境中，证明相对正熵与随机膨胀度量之间的关系，并应用它来证明如果一个随机动力学系统具有正的相对拓扑熵，那么 w 稳定类的条件度量为零。我们还证明了存在一种既不变又扩张的概率度量。此外，我们还得到了随机扩张度量概念与随机可数扩张系统概念之间的关系。
期刊介绍：
Aimed primarily at mathematicians and physicists interested in research on nonlinear phenomena, the journal''s coverage ranges from proofs of important theorems to papers presenting ideas, conjectures and numerical or physical experiments of significant physical and mathematical interest.
Subject coverage:
The journal publishes papers on nonlinear mathematics, mathematical physics, experimental physics, theoretical physics and other areas in the sciences where nonlinear phenomena are of fundamental importance. A more detailed indication is given by the subject interests of the Editorial Board members, which are listed in every issue of the journal.
Due to the broad scope of Nonlinearity, and in order to make all papers published in the journal accessible to its wide readership, authors are required to provide sufficient introductory material in their paper. This material should contain enough detail and background information to place their research into context and to make it understandable to scientists working on nonlinear phenomena.
Nonlinearity is a journal of the Institute of Physics and the London Mathematical Society.
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