{"title":"Shadowable Points of Free Semigroup Actions","authors":"Ritong Li, Dongkui Ma, Rui Kuang, Xiaojiang Ye","doi":"10.1007/s40840-024-01718-z","DOIUrl":null,"url":null,"abstract":"<p>The shadowable points of dynamical systems have been well-studied by Morales in his recent paper (2016). This paper aims to generalize the main results obtained by Morales to free semigroup actions. To this end, we introduce the notion of shadowable points of a free semigroup action. Let <i>G</i> be a free semigroup generated by finite continuous self-maps acting on compact metric space. We will prove the following results for <i>G</i> on compact metric spaces. The set of shadowable points of <i>G</i> is a Borel set. <i>G</i> has the pseudo-orbit tracing property (POTP) if and only if every point is shadowable point of <i>G</i>. The chain recurrent and non-wandering sets of <i>G</i> coincide when every chain recurrent point is shadowable point of <i>G</i>. The space <i>X</i> is totally disconnected at every shadowable point of <i>G</i> under certain condition.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40840-024-01718-z","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
The shadowable points of dynamical systems have been well-studied by Morales in his recent paper (2016). This paper aims to generalize the main results obtained by Morales to free semigroup actions. To this end, we introduce the notion of shadowable points of a free semigroup action. Let G be a free semigroup generated by finite continuous self-maps acting on compact metric space. We will prove the following results for G on compact metric spaces. The set of shadowable points of G is a Borel set. G has the pseudo-orbit tracing property (POTP) if and only if every point is shadowable point of G. The chain recurrent and non-wandering sets of G coincide when every chain recurrent point is shadowable point of G. The space X is totally disconnected at every shadowable point of G under certain condition.
莫拉莱斯在其最新论文（2016）中对动力系统的可影点进行了深入研究。本文旨在将莫拉莱斯获得的主要结果推广到自由半群作用。为此，我们引入了自由半群作用的可影点概念。设 G 是由作用于紧凑度量空间的有限连续自映射生成的自由半群。我们将为紧凑公度空间上的 G 证明以下结果。G 的可阴影点集是一个 Borel 集。当且仅当每个点都是 G 的可影点时，G 具有伪轨迹性质（POTP）。当每个链循环点都是 G 的可影点时，G 的链循环集和非游走集重合。
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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