{"title":"Topological stability for homeomorphisms with global attractor","authors":"Carlos Arnoldo Morales, Nguyen Thanh Nguyen","doi":"10.4153/s0008439523000917","DOIUrl":null,"url":null,"abstract":"<p>We prove that every topologically stable homeomorphism with global attractor of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231221035053529-0023:S0008439523000917:S0008439523000917_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbb {R}^n$</span></span></img></span></span> is topologically stable on its global attractor. The converse is not true. On the other hand, if a homeomorphism with global attractor of a locally compact metric space is expansive and has the shadowing property, then it is topologically stable. This extends the Walters stability theorem (Walters, <span>On the pseudo-orbit tracing property and its relationship to stability. The structure of attractors in dynamical systems</span>, 1978, pp. 231–244).</p>","PeriodicalId":501184,"journal":{"name":"Canadian Mathematical Bulletin","volume":"89 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Canadian Mathematical Bulletin","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4153/s0008439523000917","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We prove that every topologically stable homeomorphism with global attractor of $\mathbb {R}^n$ is topologically stable on its global attractor. The converse is not true. On the other hand, if a homeomorphism with global attractor of a locally compact metric space is expansive and has the shadowing property, then it is topologically stable. This extends the Walters stability theorem (Walters, On the pseudo-orbit tracing property and its relationship to stability. The structure of attractors in dynamical systems, 1978, pp. 231–244).
$\mathbb {R}^n$ is topologically stable on its global attractor. The converse is not true. On the other hand, if a homeomorphism with global attractor of a locally compact metric space is expansive and has the shadowing property, then it is topologically stable. This extends the Walters stability theorem (Walters, On the pseudo-orbit tracing property and its relationship to stability. The structure of attractors in dynamical systems, 1978, pp. 231–244).
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