{"title":"Shadowing of the induced map for contracting homeomorphisms","authors":"W. Jung , M. Lee , C.A. Morales","doi":"10.1016/j.jmaa.2024.128983","DOIUrl":null,"url":null,"abstract":"<div><div>Let <span><math><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></math></span> be a contracting homeomorphism of a metric space with positive diameter. We prove that the induced map <span><math><msub><mrow><mi>f</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span> in the space of probability measures equipped with the Prokhorov metric does not have the shadowing property. However, if <em>X</em> is Polish, then the restriction of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span> to the Wasserstein space has the generalized shadowing property as per Boyarsky and Gora <span><span>[4]</span></span>, concerning the Kantorovich-Rubinstein and Prokhorov metrics.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 128983"},"PeriodicalIF":1.2000,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Analysis and Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022247X24009053","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let be a contracting homeomorphism of a metric space with positive diameter. We prove that the induced map in the space of probability measures equipped with the Prokhorov metric does not have the shadowing property. However, if X is Polish, then the restriction of to the Wasserstein space has the generalized shadowing property as per Boyarsky and Gora [4], concerning the Kantorovich-Rubinstein and Prokhorov metrics.
期刊介绍:
The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions.
Papers are sought which employ one or more of the following areas of classical analysis:
• Analytic number theory
• Functional analysis and operator theory
• Real and harmonic analysis
• Complex analysis
• Numerical analysis
• Applied mathematics
• Partial differential equations
• Dynamical systems
• Control and Optimization
• Probability
• Mathematical biology
• Combinatorics
• Mathematical physics.