{"title":"SRB measures for $C^{\\infty }$ surface diffeomorphisms","authors":"","doi":"10.1007/s00222-024-01235-7","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>A <span> <span>\\(C^{\\infty }\\)</span> </span> smooth surface diffeomorphism admits an SRB measure if and only if the set <span> <span>\\(\\{ x, \\ \\limsup _{n}\\frac{1}{n}\\log \\|d_{x}f^{n}\\|>0\\}\\)</span> </span> has positive Lebesgue measure. Moreover the basins of the ergodic SRB measures are covering this set Lebesgue almost everywhere. We also obtain similar results for <span> <span>\\(C^{r}\\)</span> </span> surface diffeomorphisms with <span> <span>\\(+\\infty >r>1\\)</span> </span>.</p>","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00222-024-01235-7","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 0
Abstract
A \(C^{\infty }\) smooth surface diffeomorphism admits an SRB measure if and only if the set \(\{ x, \ \limsup _{n}\frac{1}{n}\log \|d_{x}f^{n}\|>0\}\) has positive Lebesgue measure. Moreover the basins of the ergodic SRB measures are covering this set Lebesgue almost everywhere. We also obtain similar results for \(C^{r}\) surface diffeomorphisms with \(+\infty >r>1\).
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