{"title":"双曲型全自同构的光滑局部刚性","authors":"B. Kalinin, V. Sadovskaya, Zhenqi Wang","doi":"10.1090/cams/22","DOIUrl":null,"url":null,"abstract":"<p>We study the regularity of a conjugacy <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\">\n <mml:semantics>\n <mml:mi>H</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">H</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> between a hyperbolic toral automorphism <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\n <mml:semantics>\n <mml:mi>A</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and its smooth perturbation <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f\">\n <mml:semantics>\n <mml:mi>f</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">f</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. We show that if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\">\n <mml:semantics>\n <mml:mi>H</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">H</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is weakly differentiable then it is <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript 1 plus upper H reverse-solidus quotation-mark older\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>C</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mn>1</mml:mn>\n <mml:mo>+</mml:mo>\n <mml:mtext>H\\\"older</mml:mtext>\n </mml:mrow>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">C^{1+\\text {H\\\"older}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and, if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\n <mml:semantics>\n <mml:mi>A</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is also weakly irreducible, then <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\">\n <mml:semantics>\n <mml:mi>H</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">H</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript normal infinity\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>C</mml:mi>\n <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">C^\\infty</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. As a part of the proof, we establish results of independent interest on Hölder continuity of a measurable conjugacy between linear cocycles over a hyperbolic system. As a corollary, we improve regularity of the conjugacy to <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript normal infinity\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>C</mml:mi>\n <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">C^\\infty</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> in prior local rigidity results.</p>","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"1 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Smooth local rigidity for hyperbolic toral automorphisms\",\"authors\":\"B. Kalinin, V. Sadovskaya, Zhenqi Wang\",\"doi\":\"10.1090/cams/22\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We study the regularity of a conjugacy <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper H\\\">\\n <mml:semantics>\\n <mml:mi>H</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">H</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> between a hyperbolic toral automorphism <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper A\\\">\\n <mml:semantics>\\n <mml:mi>A</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">A</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and its smooth perturbation <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"f\\\">\\n <mml:semantics>\\n <mml:mi>f</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">f</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. We show that if <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper H\\\">\\n <mml:semantics>\\n <mml:mi>H</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">H</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is weakly differentiable then it is <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper C Superscript 1 plus upper H reverse-solidus quotation-mark older\\\">\\n <mml:semantics>\\n <mml:msup>\\n <mml:mi>C</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mn>1</mml:mn>\\n <mml:mo>+</mml:mo>\\n <mml:mtext>H\\\\\\\"older</mml:mtext>\\n </mml:mrow>\\n </mml:msup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">C^{1+\\\\text {H\\\\\\\"older}}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and, if <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper A\\\">\\n <mml:semantics>\\n <mml:mi>A</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">A</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is also weakly irreducible, then <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper H\\\">\\n <mml:semantics>\\n <mml:mi>H</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">H</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper C Superscript normal infinity\\\">\\n <mml:semantics>\\n <mml:msup>\\n <mml:mi>C</mml:mi>\\n <mml:mi mathvariant=\\\"normal\\\">∞<!-- ∞ --></mml:mi>\\n </mml:msup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">C^\\\\infty</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. As a part of the proof, we establish results of independent interest on Hölder continuity of a measurable conjugacy between linear cocycles over a hyperbolic system. As a corollary, we improve regularity of the conjugacy to <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper C Superscript normal infinity\\\">\\n <mml:semantics>\\n <mml:msup>\\n <mml:mi>C</mml:mi>\\n <mml:mi mathvariant=\\\"normal\\\">∞<!-- ∞ --></mml:mi>\\n </mml:msup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">C^\\\\infty</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> in prior local rigidity results.</p>\",\"PeriodicalId\":285678,\"journal\":{\"name\":\"Communications of the American Mathematical Society\",\"volume\":\"1 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-11-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications of the American Mathematical Society\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/cams/22\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications of the American Mathematical Society","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/cams/22","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Smooth local rigidity for hyperbolic toral automorphisms
We study the regularity of a conjugacy HH between a hyperbolic toral automorphism AA and its smooth perturbation ff. We show that if HH is weakly differentiable then it is C1+H\"olderC^{1+\text {H\"older}} and, if AA is also weakly irreducible, then HH is C∞C^\infty. As a part of the proof, we establish results of independent interest on Hölder continuity of a measurable conjugacy between linear cocycles over a hyperbolic system. As a corollary, we improve regularity of the conjugacy to C∞C^\infty in prior local rigidity results.
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