{"title":"在极值附近多次混合的违例","authors":"S. Tikhonov","doi":"10.1090/mosc/322","DOIUrl":null,"url":null,"abstract":"<p>Given a mixing action <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\">\n <mml:semantics>\n <mml:mi>L</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">L</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of a group <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and a set <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> of half measure we consider the possible limits of the measures <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu left-parenthesis upper A intersection upper L Superscript m Super Subscript i Superscript Baseline upper A intersection upper L Superscript n Super Subscript i Superscript Baseline upper A right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>μ<!-- μ --></mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>A</mml:mi>\n <mml:mo>∩<!-- ∩ --></mml:mo>\n <mml:msup>\n <mml:mi>L</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:msub>\n <mml:mi>m</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>i</mml:mi>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n </mml:msup>\n <mml:mi>A</mml:mi>\n <mml:mo>∩<!-- ∩ --></mml:mo>\n <mml:msup>\n <mml:mi>L</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:msub>\n <mml:mi>n</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>i</mml:mi>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n </mml:msup>\n <mml:mi>A</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mu (A\\cap L^{m_{i}}A\\cap L^{n_{i}}A)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> as <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"i right-arrow normal infinity\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>i</mml:mi>\n <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\n <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">i\\to \\infty</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m Subscript i Baseline comma n Subscript i Baseline comma m Subscript i Baseline minus n Subscript i Baseline right-arrow normal infinity\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>m</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>i</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>n</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>i</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>m</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>i</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:msub>\n <mml:mi>n</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>i</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\n <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">m_{i},n_{i},m_{i}-n_{i}\\to \\infty</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. If the action is 3-mixing, then these limits are always equal to <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1 slash 8\">\n <mml:semantics>\n <mml:mrow>\n <mml:mn>1</mml:mn>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mn>8</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">1/8</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. In the Ledrappier example, this limit is zero for some sequences. The following question is studied: what can be said about actions if one of these limits is positive but small? In the paper we make several observations on this topic.</p>\n\n<p><italic>Bibliography</italic>: 11 titles.</p>","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"A violation of multiple mixing close to an extremal\",\"authors\":\"S. Tikhonov\",\"doi\":\"10.1090/mosc/322\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Given a mixing action <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L\\\">\\n <mml:semantics>\\n <mml:mi>L</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">L</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> of a group <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper G\\\">\\n <mml:semantics>\\n <mml:mi>G</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">G</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and a set <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> of half measure we consider the possible limits of the measures <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"mu left-parenthesis upper A intersection upper L Superscript m Super Subscript i Superscript Baseline upper A intersection upper L Superscript n Super Subscript i Superscript Baseline upper A right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>μ<!-- μ --></mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>A</mml:mi>\\n <mml:mo>∩<!-- ∩ --></mml:mo>\\n <mml:msup>\\n <mml:mi>L</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:msub>\\n <mml:mi>m</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>i</mml:mi>\\n </mml:mrow>\\n </mml:msub>\\n </mml:mrow>\\n </mml:msup>\\n <mml:mi>A</mml:mi>\\n <mml:mo>∩<!-- ∩ --></mml:mo>\\n <mml:msup>\\n <mml:mi>L</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:msub>\\n <mml:mi>n</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>i</mml:mi>\\n </mml:mrow>\\n </mml:msub>\\n </mml:mrow>\\n </mml:msup>\\n <mml:mi>A</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mu (A\\\\cap L^{m_{i}}A\\\\cap L^{n_{i}}A)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> as <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"i right-arrow normal infinity\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>i</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">→<!-- → --></mml:mo>\\n <mml:mi mathvariant=\\\"normal\\\">∞<!-- ∞ --></mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">i\\\\to \\\\infty</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"m Subscript i Baseline comma n Subscript i Baseline comma m Subscript i Baseline minus n Subscript i Baseline right-arrow normal infinity\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mi>m</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>i</mml:mi>\\n </mml:mrow>\\n </mml:msub>\\n <mml:mo>,</mml:mo>\\n <mml:msub>\\n <mml:mi>n</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>i</mml:mi>\\n </mml:mrow>\\n </mml:msub>\\n <mml:mo>,</mml:mo>\\n <mml:msub>\\n <mml:mi>m</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>i</mml:mi>\\n </mml:mrow>\\n </mml:msub>\\n <mml:mo>−<!-- − --></mml:mo>\\n <mml:msub>\\n <mml:mi>n</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>i</mml:mi>\\n </mml:mrow>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">→<!-- → --></mml:mo>\\n <mml:mi mathvariant=\\\"normal\\\">∞<!-- ∞ --></mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">m_{i},n_{i},m_{i}-n_{i}\\\\to \\\\infty</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. If the action is 3-mixing, then these limits are always equal to <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"1 slash 8\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mn>1</mml:mn>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>/</mml:mo>\\n </mml:mrow>\\n <mml:mn>8</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">1/8</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. In the Ledrappier example, this limit is zero for some sequences. The following question is studied: what can be said about actions if one of these limits is positive but small? In the paper we make several observations on this topic.</p>\\n\\n<p><italic>Bibliography</italic>: 11 titles.</p>\",\"PeriodicalId\":37924,\"journal\":{\"name\":\"Transactions of the Moscow Mathematical Society\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-03-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transactions of the Moscow Mathematical Society\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/mosc/322\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the Moscow Mathematical Society","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/mosc/322","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 1
摘要
给定群G G和半测度集合a a的混合作用L L，我们考虑测度μ (a∩L mi a∩L n i a) \mu (a \cap L^{m＿iA{}}\cap)的可能极限L^{n＿iA{)}}当i→∞i \to\infty和m i,n i,m i-n i→∞m i,n i,m i-n i {}{}{}{}\to\infty。如果动作是3混合，那么这些限制总是等于1/8 1/8。在Ledrappier的例子中，这个极限对于某些序列是零。研究了以下问题:如果这些限制中的一个是正的但很小，那么对行动可以说什么?在本文中，我们对这个话题做了一些观察。参考书目:11篇。
A violation of multiple mixing close to an extremal
Given a mixing action LL of a group GG and a set AA of half measure we consider the possible limits of the measures μ(A∩LmiA∩LniA)\mu (A\cap L^{m_{i}}A\cap L^{n_{i}}A) as i→∞i\to \infty and mi,ni,mi−ni→∞m_{i},n_{i},m_{i}-n_{i}\to \infty. If the action is 3-mixing, then these limits are always equal to 1/81/8. In the Ledrappier example, this limit is zero for some sequences. The following question is studied: what can be said about actions if one of these limits is positive but small? In the paper we make several observations on this topic.
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