在极值附近多次混合的违例

Q2 Mathematics Transactions of the Moscow Mathematical Society Pub Date : 2022-03-15 DOI:10.1090/mosc/322

S. Tikhonov

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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? 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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? 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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。

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A violation of multiple mixing close to an extremal

Given a mixing action L L of a group G G and a set A A of half measure we consider the possible limits of the measures μ ( A ∩ L m i A ∩ L n i A ) \mu (A\cap L^{m_{i}}A\cap L^{n_{i}}A) as i → ∞ i\to \infty and m i , n i , m i − n i → ∞ m_{i},n_{i},m_{i}-n_{i}\to \infty . If the action is 3-mixing, then these limits are always equal to 1 / 8 1/8 . 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.

Bibliography: 11 titles.

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