{"title":"Positive entropy implies chaos along any infinite sequence","authors":"Wen Huang, Jian Li, X. Ye","doi":"10.1090/mosc/315","DOIUrl":null,"url":null,"abstract":"<p>Let <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> be an infinite countable discrete amenable group. For any <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>-action on a compact metric space <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper X comma rho right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>X</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>ρ<!-- ρ --></mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">(X,\\rho )</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, it turns out that if the action has positive topological entropy, then for any sequence <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-brace s Subscript i Baseline right-brace Subscript i equals 1 Superscript plus normal infinity\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\n <mml:msub>\n <mml:mi>s</mml:mi>\n <mml:mi>i</mml:mi>\n </mml:msub>\n <mml:msubsup>\n <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>i</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>+</mml:mo>\n <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\n </mml:mrow>\n </mml:msubsup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\{s_i\\}_{i=1}^{+\\infty }</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> with pairwise distinct elements in <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> there exists a Cantor subset <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\n <mml:semantics>\n <mml:mi>K</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\n <mml:semantics>\n <mml:mi>X</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> which is Li–Yorke chaotic along this sequence, that is, for any two distinct points <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"x comma y element-of upper K\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>y</mml:mi>\n <mml:mo>∈<!-- ∈ --></mml:mo>\n <mml:mi>K</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">x,y\\in K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, one has <disp-formula content-type=\"math/mathml\">\n\\[\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"limit sup rho left-parenthesis s Subscript i Baseline x comma s Subscript i Baseline y right-parenthesis greater-than 0 and limit inf rho left-parenthesis s Subscript i Baseline x comma s Subscript i Baseline y right-parenthesis equals 0 period\">\n <mml:semantics>\n <mml:mrow>\n <mml:munder>\n <mml:mo movablelimits=\"true\" form=\"prefix\">lim sup</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>i</mml:mi>\n <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\n <mml:mo>+</mml:mo>\n <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\n </mml:mrow>\n </mml:munder>\n <mml:mi>ρ<!-- ρ --></mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mi>s</mml:mi>\n <mml:mi>i</mml:mi>\n </mml:msub>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>s</mml:mi>\n <mml:mi>i</mml:mi>\n </mml:msub>\n <mml:mi>y</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>></mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mtext> </mml:mtext>\n <mml:mtext>and</mml:mtext>\n <mml:mtext> </mml:mtext>\n <mml:munder>\n <mml:mo movablelimits=\"true\" form=\"prefix\">lim inf</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>i</mml:mi>\n <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\n <mml:mo>+</mml:mo>\n <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\n </mml:mrow>\n </mml:munder>\n <mml:mi>ρ<!-- ρ --></mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mi>s</mml:mi>\n <mml:mi>i</mml:mi>\n </mml:msub>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>s</mml:mi>\n <mml:mi>i</mml:mi>\n </mml:msub>\n <mml:mi>y</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>=</mml:mo>\n <mml:mn>0.</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\limsup _{i\\to +\\infty }\\rho (s_i x,s_iy)>0\\ \\text {and}\\ \\liminf _{i\\to +\\infty }\\rho (s_ix,s_iy)=0.</mml:annotation>\n </mml:semantics>\n</mml:math>\n\\]\n</disp-formula></p>","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2020-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the Moscow Mathematical Society","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/mosc/315","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: 4
Abstract
Let GG be an infinite countable discrete amenable group. For any GG-action on a compact metric space (X,ρ)(X,\rho ), it turns out that if the action has positive topological entropy, then for any sequence {si}i=1+∞\{s_i\}_{i=1}^{+\infty } with pairwise distinct elements in GG there exists a Cantor subset KK of XX which is Li–Yorke chaotic along this sequence, that is, for any two distinct points x,y∈Kx,y\in K, one has
\[
lim supi→+∞ρ(six,siy)>0andlim infi→+∞ρ(six,siy)=0.\limsup _{i\to +\infty }\rho (s_i x,s_iy)>0\ \text {and}\ \liminf _{i\to +\infty }\rho (s_ix,s_iy)=0.
\]