This works investigates the Lyapunov–Oseledets spectrum of transfer operator cocycles associated to one-dimensional random paired tent maps depending on a parameter εvarepsilon, quantifying the strength of the leakage between two nearly invariant regions. We show that the system exhibits metastability, and identify the second Lyapunov exponent λ2εlambda _2^varepsilon within an error of order ε2|logε|varepsilon ^2|log varepsilon |. This approximation agrees with the naive prediction provided by a time-dependent two-state Markov chain. Furthermore, it is shown that λ1ε=0lambda _1^varepsilon =0 and λ2εlambda _2^varepsilon are
We answer positively a question of Ryzhikov, namely we show that being a relatively weakly mixing extension is a comeager property in the Polish group of measure preserving transformations. We study some related classes of ergodic transformations and their interrelations. In the second part of the paper we show that for a fixed ergodic TT with property Amathbf {A}, a generic extension T^widehat {T} of TT also has property Amathbf {A}. Here Amathbf {A} stands for each of the following properties: (i) having the same entropy as TT, (ii) Bernoulli, (iii) K, and (iv) loosely Bernoulli.
{"title":"On some generic classes of ergodic measure preserving transformations","authors":"E. Glasner, J. Thouvenot, B. Weiss","doi":"10.1090/mosc/312","DOIUrl":"https://doi.org/10.1090/mosc/312","url":null,"abstract":"<p>We answer positively a question of Ryzhikov, namely we show that being a relatively weakly mixing extension is a comeager property in the Polish group of measure preserving transformations. We study some related classes of ergodic transformations and their interrelations. In the second part of the paper we show that for a fixed ergodic <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T\">\u0000 <mml:semantics>\u0000 <mml:mi>T</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">T</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> with property <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold upper A\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"bold\">A</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbf {A}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, a generic extension <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"ModifyingAbove upper T With caret\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mi>T</mml:mi>\u0000 <mml:mo>^<!-- ^ --></mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">widehat {T}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T\">\u0000 <mml:semantics>\u0000 <mml:mi>T</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">T</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> also has property <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold upper A\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"bold\">A</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbf {A}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Here <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold upper A\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"bold\">A</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbf {A}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> stands for each of the following properties: (i) having the same entropy as <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T\">\u0000 <mml:semantics>\u0000 <mml:mi>T</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">T</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, (ii) Bernoulli, (iii) K, and (iv) loosely Bernoulli.</p>","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48419084","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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 X
设G G是一个无限可数离散服从群。对于紧度量空间（X，ρ）（X，rho）上的任何G-作用，证明了如果作用具有正拓扑熵，则对于G G中具有成对不同元素的任何序列｛s i｝i＝1+∞｛s_ i｝_，对于任意两个不同的点x，y∈Kx，y在K中，有[lim 苏皮→ + ∞ ρ（s i x，s i y）>0和lim infi→ + ∞ ρ（s i x，s i y）=0。limsup _｛i to+infty｝rho（s_i x，s_iy
{"title":"Positive entropy implies chaos along any infinite sequence","authors":"Wen Huang, Jian Li, X. Ye","doi":"10.1090/mosc/315","DOIUrl":"https://doi.org/10.1090/mosc/315","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be an infinite countable discrete amenable group. For any <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-action on a compact metric space <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper X comma rho right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>ρ<!-- ρ --></mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(X,rho )</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, it turns out that if the action has positive topological entropy, then for any sequence <inline-formula content-type=\"math/mathml\">\u0000<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\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>s</mml:mi>\u0000 <mml:mi>i</mml:mi>\u0000 </mml:msub>\u0000 <mml:msubsup>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>i</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:mrow>\u0000 </mml:msubsup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">{s_i}_{i=1}^{+infty }</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> with pairwise distinct elements in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> there exists a Cantor subset <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation en","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48051302","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Combining intuitive probabilistic assumptions with the basic laws of classical thermodynamics, using the latter to express probabilistic parameters in terms of the thermodynamic quantities, we get a simple unified derivation of the fundamental ensembles of statistical physics avoiding any limiting procedures, quantum hypothesis and even statistical entropy maximization. This point of view also leads to some related classes of correlated particle statistics.
{"title":"On a probabilistic derivation of the basic particle statistics (Bose–Einstein, Fermi–Dirac, canonical, grand-canonical, intermediate) and related distributions","authors":"V. Kolokoltsov","doi":"10.1090/mosc/316","DOIUrl":"https://doi.org/10.1090/mosc/316","url":null,"abstract":"Combining intuitive probabilistic assumptions with the basic laws of classical thermodynamics, using the latter to express probabilistic parameters in terms of the thermodynamic quantities, we get a simple unified derivation of the fundamental ensembles of statistical physics avoiding any limiting procedures, quantum hypothesis and even statistical entropy maximization. This point of view also leads to some related classes of correlated particle statistics.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44770129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Spectral Properties of Differential Operators with Oscillating Coefficients","authors":"N. Valeev, Ya. T. Sultanaev, É. A. Nazirova","doi":"10.1090/mosc/299","DOIUrl":"https://doi.org/10.1090/mosc/299","url":null,"abstract":"","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46421482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The finiteness of the spectrum of boundary value problems defined on a geometric graph","authors":"V. Sadovnichii, Ya. T. Sultanaev, A. Akhtyamov","doi":"10.1090/mosc/293","DOIUrl":"https://doi.org/10.1090/mosc/293","url":null,"abstract":"","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42784634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Finite-dimensional approximations to the Poincaré–Steklov operator for general elliptic boundary value problems in domains with cylindrical and periodic exits to infinity","authors":"S. Nazarov","doi":"10.1090/mosc/290","DOIUrl":"https://doi.org/10.1090/mosc/290","url":null,"abstract":"","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49304375","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the existence of an operator group generated by the one-dimensional Dirac system","authors":"A. Savchuk, I. Sadovnichaya","doi":"10.1090/mosc/297","DOIUrl":"https://doi.org/10.1090/mosc/297","url":null,"abstract":"","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/mosc/297","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45922805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Andrei Andreevich Shkalikov (on his seventieth birthday)","authors":"","doi":"10.1090/mosc/300","DOIUrl":"https://doi.org/10.1090/mosc/300","url":null,"abstract":"","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41796801","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Spectral analysis and representation of solutions of integro-differential equations with fractional exponential kernels","authors":"V. V. Vlasov, N. Rautian","doi":"10.1090/mosc/298","DOIUrl":"https://doi.org/10.1090/mosc/298","url":null,"abstract":"","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/mosc/298","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47999291","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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