Ergodic Theory and Dynamical Systems最新文献

{"title":"Measure transfer and S-adic developments for subshifts","authors":"NICOLAS BÉDARIDE, ARNAUD HILION, MARTIN LUSTIG","doi":"10.1017/etds.2024.19","DOIUrl":"https://doi.org/10.1017/etds.2024.19","url":null,"abstract":"<p>Based on previous work of the authors, to any <span>S</span>-adic development of a subshift <span>X</span> a ‘directive sequence’ of commutative diagrams is associated, which consists at every level <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240309085744099-0060:S0143385724000191:S0143385724000191_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$n geq 0$</span></span></img></span></span> of the measure cone and the letter frequency cone of the level subshift <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240309085744099-0060:S0143385724000191:S0143385724000191_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$X_n$</span></span></img></span></span> associated canonically to the given <span>S</span>-adic development. The issuing rich picture enables one to deduce results about <span>X</span> with unexpected directness. For instance, we exhibit a large class of minimal subshifts with entropy zero that all have infinitely many ergodic probability measures. As a side result, we also exhibit, for any integer <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240309085744099-0060:S0143385724000191:S0143385724000191_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$d geq 2$</span></span></img></span></span>, an <span>S</span>-adic development of a minimal, aperiodic, uniquely ergodic subshift <span>X</span>, where all level alphabets <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240309085744099-0060:S0143385724000191:S0143385724000191_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal A_n$</span></span></img></span></span> have cardinality <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240309085744099-0060:S0143385724000191:S0143385724000191_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$d,$</span></span></img></span></span> while none of the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240309085744099-0060:S0143385724000191:S0143385724000191_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$d-2$</span></span></img></span></span> bottom level morphisms is recognizable in its level subshift <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240309085744099-0060:S0143385724000191:S0143385724000191_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$X_n subseteq mathcal A_n^{mathbb {Z}}$</span></span></img></span></span>.</p>","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140099758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Invariant measures for -free systems revisited","authors":"AURELIA DYMEK, JOANNA KUŁAGA-PRZYMUS, DANIEL SELL","doi":"10.1017/etds.2024.7","DOIUrl":"https://doi.org/10.1017/etds.2024.7","url":null,"abstract":"For <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000075_inline2.png\" /> <jats:tex-math> $mathscr {B} subseteq mathbb {N} $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000075_inline3.png\" /> <jats:tex-math> $ mathscr {B} $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-free subshift <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000075_inline4.png\" /> <jats:tex-math> $ X_{eta } $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is the orbit closure of the characteristic function of the set of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000075_inline5.png\" /> <jats:tex-math> $ mathscr {B} $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-free integers. We show that many results about invariant measures and entropy, previously only known for the hereditary closure of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000075_inline6.png\" /> <jats:tex-math> $ X_{eta } $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, have their analogues for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000075_inline7.png\" /> <jats:tex-math> $ X_{eta } $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> as well. In particular, we settle in the affirmative a conjecture of Keller about a description of such measures [G. Keller. Generalized heredity in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000075_inline8.png\" /> <jats:tex-math> $mathcal B$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-free systems. <jats:italic>Stoch. Dyn.</jats:italic>21(3) (2021), Paper No. 2140008]. A central assumption in our work is that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000075_inline9.png\" /> <jats:tex-math> $eta ^{*} $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> (the Toeplitz sequence that generates the unique minimal component of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000075_inline10.png\" /> <jats:tex-math> $ X_{eta } $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>) is regular. From this, we obtai","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140075464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Non-integrability of the restricted three-body problem","authors":"KAZUYUKI YAGASAKI","doi":"10.1017/etds.2024.4","DOIUrl":"https://doi.org/10.1017/etds.2024.4","url":null,"abstract":"<p>The problem of non-integrability of the circular restricted three-body problem is very classical and important in the theory of dynamical systems. It was partially solved by Poincaré in the nineteenth century: he showed that there exists no real-analytic first integral which depends analytically on the mass ratio of the second body to the total and is functionally independent of the Hamiltonian. When the mass of the second body becomes zero, the restricted three-body problem reduces to the two-body Kepler problem. We prove the non-integrability of the restricted three-body problem both in the planar and spatial cases for any non-zero mass of the second body. Our basic tool of the proofs is a technique developed here for determining whether perturbations of integrable systems which may be non-Hamiltonian are not meromorphically integrable near resonant periodic orbits such that the first integrals and commutative vector fields also depend meromorphically on the perturbation parameter. The technique is based on generalized versions due to Ayoul and Zung of the Morales–Ramis and Morales–Ramis–Simó theories. We emphasize that our results are not just applications of the theories.</p>","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140044204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bohr chaoticity of principal algebraic actions and Riesz product measures","authors":"AI HUA FAN, KLAUS SCHMIDT, EVGENY VERBITSKIY","doi":"10.1017/etds.2024.13","DOIUrl":"https://doi.org/10.1017/etds.2024.13","url":null,"abstract":"<p>For a continuous <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305151712838-0085:S0143385724000130:S0143385724000130_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {N}^d$</span></span></img></span></span> or <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305151712838-0085:S0143385724000130:S0143385724000130_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {Z}^d$</span></span></img></span></span> action on a compact space, we introduce the notion of Bohr chaoticity, which is an invariant of topological conjugacy and which is proved stronger than having positive entropy. We prove that all principal algebraic <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305151712838-0085:S0143385724000130:S0143385724000130_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {Z}$</span></span></img></span></span> actions of positive entropy are Bohr chaotic. The same is proved for principal algebraic actions of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305151712838-0085:S0143385724000130:S0143385724000130_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {Z}^d$</span></span></img></span></span> with positive entropy under the condition of existence of summable homoclinic points.</p>","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140044286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"ETS volume 44 issue 4 Cover and Front matter","authors":"","doi":"10.1017/etds.2023.83","DOIUrl":"https://doi.org/10.1017/etds.2023.83","url":null,"abstract":"","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140265367","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"ETS volume 44 issue 4 Cover and Back matter","authors":"","doi":"10.1017/etds.2023.84","DOIUrl":"https://doi.org/10.1017/etds.2023.84","url":null,"abstract":"","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140079291","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Tracial weights on topological graph algebras","authors":"JOHANNES CHRISTENSEN","doi":"10.1017/etds.2024.20","DOIUrl":"https://doi.org/10.1017/etds.2024.20","url":null,"abstract":"<p>We describe two kinds of regular invariant measures on the boundary path space <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304092108222-0628:S0143385724000208:S0143385724000208_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$partial E$</span></span></img></span></span> of a second countable topological graph <span>E</span>, which allows us to describe all extremal tracial weights on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304092108222-0628:S0143385724000208:S0143385724000208_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$C^{*}(E)$</span></span></img></span></span> which are not gauge-invariant. Using this description, we prove that all tracial weights on the C<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304092108222-0628:S0143385724000208:S0143385724000208_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$^{*}$</span></span></img></span></span>-algebra <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304092108222-0628:S0143385724000208:S0143385724000208_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$C^{*}(E)$</span></span></img></span></span> of a second countable topological graph <span>E</span> are gauge-invariant when <span>E</span> is free. This in particular implies that all tracial weights on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304092108222-0628:S0143385724000208:S0143385724000208_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$C^{*}(E)$</span></span></img></span></span> are gauge-invariant when <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304092108222-0628:S0143385724000208:S0143385724000208_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$C^{*}(E)$</span></span></img></span></span> is simple and separable.</p>","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034050","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Patterson–Sullivan theory for groups with a strongly contracting element","authors":"RÉMI COULON","doi":"10.1017/etds.2024.10","DOIUrl":"https://doi.org/10.1017/etds.2024.10","url":null,"abstract":"<p>Using Patterson–Sullivan measures, we investigate growth problems for groups acting on a metric space with a strongly contracting element.</p>","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034053","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An embedding theorem for subshifts over amenable groups with the comparison property","authors":"ROBERT BLAND","doi":"10.1017/etds.2024.21","DOIUrl":"https://doi.org/10.1017/etds.2024.21","url":null,"abstract":"<p>We obtain the following embedding theorem for symbolic dynamical systems. Let <span>G</span> be a countable amenable group with the comparison property. Let <span>X</span> be a strongly aperiodic subshift over <span>G</span>. Let <span>Y</span> be a strongly irreducible shift of finite type over <span>G</span> that has no global period, meaning that the shift action is faithful on <span>Y</span>. If the topological entropy of <span>X</span> is strictly less than that of <span>Y</span> and <span>Y</span> contains at least one factor of <span>X</span>, then <span>X</span> embeds into <span>Y</span>. This result partially extends the classical result of Krieger when <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304130334345-0052:S014338572400021X:S014338572400021X_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$G = mathbb {Z}$</span></span></img></span></span> and the results of Lightwood when <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304130334345-0052:S014338572400021X:S014338572400021X_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$G = mathbb {Z}^d$</span></span></img></span></span> for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304130334345-0052:S014338572400021X:S014338572400021X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$d geq 2$</span></span></img></span></span>. The proof relies on recent developments in the theory of tilings and quasi-tilings of amenable groups.</p>","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034056","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Invariant measures of Toeplitz subshifts on non-amenable groups","authors":"PAULINA CECCHI BERNALES, MARÍA ISABEL CORTEZ, JAIME GÓMEZ","doi":"10.1017/etds.2024.16","DOIUrl":"https://doi.org/10.1017/etds.2024.16","url":null,"abstract":"<p>Let <span>G</span> be a countable residually finite group (for instance, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${mathbb F}_2$</span></span></img></span></span>) and let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$overleftarrow {G}$</span></span></img></span></span> be a totally disconnected metric compactification of <span>G</span> equipped with the action of <span>G</span> by left multiplication. For every <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$rgeq 1$</span></span></img></span></span>, we construct a Toeplitz <span>G</span>-subshift <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$(X,sigma ,G)$</span></span></img></span></span>, which is an almost one-to-one extension of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$overleftarrow {G}$</span></span></img></span></span>, having <span>r</span> ergodic measures <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$nu _1, ldots ,nu _r$</span></span></img></span></span> such that for every <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$1leq ileq r$</span></span></img></span></span>, the measure-theoretic dynamical system <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$(X,sigma ,G,nu _i)$</span></span></img></span></span> is isomorphic to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}