Ergodic Theory and Dynamical Systems最新文献

{"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":"24 1","pages":""},"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":"&lt;p&gt;Let &lt;span&gt;G&lt;/span&gt; be a countable residually finite group (for instance, &lt;span&gt;&lt;span&gt;&lt;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\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;${mathbb F}_2$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;) and let &lt;span&gt;&lt;span&gt;&lt;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\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$overleftarrow {G}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; be a totally disconnected metric compactification of &lt;span&gt;G&lt;/span&gt; equipped with the action of &lt;span&gt;G&lt;/span&gt; by left multiplication. For every &lt;span&gt;&lt;span&gt;&lt;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\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$rgeq 1$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;, we construct a Toeplitz &lt;span&gt;G&lt;/span&gt;-subshift &lt;span&gt;&lt;span&gt;&lt;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\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$(X,sigma ,G)$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;, which is an almost one-to-one extension of &lt;span&gt;&lt;span&gt;&lt;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\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$overleftarrow {G}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;, having &lt;span&gt;r&lt;/span&gt; ergodic measures &lt;span&gt;&lt;span&gt;&lt;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\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$nu _1, ldots ,nu _r$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; such that for every &lt;span&gt;&lt;span&gt;&lt;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\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$1leq ileq r$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;, the measure-theoretic dynamical system &lt;span&gt;&lt;span&gt;&lt;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\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$(X,sigma ,G,nu _i)$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; is isomorphic to &lt;span&gt;&lt;span&gt;&lt;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":"13 1","pages":""},"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}

{"title":"Shifts of finite type on locally finite groups","authors":"JADE RAYMOND","doi":"10.1017/etds.2024.14","DOIUrl":"https://doi.org/10.1017/etds.2024.14","url":null,"abstract":"In this work we prove that every shift of finite type (SFT), sofic shift, and strongly irreducible shift on locally finite groups has strong dynamical properties. These properties include that every sofic shift is an SFT, every SFT is strongly irreducible, every strongly irreducible shift is an SFT, every SFT is entropy minimal, and every SFT has a unique measure of maximal entropy, among others. In addition, we show that if every SFT on a group is strongly irreducible, or if every sofic shift is an SFT, then the group must be locally finite, and this extends to all of the properties we explore. These results are collected in two main theorems which characterize the local finiteness of groups by purely dynamical properties. In pursuit of these results, we present a formal construction of <jats:italic>free extension</jats:italic> shifts on a group <jats:italic>G</jats:italic>, which takes a shift on a subgroup <jats:italic>H</jats:italic> of <jats:italic>G</jats:italic>, and naturally extends it to a shift on all of <jats:italic>G</jats:italic>.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979059","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":"On the stochastic bifurcations regarding random iterations of polynomials of the form","authors":"TAKAYUKI WATANABE","doi":"10.1017/etds.2024.17","DOIUrl":"https://doi.org/10.1017/etds.2024.17","url":null,"abstract":"In this paper, we consider random iterations of polynomial maps <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000178_inline2.png\" /> <jats:tex-math> $z^{2} + c_{n}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000178_inline3.png\" /> <jats:tex-math> $c_{n}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are complex-valued independent random variables following the uniform distribution on the closed disk with center <jats:italic>c</jats:italic> and radius <jats:italic>r</jats:italic>. The aim of this paper is twofold. First, we study the (dis)connectedness of random Julia sets. Here, we reveal the relationships between the bifurcation radius and connectedness of random Julia sets. Second, we investigate the bifurcation of our random iterations and give quantitative estimates of bifurcation parameters. In particular, we prove that for the central parameter <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000178_inline4.png\" /> <jats:tex-math> $c = -1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, almost every random Julia set is totally disconnected with much smaller radial parameters <jats:italic>r</jats:italic> than expected. We also introduce several open questions worth discussing.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139978846","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":"Upper, down, two-sided Lorenz attractor, collisions, merging, and switching","authors":"DIEGO BARROS, CHRISTIAN BONATTI, MARIA JOSÉ PACIFICO","doi":"10.1017/etds.2024.8","DOIUrl":"https://doi.org/10.1017/etds.2024.8","url":null,"abstract":"We present a modified version of the well-known geometric Lorenz attractor. It consists of a &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline1.png\" /&gt; &lt;jats:tex-math&gt; $C^1$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; open set &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline2.png\" /&gt; &lt;jats:tex-math&gt; ${mathcal O}$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; of vector fields in &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline3.png\" /&gt; &lt;jats:tex-math&gt; ${mathbb R}^3$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; having an attracting region &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline4.png\" /&gt; &lt;jats:tex-math&gt; ${mathcal U}$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; satisfying three properties. Namely, a unique singularity &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline5.png\" /&gt; &lt;jats:tex-math&gt; $sigma $ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;; a unique attractor &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline6.png\" /&gt; &lt;jats:tex-math&gt; $Lambda $ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; including the singular point and the maximal invariant in &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline7.png\" /&gt; &lt;jats:tex-math&gt; ${mathcal U}$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; has at most two chain recurrence classes, which are &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline8.png\" /&gt; &lt;jats:tex-math&gt; $Lambda $ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; and (at most) one hyperbolic horseshoe. The horseshoe and the singular attractor have a collision along with the union of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline9.png\" /&gt; &lt;jats:tex-math&gt; $2$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; codimension &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline10.png\" /&gt; &lt;jats:tex-math&gt; $1$ &lt;/jats:t","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"281 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139923275","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":"Measures of maximal entropy of bounded density shifts","authors":"FELIPE GARCÍA-RAMOS, RONNIE PAVLOV, CARLOS REYES","doi":"10.1017/etds.2024.6","DOIUrl":"https://doi.org/10.1017/etds.2024.6","url":null,"abstract":"We find sufficient conditions for bounded density shifts to have a unique measure of maximal entropy. We also prove that every measure of maximal entropy of a bounded density shift is fully supported. As a consequence of this, we obtain that bounded density shifts are surjunctive.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"41 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139923465","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":"On denseness of horospheres in higher rank homogeneous spaces","authors":"OR LANDESBERG, HEE OH","doi":"10.1017/etds.2024.12","DOIUrl":"https://doi.org/10.1017/etds.2024.12","url":null,"abstract":"Let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline1.png\" /&gt; &lt;jats:tex-math&gt; $ G $ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be a connected semisimple real algebraic group and &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline2.png\" /&gt; &lt;jats:tex-math&gt; $Gamma &lt;G$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be a Zariski dense discrete subgroup. Let &lt;jats:italic&gt;N&lt;/jats:italic&gt; denote a maximal horospherical subgroup of &lt;jats:italic&gt;G&lt;/jats:italic&gt;, and &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline3.png\" /&gt; &lt;jats:tex-math&gt; $P=MAN$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; the minimal parabolic subgroup which is the normalizer of &lt;jats:italic&gt;N&lt;/jats:italic&gt;. Let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline4.png\" /&gt; &lt;jats:tex-math&gt; $mathcal E$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; denote the unique &lt;jats:italic&gt;P&lt;/jats:italic&gt;-minimal subset of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline5.png\" /&gt; &lt;jats:tex-math&gt; $Gamma backslash G$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; and let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline6.png\" /&gt; &lt;jats:tex-math&gt; $mathcal E_0$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be a &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline7.png\" /&gt; &lt;jats:tex-math&gt; $P^circ $ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;-minimal subset. We consider a notion of a horospherical limit point in the Furstenberg boundary &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline8.png\" /&gt; &lt;jats:tex-math&gt; $ G/P $ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; and show that the following are equivalent for any &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline9.png\" /&gt; &lt;jats:tex-math&gt; $[g]in mathcal E_0$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;: &lt;jats:list list-type=\"number\"&gt; &lt;jats:list-item&gt; &lt;jats:label&gt;(1)&lt;/jats:label&gt; &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-gra","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139909838","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":"Sufficient conditions for non-zero entropy of closed relations","authors":"IZTOK BANIČ, RENE GRIL ROGINA, JUDY KENNEDY, VAN NALL","doi":"10.1017/etds.2024.11","DOIUrl":"https://doi.org/10.1017/etds.2024.11","url":null,"abstract":"We introduce the notions of returns and well-aligned sets for closed relations on compact metric spaces and then use them to obtain non-trivial sufficient conditions for such a relation to have non-zero entropy. In addition, we give a characterization of finite relations with non-zero entropy in terms of Li–Yorke and DC2 chaos.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"5 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139766847","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 sets and nilpotency of endomorphisms of algebraic sofic shifts","authors":"TULLIO CECCHERINI-SILBERSTEIN, MICHEL COORNAERT, XUAN KIEN PHUNG","doi":"10.1017/etds.2023.120","DOIUrl":"https://doi.org/10.1017/etds.2023.120","url":null,"abstract":"Let <jats:italic>G</jats:italic> be a group and let <jats:italic>V</jats:italic> be an algebraic variety over an algebraically closed field <jats:italic>K</jats:italic>. Let <jats:italic>A</jats:italic> denote the set of <jats:italic>K</jats:italic>-points of <jats:italic>V</jats:italic>. We introduce algebraic sofic subshifts <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001207_inline1.png\" /> <jats:tex-math> ${Sigma subset A^G}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and study endomorphisms <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001207_inline2.png\" /> <jats:tex-math> $tau colon Sigma to Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We generalize several results for dynamical invariant sets and nilpotency of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001207_inline3.png\" /> <jats:tex-math> $tau $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001207_inline4.png\" /> <jats:tex-math> $tau $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is nilpotent if and only if its limit set, that is, the intersection of the images of its iterates, is a singleton. If moreover <jats:italic>G</jats:italic> is infinite, finitely generated and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001207_inline5.png\" /> <jats:tex-math> $Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is topologically mixing, we show that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001207_inline6.png\" /> <jats:tex-math> $tau $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is nilpotent if and only if its limit set consists of periodic configurations and has a finite set of alphabet values.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"313 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139767237","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":"On a self-embedding problem for self-similar sets","authors":"JIAN-CI XIAO","doi":"10.1017/etds.2024.2","DOIUrl":"https://doi.org/10.1017/etds.2024.2","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000026_inline1.png\" /> <jats:tex-math> $Ksubset {mathbb {R}}^d$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a self-similar set generated by an iterated function system <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000026_inline2.png\" /> <jats:tex-math> ${varphi _i}_{i=1}^m$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfying the strong separation condition and let <jats:italic>f</jats:italic> be a contracting similitude with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000026_inline3.png\" /> <jats:tex-math> $f(K)subseteq K$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We show that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000026_inline4.png\" /> <jats:tex-math> $f(K)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is relatively open in <jats:italic>K</jats:italic> if all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000026_inline5.png\" /> <jats:tex-math> $varphi _i$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> share a common contraction ratio and orthogonal part. We also provide a counterexample when the orthogonal parts are allowed to vary. This partially answers a question of Elekes, Keleti and Máthé [<jats:italic>Ergod. Th. &amp; Dynam. Sys.</jats:italic>30 (2010), 399–440]. As a byproduct of our argument, when <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000026_inline6.png\" /> <jats:tex-math> $d=1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:italic>K</jats:italic> admits two homogeneous generating iterated function systems satisfying the strong separation condition but with contraction ratios of opposite signs, we show that <jats:italic>K</jats:italic> is symmetric. This partially answers a question of Feng and Wang [<jats:italic>Adv. Math.</jats:italic>222 (2009), 1964–1981].","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"31 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139767086","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}