Abstract For any primitive substitution whose Perron eigenvalue is a Pisot unit, we construct a domain exchange that is measurably conjugate to the subshift. Additionally, we give a condition for the subshift to be a finite extension of a torus translation. For the particular case of weakly irreducible Pisot substitutions, we show that the subshift is either a finite extension of a torus translation or its eigenvalues are roots of unity. Furthermore, we provide an algorithm to compute eigenvalues of the subshift associated with any primitive pseudo-unimodular substitution.
{"title":"Geometrical representation of subshifts for primitive substitutions","authors":"PAUL MERCAT","doi":"10.1017/etds.2023.101","DOIUrl":"https://doi.org/10.1017/etds.2023.101","url":null,"abstract":"Abstract For any primitive substitution whose Perron eigenvalue is a Pisot unit, we construct a domain exchange that is measurably conjugate to the subshift. Additionally, we give a condition for the subshift to be a finite extension of a torus translation. For the particular case of weakly irreducible Pisot substitutions, we show that the subshift is either a finite extension of a torus translation or its eigenvalues are roots of unity. Furthermore, we provide an algorithm to compute eigenvalues of the subshift associated with any primitive pseudo-unimodular substitution.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135819334","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}
Abstract We show that there is a distortion element in a finitely generated subgroup G of the automorphism group of the full shift, namely an element of infinite order whose word norm grows polylogarithmically. As a corollary, we obtain a lower bound on the entropy dimension of any subshift containing a copy of G , and that a sofic shift’s automorphism group contains a distortion element if and only if the sofic shift is uncountable. We obtain also that groups of Turing machines and the higher-dimensional Brin–Thompson groups $mV$ admit distortion elements; in particular, $2V$ (unlike V ) does not admit a proper action on a CAT $(0)$ cube complex. In each case, the distortion element roughly corresponds to the SMART machine of Cassaigne, Ollinger, and Torres-Avilés [A small minimal aperiodic reversible Turing machine. J. Comput. System Sci. 84 (2017), 288–301].
{"title":"Distortion element in the automorphism group of a full shift","authors":"ANTONIN CALLARD, VILLE SALO","doi":"10.1017/etds.2023.67","DOIUrl":"https://doi.org/10.1017/etds.2023.67","url":null,"abstract":"Abstract We show that there is a distortion element in a finitely generated subgroup G of the automorphism group of the full shift, namely an element of infinite order whose word norm grows polylogarithmically. As a corollary, we obtain a lower bound on the entropy dimension of any subshift containing a copy of G , and that a sofic shift’s automorphism group contains a distortion element if and only if the sofic shift is uncountable. We obtain also that groups of Turing machines and the higher-dimensional Brin–Thompson groups $mV$ admit distortion elements; in particular, $2V$ (unlike V ) does not admit a proper action on a CAT $(0)$ cube complex. In each case, the distortion element roughly corresponds to the SMART machine of Cassaigne, Ollinger, and Torres-Avilés [A small minimal aperiodic reversible Turing machine. J. Comput. System Sci. 84 (2017), 288–301].","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135405358","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}
Abstract We prove $times a times b$ measure rigidity for multiplicatively independent pairs when $ain mathbb {N}$ and $b>1$ is a ‘specified’ real number (the b -expansion of $1$ has a tail or bounded runs of $0$ s) under a positive entropy condition. This is done by proving a mean decay of the Fourier series of the point masses average along $times b$ orbits. We also prove a quantitative version of this decay under stronger conditions on the $times a$ invariant measure. The quantitative version together with the $times b$ invariance of the limit measure is a step toward a general Host-type pointwise equidistribution theorem in which the equidistribution is for Parry measure instead of Lebesgue. We show that finite memory length measures on the a -shift meet the mentioned conditions for mean convergence. Our main proof relies on techniques of Hochman.
{"title":"Some measure rigidity and equidistribution results for <i>β</i>-maps","authors":"NEVO FISHBEIN","doi":"10.1017/etds.2023.75","DOIUrl":"https://doi.org/10.1017/etds.2023.75","url":null,"abstract":"Abstract We prove $times a times b$ measure rigidity for multiplicatively independent pairs when $ain mathbb {N}$ and $b>1$ is a ‘specified’ real number (the b -expansion of $1$ has a tail or bounded runs of $0$ s) under a positive entropy condition. This is done by proving a mean decay of the Fourier series of the point masses average along $times b$ orbits. We also prove a quantitative version of this decay under stronger conditions on the $times a$ invariant measure. The quantitative version together with the $times b$ invariance of the limit measure is a step toward a general Host-type pointwise equidistribution theorem in which the equidistribution is for Parry measure instead of Lebesgue. We show that finite memory length measures on the a -shift meet the mentioned conditions for mean convergence. Our main proof relies on techniques of Hochman.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135411846","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}
Abstract Katok’s special representation theorem states that any free ergodic measure- preserving $mathbb {R}^{d}$ -flow can be realized as a special flow over a $mathbb {Z}^{d}$ -action. It provides a multidimensional generalization of the ‘flow under a function’ construction. We prove the analog of Katok’s theorem in the framework of Borel dynamics and show that, likewise, all free Borel $mathbb {R}^{d}$ -flows emerge from $mathbb {Z}^{d}$ -actions through the special flow construction using bi-Lipschitz cocycles.
{"title":"Katok’s special representation theorem for multidimensional Borel flows","authors":"KONSTANTIN SLUTSKY","doi":"10.1017/etds.2023.62","DOIUrl":"https://doi.org/10.1017/etds.2023.62","url":null,"abstract":"Abstract Katok’s special representation theorem states that any free ergodic measure- preserving $mathbb {R}^{d}$ -flow can be realized as a special flow over a $mathbb {Z}^{d}$ -action. It provides a multidimensional generalization of the ‘flow under a function’ construction. We prove the analog of Katok’s theorem in the framework of Borel dynamics and show that, likewise, all free Borel $mathbb {R}^{d}$ -flows emerge from $mathbb {Z}^{d}$ -actions through the special flow construction using bi-Lipschitz cocycles.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135405217","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}
MIHAJLO CEKIĆ, THIBAULT LEFEUVRE, ANDREI MOROIANU, UWE SEMMELMANN
Abstract Let $(M,g,J)$ be a closed Kähler manifold with negative sectional curvature and complex dimension $m := dim _{mathbb {C}} M geq 2$ . In this article, we study the unitary frame flow , that is, the restriction of the frame flow to the principal $mathrm {U}(m)$ -bundle $F_{mathbb {C}}M$ of unitary frames. We show that if $m geq 6$ is even and $m neq 28$ , there exists $unicode{x3bb} (m) in (0, 1)$ such that if $(M, g)$ has negative $unicode{x3bb} (m)$ -pinched holomorphic sectional curvature, then the unitary frame flow is ergodic and mixing. The constants $unicode{x3bb} (m)$ satisfy $unicode{x3bb} (6) = 0.9330...$ , $lim _{m to +infty } unicode{x3bb} (m) = {11}/{12} = 0.9166...$ , and $m mapsto unicode{x3bb} (m)$ is decreasing. This extends to the even-dimensional case the results of Brin and Gromov [On the ergodicity of frame flows. Invent. Math. 60 (1) (1980), 1–7] who proved ergodicity of the unitary frame flow on negatively curved compact Kähler manifolds of odd complex dimension.
{"title":"On the ergodicity of unitary frame flows on Kähler manifolds","authors":"MIHAJLO CEKIĆ, THIBAULT LEFEUVRE, ANDREI MOROIANU, UWE SEMMELMANN","doi":"10.1017/etds.2023.72","DOIUrl":"https://doi.org/10.1017/etds.2023.72","url":null,"abstract":"Abstract Let $(M,g,J)$ be a closed Kähler manifold with negative sectional curvature and complex dimension $m := dim _{mathbb {C}} M geq 2$ . In this article, we study the unitary frame flow , that is, the restriction of the frame flow to the principal $mathrm {U}(m)$ -bundle $F_{mathbb {C}}M$ of unitary frames. We show that if $m geq 6$ is even and $m neq 28$ , there exists $unicode{x3bb} (m) in (0, 1)$ such that if $(M, g)$ has negative $unicode{x3bb} (m)$ -pinched holomorphic sectional curvature, then the unitary frame flow is ergodic and mixing. The constants $unicode{x3bb} (m)$ satisfy $unicode{x3bb} (6) = 0.9330...$ , $lim _{m to +infty } unicode{x3bb} (m) = {11}/{12} = 0.9166...$ , and $m mapsto unicode{x3bb} (m)$ is decreasing. This extends to the even-dimensional case the results of Brin and Gromov [On the ergodicity of frame flows. Invent. Math. 60 (1) (1980), 1–7] who proved ergodicity of the unitary frame flow on negatively curved compact Kähler manifolds of odd complex dimension.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136113513","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}
Theodore D. Drivas, Alexei A. Mailybaev, Artem Raibekas
Abstract We study a class of ordinary differential equations with a non-Lipschitz point singularity that admits non-unique solutions through this point. As a selection criterion, we introduce stochastic regularizations depending on a parameter $nu $ : the regularized dynamics is globally defined for each $nu> 0$ , and the original singular system is recovered in the limit of vanishing $nu $ . We prove that this limit yields a unique statistical solution independent of regularization when the deterministic system possesses a chaotic attractor having a physical measure with the convergence to equilibrium property. In this case, solutions become spontaneously stochastic after passing through the singularity: they are selected randomly with an intrinsic probability distribution.
{"title":"Statistical determinism in non-Lipschitz dynamical systems","authors":"Theodore D. Drivas, Alexei A. Mailybaev, Artem Raibekas","doi":"10.1017/etds.2023.74","DOIUrl":"https://doi.org/10.1017/etds.2023.74","url":null,"abstract":"Abstract We study a class of ordinary differential equations with a non-Lipschitz point singularity that admits non-unique solutions through this point. As a selection criterion, we introduce stochastic regularizations depending on a parameter $nu $ : the regularized dynamics is globally defined for each $nu> 0$ , and the original singular system is recovered in the limit of vanishing $nu $ . We prove that this limit yields a unique statistical solution independent of regularization when the deterministic system possesses a chaotic attractor having a physical measure with the convergence to equilibrium property. In this case, solutions become spontaneously stochastic after passing through the singularity: they are selected randomly with an intrinsic probability distribution.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136058223","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}
Abstract Let $lin mathbb {N}_{ge 1}$ and $alpha : mathbb {Z}^lrightarrow text {Aut}(mathscr {N})$ be an action of $mathbb {Z}^l$ by automorphisms on a compact nilmanifold $mathscr{N}$ . We assume the action of every $alpha (z)$ is ergodic for $zin mathbb {Z}^lsmallsetminus {0}$ and show that $alpha $ satisfies exponential n -mixing for any integer $ngeq 2$ . This extends the results of Gorodnik and Spatzier [Mixing properties of commuting nilmanifold automorphisms. Acta Math. 215 (2015), 127–159].
{"title":"Exponential multiple mixing for commuting automorphisms of a nilmanifold","authors":"TIMOTHÉE BÉNARD, PÉTER P. VARJÚ","doi":"10.1017/etds.2023.73","DOIUrl":"https://doi.org/10.1017/etds.2023.73","url":null,"abstract":"Abstract Let $lin mathbb {N}_{ge 1}$ and $alpha : mathbb {Z}^lrightarrow text {Aut}(mathscr {N})$ be an action of $mathbb {Z}^l$ by automorphisms on a compact nilmanifold $mathscr{N}$ . We assume the action of every $alpha (z)$ is ergodic for $zin mathbb {Z}^lsmallsetminus {0}$ and show that $alpha $ satisfies exponential n -mixing for any integer $ngeq 2$ . This extends the results of Gorodnik and Spatzier [Mixing properties of commuting nilmanifold automorphisms. Acta Math. 215 (2015), 127–159].","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136211325","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}
Abstract For every $rin mathbb {N}_{geq 2}cup {infty }$ , we prove a $C^r$ -orbit connecting lemma for dynamically coherent and plaque expansive partially hyperbolic diffeomorphisms with one-dimensional orientation preserving center bundle. To be precise, for such a diffeomorphism f , if a point y is chain attainable from x through pseudo-orbits, then for any neighborhood U of x and any neighborhood V of y , there exist true orbits from U to V by arbitrarily $C^r$ -small perturbations. As a consequence, we prove that for $C^r$ -generic diffeomorphisms in this class, periodic points are dense in the chain recurrent set, and chain transitivity implies transitivity.
{"title":"-chain closing lemma for certain partially hyperbolic diffeomorphisms","authors":"YI SHI, XIAODONG WANG","doi":"10.1017/etds.2023.71","DOIUrl":"https://doi.org/10.1017/etds.2023.71","url":null,"abstract":"Abstract For every $rin mathbb {N}_{geq 2}cup {infty }$ , we prove a $C^r$ -orbit connecting lemma for dynamically coherent and plaque expansive partially hyperbolic diffeomorphisms with one-dimensional orientation preserving center bundle. To be precise, for such a diffeomorphism f , if a point y is chain attainable from x through pseudo-orbits, then for any neighborhood U of x and any neighborhood V of y , there exist true orbits from U to V by arbitrarily $C^r$ -small perturbations. As a consequence, we prove that for $C^r$ -generic diffeomorphisms in this class, periodic points are dense in the chain recurrent set, and chain transitivity implies transitivity.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136211303","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}
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
{"title":"ETS volume 43 issue 11 Cover and Back matter","authors":"","doi":"10.1017/etds.2022.106","DOIUrl":"https://doi.org/10.1017/etds.2022.106","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135435850","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}
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
{"title":"ETS volume 43 issue 11 Cover and Front matter","authors":"","doi":"10.1017/etds.2022.105","DOIUrl":"https://doi.org/10.1017/etds.2022.105","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135481175","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}
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