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":"27 1","pages":"0"},"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":"25 4","pages":"0"},"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":"11 1","pages":"0"},"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":"8 1","pages":"0"},"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":"49 1","pages":"0"},"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":"29 1","pages":"0"},"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":"47 1","pages":"0"},"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":"301 1","pages":"0"},"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}
Abstract In this paper, we study centrally symmetric Birkhoff billiard tables. We introduce a closed invariant set $mathcal {M}_{mathcal {B}}$ consisting of locally maximizing orbits of the billiard map lying inside the region $mathcal {B}$ bounded by two invariant curves of $4$ -periodic orbits. We give an effective bound from above on the measure of this invariant set in terms of the isoperimetric defect of the curve. The equality case occurs if and only if the curve is a circle.
{"title":"Effective rigidity away from the boundary for centrally symmetric billiards","authors":"MISHA BIALY","doi":"10.1017/etds.2023.70","DOIUrl":"https://doi.org/10.1017/etds.2023.70","url":null,"abstract":"Abstract In this paper, we study centrally symmetric Birkhoff billiard tables. We introduce a closed invariant set $mathcal {M}_{mathcal {B}}$ consisting of locally maximizing orbits of the billiard map lying inside the region $mathcal {B}$ bounded by two invariant curves of $4$ -periodic orbits. We give an effective bound from above on the measure of this invariant set in terms of the isoperimetric defect of the curve. The equality case occurs if and only if the curve is a circle.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135345083","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 study the equidistribution of orbits of the form $b_1^{a_1(n)}cdots b_k^{a_k(n)}Gamma $ in a nilmanifold X , where the sequences $a_i(n)$ arise from smooth functions of polynomial growth belonging to a Hardy field. We show that under certain assumptions on the growth rates of the functions $a_1,ldots ,a_k$ , these orbits are equidistributed on some subnilmanifold of the space X . As an application of these results and in combination with the Host–Kra structure theorem for measure-preserving systems, as well as some recent seminorm estimates of the author for ergodic averages concerning Hardy field functions, we deduce a norm convergence result for multiple ergodic averages. Our method mainly relies on an equidistribution result of Green and Tao on finite segments of polynomial orbits on a nilmanifold [The quantitative behaviour of polynomial orbits on nilmanifolds. Ann. of Math. (2) 175 (2012), 465–540].
{"title":"Pointwise convergence in nilmanifolds along smooth functions of polynomial growth","authors":"KONSTANTINOS TSINAS","doi":"10.1017/etds.2023.68","DOIUrl":"https://doi.org/10.1017/etds.2023.68","url":null,"abstract":"Abstract We study the equidistribution of orbits of the form $b_1^{a_1(n)}cdots b_k^{a_k(n)}Gamma $ in a nilmanifold X , where the sequences $a_i(n)$ arise from smooth functions of polynomial growth belonging to a Hardy field. We show that under certain assumptions on the growth rates of the functions $a_1,ldots ,a_k$ , these orbits are equidistributed on some subnilmanifold of the space X . As an application of these results and in combination with the Host–Kra structure theorem for measure-preserving systems, as well as some recent seminorm estimates of the author for ergodic averages concerning Hardy field functions, we deduce a norm convergence result for multiple ergodic averages. Our method mainly relies on an equidistribution result of Green and Tao on finite segments of polynomial orbits on a nilmanifold [The quantitative behaviour of polynomial orbits on nilmanifolds. Ann. of Math. (2) 175 (2012), 465–540].","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135815870","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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