Abstract Katok [Lyapunov exponents, entropy and periodic points of diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173] conjectured that every $C^{2}$ diffeomorphism f on a Riemannian manifold has the intermediate entropy property, that is, for any constant $c in [0, h_{mathrm {top}}(f))$ , there exists an ergodic measure $mu $ of f satisfying $h_{mu }(f)=c$ . In this paper, we obtain a conditional intermediate metric entropy property and two conditional intermediate Birkhoff average properties for basic sets of flows that characterize the refined roles of ergodic measures in the invariant ones. In this process, we establish a ‘multi-horseshoe’ entropy-dense property and use it to get the goal combined with conditional variational principles. We also obtain the same result for singular hyperbolic attractors.
微分同态的Katok [Lyapunov]指数、熵和周期点。出版。数学。Inst. Hautes Études Sci. 51(1980)， 137-173]推测黎曼流形上的每一个$C^{2}$微分同态f具有中间熵性质，即对于任意常数$C in [0, h_{ mathm}}(f))$，存在一个满足$h_{mu}(f)= C $的遍历测度$mu $。在本文中，我们得到了基本流集的一个条件中间度量熵性质和两个条件中间Birkhoff平均性质，这些性质表征了遍历测度在不变测度中的精细作用。在此过程中，我们建立了一个“多马蹄形”熵密性质，并结合条件变分原理利用它得到目标。对于奇异双曲吸引子，我们也得到了相同的结果。
{"title":"Conditional intermediate entropy and Birkhoff average properties of hyperbolic flows","authors":"XIAOBO HOU, XUETING TIAN","doi":"10.1017/etds.2023.110","DOIUrl":"https://doi.org/10.1017/etds.2023.110","url":null,"abstract":"Abstract Katok [Lyapunov exponents, entropy and periodic points of diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173] conjectured that every $C^{2}$ diffeomorphism f on a Riemannian manifold has the intermediate entropy property, that is, for any constant $c in [0, h_{mathrm {top}}(f))$ , there exists an ergodic measure $mu $ of f satisfying $h_{mu }(f)=c$ . In this paper, we obtain a conditional intermediate metric entropy property and two conditional intermediate Birkhoff average properties for basic sets of flows that characterize the refined roles of ergodic measures in the invariant ones. In this process, we establish a ‘multi-horseshoe’ entropy-dense property and use it to get the goal combined with conditional variational principles. We also obtain the same result for singular hyperbolic attractors.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"32 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134954281","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 characterize the stabilized automorphism group for odometers and Toeplitz subshifts, and then prove an invariance property of the stabilized automorphism group of these dynamical systems. Namely, we prove the isomorphism invariance of the primes for which the p -adic valuation of the period structure tends to infinity. A particular case of interest is that for torsion-free odometers, the stabilized automorphism group is a full isomorphism invariant.
{"title":"Stabilized automorphism group of odometers and of Toeplitz subshifts","authors":"JENNIFER N. JONES-BARO","doi":"10.1017/etds.2023.109","DOIUrl":"https://doi.org/10.1017/etds.2023.109","url":null,"abstract":"Abstract We characterize the stabilized automorphism group for odometers and Toeplitz subshifts, and then prove an invariance property of the stabilized automorphism group of these dynamical systems. Namely, we prove the isomorphism invariance of the primes for which the p -adic valuation of the period structure tends to infinity. A particular case of interest is that for torsion-free odometers, the stabilized automorphism group is a full isomorphism invariant.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"27 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136346330","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 that under restrictions on the fiber, any fibered partially hyperbolic system over a nilmanifold is leaf conjugate to a smooth model that is isometric on the fibers and descends to a hyperbolic nilmanifold automorphism on the base. One ingredient is a result of independent interest generalizing a result of Hiraide: an Anosov homeomorphism of a nilmanifold is topologically conjugate to a hyperbolic nilmanifold automorphism.
{"title":"Smooth models for certain fibered partially hyperbolic systems","authors":"MEG DOUCETTE","doi":"10.1017/etds.2023.102","DOIUrl":"https://doi.org/10.1017/etds.2023.102","url":null,"abstract":"Abstract We prove that under restrictions on the fiber, any fibered partially hyperbolic system over a nilmanifold is leaf conjugate to a smooth model that is isometric on the fibers and descends to a hyperbolic nilmanifold automorphism on the base. One ingredient is a result of independent interest generalizing a result of Hiraide: an Anosov homeomorphism of a nilmanifold is topologically conjugate to a hyperbolic nilmanifold automorphism.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"12 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136347397","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 To any k -dimensional subspace of $mathbb {Q}^n$ one can naturally associate a point in the Grassmannian $mathrm {Gr}_{n,k}(mathbb {R})$ and two shapes of lattices of rank k and $n-k$ , respectively. These lattices originate by intersecting the k -dimensional subspace and its orthogonal with the lattice $mathbb {Z}^n$ . Using unipotent dynamics, we prove simultaneous equidistribution of all of these objects under congruence conditions when $(k,n) neq (2,4)$ .
{"title":"Equidistribution of rational subspaces and their shapes","authors":"Menny Aka, Andrea Musso, Andreas Wieser","doi":"10.1017/etds.2023.107","DOIUrl":"https://doi.org/10.1017/etds.2023.107","url":null,"abstract":"Abstract To any k -dimensional subspace of $mathbb {Q}^n$ one can naturally associate a point in the Grassmannian $mathrm {Gr}_{n,k}(mathbb {R})$ and two shapes of lattices of rank k and $n-k$ , respectively. These lattices originate by intersecting the k -dimensional subspace and its orthogonal with the lattice $mathbb {Z}^n$ . Using unipotent dynamics, we prove simultaneous equidistribution of all of these objects under congruence conditions when $(k,n) neq (2,4)$ .","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"84 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135087569","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 12 Cover and Back matter","authors":"","doi":"10.1017/etds.2022.108","DOIUrl":"https://doi.org/10.1017/etds.2022.108","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":"280 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135475220","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 first give a necessary and sufficient condition for a factor code with an unambiguous symbol to admit a subshift of finite type restricted to which it is one-to-one and onto. We then give a necessary and sufficient condition for the standard factor code on a spoke graph to admit a subshift of finite type restricted to which it is finite-to-one and onto. We also conjecture that for such a code, the finite-to-one and onto property is equivalent to the existence of a stationary Markov chain that achieves the capacity of the corresponding deterministic channel.
{"title":"Markov capacity for factor codes with an unambiguous symbol","authors":"GUANGYUE HAN, BRIAN MARCUS, CHENGYU WU","doi":"10.1017/etds.2023.103","DOIUrl":"https://doi.org/10.1017/etds.2023.103","url":null,"abstract":"Abstract In this paper, we first give a necessary and sufficient condition for a factor code with an unambiguous symbol to admit a subshift of finite type restricted to which it is one-to-one and onto. We then give a necessary and sufficient condition for the standard factor code on a spoke graph to admit a subshift of finite type restricted to which it is finite-to-one and onto. We also conjecture that for such a code, the finite-to-one and onto property is equivalent to the existence of a stationary Markov chain that achieves the capacity of the corresponding deterministic channel.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"29 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135475739","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 12 Cover and Front matter","authors":"","doi":"10.1017/etds.2022.107","DOIUrl":"https://doi.org/10.1017/etds.2022.107","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":"285 S5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135475348","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 give a $C^1$ -perturbation technique for ejecting an a priori given finite set of periodic points preserving a given finite set of homo/heteroclinic intersections from a chain recurrence class of a periodic point. The technique is first stated under a simpler setting called a Markov iterated function system, a two-dimensional iterated function system in which the compositions are chosen in a Markovian way. Then we apply the result to the setting of three-dimensional partially hyperbolic diffeomorphisms.
{"title":"A mechanism for ejecting a horseshoe from a partially hyperbolic chain recurrence class","authors":"Bonatti, Christian, Shinohara, Katsutoshi","doi":"10.1017/etds.2023.76","DOIUrl":"https://doi.org/10.1017/etds.2023.76","url":null,"abstract":"Abstract We give a $C^1$ -perturbation technique for ejecting an a priori given finite set of periodic points preserving a given finite set of homo/heteroclinic intersections from a chain recurrence class of a periodic point. The technique is first stated under a simpler setting called a Markov iterated function system, a two-dimensional iterated function system in which the compositions are chosen in a Markovian way. Then we apply the result to the setting of three-dimensional partially hyperbolic diffeomorphisms.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"217 S696","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135775009","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 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":"32 4","pages":"0"},"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":"14 3","pages":"0"},"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}
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