The classical result of Patterson and Sullivan says that for a non-elementary convex cocompact subgroup $Gamma
Patterson和Sullivan的经典结果表明，对于一个非初等凸共紧子群$Gamma
{"title":"Hausdorff dimension of directional limit sets for self-joinings of hyperbolic manifolds","authors":"Dongryul Kim, Y. Minsky, H. Oh","doi":"10.3934/jmd.2023013","DOIUrl":"https://doi.org/10.3934/jmd.2023013","url":null,"abstract":"The classical result of Patterson and Sullivan says that for a non-elementary convex cocompact subgroup $Gamma<text{SO}^circ (n,1)$, $nge 2$, the Hausdorff dimension of the limit set of $Gamma$ is equal to the critical exponent of $Gamma$. In this paper, we generalize this result for self-joinings of convex cocompact groups in two ways. Let $Delta$ be a finitely generated group and $rho_i:Deltato text{SO}^circ(n_i,1)$ be a convex cocompact faithful representation of $Delta$ for $1le ile k$. Associated to $rho=(rho_1, cdots, rho_k)$, we consider the following self-joining subgroup of $prod_{i=1}^k text{SO}(n_i,1)$: $$Gamma=left(prod_{i=1}^krho_iright)(Delta)={(rho_1(g), cdots, rho_k(g)):gin Delta} .$$ (1). Denoting by $Lambdasubset prod_{i=1}^k mathbb{S}^{n_i-1}$ the limit set of $Gamma$, we first prove that $$text{dim}_H Lambda=max_{1le ile k} delta_{rho_i}$$ where $delta_{rho_i}$ is the critical exponent of the subgroup $rho_{i}(Delta)$. (2). Denoting by $Lambda_usubset Lambda$ the $u$-directional limit set for each $u=(u_1, cdots, u_k)$ in the interior of the limit cone of $Gamma$, we obtain that for $kle 3$, $$ frac{psi_Gamma(u)}{max_i u_i }le text{dim}_H Lambda_u le frac{psi_Gamma(u)}{min_i u_i }$$ where $psi_Gamma:mathbb{R}^kto mathbb{R}cup{-infty}$ is the growth indicator function of $Gamma$.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41816093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Summable orbits","authors":"Snir Ben Ovadia","doi":"10.3934/jmd.2023017","DOIUrl":"https://doi.org/10.3934/jmd.2023017","url":null,"abstract":"","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70085562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove an analog of the Livshits theorem for real-analytic families of cocycles over an integrable system with values in a Banach algebra$ {{mathscr G}} $or a Lie group. Namely, we consider an integrable dynamical system$ f:{{mathscr M}} equiv{{mathbb T}}^d times [-1, 1]^dto {{mathscr M}} $,$ f(theta, I) = (theta + I, I) $, and a real-analytic family of cocycles$ eta_{varepsilon} : {{mathscr M}} to {{mathscr G}} $indexed by a complex parameter$ {varepsilon} $in an open ball$ {{mathscr E}}_rho subset {{mathbb C}} $. We show that if$ eta_{varepsilon} $is close to identity and has trivial periodic data, i.e.,for each periodic point$ p = f^n p $and each$ {varepsilon} in {{mathscr E}}_{rho} $, then there exists a real-analytic family of maps$ phi_{varepsilon}: {{mathscr M}} to {{mathscr G}} $satisfying the coboundary equation$ eta_{varepsilon}(theta, I) = (phi_{varepsilon}circ f(theta, I))^{-1} cdot phi_{varepsilon} (theta, I) $for all$ (theta, I)in {{mathscr M}} $and$ {varepsilon} in {{mathscr E}}_{rho/2} $.We also show that if the coboundary equation above with an analytic left-hand side$ eta_{varepsilon} $has a solution in the sense of formal power series in$ {varepsilon} $, then it has an analytic solution.
{"title":"Noncommutative coboundary equations over integrable systems","authors":"Rafael de la Llave, Maria Saprykina","doi":"10.3934/jmd.2023020","DOIUrl":"https://doi.org/10.3934/jmd.2023020","url":null,"abstract":"We prove an analog of the Livshits theorem for real-analytic families of cocycles over an integrable system with values in a Banach algebra$ {{mathscr G}} $or a Lie group. Namely, we consider an integrable dynamical system$ f:{{mathscr M}} equiv{{mathbb T}}^d times [-1, 1]^dto {{mathscr M}} $,$ f(theta, I) = (theta + I, I) $, and a real-analytic family of cocycles$ eta_{varepsilon} : {{mathscr M}} to {{mathscr G}} $indexed by a complex parameter$ {varepsilon} $in an open ball$ {{mathscr E}}_rho subset {{mathbb C}} $. We show that if$ eta_{varepsilon} $is close to identity and has trivial periodic data, i.e.,for each periodic point$ p = f^n p $and each$ {varepsilon} in {{mathscr E}}_{rho} $, then there exists a real-analytic family of maps$ phi_{varepsilon}: {{mathscr M}} to {{mathscr G}} $satisfying the coboundary equation$ eta_{varepsilon}(theta, I) = (phi_{varepsilon}circ f(theta, I))^{-1} cdot phi_{varepsilon} (theta, I) $for all$ (theta, I)in {{mathscr M}} $and$ {varepsilon} in {{mathscr E}}_{rho/2} $.We also show that if the coboundary equation above with an analytic left-hand side$ eta_{varepsilon} $has a solution in the sense of formal power series in$ {varepsilon} $, then it has an analytic solution.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136201898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nataliya Goncharuk, Konstantin Khanin, Yury Kudryashov
The rigidity theory for circle homeomorphisms with breaks has been studied intensively in the last 20 years. It was proved [15, 21, 17, 19] that under mild conditions of the Diophantine type on the rotation number any two $C^{2+alpha}$ smooth circle homeomorphisms with a break point are $C^1$ smoothly conjugate to each other, provided that they have the same rotation number and the same size of the break. In this paper we prove that the conjugacy may not be $C^{2-nu}$ even if the maps are analytic outside of the break points. This result shows that the rigidity theory for maps with singularities is very different from the linearizable case of circle diffeomorphisms where conjugacy is arbitrarily smooth, or even analytic, for sufficiently smooth diffeomorphisms.
{"title":"Circle homeomorphisms with breaks with no $boldsymbol{C^{2-nu}}$ conjugacy","authors":"Nataliya Goncharuk, Konstantin Khanin, Yury Kudryashov","doi":"10.3934/jmd.2023019","DOIUrl":"https://doi.org/10.3934/jmd.2023019","url":null,"abstract":"The rigidity theory for circle homeomorphisms with breaks has been studied intensively in the last 20 years. It was proved [15, 21, 17, 19] that under mild conditions of the Diophantine type on the rotation number any two $C^{2+alpha}$ smooth circle homeomorphisms with a break point are $C^1$ smoothly conjugate to each other, provided that they have the same rotation number and the same size of the break. In this paper we prove that the conjugacy may not be $C^{2-nu}$ even if the maps are analytic outside of the break points. This result shows that the rigidity theory for maps with singularities is very different from the linearizable case of circle diffeomorphisms where conjugacy is arbitrarily smooth, or even analytic, for sufficiently smooth diffeomorphisms.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135651051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Regularizations of pseudo-automorphisms with positive algebraic entropy","authors":"A. Kuznetsova","doi":"10.3934/jmd.2023006","DOIUrl":"https://doi.org/10.3934/jmd.2023006","url":null,"abstract":"","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70085244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The mathematical activity of Tim Austin has been, since the very beginning, quite abundant and versatile. We will describe and comment on three of his results which were selected as most representative for the Brin Prize Award and which culminated in the proof of the weak Pinsker structure theorem.
{"title":"The Brin Prize works of Tim Austin","authors":"Jean-Paul Thouvenot","doi":"10.3934/jmd.2023024","DOIUrl":"https://doi.org/10.3934/jmd.2023024","url":null,"abstract":"The mathematical activity of Tim Austin has been, since the very beginning, quite abundant and versatile. We will describe and comment on three of his results which were selected as most representative for the Brin Prize Award and which culminated in the proof of the weak Pinsker structure theorem.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135445010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The 2021 Michael Brin Prize in Dynamical Systems","authors":"None The Editors","doi":"10.3934/jmd.2023023","DOIUrl":"https://doi.org/10.3934/jmd.2023023","url":null,"abstract":"","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135445008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the stabilized automorphism group of a subshift of finite type with a certain gluing property called the eventual filling property, on a residually finite group $ G $. We show that the stabilized automorphism group is simply monolithic, i.e., it has a unique minimal non-trivial normal subgroup—the monolith—which is additionally simple. To describe the monolith, we introduce gate lattices, which apply (reversible logical) gates on finite-index subgroups of $ G $. The monolith is then precisely the commutator subgroup of the group generated by gate lattices. If the subshift and the group $ G $ have some additional properties, then the gate lattices generate a perfect group, thus they generate the monolith. In particular, this is always the case when the acting group is the integers. In this case we can also show that gate lattices generate the inert part of the stabilized automorphism group. Thus we obtain that the stabilized inert automorphism group of a one-dimensional mixing subshift of finite type is simple.
研究了在剩余有限群$ G $上具有一定胶合性质的有限型子位移的稳定自同构群。我们证明了稳定自同构群是简单整体的，即它有一个唯一的最小非平凡正规子群-整体，它是额外简单的。为了描述整体结构，我们引入栅格，在$ G $的有限指数子群上应用(可逆逻辑)栅格。因此，该整体恰好是栅极所产生的群的换向子群。如果子移和群$ G $具有一些额外的性质，则栅格生成一个完美的群，从而生成单体。特别是，当作用群是整数时，这种情况总是存在的。在这种情况下，我们也可以证明栅格产生稳定自同构群的惰性部分。由此得出一维有限型混合子移的稳定惰性自同构群是简单的。
{"title":"Gate lattices and the stabilized automorphism group","authors":"Ville Salo","doi":"10.3934/jmd.2023018","DOIUrl":"https://doi.org/10.3934/jmd.2023018","url":null,"abstract":"We study the stabilized automorphism group of a subshift of finite type with a certain gluing property called the eventual filling property, on a residually finite group $ G $. We show that the stabilized automorphism group is simply monolithic, i.e., it has a unique minimal non-trivial normal subgroup—the monolith—which is additionally simple. To describe the monolith, we introduce gate lattices, which apply (reversible logical) gates on finite-index subgroups of $ G $. The monolith is then precisely the commutator subgroup of the group generated by gate lattices. If the subshift and the group $ G $ have some additional properties, then the gate lattices generate a perfect group, thus they generate the monolith. In particular, this is always the case when the acting group is the integers. In this case we can also show that gate lattices generate the inert part of the stabilized automorphism group. Thus we obtain that the stabilized inert automorphism group of a one-dimensional mixing subshift of finite type is simple.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135496193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $Q$ be a closed manifold with non-trivial first Betti number that admits a non-trivial $S^1$-action, and $Sigma subseteq T^*Q$ a non-degenerate starshaped hypersurface. We prove that the number of geometrically distinct Reeb orbits of period at most $T$ on $Sigma$ grows at least logarithmically in $T$.
{"title":"A Bangert–Hingston theorem for starshaped hypersurfaces","authors":"Alessio Pellegrini","doi":"10.3934/jmd.2023011","DOIUrl":"https://doi.org/10.3934/jmd.2023011","url":null,"abstract":"Let $Q$ be a closed manifold with non-trivial first Betti number that admits a non-trivial $S^1$-action, and $Sigma subseteq T^*Q$ a non-degenerate starshaped hypersurface. We prove that the number of geometrically distinct Reeb orbits of period at most $T$ on $Sigma$ grows at least logarithmically in $T$.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48747931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The goal of this article is to establish several general properties of a somewhat large class of partially hyperbolic diffeomorphisms called emph{discretized Anosov flows}. A general definition for these systems is presented and is proven to be equivalent with the definition introduced in [BFFP19], as well as with the notion of flow type partially hyperbolic diffeomorphisms introduced in [BFT20]. The set of discretized Anosov flows is shown to be $C^1$ open and closed inside the set of partially hyperbolic diffeomorphisms. Every discretized Anosov flow is proven to be dynamically coherent and plaque expansive. Unique integrability of the center bundle is shown to happen for whole connected components, notably the ones containing the time 1 map of an Anosov flow. For general connected components, a result on uniqueness of invariant foliation is obtained. Similar results are seen to happen for partially hyperbolic systems admitting a uniformly compact center foliation extending the studies initiated in [BB16].
{"title":"Global stability of discretized Anosov flows","authors":"Santiago Martinchich","doi":"10.3934/jmd.2023016","DOIUrl":"https://doi.org/10.3934/jmd.2023016","url":null,"abstract":"The goal of this article is to establish several general properties of a somewhat large class of partially hyperbolic diffeomorphisms called emph{discretized Anosov flows}. A general definition for these systems is presented and is proven to be equivalent with the definition introduced in [BFFP19], as well as with the notion of flow type partially hyperbolic diffeomorphisms introduced in [BFT20]. The set of discretized Anosov flows is shown to be $C^1$ open and closed inside the set of partially hyperbolic diffeomorphisms. Every discretized Anosov flow is proven to be dynamically coherent and plaque expansive. Unique integrability of the center bundle is shown to happen for whole connected components, notably the ones containing the time 1 map of an Anosov flow. For general connected components, a result on uniqueness of invariant foliation is obtained. Similar results are seen to happen for partially hyperbolic systems admitting a uniformly compact center foliation extending the studies initiated in [BB16].","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43382096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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