Journal of Modern Dynamics最新文献

Hausdorff dimension of directional limit sets for self-joinings of hyperbolic manifolds 双曲流形自连接方向极限集的Hausdorff维数

IF 1.1 1区数学

Journal of Modern Dynamics Pub Date : 2023-02-22 DOI: 10.3934/jmd.2023013

Dongryul Kim, Y. Minsky, H. Oh

{"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}

引用次数: 11

Summable orbits 可和轨道

IF 1.1 1区数学

Journal of Modern Dynamics Pub Date : 2023-01-01 DOI: 10.3934/jmd.2023017

Snir Ben Ovadia

引用次数: 0

Noncommutative coboundary equations over integrable systems 可积系统上的非交换共边方程

1区数学

Journal of Modern Dynamics Pub Date : 2023-01-01 DOI: 10.3934/jmd.2023020

Rafael de la Llave, Maria Saprykina

{"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}

引用次数: 0

Circle homeomorphisms with breaks with no $boldsymbol{C^{2-nu}}$ conjugacy 不带$bold符号{C^{2-nu}}$共轭的带断点的圆同胚

1区数学

Journal of Modern Dynamics Pub Date : 2023-01-01 DOI: 10.3934/jmd.2023019

Nataliya Goncharuk, Konstantin Khanin, Yury Kudryashov

引用次数: 0

Regularizations of pseudo-automorphisms with positive algebraic entropy 具有正代数熵的伪自同构的正则化

IF 1.1 1区数学

Journal of Modern Dynamics Pub Date : 2023-01-01 DOI: 10.3934/jmd.2023006

A. Kuznetsova

引用次数: 1

The Brin Prize works of Tim Austin 蒂姆·奥斯汀的布林奖作品