Ergodic Theory and Dynamical Systems最新文献

{"title":"Shifts of finite type on locally finite groups","authors":"JADE RAYMOND","doi":"10.1017/etds.2024.14","DOIUrl":"https://doi.org/10.1017/etds.2024.14","url":null,"abstract":"In this work we prove that every shift of finite type (SFT), sofic shift, and strongly irreducible shift on locally finite groups has strong dynamical properties. These properties include that every sofic shift is an SFT, every SFT is strongly irreducible, every strongly irreducible shift is an SFT, every SFT is entropy minimal, and every SFT has a unique measure of maximal entropy, among others. In addition, we show that if every SFT on a group is strongly irreducible, or if every sofic shift is an SFT, then the group must be locally finite, and this extends to all of the properties we explore. These results are collected in two main theorems which characterize the local finiteness of groups by purely dynamical properties. In pursuit of these results, we present a formal construction of <jats:italic>free extension</jats:italic> shifts on a group <jats:italic>G</jats:italic>, which takes a shift on a subgroup <jats:italic>H</jats:italic> of <jats:italic>G</jats:italic>, and naturally extends it to a shift on all of <jats:italic>G</jats:italic>.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979059","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}

{"title":"On the stochastic bifurcations regarding random iterations of polynomials of the form","authors":"TAKAYUKI WATANABE","doi":"10.1017/etds.2024.17","DOIUrl":"https://doi.org/10.1017/etds.2024.17","url":null,"abstract":"In this paper, we consider random iterations of polynomial maps <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000178_inline2.png\" /> <jats:tex-math> $z^{2} + c_{n}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000178_inline3.png\" /> <jats:tex-math> $c_{n}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are complex-valued independent random variables following the uniform distribution on the closed disk with center <jats:italic>c</jats:italic> and radius <jats:italic>r</jats:italic>. The aim of this paper is twofold. First, we study the (dis)connectedness of random Julia sets. Here, we reveal the relationships between the bifurcation radius and connectedness of random Julia sets. Second, we investigate the bifurcation of our random iterations and give quantitative estimates of bifurcation parameters. In particular, we prove that for the central parameter <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000178_inline4.png\" /> <jats:tex-math> $c = -1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, almost every random Julia set is totally disconnected with much smaller radial parameters <jats:italic>r</jats:italic> than expected. We also introduce several open questions worth discussing.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139978846","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}

{"title":"Upper, down, two-sided Lorenz attractor, collisions, merging, and switching","authors":"DIEGO BARROS, CHRISTIAN BONATTI, MARIA JOSÉ PACIFICO","doi":"10.1017/etds.2024.8","DOIUrl":"https://doi.org/10.1017/etds.2024.8","url":null,"abstract":"We present a modified version of the well-known geometric Lorenz attractor. It consists of a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline1.png\" /> <jats:tex-math> $C^1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> open set <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline2.png\" /> <jats:tex-math> ${mathcal O}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of vector fields in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline3.png\" /> <jats:tex-math> ${mathbb R}^3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> having an attracting region <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline4.png\" /> <jats:tex-math> ${mathcal U}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfying three properties. Namely, a unique singularity <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline5.png\" /> <jats:tex-math> $sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>; a unique attractor <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline6.png\" /> <jats:tex-math> $Lambda $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> including the singular point and the maximal invariant in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline7.png\" /> <jats:tex-math> ${mathcal U}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> has at most two chain recurrence classes, which are <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline8.png\" /> <jats:tex-math> $Lambda $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and (at most) one hyperbolic horseshoe. The horseshoe and the singular attractor have a collision along with the union of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline9.png\" /> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> codimension <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline10.png\" /> <jats:tex-math> $1$ </jats:t","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139923275","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}

{"title":"Measures of maximal entropy of bounded density shifts","authors":"FELIPE GARCÍA-RAMOS, RONNIE PAVLOV, CARLOS REYES","doi":"10.1017/etds.2024.6","DOIUrl":"https://doi.org/10.1017/etds.2024.6","url":null,"abstract":"We find sufficient conditions for bounded density shifts to have a unique measure of maximal entropy. We also prove that every measure of maximal entropy of a bounded density shift is fully supported. As a consequence of this, we obtain that bounded density shifts are surjunctive.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139923465","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}

{"title":"On denseness of horospheres in higher rank homogeneous spaces","authors":"OR LANDESBERG, HEE OH","doi":"10.1017/etds.2024.12","DOIUrl":"https://doi.org/10.1017/etds.2024.12","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline1.png\" /> <jats:tex-math> $ G $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a connected semisimple real algebraic group and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline2.png\" /> <jats:tex-math> $Gamma &lt;G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a Zariski dense discrete subgroup. Let <jats:italic>N</jats:italic> denote a maximal horospherical subgroup of <jats:italic>G</jats:italic>, and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline3.png\" /> <jats:tex-math> $P=MAN$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> the minimal parabolic subgroup which is the normalizer of <jats:italic>N</jats:italic>. Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline4.png\" /> <jats:tex-math> $mathcal E$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> denote the unique <jats:italic>P</jats:italic>-minimal subset of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline5.png\" /> <jats:tex-math> $Gamma backslash G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline6.png\" /> <jats:tex-math> $mathcal E_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline7.png\" /> <jats:tex-math> $P^circ $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-minimal subset. We consider a notion of a horospherical limit point in the Furstenberg boundary <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline8.png\" /> <jats:tex-math> $ G/P $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and show that the following are equivalent for any <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline9.png\" /> <jats:tex-math> $[g]in mathcal E_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>: <jats:list list-type=\"number\"> <jats:list-item> <jats:label>(1)</jats:label> <jats:inline-formula> <jats:alternatives> <jats:inline-gra","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139909838","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}

{"title":"Sufficient conditions for non-zero entropy of closed relations","authors":"IZTOK BANIČ, RENE GRIL ROGINA, JUDY KENNEDY, VAN NALL","doi":"10.1017/etds.2024.11","DOIUrl":"https://doi.org/10.1017/etds.2024.11","url":null,"abstract":"We introduce the notions of returns and well-aligned sets for closed relations on compact metric spaces and then use them to obtain non-trivial sufficient conditions for such a relation to have non-zero entropy. In addition, we give a characterization of finite relations with non-zero entropy in terms of Li–Yorke and DC2 chaos.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139766847","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}

{"title":"Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts","authors":"TULLIO CECCHERINI-SILBERSTEIN, MICHEL COORNAERT, XUAN KIEN PHUNG","doi":"10.1017/etds.2023.120","DOIUrl":"https://doi.org/10.1017/etds.2023.120","url":null,"abstract":"Let <jats:italic>G</jats:italic> be a group and let <jats:italic>V</jats:italic> be an algebraic variety over an algebraically closed field <jats:italic>K</jats:italic>. Let <jats:italic>A</jats:italic> denote the set of <jats:italic>K</jats:italic>-points of <jats:italic>V</jats:italic>. We introduce algebraic sofic subshifts <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001207_inline1.png\" /> <jats:tex-math> ${Sigma subset A^G}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and study endomorphisms <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001207_inline2.png\" /> <jats:tex-math> $tau colon Sigma to Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We generalize several results for dynamical invariant sets and nilpotency of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001207_inline3.png\" /> <jats:tex-math> $tau $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001207_inline4.png\" /> <jats:tex-math> $tau $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is nilpotent if and only if its limit set, that is, the intersection of the images of its iterates, is a singleton. If moreover <jats:italic>G</jats:italic> is infinite, finitely generated and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001207_inline5.png\" /> <jats:tex-math> $Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is topologically mixing, we show that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001207_inline6.png\" /> <jats:tex-math> $tau $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is nilpotent if and only if its limit set consists of periodic configurations and has a finite set of alphabet values.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139767237","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}

{"title":"On a self-embedding problem for self-similar sets","authors":"JIAN-CI XIAO","doi":"10.1017/etds.2024.2","DOIUrl":"https://doi.org/10.1017/etds.2024.2","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000026_inline1.png\" /> <jats:tex-math> $Ksubset {mathbb {R}}^d$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a self-similar set generated by an iterated function system <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000026_inline2.png\" /> <jats:tex-math> ${varphi _i}_{i=1}^m$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfying the strong separation condition and let <jats:italic>f</jats:italic> be a contracting similitude with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000026_inline3.png\" /> <jats:tex-math> $f(K)subseteq K$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We show that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000026_inline4.png\" /> <jats:tex-math> $f(K)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is relatively open in <jats:italic>K</jats:italic> if all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000026_inline5.png\" /> <jats:tex-math> $varphi _i$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> share a common contraction ratio and orthogonal part. We also provide a counterexample when the orthogonal parts are allowed to vary. This partially answers a question of Elekes, Keleti and Máthé [<jats:italic>Ergod. Th. &amp; Dynam. Sys.</jats:italic>30 (2010), 399–440]. As a byproduct of our argument, when <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000026_inline6.png\" /> <jats:tex-math> $d=1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:italic>K</jats:italic> admits two homogeneous generating iterated function systems satisfying the strong separation condition but with contraction ratios of opposite signs, we show that <jats:italic>K</jats:italic> is symmetric. This partially answers a question of Feng and Wang [<jats:italic>Adv. Math.</jats:italic>222 (2009), 1964–1981].","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139767086","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}

{"title":"Stable laws for random dynamical systems","authors":"ROMAIN AIMINO, MATTHEW NICOL, ANDREW TÖRÖK","doi":"10.1017/etds.2024.5","DOIUrl":"https://doi.org/10.1017/etds.2024.5","url":null,"abstract":"In this paper, we consider random dynamical systems formed by concatenating maps acting on the unit interval <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000051_inline1.png\" /> <jats:tex-math> $[0,1]$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> in an independent and identically distributed (i.i.d.) fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary measure <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000051_inline2.png\" /> <jats:tex-math> $nu $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We consider a class of non-square-integrable observables <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000051_inline3.png\" /> <jats:tex-math> $phi $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, mostly of form <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000051_inline4.png\" /> <jats:tex-math> $phi (x)=d(x,x_0)^{-{1}/{alpha }}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000051_inline5.png\" /> <jats:tex-math> $x_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a non-recurrent point (in particular a non-periodic point) satisfying some other genericity conditions and, more generally, regularly varying observables with index <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000051_inline6.png\" /> <jats:tex-math> $alpha in (0,2)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. The two types of maps we concatenate are a class of piecewise <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000051_inline7.png\" /> <jats:tex-math> $C^2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> expanding maps and a class of intermittent maps possessing an indifferent fixed point at the origin. Under conditions on the dynamics and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000051_inline8.png\" /> <jats:tex-math> $alpha $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we establish Poisson limit laws, convergence of scaled Birkhoff sums to a stable limit law, and functional stable limit laws in both the annealed and quenched case. The scaling constants fo","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139771865","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}

{"title":"Dimension estimates and approximation in non-uniformly hyperbolic systems","authors":"JUAN WANG, YONGLUO CAO, YUN ZHAO","doi":"10.1017/etds.2024.3","DOIUrl":"https://doi.org/10.1017/etds.2024.3","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000038_inline1.png\" /> <jats:tex-math> $f: Mrightarrow M$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000038_inline2.png\" /> <jats:tex-math> $C^{1+alpha }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> diffeomorphism on an <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000038_inline3.png\" /> <jats:tex-math> $m_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-dimensional compact smooth Riemannian manifold <jats:italic>M</jats:italic> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000038_inline4.png\" /> <jats:tex-math> $mu $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> a hyperbolic ergodic <jats:italic>f</jats:italic>-invariant probability measure. This paper obtains an upper bound for the stable (unstable) pointwise dimension of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000038_inline5.png\" /> <jats:tex-math> $mu $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, which is given by the unique solution of an equation involving the sub-additive measure-theoretic pressure. If <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000038_inline6.png\" /> <jats:tex-math> $mu $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a Sinai–Ruelle–Bowen (SRB) measure, then the Kaplan–Yorke conjecture is true under some additional conditions and the Lyapunov dimension of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000038_inline7.png\" /> <jats:tex-math> $mu $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> can be approximated gradually by the Hausdorff dimension of a sequence of hyperbolic sets <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000038_inline8.png\" /> <jats:tex-math> ${Lambda _n}_{ngeq 1}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. The limit behaviour of the Carathéodory singular dimension of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000038_inline9.png\" /> <jats:tex-math> $Lambda _n$ </jats:tex-math> </jats:alternatives> </jats:inl","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139771871","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}