Or Landesberg, Minju M. Lee, E. Lindenstrauss, H. Oh
Let $G=prod_{i=1}^{r} G_i$ be a product of simple real algebraic groups of rank one and $Gamma$ an Anosov subgroup of $G$ with respect to a minimal parabolic subgroup. For each $v$ in the interior of a positive Weyl chamber, let $mathcal R_vsubsetGammabackslash G$ denote the Borel subset of all points with recurrent $exp (mathbb R_+ v)$-orbits. For a maximal horospherical subgroup $N$ of $G$, we show that the $N$-action on ${mathcal R}_v$ is uniquely ergodic if $r={rank}(G)le 3$ and $v$ belongs to the interior of the limit cone of $Gamma$, and that there exists no $N$-invariant {Radon} measure on $mathcal R_v$ otherwise.
{"title":"Horospherical invariant measures and a rank dichotomy for Anosov groups","authors":"Or Landesberg, Minju M. Lee, E. Lindenstrauss, H. Oh","doi":"10.3934/jmd.2023009","DOIUrl":"https://doi.org/10.3934/jmd.2023009","url":null,"abstract":"Let $G=prod_{i=1}^{r} G_i$ be a product of simple real algebraic groups of rank one and $Gamma$ an Anosov subgroup of $G$ with respect to a minimal parabolic subgroup. For each $v$ in the interior of a positive Weyl chamber, let $mathcal R_vsubsetGammabackslash G$ denote the Borel subset of all points with recurrent $exp (mathbb R_+ v)$-orbits. For a maximal horospherical subgroup $N$ of $G$, we show that the $N$-action on ${mathcal R}_v$ is uniquely ergodic if $r={rank}(G)le 3$ and $v$ belongs to the interior of the limit cone of $Gamma$, and that there exists no $N$-invariant {Radon} measure on $mathcal R_v$ otherwise.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47919647","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 establish an extension of the Hopf-Tsuji-Sullivan dichotomy to any Zariski dense discrete subgroup of a semisimple real algebraic group $G$. We then apply this dichotomy to Anosov subgroups of $G$, which surprisingly presents a different phenomenon depending on the rank of the ambient group $G$.
{"title":"The Hopf–Tsuji–Sullivan dichotomy in higher rank and applications to Anosov subgroups","authors":"M. Burger, Or Landesberg, Minju M. Lee, H. Oh","doi":"10.3934/jmd.2023008","DOIUrl":"https://doi.org/10.3934/jmd.2023008","url":null,"abstract":"We establish an extension of the Hopf-Tsuji-Sullivan dichotomy to any Zariski dense discrete subgroup of a semisimple real algebraic group $G$. We then apply this dichotomy to Anosov subgroups of $G$, which surprisingly presents a different phenomenon depending on the rank of the ambient group $G$.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44230188","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 $G$ be a connected semisimple real Lie group with finite center, and $mu$ a probability measure on $G$ whose support generates a Zariski-dense subgroup of $G$. We consider the right $mu$-random walk on $G$ and show that each random trajectory spends most of its time at bounded distance of a well-chosen Weyl chamber. We infer that if $G$ has rank one, and $mu$ has a finite first moment, then for any discrete subgroup $Lambda subseteq G$, the $mu$-walk and the geodesic flow on $Lambda backslash G$ are either both transient, or both recurrent and ergodic, thus extending a well known theorem due to Hopf-Tsuji-Sullivan-Kaimanovich dealing with the Brownian motion.
{"title":"Some asymptotic properties of random walks on homogeneous spaces","authors":"Timoth'ee B'enard","doi":"10.3934/jmd.2023004","DOIUrl":"https://doi.org/10.3934/jmd.2023004","url":null,"abstract":"Let $G$ be a connected semisimple real Lie group with finite center, and $mu$ a probability measure on $G$ whose support generates a Zariski-dense subgroup of $G$. We consider the right $mu$-random walk on $G$ and show that each random trajectory spends most of its time at bounded distance of a well-chosen Weyl chamber. We infer that if $G$ has rank one, and $mu$ has a finite first moment, then for any discrete subgroup $Lambda subseteq G$, the $mu$-walk and the geodesic flow on $Lambda backslash G$ are either both transient, or both recurrent and ergodic, thus extending a well known theorem due to Hopf-Tsuji-Sullivan-Kaimanovich dealing with the Brownian motion.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45120858","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}
A classical Borel–Cantelli Lemma gives conditions for deciding whether an infinite number of rare events will happen almost surely. In this article, we propose an extension of Borel–Cantelli Lemma to characterize the multiple occurrence of events on the same time scale. Our results imply multiple Logarithm Laws for recurrence and hitting times, as well as Poisson Limit Laws for systems which are exponentially mixing of all orders. The applications include geodesic flows on compact negatively curved manifolds, geodesic excursions on finite volume hyperbolic manifolds, Diophantine approximations and extreme value theory for dynamical systems.
{"title":"Multiple Borel–Cantelli Lemma in dynamics and MultiLog Law for recurrence","authors":"D. Dolgopyat, B. Fayad, Sixu Liu","doi":"10.3934/jmd.2022009","DOIUrl":"https://doi.org/10.3934/jmd.2022009","url":null,"abstract":"A classical Borel–Cantelli Lemma gives conditions for deciding whether an infinite number of rare events will happen almost surely. In this article, we propose an extension of Borel–Cantelli Lemma to characterize the multiple occurrence of events on the same time scale. Our results imply multiple Logarithm Laws for recurrence and hitting times, as well as Poisson Limit Laws for systems which are exponentially mixing of all orders. The applications include geodesic flows on compact negatively curved manifolds, geodesic excursions on finite volume hyperbolic manifolds, Diophantine approximations and extreme value theory for dynamical systems.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44818180","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}
Leandro Arosio, A. Benini, J. Fornaess, Han Peters
Very little is currently known about the dynamics of non-polynomial entire maps in several complex variables. The family of transcendental Hénon maps offers the potential of combining ideas from transcendental dynamics in one variable and the dynamics of polynomial Hénon maps in two. Here we show that these maps all have infinite topological and measure theoretic entropy. The proof also implies the existence of infinitely many periodic orbits of any order greater than two.
{"title":"Dynamics of transcendental Hénon maps III: Infinite entropy","authors":"Leandro Arosio, A. Benini, J. Fornaess, Han Peters","doi":"10.3934/jmd.2021016","DOIUrl":"https://doi.org/10.3934/jmd.2021016","url":null,"abstract":"Very little is currently known about the dynamics of non-polynomial entire maps in several complex variables. The family of transcendental Hénon maps offers the potential of combining ideas from transcendental dynamics in one variable and the dynamics of polynomial Hénon maps in two. Here we show that these maps all have infinite topological and measure theoretic entropy. The proof also implies the existence of infinitely many periodic orbits of any order greater than two.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48291474","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 begin{document}$ mathscr{M} $end{document} be a geometrically finite hyperbolic manifold. We present a very general theorem on the shrinking target problem for the geodesic flow, using its exponential mixing. This includes a strengthening of Sullivan's logarithm law for the excursion rate of the geodesic flow. More generally, we prove logarithm laws for the first hitting time for shrinking cusp neighborhoods, shrinking tubular neighborhoods of a closed geodesic, and shrinking metric balls, as well as give quantitative estimates for the time a generic geodesic spends in such shrinking targets.
Let begin{document}$ mathscr{M} $end{document} be a geometrically finite hyperbolic manifold. We present a very general theorem on the shrinking target problem for the geodesic flow, using its exponential mixing. This includes a strengthening of Sullivan's logarithm law for the excursion rate of the geodesic flow. More generally, we prove logarithm laws for the first hitting time for shrinking cusp neighborhoods, shrinking tubular neighborhoods of a closed geodesic, and shrinking metric balls, as well as give quantitative estimates for the time a generic geodesic spends in such shrinking targets.
{"title":"Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds","authors":"Dubi Kelmer, H. Oh","doi":"10.3934/jmd.2021014","DOIUrl":"https://doi.org/10.3934/jmd.2021014","url":null,"abstract":"<p style='text-indent:20px;'>Let <inline-formula><tex-math id=\"M1\">begin{document}$ mathscr{M} $end{document}</tex-math></inline-formula> be a geometrically finite hyperbolic manifold. We present a very general theorem on the shrinking target problem for the geodesic flow, using its exponential mixing. This includes a strengthening of Sullivan's logarithm law for the excursion rate of the geodesic flow. More generally, we prove logarithm laws for the first hitting time for shrinking cusp neighborhoods, shrinking tubular neighborhoods of a closed geodesic, and shrinking metric balls, as well as give quantitative estimates for the time a generic geodesic spends in such shrinking targets.</p>","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70085395","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 address the problem of computing the critical regularity of groups of homeomorphisms of the interval. Our main result is that if $H$ and $K$ are two non-solvable groups then a $C^1$ actions of $Htimes K$ on a compact interval $I$ cannot be {em overlapping}, which by definition means that there must be non-trivial $hin H$ and $kin K$ with disjoint support. As a corollary we prove that the right-angled Artin group $(F_2times F_2)*mathbb{Z}$ has critical regularity one, which is to say that it admits a faithful $C^1$ action on $I$, but no faithful $C^{1,tau}$ action for $tau>0$. This is the first explicit example of a group of exponential growth whose critical regularity is finite, known exactly, and achieved. Another corollary we get is that Thompson's group $F$ does not admit a $C^1$ overlapping action on $I$, so that $F*mathbb{Z}$ is a new example of a locally indicable group admitting no faithful $C^1$--action on $I$.
{"title":"Direct products, overlapping actions, and critical regularity","authors":"Sang-hyun Kim, T. Koberda, C. Rivas","doi":"10.3934/jmd.2021009","DOIUrl":"https://doi.org/10.3934/jmd.2021009","url":null,"abstract":"We address the problem of computing the critical regularity of groups of homeomorphisms of the interval. Our main result is that if $H$ and $K$ are two non-solvable groups then a $C^1$ actions of $Htimes K$ on a compact interval $I$ cannot be {em overlapping}, which by definition means that there must be non-trivial $hin H$ and $kin K$ with disjoint support. As a corollary we prove that the right-angled Artin group $(F_2times F_2)*mathbb{Z}$ has critical regularity one, which is to say that it admits a faithful $C^1$ action on $I$, but no faithful $C^{1,tau}$ action for $tau>0$. This is the first explicit example of a group of exponential growth whose critical regularity is finite, known exactly, and achieved. Another corollary we get is that Thompson's group $F$ does not admit a $C^1$ overlapping action on $I$, so that $F*mathbb{Z}$ is a new example of a locally indicable group admitting no faithful $C^1$--action on $I$.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49506404","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}
For any accessible partially hyperbolic homogeneous flow, we show that all smooth time changes are K and hence mixing of all orders. We also establish stable ergodicity for time-one map of these time changes.
{"title":"On ergodic properties of time changes of partially hyperbolic homogeneous flows","authors":"Changguang Dong","doi":"10.3934/jmd.2023015","DOIUrl":"https://doi.org/10.3934/jmd.2023015","url":null,"abstract":"For any accessible partially hyperbolic homogeneous flow, we show that all smooth time changes are K and hence mixing of all orders. We also establish stable ergodicity for time-one map of these time changes.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44688589","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}
For any finite horizon Sinai billiard map begin{document}$ T $end{document} on the two-torus, we find begin{document}$ t_*>1 $end{document} such that for each begin{document}$ tin (0,t_*) $end{document} there exists a unique equilibrium state begin{document}$ mu_t $end{document} for begin{document}$ - tlog J^uT $end{document}, and begin{document}$ mu_t $end{document} is begin{document}$ T $end{document}-adapted. (In particular, the SRB measure is the unique equilibrium state for begin{document}$ - log J^uT $end{document}.) We show that begin{document}$ mu_t $end{document} is exponentially mixing for Hölder observables, and the pressure function begin{document}$ P(t) = sup_mu {h_mu -int tlog J^uT d mu} $end{document} is analytic on begin{document}$ (0,t_*) $end{document}. In addition, begin{document}$ P(t) $end{document} is strictly convex if and only if begin{document}$ log J^uT $end{document} is not begin{document}$ mu_t $end{document}-a.e. cohomologous to a constant, while, if there exist begin{document}$ t_ane t_b $end{document} with begin{document}$ mu_{t_a} = mu_{t_b} $end{document}, then begin{document}$ P(t) $end{document} is affine on begin{document}$ (0,t_*) $end{document}. An additional sparse recurrence condition gives begin{document}$ lim_{tdownarrow 0} P(t) = P(0) $end{document}.
{"title":"Thermodynamic formalism for dispersing billiards","authors":"V. Baladi, Mark F. Demers","doi":"10.3934/jmd.2022013","DOIUrl":"https://doi.org/10.3934/jmd.2022013","url":null,"abstract":"<p style='text-indent:20px;'>For any finite horizon Sinai billiard map <inline-formula><tex-math id=\"M1\">begin{document}$ T $end{document}</tex-math></inline-formula> on the two-torus, we find <inline-formula><tex-math id=\"M2\">begin{document}$ t_*>1 $end{document}</tex-math></inline-formula> such that for each <inline-formula><tex-math id=\"M3\">begin{document}$ tin (0,t_*) $end{document}</tex-math></inline-formula> there exists a unique equilibrium state <inline-formula><tex-math id=\"M4\">begin{document}$ mu_t $end{document}</tex-math></inline-formula> for <inline-formula><tex-math id=\"M5\">begin{document}$ - tlog J^uT $end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id=\"M6\">begin{document}$ mu_t $end{document}</tex-math></inline-formula> is <inline-formula><tex-math id=\"M7\">begin{document}$ T $end{document}</tex-math></inline-formula>-adapted. (In particular, the SRB measure is the unique equilibrium state for <inline-formula><tex-math id=\"M8\">begin{document}$ - log J^uT $end{document}</tex-math></inline-formula>.) We show that <inline-formula><tex-math id=\"M9\">begin{document}$ mu_t $end{document}</tex-math></inline-formula> is exponentially mixing for Hölder observables, and the pressure function <inline-formula><tex-math id=\"M10\">begin{document}$ P(t) = sup_mu {h_mu -int tlog J^uT d mu} $end{document}</tex-math></inline-formula> is analytic on <inline-formula><tex-math id=\"M11\">begin{document}$ (0,t_*) $end{document}</tex-math></inline-formula>. In addition, <inline-formula><tex-math id=\"M12\">begin{document}$ P(t) $end{document}</tex-math></inline-formula> is strictly convex if and only if <inline-formula><tex-math id=\"M13\">begin{document}$ log J^uT $end{document}</tex-math></inline-formula> is not <inline-formula><tex-math id=\"M14\">begin{document}$ mu_t $end{document}</tex-math></inline-formula>-a.e. cohomologous to a constant, while, if there exist <inline-formula><tex-math id=\"M15\">begin{document}$ t_ane t_b $end{document}</tex-math></inline-formula> with <inline-formula><tex-math id=\"M16\">begin{document}$ mu_{t_a} = mu_{t_b} $end{document}</tex-math></inline-formula>, then <inline-formula><tex-math id=\"M17\">begin{document}$ P(t) $end{document}</tex-math></inline-formula> is affine on <inline-formula><tex-math id=\"M18\">begin{document}$ (0,t_*) $end{document}</tex-math></inline-formula>. An additional sparse recurrence condition gives <inline-formula><tex-math id=\"M19\">begin{document}$ lim_{tdownarrow 0} P(t) = P(0) $end{document}</tex-math></inline-formula>.</p>","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42018207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the cocompact lattices begin{document}$ Gammasubset SO(n, 1) $end{document} so that the Laplace–Beltrami operator begin{document}$ Delta $end{document} on begin{document}$ SO(n)backslash SO(n, 1)/Gamma $end{document} has eigenvalues in begin{document}$ (0, frac{1}{4}) $end{document}, and then show that there exist time-changes of unipotent flows on begin{document}$ SO(n, 1)/Gamma $end{document} that are not measurably conjugate to the unperturbed ones. A main ingredient of the proof is a stronger version of the branching of the complementary series. Combining it with a refinement of the works of Ratner and Flaminio–Forni is adequate for our purpose.
{"title":"New time-changes of unipotent flows on quotients of Lorentz groups","authors":"Siyuan Tang","doi":"10.3934/jmd.2022002","DOIUrl":"https://doi.org/10.3934/jmd.2022002","url":null,"abstract":"<p style='text-indent:20px;'>We study the cocompact lattices <inline-formula><tex-math id=\"M1\">begin{document}$ Gammasubset SO(n, 1) $end{document}</tex-math></inline-formula> so that the Laplace–Beltrami operator <inline-formula><tex-math id=\"M2\">begin{document}$ Delta $end{document}</tex-math></inline-formula> on <inline-formula><tex-math id=\"M3\">begin{document}$ SO(n)backslash SO(n, 1)/Gamma $end{document}</tex-math></inline-formula> has eigenvalues in <inline-formula><tex-math id=\"M4\">begin{document}$ (0, frac{1}{4}) $end{document}</tex-math></inline-formula>, and then show that there exist time-changes of unipotent flows on <inline-formula><tex-math id=\"M5\">begin{document}$ SO(n, 1)/Gamma $end{document}</tex-math></inline-formula> that are not measurably conjugate to the unperturbed ones. A main ingredient of the proof is a stronger version of the branching of the complementary series. Combining it with a refinement of the works of Ratner and Flaminio–Forni is adequate for our purpose.</p>","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48351004","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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