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Symmetry of entropy in higher rank diagonalizable actions and measure classification 高阶可对角动作中熵的对称性与测度分类
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2018-03-21 DOI: 10.3934/jmd.2018016
M. Einsiedler, E. Lindenstrauss
{"title":"Symmetry of entropy in higher rank diagonalizable actions and measure classification","authors":"M. Einsiedler, E. Lindenstrauss","doi":"10.3934/jmd.2018016","DOIUrl":"https://doi.org/10.3934/jmd.2018016","url":null,"abstract":"An important consequence of the theory of entropy of begin{document}$ mathbb{Z}$end{document} -actions is that the events measurable with respect to the far future coincide (modulo null sets) with those measurable with respect to the distant past, and that measuring the entropy using the past will give the same value as measuring it using the future. In this paper we show that for measures invariant under multiparameter algebraic actions if the entropy attached to coarse Lyapunov foliations fail to display a stronger symmetry property of a similar type this forces the measure to be invariant under non-trivial unipotent groups. Some consequences of this phenomenon are noted.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48856668","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: 1
The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces 对称平移曲面Veech–McMullen族上的Kontsevich–Zorich共循环
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2017-10-31 DOI: 10.3934/JMD.2019002
A. Avila, C. Matheus, J. Yoccoz
{"title":"The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces","authors":"A. Avila, C. Matheus, J. Yoccoz","doi":"10.3934/JMD.2019002","DOIUrl":"https://doi.org/10.3934/JMD.2019002","url":null,"abstract":"We describe the Kontsevich--Zorich cocycle over an affine invariant orbifold coming from a (cyclic) covering construction inspired by works of Veech and McMullen. In particular, using the terminology in a recent paper of Filip, we show that all cases of Kontsevich--Zorich monodromies of $SU(p,q)$ type are realized by appropriate covering constructions.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42421215","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: 5
Entropy and quasimorphisms 熵与拟态射
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2017-07-19 DOI: 10.3934/JMD.2019017
Michael Brandenbursky, Michał Marcinkowski
{"title":"Entropy and quasimorphisms","authors":"Michael Brandenbursky, Michał Marcinkowski","doi":"10.3934/JMD.2019017","DOIUrl":"https://doi.org/10.3934/JMD.2019017","url":null,"abstract":"Let $S$ be a compact oriented surface. We construct homogeneous quasimorphisms on $Diff(S, area)$, on $Diff_0(S, area)$ and on $Ham(S)$ generalizing the constructions of Gambaudo-Ghys and Polterovich. \u0000We prove that there are infinitely many linearly independent homogeneous quasimorphisms on $Diff(S, area)$, on $Diff_0(S, area)$ and on $Ham(S)$ whose absolute values bound from below the topological entropy. In case when $S$ has a positive genus, the quasimorphisms we construct on $Ham(S)$ are $C^0$-continuous. \u0000We define a bi-invariant metric on these groups, called the entropy metric, and show that it is unbounded. In particular, we reprove the fact that the autonomous metric on $Ham(S)$ is unbounded.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46387009","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: 16
Equidistribution of saddle connections on translation surfaces 平动面上鞍座连接的均匀分布
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2017-05-30 DOI: 10.3934/JMD.2019004
B. Dozier
{"title":"Equidistribution of saddle connections on translation surfaces","authors":"B. Dozier","doi":"10.3934/JMD.2019004","DOIUrl":"https://doi.org/10.3934/JMD.2019004","url":null,"abstract":"Fix a translation surface $X$, and consider the measures on $X$ coming from averaging the uniform measures on all the saddle connections of length at most $R$. Then as $Rtoinfty$, the weak limit of these measures exists and is equal to the Lebesgue measure on $X$. We also show that any weak limit of a subsequence of the counting measures on $S^1$ given by the angles of all saddle connections of length at most $R_n$, as $R_ntoinfty$, is in the Lebesgue measure class. The proof of the first result uses the second result, together with the result of Kerckhoff-Masur-Smillie that the directional flow on a surface is uniquely ergodic in almost every direction.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46467957","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: 13
Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds Benoist 3-流形的Bowen–Margulis测度的遍历性
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2017-05-23 DOI: 10.3934/jmd.2020011
Harrison Bray
{"title":"Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds","authors":"Harrison Bray","doi":"10.3934/jmd.2020011","DOIUrl":"https://doi.org/10.3934/jmd.2020011","url":null,"abstract":"We study the geodesic flow of a class of 3-manifolds introduced by Benoist which have some hyperbolicity but are non-Riemannian, not CAT(0), and with non-C^1 geodesic flow. The geometries are nonstrictly convex Hilbert geometries in dimension three which admit compact quotient manifolds by discrete groups of projective transformations. We prove the Patterson-Sullivan density is canonical, with applications to counting, and construct explicitly the Bowen-Margulis measure of maximal entropy. The main result of this work is ergodicity of the Bowen-Margulis measure.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43177942","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: 8
Long hitting time for translation flows and L-shaped billiards 翻译流和l形台球击球时间长
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2017-05-09 DOI: 10.3934/JMD.2019011
Dong Han Kim, L. Marchese, S. Marmi
{"title":"Long hitting time for translation flows and L-shaped billiards","authors":"Dong Han Kim, L. Marchese, S. Marmi","doi":"10.3934/JMD.2019011","DOIUrl":"https://doi.org/10.3934/JMD.2019011","url":null,"abstract":"We consider the flow in direction $theta$ on a translation surface and we study the asymptotic behavior for $rto 0$ of the time needed by orbits to hit the $r$-neighborhood of a prescribed point, or more precisely the exponent of the corresponding power law, which is known as hitting time. For flat tori the limsup of hitting time is equal to the diophantine type of the direction $theta$. In higher genus, we consider a generalized geometric notion of diophantine type of a direction $theta$ and we seek for relations with hitting time. For genus two surfaces with just one conical singularity we prove that the limsup of hitting time is always less or equal to the square of the diophantine type. For any square-tiled surface with the same topology the diophantine type itself is a lower bound, and any value between the two bounds can be realized, moreover this holds also for a larger class of origamis satisfying a specific topological assumption. Finally, for the so-called Eierlegende Wollmilchsau origami, the equality between limsup of hitting time and diophantine type subsists. Our results apply to L-shaped billiards.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46462118","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: 3
Rational ergodicity of step function skew products 阶跃函数偏积的有理遍历性
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2017-03-27 DOI: 10.3934/jmd.2018012
J. Aaronson, Michael Bromberg, Nishant Chandgotia
{"title":"Rational ergodicity of step function skew products","authors":"J. Aaronson, Michael Bromberg, Nishant Chandgotia","doi":"10.3934/jmd.2018012","DOIUrl":"https://doi.org/10.3934/jmd.2018012","url":null,"abstract":"We study rational step function skew products over certain rotations of the circle proving ergodicity and bounded rational ergodicity when the rotation number is a quadratic irrational. The latter arises from a consideration of the asymptotic temporal statistics of an orbit as modelled by an associated affine random walk.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44539501","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: 1
From odometers to circular systems: A global structure theorem 从里程计到圆形系统：一个全局结构定理
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2017-03-21 DOI: 10.3934/jmd.2019024
M. Foreman, B. Weiss
{"title":"From odometers to circular systems: A global structure theorem","authors":"M. Foreman, B. Weiss","doi":"10.3934/jmd.2019024","DOIUrl":"https://doi.org/10.3934/jmd.2019024","url":null,"abstract":"The main result of this paper is that two large collections of ergodic measure preserving systems, the Odometer Based and the Circular Systems have the same global structure with respect to joinings. The classes are canonically isomorphic by a continuous map that takes factor maps to factor maps, measure-isomorphisms to measure-isomorphisms, weakly mixing extensions to weakly mixing extensions and compact extensions to compact extensions. The first class includes all finite entropy ergodic transformations with an odometer factor. By results in a previous paper, the second class contains all transformations realizable as diffeomorphisms using the strongly uniform untwisted Anosov-Katok method. An application of the main result will appear in a forthcoming paper that shows that the diffeomorphisms of the torus are inherently unclassifiable up to measure-isomorphism. Other consequences include the existence measure distal diffeomorphisms of arbitrary countable distal height.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45368108","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: 12
An effective version of Katok's horseshoe theorem for conservative $C^2$ surface diffeomorphisms Katok的马蹄铁定理在保守的C^2曲面微分同态中的一个有效版本
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2017-03-18 DOI: 10.3934/JMD.2017017
B. Fayad, Zhiyuan Zhang
{"title":"An effective version of Katok's horseshoe theorem for conservative $C^2$ surface diffeomorphisms","authors":"B. Fayad, Zhiyuan Zhang","doi":"10.3934/JMD.2017017","DOIUrl":"https://doi.org/10.3934/JMD.2017017","url":null,"abstract":"For area preserving C2 surface diffeomorphisms, we give an explicit finite information condition on the exponential growth of the number of Bowen's (n, δ)-balls needed to cover a positive proportion of the space, that is sufficient to guarantee positive topological entropy. This can be seen as an effective version of Katok's horseshoe theorem in the conservative setting. We also show that the analogous result is false in dimension larger than 3.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42609928","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: 2
Rigidity of square-tiled interval exchange transformations 方形平铺区间交换变换的刚度
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2017-02-20 DOI: 10.3934/JMD.2019006
S. Ferenczi, P. Hubert
{"title":"Rigidity of square-tiled interval exchange transformations","authors":"S. Ferenczi, P. Hubert","doi":"10.3934/JMD.2019006","DOIUrl":"https://doi.org/10.3934/JMD.2019006","url":null,"abstract":"We look at interval exchange transformations defined as first return maps on the set of diagonals of a flow of direction begin{document}$ theta $end{document} on a square-tiled surface: using a combinatorial approach, we show that, when the surface has at least one true singularity both the flow and the interval exchange are rigid if and only if begin{document}$ tantheta $end{document} has bounded partial quotients. Moreover, if all vertices of the squares are singularities of the flat metric, and begin{document}$ tantheta $end{document} has bounded partial quotients, the square-tiled interval exchange transformation begin{document}$ T $end{document} is not of rank one. Finally, for another class of surfaces, those defined by the unfolding of billiards in Veech triangles, we build an uncountable set of rigid directional flows and an uncountable set of rigid interval exchange transformations.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43094228","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: 4
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