We present a new proof of the following theorem of Benoist-Quint: Let begin{document}$ G: = operatorname{SO}^circ(d, 1) $end{document} , begin{document}$ dge 2 $end{document} and begin{document}$ Delta a cocompact lattice. Any orbit of a Zariski dense subgroup begin{document}$ Gamma $end{document} of begin{document}$ G $end{document} is either finite or dense in begin{document}$ Delta backslash G $end{document} . While Benoist and Quint's proof is based on the classification of stationary measures, our proof is topological, using ideas from the study of dynamics of unipotent flows on the infinite volume homogeneous space begin{document}$ Gamma backslash G $end{document} .
{"title":"Topological proof of Benoist-Quint's orbit closure theorem for $ boldsymbol{ operatorname{SO}(d, 1)} $","authors":"Minju M. Lee, H. Oh","doi":"10.3934/jmd.2019021","DOIUrl":"https://doi.org/10.3934/jmd.2019021","url":null,"abstract":"We present a new proof of the following theorem of Benoist-Quint: Let begin{document}$ G: = operatorname{SO}^circ(d, 1) $end{document} , begin{document}$ dge 2 $end{document} and begin{document}$ Delta a cocompact lattice. Any orbit of a Zariski dense subgroup begin{document}$ Gamma $end{document} of begin{document}$ G $end{document} is either finite or dense in begin{document}$ Delta backslash G $end{document} . While Benoist and Quint's proof is based on the classification of stationary measures, our proof is topological, using ideas from the study of dynamics of unipotent flows on the infinite volume homogeneous space begin{document}$ Gamma backslash G $end{document} .","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41852120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give effective estimates for the number of saddle connections on a translation surface that have length begin{document}$ leq L $end{document} and are in a prescribed homology class modulo begin{document}$ q $end{document} . Our estimates apply to almost all translation surfaces in a stratum of the moduli space of translation surfaces, with respect to the Masur–Veech measure on the stratum.
{"title":"Counting saddle connections in a homology class modulo begin{document}$ boldsymbol q $end{document} (with an appendix by Rodolfo Gutiérrez-Romo)","authors":"Michael Magee, René Rühr","doi":"10.3934/JMD.2019020","DOIUrl":"https://doi.org/10.3934/JMD.2019020","url":null,"abstract":"We give effective estimates for the number of saddle connections on a translation surface that have length begin{document}$ leq L $end{document} and are in a prescribed homology class modulo begin{document}$ q $end{document} . Our estimates apply to almost all translation surfaces in a stratum of the moduli space of translation surfaces, with respect to the Masur–Veech measure on the stratum.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46452740","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 $f : [0,1)rightarrow [0,1)$ be a $2$-interval piecewise affine increasing map which is injective but not surjective. Such a map $f$ has a rotation number and can be parametrized by three real numbers. We make fully explicit the dynamics of $f$ thanks to two specific functions $delta$ and $phi$ depending on these parameters whose definitions involve Hecke-Mahler series. As an application, we show that the rotation number of $f$ is rational, when the three parameters are algebraic numbers.
{"title":"Dynamics of 2-interval piecewise affine maps and Hecke-Mahler series","authors":"M. Laurent, A. Nogueira","doi":"10.3934/JMD.2021002","DOIUrl":"https://doi.org/10.3934/JMD.2021002","url":null,"abstract":"Let $f : [0,1)rightarrow [0,1)$ be a $2$-interval piecewise affine increasing map which is injective but not surjective. Such a map $f$ has a rotation number and can be parametrized by three real numbers. We make fully explicit the dynamics of $f$ thanks to two specific functions $delta$ and $phi$ depending on these parameters whose definitions involve Hecke-Mahler series. As an application, we show that the rotation number of $f$ is rational, when the three parameters are algebraic numbers.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47329273","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 show that conservative partially hyperbolic diffeomorphism isotopic to the identity on Seifert 3-manifolds are ergodic.
证明了Seifert 3-流形上恒等式的保守部分双曲微分同胚同位素是遍历的。
{"title":"Ergodicity and partial hyperbolicity on Seifert manifolds","authors":"A. Hammerlindl, J. R. Hertz, R. Ures","doi":"10.3934/jmd.2020012","DOIUrl":"https://doi.org/10.3934/jmd.2020012","url":null,"abstract":"We show that conservative partially hyperbolic diffeomorphism isotopic to the identity on Seifert 3-manifolds are ergodic.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45720471","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}
Consider a symplectic surface $Sigma$ with two properly embedded Hamiltonian isotopic curves $L$ and $L'$. Suppose $g in Ham (Sigma)$ is a Hamiltonian diffeomorphism which sends $L$ to $L'$. Which dynamical properties of $g$ can be detected by the pair $(L, L')$? We discuss two cases where one can deduce that $g$ is `chaotic': non-autonomous or even of positive entropy.
{"title":"Non-autonomous curves on surfaces","authors":"M. Khanevsky","doi":"10.3934/jmd.2021010","DOIUrl":"https://doi.org/10.3934/jmd.2021010","url":null,"abstract":"Consider a symplectic surface $Sigma$ with two properly embedded Hamiltonian isotopic curves $L$ and $L'$. Suppose $g in Ham (Sigma)$ is a Hamiltonian diffeomorphism which sends $L$ to $L'$. Which dynamical properties of $g$ can be detected by the pair $(L, L')$? We discuss two cases where one can deduce that $g$ is `chaotic': non-autonomous or even of positive entropy.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43000074","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 extend the notion of Rauzy induction of interval exchange transformations to the case of toral begin{document}$ mathbb{Z}^2 $end{document}-rotation, i.e., begin{document}$ mathbb{Z}^2 $end{document}-action defined by rotations on a 2-torus. If begin{document}$ mathscr{X}_{mathscr{P}, R} $end{document} denotes the symbolic dynamical system corresponding to a partition begin{document}$ mathscr{P} $end{document} and begin{document}$ mathbb{Z}^2 $end{document}-action begin{document}$ R $end{document} such that begin{document}$ R $end{document} is Cartesian on a sub-domain begin{document}$ W $end{document}, we express the 2-dimensional configurations in begin{document}$ mathscr{X}_{mathscr{P}, R} $end{document} as the image under a begin{document}$ 2 $end{document}-dimensional morphism (up to a shift) of a configuration in begin{document}$ mathscr{X}_{widehat{mathscr{P}}|_W, widehat{R}|_W} $end{document} where begin{document}$ widehat{mathscr{P}}|_W $end{document} is the induced partition and begin{document}$ widehat{R}|_W $end{document} is the induced begin{document}$ mathbb{Z}^2 $end{document}-action on begin{document}$ W $end{document}.
We focus on one example, begin{document}$ mathscr{X}_{mathscr{P}_0, R_0} $end{document}, for which we obtain an eventually periodic sequence of 2-dimensional morphisms. We prove that it is the same as the substitutive structure of the minimal subshift begin{document}$ X_0 $end{document} of the Jeandel–Rao Wang shift computed in an earlier work by the author. As a consequence, begin{document}$ {mathscr{P}}_0 $end{document} is a Markov partition for the associated toral begin{document}$ mathbb{Z}^2 $end{document}-rotation begin{document}$ R_0 $end{document
{"title":"Rauzy induction of polygon partitions and toral $ mathbb{Z}^2 $-rotations","authors":"S'ebastien Labb'e","doi":"10.3934/jmd.2021017","DOIUrl":"https://doi.org/10.3934/jmd.2021017","url":null,"abstract":"<p style='text-indent:20px;'>We extend the notion of Rauzy induction of interval exchange transformations to the case of toral <inline-formula><tex-math id=\"M1\">begin{document}$ mathbb{Z}^2 $end{document}</tex-math></inline-formula>-rotation, i.e., <inline-formula><tex-math id=\"M2\">begin{document}$ mathbb{Z}^2 $end{document}</tex-math></inline-formula>-action defined by rotations on a 2-torus. If <inline-formula><tex-math id=\"M3\">begin{document}$ mathscr{X}_{mathscr{P}, R} $end{document}</tex-math></inline-formula> denotes the symbolic dynamical system corresponding to a partition <inline-formula><tex-math id=\"M4\">begin{document}$ mathscr{P} $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M5\">begin{document}$ mathbb{Z}^2 $end{document}</tex-math></inline-formula>-action <inline-formula><tex-math id=\"M6\">begin{document}$ R $end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id=\"M7\">begin{document}$ R $end{document}</tex-math></inline-formula> is Cartesian on a sub-domain <inline-formula><tex-math id=\"M8\">begin{document}$ W $end{document}</tex-math></inline-formula>, we express the 2-dimensional configurations in <inline-formula><tex-math id=\"M9\">begin{document}$ mathscr{X}_{mathscr{P}, R} $end{document}</tex-math></inline-formula> as the image under a <inline-formula><tex-math id=\"M10\">begin{document}$ 2 $end{document}</tex-math></inline-formula>-dimensional morphism (up to a shift) of a configuration in <inline-formula><tex-math id=\"M11\">begin{document}$ mathscr{X}_{widehat{mathscr{P}}|_W, widehat{R}|_W} $end{document}</tex-math></inline-formula> where <inline-formula><tex-math id=\"M12\">begin{document}$ widehat{mathscr{P}}|_W $end{document}</tex-math></inline-formula> is the induced partition and <inline-formula><tex-math id=\"M13\">begin{document}$ widehat{R}|_W $end{document}</tex-math></inline-formula> is the induced <inline-formula><tex-math id=\"M14\">begin{document}$ mathbb{Z}^2 $end{document}</tex-math></inline-formula>-action on <inline-formula><tex-math id=\"M15\">begin{document}$ W $end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>We focus on one example, <inline-formula><tex-math id=\"M16\">begin{document}$ mathscr{X}_{mathscr{P}_0, R_0} $end{document}</tex-math></inline-formula>, for which we obtain an eventually periodic sequence of 2-dimensional morphisms. We prove that it is the same as the substitutive structure of the minimal subshift <inline-formula><tex-math id=\"M17\">begin{document}$ X_0 $end{document}</tex-math></inline-formula> of the Jeandel–Rao Wang shift computed in an earlier work by the author. As a consequence, <inline-formula><tex-math id=\"M18\">begin{document}$ {mathscr{P}}_0 $end{document}</tex-math></inline-formula> is a Markov partition for the associated toral <inline-formula><tex-math id=\"M19\">begin{document}$ mathbb{Z}^2 $end{document}</tex-math></inline-formula>-rotation <inline-formula><tex-math id=\"M20\">begin{document}$ R_0 $end{document","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46200568","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}
In this note we prove that if a closed monotone symplectic manifold $M$ of dimension $2n,$ satisfying a homological condition, that holds in particular when the minimal Chern number is $N>n,$ admits a Hamiltonian pseudorotation, then the quantum Steenrod square of the point class must be deformed. This gives restrictions on the existence of pseudorotations. Our methods rest on previous work of the author, Zhao, and Wilkins, going back to the equivariant pair-of-pants product-isomorphism of Seidel.
{"title":"Pseudo-rotations and Steenrod squares","authors":"E. Shelukhin","doi":"10.3934/jmd.2020010","DOIUrl":"https://doi.org/10.3934/jmd.2020010","url":null,"abstract":"In this note we prove that if a closed monotone symplectic manifold $M$ of dimension $2n,$ satisfying a homological condition, that holds in particular when the minimal Chern number is $N>n,$ admits a Hamiltonian pseudorotation, then the quantum Steenrod square of the point class must be deformed. This gives restrictions on the existence of pseudorotations. Our methods rest on previous work of the author, Zhao, and Wilkins, going back to the equivariant pair-of-pants product-isomorphism of Seidel.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43147659","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 phenomenon of robust bifurcations in the space of holomorphic maps of begin{document}$ mathbb{P}^2(mathbb{C}) $end{document} . We prove that any Lattes example of sufficiently high degree belongs to the closure of the interior of the bifurcation locus. In particular, every Lattes map has an iterate with this property. To show this, we design a method creating robust intersections between the limit set of a particular type of iterated functions system in begin{document}$ mathbb{C}^2 $end{document} with a well-oriented complex curve. Then we show that any Lattes map of sufficiently high degree can be perturbed so that the perturbed map exhibits this geometry.
{"title":"Lattès maps and the interior of the bifurcation locus","authors":"S'ebastien Biebler","doi":"10.3934/JMD.2019014","DOIUrl":"https://doi.org/10.3934/JMD.2019014","url":null,"abstract":"We study the phenomenon of robust bifurcations in the space of holomorphic maps of begin{document}$ mathbb{P}^2(mathbb{C}) $end{document} . We prove that any Lattes example of sufficiently high degree belongs to the closure of the interior of the bifurcation locus. In particular, every Lattes map has an iterate with this property. To show this, we design a method creating robust intersections between the limit set of a particular type of iterated functions system in begin{document}$ mathbb{C}^2 $end{document} with a well-oriented complex curve. Then we show that any Lattes map of sufficiently high degree can be perturbed so that the perturbed map exhibits this geometry.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44075438","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 the group of orientation-preserving isometries of a rank-one symmetric space $X$ of non-compact type. We study local rigidity of certain actions of a solvable subgroup $Gamma subset G$ on the boundary of $X$, which is diffeomorphic to a sphere. When $X$ is a quaternionic hyperbolic space or the Cayley hyperplane, the action we constructed is locally rigid.
{"title":"Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces","authors":"Mao Okada","doi":"10.3934/JMD.2021004","DOIUrl":"https://doi.org/10.3934/JMD.2021004","url":null,"abstract":"Let $G$ be the group of orientation-preserving isometries of a rank-one symmetric space $X$ of non-compact type. We study local rigidity of certain actions of a solvable subgroup $Gamma subset G$ on the boundary of $X$, which is diffeomorphic to a sphere. When $X$ is a quaternionic hyperbolic space or the Cayley hyperplane, the action we constructed is locally rigid.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49164934","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 generalize the notion of cusp excursion of geodesic rays by introducing for any $k geq 1$ the $k^{th}$ excursion in the cusps of a hyperbolic $N$-manifold of finite volume. We show that on one hand, this excursion is at most linear for geodesics that are generic with respect to the hitting measure of a random walk. On the other hand, for $k geq N-1$, the $k^{th}$ excursion is superlinear for geodesics that are generic with respect to the Lebesgue measure. We use this to show that the hitting measure and the Lebesgue measure on the boundary of hyperbolic space $mathbb{H}^N$ for any $N geq 2$ are mutually singular.
{"title":"Cusp excursion in hyperbolic manifolds and singularity of harmonic measure","authors":"Anja Randecker, G. Tiozzo","doi":"10.3934/JMD.2021006","DOIUrl":"https://doi.org/10.3934/JMD.2021006","url":null,"abstract":"We generalize the notion of cusp excursion of geodesic rays by introducing for any $k geq 1$ the $k^{th}$ excursion in the cusps of a hyperbolic $N$-manifold of finite volume. We show that on one hand, this excursion is at most linear for geodesics that are generic with respect to the hitting measure of a random walk. On the other hand, for $k geq N-1$, the $k^{th}$ excursion is superlinear for geodesics that are generic with respect to the Lebesgue measure. We use this to show that the hitting measure and the Lebesgue measure on the boundary of hyperbolic space $mathbb{H}^N$ for any $N geq 2$ are mutually singular.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43823300","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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