We study thermodynamic formalism for topologically transitive partially hyperbolic systems in which the center-stable bundle satisfies a bounded expansion property, and show that every potential function satisfying the Bowen property has a unique equilibrium measure. Our method is to use tools from geometric measure theory to construct a suitable family of reference measures on unstable leaves as a dynamical analogue of Hausdorff measure, and then show that the averaged pushforwards of these measures converge to a measure that has the Gibbs property and is the unique equilibrium measure.
{"title":"Equilibrium measures for some partially hyperbolic systems","authors":"V. Climenhaga, Y. Pesin, Agnieszka Zelerowicz","doi":"10.3934/jmd.2020006","DOIUrl":"https://doi.org/10.3934/jmd.2020006","url":null,"abstract":"We study thermodynamic formalism for topologically transitive partially hyperbolic systems in which the center-stable bundle satisfies a bounded expansion property, and show that every potential function satisfying the Bowen property has a unique equilibrium measure. Our method is to use tools from geometric measure theory to construct a suitable family of reference measures on unstable leaves as a dynamical analogue of Hausdorff measure, and then show that the averaged pushforwards of these measures converge to a measure that has the Gibbs property and is the unique equilibrium measure.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48274690","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 construct an example of a Teichmuller geodesic ray whose limit set in the Thurston boundary of Teichmuller space is a begin{document}$ d$end{document} -dimensional simplex.
We construct an example of a Teichmuller geodesic ray whose limit set in the Thurston boundary of Teichmuller space is a begin{document}$ d$end{document} -dimensional simplex.
{"title":"Teichmüller geodesics with begin{document}$ d$end{document} -dimensional limit sets","authors":"Anna Lenzhen, Babak Modami, Kasra Rafi","doi":"10.3934/jmd.2018010","DOIUrl":"https://doi.org/10.3934/jmd.2018010","url":null,"abstract":"We construct an example of a Teichmuller geodesic ray whose limit set in the Thurston boundary of Teichmuller space is a begin{document}$ d$end{document} -dimensional simplex.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42903507","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 symbolic finite-to-one extensions of the type constructed by O. Sarig for surface diffeomorphisms induce H"older-continuous conjugacies on large sets. We deduce this from their Bowen property. This notion, introduced in a joint work with M. Boyle, generalizes a fact first observed by R. Bowen for Markov partitions. We rely on the notion of degree from finite equivalence theory and magic word isomorphisms. As an application, we give lower bounds on the number of periodic points first for surface diffeomorphisms (improving a result of Sarig) and for Sinai billiards maps (building on a result of Baladi and Demers). Finally we characterize surface diffeomorphisms admitting a H"older-continuous coding of all their aperiodic hyperbolic measures and give a slightly weaker construction preserving local compactness.
{"title":"The degree of Bowen factors and injective codings of diffeomorphisms","authors":"J. Buzzi","doi":"10.3934/jmd.2020001","DOIUrl":"https://doi.org/10.3934/jmd.2020001","url":null,"abstract":"We show that symbolic finite-to-one extensions of the type constructed by O. Sarig for surface diffeomorphisms induce H\"older-continuous conjugacies on large sets. We deduce this from their Bowen property. This notion, introduced in a joint work with M. Boyle, generalizes a fact first observed by R. Bowen for Markov partitions. We rely on the notion of degree from finite equivalence theory and magic word isomorphisms. As an application, we give lower bounds on the number of periodic points first for surface diffeomorphisms (improving a result of Sarig) and for Sinai billiards maps (building on a result of Baladi and Demers). Finally we characterize surface diffeomorphisms admitting a H\"older-continuous coding of all their aperiodic hyperbolic measures and give a slightly weaker construction preserving local compactness.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41568820","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 almost every flat surface the sequence of saddle connection lengths listed in increasing order is uniformly distributed mod one.
对于几乎每一个平面，鞍座连接长度按递增顺序排列的顺序均分布于模1。
{"title":"Uniform distribution of saddle connection lengths (with an appendix by Daniel El-Baz and Bingrong Huang)","authors":"J. Chaika, D. Robertson","doi":"10.3934/jmd.2019023","DOIUrl":"https://doi.org/10.3934/jmd.2019023","url":null,"abstract":"For almost every flat surface the sequence of saddle connection lengths listed in increasing order is uniformly distributed mod one.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46477463","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 present a separation property for the gaps in the length spectrum of a compact Riemannian manifold with negative curvature. In arbitrary small neighborhoods of the metric for some suitable topology, we show that there are negatively curved metrics with a length spectrum exponentially separated from below. This property was previously known to be false generically.
{"title":"Exponential gaps in the length spectrum","authors":"E. Schenck","doi":"10.3934/jmd.2020007","DOIUrl":"https://doi.org/10.3934/jmd.2020007","url":null,"abstract":"We present a separation property for the gaps in the length spectrum of a compact Riemannian manifold with negative curvature. In arbitrary small neighborhoods of the metric for some suitable topology, we show that there are negatively curved metrics with a length spectrum exponentially separated from below. This property was previously known to be false generically.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44540724","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} $0 . We consider the one-parameter family of circle begin{document} $lambda$ end{document} -affine contractions begin{document} $f_delta:x in [0,1) mapsto lambda x + delta ; {rm mod},1 $ end{document} , where begin{document} $0 le delta . Let begin{document} $rho$ end{document} be the rotation number of the map begin{document} $f_delta$ end{document} . We will give some numerical relations between the values of begin{document} $lambda,delta$ end{document} and begin{document} $rho$ end{document} , essentially using Hecke-Mahler series and a tree structure. When both parameters begin{document} $lambda$ end{document} and begin{document} $delta$ end{document} are algebraic numbers, we show that begin{document} $rho$ end{document} is a rational number. Moreover, in the case begin{document} $lambda$ end{document} and begin{document} $delta$ end{document} are rational, we give an explicit upper bound for the height of begin{document} $rho$ end{document} under some assumptions on begin{document} $lambda$ end{document} .
{"title":"Rotation number of contracted rotations","authors":"M. Laurent, A. Nogueira","doi":"10.3934/JMD.2018007","DOIUrl":"https://doi.org/10.3934/JMD.2018007","url":null,"abstract":"Let begin{document} $0 . We consider the one-parameter family of circle begin{document} $lambda$ end{document} -affine contractions begin{document} $f_delta:x in [0,1) mapsto lambda x + delta ; {rm mod},1 $ end{document} , where begin{document} $0 le delta . Let begin{document} $rho$ end{document} be the rotation number of the map begin{document} $f_delta$ end{document} . We will give some numerical relations between the values of begin{document} $lambda,delta$ end{document} and begin{document} $rho$ end{document} , essentially using Hecke-Mahler series and a tree structure. When both parameters begin{document} $lambda$ end{document} and begin{document} $delta$ end{document} are algebraic numbers, we show that begin{document} $rho$ end{document} is a rational number. Moreover, in the case begin{document} $lambda$ end{document} and begin{document} $delta$ end{document} are rational, we give an explicit upper bound for the height of begin{document} $rho$ end{document} under some assumptions on begin{document} $lambda$ end{document} .","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48954266","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 homothety surface can be assembled from polygons by identifying their edges in pairs via homotheties, which are compositions of translation and scaling. We consider linear trajectories on a 1-parameter family of genus-2 homothety surfaces. The closure of a trajectory on each of these surfaces always has Hausdorff dimension 1, and contains either a closed loop or a lamination with Cantor cross-section. Trajectories have cutting sequences that are either eventually periodic or eventually Sturmian. Although no two of these surfaces are affinely equivalent, their linear trajectories can be related directly to those on the square torus, and thence to each other, by means of explicit functions. We also briefly examine two related families of surfaces and show that the above behaviors can be mixed; for instance, the closure of a linear trajectory can contain both a closed loop and a lamination.
{"title":"Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces","authors":"Joshua P. Bowman, Slade Sanderson","doi":"10.3934/jmd.2020005","DOIUrl":"https://doi.org/10.3934/jmd.2020005","url":null,"abstract":"A homothety surface can be assembled from polygons by identifying their edges in pairs via homotheties, which are compositions of translation and scaling. We consider linear trajectories on a 1-parameter family of genus-2 homothety surfaces. The closure of a trajectory on each of these surfaces always has Hausdorff dimension 1, and contains either a closed loop or a lamination with Cantor cross-section. Trajectories have cutting sequences that are either eventually periodic or eventually Sturmian. Although no two of these surfaces are affinely equivalent, their linear trajectories can be related directly to those on the square torus, and thence to each other, by means of explicit functions. We also briefly examine two related families of surfaces and show that the above behaviors can be mixed; for instance, the closure of a linear trajectory can contain both a closed loop and a lamination.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42477648","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 prove that if begin{document}$ G $end{document} is a countably infinite group and begin{document}$ (L, lambda) $end{document} and begin{document}$ (K, kappa) $end{document} are probability spaces having equal Shannon entropy, then the Bernoulli shifts begin{document}$ G curvearrowright (L^G, lambda^G) $end{document} and begin{document}$ G curvearrowright (K^G, kappa^G) $end{document} are isomorphic. This extends Ornstein's famous isomorphism theorem to all countably infinite groups. Our proof builds on a slightly weaker theorem by Lewis Bowen in 2011 that required both begin{document}$ lambda $end{document} and begin{document}$ kappa $end{document} have at least begin{document}$ 3 $end{document} points in their support. We furthermore produce finitary isomorphisms in the case where both begin{document}$ L $end{document} and begin{document}$ K $end{document} are finite.
{"title":"Bernoulli shifts with bases of equal entropy are isomorphic","authors":"Brandon Seward","doi":"10.3934/jmd.2022011","DOIUrl":"https://doi.org/10.3934/jmd.2022011","url":null,"abstract":"<p style='text-indent:20px;'>We prove that if <inline-formula><tex-math id=\"M1\">begin{document}$ G $end{document}</tex-math></inline-formula> is a countably infinite group and <inline-formula><tex-math id=\"M2\">begin{document}$ (L, lambda) $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M3\">begin{document}$ (K, kappa) $end{document}</tex-math></inline-formula> are probability spaces having equal Shannon entropy, then the Bernoulli shifts <inline-formula><tex-math id=\"M4\">begin{document}$ G curvearrowright (L^G, lambda^G) $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M5\">begin{document}$ G curvearrowright (K^G, kappa^G) $end{document}</tex-math></inline-formula> are isomorphic. This extends Ornstein's famous isomorphism theorem to all countably infinite groups. Our proof builds on a slightly weaker theorem by Lewis Bowen in 2011 that required both <inline-formula><tex-math id=\"M6\">begin{document}$ lambda $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M7\">begin{document}$ kappa $end{document}</tex-math></inline-formula> have at least <inline-formula><tex-math id=\"M8\">begin{document}$ 3 $end{document}</tex-math></inline-formula> points in their support. We furthermore produce finitary isomorphisms in the case where both <inline-formula><tex-math id=\"M9\">begin{document}$ L $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M10\">begin{document}$ K $end{document}</tex-math></inline-formula> are finite.</p>","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45677856","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 prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, analogous to pointwise ergodic theorems for group actions, replacing the group with a Schreier graph of the action. For any quasi-pmp graph, the theorem gives an increasing sequence of Borel subgraphs with finite connected components along which the averages of $L^1$ functions converge to their expectations. Equivalently, it states that any (not necessarily pmp) locally countable Borel graph on a standard probability space contains an ergodic hyperfinite subgraph. The pmp version of this theorem was first proven by R. Tucker-Drob using probabilistic methods. Our proof is different: it is descriptive set theoretic and applies more generally to quasi-pmp graphs. Among other things, it involves introducing a graph invariant, a method of producing finite equivalence subrelations with large domain, and a simple method of exploiting nonamenability of a measured graph. The non-pmp setting additionally requires a new gadget for analyzing the interplay between the underlying cocycle and the graph.
{"title":"Pointwise ergodic theorem for locally countable quasi-pmp graphs","authors":"A. Tserunyan","doi":"10.3934/jmd.2022019","DOIUrl":"https://doi.org/10.3934/jmd.2022019","url":null,"abstract":"We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, analogous to pointwise ergodic theorems for group actions, replacing the group with a Schreier graph of the action. For any quasi-pmp graph, the theorem gives an increasing sequence of Borel subgraphs with finite connected components along which the averages of $L^1$ functions converge to their expectations. Equivalently, it states that any (not necessarily pmp) locally countable Borel graph on a standard probability space contains an ergodic hyperfinite subgraph. \u0000The pmp version of this theorem was first proven by R. Tucker-Drob using probabilistic methods. Our proof is different: it is descriptive set theoretic and applies more generally to quasi-pmp graphs. Among other things, it involves introducing a graph invariant, a method of producing finite equivalence subrelations with large domain, and a simple method of exploiting nonamenability of a measured graph. The non-pmp setting additionally requires a new gadget for analyzing the interplay between the underlying cocycle and the graph.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46718669","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}
Using methods from symplectic topology, we prove existence of invariant variational measures associated to the flow begin{document}$ phi_H $end{document} of a Hamiltonian begin{document}$ Hin C^{infty}(M) $end{document} on a symplectic manifold begin{document}$ (M, omega) $end{document} . These measures coincide with Mather measures (from Aubry-Mather theory) in the Tonelli case. We compare properties of the supports of these measures to classical Mather measures, and we construct an example showing that their support can be extremely unstable when begin{document}$ H $end{document} fails to be convex, even for nearly integrable begin{document}$ H $end{document} . Parts of these results extend work by Viterbo [ 54 ] and Vichery [ 52 ]. Using ideas due to Entov-Polterovich [ 22 , 40 ], we also detect interesting invariant measures for begin{document}$ phi_H $end{document} by studying a generalization of the symplectic shape of sublevel sets of begin{document}$ H $end{document} . This approach differs from the first one in that it works also for begin{document}$ (M, omega) $end{document} in which every compact subset can be displaced. We present applications to Hamiltonian systems on begin{document}$ mathbb R^{2n} $end{document} and twisted cotangent bundles.
{"title":"Mather theory and symplectic rigidity","authors":"Mads R. Bisgaard","doi":"10.3934/jmd.2019018","DOIUrl":"https://doi.org/10.3934/jmd.2019018","url":null,"abstract":"Using methods from symplectic topology, we prove existence of invariant variational measures associated to the flow begin{document}$ phi_H $end{document} of a Hamiltonian begin{document}$ Hin C^{infty}(M) $end{document} on a symplectic manifold begin{document}$ (M, omega) $end{document} . These measures coincide with Mather measures (from Aubry-Mather theory) in the Tonelli case. We compare properties of the supports of these measures to classical Mather measures, and we construct an example showing that their support can be extremely unstable when begin{document}$ H $end{document} fails to be convex, even for nearly integrable begin{document}$ H $end{document} . Parts of these results extend work by Viterbo [ 54 ] and Vichery [ 52 ]. Using ideas due to Entov-Polterovich [ 22 , 40 ], we also detect interesting invariant measures for begin{document}$ phi_H $end{document} by studying a generalization of the symplectic shape of sublevel sets of begin{document}$ H $end{document} . This approach differs from the first one in that it works also for begin{document}$ (M, omega) $end{document} in which every compact subset can be displaced. We present applications to Hamiltonian systems on begin{document}$ mathbb R^{2n} $end{document} and twisted cotangent bundles.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45161718","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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