We present some works of Corinna Ulcigrai closely related to Diophantine approximations and generalizing classical notions to the context of interval exchange maps, translation surfaces and Teichmüller dynamics.
{"title":"Some arithmetical aspects of renormalization in Teichmüller dynamics: On the occasion of Corinna Ulcigrai winning the Brin Prize","authors":"S. Marmi","doi":"10.3934/jmd.2022006","DOIUrl":"https://doi.org/10.3934/jmd.2022006","url":null,"abstract":"<p style='text-indent:20px;'>We present some works of Corinna Ulcigrai closely related to Diophantine approximations and generalizing classical notions to the context of interval exchange maps, translation surfaces and Teichmüller dynamics.</p>","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70085079","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}
{"title":"The 2020 Michael Brin Prize in Dynamical Systems","authors":"The editors","doi":"10.3934/jmd.2022004","DOIUrl":"https://doi.org/10.3934/jmd.2022004","url":null,"abstract":"<jats:p xml:lang=\"fr\" />","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70085411","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}
{"title":"Urysohn-type theorem under a dynamical constraint: Non-compact case","authors":"A. Fathi","doi":"10.3934/jmd.2022015","DOIUrl":"https://doi.org/10.3934/jmd.2022015","url":null,"abstract":"","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70085214","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 and discuss C. Ulcigrai's results concerning mixing properties of locally Hamiltonian flows, spectral properties of smooth time changes of horocycle flows together with their Möbius orthogonality and the ergodicity problems of directional flows in the wind tree model of Ehrenfest.
{"title":"Ergodicity, mixing, Ratner's properties and disjointness for classical flows: On the research of Corinna Ulcigrai","authors":"M. Lemanczyk","doi":"10.3934/jmd.2022005","DOIUrl":"https://doi.org/10.3934/jmd.2022005","url":null,"abstract":"<p style='text-indent:20px;'>We present and discuss C. Ulcigrai's results concerning mixing properties of locally Hamiltonian flows, spectral properties of smooth time changes of horocycle flows together with their Möbius orthogonality and the ergodicity problems of directional flows in the wind tree model of Ehrenfest.</p>","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70085465","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 provide evidence both for and against a conjectural analogy between geometrically finite infinite covolume Fuchsian groups and the mapping class group of compact non-orientable surfaces. In the positive direction, we show the complement of the limit set is open and dense. Moreover, we show that the limit set of the mapping class group contains the set of uniquely ergodic foliations and is contained in the set of all projective measured foliations not containing any one-sided leaves, establishing large parts of a conjecture of Gendulphe. In the negative direction, we show that a conjectured convex core is not even quasi-convex, in contrast with the geometrically finite setting.
{"title":"The limit set of non-orientable mapping class groups","authors":"Sayantan Khan","doi":"10.3934/jmd.2023007","DOIUrl":"https://doi.org/10.3934/jmd.2023007","url":null,"abstract":"We provide evidence both for and against a conjectural analogy between geometrically finite infinite covolume Fuchsian groups and the mapping class group of compact non-orientable surfaces. In the positive direction, we show the complement of the limit set is open and dense. Moreover, we show that the limit set of the mapping class group contains the set of uniquely ergodic foliations and is contained in the set of all projective measured foliations not containing any one-sided leaves, establishing large parts of a conjecture of Gendulphe. In the negative direction, we show that a conjectured convex core is not even quasi-convex, in contrast with the geometrically finite setting.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45497985","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 the previous work, we introduced a method for constructing invariant probability measures of a large class of non-singular volume-preserving flows on closed, oriented odd-dimensional smooth manifolds with pseudoholomorphic curve techniques from symplectic geometry. The technique requires existence of certain pseudoholomorphic curves satisfying some weak assumptions. In this work, we appeal to Gromov-Witten theory and Seiberg-Witten theory to construct large classes of examples where these pseudoholomorphic curves exist. Our argument uses neck stretching along with new analytical tools from Fish-Hofer's work on feral pseudoholomorphic curves.
{"title":"Invariant probability measures from pseudoholomorphic curves Ⅱ: Pseudoholomorphic curve constructions","authors":"Rohil Prasad","doi":"10.3934/jmd.2023003","DOIUrl":"https://doi.org/10.3934/jmd.2023003","url":null,"abstract":"In the previous work, we introduced a method for constructing invariant probability measures of a large class of non-singular volume-preserving flows on closed, oriented odd-dimensional smooth manifolds with pseudoholomorphic curve techniques from symplectic geometry. The technique requires existence of certain pseudoholomorphic curves satisfying some weak assumptions. In this work, we appeal to Gromov-Witten theory and Seiberg-Witten theory to construct large classes of examples where these pseudoholomorphic curves exist. Our argument uses neck stretching along with new analytical tools from Fish-Hofer's work on feral pseudoholomorphic curves.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44240482","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 introduce a method for constructing invariant probability measures of a large class of non-singular volume-preserving flows on closed, oriented odd-dimensional smooth manifolds using pseudoholomorphic curve techniques from symplectic geometry. These flows include any non-singular volume preserving flow in dimension three, and autonomous Hamiltonian flows on closed, regular energy levels in symplectic manifolds of any dimension. As an application, we use our method to prove the existence of obstructions to unique ergodicity for this class of flows, generalizing results of Taubes and Ginzburg-Niche.
{"title":"Invariant probability measures from pseudoholomorphic curves Ⅰ","authors":"Rohil Prasad","doi":"10.3934/jmd.2023002","DOIUrl":"https://doi.org/10.3934/jmd.2023002","url":null,"abstract":"We introduce a method for constructing invariant probability measures of a large class of non-singular volume-preserving flows on closed, oriented odd-dimensional smooth manifolds using pseudoholomorphic curve techniques from symplectic geometry. These flows include any non-singular volume preserving flow in dimension three, and autonomous Hamiltonian flows on closed, regular energy levels in symplectic manifolds of any dimension. As an application, we use our method to prove the existence of obstructions to unique ergodicity for this class of flows, generalizing results of Taubes and Ginzburg-Niche.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41855269","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}
Dilation surfaces are generalizations of translation surfaces where the transition maps of the atlas are translations and homotheties with a positive ratio. In contrast with translation surfaces, the directional flow on dilation surfaces may contain trajectories accumulating on a limit cycle. Such a limit cycle is called hyperbolic because it induces a nontrivial homothety. It has been conjectured that a dilation surface with no actual hyperbolic closed geodesic is in fact a translation surface. Assuming that a dilation surface contains a horizon saddle connection, we prove that the directions of its hyperbolic closed geodesics form a dense subset of $mathbb{S}^{1}$. We also prove that a dilation surface satisfies the latter property if and only if its directional flow is Morse-Smale in an open dense subset of $mathbb{S}^{1}$.
{"title":"Horizon saddle connections and Morse–Smale dynamics of dilation surfaces","authors":"Guillaume Tahar","doi":"10.3934/jmd.2023012","DOIUrl":"https://doi.org/10.3934/jmd.2023012","url":null,"abstract":"Dilation surfaces are generalizations of translation surfaces where the transition maps of the atlas are translations and homotheties with a positive ratio. In contrast with translation surfaces, the directional flow on dilation surfaces may contain trajectories accumulating on a limit cycle. Such a limit cycle is called hyperbolic because it induces a nontrivial homothety. It has been conjectured that a dilation surface with no actual hyperbolic closed geodesic is in fact a translation surface. Assuming that a dilation surface contains a horizon saddle connection, we prove that the directions of its hyperbolic closed geodesics form a dense subset of $mathbb{S}^{1}$. We also prove that a dilation surface satisfies the latter property if and only if its directional flow is Morse-Smale in an open dense subset of $mathbb{S}^{1}$.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47948778","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 $Gamma$ be a countably infinite group. Given $k in mathbb{N}$, we use $mathrm{Free}(k^Gamma)$ to denote the free part of the Bernoulli shift action of $Gamma$ on $k^Gamma$. Seward and Tucker-Drob showed that there exists a free subshift $mathcal{S} subseteq mathrm{Free}(2^Gamma)$ such that every free Borel action of $Gamma$ on a Polish space admits a Borel $Gamma$-equivariant map to $mathcal{S}$. Here we generalize this result as follows. Let $mathcal{S}$ be a subshift of finite type (for example, $mathcal{S}$ could be the set of all proper colorings of the Cayley graph of $Gamma$ with some finite number of colors). Suppose that $pi colon mathrm{Free}(k^Gamma) to mathcal{S}$ is a continuous $Gamma$-equivariant map and let $mathrm{Stab}(pi)$ be the set of all group elements that fix every point in the image of $pi$. Unless $pi$ is constant, $mathrm{Stab}(pi)$ is a finite normal subgroup of $Gamma$. We prove that there exists a subshift $mathcal{S}' subseteq mathcal{S}$ such that the stabilizer of every point in $mathcal{S}'$ is $mathrm{Stab}(pi)$ and every free Borel action of $Gamma$ on a Polish space admits a Borel $Gamma$-equivariant map to $mathcal{S}'$. In particular, if the shift action of $Gamma$ on the image of $pi$ is faithful (i.e., if $mathrm{Stab}(pi)$ is trivial), then the subshift $mathcal{S}'$ is free. As an application of this general result, we deduce that if $F$ is a finite symmetric subset of $Gamma setminus {mathbf{1}}$ of size $|F| = d geq 1$ and $mathrm{Col}(F, d + 1) subseteq (d+1)^Gamma$ is the set of all proper $(d+1)$-colorings of the Cayley graph of $Gamma$ corresponding to $F$, then there is a free subshift $mathcal{S} subseteq mathrm{Col}(F, d+1)$ such that every free Borel action of $Gamma$ on a Polish space admits a Borel $Gamma$-equivariant map to $mathcal{S}$.
{"title":"Equivariant maps to subshifts whose points have small stabilizers","authors":"Anton Bernshteyn","doi":"10.3934/jmd.2023001","DOIUrl":"https://doi.org/10.3934/jmd.2023001","url":null,"abstract":"Let $Gamma$ be a countably infinite group. Given $k in mathbb{N}$, we use $mathrm{Free}(k^Gamma)$ to denote the free part of the Bernoulli shift action of $Gamma$ on $k^Gamma$. Seward and Tucker-Drob showed that there exists a free subshift $mathcal{S} subseteq mathrm{Free}(2^Gamma)$ such that every free Borel action of $Gamma$ on a Polish space admits a Borel $Gamma$-equivariant map to $mathcal{S}$. Here we generalize this result as follows. Let $mathcal{S}$ be a subshift of finite type (for example, $mathcal{S}$ could be the set of all proper colorings of the Cayley graph of $Gamma$ with some finite number of colors). Suppose that $pi colon mathrm{Free}(k^Gamma) to mathcal{S}$ is a continuous $Gamma$-equivariant map and let $mathrm{Stab}(pi)$ be the set of all group elements that fix every point in the image of $pi$. Unless $pi$ is constant, $mathrm{Stab}(pi)$ is a finite normal subgroup of $Gamma$. We prove that there exists a subshift $mathcal{S}' subseteq mathcal{S}$ such that the stabilizer of every point in $mathcal{S}'$ is $mathrm{Stab}(pi)$ and every free Borel action of $Gamma$ on a Polish space admits a Borel $Gamma$-equivariant map to $mathcal{S}'$. In particular, if the shift action of $Gamma$ on the image of $pi$ is faithful (i.e., if $mathrm{Stab}(pi)$ is trivial), then the subshift $mathcal{S}'$ is free. As an application of this general result, we deduce that if $F$ is a finite symmetric subset of $Gamma setminus {mathbf{1}}$ of size $|F| = d geq 1$ and $mathrm{Col}(F, d + 1) subseteq (d+1)^Gamma$ is the set of all proper $(d+1)$-colorings of the Cayley graph of $Gamma$ corresponding to $F$, then there is a free subshift $mathcal{S} subseteq mathrm{Col}(F, d+1)$ such that every free Borel action of $Gamma$ on a Polish space admits a Borel $Gamma$-equivariant map to $mathcal{S}$.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48795471","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 consider the derivative begin{document}$ Dpi $end{document} of the projection begin{document}$ pi $end{document} from a stratum of Abelian or quadratic differentials to Teichmüller space. A closed one-form begin{document}$ eta $end{document} determines a relative cohomology class begin{document}$ [eta]_Sigma $end{document}, which is a tangent vector to the stratum. We give an integral formula for the pairing of begin{document}$ Dpi([eta]_Sigma) $end{document} with a cotangent vector to Teichmüller space (a quadratic differential). We derive from this a comparison between Hodge and Teichmüller norms, which has been used in the work of Arana-Herrera on effective dynamics of mapping class groups, and which may clarify the relationship between dynamical and geometric hyperbolicity results in Teichmüller theory.
We consider the derivative begin{document}$ Dpi $end{document} of the projection begin{document}$ pi $end{document} from a stratum of Abelian or quadratic differentials to Teichmüller space. A closed one-form begin{document}$ eta $end{document} determines a relative cohomology class begin{document}$ [eta]_Sigma $end{document}, which is a tangent vector to the stratum. We give an integral formula for the pairing of begin{document}$ Dpi([eta]_Sigma) $end{document} with a cotangent vector to Teichmüller space (a quadratic differential). We derive from this a comparison between Hodge and Teichmüller norms, which has been used in the work of Arana-Herrera on effective dynamics of mapping class groups, and which may clarify the relationship between dynamical and geometric hyperbolicity results in Teichmüller theory.
{"title":"Hodge and Teichmüller","authors":"Jeremy A. Kahn, A. Wright","doi":"10.3934/jmd.2022007","DOIUrl":"https://doi.org/10.3934/jmd.2022007","url":null,"abstract":"<p style='text-indent:20px;'>We consider the derivative <inline-formula><tex-math id=\"M1\">begin{document}$ Dpi $end{document}</tex-math></inline-formula> of the projection <inline-formula><tex-math id=\"M2\">begin{document}$ pi $end{document}</tex-math></inline-formula> from a stratum of Abelian or quadratic differentials to Teichmüller space. A closed one-form <inline-formula><tex-math id=\"M3\">begin{document}$ eta $end{document}</tex-math></inline-formula> determines a relative cohomology class <inline-formula><tex-math id=\"M4\">begin{document}$ [eta]_Sigma $end{document}</tex-math></inline-formula>, which is a tangent vector to the stratum. We give an integral formula for the pairing of <inline-formula><tex-math id=\"M5\">begin{document}$ Dpi([eta]_Sigma) $end{document}</tex-math></inline-formula> with a cotangent vector to Teichmüller space (a quadratic differential). We derive from this a comparison between Hodge and Teichmüller norms, which has been used in the work of Arana-Herrera on effective dynamics of mapping class groups, and which may clarify the relationship between dynamical and geometric hyperbolicity results in Teichmüller theory.</p>","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70085152","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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