We prove that there are no Shimura-Teichm"uller curves generated by genus five translation surfaces, thereby completing the classification of Shimura-Teichm"uller curves in general. This was conjectured by M"oller in his original work introducing Shimura-Teichm"uller curves. Moreover, the property of being a Shimura-Teichm"uller curve is equivalent to having completely degenerate Kontsevich-Zorich spectrum. The main new ingredient comes from the work of Hu and the second named author, which facilitates calculations of higher order terms in the period matrix with respect to plumbing coordinates. A large computer search is implemented to exclude the remaining cases, which must be performed in a very specific way to be computationally feasible.
{"title":"Shimura–Teichmüller curves in genus 5","authors":"D. Aulicino, C. Norton","doi":"10.3934/jmd.2020009","DOIUrl":"https://doi.org/10.3934/jmd.2020009","url":null,"abstract":"We prove that there are no Shimura-Teichm\"uller curves generated by genus five translation surfaces, thereby completing the classification of Shimura-Teichm\"uller curves in general. This was conjectured by M\"oller in his original work introducing Shimura-Teichm\"uller curves. Moreover, the property of being a Shimura-Teichm\"uller curve is equivalent to having completely degenerate Kontsevich-Zorich spectrum. \u0000The main new ingredient comes from the work of Hu and the second named author, which facilitates calculations of higher order terms in the period matrix with respect to plumbing coordinates. A large computer search is implemented to exclude the remaining cases, which must be performed in a very specific way to be computationally feasible.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44252357","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}$mathscr{H}$end{document} denote a connected component of a stratum of translation surfaces. We show that the Siegel-Veech transform of a bounded compactly supported function on begin{document}$mathbb{R}^2$end{document} is in begin{document}$L^2(mathscr{H}, mu)$end{document} , where begin{document}$mu$end{document} is the Lebesgue measure on begin{document}$mathscr{H}$end{document} , and give applications to bounding error terms for counting problems for saddle connections. We also propose a new invariant associated to begin{document}$SL(2,mathbb{R})$end{document} -invariant measures on strata satisfying certain integrability conditions.
Let begin{document}$mathscr{H}$end{document} denote a connected component of a stratum of translation surfaces. We show that the Siegel-Veech transform of a bounded compactly supported function on begin{document}$mathbb{R}^2$end{document} is in begin{document}$L^2(mathscr{H}, mu)$end{document} , where begin{document}$mu$end{document} is the Lebesgue measure on begin{document}$mathscr{H}$end{document} , and give applications to bounding error terms for counting problems for saddle connections. We also propose a new invariant associated to begin{document}$SL(2,mathbb{R})$end{document} -invariant measures on strata satisfying certain integrability conditions.
{"title":"Siegel–Veech transforms are in begin{document}$ boldsymbol{L^2} $end{document}(with an appendix by Jayadev S. Athreya and Rene Rühr)","authors":"J. Athreya, Y. Cheung, H. Masur","doi":"10.3934/JMD.2019001","DOIUrl":"https://doi.org/10.3934/JMD.2019001","url":null,"abstract":"Let begin{document}$mathscr{H}$end{document} denote a connected component of a stratum of translation surfaces. We show that the Siegel-Veech transform of a bounded compactly supported function on begin{document}$mathbb{R}^2$end{document} is in begin{document}$L^2(mathscr{H}, mu)$end{document} , where begin{document}$mu$end{document} is the Lebesgue measure on begin{document}$mathscr{H}$end{document} , and give applications to bounding error terms for counting problems for saddle connections. We also propose a new invariant associated to begin{document}$SL(2,mathbb{R})$end{document} -invariant measures on strata satisfying certain integrability conditions.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44616222","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 class of objects which we call 'dilation surfaces'. These provide families of foliations on surfaces whose dynamics we are interested in. We present and analyze a couple of examples, and we define concepts related to these in order to motivate several questions and open problems. In particular we generalize the notion of Veech group to dilation surfaces, and we prove a structure result about these Veech groups.
{"title":"Dilation surfaces and their Veech groups","authors":"E. Duryev, C. Fougeron, Selim Ghazouani","doi":"10.3934/JMD.2019005","DOIUrl":"https://doi.org/10.3934/JMD.2019005","url":null,"abstract":"We introduce a class of objects which we call 'dilation surfaces'. These provide families of foliations on surfaces whose dynamics we are interested in. We present and analyze a couple of examples, and we define concepts related to these in order to motivate several questions and open problems. In particular we generalize the notion of Veech group to dilation surfaces, and we prove a structure result about these Veech groups.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48696185","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 the hyperbolic components of bicritical rational maps having two distinct attracting cycles each of period at least two are bounded in the moduli space of bicritical rational maps. Our arguments rely on arithmetic methods.
{"title":"Bounded hyperbolic components of bicritical rational maps","authors":"Hongming Nie, K. Pilgrim","doi":"10.3934/jmd.2022016","DOIUrl":"https://doi.org/10.3934/jmd.2022016","url":null,"abstract":"We prove that the hyperbolic components of bicritical rational maps having two distinct attracting cycles each of period at least two are bounded in the moduli space of bicritical rational maps. Our arguments rely on arithmetic methods.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45999808","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 a class of area-preserving, piecewise affine maps on the 2-sphere. These maps encode degenerating families of K3 surface automorphisms and are profitably studied using techniques from tropical and Berkovich geometries.
{"title":"Tropical dynamics of area-preserving maps","authors":"Simion Filip","doi":"10.3934/JMD.2019007","DOIUrl":"https://doi.org/10.3934/JMD.2019007","url":null,"abstract":"We consider a class of area-preserving, piecewise affine maps on the 2-sphere. These maps encode degenerating families of K3 surface automorphisms and are profitably studied using techniques from tropical and Berkovich geometries.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46138745","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 global topological rigidity theorem for locally begin{document}$ C^2 $end{document} -non-discrete subgroups of begin{document}$ {rm {Diff}}^{omega} (S^1) $end{document} .
We prove a global topological rigidity theorem for locally begin{document}$ C^2 $end{document} -non-discrete subgroups of begin{document}$ {rm {Diff}}^{omega} (S^1) $end{document} .
{"title":"Global rigidity of conjugations for locally non-discrete subgroups of begin{document}$ {rm {Diff}}^{omega} (S^1) $end{document}","authors":"Anas Eskif, J. Rebelo","doi":"10.3934/JMD.2019013","DOIUrl":"https://doi.org/10.3934/JMD.2019013","url":null,"abstract":"We prove a global topological rigidity theorem for locally begin{document}$ C^2 $end{document} -non-discrete subgroups of begin{document}$ {rm {Diff}}^{omega} (S^1) $end{document} .","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41562069","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}
Bill Veech died suddenly on August 30, 2016 at the age of 77. He was a major figure in the development of dynamical systems in the past 50 years with fundamental contributions to topological dynamics, Interval Exchange Transformations, and more generally to the field now called Teichmüller dynamics, of which he was one of the founders. According to his obituary on the Statesboro Herald, William Austin Veech “was born on Christmas Eve in 1938 in Detroit, Michigan, and obtained his BA from Dartmouth College in 1960. He earned his Ph.D. in 1963 under the supervision of Salomon Bochner at Princeton University (with a dissertation on Almost Automorphic Functions). He joined the faculty of Rice University in 1969. He served as department chair for three years between 1982 and 1986 and held an endowed chair since 1988, Milton Brockett Porter Chair, 1988-2003; Edgar Odell Lovett Chair, since 2003.” During his career Veech authored approximately 60 papers and one book on complex analysis. All of his papers are single authored. According to his obituary “he believed in the importance of developing one’s own unique perspective”. Any reader of his papers might add that he also had his own personal, idiosyncratic writing style, exacting and deep, not always easily accessible. Veech had few students, the Mathematical Genealogy Project lists five: J. Martin (Ph. D. 1971), M. Stewart (Ph. D. 1978), C. Ward (Ph. D. 1996), Y. Wu (Ph. D. 2006) and J. Fickenscher (Ph. D. 2011), all at Rice University, and we are not aware of any others. Despite the small number of students, he had broad personal influence, as he was always ready to discuss mathematics and was very generous with his time, his ideas, as well as praise and encouragement for younger researchers. He also generously gave credit to others for originating ideas and for motivating his own research, sometimes acknowledging his intellectual debt in the very title of his paper (“Boshernitzan’s criterion” [87], “Bufetov’s question” [92], . . . ). It seems only fair that several of the groundbreaking results or concepts that he introduced bear his name: in topological dynamics the Veech relation and
{"title":"Bill Veech's contributions to dynamical systems","authors":"G. Forni, H. Masur, J. Smillie","doi":"10.3934/jmd.2019v","DOIUrl":"https://doi.org/10.3934/jmd.2019v","url":null,"abstract":"Bill Veech died suddenly on August 30, 2016 at the age of 77. He was a major figure in the development of dynamical systems in the past 50 years with fundamental contributions to topological dynamics, Interval Exchange Transformations, and more generally to the field now called Teichmüller dynamics, of which he was one of the founders. According to his obituary on the Statesboro Herald, William Austin Veech “was born on Christmas Eve in 1938 in Detroit, Michigan, and obtained his BA from Dartmouth College in 1960. He earned his Ph.D. in 1963 under the supervision of Salomon Bochner at Princeton University (with a dissertation on Almost Automorphic Functions). He joined the faculty of Rice University in 1969. He served as department chair for three years between 1982 and 1986 and held an endowed chair since 1988, Milton Brockett Porter Chair, 1988-2003; Edgar Odell Lovett Chair, since 2003.” During his career Veech authored approximately 60 papers and one book on complex analysis. All of his papers are single authored. According to his obituary “he believed in the importance of developing one’s own unique perspective”. Any reader of his papers might add that he also had his own personal, idiosyncratic writing style, exacting and deep, not always easily accessible. Veech had few students, the Mathematical Genealogy Project lists five: J. Martin (Ph. D. 1971), M. Stewart (Ph. D. 1978), C. Ward (Ph. D. 1996), Y. Wu (Ph. D. 2006) and J. Fickenscher (Ph. D. 2011), all at Rice University, and we are not aware of any others. Despite the small number of students, he had broad personal influence, as he was always ready to discuss mathematics and was very generous with his time, his ideas, as well as praise and encouragement for younger researchers. He also generously gave credit to others for originating ideas and for motivating his own research, sometimes acknowledging his intellectual debt in the very title of his paper (“Boshernitzan’s criterion” [87], “Bufetov’s question” [92], . . . ). It seems only fair that several of the groundbreaking results or concepts that he introduced bear his name: in topological dynamics the Veech relation and","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48024343","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 near action of the group begin{document}$ mathrm{PC} $end{document} of piecewise continuous self-transformations of the circle. Elements of this group are only defined modulo indeterminacy on a finite subset, which raises the question of realizability: a subgroup of begin{document}$ mathrm{PC} $end{document} is said to be realizable if it can be lifted to a group of permutations of the circle. We prove that every finitely generated abelian subgroup of begin{document}$ mathrm{PC} $end{document} is realizable. We show that this is not true for arbitrary subgroups, by exhibiting a non-realizable finitely generated subgroup of the group of interval exchanges with flips. The group of (oriented) interval exchanges is obviously realizable (choosing the unique left-continuous representative). We show that it has only two realizations (up to conjugation by a finitely supported permutation): the left and right-continuous ones.
We study the near action of the group begin{document}$ mathrm{PC} $end{document} of piecewise continuous self-transformations of the circle. Elements of this group are only defined modulo indeterminacy on a finite subset, which raises the question of realizability: a subgroup of begin{document}$ mathrm{PC} $end{document} is said to be realizable if it can be lifted to a group of permutations of the circle. We prove that every finitely generated abelian subgroup of begin{document}$ mathrm{PC} $end{document} is realizable. We show that this is not true for arbitrary subgroups, by exhibiting a non-realizable finitely generated subgroup of the group of interval exchanges with flips. The group of (oriented) interval exchanges is obviously realizable (choosing the unique left-continuous representative). We show that it has only two realizations (up to conjugation by a finitely supported permutation): the left and right-continuous ones.
{"title":"Realizations of groups of piecewise continuous transformations of the circle","authors":"Yves Cornulier","doi":"10.3934/jmd.2020003","DOIUrl":"https://doi.org/10.3934/jmd.2020003","url":null,"abstract":"We study the near action of the group begin{document}$ mathrm{PC} $end{document} of piecewise continuous self-transformations of the circle. Elements of this group are only defined modulo indeterminacy on a finite subset, which raises the question of realizability: a subgroup of begin{document}$ mathrm{PC} $end{document} is said to be realizable if it can be lifted to a group of permutations of the circle. We prove that every finitely generated abelian subgroup of begin{document}$ mathrm{PC} $end{document} is realizable. We show that this is not true for arbitrary subgroups, by exhibiting a non-realizable finitely generated subgroup of the group of interval exchanges with flips. The group of (oriented) interval exchanges is obviously realizable (choosing the unique left-continuous representative). We show that it has only two realizations (up to conjugation by a finitely supported permutation): the left and right-continuous ones.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42327439","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 the number of square-tiled surfaces of genus $g$, with $n$ marked points, with one or both of its horizontal and vertical foliations belonging to fixed mapping class group orbits, and having at most $L$ squares, is asymptotic to $L^{6g-6+2n}$ times a product of constants appearing in Mirzakhani's count of simple closed hyperbolic geodesics. Many of the results in this paper reflect recent discoveries of Delecroix, Goujard, Zograf, and Zorich, but the approach considered here is very different from theirs. We follow conceptual and geometric methods inspired by Mirzakhani's work.
{"title":"Counting square-tiled surfaces with prescribed real and imaginary foliations and connections to Mirzakhani's asymptotics for simple closed hyperbolic geodesics","authors":"Francisco Arana-Herrera","doi":"10.14288/1.0385983","DOIUrl":"https://doi.org/10.14288/1.0385983","url":null,"abstract":"We show that the number of square-tiled surfaces of genus $g$, with $n$ marked points, with one or both of its horizontal and vertical foliations belonging to fixed mapping class group orbits, and having at most $L$ squares, is asymptotic to $L^{6g-6+2n}$ times a product of constants appearing in Mirzakhani's count of simple closed hyperbolic geodesics. Many of the results in this paper reflect recent discoveries of Delecroix, Goujard, Zograf, and Zorich, but the approach considered here is very different from theirs. We follow conceptual and geometric methods inspired by Mirzakhani's work.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48408936","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 a rigid, rank 1, prime transformation that is not quasi-simple and whose self-joinings form a Paulsen simplex. This seems to be the first example of a prime system whose self-joinings form a Paulsen simplex and the first example of a prime system that is not quasi-simple.
{"title":"A prime system with many self-joinings","authors":"J. Chaika, Bryna Kra","doi":"10.3934/JMD.2021007","DOIUrl":"https://doi.org/10.3934/JMD.2021007","url":null,"abstract":"We construct a rigid, rank 1, prime transformation that is not quasi-simple and whose self-joinings form a Paulsen simplex. This seems to be the first example of a prime system whose self-joinings form a Paulsen simplex and the first example of a prime system that is not quasi-simple.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44666709","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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