
-
Previous Article
Pseudo-rotations and Steenrod squares
- JMD Home
- This Volume
-
Next Article
On the non-monotonicity of entropy for a class of real quadratic rational maps
Shimura–Teichmüller curves in genus 5
1. | Department of Mathematics, Brooklyn College and CUNY Graduate Center, 2900 Bedford Avenue, Brooklyn, NY 11210-2889, USA |
2. | Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA |
We prove that there are no Shimura–Teichmüller curves generated by genus five translation surfaces, thereby completing the classification of Shimura–Teichmüller curves in general. This was conjectured by Möller in his original work introducing Shimura–Teichmüller curves. Moreover, the property of being a Shimura–Teichmüller 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.
References:
[1] |
D. Aulicino,
Teichmüller discs with completely degenerate Kontsevich-Zorich spectrum, Comment. Math. Helv., 90 (2015), 573-643.
doi: 10.4171/CMH/365. |
[2] |
D. Aulicino,
Affine invariant submanifolds with completely degenerate Kontsevich-Zorich spectrum, Ergodic Theory Dynam. Systems, 38 (2018), 10-33.
doi: 10.1017/etds.2016.26. |
[3] |
David Aulicino and Chaya Norton, Shimura–Teichmüller curves in genus 5, Sage Notebooks, https://github.com/davidaulicino/ST5. Google Scholar |
[4] |
A. Avila and M. Viana,
Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math., 198 (2007), 1-56.
doi: 10.1007/s11511-007-0012-1. |
[5] |
A. Eskin, M. Kontsevich and A. Zorich,
Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. Inst. Hautes \'Etudes Sci., 120 (2014), 207-333.
doi: 10.1007/s10240-013-0060-3. |
[6] |
A. Eskin, M. Mirzakhani and A. Mohammadi,
Isolation, equidistribution, and orbit closures for the SL $(2, \mathbb{R})$ action on moduli space, Ann. of Math. (2), 182 (2015), 673-721.
doi: 10.4007/annals.2015.182.2.7. |
[7] |
J. D. Fay, Theta Functions on Riemann Surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin-New York, 1973. |
[8] |
G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum, preprint, arXiv: 0810.0023v1 (2008). Google Scholar |
[9] |
G. Forni, C. Matheus and A. Zorich,
Square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 285-318.
doi: 10.3934/jmd.2011.5.285. |
[10] |
G. Forni,
Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103.
doi: 10.2307/3062150. |
[11] |
G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle, in Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, 2006,549–580.
doi: 10.1016/S1874-575X(06)80033-0. |
[12] |
S. Grushevsky, I. Krichever and C. Norton,
Real-normalized differentials: Limits on stable curves, Russian Math. Surveys, 74 (2019), 265-324.
doi: 10.4213/rm9877. |
[13] |
X. Hu and C. Norton, General variational formulas for Abelian differentials, Int. Math. Res. Not. IMRN (2020), no. 12, 3540–3581.
doi: 10.1093/imrn/rny106. |
[14] |
F. Herrlich and G. Schmithüsen,
An extraordinary origami curve, Math. Nachr., 281 (2008), 219-237.
doi: 10.1002/mana.200510597. |
[15] |
H. Masur,
Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J., 43 (1976), 623-635.
|
[16] |
H. Masur,
Closed trajectories for quadratic differentials with an application to billiards, Duke Math. J., 53 (1986), 307-314.
doi: 10.1215/S0012-7094-86-05319-6. |
[17] |
M. Möller,
Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32.
doi: 10.3934/jmd.2011.5.1. |
[18] |
C. Matheus and J.-C. Yoccoz,
The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis, J. Mod. Dyn., 4 (2010), 453-486.
doi: 10.3934/jmd.2010.4.453. |
[19] |
Yu. L. Rodin, The Riemann Boundary Problem on Riemann Surfaces, Mathematics and its Applications (Soviet Series), vol. 16, D. Reidel Publishing Co., Dordrecht, 1988.
doi: 10.1007/978-94-009-2885-5. |
[20] |
J. Smillie and B. Weiss,
Characterizations of lattice surfaces, Invent. Math., 180 (2010), 535-557.
doi: 10.1007/s00222-010-0236-0. |
[21] |
W. A. Veech,
The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530.
doi: 10.2307/2007091. |
[22] |
W. A. Veech,
Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583.
doi: 10.1007/BF01388890. |
[23] |
Ya. B. Vorobets,
Plane structures and billiards in rational polygons: The Veech alternative, Uspekhi Mat. Nauk, 51 (1996), 3-42.
doi: 10.1070/RM1996v051n05ABEH002993. |
[24] |
A. Yamada,
Precise variational formulas for abelian differentials, Kodai Math. J., 3 (1980), 114-143.
|
show all references
References:
[1] |
D. Aulicino,
Teichmüller discs with completely degenerate Kontsevich-Zorich spectrum, Comment. Math. Helv., 90 (2015), 573-643.
doi: 10.4171/CMH/365. |
[2] |
D. Aulicino,
Affine invariant submanifolds with completely degenerate Kontsevich-Zorich spectrum, Ergodic Theory Dynam. Systems, 38 (2018), 10-33.
doi: 10.1017/etds.2016.26. |
[3] |
David Aulicino and Chaya Norton, Shimura–Teichmüller curves in genus 5, Sage Notebooks, https://github.com/davidaulicino/ST5. Google Scholar |
[4] |
A. Avila and M. Viana,
Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math., 198 (2007), 1-56.
doi: 10.1007/s11511-007-0012-1. |
[5] |
A. Eskin, M. Kontsevich and A. Zorich,
Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. Inst. Hautes \'Etudes Sci., 120 (2014), 207-333.
doi: 10.1007/s10240-013-0060-3. |
[6] |
A. Eskin, M. Mirzakhani and A. Mohammadi,
Isolation, equidistribution, and orbit closures for the SL $(2, \mathbb{R})$ action on moduli space, Ann. of Math. (2), 182 (2015), 673-721.
doi: 10.4007/annals.2015.182.2.7. |
[7] |
J. D. Fay, Theta Functions on Riemann Surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin-New York, 1973. |
[8] |
G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum, preprint, arXiv: 0810.0023v1 (2008). Google Scholar |
[9] |
G. Forni, C. Matheus and A. Zorich,
Square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 285-318.
doi: 10.3934/jmd.2011.5.285. |
[10] |
G. Forni,
Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103.
doi: 10.2307/3062150. |
[11] |
G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle, in Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, 2006,549–580.
doi: 10.1016/S1874-575X(06)80033-0. |
[12] |
S. Grushevsky, I. Krichever and C. Norton,
Real-normalized differentials: Limits on stable curves, Russian Math. Surveys, 74 (2019), 265-324.
doi: 10.4213/rm9877. |
[13] |
X. Hu and C. Norton, General variational formulas for Abelian differentials, Int. Math. Res. Not. IMRN (2020), no. 12, 3540–3581.
doi: 10.1093/imrn/rny106. |
[14] |
F. Herrlich and G. Schmithüsen,
An extraordinary origami curve, Math. Nachr., 281 (2008), 219-237.
doi: 10.1002/mana.200510597. |
[15] |
H. Masur,
Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J., 43 (1976), 623-635.
|
[16] |
H. Masur,
Closed trajectories for quadratic differentials with an application to billiards, Duke Math. J., 53 (1986), 307-314.
doi: 10.1215/S0012-7094-86-05319-6. |
[17] |
M. Möller,
Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32.
doi: 10.3934/jmd.2011.5.1. |
[18] |
C. Matheus and J.-C. Yoccoz,
The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis, J. Mod. Dyn., 4 (2010), 453-486.
doi: 10.3934/jmd.2010.4.453. |
[19] |
Yu. L. Rodin, The Riemann Boundary Problem on Riemann Surfaces, Mathematics and its Applications (Soviet Series), vol. 16, D. Reidel Publishing Co., Dordrecht, 1988.
doi: 10.1007/978-94-009-2885-5. |
[20] |
J. Smillie and B. Weiss,
Characterizations of lattice surfaces, Invent. Math., 180 (2010), 535-557.
doi: 10.1007/s00222-010-0236-0. |
[21] |
W. A. Veech,
The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530.
doi: 10.2307/2007091. |
[22] |
W. A. Veech,
Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583.
doi: 10.1007/BF01388890. |
[23] |
Ya. B. Vorobets,
Plane structures and billiards in rational polygons: The Veech alternative, Uspekhi Mat. Nauk, 51 (1996), 3-42.
doi: 10.1070/RM1996v051n05ABEH002993. |
[24] |
A. Yamada,
Precise variational formulas for abelian differentials, Kodai Math. J., 3 (1980), 114-143.
|


Stratum | |
Stratum | |
[1] |
Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020390 |
[2] |
Joan Carles Tatjer, Arturo Vieiro. Dynamics of the QR-flow for upper Hessenberg real matrices. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1359-1403. doi: 10.3934/dcdsb.2020166 |
[3] |
Roderick S. C. Wong, H. Y. Zhang. On the connection formulas of the third Painlevé transcendent. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 541-560. doi: 10.3934/dcds.2009.23.541 |
[4] |
Akbar Mahmoodi Rishakani, Seyed Mojtaba Dehnavi, Mohmmadreza Mirzaee Shamsabad, Nasour Bagheri. Cryptographic properties of cyclic binary matrices. Advances in Mathematics of Communications, 2021, 15 (2) : 311-327. doi: 10.3934/amc.2020068 |
[5] |
Xianbo Sun, Zhanbo Chen, Pei Yu. Parameter identification on Abelian integrals to achieve Chebyshev property. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020375 |
[6] |
Qing-Hu Hou, Yarong Wei. Telescoping method, summation formulas, and inversion pairs. Electronic Research Archive, , () : -. doi: 10.3934/era.2021007 |
[7] |
Lars Grüne. Computing Lyapunov functions using deep neural networks. Journal of Computational Dynamics, 2020 doi: 10.3934/jcd.2021006 |
[8] |
Peter Giesl, Sigurdur Hafstein. System specific triangulations for the construction of CPA Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020378 |
[9] |
Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033 |
[10] |
Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392 |
[11] |
Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331 |
[12] |
Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381 |
[13] |
Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036 |
[14] |
Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185 |
[15] |
Hongguang Ma, Xiang Li. Multi-period hazardous waste collection planning with consideration of risk stability. Journal of Industrial & Management Optimization, 2021, 17 (1) : 393-408. doi: 10.3934/jimo.2019117 |
[16] |
Attila Dénes, Gergely Röst. Single species population dynamics in seasonal environment with short reproduction period. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020288 |
[17] |
Lin Jiang, Song Wang. Robust multi-period and multi-objective portfolio selection. Journal of Industrial & Management Optimization, 2021, 17 (2) : 695-709. doi: 10.3934/jimo.2019130 |
[18] |
Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127 |
[19] |
Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115 |
[20] |
Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020170 |
2019 Impact Factor: 0.465
Tools
Metrics
Other articles
by authors
[Back to Top]