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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.
|


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