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Equidistribution of saddle connections on translation surfaces
Mathematics Department, Stony Brook University, Stony Brook, NY 11794-3651, USA |
Fix a translation surface $ X $, and consider the measures on $ X $ coming from averaging the uniform measures on all the saddle connections of length at most $ R $. Then, as $ R\to\infty $, the weak limit of these measures exists and is equal to the area measure on $ X $ coming from the flat metric. This implies that, on a rational-angled billiard table, the billiard trajectories that start and end at a corner of the table are equidistributed on the table. We also show that any weak limit of a subsequence of the counting measures on $ S^1 $ given by the angles of all saddle connections of length at most $ R_n $, as $ R_n\to\infty $, is in the Lebesgue measure class. The proof of the equidistribution result uses the angle result, together with the theorem of Kerckhoff-Masur-Smillie that the directional flow on a surface is uniquely ergodic in almost every direction.
References:
[1] |
J. S. Athreya,
Quantitative recurrence and large deviations for Teichmuller geodesic flow, Geom. Dedicata, 119 (2006), 121-140.
doi: 10.1007/s10711-006-9058-z. |
[2] |
M. Boshernitzan, G. Galperin, T. Krüger and S. Troubetzkoy,
Periodic billiard orbits are dense in rational polygons, Trans. Amer. Math. Soc., 350 (1998), 3523-3535.
doi: 10.1090/S0002-9947-98-02089-3. |
[3] |
R. Bowen,
The equidistribution of closed geodesics, Amer. J. Math., 94 (1972), 413-423.
doi: 10.2307/2374628. |
[4] |
J. Chaika, Homogeneous approximation for flows on translation surfaces, preprint, 2011, arXiv: 1110.6167. |
[5] |
B. Dozier, Convergence of Siegel–Veech constants, Geometriae Dedicata, (2018), 1–12.
doi: 10.1007/s10711-018-0332-7. |
[6] |
A. Eskin and H. Masur,
Asymptotic formulas on flat surfaces, Ergodic Theory and Dynamical Systems, 21 (2001), 443-478.
doi: 10.1017/S0143385701001225. |
[7] |
A. Eskin, G. Margulis and S. Mozes,
Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141.
doi: 10.2307/120984. |
[8] |
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. |
[9] |
A. Eskin, Counting problems in moduli space, Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, 2006, 581–595.
doi: 10.1016/S1874-575X(06)80034-2. |
[10] |
R. H. Fox and R. B. Kershner,
Concerning the transitive properties of geodesics on a rational polyhedron, Duke Math. J., 2 (1936), 147-150.
doi: 10.1215/S0012-7094-36-00213-2. |
[11] |
S. Kerckhoff, H. Masur and J. Smillie,
Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2), 124 (1986), 293-311.
doi: 10.2307/1971280. |
[12] |
H. Masur, Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential, in Holomorphic Functions and Moduli, Vol. I (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ., vol. 10, Springer, New York, 1988, 215–228.
doi: 10.1007/978-1-4613-9602-4_20. |
[13] |
H. Masur,
The growth rate of trajectories of a quadratic differential, Ergodic Theory and Dynamical Systems, 10 (1990), 151-176.
doi: 10.1017/S0143385700005459. |
[14] |
L. Marchese, R. Treviño and S. Weil, Diophantine approximations for translation surfaces and planar resonant sets, preprint, 2016, arXiv: 1502.05007v2. |
[15] |
A. Nevo, Equidistribution in measure-preserving actions of semisimple groups: Case of $SL_2(\mathbb{R})$, preprint, 2017, arXiv: 1708.03886. |
[16] |
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. |
[17] |
W. A. Veech,
Siegel measures, Ann. of Math. (2), 148 (1998), 895-944.
doi: 10.2307/121033. |
[18] |
Y. Vorobets, Periodic geodesics on generic translation surfaces, in Algebraic and Topological Dynamics, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005, 205–258.
doi: 10.1090/conm/385/07199. |
[19] |
A. Wright,
Translation surfaces and their orbit closures: An introduction for a broad audience, EMS Surv. Math. Sci., 2 (2015), 63-108.
doi: 10.4171/EMSS/9. |
[20] |
A. N. Zemljakov and A. B. Katok,
Topological transitivity of billiards in polygons, Mat. Zametki, 18 (1975), 291-300.
|
[21] |
A. Zorich, Flat surfaces, in Frontiers in Number Theory, Physics, and Geometry, I, Springer, Berlin, 2006, 437–583.
doi: 10.1007/978-3-540-31347-2_13. |
show all references
References:
[1] |
J. S. Athreya,
Quantitative recurrence and large deviations for Teichmuller geodesic flow, Geom. Dedicata, 119 (2006), 121-140.
doi: 10.1007/s10711-006-9058-z. |
[2] |
M. Boshernitzan, G. Galperin, T. Krüger and S. Troubetzkoy,
Periodic billiard orbits are dense in rational polygons, Trans. Amer. Math. Soc., 350 (1998), 3523-3535.
doi: 10.1090/S0002-9947-98-02089-3. |
[3] |
R. Bowen,
The equidistribution of closed geodesics, Amer. J. Math., 94 (1972), 413-423.
doi: 10.2307/2374628. |
[4] |
J. Chaika, Homogeneous approximation for flows on translation surfaces, preprint, 2011, arXiv: 1110.6167. |
[5] |
B. Dozier, Convergence of Siegel–Veech constants, Geometriae Dedicata, (2018), 1–12.
doi: 10.1007/s10711-018-0332-7. |
[6] |
A. Eskin and H. Masur,
Asymptotic formulas on flat surfaces, Ergodic Theory and Dynamical Systems, 21 (2001), 443-478.
doi: 10.1017/S0143385701001225. |
[7] |
A. Eskin, G. Margulis and S. Mozes,
Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141.
doi: 10.2307/120984. |
[8] |
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. |
[9] |
A. Eskin, Counting problems in moduli space, Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, 2006, 581–595.
doi: 10.1016/S1874-575X(06)80034-2. |
[10] |
R. H. Fox and R. B. Kershner,
Concerning the transitive properties of geodesics on a rational polyhedron, Duke Math. J., 2 (1936), 147-150.
doi: 10.1215/S0012-7094-36-00213-2. |
[11] |
S. Kerckhoff, H. Masur and J. Smillie,
Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2), 124 (1986), 293-311.
doi: 10.2307/1971280. |
[12] |
H. Masur, Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential, in Holomorphic Functions and Moduli, Vol. I (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ., vol. 10, Springer, New York, 1988, 215–228.
doi: 10.1007/978-1-4613-9602-4_20. |
[13] |
H. Masur,
The growth rate of trajectories of a quadratic differential, Ergodic Theory and Dynamical Systems, 10 (1990), 151-176.
doi: 10.1017/S0143385700005459. |
[14] |
L. Marchese, R. Treviño and S. Weil, Diophantine approximations for translation surfaces and planar resonant sets, preprint, 2016, arXiv: 1502.05007v2. |
[15] |
A. Nevo, Equidistribution in measure-preserving actions of semisimple groups: Case of $SL_2(\mathbb{R})$, preprint, 2017, arXiv: 1708.03886. |
[16] |
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. |
[17] |
W. A. Veech,
Siegel measures, Ann. of Math. (2), 148 (1998), 895-944.
doi: 10.2307/121033. |
[18] |
Y. Vorobets, Periodic geodesics on generic translation surfaces, in Algebraic and Topological Dynamics, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005, 205–258.
doi: 10.1090/conm/385/07199. |
[19] |
A. Wright,
Translation surfaces and their orbit closures: An introduction for a broad audience, EMS Surv. Math. Sci., 2 (2015), 63-108.
doi: 10.4171/EMSS/9. |
[20] |
A. N. Zemljakov and A. B. Katok,
Topological transitivity of billiards in polygons, Mat. Zametki, 18 (1975), 291-300.
|
[21] |
A. Zorich, Flat surfaces, in Frontiers in Number Theory, Physics, and Geometry, I, Springer, Berlin, 2006, 437–583.
doi: 10.1007/978-3-540-31347-2_13. |






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