# American Institute of Mathematical Sciences

2019, 14: 87-120. doi: 10.3934/jmd.2019004

## Equidistribution of saddle connections on translation surfaces

 Mathematics Department, Stony Brook University, Stony Brook, NY 11794-3651, USA

Received  September 05, 2017 Revised  December 20, 2017 Published  March 2019

Fund Project: Supported in part by NSF grant DGE-114747.

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.

Citation: Benjamin Dozier. Equidistribution of saddle connections on translation surfaces. Journal of Modern Dynamics, 2019, 14: 87-120. doi: 10.3934/jmd.2019004
##### 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.  Google Scholar [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.  Google Scholar [3] R. Bowen, The equidistribution of closed geodesics, Amer. J. Math., 94 (1972), 413-423.  doi: 10.2307/2374628.  Google Scholar [4] J. Chaika, Homogeneous approximation for flows on translation surfaces, preprint, 2011, arXiv: 1110.6167. Google Scholar [5] B. Dozier, Convergence of Siegel–Veech constants, Geometriae Dedicata, (2018), 1–12. doi: 10.1007/s10711-018-0332-7.  Google Scholar [6] A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory and Dynamical Systems, 21 (2001), 443-478.  doi: 10.1017/S0143385701001225.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] L. Marchese, R. Treviño and S. Weil, Diophantine approximations for translation surfaces and planar resonant sets, preprint, 2016, arXiv: 1502.05007v2. Google Scholar [15] A. Nevo, Equidistribution in measure-preserving actions of semisimple groups: Case of $SL_2(\mathbb{R})$, preprint, 2017, arXiv: 1708.03886. Google Scholar [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.  Google Scholar [17] W. A. Veech, Siegel measures, Ann. of Math. (2), 148 (1998), 895-944.  doi: 10.2307/121033.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] A. N. Zemljakov and A. B. Katok, Topological transitivity of billiards in polygons, Mat. Zametki, 18 (1975), 291-300.   Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [3] R. Bowen, The equidistribution of closed geodesics, Amer. J. Math., 94 (1972), 413-423.  doi: 10.2307/2374628.  Google Scholar [4] J. Chaika, Homogeneous approximation for flows on translation surfaces, preprint, 2011, arXiv: 1110.6167. Google Scholar [5] B. Dozier, Convergence of Siegel–Veech constants, Geometriae Dedicata, (2018), 1–12. doi: 10.1007/s10711-018-0332-7.  Google Scholar [6] A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory and Dynamical Systems, 21 (2001), 443-478.  doi: 10.1017/S0143385701001225.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] L. Marchese, R. Treviño and S. Weil, Diophantine approximations for translation surfaces and planar resonant sets, preprint, 2016, arXiv: 1502.05007v2. Google Scholar [15] A. Nevo, Equidistribution in measure-preserving actions of semisimple groups: Case of $SL_2(\mathbb{R})$, preprint, 2017, arXiv: 1708.03886. Google Scholar [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.  Google Scholar [17] W. A. Veech, Siegel measures, Ann. of Math. (2), 148 (1998), 895-944.  doi: 10.2307/121033.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] A. N. Zemljakov and A. B. Katok, Topological transitivity of billiards in polygons, Mat. Zametki, 18 (1975), 291-300.   Google Scholar [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.  Google Scholar
Saddle connections of length at most $R = 7$ on a genus two translation surface (opposite sides are identified), in units where the height of the figure is approximately 2. The thickness of each saddle connection is drawn inversely proportional to its length (so the total amount of "paint" used to draw a saddle connection is independent of its length). This choice of thickness is meant to represent the measures $\mu_s$, which are all probability measures, in Theorem 1.1. That theorem says that, as the length bound $R$ goes to infinity, the picture will be uniformly colored. This picture was generated with the help of Ronen Mukamel's $\texttt{triangulated\_surfaces}$ SAGE package.
Opposite sides of the polygon are identified to give a genus two translation surface. A cylinder is shown, together with a long saddle connection contained in that cylinder.
Regions used in proof of Lemma 2.2
Proof of Lemma 4.1. The red points are group $A_1$, while the blue are $A_2$.
Adding a saddle connection to a complex, in proof of Proposition 5.4.
Comparing averages for Lemma 5.8 (Shadowing)
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