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.
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Figure 1. 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.
[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. |