Advertisement

Counting closed geodesics in strata

  • Alex Eskin
  • Maryam Mirzakhani
  • Kasra Rafi
Article
First Online:
  • 215 Downloads

Abstract

We compute the asymptotic growth rate of the number \(N({{\mathcal {C}}}, R)\) of closed geodesics of length \(\le R\) in a connected component \({{\mathcal {C}}}\) of a stratum of quadratic differentials. We prove that, for any \(0\le \theta \le 1\), the number of closed geodesics \(\gamma \) of length at most R such that \(\gamma \) spends at least \(\theta \)-fraction of its time outside of a compact subset of \({{\mathcal {C}}}\) is exponentially smaller than \(N({{\mathcal {C}}}, R)\). The theorem follows from a lattice counting statement. For points x, y in the moduli space \({{{\mathcal {M}}}(S)}\) of Riemann surfaces, and for \(0 \le \theta \le 1\) we find an upper-bound for the number of geodesic paths of length \(\le R\) in \({{\mathcal {C}}}\) which connect a point near x to a point near y and spend at least a \(\theta \)-fraction of the time outside of a compact subset of \({{\mathcal {C}}}\).

This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

We would like to thanks the referee for many useful comments that have improve the exposition of the paper at several places.

References

  1. 1.
    Arnoux, P., Yoccoz, J.: Construction de difféomorphismes pseudo-Anosov. C. R. Acad. Sci. Paris Sér. I Math. 292(1), 75–78 (1981)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Avila, A., Gouezel, S., Yoccoz, J.-C.: Exponential mixing for the Teichmüller flow. Publ. Math. IHES 104, 143–211 (2006)CrossRefGoogle Scholar
  3. 3.
    Avila, A., João, R.M.: Exponential Mixing for the Teichmüller  flow in the Space of Quadratic Differentials, preprintGoogle Scholar
  4. 4.
    Athreya, J.: Quantitative recurrence and large deviations for Teichmüller geodesic flow. Geom. Dedicata 119, 121–140 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Athreya, J., Bufetov, A., Eskin, A., Mirzakhani, M.: Lattice point asymptotics and volume growth on Teichmüller space. Duke Math. J. 161(6), 1055–1111 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Athreya, J., Forni, G.: Deviation of ergodic averages for rational polygonal billiards. Duke Math. J. 144(2), 285–319 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Avila, A., Gouëzel, S.: Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller  flow. Ann. of Math. (2) 178(2), 385–442 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Avila, A., Gouëzel, S., Yoccoz, J.-C.: Exponential mixing for the Teichmüller flow. Publ. Math. Inst. Hautes Études Sci. 104, 143–211 (2006)CrossRefGoogle Scholar
  9. 9.
    Bers, L.: An extremal problem for quasiconformal maps and a theorem by Thurston. Acta Math. 141, 73–98 (1978)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bleiler, S., Casson, A.: Automorphisms of Surfaces After Nielsen and Thurston. London Mathematical Society Student Texts, vol. 9. Cambridge University Press, Cambridge (1988)zbMATHGoogle Scholar
  11. 11.
    Bufetov, A.: Logarithmic asymptotics for the number of periodic orbits of the Teichmueller flow on Veech’s space of zippered rectangles. Mosc. Math. J. 9(2), 245–261 (2009)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Choi, Y., Series, C., Rafi, K.: Lines of minima and Teichmülcer geodesics. Geom. Funct. Anal. 18(3), 698–754 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dumas, D.: Skinning Maps are Finite-to-one, preprintGoogle Scholar
  14. 14.
    Eskin, A., Margulis, G., Mozes, S.: Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture. Ann. of Math. (2) 147(1), 93–141 (1998)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Eskin, A., Masur, H.: Asymptotic formulas on flat surfaces. Ergodic Theory Dynam. Syst. 21, 443–478 (2001)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Eskin, A., Mirzakhani, M.: Counting closed geodesics in moduli space. J. Mod. Dyn. 1, 71–105 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Farb, B., Margalit, D.: A primer on mapping class groups. In: Princeton Mathematical Series, vol. 49 (2012)Google Scholar
  18. 18.
    Fathi, A., Laudenbach, F., Poénaru, V.: Travaux de Thurston sur les surfaces, Astérisque, 66 and 67 (1979)Google Scholar
  19. 19.
    Forni, G.: Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Ann. of Math. (2) 155(1), 1–103 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Gardiner, F., Masur, H.: Extremal length geometry of Teichmüller space. Complex Var. Theory Appl. 16(2–3), 209–237 (1991)zbMATHGoogle Scholar
  21. 21.
    Hamenstädt, U.: Dynamics of the Teichmueller flow on compact invariant sets. J. Mod. Dyn. 4, 393–418 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hamenstädt, U.: Bowen’s construction for the Teichmüeller flow. J. Mod. Dyn. 7, 489–526 (2013)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Harer, J.L., Penner, R.C.: Combinatorics of Train Tracks. In: Annals of Mathematics Studies, vol. 125 (1992)Google Scholar
  24. 24.
    Hubbard, J.: Teichmüller  Theory and Applications to Geometry, Topology, and Dynamics I, Matrix Editions (2006)Google Scholar
  25. 25.
    Ivanov, N.V., Coefficients of expansion of pseudo-Anosov homeomorphisms, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 167: Issled. Topol. 6(111–116), 191 (1988)Google Scholar
  26. 26.
    Kontsevich, M., Zorich, A.: Connected components of spaces of Abelian differentials with prescribed singularities. Invent. Math. 153, 631–683 (2003)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Katok, A., Hasselblat, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  28. 28.
    Kerckhoff, S.: The asymptotic geometry of Teichmüller space. Topology 19, 23–41 (1980)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Lanneau, E.: Connected components of the strata of the moduli spaces of quadratic diferentials with prescribed singularities. Ann. Sci. Ecole Norm. Sup. (4) 41, 1–56 (2008)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Margulis, G.A.: On Some Aspects of the Theory of Anosov Flows, Ph.D. Thesis (1970), Springer, Berlin (2003)Google Scholar
  31. 31.
    Maskit, B.: Comparison of hyperbolic and extremal lengths. Ann. Acad. Sci. Fenn. 10, 381–386 (1985)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Masur, H.: Interval exchange transformations and measured foliations. Ann. of Math. (2) 115(1), 169–200 (1982)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Masur, H., Minsky, Y.: Geometry of the complex of curves. II. Hierarchical structure. Geom. Funct. Anal. 10(4): 902–974 (2000)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Masur, H., Smillie, J.: Hausdorff Dimension of Sets of Nonergodic Measured Foliations. Ann. Math. 134(3): 455–543 (1991)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Masur, H., Smillie, J.: Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms. Commentarii Mathematics Helevetici 68(1), 289–307 (1993)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Minsky, Y.: Extremal length estimates and product regions in Teichmüller space. Duke Math. J. 83(2), 249–286 (1996)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Papadopoulos, A., Penner, R.: A characterization of pseudo-Anosov foliations. Pac. J. Math. 130, 359–377 (1987)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Papadopoulos, A., Penner, R.: Enumerating pseudo-Anosov foliations. Pac. J. Math. 142(1), 159–173 (1990)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Penner, R.: A construction of pseudo-Anosov homeomorphisms. Trans. Am. Math. Soc. 310(1), 179–197 (1988)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Rafi, K.: A characterization of short curves of a Teichmüller geodesic. Geom. Topol. 9, 179–202 (2005)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Rafi, K.: A combinatorial model for the Teichmüller metric. Geom. Funct. Anal 17(3), 936–959 (2007)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Rafi, K.: Thick–thin decomposition of quadratic differentials. Math. Res. Lett. 14(2), 333–341 (2007)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Rafi, K.: Hyperbolicity in Teichmüller space. Geom. Topol. 18–5, 3025–3053 (2014)CrossRefGoogle Scholar
  44. 44.
    Rafi, K.: Closed geodesics in the thin part of moduli space. In preparationGoogle Scholar
  45. 45.
    Strebel, K.: Quadratic differentials. Ergebnisse der Math 5, Springer, Berlin (1984)CrossRefGoogle Scholar
  46. 46.
    Thurston, W.: On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Am. Math. Soc. (N.S.) 19(2), 417–431 (1988)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Thurston, W.: Shapes of polyhedra and triangulations of the sphere, (English summary) The Epstein birthday schrift, 511Đ549, Geom. Topol. Monogr., 1, Geom. Topol. Publ., Coventry (1998)Google Scholar
  48. 48.
    Veech, W.: The Teichmüller geodesic flow. Ann. Math. (2) 124(3), 441–530 (1986)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Vorobets, Y.: Periodic geodesics on generic translation surfaces, (English summary) Algebraic and topological dynamics, 205Đ258, Contemp. Math., 385. Am. Math. Soc., Providence, RI (2005)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Department of MathematicsStanford UniversityStanfordUSA
  3. 3.Department of MathematicsUniversity of TorontoTorontoCanada

Personalised recommendations