Inventiones mathematicae

, Volume 97, Issue 3, pp 553–583 | Cite as

Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards

  • W. A. Veech


There exists a Teichmüller discΔ n containing the Riemann surface ofy2+x n =1, in the genus [n−1/2] Teichmüller space, such that the stabilizer ofΔ n in the mapping class group has a fundamental domain of finite (Poincaré) volume inΔ n . Application is given to an asymptotic formula for the length spectrum of the billiard in isosceles triangles with angles (π/n, π/n,n−2/nπ) and to the uniform distribution of infinite billiard trajectories in the same triangles.


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  1. 0.
    Fathi, A., Laudenbach, F., Poenaru, V.: Travaux de Thurston sur Les Surfaces. Asterisque.66–67, (1979)Google Scholar
  2. 1.
    Kerckhoff, S., Masur, H., Smillie, J.: Ergodicity of billiard flows and quadratic differentials. Ann. Math.124, 293–311 (1986)Google Scholar
  3. 2.
    Kubota, T.: Elementary theory of Eisenstein series. New York: Halsted Books 1973Google Scholar
  4. 3.
    Lehner, J.: Discontinuous groups and automorphic functions. Am. Math. Soc. Surv. No.8, (1964)Google Scholar
  5. 4.
    Masur, H.: Interval exchange transformations and measured foliations. Ann. Math.115, 169–200 (1982)Google Scholar
  6. 5.
    Masur, H.: Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential. PreprintGoogle Scholar
  7. 6.
    Selberg, A.: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Ind. Math. Soc.XX, 47–87 (1956)Google Scholar
  8. 7.
    Strebel, K.: Quadratic differentials. Berlin-Heidelberg-New York: Springer 1984Google Scholar
  9. 8.
    Thurston, W.: On the dynamics of diffeomorphisms of surfaces. PreprintGoogle Scholar
  10. 9.
    Veech, W.A.: Moduli spaces of quadratic differentials. PreprintGoogle Scholar
  11. 10.
    Veech, W. A.: The Teichmüller geodesic flow. Ann. Math.124, 441–530 (1986)Google Scholar
  12. 11.
    Zemlyakov, A. N., Katok, A. B.: Topological transitivity of billiards in polygons. Mat. Zametki18, 291–300 (1975) (Russian)Google Scholar
  13. 12.
    Boshernitzan, M.: A condition for minimal interval exchange maps to be uniquely ergodic. Duke Math. J.52, 723–752 (1985)Google Scholar
  14. 13.
    Gutkin, E.: Billiards on almost integrable polyhedral surfaces. Ergodic Theory Dynam. Syst.4, 569–584 (1984)Google Scholar
  15. 14.
    Kra, I.: On the Nielson-Thurston-Bers type of some self-maps of Riemann surfaces. Acta Math.146, 231–270 (1981)Google Scholar
  16. 15.
    Widder, D. V: The Laplace Transform, Princeton University Press., Princeton, N.J.: Princeton University Press 1946Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • W. A. Veech
    • 1
  1. 1.Department of MathematicsRice UniversityHoustonUSA

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