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Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. (English) Zbl 0676.32006
Invent. Math. 97, No. 3, 553-583 (1989); erratum ibid. 103, No. 2, 447 (1991).
This paper is concerned with some special aspects of Teichmüller theory and applications to billiards. From the summary: “There exists a Teichmüller disc \(\Delta_ n\) containing the Riemann surface of \(y^ 2+x^ n=1,\) in the genus [(n-1)/2] Teichmüller space, such that the stabilizer of \(\Delta_ n\) in the mapping class group has a fundamental domain of finite (Poincaré) volume in \(\Delta_ n\). Application is given to an asymptotic formula for the length spectrum of the billiard in isosceles triangles with angles (\(\pi\) /n,\(\pi\) /n,((n-2)/n)\(\pi)\) and to the uniform distribution of infinite billiard trajectories in the same triangles.”
Reviewer: S.Kosarew

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F30 Differentials on Riemann surfaces
Full Text: DOI EuDML
[1] Fathi, A., Laudenbach, F., Poenaru, V.: Travaux de Thurston sur Les Surfaces. Asterisque.66-67, (1979)
[2] Kerckhoff, S., Masur, H., Smillie, J.: Ergodicity of billiard flows and quadratic differentials. Ann. Math.124, 293-311 (1986) · Zbl 0637.58010
[3] Kubota, T.: Elementary theory of Eisenstein series. New York: Halsted Books 1973 · Zbl 0268.10012
[4] Lehner, J.: Discontinuous groups and automorphic functions. Am. Math. Soc. Surv. No.8, (1964) · Zbl 0178.42902
[5] Masur, H.: Interval exchange transformations and measured foliations. Ann. Math.115, 169-200 (1982) · Zbl 0497.28012
[6] Masur, H.: Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential. Preprint · Zbl 0661.30034
[7] 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) · Zbl 0072.08201
[8] Strebel, K.: Quadratic differentials. Berlin-Heidelberg-New York: Springer 1984 · Zbl 0547.30001
[9] Thurston, W.: On the dynamics of diffeomorphisms of surfaces. Preprint · Zbl 0674.57008
[10] Veech, W.A.: Moduli spaces of quadratic differentials. Preprint · Zbl 0722.30032
[11] Veech, W. A.: The Teichmüller geodesic flow. Ann. Math.124, 441-530 (1986) · Zbl 0658.32016
[12] Zemlyakov, A. N., Katok, A. B.: Topological transitivity of billiards in polygons. Mat. Zametki18, 291-300 (1975) (Russian) · Zbl 0315.58014
[13] Boshernitzan, M.: A condition for minimal interval exchange maps to be uniquely ergodic. Duke Math. J.52, 723-752 (1985) · Zbl 0602.28009
[14] Gutkin, E.: Billiards on almost integrable polyhedral surfaces. Ergodic Theory Dynam. Syst.4, 569-584 (1984) · Zbl 0569.58028
[15] Kra, I.: On the Nielson-Thurston-Bers type of some self-maps of Riemann surfaces. Acta Math.146, 231-270 (1981) · Zbl 0477.32024
[16] Widder, D. V: The Laplace Transform, Princeton University Press., Princeton, N.J.: Princeton University Press 1946
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