Abstract
There are two rather separate sections to this paper. In the first part we indicate how the geometry of Teichmüller space and moduli space can be used to study the dynamics of rational billiards and more generally the dynamics of foliations defined by flat structures or quadratic differentials. In the second part of the paper we study random walks on the mapping class group of a surface and on Teichmüller space and show how the sphere of foliations defined by Thurston can be realized as the boundary of the random walks.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
C. Boldrighini, C. M. Keane, and F. Marchetti,Billiards in polygons, Ann. of Probab.6 (1978), 532–540.
A. Fathi, F. Laudenbach, and V. Poenaru, Travaux de Thurston sur les surfaces, Astérisque66–67 (1979).
A. Fathi, F. Laudenbach, and V. Poenaru, Travaux de Thurston sur les surfaces, Astérisque66–67 (1979).
W. Floyd,Group completions and limit sets of Kleinian groups, Invent. Math.57 (1981), 205–218.
H. Furstenberg,Strict ergodicity and transformations of the torus, Amer J. Math.88 (1961), 573–601.
H. Furstenberg,Poisson boundaries and envelopes of discrete groups, Bull. Amer. Math. Soc.73 (1967), 350–356.
H. Furstenberg,Random walks and discrete subgroups of Lie groups, Adv. Probab. Related Topics, vol. 1, Dekker, New York, 1971, pp. 3–63.
N. Ivanov,Rank of Teichmüller modular groups, Mat. Zametki44 (1988), 636–644.
V. Kaimanovich,The Poisson boundary of hyperbolic groups, C.R. Acad. Sci. Paris318 (1994), 59–64.
V. Kaimanovich and H. Masur,The Poisson boundary of the mapping class group and of Teichmüller space, submitted.
S. Kerckhoff, H. Masur, and J. Smillie,Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2)124 (1986), 293–311.
H. Masur,Closed trajectories of a quadratic differential with an application to billiards, Duke Math. J.53 (1986), 307–313.
H. Masur,Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential, Holomorphic Functions and Moduli, vol. 2, Springer-Verlag, Berlin and New York, 1988.
H. Masur,The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynamical Systems10 (1990), 151–176.
H. Masur,Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J.66 (1992), 387–442.
H. MasurRandom walk on Teichmüller space and the mapping class group, to appear in Journal d’Analyse Mathématique.
H. Masur and J. Smillie,Hausdorff dimension of sets of nonergodic measured foliations, Ann. of Math. (2)134 (1991), 455–543.
H. Masur and J. Smillie,Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms, Comm. Math. Helv.68 (1993), 289–307.
Y. Sinai,Hyperbolic billiards, Proc. Internat. Congress Math., 1990, vol. 1, Kyoto, pp. 249–260.
K. Strebel, Quadratic Differentials, Springer-Verlag, Berlin and New York, 1984.
W. P. Thurston,On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc.18 (1988), 417–431.
W. Veech,Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem mod 2, Trans. Amer. Math. Soc.140 (1969), 1–34.
W. Veech,The Teichmüller geodesic flow, Ann. of Math. (2)124 (1986), 441–530.
W. Veech,Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math.97 (1989), 553–583.
A. Zemylakov and A. Katok,Topological transitivity of billiards in polygons, Mat. Zametki18 (1975), 291–300.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Birkhaäser Verlag
About this paper
Cite this paper
Masur, H. (1995). Teichmüller Space, Dynamics, Probability. In: Chatterji, S.D. (eds) Proceedings of the International Congress of Mathematicians. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-9078-6_77
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9078-6_77
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9897-3
Online ISBN: 978-3-0348-9078-6
eBook Packages: Springer Book Archive