Hausdorff dimension of divergent Teichmüller geodesics

Author:
Howard Masur

Journal:
Trans. Amer. Math. Soc. **324** (1991), 235-254

MSC:
Primary 30F30; Secondary 28A75, 32G15, 58F17

MathSciNet review:
984857

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be given and let be an -tuple of positive integers whose sum is . Denote by the set of all holomorphic quadratic differentials on compact Riemann surfaces of genus whose zeros have orders . is called a stratum inside the cotangent space of all holomorphic quadratic differentials over the Teichmüller space of genus . Let be the moduli space where is the mapping class group. Each defines a Teichmüller geodesic.

**Theorem**. *There exists* *so that for almost all* , *the set of* , *such that the geodesic defined by* *eventually leaves every compact set in* , *has Hausdorff dimension* .

**[B]**A. F. Beardon,*The Hausdorff dimension of singular sets of properly discontinuous groups*, Amer. J. Math.**88**(1966), 722–736. MR**0199385****[Ber]**Lipman Bers,*Finite-dimensional Teichmüller spaces and generalizations*, Bull. Amer. Math. Soc. (N.S.)**5**(1981), no. 2, 131–172. MR**621883**, 10.1090/S0273-0979-1981-14933-8**[F]**K. J. Falconer,*The geometry of fractal sets*, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR**867284****[Fa]***Travaux de Thurston sur les surfaces*, Astérisque, vol. 66, Société Mathématique de France, Paris, 1979 (French). Séminaire Orsay; With an English summary. MR**568308****[G]**Frederick P. Gardiner,*Teichmüller theory and quadratic differentials*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. A Wiley-Interscience Publication. MR**903027****[KMS]**Steven Kerckhoff, Howard Masur, and John Smillie,*Ergodicity of billiard flows and quadratic differentials*, Ann. of Math. (2)**124**(1986), no. 2, 293–311. MR**855297**, 10.2307/1971280**[K]**S. Kerckhoff (to appear).**[M1]**Howard Masur,*Closed trajectories for quadratic differentials with an application to billiards*, Duke Math. J.**53**(1986), no. 2, 307–314. MR**850537**, 10.1215/S0012-7094-86-05319-6**[M2]**Howard Masur,*Interval exchange transformations and measured foliations*, Ann. of Math. (2)**115**(1982), no. 1, 169–200. MR**644018**, 10.2307/1971341**[M3]**-,*The asymptotics of saddle connections for a quadratic differential*, Ergodic Theory Dynamical Systems**10**(1990), 151-176.**[MS]**Howard Masur and John Smillie,*Hausdorff dimension of sets of nonergodic measured foliations*, Ann. of Math. (2)**134**(1991), no. 3, 455–543. MR**1135877**, 10.2307/2944356**[R]**Mary Rees,*An alternative approach to the ergodic theory of measured foliations on surfaces*, Ergodic Theory Dynamical Systems**1**(1981), no. 4, 461–488 (1982). MR**662738****[S]**Kurt Strebel,*Quadratic differentials*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. MR**743423****[V1]**William A. Veech,*Gauss measures for transformations on the space of interval exchange maps*, Ann. of Math. (2)**115**(1982), no. 1, 201–242. MR**644019**, 10.2307/1971391**[V2]**William A. Veech,*The Teichmüller geodesic flow*, Ann. of Math. (2)**124**(1986), no. 3, 441–530. MR**866707**, 10.2307/2007091**[V3]**W. A. Veech,*Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards*, Invent. Math.**97**(1989), no. 3, 553–583. MR**1005006**, 10.1007/BF01388890

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
30F30,
28A75,
32G15,
58F17

Retrieve articles in all journals with MSC: 30F30, 28A75, 32G15, 58F17

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1991-0984857-0

Article copyright:
© Copyright 1991
American Mathematical Society