July  2013, 7(3): 395-427. doi: 10.3934/jmd.2013.7.395

Winning games for bounded geodesics in moduli spaces of quadratic differentials

1. 

Mathematics Department, 155 S 1400 E Room 233, University of Utah, Salt Lake City, UT 84112-0090, United States

2. 

Mathematics Department, San Francisco State University, 1600 Holloway Avenue, San Francisco, CA 94132, United States

3. 

Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Il 60637, United States

Received  December 2012 Published  December 2013

We prove that the set of bounded geodesics in Teichmüller space is a winning set for Schmidt's game. This is a notion of largeness in a metric space that can apply to measure $0$ and meager sets. We prove analogous closely related results on any Riemann surface, in any stratum of quadratic differentials, on any Teichmüller disk and for intervals exchanges with any fixed irreducible permutation.
Citation: Jonathan Chaika, Yitwah Cheung, Howard Masur. Winning games for bounded geodesics in moduli spaces of quadratic differentials. Journal of Modern Dynamics, 2013, 7 (3) : 395-427. doi: 10.3934/jmd.2013.7.395
References:
[1]

M. Boshernitzan, A condition for minimal interval exchange maps to be uniquely ergodic, Duke Math. J., 52 (1985), 723-752. doi: 10.1215/S0012-7094-85-05238-X.

[2]

M. Boshernitzan, Rank two interval exchange transformations, Ergod. Theory Dynam. Systems, 8 (1988), 379-394. doi: 10.1017/S0143385700004521.

[3]

S. G. Dani, Bounded orbits of flows on homogeneous spaces, Comment. Math. Helv., 61 (1986), 636-660. doi: 10.1007/BF02621936.

[4]

A. Fathi, F. Laudenbach and V. Poenaru, Travaux de Thurston sur les surfaces, Asterisque, No. 66-67, Société Mathématique de France, Paris, 1979.

[5]

J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta. Math., 142 (1979), 221-274. doi: 10.1007/BF02395062.

[6]

S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Annals of Math. (2), 124 (1986), 293-311. doi: 10.2307/1971280.

[7]

D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinaĭ's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, 171, Amer. Math. Soc., Providence, RI, 1996, 141-172.

[8]

D. Kleinbock and B. Weiss, Bounded geodesics in moduli space, Int. Math. Res. Not., 2004 (2004), 1551-1560. doi: 10.1155/S1073792804133412.

[9]

D. Kleinbock and B. Weiss, Modified Schmidt games and Diophantine approximation with weights, Advances in Math., 223 (2010), 1276-1298. doi: 10.1016/j.aim.2009.09.018.

[10]

C. T. McMullen, Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal., 20 (2010), 726-740. doi: 10.1007/s00039-010-0078-3.

[11]

C. T. McMullen, Diophantine and ergodic foliations on surfaces, J. Topol., 6 (2013), 349-360. doi: 10.1112/jtopol/jts033.

[12]

H. Masur and S. Tabachnikov, Rational billiards and flat structures, in Handbook of Dynamical Systems, Vol. 1A (eds. B. Hasselblatt and A. Katok), North-Holland, Amsterdam, 2002, 1015-1089. doi: 10.1016/S1874-575X(02)80015-7.

[13]

Y. Minsky and B. Weiss, Nondivergence of horocyclic flows on moduli space, I. Reine Angew. Math., 552 (2002), 131-177. doi: 10.1515/crll.2002.088.

[14]

R. Penner and J. Harer, Combinatorics of Train Tracks, Annals of Math. Studies, {125}, Princeton University Press, Princeton, NJ, 1992.

[15]

W. Schmidt, On badly approximable numbers and certain games, Trans. Amer. Math. Soc., 123 (1966), 178-199. doi: 10.1090/S0002-9947-1966-0195595-4.

[16]

W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391.

show all references

References:
[1]

M. Boshernitzan, A condition for minimal interval exchange maps to be uniquely ergodic, Duke Math. J., 52 (1985), 723-752. doi: 10.1215/S0012-7094-85-05238-X.

[2]

M. Boshernitzan, Rank two interval exchange transformations, Ergod. Theory Dynam. Systems, 8 (1988), 379-394. doi: 10.1017/S0143385700004521.

[3]

S. G. Dani, Bounded orbits of flows on homogeneous spaces, Comment. Math. Helv., 61 (1986), 636-660. doi: 10.1007/BF02621936.

[4]

A. Fathi, F. Laudenbach and V. Poenaru, Travaux de Thurston sur les surfaces, Asterisque, No. 66-67, Société Mathématique de France, Paris, 1979.

[5]

J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta. Math., 142 (1979), 221-274. doi: 10.1007/BF02395062.

[6]

S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Annals of Math. (2), 124 (1986), 293-311. doi: 10.2307/1971280.

[7]

D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinaĭ's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, 171, Amer. Math. Soc., Providence, RI, 1996, 141-172.

[8]

D. Kleinbock and B. Weiss, Bounded geodesics in moduli space, Int. Math. Res. Not., 2004 (2004), 1551-1560. doi: 10.1155/S1073792804133412.

[9]

D. Kleinbock and B. Weiss, Modified Schmidt games and Diophantine approximation with weights, Advances in Math., 223 (2010), 1276-1298. doi: 10.1016/j.aim.2009.09.018.

[10]

C. T. McMullen, Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal., 20 (2010), 726-740. doi: 10.1007/s00039-010-0078-3.

[11]

C. T. McMullen, Diophantine and ergodic foliations on surfaces, J. Topol., 6 (2013), 349-360. doi: 10.1112/jtopol/jts033.

[12]

H. Masur and S. Tabachnikov, Rational billiards and flat structures, in Handbook of Dynamical Systems, Vol. 1A (eds. B. Hasselblatt and A. Katok), North-Holland, Amsterdam, 2002, 1015-1089. doi: 10.1016/S1874-575X(02)80015-7.

[13]

Y. Minsky and B. Weiss, Nondivergence of horocyclic flows on moduli space, I. Reine Angew. Math., 552 (2002), 131-177. doi: 10.1515/crll.2002.088.

[14]

R. Penner and J. Harer, Combinatorics of Train Tracks, Annals of Math. Studies, {125}, Princeton University Press, Princeton, NJ, 1992.

[15]

W. Schmidt, On badly approximable numbers and certain games, Trans. Amer. Math. Soc., 123 (1966), 178-199. doi: 10.1090/S0002-9947-1966-0195595-4.

[16]

W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391.

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