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. doi: 10.1215/S0012-7094-85-05238-X. Google Scholar

[2]

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

[3]

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

[4]

A. Fathi, F. Laudenbach and V. Poenaru, Travaux de Thurston sur les surfaces,, Asterisque, (1979), 66. Google Scholar

[5]

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

[6]

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

[7]

D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces,, in Sinaĭ's Moscow Seminar on Dynamical Systems, (1996), 141. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in Handbook of Dynamical Systems, (2002), 1015. doi: 10.1016/S1874-575X(02)80015-7. Google Scholar

[13]

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

[14]

R. Penner and J. Harer, Combinatorics of Train Tracks,, Annals of Math. Studies, (1992). Google Scholar

[15]

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

[16]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

A. Fathi, F. Laudenbach and V. Poenaru, Travaux de Thurston sur les surfaces,, Asterisque, (1979), 66. Google Scholar

[5]

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

[6]

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

[7]

D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces,, in Sinaĭ's Moscow Seminar on Dynamical Systems, (1996), 141. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in Handbook of Dynamical Systems, (2002), 1015. doi: 10.1016/S1874-575X(02)80015-7. Google Scholar

[13]

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

[14]

R. Penner and J. Harer, Combinatorics of Train Tracks,, Annals of Math. Studies, (1992). Google Scholar

[15]

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

[16]

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

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