July  2013, 7(3): 429-460. doi: 10.3934/jmd.2013.7.429

Modified Schmidt games and a conjecture of Margulis

1. 

Goldsmith 207, Brandeis University, Waltham, MA 02454-9110

2. 

Ben Gurion University, Be'er Sheva, Israel 84105

Received  December 2012 Published  December 2013

We prove a conjecture of G.A. Margulis on the abundance of certain exceptional orbits of partially hyperbolic flows on homogeneous spaces by utilizing a theory of modified Schmidt games, which are modifications of $(\alpha,\beta)$-games introduced by W. Schmidt in mid-1960s.
Citation: Dmitry Kleinbock, Barak Weiss. Modified Schmidt games and a conjecture of Margulis. Journal of Modern Dynamics, 2013, 7 (3) : 429-460. doi: 10.3934/jmd.2013.7.429
References:
[1]

J. An, Two dimensional badly approximable vectors and Schmidt's game, preprint, (2012), arXiv:1204.3610.

[2]

V. Beresnevich, Badly approximable points on manifolds, preprint, (2013), arXiv:1304.0571.

[3]

A. Borel, Linear Algebraic Groups, Second edition, Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0941-6.

[4]

A. Borel, Introduction aux Groupes Arithmétiques, (French) Publications de l'Institut de Mathématique de l'Université de Strasbourg, XV, Actualités Scientifiques et Industrielles, No. 1341, Hermann, Paris, 1969.

[5]

D. Badziahin, A. Pollington and S. Velani, On a problem in simultaneous Diophantine approximation: Schmidt's conjecture, Ann. of Math. (2), 174 (2011), 1837-1883. doi: 10.4007/annals.2011.174.3.9.

[6]

D. Badziahin and S. Velani, Badly approximable points on planar curves and a problem of Davenport, preprint, (2013), arXiv:1301.4243.

[7]

S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math., 47 (1978), 101-138. doi: 10.1007/BF01578067.

[8]

_______, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math., 359 (1985), 55-89. doi: 10.1515/crll.1985.359.55.

[9]

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

[10]

_______, On badly approximable numbers, Schmidt games and bounded orbits of flows, in Number Theory and Dynamical Systems (York, 1987), London Math. Soc. Lecture Note Ser., 134, Cambridge Univ. Press, Cambridge, 1989, 69-86. doi: 10.1017/CBO9780511661983.006.

[11]

D. Dolgopyat, Bounded orbits of Anosov flows, Duke Math. J., 87 (1997), 87-114. doi: 10.1215/S0012-7094-97-08704-4.

[12]

M. Einsiedler, A. Katok and E. Lindenstrauss, Invariant measures and the set of exceptions to Littlewood's conjecture, Ann. Math. (2), 164 (2006), 513-560. doi: 10.4007/annals.2006.164.513.

[13]

M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010.

[14]

K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, Second edition, John Wiley & Sons, Inc., Hoboken, NJ, 2003. doi: 10.1002/0470013850.

[15]

H. Garland and M. S. Raghunathan, Fundamental domains for lattices in (R-)rank 1 semisimple Lie groups, Ann. of Math. (2), 92 (1970), 279-326. doi: 10.2307/1970838.

[16]

D. Kleinbock, Nondense orbits of flows on homogeneous spaces, Ergodic Theory Dynam. Systems, 18 (1998), 373-396. doi: 10.1017/S0143385798100408.

[17]

________, Flows on homogeneous spaces and Diophantine properties of matrices, Duke Math. J., 95 (1998), 107-124. doi: 10.1215/S0012-7094-98-09503-5.

[18]

D. 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.

[19]

D. Kleinbock, N. Shah and A. Starkov, Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory, in Handbook on Dynamical Systems, Vol. 1A, North Holland, Amsterdam, 2002, 813-930. doi: 10.1016/S1874-575X(02)80013-3.

[20]

D. Kleinbock and B. Weiss, Badly approximable vectors on fractals. Probability in mathematics, Israel J. Math., 149 (2005), 137-170. doi: 10.1007/BF02772538.

[21]

_______, Dirichlet's theorem on diophantine approximation and homogeneous flows, J. Mod. Dyn., 4 (2008), 43-62.

[22]

_______, Modified Schmidt games and diophantine approximation with weights, Adv. Math., 223 (2010), 1276-1298. doi: 10.1016/j.aim.2009.09.018.

[23]

G. A. Margulis, Dynamical and ergodic properties of subgroup actions on homogeneous spaces with applications to number theory, in Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, 193-215.

[24]

D. Morris, Ratner's Theorems on Unipotent Flows, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2005.

[25]

E. Nehsarim, Badly approximable vectors on a vertical Cantor set, preprint, (2012), arXiv:1204.0110.

[26]

Y. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.

[27]

A. Pollington and S. Velani, On simultaneously badly approximable numbers, J. London Math. Soc. (2), 66 (2002), 29-40. doi: 10.1112/S0024610702003265.

[28]

M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972.

[29]

M. Ratner, Interactions between ergodic theory, Lie groups, and number theory, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 157-182.

[30]

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

[31]

_______, Badly approximable systems of linear forms, J. Number Theory, 1 (1969), 139-154. doi: 10.1016/0022-314X(69)90032-8.

[32]

_______, Open problems in Diophantine approximation, in Diophantine Approximations and Transcendental Numbers (Luminy, 1982), Progr. Math., 31, Birkhäuser, Boston, 1983, 271-287.

[33]

A. N. Starkov, The structure of orbits of homogeneous flows and the Raghunathan conjecture, (Russian) Uspekhi Mat. Nauk, 45 (1990), 219-220; translation in Russian Math. Surveys, 45 (1990), 227-228. doi: 10.1070/RM1990v045n02ABEH002338.

[34]

G. Tomanov and B. Weiss, Closed orbits for actions of maximal tori on homogeneous spaces, Duke Math. J., 119 (2003), 367-392. doi: 10.1215/S0012-7094-03-11926-2.

[35]

H. Wegmann, Die Hausdorff-Dimension von kartesischen Produkten metrischer Räume, (German) J. Reine Angew. Math., 246 (1971), 46-75.

[36]

B. Weiss, Finite-dimensional representations and subgroup actions on homogeneous spaces, Israel J. Math., 106 (1998), 189-207. doi: 10.1007/BF02773468.

[37]

R. J. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Mathematics, 81, Birkhäuser Verlag, Basel, 1984.

show all references

References:
[1]

J. An, Two dimensional badly approximable vectors and Schmidt's game, preprint, (2012), arXiv:1204.3610.

[2]

V. Beresnevich, Badly approximable points on manifolds, preprint, (2013), arXiv:1304.0571.

[3]

A. Borel, Linear Algebraic Groups, Second edition, Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0941-6.

[4]

A. Borel, Introduction aux Groupes Arithmétiques, (French) Publications de l'Institut de Mathématique de l'Université de Strasbourg, XV, Actualités Scientifiques et Industrielles, No. 1341, Hermann, Paris, 1969.

[5]

D. Badziahin, A. Pollington and S. Velani, On a problem in simultaneous Diophantine approximation: Schmidt's conjecture, Ann. of Math. (2), 174 (2011), 1837-1883. doi: 10.4007/annals.2011.174.3.9.

[6]

D. Badziahin and S. Velani, Badly approximable points on planar curves and a problem of Davenport, preprint, (2013), arXiv:1301.4243.

[7]

S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math., 47 (1978), 101-138. doi: 10.1007/BF01578067.

[8]

_______, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math., 359 (1985), 55-89. doi: 10.1515/crll.1985.359.55.

[9]

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

[10]

_______, On badly approximable numbers, Schmidt games and bounded orbits of flows, in Number Theory and Dynamical Systems (York, 1987), London Math. Soc. Lecture Note Ser., 134, Cambridge Univ. Press, Cambridge, 1989, 69-86. doi: 10.1017/CBO9780511661983.006.

[11]

D. Dolgopyat, Bounded orbits of Anosov flows, Duke Math. J., 87 (1997), 87-114. doi: 10.1215/S0012-7094-97-08704-4.

[12]

M. Einsiedler, A. Katok and E. Lindenstrauss, Invariant measures and the set of exceptions to Littlewood's conjecture, Ann. Math. (2), 164 (2006), 513-560. doi: 10.4007/annals.2006.164.513.

[13]

M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010.

[14]

K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, Second edition, John Wiley & Sons, Inc., Hoboken, NJ, 2003. doi: 10.1002/0470013850.

[15]

H. Garland and M. S. Raghunathan, Fundamental domains for lattices in (R-)rank 1 semisimple Lie groups, Ann. of Math. (2), 92 (1970), 279-326. doi: 10.2307/1970838.

[16]

D. Kleinbock, Nondense orbits of flows on homogeneous spaces, Ergodic Theory Dynam. Systems, 18 (1998), 373-396. doi: 10.1017/S0143385798100408.

[17]

________, Flows on homogeneous spaces and Diophantine properties of matrices, Duke Math. J., 95 (1998), 107-124. doi: 10.1215/S0012-7094-98-09503-5.

[18]

D. 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.

[19]

D. Kleinbock, N. Shah and A. Starkov, Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory, in Handbook on Dynamical Systems, Vol. 1A, North Holland, Amsterdam, 2002, 813-930. doi: 10.1016/S1874-575X(02)80013-3.

[20]

D. Kleinbock and B. Weiss, Badly approximable vectors on fractals. Probability in mathematics, Israel J. Math., 149 (2005), 137-170. doi: 10.1007/BF02772538.

[21]

_______, Dirichlet's theorem on diophantine approximation and homogeneous flows, J. Mod. Dyn., 4 (2008), 43-62.

[22]

_______, Modified Schmidt games and diophantine approximation with weights, Adv. Math., 223 (2010), 1276-1298. doi: 10.1016/j.aim.2009.09.018.

[23]

G. A. Margulis, Dynamical and ergodic properties of subgroup actions on homogeneous spaces with applications to number theory, in Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, 193-215.

[24]

D. Morris, Ratner's Theorems on Unipotent Flows, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2005.

[25]

E. Nehsarim, Badly approximable vectors on a vertical Cantor set, preprint, (2012), arXiv:1204.0110.

[26]

Y. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.

[27]

A. Pollington and S. Velani, On simultaneously badly approximable numbers, J. London Math. Soc. (2), 66 (2002), 29-40. doi: 10.1112/S0024610702003265.

[28]

M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972.

[29]

M. Ratner, Interactions between ergodic theory, Lie groups, and number theory, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 157-182.

[30]

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

[31]

_______, Badly approximable systems of linear forms, J. Number Theory, 1 (1969), 139-154. doi: 10.1016/0022-314X(69)90032-8.

[32]

_______, Open problems in Diophantine approximation, in Diophantine Approximations and Transcendental Numbers (Luminy, 1982), Progr. Math., 31, Birkhäuser, Boston, 1983, 271-287.

[33]

A. N. Starkov, The structure of orbits of homogeneous flows and the Raghunathan conjecture, (Russian) Uspekhi Mat. Nauk, 45 (1990), 219-220; translation in Russian Math. Surveys, 45 (1990), 227-228. doi: 10.1070/RM1990v045n02ABEH002338.

[34]

G. Tomanov and B. Weiss, Closed orbits for actions of maximal tori on homogeneous spaces, Duke Math. J., 119 (2003), 367-392. doi: 10.1215/S0012-7094-03-11926-2.

[35]

H. Wegmann, Die Hausdorff-Dimension von kartesischen Produkten metrischer Räume, (German) J. Reine Angew. Math., 246 (1971), 46-75.

[36]

B. Weiss, Finite-dimensional representations and subgroup actions on homogeneous spaces, Israel J. Math., 106 (1998), 189-207. doi: 10.1007/BF02773468.

[37]

R. J. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Mathematics, 81, Birkhäuser Verlag, Basel, 1984.

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