Bounded orbits of certain diagonalizable flows on $SL_{n}(R)/SL_{n}(Z)$
HTML articles powered by AMS MathViewer
- by Lifan Guan and Weisheng Wu PDF
- Trans. Amer. Math. Soc. 370 (2018), 4661-4681 Request permission
Abstract:
We prove that the set of points that have bounded orbits under certain diagonalizable flows is a hyperplane absolute winning subset of $SL_{n}(\mathbb {R})/SL_{n}(\mathbb {Z})$.References
- J. An, Badziahin-Pollington-Velani’s theorem and Schmidt’s game, The Bulletin of the London Mathematical Society 45 (2013), no. 4, 721–33.
- Jinpeng An, 2-dimensional badly approximable vectors and Schmidt’s game, Duke Math. J. 165 (2016), no. 2, 267–284. MR 3457674, DOI 10.1215/00127094-3165862
- Jinpeng An, Lifan Guan, and Dmitry Kleinbock, Bounded orbits of diagonalizable flows on $\mathrm {SL}_3(\Bbb {R})/\mathrm {SL}_3(\Bbb {Z})$, Int. Math. Res. Not. IMRN 24 (2015), 13623–13652. MR 3436158, DOI 10.1093/imrn/rnv120
- Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
- Dzmitry Badziahin, Andrew Pollington, and Sanju Velani, On a problem in simultaneous Diophantine approximation: Schmidt’s conjecture, Ann. of Math. (2) 174 (2011), no. 3, 1837–1883. MR 2846492, DOI 10.4007/annals.2011.174.3.9
- Victor Beresnevich, Badly approximable points on manifolds, Invent. Math. 202 (2015), no. 3, 1199–1240. MR 3425389, DOI 10.1007/s00222-015-0586-8
- Ryan Broderick, Lior Fishman, Dmitry Kleinbock, Asaf Reich, and Barak Weiss, The set of badly approximable vectors is strongly $C^1$ incompressible, Math. Proc. Cambridge Philos. Soc. 153 (2012), no. 2, 319–339. MR 2981929, DOI 10.1017/S0305004112000242
- J. W. S. Cassels, An introduction to the geometry of numbers, Classics in Mathematics, Springer-Verlag, Berlin, 1997. Corrected reprint of the 1971 edition. MR 1434478
- S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math. 359 (1985), 55–89. MR 794799, DOI 10.1515/crll.1985.359.55
- S. G. Dani, Bounded orbits of flows on homogeneous spaces, Comment. Math. Helv. 61 (1986), no. 4, 636–660. MR 870710, DOI 10.1007/BF02621936
- L. Fishman, D. S. Simmons and M. Urbański, Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces (extended version), Memoirs of the American Mathematical Society, to appear.
- L. Guan and J. Yu, Badly approximable vectors in higher dimension, arXiv preprint arXiv:1509.08050, 2015.
- Dmitry Y. Kleinbock, Flows on homogeneous spaces and Diophantine properties of matrices, Duke Math. J. 95 (1998), no. 1, 107–124. MR 1646538, DOI 10.1215/S0012-7094-98-09503-5
- D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, Sinaĭ’s Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, vol. 171, Amer. Math. Soc., Providence, RI, 1996, pp. 141–172. MR 1359098, DOI 10.1090/trans2/171/11
- Dmitry Kleinbock and Barak Weiss, Values of binary quadratic forms at integer points and Schmidt games, Recent trends in ergodic theory and dynamical systems, Contemp. Math., vol. 631, Amer. Math. Soc., Providence, RI, 2015, pp. 77–92. MR 3330339, DOI 10.1090/conm/631/12597
- Grigorii A. Margulis, Dynamical and ergodic properties of subgroup actions on homogeneous spaces with applications to number theory, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 193–215. MR 1159213
- Dave Witte Morris, Ratner’s theorems on unipotent flows, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2005. MR 2158954
- Calvin C. Moore, Ergodicity of flows on homogeneous spaces, Amer. J. Math. 88 (1966), 154–178. MR 193188, DOI 10.2307/2373052
- Erez Nesharim and David Simmons, $\textbf {Bad} (s,t)$ is hyperplane absolute winning, Acta Arith. 164 (2014), no. 2, 145–152. MR 3224831, DOI 10.4064/aa164-2-4
- Marina Ratner, Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), no. 1, 235–280. MR 1106945, DOI 10.1215/S0012-7094-91-06311-8
- Wolfgang M. Schmidt, On badly approximable numbers and certain games, Trans. Amer. Math. Soc. 123 (1966), 178–199. MR 195595, DOI 10.1090/S0002-9947-1966-0195595-4
- Wolfgang M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. MR 568710
Additional Information
- Lifan Guan
- Affiliation: Beijing International Center for Mathematical Research, Peking University, Beijing 100871, People’s Republic of China
- Address at time of publication: Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom
- MR Author ID: 1141210
- Email: lifan.guan@york.ac.uk
- Weisheng Wu
- Affiliation: Department of Applied Mathematics, College of Science, China Agricultural University, Beijing, 100083, People’s Republic of China
- Email: wuweisheng@math.pku.edu.cn
- Received by editor(s): April 28, 2016
- Received by editor(s) in revised form: September 20, 2016
- Published electronically: December 20, 2017
- Additional Notes: The research of the second author was supported by CPSF (#2015T80010)
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 4661-4681
- MSC (2010): Primary 11J04; Secondary 22E40, 28A78
- DOI: https://doi.org/10.1090/tran/7082
- MathSciNet review: 3812091