Hostname: page-component-766f47fc8c-wjxcd Total loading time: 0 Render date: 2025-03-03T09:15:40.325Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-dense orbits on homogeneous spaces and applications to geometry and number theory

Part of: Dynamical systems with hyperbolic behavior Geometry of numbers Ergodic theory

Published online by Cambridge University Press:  18 March 2021

JINPENG AN ,
LIFAN GUAN  and
DMITRY KLEINBOCK
JINPENG AN
Affiliation:
School of Mathematical Sciences, Peking University, Beijing100871, China (e-mail: [email protected])
LIFAN GUAN
Affiliation:
Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstrasse 3-5, D-37073Gottingen, Germany (e-mail: [email protected])
DMITRY KLEINBOCK*
Affiliation:
Department of Mathematics, Brandeis University, WalthamMA02454-9110, USA
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let G be a Lie group, let $\Gamma \subset G$ be a discrete subgroup, let $X=G/\Gamma $ and let f be an affine map from X to itself. We give conditions on a submanifold Z of X that guarantee that the set of points $x\in X$ with f-trajectories avoiding Z is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is proved for one-parameter actions on X. This has applications in constructing exceptional geodesics on locally symmetric spaces and in non-density of the set of values of certain functions at integer points.

Keywords

flows on homogeneous spacesnondense orbitshyperplane absolute winningtransversality

MSC classification

Primary: 37A17: Homogeneous flows
Secondary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 11H06: Lattices and convex bodies
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 4 , April 2022 , pp. 1327 - 1372
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

An, J., Ghosh, A., Guan, L. and Ly, T.. Bounded orbits of diagonalizable flows on finite volume quotients of products of ${\textrm{SL}}_2\left(\mathbb{R}\right)$ . Adv. Math. 354 (2019), 106743, p. 18.CrossRefGoogle Scholar
An, J., Guan, L. and Kleinbock, D.. Bounded orbits of diagonalizable flows on ${\textrm{SL}}_3\left(\mathbb{R}\right)/{\textrm{SL}}_3\left(\mathbb{Z}\right)$ . Int. Math. Res. Not. 2015(24) (2015), 13623–13652.CrossRefGoogle Scholar
Broderick, R., Fishman, L. and Kleinbock, D.. Schmidt’s game, fractals, and orbits of toral endomorphisms. Ergod. Th. & Dynam. Sys. 31(4) (2011), 1095–1107.CrossRefGoogle Scholar
Broderick, R., Fishman, L., Kleinbock, D., Reich, A. and Weiss, B.. The set of badly approximable vectors is strongly ${\textrm{C}}^1$ incompressible. Math. Proc. Cambridge Philos. Soc. 153(2) (2012), 319–339.CrossRefGoogle Scholar
Broderick, R., Fishman, L. and Simmons, D.. Badly approximable systems of affine forms and incompressibility on fractals. J. Number Theory 133(7) (2013), 2186–2205.CrossRefGoogle Scholar
Burns, K. and Pollicott, M.. Self-intersections of geodesics and projecting flow invariant sets. Preprint, 1994.Google Scholar
Buyalo, S. and Schroeder, V.. Invariant subsets of rank $1$ manifolds. Manuscripta Math. 107 (2002), 73–88.CrossRefGoogle Scholar
Buyalo, S., Schroeder, V. and Walz, M.. Geodesics avoiding open subsets in surfaces of negative curvature. Ergod. Th. & Dynam. Sys. 20 (2000), 991–1006.CrossRefGoogle Scholar
Cassels, J. W. S. and Swinnerton-Dyer, H. P. F.. On the product of three homogeneous linear forms and the indefinite ternary quadratic forms. Philos. Trans. Roy. Soc. Lond. Ser. A 248 (1955), 73–96.Google Scholar
Dani, S. G.. Divergent trajectories of flows on homogeneous spaces and Diophantine approximation. J. Reine Angew. Math. 359 (1985), 55–89.Google Scholar
Dani, S. G.. Bounded orbits of flows on homogeneous spaces. Comment. Math. Helv. 61 (1986), 636–660.CrossRefGoogle Scholar
Dani, S. G.. On orbits of endomorphisms of tori and the Schmidt game. Ergod. Th. & Dynam. Sys. 8 (1988), 523–529.CrossRefGoogle Scholar
Dolgopyat, D.. Bounded orbits of Anosov flows. Duke Math. J. 87(1) (1997), 87–114.CrossRefGoogle Scholar
Duvall, J.. Schmidt’s game and nonuniformly expanding interval maps. Nonlinearity 33 (2020), 5611–5628.CrossRefGoogle Scholar
Einsiedler, M., Katok, A. and Lindenstrauss, E.. Invariant measures and the set of exceptions to Littlewood’s conjecture. Ann. of Math. 164(2) (2006), 513–560.CrossRefGoogle Scholar
Guan, L. and Wu, W.. Bounded orbits of certain diagonalizable flows on ${SL}_n\left(\mathbb{R}\right)/{SL}_n\left(\mathbb{R}\right)$ . Trans. Amer. Math. Soc. 370 (7) (2018), 4661–4681.CrossRefGoogle Scholar
Hussain, M., Kristensen, S. and Simmons, D.. Metrical theorems on systems of affine forms. J. Number Theory 213 (2020), 67–100.CrossRefGoogle Scholar
Hu, H. and Yu, Y.. On Schmidt’s game and the set of points with non-dense orbits under a class of expanding maps. J. Math. Anal. Appl. 418(2) (2014), 906–920.CrossRefGoogle Scholar
Kleinbock, D.. Nondense orbits of flows on homogeneous spaces. Ergod. Th. & Dynam. Sys. 18 (1998), 373–396.CrossRefGoogle Scholar
Kleinbock, D. and Margulis, G.A.. Bounded orbits of nonquasiunipotent flows on homogeneous spaces. Amer. Math. Soc. Transl. 171 (1996), 141–172.Google Scholar
Kleinbock, D. and Margulis, G.A.. Logarithm laws for flows on homogeneous spaces. Inv. Math. 138 (1999), 451–494.CrossRefGoogle Scholar
Kleinbock, D. and Weiss, B.. Modified Schmidt games and a conjecture of Margulis. J. Mod. Dyn. 7(3) (2013), 429–460.CrossRefGoogle Scholar
Kleinbock, D. and Weiss, B.. Values of binary quadratic forms at integer points and Schmidt games. Recent Trends in Ergodic Theory and Dynamical Systems (Vadodara, 2012) (Contemporary Mathematics, 631). American Mathematical Society, Providence, RI, 2015, pp. 77–92.Google Scholar
Margulis, G. A.. Indefinite quadratic forms and unipotent flows on homogeneous spaces. Dynamical Systems and Ergodic Theory (Warsaw, 1986) (Banach Center Publications, 23). PWN, Warsaw, 1989, pp. 399–409.Google Scholar
Margulis, G. A.. Discrete subgroups and ergodic theory. Number Theory, Trace Formulas and Discrete Groups (Oslo, 1987). Academic Press, Boston, MA, 1989, pp. 377–398.Google Scholar
Margulis, G. A.. Problems and conjectures in rigidity theory. Mathematics: Frontiers and Perspectives. American Mathematical Society, Providence, RI, 2000, pp. 161–174.Google Scholar
Mautner, F. I.. Geodesic flows on symmetric Riemann spaces. Ann. of Math. 65(3) (1957), 416–431.CrossRefGoogle Scholar
McMullen, C.. Winning sets, quasiconformal maps and Diophantine approximation. Geom. Funct. Anal. 20(3) (2010), 726–740.CrossRefGoogle Scholar
Moshchevitin, N. G.. A note on badly approximable affine forms and winning sets. Mosc. Math. J. 11 (2011), 129–137.CrossRefGoogle Scholar
Reinold, B.. Flow invariant subsets for geodesic flows of manifolds with non-positive curvature. Ergod. Th. & Dynam. Sys. 24(6) (2004), 1981–1990.CrossRefGoogle Scholar
Schmidt, W. M.. On badly approximable numbers and certain games. Trans. Amer. Math. Soc. 123 (1966), 178–199.CrossRefGoogle Scholar
Schroeder, V.. Bounded geodesics in manifolds of negative curvature. Math. Z. 235 (2000), 817–828.CrossRefGoogle Scholar
Tseng, J.. Nondense orbits for Anosov diffeomorphisms of the $2$ -torus. Real Anal. Exchange 41(2) (2016), 307–314.CrossRefGoogle Scholar
Urbanski, M.. The Hausdorff dimension of the set of points with nondense orbit under a hyperbolic dynamical system. Nonlinearity 2 (1991), 385–397.CrossRefGoogle Scholar
Weil, S.. Schmidt games and conditions on resonant sets. Preprint, 2012, arXiv:1210.1152.Google Scholar
Wu, W.. Schmidt games and non-dense forward orbits of certain partially hyperbolic systems. Ergod. Th. & Dynam. Sys. 36(5) (2016), 1656–1678.CrossRefGoogle Scholar
Wu, W.. Modified Schmidt games and non-dense forward orbits of partially hyperbolic systems. Discrete Contin. Dyn. Syst. 36(6) (2016), 3463–3481.CrossRefGoogle Scholar