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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: anjinpeng@gmail.com)
LIFAN GUAN
Affiliation:
Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstrasse 3-5, D-37073Gottingen, Germany (e-mail: guanlifan@gmail.com)
DMITRY KLEINBOCK*
Affiliation:
Department of Mathematics, Brandeis University, WalthamMA02454-9110, USA
*
e-mail: kleinboc@brandeis.edu
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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
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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