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

Published online by Cambridge University Press:  18 March 2021

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

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.

Type
Original Article
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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