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Let X =G/Γ, where G is a connected Lie group and Γ is a lattice in G. Let O be an open subset of X, and let F = {g_t : t ≥ 0} be a one-parameter subsemigroup of G. Consider the set of points in X whose F-orbit misses O; it has measure zero if the flow is ergodic. It has been conjectured that, assuming ergodicity, this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture has been proved when X is compact or when G is a simple Lie group of real rank 1, or, most recently, for certain flows on the space of lattices. In this paper we prove this conjecture for arbitrary Addiagonalizable flows on irreducible quotients of semisimple Lie groups. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on G/Γ. We also derive an application to jointly Dirichlet-Improvable systems of linear forms.
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Non-dense orbits on homogeneous spaces and applications to geometry and number theory
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.
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- Award ID(s):
- 1900560
- PAR ID:
- 10336682
- Date Published:
- Journal Name:
- Ergodic Theory and Dynamical Systems
- Volume:
- 42
- Issue:
- 4
- ISSN:
- 0143-3857
- Page Range / eLocation ID:
- 1327 to 1372
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/etds.2021.4
