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  1. ; ; (, Ergodic Theory and Dynamical Systems)
    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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