Abstract
Let G be a Lie group, let be a discrete subgroup, let 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 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.