  Let $G$ be a Lie group, $\Gamma\subset G$ a discrete subgroup, $X=G/\Gamma$,
and $f$ an affine map from $X$ to itself. We give conditions on a submanifold
$Z$ of $X$ guaranteeing 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 to
constructing exceptional geodesics on locally symmetric spaces, and to
non-density of the set of values of certain functions at integer points.
