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.

Let $g:M\rightarrow M$ be a $C^{1+\unicode[STIX]{x1D6FC}}$-partially hyperbolic diffeomorphism preserving an ergodic normalized volume on $M$. We show that, if $f:M\rightarrow M$ is a $C^{1+\unicode[STIX]{x1D6FC}}$-Anosov diffeomorphism such that the stable subspaces of $f$ and $g$ span the whole tangent space at some point on $M$, the set of points that equidistribute under $g$ but have non-dense orbits under $f$ has full Hausdorff dimension. The same result is also obtained when $M$ is the torus and $f$ is a toral endomorphism whose center-stable subspace does not contain the stable subspace of $g$ at some point.