  We show that set of points nondense under the $\times n$-map on the circle
and dense for the geodesic flow under the induced map on the circle
corresponding to the expanding horospherical subgroup has full Haudorff
dimension. We also show the analogous result for toral automorphisms on the
$2$-torus and a diagonal flow. Our results can be interpreted in
number-theoretic terms: the set of well approximable numbers that are nondense
under the $\times n$-map has full Hausdorff dimension. Similarly, the set of
well approximable $2$-vectors that are nondense under a hyperbolic toral
automorphism has full Hausdorff dimension. Our result for numbers is the
counterpart to a classical result of Kaufmann and gives a comprehensive
understanding.
