  For a surface diffeomorphism, a compact invariant locally maximal set $W$ and
some subset $A\subset W$ we study the $A$-exceptional set, that is, the set of
points whose orbits do not accumulate at $A$. We show that if the Hausdorff
dimension of $A$ is smaller than the Hausdorff dimension $d$ of some ergodic
hyperbolic measure, then the topological entropy of the exceptional set is at
least the entropy of this measure and its Hausdorff dimension is at least $d$.
Particular consequences occur when there is some a priori defined hyperbolic
structure on $W$ and, for example, if there exists an SRB measure.
