  Given a rational map of the Riemann sphere and a subset $A$ of its Julia set,
we study the $A$-exceptional set, that is, the set of points whose orbit does
not accumulate at $A$. We prove that if the topological entropy of $A$ is less
than the topological entropy of the full system then the $A$-exceptional set
has full topological entropy. Furthermore, if the Hausdorff dimension of $A$ is
smaller than the dynamical dimension of the system then the Hausdorff dimension
of the $A$-exceptional set is larger than or equal to the dynamical dimension,
with equality in the particular case when the dynamical dimension and the
Hausdorff dimension coincide.
  We discuss also the case of a general conformal $C^{1+\alpha}$ dynamical
system and, in particular, certain multimodal interval maps on their Julia set.
