  For a geodesic flow on a negatively curved Riemannian manifold $M$ and some
subset $A\subset T^1M$, we study the limit $A$-exceptional set, that is the set
of points whose $\omega$-limit do not intersect $A$. We show that if the
topological $\ast$-entropy of $A$ is smaller than the topological entropy of
the geodesic flow, then the limit $A$-exceptional set has full topological
entropy. Some consequences are stated for limit exceptional sets of invariant
compact subsets and proper submanifolds.
