  In this article we prove that for a $C^{1+\alpha}$ diffeomorphism on a
compact Riemannian manifold, if there is a hyperbolic ergodic measure whose
support is not uniformly hyperbolic, then the topological entropy of the set of
irregular points that are not uniformly hyperbolic is larger than or equal to
the metric entropy of the hyperbolic ergodic measure. In the process of proof,
we give an abstract general mechanism to study topological entropy of irregular
points provided that the system has a sequence of nondecreasing invariant
compact subsets such that every subsystem has shadowing property and is
transitive.
