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

Entropy of irregular points that are not uniformly hyperbolic