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In this paper we prove that for an ergodic hyperbolic measure $\omega$ of a$C^{1+\alpha}$ diffeomorphism $f$ on a Riemannian manifold $M$, there is an$\omega$-full measured set $\widetilde{\Lambda}$ such that for every invariantprobability $\mu\in \mathcal{M}_{inv}(\widetilde{\Lambda},f)$, the metricentropy of $\mu$ is equal to the topological entropy of saturated set $G_{\mu}$consisting of generic points of $\mu$: $$h_\mu(f)=h_{\top}(f,G_{\mu}).$$Moreover, for every nonempty, compact and connected subset $K$ of$\mathcal{M}_{inv}(\widetilde{\Lambda},f)$ with the same hyperbolic rate, wecompute the topological entropy of saturated set $G_K$ of $K$ by the followingequality: $$\inf\{h_\mu(f)\mid \mu\in K\}=h_{\top}(f,G_K).$$  In particular these results can be applied (i) to the nonuniformy hyperbolicdiffeomorphisms described by Katok, (ii) to the robustly transitive partiallyhyperbolic diffeomorphisms described by ~Ma{\~{n}}{\'{e}}, (iii) to therobustly transitive non-partially hyperbolic diffeomorphisms described byBonatti-Viana. In all these cases $\mathcal{M}_{inv}(\widetilde{\Lambda},f)$contains an open subset of $\mathcal{M}_{erg}(M,f)$.

Variational equalities of entropy in nonuniformly hyperbolic systems