Variational equalities of entropy in nonuniformly hyperbolic systems

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In this paper we prove that for an ergodic hyperbolic measure ω of a C¹⁺α diffeomorphism f on a Riemannian manifold M, there is an ω-full measured set Λ such that for every invariant probability μ∈ Minv(Λ,f), the metric entropy of μ is equal to the topological entropy of saturated set Gμ consisting of generic points of μ: hμ(f)=h(f,Gμ). Moreover, for every nonempty, compact and connected subset K of Minv(Λ,f) with the same hyperbolic rate, we compute the topological entropy of saturated set Gₖ of K by the following equality: {hμ(f) μ∈ K}=h(f,Gₖ). In particular these results can be applied (i) to the nonuniformy hyperbolic diffeomorphisms described by Katok, (ii) to the robustly transitive partially hyperbolic diffeomorphisms described by Mañé, (iii) to the robustly transitive non-partially hyperbolic diffeomorphisms described by Bonatti-Viana. In all these cases Minv(Λ,f) contains an open subset of Merg(M,f).