Exponential rate of periodic points and metric entropy in nonuniformly hyperbolic systems

For an ergodic hyperbolic measure ω of a C1+r diffeomorphism f there is an ω-full measured set Λ˜ such that for every invariant measure μ∈Minv(Λ˜,f) the exponential growth rate of the number of periodic points whose atomic measures approximate μ equals to the metric entropy hμ(f).