Uniform hyperbolic approximations of measures with non zero Lyapunov   exponents
  We show that for any C^1+alpha diffeomorphism of a compact Riemannian manifold, every non-atomic, ergodic, invariant probability measure with non-zero Lyapunov exponents is approximated by uniformly hyperbolic sets in the sense that there exists a sequence Omega_n of compact, topologically transitive, locally maximal, uniformly hyperbolic sets, such that for any sequence mu_n of f-invariant ergodic probability measures with supp (mu_n) in Omega_n we have mu_n -> mu in the weak-* topology. 
Stefano Luzzatto and Fernando J S\'anchez-Salas
math.DS