Uniform hyperbolic approximations of measures with non-zero Lyapunov exponents

Abstract

We show that for any C 1 + α 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 Ω n \Omega _{n} of compact, topologically transitive, locally maximal, uniformly hyperbolic sets such that for any sequence { μ n } \{\mu _{n}\} of f f -invariant ergodic probability measures with s u p p ( μ n ) ⊆ Ω n supp (\mu _{n}) \subseteq \Omega _{n} we have μ n → μ \mu _{n}\to \mu in the weak- ∗ * topology.