Uniform hyperbolic approximations of measures with non-zero Lyapunov exponents

Abstract:

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 $supp (\mu _{n}) \subseteq \Omega _{n}$ we have $\mu _{n}\to \mu$ in the weak-$*$ topology.