Uniform hyperbolic approximations of measures with non zero Lyapunov exponents

TL;DR

This paper demonstrates that for certain dynamical systems, measures with non-zero Lyapunov exponents can be approximated by uniformly hyperbolic sets, providing a bridge between non-uniform and uniform hyperbolicity.

Contribution

It introduces a method to approximate non-atomic ergodic measures with non-zero Lyapunov exponents using sequences of uniformly hyperbolic sets in smooth dynamical systems.

Findings

Non-atomic ergodic measures with non-zero Lyapunov exponents can be approximated by hyperbolic sets.

Construction of sequences of hyperbolic sets approximating given measures.

Weak-* convergence of invariant measures supported on hyperbolic sets to the target measure.

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) in Omega_n we have mu_n -> mu in the weak-* topology.