Approximation of Oseledets Splittings

TL;DR

This paper demonstrates that Oseledets splittings for ergodic hyperbolic measures can be approximated by those of atomic measures on periodic orbits, removing previous spectral assumptions and strengthening existing lemmas.

Contribution

It introduces a new approximation method for Oseledets splittings that does not require simple spectrum assumptions, enhancing the understanding of hyperbolic dynamics.

Findings

Oseledets splittings can be approximated by atomic measures on periodic orbits.

The approximation removes the need for simple spectrum assumptions.

Strengthens Katok's closing lemma for hyperbolic measures.

Abstract

We prove that the Oseledets splittings of an ergodic hyperbolic measure of a $C^{1+ r}$ diffeomorphism can be approximated by that of atomic measures on hyperbolic periodic orbits. This removes the assumption on simple spectrum in \cite{Liang} and strengthens Katok's closing lemma.