The Lyapunov Exponents of Hyperbolic Measures for $C^1$ Vector Fields with Dominated Splitting

TL;DR

This paper proves that for certain $C^1$ vector fields with dominated splitting, ergodic hyperbolic measures can be approximated by periodic measures, and their Lyapunov exponents can be similarly approximated.

Contribution

It establishes the approximation of ergodic hyperbolic measures and their Lyapunov exponents by periodic measures under dominated splitting conditions.

Findings

Ergodic hyperbolic measures can be approximated by periodic measures.

Lyapunov exponents of these measures can be approximated by those of periodic measures.

Results apply to $C^1$ vector fields with dominated splitting.

Abstract

In this paper, we prove that for every $C^1$ vector field preserving an ergodic hyperbolic invariant measure which is not supported on singularities, if the Oseledec splitting of the ergodic hyperbolic invariant measure is a dominated splitting, then the ergodic hyperbolic invariant measure can be approximated by periodic measures, and the Lyapunov exponents of the ergodic hyperbolic invariant measure can also be approximated by the Lyapunov exponents of those periodic measures.