The Lyapunov Exponents of Hyperbolic Measures for $C^1$ Star Vector Fields on Three-dimensional Manifolds

TL;DR

This paper proves that for certain three-dimensional dynamical systems, hyperbolic measures can be approximated by periodic measures, including their Lyapunov exponents, enhancing understanding of system stability.

Contribution

It establishes the approximation of ergodic hyperbolic measures and their Lyapunov exponents by periodic measures for $C^1$ star vector fields on 3D manifolds.

Findings

Hyperbolic measures can be approximated by periodic measures.

Lyapunov exponents of hyperbolic measures can be approximated.

Results apply to $C^1$ star vector fields on three-dimensional manifolds.

Abstract

In this paper, we proved that for every $C^1$ star vector fields on three-dimensional manifolds, every ergodic hyperbolic invariant measure which is not supported on singularities 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.