On the Density of Periodic Measures for Star Vector Fields

TL;DR

This paper proves that for star vector fields, ergodic hyperbolic measures can be approximated by periodic measures, extending classical results from diffeomorphisms to vector fields in the $C^1$ setting.

Contribution

It extends Katok's approximation result from $C^{1+ ext{alpha}}$ diffeomorphisms to $C^1$ star vector fields of any dimension.

Findings

Ergodic hyperbolic measures are approximable by periodic measures in $C^1$ star vector fields.

The result generalizes classical approximation theorems to a broader class of dynamical systems.

The proof applies to vector fields of any dimension.

Abstract

In this paper, we prove that every ergodic hyperbolic invariant measure of a $C^1$ star vector field can be approximated by periodic measures in weak$^*$ topology. This extends a classical result of Katok \cite{Ka} for $C^{1+\alpha}(\alpha>0)$ diffeomorphisms to $C^1$ star vector field of any dimension.