On the Density of Periodic Measures for Star Vector Fields

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${}^{\ast }$ 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.