Periodic shadowing of vector fields

Abstract

A vector field has the periodic shadowing property if for any ε > 0 there is d > 0 such that, for any periodic d-pseudo orbit g there exists a periodic orbit or a singularity in which g is ε-shadowed. In this paper, we show that a vector field is in the C 1 interior of the set of vector fields satisfying the periodic shadowing property if and only if it is Ω-stable. More precisely, we prove that the C 1 interior of the set of vector fields satisfying the orbital periodic shadowing property is a subset of the set of Ω-stable vector fields.