Oriented shadowing property and $\Omega$-stability for vector fields

Abstract

We call that a vector field has the oriented shadowing property if for any $\epsilon >0$ there is $d>0$ such that each $d$-pseudo orbit is $\epsilon$-oriented shadowed by some real orbit. In this paper, we show that the $C^1$-interior of the set of vector fields with the oriented shadowing property is contained in the set of vector fields with the $\Omega$-stability.