Robustly measure expansiveness for C¹ vector fields

Abstract

Let X be a C¹ vector field of a compact connected boundaryless smooth smooth Riemannian manifold M with dimM ≥ 3. We show the following: (i) if the chain recurrence set ????(X) is robustly measure expansive, then it is Axiom A without cycles; and (ii) if the homoclinic class H(Orb(p), X) that contains a hyperbolic periodic orbit Orb(p) is generic-robustly measure expansive, then it is hyperbolic.