Robustly measure expansiveness for C¹ Vector fields

Abstract

Let X be a C¹ vector eld of a compact connected boundaryless smooth smooth Riemannian manifold M with dimM 3. We show the following: (i) if the
chain recurrence set R(X) is robustly measure expansive, then it is Axiom A with- out 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.

Key words: Expansive, measure expansive, chain recurrent set, homoclinic class, robust, R-robust, hyperbolic.