Asymptotic Measure Expansive Flows

Abstract

In this paper, we extend the notion of asymptotic measure expansivity for diffeomorphisms to flows on a compact metric space, and prove that there exists an asymptotic measure expansive flow in that space. Then we consider the hyperbolicity of the asymptotic measure expansive vector fields on a closed smooth Riemannian manifold M. More precisely, we prove that any C-1 stably asymptotic measure expansive vector field on M satisfy Axiom A without cycles, and it is quasi-Anosov. On the other hand, we show that C-1 generically, every asymptotic measure expansive vector field on M satisfies Axiom A without cycles. Moreover, we study the asymptotic measure expansivity of the homoclinic classes which are the delegate of the invariant subsets for given systems, prove that C-1 stably and C-1 generically, the asymptotic measure expansive homoclinic class is hyperbolic. Furthermore, we consider the hyperbolicity of asymptotic measure expansive divergence-free vector fields for C-1 stably and C-1 generic point of view. We also apply the our results to the divergence-free vector fields.