Weak measure expansive flows

A notion of measure expansivity for flows was introduced by Carrasco-Olivera and Morales in [3] as a generalization of expansivity, and they proved that there were no measure expansive flows on closed surfaces. In this paper we introduce a concept of weak measure expansivity for flows which is really weaker than that of measure expansivity, and show that there is a weak measure expansive flow on a closed surface. Moreover we show that any C1C1 stably weak measure expansive flow on a C∞C∞ closed manifold M   is Ω-stable, and any C1C1 stably measure expansive flow on M satisfies both Axiom A and the quasi-transversality condition.