Journal of the
Korean Mathematical Society JKMS

A notion of measure expansivity for homeomorphisms was introduced by Morales recently as a generalization of expansivity, and he obtained many interesting dynamic results of measure expansive homeomorphisms in \cite{M}. In this paper, we introduce a concept of weak measure expansivity for homeomorphisms which is really weaker than that of measure expansivity, and show that a diffeomorphism $f$ on a compact smooth manifold is $C^1$-stably weak measure expansive if and only if it is $\Omega$-stable. Moreover we show that $C^1$-generically, if $f$ is weak measure expansive, then $f$ satisfies both Axiom $A$ and the no cycle condition.

Keywords: expansive, measure expansive, weak measure expansive, $\Omega$-stable, Axiom $A$, generic

MSC numbers: 37A05, 37C20, 37C05, 37D20