Journal of the
Korean Mathematical Society JKMS

It is shown that every continuum-wise expansive $C^1$ generic vector field $X$ on a compact connected smooth manifold $M$ satisfies Axiom A and has no cycles, and every continuum-wise expansive homoclinic class of a $C^1$ generic vector field $X$ on a compact connected smooth manifold $M$ is hyperbolic. Moreover, every continuum-wise expansive $C^1$ generic divergence-free vector field $X$ on a compact connected smooth manifold $M$ is Anosov.

Keywords: Expansive, continuum-wise expansive, Axiom A, homoclinic class, hyperbolic, generic

MSC numbers: 37C10, 37C20, 34D10, 37C27