Measure-expansive homoclinic classes

Abstract

Let $p$ be a hyperbolic periodic point of a diffeomorphism $f$ on a compact $C^{\infty}$ Riemannian manifold $M$. In this paper we introduce the notion of $C^{1}$ stably measure expansiveness of closed $f$-invariant sets, and prove that (i) the chain recurrent set $\mathcal{R}(f)$ of $f$ is $C^{1}$ stably measure expansive if and only if $f$ satisfies both Axiom A and no-cycle condition, and (ii) the homoclinic class $H_{f}(p)$ of $f$ associated to $p$ is $C^{1}$ stably measure expansive if and only if $H_{f}(p)$ is hyperbolic.