Shadowable Chain Recurrence Classes for Generic Diffeomorphisms

Abstract

Let $\operatorname{Diff}(M)$ be the space of diffeomorphisms of a closed $C^{\infty}$ manifold $M$ $(\dim M \geq 2)$ endowed with the $C^1$ topology. In this paper we show that for $C^1$ generic $f \in \operatorname{Diff}(M)$, any shadowable chain recurrence class $C_f$ is hyperbolic if it contains a hyperbolic periodic point.