$C^1$-stably shadowable chain components are hyperbolic

Abstract

Let f be a diffeomorphism on a closed manifold, and p be a hyperbolic periodic point of f. Denote $C_f(p)$ the chain component of f that contains p. We say $C_f(p)$ is $C^1$-stably shadowable if there is a $C^1$-neighborhood $\mathcal{U}$ of f such that for every $g\in \mathcal{U}$, $C_g(p_g)$ has the shadowing property, where $p_g$ is the continuation of p. We prove in this paper that if $C_f(p)$ is $C^1$-stably shadowable, then $C_f(p)$ is hyperbolic.