Communications of the
Korean Mathematical Society CKMS

Let $f$ be a diffeomorphism of a compact $C^\infty$ manifold, and let $p$ be a hyperbolic periodic point of $f$. In this paper we introduce the notion of $C^1$-stable inverse shadowing for a closed $f$-invariant set, and prove that (i) the chain recurrent set ${\mathcal R}(f)$ of $f$ has $C^1$-stable inverse shadowing property if and only if $f$ satisfies both Axiom A and no-cycle condition, (ii) $C^1$-generically, the chain component $C_f(p)$ of $f$ associated to $p$ is hyperbolic if and only if $C_f(p)$ has the $C^1$-stable inverse shadowing property.

Keywords: homoclinic class, $C^1$-stable inverse shadowing, residual, generic, chain recurrent, chain component, hyperbolic, axiom A

MSC numbers: 37B20, 37C50, 37D30