C1-stable shadowing diffeomorphisms

Let f be a diffeomorphism of a closed C-infinity manifold. In this paper, we define the notion of the C-1-stable shadowing property for a closed f-invariant set, and prove that (i) the chain recurrent set R(f) of f has the C-1-stable shadowing property if and only if f satisfies both Axiom A and the no-cycle condition, and (ii) for the chain component C-f (p) of f containing a hyperbolic periodic point p, C-f (p) has the C-1-stable shadowing property if and only if C-f (p) is the hyperbolic homoclinic class of p.