S-limit shadowing implies the denseness of chain components with the shadowing property

Abstract

For any continuous self-map of a compact metric space, we consider the space of chain components and prove that the s-limit shadowing implies the denseness of chain components with the shadowing property. It gives a partial answer to a question raised by Moothathu (2011, [15]). We also prove that the s-limit shadowing is not generic in the space of homeomorphisms of a closed differentiable manifold.