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

TL;DR

This paper shows that s-limit shadowing ensures the density of chain components with shadowing in compact metric spaces and explores its non-generic nature among homeomorphisms of manifolds.

Contribution

It establishes a link between s-limit shadowing and the denseness of shadowing chain components, partially answering a question by Moothathu.

Findings

s-limit shadowing implies dense chain components with shadowing

not generic among homeomorphisms of closed manifolds

provides partial answer to Moothathu's question

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 [Topology Appl. 158 (2011) 2232--2239]. We also prove that the s-limit shadowing is not generic in the space of homeomorphisms of a closed differentiable manifold.