Shadowable points

TL;DR

This paper introduces shadowable points for homeomorphisms on metric spaces, establishing their properties and connections to pseudo-orbit tracing, chain recurrence, and space structure, especially in compact connected spaces.

Contribution

It defines shadowable points and explores their properties, linking them to key dynamical concepts and improving existing theorems on pseudo-orbit tracing.

Findings

Shadowable points form an invariant set, possibly nonempty or noncompact.

Homeomorphisms with the pseudo-orbit tracing property have all points shadowable.

Minimal or distal homeomorphisms of compact connected spaces have no shadowable points.

Abstract

We define shadowable points for homeomorphism on metric spaces. In the compact case we will prove the following results: The set of shadowable points is invariant, possibly nonempty or noncompact. A homeomorphism has the pseudo-orbit tracing property if and only if every point is shadowable. The chain recurrent and nonwandering sets coincides when every chain recurrent point is shadowable. Minimal or distal homeomorphisms of compact connected metric spaces have no shadowable points. The space is totally disconnected at every shadowable point for distal homeomorphisms (and conversely for equicontinuous homeomorphisms). A distal homeomorphism has the pseudo-orbit tracing property if and only if the space is totally disconnected (this improves Theorem 4 in \cite{mo}).