Shadowable points

Abstract

ABSTRACTA shadowable point is a point where the shadowing lemma holds for pseudo-orbits passing through it. We will prove the following results for homeomorphisms on compact metric spaces. The set of shadowable points is invariant, empty or non-empty, possibly non-compact. A homeomorphism has the pseudo-orbit tracing property (POTP) if and only if every point is shadowable. The chain recurrent and non-wandering sets coincide when every chain recurrent point is shadowable. The space is totally disconnected at every shadowable point for pointwise-recurrent homeomorphisms (and conversely for equicontinuous ones). In particular, pointwise-recurrent (like minimal or distal) homeomorphisms of non-degenerated continua have no shadowable points. The space admits a pointwise-recurrent homeomorphism with the POTP if and only if it is totally disconnected. A distal homeomorphism has the POTP if and only if the space is totally disconnected. We also exhibit homeomorphisms without the POTP for which the set of shadowable points is dense.