Shadowing, internal chain transitivity and α-limit sets

[摘要] Let f : X -> X be a continuous map on a compact metric space X and let alpha(f), omega(f) and ICTf denote the set of alpha-limit sets, omega-limit sets and nonempty closed internally chain transitive sets respectively. We show that if the map f has shadowing then every element of ICTf can be approximated (to any prescribed accuracy) by both the alpha-limit set and the omega-limit set of a full-trajectory. Furthermore, if f is additionally expansive then every element of ICTf is equal to both the alpha-limit set and the omega-limit set of a full-trajectory. In particular this means that shadowing guarantees that (alpha(f)) over bar = (omega(f)) over bar = ICTf (where the closures are taken with respect to the Hausdorff topology on the space of compact sets), whilst the addition of expansivity entails alpha(f) = omega(f) = ICTf. We progress by introducing novel variants of shadowing which we use to characterise both maps for which (alpha(f)) over bar = ICTf and maps for which alpha(f) = ICTf. (C) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-11-01

[关键词] Shadowing;alpha-limit set;omega-limit set;Internally chain transitive;Expansive;Pseudo-orbit