Shadowing, internal chain transitivity and α-limit sets

Let f:X→X be a continuous map on a compact metric space X and let α _f, ω _f and ICT _f denote the set of α-limit sets, ω-limit sets and nonempty closed internally chain transitive sets respectively. We show that if the map f has shadowing then every element of ICT _f can be approximated (to any prescribed accuracy) by both the α-limit set and the ω-limit set of a full-trajectory. Furthermore, if f is additionally expansive then every element of ICT _f is equal to both the α-limit set and the ω-limit set of a full-trajectory. In particular this means that shadowing guarantees that α _f‾=ω _f‾=ICT _f (where the closures are taken with respect to the Hausdorff topology on the space of compact sets), whilst the addition of expansivity entails α _f=ω _f=ICT _f. We progress by introducing novel variants of shadowing which we use to characterise both maps for which α _f‾=ICT _f and maps for which α _f=ICT _f.