LAUSR

Abstract Let f : X → X be a continuous map on a compact metric space X and let α f , ω f and I C T 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 I C T 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 I C T 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 ‾ = I C T 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 = I C T f . We progress by introducing novel variants of shadowing which we use to characterise both maps for which α f ‾ = I C T f and maps for which α f = I C T f .