When is the beginning the end? On full trajectories, limit sets and internal chain transitivity.

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. In this paper we characterise, by introducing novel variants of shadowing, maps for which every element of I C T f is equal to (resp. may be approximated by) the α -limit set and the ω -limit set of the same full trajectory. We construct examples highlighting the difference between these properties. [ABSTRACT FROM AUTHOR]