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

Let $f\text{\hspace{0.17em}⁣}:X\to X$ be a continuous map on a compact metric space $X$ and let ${\alpha }_f$, ${\omega }_f$ and $IC{T}_f$ denote the set of $\alpha$-limit sets, $\omega$-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 $IC{T}_f$ is equal to (resp. may be approximated by) the $\alpha$-limit set and the $\omega$-limit set of the same full trajectory. We construct examples highlighting the difference between these properties.