Orbital shadowing, internal chain transitivity and omega-limit sets

Let $f:X\to X$ be a continuous map on a compact metric space, let $\ols{f}$ be the collection of $\omega$-limit sets of $f$ and let $ICT(f)$ be the collection of closed internally chain transitive subsets. Provided that $f$ has shadowing, it is known that the closure of $\ols{f}$ in the Hausdorff metric coincides with $ICT(f)$.

In this paper, we prove that $\ols{f}=ICT(f)$ if and only if $f$ satisfies Pilyugin's notion of orbital limit shadowing. We also characterize those maps for which $\overline{\ols{f}}=ICT(f)$ in terms of a variation of orbital shadowing.