Orbital shadowing, internal chain transitivity and -limit sets

Abstract

Let $f:X\rightarrow X$ be a continuous map on a compact metric space, let $\unicode[STIX]{x1D714}_{f}$ be the collection of $\unicode[STIX]{x1D714}$-limit sets of $f$ and let $\mathit{ICT}(f)$ be the collection of closed internally chain transitive subsets. Provided that $f$ has shadowing, it is known that the closure of $\unicode[STIX]{x1D714}_{f}$ in the Hausdorff metric coincides with $\mathit{ICT}(f)$. In this paper, we prove that $\unicode[STIX]{x1D714}_{f}=\mathit{ICT}(f)$ if and only if $f$ satisfies Pilyugin’s notion of orbital limit shadowing. We also characterize those maps for which $\overline{\unicode[STIX]{x1D714}_{f}}=\mathit{ICT}(f)$ in terms of a variation of orbital shadowing.