Shadowing, internal chain transitivity and $\alpha$-limit sets

TL;DR

This paper investigates the relationship between shadowing, chain transitivity, and limit sets in dynamical systems, showing how shadowing and c-expansivity influence the approximation and equality of various limit sets.

Contribution

It introduces new variants of shadowing to characterize maps where limit sets coincide or are approximated by full-trajectory limit sets.

Findings

Shadowing ensures approximation of chain transitive sets by limit sets.

C-expansivity guarantees equality of limit sets and chain transitive sets.

New shadowing variants characterize when limit sets and chain transitive sets coincide.

Abstract

Let $f \colon X \to X$ be a continuous map on a compact metric space $X$ and let $\alpha_f$, $\omega_f$ and $ICT_f$ denote the set of $\alpha$-limit sets, $\omega$-limit sets and nonempty closed internally chain transitive sets respectively. We show that if the map $f$ has shadowing then every element of $ICT_f$ can be approximated (to any prescribed accuracy) by both the $\alpha$-limit set and the $\omega$-limit set of a full-trajectory. Furthermore, if $f$ is additionally c-expansive then every element of $ICT_f$ is equal to both the $\alpha$-limit set and the $\omega$-limit set of a full-trajectory. In particular this means that shadowing guarantees that $\overline{\alpha_f}=\overline{\omega_f}=ICT(f)$ (where the closures are taken with respect to the Hausdorff topology on the space of compact sets), whilst the addition of c-expansivity entails $\alpha_f=\omega_f=ICT(f)$. We progress by introducing novel variants of shadowing which we use to characterise both maps for which $\overline{\alpha_f}=ICT(f)$ and maps for which $\alpha_f=ICT(f)$.