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

Joel Mitchell

paper

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

arXiv:2004.02878

Abstract

Let $f : X \to X$ 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.

15 pages, 3 figures. arXiv admin note: text overlap with arXiv:1909.02881