On the structure of $\alpha$-limit sets of backward trajectories for graph maps

TL;DR

This paper investigates the structure of alpha-limit sets of backward trajectories in graph maps, revealing their relation to omega-limit sets and how entropy influences their properties.

Contribution

It provides a detailed characterization of alpha-limit sets in graph maps, distinguishing between zero and positive entropy cases and their relation to omega-limit sets.

Findings

In mixing maps, all alpha-limit sets are omega-limit sets.

For almost all points, every omega-limit set can be realized as an alpha-limit set.

In zero entropy maps, alpha-limit sets are minimal sets.

Abstract

In the paper we study what sets can be obtained as $\alpha$-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those $\alpha$-limit sets are $\omega$-limit sets and for all but finitely many points $x$, we can obtain every $\omega$-limits set as the $\alpha$-limit set of a backward trajectory starting in $x$. For zero entropy maps, every $\alpha$-limit set of a backward trajectory is a minimal set. In the case of maps with positive entropy, we obtain a partial characterization which is very close to complete picture of the possible situations.