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

Abstract

In the paper we study what sets can be obtained as \begin{document}$ \alpha $\end{document}-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those \begin{document}$ \alpha $\end{document}-limit sets are \begin{document}$ \omega $\end{document}-limit sets and for all but finitely many points \begin{document}$ x $\end{document}, we can obtain every \begin{document}$ \omega $\end{document}-limits set as the \begin{document}$ \alpha $\end{document}-limit set of a backward trajectory starting in \begin{document}$ x $\end{document}. For zero entropy maps, every \begin{document}$ \alpha $\end{document}-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.