On the structure of \begin{document}$ \alpha $\end{document}-limit sets of backward trajectories for graph maps.

In the paper we study what sets can be obtained as α α -limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those α α -limit sets are ω ω -limit sets and for all but finitely many points x x , we can obtain every ω ω -limits set as the α α -limit set of a backward trajectory starting in x x. For zero entropy maps, every α α -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. [ABSTRACT FROM AUTHOR]