On negative limit sets for one-dimensional dynamics

In this paper we study the structure of negative limit sets of maps on the unit interval. We prove that every αα-limit set is an ωω-limit set, while the converse is not true in general. Surprisingly, it may happen that the space of all αα-limit sets of interval maps is not closed in the Hausdorff metric (and thus some ωω-limit sets are never obtained as αα-limit sets). Moreover, we prove that the set of all recurrent points is closed if and only if the space of all αα-limit sets is closed.