Special α-limit sets

We investigate the notion of the special α-limit set of a point. For a continuous selfmap of a compact metric space, it is defined as the union of the sets of accumulation points over all backward branches of the map. The main question is whether a special α-limit set has to be closed. We show that it is not the case in general. It is unknown even whether a special α-limit set has to be Borel or at least analytic (it is in general an uncountable union of closed sets). However, we answer this question affirmatively for interval maps for which the set of all periodic points is closed. We also give examples showing how those sets may look like and we provide some conjectures and a problem