Special α-limit sets
Sergiy Kolyada, Michał Misiurewicz and Ľubomír Snoha
Abstract
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.