This is an unfinished work, started last year with Sergiy Kolyada and

Lubomir Snoha. I hoped all three of us would complete it during this

meeting...

We investigate the notion of the special alpha-limit set of a point.

For a given map of a compact space to itself, it is defined as the

union of the sets of accumulation points over all backward branches of

the map. We consider mainly the case of interval maps. We give many

examples showing how those sets may look like. The main question is

whether a special alpha-limit set has to be closed. We answer it

affirmatively for interval maps for which the set of periods of

periodic points is finite. However, in general it is unknown even

whether a special alpha-limit set has to be Borel (it is in general an

uncountable union of closed sets).

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