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Special alpha-limit sets

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Michal Misiurewicz
IUOUI, Indianapolis
Tue, 03/07/2018 - 10:00 - 10:50
MPIM Lecture Hall

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


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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