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Session 35. Topological fixed point theory and related topics
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The period-set of a map from the Cantor-Set to itself |
Andreas Zastrow, Institute of Mathematics, University of Gdańsk, Poland
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The talk is based on the joint work with James W. Cannon & Mark Meilstrup
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This piece of research was motivated by Sharkovskiĭ's Theorem,
which shows that the periods of the periodic points of a self-map
of the unit-interval are severely restricted in the sense,
that only tails of a non-standard linear order of the natural
numbers can be realized as period-sets. In this research project
we asked the analogous question for the Cantor-Set: Which
period-sets can be realized by a (continuous) self-map \(f\) of
the Cantor-Set to itself? \(-\) Although the proof of Sharkovskiĭ's
Theorem heavily relies on using the Intermediate Value Theorem
which does by no means apply to the Cantor-Set, we came
to the conclusion that some kind of restriction does even
hold in case of the Cantor-Set:
The talk will be devoted to state and sketch the proof of a
Theorem confirming that, while an arbitrary subset of the natural
numbers can occur as period-set of \(f\) as long as \(f\) is allowed
to have aperiodic points or preperiodic points, a necessary and
sufficient restriction for a set to become a period set
of some \(f\) will be described for those \(f\) where
each point of the Cantor set belongs to some period.
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