Combinatorics of Periodic Points of Interval Maps.
London, Tyler.
2009
- In 1964, Sharkovsky showed that there is a linear ordering < of the natural numbers such that m < n if every continuous interval map having an n-periodic point has an m-periodic point. This idea is generalized to get a partial ordering of cyclic permutations called the forcing relation by taking into consideration the pattern of the periodic orbit. We show that forcing can be reduced to the study ... read moreof certain canonical piecewise linear maps. Combinatorial algorithms of Baldwin and Coven et. al. that decide forcing are presented, as well as a diagram of the forcing relation on over 190 cycles of length up to 8. The global structure of forcing is described by considering flips of cycles and unimodal cycles. It is shown that the forcing relation restricted to unimodal cycles is a total ordering, and this relation is diagrammed on 115 unimodal cycles. The local structure of forcing is also explored by studying doublings and extensions. A new combinatorial proof of a theorem describing the structure of doublings in the forcing order is given. Forcing-minimal properties are examined by considering primary and simple cycles. Finally, using the results proven about the forcing relation, we derive Sharkovsky's theorem as a consequence of the forcing theorem.read less
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