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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Some results on the Šarkovskiĭ partial ordering of permutations


Author: Irwin Jungreis
Journal: Trans. Amer. Math. Soc. 325 (1991), 319-344
MSC: Primary 58F08; Secondary 58F03, 58F10
MathSciNet review: 998354
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Abstract: If $ \pi $ is a cyclic permutation and $ x$ is a periodic point of a continuous function $ f:{\mathbf{R}} \mapsto {\mathbf{R}}$ with $ {\text{period}}(x) = {\text{order}}(\pi) = n$, then we say that $ x$ has type $ \pi $ if the orbit of $ x$ consists of points $ {x_1} < {x_2} < \cdots < {x_n}$ with $ f({x_i}) = {x_{\pi (i)}}$. In analogy with Sarkovskii's Theorem, we define a partial ordering on cyclic permutations by $ \theta \to \pi $ if every continuous function with a periodic point of type $ \theta $ also has a point of type $ \pi $. In this paper we examine this partial order form the point of view of critical points, itineraries, and kneading sequences. We show that $ \theta \to \pi $ if and only if the maxima of $ \theta $ are "higher" and the minima "lower" than those of $ \pi $, where "higher" and "lower" are precisely defined in terms of itineraries. We use this to obtain several results about $ \to$: there are no minimal upper bounds; if $ \pi $ and $ \theta $ have the same number of critical points (or if they differ by $ 1$ or sometimes $ 2$), then $ \theta \to \pi $ if and only if $ \theta \to {\pi_\ast}$ for some period double $ {\pi_\ast}$ of $ \pi $; and finally, we prove a conjecture of Baldwin that maximal permutations of size $ n$ have $ n - 2$ critical points, and obtain necessary and sufficient conditions for such a permutation to be maximal.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1991-0998354-X
PII: S 0002-9947(1991)0998354-X
Article copyright: © Copyright 1991 American Mathematical Society