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 is a cyclic permutation and is a periodic point of a continuous function with , then we say that has type if the orbit of consists of points with . In analogy with Sarkovskii's Theorem, we define a partial ordering on cyclic permutations by if every continuous function with a periodic point of type also has a point of type . In this paper we examine this partial order form the point of view of critical points, itineraries, and kneading sequences. We show that if and only if the maxima of are "higher" and the minima "lower" than those of , where "higher" and "lower" are precisely defined in terms of itineraries. We use this to obtain several results about : there are no minimal upper bounds; if and have the same number of critical points (or if they differ by or sometimes ), then if and only if for some period double of ; and finally, we prove a conjecture of Baldwin that maximal permutations of size have critical points, and obtain necessary and sufficient conditions for such a permutation to be maximal.

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DOI:
https://doi.org/10.1090/S0002-9947-1991-0998354-X

Article copyright:
© Copyright 1991
American Mathematical Society