Some results on the Šarkovskiĭ\ partial ordering of permutations

Irwin Jungreis
1991 Transactions of the American Mathematical Society  
If n is a cyclic permutation and x is a periodic point of a continuous function /:R •-► R with period(x) = order(Tt) = n , then we say that x has type n if the orbit of x consists of points xx < x2 < •■■ < xn with f(x¡) = x"/« . In analogy with Sarkovskii's Theorem, we define a partial ordering on cyclic permutations by 8 -» n if every continuous function with a periodic point of type 0 also has a point of type n . In this paper we examine this partial order form the point of view of critical
more » ... ints, itineraries, and kneading sequences. We show that 0 -► 7t if and only if the maxima of 0 are "higher" and the minima "lower" than those of n , 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 it and 0 have the same number of critical points (or if they differ by 1 or sometimes 2), then 6 -► n if and only if 6 -► nm for some period double nt of n ; 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.
doi:10.1090/s0002-9947-1991-0998354-x fatcat:lnlaz6hjkvbsjlm53l3avgoec4