Some results on the Šarkovskiĭ partial ordering of permutations

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.