Necklaces, permutations, and periodic critical orbits for quadratic polynomials

Abstract

Let $G_n$ denote the $n^{\rm th}$ Gleason polynomial, whose roots correspond to parameters $c$ such that the critical point $0$ is periodic of exact period $n$ under iteration of $z^2 + c$, and let $\bar{G}_n$ denote the reduction of $G_n$ modulo $2$. Buff, Floyd, Koch, and Parry made the surprising observation that the number of real roots of $G_n$ is equal to the number of irreducible factors of $\bar{G}_n$ for all $n$. We provide a bijective proof for this result by first providing explicit bijections between (a) the set of real roots of $G_n$ and the set $\bar{N}(n)$ of equivalence classes of primitive binary necklaces of length $n$ under the inversion map swapping $0$ and $1$; and (b) the set of irreducible factors of $G_n$ modulo 2 and the set $\tilde{N}^+(n)$ of binary necklaces which are either primitive of length $n$ with an even number of $1$'s or primitive of length $n/2$ with an odd number of $1$'s. We then provide an explicit bijection, closely related to Milnor and Thurston's kneading theory, between $\bar{N}(n)$ and $\tilde{N}^+(n)$. In addition, we provide explicit bijections between $\bar{N}(n)$, the set ${\rm CUP}(n)$ of cyclic unimodal permutations of $\{ 1,\ldots,n \}$, and the set $N^-(n)$ of primitive binary necklaces of length $n$ with an odd number of $1$'s.