标题： 项链、排列和二次多项式的周期临界轨道 显示英文标题

标题： Necklaces, permutations, and periodic critical orbits for quadratic polynomials

摘要： 设$G_n$表示一个$n^{\rm th}$的Gleason多项式，其根对应于参数$c$，使得在$z^2 + c$迭代下临界点$0$是精确周期为$n$的周期点，令$\bar{G}_n$表示$G_n$对$2$的约简。 Buff、Floyd、Koch 和 Parry 惊讶地观察到，对于所有$n$，$G_n$的实根数目等于$\bar{G}_n$的不可约因子数目。 我们通过首先提供显式的双射来证明这个结果，(a) 在$G_n$的实根集合与在反转映射交换$0$和$1$下的长度为$n$的原始二元项链等价类集合$\bar{N}(n)$之间的显式双射；以及 (b) 在模 2 下的$G_n$的不可约因子集合与集合$\tilde{N}^+(n)$之间的显式双射，该集合中的二元项链要么是长度为$n$且有偶数个$1$的原始项链，要么是长度为$n/2$且有奇数个$1$的原始项链。 我们然后提供一个显式的双射，与米尔诺和塔尔斯基的折叠理论密切相关，介于$\bar{N}(n)$和$\tilde{N}^+(n)$之间。 此外，我们提供了$\bar{N}(n)$与集合${\rm CUP}(n)$的显式双射，该集合为$\{ 1,\ldots,n \}$的循环单峰排列，以及与长度为$n$且有奇数个$1$的原始二进制项链的集合$N^-(n)$。 显示英文摘要

摘要： 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.