Counting the number of $\mathcal{O}_{K}$-fixed points of a discrete dynamical system with applications from arithmetic statistics, II

Abstract

In this follow-up paper, we again inspect a surprising connection between the set of fixed points of a polynomial map $\varphi_{d,c}$ defined by $\varphi_{d,c}(z) = z^d + c$ for all $c, z \in \mathcal{O}_{K}$ and the coefficient $c$, where $K$ is any number field of degree $n > 1$ and $d > 2$ is an integer. As before, we wish to study counting problems which are inspired by exciting advances in arithmetic statistics, and again partly by point-counting result of Narkiewicz on real $K$-rational periodic points of any odd degree map $\varphi_{d,c}$ in arithmetic dynamics. In doing so, we then first prove that for any real algebraic number field $K$ of degree $n \geq 2$, and for any prime $p \geq 3$ and integer $\ell \geq 1$, the average number of distinct integral fixed points of any $\varphi_{p^{\ell},c}$ modulo prime ideal $p\mathcal{O}_{K}$ is $3$ or $0$ as $c\to \infty$. Motivated further by $K$-rational periodic point-counting result of Benedetto on any $\varphi_{(p-1)^{\ell},c}$ for any prime $p \geq 5$ and integer $\ell \in \mathbb{Z}_{\geq 1}$ in arithmetic dynamics, we then also prove unconditionally that for any number field (not necessarily real) $K$ of degree $n \geq 2$, the average number of distinct integral fixed points of any $\varphi_{(p-1)^{\ell},c}$ modulo prime $p\mathcal{O}_{K}$ is $1$ or $2$ or $0$ as $c\to \infty$. Finally, we then apply density and number field-counting results from arithmetic statistics, and as a result obtain counting and statistical results on irreducible polynomials and number fields arising naturally in our polynomial discrete dynamical settings.