Portraits of preperiodic points for rational maps

TL;DR

This paper proves that for a broad class of rational functions over function fields, preperiodic points with specified properties exist at all but finitely many pairs, confirming conjectures and enabling construction of polynomials with prescribed dynamics.

Contribution

It establishes the existence of preperiodic points with given preperiod and period for almost all pairs, confirming conjectures by Ingram-Silverman and Faber-Granville, and extends results to functions with totally ramified points.

Findings

Almost all pairs (m,n) have a place where α has preperiod m and period n.

Confirmed conjectures on the distribution of preperiodic points.

Constructed polynomials with prescribed preperiodic behavior for multiple points.

Abstract

Let $K$ be a function field over an algebraically closed field $k$ of characteristic $0$, let $\varphi\in K(z)$ be a rational function of degree at least equal to $2$ for which there is no point at which $\varphi$ is totally ramified, and let $\alpha\in K$. We show that for all but finitely many pairs $(m,n)\in \mathbb{Z}_{\ge 0}\times \mathbb{N}$ there exists a place $\mathfrak{p}$ of $K$ such that the point $\alpha$ has preperiod $m$ and minimum period $n$ under the action of $\varphi$. This answers a conjecture made by Ingram-Silverman and Faber-Granville. We prove a similar result, under suitable modification, also when $\varphi$ has points where it is totally ramified. We give several applications of our result, such as showing that for any tuple $(c_1,\dots , c_{d-1})\in k^{n-1}$ and for almost all pairs $(m_i,n_i)\in \mathbb{Z}_{\ge 0}\times \mathbb{N}$ for $i=1,\dots, d-1$, there exists a polynomial $f\in k[z]$ of degree $d$ in normal form such that for each $i=1,\dots, d-1$, the point $c_i$ has preperiod $m_i$ and minimum period $n_i$ under the action of $f$.