Portraits of preperiodic points for rational maps

Abstract

Let $K$ be a function field over an algebraically closed field $k$ of characteristic $0$, let $\phi \in K(z)$ be a rational function of degree at least equal to $2$ for which there is no point at which $\phi$ 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 $\phi$. This answers a conjecture made by Ingram-Silverman and Faber-Granville. We prove a similar result, under suitable modification, also when $\phi$ 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 $f$.