Portraits of preperiodic points for rational maps

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 \(f\).