Preperiodic portraits for unicritical polynomials over a rational function field

Abstract

Let K K be an algebraically closed field of characteristic zero, and let K := K ( t ) \mathcal {K} := K(t) be the rational function field over K K . For each d ≥ 2 d \ge 2 , we consider the unicritical polynomial f d ( z ) := z d + t ∈ K [ z ] f_d(z) := z^d + t \in \mathcal {K}[z] , and we ask the following question: If we fix α ∈ K \alpha \in \mathcal {K} and integers M ≥ 0 M \ge 0 , N ≥ 1 N \ge 1 , and d ≥ 2 d \ge 2 , does there exist a place p ∈ S p e c K [ t ] \mathfrak {p} \in \mathrm {Spec} K[t] such that, modulo p \mathfrak {p} , the point α \alpha enters into an N N -cycle after precisely M M steps under iteration by f d f_d ? We answer this question completely, concluding that the answer is generally affirmative and explicitly giving all counterexamples. This extends previous work by the author in the case that α \alpha is a constant point.