Preperiodic portraits for unicritical polynomials over a rational function field

Abstract

Let $K$ be an algebraically closed field of characteristic zero, and let $\mathcal{K}:=K(t)$ be the rational function field over $K$. For each $d\ge 2$, we consider the unicritical polynomial $f_d(z):=z^d+t\in \mathcal{K}[z]$, and we ask the following question: If we fix $\alpha \in \mathcal{K}$ and integers $M\ge 0$, $N\ge 1$, and $d\ge 2$, does there exist a place $\mathfrak{p}\in \mathrm{Spec}K[t]$ such that, modulo $\mathfrak{p}$, the point $\alpha$ enters into an $N$-cycle after precisely $M$ steps under iteration by $f_d$? 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 $\alpha$ is a constant point.

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