Preperiodic portraits for unicritical polynomials

Abstract

Let $K$ be an algebraically closed field of characteristic zero, and for $c\in K$ and an integer $d\ge 2$, define $f_{d,c}(z):=z^d+c\in K[z]$. We consider the following question: If we fix $x\in K$ and integers $M\ge 0$, $N\ge 1$, and $d\ge 2$, does there exist $c\in K$ such that, under iteration by $f_{d,c}$, the point $x$ enters into an $N$-cycle after precisely $M$ steps? We conclude that the answer is generally affirmative, and we explicitly give all counterexamples. When $d=2$, this answers a question posed by Ghioca, Nguyen, and Tucker.

arXiv commentUpdated references; fixed minor error in Lemma 3.6