Preperiodic portraits for unicritical polynomials

TL;DR

This paper investigates the existence of parameters in unicritical polynomials over algebraically closed fields that produce points with specified preperiodic portraits, providing a comprehensive answer and explicit counterexamples, especially for quadratic cases.

Contribution

It characterizes when points can have prescribed preperiodic portraits under unicritical polynomial iteration, answering a previously posed question for quadratic polynomials.

Findings

Most preperiodic portraits are realizable with appropriate parameters.

Explicit counterexamples are provided where such portraits cannot be realized.

The results apply to all degrees d ≥ 2, with special emphasis on the quadratic case.

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.