Preperiodic portraits for unicritical polynomials

Abstract

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