Misiurewicz polynomials and dynamical units, Part I

TL;DR

This paper explores the arithmetic properties of Misiurewicz parameters in unicritical polynomial dynamics, focusing on relations between postcritical points and conditions under which certain algebraic integers are units.

Contribution

It introduces new arithmetic relations for Misiurewicz parameters when the degree is a prime power and investigates when evaluated Misiurewicz polynomials yield algebraic units, providing partial proofs of conjectures.

Findings

Arithmetic relations between postcritical points for prime power degrees

Conditions under which evaluated Misiurewicz polynomials are algebraic units

Partial proofs supporting conjectures on algebraic units in this context

Abstract

We study the dynamics of the unicritical polynomial family $f_{d,c}(z)=z^d+c\in \mathbb{C}[z]$. The $c$-values for which $f_{d,c}$ has a strictly preperiodic postcritical orbit are called Misiurewicz parameters, and they are the roots of Misiurewicz polynomials. The arithmetic properties of these special parameters have found applications in both arithmetic and complex dynamics. In this paper, we investigate some new such properties. In particular, when $d$ is a prime power and $c$ is a Misiurewicz parameter, we prove certain arithmetic relations between the points in the postcritical orbit of $f_{d,c}$. We also consider the algebraic integers obtained by evaluating a Misiurewicz polynomial at a different Misiurewicz parameter, and we ask when these algebraic integers are algebraic units. This question naturally arises from some results recently proven by Buff, Epstein, and Koch and by the second author. We propose a conjectural answer to this question, which we prove in many cases.