A note on Misiurewicz polynomials

TL;DR

This paper investigates the properties of Misiurewicz points in polynomial dynamics, providing new results for prime degrees and periods greater than three, advancing understanding of their algebraic and Galois-theoretic structure.

Contribution

It offers the first provable results for Misiurewicz points with period sizes greater than three when the degree is prime.

Findings

New results for prime degree polynomials with period > 3

Progress on the Galois conjugacy question for Misiurewicz points

Extension of known cases beyond period size 3

Abstract

Let $f_{c,d}(x)=x^d+c\in \mathbb{C}[x]$. The $c_0$ values for which $f_{c_0,d}$ has a strictly pre-periodic finite critical orbit are called Misiurewicz points. Any Misiurewicz point lies in $\bar{\mathbb{Q}}$. Suppose that the Misiurewicz points $c_0,c_1\in \bar{\mathbb{Q}}$ are such that the polynomials $f_{c_0,d}$ and $f_{c_1,d}$ have the same orbit type. One classical question is whether $c_0$ and $c_1$ need to be Galois conjugates or not. Recently there has been a partial progress on this question by several authors. In this note, we prove some new results when $d$ is a prime. All the results known so far were in the cases of period size at most $3$. In particular, our work is the first to say something provable in the cases of period size greater than $3$.