Misiurewicz polynomials and dynamical units, Part II

TL;DR

This paper explores the arithmetic properties of Misiurewicz parameters in unicritical polynomials, revealing connections to algebraic integers and multipliers, and introduces a new class of polynomials called p-special.

Contribution

It advances understanding of the algebraic and dynamical properties of Misiurewicz parameters, including the introduction of p-special polynomials and their number theoretic significance.

Findings

Connection between algebraic integers and multipliers of periodic points.

Introduction of p-special polynomials with potential number theoretic interest.

Further arithmetic properties of Misiurewicz parameters analyzed.

Abstract

Fix an integer $d\geq 2$. The parameters $c_0\in \bar{\mathbb{Q}}$ for which the unicritical polynomial $f_{d,c}(z)=z^d+c\in \mathbb{C}[z]$ has finite postcritical orbit, also known as Misiurewicz parameters, play a significant role in complex dynamics. Recent work of Buff, Epstein, and Koch proved the first known cases of a long-standing dynamical conjecture of Milnor using their arithmetic properties, about which relatively little is otherwise known. Continuing our work in a companion paper, we address further arithmetic properties of Misiurewicz parameters, especially the nature of the algebraic integers obtained by evaluating the polynomial defining one such parameter at a different Misiurewicz parameter. In the most challenging such combinations, we describe a connection between such algebraic integers and the multipliers of associated periodic points. As part of our considerations, we also introduce a new class of polynomials we call $p$-special, which may be of independent number theoretic interest.