A survey on abelian dynamical Galois groups

TL;DR

This survey reviews three different approaches to understanding when dynamical Galois groups are abelian, focusing on a conjecture linking this property to specific polynomial types and roots of unity.

Contribution

It compiles and analyzes three distinct methods that prove various cases of the conjecture relating abelian dynamical Galois groups to Chebyshev and power maps.

Findings

Several cases of the conjecture are proven using different approaches.

The approaches connect dynamical Galois groups to roots of unity and specific polynomial forms.

The survey clarifies the current state and progress in this area of number theory.

Abstract

Let $K$ be a number field, $f\in K[x]$ and $\alpha\in K$. A recent conjecture of Andrews and Petsche predicts that the dynamical Galois group of the pair $(f,\alpha)$ is abelian if and only if the pair $(f,\alpha)$ is $K^{\text{ab}}$-conjugated to $(g,\beta)$, where $g$ is a power or a Chebyshev map and $\beta$ is $\zeta$ or $\zeta+\zeta^{-1}$, respectively, and $\zeta$ is a root of unity. We review three completely different approaches that allow to prove several cases of the conjecture.