A survey on abelian dynamical Galois groups

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.