Abelian extensions in dynamical Galois theory

TL;DR

This paper conjectures a criterion for when the dynamical Galois group of a polynomial is abelian, proving it in several cases including stable quadratic polynomials over rationals, using algebraic and height techniques.

Contribution

It introduces a conjectural characterization of abelian dynamical Galois groups and proves it in specific cases, advancing understanding in dynamical Galois theory.

Findings

Conjectural criterion for abelian dynamical Galois groups.

Proof of the conjecture in stable quadratic cases over ${f Q}$.

Use of algebraic and height methods in different cases.

Abstract

We propose a conjectural characterization of when the dynamical Galois group associated to a polynomial is abelian, and we prove our conjecture in several cases, including the stable quadratic case over ${\mathbb Q}$. In the postcritically infinite case, the proof uses algebraic techniques, including a result concerning ramification in towers of cyclic $p$-extensions. In the postcritically finite case, the proof uses the theory of heights together with results of Amoroso-Zannier and Amoroso-Dvornicich, as well as properties of the Arakelov-Zhang pairing.