  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.
