On the size and structure of dynamical Galois groups.

Following an analogy with Serre's open image theorem, dynamical Galois groups are expected to be large and complicated, unless the underlying rational function is "special" in that it features some exceptional amount of extra-structure (the amount depending on the property at hand). I will review conjectures and expectations along these lines and present past and ongoing joint work with Andrea Ferraguti, where the property at hand for the dynamical Galois group is "being abelian". Here "special" has been explicitly conjectured by Andrews--Petsche. In particular I will present partial results we obtained in the unicritical case over number fields, along with an ongoing work where we obtain a full proof of the conjecture in the case of global function fields of positive characteristic for maps of degree smaller than the characteristic, having a super-attractive fixed point (in particular any polynomial). The ingredients for the former past results are a blend of group theory, height estimates (in the spirit of Amoroso--Zannier) and explicit rational/integral points on curves (using the Chabauty method). The ingredients for the latter ongoing work are a height estimate of Ferraguti--Ostafe--Zannier (to show the map is PCF), a version of Thurston rigidity theorem in positive characteristic (to show the map is constant) with a finishing move provided by exploiting Böttcher coordinates locally at every place (forcing the base-point to be constant as well).