A Dynamical Analogue of Sen’s Theorem

Abstract

Abstract We study the higher ramification structure of dynamical branch extensions and propose a connection between the natural dynamical filtration and the filtration arising from the higher ramification groups: each member of the former should, after a linear change of index, coincide with a member of the latter. This is an analogue of Sen’s theorem on ramification in $p$-adic Lie extensions. By explicitly calculating the Hasse–Herbrand functions of such branch extensions, we are able to show that this description is accurate for some families of polynomials, in particular post-critically bounded polynomials of $p$-power degree. We apply our results to give a partial answer to a question of Berger [8] and a partial answer to a question about wild ramification in arboreal extensions of number fields [1, 9].