A Dynamical Analogue of Sen's Theorem

TL;DR

This paper explores the higher ramification structure of dynamical branch extensions, proposing a connection between dynamical and classical ramification filtrations, with explicit calculations confirming the analogy for certain polynomial families.

Contribution

It introduces a dynamical analogue of Sen's theorem, linking dynamical and ramification filtrations, supported by explicit calculations for specific polynomial families.

Findings

The dynamical filtration coincides with ramification groups after a linear change of index for some polynomial families.

Explicit Hasse-Herbrand functions are computed for dynamical branch extensions.

Partial answers are provided to questions on wild ramification in arboreal extensions and a conjecture by Berger.

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 (in arXiv:1411.7064) and a partial answer to a question about wild ramification in arboreal extensions of number fields (raised in both arXiv:math/0408170 and arXiv:1511.00194).