Local fields, iterated extensions, and Julia Sets

Abstract

Let $K$ be a field complete with respect to a discrete valuation $v$ of residue characteristic $p$. For $\hat{I}\pm \in K$, let $K_{\infty }$ be the extension obtained by adjoining all iterated preimages of $\hat{I}\pm$ under a unicritical polynomial $f_c(z)=z^{\ell }-c\in K[z]$. We study the extension $K_{\infty }/K$ and show that its qualitative behavior depends only on the valuation of $c$. This removes the previous restrictions on $\ell$ in work of Anderson--Hamblen--Poonen--Walton and completes the classification for all $\ell \ge 2$. We also relate the ramification to the structure of the Berkovich Julia set of $f_c$.