Local fields, iterated extensions, and Julia Sets

Abstract

Let $K$ be a field complete with respect to a discrete valuation $v$ of residue characteristic $p$, and let $f(z) = z^\ell - c \in K[z]$ be a separable polynomial. We explore the connection between the valuation $v(c)$ and the Berkovich Julia set of $f$. Additionally, we examine the field extensions generated by the solutions to $f^n(z) = \alpha$ for a root point $\alpha \in K$, highlighting the interplay between the dynamics of $f$ and the ramification in the corresponding field extensions.