Geometric location of periodic points of 2-ramified power series

Abstract

In this paper we study the geometric location of periodic points of power series defined over fields of prime characteristic $p$. More specifically, we find a lower bound for the absolute value of all periodic points in the open unit disk of minimal period $p^n$ of 2-ramified power series. We prove that this bound is optimal for a large class of power series. Our main technical result is a computation of the first significant terms of the $p^n$th iterate of 2-ramified power series. As a by-product we obtain a self-contained proof of the characterization of 2-ramified power series.