In this talk we study lower ramification numbers of power series tangent to the identity that are defined over fields of positive characteristic \(p\). Let \(g\) be such a series, then \(g\) has a fixed point at the origin and the corresponding lower ramification numbers of \(g\) are then, up to a constant, the degree of the first non-linear term of \(p\)-power iterates of \(g\). The result is a characterization of power series \(g\) having ramification numbers of the form \(2(1 + p + \dots + p^n)\); such series are henceforth called \(2\)-ramified. The results rely on finding the first significant terms of \(g\) at its \(p^n\)th iterate.

We also discuss the relation between these results and ultrametric dynamics. Recent results by Lindahl and Rivera-Letelier show a connection between the geometric location of periodic points of power series over ultrametric fields and its lower ramification numbers. Using our results it is possible to find the exact bound of the norm of periodic points of 2-ramified power series.