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 positive characteristic p. We find a lower bound for the norm of all nonzero periodic points in the open unit disk 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 p-power iterates of 2-ramified power series. As a by-product we obtain a self-contained proof of the characterization of 2-ramified power series.

Residue fixed point index and wildly ramified power series

“…This last result also applies to . In the case , and for power series with integer coefficients, Theorem 3 was shown by Lindahl and the first author [14, Theorem A].…”

“…Such power series are called wildly ramified . See, for example, [9, 12, 23, 25] for background on wildly ramified power series, [8, 11, 14–16, 19, 21] for results related to this paper, and [6, 13, 17, 22] and references therein for local dynamics of analytic germs in positive characteristic. See also, for example, [4, 7] and references therein, for the myriad of group‐theoretic results about the ‘Nottingham group’, which is the group under composition formed by the wildly ramified power series.…”

“…For a wildly ramified power series , the lower ramification numbers of are defined by See, for example, [9, 12, 23, 25] and references therein for background on wildly ramified power series and their lower ramification numbers. Due to their relation to ultrametric dynamics, they have been studied in, for example, [14–16; 21, § 3.2]. Note that the lower ramification numbers are invariant under coordinate changes.…”

Ramification of Wild Automorphisms of Laurent Series Fields

“…In Section 3, we give an explicit computation of the first nontrivial coefficient of the p-th iterate of a power series f ∈ N (k) with i(f ) = b, which yields the desired necessary and sufficient condition for f being b-ramified. In Section 4, we describe the implications this computation has for a generalization of Lindahl-Nordqvist's work in [13] on the locations of periodic points in the open unit disc of a nonarchimedean field of characteristic p under iteration of power series.…”

“…Lindahl and Nordqvist, 2018). Suppose p ≥ 5 and letf (z) = z + ∞ i=1 a i z i+2 ∈ O k [[z]].If z 0 ∈ m k is a periodic point of period p n under the action of f , then|z 0 | ≥ a p−3…”