Characterization of 2-ramified power series

Abstract

Abstract. In this paper we study lower ramification numbers of power series tangent to the identity that are defined over fields of positive characteristics 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 complete characterization of power series g having ramification numbers of the form 2(1 + p + · · · + p n ). Furthermore, in proving said characterization we explicitly compute the first significant terms of g at its pth iterate.

Geometric location of periodic points of 2-ramified power series

“…The main technical result of this paper is the computation of the first significant terms of f at its pth iterate which is done in the Main Lemma in §4. This result is an extension of Proposition 1 in [Nor17]. The setup for this proof is discussed in §3, and we state and prove the Main Lemma in §4 together with the proof of Theorem B.…”

“…Proof. All parts of the proof except for the evaluation of R n and T n at n = p − 1 and n = p − 2 are given in [Nor17]. Thus to complete the proof we compute R p−1 , R p−2 , T p−1 and T p−2 .…”

“…Proof. As for the proof of Lemma 4 the proof is given in [Nor17] except for the calculations of S p−1 and S p−2 . Note that for j in The main idea behind finding the reductions of the sums of rational functions we set out to find, is to compute and sum over its residues.…”

“…The main work of the proof is the Main Lemma given in §4 where we compute the first significant terms of f at its pth iterate by solving systems of difference equations in characteristic p. This result is an extension of Proposition 1 in [Nor17]. We also note that due to Theorem B we have a self-contained proof of Theorem 1 in [Nor17].…”

Pro‐p groups of positive rank gradient and Hausdorff dimension

“…Thus, and, for , it is known that satisfies and ; compare [, Lemma 1, Theorem 6]. (In fact, for all , in other words: is 2‐ramified; compare .) For , we observe that , hence and we conclude that , where denotes the standard map.…”