Optimal cycles in ultrametric dynamics and minimally ramified power series

Abstract

Abstract. We study ultrametric germs in one variable having an irrationally indifferent fixed point at the origin with a prescribed multiplier. We show that for many values of the multiplier, the cycles in the unit disk of the corresponding monic quadratic polynomial are "optimal" in the following sense: They minimize the distance to the origin among cycles of the same minimal period of normalized germs having an irrationally indifferent fixed point at the origin with the same multiplier. We also give examples of multipliers for which the corresponding quadratic polynomial does not have optimal cycles. In those cases we exhibit a higher degree polynomial such that all of its cycles are optimal. The proof of these results reveals a connection between the geometric location of periodic points of ultrametric power series and the lower ramification numbers of wildly ramified field automorphisms. We also give an extension of Sen's theorem on wildly ramified field automorphisms, and a characterization of minimally ramified power series in terms of the iterative residue.

Characterization of 2-ramified power series

“…[RL03], [LRL16b], and find the difference equations that defines the first three significant terms in ∆ m . In the second part we solve these difference equations for an arbitrarily chosen m, and in the last part we determine the coefficient of the first significant term in ∆ p and hence of g p (ζ)−ζ.…”

“…Finding the difference equations. Analogous to [LRL16b] for m = 1 we define the recurrence relation ∆ 1 (ζ) := g(ζ) − ζ and for m ≥ 2…”

“…In this article we are interested in power series defined over fields of positive characteristic p. In particular, the degree of the first non-linear term of p-power iterates of power series tangent to the identity, the so called lower ramification numbers of such series. Recent results by Lindahl and Rivera-Letelier [Lin13,LRL16a,LRL16b] show that there is a connection between the geometric location of periodic points of power series over ultrametric fields and lower ramification numbers.…”

“…Our approach of calculating the pth iterate of g given in Proposition 1 is similar to methods used in [LRL16a,LRL16b]. Proofs of Theorem 1 and Proposition 1 will be given in §3.…”

“…Implication. In [LRL16a,LRL16b] the authors study the relation between the lower ramification numbers and the geometric location of periodic points of ultrametric dynamical systems. In their results several important discoveries concerning lower ramification numbers have been made.…”