Generic parabolic points are isolated in positive characteristic

TL;DR

This paper proves that generic parabolic fixed points are isolated in positive characteristic fields by analyzing minimally ramified power series and their periodic points, confirming a key conjecture in non-Archimedean dynamics.

Contribution

It provides an explicit proof that generic parabolic germs with prescribed multipliers have isolated fixed points in positive characteristic, using detailed computations of minimally ramified power series.

Findings

Confirmed the conjecture for generic germs with prescribed multipliers

Computed the first significant terms of minimally ramified power series

Established a lower bound for the norm of nonzero periodic points

Abstract

We study analytic germs in one variable having a parabolic fixed point at the origin, over an ultrametric ground field of positive characteristic. It is conjectured that for such a germ the origin is isolated as a periodic point. Our main result is an affirmative solution of this conjecture in the case of a generic germ with a prescribed multiplier. The genericity condition is explicit: That the power series is minimally ramified, i.e., that the degree of the first non-linear term of each of its iterates is as small as possible. Our main technical result is a computation of the first significant terms of a minimally ramified power series. From this we obtain a lower bound for the norm of nonzero periodic points, from which we deduce our main result. As a by-product we give a new and self-contained proof of a characterization of minimally ramified power series in terms of the iterative residue.