Generic parabolic points are isolated in positive characteristic
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.
Cite
@article{arxiv.1501.03965,
title = {Generic parabolic points are isolated in positive characteristic},
author = {Karl-Olof Lindahl and Juan Rivera-Letelier},
journal= {arXiv preprint arXiv:1501.03965},
year = {2016}
}
Comments
Minor changes in the presentation. arXiv admin note: text overlap with arXiv:1311.4478