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arXiv · 1501.03965

Generic parabolic points are isolated in positive characteristic

Karl-Olof Lindahl·Juan Rivera-Letelier
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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.

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Karl-Olof Lindahl, Juan Rivera-Letelier. 2016-02-25. Generic parabolic points are isolated in positive characteristic. https://doi.org/10.1088/0951-7715%2F29%2F5%2F1596

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