In this paper we study the geometric location of periodic points of power series defined over fields of prime characteristic @math . More specifically, we find a lower bound for the absolute value of all periodic points in the open unit disk of minimal period @math of 2-ramified power series. We prove that this bound is optimal for a large class of power series. Our main technical result is a computation of the first significant terms of the @math th iterate of 2-ramified power series. As a by-product we obtain a self-contained proof of the characterization of 2-ramified power series.
In @cite_5 the authors give a corresponding result of Theorem for minimally ramified power series, where @math is expressed in terms of the coefficients of the first two non-linear terms. Provided the information from Theorem we can make a corresponding version of Corollary for minimally ramified power series, where the conditions for optimality are expressed in terms of the four lowest degree non-linear terms.
@cite_5
2135702680
We study analytic germs in one variable with 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: the power series is minimally ramified, i.e. the degree of the first nonlinear 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.