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Optimal cycles in ultrametric dynamics and minimally ramified power series

Published online by Cambridge University Press:  07 September 2015

Karl-Olof Lindahl
Affiliation:
Department of Mathematics, Linnæus University, 351 95, Växjö, Sweden email karl-olof.lindahl@lnu.se
Juan Rivera-Letelier
Affiliation:
Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile email riveraletelier@mat.puc.cl
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Abstract

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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.

Type
Research Article
Copyright
© The Authors 2015 

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