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Optimal cycles in ultrametric dynamics and minimally ramified power series

Part of: Algebraic number theory: local and $p$-adic fields Complex dynamical systems Arithmetic and non-Archimedean dynamical systems

Published online by Cambridge University Press:  07 September 2015

Karl-Olof Lindahl  and
Juan Rivera-Letelier
Karl-Olof Lindahl
Affiliation:
Department of Mathematics, Linnæus University, 351 95, Växjö, Sweden email [email protected]
Juan Rivera-Letelier
Affiliation:
Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile email [email protected]
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Abstract

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.

Keywords

non-Archimedean dynamical systemsperiodic pointsrotation domainsramification theory

MSC classification

Primary: 37P05: Polynomial and rational maps 11S15: Ramification and extension theory 37F50: Small divisors, rotation domains and linearization; Fatou and Julia sets
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 1 , January 2016 , pp. 187 - 222
Copyright
© The Authors 2015 

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