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Optimal cycles in ultrametric dynamics and minimally ramified power series

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

Berkovich, V. G., Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33 (American Mathematical Society, Providence, RI, 1990).Google Scholar
Écalle, J., Théorie itérative: introduction à la théorie des invariants holomorphes, J. Math. Pures Appl. (9) 54 (1975), 183–258.Google Scholar
Einsiedler, M., Everest, G. and Ward, T., Morphic heights and periodic points, in Number theory (New York, 2003) (Springer, New York, 2004), 167–177.Google Scholar
Einsiedler, M., Everest, G. and Ward, T., Periodic points for good reduction maps on curves, Geom. Dedicata 106 (2004), 29–41.CrossRefGoogle Scholar
Herman, M.-R., Recent results and some open questions on Siegel’s linearization theorem of germs of complex analytic diffeomorphisms of Cn near a fixed point, in VIIIth International congress on mathematical physics (Marseille, 1986) (World Scientific, Singapore, 1987), 138–184.Google Scholar
Herman, M. and Yoccoz, J.-C., Generalizations of some theorems of small divisors to non-Archimedean fields, in Geometric dynamics (Rio de Janeiro, 1981), Lecture Notes in Mathematics, vol. 1007 (Springer, Berlin, 1983), 408–447.CrossRefGoogle Scholar
Keating, K., Automorphisms and extensions of k ((t)), J. Number Theory 41 (1992), 314–321.CrossRefGoogle Scholar
Lang, S., Algebra, Graduate Texts in Mathematics, vol. 211, third edition (Springer, New York, 2002).CrossRefGoogle Scholar
Laubie, F., Movahhedi, A. and Salinier, A., Systèmes dynamiques non archimédiens et corps des normes, Compositio Math. 132 (2002), 57–98.CrossRefGoogle Scholar
Laubie, F. and Saïne, M., Ramification of some automorphisms of local fields, J. Number Theory 72 (1998), 174–182.CrossRefGoogle Scholar
Li, H.-C., p-adic periodic points and Sen’s theorem, J. Number Theory 56 (1996), 309–318.CrossRefGoogle Scholar
Lindahl, K.-O., On Siegel’s linearization theorem for fields of prime characteristic, Nonlinearity 17 (2004), 745–763.CrossRefGoogle Scholar
Lindahl, K.-O., Divergence and convergence of conjugacies in non-Archimedean dynamics, in Advances in p-adic and non-Archimedean analysis, Contemporary Mathematics, vol. 508 (American Mathematical Society, Providence, RI, 2010), 89–109.Google Scholar
Lindahl, K.-O., The size of quadratic p-adic linearization disks, Adv. Math. 248 (2013), 872–894.CrossRefGoogle Scholar
Lindahl, K.-O. and Rivera-Letelier, J., Generic parabolic points are isolated in positive characteristic, Preprint (2015), arXiv:1501.03965v1.Google Scholar
Lubin, J., Non-Archimedean dynamical systems, Compositio Math. 94 (1994), 321–346.Google Scholar
Lubin, J., Sen’s theorem on iteration of power series, Proc. Amer. Math. Soc. 123 (1995), 63–66.Google Scholar
Milnor, J., Dynamics in one complex variable, Annals of Mathematics Studies, vol. 160, third edition (Princeton University Press, Princeton, NJ, 2006).Google Scholar
Pérez-Marco, R., Fixed points and circle maps, Acta Math. 179 (1997), 243–294.CrossRefGoogle Scholar
Rivera-Letelier, J., Dynamique des fonctions rationnelles sur des corps locaux, Astérisque 287 (2003), 147–230; Geometric methods in dynamics. II.Google Scholar
Sen, S., On automorphisms of local fields, Ann. of Math. (2) 90 (1969), 33–46.CrossRefGoogle Scholar
Serre, J.-P., Corps Locaux, Publications de l’Institut de Mathématique de l’Université de Nancago, No. 8, second edition (Hermann, Paris, 1968).Google Scholar
Wintenberger, J.-P., Automorphismes des corps locaux de caractéristique p, J. Théor. Nombres Bordeaux 16 (2004), 429–456.CrossRefGoogle Scholar
Yoccoz, J.-C., Centralisateurs et conjugaison différentiable des difféomorphismes du cercle, Astérisque 231 (1995), 89–242; Petits diviseurs en dimension 1.Google Scholar