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Uniform perfectness of the Berkovich Julia sets in non-archimedean dynamics

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

Published online by Cambridge University Press:  21 December 2021

YÛSUKE OKUYAMA
YÛSUKE OKUYAMA*
Affiliation:
Division of Mathematics, Kyoto Institute of Technology, Sakyo-ku, Kyoto 606-8585, Japan. e-mail: okuyama@kit.ac.jp
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Abstract

We show that a rational function f of degree $>1$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value has no potentially good reductions if and only if the Berkovich Julia set of f is uniformly perfect. As an application, a uniform regularity of the boundary of each Berkovich Fatou component of f is also established.

MSC classification

Primary: 37P50: Dynamical systems on Berkovich spaces
Secondary: 11S82: Non-Archimedean dynamical systems 31C15: Potentials and capacities
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 3 , November 2022 , pp. 573 - 590
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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