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Configuration of the crucial set for a quadratic rational map. (English) Zbl 1391.37086

Summary: Let \(K\) be a complete, algebraically closed non-Archimedean valued field, and let \(\varphi (z) \in K(z)\) have degree two. We describe the crucial set of \(\varphi \) in terms of the multipliers of \(\varphi \) at the classical fixed points, and use this to show that the crucial set determines a stratification of the moduli space \(\mathcal {M}_2(K)\) related to the reduction type of \(\varphi \). We apply this to settle a special case of a conjecture of Hsia regarding the density of repelling periodic points in the classical non-Archimedean Julia set.

MSC:

37P50 Dynamical systems on Berkovich spaces
11S82 Non-Archimedean dynamical systems
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
11Y40 Algebraic number theory computations
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