Configuration of the crucial set for a quadratic rational map

摘要

Let K be a complete, algebraically closed non-Archimedean valued field, and let $$arphi (z) in K(z)$$ φ ( z ) ∈ K ( z ) have degree two. We describe the crucial set of $$arphi $$ φ in terms of the multipliers of $$arphi $$ φ at the classical fixed points, and use this to show that the crucial set determines a stratification of the moduli space $$mathcal {M}_2(K)$$ M 2 ( K ) related to the reduction type of $$arphi $$ φ . 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.