Configuration of the Crucial Set for a Quadratic Rational Map

Abstract

Let $K$ be a complete, algebraically closed non-archimedean valued field, and let $\phi (z)\in K(z)$ have degree two. We describe the crucial set of $\phi$ in terms of the multipliers of $\phi$ 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 $\phi$. We apply this to settle a special case of a conjecture of Hsia regarding the density of repelling periodic points in the non-archimedean Julia set.