Configuration of the Crucial Set for a Quadratic Rational Map

TL;DR

This paper characterizes the crucial set of quadratic rational maps over non-archimedean fields using fixed point multipliers, linking it to moduli space stratification and addressing a conjecture on Julia set dynamics.

Contribution

It provides a new description of the crucial set in terms of fixed point multipliers and connects this to the structure of the moduli space of quadratic maps.

Findings

Crucial set determined by fixed point multipliers.

Stratification of moduli space based on reduction type.

Partial resolution of Hsia's conjecture on Julia sets.

Abstract

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 non-archimedean Julia set.