A new equivariant in nonarchimedean dynamics

Abstract

Let $K$ be a complete, algebraically closed nonarchimedean valued field, and let $\phi \left(z\right)\in K\left(z\right)$ have degree $d\ge 2$. We show there is a canonical way to assign nonnegative integer weights $w_{\phi }\left(P\right)$ to points of the Berkovich projective line over $K$ in such a way that ${\sum }_P{w}_{\phi }\left(P\right)=d-1$. When $\phi$ has bad reduction, the set of points with nonzero weight forms a distributed analogue of the unique point which occurs when $\phi$ has potential good reduction. Using this, we characterize the minimal resultant locus of $\phi$ in analytic and moduli-theoretic terms: analytically, it is the barycenter of the weight-measure associated to $\phi$; moduli-theoretically, it is the closure of the set of points where $\phi$ has semistable reduction, in the sense of geometric invariant theory.