Quantitative Logarithmic Equidistribution of the Crucial Measures

Abstract

Let $K$ be a algebraically closed field of characteristic 0 that is complete with respect to a non-Archimedean absolute value. Let $\varphi \in K(z)$ with $\text{deg}(\varphi )\ge 2$. In this paper we establish uniform logarithmic equidistribution of the crucial measures ${\nu }_{{\varphi }^n}$ attached to the iterates of $\varphi$. These measures were introduced by Rumely in his study of the Minimal Resultant Locus of $\varphi$. Our equidistribution result comes from a bound on the diameter of points in $\text{supp}({\nu }_{{\varphi }^n})$ that depends only on $n$ and $\varphi$. We also show that the sets $\text{MinResLoc}({\varphi }^n)$ are bounded independent of $n$, and we give an explicit bound for the radius of a ball about ${\zeta }_{\text{Gauss}}$ containing $\text{Bary}({\mu }_{\varphi })$.