Quantitative Logarithmic Equidistribution of the Crucial Measures

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