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 $ϕ\in K(z)$ with $\textrm{deg}(ϕ)\geq 2$. In this paper we establish uniform logarithmic equidistribution of the crucial measures $ν_{ϕ^n}$ attached to the iterates of $ϕ$. These measures were introduced by Rumely in his study of the Minimal Resultant Locus of $ϕ$. Our equidistribution result comes from a bound on the diameter of points in $\textrm{supp}(ν_{ϕ^n})$ that depends only on $n$ and $ϕ$. We also show that the sets $\textrm{MinResLoc}(ϕ^n)$ are bounded independent of $n$, and we give an explicit bound for the radius of a ball about $ζ_{\textrm{Gauss}}$ containing $\textrm{Bary}(μ_ϕ)$.

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