Quantitative Logarithmic Equidistribution of the Crucial Measures

TL;DR

This paper proves uniform logarithmic equidistribution of crucial measures associated with iterates of a rational function over a non-Archimedean field, providing bounds on measure support and minimal resultant loci.

Contribution

It establishes the first uniform logarithmic equidistribution results for crucial measures of iterates of rational functions over non-Archimedean fields, with explicit bounds.

Findings

Bound on the diameter of support points depending only on n and φ

Sets MinResLoc(φ^n) are uniformly bounded independently of n

Explicit radius bound for a ball containing the barycenter of μ_φ

Abstract

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 $\textrm{Bary}(\mu_\phi)$.