Fekete configuration, quantitative equidistribution and wandering critical orbits in non-archimedean dynamics

TL;DR

This paper studies the distribution of points under iteration of rational functions over non-archimedean fields, establishing quantitative equidistribution results and error estimates related to wandering critical orbits and their proximity.

Contribution

It introduces the asymptotic Fekete property for pullbacks of points and provides new error bounds for equidistribution in non-archimedean dynamics, extending previous results.

Findings

Error estimate of order O(√(k/d^k)) for equidistribution

Identification of exceptional sets of initial points with capacity zero

Connection to Favre and Rivera-Letelier's local equidistribution results

Abstract

Let $f$ be a rational function of degree $d>1$ on the projective line over a possibly non-archimedean algebraically closed field. A well-known process initiated by Brolin considers the pullbacks of points under iterates of $f$, and produces an important equilibrium measure. We define the asymptotic Fekete property of pullbacks of points, which means that they mirror the equilibrium measure appropriately. As application, we obtain an error estimate of equidistribution of pullbacks of points for $C^1$-test functions in terms of the proximity of wandering critical orbits to the initial points, and show that the order is $O(\sqrt{kd^{-k}})$ upto a specific exceptional set of capacity 0 of initial points, which is contained in the set of superattracting periodic points and the omega-limit set of wandering critical points from the Julia set or the presingular domains of $f$. As an application in arithmetic dynamics, together with a dynamical Diophantine approximation, these estimates recover Favre and Rivera-Letelier's quantitative equidistribution in a purely local manner.