A Criterion for Weak Convergence on Berkovich Projective Space

TL;DR

This paper establishes a criterion for weak convergence of measures on Berkovich projective space, with applications to equidistribution, ergodic theory, and non-archimedean dynamics, extending classical complex results to non-archimedean settings.

Contribution

It introduces a new criterion for weak convergence of measures on Berkovich projective space and applies it to equidistribution and ergodic theorems in non-archimedean geometry.

Findings

Provides a sufficient condition for equidistribution based on Zariski-density.

Establishes an ergodic-theoretic equidistribution result in residue characteristic zero.

Extends classical complex equidistribution results to non-archimedean contexts.

Abstract

We give a criterion for the weak convergence of unit Borel measures on the N-dimensional Berkovich projective space over a complete non-archimedean field. As an application, we give a sufficient condition for equidistribution in terms of a strong Zariski-density property on the scheme-theoretic projective space over the residue field. As a second application, in the case of residue characteristic zero we give an ergodic-theoretic equidistribution result for the powers of a point in the N-dimensional unit torus. This is a non-archimedean analogue of a well-known complex equidistribution result of Weyl, and its proof makes essential use of a theorem of Mordell-Lang type due to Laurent.