A criterion for weak convergence on Berkovich projective space

Abstract

We give a criterion for the weak convergence of unit Borel measures on the N-dimensional Berkovich projective space \({{\bf P}^{N}_K}\) over a complete non-archimedean field K. As an application, we give a sufficient condition for a certain type of equidistribution on \({{\bf P}^{N}_K}\) in terms of a weak Zariski-density property on the scheme-theoretic projective space \({{\mathbb P}^N_{\tilde{K{}_{\vphantom{0}}}}}\) over the residue field \({\tilde{K}}\) . As a second application, in the case of residue characteristic zero we give an ergodic-theoretic equidistribution result for the powers of a point a in the N-dimensional unit torus \({{\mathbb T}^N_K}\) over K. This is a non-archimedean analogue of a well-known result of Weyl over \({\mathbb C}\) , and its proof makes essential use of a theorem of Mordell-Lang type for \({{\mathbb G}_m^N}\) due to Laurent.