Equidistribution of preimages over nonarchimedean fields for maps of good reduction

TL;DR

This paper extends the concept of equidistribution of preimages from complex dynamics to nonarchimedean fields, providing partial and complete analogues depending on the map's properties, with strengthened results in trivial absolute value cases.

Contribution

It establishes an analogue of the equidistribution of preimages theorem for maps of good reduction over nonarchimedean fields, including a strengthened version for trivial absolute value fields.

Findings

Partial analogue of complex equidistribution theorem for nonarchimedean fields.

Complete analogue for most maps of good reduction.

Preimages of tame valuations equidistribute to a canonical measure in trivial absolute value cases.

Abstract

In this article we prove an analogue of the equidistribution of preimages theorem from complex dynamics for maps of good reduction over nonarchimedean fields. While in general our result is only a partial analogue of the complex equidistribution theorem, for most maps of good reduction it is a complete analogue. In the particular case when the nonarchimedean field in question is equipped with the trivial absolute value, we are able to supply a strengthening of the theorem, namely that the preimages of any tame valuation equidistribute to a canonical measure.