Equidistribution non-archim\'edienne et actions de groupes sur les arbres

TL;DR

This paper establishes equidistribution results for elements of function fields and quadratic irrationals over finite fields within their local fields, using ergodic theory on trees and group actions.

Contribution

It introduces new equidistribution theorems derived from ergodic properties of geodesic flows on trees, connecting algebraic elements with geometric group actions.

Findings

Equidistribution of function field elements in local fields.

Equidistribution of quadratic irrationals in local fields.

Application of ergodic theory to non-Archimedean group actions.

Abstract

Nous donnons des r\'esultats d'\'equidistribution d'\'el\'ements de corps de fonctions sur des corps finis, et d'irrationnels quadratiques sur ces corps, dans leurs corps locaux compl\'et\'es. Nous d\'eduisons ces r\'esultats de th\'eor\`emes d'\'equidistribution de perpendiculaires communes dans des quotients d'arbres par des r\'eseaux de leur groupe d'automorphismes, d\'emontr\'es \`a l'aide de propri\'et\'es ergodiques du flot g\'eod\'esique discret. Non-Archimedean equidistribution and group actions on trees. We give equidistribution results of elements of function fields over finite fields, and of quadratic irrationals over these fields, in their completed local fields. We deduce these results from equidistribution theorems of common perpendiculars in quotients of trees by lattices in their automorphism groups, proved by using ergodic properties of the discrete geodesic flow.