Equidistribution non-archimédienne et actions de groupes sur les arbres
Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
TL;DR
The paper addresses equidistribution phenomena in non-archimedean local fields attached to function fields, linking arithmetic questions about rationals and quadratic irrationals with dynamics on trees. It develops a unified dynamical framework based on ergodic properties of the geodesic flow on Bruhat–Tits buildings and the associated quotient spaces to derive explicit equidistribution results with effective constants. Key contributions include (i) quantitative non-archimedean equidistribution for orbits of $G$ on $P^1(K_v)$ and for $G$-orbits of quadratic irrationals with height constraints; (ii) a central geometric equidistribution theorem for common perpendiculars in tree quotients, with exponential error under geometric finiteness; and (iii) arithmetic corollaries for function fields, expressed as non-archimedean Mertens-type density theorems and quadratic irrational distribution in $K_v$. This work bridges dynamics on trees with function field arithmetic via Bruhat–Tits buildings and Nagao lattices, providing a coherent methodology for non-archimedean equidistribution and explicit constants.
Abstract
Nous donnons des résultats d'équidistribution d'éléments de corps de fonctions sur des corps finis, et d'irrationnels quadratiques sur ces corps, dans leurs corps locaux complétés. Nous déduisons ces résultats de théorèmes d'équidistribution de perpendiculaires communes dans des quotients d'arbres par des réseaux de leur groupe d'automorphismes, démontrés à l'aide de propriétés ergodiques du flot géodésique 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.