Equidistribution of divergent diagonal orbits in positive characteristic

Abstract

Given a local field $\hat{K}$ with positive characteristic, we study the dynamics of the diagonal subgroup of the linear group ${\mathrm{GL}}_n(\hat{K})$ on homogeneous spaces of discrete lattices in ${\hat{K}}^n$. We first give a function field version of results by Margulis and Tomanov-Weiss, characterizing the divergent diagonal orbits. When $n=2$, we relate the divergent diagonal orbits with the divergent orbits of the geodesic flow in the modular quotient of the Bruhat-Tits tree of ${\mathrm{PGL}}_2(\hat{K})$. Using the (high) entropy method by Einsiedler-Lindentraus et al, we then give a function field version of a result of David-Shapira on the equidistribution of a natural family of these divergent diagonal orbits, with height given by a new notion of discriminant of the orbits.

54 pages, 5 figures