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arXiv · 2503.13995

Equidistribution of divergent diagonal orbits in positive characteristic

Nguyen-Thi Dang·Frédéric Paulin·Rafael Sayous
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Abstract

Given a local field $\widehat K$ with positive characteristic, we study the dynamics of the diagonal subgroup of the linear group $\operatorname{GL}_n(\widehat K)$ on homogeneous spaces of discrete lattices in ${\widehat K}^{\,n}$. 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 $\operatorname{PGL}_2(\widehat K)$. 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.

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Nguyen-Thi Dang, Frédéric Paulin, Rafael Sayous. 2025-03-18. Equidistribution of divergent diagonal orbits in positive characteristic. https://arxiv.org/abs/2503.13995

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