Equidistribution of divergent diagonal orbits in positive characteristic

2025/03/18 by Dang, Nguyen-Thi, Paulin, Frédéric, Sayous, Rafael · 1 citation
#11J70 #11N45 #11P21 #14G17 #20G30 #22F30 #28C10 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2503.13995

Abstract

Given a local field \widehat K with positive characteristic, we study the dynamics of the diagonal subgroup of the linear group 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 PGL₂(\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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