Escape of mass in homogeneous dynamics in positive characteristic

TL;DR

This paper investigates how probability measures supported on periodic orbits in positive characteristic can escape mass when varied along rays of Hecke trees, contrasting with zero characteristic behavior.

Contribution

It demonstrates that in positive characteristic, measures can escape or accumulate on non-uniform measures along certain rays, revealing new phenomena in homogeneous dynamics.

Findings

Rational rays exhibit uniform escape of mass.

Uncountably many rays cause escape of mass.

Some rays lead to accumulation on non-absolutely continuous measures.

Abstract

We show that in positive characteristic the homogeneous probability measure supported on a periodic orbit of the diagonal group in the space of 2-lattices, when varied along rays of Hecke trees, may behave in sharp contrast to the zero characteristic analogue; that is, that for a large set of rays the measures fail to converge to the uniform probability measure on the space of 2-lattices. More precisely, we prove that when the ray is rational there is uniform escape of mass, that there are uncountably many rays giving rise to escape of mass, and that there are rays along which the measures accumulate on measures which are not absolutely continuous with respect to the uniform measure on the space of 2-lattices.