  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.
