  Several problems in number theory when reformulated in terms of homogenous
dynamics involve study of limiting distributions of translates of algebraically
defined measures on orbits of reductive groups. The general non-divergence and
linearization techniques, in view of Ratner's measure classification for
unipotent flows, reduce such problems to dynamical questions about linear
actions of reductive groups on finite dimensional vectors spaces. This article
provides general results which resolve these linear dynamical questions in
terms of natural group theoretic or geometric conditions.
