Institute for Mathematical Sciences
"Unique Ergodicity on Compact Homogeneous Spaces"
Abstract: Extending results of a number of authors, we prove that if $U$ is the unipotent radical of a solvable epimorphic subgroup of
an algebraic group $G$, then the action of $U$ on $G/\Gamma$ is
uniquely ergodic for every cocompact lattice $\Gamma$ in
$G$. This gives examples of uniquely ergodic and minimal
two-dimensional flows on homogeneous spaces of arbitrarily high
dimension. Our main tools are Ratner classification of ergodic
invariant measures for the action of a unipotent subgroup on a
homogeneous space, and a simple lemma (the `Cone Lemma') about
representations of epimorphic subgroups.
(revised version of July 1999)
View ims99-7c (PDF format)