Abstract:
Let G be a Lie group, X be a Ghomogeneous space
of finite volume, and H be a closed subgroup of G.
What are the Hinvariant probabilities on X?
What are the Hinvariant closed subsets in X?
I will first survey Ratner's results addressing
the case when H is generated by unipotent elements.
I will then focus on a joint work with J.F. Quint
addressing the case when G is simple and
H is Zariski dense. In this case,
the Haar probability on X is the only
atomfree Hinvariant probability on X.
Moreover every Hinvariant
subset of X is either finite or dense.
The proof uses random walks on X.
