On stationary measure rigidity and orbit closures for actions of non-abelian groups

I will describe joint work in progress with Aaron Brown, Federico Rodriguez-Hertz and Simion Filip.
Our aim is to find some analogue, in the context of smooth dynamics, of Ratner's theorems on
unipotent flows. This would be a (partial) generalization of the results of Benoist-Quint and my work
with Elon Lindenstrauss in the homogeneous setting, the results of Brown and Rodriguez-Hertz in
dimension 2, and my results with Maryam Mirzakhani in the setting of Teichmuller dynamics.