Appears in collection : Homogeneous Dynamics and Geometry in Higher-Rank Lie Groups

Orbit closure and measure classification results are quite central results and tools in homogeneous dynamics. Recently, new techniques provided more 'robust' approaches to this kind of results and it makes sense to try to extend some results to the non-homogeneous setting. I will try to explain what would be the natural questions in the non-linear setting and report some progress in this direction.