On the higher ergodic theory of certain non-discrete actions

Quasi-invariant measures for non-discrete groups of diffeomorphisms containing a Morse–Smale dynamics are studied. The assumption concerning the presence of a Morse–Smale dynamics allows us to extend to higher dimensions a number of recently established results for non-discrete groups acting on the circle. The last section of this paper discusses the connection between these results and a few interesting questions about rigidity of continuous and non-continuous orbit equivalences for many groups as above.