  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. These results are also applied to show that, for many groups as above,
every continuous orbit equivalence must coincide almost everywhere with a
diffeomorphism of the corresponding manifold.
