  We present some rigorous results on the absence of a wide class of invariant
measures for dynamical systems possessing attractors. We then consider a
generalization of the classical nonholonomic Suslov problem which shows how
previous investigations of existence of invariant measures for nonholonomic
systems should necessarily be extended beyond the class of measures with
strictly positive $C^1$ densities if one wishes to determine dynamical
obstructions to the presence of attractors.
