Invariant measures as obstructions to attractors in dynamical systems and their role in nonholonomic mechanics

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.