Equidistribution of singular measures on nilmanifolds and skew products

We prove that for a minimal rotation T on a 2-step nilmanifold and any measure mu, the push-forward T^n(mu) of mu under T^n tends toward Haar measure if and only if mu projects to Haar measure on the maximal torus factor. For an arbitrary nilmanifold we get the same result along a sequence of uniform density 1. These results strengthen Parry's result that such systems are uniquely ergodic. Extending the work of Furstenberg, we prove an analogous theorem for a large class of iterated skew products. Additionally we prove a multiplicative ergodic theorem for functions taking values in the upper unipotent group. Finally, we characterize limits of T^n(mu) for some skew product transformations with expansive fibers. All results are presented in terms of twisting and weak twisting, properties which strengthen unique ergodicity in a way analogous to how mixing and weak mixing strengthen ergodicity for measure preserving systems.