  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.
