Equidistribution of singular measures on nilmanifolds and skew products

We prove that for a minimal rotation T on a two-step nilmanifold and any measure μ, the push-forward Tn⋆μ of μ under Tn tends toward Haar measure if and only if μ projects to Haar measure on the maximal torus factor. For an arbitrary nilmanifold we get the same result along a sequence of uniform density one. These results strengthen Parry’s result [Ergodic properties of affine transformations and flows on nilmanifolds. Amer. J. Math.91 (1968), 757–771] that such systems are uniquely ergodic. Extending the work of Furstenberg [Strict ergodicity and transformations of the torus. Amer. J. Math.83 (1961), 573–601], we get the same result 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 Tn⋆μ for some skew product transformations with expansive fibers. All results are presented in terms of twisting and weak twisting, properties that strengthen unique ergodicity in a way analogous to that in which mixing and weak mixing strengthen ergodicity for measure-preserving systems.