Equidistribution of singular measures on nilmanifolds and skew products

TL;DR

This paper investigates the convergence behavior of measures under minimal rotations on nilmanifolds and skew products, establishing conditions for equidistribution and extending classical ergodic theorems.

Contribution

It proves new criteria for measure convergence on nilmanifolds and extends Furstenberg's work to a broad class of skew products, including a multiplicative ergodic theorem.

Findings

Measures converge to Haar measure under certain conditions

Results hold along sequences of density 1 for general nilmanifolds

Characterization of limits for skew product transformations

Abstract

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.