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arXiv · 1003.3112

Equidistribution of singular measures on nilmanifolds and skew products

Fabrizio Polo
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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.

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Fabrizio Polo. 2010-03-24. Equidistribution of singular measures on nilmanifolds and skew products. https://arxiv.org/abs/1003.3112

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