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Equidistribution of dilations of polynomial curves in nilmanifolds

Authors: Michael Björklund and Alexander Fish
Journal: Proc. Amer. Math. Soc. 137 (2009), 2111-2123
MSC (2000): Primary 60B15
Published electronically: January 27, 2009
MathSciNet review: 2480293
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Abstract: In this paper we study the asymptotic behaviour under dilations of probability measures supported on polynomial curves in nilmanifolds. We prove, under some mild conditions, the effective equidistribution of such measures to the Haar measure. We also formulate a mean ergodic theorem for $\mathbb {R}^n$-representations on Hilbert spaces, restricted to a moving phase of low dimension. Furthermore, we bound the necessary dilation of a given smooth curve in $\mathbb {R}^n$ so that the canonical projection onto $\mathbb {T}^n$ is $\varepsilon$-dense.

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Additional Information

Michael Björklund
Affiliation: Department of Mathematics, KTH - Royal Institute of Technology, SE-100 44 Stockholm, Sweden

Alexander Fish
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210

Received by editor(s): September 5, 2008
Published electronically: January 27, 2009
Additional Notes: The research of the second author was partly done during his visit to MSRI, Berkeley
Communicated by: Bryna Kra
Article copyright: © Copyright 2009 American Mathematical Society