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Universal characteristic factors and Furstenberg averages

Author: Tamar Ziegler
Journal: J. Amer. Math. Soc. 20 (2007), 53-97
MSC (2000): Primary 37Axx
Published electronically: March 17, 2006
MathSciNet review: 2257397
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Abstract: Let $ X=(X^0,\mathcal{B},\mu,T)$ be an ergodic probability measure-preserving system. For a natural number $ k$ we consider the averages

$\displaystyle \tag{*} \frac{1}{N}\sum_{n=1}^N \prod_{j=1}^k f_j(T^{a_jn}x) $

where $ f_j \in L^{\infty}(\mu)$, and $ a_j$ are integers. A factor of $ X$ is characteristic for averaging schemes of length $ k$ (or $ k$-characteristic) if for any nonzero distinct integers $ a_1,\ldots,a_k$, the limiting $ L^2(\mu)$ behavior of the averages in (*) is unaltered if we first project the functions $ f_j$ onto the factor. A factor of $ X$ is a $ k$-universal characteristic factor ($ k$-u.c.f.) if it is a $ k$-characteristic factor, and a factor of any $ k$-characteristic factor. We show that there exists a unique $ k$-u.c.f., and it has the structure of a $ (k-1)$-step nilsystem, more specifically an inverse limit of $ (k-1)$-step nilflows. Using this we show that the averages in (*) converge in $ L^2(\mu)$. This provides an alternative proof to the one given by Host and Kra.

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Additional Information

Tamar Ziegler
Affiliation: Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, Ohio 43210
Address at time of publication: School of Mathematics, The Institute of Advanced Study, Princeton, New Jersey 08540

Received by editor(s): October 18, 2004
Published electronically: March 17, 2006
Article copyright: © Copyright 2006 American Mathematical Society

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