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Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Universal characteristic factors and Furstenberg averages


Author: Tamar Ziegler
Journal: J. Amer. Math. Soc. 20 (2007), 53-97
MSC (2000): Primary 37Axx
DOI: https://doi.org/10.1090/S0894-0347-06-00532-7
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 \begin{equation*} \tag {*} \frac {1}{N}\sum _{n=1}^N \prod _{j=1}^k f_j(T^{a_jn}x) \end{equation*} 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
Email: tamar@math.ohio-state.edu, tamar@math.ias.edu

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