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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Universal characteristic factors and Furstenberg averages
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by Tamar Ziegler PDF
J. Amer. Math. Soc. 20 (2007), 53-97 Request permission

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
  • © Copyright 2006 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 20 (2007), 53-97
  • MSC (2000): Primary 37Axx
  • DOI: https://doi.org/10.1090/S0894-0347-06-00532-7
  • MathSciNet review: 2257397