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Convergence of weighted polynomial multiple ergodic averages

Author(s): Qing Chu
Journal: Proc. Amer. Math. Soc. 137 (2009), 1363-1369.
MSC (2000): Primary 37A05, 37A30
Posted: October 16, 2008
MathSciNet review: 2465660
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Abstract: In this article we study weighted polynomial multiple ergodic averages. A sequence of weights is called universally good if any polynomial multiple ergodic average with this sequence of weights converges in $L^{2}$ . We find a necessary condition and show that for any bounded measurable function $\phi$ on an ergodic system, the sequence $\phi(T^{n}x)$ is universally good for almost every . The linear case was covered by Host and Kra.

References:

1.: V. Bergelson, B. Host and B. Kra, with an appendix by I. Ruzsa. Multiple recurrence and nilsequences. Inventiones Math. 160 (2005), 261-303. MR 2138068 (2007i:37009)
2.: B. Host and B. Kra. Nonconventional ergodic averages and nilmanifolds. Ann. Math. 161 (2005), 397-488. MR 2150389 (2007b:37004)
3.: B. Host and B. Kra. Convergence of polynomial ergodic averages. Israel J. Math. 149 (2005), 1-19. MR 2191208 (2007c:37004)
4.: B. Host and B. Kra. Uniformity seminorms on $\ell^{\infty}$ and applications. Available at http://arxiv.org/abs/0711.3637v1.
5.: A. Leibman. Convergence of multiple ergodic averages along polynomials of several variables. Israel J. Math. 146 (2005), 303-315. MR 2151605 (2006c:28016)
6.: A. Leibman. Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. Ergodic Th. and Dyn. Sys. 25 (2005), 201-213. MR 2122919 (2006j:37004)
7.: E. Lesigne. Un théorème de disjonction de systèmes dynamiques et une généralisation du théorème ergodique de Wiener-Wintner. Ergodic Th. and Dyn. Sys. 10 (1990), 513-521. MR 1074316 (91h:28016)
8.: E. Lesigne. Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes. Ergodic. Th. and Dyn. Sys. 13 (1993), 767-784. MR 1257033 (95e:28012)
9.: N. Wiener and A. Wintner. Harmonic analysis and ergodic theory. Amer. J. Math. 63 (1941), 415-426. MR 0004098 (2:319b)
10.: T. Ziegler. A non-conventional ergodic theorem for a nilsystem. Ergodic Th. and Dyn. Sys. 25 (2005), 1357-1370. MR 2158410 (2006d:37009)

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Additional Information:

Qing Chu
Affiliation: Université Paris-Est, Laboratoire d'Analyse et de Mathématiques Appliquées, UMR CNRS 8050, 5 bd Descartes, 77454 Marne la Vallée Cedex 2, France
Email: qing.chu@univ-mlv.fr

DOI: 10.1090/S0002-9939-08-09614-7
PII: S 0002-9939(08)09614-7
Keywords: Weighted ergodic averages, universally good sequences, Wiener-Wintner ergodic theorem, nilsequences
Received by editor(s): February 21, 2008,
Received by editor(s) in revised form: April 14, 2008
Posted: October 16, 2008
Communicated by: Bryna Kra
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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