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Convergence of diagonal ergodic averages

Published online by Cambridge University Press:  01 August 2009

HENRY TOWSNER
HENRY TOWSNER*
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (email: [email protected])
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Abstract

Tao has recently proved that if T1,…,Tl are commuting, invertible, measure-preserving transformations on a dynamical system, then for any L∞ functions f1,…,fl, the average (1/N)∑ n=0N−1∏ i≤lfi∘Tin converges in the L2 norm. Tao’s proof is unusual in that it translates the problem into a more complicated statement about the combinatorics of finite spaces by using the Furstenberg correspondence ‘backwards’. In this paper, we give an ergodic proof of this theorem, essentially a translation of Tao’s argument to the ergodic setting. In order to do this, we develop two new variations on the usual Furstenberg correspondence, both of which take recurrence-type statements in one dynamical system and give equivalent statements in a different dynamical system with desirable properties.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 4 , August 2009 , pp. 1309 - 1326
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Austin, T.. On the norm convergence of nonconventional ergodic averages, 2008.http://arXiv.org/abs/0805.0320.Google Scholar
[2]Conze, J.-P. and Lesigne, E.. Théorèmes ergodiques pour des mesures diagonales. Bull. Soc. Math. France 112(2) (1984), 143–175.CrossRefGoogle Scholar
[3]Elek, G. and Szegedy, B.. Limits of hypergraphs, removal and regularity lemmas. A non-standard approach, 2007. http://arXiv.org/abs/0705.2179.Google Scholar
[4]Frantzikinakis, N. and Kra, B.. Convergence of multiple ergodic averages for some commuting transformations. Ergod. Th. & Dynam. Sys. 25(3) (2005), 799–809.CrossRefGoogle Scholar
[5]Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. M. B. Porter Lectures.. Princeton University Press, Princeton, NJ, 1981.CrossRefGoogle Scholar
[6]Glasner, E.. Ergodic Theory Via Joinings (Mathematical Surveys and Monographs, 101). American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
[7]Goldblatt, R.. An introduction to nonstandard analysis.. Lectures on the Hyperreals (Graduate Texts in Mathematics, 188). Springer, New York, 1998.CrossRefGoogle Scholar
[8]Host, B. and Kra, B.. Nonconventional ergodic averages and nilmanifolds. Ann. of Math. (2) 161(1) (2005), 397–488.CrossRefGoogle Scholar
[9]Jerome Keisler, H.. Hyperfinite model theory. Logic Colloquium 76, (Oxford, 1976) (Studies in Logic and Foundations of Mathematics, 87). North-Holland, Amsterdam, 1977, pp. 5–110.Google Scholar
[10]Keisler, H. J.. An Infinitesimal Approach to Stochastic Analysis. American Mathematical Society, Providence, RI, 1984.CrossRefGoogle Scholar
[11]Lesigne, E.. Équations fonctionnelles, couplages de produits gauches et théorèmes ergodiques pour mesures diagonales. Bull. Soc. Math. France 121(3) (1993), 315–351.CrossRefGoogle Scholar
[12]Tao, T.. Norm convergence of multiple ergodic averages for commuting transformations, 2007.CrossRefGoogle Scholar
[13]Tao, T.. Norm convergence of multiple ergodic averages for commuting transformations.http://terrytao.wordpress.com/2007/07/10/norm-convergence-of-multiple-ergodic-averages-for-commuting-transformations/, 2007.Google Scholar
[14]Towsner, H.. A general correspondence between averages and integrals. Draft, 2008.http://arXiv.org/abs/0804.2773.Google Scholar
[15]Wiener, N.. The ergodic theorem. Duke Math. J. 5(1) (1939), 1–18.CrossRefGoogle Scholar
[16]Zhang, Q.. On convergence of the averages (1/N)∑ Nn=1f1(Rnx)f2(Snx)f3(Tnx). Monatsh. Math. 122(3) (1996), 275–300.CrossRefGoogle Scholar
[17]Ziegler, T.. Universal characteristic factors and Furstenberg averages. J. Amer. Math. Soc. 20(1) (2007), 53–97 (electronic).CrossRefGoogle Scholar