Hostname: page-component-77c89778f8-vsgnj Total loading time: 0 Render date: 2024-07-18T09:09:20.088Z Has data issue: false hasContentIssue false
  • English
  • Français

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: henryt@cmu.edu)
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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