Abstract
We prove a version of Furstenberg’s ergodic theorem with restrictions on return times. More specifically, for a measure preserving system (X, B, μ,T), integers 0 ≤j <k, andE ⊂X with μ(E) > 0, we show that there existsn ≡ j (modk) with ώ(E ∩T -nE ∩T -2nE ∩T -3nE) > 0, so long asT k is ergodic. This result requires a deeper understanding of the limit of some nonconventional ergodic averages and the introduction of a new class of systems, the ‘Quasi-Affine Systems’.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.-P. Conze and E. Lesigne,Théorèmes ergodiques pour des mesures diagonales, Bull. Soc. Math. France112 (1984), 143–175.
J.-P. Conze and E. Lesigne,Sur un théorème ergodique pour des mesures diagonales, Publications de l’Institut de Recherche de Mathématiques de Rennes, Probabilités, 1987.
J.-P. Conze and E. Lesigne,Sur un théorème ergodique pour des mesures diagonales, C. R. Acad. Sci. Paris, Série I306 (1988), 491–493.
H. Furstenberg,Ergodic behavior of diagonal measures and a theorem of Szeméredi on arithmetic progressions, J. Analyse Math.31 (1977), 204–256.
H. Furstenberg and B. Weiss, A mean ergodic theorem for\(\frac{1}{N}\sum\nolimits_{n = 1}^n f \left( {T^n x} \right)g\left( {T^{n^2 } x} \right)\), inConvergence in Ergodic Theory and Probability (V. Bergelson, P. March and J. Rosenblatt, eds.), Walter de Gruyter & Co, Berlin, New York, 1996, pp. 193–227.
B. Host and B. Kra,Convergence of Conze-Lesigne averages, Ergodic Theory Dynam. Systems21 (2001), 493–509.
E. Lesigne,Résolution d’une équation fonctionelle, Bull. Soc. Math. France112 (1984), 177–196.
E. Lesigne,Théorèmes ergodiques pour une translation sur une nilvariété, Ergodic Theory Dynam. Systems9 (1989), 115–126.
E. Lesigne,Équations fonctionelles, couplages de produits gauches et théorèmes ergodiques pour mesures diagonales, Bull. Soc. Math. France121 (1993), 315–351.
D. J. Rudolph,Eigenfunctions of T x Sand the Conze-Lesigne algebra, inErgodic Theory and its Connections with Harmonic Analysis (K. Petersen and I. Salama, eds.), Cambridge University Press, New York, 1995, pp. 369–432.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially carried out while the second author was visiting the Université de Marne la Vallée, supported by NSF grant 9804651.
Rights and permissions
About this article
Cite this article
Host, B., Kra, B. An odd Furstenberg-Szemerédi theorem and quasi-affine systems. J. Anal. Math. 86, 183–220 (2002). https://doi.org/10.1007/BF02786648
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02786648