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Wiener-Wintner return-times ergodic theorem. (English) Zbl 0843.28007
The authors consider ergodic averages of the form ${1\over N} \sum e^{in\lambda} f'(S^n y). f(T^n x)$. They investigate how these averages are related to (and characterize) a new factor of the system (and its orthogonal) -- the maximal factor of the type “Abelian group extension of a group rotation” which satisfies a certain functional equation.

28D05Measure-preserving transformations
37A99Ergodic theory
Full Text: DOI
[1] [A1] I. Assani,A Wiener-Wintner property for the helical transform, Ergodic Theory and Dynamical Systems12 (1992), 185--194. · Zbl 0767.58028
[2] [A2] I. Assani,Uniform Wiener-Wintner theorems for weakly mixing dynamical systems, preprint, unpublished.
[3] [B1] J. Bourgain,Return times sequences of dynamical systems, preprint (3/1988), IHES.
[4] [B2] J. Bourgain,Double recurrence and almost sure convergence, Journal für die reine und angewandte Mathematik404 (1990), 140--161. · Zbl 0685.28008 · doi:10.1515/crll.1990.404.140
[5] [BFKO] J. Bourgain, H. Furstenberg, Y. Katznelson and D. Ornstein,Return times of dynamical systems, Appendix to J. Bourgain’s ”Pointwise Ergodic Theorems For Arithmetic Sets”, Publications IHES69 (1990), 5--45.
[6] [CL1] J. P. Conze and E. Lesigne,Théorèmes ergodiques pour des mesures diagonales, Bulletin de la Société Mathématique de France112 (1984), 143--175. · Zbl 0595.28018
[7] [CL2] J. P. Conze and E. Lesigne,Sur un théorème ergodique pour des measures diagonales, Comptes Rendus de l’Academie des Sciences, Paris306, serie I (1988), 491--493. · Zbl 0641.28010
[8] [F] H. Furstenberg,Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, NJ, 1981. · Zbl 0459.28023
[9] [KN] L. Kuipers and H. Niederreiter,Uniform Distribution of Sequences, J. Wiley and Sons, New York, 1974. · Zbl 0281.10001
[10] [L1] E. Lesigne,Equations fonctionnelles, couplages de produits gauches et théorèmes ergodiques pour measures diagonales, Bulletin de la Société Mathématique de France121 (1993), 315--351.
[11] [L2] E. Lesigne,Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes, Ergodic Theory and Dynamical Systems, to appear.
[12] [R1] D. Rudolph,A joining proof of J. Bourgain’s return times theorem, Ergodic Theory and Dynamical Systems14 (1994), 197--203. · Zbl 0799.28010 · doi:10.1017/S014338570000780X
[13] [R2] D. Rudolph,Eigenfunctions of T{$\times$}S and the Conze-Lesigne algebra, preprint. · Zbl 0877.28012
[14] [WW] N. Wiener and A. Wintner,Harmonic analysis and ergodic theory, American Journal of Mathematics63 (1941), 415--426. · Zbl 0025.06504 · doi:10.2307/2371534