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Multiple recurrence and almost sure convergence for weakly mixing dynamical systems. (English) Zbl 0920.28011

One of the basic multiple recurrence results due to H. Furstenberg, Y. Katznelson and D. Ornstein [Bull. Am. Math. Soc., New Ser. 7, 527-552 (1982; Zbl 0523.28017)] is that, for a weakly mixing transformation \(T\), and \(L^{\infty}\) functions \(f_1,\dots,f_k\), the non-conventional ergodic averages \(({1}/{N})\sum_{n=1}^{N}T^nf_1\cdot T^{2n}f_2\dots T^{kn}f_k\) converge in the \(L^{2}\) norm to the product \(\prod_{i=1}^k\int f_i\) as \(N\to\infty\). The question of a.e. convergence of such averages is delicate; in this paper the above convergence in norm is shown to hold a.e. under the assumption that the restriction of \(T\) to its Pinsker algebra has singular spectrum. For \(k=3\) a more precise result is found. The method of proof is to first generalize a result of E. Lesigne [C. R. Acad. Sci., Paris, Sér. I 298, 425-428 (1984; Zbl 0579.60024)] to allow reduction to the Pinsker algebra, and then to extend work of B. Host [Isr. J. Math. 76, No. 3, 289-298 (1991; Zbl 0790.28010)] on the structure of pairwise independent joinings under a spectral condition.

MSC:

28D05 Measure-preserving transformations
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References:

[1] [A] I. Assani,Multiterm return times theorem for weakly mixing dynamical systems, preprint. · Zbl 0985.37004
[2] Bourgain, J., Double recurrence and almost sure convergence, Journal für die reine und angewandte Mathematik, 404, 140-161 (1990) · Zbl 0685.28008 · doi:10.1515/crll.1990.404.140
[3] Furstenberg, H.; Katznelson, Y.; Ornstein, D., The ergodic theoretical proof of Szemerdi’s theorem, Bulletin of the American Mathematical Society, 7, 527-552 (1982) · Zbl 0523.28017
[4] Furstenberg, H., Recurrence in Ergodic Theory and Combinatorial Number Theory (1981), Princeton, NJ: Princeton University Press, Princeton, NJ · Zbl 0459.28023
[5] Host, B., Mixing of all orders and pairwise independent joining, Israel Journal of Mathematics, 76, 289-298 (1991) · Zbl 0790.28010
[6] Lesigne, E., Sur la convergence ponctuelle des certaines moyennes ergodiques, Comptes Rendus de l’Académie des Sciences, Paris, 17, 298-298 (1984)
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