Multiple recurrence and almost sure convergence for weakly mixing dynamical systems

Abstract

We prove the following: Let (X, β, μ,T) be a weakly mixing dynamical system such that the restriction ofT to its Pinsker algebra has singular spectrum, then for all positive integersH, for allf _i ∈L ^∞, 1≤i≤H, the averages

$$\frac{1}{N}\sum\limits_{n = 1}^N {f_1 (T^n x)f_2 (T^{2n} x) \cdot \cdot \cdot f_H (T^{Hn} x)} converge a.e. to \prod\limits_{i = 1}^H {\int {f_i d\mu } } $$

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