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 Publish in OALib Journal ISSN: 2333-9721 APC: Only $99  Views Downloads  Relative Articles A good universal weight for multiple recurrence averages with commuting transformations in norm Multiple recurrence for two commuting transformations A pointwise cubic average for two commuting transformations Pointwise convergence for cubic and polynomial ergodic averages of non-commuting transformations Pointwise multiple averages for systems with two commuting transformations Multiple recurrence for non-commuting transformations along rationally independent polynomials Averages along cubes for not necessarily commuting measure preserving transformations On multiple recurrence and other properties of "nice" infinite measure preserving transformations On Multiple and Polynomial Recurrent extensions of infinite measure preserving transformations Pointwise convergence along cubes for measure preserving systems More... Mathematics 2013 # Pointwise recurrence for commuting measure preserving transformations  Full-Text Cite this paper Abstract: Let$(X,\mathcal{A}, \mu)$be a probability measure space and let$T_i,1\leq i\leq H,$be commuting invertible measure preserving transformations on this measure space. We prove the following pointwise results; The averages $$\frac{1}{N}\sum_{n=1}^N f_1(T_1^nx)f_2(T_2^nx)\cdots f_H(T_H^nx)$$ converge a.e. for every function$f_i \in L^{\infty}(\mu)$.\\ As a consequence if$T_i = T^i$for$1\leq i \leq H$where$T$is an invertible measure preserving transformation on$(X, \mathcal{A}, \mu)$then the averages $$\frac{1}{N}\sum_{n=1}^N f_1(T^nx)f_2(T^{2n}x)...f_H(T^{Hn}x)$$ converge a.e. This solves a long open question on the pointwise convergence of nonconventional ergodic averages. For$H=2\$ it provides another proof of J. Bourgain's a.e. double recurrence theorem.

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