Pointwise Convergence of Some Multiple Ergodic Averages
نویسندگان
چکیده
We show that for every ergodic system (X, μ,T1, . . . ,Td) with commuting transformations, the average 1 Nd+1 ∑ 0≤n1,...,nd≤N−1 ∑ 0≤n≤N−1 f1(T n 1 d ∏ j=1 T n j j x) f2(T n 2 d ∏ j=1 T n j j x) · · · fd(T n d d ∏ j=1 T n j j x). converges for μ-a.e. x ∈ X as N → ∞. If X is distal, we prove that the average 1 N N ∑ i=0 f1(T n 1 x) f2(T n 2 x) · · · fd(T n d x) converges for μ-a.e. x ∈ X as N → ∞. We also establish the pointwise convergence of averages along cubical configurations arising from a system commuting transformations. Our methods combine the existence of sated and magic extensions introduced by Austin and Host respectively with ideas on topological models by Huang, Shao and Ye.
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