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We study the limiting behavior of multiple ergodic averages involving severalnot necessarily commuting measure preserving transformations. We work on twotypes of averages, one that uses iterates along combinatorial parallelepipeds,and another that uses iterates along shifted polynomials. We prove pointwiseconvergence in both cases, thus answering a question of I.Assani in the formercase, and extending results of B.Host-B.Kra and A.Leibman in the latter case.Our argument is based on some elementary uniformity estimates of generalbounded sequences, decomposition results in ergodic theory, andequidistribution results on nilmanifolds.

Pointwise convergence for cubic and polynomial ergodic averages of non-commuting transformations