Abstract. We prove mean convergence, as N → ∞, for the multiple ergodic averages

This paper is devoted to a study of the multiple recurrence of two commuting transformations. We derive a result which is similar but not identical to that of one single transformation established by Bergelson, Host and Kra. We will use the machinery of "magic systems" established recently by B. Host for the proof.

We study the limiting behavior of multiple ergodic averages involving several not necessarily commuting measure preserving transformations. We work on two types of averages, one that uses iterates along combinatorial parallelepipeds, and another that uses iterates along shifted polynomials. We prove pointwise convergence in both cases, thus answering a question of I. Assani in the former case, 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 general bounded sequences, decomposition results in ergodic theory, and equidistribution results on nilmanifolds.