Abstract We prove a polynomial multiple recurrence theorem for finitely many commut- ing measure preserving transformations of a probability space, extending a polyno- mial Szemeredi theorem appearing in [BL1]. The linear case is a consequence of an ergodic IP-Szemeredi theorem of Furstenberg and Katznelson ([FK2]). Several applications to the fine structure of recurrence in ergodic theory are given, some of which involve weakly mixing systems, for which we also prove a multiparameter weakly mixing polynomial ergodic theorem. The techniques and apparatus em- ployed include a polynomialization of an IP structure theory developed in [FK2], an extension of Hindman's theorem due to Milliken and Taylor ([M], [T]), a polyno- mial version of the Hales-Jewett coloring theorem ([BL2]), and a theorem concerning limits of polynomially generated IP-systems of unitary operators ([BFM]). 1991 Mathematics Subject Classification: Primary: 28D05 Secondary: 05A17, 05D10, 11B05, 11B83. Keywords and phrases: Ergodic Ramsey theory Furstenberg correspondence principle IP-sets Hindman's theorem Mild mixing Multiple recurrence Polyno- mial Hales-Jewett theorem Polynomial Szemeredi theorem Weak mixing. vm
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