Multiple recurrence and the structure of probability-preserving systems

Details Multiple recurrence and the structure of probability-preserving systems Multiple recurrence and the structure of probability-preserving systems

2010-06-02

arxiv.org/abs/1006.0491v2

Mathematics

Dynamical Systems

81 pages. This was originally submitted as my UCLA Ph.D. dissertation. [TDA Jun 8th 2010:] New version with slight textual cor

Scientific paper

In 1975 Szemer\'edi proved the long-standing conjecture of Erd\H{o}s and Tur\'an that any subset of $\bbZ$ having positive upper Banach density contains arbitrarily long arithmetic progressions. Szemer\'edi's proof was entirely combinatorial, but two years later Furstenberg gave a quite different proof of Szemer\'edi's Theorem by first showing its equivalence to an ergodic-theoretic assertion of multiple recurrence, and then bringing new machinery in ergodic theory to bear on proving that. His ergodic-theoretic approach subsequently yielded several other results in extremal combinatorics, as well as revealing a range of new phenomena according to which the structures of probability-preserving systems can be described and classified. In this work I survey some recent advances in understanding these ergodic-theoretic structures. It contains proofs of the norm convergence of the `nonconventional' ergodic averages that underly Furstenberg's approach to variants of Szemer\'edi's Theorem, and of two of the recurrence theorems of Furstenberg and Katznelson: the Multidimensional Multiple Recurrence Theorem, which implies a multidimensional generalization of Szemer\'edi's Theorem; and a density version of the Hales-Jewett Theorem of Ramsey Theory.

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