Recurrent Ergodic Structures and Ramsey Theory

Ramsey theory is best defined by example and the classic example of a Ramsey type theorem is the result of van der Waerden: if the integers Z are partitioned into finitely many sets, one of these contains arbitrarily long arithmetic progressions. An equivalent version states: given natural numbers r, 1, 3N(r,l) so that if N > N(r} 1) and the integers {1,2,...,N} are partitioned into r sets, one of these contains an /-term arithmetic progression. Erdos and Turân realized that this result would follow if it were the case that any subset of {1,2,..., N} comprising ON elements contains an /-term arithmetic progression provided N is sufficiently large. This result, conjectured in 1936 [ETI] was proved by E. Szemeredi in 1973 [Szl], and was reproved using ergodic theoretic methods in 1976 [Fui]. The theorems of van den Waerden and Szemeredi illustrate the two sides of Ramsey theory: coloring or partition results, and density results. The latter are clearly stronger than the former, and the proofs are typically more recondite. The role of ergodic theory in density theorems of this type stems from the fact that in a number of situations theorems about patterns found in sets having "density" bounded from below are equivalent to theorems about the "return", or "recurrence", patterns for measure preserving transformations acting on a measure space. It would appear that the ubiquity of certain patterns in the combinatorics of large sets reflects the phenomenon of recurrence in ergodic theoretic contexts and the latter has to be studied to gain insight into the former. We shall be examining three different contexts for recurrence results in ergodic theory. We shall mention the combinatorial (Ramsey theoretic) equivalents of these results, although we shall have to refer the reader elsewhere for the proof of the equivalence. Our purpose here is to display the common features of these recurrence phenomena. We shall find that in each setting there is a notion of rigidity and the notion of a special system constructed from scratch by a finite succession of rigid extension. For these systems (distal and quasi-distal) it will be possible to deduce recurrence properties directly, the key tool being Ramsey theory of the van der Waerden sort. For arbitrary systems we shall find a method to "factor out" the "independent" component, reducing the recurrence property