Stationary Dynamical Systems and a Szemerédi Theorem for the Free Group

If G is a group acting on a space X and \mu is a probability measure, we can talk of a measure \nu on X as being "stationary" relative to \mu, the case of an invariant measure being a very special case. Much of ergodic theory can be carried over to this situation. In particular the phenomenon of recurrence - even multiple recurrence - can be established. Using a "correspondence principle" one can derive combinatorial consequences,. In analogy with Szemeredi's theorem for sets of integers of positive density one can formulate the notion of a non-negligible subset of the free group (on finitely many generators) necessarily containing arbitrarily long non-trivial geometric progressions.