Maximal abelian subalgebras of the hyperfinite factor, entropy and ergodic theory

Given a free ergodic action of a discrete abelian group G on a measure space (X, mu), the crossed product L-infinity (X, mu) x G contains two distinguished maximal abelian subalgebras. We discuss what kind of information about the action can be extracted from the positions of these two subalgebras inside the crossed product algebra.