An anti-classification theorem for ergodic measure preserving transformations

Despite many notable advances the general problem of classifying ergodic measure preserving transformations (MPT) has remained wide open. We show that the action of the whole group of MPT's on ergodic actions by conjugation is turbulent in the sense of G. Hjorth. The type of classifications ruled out by this property include countable algebraic objects such as those that occur in the Halmos-von Neumann theorem classifying ergodic MPT's with pure point spectrum. We treat both the classical case of Z as well as the case of general countable amenable groups.