Groups and Dynamics Seminar

A classical theorem of Dye, extended by Ornstein and Weiss, states that any two free p.m.p. actions of (infinite) amenable groups are orbit equivalent. In other words, the relation of being orbit equivalent is trivial for this class of actions. The picture becomes much more interesting, however, if one additionally asks that the cocycles associated with such an orbit equivalence satisfy some integrability conditions. For instance, it has been shown that, in certain cases, the entropy of the action and the growth of the group must be preserved. On the other hand, not much is known about the positive direction -- that is, given a pair of actions and an integrability condition, can one construct an orbit equivalence between the actions with cocycles satisfying the given condition? We completely resolve this question for the class of integer odometers by showing that any two of them are sub-$L^1$-orbit equivalent. Based on a joint work with Spyros Petrakos.