Existentially closed measure-preserving actions of free groups

I will discuss a joint work with Alexander Berenstein and Ward Henson, in which we show that the theory of probability algebras with two automorphisms has a model completion, which moreover has quantifier elimination and is stable. We also exhibit two non-isomorphic (but approximately isomorphic) models of the model completion.
More generally, we give a sufficient set of conditions for the axiomatizability (in continuous logic) of the existentially closed actions of a free group on a separably categorical, stable structure.
I will also mention a number of open questions.