Questions

1. Is there a version of Furstenber-Zimmer Theorem for non-invertible measure preserving systems?

2. Where can I find it?

3. What is the precise statement?

Background

In many works that reference the Furstenberg-Zimmer Theorem, the theorem itself is not stated. Authors usually cite the works of Furstenberg (The structure of distal flows and/or Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions) and Zimmer (Extensions of ergodic group actions and/or Extensions of ergodic actions and generalized discrete spectrum). The point is that in many places, the theorem is being used for non-invertible systems. This happens, for instance in On Li-Yorke Pairs, where the systems are assumed to be surjective, but not necessarily invertible. In this paper, for the proof of Theorem 2.1, the authors use Furstenber-Zimmer Theorem.

As far as I understood, Zimmer's work deals with group actions. That is, invertible systems. And for Furstenberg's Ergodic behaviour of diagonal measures [...], he deals with regular measure preserving systems.

Unfortunately, Furstenberg and Zimmer (obviously) did not call their result the Furstenberg-Zimmer Theorem. In fact, it seems to me that Furstenberg didn't even call it a theorem. :-P

I could find a precise statement of the theorem for the invertible case at a Terry Tao's post. But I could not find any precise statement for the non-invertible case.