## Questions

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

Where can I find it?

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.

followthe proof of Theorem 8.3 you can define adistal serieswhich can also be defined transfinitely, allowing one to reach at amaximal distal factor. Furstenberg continues:These notions were referred to in the Introduction, but we shall not actually make use of them in the sequel. – André Caldas Oct 3 '11 at 15:22according to Furstenberg Theorem (see blah blah)without giving a reference to where it is stated. :-( – André Caldas Oct 3 '11 at 18:42