Questions
---------


1. Is there a version of the Furstenberg-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*](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-20/issue-3/Extensions-of-ergodic-group-actions/10.1215/ijm/1256049780.full "Zimmer's Paper")
and/or
[*Extensions of ergodic actions and generalized discrete spectrum*](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-81/issue-3.P1/Extensions-of-ergodic-actions-and-generalized-discrete-spectrum/bams/1183536891.full "Extended Zimmer's Paper")).
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](https://www.imath.kiev.ua/%7Eskolyada/LY.pdf "PDF: 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](https://terrytao.wordpress.com/2008/03/05/254a-lecture-15-the-furstenberg-zimmer-structure-theorem-and-the-furstenberg-recurrence-theorem/ "Terry Tao's statement of Furstenberg-Zimmer Theorem").
But I could not find any precise statement for the non-invertible case.



Furstenberg-Zimmer theorem: non-invertible systems
Posted as requested - consult the book by Manfred Einsiedler and Tom Ward - "Ergodic Theory with a view towards number theory" - published in GTM, especially in ch 7.


