## 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.

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 Not explicitly calling it a theorem is not a crime :) They may have called it a proposition, and others promoted it to a theorem. Or even a lemma... But I like this question. – David Roberts Oct 3 2011 at 3:54 Thank you for your comment, David. Furstenberg's paper is too difficult for me. It is possible I am totally wrong, but what I found closer to the theorem was a comment after definition 8.4. He says that if you follow the proof of Theorem 8.3 you can define a distal series which can also be defined transfinitely, allowing one to reach at a maximal 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 2011 at 15:22 @David: Only now I understood that I sounded like complaining about Furstenberg. I am not. He was looking for something else. But since it is not stated as a theorem in his paper, I am complaining about authors that say: according to Furstenberg Theorem (see blah blah) without giving a reference to where it is stated. :-( – André Caldas Oct 3 2011 at 18:42 @Andre,I didn't see where Tao assumed that T is invertible, anyways,a good recent ref. would be Manfred's book-Chapter 7,and maybe even Furstenberg's book. For every system there's a compact (Borel, standrad, whatever kind of space you like) extension making it bi-invarient (see Manfred's book) and you usually can analyze this situation and then project it down to the original system without too much of a trouble (because the fibers are compact). For example in the SZ theorem, you can basically work with one-sided shift and not two-sided, this is enough for proving SZ by Furstenberg's method. – Asaf Oct 4 2011 at 10:43 @Asaf: Tao assumes it in Lecture 1: terrytao.wordpress.com/2008/01/08/… For the reference you pointed (Manfred), I think this is exactly what I needed. I didn't know about this book. If I may, I'd like to suggest you to post it as an answer. :-) Manfred's book is what I should be reading!! Thank you very very much! – André Caldas Oct 4 2011 at 17:43

## 1 Answer

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.

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Thank you very much, Asaf! This is my new canonical reference for ergodic theory. I am ordering this book. I had a pick, and what I really needed was Theorem 7.21. It is stated for not necessarily invertible "Borel Probability Spaces" (Definition 5.13). – André Caldas Oct 4 2011 at 19:09