Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
    
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 '11 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 '11 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 '11 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 '11 at 10:43
    
@Asaf: Tao assumes it in Lecture 1: terrytao.wordpress.com/2008/01/08/254a-lecture-1-overview 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 '11 at 17:43

1 Answer 1

up vote 2 down vote accepted

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.

share|improve this answer
1  
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 '11 at 19:09

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.