Furstenberg-Zimmer Theorem: non-invertible systems. - MathOverflow most recent 30 from http://mathoverflow.net2013-06-20T09:04:38Zhttp://mathoverflow.net/feeds/question/77007http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/77007/furstenberg-zimmer-theorem-non-invertible-systemsFurstenberg-Zimmer Theorem: non-invertible systems.André Caldas2011-10-03T03:26:48Z2011-10-04T19:24:34Z
<h2>Questions</h2>
<ol>
<li><p>Is there a version of Furstenber-Zimmer Theorem for
non-invertible measure preserving systems?</p></li>
<li><p>Where can I find it?</p></li>
<li><p>What is the precise statement?</p></li>
</ol>
<h2>Background</h2>
<p>In many works that reference the Furstenberg-Zimmer Theorem,
the theorem itself is not stated.
Authors usually cite the works of Furstenberg
(<em>The structure of distal flows</em>
and/or
<em>Ergodic behavior of diagonal measures and a
theorem of Szemerédi on arithmetic progressions</em>)
and Zimmer
(<a href="http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.ijm/1256049780&page=record" rel="nofollow" title="Zimmer's Paper"><em>Extensions of ergodic group actions</em></a>
and/or
<a href="http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183536891" rel="nofollow" title="Extended Zimmer's Paper"><em>Extensions of ergodic actions and generalized discrete spectrum</em></a>).
The point is that in many places, the theorem is being used
for non-invertible systems.
This happens, for instance in
<a href="http://www.imath.kiev.ua/~skolyada/LY.pdf" rel="nofollow" title="PDF: On Li-Yorke Pairs">On Li-Yorke Pairs</a>,
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.</p>
<p>As far as I understood, Zimmer's work deals with
<em>group actions</em>.
That is, invertible systems.
And for Furstenberg's <em>Ergodic behaviour of diagonal measures [...]</em>,
he deals with <em>regular measure preserving systems</em>.</p>
<p>Unfortunately, Furstenberg and Zimmer (obviously) did not call their result
<em>the Furstenberg-Zimmer Theorem</em>.
In fact, it seems to me that
Furstenberg didn't even call it a <em>theorem</em>. :-P</p>
<p>I could find a precise statement of the theorem
for the invertible case at a
<a href="http://terrytao.wordpress.com/2008/03/05/254a-lecture-15-the-furstenberg-zimmer-structure-theorem-and-the-furstenberg-recurrence-theorem/" rel="nofollow" title="Terry Tao's statement of Furstenberg-Zimmer Theorem">Terry Tao's post</a>.
But I could not find any precise statement for the non-invertible case.</p>
http://mathoverflow.net/questions/77007/furstenberg-zimmer-theorem-non-invertible-systems/77160#77160Answer by Asaf for Furstenberg-Zimmer Theorem: non-invertible systems.Asaf2011-10-04T18:27:32Z2011-10-04T18:27:32Z<p>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.</p>