Furstenberg-Zimmer Theorem: non-invertible systems. - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T09:04:38Z http://mathoverflow.net/feeds/question/77007 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77007/furstenberg-zimmer-theorem-non-invertible-systems Furstenberg-Zimmer Theorem: non-invertible systems. André Caldas 2011-10-03T03:26:48Z 2011-10-04T19:24:34Z <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&amp;version=1.0&amp;service=UI&amp;handle=euclid.ijm/1256049780&amp;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&amp;version=1.0&amp;verb=Display&amp;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#77160 Answer by Asaf for Furstenberg-Zimmer Theorem: non-invertible systems. Asaf 2011-10-04T18:27:32Z 2011-10-04T18:27:32Z <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>