Poincaré recurrence; Time Return - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T04:36:02Z http://mathoverflow.net/feeds/question/48381 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48381/poincare-recurrence-time-return Poincaré recurrence; Time Return Juan Valdez 2010-12-05T18:12:55Z 2010-12-05T23:47:29Z <p>Hello everybody! Recently I start a reading of a survey by Benoit Saussol, AN INTRODUCTION TO QUANTITATIVE POINCARE RECURRENCE IN DYNAMICAL SYSTEMS, I am interested in references (Papers) Basics Poincare Recurrence. I know that this survey is already basic, but wanted to know more references of this kind and I would also references to where I can find open problems in this matter. I would also references to applications of this theory in other fields in mathematics.</p> http://mathoverflow.net/questions/48381/poincare-recurrence-time-return/48392#48392 Answer by Joseph O'Rourke for Poincaré recurrence; Time Return Joseph O'Rourke 2010-12-05T21:05:16Z 2010-12-05T21:11:10Z <p>I like this 2006 paper by Luis Barreira (cited in <a href="http://en.wikipedia.org/wiki/Poincar%25C3%25A9_recurrence_theorem" rel="nofollow">the Wikipedia article</a>), which I encountered when pursuing <a href="http://mathoverflow.net/questions/46414/how-quickly-will-billiard-trajectories-cluster" rel="nofollow">this MO question on billiard trajectories</a> (which you might visit): "Poincaré recurrence: Old and new" (in Zambrini, Jean-Claude, <em>XIVth International Congress on Mathematical Physics</em>, World Scientific, pp. 415â€“422.) You can get a preliminary version from <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.101.3744" rel="nofollow">citeseer here</a>. Here is the Abstract:</p> <blockquote> <p>The classical theorem of Poincaré on recurrence only gives information of qualitative nature. On the other hand it is clearly a matter of intrinsic difficulty and not of lack of interest that less is known concerning the quantitative behavior of recurrence. Here we discuss recent developments that include the almost everywhere coincidence between the recurrence rate and the pointwise dimension in the case of hyperbolic dynamics. We also discuss the almost product structure of recurrence, which closely imitates the product structure provided by the families of stable and unstable manifolds as well as the almost product structure of hyperbolic measures.</p> </blockquote> http://mathoverflow.net/questions/48381/poincare-recurrence-time-return/48405#48405 Answer by Steve Huntsman for Poincaré recurrence; Time Return Steve Huntsman 2010-12-05T23:47:29Z 2010-12-05T23:47:29Z <p>Joseph's answer is the first place I would (and did) look for information on this topic. However there are a couple of recent ancillary references along these lines that may be helpful. For instance, see </p> <p>M. S. Baptista et al., "Kolmogorovâ€“Sinai entropy from recurrence times". Phys. Lett. A 374, 1135 (2010)</p> <p>the obvious cite </p> <p>L. Barreira and B. Saussol, "Product structure of Poincar\'e recurrence". Ergodic Th. Dyn. Sys. 22, 33 (2002)</p> <p>and finally</p> <p>G. Robinson and M. Thiel, "Recurrences determine the dynamics". Chaos 19, 023104 (2009).</p>