# Poincaré recurrence; Time Return

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.

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See Karl Petersen's book "Ergodic Theory". After proving the recurrence theorem he comments that the expected return times are enormous and proves a simple theorem (of M. Kac) about the expected value of the return time for a two-urn experiment. – KConrad Dec 5 '10 at 18:20
Hi KConrad!I would also references to where I can find open problems in this matter Thanks! – Juan Valdez Dec 5 '10 at 18:41
The book by Barreira "Dimension and recurrence in hyperbolic dynamics" has a chapter on recurrence, with several modern results and lots of references. I don't think it explicitly lists any open problems, but it can give you an idea of the current research being done in this area. – Pablo Shmerkin Dec 5 '10 at 20:41

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

M. S. Baptista et al., "Kolmogorov–Sinai entropy from recurrence times". Phys. Lett. A 374, 1135 (2010)

the obvious cite

L. Barreira and B. Saussol, "Product structure of Poincar\'e recurrence". Ergodic Th. Dyn. Sys. 22, 33 (2002)

and finally

G. Robinson and M. Thiel, "Recurrences determine the dynamics". Chaos 19, 023104 (2009).

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I like this 2006 paper by Luis Barreira (cited in the Wikipedia article), which I encountered when pursuing this MO question on billiard trajectories (which you might visit): "Poincaré recurrence: Old and new" (in Zambrini, Jean-Claude, XIVth International Congress on Mathematical Physics, World Scientific, pp. 415–422.) You can get a preliminary version from citeseer here. Here is the Abstract:

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.

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