Welcome to MathOverflow

Question

4

2

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.

edited Dec 5 2010 at 20:43

asked Dec 5 2010 at 18:12

Juan Valdez
571●2●7

1

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 2010 at 18:20

Hi KConrad!I would also references to where I can find open problems in this matter Thanks! – Juan Valdez Dec 5 2010 at 18:41

1

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 2010 at 20:41

Steve Huntsman · Accepted Answer · 2010-12-05 23:47:29Z UTC

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).

Joseph O'Rourke · Answer 2 · 2010-12-05 21:11:10Z UTC

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.

Poincaré recurrence; Time Return

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

2 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.