# Importance of Poincaré recurrence theorem? Any example?

Recently I am learning ergodic theory and reading several books about it.

Usually Poincaré recurrence theorem is stated and proved before ergodicity and ergodic theorems. But ergodic theorem does not rely on the result of Poincaré recurrence theorem. So I am wondering why the authors always mention Poincaré recurrence theorem just prior to ergodic theorems.

I want to see some examples which illustrate the importance of Poincaré recurrence theorem. Any good example can be suggested to me?

Books I am reading: Silva, Invitation to ergodic theory. Walters, Introduction to ergodic theory. Parry, Topics in ergodic theory.

A few day ago I put this question in mathoverflow. I now realize that it would also be appropriate to ask here since my question is quite general.

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Since you already asked the same thing on MathOverflow, you should mention that so people can see the other answers you have received and so people do not write duplicates. – Douglas Zare Mar 26 '11 at 19:31
Thanks for your suggestion. – Choi Mar 26 '11 at 19:41
A physics application: If you have a physical system for which the assumptions of the Poincaré recurrence theorem hold, then we expect that at some point it will violate the 2nd law of thermodynamics by returning to a low-entropy state after visiting higher-entropy states. There is a nice discussion of this in a recent popularization by Sean Carroll, called From Eternity to Here. – Ben Crowell Feb 25 '12 at 23:51

There is a whole field of research focused on the consequences of Poincare recurrence theorem.

The link below provides one of the first articles, critical theory at the beginning of this

Two professionals who work extensively with these issues are: Suassol Benoit and Luis Barreira, below is a link to the page the first author:

http://www.math.univ-brest.fr/perso/benoit.saussol/articles.html

Containing several articles on this subject, with interesting results.

I hope this information is useful.

Best,

Eduardo.

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A very simple reason exists for that. If you had a transformation that is not recurrent, then you do not have an invariant measure and therefore do not have an ergodic measure. This is just an example.

I'm sorry if my English is wrong, but I'm not a native speaker.

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