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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](http://mathoverflow.net/questions/59222/importance-of-poincare-recurrence-theorem-any-example), you should mention that so people can see the other answers you have received and so people do not write duplicates.2011-03-26
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    Thanks for your suggestion.2011-03-26
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    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.2012-02-25

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