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.
A very simple reason exists for introducing "Poincaré recurrence theorem just prior to ergodic theorems." 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.