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.
Best Answer
Part of the importance of the Poincare recurrence theorem is in the follow-up questions it legitimizes. Knowing that for any set $A$ of positive measure we can find a positive $n$ such that $\mu(A \cap T^{-n} A) > 0$, one could ask whether
One example of "nice" is "a square": one can always find a positive $n$ such that $\mu(A \cap T^{-n^2}A) > 0$. See for example theorem 2.1 in part 6 of these notes.
An example of "multiple" is that one can always find positive integers $m$ and $n$ such that $\mu(A \cap T^{-n}A \cap T^{-m}A \cap T^{-(m+n)}A) > 0$. To prove this, iterate the Poincare recurrence theorem. A more involved example of "multiple" is given by requiring that $m = n$ in the previous expression.
Lastly, an example of "many" is given by "syndetic": it follows from Khintchine's recurrence theorem (theorem 3.3 in Petersen's "Ergodic Theory") that the set of such $n$ has bounded gaps.