A very important theorem in mathematical physics is Poincaré’s recurrence theorem.

As you recall, this theorem states that given a dynamical system $(M , \phi , \mu)$ with $ \mu M < +\infty$, for every $A$ measurable the set of points of $A$ for which it exists some $N>0$ such that $\phi^Na \notin A$ for $n \geq N$ has measure zero.

This theorem applies, for example, to Hamiltonian dynamics, hence it has non trivial physical implications. For example a gas expanding in an empty room and then going back to the initial position.

What I would like to ask is: what are the physical implications of Poincaré theorem especially in statistical physics?

Is it possibile to use Poincaré’s recurrence in order to argue for a sort of circularity of time? In particular, can the evolution of the universe from its initial state be seen as dynamical system satisfying the hypotheses of Poincaré’s recurrence theorem?

I am writing it here on mathoverflow, because i) I see no difference between mathematics and physics ii) I am under the (possibly wrong) impression that not many physicists care much about Poincaré recurrence, while mathematicians do.

**edit** I have found this physics stackexchange quesiton of 8 years ago which asks one part of my question https://physics.stackexchange.com/questions/94122/is-poincare-recurrence-relevant-to-our-universe

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