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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.

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The ergodic theorems can be seen as improvements of the Poincaré recurrence theorem, because they relate the time average with the space average. However, the Poincaré theorem is treated first because it is not hard to prove and it is very nice. – subshift Mar 22 '11 at 19:33
This isn't very related to your question, but I warmly recommend Einsiedler's & Ward's book too ("Ergodic Theory with a view towards Number Theory"). – Mark Mar 22 '11 at 19:56
The Poincare recurrence theorem has counterintuitive (or paradoxical) consequences in the development of statistical mechanics. See the discussion of the theorem and why it is famous in Petersen's Ergodic Theory (p. 34 ff). Imagine a wall divides an empty chamber in two and a gas is pumped into one side. Remove the wall and the gas diffuses. Would you expect the gas at a future point to all recollect in one half of the chamber (infinitely often)? What does the Poincare recurrence theorem say? – KConrad Mar 22 '11 at 21:09
The proof of Poincare recurrence is intuitive and quantitative. The first property for the Birkhoff ergodic theorem is debatable, the second property wrong. Also the assumption of being measure preserving is more natural than ergodicity! Hence, the Poincare recurrence theorem is the better theorem. – Helge Mar 23 '11 at 4:25

11 Answers 11

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

  • we can choose $n$ from some "nice" set;
  • it is possible to observe "multiple" recurrence;
  • there are "many" $n$ for which the result holds.

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.

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That the Poincare recurrence theorem is just the tip of the iceberg for more general multiple recurrence theorems is a good example. One role for multiple recurrence theorems can be found in the book review for Furstenberg's Recurrence in Ergodic Theory and Combinatorial Number Theory at (But note the typo in the heading of the review: Princeton is not in the state of New Yersey.) – KConrad Mar 23 '11 at 2:01
The start of the notes at by Soundararajan, which I learned about from an answer by Frank Thorne, traces a path from the Poincare recurrence theorem to multiple recurrence theorems to applications in additive combinatorics. – KConrad Mar 25 '11 at 8:17
Does the celebrated Green-Tao Theorem dependences on the result of multiple recurrence (Furstenberg's Theorem)? – Po C. Apr 4 '11 at 10:54
The Green-Tao theorem relies on Szemeredi's theorem, which is logically equivalent to Furstenberg's proof of the aforementioned multiple recurrence theorem in ergodic theory. – Mark Apr 16 '11 at 23:07

If you are number-theoretically inclined, you might want to have a look at Harry Furstenberg, Poincare recurrence and number theory, Bulletin of the American Mathematical Society 5 (1981) 211-234, which seems to be freely available on the web. Among other things, Furstenberg uses Poincare recurrence to prove van der Waerden's Theorem on arithmetic progressions: given any partition of the integers into finitely many subsets, at least one of the subsets contains arbitrarily long arithmetical progressions.

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There are some excellent examples of the applications of Poincare recurrence in that article, but if you'll forgive my pickiness I don't think that the proof of van der Waerden's theorem is one of them. Furstenberg's proof of van der Waerden uses a generalisation of the Birkhoff recurrence theorem, not the Poincare recurrence theorem. – Ian Morris Mar 24 '11 at 9:48
Guess I skimmed it too lightly. Glad to see confirmation that the article's relevant to the question, even if I got the details wrong. Thanks. – Gerry Myerson Mar 24 '11 at 11:13

First of all Poincare's Recurrence Theorem is important historically because it poses a serious obstacle to modeling an ideal gas in a box as a mechanical system consisting of hard spheres bouncing around. See The wikipedia article on the second law of thermodynamics but the basic idea is that such a mechanical system preserves a natural volume on its phase space (Liouville's theorem) but this is in contradiction (by Poincare's Theorem) with the fact that entropy decreases along all trajectories (assuming that entropy is a continuous function in phase space).

Also, I like this proof that any harmonic function of a compact Riemannian manifold is constant:

Take the gradient flow of your harmonic function and notice that it's volume preserving (more or less by the definition of harmonicity since divergence of the gradient is zero). Now apply Poincare's Recurrence Theorem so that you now have a dense set of recurrent orbits. However since this is a gradient flow, the value of the function decreases strictly along any non-singular orbit, this means all the recurrent orbits are singular so that the gradient has a dense set of zeros. Conclusion: the function is constant.

Last but not least Poincare's Recurrence Theorem is more or less just the Pigeonhole Principle (finite space + lots of stuff = lots of overlap) which is certainly a fundamental mathematical fact even if not so many theorems follow directly from it. See Terrence Tao's course notes on Ergodic Theory where he emphasized this point very beautifully.

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+1 for harmonic functions! – Margaret Friedland Feb 16 '12 at 21:50

This may not be a very dynamical application but still very interesting. Poincaré recurrence theorem was used in this paper

to show that every amenable left-orderable group is locally indicable.

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An interesting application of Poincare recurrence theorem is provided by Masur-Veech theorem on the unique ergodicity of almost every interval exchange maps. Indeed, very roughly speaking, a proof of this result goes like this. There is a natural renormalization dynamics of interval exchange maps known as Rauzy-Veech induction. By construction of an appropriate invariant finite mass measure (sometimes called Masur-Veech measure), it follows from Poincare recurrence theorem that almost every interval exchange map is recurrent with respect to the Rauzy-Veech induction and this information can be shown to imply unique ergodicity.

For more details on this, see this survey by J.-C. Yoccoz.

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The Poincaré recurrence theorem is sometimes useful because of the way it translates into recurrence in metric spaces. For example, a corollary of the Poincaré theorem is that for a measure-preserving transformation of a separable metric space - which need not be continuous - almost every point is recurrent in the topological sense. To see this, choose a sequence which is dense in the metric space, and consider the cover of the metric space by balls of radius $1/n$ around points in this sequence. By the Poincaré theorem almost every point belonging to one of these balls returns to that ball infinitely often, and hence returns to within distance $2/n$ of itself. Now take the intersection over $n$ to see that almost every point is recurrent.

A more general corollary which is also sometimes useful is the following: if $T \colon X \to X$ is measure-preserving, and $f$ is a measurable function from $X$ to a separable metric space, then $\liminf_{n \to \infty} d(f(T^nx),f(x))=0$ almost everywhere. This result naturally can be very useful in circumstances where one knows that a function is measurable, but no more. An example which springs to mind is the proof of Theorem 15 in "A formula with some applications to the theory of Lyapunov exponents" by Avila and Bochi, where the Poincaré theorem is applied to prove the recurrence of the measurable splittings in the multiplicative ergodic theorem.

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Instead of comparing the Poincare recurrence theorem with ergodic theorems one should rather look at the underlying notions of conservativity and ergodicity in the general context of a measure class preserving action of a countable group. If the Poincare theorem says precisely that actions with a finite invariant measure are conservative, it is somewhat misleading to identify ergodicity with presence of an ergodic theorem (as this is the case for a very limited class of actions only).

In terms of the ergodic decomposition of an action, ergodicity (of course) means that there is only one ergodic component which coincides with the whole action. On the other hand, conservativity means that there are no discrete ergodic components (i.e., ones which coincide just with action orbits) - indeed, any wandering set obviously gives rise to discrete ergodic components, and it is not hard to see that the converse is also true. From this point of view ergodicity is a strengthening of conservativity.

However, there are several classes of dynamical systems (actions), for which conservativity and ergodicity are equivalent, i.e., any system from this class is either ergodic or completely dissipative (all ergodic components are discrete $\equiv$ the whole state space is the union of translates of a certain ``fundamental domain"). This phenomenon is called "Hopf dichotomy", the most famous example of which (precisely the one originally studied by Hopf) is the case of geodesic flows on negatively curved manifolds (and of the associated boundary actions).

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You can also use it to prove Borel-Harish Chandra theorem that an irreducible lattice in a semi-simple Lie group (without compact factors) is Zariski dense. Here is an elementary case: Take $G=SL_n({\mathbb R})$ and $\Gamma=SL_n({\mathbb Z})$. Let $H$ be the Zariski closure of $H$ and let $f: G \to SL_m({\mathbb R})$ be a representation such that the stabilizer of $v \in {\mathbb R}^m$ is exactly $H$. The existence of $f$ follows from a general fact known as Chevalley's theorem. This allows you to consider the projective space $P({\mathbb R}^m)$ as a $G$-space. Consider the map $G/ \Gamma \to P({\mathbb R}^m)$ defined by $g \Gamma \mapsto gv$ and let $\mu$ be the push-forward of the finite $G$-invariant measure on $G/ \Gamma $, which will be $G$-invariant. Now,using the Jordan canonical form, you can see that for any unipotent element $u \in \Gamma$ $u^n \cdot w$ converges to a point in projective space. One the other hand, by Poincare recurrence, the only way you can have this is when $\mu$ charges only one point. Now since $\Gamma$ is generated by one-parameter (discrete) unipotent subgroups, you will see that the orbit has to be only one point, hence $H=G$.

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The Poincare recurrence lemma is used in the theory of measure foliations (or measured geodesic laminations) on surfaces, to construct combinatorial approximations of the foliation, as explained in "Thurston's works on surfaces" by Fathi, Laudenbach, and Poenaru. In terminology that came a little after that book, one is constructing train track approximations. This is related to the comment of Matheus regarding Rauzy induction, and the comment of Stephane Laurent about Rokhlin towers.

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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

this article is really interesting results, and worth being read.

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

Containing several articles on this subject, with interesting results.

The following link contains a collection of articles accordingly:

I hope this information is useful.



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Poincaré's lemma also allows to define the induced transformation on a Borel set. And then to derive results on perodic approximations based on Kakutani-Rokhlin towers. (I wanted to post my remark as a comment but the "add coment" box does not appear here...)

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