Importance of Poincaré recurrence theorem? Any example?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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

http://arxiv.org/pdf/math/0606232

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

Answer 5

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.

Answer 6

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.

Answer 7

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.

Answer 8

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

http://www.springerlink.com/content/r415482r0112g418/

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:

http://www.math.univ-brest.fr/perso/benoit.saussol/articles.html

Containing several articles on this subject, with interesting results.

The following link contains a collection of articles accordingly:

http://www.im.ufrj.br/~arbieto/ensino/20091/20091.html

I hope this information is useful.

Best,

Eduardo.

Answer 9

This is an old question but I don't see the obvious answer, so here we go.

A huge field of research in mathematical physics during the XIXe century revolved around giving explicit solutions to the equations of classical mechanics, using brute computation. At that time, finding a new first integral in the equations of motions of some physical system was a sure path to academic fame. If sufficiently many first integrals are found, the system is integrable. If moreover the motion is constrained to a bounded domain of the phase space , it is quasi-periodic and thus both regular and recurrent. The long term behavior of the system is thoroughly described.

There was certainly some hope at first to show the stability of the solar system, or at least the three body problem, by finding these first integrals. Huge efforts went into that line of research. A century later, we know that the three body problem is not integrable in general and integrability is a pretty rare property of dynamical systems. In particular it is not stable by perturbation.

But wait, we know that Earth won't suddenly fly towards Pluto and stay there forever. Actually, we are pretty sure that it will come back to its current position, and this follows from Poincaré recurrence theorem, once you note that the Liouville measure is left invariant by the motion. No need to explicitly solve the equations of motion, and the result is so general that it applies to all hamiltonian systems in restriction to a compact level of energy. And the proof is short and elegant! To be pedantic, this was a paradigm shift and the birth of a new method in the field of mathematical physics, best described by Poincaré himself in his numerous books.

We now consider the Poincaré recurrence theorem as marking the birth of a new mathematical discipline called ergodic theory, with striking applications to arithmetic, Lie groups, foliations, moduli spaces etc. To the point that people seem to have forgotten its celestial origins.

Answer 10

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 $\Gamma$ 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 $H=G$.

Answer 11

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).

Answer 12

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...)