Two weeks ago (October 29), Jean-Christophe Yoccoz, Artur Avila and I talked about the action on the moduli space of Abelian differentials and the spectral gap of the related unitary representations. The outcome of our conversation was the following result:

Theorem 1 There are Teichmüller curves (associated to square-tiled surfaces) with complementary series.

Since the short proof of this simple result depends on some facts from the theory of unitary representations (e.g., Ratner’s results about rates of mixing), we’ll divide this post into 5 sections:

• the first 4 sections covers some well-known useful facts about unitary representation such as Bargmann’s classification, examples of (regular) unitary representations coming from dynamical systems and Ratner’s estimates of rates of mixing; the basic reference for the facts stated in these sections is Ratner’s paper “The rate of mixing for geodesic and horocyclic flows” (besides the references therein).
• the last section contains the proof of theorem 1.

For the specialists, we can advance a few keywords of the proof of theorem 1: by a recent theorem of J. Ellenberg and D. McReynolds, given any finite index subgroup of the congruence subgroup , we can find a square-tiled surface whose Teichmüller curve is ; in particular, it suffices to find a subgroup such that the regular representation of on has complementary series; however, the existence of such subgroups can be easily derived from certain cyclic covering constructions plus a “reverse Ratner estimate” argument.

Remark 1 After we performed this cyclic covering construction, Pascal Hubert and Nicolas Bergeron informed us that this procedure was already known by A. Selberg. In fact, Selberg’s argument (see the subsection \textrm{“Petites valeurs propres I. Critere geometrique d’existence”} of this book project of N. Bergeron) and our argument are completely equivalent: Selberg uses that sufficiently small eigenvalues () of the Laplacian on lead to complementary series and we use that slow decay of correlations of the geodesic flow of lead to complementary series, which are two equivalent ways to state the same fact in view of Ratner’s paper.

Continuing ERT2, we’ll discuss other results related to the convergence of polynomial ergodic averages. Given a probability space , there are two notions of convergence of functions defined on . The first one is norm convergence. Given , let denote the space of functions such that

We say that the sequence converges in the -norm if there exists such that

The other is pointwise convergence: a sequence converges pointwise if there are and a set such that and

These notions relate to each other in the following way.

Theorem 1 If and converges in the -norm to , then there is a subsequence which converges pointwise to .

Proof: Look at Rudin’s book Real and Complex Analysis.

Theorem 1 might infer that norm convergence is stronger than pointwise convergence. This is not the case. In fact, we are interested in proving convergence along the whole sequence, so that norm convergence does not guarantee pointwise convergence. These are two distinct notions and neither of them is stronger than the other: they are just different!

Exercise 1 Consider the sequence of functions , , defined in the following way: given , ,

Prove that, for any , converges in the norm to the zero function but does not converge pointwise.

Given a mps and , , consider the sequence of functions

Von Neumann’s Theorem proves -norm convergence. A more general and deeper result is Birkhoff’s Ergodic Theorem, which is one of the main pilars in Ergodic Theory.

Theorem 2 (Birkhoff, 1931) Given any , the sequence converges pointwise to a -invariant function .

Furstenberg’s Theorem on -norm convergence of polynomial ergodic averages (see ERT2) induces a natural question: does norm/pointwise convergence holds for polynomial ergodic averages of -functions? This was solved partially (and in a very satisfactory way) by the Fields medalist Jean Bourgain.

Theorem 3 (Bourgain, 1988) Let be a mps and a polynomial such that , for every , and . Then, for every , , the limit

converges pointwise to a function .

The case remained open until 2005, when Daniel Mauldin and Zoltan Buczolich published the paper Divergent Square Averages in which they construct a mps and a function for which the quadratic ergodic averages do not converge pointwise.

After Bourgain’s paper, other beautiful results were published using his sharp estimate methods. One of them proves pointwise convergence along the prime numbers.

Theorem 4 (Wierdl, 1988) Let be a mps and represent the set of prime numbers. For every , , the limit

converges pointwise to a function .

At first sight, this theorem is very restrictive: the prime numbers form a very special sequence. The fact is that, in constrast to polynomials, prime numbers have a randomic distribution, so that methods used to prove prime number’s ergodic theorems may be applied to other non-structured situations. In addition, they help to better understand the dichotomy between structure and ramdomness (see ERT2). The work of Ben Green and Terence Tao about the existence of arbitrarily long arithmetic progressions in the prime numbers is an important example of this situation (Terence Tao received his Fields medal in part because of this result).

We end this note with a recent result of Elon Lindenstrauss about the convergence of ergodic averages on amenable groups, which generalizes Birkhoff’s Ergodic Theorem.

Theorem 5 (Lindenstrauss, 2001) Let be a locally compact amenable group acting on a probability space and a tempered Flner sequence. For any , there is a -invariant such that

for -almost every , where denotes the (unique) left Haar probability on .

Remark 1 The interested reader not used to the above notions is encouraged to see this note of Alexander Gorodnik and Vitaly Bergelson.

Previous posts: ERT0, ERT1, ERT2.

As proved in ERT1, the existence of ergodic averages implies recurrence results for measure-preserving system (from now on, denoted by mps). A natural question is to ask about some kind of generalized ergodic averages and its implications in recurrence. By generalized ergodic averages we mean expressions like

where is a sequence of positive integers. Von Neumann’s Theorem shows that convergence holds if , where are positive integers: just apply the result to the transformation and the function . In this post, we prove that the same result holds if , where is a polynomial such that , for every . We can assume, without lost of generality, that . In fact, and satisfies the required condition.

Theorem 1 (H. Furstenberg) If is a mps and is a polynomial such that , for every , and , then the limit

converges in for every .

Again, this theorem is Hilbertian in nature and follows from a more general version for unitary operators.

Theorem 2 If is a unitary operator on a Hilbert space and is a polynomial such that , for every , and , then the sequence of operators

converges pointwise in norm.

Proof: The idea is the same as in Von Neumann’s Theorem: we look for an orthogonal decomposition such that the behaviour of is understood in each component. will represent the structured component of and the randomic one in the following sense:

• the long-time behaviour of elements is (almost-)periodic.
• the long-time behaviour of elements of self-amortizes and converges to zero.

Unfortunately, the decomposition of Von Neumann’s Theorem does not work here. In fact, let be periodic with respect to , say , . If we write , ,

converges to , because

This means that every periodic point of has a structured behaviour with respect to . For this reason, let

Exercise 1 Prove that the set of for which the sequence converges is a closed subspace of .

By Exercise 1, the sequence converges whenever . By linearity, it remains to prove convergence for . Such subspace is characterized by

This follows from Von Neumann’s Theorem: if is the decomposition with respect to , , then is equal to and its orthogonal complement is given by

which proves (1). This means that for every and for every degree-one polynomial , . The proof will be complete if we show that the same happens for larger-degree polynomials. By induction, suppose that

for every polynomial such that , and . Consider such that , and . We wish to reduce the convergence to one of the form , with (which we know to be true, by the induction hypothesis). This is done with the use of Van der Corput’s Trick (see this lecture of Terry Tao for a broader discussion on this trick).

Theorem 3 (Van der Corput Trick) If is a bounded sequence such that

for every , then

Exercise 2 Prove the above theorem. (Hint: this is Theorem 2.2 of this survey of Vitaly Bergelson.)

We’re done if the sequence , , satisfies the conditions of Theorem 3. In fact, as is unitary,

where is a polynomial of smaller degree, and so (2) is satisfied. This concludes the proof of Theorem 2.

The method used above is one of the main principles Ergodic Ramsey Theory: the dichotomy between structure and randomness, decomposing the object of study into these two components. Usually, we first define the structured one, in terms of the desired ergodic averages, so that convergence follows almost directly from the definition. Its orthogonal complement is the randomic component and convergence along it is proved using Van der Corput like theorems. For a further discussion on this dichotomy, the reader is referred to this paper of Terence Tao. Observe that the same method applies to prove the following

Theorem 4 If is a unitary operator on a Hilbert space and is a polynomial such that , for every , and , then the sequence of operators

converges pointwise in norm.

Now it’s time to obtain the recurrence consequences (which, as expected, will be stronger than those in ERT1). Let be the orthogonal projection. We’ll proceed exactly as in Proposition 6 of ERT1, except that the notation will be heavier.

Proposition 5 Let be such that . Then and .

Proof: Consider the subspaces , . By approximation, if each projection of into satisfies and , the same happens to . Fix and consider the function . Then (Exercise 3) and . Because minimizes the distance of to , we have . In addition, if we had , then

implying that for some . Integrating, we conclude

a contradiction.

Theorem 6 If is a mps, is a polynomial such that , for every , , and such that , then the set

is syndetic.

Proof: If , then from (2) the expression converges to as . Since

Theorem guarantees the conclusion.

Previous posts: ERT0, ERT1.

Le prix Fermat 2009 de Recherche en Mathématiques a

By taking a look at the French blog Images des mathématiques, I learned that Elon Lindenstrauss and Cédric Villani were awarded the 2009 edition of the Fermat prize. Below you can see a description of the prize (taken from Fermat prize page):

[Update (Oct. 28): Brief descriptions (in French) of the works of E. Lindenstrauss and C. Villani (also taken from Fermat prize page) were included]

The FERMAT PRIZE rewards research works in fields where the contributions of
Pierre de FERMAT have been decisive :

* Statements of Variational Principles
* Foundations of Probability and Analytical Geometry
* Number theory.

The spirit of the prize is focused on rewarding the results of researches accessible to the greatest number of professional mathematicians within these fields.

The amount of the Fermat prize has been fixed at 20 000 Euros. The FERMAT prize is
awarded once every two years in Toulouse ; the eleventh award will be announced in
October 2009.

Winners of the preceding editions: A. Bahri, K.A. Ribet (1989) – J.-L. Colliot-Thélène
(1991) – J.-M. Coron (1993) – A.J. Wiles (1995) – M. Talagrand (1997) – F. Bethuel,
F. Hélein (1999) – R. L. Taylor, W. Werner (2001) – L. Ambrosio (2003) – P. Colmez,
J.F. Le Gall (2005) – C. Khare (2007).

————————————————————————————————————-

été attribué à :

Elon LINDENSTRAUSS (Princeton University)

pour ses travaux en théorie ergodique et leurs applications en théorie des nombres,

et à

Cédric VILLANI (ENS de Lyon)

pour ses contributions à la théorie du transport optimal et à l’étude des équations d’évolution non linéaires.

Hello! Yesterday I visited IMPA’s webpage and I discovered that my friend Enrique Pujals won the TWAS 2009 prize in Mathematics. Congratulations to him!

Below you find brief descriptions of TWAS objectives and Enrique’s works (both taken from TWAS webpage):

“TWAS is an autonomous international organization, based in Trieste, Italy, that promotes scientific excellence for sustainable development in the South.“

“Enrique Pujals, a native Argentinean who serves as professor at the Institute of Pure and Applied Mathematics (IMPA) in Rio de Janeiro, Brazil, has been named the winner of the 2009 TWAS Prize in mathematics for his contribution to develop a theory about robust dynamics and about the role of homoclinic bifurcation as a universal mechanism to describe the way to produce very rich and complex dynamics.“

Let’s define the settings: we will always consider a measure-preserving system , meaning that is a probability space and is a measurable transformation that preserves :

If , then there exists such that . This is easy to see because the family , , satisfies a stationary condition

So, if , two of the sets have positive-measure intersection. In fact, if this is not the case, then

a contradiction. We then get the original Poincaré’s Recurrence Theorem:

Theorem 1 (PRT) If , then there exists such that .

Remark 1 The modern statements of PRT are: if , then a.e. point returns to . This means that , which obviously implies the above theorem.

This proves more: call a set syndetic if it has bounded gaps, i.e., if there exists such that .

Exercise 1 Prove that if , then the set is syndetic.

For further discussions about PRT, the reader may consult this paper of Vitaly Bergelson. Altought its simplicity, this is a remarkable result. It implies, for example, that almost every has infinitely many ’s in its decimal representation, and the same happens for any finite sequence of digits.

As preserves , it defines a unitary operator on the Hilbert space by , for simplicity denoted from now on as . With this notation, if , then

where is the inner product in , so

such that the sequence , , may give more general results than PRT. This is what happens.

Theorem 2 (Von Neumann) If , then the sequence , , converges in .

This theorem, also known as Mean Ergodic Theorem, is in fact a spectral theoretical result and a more general version holds, given by

Theorem 3 If is a unitary operator on a Hilbert space , then the sequence of operators , , converges pointwise in norm to the orthogonal projection onto the subspace of -fixed elements .

Proof: When is unitary, . From the general orthogonal decomposition

we obtain

For , the convergence is obvious. If , then

By approximation and applying the triangle inequality, the same happens in , which concludes the proof.

Remark 2 Being, as we said, Hilbertian in nature, Theorem 2 also holds when .

Exercise 2 Under the same conditions of Theorem 3, prove that the same conclusion happens for a sequence such that .

Let’s show how to use these convergences to obtain recurrence results.

Proposition 4 Let be such that . Then and .

Note that satisfies the above conditions.

Proof: Consider the function . Then and . Because minimizes the distance of to , we have . In addition, if we had , then , such that . Integrating, we conclude

a contradiction.

Exercise 3 Using the above proposition, prove that if , then the set is syndetic. (Hint: if , then converges to as .)

So, expressions of the type , from now on called ergodic averages, are important when dealing with recurrence. This will be our main interest in the next posts. For another perspective on Von Neumann’s Theorem and related results, the reader is referred to this Terence Tao’s lecture.

Previous posts: ERT0.

Hi! Today I’m posting an expanded version of an informal talk (directed to PhD students at IMPA) I gave in January 21, 2005 about the so-called ADM mass in General Relativity and its applications. The spirit of the talk was strongly inspired by the famous article “The unreasonable effectiveness of Mathematics in the Natural Sciences” of the Nobel laureate Eugene Wigner. In fact, my goal was to present a beautiful chapter of the interaction between Differential Geometry (Mathematics) and General Relativity (Physics).

-Introduction-

The “unreasonably effective” relationship between Mathematics and Physics is widely known: for instance, it was the lack of an adequate language to understand the so-called Classical Mechanics (Physics) lead Isaac Newton and Gottfried Leibniz (independently) to the foundations of Differential and Integral Calculus.

The bulk of the current discussion is the non-technical presentation of the beautiful interaction between Mathematics and Physics appearing in the definition of the ADM mass in General Relativity and its application (by Richard Schoen) to the solution of the Yamabe problem in Differential Geometry. More precisely, our general plan is the following:

• in the next section, we’ll see how Mathematics helped Physics with the rigorous description of a global definition of mass in General Relativity; in order to do so, we’ll briefly review some of the history of Newtonian Mechanics, Maxwell’s theory of Electromagnetism and Einstein’s (Special and General) theory of Relativity; after that, we’ll introduce Schwarzschild solution to Einstein’s equation (modelling a black hole) and the concept of ADM mass (named after the three physicists Arnowitt, Deser and Misner); finally, we’ll illustrate the effectiveness of Mathematical tools in Physics with some comments about R. Schoen and S.T. Yau proof of the positivity of the ADM mass (via some arguments from the geometry of minimal surfaces);
• in the last section, we’ll illustrate the effectiveness of Physical tools in Mathematics with a rough sketch of R. Schoen solution to the Yamabe problem (via the positivity of the ADM mass).

Before closing the introduction, let me say that this particularly beautiful interaction between Differential Geometry and General Relativity certainly motivates the following extension of the title of Wigner’s article:

“The unreasonable effectiveness of Mathematics in the Natural Sciences and vice-versa“

Also I would like to acknowledge my friend Fernando Coda Marques who patiently explained me the ideas and technical details appearing in R. Schoen’s solution to Yamabe problem and its relationship with the ADM mass in General Relativity. Of course, this talk is an outcome of our discussions (although the mistakes and errors below are my sole responsibility, of course).

Note by C.M.: After talking with my friend Yuri Lima (a 3rd year PhD student at IMPA, currently at Columbus, Ohio, working with Vitaly Bergelson), I proposed to him to write some posts for this blog about the topics of his interest. He accepted my invitation and started a post (see below) containing an overview of his plans. Enjoy it!

We begin with a question: what conditions a set must have to possess arbitrarily long arithmetic progressions? Well, if this set is very sparse (such as the powers of ), there is no chance for such thing. On the other hand, a set with arbitrarily large intervals trivially satisfies it. Althought the precise condition is not known, there is one of great interest which is sufficient. Define the density of as

(Here, stands for the cardinality of the set ). Such limit not always exists, so that it is more convenient to consider the upper density of :

This is a well-defined number between and . In , Erdös and Turán conjectured that if , then has arbitrarily long arithmetic progressions. It remained wide open until , when Roth proved that such sets contain progression of lenght three. Later, Szemerédi, in , proved that they also have progressions of lenght four and, finally, in he solved the conjecture.

Theorem (Szemerédi, ). If has positive upper density, then it contains arbitrarily long arithmetic progressions.

His proof is a very hard combinatorial argument and relies in the Szemerédi’s Regularity Lemma (which we intend to talk in the future).

Breakthrough and the birth of a new area.

Two years later, Hillel Furstenberg gave another proof of Szemerédi’s Theorem, based on an deep analysis of the structure of general measure-preserving systems, known as Furstenberg’s Structural Theorem (see this lecture of Terence Tao for a discussion of this result in the case of distal systems). This gave birth to a new area, called Ergodic Ramsey Theory. As the name suggests, Ergodic Ramsey Theory deals with the use of Ergodic Theory (and related areas, such as topological dynamics) machinery to prove Ramsey Theory (and related combinatorial) problems.

In the next posts, we plan to discuss this interaction. Here is a sketch:

1. Poincaré’s Recurrence Theorem.

2. Classical Von Neumann’s Theorem.

3. Polynomial Von Neumann’s Theorem.

4. Multiple Poincaré’s Recurrence Theorem.

5. Furstenberg’s Correspondence Principle.

6. Szemerédi’s Theorem.

7. Topological Dynamics and Van der Waerden’s Theorem.

8. Two simple models of measure-preserving systems: compact and weak mixing systems.

9. Compact and weak-mixing extensions.

10. A glance at Furstenberg’s Structural Theorem and the proof of Multiple Poincaré’s Recurrence Theorem.

11. Generalized ergodic avergares: and a.e. convergence.

12. Green-Tao’s Theorem on the existence of arbitrarily long arithmetic progressions of primes.

The posts will be tagged by ERT+(number of the lecture).

Hi! A few months ago, my friend Carlos Gustavo (Gugu) Moreira posted at the IMPA’s preprint server an article entitled “Geometric properties of the Markov and Lagrange spectra” explaining the proofs of his results on the Markov and Lagrange spectrum (see this previous post for an introduction of these spectra and the statements of Gugu’s results). Today, we’ll discuss the dynamical aspects of Gugu’s results. However, before starting the discussion of this preprint, let me take the opportunity to congratulate Gugu: he managed to post the first version of his interesting article at the same time of his first son’s birth! Moreover, let me thank my wife Aline Gomes Cerqueira whose nice comments helped me to clarify my thoughts about Gugu’s argument (and also helped the improvement of this work by her PhD advisor) .

[Update: Ops, I forgot to congratulate Gugu also for his recent UMALCA Prize 2009!]

[Update (August 11, 2009): The mistakes pointed out by Yuri are now fixed.]

A few weeks ago (from July 5 to July 11) I attented the interesting conference Dynamische Systeme held at the MFO (Oberwolfach). The talks concerned several topics in Dynamical Systems and Symplectic Geometry, so that it was a nice opportunity to learn some useful techniques and theorems from (directly or indirectly related) diverse subareas.

Also, the organizers (Jean-Christophe Yoccoz, Hakan Eliasson and Eduard Zehnder) gave an opportunity to speak about a work in progress (joint with C. Gugu Moreira and E. Pujals) towards the so-called Smale conjecture. The title of the talk was “ density of hyperbolicity for Benedicks-Carleson toy models” and the goal of the talk was the proof of Smale’s conjecture in a “toy” version of the Henon dynamics introduced by M. Benedicks and L. Carleson in their seminal paper “The dynamics of Henon map“.

More precisely, Smale’s conjecture claims that hyperbolic (Axiom A) diffeomorphisms of compact surfaces are dense. Firstly, let me make some comments about this conjecture: from the works of M. Shub and R. Mane we know that Smale’s conjecture is false in higher dimensions (i.e., among diffeomorphisms of compact manifolds with dimension ). Also, S. Newhouse showed that Smale’s conjecture is also false when we replace by , : in fact, he used clever arguments involving dynamically defined Cantor sets in order to prevent hyperbolicity in a robust way via the so-called homoclinic tangencies (this is the known as Newhouse phenomena). However, after a recent work of Gugu (Moreira), we know that Newhouse’s argument (using the existence of stable intersections of Cantor sets) doesn’t work in the topology. In particular, this gives some hope towards the validity of Smale’s conjecture. For a more detailed exposition of these topics, one can see my previous posts (part I and part II) on Gugu’s theorem.

In this direction, Gugu (Moreira), Enrique (Pujals) and I decided to test some ideas around Smale’s conjecture in some toy models. Of course, we wanted some nice class of examples such that: the class is rich enough so that the Newhouse phenomena occurs in the world, but the class is simple enough so that one can apply Gugu’s results (and some geometrical arguments) to prove Smale’s conjecture in the world. Inspired by the Benedicks-Carleson work, we consider their toy models for the Henon dynamics. This class is quite promising because the geometry is simple (it is a skew-product of the quadratic family and some contracting maps on a fan of lines), although it is rich enough to exhibit the Newhouse phenomena (by the works of R. Ures).

During the talk, I discussed our ideas (related to the notion of dynamical critical points of F. Rodriguez-Hertz and E. Pujals) and how they eventually lead to a proof of Smale conjecture in this particular class.

For more details and references, please see the report of my talk appearing in this preliminary version of the Oberwolfach Report N. 32 containing the reports of all lectures of the conference (these reports are going to appear in the Oberwolfach Reports in a near future).

Finally, you can see some photos of this workshop here.