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.

-Introduction-

Let be an unitary representation of , i.e., is a homomorphism from into the group of unitary transformations of the complex separable Hilbert space . We say that a vector is a -vector of if is . Recall that the subset of -vectors is dense in .

The Lie algebra of (i.e., the tangent space of at the identity) is the set of all matrices with zero trace. Given a -vector of and , the Lie derivative is

where is the exponential map (of matrices).

Exercise 1 Show that for any -vectors of and .

An important basis of is

Exercise 2 Show that , and . Furthermore, , and where is the Lie bracket of (i.e., is the commutator).

The Casimir operator is on the dense subspace of -vectors of . It is known that for any -vectors , the closure of is self-adjoint, commutes with on -vectors for any and commutes with for any .

Furthermore, when the representation is irreducible, is a scalar multiple of the identity operator, i.e., for some and for any -vector of .

In general, as we’re going to see below, the spectrum of the Casimir operator is a fundamental object.

-Bargmann’s classification-

We introduce the following notation:

Note that satisfies the quadratic equation when .

Bargmann’s classification of irreducible unitary says that the eigenvalue of the Casimir operator has the form

where falls into one of the following three categories:

• Principal series: is purely imaginary, i.e., ;
• Complementary series: and is isomorphic to the representation , where belongs to the Hilbert space ;
• Discrete series: .

In other words, belongs to the principal series when , belongs to the complementary series when and belongs to the discrete series when for some natural number .

Note that, when (i.e., belongs to the complementary series), we have .

-Some examples of unitary representations-

Given a dynamical system consisting of a action (on a certain space ) preserving some probability measure (), we have a naturally associated unitary representation on the Hilbert space of functions of the probability space . More concretely, we’ll be interested in the following two examples.

Hyperbolic surfaces of finite volume. It is well-known that is naturally identified with the unit cotangent bundle of the upper half-plane . Indeed, the quotient is diffeomorphic to via

Let be a lattice of , i.e., a discrete subgroup such that has finite volume with respect to the natural measure induced from the Haar measure of . In this situation, our previous identification shows that is naturally identified with the unit cotangent bundle of the hyperbolic surface of finite volume with respect to the natural measure .

Since the action of on and preserves the respective probability measures and (induced from the Haar measure of ), we obtain the following (regular) unitary representations:

and

Observe that is a subrepresentation of because the space can be identified with the subspace . Nevertheless, it is possible to show that the Casimir operator restricted to -vectors of coincides with the Laplacian on . Also, we have that a number belongs to the spectrum of (on ) if and only if belongs to the spectrum of on .

Moduli spaces of Abelian differentials. An interesting space philosophically related to the hyperbolic surfaces of finite volumes are the moduli spaces of Abelian differentials: we consider the space of Riemann surfaces of genus equipped with Abelian (holomorphic) 1-forms of unit area modulo the equivalence relation given by biholomorphisms of Riemann surfaces respecting the Abelian differentials. By writing , we can make act naturally on by

See this excellent survey of Anton Zorich for more details about the action on the moduli spaces of Abelian differentials.

The case of is particularly clear: it is well-known that is isomorphic to the unit cotangent bundle of the modular curve. In this nice situation, the action has a natural absolutely continuous (w.r.t. Haar measure) invariant probability , so that we have a natural unitary representation on .

After the works of H. Masur and W. Veech, we know that the general case has some similarities with the genus 1 situation, in the sense that, after stratifying by listing the multiplicities of the zeroes of and taking connected components of these strata, this action has an absolutely continuous (with respect to a natural “Lesbegue” class induced by the period map) invariant probability . In particular, we get also an unitary representation on .

-Rates of mixing and size of the spectral gap-

Once we have introduced two examples (coming from Dynamical Systems) of unitary representations, what are the possible series (in the sense of Bargmann classification) appearing in the decomposition of our representation into its irreducible factors.

In the case of hyperbolic surfaces of finite volume, we understand precisely the global picture: the possible irreducible factors are described by the rates of mixing of the geodesic flow on our hyperbolic surface.

More precisely, let

be the 1-parameter subgroup of diagonal matrices of . It is not hard to check that the geodesic flow on a hyperbolic surface of finite volume is identified with the action of the diagonal subgroup on .

Ratner showed that the Bargmann’s series of the irreducible factors of the regular representation of on can be deduced from the rates of mixing of the geodesic flow along a certain class of observables. In order to keep the exposition as elementary as possible, we will state a very particular case of Ratner’s results (referring the reader to Ratner’s paper for more general statements). We define equipped with the usual inner product . In the sequel, we denote by

the intersection of the spectrum of the Laplacian with the open interval ,

with the convention when ) and

We remember the reader that the subset detects the presence of complementary series in the decomposition of into irreducible representations. Also, since is a lattice, it is possible to show that is finite and, a fortiori, . Because essentially measures the distance between zero and the first eigenvalue of on , it is natural to call the spectral gap.

Theorem 2 For any and , we have

 

• when ;
• when , and is not an eigenvalue of the Casimir operator ;
• otherwise, i.e., when and either or is an eigenvalue of the Casimir operator .

Here is a constant such that is uniformly bounded when varies on compact subsets of .

In other words, Ratner’s theorem relates the (exponential) rate of mixing of the geodesic flow with the spectral gap: indeed, the quantity roughly measures how fast the geodesic flow mixes different places of phase space (actually, this is more clearly seen when and are characteristic functions of Borelian sets), so that Ratner’s result says that the exponential rate of mixing of is an explicit function of the spectral gap of .

In the case of moduli spaces of Abelian differentials, our knowledge is less complete than the previous situation: as far as I know, the sole result about the “spectral gap” of the representation on the space of zero-mean -functions (wrt to the natural measure ) on a connected component of the moduli space is:

Theorem 3 (A. Avila, S. Gouezel, J.C. Yoccoz) The unitary representation has spectral gap in the sense that it is isolated from the trivial representation, i.e., there exists some such that all irreducible factors of in the complementary series are isomorphic to the representation with .

In the proof of this result, Avila, Gouezel and Yoccoz proves firstly that the Teichmüller geodesic flow (i.e., the action of the diagonal subgroup on the moduli space of Abelian differentials is exponentially mixing (indeed this is the main result of their paper) and they use a reverse Ratner estimate to derive the previous result from the exponential mixing.

Observe that, generally speaking, the result of Avila, Gouezel and Yoccoz says that doesn’t contain all possible irreducible representations of the complementary series, but it is doesn’t give any hint about quantitative estimates of the “spectral gap”, i.e., how small can be in general. In fact, at the present moment, it seems that the only situation where one can say something more precise is the case of the moduli space :

Theorem 4 (Selberg/Ratner) The representation has no irreducible factor in the complementary series and it holds .

In fact, using the notation of Ratner’s theorem, Selberg proved that . Since we already saw that , the first part of the theorem is a direct consequence of Selberg’s result, while the second part is a direct consequence of Ratner’s result.

In view of the previous theorem, it is natural to make the following conjecture:

Conjecture (J.-C. Yoccoz) The representations don’t have complementary series.

While we are not going to discuss this conjecture here, we see that the goal of the theorem 1 is to show that this conjecture is false if we replace the invariant natural measure by other invariant measures supported on smaller loci.

-Teichmüller curves with complementary series-

After this long revision of the basic facts around unitary -representations, we focus on the proof of theorem 1. Before starting the argument, let us briefly recall the definition of Teichmüller curves of square-tiled surfaces and the definition of the associated representation. Firstly, we remind that a square-tiled surface is a Riemann surface obtained by gluing the sides of a finite collection of unit squares of the plane so that a left side (resp., bottom side) of one square is always glued with a right side (resp., top) of another square together with the Abelian differential induced by the quotient of under these identifications. It is known that square-tiled surfaces are dense in the moduli space of Abelian differentials (because is square-tiled iff the periods of are rational) and the -orbit of any square-tiled surface is a nice closed submanifold of which can be identified with for an appropriate choice of a finite-index subgroup of (in the literature is called the Veech group of our square-tiled surface). Furthermore, the Teichmüller geodesic flow is naturally identified with the geodesic flow on . In other words, the Teichmüller flow of the -orbit of a square-tiled surface corresponds to the geodesic flow of where is a finite-index subgroup of (hence is a lattice of ). In the converse direction, J. Ellenberg and D. McReynolds recently proved that:

Theorem 5 (Ellenberg and McReynolds) Any finite-index subgroup of the congruence subgroup gives rise to a Teichmüller curve of a square-tiled surface.

For a nice introduction to the Teichmüller flow and square-tiled surfaces see this survey of A. Zorich. In any case, the fact that the -orbits of square-tiled surfaces are identified to the unit cotangent bundle of permit to introduce the regular unitary -representation on the space of zero-mean -functions (wrt the natural measure on ) of .

In this notation, the theorem 1 says that there are square-tiled surfaces such that the representation associated to its -orbit has irreducible factors in the complementary series.

In view of Ellenberg and McReynolds theorem, it suffices to find a finite-index subgroup such that has complementary series. As we promised, this will be achieved by a cyclic covering procedure. Firstly, we fix a congruence subgroup such that the corresponding modular curve has genus , e.g., . Next, we fix a homotopically non-trivial closed geodesic of after the compactification of its cusps and we perform a cyclic covering of (i.e., we choose a subgroup ) of high degree such that a lift of satisfies . Now, we select two small open balls and of area whose respective centers are located at two points of belonging to very far apart fundamental domains of the cyclic covering , so that the distance between the centers of and is . Define , . Take and the zero mean parts of and .

We claim that has complementary series, i.e., (i.e., ) for a sufficiently large . Actually, we will show a little bit more: is arbitrarily close to for large (i.e., the spectral gap of can be made arbitrarily small).

Indeed, suppose that there exists some such that for every . By Ratner’s theorem 2, it follows that

for any . On the other hand, since the distance between the centers of and is , the support of is disjoint from the image of the support of under the geodesic flow for a time . Thus, , and, a fortiori,

Putting these two estimates together and using the facts that and , we derive the inequality

In particular, , a contradiction for a sufficiently large .

Remark 2 As we pointed out in the introduction, the algebraic part of the proof of theorem 1 was already known by Selberg: in fact, as pointed out to us by N. Bergeron and P. Hubert, the same cyclic covering construction giving arbitrarily small first eigenvalue of (i.e., arbitrarily small spectral gap) was found by Selberg and the reader can find an exposition of this argument in the subsection 3.10.1 of Bergeron’s book project. In particular, although the “difference” between Selberg argument and the previous one is the fact that the former uses the first eigenvalue of the Laplacian while we use the dynamical properties of the geodesic flow (more precisely the rates of mixing) and a reverse Ratner estimate, it is clear that both arguments are essentially the same.

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.

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).

————————————————————————————————————-

Le prix Fermat 2009 de Recherche en Mathématiques a

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).

-Mathematics helping Physics-

Newtonian Mechanics. In 1642, Sir Isaac Newton proposed a theory (nowadays called Newtonian Mechanics in his honor) whose central goal was the description of the trajectory of a given body via certain Ordinary Differential Equations (ODEs): for instance, given a certain body and denoting by the time variable, its position at time , its velocity at time and its acceleration at time , we have

Furthermore, Newton’s second law allows to understand the trajectory of a body via its mass and the net external force :

In fact, the knowledge of the mass and the force permits to use Newton’s second law to compute the acceleration (and, a posteriori, the velocity and the position ).

Of course, this is a very crude resume of Newton’s theory (which contains several other fundamental principles such as the law of conservation of energy), but one of the basic philosophy is that Newton’s theory gives a precise description about how the presence of an external force interferes with the movement of a given object.

However, even after the consolidation of Newtonian Mechanics, a serious drawback was the fact that this theory was unable to answer the following question:

What is the origin of the forces?

Newton’s law of universal gravitation. In 1687, after important contributions of Galileo Galillei, Isaac Newton answered the previous question with the introduction of Newton’s law of universal gravitation: it says that two objects of masses and at distance are mutually attracted by a force (called gravitational force) whose intensity is

where is a numerical constant (of known value) called gravitational constant.

Again, Isaac Newton obtained a great success because his universal gravitation theory was very accurate (specially concerning its applications to the movement of celestial bodies, such as the Sun, Earth, etc.). However, Newton’s universal gravitation theory left open the following pertinent question:

How the gravitational forces are transmitted?

Before passing to this important question, let us see another (even more serious) problem with the Newtonian Mechanics which became apparent with the advent of Maxwell’s Electromagnetic theory

Maxwell’s Electromagnetism. In 1895, after the advent of new phenomena (namely, electricity and magnetism) without any explanation in Newton’s theory lead the Scottish physicist James C. Maxwell to the development of his Electromagnetic theory. In crude terms, this theory says that electricity and magnetism can be understood as an unique object called electromagnetic wave whose dynamics is described by a set of 4 Partial Differential Equations (known as Maxwell equations). An astonishing consequence of Maxwell equations is the fact that electromagnetic waves (such as light) in the vacuum always travels with constant speed (called speed of light). Evidently, Maxwell theory was a great achievement (due to its accurate description of electromagnetic phenomena), but, after the initial excitement, a serious dilemma between Newton and Maxwell theories emerged, as we are going to see below.

A paradox between Newton and Maxwell. Let’s see what happens when we try to measure the velocity of light of a laser beam using Newton and Maxwell theories. From Newton’s point of view, the speed of light measured by an inertial referential, i.e., a referential moving with constant velocity should be (depending whether the referential moves in an opposite direction or in the same direction of the laser beam). But, from Maxwell’s point of view, the speed of light should be in any inertial referential!

A frustrating aspect of this paradox is the fact that Newton theory is very accurate in Celestial Mechanics and Maxwell theory is very accurate in Electromagnetism. Thus, the solution of this paradox became a basilar problem in Physics. Here, an young brilliant German man working at the patent office in Bern (Swiss) comes to the rescue: of course, I’m talking about Albert Einstein.

Special Relativity of Albert Einstein. In 1905, Albert Einstein solved the paradox between Newton and Maxwell with the introduction of the notion of space-time: the basic idea is that the time and the space can’t be considered as two distinct entities because they are an unique object. In fact, using the point of view of space-time, Einstein postulated that any object moves at the speed of light in space-time! At a first glance, you can imagine that this is a misprint since you’re probably sit down comfortably in your chair reading this post and it is clear that you’re not moving at the speed of light. However, you should read Einstein’s postulate more carefully: he says that we are moving at light speed in space-time. In other words, denoting by your velocity vector in space-time, it has coordinates where are the spatial part of and is the temporal part of and Einstein’s postulate says that

In particular, while you may not be moving in space (i.e., ), the fact that you’re moving with the speed of light in space-time simply means that , i.e., from your point of view the time passes with the speed of light. An immediate (and interesting) consequence of this postulate is the fact that the time slows down when we move fast in space (i.e., decreases if we increase ): a popular way of quoting this consequence is “time is relative” (since the way you feel its passage changes depending on the modulus of your spatial velocity).

Furthermore, Einstein formulated his famous formula

relating mass and energy (this is the analog of in Newtonian Mechanics), etc. In resume, Einstein solved the paradox between Newton and Maxwell via a profound (and highly non-trivial) modification of Newton’s laws. However, the attentive reader noticed that we intentionally skipped one question: although Einstein solved the previous paradox, his Special Relativity theory doesn’t explain how the gravitational forces are transmitted. As we are going to see below, the final answer to this subtle question came only after 10 years of hard work of Einstein.

General Relativity of Albert Einstein. In 1915, Einstein formulated the crucial postulate explaining the transmission of gravitational forces: the presence of mass changes the geometry of the space-time — more precisely, it becomes “curved” around massive objects. A nice pictorial description of this phenomena can be found here. Roughly speaking, the idea is that the space-time is a thin rubber leaf: when we place a bowling ball in this thin rubber leaf, it will become curved in its vicinity and any small object (a tennis ball say) nearby will be attracted to the bowling ball (i.e., the attraction [gravitational force] occurs due to the curvature of the rubber leaf [space-time]). Of course, the analogy space-time versus rubber leaf is only a crude comparison (with plenty of defects), but it illustrates the basic idea in a first approach.

Mathematically speaking, the correct way to formalize this intuition passes through the notion of Riemannian (and Lorentzian) metrics. Again roughly speaking, a Riemannian metric is a way of measuring distances and angles in general (maybe curved) ambients (called manifolds). As it is well-known in Differential Geometry, any Riemannian metric naturally introduces a notion of curvature. Although the definition of curvature (tensor) is rather technical in general, it is not hard to understand the notion of curvature of plane curves (such as Formula 1 circuits). In fact, suppose that Michael Schumacher is driving his Ferrari during the Brazilian Grand Prix (at Interlagos, Sao Paulo). Here you can see a picture of this circuit. There you can identify three special places: ‘Reta Oposta’ (meaning Opposite Straight Road since it is opposite to the straight road containing the starting grid), ‘S do Senna’ (named after Ayrton Senna) and ‘Bico do Pato’. Suppose that Schumacher keeps his Ferrari at constant speed (say 300 km/h). Of course, he will not have any trouble with ‘Reta Oposta’ because it is ’straight’, however it will be difficult (even to him!) to keep the control of his Ferrari during the ‘curved’ places such as ‘S do Senna’ and ‘Bico do Pato’. But, what’s the difference between ‘Reta Oposta’ and ‘S do Senna’/'Bico do Pato’ that lead us to say that ‘Reta Oposta’ is less curved than ‘S do Senna’ and ‘Bico do Pato’? Well, looking at the corresponding velocity vectors, we see that it is (almost) constant at ‘Reta Oposta’ while it changes drastically at ‘S do Senna’/'Bico do Pato’ (i.e., the modulus of the acceleration vector is almost zero in ‘Reta Oposta’ while it is large at ‘S do Senna’). Thus, this means that the modulus of the acceleration vector gives a nice measure of curvature of this circuit. Notice that we obliged Schumacher to perform the entire circuit with constant speed (300 km/h) in order to measure the curvature at different places. The curious reader may ask why we did so and the answer is simple: although ‘Bico do Pato’ is more curved than ‘S do Senna’ (as you can see in the picture), it would be fairly easy to Schumacher to perform ‘Bico do Pato’ with a speed of 20 km/h while it would be incredibly difficult to him to perform ‘S do Senna’ with a speed of 400 km/h (i.e., a variable speed leads to a non-intuitive notion of curvature). This explains (I guess) the notion of curvature of plane curves. In general, the differential geometers have a recipe to generalize the notion of curvature of plane curves to arbitrary Riemannian manifolds (i.e., manifolds with Riemannian metrics), but we’ll omit this technical part.

Coming back to Physics, Einstein’s General Relativity says that our universe is modeled by where is the space-time and is a Lorentzian metric. Here is a manifold of dimension 4 (where 3 dimensions corresponds to the space and 1 dimension corresponds to the time). However, in order to simplify our discussion, we’ll assume that (and is a product metric) for two reasons: this case is more easy to treat (since during most part of the discussion we can concentrate in the spatial part ) and a great part of the discussion can be generalized (with a little bit of technical work) to any . In this particular situation, Einstein’s equation relating Differential Geometry (curvature) and Physics (energy and momentum) is

where is the Ricci curvature of and is the stress-energy tensor. Unfortunately, due to the usual limitations of space and time (sorry for the implicit joke! ), I’ll omit a detailed explanation of the terms appearing in this beautiful formula, but I hope that the reader will remember its main feature: it unifies the geometrical and physical aspects of space-time in a single equation. The relevant fact here is: the knowledge of the geometrical and physical aspects of the space-time are encoded by the metrics satisfying Einstein’s equation (for instance, the ’straight lines’ of the metric geodesics are all possible trajectories in view of the minimal action principle, etc.)

In any case, once we arrived at Einstein’s equation, it is natural to ask about the existence of solutions. The first solution was discovered by K. Schwarzchild in 1916: it is the metric

on , where is a real parameter and is the standard Euclidean metric on . Physically speaking, this solution models a famous entity called black hole with a mass placed at the origin . An interesting property of these black holes is the fact that the 2-dimensional sphere of radius works as an event horizon: any object (e.g., light) entering the ball can’t escape back to (actually, after entering the event horizon, they are attracted towards the origin).

A nice feature of the Schwarzschild’s solution is the natural identification of the real parameter with the mass of the black hole: in fact, the deviation of geodesics (i.e., curvature) increases proportionally to . Of course, this leads to the following natural problem in General Relativity: can we define a ‘good’ notion of mass of arbitrary space-times?

ADM mass in General Relativity. In general, the definition of mass in General Relativity (i.e., a natural notion which is invariant under change of coordinates [inertial referentials]) is a delicate issue. Nowadays, as we are going to see, one has such a notion only for a point or the entire Universe, but we don’t have a consensus (besides the several recent efforts) about a natural notion of a fixed non-trivial part of the Universe (i.e., a ”local mass”).

In any case, since we have a good notion of mass in the case of Schwarzschild black holes (namely, the parameter ), it is clear that any natural notion of mass should coincide with in Schwarzschild case. On the other hand, in order to introduce a definition of mass, we’ll make the physically reasonable hypothesis (based on the accepted theory that the entire Universe started with the Big-Bang, etc.) that the whole mass of the Universe (or more precisely massive objects) are confined to a bounded (i.e., compact) region, so that the gravitational effects tend to decay at infinity. Technically speaking, we are assuming that the metric is asymptotically flat (i.e., we are supposing that there is a compact set such that is diffeomorphic to and tends to the Euclidean metric [at a certain rate] at infinity).

Under these assumptions, the physicists R. Arnowitt, S. Deser and C. Misner introduced (in 1960) the following definition of mass (called ADM mass in their honor)

inspired by variational arguments with the action functional of Einstein-Hilbert . Here, is the ‘’sphere at infinity” (i.e., the previous integral should be interpreted as a limit), is the area element of , is the metric written in geodesic coordinates, are the derivates of (while is the scalar curvature and is the volume element).

Of course, this is a good definition of mass in the sense that the Australian mathematician R. Bartnik proved that the ADM mass is independent of the choice of coordinates (i.e., inertial referentials) and, after a straightforward calculation, one can show that in the case of Schwarzschild’s black holes.

One of the central theorems about the ADM mass (in General Relativity) is:

Theorem 1 (Positive mass theorem) Let be an asymptotically flat Riemannian manifold of scalar curvature (at every point). Then, . Furthermore, if and only if and (i.e., the ADM mass is zero exactly for the vacuum space-time).

Remark 1 Of course, the term can have any sign (usually it has negative sign when the ”potential” energy surpasses the ”kinetic” energy) so that the positivity of the ADM mass is very far from obvious. On the other hand, the previous theorem says that when the local density of energy (measured by the scalar curvature) is non-negative everywhere (i.e., ).

The first proof of this theorem was given by R. Schoen and S.T. Yau in 1979 using variational methods based on the geometry of minimal surfaces. Therefore, this permits to say that the Mathematical tools helped the understanding of the notion of mass in General Relativity (Physics). However, since the proof of this beautiful result is beyond the scope of this post, we’ll pass to the next topic: how the Physics helps the advance of important Mathematical problems.

-Physics helping Mathematics-

Momentarily, we’ll apparently change the focus of our discussion in order to discuss the famous Yamabe problem. In simple terms, Yamabe problem concerns the existence of very ”round” metric in a given Riemannian manifold. More precisely, given a compact boundaryless Riemannian manifold of dimension , we search for a metric conformal to (i.e., where is a positive function) whose scalar curvature is constant. This problem was coined ‘Yamabe problem’ because H. Yamabe considered this question (as a preliminary step of a program to attack Poincare’s conjecture [whose complete solution was given by G. Perelman pursuing different techniques]) and he claimed in 1960 to have positively solved the problem, although, as it was pointed out by the Australian mathematician N. Trudinger, Yamabe’s solution was incorrect.

Despite a crucial error in his solution, Yamabe introduced a nice idea to attack the problem, namely, he observed that the equation is equivalent to solving a certain Partial Differential Equation (PDE) in terms of . More precisely, after changing the factor by (the exponent is chosen to get the simplest equation possible), one can perform some computations to see that is equivalent to

This PDE belongs to the well-known class of (non-linear) elliptic PDEs. At this point, Yamabe made his mistake: he believed that the existence of solutions of this PDE was a direct consequence of the theory of elliptic PDEs. However, this geometrically motivated elliptic PDE interestingly lies at the frontier of the theory of elliptic PDE (in other words, it is a critical PDE). More precisely, if the exponent of nonlinear term of the right-hand side of the equation were any number strictly less than , the usual (variational) methods of elliptic PDEs provide the desired solutions. But, since the exponent is exactly , we should work more. On the other hand, although the variational methods don’t solve directly the equation, it provides a simple criterion for the solvability of this PDE: it suffices to show that where

is the so-called Yamabe quocient of (and is the canonical sphere with the round metric of constant curvature ). Therefore, this reduces the Yamabe problem to the following problem: show that for any not globally conformal to .

Remark 2 Since is conformal to the standard Euclidean space by stereographic projection, we call conformally flat any manifold globally conformal to . In this notation, Yamabe problem is reduced to prove that non-conformally flat implies .

The case of and is not locally conformally flat (i.e., there is some region of where there are no conformal deformations of equal to the Euclidean metric) was treated by T. Aubin using adequate (depending on the Weyl tensor) local cut-offs of the functions

These functions are naturally associated to the problem because they realize the infimum of the expression defining . After a long series of computations, Aubin was able to show that in this situation, so that the Yamabe problem can be solved in the corresponding cases.

Therefore, this leaves the cases or locally conformally flat (but not globally conformally flat) open. Here, Richard Schoen had a brilliant idea: instead of performing local cut-offs of directly, he decided to glue these functions with certain Green functions of the Yamabe PDE (here is a term whose behaviour is equal to up to second order). After a long calculation, R. Schoen showed that

Hence, Yamabe problem can be positively solved in the remaining case once we can show that . Exactly at this moment, the Physics (General Relativity) comes to our rescue (another brilliant idea of Schoen): using the (properties of the) Green function (or more precisely its power ) as a conformal factor of , we can define a metric such that is asymptotically flat and . Thus, the positive mass theorem can be applied to conclude that (in fact, can’t happen since implies that is Euclidean, i.e., is globally conformally flat).

Now, after some more or less direct computations, R. Schoen shows an almost magical fact , so that the Yamabe problem is solved! In other words, the crucial point of the final part of Schoen’s argument is the use of the concept of ADM mass (Physics) to solve Yamabe problem (Mathematics).

Before closing the post, let me take the opportunity to say that, of course, this is not the end of the history of the fruitful (and unreasonably effective) interaction between Mathematics and Physics. In fact, some of examples of successful mutual feed-back are, for instance, Quantum Mechanics and Operator Algebra, Yangs-Mills theory and Donaldson construction of exotic structures of , String Theory and calculation of rational points of algebraic varieties in Enumerative Geometry. In any case, I hope you enjoyed this post! See you!

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.]

-Quick review of the Markov and Lagrange spectrum-

During this section, we review the discussion of the first section of the this post. Given an irrational number , Dirichlet’s theorem claims that the inequality

has infinitely many rational solutions (in fact, it is a simple exercise to the reader to check that this is a direct consequence of Dirichlet’s pigeonhole principle). Furthermore, Markov and Hurwitz proved that the inequality

has infinitely many rational solutions for all irrational and is the biggest constant with this property, namely, for every ,

has only a finite number of rational solutions.

Nevertheless, during the study of Diophantine properties of specific irrational numbers , it is interesting to introduce the function

assigning to each irrational number its best rational approximation constant . Observe that Khintchine’s theorem says that the set of irrational numbers with has full Lebesgue measure.

In any case, we can consider the Lagrange spectrum

formed by the collection of finite best constants .

Concerning the structure of , Markov (1879) showed that

where are explicit quadratic irrationals (i.e., , , i.e., by Lagrange theorem, has periodic continued fraction expansion). In particular, the beginning of is discrete. On the other hand, M. Hall (1947) showed that and G. Freiman (1975) determined the biggest half-line contained , namely, he proved that is

Therefore, it remains only to understand the structure of middle of . This is the content of Gugu’s preprint.

However, before entering this issue, let me mention that the arguments of Hall and Freiman use the study of arithmetic sums of dynamically defined Cantor sets. The interaction between the dynamically defined (regular) Cantor sets and the Lagrange spectrum occurs via the continued fraction expansion. We’ll explain this relationship later (when we introduce the dynamically defined Cantor sets associated to the Gauss map). In any case, we can reformulate the Lagrange spectrum in the language of Dynamical Systems as follows. We observe that, for any irrational number (where () is the usual continued fraction notation), it holds

where and . Indeed, this fact follows from the identity (which can be checked by induction). Keeping this notation in mind, we can give an alternative definition of the Lagrange spectrum:

Definition 1 Denote by the set of bi-infinite sequences of natural numbers. For each , we put

and we introduce the function given by

In this setting, the Lagrange spectrum is

where is the shift dynamics .

Of course, this alternative definition of the Lagrange spectrum motivates the introduction of the Markov spectrum :

Definition 2 .

Remark 1 admits the following arithmetic characterization:

Remark 2 It is possible to prove that and are closed subsets of such that .

A nice reference for the theory of continued fractions and the Markov and Lagrange spectrum containing the proofs of all facts quoted above is the book of Cusick and Flahive.

At this point, let us state the main theorems of Gugu’s preprint (concerning about the middle part of ).

-Statements of the main results-

The main three results of Gugu’s preprint are:

Theorem 3 Given , we have (where stands for the Hausdorff dimension of the set ). Furthermore, is a continuous and surjective function from to . Moreover,

• where is a continuous function from to ;
• ;
• .

Remark 3 A very interesting consequence of theorem 3 is the fact that is not Hölder continuous (with respect to any exponent ). Indeed, sends the subset of Hausdorff dimension onto the interval for every . Thus, if is -Hölder continuous for some , one would have for all , a contradiction (for a sufficiently small ).

Theorem 4 .

Theorem 5 The set of accumulation points of is a perfect set, i.e., .

In the sequel, we’ll content ourselves with the discussion of the first part of theorem 3 only (i.e., the continuity of ). The starting point of Gugu’s argument is inspired by the proofs of the theorems of M. Hall and G. Freiman, namely, one consider the dynamically defined Cantor sets related to the Gauss map. Then, the basic idea during the proof of the continuity properties of the function is the combination of some approximation arguments of the Lagrange spectrum by the arithmetic sum of two regular Cantor sets with the fact that the Hausdorff dimension of these arithmetic sums are well-behaved due to the so-called dimension formula. Now, let’s turn to the details.

-The dimension formula-

A fundamental tool in Gugu’s article is the so-called dimension formula (which will appear in a forthcoming paper by Gugu). Basically, it says that the Hausdorff dimension of the arithmetic sum of generic Cantor sets , is . In order to properly state this theorem, we introduce the notion of non essentially affine regular Cantor sets. We recall that a -regular (i.e., dynamically defined) Cantor set is the maximal invariant set of a transitive expanding map from a disjoint finite union of compact intervals to the real line verifying the Markov partition property (see this post for more details). Observe that, for every periodic point of period of , one can find a diffeomorphism of the convex hull of such that is affine in , where is the connected component of the domain of containing .

Definition 6 We say that is non essentially affine whenever we can find a periodic point (as above) such that the map verifies for some .

In other words, is non essentially affine if it is not possible to perform a conjugation of the dynamics so that it becomes affine near the corresponding Cantor set. Using this notion, Gugu will show (in a forthcoming article) the following theorem:

Theorem 7 (Dimension formula) If and are -regular Cantor sets and is non essentially affine, then

Remark 4 The proof of this result uses in a crucial way the previous work of Gugu and J.C. Yoccoz about the stable intersections of -Cantor sets. More precisely, Gugu plans to use the so-called Scale Recurrence Lemma of this paper to prove the dimension formula. In fact, the curious reader can consult the article of P. Shmerink (15 pages) for a complete proof (along these lines) of Gugu’s dimension formula under slightly stronger assumptions (of non-resonance).

Remark 5 The appearance of the term on the right-hand side of the dimension formula has a natural explanation: as we saw in a previous post about Marstrand’s theorem, the arithmetic sum is (essentially) the orthogonal projection of the product set into the diagonal of (making angle of with the -axis). Hence, the dimension formula says that, for generic Cantor sets, this projection has the correct dimension, i.e., (the Hausdorff dimension of ).

Assuming the validity of the dimension formula, we can begin the discussion of the proof of theorem 3.

-Regular Cantor sets associated to the Gauss map-

In Number Theory, the Gauss map is the fractional part of (this should not be confused with the Gauss map from the Differential Geometry). Of course, the iterates of are intimately related to the continued fraction algorithm. A central role in the subsequent arguments will be played by the regular Cantor sets associated to complete shifts via the Gauss map. More precisely, given a finite set () of finite sequences () of positive integers such that doesn’t start with for every , we introduce

Definition 8 The complete shift associated to is the subset of formed by the sequences obtained by concatenations of the elements for every .

Using the complete shift and the Gauss map, we can construct a regular Cantor set:

Definition 9 We denote by the Cantor set of real numbers whose continued fractions are sequences of , that is,

Exercise 1 Show that is a regular Cantor set. (Hint: For each , , consider the interval where and and define )

In view of the shape of the graph of the Gauss map, the following proposition is very natural:

Proposition 10 is non essentially affine.

Proof: For each (say ), let us denote by be the fixed point of . From the theory of continued fractions (see Cusick and Flahive), we know that

where is the -th rational approximation of (for ). It follows that is a positive solution of .

On the other hand, since is a Möbius transformation with an expanding fixed point , we can find a Möbius function such that , and is affine. This reduces our task to show that is not affine (because the second derivative of a non-affine Möbius transformation never vanishes).

Assuming that is affine, we see that is a common fixed point of and (since the point at infinity is a common fixed point of the affine maps and ) and, a fortiori, it is a common solution of the quadratic equations for . Because these two quadratic polynomials of are irreducible (since and are irrational numbers), they must coincide. In particular, this forces , a contradiction with .

Once we know that the regular Cantor sets are non essentially affine, we’ll try to combine this information with the dimension formula (for the arithmetic sum of Cantor sets) in order to show the continuity of . Note that this is a natural approach because the Lagrange and Markov spectra are naturally related to the values of a certain (sum) function over the orbits of bi-infinite sequences by the shift map. Before starting the discussion of this topic, let us close this section with some remarks and interesting facts.

Remark 6 Given a sequence , we define its transposition by . Of course, this notion extends to finite sets of sequences (via . In this notation, we have the classical fact about the continuants of the continued fractions associated to and (see the appendix of Cusick and Flahive). Using this fact, it is not hard to prove that .

Putting together the previous remark 6 with the proposition 10, we conclude:

Corollary 11 .

Remark 7 This corollary is related to the fact that (see the theorem 3).

-Lagrange/Markov spectrum of complete shifts-

Denote by where and and let be the shift operator. Recall that the Lagrange spectrum is and the Markov spectrum is , where and (see the definition 1 for the definition of and ).

Remark 8 If , then . In particular, if , we have that belongs to M. Hall’s ray (since ).

Therefore, since we are interested in the continuity properties of the function , the previous remark allows us to make the following assumption:

From now on, we’ll always assume that for every , and .

In our approach to and , it is convenient to introduce the Lagrange and Markov spectrum associated to a complete shift :

Definition 12 Let be a complete shift. Its Lagrange spectrum is and its Markov spectrum is .

A nice feature of the Lagrange and Markov spectrum of complete shifts is:

Lemma 13 (lemma 2 of Gugu’s paper) Let be a complete shift. Then, it holds

Proof: The reader clearly sees that , where $R$ is the length of the biggest word of $B$, so that

On the other hand, given , it suffices to construct two regular Cantor sets and such that and . In this direction, we note that, up to replacing by for a large , one can assume that for any with . Next, we order the elements of (resp. ) as follows: given (resp. ), we say that if and only if . Using this total order, we can select and (resp. and ) the minimal and maximal elements of (resp. ). We define

and

Observe that and (here we used the remark 6). Of course, since we removed the minimal and maximal elements of and , we expect that the values of on decrease in view of the following classical comparison result of the theory of continued fractions:

if and only if , where is the smallest integer such that (this parity issue during the comparison of two continued fractions justifies the exclusion of the minimal and maximal elements of ).

Unfortunately, the exclusion of the minimal and maximal elements of and is not sufficient to ensure that (in fact, although the values of on decrease, this doesn’t guarantees that they belong to . However, this technical problem can be solved by considering some smaller replicas of and . Pick , , a point of where the maximum of is attained (in a position associated to ). Using , for each , we construct the subsets formed of the sequences

where for every and for every . By a compactness argument (involving the fact that we can eventually increase the value of by replacing some of its elements by or ), Gugu shows that there exists and a large such that, for every , the value is attained at a position associated to the central

block

and at any position outside . Next, we pick an arbitrary such that and we associated to each the sequence

Observe that, for each position of the central block, doesn’t depend on , so that , the function is a Möbius transformation (for such positions ). Moreover, these functions are mutually distinct. In particular, the values (for the positions of ) are distinct except for a finitely many choices of . This allows us to take such that for every two distinct positions and of . Let’s denote by the position of where the value is maximum.

Now, we take large such that the enlarged block verifies the following property: we have

for any with and (), where

The existence of is a consequence of the choice of (so that the is attained at some position of ), the fact that the values of are mutually distinct at the positions associated to (so that for every position of ) and the fact that the values of are very close to for a sufficiently large .

Finally, we define

and

Observe that is a (diffeomorphic) copy of and is a (diffeomorphic) copy of , so that and .

We claim that . Indeed, given

and

it is not hard to check that where

and where

and . This ends the proof of the lemma.

-Approximation of Lagrange/Markov spectrum by complete shifts-

During the study of the function , it is interesting to introduce the subshifts given by

and the associated Cantor sets

where is the natural projection. For later use (in the arguments related to Hausdorff dimension), we’ll say that the size of a finite sequence of strictly positive integers is the length of the interval

Note that the extremities of are and , so that .

We also introduce ,

for each , and

The cardinatily of is denoted by . In this setting, it is possible to show that the box dimension of is

This is expected since the collection corresponds to the natural intervals (with respect to the continued fraction algorithm) of scale covering the set . For more details see Gugu’s preprint.

Proposition 14 is a continuous function.

Proof: In fact, from the theory of continued fractions, we have that for any finite sequences and any . It follows that one can cover using the intervals of the form , where , , . Furthermore, these intervals verifies . In particular, , so that

In other words, the sequence is sub-aditive. Hence,

Using this information, we can conclude that is continuous: otherwise, one can find some and such that for every (since is monotone). In view of the previous identity, it follows that we can select some large integer such that

for every and . On the other hand, because for any and (by a simple compactness argument), by taking the limit when , we see that the previous estimate implies

a contraction. This completes the proof of the proposition.

Now, we are able to state the following lemma about the approximation of the Lagrange/Markov spectrum by regular Cantor sets associated to complete shifts:

Lemma 15 (lemma 1 of Gugu’s paper) Given and , we can find and a regular Cantor set associated to a complete shift such that

• and
• .

Proof: Of course, it is not hard to approximate by a complete shift (in order to ensure is close to ). Indeed, consider and fix large so that

for any . Now, define and put . Take and we introduce

It is not hard to check that the Hausdorff dimension of is close to : firstly, a simple counting argument shows that ; in particular, since the intervals , are a covering of such that (exercise), we see that , but the main problem is the fact that we don’t have any control on the values of on (since we would like to decrease them from to ). In order to overcome this difficulty, Gugu introduces the notion of lef-good and right-good positions of a given .

More precisely, we say that is a right-good position of whenever one can find two elements () of such that the following inequality holds

Similarly, is a left-good position of whenever one can find two elements () of such that

Finally, we say that is a good position of if it is both a left-good and a right-good position.

By a crude bound on the number of bad positions (namely, given a position, there are only two choices of making a bad position: ), it is possible to prove that the quantity of words of with good positions is . Such words are called excellent words by Gugu. Of course, we can hope to decrease the values of on any excellent words (by some fixed amount ), but a new problem emerges: it may happen that the subset of formed by excellent words doesn’t give a complete subshift (e.g., it may be not possible to concatenate excellent words). At this stage, Gugu employs a exclusion procedure combined with a double counting argument to show that we can build a complete shift by focusing at a large portion of the good positions of and fixing the words of appearing into these good positions. Namely, Gugu shows that there are special good positions (i.e., they are good positions such that are also good positions) such that the subset of excellent words whose entries at these good positions are equal to conveniently chosen words (i.e., and verifies

Furthermore, Gugu proves that there are two integers such that the image of the projection of the form

is the desired subset of words such that (essentially by the definition of good position) and (since , , is a covering of composed of intervals verifying ).

This completes the proof of the lemma.

Once we have the lemmas 13, 15 and the proposition 14 in our toolbox, we can prove the first part of theorem 3.

-End of the proof of theorem 3-

We claim that . Indeed, using the lemma 13 to the complete shift constructed in the lemma 15, we see that

Nevertheless,

On the other hand,

Also,

Furthermore,

Putting these four estimates together and letting , we conclude the proof of the claim. Observe that this argument also shows that .

Finally, the continuity of follows from the formula and the fact that is continuous (see the proposition 14).

Remark 9 From the identity , we see that the first item of theorem 3 is equivalent to (where ). Although this is not terribly difficult to show, we are not going to prove this fact here.

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.