In last April 15 (two weeks ago), the Brazilian president Luis Inacio “Lula” da Silva visited IMPA – Instituto de Matematica Pura e Aplicada to discuss some aspects of mathematical research and teaching in Brazil (this was the first time a Brazilian president visited IMPA). The reader can find several photos of this event here.

Being both a Brazilian and a former student at IMPA (where I got my PhD), I’m happy to see that the Brazilian government is really paying the deserved attention to the fundamental sciences, specially Mathematics (although there are still several weak points concerning the distribuition of permanent positions in public universities, etc.)

In any case, I enjoyed a lot Lula’s speech while I got a little upset with some demagogic phrases of Sergio Cabral, the governor of the state of  Rio de Janeiro, who said:

“… Impa already organized a Math. Olympiad focused to private schools and what the president did? He moves IMPA, a world-wide recognized mathematical institution, from its academic pedestal to show it the real life of the Brazilian population by realizing the OBMEP [Math. Olympiads for Public Schools]…”

Let me say that you can see a resume (in Portuguese) of Lula and Cabral speechs here. Also, let me make two comments about the three points I marked (in bold) in Cabral’s phrase

• concerning the organization of Math. Olympiads for private schools: firstly, IMPA’s main goal is the mathematical research and the training of PhD students; of course, IMPA supported other mathematical activities such as Math. Olympiads by providing physical space, etc., but evidently this isn’t its main objective; secondly, the Math. Olympiad organized at IMPA (called OBM – Brazilian Math. Olympiad) is completely open: anyone is able to make the inscription to participate the competitions; however, the fact that most of the “successful” students (meaning the ones making their way to the IMO) comes from private schools has nothing to do with the Cabral’s claim that OBM is focused on private schools! It has to do with the simple fact that public schools are fairly weak in Brazil!
• concerning “academic pedestal” and “real life”: this is a repetition of the classical comparision between Mathematics and an Ivory Tower -  the image of an austere science closed into itself, etc., so that I think this absurd point doesn’t need any further explanations.

Finally, let me mention that Lula’s visit to IMPA takes place in an important moment for the Brazilian mathematical community: for instance, Brazilian mathematicians are consolidating their presence in the international community as the invitation of my friends Fernando Coda Marques and Artur Avila (two young Brazilian mathematicians) to give talks in the ICM 2010 confirms (Fernando was invited for the Geometry Session and Artur was invited for a Plenary Talk).

Hi! Firstly, let me apologize for the 1 month and half hiatus: while I plan to close up my series on the talks and courses of the Dynamical Systems conference (held at ICTP, Italy), I wasn’t able to do so during April because I was learning some interesting facts about square-tiled surfaces and Teichmuller dynamics from my new friends Pascal Hubert and Anton Zorich. Of course, I plan to share these facts in some future posts, although I can’t promise to do so in the next few months. On the other hand, I can give a clue to the curious reader by providing a copy of the slides of my recent talk at Orsay entitled: Sur le cocycle de Kontsevich-Zorich au-dessus de deux origamis spéciaux.

Warning. While the title is written in French (and the actual talk was my third presentation speaking in French), the slides are written in English (because I don’t feel comfortable to write in French yet).

Before giving the link to the slides, let me make a brief description of the talk.

The basic objective was the discussion of the Teichmuller geodesic flow and the Kontsevich-Zorich cocycle. Nowadays, after the works of Forni (2002), Avila and Viana (2006), we know that the Kontsevich-Zorich conjecture is true: the Lyapunov spectrum of the Kontsevich-Zorich (KZ) cocycle with respect to the absolutely continuous -invariant ergodic probability is non-uniformly hyperbolic (i.e., 0 doesn’t belong to the spectrum) and simple (i.e., all exponents have multiplicity one). In other words, the ergodic behavior of generic points with respect to the Lebesgue measure is more or less well-understood.

However, one can follow Veech and pose the question: what about the ergodic behavior of non-generic points? In this direction, Forni (2005) and Forni, Matheus (2008) showed that there are examples of orbits with totally degenerate Kontsevich-Zorich  spectrum (see this previous post). An interesting feature of the KZ cocycle over totally degenerate orbits is the fact that it is isometric (this follows from Forni’s 2002 work). In the talk, I discussed an improvement of this fact (obtained jointly with Jean-Christophe Yoccoz): by computing the action of the group of affine diffeomorphisms on the relative homology of these examples, we are able to show that the Kontsevich-Zorich cocycle is represented by a subgroup of the Weyl group of the root system . A nice consequence is the fact that the KZ cocycle acts by a finite group of rigid matrices, so that this gives a new proof of Forni and Matheus results by more explicit methods.

Finally, the slides can be found here.

After my previous post on a Fourier-analysis approach to Furstenberg’s problem, I would like to present another application of the Fourier transform of measures: a simple proof of Marstrand theorem about projections of Cantor sets. Before entering the details, let me tell you what are my motivations on this topic: besides the beauty of the Fourier-analysis approach to a relevant geometrical problem, this particular proof of Marstrand theorem was generalized by C. Moreira and J. C. Yoccoz in their work on stable intersections of regular Cantor sets so that it could be applied to the study of homoclinic tangencies (Dynamical Systems) and the Markov/Lagrange spectrum (Number Theory) (as we briefly described in this post). In particular, since I’m planning to make a series of posts around these works of C. Moreira and J. C. Yoccoz, I think that a discussion of Kaufmann’s proof of Marstrand theorem is the best place to start this project.

-Statement of Marstrand’s theorem-

Define , and denote by the orthogonal projection of the plane onto the line . In other words, if we identify with the real line (via , then, given , we have .

Theorem 1 (J. Marstrand’s theorem) Let be a subset with Hausdorff dimension . Then, its projection has positive Lebesgue measure for (Lebesgue) almost every .

An interesting consequence of this theorem (in the context of stable intersections of Cantor sets of the real line) is the following fact:

Corollary 2 Let and be two Cantor sets of the real line such that

Then, has positive Lebesgue measure for almost every .

Proof: Firstly, we recall the following well-known property of the Hausdorff dimension: (see K. Falconer’s book for more details). Secondly, we notice that the projection of is where . Since for and we can parametrize via with , we get the desired corollary from Marstrand’s theorem.

Now we turn to the proof of Marstrand’s theorem following the exposition of the book of Palis and Takens (pages 64, 65, 66 and 67).

During my conversations with my friend Artur Avila around the 2x, 3x mod 1 problem and its applications, Artur explained me a cute argument (using the Fourier transform of measures) to show a weak version of Rudolph’s theorem (which is a partial answer to this problem). Of course, I believe it is a good idea to share it here, but, in order to keep this post more self-contained, I’ll not proceed directly to the argument. In fact, I’ll divide the exposition into 3 sections: the first one is a general introduction to Furstenberg problem, Rudolph’s theorem and the weak version of Rudolph’s theorem, the second section contains the proof of this weak version of Rudolph result and the third section briefly discusses Einsiedler, Katok and Lindenstrauss’ application of a (highly non-trivial) analog of Rudolph’s theorem to the so-called Littlewood conjecture (about simultaneous Diophantine approximations).

Disclaimer. While Artur’s explanation was clear and correct, any possible mistakes and errors appearing below are my responsability (of course).

-Furstenberg’s problem: measure rigidity of expanding rank-two semi-group actions-

Consider the following (uniformly expanding) dynamical system on the circle :

,

where is an integer. Despite its simple definition, is rich from the dynamical point of view: possesses plenty of ergodic invariant measures, e.g., the Lebesgue measure and the Dirac measures supported on its periodic points (in fact, due to the expanding features of , one can use the so-called thermodynamical formalism [explained in Bowen's book] to construct a whole family of ergodic invariant measure “interpolating” the Lebesgue measure and the Dirac measures); since, by Birkhoff’s theorem, the ergodic invariant measures captures the statistical features of most orbits in its support, we see that the existence of several distinct ergodic measures for indicates the presence of several distinct statistical behaviours of the orbits of (which justifies the adjective “rich”). Of course, the dynamics of an individual uniformly expanding (and, more generally, hyperbolic) dynamical systems is well-understood nowadays (after the works of Anosov, Smale, Sinai, Ruelle and Bowen among several other authors) and we do not plan to discuss further this issue today. Instead, let us investigate the dynamics of the joint action of two expanding maps, say and . In other words, denoting by and , we are replacing the action

of the rank-one semi-group (i.e., the individual action of ) by the action

of the rank-two semi-group (i.e., the joint action of and ).

Remark. In general, a similar action by a rank-two semi-group can be obtained replacing by any pair of multiplicatively independent non-negative integers.

The basic topological result about the dynamics of is:

Theorem (H. Furstenberg). The sole infinite closed -invariant subset of the circle is the circle itself.

This theorem should be constrasted with the rank-one situation: while has only one closed invariant set, it is not hard to see that has a lot of invariant Cantor sets. In fact, since is semi-conjugated to a full Bernoulli shift with symbols (via a codification of the orbits using the Markov partition ), any subshift gives you an invariant Cantor set. More concretely, you can take a finite collection of disjoint closed intervals whose union doesn’t coincide with the whole circle and define the set of points of the circle whose -orbit never enters . Of course, is -invariant and, from the expanding features of , it follows that is a Cantor set (exercise).

In resume, Furstenberg’s result says that is more rigid than (dynamically speaking), which is quite unexpected because is the combination of the dynamics of two strongly chaotic endomorphisms, namely, and . Taking this theorem as a motivation, Furstenberg posed the following problem of the measure (i.e., ergodic-theoretical) rigidity of :

Furstenberg’s 2x,3x (mod 1) problem. Is it true that the Lebesgue measure is the sole non-atomic ergodic -invariant measure? Equivalently, is it true that the Lebesgue measure is the unique non-atomic ergodic measure which is both and -invariant?

In other words, this problem asks whether the intersection of two enormous sets of measures (the set -invariant probabilities and the set of -invariant probabilities) is relatively small (namely, the Lebesgue measure and the Dirac measures supported on periodic orbits).

While Furstenberg’s problem is still open (to the best of my knowledge), some important partial results are available nowadays. In particular, let me quote the following theorem of D. Rudolph:

Theorem (D. Rudolph). Let be a and invariant ergodic measure such that

.

Then, is the Lebesgue measure. Equivalently, if a invariant measure is not the Lebesgue measure, then the semigroup

is a group for almost every .

In this statement, denotes the metric (Kolmogorov-Sinai) entropy of the -invariant measure .

Remark. The motivation of the metric entropy condition in Rudolph’s theorem comes from the fact that the metric entropy of Dirac measures supported on periodic orbits is zero. Of course, Rudolph theorem is just a partial answer to Furstenberg question because non-atomic invariant measures can have zero entropy.

While we do not pretend to give a complete proof of Rudolph’s theorem (which is not very hard but involves some amount of abstract ergodic theory), we do plan to show in the next section the following fact:

Theorem 1 (weak version of Rudolph’s theorem). Let be a and invariant measure. Assume that and is ergodic. Then, is the Lebesgue measure.

-Weak Rudolph theorem: proof of theorem 1-

Take a probability measure verifying the assumptions of theorem 1 and let be the translation of on the circle .

Note that and are commuting maps. In particular, since is invariant, it follows that is invariant. Moreover, the assumption of ergodicity of with respect to implies that and are -ergodic.

On the other hand, the positive entropy assumption implies that the restriction of to the support of is not invertible. Equivalently, we can always find some pairs of generic points with the same image under . However, such a pair of points is always permuted by the translation . In particular, we have that the probability measures and aren’t mutually singular.

Putting all these facts together, we have two invariant ergodic measures ( and ) such that and aren’t mutually singular. It follows that , i.e., is -invariant.

At this stage, one has enough information to conclude that is the Lebesgue measure. Indeed, we claim that all of the non-zero Fourier modes of vanish, that is,

(1) for any .

In fact, let us start with the odd Fourier modes (i.e., where ). Using the -invariance of , we obtain

.

Since , we get

.

In particular, it follows that whenever is odd. Finally, we can use the -invariance of (with a similar argument) to obtain

for every ,

that is, the even Fourier modes can be deduced from the odd ones (more precisely, writing a non-zero even number as where is odd, we have ). Consequently, our claim (1) about the vanishing of the Fourier modes of is proved. As we know (1) forces to be the Lebesgue measure, so that the proof of theorem 1 is complete.

-Measure rigidity of higher-rank hyperbolic actions: Einsiedler-Katok-Lindenstrauss theorem-

In this final section, we briefly outline a recent application of the measure rigidity of higher-rank hyperbolic actions to a partial solution of a number-theoretical problem.

A well-known fruitful interaction in Mathematics occurs between Dynamical Systems and Number Theory. Among the several applications of dynamical ideas to number-theoretical problems, one finds Margulis’ solution to Oppenheim conjecture (see e.g. this blog post of Terence Tao). Here, the basic idea is to convert the study of the values of indefinite quadratic forms in variables () into the study of the dynamics of an action of a higher-rank group in the space of unimodular lattices in . Another (more recent) application of these ideas was performed by M. Einsiedler, A. Katok and E. Lindenstrauss in the study of the so-called Littlewood conjecture:

Littlewood conjecture. For every , it holds

.

Here denotes the fractional part of .

Again, the basic idea is to convert this problem into the study of the dynamics of the action of the group of positive diagonal matrices in .

However, although the basic strategy of Margulis and Einsiedler-Katok-Lindenstrauss is the same, there is a subtle and important difference: while in Margulis’ context the acting group contains only unipotent (parabolic) elements, in Einsiedler-Katok-Lindenstrauss’s setting the acting group contains only hyperbolic elements.

In particular, at the present moment, we have the following picture: one can completely classify the invariant measures of the -action (by Ratner’s theorems), but one can say something interesting about the invariant measures of the -action only when they have positive entropy. In other terms, it is “morally” more easy to show measure rigidity statements (in the spirit of Furstenberg’s problem) for the -action than -action because the individual elements of the group are already parabolic, i.e., rigid. Here, the positive entropy condition for the measure rigidity statement (for the action) is certainly motivated by Rudolph’s result.

In any case, the moral philosophy here is the following: the number-theoretical problems quoted above (Oppenheim and Littlewood conjectures) can be reduced to the complete classification of the invariant measures of adequate higher-rank group actions. Since such a complete classification is possible for the -action, Margulis completely solved Oppenheim conjecture; but because our current technology only allows us to classify invariant measures of positive entropy, Einsiedler-Katok-Lindenstrauss partially (almost) solved Littlewood conjecture. More precisely, they proved the following result:

Theorem (Einsiedler, A. Katok and E. Lindenstrauss). The set of (possible) exceptions to Littlewood conjecture has Hausdorff dimension 0. 

Closing this post, let me say that this last result deserves further explanation and I plan to discuss it a little bit more in a future post (probably after the series of posts around Trieste’s conference). By the way, the curious reader may consult (besides the original article) these expository notes of Akshay Venkatesh for a nice introduction.  Ciao!

Today we’ll close our current discussion of the standard map with the proof of Duarte’s theorem. As we mentioned earlier, the basic strategy consists into three steps: the construction of a dynamically increasing family of hyperbolic basic sets of saddle-type (“horseshoes”), the existence of a dense set of parameters where  a quadratic tangency is generically unfolded and the construction of elliptic islands from the bifurcation of quadratic tangencies via a conservative version of Newhouse phenomena. More precisely, we are going to show the statements of theorems 1, 2 and 3 of the previous post.

-An “increasing” family of hyperbolic basic sets-

In order to find hyperbolic sets, we’ll use the well-known invariant cone-field criterion:

Theorem (invariant cone-field criterion). Let be a -diffeomorphism. Consider a compact -invariant set. Assume that there are , a decomposition (not necessarily -invariant) and two families , of (closed) cones such that

• and for all and ;
• and for all and .

Then, is a hyperbolic set.

Proof. See corollary 6.4.8 of Katok-Hasselblat book. A sketch of proof goes as follows. Fix a point and consider the following two sequences of cones on the tangent space : and . Our assumptions implies that these two sequences are nested sequences of closed cones, so that the intersections

and

are closed -invariant closed cones. In fact, one can work a bit more (with the facts and ) to see that and are vector subspaces with dimensions and resp.. Finally, once we know that and are -invariant subspaces, our assumptions of expansion (resp. contraction) of vectors belonging to the cone (resp. ), we see that is the unstable subspace and is the stable subspace. This completes the sketch.

Proposition 1. Consider the invertible area-preserving map

of the 2-torus and let be a -invariant compact set. Assume that there exists such that for all . Then, is a hyperbolic set (of saddle type).

Proof. Note that

In particular, the trace of verifies so that the matrices are uniformly hyperbolic. In fact, this follows from the fact that the constant cone-field

is an unstable cone-field whenever (note that such a choice is possible since ). Indeed, if we write , we see that

so that where by the choice of the parameter , i.e., is -invariant. Furthermore, denoting by , we get, for any ,

with , i.e., (uniformly) expands any vector inside . On the other hand, it is not hard to see that implies that the same argument can be applied to in order to get a stable cone-field. Using the invariant cone-field criterion, the proof is complete.

An immediate consequence of this proposition is the following result:

Corollary 1. For the standard family

,

given any , the maximal invariant set

is hyperbolic.

Remark. It is worth to note that this result gives a clue about the location of the critical region of non-hyperbolicity: for a given , the set of points   converges to the union of the two circles when .

Of course, this corollary says that the hyperbolic sets are a family of dynamically increasing basic sets. In fact, it turns out that this can be checked by hand (see section 4.2 of Duarte’s paper), but we’ll skip this fact for sake of brevity of the exposition.

-Global dynamical foliations and their tangency lines-

After the description of the (“big”) hyperbolic sets of the standard map , we proceed to the study of the tangencies between their invariant foliations. In order to do so, we need to extend these foliations to some uniform neighborhood of (since we want to perform an analysis for several large parameters) while keeping good estimates of distortion of the holonomy maps. At this point, our first technical problem arises: from the general theory of uniformly hyperbolic sets (see the book of Palis-Takens), we know that admits some neighborhood so that the stable and unstable foliations of can be extended to (while keeping good estimates), but a priori the region where the good estimates are ensured can deteriorate when . To overcome this problem, Duarte takes the following point of view. Near the critical region , he replaces by a function having two poles at and he tries to compare the dynamics of with the dynamics of the singular diffeomorphism .

More precisely, where outside a -neighborhood of and . Then, after the somewhat tedious work of redoing the theory of invariant manifolds (following the exposition of Hirsch-Pugh-Shub), he checks that has global stable and unstable foliations on verifying uniform distortion estimates (i.e., their holonomy maps have uniformly bounded -norm). Here, the uniform control of distortion comes from the choice of : indeed, assuming that coincides with outside a -neighborhood of , it is not hard to see that the Schwartzian derivate of is bounded from below (in the critical region ) by

.

In particular, since we want to take the largest possible so that coincides with in and with bounded Schwartzian derivative (because it is well-known that bounded Schwartzian derivative implies bounded distortion), it is natural to take outside (i.e., ).

Of course, once we performed this work (which takes 21 pages of Duarte’s paper), we have to compare the dynamics of and . However, this is not hard: the maximal invariant set is the same for both and , and, using the strong hyperbolic features of the singular diffeomorphism , it is possible to show that the dynamics of and on are conjugated to a full (Bernoulli) shift (where ); furthermore, ,  its stable and unstable thickness (where , resp. , is the thickness of the Cantor set obtained by projection of along the stable, resp. unstable, foliation on an arbitrarily fixed transversal section) and, as a consequence, its Hausdorff dimension satisfies .

Next, we analyse the relative positions of the (-invariant) foliations and . Applying to , we obtain a new foliation (recall that is -invariant but it is not -invariant). It is not hard to see that the set of tangencies between and are two circles close to . Moreover, the projection of along and into these two circles gives rise to two Cantor sets and satisfying and (here we are using the previous thickness estimate and the fact that the application of to the foliation doesn’t change very much the thickness). The picture below (borrowed from Duarte’s paper) summarizes our discussion about the relative position of and :

Here is the almost vertical foliation and is the foliation folding along the two dotted circles. Now, using the fact that (for any large ), we can apply Newhouse’s gap lemma to obtain that . In other words, we get that and exhibits persistent tangencies.

Remark. In proposition 16 of Duarte’s paper, a version of Newhouse’s gap lemma in the circle is wrongly stated: indeed, Duarte claims that the fact that the two Cantor sets are contained in the circle automatically implies that the two Cantor sets are linked. However, this is not correct (as the example of two thick Cantor sets supported by two disjoint compact intervals shows), although this is not a serious problem for this argument: from a careful checking of the geometry of (via the features of the singular diffeomorphism ), it is not hard to see that and are linked (this follows from Duarte’s argument in section 4.2 of his paper).

Finally, closing this section, we claim that these persistent tangencies are quadratic and unfold generically with the parameter (as the picture above indicates). While a complete proof of this result takes 7 pages of technical calculations, we’ll provide a convincingly enough (I hope! ) heuristic argument. We know that the -invariant foliations and are almost vertical and horizontal (resp.). In particular, it is reasonable to expect that the circles of tangencies between and are close to the circles of tangencies between the horizontal foliation and the image of the vertical foliation under . On the other hand, since where is the counterclockwise rotation and is a shear (of variable intensity) along the horizontal foliation, we can compute the image of the vertical foliation under as follows: the image of the horizontal foliation by is the vertical foliation and the image of the vertical foliation by is a foliation by the family of curves which are parallel to the graph . In particular, the circle of tangencies between these two foliations are exactly the critical circles . At such points, the difference between the curvatures is measured by , so that the tangencies between the horizontal foliation and the -image of the vertical foliation are quadratic (i.e., locally you are seeing the intersections between straight lines and parabolas) and, a fortiori, the same holds for the tangencies between and . Also, when the parameter increases, the -invariant foliations and doesn’t change very much (they are almost constant), while the -coordinates of the tangency points between and (which are close to ) move with velocity (close to) 1 (indeed, implies ). Hence, these tangencies are unfolded generically.

At this point, the reader noticed that this discussion gives the theorems 1 and 2 of the previous post. Now, we proceed to discuss the conservative version of the Newhouse phenomena.

-Conservative version of Newhouse phenomena: proof of theorem 3-

Before entering into the proof of the abundance of elliptic islands close to a generically unfolded quadratic tangency, let me review a little bit some facts around the proof of the “classical” Newhouse phenomena.

Given a 1-parameter family of surface diffeomorphisms generically unfolding a quadratic homoclinic tangency associated to a hyperbolic periodic point of saddle-type (at the parameter say), Newhouse manage to define a renormalization scheme near as follows: for every large , one can select small boxes near which are mapped by near itself  with the shape of a parabola so that their relative positions resembles a horseshoe; next, we compose this dynamics with appropriate rescalings of these boxes in order to put these very small boxes into a fixed scale (e.g., a unit square) so that we obtain the families of dynamical systems given by (these are called the successive renormalizations of the dynamics near the homoclinic tangency). The usefulness of idea is more or less clear: assuming that there exists some limiting object , any stable dynamical property of will be shared by and a fortiori .  It turns out that Newhouse showed that this renormalization scheme converges (i.e., the limiting object exists) when the periodic point is dissipative (i.e., ).  Moreover, the limit in this case is the quadratic family

.

Using this information, the existence of sinks near the homoclinic tangency follows directly. A detailed exposition of Newhouse’s argument can be found in the excelent book of Palis and Takens.

After this quick review of Newhouse arguments, let us consider again the situation of the standard map: in the conservative setting, given a family of area-preserving diffeomorphisms generically unfolding a quadratic homoclinic tangency associated to a hyperbolic periodic point, one can repeat the renormalization scheme of Newhouse to get as a limit object the conservative Hénon family

.

Next, it is possible (exercise) to show the presence of elliptic fixed points for this family when (with the eigenvalue of this point running from 1 to -1). Because generic (i.e., non-resonant) elliptic periodic point is stable by conservative perturbations (this follows from the so-called KAM theory; see e.g. this monograph of J. Moser), we conclude the existence of (generic) elliptic periodic points nearby the homoclinic tangency. Combining this result with the theorems 1 and 2 proved in the previous sections, we see that, similarly to the proof of Newhouse theorem, the proof of theorem 3 is complete!

My next topic on my series of posts about some lectures of the School and Workshop of Dynamical held at the ICTP, Trieste (Italy) is a beautiful theorem of Pedro Duarte quoted by Anton Gorodestki during his talk ”On the size of the stochastic layer of the standard map“. I should say that I am not planning to discuss Gorodetski’s theorem here but I will focus only on Duarte’s result. My main reason to do so is the fact that Gorodetski result is strongly based on Duarte’s techniques, so that it would be very hard for me to explain the former theorem without the latter one.

The plan of this post is the following:

• in next section, we introduce the standard map and briefly discuss the some of its well-known dynamics (due to the KAM theory);
• after that, we recall the positive entropy conjecture for the standard map;
• finally, we state Duarte’s theorem showing that the positive entropy conjecture for the standard map is a subtle problem due to the presence of ‘elliptic islands‘.

Closing this introduction, let me point out that I will postpone the proof of Duarte’s theorem to a subsequent discussion.

-Standard map: a discrete-time model of the pendulum dynamics-

During our undergratude studies in Mathematics (or Physics perhaps), we certainly faced at some point the so-called pendulum equation:

.

This ODE models an idealized pendulum: the chord is massless and inextensible, the system moves on a 2-dimensional plane due only to the gravitational force and the total energy of the system is conserved (i.e., there is no friction). For more details, see the link above (for the Wikipedia article on the mathematical pendulum).

It is known that the dynamics of the pendulum is very rich. In order to approach it, one can look at the discrete-in-time model provided by the solutions of the following difference equation:

(1)

obtained by the the substitution of the second order differential operator by the second order difference operator . A simple way to encode the pairs satisfying the difference equation (1) is provided by the orbits of the so-called standard family (of area-preserving maps acting on the 2-torus ):

.

In fact, it is easy to check that (exercise).

Remark. Observe that the maps and are conjugated via the translation . Hence, we can restrict ourselves to the standard family with parameters .

On the other hand, we should point out that the dynamics of the standard map is a good aproximation of the pendulum dynamics only for small values of the parameter . Indeed, the pendulum’s flow is always integrable (since the total energy, i.e., the kinetic energy plus the potential energy is a conserved quantity for this flow), the same is not exactly true for the standard family: while the standard map is integrable, that is, is foliated by completely invariant (KAM) circles (whose dynamics is conjugated to a rotation), these circles starts to break up as the parameter increases and the orbit behavior becomes “chaotic”. In fact, this informal description of the dynamics of the standard family for large is strongly supported by the computer experiments. For instance, one can see below a computer simulation ( borrowed from this Wikipedia article about the standard map) of the 10 orbits of the standard map :

The orbit space of the standard map with parameter k=2. The green region is the ``chaotic

The reader can found further beautiful pictures (for other choices of parameters) in this Scholarpedia article. Of course, one can ask (motivated by this conjectural description) the following question:

Question (Positive entropy conjecture). Is the Kolmogorov-Sinai entropy of the Lebesgue measure positive for large values of ?

While the computer experiments seem to show a lower bound , the positive entropy conjecture is not known to be true even for a single value of (despite the efforts of many mathematicians, e.g., Oliver Knill who tried an operator theory strategy based on the so-called Herman’s subharmonicity method). Anatole Katok describes the positive entropy conjecture as one of the five most resistant problems in dynamics!

The positivity of the (metric) entropy of the standard map for typical large values of has several interesting consequences:

• from the Pesin’s entropy formula, we know that the positivity of the entropy of the Lebesgue measure (area) implies the existence of an invariant set of positive area such that every has a positive (maximal) Lyapounov exponent (and, due to the conservation of mass,  a negative minimal Lyapounov exponent also).
• in particular, from Pesin’s theory, the non-vanishing of the (two) Lyapounov exponents of on (i.e., its non-uniform hyperbolicity) implies that the dynamics of restricted to the set is chaotic (for instance, the system exhibits sensitivity to initial conditions and Katok’s theorem allows us to infer the existence of ‘large’ horseshoes contained inside ).

At this point, we are ready to state the main theorem of this post:

Theorem (Duarte). There exists a large parameter and a residual subset such that set of elliptic periodic points of the standard map is -dense set (i.e., for any ) for all .

It is worthwhile to point out that Duarte’s theorem goes into the opposite direction of the positive entropy conjecture. In fact, the presence of elliptic periodic points is potentially dangerous for the positivity of the entropy because the KAM theory says that a typical elliptic periodic point (i.e., an elliptic point satisfying a certain non-resonance condition) can be locally linearized (that is, the initial diffeomorphism can be locally conjugated to the its derivative at the periodic point). Since the derivative at an elliptic point is essentially a rotation, the entropy of the initial diffeomorphism (restricted to the basin of the elliptic point) is zero. In other words, the presence of these ‘elliptic islands’ (i.e., open invariant regions where the diffeomorphism behaves like a rotation) are a serious obstruction to the positivity of the entropy.

Remark. Observe that Duarte’s theorem doesn’t contradicts the positive entropy conjecture. In fact, this follows from the ‘usual reason’: residual sets (in Baire sense) may have zero Lebesgue measure. However, this result certainly shows how subtle the positive entropy conjecture is.

Ending this post, let me outline the strategy of the proof of Duarte’s theorem. Roughly speaking, the idea is to get the result from a conservative version of Newhouse phenomena. More precisely, one begins with the following result:

Theorem 1. For each parameter , there exists a basic set of (i.e., a compact -invariant transitive hyperbolic set) with the following properties:

• the family is dynamically increasing: given , for any sufficiently small , the set contains the dynamical continuation of (that is, the unique compact -invariant hyperbolic set close to in the Hausdorff topology);

• the thickness of grows to infinity as : for all sufficiently large parameter , the local stable and unstable thickness are

;

• the Hausdorff dimension of goes to as : for all sufficiently large parameter , it holds

;

• is topologically conjugated to a (full) Bernoulli shift in symbols where

as ;

• for all large , the set is -dense in the torus .

Starting with this theorem, Duarte shows that the basic set generically unfolds a quadratic homoclinic tangency for a dense set of (large) parameters:

Theorem 2. There exists such that, for any and any periodic point , the set of parameters such that the invariant manifolds and of the continuation of generically unfold a quadratic homoclinic tangency is dense in .

In resume, the theorems 1 and 2 put us in the framework of the (conservative version of) Newhouse phenomena: for a large parameter in the dense set , we have a thick basic set unfolding (generically) a quadratic tangency; this permits to use some of the ideas employed by Newhouse (as discussed in a previous post): of course, it is impossible to obtain the same conclusion of Newhouse (namely, the coexistence of infinetely many sinks/sources for a generic nearby system) because we are dealing exclusively with area-preserving diffeomorphisms; but, the techniques of Newhouse gives us the coexistence of many elliptic periodic points. Indeed, this is the content of the following result of Duarte:

Theorem 3. There are large and a residual subset such that, for every , the closure of the elliptic periodic points of contains the -dense basic set .

Of course, Duarte’s theorem is an immeadiate consequence of theorem 3. In the next post, we will concentrate on the proofs of theorems 1, 2 and 3. Ja ne!

Firstly, I would like to apologize for the long hiatus between the first and the second post on Asaoka’s theorem. Basically, my blog schedule was somewhat affected by the World Chess Championship! By the way, the third and fifth matches between V. Anand and V. Kramnik were really interesting: Anand won the third match with a variant of the semi-slav defense and he managed to use the SAME move (ok not exactly the same because he interposed the rook before the bishop) three days after (in the fifth match) but Kramnik (and his mates) were still unable to find an adequate defense!

Anyway, since I’m not an expert in chess, let me return to the main purpose of this post. Following the plan of the previous post, today we’ll discuss the proof of Asaoka’s theorem:

Theorem (Asaoka). Any transitive codimension-one Anosov flow is topologically conjugated to a smooth volume-preserving Anosov flow.

Remark. It is not hard to show that volume-preserving Anosov systems are transitive (exercise). Thus, Asaoka’s theorem is a converse of this fact in the case of codimension-one Anosov flows. In fact, it is a good open problem to know whether a similar statement is true in general.

Roughly speaking, the proof of this theorem has two steps:

• 1st step: a generalization of Elise Cawley’s work in order to show that one can deform any codimension-one Anosov flow inside its topological conjugacy class so that the derivative along the stable and unstable fits any ‘coherently’ prescribed dynamical behaviour; in particular, one can deform the flow in order to get a topologically equivalent flow preserving a Hölder continuous volume form;
• 2nd step: by a -small perturbation of a flow preserving a Hölder continuous volume form, one can get a flow preserving a smooth volume form (via the so-called ‘pasting lemma‘ technique developed by Alexander Arbieto and myself);

Since it is well-known that Anosov flows are structurally stable (i.e., they are topologically conjugated to any -nearby flow), these two steps complete the proof of Asaoka’s theorem.

Remark 1. Concerning the 1st step, it turns out that the resulting flow can’t be expected to preserve a smooth volume form in general. Indeed, this happens due to the lack of regularity of invariant foliations of Anosov flows: typically, they are only even when the codimension-one Anosov flow is .

Now let us turn to the details.

-Generalization of Cawley’s deformation arguments-

Consider a codimension-one Anosov flow with a hyperbolic splitting . For sake of concreteness, we assume that the unstable direction is one-dimensional (of course this can always be achieved by a time change ). Denote by the associated strong-unstable foliation and let the holonomy map of the weak stable foliation (i.e., the foliation tangent to the subbundle ).

Definition 1. We denote by the set of families of measures such that

• is a Borel, non-atomic, positive on open sets, locally finite measure on for all ;
• implies ;
• the Radon-Nikodym derivative at is well-defined for any and close to ; morever is Hölder continuous on both variables and .

The next result is a precise statement of the generalization of Cawley’s work (quoted in the 1st step of the outline) about the realization of any dynamical behaviour along the unstable direction:

Theorem 1 (Radon-Nikodym realization theorem). Given any positive Hölder continuous function , there are and such that

Remark 2. In fact, the reader should notice that we are not deforming the flow in order to obtaining the desired Radon-Nikodym derivative (although we promised to do so in the 1st step); of course, we could do it, but this would lead to technical problems which we are going to avoid in the following way: instead of deforming directly the flow (in order to get a desired dynamical behaviour), we keep the same flow at the cost of using the family of measures to deform the differentiable structure of the manifold. In other words, we make a little change of point of view: a deformed flow on a given manifold corresponds to a fixed flow on a deformed manifold (with the advantage that the deformation of the differentiable structure is easier to control than the deformation of the flow).

Proof of theorem 1. The basic idea here is the following: Anosov flows can be accurately modeled by the suspension flow over a subshift of finite type: more precisely, one can consider a Markov partition so that the dynamics of any point (outside the boundaries of ) can be ‘codified’ by its itinerary with respect to the elements of this partition (i.e., at each time we look at the number indexing the rectangle containing our point). For more details on the suspension flow (over an arbitrary discrete time dynamical system) the reader can consult my previous post and for more explanations about Markov partitions for hyperbolic systems the reader can see the excelent classical book of R. Bowen. It follows that the original Anosov flow and the suspension of the subshift are Hölder conjugated, so that the replacement of the Anosov system by this ‘toy model’ is harmless for our purposes.

Next, one look at the toy model provided by the suspension of the subshift with a mixing transition matrix and it turns out that the similar statement is true by the theory of equilibrium states (essentially the measure we are searching is the Gibbs measure obtained by maximizing the pressure of a certain potential). I.e., given a Hölder continuous function , take the reference measure for the Gibbs (-invariant) measure attaining the topological pressure of among all -invariant measures (that is, ). It holds . Moreover, if verifies , we can find a constat such that the Bowen’s equation holds. On the other hand, after doing the translation of the Radon-Nikodym problem for to the Radon-Nikodym problem for the subshift with transition matrix A that it suffices to find a measure such that

where is a Hölder continuous function with . As we saw above, it suffices to use the theory of equilibrium states with the potential to obtain the desired measure.

Unfortunately, since the theory of equilibrium states for subshifts of finite type takes an entire post by itself, I’m unable to give further details of this argument here. I refer the reader to the sections 2 and 3 of Asaoka’s paper for further details.

Remark 3. Although it was not explicitly stated in the ‘proof’ of theorem 1, we are strongly using the codimension-one hypothesis here (as Cawley did when dealing with Anosov diffeomorphisms of ).

Now we proceed to investigate the deformation of the Anosov flow versus the deformation of the differentiable structure of the manifold. Recall that the weak stable foliation is . So, if we fix a (Hölder) Riemannian metric , it makes sense to consider an orthogonal splitting and the Hölder continuous flows and where (resp. ) denotes the orthogonal projection onto (resp. ). Also, we introduce the cocycles and over : the name cocycle comes from the property

and

We apply the Radon-Nikodym realization theorem with , so that we get a constant and a family of measures verifying

.

Lemma 1. .

Proof. Take a one-dimensional foliation transverse to . Note that this is possible since is a codimension-one Anosov flow. Fix an atlas on and a parametrization such that

• and ;
• and (where );
• and .

Using this atlas , the parametrizations and the family of measures provided by theorem 1, we can change the differentiable structure of in the following way: we define and we declare that the atlas given by

is our new differentiable structure. In order to avoid some confusion, we denote by the manifold equipped with this new atlas. It is not hard to check that is a manifold (although the ‘identity’ map is only a bi-Hölder homeomorphism). See lemmas 4.2 and 4.3 of Asaoka’s paper. In any case, it turns out that the flow is (despite the fact that it is generate by a Hölder continuous vector field). Furthermore, one can directly check that

and ,

where and (observe that is a Hölder continuous metric).

Assume momentarily that is generated by a (Anosov) vector field (recall that it is only generated by a Hölder vector field at this point). Then, we have , so that

.

In particular, there exists such that , i.e., . On the other hand, we know that . Hence, by computing this last expression at , we get

.

Since is a Anosov flow, it follows that (because contracts any direction tangent to which is transverse to the flow direction). Hence, .

Finally, it remains to get rid of the assumption that the vector field generating is . This can be accomplish by perturbation of the vector field (in the local coordinates provided by ). It follows that this nearby (deformed) vector field is Anosov (and topologically conjugated to the initial system) so that the previous argument applies: indeed, although the initial vector field associated to is only Hölder (and it does not make sense to say that it is Anosov), it ‘remembers’ that it was constructed from the Anosov smooth system . In particular, after a little bit of technical work (see lemma 4.4 of Asaoka’s paper), we can show that the initial Hölder vector field behaves as an Anosov system (in the sense that nearby smooth systems are Anosov and topologically conjugated to ). This completes the proof.

Theorem 2. is topologically conjugated to an Anosov flow preserving a Hölder continuous volume form.

Proof. We have that is topologically conjugated to an Anosov system such that . In particular, it follows that preserves the volume form associated to the Hölder metric . Sending back this structure (from to ) via the Hölder homeomorphism gives the desired conclusion.

At this point, the proof of Asaoka’s theorem is essentially complete: up to the somewhat boring fact that the volume form is Hölder continuous (although one would like to get smooth volume form for the application to Verjovsky conjecture because S. Simic’s method needs some regularity). We overcome this technical problem in the next section.

-‘Pasting lemma’ technique and the regularization of the volume form-

A basic problem in order to get a smooth volume form is the following: since the differentiable structure on is only , our volume forms can’t be more regular than . However, one can bypass this problem with the following result of Hart:

Theorem 3 (Hart). For any foliation on a manifold , there exists a diffeomorphism on such that is a subbundle of . Moreover, can be taken arbitrarily close to the identity.

The proof of this result is not very difficult (for a short proof [of 1 page] of it see the appendix A of Asaoka’s paper), but we will skip it here. An interesting direct corollary of this theorem is the following fact:

Corollary 1. Given a manifold and an oriented foliation , we can find a differentiable structure on which is compatible with the initial structure such that is generated by a vector field.

We apply this corollary in the context of the theorem 2. We consider the orbit foliation of on the manifold . Since is , the corollary 1 gives a smooth structure on compatible with the initial structure and a nearby Anosov vector field generating such that preserves a Hölder volume form.

In order to conclude the proof of Asaoka’s theorem, it suffices to construct a vector field preserving a smooth volume form such that is nearby to (in the topology). To do so, we fix a Riemannian metric on and we consider a Hölder function such that for all (where is the flow generated by ). Note that this implies

.

Thus, although is only Hölder, we have that is differentiable along and

.

In particular, is a Hölder function. Now, we consider a function $h$ such that $h$ is close to and is close to , and a vector field $Y$ nearby in the topology.  Of course, this can be achieved by a standard use of mollifiers. It follows that is close to .

Next, we recall the formula

for any vector field , any (continuous) Riemannian metric and any function differentiable along the direction . Using this formula and the fact , we obtain

.

Therefore, is close to 0. In other words, we are almost done since the condition of the type means that the vector field preserves the volume form associated to .

At this point, we ‘adjust’ the vector field by a technique called ‘Pasting Lemma‘ (due to my coauthor A. Arbieto and myself in this paper). Initially, we developed the Pasting Lemma to deal with questions related to volume preserving diffeomorphisms (where the corresponding equation is ): the basic idea going back to the seminal paper of Dacorogna and Moser is the simple and powerful remark that one can adjust the diffeomorphism (or vector field) via the resolution of a very well-know PDE, namely, (perhaps with Dirichlet or Neumann boundary condition), but it turns out that this pasting lemma has some applications by other authors, e.g., Bochi, Fayad and Pujals, Araujo and Bessa (besides Asaoka himself as we are going to see).

In the present case, the application is quite direct because the PDE is linear on (while the case of diffeomorphisms is technically more subtle since is highly non-linear). In any case, we fix a reference point and we take the unique solution of the PDE with . Note that exists because is (and, a fortiori, ) and (since is compact and boundaryless). Moreover, the Schauder estimates say that is close to 0 because is close to 0. In particular, the vector field is a vector field $C^{1+}$ close to such that

,

i.e., preserves the (smooth) volume form of . This ends the proof of Asaoka’s theorem.

My main motivation for my posts in this blog is certainly the exposition of mathematical topics of my own interest. In particular, I try to avoid the discussion of political issues here. However, two friends and Iranian mathematicians working in Brazil (Hossein Movasati and Ali Tahzibi) were unable to get short visa in order to take part into scientific activities (a conference in Germany and an one-year post-doc in France, resp.) due to the fragile political relationship between Iran and other countries. Of course, each country is free to control its borders, but in the specific case of these two mathematicians, we can clearly say that these decisions were completely arbitrary at least for two reasons:

• both of them have permanent visa and job in Brazil (so that, as Hossein wrote in this post, they could prove that they are ‘good men’);
• both of them had the support of the local mathematicians (German and French, resp.) for these scientific visits;

As Amie pointed out in this comment here, there are certainly stories of this flavor involving the US government, but I was shocked to heard Ali’s history because I was unaware of such positions from the French government (specially because I’m living in France now).

In any case, if you want to express your opinion on these shameful political decisions and/or share similar histories, I invite you to joint us in this blog

http://iranianmath.blogspot.com/

begun by Hossein and Ali. I hope we can succeed against this kind of arbitrary politicial decision which is so harmful for the scientific (and in particular mathematical) community. In this direction, Ali proposed here the interesting idea of asking some help of important mathematical organizations such as IMU (and TWAS perhaps).

Thanks in advance for any support you can give in this issue! I promise my next topic will be on Asaoka’s proof of Verjovsky conjecture!

Yesterday, I gave a talk on the “Seminaire de Theorie Ergodique” at the Universite de Paris 13 (Villetaneuse) about a short note entitled “An example of a Teichmuller disk in genus 4 with degenerate Kontsevich-Zorich spectrum” that G. Forni and me just posted on the arXiv.

The Teichmuller geodesic flow acts (as any respectable geodesic flow) on the unit cotangent bundle of the Teichmuller space by parallel transport of points and vectors along geodesics of the Teichmuller metric (here one should be careful because the Teichmuller metric is not a Riemannian metric but only a Finsler metric).

Using the results of Alhfors and Bers in Riemann surface theory, it is possible to identify the unit cotangent bundle of the Teichmuller space with the space of holomorphic quadratic differentials on a Riemann surface of genus with area 1 modulo the group of orientation-preserving diffeomorphisms of which are isotopic to the identity. Note that acts naturally on by linear transformations on the pairs of Abelian differentials. The interesting consequence of this point of view is that it turns out that the Teichmuller geodesic flow on is simply the action of the diagonal subgroup on .

Now let me make a few general remarks about the fine structure of :

• is stratified into analytic spaces obtained by fixing the multiplicities of the zeroes of the quadratic differentials (here in view of the Riemann-Hurwitz theorem);
• can be endowed with a natural notion of ‘Lebesgue measure’: for each stratum there is a -invariant probability measure in the same class of the Lebesgue measure on the local charts given by the period map;

Once we know that has a good structure, we can start doing some Ergodic Theory. We consider the Teichmuller flow on a stratum . Although Veech showed that the strata are not always connected (see the works of Kontsevich, Zorich and Lanneau for the complete classification of the connected components), Masur and Veech (independently) managed to show that is ergodic on each connected component of and, more recently, Avila, Gouezel and Yoccoz proved that is exponentially mixing.

Next, we ask about the Lyapounov spectrum (i.e., the collection of Lyapounov exponents) of . In order to adress properly this question, Kontsevich and Zorich introduced the so-called Kontsevich-Zorich cocycle given by the quotient of the trivial cocycle by the mapping class group. It is know that is symplectic so that its Lyapounov spectrum with respect to any ergodic measure is symmetric:

.

Moreover, one can show that the Lyapounov exponents of are determined by the first (non-negative) Lyapounov exponents of .

After several numerical experiments, Kontsevich and Zorich conjectured that the Lyapounov exponents of with respect to the canonical ‘Lebesgue’ measure are all non-zero (i.e., is non-uniformly hyperbolic) and simple (i.e., multiplicity 1). Nowadays, we know that this conjecture is true due to the results of G. Forni (who proved the non-uniform hyperbolicity of ) and A. Avila, M. Viana (who showed that the simplicity of the Lyapounov spectrum). In particular, the Kontsevich-Zorich spectrum (i.e., the Lyapounov spectrum of ) of a generic point is well-understood.

A natural question (posed by Veech) related to this result concerns the Kontsevich-Zorich spectrum of non-generic points: how bad (or ‘degenerate’) can it be? This question was firstly answered by G. Forni who showed the existence of a Veech surface of genus 3 so that any -invariant measure supported on the -orbit of this surface has (moreover, it seems that there are no such examples in genus 2).

At this point, we are ready to state the following result:

Theorem(G. Forni, –). Any -invariant measure supported on the -orbit of the genus 4 Riemann surface associated to the algebraic equation

has ‘degenerate’ Kontsevich-Zorich spectrum: .

Roughly speaking, the basic idea here is: this Riemann surface is ’sufficiently symmetric’ (i.e., it has a ‘good’ automorphism group); on the other hand, by Forni’s method, the presence of symmetries to show some cancellations of the Lyapounov exponents (and the more symmetries you have, the more cancellations you get); finally, in the case of this surface, it turns out that the cyclic group of automorphisms generated by the symmetry (where is a 6-th root of unity) suffices to completely annilihate the Lyapounov spectrum (except for the ‘trivial’ exponent ).

Remark 1. It turns out that Forni’s method automatically implies that the cocycle along the -orbit of this surface is isometric (and moreover the cocycle is trivial in the sense that it is conjugated to constant). This is an interesting phenomenon if you compare with the ‘chaotic‘ behavior exhibited by generic points.

Remark 2. It is not hard to see (via simple arithmetic arguments) that the method of our paper does not produce any new examples of the type

where , and (the first two conditions are imposed to guarantee good symmetries while the third condition is necessary to get a connected Riemann surface). On the other hand, M. Möller told us that there are no further such examples among Veech surfaces (in any genus).

Of course, this theorem is just the tip of the iceberg: for instance, it would be interesting to know whether one can find examples of surfaces with a prescribed number of non-zero Lyapounov exponents (that is, given any , there is a surface with ?). It is worth to observe that Forni’s method has a weak point: while we can detect surfaces with all (but one) Lyapounov exponents equal to zero, we can’t prove that a prescribed part of the spectrum vanishes for a given surface and the basic reason is the lack of an explicit formula for the sum of the first exponents although Forni has an explicit formula for the sum of all exponents. Currently, I’m working with my post-doc advisor J.-C. Yoccoz in order to come around this problem by the usage of other methods, but this is still a work in progress… So, I think here is a good point to end this post!

I hope to see you soon in the proof of Asaoka’s theorem! Bye!

My third post on the series of discussions of lectures held at ICTP (Trieste, Italy) concerns Asaoka’s recent solution of a conjecture of Verjovsky.

Let me start by recalling some basic definitions and facts.

Definition 1. A flow on a compact manifold is called an Anosov flow whenever there are a -invariant splitting of the tangent bundle of and a constant such that

• is the one-dimensional subbundle of of tangents to the orbits of ,
• is uniformly contracted: for any and ,
• is uniformly expanded: for any and .

We say that is a codimension-one Anosov flow if is an Anosov flow with .

For sake of comparision, let me remind you the definition of Anosov diffeomorphisms:

Definition 2. A -diffeomorphism is Anosov if there are a $Df$-invarinat splitting and some constants , such that

• is uniformly contracted: for any and ,
• is uniformly expanded: for any and .

Remark 1. As one can expect, there are some “dynamical” differences between these two definitions: Anosov diffeomorphism are discrete-time systems while Anosov flows are continuous-time systems. At a first glance this seems to be of little relevance, but there are fundamental distinct dynamical behaviors hidden here. In fact, given an Anosov flow , one can infer some of its properties by the so-called time-one diffeomorphism associated to this flow. It turns out that is not an Anosov diffeomorphism: indeed, although we can decompose the tangent bundle into three -invariant subbundles, since is an isometry along the direction (exercise), can’t be Anosov due to the presence of this ”neutral” (or central) direction. We will come back to this point later in the discussion.

Now, let us see some basic examples of Anosov systems:

Example 1 (Hyperbolic linear tori automorphisms). Take a matrix with integer coefficients and determinant 1, i.e., . It is not hard to see that induces a diffeomorphism on the torus . Moreover, one can show that is an Anosov diffeomorphism whenever the spectrum of doesn’t intersect the unit circle (i.e., there are no eigenvalues with modulus 1): indeed, is the sum of the eigenspaces associated to eigenvalues of modulus and is the sum of the eigenspaces associated to eigevalues of modulus . In this situation, is called a hyperbolic toral automorphism.

Example 2 (Suspension of an Anosov diffeomorphism). Given a map and a “roof” function , one can define a suspension flow associated to in the following way: consider the space

and the flow

where is the unique integer such that

.

An interesting feature of this construction is the following fact: the suspension flow of an Anosov diffeomorphism is an Anosov flow (exercise). In particular, we see that the suspension flow of an Anosov diffeomorphism shares the good dynamical properties of original diffeomorphism.

From the example 2, we see that the study of Anosov flows would be significantly more easy provided that a general Anosov flow possesses the dynamical structure of a suspension of an Anosov diffeomorphism. More precisely, our life would be easier if every Anosov flow is topologically conjugated to a suspension of an Anosov diffeomorphism since the dynamics of Anosov diffeomorphisms are well-studied, e.g.:

Theorem (Franks and Newhouse). A codimension-one Anosov diffeomorphism is always topologically conjugated to a hyperbolic toral automorphism.

However, our life is not always easy: Franks and Williams constructed an Anosov flow in a 3-manifold whose non-wandering set is not the whole manifold. In particular, this flow is not conjugated to a suspension flow. Indeed, since our manifold is 3-dimensional, the Anosov flow associated to the suspension is forced to be of codimension one. By the result of Newhouse, the diffeomorphism and, a fortiori, the suspension flow is transitive and, therefore, it can’t be topologically conjugated to a non-transitive system.

Roughly speaking, the idea of Franks and Williams is the following: Anosov flows possesses a central direction where the dynamics is isometric; taking advantage of this neutral direction, they construct a Reeb component preserved by the flow; since the 2-torus of the Reeb foliation separates the 3-torus into two distinct connected components, the flow isn’t transitive.

On the other hand, Verjovsky showed that this phenomenon is purely 3-dimensional: any codimension-one Anosov flow on a n-manifold with is transitive. Motivated by this result, he made the following conjecture:

Verjovsky conjecture. Any codimension-one Anosov flow on a compact manifold of dimension is topologically equivalent to a suspension flow over an Anosov diffeomorphism of the torus.

As we already mentioned, Asaoka solved this conjecture based on a preprint of Simic (which is still under revision). More precisely, Asaoka’s strategy is the following:

• by the theorem of Verjovsky, we know that any codimension-one Anosov flow is transitive (when );
• by the results of Simic, a codimension-one volume-preserving Anosov is topologically equivalent to the suspension of a hyperbolic toral automorphism;

Thus, the proof of the Verjovsky conjecture is complete once we show the following theorem:

Theorem (Asaoka). Any transitive codimension-one Anosov flow is topologically conjugated to a smooth codimension-one volume-preserving Anosov flow.

Of course, the details of the proof of this result deserves another entire post, so that this is probably the best place to close the current discussion! Ja ne!