During the last 10 days, I was attending J. Palis’ 70th birthday conference held at Atlantico Hotel, Buzios, Brazil. As one could expect from a conference of this size (~220 participants), it was a nice opportunity to talk to coauthors/friends and to learn several interesting tools. Also, since Buzios has plenty of superb beaches (very close to the hotel) and we had 2 hours of lunch time break, the conference atmosphere was an interesting mixture of intense and relaxing moments. Furthermore, I guess that the fact that the young and senior participants were together in the same place provided a great opportunity to the young generations to meet (and talk directly) with the senior generations, so that they could start new collaborations. In particular, I strongly believe that Jacob Palis and the organizers are very proud of this beautiful conference (which was extremely successful in many ways).

Concerning the (intense) conference program (with 5 plenary talks and 2 parallel sessions per day), you can find the details here, but I should advance that I’m planning to write some posts related to some talks (e.g., J.C. Yoccoz, A. Zorich, A. Avila, etc.).

In any case, this post is intended to accomplish two goals: firstly, I’m making the slides of Enrique Pujals’ talk (on Palis’ work) available here (with his permission, of course), and secondly, I’m fulfilling my (public) promise of putting the slides of my talk here. Remark: Please notice that these pdf files contain photos, so that their sizes are relatively big (~46MB and ~34MB resp.)

About Enrique Pujals’ talk, let me mention that it was a very touching moment: during the preparation of his slides, Enrique had the nice idea of inviting some of Jacob’s mathematical sons and nephews to draw some pictures in order to show how Palis influence appears in several contexts; in particular, each slide of Enrique’s talk was a small tribute of some son/nephew of Jacob (whose signature always appears in the corresponding slide). Also, Enrique putted a lot of effort in making the exposition as funny as possible (e.g., when he showed a picture of Gugu’s son illustrating Newhouse phenomena with his hands, Enrique said that Jacob is still recruiting young people to Dynamical Systems). Finally, the talk ended with Enrique embracing Jacob and saying to him that student isn’t a word meant to refer to a past event, but to a present one.

About my talk, I have only to acknowledge the organizers for the imense honor to give the final talk of the conference: in fact, I was extremely touched (and nervous) with this invitation, specially because 10 years ago (during Palis 60th birthday conference), I was a 1st year graduate student trying to choose my primary research area (and of course my choice was particularly influenced by J. Palis). During the talk, I started by telling the audience about my first meeting with Palis. I guess that the first time Palis heard of me was during a visit of Jean-Pierre Serre. In fact, I was so excited to learn that J.P. Serre was at IMPA that I interrupted him in the middle of a calculation (in his office) to ask in Portuguese to take a photo with him. Of course, Serre didn’t understand my request and told to Jacob (IMPA’s director in 1997) about a strange boy who went into his office like a crazy. After this initial bad impression, during Palis 60th birthday conference (in 2000), I talked to him asking for some advice concerning my research career. Instead of giving me a long speech, Palis saw that I was planning to read his joint book with Floris Takens and he told me to take read the book to decide if I liked the subject or not. I acknowledge his advice and asked for an autograph in this book. Again, instead of writing a long dedicatory, he simply put one decisive word of encouragement: Sucess (this appears as the second slide of my talk). After that, I talked about two mathematical results (joint with J.C. Yoccoz, and G. Forni and A. Zorich) and I closed the exposition with a selection of key (serious and funny) moments of the conference. I hope you will enjoy it!

Remember one of the characterizations we got in ERT8 of weak mixing: a mps is weak mixing if and only if, for any ,

that is, if and only if the sequence of functions converges in the -norm to .

For our interests, the notion of weak mixing will be useful only if the above property extends to multiple functions, because, as we saw in ERT4, the convergence of these sequence corresponds to multiple recurrence. This actually holds, as we will see below, and is called weak mixing implies weak mixing of all orders. Such property will follow from two main ingredients:

1. The product characterization of weak mixing can be extended to multiple products.
2. The van der Corput trick allows an inductive argument.

It is worth mentioning that this is the second time van der Corput trick appears in these lectures (the first one was in ERT2). The reader not used to it might think it is just a technical step in the proof, but it is actually an ingredient present in many situations of Ergodic Ramsey Theory when dealing with random components of mps. It has many versions, each of them to the purpose of particular notions of multiple recurrence. We first discuss the one we need.

1. The van der Corput trick

Theorem 1 (van der Corput trick) If is a bounded sequence in a Hilbert space and if

then .

Proof: Take such that . Notice that, for a fixed ,

goes to zero as and so the assertion is equivalent to

By the triangle inequality,

which goes to zero as .

The significance of this inequality is that it replaces the task of bounding a sum of coefficients by that of bounding a sum of “differentiated” coefficients . This trick is thus useful in “polynomial” type situations when the differentiated coefficients are often simpler (have smaller order) than the original coefficients.

The above theorem is written in a modern fashion. The original van der Corput trick is actually known as van der Corput difference theorem and comes from the theory of uniform distributions.

Definition 2 We say that a sequence is uniformly distributed if, for every interval ,

Exercise 1 Given a sequence , prove that the following assertions are equivalent.

1. is uniformly distributed.
2. For every continuous , where is the Lebesgue measure.
3. (Weyl criterion) For any ,

Theorem 3 (van der Corput difference theorem) A sequence is uniformly distributed if is uniformly distributed for every .

Proof: By Weyl criterion and the assumption, . Fix , and define . Then

converges to zero as . By van der Corput trick,

converges to zero which, again by Weyl criterion, proves that is uniformly distributed.

The previous result implies Weyl equidistribution theorem, which is a generalization of Kronecker theorem on the uniform distribution of , for irrational .

Theorem 4 (Weyl) If is a polynomial with at least one of its coefficients irrational, then , , is uniformly distributed.

Proof: Follows from Kronecker theorem by sucessive applications of Theorem 3.

2. Weak mixing implies weak mixing of all orders

Remember that is weak mixing if and only if is ergodic whenever is ergodic. For each , let .

Proposition 5 If is weak mixing, then is ergodic, for every integers . In fact, it is weak mixing.

Proof: Note that is weak mixing, for any . To prove this, consider and of zero density such that

Let . This set has zero density, as

and the last fraction converges to zero.

We proceed by induction on . The case was proved in the last paragraph. Suppose that, for every integers , is ergodic. If ,

is the product of an ergodic and a weak mixing system, which proves our assertion.

The main result of this section is

Theorem 6 Let be weak mixing. Then, for any ,

in the -norm.

In other words, this means that, for any ,

The proof will consist in an inductive argument, with the use of van der Corput trick to reduce the case to and so on. By linearity of the expression, we assume that . In this setting, we want to prove that

where represents the norm of .

Proof: The case follows from the ergodicity of :

Suppose the result is true for and take . Define

Given , let . Then

By hypothesis,

Rewrite the above expression as

where and is given by

As is ergodic,

By the van der Corput trick,

which concludes the proof.

Corollary 7 (Multiple Poincaré Recurrence for weak mixing mps) If is weak mixing and , then

for every .

Proof: Just consider in (1).

Previous posts: ERT0, ERT1, ERT2, ERT3, ERT4, ERT5, ERT6, ERT7, ERT8.

1. The dichotomy between structure and randomness

The main tool used in ERT1 and ERT2 was: given a mps , we decomposed the space into two pieces: one structured, formed by the fixed (or periodic, depending on the case) functions, and other random, formed by the functions for which the Cesàro averages converge to zero. This represents an example of the dichotomy that surrounds Ergodic Ramsey Theory: structure vs. randomness. This is actually briefly discussed in the end of ERT2 and in a broader way in this paper of Terence Tao.

This idea is also used in other branches of Mathematics, specially in combinatorics, harmonic analysis, number theory, etc. We can cite many situations:

1. Theorems about the existence of ergodic averages.
2. Szemerédi regularity lemma.
3. Roth theorem on the existence of arithmetic progressions of length three in sets of positive density.
4. Gowers norms.
5. All proofs (up to my knowledge) of Szemerédi theorem.
6. Green-Tao theorem on the existence of arithmetic progressions in the primes.
7. The Julia set of holomorphic functions is either connected or a Cantor set.
8. Mané-Bochi ‘02: a -generic conservative diffeomorphim in surfaces either has all Lyapunov exponents zero almost everywhere or is Anosov.
9. Avila-Moreira ‘03: For almost every , the quadratic function is either regular (has a periodic attractor) or stochastic (has an invariant absolutely continuous probability with positive Lyapunov exponent).
10. Avila-Forni ‘07: almost every interval exchange transformation is either an irrational rotation or weak mixing.
11. Avila ‘10: for Schrodinger operators with a one-frequency and typical real analytic potential, the spectrum is either subcritical or supercritical.

Each situation has a notion of structure/randomness. The one we are interested is multiple recurrence of mps. Let us see this from the spectral theory point of view.

Given a mps , denote also by the Koopman-von Neumann operator, defined by

When necessary, we use the notation to denote this operator. Many of its spectral properties are related to ergodic properties of . We investigate the eigenvalues/eigenfunctions of . If the eigenfunctions form a basis of , then is determined. In fact, let be the multiset (repeated with multiplicity) of eigenvalues of and, for each , the eigenfunction associated to . If

then

In this case, we say that has pure point spectrum and is a compact system. This constitutes the structured notion we were looking for.

In contrast, when has no eigenvalues other than and it is simple, we say that has continuous spectrum and is a weak mixing system. It forms the random part.

As pure point/continuous spectrum are opposite notions, there is a hope that every mps can be decomposed into two parts: one compact and other weak mixing. This is not true at all. Instead, can be decomposed in several parts in such a way that every part is an extension of the previous one and it is a compact or weak mixing extension of the smaller one. In other words, the dynamics of is broken into many parts in which every braking is obtained from the previous one by adding one of the two dynamical prototypes we discussed above.

Formally speaking, given two mps and , we say that is an extension of if there is a surjective measurable map such that

We denote this by and is called a factor of .

Theorem 1 (Furstenberg structural theorem) Given a mps , there exists an ordinal and a family of factors of , for every , such that:

• is a single point.
• is a compact extension of for every successor ordinal .
• is the inverse limit of , for every limit ordinal .
• is a weak mixing extension of .

Above, inverse limit is in the sense that . This result will be discussed in Lebesgue-full detail in the last post. In order to understand it, we need to study four concepts:

1. Weak mixing systems.
2. Compact systems.
3. Weak mixing extensions.
4. Compact extensions.

These will be the topics of this and the next 2 or 3 lectures.

2. Weak mixing systems

The definition used below is different from the one we assumed above, but don’t worry: they will be shown to coincide. Actually, we will obtain various equivalent definitions of weak mixing.

Definition 2 A mps is weak mixing if, for every ,

Exercise 1 Consider a bounded sequence of nonnegative real numbers. Prove:

1. If , then Conclude that strong mixing implies weak mixing.
2. If , then Conclude that weak mixing implies ergodicity.

Exercise 2 is weak mixing if and only if

for every .

Proposition 3 is weak mixing if and only if

for every such that .

We leave the proof to the reader, which may be found in the book Topics in ergodic theory of W. Parry. The notion of weak mixing means that, in some sense, almost all the system behaves in a strong mixing way. This is what says the following lemma.

Lemma 4 Consider a bounded sequence of nonnegative real numbers. Then

if and only if there exists a set of zero density such that

Proof: () Define, for each , the set

is a ascending chain of subsets of . They are the sets that may give problems in the convergence of to zero. Each of them has zero density, because

and then

In this way, take an increasing sequence of integers such that . Define and

By definition,

It remains to prove that has zero density. Consider an integer , let us say, with . As ,

and then

which, by (2), implies that

Then, has zero density.

() Let such that , for every . Given , we want to prove that

for every large enough. By hypothesis, there is for which

and

Then, for ,

Taking such that

we get

which concludes the proof.

Taking and , Lemma 4 implies that is weak mixing if and only if, for every , there exists of zero density such that

By approximation, (4) is equivalent to the existence, for every , of a set of zero density such that

Another characterization comes from Proposition 3: is weak mixing if and only if

for every such that . In fact, by Lemma 4, converges a Cesàro to zero if and only if the same happens to .

Weak mixing, at first impression, seems an artificial notion, obtained by the relaxation of strong mixing. This is not the case: first because it also has the natural spectral characterization discussed in section 1 and second, as was already discussed above, weak mixing is an important part of every mps. In other contexts, it is abundant. For example, Avila and Forni proved that almost every interval exchange transformation is either an irrational rotation or weak mixing, in contrast to an older result of Katok which proved that these are never strong mixing.

3. Product characterization of weak mixing

Consider two mps and .

Definition 5 The product mps of and is the quadruple

where is the -algebra generated by and is the probability measure on defined by

Theorem 6 Given a mps , the following are equivalent.

1. is weak mixing.
2. is weak mixing.
3. is ergodic, for every ergodic mps .
4. is ergodic.

Proof: (i) (ii). It is enough to check (4) for a generating algebra of . Let . By assumption, there exist of zero density such that

and

The set has zero density and satisfies

proving that is weak mixing.

(ii) (i). Given , there exists of zero density such that

that is,

(i) (iii). Follows from the exercise below.

Exercise 3 Consider two bounded sequences and of real numbers. If and converge a Cesàro to and , respectively, then converges a Cesàro to .

(iii) (iv). If is the trivial mps with consisting of a single point, we conclude that is ergodic. Then, takin , it follows that is ergodic.

(iv) (i). First, note that is ergodic. Given ,

converges a Cesàro to

proving the assertion. Then

which converges to

4. Spectral characterization of weak mixing

We now characterize weak mixing in terms of spectral properties. At this point, it is interesting to introduce the

Theorem 7 If is an unitary operator on the Hilbert space and , then there is a unique finite Borel measure on the circle such that

When has continuous spectrum, is a continuous measure (it has no atoms), for every such that .In this case, Fubini theorem guarantees that gives zero measure to the diagonal . This in turn implies the

Theorem 8 is weak mixing if and only if has continuous spectrum.

Proof: () Suppose is an eigenfunction associated to . The function defined by is an eigenfunction of associated to . By Theorem 6, is constant and the same happens to .

() Let us check (6). Take such that . Using Theorem 7,

Decompose , where is the diagonal. For , the summand

converges to zero as uniformly in . Since assigns zero measure to , we’re done.

5. Conditions for weak mixing

In this section we resume all conditions obtained above for a a mps be weak mixing.

1. For any ,
2. For any ,
3. For any such that ,
4. For any such that ,
5. For any , there exists of zero density such that
6. For any , there exists of zero density such that
7. is ergodic.
8. is ergodic, for every ergodic.
9. is weak-mixing.
10. has continuous spectrum.

Previous posts: ERT0, ERT1, ERT2, ERT3, ERT4, ERT5, ERT6, ERT7.

As promissed in ERT6, this post intends to prove

Theorem 1 Let be a homeomorphism of the compact metric space . Then, for any and , there exist and such that

Moreover, given any dense subset , we can take .

Actually, we prove the more general version

Theorem 2 Let be a compact metric space and commutative homeomorphisms of . Then some is multiply recurrent, that is: there exists a sequence such that

1. Homogeneity

Consider a homeomorphism .

Definition 3 is homogeneous if there is a group of homeomorphisms of , each element of commuting with , such that the action is minimal.

We remark that the notion of minimality of a group action is the same as that of a single homeomorphism:

Definition 4 is homogeneous if there is a group of homeomorphisms of , each element of commuting with , such that and the action is minimal.

In particular, is closed. Note that does not need to be -invariant. This is the good point of the above definition: we analyse the dynamics of not by the original map , but through the action of . We now proceed to the technical details of the proof.

Proposition 5 Let be a homeomorphism and homogeneous. Suppose that, for each , there exist and such that

Then, for every , there exist and such that

In other words, the property assumed also holds for a pair on the diagonal .

Proof: Let be the group associated to the homegeneity of and . The -orbit of every is dense in and then becomes as close as we want to any other .

Let us prove that, in a scale of , the -orbits in are finite. More specifically, we prove the existence of for which

Consider an open cover of by , each of them with diameter at most . For each , the family is an open cover of (in fact, if , there exists such that , that is, ). Reduce this cover to a finite one:

The finite set satisfies the required property: if , then , for some , and , for some , implying that

Just rename the .

We now prove that, for every , there exist and such that

Let be such that

and , such that

Taking for which

and , we conclude the inequalities

Finally, we prove the proposition. Take and such that

By continuity, there is for which

Take and satisfying

Proceeding by induction, if

take such that

and , for which

Then, if ,

But may be chosen such that and then, taking , (1) implies the inequality

Proposition 6 Under the same conditions of Proposition 5, there exists recurrent for .

Proof: Define by

From Proposition 5, the image of has values arbitrarily close to zero. In addition, is upper semicontinuous. In fact, given and , let such that

By continuity of , there is for which

Then

and so, as is arbitrary,

Then the set of continuity points of is residual (in particular, it is non-empty). Let us show that . This will conclude the proof.

By contradiction, suppose that for some . Take a neighbourhood of and such that

We now use the homogeneity of to prove that a similar inequality to (1) holds in all of .

Let be the group associated to the homogeneity of . By compactness of and minimality of , there are such that

Take for which

Let us prove that , for every . If , there exists such that

and then, taking for which , (2) guarantees that

which contradicts (1).

2. Proof of Theorem 2

We proceed by induction. For , it is sufficient to take any point of a minimal set (which exists by Zorn’s Lemma). Suppose the result is true for and consider commutative homeomorphisms of . Let be the group generated by . We can assume that acts minimaly on . If this is not the case, we restrict to a -invariant closed subset for which is minimal (such minimal set exists again by Zorn’s Lemma).

Let be the diagonal and consider the product transformation . We wish to show that there exists recurrent for . To this matter, it suffices to check the hypotheses of Proposition 5:

I. is homogeneous for .

II. For each , there exist and such that

The action of can be induced in in a natural way, associating to the map . In this setting, if , the pairs and are isomorphic and then acts minimaly on . In particular, is homogeneous for , establishing I.

To prove II, define , . By the induction hypothesis, there are and such that

Taking

we get

concluding the proof of the theorem.

Previous posts: ERT0, ERT1, ERT2, ERT3, ERT4, ERT5, ERT6.

Hello! Since everybody (including myself) in Rio de Janeiro is in Carnival mood, this post will be very brief. Two weeks ago, Vitaly Bergelson and I shared a mini-course entitled Ergodic Ramsey Theory: where Combinatorics and Number Theory meet        Dynamics. This mini-course was a part of actvities of the first Brazilian School of Dynamical Systems held at Maceió from February 1 to Februrary 11 (my lectures were in the first week and Vitaly´s lecture were in the second week). During my lectures, I focused on the basic aspects of Ergodic Ramsey Theory, so that Vitaly could touch upon the more sophisticated (and recent) results. In particular, I talked about Ramsey Theorem, van der Waerden theorem, Szemeredi theorem, Furstenberg´s proof of Szemeredi theorem via his correspondence principle and multiple recurrence theorem, multiple recurrence for weak-mixing and compact systems, weak-mixing and compact extensions and Furstenberg-Zimmer structural theorem. Of course, the material of my part of the mini-course is largely covered by Furstenberg´s book and Terence Tao´s lecture notes, but I thought it could be useful for any interested student to take a look at the following set of notes prepared jointly with my friend Yuri Lima:

• First Lecture: pigeonhole principle, friendship theorem and Ramsey theorem; van der Waerden theorem and Hindman theorem; Szemeredi theorem; basic principles in Ramsey theory and Ergodic Ramsey theory; dynamical proofs of van der Wander and Szemeredi theorem modulo appropriate multiple recurrence results; Furstenberg´s multiple recurrence theorem; (there is also three appendices about the combinatorial proof of van der Waerden theorem, proof of topological multiple recurrence theorem of Furstenberg and Weiss and Furstenberg´s correspondence principle)
• Second Lecture: weak-mixing systems; characterizations of weak-mixing (inclunding its stability under products with ergodic systems and its spectral counterpart); proof of Furstenberg´s multiple recurrence theorem for weak-mixng systems; van der Corput trick;
• Third Lecture: compact systems; proof of Furstenberg´s multiple recurrence theorem for compact systems; Kronecker systems, topological classification of minimal equicontinuous systems, metrical classification of ergodic compact systems; spectral characterization of compact systems; compact and weak-mixing extensions; multiple recurrence results for weak-mixing and compact extensions; dichotomy between structure and ramdomness; Furstenberg-Zimmer structural theorem and “completion´´ of the proof of Furstenberg´s multiple recurrence theorem;
• Fourth Lecture: Harmonic Analysis proof of Roth theorem and its relationship with Furstenberg-Zimmer structural theorem.
• Fifth Lecture: Ultrafilters and Hindman theorem.

The previous post showed how to connect sets with ergodic theory, namely a measure-preserving system , where is the symbolic space and is the shift map. As the reader can check in ERT5, the measure is an accumulation point of Dirac probability measures along increasing intervals of orbits of the point associated to . For this reason, is supported in the orbit of . Then we could take as the orbit closure

instead of the whole space . The set , in addition of composing the mps , has the natural metric induced by . More precisely, endow with the discrete metric and with the product topology. By Tychonoff Theorem, is a compact topological space, and the distance defined as

generates the topology of (to see this, just note that the cylinders – sets of elements with fixed entries in a finite number of positions – form a basis of topology of and each of them is a ball with respect to the metric ). Also, is homeomorphism. In fact, we leave as exercise to the reader to prove that

It is natural to wonder how general results of topological dynamics can be obtained and applied to this setting. This is what we are going to do in this post. The first section consists of the relations between arithmetic properties of and topological properties of . The second section is deeper and we prove van der Waerden theorem assuming a topological multiple recurrence theorem, which will be proved in ERT7. The main result of this post is, then,

Theorem 1 (van der Waerden) If , then some contains arbitrarily long arithmetic progressions.

1. Combinatorics of vs Topology of

The set is always a compact, totally disconnected set (because is) and transitive with respect to (the orbit of is dense in ).

Proposition 2 Let and its associated set.

(i) is finite if and only if there exist a finite set and an integer such that is the disjoint union

(ii) is thick if and only if .

(iii) if and only if .

Proof: (i) is finite if and only if is periodic for , that is, if and only if there exists such that . Considering , we obtain the desired conclusion.

(ii) If is thick, there are intervals , , such that . Then

which converges to if . For the opposite implication, the same argument works: if then, for every , there exists such that

that is, . As is arbitrary, is thick.

(iii) If , there exist intervals , , such that

Fix an integer and decompose as the union of intervals of length (except, at most, the last one). That is, write , , and into intervals of lenght and one of lenght ( is possibly empty). If is large, and then some is contained in , so that

Again, as is arbitrary, . Reciprocally, if , there exists, for each , an integer such that

that is, . This proves that .

The situation (i) happens if , where . These sets have low complexity and are highly structured sets formed by infinite arithmetic progressions.

Proposition 3 If is minimal, then is syndetic.

Proof: Take any and consider the clopen cylinder

By minimality, the set of return times of to is syndetic. But

and so , implying that is syndetic.

The converse is false. For example,

is syndetic, but contains the fixed point (this follows from Proposition 2, because ). This means we have to look for more conditions about to characterize minimality of .

Question. What are these conditions?

Given , consider the -limit of , defined as

We say that is recurrent if .

Definition 4 A set is called IP-set if it there exists an increasing sequence such that contains the set

Theorem 5 If is recurrent, then contains the translate of an IP-set.

Proof: Construct inductively an increasing sequence as follows: is any element of and, for every , is a positive integer greater than such that the first entries of and are equal, that is

This last condition means that

We’ll prove that . This is equivalent to , for every . The case is obvious:

Suppose for some . Then

which concludes the proof.

Corollary 6 Choose ramdomly, that is, each is in with probability . Then almost surely contains the translate of an IP-set.

Proof: Consider the probability in induced by the vector . By Poincaré Recurrence Theorem (see ERT1), almost-every is recurrent.

2. Proof of van der Waerden theorem

We know how to translate the notion of subsets of to symbolic spaces. How to encode a partition

to topology? Well, instead if considering , we take and, for the partition given in (1), associate the element defined as

Such association follows the same philosophy of the previous section: to , there is a natural partition . By the same reasons described in Section 1, is a compact metric space and the same happens to the orbit closure of with respect to the shift . In this encoding, by definition of the distance ,

Therefore, the existence of a monochromatic arithmetic progression is equivalent to , that is,

This condition is guaranteed by the

Theorem 7 (Furstenberg-Weiss topological multiple recurrence) Let be a continuous map of the compact metric space . Then, for any and , there exist and such that

Moreover, given any dense subset , we can take .

Consider the transformation . By Theorem 7, we can fix such that, for some , the element satisfies

which is exactly (2). This concludes the proof of Theorem 1.

Previous posts: ERT0, ERT1, ERT2, ERT3, ERT4, ERT5.

From now on, instead of upper density, we will consider upper Banach density. This notion reflects better the the asymptotic behavior of the set. For example, the union of intervals , , has zero upper density, but has a rich combinatorics.

Definition 1 The upper Banach density of a set is

Note that , which will reveal to be good for our applications.

In the previous posts, we developed some machinery in Ergodic Theory and now we will relate them to combinatorics. The main tool is a correspondence principle due to Furstenberg. Before discussing it, let us restate a theorem of the last post.

Theorem 2 (Furstenberg) If is a mps, such that and is a positive integer, then there exists a positive integer such that

This is the main result of Furstenberg’s 1977 paper and was used to prove Szemerédi’s Theorem.

Theorem 3 (Szemerédi, 1975) If has positive upper Banach density, then it contains arbitrarily large arithmetic progressions.

1. Furstenberg’s Correspondence Principle

The question is: given such set , how to create a mps somehow related to ?

Theorem 4 (Furstenberg’s Correspondence Principle) Given a subset of positive upper Banach density, there exist a mps and a set such that and

for any integers .

Theorems 2 and 4 prove Theorem 3: note that the arithmetic progression belongs to if and only if . So, positive denseness of the set , in particular, proves its non-emptyness. Apply Theorem 2 to the mps obtained by Theorem 4: given , there exists such that

and then

As is arbitrary, Szemerédi’s Theorem is established.

The upper Banach density gives the same weight to all integers. It might happen (altough I still did not find any examples) that a set might have positive density if some subsets of are heavier than others. For example, if we give no weight to the even numbers, then the odd numbers have density one. In this post, I will give a general version of the correspondence principle. It follows the same ideas of the original proof of Theorem 4. I hope this extension, in addition of proving the classical correspondence principle, give rise to new applications of Theorem 2 for some classes of sets of zero upper Banach density that are big in some other sense.

2. A general correspondence principle

Consider a sequence of non-negative real numbers such that

For example, every constant sequence satisfies this condition. Given a set , define the upper Banach density with respect to as:

where stands for the usual Dirac measure. This density is a well defined number between and . We will prove the following

Theorem 5 If satisfies , then there exist a mps and a set such that and

for every .

Proof: We consider the most natural system: the set of characteristic functions of sets of integers.

The dynamics considered is the bilateral shift defined by

We apply a Krylov-Bogolubov argument to create a -invariant measure . Take , that is, with .

Let , be sequences of integers for which

We may assume (restricting , to subsequences, if necessary) that the probabilities defined by

converge in the weak- topology to a probability , that is,

for every continuous  . Under the assumption (2), is -invariant. In fact,

and so

implying that

which, by hypothesis, converges to zero as . Good: we have our probability space!

Continue the construction taking . Then

that is,

This is the connection between and we were looking for. It implies that

and, as is a clopen set (all cylinders are clopen),

It remains to verify (3), which actually follows from the last argument:

and then

This concludes the proof.

It is established the connection between combinatorics of sets of integers and ergodic theory.

Previous posts: ERT0, ERT1, ERT2, ERT3, ERT4.

In this post we discuss, without proofs, convergence of multiple ergodic averages to give the reader a broader notion of the flavour of the results. The last two posts showed that recurrence is a natural phenomenon and occurs in a regular way. The next question is to ask for multiple recurrence: given a mps , such that and a positive integer , there exist positive integers such that

Formulated as it is, this question follows simply by multiple applications of Poincaré’s Recurrence Theorem (see ERT1): there exists such that . Letting , there exists such that , which is the same as

Repeating the argument times, we obtain positive integers such that

where . This is much more than we wanted. In fact, applying the argument infinitely many times, we construct a sequence of positive integers such that

for every finite family of subsets of . Unfortunately, we have no control in the ’s, so that combinatorial applications are harder. It would be interesting if we had some regularity in them. For example, can they form an arithmetic progression? The answer is YES and this constitutes one of the pilars of Ergodic Ramsey Theory.

Theorem 1 (Furstenberg) If is a mps, such that and is a positive integer, then there exists a positive integer such that

Obviously, the existence of such is equivalent to the existence of such that

Taking , the characteristic function of ,

where , so that (2) is equivalent to

This inquires the analysis of the averages

Due to its nonsymmetry, instead of (3) we consider commuting transformations , all of them preserving , and the averages

from now on called multiple ergodic averages. Clearly, (3) is a special case of (4) considering , . Although the purpose is the full generality of (4), it is natural first to investigate (3). Four situations deserve attention:

• If and , does (4) have positive ?
• -norm convergence.
• Pointwise convergence.
• What about convergence of multiple polynomial ergodic averages?

The first one was solved affirmatively by H. Furstenberg in the 1977 seminal paper Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions.

Theorem 2 (Furstenberg) Let be a mps and be non-negative and satisfy . Then, for any ,

As we’ll see in forecoming posts, this actually proves Szemerédi’s Theorem, via a correspondence principle between sets of integers of positive density and measure-preserving systems. One year after, motivated by a topological analogue due to B. Weiss, Furstenberg and Y. Katznelson established in An ergodic Szemerédi theorem for commuting transformations an extension to the commutative case.

Theorem 3 (Furstenberg and Katznelson) Let be a probability measure space and commuting transformations, all of them preserving . Then, for any non-negative such that ,

This result, in addition to extending Furstenberg’s Theorem, implies a purely combinatorial multidimensional version of Szemerédi’s Theorem.

Theorem 4 (Multidimensional Szemerédi’s Theorem) Let be a subset with positive upper-Banach density and be any finite configuration. Then there are an integer and a vector such that .

An interesting feature is that, until 2007, when hypergraph versions of Szemerédi’s Regularity Lemma were developed by T. Gowers, there was no combinatorial proof of this result.

After establishing positivity, we discuss -norm convergence, solved in Nonconventional ergodic averages and nilmanifolds by B. Host and B. Kra.

Theorem 5 (Host and Kra) Let be a mps and be bounded measurable functions on . Then

exists in .

Observe that we no longer have only one function, but . Three years later, in 2008, T. Tao extended it to the commuting setup in the work Norm convergence of multiple ergodic averages for commuting transformations.

Theorem 6 (Tao) Let be a probability measure space, measure-preserving commuting transformations and be bounded measurable functions on . Then

exists in .

It is worth mentioning that this year T. Austin gave a new proof of it using classical ergodic theory (On the norm convergence of  nonconventional ergodic averages). A few is known about pointwise convergence, only that

converges almost surely, for any and . This was obtained by J. Bourgain in Double recurrence and almost sure convergence.

Now consider polynomials with integers coefficients and the limits

What is known? In terms of combinatorial appications, does it at least have positive whenever is a non-negative bounded function such that ? Yes… due to V. Bergelson and A. Leibman in the work Polynomial extensions of van der Waerden’s and Szemerédi’s theorems.

Theorem 7 (Bergelson and Leibman) Let be a probability measure space, measure-preserving commuting transformations, polynomials with integer coefficients and a bounded measurable functions on such that . Then

In fact, they proved a more general result.

Theorem 8 (Bergelson and Leibman) Let be a probability measure space, measure-preserving commuting transformations, , ,, polynomials with integer coefficients and a bounded measurable functions on such that . Then

I could not find any result about -norm convergence. Two works, Convergence of polyomial ergodic averages by Host and Kra and Pointwise convergence of ergodic averages for polynomial sequences of rotations of a nilmanifold by Leibman, both in 2005, proved the situation of multiple polynomial ergodic averages along one transformation.

Theorem 9 (Host, Kra and Leibman) Let be a mps, polynomials with integer coefficients and bounded measurable functions on . Then

exists in .

Last News: a few hours ago this paper was posted in arXiv by Q. Chu, N. Frantzikinakis and B. Host stating many cases of the -norm convergence of multiple ergodic polynomial averages for commuting transformations.

Theorem 10 (Chu, Frantzikinakis and Host) Let be a probability measure space, measure-preserving invertible commuting transformations, polynomials with different degrees and bounded measurable functions on . Then

exists in .

As every fresh result, it first needs to be checked in full details.

Previous posts: ERT0, ERT1, ERT2, ERT3.

Today Jean-Christophe Yoccoz and I uploaded to the arXiv our joint paper “The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis“. Since the goal of this paper was already explained in this previous post (where the reader can also find some slides of a talk I gave at Orsay a few months ago), here I will only reproduce the abstract of the paper:

We compute explicitly the action of the group of affine diffeomorphisms on the relative homology of two remarkable origamis discovered respectively by Forni (in genus 3) and Forni-Matheus (in genus 4). We show that, in both cases, the action on the non trivial  part of the homology is through finite groups. In particular, the action on some 4-dimensional invariant subspace of the homology leaves invariant a root system of type. This provides as a by-product a new proof of (slightly stronger versions of) the results of Forni and Matheus: the non trivial Lyapunov exponents of the Kontsevich-Zorich cocycle for the Teichmuller disks of these two origamis are equal to zero.