In this post, I’ll transcript my notes of this nice talk (while taking full responsibility for any errors/mistakes in what follows).

**1. Introduction**

**1.1. Stationary measures**

Consider the linear action of on induces an action on the projective space . For later use, recall that via

Given a probability measure on , we can build a Markov chain / random walk whose steps consist into taking points into where is chosen accordingly with the law of .

The absence of hypothesis on might lead to uninteresting random walks: in fact, if a point is stabilized by two elements , then the random walk starting at associated to is not very interesting.

For this reason, we shall assume that

Hypothesis (i): the support of generates a *Zariski-dense* semigroup .

Remark 1By Tits alternative, in our current setting of , the hypothesis (i) can be reformulated by replacing “Zariski-dense” with “not solvable”.

As it was famously established by Furstenberg, the random walks associated to have a well-defined asymptotic behaviour whenever (i) is fulfilled:

Theorem 1 (Furstenberg)Under (i), there exists (an unique) probability measure on such that, for all ,

as . Here, the convolution of with a probability measure on is a probability measure on defined as

so that is the distribution of points obtained from after steps of the Markov chain associated to .

In the literature, is called *Furstenberg measure*, and it is an important example of –*stationary measure*, i.e., a probability measure on which is “invariant on average”:

**1.2. Lyapunov exponents**

The stationary measure can be used to describe the growth of the norms of random products associated to -almost every whenever satisfies (i) *and* its first moment is finite:

Theorem 2 (Furstenberg, Guivarch–Raugi)If has finite first moment, i.e.,

and satisfies (i), then

for -almost every .

The quantity is called *Lyapunov exponent*.

**1.3. Regularity of stationary measures**

The Furstenberg measure dictates the distribution of the Markov chains associated to and, for this reason, it is natural to inquiry about the *regularity* properties of stationary measures.

In this direction, Guivarch showed that the Furstenberg measures have a certain regularity when satisfies (i) and its exponential moment is finite:

Hypothesis (ii): there exists with .

Theorem 3 (Guivarch)Under (i) and (ii), there are and such that

for all and (where is the interval of radius centered at ). In particular, has no atoms.

More recently, Jialun Li established in this article here another regularity result by showing the decay of the *Fourier coefficients*

(where ). More concretely, he proved that:

Theorem 4 (Li)Under (i) and (ii), we have . In other words, is a Rajchman measure.

In a certain sense, the role of assumption (i) in the previous theorem is to avoid the following kind of example:

Example 1Let with

Note that the semigroup generated by is not Zariski dense in (as and are upper-triangular).We affirm that there is no decay of Fourier coefficients in this situation. Indeed, recall that if we identify with via , then acts on via Möbius transformations, i.e., an element acts on as

In particular, , , and the Fourier coefficients of the stationary measure given by the standard Hausdorff measure on middle-third Cantor set do not decay to zero.In a similar vein, if is a real number such that is a Pisot number, then with

admits a stationary measure (called the Bernoulli convolution of parameter describing the distribution of the points where with probability ) whose Fourier coefficients do not decay.

The proof of Theorem 4 is based on a renewal theorem. More concretely, given a function , let

By thinking of as a smooth version of the characteristic function of an interval , we see that is “counting random products with norm in the interval ”. In this context, Guivarch and Le Page established the following renewal theorem:

Theorem 5 (Guivarch–Le Page)Under (i) and (ii), one has

as .

Remark 2Another important fact in the proof of Theorem 4 is the non-arithmeticity of the Jordan projections of the elements of , i.e., the fact that these Jordan projections generate a dense subgroup of (whenever (i) is satisfied).

Since we will come back later to the discussion of deriving the decay of Fourier coefficients (e.g., Theorem 4) from a renewal theorem, let us now move forward in order to introduce the main result of this post, namely, a *quantitative* version of Theorem 4.

**2. Quantitative decay of Fourier coefficients**

The central result of this post is inspired by the following theorem of Bourgain and Dyatlov.

Theorem 6 (Bourgain–Dyatlov)If is the Patterson–Sullivan measure associated to a Schottky subgroup of , then there exists (depending only on the dimension of , i.e., the Hausdorff dimension of the limit set of the Schottky subgroup) such that

for all .

The method of proof of this result is based on the so-called discretized sum-product estimates from additive combinatorics.

Interestingly enough, this result can be interpreted as a decay of Fourier coefficients of certain *stationary* measures thanks to the following theorem:

Theorem 7 (Furstenberg, Sullivan, …)The Patterson–Sullivan measure of a Schottky subgroup coincides with the stationary measure of some probability measure on satisfying (i) and (ii).

Remark 3We saw the proof of a version of this result for cocompact lattices of in Proposition 14 of this blog post here.

The previous theorems suggest that a decay of Fourier coefficients of the Furstenberg measure associated to a probability measure on satisfying (i) and (ii). This statement was recently proved by Jialun Li in this article here.

Theorem 8 (Li)If is a probability measure on satisfying (i) and (ii), then there exists such that the Furstenberg measure associated to verifies

for all .

Remark 4Actually, Li’s theorem is stated in his article for any real split semisimple Lie group .

The proof of this result is also based on a discretized sum-product estimate. Moreover, this statement is closely related to spectral gap of transfer operators and a renewal theorem:

Theorem 9 (Li)Let be a probability measure on verifying (i) and (ii). Given , consider the transfer operator

acting on (with small enough). Then,we have the following spectral gap property: there exists such that the spectral radius of satisfies

for all .

Theorem 10 (Li)Under (i) and (ii), there exists such that the renewal operator satisfies

for all .

In his article, Li establishes first Theorem 8 from a discretized sum-product estimate, and subsequently Theorems 9 and 10 are deduced from Theorem 8.

Nevertheless, Li pointed out in his talk that Theorems 8, 9 and 10 are “morally equivalent” to each other. In fact,

- Theorem 8 Theorem 9: the Fourier decay can be used to prove spectral gap for transfer operators via the so-called
*Dolgopyat method*(which was discussed in this blog post here); - Theorem 9 Theorem 10: the spectral gap for transfer operators allows to deduce the renewal theorem because some elementary calculations reveal that is related to ;
- Theorem 10 Theorem 8: let us finally fulfil our promise made in the end of the previous section by briefly explaining the idea of the derivation of the Fourier decay in Theorem 8 from the renewal theorem in Theorem 10; since , the th Fourier coefficient of the Furstenberg measure is
by Cauchy–Schwarz inequality, the control of is reduced to the study of

since , we see that the size of the integral above depends on the “number of random products with norm in a given interval”, and the answer to this kind of “counting problem” is encoded in the asymptotic property of the renewal operator provided by Theorem 10.

]]>

Remark 5The analog of Theorem 10 in Abelian settings is false: the random walks driven by a finitely supported law on which is not arithmetic (i.e., its support generates a dense subgroup) verify a renewal theorem

for , but the error term is never exponential because grows polynomially with . (Of course, this phenomenon is avoided in the context of thanks to the fact that the Zariski-density assumption (i) on ensures an exponential growth of with .)

Today, I’ll transcript below my notes of a talk by Romain Dujardin explaining to the participants of our *groupe de travail* some basic convexity and continuity properties of the joint spectrum. After that, we close the post with a brief discussion of the question of prescribing the joint spectrum.

As usual, all mistakes in what follows are my sole responsibility.

**1. Preliminaries**

Let us warm up by reviewing the setting of the previous posts of this series.

Let be a reductive real linear algebraic group and denote its rank by . By definition, a maximal torus is isomorphic to .

The Cartan decomposition (with a maximal compact subgroup of ) allows to write any as for an unique where is a choice of Weyl chamber in the Lie algebra of . The interior of the Weyl chamber is denoted by .

Example 1For , we can take in , so that .

The element is called the Cartan projection of .

Example 2For , , where are the singular values of .

Similarly, the Jordan projection is defined in terms of the Jordan-Chevalley decomposition. For , this amounts to write the Jordan normal form with diagonalisable and nilpotent, so that with unipotent, , and has eigenvalues where are the eigenvalues of (ordered by decreasing sizes of their moduli).

The group has a family of distinguished representations such that the components of the vectors , resp. , are linear combinations of , resp. . In particular, the usual formula for the spectral radius implies that as (and, as it turns out, this fact is important in establishing the coincidence of the limits of the sequences and ).

Example 3For , the representations of on , , have the property that the eigenvalue of with the largest modulus is .

The rank of can be written as where is the dimension of the center of . In the literature, is called the *semi-simple rank* of . In general, we have “truly” distinguished representations which are completed by a choice of characters of .

Example 4For , , and the representations from the previous example have the property that with is “truly” distinguished and the determinant representation comes from the center.

Remark 1Recall that a weight of a representation of is a generalized eigenvalue associated to a non-trivial -invariant subspace, i.e., is a weight whenever

The weights are partially ordered via if and only if for all , and any irreducible representation possesses an unique maximal weight (and, as it turns out, is one-dimensional).In this context, the distinguished representations form a family of representations whose maximal weights provide a basis of .

A matrix is proximal when its projective action on possesses an attracting fixed point and a repulsive hyperplane . Also, an element is called –*proximal* if and only if the matrices are proximal for all (or, equivalently, ).

A matrix is -proximal whenever is proximal, , and for all , (where is the Fubini-Study on the projective space ). Moreover, is –*proximal* if and only if the matrices are -proximal for all .

A beautiful theorem of Abels–Margulis–Soifer asserts that -proximal elements are really abundant: given a Zariski-dense monoid of , there exists such that for all , one can find a finite subset with the property that for any , one can find with -proximal.

In the previous post of this series, we saw that Abels–Margulis–Soifer was at the heart of Breuillard–Sert proof of the following result:

Theorem 1If is compact and the monoid generated by is Zariski-dense in , then the sequences and converge in Hausdorff topology to a compact subset called the joint spectrum of .

After this brief review of the definition of the joint spectrum, let us now study some of its basic properties.

**2. Convexity of the joint spectrum**

Remark 2Later, we will see some sufficient conditions to get .

Similarly to the proof of Theorem 1, some important ideas behind the proof of Theorem 2 are:

- the Jordan projection behaves well under powers: ;
- the Cartan projection is subadditive: ;
- the Cartan and Jordan projections of proximal elements are comparable: there is a constant such that for all -proximal;
- Abels–Margulis–Soifer provides a huge supply of proximal elements.

We start to formalize these ideas with the following lemma:

Lemma 3If and are -proximal elements, then there are and such that for all .

*Proof:* After replacing by the matrix , our task is reduced to study the behaviours of the eigenvalues of largest moduli of proximal matrices .

By definition of proximality, the matrices converge to a projection on parallel to as . Also, an analogous statement is valid for . In particular, for any , one has

as .

It is not hard to show that there exists such that is not nilpotent: in fact, this happens because is Zariski-dense and the nilpotency condition can be describe in polynomial terms. In particular, and, by continuity, there exists with

for all . This ends the proof.

At this point, we are ready to prove Theorem 2. Since is a compact subset of , the proof of its convexity is reduced to show that for all .

For this sake, we begin by applying Abels–Margulis–Soifer theorem in order to fix and a finite subset so that for any we can find with -proximal. By definition, there exists such that any satisfies for some .

Next, we consider and we recall that . Hence, given , we have that for all sufficiently large, there are with

Now, we select with and -proximal. Recall that, by proximality, there exists a constant with

(and an analogous statement is also true for ). Furthermore, by Lemma 3, there are , say , and with

for all . Observe that .

By dividing by , by taking large (so that ) and by letting (so that ), we see that

for and sufficiently large.

Since is arbitrary and is closed, this proves that . This completes the proof of Theorem 2.

**3. Continuity properties of the joint spectrum**

**3.1. Domination and continuity**

Definition 4We say that is -dominated if there exists such that

for all sufficiently large and . (Recall that are the singular values of .)

Definition 5We say that is -dominated if is -dominated for all .

Remark 3If is -dominated, then it is possible to show that the joint spectrum is well-defined even when is not Zariski dense in .

The next proposition asserts that the notion of -domination generalizes the concept of matrices with simple spectrum (i.e., all of its eigenvalues have distinct moduli and multiplicity one).

Proposition 6is -dominated if and only if .

On the other hand, the notion of -domination is related to Schottky families.

Definition 7We say that is a -Schottky family if

- (a) any is -proximal;
- (b) for all .

Proposition 8is -dominated there are and so that is a -Schottky family.

*Proof:* Let us first establish the implication . It is not hard to see that if is -dominated, then is -dominated. Therefore, we can assume that is a -Schottky family. At this point, we invoke the following lemma due to Breuillard–Gelander:

Lemma 9 (Breuillard–Gelander)If is -Lipschitz on an non-empty open subset of , then .

*Proof:* Thanks to the decomposition, we can assume that . Given and sufficiently small, our assumption on implies that and . These inequalities imply the desired fact that after some computations with the Fubini-Study metric .

If is a -Schottky family, then all elements of are -Lipschitz on a neighborhood of any fixed , for all . By the previous lemma, we conclude that for all sufficiently large and . Thus, is -dominated.

Let us now prove the implication . For this sake, we use a result of Bochi–Gourmelon (justifying the nomenclature “domination”): is -dominated if and only if there is a dominated splitting for a natural linear cocycle over the full shift dynamics on , i.e.,

*Splitting condition*: there are continuous maps and such that for all (here, is the Grassmannian of hyperplanes of );*Invariance condition*: and for all (here, denotes the left shift dynamics );*Domination condition*: the weakest contraction along dominates the strongest expansion along , that is, there are and such that .

Remark 4For , the equivalence between -domination and the presence of dominated splittings was established by Yoccoz.

An important metaprinciple in Dynamics (going back to the classical proofs of the stable manifold theorem) asserts that “stable spaces depend only on the future orbit”. In our present context, this is reflected by the fact that one can show that depends only on and depends only on for all .

An interesting consequence of this fact is the following statement about the “non-existence of tangencies between and ”: if is -dominated, then for all . Indeed, this statement can be easily obtained by contradiction: if for some and , then has the property that and . Hence, , a contradiction with the splitting condition above.

At this stage, we are ready to show that if is -dominated, then is a -Schottky family for some and . In fact, given , let be the periodic sequence obtained by infinite concatenation of the word . We affirm that, for sufficiently large, is proximal with and , and -Lipschitz outside the -neighborhood of . This happens because the compactness of and the non-existence of tangencies between and provide an uniform transversality between and . By combining this information with the domination condition above (and the fact that for sufficiently large), a small linear-algebraic computation reveals that any is proximal and -Lipschitz outside the -neighborhood of for adequate choices of and .

The proof of the previous proposition gave a clear link between -domination and the notion of dominated splittings. Since a dominated splitting is robust under small perturbations (because they are detected by variants of the so-called cone field criterion), a direct consequence of the proof of the proposition above is:

Corollary 10The -domination property is open: if is -dominated, then any included in a sufficiently small neighborhood of is also -dominated.

The previous proposition also links -domination to Schottky families and, as it turns out, this is a key ingredient to obtain the continuity of the joint spectrum in the presence of domination.

Theorem 11If is -dominated, then the map is continuous at .

Very roughly speaking, the proof of this result relies on the fact that if a matrix is “very Schottky” (like a huge power of a proximal matrix), then this matrix is quite close to a rank 1 operator and, in this regime, the Jordan projection behaves in an “almost additive” way.

**3.2. Examples of discontinuity**

**3.2.1. Calculation of a joint spectrum in**

Recall that acts on Poincaré disk by isometries of the hyperbolic metric. Consider , where and are loxodromic elements of acting by translations along disjoint oriented geodesic axis and on from to and from to . We assume that the endpoints of the axes and are cyclically order on as , and we denote by and the translation lengths of and along and .

In the sequel, we want to compute and, for this sake, we need to understand where is a word of length on and .

Proposition 12If and are elements of as above, denotes the distance between the axes of and , and , then is the interval

*Proof:* One can show (using hyperbolic geometry) that is a loxodromic element whose axis stays between the axes of and while going from a point in to a point in , and the translation length of satisfies

In particular, .

We affirm that if is a word on and and is a word obtained from by replacing some letter by , then . In fact, by performing a conjugation if necessary, we can assume that and , so that and .

Therefore, if we start in with and we successively replace by until we reach , then we see from the claim in the previous paragraph that becomes denser in as . This proves that .

**3.2.2. Some joint spectra in**

Let as above and fix . We assume that there exists such that where is the rotation by .

The joint spectrum of in the plane with axis and is a triangle with vertex at , intersecting the -axis on the interval , and the side opposite to the vertex contained in the line . Indeed, one eventually get this description of because , , can be computed explicitly in terms of the joint spectrum of thanks to the fact that commutes with and . Note that .

Let us now consider , where denotes the rotation by . We affirm that for all and, *a fortiori*, is *discontinuous* at (because as ). In fact, given , since , the word

equals to . Therefore,

and, by letting , we conclude that , as desired.

**4. Prescribing the joint spectrum**

We close this post with a brief sketch of the following result:

Theorem 13

- (1) If is a convex body dans , there exists a compact subset of generating a Zariski-dense monoid such that .
- (2) Moreover, if is a convex polyhedron with a finite number of vertices, then there exists a finite subset generating a Zariski-dense monoid such that .

*Proof:* (1) If we forget about the Zariski-denseness condition, then we could take simply . In order to respect the Zariski-density constraint, we fix and we set where is a small neighborhood of the identity. In this way, the monoid generated by is Zariski-dense and it is possible to check that whenever is sufficiently small.

(2) Given a finite set whose convex hull is , we can take where is a finite set with sufficiently many points so that the monoid generated by is Zariski-dense.

]]>The links to the titles, abstracts, slides and videos for the talks of this excellent meeting can be found here.

In this blog post, I would like to transcript my notes for the amazing survey talk “On ellipsephic integers” by Cécile Dartyge on one of Christian’s favorite topics in Analytic Number Theory, namely, the statistics of integers missing some digits.

Of course, all mistakes in the sequel are my sole responsibility.

**1. Introduction**

*Ellipsephic integers* refers to a collection of integers with missing digits in a certain basis (e.g., all integers whose representation in basis 10 doesn’t contain the digit 9). Christian Mauduit proposed this nomenclature partly because ellipsis = missing and psiphic = digit in Greek.

Formally, we consider a basis , , and a subset of of cardinality . The corresponding set of ellipsephic integers is

The subset of ellipsephic integers below a certain threshold is denoted by

For the sake of exposition, we shall assume from now on that and

unless it is explicitly said otherwise.

**2. Ellipsephic integers on arithmetic progressions**

Let . Despite their sparseness, it was proved by Erdös, Mauduit and Sárközy that ellipsephic integers behave well (i.e., “à la Siegel-Walfisz”) along arithmetic progressions:

Theorem 1 (Erdös–Mauduit–Sárközy)There are two constants and such that

for all , , and sufficiently large.

*Proof:* As it is usual in this kind of counting problem, one relies on exponential sums. More precisely, note that

where . The “main term” comes from the case , so that our task consists into estimating the “error term”. For this sake, one has essentially to study

where . Observe that

The terms are controlled thanks to the following lemma (giving some saving over the trivial bound for all ):

Lemma 2 (Erdös–Mauduit–Sárközy)Let . For any , one has

where .

In order to take full advantage of the saving on the right-hand side of the inequality, one needs the following lemma:

Lemma 3 (Mauduit–Sárközy)For any and , one has

The details of the derivation of the desired theorem from the two lemmas above is explained in Section 4 of Erdös–Mauduit–Sárközy paper.

The methods of Erdös–Mauduit–Sárközy above paved the way to further results about ellipsephic integers. For instance, similarly to Bombieri–Vinogradov theorem, it is natural to expect that the distribution result of Erdös–Mauduit–Sárközy gets better on average: as it turns out, this was done independently by C. Dartyge and C. Mauduit, and S. Konyagin (circa 2000):

Theorem 4 (Dartyge–Mauduit, Konyagin)There exists such that for all there exists with the property that

*Proof:* One uses Lemmas 2 and 3 above, a large sieve method, and some bounds on the moments

of the function .

Remark 1More recently, K. Aloui, C. Mauduit and M. Mkaouar improved (in 2017) some of the results of Erdös–Mauduit–Sárközy to obtain some distribution results for ellipsephic and palindromic integers.

**3. Ellipsephic primes and almost primes**

By pursuing sieve methods, Dartyge and Mauduit obtained in 2001 the following result about ellipsephic almost primes:

Theorem 5 (Dartyge–Mauduit)There exists such that

where stands for the number of prime factors of (counted with multiplicity).

A natural question motivated by this theorem concerns the determination of explicit values of in the previous statement. The answer to this question is somewhat related to the value of in the last theorem of the previous section and, in this direction, it is possible to show that

- if , then one can take
- (and ) for ,
- (and ) for , …,
- for , and
- as

- if , , then one can take .

In 2009 and 2010, C. Mauduit and J. Rivat proved two conjectures of Gelfond on sums of digits of primes and squares. The methods in these articles gave hope to reach the case (of ellipsephic primes) in Dartyge–Mauduit theorem above. This was recently accomplished by J. Maynard in 2016: if and , then

Remark 2In his thesis, A. Irving got analogous results for palindromic ellipsephic integers with digits in basis with two prime factors.

After this brief discussion of ellipsephic almost primes, let us now talk about ellipsephic integers possessing only small prime factors.

**4. Friable ellipsephic integers**

Recall that a friable integer is an integer without large prime factors. For later reference, we denote the largest prime factor of by

It was shown by Erdös–Mauduit–Sárközy that, for any fixed and for all , there are *infinitely many* ellipsephic integers of the form whose largest prime factor is .

Logically, this results motivates the question to establish the existence of a *positive proportion* of friable ellipsephic integers. This seems a hard task for *arbitrary* , but this problem becomes more tractable for small values of when the basis is large enough.

In fact, S. Col showed that there exists such that

Moreover, if , then it is possible to take (which is close to one for large). On the other hand, can be taken very small when and sufficiently large.

**5. Ellipsephic solutions to Vinogradov systems**

A Vinogradov system is a system of equations on the variables of the form:

A major breakthrough on counting solutions to Vinogradov systems was famously obtained by J. Bourgain, C. Demeter and L. Guth (see also the text and the video of L. Pierce’s Bourbaki seminar talk on this subject).

Concerning ellipsephic solutions to Vinogradov systems (i.e., solutions with for all ), Kirsty Briggs showed that for , prime and , the trivial bound on the number of solutions of the Vinogradov system

with can be improved into whenever . (In particular, this result is saying that in the case , the main contribution to the number of ellipsephic solutions of the corresponding Vinogradov system comes from the trivial solutions .)

**6. Ellipsephic numbers in finite fields**

The notion of finite-field analogs of ellipsephic numbers was studied by several authors including Dartyge, Mauduit and Sárközy.

In order to explain some results in this direction, let us setup some notations. Let be the power of a prime number and denote by a primitive element generating a basis of over . In this way, we can represent a number as

Given a set of digits with , the associated subset of ellipsephic numbers is

Given a polynomial , we can study the set of its ellipsephic values via the set

The size of is described by the following theorem:

Theorem 6 (Dartyge–Mauduit–Sárközy)If , then

This result is specially interesting when contains a positive proportion of . Moreover, it can be improved when contains consecutive digits.

More recently, a better result was obtained by R. Dietmann, C. Elsholtz and I. Shparlinski for the case . Finally, the reader can consult the work of C. Swaenepoel for further results.

]]>Today, after a long hiatus, I’ll try to accomplish part of this promise. More precise, I’ll transcript below my notes for the two talks (by Rodolfo Gutiérrez-Romo and myself) aiming to explain to the participants of our *groupe de travail* the proof of the first portion of Theorem 5 in the previous post, i.e., the convergence of (cf. Theorem 5 below), the convergence of (cf. Theorem 7 below), and the equality of the limits (cf. Theorem 9 below).

Evidently, all mistakes in what follows are my sole responsibility.

**1. Spectral radius formula revisited**

Let be a reductive linear algebraic group. Recall that the Cartan projection and Jordan projection were defined in the previous post via the Cartan decomposition and the Jordan–Chevalley decomposition with elliptic, unipotent, and “hyperbolic” conjugated to .

The *semisimple rank* of is where is a maximal torus of and is the center of . We denote by , a system of roots such that is a base of simple roots.

Each induces a weight satisfying and for all , where is the Lie subalgebra of and is a fixed extension of the Killing form on the Lie subalgebra of to the Lie algebra of such that becomes an orthogonal decomposition.

The weights , , are the highest weights of distinguished representations , of . One has

where is a choice of -invariant norm with diagonalisable in an orthonormal basis and stable under the adjoint operation, and denotes the top eigenvalue of a matrix . In particular, the Cartan projection is represented by a vector of logarithms of norms of matrices, the Jordan projection is represented by a vector of the logarithms of the moduli of top eigenvalues of matrices, and, *a fortiori*, the usual formula for the spectral radius implies that:

for every .

**2. Proximal elements and Cartan projections**

As we indicated in § 2.1 of the previous post of this series, the convergence of relies on the notion of *proximal matrices*.

Definition 2Let be the Fubini-Study metric on the projective space of a finite-dimensional real vector space equipped with an Euclidean norm .Given , we say that is a -proximal matrix whenever:

- has an unique eigenvalue of maximal modulus with eigendirection and -invariant supplementary hyperplane ;
- ;
- for all with and .

In general, we say that an element of a reductive linear algebraic group with distinguished representations , , is –*proximal* when the matrices are -proximal for all .

A basic feature of proximal elements is the fact their Cartan and Jordan projections are comparable (cf. Lemmas 2.15 and 2.16 of Breuillard–Sert paper extracted from Benoist’s paper).

Lemma 3There is a constant such that

and

for all .For each , there is a constant such that the Cartan and Jordan projections of any -proximal element satisfy

Another crucial feature of proximal elements (discovered by Abels, Margulis and Soifer, see Theorem 4.1 of their paper) is their ubiquity in Zariski dense monoids:

Theorem 4 (Abel–Margulis–Soifer)Let be a connected, reductive, real Lie group. Suppose that is a Zariski dense monoid. Then, there exists such that, for all , there exists a finite subset with the property: given , there exists so that is -proximal.

At this point, we are ready to prove the convergence of Cartan projections:

Theorem 5Let be a connected reductive real Lie group and suppose that is a compact subset generating a Zariski dense subgroup. Then,

converges in the Hausdorff topology as .

*Proof:* By Lemma 2 of the previous post, our task is reduced to show that , , stays in a compact region of , and for each , there exists such that for all and .

By Lemma 3, there exists a constant such that

for all . It follows that for all , that is, is confined in a compact region of .

Let us now estimate for , say with . By Abels–Margulis–Soifer theorem 4, we can select a *finite* subset of the monoid generated by such that for each , there exists so that is -proximal. In particular, we can take such that is -proximal for all . By Lemma 3, we have

and

Since , it follows from the triangular inequality that

Therefore, if we fix and we write , , we can use the Euclidean division , to obtain an element of via the formula

It follows from the definitions and Lemma 3 that

Since

by taking (or equivalently ) we derive that

Hence, given , there exists such that

for all , . This completes the proof.

**3. Twisting and Jordan projections**

A Zariski dense monoid of matrices is *twisting* in the sense that it always contains an element putting a finite configuration of lines and hyperplanes in general positions:

Lemma 6Let be a connected, reductive Lie group and suppose that is a Zariski dense monoid. Given a finite collection , , of irreducible representations of and finite configurations and , , of points and hyperplanes in , there is an element such that

for all and .

*Proof:* Since are irreducible, the sets are non-empty and Zariski open in . Thus,

is Zariski open in and non-empty (because is connected). Since is Zariski dense,

This completes the argument.

Remark 1The conclusion of this lemma can be reinforced as follows (cf. Remark 2.22 of Breuillard–Sert paper): it is possible to select from a finite subset of depending only on and (but not on and ).

At this stage, we can start the discussion of the convergence of Jordan projections:

Theorem 7Let be a connected reductive real Lie group and suppose that is a compact subset generating a Zariski dense subgroup. Then,

converges in the Hausdorff topology as .

*Proof:* Similarly to the previous section (on convergence of Cartan projections), our task consists into showing that for each , there exists such that

for all , . In this direction, let us fix and let us take , . By the formula for the spectral radius (cf. Lemma 1), we can fix with

By Abels–Margulis–Soifer theorem 4, we can fix a finite subset of the monoid generated by such that for some , we have that is -proximal.

By Lemma 3,

where , and

for all .

Consider the distinguished representations , , of . Note that the dominant eigendirection and the dominated hyperplane for the actions of the proximal matrices on are the same for all .

We fix . By the twisting property in Lemma 6, there exists , say , such that

for all and .

The dynamics of projective actions of the iterates of a proximal matrix is easy to describe: any direction transverse to is attracted towards . By rendering this argument slightly more quantitative (with the aid of the so-called *Tits proximality criterion*), Breuillard and Sert proved in Lemma 3.6 of their paper that

Lemma 8If is -proximal and is a finite subset such that

for all and , then there exists such that for all and , one has that is -proximal for all .

By applying this lemma with , we can select such that is -proximal for all , and .

Once again, it follows from Lemma 3 that

and

for all and .

By Euclidean division, we can write with and define

From our discussion above, we derive that

Since and

by letting (or equivalently, ) we conclude that

for all . This completes the proof.

**4. Coincidence of the limits**

Let be a connected, reductive real Lie group and let be a compact subset generating a Zariski dense monoid. By Theorems 5 and 7, we have that

as .

*Proof:* By the formula for the spectral radius (cf. Lemma 1), for all , one has as . In particular, .

In order to derive the other inclusion, we recall that the proof of Theorem 5 about the convergence of Cartan projections revealed that there exists and a constant such that for all and , there exists with and

Therefore,

Since , we have that is bounded and, *a fortiori*,

as . This shows that , as desired.

**5. Realization of the joint spectrum by sequences**

Closing this post, let us further discuss Theorem 5 from the previous post by showing that with is realized by a single sequence in the sense that .

For this sake, we use Abels–Margulis–Soifer theorem 4 and the strong version in Remark 1 of the twisting property in Lemma 6 to select a finite subset of and some constants such that for each there are with the property that is a *Schottky family* in the sense that is -proximal, and and for all . (This nomenclature comes from the fact that the projective actions of the elements in a Schottky family resemble the classical Schottky groups.) Note that where .

Let us now choose a rapidly increasing sequence so that

for all , and we define by

By definition, any finite word has the form where and is a prefix of . Observe that

By Lemma 3, . Moreover, Lemma 2.17 in Breuillard–Sert paper ensures that the Schottky property for the family makes that is a -proximal element with

Therefore, it follows from Lemma 3 that

Since converges to , we conclude that converges to as .

]]>**Serrapilheira Postdoctoral Fellowship – UFC**

The Department of Mathematics at Universidade Federal do Ceará (UFC) invites applications for a Serrapilheira Postdoctoral Fellowship in dynamical systems and ergodic theory. The position is for one year with start date at any moment between March 2020 and September 2020, with possibility of extension for another year.

**Qualifications and expectations**

The position is part of the project “Jangada Dinâmica – boosting dynamical systems in Brazil’s Northeastern region”, which is funded by Instituto Serrapilheira and aims to boost dynamical systems and ergodic theory in the mathematical community of universities located in the Northeastern region of Brazil. The applicant must have completed a PhD and be qualified for conducting research in either dynamical systems and/or ergodic theory. There are NO teaching duties. As part of the program, and to foster interaction, the fellow shall visit another department of Mathematics in the Northeast for one month each semester or two months per year. Applications from underrepresented groups in Mathematics are highly encouraged.

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The salary will range from 5000–6000 Brazilian Reais monthly, tax free, in a twelve month-base calendar, according to the applicant’s qualifications. There will be an extra 5000 Brazilian Reais for each of the two months of visits to another institution in the Northeast. The salary is more attractive than those offered by regular Brazilian funding agencies.

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**More information:** jangadadinamica@gmail.com.

Very roughly speaking, this heuristic computation went as follows: the WP sectional curvature of any -plane can be written as the sum of three terms; for the -planes considered in the previous post, the main term among those three *seemed* to be a kind of -norm of Beltrami differentials with essentially disjoint supports; finally, this -type norm was shown to be really small once a certain Green propagator is *ignored*.

Last April 2019, I met Scott during an event at Simons Center for Geometry and Physics, and I took the opportunity to tell him that one could perhaps show that the measure of the set of -planes leading to tiny WP curvatures is very small using the real-analyticity of the WP metric.

More concretely, my idea was very simple: since the Grassmannian of -planes tangent to a point is a compact space, the WP sectional curvature defines a real-analytic function , and we dispose of good upper bounds for and all of its derivatives in terms of the distance of to the boundary (see this article here), we can hope to get reasonable estimates for the measure of the sets using the techniques of these articles here and here (which are close in spirit to the classical fact [explained in Lemma 3.2 of Kleinbock–Margulis paper, for instance] that the measure of the sets are small whenever is a polynomial function on whose degree and -norm are bounded).

As it turns out, Scott thought that this strategy made some sense and, in particular, he promised to use my suggestion as a motivation to review his arguments concerning WP sectional curvatures.

After several email exchanges with Howard Masur and I, Scott announced that there were some *mistakes* in the construction of tiny WP sectional curvature: in a nutshell, one should not restrict the analysis to a single “main term” in the formula for WP sectional curvatures as a sum of three expressions, and one can not ignore the effect of the Green propagator. More importantly, Scott made a detailed study of these mistakes which ultimately led him to establish polynomial upper bounds for WP sectional curvatures at the heart of his newest preprint available here.

In this post, we will follow closely Scott’s preprint in order to give an outline of the proof of a polynomial upper bound for WP sectional curvatures:

Theorem 1 (Wolpert)Given two integers and with , there exists a constant with the following property.If denotes the product of the lengths of the short geodesics of a hyperbolic surface of genus with cusps whose systole is sufficiently small, then the sectional curvatures of the Weil-Petersson metric at are at most

Remark 1As it was pointed out by Scott in his preprint, it is likely that this estimate is not optimal: indeed, one expects that the best exponent should be rather than .

In what follows, we’ll assume some familiarity with some basic aspects of the geometry of the Weil–Petersson metric (such as those described in these posts here and here).

**1. Weil–Petersson sectional curvatures**

Let be a hyperbolic surface of genus with . If we write , where is the usual hyperbolic plane and is a group of isometries of describing the fundamental group of , then the holomorphic *tangent* space at to the moduli space of Riemann surfaces of genus with punctures is naturally identified with the space of harmonic Beltrami differentials on (and the *cotangent* space is related to quadratic differentials).

In this setting, the Weil–Petersson metric is the Riemannian metric induced by the Hermitian inner product

where and is the hyperbolic area form on .

Remark 2Note that is well-defined: if and are Beltrami differentials, then is a function on .

The Riemann tensor of the Weil–Petersson metric was computed by Wolpert in 1986:

where and is an operator related to the Laplace–Beltrami operator on .

Remark 3Our choice of notation here differs from Wolpert’s preprint! Indeed, he denotes the Laplace–Beltrami operator by and he writes .

The Riemann tensor gives access to nice formulas for the sectional curvatures thanks to the work of Bochner. More concretely, given and span a -plane in the real tangent space to at , let us take Beltrami differentials and such that , , and is orthonormal. Then, Bochner showed that the sectional curvature of is

Hence, by Wolpert’s formula for the Riemann tensor of the WP metric, we see that

**2. Spectral theory of **

Wolpert’s formula for the Riemann tensor of the WP metric hints that the spectral theory of plays an important role in the study of the WP sectional curvatures.

For this reason, let us review some key properties of (and we refer to Section 3 of Wolpert’s preprint for more details and references). First, is a positive operator on whose norm is : these facts follow by integration by parts. Secondly, is essentially self-adjoint on , so that is self-adjoint on . Moreover, the maximum principle permits to show that is also a positive operator on with unit norm. Finally, has a positive symmetric integral kernel: indeed,

where the Green propagator is the Poincaré series

associated to an appropriate Legendre function . (Here, stands for the hyperbolic distance on .) For later reference, we recall that has a logarithmic singularity at and whenever is large.

**3. Negativity of the WP sectional curvatures**

Interestingly enough, as it was first noticed by Wolpert in 1986, the spectral features of described above are sufficient to derive the negativity of WP sectional curvatures from Cauchy-Schwarz inequality. More precisely, since is self-adjoint, i.e.,

and its integral kernel is a *real* function, a straightforward computation reveals that the equation (1) for the sectional curvature of a -plane can be rewritten as

If we decompose the function into its real and imaginary parts, say , then we see that

Since is a positive operator, we conclude that and, *a fortiori*,

The non-positivity of the right-hand side of (2) can be established in three steps. First, the positivity of also implies that

Secondly, the fact that has a positive integral kernel allows to apply the Cauchy–Schwarz inequality to get that . Therefore,

Finally, the Cauchy–Schwarz inequality also says that

In particular, , so that it follows from (2) that all sectional curvatures of the WP metric are non-positive, i.e., .

Actually, it is not hard to derive that at this stage: indeed, would force a case of equality in Cauchy-Schwarz inequality and this is not possible in our context because is orthonormal.

Remark 4Philosophically speaking, the “analog” to this argument in the realm of Teichmüller dynamics is Forni’s proof of the spectral gap property for the Lyapunov exponents of the Teichmüller geodesic flow. In fact, after some computations with variational formulas for the so-called Hodge norm, Forni establishes that by ruling out an equality case in a certain Cauchy-Schwarz estimate.

**4. Reduction of Theorem 1 to bounds on ‘s kernel**

The discussion in the previous section says that small WP sectional curvatures correspond to almost equalities in certain Cauchy-Schwarz inequalities.

Hence, a natural strategy towards the proof of Theorem 1 consists into showing that an almost equality in (3) is impossible. In this direction, Wolpert establishes the following result:

Theorem 2 (Wolpert)There are two constants and with the following property. If we have an almost equality

between the terms and in (3), then and can not be almost equal:

Of course, Theorem 1 is an immediate consequence of Theorem 2 (in view of (2) and the estimate [implied by (3)]).

Thus, it remains only to prove Theorem 2. For this sake, we need further spectral information on , namely, some lower bounds on its the kernel . In order to illustrate this point, let us now show Theorem 2 *assuming* the following statement.

Proposition 3There exists a constant such that

whenever and do not belong to the cusp region of .

Remark 5We recall that the cusp region of is a finite union of portions of which are isometric to a punctured disk (equipped with the hyperbolic metric ).

For the sake of exposition, let us first establish Theorem 2 when is compact, i.e., , before explaining the extra ingredient needed to treat the general case.

**4.1. Proof of Theorem 2 modulo Proposition 3 when **

Suppose that for a constant to be chosen later. In this regime, our goal is to show that is “big” and is “small”, so that is necessarily “big”.

We start by quickly showing that is “big”. Since and are unitary tangent vectors, it follows from Proposition 3 that

Let us now focus on proving that is “small”. Since (cf. (3)), if we write (where and are the positive and negative parts of the real part of , then we obtain that

Since is positive, we derive that . Thus, if is compact, i.e., , then Proposition 3 says that for all . It follows that

By orthogonality of , we have that , i.e., . By plugging this information into the previous inequality, we obtain the estimate

Next, we observe that (cf. (3)) in order to obtain that

On the other hand, Proposition 3 ensures that for . Since and are unitary tangent vectors, one has for . By inserting this inequality into the previous estimate, we derive that

whenever has a sufficiently small systole.

This bound on can be converted into a bound thanks to Cauchy integral formula. More concrentely, as it is explained in Section 2 of Wolpert’s preprint, after observing that and replacing Beltrami differentials and by the dual objects and (namely, quadratic differentials), we are led to study quartic differentials . By Cauchy integral formula on , one has

On the other hand, if has systole and the cusp region is empty, then the injectivity radius at any is . Thus, there exists an universal constant such that

for all . By plugging this inequality into (7), we conclude that

for all .

Since is a positive operator on with unit norm (cf. Section 2 above) and , we have that the previous inequality implies the following bound on :

for all . By combining this estimate with (7), we conclude that

In summary, (4) and (8) imply that

for the choice of constant . This proves Theorem 2 in the absence of cusp regions.

**4.2. Proof of Theorem 2 modulo Proposition 3 when **

The arguments above for the case also work in the case because the cusp regions carry only a tiny fraction of the mass of the relevant functions, Beltrami differentials, etc.

More precisely, as it is explained in Section 2 of Wolpert’s preprint, if the constant is chosen correctly, then the Cauchy integral formula and the Schwarz lemma can be used to prove that

for all holomorphic quartic differentials .

In particular, we do not lose too much information after truncating , , etc. to and this allows us to repeat the arguments of the case to the corresponding truncated objects , , etc. without any extra difficulty: see Section 5 of Wolpert’s preprint for more details.

**5. Proof of Proposition 3**

Closing this post, let us give an idea of the proof of Proposition 3 (and we refer the reader to Section 4 of Wolpert’s preprint for more details).

Since and (cf. Section 2 above), our task is reduced to give lower bounds on the Poincaré series

For this sake, let us first recall that a hyperbolic surface has thick-thin decomposition: the thick portion is the region where the injectivity radius is bounded away from zero by a uniform constant and the thin portion is the complement of the thick region. Geometrically, the thin region is the disjoint union of the cusp region and a finite number of *collars* around simple closed short geodesics: roughly speaking, a collar consisting of the points at distance of a short simple closed geodesic of length .

We can provide lower bounds on in terms of the behaviours of simple geodesic arcs connecting and on .

More concretely, let be the shortest geodesic connecting and . Since is simple, we have that, for certain adequate choices of the constants defining the collars, one has that can not “back track” after entering a collar, i.e., it must connect the boundaries (rather than going out via the same boundary component). Furthermore, can not go very high into a cusp. Thus, if we decompose according to its visits to the thick region, the collars and the cusps, then the fact that permits to check that it suffices to study the passages of through collars in order to get a lower bound on .

Next, if is a subarc of crossing a collar around a short closed geodesic , then we can apply Dehn twists to to get a family of simple arcs indexed by giving a “contribution” to of

for some constant depending only on the topology of . In this way, the desired result follows by putting all “contributions” together.

]]>- each -diffeomorphism of the circle has a well-defined rotation number (which can be defined using the cyclic order of its orbits, for instance);
- is
*topologically semi-conjugated*to the rigid rotation (i.e., for a surjective continuous map ) whenever its rotation number is irrational; - if has irrational rotation number , then is
*topologically conjugated*to (i.e., there is an*homeomorphism*such that ); - if , , has an irrational rotation number satisfying a
*Diophantine condition*of the form for some , , and all , then there exists conjugating and (i.e., ); - etc.

In particular, if has *Roth type* (i.e., for all , there exists such that for all ), then any with rotation number is conjugated to whenever . (The nomenclature is motivated by Roth’s theorem saying that any irrational algebraic number has Roth type, and it is well-known that the set of Roth type numbers has full Lebesgue measure in .)

In the last twenty years, many authors gave important contributions towards the extension of this beautiful theory.

In this direction, a particularly successful line of research consists into thinking of circle rotations as standard interval exchange transformations on 2 intervals and trying to build smooth conjugations between generalized interval exchange transformations (g.i.e.t.) and standard interval exchange transformations. In fact, Marmi–Moussa–Yoccoz studied the notion of standard i.e.t. of *restricted Roth type* (a concept designed so that the circle rotation has restricted Roth type [when viewed as an i.e.t. on 2 intervals] if and only if has Roth type) and proved that, for any , the g.i.e.t.s close to a standard i.e.t. of restricted Roth type such that is -conjugated to form a -submanifold of codimension where is the first return map to an interval transverse to a translation flow on a translation surface of genus and is an i.e.t. on intervals.

An interesting consequence of this result of Marmi–Moussa–Yoccoz is the fact that local conjugacy classes behave differently for circle rotations and arbitrary i.e.t.s. Indeed, a circle rotation is an i.e.t. on 2 intervals associated to the first return map of a translation flow on the torus , so that has genus and also . Hence, Marmi–Moussa–Yoccoz theorem says that its local conjugacy class of with of Roth type has codimension *regardless* of the differentiability scale . Of course, this fact was previously known from the theory of circle diffeomorphisms: by the results of Herman and Yoccoz, the sole *obstruction* to obtain a smooth conjugation between and (with of Roth type) is described by a single parameter, namely, the rotation number of . On the other hand, Marmi–Moussa–Yoccoz theorem says that the codimension

of the local conjugacy class of an i.e.t. of restricted Roth type with genus *grows* linearly with the differentiability scale .

Remark 1This indicates that KAM theoretical approaches to the study of the dynamics of g.i.e.t.s might be delicate because the “loss of regularity” in the usual KAM schemes forces the analysis of cohomological equations (linearized versions of the conjugacy problem) in several differentiability scales and Marmi–Moussa–Yoccoz theorem says that these changes of differentiabilty scale produce non-trivial effects on the numbers of obstructions (“codimensions”) to solve cohomological equations.

In any case, this interesting phenomenon concerning the codimension of local conjugacy classes of i.e.t.s of genus led Marmi–Moussa–Yoccoz to make a series of conjectures (cf. Section 1.2 of their paper) in order to further compare the local conjugacy classes of circle rotations and i.e.t.s of genus .

Among these fascinating conjectures, the second open problem in Section 1.2 of Marmi–Moussa–Yoccoz paper asks whether, for *almost all* i.e.t.s , any g.i.e.t. with trivial conjugacy invariants (e.g., “simple deformations”) and conjugated to is also conjugated to . In other words, the and conjugacy classes of a *typical* i.e.t. coincide.

In this short post, I would like to transcript below some remarks made during recent conversations with Pascal Hubert showing that the hypothesis “for *almost all* i.e.t.s ” can *not* be removed from the conjecture above. In a nutshell, we will see in the sequel that the self-similar standard interval exchange transformations associated to two special translation surfaces (called *Eierlegende Wollmilchsau* and *Ornithorynque*) of genera and are but not conjugated to a rich family of piecewise affine interval exchange transformations. Of course, I think that these examples are probably well-known to experts (and Jean-Christophe Yoccoz was probably aware of them by the time Marmi–Moussa–Yoccoz wrote down their conjectures), but I’m including some details of the construction of these examples here mostly for my own benefit.

**Disclaimer:** As usual, even though the content of this post arose from conversations with Pascal, all mistakes/errors in the sequel are my sole responsibility.

**1. Preliminaries**

**1.1. Rauzy–Veech algorithm**

The notion of “irrational rotation number” for generalized interval exchange transformations relies on the so-called *Rauzy–Veech algorithm*.

More concretely, given a -g.i.e.t. sending a finite partition (modulo zero) of into closed subintervals disposed accordingly to a bijection to a finite partition (modulo zero) of into closed subintervals disposed accordingly to a bijection (via -diffeomorphisms ), an elementary step of the *Rauzy–Veech algorithm* produces a new -g.i.e.t. by taking the first return map of to the interval where , resp. whenever , resp. (and is not defined when ).

We say that a -g.i.e.t. has *irrational rotation number* whenever the Rauzy–Veech algorithm can be iterated indefinitely. This nomenclature is partly justified by the fact that Yoccoz generalized the proof of Poincaré’s theorem in order to establish that a -g.i.e.t. with irrational rotation number is *topologically semi-conjugated* to a standard, minimal i.e.t. .

**1.2. Denjoy counterexamples**

Similarly to Denjoy’s theorem in the case of circle diffeomorphisms, the *obstruction* to promote topological semi-conjugations between and as above into -conjugations is the presence of *wandering intervals* for , i.e., non-trivial intervals whose iterates under are pairwise disjoint (i.e., for all , ).

Moreover, as it was also famously established by Denjoy, a little bit of smoothness (e.g., with derivative of bounded variation) suffices to preclude the existence of wandering intervals for circle diffeomorphisms, and, actually, some smoothness is needed because there are several examples of -diffeomorphisms with any prescribed irrational rotation number and possessing wandering intervals. Nevertheless, it was pointed out by several authors (including Camelier–Gutierrez, Bressaud–Hubert–Maas, Marmi–Moussa–Yoccoz, …), a high amount of smoothness is *not* enough to avoid wandering intervals for arbitrary -g.i.e.t.: indeed, there are *many* examples of piecewise *affine* interval exchange transformations possessing wandering intervals.

Remark 2The facts mentioned in the previous two paragraphs partly justifies the nomenclature Denjoy counterexample for a -g.i.e.t. with irrational rotation number possessing wandering intervals.

In the context of piecewise affine i.e.t.s, the Denjoy counterexamples are also characterized by the behavior of certain *Birkhoff sums*. More concretely, let be a piecewise affine i.e.t. with irrational rotation number, say is semi-conjugated to a standard i.e.t. . By definition, the logarithm of the slope of is constant on the continuity intervals of and, hence, it allows to naturally define a function taking a constant value on each continuity interval of . In this setting, it is possible to prove (see, e.g., the subsection 3.3.2 of Marmi–Moussa–Yoccoz paper) that has wandering intervals if and only if there exists a point with bi-infinite -orbit such that

where the Birkhoff sum at a point with orbit for all is defined as , resp. for , resp. .

For our subsequent purposes, it is worth to record the following interesting (direct) consequence of this “Birkhoff sums” characterization of piecewise affine Denjoy counterexamples:

Proposition 1Let be a piecewise affine i.e.t. topologically semi-conjugated to a standard, minimal i.e.t. . Denote by the piecewise constant function associated to the logarithms of the slopes of .If for all with bi-infinite -orbit, then is topologically conjugated to (i.e., is not a Denjoy counterexample).

**1.3. Special Birkhoff sums and the Kontsevich–Zorich cocycle**

An elementary step of the Rauzy–Veech algorithm replaces a standard, minimal i.e.t. on an interval by a standard, minimal i.e.t. given by the first return map of on an appropriate subinterval .

The *special Birkhoff sum* associated to an elementary step is the operator mapping a function to a function , , where stands for the first return time to .

The special Birkhoff sum operator preserves the space of piecewise constant functions in the sense that is constant on each whenever is constant on each . In particular, the restriction of to the space of such piecewise constant functions gives rise to a matrix . The family of matrices obtained from the successive iterates of the Rauzy–Veech algorithm provides a concrete description of the so-called *Kontsevich–Zorich cocycle*.

In summary, the behaviour of special Birkhoff sums (i.e., Birkhoff sums at certain “return” times) of piecewise constant functions is described by the Kontsevich–Zorich cocycle. Therefore, in view of Proposition 1, it is probably not surprising to the reader at this point that the Lyapunov exponents of the Kontsevich–Zorich cocycle will have something to do with the presence or absence of piecewise affine Denjoy counterexamples.

**1.4. Eierlegende Wollmilchsau and Ornithorynque**

The Eierlegende Wollmilchsau and Ornithorynque are two remarkable translation surfaces and of genera and obtained from finite branched covers of the torus . Among their several curious features, we would like to point out that the following fact proved by Jean-Christophe Yoccoz and myself: if is a standard i.e.t. on or intervals (resp.) associated to the first return map of the translation flow in a typical direction on or (resp.), then there are vectors , and a -dimensional vector subspace such that is an equivariant decomposition with respect to the matrices of the Kontsevich–Zorich cocycle with the following properties:

- (a) generates the Oseledets direction of the top Lyapunov exponent ;
- (b) generates the Oseledets direction of the smallest Lyapunov exponent ;
- (c) the matrices of the Kontsevich–Zorich cocycle act on through a
*finite*group.

In the literature, the Lyapunov exponents are usually called the *tautological* exponents of the Kontsevich–Zorich cocycle. In this terminology, the third item above is saying that all non-tautological Lyapunov exponents of the Kontsevich–Zorich associated to and vanish.

In the next two sections, we will see that this curious behaviour of the Kontsevich–Zorich cocycle of or along allows to construct plenty of piecewise affine i.e.t.s which are but not conjugated to standard (and uniquely ergodic) i.e.t.s.

**2. “Il n’y a pas de contre-exemple de Denjoy affine par morceaux issu de et ”**

In this section (whose title is an obvious reference to a famous article by Jean-Christophe Yoccoz), we will see that the Eierlegende Wollmilchsau and Ornithorynque never produce piecewise affine Denjoy counterexamples with irrational rotation number of “bounded type”.

More precisely, let us consider is a piecewise affine i.e.t. topologically semi-conjugated to coming from (the first return map of the translation flow in the direction of a pseudo-Anosov homeomorphism of) or . It is well-known that the piecewise constant function associated to the logarithms of the slopes of belongs to (see, e.g., Section 3.4 of Marmi–Moussa–Yoccoz paper). In order to simplify the exposition, we assume that the “irrational rotation number” has “bounded type”, that is, is self-similar in the sense that some of its iterates under the Rauzy–Veech algorithm actually coincides with up to scaling.

If , then the item (c) from Subsection 1 above implies that all special Birkhoff sums of (in the future and in the past) are bounded. From this fact, we conclude that for all with bi-infinite -orbit: indeed, as it is explained in details in Bressaud–Bufetov–Hubert article, if is self-similar, then the orbits of can be described by a *substitution* on a finite alphabet and this allows to select a bounded subsequence of thanks to the repetition of certain words in the *prefix-suffix decomposition*.

In particular, it follows from Proposition 1 above that there is *no* Denjoy counterexample among the piecewise affine i.e.t.s topologically semi-conjugated to a self-similar standard i.e.t. coming from or such that .

Remark 3Actually, it is possible to explore the fact that is a stable vector (i.e., it generates the Oseledets space of a negative Lyapunov exponent) to remove the constraint “” from the statement of the previous paragraph.

In other words, we showed that any *always* provides a piecewise affine i.e.t. -conjugated to . Note that this is a relatively rich family of piecewise affine i.e.t.s because is a vector space of dimension , resp. , when is a self-similar standard i.e.t. coming from , resp. .

**3. Cohomological obstructions to conjugations**

Closing this post, we will show that the elements always lead to piecewise affine i.e.t.s which are *not* conjugated to self-similar standard i.e.t.s of or . Of course, this shows that the and conjugacy classes of a self-similar standard i.e.t. of or are distinct and, *a fortiori*, the Marmi–Moussa–Yoccoz conjecture about the coincidence of and conjugacy classes of standard i.e.t.s becomes false if we remove “for almost all standard i.e.t.s” from its statement.

Suppose that is a piecewise affine i.e.t. -conjugated to a self-similar standard i.e.t. of or , say for some -diffeomorphism . By taking derivatives, we get

since is an isometry. Of course, we recognize the slope of on the left-hand side of the previous equation. So, by taking logarithms, we obtain

where is a function. In other terms, is a solution of the cohomological equation and is a -coboundary. Hence, the Birkhoff sums are bounded and, by continuity of , the special Birkhoff sums of converge to zero. Equivalently, belongs to the *weak stable space* of the Kontsevich–Zorich cocycle (compare with Remark 3.9 of Marmi–Moussa–Yoccoz paper).

However, the item (c) from Subsection 1.4 above tells that the Kontsevich–Zorich cocycle acts on through a finite group of matrices and, thus, can *not* converge to zero under the Kontsevich–Zorich cocycle.

This contradiction proves that is not -conjugated to , as desired.

]]>Their beautiful paper was motivated by the quest of finding minimal homeomorphisms on punctured spheres . More concretely, the non-existence of such homeomorphism was previously known when (as an easy application of the features of Lefschetz indices), (thanks to the works of Brouwer and Guillou), and (thanks to the work of Handel), so that the main result in Jean-Christophe and Patrice paper ensures the non-existence of minimal homeomorphisms in the remaining (harder) case of .

A key step in Jean-Christophe and Patrice proof of their theorem above is to establish the following result about the sequence of Lefschetz indices of iterates of a local homeomorphism of the plane at a fixed point of : if is not a sink nor a source, then there are integers such that

As it turns out, Jean-Christophe and Patrice planned a sequel to this paper with the idea of extending their techniques to compute the sequences of Lefschetz indices of periodic points of belonging to any given Jordan domain with is compact.

In fact, this plan was already known when the review of Jean-Christophe and Patrice paper came out (see here), and, as Patrice told me, some arguments from this promised subsequent work were used in the literature as a sort of folklore.

Nevertheless, a final version of this preprint was never released, and, even worse, some portions of the literature were invoking some arguments from a version of the preprint which was available only to Jean-Christophe (but not to Patrice).

Of course, this situation became slightly problematic when Jean-Christophe passed away, but fortunately Patrice and I were able to locate the final version of the preprint in Jean-Christophe’s mathematical archives. (Here, the word “final” means that all mathematical arguments are present, but the preprint has no abstract, introduction, or other “cosmetic” details.)

After doing some editing (to correct minor typos, add better figures [with the aid of Aline Cerqueira], etc.), Patrice and I are happy to announce that the folklore preprint by Jean-Christophe and Patrice (entitled “*Suite des indices de Lefschetz des itérés pour un domaine de Jordan qui est un bloc isolant*“) is finally publicly available here. We hope that you will enjoy reading this text (written in French)!

This beautiful gesture of donating the books of a great mathematician to a developing country helped in the training of several mathematicians. In particular, I remember that reading Herman’s books during my PhD at IMPA was a singular experience in two aspects: intellectually, it gave me access to many high level mathematical topics, and olfactively, it was curious to get a smell of cigarette smoke out of old books (rather than the “usual” smell). (As I learned later, this experience was fully justified by the facts that Herman was an avid reader and a heavy smoker.)

Of course, this attitude of Jean-Christophe prompted me to discuss with Stefano Marmi about an appropriate destination in Africa to send Yoccoz’s mathematical books. After some conversations, we contacted ICTP (and, in particular, Stefano Luzzatto) to inquire about the possibility of sending Yoccoz’s books to Senegal (as a sort of “retribution” for the good memories that Jean-Christophe had during his visit to AIMS-Senegal and University of Dakar in December 2011) or Rwanda.

Unfortunately, some organisational difficulties made that we were obliged to split this plan into two parts. More concretely, rather than taking unnecessary risks by rushing to send Yoccoz’s books directly to Africa, last Thursday I sent all of them (a total of 13 boxes weighting approximately 35kg each) to ICTP library, so that they can already be useful to all ICTP visitors — in particular those coming from developing countries — instead of staying locked up in my office (where they were only sporadically read by me). In this way, we get some extra time to carefully think the definitive transfer of Yoccoz’s books to Africa while making them already publicly available.

Anyhow, the next time you visit ICTP, I hope that Yoccoz’s books will help you in some way!

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The mapping class group of isotopy classes of orientation-preserving diffeomorphisms of acts naturally on .

The dynamics of mapping class groups on character varieties was systematically studied by Goldman in 1997: in his landmark paper, he showed that the -action on is ergodic with respect to Goldman–Huebschmann measure whenever .

Remark 1This nomenclature is not standard: we use it here because Goldman showed here that has a volume form coming from a natural symplectic structure and Huebschmann proved here that this volume form has finite mass.

The ergodicity result above partly motivates the question of understanding the dynamics of individual elements of mapping class groups acting on -character varieties.

In this direction, Brown studied in 1998 the actions of elements of on the character variety . As it turns out, if is a small loop around the puncture, then the -action on preserves each level set , , of the function sending to the trace of the matrix . Here, Brown noticed that the dynamics of elements of on level sets with close to fit the setting of the celebrated KAM theory (assuring the stability of non-degenerate elliptic periodic points of smooth area-preserving maps). In particular, Brown tried to employ Moser’s twisting theorem to conclude that no element of can act ergodically on all level sets , .

Strictly speaking, Brown’s original argument is not complete because Moser’s theorem is used without checking the twist condition.

In the sequel, we revisit Brown’s work in order to show that his conclusions can be derived once one replaces Moser’s twisting theorem by a KAM stability theorem from 2002 due to Rüssmann.

**1. Statement of Brown’s theorem**

**1.1. -character variety of a punctured torus**

Recall that the fundamental group of an once-punctured torus is naturally isomorphic to a free group on two generators and such that the commutator corresponds to a loop around the puncture of .

Therefore, a representation is determined by a pair of matrices , and an element of the -character variety of is determined by the simultaneous conjugacy class , , of a pair of matrices .

The traces , and of the matrices , and provide an useful system of coordinates on : algebraically, this is an incarnation of the fact that the ring of invariants of is freely generated by the traces of , and .

In particular, the following proposition expresses the trace of in terms of , and .

*Proof:* By Cayley–Hamilton theorem (or a direct calculation), any satisfies , i.e., .

Hence, for any , one has

so that

It follows that, for any , one has

and

Since and , the proof of the proposition is complete.

**1.2. Basic dynamics of on character varieties**

Recall that the mapping class group is generated by Dehn twists and about the generators and of . In appropriate coordinates on the once-punctured torus , the isotopy classes of these Dehn twists are represented by the actions of the matrices

on the flat torus . In particular, at the homotopy level, the actions of and on are given by the Nielsen transformations

Since the elements of fix the puncture of , they preserve the homotopy class of a small loop around the puncture. Therefore, the -action on the character variety respects the level sets , , of the function given by

Furthermore, each level set , , carries a finite (*Goldman*—*Huebschmann*) measure coming from a natural -invariant symplectic structure.

In this context, the level set corresponds to impose the restriction , so that is naturally identified with the character variety .

In terms of the coordinates , and on , we can use Proposition 1 (and its proof) and (1) to check that

Hence, we see from (2) that:

- the level set consists of a single point ;
- the level sets , , are diffeomorphic to -spheres;
- the character variety is a -dimensional orbifold whose boundary is a topological sphere with 4 singular points (of coordinates with ) corresponding to the character variety .

After this brief discussion of some geometrical aspects of , we are ready to begin the study of the dynamics of . For this sake, recall that the elements of are classified into three types:

- is called elliptic whenever ;
- is called parabolic whenever ;
- is hyperbolic whenever .

The elliptic elements have finite order (because and ) and the parabolic elements are conjugated to for some .

In particular, if is elliptic, then leaves invariant non-trivial open subsets of each level set , . Moreover, if is parabolic, then preserves a non-trivial and non-peripheral element and, *a fortiori*, preserves the level sets of the function , . Since any such function has a non-constant restriction to any level set , , Brown concluded that:

Proposition 2 (Proposition 4.3 of Brown’s paper)If is not hyperbolic, then its action on is not ergodic whenever .

On the other hand, Brown observed that the action of any hyperbolic element of on can be understood via a result of Katok.

Proposition 3 (Theorem 4.1 of Brown’s paper)Any hyperbolic element of acts ergodically on .

*Proof:* The level set is the character variety . In other words, a point in represents the simultaneous conjugacy class of a pair of *commuting* matrices in .

Since a maximal torus of is a conjugate of the subgroup

we have that is the set of simultaneous conjugacy classes of elements of . In view of the action by conjugation

of the element of the Weyl subgroup of , we have

In terms of the coordinates given by the phases of the elements

the element acts by , so that is the topological sphere obtained from the quotient of by its hyperelliptic involution (and has only four singular points located at the subset of fixed points of the hyperelliptic involution). Moreover, an element acts on by mapping to .

In summary, the action of on is given by the usual -action on the topological sphere induced from the standard on the torus .

By a result of Katok, it follows that the action of any hyperbolic element of on is ergodic (and actually Bernoulli).

**1.3. Brown’s theorem**

The previous two propositions raise the question of the ergodicity of the action of hyperbolic elements of on the level sets , . The following theorem of Brown provides an answer to this question:

Theorem 4Let be an hyperbolic element of . Then, there exists such that does not act ergodically on .

Very roughly speaking, Brown establishes Theorem 4 along the following lines. One starts by performing a blowup at the origin in order to think of the action of on as a one-parameter family , , of area-preserving maps of the -sphere such that is a finite order element of . In this way, we have that is a non-trivial one-parameter family going from a completely elliptic behaviour at to a non-uniformly hyperbolic behaviour at . This scenario suggests that the conclusion of Theorem 4 can be derived via KAM theory in the elliptic regime.

In the next (and last) section of this post, we revisit Brown’s ideas leading to Theorem 4 (with an special emphasis on its KAM theoretical aspects).

**2. Revisited proof of Brown’s theorem**

**2.1. Blowup of the origin**

The origin of the character variety can be blown up into a sphere of directions . The action of on factors through an octahedral subgroup of : this follows from the fact that (3) implies that the generators and of act on as

In this way, each element is related to a root of unity

of order coming from the eigenvalues of the derivative of at any of its fixed points.

Example 1The hyperbolic element acts on via the element of of order .

**2.2. Bifurcations of fixed points**

An hyperbolic element induces a non-trivial polynomial automorphism of whose restriction to describe the action of on . In particular, the set of fixed points of this polynomial automorphism in is a semi-algebraic set of dimension .

Actually, it is not hard to exploit the fact that acts on the level sets , , through area-preserving maps to compute the Zariski tangent space to in order to verify that is one-dimensional (cf. Proposition 5.1 in Brown’s work).

Moreover, this calculation of Zariski tangent space can be combined with the fact that any hyperbolic element has a discrete set of fixed points in and, a fortiori, in to get that is transverse to except at its discrete subset of singular points and, hence, is discrete for all (cf. Proposition 5.2 in Brown’s work).

Example 2The hyperbolic element acts on via the polynomial automorphism (cf. (3)). Thus, the corresponding set of fixed points is given by the equations

describing an embedded curve in .

In general, the eigenvalues of the derivative at of the action of an hyperbolic element on can be continuously followed along any irreducible component of .

Furthermore, it is not hard to check that is not constant on (cf. Lemma 5.3 in Brown’s work). Indeed, this happens because there are only two cases: the first possibility is that connects and so that varies from to the unstable eigenvalue of acting on ; the second possibility is that becomes tangent to for some so that the Zariski tangent space computation mentioned above reveals that varies from (at ) to some value (at any point of transverse intersection between and a level set of ).

**2.3. Detecting Brjuno elliptic periodic points**

The discussion of the previous two subsections allows to show that the some portions of the action of an hyperbolic element fit the assumptions of KAM theory.

Before entering into this matter, recall that is Brjuno whenever is an irrational number whose continued fraction has partial convergents satisfying

For our purposes, it is important to note that the Brjuno condition has full Lebesgue measure on .

Let be an hyperbolic element. We have three possibilities for the limiting eigenvalue : it is not real, it equals or it equals .

If the limiting eigenvalue is not real, then we take an irreducible component intersecting the origin . Since is not constant on implies that contains an open subset of . Thus, we can find some such that has a Brjuno eigenvalue , i.e., the action of on has a Brjuno fixed point.

If the limiting eigenvalue is , we use Lefschetz fixed point theorem on the sphere with close to to locate an irreducible component of such that is a fixed point of positive index of for close to . On the other hand, it is known that an isolated fixed point of an orientation-preserving surface homeomorphism which preserves area has index . Therefore, is a fixed point of of index with multipliers close to whenever is close to . Since a hyperbolic fixed point with positive multipliers has index , it follows that is a fixed point with when is close to . In particular, contains an open subset of and, hence, we can find some such that has a Brjuno multiplier .

If the limiting eigenvalue is , then is an hyperbolic element with limiting eigenvalue . From the previous paragraph, it follows that we can find some such that contains a Brjuno elliptic fixed point of .

In any event, the arguments above give the following result (cf. Theorem 4.4 in Brown’s work):

Theorem 5Let be an hyperbolic element. Then, there exists such that has a periodic point of period one or two with a Brjuno multiplier.

**2.4. Moser’s twisting theorem and Rüssmann’s stability theorem**

At this point, the idea to derive Theorem 4 is to combine Theorem 5 with KAM theory ensuring the stability of certain types of elliptic periodic points.

Recall that a periodic point is called stable whenever there are arbitrarily small neighborhoods of its orbit which are invariant. In particular, the presence of a stable periodic point implies the non-ergodicity of an area-preserving map.

A famous stability criterion for fixed points of area-preserving maps is Moser’s twisting theorem. This result can be stated as follows. Suppose that is an area-preserving , , map having an elliptic fixed point at origin with multipliers , such that for . After performing an appropriate area-preserving change of variables (tangent to the identity at the origin), one can bring into its Birkhoff normal form, i.e., has the form

where , , are uniquely determined Birkhoff constants and denotes higher order terms.

Theorem 6 (Moser twisting theorem)Let be an area-preserving map as in the previous paragraph. If for some , then the origin is a stable fixed point.

The nomenclature “twisting” comes from the fact when is a twist map, i.e., has the form in polar coordinates where is a smooth function with . In the literature, the condition “ for some ” is called twist condition.

Example 3The Dehn twist induces the polynomial automorphism on . Each level set , , is a smooth -sphere which is swept out by the -invariant ellipses obtained from the intersections between and the planes of the form .Goldman observed that, after an appropriate change of coordinates, each becomes a circle where acts as a rotation by angle . In particular, the restriction of to each level set is a twist map near its fixed points .

In his original argument, Brown deduced Theorem 4 from (a weaker version of) Theorem 5 and Moser’s twisting theorem. However, Brown employed Moser’s theorem with while checking only the conditions on the multipliers of the elliptic fixed point but not the twist condition .

As it turns out, it is not obvious to check the twist condition in Brown’s setting (especially because it is not satisfied at the sphere of directions ).

Fortunately, Rüssmann discovered that a Brjuno elliptic fixed point of a real-analytic area-preserving map is always stable (independently of twisting conditions):

Theorem 7 (Rüssmann)Any Brjuno elliptic periodic point of a real-analytic area-preserving map is stable.

Remark 2Actually, Rüssmann obtained the previous result by showing that a real-analytic area-preserving map with a Brjuno elliptic fixed point and vanishing Birkhoff constants (i.e., for all ) is analytically linearisable. Note that the analogue of this statement in the category is false (as a counterexample is given by ).

In any case, at this stage, the proof of Theorem 4 is complete: it suffices to put together Theorems 5 and 7.

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