**1. Introduction **

A classical problem in Dynamical Systems is the investigation of closed trajectories in billiards in polygons.

In the case of rational polygons (i.e., polygons whose angles belong to ), it is known that closed trajectories are abundant. A popular way to establish this fact passes through the procedure of unfolding a rational polygon into a flat surface: roughly speaking, instead of letting the trajectories reflect on the boundary of the polygon, we reflect (finitely many times) the boundaries of the polygon in order to obtain straight line trajectories on a flat surface. See this excellent survey of Masur-Tabachnikov for more details.

For our purposes, we recall that a *flat surface* is the data of a Riemann surface , a non-trivial holomorphic -form and the flat metric thought as the Kähler form . Note that has zeros, i.e., has (conical) singularities, whenever has genus (by Riemann-Roch theorem).

Some of the key features concerning closed trajectories in flat surfaces are:

- closed geodesics of the flat metric come in families of parallel trajectories called
*cylinders*in the literature; - such closed geodesics occur in a dense set of directions in the unit circle ;
- Eskin and Masur showed the following asymptotics for the problem of counting cylinders: there exists a constant such that the number of (cylinders of) closed trajectories of length is .

The goal of this post is to generalize this picture to higher dimensions or, more precisely, to K3 surfaces.

**2. K3 surfaces **

Definition 1A compact complex -dimensional manifold is a K3 surface if

- (i) admits a (global) nowhere vanishing holomorphic -form ;
- (ii) the first Betti number is zero (this is equivalent to in this context).

Two basic examples of K3 surfaces are:

Example 1Quartic surfaces in , i.e., , is a polynomial of degree .

Example 2 (Kummer examples)Let be a complex torus (i.e., is a lattice of ). Then, has a subset of singular points, and the blow-up is a K3 surface.

Some basic properties of K3 surfaces include:

- all K3’s are diffeomorphic;
- all K3’s are Kähler (Siu);
- has rank , the Hodge intersection form has signature on , and is an even unimodular lattice;
- the Hodge decomposition of is , where has rank and signature , has rank and signature , and has rank and signature ;
- the data in the previous two items determine the K3 surface (by Torelli theorem).

See, e.g., the lecture notes of D. Huybrechts for more details.

**3. Special Lagrangian submanifolds **

The natural generalization of closed trajectories on flat surfaces are *special Lagrangian submanifolds* (SLags).

Definition 2Let , is a compact complex -dimensional manifold, is a Kähler form, is a holomorphic -form. A real -dimensional submanifold is a special Lagrangian submanifold (SLag) if

- is Lagrangian, i.e., ;
- is special, i.e., .

Remark 1Special Lagrangians are minimal submanifolds (in the sense that they minimize the volume in their cohomology class).

The next example justifies the claim that special Lagrangian submanifolds are the analog of closed trajectories on flat surfaces.

Example 3Consider the case , i.e., is a flat surface. In this situation, all real -dimensional submanifolds are Lagrangian. On the other hand, since is locally (away from its divisors) in this setting, we see that a special Lagrangian is a horizontal geodesic of the flat metric. In particular, if we replace by , then the SLags become the straight line trajectories at angle on .

In a similar vein, special Lagrangian fibrations are the analog of cylinders of closed horizontal trajectories on flat surfaces.

**4. Special Lagrangian fibrations **

Definition 3A fibration of over a real -dimensional base is SLag if its fibers are compact -dimensional SLags submanifolds of . The volume of such a fibration is where is any fiber of .

Remark 2The fibers of SLag fibrations are torii. One can compare this with the Arnold-Liouville theorem saying that the fibers of a fibration of a symplectic manifold by compact Lagrangian submanifolds are necessarily torii. In particular, the base has an integral affine structure whose structural group is the semi-direct product of by .

Remark 3Similarly to the case , a typical K3 surface doesn’t admit a SLag fibration.

K3 surfaces possess a significant amount of relevant structures. For example, a particular case of Yau’s solution to Calabi’s conjecture says that:

Theorem 4 (Yau)Let be a K3 surface equipped with a Kähler form . Then, there exists an unique in the same cohomology class of in such that induces a Ricci-flat metric.

Moreover, K3 surfaces with Ricci-flat metrics are hyperKähler manifolds, i.e., they admit three complex structures such that has the following properties:

- is a Ricci-flat Riemannian metric;
- are complex structures satisfying the usual quaternionic relations: ;
- are compatible with : the forms are closed (i.e. ) for .

Equivalently, we can write the data of a hyperKähler manifold as where and . In this way, we obtain a presentation of K3 surfaces bearing some similarities with our definition of flat surfaces.

**5. Statement of the main result **

In this setting, we change direction of SLag fibrations (in analogy with the case of closed trajectories in flat surfaces) through the notion of *twistor families*. More concretely, we consider the sphere and we denote by

where the fiber is equipped with the complex structure for .

At this point, we are almost ready to state the main result of this post: for technical reasons, we will give an impressionistic statement before explaining the true theorem in Remark 4 below.

Theorem 5 (Filip)Fix a (generic) twistor family . Then, there exists and such that

as .

Remark 4This statement is not quite true in the sense that one should count “equators in the twistor sphere” rather than counting points . Indeed, this is so because if a complex structure admits a SLag fibration at some point in the equator , then one also has SLag fibrations with the same “angle” as varies along the equator .

At this point, Filip started running out of time, and, for this reason, he offered the following sketch of proof of his theorem.

The first step is to reduce to counting elliptic fibrations (i.e., holomorphic fibrations whose fibers are elliptic curves).

The second step is to show that if the twistor family is not too special, then the counting problem reduces to the “linear algebra level” of the rank vector space equipped with a form of signature .

Finally, this last counting problem can be solved through quantitative equidistribution results on for the action of a certain -parameter subgroup on the quotient of the stabilizer in of a null vector in by a lattice (that is, the quotient of the semidirect product of and by).

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How frequent are *thin groups* among *Kontsevich-Zorich monodromies*?

Instead of explaning the meaning of Sarnak’s question in general, we shall restrict ourselves to the case of Kontsevich-Zorich (KZ) monodromies associated to *square-tiled surfaces*.

More concretely, let be a *square-tiled surface* (also called *origamis*) of genus , i.e., is a finite branched covering which is unramified off and is the pullback of on . We have a natural representation

from the group of affine homeomorphisms of to the group of symplectic matrices of the subspace of integral homology classes of projecting to zero under . In this setting, the *Kontsevich-Zorich monodromy* (associated to the -orbit of in the moduli space of translation surfaces) is the image of , i.e.,

(See e.g. these posts here for more background material on square-tiled surfaces.)

By following Sarnak’s terminology, we will say that is a *thin* group if is an infinite index subgroup of whose Zariski closure is

In the particular case of square-tiled surfaces, Sarnak’s question above is related to the following two problems:

- (a) find examples of square-tiled surfaces whose KZ monodromies are thin;
- (b) decide whether the “majority” of square-tiled surfaces in a given connected component of a stratum of the moduli spaces of unit area translation surfaces has thin KZ monodromy (here, “majority” could mean “all but finitely many” or “almost full probability as the number of squares/tiles grows”.)

The goal of this post is to record (below the fold) some discussions with Vincent Delecroix and certain participants of MathOverFlow around item (a).

Remark 1While we willnotgive answers to items (a) and/or (b) in this post, we decided to write it down anyway with the hope that it might be of interest to some readers of this blog: in fact, by the end of this post, we will show the followingconditional statement: if the group generated by the matrices

has infinite-index in , then a certain square-tiled surface of genus answers item (a) affirmatively.

Remark 2Some “evidence” supporting a positive answer to item (b) is provided by this recent paper of Fuchs-Rivin where it is shown that two “randomly chosen” elements (in ) “tend” to generate thin groups.

**1. Faithfulness and thinness **

Let be a square-tiled surface of genus . For the sake of simplicity, suppose that has no non-trivial automorphisms. In this case, the group is naturally identified with a finite-index subgroup of called *Veech group*, and, therefore, we get a representation

The following proposition produces a certificate of thinness for .

Proposition 1Assume that is virtually free and is faithful. Then, has infinite index in .

*Proof:* If is virtually free, is faithful and has finite index in , then the group of higher rank would contain a lattice isomorphic to a free group.

However, this is impossible because higher rank linear groups do not contain lattices isomorphic to free groups: on one hand, a higher rank linear group satisfies Kazhdan’s property (T), so that its lattices also have this property (by Kazhdan’s theorem), and, on the other hand, a free group does not have Kazhdan’s property (T).

This proposition says that a positive answer to item (a) would follow from a positive answer to the following problem:

- (c) find a square-tiled surface of genus (without non-trivial automorphisms) such that the Veech group is virtually free, the image of is Zariski dense in and is faithful.

As we are going to see in the next section, it is not hard to construct explicit examples of square-tiled surfaces (of genus without non-trivial automorphisms) such that the Veech group is virtually free. In other words, a positive solution of (c) is somewhat related to a positive answer to the following problem of independent interest (for the experts on translation surfaces):

- (d) find a square-tiled surface of genus such that the monodromy representation is faithful.

Remark 3If a square-tiled surface has a rational direction of homological dimension one (i.e., the waist curves of maximal cylinders of in a direction of rational slope span a one-dimensional subspace in absolute homology), then isnotfaithful: indeed, this happens because Dehn twists in such a direction act by the identity matrix on (see, e.g., Lemma 5.3 of this paper here), that is, they induce elements of .In particular, since any square-tiled surface of genus always have rational directions of one-cylinder decompositions (by the works of Hubert-Lelièvre and McMullen), it follows that the answer for the analog of item (d) in genus is always negative.

This remark suggests to try to answer items (c) and (d) by looking at square-tiled surfaces of genus with no one-cylinder rational directions. After a computer search (using Sage), Vincent Delecroix found a beautiful example of such a square-tiled surface whose description occupies the next section.

Remark 4For any prescribed , one can construct infinite families of square-tiled surfaces without -cylinders rational directions (via the Hubert-Lelièvre-Kani invariant described in this paper here).

**2. An origami without one-cylinder decompositions **

Consider the square-tiled surface associated to the pair of permutations

The commutator is

so that is a genus square-tiled surface.

The -orbit of consists of four elements. Indeed, this fact can be checked as follows. We recall that:

- the generators and of act on pairs of permutations by the rules and ;
- the pairs of permutations and give rise to the same square-tiled surface.

Therefore, the -orbit of is where is given by the pair of permutations with

As it turns out, the -orbit of accounts for its entire -orbit because

where

and

Remark 5For later use, observe that the matrix acts on pairs of permutations by . In particular, the action of on is completely described by the formulas

where and .

In summary, the -orbit of can be depicted as follows.

Since the cylinder decompositions of in any rational direction described by the horizontal cylinder decomposition of some element of , we have that all cylinder decompositions in rational directions of have exactly three cylinders.

Remark 6It follows from this discussion that has a single cusp (i.e., single -orbit in ). Also, the homological dimension of in the sense of Forni is three. Thus, by the results in this paper of Forni, the Lyapunov spectrum of the Kontsevich-Zorich cocycle over with respect to the Haar measure has the form

Moreover, the Eskin-Kontsevich-Zorich formula for the sum of non-negative Lyapunov exponents of the Kontsevich-Zorich cocycle imply that , i.e.,

Some numerical experiments (with Sage) indicate that and

**3. The affine homeomorphisms of **

The group of affine homeomorphisms of is the stabilizer of in the moduli space of translation surfaces.

It is not hard to see that the subgroup of automorphisms of is trivial. It follows that the elements of are determined by their linear parts in , that is, the natural map

is injective. Hence, is isomorphic to its image under this map.

The group is the finite-index subgroup of consisting of all elements of stabilizing : in the literature, is called the Veech group of .

We saw in the previous section that . Thus, is an index four subgroup of . Furthermore, is a congruence subgroup of level , and the Teichmüller curve has genus zero. Thus, is generated by elliptic and parabolic elements: indeed, one can check that is generated by the following two elliptic matrices

of orders .

The group structure of is provided by the following lemma:

*Proof:* Consider the twelve cones defined by the following properties:

- for each ;
- each , , consists of the convex combinations of positive multiples of the vectors and , where , , , , , and .

A simple calculation shows that

- for each ;
- , , , , , and .

It follows that and play ping-pong with the tables

and

in the sense that and are disjoint subsets of such that

- , ;
- , .

By the ping-pong lemma, we conclude that .

Remark 7The construction of the cones above is similar to the cones studied in Section 2 of this paper of Brav-Thomas.

**4. The Kontsevich-Zorich monodromy of **

The natural representation is called Kontsevich-Zorich cocycle over the arithmetic Teichmüller curve . In the sequel, we will compute the image under of the generators and of .

** 4.1. The relative homology groups of , **

Given , , let us denote by , resp., the relative cycles on consisting of the bottommost horizontal and leftmost vertical sides of the square numbered .

Note that each square of gives a relation , that is,

- , , , , , , , , ;
- , , , , , , , , ;
- , , , , , , , , ;
- , , , , , , , , .

** 4.2. The action of on the relative homology groups **

The matrix takes to , and it acts on the corresponding relative homology groups by the matrix such that

Similarly, the matrix takes to , and it acts on the corresponding relative homology groups by the matrix such that

Finally, exchange and , resp. and , and it acts on the corresponding relative homology groups by the matrices and such that

and

** 4.3. The absolute homology groups of , **

The absolute homology group has a basis where

and

Note that this basis is adapted to the decomposition in the sense that this decomposition corresponds to the partition where and , i.e.,

and

Moreover, it is worth to point out that, in the basis , the matrix of the restriction to of the intersection form is

** 4.4. The action of on the absolute homology group **

The matrices of , and with respect to the basis are

This allows us to compute the images and of the generators and of under the KZ cocycle . Indeed,

so that

For later use, we observe that these formulas give that the non-tautological subrepresentation of takes values

(with respect to the basis of ) at the two generators and of . Moreover, if we denote by , , then the characteristic polynomials and of the matrices and are

and

**5. Zariski density of in **

The matrices and are *Galois-pinching*, i.e., all roots of their characteristic polynomials and are real and simple, and the Galois groups of and have order (that is, the largest possible for symplectic matrices). Furthermore, the splitting fields of their characteristic polynomials are disjoint.

Indeed, these facts follow from the analysis of the following discriminants

and

related to the quadratic subfields of the splitting fields of and : see Proposition 6.14, Remark 6.15 and Proposition 6.16 in this paper here for more details.

By the Zariski density criterion of Prasad-Rapinchuk (see also page 3 of Rivin’s paper or this blog post here), we deduce that:

Proposition 3The Kontsevich-Zorich monodromy is Zariski-dense in .

Remark 8By the main result in this paper here, the Zariski-density of in implies that the Lyapunov spectrum in Remark 6 is simple, i.e.,

**6. Non-faithfulness of the representation **

Once we have Proposition 3 in our toolkit, it is natural to investigate the thinness of . Here, it is tempting to try to use the thinness certificate discussed in Proposition 1 from Section 1 in order to give a positive (partial) answer to Sarnak’s question.

By definition, the faithfulness of would amount to show that has the same group structure of the Veech group (described in Lemma 2), i.e., .

After some numerical experiments with *all* non-trivial words of length on and and *some* non-trivial words of length on and , I thought that could be faithful.

As it turns out, after I asked about the faithfulness of on MathOverflow, Stefan Kohl noticed that is *not* faithful because (a computer-assisted calculation shows that) the kernel of contains a certain non-trivial word of length in and .

Proposition 4 (S. Kohl)The representation is not faithful because

Remark 9The matrices and appearing in my MathOverflow question generate a conjugate of : indeed,

and

where

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Remark 1Still concerning beautiful talks delivered in this conference, I strongly recommend taking a look at this video of Peter Scholze’s talk on cohomology of algebraic varieties: indeed, I think that he was extremely sucessful in communicating his results to a broad audience of non-experts in Algebraic Geometry (such as myself).

This post is a transcription of my notes for Yves’ lecture and, as usual, all errors/mistakes below are my entire responsibility.

**1. Harmonic maps and quasi-isometries **

Definition 1Let and be Riemannian manifolds. An harmonic map is a critical point of Dirichlet energy

Equivalently, is harmonic whenever it satisfies the Euler-Lagrange equation

associated to Dirichlet energy.

Example 1Constant maps, geodesics ( interval) and, more generally, isometries with totally geodesic images are harmonic maps.

The literature dedicated to the questions of existence, uniqueness and regularity of harmonic maps is vast: see for instance these references here. In particular, we know that if is a simply connected non-positively curved Riemannian manifold, then any harmonic map is smooth () and attains the minimum of Dirichlet energy among all maps coinciding with outside a compact subset of .

Definition 2Given , a map between two metric spaces and is a –quasi-isometry whenever

for all .

A conjecture of Schoen-Li-Wang predicts the existence and uniqueness of an harmonic map within bounded distance to any given quasi-isometric self-map from a symmetric space of rank one.

**Conjecture (Schoen-Li-Wang).** Let be a non-compact symmetric space of rank one, i.e., is a hyperbolic space , , or (where is Hamilton quaternion algebra), or (where is Cayley octonion algebra).

Given a quasi-isometry , there exists an unique harmonic map within bounded distance to in the sense that

Remark 2Schoen made this conjecture for . Subsequently, Li-Wang proposed the generalized version of Schoen’s conjecture described above and they proved the uniqueness part of this conjecture.

Remark 3Before these conjectures were formulated, Pansu had already established them for , and .

Remark 4More recently, Markovic solved the case of Li-Wang conjecture and the initial conjecture of Schoen (i.e., the case ) in these papers here and here. Also, Lemm-Markovic confirmed the Li-Wang conjecture for the case , , in this paper here.

The purpose of this post is to discuss the following theorem of Benoist-Hulin establishing a *generalized* version of Schoen-Li-Wang conjecture:

Theorem 3 (Benoist-Hulin)Let and be non-compact symmetric spaces of rank one. Given a quasi-isometry , there exists an unique harmonic map such that

This result brings (at least) two novelties in comparison with previous theorems in the literature because:

- it settles Schoen-Li-Wang conjecture for (the remaining cases of) quasi-isometries of complex hyperbolic spaces, and
- it deals with quasi-isometries between and with different dimensions such as .

**2. Sketch of proof of Benoist-Hulin theorem **

For the sake of exposition, we will sketch the proof of Theorem 3 for quasi-isometries , i.e., **from now on** we will take and .

We begin by noticing that it is sufficient to show the *existence* of an harmonic map within bounded distance from : indeed, as we already told in Remark 2, the uniqueness of follows from the work of Li-Wang.

** 2.1. Regularization of quasi-isometries **

The first step in the proof of Theorem 3 is to *regularize* : by using bump functions, Benoist-Hulin show (with a 2 pages straighforward calculation) that is within bounded distance to a quasi-isometry whose covariant derivatives are bounded on for all (cf. Proposition 3.4 in Benoist-Hulin paper).

In other words, by replacing by if necessary, we can assume that is a quasi-isometry with and for some constant .

** 2.2. Reduction to a priori -estimates **

The second step is to reduce the construction of to an *a priori* -estimate (through a standard *compactness argument*).

More precisely, let us fix an origin , and let us consider the closed balls in . Next, we take the unique harmonic map satisfying the Dirichlet boundary condition on : the map minimizes the Dirichlet energy

among all maps with on . (These facts were proved by Schoen [see here] and Schoen-Uhlenbeck here and here.)

In this context, the existence of in Theorem 3 can be reduced to an uniform estimate on the distances between and :

Proposition 4Suppose that there exists such that

for all . Then, the sequence converges uniformly on compact subsets of to an harmonic map with

Before showing this proposition, we need to recall a (particular case of a) key lemma due to Cheng (see also Lemma 3.3 in Benoist-Hulin paper) allowing to “upgrade” estimates on harmonic maps into bounds on them.

Lemma 5 (Cheng)Let and a harmonic map such that the following estimate holds:

for some . Then, we have the following estimate on at :

Let us now use this lemma to prove Proposition 4.

*Proof:* By assumption, there exists such that for all , and, furthermore, for some . It follows that, given any increasing sequence , we have

for all large enough.

By Cheng’s lemma 5, this estimate leads to the following bound

for all and sufficiently large.

By Arzela-Ascoli theorem, we can find a subsequence converging uniformly on every ball , , to a continuous map with

Moreover, the harmonic maps minimize the Dirichlet energy and, on each ball , their energies are uniformly bounded

thanks to (1). By a compactness theorem of Luckhaus, this implies that the limiting map is harmonic and minimizes the Dirichlet energy.

Finally, the convergence , , of *whole* sequence follows from the convergence along subsequences and the uniqueness theorem of Li-Wang (see Remark 2) ensuring that the harmonic map with is unique.

In summary, Proposition 4 reduces the proof of Theorem 3 to

Theorem 6 (Benoist-Hulin)There exists such that

for all .

The proof of this theorem has two components:

*Boundary estimates*: one estimates for near ;*Interior estimates*: one estimates for far from .

Remark 5For the interior estimates, Benoist-Hulin use a proof by contradiction, i.e., they rule out the situation where as . Nevertheless, a careful inspection of their arguments shows that can be quantitavely bounded in terms of : see Remark 6 below.

** 2.3. Boundary estimates **

The behavior of for near is controlled by the following result:

Proposition 7Then,Let be a smooth map with and for some . Fix and denote by the harmonic map from the closed ball to with on .

for all .

*Proof:* Given , take closest to it, i.e., . Since and , one has

In order to estimate , we will use a barrier function. More precisely, we consider the geodesic passing through and , and we select a point in this geodesic with the following properties:

- is very far from , i.e., for all ;
- .

Since is harmonic, the function is subharmonic. Moreover, this function coincides with the smooth function on . Therefore, the maximum principle tells us that

for all (with equality for ), where is the smooth harmonic function coinciding with on .

In particular, our choice of implies

The barrier function can be controlled thanks to an estimate of Anderson-Schoen. More concretely, the function vanishes on and a short computation (at page 15 of Benoist-Hulin paper) reveals that its Laplacian verifies

In this context, the estimate of Anderson-Schoen (see Proposition 2.4 in Benoist-Hulin paper) asserts that

Therefore, this estimate together with (2) gives

This proves the proposition.

** 2.4. Interior estimates **

The boundary estimate in Proposition 7 says that a point with

For the sake of contradiction, let us suppose that as . Consider the polar exponential coordinates centered at . We will use these coordinates to study and on a ball of radius . More concretely, if denotes the angle between , then we will contradict the triangle inequality

by adapting an idea of Markovic to find in the set

(where is the geodesic connecting to ) such that

In fact, this will be a contradiction to (4) because we will prove that

as for all .

Pictorially, the previous paragraph is summarized in Figure 1 of Benoist-Hulin paper.

Formally, we proceed as follows:

- first, we give upper bounds for when and ;
- secondly, we give upper bounds for when and for all ;
- finally, we give lower bounds on for some .

**2.4.1 Upper bounds on **

for all with .

*Proof:* Consider as above. The triangle of vertices has sides of lengths:

- (by definition of );
- (because and is -quasi-isometric);
- (by our assumption on ).

In particular, the sum of the lengths of the sides of which are adjacent to the angle is much *bigger* than the length of the opposite side to this angle:

By “elementary hyperbolic geometry”, i.e., the fact that the angle is controlled by the Gromov product (i.e., the “excess” in the triangle inequality)

through the relation

(cf. Lemma 2.1 in Benoist-Hulin paper), we deduce from (5) that

This proves the lemma.

**2.4.2 Upper bounds on **

for all with for all (where is the geodesic from to ).

*Proof:* Note that

by our choice of the polar exponential coordinates at .

In order to estimate , we use a lemma of Gauss saying that

for all .

From the previous two estimates and our assumption that , we deduce that

Finally, we control with the aid of Cheng’s lemma. More precisely, because

for all (since and by hypothesis). Thus, Cheng’s lemma 5 ensures that

In summary, the previous estimate show that

because .

**2.4.3 Lower bounds on **

We start our quest for a point with far away from zero by estimating the Lebesgue measure of .

Lemma 10Suppose that and denote by the normalized spherical Lebesgue measure on . Then,

*Proof:* We write where

and

(where is the geodesic path between and ).

For the sake of exposition, we will discuss *only* the estimate of : indeed, the estimate of in Lemma 4.4 of Benoist-Hulin is very similar (up to a minor technicality which is treated with the aid of Cheng’s lemma) and we refer the curious reader to pages 18 and 19 of the original paper for more details.

The key tool to bound is the following Green formula:

The proof of this Green formula uses the subharmonicity of (which is a consequence of the subharmonicity of the distance function of and the harmonicity of ). More precisely, denote by the compact group of isometries of ) fixing , and let be its Haar measure.

Note that acts transitively on each sphere , . Thus, the subharmonic function takes a constant value on , so that

for all . Because , this proves (6).

Next, we observe that for all : in fact,

Therefore, by combining this estimate with (6), we see from the definition of that

i.e.,

. This completes our sketch of proof of the lemma (modulo the verification of the bound ).

Using Lemma 10, we can find with the property that is far away from zero:

*Proof:* Let us set . By Lemma 10,

(because .)

Since any subset of the Euclidean sphere with normalized Lebesgue measure has diameter , for sufficiently large, we can find such that

where is the angle between the vectors such that

Before proceeding further, we need to recall some facts about the relationship between angles of a triangle in a hyperbolic space and Gromov products (cf. Lemmas 2.1 and 2.2 in Benoist-Hulin paper). Let be a geodesic triangle in with vertices and denote by the corresponding angle at . Then,

- (a) ;
- (b) ;
- (c) whenever ;
- (d) if is a -quasi-isometry, then
where is a constant depending only on .

We use these facts as follows. Note that

On the other hand, *if* we can show that

*then* the items (c), (d), (b) and the estimate (7) ensure that

which would complete the proof of the lemma (by taking or ) in view of (8) (because and are independent of ).

In order to check (9), it is sufficient to verify that

thanks to item (d) above. For this sake, we observe that the item (a) and (7) imply that

because . It follows that the desired estimate (10) holds whenever is sufficiently large so that

This proves the lemma.

At this point, the proof of Theorem 6 is complete: indeed, it suffices to put together the estimate (4) with Lemmas 8, 9 and 11 to derive a contradiction with a scenario where as .

We close this post with the following comment about the arguments of Benoist-Hulin.

Remark 6As we already mentioned, the arguments of Benoist-Hulin provide a quantitative bound on in terms of with , . Indeed, the proofs of the lemmas above show that we get a contradiction when the quantities and satisfy the following inequalities:

- ;
- , and

where is the constant provided by item (d) in the proof of Lemma 11.Since these numerical constraints are fulfilled by any for an appropriate (“explicit”) choice of , one gets the following quantitative version of Benoist-Hulin theorem 6: for all .

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This post is a transcription of my notes for Harald’s talk, and, evidently, all mistakes/errors are my responsibility.

**1. Some notations **

We denote by the group of all permutations of .

For , we define their distance by

Definition 1 (Gromov; Weiss)Let be a group. Given and finite, we say that is a -sofic representation whenever

- (a) is an “approximate homomorphism”: for all with ;
- (b) has “few” fixed points for : for all .

We say that a group is sofic if it has -sofic representation for all finite, all (and some ).

Basic examples of sofic groups are: finite groups, amenable groups, etc. In general, it is known that several families of groups are sofic, but it is an important open problem to construct (or show the existence of) non-sofic groups.

The goal of this post is to discuss a candidate for non-sofic group and its connections to Number Theory.

** 1.1. Higman groups **

For , let

The groups and are trivial, and the group is the so-called Higman group.

Remark 1Several statements in this post can be generalized for for all , but for the sake of exposition we will stick to .

Theorem 2 (Helfgott-Juschenko)Then, for every , there exists and a bijection such thatAssume that is sofic.

- (a) is an “almost exponential function”: for all where is a subset of cardinality .
- (b) for all .

Remark 2The existence of functions as above is “unlikely” when is small. More precisely, it is possible to show that there are no bijections satisfying item (a) with and item (b) when is the fifth power of a prime (cf. Remark 4 below for a more precise statement).

In other words, if we could take and in the statement of Theorem 2, the non-soficity of the Higman group would follow.

Unfortunately, the techniques of Helfgott and Juschenko do not allow us to take in Theorem 2, but they permit to control the integer . More concretely, as we are going to see in Theorem 4 below, the integer can be chosen from any fixed sequence which is *thick* in the following sense:

Definition 3A sequence of positive integers isthickif for every there exists such that

for all .

Remark 3It does not take much to be a thick sequence: for example, the sequences and are thick.

As we already announced, the main result of Helfgott-Juschenko is the following improvement of Theorem 2:

Theorem 4 (Helfgott-Juschenko)Then, for every , there exists and a bijection such thatAssume that is sofic and let be a thick sequence.

- (a) for all where is a subset of cardinality .
- (b) for all .

Remark 4The non-soficity of Higman group would follow from this theorem if the bijection , , provided by this statement could be taken so that

- (a*) for all where is a subset of cardinality .
- (b) for all .

Indeed, this is so because Glebsky and Holden-Robinson proved that there is no verifying (a*) and (b).

Remark 5A natural question related to Theorem 4 is: what happens with fewer iterations in item (b)? In this situation, it is possible to use the fact that the group is trivial to show that, for each , there exists such that for any there is no bijection such that

- (a) for all for .
- (b) for all for .

This last remark can be generalized as follows.

Theorem 5 (Helfgott-Juschenko)Let be coprime. Consider the function given by

Then, the equation

has at most solutions .

Remark 6This theorem improves on a result of Glebsky-Shparlinski.

** 1.2. Main ideas in the proofs of Theorems 4 and 5**

As we are going to see in a moment, the key ingredient in the proof of Theorem 4 is the following result of Kerr-Li and Elek-Szabo

Theorem 6Sofic representations of an amenable group are “conjugated”: if , are -sofic representations of with large enough and small enough in terms of a parameter , then there exists a bijection such that

for all and for -almost all (i.e., for at least values of ).

while the idea of the proof of Theorem 5 is to use a finitary version of Poincaré recurrence theorem.

Let us now see how these ingredients are employed by Helfgott-Juschenko in their proofs of Theorems 4 and 5.

**2. Proof of Theorem 4**

** 2.1. Amenable groups and Baumslag-Solitar groups **

Let be a finitely generated group, say , finite. We say that is amenable if there exists a countable sequence of finite sets exhausting (i.e., ) consisting of almost invariant subsets in the sense that:

for all and .

The Baumslag-Solitar groups are

Example 1is amenable.

It follows from its amenability that is a sofic group. As it turns out, one can give a direct proof of this last fact along the following lines. Given , let be an integer such that

(e.g., let be a sufficiently large prime number), and is coprime with .

Define by

(so that ). One can check that is a -sofic representation.

** 2.2. End of proof of Theorem 4 **

Suppose is sofic. Then, the semi-direct product of and is also sofic (because it is an amenable extension of a sofic group). Here, the semi-direct product is defined by letting a generator act on by conjugation as follows: , .

Denote by the set of words of length on , and let be a -sofic representation.

We think of the Baumslag-Solitar group as sitting inside and , and we consider the restriction as an -sofic representation.

By the theorem of Kerr-Li and Elek-Szabo (cf. Theorem 6 above), is “conjugated” to the representation constructed above, i.e., there exists such that, for all ,

for at least values of .

Since is an almost homomorphism, and , we have that

for almost all () values of . Thus,

and

for almost all values of . Furthermore, since and are conjugated, the last equation is also true for .

Therefore, by adjusting the values of on a subset of values of of cardinality , we obtain such that and

for values of , and for all values of . This completes the proof of Theorem 4.

**3. Idea of proof of Theorem 5**

This short section contains an oversimplified argument because we will implicitly assume that the exponential maps are well-defined in . Nevertheless, the discussion below can be adapted to produce an actual proof of Theorem 5.

Let be a generator of , and suppose that

for a positive proportion of values of .

Then, there exists (bounded in terms of the proportion above) such that (1) holds for a both and for a positive proportion of values of : this is an incarnation of Poincaré recurrence theorem.

In other terms, for a positive proportion of values of , one has

By setting , one deduces that the equation

has a set of solutions with positive proportion.

Using Poincaré recurrence theorem once more, we can find such that (2) holds for a both and for a positive proportion of values of .

By writing , we have a positive proportion of values of satisfying the equation

However, this is a contradiction because , and are fixed, so that the previous equation is a polynomial equation on with a bounded number of solutions (and, thus, it can’t be satisfied for a positive proportion of values of ).

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As it turns out, Sophie Morel was kind towards the non-experts in this subject (like myself): indeed, a large part of the talk (see the video here) was introductory, while the more advanced material was delegated to the lecture notes (available here).

In the remainder of this post, I’ll try to summarize some of the topics discussed by Sophie Morel’s talk (using the video in the link above as the main source).

**Disclaimer:** Since I’m not an expert on this subject, all mistakes in this post are my responsibility.

Remark 1If you are in the Oxford area this week, then you will have the opportunity to learn about the main results in Morel’s talk directly from the authors: indeed, Boxer and Scholze (resp.) will give talks tomorrow, resp. Wednesday on their works (see this schedule here of the corresponding 2015 Clay research conference.)

**1. Motivations **

The basic references for this section are Zagier’s text on modular forms and the lecture notes from Deligne’s Bourbaki seminar talk in 1968–1969.

** 1.1. Modular forms **

Let be the upper-half plane, and consider the action of on by homographies:

A modular form of weight and level is a holomorphic function such that

- (i) for all and ;
- (ii) is holomorphic as .

A modular form is called cuspidal if satisfies (i), (ii) and

- (iii) vanishes at infinity.

Note that the condition (i) implies that any modular form is -periodic: for all . In particular, any modular form has a Fourier series

Since as , we see that the condition (ii) is equivalent to

- (ii)’ if

Also, the condition (iii) ( is cuspidal) is equivalent to .

** 1.2. Hecke operators **

The modular forms of weight and level form a finite-dimensional vector space. For each , the Hecke operator

acts on this vector space.

Remark 2This formula gets significantly simpler when is a prime number.

Among the basic properties of Hecke operators, it is worth to mention that and commute, and is cuspidal if the modular form is cuspidal.

These properties suggest the study of the following objects:

Definition 1is called a proper normalized cuspidal modular form if

- is proper (i.e., an eigenvector) for all Hecke operators ;
- the Fourier coefficient in (1) is normalized: .

Remark 3One can show that a proper normalized cuspidal modular form satisfies for all is prime. In other terms, the eigenvalues of the Hecke operators can be read through Fourier coefficients of proper normalized cuspidal modular forms.

** 1.3. Deligne’s proof of Ramanujan-Petersson conjecture **

In this context, the Ramanujan-Petersson conjecture states that the Fourier coefficients (1) of a proper normalized cuspidal modular form of weight satisfy the following bound:

for all prime.

As it is well-known, this conjecture was settled by Eichler-Shimura in the case , and in full generality by Deligne.

The strategy of Deligne was to “reduce” the Ramanujan-Petersson conjecture to Weil conjectures. Very roughly speaking, given a proper normalized cuspidal modular form , we can divide Deligne’s strategy into two steps:

- (1) for each prime , there exists a continuous irreductible representation such that
- is non-ramified at all prime;
- if , the characteristic polynomial of is ;
- appears in the (Betti) cohomology of a (proper, smooth) algebraic variety over .

- (2) the Riemann hypothesis part of Weil’s conjectures (also established by Deligne) provides useful information on the eigenvalues of acting on the cohomology of algebraic varieties, and this can be exploited to establish the Ramanujan-Petersson conjecture because is the sum of eigenvalues of (cf. the expression for the characteristic polynomial of ).

Before proceeding further, let us give some explanations about (1).

The absolute Galois group is the group of automorphisms of the algebraic closure of .

Given a prime number , we can choose an algebraic closure of the field of –adic numbers and a morphism such that the diagram associated to the arrows , , and commute.

In this way, we obtain an embedding where is the group of continuous automorphisms of .

Remark 4The embedding depends on the choice of . In particular, this embedding is only well-defined modulo conjugation (but, as it turns out, this is sufficient for our purposes).

The group is part of an exact sequence

where the arrow is defined by reduction modulo , or, more precisely, by considering a commutative diagram whose arrows are:

- ,
- ,
- , , .

Here, , resp. is the ring of integers of , resp. , is the finite field of elements, and is the algebraic closure of .

The group is topologically generated by a single element, namely, Frobenius endomorphism .

The kernel of the arrow is called inertia group (but its structure is not need for the sake of this post).

Definition 2We say that a representation of is non-ramified at if and only if . In particular, is well-defined modulo conjugation whenever is non-ramified at (cf. Remark 4).

Remark 5The fact that is well-defined modulo conjugation is not a serious issue for the first step of Deligne’s strategy: indeed, we imposed only a condition on the characteristic polynomial of (and this polynomial depends only on the conjugacy class of ).

Closing this subsection, let us outline the construction of the representation as in the first step of Deligne’s strategy in the particular case of proper normalized cuspidal modular forms of weight .

We begin by introducing the modular surface . The modular form is *not* a function on the non-compact Riemann surface (because it is not -invariant): in fact, the modularity condition (i) (with ) implies that is a section of the line bundle of holomorphic differentials on .

Note that the one-point compactification of is an orbifold topologically isomorphic to a sphere. The fact that is holomorphic at infinity (by condition (ii)) implies that can be extended to (i.e., ).

Next, we consider the Hodge decomposition

of the first (Betti) cohomology group of .

In these terms, the Hecke operators admit the following *geometrical* interpretation. For each , we can choose a *finite-index* subgroup and an element such that the action of on (or or ) consists into taking the pullback under the natural arrows

and then taking the “trace of the operator” induced by the arrows

where is the multiplication by . (See, e.g., the subsection 2.3 of this PhD thesis here for more details.)

In particular, a normalized proper cuspidal modular form

gives rise to a non-trivial simultaneous eigenspace of the operators on and, *a fortiori*, , with eigenvalues .

This geometrical interpretation of permits to construct along the following lines.

First, we observe that and are algebraic varieties over in a “canonical way” (because , resp., are moduli spaces of elliptic curves, resp. elliptic curves with extra (“level”) structure).

This implies that, for each prime, we have an (algebraically defined) isomorphism

from to the étale cohomology group of the algebraic variety over . Moreover, this isomorphim behaves “equivariantly” with respect to the Hecke operators .

In particular, the non-trivial simultaneous eigenspace of associated to can be transferred to .

Secondly, we note that this isomorphism is interesting because acts on . Furthermore, since is defined over (and behaves “equivariantly” with respect to this isomorphism), the transferred non-trivial simultaneuous eigenspace of in is *invariant* under this action of .

In summary, the normalized proper cuspidal modular form induces a representation of in a transferred non-trivial simultaneous eigenspace of in , and, as it turns out, this is (more or less) the representation that we were looking for.

** 1.4. Hecke operators from the adelic point of view **

Before closing this introductory section, let us make the following observation about the geometrical interpretation of the Hecke operators.

Our geometrical construction of Hecke operators involved pullback and “trace” operators related to the varieties (for appropriate choices of ).

Nevertheless, for the sake of *generalizing* this discussion, it is helpful to replace by

where is the ring of adeles (with denoting the “finite adeles”),

(with ),

and

The fact that is closely related to comes from the equality

where and .

The key points of this complicated definition of are:

- each operator acts only on the -th component of ;
- this definition can be generalized to other groups (see below).

**2. The works of Boxer and Scholze **

** 2.1. Hecke algebras **

Let us quickly indicate how the discussion of the previous section for can be generalized to , .

We start by introducing

for (where all implied groups are defined by their natural counterparts in ).

Similarly to the case , we have that is a real-analytic variety (given by the disjoint union of two symmetric spaces for ).

In this setting, the Hecke operators are obtained by taking a prime number and an adequate element , and considering the composition of the pullback and trace operators induced on by the natural arrows (inclusion and multiplication by ):

This construction can be slightly generalized by noticing that, if , then we have an action on of the *Hecke algebra*

equipped with the product given by the convolution with respect to the Haar measure with volume one.

Remark 6For , we studied exclusively the action of certain elements , but from now on we will consider the whole action of .

After the works of Satake, we know that is a commutative algebra whose characters correspond to the conjugations classes of under a *canonical* bijection called Satake isomorphism.

For the sake of convenience, we will put “together” the Hecke algebras for by defining the Hecke algebra unramified off as:

** 2.2. Characters of Hecke algebras and Galois representations **

The following two conjectures are part of the so-called Langlands program.

\noindent**Conjecture.** Let be a character such that the corresponding eigenspace in is non-trivial (i.e., is a sort of “normalized proper cuspidal modular form”). Then, there exists a (continuous) Galois representation such that

- is not ramified on (i.e., is trivial on the inertia group );
- corresponds to under Satake’s isomorphism.

Remark 7The reader certainly noticed the similarity between the statement of this conjecture and the item (1) of Deligne’s strategy of proof of Ramanujan-Petersson’s conjecture. Of course, this is not a coincidence, but this conjecture is somewhat “surprising” in comparison with Deligne’s setting because isnotalgebraic when .

\noindent**Conjecture’.** The same conjecture as above is true when is replaced by the *cohomology with torsion* .

The conjecture above was proved to be true by Harris-Lan-Taylor-Thorne. More recently, George Boxer and Peter Scholze proved (independently) the following result

Theorem 3 (Boxer; Scholze)The Conjecture’ is also true.

Remark 8It is known that (a version of) Conjecture’ (for ) implies Conjecture (by taking the limit ). In particular, the methods of Boxer and Scholze are able to recover the theorem of Harris-Lan-Taylor-Thorne.

** 2.3. Some words about the results of Boxer and Scholze **

We stated Conjecture and Conjecture’ for , , but we can also study them for other reductive connected groups such as , etc.

An interesting feature of this generalization of Conjecture/Conjecture’ is that sometimes becomes an algebraic (Shimura)variety (e.g., for , etc.), and this gives us a *hope* of mimicking the arguments from the first section of this post.

Remark 9For example, if is an imaginary quadratic extension of and act on , then the strategy used for doesThe fact that is algebraic isnotsufficient in general to reproduce the strategy employed for in the first section of this post.notproduce a “good” representation attached to a character .In fact, one gets a good representation

onlyafter applying Langlands functoriality principle to “transfer” the problem from to , and then using the strategy for in this new setting.

In summary, even if is algebraic, one can’t apply the strategy for in a simple-minded way in order to deduce Conjecture/Conjecture’.

On the other hand, the fact that is algebraic for certain choices of does not seem to help us in the context of the results of Boxer and Scholze because we know that is not algebraic in their setting.

Nevertheless, Clozel noticed that the non-algebraic varieties usually are strata in the compactification of an algebraic variety. In particular, one can try to exploit this to build the desired representations from characters appearing in .

Let us illustrate this idea of Clozel in the first non-trivial case of the generalization of the Conjecture, i.e., , imaginary quadratic extension (where is a non-algebraic variety of real dimension 3).

We take , so that is algebraic (but not compact). Its Borel-Serre compactification is not algebraic but its boundary has some components associated to parabolic subgroups of such as Levi’s parabolic subgroup

which turns out to be isomorphic to .

In this way, we get two arrows allowing to relate to with the advantage that is algebraic. In particular, this allows to transfer the problem of showing Conjecture/Conjecture’ from non-algebraic settings to algebraic settings (but this is not the end of the history: cf. Remark 9 above!)

At this point, Sophie Morel runned out of time and she decided to conclude her talk by mentionning that after transferring the problem from to as above, an important ingredient in Boxer and Scholze proof of Conjecture’ is the following theorem:

Theorem 4 (Boxer; Scholze)If appears in , then there exists a (cuspidal) in characteristic zero appearing in such that (mod ).

Then, she told that a “one-sentence proof” of this result is the following: one uses a comparison theorem to relate to cohomology groups of affinoid spaces.

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given by

where and is the shift map equipped with the Bernoulli measure .

By Oseledets multiplicative ergodic theorem, the Lyapunov exponents of the random product of and (i.e., the linear cocycle ) are well-defined quantities (depending only on and ) describing the exponential growth of the singular values of the random products

for any -typical choice of .

Moreover, the fact that and are symplectic matrices implies that the Lyapunov exponents are symmetric with respect to the origin, i.e., for each . In other words, the Lyapunov exponents of the symplectic linear cocycle have the form:

In fact, this structure of the Lyapunov exponents of a symplectic linear cocycle reflects the fact that if is an eigenvalue of a symplectic matrix , then is also an eigenvalue of .

A natural *qualitative* question about Lyapunov exponents concerns their *simplicity* in the sense that there are no repeated numbers in the list above (i.e., for all ).

The simplicity property for Lyapunov exponents is the subject of several papers in the literature: see, e.g., the works of Furstenberg, Goldsheid-Margulis, Guivarch-Raugi, and Avila-Viana (among *many* others).

Very roughly speaking, the basic philosophy behind these papers is that the simplicity property holds whenever the monoid generated by and is *rich*. Of course, there are several ways to formalize the meaning of the word “rich”, for example:

- Goldsheid-Margulis and Guivarch-Raugi asked to be Zariski-dense in ;
- Avila-Viana required to be
*pinching*: there exists whose eigenvalues are all real with distinct moduli; such a is called a*pinching matrix*;*twisting*: there exists a pinching matrix and a*twisting matrix*with respect to in the sense that for*all*isotropic -invariant subspaces and*all*coisotropic -invariant subspaces with .

Of course, these notions of “richness” of a monoid are “close” to each other, but they *differ* in a subtle detail: while the Zariski-density condition on is an *algebraic* requirement, the pinching and twisting condition on makes *no* reference to the algebraic structure of the linear group .

In particular, this leads us to the main point of this post:

How the *Zariski-density* and *pinching and twisting* conditions relate to each other?

The first half of this question has a positive answer: a Zariski-dense monoid is also pinching and twisting. Indeed:

- (a) a modification of the arguments in this blog post here (in Spanish) permits to prove that any Zariski-dense monoid contains a pinching matrix , and
- (b) the twisting condition on a matrix with respect to a pinching matrix can be phrased in terms of the non-vanishing of certain (isotropic) minors of the matrix of written in a basis of eigenvectors of ; thus, a Zariski-dense monoid contains a twisting matrix with respect to any given pinching matrix.

On the other hand, the second half of this question has a negative answer: we exhibit below a pinching and twisting monoid which is not Zariski dense.

Remark 1The existence of such examples of monoids is certainly known among experts. Nevertheless, I’m recording it here because it partly “justifies” a forthcoming article joint with Artur Avila and Jean-Christophe Yoccoz in the following sense.

The celebrated paper of Avila-Viana quoted above (on Kontsevich-Zorich conjecture) shows that the so-called “Rauzy monoids” are pinching and twisting (and this is sufficient for their purposes of proving simplicity of Lyapunov exponents of the Kontsevich-Zorich cocycle for Masur-Veech measures).

On the other hand, since a pinching and twisting monoid is not necessarily Zariski dense (as we are going to see below), the results of Avila-Viana (per se) can not answer a question of Zorich (see also Remark 6.12 in Avila-Viana paper) about the Zariski density of Rauzy monoids.

In this direction, Artur, Jean-Christophe and I solve (in an article still in preparation) Zorich’s question about Zariski density of Rauzy monoids in the special case of hyperelliptic Rauzy diagrams, and the main example of this post (which will be included in our forthcoming article with Artur and Jean-Christophe) serves to indicate that the results obtained by Artur, Jean-Christophe and myself can not be deduced as “abstract consequences” of the arguments in Avila-Viana paper.

Remark 2The main example of this post also shows that (a version of) Prasad-Rapinchuk’s criterion for Zariski density (cf. Theorem 9.10 of Prasad-Rapinchuk paper or Theorem 1.5 in Rivin’s paper) based on Galois-pinching (in the sense of this paper here) and twisting properties is “sharp”: indeed, an important feature of the main example of this post is the failure of the Galois-pinching property (cf. Remark 4 below for more comments).

**1. A monoid of 4×4 symplectic matrices **

Let be the third symmetric power of the standard representation of . In concrete terms, is constructed as follows. Consider the basis of the space of homogenous polynomials of degree on two variables and . By letting act on and as and , we get a linear map on whose matrix in the basis is

Remark 3The faithful representation is the unique irreducible four-dimensional representation of .

The matrices preserve the symplectic structure on associated to the matrix

Indeed, a direct calculation shows that if , then

where stands for the transpose of .

Therefore, the image is a linear algebraic subgroup of the symplectic group , and the Zariski closure of the monoid generated by the matrices

and

is precisely .

Remark 4Coming back to Remark 2, observe that does not contain Galois-pinching elements of in the sense of this paper here (i.e., pinching elements of with integral entries whose characteristic polynomial has the largest possible Galois group for a reciprocal polynomial [namely, the hyperoctahedral group]) because its rank is . Alternatively, a straightforward computation reveals that the characteristic polynomial of is

and, consequently, the eigenvalues of are

and

In particular, since the characteristic polynomial of always splits, it is never the case that is Galois-pinching.

On the other hand, the element is pinching because its eigenvalues are

Also, the matrix is twisting with respect to . Indeed, the columns of the matrix

consist of eigenvectors of . Thus, is the matrix of in the corresponding basis of eigenvectors of . Moreover, is twisting with respect to if and only if all entries of and all of its minors associated to isotropic planes are non-zero (cf. Lemma 4.8 in this paper here). Finally, this last fact is a consequence of the following exact calculation (see also the numerical approximations) for and its matrix of minors:

and

In summary, the monoid is pinching and twisting, but not Zariski dense in .

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In the sequel, I will transcript my notes from Shen’s talk.

**1. Introduction **

In Real Analysis, the classical Weierstrass function is

with .

Note that the Weierstrass functions have the form

where is a -periodic -function.

Weierstrass (1872) and Hardy (1916) were interested in because they are concrete examples of continuous but nowhere differentiable functions.

Remark 1The graph of tends to be a “fractal object” because is self-similar in the sense that

We will come back to this point later.

Remark 2is a -function for all . In fact, for all , we have

so that

whenever , i.e., .

The study of the graphs of as fractal sets started with the work of Besicovitch-Ursell in 1937.

Remark 3The Hausdorff dimension of the graph of a -function isIndeed, for each , the Hölder continuity condition

leads us to the “natural cover” of by the family of rectangles given byNevertheless, a direct calculation with the family

does notgive us an appropriate bound on . In fact, since for each , we have

Fortunately, we canfor . Because is arbitrary, we deduce that . Of course, this bound is certainly suboptimal for (because we know that anyway).refinethe covering by taking into account that each rectangle tends to be more vertical than horizontal (i.e., its height is usually larger than its width ). More precisely, we can divide each rectangle into squares, say

such that every square has diameter . In this way, we obtain a covering of such that

for . Since is arbitrary, we conclude the desired bound

A long-standing conjecture about the fractal geometry of is:

**Conjecture** (Mandelbrot 1977): The Hausdorff dimension of the graph of is

Remark 4In view of remarks 2 and 3, the whole point of Mandelbrot’s conjecture is to establish the lower bound

Remark 5The analog of Mandelbrot conjecture for the box and packing dimensions is known to be true: see, e.g., these papers here and here).

In a recent paper (see here), Shen proved the following result:

Theorem 1 (Shen)For any integer and for all , the Mandelbrot conjecture is true, i.e.,

Remark 6The techniques employed by Shen also allow him to show that given a -periodic, non-constant, function, and given integer, there exists such that

for all .

Remark 7A previous important result towards Mandelbrot’s conjecture was obtained by Barańsky-Barány-Romanowska (in 2014): they proved that for all integer, there exists such that

for all .

The remainder of this post is dedicated to give some ideas of Shen’s proof of Theorem 1 by discussing the *particular* case when and is large.

**2. Ledrappier’s dynamical approach **

If is an *integer*, then the self-similar function (cf. Remark 1) is also -periodic, i.e., for all . In particular, if is an integer, then is an invariant repeller for the endomorphism given by

This dynamical characterization of led Ledrappier to the following criterion for the validity of Mandelbrot’s conjecture when is an integer.

Denote by the alphabet . The unstable manifolds of through have slopes of the form

where , , , and

In this context, the push-forwards of the Bernoulli measure on (induced by the discrete measure assigning weight to each letter of the alphabet ) play the role of *conditional measures along vertical fibers* of the unique *Sinai-Ruelle-Bowen (SRB) measure* of the expanding endomorphism ,

where and . In plain terms, this means that

where is the unique -invariant probability measure which is absolutely continuous along unstable manifolds (see Tsujii’s paper).

As it was shown by Ledrappier in 1992, the fractal geometry of the conditional measures have important consequences for the fractal geometry of the graph :

Theorem 2 (Ledrappier)Suppose that for Lebesgue almost every the conditional measures have dimension , i.e.,

Then, the graph has Hausdorff dimension

Remark 8Very roughly speaking, the proof of Ledrappier theorem goes as follows. By Remark 4, it suffices to prove that . By Frostman lemma, we need to construct a Borel measure supported on such that

where . Finally, the main point is that the assumptions in Ledrappier theorem allow to prove that the measure given by the lift to of the Lebesgue measure on via the map satisfies

An interesting consequence of Ledrappier theorem and the equation 1 is the following criterion for Mandelbrot’s conjecture:

Corollary 3If is absolutely continuous with respect to the Lebesgue measure , then

*Proof:* By (1), the absolute continuity of implies that is absolutely continuous with respect to for Lebesgue almost every .

Since for almost every implies that for almost every , the desired corollary now follows from Ledrappier’s theorem.

**3. Tsujii’s theorem **

The relevance of Corollary 3 is explained by the fact that Tsujii found an explicit *transversality condition* implying the absolute continuity of .

More precisely, Tsujii firstly introduced the following definition:

Definition 4

- Given , and , we say that two infinite words are -transverse at if either
or

- Given , , and , we say that two finite words are -transverse at if , are -transverse at for all pairs of infinite words ; otherwise, we say that and are -tangent at ;
- ;
- .

Next, Tsujii proves the following result:

Theorem 5 (Tsujii)If there exists integer such that , then

Remark 9Intuitively, Tsujii’s theorem says the following. The transversality condition implies that the majority of strong unstable manifolds are mutually transverse, so that they almost fill a small neighborhood of some point (see the figure below extracted from this paper of Tsujii). Since the SRB measure is absolutely continuous along strong unstable manifolds, the fact that the ‘s almost fill implies that becomes “comparable” to the restriction of the Lebesgue measure to .

Remark 10In this setting, Barańsky-Barány-Romanowska obtained their main result by showing that, for adequate choices of the parameters and , one has . Indeed, once we know that , since , they can apply Tsujii’s theorem and Ledrappier’s theorem (or rather Corollary 3) to derive the validity of Mandelbrot’s conjecture for certain parameters and .

For the sake of exposition, we will give just a flavor of the proof of Theorem 1 by sketching the derivation of the following result:

Proposition 6Let . If and is sufficiently large, then

In particular, by Corollary 3 and Tsujii’s theorem, if and is sufficiently large, then Mandelbrot’s conjecture is valid, i.e.,

Remark 11The proof of Theorem 1 in full generality (i.e., for integer and ) requires the introduction of a modified version of Tsujii’s transversality condition: roughly speaking, Shen defines a function (inspired from Peter-Paul inequality) and he proves

- (a) a variant of Proposition 6: if integer and , then for some integer ;
- (b) a variant of Tsujii’s theorem: if for some integer , then .

See Sections 2, 3, 4 and 5 of Shen’s paper for more details.

We start the (sketch of) proof of Proposition 6 by recalling that the slopes of unstable manifolds are given by

for , , so that

Remark 12Since , the series defining converges faster than the series defining .

By studying the first term of the expansion of and (while treating the remaining terms as a “small error term”), it is possible to show that if , then

(cf. Lemma 3.2 in Shen’s paper).

Using these estimates, we can find an upper bound for as follows. Take with , and let be such that distinct elements listed in such a way that

for all , where .

From (3), we see that

for all .

Since

for large enough. Indeed, this happens because

- if ;
- if ;
- as , and as (here we used ).

By combining (4) and (5), we deduce that

for all .

Since , the previous estimate implies that

i.e.,

Thus, it follows from our assumptions (, large) that

This completes the (sketch of) proof of Proposition 6 (and our discussion of Shen’s talk).

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- The Third Palis-Balzan International Symposium on Dynamical Systems will be held at
*amphithéâtre Hermite*in Institut Henri Poincaré (Paris, France) next week. - The Workshop on Combinatorics, Number Theory and Dynamical Systems will be held at IMPA (Rio de Janeiro, Brazil) from August 24th to August 28th, 2015.

The Third Palis-Balzan International Symposium on Dynamical Systems closes the five-year long *Project Palis-Balzan – Dynamical Systems, Chaotic Behaviour-Uncertainty*, sponsored by the Balzan Foundation, related to the prestigious award conferred to Jacob Palis (and IMPA) by the Balzan Foundation in 2010.

A detailed description of the program and the titles and abstracts of talks of this conference can be found here and here.

The Workshop on Combinatorics, Number Theory and Dynamical Systems is the second edition of an event organized by C. Mauduit, C. G. Moreira, Y. Lima, J.-C. Yoccoz and myself back in 2013.

The full list of speakers for the 2015 edition of this workshop can be found here.

I guess that this is all I have to say for now (but you can look at their respective webpages for updated information). See you in Paris or Rio!

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Proposition 1Then, there exist , , and such that the iterates of the weighted transfer operator satisfyLet be an uniformly expanding Markov map on and let be a good roof function with exponential tails.

for all and with , .

Remark 1We use the same terminology from the previous post of this series.

Roughly speaking, the basic idea behind the exponential contraction property in Proposition 1 is that “*oscillations produce cancellations*”. In particular, the analysis of the “size” of is divided into two regimes:

- (A) If exhibits a high oscillation at scale (in the sense that ), then we will have a “cancelation” (significant reduction of the size of ) thanks to classical methods (Lasota-Yorke inequality);
- (B) If the oscillation of at scale is not high, then we will have a “cancelation” thanks to Dolgopyat’s mechanism, i.e., a combination of high oscillations of Birkhoff sums of the roof function (coming from the fact that is not a -coboundary) and the big phases , , of the terms in the formula defining .

In the remainder of this post, we will formalize this outline of proof of Proposition 1. More precisely, the next section contains a discussion of Lasota-Yorke inequality and the regime (A), and the last section is devoted to Dolgopyat’s cancelation mechanism and the regime (B).

**1. Lasota-Yorke inequality and regime (A) **

Before attacking Proposition 1, let us warm up with a digression on the spectral properties of the weighted transfer operators .

The usual transfer operator acts on the space of functions. This action has a simple isolated eigenvalue at . The eigenfunction associated to the eigenvalue and normalized so that is the density of the unique absolutely continuous invariant probability measure of . Furthermore, the essential spectral radius of is and has no eigenvalues of modulus except for . (See Aaronson’s book for more explanations.)

For close to , the operator is a small perturbation of . In particular, has an eigenfunction associated to its unique eigenvalue close to such that and converges to in the -topology as .

From now on, let us fix such that is well-defined and bounded away from zero for . (Here, is the constant appearing in the exponential tails condition for the roof function .)

From the technical point of view, it is convenient to “uniformize” this spectral picture by normalizing the operators (for with ) as follows:

The *normalized weighted transfer operator* satisfy and . In other words, if we replace by , we normalize both the eigenvalue and the eigenfunction to . Moreover, the proof of Proposition 1 can reduced to the analogous statement for the normalized operators:

Proposition 2Suppose that there exist , , , and such that

for all and with and .Then, the conclusion of Proposition 1 is valid.

The proof of this proposition is based on *Lasota-Yorke inequality* (cf. Lemmas 7.8 and 7.9 of AGY paper):

Lemma 3There exists a constant such that, for all and with , , we have

- for all ;
- if , then ;
- if , then ,

where is the expansion constant of .

Before proving this lemma, let us use it to show Proposition 2. Recall that the assumption of this proposition is that

for some fixed , , and for all and with , , and our task is to prove the analogous statement in Proposition 1 for .

The idea is very simple: from the spectral discussion above, it is not hard to see that we introduce a factor of the order of when replacing by ; since is close to , this factor does not significantly affect definite contraction on the size of provided by the hypothesis of Proposition 2. Let us now turn into the details.

Given , we write with . Since and is uniformly close to , the previous estimate for the normalized operator gives that

for all with , . By the last item of Lemma 3, it follows that

Because is close to for small, we can choose such that

we deduce from the previous estimate that

for all and with , . This completes the proof of Proposition 2 modulo Lemma 3.

** 1.1. Proof of Lasota-Yorke inequality **

Let us now prove Lemma 3. The first item follows from a computation similar to the proof of Lemma 8 of the previous post, and its half-page proof is given in Lemma 7.8 of AGY paper.

For the sake of convenience, we provide just a sketch of proof. We write

where is the set of inverse branches of , and (with ). By taking the derivative , we obtain five terms , , , , depending by differentiating , , , or .

The terms and are easy to deal with: the uniform bounds on and , and the contraction of inverse branches of imply that and . Thus, .

Similarly, the distortion bound (Renyi condition) on (see the previous post) implies that , so that .

Since and (because is a good roof function and the inverse branches of contract exponentially), we see that and, *a fortiori*, .

Finally, the exponential contraction of inverse branches of says that , so that . This proves the first item of Lemma 3.

The second item is an immediate consequence of the estimate and the first item just proved. Indeed, if , then

Closing the proof of Lemma 3, we observe that the third item is a direct corollary of the second item.

** 1.2. First part of Proposition 1: study of regime (A)**

By Proposition 2, one can establish Proposition 1 by showing a contraction property for the normalized weighted transfer operators .

As we already mentioned, the first step in this direction is the treatment of the regime (A) of functions displaying high oscillations via Lasota-Yorke inequality:

Lemma 4Let with and and suppose that exhibits a high oscillation at scale in the sense that (where is the constant from Lemma 3). Then,There exists such that any has the following property.

*Proof:* The facts that is normalized and has high oscillations imply that

Furthermore, the Lasota-Yorke inequality in the first item of Lemma 3 says that

Since and , we get that

for and sufficiently large (so that ).

Of course, we can iterate this lemma to establish the contraction property in Proposition 2 *while* the high oscillation property is not destroyed by :

Corollary 5Fix where is the integer provided by Lemma 4. Suppose that is a function on and is an integer such that the high oscillation property is not destroyed by the first iterates of , i.e.,

for all and for some with , . Then,

*Proof:* By the third item of Lemma 3, we have that

By iterating Lemma 4, it follows that

This proves the corollary.

Remark 2The proof of the corollary shows that a strong (pointwise/) form of cancellation occurs in the high oscillation case (regime (A)). As we are going to see in the next section, one has a much weaker () form of cancellation in the low oscillation case (regime (B)).

**2. Dolgopyat’s mechanism and regime (B) **

After our success in dealing with the regime (A) (cf. Corollary 5), let us analyze the regime (B) of functions whose oscillation is not high.

** 2.1. UNI condition **

As we told in the introduction, the cancellation mechanism in regime (B) originates from the oscillations of the Birkhoff sums of the roof function . Here, by “oscillations of Birkhoff sums” we mean the following:

Proposition 6Let be an uniformly expanding Markov map. Let be a function such that

Then, the following conditions are equivalent:(where is the set of inverse branches of .)

- is not a coboundary: it is not possible to write with which is constant on each and .
- Uniform non-integrability (UNI) condition (“oscillation of Birkhoff sums”): there exists a constant such that
for some arbitrarily large and some inverse branches of .

This proposition is (contained in) Proposition 7.4 of AGY paper and we refer to it for the one page proof of this result.

Remark 3The main point of this proposition is that a qualitative property (“not a coboundary”) in our definition of good roof function in the previous post turns out to be equivalent to a quantitative property (“definite oscillation of Birkhoff sums”).The nomenclature “uniform non-integrability” (UNI) comes from the fact that this is a (uniform) quantitative property issued from the fact that the suspension flow associated to and is not integrable (conjugated to a suspension flow with piecewise constant roof function) when is not a coboundary.

** 2.2. Dolgopyat’s cancellation mechanism **

Let us now use UNI condition to produce non-trivial cancellations in some regions of the phase space for functions with “low oscillations”. In other terms, we want to study , , for such that .

For technical reasons, we will keep track of *and* an appropriate -bound for :

Definition 7Given , we say that if is , is a function providing an adequate bound for in the sense that

- , and

for all .

Remark 4is a cone and we are going to show in Lemma 10 that sends “strictly inside itself” in a certain sense.

Moreover, we need to anticipate the fact in Remark 3 that the cancellations in regime (B) occur only at certain spots of the phase space. In particular, the following “localization tool” (cf. Lemma 7.12 in AGY paper) will be helpful in the subsequent cancellation discussion.

Lemma 8There exists an integer such that any has the following property. Let with , , let , and let with , . Suppose that, for all , we have

Then, .

This lemma is a consequence of the Lasota-Yorke inequality in Lemma 3 and its half-page proof can be found after the statement of Lemma 7.12 in AGY paper.

Remark 5We call this lemma a “localization tool” for the following reason. We think of as a bump function supported on disjoint intervals of size . Next, we use to “localize” anappropriatebound in a pair to the support of by considering . In this setting, the lemma says that if the iterate of this localization gives a bound to , then is actually anappropriatebound for (i.e., ).

**From now on**, we use UNI condition (cf. Proposition 6) to **fix** and inverse branches such that

where , and are the constants in Lemmas 3 and 8. (This is possible because .)

In this language, Dolgopyat’s cancellation mechanism can be stated as:

Lemma 9Then, for every interval , we can find a point such that one of the following possibilities holds:There exist a small constant and a large constant with the following property. Let with , , and .

- Cancellation of type : for all , we have
- Cancellation of type : for all , we have

In other words, this lemma says that, for each interval of of size of order , we can find a subinterval of size of the *same* order where the two terms of

associated to the inverse branches and fixed above exhibits a *significant cancellation* w.r.t. the trivial bound (i.e., one gets a factor of instead of ).

This lemma is exactly Lemma 7.13 in AGY paper and we provide a sketch of its proof in the sequel.

*Proof:* Let , resp. be a small, resp. large constant. Consider and . Our task is to find with so that the conclusion of the lemma is valid.

The argument is divided into **two cases**. In the **first** (easy) **case**, we assume that the appropriate bound for is not “tight”, i.e., there exists such that or . In this situation, we want to take advantage of the non-tightness (and the fact that does not have high oscillations [as ]) to get some cancellation in an interval of size centered at .

For this sake, up to exchanging the roles of and , we can assume that . Since , we know that . Thus, because is a contraction. By Gronwall’s inequality, it follows that . Therefore,

for all . By integrating this estimate, we see that

for all . Since (by “non-tightness” assumption), we obtain from the estimates above that, if is small enough, then

for all . This proves the lemma in this “non-tight” case.

In the **second case**, we assume that the bound for is “tight”, i.e., and for all . In this context, we want to find such that the complex numbers

have *opposite* phases.

For this sake, we put for and we denote by the difference of the phases (arguments) of the complex numbers

Using UNI condition (2), the fact that does not have high oscillations (i.e., ) and the “tightness” of the bound for , it is possible to prove with a short calculation that

where . (See page 192 of AGY paper.)

It follows that if is sufficiently large (e.g., ), then there exists such that and have opposite phases.

We set . By exchanging the roles of and if necessary, we can assume that . By exploiting the fact that does not have high oscillations and the tightness of the bound for (via Gronwall inequality), it is possible to show (through another short calculation, cf. page 193 in AGY paper) that if is sufficiently small, then for all one has

and

where and denote the phases of the complex numbers and . In other terms, for any in the interval ,

- the size of does not drop too much in comparison with the size of : except for a factor , one still has the same comparison from the case ;
- and have almost opposite phases.

On the other hand, an elementary trigonometry lemma (cf. Lemma 7.14 in AGY paper) says that two complex numbers and such that

and

verify the “cancellation estimate”

By applying this lemma to and for , we deduce that if , then

that is, one has a cancellation of type .

This completes the sketch of proof of Lemma 9.

** 2.3. Second part of Proposition 1: study of regime (B)**

By Proposition 2 and Corollary 5, one can establish Proposition 1 by proving a -contraction property in regime (B) for the normalized weighted transfer operators.

Evidently, the key tool to get this -contraction property is Dolgopyat’s cancellation lemma 9 because it says that, given any interval of size , a non-trivial amount of cancellation for must happen inside a subinterval of comparable size . In particular, for the regime (B), even though we do *not* get cancellation *everywhere* in phase space, we still have cancellation in large chunks of the phase space. Thus, it is reasonable to expect a contraction property for in the norm (but not for the norm) in the regime (B).

Let us now try to formalize this heuristic argument. Fix such that where and are the constants in Lemma 9.

Lemma 10There exists and with the following property. Let with , , and let . Then, there exists such that and

In other words, this lemma says that for any in regime (B) (i.e., ), we can find an appropriate bound for whose -norm got contracted by a definite factor in comparison with the -norm of an appropriate -bound for .

This lemma is exactly Lemma 7.15 in AGY paper. For the sake of convenience, let us now sketch its proof.

*Proof:* We take a maximal set of points such that the intervals are pairwise disjoint and compactly contained in . Note that the intervals cover .

By Dolgopyat’s cancellation lemma 9, we know that each contains a subinterval such that a cancellation of type or occurs for the pair in regime (B). We say that has type , resp. type depending on the type of cancellation occurring in .

We want to use our knowledge of the *cancellation* on each to *modify* the *trivial* bound for : roughly speaking, we want to insert a factor in front of the terms of associated to or whenever . At this point, the localization tool in Lemma 8 will prove itself useful.

From the technical point of view, we implement the idea from the previous paragraph as follows. We consider a bump function localized on each : we impose on , outside and (for some universal constant ). Next, we construct on by putting

Note that for some constant depending only on the *fixed* objects , and . In particular, is *independent* of and the pair . The function was built up so that is a function taking values in such that and

has a definite contraction factor in front of the term associated to or whenever for some .

In summary, we manipulated some bump functions associated to to get a function such that:

- , ,
- for and given by the type of ,
- for or and not of the type of .

From the properties in the last two items, we see that the statement of Dolgopyat’s cancellation lemma 9 implies that

that is, is a bound for . By combining this fact with the property in the first item and the localization tool in Lemma 8, we have that is actually an *appropriate* bound for , i.e.,

Therefore, our task is reduced to prove that for some constant (independent of , and ). In this direction, a short (half-page) calculation (cf. page 193 in AGY paper) exploiting the cancellation mechanism for any shows that

and

Note that the estimate (3) goes in the *good* direction: it implies that

by the definition of the normalized transfer operator (and the fact that ), and we know that for sufficiently small.

On the other hand, the estimate (4) by itself is *not* sufficient to control in the desired way.

Fortunately, we know that covers “most” of the phase space and does not oscillate too much: more precisely, we have that is formed of intervals of size contained in some intervals covering the whole phase space (as it was said in the beginning of the proof) and .

By combining this facts with Gronwall’s inequality, it is possible to prove (after a short calculation, cf. pages 195 and 196 of AGY paper) that there exists a constant such that

Because the intervals are pairwise disjoint, the intervals cover , the density is bounded away from and , and is uniformly bounded (since the size of *both* and is ), the estimate above implies that there exists a constant such that

This new information (coming from the low oscillations of and the fact that covers almost all of ) permits to get the desired -contraction for . Indeed, this is a consequence of the following simple computation (resembling to “Peter-Paul inequality”). Denote by . By (3) and (4), we have

for any parameter . By applying (5) to the right-hand side of the previous inequality, we deduce that

for each . By taking so that , we see that the estimate above implies that

where .

Therefore, if we take small enough so that , then

This completes the proof of the lemma.

Evidently, the main point of Lemma 10 is that one can iterate it to obtain the contraction property in Proposition 2 in regime (B):

Corollary 11Let and . Suppose that is the first time such that does not exhibit high oscillation at scale , i.e.,Consider the integer fixed above. Let , and be the constants provided by Lemma 10, and denote by with , .

Then,

*Proof:* We set . By definition, where is the constant function . By successively applying Lemma 10, we obtain a sequence of pairs such that

By setting and by recalling that for all , we conclude from the previous estimate that

This proves the corollary.

At this point, the proof of Proposition 1 is complete. Indeed, this is so because the hypothesis of Proposition 2 is always satisfied in both regimes (A) (by Corollary 5) and (B) (by Corollary 11).

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Here, our main goal is to provide a little bit more of details on how this technique works by offering a “guided tour” through Sections 2 and 7 of a paper of Avila-Goüezel-Yoccoz.

For this sake, we organize this short series of posts as follows. In next section, we introduce a prototypical class of semiflows exhibiting exponential mixing. After that, we state the main exponential mixing result of this post (an analog of Theorem 7.3 of Avila-Goüezel-Yoccoz paper) for such semiflows, and we reduce the proof of this mixing property to a Dolgopyat-like estimate on weighted transfer operators. Finally, the next post of the series will be entirely dedicated to sketch the proof of the Dolgopyat-like estimate.

**1. Expanding semiflows **

Recall that a *suspension flow* is a semiflow , , associated to a *base dynamics* (discrete-time dynamical system) and a *roof function* in the following way. We consider where is the equivalence relation induced by , and we let be the semiflow on induced by

Geometrically, , flows up the point , , linearly (by translation) in the fiber until it hits the “roof” (the graph of ) at the point . At this moment, one is sent back (by the equivalence relation ) to the basis at the point , and the semiflow restarts again.

A more concise way of writing down is the following: denoting by , one defines where is the Birkhoff sum

and is the unique integer such that

In this post, we want to study the decay of correlations of *expanding semiflows*, that is, a suspension flow so that the base dynamics is an *uniformly expanding Markov map* and the roof function is a *good roof function with exponential tails* in the following sense.

Remark 1Avila-Gouëzel-Yoccoz work in greater generality than the setting of this post: in fact, they allow to be aJohn domainand they prove results forexcellent hyperbolic semiflows(which are more common in “nature”), but we will always take and we will study exclusively expanding semiflows (which are obtained from excellent hyperbolic semiflows by taking the “quotient along stable manifolds”) in order to simplify our exposition.

Definition 1Let , be the Lebesgue measure on , and be a finite or countable partition of modulo zero into open subintervals. We say that is anuniformly expanding Markov mapif

- is a
Markov partition: for each , the restriction of to is a -diffeomorphism between and ;- is
expanding: there exist a constant and, for each , a constant such that for each ;- has
bounded distortion: denoting by the inverse of the Jacobian of and by the set of inverse branches of , we require that is a function on each and there exists a constant such thatfor all and . (This condition is also called

Renyi conditionin the literature.)

Remark 2Araújo and Melbourne showed recently that, for the purposes of discussing exponential mixing properties (for excellent hyperbolic semiflows with one-dimensional unstable subbundles), the bounded distortion (Renyi condition) can be relaxed: indeed, it suffices to require that is a Hölder function such that the Hölder constant of is uniformly bounded for all .

Example 1Let be the finite partition (mod. ) of provided by the two subintervals , .The map given by for is an uniformly expanding Markov map (preserving the Lebesgue measure ).

An uniformly expanding map preserves an *unique* probability measure which is absolutely continuous with respect to the Lebesgue measure . Moreover, the density is a function whose values are bounded away from and , and is ergodic and mixing.

Indeed, the proof of these facts can be found in Aaronson’s book and it involves the study of the spectral properties of the so-called *transfer* (Ruelle-Perron-Frobenius) *operator*

(as it was discussed in Theorem 1 of Section 1 of this post in the case of a *finite* Markov partition , )

Definition 2Let be an uniformly expanding Markov map. A function is agood roof functionif

- there exists a constant such that for all ;
- there exists a constant such that for all and all inverse branch of ;
- is
nota –coboundary: it is not possible to write where is constant on each and is .

Remark 3Intuitively, the condition that is not a coboundary says that it is not possible to change variables to make the roof function into a piecewise constant function. Here, the main point is that we have to avoid suspension flows with piecewise constant roof functions (possibly after conjugation) in order to have a chance to obtain nice mixing properties (see this post for more comments).

Definition 3A good roof function hasexponential tailsif there exists such that .

The suspension flow associated to an uniformly expanding Markov map and a good roof function with exponential tails preserves the probability measure

on . Note that is absolutely continuous with respect to (because is absolutely continuous with respect to ).

Remark 4All integrals in this post are always taken with respect to or unless otherwise specified.

Remark 5In the sequel, AGY stands for Avila-Gouëzel-Yoccoz.

**2. Statement of the exponential mixing result **

Let be an expanding semiflow.

Theorem 4There exist constants , such that

for all and for all .

Remark 6By applying this theorem with in the place of , we obtain theclassicalexponential mixing statement:

Remark 7This theorem is exactly Theorem 7.3 in AGY paperexceptthat they work with observables and belonging to Banach spaces and which are slightly more general than (in the sense that ). In fact, AGY need to deal with these Banach spaces because they use their Theorem 7.3 to deduce a more general result of exponential mixing for excellent hyperbolic semiflows (see their paper for more explanations), but we will not discuss this point here.

The remainder of this post is dedicated to the proof of Theorem 4.

** 2.1. Reduction of Theorem 4 to Paley-Wiener theorem**

From now on, we fix two observables such that

Of course, there is no loss in generality here because we can always replace by if necessary.

In this setting, we want to show that the correlation function

For this sake, we will use the following classical theorem in Harmonic Analysis (stated as Theorem 7.23 in AGY paper):

Theorem 5 (Paley-Wiener)Suppose that can be analytically extended to a function defined on a strip in such a way thatLet be a bounded measurable function and denote by (defined for with ) the Laplace transform of .Then, there exists a constant and a full measure subset such that

for all .

In other words, the Paley-Wiener theorem says that a bounded measurable function decays exponentially whenever its Laplace transform admits a nice analytic extension to a vertical strip to the left of the imaginary axis .

In our context, we will produce such an analytic extension by writing the Laplace transform of the correlation function as an appropriate geometric series of terms depending on . In fact, the th term of this series will have a clear dynamical meaning: it will be related to the pieces of orbit hitting times the graph of the roof function.

More concretely, for each , we decompose the phase space as

where

- is the subset of points whose piece of orbit hits the graph of the roof function at least once,
- and is the subset of points whose piece of orbit does not hit the graph of the roof function.

Let us denote by

the corresponding decomposition of the correlation function (3).

The fact that the roof function has exponential tails implies that the probability of the event that the piece of orbit does not hit the graph of becomes exponentially small for large.

Thus, it is not surprising that the following calculation shows that decays exponentially fast as grows:

where (by the exponential tails condition on ).

In particular, the proof of Theorem 4 is reduced to show that decays exponentially. As we already mentioned, the basic idea to achieve this goal is to use the Paley-Wiener theorem and, for this reason, we want to write the Laplace transform as an appropriate series by decomposing the phase space accordingly to the number of times that a certain piece of orbit hits the graph of the roof function:

where is defined by (1). Note that this is valid for any with .

An economical way of writing down this series uses the *partial Laplace transform* of a function defined by the formula for . In this language, the previous identity gives that

By a change of variables (i.e., “duality”), we obtain that

where is the th iterate of the *weighted transfer operator*

Remark 8In general, we expect the terms of the series (4) to decay exponentially because is expanding and is “dual” to the (Koopman-von Neumann) operator defined by composing functions with .

This formula (stated as Lemma 7.17 in AGY paper) is the starting point for the application of Paley-Wiener theorem to . More precisely, we can exploit it to analytically extend into three steps:

- (a) Lemma 7.21 in AGY paper: can be extended to a neighborhood of any ;
- (b) Lemma 7.22 in AGY paper: can be extended to a neighborhood of the origin ;
- (c) Corollary 7.20 in AGY paper: can be extended to a strip for some small and large in such a way that the extension satisfies (for some constant depending on and ).

Geometrically, the first two steps permit to extend to *rectangles* of the form

for . Indeed, this is a consequence of (a), (b) and the compactness of the segment .

Of course, these steps are *not* sufficient to extend to a *whole* strip to the left of the imaginary axis (because there is *nothing* preventing as ), and this is why the third step is *crucial*.

In summary, the combination of the three steps (a), (b) and (c) (and the fact that the function is integrable) says that has an analytic extension to a strip satisfying the hypothesis of Paley-Wiener theorem 5.

Therefore, we have that decays exponentially (and, a fortiori, the correlation function also decays exponentially) *if* we can establish (a), (b) and (c).

** 2.2. Implementation of (a), (b) and part of (c) **

Observe that the series in (4) is bounded by

This hints that we should first compare the sizes of and to the sizes of and before trying to show the geometric convergence of this series (for certain values of ). In this direction, we have the following pointwise bound (compare with the equation (7.66) in AGY paper):

Lemma 6There exists a constant such that

for any function and for all with .

*Proof:* Recall that . Hence, the desired estimate is trivial for . The remaining case is dealt with by integration by parts. Indeed, we have

Since the right-hand side has boundary terms bounded by and an integral term bounded by

(as ), we see that the proof of the lemma is complete.

This pointwise bound permits to control (cf. Lemma 7.18 in AGY paper):

Corollary 7There exists a constant such that

for all with .

*Proof:* This is an immediate consequence of Lemma 6 and the fact that the function is integrable (by the exponential tails condition on ).

On the other hand, this pointwise bound is *not* adequate to control in terms of a geometric series. In fact, it is well-known that the weighted transfer operators *only* exhibit some contraction property when one *also* works with stronger norms than . In our current setting, it might be tempting to try to control the norm of in terms of the norm of . As we are going to see now, this does not quite work for directly (as the pointwise bound in Lemma 6 also involves the function which might be unbounded), but it does work for (compare with Lemma 7.18 of AGY paper):

Lemma 8There exists a constant such that

- ;
- ,

for all with .

*Proof:* Recall that .

It follows from Lemma 6 that

where denotes the constant function of value one. Since it is not hard to check that the operator acting on the space of functions is bounded for , the first item of the lemma is proved.

Let us now prove the second item of the lemma. For this sake, we write . In particular, the derivative has four terms: one can differentiate the term or or the limit of integration or (resp.). Let us denote by , , and (resp.) the terms obtained in this way.

The fourth term is bounded by

Similarly, the bounded distortion property for implies that

Finally, since (by definition of good roof function) and , we see that the third term is bounded by

and the first term is bounded by

(thanks to the estimate from the first item.)

This completes the proof of the lemma.

Remark 9This suggests that we should measure functions using the normAn important point in this lemma is that the norm of behavesdifferentlyfrom the norm of .

in order to get some uniform control on the operator : indeed, this norm allows to rewrite the previous lemma as

and this is exactly the statement of Lemma 7.18 in AGY paper.This norm will show up again in the statement of the Dolgopyat-like estimate.

Once we have Corollary 7 and Lemma 8, we are ready use the estimate (6) and some classical properties (namely, *Lasota-Yorke inequality* and *weak mixing for* ) in order to implement the step (a) of the “Paley-Wiener strategy”:

Lemma 9For any , there exists an open disk (of radius independent of and ) centered at such that has an analytic extension to .

*Proof:* It is well-known (cf. Lemma 7.8 in AGY paper) that the weighted transfer operator acting on satisfies a *Lasota-Yorke inequality*. We will come back to this point in the next post of this series, but for now let us just mention a key *spectral* consequence of a Lasota-Yorke inequality for .

By Hennion’s theorem (cf. Baladi’s book), a Lasota-Yorke inequality for implies that its essential spectral radius is and its *spectral radius* . In concrete terms, this means that there exists a constant such that the spectrum of is entirely contained in the ball *except* for possibly finitely many eigenvalues (counted with multiplicity) located in the annulus .

In other words, *if* one can show that has *no* eigenvalues of modulus , *then* the previous description of the spectrum of gives that for some constants and , and for all . Of course, since is a small analytic perturbation of for any in a small disk centered at , this implies that

for some constants , and for all , . In particular, by combining this estimate with Corollary 7, Lemma 8 and (6), we obtain that in this setting the series

defines an analytic extension to of .

In summary, we have reduced the proof of the lemma to the verification of the fact that has no eigenvalues of modulus (when ). As it turns out, this is an easy consequence of the spectral characterization of the weak-mixing property for the expanding semiflow : indeed, this property says that the Koopman-von Neumann operator given by composition with has no eigenvalues of modulus , and it is not difficult to see that this means that has no eigenvalues of modulus .

This proves the lemma.

Remark 10A more direct proof of this lemma (without relying on the weak-mixing property for ) can be found in Lemma 7.21 of AGY paper.

Next, let us adapt the argument above to perform the step (b) of the Paley-Wiener strategy:

Lemma 10There exists an open disk (of radius independent of and ) centered at such that has an analytic extension to .

*Proof:* The transfer operator has a simple eigenvalue (cf. Aaronson’s book). In particular, the argument used to prove the previous lemma does not work (i.e., it is simply *false* that decays exponentially as ).

Nevertheless, we can overcome this difficulty as follows. For in a small open disk centered at , is an analytic perturbation of . Thus, has an eigenvalue close to , and we can write

where is the spectral projection to the eigenspace generated by the normalized eigenfunction (with ) associated to , and . Furthermore, the spectral properties of mentionned during the proof of Lemma 9 also tell us that there exist uniform constants and such that for all and .

In other terms, after we remove from the component associated to the eigenvalue , we obtain an operator with nice contraction properties.

At this point, the basic idea is to “repeat” the argument of the proof of Lemma 9 with replaced by . In this direction, we rewrite the series (4) as

(Here, we used that , , and )

Observe that the series converges for all thanks to Corollary 7, Lemma 8 and the fact that with for all .

It follows that we can use the previous equation to define an analytic extension of to *if* we can control the term . In other words, the proof of the lemma is reduced to show that

Note that this is a completely obvious task for because , i.e., the analytic function

has a *pole* at .

Fortunately, the order of pole at of this function can be shown to be one by the following calculation. Since is an analytic perturbation of , we have that and . In particular,

where is the absolutely continuous invariant probability measure of . This means that has derivative at , and, *a fortiori*, the pole of at has order one.

Thus, the function (8) is analytic on . Moreover, it also follows that the function (8) can be analytically extended to if we show that

has a zero at . This last fact is not hard to check: by definition, is a constant multiple of the function and , so that

and our assumption (2) was precisely that .

This proves the lemma.

Closing this post, let us reduce the step (c) to the following *Dolgopyat-like estimate* (compare with Proposition 7.7 in AGY paper):

Proposition 11There exist , , and such that

for all with , , . (Here, is the norm introduced in Remark 9.)

The proof of this proposition will occupy the next post of this series. For now, let us implement the step (c) of the Paley-Wiener strategy assuming Proposition 11.

We want to use the formula (4) to define a suitable analytic extension

of to a strip of the form .

By (6), Proposition 11, Corollary 7 and Lemma 8, we have

for all with and .

This proves that is an analytic extension of to such that , which are exactly the properties required in the step (c).

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