Arnaud Ch Éritat

TAN LEI AND SHISHIKURA'S EXAMPLE OF NON-MATEABLE DEGREE 3 POLYNOMIALS WITHOUT A LEVY CYCLE

After giving an introduction to the procedure dubbed slow polynomial mating and quickly recalling known results about more classical notions of polynomial mating, we show conformally correct pictures of the slow mating of two degree 3 post critically finite polynomials introduced by Shishikura and Tan Lei as an example of a non matable pair of polynomials without a Levy cycle. The pictures show a limit for the Julia sets, which seems to be related to the Julia set of a degree 6 rational map. We give a conjectural interpretation of this in terms of pinched spheres and show further conformal representations.

1. Introduction 1.1. Context 1.2. Post-critically finite polynomials 1.3. Topological mating vs formal mating 1.4. Levy cycles 2. Conformal limit of the example of Shishikura and Tan Lei. 2.1. Conformally correct computer generated pictures. 2.2. Interpretation of the pictures. 2.3. Zooming on a tunnel.

Acknowledgments References 1. Introduction

In [START_REF] Shishikura | A family of cubic rational maps and matings of cubic polynomials[END_REF], Tan Lei and Shishikura gave an example of two post critically finite polynomials of degree 3 whose "formal" mating has an obstruction that is not a Levy cycle.

Let us recall the context:

1.1. Context. One thing making the discussion difficult, but also interesting, is that there are several non-equivalent definitions of polynomial matings. Let us recall the definition of the slow variant (see [START_REF] Milnor | Geometry and dynamics of quadratic rational maps[END_REF] p. 54):

1.1.1. Definition of slow polynomial mating. Consider two polynomials P 1 , P 2 of the same degree d ≥ 2. For the construction to make sense we need that their Julia sets be connected.

Remark. No other assumption is done. In other constructions it is either required that the Julia sets are locally connected or even stronger: that the polynomials are post-critically finite.

Recall that a polynomial of degree d ≥ 2 with connected Julia set has d -1 fixed external rays (i.e. of period 1). The construction that follows depends on the choice of such an external ray for P 1 , and of one for P 2 . In the degree 2 case, since there is only one fixed external ray, there is no choice.

For any real number V > 0 let R = e V and let U i (R) be the simply connected open subset of C bounded by the equipotential of P i with potential V . Bascially we would like to build a Riemann surface S R by gluing the closure of U 1 (R) and U 2 (R) along their boundaries in some specific fashion. Provided the gluing map is analytic, we get a Riemann surface. However we prefer to present the construction slightly differently, by gluing along open regions.

So choose V ∈ ]V, 2V ], and let R = e V and

U i = U i (R ).
Now let φ 1 be the unique Böttcher coordinate for P 1 that maps the chosen fixed external ray to the interval ] 1, +∞ [, let φ 2 be the same for P 2 and glue z ∈ U 1 with w ∈ U 2 whenever

φ 1 (z) R • φ 2 (w) R = 1.
Note that this relation restricts to an orientation reversing bijection between the equipotentials of level V of P 1 and P 2 . Note also that if z ∼ w then their external angles 1 are opposite. The gluing can equivalently be expressed in

U 1 U 2
Figure 1: Example of construction of the Riemann surface S R . The images depict the Julia sets nicknamed the Basilica and the Rabbit. Some value of R has been chosen and the domain bounded by the equipotential V > log R is indicated by shades of gray. The respective external rays of argument 0 have also been drawn. The parts of U i that get glued together are hatched.

In the image below we presented the two charts facing each other in a 3D space so that points glued together are close.

φ 1 φ 2 log -log U 1 (R) U 2 (R)
Figure 2: Left: gluing U 1 (R) and U 2 (R) amounts to gluing their Böttcher coordinates along the circle |z| = R by an inversion. Better, it is equivalent to gluing the logarithms of the Böttcher coordinates with an isometry. Right: in fact to define a Riemann surface we take slightly larger domains so that open sets are glued together. This depends on another parameter V (with V < V ≤ 2V ) which determines the overlap (the hatched parts on the tubes are glued together by an isometry). However the resulting surfaces are the same (canonically isomorphic).

the coordinates log φ i :

log φ 1 (z) + log φ 2 (z) 2 = V (mod 2πi).
It can be checked that the quotient topological space under this equivalence relation is indeed a topological manifold. Moreover, U 1 and U 2 provide an atlas for which the change of coordinates is holomorphic and thus we have defined a Riemann surface. It does not depend on the choice of V ∈ ]V, 2V ] in the sense that there are canonical isomorphisms between two such surfaces (see Figure 2) and will be designated as S R regardless of the value of V .

Figure 3: Here we drew the domains bounded by the potential V together with more equipotentials. The image on the right has been rotated 180 degrees so that now the external rays of argument 0 are facing each other. On S R , the boundaries of these two domains are glued together and the point on the external ray 0 match. On Figure 4, we see what these pictures become after being glued to form S R .

All these surfaces are homeomorphic to the sphere, so they are isomorphic to the Riemann sphere C. Recall that the latter has a group of automorphisms which is not small 2 so the isomorphism to C is not unique. Its choice will play a determinant role when we look at degenerating cases. The choice of such an isomorphism is called a normalization.

There are several objects of interest than can be defined on S R . First the Julia sets of P 1 and P 2 correspond to two compact subsets of S R . The same holds for the filled-in Julia sets. Equipotentials and external rays are also well-defined, together with a conformal map Φ from A to the round annulus R -1 < |z| < R where A is the the complement of the union of the two filled-in Julia sets:

Φ : A → z R -1 < |z| < R .
Let us designate these subsets of S R by J 1 (R), K 1 (R), J 2 (R), K 2 (R) and A(R). The equator of A(R) is the preimage of the unit circle by Φ and corresponds in either chart U i to the equipotential log R. More complicated objects can be defined. For instance the measure µ R on the equator of A(R) which is the pull-back by Φ of the uniform probability measure on the unit circle.

There is an important subtlety about the transfer to S R of the dynamical system. It turns out that P 1 together with P 2 define a holomorphic map from S R to S R d , not to S R :

S R F R -→ S R d . To explain this, let us recall that R = e V and V ∈]V, 2V ]. Let R = e V . Then P i maps U i (R ) to U i (R d ) and if z ∈ U i \ K(P i ) then the Böttcher coordinate satisfies φ i (P i (z)) = φ i (z) d . Hence if the product φ 1 (z)φ 2 (w) = R 2 then φ 1 (P 1 (z))φ 2 (P 2 (z)) = R 2d . 3 If isomorphisms from S R and S R d to
C have been chosen then the holomorphic map F R becomes a rational map of degree d.

Hence we get a one real parameter family of rational maps F R of degree d between spheres with distinguished compact subsets. Note that F R (K i (R)) = K i (R d ), and the same holds for J i (R) and A(R). Equipotentials and external rays are mapped to equipotentials and external rays.

1.1.2. Limits of slow mating? Now, as R -→ 1 by values > 1, do these objects have limits? Are these limits compatible? It depends on the initial maps P 1 and P 2 , but also on the meaning we give to having a limit and to compatibility. One thing we could ask for is to find a normalization of the spheres S R such that the maps F R tend to a rational map F of the same degree (which is d). We would then like both sets J i (R) to tend to the Julia set of F . We could ask more, for instance 4 that the equatorial measure µ R tends to the Brolin-Lyubich measure of F . We could also ask for less, for instance that F Rn converges for some sequence R n -→ 1, or even allow F R to tend to a lower degree map (convergence occuring on the complement of a finite set). The present article does not study these questions, it just shows an example that degenerates in a specific way.

By slow mating we mean the definition of the Riemann surfaces S R and the objects J i (R), . . . , F R that we defined on or between them. We do not designate the (hypothetical) limit map F . Let us stress out once more that the slow mating of two polynomials of the same degree d > 1 is well-defined whenever both have connected Julia sets. They do not need to be postcritically finite, and the Julia sets do not even need to be locally connected. The question of the convergence of F R can be posed in the non-locally connected case and is non-trivial. In the locally connected case, there is a notion of topological mating 5 obtained by gluing K 1 and K 2 along their boundaries respecting the Caratheodory loop (see Ref to Meyer-Petersen in the present volume for a definition), which been proven in some cases 6 to be topologically conjugate to the limit of F R . In the non-locally connected case, even if there is a limit, there is no notion of topological mating to which the the limit could be compared.

Remark: A twist can be added to the gluing: see the end of the article by Buff, Epstein and Koch in the present volume, Ref to slow matings and twisted matings. The real parameter R > 1 then becomes a complex 3 In the degree 2 case it would be tempting to try to define a map from S R to itself by deciding to map each half-annulus to the whole annulus. However it would be discontinuous at the equator. 4 The author ignores if this particular requirement is implied by the previous ones.

5 It may also be called the instant mating by opposition to the slow mating.

6 A detailed discussion of this is out of the scope of the present article.

parameter ρ = Re iθ ∈ C \ D and we get Riemann surfaces S ρ , objects J i (ρ), etc. . . and maps F ρ : S ρ → S ρ d . We will not study this here. Remark: Traditionally the mating of post-critically finite polynomials is not defined in terms of the slow mating but either with the topological mating (Post critically finite polynomials have locally connected Julia sets), see for instance [START_REF] Rees | A partial description of parameter space of rational maps of degree two[END_REF] and [START_REF] Shishikura | On a theorem of M. Rees for matings of polynomials[END_REF], or in terms of corrected formal matings, see [START_REF] Shishikura | A family of cubic rational maps and matings of cubic polynomials[END_REF]. Relations between these notions have been studied but not completely yet. See also ref to slow matings and twisted matings and ref to Questions about polynomial matings in the present volume.

1.2. Post-critically finite polynomials. From now on, with a few explicitly indicated exceptions, we focus on the case where P 1 and P 2 are post-critically finite, abbreviated as PCF throughout this article.

Remark. In this article, mutlicurves will be mentioned. See [START_REF] Douady | A proof of Thurston's topological characterization of rational functions[END_REF] for a definition, or ref to Pilgrim's article in the present Volume. Many authors use the word "multicurve" for isotopy classes of these. We will not. 1.2.1. About the Thurston machinery. We recall that a PCF orientation preserving ramified self cover (continuous but not assumed holomorphic) of the (oriented) sphere is called a Thurston map. We assume the reader knows the definition of Thurston equivalence of Thurston maps, the definition of a Thurston map with hyperbolic orbifold, the definition of Thurston obstructions and the related existence and uniqueness theorem of rational realizations of classes of Thurston maps with hyperbolic orbifold. We do not assume the reader knows the proof, but at some point we will use Thurston's pull-back map (a.k.a. Thurston's algorithm). These subjects are discussed in details in [START_REF] Douady | A proof of Thurston's topological characterization of rational functions[END_REF]. Note that in the present article, we do not require an obstruction to be a stable multicurve. 7 1.2.2. The marked set. In the case where P 1 and P 2 are post-critically finite, one may map the post-critical set 8 of each polynomial to S R (let us call P R ⊂ S R this finite collection of points) and try to see if these points remain at some distance from each other as R -→ 1 or if they will gather in different groups. It is important to stress that P R is not 9 the post-critical set 10 of some F R . This is why we prefer to call P R the marked set. One may try to guess the Thurston equivalence class of the limit map F of the family 7 Also called invariant multicurve. A multicurve Γ for f is called stable if all components of f -1 (Γ) are either peripheral or homotopic rel. the post-critical set to a curve in Γ.

8 the union of critical values and their orbits 9 unless accidentally 10 However, it is the union over n > 0 of the critical values of the compositions

F R 1/d • F R 1/d 2 • • • • • F R 1/d n
F R , if any limit exists. A candidate is provided by the so-called formal mating, which is a Thurston map built from P 1 and P 2 (see the description in other articles in this volume, like ref to Meyer-Petersen and ref to slow matings and twisted matings; we also give a description in Section 1.2.5 of the present article). In this article, it will be called the candidate. However there are some cases where the map F R has a post-critically finite limit of degree d but of a different Thurston class: just because some points in P R collapse at the limit does not mean that the degree drops. This case has been acknowledged and a way to correct the formal mating is known (see Section 1.3), though the author does not know if the link between corrected formal mating and limit of slow mating has been completely proved to hold.

1.2.3. Relation between slow mating and Thurston's algorithm. Recall that the set Aut( C) of automorphisms of the Riemann sphere C consists of Möbius maps. Let f be the formal mating of P 1 and P 2 . It is a Thurston map defined on some oriented topological sphere S. This sphere and the post-critical set P f ⊂ S serve as a marker for a Teichmüller space T , that plays an important role in Thurston's algorithm. We remind the reader that T is the set of equivalence classes, denoted [φ], of orientation preserving homeomorphisms φ : S → C for the relation φ ∼ φ iff ∃µ ∈ Aut( C) such that µ • φ coincides with φ on P f and is isotopic rel P g to φ on the rest of S. Thurston's algorithm consists in the iteration of a map, the Thurston pullback map, which will be denoted σ f : T → T . Recall that for f PCF with hyperbolic orbifold, σ f is weakly contracting 11 and σ 2 f is locally strictly contracting, for some metric on T called the Teichmüller metric. Also, f is realizable by a rational map if and only if σ f has a fixed point. Therefore, f is realizable if and only if σ n f ([φ]) has a limit in T , and this is independent of the starting point [φ]. Now, the Riemann surfaces S R defined by the slow mating come with natural 12 markings by S, i.e maps φ R : S → S R .

To specify such a marking we need to specify a construction of S and f first, and there is some flexibility in the choice. Even with a given choice, there is still some flexibility in the definition of φ R . In Section 1.2.5, we give one example of construction with the following nice property:

F R • φ R = φ R d • f
11 distances are not increasing 12 unique up to isotopy rel P f i.e. the following diagram commutes:

S φ R d ----→ S R d f     F R S φ R ----→ S R
It has the following consequence: let T R be the point in the Teichmüller space T defined by φ R . Then:

13 σ f (T R d ) = T R .
In words, this says that the slow mating defines a path R → T R in the Teichmüller space, parameterized by R ∈]1, +∞[, and that Thurston's pullback map associated to the formal mating acts on it as the d-th root on the parameter.

1.2.4. Squeezing the annulus. In this section we do not need the polynomials P 1 and P 2 to be PCF, we just require them to have connected Julia sets. Given two values R, R it is quite natural to define the following nonholomorphic map: 14 Ψ R ,R : S R → S R defined as folows: map z ∈ K 1 in the chart U 1 (R) to the point of S R with the same coordinate z in U 1 (R ). Do the same for K 2 . Map a point in the annulus A(R) to the point on the same external ray but with potential multiplied by log R / log R (the potential may be measured in either the Böttcher coordinates of P 1 or that of P 2 , or also with log |Φ| that assigns 0 to the equator, where Φ is the isomorphism from A(R) to 1/R < |z| < R; in all cases this gives the same result). This map is continuous and better: it is quasiconformal.

These maps are compatible with the dynamics:

F R • Ψ R ,R = Ψ R d ,R d • F R
i.e. the following diagram commutes:

S R d Ψ R d ,R d -----→ S R d F R     F R S R Ψ R ,R ----→ S R
13 Note that such a strong property as

F R • φ R = φ R d • f is not necessary to get σ f (T R d ) = T R : a Thurston equivalence φ -1 R d • F R • φ R ∼ f would have been enough
. However, it is as easy to directly get a φ R satisfying the strong assumption.

14 Note the inversion: it will make equations described later easier to handle; it is regrettable but almost imposed on us by the fact that the composition notation f • g reads backwards.

and Ψ

R ,R = Ψ R ,R • Ψ R ,R .
On the annuli A(R), A(R ), . . . , these two identities basically follow from the fact that F and Ψ act as multiplications on the potential and on the external angle, and multiplications commute.

1.2.5. A construction of the formal mating and of markings of S R . In fact we'll give two. In this section we do not need the polynomials P 1 and P 2 to be PCF, we just require them to have connected Julia sets.

The quick and dirty way consists in choosing a particular value of R, say R = exp(1) = e and using S = S e , f = Ψ e,e d • F e , and letting the marking be: φ R = Ψ R,e . The claims of Section 1.2.3 follow at once.

For the second construction, map C to the unit disk by the non-conformal homeomorphism ψ : z → z/(1 + |z|). Let S be the quotient of the disjoint union of two copies D 1 and D 2 of the closed unit disk, glued along their boundaries with e iθ ∈ ∂D 1 identified with e -iθ ∈ ∂D 2 . We will identify D 1 and D 2 with their image in S. We first define a modification of P 1 , call it P 1 as follows: P 1 = P 1 on K 1 and for z ∈ C \ P 1 , P 1 (z) is the point of C \ K 1 with the same potential as z, but with external angle multiplied by d. The map P 2 is defined similarly. Then let f be the conjugate of P 1 by ψ on D 1 , the conjugate of P 2 by ψ on D 2 an be the multiplication by d of the argument on the circle bounding D 1 and D 2 . Now for z ∈ S let φ R (z) be defined as follows. If z ∈ D 1 then let w = ψ -1 (z) ∈ C; if w ∈ K(P 1 ) then let φ R (z) be the point of coordinates w the chart U 1 ; if w / ∈ K(P 1 ) then let V be the potential of w (it can be any real number in ]0, +∞[ since w can be any complex number in C \ K(P 1 )) and let φ R (z) be the point in the chart U 1 (R) whose coordinate is the point on the same external ray of P 1 as w, but with potential log(R)V /(1 + V ).

Remark. In both cases we chose the markings so that

φ R = Ψ R ,R • φ R .
Such a strong property was not necessary. Note also that, as far as the Thurston characterization of rational functions is concerned, the particular dynamics of f is not relevant, only its Thurston equivalence class is. Other constructions exist, for instance one where the map f is C ∞ on a C ∞ sphere, and has an attracting equator. In the latter case, the markings cannot satisfy both

φ R = Ψ R ,R • φ R and φ -1 R d • F R • φ R = f , but they still satisfy the following weaker form: φ -1 R d • F R • φ R
is isotopic to f by an isotopy that does not move points in K 1 and K 2 . If P 1 and P 2 are PCF, this weak form nonetheless implies that φ -1

R d • F R • φ R and f are Thurston equivalent.
1.3. Topological mating vs formal mating. This section will mention the notions of topological mating and ray equivalence classes. They are discussed in details in the present volume in ref to Meyer-Petersen. Note that when the topological quotient space S/ ∼ in the definition of topological mating is homeomorphic to the sphere, there is a well-defined preferred orientation: indeed the choice of an orientation on a space X homeomorphic to the sphere induces a choice of isomorphism of its first singular homology group H 1 (X) to Z. Having chosen an orientation on S, we choose the orientation on the quotient so that the quotient map S → S/ ∼ induces the identity Z → Z.

Assume that P 1 and P 2 are PCF. It may happen that some ray equivalence classes contain several marked or critical points. Then the topological mating cannot be Thurston equivalent to the formal mating: the formal mating has to be corrected. Rees, Shishikura and Tan Lei have devised at least three ways of doing it, which probably have been proven equivalent (though the author could not find a written proof): see [START_REF] Rees | A partial description of parameter space of rational maps of degree two[END_REF], [START_REF] Shishikura | On a theorem of M. Rees for matings of polynomials[END_REF] and [START_REF] Shishikura | A family of cubic rational maps and matings of cubic polynomials[END_REF]. In [START_REF] Shishikura | On a theorem of M. Rees for matings of polynomials[END_REF], it is proven that if the corrected formal mating in the sense of [START_REF] Shishikura | On a theorem of M. Rees for matings of polynomials[END_REF] exists and is realizable (Thurston-equivalent to a rational map) then the topological mating is conjugate to the same rational map. The converse of this statement reads: if the topological mating of two post-critically finite polynomials is conjugate to a rational map F then the corrected formal mating in the sense of [START_REF] Shishikura | On a theorem of M. Rees for matings of polynomials[END_REF] exists and is Thurstonequivalent to F . It has been proven by Rees in the case that the two polynomials are hyperbolic (see [START_REF] Rees | A partial description of parameter space of rational maps of degree two[END_REF]), in which case the formal mating does not need correction. The author thinks that the converse also holds for any pair of post-critically finite polynomials, though he could not find a written proof of it. It should follow from the following theorem of Moore (see ref to Meyer-Petersen for instance): if S is the 2-dimensional sphere and φ : S → S is continuous, surjective, with connected fibers, then φ is the uniform limit of a sequence of homeomorphisms.

Even in degree 2, there are pairs of post-critically finite polynomials which have an obstructed formal mating but still have a topological mating conjugated to a rational map: for instance the mating, studied in [Mil04], of P = z 2 +c with itself where c is at the end of the Mandelbrot set external ray of argument 1/4. The post-critical set of P 1 = P 2 = P has three elements: P (c), P 2 (c), P 3 (c) = P 4 (c). The ray equivalence identifies P 2 1 (c) with P 2 2 (c) and P 3 1 (c) with P 3 2 (c), but no other point in the post-critical set of P 1 and P 2 . Their topological mating is conjugate to the non-flexible Lattès map associated to multiplication by 1 + i on the square lattice.

Note that the slow mating is a realization of the Thurston algorithm for the formal mating but not for the corrected formal mating. In the case of a mating that needs and has a correction, it is likely that under the slow mating, the marked points belonging to a same ray class will get closer and closer and tend to a single point 15 , and that the maps F R will converge to a map of the same degree (i.e. without loss of degree). We do not know if it has been proved completely.

1.4. Levy cycles. In his thesis [START_REF] Levy | Critically Finite Rational Maps[END_REF], Silvio Levy proved the following theorem. Let f be a post-critically finite topological ramified self-cover of the sphere:

Theorem 1 (Levy). If f has degree 2 then it has a Thurston obstruction if and only if it has a Levy cycle.

A Levy cycle is a numbered multicurve γ 0 , γ 1 , . . . , γ n = γ 0 such that for all k, γ k+1 is isotopic rel. P f to a component of f -1 (γ k ) on which f has degree 1, see ref to slow Matings and twisted matings in this volume The curves of a Levy cycle form an obstruction so one implication of the theorem is trivial. Note that the Levy character of a given numbered multicurve cannot be read off from its Thurston matrix. 16 1.4.1. Polynomial matings and Levy cycles. Formal matings which need and have a correction have a special class of Levy cycles called removable Levy cycles.

Concerning matings of post-critically finite polynomials, the situation is completely understood in degree 2: using Levy's theorem, Tan Lei proved:

Theorem 2 (Tan Lei). The pairs of post-critically finite polynomials of degree 2 whose topological mating is not equivalent to a rational map are those who belong to conjugate limbs of the Mandelbrot set. Moreover, their ray equivalence classes contain loops and the quotient topological space in the definition of the topological mating is not a sphere.

However, not only Levy's theorem has counter examples in degree ≥ 3 but it is has counter examples among formal matings:

Theorem 3 (Shishikura, Tan Lei, [START_REF] Shishikura | A family of cubic rational maps and matings of cubic polynomials[END_REF]). There exist a pair of postcritically finite polynomials of degree 3 whose formal mating is obstructed but has no Levy cycles. Moreover, the corresponding quotient topological space is homeomorphic to a sphere. 15 In the sense that they will be separated from the other marked points by an annulus of modulus tending to +∞ separating them from the others, or equivalently by a geodesic of length tending to 0, for the hyperbolic metric on the complement of the marked points: see Section 1.4.4

16 There exists a pair of maps each with a multicurve with the same matrix, but one multicurve is a Levy cycle and the other is not. This is because the coefficients in the matrix are sums of inverse of integers, and one cannot deduce the presence of the term 1 from the value of the sum.

Having no Levy cycles, this mating is not correctable, so the topological mating of P 1 and P 2 is not conjugate to a rational map. The author believes one can prove it implies there is no way to normalize the spheres S R so that the rational maps F R converge uniformly as R -→ 1, even for a subsequence R n -→ 1. In this particular example, this is supported by the experiments presented in Section 2. It is remarkable, though, that the quotient topological space is still a sphere: this is basically because the ray equivalence relation is closed and the classes contain no loop (see [START_REF] Shishikura | A family of cubic rational maps and matings of cubic polynomials[END_REF]). 1.4.2. Description of the example of Tan Lei and Shishikura. Their example is an explicit pair of post critically finite polynomials (P 1 , P 2 ) that we will define now.

Post-critically finite polynomials can be characterized, up to complexaffine conjugacy, by the augmented Hubbard trees. Recall that the augmented Hubbard tree is the Hubbard tree together with some angle information at some vertices. 17 Remember also that to mate polynomials, we must align the external angles. This can be done by choosing which fixed external ray is the one of angle 0 (this amounts to choosing monic centered polynomials). In terms of the combinatorial information, this means giving some enough supplementary information to distinguish it. 18 It turns out that less information is often sufficient. Here the polynomials P 1 and P 2 in the example of Tan Lei and Shishikura can be charaterized up to C-affine conjugacy by the data on Figure 5. The monic centered polynomials are uniquely determined by the data on Figure 6. In particular P 1 has critical points x, y and a 3-cycle x → y → z → x, and P 2 has a double critical point (thus of local degree 3) of period 3. The notations x, y, z are that of [START_REF] Shishikura | A family of cubic rational maps and matings of cubic polynomials[END_REF]; in the present article, these points will be denoted x 0 , y 0 and y 1 .

The polynomials are

P 1 (z) = z 3 + az + b P 2 (z) = z 3 + c
17 Explanation: by uniqueness in Thurston's theorem, a polynomial will be characterized, up to C-affine conjugacy, by its Thurston class. It is well known that the Hubbard tree is not enough for characterizing the Thurston class. In fact, once one has the tree H, one can characterize its first preimage H by P up to homeomorphism of the plane. The problem is then to determine how H is embedded in H , and there may be several inequivalent ways for a fixed H, even taking the dynamics on H into account. Supplementary information is needed to determine this. The augmented Hubbard tree gives sufficient information.

18 This is not very important, but note that for polynomials commuting with a rotation, one only needs to characterize a class of fixed external ray modulo this rotation. This data is enough, at least on these examples, to uniquely determine the monic centered polynomials P 1 and P 2 . We did not indicate the other fixed external ray, which has angle 1/2. with

a = -3x 2 0 , b = 2x 3 0 -x 0 , 32x 8 0 -24x 6 0 + 2x 2 0 -1 = 0, x 0 ≈ 0.8445, c 8 + 3c 6 + 3c 4 + c 2 + 1 = 0, c ≈ -0.264 + 1.260i.
And their Julia sets are illustrated on Figure 7. 1.4.3. An obstruction for the example. The formal mating f has a Thurston obstruction, a multicurve Γ = {a, b}, illustrated on Figure 8. On Figure 9 we put the preimage of Γ. From this it is easily seen that the Thurston y 0

x 0 y 1 c = P 2 (0) The moduli space M(P f ), more tractable in many respects, is the set of maps from P f to C modulo post-composition by a Möbius map. We will call its elements configurations. There is a projection π : T (S, P f ) → M(P f ) Figure 11: Same object but with shorter geodesics. The hyperbolic version is represented with a bigger scale than on the previous figure. This picture and the previous one are not meant to be precise, but to give a sense of the hyperbolic metric.

0 = P 3 2 (0) P 2 2 (0)
that maps the class of φ to the class of its restriction to P f . The following proposition is important in the theory of Thurston's pull-back map: if a Thurston map f has hyperbolic orbifold, then

σ n f ([φ]) diverges in T (S, P f ) if and only if π(σ n f ([φ]
)) leaves every compact of M(P f ). Thus, no matter what normalization one chooses, for the configuration of points on C given by π(σ n f ([φ])), after passage to a subsequence, there will be at least two points with the same limit.

The fact that one has normalizations to choose makes the discussion difficult and it is much more pleasant to work with the more rigid structure provided by the hyperbolic metric, a.k.a. Poincaré metric, of the complement of the marked points, i.e. of U = C \ φ(P f ). We will explain why.

Before, let us state here a few facts concerning the geometry of the Poincaré metrics (see [START_REF] Hamal | Teichmüller theory and applications to geometry[END_REF]).

First fact. In each of the two components of the complement in C of a simple closed geodesic in U , there are at least two marked points. Conversely, for each simple closed curve γ in U such that both components of C \ γ contain at least two marked points (such curves are called non-peripheral ), there is a unique geodesic homotopic to γ (and it is the shortest curve homotopic to γ).

Second fact. Endow temporarily C with a spherical metric, using for instance a stereographic projection from the Euclidean 2-sphere. There exists a map h(ε) -→ ε→0 0 such that for a configuration C ∈ M(P f ) with a simple closed geodesic of hyperbolic length ≤ ε, whichever representative (normalization) φ ∈ C is chosen, there is at least one side of the geodesic where the marked points are grouped in a cluster of spherical diameter at most h(ε), and there exists φ ∈ C such that both groups live at spherical distance ≤ h(ε) of respectively 0 and ∞. Conversely there is a map g(ε) -→ ε→0 0 such that if C ∈ M(P f ) has a representative for which the marked points are split in two groups, each with at least two points, and one at spherical distance ≤ g(ε) from 0 and the other at spherical distance ≤ g(ε) from ∞, then there is a simple closed geodesic separating the groups and of hyperbolic length < ε in U .

As a corollary: a sequence of configurations [φ n ] ∈ M(P f ) will leave every compact in M(P f ) if and only if there exists a sequence of simple closed geodesics γ n in U n = C \ φ n (P f ) whose lengths tend to 0.

Third fact. Two disjoint non-peripheral curves are homotopic to geodesics which are either disjoint or equal. In particular a multicurve in U = C \ φ(P f ) has a privileged representative, which is the collection of simple closed geodesics homotopic to its curves.

Fourth fact. There is a universal constant L 0 > 0 such that no two different geodesics of length < L 0 can intersect (also, but we will not use this fact, a closed geodesic with length < L 0 is necessarily simple). If P f has k points then there are at most k -2 different simple closed geodesics of length < L 0 .

Pieces: Hence endowing C \ φ(P f ) with its hyperbolic metric, gives us a way to separate the marked points into groups, by cutting the sphere along those few simple closed geodesics that are small (with a notion of small chosen according to the use, for instance shorter than L 0 ). In fact we get more than just groups of points: all pieces do not necessary contain a marked point; the graph whose vertices are pieces and edges are curves separating two pieces is a tree. See Figures 10 and11.

Figure 12: Another (schematic) conformal model of the punctured sphere with short geodesics: a tree of spheres. A Möbius map will zoom on each piece, represented here as a sphere. By removing carefully chosen small disks from each sphere and adding tubes to link them together, one can recover a punctured Riemann surface that is isomorphic to the punctured sphere under consideration. Note that there is flexibility in the choice of the sizes of the removed disks and in the choice of the shapes of the tubes. Tree: More generally cut a sphere with marked points along a multicurve. The sphere is split into open pieces U i . Let U i be the pieces minus the marked points. Each U i has at least three boundary components, which may be points or curves. Sometimes, there are no points in some pieces U i . Nevertheless, it is possible to single out any U i by choosing three marked points. More precisely, for any three marked points, there is a unique U i which, when removed from the sphere, disconnects the three points (this is related to the tree structure associated to the pieces). For every component, there is a (not necessarily unique) set of three points that it disconnects.

Zoom: Let us get back to C \ φ(P f ) with its conformal structure and its hyperbolic geometry. Cut it along small simple closed geodesics (the meaning of the word "small" depends on the practical application). Select three marked points, with some order, and compose φ with the unique Möbius map sending them respectively to 0, 1, ∞. This normalization gives a new representative of the configuration which focuses (or kind of zooms) on the piece associated to the selected triple. The smaller the geodesics bounding the piece, the more concentrated the marked point on the other side of the geodesic, viewed on the normalized Riemann sphere.

Figure 12 sums this up by a conformal presentation of the Riemann surface taking pinching curves into account. These trees of spheres are the usual point of view used for the definition of a standard compactification of the moduli space, see [BFL + 03]. Figure 13 gives yet another way to conformally represent things. It puts the emphasis both on the sphere and on the funnels between them. The latter are annuli with big moduli, so they can also be viewed as tunnels. Later we will present some conformally accurate images that focus on a tunnel instead of focusing on the big spheres. 1.4.5. The canonical obstruction. Let us say that two simple closed curves γ and γ in C \ P f are homotopically transverse if there is no curve homotopic to γ and disjoint from γ . Kevin Pilgrim in [START_REF] Pilgrim | Canonical Thurston obstructions[END_REF] precised how the curves pinch under iteration of Thurston's pull-back map:

Theorem 4 (Pilgrim). If f is Thurston map with hyperbolic orbifold and is not realizable (so it has obstructions), then there exists a multicurve Γ, whose isotopy class is called the canonical obstruction, such that ∀

[φ] ∈ T , let [φ n ] = σ n ([φ]) and U n = C \ φ n (P f ):
• Γ is not empty,

• for all curve γ ∈ Γ, the (simple closed) geodesic in U n homotopic to φ n (γ) has a length tending to 0 as n -→ +∞, • ∃N ∈ N and L > 0 such that ∀n ≥ N , these geodesics are the only simple closed geodesics shorter than L, • Γ is an obstruction, • Γ is f -stable and surjective 19 , • no curve in any obstruction can be homotopically transverse to a curve of Γ, The canonical obstruction is unique (as an isotopy class).

It shall be emphasized that there exists examples with hyperbolic orbifold having obstructions that do not pinch. Still, these examples have canonical obstructions that do pinch.

19 Surjective (non official terminology): It means that all curve in γ is homotopic to at least one curve in f -1 (Γ). To a multicurve is associated a directed graph with vertices the curves and arrows a → b whenever b is homotopic to a component of f -1 (a). The multicurve is called surjective whenever all b have an incoming arrow, in analogy to functions. It does not imply that the matrix is surjective. One last remark: even though the hyperbolic geometry is a pleasant way to formulate things, the core of the proofs of Thurston's and Pilgrim's theorems use moduli of annuli, and this language could have been used instead.

1.4.6. Does the obstruction pinch? We prove in this section that in the example of Shishikura and Tan Lei, the given obstruction Γ = {a, b} is the canonical obstruction.

In all this section, homotopic means homotopic in the complement of P f and isotopic means isotopic rel. P f . Lemma 1. Let γ be a non-peripheral simple closed curve disjoint from a and b. Then all components γ of f -1 (γ) are either peripheral or homotopic to a or b.

Proof. The multicurve Γ = {a, b} cuts the sphere into three parts A 1 , A 2 , A 3 , each of which contains exactly two marked points. The curve a separates A 1 from A 2 and b separates A 2 from A 3 . See Figure 14. The curve γ is contained in one of these components.

(1) If γ ⊂ A 1 , since A 1 \ P f only has three boundary components, which are two marked points and the curve a, all non homotopically trivial simple closed curve in A 1 \ P f is either homotopic to a small loop around a marked point, and thus peripheral, or homotopic to a. In particular γ is homotopic to a. The preimage of a has two components, one homotopic to a and the other to b.

(2) Similarly, if γ ⊂ A 3 , then it is homotopic to b, which has two preimages, one homotopic to a and the other which is homotopically trivial.

(3) Last, if γ ⊂ A 2 , note that A 2 has two preimages by f , one which is an annulus containing no marked points and whose equator is homotopic to a, and one which is an annulus containing two marked point and isotopic to a subset of A 3 . If γ is contained in the first one, it is either null-homotopic or homotopic to a. If γ is contained in the second one, then it is homotopic via the isotopy to a curve contained in A 3 , and by the same arguments as above, it is thus either peripheral or homotopic to b.

Let C be the union of the homotopy classes in the canonical obstruction. Denote γ → γ whenever γ is homotopic to a component of f -1 (γ). Note that the arrow goes in the opposite direction to that of f . Surjectivity of the canonical obstruction tells that all curves γ ∈ C are homotopic to a component of the preimage of a curve in C. By the last item of Theorem 4, this component has a representative disjoint from a and b. Since γ is not peripheral, the lemma tells us γ ∼ a or γ ∼ b. So the canonical obstruction is homotopically contained in Γ. Moreover since the canonical obstruction is not empty, a or b must be in C. Last, by f -stability, and since a → b and b → a, we know that a and b belong to C. So Γ is the canonical obstruction.

Conformal limit of the example of Shishikura and Tan Lei.

In this section we show conformally accurate pictures of the slow mating on the Riemann sphere. We then give a conjectural explanation of what is going on, and finish with a few more pictures.

2.1. Conformally correct computer generated pictures. We need to give practical labels to the post-critical points of the formal mating f and to the marked points. Let x 0 and y 0 denote the critical points of f corresponding to the critical points x 0 and y 0 of P 1 . Let y 1 be the point corresponding to P 1 (y 0 ). Recall that x 0 → y 0 → y 1 → x 0 . Let c 0 denote the critical point of f corresponding to the critical point 0 of P 2 . Let c 1 and c 2 correspond to c = P 2 (0) and P 2 2 (0). Recall that the marked points on S R are the images of the post-critical points of f by the marking φ R : S → S R . We will use the same labels for the marked points as for the post-critical points of f .

c 0 c 1 c 2 x 0 y 0 y 1 a b A 1 A 2 A 3
Figure 16: A pinched sphere model of S R for R close to 1.

On Figure 15 we drew the spheres S R for a pair of values R and R 3 , together with the Julia sets and the equator. We also drew a multicurve 20 corresponding to the obstruction Γ = {a, b} and its preimage by F R . 20 Multicurves are usually defined for Thurston maps f : S → S but in the proof of Thurston's theorem one also considers the image of multicurves by markings φ : S → C. See also Section 1.4.4.

In Section 1.4.6 we proved that the canonical obstruction is Γ = {a, b}. Thus as R -→ 1, the marked sphere will be conformally equivalent to something looking like Figure 16.

On Figure 17 we show a sequence of twelve spheres S R1 to S R12 with R n = 10 4/3 n , which satisfies R 3 n+1 = R n , so that each sphere maps to the previous by F Rn+1 ; it ranges from R 1 = 21.54 . . . to R 12 = 1.0000173 . . . The marked points and the point on the equator of argument 0, call it e 0 , are highlighted as red and green dots according to a convention that we explain now: the green dots indicate the points used for the normalization of the sphere, and the red dots indicate those marked points that are not green. The normalization was chosen as follows: c 0 is sent to 0 ∈ C; x 0 is mapped to ∞ ∈ C; the point e 0 is mapped to 1 ∈ C. Hence two marked points are green. The conformal projection from C to the Euclidean sphere has been chosen so that the three green points are all visible, with 0 near the bottom, 1 on the right and ∞ near the top.

This choice of normalization is different from the ones proposed in Section 1.4.4. Instead of mapping three marked points to 0, 1 and ∞, we normalize so that only two marked points plus a point on the equator are mapped to 0, 1 and ∞. The reason is that for values of R that are not small, this gives a more balanced picture: indeed, if the three marked were normalized, then at least two of them would belong to one of the two polynomial. As a consequence, on the picture, only that polynomial would be visible. The drawback is that it is not obvious on which piece of the sphere cut along the short geodesics we are zooming in.

On the sequence on Figure 17, it appears that we are zooming on the middle sphere, A 2 : the points y 0 and c 1 seem to gather together, like do the points y 1 , c 2 and e 0 . The explanation would be that in the normalizations focusing on A 3 (sending the triple y 1 , c 2 and z = any of the remaining four marked points to ∞,0,1) e 0 converges to a point distinct from the limit of the marked points (i.e. from 0, 1 and ∞).

Question 1. Prove this assertion. 21 On Figures 18, 19 and 20 we drew (still conformally correct) flat views, where the green point x 0 is sent to infinity, for R 10 , R 15 and R 20 .

On Figure 21 we show in parallel three sequences of four vertically stacked images, each sequence has a different normalization, each row has the same value of R. Each normalization focuses on a different piece: A 1 , A 2 , A 3 . Notice how the middle and right column converge to a similar limit. 2.2. Interpretation of the pictures. In the three normalizations focusing on respectively A 1 , A 2 and A 3 , even though the family of maps F R is 21 For instance e 0 could be added to the set of marked points and Thurston's algorithm extended to include cycles. See the work of Selinger in [Sel]. supposed to diverge, the picture seems still to converge to a well defined limit, very reminiscent of a Julia set, but with mess in some Fatou components. We propose the following (conjectural) interpretation. Work of Selinger in [Sel] may help in giving proofs.

As R -→ 1, the marked sphere tends to a pack of three Riemann spheres: left, middle and right, limits of respectively A 1 , A 2 and A 3 , touching each other at two singular points. The map F R has a limit F on this pack minus the contact points, which maps the three spheres between themselves, as rational maps. More precisely the limit F sends the middle sphere to the left, the left to the right, and the right back to the middle. Let us write F 1 , F 2 and F 3 for the three rational maps between these spheres (the domains of definition of the F i include the contact points but the maps will not necessarily agree on these points, and the convergence will occur on the spheres minus these points).

On the middle sphere, the limit F 2 has a critical point at x 0 , and a double critical point at c 0 . There cannot be an odd number of critical points counted with multiplicity so there must be a supplementary one. It is hiding at the contact point between the left and middle spheres. Indeed, recall that the multicurve component a has a preimage component which is homotopic to a and which is mapped 2 : 1 to a (in an orientation reversing way). There is no critical point at the singular point between the middle and the right spheres, because the component of the preimage of a that is homotopic to b is sent 1 : 1 to a.

The same analysis on the left sphere indicates that F 1 has two critical points, so it has only degree 2. The critical points are y 0 and the contact point, and they respectively map to y 1 and the other contact point. Finally, the map F 3 has no critical point. Thus the respective degrees of F 1 , F 2 , F 3 , are 2, 3, 1. Another way of seeing that would have been to look at the preimage of the nearly pinched sphere by F R : this gives a pinched sphere with five bubbles, as on the second row of Figure 26, three of which contain the marked points (thus excluding e 0 , which is not marked). Counting degrees then gives the same result.

Knowing this and the way marked points and contact points are mapped to each other, is enough, at least on the example we are studying, to completely determine the maps F 1 , F 2 , F 3 . Let us stick to the normalization used in the sequence of pictures previously shown: before the limit, map e 0 to 1 on each sphere, and the two non collapsing marked point in each sphere to 0 and ∞ for respectively P 2 and P 1 . This implies in the limit that the contact point on the left sphere, the right contact point on the middle sphere, and the point e 0 on the right sphere, all map to 1 under the normalization, and that the two marked points in each sphere map to 0 and ∞ under the normalization. Then the three rational maps F i must send 0 to 0, 1 to 1 and ∞ to ∞. Since moreover F 3 has degree 1, it is the identity:

F 3 (z) = z.
Note that F 1 is a polynomial and that the conjugate G 2 of F 2 by z → 1/z is polynomial (recall that a rational map is polynomial if and only if the preimage of ∞ is itself, no more, no less). The degree of G 2 is 3 and G 2 (0) = 0, G 2 (0) = 0, so G(z) = bz 3 + az 2 with b = 0. Also G 2 (1) = 1 so a + b = 1, and G 2 maps the remaining critical point c = -2a/3b (whose coordinate in the middle sphere we do not know yet) to 1. This gives 4a 3 = 27b 2 . So b = 1 -a and 4(a/3) 3 -a 2 + 2a -1 = 0. The latter equation has a double root a = 3 and a single root a = 3/4. In the case a = 3, we get that the other critical point c = 1, which we exclude: the two contact points cannot be equal. Thus a = 3/4, b = 1/4, c = -2 and G 2 (z) = z 2 (z + 3)/4:

F 2 (z) = 4z 3 3z + 1 .
We also got the coordinates of the left contact point on the middle sphere, which is a critical point of F 2 and is 1/c = -1/2. Finally, F 1 is a degree

(0, 1, ∞). ∞ 2 ∞ 2 ∞ 0 0 3 0 1 -1/2 1 -1/2 2 1 2 F 3 (z) = z F 2 (z) = 4z 3 3z + 1 F 1 (z) = z 2 /2 -z ∞ ∞ ∞ 2 2 -1/2 1 1 -1/2 2 2 0 0 0 3 F 2 F 1 middle left right
Figure 22: Summary. Above: the subscript indicates the multiplicity of the critical points. Cross ratio of complex numbers is not respected on this schematic illustration. Below: who maps where.

2 polynomial with a critical point at z = 1, fixing 0 and mapping 1 to the contact point on the right sphere. The latter being mapped to the contact point -1/2 on the middle sphere by F 3 = id we get that F 1 (1) = -1/2.

F 1 (z) = ((z -1) 2 -1)/2 = z 2 /2 -z.
This is summarized on Figure 22.

Question 2. Prove that the limit of F R in the pinching spheres representation is indeed the map

F = F 1 ∪ F 2 ∪ F 3 .
Now, the third iterate of the map F is a map from the pack to itself which sends each sphere to itself, by three degree 6 maps that are semi-conjugate to each other because they all consist in making one turn in the following non-commutative diagram:

F 2 F 1 F 3
The map in the middle is

H 2 = F 3 • F 1 • F 2 , the map on the left is H 1 = F 2 • F 3 • F 1 and the map on the right is H 3 = F 1 • F 2 • F 3 . Since F 3
is the identity in this example and with our choices of coordinates, we only get two different degree 6 rational maps. Let us characterize further these two maps. They both have three critical values 22 : 0, 1, ∞. The map H 2 from the middle sphere to itself has 5 critical points: and0 3 -→ 0, the number above the arrow indicates the local degree. The sum of "local degrees minus one" is 10, which agrees with the formula 2d -2 counting the number of critical points with multiplicity for a degree d = 6 map. The map H 1 on the left sphere has 4 critical points:

∞ 4 -→ ∞, 1 4 -→ 1, -1/2 2 -→ -1/2, -1/3 2 -→ ∞
∞ 4 -→ ∞, 1 4 -→ 1, 0 3 -→ 0, 2 3 -→ ∞. This is better presented as a diagram: 0 ∞ 1 2 3 3 4 4 F 2 • F 3 • F 1 0 ∞ 1 -1/3 -1/2 3 2 4 2 4 F 3 • F 1 • F 2 (= F 1 • F 2 • F 3 )
On Figure 23 are shown pictures of the Julia and Fatou sets of these two maps H 1 and H 2 . For both, the behavior of the critical points and the classification of Fatou components imply that every Fatou component eventually falls under iteration in the immediate basin of one of the three fixed critical points of H i . The picture suggests that there may be better coordinates in which to express the H i . For instance if one takes the coordinates w = z -1 in the left sphere and u = 1/(2z + 1) in the middle sphere, then we get

F 3 • F 1 : w → u = 1/w 2 and F 2 : u → w = - 1 u 2 • u -1/3 1 -u/3 .
These are two real Blaschke fractions. So they commute with z → z and z → 1/z and preserve both the real axis and the unit circle. Their compositions H 1 and H 2 have relatively simple expressions:

H 2 = F 3 • F 1 • F 2 reads u → u 4 1 -u/3 u -1/3 2 22
In this respect they are rigid in the following sense: any rational map topologically equivalent to one of them as a ramified cover, by a pair of orientation preserving homeomorphisms, must me equivalent to it by a pair of homographies.

-1/2 -1/3 0 1 z → z 2 /2 -z z → 4z 3 3z+1 ∞ 2 1 -1/3 -1/2 ∞ 1
Figure 23: Julia sets of the maps H i (third iterate of the presumed limit on the tree of spheres). The three basins of attraction are drawn in white and shades of gray. We have put a mark " * " in each immediate basin in the first row. Compare with Figure 21. The column on the left represents H 1 (left sphere). The column on the right represents H 2 and H 3 (middle and right spheres). The first row is the sphere view, the second row is the flat view, and the last is the flat view followed by an inversion putting 0 at infinity. Putting the third fixed critical point at infinity would give an image identical to the second row but with a permutation of light and dark gray.

Figure 24: The three spheres at the bottom of Figure 21, with tubes between them symbolized by blue lines.

and

H 1 = F 2 • F 3 • F 1 reads w → -w 4 1 -w 2 /3 w 2 -1/3 .
Another presentation would take s = 1-2/z in the left sphere and t = 1+1/z in the middle sphere, and then

F 3 • F 1 : s → t = s + s -1 2 and F 2 : t → s = 3t -t 3 2 .
These are odd real maps, so they commute with z → z and z → -z and preserve the real and the imaginary axis. The composition

H 1 = F 2 •F 3 •F 1 reads s → -1 8 s 3 -3s - 3 s + 1 s 3 .
Let us show again the bottom row of Figure 21, and add blue lines symbolizing the tunnels between the three spheres (one can cut little openings in each sphere and add tunnels to glue them together, and recover the inital Riemann sphere S R ): this gives Figure 24. Comparing Figures 24 and23, the reader will have noticed that one of the basins of the limit maps H i is replaced by some sort of mess, that gets denser and denser as R → 1. In any of the three normalizations, the Hausdorff limit of the two Julia sets seems to contain the whole basin that is white in Figure 23.

Question 3. Prove that, in these three normalizations, the Hausdorff limit of the Julia set is indeed the union of the Julia set of H i and of the basin of attraction that is painted white in Figure 23.

In the next section we will focus on what happens in the tunnels between the three spheres. This may help to prove the assertion above.

Figure 25: A zoom on the messy part. This is a traversal of the tube starting from the middle sphere and ending at the left sphere. In the center of the last picture, we can see two red dots, which are the two marked points in the left sphere. On one picture, we also drew loops linking chains of light and dark gray components, which are approximately iterated square roots of the pictures in the middle sphere for bigger values of R. 2.3. Zooming on a tunnel. Up to now we observed the limits of the Julia sets in the three normalizations associated to the way the marked set degenerates. However, we could look at many other normalizations. For instance, we may focus on the tube linking the left sphere and the middle one (see Figure 16). Call it the a-tube, and call b-tube the one between the middle and right sphere. Since the left sphere is mapped to the right one and the middle to the left, the image of the a-tube has to span over the middle sphere. Recall that the map F R is 2 : 1 on the a-tube. When Figure 27: Continutation of Figure 26. The spheres separating the left and middle sphere map 2 : 1 to each other by F R or F R 3 •F R , and repeating this, eventually to the middle sphere. At the limit these separating spheres could be thought of a countable collection filling the gaps in the complement of a Cantor set. the a-tube is long (i.e. R is close to 1), this map is close to the squaring map z → z 2 in appropriate coordinates. This is why we see these square rooted copies of the central sphere image: see Figure 25. Moreover, this get replicated by futher preimages and explains the other rings visible in Figure 25, which are more or less copies of each other by more and more square roots. Let us explain this (see also Figures 26 and27). For a given R close to 1, the b-tube in S R 1/3 must contain a nearly 1 : 1 copy of the a-tube for R. The square root copy of the central sphere for R found in the a-tube for R 1/3 is therefore also present in the b-tube for R 1/3 2 . Now the image of the a-tube for R 1/3 not only spans over the middle sphere but also over the a-tube and b-tube for R: in fact the middle sphere represents a smaller and smaller part as R -→ 1. It follows that for each R, two fourth root copies of the central sphere must be found on the a-tube and two on the b-tube: these are roots of the central sphere for R 3 k with different values of k; however all these powers of R are close to 1 since R is. This goes on as on Figure 27 and accounts for all these rings we saw on the zoom.

On Figure 28 we show another conformal representation of the Riemann surfaces S R . These are sticks consisting of a cylinder capped with two half spheres. The length of the stick increases with n and is chosen so that we have limit pictures at the left, at the right and in the center, that are the limits of the picture in the three main normalizations as on Figure 21. The chosen normalization here puts the points on the equator with angle 1/3, 1/9, 0 respectively to the following points on the stick: the extremal left point, the central point facing the viewer, and the extremal right point. As n increases, you will note that each end has an apparent rotation, while the central part has a non-rotating limit. Note that we also get (rotating) Figure 28: Conformally correct pictures of S R mapped to tubes, with R = R n = 10 4/3 n and n =5, 6, 7, 8, 9, 10, 13 and 16. The last two tubes were cut so as to fit in the frame. See the text for a more detailed description. limits at 1/4 and 3/4 of the stick, and in fact at each p/2 q : these are the aforementioned rings of 2 q -fold preimages of the central sphere. This stick representation makes the rings and their relationships more visible. As R -→ 1, the rings get further and further appart from each other.

n R n v n v n /v n-1 u n /v n 2
If the tunnels on Figure 28 have a length that grows at a linear pace, which seems to be the case, then the rings also separate at a linear pace. The closer the rings, the slower the pace. When the tube is conformally mapped to a geometric sphere, say according to the central sphere normalization, this linear growth becomes an exponential shrinking: each ring in the white basin has a size that shrinks to 0 exponentially, with rings close to the boundary having a slower rate of convergence. More precisely for R n = e const/3 n , the ring at p/2 q should become visible when n ≥ const q and its diameter should shrink like e -const np/2 q where const designates different constants. For each q take the first n when the p/2 q are visible: this makes rings of diameter e -const pq/2 q for p from 1 to q. In particular for all ε > 0, when q gets big, there should be a lot of rings whose diameter is > ε and with p odd, so that these rings are thin. This explains the bigger and bigger density of the distorted Julia sets K 1 (R) and K 2 (R) in the white basin as n grows. To finish, we would like to show a computer experiment measuring the rate at which the marked points group together, which is related to the rate of growth of the tubes, and the rate of shrinking of the geodesics homotopic to the canonical obstruction. Let us choose our central sphere normalization, where the points labeled c 0 , e 0 , x 0 are mapped respectively to 0, 1, ∞. Let R n = 10 4/3 n and v n be difference of complex coordinates from y 0 to c 1 (in the chosen normalization) and u n from y 1 to c 2 . 23 On Table 1 we list, as a function of n with R = R n , the value of v n , of the rate v n /v n-1 and of the ratios u n /v n . The ratio u n /v n seems to converge: the two bunches of points converge at the same rate, which is coherent with the observation that the middle sphere is near the middle of the tubes on Figure 28 and coherent too with the fact that the tunnel in S R between the middle and right sphere is a 1 : 1 preimage by F R of the other tunnel for S R 3 (growing linearly). The latter remark suggests to look at the limit of the ratio u n /v n-1 and see if it is close to have absolute value one: indeed, according to Table 2, it seems to converge to -8/9. The rate v n /v n-1 seems to tend to a complex number ρ ≈ 0.1602499522 + i0.277561059.

Putting the square of its norm in Simon Plouffe's inverter24 suggests it might be ρ ? = 2e iπ/3 /3 5/3 .

It shall not come as a surprise that the limit rate of convergence be an algebraic number: it is probably one of the eigenvalues of the fixed point of a holomorphic map with an algebraic expression. Let us explain this in more details: McMullen proved that the graph of Thurston's pull-back map projects to a subset of the graph of an algebraic correspondence on the moduli space, with algebraic coefficients, see [Koc08]. Selinger proved that Thurston's pull-back map extends continuously to the Weil-Petersson completion of the Teichmüller space, see [Sel]. The latter projects to a natural compactification of moduli space: the Deligne-Mumford compactification, which is a complex smooth algebraic variety. The correspondence extends to this compactification (though it may have an indeterminacy locus in some parts); it has a fixed point corresponding to the tree of three spheres; it has a branch defined near this fixed point, that is possibly ramified, in which case: a blow up is necessary, the fixed point on the exceptional divisor must be identified and the map must be studied near this new fixed point.

Then we expect the map to be regular near the fixed point and one of its eigenvalues to be the limit rate of convergence.

Question 4. Carry out these computations, compute this eigenvalue, prove it is equal to the limit rate of convergence of the points, and compare its value to the one conjectured above.

Even though from Table 1, R n -1 and v n seem to decrease to 0 at the same rate, we expect one to tend slightly faster: R n -1 ∼ ln(10)/3 n whereas v n ∼ constant ρ n for the value of ρ given above, and |ρ| = 0.320 . . . = 1/3 = 0.333 . . .

Acknowledgments

The author would like to thank the referee for suggesting many improvements, both on the exposition of the mathematics and on the English language. After I posted on arXiv the preprint version of the present article, Nikita Selinger pointed out to me that his thesis [Sel] includes theorems that prove the statements in Questions 1 and 2 and Wolf Jung told me how he can compute the eigenvalues in Question 4.

2

  It is the group of homographies a.k.a. Möbius transformations or projective self maps, i.e. those maps of the form z → (az + b)/(cz + d) that are not constant. The group is also sharply 3-transitive: given a triple of pairwise distinct points a 1 , a 2 , a 3 in C an another one b 1 , b 2 , b 3 there is a unique Möbius map sending each a i to b i .

Figure 4 :

 4 Figure 4: The Riemann surface S R conformally mapped to the Euclidean sphere, painted with the drawings of Figure 3. The method for producing such a picture is interesting and explained in another article in the present volume: it does not work by computing the conformal map, but instead by pulling-back Julia sets by a series of rational maps. It has connections with Thurston's algorithm.

Figure 5 :Figure 6 :

 56 Figure 5: Hubbard trees H of P 1 and P 2 . The number over the arrow indicates the local degree at the starting point when it is not one. The dotted line indicates how some particular edge of the first preimage of H branches on H.2

Figure 7 :Figure 8 :

 78 Figure 7: The Julia sets of P 1 and P 2 .

Figure 9 :

 9 Figure9: Preimage by the formal mating of the multicurve {a, b} and of the Hubbard trees. The external ray of angle 0 is also indicated. Big dots represent the marked points, i.e. the post-critical set of the formal mating f . Here, the preimage of the big dots by f is the union of the big and the small dots.

Figure 10 :

 10 Figure 10: A configuration of points on the Riemann sphere C, together with a multicurve, supposedly homotopic to the short geodesics. Below it, a representation of what the hyperbolic metric should look like; the 8 cusps correspond to the 8 marked points, and they are infinitely long.

Figure 13 :

 13 Figure 13: Yet another conformal model.

Figure 14 :

 14 Figure 14: Illustration for Lemma 1.

Figure 15 :

 15 Figure 15: Conventions: In both pictures, the gray line is the external ray of argument 0 and the black dots are the marked points. On the right, the multicurve {a, b} is drawn with a in white and b in black. On the left we drew the (accurate) preimage of the right by F R . The white dots indicate those preimages of the black dots that are not themselves marked. Here, R ≈ 1.41, R 3 ≈ 2.78.

Figure 17 :

 17 Figure 17: Sequence, centered on the middle sphere A 2 . See the text for a description.

Figure 18 :

 18 Figure 18: Middle sphere centered flat view of S R for R = R 10 .

Figure 19 :

 19 Figure 19: Middle sphere centered flat view of S R for R = R 15 .

Figure 20 :

 20 Figure 20: Middle sphere centered flat view of S R for R = R 20 .

Figure 21 :

 21 Figure 21: Each row shows the same sphere S R for R = respectively R 2 , R 4 , R 8 and R 16 , with R n = 10 4/3 n . Each column has a different normalization: from left to right, (c 1 , e, y 0 ), (c 0 , e, x 0 ) and (c 2 , e, y 1 ) are mapped to (0, 1, ∞).

Figure 26 :

 26 Figure26: Pulling back the (partially) pinched sphere tree by the maps F R defines more and more pinchings. The tube between the left and the middle initial spheres gets more and more spheres, some of which separate the two inital spheres.

  88888889199 -i1.46e -9 25 -0.88888888888985 -i1.158e -11 30 -0.8888888888888565 -i2.24e -14 Table2. Computer experiment. Numbers rounded by truncation.

Figure 29 :

 29 Figure 29: Reminder of Figure 16.

23

  To be very precise, if φ 1 [R], φ 2 [R] and φ 3 [R] from S to C denote the three normalization, with φ 1 for the left normalization, φ 2 for the middle one and φ 3 for the right one, then vn = φ 2 [Rn](c 1 ) -φ 2 [Rn](y 0 ) and un = φ 2 [Rn](c 2 ) -φ 2 [Rn](y 1 ).

Table 1 .

 1 Computer experiment. Numbers rounded by truncation.

		2.78	1.15 +i0.35	0.0943 +i0.0286	-2.19 +i0.48
	3	1.40	0.081 +i0.383	0.158 +i0.283	-1.49 +i1.33
	4	1.120	-0.113 +i0.064	0.099 +i0.316	-1.15 +i2.03
	5	1.038	-0.0324 -i0.0230	0.129 +i0.276	-1.202 +i2.390
	10 1+1.55e -4	-1.16e -4 +i0.55e -4	0.16016 +i0.27763	-1.38658 +i2.40163
	15 1+6.41e -7	-0.36e -7 +i4.35e -7	0.1602506 +i0.2775626	-1.386716 +i2.401863
	20 1+2.64e -9	1.21e -9 +i0.84e -9	0.160249937 +i0.277561021	-1.38672276 +i2.40187419
	25 1+1.08e -11	4.51e -12 -i2.12e -12	0.1602499527 +i0.2775610603	-1.386722542 +i2.401873901
	30 1 + 4.47e -14	0.14e -15 -i1.68e -14	0.1602499522 +i0.2775610592	-1.386722548 +i2.401873910

Available on the Internet at http://pi.lacim.uqam.ca/ as of January 2012.

Centre National de la Recherche Scientifique, Institut de Mathématiques de Toulouse, Université Paul Sabatier 118 Route de Narbonne 31062 Toulouse Cedex 9, France E-mail address: arnaud.cheritat@math.univ-toulouse.fr

This research was funded by the grant ANR-08-JCJC-0002 of the Agence Nationale de la Recherche.


Jean-Claude Weill
email: weill@necker.fr

Sandra Weller

Claude-Agnès Reynaud
email: reynaud@necker.fr

A bird's eye view on human B cells

We show in this review that there is a continuum between the chicken B cell system classified as the first GALT model described and the human B cell system. We propose that humans have conserved for one B-cell subpopulation, the marginal zone B-cell subset in charge of T-independent responses, the strategies of diversification used by GALT species to generate their pre-immune repertoire.

When the chicken model of B cell development was described at the molecular level, it provided a different version of the GOD's (generation of diversity) solution, introducing gut-associated lymphoid tissues (GALT) as the site where the all pre-immune B cell repertoire is produced by post-rearrangement diversification processes [START_REF] Weill | The chicken B cell compartment[END_REF] .

At that time the GOD's enigma seemed more or less solved by the mouse model which had showed that Ig diversity was generated by the « roulette of rearrangement »: a few hundred V genes recombining to a small number of D and/or J elements in a random fashion, allowing the generation throughout life in the bone marrow of an infinite number of B cells, each of them carrying theoretically a different Ig receptor [START_REF] Tonegawa | Somatic generation of antibody diversity[END_REF] .

In the chicken several steps appeared strikingly different 3 :

1) Rearrangement is used to produce a quasi-monoclonal population of B cells, each of them having rearranged a unique functional V H -V L pair.

2) Ig gene rearrangement occurs during a short period of time during embryonic development and targets a few million progenitors. A few hundred thousands B cell progenitors having performed a productive H-L rearrangement localize in the epithelium of the bursa of Fabricius and proliferate within bursal follicles (10,000 follicles with 1-10 founder cells). A few days before hatching and several weeks after, bursal B cells start to migrate to the periphery, generating the B cell compartment of the animal.

3) Diversity is generated on this VH-VL pair by ongoing gene conversion involving upstream pseudogenes, starting during embryonic development and thereafter during the first months of life in the bursa, a primary lymphoid organ that involutes and disappears around 6 months after hatching.

GALT models of B cell development were further discovered in mammalian species with the molecular description of the sheep and rabbit B cell immune system (table 1). The search for a « mammalian bursa» had been the goal of the pioneers who described the chicken immune system. Their first choice was the rabbit (rightfully, see below), having shown that removal at birth of appendix and other GALT (Sacculus rotundus and Peyer's patches) reduced considerably the B cell response in this species [START_REF] Cooper | Production of an antibody deficiency syndrome in rabbits by neonatal removal of organized intestinal lymphoid tissues[END_REF] . However at the same time, it was shown that, in the mouse and human models, transfer of bone marrow precursors could restore the B cell lineage of the receiver [START_REF] Claman | Thymus-marrow cell combinations. Synergism in antibody production[END_REF][START_REF] Mitchell | Cell to cell interaction in the immune response. II. The source of hemolysin-forming cells in irradiated mice given bone marrow and thymus or thoracic duct lymphocytes[END_REF] . The description of an ontogenetic process leading to the generation of B cells in the mouse bone marrow consolidated further the model and the idea of a mammalian bursa-equivalent was abandoned [START_REF] Osmond | Differentiation of lymphocytes in mouse bone marrow. II. Kinetics of maturation and renewal of antiglobulin-binding cells studied by double-labelling[END_REF] .

Bursa-equivalent primary organs in mammals

In readdressing the question of the existence of a mammalian bursa, we were motivated by the work of Reynolds and Morris who showed that Peyer's patches were of two sorts in sheep: the ileal Peyer 's patches, which morphologically and functionally behaved like a primary lymphoid organ, and the jejunal Peyer's patches, which on the contrary possessed all the properties of secondary lymphoid organs [START_REF] Reynolds | The evolution and involution of Peyer's patches in fetal and postnatal sheep[END_REF] .

The study of the light chain locus during development indeed showed that the sheep used its ileal Peyer's patches to generate its B cell pre-immune repertoire with similarities and differences with the chicken model [START_REF] Reynaud | Somatic generation of diversity in a mammalian primary lymphoid organ: the sheep ileal Peyer's patches[END_REF][START_REF] Reynaud | Hypermutation generating the sheep immunoglobulin repertoire is an antigen-independent process[END_REF] .

1) Rearrangement occurs in sheep during a short window of development during fetal life, mostly in the fetal liver and in the spleen. The light chain lambda locus (the major light chain isotype) contains around 100 genes among which approximately half are functional, but 2 Vλ genes contribute 50% of the rearrangements. A few million cells having performed a productive rearrangement reach the ileal epithelium during fetal life and induce the formation of 100,000 follicles in which they proliferate for several months after birth.

2) During proliferation in ileal Peyer's patch follicles, all B cells diversify their Ig receptor by somatic hypermutation, the very process used by most species during affinity maturation for the response to T-dependent (TD) antigens.

3) Similarly to what occurs in the chicken, Ig diversification starts in the antigen-free environment of the embryo and goes on for several months after birth as an antigen-and T-independent process. Diversified B cells migrate to the periphery and install the B cell compartment for the whole life of the animal. Ileal Peyer's patches involute and completely disappear after one year while the jejunal Peyer's patches remain throughout life.

The rabbit model was shown by the group of K. Knight to be another variation on the GALT theme, confirming that this mode of B cell development had been conserved in the mammalian kingdom. At the genomic level, the heavy chain locus was very « mouselike », as it includes a large family of functional V H genes, but all V H genes, as in the chicken, belong to a single V H family (V H 3). Instead of rearranging at the same frequency anyone of these genes, the most proximal V H 1 gene is rearranged in 80% of its B cells. This V H gene is thereafter diversified by a « chicken-like » ongoing gene conversion process involving upstream functional or non-functional V H genes while the CDR3 region is diversified by a« sheep-like » hypermutation mechanism [START_REF] Becker | Somatic diversification of immunoglobulin heavy chain VDJ genes: evidence for somatic gene conversion in rabbits[END_REF] .

As in the chicken, rearrangement occurs essentially during early development while diversification by gene conversion occurs mainly in appendix follicles [START_REF] Weinstein | Rabbit IgH sequences in appendix germinal centers: VH diversification by gene conversion-like and hypermutation mechanisms[END_REF] . Contrarily to the chicken and to the sheep, B cell proliferation and diversification only start after birth and the GALT remain once the B cell compartment is installed after 2-3 months, shifting their activities from that of a primary organ of diversification to that of a secondary lymphoid organ with clear germinal centers involved in T-dependent immune responses.

As for the lymphoid organs in which these events take place, we have proposed, on the basis of experiments performed on germ free sheep and on ileal loops realized on fetal sheep, that gut bacterial constituents essentially provide the B cells present in follicles with a proliferative signal rather than with an antigenic stimuli [START_REF] Reynaud | Hypermutation generating the sheep immunoglobulin repertoire is an antigen-independent process[END_REF] . A similar proposition was recently formulated for the rabbit model in which it was shown that it is the presence of these bacterial constituents in the gut that permits the initiation of this proliferative phase in the GALT, most probably again due to their mitotic enhancement capacity [START_REF] Sehgal | Distinct clonal Ig diversification patterns in young appendix compared to antigen-specific splenic clones[END_REF][START_REF] Lanning | Immunoglobulin genes and generation of antibody repertoires in higher vertebrates: a key role for GALT[END_REF] . It is therefore tempting to associate at the evolutionary level these post-rearrangement diversification processes requiring many cycles of proliferation and selection with an organ in which this proliferation can be naturally maintained. Others GALT models have since been described, in cattle, pigs, dogs, and it is highly probable that a large number of mammalian species uses this strategy to generate their B cell repertoire [START_REF] Lanning | Immunoglobulin genes and generation of antibody repertoires in higher vertebrates: a key role for GALT[END_REF] . Another surprising property of the GALT species is that once their B cell compartment is installed at the periphery at the end of development, there is no more de novo production of new B cells, implying that this peripheral compartment is sufficient to protect the animal for its life time. How is this peripheral compartment maintained is not known at the moment. It must be composed of memory and naive B cells but whether the latter ones can self-renew or are maintained by a pool of progenitors with stem cell properties is an open question.

Human and mice would have thus evolved this luxury to continually produce B cells with the possibility that any new antigen being encountered (Mel Cohn's moon dust [START_REF] Cohn | The protecton: the unit of humoral immunity selected by evolution[END_REF] ) may be taken care of by any one of these newly produced VDJ recombinations.

At the evolutionary level, whether diversity is stored in V pseudogenes or functional genes and whether the mecanism of diversification of the pre-immune repertoire is ongoing rearrangement or ongoing gene conversion and hypermutation does not make much difference. In such sense the chicken and the rabbit pseudoV genes are indeed functional donors of diversity for an acceptor V gene. Nevertheless there is a great difference between an ongoing rearrangement process involving mainly a variability at the CDR3 level and the variability generated on the three CDRs by gene conversion or hypermutation. Surprisingly enough, hypermutation appears to be a fully efficient mechanism to generate a pre-immune repertoire. This comes from two different processes:

first, a co-evolution of the mutation mechanism and of the target V gene sequence, by which mutation hotspots leading to replacement mutations are specifically clustered in CDRs, thus allowing a strong concentration and a high R/S ratio of CDR mutations [START_REF] Reynaud | Hypermutation generating the sheep immunoglobulin repertoire is an antigen-independent process[END_REF][START_REF] Wagner | Codon bias targets mutation[END_REF] .

Second, a constant selection for cells with a functional BCR at the surface that can provide the appropriate tonic signal, ensuring the counter-selection of deleterious mutations in framework regions (and a low R/S ratio of framework mutations). Both of these features result in a mutation selection pattern very similar to the one observed during antigenselected immune responses [START_REF] Reynaud | Hypermutation generating the sheep immunoglobulin repertoire is an antigen-independent process[END_REF] .

A T-independent diversification pathway in humans

In mice and humans, germline antibodies display a rather low affinity for their specific antigens and show some polyspecificity. It has been proposed that this loose specificity could be explained by the flexibility of their combining site. According to this model, affinity maturation and selection, after encounter with its specific protein antigen, should allow for the fixation of this antibody-combining site to its ligand, thus suppressing its flexibility and polyspecificity [START_REF] Wedemayer | Structural insights into the evolution of an antibody combining site[END_REF] . The most obvious explanation for this strategy would be to preserve the pre-immune repertoire from the holes of negative selection. Included in this scheme is the fact that immune responses to non-protein antigens will only raise germ-line antibodies responses.

In the GALT models all types of affinities will emerge from the primary repertoire and in fact most de novo produced B cells will die in the GALT and only around 5% will migrate to the periphery. Immune responses to proteins antigens will still be improved by an additional diversification step in germinal centers by hypermutation and/or gene conversion. Immune responses to T-independent antigens on the other hand will theoretically generate high affinity responses with mutated antibodies. In human the Tindependent responses represent a paradox since they are by definition against nonprotein antigens and therefore germinal center-independent. Strikingly, antibodies raised by T-independent vaccines such as bacterial capsular polysaccharides are usually mutated and, to explain this paradox, it is proposed the rather unlikely scenario that these bacterial polysaccharides in each vaccine preparation may be coated with bacterial proteins that would drive them into a T-dependent hypermutation process in germinal centers [START_REF] Lucas | Polysaccharide vaccines as probes of antibody repertoires in man[END_REF] .

Another explanation could be that some specialized B cells may in human mutate their Ig receptor after stimulation by T-independent antigens. A third explanation which we favor and which does not exclude the latter one is that the B cells in charge of T-I responses in humans already mutate their Ig receptor during their postnatal development, as it occurs in rabbit, before their immune response to external antigens. In this last scheme, the TIantigens would select for B cell clones with the best affinity in the splenic marginal zone and induce a second round of hypermutation after stimulation.

We have come to this proposition because of the following observations [START_REF] Weller | CD40-CD40L independent Ig gene hypermutation suggests a second B cell diversification pathway in humans[END_REF][START_REF] Weller | Human blood IgM "memory" B cells are circulating splenic marginal zone B cells harboring a pre-diversified immunoglobulin repertoire[END_REF] there is an expansion of the circulating IgM-only CD27 + cells, which can amount to 10% of their total B cells whereas such cells are extremely minor in normal individuals.

2) Phenotypic analysis and gene expression profiling indicate that blood IgM + IgD + CD27 + cells are circulating marginal zone B cells.

3) After vaccination with a non-conjugated pneumococcal vaccine, blood and splenic IgM + IgD + CD27 + B cells are specifically mobilized in T-independent responses. This response is taken care of by mutated antibodies. Upon expansion of these cells in the splenic marginal zone during the week following vaccination, the hypermutation process seems to be reactivated.

4) Most strikingly, the blood IgM + IgD + CD27 + B cell population was well expanded and mutated in children below two years, an age at which toddlers do not respond to Tindependent antigens, thus strongly suggesting that this diversification results at this stage from a developmental program and not from an immune response.

Our conclusion from these preliminary investigations was, as proposed in our first report [START_REF] Weller | CD40-CD40L independent Ig gene hypermutation suggests a second B cell diversification pathway in humans[END_REF] , that this so-called memory IgM + IgD + CD27 + B cell population, which represent 10-30% of B cells in blood and spleen, belonged to a different pathway of differentiation and was linked to the response against TI-antigens. A similar conclusion was reached by

Carsetti and her colleagues in a recent report, based on their study on the responding capacity of normal individuals and asplenic or splenectomized patients to T-independent pneumococcal antigens [START_REF] Kruetzmann | Human IgM memory B cells controlling Streptococcus pneumoniae infections are generated in the spleen[END_REF] .

What can be concluded from this first series of results? We would like to propose that during early ontogeny blood IgM + IgD + CD27 + B cells develop and mutate their Ig receptor before 2 years of age, at which stage they become functionally competent against TI antigens, this competence being probably caused by the differentiation of the appropriate accessory cells in the splenic marginal zone microenvironment (fig. 1).

But how is this population produced and diversified throughout life? It may either be maintained in the periphery after its initial production and thus constitute like mouse B1 cells a separate self-renewing B cell lineage with a large repertoire of specificities and affinities [START_REF] Carvalho | Arrested B lymphopoiesis and persistence of activated B cells in adult interleukin 7(-/-) mice[END_REF][START_REF] Hao | Homeostasis of peripheral B cells in the absence of B cell influx from the bone marrow[END_REF] . Alternatively and more likely, it may be continuously produced, sharing up to a certain stage a common pathway of differentiation with conventional B cells. At the transitional bifurcation step, it could be driven by self or bacterial antigens and by specific interactions with neighboring cells to its final marginal zone B cell phenotype [START_REF] Saito | Notch2 is preferentially expressed in mature B cells and indispensable for marginal zone B lineage development[END_REF][25][26] . As shown for GALT species, these ligands, whether internal or provided by commensal bacteria, would only induce the proliferation and Ig diversification of these B cells and would not trigger any effector function, i.e. the production of antibodies. This prediversification step, rather than in B cell follicles along the gut, could take place in the splenic marginal zone, or in the marginal zone-like regions of tonsils, lymph nodes, and Peyer's patches. Asplenic patients indeed harbor IgM + IgD + CD27 + B cells with normally mutated genes but their frequency appears somewhat reduced in the adults, suggesting that such marginal zone-like sites, if they can compensate for the diversification function of the spleen, are not able to support the production and/or survival of these cells to the same extent.

There are some apparent differences among GALT species that may confuse the picture. In chickens and sheep, diversification of the pre-immune repertoire starts in the sterile environment of the embryo and then goes on for several months under the proliferative stimulation of the gut bacterial constituents. In rabbits, diversification of the B cell repertoire requires the gut constituents since it starts only after birth, but here again their role is essentially to drive B cell proliferation. In humans, as it occurs in rabbits, there is little expansion and diversification of the IgM + IgD + CD27 + B cells at birth, and by 2 years of age, the frequency of mutations carried by these cells is more or less equivalent to the one found in adults, suggesting that each new wave of cells undergoes the same diversification process. What drives pre-diversification of marginal zone B cells in humans and how does it occur is one of the next burning question.

outside the strict engagement of their B-cell receptor. Strikingly, this has been shown recently to occur in the mouse under specific experimental conditions: one of these settings corresponds to a situation of strong germline repertoire restriction, leading to preimmune diversification by hypermutation 27 ; the second one to a constitutive B cell signaling, turning on hypermutation in the context of a microbial-driven, non BCRmediated, stimulation 28 .

In conclusion, this model of differentiation for human marginal zone B cells that includes a pre-diversification step provides an explanation for the surprising fact that Tindependent responses in humans are taken care by mutated antibodies. It also shows that humans may have conserved for one arm of their B cell system, in its normal physiological function, the strategies used by GALT species to diversify their pre-immune repertoire.

Figure legend Figure 1 .

 legend1 Figure legend

: 1 )

 1 Hyper-IgM patients with either CD40L or CD40 deficiency, which do not have germinal centers, possess a population of circulating IgM + IgD + CD27 + B cells with a mutated Ig receptor, in the complete absence of CD27 + IgM-only and switched memory B cells. In both type of patients the frequency of Ig mutations observed in these cells is very similar to the one observed in controls although the number of IgM + IgD + CD27 + blood cells is in general lower in the patients as compared to normal children. The absence of CD40 or CD40L may alter B cell development by hampering the network of cytokines interactions involving dendritic cells, activated T cells and natural killer cells. Nevertheless the fact that these patients never display any CD27 + IgM-only B cells indicate that the IgM + IgD + CD27

+

B cells present are not produced in some cryptic germinal center structures, but rather belong to a separate population. Conversely, in AID-deficient patients in which germinal centers are formed but B cells are unable to activate hypermutation and isotype switching,

Chicken