Amaury Bittmann

DOUBLY-RESONANT SADDLE-NODES IN C 3 , 0 AND THE FIXED SINGULARITY AT INFINITY IN THE PAINLEVÉ EQUATIONS. PART III: LOCAL ANALYTIC CLASSIFICATION

Keywords: Painlevé equations, singular vector field, irregular singularity, resonant singularity, analytic classification, Stokes diffeomorphisms. 1

.

Introduction

As in [START_REF] Bittmann | Doubly-resonant saddle-nodes in C 3 and the fixed singularity at infinity in the painlevé equations: Formal classification[END_REF][START_REF] Bittmann | Doubly-resonant saddle-nodes in (C 3 , 0) and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization[END_REF], we consider (germs of) singular vector fields Y in C 3 which can be written in appropriate coordinates (x, y) := (x, y 1 , y 2 ) as Y = x 2 ∂ ∂x +λy 1 + F 1 (x, y) ∂ ∂y 1 + λy 2 + F 2 (x, y) ∂ ∂y 2 , (1.1) where λ ∈ C * and F 1 , F 2 are germs of holomorphic function in C 3 , 0 of homogeneous valuation (order) at least two. They represent irregular two-dimensional differential systems having two opposite nonzero eigenvalues and a vanishing third eigenvalue. These we call doubly-resonant vector fields of saddlenode type (or simply doubly-resonant saddle-nodes). For a historical context, a presentation of the main motivations (the study of the irregular singularity at infinity in Painlevé equations (P j ) j=I...V ), and a review of some results linked with this study, we refer to [START_REF] Bittmann | Doubly-resonant saddle-nodes in C 3 and the fixed singularity at infinity in the painlevé equations: Formal classification[END_REF].

Several authors studied the problem of convergence of formal transformations putting vector fields as in (1.1) into "normal forms". Shimomura, improving on a result of Iwano [START_REF] Iwano | On a general solution of a nonlinear 2-system of the form x 2 dw/dx = Λw + xh(x, w) with a constant diagonal matrix Λ of signature (1, 1)[END_REF], shows in [START_REF] Shimomura | Analytic integration of some nonlinear ordinary differential equations and the fifth Painlevé equation in the neighbourhood of an irregular singular point[END_REF] that analytic doubly-resonant saddle-nodes satisfying more restrictive conditions are conjugate (formally and over sectors) to vector fields of the form

x 2 ∂ ∂x + (-λ + a 1 x) y 1 ∂ ∂y 1 + (λ + a 2 x) y 2 ∂
∂y 2 via a diffeomorphism whose coefficients have asymptotic expansions as x → 0 in sectors of opening greater than π.

Stolovitch then generalized this result to any dimension in [START_REF] Stolovitch | Classification analytique de champs de vecteurs 1-résonnants de (C n , 0)[END_REF]. More precisely, Stolovitch's work offers an analytic classification of vector fields in C n+1 with an irregular singular point, without further hypothesis on eventual additional resonance relations between eigenvalues of the linear part. However, as Iwano and Shimomura did, he needed to impose other assumptions, among which the condition that the restriction of the vector field to the invariant hypersurface {x = 0} is a linear vector field. In [START_REF] Bonckaert | Gevrey normal forms of vector fields with one zero eigenvalue[END_REF], the authors obtain a Gevrey-1 summable "normal form", though not as simple as Stolovitch's one and not unique a priori, but for more general kind of vector field with one zero eigenvalue. However, the same assumption on hypersurface {x = 0} is required (the restriction is a linear vector field). Yet from [START_REF] Yoshida | 2-parameter family of solutions for Painlevé equations (I)-(V) at an irregular singular point[END_REF] (and later [START_REF] Bittmann | Doubly-resonant saddle-nodes in C 3 and the fixed singularity at infinity in the painlevé equations: Formal classification[END_REF]) stems the fact that this condition is not met in the case of Painlevé equations (P j ) j=I...V .

In comparison, we merely ask here that this restriction be orbitally linearizable (see Definition 1.6), i.e. the foliation induced by Y on {x = 0} (and not the vector field Y |{x=0} itself) be linearizable. The fact that this condition is fulfilled by the singularities of Painlevé equations formerly described is well-known. As discussed in Remark 1.16, this more general context also introduces new phenomena and technical difficulties as compared to prior classification results.

1.1. Scope of the paper.

The action of local analytic / formal diffeomorphisms Ψ fixing the origin on local holomorphic vector fields Y of type (1.1) by change of coordinates is given by

Ψ * Y := (DΨ.Y ) • Ψ -1 .
In [START_REF] Bittmann | Doubly-resonant saddle-nodes in C 3 and the fixed singularity at infinity in the painlevé equations: Formal classification[END_REF] we performed the formal classification of such vector fields by exhibiting an explicit universal family of vector fields for the action of formal changes of coordinates at 0 (called a family of normal forms). Such a result seems currently out of reach in the analytic category: it is unlikely that an explicit universal family for the action of local analytic changes of coordinates be described anytime soon. If we want to describe the space of equivalent classes (of germs of a doubly-resonant saddle-node under local analytic changes of coordinates) with same formal normal form, we therefore need to find a complete set of invariants which is of a different nature. We call moduli space this quotient space and give it a (non-trivial) presentation based on functional invariants à la Martinet-Ramis [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF][START_REF] Martinet | Classification analytique des équations différentielles non linéaires résonnantes du premier ordre[END_REF].

In this paper we will therefore present only the x-fibered local analytic classification for vector fields of the form (1.1), with some additional assumptions detailed further down (see Definitions 1.1, 1.2 and 1.6). Importantly, these hypothesis are met in the case of Painlevé equations mentioned above. The full analytic classification (under the action of all local diffeomorphisms, not necessarily x-fibered) will be done in a forthcoming work.

In [START_REF] Bittmann | Doubly-resonant saddle-nodes in (C 3 , 0) and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization[END_REF], we have proved the existence of analytic sectorial normalizing maps (over a pair of opposite "wide" sectors of opening greater than π whose union covers a full punctured neighborhood of {x = 0}). Then we attach to each vector field a complete set of invariants given as transition maps (over "narrow" sectors of opening less than π) between the sectorial normalizing maps. Although this viewpoint has become classical since the work of Martinet and Ramis, and has latter been generalized by Stolovitch as already mentioned, our approach has a more geometric flavor (for instance, we perform a precise study of the Stokes diffeomorphisms in the space of leaves).

As a by-product, we deduce that the normalizing sectorial diffeomorphisms of [START_REF] Bittmann | Doubly-resonant saddle-nodes in (C 3 , 0) and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization[END_REF] are Gevrey-1 asymptotic to the normalizing formal power series of [START_REF] Bittmann | Doubly-resonant saddle-nodes in C 3 and the fixed singularity at infinity in the painlevé equations: Formal classification[END_REF], retrospectively proving their 1summability. When the vector field additionally supports a symplectic transverse structure (which is again the case of Painlevé equations) we prove a theorem of analytic classification under the action of transversally symplectic diffeomorphisms.

Definitions and previous results.

To state our main results we need to introduce some notations and nomenclature.

• For n ∈ N >0 , we denote by (C n , 0) an (arbitrary small) open neighborhood of the origin in C n .

• We denote by C {x, y}, with y = (y 1 , y 2 ), the C-algebra of germs of holomorphic functions at the origin of C 3 , and by C {x, y} × the group of invertible elements for the multiplication (also called units), i.e. elements U such that U (0) = 0.

• χ C 3 , 0 is the Lie algebra of germs of singular holomorphic vector fields at the origin C 3 . Any vector field in χ C 3 , 0 can be written as where DΦ is the Jacobian matrix of Φ. • Diff fib C 3 , 0 is the subgroup of Diff C 3 , 0 of fibered diffeomorphisms preserving the xcoordinate, i.e. of the form (x, y) → (x, φ (x, y)). • We denote by Diff fib C 3 , 0, Id the subgroup of Diff fib C 3 , 0 formed by diffeomorphisms tangent to the identity. All these concepts have formal analogues, where we only suppose that the objects are defined with formal power series, not necessarily convergent near the origin.

Y = b (x, y 1 , y 2 ) ∂ ∂x + b 1 (x, y 1 , y 2 ) ∂ ∂y 1 + b 2 (x,
Definition 1.1. A diagonal doubly-resonant saddle-node is a vector field Y ∈ χ C 3 , 0 of the form Y = x 2 ∂ ∂x + -λy 1 + F 1 (x, y) ∂ ∂y 1 + λy 2 + F 2 (x, y) ∂ ∂y 2 , (1.3) with λ ∈ C * and F 1 , F 2 ∈ C {x,
y} of order at least two. We denote by SN diag the set of such vector fields.

Based on this expression, and considering the expansion

F j (x, y) = k=(k0,k1,k2) F j,k x k0 y k1
1 y k2 2 for j = 1, 2, we state:

Definition 1.2. The residue of Y ∈ SN diag as in (1.3) is the complex number res (Y ) := F 1,(1,1,0) + F 2,(1,0,1) .
We say that Y is non-degenerate (resp. strictly non-degenerate

) if res (Y ) / ∈ Q ≤0 (resp. ℜ (res (Y )) > 0).
Remark 1.3. It is obvious that there is an action of Diff fib C 3 , 0, Id on SN diag . The residue is an invariant of each orbit of SN fib under the action of Diff fib C 3 , 0, Id by conjugacy (it is actually invariant by formal conjugacies, see [START_REF] Bittmann | Doubly-resonant saddle-nodes in C 3 and the fixed singularity at infinity in the painlevé equations: Formal classification[END_REF]).

The main result of [START_REF] Bittmann | Doubly-resonant saddle-nodes in C 3 and the fixed singularity at infinity in the painlevé equations: Formal classification[END_REF] can now be stated as follows:

Theorem 1.4. [START_REF] Bittmann | Doubly-resonant saddle-nodes in C 3 and the fixed singularity at infinity in the painlevé equations: Formal classification[END_REF] Let Y ∈ SN diag be non-degenerate. Then there exists a unique formal fibered diffeomorphism Φ tangent to the identity such that:

Φ * (Y ) = x 2 ∂ ∂x + (-λ + a 1 x + c 1 (y 1 y 2 )) y 1 ∂ ∂y 1 + (λ + a 2 x + c 2 (y 1 y 2 )) y 2 ∂ ∂y 2 , (1.4)
where λ ∈ C * , c 1 , c 2 ∈ vC v are formal power series in v = y 1 y 2 without constant term and a 1 , a 2 ∈ C are such that a 1 + a 2 = res (Y ) ∈ C\Q ≤0 .

Definition 1.5. The vector field obtained in (1.4) is called the formal normal form of Y . The formal fibered diffeomorphism Φ is called the formal normalizing map of Y .

The above result is valid for formal objects, without considering problems of convergence. The main result in [START_REF] Bittmann | Doubly-resonant saddle-nodes in (C 3 , 0) and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization[END_REF] states that this formal normalizing map is analytic in sectorial domains, under some additional assumptions that we are now going to precise. Definition 1.6.

• We say that a germ of a vector field X in C 2 , 0 is orbitally linear if

X = U (y) λ 1 y 1 ∂ ∂y 1 + λ 2 y 2 ∂ ∂y 2 , for some U (y) ∈ C {y} × and (λ 1 , λ 2 ) ∈ C 2 .
• We say that a germ of vector field X in C 2 , 0 is analytically (resp. formally) orbitally linearizable if X is analytically (resp. formally) conjugate to an orbitally linear vector field.

• We say that a diagonal doubly-resonant saddle-node Y ∈ SN diag is div-integrable if Y |{x=0} ∈ χ C 2 , 0 is (analytically) orbitally linearizable.
Remark 1.7. Alternatively we could say that the foliation associated to Y |{x=0} is linearizable. Since Y |{x=0} is analytic at the origin of C 2 and has two opposite eigenvalues, it follows from a classical result of Brjuno (see [START_REF] Martinet | Normalisation des champs de vecteurs holomorphes (d'après A.-D. Brjuno)[END_REF]), that Y |{x=0} is analytically orbitally linearizable if and only if it is formally orbitally linearizable.

Definition 1.8. We denote by SN diag,0 the set of strictly non-degenerate diagonal doubly-resonant saddle-nodes which are div-integrable.

The main result of [START_REF] Bittmann | Doubly-resonant saddle-nodes in (C 3 , 0) and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization[END_REF] can now be stated (we refer to section 2. for precise definitions on weak 1-summability)).

Theorem 1.9. [START_REF] Bittmann | Doubly-resonant saddle-nodes in (C 3 , 0) and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization[END_REF]Let Y ∈ SN diag,0 and let Φ (given by Theorem 1.4) be the unique formal fibered diffeomorphism tangent to the identity such that

Φ * (Y ) = x 2 ∂ ∂x + (-λ + a 1 x + c 1 (y 1 y 2 )) y 1 ∂ ∂y 1 + (λ + a 2 x + c 2 (y 1 y 2 )) y 2 ∂ ∂y 2 =: Y norm ,
where λ = 0 and c 1 (v) , c 2 (v) ∈ vC v are formal power series without constant term. Then:

(1) the normal form Y norm is analytic (i.e. c 1 , c 2 ∈ C {v}), and it also is div-integrable, i.e. c 1 +c 2 = 0;

(2) the formal normalizing map Φ is weakly 1-summable in every direction θ = arg (±λ);

(3) there exist analytic sectorial fibered diffeomorphisms Φ + and Φ -, (asymptotically) tangent to the identity, defined in sectorial domains of the form S + × C 2 , 0 and S -× C 2 , 0 respectively, where

S + := x ∈ C | 0 < |x| < r and arg x iλ < π 2 + ǫ S -:= x ∈ C | 0 < |x| < r and arg -x iλ < π 2 + ǫ
(for any ǫ ∈ 0, π 2 and some r > 0 small enough), which admit Φ as weak Gevrey-1 asymptotic expansion in these respective domains, and which conjugate Y to Y norm . Moreover Φ + and Φ - are the unique such germs of analytic functions in sectorial domains (see Definition 2.2). Definition 1.10. We call Φ + and Φ -the sectorial normalizing maps of Y ∈ SN diag,0 .

They are the weak 1-sums of Φ along the respective directions arg (iλ) and arg (-iλ). Notice that Φ + and Φ -are germs of analytic sectorial fibered diffeomorphisms, i.e. they are of the form

Φ + : S + × C 2 , 0 -→ S + × C 2 , 0 (x, y) -→ (x, Φ +,1 (x, y) , Φ +,2 (x, y))
and

Φ -: S -× C 2 , 0 -→ S -× C 2 , 0 (x, y) -→ (x, Φ -,1 (x, y) , Φ -,2 (x, y))
(see section 2. for a precise definition of germ of analytic sectorial fibered diffeomorphism). The fact that they are also (asymptotically) tangent to the identity means that we have:

Φ ± (x, y) = Id + O (x, y) 2 .
Another result proved in [START_REF] Bittmann | Doubly-resonant saddle-nodes in (C 3 , 0) and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization[END_REF], is that the uniqueness of the sectorial normalizing maps holds in fact under weaker assumptions.

Proposition 1.11. Let ϕ + and ϕ -be two germs of sectorial fibered diffeomorphisms in S + × C 2 , 0 and S -× C 2 , 0 respectively, where S + and S -are as in Theorem 1.9, which are (asymptotically) tangent to the identity and such that

(ϕ ± ) * (Y ) = Y norm .
Then, they necessarily coincide with the weak 1-sums Φ + and Φ -defined above.

Main results.

The first main result of this paper is the following.

Theorem 1.12. Let Y ∈ SN diag,0 and let Φ (given by Theorem 1.4) be the unique formal fibered diffeomorphism tangent to the identity such that

Φ * (Y ) = x 2 ∂ ∂x + (-λ + a 1 x -c (y 1 y 2 )) y 1 ∂ ∂y 1 + (λ + a 2 x + c (y 1 y 2 )) y 2 ∂ ∂y 2 =: Y norm ,
where λ = 0 and c (v) ∈ vC {v}. Then Φ is 1-summable (with respect to x) in every direction θ = arg (±λ), and Φ + , Φ -in Theorem 1.9 are the 1-sums of Φ in directions arg (iλ) , arg (-iλ) respectively.

Since two analytically conjugate vector fields are also formally conjugate, we fix now a normal form According to Theorem 1.9, to any Y ∈ [Y norm ] we can associate two sectorial normalizing maps Φ + , Φ -, which can in fact extend analytically in domains S + × C 2 , 0 and S -× C 2 , 0 , where S ± is an asymptotic sector in the direction arg (±iλ) with opening 2π (see Definition 2.3):

Y norm = x 2 ∂ ∂x + (-λ + a 1 x -c (v)) y 1 ∂ ∂y 1 + (λ + a 2 x + c (v)) y 2 ∂ ∂y 2 , with λ ∈ C * , ℜ (a 1 + a 2 ) >
(S + , S -) ∈ AS arg(iλ),2π × AS arg(-iλ),2π .
Then, we consider two germs of sectorial fibered diffeomorphisms Φ λ , Φ -λ analytic in S λ , S -λ , with

S λ := S + ∩ S -∩ ℜ x λ > 0 ∈ AS arg(λ),π (1.5) S -λ := S + ∩ S -∩ ℜ x λ < 0 ∈ AS arg(-λ),π ,
defined by: Our second main result gives the moduli space for the analytic classification that we are looking for.

Φ λ := Φ + • Φ -1 -|S λ ×(C 2 ,0) ∈ Diff fib S arg(λ),ǫ ; Id , ∀ǫ ∈ [0, π[ Φ -λ := Φ -• Φ -1 + |S -λ ×(C 2 ,
Theorem 1.15. The map

[Y norm ] Diff fib C 3 , 0, Id -→ Λ λ (Y norm ) × Λ -λ (Y norm ) Y -→ (Φ λ , Φ -λ )
is well-defined and bijective.

In particular, the result states that Stokes diffeomorphisms only depend on the class of

Y ∈ [Y norm ]
in the quotient [Y norm ] Diff fib C 3 , 0, Id . We will give a description of this set of invariants in terms of power series in the space of leaves in section 4.

Remark 1.16. In [START_REF] Bittmann | Doubly-resonant saddle-nodes in (C 3 , 0) and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization[END_REF] we have proved a theorem of sectorial normalizing map analogous to the classical one due to Hukuhara-Kimura-Matuda for saddle-nodes in C 2 , 0 [HKM61], generalized later by Stolovitch in any dimension in [START_REF] Stolovitch | Classification analytique de champs de vecteurs 1-résonnants de (C n , 0)[END_REF]. Unlike the method based on a fixed point theorem used by these authors, we have used a more geometric approach (following the works of Teyssier [START_REF] Teyssier | Analytical classification of saddle-node vector fields[END_REF][START_REF] Teyssier | Équation homologique et cycles asymptotiques d'une singularité noeud-col[END_REF]) based on the resolution of an homological equation, by integrating a well chosen 1-form along asymptotic paths. This latter approach turned out to be more efficient to deal with the fact that Y |{x=0} is not necessarily linearizable. Indeed, if we try to adapt the proof of [START_REF] Stolovitch | Classification analytique de champs de vecteurs 1-résonnants de (C n , 0)[END_REF], one of the first new main difficulties is that in the irregular systems that needs to be solved by a fixed point method (for instance equation (2.7) in the cited paper), the non-linear terms would not be divisible by the independent variable (i.e. the time) in our situation. This would complicate the different estimates that are done later in the cited work. This was the first main new phenomena we have met.

In contrast to a result of [START_REF] Bittmann | Doubly-resonant saddle-nodes in (C 3 , 0) and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization[END_REF] which states that the only sectorial isotropy (tangent to the identity) of the normal form over wide sectors (of opening > π) is the identity, we will see here that the situation is rather different over sector with narrow opening. In order to prove both Theorems 1.12 and 1.15, we will show that the Stokes diffeomorphisms Φ λ and Φ -λ obtained from the germs of sectorial normalizing maps Φ + and Φ -,admit the identity as Gevrey-1 asymptotic expansion. In the cited reference we were only able to establish that fact with a weaker notion of Gevrey-1 expansion. The main difficulty is to prove that such a sectorial isotropy of Y norm over the "narrow" sectors described above is necessarily exponentially close to the identity (see Proposition 3.5). This will be done via a detailed analysis of these maps in the space of leaves (see Definition 4.4). In fact, this is the second main new difficulties we have met, which is due to the presence of the "resonant" term

c m (y 1 y 2 ) m log (x)
x in the exponential expression of the first integrals of the vector field (see (4.1)). In [START_REF] Stolovitch | Classification analytique de champs de vecteurs 1-résonnants de (C n , 0)[END_REF], similar computations are done in subsection 3.4.1. In this part of the paper, infinitely many irregular differential equations appear when identifying terms of same homogeneous degree. The fact that Y |{x=0} is linear implies that these differential equations are all linear and independent of each others (i.e. they are not mixed together). In our situation, this is not the case and then more complicated. Using a "non-abelian" version of the Ramis-Sibuya theorem due to Martinet and Ramis [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF], we prove both sectorial normalizing maps Φ + and Φ -admit the formal normalizing map Φ as Gevrey-1 asymptotic expansion in the corresponding sectorial domains. This establishes the Gevrey-1 summability of Φ.

1.4. The transversally Hamiltonian case.

In order to motivate the following definition, we refer to [START_REF] Bittmann | Doubly-resonant saddle-nodes in (C 3 , 0) and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization[END_REF] where the study of the Painlevé case is performed.

Definition 1.17. Consider the rational 1-form

ω := dy 1 ∧ dy 2 x .
We say that vector field Y is transversally Hamiltonian (with respect to ω and dx) if

L Y (dx) ∈ dx and L Y (ω) ∈ dx .
For any open sector S ⊂ C * , we say that a germ of sectorial fibered diffeomorphism Φ in S × C 2 , 0 is transversally symplectic (with respect to ω and dx) if

Φ * (ω) ∈ ω + dx
(here Φ * (ω) denotes the pull-back of ω by Φ).

We denote by Diff ω C 3 , 0; Id the group of transversally symplectic diffeomorphisms which are tangent to the identity.

Remark 1.18. A fibered sectorial diffeomorphism Φ is transversally symplectic if and only if det (DΦ) = 1.

Definition 1.19. A transversally Hamiltonian doubly-resonant saddle-node is a transversally Hamiltonian vector field which is conjugate, via a (fibered) transversally symplectic diffeomorphism, to one of the form

Y = x 2 ∂ ∂x + -λy 1 + F 1 (x, y) ∂ ∂y 1 + λy 2 + F 2 (x, y) ∂ ∂y 2 ,
with λ ∈ C * and F 1 , F 2 analytic in C 3 , 0 and of order at least 2.

Notice that a transversally Hamiltonian doubly-resonant saddle-node is necessarily strictly nondegenerate (since its residue is always equal to 1), and also div-integrable (see section 3).

We recall the second main result from [START_REF] Bittmann | Doubly-resonant saddle-nodes in C 3 and the fixed singularity at infinity in the painlevé equations: Formal classification[END_REF].

Theorem 1.20. [START_REF] Bittmann | Doubly-resonant saddle-nodes in C 3 and the fixed singularity at infinity in the painlevé equations: Formal classification[END_REF] Let Y be a diagonal doubly-resonant saddle-node which is transversally Hamiltonian. Then, there exists a unique formal fibered transversally symplectic diffeomorphism Φ tangent to identity such that:

Φ * (Y ) = x 2 ∂ ∂x + (-λ + a 1 x -c (y 1 y 2 )) y 1 ∂ ∂y 1 + (λ + a 2 x + c (y 1 y 2 )) y 2 ∂ ∂y 2 =: Y norm , (1.6)
where λ ∈ C * , c (v) ∈ vC v a formal power series in v = y 1 y 2 without constant term and a 1 , a 2 ∈ C are such that a 1 + a 2 = 1.

The second main result in [START_REF] Bittmann | Doubly-resonant saddle-nodes in (C 3 , 0) and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization[END_REF] is the following.

Theorem 1.21. Let Y be a transversally Hamiltonian doubly-resonant saddle-node. Let Φ be the unique formal normalizing map given by Theorem 1.20. Then the associate sectorial normalizing maps Φ + and Φ -given by Theorem 1.9 are also transversally symplectic.

Let us fix a normal form Y norm as in Theorem 1.21, and consider two sectorial domains S λ × C 2 , 0 and S -λ × C 2 , 0 as in (1.5). Then, the Stokes diffeomorphisms (Φ λ , Φ -λ ) defined in the previous subsection as

Φ λ := Φ + • Φ -1 -|S λ ×(C 2 ,0) Φ -λ := Φ -• Φ -1 + |S -λ ×(C 2 ,0)
, are also transversally symplectic.

Definition 1.22. We denote by Λ ω λ (Y norm ) resp. Λ ω -λ (Y norm ) the group of germs of sectorial fibered isotropies of Y norm , admitting the identity as Gevrey-1 asymptotic expansion in sectorial domains of the form S λ × C 2 , 0 resp. S -λ × C 2 , 0 , and which are transversally symplectic.

Let us denote by [Y norm ] ω the set of germs of vector fields which are formally conjugate to Y norm via (formal) transversally symplectic diffeomorphisms tangent to the identity. As a consequence of Theorems (1.15) and (1.21), we can now state our third main result:

Theorem 1.23. The map

[Y norm ] ω Diff ω C 3 , 0; Id -→ Λ ω λ (Y norm ) × Λ ω -λ (Y norm ) Y -→ (Φ λ , Φ -λ )
is a well-defined bijection.

Outline of the paper.

In section 2, we recall the different tools, notations and nomenclature we will need regarding asymptotic expansion, Gevrey-1 series, 1-summability and sectorial germs.

In section 3, we prove the main theorems presented above, assuming the Proposition 3.5 holds.

In section 4, we prove the key Proposition 3.5 by studying the automorphisms of the space of leaves.

In section 5, we give a description of the moduli space in Theorem 1.15 in terms of power series in the space of leaves, and present some applications.

In section 6, we present a generalization of Theorem 1.15 where we study the action of Diff C 3 , 0 instead of Diff fib C 3 , 0 .

Background

We refer the reader to [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF], [START_REF] Malgrange | Sommation des séries divergentes[END_REF] and [START_REF] Pierre | Divergent series and holomorphic dynamical systems[END_REF] for a general and detailed introduction to the theory of asymptotic expansion, Gevrey series and summability (see also [START_REF] Stolovitch | Classification analytique de champs de vecteurs 1-résonnants de (C n , 0)[END_REF] for a useful discussion of these concepts). We refer more precisely to [START_REF] Bittmann | Doubly-resonant saddle-nodes in (C 3 , 0) and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization[END_REF] when it comes to the notion of weak 1-summability.

We call x ∈ C the independent variable and y := (y 1 , . . . , y n ) ∈ C n , n ∈ N, the dependent variables. As usual we define

y k := y k1 1 . . . y kn n for k = (k 1 , . . . , k n ) ∈ N n , and |k| = k 1 + • • • + k n .
The notions of asymptotic expansion, Gevrey series and 1-summability presented here are always considered with respect to the independent variable x living in (open) sectors

S (r, α, β) = {x ∈ C | 0 < |x| < r and α < arg (x) < β} ,
the dependent variable y belonging to poly-discs

D (0, r) := {y = (y 1 , . . . , y n ) ∈ C n | |y 1 | < r 1 , . . . |y n | < r n } , of poly-radius r = (r 1 , . . . , r n ) ∈ (R >0 ) n .
Given an open subset U ⊂ C n+1 , we denote by O (U) the algebra of holomorphic function in U.

Sectorial germs.

Let θ ∈ R, η ∈ R ≥0 and n ∈ N.

Definition 2.1.

(1) An x-sectorial neighborhood (or simply sectorial neighborhood ) of the origin

(in C n+1 ) in the direction θ with opening η is an open set S ⊂ C n+1 such that S ⊃ S r, θ - η 2 -ǫ, θ + η 2 + ǫ × D (0, r)
for some r > 0, r ∈ (R >0 ) n and ǫ > 0. We denote by (S θ,η , ≤) the directed set formed by all such neighborhoods, equipped with the order relation

S 1 ≤ S 2 ⇐⇒ S 1 ⊃ S 2 .
(2) The algebra of germs of holomorphic functions in a sectorial neighborhood of the origin in the direction θ with opening η is the direct limit

O (S θ,η ) := lim -→ O (S)
with respect to the directed system defined by {O (S) : S ∈ S θ,η }.

We now give the definition of a (germ of a) sectorial diffeomorphism.

Definition 2.2.

(1) Given an element S ∈ S θ,η , we denote by Diff fib (S, Id) the set of holomorphic diffeomorphisms of the form

Φ : S → Φ (S) (x, y) → (x, φ 1 (x, y) , φ 2 (x, y)) , such that Φ (x, y) -Id (x, y) = O x, y 2 , as (x, y) → (0, 0) in S. 1
(2) The set of germs of (fibered) sectorial diffeomorphisms in the direction θ with opening η, tangent to the identity, is the direct limit

Diff fib (S θ,η ; Id) := lim -→ Diff fib (S, Id)
with respect to the directed system defined by {Diff fib (S, Id) : S ∈ S θ,η }. We equip Diff fib (S θ,η ; Id) of a group structure as follows: given two germs Φ, Ψ ∈ Diff fib (S θ,η ; Id) we chose corresponding representatives Φ 0 ∈ Diff fib (S, Id) and Ψ 0 ∈ Diff fib (T , Id) with S, T ∈ S θ,η such that T ⊂ Φ 0 (S) and let Ψ • Φ be the germ defined by

Ψ 0 • Φ 0 . 2
We will also need the notion of asymptotic sectors.

Definition 2.3. An (open) asymptotic sector of the origin in the direction θ and with opening

η is an open set S ⊂ C such that S ∈ 0≤η ′ <η S θ,η ′ .
We denote by AS θ,η the set of all such (open) asymptotic sectors.

2.2. Gevrey-1 power series and 1-summability.

In this subsection we fix a formal power series

f (x, y) = k≥0 f k (y) x k = (j0,j)∈N n+1 f j0,j x j0 y j ∈ C x, y .
Definition 2.4.

• An analytic (and bounded) function f in a sectorial domain S (r, α, β) × D (0, r) admits f as Gevrey-1 asymptotic expansion in this domain, if for all closed sub-sector S ′ ⊂ S (r, α, β), there exists A, C > 0 such that:

f (x, y) - N -1 k=0 f k (y) x k ≤ AC N (N !) |x| N
for all N ∈ N and (x, y) ∈ S ′ × D (0, r). • A formal power series f ∈ C x, y is 1-summable in the direction θ if and only if there exists a germ of a sectorial holomorphic function f θ ∈ O (S θ,π ) which admits f as Gevrey-1 asymptotic expansion in some S ∈ S θ,π .

Remark 2.5. In the definition above, f θ is unique as a germ in O (S θ,π ) , and is called the 1-sum of f in the direction θ.

Lemma 2.6. The set Σ θ ⊂ C x, y of 1-summable power series in the direction θ is an algebra closed under differentiation. Moreover the map

Σ θ -→ O (S θ,π ) f -→ f θ
is an injective morphism of differential algebras.

Proposition 2.7. Let Φ (x, y) ∈ C x, y be 1-summable in directions θ and θπ, and let Φ + (x, y) and Φ -(x, y) be its 1-sums directions θ and θπ respectively. Let also f1 (x, z) , . . . , fn (x, z) be 1summable in directions θ, θπ, and f 1,+ , . . . , f n,+ , and f 1,-, . . . , f n,-be their 1-sums in directions θ and θπ respectively. Assume that (2.1) fj (0, 0) = 0, for all j = 1, . . . , n .

Then Ψ (x, z) := Φ x, f1 (x, z) , . . . , fn (x, z) is 1-summable in directions θ, θπ, and its 1-sum in the corresponding direction is

Ψ ± (x, z) := Φ ± (x, f 1,± (x, z) , . . . , f n,± (x, z)) ,
which is a germ of a sectorial holomorphic function in directions θ and θπ.

Proof. See [START_REF] Bittmann | Doubly-resonant saddle-nodes in C 3 and the fixed singularity at infinity in the painlevé equations: Formal classification[END_REF].

Consider Ŷ a formal singular vector field at the origin and a formal fibered diffeomorphism φ : (x, y) → x, φ (x, y) . Assume that both Ŷ and φ are 1-summable in directions θ and θπ, for some θ ∈ R, and denote by Y + , Y -(resp. ϕ + , ϕ -) their 1-sums in directions θ and θπ respectively. As a consequence of Proposition 2.7 and Lemma 2.6, we can state the following:

Corollary 2.8. Under the assumptions above, φ * Ŷ is 1-summable in both directions θ and θπ, and its 1-sums in these directions are ϕ + (Y + ) and ϕ -(Y -) respectively.

An important result by Martinet and Ramis.

We are going to make an essential use of an isomorphism theorem proved in [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF]. This result is of paramount importance in the present paper since it will be a key tool in the proofs of both Theorems 1.9 and 1.15 (see section 3).

Let us consider two open asymptotic sectors S and S ′ at the origin in directions θ and θπ respectively, both of opening π:

S ∈ AS θ,π S ′ ∈ AS θ-π,π
(see Definition 2.3). In this particular setting, the cited theorem can be stated as follows.

Theorem 2.9. [MR82, Théorème 5.2.1] Consider a pair of germs of sectorial diffeomorphisms (ϕ, ϕ ′ ) ∈ Diff fib (S θ,0 ; Id) × Diff fib (S θ-π,0 ; Id) such that ϕ and ϕ ′ extend analytically and admit the identity as Gevrey-1 asymptotic expansion in S × C 2 , 0 and S ′ × C 2 , 0 respectively. Then, there exists a pair (φ + , φ -) of germs of sectorial fibered diffeomorphisms

(φ + , φ -) ∈ Diff fib S θ+ π 2 ,η ; Id × Diff fib S θ-π 2 ,η ; Id with η ∈ ]π, 2π[, which extend analytically in S + × C 2 , 0 and S -× C 2 , 0 respectively, for some S + ∈ AS θ+ π 2 ,2π and S -∈ AS θ-π 2 ,2π , such that: φ + • (φ -) -1 |S×(C 2 ,0) = ϕ φ + • (φ -) -1 |S ′ ×(C 2 ,0) = ϕ ′ .
There also exists a formal diffeomorphism φ which is tangent to the identity, such that φ + and φ -both admit φ as Gevrey-1 asymptotic expansion in S + × C 2 , 0 and S -× C 2 , 0 respectively. In particular, in the theorem above φ is 1-summable in every direction except θ and θπ, and its 1-sums in directions θ + π 2 and θ -π 2 respectively are φ + and φ -. For future use, we are going to prove a "transversally symplectic" version of the above theorem.

Corollary 2.10. With the assumptions and notations of Theorem 2.9, if ϕ and ϕ ′ both are transversally symplectic (see Definition 1.17), then there exists a germ of an analytic fibered diffeomorphism ψ ∈ Diff fib C 3 , 0, Id (tangent to the identity), such that σ + := φ + • ψ and σ -:= φ -• ψ both are transversally symplectic. Moreover we also have:

σ + • (σ -) -1 |S×(C 2 ,0) = ϕ σ + • (σ -) -1 |S ′ ×(C 2 ,0) = ϕ ′ .
Proof. We recall that for any germ ϕ of a sectorial fibered diffeomorphism which is tangent to the identity, ϕ is transversally symplectic if and only if det (Dϕ) = 1.

First of all, let us show that

det (Dφ + ) = det (Dφ -) in (S + ∩ S -) × C 2 , 0 .
Since φ + and φ -both are sectorial fibered diffeomorphism which are tangent to the identity and transversally symplectic, then

det φ + • (φ -) -1 |(S+∩S-)×(C 2 ,0) = 1 . The chain rule implies immediately that det (Dφ + ) = det (Dφ -) in (S + ∩ S -) × C 2 , 0 .
Thus, this equality allows us to define (thanks to the Riemann's Theorem of removable singularities) a germ of analytic function f ∈ O C 3 , 0 . Notice that f (0, 0, 0) = 1 because φ + and φ -are tangent to the identity. Now, let us look for an element ψ ∈ Diff fib C 3 , 0, Id of the form

(2.2) ψ : (x, y 1 , y 2 ) → (x, ψ 1 (x, y) , y 2 ) such that σ + := φ + • ψ and σ -:= φ -• ψ
both be transversally symplectic. An easy computation gives:

det (σ ± ) = (det (Dφ ± ) • ψ) det (Dψ) = 1 i.e. (f • ψ) det (Dψ) = 1 .
According to (2.2), we must have:

(2.3) (f • ψ) ∂ψ 1 ∂y 1 = 1 .
Let us define

F (x, y 1 , y 2 ) := ˆy1 0 f (x, z, y 2 ) dz ,
so that (2.3) can be integrated as

F • ψ = y 1 + h (x, y 2 ) , for some h ∈ C {x, y 2 }. Notice that ∂F ∂y 1 (0, 0, 0) = 1 since f (0, 0, 0) = 1.
Let us chose h = 0. Then, we have to solve

F • ψ = y 1 , with unknown ψ ∈ Diff fib C 3 , 0, Id as in (2.3). If we define Φ : (x, y) → (x, F (x, y) , y 2 ) ,
the latter problem is equivalent to find ψ as above such that:

Φ • ψ = Id .
Since DΦ 0 = Id, the inverse function theorem gives us the existence of the germ ψ = Φ -1 ∈ Diff fib C 3 , 0, Id .

U × D (0, r ′ ) with r ′ 1 < r 1 , . . . , r ′ n < r n , can be written (2.4) f (x, y) = j∈N n F j (x) y j ,
where for all j ∈ N n , F j is analytic and bounded on U, and defined via the Cauchy formula:

F j (x) = 1 (2iπ) n ˆ|z1|=r ′ 1 . . . ˆ|zn|=r ′ n f (x, z) (z 1 ) j1+1 . . . (z n ) jn+1 dz n . . . dz 1 .
Notice that the convergence of the series above is uniform on every compact with respect to x and y.

In the same way, any formal power series f (x, y) ∈ C x, y can be written as

f (x, y) = j∈N n Fj (x) y j .
We present here a weaker notion of 1-summability that we will also need.

Definition 2.11.

• A function f (x, y) = j∈N n F j (x) y j
analytic and bounded in a domain S (r, α, β) × D (0, r), admits f as weak Gevrey-1 asymptotic expansion in x ∈ S (r, α, β), if for all j ∈ N n , F j admits Fj as Gevrey-1 asymptotic expansion in S (r, α, β). • The formal power series f is said to be weakly 1-summable in the direction θ ∈ R, if the following conditions hold:

for all j ∈ N n , Fj (x) ∈ C x is 1-summable in the direction θ, whose 1-sum in the direction θ is denoted by F j ; the series f θ (x, y) := In this case, f θ (x, y) is called the weak 1-sum of f in the direction θ.

The following proposition is an analogue of Proposition 2.7 for weak 1-summable formal power series, with the a stronger condition instead of (2.1).

Proposition 2.12.

Let Φ (x, z) = j∈N n Φj (x) z j ∈ C x, y and
f (k) (x, y) = j∈N n F (k) j (x) y j ∈ C x, y ,
for k = 1, . . . , n, be n+1 formal power series which are weakly 1-summable in directions θ and θπ, and let us denote by Φ + , f

+ , . . . , f

(n) + resp. Φ -, f (1)
-, . . . , f

(n) -their respective weak 1-sums in the direction θ ( resp. θπ). Assume that F (k) 0 = 0 for all k = 1, . . . , n. Then,

Ψ (x, y) := Φ x, f (1) (x, y) , . . . , f (n) (x, y)
is weakly 1-summable directions θ and θπ, and its 1-sum in the corresponding direction is

Ψ ± (x, y) = Φ ± x, f (1)
± (x, y) , . . . , f

± (x, y) , which is a germ of a sectorial holomorphic function in the direction θ ( resp. θπ) with opening π.

Proof. See [START_REF] Bittmann | Doubly-resonant saddle-nodes in (C 3 , 0) and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization[END_REF].

As proved in [START_REF] Bittmann | Doubly-resonant saddle-nodes in (C 3 , 0) and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization[END_REF], the next corollary gives the link between 1-summability in some direction and weak 1-summability in the same direction (we refer to [START_REF] Bittmann | Doubly-resonant saddle-nodes in (C 3 , 0) and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization[END_REF], Definition 2.8, or to [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF], section IV, for a definition of the norm • λ,θ,δ,ρ associate to the space of 1-summable formal power series in the direction θ).

Corollary 2.13. Let f (x, y) = j∈N n Fj (x) y j ∈ C x, y be a formal power series. Then, f is 1-summable in the direction θ ∈ R, of 1-sum f ∈ O (S θ,π ), if and only if the following two conditions hold:

• f is weakly 1-summable in the direction θ, i.e. there exists λ, δ, ρ such that ∀j ∈ N n , Fj λ,θ,δ,ρ < ∞

• the power series

j∈N n Fj λ,θ,δ,ρ
y j is convergent in some polydisc D (0, r).

Proof. See [START_REF] Bittmann | Doubly-resonant saddle-nodes in (C 3 , 0) and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization[END_REF].

Proofs of the main theorems

The aim of this section is to prove the main results of this paper, assuming Proposition 3.5 below holds.

Analytic invariants: Stokes diffeomorphisms.

From now on, we fix a normal form

Y norm = x 2 ∂ ∂x + (-λ + a 1 x -c (y 1 y 2 )) y 1 ∂ ∂y 1 + (λ + a 2 x + c (y 1 y 2 )) y 2 ∂ ∂y 2 ,
with λ ∈ C * ,ℜ (a 1 + a 2 ) > 0 and c ∈ vC {v} vanishing at the origin. We denote by [Y norm ] the set of germs of holomorphic doubly-resonant saddle-nodes in C 3 , 0 , formally conjugate to Y norm by formal fibered diffeomorphisms tangent to the identity. We refer the reader to Definition 2.1 for notions relating to sectors.

Definition 3.1.

• We define Isot fib (Y ; S θ,η ; Id), for all θ ∈ R and η ∈ [0, 2π], as the group of germs of sectorial fibered isotropies of Y norm in sectorial domains in S θ,η (see Definition 2.3), which are tangent to the identity.

• We define Λ (weak) λ (Y norm ) resp. Λ (weak)
-λ (Y norm ) as the group of germs of sectorial fibered isotropies of Y norm , admitting the identity as weak Gevrey-1 asymptotic expansion in sectorial domains of the form S λ × C 2 , 0 resp. S -λ × C 2 , 0 , where:

S λ ∈ AS arg(λ),π S -λ ∈ AS arg(-λ),π (see Definition 2.3).
We recall the notations given in the introduction: we have defined Λ λ (Y norm ) (resp. Λ -λ (Y norm )) as the group of germs of sectorial fibered isotropies of Y norm , admitting the identity as Gevrey-1 asymptotic expansion in sectorial domains of the form S λ × C 2 , 0 resp. S -λ × C 2 , 0 . It is clear that we have:

Λ ±λ (Y norm ) ⊂ Λ (weak) ±λ (Y norm ) ⊂ Isot fib Y ; S arg(±λ),η ; Id , ∀η ∈ ]0, π[ .
According to Theorem 1.9, to any Y ∈ [Y norm ], we can associate a pair of germs of sectorial fibered isotropies in S λ × C 2 , 0 and S -λ × C 2 , 0 respectively, denoted by (Φ λ , Φ -λ ):

Φ λ := Φ + • Φ -1 -|S λ ×(C 2 ,0) ∈ Isot fib Y ; S arg(λ),η ; Id , ∀η ∈ ]0, π[ Φ -λ := Φ -• Φ -1 + |S -λ ×(C 2 ,0) ∈ Isot fib Y ; S arg(-λ),η ; Id , ∀η ∈ ]0, π[ ,
where (Φ + , Φ -) is the pair of the sectorial normalizing maps given by Theorem 1.9.

Proposition 3.2. For any given η ∈ ]0, π[ the map

[Y norm ] -→ Isot fib Y ; S arg(λ),η ; Id × Isot fib Y ; S arg(-λ),η ; Id Y -→ (Φ λ , Φ -λ ) , actually ranges in Λ (weak) λ (Y norm ) × Λ (weak) -λ (Y norm ).
Proof. The fact that the sectorial normalizing maps Φ + , Φ -given by Theorem 1.9 both conjugate Y ∈ [Y norm ] to Y norm in the corresponding sectorial domains proves that the arrow above is welldefined, with values in Isot fib Y ; S arg(λ),η ; Id × Isot fib Y ; S arg(-λ),η ; Id , for all η ∈ ]0, π[. The fact that Φ ±λ admits the identity as weak Gevrey-1 asymptotic expansion in S ±λ × C 2 , 0 comes from the fact that Φ + and Φ -admits the same weak Gevrey-1 asymptotic expansion in S λ × C 2 , 0 and S -λ × C 2 , 0 , and from Proposition 2.12.

The subgroup Diff fib C 3 , 0, Id ⊂ Diff fib C 3 , 0 formed by fibered diffeomorphisms tangent to the identity acts naturally on [Y norm ] by conjugacy. Now we show that the uniqueness of germs of sectorial normalizing maps (Φ + , Φ -) implies that the Stokes diffeomorphisms (Φ λ , Φ -λ ) of a vector field Y ∈ [Y norm ] is invariant under the action of Diff fib C 3 , 0, Id . Furthermore, this map is one-to-one.

Proposition 3.3. The map

[Y norm ] -→ Λ (weak) λ (Y norm ) × Λ (weak) -λ (Y norm ) Y -→ (Φ λ , Φ -λ )
factorizes through a one-to-one map By assumption, Φ± • Θ is also a germ of a sectorial fibered normalization of Y in S ± × C 2 , 0 , which is tangent to the identity. Thus, according to the uniqueness statement in Theorem 1.9:

[Y norm ] Diff fib C 3 , 0, Id -→ Λ (weak) λ (Y norm ) × Λ (weak) -λ (Y norm ) Y -→ (Φ λ , Φ -λ ) .
Φ ± = Φ± • Θ .
Consequently, in S ±λ × C 2 , 0 we have

Φ λ = Φ + • Φ -1 -|S λ ×(C 2 ,0) = Φ+ • Θ • Θ -1 • Φ- = Φλ ,
and similarly

Φ -λ = Φ -• Φ -1 + |S -λ ×(C 2 ,0) = Φ-• Θ • Θ -1 • Φ+ -1 = Φ-λ .
Let us prove that the map is one-to-one. Let Y, Ỹ ∈ [Y norm ] share the same Stokes diffeomorphisms (Φ λ , Φ -λ ). We denote by Φ ± (resp. Φ± ) the germ of a sectorial fibered normalizing map of Y (resp. Ỹ ) S ± × C 2 , 0 . We have:

     Φ + • (Φ -) -1 = Φ λ = Φ+ • Φ- -1 in S λ × C 2 , 0 Φ -• (Φ + ) -1 = Φ -λ = Φ-• Φ+ -1 in S -λ × C 2 , 0 . Thus:      Φ+ -1 • Φ + = Φ- -1 • Φ -in S λ × C 2 , 0 Φ+ -1 • Φ + = Φ- -1 • Φ -in S -λ × C 2 , 0 .
We can then define a map ϕ analytic in a domain of the form (D (0, r) \ {0}) × D (0, r) by setting:

     ϕ |S+ = Φ+ -1 • Φ + in S + ϕ |S-= Φ- -1 • Φ -in S -.
This map is analytic and bounded in (D (0, r) \ {0}) × D (0, r), and the Riemann singularity theorem tells us that this map can be analytically extended to the entire poly-disc D (0, r) × D (0, r). As a conclusion, ϕ ∈ Diff fib C 3 , 0, Id , Φ ± = Φ± • ϕ and ϕ * (Y ) = Ỹ .

3.2. Proof of Theorem 1.12: 1-summability of the formal normalization.

We fix a normal form

Y norm = x 2 ∂ ∂x + (-λ + a 1 x -c (y 1 y 2 )) y 1 ∂ ∂y 1 + (λ + a 2 x + c (y 1 y 2 )) y 2 ∂ ∂y 2 ,
with λ ∈ C * ,ℜ (a 1 + a 2 ) > 0 and c ∈ vC {v} vanishing at the origin. In section 4 we will prove the following result.

Proposition 3.5. Any ψ ∈ Λ (weak) ±λ (Y norm ) admits the identity as Gevrey-1 asymptotic expansion in S ±λ × C 2 , 0 . In other words:

Λ (weak) ±λ (Y norm ) = Λ ±λ (Y norm ) .
As a first consequence of Proposition 3.5, we obtain Theorem 1.12 which states that the formal normalizing map from [START_REF] Bittmann | Doubly-resonant saddle-nodes in C 3 and the fixed singularity at infinity in the painlevé equations: Formal classification[END_REF] is in fact 1-summable.

Proof of Theorem 1.12.

Let us consider the unique germs of a sectorial normalizing map Φ + and Φ -in S + × C 2 , 0 and S -× C 2 , 0 respectively, and their associated Stokes diffeomorphisms:

Φ λ = Φ + • Φ -1 -|S λ ×(C 2 ,0) ∈ Λ (weak) λ (Y norm ) Φ -λ = Φ -• Φ -1 + |S -λ ×(C 2 ,0) ∈ Λ (weak) -λ (Y norm ) .
According to Proposition 3.5,

Λ (weak) ±λ (Y norm ) = Λ ±λ (Y norm ) ,
so that Φ λ and Φ -λ both admit the identity as Gevrey-1 asymptotic expansion, in S λ × C 2 , 0 and S -λ × C 2 , 0 respectively. Then, Theorem 2.9 gives the existence of (φ + , φ -) ∈ Diff fib S arg(iλ),η ; Id × Diff fib S arg(-iλ),η ; Id for all η ∈ ]π, 2π[, such that:

   φ + • (φ -) -1 |S λ ×(C 2 ,0) = Φ λ φ -• (φ + ) -1 |S -λ ×(C 2 ,0) = Φ -λ ,
and the existence of a formal diffeomorphism φ which is tangent to the identity, such that φ + and φ - both admit φ as Gevrey-1 asymptotic expansion in S + × C 2 , 0 and S -× C 2 , 0 respectively. In particular, we have:

(Φ + ) -1 • φ + |(S λ ∪S -λ )×(C 2 ,0) = (Φ -) -1 • φ - |(S λ ∪S -λ )×(C 2 ,0)
. This proves that the function Φ defined by

(Φ + ) -1 • φ + in S + × C 2 , 0 and by (Φ -) -1 • φ -in S -× C 2 ,
0 is well-defined and analytic in D (0, r) \ {0} × D (0, r). Since it is also bounded, it can be extended to an analytic map Φ in D (0, r) × D (0, r) by Riemann's theorem. Hence:

φ + = Φ + • Φ φ -= Φ -• Φ .
In particular, by composition, Φ + and Φ -both admit φ • Φ -1 as Gevrey-1 asymptotic expansion in S λ × C 2 , 0 and S -λ × C 2 , 0 respectively. Since Φ + and Φ -conjugates Y to Y norm and since the notion of asymptotic expansion commutes with the partial derivative operators, the formal diffeomorphism φ • Φ -1 formally conjugates Y to Y norm . Finally, notice that φ • Φ -1 is necessarily tangent to the identity. Hence, by uniqueness of the formal normalizing map given by Theorem 1.4, we deduce that φ • Φ -1 = Φ, the unique formal normalizing map tangent to the identity.

Proofs of Theorems 1.15 and 1.23.

Let us now present the proofs of Theorems 1.15 and 1.23, assuming Proposition 3.5.

3.3.1.

Proof of Theorem 1.15.

Proof of Theorem 1.15.

Propositions 3.3, together with Proposition 3.5, tell us that the considered map is well-defined and one-to-one. It remains to prove that this map is onto. Let

Φ λ ∈ Λ λ (Y norm ) Φ -λ ∈ Λ -λ (Y norm )
.

According to Theorem 2.9, there exists (φ + , φ -) ∈ Diff fib S arg(iλ),η ; Id × Diff fib S arg(-iλ),η ; Id with η ∈ ]π, 2π[, which extend analytically to S + × C 2 , 0 and S -× C 2 , 0 respectively, such that:

φ ± • (φ ∓ ) -1 |S ±λ ×(C 2 ,0) = Φ ±λ
and there also exists a formal diffeomorphism φ which is tangent to the identity, such that φ ± both admit φ as asymptotic expansion in S ± × C 2 , 0 . Let us consider the two germs of sectorial vector fields obtained as

Y ± := φ -1 ± * (Y norm )
In particular, since φ is the Gevrey-1 asymptotic expansion of φ ± , the vector fields Y ± both admit φ * (Y norm ) as Gevrey-1 asymptotic expansion. The fact that φ

+ • (φ -) -1 is an isotropy of Y norm implies immediately that Y + = Y -on (S + ∩ S -) × C 2 , 0 = (S λ ∪ S -λ ) × C 2 , 0 .
Then, the vector field Y , which coincides with Y ± in S ± × C 2 , 0 , defines a germ of analytic vector field in C 3 , 0 by Riemann's theorem. By construction, Y ∈ Diff fib C 3 , 0, Id * (Y norm ) and admits (Φ λ , Φ -λ ) as Stokes diffeomorphisms.

3.3.2. Proof of Theorem 1.23.

In a similar way, we prove now Theorem 1.23.

Proof of Theorem 1.23.

Let Y norm ∈ SN diag,0 be a normal form which is also transversally symplectic. We refer to subsection 1.4 for the notations. It is clear from Theorems 1.15 and 1.21 that the mapping is well-defined and one-to-one. It remains to prove that it is also onto. Let

Φ λ ∈ Λ ω λ (Y norm ) Φ -λ ∈ Λ ω -λ (Y norm ) . Since Λ ω λ (Y norm ) ⊂ Λ λ (Y norm ) and Λ ω -λ (Y norm ) ⊂ Λ -λ (Y norm )
, according to Theorem 2.9 there exists (φ + , φ -) ∈ Diff fib S arg(iλ),η ; Id × Diff fib S arg(-iλ),η ; Id with η ∈ ]π, 2π[, which extend analytically in S + × C 2 , 0 and S -× C 2 , 0 respectively, such that:

φ ± • (φ ∓ ) -1
|S ±λ ×(C 2 ,0) = Φ ±λ and there also exists a formal diffeomorphism φ which is tangent to the identity, such that φ ± both admit φ as Gevrey-1 asymptotic expansion in S ± × C 2 , 0 . According to Corollary 2.10, there exists a germ of an analytic fibered diffeomorphism ψ ∈ Diff fib C 3 , 0, Id (tangent to the identity), such that σ ± := φ ± • ψ both are transversally symplectic. Then, we have:

σ ± • (Ψ ∓ ) -1 |S ±λ ×(C 2 ,0) = Φ ±λ .
The end of the proof goes exactly as at the end of the proof of the previous theorem.

Sectorial isotropies and space of leaves: proof of Proposition 3.5

A normal form

Y norm = x 2 ∂ ∂x + (-λ + a 1 x -c (y 1 y 2 )) y 1 ∂ ∂y 1 + (λ + a 2 x + c (y 1 y 2 )) y 2 ∂
∂y 2 is fixed for some λ ∈ C * , ℜ (a 1 + a 2 ) > 0 and c ∈ vC {v} (vanishing at the origin). The aim of this section is to prove Proposition 3.5 stated in Section 3.

Let us denote

a := res (Y norm ) = a 1 + a 2 , m := 1 a and c (v) = +∞ k=1 c k v k .
If m / ∈ N >0 , we set c m := 0. We also define the following power series

c (v) = m k =m c k k -m v k ,
and we notice that c (v) ∈ vC {v}.

Sectorial first integrals and the space of leaves.

In a sectorial neighborhood of the origin of the form S λ × C 2 , 0 resp. S -λ × C 2 , 0 ,with S ±λ ∈ S arg(±λ),ǫ and ǫ ∈ ]0, π[, we can give three first integrals of Y norm which are analytic in the considered domain. Let us start with the following proposition.

Proposition 4.1. The following quantities are first integrals of Y norm , analytic in S ±λ × C 2 , 0 :

(4.1)              w ±λ := y 1 y 2 x a h 1,±λ (x, y) := y 1 exp -λ x + c m (y 1 y 2 ) m log (x) x + c (y 1 y 2 ) x x -a1 h 2,±λ (x, y) := y 2 exp λ x - c m (y 1 y 2 ) m log (x) x - c (y 1 y 2 ) x x -a2
(we fix here a branch of the logarithm analytic in S ±λ , and we write simply h j and w instead of h j,±λ and w ±λ respectively, if there is no ambiguity on the sector S ±λ ). Moreover, we have the relation:

h 1 h 2 = w .
Proof. It is an elementary computation.

Remark 4.2. In other words, in a sectorial domain, we can parametrize a leaf (which is not in {x = 0}) of the foliation associated to Y norm by:

       y 1 (x) = h 1 exp λ x -c m (h 1 h 2 ) m log (x) - c (h 1 h 2 x a ) x x a1 y 2 (x) = h 2 exp - λ x + c m (h 1 h 2 ) m log (x) + c (h 1 h 2 x a ) x x a2 (4.2) (h 1 , h 2 ) ∈ C 2 .
Corollary 4.3. The map

H ±λ : S ±λ × C 2 , 0 → S ±λ × C 2 (x, y) → (x, h 1,±λ (x, y) , h 2,±λ (x, y)) ,
(where h 1,±λ , h 2,±λ are defined in (4.1)) is a sectorial germ of a fibered analytic map in S ±λ × C 2 , 0 , which is into. Moreover, there exists an open neighborhood of the origin in C 2 , denoted by Γ ±λ ⊂ C 2 , such that

H ±λ S ±λ × C 2 , 0 = S ±λ × Γ ±λ .
In particular, H ± induces a fibered biholomorphism

S ±λ × C 2 , 0 H ±λ -→ S ±λ × Γ ±λ which conjugates Y norm to x 2 ∂ ∂x , i.e. (H ±λ ) * (Y norm ) = x 2 ∂ ∂x .
Definition 4.4. We call Γ ±λ the space of leaves of Y norm in S ±λ × C 2 , 0 .

Remark 4.5. The set Γ ±λ depends on the choice of the neighborhood C 2 , 0 , but also on the choice of the sectorial neighborhood S ±λ ∈ S arg(±λ),ǫ .

4.2. Sectorial isotropies in the space of leaves. Now, we consider a germ of a sectorial isotropy ψ ±λ ∈ Λ (weak) ±λ (Y norm ) and we denote by Γ ′ ±λ the (germ of an) open subset of C 2 such that:

H ±λ • ψ ± S ±λ × C 2 , 0 = S ±λ × Γ ′
±λ . Proposition 4.6. With the notations and assumptions above, the map

Ψ ±λ := H ±λ • ψ ± • H -1 ±λ : S ±λ × Γ ±λ -→ S ±λ × Γ ′ ±λ
is a sectorial germ of a fibered biholomorphism from S ±λ × Γ ±λ to S ±λ × Γ ′ ±λ , which is of the form:

Ψ ±λ (x, h 1 , h 2 ) = (x, Ψ 1,±λ (h 1 , h 2 ) , Ψ 2,±λ (h 1 , h 2 )) .
In particular, Ψ 1,±λ and Ψ 2,±λ are analytic and depend only on (h 1 , h 2 ) ∈ Γ ±λ , while Ψ ±λ induces a biholomorphism (still written Ψ ±λ ):

Ψ ±λ : Γ ±λ → Γ ′ ±λ (h 1 , h 2 ) → (Ψ 1,±λ (h 1 , h 2 ) , Ψ 2,±λ (h 1 , h 2 )) .
Proof. We only have to prove that Ψ 1,±λ and Ψ 2,±λ depend only on (h 1 , h 2 ) ∈ Γ ±λ . By assumption, Ψ ±λ is an isotropy of x 2 ∂ ∂x :

(Ψ ±λ ) * x 2 ∂ ∂x = x 2 ∂ ∂x .
We immediately obtain:

∂Ψ 1,±λ ∂x = ∂Ψ 2,±λ ∂x = 0 .
In the space of leaves Γ ±λ equipped with coordinates (h 1 , h 2 ), we denote by w the product of h 1 and h 2 :

w (h 1 , h 2 ) := h 1 h 2 . We define the two following quantities:

(4.3)    f 1 (x, w) := exp λ x -c m w m log (x) -c(wx a ) x x a1
f 2 (x, w) := exp -λ x + c m w m log (x) + c(wx a )

x

x a2 , such that the leaves of the foliations are parametrized by:

y 1 (x) = h 1 f 1 (x, h 1 h 2 ) y 2 (x) = h 2 f 2 (x, h 1 h 2 ) , (h 1 , h 2 ) ∈ C 2 .
Notice that:

f 1 (x, w) f 2 (x, w) = x a .
Moreover, one checks immediately the following statement.

Fact 4.7. For all w ∈ C:

       lim x→0 x∈S λ |f 1 (x, w)| = lim x→0 x∈S -λ |f 2 (x, w)| = +∞ lim x→0 x∈S -λ |f 1 (x, w)| = lim x→0 x∈S λ |f 2 (x, w)| = 0 .
Using notations of Proposition 4.6, we also assume from now on that C 2 , 0 = D (0, r), with r = (r 1 , r 2 ) ∈ (R >0 )

2 and r 1 , r 2 > 0 small enough so that

ψ ±λ (S ±λ × D (0, r)) ⊂ S ±λ × D (0, r ′ ) for some r ′ = (r ′ 1 , r ′ 2 ) ∈ (R >0 ) 2 .
Let us now define in a general way the following set associated to the sector S ±λ and to a polydisc D (0, r), with r := (r 1 , r2 ). 2 , we define

Γ ±λ (x, r) := (h 1 , h 2 ) ∈ C 2 | |h j | ≤ rj |f j (x, h 1 h 2 )|
, for j ∈ {1, 2} .

We also consider the:

Γ ±λ (r) := x∈S ±λ Γ ±λ (x, r) = (h 1 , h 2 ) ∈ C 2 | ∃x ∈ S ±λ s.t. |h j | ≤ rj |f j (x, h 1 h 2 )|
, for j ∈ {1, 2}

(cf. figure 4.1).

Since we assume now that C 2 , 0 = D (0, r), then we have:

Γ ±λ = Γ ±λ (r) , and
Γ ′ ±λ ⊂ Γ ±λ (r ′ ) .
Remark 4.9.

(1) It is important to notice that the particular form of Ψ ±λ implies that the image of any fiber

{x = x 0 } × Γ ±λ (x 0 , r)
by Ψ ±λ is included in a fiber of the form

{x = x 0 } × Γ ±λ (x 0 , r ′ ) . (2) If (h 1 , h 2 ) ∈ Γ ±λ (x, r), then |h 1 h 2 | < r 1 r 2 |x a | .
(3) As (h 1 , h 2 ) ∈ Γ ±λ varies the values of w = h 1 h 2 cover the whole C.

Action on the resonant monomial in the space of leaves.

Let us study the the action of Ψ ±λ on the resonant monomial w = h 1 h 2 in the space of leaves.

Lemma 4.10. We consider a biholomorphism

Ψ ±λ : Γ ±λ → Γ ′ ±λ (h 1 , h 2 ) → (Ψ 1,± (h 1 , h 2 ) , Ψ 2,± (h 1 , h 2 )) ,
such that for all x ∈ S ±λ , we have

Ψ ±λ (Γ ±λ (x 0 , r)) ⊂ Γ ±λ (x 0 , r ′ ) .
We also define Ψ w,±λ := Ψ 1,±λ Ψ 2,±λ . Then, for all n ∈ N, there exists entire ( i.e. analytic over C) functions Ψ w,λ,n and Ψ w,-λ,n such that

       Ψ w,λ (h 1 , h 2 ) = n≥0 Ψ w,λ,n (h 1 h 2 ) h n 1 Ψ w,-λ (h 1 , h 2 ) = n≥0 Ψ w,-λ,n (h 1 h 2 ) h n 2 .
Moreover, the series above uniformly converge (for the sup-norm) in every subset of Γ ±λ of the form Γ ±λ (r), with r := (r 1 r2 ) and 0 < rj < r j , j ∈ {1, 2} ( cf. Definition 4.8). More precisely, for all r1 , r2 , δ > 0 such that

0 < rj + δ < r j , j ∈ {1, 2}
for all x ∈ S ±λ and w ∈ C we have

|wx a | ≤ r1 r2 =⇒    |Ψ w,λ,n (w)| ≤ r ′ 1 r ′ 2 |x a | f1(x,w) r1+δ n |Ψ w,-λ,n (w)| ≤ r ′ 1 r ′ 2 |x a | f2 (x,w) r2+δ n
, ∀n ≥ 0 .

Proof. Let us give the proof for Ψ w,λ , Ψ 1,λ and Ψ 2,λ in Γ λ (the same proof applies also for Ψ w,-λ in Γ -λ by exchanging the role played by h 1 and h 2 ). We fix some 0 < rj < r j , j ∈ {1, 2}, and δ > 0 such that 0 < rj + δ < r j , j ∈ {1, 2} . For a fixed value w ∈ C, we consider the restriction of Ψ w,λ to the hypersurface M w := {h 1 h 2 = w}∩Γ λ : this restriction is analytic in M w . The map

ϕ w : h 1 → Ψ w,λ h 1 , w h 1 is analytic in M w,1 := x∈S λ |wx a |<r1r2 Ω x,w ,
where for all x ∈ S λ with |wx a | < r 1 r 2 , the set Ω x,w is the following annulus:

Ω x,w := h 1 ∈ C | wf 2 (x, w) r 2 < |h 1 | < r 1 f 1 (x, w) .
In particular, ϕ w admits a Laurent expansion

ϕ w (h 1 ) = Ψ w,+ h 1 , w h 1 = n≥-L Ψ w,+,n (w) h n 1
in every annulus Ω x,w , with x ∈ S λ such that |wx a | < r 1 r 2 . Moreover for all x ∈ S λ such that |wx a | < r 1 r 2 , Cauchy's formula gives

Ψ w,λ,n (w) = 1 2iπ ˛γ(x,w) Ψ w,λ h 1 , w h1 h n+1 1 dh 1 , for all n ∈ Z,
where γ (x, w) is any circle (oriented positively) centered at the origin with a radius ρ (x, w) satisfying

wf 2 (x, w) r 2 < ρ (x, w) < r 1 f 1 (x, w)
.

If |wx a | < (r 1 + δ) (r 2 + δ), we can take for instance ρ (x, w) = r1 + δ f 1 (x, w) .
Therefore, for all x ∈ S λ and all w ∈ C such that |wx a | ≤ r1 r2 , for all ξ ∈ C with |ξ| < δ, we also have:

Ψ w,λ,n (w + ξ) = 1 2iπ ˛γ(x,w) Ψ w,λ h 1 , w+ξ h1 h n+1 1 dh 1 , for all n ∈ Z, where γ (x, w) is the same circle of radius ρ (x, w) = r1 + δ f 1 (x, w)
for all |ξ| < δ. Moreover, since for all x ∈ S λ , we have

Ψ λ (Γ λ (x, r)) ⊂ Γ λ (x, r ′ ) , and since for all (h ′ 1 , h ′ 2 ) ∈ Γ λ (x, r ′ ) we have |h ′ 1 h ′ 2 | ≤ r ′ 1 r ′ 2 |x a | ,
then for all x ∈ S λ and w ∈ C such that |wx a | ≤ r1 r2 , the following inequality holds for all h 1 with |h 1 | < r1 f1(x,w) :

Ψ w,λ h 1 , w h 1 < r ′ 1 r ′ 2 |x a | .
The well-known theorem regarding integrals depending analytically on a parameter asserts that for all n ∈ Z the mapping Ψ w,λ,n is analytic near any point w ∈ C. Hence, it is an entire function (i.e. analytic over C). Moreover, the inequality above and the Cauchy's formula together imply that for all n ∈ Z and for all (x, w) ∈ S λ × C such that |wx a | ≤ r1 r2 , we have:

|Ψ w,λ,n (w)| < r ′ 1 r ′ 2 |x a | ρ (x, w) n = r ′ 1 r ′ 2 |x a | f 1 (x, w) r1 + δ n .
According to Fact 4.7, for a fixed value w ∈ C, if n < 0, the right hand-side tends to 0 as x tends to 0 in S λ . This implies in particular that Ψ w,λ,n = 0 for all n < 0. Consequently:

Ψ w,λ h 1 , w h 1 = n≥0 Ψ w,λ,n (w) h n 1 .
Moreover, for all w ∈ C the series converges normally in every domain of the form

Ω x,w := h 1 ∈ C | |h 1 | ≤ r1 f 1 (x, w) , for all x ∈ S λ , 0 < r1 < r 1 ,
since the Laurent expansion's range is n ≥ 0. This actually means that the series converges normally in an entire neighborhood of the origin in C. In particular, for all fixed w ∈ C, the map

h 1 → Ψ w,λ h 1 , w h 1 = n≥0 Ψ w,λ,n (w) h n 1
is analytic in a neighborhood of the origin. Finally, the series

Ψ w,λ (h 1 , h 2 ) = n≥0 Ψ w,λ,n (h 1 h 2 ) h n 1
converges normally, and hence its sum is analytic in every domain of the form Γ λ (r), with 0 < r1 < r 1 and 0 < r2 < r 2 .

Action on the resonant monomial.

Since ψ ±λ ∈ Λ (weak) ±λ (Y norm ), the mapping ψ ±λ is of the form

ψ ±λ (x, y) = (x, ψ 1,±λ (x, y) , ψ 2,±λ (x, y)) ,
with ψ 1,±λ , ψ 2,±λ analytic and bounded in S ±λ × D (0, r). Moreover, by assumption ψ ±λ admits the identity as weak Gevrey-1 asymptotic expansion, i.e. we have a normally convergent expansion:

ψ i,±λ (x, y) = y i + k∈N 2 ψ i,±λ,k (x) y k ,
where ψ i,±λ,k is holomorphic in S ±λ and admits 0 as Gevrey-1 asymptotic expansion, for i = 1, 2 and

all k = (k 1 , k 2 ) ∈ N 2 .
Lemma 4.11. With the notations and assumptions above, let us define ψ v,±λ := ψ 1,±λ ψ 2,±λ . Then ψ v,λ and ψ v,-λ can be expanded as the series

           ψ v,λ (x, y) = y 1 y 2 + x a n≥1 Ψ w,λ,n y 1 y 2 x a y 1 f 1 x, y1y2 x a n ψ v,-λ (x, y) = y 1 y 2 + x a n≥1 Ψ w,-λ,n y 1 y 2 x a y 2 f 2 x, y1y2
x a n which are normally convergent in every subset of S ±λ × D (0, r) of the form S ±λ × D (0, r), where D (0, r) is a closed poly-disc with r = (r 1 , r2 ) such that 0 < rj < r j , j ∈ {1, 2} .

Here Ψ w,λ,n and Ψ w,-λ,n , for n ∈ N, are the ones appearing in Lemma 4.10. Moreover, for all closed sub-sector S ′ ⊂ S ±λ and for all closed poly-disc D ⊂ D (0, r), there exists A, B > 0 such that:

|ψ v,±λ (x, y 1 , y 2 ) -y 1 y 2 | ≤ A exp - B |x| , ∀ (x, y) ∈ S ′ × D .
In particular, ψ v,±λ admits y 1 y 2 as Gevrey-1 asymptotic expansion in S ±λ × D (0, r).

Proof. By definition, we have

Ψ ±λ • H ±λ = H ±λ • ψ ±λ .
In particular, for all (x, y) ∈ S ±λ × D (0, r):

Ψ w,± x, y 1 f 1 x, y1y2 x a , y 2 f 2 x, y1y2 x a = ψ v,± (x, y 1 , y 2 ) x a .
Thus, according to Lemma 4.10 we have:

(4.4)            ψ v,λ (x, y) = x a n≥0 Ψ w,λ,n y 1 y 2 x a y 1 f 1 x, y1y2 x a n ψ v,-λ (x, y) = x a n≥0 Ψ w,-λ,n y 1 y 2 x a y 2 f 2 x, y1y2 x a n .
Besides we know that ψ v,±λ admits y 1 y 2 as weak Gevrey-1 asymptotic expansion in S ±λ × D (0, r):

ψ v,±λ (x, y 1 , y 2 ) = y 1 y 2 + k∈N 2 ψ v,±λ,k (x) y k , (4.5)
where for all k = (k 1 , k 2 ) ∈ N 2 the mapping ψ v,±λ,k is holomorphic in S ±λ and admits 0 as Gevrey-1 asymptotic expansion. Let us compare both expressions of ψ v,±λ above. Looking at monomials y k with k 1 = k 2 in (4.5), and at terms corresponding to n = 0 on the right-hand side of (4.4), we must have for all x ∈ S ±λ and v ∈ C with |v| < r 1 r 2 :

v + k≥0 ψ v,λ,k(k,k) (x) v k = x a Ψ w,λ,0 v x a .
Since Ψ w,±λ,0 is analytic in C, there exists (α ±λ,k ) k∈N ⊂ C such that

Ψ w,±λ,0 v x a = k≥0 α ±λ,k v x a k .
This can only happen if α ±λ,k = 0 whenever k = 1, for ψ v,±λ,k is holomorphic in S ±λ and admits 0 as Gevrey-1 asymptotic expansion. A further immediate identification yields

Ψ v,±λ,0 (w) = w . Thus            ψ v,λ (x, y) = y 1 y 2 + x a n≥1 Ψ w,λ,n y 1 y 2 x a y 1 f 1 x, y1y2 x a n ψ v,-λ (x, y) = y 1 y 2 + x a n≥1 Ψ w,-λ,n y 1 y 2 x a y 2 f 2 x, y1y2 x a n .
Let us prove that ψ v,±λ admits y 1 y 2 as Gevrey-1 asymptotic expansion in S ±λ × C 2 , 0 . We have to show that |ψ v,±λ (x, y 1 , y 2 )y 1 y 2 | is exponentially small with respect to x ∈ S ±λ , uniformly in y ∈ D (0, r). As for the previous lemma, we perform the proof for ψ v,λ only (the same proof applies for ψ v,-λ by exchanging y 1 and y 2 ).

From the computations above we derive

|ψ v,λ (x, y 1 , y 2 ) -y 1 y 2 | ≤ n≥1 x a Ψ w,λ,n y 1 y 2 x a y 1 f 1 x, y1y2 x a n .
Let us fix r1 , r2 , δ > 0 in such a way that 0 < rj + δ < r j , j ∈ {1, 2} .

Let us take |x|, |y 1 | and |y 2 | small enough so that 2x ∈ S λ and

|y 1 y 2 | < r1 r2 |2 a | < r 1 r 2 .
According to Lemma 4.10, for all x ∈ S λ and all w ∈ C:

|wx a | ≤ r1 r2 =⇒ |Ψ w,λ,n (w)| ≤ r ′ 1 r ′ 2 |x a | f 1 (x, w) r1 + δ n .
In particular for x = 2x and w = y1y2 x a we derive |wx a | < r1 r2 , from which we conclude

Ψ w,λ,n y 1 y 2 x a ≤ r ′ 1 r ′ 2 |2 a x a | f 1 2x, y1y2 x a r1 + δ n .
Consequently, for all (x, y 1 , y 2 ) ∈ S λ × D (0, r) with

2x ∈ S λ |y 1 y 2 | < r1 r2 |2 a | < r 1 r 2 ,
we have

|ψ v,λ (x, y 1 , y 2 ) -y 1 y 2 | ≤ n≥1 x a r ′ 1 r ′ 2 2 a x a f 1 2x, y1y2 x a r1 + δ n y 1 f 1 x, y1y2 x a n ≤ r ′ 1 r ′ 2 |2 a | n≥1 y 1 r1 + δ n f 1 2x, y1y2 x a f 1 x, y1y2 x a n .
Since c (v) is the germ of an analytic function near the origin which is null at the origin, we can take r 1 ,r 2 > 0 small enough in order that for all closed sub-sector S ′ ⊂ S λ , for all r1 ∈ ]0, r 1 [ and r2 ∈ ]0, r 2 [, there exist A, B > 0 satisfying:

(x, y 1 , y 2 ) ∈ S ′ × D (0, r) =⇒ |ψ v,λ (x, y 1 , y 2 ) -y 1 y 2 | A exp - B |x| .
Let us prove this. We need here to estimate the quantity:

f 1 2x, y1y2 x a f 1 x, y1y2 x a = 2 a1 exp - λ 2x -c m (y 1 y 2 ) m x log (2) - c (y 1 y 2 2 a ) 2x + c (y 1 y 2 )
x .

On only have tot deal with the case where x ∈ S ′ is such that 2x ∈ S ′ (otherwise, x is "far from the origin", and we conclude without difficulty). We have:

(x, y 1 , y 2 ) ∈ S ′ × D (0, r) et 2x ∈ S =⇒ f 1 2x, y1y2 x a f 1 x, y1y2 x a ≤ |2 a1 | exp - B |x| < 1 . Hence |ψ v,λ (x, y 1 , y 2 ) -y 1 y 2 | ≤ r ′ 1 r ′ 2 |2 a | n≥1 2 a1 y 1 r1 + δ exp - B |x| n ≤ r ′ 1 r ′ 2 |2 a | 2 a 1 y1 r1+δ exp -B |x| 1 -2 a 1 y1 r1+δ exp -B |x| ≤ A exp - B |x| ,
for a convenient A > 0.

The latter lemma implies Ψ v,±λ,0 (w) = w, having for consequence the next result.

Corollary 4.12. For all closed sub-sector S ′ ⊂ S ±λ and for all r1 ∈ ]0, r 1 [ and r2 ∈ ]0, r 2 [, there exists A, B > 0 such that for all x ∈ S ′ :

|h 1 | ≤ r1 |f 1 (x, h 1 h 2 )| |h 2 | ≤ r2 |f 2 (x, h 1 h 2 )|      =⇒ |Ψ w,± (x, h 1 , h 2 ) -h 1 h 1 | ≤ A exp -B |x| |x a | .
In particular, there exists C > 0 such that:

|h 1 | ≤ r1 |f 1 (x, h 1 h 2 )| |h 2 | ≤ r2 |f 2 (x, h 1 h 2 )|      =⇒ exp c m (h 1 h 2 ) m log (x) + c(x a (h1h2) m )
x exp c m (Ψ w (x, h 1 , h 2 )) m log (x) + c(x a (Ψw (x,h1,h2)) m )

x < C .

4.5.

Power series expansion of sectorial isotropies in the space of leaves. Now, we give a power series expansion of Ψ 1,±λ and Ψ 2,±λ in the space of leaves. Let us introduce the following notations:

N (1, +) := N (2, -) := 1 N (1, -) := N (2, +) := -1 .
Lemma 4.13. With the notations and assumptions above, there exists entire functions ( i.e. analytic over C) denoted by Ψ j,±λ,n , j ∈ {1, 2}, n ≥ N (j, ±), such that for j ∈ {1, 2} :

       Ψ j,λ (h 1 , h 2 ) = n≥N (j,+) Ψ j,λ,n (h 1 h 2 ) h n 1 Ψ j,-λ (h 1 , h 2 ) = n≥N (j,-) Ψ j,λ,n (h 1 h 2 ) h n 2 .
These series converge normally in every subset of Γ ±λ of the form Γ ±λ (r) with 0 < r1 < r 1 and 0 < r2 < r 2 ( cf. Definition 4.8). More precisely, for all r1 , r2 , δ > 0 such that 0 < rj + δ < r j , j ∈ {1, 2} there exists C > 0 such that for all x ∈ S ±λ and for all w ∈ C, we have:

|wx a | ≤ r1 r2 =⇒                          |Ψ 1,λ,n (w)| < Cr ′ 1 |f 1 (x, w)| n-1 (r 1 + δ) n , n ≥ 1 |Ψ 2,λ,n (w)| < Cr ′ 2 |x a | |f 1 (x, w)| n+1 (r 1 + δ) n , n ≥ -1 |Ψ 1,-λ,n (w)| < Cr ′ 1 |x a | |f 2 (x, w)| n+1 (r 2 + δ) n , n ≥ -1 |Ψ 2,-λ,n (w)| < Cr ′ 2 |f 2 (x, w)| n-1 (r 2 + δ) n , n ≥ 1 .
Moreover: Ψ 1,-λ,-1 (0) = Ψ 2,λ,-1 (0) = 0 .

Proof. We use the same notations as in the proof of Lemma 4.10, and as usual, we give the proof only for Ψ λ (the proof for Ψ -λ is analogous, by exchanging the role played by h 1 and h 2 ). For fixed w ∈ C, the maps

ϕ 1 : h 1 → Ψ 1,λ h 1 , w h 1 and ϕ 2 : h 1 → Ψ 2,λ h 1 , w h 1 are analytic in M w,1 = x∈S λ |wx a |<r1r2 Ω x,w
(see the proof of Lemma 4.10). In particular, ϕ 1 and ϕ 2 admit Laurent expansions

         ϕ 1 (h 1 ) = Ψ 1,λ h 1 , w h 1 = n≥-L1 Ψ 1,λ,n (w) h n 1 ϕ 2 (h 1 ) = Ψ 2,λ h 1 , w h 1 = n≥-L2
Ψ 2,λ,n (w) h n 1 in every annulus Ω x,w , with x ∈ S λ such that |wx a | < r 1 r 2 . Using the same method as in the proof of Lemma 4.10, we prove without additional difficulties that for all n ∈ Z, Ψ 1,λ,n and Ψ 2,λ,n are analytic in any point w ∈ C, and thus are entire functions (i.e. analytic over C). Moreover, we also show in the same way as earlier that for all r1 , r2 , δ > 0 with 0 < rj + δ < r j , j ∈ {1, 2} , for all n ∈ Z and for all (x, w) ∈ S λ × C such that |wx a | ≤ r1 r2 , we have:

           |Ψ 1,λ,n (w)| < r ′ 1 f 1 x, Ψ w,λ x, h 1 , w h1 f 1 (x, w) r1 + δ n |Ψ 2,λ,n (w)| < r ′ 2 f 2 x, Ψ w,λ x, h 1 , w h1 f 1 (x, w) r1 + δ n .
According to Corollary 4.12, there exists C > 0 such that for all (x, w) ∈ S λ × C with |wx a | ≤ r1 r2 , we have:

       |Ψ 1,λ,n (w)| < Cr ′ 1 |f 1 (x, w)| n-1 (r 1 + δ) n |Ψ 2,λ,n (w)| < Cr ′ 2 |x a | |f 1 (x, w)| n+1 (r 1 + δ) n .
According to the statement in Fact 4.7, for a fixed value w ∈ C, if we look at the limit as x tends to 0 in S λ of the right hand-sides above we deduce that:

|Ψ 1,λ,n (w)| = 0 , ∀n ≤ 0 |Ψ 2,λ,n (w)| = 0 , ∀n ≤ -2 . Consequently:        Ψ 1,λ (h 1 , h 2 ) = n≥1 Ψ 1,λ,n (h 1 h 2 ) h n 1 Ψ 2,λ (h 1 , h 2 ) = n≥-1 Ψ 2,λ,n (h 1 h 2 ) h n 1 .
These function series converges normally (and are analytic) in every domain of the formΓ λ (r) with r := (r 1 , r2 ) and 0 < rj + δ < r j , j ∈ {1, 2}

(cf. Definition 4.8). Moreover, for any fixed value of h 2 , on the one hand the function series

h 1 → Ψ 2,λ (h 1 , h 2 ) = n≥-1 Ψ 2,λ,n (h 1 h 2 ) h n 1 is analytic in a punctured disc, since |f 2 (x, h 1 , h 2 )| -→ x→0 x∈S λ 0 ,
and on the other hand, we already know that the function h 1 → Ψ 2,λ (h 1 , h 2 ) is analytic in a neighborhood of the origin. Thus, we must have Ψ 2,λ,-1 (0) = 0.

4.6. Sectorial isotropies: proof of Proposition 3.5.

The following lemma is a more precise version of Proposition 3.5. We recall the notations:

N (1, +) = N (2, -) = 1 N (1, -) = N (2, +) = -1 .
Lemma 4.14. With the notations and assumptions above, we consider ψ ±λ ∈ Λ (weak) ±λ (Y norm ), with ψ ±λ (x, y) = (x, ψ 1,±λ (x, y) , ψ 2,±λ (x, y)) .

Then, for i ∈ {1, 2}, ψ i,λ and ψ i,-λ can be written as power series as follows:

             ψ i,λ (x, y) = y i + f i x, ψ v,λ (x, y) x a n≥N (i,+)+1 Ψ i,λ,n y 1 y 2 x a y 1 f 1 x, y1y2 x a n ψ i,-λ (x, y) = y i + f i x, ψ v,-λ (x, y) x a n≥N (i,-)+1 Ψ i,-λ,n y 1 y 2 x a y 2 f 2 x, y1y2 x a n .
which are normally convergent in every subset of S ±λ × D (0, r) of the form S ±λ × D (0, r), where D (0, r) is a closed poly-disc with r = (r 1 , r2 ) such that 0 < rj < r j , j ∈ {1, 2} .

Here Ψ i,λ,n , Ψ i,-λ,n (for i = 1, 2 and n ∈ N) are given in Lemma 4.13. Moreover, for all closed sub-sector S ′ ⊂ S ±λ and for all closed poly-disc D ⊂ D (0, r), there exists A, B > 0 such that for j = 1, 2:

|ψ j,±λ (x, y 1 , y 2 ) -y j | ≤ A exp - B |x| , ∀ (x, y) ∈ S ′ × D .
As a consequence, ψ j,±λ admits y j as Gevrey-1 asymptotic expansion in S ±λ × D (0, r).

Remark 4.15. In particular, Ψ 1,λ,1 (w) = Ψ 2,-λ,1 (w) = 1 and Ψ 1,-λ,-1 (w) = Ψ 2,λ,-1 (w) = w.

Proof. By definition, we have

Ψ ±λ • H ±λ = H ±λ • ψ ± .
In particular, for j = 1, 2 and all (x, y) ∈ S ±λ × D (0, r):

Ψ j,±λ x, y 1 f 1 x, y1y2 x a , y 2 f 2 x, y1y2 x a = ψ j,±λ (x, y 1 , y 2 ) f j x, ψv,±(x,y1 ,y2) x a
. Thus, according to Lemma 4.13 we have for i = 1, 2:

(4.6)              ψ i,λ (x, y) = f i x, ψ v,λ (x, y) x a n≥N (i,+) Ψ i,λ,n y 1 y 2 x a y 1 f 1 x, y1y2 x a n ψ i,-(x, y) = f i x, ψ v,-λ (x, y) x a n≥N (i,-) Ψ i,-λ,n y 1 y 2 x a y 2 f 2 x, y1y2
x a n , and these series are normally convergent (and then define analytic functions) in any domain of the form S ′ × D (0, r), where S ′ is a closed sub-sector of S ±λ and D (0, r) is a closed poly-disc with r = (r 1 , r2 ) such that 0 < rj < r j , j ∈ {1, 2} . Let us compare the different expressions of ψ j,±λ , j = 1, 2. We know that ψ j,±λ (x, y 1 , y 2 ) admits y j as weak Gevrey-1 asymptotic expansion in S ±λ × D (0, r). Thus, we can write:

ψ j,±λ (x, y 1 , y 2 ) = y j + k∈N 2 ψ j,±λ,k (x) y k ,
where for all k = (k 1 , k 2 ) ∈ N 2 , ψ j,±λ,k is analytic in S ±λ and admits 0 as Gevrey-1 asymptotic expansion. As usual, let us deal with the case of ψ 1,λ and ψ 2,λ (the other one being similar by exchanging y 1 and y 2 ).

According to the expressions given by Lemmas 4.10 and 4.13, we can be more precise on the index sets in the sums above:

(4.7)          ψ 1,λ (x, y 1 , y 2 ) = y 1 + k=(k1,k2)∈N 2 k1≥k2+1 ψ 1,λ,k (x) y k1 1 y k2 2 ψ 2,λ (x, y 1 , y 2 ) = y 2 + k=(k1,k2)∈N 2 k1≥k2 ψ 2,λ,k (x) y k1 1 y k2 2 .
Let us deal with ψ 1,λ (a similar proof holds for ψ 2,λ ). Looking at terms for n = 1 in (4.6) and at monomials terms y k such that k 1 ≤ k 2 + 1 in (4.7), we must have for all x ∈ S λ , y 1 , y 2 ∈ C with

|y 1 | < r 1 , |y 2 | < r 2 : 1 + k≥0 ψ 1,λ,(k+1,k) (x) y k 1 y k 2 = f 1 x, ψ v,λ (x,y) x a f 1 x, y1y2 x z Ψ 1,λ,1 y 1 y 2 x a .
According to Lemma 4.11 and Corollary4.12, we have:

f 1 x, ψ v,λ (x,y) x a f 1 x, y1y2 x z = 1 + j1≥j2+1≥1 F j1,j2 (x) y j1 1 y j2 2 = 1 + O (x,y)-→0 (x,y)∈S λ ×D(0,r) (|y 1 |) ,
for some analytic and bounded functions F j1,j2 (x), j 1 ≥ j 2 . As in the proof of Lemma 4.11, using the fact that ψ λ admits the identity as weak Gevrey-1 asymptotic expansion, we deduce that Ψ 1,λ,1 (w) = 1, and then:

ψ 1,λ (x, y) = y 1 + f 1 x, ψ v,λ (x, y) x a n≥2 Ψ 1,λ,n y 1 y 2 x a y 1 f 1 x, y1y2 x a n = y 1 + k=(k1,k2)∈N 2 k1≥k2+2 ψ 1,λ,k (x) y k1 1 y k2 2 .
It remains to show that ψ 1,λ admits y 1 as Gevrey-1 asymptotic expansion in S λ × D (0, r). From the computations above, we deduce:

|ψ 1,λ (x, y 1 , y 2 ) -y 1 | ≤ n≥2 Ψ 1,λ,n y 1 y 2 x a y 1 f 1 x, y1y2 x a n-1 f 1 x, ψ v,λ (x,y) x a f 1 x, y1y2
x a y 1 .

Using Lemma 4.13, Corollary 4.12 and the same method as at the end of the proof of Lemma 4.11, we can show the following: we can take r 1 , r 2 > 0 small enough such that for all closed sub-sector S ′ of S λ for all r1 ∈ ]0, r 1 [ and r2 ∈ ]0, r 2 [, there exists A, B > 0 satisfying:

(x, y 1 , y 2 ) ∈ S ′ × D (0, r) =⇒ |ψ 1,λ (x, y 1 , y 2 ) -y 1 | ≤ A exp - B |x| .
A similar proof holds for ψ 2,λ , ψ 2,-λ and ψ 1,-λ .

Remark 4.16. It should be noticed that in the expressions

           ψ 1,λ (x, y) = y 1 + f 1 x, ψ v,λ (x, y) x a n≥2 Ψ 1,λ,n y 1 y 2 x a y 1 f 1 x, y1y2 x a n ψ 1,-λ (x, y) = y 1 + f 1 x, ψ v,-λ (x, y) x a n≥0 Ψ 1,-λ,n y 1 y 2 x a y 2 f 2 x, y1y2
x a n given by Lemma 4.14, the expansion of ψ 1,λ with respect to y = (y 1 , y 2 ) starts with a term of order 1, namely y 1 , followed by terms of order at least 2, while in the expansion of ψ 1,-λ , the term of lowest order is a constant, namely Ψ 1,-λ,0 (0). Similarly, the expansion of ψ 2,-λ (with respect to y = (y 1 , y 2 )) starts with y 2 , while the expansion of ψ 1,-λ starts with the constant Ψ 2,λ,0 (0).

Description of the moduli space and some applications

From Lemmas 4.13 and 4.14, we can give a description of the moduli space Λ λ (Y norm )× Λ -λ (Y norm ) of a fixed analytic normal form Y norm .

A power series presentation of the moduli space.

We use the notations introduced in section 4. We denote by O (C) the set of entire functions, i.e. of functions holomorphic in C. We consider the functions f 1 and f 2 defined in (4.3) and introduce four subsets of (O (C))

N , denoted by E 1,λ (Y norm ), E 2,λ (Y norm ), E 1,-λ (Y norm ) and E 2,-λ (Y norm ), defined as follows. On remind the notations

N (1, +) = N (2, -) = 1 N (1, -) = N (2, +) = -1 . Definition 5.1. For j ∈ {1, 2}, a sequence (ψ n (w)) n≥N (j,±)+1 ∈ (O (C)) N belongs to E j,±λ (Y norm ) if
there exists an open polydisc D (0, r) and an open asymptotic sector

S ±λ ∈ AS arg(±λ),2π
such that for all r1 , r2 , δ > 0 with

0 < ri + δ < r i , i ∈ {1, 2}
there exists C > 0 such that for all x ∈ S λ (resp. x ∈ S -λ ) and for all w ∈ C: As explained in section 4, we can associate to any pair

|wx a | ≤ r1 r2 =⇒                          |ψ n (w)| < C |f 1 (x,
(ψ λ , ψ -λ ) ∈ Λ λ (Y norm ) × Λ -λ (Y norm )
two germs of sectorial biholomorphisms of the space of leaves corresponding to each "narrow" sector, which we denote by Ψ λ and Ψ -λ , defined by:

(5.1) Ψ ±λ := H ±λ • ψ ±λ • H -1 ±λ , where H ±λ is given by Corollary 4.3. According to Lemmas 4.13 and 4.14, if we write Ψ ±λ = (x, Ψ 1,±λ , Ψ 2,±λ ), then for j = 1, 2 we have: Ψ j,λ (h 1 , h 2 ) = h j + n≥N (j,+)+1 Ψ j,λ,n (h 1 h 2 ) h n 1 (5.2) Ψ j,-λ (h 1 , h 2 ) = h j + n≥N (j,-)+1 Ψ j,-λ,n (h 1 h 2 ) h n 2 (Ψ j,±λ,n ) n ∈ E j,±λ . Conversely, given (Ψ j,±λ ) n ∈ E j,±λ for j = 1, 2, the estimates made in section 4 show that ψ ±λ := H -1 ±λ • Ψ ±λ • H ±λ , where Ψ ±λ (x, h) = (x, Ψ 1,±λ (h) , Ψ 2,±λ (h)), belongs to Λ ±λ (Y norm ). Consequently, we can state: Proposition 5.2. We have the following bijections:

Λ λ (Y norm ) → E 1,λ (Y norm ) × E 2,λ (Y norm ) ψ λ → (Ψ 1,λ , Ψ 2,λ )
and

Λ -λ (Y norm ) → E 1,-λ (Y norm ) × E 2,-λ (Y norm ) ψ -λ → (Ψ 1,-λ , Ψ 2,-λ )
we identify here Ψ ±λ (x, h) = (x, Ψ 1,±λ (h) , Ψ 2,±λ (h)) with (Ψ 1,±λ (h) , Ψ 2,±λ (h)) .

Analytic invariant varieties and two-dimensional saddle-nodes.

We can give a necessary and sufficient condition for the existence of analytic invariant varieties in terms of the moduli space described above.

We recall that for any vector field Y ∈ [Y norm ] as in (1.1) (cf. Definition 1.13), there always exist three formal invariant varieties: C = {(y 1 , y 2 ) = (g 1 (x) , g 2 (x))}, H 1 = {y 1 = f 1 (x, y 2 )} and H 2 = {y 2 = f 2 (x, y 1 )}, where g 1 , g 2 , f 1 , f 2 are formal power series with null constant term. The first one is classically called the center variety, and we have C = H 1 ∩ H 2 . If Y = Y norm , then:

     C = {y 1 = y 2 = 0} H 1 = {y 1 = 0} H 2 = {y 2 = 0} .
Proposition 5.3. Let Y ∈ [Y norm ] and (Φ λ , Φ -λ ) ∈ Λ λ (Y norm ) × Λ -λ (Y norm ) be its Stokes diffeomorphisms. We consider Ψ ± = H ±λ • Φ ±λ • H -1 ±λ as above. Then: (1) the center variety C is convergent (analytic in the origin) if and only if Ψ 2,λ,0 (0) = Ψ 1,-λ,0 (0) = 0;

(2) the invariant hypersurface H 1 is convergent (analytic in the origin) if and only if for all n ≥ 0, we have Ψ 1,-λ,n (0) = 0; (3) the invariant hypersurface H 2 is convergent (analytic in the origin) if and only if for all n ≥ 0, we have Ψ 2,λ,n (0) = 0.

Proof. It is a direct consequence of the power series representation (5.2) of the Stokes diffeomorphisms (Φ λ , Φ -λ ). Let us explain item 2. (the same arguments hold for 1. and 3. with minor adaptation).

The fact that Ψ 1,-λ,n (0) = 0 for all n ≥ 0 means that Ψ 1,-λ is divisible by h 1 . Equivalently, both Φ 1,λ and Φ 1,-λ are divisible by y 1 , so that the analytic hypersurface {y 1 = 0} has the same pre-image by the sectorial normalizing maps Φ + and Φ -. These pre-images glue together in order to define an analytic invariant hypersurface H 1 .

Notice that if we consider the restriction of a formal normal form Y norm to one of the formal invariant hypersurfaces, we obtain precisely the normal form for two-dimensional saddle-nodes as given in [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF]. When one of these hypersurfaces is convergent (i.e. analytic), we recover the Martinet-Ramis invariants by restriction to this hypersurface, as we present below.

Proposition 5.4. Suppose that the formal invariant hypersurface H 1 is convergent ( i.e. analytic in the origin). Then, the Martinet-Ramis invariants for the saddle-node Y |H1 are given by:

   Ψ 2,λ (0, h 2 ) = h 2 + Ψ 2,λ,0 (0) ∈ Af f (C) Ψ 2,-λ (0, h 2 ) = h 2 + n≥2 Ψ 2,-λ,n (0) h n 2 ∈ Diff (C, 0).
Similar result holds for the hypersurface H 2 . given in (5.1) is a bijection we identify here Ψ ±λ (x, h) = (x, Ψ 1,±λ (h) , Ψ 2,±λ (h)) with (Ψ 1,±λ (h) , Ψ 2,±λ (h)) . An easy computation based on (4.1) gives:

H -1 ±λ * dy 1 ∧ dy 2 x = dh 1 ∧ dh 2 + dx .
This means in particular that ψ ±λ is transversally symplectic with respect to ω = dy 1 ∧ dy 2 x , i.e.

(ψ ±λ ) * (ω) = ω + dx , if and only if Ψ ±λ = (Ψ 1,±λ , Ψ 2,±λ ) preserves the standard symplectic form dh 1 ∧ dh 2 in the space of leaves, i.e. det (DΨ ±λ ) = 1. In other words:

Proposition 5.5. We have the following bijections:

Λ ω λ (Y norm ) → (E 1,λ (Y norm ) × E 2,λ (Y norm )) ω ψ λ → (Ψ 1,λ , Ψ 2,λ )
and

Λ ω -λ (Y norm ) → (E 1,-λ (Y norm ) × E 2,-λ (Y norm )) ω ψ -λ → (Ψ 1,-λ , Ψ 2,-λ )
we identify here Ψ ±λ (x, h) = (x, Ψ 1,±λ (h) , Ψ 2,±λ (h)) with (Ψ 1,±λ (h) , Ψ 2,±λ (h)) .

5.4. Quasi-linear Stokes phenomena in the first Painlevé equation.

In [START_REF] Bittmann | Analytic classification of germs of three-dimensional doubly-resonant vector fields and applications to Painlevé equations[END_REF], we link the study of quasi-linear Stokes phenomena (see [START_REF] Kapaev | Quasi-linear stokes phenomenon for the Painlevé first equation[END_REF] for the first Painlevé equation) to our Stokes diffeomorphisms. For instance, in the case of the first Painlevé equation, we show that the quasi-linear Stokes phenomena formula found by Kapaev in [START_REF] Kapaev | Quasi-linear stokes phenomenon for the Painlevé first equation[END_REF] allows to compute the terms Ψ 2,λ,0 (0) and Ψ 1,-λ,0 (0) in (5.2). More precisely, elementary computations (using Kapaev's connection formula) give: Moreover, our description of the Stokes diffeomorphisms implies a more precise estimate of the order of the remaining terms in Kapaev's formula. In a forthcoming paper, we will use the study of some non-linear Stokes phenomena for the second Painlevé equations (see e.g. [START_REF] Clarkson | A connection formula for the second Painlevé transcendent[END_REF]) in order to compute coefficients of the Ψ i,±λ 's.

  y 1 , y 2 ) ∂ ∂y 2 with b, b 1 , b 2 ∈ C {x, y 1 , y 2 } vanishing at the origin. • Diff C 3 , 0 is the group of germs of holomorphic diffeomorphisms fixing the origin of C 3 . It acts on χ C 3 , 0 by conjugacy: for all (Φ, Y ) ∈ Diff C 3 , 0 × χ C 3 , 0 we define the push-forward of Y by Φ by (1.2) Φ * (Y ) := (DΦ • Y ) • Φ -1 ,

  0 and c ∈ vC {v} vanishing at the origin. Definition 1.13. We denote by [Y norm ] the set of germs of holomorphic doubly-resonant saddle-nodes in C 3 , 0 which are formally conjugate to Y norm by formal fibered diffeomorphisms tangent to the identity, and denote by [Y norm ] Diff fib C 3 , 0, Id the set of orbits of the elements in this set under the action of Diff fib C 3 , 0, Id .

  Gevrey-1 power series and 1-summability 10 2.3. An important result byMartinet and Ramis 11 2.4. 1-summability implies weakly 1-summability 13 3. Proofs of the main theorems 14 3.1. Analytic invariants: Stokes diffeomorphisms 14 3.2. Proof of Theorem 1.12: 1-summability of the formal normalization 16 3.3. Proofs of Theorems 1.15 and 1.23 17 4. Sectorial isotropies and space of leaves: proof of Proposition 3.5 18 4.1. Sectorial first integrals and the space of leaves 19 4.2. Sectorial isotropies in the space of leaves 20 4.3. Action on the resonant monomial in the space of leaves 22 4.4. Action on the resonant monomial 24 4.5. Power series expansion of sectorial isotropies in the space of leaves 27 4.6. Sectorial isotropies: proof of Proposition 3.5 28 5. Description of the moduli space and some applications 30 5.1. A power series presentation of the moduli space 31 5.2. Analytic invariant varieties and two-dimensional saddle-nodes 32 5.3. The transversally symplectic case and quasi-linear Stokes phenomena in the first Painlevé equation 32 5.4. Quasi-linear Stokes phenomena in the first Painlevé equation 33 References 33

  2.4. 1-summability implies weakly 1-summability.Any function f (x, y) analytic in a domain U × D (0, r), where U ⊂ C is open, and bounded in any domain

F

  j (x) y j defines a germ of a sectorial holomorphic function in a domain of the form S r, θ -π 2 ǫ, θ + π 2 + ǫ × D (0, r) .

Remark 3. 4 .

 4 This very result means that the Stokes diffeomorphisms encode completely the class of Y in the quotient [Y norm ] Diff fib C 3 , 0, Id as they separate conjugacy classes. Proof. First of all, let us prove that the latter map is well-defined. Let Y, Ỹ ∈ [Y norm ] and Θ ∈ Diff fib C 3 , 0, Id be such that Θ * (Y ) = Ỹ . We denote by Φ ± (resp. Φ± ) the sectorial normalizing maps of Y (resp. Ỹ ), and (Φ λ , Φ -λ ) resp. Φλ , Φ-λ the Stokes diffeomorphisms of Y (resp. Ỹ ).

Figure 4 . 1 .

 41 Figure 4.1. Representation of the space of leaves in terms of |h 1 | and |h 2 | when c = 0: in this case, it is a Reinhardt domain (cf. [Hor73]).

5. 3 .

 3 The transversally symplectic case and quasi-linear Stokes phenomena in the first Painlevé equation.Let us now focus on the transversally symplectic case studied in Theorem 1.23. Let Y norm ∈ SN diag,0 be transversally symplectic (i.e. its residue is res (Y norm ) = 1). Using the notations introduced in paragraph 5.1, we define the following sets:(E 1,λ (Y norm ) × E 2,λ (Y norm )) ω := Ψ λ = (Ψ 1,λ , Ψ 2,λ ) ∈ E 1,λ (Y norm ) × E 2,λ (Y norm ) such that: det (DΨ λ ) = 1 (E 1,-λ (Y norm ) × E 2,-λ (Y norm )) ω := Ψ -λ = (Ψ 1,-λ , Ψ 2,-λ ) ∈ E 1,-λ (Y norm ) × E 2,-λ (Y norm ) such that: det (DΨ -λ ) = 1 .According to Proposition 5.2, the mapΛ ±λ (Y norm ) -→ E 1,±λ (Y norm ) × E 2,±λ (Y norm ) ψ ±λ → Ψ ±λ := H ±λ • ψ ±λ • H -1 ±λ

Ψ

  2,λ,0 (0) = iΨ 1,-λ,0 (0) = e

  0) Diff fib S arg(-λ),ǫ ; Id , ∀ǫ ∈ [0, π[ . Notice that Φ λ , Φ -λ are isotropies of Y norm , i.e. they satisfy:(Φ ±λ ) * (Y norm ) = Y norm .With the above notations, we define Λ λ (Y norm ) (resp. Λ -λ (Y norm )) as the group of germs of sectorial fibered isotropies of Y norm , tangent to the identity, and admitting the identity as Gevrey-1 asymptotic expansion (see Definition 2.4) in sectorial domains of the formS λ × C 2 , 0 resp. S -λ × C 2 ,0 , with S ±λ ∈ AS arg(±λ),π . The two sectorial isotropies Φ λ and Φ -λ defined above are called the Stokes diffeomorphisms associate to Y ∈ [Y norm ].

	Definition 1.14.

This condition implies in particular that Φ (S) ∈ S θ,η .


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Identification of developmental zones in maize lateral roots on the basis of longitudinal epidermis cell length profiles

Yann Guédon, Beatriz Moreno-Ortega, Bertrand Muller

To cite this version:

Yann Guédon, Beatriz Moreno-Ortega, Bertrand Muller. Identification of developmental zones in maize lateral roots on the basis of longitudinal epidermis cell length profiles.

Identification of developmental zones in maize lateral roots on the basis of longitudinal epidermis cell length profiles

Yann Guédon, Beatriz Moreno Ortega, Bertrand Muller yann.guedon@cirad.fr

Research focus: Cell division and elongation are key processes in root development. A growing root tip is assumed to be made of three contiguous zones: the division zone (DZ), the elongation zone (EZ) and the mature zone (MZ). Because of the concentric organization of tissues in roots, we here focus on epidermis cell length profiles for delimiting these three developmental zones. We assume that DZ is characterized by small cells, EZ by cells of gradually increasing length and MZ by rather large cells. These zones were identified on 42 maize lateral roots belonging to three genotypes and, for each genotype, to a rather large diversity in terms of development.

Methods: We built two types of Gaussian multiple change-point models for each root: heteroscedastic piecewise Gaussian linear models built on the basis of the measured epidermis cell length profiles and Gaussian change in the variance model built on the basis of the residual series deduced from the optimal piecewise linear function. The number of segments was selected using the slope heuristic, a nonasymptotic penalized likelihood criterion. The outputs of the analysis of each root using multiple changepoint models (e.g. lengths of DZ and EZ, cell lengths predicted at the limits between DZ and EZ and between EZ and MZ) combined with some morphological features (first root hair position and root diameter in MZ) were then globally analyzed using principal component analysis (PCA).

Results: We identified at most three zones corresponding to DZ, EZ and MZ except in arrested roots where DZ and in some cases EZ were missing. In our modeling framework, the limit between two successive zones corresponds to a marked change in slope and in residual standard deviation. In more than 90% of cases, the limits founds on the residual series on the basis of the sole change in residual standard deviation co-localized with the limit found on the original series. In most cases, the limit between EZ and MZ was consistent with the first hair position. In PCA, the first axis accounting for 49% of variance corresponds to the longitudinal variables (mainly DZ and EZ lengths but also first root hair position) while the second axis accounting for 29% of variance corresponds to the cell length variables.

Conclusions: We here provide the first pipeline of methods combining image analysis and statistical modeling for identifying developmental zones in root tips. This pipeline enabled in particular to identify emerging properties such as a jump in residual standard deviation between DZ and EZ suggesting that the distal boundary of the meristem corresponds to the onset of rapid cell expansion and a jump in residual standard deviation of high amplitude between EZ and MZ co-localized with a change in slope that could be due to the differentiation of trichoblast and atrichoblast cell files.