Amaury Bittmann
email: bittmann@math.unistra.fr

Doubly-resonant saddle-nodes in C 3 and the fixed singularity at infinity in the Painlevé equations: formal classification

Keywords: Painlevé equations, singular vector fields, irregular singularity, resonant singularity, normal form

In this work we consider formal singular vector fields in C 3 with an isolated and doubly-resonant singularity of saddle-node type at the origin. Such vector fields come from irregular two-dimensional systems with two opposite non-zero eigenvalues, and appear for instance when studying the irregular singularity at infinity in Painlevé equations (Pj ), j ∈ {I, II, III, IV, V }, for generic values of the parameters. Under generic assumptions we give a complete formal classification for the action of formal diffeomorphisms (by changes of coordinates) fixing the origin and fibered in the independent variable x. We also identify all formal isotropies (self-conjugacies) of the normal forms. In the particular case where the flow preserves a transverse symplectic structure, e.g. for Painlevé equations, we prove that the normalizing map can be chosen to preserve the transverse symplectic form.

Introduction

Definition and main result

We consider 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 formal power series of order at least two. They represent singular irregular 2-dimensional systems having two opposite non-zero eigenvalues and a vanishing third eigenvalue.

Our main motivation is the study of the irregular singularity at infinity in Painlevé equations (P j ), j ∈ {I, II, III, IV, V }, for generic values of the parameters [START_REF] Yoshida | 2-parameter family of solutions for Painlevé equations (I)-(V) at an irregular singular point[END_REF]. These equations, discovered (mainly) by Paul Painlevé [START_REF] Painlevé | Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme[END_REF], share the property that the only movable singularities of their solutions are poles (the so-called Painlevé property); this is the complete list of all such equations up to changes of variables. They have been intensively studied since the important work of Okamoto [START_REF] Okamoto | Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé[END_REF]. The study of fixed singularities, and more particularly those at infinity, started to be investigated by Boutroux with his famous tritronquées solutions [START_REF] Painlevé | Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme[END_REF]. Recently, several authors provided more complete information about such singularities, studying the so-called quasi-linear Stokes phenomena and also giving connection formulas ( [START_REF] Joshi | On Boutroux's tritronquée solutions of the first Painlevé equation[END_REF], [START_REF] Kapaev | Connection formulae for the first Painlevé transcendent in the complex domain[END_REF] and [START_REF] Kapaev | Quasi-linear stokes phenomenon for the Painlevé first equation[END_REF]). However, to the best of our knowledge there are no general analytic classification for this kind of doubly-resonant saddle-nodes yet (using normal form theory).

More precisely, we would like to understand the action of germs of analytic diffeomorphisms on such vector fields by changes of coordinates. If one tries to do this, a first step would be to provide a formal classification, that is to study the action of formal changes of coordinates on these vector fields. This is the aim of this paper. Based on the usual strategy employed for the classification of resonant vector fields [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF] in dimension 2, we give in a forthcoming paper a complete analytic classification for a more specific class of vector fields, by studying the non-linear Stokes phenomena.

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

• C x is the C-algebra of formal power series in the (multi)variable x = (x 1 , . . . , x n ) with coefficients in C. We denote by m its unique maximal ideal: it is formed by formal series with null constant term. For any formal series f 1 , . . . , f m in C x , we denote by f 1 , . . . , f m the ideal generated by these elements.

• D (1) is the Lie algebra of formal vector fields at the origin of C 3 which are singular (i.e. vanish at the origin). Any formal vector field in D (1) can be written

Y = b ∂ ∂x + b 1 ∂ ∂y 1 + b 2 ∂ ∂y 2 with b, b 1 , b 2 ∈ m.
• Diff is the group of formal diffeomorphisms fixing the origin of C 3 . It acts on D (1) by conjugacy: if (Φ, Y ) ∈ Diff × D (1) ,

Φ * (Y ) := (DΦ • Y ) • Φ -1 , (1.2)
where DΦ is the Jacobian matrix of Φ.

• Diff fib is the subgroup of Diff of diffeomorphisms fibered in the x-coordinate, i.e. of the form (x, y) → (x, φ (x, y)).

Definition 1.1. A doubly-resonant saddle-node is a vector field Y ∈ D (1) which is Diff fib -conjugate 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 ∈ m 2 . We will denote by SN the set of all such formal vector fields.

By Taylor expansion up to order 1 with respect to y, given a vector field Y ∈ SN written as above we can consider the associated 2-dimensional differential system:

x 2 dy dx = α (x) + A (x) y (x) + f (x, y (x)) , (1.3)
where y = (y 1 , y 2 ), such that the following conditions hold:

• α (x) = α 1 (x) α 2 (x) , with α 1 , α 2 ∈ x 2 ⊂ C x • A (x) ∈ Mat 2,2 (C x ) with A (0) = Diag (-λ, λ), λ ∈ C * • f (x, y) = f 1 (x, y) f 2 (x, y) , with f 1 , f 2 ∈ y 1 , y 2 2 ⊂ C x, y .
Based on this expression, we state: We say that Y is non-degenerate if res (Y ) ∈ C\Q ≤0 , and we denote by SN nd ⊂ SN the subset of non-degenerate vector fields.

We will prove in subsection 3.1 that the residue of Y ∈ SN is invariant under the action of Diff fib by conjugacy. We can state now our first main result.

Theorem 1.3. Let Y ∈ SN nd be a non-degenerate doubly-resonant saddlenode. Then there exists a fibered diffeomorphism Φ ∈ Diff fib such that:

Φ * (Y ) = x 2 ∂ ∂x + (-λ + a 1 x + c 1 (v)) y 1 ∂ ∂y 1 + (λ + a 2 x + c 2 (v)) y 2 ∂ ∂y 2 , (1.4)
where we put v := y 1 y 2 . Here, c 1 , c 2 in v = vC v are formal power series with null constant term and a 1 , a 2 ∈ C are such that a 1 + a 2 = res (Y ).

Remark 1.4. We will see in Corollary 3.4 and Proposition 3.2 that Φ as above is essentially unique (that is, unique up to pre-composition by linear transforms).

Definition 1.5. The parameter space for SN nd is the set

P := p = (λ, a 1 , a 2 , c 1 , c 2 ) ∈ C * × C 2 \∆ × (vC v ) 2 where ∆ = (a 1 , a 2 ) ∈ C 2 | a 1 + a 2 ∈ Q ≤0
is the locus of degeneracy. A vector field in the form (1.4) will be called a normal form of SN nd with parameters (λ, a 1 , a 2 , c 1 , c 2 ) in P.

Let us consider the quotient space

P (C * × Z /2Z)
where the group (C * × Z /2Z) acts on P as follows. Given p = (λ,

a 1 , a 2 , c 1 , c 2 ) ∈ P, θ ∈ C * and ǫ ∈ Z /2Z we define θ • (λ, a 1 , a 2 , c 1 , c 2 ) = (λ, a 1 , a 2 , c 1 • ϕ θ , c 2 • ϕ θ ) ǫ • (λ, a 1 , a 2 , c 1 , c 2 ) = (λ, a 1 , a 2 , c 1 , c 2 ) , if ǫ = 0 (-λ, a 2 , a 1 , c 2 , c 1 ) , if ǫ = 1 ,
where ϕ θ is the homothecy v → θv. If two parameters p, p ′ ∈ P are in the same orbit for this action we write p ∼ p ′ . Our second main result shows the uniqueness of the normal forms up to this action.

Theorem 1.6. Suppose Z and Z ′ are two normal forms of SN nd with respective parameters p = (λ, a 1 , a 2 , c 1 , c 2 ) ∈ P and

p ′ = (λ ′ , a ′ 1 , a ′ 2 , c ′ 1 , c ′ 2 ) ∈ P.
Then Z and Z ′ are Diff fib -conjugate if and only p ∼ p ′ .

One can rephrase the above results in terms of group actions as follows.

Corollary 1.7. There exists a bijection

SN nd Diff fib ≃ P (C * × Z /2Z) ,
where Diff fib acts on SN nd by conjugacy.

Let us make some remarks. Remark 1.8.

1. The condition of non-degeneracy is necessary to obtain such normal forms.

For instance for any

a 1 , a 2 ∈ C such that a 1 + a 2 = -p q ∈ Q ≤0 , with (p, q) ∈ N × N * , the vector field Y = x 2 ∂ ∂x + -λ + a 1 x + x p+1 (y 1 y 2 ) q y 1 ∂ ∂y 1 + (λ + a 2 x) y 2 ∂ ∂y 2 ,
with residue res (Y ) = -p q is not Diff fib -conjugate to a normal form as in Theorem 1.3. Indeed, the resonant term x p+1 (y 1 y 2 ) q cannot be eliminated by the action of Diff fib .

2. Notice that the above two results are not immediate consequences of Poincaré-Dulac normal form theory. In fact, the usual Poincaré-Dulac normal form possibly contains several additional resonant terms of the form

x k (y 1 y 2 ) l k,l∈N
, and is far from being unique.

Painlevé equations and the transversally symplectic case

In [START_REF] Yoshida | 2-parameter family of solutions for Painlevé equations (I)-(V) at an irregular singular point[END_REF] Yoshida shows that a vector field in the class SN nd naturally appears after a suitable compactification (given by the so-called Boutroux coordinates [START_REF] Boutroux | Recherches sur les transcendantes de M. Painlevé et l'étude asymptotique des équations différentielles du second ordre[END_REF]) of the phase-space of Painlevé equations (P j ), j ∈ {I, II, III, IV, V } (for generic values of the parameters). In these cases the vector field presents an additional Hamiltonian structure that will interest us. Let us illustrate these computations in the case of the first Painlevé equation:

(P I ) d 2 z 1 dt 2 = 6z 2 1 + t .
As is well known since Okamoto [START_REF] Okamoto | Polynomial Hamiltonians associated with Painlevé equations[END_REF], (P I ) can be seen as a non-autonomous Hamiltonian system

∂z1 ∂t = -∂H ∂z2 ∂z2 ∂t = ∂H ∂z1
with Hamiltonian

H (t, z 1 , z 2 ) := 2z 3 1 + tz 1 - z 2 2 2 .
More precisely, if we consider the standard symplectic form ω st := dz 1 ∧ dz 2 and the vector field

Z := ∂ ∂t - ∂H ∂z 2 ∂ ∂z 1 + ∂H ∂z 1 ∂ ∂z 2
induced by (P I ), then the Lie derivative

L Z (ω st ) = ∂ 2 H ∂t∂z 1 dz 1 + ∂ 2 H ∂t∂z 2 dz 2 ∧ dt = dz 1 ∧ dt
belongs to the ideal dt generated by dt in the exterior algebra Ω * C 3 of differential forms in variables (t, z 1 , z 2 ). Equivalently, for any t 1 , t 2 ∈ C the flow of Z at time (t 2t 1 ) acts as a symplectomorphism between fibers {t = t 1 } and {t = t 2 }.

The weighted compactification given by the Boutroux coordinates [START_REF] Boutroux | Recherches sur les transcendantes de M. Painlevé et l'étude asymptotique des équations différentielles du second ordre[END_REF] (see also [START_REF] Chiba | The first, second and fourth Painlevé equations on weighted projective spaces[END_REF]) defines a chart near {t = ∞} as follows:

     z 2 = y 2 x -3 5 z 1 = y 1 x -2 5 t = x -4 5 .
In the coordinates (x, y 1 , y 2 ), the vector field Z is transformed, up to a translation

y 1 ← y 1 + ζ with ζ = i √ 6
, into the vector field

Z = - 5 4x 1 5 Y (1.5)
where

Y = x 2 ∂ ∂x + - 4 5 y 2 + 2 5 xy 1 + 2ζ 5 x ∂ ∂y 1 + - 24 5 y 2 1 - 48ζ 5 y 1 + 3 5 xy 2 ∂ ∂y 2 . (1.6)
We observe that Y is a non-degenerate doubly-resonant saddle-node Y as in Definitions 1.1 and 1.2 with residue res (Y ) = 1. Furthermore we have:

     dt = -4 5 5 4 5 x -9 5 dx dz 1 ∧ dz 2 = 1 x (dy 1 ∧ dy 2 ) + 1 5x 2 (2y 1 dy 2 -3y 2 dy 1 ) ∧ dx ∈ 1 x (dy 1 ∧ dy 2 ) + dx
, where dx denotes the ideal generated by dx. We finally obtain

   L Y dy 1 ∧ dy 2 x = 1 5x (3y 2 dy 1 -(2ζ + 2y 1 ) dy 2 ) ∧ dx L Y (dx) = 2xdx . Therefore, both xL Y dy 1 ∧ dy 2 x
and L Y (dx) are differential forms which lie in the ideal dx . This motivates the following definition.

Definition 1.9. Consider the rational 1-form

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

L Y (dx) ∈ dx and xL Y (ω) ∈ dx .
We say a formal diffeomorphism Φ ∈ Diff is transversally symplectic (with respect to ω and dx) if

Φ * (x) = x and xΦ * (ω) ∈ xω + dx .
(Here Φ * (ω) denotes the pull-back of ω by Φ.)

We denote respectively by D ω and Diff ω the sets of transversally Hamiltonian vector fields and transversally symplectic diffeomorphisms.

Remark 1.10.

• The flow of a transversally Hamiltonian X defines a map between fibers {x = x 1 } and {x = x 2 } which sends ω |x=x1 onto ω |x=x2 , since

(exp (X)) * (ω) ∈ ω + dx .
• By our definition, a transversally symplectic diffeomorphism Φ ∈ Diff ω is necessarily a fibered diffeomorphism. In other words: Diff ω ⊂ Diff fib .

Definition 1.11. A transversally Hamiltonian doubly-resonant saddlenode is a vector field Y ∈ D ω which is Diff ω -conjugate 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 ∈ m 2 . We will denote by SN ω the set of all such formal vector fields.

Notice that a transversally Hamiltonian doubly-resonant saddle-node Y ∈ SN ω is necessarily non-degenerate since its residue is always equal to 1. In other words: SN ω ⊂ SN nd . Theorem 1.12. Let Y ∈ SN ω be a transversally Hamiltonian doubly-resonant saddle-node. Then, there exists a transversally symplectic diffeomorphism Φ ∈ Diff ω such that:

Φ * (Y ) = x 2 ∂ ∂x + (-λ + a 1 x -c (v)) y 1 ∂ ∂y1 + (λ + a 2 x + c (v)) y 2 ∂ ∂y2 . (1.7)
where we put v := y 1 y 2 . Here, c (v) in vC v is a formal power series with null constant term and a 1 , a 2 ∈ C are such that a 1 + a 2 = res (Y ) = 1. Furthermore this normal form is unique with respect to the action of Diff ω .

One can rephrase the theorem above in terms of group action.

Corollary 1.13. There exists a bijection

SN ω Diff ω ≃ C * × (a 1 , a 2 ) ∈ C 2 | a 1 + a 2 = 1 × vC v .
Remark 1.14.

1. As for Theorem 1.3, Φ is essentially unique (Corollary 3.4). This is an immediate consequence of Theorem 1.6. However, the fact that the normalizing diffeomorphism Φ in Theorem 1.12 is transversally symplectic is not an immediate consequence of Theorem 1.6.

2. The above normalization theorem can be interpreted as defining local action-angle coordinates for vector fields in SN ω . More precisely, if we consider the successive symplectic changes of coordinates   

y 1 = e i π 4 √ 2 (u 1 + iu 2 ) y 2 = e i π 4 √ 2 (u 1 -iu 2 )
and

u 1 = √ 2ρ cos ϕ u 2 = √ 2ρ sin ϕ ,
then the vector field (1.7) becomes:

x 2 ∂ ∂x + e -i π 4 x √ ρ ∂ ∂ρ + i λ + c (iρ) + (a 2 -a 1 ) 2 x ∂ ∂ϕ .
Notice that the corresponding differential equation can be explicitly integrated by quadratures in terms of an anti-derivative of c.

We will explain in section 4 how to compute inductively any finite jet of c (v) in the case of the Painlevé equations (for which c (v) is a germ of an analytic function at the origin).

Corollary 1.15. Let Y be as in (1.6). Then

a 1 = a 2 = 1 2 , λ = 8 √ 3ζ 5 = 4•2 3 4 •3

Known results

In [START_REF] Yoshida | 2-parameter family of solutions for Painlevé equations (I)-(V) at an irregular singular point[END_REF], [START_REF] Yoshida | A general solution of a nonlinear 2-system without Poincaré's condition at an irregular singular point[END_REF] Yoshida shows that the doubly-resonant saddle-nodes arising from the compactification of Painlevé equations (P j ), j ∈ {I, II, III, IV, V } (for generic values for the parameters) is conjugate to polynomial vector fields of the form

Z = x 2 ∂ ∂x + -(1 + γy 1 y 2 ) + a 1 x y 1 ∂ ∂y 1 + 1 + γy 1 y 2 + a 2 x y 2 ∂ ∂y 2 , (1.8)
with γ ∈ C * and (a 1 , a 2 ) ∈ C 2 such that a 1 +a 2 = 1. One drawback of this result is that Yoshida admits fibered transformations Ψ (x, y) = (x, ψ 1 (x, y) , ψ 2 (x, y))

of a more general form:

ψ i (x, y) = y i     1 + (k0,k1,k2)∈N 3 k1+k2≥1 q i,k (x) x k0 y k1+k0 1 y k1+k0 2     , (1.9)
where each q i,k is a formal power series. Notice that x can appear with negative exponents and therefore Ψ / ∈ Diff. As we will see in the next subsection, the problem (when seen from the viewpoint of analytic classification) is that the transformations used by Yoshida have "small" regions of convergence, in the sense that one cannot cover an entire neighborhood of the origin in C 3 by taking the union of these regions. On the contrary, we prove in an upcoming work that the formal normalizations presented here can be embodied by diffeomorphisms analytic on finitely many sectors whose union is a neighborhood of the origin. This entails the classical theory of summability of formal power series.

Analytic results

Several authors studied the problem of convergence of the conjugating transformations described above. Some results (that we recall soon) will hold not only in the class of formal objects, but also for Gevrey (and even summable) ones, or more generally for holomorphic functions with asymptotic expansions in sectorial domains. We refer to [START_REF] Malgrange | Sommation des séries divergentes[END_REF] and [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF] for details on asymptotic expansions, Gevrey and summability theory.

Assuming that the initial vector field is analytic, Yoshida proves in [START_REF] Yoshida | A general solution of a nonlinear 2-system without Poincaré's condition at an irregular singular point[END_REF] that he can chose a conjugacy of the form (1.9) which is the asymptotic expansion of an analytic function in a domain

(x, z) ∈ S × D (0, r) | |z 1 z 2 | < ν |x|
for some small ν > 0, where S is a sector of opening less than π with vertex at the origin and D (0, r) is a polydisc of small poly-radius r = (r 1 , r 2 ). Moreover, the (q i,k (x)) i,k are in fact Gevrey-1 series.

Under more restrictive conditions (which correspond to c 1 = c 2 = 0 and Re (a 1 + a 2 ) > 0 in Theorem 1.3), Shimomura, improving on a result by 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 these conditions are conjugate to:

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 generalized this result for any dimension in [START_REF] Stolovitch | Classification analytique de champs de vecteurs 1résonnants de (C n , 0)[END_REF]. Unfortunately, as shown by Yoshida in [START_REF] Yoshida | 2-parameter family of solutions for Painlevé equations (I)-(V) at an irregular singular point[END_REF], the hypothesis c 1 = c 2 = 0 is not met in the case of Painlevé equations.

In a forthcoming series of papers we will prove an analytic version of Theorem 1.12, valid in sectorial domains with sufficiently large opening, which in turn will help us to provide an analytic classification. Let us insist once more on the key fact that the union of these sectorial domains forms a whole neighborhood of the origin.

Outline of the paper

• In section 2 we recall some basic concepts and results from the theory of formal vector fields and differential forms.

• In section 3 we prove Theorems 1.3, 1.6 and 1.12, and compute the isotropies of the associated normal forms.

• In section 4 we explain how to compute any finite jet of the formal invariant c in Theorem 1.12 in the case of the Painlevé equations.

Background

We refer the reader to [START_REF] Ilyashenko | Lectures on analytic differential equations[END_REF], [START_REF] Martinet | Normalisation des champs de vecteurs holomorphes (d'après A.-D. Brjuno)[END_REF] and [START_REF] Brjuno | Analytic form of differential equations. I, II[END_REF] for a detailed introduction to formal vector fields and formal diffeomorphisms. Although these concepts are well-known by specialists, we will recall briefly the needed results and nomenclature.

Formal power series, vector fields and diffeomorphisms

As usual, we will denote a formal power series as

f (x) = k f k x k where, for all k = (k 1 , . . . , k n ) ∈ N, f k ∈ C and x k = x k1 1 . . . x kn n .
We will also use the notation

|k| = k 1 + • • • + k n for the degree of a monomial x k (which is of homogenous degree k = (k 1 , . . . , k n )).
We denote respectively by C x , D, Diff the sets of formal power series (equipped with an algebra structure), vector field (equipped with a Lie algebra structure), diffeomorphisms (equipped with a group structure). The maximal ideal of the algebra C x formed by formal power series with null constant term is denoted by m.

A vector field will be seen either as a an element of (C x )

n or as a derivation on C x : for any vector field

X = α 1 ∂ ∂x 1 + • • • + α n ∂ ∂x n ∈ (C x ) n , (2.1)
its Lie derivative is defined as the operator

L X (f ) = α 1 ∂f ∂x 1 + • • • + α n ∂f ∂x n , (2.2)
for any formal power series f ∈C x . The Lie bracket [X, Y ] of two vector fields X, Y ∈ D is defined by

L [X,Y ] (f ) = L X (L Y (f )) -L Y (L X (f )) for all f ∈ C x .
Similarly, a formal diffeomorphism will be seen either as an element of

Φ (x) ∈ (C x ) n such that Φ (0) = 0 and D 0 Φ = Jac (Φ (0)) ∈ Gl n (C), or as an algebra automorphism of C x : given a formal series f = k∈N n a k x k ∈ C x , we denote by Φ * (f ) = k∈N n a k φ k1 1 • • • φ kn n , (2.3)
the pull-back of f by Φ ∈ Diff, where

φ 1 = Φ (x 1 ) , . . . , φ n = Φ (x n ) .
The Jacobian matrix (or the linear part) of Φ in the basis (x 1 , . . . , x n ) is the

matrix ∂φ i ∂x j (0, . . . , 0) i,j
.

The order ord (f ) (resp. ord (X), resp. ord (Φ)) of a non-zero formal power series

f ∈ C x (resp. vector field X ∈ D, resp. diffeomorphism Φ ∈ Diff) is the maximal integer k ≥ 0 such that f ∈ m k (resp. L X (m) ⊂ m k , resp. Φ * (m) ⊂ m k ).
The notion of order allows to define the classical Krull topology on C x , D and Diff. The set of formal vector field of order at least k is a submodule denoted by D (k) ⊂ D. In particular, D (1) is the submodule of singular formal vector fields. We denote by A (k) ⊂ Diff the normal subgroup formed by those automorphisms Φ such that

Φ (x i ) -x i ∈ m k+1
for each i = 1, . . . , n. Each element of A (k) will be called a formal diffeomorphism tangent to the identity up to order k.

Given a subgroup G ⊂ Diff, we say that two vector fields Y 1 , Y 2 in D are G-conjugate if there exits a Φ∈G such that:

L Y1 • Φ = Φ • L Y2 .
The following two lemmas will be important in the proof of Theorem 1.3. Lemma 2.1. Let X, Y ∈ D (1) be two singular formal vector fields. Then:

ord ([X, Y ]) ≥ ord (X) + ord (Y ) -1 . Lemma 2.2. Let (d n ) n≥0 ⊂ N >0 be
a strictly increasing sequence of positive integers, and (Φ n ) n≥0 a sequence of formal diffeomorphisms. Assume that for all n ≥ 0, Φ n (x) = x + P dn (x) mod m dn+1 , where P dn (x) is a vector homogenous polynomial of degree d n . Then the sequence

Φ [n] n≥0 , defined by Φ [n] = Φ n • • • • • Φ 0 , for all n ≥ 0, is convergent, of limit Φ ∈ Diff.
Moreover, if each Φ n is fibered then Φ is fibered too.

Proof. It suffices to prove by induction that for all n ≥ 0:

Φ n • • • • • Φ 0 (x) = x + P d0 (x) + • • • + P dn (x) mod m dn+1 ,
because the sequence (d n ) n≥0 ⊂ N >0 is strictly increasing.

Exponential map and logarithm

Given formal vector field X ∈ D (1) and a formal power series f ∈ C x and we set

L •0 X (f ) = f L •(k+1) X (f ) := L X L •k X (f )
, for all k ≥ 0 so that we can consider the algebra homomorphism given by:

exp (X) * : f → k≥0 1 k! L •k X (f ) . (2.4)
This series is convergent in the Krull topology and defines in fact a formal diffeomorphism, which is called the time 1 formal flow of X ∈ D (1) or the exponential of X. (see e.g. section 3 in [START_REF] Ilyashenko | Lectures on analytic differential equations[END_REF]).

For any vector field X ∈ D (1) , we consider also the adjoint map

ad X : D (1) → D (1) Y → [X, Y ]
and define

(ad X ) •0 := Id (ad X ) •(k+1) := ad X • (ad X ) •k , ∀k ∈ N .
We will need the following classical formula (see [START_REF] Martinet | Normalisation des champs de vecteurs holomorphes (d'après A.-D. Brjuno)[END_REF]).

Proposition 2.3. Given X, Y ∈ D (1) :

exp (X) * (Y ) = exp (ad X ) (Y ) ,
where

exp (ad X ) (Y ) = k≥0 1 k! (ad X ) •k (Y ) = Y + 1 1! [X, Y ] + 1 2! [X, [X, Y ]] + . . . .
We also recall the existence of a logarithm for all formal diffeomorphisms tangent to the identity (see [START_REF] Ilyashenko | Lectures on analytic differential equations[END_REF], section 3). Proposition 2.4. For any formal diffeomorphism Φ ∈ Diff, there exists a unique vector field F ∈ D (2) such that Φ = ϕ • exp (F ), where ϕ ∈ Diff is the linear change of coordinate given by D 0 Φ. Moreover, for each k ≥ 2, the exponential map defines a bijection between D (k) and A (k-1) .

Jordan decomposition and Dulac-Poincaré normal forms

According to [START_REF] Martinet | Normalisation des champs de vecteurs holomorphes (d'après A.-D. Brjuno)[END_REF], any singular formal vector field X ∈ D (1) admits a unique Jordan decomposition: X = X S + X N , with X S , X N ∈ D (1) and [X S , X N ] = 0 , (

where the restriction of X S (resp. X N ) to each k-jet vector space J k = m /m k (which is finite dimensional), k ≥ 0, is semi-simple (resp. nilpotent). This decomposition is compatible with truncation and invariant by conjugacy: if

X = X S + X N is the Jordan decomposition of X then 1. for all k ≥ 0, j k (X) = j k (X S ) + j k (X N )
is the Jordan decomposition of j k (X)(here, for any singular vector field Y ∈ D (1) , j k (Y ) is the endomor-

phism J k → J k induced by L Y ); 2. for any formal diffeomorphism ϕ ∈ Diff, ϕ * (X) = ϕ * (X S ) + ϕ * (X N ) is the Jordan decomposition of ϕ * (X).
Definition 2.5. We say that X ∈ D (1)

is in Poincaré-Dulac normal form if its Jordan decomposition X = X S + X N is such that X S is in diagonal form, i.e. X S = S (λ), where λ := (λ 1 , . . . , λ n ) ∈ C n and S (λ) := λ 1 x 1 ∂ ∂x 1 + • • • + λ n x n ∂ ∂x n . (2.6)
As mentioned in the introduction, according to Poincaré-Dulac Theorem, any singular vector field is conjugate to a Poincaré-Dulac normal form, but this normal form is far from being unique: every vector field is conjugate to many distinct Poincaré-Dulac normal forms.

Definition 2.6. A monomial vector field is a vector field in D of the form x k S (µ) for some k ∈ I, where I is the index set

I := {k = (k 1 , . . . k n ) ∈ (Z ≥-1 ) n | at most one of the k j 's is -1} ,
and some µ ∈ C n with the condition that µ = (0, . . . , 0, µ j , 0, . . . 0)

↑ j if k j = -1.
Fixing λ ∈ C n , each monomial vector field x k S (µ) is an eigenvector for ad S(λ) with eigenvalue

λ, k := λ 1 k 1 + • • • + λ n k n .
This is a consequence of the following elementary lemma. Lemma 2.7. For all λ, µ ∈ C n , and for all l, m ∈ Z n :

x l S (λ) , x m S (µ) = x l+m ( λ, m S (µ) -µ, l S (λ)) .
Remark 2.8. Notice that each X ∈ D can be uniquely written as an infinite sum of monomial vector fields of the form

X = k∈I x k S (µ k ) ,
which is a Krull-convergent series in D. We will call this expression the monomial expansion of X.

Assume now that X = S (λ) + X N is in Poincaré-Dulac normal form and let us consider the monomial expansion of X N :

X N = k∈I x k S (µ k ) . The condition [X S , X N ] = 0 is equivalent to require ∀k ∈ I, λ, k = 0 =⇒ µ k = 0 ;
in other words, each x k in the monomial expansion of X N is a so-called resonant monomial.

Proposition 2.9. Let X, Y ∈ D (1) be two vector fields in Poincaré-Dulac normal form with the same semi-simple part S (µ) for some µ ∈ C n , and with nilpotent parts in D (2) :

X = S (µ) + X N
, with X N ∈ D (2) , nilpotent, and

[S (µ) , X N ] = 0 Y = S (µ) + Y N , with Y N ∈ D (2) , nilpotent, and [S (µ) , Y N ] = 0 .
Assume X and Y are conjugate by a formal diffeomorphism Φ such that D 0 Φ = diag (λ 1 , . . . , λ n ) for some λ 1 , . . . , λ n ∈ C * . If we write Φ = ϕ•exp (F ) for some vector field F ∈ D (2) , where ϕ ∈ Diff is the linear diffeomorphism associated to D 0 Φ = diag (λ 1 , . . . , λ n ), then necessarily [S (µ) , F ] = 0.

Remark 2.10. Recall that the condition [S (µ) , F ] = 0 means that if we write

F = k∈I x k S (λ k ), then µ, k = 0 =⇒ λ k = 0.
Proof. We can assume without loss of generality that Φ is tangent to the identity. Indeed by setting P := (D 0 Φ) -1 , we obtain that P • Φ is tangent to the identity and conjugates X to Ỹ = DP • Y • P -1 . Since DP is diagonal, the assumptions made on Y are also met by Ỹ . Moreover, it is clear that the property we have to prove is true for Φ if and only if it is true for P • Φ. Therefore we may suppose that Φ is tangent to the identity. According to Proposition 2.4, there exists F ∈ D (2) such that exp (F ) = Φ, while according to Proposition 2.3 we have:

exp (F ) * (S (µ)) = S (µ) + [F, S (µ)] + 1 2! [F, [F, S (µ)]] + . . . (2.7)
Since exp (F ) * (S (µ)) = S (µ) by uniqueness of the Jordan decomposition, we have

[F, S (µ)] + 1 2! [F, [F, S (µ)]] + . . . = 0 . (2.8)
This implies that [F, S (µ)] = 0, using Lemma 2.1 and the fact that ord (F ) ≥ 2.

Remark 2.11. The assumption that D 0 Φ is in diagonal form necessarily holds if µ i = µ j , for all i = j.

Formal differential forms

Definition 2.12. We denote by Ω 1 (C x ) (or just Ω 1 for simplicity) the set of formal 1-forms in C n . It is the dual of Der (C x ) as C x -module.

Fixing the dual basis (dx 1 , . . . One can also extend the Krull topology to Ω. We can define the action of Diff by pull-back on Ω (C x ) thanks to the following properties:

dx n ) of (C n ) * , Ω 1 (C x ) is a free C x - module of
1. C-linearity 2. for all f ∈ C x , Φ * (f ) is defined as in (2.3) 3. ∀α, β ∈ Ω (C x ), ∀Φ ∈ Diff, Φ * (α ∧ β) = Φ * (α) ∧ Φ * (β) 4. ∀Φ ∈ Diff, Φ * • d = d • Φ * .
For any X ∈ D (1) and α ∈ Ω (C x ), we denote by L X (α) the Lie derivative of α with respect to X. We recall that L X is uniquely determined by the following properties:

1. for all k ≥ 0, L X : Ω k (C x ) -→ Ω k (C x ) is linear 2. for all f ∈ C x (i.e. f is a 0-form ), L X (f ) is as in Definition (2.2)
3. L X is a derivation of Ω (C x ), i.e. for all α, β ∈ Ω (C x ):

L X (α ∧ β) = L X (α) ∧ β + α ∧ L X (β) (Leibniz rule) 4. L X • d = d • L X .
We will need the following classical formula, which extends (2.4).

Proposition 2.15. ∀α ∈ Ω (C x ) , X ∈ D (1) :

exp (X) * (α) = exp (L X ) (α) = k≥0 1 k! L •k X (α) .
Proof. (Sketch) This formula is true for 0-forms, and we just has to prove it for 1-forms, because we can then it extend to any p-form using the exterior product and the Leibniz formula. In order to prove the result for 1-forms, one has to use the fact that

L X • d = d • L X .
With the same arguments, and using formulas

Φ * •d = d•Φ * and Φ * (α ∧ β) = Φ * (α) ∧ Φ * (β)
, we can prove the following Proposition. Proposition 2.16. For all Φ ∈ Diff, X ∈ D (1) and θ ∈ Ω (C x ), we have;

Φ * L Φ * (X) (ω) = L X (Φ * (ω)) .
In other words, the following diagram is commutative for all p ≥ 0:

Ω p Φ * / / L Φ * (X) Ω p LX Ω p Φ * / / Ω p
From now on, we set n = 3, we denote by x = (x, y) = (x, y 1 , y 2 ) the coordinates in C 3 . Definition 2.17. We denote by dx the ideal generated by dx in Ω = Ω (C x, y ): it is the set of forms θ ∈ Ω such that θ = dx ∧ η, for some η ∈ Ω.

Proposition 2.18. Let θ ∈ Ω, X ∈ D (1) and set Φ := exp (X) ∈ Diff. Then the following assertions are equivalent:

1. L X (x) = 0 and L X (θ) ∈ dx 2. Φ * (x) = x and Φ * (θ) ∈ θ + dx .
Proof. It is just a consequence of Propositions 2.15 and 2.3.

The next Lemma is proved by induction, as Lemma 2.2.

Lemma 2.19. In the situation described in Lemma 2.2, if we further assume that there exists a form θ ∈ Ω such that Φ * n (θ) ∈ θ + dx , for all n ≥ 0, then Φ * (θ) ∈ θ + dx .

Transversal Hamiltonian vector fields and transversal symplectomorphisms

We will need in fact to deal with forms with rational coefficients, and more precisely with

ω := dy 1 ∧ dy 2 x .
Given a formal vector field X such that L X (x) ∈ x we can easily extend its Lie derivative action to the set x -1 Ω (C x, y ) by setting:

L X 1 x θ := - L X (x) x 2 θ + 1 x L X (θ) , θ ∈ Ω (C x, y ) ∈ x -1 Ω (C x, y ) , because L X (x) ∈ x .
In particular we have

xL X 1 x θ ∈ Ω (C x, y ) .
Notice that if a vector field X satisfy L X (dx) ∈ dx , then L X (x) ∈ x . Similarly, we naturally extend the action of fibered diffeomorphisms by pullback on rational forms in x -1 Ω (C x, y ) by:

Φ * 1 x θ = 1 x Φ * (θ) , for (Φ, θ) ∈ Diff fib × Ω (C x, y ) so that xΦ * 1 x θ = Φ * (θ) .
Recalling Definition 1.9, we can now state a result analogous to Proposition 2.18.

Proposition 2.20. Let F ∈ D (1) be a singular vector field. The following two statements are equivalent:

1. exp (F ) is a transversally symplectic diffeomorphisms , Proof. This comes from Proposition 2.16: Φ * L Φ * (X) (ω) = L X (Φ * (ω)), and from the fact that Diff ω is a group, so Φ -1 ∈ Diff ω . Consequently we have:

xL Φ * (X) (ω) = x Φ -1 * L X (Φ * (ω)) = x Φ -1 * L X (ω + dx ) = x Φ -1 * (L X (ω)) + x Φ -1 * (L X ( dx )) = x Φ -1 * ( dx ) + x Φ -1 * ( dx ) ∈ dx .
Remark 2.22. In other words, we have an action of the group Diff ω on D ω , and then on SN ω .

We would like now to give a characterization of transversally Hamiltonian vector fields in terms of its monomial expansion (see Remark 2.8). Consider a monomial vector field X = x k0 y k1 1 y k2 2 S (µ) , with µ = (µ 0 , µ 1 , µ 2 ) ∈ C 3 \ {0}, such that L X (x) ∈ x . We necessarily have either µ 0 = 0 or k 0 ≥ 0. Let us compute its Lie derivative applied to ω:

L X (ω) = - L X (x) x 2 dy 1 ∧ dy 2 + 1 x d (L X (y 1 )) ∧ dy 2 + 1 x dy 1 ∧ d (L X (y 2 )) = -µ 0 x k0-1 y k1 1 y k2 2 dy 1 ∧ dy 2 + µ 1 x d x k0 y k1+1 1 y k2 2 ∧ dy 2 + µ 2 x dy 1 ∧ d x k0 y k1 1 y k2+1 2 = (µ 1 (k 1 + 1) + µ 2 (k 2 + 1) -µ 0 ) x k0-1 y k1 1 y k2 2 dy 1 ∧ dy 2 + dx .
Moreover:

L X (dx) = d (L X (x)) = d µ 0 x k0+1 y k1 1 y k2 2 = µ 0 (k 0 + 1) x k0 y k1 1 y k2 2 dx + k 1 x k0+1 y k1-1 1 y k2 2 dy 1 + k 2 x k0+1 y k1 1 y k2-1 2 dy 2 .
Thus, we see that X is transversally Hamiltonian if and only if the following two conditions hold:

1. µ 1 (k 1 + 1) + µ 2 (k 2 + 1) = µ 0 2. either µ 0 = 0 or k 1 = k 2 = 0.
So we have the following:

Proposition 2.23. Let X ∈ D (1) be a singular vector field and let

X = k∈I x k S (µ k ) be its
L x l S(µ l ) (dx) / ∈ dx
to obtain a contradiction, by looking at the terms of higher order. Similarly, according to the computation above, for each k ∈ I:

L x k S(µ k ) (ω) = (µ 1 (k 1 + 1) + µ 2 (k 2 + 1) -µ 0 ) x k0-1 y k1 1 y k2
2 dy 1 ∧ dy 2 + dx If one of the two conditions 1. or 2. above were not satisfied by a couple (k, µ k ) with |k| minimal, then we could not have xL X (ω) ∈ dx (just consider the terms of higher order).

3 Formal classification under fibered transformations

Invariance of the residue by fibered conjugacy

We start this section by proving that the non-degenerate condition defined in the introduction only depends on the conjugacy class of the vector field under the action of fibered diffeomorphisms. More precisely, the following proposition states that the residue is an invariant of a doubly-resonant saddle-node under the action of Diff fib . Proof. Consider the system

x 2 dy dx = α (x) + A (x) y (x) + f (x, y (x)) , (3.1)
with y = (y 1 , y 2 ) and where the following conditions hold:

• α (x) = α 1 (x) α 2 (x) , with α 1 , α 2 ∈ x 2 ⊂ C x • A (x) ∈ Mat 2,2 (C x ) with A (0) = Diag (-λ, λ), λ ∈ C * • f (x, y) = f 1 (x, y) f 2 (x, y) , with f 1 , f 2 ∈ y 1 , y 2 2 ⊂ C x, y .
Perform the change of coordinates given by y = β (x) + P (x) z + h (x, z), with z = (z 1 , z 2 ) and where:

• β (x) = β 1 (x) β 2 (x) , with β 1 , β 2 ∈ x ⊂ C x • P (x) ∈ Mat 2,2 (C x ) such that P (0) ∈ GL 2 (C) • h (x, y) = h 1 (x, y) h 2 (x, y) , with h 1 , h 2 ∈ z 1 , z 2 2 ⊂ C x, z .
Then one obtain the following system satisfied by z (x):

x 2 dz dx = P (x) -1 α (x) + A (x) β (x) + f (x, β (x)) -x 2 dβ dx (x) +P (x) -1 A (x) P (x) -x 2 dP dx (x) + ∂f ∂y (x, β (x)) P (x) z + z 1 , z 2 2 , Since A (0) ∈ GL 2 (C), f (x, y) ∈ y 1 , y 2 2
and ord (β) ≥ 1, the order of

P (x) -1 α (x) + A (x) β (x) + f (x, β (x)) -x 2 dβ dx (x)
is at least 2 if and only if ord (β) ≥ 2. Then:

Tr P (x) -1 A (x) P (x) -x 2 dP dx (x) + ∂f ∂y (x, β (x)) P (x) ∈ Tr (A (x)) + x 2 . So Tr (A (x)) x |x=0
is invariant by fibered change of coordinates on system of the form (3.1) with ord (α) ≥ 2.

Proof of Theorems 1.3 and 1.12

We will use the tools described in Section 2.

Proof. Let Y ∈ SN nd (resp. in SN ω ) be a non-degenerate (resp. transversally Hamiltonian) doubly-resonant saddle-node:

Y = x 2 ∂ ∂x + (-λy 1 + F 1 (x, y 1 , y 2 )) ∂ ∂y 1 + (λy 2 + F 2 (x, y 1 , y 2 )) ∂ ∂y 1 ,
with λ ∈ C * , and F ν (x, y) ∈ m 2 , for ν = 1, 2 . As seen in the previous subsection, we can assume that F 1 (x, 0, 0) = F 2 (x, 0, 0) = 0. The general idea is to apply successive (infinitely many) diffeomorphisms of the form exp x j0 y j1 1 y j2 2 S (0, µ 1,j , µ 2,j )

for convenient choices of j, µ 1,j , µ 2,j , in order to remove all the terms we want to. Let us consider the monomial expansion of Y :

Y = λS (0, -1, 1) + xS (1, 0, 0) + k∈I, |k|≥1 x k0 y k1 1 y k2 2 S (0, µ 1,k , µ 2,k ) . (3.2)
Since Y in non-degenerate we necessarily have

µ 1,(1,00) + µ 2,(1,0,0) = res (Y ) ∈ C\Q ≤0 .
In the transversally Hamiltonian case, each term in the sum k∈I, |k|≥1

x k0 y k1 1 y k2 2 S (0, µ 1,k , µ 2,k ) must satisfy µ 1,k (k 1 + 1) + µ 2,k (k 2 + 1) = 0 , if k = (1, 0, 0) and µ 1,(1,0,0) + µ 2,(1,0,0) = 1.
The normalizing conjugacy Φ is constructed in two steps.

1. The first step is aimed at removing all non-resonant monomial terms, i.e. those of the form

x k0 y k1 1 y k2 2 S (0, µ 1,k , µ 2,k ) , with k ∈ I, |k| ≥ 1 and k 1 = k 2 .
2. The second step is aimed at removing certain resonant monomial terms, and more precisely those of the form

x k0 (y 1 y 2 ) k S (0, η 1,i , η 2,i ) , except for (k 0 , k) = (1, 0) and k 0 = 0 .

We will see that each one of these steps allows us to define a fibered diffeomorphism Φ j (transversally symplectic in the transversally Hamiltonian case), for j = 1, 2. Finally we define Φ := Φ 2 • Φ 1 . The main tool used at each step is Proposition 2.15. Moreover, each Φ j will be constructed using Corollary 2.2. The fact that each Φ j is a fibered diffeomorphism (transversally symplectic in the transversally Hamiltonian case) will again come from Lemma 2.2 (and Lemma 2.19 in the transversally symplectic case, and each Y j = (Φ j ) * (Y j-1 ), j = 1, 2 with Y 0 := Y , will be transversally Hamiltonian according to Lemma 2.21).

1. First step: we remove all non-resonant monomial terms, using diffeomorphisms of the form

exp x i0 y i1 1 y i2 2 S (0, η 1,i , η 2,i ) ,
with i ∈ I, |i| ≥ 1, i 1 = i 2 and η 1,i , η 2,i to be determined. We have, thanks to Proposition 2.15:

exp x i0 y i1 1 y i2 2 S (0, η 1,i , η 2,i ) * (Y 0 ) = Y 0 + 1 1! x i0 y i1 1 y i2 2 S (0, η 1,i , η 2,i ) , Y 0 +. . . ,
where (. . . ) are terms computed via successive nested brackets, and they are all of order at least |i| + 1. Let us compute the first bracket:

x i 0 y i 1 1 y i 2 2 S (0, η 1,i , η 2,i ) , Y0 = λ (i1 -i2) x i 0 y i 1 1 y i 2 2 S (0, η 1,i , η 2,i ) -i0x i 0 +1 y i 1 1 y i 2 2 S (0, η 1,i , η 2,i ) + k∈I, |k|≥1 x i 0 +k 0 y i 1 +k 1 1 y i 2 +k 2 2 (k1η1,i 0 + k2η2,i 2 ) S (0, µ 1,k , µ 2,k ) - k∈I, |k|≥1 x i 0 +k 0 y i 1 +k 1 1 y i 2 +k 2 2 (i1µ 1,k + i2µ 2,k ) S (0, η 1,i , η 2,i ) .
Then one can remove all terms of the form x i0 y i1 1 y i2 2 S (0, µ 1,i , µ 2,i ) with |i| ≥ 1 and i 1 = i 2 by induction on |i| ≥ 1. We then define (using Lemma 2.2) a fibered diffeomorphism Φ 1 , such that Y 1 := (Φ 1 ) * (Y 0 ) is still of the form (3.2), but without non-resonant terms:

Y 1 = λS (0, -1, 1) + xS (1, a 1 , a 2 ) + k0+k≥1 (k0,k) =(1,0) x k0 y k 1 y k 2 S (0, µ 1,k , µ 2,k )
for maybe different µ j,k . Notice that a 1 , a 2 here are necessarily such that a 1 + a 2 / ∈ Q ≤0 since the vector field is supposed to be non-degenerate, and this condition is invariant under fibered change of coordinates. Remark. In the transversally Hamiltonian case, the terms x i0 y i1 1 y i2 2 S (0, η 1,i , η 2,i ) to be removed at this stage satisfy η 1,i (i 1 + 1) + η 2,i (i 2 + 1) = 0, so that Φ 1 is transversally symplectic according to Proposition 2.20 and Lemma 2.19. Moreover, in this case, we necessarily have a 1 + a 2 = 1.

2.

Second step: we finally remove all the terms of the form x i0 (y 1 y 2 ) i S (0, η 1,i , η 2,i ) , except for (i 0 , i) = (1, 0) and i 0 = 0 , using diffeomorphisms of the form

exp x i0 (y 1 y 2 ) i S (0, η 1,i , η 2,i ) ,
with i 0 + i ≥ 1, and η 1,i , η 2,i to be determined. We have, thanks to Proposition 2.15:

exp x i0 (y 1 y 2 ) i S (0, η 1,i , η 2,i ) * (Y 1 ) = Y 1 + 1 1! x i0 (y 1 y 2 ) i S (0, η 1,i , η 2,i ) , Y 1 +. . . ,
where (. . . ) are terms computed via successive nested brackets, and they are all of order strictly greater than the order of the first bracket. Let us compute the first bracket:

x i 0 (y1y2) i S (0, η 1,i , η 2,i ) , Y3 = -(i0 + i (a1 + a2)) x i 0 +1 (y1y2) i S (0, η 1,i , η 2,i ) + k 0 +2k≥1 (k 0 ,k) =(1,0) x i 0 +k 0 (y1y2) i+k k (η1,i 0 + η2,i 2 ) S (0, µ 1,k , µ 2,k ) - k 0 +2k≥1 (k 0 ,k) =(1,0) x i 0 +k 0 (y1y2) i+k i (µ 1,k + µ 2,k ) S (0, η 1,i , η 2,i ) .
Then we see that one can remove all terms of the form x i0 (y 1 y 2 ) i S (0, η 1,i , η 2,i ) except for (i 0 , i) = (1, 0) and for i 0 = 0, without creating non-resonant terms, since (a 1 + a 2 ) / ∈ Q ≤0 . We do this by induction on I := i 0 + i ≥ 1, and for fixed I ≥ 1, we remove the terms with i increasing and i 0 decreasing. Notice that at each step we do not create terms already removed earlier in the process. We then define (using Lemma 2.2) a fibered diffeomorphism Φ 2 , such that

Y 2 := (Φ 2 ) * (Y 1 ) is of the form Y 2 = λS (0, -1, 1) + xS (1, a 1 , a 2 ) + k≥1 (y 1 y 2 ) k S (0, µ 1,k , µ 2,k ) .
Remark. In the transversally Hamiltonian case, the terms x i0 (y 1 y 2 ) i S (0, η 1,i , η 2,i ) to be removed at this stage satisfy (η 1,i + η 2,i ) = 0, so that Φ 2 is transversally symplectic, according to Proposition 2.20 and and Lemma 2.19.

Finally, we define Φ := Φ 2 • Φ 1 , so that Φ * (Y ) = Y 2 and Φ is a fibered diffeomorphism (transversally symplectic in the Hamiltonian case).

Uniqueness: proof of Theorem 1.6

We now prove Theorem 1.6.

Proof. Let Z = x 2 ∂ ∂x + (-λ + a 1 x + c 1 (v)) z 1 ∂ ∂z 1 + (λ + a 2 x + c 2 (v)) z 2 ∂ ∂z 2 Z ′ = x 2 ∂ ∂x + (-λ ′ + a ′ 1 x + c ′ 1 (v)) z 1 ∂ ∂z 1 + (λ ′ + a ′ 2 x + c ′ 2 (v)) z 2 ∂ ∂z 2 , where (λ, λ ′ , a 1 , a 2 , a ′ 1 , a ′ 2 ) ∈ (C * ) 2 × C 4 , (a 1 + a 2 , a ′ 1 + a ′ 2 ) ∈ (C\Q ≤0 ) 2 and (c 1 , c 2 , c ′ 1 , c ′ 2 ) ∈ (vC v )
4 are formal power series in v = z 1 z 2 of order at least one.

• It is clear that if there exists ϕ : v → θv with θ ∈ C * such that

(λ, a 1 , a 2 , c 1 , c 2 ) = (λ ′ , a ′ 1 , a ′ 2 , c ′ 1 • ϕ, c ′ 2 • ϕ) resp. (λ, a 1 , a 2 , c 1 , c 2 ) = (-λ ′ , a ′ 2 , a ′ 1 , c ′ 2 • ϕ, c ′ 1 • ϕ) then Z is Diff fib -conjugate to Z ′ .
• Now assume that Z is Diff fib -conjugate to Z ′ . First of all, studying the terms of degree 1 with respect to z, we see that we either have

(λ, a 1 , a 2 ) = (λ ′ , a ′ 1 , a ′ 2 ) or (λ, a 1 , a 2 ) = (-λ ′ , a ′ 2 , a ′ 1 )
. Up to perform a linear change of coordinates beforehand, let us assume that (λ, a 1 , a 2 ) = (λ ′ , a ′ 1 , a ′ 2 ). In the following, and for convenience, we will use the notations:

Z = Z (c,r) := xS (1, a 1 , a 2 ) + (λ + c (v)) S (0, -1, 1) + r (v) S (0, a 1 , a 2 ) Z ′ = Z (c ′ ,r ′ ) := xS (1, a 1 , a 2 ) + (λ + c ′ (v)) S (0, -1, 1) + r ′ (v) S (0, a 1 , a 2 ) ,
where:

c 1 = -c + r , c 2 = c + r c ′ 1 = -c ′ + r ′ , c ′ 2 = c ′ + r ′ , so that ord (c) ≥ 1, ord (r) ≥ 1. Now we have to prove that if Z (c,r) is Diff fib -conjugate to Z (c ′ ,r ′ ) , then (c, r) = (c ′ , r ′ ).
By assumption, there exists Φ ∈ Diff fib such that

Φ * Z (c,r) = Z (c ′ ,r ′ ) . By Remark 2.11, D 0 Φ = diag (1, θ 1 , θ 2 ) is diagonal. Now, set Ψ := (D 0 Φ) -1 • Φ , ϕ : v → (θ 1 θ 2 ) v, and (c, r) := (c ′ • ϕ, r ′ • ϕ)
, so that:

Ψ * Z (c,r) = Z (c,r) .
We are going to prove that Ψ = Id. By Proposition 2.9, there exists G ∈ D (1) such that Ψ = exp (G) and:

G = g 0 (x, v) ∂ ∂x + g 1 (x, v) z 1 ∂ ∂z 1 + g 2 (x, v) z 2 ∂ ∂z 2 ,
where

g i ∈ m ⊂ C x, v for i = 1, 2 and g 0 ∈ m 2 ⊂ C x, v
is of order at least two. Since Ψ is fibered in x we deduce that g 0 = 0. Therefore, using the notation (2.6), we can write:

G = A (x, v) S (0, -1, 1) + B (x, v) S (0, a 1 , a 2 ) ,
where

     A = A i,j x i v j B = i,j≥0 i+j≥1 B i,j x i v j .
Let us prove that A = B = 0 (hence G = 0) so that Ψ = Id. We consider the Jordan decompositions of Z := Z (c,r) and Z := Z (c,r) :

Z = Z S + Z N , Z S semi-simple, Z N nilpotent, [Z S , Z N ] = 0 Z = Z S + Z N , Z S semi-simple, Z N nilpotent, Z S , Z N = 0 .
By uniqueness of this decomposition we clearly have:

     Z S = Z S = S (0, -λ, λ) Z N = xS (1, a 1 , a 2 ) + c (v) S (0, -1, 1) + r (v) S (0, a 1 , a 2 ) Z N = xS (1, a 1 , a 2 ) + c (v) S (0, -1, 1) + r (v) S (0, a 1 , a 2 )
, and we also know that:

Ψ * (Z) = Z ⇒ Ψ * (Z S ) = Z S Ψ * (Z N ) = Z N .
Let us now consider the associated two-dimensional vector fields in the variables (x, v). In the "chart" (x, v) the vector field G is given by F = B.S (0, a), with a = a 1 + a 2 . Z and Z correspond respectively to:

Y := xS (1, a) + r (v) S (0, a) , Y := xS (1, a) + r (v) S (0, a) .
Thus we have exp (F ) * (Y ) = Y . By Proposition 2.3 we derive

exp (F ) * (Y ) = Y + [F, Y ] + 1 2! [F, [F, Y ]] + . . . and r (v) S (0, a) + [F, Y ] + 1 2! [F, [F, Y ]] + . . . = r (v) S (0, a) . (3.3)
We compute next

[F, Y ] = -x L S(1,a) (B) + B L S(0,a) (r) -r L S(0,a) (B) S (0, a)
and, setting

C (1) (x, v) := -x L S(1,a) (B) + B L S(0,a) (r) (3.4)
-r L S(0,a) (B) ,

we obtain

[F, Y ] = C (1) (x, v) S (0, a) .
Now, it is easy to see that for all l ∈ N, ad •l F (Y ) ca be written

ad •l F (Y ) = C (l) (x, v) S (0, a) ,
where C (l) is determined by the recursive relation:

C (l+1) (x, v) = B (x, v) L S(0,a) C (l) -C (l) (x, v) L S(0,a) (B) .
In particular, we see that for all l ≥ 2, C (l) (x, 0) = 0. Equation (3.3) can now be rewritten:

r (v) + C (1) (x, v) + l≥2 C (l) (x, v) = r (v) . (3.5) Let us set r (v) = k≥1 r k v k and r (v) = k≥1 r k v k . Looking at terms inde- pendent of v in (3.5) (i.e. by taking v = 0), we see that C (1) (x, 0) = 0.
Taking (3.4) into account we obtain that ∂B (x, 0) ∂x = 0. Since ord (B) ≥ 1

(by assumption) this means that B (x, 0) = 0. Let us prove the properties B i,k = 0 and r k = r k for all i, j ∈ N and k ≤ j by induction on j ≥ 0.

j = 0. This corresponds to the case described above: for all i ≥ 0, B i,0 = 0 (and r 0 = r 0 ).

-If the property holds at a rank j ≥ 0, if we consider for all i ≥ 0 terms of homogenous degree (i + 1, j + 1) in (3.5), we obtain:

(i + a (j + 1)) B i,j+1 = 0 by induction, and because for all l ≥ 2 the relation C (l) (x, 0) = 0 also holds. Since a / ∈ Q ≤0 we have B i,j+1 = 0. On the other hand, if we look at terms of homogeneous degree (0, j + 1), we obtain: r j+1 = r j+1 .

We conclude that B = 0, so that F = 0 and r = r. Finally, we have G = AL (0, -1, 1). Taking the relation exp (G) * (Z N ) = Z N into account, we have:

Z N + [G, Z N ] + 1 2! [G, [G, Z N ]] + . . . = Z N if and only if c (v) S (0, -1, 1) + [G, Z N ] + 1 2! [G, [G, Z N ]] + . . . = c (v) S (0, -1, 1) . Let us compute [G, Z N ]: [G, Z N ] = -xL S(1,a1,a2) (A) + r (v) L S(0,a1,a2) (A) S (0, -1, 1) .
All other Lie brackets vanish. There only remains

c (v) -xL S(1,a1,a2) (A) -r (v) L S(0,a1,a2) (A) = c (v) ,
which becomes a system of identities between terms of same degree:

           c j - j-k1 k=0 akA 0,k r j-k = c j , j ≥ 0 (i + aj) A i,j + j-k1 k=0 akA i+1,k r j-k = 0 , i ≥ 0, j ≥ 0 .
Once again, we prove by induction on j ≥ 0 that for all i ≥ 0 and all 0 ≤ k ≤ j the relations A i,k = 0 and c k = c k hold. Thus A = 0 and c = c. As a conclusion Ψ = Id and (c, r) = (c, r), so that

Φ = D 0 Φ = diag (1, θ 1 , θ 2 ) and (c, r) := (c ′ • ϕ, r ′ • ϕ) where ϕ : v → (θ 1 θ 2 ) v.

Fibered isotropies of the formal normal form

Looking back at the uniqueness proof in the previous paragraph, we immediately obtain all formal fibered isotropies of the normal form given by Theorems 1.3 and 1.12. We recall that an isotropy of a vector field is a self-conjugacy. For a vector field X ∈ D (1) , we set:

Isot fib (X) := Φ ∈ Diff fib | Φ * (X) = X . Proposition 3.2. Consider a normal form of SN nd Z = 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 ,
with parameters (λ, a 1 , a 2 , c 1 , c 2 ) ∈ P. Then:

Isot fib (Z) = diag (1, θ 1 , θ 2 ) , (θ 1 , θ 2 ) ∈ (C * ) 2 (c 1 , c 2 ) (θ 1 θ 2 v) = (c 1 , c 2 ) (v) . Remark 3.3. If (c 1 , c 2 ) = (0, 0) the condition c i (θ 1 θ 2 v) = c i (v)
for each i ∈ {1, 2} is equivalent to requiring that each c i lie in C v q , for some q ∈ N >0 , and that θ 1 θ 2 be a q th root of unity.

This proposition has for immediate consequence the (almost) uniqueness of the normalizing conjugacy Φ ∈ Diff in Theorem 1.3. More precisely: Corollary 3.4. Let Y ∈ SN nd be a non-degenerate doubly-resonant saddlenode such that D 0 Y = diag (0, -λ, λ), with λ = 0. Then there exists a unique fibered diffeomorphism Φ ∈ Diff fib tangent to the identity such that:

Φ * (Y ) = x 2 ∂ ∂x + (-λ + a 1 x + c 1 (v)) y 1 ∂ ∂y 1 + (λ + a 2 x + c 2 (v)) y 2 ∂ ∂y 2 , (3.6)
where we put v := y 1 y 2 . Here, c 1 , c 2 belong to v = vC v and a 1 , a 2 ∈ C are such that a 1 + a 2 = res (Y ).

Z = x 2 ∂ ∂x + (-(λ + c (v)) + a 1 x) y 1 ∂ ∂y 1 + (λ + c (v) + a 2 x) y 2 ∂ ∂y 2 .
Then:

Isot ω (Z) = diag 1, α, 1 α , α ∈ C\ {0} ≃ C\ {0} .

Applications to Painlevé equations

In this section we investigate the study of the irregular singularity at infinity in the first Painlevé equation

(P I ) d 2 z 1 dt 2 = 6z 2 1 + t
in terms of Theorem 1.12. More precisely, we are going to explain that the formal invariant c ∈ C v of a doubly-resonant, transversally symplectic saddle-node Y ∈ SN ω is in fact a germ of an analytic function at the origin, whenever Y is analytic at the origin (and not merely a formal vector field). Moreover, we show how to compute recursively this invariant in some specific cases, including Painlevé equations.

Asymptotically Hamiltonian vector fields

We deal here with the case of asymptotically Hamiltonian vector fields.

Definition 4.1.

• We say that a formal 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 unity U (y) ∈ C y × (i.e. U (0, 0) = 0) and (λ 1 , λ 2 ) ∈ C 2 .

• We say that a formal (resp. germ of an analytic) vector field X in C 2 , 0 is formally (resp. analytically) orbitally linearizable if X is formally (resp. analytically) conjugate to an orbitally linear vector field.

• We say that a doubly-resonant saddle-node Y ∈ SN is formally/analytically asymptotically orbitally linearizable if the formal/analytic vector field Y |{x=0} in C 2 , 0 is formally/analytically orbitally linearizable.

Remark 4.2.

1. If a vector field X 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 X is analytically orbitally linearizable if and only if it is formally orbitally linearizable.

2. The fact of being orbitally linearizable is naturally invariant under orbital equivalence, and then, by (almost) uniqueness of c 1 , c 2 in Theorem 1.6, if Y ∈ SN nd is asymptotically linearizable, then its formal invariants c 1 , c 2 satisfy c 1 + c 2 = 0. In this case, we write c := c 2 = -c 1 .

The two remarks above imply the following corollary.

Φ * (Y ) = x 2 ∂ ∂x + (-λ + a 1 x -c (v)) y 1 ∂ ∂y 1 + (λ + a 2 x + c (v)) y 2 ∂ ∂y 2 ,
where we put v := y 1 y 2 . Here, c (v) ∈ vC {v} is a germ of an analytic function vanishing at the origin, and a 1 , a 2 ∈ C are such that a 1 +a 2 = res (Y ). Moreover, Φ is unique up to linear transformations.

It is important to notice that the following property holds. Proof. The facts that L Y (ω) ∈ dx and L Y (x) = x 2 imply that a 1 + a 2 = 1 and then:

L Y (dy 1 ∧ dy 2 ) = x (dy 1 ∧ dy 2 ) + dx .
Consequently, if we denote Y 0 := Y |x=0 the restriction of Y to the invariant hypersurface {x = 0}, we have:

L Y0 (dy 1 ∧ dy 2 ) = 0 .
This means that Y 0 is a Hamiltonian vector field, i.e. there exists H (y) ∈ C y such that:

Y 0 (y 1 , y 2 ) = - ∂H ∂y 2 (y 1 , y 2 ) ∂ ∂y 1 + ∂H ∂y 1 (y 1 , y 2 ) ∂ ∂y 2 .
Possibly by performing a linear change of coordinate, we can assume that H (y) ∈ λy 1 y 2 + m 3 , therefore we can write:

Y = x 2 ∂ ∂x + - ∂H ∂y 2 + xF 1 (x, y) ∂ ∂y 1 + ∂H ∂y 1 + xF 2 (x, y) ∂ ∂y 2 ,
where F 1 , F 2 ∈ C x, y vanish at the origin. If we define J := 0 -1

1 0 ∈ M 2 (C)
and ∇H := t (DH), then Y |{x=0} = J∇H. According to the Morse lemma for holomorphic functions, there exists an analytic change of coordinates ϕ ∈ Diff in C 2 , 0 tangent to the identity such that H (y) := H ϕ -1 (y) = y 1 y 2 . Let us now recall a trivial result from linear algebra.

Fact. Let J := 0 -1 1 0 ∈ M 2 (C), and P ∈ M 2 (C). Then, P JP t = det (P ) J.

We deduce the next result.

Lemma. Let H ∈ m 2 ⊂ C y , Y 0 := J∇H the associated Hamiltonian vector field in C 2 (for the standard symplectic form dy 1 ∧ dy 2 ), and an analytic diffeomorphism near the origin denoted by ϕ. Then:

ϕ * (Y 0 ) := Dϕ • ϕ -1 • Y 0 • ϕ -1 = det Dϕ • ϕ -1 J∇ H ,
where

H := H • ϕ -1 .
As a conclusion,the previous lemma shows that Y is asymptotically orbitally linearizable.

The next property is a straightforward consequence of Corollary 4.3, Proposition 4.4 and Theorem 1.12.

Corollary 4.5. Let Y ∈ SN ω be a transversally Hamiltonian doubly-resonant saddle-node. Then, there exists a transversally symplectic diffeomorphism Φ ∈ Diff ω such that Φ |{x=0} be a germ an analytic diffeomorphism in C 2 , 0 and:

Φ * (Y ) = x 2 ∂ ∂x + (-λ + a 1 x -c (v)) y 1 ∂ ∂y1 + (λ + a 2 x + c (v)) y 2 ∂ ∂y2 . (4.
1) where we put v := y 1 y 2 . Here, c (v) ∈ vC {v} is a germ of an analytic function vanishing at the origin, and a 1 , a 2 ∈ C are such that a 1 + a 2 = res (Y ) = 1. Moreover, Φ is unique up to linear symplectic transformations, and:

Φ |{x=0} * (dy 1 ∧ dy 2 ) = dy 1 ∧ dy 2 .

Periods of the Hamiltonian on {x = 0}

From now on, we consider a vector field

Y = x 2 ∂ ∂x + - ∂H ∂y 2 + xF 1 (x,
Y 0 = - ∂H ∂y 2 ∂ ∂y 1 + ∂H ∂y 1 ∂ ∂y 2 .
We fix a small polydisc D (0, r) ⊂ C 2 on which H is analytic with r = (r 1 , r 2 ).

The leaves of the foliation defined by Y 0 in D (0, r) are given by the level curves L a := {H = a} ∩ D (0, r), a ∈ D (0, r), with r > 0 small enough. The Morse Lemma for holomorphic functions tells us that L a is topologically a cylinder for a = 0, and r,r 1 , r 2 small enough. Thus we can consider a generator γ a of the first homology group of L a . We also consider a time-form for Y 0 , which is a meromorphic 1-form τ Y0 in D (0, r) with a unique pole at the origin and such

that τ Y0 • (Y 0 ) = 1. For instance, take τ Y0 = - dy 1 ∂H ∂y2
. Now we define the associated period map:

T H : D (0, r) \ {0} -→ C a -→ T H (a) := 1 2iπ ˛γa τ Y0 .
This mapping is a well-defined meromorphic function of a ∈ D (0, r).

Proposition 4.6. For r > 0 small enough, and a ∈ D (0, r) \ {0}, T Y0 (a) only depends on the class of γ a in H 1 (L a , Z). In other words, if τ ′ Y0 is another time-form of Y 0 and γ ′ a is any loop in L a homologous to γ a , then

˛γa τ Y0 = ˛γ′ a τ ′ Y0 .
Proof. The fact that this quantity does not depend on a specific choice of a representative of γ a in its homology class comes from Stokes Theorem. The fact that it does not depend on the choice of a specific time-form comes from the fact that γ a lies in a leaf of the foliation generated by Y 0 . If

γ a : [0, 1] → L a t → (γ a,1 (t) , γ a,2 (t)) , then d dt (γ a ) (t) = v a (t) Y 0 (γ a (t)), where v a (t) = 1 -∂H ∂y2 (γ a (t)) dγ a,1 (t) dt = 1 ∂H ∂y1 (γ a (t))
dγ a,2 (t) dt .

Then:

˛γa τ Y0 = ˆ1 0 τ Y0 (γ a (t)) • d dt (γ a ) (t) dt = ˆ1 0 τ Y0 (γ a (t)) • (v a (t) Y 0 (γ a (t))) dt = ˆ1 0 v a (t) dt since τ Y0 • (Y 0 ) = 1.
Definition 4.7. We call T H the period map of H near the origin. Now, consider a germ of an analytic diffeomorphism Ψ fixing the origin of C 2 . Then:

T H (a) = 1 2iπ ˛γa τ Y0 = 1 2iπ ˛Ψ-1 (γa) Ψ * (τ Y0 ) .
Notice that if we write X 0 := Ψ -1 * (Y 0 ) and τ X0 := Ψ * (τ Y0 ), then: Consequently, taking for instance τ X0 = -dy 1 y 1 (λ + c (v))

τ X0 • (X 0 ) = (Ψ * (τ Y0 )) • Ψ -1 * (Y 0 ) = τ Y0 • (Y 0 ) = 1 .
, we see that: As a consequence, for a = h (v), we have the following relation:

dh dv (v) .T H (h (v)) = -1 .
If we consider the antiderivative S H of T H such that S H (0) = 0, we have

S H (h (v)) = -v ,
and in particular S H = -g .

Let us summarize this study in the following proposition. given by Theorem 1.12. Consider the period map T H as defined above. Then the following holds:

1. c is the germ of an analytic function at the origin.

2. T H defines the germ of an analytic function in a neighborhood of 0 ∈ C 2 , such that T H (0) = -1 λ . 3. If S H is the primitive of T H such that S H (0) = 0, then (-S H ) is invertible (for the composition), and its inverse h satisfy:

dh dv (v) = λ + c (v) .
The conclusion is that if one is able to compute the period map of the original Hamiltonian vector field on {x = 0}, then one can compute the formal invariant c in the normal form given in Theorem 1.12, which is in fact even analytic in this case. Remark 4.9. The Hamiltonian function h (y 1 y 2 ) = λy 1 y 2 + ´y1y2 c (v) dv is in fact the symplectic normal form of the original Hamiltonian function H (y 1 , y 2 ) = λy 1 y 2 + o z→0 y 2 , as described in [START_REF] Chow | Normal forms and bifurcation of planar vector fields[END_REF] (section 2.7). , according to equation (1.6). These are not the system of coordinates which diagonalizes the linear part of the vector field, but the value of the period does not changes by symplectic changes of coordinates (those which preserve dy 1 ∧ dy 2 ). Now, if we fix a = 0 with |a| small enough and look at the level curve {H = a} near the origin in C 2 , we can compute the associated period: (see e.g. [START_REF] Bryuno | Boutroux asymptotics of solutions of Painlevé equations and power geometry[END_REF], [START_REF] Abenda | Singularity analysis of 2D complexified Hamiltonian systems[END_REF]). To compute it we can chose for instance γ a,1 : [0, 2π] -→ C t -→ ρ a e it where ρ a > 0 is such that 12ζy 2 1 + 4y Notice that y 1 → 24ζy 2 1 + 8y 3 1 (k+ 1 2 ) is in fact analytic in a neighborhood of the origin, with a zero of order 2k + 1. Hence we can compute the integral above using the residue theorem. As we have for k ≥ 1. Since S H (0) = 0 and dSH da (0) = T H (0) = 0, the mapping (-S H ) is invertible for the composition and we can compute recursively its inverse (denoted by h):

Example: the case of the first Painlevé equation

h (v) = k≥1 h k v k .
For all k ≥ 1, the coefficient h k is uniquely determined by the coefficients S H,j , j ≤ k. Finally, we have

λ + c (v) = dh dv (v) = k≥0 (k + 1) h k+1 v k = λ + k≥1 c k v k .

Definition 1 . 2 .

 12 The residue of Y ∈ SN is the complex number res (Y ) := Tr (A (x)) x |x=0 .

4 v 3 +

 43 O v 4 .

1 1 . 3 1 . 4 1 . 5 18 3

 113141518 Definition and main result . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Painlevé equations and the transversally symplectic case . . . . . 5 Known results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Analytic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Background 11 2.1 Formal power series, vector fields and diffeomorphisms . . . . . . 11 2.2 Exponential map and logarithm . . . . . . . . . . . . . . . . . . . 13 2.3 Jordan decomposition and Dulac-Poincaré normal forms . . . . . 14 2.4 Formal differential forms . . . . . . . . . . . . . . . . . . . . . . . 16 2.5 Transversal Hamiltonian vector fields and transversal symplectomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formal classification under fibered transformations 20 3.1 Invariance of the residue by fibered conjugacy . . . . . . . . . . . 20 3.2 Proof of Theorems 1.3 and 1.12 . . . . . . . . . . . . . . . . . . . 21 3.3 Uniqueness: proof of Theorem 1.6 . . . . . . . . . . . . . . . . . 24 3.4 Fibered isotropies of the formal normal form . . . . . . . . . . . 28 4 Applications to Painlevé equations 29 4.1 Asymptotically Hamiltonian vector fields . . . . . . . . . . . . . . 29 4.2 Periods of the Hamiltonian on {x = 0} . . . . . . . . . . . . . . . 32 4.3 Example: the case of the first Painlevé equation . . . . . . . . . . 35 5 Acknowledgement 37

  rank n, generated by dx 1 , . . . , dx n . Definition 2.13. For any p ∈ N, we denote the p-exterior product of Ω 1 (C x ) byΩ p (C x ) := p Ω 1 (C x )(or just Ω p ). Its elements will be called formal p-forms.The set of 0-forms is the set of formal series: Ω 0 (C x ) := C x . Definition 2.14. We denote byΩ (C x ) := +∞ ⊕ p=0 Ω p (C x )(or just Ω for simplicity) the exterior algebra of the formal forms in C n , and by d the exterior derivative on it.

2 .

 2 L F (x) = 0 and F is transversally Hamiltonian. Proof. It is just a consequence of Proposition 2.18. Lemma 2.21. Let Φ ∈ Diff ω and X ∈ D ω . Then, Φ * (X) ∈ D ω .

Proposition 3 . 1 .

 31 Let X, Y ∈ SN . If X and Y are Diff fib -conjugate, then res (X) = res (Y ).

Definition 3 . 5 .

 35 Let Z ∈ SN ω . We denote by Isot ω (Z) the subgroup of elements Φ ∈ Diff ω such that Φ * (Z) = Z. Proposition 3.6. Let (λ, a 1 , a 2 ) ∈ C * ×C 2 such that a 1 +a 2 = 1, and c ∈ vC v with v = y 1 y 2 . Consider

Corollary 4 . 3 .

 43 Let Y ∈ SN nd be a doubly-resonant saddle-node asymptotically orbitally linearizable such that Y 0 := Y |{x=0} be a germ of an analytic vector field in C 2 , 0 . Then, there exists Φ ∈ Diff fib such that Φ |{x=0} be a germ an analytic diffeomorphism in C 2 , 0 and:

Proposition 4 . 4 .

 44 If Y ∈ SN ω is doubly-resonant transversally Hamiltonian saddle-node, then Y is asymptotically orbitally linearizable.

Now, let us+ y 2 ∂ ∂y 2 ,

 2 take Ψ -1 = Φ |{x=0} as in Corollary 4.5 such that X 0 = (λ + c (v)) -y 1 ∂ ∂y 1 with v = y 1 y 2 , c ∈ C {v} and c (0) = 0. Then γa := Ψ -1 (γ a ) = Φ |{x=0} (γ a ) is a loop generating the homology of the leaf Φ |{x=0} (L a ). Consider h := H • Ψ near the origin. Then, Φ |{x=0} (L a ) = {h = a} in a neighborhood of the origin. Notice that h depends in fact only on v = y 1 y 2 , and h (v) = λv + o |v|→0 (|v|). Since λ = 0, the inverse function theorem ensures the existence of an analytic function g ∈ C {v} such that g (0) = 0 and h • g (v) = g • h (v) = v in a neighborhood of 0. Thus, {h (v) = a} = {v = g (a)}.

  + c (g (a)) according to the orientation chosen for γ a . In particular, we see that T H is analytic at the origin, and T H can be extend at 0 by-1 λ . The fact that Φ |{x=0} satisfiesΦ |{x=0} * (dy 1 ∧ dy 2 ) = dy 1 ∧ dy 2implies that det Φ |{x=0} = 1, so thatX 0 =c (v) .

2 is analytic at the origin of C 2 .

 22 on {x = 0} of a transversally Hamiltonian doubly-resonant saddle-node Y ∈ SN ω , where H (y) = λy 1 y 2 + o z→0 y Consider its unique transversally Hamiltonian normal formX = x 2 ∂ ∂x + (-(λ + c (v)) + a 1 x) y 1 ∂ ∂y 1 + (λ + c (v) + a 2 x) y 2 ∂ ∂y 2

  In the case of the first Painlevé equation, in appropriate coordinates, we are working with the Hamiltonian H (y 1 , y 2 )

where γ a, 1

 1 is the component of γ a with respect to ∂ ∂y1 . Remark 4.10. The period T H (a) is one of the periods of the Weierstrass function ℘ associated to the cubic H (y 1 , y 2 ) = a

1 -. 2 )

 12 As we have normal convergence, we can swap the order of summation and integration: dy 1 a k .

1 ,S

 1 we see that the associated residue at 0 is equal to8 2k (24ζ) H,k a k with S H,k = T H,k-1 k

  monomial expansion. X is transversally Hamiltonian if and only if for all k ∈ I, x k S (µ k ) is transversally Hamiltonian.Proof. Clearly if x k S (µ k ) is transversally Hamiltonian for all k ∈ I, then X is transversally Hamiltonian is obvious, by convergence of the above series in the Krull topology. Assume conversely that X is transversally Hamiltonian.

First of all, notice that we necessarily have, for all k ∈ I, L x k S(µ k ) (dx) ∈ dx . Indeed, if it were not the case, consider k with |k| minimum among the set of multi-index l satisfying

  analytic at the origin of C 2 , and F 1 , F 2 ∈ C x, y vanishing at the origin. Let us consider the restriction Y 0 := Y |{x=0} : it is an analytic Hamiltonian vector field in C 2 , 0 :

		y)	∂ ∂y 1	+	∂H ∂y 1	+ xF 2 (x, y)	∂ ∂y 2	,
	with H (y) = λy 1 y 2 + O y→0	y 3

  3 1 > 5|a| 2 , for all y 1 = γ a,1 (t), t ∈ [0, 2π]. Now we write:

As a conclusion, the jet of order k of T H gives us the jet of order k of c. After computations performed with Maple, we obtain for instance:

.

One can in fact compute any finite jet of c. Remark 4.11. Similar computations can be performed for any Hamiltonian of the form H (y 1 , y 2 ) = βy 2 2 + αy 2 1 + f (y 1 ), where α, β ∈ C\ {0} and f ∈ C {y 1 }.

Acknowledgement

I would like to express my gratitude to my supervisors Daniel Panazzolo and Loïc Teyssier for their useful comments and suggestions. The many discussions we had played an important role in the improvement of this paper.


Professor N Baron
email: nbaron@univ-mlv.fr

Chapter 2: Railways and demographic change

come   L'archive ouverte pluridisciplinaire

Introduction

An attempt to understand the interactions between the spread of railways and the dynamics of settlement, demographic change and urban growth is part of a body of research that has a long history. Few researchers have summarised this researchin termsas straightforward as those of the geographer Pierre George, who wrote, in 1968: 'The layoutof the railway network determines the rates of growth of cities [page 233]'. This statement, while obviously simplistic, provides us with a possible starting point as it refers to the more or less general belief that a technological systemsuch as the train encourages the population to group together and, hence, provides an impetus for trade flows, and hence business activities and wealth [START_REF] Ribeill | Au temps de la révolutionferroviaire:L'utopiqueréseau[END_REF]. 1 As this George's idea has determined the nature of this field of research, and is part of a long intellectual tradition, the first part of this chapter sets out to highlight not itsthe truth of the statement, but rather its continual presence during different phases of history or, to use David Banister's expression, during the different ages of the train [START_REF] Banister | The age of the train[END_REF].

We will then present the mechanisms of interaction between the rail network and demographic dynamics, before explaining how demographic changes have affected the overall performance of the European rail system.

The railway network and urban growth: a constant factor in geographical thought

Correlation between the network construction and the urban changes of the nineteenth century

When making his simple statement, Pierre George was speaking in the tradition of nineteenth-century engineers who attempted to optimise the location of the branches of the railway network on the basis of geographical criteria. For example, in 1863 the civil engineer LéonLalanne presented the Académie des Sciences with a design rule that exploiteddensity measurements of towns and villages,along with calculations of their distance from centres in order to establishwhat he referred to as ademographicallybased 'theory of railways' [START_REF] Lalanne | Essaid'unethéorie des réseaux de chemins de ferfondéesurl'observation des faits et sur les loisprimordiales qui président au groupement des populations[END_REF]. Another civil engineer, Charles-Joseph Minard (see Figure 1) provided maps that showed the clear spatial connections between urbanisation and the railway network [START_REF] Friendly | Visions and re-visions of Charles Joseph Minard[END_REF], Robinson AH, 1967).Therefore, Minard's maps do not show the lines themselves, but the relative size of the passenger flows on each line. This proves that contemporaries were aware that urban growth was taking place at the same time as, andin interaction with,strong industrial development, human migration (from the countryside into the cities, but also from one city to another) and the extension of the railway network.

PLEASE ADD A FRAME WITH MAP AND FOLLOWING CAPTION

When the railway network and the network of cities form a single entity: territorial modelsand regional typologies on the threshold of the 1960s

If we now move a century forward, to the time during which Pierre Georgemade his statement, the point at issue was not so much to build new lines, as the European network was considered to be 'mature', but to modernise the railway system in order to adapt to a new economic and demographic situation.The rural exodus was petering out, and cities were diversifying their industrial and service bases. Consequently,Pierre Georgemeant something different by the term 'urban growth'thanLalanne orMinard did. He was no longer attempting to optimise the design of the network in relation to a given pattern of human settlement, nor was he seeking to provide the best possible services between mining areas and industrial centres.

In the 1960s, the car was already providing stiff competition for passenger rail services, and lorries were transporting a growing proportion of freight traffic. Consequently,what was at stakein the interaction between railways and demographic change was regional planning.

The idea was to allow the rail system to give structure toan efficient network of cities that would act as regional centres, create growth and attract a young, qualified and skilled workforce. Pierre George's research was therefore concerned with the link between demographic change, the network of cities and the railway system,with a view to creatingan 'urban backbone'that would fulfil the role of wealth creation with varying degrees of success [START_REF] Juillard | Croissanceurbaine et accessibilité des villes[END_REF].This research was very much influenced by an updated form of central place theory that was known by the name of 'regional science', and which generated models that explained the effects of the network on the urban system and human settlement.Among the research conducted during this period, studies that examined'the demography of railway networks' [START_REF] Chatelain | Démogéographie du grand troncferrésud-est (Paris-Lyon-Méditerranée)[END_REF]highlighted the success of some cities and the failure of others as a result of traffic conditions, service frequencies, passenger numbers and the economic specialisation of different areas.

High-speed rail's re-organisation of the network of cities

There was something of a hiatus in scientific output between the 1960s and the advent of a third phase of research, which went together with the third age of the train, namely highspeed rail (HSR),and the third urban age, that of the globalised metropolis [START_REF] Claval | The European system of capital cities[END_REF]. In this new phase, the shortening of journey times, the deindustrialisation of cities, the digital revolution and the globalisation of the economy workedtogether with demographic changesand, for a number of reasons, led to the adoption of a radically different approach to the issue. First, the high-speed network marked out some parts of the country from the remaining network of cities, thereby creating preferential accessibility conditions between (frequently large) cities and omitting those cities that lie between them(the so-called tunnel effect) [START_REF] Hall | High speed rail: Shrinking spaces, shaping places[END_REF]. HSR is also responsible for a change in commuting patterns and leisure travel (Dieleman and Faludi, 1998).

Thus,the three historical phases we have proposed provide us with a useful analysis framework, as they show the holistic nature of the human, urban and economic changes that have gripped the European continent since the middle of the nineteenth century, and the way in which rail can be seen as both a product of these changes, and a factor that has accelerated them [START_REF] Labasse | Chemin de fer et Urbanisme[END_REF]. The train in itself has not directly set in motion or affected any demographic events, nor triggeredexnihiloany processes of urbanisation, but it has intensified or impeded broader mechanisms which, when analysed, reveal the singular nature of the regional profiles in question. This also shows that the interactions involved are highly diverse, as are the scientific approaches to their study,which depend on the nature and goals of research and the socioeconomic context (Jensen and Richarson, 2007)

2.Retrospective analysis of the interactions between railways, human flows and urbanisation from 1830s to the dawn of the 21st century

What is the point of a retrospective analysis?

Some readers may doubt the value of studying past interactions between demographic change and railways. How, for example, does an explanation of the way the railway network emptied the large agricultural estates of eastern Prussiahelp us understand the rail strategies implemented today?In our view, operators frequently make the mistake of considering demographic factors as an externality (Accenture, 2010), in the same terms as the main environmental or energy-related issues. However, the history of the location of settlement is an internality, as the railway network has been, to a large extent, inherited from earlier phases of the interaction between settlement, geography and the economy. Comparing the settlement modelin relation to models of the entire European railway network, and considering with equanimity the issue of the 'path dependency'generated by the excessive provision of railways in the first half of the twentieth century may prove useful in making us aware of issues that are very much of current concern (Merger and Polino, 2004).

What we know about the interactions between the rail network and settlement

Few studies haveanalysed demographic change and change in the rail network at European level since the creation of the latter [START_REF] Martí-Henneberg | European integration and national models for railway networks (1840-2010)[END_REF][START_REF] O'brien | Railways and the Economic Development of Western Europe 1830-1914[END_REF], which contrasts with the situation relating tothe creation of each country's network [START_REF] Caron | Histoire des chemins de fer en France[END_REF][START_REF] Cobb | The Railways of Great Britain: A Historical Atlas[END_REF][START_REF] Del Río | La evolución de la red ferroviaria en el ReinoUnido (1825-2000)[END_REF][START_REF] Turnock | An Historical Geography of Railways in Great Britain and Ireland[END_REF][START_REF] Schram | Railways and the Formation of the Italian State in the Nineteenth Century[END_REF][START_REF] Silveira | Population and railways in Portugal, 1801-1930[END_REF]. Nevertheless, irrespective of where the focus is placed, research in this area potentially targets two different goals.The first is to discuss what rail owes to demography and what it does not; that is,to seehow well, or how badly, the geographical distribution of the population matched the railway infrastructure when it was designed. This field of research reveals the 'overprovision'or 'under-provision' ofrailways in different regions.

This type of work also allows us to discuss how important demographic factors are in the creation of the networkin comparison with other criteria of a political, economic or environmental nature. Railways have not always been built with reference to settlement, they also have to fit in with the physical geography (the first lines could not climb steep gradients, for example). Moreover, the network was created as the result of a set of decisions made by the State and/or private companies that aimed to satisfy a wide range of political and economic interests. Consequently, France and Spain built a radial network around the capital by which a centralizing authority asserted itself. Other countries,such as the Netherlands [START_REF] Schmal | Cities and railways in the Netherlands between 1830 and 1860[END_REF], pursued a mercantile policy and designed a rail network that provided good port/hinterland links.

The second direction for research into historical demography sets out to identify and measure the effects of the railway on demographic concentration [START_REF] Martí-Henneberg | Empirical evidence of regional population concentration in Europe (1870-2000)[END_REF],urban growth and the hierarchy of cities.Several parameters are measured in this context: the demographic impact of rail as a result of population displacement (attraction, concentration, the structuring of space around stations, and the creation of rail hubs 2 ).Studies are generally concerned with functional changes in the focal city and modification of the relationship between it and other cities, as a result of changes in its size or rank in the urban pyramid of the region. In this area, researchers have revealed the effects of demographic thresholds in terms of both density and city size. 3 2 An in-depth study of this topic has been conducted in Wales (Gregory andMartí-Henneberg, 2011) Also of note is Felis-Rota, Martí-Henneberg and Mojica(2012).

3 Denise Pumain(1982) researched the spread of railways and the urban hierarchy in France between 1831 and 1911. Although urban growth was contemporary with the creation of the network, she detailed the correlations between the position of cities in the hierarchy (the rank-size rule), the way

Rail and the growth of cities: some convincing statistical demonstrations

The findings of this interdisciplinary research, which combines history, demography and spatial analysis (Martí-Henneberg2008),clearly reveal the role played by rail networks in urban growth and highlight the influence of demographic changes on the development of railways.

The research generates three findings. First, railways tend to encourage geographical concentration, which leads to very major changes in the structure and hierarchy of the network of cities (Mojica and Martí-Henneberg, 2011). Railways primarily acted as a stimulus to the cities that were served early on, were located at junctions, and had populations of more than 10,000 inhabitants. However, a disaggregate analysis of the demographic variables casts doubt on the commonly held view that the railways hastened rural depopulation and the decline of village communities. As [START_REF] Schwartz | Spatial history: railways, uneven development, and population change in France and Great Britain, 1850-1914[END_REF] explained'it is more the case that rail gave a second chance to rural communities with stable or restricted growth and opened up new cultural and economic horizons for them (for their agriculture, for the export of raw materials, and for timber or firewood)'[page 55]This being so, locations that could not be accessed by train became increasingly marginalised, so that rail did tend to exacerbate geographical inequalities. Finally, recent research has explained that line closures (from the beginning of the twentieth century) did not result in a general reduction in accessibility because the population was becoming concentrated in larger centres that retained their access to the network.

Rail and migration: railways as one factor of rural desertification

Another important topic is the link between the existence of the railway and urban-rural migration. The layman's view is that the railway, whose arrival coincided with the large-scale rural exodus, accentuated the process; however, thorough research involving geographically weighted regression4 shows the contrary. In reality, the more the accessibility of the rural zone by railway increased, the more net migration tended to fall. In France,the migratory dynamic was highly dependent on the station density of the canton and its distance from the they are integrated with the network and the intensity of their growth. Her research shows the importance of the threshold of 10,000 inhabitants:'The larger the city, and the earlier the railway arrived there, the more important it became as a rail hub and the greater its growth'(p. 539). This work also shows that being positioned at the terminus or a junction is more advantageous than being simply an intermediate station.

urban centre. 5 Railways explain approximately 40 per centof migration, and statistical regression shows a wide range of interactions between demographic change, accessibility by rail, and the rural environment.

We therefore refer the reader to a large body of recently published research [START_REF] Morillas | Creation of a geo-spatial database to analyse railways in Europe (1830-2010). A historical GIS approach[END_REF]which emphasises that many methodological difficulties still remain:the availability of data, the stability of municipal boundaries and the historical continuity of statistical series all needattention in order to enhance the general quality of the databases [START_REF] Siebert | Using GIS to map rail network history[END_REF]. The explanatory power of the regressions between demographic and railway data seem to be stronger during the periods when the network was being built and when major human migrations took place (generally, between 1860 and the 1920s) (Martí-Henneberg, 2011).In addition, few studies have examined the redrawing of national boundaries, the reconstruction of the network and the movements of populations (or peoples) that occurred in the aftermath of the Second World War [START_REF] Carter | The German railways an analysis of economic adaptation[END_REF][START_REF] Mellor | Railways and German Unification[END_REF].

Demographic changes and the efficiency of European railways

A caveat: the importance of not confusing demographic projection and long-term forecasting

Let us now turn to the present. Much research has examined the effects of demographic and sociological change on the railway system, but little of this exploits all of the available data.Nevertheless, there are harmonised databases that use a fine-grained spatial scale (at municipal level with global coverage, as in the case of the work performed by Denise Pumain and FrançoisMoriconiEbrard(1997). These allow us to build spatial models of changes in the total population size and make demographic projections on the assumption that observed trends will continueinto the present and the medium-term,and provide information on the processes in question for the next 30 years or so. This data gives a general idea of the number of people who will be living in a given area. Theprojections highlight Europe's low overall demographic growth, and even a clear reduction in the total population, which is obviously unfavourable for the demand for passenger rail transport. Such projections are based on the continuation of the observed trend in the rate of natural increase (the birth rate minus the death rate),and a set of assumptions regarding the arrival of immigrants from outside Europe to make up the natural deficit.

5

'Multiple regression analysis shows that 31 per cent of the variation in the rate of growth between 1831 and 1911 depended on the importance of the city's railway junction, the date of the arrival of the railway and the city's size at that date' (Pumain 1982, pp. 539).

Demographic projections extend dynamics that already exist,without adding new factors. However, they are based on assumptions about the net migration rate, which means that the process is effectively one of forecastingthat is,based on conjectures, which are often linked together to form contrasting narratives or scenarios. The future of railways necessarily involves both projections and scenarios: it requires us to make a forecast about migration, which is a topic that is obviously highly politically sensitive. Moreover, it requires us to work from demographic projections and to enrich these with socioeconomic characteristics: the propensity of future generations to travel, and the extent to which lifestyles will be mobile or immobile, which is in the domain of pure forecasting. Thus, in spite of the apparent freedom the forecasting narrative allows, such work tends to focus on the 27-member European Union,and rarely considersthe space formed by Europe and its neighbours (Banister and Givoni, 2013).It consequently remains very cautious on the topic of future migration.

Demographic change and rail forecasting: benefits and limitations of some scenarios

We have already mentioned the considerable uncertainty that reigns with regard to the ways in which societies and individuals may adapt to new demographic situations. However, we have no alternative but to try to link these masses of data, which are heterogeneous for the systems-dependent reasons described in the first section: the major models of mobility are influenced by the general economic climate and other major political (the distribution of wealth) and environmental (energy and climate) challenges.

The discipline of geographical forecasting, which is currently gaining prominence, occupies the middle ground between scienceand decision-making support for rail operators, and is responsible for two types of overarching narrative [START_REF] Crozet | The Prospects for Inter-Urban Travel Demand: Three scenario families for inter-urban mobility in France to 2050[END_REF]. The first is the idea of 'contented mobility':in this scenario, Europe will open up its metropolises to a wave of immigration that will have the potential to mitigate the effects of population ageing on the jobs market and repopulate rural zones.The intensive use of high-speed transport systems [START_REF] Gutiérrez | The European high-speed train network, predicted effects on accessibility patterns[END_REF]is essential for this scenario, as high-speed corridors, which will have become transnational, will connect capital cities and compete successfully with air transport, whose fuel will have become too expensive [START_REF] Gutiérrez | Location, economic potential and daily accessibility: an analysis of the accessibility impact of the high-speed line Madrid-Barcelona-French border[END_REF]. Some lines will carry businessmen between banking and administrative centres, while others will carry tourists and individuals engaging in leisure activities, which will no longer be seasonal butdesynchronised.Some groups will even manage to 'live within the flows' (Cattan, Frétigny, 2012)and, using new technologies, construct their family lives and interpersonal relations around multiple homes, activities and national affiliations.For these individuals, the new hubs will be intermodal metropolitan stations containing a series of high-value-added business services, connected work areas and luxury hotels.

The other view, which provides a counterpoint to the first, is that of an ageing, impoverished Europe with ever-decreasing levels of mobility (Hoedemaeker2013; [START_REF] Harper | The Implications of an Ageing Population for the Railway :Scoping Research on Ageing for Passengers and the Railway Workforce, Rail Safety and Standards Board Ltd[END_REF]. This is founded on the rapid increase in the percentage of the population aged over 65 years (30 per cent in 2060, compared with 17 per centtoday), the pandemic of obesity, which raises doubts about an increase in life expectancy in good health, increasing solitude and social fragmentation, lacklustre long-term economic growth, rising energy prices, the impoverishment of lower-and middle-income groups, and youth unemployment (Tetraplan A/S Copenhagen, 2007). These factors will combine to lower the total number of trips, as well as leading to a continuous and silent drop in the number of people going away on holiday,as has already been apparent for almost a decade [START_REF] Schäfer | Long-term trends in global passenger mobility[END_REF]. This scenario will have an impact on railway traffic by reducing the number of journeys and the profitability of railway undertakings, which become unable to adequately finance network maintenance.

We have no need to choose between these two meta-scenarios,as they are, above all, political in nature, and as such are simply intended to elicit reactions [START_REF] Givoni | Speed rail: catalyst for european integration?[END_REF].

Essentially, these scenarios apply everywhere and are a-spatial, and, unlike those we shall describe below, take little account of concrete local factors and geographical differentiation.

Demographic change and railway performance: a regional typology

A second way of approaching the interaction between demographic forecasting and the performance of the railway system is to use spatial modelling (Kunz and Buiter, 2010). The principle here involves using geographical images to help us understandthis interaction. The images are produced not only by display systems that represent 'big data' but also narratives based on the statements of experts and varied conjectures.This type of approach allows us to identify three distinct geographical cases that are, to varying degrees,compatible with railway connectivity and accessibility. AsEllwangerandGoerger (2001) explained,'high speed rail would be able to cope extremely well with a non-uniform reduction in density, with dense zones separated by deserted areas '[page 140].This quotation from a UIC study brings us to our first case, that of the enormous urbanised corridor that stretches from Italy to the Rhineland and contains the world's highest population densities, the most integrated metropolitan archipelagos and the highest density of railway lines, both per hectare and per thousand of the population.

ThisEuropean 'heartland' is made up of a set of economically integrated (Euro zone) metropolisesthat are connected together within a robust HSR system with regular services.

The challenges facing this heartland are to forecast increases in traffic, and to modernise in order to be able to cope with mass transit flows [START_REF] Grasland | Study Programme on European Spatial Planning, Final Report, BundesamtfürBauwesen und Raumordnung (BBR) Forschungen[END_REF].The fact that it is impossible for these urban mega-regions to become more urbanised is rarely mentioned (Belgium already has an urbanisation rate of 98 per cent [START_REF] Vanoutrive | Rail commuting to workplaces in Belgium: a multilevel approach[END_REF]).In addition, this heartland contains the regions of Europe that will suffer the most from net population decline due to low fertility. This central area is marked out from a system of marginal zones with distance-related gradation,and the presence of mountain and maritime barriers. The inner marginal zones consist of archipelagos of medium-sized cities, which often receive little in the way of demographic and economic stimulation, for which the efficiency of railways has been brought into question [START_REF] Taylor | Railway closures to passenger traffic in Poland and their social consequences[END_REF].The outermarginal zonescorrespond to non-metropolised rural regions, where the presence of railways is either a historical relic or the outcome of the need to travel through the area or change trains. The areas that contain cities are frequently located in a circle around the Rhineland (the north-eastern part of France, the Polish plain, and the Czech and Hungarian systems of cities (Kosinski-Leszek, 1971)),while the rural marginal zonesare largely empty and provide a location fortranscontinental routes (Portugal-Ukraine, Ireland-Mezzogiorno, the Baltic-Black Sea, etc.).

The influence of the centre-periphery model is evident in this research, from which one main factoremerges: the central position of the re-unified Germany (Buckwalter, 2005), the combination of its industrial and demographic power, the importance of its railways and the implementation of a model that is both centralised (a radial network around Berlin) and polycentric cities regularly dispalyed in a distance of an hour away from one another by train).

Germany is also the heart of european railway system thanks to the grid structure of its network, with lines running east-west (from the northern range ports to Poland and Russia [START_REF] Friebel | Railroad restructuring in Russia and Central and Eastern Europe: one solution for all problems?[END_REF]) and north-south (from Scandinavia to the Mediterranean through the Swiss and Austrian tunnels). These cartographic exercises demonstrate the real advantages that stem from a central position, but they run counter with other analyses that show the concentration of the population in the margins of Europe along the Atlantic and Mediterranean coastal corridors,which is in keeping with the globalisation of the economy and the emergence of city-state port models that are fully integrated within financial flows.6

3.4.Sociodemographic variables and railway performance indicators

Apart from these spatial typologies, which are clearly highly dependent on systems ofrepresentation, there are a certain number of studies (Ellwanger and Georger, 2001)that use demographic indicators to construct railway-performance models. These demographic indicators are not disconnected from the other key indicators of the profitability of railway companies, and always involve the four vital factors of low interest rates, improved services, acceptable rail access charges and dynamic demand [START_REF] Beck | Railway Efficiency, Discussion Paper,International Transport Forum[END_REF]. However, if we look uniquely at demographics, the trends that are identified are double-edged. These trends have the potential either to encourage or discourage general levels of travel, the use of railways (for intercity travel and in Metropolitan zones) and HSR (Francisco and Martí-Henneberg, 2009).

On one hand, considering the lengthening of life in good health provides us with between 150 and 200 million travellers who could potentially use the train for their work-related and leisure trips. These cohorts, who are already alive, are used to travelling, but are also used to cars and using air travel for tourism. It is reasonable to suggest that providing services (in terms of locations, times and pricing) that suit the lifestyles of these people will encourage them to travel by train. Furthermore, we can assume that energy constraints will foster environmentally friendly travel strategies that will include metropolitan, intercity and HSR travel. The growth in tourist flows to European destinations, in particular from emerging countries, is another potential source of rail passengers, with the future number of international visitors estimated at 800 million (twice today's level)(UNWTO, 2010).

There are of course other demographic factors that are less favourable to the increase in travel in general, and railway travel and high-speed travel in particular. The value of this type of exercise is to discuss the way in which demography combines with performance indicators [START_REF] Hooghiemstra | Decision support systems support the search for win-win solutions in railway network design[END_REF] such as capacity and locations served, speeds and time savings, punctuality, the travel environment and the quality of the experience, additional commercial services, etc. [START_REF] Beck | Railway Efficiency, Discussion Paper,International Transport Forum[END_REF].Thus, a given factor is not in itself either positive or negative,but rather is part of a strategic approach. Let us consider the example of the desynchronisation of working hoursand the rhythms of work and leisure. This factor appears to be favourable to the train. At the present time, the success of HSR depends on two contrasting types of use: business trips between Monday and Thursday and leisure trips at weekends. Interaction between the two travel purposes and their spread over the entire week has the potential either to increase passenger load factors or create friction (caused by one type of passenger interfering with the other, inappropriate fares, provision of incompatible services, etc.),which will reduce the quality of service [START_REF] Caldas De Castro | Spatial demography: an opportunity to improve policy making at diverse decision levels[END_REF].

Conclusion and outlook

This chapter shows the value of taking into account the European level when attempting to understand the links between railways, settlement and urban growth. A number of key ideas emerge from this exercise. First, these links were built up over a long period of time, and they have not only shaped a considerable diversity of geographical systems, but also constructed very powerful representations that still have a strong influence on debates on the efficiency of rail transport. Next, we have shown that the interaction between railways and demographic change is extremely dynamic, as the two have evolved together over decades, with periods of rapid readjustment and differences in both the chronology and nature of changes between one country or one region and another.Finally, knowledge about the future is vital for the rail sector, as it takes approximately 10 years to introduce new rolling stock and at least 20 to design and build a high speed line.In spite of its faults and weaknesses, we therefore still need forecasting [START_REF] De Palma | Assessing transport investmentstowards a multi-purpose tool[END_REF].However, it must be improved in two respects:first, greater transparency is required with regard to the construction and handling of the fundamental indicators and variables; andsecond, experts need to make greater use of geographic information systems in models that analyse the performance, vulnerability and resilience of the railway system [START_REF] Tschopp | Accessibiltity, spatial organisation and demography in Switzerland through 1920 to 2000: First Results[END_REF] in order to promote a more geographicallybased approach.

Figure 1

 1 Figure 1 European Railways Passenger's flows in Europe in 1862

Geographically weighted regression (GWR) is an extension of a classical regression method that describes the relation between net migration and one or more explanatory factors. However, while regression produces an estimate of the relation between the variables in the entire study zone, GWR produces a set of estimates derived from the study of spatial units that are adjacent to each other. Thus, instead of giving a single, mean, result GWR produces an estimate of this relation for each spatial unit and shows the geographical variation between the variable and its explanatory factors.

This internal segmentation of the national rail system means that it is difficult to picture the overall competitiveness of a national railway network (see[START_REF] Haywood | Britain's national railway network: fit for purpose in the 21st century?[END_REF] Lapsley,1984).