Keywords: AMS Subject Classification: 34A26, 34C20, 34C45, 34M35, 37C15, 37C85, 37D10, 37D15, 37D30, 37F75, 37G05, 01-02, 01A80 Hyperbolic, action germ, conjugacy, invariant manifold, normal form, flow, Lyapunov function, Cauchy problem, holomorphic vector field, first integral

Two weakly hyperbolic smooth Z k × R m -action germs are smoothly conjugate if and only if they are formally conjugate, and such.

A forgotten theorem on Z k × R m -action germs and related questions Marc Chaperon * -Qu'as tu fait de ta vie, pitance de roi ?

-J'ai vu l'homme.

Je n'ai pas vu l'homme comme la mouette, vague au ventre, qui file rapide sur la mer indéfinie. J'ai vu l'homme à la torche faible, ployé et qui cherchait. Il avait le sérieux de la puce qui saute, mais son saut était rare et réglementé. Sa cathédrale avait la flèche molle. Il était préoccupé.

Henri Michaux

How I met Alain Chenciner and what I did under his supervision

In my first year at the École Normale Supérieure, I was disappointed by mathematicians, who got excited about futile problems instead of sticking to meaningful ones 1 . Hence, I studied computer programming during the second year and mathematical economics during the third. The ensuing boredom 2 took me back to mathematics, which after all I loved or at least loved doing. Then, René Thom's Stabilité structurelle et morphogénèse appeared. Even though I did not understand half of it 3 , I felt strong affinities with the underlying vision of the world and chose to go that way.

After one year of very well paid purgatory as the mathematics teacher of a classe préparatoire 4 , I got-thanks to Thom-a temporary research position at the Centre de mathématiques de l' École Polytechnique, directed by its founder Laurent Schwartz. There, in a nice stimulating atmosphere, I met other members of my mathematical family: Michael R. Herman, who had not yet proven the Arnold conjecture [START_REF] Herman | Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations[END_REF][START_REF] Yoccoz | Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne[END_REF], François Laudenbach, Bernard Teissier and Alain Chenciner; closest to Thom, Alain was to become my doctoral supervisor 5 .

Thom having advised me to start with a thesis in pure mathematics ("much easier than applied mathematics"), my first work was to study, complete and extend his wonderful little paper [START_REF] Thom | Sur les équations différentielles multiformes et leurs intégrales singulières[END_REF] on implicit differential equations, outstanding propaganda for contact geometry and singularity theory. My contribution concerned the singularities of the 1-dimensional characteristic foliation of the submanifold defined by the equation in 1-jet space 6 . Most of it remained unpublished 7 due to work by Takens on constrained systems [START_REF] Takens | Constrained equations; a study of implicit differential equations and their discontinuous solutions. Structural stability, the theory of catastrophes, and applications in the sciences[END_REF], not the same question but quite the same structure. However, thanks to Herman, I had learned much about the analytic or smooth local classification of dynamical systems, widely ignored at the time because of Thom's structural stability program.

A more permanent position at the CNRS enabled me, after an excursion in partial differential equations [START_REF]Modèles microlocaux pour certains opérateurs pseudodifférentiels fuchsiens[END_REF][START_REF]-Singularités en géométrie de contact[END_REF], to work on a conjecture of Camacho, Kuiper and Palis [START_REF] Camacho | The topology of holomorphic flows with singularity[END_REF] on the topological classification of the singular 1-dimensional complex foliations defined by holomorphic vector fields near their zeros.

Dumortier and Roussarie had just proven it to be "almost always" true 8 as a consequence of their smooth simultaneous linearisation result for pairs of commuting smooth vector fields near a common zero [START_REF] Dumortier | Smooth linearization of germs of R 2 -actions and holomorphic vector fields[END_REF]. Since this was close to my brand new competence domain, my program was to simplify their proof, extend it from formally linearisable germs of R 2 -actions to more general germs of Z k × R m -actions 9 and use the extension to prove the conjecture.

Two years later, I had fulfilled this program and defended my thesis [START_REF]Propriétés génériques des germes d'actions différentiables de groupes de Lie commutatifs élémentaires[END_REF]. Alain's help had been unvaluable: though not a specialist, he could understand very quickly what I was doing and give amazingly good advice.

After the excitement of discovery came the much duller task of making my work known. This took me nearly five years 10 and resulted in two publications:

• the book [START_REF]-Géométrie différentielle et singularités de systèmes dynamiques[END_REF], which contains much background material, many unpublished novelties (probably ignored up to now) and my generalisation of the Dumortier-Roussarie linearisation theorem to germs of Z k × R m -actions

• the article [START_REF]-C k -conjugacy of holomorphic flows near a singularity[END_REF], where-among other things-the Camacho-Kuiper-Palis conjecture is proven.

One of the main results of [START_REF]Propriétés génériques des germes d'actions différentiables de groupes de Lie commutatifs élémentaires[END_REF] was not included 11 , hence the present article. As the chief reason for publishing it lies in current work on complete integrability, theorems on first integrals are stated in the end and their proofs are sketched.

1 Statement of the main theorem 1.1 Germs of Z k × R m -actions and their linear part

Given a finite dimensional smooth manifold M and a ∈ M , a smooth germ at a of Z k × R m -action on M is defined by the datum of k smooth diffeomorphism germs g 1 , . . . , g k : (M, a) → (M, a) and m germs X 1 , . . . , X m at a of smooth (i.e. C ∞ ) vector fields vanishing at a, each of which commutes with the others in the sense that   

g i • g j = g j • g i for 1 ≤ i < j ≤ k, g * i X j = X j for 1 ≤ i ≤ k and 1 ≤ j ≤ m, [X i , X j ] = 0 for 1 ≤ i < j ≤ m. (1)
Setting r = k + m and denoting by g t k+j the flow of X j , this defines the homomorphism g : t → g t of Z k × R m into the group of smooth diffeomorphism germs (M, a) → (M, a) given by

g t = g t1 1 • • • • • g tr r for t = (t 1 , . . . , t r ) ∈ Z k × R m .
The endomorphisms L i = dg i (a) and Λ j = dX j (a) of the tangent space E = T a M all commute; they define the linear part of the action germ. Setting L s k+j = e sΛj , this linear part can be viewed as the linear representation L : t → L t of Z k ×R m on E defined by

L t = L t1 1 • • • • • L tr r for t = (t 1 , . . . , t r ) ∈ Z k × R m .
The following proposition will be made precise in subsection 2.1: Proposition 1.1 There exist continuous group homomorphisms a 1 , . . . , a n of Z k × R m into C * and a decomposition of E as the direct sum of subspaces E 1 , . . . , E n , the characteristic subspaces of L, with the following properties: i) For 1 ≤ ≤ n, all the automorphisms L t preserve E and either a is real-valued, in which case each L t induces an automorphism of E having a (t) as its single eigenvalue, or a is not real, and then each L t induces an automorphism of E with the sole eigenvalues a (t), a (t) (which can be real for some t's).

ii) One has a , a = a , a for 1 ≤ < ≤ n.

The set {E 1 , . . . , E n } is determined by L in a unique way.

Note. If (δ 1 , . . . , δ r ) denotes the canonical basis of R r , then:

• for 1 ≤ j ≤ k, the numbers a (δ j ), a (δ j ) are the eigenvalues of L j ;

• for k < j ≤ r, one has a (sδ j ) = e α ,j s for all s ∈ R, where the numbers α ,j , α ,j are the eigenvalues of Λ j-k .

Hyperbolicity, conjugacy, the main theorem

Under the same hypotheses and with the same notation, the group homomorphisms t → ln |a (t)| of Z k × R m into R, intrinsically associated to L, can be extended to linear forms c 1 , . . . , c n on R r = R k+m ; following [START_REF]Linéarisation des germes hyperboliques d'actions différentiables de R k ×Z m : le domaine de Poincaré[END_REF][START_REF]Linéarisation C ∞ des germes d'actions C ∞ de Z k × R m[END_REF], we call the linear action L (and the action germ g)

• weakly hyperbolic if, for 1 ≤ s ≤ r and 1 , . . . , s ∈ {1, . . . , n}, the convex hull conv{c 1 , . . . , c s } does not contain the origin;

• hyperbolic if, for 1 ≤ s ≤ r and 1 ≤ 1 < • • • < s ≤ n, the linear forms c 1 , . . . , c s are linearly independent;

• strongly hyperbolic if it is hyperbolic and, moreover, the a 's are simple in the sense that E is a line when a is real, a 2-plane otherwise.

Still denoting by (δ 1 , . . . , δ r ) the canonical basis of R r , the numbers c (δ j ) are the logarithms of the moduli of the eigenvalues of L j for 1 ≤ j ≤ k, the real parts of the eigenvalues of Λ j-k for k < j ≤ r.

Examples. For r = 1, weak hyperbolicity is just hyperbolicity in the usual sense: if (k, m) = (1, 0) (resp. (0, 1)) the automorphism L 1 (resp. the endomorphism Λ 1 ) has no eigenvalue on the unit circle (resp. the imaginary axis).

A holomorphic vector field germ vanishing at a defines a holomorphic Caction germ, which is the R 2 -action germ defined by X 1 = X and X 2 = iX viewed as real vector fields; if λ 1 , . . . , λ n are the eigenvalues of the complex endomorphism L = dX(a), then E 1 , . . . , E n are the characteristic subspaces of L and the isomorphism λ → (t 1 , t 2 ) → e λ(t 1 + it 2 ) of C onto L(R 2 , R) identifies the λ 's to the c 's; hence, weak hyperbolicity means that no segment [λ , λ ] contains the origin, whereas hyperbolicity means that the eigenvalues λ are two by two R-independent.

Conjugacy. Smooth Z k × R m -action germs g at a ∈ M and g at a ∈ M , defined by g 1 , . . . , g k , X 1 , . . . , X m and g 1 , . . . , g k , X 1 , . . . , X m respectively, are

C α -conjugate, 0 ≤ α ≤ ∞, when there exists a C α diffeomorphism germ h : (M, a) → (M , a ) such that g j = h * g j := h -1 • g j • h for 1 ≤ j ≤ k and X j = h * X j for 1 ≤ j ≤ m (if r = 0, the relation X j = h * X j means that, for suitable representatives, h maps the integral curves of X j to those of X j ). Hence h * g t = (g ) t for all t ∈ Z k × R .
The two action germs are formally conjugate when there exists a smooth diffeomorphism germ h : (M, a) → (M , a ) such that g j and h * g j have infinite contact at a for 1 ≤ j ≤ k and so have X j and h * X j for 1 ≤ j ≤ m. In other words, h * g t and (g ) t have infinite contact at a for all t ∈ Z k × R . Theorem 1.2 Two weakly hyperbolic smooth Z k ×R -action germs are smoothly (i.e. C ∞ -) conjugate if and only if they are formally conjugate.

If r = 1, this is the Sternberg-Chen theorem [START_REF]On the structure of local homeomorphisms of euclidean n-space[END_REF][START_REF] Chen | Equivalence and decomposition of vector fields about an elementary critical point[END_REF]. When L is not weakly hyperbolic, g can be formally but not C 0 -conjugate to L ([10], 6.3, Théorème).

Proof of Theorem 1.2

The "only if" is obvious. To establish the "if", we can fortunately rely on [START_REF]-Géométrie différentielle et singularités de systèmes dynamiques[END_REF][START_REF]-C k -conjugacy of holomorphic flows near a singularity[END_REF].

Hypotheses and notation Same as in section 1. Weak hyperbolicity is not assumed in 2.1 and 2.2, extracted from [START_REF]-Géométrie différentielle et singularités de systèmes dynamiques[END_REF] (to which we refer for details).

Algebraic part of the proof: normal forms

The following result is not so widely known even for one endomorphism:

x = (x 1 , . . . , x n ) = (x 1,p ) 1≤p≤d1 , . . . , (x n,p ) 1≤p≤dn : E → 1≤ ≤c C d × c< ≤n R d such that, for every t ∈ Z k × R m , the real automorphism x * L t = x • L t • x -1 of 1≤ ≤c C d × c< ≤n R d is given by a block-diagonal matrix       T 1 (t) 0 • • • 0 0 . . . . . . . . . . . . . . . . . . 0 0 • • • 0 T n (t)      
and the following properties are satisfied:

i) Each diagonal block T (t) is a lower triangular d × d matrix, with coef- ficients in C for ≤ c, in R otherwise.
ii) For 1 ≤ ≤ n and t ∈ Z k × R m , the diagonal elements of T (t) all equal the same number a (t) ∈ C * .

iii) The continuous group homomorphisms a : Z k ×R m → C * so defined take their values in R * only for > c.

iv) For 1 ≤ < ≤ n, one has a = a = a . v) For 1 ≤ ≤ n, one has T (t) = a (t) exp ε (t)
, where ε is a linear map of R r into the space of lower triangular d × d nilpotent complex matrices, with real coefficients for > c.

In particular, Proposition 1.1 holds with E = = ker x .

A proof. Setting Aj = Lj for 1 ≤ j ≤ c and Aj = Λ j-k for c < j ≤ r, the complex vector space E * C := L(E, C) is the direct sum of the characteristic subspaces F 1, of the C-linear map A * 1 : u → u • A1, each of which is stable under every A * j because AjA1 = A1Aj; hence, each F 1, is the direct sum of the characteristic spaces of the endomorphism induced by A * 2 , yielding a decomposition of E * C as the direct sum of A * j -invariant subspaces F 2, on which the endomorphisms induced by A * 1 and A * 2 have a single eigenvalue. Iterating this procedure, one gets a decomposition of E * C as the direct sum of A * j -invariant subspaces F = F r, on which the endomorphism B j, induced by every A * j has a single eigenvalue λ j, and, moreover, to each corresponds a different (co)vector (λ 1, , . . . , λ r, ).

For each , if d = dim F , the commuting endomorphisms B j, have a common eigenvector x ,1 , which (replacing it by its real or imaginary part) can be chosen real (i.e. in E * := L(E, R)) when every λ j, is real. For d > 1 the same argument, applied in F /Cx ,1 , shows that there exists x ,2 ∈ F , real if every λ j, is, such that each (A * j -λ j, )x ,2 lies in Cx ,1 , and in Rx ,1 if every λ j, is real, etc. This standard triangulation procedure yields Proposition 2.1, except assertion (v), as follows:

• number the F 's so that every λ j, is real for c < ≤ n and λ j, = λ j,n+ , hence

F = F n+ , for 1 ≤ ≤ c
• notice that one can take x n+ ,p = x ,p for 1 ≤ ≤ c

• remark that x = (x1,p) 1≤p≤d 1 , . . . , (xn,p) 1≤p≤dn must be an isomorphism

since ker x = {0} and dim C E * C = dim R E = 2 ≤c d + c< ≤n d .
Finally, Proposition 2.1 v) follows from the fact ([10], 4.3.1, Lemme 2) that for each real vector space F , the map ν → -Id + exp ν is a bijection of the set of nilpotent endomorphisms of F onto itself, whose inverse map is N → ln(Id + N ).

Comments. What is not so widely known is the complex part of the normal form, for which one has to choose one element in each pair (a , a ).

The notation ε indicates that this is the "small" part of L: indeed, one can make it arbitrarily small by multiplying the x ,p 's by positive constants.

Notation. Recall that each L t has a unique Jordan decomposition as the sum of a semi-simple endomorphism S t (meaning that the complexified endomorphism is diagonalisable) and a nilpotent one, commuting with each other. Proposition 2.1 states in particular that the semi-simple parts S t commute with each other and that the isomorphism x diagonalizes them simultaneously. This is the viewpoint we now adopt on normal forms, first in a somewhat cryptic way: Proposition 2.2 ([10], 4.3.2, Théorème 4) The action germ g can be put formally into normal form in the following sense: there exists a smooth diffeomorphism germ h : (M, a) → (E, 0), tangent to the identity at a, such that, for all t , t ∈ Z k × R m , the map germs S t • (h * g t ) and (h * g t ) • S t have infinite contact at 0.

Idea of the proof. To each smooth diffeomorphism germ g : (M, a) → (M, a) is associated the automorphism g * : f → f •g -1 of the algebra of smooth germs f : (M, a) → R. The map g → g * induces for every positive integer s an isomorphism ǧ → ǧ * of the group Ďs of s th order jets (Taylor expansions) at 0 of diffeomorphism germs (M, a) → (M, a) onto the group of automorphisms of the real algebra Ěs of s th order jets at 0 of germs (M, a) → R. Elementary arguments ([10], 4.3.2, proof of Theorème 3) or the Jordan-Chevalley theorem [START_REF] Humphrey | Introduction to Lie algebras and representation theory[END_REF] show that, for each ǧ ∈ Ďs , the semi-simple part of the automorphism ǧ * is itself an automorphism of the algebra Ěs and therefore of the form σ * for a unique σ ∈ Ďs , the semi-simple part of ǧ. Now, the very definition of a semi-simple automorphism of Ěs implies almost immediately ([10], 4.3.2, Théorème 2) that there exists a germ h : (M, a) → (E, 0), tangent to the identity at a, whose s-jet ȟ linearises σ: denoting by S the (semi-simple) differential (1-jet) of σ at a, identified to its s-jet, one has that ȟ * σ = S.

As the semi-simple parts of commuting endomorphisms commute, so do the semisimple parts σt of the s-jets ǧt at a of the germs g t ; hence, as in Proposition 2.1, one can take the same h for all t, so that ȟ * σt = S t (identified to its s-jet) for all t. This passes to the projective limit when s → ∞ and yields (via the Borel extension theorem) the required diffeomorphism germ h: indeed, the infinite jets at 0 satisfy ȟ * σt = S t for all t , hence

S t • ( ȟ * ǧ t ) = ȟ * (σ t • ǧt ) = ȟ * (ǧ t • σt ) = (h * g t ) • S t .
We

a n+ := a : t → a (t) , 1 ≤ ≤ c , 1 ≤ p ≤ d =: d n+ x n+ ,p := x ,p , 1 ≤ ≤ c , 1 ≤ p ≤ d =: d n+ x α = x α1 1 • • • x αn+c n+c := ,p x α ,p ,p , α = (α ) = (α ,p ) 1≤p≤d 1≤ ≤n+c ∈ N d
and, for 1 ≤ ≤ n,

Π := β ∈ N n+c : a = a β := a β1 1 • • • a βn+c n+c , β 1 + • • • + β n+c ≥ 2 P := α ∈ 1≤ ≤n+c N d : µ(α) := (|α | 1 ) 1≤ ≤n+c := p α ,p 1≤ ≤n+c ∈ Π , then, for 1 ≤ ≤ n, 1 ≤ p ≤ d and t ∈ Z k × R m , the Taylor series of x ,p • S -t • h * g t at 0 is of the form x ,p + 1≤q<p e ,p,q (t)x ,q + α∈P f ,p,α (t)x α
where the e ,p,q 's and f ,p,α 's are polynomial functions on R r .

Proof. As h is tangent to the identity, the differential of h * g t at 0 is L t , whose semisimple part is S t ; hence, the linear terms x ,p + q<p e ,p,q (t)x ,q come from exp ε (t) and are of the required form.

Π 0 := β ∈ N n+c {0} : a β = 1};
then, if N n+c is endowed with the ordering "β ≤ β if and only if β ≤ β for all ", one has the following: i) For 0 ≤ ≤ n, the set min Π of minimal elements of Π is finite and every element of Π is comparable to an element of min Π .

ii) Hence, for each β ∈ Π , either β lies in min Π , or it is the sum of an element of min Π and an element of Π 0 .

iii) Thus, if Π 0 = ∅ (which means that L has no non-constant formal first integral), the subset Π = min Π is finite for 1 ≤ ≤ n. In that case, with the notation of Corollary 2.3, the formulae

x ,p • u t = x ,p + 1≤q<p e ,p,q (t)x ,q + α∈P f ,p,α (t)x α , t ∈ R r (2)
define an algebraic action u of R r on E satisfying u t • S t = S t • u t for all t ∈ R r and t ∈ Z k × R m , and Corollary 2.3 states that the action germ g is formally conjugate to the (germ at 0 of the) analytic action g 0 of Z k × R m on E defined by

g t 0 := S t • u t = u t • S t , i.e. x ,p • g t 0 = a (t) x ,p • u t . (3)
Unfortunately, for r > 1 and Π 0 = ∅, formal normal forms do not define model action germs in such an obvious way. However, they do provide something interesting along germs at 0 of special submanifolds of E, to which we now turn.

Strongly invariant manifolds. A preparation lemma

A strongly invariant manifold [START_REF]Linéarisation C ∞ des germes d'actions C ∞ de Z k × R m[END_REF] (abbreviated SIM in the sequel) of the action germ g is a germ W at a of C 1 submanifold which is the (strong) unstable manifold of g t for some t ∈ Z k × R m . Now, this unstable manifold W is smooth, its tangent space T a W is the unstable subspace c (t)>0 E of L t , and W is the only g t -invariant submanifold germ having this tangent space.

It follows that every SIM of g is invariant by all the germs g t and that the SIM's of g form a finite set of smooth submanifold germs.

The Poincaré domain. The following two properties, expressed by saying that L (or g) is in the Poincaré domain, are equivalent: a) the germ at a of the ambient manifold M is a SIM of g b) the convex hull conv{c 1 , . . . , c n } in R r * does not contain the origin (in particular, g is weakly hyperbolic).

Indeed, (b) means that there exists t ∈ R r such that every c (t) is positive and, as Q r is dense in R r , one can take t ∈ Z r .

Theorem 2.5 ([10], 4.4.2b, Théorème 2) If g is in the Poincaré domain then, with the notation of Proposition 2.4, the set Π 0 is empty and g is smoothly conjugate to the analytic action germ g 0 .

Idea of the proof. For α ∈ N n+c {0}, the relation a α = 1 expressing that α lies in Π0 yields j≤c (αj + αn+j)cj + j>c αjcj = 0 and therefore 0 ∈ conv{c1, . . . , cn}, hence Π0 = ∅ in the Poincaré domain. The rest follows from fixed point arguments:

• if E is the unstable subspace of L t 0 then there exists an integer q0 determined by L t 0 such that, for every integer q > q0, the map f → h * g t 0 • f • g -t 0 0 has a unique fixed point h1 in the space of C q germs f : (E, 0) → (E, 0) having q th 0 order contact with the identity at 0; this h1 is smooth, has infinite contact with the identity at 0 and clearly satisfies (h -1 1 • h) * g t 0 = g t 0 0 ; • for all t ∈ Z k × R m , the two germs (h -1 1 • h) * g t and g t 0 have infinite contact at 0 and commute with g t 0 0 ; hence, they coincide, for the germ (h

-1 1 • h) * g t • g -t 0 is the unique fixed point of the map f → g t 0 0 • f • g -t 0 0
in the previous space, namely the germ of the identity;

• similarly, for 1 ≤ k ≤ m, the germ (h -1 1 •h) * Xj and the germ at 0 of d dt g

tδ k+j 0 |t=0 have infinite contact at 0 and are g t 0 0 -invariant; hence, they coincide. This shows that h -1 1 • h is a conjugacy between g and g0.

As two formally conjugate action germs have the same normal form, this yields Corollary 2.6 Theorem 1.2 is true in the Poincaré domain.

Corollary 2.7 If g is hyperbolic and satisfies n ≤ r, then it is C ∞ -linearisable: there exists a C ∞ -conjugacy between g and the germ of L at 0.

Notes. The idea of this proof of Theorem 2.5 is essentially due to Sternberg [START_REF] Sternberg | Local contractions and a theorem of Poincaré[END_REF]. A key point is that infinite contact between h * g t and g t 0 can be replaced by q th 0 order contact; for holomorphic germs of vector fields, this h can be taken holomorphic, hence h 1 is, yielding the holomorphic linearisation theorem in Poincaré's thesis. The same fact, applied to the complexified maps of h * g t and g t 0 with h analytic, implies that the conjugacy in Theorem 2.5 and Corollaries 2.6-2.7 is analytic when g is. The method also provides a C q conjugacy between g and g 0 when g is C q with q > q 0 . All this follows at once from a very simple invariant manifold theorem obtained later [START_REF]-Variétés stables et formes normales[END_REF].

We refer to [START_REF]Linéarisation des germes hyperboliques d'actions différentiables de R k ×Z m : le domaine de Poincaré[END_REF] for a rather explicit proof of the following more difficult result: Corollary 2.8 If g is in the Poincaré domain and strongly hyperbolic, it is C 0 -linearisable: there exists a C 0 -conjugacy between g and the germ of L at 0.

Hypothesis. From now on, L is assumed to be in the Siegel domain, i.e. not in the Poincaré domain 12 .

If Π 0 = ∅, Corollary 2.3 states that formulae (2) and ( 3) define respectively an algebraic action u of R r and an analytic action g 0 of Z k × R m on E. Otherwise, they define formal actions, i.e. homomorphisms R r t → u t and Z k × R m t → g t 0 into the group of infinite jets at 0 of diffeomorphisms (E, 0) → (E, 0).

Proposition 2.9 ([10], 4.4.2b, Proposition 1) For each strongly invariant manifold W of L, the formal actions u and g 0 define formal actions u W and g

W of Z k × R m along W leaving W invariant, i.e. homomorphisms R r t → u t W and Z k × R m t → g t W into
the group of infinite jets at W of diffeomorphisms (E, W ) → (E, W ), as follows:

i) The subspace W of E is the unstable subspace of L t0 for some t 0 ; hence,

W = c (t0)>0 E ; if Ŵ = c (t0)≤0 E , then E = W ⊕ Ŵ ∼ = W × Ŵ so
we can write the elements of E under the form (y, z) ∈ W × Ŵ .

ii) For all s ∈ N, the s th partial derivative

∂ s z u t (y, 0) is a polynomial function of (t, y) ∈ R r × W , hence 13 u t = s∈N 1 s! ∂ s z u t (y, 0)z s and g t 0 = S t s∈N 1 s! ∂ s z u t (y, 0)z s do define infinite jets along W .
iii) The Ŵ -component of u t (y, 0) is zero for all (t, y) ∈ R r × W , implying the rest of our statement since every S t preserves W and Ŵ .

Proof. ii) If Π0 is empty, this is obvious. Otherwise,

• for s = 0, as 0 / ∈ conv{c : c (t0) > 0}, every x α with α ∈ P0 vanishes on W ; hence, by Proposition 2.4 i)-ii), every x ,p • u t coincides with the polynomial x ,p + 1≤q<p e ,p,q (t)x ,q + µ(α)∈min Π f ,p,α (t)x α on W .

• for arbitrary s, again by Proposition 2.4 i)-ii), for 1 ≤ ≤ n and α ∈ P , one has that µ(α) = β + γ1 + • • • + γ s with β ∈ min Π and γ1, . . . , γ s ∈ min Π0 (this decomposition is not unique in general but there are finitely many of them and we can choose s maximal); it follows that ∂ s y x α = 0 on W for s > s, which agains leaves only a finite number of multi-indices α ∈ P with ∂ s y x α = 0. iii) For c (t0) ≤ 0, α ∈ P and β = µ(α), as c = ≤c (β +β n+ )c + c< ≤n β c , some with c (t0) ≤ 0 must satisfy (β + β n+ ) > 0 or β > 0, hence x α |W = 0; as the x ,q 's also vanish on W for c (t0) ≤ 0, this proves what we wanted.

Preparation Lemma 2.10 ([10], 4.4.2b, Théorème 1) The action germ g can be put into normal form along the union V of the SIM's of L: there exists a smooth diffeomorphism germ h ∞ : (E, 0) → (E, 0) having infinite contact with the identity at 0 such that, for every SIM W of L, the infinite jet of (h

-1 ∞ • h) * g t along W is the germ at 0 of g t W for all t ∈ Z k × R m ; it follows that the infinite jet of (h -1 ∞ • h) * X j along W is the germ at 0 of d dt g tδ k+j W | t=0 for 1 ≤ k ≤ m. In particular, the SIM's of (h -1 ∞ • h) * g are the germs at 0 of the SIM's of L.
Idea of the proof. If W is the unstable subspace of L t 0 , let gt 0 W : (E, W ) → (E, W ) be a smooth extension of g t 0 W (its only role is to simplify notation). For every integer s, • taking representatives, the sequence of jets j s (h * g t 0 ) p • (g t 0 W ) -p W p∈N converges in the C ∞ sense near 0 to the s-jet along W of a local diffeomorphism hs,W : (E, 0) → (E, 0), obviously such that j s (h -1 s,W • h) * g t 0 = j s gt 0 W along W : this follows from [START_REF]-Géométrie différentielle et singularités de systèmes dynamiques[END_REF], 4.2.2, Théorème 1 (see the proof of Corollary 2.12);

• for all t ∈ Z k × R m , the two local maps j s (h -1 s,W • h) * g t |W and (j s gt W )|W coincide near 0 and so do, for 1 ≤ k ≤ m, the germ j s (h When s → ∞, the definition domain of (j s hs,W )|W does not shrink, hence in the limit the infinite jet along W of a smooth local diffeomorphism h∞,W : (E, 0) → (E, 0), which has infinite contact with the identity at 0.

One can take the same h∞,W = h∞ for every SIM W of L: indeed, the jets (j ∞ h∞,W )|W corresponding to the various SIM's W of L coincide on their intersections because the previous argument applies when W is the intersection of two SIM's; hence, a mild version of Whitney's extension theorem yields our result.

Notes. For finite s and fixed W , this proof is essentially that of Theorem 2.5 and [START_REF]-Variétés stables et formes normales[END_REF] could be used. In [START_REF]Invariant manifolds and a preparation lemma for local holomorphic flows and actions[END_REF] we show that, for finite s, an analytic action germ can be put into normal form to order s along V by an analytic diffeomorphism germ (this simplifies the exposition in [START_REF]-C k -conjugacy of holomorphic flows near a singularity[END_REF]). Of course, infinite contact is where the analytic and C ∞ theories split apart completely. If r = 1, the result is essentially Sternberg's [START_REF]On the structure of local homeomorphisms of euclidean n-space[END_REF]; the case r > 1 was a novelty with respect to [START_REF] Dumortier | Smooth linearization of germs of R 2 -actions and holomorphic vector fields[END_REF]. The technology of 2.3.2-2.3.3 hereafter makes it possible to obtain normal forms that are the product of S and an action germ commuting with it as in [START_REF] Chen | Equivalence and decomposition of vector fields about an elementary critical point[END_REF], but they are not strikingly good in general for r > 1 and P 0 = ∅.

End of the proof

Hypothesis We now assume g weakly hyperbolic, still in the Siegel domain.

An extension lemma. Proof of Theorem 1.2 for Z-action germs

Extension lemma 2.11 ([10], 4.2.3, théorème 2) 14 Let N be a compact manifold without boundary, E + , E -two nontrivial Euclidean spaces, and let W + , W -, Σ be the submanifolds of Q := N × E + × E -defined by

W + := N × E + × {0}, W -:= N × {0} × E -, Σ := W + ∩ W -= N × {(0, 0)}. Writing (x 0 , x + , x -) the points of Q := N × E + × E -, assume that the smooth diffeomorphism germ ϕ = (ϕ 0 , ϕ + , ϕ -) : (Q, Σ) → (Q, Σ
) leaves invariant the germs of W + and W -at Σ and satisfies the weak normal hyperbolicity condition max max

x∈Σ ∂ϕ + ∂x + (x) , max x∈Σ ∂ϕ - ∂x - (x) -1 < 1. If ψ : (Q, Σ) → (Q, Σ) has infinite contact with ϕ along W + ∪ W -, then every smooth germ 15 h : (Q W -, Σ) → Q W -having infinite contact with the identity along W + Σ and conjugating ψ| Q W -to ϕ| Q W -extends to a smooth germ (Q, Σ) → (Q, Σ
) having infinite contact with the identity along W + ∪ W - and conjugating ψ to ϕ.

Corollary 2.12 ([10], 4.2.3, théorème 2) If ϕ is as in the extension lemma then, for every smooth germ ϕ 1 : (Q, Σ) → (Q, Σ) having infinite contact with ϕ along Σ, i) there exists a smooth conjugacy H : (Q, Σ) → (Q, Σ) of ϕ 1 to ϕ having infinite contact with the identity along Σ

ii) for each smooth diffeomorphism germ

H 0 : (Q, Σ) → (Q, Σ), if H 0 • ϕ and ϕ • H 0 have infinite contact along Σ, then the smooth germs H : (Q, Σ) → (Q, Σ)
conjugating ϕ 1 to ϕ and having infinite contact with H 0 along Σ form an infinite dimensional space-in particular, so do the conjugacies H in (i).

Proof. i) There exists a smooth diffeomorphism germ h1 : (Q, Σ) → (Q, Σ) having infinite contact with the identity along Σ and such that ψ := h * 1 ϕ1 satisfies the hypotheses of the extension lemma: taking representatives of our germs, the jet j ∞ W ± h1 is the limit when

p → ±∞ of j ∞ W ± (ϕ -p 1 • ϕ p ) ([ 10
], 4.2.2, théorème 1). Let us find h as in the extension lemma: taking representatives, we may assume ϕ, ψ defined for some η > 0 on B := {x ∈ Q : max{|x+|, |x-|} ≤ η} and such that sup

B W - max |ϕ+(x)| |x+| , |ψ+(x)| |x+| < 1 < inf B W + min |ϕ-(x)| |x-| , |ψ-(x)| |x-| =: c -1 . (4)
taking a smaller η, we may assume that ϕ, ψ are embeddings and define h by a "Cauchy problem" 16 , i.e. by its restriction h0 to the closure D0 of D0 := B ϕ(B), which must have infinite contact with the identity along W + ϕ(W + ) and have infinite contact with ψ • h0 • ϕ -1 along the inner boundary ϕ(∂ + B) ∩ B, where ∂ + B is the outer boundary {x ∈ B : |x+| = η}; thus, if we look for an h0 having infinite contact with the identity along ∂ + B, it must have infinite contact with ψ • ϕ -1 along ϕ(∂ + B) ∩ B; this defines a Whitney-extendable jet along

∂ + B ∪ W + ϕ(W + ) ∪ ϕ(∂ + B) ∩ B ,
which is indeed the jet of a smooth map h0 : D0 → Q.

We now extend h0 to a smooth conjugacy h of ψ to ϕ defined in B W -∩{|x-| ≤ ρ} for some small enough positive ρ ≤ η:

• by ( 4), the sets D0, D1 := B ∩ ϕ(D0), D2 := B ∩ ϕ(D1), . . . form a partition of B W -, and so do the corresponding sets ∆0, ∆1, . . . for ψ;

• if Dn,ρ := Dn ∩ {|x-| ≤ ρ}, the map hn := h|D n,ρ should satisfy

hn(x) = ψ n • h0 • ϕ -n (x) = ψ • hn-1 • ϕ -1 (x) for n > 0; (5)
• we claim that this does define the required h for small enough ρ.

Indeed, if we equip N with a riemannian metric and B with the product metric, given

α > 1 such that c α Lip ψ < 1, the number R := sup x∈B 0 W + |x-| -α d h0(x), x
is finite since h0 has infinite contact with the identity along W + ; similarly, as ϕ -1 , ψ -1 have infinite contact along W + , the number bρ := sup

0<|x -|≤ρ |x-| -α d ϕ -1 (x), ψ -1 (x)
tends to 0 when ρ → 0. Hence, for small enough ρ ≤ η, the map h0| D0,ρ is an embedding and one has the following:

c + Rc α ρ α-1 ≤ 1 (6) c α + R -1 bρ Lip ψ ≤ 1 (7) ρ + Rρ α ≤ η. (8)
If all this holds, let us prove that each hn is a well-defined map of Dn,ρ into ∆n and sup

x∈Dn,ρ W + |x-| -α d (hn(x), x) ≤ R : ( 9
)
• this is the case if n = 0: indeed, (9) follows from the definition of R; moreover, h0 preserves the outer boundary ∂ + B and maps the inner boundary ϕ( [START_REF]Quelques questions de géométrie symplectique [d'après, entre autres[END_REF] and the definition of R yield

∂ + B)∩B onto ψ(∂ + B) ∩ ψ • ϕ -1 (B); now,
| h0(x)-| ≤ | h0(x)--x-| + |x-| ≤ R|x-| α + |x-| ≤ Rρ α + ρ ≤ η
for all x ∈ D0,ρ; hence, the embedding h0| D0,ρ takes its values in |x-| ≤ η and maps

∂ + B ∪ ϕ(∂ + B) ∩ {|x-| ≤ ρ} into ∂ + B ∪ ψ(∂ + B) ∩ B, yielding h0(D0,ρ) ⊂ ∆0;
let us now prove the same for n > 1, assuming it true for n -1,

• hn is well-defined by ( 5) since ( 4), ( 6) and ( 9) for n -1 yield, for x ∈ Dn,ρ,

|hn-1 • ϕ -1 (x)-| ≤ |hn-1 • ϕ -1 (x)--ϕ -1 (x)-| + |ϕ -1 (x)-| ≤ R|ϕ -1 (x)-| α + |ϕ -1 (x)-| ≤ Rc α |x-| α + c|x-| ≤ Rc α ρ α + cρ = ρ Rc α ρ α-1 + c ≤ ρ ≤ η;
• it does satisfy ( 9) since ( 4), ( 6) and ( 9) for n -1 yield, for x ∈ Dn,ρ,

|x-| -α d (hn(x), x) = |x-| -α d ψ • hn-1 • ϕ -1 (x), ψ • ψ -1 (x) ≤ Lip ψ |x-| α d hn-1 • ϕ -1 (x), ψ -1 (x) ≤ Lip ψ |x-| α d hn-1 • ϕ -1 (x), ϕ -1 (x) + d ϕ -1 (x), ψ -1 (x) ≤ R|x-| -α |ϕ -1 (x)-| α + bρ Lip ψ ≤ R c α + R -1 bρ Lip ψ ≤ R;
• it does map Dn,ρ into ∆n: indeed, as this is assumed true for n -1, it takes its values in ψ(∆n-1) by ( 5), and ( 8)-( 9) yield

|hn(x)-| ≤ |hn(x)--x-| + |x-| ≤ R|x-| α + |x-| ≤ Rρ α + ρ ≤ η
for all x ∈ Dn,ρ, hence our result since ∆n = ψ(∆n-1) ∩ B. As the definition of h0 implies that each hn has infinite contact with the identity along Dn ∩ W + and that hn-1 and hn fit together smoothly along the common boundary of Dn-1,ρ and Dn,ρ, this does define the required h.

ii) Since the germs along W + of the solutions h0 of our extension procedure form an infinite dimensional space, the result is true when H0 is the germ of the identity. Otherwise, notice that H conjugates ϕ1 to ϕ if and only if

H -1 0 • H conjugates ϕ1 to H -1 0 • ϕ • H0 and apply (i) with ϕ := H -1 0 • ϕ • H0.
Notes. The ingredients of this proof are extracted from that of 4.2.2, Théorème 1 in [START_REF]-Géométrie différentielle et singularités de systèmes dynamiques[END_REF], where the result is not proven (nor used) in such generality. It contrasts sharply with the analytic case: if ϕ, ϕ 1 , H 0 are analytic, then H 0 is the only analytic H in our infinite dimensional space: the problem is whether it exists. Corollary 2.13 (Sternberg-Chen) Theorem 1.2 is true for Z-action germs.

Proof. With the notation of the preparation lemma 2.10, identifying each c to c (1) as usual, we can apply Corollary 2.12 to ϕ :

= (h -1 ∞ • h) * g 1 with Σ = {0} and E ± = ±c <0 E , equipped with the Euclidean norm | • | 2 δ := ±c <0 p δ 2p |x ,p | 2 :
indeed, ϕ satisfies the hypothesis of the extension lemma for small enough δ > 0 since

lim δ→0 max ∂ x+ ϕ + (0) δ , ∂ x-ϕ -(0) -1 δ = max 1≤ ≤n e -|c | < 1;
thus, if the action germ g is formally conjugate to g, then g 1 is smoothly conjugate to a germ ϕ 1 : (E, 0) → (E, 0) to which Corollary 2.12 applies. The case of flows, though accessible to this local approach, will now be treated in a somewhat different spirit. The case of real flows If (k, m) = (0, 1) then F is a real-valued Lyapunov function for the linear flow S t , as17

The Lyapunov map

Let F := 1 2 c 1 |x 1 | 2 + • • • + c n |x n | 2 : E → R r * ,
F • S t = 1 2 c 1 |x 1 | 2 e 2c1t + • • • + c n |x n | 2 e 2cnt (10)
and therefore

d dt F • S t t=0 = c 2 1 |x 1 | 2 + • • • + c 2 n |x n | 2 .
Weak hyperbolicity means that none of the c 's is zero, hence the only critical point of F is the origin. As S is in the Siegel domain, the c 's do not all have the same sign; therefore, [START_REF]-Géométrie différentielle et singularités de systèmes dynamiques[END_REF] shows that, along every orbit of the flow, F tends to +∞ (resp. -∞) when t → +∞ (resp. -∞), except when the orbit lies in the stable (resp. unstable) manifold E + := c <0 E (resp. E -:= c >0 E ) of the flow S at 0, in which case the limit of F is 0.

Thus, if b is a regular value of F (i.e. b = 0) and Indeed, there is a unique smooth conjugacy H : E b → E b of ρ to S equal to the identity on Q b : if the smooth functions T, τ : E b → R are defined by S -T (P ) (P ) ∈ Q b and ρ -τ (P ) (P ) ∈ Q b , then H(P ) = ρ T (P ) (S -T (P ) (P )) and H -1 (P ) = S τ (P ) (ρ -τ (P ) (P )), since T • S t = t + T and τ

Q b := F -1 (b), then
• ρ t = t + τ in E b .
In general, this conjugacy H between S and ρ does not extend continuously to the missing invariant subspace E ± = E E b (dramatic example: ρ t = S -t ), but the same idea and the extension lemma yield Corollary 2.14 (Sternberg-Chen) Theorem 1.2 is true for R-action germs.

Proof. If we replace each x ,p by δ p-1 x ,p with δ > 0 small enough, F is a Lyapounov function for the linear flow L and not only for S.

Given an R-action germ g formally conjugate to g, it is formally conjugate to the same normal form g0 as g and therefore, by the preparation lemma, smoothly conjugate to an action germ on E whose infinitesimal generator Y1 has infinite contact with the infinitesimal generator Y0 of (h -1 ∞ • h) * g along E + ∪ E -. Denote again by Yj, j = 0, 1, a smooth local vector field with that germ at 0, by Λ the infinitesimal generator of L and by θ : E → [0, 1] a compactly supported smooth function equal to 1 near 0. For small enough positive η, the formulae Ỹj(y) = Λ(y) + θ(η -1 y) Yj(y) -Λ(y) for η -1 y ∈ supp θ Λ(y) otherwise define two smooth vector fields Ỹ0, Ỹ1 on E with the following properties:

• they have infinite contact along E + ∪ E -;

• their flows φt 0 , φt 1 have E + , E -as stable and unstable manifolds at 0; • the function F is a Lyapunov function for both.

For negative b, it follows as before that there is exactly one smooth conjugacy H :

E b = E E -→ E b of
Ỹ1 to Ỹ0 equal to the identity on Q b ; as it has infinite contact with the identity along E + {0}, the extension lemma applies to the germs h, ϕ, ψ of H, φ1 0 , φ1 1 respectively at 0 since ϕ has the required properties, hence our corollary.

Notes. Via the conjugacy relation, this local extension can then be globalised, a remark that will prove useful in the sequel. If b is negative enough, then φt 0 (y) = φt 1 (y) = L t (y) for all y ∈ Q b and t ≤ 0, implying that the extended H is the limit of φt 0 • φ-t 1 when t → +∞ (of course, t can be restricted to the integers): this is Nelson's approach [START_REF] Nelson | Topics in dynamics. I: Flows. Mathematical Notes[END_REF].

Off W -, the Lyapunov function Proof. An occasion to recall definitions. A critical point of F is an y ∈ E such that DF (y) :

E → R r * is onto. As DF (y)v = c x (y) • x (v) (scalar product) for all y, v ∈ E, it is clear that DF (y)E = DF (y)EI = D(F |E I )(y)
EI for y ∈ EI , implying the first assertion. The rest is pure terminology: if y is a critical point of F , then F (y) is called a critical value of F , hence the second assertion. A regular value is a point in the target space which is not a critical value, hence the last assertion. Formula (10) still holds for r > 1 and has about the same consequences as before:

Proposition 2. [START_REF] Herman | Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations[END_REF] The map F is a "Lyapunov map" for S:

for each y ∈ E, i) the map Ψ y : t → F • S t (y) of R r into R r * is the derivative of the real function f y : t → 1 4 |x (y)| 2 e 2c t , which is convex since its second derivative D 2 f y (t) = |x (y)| 2 e 2c t c ⊗ c is nonnegative for all t;
ii) the bilinear form D 2 f y (t) is positive definite for all t if and only if y does not belong to the S-invariant set

Crit F = {y ∈ E : V J(y) = R r * }, which is indeed the critical set of F since DF (y)R r = V J(y) ; iii) for y ∈ Crit F , it follows that Ψ y = Df y is a diffeomorphism of R r onto
an open subset of R r * , which turns out to be the interior CJ(y) of the closed convex cone C J(y) ; iv) either y lies in the union V of the SIM's of S (which contains Crit F ), or Ψ y is a diffeomorphism of R r onto R r * ; v) the restriction of F to each SIM of S and therefore to V is a proper map.

Proof. i) is obvious.

ii) The nonnegative bilinear form D 2 fy(t) is degenerate if and only if there exists s ∈ R r {0} such that D 2 fy(t)s 2 = 0, i.e. c (s) = 0 for x (y) = 0; this does mean that the c 's with x (y) = 0 belong to the hyperplane {c ∈ R r * : c(s) = 0} for some nonzero s ∈ R r , i.e. that they do not span R r * . To see that DF (y) : δy →

x (δy) • x (y)c maps R r onto V J(y) , notice that DF (y)R r ⊂ V J(y) and that every linear combination ∈J(y) λ c is of the form DF (y)δy with x (δy) = λ x (y)/|x (y)| 2 for ∈ J(y).

iii) The first assertion is classical and the inclusion Ψy(R r ) ⊂ CJ(y) clear; for the whole, see [START_REF]-Géométrie différentielle et singularités de systèmes dynamiques[END_REF], 5.1, proof of Théorème 1. iv) One shoud prove that y ∈ V if and only if CJ(y) = R r * . The "only if" is clear: if y is in the unstable manifold of S t , then either t = 0, yielding y = 0 and CJ(y) = ∅, or one has c (t) > 0 for all ∈ J(y), hence C J(y) is contained in {c : c(t) ≥ 0}; to prove the "if", notice that CJ(y) is then contained in the half-space {c : c(t) > 0} for some nonzero t; now, by weak hyperbolicity, the convex hull of the c 's lying in the hyperplane {c : c(t) = 0} does not contain the origin, hence t can be changed a bit so that they satisfy c (t) > 0, which will in particular be the case for all ∈ J(y).

v) If W is the unstable manifold of S t , then W y → F (y)t is a positive definite quadratic form.

Notes. Proposition 2.16 holds in the Poincaré domain. In the Siegel domain, (iv) implies that F is onto, hence every regular value of F is a value of F .

By weak hyperbolicity, the orbits of S contained in V are precisely those adherent to 0, called the Poincaré leaves of the singular foliation defined by S. ∈ CJ(y) then b / ∈ C J(y) by (i), hence there exists t ∈ R r such that bt < 0 and c t ≥ 0 for all ∈ J(y), implying that F does not take the value b on W := c t≥0 E ; now W contains y and is a SIM of S since, by weak hyperbolicity, the inequalities c t ≥ 0 can be made strict.

iii) By (ii), one has y ∈ E b if and only if b lies in CJ(y) , which is the image of the diffeomorphism Ψy by Proposition 2.16 iii).

2.3.3

Untangling the R r -action S when k = 0 Notation. For 0 ≤ j ≤ r, we identify R j to the subgroup R j × {0} of R r and denote by π j : R r * → R j * the restriction map c → c| R j . The canonical basis of R r is still denoted by (δ 1 , . . . , δ r ).

Proposition 2.18 ([10], 5.2, Propositions 1 and 2) Replacing S t by S At for a suitable automorphism A of R r , one may assume that, for all j ∈ {0, . . . , r} and I ⊂ {1, . . . , n}, the restricted projection π j | V I has maximal rank, which implies the following: i) For 0 ≤ j ≤ r the action S j := S| R j ×E is weakly hyperbolic and the previous properties of S, F hold for S j and F j := π j • F .

ii) For 0 ≤ j < r, if I belongs to

K j := {I ⊂ {1, . . . , n} : dim V I = j and c ∈ V I ⇒ ∈ I} ,
(when S is hyperbolic, I ∈ K j if and only if I has j elements), then there exists one g I ∈ R j such that

π j+1 V I = {c ∈ R j+1 * : c(δ j+1 -g I ) = 0}.
iii) For almost all b ∈ R r , every F j admits b j := π j (b) as a regular value. Hence, if Vbj denotes the union of those SIM's of S j on which F j does not take the value b j , the submanifold Q bj := F -1 j (b j ) is a quotient by S j of the dense S j -invariant open subset E bj := E Vbj ; in other words, the map ϕ bj : Q bj × R j → E bj defined by ϕ bj (y, t) := S t j (y) is a diffeomorphism.

Idea of the proof. This relies on simple general position arguments: see [START_REF]-Géométrie différentielle et singularités de systèmes dynamiques[END_REF], where the subgroups Hj = AR j are considered rather than A itself.

Hypothesis and notation

We assume that S and b have the properties stated in Proposition 2.18. As in [START_REF]-Géométrie différentielle et singularités de systèmes dynamiques[END_REF], for 0 ≤ j < r, we let

I bj := I ∈ K j : Q bj ∩ E I = ∅ .
We denote by ξ : R r → L(E, E) the homomorphism of Lie algebras which to u ∈ R r associates the infinitesimal generator of the linear flow (t, y) → S tu y.

Proposition 2.19 ([10], 5.2, Proposition 3) For 0 ≤ j < r, the action S j+1 | E b j ×R j+1 reads as follows via the diffeomorphism ϕ bj : for all y ∈ Q bj , s, s ∈ R j and t ∈ R,

ϕ * bj S (s,t) j+1 (y, s ) = Φ t bj (y), s + s + t 0 g bj • Φ τ bj (y) dτ ,
where Φ t bj : Q bj → Q bj and g bj : Q bj → R j are defined in the following way:

• Φ t bj is the flow of the vector field on Q bj whose value at y ∈ Q bj is the image of ξ(δ j+1 )y by the projection of E onto T y Q bj along the tangent space ξ(R j )y of the orbit of y by S j ;

• this vector field is of the form y → ξ δ j+1 -g bj (y) y with g bj (y) ∈ R j , which defines g bj .

It follows that the flow Φ t bj is complete and that its orbits are the intersections of Q bj with the orbits of S j+1 .

Proof. Natural.

We now describe the structure of the flow Φ t bj :

Proposition 2.20 ([10], 5.2, Théorèmes 1-2) For 0 ≤ j < r, the real function f bj : y → F (y)δ j+1 on Q bj and the flow Φ t bj have the following properties:

i) The critical set of f bj is Σ bj = {y ∈ Q bj : dim V J(y) = j}, disjoint union of the compact Φ t bj -invariant submanifolds Σ bj ,I := Q bj ∩ E I with I ∈ I bj , which satisfy g bj (Σ bj ,I ) = {g I } and f bj (Σ bj ,I ) = {b j (g I )}.

ii) The restriction of

f bj to Q bj Σ bj is a Lyapunov function for Φ t bj | Q b j Σ b j .
iii) For each y ∈ Q bj , the function ψ y : t → f bj • Φ t bj (y) is bounded from above (resp. below) if and only if y belongs to the stable (resp. unstable) manifold W + bj ,I (resp. W - bj ,I ) 19 of Σ bj ,I for some I ∈ I bj .

I ± 0 := { ∈ {1, . . . , n} : ±c (δ j+1 -g I ) ≤ 0}.
v) The set I∈I b j W + bj ,I ∪W - bj ,I is the intersection of Q bj with the union V j+1 of the SIM's of S j+1 ; hence, it is closed and contains

Q bj ∩ V j . Moreover, f bj | Q b j ∩Vj+1 is proper, Q bj ∩ Vbj+1 = b(δj+1-g I )>0 W + bj ,I ∪ b(δj+1-g I )<0 W - bj ,I (11)
and Q bj+1 is a quotient of Q bj Vbj+1 by the flow Φ t bj .

vi) The set of those y ∈ Q bj for which ψ y :

t → f bj • Φ t bj (y) is bounded (bounded orbits of the flow) is Q bj ∩ V j ; it is compact, contains Σ bj and min f bj (Q bj ∩ V j ) = min I∈I b j b(g I ) , max f bj (Q bj ∩ V j ) = max I∈I b j b(g I )
Proof. i) As bj is a regular value of Fj, every y ∈ Q b j satisfies dim(πjV J(y) ) = j, and the critical set of f b j is Q b j ∩ Crit Fj+1 = {y ∈ Q b j : dim(πj+1V J(y) ) ≤ j}; by the general position hypothesis stated in Proposition 2.18, this is Σ b j . Of course, every y ∈ Σ b j lies in only one Σ b j ,I , defined by I = { : c ∈ V J(y) }; in that case, the formula

DF (y) (ξ(u)y) = ∈J(y) |x (y)| 2 c (u)c , u ∈ R r , (12)
shows that DF (y) (ξ(δj+1 -gI )y) = 0, hence ξ(δj+1 -gI ) ∈ ker DFj(y) = TyQ b j and therefore g b j (Σ b j ,I ) = {gI }. For y ∈ Σ b j ,I , one does have

f b j (y) = 1 2 ∈I |x (y)| 2 c (δj+1) = 1 2 ∈I |x (y)| 2 c (gI ) = Fj(y)gI = bj(gI ).
Finally, each Σ b j ,I is a submanifold since bj is a regular value of Fj|E I by Propositions 2.15 and 2.18; it is compact because it lies in Q b j ∩ Vj, see the proof of (vi). ii) For y ∈ Q b j , the relation ξ δj+1 -g b j (y) y ∈ TyQ b j and ( 12) yield

0 = DFj(y) ξ δj+1 -g b j (y) y g b j (y) = ∈J(y) |x (y)| 2 c δj+1 -g b j (y)c c g b j (y) ,
hence, again by [START_REF]-C k -conjugacy of holomorphic flows near a singularity[END_REF],

Df b j (y) ξ δj+1 -g b j (y) y = ∈J(y) |x (y)| 2 c δj+1 -g b j (y) c (δj+1) = ∈J(y) |x (y)| 2 c δj+1 -g b j (y) 2 ;
this is positive unless πj+1c vanishes on δj+1 -g b j (y) for all ∈ J(y), in which case dim(πj+1V

J(y) ) ≤ j, that is, y ∈ Q b j ∩ Crit Fj+1 = Σ b j .
iii) The "only if" being clear, we prove the "if". If ψy(t) = f b j • Φ t b j (y) is bounded from above (resp. below) then, by Proposition 2.19, the map R j+1 s → Fj+1(S s y) is not onto (it does not take every value c ∈ R j+1 * with c| R j = bj) and therefore y ∈ Q b j ∩ Vj+1 by Proposition 2.16 iv). As Fj+1|V j+1 is proper by Proposition 2.16 v), so is f b j |Q b j ∩V j+1 ; hence, there exists a real sequence (tp) tending to +∞ (resp. -∞) such that Φ tp b j (y) tends to a limit z ∈ Q b j . We now use the following obvious result: Lemma 2. [START_REF] Sternberg | Local contractions and a theorem of Poincaré[END_REF] One has E bj = I∈I b j E bj ,I . For all I ∈ I bj , if

I ± := I ± 0 I, i) a quotient of the S-invariant subset E bj ,I by S j is the submanifold Q bj ,I that reads Σ bj ,I × E I + × E I -in the identification of E = E I ⊕ E I + ⊕ E I - to E I × E I + × E I -; ii) the image of Φ t bj | Q b j ∩E b j ,I by the diffeomorphism C bj ,I : Q bj ∩E bj ,I → Q bj ,I obtained by following the orbits of S j is Φ t bj ,I := S t(δj+1-g I ) | Q b j ,I ;
iii) In the identification (i), the flow Φ t bj ,I splits:

Φ t bj ,I (y 0 , y + , y -) = Φ t bj (y 0 ), S t(δj+1-g I ) y + , S t(δj+1-g I ) y -, and the linear endomorphism y ± → S t(δj+1-g I ) y ± of E I ± is a strict contraction for ±t > 0; in particular, the stable and unstable manifolds Y +

j ,I = C -1 b j ,I (Y ± b j ,I ) = Q b j ∩ E b j ,I ∩ E I ± 0 . v)
We have just proven the inclusion I∈I b j W + b j ,I ∪W - b j ,I ⊂ Q b j ∩Vj+1; to establish equality, remember that if y ∈ Q b j belongs to some v.f.i. of Sj+1, then the interior of πj+1C J(y) differs from R j+1 * by Proposition 2.16 iv). If this interior is empty, i.e. dim πj+1V J(y) ≤ j, then y ∈ Σ b j . Otherwise, as Fj(y) = bj and y / ∈ Crit Fj, the interior of πjC J(y) contains bj, hence one of the j-dimensional faces πj+1CK , K ⊂ J(y), of the closed convex cone πj+1C J(y) must satisfy bj ∈ πjCK ; if I is the element of I b j such that VI = VK then, as πj+1C J(y) lies in one of the closed half-spaces bounded by πj+1VI , we have y , i.e. c (δj+1 -gJ ) ≤ 0 ≤ c (δj+1 -gI ) for all ∈ J(y), implying that y belongs to the SIM c (g J -g I )≥0 of Sj; in the first case I = J, one has y ∈ Σ b j ,I and therefore the c 's with ∈ J(y) lie in the j-dimensional subspace VI where, by weak hyperbolicity, they are contained in an open half-space H with 0 ∈ ∂H; as πj|V I has maximal rank, every πjc with ∈ J(y) lies in the open half-space πjH of R j * , proving that y ∈ Vj. The bounds of f b j (Q b j ∩ Vj) follow at once from (iii).

∈ Q b j ∩ E b j ,I ∩ E I ± 0 = W ± b j ,I . The function f b j |Q b j ∩V j+1 is proper because Fj+1|V j+1 is. Finally, for y ∈ Q b j , it

Proof of Theorem 1.2 for R r -action germs

Even though we could remain at the local level as in [START_REF]-C k -conjugacy of holomorphic flows near a singularity[END_REF] or in the proof of Corollary 2.12, it will be more comfortable to extend our action germs into genuine actions with properties very close to those of S; the required conjugacy will then be the solution of a Cauchy problem as in the proof of Corollary 2.14.

Hypothesis and notation. We still assume that S and b have the properties stated in Proposition 2.18. For 0 ≤ j < r, we let

m j := min b(δ j+1 ), min I∈I b j b(g I )
and M j := max b(δ j+1 ), max

I∈I b j b(g I ) , hence Q bj ∩ V j ⊂ f -1 bj ([m j , M j ]
) by Proposition 2.20 vi).

Lemma 2.22 Given a smooth function θ r :

Q br = Q b → [0, 1] with compact support, equal to 1 near Q b ∩ V, one defines inductively a compactly supported smooth function θ = θ 0 : E → [0, 1] as follows: i) Its restriction to Q br is θ r .
ii) For 0 ≤ j < r, its restriction θ j to Q bj is determined from θ j+1 by the formula

∀y ∈ Q bj +1 ∀t ∈ R θ j Φ t bj (y) = κ j f bj Φ t bj (y) θ j+1 (y)
where κ j ∈ C ∞ (R, [0, 1]) has compact, connected support and equals 1 on a compact interval K j containing [m j , M j ] in its interior.

For 0 ≤ j < r, the function θ j is equal to 1 in a neighbourhood of Q bj ∩ V j ; in particular, θ = 1 near 0.

Proof. We use the following

Fact For 0 ≤ j < r, if N is a compact neighbourhood of Q b j+1 ∩ Vj+1 in Q b j+1 and K a compact interval with bj+1 ∈ K, then the closure f -1 b j (K) ∩ t Φ t b j (N ) ∪ Vb j+1 of f -1 b j (K) ∩ t Φ t b j (N ) is a compact neighbourhood of f -1 b j (K ) ∩ Vj+1 in Q b j for every compact interval K ⊂ K; in particular, for [mj, Mj] ⊂ K, it is a compact neighbourhood in Q b j of Q b j ∩ Vj.
This fact implies our lemma: indeed, for N = supp θj+1 and K = supp κj, it shows that supp θj is compact; for N = θ -1 j+1 (1) and supp κj ⊂ K, it shows that θj is smooth, being equal to κj • f b j near the only litigious part, namely Vj+1 ∩ Q b j ; for N = θ -1 j+1 (1) and K = Kj, one gets that θj = 1 near Q b j ∩ Vj.

The previous fact follows immediately from Proposition 2.20 ii-iii) if K does not contain any critical value of f b j , but it remains true otherwise because Φ t b j is (normally) hyperbolic at each Σ b j ,I (see step 3 in the proof of the globalisation lemma 2.24).

Notation

• We fix θ as in Lemma 2.22 and let B be a bounded open subset of E containing supp θ, hence 0 ∈ B.

• With the notation of the preparation lemma 2.10, we let Y 1 , . . . , Y r be commuting smooth vector fields on an open neighbourhood of 0 in E, whose germs at 0 are (h

-1 ∞ • h) * X 1 ,. . . , (h -1 ∞ • h) * X r ;
if ξ is as in the previous paragraph, the semi-simple part of Λ j = DY j (0) is ξ j := ξ(δ j ).

• For η > 0, we define the weighted zooming map ζ η : E → E by x ,p • ζ η := η p-2d-1 x ,p , where d := max d . Lemma 2.23 When η tends to 0, the vector fields ζ η * Y j -ξ j tend uniformly to 0 on B and so do their derivatives at all orders.

Proof. Since ζη * ξj = ξj and Yj = Λj + Rj with j 1 Rj(0) = 0, it is enough to check that a) ζη * Rj and its derivatives tend uniformly to 0 on B and b) each linear vector field ζη * (Λj -ξj) tends to 0 in L(E, E).

As Λj -ξj is in triangular form x ,p • (Λj -ξj) = 1≤q<p ε ,p,q x ,q , x ,p • ζη • (Λj -ξj) = η p-2d-1 1≤q<p
ε ,p,q x ,q = 1≤q<p η p-q ε ,p,q x ,q • ζη, hence x ,p • ζη * (Λj -ξj) = 1≤q<p η p-q ε ,p,q x ,q , proving b). By Taylor's formula, Rj = m,q,m ,q xm,qx m ,q a m,q,m ,q near 0 ∈ E with a m,q,m ,q smooth, hence

x ,p • ζη * Rj (y) = η p-2d-1 m,q,m ,q η 4d+2-q-q xm,q (y) x m ,q (y) a m,q,m ,q ζ -1 η (y) = m,q,m ,q η 2d+1+p-q-q xm,q (y) x m ,q (y) a m,q,m ,q ζ -1 η (y) ,

which tends uniformly to 0 on B as well as its derivatives, since ζ -1 η (y) does and q + q -p ≤ dm + d m -0 ≤ 2d, in other words 2d + 1 + p -q -q ≥ 1.

The following result shows in particular that every weakly hyperbolic smooth R r -action germ on E is the germ of a smooth R r -action.

Globalisation lemma 2.24 For small enough η > 0, the formulae

∀y ∈ Q bj-1 Ỹj (y) = ξ j (y) + θ(y)(ζ η * Y j -ξ j )(y) for y ∈ supp θ ξ j (y) otherwise ( 13
)
define the generators Ỹj :

= d dτ gτδj | τ =0 of a unique smooth action g of R r on E, whose germ at 0 is (ζ η • h -1 ∞ • h) * g: for each j, the germ of Ỹj at 0 is the germ (ζ η • h -1 ∞ • h) * X j of ζ η * Y j .
This action possesses (for our fixed b and small enough η) all the properties of S stated in Propositions 2.18 iii), 2.19 and 2.20-which will be detailed in the proof.

Proof. For small enough η, every vector field ζη * Yj is well-defined on B, hence (13) does define a vector field Ỹj on E over Q b j-1 .

The expression"For small enough η > 0 and 0 ≤ j < r," is implicit at each of the following steps.

Step 1 One defines a smooth vector field Z b j on Q b j as follows: each Z b j (y) is the projection of Ỹj+1(y) in TyQ b j along 1≤i≤j R Ỹi(y). The flow Ψ t b j of Z b j is complete and preserves the intersection of Q b j with every SIM W of S.

Indeed, with the notation of Lemma 2.22, recall that supp θj := Q b j ∩ supp θ; for y ∈ Q b j supp θj, one has Ỹi(y) = ξj(y) for 1 ≤ i ≤ j + 1, hence Z b j (y) = ξ δj+1 -g b j (y) ; on supp θj, every Ỹi with 1 ≤ i ≤ j converges uniformly to ξi when η → 0; now, as Q b j is a quotient of E b j by Sj, the vectors fields ξi with 1 ≤ i ≤ j are linearly independent at every point of supp θj, hence so are, for small enough η, the vectors fields Ỹi with 1 ≤ i ≤ j, implying that Z b j is well-defined and smooth. The flow Ψ t b j , having the same generator as the complete flow Φ t b j off the compact subset supp θj, is complete. It preserves Q b j ∩ W for each SIM W of S because every ξi and every ζη * Yi| supp θ j with i ≤ j + 1 is tangent to W .

Step 2 The restriction of

f b j to the Ψ t b j -invariant subset Q b j Σ b j is a Lyapunov function for the flow Ψ t b j |Q b j Σ b j . Indeed, by Step 1, every Σ b j ,I = Q b j ∩ E I + 0 ∩ E I - 0 with I ∈ I b j is Ψ t b j -invariant since E I + 0 , E I - 0 are SIM's of S. We should verify that LZ b j f b j (y) = df b j (y)Z b j (y)
is positive off Σ b j . By Proposition 2.20 ii), this is true in Q b j supp θj, where Z b j generates Φ t b j ; by Lemma 2.23 and Proposition 2.20 ii), for every open subset V ⊃ Σ b j of Q b j , the function LZ b j f b j is positive on supp θj V for small enough η; hence, the problem is to find an open subset V ⊃ Σ b j of Q b j such that LZ b j f b j is positive on V Σ b j for small enough η. This follows essentially from the fact that each Σ b j ,I is a critical submanifold of LZ b j f b j along which the Hessian of LZ b j f b j is positive definite in the normal direction E I + ⊕ E I -for small enough η.

Let us prove this: for y ∈ Q b j , since Fj(y) = bj and c (δj+1 -gI ) = 0 for ∈ I,

f b j (y) = bj(gI ) -Fj(y)gI + f b j (y) = bj(gI ) + 1 2 / ∈I c (δj+1 -gI )|x (y)| 2 (14)
hence, denoting the standard Euclidean scalar product by a dot,

LZ b j f b j (y) = / ∈I c (δj+1 -gI ) x (y) • x Z b j (y) .
For y ∈ Σ b j ,I , as Z b j (y) is tangent to Σ b j ,I , every x Z b j (y) (and of course x (y)) with / ∈ I vanishes, yielding

d LZ b j f b j (y) = / ∈I c (δj+1 -gI ) x (dy) • x Z b j (y) + x (y) • d x • Z b j (y) = 0 and implying that the Hessian D 2 (LZ b j f b j )(y) is the quadratic form v → 2 / ∈I c (δj+1 -gI ) x (v) • d x • Z b j (y)v;
when η → 0, this quadratic form tends uniformly on Σ b j ,I to its analogue for the flow Φ t b j , namely v → 2 / ∈I c (δj+1 -gI ) 2 |x (v)| 2 ; as this is positive definite on E I + ⊕ E I -, so is D 2 (LZ b j f b j )(y) for all y ∈ Σ b j ,I when η is small enough.

To conclude, writing the points of E b j ,I under the form (y, z) with y ∈ Σ b j ,I and z ∈ E I + ⊕ E I -and remembering that LZ b j f b j vanishes to order 1 along Σ b j ,I , Taylor's formula writes C b j ,I * LZ b j f b j (y, z) = χy,z(z) where χy,z is the quadratic form

1 0 (1 -t)D 2 C b j ,I * LZ b j f b j (y,
tz)dt; when η → 0, this quadratic form tends uniformly near Σ b j ,I to its analogue for S, which is positive definite if the neighbourhood is small enough since, by what we have just done, it is positive definite on Σ b j ,I .

Step 3 With the notation of Lemma 2.21, the germ of C b j ,I * Ψ t b j at Σ b j ,I satisfies the hypotheses of the extension lemma 2.11 for all I ∈ I b j and t = 0, with

W + = Y ± b j ,I and W -= Y ∓ b j ,I for ±t > 0. Hence, a neighbourhood basis of Σ b j ,I in Q b j consists of the compact subsets Bε,V := f -1 b j ([b(gI ) -ε, b(gI ) + ε]) ∩ t Ψ t b j (V ) ∪ f -1 b j ([b(gI ), b(gI ) + ε]) ∩ W - b j ,I
with ε > 0 small and V a small compact neighbourhood of

W + b j ,I ∩ f -1 b j (b(gI ) -ε) in f -1 b j (b(gI ) -ε). For y ∈ Bε,V , the function t → f b j • Ψ t b j (y) takes the value b(gI ) ± ε if and only if y / ∈ W ± b j ,I .
Indeed, since Ψ t b j preserves the intersections of Q b j with the SIM's E I ± 0 of S, the germs at Σ b j ,I of their images by C b j ,I are preserved by C b j ,I * Ψ t b j . Hence, as Ψ t b j tends in the C 1 sense to Φ t b j near Σ b j ,I when η → 0, the first part of our statement follows from the analogous fact for Φ t b j (last assertion of Lemma 2.21). We now deduce the second part in a more or less standard way: the boxes

Bρ = C -1 b j ,I y ∈ Q b j ,I : max ∈I + |x (y)| 2 , ∈I - |x (y)| 2 ≤ ρ 2
with ρ > 0 small enough form a neighbourhood basis of Σ b j ,I in Q b j and, for all y ∈ Bρ, the orbit Ψ t b j (y) leaves Bρ for some t with ±t ≥ 0 through

∂ ± Bρ = y ∈ Bρ : ∈I ∓ |x • C b j ,I (y)| 2 = ρ 2
unless y ∈ W ± b j ,I , in which case Ψ t b j (y) remains in Bρ for ±t ≥ 0 and tends to Σ b j ,I . Moreover 20 , the closures Cρ(N ) of the subsets

Bρ ∩ t Ψ t b j (N ) with N a neighbourhood of ∂ -Bρ∩W + b j ,I in ∂ -Bρ form a neighbourhood basis of (W + b j ,I ∪W - b j ,I )∩Bρ in Bρ; thus, when N is small enough, one has max f b j (N ) < b(gI ) < min f b j Cρ(N ) ∩ ∂ + Bρ , since max f b j (W + b j ∩ ∂ -Bρ) < b(gI ) < min f b j (W - b j ∩ ∂ + Bρ)
by, e.g., [START_REF] Chen | Equivalence and decomposition of vector fields about an elementary critical point[END_REF]; it follows that, for max

f b j (N ) < b(gI ) -ε < b(gI ) + ε < min f b j Cρ(N ) ∩ ∂ + Bρ the subset Cρ(N ) ∩ f -1 b j ([b(gI ) -ε, b(gI ) + ε]) of Bρ is a neighbourhood of Σ b j ,I of the form Bε,V with V = Cρ(N ) ∩ f -1 b j (b(gI ) -ε).
Step 4 For each y ∈ Q b j , the function χy : t → f b j • Ψ t b j (y) is bounded from above (resp. below) if and only if y belongs to the stable (resp. unstable) manifold of Ψ t b j at Σ b j ,I for some I ∈ I b j . These global stable and unstable manifolds are the same as for

Φ t b j , namely W + b j ,I = Q b j ∩ E b j ,I ∩ E I + 0 and W - b j ,I = Q b j ∩ E b j ,I ∩ E I - 0 . Hence, Q b j+1 is a quotient of Q b j Vb j+1 by the flow Ψ t b j .
Indeed, if χy is bounded, say, from above, then Ψ t 0 b j (y) ∈ supp θj for some t0 since, otherwise, Ψ t b j (y) = Φ t b j (y) could not tend to any Σ b j ,I , which lies inside supp θj, yielding the contradiction χy(t) → +∞ by Proposition 2.20 iii); we claim that Ψ t b j (y) remains in supp θj for all t ≥ t0: it cannot escape through the "lateral" part of the boundary of supp θj that consists of segments of orbits of Φ t b j coming from the boundary of supp θj-1, which are segments of orbits of Ψ t b j because there θ = 0; the only remaining escape, through the part of the boundary where f b j is maximal, is also excluded because Ψ t b j (y) would never come back and therefore χy(t) → +∞ as before. Now χ y (t) = LZ b j f b j Ψ t b j (y) accumulates for t ≥ t0 to 0 hence, by Proposition 2.20 i), Ψ t b j (y) accumulates to Σ b j ,I and therefore limt→+∞ χy(t) = b(gI ) for some I ∈ I b j ; it does follow that Ψ t b j (y) ∈ W + b j ,I for all large enough t, since it must enter some neighbourhood Bε,V as in Step 3 and χy(t) cannot take the value b(gI ) + ε.

As Ψ t b j (y) ∈ W + b j ,I for large t and W + b j ,I is included in the Ψ t b j -invariant manifold

E I + 0 ∩ Q b j , we have y ∈ E I + 0 ∩ Q b j
and there remains to show that y ∈ E b j ,I ; now, by 20 See for example [START_REF]-C k -conjugacy of holomorphic flows near a singularity[END_REF], (2.2), Isolating Block Lemma (iii).

Proposition 2.20 v), the closed subset E

I + 0 ∩ Q b j E b j ,I = E I + 0 ∩ Q b j W + b j ,I equals E I + 0 ∩ J =I W + b j ,J ∪ J W - b j ,J W + b j ,I
; as it is closed, this yields

E I + 0 ∩ Q b j W + b j ,I = E I + 0 ∩ J =I Q b j ∩ E J + 0 ∪ J Q b j ∩ E J - 0 (since W + b j ,J is dense in Q b j ∩ E J + 0 and W - b j ,J W + b j ,I dense in Q b j ∩ E J - 0 for all J ∈ I b j ), implying that E I + 0 ∩ Q b j W + b j ,I is Ψ t b j -invariant and therefore y ∈ W + b j ,I .
Step 5 For small η, if Nr is a compact neighbourhood of Q b ∩V in Q b (which we choose to be a submanifold with boundary), one defines inductively a compact neighbourhood Nj of Q b j ∩ Vj in Q b j for 0 ≤ j < r, which is a submanifolds with corners, as follows:

Nj is the closure f -1 b j (Kj)∩ t Ψ t b j (Nj+1) ∪ Vb j+1 of f -1 b j (Kj)∩ Ψ t b j (Nj+1)
. When η and Nr are small enough, the compact neighbourhood N0 of 0 in E is contained in θ -1 (1) and therefore so is

Nj = N0 ∩ Q b j for 0 ≤ j ≤ r.
Indeed, the first part of the statement is the analogue for Ψ t b j of the fact used in the proof of Lemma 2.22 and its proof is the same. The second part follows from the fact that every Nj tends to its analogue for Φ t b j when η → 0.

Hypothesis. We fix η, Nr as in step 5 so that N0 ⊂ θ -1 (1) and conclude by induction:

Step 6 For 1 ≤ j ≤ r, the vector fields Ỹi defined by (13) on Q b i-1 for 1 ≤ i ≤ j extend in a unique fashion to commuting smooth vector fields on E, whose germs at 0 are

(ζη • h -1 ∞ • h) * X1, . . . , (ζη • h -1 ∞ • h) * Xj.
More precisely, Ỹi = ζη * Yi on N0. The vector fields Ỹ1, . . . , Ỹj define a smooth action gj of R j on E preserving every SIM of S and possessing (for our fixed b) all the properties of Sj stated in Propositions 2.18 iii), 2.19 and 2.20. In particular, the map ψ b j :

Q b j × R j → E defined by ψ b j (y, t) := gt j (y) is a diffeomorphism onto E b j .
This is obvious if j = 1, as (13) defines a vector field Z b 0 := Ỹ1 on Q b 0 = E equal to ζη * Y1 on θ -1 (1) ⊃ N0 and for which we have just proven our statement.

Assuming the result established for some j < r, let us prove it for j + 1. The action gj+1 must read as follows via ψ b j : for all y ∈ Q b j , s, s ∈ R j and t ∈ R,

ψ * b j g(s,t) j+1 (y, s ) = Ψ t b j (y), s + s + t 0 s b j • Ψ τ b j (y) dτ (15)
where Ψ t b j is the flow of Z b j and s b j (y) is the element (σ1, . . . σj) of R j such that Z b j (y) = Ỹj+1(y) -1≤i≤j σi Ỹi(y). The formula [START_REF] Dumortier | Smooth linearization of germs of R 2 -actions and holomorphic vector fields[END_REF] defines an action gj+1 of R j+1 on the dense open subset E b j (its extension to R j+1 × E will therefore be unique), whose generators are Ỹ1, . . . Ỹj (already extended to E) and the vector field Ỹj+1 on E b j given by ψ * b j Ỹj+1(y, s ) = Z b j (y), s b j (y) , i.e.

∀y ∈ Q b j ∀s ∈ R j gs * j Ỹj+1(y) = Ỹj+1(y). [START_REF] Herman | Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations[END_REF] we should show that it extends smoothly to E so that Ỹj+1 = ζη * Yj+1 on N0. We claim that (16) defines a vector field Ỹj+1 on E b j equal to ζη * Yj+1 on N0 ∩ E b j = N0 Vb j ;

this will prove our result since Ỹj+1 will extend by density to a smooth vector field Ỹj+1 equal to ζη * Yj+1 on N0, which will extend smoothly to the rest of Vb j because it is invariant by gj.

To establish our claim, first recall that Ỹj+1 = ζη * Yj+1 on θ -1 j (1) ⊃ Nj by (13) with j := j + 1. Now, by definition of N0, the subset N0 ∩ E b j consists of points y0, each of which is obtained from a unique yj ∈ Nj by concatenating the paths [0, ti] t → Ψ t b i-1 (yi) ∈ Ni from yi ∈ Ni to yi-1 := Ψ t i b i-1 (yi) ∈ Ni-1 for j ≥ i ≥ 1; as [START_REF] Dumortier | Smooth linearization of germs of R 2 -actions and holomorphic vector fields[END_REF] with j := i -1 yields

Ψ t b i-1 (yi) = g(-t 0 s b i-1 •Ψ τ b i-1 (y i ) dτ , t) i (

End of the proof

Hypotheses and notation. Given an R r -action germ g formally conjugate to g, it is formally conjugate to the same normal form g 0 as g and therefore, by the preparation lemma, smoothly conjugate to an action germ on E whose infinitesimal generators possess representatives Y 1 , . . . , Y r having infinite contact with Y 1 , . . . , Y r along V. If we apply to them the globalisation lemma 2.24 with the same η and θ as for Y 1 , . . . , Y r (which is possible if η is chosen small enough), we get an R r -action ρ on E with the same properties as g, having infinite contact with g along R r × V.

Lemma 2.25 There exists a unique smooth conjugacy h : E → E of g to ρ equal to Id E on Q b , and it has infinite contact with the identity along V. In particular, Theorem 1.2 is true for R r -action germs.

χ b (y, t) = ρ t (y), the conjugacy relation h • gt (y) = ρ t • h(y) reads h • gt (y) = ρ t (y) for y ∈ Q b , i.e. h • ψ b = χ b , hence h|E b = χ b • ψ -1 b .
As for flows, this defines a smooth diffeomorphism of E b onto itself, conjugating g| R r ×E b to ρ| R r ×E b . Moreover, as g and ρ have infinite contact along R r × V, the diffeomorphism h|E b has infinite contact with the identity along V ∩ E b . Since E b is dense, uniqueness follows and all we have to show is that the bijection h : E → E defined by

h(y) = χ b • ψ -1 b (y) for y ∈ E b y
for y ∈ Vb is smooth and has infinite contact with the identity along V. We will prove inductively that h|E b j is smooth, maps E b j onto itself and has infinite contact with the identity along V ∩ E b j = V Vb j for 0 ≤ j ≤ r, hence Lemma 2.25 for j = 0, as

E b 0 = E.
By what we have just done, the result is true if j = r; given j < r, we now prove it for j assuming it true for j + 1. First note that, as h is a bijection equal to the identity on Vb ⊃ Vb j , we do have h(E b j ) = E b j .

Let ψ b j , χ b j : Q b j × R j → E b j be the diffeomorphisms given by ψ b j (y, t) = gt (y) and χ b j (y, t) = ρ t (y). If we set

h b j := χ -1 b j • h • ψ b j and, for y ∈ Q b j , h 0 (y), h 1 (y) := h b j (y, 0) = χ -1 b j • h(y), then for s ∈ R j the relation h • gs (y) = ρ s • h(y) reads h b j (y, s) = h 0 (y), s + h 1 (y) (17)
and, for t ∈ R, the relation h • g(s,t) (y) = ρ (s,t) • h(y) writes

h b j • ψ * b j g(s,t) (y, 0) = χ * b j ρ (s,t) • h b j (y, 0).
By [START_REF] Dumortier | Smooth linearization of germs of R 2 -actions and holomorphic vector fields[END_REF], ψ * b j g(s,t) (y, 0) = ψ * b j g(0,t) (y, s); similarly, χ * b j ρ (s,t) (y , s ) = χ * b j ρ (0,t) (y , s + s ) hence, by [START_REF] Humphrey | Introduction to Lie algebras and representation theory[END_REF]

, χ * b j ρ (s,t) • h b j (y, 0) = χ * b j ρ (0,t) • h b j (y, s); setting Ψt b j := ψ * b j g(0,t) and Ξt b j := χ * b j ρ (0,t)
it follows that the relation h

• g(s,t) |E b j = ρ (s,t) • h|E b j is equivalent to h b j • Ψt b j = Ξt b j • h b j . (18)
Now, by [START_REF] Dumortier | Smooth linearization of germs of R 2 -actions and holomorphic vector fields[END_REF],

Ψt b j (y, s) = Ψ t b j (y), s + t 0 s b j • Ψ τ b j (y) dτ (19)
and similarly, if Ξ t b j , σ b j denote the analogues of Ψ t b j , s b j for ρ,

Ξt b j (y, s) = Ξ t b j (y), s + t 0 σ b j • Ψ τ b j (y) dτ . (20)
By ( 17)-( 19)-( 20), both h b j and the flows Ψt b j , Ξt b j commute with the action of Z j on Q b j × R j by translation on the second factor; hence, they induce a bijection hb j of Q b j × T j onto itself and two flows Ψt b j , Ξt b j on Q b j × T j .

Our induction hypothesis is that h is smooth on E b j+1 and has infinite contact with the identity along V Vb j+1 , i.e. that h b j is smooth on

ψ -1 b j (E b j+1 ) = (Q b j ∩ E b j+1 ) × R j = (Q b j Vb j+1 ) × R j
and has infinite contact with the identity along

ψ -1 b j (V Vb j+1 ) = (Q b j ∩ V Vb j+1 ) × R j ;
we wish to show that h b j is smooth and has infinite contact with the identity along (Q b j ∩ Vb j+1 ) × R j . Using the extension lemma 2.11, we will now prove that hb j , which is smooth on (Q b j Vb j+1 ) × T j , is smooth on the whole of Q b j × T j and has infinite contact with the identity along (Q b j ∩ Vb j+1 ) × T j ; this will imply what we want.

The flows Ψt b j , Ξt b j have all the properties of Ψ t b j stated in the proof of the globalisation lemma 2.24 if Q b j , Σ b j ,I , W ± b j ,I are replaced respectively by their cartesian products Qb j , Σb j ,I , W ± b j ,I with T j and f b j by fb j : (y, s) → f b j (y). By [START_REF]Invariant manifolds and a preparation lemma for local holomorphic flows and actions[END_REF],

(Q b j ∩ Vb j+1 ) × T j = b(δ j+1 -g I )>0 W + b j ,I ∪ b(δ j+1 -g I )<0
Wb j ,I ; let us explain how to "fill in the gap" along b(δ j+1 -g I )<0 Wb j ,I by showing that there, hb j is smooth and has infinite contact with the identity: ; hence, the latter extends to a smooth germ having infinite contact with the identity along the germ of Wb j ,I , which of course is the germ of hb j at Σb j ,I ; • now, (18) yields hb j • Ψt b j = Ξt b j • hb j for all positive t; as Ψt b j and Ξt b j have infinite contact along their common unstable manifold Wb j ,I at Σb j ,I , it follows that hb j is smooth and has infinite contact with the identity along Wb j ,I for all I ∈ I b j with b(gI ) = v1;

• if b(δj+1)
• if there is no I ∈ I b j with b(gI ) > v1, the gap is filled;

• otherwise, if v2 denotes the smallest b(gI ) > v1 then, for all I ∈ I b j with b(gI ) = v2, as every possible gap along W + b j ,I has just been filled, the same argument shows that hb j is smooth and has infinite contact with the identity along Wb j ,I ; • iterating this procedure, the gap is filled.

The gap along b(δ j+1 -g I )>0

W + b j ,I is filled in the same way by considering first the largest b(gI ) < b(δj+1) (if any), calling it v-1, applying the extension lemma for some negative t to the germs at Σb j ,I of Ψt On the "quotient" N of N Vb 1 by ζη * g1, the two embeddings ζη * g2 and ζη * g 2 induce (partially defined) embeddings ḡ2 and ḡ 2 ; if we can construct a conjugacy h of ḡ2 to ḡ 2 defined near the quotient V1 of V1 ∩ D0 and having infinite contact with the identity along it then, by the extension lemma 2.11, it will extend to a local diffeomorphism h preserving ζη * g1 and having infinite contact with the identity along V1 near 0, which will conjugate ζη * g2 to ζη * g 2 , hence our result.

To obtain h, we solve the same kind of Cauchy problem as in the proof of Corollary 2.12: using a reasonable presentation of N as a circle bundle (for which the quotient W of W ∩ D0 is a union of fibres for every SIM W of S), we extend f b 1 to the function fb 1 on N constant on the fibres. Denoting by V the quotient of V ∩D0, we then choose h to have infinite contact with the identity along a compact neighbourhood N of V ∩ { fb 1 = b(δ2)} in { fb 1 = b(δ2)}, to have the corresponding jet along ḡ2(N ) and to have the same jet as the identity in between along V; the Whitney extension theorem then yields a "fattened" version of the Cauchy problem in the proof of Lemma 2.25, which can be solved by the same repeated use of the extension lemma.

Consequences and variants

Corollary 2.26 With the notation of Proposition 2.4, if g is weakly hyperbolic and Π 0 empty, then g is smoothly conjugate to the analytic action germ g 0 .

Proof. In the Poincaré domain, this is Theorem 2.5. Otherwise, as g and g0 are formally conjugate, they are smoothly conjugate by Theorem 1.2.

Theorem 2.27 Under the hypotheses of Corollary 2.26, if g is strongly hyperbolic, then it is C 0 -linearisable, i.e. C 0 -conjugate to S.

Idea of the proof. Similar to that of Theorem 3.1 hereafter but much easier analytically, as the action u of Proposition 2.4 is algebraic. The additional fact needed (because of the commutation relation) is that one can define an ordering on {1, . . . , n} as follows: i is strictly less than j if and only if there exists p ∈ Pj with pi = 0; this enables one to "kill" first the monomials x p with p ∈ Pi and i maximal, etc.

Theorem 2.28 If g is weakly hyperbolic, in the Siegel domain and formally conjugate to a smooth Z k × R m -action germ g at a ∈ M , meaning that there exists a smooth diffeomorphism germ h 0 : (M, a) → (M , a ) such that h * 0 g and g have infinite contact at a, then, for each such h 0 , the smooth conjugacies of g to g having infinite contact with h 0 at a form an infinite dimensional space.

Idea of the proof. As in the proof of Corollary 2.13, we may assume h0 = IdE.

If k = 0, the result follows from the fact that, in Lemma 2.25, the Cauchy problem h|Q b can be any C 1 -small enough perturbation of IdE|Q b , equal to IdE|Q b off a compact subset and having infinite contact with IdE|Q b along V ∩ Q b . The germs at V ∩ Q b of such perturbations obviously form an infinite dimensional space and identify to the conjugacies between the action germs g and g .

For k > 0, the result is more obvious since the solutions of our Whitney extension problem form an infinite dimensional space, as in the proof of Corollary 2.13 ii).

Theorem 2.29 All the previous results remain true if Z k × R m is replaced by an elementary Abelian group, i.e. a Lie group G isomorphic to G 0 × Z k × R m , where G 0 is the product of a finite Abelian group and a torus 23 .

Idea of the proof [START_REF]Propriétés génériques des germes d'actions différentiables de groupes de Lie commutatifs élémentaires[END_REF][START_REF]-Géométrie différentielle et singularités de systèmes dynamiques[END_REF]. By a theorem of Bochner ([10], 3.1.4, théorème), the action germ restricted to the maximal compact subgroup G0 of G can be linearised smoothly and then all the proofs can be made invariant by this linear G0-action.

Remark. This might be helpful in the study of smooth completely integrable systems via Nguyen Tien Zung's moto: "Always look for the torus action".

3 Related results and questions 3.1 Germs of holomorphic vector fields Theorem 3.1 ([12], Theorem 2) Let X be a germ at a ∈ M of holomorphic vector field with X(a) = 0, generating a weakly hyperbolic C-action germ g, and let σ be the semi-simple part of the endomorphism dX(a) of the complex vector space E = T a M . For every positive integer k, the action germ g is C k -conjugate to the holomorphic C-action germ generated by a polynomial normal form, i.e. a complex polynomial vector field σ + ν on E with dν(0) nilpotent and [σ, ν] = 0.

Proof. In this case, the algebraic part of the proof of Theorem 1.2 reads as follows: for every integer s, there exists a holomorphic conjugacy (M, a) → (E, 0) of X to a holomorphic vector field germ having s th order contact at 0 with a polynomial normal form σ + νs of degree s; passing to the projective limit, one gets a smooth diffeomorphism germ h : (M, a) → (E, 0) such that h * g has s th order contact at 0 with the C-action generated σ + νs for all s; hence, for every integer q, the smooth action germ (h -1 ∞ • h) * g of the preparation lemma 2.10 has q th order contact along V with the C-action germ g (q) generated by a polynomial normal form σ + νs q ; globalising (h -1 ∞ • h) * g and g (q) as in lemma 2.24, one gets two R 2 -actions having q th order contact along V. For large enough q, the C p version of the extension lemma 2.11, used repeatedly as in the proof of lemma 2.25, yields a C k conjugacy between them.

Notes. The proof in [START_REF]-C k -conjugacy of holomorphic flows near a singularity[END_REF] is much the same but remains holomorphic as long as possible: by [START_REF]Invariant manifolds and a preparation lemma for local holomorphic flows and actions[END_REF], for every integer q, there exists a holomorphic conjugacy of X to a holomorphic vector field germ having q th order contact along V with σ + νs q ; the end of the proof is restricted to something like N0 instead of globalising the actions.

In the Poincaré domain, it has been known since Poincaré and Dulac that X is holomorphically conjugate to a polynomial normal form. Theorem 3.1 shows that in the weakly hyperbolic case small denominators have no C k meaning for any k. Finding the best (least) possible degree for ν seems very difficult.

For general smooth Z k × R m -action germs with k + m > 1, I do not see how to get nice normal forms like σ + ν; this prevented me from obtaining the general version of the following result: Theorem 3.2 ([12], Theorem 1) Under the hypotheses of Theorem 3.1, if g is strongly hyperbolic, it is C 0 -conjugate to the germ of the linear C-action e tσ .

Idea of the proof. By Theorem 3.1, it is enough to C 0 -conjugate the complex flow generated by the normal form σ + ν to e tσ . This is done by solving the same Cauchy problem as before, but several times and with much more care: using a first b, one can C 0 -conjugate the complex flow of σ + ν to that of the vector field obtained from σ + ν by killing the monomials of ν which vanish on Vb ; one can then kill successively the other monomials by using different b's. The key remark is that the formal flow defined by a normal form converges in a domain large enough to define its complex flow where needed for the proof.

Notes. This explicit method yields conjugacies that are Hölder continuous of every exponent less than 1 (but not Lipschitzian in general). Despite superficial analogy, Theorem 3.2 is much more difficult than the Grobman-Hartman theorem, as it is shown in [START_REF] Camacho | The topology of holomorphic flows with singularity[END_REF] that there are moduli already for topological equivalence between germs of complex linear vector fields. The Camacho-Kuiper-Palis conjecture was the weaker version of Theorem 3.2 where C 0 -conjugacy is replaced by topological equivalence. It seems that no simpler proof has been found.

First integrals

Hypotheses, notation and definition. We go back to the hypotheses and notation of section 1. A first integral of g is a smooth function germ J : (M, a) → R such that J • g j = J for 1 ≤ j ≤ k and L Xj J = 0 for 1 ≤ j ≤ m, hence J • g t = J for all t ∈ Z k × R m . A formal first integral of g is a smooth function germ J 0 : (M, a) → R such that J 0 • g t and J 0 have infinite contact at a for all t ∈ Z k × R m .

Here again, the contrast between the Poincaré and Siegel domains is striking: Theorem 3.3 If g is in the Poincaré domain, its only first integrals are the germs at a of constant functions.

If g is in the Siegel domain and weakly hyperbolic, it possesses the following property: for every formal first integral J 0 of g, the first integrals of g having infinite contact with J 0 at a form an infinite dimensional space.

Idea of the proof. When g is in the Poincaré domain, taking representatives, some g t satisfies limn→∞ g nt (x) = a for every x close enough to a; if J is a first integral of g, as J (x) = J • g nt (x) for all n ∈ N, it follows that J is the constant J (a) (note that this holds assuming only that J is continuous at a).

In the Siegel domain, with the notation of the preparation lemma 2.10, the germ (h -1 ∞ •h) * J0 defines a formal integral JV of (h -1 ∞ •h) * g along V, whose jet along the unstable manifold W of S t is (taking representatives) lim n→∞ (j ∞ (h -1 ∞ • h) * (J0 • g -nt ) |W .

One can then conclude as in the proof of Theorem 1.2: the infinite jet of JV along V yields a unique jet along the whole of V of first integral of g; then, if for example k = 0, every smooth function J b : Q b → R having the induced jet along Q b ∩ V extends to a unique first integral J of g, having the same jet as the extended JV along V: this follows from the analogue for first integrals of the extension lemma 2.11 ([10], 4.2.4, Théorème 2). The case k = 0 is similar.

Notes. "In general" Π 0 = ∅, hence the only formal first integrals of g are the smooth function germs J 0 having infinite contact with a constant at a. However, this proof shows that, in the C ∞ sense, every weakly hyperbolic R r -action germ g in the Siegel domain is more or less completely integrable, as it is possible to find dim(M ) -r first integrals functionally independent off V. Theorem 3.3 remains true for elementary Abelian group action germs. The same methods apply to various problems, for example the solution of "(co)homological equations", all of which (including the conjugacy problem and the problem of first integrals) are particular cases of invariant manifold problems [START_REF]-Variétés stables et formes normales[END_REF], as I plan to show in a forthcoming book. In the Poincaré domain, every formal solution yields a unique smooth solution whereas, in the Siegel domain, there is a huge amount of flexibility.

Proposition 2 . 1 (

 21 [START_REF]-Géométrie différentielle et singularités de systèmes dynamiques[END_REF], 4.3.1, Proposition 5) There exist positive integers n, d 1 ,. . . , d n , a nonnegative integer c ≤ n and an isomorphism of real vector spaces

  -1 s,W • h) * Xj |W and the germ at 0 of (j s ( d dt gtδ k+j W |t=0))|W : this follows from [10], 4.2.2, Théorème 2.

  where |x | 2 := p |x ,p | 2 and the coordinates x ,p are as in Proposition 2.1.

•

  either b is positive and Q b is a quotient by the flow S of the invariant open subset E b := E E + , meaning that every orbit of S contained in E b intersects Q b transversally and exactly once, • or b is negative and Q b is a quotient by S of the invariant open subset E b := E E -. If ρ is another smooth R-action on E leaving E b invariant and for which Q b is a quotient of E b , the two flows on E b induced by ρ and S are smoothly conjugate.

F

  and the "curved" hypersurface Q b can be replaced respectively by x → |x + | 2 and by the cylinder {|x + | = 1}. Basic facts about the R r -action S when k = 0 Notation. As in [10], for y ∈ E, we let J(y) := { : x (y) = 0} and, for I ⊂ {1, . . . , n}, E I := ∈I E , V I := ∈I Rc and C I := ∈I R + c . Proposition 2.15 For I ⊂ {1, . . . , n}, a point y ∈ E I is a critical point of F | E I if and only if it is a critical point of F ; hence, every critical value of F | E I is a critical value of F , and every regular value of F is a regular value of F | E I .

Proposition 2 . 17

 217 Let b ∈ R r * . i) It is a critical value of F if and only if it lies in some C I with dim V I < r. ii) Otherwise, the union Vb of the SIM's of S on which F does not take the value b consists of all y ∈ E with b / ∈ CJ(y) and therefore contains Crit F . iii) Still assuming that b is a regular value of F , the submanifold Q b := F -1 (b) is a quotient by S of the dense S-invariant open subset E b := E Vb in the same sense as for flows; in other words, the map ϕ b : Q b × R r → E b defined by ϕ b (y, t) := S t (y) is a diffeomorphism 18 . Proof. i) If y ∈ F -1 (b) is a critical point of F , one can take I = J(y) since V J(y) = R r * and b = F (y) ∈ C J(y) . If b ∈ CI with VI = R r * , there clearly exists y ∈ F -1 (b) with J(y) ⊂ I, hence y ∈ Crit F . ii) Given I ⊂ {1, . . . , n}, one clearly has F (EI ) ⊂ CI hence, by (i), b / ∈ F (EI ) for dim VI < r; if now VI = R r * , then either b / ∈ F (EI ), or If y ∈ Vb then of course b does not lie in Ψy(R r ) which, by Proposition 2.16 ii)-iii), equals CJ(y) unless CJ(y) = ∅; conversely, if b /

  19 W ± b j ,I is the set of all y ∈ Q b j such that the distance of Φ t b j (y) to Σ b j ,I tends to 0 when t → ±∞. iv) For each I ∈ I bj , the Φ t bj -invariant subset W ± bj ,I equals Q bj ∩ E bj ,I ∩ E I ± 0 , where E bj ,I is the S-invariant open subset E bj ,I := y ∈ E : b j ∈ π j C J(y)∩I and E I ± 0 is the SIM of S j+1 defined by

  bj ,I of Σ bj ,I for the flow Φ t bj ,I are Y ± bj ,I := Q bj ,I ∩ E I ± 0 and the germ of Φ t bj ,I at Σ bj ,I satisfies the hypotheses of the extension lemma 2.11. Back to the proof of Proposition 2.20 iii). The first assertion of the lemma shows that z ∈ E b j ,I for some I ∈ I b j , hence Φ tp b j (y) ∈ Q b j ∩ E b j ,I for large enough p and in fact y ∈ Q b j ∩ E b j ,I ; as C b j ,I • Φ tp b j (y) = Φ tp b j ,I • C b j ,I (y) tends to C b j ,I (z), the form of the flow Φ tp b j ,I implies C b j ,I (y) ∈ Y ± b j ,I and C b j ,I (z) ∈ Σ b j ,I , hence y ∈ W ± b j ,I and z ∈ Σ b j ,I . iv) What we have just done shows that W ± b

  follows from Proposition 2.19 that y ∈ Vb j+1 if and only if t → f b j • Φ t b j (y) does not take the value bj+1(δj+1) = b(δj+1); now, by (iii), this means that either y ∈ W + b j ,I for some I such that the value bj(gI) = b(gI ) of f b j on Σ b j ,I is less than b(δj+1), or y ∈ W - b j ,I for some I with b(gI ) > b(δj+1), hence (11). vi) As Fj|V j is proper, Q b j ∩ Vj is compact; since Vj is S-invariant,this compact subset is invariant by the flow Φ t b j and therefore contained in the set of y ∈ Q b j for which ψy is bounded. Conversely, given such an y, it follows from (iii) that it belongs to W - b j ,I ∩ W + b j ,J for uniquely determined I, J ∈ I b j which, by (i)-(ii), satify either I = J, or b(gI ) < b(gJ ); in the second case, by (iv), one has y ∈ E I - 0 ∩ E J + 0

  yi), it follows that y0 is the endpoint of a path t → gγ(t) (yj) in N0 with γ : [0, T ] → R j continuous, piecewise smooth and γ(0) = 0; now, if γ = (γ1, . . . , γj),d dt gγ(t) * j ( Ỹj+1 -ζη * Yj+1)(yj) = gγ(t) * j j i=1 γ i (t) Ỹi , Ỹj+1 -ζη * Yj+1 (yj) = j i=1 γ i (t) gγ(t) * j Ỹi , Ỹj+1 -ζη * Yj+1 (yj) Ỹj+1] -[ζη * Yi, ζη * Yj+1] (yj) = 0 since Ỹi and ζη * Yi coincide on the submanifold with corners N0 and therefore have the same 1-jet at gγ(t) j (yj) for 1 ≤ i ≤ j and 0 ≤ t ≤ T ; hence, gγ(T ) * j ( Ỹj+1 -ζη * Yj+1)(yj) = gγ(0) * j ( Ỹj+1 -ζη * Yj+1)(yj) = ( Ỹj+1 -ζη * Yj+1)(yj) = 0 i.e. Ỹj+1(y0) = ζη * Yj+1(y0), as claimed.

  is larger than b(gI ) for all I ∈ I b j , there is no gap to fill; • otherwise, there is a smallest b(gI ) > b(δj+1); if we call it v1 then, for every 21 I ∈ I b j with b(gI ) = v1, the hypotheses of the extension lemma 2.11 are satisfied for positive t by the germs at Σb j ,I of Ψt b j , Ξt b j and hb j | Qb j Wb j ,I

  b j , Ξt b j and hb j | Qb j W + b j ,I for all I ∈ I b j with b(gI ) = v-1, concluding that hb j is smooth and has infinite contact with the identity along b(g I )=v -1 W + b j ,I , passing to the largest b(gI ) < v-1 (if any), etc.

Index CI , 16 EI 18 F

 1618 direct sum of the E 's with ∈ I, 16 E characteristic subspace of L, 3 E b union E Vb of the orbits of S intersecting Q b , 15 E b j ,I , 20 E b j union E Vb j of the orbits of Sj intersecting Q b j , Lyapunov map of S, 15 Fj Lyapunov map of Sj, 18 I ± , 21 I ± 0 , 20 J(y) set of indices with x (y) = 0, 16 L linear part of g, 3 Li generators dgi(a) of L, 3Q b quotient F -1 (b) of E b by S, 15 Q b j ,I , 21 Q b j quotient F -1 j (bj) of E b j by Sj, 18 S semi-simple part of L, 6 Sj restriction of S to R j × E, 18 VI , 16 W + b j ,I , W - b j ,Istable and unstable manifolds of Σ b j ,I for Φ t b j , 19 Xj infinitesimal generators of g, 3 Yj, 23 Z b j infinitesimal generator of Ψ t b j , 24 ϕ b j , 18 Λj infinitesimal generators dXj(a) of L, 3 Φ t b j flow on Q b j induced by Sj+1, 19 Π , P resonance subsets, 7 Ψ t b j flow on Q b j induced by gj+1, 24 Σ b j ,I compact invariant manifold of Φ t b j , 19 Σ b j critical set of f b j , union of the Σ b j ,I 's, 19 δj vectors of the canonical basis of R r , 3 Vb union of the SIM's of S on which F does not take the value b, 17 Vb j union of the SIM's of Sj on which Fj does not take the value bj, 18 κj, 22 gI , 18 g b j , 19 s b j , 27 C b j ,I , 21 I b j , 19 Kj, 18 V union of the SIM's of S, 11 Vj union of the SIM's of Sj, 20 µ, 7 ψ b j , 27 θ, θj, 22 Ỹj, 24 g, gj, 24 ξ, 19 ξj, 23 ζη weighted zooming map, 23 a characters defined by L, 3 c "Lyapunov exponents" of L and S, 4 d real or complex dimension of E , 5 f b j Lyapunov function of Φ t b j , 19 g, g smooth germs of Z k × R m -actions, 3 g0 normal form of g, 8 gW , 10 gi generators of g, 3 h formal conjugacy of g to a normal form, 6 h -1 ∞ conjugacy of h * g to a normal form along V, 11 mj, Mj, 22 u formal action of R r on E, 8 x isomorphism triangulating L, 5 x , x ,p components of x, 5

  can now see what this means in the coordinate system x of Proposition 2.1:

	Corollary 2.3 ([10], 4.3.2, Corollaire 3) With the notation of Propositions
	2.1 and 2.2, if we set

  The rest of the Taylor series reads α f ,p,α (t)x α with α ∈ 1≤s≤n+c N ds and s |αs|1 ≥ 2; for every t ∈ Z k ×R m , as xs,p •S t = as(t )xs,p,

	hence f ,p,α (t)(a -a µ(α) ) = 0 for all α, proving that f ,p,α (t) = 0 for α ∈ P .
	Finally, every f ,p,α is polynomial for the following reason: for each positive integer
	s, as S t is the semi-simple part of the s-jet ȟ * ǧ	t of h * g t at 0 for all t, the automor-
	phism (S -t • ȟ * ǧ	t ) * of the algebra Fs of s-jets of germs (E, 0) → R is of the form
	exp ν(t), where ν is a linear map of R r into the set of nilpotent derivations of Fs ([10],
	Théorème 4 (iii)); in particular, the coefficients of the polynomial (S -t • ȟ * g	t ) * x ,p of
	the variables x α are polynomial functions of t.
	Remarks. For a dense open set of linear representations L, the multiplicities
	d all equal 1, hence Π = P ; moreover, P = ∅ for almost every L (not an open
	condition in general) and then what we get is a formal linearisation result.
	The Jordan decomposition is not the only way to obtain normal forms, but
	it is probably the most understandable for r > 1.
	Proposition 2.4 ([10], 4.3.2, Corollaire 4) Let
	the fact that x ,p • S t • S -t • h * g t and x ,p • S -t • h * g t • S t have infinite contact at
	0 writes
		a (t )	f ,p,α (t)x α =	f ,p,α (t)a µ(α) (t )x α ,
		α	α

Proof. The linear forms c1, . . . , cn are independent, hence 0 ∈ conv{c1, . . . , cn} and P1 = • • • = Pn = ∅.

The terminology, due to Arnold, refers to Poincaré's holomorphic linearisation theorem and its analogue in the Siegel domain[START_REF] Siegel | Über die Normalformen analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung[END_REF], a celebrated triumph over "small denominators".

With the usual convention z s = (z, . . . , z) repeated s times.

We state only the C ∞ version of this key result, of which two proofs are given in[START_REF]-Géométrie différentielle et singularités de systèmes dynamiques[END_REF].

A germ (Q W -, Σ) → Q W -is an equivalence class for the relation "there is an open subset U ⊃ Σ in U 0 ∩U 1 such that f 0 = f 1 on U W -" between maps f j : U j W -→ Q W - with U j ⊃ Σ open in Q.

See the proof of Corollary 2.14 and the subsequent notes.

When all the eigenvalues of dX 1 (0) are real, S t is the gradient flow of F with respect to the standard Euclidean metric dx 2 ,p .

When S is a linear holomorphic C s -action in the Poincaré domain, this provides examples of compact holomorphic manifolds with no real symplectic structure[START_REF] López De Medrano | A new family of complex, compact, non symplectic manifolds[END_REF].

Proof. The last assertion is clear since h* Y j = Yj near 0 for 1 ≤ j ≤ r. If the diffeomorphisms ψ b , χ b : Q b × R r → E bare given by ψ b (y, t) = gt (y) and

One could add to Proposition 2.18 the general position hypothesis that b(g I ) = b(g J ) for I = J but, for j > 1, this strict ordering of I b j would not mean much dynamically.

This is why Theorem 1.2 in the Siegel domain was proven in[START_REF]-Géométrie différentielle et singularités de systèmes dynamiques[END_REF] only for linearisations.

(Weak) hyperbolicity is that of the action germ restricted to Z k × R m , a notion independent of the isomorphism chosen[START_REF]-Géométrie différentielle et singularités de systèmes dynamiques[END_REF].

Acknowledgements As explained in the introduction, this rather heavy birthday cake owes much to Alain Chenciner, whose friendship has illuminated my life in the last thirty-nine years. Whatever the interest of this work, it required concentration and freedom, available because Laurent Schwartz valued results above office hours. Had not Ivan Kupka encouraged me, I might have given up just before finding the missing piece of the puzzle, namely the Lyapunov map F . The initial impulse came from Robert Roussarie's talk about [15] at the Thom seminar; a few years later, he was the courageous referee of [10] and I am very sorry some of his excellent suggestions had to be ignored by fear of a riot among the secretaries in charge of typing the manuscript (those were hard, pre-T E X times). Even though the ideology of this paper comes from [7], many details are new; I am grateful to Alain Albouy and Alexey Borisov for offering me the opportunity for such improvements. Finally, I thank Eva Miranda and Nguyen Tien Zung, whose interest in the domain caused the renewal of mine.

Idea of the proof of Theorem 1.2 for k > 0

In that case, the dispensable analogue of the globalisation lemma 2.24 [START_REF]Propriétés génériques des germes d'actions différentiables de groupes de Lie commutatifs élémentaires[END_REF] is lengthy 22 .

The linear action S| 2(Z k ×R m )×E is the restriction of a linear action σ of R r on E with x ,p • σ t = u (t)c (t)x ,p , |u (t)| = 1 ([10], 6.1.1, Proposition 1), possessing all the properties described in 2.3.2. Moreover, the automorphism A in Proposition 2.18 can be obtained from an automorphism A of Z k × R m ([10], 6.1.1, Proposition 2, where this is expressed in terms of subgroups), so that σ has all the properties stated in 2.3.3 once the action germs composed with A, for every b as in Proposition 2. [START_REF] López De Medrano | A new family of complex, compact, non symplectic manifolds[END_REF].

We now explain how to prove that two weakly hyperbolic formally conjugate smooth Z 2 -action germs g, g are C ∞ -conjugate: after applying the preparation lemma, we get two pairs g1, g2 and g 1 , g 2 of commuting local diffeomorphisms (E, 0) → (E, 0) defining the same formal normal form along V; by our general position hypothesis, g1 and g 1 are hyperbolic and therefore conjugate by Corollary 2.13; moreover, this conjugacy can be chosen with the same infinite jet as the identity along V: just impose this as part of the Whitney extension problem in the proof of Corollary 2.12 i).

Assuming therefore g 1 = g1, we should now construct a smooth local conjugacy (E, 0) → (E, 0) of g2 to g 2 preserving g1. With the notation of 2.3.4,

• we choose a positive number µ1 so that

), and set K0 := [b1, µ1];

• we fix a bounded open neighbourhood B 0 in E of the compact subset

). When η → 0, the map ζη * g1 tends in the C ∞ sense to S δ 1 on B and ζη * g2, ζη * g 2 , to S δ 2 . It follows that, for small enough η,

• on B {0}, the function f b 0 = F1 is "Lyapunov" for ζη * g1, i.e. F1 • ζη * g1 > F1;