Dominique Cerveau

Julie Déserti

ALGEBRAIC PROPERTIES OF THE GROUP OF GERMS OF DIFFEOMORPHISMS

Keywords: 2020 Mathematics Subject Classification, 57S05, 20E99, 1

We establish some algebraic properties of the group Diff(C n , 0) of germs of analytic diffeomorphisms of C n , and its formal completion Diff(C n , 0). For instance we describe the commutator of Diff(C n , 0), but also prove that any finitely generated subgroup of Diff(C n , 0) is residually finite; we thus obtain some constraints of groups that embed into Diff(C n , 0). We show that Diff(C n , 0) is an Hopfian group, and that Diff(C n , 0) and Diff(C n , 0) are not co-Hopfian. We end by the description of the automorphisms groups of Diff(C, 0), and Diff(C, 0).

INTRODUCTION

Let F be a codimension p holomorphic foliation on the complex manifold M n of dimension n. Suppose that N ⊂ M is a submanifold of dimension np invariant by F . Then there exists a natural morphism Π 1 (N ∖ Sing(F ), * ) → Diff(C p , 0) the so-called holonomy representation. As a consequence the study of such representations is an interesting problem, and requires the knowledge of algebraic properties of the groups Diff(C p , 0). It is the purpose of this note in the same spirit of the description of the classical groups GL(n, k) (see [START_REF] Dieudonné | La géométrie des groupes classiques[END_REF]).

Structure of the paper. If n is an integer, then G n 0 denotes Diff(C n , 0), and G n 0 its formal completion. Furthermore, G n k denotes the set of elements of G n 0 tangent to the identity at order k, and G n k its formal completion. In §2 we establish some consequences of Poincaré and Siegel theorems:

Theorem A. For any k ≥ 1 we have [G n 0 , G n k ] = G n k , and
[ G n 0 , G n 1 ] = G n 1 .
As a consequence, we obtain:

[G n 0 , G n 0 ] = f ∈ G n 0 | det D f (0) = 1 .
Finding the finitely generated subgroups that embed into G n 0 is a problem related to the foliations theory; in §3 we deal with this question and get: Theorem B. Any finitely generated subgroup of G n 0 (resp. G n 0 ) is residually finite. Hence, if H is a finitely generated and non residually finite group, then H does not embed into G n 0 .

But any finitely generated residually finite group is a Hopfian group ( [START_REF] Malcev | On isomorphic matrix representations of infinite groups[END_REF][START_REF] Mal'tsev | On the faithful representation of infinite groups by matrices[END_REF]); as a result any finitely generated subgroup of G n 0 (resp. G n 0 ) is a Hopfian group. In §4 we refine this result, and also look at the co-Hopfian notion:

Theorem C. The group G 1 0 is a Hopfian group. The groups G n 1 and G n 1 are not co-Hopfian groups.

Inspired by [D 06] we study in §6 the automorphisms groups of G 1 0 and G 1 0 :

Theorem D. The group Aut( G 1 0 ) is generated by the inner automorphisms and the automorphisms of the field C. In other words

Out( G 1 0 ) ≃ Aut(C, +, •) where Out( G 1 0 ) denotes the non-inner automorphisms of G 1 0 . The group Out(G 1 0 ) is isomorphic to Z ⧸ 2Z .

Finally we try to generalize this result in higher dimension, and give some partial results ( §7).

NOTATIONS AND DEFINITIONS

Let n be an integer; consider

G n 0 = Diff(C n , 0) = f : C n ,0 → C n ,0 holomorphic | D f (0) ∈ GL(C n ) .
Denote by G n 0 the formal completion of G n 0 ; in other words G n 0 is the set of formal diffeomorphisms f = ( f 1 , f 2 , . . . , f n ) where

• f i is a formal serie, • f (0) = 0 • and D f (0) belongs to GL(C n ).
Let k be an integer; consider G n k the set of elements of G n 0 tangent to the identity at order k (i.e. f belongs to G n k if and only if f = id+ terms of order ≥ k + 1). Denote by G n k the formal completion of G n k . The G n k 's (resp. G n k 's) are normal subgroups of G n 0 (resp. G n 0 ). The quotients G n 0 ⧸ G n k and G n 0 ⧸ G n k are isomorphic, and can be identified with the group of polynomial maps

Pol n k = f : C n → C n polynomial | f (0) = 0, deg f ≤ k, D f (0) ∈ GL(C n )
whose the law group is the law of composition truncated to order k + 1.

Remark that G n 0 ⧸ G n k acts faithfully on the space C[x 1 , x 2 , . . . , x n ] k of polynomials of degree less or equal than k (still by truncated composition). Therefore the

G n 0 ⧸ G n k 's can be identified with subgroups of GL(C[x 1 , x 2 , . . . , x n ] k ).
The group G 1 0 is quite amazing: it is not solvable, contains free subgroups ([BCLN96]) and is "limit" of solvable groups; indeed, the quotients G 1 0 ⧸ G 1 k are solvable.

Denote by p

k : G n 0 → G n 0 ⧸ G n k the projection.
Let M n be the maximal ideal of O(C n , 0) given by

M n = f ∈ O(C n , 0) | f (0) = 0 .
Denote by χ n 0 the set of germs of holomorphic vector fields at the origin of C n , and by χ n 0 its formal completion. Let χ n k be the subspace of χ n 0 made up of vector fields that vanish at 0 at order k -1, that is

χ n k = M k n χ n 0 .
Finally, let us denote by χ n k the formal completion of χ n k .

POINCARÉ AND SIEGEL THEOREMS AND THEIR CONSEQUENCES

Let λ = (λ 1 , λ 2 , . . . , λ n ) be a n-uplet of non-zero complex numbers. We say that λ is without resonance if the equality λ p 1 1 λ p 2 2 . . . λ p n n = λ j , p ℓ ∈ N, implies p j = 1, and p ℓ = 0 for ℓ ̸ = j.

Theorem 2.1 (formal Poincaré Theorem). Let φ be an element of G n 0 . Set A = Dφ (0) . Assume that Spec(A) is without resonance. Then φ is formally conjugate to A, i.e. there exists ϕ ∈ G n 1 such that φ = ϕAϕ -1 .

Theorem 2.2 (holomorphic Poincaré Theorem). Let φ be an element of G n 0 . Set A = Dφ (0) . Assume that Spec(A) is without resonance, and that Spec(A) is contained either in the open unit disk D(0, 1), or in the complement ∁D(0, 1) of D(0, 1).

Then φ is holomorphically conjugate to A, i.e. there exists ϕ ∈ G n 1 such that φ = ϕAϕ -1 .

We say that φ is formally linearizable (resp. holomorphically linearizable) in the formal (resp. holomorphic) case. In both cases the diffeomorphism ϕ is called the linearizing map.

Remark 2.3. We do not use explicitely the Siegel linearization Theorem when Spec(A) = (λ 1 , λ 2 , . . . , λ n ) is contained neither in D(0, 1), nor in ∁D(0, 1); it requires diophantine conditions controlling λ p 1 1 λ p 2 2 . . . λ p n nλ j that produce the convergence of the linearizing maps ([Sie42, Sie52]).

Remark 2.4. In the Poincaré Theorems assume that φ can be written as Ah where h = id + . . . ∈ G n k ; then the linearizing map ϕ can be chosen in G n k (resp. G n k in the formal case).

Curiously while the proof of Poincaré Theorem comes from analysis (in its holomorphic version) we deduce from it algebraic properties. Let us recall that if H is a group, the derived subgroup (or commutator group) of H is the subgroup of H generated by all the commutators; we denote it by [H, H].

Theorem 2.5. We have

[G n 0 , G n 1 ] = G n 1 , and similarly [ G n 0 , G n 1 ] = G n 1 .
Remark 2.6. Such a result has been proved in [START_REF] Cano | Théorie élémentaire des feuilletages holomorphes singuliers[END_REF] for n = 1.

Proof. Let A be an element of GL(n, C) that satisfies the assumptions of Poincaré Theorem, for instance

A = λ id with 0 < |λ| < 1. If h belongs to G n 1 , then hA is linearizable, that is there exists ϕ ∈ G n 1 such that ϕAϕ -1 = hA.
As a result [ϕ, A] = h, and we get the result.

A similar proof works in the formal case. □ Corollary 2.7. We can describe the commutator of G n 0 :

[G n 0 , G n 0 ] = f ∈ G n 0 | det D f (0) = 1 .
Proof. Let f be an element of G n 0 ; denote by A ∈ SL(n, C) its linear part. We can write A as [α, β] with α, β ∈ GL(n, C). Note that one can choose β without resonance, and Spec(β) ⊂ D(0, 1). Let h be the element of G n 1 defined by: h = αβ -1 α -1 f β; in particular f = αβα -1 hβ -1 . According to Poincaré Theorem there exists ϕ ∈ G n 1 such that

αβα -1 h = ϕβϕ -1 ; then f = [ϕ, β]. □
Theorem 2.5 can be generalized as follows:

Theorem 2.8. Any element of G n k is the commutator of an element of G n 0 and an element of

G n k , i.e. [G n 0 , G n k ] = G n k .
We deduce from it the following statement that can be useful:

Proposition 2.9. Let H be a group, and let τ : G n 0 → H be a group homomorphism. Assume that there exists f in ker τ such that D f (0) = A satisfies the assumptions of Poincaré (resp. Siegel) Theorem. Then G n 1 is contained in ker τ.

Proof. Let h be an element of G n 1 . Since A satisfies the assumptions of Poincaré (resp. Siegel) Theorem, then h f and f are conjugate: there exists ϕ in G n 1 such that h f = ϕ f ϕ -1 . As a consequence, h = [ϕ, f ] and h belongs to ker τ. □ Remark 2.10. Since A and f ∈ ker τ are conjugate, A belongs to ker τ, and so does the normal subgroup of GL(n, C) generated by A. The quotient G n 0 ⧸ ker τ ≃ im τ can thus be identified with the quotient GL(C n ) ⧸ GL(C n ) ∩ ker τ .

RESIDUALLY FINITE GROUPS

There are a number of equivalent definitions of residually finite groups; we will use the following one: Definition 3.1. A group H is residually finite if for every element h in H ∖ {id H there exists a group morphism ϕ : H → F from H to a finite group F such that ϕ(h) ̸ = id F .

The groups Z and SL(n, Z) are residually finite (by reduction modulo p). Subgroups of residually finite groups are residually finite. Conversely, a non-finite simple group is not residually finite, and the Baumslag-Solitar group

BS(2, 3) = ⟨a, b | a -1 b 2 a = b 3 ⟩ is not residually finite ([BS62]). A group is linear if it is isomorphic to a subgroup of GL(n, k)
where k is a field. [START_REF] Malcev | On isomorphic matrix representations of infinite groups[END_REF][START_REF] Mal'tsev | On the faithful representation of infinite groups by matrices[END_REF]). A finitely generated linear group is residually finite.

Theorem 3.2 ([
In particular there is no faithful linear representation of a non-residually finite, finitely generated group.

Remark 3.3. The assumption "finitely generated" turns out to be essential. For instance Q is not residually finite. If ξ is a positive transcendental number, then ξ

Q = ξ p q | p q ∈ Q is isomorphic to Q.
As a subgroup of a residually finite group is residually finite, we thus get that GL(1, C) ≃ C * , GL(n, C), G n 0 and G n 0 are not residually finite.

The following statement, that is mentioned in [START_REF] Claudon | Holonomy representation of quasi-projective leaves of codimension one foliations[END_REF] without detail in the 1-dimensional and formal case, is a direct consequence of Theorem 3.2: Theorem 3.4. Any finitely generated subgroup of G n 0 (resp. G n 0 ) is residually finite.

Proof. Let H be a finitely generated subgroup of G n 0 . Let h be an element of G n 0 ∖ {id}. There exists an integer k such that p k (h) is non-trivial in the quotient group

G n 0 ⧸ G n k . Let us recall that G n 0 ⧸ G n k is isomorphic to a subgroup of a linear group; Theorem 3.2 applied to the group p k (H) asserts the existence of a morphism ϕ k : p k (H) → F k from p k (H) to a finite group F k such that ϕ k (p k (h)) ̸ = id F k . Then, the morphism ϕ k • p k|H : H → F k suits. □
Finding the finitely generated subgroups that embed into G n 0 is an important problem, in particular related to the theory of foliations (representations of holonomy, and cycles in leaves). For instance in [START_REF] Cantat | Surface groups in the group of germs of analytic diffeomorphisms in one variable[END_REF] one can find:

Theorem 3.5 ([CCGS20]
). The fundamental group of the compact surface of genus g embeds into G 1 0 , and so into G n 0 .

Conversely we get the following statement:

Corollary 3.6. The Baumslag-Solitar group BS(2, 3) does not embed into G n 0 . More generally, if H is a finitely generated and non residually finite group, then H does not embed into G n 0 .

Theorem 3.4 has direct applications in the theory of holomorphic foliations. Toledo constructs smooth complex projective varieties with fundamental groups which are not residually finite, answering to Serre's question ( [START_REF] Toledo | Projective varieties with non-residually finite fundamental group[END_REF]). Assume that F is, for instance, a codimension one holomorphic foliation on the complex manifold M having an invariant variety N ⊂ M satisfying Toledo's property. Then the holonomy representation ([CCD13])

Hol : Π 1 (N, * ) → Diff(C, 0)
is not faithfull. As a consequence, there exist families of cycles in the leaves of F near the invariant manifold N.

In [START_REF] Druţu | Non-linear residually finite groups[END_REF] Drutu and Sapir construct residually finite groups that are not linear. One of their examples

is the group G = ⟨a, b | b 2 ab -2 = a 2 ⟩; we prove that G can not be embeded into G 1 0 (or G 1 0 ): Proposition 3.7. There is no faithfull representation of G = ⟨a, b | b 2 ab -2 = a 2 ⟩ into G 1 0 (resp. G 1 0 ).
Proof. Assume by contradiction that there exists a faithfull representation ϕ of G into G 1 0 . Set A = ϕ(a), and B = ϕ(b). From B 2 AB -2 = A 2 we get that A and A 2 are conjugate (by B 2 ). In particular A is tangent to the identity, that is A belongs to G 1 1 ; so there exists a formal vector field X of order at least 2 such that A = exp X. Since exp X and exp 2X are conjugate, the vector fields X and 2X are conjugate (by B 2 ). One can assume, up to conjugacy, that X = x ν+1 1-λx ∂ ∂x with ν ≥ 1 and λ ∈ C * . Let µx be the linear part of B 2 ; note that µx has to conjugate the first non-zero jet x ν+1 ∂ ∂x of X to the first non-zero jet 2x ν+1 ∂ ∂x of 2X. Hence µ ν = 2, and the linear part of B is a 2ν-th root of 2; we thus can linearize B, i.e. assume that B = αx where α = 2

1 2ν . Let Y = a(x) ∂ ∂x be a vector field of order ν + 1 such that B 2 * Y = 2Y , that is such that α -2 a α 2 x = 2a, i.e. in other words a(α 2 x) = 2α 2 a. (3.1) Write a as a = ∑ ℓ≥ν+1 a ℓ x ℓ ; then (3.1) yields to α 2ℓ a ℓ = 2α 2 a ℓ , i.e. α 2(ℓ-1) a ℓ = 2a ℓ . For any a ℓ ̸ = 0 we get α 2(ℓ-1) = 2; in other words 2 ℓ-1 ν -1 = 1, and so 2 ℓ-(ν+1) ν = 1; as a consequence, ℓ = ν + 1.
As a result, in the linearizing coordinate for B, we have: B = αx and A = exp cx ν+1 ∂ ∂x for some c. In particular the group generated by A and B is linear, whereas G is not ([DS05]): contradiction.

□

In [START_REF] Cerveau | Un théorème de Frobenius singulier via l'arithmétique élémentaire[END_REF] the authors prove the following curious result. Let γ be an irreducible curve in P 2 C of degree p s with p prime number. If ϕ : Let us now establish the following statement:

Π 1 (P 2 C ∖ γ, * ) → G 1 0 or ϕ : Π 1 (P 2 C ∖ γ, * ) → G 1 0 is a morphism, then
Theorem 4.5. The group G 1 0 is a Hopfian group.

To prove it we will use the following result of finite determination, result specific to the 1-dimensional and formal case: Lemma 4.6. Let h be an element of G 1 1 . There exists an integer ℓ such that if g belongs to G 1 ℓ , then h and hg are conjugate in the group G 1 1 . In other words, if two elements of G 1 1 coincide up to a sufficiently large order, then they are conjugate.

The proof is part of folklore and it is difficult to determine the "first author", maybe F. Takens. It is based on the following two facts:

• If X = a(x) ∂
∂x is a formal vector field such that a(0) = a ′ (0) = 0, then X is formally conjugate to a vector field of the type X p,λ = x p+1 1+λx p ∂ ∂x for some λ ∈ C and some p. It is a smart consequence of the local inversion theorem ([CCD13]). Such vector fields are thus finitely "determinated": if Y is of order sufficiently large, then X and X +Y are conjugated.

• If h belongs to G 1 1 , then there exists a formal vector field X = a(x) ∂ ∂x with a(0) = a ′ (0) = 0 such that h = exp X, i.e. h is the flow at time 1 of the vector field X.

Remark 4.7. The first fact is still true in the holomorphic case whereas the second is not ( [É75]): if h belongs to G 1 1 and h = exp X, then X is most of the time divergent.

Proof of Theorem 4.5.

Let ϕ : G 1 0 → G 1 0 be a surjective morphism. Assume that ϕ is not injective. Let f ∈ G 1 0 ∖ {id} such that ϕ( f ) = id.
Replacing f by a non-trivial commutator [g, f ] (that also belongs to ker ϕ) if needed we can assume that f belongs to G 1 1 ∖ {id}. Consider h in G 1 ℓ for ℓ sufficiently large. According to Lemma 4.6 the elements f and h f are conjugate, i.e. there exists g ∈ G 1

1 such that g f g -1 = h f . As a consequence, h = [g, f ] belongs to ker ϕ. Hence, ker ϕ contains G 1 ℓ for ℓ sufficiently large. Since ϕ is surjective, G 1 0 and G 1 0 ⧸ ker ϕ are isomorphic. As G ℓ 1 is contained in ker ϕ the morphism

G 1 0 ⧸ G 1 ℓ → G 1 0 ⧸ ker ϕ is surjective. The group G 1 0 ⧸ G 1 ℓ is solvable, so does G 1 0 ⧸ ker ϕ ≃ G 1 0 : contradiction
with the fact that G 1 0 contains free subgroups ([BCLN96]). □ Problems 4.8. 1) Is the group G 1 0 a Hopfian group ? One way to answer to this question is to show that if τ : G 1 0 → G 1 0 is surjective, then τ can be extended to a morphism τ : G 1 0 → G 1 0 still surjective.

2) Are the groups G n 0 and G n 0 Hopfian groups ?

Unfortunately the method used for the proof of Theorem 4.5 turns out to be ineffective for Problems 4.8.

Let us now deal with the notion of co-Hopfian group. Note that if τ : C → C is a non-surjective morphism of the field C, then τ induces an injective and non-surjective homomorphism from G n 0 into itself defined by

∑ A I x I → ∑ τ(A I )x I
where A I belongs to C n . In particular G n 0 is not co-Hopfian.

Theorem 4.9. The groups G n 1 and G n 1 are not co-Hopfian groups.

Proof. Let us first assume that n = 1. The morphism τ 1 :

f → τ 1 ( f ) defined by τ 1 ( f )(x) = f (x 2 ) 1/2
is injective but not surjective; indeed any τ 1 ( f ) commutes with the involution x → -x (we choose the determination √ 1 = 1).

Suppose now that n > 1. We will use a similar idea considering the application

E : (x 1 , x 2 , . . . , x n ) → (x 2 1 , x 1 x 2 , x 1 x 3 , . . . , x 1 x n ) whose inverse is E -1 : (x 1 , x 2 , . . . , x n ) → √ x 1 , x 2 √ x 1 , x 3 √ x 1 , . . . , x n √ x 1 .
Let us choose the determination of E -1 associated to the principal determination of √ ; the application τ n defined by

τ n ( f ) = τ n ( f 1 , f 2 , . . . , f n )(x) = E -1 ( f • E)(x)
is an injective morphism that is not surjective; indeed the τ n ( f ) commute with the involution x → -x. □ Problem 4.10. Is the group G n 0 a co-Hopfian group ?

TITS ALTERNATIVE

A group G satisfies Tits alternative if for every finitely generated subgroup H of G

• either H is virtually solvable (i.e. H contains a solvable subgroup of finite index),

• or H contains a non-abelian free subgroup.

Tits proved in [START_REF] Tits | Free subgroups in linear groups[END_REF] that linear groups satisfy Tits alternative.

Problem 5.1. Do the groups G n 0 and G n 0 satisfy Tits alternative ?

This important question is related to the Galois theory of holomorphic foliations. Note that, as it can be seen in [START_REF] Berthier | Sur les feuilletages analytiques réels et le problème du centre[END_REF], the group G n 0 contain free subgroups of rank ≥ 2.

6. AUTOMORPHISMS GROUPS OF G 1 0 AND G 1 0

In [START_REF] Whittaker | On isomorphic groups and homeomorphic spaces[END_REF] Whittaker proves the following statement: let X and Y be compact manifolds, with or without boundary, and ϕ be a group isomorphism between the group Hom(X) of all homeomorphisms of X into itself and Hom(Y ), then there exists an homeomorphism ψ of X onto Y such that ϕ( f ) = ψ f ψ -1 for all f ∈ Hom(X). When X = Y we get that every automorphism of Hom(X) is an inner one. In [START_REF] Filipkiewicz | Isomorphisms between diffeomorphism groups. Ergodic Theory Dynam[END_REF] Filipkiewicz gives a similar result in the context of differentiable manifolds: let M and N be smooth manifolds without boundary, and let Diff p (M) denote the group of C p -diffeomorphisms of M.

The author proves that if Diff p (M) and Diff q (N) are isomorphic as abstract groups, then p = q, and the isomorphism induced by a C p -diffeomorphism from M to N. Let us mention that there are similar results in different contexts: see for instance [Ban86, Ban97, D 06]... in particular in [D 06] the author proves that any automorphism of the Cremona group Bir(P 2 C ) of birational self-maps of the complex projective plane is the composition of an inner automorphism and an automorphism of the field of complex numbers. The proof is based on the study of maximal abelian uncountable subgroups of Bir(P 2 C ). Let G 1 , G 2 , . . ., G ℓ be some of such subgroups, and let ϕ be an automorphism of Bir(P 2 C ); the author proves that up to inner conjugacy and the action of an automorphism of the field C we have ϕ |G k = id for 1 ≤ k ≤ ℓ, and deduce from it that ϕ |Bir(P 2 C ) = id. A similar strategy will be used in this section.

6.1. Automorphisms groups of G 1 0 . The study of the maximal abelian subgroups of G n 0 (resp. G n 0 ) is essential for the understanding of automorphism groups of G n 0 (resp. G n 0 ). Unfortunately the comprehension of these groups can only be apprehended in the case G 1 0 . Once again the essential argument is the following one: if

f belongs to G 1 k ∖ G 1 k+1 , then f is conjugate to exp X k,λ = exp x k+1 1+λx k ∂ ∂x for a certain λ. A computation ([CM88, CCD13]) shows that the centralizer Cent(exp X k,λ , G 1 0 ) = f ∈ G 1 0 | f exp X k,λ = exp X k,λ f of exp X k,λ in G 1 0 coincides with the group A k,λ = exptX k,λ |t ∈ C × x → ξx | ξ k = 1 .
This group, which is abelian and so maximal abelian, contains exactly (k -1) non-trivial torsion elements.

Let λ be in C * ; denote by λ : x → λx the homothety of ratio λ. If λ is not a root of unity, then

Cent(λ, G 1 0 ) = A 0 = µ | µ ∈ C * which is also a maximal abelian subgroup. A subgroup of G 1
0 whose all elements are periodic is abelian and conjugate to a subgroup of A 0 (see [CCD13, Corollary 7.21]); in particular a maximal abelian subgroup of G 1 0 contains a non-periodic element. As a consequence we get:

σ τ ( f ) = ∑ ℓ≥1 τ(a ℓ )x ℓ .
Note that the image of a maximal abelian subgroup of G 1 0 by σ is still a maximal abelian subgroup of G 1 0 .

Theorem 6.2. Let σ be an automorphism of G 1 0 . Then, up to a suitable conjugacy, σ(A 0 ) = A 0 and σ(A k,0 ) = A k,0 for any integer k ≥ 1.

Furthermore, if λ is non-zero, then σ(A k,λ ) is conjugate to A k,µ for some µ in C * .

Proof. The group A 0 has an infinite number of torsion elements whereas the A k,λ 's don't; this gives the first assertion. We can thus assume that σ(A 0 ) = A 0 . Let us note that A 0 acts by conjugacy on the groups A k,0 : if µ belongs to A 0 , then

µ exptX k,0 µ -1 = exp µ -k tX k,0 .
However, the conjugate of A k,λ by µ is A k,µ -k λ , and A 0 does not act on the A k,λ . As a result A 0 = σ(A 0 ) acts by conjugacy on σ(A k,0 ). Counting the torsion elements we get that σ(A k,0 ) is conjugate to A k,λ for some λ in C. In particular

σ exptx k+1 ∂ ∂x = exptX |t ∈ C
where X is a formal vector field conjugate to X k,λ for some λ in C.

Lemma 6.3. One has: X = a k x k+1 ∂ ∂x for some a k ∈ C * . Proof of Lemma 6.3. The action of µ : x → µx on exptX can be written as

µ exptXµ -1 = exptµ * X. If X = ∑ ℓ≥k a ℓ x ℓ ∂ ∂x , a k ̸ = 0, then µ * X = ∑ ℓ≥k a ℓ µ 1-ℓ x ℓ ∂ ∂x .
Let µ be a non-zero complex number; assume that µ is not a root of unity. As we have seen

µ exptXµ -1 = expt 0 X.
for some t 0 . As a consequence µ * X = t 0 X (see [START_REF] Cerveau | Groupes d'automorphismes de (C, 0) et équations différentielles ydy + • • • = 0[END_REF]) and

a ℓ µ 1-ℓ = t 0 a ℓ for ℓ ≥ k t 0 = µ 1-k
If for ℓ > k one of the a ℓ is non-zero, then µ 1-ℓ = t 0 = µ 1-k , and µ is a root of unity: contradiction. □

As a result σ(A k,0 ) = A k,0 for any k; the torsion elements can optionally be permuted and exptx

k+1 ∂ ∂x is sent onto expt σ k x k+1 ∂ ∂x . Let us remark that C ∋ t → t σ k ∈ C is an additive homomorphism group. □ Hence an automorphism σ of G 1 0 induces a multiplicative isomorphism of C * σ(λ) = λ σ
and an additive isomorphism σ k of C for any k

σ k (t) = t σ k .

Let us come back to the action of A

0 on A k,0 λ, exptx k+1 ∂ ∂x → exp λ k tx k+1 ∂ ∂x that corresponds to the action of C * on C (λ,t) → λ k t.
The action is transformed by the automorphism σ into

λ σ , expt σ k x k+1 ∂ ∂x → exp(λ σ ) k t σ k x k+1 ∂ ∂x but also into λ σ , expt σ k x k+1 ∂ ∂x → exp(λ k t) σ k x k+1 ∂ ∂x . Therefore (λ σ ) k t σ k = (λ k t) σ k ;
in particular for t = 1 we get

s σ 1 σ k = s σ k .
Hence s → s σ is an automorphism of the field C, and the additive morphisms s → s σ k differ from s σ only by a multiplicative constant 1 σ k . Up to the action of the automorphism of G 1 0 associated to this field automorphism we can assume that s → s σ is the identity and that s σ k = ε k s where ε k denotes a non-zero constant.

Lemma 6.4. If σ belongs to Aut( G 1 0 ), then σ( G 1 k ) = G 1 k .
Proof. Let h be an element of G 1 k ; then there exist ϕ in G 1 0 , p ≥ k, and λ in C such that h = ϕ exp X p,λ ϕ -1 .

Recall that σ(exp X p,λ ) = exp aX p,λ ′ for some a ∈ C * and λ ′ ∈ C. Hence

σ(h) = σ(ϕ) exp aX p,λ ′ σ(ϕ -1 )
belongs to G 1 k ; similarly σ -1 (h) belongs to G 1 k . □ Remark 6.5. From Lemma 6.4 we get that σ is a continuous automorphism of G 1 0 endowed with the Krull topology.

Let us recall the Baker-Campbell-Hausdorff formula applied to the formal vector fields

X = a(x) ∂ ∂x and Y = b(x) ∂ ∂x of χ 1 2 . If Z ∈ χ 1 2 is a solution of exp Z = exp X expY then Z = X +Y + 1 2 [X,Y ] + 1 12 [X, [X,Y ]] - 1 12 [Y, [X,Y ]] + . . . In particular exp X expY exp -X exp -Y = exp [X,Y ] + h.o.t. (6.1)
where h.o.t. denotes terms of order ≥ 2 in the algebra generated by X and Y . We thus get

exp x k+1 ∂ ∂x exp x ℓ+1 ∂ ∂x exp -x k+1 ∂ ∂x exp -x ℓ+1 ∂ ∂x = exp exp x k+1 ∂ ∂x , exp x ℓ+1 ∂ ∂x + χ
where χ is a vector field of the form

χ = x ℓ+k+1+inf(k,ℓ) a(x) ∂ ∂x ; in other words exp x k+1 ∂ ∂x exp x ℓ+1 ∂ ∂x exp -x k+1 ∂ ∂x exp -x ℓ+1 ∂ ∂x = exp (ℓ -k)x k+ℓ+1 ∂ ∂x + χ = exp (ℓ -k)x k+ℓ+1 ∂ ∂x h
where h denotes an element of G 1 ℓ+k+inf(k,ℓ) (still by Baker-Campbell-Hausdorff formula). Applying σ we get

exp ε k x k+1 ∂ ∂x exp ε ℓ x ℓ+1 ∂ ∂x exp -ε k x k+1 ∂ ∂x exp -ε ℓ x ℓ+1 ∂ ∂x = exp ε k ε ℓ x k+1 ∂ ∂x , x ℓ+1 ∂ ∂x + χ = exp ε k+ℓ x k+ℓ+1 ∂ ∂x σ(h)
where χ is given by (6.1) and σ(h) is controlled by Lemma 6.4. By truncating to a suitable order we see that ε k ε ℓ = ε k+ℓ . Up to the action by an homothety we can assume that ε 1 = 1; as a result ε ℓ = ε ℓ+1 .

Hence by induction ε k = 1 for any k. The automorphism σ thus fixes the homotheties and the exptx k+1 ∂ ∂x . The quotient groups G 1 0 ⧸ G 1 k are generated by the projections of the homotheties and by the projections of the exptx k+1 ∂ ∂x , t ∈ C, ℓ ≤ k -1. According to Lemma 6.4 the automorphism σ induces an automorphism of the quotient groups G 1 0 ⧸ G 1 k that coincides with the identity on any G 1 0 ⧸ G 1 k . Therefore, σ coincides with the identity:

Theorem 6.6. The group Aut( G 1 0 ) is generated by the inner automorphisms and the automorphisms of the field C. In other words Out( G 1 0 ) ≃ Aut(C, +, •) where Out( G 1 0 ) denotes the non-inner automorphisms of G 1 0 .

6.2. Automorphisms groups of G 1 0 . We are going to adapt the above approach to the holomorphic case, i.e. to the description of Out(G 1 0 ). Theorem 6.1 has no analogue but [É75] and [START_REF] Cerveau | Groupes d'automorphismes de (C, 0) et équations différentielles ydy + • • • = 0[END_REF] imply:

Theorem 6.7. The uncountable maximal abelian subgroups A of G 1 0 such that A ∩ G 1 1 is uncountable are the conjugate of the groups A 0 and A k,λ where k ≥ 1 denotes an integer, and λ a complex number.

The group A 0 is a maximal abelian subgroup of G 1 0 .

Remarks 6.8.

(1) The group G 1 1 contains many countable abelian maximal subgroups; in fact generically the group generated by a generic element of G 1 1 is maximal ([É75]). (2) There are diffeomorphisms of type λz + h.o.t, λ = exp(2iπγ), γ ∈ R ∖ Q, which are not holomorphically linearizable f . The maximal abelian group that contains f is not necessarily of the previous type. It can be uncountable (see [START_REF] Marco | Nonlinearizable holomorphic dynamics having an uncountable number of symmetries[END_REF]).

Let σ be an element of Aut(G 1 0 ). We can, as in the formal case, characterize the groups A 0 and A k,λ by their torsion elements. Recall that if f belongs to G 1 1 , then f is a commutator, and σ( f ) also. As a consequence, σ(G 1 1 ) = G 1 1 , and if H is a maximal abelian subgroup of G 1 1 , then σ(H) also. Hence, as in the formal context, σ(A k,0 ) is holomorphically conjugate to one of the A k,λ . We want to prove that σ(A 0 ) is conjugated to A 0 . Lemma 6.9. Let f be an element of G 1 0 . Assume that f acts by conjugation on A k,λ , i.e. f A k,λ f -1 = A k,λ . Then

• if λ ̸ = 0, then f = ξ exp ε x p+1 1+λx p ∂ ∂x for some ε, ξ in C such that ξ p 1; • if λ = 0, then f = µ exp εx p+1 ∂
∂x for some µ in C * , and ε in C.

Proof. Left to the reader. □ Now, let us fixed k; we can assume, up to conjugacy, that σ(A k,0 ) = A k,λ for some λ. But σ(A 0 ) contains periodic elements of all periods so λ = 0 and σ(A 0 ) is an abelian subgroup of the "affine" group µ exp εx p+1 ∂ ∂x | µ ∈ C * , ε ∈ C (Lemma 6.9). Such a group is either conjugate to a subgroup of A 0 , or conjugate to a subgroup of

ξ exp ε x p+1 1 + λx p ∂ ∂x | ε, ξ ∈ C, ξ p = 1 .
The fact that σ(A 0 ) contains an infinite number of periodic elements implies that σ(A 0 ) is a subgroup of A 0 . As σ(A 0 ) is maximal, one gets: σ(A 0 ) = A 0 .

Recall that σ(A 0 ) = A 0 acts on all the σ(A ℓ,0 ); an argument similar to that used in the formal case ( §6.1) implies that σ(A ℓ,0 ) = A ℓ,0 for all ℓ ∈ N.

As before σ : A 0 → A 0 corresponds to an automorphism of the field C.

Lemma 6.10. Any σ ∈ Aut(G 1 0 ) extends into an automorphism σ :

G 1 0 → G 1 0 . Moreover, σ(G 1 k ) = G 1 k .
Proof. The projections of the homotheties and the exp

x ℓ ∂ ∂x 's, ℓ ≤ k -1, generate G 1 0 ⧸ G k 0 ≃ G 1 0 ⧸ G k 0 .
Since σ preserves A 0 and the A ℓ,0 's, it induces an automorphism

σ k : G 1 0 ⧸ G k 0 → G 1 0 ⧸ G k 0
for any k. By construction these automorphisms are compatible with the filtration induced by the G k 0 's

G 1 0 ⧸ G k 0 σ k / / G 1 0 ⧸ G k 0 G 1 0 ⧸ G k+ℓ 0 O O σ k+ℓ / / G 1 0 ⧸ G k+ℓ 0 O O
As a result, the σ k 's determine an automorphism σ : G 1 0 → G 1 0 . This automorphism extends σ in the following sense: if we fix a coordinate z, we get an embedding G 1 0 → G 1 0 that associates to the convergent element f = ∑ a n z n the element f = ∑ a n z n seen as a formal serie. By construction σ( f ) = σ( f ), i.e. σ sends a holomorphic diffeomorphism onto a holomorphic diffeomorphism.

□

According to what we have done before we can assume that σ is associated to an automorphism τ of the field C: if f = ∑ a n z n , then

σ( f ) = ∑ n≥1 τ(a n )z n .
The automorphism τ satisfies the following property: if ∑ n≥1 a n z n converges, then ∑ n≥1 τ(a n )z n also converges.

Lemma 6.11. Either τ is the identity z → z, or τ is the complex conjugation z → z.

Proof. If τ preserves R, that is if τ(R) = R, then τ is either the identity, or the complex conjugation.

If τ(R) ̸ = R, then the image of the unit disk D(0, 1) is dense in C (see [START_REF] Kestelman | Automorphisms of the field of complex numbers[END_REF]). Suppose that τ is neither the identity, nor the complex conjugation. By density there exists a n ∈ D(0, 1), n ≥ 2, such that |τ(a n )n!| < 1. If f = z + ∑ a n z n , then on the one hand f belongs to G 1 0 , and on the other hand z + ∑ a n z n diverges: contradiction. □

We finally can state:

Theorem 6.12. The group Out(G 1 0 ) is isomorphic to Z ⧸ 2Z . Let σ be an element of Aut( G n 0 ). Denote by j

AUTOMORPHISMS GROUPS

1 : G n 0 → GL(C n ), f → D f (0)
the map that associates to f its linear part at 0. Set ϕ = j 1 • σ, i.e.

ϕ : G n 0 σ -→ G n 0 j 1 -→ GL(C n ) f → σ( f ) → D(σ( f )) (0)
and consider ϕ = ϕ |GL(C n ) the restriction of ϕ to GL(C n ).

Lemma 7.3. The map

ϕ : GL(C n ) → GL(C n ), A → D(σ(A)) (0)
is an injective homomorphism.

Proof. Denote by H the subgroup of homotheties C * id of GL(C n ).

Lemma 7.4. The periodic elements of GL(C n ) ∖ {id} do not belong to ker ϕ.

Proof. Let A ∈ GL(C n ) be a non-trivial periodic element of period p, i.e. A p = id, and A k ̸ = id for any 1 ≤ k ≤ p -1. Then σ(A) is also a non-trivial periodic element of GL(C n ), and its period is p. As a consequence σ(A) is conjugate to its linear part ϕ(A), and ϕ(A) is a periodic element of GL(C n ) of period p. Therefore A does not belong to ker ϕ. □

The normal subgroups of GL(C n ) are either subgroups of H , or overgroups of SL(C n ). But SL(C n ) contains periodic elements, so Lemma 7.4 implies that the normal subgroup ker ϕ of GL(C n ) is a subgroup of H .

Let us consider an element B of ker ϕ. Then σ(B) = id + h.o.t., and σ(B) = exp X where X denotes a formal vector field of order ν ≥ 2. Let us remark that exptX = id + ∑ P I (t)x I where the P I (t)'s are polynomial transformations. Hence if f commutes to exp X, then f commutes to all exptX, and f * X = X. Since B belongs to ker ϕ ⊂ H , B commutes to GL(C n ), and σ(B) commutes to σ(GL(C n )): if ψ belongs to σ(GL(C n )), then ψ * X = X. As a result, if X ν is the homogeneous part of lower degree of X, then for any φ * ∈ ϕ(GL(C n )) we have φ * X ν = X ν .

Let C = µ id be an homothety such that µ is a p-th root of unity with the following property: µ ν-1 ̸ = 1. As we have seen σ(C) is periodic, non-trivial, of period p; hence σ(C) is conjugate to ϕ(C). As a consequence ϕ(C) is a periodic element of GL(C n ). i) Assume that ϕ(C) is in H , i.e. ϕ(C) = α id with α ∈ C * such that α p = 1 and α ν-1 ̸ = 1. On the one hand φ * X ν = X ν , and on the other hand

φ * X ν = (α id) * X ν = α ν-1 X ν . As a consequence α ν-1 = 1: contradiction. ii) Suppose that ϕ(C) is not in H . Since ϕ(C) is periodic, ϕ(C) is semi-simple. We can thus write ϕ(C) as follows                     λ 1 . . . λ 1       λ 2 . . . λ 2    . . .                 
with λ i ̸ = λ j as soon as i ̸ = j. Note that we have the following diagram:

GL(C n ) ϕ / / ϕ(GL(C n )) GL(C n ) ⧸ ker ϕ ≃ 5 5
As ker ϕ is a group of homotheties we have a surjection

GL(C n ) ⧸ ker ϕ ↠ PGL(C n )
and so a surjection ϕ(GL(C n )) ↠ PGL(C n ). Since C commutes with GL(C n ), ϕ(C) commutes with ϕ(GL(C n )). As a result ϕ(GL(C n )) is a group of block matrices. But there is no surjection from a group of block matrices into PGL(C n ); indeed, such a group contains non-trivial proper normal subgroups whereas PGL(C n ) is simple.

□

According to Lemma 7.3, ker ϕ = {id}. Consider a non-periodic homothety λ id. Then, ϕ(λ id) is non-periodic, and as λ id commutes to GL(C n ), ϕ(λ id) commutes to ϕ(GL(C n )) ≃ GL(C n ).

Claim 7.5. The image ϕ(λ id) of λ id by ϕ is an homothety.

Proof of Claim 7.5. Assume by contradiction that ϕ(id) is not an homothety. Then, Cent(ϕ(λ id), GL(C n )) is isomorphic to a block matrices group H. In particular, ϕ(GL(C n )) is a subgroup of H, and thus can not be isomorphic to GL(C n ): contradiction. □ Since ϕ(λ id) is a non-periodic homothety, ϕ(λ id) can be written as µ id for some µ ∈ C * that is not a root of unity, and σ(λ id) = µ id + h.o.t. is (maybe only formally if |µ| = 1) conjugate to µ id. After conjugacy let us assume that σ(λ id) = ϕ(λ id). As σ(GL(C n )) commutes to µ id, the inclusion σ(GL(C n )) ⊂ GL(C n ) holds after conjugacy. Let us now prove the reciprocal inclusion, that is GL(C n ) ⊂ σ(GL(C n )). Let B be an element of GL(C n )∖ σ(GL(C n )). But B commutes to µ id, so σ -1 (B) commutes to σ -1 (µ id) = λ id. Since λ is not a root of unity, σ -1 (B) is linear, i.e. belongs to GL(C n ): contradiction with GL(C n ) ∖ σ(GL(C n )). Finally GL(C n ) = σ(GL(C n )), and one can state: Lemma 7.6. If σ belongs to Aut( G n 0 ), then up to conjugacy σ |GL(C n ) is an automorphism of GL(C n ).

Proposition 7.7. Up to conjugacy and up to the action of an element of Aut(C, +, •), the restriction of an element of Aut( G n 0 ) to GL(C n ) is the identity map.

Proof. Let σ be an element of Aut( G n 0 ) According to Lemma 7.6, up to the action of an automorphism of the field C, either σ |GL(C n ) is the identity map, or σ |GL(C n ) is the involution A → t A -1 .

Assume that σ |GL(C n ) is the involution A → t A -1 . Set f = (x 1 + x 2 2 , x 2 , x 3 , . . . , x n ), As a consequence, σ( f ) commutes to both (x 1 , x 2x 1 , x 3 , x 4 , . . . , x n ) and x 1 a 2 , x 2 a , x 3 b 3 , x 4 b 4 , . . . , x n b n . From σ( f )σ(h a,b ) = σ(h a,b )σ( f ), we get that there exist ε in C and λ i in C * such that σ( f ) = (λ 1 x 1 + εx 2 2 , λ 2 x 2 , λ 3 x 3 , . . . , λ n x n ); furthermore, from σ( f )σ(g) = σ(g)σ( f ), we obtain ε = 0 and λ 1 = λ 2 , i.e.

g = (x 1 + x 2 , x 2 , x 3 , . . . , x n ),
σ( f ) = (λ 1 x 1 , λ 1 x 2 , λ 3 x 3 , . . . , λ n x n ).

In particular σ( f ) = σ x 1 λ 1 , x 2 λ 1 , x 3 λ 3 , x 4 λ 4 , . . . , x n λ n : contradiction. □ Proposition 7.7 can used as follows. Take an element A in GL(n, C), typically A 0 = (λx 1 , x 2 , x 3 , . . . , x n ), and consider the group Cent(A, G n 0 ) = f ∈ G n 0 | f • A = A • f . In the case of the example of A 0 , for some generic λ, we have Cent(A 0 , G n 0 ) = a(x 2 , x 3 , . . . , x n )x 1 , g(x 2 , x 3 , . . . ,

Theorem 6. 1 .

 1 The maximal abelian subgroups of G 1 0 are the conjugate of the groups A 0 and A k,λ where k ≥ 1 is an integer and λ an element of C. Let us now consider σ : G 1 0 → G 1 0 an automorphism of G 1 0 . For instance the inner automorphisms of G 1 0 f → τ( f ) = ϕ f ϕ -1 are such examples. Let Aut(C, +, •) be the automorphisms group of the field C. Any element τ ∈ Aut(C, +, •) is an automorphism of the field C, then τ induces an automorphism σ τ that sends f = ∑ ℓ≥1 a ℓ x ℓ to

  OF G n 0 , AND G n 0 We have in mind the following two conjectures: Conjecture 7.1. The automorphisms group of G n 0 is generated by the inner automorphisms, and the automorphisms of the field C. Conjecture 7.2. The group Out(G n 0 ) is isomorphic to Z ⧸ 2Z .

  h a,b = (a 2 x 1 , ax 2 , b 3 x 3 , b 4 x 4 , . . . , b n x n ) with a i , b i ∈ C * . Note that • g belongs to GL(C n ), so σ(g) = (x 1 , x 2x 1 , x 3 , x 4 , . . . , x n ); • h a,b belongs to GL(C n ), hence σ(h) = x 1 a 2 , x 2 a , x 3 b 3 , x 4 b 4 , . . . , x n b n ; • f g = g f ; • f h a,b = h a,b f .

0 .

 0 x n ) | a ∈ C[[x 2 , x 3 , . . . , x n ]] * , g ∈ G n-1 If σ ∈ Aut( G n 0 ) is such that σ |GL(n,C) = id GL(n,C) , then σ Cent(A, G n 0 ) = Cent(A, G n 0 ), and we can expect for instance with A of type A 0 to use an induction on the dimension to prove Conjecture 7.1.

  Definition 4.1. A group G is Hopfian if every surjective homomorphism group from G to G is an isomorphism. Equivalently, a group is Hopfian if and only if it is not isomorphic to any of its proper quotients. Definition 4.2. A group G is co-Hopfian if every injective homomorphism group from G to G is an isomorphism. Equivalently, a group is co-Hopfian if and only if it is not isomorphic to any of its proper subgroups.Every finite group and every simple group are Hopfian groups. The group Z of integers and the group Q of rationals are Hopfian groups. However, C * is not a Hopfian group (the morphisms C * → C * , x → x p are not injective), and R * is not a Hopfian group (the morphisms R * → R * , x → x p , p even, are not injective). In [D 07] the author shows that the group Bir(P 2 C ) of birational self-maps of the complex projective plane P 2 C is Hopfian. Let us mention an other statement due to Malcev: Any finitely generated residually finite group is a Hopfian group.

	Problem 3.8. Let γ be a curve in P 2 C ; are the groups Π 1 (P 2 C ∖ γ, * ) linear groups ? are the groups
	Π 1 (P 2 C ∖ γ, * ) residually finite groups ?
	4. HOPFIAN AND CO-HOPFIAN GROUPS
	Theorem 4.3 ([Mal40, Mal65]). Corollary 4.4. Any finitely generated subgroup of G n 0 (resp. G n 0 ) is a Hopfian group.

the image of ϕ is a finite group (conjugate to a group of linear rotations); moreover, there are some γ such that Π 1 (P 2 C ∖ γ, * ) contains a free group of rank 2.


M I Yue

Laurent Stolovitch

ON LINEARIZATION OF BIHOLOMORPHISM WITH NON-SEMI-SIMPLE LINEAR PART AT A FIXED POINT

We prove the holomorphic linearizability of germs of biholomorphisms of (C n , 0), fixing the origin, point at which the linear part has nontrivial Jordan blocks under the following assumptions : We first assume the eigenvalues are of modulus less or equal than 1, and that they are non-resonant. We also assume that they satisfied not only a classical Diophantine condition but also new Diophantine-like conditions related to quasi-resonance phenomena.

Introduction and main results

Let F be a germ of biholomorphism of (C n , 0) fixing the origin and let A = F ′ (0) be its linear part (Jacobian matrix) at the origin. Let spec(A) = {λ 1 , λ 2 , • • • , λ n } be eigenvalues of A. We define the set of resonant multi-indices of spec(A) as follows :

Res[spec(A)] = n j=1
Res j [spec(A)],

Res j [spec(A)] = {α ∈ Z n + (2) : λ αλ j = 0}, where

Z n + (k) = {α ∈ Z n + : |α| ≥ k}, λ = (λ 1 , λ 2 , • • • , λ n ) and λ α = λ α 1 1 • • • λ αn n for α = (α 1 , α 2 , • • • , α n ) ∈ Z n + and |α| = n i=1 |α i |.
We say that F is formally linearizable at the origin if there exists a formal power series transformation, fixing the origin, which is tangent to the identity Φ(z

) = z + ϕ ≥2 (z) ∈ (C[[z]]) n such that (1.1) Φ -1 • F • Φ(z) = Az.
Here, ϕ ≥2 vanishes at order ≥ 2 at the origin. If A is diagonal, the well known Poincaré-Dulac theorem [START_REF] Arnold | Geometrical methods in the theory of ordinary differential equations, volume 250 of Grundlehren der Mathematischen Wissenschaften[END_REF] ensures that if there is no resonance (i.e., Res[spec(A)] = ∅), then F is formally linearizable. As the coefficients of the formal solution of (1.1) involves products of form (λ αλ j ) -1 , when the degree |α| goes to infinity, its convergence in a neighborhood of the origin is very related to the so-called small divisors problem occurring when inf α |λ αλ j | = 0 for some j ∈ {1, • • • , n}.

The pioneering work of C.L.Siegel [START_REF] Carl | Iteration of analytic functions[END_REF], followed by E.Zehnder [START_REF] Zehnder | A simple proof of a generalization of a theorem by C[END_REF], shows that the Diophantine condition (for some fixed positive C 0 , σ) is sufficient to ensure the analyticity of Φ at the origin. This Diophantine condition has been weakened by H.Rüssmann [START_REF] Rüssmann | Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition[END_REF] (and by A.D.Brjuno [START_REF] Brjuno | Analytic form of differential equations. I, II[END_REF] for vector fields) to : for all m ≥ 2,

(1.2) |λ α -λ j | > C 0 |α| -σ for all j = 1, 2, • • • , n, α ∈ Z n + (
|λ α -λ j | ≥ 1 Ω(m)
for all 1 ≤ j ≤ n and |α| = m,

where Ω : N → R is a function satisfying :for all m ∈ N, m ≤ Ω(m) ≤ Ω(m + 1),

∞ m=1
Log Ω(m) m 2 < ∞.

It is a major achievement due to J.-C. Yoccoz who proved the necessity of this condition for holomorphic linearization of non-resonant biholomorphism of (C, 0) [START_REF] Yoccoz | Théorème de Siegel, nombres de Bruno et polynômes quadratiques[END_REF].

While previous results concern the linearization problem of a single holomorphic map, T.Gramchev and M.Yoshino [START_REF] Gramchev | Rapidly convergent iteration method for simultaneous normal forms of commuting maps[END_REF], L. Stolovitch [START_REF] Stolovitch | Family of intersecting totally real manifolds of (C n , 0) and germs of holomorphic diffeomorphisms[END_REF] also obtained results on simultaneous linearization of a family of commuting biholomorphisms of (C n , 0). All of results above require the linear part of the holomorphic map to be semi-simple. Little is known about the (non)linearizability of F in the analytic category. We mention the work of T.Ueda [START_REF] Ueda | Normal forms of attracting holomorphic maps[END_REF] in which a new proof of the holomorphic conjugacy to a "lower triangular polynomial map" with contracting linear parts is given, that is when

(1.3) max 1≤j≤n |λ j | < 1.
In particular, this ensures the holomorphic linearization in the non-resonnant case.

In dimension n = 2, J.-C.Yoccoz [START_REF] Yoccoz | Théorème de Siegel, nombres de Bruno et polynômes quadratiques[END_REF][pp. [86][87] proved that in general the analytic linearization can not be achieved when the linear part is a single Jordan block associated to an eigenvalue on the unit circle. In dimension n = 3 and n = 4, D.Delatte and T.Gramchev [START_REF] Delatte | Biholomorphic maps with linear parts having Jordan blocks: linearization and resonance type phenomena[END_REF] gave a positive answer for biholomorphic germs whose linear parts have one nontrivial 2-dimensional Jordan block, provided that their eigenvalues satisfy some non-resonant and Diophantine-like conditions. They also gave an example to show the need of an arithmetic condition for the convergence to hold. Their proofs rely on very explicit computations of the solution of the homological equation.

The main purpose of the present paper is to extend their positive answers to any dimension with linear parts having possibly nontrivial Jordan blocks of any dimension. In this situation, the explicit computations of Delatte-Gramchev cannot be carried on and a more conceptual framework had to be developed for that purpose. In what follows, we assume

(1.4) max 1≤i≤n |λ i | ≤ 1.
Let us state our main theorem. If the linear part F ′ (0) of F is not semi-simple, then a preliminary linear change of variables allows us to assume that it is a typical Jordan normal form :

(1.5) F ′ (0) = Λ ǫ = Λ + ǫN,
where Λ = diag{λ 1 , • • • , λ n } is the diagonal matrix, N is an upper triangular nilpotent matrix where all non-zero entries are 1 and lie on the upper-diagonal. By dilation of coordinates, ǫ = 0 can be made arbitrarily small. Let {µ i } m i=1 be the set of distinct modulus of its eigenvalues, and without loss of generality we can suppose that :

(

1.6) 1 = µ 1 > µ 2 > • • • > µ m .
We emphasize that 1 is a modulus since Λ ǫ is a not a strict contraction. Denote {1, 2,

• • • , n} = m i=1 I i , for each 1 ≤ i ≤ m, j ∈ I i when |λ j | = µ i . Thus I i I j = ∅ if i = j. For a fixed multi-index α ∈ N n , we write (α 1 , α 2 , • • • , α n ) = (A 1 , A 2 , • • • , A m ) and A i = (α l i +1 , α l i +2 , • • • , α l i +n i ),
where

l 1 := 0, l i+1 := l i + n i , n i := |I i |, for i ≤ m.

Thus we have

|A i | = n i j=1 α l i +j and n = m i=1 n i . Correspondingly, we can also denote (λ 1 , • • • , λ n ) = (Λ 1 , • • • , Λ m ) and (z 1 , • • • , z n ) = (Z 1 , • • • , Z m ) in a similar way. Thus for all i = 1, • • • , m we have: (1.7) Λ A i i = n i j=1 λ α l i +j l i +j , Z A i i = n i j=1 z α l i +j l i +j .
If there is no confusion, for a fixed 1 ≤ i ≤ m, we will usually write i j for l i + j so that α i j stands for

α l i +j for j = 1, 2, • • • , n i . Remark 1.1. Notice that if there is a tuple of non-negative integers {κ i } 1≤i≤m such that m i=1 κ i ≥ 2 and m i=1 µ κ i i = µ d for some 3 ≤ d ≤ m, then we must have κ j = 0 for all j ≥ d.
With these notations, we can introduce the definition of the quasi-resonance : Definition 1.1. Let {µ i } m i=1 be distinct positive numbers satisfying (1.6). For each 3 ≤ i ≤ m, we call Quasi-Resonance (w.r.t µ i ) a relation of the form if :

(1.8) i-1 j=2 µ κ j j = µ i ,
for some tuple of non-negative integers {κ j } 2≤j≤i-1 . We also assume that for all k ∈ I i , for all

A j ∈ N n j such that |A j | = κ j and i-1 j=1 |A j | ≥ 2, 2 ≤ j ≤ i -1, we have Λ A 1 1 • i-1 j=2 Λ A j j = λ k .
In order to achieve the convergence of the linearization, we need to give the Diophantine-like assumptions for each quasi-resonance. Definition 1.2. We say that Λ ǫ satisfies a QR-Diophantine condition (QR stands for Quasi-Resonant) if the following holds : For each 3 ≤ i ≤ m where there is a quasi-resonance (w.r.t µ i ), we have :

(1.9) Λ A 1 1 • i-1 j=2 Λ A j j -λ k ≥ C 0   |A 1 | + i-1 j=2 κ j   -σ , for all k ∈ I i , i-1 j=1 |A j | ≥ 2 and |A j | = κ j , 2 ≤ j ≤ i -1.
Theorem 1.1. Let F (z) = Λ ǫ z + f (z) be a holomorphic map in a neighborhood of 0 in C n , f has the order greater or equal than 2 at the origin. Assume the linear part (1.5) is non-resonant and ǫ = 0 is sufficiently small, then there is a unique transformation

z = Φ(ζ) = ζ + φ ≥2 (ζ)
, biholomorphic in a neighborhood of 0, which solves the linearization problem F •Φ = Φ•Λ ǫ near the origin, if one of the following conditions is satisfied : (I) If there is no quasi-resonance, then there exists two positive numbers C 0 , σ > 0 such that for all λ i , i ∈ I 1 ,

(1.10) Λ A 1 1 -λ i ≥ C 0 (|A 1 |) -σ , ∀ |A 1 | ≥ 2
, and for any i, j ∈ I k , k ≥ 2, we have

(1.11) Λ A 1 1 λ i -λ j ≥ C 0 (|A 1 | + 1) -σ , ∀ |A 1 | ≥ 1.
(II) If there is a quasi-resonance, then besides (1.10) and (1.11), we also assume Λ ǫ satisfies the QR-Diophantine condition (1.9).

Remark 1.2. Assuming we have (1.6), then there are only finite numbers of quasiresonance. In fact, for each i = 3, 4, • • • , m, there are at most finite tuples of non-negative integers {κ (i) :

κ (i) ∈ N i-2 } satisfying (1.8).
The main idea for proof of the convergence is to estimate the solutions of homological equations L(f ) = g, that is the "linearized version" of the conjugacy equation. In order to do that, we decompose the set of monomials along two sets called Poincaré slice and Siegel slice respectively. We then decompose the Taylor expansion of any germ of holomorphic function at the origin accordingly. We show that these slices are invariant sets of the homological operator. This enables to solve the projected equations L(f P ) = g P and L(f S ) = g P . The aim of this decomposition is that on the Siegel slice we will encounter small divisors but we have arithmetic conditions to control it, while on the Poincaré slice, we have the complicated remainders brought by Jordan blocks but we do not encounter small divisors so that we have a uniform bound for the solution to the homological equation restricted to that set. The details will be seen in section 3.

The paper is organized as follows: In section 2 we introduce some basic notations and definitions of norms that we would need to use in the KAM scheme. Section 3 introduces the homological operator as well as Poincaré slice and Siegel slice.

In section 4, we deal with the estimation of the remainders in the homological equation which is brought by nontrivial Jordan blocks. In section 5, we give the most important estimation of the solution to the homological equation, which will be used several times in the proof of the convergence in the section 6.

Notations and preliminaries

For an integer k ≥ 1, and ρ > 0, 0 < δ < r ≤ 1,

• P n k denotes the C-space of homogeneous polynomial vector fields on C n and of degree k, • p n k denotes the C-space of homogeneous polynomials on C n of degree k, • O n denotes the ring of germs at 0 of holomorphic functions in

C n , • C[[z]] ≥k = f | f (z) = α∈Z n + (k) f α z α , f α ∈ C . When z ∈ C n , |z| := max 1≤i≤n |z i |. Denote B R = {z ∈ C n : |z i | ≤ r i , ∀ 1 ≤ i ≤ n} for R = (r 1 , r 2 , • • • , r n ) with positive numbers r i . For a formal power series g(z) ∈ C[[z]] denote ḡ(z) = α∈Z n + |g α |z α , the common polydisc norm is |g| R = ḡ(r 1 , r 2 , • • • , r n ) corresponding with B R .
We set an asymmetric ball: for two disjoint sets D and J such that D J = {1, 2, • • • , n}, where actually λ j , j ∈ J corresponds to the Jordan block. Denote We introduce two Banach algebras :

H(B r,ρ ) = g(z) = α∈Z n + g α z α , g α ∈ C : |g| r,ρ < +∞ ,
with the polydisc norm corresponding to B r,ρ :

|g| r,ρ := ḡ(r, ρ), z i = r if i ∈ I, z j = ρ if j ∈ J.
Let D ⊇ B r,ρ be an open set. Let O 2 (D) be the space of all holomorphic functions in D such that f (0

) = 0, ∂ z j f (0) = 0, j = 1, 2, • • • , n. Let us define H(D) = f ∈ C(D) ∩ O n 2 (D) : f D < +∞ , f D = sup z∈D |f (z)|.
For all vector-valued functions F = (f 1 , . . . , f n ), the norm |F | r,ρ (resp. F ) denote the maximum of the norm of its components. To simplify the notation, we also denote F r = F Br . By Cauchy estimates, we have :

(2.2) g r-δ ≤ |g| r-δ ≤ r δ n g r .
For an n × n matrix A, we set |A| := max 1≤i,j≤n |a ij |. For z ∈ C n , we have :

(2.3) |Az| ≤ n|A||z|.
We introduce two basic inequalities (e.g. [START_REF] Stolovitch | Singular complete integrability[END_REF][p.149]) which will be used in section 4:

(2.4)

|f | r-δ ≤ r -δ r k |f | r , if ord(f) ≥ k, ∂f ∂z i r ≤ k r |f | r , if f ∈ P k n .
We shall write f ≥k to emphasize that f has order ≥ k at the origin. At last, we introduce a domain and a norm which is need in the proof of convergence (here ǫ is fixed) :

D r := B r ∪ Λ ǫ (B r ),
g C 1 ,r := max{ g Dr , Dg Dr }.

Representation of the homological operator and its decomposition

Let F (z) = Λ ǫ z + f ≥2 (z) be a germ of biholomorphism of (C n , 0). Let Φ(y) = y + φ(y) be a linearizing change of variables : F • Φ = Φ(Λ ǫ z). Hence, φ, which vanishes at order ≥ 2 at the origin, solves φ • Λ ǫ -Λ ǫ φ = f ≥2 (I + φ). Hence, it is natural to introduce the homological equation/operator :

(3.1) L(ϕ) := ϕ • Λ ǫ -Λ ǫ ϕ = g, g ∈ H(B r ) n .
We want first to find a formal solution ϕ ∈ C[[z]] n and then we want to estimate ϕ with respect to a given g. With the above settings, we can rewrite L(ϕ) in (3.1) as :

(3.2) L(ϕ) = (Ω -ǫN )ϕ + R ǫ ϕ := L ǫ ϕ + R ǫ ϕ,
where

(3.3) Ω(ϕ) = ϕ • Λ -Λϕ, R ǫ (ϕ) = ϕ • (Λ + ǫN ) -ϕ • Λ.
Given a formal power series f (z) = α f α z α , we have

(3.4) Ω(f )(z) = α (Ω α f α )z α , N (f )(z) = α (N f α )z α ,
where

Ω α = diag{λ α -λ 1 , • • • , λ α -λ n } is a diagonal matrix.
As Λ ǫ is Jordan matrix, the matrix Ω α and N are commuting for each α ∈ Z n + (2). Thus due to the nilpotency of N and the non-resonance of the eigenvalues of Λ ǫ , (Ω -ǫN ) is invertible at the formal level (i.e. on C[[z]] n ≥2 ), we have

(3.5) (Ω α -ǫN ) -1 = Ω -1 α + d-1 s=1 ǫ s Ω -s-1 α N s ,
where d is the nilpotency order of N . Hence, emphasizing the remainder operator R ǫ , equation (3.1) reads:

(3.6) ϕ = (Ω -ǫN ) -1 g -(Ω -ǫN ) -1 R ǫ ϕ.
The abstract equation above plays an important role in the estimation of the solution to the homological equation. The main difficulty in the Jordan block case is to estimate the remainder term R ǫ . To overcome this difficulty, we introduce the most important idea : (S,P)-decomposition, which is inspired by D.Delatte and T.Gramchev [START_REF] Delatte | Biholomorphic maps with linear parts having Jordan blocks: linearization and resonance type phenomena[END_REF][p.10, Definition 2.4]. The aim of this decomposition is that on the Siegel slice we will encounter small divisors but we have arithmetic conditions to control it, while on the Poincaré slice, although we have complicated remainders brought by Jordan blocks, we do not encounter small divisors and we will have a uniform bound for the solution to the homological equation restricted on it. Briefly speaking, we want to find two disjoint subsets S and P of Z n + (2) such that the following conditions holds: i)

Z n + (2) = S P, S P = ∅, ii) Invariance: L ±1 S ⊂ S, L ±1 P ⊂ P .
iii) The inverse L -1 S of restriction of the operator on Siegel slice L S := L| S , is related to small divisors problem, but Diophantine conditions for α ∈ S gives a control; iv) Poincaré slice L P := L| P does not involve in any small divisors, thus we have a uniform estimate of its inverse L -1

P . We rewrite ϕ ∈ C[[z]] n along such a decomposition : (3.7) ϕ(z) = α∈Z n + (2) ϕ α z α = ϕ S (z) + ϕ P (z), ϕ S (z) = α∈S ϕ α z α , ϕ P (z) = α∈P ϕ α z α .
By abuse of notations, we shall write that ϕ(z) ∈ S if for all all monomials (of the Taylor expansion at 0) of ϕ(z) belong to S. Let us define the sets S and P according to the diffrent cases of Theorem 1.1 :

In case (I), as there is no quasi-resonance, we decompose Z n + (2) for Λ ǫ in (1.5) as follows :

(3.8)

S := {α ∈ Z n + (2) : m i=2 |A i | ≤ 1}, P := {α ∈ Z n + (2) : m i=2 |A i | ≥ 1}.
In case (II), there are quasi-resonances. Let us denote

QR i =    κ (i) = (κ (i) 2 , κ (i) 3 , • • • , κ (i) i-1 ) ∈ N i-2 : i-1 j=2 µ κ (i) j j = µ i  
 the set of quasi-resonances w.r.t µ i . According to (1.6), then for each 3 ≤ i ≤ m, there exists at most finite quasi-resonances, i.e., ♯|QR i | < ∞. By non-resonance condition we also have:

Λ A 1 1 • i-1 j=2 Λ A j j = λ k , for all k ∈ I i , i-1 j=1 |A j | ≥ 2 and |A j | = κ (i) j , 2 ≤ j ≤ i -1. Denote the subsets of Z n + (2)
as the following:

(3.9)

S 1 := {α ∈ Z n + (2) : |A j | = 0 , for 2 ≤ j ≤ m}, S 2 := {α ∈ Z n + (2) : m l=2 |A j | = 1},
and for any κ ∈ QR i , (3.10)

S i,κ := {α ∈ Z n + (2) : |A j | = κ j for 2 ≤ j ≤ i -1, |A j | = 0 for i ≤ j ≤ m}, S i := κ∈QR i S i,κ , 3 ≤ i ≤ m.
We define the (S, P )-decomposition to be :

(3.11) S := m i=1 S i , P := m i=1 S c i ,
where

S c i denotes the complement of S i in Z n + (2). Remark 3.1. The idea for defining these slices is to select α ∈ Z n + (2) such that |λ α | = |λ i |.
Let us check that the first two properties i), ii) of our decomposition. The last two properties will be seen in the following sections.

Lemma 3.1. The homological operator (3.2) as well as decomposition (3.3) have the following invariance property : For each Siegel slice S i , i = 1, 2, • • • , m, we have :

L ±1 S i ⊂ S i .
On Poincaré slice we have :

L ±1 P ⊂ P.
Proof. Recalling (3.4), we observe that Ω(ϕ) and N (ϕ) preserve each monomial of ϕ. Hence the operators Ω and N satisfy the invariance property. For the remainder part, first we need to know the specific form of the linear part (1.5). Assume there is only one Jordan block corresponding to Λ t , here we use the notations of (1.7) and replace l t + j by t j for simplicity. Let us write

ϕ(z) = α∈Z n + (2) ϕ α z α = A∈Z n + (2) ϕ A Z A .
By (3.3), we have :

(3.12) R ǫ ϕ(z) = ϕ • (Λ + ǫN ) -ϕ • Λ = A ϕ A 1≤j≤m j =t (Λ j Z j ) A j nt-1 l=1 (λ t l z t l + ǫz t l+1 ) αt l • (λ tn t z tn t ) αt n t - nt l=1 (λ t l z t l ) αt l .
We claim that the operator R ǫ does not change the total degree of those variables corresponding to the Jordan block and it keeps each monomial of those variables corresponding to the diagonal eigenvalues.

In fact, for a monomial z

αt 1 t 1 • • • z αt n t
tn t the variables of which correspond to Λ t , assume its total degree |A t | = nt l=1 α t l = κ for some κ. By binomial expansion, for each 1 ≤ l ≤ n t -1 we have :

(3.13) (λ t l z t l + ǫz t l+1 ) αt l = αt l s l =0 α t l s l (λ t l z t l ) s l (ǫz t l+1 ) αt l -s l .
Let Ãt = (α t 1 , • • • , αtn t ) be the exponent of one of the new monomials in the variables z t 1 , • • • , z tn t appearing in the bracket of the formula (3.12). Each new monomial coming from the product in the bracket in (3.12), is obtained by choosing different s l in (3.13), that is,

αt 1 = s 1 , αt l+1 = α t l -s l + s l+1 , αtn t = α t n t -1 -s nt-1 + α tn t , for 1 ≤ l ≤ n t -2.
This directly shows :

| Ãt | = nt l=1 αt l =s 1 + nt-2 l=1 (α t l -s l + s l+1 ) + α t n t -1 -s nt-1 + α tn t = nt l=1 α t l = |A t | = κ.
Thus the total degree of all of the new monomials corresponding to Λ t does not change and the degree of each one of the other variables is kept fixed, which proves our claim.

Similarly, if there are two blocks corresponding to Λ t and Λ s , then

R ǫ ϕ(z) = ϕ • (Λ + ǫN ) -ϕ • Λ = α ϕ α 1≤j≤m j =t,s (Λ j Z j ) A j nt-1 l=1 (λ t l z t l + ǫz t l+1 ) αt l • (λ tn t z tn t ) αt n t • ns-1 l=1 (λ s l z s l + ǫz s l+1 ) αs l • (λ sn s z sn s ) αs ns - α ϕ α 1≤j≤m j =t,s (Λ j Z j ) A j nt l=1 (λ t l z t l ) αt l • ns l=1 (λ s l z s l ) αs l .
Observe that if we restricted the operator on

(3.14) |A t | = κ t , |A s | = κ s ,
for some fixed integers κ t , κ s , then for the new exponents of monomials produced, | Ãt |, | Ãs | will be maintained, i.e.,

(3.15)

| Ãt | = κ t , | Ãs | = κ s .
Actually we find that once the linear part has a Jordan block, the total degree of the corresponding monomials will always be maintained, moreover, each monomial of the variables corresponding to diagonal part is kept fixed. Considering the way of choosing the Siegel slice (3.9) and (3.10), we can deduce that for each 1 ≤ i ≤ m, (R ǫ ) ±1 S i ⊂ S i . Thus actually we have :

L ±1 S i ⊂ S i .
On Poincaré slice, considering the choice of the Siegel slice (3.9) and (3.10) are all based on |A j | = a for some fixed a = 0, 1 or a = κ j in (3.

R ǫ ( κ∈QR i S c i,κ ) ⊂ κ∈QR i R ǫ (S c i,κ ), thus R ǫ ( κ∈QR i S c i,κ ) ( κ∈QR i S i,κ ) = ∅, i.e. R ǫ (S c i ) (S i ) = ∅.
Similarly we have

(3.17) R ǫ ( m i=3 S c i ) ( m i=3 S i ) = ∅.
Combining (3.16) and (3.17) we have

(3.18) R ǫ ( m i=1 S c i ) ( m i=1 S i ) = ∅.
This directly shows that

R ǫ (P ) = R ǫ ( m i=1 S c i ) ⊂ m i=1 S c i = P.
Thus the lemma is proved.

This shows that our (S, P )-decomposition satisfies the invariance property i), ii). The left two properties will be proved in the following sections. To summarize, (S, P )-decomposition allows us to consider it separately within the two slices in order to get a final convergence of the solution to the convergent linearization problem.

Estimate of the remainder operator R ǫ

In order to simplify the notations and statements, we will first consider the case of a single Jordan block. We assume it corresponds to Λ d for some fixed integer d ≥ 2. In this case, the asymmetric ball (2.1) reads :

(4.1) B r,ρ = {z ∈ C n : |z i | < r for i / ∈ I d , |z j | < ρ for j ∈ I d }.
Using notations from (1.7), we will replace l i + j by i j .

In this section we give a crucial estimate for the homological operator restricted to the Poincaré slice : Lemma 4.1. Let ǫ be small enough (depending only on Λ). Then there is a constant C which depends only on Λ, such that, for any ϕ ∈ H(B r,ρ ), (4.2)

|R ǫ ϕ| r,ρ ≤ ǫ C|ϕ| r,ρ .

Proof. First suppose there is only one single Jordan block in (1.5). Then by Taylor formula we have

(4.3) R ǫ (ϕ)(z) = ϕ(Λz + ǫN z) -ϕ(Λz) = 1 0 Dϕ(Λz + tǫN z) • (ǫN z)dt. Let us write ϕ = (ϕ i ) i=1,••• ,n with ϕ i = k≥2 ϕ i k where ϕ i k ∈ p k n .
Since N is nilpotent, only those derivatives of the variables corresponding to Λ d make contributions to the integral in (4.3). Since |λ j | < 1, j ∈ I d , we may choose ǫ sufficiently small such that |λ j | + ǫ < 1, j ∈ I d . For all 0 < t < 1 we have

(Λ + tǫN )B r,ρ ⊂ Λ ǫ (B r,ρ ) ⊂ B r,(µ d +ǫ)ρ .
Applying the Cauchy estimates for polydisc norm (2.4) to each partial derivative

∂ϕ i k ∂z j , j ∈ I d from B r,(µ d +ǫ)ρ to B r,ρ , for all i = 1, 2, • • • , n and k ≥ 2 : |R ǫ ϕ i k | r,ρ = 1 0 Dϕ i k (Λz + tǫN z) • (ǫN z)dt r,ρ ≤ 1 0 n d -1 j=1 ∂ϕ i k ∂z d j (Λz + tǫN z) • ǫz d j+1 dt r,ρ ≤ n d -1 j=1 ∂ϕ i k (Λ ǫ ) ∂z d j r,ρ • ǫz d j+1 r,ρ ≤ n d -1 j=1 ∂ϕ i k ∂z d j r,(µ d +ǫ)ρ • ǫρ (4.4) ≤ ǫ(n d -1)ρ k (µ d + ǫ)ρ |ϕ i k | r,(µ d +ǫ)ρ ≤ ǫ(n d -1) k µ d + ǫ (µ d + ǫ)ρ ρ k |ϕ i k | r,ρ = ǫ(n d -1)(µ d + ǫ) k-1 k • |ϕ i k | r,ρ .
For a constant a, 0 < a < 1, we consider a function h a (x) = ln(a x-1 x) = (x -1) ln a + ln x, its derivative is h ′ a (x) = ln a + 1 x , so h a (x) has maximum at x 0 = -1 ln a . Let k 0 be the one who satisfies h a (k 0 ) = max{h a ([x 0 ]), h a ([x 0 ] + 1), h a (2)}, thus for all k ≥ 2 we have h a (k) ≤ h a (k 0 ). Now if we suppose

(4.5) ǫ < 1 -µ d 2 ,
then for µ d + ǫ < 1+µ d 2 := a < 1, we have for all k ≥ 2 :

(µ d + ǫ) k-1 k ≤ exp h a (k 0 ),
where k 0 and a depend only on Λ. By this estimate and observing that

ϕ i = k≥2 ϕ i k , for each i = 1, 2, • • • , n we obtain (4.6) |R ǫ ϕ i | r,ρ ≤ ǫ C|ϕ i | r,ρ ,
where the constant C = C(n d , λ d ) which depends only on n d and λ d . With the estimates above, it can be seen easily that, in case of several Jordan blocks, we consider in (2.1) with |z i | < r corresponding to variables of diagonal part and |z j | < ρ corresponding to variables in Jordan blocks. Then there will be several summations in equation (4.4), and C will depend on the dimension and eigenvalues of those corresponding blocks. Thus the lemma is proved.

Remark 4.1. Although ϕ does not need to be restricted on the Poincaré slice in the previous lemma, we shall only use this estimate on Poincaré slice.

Estimate of the solution to the homological equation

In this section, for the convenience of convergence proof we will consider the ball B r = B r,r . Our main purpose is to solve and estimate the solution of the homological equation (3.1). According to invariance property ii) of (S, P )-decomposition, we consider it along the two separate slices :

(5.1) L(ϕ P + ϕ S ) = g P + g S ⇒ L(ϕ P ) = g P , L(ϕ S ) = g S .

Suppose that the linear part (1.5) satisfies the corresponding QR-Diophantine conditions, our main goal is to prove the following proposition :

Proposition 5.1. Suppose ǫ is sufficiently small and Λ ǫ satisfies the conditions in Theorem 1.1, then there exists a constant C 2 which depends on κ, σ, Λ, n, C 0 so that for a holomorphic map g ≥2 ∈ H(B r ) n , there is a unique solution ϕ ∈ H(B r-δ ) n of the homological equation (3.1) which satisfies the following estimate :

(5.2) ϕ r-δ ≤ C 2 • δ -(ϑ+n) g r , for all 0 < δ < r ≤ 1,
where ϑ is the positive constant in equation (5.25) which depends only on κ, σ, n.

We need to state some basic lemmas first for its proof. First, on the Poincaré slice we have a formal solution and we estimate it.

Lemma 5.1. The restriction to the Poincaré slice of the homological equation (5.1) has a unique solution ϕ P ∈ H(B r ) n w.r.t any given g P ∈ H(B r ) n . Suppose ǫ is sufficiently small, then we have the following estimate :

(5.3) |ϕ P | r ≤ C P |g P | r ,
where C P = C(Λ, n, κ) depends only on Λ, n and κ.

Proof. First we suppose there is only one single Jordan block. The restriction of the homological equation (3.2) to the Poincaré slice is :

(5.4) L(ϕ P ) = (Ω -ǫN )ϕ P + R ǫ ϕ P = g P .

Throughout this proof, let α ∈ P ⊂ Z n + (2). We claim that for each i = 1, 2, • • • , n, there exists a constant γ i > 0 depending on Λ and κ such that (5.5)

|λ α -λ i | ≥ γ i .
In fact, by the (S, P )-decomposition (3.8) and (3.11), we have :

(5.6)

|λ α | = |λ i | for all i = 1, 2, • • • , n.
Equation (1.6) indicates that |λ α | tends to zero as | j / ∈I 1 α j | goes to infinity. Combining the non-resonant condition (5.7)

|λ α -λ i | = 0 for all i = 1, 2, • • • , n,
and (5.6), there exists an integer M i such that we have

|λ α -λ i | ≥ 1 2 |λ i | when |α| ≥ M i . Denote γ i := min{ inf |α|<M i |λ α -λ i |, 1 2 |λ i |},
by (5.7) we have γ i > 0, thus the claim (5.5) is proved.

The n × n-dimension diagonal matrix Ω α on Poincaré slice is :

Ω α = diag{λ α -λ 1 , λ α -λ 2 , • • • , λ α -λ n }.

Denote

(5.8) γ p := max 1≤i≤n γ -1 i , by (5.5) we have :

|Ω -1 α | ≤ max 1≤i≤n {|λ α -λ 1 | -1 , |λ α -λ 2 | -1 , • • • , |λ α -λ n | -1 } ≤ max 1≤i≤n γ -1 i = γ p .
According to the equation (3.5), we suppose (5.9)

ǫγ p ≤ 1 2 .
Then for all α ∈ P we have :

(5.10)

|(Ω α -ǫN ) -1 | = n d -1 s=0 Ω -s-1 α (ǫN ) s ≤ n d -1 s=0 |Ω -1 α | s+1 ǫ s ≤ 2γ p (1 -2 -n d ) := Γ(Λ, κ),
For a given g P ∈ H(B r ) n on the Poincaré slice, by (S, P )-decomposition (3.11) and (2.3), using (5.10) on P we have :

(5.11)

(Ω -ǫN ) -1 g P r = (Ω -ǫN ) -1 α∈P g α z α r ≤ α∈P |g α | (Ω α -ǫN ) -1 r |α| ≤ Γ(Λ, κ) α∈P |g α |r |α| = Γ(Λ, κ)|g P | r .
The equation (5.4) reads as :

I + (Ω -ǫN ) -1 R ǫ ϕ P = (Ω -ǫN ) -1 g P .
By the estimates above and Lemma 4.1, if we choose ǫ sufficiently small such that |(Ω -ǫN ) -1 R ǫ | < 1, then the operator I + (Ω -ǫN ) -1 R ǫ is invertible at a formal level and we have :

ϕ P = I + (Ω -ǫN ) -1 R ǫ -1 (Ω -ǫN ) -1 g P ,
thus the existence is proved. Let us estimate the solution. We rewrite (5.4) as :

ϕ P = (Ω -ǫN ) -1 g P -(Ω -ǫN ) -1 R ǫ ϕ P ,
taking the polydisc norm of the ball B r on both sides and by Lemma 4.1 we obtain :

|ϕ P | r = (Ω -ǫN ) -1 g P -(Ω -ǫN ) -1 R ǫ ϕ P r ≤ (Ω -ǫN ) -1 g P r + (Ω -ǫN ) -1 R ǫ ϕ P r ≤ Γ(Λ, κ)|g P | r + Γ(Λ, κ) |R ǫ ϕ P | r ≤ Γ(Λ, κ) |g P | r + ǫΓ(Λ, κ) C |ϕ P | r ,
where C = C(n d , λ d ) comes from (4.2) which depends only on Λ.

To summarize, on the Poincaré slice, if ǫ is sufficiently small such that (5.12) ǫΓ(Λ, κ) C < 1 and (5.9) are satisfied, then there is a constant C(Λ, κ) = Γ(Λ, κ) C which depends on Λ and κ such that :

|ϕ P | r ≤ Γ(Λ, κ) 1 -ǫ • C(Λ, κ) |g P | r .
If there is more than one Jordan block in Λ ǫ , then the estimate in (5.10) will depend on the maximum of n d where I d corresponds to Jordan blocks. Thus the lemma is proved.

Remark 5.1. We need to mention that this idea can not be applied on Siegel slice, because we will encounter small divisor there. In fact, on Siegel slice the estimate of |(Ω -ǫN ) -1 | in (5.10) is related to the multi-indices α, which tends to infinity as |α| goes to infinity. So we can not choose ǫ sufficiently small to satisfy (5.12).

Before we start to estimate the solution of the homological equation restricted on the Siegel slice, we need to introduce the lexicographic order for monomials.

Definition 5.1. Given two exponent vectors α = (α 1 , α 2 , • • • , α n ), β = (β 1 , β 2 , • • • , β n ). One has (α 1 , α 2 , • • • , α n ) ≺ (β 1 , β 2 , • • • , β n ) if either |α| < |β|, or |α| = |β| but α i < β i for the smallest i for which α i = β i .
Then naturally we have a relation order "less than:

≺" in Z n + (2). The lexicographic order of a monomial z α 1 1 • • • z αn n is defined as : LO(z α 1 1 • • • z αn n ) := α = (α 1 , • • • , α n ). For any finite subset W ⊂ Z n + (2) and a power series f W (z) = α∈W f α z α , define LO(f W ) := max α∈W LO(z α ),
where the maximum is taken under the relation order ≺.

With this definition, we have the following lemma :

Lemma 5.2. On Siegel slice (3.11), the operator R ǫ is nilpotent. i.e., there exists a finite integer η = η(κ, n) which depends only on κ and n such that :

(R ǫ ) η g S = 0 , for all g S ∈ H(B r ) n .

Proof. First we consider the case of a single Jordan block Λ d . Recall that each Siegel slice is (3.10), on S 1 and S 2 we have :

g S 1 (z) = ∞ |A 1 |=2 g A 1 Z A 1 1 , g S 2 (z) = ∞ |A 1 |=1 m i=2 |A i |=1 g A 1 ,A i Z A 1 1 Z A i i .
By (3.12) and direct calculation we obtain :

(5.13) R ǫ (g S 1 ) = 0

For any i = d we have R ǫ (Z A 1 1 Z A i i ) = 0.
Note that the operator R ǫ acts only on monomials, thus

R ǫ g S 2 (z) = ∞ |A 1 |=1 |A d |=1 g A 1 ,A d (R ǫ )(Z A 1 1 Z A d d ).
For each j ∈ I d , j = n d and |A j | = 1, by observing that

R ǫ (Z A 1 1 Z A j j ) = R ǫ (Z A 1 1 z j ) = ǫZ A 1 1 z j+1 , we have : (5.14) (R ǫ ) n d g S 2 (z) = ∞ |A 1 |=1 g A 1 ,A d 1 ǫ n d -1 R ǫ (Z A 1 1 z n d ) = 0.
When there is a quasi-resonance κ ∈ QR i , we have :

(5.15)

g S i,κ (z) = α∈S i,κ g α Z A 1 1 Z A 2 2 • • • Z A i-1 i-1 .
For each quasi-resonance, we consider the monomial

Z A 1 1 Z A 2 2 • • • Z A i-1 i-1 for a fixed α = (A 1 , A 2 , • • • , A i-1 , 0) which belongs to the slice S i,κ . First we observe that R ǫ (Z A 1 1 Z A 2 2 • • • Z A i-1 i-1 ) = 0 iff 2 ≤ d ≤ i -1. Actually we have : R ǫ (Z A 1 1 Z A 2 2 • • • Z A i-1 i-1 ) = 1≤j≤i-1 j =d (Λ j Z j ) A j n d -1 i=1 (λ d i z d i + ǫz d i+1 ) α d i • (λ dn d z dn d ) α dn d - n d i=1 (λ d i z d i ) α d i .
(5.16)

We only expand terms of non zero exponent α d i = 0 through binomial expansion.

In order to simplify notations, in the following we suppress d and substitute n d , d i by n, i respectively. For the first product in the bracket of (5.16), we distribute the first parentheses of the product as :

n-1 i=1 (λ i z i ) α i + α i -1 s i =0 α i s i (λ i z i ) s i (ǫz i+1 ) α i -s i • (λ n z n ) αn (5.17) =(λ 1 z 1 ) α 1 n-1 i=2 (λ i z i + ǫz i+1 ) α i • (λ n z n ) αn + (5.18) α 1 -1 s 1 =0 α 1 s 1 (λ 1 z 1 ) s 1 (ǫz 2 ) α 1 -s 1 • n-1 i=2 (λ i z i + ǫz i+1 ) α i • (λ n z n ) αn . (5.19)
We notice that all of the lexicographic orders of those terms in (5.19), by Definition 5.1 are strictly less than LO(z

α 1 1 z α 2 2 • • • z αn n )
, because the degree of z 1 decreases at least 1. Let us focus on the term (5.18), we also expand the first parentheses of the product in (5.18) :

(λ 1 z 1 ) α 1 n-1 i=2 (λ i z i + ǫz i+1 ) α i • (λ n z n ) αn =(λ 1 z 1 ) α 1 n-1 i=2 (λ i z i ) α i + α i -1 s i =0 α i s i (λ i z i ) s i (ǫz i+1 ) α i -s i • (λ n z n ) αn =(λ 1 z 1 ) α 1 (λ 2 z 2 ) α 2 n-1 i=3 (λ i z i + ǫz i+1 ) α i • (λ n z n ) αn + (5.20) (λ 1 z 1 ) α 1 α 2 -1 s 2 =0 α 2 s 2 (λ 2 z 2 ) s 2 (ǫz 3 ) α 2 -s 2 • n-1 i=3 (λ i z i + ǫz i+1 ) α i • (λ n z n ) αn . (5.21)
Similarly we find that all of the lexicographic orders of those monomials in the term of (5.21), by Definition 5.1 are strictly less than LO(z

α 1 1 z α 2 2 • • • z αn n )
, because the degree of z 2 decreases at least 1 and the degree of z 1 is the same. As for term (5.20), we continue to expand the first parentheses in the product left. Thus when we expand all of the n -1 parentheses one by one, we can find that except from the first term

(5.22) n i=1 (λ i z i ) α i ,
the lexicographic order of all the other terms produced in (5.17) decreases, for that there is always one i in {1, 2, • • • , n -1} such that the degree of z i decreases at least 1 and the degree of the previous z j 's, j ≤ i -1 keep the same. And the term (5.22) of the original degree is eliminated in the bracket in (5.16). With these observations above, recall that we suppress d before, by Definition 5.1 we conclude that:

LO R ǫ (Z A 1 1 Z A 2 2 • • • Z A i-1 i-1 ) < LO(Z A 1 1 Z A 2 2 • • • Z A i-1 i-1 ).
for a fixed α = (A 1 , A 2 , • • • , A i-1 , 0). Since the formula (5.16) also tells that the new monomials produced by R ǫ still have the same indices of A j , j = d, the remainder operator only changes the exponents of the variables corresponding with Λ d . Thus we apply the remainder operator R ǫ at most for a finite time, we will get the only one term which has the smallest lexicographic order :

(5.23)

Z A 1 1 • • • z κ (i) d n d • • • Z A i-1
i-1 , for an index α ∈ S i,κ with fixed A j , j = d. It is obvious that if we apply the remainder operator R ǫ to (5.23) once again, it turns into zero. Thus for each A d , there exists an integer η A d such that

(R ǫ ) η A d (Z A 1 1 Z A 2 2 • • • Z A i-1 i-1 ) = 0.
While on each Siegel slice S i,κ , we have a finite

A d such that |A d | = κ (i) d , thus denote η i,κ := max |A d |=κ (i) d η A d ,
and we have (5.24) (R ǫ ) η i,κ g S i,κ = 0.

Since we have proved the nilpotency of the remainder operator R ǫ on each Siegel slice S i,κ , note that the whole Siegel slice S in (3.11) is a finite and disjoint union of S i,κ , κ ∈ QR i . Considering ♯|QR i | < ∞ and

g S = m i=1 g S i = g S 1 + g S 2 + m i=3 κ∈QR i g S i,κ ,
combining (5.13),(5.14) and (5.24), there exists a finite number

η(κ, n) := max{1, n d , max κ∈QR i 3≤i≤m η i,κ } such that :
(R ǫ ) η g S = 0. Thus we have proved our conclusion for only one single Jordan block. When there are several Jordan blocks, the monomial having the smallest lexicographic order in (5.23) will be the product of several monomials with the smallest lexicographic orders which corresponds to different blocks. Thus the lemma is proved. With this lemma above, we can solve and estimate the homological equation restricted on Siegel slice.

Lemma 5.3. The restriction to the Siegel slice of the homological equation (5.1) has a unique solution ϕ S w.r.t g S for any given g S ∈ H(B r ) n . Moreover, we have the following estimate :

(5.25)

|ϕ S | r-δ ≤ C S • δ -ϑ |g S | r , for all 0 < δ < r ≤ 1,
where ϑ is a positive constant which depends only on κ, σ, n and C S is a constant which depends only on κ, σ, Λ, n, C 0 .

Proof. First we consider the linear part having only one single Jordan block Λ d . Without loss of generality, in order to make this proof simpler we will suppose that only the first eigenvalue belongs to the unit circle, i.e., I 1 = {1}. If I 1 has more that one element, we give the similar assumption of Diophantine conditions and this makes no essential difference to our proof. Similar to Poincaré slice case, by (S, P )-decomposition (3.11), on S 1 we have :

Ω α = diag{λ α 1 1 -λ 1 , λ α 1 1 -λ 2 , • • • , λ α 1 1 -λ n }. By condition (1.10), we have |λ α 1 1 -λ 1 | ≥ C 0 |α 1 | -σ , and |λ α 1 1 -λ i | ≥ 1 -|λ i | for 2 ≤ i ≤ n. Recall that S 2 = {α ∈ Z n + (2) : m l=2 |A j | = 1}, if α ∈ S 2 then in the diagonal matrix Ω α = diag{λ α -λ 1 , λ α -λ 2 , • • • , λ α -λ n },
α has and only has one |A j | = 1, j ≥ 2, thus for i, j ≥ 2 by (1.11) we have :

|λ α 1 1 λ i -λ j | ≥ |λ j -λ i | for |λ i | = |λ j |, |λ α 1 1 λ i -λ j | ≥ C 0 (α 1 + 1) -σ for λ i = λ j but |λ i | = |λ j |, |λ α 1 1 λ j -λ i | ≥ |λ j ||λ α 1 1 -1| ≥ |λ j | • C 0 |α 1 + 1| -σ for λ i = λ j .
Finally, for each 3 ≤ i ≤ m, we fix a κ (i) ∈ N i-2 . Let α ∈ S i,κ , then by the QR-Diophantine condition of quasi-resonance (1.9) we have :

λ α 1 1 • i-1 j=2 Λ A j j -λ k ≥ C 0 (|α 1 | + i-1 j=2 κ (i) j ) -σ , for all k ∈ I i , i-1 j=1 |A j | ≥ 2 and |A j | = κ (i) j , 2 ≤ j ≤ i -1.
Thus we can get the estimate |Ω -1 α | on each Siegel slice S i,κ , which will be used below. With the observations above, let us estimate |(Ω α -ǫN ) -1 | when α ∈ S. In order to simplify the following estimate, we denote ω(α 1 ) := C -1 0 |α 1 | σ . We can assume ω(α 1 ) ≥ 1 even if this means |α 1 | is large enough. According to (3.5), by our previous choice (5.9) there exists a constant γ s which depends on Λ, κ such that when α ∈ S 1 we have :

(5.26) |w -1 0 | := |(Ω α -ǫN ) -1 | = n-1 k=0 Ω -k-1 α (ǫN ) k ≤ n-1 k=0 |Ω -1 α | k+1 ǫ k ≤ γ s n•ω(α 1 ) n .
On the other hand, when α ∈ S 2 , we have:

(5.27)

|w -1 1 | := |(Ω α -ǫN ) -1 | = n-1 k=0 Ω -k-1 α (ǫN ) k ≤ n-1 k=0 |Ω -1 α | k+1 ǫ k ≤ γ s n • ω(α 1 + 1) n .
Although this means increasing γ s , we can assume that γ s n ≥ 1 so that the right hand side above is greater that 1. For 3 ≤ i ≤ m, suppose there is quasi-resonance κ ∈ QR i with α ∈ S i,κ . Then, we have : (5.28)

|w -1 κ | := |(Ω α -ǫN ) -1 | = n-1 k=0 Ω -k-1 α (ǫN ) k ≤ n-1 k=0 |Ω -1 α | k+1 ǫ k ≤ γ s n•ω(α 1 + i-1 j=2 κ j ) n .
Similar to the Poincaré slice, the homological equation restricted on Siegel slice is :

(5.29)

L(ϕ S ) = (Ω -ǫN )ϕ S + R ǫ ϕ S = g S , for a given g S ∈ H(B r ) n .
We write the formal solution of it :

ϕ S = I + (Ω -ǫN ) -1 R ǫ -1 (Ω -ǫN ) -1 g S .
By (3.3) and (3.4), for a given g S = α∈S g α z α , we have ΩR ǫ = R ǫ Ω. Indeed, we have

ΩR ǫ g S = g S (Λ 2 + ǫN Λ) -g S (Λ 2 ) -Λ (g S (Λ + ǫN ) -g S (Λ)) R ǫ Ωg S = g S (Λ 2 + ǫN Λ) -Λg S (Λ + ǫN ) -g S (Λ 2 ) -Λg S (Λ).
On the other hand, R ǫ N (ϕ) = N ϕ(Λ+ǫN )-N ϕ(Λ) = N R ǫ (ϕ). Hence, by observing (3.5), (Ω -ǫN ) -1 and R ǫ are pairwise commuting. Then according to Lemma 5.2, the inverse of the operator I + (Ω -ǫN ) -1 R ǫ does exist as a finite sum :

I + (Ω -ǫN ) -1 R ǫ -1 = j≥0 (-1) j (Ω -ǫN ) -1 R ǫ j = η-1 j=0 (-1) j (Ω -ǫN ) -j (R ǫ ) j .
Thus we have proved the existence of the formal solution to the homological equation restricted on Siegel slice (5.29). By Lemma 3.1 we can estimate it on each Siegel slices S i separately. First on S 1 , by equation (5.26) we have:

|ϕ S 1 | r-δ = I + (Ω -ǫN ) -1 R ǫ -1 (Ω -ǫN ) -1 g S 1 r-δ = ∞ α 1 =2 (Ω α -ǫN ) -1 g α 1 ,0 z α 1 1 r-δ ≤ ∞ α 1 =2 |w 0 | -1 |g α 1 ,0 |(r -δ) α 1 ≤ C S 1 • ∞ α 1 =2 ω(α 1 ) n r -δ r α 1 |g α 1 ,0 |r α 1 .
And we have

(5.30) 1 - δ r ≤ 1 -δ ≤ e -δ when 0 < δ < r ≤ 1. Let h(x) := x n e -δx , x ≥ 2, then h ′ (x) = (n -δx)
x n-1 e -δx , thus h(x) achieves its maximum at x 0 = n δ , i.e.,

(5.31)

h(x) ≤ h( n δ ) = δ -n • n n e -n .
Combining (5.30) and (5.31) we obtain:

(5.32)

|ϕ S 1 | r-δ ≤ C S 1 ∞ α 1 =2 α σn 1 e -δα 1 |g α 1 ,0 |r α 1 ≤ C S 1 • δ -σn |g S 1 | r .
Note that among the estimates above we do not change the notation C S 1 , because it is constant which only depends on σ, Λ, n, κ, C 0 and it does not influence the convergence proof in section 6. On S 2 , according to Lemma 5.2 we have :

|ϕ S 2 | r-δ = I + (Ω -ǫN ) -1 R ǫ -1 (Ω -ǫN ) -1 g S 2 r-δ = n-1 j=0 ∞ α 1 =1 n i=2 (-1) j (Ω α -ǫN ) -(j+1) g α 1 ,e i (R ǫ ) j (z α 1 1 z i ) r-δ .
Since the sum for index j is a finite sum, it will produce at most finite monomials of form z α 1 1 z i for a fixed α 1 , we also assume |(Ω α -ǫN )| -1 ≥ 1, by (5.27), (5.30) and Cauchy estimates we have :

|ϕ S 2 | r-δ ≤ C S 2 • ∞ α 1 =1 n i=2 |w 1 | -n |g α 1 ,e i |(r -δ) α 1 +1 ≤ C S 2 ∞ α 1 =1 ω(α 1 + 1) n 2 r -δ r α 1 +1 |g α 1 ,e i |r α 1 +1 ,
where C S 2 is a constant which only depends on σ, Λ, n, κ, C 0 . Thus by (5.31) we obtain :

(5.33)

|ϕ S 2 | r-δ ≤ C S 2 • ∞ α 1 =2
α σn 2 1 e -δα 1 |g α 1 ,e i |r α 1 +1 ≤ C S 2 • δ -σn 2 |g S 2 | r .

Finally, let us consider the slice S i,κ . For a fixed κ ∈ QR i , by Lemma 5.2 and (5.30) we have :

|ϕ S i,κ | r-δ = I + (Ω -ǫN ) -1 R ǫ -1 (Ω -ǫN ) -1 g S i r-δ = η-1 j=0 (-1) j (Ω -ǫN ) -(j+1) (R ǫ ) j ∞ α 1 =0 2≤j≤i-1 |A j |=κ j g α z α 1 1 Z A 2 2 • • • Z A i-1 i-1 r-δ = η-1 j=0 ∞ α 1 =0 2≤j≤i-1 |A j |=κ j (Ω α -ǫN ) -(j+1) g α (R ǫ ) j z α 1 1 Z A 2 2 • • • Z A i-1 i-1 r-δ
.

Similar to the estimate on S 2 , since the sum for index j is finite and is related to η, it will produce at most finite monomials of form

z α 1 1 Z A 2 2 • • • Z A i-1
i-1 for a fixed α 1 , we can also assume |(Ω α -ǫN )| -1 ≥ 1, by (5.28), (5.30) and (5.31) we have: (5.34)

|ϕ S i,κ | r-δ ≤ C S i,κ ∞ α 1 =0 2≤j≤i-1 |A j |=κ j ω   α 1 + i-1 j=2 κ j   ηn |g α |(r -δ) α 1 + i-1 j=2 κ j ≤ C S i,κ • ∞ α 1 =0   α 1 + i-1 j=2 κ j   σηn r -δ r α 1 + i-1 j=2 κ j |g α |r α 1 + i-1 j=2 κ j ≤ C S i,κ • δ -σηn |g S i,κ | r .
With the above estimates on these Siegel slices, combining (5.32), (5.33) and (5.34), by the invariance Lemma 3.1 we have :

|ϕ S | r-δ = m i=1 ϕ S i r-δ = ϕ S 1 r-δ + ϕ S 2 r-δ + m i=3 κ∈QR i ϕ S i,κ r-δ ≤ C S 1 • δ -σn |g S 1 | r + C S 2 • δ -σn 2 |g S 2 | r + m i=3 κ∈QR i C S i,κ • δ -σηn |g S i,κ | r ≤ C S • δ -ϑ |g S | r ,
where C S depends only on κ, σ, Λ, n, C 0 and ϑ = max{σn, σn 2 , σηn}, which depends only on κ, η, σ, n. Thus the lemma is proved. 6.1. Idea. We consider the linearization problem with Φ = Id + φ ≥2 , where φ ≥2 ∈ H(B r ) n . We try to solve the equation F(Φ) = 0, where

F(Φ) := F • Φ -Φ • Λ ǫ .
Since F(Id) = f , which is small near 0 , we are dealing with a perturbation problem. Since f contains only terms of order ≥ 2, then by eventually conjugating by an homothety, we can assume without loss of generality, that f is holomorphic on |z| < 1, and that (6.2) f 1 < δ 0 , for δ 0 as small as we want, to be chosen later on. Assuming F(Φ) to be small, we are looking for a better approximation Φ + v, which makes F(Φ + v) smaller. By Taylor expansion on Banach space we have :

F(Φ + v) = F(Φ) + F ′ (Φ)v + R(Φ, v),
where (6.3)

F ′ (Φ)v := d dt F(Φ + tv) | t=0 = DF • Φ • v -v • Λ ǫ ,
and the high order term R(Φ, v) is given by (6.4) R(Φ, v)

= 1 0 (1 -t) d 2 dt 2 F(Φ + tv) • v 2 dt = 1 0 (1 -t) d 2 dt 2 f (Φ + tv) • v 2 dt.
We would have to solve

F(Φ) + F ′ (Φ)v = 0 such that F(Φ + v) = O 2 (F(Φ)).
Unfortunately, because of small divisors the linear operator F ′ (Φ) given by (6.3) has no right-inverse on the space of holomorphic map on a fixed domain. We need to construct a sufficiently good approximating right-inverse of F ′ (Φ). Following Rüssmann [START_REF] Helmut Rüssmann | II. Bemerkungen zur Newtonschen Methode[END_REF] we have Let us set : v := DΦ • ϕ. Combining (6.3) and (6.5) we obtain

F ′ (Φ)v = DF(Φ) • ϕ + DΦ • Λ ǫ (Λ ǫ ϕ -ϕ • Λ ǫ ). Consequently (6.6) F(Φ + v) = F(Φ) + DΦ • Λ ǫ (Λ ǫ ϕ -ϕ • Λ ǫ ) + R(Φ, v) + DF(Φ) • ϕ.
By Lemma 6.1, we are able to solve the equation F(Φ)+DΦ•Λ ǫ (Λ ǫ ϕ-ϕ•Λ ǫ ) = 0 in case that DΦ•Λ ǫ is invertible. According to (6.6) we still have F(Φ+v) = O 2 (F(Φ)).

6.2. Set up. With the linear operator defined in (3.1), we shall define inductively the iteration Φ ν , for ν = 0, 1, 2, • • • as follows: Φ 0 = Id, and for ν ≥ 0, set (6.7)

Φ ν+1 = Φ ν + v ν , v ν = DΦ ν • ϕ ν , ϕ ν = L -1 ((DΦ ν • Λ ǫ ) -1 F(Φ ν )).
With this formula and (6.6) we then have :

(6.8) F(Φ ν+1 ) = DF(Φ ν ) • ϕ ν + R(Φ ν , v ν ),

  2) † Y.Mi was supported by China Scholarship Council(CSC) (No.202006190129). † †The research of L. Stolovitch has been supported by the French government through the UCAJEDI Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-01.

  (2.1) B r,ρ = {z ∈ C n : |z d | < r, d ∈ D and |z j | < ρ, j ∈ J}, B r := B r,r .

  10). Combining (3.14) and (3.15), when an α ∈ S c 1 S c 2 , we have the corresponding indices| Ãj | = 0, 1 for one of j ∈ {2, • • • , m}, thus (3.16) R ǫ (S c 1 S c 2 ) (S 1 S 2 ) = ∅.On the other hand, for each 3 ≤ i ≤ m and α ∈ S c i,κ , we have either | Ãj | = κ j for some j ∈ {2, • • • , i -1}, or | Ãj | = 0 for some j ≥ i. This means R ǫ (S c i,κ ) S i,κ = ∅, and naturally we have

(6. 5 )

 5 DF(Φ)(z) = DF • Φ(z) • DΦ(z) -DΦ • Λ ǫ (z) • Λ ǫ .

With those lemmas above, eventually we obtain the estimate of the solution to the homological equation (3.1).

Proof of Proposition 5.1. Suppose ǫ sufficiently small so that it satisfies (5.9) and (5.12), i.e., (5.35) ǫ < min{(2γ p ) -1 , (nΓ(Λ, κ) C ) -1 }.

Combining (2.2),(5.3) and (5.30), from Lemma 5.1 and 5.3 we conclude that :

here we use r ≤ 1 and we also keep the notation of the constant C 2 , which depends only on κ, σ, Λ, n, C 0 . Thus the proposition is proved.

Convergence proof

With all of the preparations above, in order to proof the holomorphic linearization, we can use verbatim the proof by the Newton iteration method as introduced by E.Zehnder [START_REF] Zehnder | A simple proof of a generalization of a theorem by C[END_REF]. We shall recall its main points without proof.

We first give the most important lemma in our convergence proof. The main idea comes from E.Zehnder[Zeh77]: Lemma 6.1. Let ǫ satisfies (5.35), g ∈ H(B r ) n for some 0 < r ≤ 1, then the homological equation (3.1) has a unique solution ϕ ∈ H(D r ) n , moreover, the following estimate holds for all 0 < δ < r ≤ 1 :

where τ = ϑ + n + 1, which depends only on κ, σ, n and

Proof. Since ϕ is holomorphic on B r-δ and solves there the holomological equation (5.1), then ϕ • Λ ǫ is also holomorphic there. According to the estimate (5.2), we have

And by Cauchy estimate we have

where C 2 comes from the estimate (5.2) which depends only on κ, σ, Λ, n, C 0 . Thus we can choose a suitable constant C 3 which is much greater than the coefficients above, which concludes the proof.

which is O 2 (F(Φ ν )). The domains B rν are defined with

Clearly lim ν→∞ r ν = 1 2 , and B r ν+1 ⊂ B rν for all ν ≥ 0. The sequence of small numbers, ε ν , is defined by ε ν+1 = C ν+1 ε 2 ν , ν ≥ 0, where C is a large constant depending on κ, σ, C 0 , n, |Λ ǫ |, |(Λ ǫ ) -1 | which will be determined later on. For ε 0 sufficiently small, the sequence ε ν tends rapidly to 0, actually for all ν ≥ 0, (refer to the explicit calculation in [START_REF] De La Llave | A tutorial on KAM theory[END_REF]) we have: (6.9)

In particularly we then have (6.10)

With all of these preparations above, we can get the induction lemma.

6.3. Induction. We shall prove that if ε 0 is sufficiently small (that is to say C is sufficiently large), then the following inductive lemma holds true for all ν ≥ 0, defined by (6.7) inductively. It follows verbatim from E.Zehnder [START_REF] Zehnder | A simple proof of a generalization of a theorem by C[END_REF]: