L’INSTITUTFOURIER DE ANNALES

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A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Javier RIBÓN

Non-embeddability of general unipotent diffeomorphisms up to formal conjugacy

Tome 59, n^o3 (2009), p. 951-975.

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NON-EMBEDDABILITY OF GENERAL UNIPOTENT DIFFEOMORPHISMS UP TO FORMAL CONJUGACY

by Javier RIBÓN (*)

Abstract. — The formal class of a germ of diffeomorphismϕis embeddable in a flow ifϕis formally conjugated to the exponential of a germ of vector field. We prove that there are complex analytic unipotent germs of diffeomorphisms at Cⁿ (n >1) whose formal class is non-embeddable. The examples are inside a family in which the non-embeddability is of geometrical type. The proof relies on the properties of some linear functional operators that we obtain through the study of polynomial families of diffeomorphisms via potential theory.

Résumé. — La classe formelle d’un difféomorphisme localϕest plongeable dans un flot si ϕ est formellement conjugué à l’exponentielle d’un germe de champs de vecteurs. On prouve qu’il existe des difféomorphismes unipotents analytiques complexes définis au voisinage de l’origine dansCⁿ(n >1) dont la classe formelle n’est pas plongeable. Les exemples appartiennent à une famille où le manque de plongeabilité est une propriété de type géométrique. La preuve est basée sur les propriétés de certains opérateurs fonctionnels linéaires qu’on obtient grâce à l’étude des familles polynomiales de difféomorphismes via la théorie du potentiel.

1. Introduction

In this paper we prove

Main Theorem. — There exists a unipotent germ of complex analytic diffeomorphism at (C²,0) whose formal class is non-embeddable.

Denote by Diff(Cⁿ,0) the set of germs of complex analytic diffeomorphisms at (Cⁿ,0) whereasDiff (d Cⁿ,0)is the formal completion of Diff(Cⁿ,0).

Keywords: Holomorphic dynamical systems, diffeomorphisms, vector fields, potential theory.

Math. classification:37F75, 32H02, 32A05, 40A05.

(*) The author thanks the referee for the helpful remarks.

The work of the author has been partly financed by Junta de Castilla y León, Project VA059A07 and CNPQ, Project PRONEX-Teoria Geométrica das Equações Diferenciais Complexas.

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A “normal form” forϕ∈Diff(Cⁿ,0) should be a diffeomorphism formally conjugated to ϕ but somehow simpler. Every ϕ ∈ Diff(Cⁿ,0) admits a unique Jordan decomposition

ϕ=ϕs◦ϕu=ϕu◦ϕs

where ϕs ∈ Diff (d Cⁿ,0) is semisimple and ϕu ∈ Diff (d Cⁿ,0) is unipotent, that isj¹ϕu−Idis nilpotent. Thenϕsis formally linearizable and has the natural normal formj¹ϕs. Note thatϕuis not formally linearizable unless ϕ_u ≡Id. Thus, a different approach is required to obtain simple models, up to formal conjugacy, for unipotent diffeomorphisms.

A unipotent ϕ∈Diff(Cⁿ,0) is the exponential of a unique formal nilpotent vector fieldX, the so called infinitesimal generator ofϕ (see [3] and [8]), which is geometrically significant even if it diverges in general. We denoteX bylogϕ. We say that theformal class ofϕis embeddableiflogϕ is formally conjugated to a germ of convergent vector fieldY. The diffeomorphisms of the formexp(Y)provide continuous modelsexp(tY)(t∈C) to compare with the discrete dynamics ofϕ.

The existence of embeddable elements in the formal class of a diffeomorphism has useful implications. For instance, for n = 1, a unipotent ϕ∈Diff(C,0)satisfiesj¹ϕ=Idandlogϕis always conjugated to a germ of analytic vector fieldY. The normalizing transformation is in general divergent. Anyway, there exist regions (Fatou petals) where it is the asymptotic development of an analytic mapping conjugatingexp(Y)andϕ. This phe- nomenon provides the basis (see Il’yashenko’s paper in [4]) to construct the Ecalle-Voronin complete system of analytic invariants (Ecalle [2], Voronin [18], Martinet-Ramis [8], Malgrange [7]). The same strategy can be applied in some cases in higher dimension. For instance Voronin [17] classifies an- alytically the unipotent diffeomorphisms in Diff(C²,0) which are formally conjugated to a diffeomorphism of the form(x, y+x^k) = exp(x^k∂/∂y)for somek∈N.

In the previous cases the analytic classification relies on the study of the normalizing transformation to a simpler model. Normalizing transformations are quite well understood whereas it is more difficult to describe the possible final models up to formal conjugacy and their properties. A good example is provided by Birkhoff normal forms attached to analytic Hamiltonian vector fields (see [12], see [6] for properties of embeddability of symplectic diffeomorphisms in the flow of an analytic Hamiltonian vector field). On the one hand the properties of the normalizing transformations are well-documented. On the other hand the Birkhoff normal forms are either always convergent or generically divergent [12]. Nevertheless Pérez

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Figure 1.1. Real picture of the transport mapping

Marco stresses that whether or not there exists a divergent Birkhoff normal form is still an open problem. Our Main Theorem claims that examples of unipotent diffeomorphisms whose formal class only contains diffeomorphisms with divergent infinitesimal generator can be provided. They can be chosen of the form

ϕ_∆,w(x, y) = (x+y(y−x)∆(x, y), y+y(y−x)w(x, y)).

Moreover, for this kind of diffeomorphisms the obstruction to the embeddability of the formal class is of geometrical type.

The proof of the Main Theorem is based on the study of the transport mapping. Supposelogϕ∆,w is a germ of convergent vector field. Then

L_∆,w^def= 1

y(y−x)logϕ_∆,w

is regular, i.e.L∆,w(0)6= 0, and transversal to both y= 0 andy =x. We can define a correspondence T r_∆,w associating to each point P in y = 0 the unique point in y = x contained in the trajectory of L∆,w passing throughP. This correspondence is the transport mapping. Even iflogϕ∆,w

is divergent we manage to defineT r_∆,w; it is a formal invariant. We prove the following characterization for the non-embeddability of the formal class:

Proposition 1.1. — If the formal class of ϕ∆,w is embeddable then T r_∆,w is an analytic mapping.

Fix a unitw∈C{x, y},w(0)6= 0, such thatlnϕ_0,wis not convergent. We claim that there exists∆ in C{x, y}, ∆(0) = 0, such thatT r∆,w diverges.

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We argue by contradiction. Note that T r_∆,w can be analytic (for exam- pleT r∆,w(x,0)≡(x, x)) whereaslogϕ∆,w is divergent. The divergence of T r_∆,w is even stronger than the non-existence of a germ of convergent vector field collinear tologϕ∆,w. It is obtained through a fine analysis of the nature of the family(ϕ∆,w)_∆.

We consider polynomial families on λ∈Cof the form

ϕλ∆,w(x, y) = (x+λy(y−x)∆(x, y), y+y(y−x)w(x, y)).

The embeddability of the formal class ofϕλ∆,w for any λ ∈ C allows to find a linear equation

ˆ

−ˆ◦ϕ0,w=y(y−x)∆ (homological equation)

such thatˆ∆(x, x)−ˆ∆(x,0)∈C{x}for every solutionˆ∆∈C[[x, y]]. The proof of the convergence ofˆ_∆(x, x)−ˆ_∆(x,0)is based on potential theory techniques. The homological equation has a formal solution ˆ∆ for any

∆∈C[[x, y]], moreoverˆ_∆(x, x)−ˆ_∆(x,0) does not depend on the choice ofˆ∆. Then the operatorSw:C[[x, y]]→C[[x]]given by

Sw(∆) = ˆ∆(x, x)−ˆ∆(x,0)

is linear, well-defined andSw(mx,y)⊂ C{x} where mx,y ⊂C{x, y} is the maximal ideal. Now it suffices to study a linear operator attached to the dynamically simple diffeomorphismϕ0,w, in particularx◦ϕ0,w =xwill be a key property to prove thatS_w(m_x,y)contains divergent elements.

The operator Sw was defined in terms of a difference equation. We can replace the difference equation with a differential equation easier to handle.

More preciselySw(∆) = ˆΓ(x, x)−Γ(x,ˆ 0)for every solutionΓˆ ∈C[[x, y]]of (logϕ0,w)(ˆΓ) =−y(y−x)∆.

Since logϕ_0,w diverges and x◦ϕ_0,w = xthen logϕ_0,w = ˆwy(y−x)∂/∂y for some divergentwˆ ∈C[[x, y]]. The collinearity of logϕ0,w and ∂/∂y, in addition to some functional analysis techniques, can be used to prove that the propertyS_w(m_x,y)⊂C{x}implies thatwˆ ∈C{x, y}. Here we have our contradiction.

We do not describe the nature of the transport mapping, besides the fact that it is generically divergent. It would be interesting to know what divergent mappings can be obtained as transport mappings of diffeomorphisms of typeϕ∆,w.

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2. Basic facts

In this section we introduce some well-known facts about diffeomorphisms and vector fields for the sake of completeness.

We denote byX(Cⁿ,0) the set of germs of complex analytic vector fields at0∈Cⁿ. Aformal vector fieldXˆ is a derivation of the ringC[[x₁, . . . , x_n]].

We also expressXˆ in the more conventional form Xˆ =

n

X

j=1

X(xˆ _j) ∂

∂x_j.

GivenX∈ X(Cⁿ,0) the derivation associated toXrestricts to a derivation of the ringC{x1, . . . , xn}. Indeed X(g)is the Lie derivative LXg for any g∈C{x₁, . . . , x_n}.

We say that a formal vector field Xˆ =Pn

j=1ˆaj(x1, . . . , xn)∂/∂xj where ˆ

aj ∈ C[[x1, . . . , xn]] for any 1 6j 6n is nilpotent ifXˆ(0) = 0and j¹Xˆ is nilpotent. We denote by XˆN(Cⁿ,0) and XN(Cⁿ,0) the sets of formal nilpotent vector fields and germs of nilpotent vector fields respectively.

Let mˆ be the maximal ideal of C[[x1, . . . , xn]]. We say that a formal diffeomorphismis an automorphismσˆ:C[[x1, . . . , xn]]→C[[x1, . . . , xn]]of C-algebras such thatσ( ˆˆ m) = ˆm. Equivalently we can expressσˆ in the form

ˆ

σ= (ˆσ(x₁), . . . ,ˆσ(x_n))∈C[[x₁, . . . , x_n]]ⁿ

wherej¹σˆis a linear isomorphism. We denote byDiff (d Cⁿ,0)and Diff(Cⁿ,0) the set of formal diffeomorphisms and germs of diffeomorphisms respectively. Ifj¹σˆ is unipotent (i.e. if1 is the only eigenvalue of j¹σ) then weˆ say thatˆσis unipotent. We denote byDiffd_u(Cⁿ,0)the set of formal unipotent diffeomorphisms.

LetX∈ X(Cⁿ,0); suppose thatXis singular at0. We denote byexp(tX) (fort ∈C) the flow of the vector fieldX, it is the unique solution of the differential equation

∂

∂texp(tX) =X(exp(tX))

with initial conditionexp(0X) =Id. We define the exponential exp(X)of X asexp(1X). We define the exponential operator

exp( ˆX) : C[[x₁, . . . , x_n]] → C[[x₁, . . . , x_n]]

g → P∞

j=0 Xˆ^j

j! (g).

for any elementXˆ ofXˆ_N(Cⁿ,0). The definitions of exponential coincide if Xˆ ∈ XN(Cⁿ,0), i.e. the image ofgby the operatorexp( ˆX)isg◦exp( ˆX)for

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anyg∈C{x1, . . . , x_n}. Givent∈Cthe formal flowexp(tXˆ)is an operator such that

(2.1) exp(tXˆ)(g) =

∞

X

j=0

(tX)ˆ ^j j! (g) =

∞

X

j=0

t^j Xˆ^j

j! (g) for anyg∈C[[x₁, . . . , x_n]]. The property

(2.2) exp((t+s) ˆX) = exp(tX)ˆ ◦exp(sXˆ)∀s, t∈C

follows from equation (2.1) and the formal identitye^t+s = e^te^s since tXˆ andsXˆ commute.

The nilpotent character of Xˆ implies that the power series exp( ˆX)(g) converges in the Krull topology for anyg∈C[[x₁, . . . , x_n]]. Moreover, since Xˆ is a derivation thenexp( ˆX)acts as a diffeomorphism, i.e.

exp( ˆX)(g1g2) = exp( ˆX)(g1)exp( ˆX)(g2) ∀g1, g2∈C[[x1, . . . , xn]].

Moreoverj¹exp( ˆX) = exp(j¹X), thusˆ j¹exp( ˆX) is a unipotent linear isomorphism. The following proposition is classical.

Proposition 2.1. — (see [3], [8]) The mapping Xˆ 7→ exp(1 ˆX) maps bĳectivelyXˆ_N(Cⁿ,0) ontoDiffd_u(Cⁿ,0). Moreover, ifXˆ ∈Xˆ_N(Cⁿ,0) then exp(tX)(g)ˆ belongs toC[t][[x1, . . . , xn]]for anyg∈C[[x1, . . . , xn]].

Consider the inverse mapping log : Diffdu(Cⁿ,0) → XˆN(Cⁿ,0). An ele- mentϕofDiffd_u(Cⁿ,0)defines an isomorphism

ϕ:C[[x1, . . . , xn]]→C[[x1, . . . , xn]]

of C-algebras such that ϕ(g) = g◦ϕ for any g ∈ C[[x1, . . . , xn]]. Denote byΘ : C[[x₁, . . . , x_n]] →C[[x₁, . . . , x_n]] the operator ϕ−Id, i.e. we have Θ(g) =ϕ(g)−Id(g) =g◦ϕ−gfor any g∈C[[x1, . . . , xn]]. The operator ϕ−Idis not associated to a diffeomorphism. We have

(2.3) (logϕ)(g) = (log(Id+ Θ))(g) =

∞

X

j=1

(−1)^j+1Θ^j(g) j

forg ∈ C[[x1, . . . , xn]]. The series in the right hand side converges in the Krull topology sinceϕis unipotent. Moreoverj¹(logϕ) = log(j¹ϕ)is nilpotent andlogϕsatisfies the Leibnitz rule. We say thatlogϕis theinfinites- imal generatorof ϕ. Even ifϕ∈Diffdu(Cⁿ,0)∩Diff(Cⁿ,0) in generalϕ is divergent.

Consider an ideal Iˆ ⊂ C[[x1, . . . , xn]]. We denote by Z( ˆI) the set of formal curvesˆγ∈(tC[[t]])ⁿ such that ˆh◦ˆγ= 0for any ˆh∈I. Conversely,ˆ for ∆ˆ ⊂ (tC[[t]])ⁿ we define I( ˆ∆) as the set of series ˆh ∈ C[[x1, . . . , xn]]

such thatˆh◦γˆ= 0for anyγˆ∈∆. We haveˆ

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Proposition 2.2(Formal theorem of zeros [16], pages 49-50). — LetIˆ be an ideal ofC[[x1, . . . , xn]]. Then

I(Z( ˆI)) =p I.ˆ

LetYˆ = ˆa(x, y)∂/∂x+ ˆb(x, y)∂/∂y be a formal vector field. We consider the set

F( ˆY) ={ˆg∈C[[x, y]] : ˆY(ˆg) = 0}

of first integrals ofYˆ. We say that fˆ∈ F( ˆY)is primitive if ^k

qfˆdoes not belong toC[[x, y]] fork >1. IfF( ˆY)6=Cthere exists a primitive formal first integralfˆ; moreover we have

F( ˆY) =C[[z]]◦fˆ

(Mattei-Moussu [9]), and the primitive first integral can be chosen inC{x, y}

ifYˆ is a germ of holomorphic vector field.

We can give an alternative characterization for the first integrals of the infinitesimal generator of a unipotent diffeomorphism.

Lemma 2.3. — Letσ∈Diffdu(Cⁿ,0)andfˆ∈C[[x1, . . . , xn]]. Then (logσ)( ˆf) = 0⇔fˆ◦σ= ˆf .

Proof. — We have

fˆ◦exp(logσ) = ˆf+ (logσ)( ˆf) +(logσ)²( ˆf) 2! +. . . Therefore(logσ)( ˆf) = 0impliesfˆ◦σ= ˆf.

Supposefˆ◦σ= ˆf. Denote byΘthe operatorσ−Id. We haveΘ( ˆf) = 0 by hypothesis and thenΘ^j( ˆf) = 0for anyj∈N. We obtain(logσ)( ˆf) = 0

by equation (2.3).

3. Formal conjugacy

Throughout this paper we work with germs of diffeomorphisms in (C²,0) of the form

ϕ∆,w(x, y) = (x+y(y−x)∆(x, y), y+y(y−x)w(x, y))

wherew,∆∈C{x, y}andw(0,0)6= 0 = ∆(0,0). In this section we describe a geometrical condition for the formal class ofϕ∆,w not to be embeddable.

Next, we describe the structure of logϕ_∆,w.

Definition 3.1. — Let m_x,y and mˆ_x,y be the maximal ideals of the ringsC{x, y}andC[[x, y]]respectively.

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Lemma 3.2. — The formal vector fieldlogϕ_∆,w is of the form logϕ∆,w =y(y−x)

w(0,0) ∂

∂y +h.o.t.

where h.o.t stands for a formal vector field whose coefficients belong to mˆx,y.

Proof. — We denote by Θ the operator ϕ_∆,w−Id. Let I be the ideal (y(y−x))ofC[[x, y]]. We denoteX = logϕ∆,w. It is clear thatΘ(C[[x, y]])is contained inI. ThusX(C[[x, y]])is contained inIby equation (2.3). Since we haveX =X(x)∂/∂x+X(y)∂/∂yandI⊂mˆ²_x,ywe obtainX(I)⊂Imˆx,y. Giveng∈Iwe have

Θ(g) =X



g+X





∞

X

j=2

X^j−2(g) j!







∈X(I+X(C[[x, y]])) =X(I).

Therefore we getΘ(I)⊂X(I)⊂Imˆ_x,y. The equation (2.3) implies X(x) =y(y−x)∆(x, y) + Θ



Θ





∞

X

j=2

(−1)^j+1Θ^j−2(x) j









and then

X(x)∈(y(y−x)∆(0,0) +Imˆx,y) + Θ(I) =y(y−x)∆(0,0) +Imˆx,y. Analogously we obtainX(y)−y(y−x)w(0,0) ∈ Imˆx,y. The lemma is a consequence of the equationX=X(x)∂/∂x+X(y)∂/∂y.

We denote the formal vector fieldlogϕ∆,w/(y(y−x))byL∆,w.

Lemma 3.3. — For anyϕ∆,w and ˆg∈C[[x]] there exists a uniquefˆin C[[x, y]]such that(logϕ_∆,w)( ˆf) = 0andfˆ(x,0) = ˆg(x).

Proof. — By lemma 3.2 we have thatL∆,w(y)is a unit. Then (logϕ∆,w)( ˆf) = 0⇔ ∂fˆ

∂y =−L∆,w(x) L∆,w(y)

∂fˆ

∂x.

As a consequence there is a unique formal solution of the previous equation fulfilling the initial conditionf(x,ˆ 0) = ˆg(x).

We want to introduce the formal invariants of ϕ_∆,w. The first formal invariant is the fixed points setF ix(ϕ∆,w).

Proposition 3.4. — Let τ₁, τ₂ ∈Diff(Cⁿ,0) and σˆ ∈Diff (d Cⁿ,0) such thatσˆ◦τ1=τ2◦ˆσ. Then we haveσ(F ix(τˆ 1)) =F ix(τ2).

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An equivalent statement is the following: LetIˆ_j=I(F ix(τ_j))forj= 1,2.

Then we haveIˆ₂◦σˆ = ˆI₁.

Proof. — Letˆγ∈C[[t]]ⁿ∩Z( ˆI1). We have τ1◦γ(t) = ˆˆ γ(t); we obtain ˆ

σ◦τ1(ˆγ(t)) =τ2◦σ(ˆˆ γ(t))⇒τ2◦σ(ˆˆ γ(t)) = ˆσ(ˆγ(t)).

Henceσˆ◦γ(t)ˆ belongs to Z( Î2)and thenσ(Zˆ ( Î1))⊂Z( Î2). By the analogous argument applied to σˆ⁽⁻¹⁾ we obtain Z( Î2) ⊂ σ(Z( ˆˆ I1)) and then ˆ

σ(Z( Î₁)) =Z( Î₂). This is equivalent to I(Z( Î₂))◦σˆ =I(Z( Î₁)). SinceIˆ₁ and Iˆ2 are radical ideals thenIˆ2◦σˆ = Î1 is a consequence of the formal

theorem of zeros.

Remark 3.5. — It is not required to use the formal theorem of zeros to prove the previous proposition but this proof makes clear that the image byσˆ of a parametrizationγ(t)ˆ of a formal curve contained inF ix(τ1)is a parametrizationσˆ◦ˆγ(t)of a formal curve contained in F ix(τ2).

Definition 3.6. — Leta∈C[[x]]. We define ν(a) = sup{b∈N∪ {0}:a∈(x^b)}.

Lemma 3.7. — Letϕ∆,w,τ ∈Diff(C²,0)andσˆ ∈Diff (d C²,0) such that ˆ

σ◦ϕ∆,w =τ◦σ. Thenˆ

(1) F ix(τ)is an analytic set.

(2) F ix(τ)has two irreducible componentsf1= 0 andf2= 0, both of them are smooth curves.

(3) ˆσ_∗(logϕ_∆,w) = logτ.

(4) j⁰(logτ /(f1f2))is transversal to bothf1= 0andf2= 0.

(5) Letfˆbe a primitive element of F(logϕ_∆,w). Then fˆ◦ˆσ⁽⁻¹⁾ is a primitive element ofF(logτ).

(6) fˆ◦σˆ⁽⁻¹⁾_|f

1=0 andfˆ◦σˆ⁽⁻¹⁾_|f

2=0 are “injective". In other words, ifˆγ(t)is a minimal parametrization off_j= 0we have ν( ˆf◦σˆ⁽⁻¹⁾◦ˆγ) = 1.

Proof. — Condition (1) is obvious. Conditions (3) and (5) can be de- duced of the uniqueness of the infinitesimal generator.

Denote σ(x, y) = (ˆˆ σ1(x, y),σˆ2(x, y))and ∂ˆσ/∂z= (∂σˆ1/∂z, ∂ˆσ2/∂z)for z = x or z = y. Since by prop. 3.4 we have σ(F ix(ϕˆ ∆,w)) = F ix(τ) thenτ has two irreducible components f₁ = 0 and f₂ = 0 corresponding respectively toσ(yˆ = 0)andσ(yˆ =x). Moreover

ˆ

σ(t,0) = (ˆσ1(t,0),ˆσ2(t,0)) and ˆσ(t, t) = (ˆσ1(t, t),ˆσ2(t, t))

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are formal parametrizations off₁= 0andf₂= 0respectively. Sincej¹σˆ is a linear isomorphism then the vectors

∂(ˆσ(t,0))

∂t (0) =∂σˆ

∂x(0,0) and ∂(ˆσ(t, t))

∂t (0) = ∂σˆ

∂x(0,0) +∂σˆ

∂y(0,0) are linear independent and then different than(0,0). An analytic minimal parametrization off1= 0is of the form ˆσ(â(t),0)for some â∈C[[t]]such thatν(â) = 1. We obtain

∂(ˆσ(ˆa(t),0))

∂t (0) = ∂σˆ

∂x(0,0)∂ˆa

∂t(0)6= (0,0).

Thusf1 = 0is a smooth curve which is tangent at the origin to the line generated by(∂ˆσ/∂x)(0,0). Analogously the curve f2 = 0is smooth and tangent at the origin to the line generated by(∂σ/∂x)(0,ˆ 0)+(∂ˆσ/∂y)(0,0).

Condition (3) and lemma 3.2 imply logτ = [y(y−x)]◦σˆ⁻¹

w(0,0)

∂σˆ1

∂y (0,0) ∂

∂x +∂σˆ2

∂y (0,0) ∂

∂y

+h.o.t.

where h.o.t stands for a formal vector field whose coefficients belong to mˆx,y. The power series[y(y−x)]◦σˆ⁻¹ is of the form f1f2uˆ where uˆ is a unit ofC[[x, y]]. We have

j⁰ logτ

f1f2

= ˆu(0,0)w(0,0) ∂ˆσ1

∂y (0,0) ∂

∂x+∂ˆσ2

∂y (0,0) ∂

∂y

. We obtain condition (4) since the vectors

∂σˆ

∂y(0,0), ∂σˆ

∂x(0,0) and ∂σˆ

∂x(0,0) +∂σˆ

∂y(0,0) are pairwise non-collinear.

Condition (6) is equivalent to prove that ν( ˆf(x,0)) = ν( ˆf(x, x)) = 1 for every primitive fˆin F(logϕ∆,w). We can suppose that fˆ(x,0) = x since thenfˆis primitive and the set of primitive elements ofF(logϕ∆,w) is Diff (d C,0)◦ f. The relationˆ L∆,w( ˆf) = 0 implies j¹fˆ = x. Therefore

ν( ˆf(x,0)) =ν( ˆf(x, x)) = 1.

Proposition 3.8. — The couples

(y= 0,(J∆,w)_|y=0) and (y=x,(J∆,w)_|y=x) are formal invariants ofϕ_∆,w.

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Proof. — Supposeσ◦ϕˆ ∆,w =τ◦ˆσforτ∈Diff (d C²,0)andσˆ∈Diff (d C²,0).

We have

(Jσˆ◦ϕ_∆,w)J∆,w= (Jτ◦σ)Jˆ ˆσ

by the chain rule. Letγ(t)∈(tC{t})² be a parametrization of eithery= 0 ory=x. We have ϕ∆,w◦γ(t) =γ(t); that implies

J∆,w◦γ(t) =Jτ◦(ˆσ◦γ(t))

as we wanted to prove.

Definition 3.9. — We defineT r∆,w : (y= 0)→(y=x)as the unique formal mapping such thatfˆ∆,w◦T r∆,w= ˆf∆,w where fˆ∆,w is a primitive formal first integral of logϕ_∆,w. By condition (6) in lemma 3.7 we have thatfˆ_∆,w(x,0) andfˆ_∆,w(x, x)belong toDiff (d C,0). As a consequence

T r_∆,w(x,0) = (( ˆf_∆,w(x, x))⁽⁻¹⁾◦fˆ_∆,w(x,0),( ˆf_∆,w(x, x))⁽⁻¹⁾◦fˆ_∆,w(x,0)) is the expression of T r∆,w in coordinates. The mapping T r∆,w does not depend on the choice of fˆ∆,w. We call T r∆,w the transport mapping. If logϕ∆,w is a germ of vector field then T r∆,w(x,0) is the only point in y=xcontained in the same trajectory ofL_∆,w than(x,0).

Proposition 3.10. — The transport mapping T r∆,w associated to a diffeomorphismϕ_∆,w is a formal invariant.

Suppose ˆσ◦ϕ∆,w = τ ◦σˆ for τ ∈ Diff(C²,0) and σˆ ∈ Diff (d C²,0). In general the mappingτis not necessarily of the formϕ∆⁰,w⁰; it is necessary to explain what we mean by the transport mapping ofτ. By proposition 3.4 the formal curvesγ1= ˆσ(y= 0)andγ2= ˆσ(y=x)are in fact analytic. We defineT r_τ :γ₁→γ₂ as the unique formal mapping such thatgˆ◦T r_τ = ˆg for every primitiveˆg inF(logτ). ThenT rτ is well-defined by lemma 3.7.

Proof. — We keep the notations in the previous paragraph. We want to prove the equality(T rτ◦σ)(x,ˆ 0) = (ˆσ◦T r∆,w)(x,0). We choose a primitive fˆ∆,w ∈ F(logϕ∆,w); the series gˆ= ˆf∆,w◦σˆ⁽⁻¹⁾ is a primitive element of F(logτ)(lemma 3.7). Since fˆ_∆,w(x,0) = ˆf_∆,w(T r_∆,w(x,0)) by definition ofT r∆,w then we obtain

( ˆf_∆,w◦σˆ⁽⁻¹⁾)(ˆσ(x,0)) = ( ˆf_∆,w◦ˆσ⁽⁻¹⁾)(ˆσ(T r_∆,w(x,0))).

The definition ofT rτ impliesT rτ(ˆσ(x,0)) = ˆσ(T r∆,w(x,0)) as we wanted

to prove.

Definition 3.11. — The formal class of a unipotent diffeomorphism τ is embeddable iflogτ is formally conjugated to a convergent germ of vector field.

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Next we introduce an obstruction to the embeddability of a formal class.

Proposition 3.12. — Suppose that there existX ∈ XN(C²,0) and ˆσ inDiff (d C²,0)such thatˆσ◦ϕ∆,w= exp(X)◦σ. Thenˆ T r∆,w is a convergent mapping.

Proof. — The transport mapping T r∆,w maps y = 0 to y =x, thus it satisfiesT r∆,w(x,0)≡(â(x),â(x))for someaˆ∈C[[x]]. It suffices to prove thatâbelongs toC{x}. The transport mapping is a formal invariant (prop.

3.10), hence we have σˆ ◦T r∆,w(x,0) = T r_exp(X)◦σ(x,ˆ 0). The previous equation implies

ˆ

a(x) = (ˆσ(x, x))⁻¹◦T rexp(X)◦σ(x,ˆ 0).

SinceT r_exp(X)is convergent then it suffices to prove thatσ(x,ˆ 0)andσ(x, x)ˆ belong toC{x}².

We have J∆,w = 1 + (2y −x)w(0,0) +h.o.t. Therefore the function (J_∆,w)_|y=0 is injective. Consider a convergent minimal parametrization η(x)of ˆσ(y= 0); there existshˆ ∈Diff (d C,0) such thatσ(x,ˆ 0) =η◦ˆh(x).

SinceJ_exp(X)◦σ(x,ˆ 0) =J∆,w(x,0)then

∂

∂x J_exp(X)◦η(x)

(0) =− w(0,0)

∂ˆh/∂x(0) 6= 0.

As a consequence

ˆh= (J_exp(X)◦η(x)−1)⁽⁻¹⁾◦(J∆,w(x,0)−1)

belongs to Diff(C,0). That impliesσ(x,ˆ 0) =η◦ˆh∈C{x}². The proof for ˆ

σ(x, x)is analogous.

Remark 3.13. — In order to find a unipotent diffeomorphism whose formal class is non-embeddable it suffices to exhibitϕ∆,w such thatT r∆,w

is divergent.

Remark 3.14. — We do not prove it in this paper but a diffeomorphism ϕ∆,wsuch thatT r∆,w is an analytic mapping has embeddable formal class.

In particular a diffeomorphismϕ0,w= (x, y+y(y−x)w(x, y))has embeddable formal class.

4. Polynomial families Consider the family

ϕλ∆,w= (x+λy(y−x)∆(x, y), y+y(y−x)w(x, y))

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where w(0,0) 6= 0 = ∆(0,0) and λ ∈ C. It is affine on λ. We denote by fˆλthe only element ofF(logϕλ∆,w)such thatfˆλ(x,0) =x. The transport mappingT r_λ∆,w satisfies

T r_λ∆,w(x,0) = (( ˆf_λ(x, x))⁽⁻¹⁾◦fˆ_λ(x,0),( ˆf_λ(x, x))⁽⁻¹⁾◦fˆ_λ(x,0)) and thenT r_λ∆,w(x,0) = (( ˆf_λ(x, x))⁽⁻¹⁾,( ˆf_λ(x, x))⁽⁻¹⁾). As a consequence T r_λ∆,w is convergent if and only iffˆ_λ(x, x)∈C{x}.

Lemma 4.1. — We have logϕλ∆,w

y(y−x) =



 X

06k,l

a¹_k,l(λ)x^ky^l





∂

∂x +



 X

06k,l

a²_k,l(λ)x^ky^l





∂

∂y

wherea^j_k,l∈C[λ]for allk, l>0andj∈ {1,2}.

Proof. — DenoteX_λ= logϕ_λ∆,w. Denote byΘ_λthe operatorϕ_λ∆,w−Id forλ∈C. Givenj>0 consider an elementg of(y(y−x)) ˆm^j_x,y. The series gbelongs tomˆ^j+2_x,y; we obtain

Xλ(g) =Xλ(x)∂g

∂x +Xλ(y)∂g

∂y ∈(y(y−x)) ˆm^j+2−1_x,y = (y(y−x)) ˆm^j+1_x,y by lemma 3.2 for anyλ∈C. We iterate the argument to get

(4.1) X_λ^k((y(y−x)) ˆm^j_x,y)⊂(y(y−x)) ˆm^j+k_x,y for allj, k >0 and λ ∈ C. Moreover, since Θ_λ(g) =P∞

k=1X_λ^k(g)/j! the equation (4.1) impliesΘλ((y(y−x)) ˆm^j_x,y)⊂(y(y−x)) ˆm^j+1_x,y for anyλ∈C. We obtain

(4.2) Θ^k_λ((y(y−x)) ˆm^j_x,y)⊂(y(y−x)) ˆm^j+k_x,y

for allj>0,k>1andλ∈C. The seriesΘ_λ(g)belongs to(y(y−x)), thus the equation (4.2) impliesΘ^j_λ(g) = Θ^j−1_λ (Θλ(g))∈(y(y−x)) ˆm^j−1_x,y for all g∈C[[x, y]],j>1andλ∈C. SinceXλ(g) =P∞

j=1(−1)^j+1Θ^j_λ(g)/jthen a¹_k,l(λ) = 1

k!l!

k+l+1

X

j=1

(−1)^j+1 j

∂^k+l[Θ^j_λ(x)/(y(y−x))]

∂x^k∂y^l (0,0).

An analogous expression is obtained fora²_k,l for all k, l >0. It suffices to prove thatΘλ(C[λ][[x, y]])⊂C[λ][[x, y]]; then a¹_k,l anda²_k,l are finite sums of polynomials for allk, l>0.

Given h=P

k,l>0hk,l(λ)x^ky^l∈C[λ][[x, y]]we have Θλ(h) = X

k,l>0

hk,l(λ)Θλ(x^ky^l)

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and then

(4.3) Θλ(h) = X

k,l>0

hk,l(λ)[(x+y(y−x)λ∆)^k(y+y(y−x)w)^l−x^ky^l].

Since Θλ(x^ky^l) ∈ mˆ^k+l_x,y for all k, l > 0 and λ ∈ C then the series (4.3) converges in the Krull topology to an element ofC[λ][[x, y]].

Proposition 4.2. — Letfˆλ be the unique formal first integral of log ϕ_λ∆,w such that fˆ_λ(x,0) =x. Then fˆ_λ can be expressed in the form

fˆλ=x+y X

j+k>1

fj,k(λ)x^jy^k

wheref_j,k∈C[λ]anddegf_j,k6j+k for anyj+k>1.

Proof. — Letm˜ be the ideal(x, y)of the ringC[λ][[x, y]]. The property (logϕ_λ∆,w)( ˆf_λ) = 0is equivalent toL_λ∆,w( ˆf_λ) = 0. We denote

Lλ∆,w=a(λ, x, y)∂/∂x+b(λ, x, y)∂/∂y.

Thenaandbbelong toC[λ][[x, y]]by lemma 4.1. Moreover, we havea∈m˜ andb(λ,0,0)≡w(0,0)by lemma 3.2. The seriesbis of the formw(0,0)(1−

b₀)where b₀∈m. Hence we obtain˜ (4.4) a∂fˆλ

∂x +b∂fˆλ

∂y = 0⇒ ∂fˆλ

∂y =−aw(0,0)⁻¹(

∞

X

j=0

b^j₀)∂fˆλ

∂x. Since b^j₀ ∈ m˜^j for any j >0 then P∞

j=0b^j₀ converges in the Krull topology to an element ofC[λ][[x, y]]. Denotec=−aw(0,0)⁻¹(P∞

j=0b^j₀). Then c belongs to C[λ][[x, y]]∩m. The series˜ c can be expressed in the form P∞

j=0cj(λ, x)y^j wherecj ∈C[λ][[x]] for anyj>0. Moreoverc0 belongs to the ideal(x)ofC[λ][[x]]. Denotefˆλ=P∞

j=0fj(λ, x)y^j. We havef0≡xby choice. We obtain

(4.5)

∞

X

j=1

jfj(λ, x)y^j−1= (

∞

X

j=0

cj(λ, x)y^j)





∞

X

j=0

fj(λ, x)

∂x y^j





by developing equation (4.4). By comparing the coefficients ofy⁰ in both sides of equation (4.5) we obtainf1 =c0(∂f0/∂x) =c0 ∈(x)⊂C[λ][[x]].

Suppose thatf0, . . . , fk belong toC[λ][[x]]for some k>0. Since (k+ 1)f_k+1=

k

X

j=0

c_j(∂f_k−j/∂x)

thenfk+1 belongs toC[λ][[x]]. We deduce thatfj belongs toC[λ][[x]] for anyj >0 by induction. Therefore fˆλ belongs to C[λ][[x, y]] and it can be

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expressed in the form x+yP

j+k>0f_j,k(λ)x^jy^k where f_j,k ∈ C[λ] for all j, k>0. Moreoverf0,0is identically zero since f1(λ,0)≡0.

Consider

τ_∆,w,λ(x, y) =x λ,y

λ

◦ϕ_∆/λ,w◦(λx, λy).

We have

τ_∆,w,λ(x, y) = (x+y(y−x)∆(λx, λy), y+λy(y−x)w(λx, λy)).

We can proceed as in lemma 3.2 to prove logτ∆,w,λ(x, y) =λy(y−x)

w(0,0) ∂

∂y +h.o.t.

.

There is an analogue of lemma 4.1 forlogτ_∆,w,λ/(λy(y−x)). Again such an expression can be used to prove that the unique first integral

ˆ

g_λ=x+y X

j+k>1

g_j,k(λ)x^jy^k

oflogτ∆,w,λ such that ˆgλ(x,0) =xsatisfiesgj,k ∈C[λ] for anyj+k>1.

The relation betweenϕ_λ∆,w andτ_∆,w,λ implies fˆ_1/λ(λx, λy) =λˆgλ(x, y).

We obtainλfj,k(1/λ)λ^j+k=λgj,k(λ)for anyj+k>1. Sincefj,k andgj,k

are polynomials we deduce thatf_j,kis a polynomial of degree at mostj+k

for anyj+k>1.

We have fˆ_λ(x, x) =x+P

j+k>1f_j,k(λ)x^j+k+1. Next result is crucial.

Proposition 4.3 ([11, 10]). — LetPˆ=P

j>0P_j(λ)x^jwhereP_j ∈C[λ]

anddegP_j 6Aj+Bfor someA, B∈Rand anyj∈N. Then eitherP(λ, x)ˆ is convergent in a neighborhood ofx= 0 or P(λ)ˆ ∈C[[x]]\C{x} for any λ∈Coutside a polar set.

The series Pˆ(λ, x) is convergent in a neighborhood of x = 0 if there exists a neighborhoodV ofx= 0in C² and a functionP holomorphic in V such that Pˆ(λ, x) is the power series development of P. This property impliesPˆ(λ, x)∈ C{x} for any λ ∈C. The reciprocal is false if we drop the condition on the linear growth ofdegPj.

A polar set (see [13]) has measure zero as well as Haussdorff dimension zero. Moreover, it is totally disconnected.

Corollary 4.4. — Fix ∆ ∈ mx,y and w ∈ C{x, y} with w(0,0) 6= 0.

Either (λ, x) 7→ T rλ∆,w(x) is convergent in a neighborhood of x = 0 or x7→T r_λ∆,w(x)is divergent for anyλ∈Coutside a polar set.

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Proposition 4.5. — Let w ∈ C{x, y} \m_x,y and ∆ ∈ m_x,y (see def.

3.1). Suppose that the formal class ofϕλ∆,w is embeddable for anyλ∈C. Then the equation

ˆ

−ˆ◦ϕ0,w =y(y−x)∆(x, y) (homological equation) has a solutionˆ∆∈C[[x, y]] such thatˆ∆(x, x)−ˆ∆(x,0)∈C{x}.

Proof. — Letfˆλ be the first integral oflogϕλ∆,w such thatfˆλ(x,0) =x.

It satisfiesfˆλ◦ϕλ∆,w= ˆfλ for anyλ∈Cby lemma 2.3. We obtain

∂( ˆfλ◦ϕλ∆,w)

∂λ |λ=0=∂fˆλ

∂λ |λ=0. Sincefˆ0=xthen

y(y−x)∆(x, y) + ∂fˆ_λ

∂λ|λ=0

!

◦ϕ_0,w=∂fˆ_λ

∂λ|λ=0

.

We defineˆ∆(x, y) = (∂fˆλ/∂λ)(0, x, y). We haveˆ∆(x,0) = 0by definition offˆ_λ. The corollary 4.4 and proposition 3.12 imply that(λ, x)7→fˆ_λ(x, x)is convergent in a neighborhood of(λ, x) = (0,0). Thereforeˆ∆(x, x)∈C{x}

and thenˆ∆(x, x)−ˆ∆(x,0)∈C{x}.

Note that in this section we related the existence of a diffeomorphism ϕ_∆,w with ∆ 6= 0and no embeddable formal class with the properties of ϕ0,w (which by the way has embeddable formal class).

The rest of the paper is devoted to prove the existence of a homological equation such thatˆ(x, x)−ˆ(x,0)diverges for any formal solutionˆ(x, y).

5. The homological equation

The main result in this section is proving that the homological equation can be replaced by a differential equation.

Lemma 5.1. — logϕ_0,w is of the formy(y−x)(w(0,0) +h.o.t.)∂/∂y.

Proof. — By lemma 3.2logϕ0,w is of the form y(y−x)(a(x, y)∂/∂x+b(x, y)∂/∂y)

wherea, b∈C[[x, y]],a(0,0) = 0and b(0,0) =w(0,0). It suffices to prove that a≡0 or equivalently (logϕ0,w)(x) = 0. Since x◦ϕ0,w =x then we

have(logϕ_0,w)(x) = 0by lemma 2.3.

Lemma 5.2. — The equationˆ−ˆ◦ϕ_0,w =y(y−x)∆(x, y)has a solution ˆ

= ˆ∆∈C[[x, y]]for any∆∈C[[x, y]].

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Proof. — We defineν(A) = sup{j˜ ∈N∪{0}:A∈mˆ^j_x,y}forA∈C[[x, y]].

We define∆0= ∆. Consider the equation

(5.1) (logϕ_0,w)(˜₀) =−y(y−x)∆₀.

SinceL0,wis non-singular at(0,0)there exists a solution˜0∈C[[x, y]]such thatν(˜˜ 0)>˜ν(∆0) + 1. We consider

(5.2) (˜₀+₁)−(˜₀+₁)◦ϕ_0,w=y(y−x)∆₀.

A solution 1 ∈C[[x, y]] of equation (5.2) provides a solution of the orig- inal equation. By plugging the equation (5.1) in the development of the exponential˜0◦ϕ0,w = ˜0+P∞

j=1(logϕ0,w)^j(˜0)/j!we obtain

˜

0◦ϕ0,w= ˜0−y(y−x)∆0+

∞

X

j=2

(logϕ_0,w)^j−1(−y(y−x)∆₀)

j! .

As a consequence the equation (5.2) is equivalent to 1−1◦ϕ0,w =X

k>2

1

k!(logϕ0,w)^k−1(−y(y−x)∆0(x, y)).

We denote the term in the right-hand side byy(y−x)∆1. The series ∆1

satisfiesν˜(∆₁)>ν˜(∆₀) + 1. We have˜₀−˜₀◦ϕ_0,w=y(y−x)(∆−∆₁)by construction. We proceed by induction. Given∆j ∈ C[[x, y]] there exists

˜

_j∈C[[x, y]]such that(logϕ_0,w)(˜_j) =−y(y−x)∆jandν(˜˜ _j)>ν˜(∆_j)+1.

As previously we define y(y−x)∆j+1=X

k>2

1

k!(logϕ0,w)^k−1(−y(y−x)∆j(x, y)).

We obtainν˜(∆_j+1) >˜ν(∆_j) + 1 for anyj >0. The construction implies that˜j−˜j◦ϕ0,w=y(y−x)(∆j−∆j+1)and then

(5.3) (˜0+. . .+ ˜j)−(˜0+. . .+ ˜j)◦ϕ0,w=y(y−x)(∆−∆j+1) for anyj ∈N∪ {0}. Since ˜ν(∆_j+1)>ν(∆˜ _j) + 1forj>0thenν˜(∆_j)>j andν(˜˜ j)>j+ 1for anyj >0. The series∆−∆j+1converges to∆in the Krull topology whenj→ ∞. AnalogouslyPj

k=0˜j converges in the Krull topology to someˆ_∆∈C[[x, y]] whenj → ∞. By taking limits in equation (5.3) whenj→ ∞we obtainˆ∆−ˆ∆◦ϕ0,w =y(y−x)∆.

Lemma 5.3. — Fix∆∈C[[x, y]]. The seriesˆ∆(x, x)−ˆ∆(x,0)does not depend on the solutionˆ_∆∈C[[x, y]] ofˆ−ˆ◦ϕ_0,w=y(y−x)∆(x, y).

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Proof. — It suffices to prove ˆ(x, x)−ˆ(x,0) = 0 for any solution ˆ in C[[x, y]]of ˆ−ˆ◦ϕ0,w= 0. The series ˆbelongs to F(logϕ0,w) by lemma 2.3. Moreover, lemma 5.1 implies∂ˆ/∂y= 0. Henceˆbelongs toC[[x]]and

clearlyˆ(x, x)−ˆ(x,0) = 0.

Given ∆∈C[[x, y]]and a solutionˆ∆∈C[[x, y]]of the equation ˆ

−ˆ◦ϕ_0,w=y(y−x)∆(x, y)

we defineS_w(∆) = ˆ_∆(x, x)−ˆ_∆(x,0). The lemmas 5.2 and 5.3 imply that Sw : C[[x, y]] → C[[x]] is a well-defined linear functional. Proposition 4.5 implies that ifϕ_∆,w has embeddable formal class for any ∆ ∈ m_x,y then Sw(mx,y)⊂C{x}.

Lemma 5.4. — Letw∈C{x, y} \m_x,y. Then we have Sw

logϕ0,w

y(y−x)[y(y−x)∆(x, y)]

= 0

for any∆∈C[[x, y]].

Proof. — Letˆ0∈C[[x, y]]be a solution ofˆ−ˆ◦ϕ0,w =y(y−x)∆(x, y).

Lett∈C. By pre-composing the previous equation withexp(tlnϕ_0,w)we obtain

ˆ

0◦exp(tlnϕ0,w)−ˆ0◦ϕ0,w◦exp(tlnϕ0,w) = [y(y−x)∆]◦exp(tlnϕ0,w).

Denote ˆ_t = ˆ₀ ◦ exp(tlnϕ_0,w). The formal diffeomorphisms ϕ_0,w and exp(tlnϕ0,w) commute for any t ∈ C (see equation (2.2) in section 2).

Thusˆ_t satisfies the equation ˆ

_t−ˆ_t◦ϕ_0,w = [y(y−x)∆(x, y)]◦exp(tlnϕ_0,w) for anyt∈C. Moreover, we have

ˆ

0◦exp(tlnϕ0,w)(x, x)−ˆ0◦exp(tlnϕ0,w)(x,0) = ˆ0(x, x)−ˆ0(x,0).

That implies Sw

[y(y−x)∆(x, y)]◦exp(tlnϕ0,w)−[y(y−x)∆(x, y)]

ty(y−x)

= 0 for any t ∈ C^∗. By taking the limit att = 0 we obtain the thesis of the

lemma.

Next we replace our difference equation with a differential equation.

Proposition 5.5. — Letw∈C{x, y}\m_x,yand∆∈C[[x, y]]. Consider a solutionˆΓ_∆of(logϕ_0,w)(ˆΓ) =−y(y−x)∆. Then we have

Sw(∆) = ˆΓ∆(x, x)−Γˆ∆(x,0).

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Proof. — We defineΠ∈C[[x, y]]such that y(y−x)Π =X

k>2

(logϕ0,w)^k−2(y(y−x)∆(x, y))

k! .

The definition is correct since the right hand side belongs to the ideal (y(y−x))ofC[[x, y]]. By usingΓˆ∆◦ϕ0,w= ˆΓ∆+P∞

j=1(logϕ0,w)^j(ˆΓ∆)/j!

and(logϕ0,w)(ˆΓ∆) =−y(y−x)∆ we obtain

Γˆ∆−Γˆ∆◦ϕ0,w=y(y−x)∆ + (logϕ0,w)(y(y−x)Π).

Consider a solutionˆ∆ ∈C[[x, y]] ofˆ−ˆ◦ϕ0,w =y(y−x)∆(x, y). Then ˆ

α= ˆΓ∆−ˆ∆ is a solution of ˆ

α−αˆ◦ϕ0,w=y(y−x)logϕ_0,w

y(y−x)[y(y−x)Π].

The right hand side is a formal power series. We obtain (ˆΓ_∆−ˆ_∆)(x, x)−(ˆΓ_∆−ˆ_∆)(x,0) = 0

by lemma 5.4.

Corollary 5.6. — Suppose logϕ_0,w ∈ X_N(C²,0). Then S_w(C{x, y}) is contained inC{x}.

Remark 5.7. — Given logϕ_0,w ∈ X_N(C²,0) there is no a convergent solution ˆ∆ of the equation ˆ−ˆ◦ ϕ0,w = y(y −x)∆(x, y) for general

∆ ∈ C{x, y}. The divergence of Sw(∆) is a stronger property than the divergence of everyˆ∆.

6. The induced differential equation

Let v ∈ C[[x, y]]. Let D_v : C{x, y} → C[[x]] be the operator defined byDv(H) = ˆH(x, x)−ˆH(x,0) where ˆH ∈ C[[x, y]] is a solution of the equation ∂ˆ/∂y = vH. The definition of Dv(H) does not depend on the choice ofˆ_H. This section is devoted to prove

Proposition 6.1. — Letv ∈C[[x, y]]. If Dv(C{x, y}) ⊂C{x} then v belongs toC{x, y}.

Fix , δ > 0. We define the Banach space B_,δ whose elements are the seriesH=P

06j,kHj,kx^jy^k ∈C[[x, y]] such that

||H||,δ= X

06j,k

|Hj,k|^jδ^k<+∞.

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We haveB_,δ ⊂C{x, y}. Moreover, a functionH ∈B_,δ is holomorphic in B(0, )×B(0, δ) and continuous in B(0, )×B(0, δ). Given v in C[[x, y]]

we can define forj>1the linear functionals D^j_v:B_,δ→Csuch that D_v(H) =X

j>1

D^j_v(H)x^j

for anyH∈B,δ.

Lemma 6.2. — Let v ∈C[[x, y]]. Then D^j_v : B,δ →C is a continuous linear functional for anyj ∈N.

Proof. — We denoteH =P

06k,lHk,l(H)x^ky^l. Denote v= X

a,b>0

va,bx^ay^b. A solution of

∂ˆ

∂y =vH= X

a,b,k,l>0

va,bHk,l(H)x^a+ky^b+l

is obtained by makingˆ=P

a,b,k,l>0(b+l+ 1)⁻¹va,bHk,l(H)x^a+ky^b+l+1. Hence we have

Dv(H) = X

a,b,k,l>0

va,bHk,l(H)

b+l+ 1 x^a+k+b+l+1.

and then

D^j_v= X

k+l<j





X

a+b=j−k−l−1

va,b

b+l+ 1



Hk,l ∀j ∈N.

It suffices to prove thatH_k,l :B_,δ →C is a continuous functional for all 0 6 k, l. Since |Hk,l(H)|^kδ^l 6 ||H||,δ for any H ∈ B,δ then we obtain

||H_k,l||= sup{|H_k,l(H)|:H ∈B_,δ with||H||_,δ61}6^−kδ^−l. Lemma 6.3. — Letv∈C[[x, y]]. Consider the operatorsD^j_v:B,δ→C forj∈N. Either lim sup_j→∞ ^j

q

||Dv^j||<+∞or Dv(H)6∈C{x} for anyH in a dense subset ofB,δ.

Proof. — Suppose lim sup_j→∞ ^j q

||Dv^j|| = +∞. We choose a sequence (a_j)of positive numbers such thata_j→ ∞and

lim sup

j→∞

j

q

||Dv^j||

aj

= +∞.

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Hencelim sup_j→∞||D^j_v/a^j_j||= +∞. We deduce that lim sup

j→∞

|D_v^j(H)|/a^j_j= +∞

for anyHin a dense subsetEofB,δby the uniform boundedness principle.

Moreover, since

lim sup

j→∞

j

q

|Dv^j(H)|>lim inf

j→∞aj= +∞

thenD_v(H)6∈C{x} for anyH ∈E.

Proposition 6.4. — Consider v ∈ C[[x, y]] and Dv : B,δ → C[[x]].

Suppose D_v(B_,δ) ⊂ C{x}. Then there exists η_,δ > 0 such that D_v(H) belongs toO(B(0, η,δ))for anyH ∈B,δ.

Proof. — There exists η_,δ >0 such that lim sup_j→∞ ^j q

||D^jv|| 61/η_,δ by lemma 6.3. As a consequence

lim sup

j→∞

j

q

|Dv^j(H)|6lim sup

j→∞

j

q

||Dv^j||q^j

||H||,δ

61/η,δ. That implies thatDv(H)∈ O(B(0, η,δ))for anyH ∈B,δ.

Proof of prop. 6.1. — Consider the functionals Dv :B1,1→C[[x]] and D^j_v : B1,1 → C for any j > 1. Since Dv(B1,1) ⊂C{x} then there exists C > 1 such that ||D^j_v|| 6 C^j for any j > 1 by lemma 6.3. We denote v=P

06a,bva,bx^ay^b. We have

D¹_v(1) =v0,0⇒ |v0,0|6||D¹_v||||1||1,16C.

We want to estimatev_k,0,v_k−1,1,. . .,v_0,k for anyk>0. Let us calculate Dv(y^r−1)forr ∈N. The equation∂ˆ/∂y =vy^r−1 =P

a,b>0va,bx^ay^b+r−1 has a solutionˆ=P

a,b>0(b+r)⁻¹v_a,bx^ay^b+r. We deduce that D_v(y^r−1) = X

a,b>0

(b+r)⁻¹v_a,bx^a+b+r.

In particularD_v^k+r(y^r−1) =P

a+b=kva,b/(b+r) =Pk

b=0v_k−b,b/(b+r)for anyr∈N. We obtain

Hilb^k





 vk,0

v_k−1,1 ... v0,k







=







D^k+1_v (1) D^k+2_v (y)

... D^2k+1_v (y^k)







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where Hilb^k is the(k+ 1)×(k+ 1)Hilbert matrix; this is a real symmetric matrix such that Hilb^k_a,b = 1/(a+b−1) for 1 6a, b 6k+ 1. Moreover Hilb^k is positive definite and following [5] we obtain that

||(Hilb^k)⁻¹||2= ρ^4k K√

k(1 +o(1)) whereK= (8π^3/22^3/4)/(1 +√

2)⁴,ρ= 1 +√

2 and||. . .||₂ is the spectral norm. We have |D^k+l+1_v (y^l)| 6 ||D^k+l+1_v ||||y^l||1,1 6 C^k+l+1. As a consequence we obtain

||vk,0, . . . , v0,k||26 ρ^4k K√ k

√

k+ 1C^2k+1(1 +o(1)).

where||vk,0, . . . , v0,k||2is the euclidean norm. Then

|vl,m|6 ρ^4(l+m) K√

l+m

√

l+m+ 1C^2(l+m)+1(1 +o(1))

for 0 6 l, m where lim_l+m→∞o(1) = 0. We deduce that v belongs to O(B(0,1/(ρ⁴C²))×B(0,1/(ρ⁴C²))).

7. End of the proof of the Main Theorem The following proposition will imply the Main Theorem.

Proposition 7.1. — Let w ∈ C{x, y} \m_x,y. Suppose that logϕ_0,w is not convergent. Then there exists ∆ ∈ mx,y such that ϕ∆,w has non- embeddable formal class.

Proof. — Suppose the result is false. HenceSw(mx,y)⊂C{x}by proposition 4.5. Let wˆ ∈ C[[x, y]] be the unit such that L_0,w = ˆw(x, y)∂/∂y (see lemma 5.1). By hypothesis wˆ is a divergent power series. By proposition 5.5 the seriesΓˆ∆(x, x)−Γˆ∆(x,0) belongs toC{x} for any solution ˆΓ∆∈C[[x, y]]of

∂Γˆ

∂y =−∆(x, y) ˆ w(x, y)

and any ∆ ∈ mx,y. Since D_−x/_w_ˆ(C{x, y}) ⊂ C{x} then −x/wˆ belongs to C{x, y} by proposition 6.1. We deduce that wˆ ∈ C{x, y}; that is a

contradiction.

To end the proof of the Main Theorem it suffices to exhibit an example of a diffeomorphismϕ_0,w such thatlogϕ_0,wis divergent by proposition 7.1.

Ifw(0, y)∈ O(C)then(y◦ϕ0,w)(0, y)is an entire function different than y. Then(logϕ0,w)_|x=0is divergent (see [1]). Thereforelogϕ0,wis divergent.

In particular we can choosew= 1.