Normal forms and Sternberg conjugation theorems for infinite dimensional coupled map lattices

In this paper we present local Sternberg conjugation theorems near attracting fixed points for lattice systems. The interactions are spatially decaying and are not restricted to finite distance. The conjugations obtained retain the same spatial decay. In the presence of resonances the conjugations are to a polynomial normal form that also has decaying properties.


Introduction
Coupled map lattices are used to model many systems in physics, chemistry and biology. They are formed by sequences of nodes, each one having its own internal dynamics and being influenced by the dynamics of other nodes through some interactions.
Its origin can be found in the first models for the dynamics of chains of particles under the action of a potential, with a nearest neighbours interaction, models which were first considered by Prandtl [29] and Dehlinger [9]. Later, these models were also considered by Frenkel and Kontorova for specific cases in dislocation models of solids in [16] and [17].
Several problems can be studied under the Frenkel-Kontorova model (or some generalization), ranging from chains of coupled pendula, dislocation dynamics and surface physics to DNA and neural dynamics. See [4] for a modern description and many applications of this model.
One can consider higher dimensional lattices with interactions among all particles. In this case, we have to require some decay in the strength of the interaction, because as it is physically natural, the larger the separation between particles is, the smaller the force of interaction should be.
Assuming each node is represented by R n , in this paper we consider m-dimensional lattices modelled as We allow each node to interact with every other node, but the strength of the interactions decay with the distance between them with a spatial decay controlled by a function Γ : Z m → [0, ∞) satisfying certain properties. To this end, we will use the decay functions introduced in [22] and presented in Section 2.1. We will work with differentiable maps F : ℓ ∞ → ℓ ∞ such that the derivative of the component of F corresponding to the i-th node with respect to the variable x j of the j-th node satisfies ∂F i ∂x j ≤ CΓ(i − j).
More generally, we will work with spaces C r Γ (ℓ ∞ , ℓ ∞ ) of C r functions having decay properties. To be able to work with them, first we have to introduce linear and multilinear maps with decay in ℓ ∞ spaces. See Sections 3 and 4 for the precise definitions and properties. These spaces or similar constructs were introduced in [22], [13], [14], [15].
In this paper we describe normal forms and Sternberg theorems [32] around fixed points in the setting described above, which give differentiable conjugations of the map to their normal forms in neighbourhoods of attracting fixed points. In absence of resonances, the normal forms reduce to the linearisation of the map at the fixed point. For more general fixed points, even in finite dimensions, the study seems to require the use of differentiable bump functions in the ambient space, as it is the case in other settings we are aware of [33], [3], [20], [7]. However, such bump functions do not exist in ℓ ∞ (R n ).
One important consequence of our results is that the normal forms and the obtained conjugations have the same kind of decay as the original map.
From the differentiable conjugation to the linear map we can obtain several invariant manifolds: if we can linearise the map, we can find as many manifolds as linear invariant subspaces the linear map has. Among them, the slow manifolds. These define the motion which converges to the attractor the slowest, and contain the dynamics that can be observed in simulations or physical systems. These manifolds have parameterisations that decay in the aforementioned sense. This property is important in the study of statistical mechanics, see [22].
In the study of normal forms and the linearisation procedure we have to deal with cohomological equations in ℓ ∞ (R n ), in the setting of linear maps with decay. For this, we use the so-called Sylvester operators (see [6]) and we adapt the theory to work in the space of k-linear maps with decay and study their invertibility properties. To that end we introduce the Γ-spectrum in Section 5, a tool enabling us to study these operators in this setting.
We obtain Sternberg theorems for the conjugation of a map to its linear part or to its normal form, in the case that the linear part is a contraction (Poincaré domain). Assuming decay properties for the map we obtain decay properties for the conjugating map. For the results where we allow the existence of resonances, we use a normal form theory with decay which we develop here (analogous to the standard normal form theory around a fixed point) and based on the use of Sylvester operators in spaces of k-linear maps in ℓ ∞ (R n ), introduced in Section 6.
We study two cases, the first one for maps that are small perturbations of an uncoupled map with equal dynamics in each node. For this class of maps we add conditions on the eigenvalues of the linearisation of the unperturbed map at the fixed point restricted to a node (all maps are the same in each node).
In the absence of resonances among eigenvalues we have the following result that gives differentiable conjugation to the linear part of the map. The norm · Γ is introduced in Section 3 and the space of C r functions with decay C r Γ is introduced in Section 4.
If we allow for the existence of resonances, i.e. omitting Hypothesis (H2) above, we have an analogous result giving a differentiable conjugation to a polynomial normal form (Theorem 7 in Section 8).
The second case we consider is non-perturbative and requires conditions over the Γ-spectrum of the linear part, introduced in Section 5.
The paper is structured as follows. Section 2.1 provides the definition of the class of decay functions we will work with and the main examples of them. Section 3 deals with linear and multilinear maps with decay while Section 4 deals with C k maps with decay. In Section 5 we introduce the Γ-spectrum of linear maps with decay. In Section 6 we study some spectral properties of Sylvester operators. Finally, Sections 7 and 8 provide the normal forms and the conjugation results respectively.

Lattices, decay functions and dynamical systems
In this work we will consider dynamical systems in the space of bounded sequences of points of R n with indices in Z m . That is, we will work in the infinite product space (R n ) Z m , where as usual we will call node each individual R n in the lattice. Associated with this space we will consider a decay function, which will control the strength of the interactions between different nodes.
The space of bounded sequences in the infinite product space (R n ) Z m is denoted by ℓ ∞ (R n ) and formally defined as where · is a given norm in R n . We endow ℓ ∞ (R n ) with the norm x ∞ = sup i∈Z m x i as usual. Note that if we change the norm in R n we end up with an equivalent norm in ℓ ∞ (R n ). We denote by proj i : ℓ ∞ (R n ) → R n the projection onto the i-th component, and the related function, emb i : R n → ℓ ∞ (R n ) the i-th embedding. They satisfy proj j (emb i (u)) = 0, i = j, and proj i (emb i (u)) = u for every u ∈ R n . This embedding is an isometry if the norm in ℓ ∞ (R n ) is induced by the norm considered in R n .

Decay functions in lattices.
To be able to define meaningful localised perturbations in ℓ ∞ (R n ), we consider an appropriate set of weighted Banach spaces. The main idea is that the coupling term in the perturbed system belongs to a weighted space, which controls the strength of the interaction between nodes. We should note that nearest-neighbour coupling (or any other finite rank coupling) will satisfy these hypotheses.
We will make use of the following decay functions, originally introduced in [22].
Definition 1. We say that a function Γ : Z m → R + is a decay function when it satisfies: The first property ensures that interaction propagation related to such a decay function is finite, while the second property is akin to a triangular inequality in a discrete lattice. As pointed out by Prof. L. Sadun, the second property can be interpreted as that the sum of the interactions between two nodes through the interactions involving third nodes is dominated by the direct interaction between them.
The following proposition can be found in [22] and provides a family of examples of decay functions satisfying Definition 1. Note that the standard exponential function Γ(j) = Ce −θ|j| is not a decay function for any C, θ > 0, as proved in [22].

Linear and multilinear maps with decay
To define C r maps with decay properties in lattices we need to first introduce spaces of linear and multilinear mappings with suitable decay properties. Then we can use these definitions to introduce spaces of C r maps with these predefined decay properties for its derivatives. In this section we will define linear maps with decay and its related norm · Γ . From now on we will use · to denote the norm induced in the space of linear or multilinear maps by the same norm in ℓ ∞ (R n ). All decay functions will satisfy Definition 1. We reproduce some statements from [22] and [13], and provide some details and some additional results for the convenience of the reader.
3.1. The space of linear maps with decay. The most natural way to define linear maps with decay is to require the components of "infinite matrices" to have decay properties with respect to their indices. This is formalised as follows. Given · a norm in R n we define with A the operator norm of A and Remark 1. We will use L Γ as an abbreviation for L Γ (ℓ ∞ (R n ), ℓ ∞ (R n )). Although the definition has been written for ℓ ∞ (R n ), we can define linear maps with decay among arbitrary vector subspaces E, F of ℓ ∞ (R n ) as All results stated for L Γ (ℓ ∞ (R n ), ℓ ∞ (R n )) extend in a straightforward way to L Γ (E, F ).
We can provide an interpretation of γ(A) in terms of the elements of the infinite dimensional matrix. If we denote A ij = proj i A emb j then we have It is worth emphasizing that the elements A ij do not determine A. We can find a specific counter-example in [13, p.2843]

Remark 2. Observe that given an uncoupled linear map
Several basic properties follow.
Remark 3. Proposition 2 and Proposition 3 imply L Γ (ℓ ∞ (R n ), ℓ ∞ (R n )) is a Banach algebra. It has a unit element Id but Id Γ = 1. This makes spectral theory in L Γ less straightforward, since the classic results (cf. [30]) require unit elements with norm 1 in most proofs. There is however a standard trick to overcome this difficulty, see Section 5. and

as a Neumann series as
3.2. The space of k-linear maps with decay. To characterise higher order differentiable functions with decay we inductively define multilinear maps with decay. Recall that we can define the space of k-linear maps There are k possible identifications defined by the isomorphisms ι j as follows. Given the map The maps ι j , 1 ≤ j ≤ k, are isometries in the corresponding operator norms. We define

Remark 4. Note that this definition is consistent because we can identify
With the definition of this norm we can prove that L k Γ (ℓ ∞ (R n ), ℓ ∞ (R n )) is a Banach space using the same tools as in the proof of Proposition 2.
The following proposition gives bounds to the norm of multilinear contractions. These bounds are fundamental later on, since multilinear contractions appear naturally when differentiating repeatedly invariance equations, a basic step in the study of normal form equations and the parameterisation method.
Then, for any permutation of k elements τ ∈ S k , the map Proposition 5 can be used to bound Γ-norms of contractions in such a way that decay properties can be ignored except for the bound of just one component, as the next proposition shows.
From Propositions 3 and 5 we also obtain the following composition property, which will prove crucial for later developments. Given . . , k and w lj ∈ ℓ ∞ (R n ) lj , we define the composition AB 1 · · · B k by A consequence of the proof is that if B p = B q for all q = p the bounds can be written instead as An important special case of the previous proposition is the following result, which follows from the proofs of Propositions 3 and 6.

Spaces of differentiable functions with decay
With the definitions of linear and multilinear applications with decay from the previous section we are prepared to define spaces of differentiable functions with decay whose domain is an open set in ℓ ∞ (R n ).
where F C 0 = sup x∈U F (x) ∞ as usual. We can also define This definition is consistent since Based on the above, we define spaces of C r Γ functions as: is a Banach space. We have the following result concerning the composition of maps, which can be found in [13].
In this case the estimates in the proof are much easier and the bound is We defer the proof of this result until Section 5.

Spectral theory for Γ-coupled linear maps
In this section we recall some results in spectral theory of linear operators and also introduce a new notion, the Γ-spectrum of a linear operator in a lattice, associated to a decay function Γ satisfying the definition introduced in Section 2.1.
Given a Banach space E the space of continuous linear maps L(E, E) is also a Banach space with the standard operator norm. Moreover, it is a Banach algebra with the product given by the composition of maps.
Let E = ℓ ∞ (R n ). Given a decay function Γ as in Definition 1, the space L Γ (E, E) introduced in Section 3.1 is a Banach algebra (see Proposition 3).
The inclusion L Γ (E, E) ⊂ L(E, E) holds considering both spaces as sets, but L Γ (E, E) is not a closed subalgebra of L(E, E), hence it is not a Banach subalgebra of L(E, E). Indeed, consider a specific decay function: with α > m, θ > 0 and a > 0 small enough. Consider the sequence of linear maps {A k } k∈N defined by which goes to zero as k → ∞ because l∈Z m |l|Γ(l) is convergent provided either θ > 0 or θ = 0 and α > m + 1. However The space L Γ (E, E) is a Banach algebra with the identity as unit, but Id Γ = Γ(0) −1 = 1. To be able to apply the general results of Banach algebras with unit, we can introduce an equivalent norm in L Γ (E, E), say · ′ , such that Id ′ = 1. The procedure is standard (see [24]). We define The properties of norm are easily checked from the definition, proving the equivalence requires the following. On one hand, on the other hand, Finally, To illustrate some features of L Γ (E, E) we present an example of an invertible linear map in L Γ (ℓ ∞ (C n ), ℓ ∞ (C n )) such that its inverse may not be in L Γ (ℓ ∞ (C n ), ℓ ∞ (C n )) depending on the decay function Γ considered.
Let ℓ ∞ (C) be a one dimensional lattice (m = 1), r ∈ N, a 0 , . . . , a r ∈ C and A ∈ L(ℓ ∞ (C n ), ℓ ∞ (C n )) determined by We look for the inverse B of A assuming a priori that the inverse is upper triangular and a band matrix. That is, Imposing the condition AB = Id, or equivalently When i = j we have a 0 b 0 = 1. This condition implies a 0 = 0. We assume it from now on. Then we proceed by induction and recursively obtain b j for j > 0. Actually b j satisfies the r-th order linear difference equation Using the theory of linear difference equations we can compute b j in terms of the zeros of the characteristic polynomial of this equation, a 0 x r + a 1 x r−1 + . . . + a r = 0.
Once we have determined b j and hence B, we can check that indeed For this to hold we strongly use that A ik ∈ L(R, R) ∼ R. It remains to check that B sends ℓ ∞ (R) to itself. This will depend on the choice of the values of a i .
To work with a specific example, assume r = 2 and a 0 = 1. Hence we can determine the zeros of the characteristic polynomial and write the general solution of the difference equation as for suitable values β 1 , β 2 . Now we can choose a 1 , a 2 to adjust the growth of the coefficients b j . For instance, taking With the choice of decay function Γ(j) = a|j| −α e −θ|j| we have that which is finite when θ < log 2. Hence B ∈ L Γ (ℓ ∞ (R), ℓ ∞ (R)) (with this particular choice of decay function Γ) if and only if θ < log 2.
5.1. Γ-spectrum of linear maps on lattices. Consider the lattice ℓ ∞ (R n ) and a decay function Γ. Also consider the complexified space Let E be a linear subspace of ℓ ∞ (C n ). Given A ∈ L Γ (E, E) we define: • Γ-spectrum of A as • Γ-spectral radius of A as From the definitions above it is immediate that and therefore Spec(A) ⊂ Spec Γ (A), r(A) ≤ r Γ (A). Also from the definitions, it is clear that Spec Γ (A) is the spectrum of A as an element of the Banach algebra L Γ (E, E). Hence all general properties of spectra in unitary Banach algebras apply to Spec Γ .

Operational calculus.
The results in this section are similar to those for the spectrum of a linear operator, a standard reference is [31]. For the remaining of this section let E denote a complex Banach space.
Let A ∈ L Γ (E, E) and Ω be an open set such that Spec Γ (A) ⊂ Ω. Let ω be an open set such that Spec Γ (A) ⊂ ω ⊂ ω ⊂ Ω (5) and ∂ω is a finite union of closed curves.
Then, given f : Ω → C analytic we define This definition is independent of the choice of ω provided it satisfies the conditions above, and we have that In the case that f is a polynomial, f (z) = m k=0 a k z k , the previous definition gives f (A) = m k=0 a k A k . The following proposition proves upper semicontinuity of Spec Γ (A) with respect to A.
Moreover, if f, g : Ω → C are analytic functions and h(z) = f (z)g(z), hence As a consequence, under these conditions if we assume f : Ω → C is not zero, thus where ω satisfies (5).
With these results on spectra of Γ-linear maps we can now prove the inverse function theorem in lattices with spatial decay.
Proof of Theorem 3. From the standard inverse function theorem in Banach spaces, F is locally invertible and F −1 is defined in a neighbourhood V of q. Moreover, DF −1 (q) = DF (p) −1 and by the continuity of DF and continuity of Spec Γ , we can obtain the higher order derivatives of F −1 by taking derivatives in the right hand side of (8). For instance, Then, by Proposition 6, we have D 2 F −1 (x) ∈ L 2 Γ . Proceeding in the same way for the other derivatives we get that F −1 ∈ C r Γ (V, ℓ ∞ (R n )). Alternatively, we can use (9) to prove inductively that F −1 ∈ C i Γ assuming F −1 ∈ C i−1 Γ , for i ≤ r.

5.3.
Spectral projections associated to a gap in the Γ-spectrum. We can adapt the spectral projection theorem (see for instance [31]) to the setting of Γspectrum. The statements and proofs are very similar to the ones corresponding to L(E, E) but we give them here for the sake of completeness. Assume that where Ω i are disjoint open sets and ω i are open sets such that ∂ω i are finite union of simple closed curves.
We define Results on integration of elements of L Γ can be found in [13].
Proof. Part (i) follows from the properties of integrals of functions in the Banach algebra L Γ in [13]. Part (ii) follows from the fact that P can be written as proving P is a projection. For Part (iii), P (E) and Ker(P ) are invariant when P is a projection, and E = P (E) ⊕ Ker(P ). Moreover since P (E) = Ker(Id −P ) and P is continuous, both P (E) and Ker(Id −P ) are closed.

Theorem 4. We have that
Proof. Let Moreover, let λ / ∈ σ 1 and g 1 (z) = f (z) λ−z . The function g 1 is analytic in a neighbourhood U of Spec Γ (A) and satisfies By (6), f (A)g 1 (A) = g 1 (A), and hence and Then λ ∈ ρ Γ (A), which implies that 6. Sylvester operators in L k Γ In this section we will introduce Sylvester operators and prove some results in spaces of k linear maps with decay.

respectively.
Note that by Proposition 6, if W ∈ L k Γ (E, E) then R j,A (W ), L B (W ) and S B,A (W ) are in L k Γ (E, E) so that the operators are well defined. Given two subsets X, Y of C we denote by X · Y the set Analogously, we define X k = X· k . . . ·X.
Proposition 9. We have The proof of this proposition is a consequence of the following theorem and the next lemma.

Lemma 2. Given
To study the invertibility of and hence W ∈ L k Γ (E, E) and λ ∈ ρ(R j,A ). The proof of the result for L B is completely analogous.
Proof of Proposition 9. It follows directly from the fact that and the fact that the operators on the r.h.s. of (13) commute. Then Theorem 5 proves the result.

Normal forms of maps in lattices
In this section we consider the computation of normal forms around a fixed point of a map in a lattice, assuming the map has decay properties. We estimate the decay properties of the normal form and the transformation leading to it.
To study the decay properties of normal forms we will use Sylvester operators in spaces with decay, introduced in the previous section.
We consider an open set U of ℓ ∞ (R n ) such that 0 ∈ U and a map , invertible and consider its Γ-spectrum Spec Γ (A). Theorem 6. In the previously described setting there exist polynomials K ∈ C ∞ Γ (ℓ ∞ (R n ), ℓ ∞ (R n )) and H ∈ C ∞ Γ (ℓ ∞ (R n ), ℓ ∞ (R n )) of degree at most r such that K(0) = 0, DK(0) = Id and and

Corollary 2. Under the conditions of the previous theorem, if
Proof of Theorem 6. We look for K and H in the form . Taking derivatives on both sides of and evaluating at 0 we have A K 1 = K 1 H 1 . This equation has the obvious solution K 1 = Id, H 1 = A, although other solutions are possible, for instance taking K 1 as any linear map which commutes with A, like K 1 = α Id, α ∈ R and H 1 = A. However, in what follows we make the choice K 1 = Id and H 1 = A.
Taking k-th order derivatives on both sides of (14), using the Faà di Bruno formula, (where for the sake of simplicity we have not written the dependence of C on the indices), and evaluating the derivatives at 0 we can write where Observe that G k consists of sums and contractions of multilinear operators.
Using Sylvester operators (introduced in Section 6) we rewrite Equation (15) as Now we proceed inductively from k = 2 up to k = r. Assume that for j up to the (k − 1)-th step we have obtained K j and H j in L j Γ (ℓ ∞ (R n ), ℓ ∞ (R n )) by solving Equation (16). From the way G k is defined, Proposition 6 proves that G k ∈ L k Γ (ℓ ∞ (R n ), ℓ ∞ (R n )) and hence the operator S A −1 ,A may not be invertible and we set H k = G k and K k = 0. This is not the only possible choice, it is only the simplest one and also has K k , Another standard possibility is to decompose L k Γ (E, E) = Im S A −1 ,A ⊕ V , where Im stands for the range of S A −1 ,A and V is a complementary subspace in L k Γ (E, E). Then one decomposes A −1 G k according to this splitting of the space as Of course this choice also depends on the choice of the complementary space V .
By the choices of K k , H k , 1 ≤ k ≤ r, and hence, by Taylor's theorem,

Sternberg theorems in lattices
In this section we will prove several Sternberg conjugation theorems for contractions under several non-resonance hypotheses. All of them are adaptations to our setting of the classical proof in [32], using the normal form theory developed in the previous section.
We will begin by proving Theorem 1. First, two remarks from the requirements.
Remark 7. Note that we do not require F 1 to be small but only B = DF 1 (0) to be small in the Γ-norm.
Remark 8. Since r 0 is finite, assumption (H2) involves only a finite set of conditions.
Before starting the proof we perform a rescaling of F in order to transfer the smallness conditions on the domain of definition to smallness of an auxiliary parameter.
Let δ ∈ R, δ > 0, and the rescaling map T δ x = δx. We define . From now on we will not write the dependence on δ of F δ (x) and N δ (x) and we will assume that F is defined on B(0, 1) ⊂ ℓ ∞ (R n ) and δ is as small as needed. In particular, if F is at least of class C 2 , we have that and moreover N C r Γ = O(δ). Given r, r 0 ∈ N, r ≥ r 0 , we introduce the spaces χ r,r0 = g ∈ C r (B(0, 1), ℓ ∞ (R n )) | D j g(0) = 0, 0 ≤ j ≤ r 0 , g C r < ∞ , Γ is a closed subspace of C r Γ . Lemma 3. Assume the hypotheses of Theorem 1.
Then, if r ≥ r 0 , F ∈ C r Γ (U, ℓ ∞ (R n )) and B Γ and the rescaling parameter δ are small enough, the linear operator G m : χ r,r0 is well defined and is a contraction in the C r Γ -norm. Proof. First, we fix some quantities to be used throughout the proof. From the definition of r 0 and the fact that β < 1 we have Then there exists m ∈ N such that Γ(0) −2 α −1 β r0 m < 1 and also there exist positive numbers ε 1 , ε 2 and ε 3 such that Note that this condition requires β + ε 1 < 1. There exists a norm in R n such that where · is the associated operator norm. Clearly, Moreover, In the same way, now using Proposition 4, where C is a combinatorial coefficient which depends on all indices in the sum. From (18) it is clear that D p (g • F m )(0) = 0 for 1 ≤ p ≤ r 0 , since F (0) = 0. Since g ∈ χ r,r0 , by Taylor's theorem in integral form (see [1]), Using the previous formulas, Proposition 5 and usual results about integration in Banach spaces (see [1] for the theory of Cauchy-Bochner integration on Banach spaces) we have As a consequence of the two previous bounds, we can write the more compact form Then, using Proposition 6 By the rescaling, if x ∈ B(0, 1), for 1 ≤ p ≤ r, and finally, Then if δ is small enough, by (17) the factor in front of g C r Γ is strictly less than 1 and hence G is a contraction in χ r,r0 Γ .
Now we use the normal form theory in the previous section to find a decay map which linearises our map F up to order r 0 . The form of A implies that Spec(A) = {λ 1 , . . . , λ n } and Spec(A m ) = {λ m 1 , . . . , λ m n }. Since A m is uncoupled, Spec Γ (A m ) = Spec(A m ). Moreover the non-resonance condition (H2) implies that λ m i = λ mk1 1 · · · λ mkn n , k ∈ (Z + ) n , 2 ≤ |k| ≤ r 0 , and therefore (Spec Γ (A m )) j ∩ Spec Γ (A m ) = ∅, j ≥ 2. Taking B Γ sufficiently small, since Spec Γ is upper semicontinuous by Proposition 8, we have because we are only dealing with a finite set of conditions. Hence Corollary 2 gives us that there exists a polynomial K ∈ C ∞ Γ (ℓ ∞ (R n ), ℓ ∞ (R n )) of degree (at most) r 0 such that K(0) = 0, DK(0) = Id and Let S 0 = K −1 be the local inverse. Taking the rescaling parameter δ smaller if necessary we can assume that S 0 is defined in B(0, 1) ⊂ ℓ ∞ (R n ). By Theorem 3, we have S 0 ∈ C r Γ and satisfies S 0 (0) = 0, DS 0 (0) = Id, Starting with this approximate conjugation we define the sequence The next lemma proves that S n converges to a well-defined conjugation in the space C r Γ . Lemma 4. The sequence {S n } n∈N defined above converges to a function S ∈ C r Γ (B(0, 1), ℓ ∞ (R n )) satisfying S(0) = 0, DS(0) = Id and Proof. Since m is fixed we will drop the dependence of G m on m. First, we prove the following relation where G 0 = Id and G j = G • G j−1 , j ≥ 1. Observe that M −m S 0 • F m − S 0 ∈ χ r,r0 , since S 0 solves the conjugation equation formally up to order r 0 . Moreover We prove (19) by induction. When n = 1, we use the definition S 1 = G(S 0 ): (19) is true up to index n, then

Now assume Equation
By Lemma 3, G is a contraction in χ r,r0 Γ and therefore the series arising from (19) converges and lim n→∞ S n exists and belongs to χ r,r0 Γ . Finally, we check the conjugacy property. Indeed, and since DF mn (0) = DF (F mn−1 (0)) · · · DF (0) = M mn and DS 0 (0) = Id, Thus, S conjugates F m to M m . The final step is to show that S also conjugates F to M .
By the spectral properties and Corollary 2, there exists a polynomial K ∈ C ∞ (ℓ ∞ (R n ), ℓ ∞ (R n )) such that Let R 0 = K −1 , which similarly to S 0 , we can assume is defined in ) and as a consequence Lemma 5. Under the hypotheses of Theorem 1, if B and the rescaling parameter δ are small enough, the operator defined by G(g) = M −1 g • F is well defined and is a contraction in the C r -norm.
We omit the proof of this lemma since it is completely analogous to the proof of Lemma 3 but the estimates are much simpler, since they do not involve decay functions.
We define the sequence The same arguments as the ones used in the proof of Lemma 4 but now in the space Now consider the iteration S n introduced in the proof of Lemma 4 with S 0 = R 0 ∈ C r Γ . Since R 0 satisfies (20), we see that S n is a subsequence of R n , being both sequences convergent in the larger space C r . Then R = lim n→∞ R n = lim n→∞ S n = S ∈ C r Γ and therefore S also conjugates F with M , proving Theorem 1.
An improvement of Theorem 1 consists of not assuming the non-resonance condition (H2). In such case we obtain a C r Γ local conjugation to a normal form of F instead to a conjugation to its linear part.
Proof. We will only comment on the differences of this proof with the proof of Theorem 1. We rescale the map, we consider the spaces χ r,r0 and χ r,r0 Γ and we use the same integer m as in the proof of that theorem. We take a normal form H provided by Theorem 6 and define the operator Now the estimates on G m become more involved because in this case H is not linear. This implies that G m is not linear anymore. However, because of the rescaling, H −1 is very close to M −1 in C r Γ (and C r ) norm, a fact which gives similar estimates and thus proves Lip G m < 1. The remaining part of the proof is analogous.
The previous theorems assume that the linear part of the maps is close to an uncoupled map with identical dynamics on each node. This gives sufficient conditions for the conjugation in terms of the eigenvalues of the projections to the nodes.
Theorem 2 requires instead conditions on the Γ-spectrum of the linear part of the map.
For the uniqueness arguments needed at the end of the proof, we consider the operator G(g) = A −1 g • F in C r (B(0, 1), ℓ ∞ (R n )). Since Spec(A) ⊂ Spec Γ (A), condition (H2) implies that there are also no resonances among the elements of Spec(A) and that r(A) < β Γ and r(A −1 ) < α −1 Γ . Hence we can find a norm in the space ℓ ∞ (R n ), equivalent to the original one, such that where in the previous expression · stands for the corresponding operator norm. The bound (22) allows us to prove the estimates needed to show that G is a contraction. With these ingredients we can finish the proof in this setting in the same way as in Theorem 1.
The analogous version of Theorem 7 in this setting is the following.
Theorem 8. Under the conditions of Theorem 2, except condition (H2), if F ∈ C r Γ (ℓ ∞ (R n ), ℓ ∞ (R n )) with r ≥ r 0 there exists a polynomial H ∈ C ∞ Γ (ℓ ∞ (R n ), ℓ ∞ (R n )) of degree not larger than r 0 and R ∈ C r Γ (ℓ ∞ (R n ), ℓ ∞ (R n )) such that R(0) = 0, DR(0) = Id The proof of this theorem is a combination of the arguments in the proofs of Theorem 7 and Theorem 2.