Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points

We study the Gevrey character of a natural parameterization of one dimensional invariant manifolds associated to a parabolic direction of fixed points of analytic maps, that is, a direction associated with an eigenvalue equal to $1$. We show that, under general hypotheses, these invariant manifolds are Gevrey with type related to some explicit constants. We provide examples of the optimality of our results as well as some applications to celestial mechanics, namely, the Sitnikov problem and the restricted planar three body problem.

(Communicated by the associate editor name)

Introduction
Let us consider a dynamical system defined through a map F ∶ R m → R m with a fixed point at the origin. To each invariant subspace E of DF (0) one can try to identify its corresponding counterpart for F , that is, a manifold tangent to E at the origin invariant by F , if it exists. Of course, these invariant manifolds need not be unique, or even if they do exist, they can be less regular than the map F , depending on the resonance relations in Spec DF (0) E . In the case that F is analytic or C ∞ , one can even ask if there exists a formal invariant manifold tangent to E, that is, a formal power series which solves at all orders an appropriate invariance equation.
One way to obtain manifolds invariant by F is by using the parameterization method. A brief description is the following. If E ⊂ R m is a subspace of dimension n, invariant by DF (0), one can try to find an invariant manifold by F tangent to E at the origin as an embedding K ∶ B ρ ⊂ R n → R m (here B ρ denotes the ball of radius ρ) such that K(0) = 0, DK(0)R n = E and a reparameterization R ∶ B ρ → R n , R(0) = 0, satisfying the invariance equation Well known examples of invariant manifolds are the strong stable and unstable manifolds, which, roughly speaking, are associated to the eigenvalues λ of DF (0) such that λ > µ > 1 and λ < ν < 1, respectively, for given constants µ and ν. See, for instance, [HP70,Irw70,Irw80] and [CFdlL03, HCF + 16, CFdlL05] and the references therein. These manifolds are as regular as the map in a neighborhood of the fixed point. In particular, analytic if so is the map. Their expansions in power series are convergent.
When one considers invariant manifolds tangent to subspaces associated to subsets of non-resonant eigenvalues the situation becomes more interesting. The invariance equation can be solved at all orders, due to the non-resonant character of the eigenvalues. This solution provides a formal invariant manifold. In general, this formal series corresponds to a regular meaningful object if one imposes the non-resonant eigenvalues to be of modulus larger (resp. smaller) than one. That is, when the non-resonant manifolds are submanifolds of the strong unstable (resp. stable) manifold. See for instance [CFdlL03]. If the map is analytic, these nonresonant manifolds are also analytic and, again, their expansions are convergent.
Here we consider the totally resonant case, that is, manifolds tangent to subspaces associated to the eigenvalue 1 and, thus, submanifolds of the center manifold. We call these manifolds parabolic.
When the map is tangent to the identity at the fixed point, that is, DF (0) = Id, any subspace of R m is invariant by DF (0). In order to identify the subspaces which are susceptible to have an invariant manifold tangent to them it is necessary to pay attention to the first next non-vanishing terms of the Taylor expansion of F at the origin. This is the case considered in [BFdlLM07], when one looks for one dimensional manifolds. See also [McG73]. In the latter, only analytic manifolds where considered, while the former includes the case of finite differentiability. The former also includes the construction of formal solutions of the invariance equation (1). Under the conditions in [BFdlLM07], if the map F is analytic or C ∞ , the parabolic invariant manifolds exist and are C ∞ at the fixed point. See also [Hak98] and the survey [Aba15] in the setting of complex dynamics.
In [BF04] it is studied the case of F analytic, tangent to the identity and with invariant manifolds of dimension two or greater. These manifolds are analytic in their domain, although in general the fixed point is only at their boundary. In this case, however, it is easy to see that in general there are no formal solutions (in the sense of power series) of the invariance equation (1). In the same setting, in [BFM15] it is shown that the invariant manifolds can be approximated by sums of homogeneous functions of increasing order.
In the present paper we assume that F is an analytic local diffeomorphism in a neighborhood of the origin in R × R d × R d ′ and satisfies with 1 ∈ Spec C and Id d is the identity matrix in R d . When d = 0, that is, when 1 is a simple eigenvalue of DF (0), this class of maps was studied in [BH08]. There the authors proved that if the map F has the form where O j stands for O( (x, z) j ), with a ≠ 0, N ≥ 2, the invariance equation admits a formal solutionK(t) = ∑ k≥1 K j t j , K j ∈ R 1+d ′ , with some polynomial reparameterization R, and that the series is α-Gevrey with α = 1 (N − 1), that is, there exist constants c 1 , c 2 > 0 such that Furthermore, if a > 0 and Spec C ⊂ {z ∈ C z ≥ 1, z ≠ 1}, there is an analytic solution K of the invariance equation, defined in some convex set V with 0 ∈ ∂V , that is α-Gevrey asymptotic toK, that is, there exist constants c 1 , c 2 > 0 such that K j t j ≤ c 1 c n 2 n! α t n , n ≥ 1, t ∈ V.
Here we generalize these results to the case d > 0, d ′ ≥ 0. That is, if the map F has the linear part (2) and certain conditions on the nonlinear terms are met (see Theorem 3.1), the invariance equation (1) for the map F admits a formal solutionK(t), which is γ-Gevrey for a precise γ (defined in (4)). We provide examples for which this value of γ is sharp, that is,K(t) is not γ ′ -Gevrey for any 0 ≤ γ ′ < γ. These conditions can be seen as non-resonances, because they allow to solve some cohomological equations (see also Claim 4.2). Also they would imply the existence of a characteristic direction, if the map was truly tangent to the identity at the fixed point.
Adding some additional conditions (see Theorem 3.3), we also prove that there is a true invariant manifold given by an analytic parameterization K which is γ-Gevrey asymptotic the the formal seriesK in some complex convex set with 0 at its boundary. We will refer to this manifold as a parabolic manifold and we notice that the information about its internal dynamics is given by R(t) which, in our case, turns out to be a polynomial. Depending on R the parabolic manifold may behave as a (weak) stable manifold (in the sense that the iterates of its points converge to de origin) or a (weak) unstable manifold. In those cases we will denote them by parabolic stable/unstable manifolds.
Of course, the conditions that allow the existence of a formal solution are weaker than the ones we need to impose in order to have a true invariant manifold. However, we prove that if the map possesses a one dimensional parabolic stable invariant manifold to the origin, tangent to a particular direction associated to an eigenvalue equal to 1, and it is non-degenerate (in the sense of Proposition 3.2), then there are suitable coordinates in which the map satisfies our conditions (listed in (3) and hypotheses below).
Our results provide upper bounds for the coefficients of the asymptotic expansion of the invariant manifold. The existence of lower bounds remains open. Although we provide examples that show the optimality of our results, we also prove that if the map is the time one map of an autonomous analytic vector field, satisfying our hypotheses, the invariant manifold, when written as a graph, extends analytically to a neighborhood of the origin (see Claim 4.2). That is, the invariant manifolds can be more regular than what we claim. This is no longer true for the stroboscopic Poincaré map of time-periodic equations (Claim 4.3). However, although obtaining lower bounds is out of the scope of the present work, we show in Proposition 3.8 that the conditions to obtain such lower bounds cannot depend on a finite number of coefficients of the Taylor expansion of the map F .
An important consequence of our present results is the Gevrey character of some invariant manifolds in some problems of Celestial Mechanics. In several instances of the restricted three body, like the Sitnikov problem or the restricted planar three body problem, the parabolic infinity is foliated by periodic orbits. The associated stroboscopic Poincaré map satisfies the conditions of our existence result (Theorem 3.3) with d ′ = 0, which implies that the manifolds are at least 1 3-Gevrey at the origin. See Section 4.3 for more details. Simó and Martínez announced in 2009 [MS09] that, in the case of the Sitnikov problem, the manifolds are precisely 1 3-Gevrey, which would imply the optimality of our result in the sense that these manifolds are not more regular. The numerical experiments in [MS14] strongly support the same claim for the restricted circular planar three body problem. These computations and the example we provide in Claim 4.3 move us to conjecture that the invariant manifolds of infinity of the restricted three body problem are exactly 1 3-Gevrey (see Conjecture 4.5 for the precise statement).
The structure of the paper is as follows. In Section 2 we introduce the definitions and notations we will use along the paper. In Section 3 we collect the main results of the paper. Section 4 is devoted to present some examples that show that the Gevrey order we find is optimal. We also show how our theorems apply to the restricted three body problem. The rest of the paper contains the proofs of the results on Section 3. In Section 5 we obtain the formal solution of the invariance equation. Its Gevrey character is studied in Section 6. The existence of the true manifold is proved in Section 7. The appendix contains the proofs of Propositions 3.2 and 3.8.

Set up and notation
where: • N, M ≥ 2 are integer numbers; • the constant a is non-zero; is a homogeneous polynomial of degree M such that g M (x, 0, 0) = 0, D y g M (x, 0, 0) = 0 and D z g M (x, 0, 0) = 0; • f ≥N +1 has order N + 1 (the function and its derivatives vanish up to order N at (0, 0, 0)), g ≥M+1 has order M + 1 and h ≥2 has order 2.
Since F is real analytic, it can be extended to a complex neighborhood U C of U. For simplicity, we will denote also by F this complex extension.
We introduce the following notational conventions we use throughout the paper. We denote byŴ (t) = ∑ k≥0 W k t k any formal series in t and if W (t) is a map, we denote by W k = 1 k! D k W (0), if the derivatives are defined. The expressions W ≤l , W ≥l+1 , etc. will mean ∑ l k=0 W k t k , ∑ k≥l+1 W k t k , etc., and we will use them without further mention. The projection over the x, y or z-component is denoted by π x , π y and π z . If W (⋅) ∈ C 1+d+d ′ (or if W is a map taking values in C 1+d+d ′ , or a power series with coefficients in C 1+d+d ′ ), we write W x = π x W , W y = π y W and W z = π z W . We also use π x,y W = W x,y = (W x , W y ), or any other combination of the variables.
We finally introduce the constants which will play a capital role in our results.

Main Results
We start dealing with formal solutions of the invariance equation F ○ K = K ○ R. We provide conditions that ensure the existence of a formal solution as a power series, which turns out to be γ-Gevrey.
(in the sense of formal series composition). More precisely, under these conditions, there exists a unique polynomial R(t) = t − at N + bt 2N −1 such that for any c ∈ R, there is a unique formal power serieŝ This expansion is γ-Gevrey, that is, there exist constants c 1 , c 2 > 0 such that We prove Theorem 3.1 along Sections 5 and 6. First, in Proposition 5.1 we prove the existence of the formal solution of (5) and provide formulas to compute it. Then, with the aid of some technical lemmas, we prove in Proposition 6.6 that this formal solution is γ-Gevrey.
The following proposition emphasizes the conditions on our map given by (3) are not too restrictive when considering parabolic-hyperbolic fixed points.
with 1 ∈ Spec C, having an invariant curve associated to the origin of the form (y, z) = ϕ(x). Assume that there exist N ≥ 2 and a ≠ 0 such that and that ϕ is C r with r ≥ N . Then, by means of changes of variables and a blow up, F can be expressed as in the form (3) for some M ≥ 2.
The proof of this result is elementary. We defer it to Appendix A.
The following result assures that, under additional conditions, the formal ex-pansionK given by Theorem 3.1 is the asymptotic series of a true solution of the invariance equation, analytic in some domain with 0 at its boundary.
Theorem 3.3. Let F be a map of the form (3). Assume that a > 0 and Then, for any 0 < β < απ, there exist ρ small enough and a real analytic function K defined on the open sector Moreover, K is γ-Gevrey asymptotic to the γ-Gevrey formal solutionK. That is, for any 0 <β < β and 0 <ρ < ρ, there exist constants c 1 , c 2 such that, for any n ∈ N, for all t ∈S 1 ∶= {t = re iϕ ∈ C ∶ 0 < r ≤ρ, ϕ ≤β 2}.
In particular, K can be extended to a C ∞ function in [0, ρ).
The proof of this theorem is given in Section 7. Now we give conditions that ensure that the manifold given by Theorem 3.3 is unique (in a suitable open set).
Theorem 3.4. Under the same assumptions of Theorem 3.3, if the matrices C and B 1 satisfy that and there exists a unique right hand side branch of a curve in the center manifold which is a parabolic stable manifold to the origin. That is, if we denote by B(ρ) ⊂ R 1+d+d ′ the open ball of radius ρ, the following local stable manifold . This theorem is proven by using the same geometrical arguments in [BH08]. We omit the proof.
Remark 3.5. In the last two theorems we have assumed a > 0. Clearly, if a < 0, the map F −1 has the form in (3) substituting a, B 1 , B 2 and C by −a, −B 1 , −B 2 and C −1 respectively. Therefore, if a < 0, we can apply (if the other conditions are satisfied) Theorems 3.3 and 3.4 to F −1 obtaining a local unstable parabolic invariant manifold.
A straightforward consequence of Theorem 3.3 is the following.
Corollary 3.6. If a ≠ 0, there exists a unique constant b such that the real analytic maps , with 0 < β < απ and ρ small, by means of an analytic function h ∶ S(β, ρ) → C which is α-Gevrey asymptotic to a α-Gevrey formal series at 0.
In the next section we will provide examples and describe the parabolic manifolds as graphs of functions. We remark that Remark 3.7. Let F be a map of the form (3) satisfying the hypotheses of Theorem 3.3. Then, the graph invariance equation with the condition Φ(0) = 0, DΦ(0) = 0, has a γ-Gevrey solution if and only if the Let h be the γ-Gevrey conjugation provided by Corollary 3.6. Then is invertible around the origin and its inverse is Gevrey.
The same happens at a formal level. In this case, only the hypotheses of Theorem 3.1 are required.
The statements in this section provide upper bounds to the coefficients of the formal solution of the invariance equation. In the next section we will give examples that show that our results are sharp but also examples that show that a map satisfying our hypotheses can have an analytic invariant manifold. To provide conditions that ensure the existence of lower bounds of the coefficients remains an open problem. The following proposition shows that these conditions cannot depend only on a finite number of coefficients of the Taylor expansion of F at the origin.
Then, for any p ≥ 2, there exists an analytic map G such that with graph ϕ ≤p as invariant manifold to the origin. If p ≥ max{N, M }, G satisfies the hypotheses of Theorem 3.1 and, consequently, graph ϕ ≤p is a parabolic invariant manifold.
We defer the proof of this proposition to Appendix B.
In the following section we consider some examples. It is often easier to provide examples of maps arising from flows. The following remark is straightforward, but allows us to apply our results directly to flows.

⎞ ⎟ ⎠
Assume that the functions f N , f ≥N +1 , g M , g ≥M+1 , h ≥2 satisfy the hypotheses in Section 2 for all t ∈ [0, T ] and that 0 ∉ Spec D. Then, any stroboscopic Poincaré map ofξ = X(ξ, t) has the form in (3) with the same a, B 1 , B 2 and C = e T D .

Examples
In this section we provide several examples. In particular we show that, under the hypotheses of Theorem 3.3, the parabolic manifold (and, consequently, the formal solution) is indeed γ-Gevrey and not more regular, that is, it is not γ ′ -Gevrey for 0 ≤ γ ′ < γ.
It is more convenient to work with differential equations and manifolds represented as graphs. That is, for a given periodic in time system X(x, y, z, t) of the form (9), we look for formal solutions (y, z) =Φ(x, t), depending periodically on t, of the invariance equation: It is often useful to use the following equivalent definition of a s-Gevrey series (see [Bal94]): a formal series ∑ n≥0 a n z n is s-Gevrey if there exist c 1 , c 2 > 0 such that a n ≤ c 1 c n 2 Γ(1 + sn), for all n ≥ 0. 4.1. Some elementary examples. The first one is a generalization of the ones in [BH08]. Here we add the variables corresponding to the eigenvalue equal to 1 but still require the presence of the hyperbolic directions.
Claim 4.1. Let X(x, y, z) be the autonomous vector field Assume that a, B 1 satisfy their corresponding conditions in Theorem 3.1. Then Then ψ n = 0 if n ≠ ℓ + k(N − 1) and , then,ψ is exactly of the Gevrey order claimed in Theorem 3.1. Therefore, no matter what is the Gevrey order ofφ, the asymptotic seriesΦ is γ-Gevrey.
Now we consider the case M < N . The invariance of the formal solutionφ( In the same way as in (12), it follows thatφ is Gevrey of order γ = 1 (N − M ).
We emphasize that, when M < N , the map defined by (11) has a Gevrey formal solution of order precisely γ = 1 (N − M ) even if d ′ = 0, that is, even if F is tangent to identity, but the same claim (for this particular example) only holds for M ≥ N if d ′ ≥ 1. In the next subsection we will deal with the case M ≥ N and d ′ = 0, which is the relevant one in the problems of celestial mechanics we will consider in Section 4.3.
4.2. The tangent to the identity case (d ′ = 0) when M ≥ N . In this section we present a family of differential equations of the form (9) having a formal solution of the invariance equation (10). We check that this formal solution is precisely γ-Gevrey. Recall that in this case γ = 1 (N − 1).
The example we will consider will be given by a non autonomous time periodic vector field. The reason is because if the vector field is autonomous, the parabolic invariant manifold is analytic (when written as a graph), as the following claim shows.
Claim 4.2. Assume M ≥ N . Let X be an analytic vector field of the form ), a ≠ 0 and B 1 satisfying the condition stated in Theorem 3.1. Then, the invariance equation (10) has a real analytic solution ϕ ∶ B ρ ⊂ C → C d tangent to the x-axis at the origin. As a consequence the real analytic maps with a ≠ 0, which are the time 1 map of systems like (13), have an analytic solution of the graph invariance equation (8).
Proof. The one dimensional invariant manifold we are looking for is the graph of a function y = ϕ(x) satisfying the equation We introduce the new variable u by y = xu. The system becomes In addition, since f and g are analytic functions at (x, y) = (0, 0) and M ≥ N , so are the functions We consider now the system The origin is a fixed point, having a single hyperbolic direction corresponding to the eigenvalue −a. Indeed, when M > N , the linear part of the field in (17) at which may be not diagonal. Using the non-resonance condition −a ∉ Spec (B 1 +laId) if M = N and that a ≠ 0 if M > N , one deduces from the theory of nonresonant invariant manifolds ( [CFdlL03]) that the corresponding to the eigenvalue −a onedimensional invariant manifold is the graph of a real function h, analytic at x = 0, which is a solution of (16). Let h( is a real analytic solution of (14) tangent to the x axis.
Claim 4.3. Let X be the 2π-periodic vector field defined by with a, b > 0. The parabolic stable manifold has a formal Taylor expansion at 0 which is Gevrey of order exactly γ = 1 (N − 1).
Proof. We first note that Theorem 3.3 and 3.4 assure the existence and uniqueness of the parabolic stable manifold when a, b > 0. For any initial conditions x 0 , y 0 , t 0 , the associated flow is given by where we have introduced β = b a. Since we are looking for the stable invariant manifold, we want the solution such that (x(t), y(t)) → (0, 0) as t → ∞. Hence, since β > 0, we need to impose Therefore the stable invariant manifold is described by Notice that ϕ is 2π-periodic with respect to t. Now we will prove that the series of . We take θ ∈ (0, π) and change the integration path in the above integrals as: It is well known that these integrals define the confluent hypergeometric functions Ψ (see [EMOT53,p. 280 By [EMOT53,p. 302], an asymptotic expansion of Ψ(a, c, z) for large z is with (a) n = Γ(a + n) Γ(a). Therefore, the Taylor formal series at 0 of ϕ iŝ This formal series is Gevrey of order γ = 1 (N − 1). Indeed, comparing Γ(k + σ) with Γ(1 + γ(N − 1)(k + 1)) = Γ(k + 2) we conclude thatφ is a Gevrey formal series of order exactly γ.

4.3.
Aplications to Celestial Mechanics. The three body problem describes the motion of three point bodies evolving under their mutual Newtonian gravitational attraction. The restricted three body problem is the simplification of the three body problem obtained by assuming that one of the bodies has zero mass. Consequently, the bodies with mass, usually called primaries, describe Keplerian orbits. See, for instance, [MH92]. Among the several instances of the restricted three body problem one finds the Sitnikov problem, which is the special case when the primaries move in ellipses and the massless body in the line orthogonal to the plane of the primaries through their center of mass. The relevant parameter in the Sitnikov problem is the eccentricity e of the orbits of the primaries. When e = 0, the Sitnikov problem is integrable. Another important subproblem is the so called restricted planar three body problem (RPTBP), when the massless body moves in the plane where the primaries lie, while the latter describe Keplerian ellipses. In this case, a relevant parameter is the mass ratio of the primeries, µ, which can be assumed to be in [0, 1 2]. When µ = 0, the RPTBP is integrable.
In both cases, the parabolic infinity can be written as where (x, y 1 ,ỹ) ∈ R × R × R n and O k stands for a function in (x, y 1 ,ỹ, t), 1-periodic with respect to t, analytic in a neighborhood of x = y 1 = 0,ỹ = 0 and of order O( (x, y 1 ,ỹ) k ). In the case of the Sitnikov problem, n = 0, while n = 2 in the RPTBP. See [Mos73] for the derivation of the above equations in Sitnikov problem and [GM + 17] in the planar restricted three body problem. It is immediate to check that any stroboscopic Poincaré map of the system (18) has the form which has the form (3) with Consequently α = 1 3. Since the eigenvalues of B 1 are positive, Theorems 3.1, 3.3 and 3.4 apply. Hence we have Corollary 4.4. The parabolic infinity in the Sitnikov problem (for any e ∈ [0, 1)) and in the RPTBP (for any µ ∈ [0, 1 2]) possesses invariant manifolds which are 1 3-Gevrey.
As we have already mentioned, Theorems 3.1, 3.3 only provide upper bounds on the coefficients of the expansion of the invarariant manifold. However, in view of Martínez and Simó's numerical computations [MS14] and the example in Claim 4.3, where a time periodic perturbation of a system with a parabolic fixed point is considered, we present the following conjecture.

Formal parameterization of the manifold
In this section we obtain a formal solution of the equation F ○K = K○R, that is, a formal series which solves the equation at all orders. We will need also a precise expression of the coefficients in order to obtain Gevrey estimates for them.
We will use the following notation, that arises from the Faà-di-Bruno formula. Assuming that f and g are two C ∞ functions such that f ○ g makes sense, f (0) = 0 and g(0 Here f k and g k are k-multilinear symmetric maps. This expression also holds when dealing with the composition of formal power seriesf (w) = ∑ l≥1 f l w l and g(v) = ∑ l≥1 g l v l . The coefficient of the l order term of the formal composition f ○g is given by (19). It depends only on f ≤l (w) = ∑ l k=1 f k w k and g ≤l (v) = ∑ l k=1 g k v k . The only term of (f ○ĝ) l in which f l appears is f l g l 1 , and the only term in which g l appears is f 1 g l .
We introduce the maps for l ≥ 2, the family of operators Proposition 5.1. There exists a unique b ∈ R such that for any c ∈ R there exists a unique formal power The coefficients of K and R can be given inductively. For l > 1 we have In addition, if 1 ≤ l ≤ M − L + 1, K y l = 0. Proof. First we prove by induction that there exist a formal series K = ∑ n≥1 K n t n , K n = (K x n , K y n , K z n ) ⊺ ∈ R 1+d+d ′ and a polynomial R(t) = ∑ n0 n≥1 R n t n , R n ∈ R, with as much as possible coefficients equal to 0, such that the error To deal simultaneously with both cases we introduce P (l) as Note that P (l − 1) + 1 ≥ P (l) and that P (l) = max{M + 1, l + L}.
We can write E l (t) = ∑ n≥1 E l n t n , with E l n ∈ R × R d × R d ′ . We denote by E l,x l+N −1 , E l,y P (l) and E l,z l the first non-zero terms of (E l,x , E l,y , E l,z ) respectively. From the proof it will become clear that E m,x l+N −1 , E m,y P (l) and E m,z l actually do not depend on m provided m ≥ l − 1. We will simply denote them by E x l+N −1 , E y P (l) and E z l respectively. These values are the ones which appear in the statement. Taking R(t) = t − at N + O(t N +1 ) and K 1 = (1, 0, 0) ⊺ the claim holds true for l = 1 because Now, let l ≥ 2 and assume that there exist polynomials K ≤l−1 of degree at most l − 1 and R ≤l+N −2 of degree at most l + N − 2 such that We remark that the value of the constant b = R 2N −1 will be determined at the step l = N . In addition, we assume that K y By the induction hypothesis, Now we identify the lowest order terms in (26), (27), (28) and (29). Using that l ≥ 2 we easily estimate (26) Concerning (27), taking into account that K 1 = (1, 0, 0) ⊺ , As for (28), taking into account that K y j = 0 if 1 ≤ j ≤ l − 1 ≤ M − L + 1, which implies that Finally we evaluate (29) From the above calculations, since l ≥ 2, we have This expression permits to choose (K x l , K y l , K z l ) and R l+N −1 in order to E l has the claimed order. We start dealing with the third component. We have to take We write c = K x N which can be chosen arbitrarily. We recall that R 2N −1 corresponds to the coefficient b.
Now we come to compute E x l+N −1 , E y l+L−1 and E z l+L−1 . By definition, E z l is the term of order l of π z E l−1 = F z ○ K ≤l−1 − K z ≤l−1 ○ R ≤l+N −2 , that is,

By the Faà di Bruno formula (19),
In the first term of (30) the addend with k = 1 vanishes because K z 1 = 0. Moreover, for k ≥ 2, F z k = G z k . In the second term, the addend with k = 1 also vanishes.
the addend with this k vanishes because then l < k + N − 1 and the next non-zero term after order k is of order k + N − 1. This proves formula (24). Analogously, Applying again Faà di Bruno's formula we obtain We begin by determining the indices in (31) that provide non-zero terms in E y l+L−1 . The term with k = 1 in the first addend (31) vanishes because it would be F y 1 K y l+L−1 , but for E l we are working with K ≤l−1 . Moreover, since F y k = 0 if 2 ≤ k ≤ M − 1, the sum must start with k = M . Also, M ≤ k ≤ l 1 + ⋯ + l k = l + L − 1 implies l ≥ M − L + 1. In addition, when l = M − L + 1, we always have that Therefore, if l = M − L + 1, k = M and l 1 = ⋯ = l M = 1. Since K 1 = (1, 0, 0) ⊺ , the corresponding term is Then if l ≤ M − L + 1 the first term is void. To finish with the first term, we note that for all i, using again that k ≥ M , that is, the first term of (31) has the form claimed in formula (23). With respect to the second term of (31), we only need to note that K y k = 0 for 1 ≤ k ≤ M − L + 1, and analogously as before, that This ends the proof of formula (23) for E y l . To check formula (22) for E x l+N −1 we use the form of F x (x, y, z) = x − ax N + G x (x, y, z) and the proof follows the same lines as the one for E y l+L−1 .

Gevrey estimates
Before starting to obtain the Gevrey estimates of the formal solution K we perform two change of coordinates. The first one is a close to the identity change that uses the (N − 1)-degree approximation of the formal parabolic curve obtained in Proposition 5.1 to put it closer to the x-axis. In the new variables the parameterization will be the embedding to the x-axis plus terms of order at least N .
The structure of this section is quite similar to the counterpart in [BH08], however, there are some differences to take into account.
(2) The formal solutionK andR ofF ○K −K ○R = 0 obtained applying Proposition 5.1 toF satisfies (3) The Gevrey character is not affected by this change, i.e., if one of K orK is Gevrey of some order the other is also Gevrey of the same order.
The proof of this lemma depends on cumbersome but straightforward computations and uses, among other properties, that K y 1 = ⋯ = K y M−L+1 = 0. Next we perform a rescaling of parameter λ to achieve a good control on the growth of the terms K l of the formal solution up to some suitable order l 0 so that we can start an induction procedure to estimate the terms K j from l 0 on and obtain a significantly simpler bound from them.
Let U ⊂ C 1+d+d ′ be the domain of a complex extension ofF . Let B(δ) be a ball of radius δ > 0 such that B(δ) ⊂ U.
We remark that σ ∶=b a 2 does not depend on the rescaling parameter λ.
Lemma 6.3. The matricesÃ l defined as in (21) withB 1 instead of B 1 , after the changes of variables in Lemmas 6.1 and 6.2, The following technical lemmas are slight variations of lemmas in [BH08]. For the reader's convenience, we state and prove them.
Lemma 6.4. Let k, ν ∈ N, ν ≥ k and β ≥ 1 N −1 . Let also Proof. Note that R k,ν is the coefficient of t ν of the polynomial (t −ãt N +bt 2N −1 ) k . Then, we can rewrite it as The conditions on the indices m 2 , m 3 in the previous formula imply (N − 1)m 2 + 2(N − 1)m 3 = ν − k, that is, When ν = k, m 2 = m 3 = 0 and m = 0. Then R k,k = 1 and J 1 k,k = k! β . If m ≥ 1, we reduce (33) to a sum with a single index as Using Finally, since k = ν − (N − 1)m, using that m ≥ 1 and that where we use that (N − 1)β ≥ 1.
The next lemma collects two technical results on bounds of some products of factorials.
Lemma 6.5. Let N ≥ 2, β ≥ 1 N −1 and N β = N β(N −1) . i) Let k ≥ 1, ν ≥ kN and If ν < kN , the sum in (34) is void and we define M k,ν = 0. We have ii) Let k ≥ 1, ν ≥ k and If ν < k the sum in (35) is void and we define J 2 k,ν = 0. We have Proof. i) If kN > ν, one has that M k,ν = 0. Let us assume that kN ≤ ν. One can check that, if a, b, c ∈ N with b ≤ c, then (a + b)!c! ≤ b!(a + c)!. Therefore, for l 1 , l 2 , ⋯, l k ≥ N such that l 1 + ⋯ + l k = ν one has that l 1 !l 2 ! ≤ N !(l 1 + l 2 − N )!, that and applying this procedure recursively we get On the other hand it is clear that Finally the bound in i) follows because β(N − 1) ≥ 1.
ii) For k = ν, J 2 k,ν = 1 and the bound is obvious. Assume that ν > k. Then, and the proof is complete. Now we are going to prove thatK(t) = ∑ l≥1Kl t l is Gevrey of order γ. Recall that γ was defined in (4).

A solution of the invariance equation
In this section we prove Theorem 3.3, that is, there exists a real analytic function which is a true solution of the invariance equation in an appropriate domain and it is γ-Gevrey asymptotic to the formal solutionK found in the previous section.
We will use some basic properties about Gevrey functions. A summary of these properties can be found in [BH08]. See also [Bal94].
We begin by applying Borel-Ritt's theorem for Gevrey functions to the formal solutionK ([Bal94, p.17]). Let 0 < β < απ be an opening of a sector. Then there exist ρ small enough and a γ-Gevrey real analytic function, K e , defined on the sector S(β, ρ), which is γ-Gevrey asymptotic to the formal solutionK (see (7) for the definition of the sector). Then, being E a real analytic function on S(β, ρ) γ-Gevrey asymptotic to the identically zero formal series. As a consequence, for any closed sector there exist c 0 , c such that We look for a real analytic function H defined onS 1 such that For that we rewrite (37) as a fixed point equation. Let us to introduceĈ(t) and N as: Then the equation (37) becomes We introduce S 1 (β,ρ) = int(S 1 (β,ρ)) and the Banach spaces It is straightforward to check that, if H 1 , H 2 are C 0 functions inS(ρ,β) ∪ {0}, satisfying that H 1 (0) = H 2 (0) = 0, then, denoting ∆H(t) = H 1 − H 2 , To prove the above inequalities we take into account that K y e , K z e = O( t 2 ) as well as the form (3) of F . We observe that, by scaling the variable z, the norm B 2 is as small as we need. In addition, the matrix C does not change with this scaling.
We are forced to distinguish two cases according to the different values of M and N . 7.1. The case M ≥ N . Recall that we are assuming that Spec C ⊂ {z ∈ C ∶ z ≥ 1}. In this case we reinterpret (38) as the fixed point equation which is, essentially, the same as the one considered in [BH08]. A crux point is that, if H ∈ X 0,1+d+d ′ , then cos λ H 0 so that this term is contracting. Following the steps in the mentioned work, one can easily check that, taking 0 < β < απ and ρ small enough, the fixed point equation (40) has a unique solution belonging to the Banach space X 0,1+d+d ′ for anȳ ρ,β such thatS 1 (β,ρ) ⊂ S(β, ρ). As a consequence, the invariance condition (37) can be solved and the solution K e +H is analytic in the sector S(β, ρ) and α-Gevrey asymptotic to the formal solutionK.
7.2. The case M < N . When M < N the strategy developed for the case M ≥ N can not be applied. In this case bound (41) is not longer true. Indeed, as shown in Lemma 7.1 below (see also [BH08] . This implies that the term H ○ R is not contracting. For this reason we rewrite (38) as another fixed point equation. We recall that, when M < N , γ = 1 (N − M ) and we are assuming that Spec B 1 ⊂ {z ∈ C ∶ Re z > 0} and that Spec C ⊂ {z ∈ C ∶ z > 1}.
First we define an appropriate norm in C 1+d+d ′ . We take a norm in C d ′ such that C −1 d ′ < 1. Notice that, since Spec B 1 ⊂ {z ∈ C ∶ Re z > 0}, there exists a norm in C d such that Id − B 1 t M−1 d < 1 − µ t M−1 ≤ 1. This follows from the fact that Id − B 1 t M−1 is in Jordan form if B 1 is in Jordan form as well. Therefore, since K x e (t) = t + O(t 2 ), taking ρ small enough, If necessary, we will write (x, y) = max{ x , y d }.
We rewrite equation (38) as: 0). The usual way to proceed is: i) to find a formal inverse, S, of the linear operator G, ii) to prove that S is continuous in appropriate Banach spaces and iii) to write equation (42) as a fixed point equation and to apply the fixed point theorem.
The formal operator S = (S x , S y , S z ), acting on analytic functions T , defined by is the formal inverse of G. The proof of this fact is straightforward. We (formally) rewrite equation (42) as: To obtain accurate bounds for S, we need precise estimates on the convergence of the iterates R k (t) for t ∈ S(β, ρ).
Since R ν is the flow time 1 of the one dimensional equationu = −ανu N , i.e. R ν (u) = ϕ(1, u), then R k ν is the flow time k of the same equation, that is: Using that d du R ν (u) > 0, it is easy to prove by induction that R k (t) ≤ R k ν ( t ). To prove that R(S(β, ρ)) ⊂ S(β, ρ) is straightforward, see [BH08]. Now we deal with the linear operator S.
Lemma 7.2. Let 0 < β < απ 2, 0 < ν < a(N − 1) cos λ and ℓ, ℓ ′ ∈ R. If ρ is small enough, then, for anyβ ∈ (0, β) andρ ∈ (0, ρ), S is a well defined, linear and bounded operator from X ℓ,1+d × X ℓ ′ ,d ′ to X ℓ−M+1,1+d × X ℓ ′ ,d ′ . In addition, Proof. Let T be a function belonging to X ℓ,1+d+d ′ . Since S z (T z ) = C −1 T z , the claim is clear. We have that Let I(t) be the integral in the right hand side of the last inequality. By performing the change of variables we have that Integrating by parts we easily obtain and the claim is proven since we can take t < ρ small enough.
We deduce from this lemma that equation (37) is satisfied for H ∈ X −M+1,1+d+d ′ . Therefore, the function K = K e + H is a solution of F ○ K = K ○ R, analytic in S(β, ρ) and withK as its asymptotic γ-Gevrey series. This proves Theorem 3.3 for the case M < N .
Appendix A. Proof of Proposition 3.2 We write ϕ = ϕ ≤r + ϕ >r being ϕ ≤r the Taylor decomposition of ϕ up to order r. Note that ϕ(0) = 0. We also will use N ≤r and the notation introduce in Section 2.
Secondly, we observe that, using the mean's value theorem F v (u, v, w) = v (1 + f u (u, v, w)) m +B 1 (u, v, w)v + u n−mB 2 (u, v, w)w + o( u r−m ), B 1 ,B 2 being matrices with every entry of order at least O( (u, u m v, u n w) ). Note B 2 (u, u m v, u n w)w =B 2 (u, 0, 0)w + O( u m v 2 ) + O( u n w 2 ).
We have then that The result follows with M = min{M 1 , N } taking m + 2 ≥ max{M 1 , N }, n = m + max{0, M − M 2 } ≥ m and B 1 and B 2 adequately.
Since the matrix in the left hand side is invertible if x is small enough, ϕ z (x) = O( x ) N .
Performing analogous computations as the ones for F v and taking into account that ϕ z ≤r (x) = O( x N ) and that N z (x, y, 0) = O( (x, y) N ) one obtains that and the proof is complete since N + m − n − 1 ≥ 1.
Remark A.1. As a consequence of the proof, Proposition 3.2 holds true if F ∈ C r with r big enough (including the C ∞ case). In addition the map of the form (3) is also C r .