Convergence of the Birkhoff normal form sometimes implies convergence of a normalizing transformation

Consider an analytic Hamiltonian system near its analytic invariant torus $\mathcal T_0$ carrying zero frequency. We assume that the Birkhoff normal form of the Hamiltonian at $\mathcal T_0$ is convergent and has a particular form: it is an analytic function of its non-degenerate quadratic part. We prove that in this case there is an analytic canonical transformation -- not just a formal power series -- bringing the Hamiltonian into its Birkhoff normal form.


Introduction
The goal of this paper is to study the convergence of the transformations of an analytic Hamiltonian system in a neighborhood of an invariant torus to the Birkhoff normal form.
Here we assume that the frequency vector at the invariant torus is very resonant, hence already at the formal level, the existence of the Birkhoff normal form has obstructions. The main result, Theorem 1 below, will show that if the obstructions for the formal equivalence between the system and its Birkhoff normal form vanish and the normal form is convergent and has a particular form, then the system is analytically equivalent to its normal form. Hence, this result can be considered as a part of the rigidity program: identifying obstructions for a weak form of equivalence whose vanishing implies a stronger form of equivalence.
1.1 Classical theory of normal forms: existence and uniqueness. where θ ∈ T d = R d /Z d , I ∈ (R d , 0), ·, · denotes the usual scalar product in R d , and λ 0 ∈ R d is a constant vector called the frequency vector. The Hamiltonian system associated to it isİ = ∂ θ H(I, θ),θ = −∂ I H(I, θ). Note that we are assuming the standard symplectic form. In particular, the set T 0 := {0} × T d is an invariant torus of this system. We say that H(I, θ) has a Birkhoff normal form (BNF) N(I) in a neighborhood of T 0 if N(I) is a formal power series, and there exists a formal symplectic transformation Ψ(I, θ), tangent to the identity Ψ(I, θ) = (I + O 2 (I), θ + O(I)) such that H • Φ(I, θ) = N(I) in the sense of formal power series. Any canonical coordinate change Φ(I, θ) as above is called a normalizing transformation. The following fundamental result is called the Birkhoff normal form [SM71,MHO]. For H(I, θ) as above, assume that λ 0 satisfies a Diophantine condition: there exist constants (C, τ ) such that for all k ∈ Z d \ {0} we have (1.2) Then H(I, θ) has a (formal) Birkhoff normal form. Moreover, if a normal form exists and λ 0 is rationally independent, then the Birkhoff normal form is unique (up to trivial changes relabelling the actions). Note that the normalizing transformations are not unique, since composing Φ(I, θ) with any transformation that preserves I gives a normalizing transformation.
Birkhoff normal form is an important tool in the study of Hamiltonian systems. Already the assumption of existence and nondegeneracy of the normal form has strong dynamical consequences (see, e.g., [EFK15] Th.C). The importance of the BNF becomes even stronger if the normal form is convergent, and even more so if there exists an analytic normalizing transformation.
The standard way of constructing BNF, which we will review in more detail later, is to proceed iteratively, devising transformations that normalize H(I, θ) up to the coefficients of order I n . The normalization step involves solving differential equations with analytic conditions. The Diophantine conditions (1.2) can be somewhat weakened to subexponential growth (lim N →∞ 1 N log sup |k|≤N | λ 0 , k | −1 = 0). If λ 0 is resonant, one cannot guarantee the existence of the Birkhoff normal form even at the level of formal power series, since there may be some terms in the formal power series of H that cannot be eliminated by a canonical transformation. On the other hand, there are, of course, systems (e.g the BNF itself, or changes of variables from it) for which one can construct a BNF even in the resonant case. Then one speaks of the Birkhoff-Gustavson normal form [Gu66].  [Kri].
Up to very recently it was unclear which of the possibilities is actually realized. A large progress has been made by R. Krikorian [Kri], who proved that there exists a real analytic symplectic diffeomorphism f of a two-dimensional annulus such f (T × {0}) = (T × {0}), f (θ, 0) = (θ + ω 0 , 0) with ω 0 Diophantine and having a non-degenerate divergent Birkhoff normal form. Combined with the aforementioned result of Perez-Marco, this implies that Birkhoff Normal Form of an analytic Hamiltonian is "in general" divergent.
Concerning the normalizing transformations, H. Poincaré proved that they are divergent for a generic Hamiltonian. C. L. Siegel proved the same statement in a neighborhood of an elliptic fixed point (in fact, for a larger class of Hamiltonians than just generic, [Si54]). This is implied by showing that the orbit structure of the map in any neighborhood is very different from that of the Birkhoff normal form (which is integrable). Analogous results for symplectic maps near an elliptic fixed point appear in [Rü59 ]. Very different arguments showing divergence of normalizing transformations for generic systems appear in [Ze73] and for some concrete polynomial mappings in [Mo60].

Convergence of the transformations under the Diophantine conditions for some particularly simple BNF
There are classes of Hamiltonians for which we can guarantee the convergence of the normalizing transformation. The following influential rigidity result was proved independently by A. D. Bruno [Br71] and H. Rüssmann [Rü67]. Note that the main assumption is that the (in principle only formal) BNF is of a particular kind.
Then there exists an analytic normalizing transformation, and the BNF is, in fact, analytic.
We remark that Bruno proves the above result under a weaker condition on λ 0 than (1.2). For analogous statements in the case of invariant tori see [Br89]. Other modifications can be found in [Rü02,Rü04]. This result has been recently generalised to a much more general context by Eliasson, Fayad and Krikorian [EFK13,EFK15]. We stress that in all these works mentioned above, λ 0 is assumed to be non-zero and the crucial assumption is that λ 0 satisfies a Diophantine-type condition and that the BNF is of a very simple form.

"Sometimes" convergence of the BNF implies convergence of a normalizing transformation.
Our main result is close in spirit to the above works, but it does not rely on a Diophantine condition. In fact, we consider a special class of diffeomorphisms such that the frequency λ 0 is zero. Thus, the BNF is degenerate in the previous sense. But within this class of Hamiltonians we just use a standard non-degeneracy assumption on the quadratic part. Namely, we prove the following. Note that we start from a resonant torus, so that the existence of a BNF of the form we assume, requires vanishing of (formal) obstructions. Hence, our main result can be reformulated as saying that the formal assumptions imply convergence of the normalizing transformation.
Similar rigidity statements have appeared in other contexts. In [Po92, Ch. 5], H.Poincaré studied the formal power series of canonical transformations, which send a family of Hamiltonian systems into a family of integrable systems (in the sense of power series). In [Po92] it was shown that these formal power series do not exist unless there are some conditions (which are not met in the three body problem for arbitrary masses). The nonexistence of formal power series, a fortiori implies the non-existence of analytic families of analytic transformations integrating the three body problem.
The paper [Ll] proved a converse to the result in [Po92]: if the system satisfies a very specific and generic non-degeneracy condition, then, existence of a formal power series that integrates the family of transformations in the sense of power series implies existence of a convergent one.
Assumption A 3 is there for technical purposes, see Sec. 3.3. Note that it is trivial for d = 1. This assumption reminds of that of Rüssmann in [Rü67,Rü02,Rü04].
The assumption that the Birkhoff normal form is a function of N 0 has been discussed in [Ga] under the name of relative integrability. Two Hamiltonian dynamical systems are relatively integrable when one of them can be obtained from the other by a symplectic change of coordinates and a reparameterization of the time which only depends on the total energy. That is, the orbit structures of the two systems in an energy surface are equivalent up to a change of scale of time. The paper [Ga] includes several arguments for why the notion of relative integrability is natural when discussing formal equivalence. In the present paper, however, the focus lies on the notion of equivalence under a symplectic change of variables. We show that, for a certain class of systems, equivalence in the sense of formal power series implies equivalence in the sense of analytic canonical changes of variables. Hence, our main result can be understood as a rigidity result. The class of systems for which this rigidity result holds can be succinctly described as the set of systems that relatively integrable with respect to the main term.
In the context of formal equivalence implying analytically convergent equivalence, it is natural to formulate: Conjecture. Assume that an analytic Hamiltonian H(I, θ) as in (1.1) has a convergent BNF that satisfies the non-degeneracy assumption that the frequency map is a local diffeomormphism. Then there is a convergent normalizing transformation.
Note that the problems studied in [Rü67] and [Br71] do not satisfy the hypothesis of the conjecture, even though they satisfy the conclusions.
In the other direction one can construct examples [S] of analytic maps near a hyperbolic fixed point such that the Birkhoff normal form is quadratic (in the above notations, N = Λ 0 ) with a non-resonant set of eigenvalues, and any normalizing transformation to the normal form diverges. In these examples, the eigenvalues form carefully chosen Liouville vectors. That is, the paper [S] shows that, depending on the Diophantine conditions, quadratic normal forms may be rigid or not. The models in [S] do not satisfy the hypothesis of the conjecture above.

Overview of the proof.
The standard method of obtaining the Birkhoff Normal form is an iterative procedure in which we construct the transformations order by order: at the n-th step of the procedure one computes the n-th order terms in the Taylor expansions, assuming that all the terms of lower orders are computed. It would appear natural to follow this scheme and try to estimate the transformations at each step of the recursive procedure. Unfortunately, this seems technically unfeasible. One of the main complications in any possible proof of convergence of the transformations is that even if the BNF is unique, the formal transformations Φ N are very far from unique (Since the BNF depeds only on the actions, the Φ N can be composed with any canonical transformation which moves the angles but preserves the actions. So, an essential ingredient of any proof of convergence should be a especification of how to choose the normalizing transformations.
In this paper we use a quadratically convergent method in which we double the number of known coefficients at each step. Roughly -see more details in the next paragraphswe will show that if the formal obstructions vanish we can choose a sequence of canonical transformations that proceed to converge quadratically: doubling the order of the BNF at every step of the construction. More importantly, there is a specific choice of the transformation that satisfies very explicit bounds. The bounds on the new transformation in terms of the remainder turn out to involve a loss of derivatives. Therefore we need to implement a Nash-Moser scheme to estimating the important objects in a sequence of domains which decrease slowly.
Here is a short overview of the proof; all the necessary notations are introduced in the next section. At the n-th step of the iterative procedure we will start with a Hamiltonian of the form where N n (I) is a polynomial in I of degree m n = 2 n + 1, and the remainder term R n is small in the following sense: for a certain domain-dependent norm, introduced in Sec. 2.1.1, for a certain small δ n (we assume δ n → 0 with n → ∞) and κ > 0 the remainder term satisfies | R n | ρn,ρn ≤ δ κ n . At this step we construct a symplectic change of coordinates Φ n , such that where N n+1 has degree m n+1 = 2m n − 1, and | R n+1 | ρ n+1 ,ρ n+1 ≤ δ κ n+1 = 2 −κ δ κ n . We construct Φ n as a time one map of a the flow of a Hamiltonian vector field F n . The main ingredient consists in constructing and estimating the norm of F n (and thus Φ n ), which is found as a solution of a certain homological equation (see (3.1) and in a simplified form (4.1)). In general, this equation may not have even a formal solution unless some constraints are met. However, the assumption of Theorem 1 implies that this equation does have a formal solution. The key observation in this paper is the following: if this homological equation has a formal solution, then it also has an analytic solution with tame estimates for it (in the sense of Nash-Moser theory). This statement is the contents of Lemma 6. We note that the tame estimates use an argument different from the matching of powers.
The procedure can be repeated, because the main assumption used to show the existence of solutions of the Newton equation is that there is a formal solution to all orders. This assumption is clearly preserved if we make any analytic change of variables. Once we know that the Newton procedure can be repeated infinitely often, the convergence is more or less standard.
2 Notations and a step of induction.

Norms and majorants.
Let Let O(A ρ,σ ) be the set of functions holomorphic in A ρ,σ that are real symmetric, i.e., such that f (Ī,θ) = f (I, θ) (where the bar stands for the complex conjugate). We use supremum norms over A ρ,σ , denoted by f ρ,σ . In the same way we define the set O(D ρ ) with the corresponding norm f ρ being the sup-norms over the disc D d ρ .
For a function f ∈ O(A ρ,σ ) consider its Taylor-Fourier representation in the powers of I: Consider a majorant for f of the form We denote by |f | ρ,σ the norm of the corresponding majorant f (I): Clearly, f ρ,σ ≤ |f | ρ,σ . Analogous notation |f | ρ corresponds to the norm f ρ above.
In what follows we will mostly have σ = ρ.
2.1.2 Important constants for the iterative procedure.
The order of polynomials involved in the n-th step of the iterative procedure is m n = 2 n + 1.

Polynomials.
In the iterative procedure we will work with polynomials in I whose coefficients depend on θ.
We will say that a monomial M k,l = I k e 2πi l,θ is resonant if it satisfies {N 0 , M} = 0.
R [j] (I, θ) stands for a homogeneous polynomial in I of degree j with coefficients depending on θ: We also use notation R [m,n] to denote the range of degrees in I: Let m n be as above. The following functions will be of special importance.
The normal form N(I) is assumed to have the form 0 is quadratic. The rest term at the n-th inductive step is R n (I, θ): (2.4) We will also need polynomials in I with θ-dependent coefficients: R n (I, θ) and F n (I, θ) of the following degrees: such that Then for any a > 0 there exists a Hamiltonian H(I, θ) and a formal (resp., analytic) symplectic transformation Ψ(I, θ) = (I + O 2 (I), θ + O(I)) such that where | R 0 | 1 a ρ,σ ≤ aδ, and Proof. Define H(I, θ) = 1 a 2 H(aI, θ), and Ψ(I, θ) = 1 a φ(aI, θ), ψ(aI, θ) . It can be verified directly that Ψ is symplectic and tangent to the identity. Moreover,

Induction step.
While the base of induction is given by formula (2.12), the step of the iterative procedure is provided by the following proposition.
Proposition 1. For a fixed n > 0, let m n , ρ n and δ n be as in Sec. 2.1.2 above. Suppose that H n (I, θ) is formally conjugated to the BNF of the form (2.2): and the normal form satisfies:

Proof of Theorem 1.
Lemma 1 permits us to assume without loss of generality that for the given Hamiltonian H 0 (I, θ) := H(I, θ) = N 0 (I) + R 0 (I, θ), Since the function B is analytic, the same lemma permits us to assume that (2.6) and (2.7) hold for each n.
The step of induction is provided by Proposition 1. Since H n is formally reducible to the normal form N, the same can be said about H n+1 .
Repetition of this process leads to a sequence of transformations Let us show that T n converges to the desired coordinate change Φ = T ∞ , analytic in the polydisc A ρ∞,ρ∞ , where ρ 0 b < ρ ∞ < ρ 0 . Indeed, with the notations of Sec. 2.1.2, Then for any n we have It is left to prove that T n converges of to an analytic function T ∞ , satisfying (1.3). Denote the variables, involved in the n-th step of the induction by w n−1 = (I, θ) and w n = (I ′ , θ ′ ), where w n = Φ −1 n−1 w n−1 . In these notations, Now, for w n = (I ′ , θ ′ ) we have H • T n (I ′ , θ ′ ) = N n (I ′ ) + R n (I ′ , θ ′ ).
Since (Φ n (I ′ , θ ′ ) −(I ′ , θ ′ )) starts with the terms of degree 2 n in I ′ , for each j the expansion of (T n (I ′ , θ ′ ) − T n+j (I ′ , θ ′ )) starts with the terms of degree 2 n in I ′ . This implies that the sequence of maps T n formally converges, when n → ∞, to a formal map T ∞ such that (1.3) holds: We still need to show that T ∞ is analytic. It is more convenient to prove that the maps converge to an analytic map T −1 ∞ . By Proposition 1, the map w n+1 = Φ −1 n w n is analytic in A ρ 0 b/2,ρ 0 b/2 , and for all n we have: since ρ n − 3δ ≥ ρ n+1 > ρ 0 b for all n. Therefore, the map T −1 n such that w n = T −1 n w 0 is analytic in A ρ 0 b/4,ρ 0 b/4 , and for such w 0 we have implies the convergence of the sequence of maps T −1 n to an analytic map T −1 ∞ in A ρ 0 b/4,ρ 0 b/4 . Since the formal inverse of T −1 ∞ is the series T ∞ , the latter also defines an analytic function, providing the desired coordinate change. We set Φ = T ∞ in the notations of Theorem 1. ✷

Formal analysis.
Here we start the proof of Proposition 1 by the formal analysis of the iterative procedure.

Iterative Procedure.
Given H n as in Proposition 1, we will construct Φ n as the time one map of the flow of a Hamiltonian F n , i.e., Φ n = X 1 Fn where X t Fn is the flow defined bẏ I = F θ (I, θ),θ = −F I (I, θ).
In this case, Φ n is automatically symplectic.
Notice that the normalising transformation Φ n , as well as the corresponding generating function F n , is not unique (one can compose with rotations in the angles which preserve the actions, for example). Clearly, the transformation that converges has to be very carefully chosen.
In the following Lemma 2 we show that if a (formal) normalizing transformation exists, then there exists (another) normalizing transformation of a special kind. Namely, such that the corresponding generating function is a polynomial (in the sense of section 2. The idea of the proof is that we can always move the formal normalizing transformation by composing with some transformations that do not change the normal form. Therefore, we can ensure that the normalizing transformations belong to a space which is transversal to the space spanned by resonant monomials. Note that in the proof of Lemma 2 we use crucially the fact that the normal form is a function of N 0 so that the resonant terms are the same at all orders.
There are some analogies between Lemma 2 and Proposition 2.6 in [Ll], but that result is significantly less delicate since there is an extra parameter that controls the smallness. In our case, the variable I controls both the smallness and the distance to the origin at the same time.
Let {·, ·} denote the standard Poisson bracket. Recall that for a differentiable function G it holds: It is a classical fact that the composition Ψ • k in the sense of formal power series is the time-one map of another Hamiltonian given by the Cambell-Baker-Dynkin formula [Dragt, Appendix C], [LlMM, Appendix]; here we denote it by CBD formula. Note that in these references the usual notation for the Hamiltonian vector field defined by G is L G , and exp(L G ) stands for its time one map. In the present paper the same map is denoted by X 1 G . Now, suppose that Ψ = X 1 G and k = X 1 K . CBD formula implies that the composition of these maps satisfies: The last sum is to be understood in the sense of formal power series in I.
To prove Lemma 2, we use CBD formula, and choose K recursively (order by order in I) so thatG has no resonant terms up to order 2m. At each step of the recursion we choose (−K(I, θ)) to be equal to the lowest order resonant term of G, and setG to be the new G. As we saw above, the mapΨ = Ψ • K, used as a normalization map, brings H to the same normal form as Ψ did. But its generating HamiltonianG has no lower order resonant monomials. Iterating this procedure, we get a normalization with the desired property.
(2). Since we can normalise H = N m + R [>m] to N 2m with the help of the generating function G = O 2 (I), then, by (1), we can also achieve the normalization using the transformationΨ generated by a resonance-free HamiltonianG. Note thatG = O 2 (I).
By the Taylor formula for power series, we have: SinceG is resonance-free, any monomial P inG gives a non-zero impact {N 0 , P } to the sum above, whose order in I is strictly larger than the order of P . By comparing the orders of the coefficients in I we see that the lowest possible order of a monomial in {N 0 ,G} is the same as that in R [>m] , and henceG =G [≥m] . Finally notice that the reduced generating function F :=G [m,2m−2] produces the same normal form.

✷
The following lemma introduces the notations used in the proof of the Main Theorem.
Here we use the results of Lemma 2 to relate the conjugating function to the solutions of the homological equation (3.1) below.
Notice that the expressions for A n , B n , C n start with terms of order m n+1 + 1, and hence, , as needed.
Proof. Let m = m n = 2 n + 1. Then m n+1 = 2m − 1. With the notations for the degrees of polynomials from Sec. 2.1.3, Lemma 2 implies that there exists a polynomial F n = F [mn,m n+1 −1] n such that Φ n := X 1 Fn satisfies H n • Φ n = N n+1 + R n+1 . By the Taylor formula we have: (3.4) Notice that by extracting all the terms of orders m n + 1, . . . , m n+1 from the equation above, one gets the cohomological equation (3.1).

Homological equation order by order.
Here we rewrite equation (3.1) as a (finite) set of equations for each degree of I. Equations corresponding to degrees m n + 1, . . . , m n+1 will formally determine F n (they are written out explicitly in (3.5)). The rest of equations define C n (which is a part of the new remainder term). Equating coefficients with the same homogeneous degree in I in both sides of (3.4) we obtain for the degrees from m n +1 to m n+1 the following recursive formula (we write m instead of m n for typographic reasons): (3.5) Recall that 2m n − 1 = m n+1 , see Sec. 2.1.2. From the formal solvability we know that each of these equations has a formal solution F [m+j] n . Of course, such a solution is not unique. We will make the solution unique by prescribing the condition n (I, θ) = 0.
As we will see, this normalization will allow us to get the estimates needed for the proof of the convergence. The sum of the terms of orders m n+1 + 1, . . . , m n+1 + m n − 2 (i.e., 2m n , . . . , 3m n −3) that appear in equation (4.1) is denoted by C n . In the notation m = m n , we have: C n = C  This can be written more compactly as (3.7) This should be viewed as a definition of the remainder term C n .

An important simplification.
In the case when the normal form is an analytic function of N 0 (I) as in (2.2), we have an important simplification. Denote g 2j (I) := jb j (N 0 (I)) j−1 and g 2j+1 (I) ≡ 0. (3.8) Then for j ∈ N we have: We formulate this as a lemma: Lemma 4. If the normal form is an analytic function of N 0 (I) as in (2.2), then equation (3.5) is equivalent to (3.10) and (3.11)

Homological equations in majorants.
Here we study a simple recursive formula and estimate its terms. Later it will provide an important estimate of |{N 0 , F j }| ρn,ρn . Here is the idea: suppose that in the Lemma above for some ǫ > 0, for all j = 0, . . . , m we have: Define S j by the relations (3.12) below. Then, by Lemma 4, for all j = 0, . . . , m we have Lemma 5. Given ǫ > 0, suppose that for all j = 1, . . . , m − 1 the numbers P j satisfy 0 < P j ≤ ǫ.
Let S j be defined recursively by equations (3.12) Then for each j we have S j ≤ 2ǫ, j = 1, . . . , m − 1.
Proof. By the formula for S [j] above,

This implies
Formal solution provides analytic with estimates.
In this section we study a homological equation (4.1) below with an analytic right-hand side Q(I, θ). Assuming that it has a formal solution, we will find an analytic one, and estimate it in terms of the right hand side. Similar procedures appear in [Ll].
Lemma 6. Let N 0 (I) = I tr ΩI where Ω is a symmetric matrix with det Ω = 0, and let Q(I, θ) be analytic in an annulus A ρ,σ for some ρ, σ > 0. Suppose that the following equation has a formal solution F (I, θ): Then equation (4.1) has an analytic solution F (I, θ), defined in A ρ,σ , and for any 0 < δ < ρ, 0 < γ < σ we have: where c(d, Ω) is a constant only depending on d and Ω.
Moreover, if Q(I, θ) is a homogeneous polynomial in I with coefficients depending on θ, then so is F (I, θ).
Proof. Expanding F formally into a Fourier series: F = k∈Z d F k (I)e 2πi k,θ , we get: Recall that Ω is symmetric, so k, ΩI = Ωk, I . Expressing Q = k∈Z d Q k (I)e 2πi k,θ , we can rewrite equation (4.1) as a series of equations indexed by k: Q k (I) = 4πi Ωk, I F k (I). (4.2) If k, ΩI = 0, we can express F k = Q k (I)/(4πi Ωk, I ).
Since we have assumed existence of a formal solution of the homological equation (4.1) (and hence, a solution of (4.2) for each k), we have: Hence, for Ωk, I = 0, the equation is satisfied for any value of F k (I). We define F k at these points by continuity. A way to do it is the following. Differentiate equation (4.2) in the direction of Ωk: Ωk, ∇ Q k (I) = 4πi |Ωk| 2 F k (I) + Ωk, I Ωk, where for a vector v ∈ R d we denote |v| 2 = d j=1 v 2 j . For Ωk, I = 0, define F k (I) = Ωk, ∇ Q k (I) /(4πi|Ωk| 2 ). Summing up, we have defined a continuous function F k (I) by Now let us estimate the norm of the solution. Fix 0 < δ < ρ/2, 0 < γ < σ. For each fixed k ∈ Z d , we will estimate the corresponding F k (I) in two steps: first 'δ/2-close" to the resonant plane Ωk, I , and then in the rest of D ρ−δ .
By Cauchy estimates, we have: Since det Ω = 0, there exists a constant c(Ω) such that |Ωk| ≥ |k|/c(Ω) for all k. Then Finally, for small δ and γ we have: where c(d, Ω) is a constant only depending on d and Ω. The estimates above are very wasteful, but they are enough for our purposes. ✷ 5 Proof of Proposition 1.
Here we summarize the preparatory work to complete the proof of Proposition 1. Let us return to the original problem. For a fixed n, let the necessary constants be as in Sec.2.1.2, | R n | ρn ≤ δ κ n , and let g 2j (I) = j b j (N 0 (I)) j−1 as in (3.8).

Estimates for Φ n .
Here we prove that with F n as above, estimates (2.10) and (2.11) hold true. Indeed, the coordinate change Φ n = X 1 Fn is the time one map of the flow X t Fn defined by the equationṡ I = ∂ θ F n (I, θ),θ = −∂ I F n (I, θ).
5.4 Estimate of the new remainder R n+1 .
Proof. By Lemma 3, where A n , B n and C n are defined by (3.2) and (3.3).
Estimate of C n : We showed in section 5.1 that |C n | ρn,ρn ≤ δ κ n .
Notice that, by formulas (3.1) and (3.3), we have {N n , F n } = R n + N n − N n−1 + C n .