Perihelion librations in the secular three--body problem

A normal form theory for non--quasi--periodic systems is combined with the special properties of the partially averaged Newtonian potential pointed out in [15] to prove, in the averaged, planar three--body problem, the existence of a plenty of motions where, periodically, the perihelion of the inner body affords librations about one equilibrium position and its ellipse squeezes to a segment before reversing its direction and again decreasing its eccentricity (perihelion librations).


Introduction
This paper deals with certain motions of three point masses undergoing Newtonian attraction. More precisely, we study the case of two light bodies orbiting their common center of mass (a "binary asteroid system") while interacting with a heavier mass (a "planet"), whose position is external to their trajectories. As no Newtonian interaction can be neglected, there is no reason to claim that the system undergoes the "Keplerian" approximation, successfully used in the socalled "planetary" model 1 [4,7,11,13,5]. We think to the case that the time of mutual revolution of the lighter bodies is much shorter than the time scale of the motions of the massive body and that the ratio ε between the semi-axis of the ellipse of the asteroids and the position ray of the heavier one keeps to be less than 1 2 , so that collisions between the asteroids and the planet are not possible (a body moving on a Keplerian ellipse does not go beyond twice the semi-major axis from its focus). We look at the system from a reference frame centred with one of the asteroids or with their center of mass, so as to deal with an effective two-particle system, given by the other asteroid and the planet. We shall refer as "asteroidal ellipse" the instantaneous ellipse of this asteroid, focused on the center of the reference frame. We are interested to its motions. To simplify the analysis a little bit, we introduce three assumptions. The main one is based on the belief that, as long as the difference between the time scales persists, the system is "well represented" by a a certain "averaged" problem, which we call call secular problem. We remark that such average is meant with respect to the proper time of the asteroid, so it should not be confused with the homonymous procedure often studied in the literature (e. g. [8]), where the Keplerian approximation is used for two particles about their common sun, and the average is done with respect to both their mean anomalies. The secular system is simpler than the original problem, as we loose information concerning the position of the asteroid. In particular, collisions between the lighter particles are not observable. The degrees of freedom of the system are the motions of the eccentricity (or of the angular momentum) and of the pericenter direction of the asteroidal ellipse and the motions of the massive body. We associate to such system a certain "limiting system" which is similarly defined, but with the massive body being firm. From now on, we refer to such limiting problem as "unperturbed", and to the full secular problem as "perturbed". The terminology is here used with abuse, as we do not assume that the massive body has slow velocity in the full problem. For the unperturbed system, only movements of the asteroidal ellipse occur. By [15] the unperturbed problem turns out to be integrable, in the sense that it possesses a complete family of independent and commuting first integrals. More importantly, it reveals a surprising property, which we named renormalizable integrability. Such property (recalled for completeness in Section 2.2 below) offers a remarkable shortcut to the knowledge of movements of the asteroidal ellipse. By [16], in the case that the interacting particles are constrained on a plane (this is actually our second assumption), and ε < 1 2 , there are two stable equilibria such that the pericenter direction of the asteroidal ellipse in the unperturbed problem affords small oscillations about them, while its angular momentum oscillates about zero, affording a periodic change of sign. Physically, this means that the asteroidal ellipse is highly eccentric at any time and, moreover, there are two times ("squeezing times") in a period of oscillation of the pericenter when the eccentricity is equal to one. Namely, at those times, the ellipse becomes a segment. After the squeezing times, the eccentricity of the ellipse decreases while the sense of the motion is reversed. We call "perihelion librations" such kind of motions. The question which here we address is whether perihelion librations do persist in the full perturbed problem, when the massive body moves. We are able to give a positive answer to this question under our third assumption, which consists in taking the total angular momentum of the system (which is preserved during the motion) equal to zero. Under this assumption, the symmetries of the Hamiltonian ensure that the equilibria persist also in the perturbed problem, even though the integrability is lost. On the other side, such assumption is a source of difficulty, as the angular momenta of the two particles will simultaneously vanish, and hence collisions of the heavy body with the center of the system are to be controlled. We shall formulate our result below, after we have introduced some mathematical tool.
In terms of Jacobi coordinates 2 the three-body problem Hamiltonian with masses m 0 , µm 0 , κm 0 is the translation-free function Here, (y ′ , y, x ′ , x) ∈ (R 3 ) 4 (or (R 2 ) 4 ), · denotes Euclidean distance and the gravity constant has been taken equal to one, by a proper choice of the units system. We rescale impulses and positions multiply the Hamiltonian by 1+µ µ (by a rescaling of time) and obtain Likewise, one might consider the problem written in the so-called m 0 -centric 3 coordinates, and in this case the Hamiltonian is We apply an analogue rescaling, but with We arrive at We remark that in the case, of our interest, that κ ≫ µ ∼ 1, the above definition of Jacobi coordinates differs substantially from the usual one, because the barycentric reduction begins with one of the two lighter masses, rather than with the heavier one. A similar observation holds for the the m 0 -centric reduction, which here is not centred on the most massive body, contrarily to the usual convention. We look at the Hamiltonians H i in (2) and (5). As mentioned above, we make three assumptions.
2 If (y 0 , x 0 ) (y 1 , x 1 ), (y 2 , x 2 ) are impulse-position coordinates of m 0 , µm 0 , κm 0 , respectively, by "Jacobi coordinates" one usually means a linear, canonical change (y 0 , y 1 , y 2 , x 0 , x 1 , x 2 ) ∈ (R 3 ) 6 → (yc, y, y ′ , xc, x, x ′ ) ∈ (R 3 ) 6 defined so that xc = (x 0 + µx 1 + κx 2 )(1 + µ + κ) −1 is the center of mass of the system, while x, x ′ are, respectively, the mutual position of x 1 with respect to x 0 and the position of x 2 with respect to the center of mass of x 0 and x 1 : The new impulses (ycm, y, y ′ ) are uniquely defined by the constraint of symplecticity. The new Hamiltonian turns to be xc-independent, due to the conservation of yc. The dependence on yc can be eliminated choosing (as it is always possible to do) a reference frame where xc ≡ 0. See e.g., [10, §5.2-5.3] for more details.
A 1 ) If ℓ is the mean anomaly associated to the Keplerian motions of the term we replace the Hamiltonians (2) and (5) with their respective ℓ-averages A 2 ) The coordinates x, x ′ and the impulses y, y ′ are constrained on the plane R 2 ; A 3 ) The total angular momentum C = x ′ × y ′ + x × y of the system vanishes.
As mentioned above, the main assumption is A 1 ). It allows us to exploit facts highlighted in [15,16], as now we describe.
Since the H i 's in (7) are ℓ-independent, Λ is a first integral, hence the term − m 5 0 2Λ 2 may be neglected. After a further rescaling of time t → γt, we are led to look at the Hamiltonians H i in (7), which are given by is the ℓ-average of the Newtonian potential. Remark that y(ℓ) has vanishing ℓ-average 4 , so that the last term in (5) does not survive. In the case of the planar problem, after the reduction of rotation invariance, the Hamiltonians H i have two degrees of freedom. We use the following canonical coordinates R = y ′ · x ′ y ′ , r = x ′ G = x × y , g = anomaly of P with respect to a fixed direction a where P is the perihelion of (6), and the direction a, orthogonal to x × y, will be specified later. Note the coordinates above are fit to describe motions of Keplerian elements for (y, x), but not of (y ′ , x ′ ). If y ′ is set to zero, the H i 's reduce to sums of averaged Newtonian potentials, which are integrable, as do not depend on R. The function U β | β=1 has been thoroughly studied in [15]. Its phase portrait in the plane (g, G), while the ratio ε = a/r (where a = Λ 2 /m 3 0 is the semi-major axis associated to (6)) varies, is as follows.
(iii) Case ε > 1. The equilibria on the G-axis and the saddle persist. There is the birth of rotational motions (Figure 3).
The purpose of this paper is to prove that motions of the kind (i) do persist in H i , when y ′ = 0. Note that we shall not require y ′ small. To state the result, we introduce the following quantities, which will be used as mass parameters, at the place of µ and κ: where β and β are as in (3), (4), respectively. Observe that β * < β * and the case, of our interest, κ ≫ µ ∼ 1 corresponds to β * ∼ β * /2 ≫ 1. We shall prove the following result.
Moreover, the angle γ(t) between the position ray of Γ 0 (t) and the g-axis affords a variation larger than 2π during the time T .
A similar statement concerning perihelion librations about (0, π) holds true. The statement of Theorem 1.1 deserves two remarks. The former regards the motions involved, which are quasirectilinear, hence, close to be collisional. Generally speaking, for a three-body system composed of two asteroids and one planet, three kinds of collisions are possible: (1) collisions between the two asteroids; (2) collision between one of the asteroids and the planet; (3) triple collision. The system under investigation is, as stressed above, an averaged problem derived from the full above problem. For this averaged problem collisions of kind (1) or (3) do not exist, as the position of the asteroids is treated only in averaged meaning. Collisions of the kind (2) may exist, but they are to be intended as collisions between the planet and the average ellipse, rather than with a single particle on this ellipse. They are prevented by the assumption that the orbit of the planet is sufficiently far from the orbit of the asteroids. Namely, with a careful choice of the domain of the coordinates. Under this assumption, the averaged Hamiltonians H i keep finite. Incidentally, their regularity is studied in Proposition 3.3. During the proof of Theorem 1.1, in Section 5, the trajectory of the massive planet is controlled to keep outside the trajectory of the asteroid for all the time of a perihelion libration. The second remark concerns the thesis of Theorem 1.1. It holds in an open subset of phase space. In a sense, it recalls the statement of Nekhorossev's Theorem [12]. However, differently from it, Theorem 1.1 is not an application of perturbation theory, nor it uses trapping arguments. The reason is the following. In Section 4, we shall see that the manifolds are in fact invariant for H i . On such invariant manifold, by A 3 ), we have so Moreover, the functions U β , U −β , U β+β in (8) are asymptotic (as ε → 0) to 1 r . Hence, the motion of the coordinates (R, r) on M 0 and M π is ruled by an Hamiltonian asymptotic to This Hamiltonian generates unbounded (hence, non quasi-periodic) motions, for both positive and negative energies: For positive energies, both R and r are unbounded; for negative energies, only R is so. In any case, these motions are not quasi-periodic and hence we cannot apply the machinery of perturbation theory. In Section 4.1 we develop a theory suited to the case (see [9] for a result of the same kind). In this theory, no small denominators will arise, which is the reason why no trapping argument is needed. Before switching to full statements and proofs, we quote three open questions arising from the present setting.
Q 1 ) Let us consider the cases 1 2 < ε < 1 or ε > 1 (Figures 2, 3, respectively). Does the separatrix split so as to produce chaotic dynamics in the partially averaged planar problem?
Q 2 ) Again in the cases above, let us consider the full three-body problem. It has three degrees of freedom. Does the separatrix split so as to produce Arnold instability [2, 6]?
Q 3 ) What is the scenario in the case of the spatial problem?
This paper is organised as follows. In Section 2 we provide a review of the main results of [15,16]. In particular, we recall the mathematical formulation of the mentioned renormalizable integrability and we carry from [16] a set of action-angle like coordinates suited to our needs. In Section 3 we refine the analysis of [15] to the case of the planar secular problem, in the region of phase space where 0 < ε < 1 2 . In this case we are able to obtain simpler formulae compared to [15] and hence to study the regularity region of H i completely. In Section 4 we state a normal form theory without small divisors (Theorem 4.1), suited for a-periodic systems. The proof of Theorem 4.1 is deferred to Appendix A. In Section 5 we provide the proof of Theorem 1.1, as well as of a more precise version of it (Theorem 5.1 below), as an application of Theorem 4.1.
2 Review of the results of [15,16]

K coordinates
We describe canonical coordinates suited to our problem. We fix an arbitrary orthonormal frame For given m 0 > 0, fix a region of phase space (i.e., a set of values of (y ′ , y, x ′ , x)) where the Kepler Hamiltonian (6) takes negative values. Consider the motion generated by (6) with initial datum (y, x), and denote:   • a the semi-major axis; • P, with P = 1, the direction of perihelion, assuming the ellipse is not a circle; • ℓ: the mean anomaly, defined, mod 2π, as the area of the elliptic sector spanned by x from P, normalized to 2π.
Denote also: where "×" denotes skew-product in R 3 . Observe the following relations Let and assume Given three vectors i, i ′ and k, with i, i ′ ⊥ k, we denote as α k (i, i ′ ) the oriented angle from i to i ′ relatively to the positive orientation established by k. We define the coordinates The canonical character of K has been discussed in [15], based on [14].
Using the formulae in the previous section, we provide the expressions of the following functions ) which will be mentioned in the next section. They are: where a = a(Λ) the semi-major axis; e = e(Λ, G), the eccentricity of the ellipse, ̺ = ̺(Λ, G, ℓ), p = p(Λ, G, ℓ, g) are defined as with ξ = ξ(Λ, G, ℓ) the eccentric anomaly, defined as the solution of Kepler equation Also these formulae have been discussed in [16].

Renormalizable integrability
We recall some results concerning the functions U and E in (16). We refer to [15] for full details.

Proposition 2.1 ([15, Proposition 4])
If f is renormalizably integrable by g, then: (i) I 1 , · · · , I n are first integrals to f and g; (ii) f and g Poisson commute.
Observe that, if f is renormalizably integrable via g, then, generically, their respective time laws for the coordinates (y, x) are the same, up to rescaling the time. Formally:

Proposition 2.2 ([15, Proposition 5])
Let f be renormalizably integrable via g. Fix a value I 0 for the integrals I and look at the motion of (y, x) under f and g, on the manifold I = I 0 .
For any fixed initial datum (y 0 , x 0 ), let In particular, under this condition, all the fixed points of g in the plane (y, x) are fixed point to f . Values of (I 0 , g 0 ) for which ø(I 0 , g 0 ) = 0 provide, in the plane (y, x), curves of fixed points for f (which are not necessarily curves of fixed points to g).
The phase portrait of E in the planar case is as in Figures 1, 2 and 3, accordingly to the values of ε.

Asymptotic action-angle coordinates
In this section, we focus on the planar case, i.e., when y ′ , y, x ′ , x ∈ R 2 . In that case, the following the 8-dimensional diffeomorphism replaces K in (15): K 0 may be regarded as the natural limit of K, once Θ and ϑ are fixed to (0, 0), (0, π) (which are the values they take in the planar case), respectively, and (Z, z) are neglected. The functions U and E in (16) become where are a(Λ), ̺(Λ, G, ℓ), p(Λ, G, ℓ, g) are as in (17). Unfortunately, the action-angle coordinates associated to E are not explicit, since they are defined via inversion of elliptic integrals. However, it is possible to define, explicitly, action-angle coordinates associated to the leading part of E in the case of of large r: As discussed in [16], these coordinates, denoted as 5 (G, γ), are defined via the canonical 6 change for any fixed value of Λ. Observe that positive values of G (hence, k = 0) provide coordinates with the image (G, g) in a neighborhood of (0, 0); negative values (k = 1) are for (G, g) in a neighborhood of (0, π). Using these "approximate" coordinates, one obtains the expression of E as a close-to-be-integrable system for large r: The coordinates (G, γ) will be used in Section 5.

A deeper look at the planar case
In the planar case, the relation (23) becomes very special. Instead of U and E, it is convenient to switch to the functions  (15). We rewrite relation (23) as Here we have used that F ε does not depend explicitly on Λ, since both U and E depend on Λ only via G Λ . We claim that 5 Beware not to confuse the coordinate γ in (26) with its homonymous in (15). The latter is a cyclic coordinate for the Hamiltonians H i in (2) and (5), and hence has no rôle in the paper. 6 The change of coordinates discussed in [16] is a little more general than (26), since it is a four-dimensional map composed by (26) and In this paper we shall only use the two-dimensional projection (26).
To prove Proposition 3.1, we need to recall the following result from [15]. Then f renormalizably integrable by g through f , and f can be chosen to be for some fixed x i , y j .

t) looses its holomorphy if and only if
Proof We equivalently write Then we change variable, letting x = 1 − cos ξ. The integral becomes The only possibility it diverges is that there are two coinciding roots of the denominator on the real interval [0, 2]. The polynomial under the second square root has not coinciding roots on [0, 2] for |ε| < 1 2 and never has the root x = 0. The only possibility is that it has the root x = 2. This happens when t = ε + 1 4ε .

Remark 3.2
The formula in (31) (and its consequences below) is pretty specific for the planar case. In [15,Equation (49)] we proposed a general formula (holding for the planar and the spatial case), which, unfortunately, does not seem equally exploitable.

Set out and analytic tools
For definiteness, we refer to perihelion librations about (0, 0); the case (0, π) being specular. Using the identity in (30), the assumption A 3 in the introduction, which gives and the relation we rewrite the Hamiltonians (8) as having neglected to write (as well as we shall do below) the dependence on Λ.
We suddenly remark that H 1 and H 2 are both even with respect to G and g separately, because so are the functions E ε (G, g) and the 7 term G 2 2m0r 2 . Then the manifolds 7 Note that this circumstance would not hold without assuming A 3 , since, in that case, instead of the term G 2 r 2 , we would have which, by the discussions of Section 2, are invariant for E ε (G, g), keep to be so also for H i . We focus on M 0 . On M 0 , the motions of the coordinates (R, r) are governed by the Hamiltonians where (using (33) and dehomogenizating) The "potentials" V 1 and V 2 are well defined and increasing from −∞ to 0 for r > 2ā, r > 2(β +β), respectively, so, action-angle coordinates do not exist. In other words, closely to M 0 , H i has not close to an integrable system in the sense of Liouville-Arnold [1] and hence standard perturbation theory does not apply. In the next section, we develop an analytic theory suited to this case. It will be used to "decouple" the the Hamiltonians.

A normal form theory without quasi-periodic unperturbed motions
In this section we describe a procedure for eliminating the angles 8 ϕ at high orders, given Hamiltonian of the form H(I, ϕ, p, q, y, x) = h(I, J(p, q), y) + f (I, ϕ, p, q, y, x) which we assume to be holomorphic on the neighbourhood P ρ,s,δ,r,ξ = I ρ × T n s × B δ × Y r × X ξ ⊃ P = I × T n × B × Y × X , and J(p, q) = (p 1 q 1 , · · · , p m q m ) .
If φ is independent of x, we simply write φ ρ,r for φ ρ,r,ξ .
where Z ρ,s,δ,r,ξ , N ρ,s,δ,r,ξ are the "zero-average" and the "normal" classes respectively. We finally let ω y,I,J := ∂ y,I,J h. We shall prove the following result, where no-resonance condition is required on the frequencies ω I , which, as a matter of fact, might also be zero.

Remark 4.1 (Extensions)
(i) There is an obvious extension to the case that I ρ , T n s are replaced with (I 1 ) ρ1 × · · · × (I n ) ρn , T s1 × · · · × T sn . In this case, the number s in the former equation in (44) is to be replaced with min i {s i }. Moreover, the product ρ s in the definition of d is to be replaced with min i {ρ i s i }. Finally, the bound in (47) is to be changed taking into account the different sizes.

Outline of the proof
The complete proof of Theorem 4.1 is provided in the Appendix A, but here we aim spend some word, so as to highlight the main ideas. We proceed by recursion. We assume that, at a certain step, we have a system of the form H(I, ϕ, J(p, q), y) = h(I, J(p, q), y) + g(I, J(p, q), y, x) + f (I, ϕ, J(p, q), y, x) (48) where f ∈ O ρ,s,δ,r,ξ , g ∈ N ρ,s,δ,r,ξ . At the first step, just take g ≡ 0.
After splitting f on its Taylor-Fourier basis one looks for a time-1 map generated by a small Hamiltonian φ which will be taken in the class Z ρ,s,δ,r,ξ in (42). Here, denotes the Poisson parentheses of φ and f . One lets The operation φ → {φ, h} acts diagonally on the monomials in the expansion (49), carrying Therefore, one defines The formal application of Φ = e L φ yields: where the Φ h 's are the tails of e L φ , defined in Appendix A.
Next, one requires that the residual term −D ω φ + f lies in the class N ρ,s,δ,r,ξ in (43) for φ.
Since we have chosen φ ∈ Z ρ,s,δ,r,ξ , by (50), we have that also D ω φ ∈ Z ρ,s,δ,r,ξ . So, Equation (52) becomes In terms of the Taylor-Fourier modes, the equation becomes In the standard situation, one typically proceeds to solve such equation via Fourier series: so as to find φ khjℓ = f khjℓ µ khjℓ with the usual denominators µ khjℓ := l khj +iℓω y which one requires not to vanish via, e.g., a "diophantine inequality" to be held for all (k, h, j, ℓ) with (k, h − j) = (0, 0). In this standard case, there is not much freedom in the choice of φ. In fact, such solution is determined up to solutions of the homogenous equation which, in view of the Diophantine condition, has the only trivial solution φ 0 ≡ 0. The situation is different if f is not periodic in x, or φ is not needed so. In such a case, it is possible to find a solution of (53), corresponding to a non-trivial solution of (54), where small divisors do not appear. This is Multiplying by e ikϕ and summing over k, h and j, we obtain φ(I, ϕ, p, q, y, x) = 1 ω y In Appendix A, we shall prove that, under the assumptions (44), this function can be used to obtain a convergent time-one map and that the construction can be iterated so as to provide the proof of Theorem 4.1. The construction of the iterations and the proof of its convergence is obtained adapting the techniques of [17] to the present case.

Proof of Theorem 1.1
The purpose of this section is state and prove a more precise version of Theorem 1.1 (Theorem 5.1 below). We shall obtain it as an application of Theorem 4.1 to the Hamiltonians H i . Therefore, we need to introduce a change new coordinates C which put the H i 's in the suited form (41).
Since the potentials V i in (40) are, for large r, asymptotic to − m 2 0 r , it is convenient to rewrite the functions H i in (37) as and take C as the composition of two independent and canonical changes where C 1 is defined as in (26), with k = 0, while C 2 is defined via the formulae where ξ ′ (x) solves C 2 has been chosen so that Using the new coordinates, we have having abusively denoted as ε(y, x) the function ε(r(y, x)), and (using the formulae in (27)-(29) We now determine a domain of holomorphy of H i . Recall that we use, as mass parameters, the numbers β * , β * in (10). For the coordinates (y, x), we choose the complex domains Y √ with β * as in (10) and c 0 being such that for any 0 < ε 0 < 1 and for any x ∈ X √ ε0 , Equation (58) has a unique solution ξ ′ (x) which depends analytically on x and and verifies (the existence of such a number c 0 is well known). For the coordinates G, γ, we choose the domains G δ , T s0 , with 0 < δ < Λ, s 0 fixed, and G := G ∈ R : Λ − δ < G < Λ . Remark that, since, in H i , there is no dependence of G, but only on G 2 , G = Λ is a regular point for H i . Then we let We now check that, under the further assumptions H i are holomorphic functions on the domain By (62), the first equation in (60), and the expression of r(y, x) in (57), we have and hence, because of (61), By inequalities (65)-(66) and Proposition 3.3, we only need to check that, if ε * ∈ {βε, −βε} for i = 1 and ε * = (β + β)ε for i = 2, then Equation (34) with ε = ε * and t = E ε * (G, γ) has not solutions in D δ,s0, √ m 3 0 α−, √ ε0 . We prove that any such solution would verify |ε * | ≥ 1 4 , which would contradict (66), as |β * ε| ≥ |ε * |. Using (59), Equation (34) with ε = ε * and t = E ε * (G, γ) is We solve for ε * : We are now ready to state the result. Observe that the motions correspond, using the coordinates (G, g), to librations about (0, 0) if 0 < G 0 < Λ; about (0, π) if −Λ < G 0 < 0. We shall prove the existence, in H i 's, of motions close to these ones.
Theorem 5.1 Let α − , α + , β, β, δ, ε 0 and s 0 be fixed; β * , β * as in (10). There exist two numbers C * > C * > 1, both independent of α − , α + , β, β, δ, ε 0 , with C * possibly depending on s 0 , while C * independent of s 0 , such that, if the following inequalities are satisfied it is possible to find new coordinates (G * , γ * , y * , x * ) and a time T such that any solution Γ stays in D for all 0 ≤ t ≤ T and G * varies a little in the course of such time: If, in addition, then motions close to librations occur, in the sense that, also The time T can be taken to be Finally, the change is real-analytic and close to the identity, in the sense that Remark 5.1 (Proof of Theorem 1.1) Inequalities (67), (68) and (69) are simultaneously satisfied provided that the following holds. Fix 0 < ε 0 < 1 and 0 < δ ≤ Λ 4 once forever. Then identify δ as the size of U 0 and 2 −N0 as the size of V 0 . Take For short, we have written "a ✄ b" if there exist c > 1, independent of δ, Λ, α − , α + , a and s 0 such that a > cb. Note that here it is essential that α − , s 0 , β * and β * can be chosen arbitrarily large.
Proof During the proof we shall make extensive use of Cauchy 10 inequalities. We aim to apply Theorem 4.1, with I = G, ϕ = γ, (y, x) as in (57), h(y) = − m 5 0 2y 2 and, finally In our case, p, q do not exist and the unperturbed term h does not depend on I = G. Therefore, we have only to verify the last condition in (44). We have (having used ε 0 < 1) and with 11 C 1 , C 2 , C * independent of α − , α + , δ, β * , β * but C 1 possibly depending on s 0 while C 2 , C * independent of s 0 . We have choosen the number C * in (67) larger or equal than 2C 1 /C 2 and the number C * larger or equal than 3C 2 /2, so that C 1 Proceeding as in (72) and using Cauchy inequalities, one sees that inf |∂ G * f | ≥ c * β * m 2 0 a Λα 2 + . So, using (72) and N −1 with c * , c • independent of α − , α + , β, β, δ, ε 0 and s 0 . We then see that |γ * (T ) − γ * (0)| is lower bounded by 3π as soon as the condition in (69) is satisfied.

A Proof of Theorem 4.1
In this section we use the definitions and the notations introduced in Section 4.1, plus the following further ones.
Remark A.2 The right hand side of (81) benefits of the absence of small divisors, as no "ultraviolet" (i.e, with size ∼ e −N s f ) term appears.
Now we can proceed with the 13 The three first inequalities in (86) are immediately seen to be stronger that the corresponding three first inequalities in (79). On the other hand, rewriting the second inequality in (86) as δ ′ δ < 1 − 2δ ′ δ and using the inequality (which holds for all x ≥ 1) log x ≥ 1 − 1 x with x = δ 2δ ′ , we have also δ ′ δ < log δ 2δ ′ .
X i ℑ ω I ω y ρi,ri < s ′ i , X i