Chaotic scattering of He atoms off a Cu surface with corrugated Morse potential

arXiv:2401.05795v1 [math.DS] 11 Jan 2024 Chaotic scattering of He atoms off a Cu surface with corrugated Morse potential Florentino Borondo∗ , Ernest Fontich† , and Pau Martín‡ . January 12, 2024 Abstract We consider a Hamiltonian system that models the scattering of helium atoms off a copper surface. The interaction between the He and the Cu atoms is described by a corrugated Morse potential. Using corrugation coefficients values in the potential obtained by fitting to experimental values, we prove that, provided some coefficient of an auxiliary function is different from 0, there are regions of the phase space, corresponding to sufficiently large energy of the incident atom, where the scattering is chaotic. Furthermore, we prove that the system has oscillatory motions. Keywords: chaotic scattering, Hamiltonian systems, oscillatory orbits, exponentially small splitting, inner equation. Contents 1 Introduction 2 1.1 The He-Cu scattering problem. Main statement . . . . . . . . . . . . . . . . . . . . 2 1.2 Exponentially small splitting of invariant manifolds . . . . . . . . . . . . . . . . . . 4 1.3 Structure of the paper .................................. 4 2 Hamiltonian formulation of the He-Cu scattering problem and splitting of separatrices 5 2.1 McGehee-like coordinates and fast dynamics ...................... 5 2.2 Dynamics of H0...................................... 6 2.3 The Melnikov potential ................................. 8 2.4 Dynamics of the full system. Splitting of the invariant manifolds . . . . . . . . . . 9 3 Chaotic dynamics 9 3.1 Poincaré-Cartan reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Local coordinates around q=p= 0 and a parabolic λ-lemma . . . . . . . . . . . . 10 3.3 The local and the global maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.4 Symbolic dynamics. Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . 12 4 Hamilton-Jacobi equation 14 4.1 Notation .......................................... 14 4.2 Deriving the Hamilton-Jacobi equation . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.3 Definitions and technical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.4 Solution of the Hamilton-Jacobi equation (4.3) . . . . . . . . . . . . . . . . . . . . 18 ∗f.b[email protected], Departamento de Química, Universidad Autónoma de Madrid. †[email protected], Departament de Matemàtiques i Informàtica, Universitat de Barcelona (UB), and Centre de Recerca Matemàtica (CRM). ‡[email protected], Departament de Matemàtiques, Universitat Politècnica de Catalunya (UPC), and Centre de Recerca Matemàtica (CRM) 1 5 First extension of the invariant manifold 20 5.1 From Hamilton-Jacobi parametrization to flow parametrization . . . . . . . . . . . 20 5.1.1 Preliminaries and technical lemmas to solve equation (5.6) . . . . . . . . . . 21 5.1.2 Solution of equation (5.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Extension of the flow parametrization . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.3 From flow parametrization to Hamilton-Jacobi . . . . . . . . . . . . . . . . . . . . 29 6 The inner equation 31 6.1 Spaces and technical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.2 Fixed point equation ................................... 33 7 Approximation of the manifold in the inner domain 35 7.1 Spaces and technical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7.2 The fixed point equation ................................. 39 8 Difference between solutions of the inner equation 41 9 Difference of the solutions of the Hamilton-Jacobi equation 47 9.1 Straightening the linear operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 9.2 The exponentially small formula for Φ+−Φ−. . . . . . . . . . . . . . . . . . . . . 52 10 Acknowledgements 55 A Proof of Theorem 3.2 55 B Proofs of claims in Appendix A 59 B.1 Proof of Lemma A.1 ................................... 59 B.2 Proof of Lemma A.4 ................................... 60 B.3 Proof of Lemma A.5 ................................... 61 1 Introduction 1.1 The He-Cu scattering problem. Main statement We consider the motion of a helium atom bouncing off a copper surface. The problem arises from experimental techniques, where the scattering of noble gas atoms after collisions with a surface is used to characterize surface structures in a non-destructive way [Hul92]. The corrugation of the surface is modeled by the effective potential seen by the atoms as they come near the surface. Corrugation depends on the incident energy: the higher the energy of the He atom, the closer will get to the surface. Numerical evidence of the existence of chaotic scattering was given in [GBMA97,BGB+99], where it was related to some invariant manifolds in the system. The purpose of the present paper is to provide an actual rigorous proof of such chaotic behavior, thus establishing a link between experiments and theory on firm grounds. We simplify the problem assuming that the He atom motion takes place on a plane; actually the out-of-plane scattering is relatively small for the values of the energies and incident angles considered in the experiments. We denote by (x, z)∈R2the position of the He atom, where xand zare the horizontal and vertical displacements, respectively. Let (px, pz)be the conjugate momenta. We model the interaction of the He atom with the copper surface by a corrugated Morse potential, where the corrugation represents the presence of the copper atoms in the surface. More concretely, we consider the Hamiltonian HCM(x, z, px, pz) = 1 2m(p2 x+p2 z) + VM(z) + VC2πx a, z,(1.1) where VM(z) = De−αz(e−αz −2), VC(θ, z) = De−2αz V(θ), V(θ) = X n≥1 (rncos(nθ) + snsin(nθ)) . (1.2) 2 The coefficients rnand snare determined experimentally. Their values are r1= 0.06, r2= 0.008, rn= 0, n ≥3, sn= 0, n ≥1, D= 6.35 meV, a= 3.6Å, α= 1.05 Å−1. See [GBMA97]. In particular, r16= 0. In the present paper, our only requirement on Vis that it is an analytic even function, that is, sn= 0, for all n. We emphasize that we do not require Vto be a trigonometric polynomial. Evenness is not an important requirement and the results will hold with the same techniques for non-even V, but the added symmetry will simplify some of the details, particularly the numerical computations performed in the problem that are carried out in [BBF+23]. Since θ= 2πx/a appears in the equation through V, we will take θ∈T=R/2πZ. The purpose of this paper is twofold. On the one hand, we want to prove the presence of chaos in some parts of the phase space of (1.1). Here, the notion of chaos is the one introduced by Smale in [Sma65] and it is based on the presence of a Smale horseshoe and a conjugation with the shift on a space of sequences of symbols. It is worth to remark that the set where chaos takes place is a hyperbolic set. We recall the definition of symbolic dynamics with an infinite number of symbols, as introduced by Moser in [Mos73]. Let S={s= (...,s−1, s0, s1,...)|si∈N}be the space of two sided sequences of infinite symbols, with the topology induced by the neighborhood basis of s∗= (...,s∗ −1, s∗ 0, s∗ 1,...), Ij(s∗) = {s∈S|sk=s∗ k,|k|< j}, s∗∈S. It is well known that the shift σ:S→Sdefined by σ(s)i=si+1 is a homeomorphism. The shift σis transitive and the set of its periodic orbits is dense in S. It has sensitive dependence on initial conditions. It is one of the paradigms of chaos. The second goal of the paper is to prove the existence of oscillatory orbits, that is, solutions (x(t), z(t)) of (1.1) with the property that lim suptz(t) = ∞and lim inftz(t)<∞, that is, solutions that go higher and higher but always go back again to a finite distance of the Cu surface. The claims in the present paper follow the scheme proposed by Moser in [Mos73], which was also used in the restricted planar three body problem in [LS80]. In this last paper, an important technicality, not present in Moser’s work, appeared: the exponentially small behavior of the difference between the manifolds which give rise to the interesting dynamics. This problem was later overcome in [GMS16]. This exponentially small behavior also appears in the model under consideration here and dealing with it will represent an important part of our work. Our main theorem is the following. Theorem 1.1. There exists a function f1of the coefficients r1, r2,... such that if f16= 0, then, for any hlarge enough, there exists a section Σ⊂ {HCM =h}of the vector field associated to HCM and a subset I ⊂ Σsuch that the Poincaré map Ψ : I → I is a homeomorphism and is conjugate to the shift σof infinite symbols. The set Iis a hyperbolic set for Ψ. Furthermore, for any hlarge enough, the Hamiltonian HCM in (1.1)possesses oscillatory orbits. Remark 1.2. The function f1appears in a problem independent of the energy h. This equation is usually known in the literature as inner equation. The numerical evidence in [BBF+23] strongly suggests that for the experimental values mentioned above, f1is non-zero. Furthermore, we prove that if one replaces riby ε˜riin the definition of V, in (1.2), that is, if we consider the term VCas a small perturbation, then f1=επ 4˜r1+O(ε2). The rest of the paper is devoted to prove Theorem 1.1. In particular, it will be an immediate consequence of Theorem 3.5. We would like to remark that the hyperbolic set, as is the case of the Sitnikov problem considered in [Mos73] or the restricted planar three body problem in [LS80,GMS16], is related to the invariant manifolds of a certain periodic orbit at z=∞. In our case, since the equations of motion are ˙x=1 mpx, ˙z=1 mpz, ˙px=−2π aDe−2αzV′2πx a, ˙pz=−2Dα e−αz 1−e−αz 1 + V2πx a, (1.3) 3 we have that the set {(x, z, px, pz)∈R/aZ×R×R2|z=∞, pz= 0, px= (2mh)1/2}(1.4) is a periodic orbit of (1.3) at infinity in the energy level h. The proof of Theorem 1.1 consists in checking that this periodic orbit possesses invariant stable and unstable manifolds, that these invariant manifolds intersect transversally if his large enough and then prove a suitable λ-lemma that ensures that this transversal intersection gives rise to the standard isolating blocks with cone conditions. This last part is due to the fact that the periodic orbit (1.4) is neither hyperbolic nor elliptic, but degenerate. However, although degenerate, it possesses stable and unstable invariant manifolds. The study of the invariant manifolds of these type of degenerate objects goes back to [McG73]. See also [BFdlLM07], where the parametrization method is used, and the subsequent works, [BFM20a,BFM20b,BFM17,BFM20c]. In this case, when the periodic orbit is degenerate, the standard λ-lemma does not hold. In particular, it is not true that the forward images of a manifold intersecting transversally the stable manifold of the orbit accumulate to the whole unstable manifold. Hence, a different argument is needed to control the passage close to the periodic orbit. This was already known by Moser in [Mos73]. However, his proof does not directly apply to the present case, because the degree of degeneracy of our case is different from the one of the Sitnikov problem. We present another proof, based in the ideas in [GMPS22]. 1.2 Exponentially small splitting of invariant manifolds One of the main difficulties of the present work is to establish that the angle of intersection between the stable and unstable invariant manifolds of certain periodic orbits is non-zero. Indeed, since our goal is to deal with the physical problem, in which the corrugation, modelled by the function V in (1.2), is fixed, the only parameter we will have to deal with the problem will be the energy of the system. We will see that the angle of intersection is in fact exponentially small in the energy hence precluding the use of the standard Melnikov theory. Starting with the seminal paper of Lazutkin [Laz84] (see the English translation in [Laz03]), there is a long list of works in the literature concerning the exponentially small splitting of separatrices. In particular, the method introduced in [Sau01] has been essential in posterior developments of the field. See [BFGS12] and the references therein. Our approach here is similar to the the one established in [BFGS12] and [Bal06]. It is important to remark that the results in these last two papers do not apply to our setting. As a matter of fact, if one tries to write Hamiltonian (1.1), which has 2 degrees of freedom, as a 11 2degrees of freedom one using the Poincaré-Cartan reduction, the reduced Hamiltonian does not satisfy some of the hypotheses in [BFGS12,Bal06]. Particularly, when written as an integrable Hamiltonian plus a perturbation, the perturbation is not polynomial and has branching singularities when extended to the complex domain. This is true even in the McGehee variables that are introduced in Section 2. Since we are not assuming the perturbation to be small, we need to study the inner equation associated to the problem in order to capture the leading term of the exponentially small behavior of the splitting. This inner equation was used by Gelfreich in [Gel99] to prove the splitting of sepatrices in the Chirikov standard map. See also [MSS11a,MSS11b], where the McMillan map was studied by using resurgence theory. The study of the inner equation for the Hénon map was performed in [GS01]. The exponentially small splitting of invariant manifolds also appears in other physical problems, since it is related to the existence of some fast frequencies on the system. In particular, it has been dealt with successfully in some problem of celestial mechanics [GMS16,GMPS22,GPS23]. In these last cases, however, the Melnikov function predicts correctly the first order of the splitting. 1.3 Structure of the paper The structure of the paper is as follows. In Section 2we write the system in suitable McGehee coordinates, describe the geometric behaviour of it and claim the main theorem of the splitting of separatrices. In Section 3we apply the splitting theorem to deduce the existence of chaotic dynamics for the system. To do so, we need to cope with the problem that the periodic orbits under consideration are not hyperbolic but degenerate, which implies that the standard lambda lemma does not apply. 4 Here we use a version of the parabolic lambda lemma in [GMPS22] to obtain the conjugation to the Bernoulli shift. Sections 4to 9deal with the actual splitting of the invariant manifolds of the periodic orbit at infinity. In Sections 4and 5we obtain suitable approximations of the invariant manifolds by solving an appropriate Hamilton-Jacobi equation. These approximations are valid in a certain complex domain. However, we are not able to obtain enough information of their difference. To do so, we introduce the inner equation in Section 6, from which we obtain two solutions. In Section 7, we check that these solutions are also good approximations of the invariant manifolds in some region of their complex domain of definition. In Section 8we obtain an exponentially small formula for the difference of solutions of the inner equation which, in turn, we use in Section 9to obtain the exponentially small formula for the difference of the invariant manifolds. We have left to Appendix Athe proof of the parabolic lambda lemma (Theorem 3.2). Appendix B.1 contains the proofs of some technical lemmas used in the proof of Theorem 3.2. 2 Hamiltonian formulation of the He-Cu scattering problem and splitting of separatrices 2.1 McGehee-like coordinates and fast dynamics We are interested in motions for which the He particle arrives to z=∞with zero momentum. For this reason, we introduce the McGehee-like coordinates    Aq2=e−αz,a 2πθ=x, Bp =pz, CI =px, where A= 2, B =aα 4πC, C2= 2mD 8π aα2 . Without loss of generality, we take C > 0. This change transforms the standard 2-form dx ∧dpx+ dz ∧dpzinto the b-symplectic form (aC/2π)ω, where ω=dθ ∧dI −1 qdq ∧dp (2.1) and the Hamiltonian function HCM in (1.1) becomes 8D·H, where H(q, p, θ, I) = 1 2(νI2+p2)−1 2q2+1 2q4+1 2q4V(θ)(2.2) and ν=4π aα2 . For the experimental values of the He-Cu problem, the value of νis ν= 11.051879175935... Rescaling time, the dynamics of HCM is equivalent to the one generated by Hwith respect to the form ω. To study the behavior of the system for large values of I0, we introduce the new variable J by I=I0+Jin (2.2) with I0≫1. This change preserves the 2-form ω. We will denote the Hamiltonian in these new variables with the same letter, namely, H(q, p, θ, J) = H0(q, p, J) + H1(q, p, θ, J),(2.3) where H0(q, p, J) = 1 2(ν(I0+J)2+p2)−1 2q2+1 2q4, H1(q, p, θ, J) = 1 2q4V(θ). (2.4) 5 Since H0does not depend on θ, it is integrable. We remark that Hlooks like a periodically perturbed Duffing equation. However, since the vector field generated by His obtained through the non-standard 2-form ω, the system is not equivalent to the Duffing equation. Indeed, the equations of motion generated by Hare ˙q=−q∂H ∂p =−qp, ˙ θ=∂H ∂I =νI0+νJ, ˙p=−q−∂H ∂q =−q2+ 2q4+ 2q4V(θ),˙ J=−∂H ∂θ =−q4 2V′(θ), (2.5) which is not a Duffing oscillator due to the presence of the factor qin the (q, p)components of the vector field. We consider, at the energy level νI2 0/2of Hin (2.3), the set ΛνI0={q=p=J= 0, θ ∈T}. From (2.5), it is invariant and the dynamics on ΛνI0is given by θ=θ0+νI0t, that is, it is periodic with frequency νI0/(2π), which is fast when νI0is large. That is, at the energy level νI2 0/2, for large values of νI0,Hcan be seen as a fast periodic perturbation of the integrable Hamiltonian H0. We emphasize that the perturbation H1is not small since we are assuming that r1is fixed and we will consider motions in which qwill reach size of O(1). The proof of Theorem 1.1 consists in proving that each of these periodic orbits possesses invariant manifolds and that they intersect transversally. From this intersection we will deduce the existence of a horseshoe with infinitely many symbols. However, since the system is a fast periodic perturbation of an integrable system, it is well known that the angle of intersection of the manifolds will be smaller than any power of the inverse of the frequency which makes the question of computing it a beyond all orders phenomenon. Remark 2.1. Although H1is not small, it is well known that the fact that depends on a fast angle implies that its contribution averages out up to an exponentially small remainder. Indeed, we can apply a step of averaging and still obtain an explicit expression for the remainder, which is of size O((νI0)−1). It is immediate to check that the change θ= Θ, J =K+A′(Θ)Q4, q =Q, p =P+ 4A(Θ)Q4, preserves the 2-form ωin (2.1). In these new variables, Hbecomes e H(Q, P, Θ, K) =1 2(ν(I0+K−A′(Θ)Q4)2+ (P+ 4A(Θ)Q4)2)−1 2Q2+1 2Q4+1 2Q4V(Θ) =1 2(ν(I0+K)2+P2)−1 2Q2+1 2Q4+1 2V(Θ) −νI0A′(Θ)Q4 −νA′(Θ)KQ4+ 4A(Θ)PQ4+ 8A(Θ)2Q8+1 2νA′(Θ)2Q8. Taking Asuch that A′(Θ) = −V(θ)/(2νI0)and introducing W(Θ) = RΘV(ϑ)dϑ, we obtain e H(Q, P, Θ, K) = H0(Q, P, K) + e H1(Q, P, Θ, K), where e H1(Q, P, Θ, K) = 1 2νI0−νV (Θ)KQ4+ 4W(Θ)PQ4+1 4I0 V(Θ)2Q8+4 νI0 W(Θ)2Q8. The computations of the Melnikov potential in Section 2.3 below lead to a formula of the same order. We will not follow this approach. On the contrary, we will work directly with the Hamiltonian H=H0+H1in (2.3). 2.2 Dynamics of H0 In view of Remark 2.1, although H1is not small, we can see H0as a reference system for H. Here we describe its dynamics. First we observe that, since H0does not depend on θ,Jis a conserved quantity. Hence, H0is integrable. Using the 2-form ωin (2.1), the equations of motion of H0are ˙q=−qp, ˙ θ=νI0+νJ, ˙p=−q2+ 2q4,˙ J= 0.(2.6) 6 The set ΛνI0={q=p=J= 0, θ ∈T} is a periodic orbit in {H0=νI2 0/2}. From (2.6), as we have already mentioned, it is clear that the restriction of ΛνI0to {H0=νI2 0/2, J = 0}is not hyperbolic. However, it possesses invariant stable and unstable manifolds, that generate two homoclinic loops, given by W0 νI0=(q, p, θ, J)|J= 0, θ ∈T, H0(q, p, 0) = 1 2νI2 0. See Figure 1. We will focus on the loop with q≥0. The figure is slightly misleading: for any J, the set {q= 0, p =p0}is a periodic orbit. In particular, {q= 0}is invariant, even for the full Hamiltonian H. This means that, in fact, in the (q, p)-plane, (q, p) = (0,0) is not a topological saddle because, in particular, is not an isolated equilibrium point because q= 0 is a line of equilibria. As a consequence, the standard λ-lemma is no longer true and another statement will be necessary. The needed result is Theorem 3.2 below. We will be interested in the part of the phase space where q > 0. Figure 1: The projection of the invariant manifolds W0 νI0onto the (q, p)-plane. For each p=p0, {(0, p0, θ, 0) |θ∈T}is a periodic orbit. The following lemma, whose proof is a straightforward computation, provides the time parametrization of the right hand side homoclinic orbit. Lemma 2.2. The time parametrization of the homoclinic orbit to (0,0) of (H0)|J=0 that passes through (1,0) at t= 0 is given by qh(t) = 1 √1 + t2, ph(t) = −˙qh(t) qh(t)=t 1 + t2. As a consequence, Γ0(u, θ) =     qh(u) ph(u) θ 0    (2.7) parametrizes W0 νI0. Denoting φ0 tthe flow generated by H0, this parametrization satisfies φ0 t◦ Γ0(u, θ) = Γ0(u+t, θ +νI0t). 7 2.3 The Melnikov potential As usual, we define the Melnikov potential associated to the Hamiltonian Has L(u, θ) = −Z∞ −∞ H1(qh(u+t), ph(u+t), θ +νI0t, 0) dt. (2.8) Expanding Vin Fourier series, V(θ) = X k∈Z V[k]eikθ, by (2.4), we can write H1(q, p, θ, 0) = q4 2V(θ) = q4 2X k∈Z V[k]eikθ. Since Vis real analytic, even and has zero average, we have that V[k]∈R, V [0] = 0, V [k]=V[−k], k ∈Z\{0} and there exists σ0>0and K > 0such that, for all k∈Z, |V[k]| ≤ Ke−|k|σ0.(2.9) Hence, L(u, θ) = −Z∞ −∞ H1(qh(t), ph(t), θ −νI0u+νI0t, 0) dt =−X k∈Z eik(θ−νI0u)Z∞ −∞ q4 h(t) 2V[k]eikνI0tdt. That is, L(u, θ) = X k∈Z L[k](νI0)eik(θ−νI0u), where L[k](νI0) = −1 2V[k]Z∞ −∞ q4 h(t)eikνI0tdt. Proposition 2.3. The Melnikov potential Lsatisfies L[k](νI0) = −πνI0V[k] 4e−|k|νI0|k|+1 νI0, k ∈Z. In particular, since V[0] = 0 then L[0](νI0) = 0. Proof. Since, in view of Lemma 2.2,q4 his a meromorphic function that has poles of order 2at ±i, the claim follows from a straightforward residue computation. Corollary 2.4. If r16= 0 in (1.2), then L(u, θ) = −π 4νI0r1e−νI01 + 1 νI0cos(θ−νI0u) + O(e−νI0). If the coefficients riin the definition of Vin (1.2) satisfy ri=ε˜riand ˜r16= 0, the standard Melnikov theory implies that the distance between the unstable and stable invariant manifolds of ΛνI0is given by the derivatives of εL plus an error of size O(ε2). However, since Lhas size O(e−νI0), this means that at the energy level νI2 0/2,εL gives the leading order of the distance only if ε < O(e−νI0), that is, for a fixed value of the energy, the invariant manifolds of infinity split only if the corrugation is exponentially small in the energy. We will see that εand νI0can be taken as independent parameters while the exponentially small formula for the distance will remain valid. The physical implications are important: we will see that, for a given corrugation, if certain coefficient is different from 0(that depends on the coefficients of the corrugation and is generically non-zero), for any large enough energy, the invariant manifolds of infinity split. Moreover, this coefficient depends on first order on L[1] if εis small. It is important to remark that if r1= 0, the Melnikov function is of order e−2νI0. However, in this case, the true behavior, given by formula (2.11) in Theorem 2.5, below, can be of order e−νI0, if certain coefficient f1is different from 0. See [BMS23]. 8 2.4 Dynamics of the full system. Splitting of the invariant manifolds The periodic orbit ΛνI0={q=p=J= 0, θ ∈T}of H0remains when we consider the full system Hin (2.3), with the same frequency νI0. Although it is not hyperbolic, we will prove that it has invariant unstable and stable manifolds, which, if certain non-degeneracy condition holds, no longer coincide. Next theorem summarizes the claim. It is the first step in the proof of Theorem 1.1. Theorem 2.5. If νI0is large enough, the periodic orbit ΛνI0={q=p=J= 0, θ ∈T}possesses invariant unstable and stable manifolds, W± νI0. Moreover, for any 0< u0< u1, there exist analytic functions Φ±: [u0, u1]×T→Rsuch that Γ±(u, θ) =     qh(u) ph(u)−1∂uΦ±(u, θ) θ ∂θΦ±(u, θ)    (2.10) are parametrizations of pieces of W± νI0satisfying (1) Γ±= Γ0+O((νI0)−1), where Γ0is given in (2.7), (2) there exists f1,Λ0∈Rsuch that, for any j, k ≥0,0≤j+k≤3, ∂j u∂k θ(Φ+(u, θ)−Φ−(u, θ)−Λ0) = (−1)j(νI0)j+1e−νI02f1∂j+k αcos α|α=(θ−νI0u)+O1 log(νI0).(2.11) The coefficients f1and Λ0do not depend on νI0. They only depend on the coefficients riof the function Vin H1. If one assumes that ri=ε˜ri, i.e. the corrugation term H1is a small perturbation of H0, then f1=π 2˜r1ε+O(ε2).(2.12) Remark 2.6. For the experimental values of Vcorresponding to the interaction between helium and copper atoms, numerical experiments in [BBF+23] suggest that f16= 0. However, if ri=ε˜ri, we will see that f1is an analytic function of ε. Then, (2.12)implies that, if ˜r16= 0,f1can only vanish for a discrete number of values of ε. Corollary 2.7. If νI0is large enough, the invariant manifolds W± νI0of ΛνI0intersect transversely along two primary homoclinic orbits. Proof. Since W± νI0are contained in the level νI2 0/2of H, the expression of Γ±in (2.10) implies that their intersections are given by ∂uΦ+(u, θ)−∂uΦ−(u, θ) = 0. Then, (2.11) allows us to apply the standard implicit function theorem to obtain that the primary homoclinic orbits are given by θ−νI0u+O((νI0)−1) = kπ. Theorem 2.5 provides an exponentially small formula for the difference of the invariant manifolds W± νI0, with the energy of the system as parameter. We will find suitable approximations of the invariant manifolds of ΛνI0which extend to complex values of their variables. The leading term of these approximations, for both the stable and unstable manifolds, will be the unperturbed separatrix Γ0. This is done in Sections 4and 5. In many cases, this is enough to obtain the exponentially small formula, whose leading term is given by the Melnikov potential (2.8). Here, however, in order to capture the leading term of the exponentially small difference between the manifolds, we need to resort to what is often known as the inner equation. This is done in Section 6 and 8. In Section 7the manifolds are compared with the solutions of the inner equation and, finally, all the information is gathered in Section 9, completing the proof of Theorem 2.5. 3 Chaotic dynamics Here we define a return map on a suitable section such that it possesses an invariant hyperbolic set and when restricted to this set, it is topologically conjugate to the shift of infinite symbols. The construction is analogous to the one of Moser in [Mos73]. 9 equation (4.6) becomes L(Φ1) = F(Φ1).(4.9) We remark that F(0)(u, θ) = −H1(qh(u), θ) = −1 2 1 (1 + u2)2V(θ).(4.10) To solve (4.9), we will rewrite it as a fixed point equation by means of a suitable right inverse Gof L, to be defined later on, that is, Φ1=G ◦F(Φ1).(4.11) We will devote the rest of the section to prove the existence of a solution of (4.6) with the boundary conditions as u→ −∞ in (4.4). 4.3 Definitions and technical lemmas Let 0< β1< β2< π/2be fixed. For κ > 1and δ∈(0,1/2), we consider the complex domain D+ κ,δ ={u∈C| |ℑu|<−tan β1ℜu+ 1 −κ(νI0)−1,|ℑu|>tan β2ℜu+ 1 −δ.(4.12) We will be interested in the case where I0and κare big but satisfy κ(νI0)−1< δ. Observe that, if κ(νI0)−1< δ, then d(D+ κ,δ,±i) = Cκ(νI0)−1with C= cos β1. See Figure 2. Figure 2: The domain D+ κ,δ defined in (4.12). To solve equation (4.9), for r, s ∈Rwe introduce the Banach space of real analytic, 2π-periodic in θ, functions Xr,s ={Ψ : D+ κ,δ ×Tσ→C|Ψreal analytic,kΨkr,s <∞} and, taking into account the Fourier expansion Ψ(u, θ) = Pk∈ZΨ[k](u)eikθ, the norm kΨkr,s =X k∈ZkΨ[k]kr,se|k|σ, where, for an analytic function f:D+ κ,δ →C, kfkr,s = max   sup u∈D+ κ,δ,ℜu≤−u0|urf(u)|,sup u∈D+ κ,δ,ℜu>−u0|(1 + u2)sf(u)|  , where u0∈Ris chosen such that u0≥max{1,(1 −δ)/tan β2}. 16 Lemma 4.2. Let r, r1, r2, s, s1, s2∈R. We have (1) If Ψ∈ Xr+t1,s+t2with t1, t2≥0then Ψ∈ Xr,s and kΨkr,s ≤KνI0 κt2 kΨkr+t1,s+t2, where K > 0is independent of νI0and κ. (2) If Ψ1∈ Xr1,s1and Ψ2∈ Xr2,s2then Ψ1Ψ2∈ Xr1+r2,s1+s2and kΨ1Ψ2kr1+r2,s1+s2≤ kΨ1kr1,s1kΨ2kr2,s2. The proof of the previous lemma is straightforward from the ideas in [Sau01]. We also introduce e Xr,s ={Ψ∈ Xr,s |∂uΨ∈ Xr+1,s+1, ∂θΨ∈ Xr+1,s+1,TΨUr,s <∞}, where TΨUr,s =kΨkr,s +k∂uΨkr+1,s+1 +νI0k∂θΨkr+1,s+1. Given Ψ∈ Xr,s, with r > 0,s≥0, we formally define Gu(Ψ)(u, θ) = Z0 −∞ Ψ(u+ξ, θ +νI0ξ)dξ =Zu −∞ Ψ(ξ, θ +νI0(ξ−u)) dξ. (4.13) Clearly, when Gu(Ψ) is well defined, if Ψis real analytic, so is Gu(Ψ). The operator Guformally satisfies L◦Gu(Ψ) = Ψ, where Lis the differential operator introduced in (4.7). The next technical lemma will be the main tool to find the desired solutions of equation (4.11). Its proof is an immediate variation of the arguments in [GOS10]. Lemma 4.3. Let Gube the operator defined in (4.13). (1) If Ψ∈ Xr,s, with r > 1,s≥0, then Gu(Ψ) ∈ Xr−1,s and kGu(Ψ)kr−1,s ≤KkΨkr,s. If r, s > 0and hΨi= 0, then Gu(Ψ) ∈ Xr,s and kGu(Ψ)kr,s ≤K(νI0)−1kΨkr,s. (2) If Ψ∈ Xr,s, with r, s > 1,Gu(Ψ) ∈ Xr−1,s−1and kGu(Ψ)kr−1,s−1≤KkΨkr,s. (3) If Ψ∈ Xr,s, with r≥1,s > 0,∂uGu(Ψ), ∂θGu(Ψ) ∈ Xr,s and k∂uGu(Ψ)kr,s ≤KkΨkr,s,k∂θGu(Ψ)kr,s ≤K(νI0)−1kΨkr,s. As a consequence, if Ψ∈ Xr,s, with r, s > 1,Gu(Ψ) ∈e Xr−1,s−1and TGu(Ψ)Ur−1,s−1≤KkΨkr,s. The constant Konly depends on r, s and the constants involved in the definition of D+ κ,δ but it is independent of νI0and κ. 17 4.4 Solution of the Hamilton-Jacobi equation (4.3) Let Fand Gube the operators defined by (4.8) and (4.13), respectively. We formally introduce L+ out(u, θ) = Gu◦F(0)(u, θ) = −Zu −∞ H1(qh(s), θ +νI0(s−u)) ds =−Zu −∞ 1 2(1 + s2)2V(θ−νI0(s−u)) ds. (4.14) For later use we also introduce L− out(u, θ) = −L+ out(−u, −θ) = Z∞ u 1 2(1 + s2)2V(θ+νI0(s−u)) ds and Lout(u, θ) = L+ out(u, θ)−L− out(u, θ) = −Z∞ −∞ 1 2(1 + s2)2V(θ+νI0(s−u)) ds. Observe that Lout =L, where Lis the Melnikov potential introduced in (2.8). Proposition 4.4. The function L+ out ∈ X4,2in D+ κ,δ and satisfies kL+ outk4,2,k∂θL+ outk4,2,k∂uL+ outk5,3≤K νI0 , for some K > 0, independent of νI0and κ. Furthermore, hL+ outi= 0. Proof. We recall that F(0)(u, θ) = −H1(qh(u), θ) = −1 2q4 h(u)V(θ) = −1 2 1 (1 + u2)2V(θ), where H1was introduced in (2.4) and the function Vin (1.2). Clearly, F(0) ∈ X4,2and hF(0)i= 0. The bounds on kL+ outk4,2and on k∂θL+ outk4,2follow from (1) and (3) of Lemma 4.3, respectively. To obtain the bound on k∂uL+ outk5,3we observe that, by integration by parts, if k6= 0, ∂uZu −∞ 1 2 1 (1 + s2)2eikνI0(s−u)eikθ ds =−2Zu −∞ s (1 + s2)3eikνI0(s−u)eikθ ds. Hence, defining g(u, θ) = −2u (1 + u2)3V(θ), we have that ∂uL+ out =Gu(g). Since hgi= 0 and g∈ X5,3, using again (1) of Lemma 4.3, k∂uL+ outk5,3=kGu(g)k5,3≤K(νI0)−1kgk5,3. The last claim follows from the fact that hF(0)i= 0. We use L+ out to rewrite equation (4.11) as a new fixed point equation, with better control of its solution. To do so, we introduce Φ2by Φ1=L+ out + Φ2. Then, Φ1is a solution of (4.11) if and only if Φ2satisfies Φ2=Gu◦e F(Φ2), where e F(Φ2) = F(L+ out + Φ2)−F(0).(4.15) Proposition 4.5. There exists K∗>0such that, if κis big enough, the operator Gu◦e F: BK∗(νI0)−2⊂e X5,3→ BK∗(νI0)−2is well defined and a contraction. Let Φ+ 2be its fixed point. As a consequence, since Φ+ 2is the fixed point of Gu◦e F, then Φ+ 1=L+ out + Φ+ 2is a solution of equation (4.9). 18 Proof. We first claim that e F(0) ∈ X6,4and ke F(0)k6,4≤K1(νI0)−2, for some K1independent of νI0. Indeed, by the definitions of Fin (4.8) and e Fin (4.15), using Lemma 4.2, Proposition 4.4, the fact that p−2 h∈ X−2,−2and k∂θL+ outk2,2≤ k∂θL+ outk4,2≤K νI0 , we have that ke F(0)k6,4= 1 2p2 h (∂uL+ out)2+ν 2(∂θL+ out)26,4 ≤ 1 2p2 h ∂uL+ out1,1k∂uL+ outk5,3+ν 2k∂θL+ outk2,2k∂θL+ outk4,2 ≤K 2kp−2 hk−2,−2k∂uL+ outk3,3 1 νI0 +νK 2k∂θL+ outk2,2 1 νI0 ≤K2 2kp−2 hk−2,−2 1 (νI0)2+νK2 2(νI0)2 ≤K1 (νI0)2. Hence, by the last claim of Lemma 4.3,TGu◦e F(0)U5,3≤Kke F(0)k6,4≤KK1 (νI0)2<∞. We take K∗= 2KK1. Let Ψ∈ BK∗(νI0)−2⊂e X5,3. In view of (4.8) and (4.15), we write e F(Ψ) = e F(0) + e F(Ψ) −e F(0) = F0−F1−F2, where F0=e F(0), F1=1 2p2 h(∂uL+ out +∂uΨ)2−(∂uL+ out)2, F2=ν 2(∂θL+ out +∂θΨ)2−(∂θL+ out)2. We start with F1. Using that kp−2 hk−2,−2<∞,k∂uL+ outk5,3≤K(νI0)−1,Ψ∈ BK∗(νI0)−2⊂ e X5,3and Lemma 4.2, we have that kp−2 h(2∂uL+ out +∂uΨ)k0,0≤ kp−2 hk−2,−2k2∂uL+ out +∂uΨk2,2 ≤K2νI0 κk∂uL+ outk5,3+(νI0)2 κ2k∂uΨk6,4≤K1 κ+K∗ κ2(4.16) for some constants Kindependent of νI0and κ. Then, using again Lemma 4.2, kF1k6,4=1 2kp−2 h(2∂uL+ out +∂uΨ)∂uΨk6,4 ≤1 2kp−2 h(2∂uL+ out +∂uΨ)k0,0k∂uΨk6,4 ≤K 21 κ+K∗ κ2K∗ (νI0)2. Now we deal with F2. We observe that, since kL+ outk4,2=K(νI0)−1and Ψ∈ BK∗(νI0)−2⊂e X5,3, by Lemma 4.2, k2∂θL+ out +∂θΨk0,0≤2K(νI0)2 κ2k∂θL+ outk4,2+K(νI0)4 κ4k∂θΨk6,4≤KνI0 κ2+K∗νI0 κ4.(4.17) Then, kF2k6,4≤ν 2k2∂θL+ out +∂θΨk0,0k∂θΨk6,4≤ν 2K1 + K∗ κ21 κ2(νI0)2. 19 Then, e F(Ψ) ∈ X6,4and, by Lemma (4.3), taking κlarge enough, Gu◦e F(Ψ) ∈ BK∗(νI0)−2⊂e X5,3. Now we check that e Fis Lipschitz with Lip e F ≤ Kκ−1. Let Ψ,Ψ′∈ BK∗(νI0)−2⊂e X5,3. Observe that kp−2 h(2∂uL+ out +∂uΨ + ∂uΨ′)k0,0and k2∂θL+ out +∂θΨ + ∂θΨ′k0,0are bounded as in (4.16) and (4.17). Then, ke F(Ψ) −e F(Ψ′)k6,4≤1 2kp−2 h(2∂uL+ out +∂uΨ + ∂uΨ′)k0,0k∂uΨ−∂uΨ′k6,4 +νk2∂θL+ out +∂θΨ + ∂θΨ′k0,0k∂θΨ−∂θΨ′k6,4 ≤K1 κ+2K∗ κ2TΨ−Ψ′U5,3+νK νI0 κ2+ 2K∗νI0 κ4(νI0)−1TΨ−Ψ′U5,3 ≤K κTΨ−Ψ′U5,3, for some constant K2>0. Then, by the last claim of Lemma 4.3, TGu◦e F(Ψ) −Gu◦e F(Ψ′)U5,3≤Kke F(Ψ) −e F(Ψ′)k6,4≤K κTΨ−Ψ′U5,3. The claim follows taking κlarge enough. Then, we easily check that Gu◦e Fsends the ball BK∗(νI0)−2 into itself and has a unique fixed point there. 5 First extension of the invariant manifold In Section 4we have found the extension of the local unstable invariant manifold to the complex domain D+ κ,δ ×Tσ(see Figure 2) by means of the function Φ+= Φ0+L+ out + Φ+ 2, where Φ0is introduced in (4.5), L+ out in (4.14) and Φ+ 2in Proposition 4.5. Hence, as commented in Remark 4.1, the parametrization of the stable manifold is given by Φ−(u, θ) = −Φ+(−u, −θ). It is defined in D− κ,δ ×Tσ, where D− κ,δ =−D+ κ,δ. This extension of Φ+is not enough for our purposes, because D+ κ,δ ∩ D− κ,δ ∩R=∅and we cannot compute the difference between the manifolds in the reals. In this section we will extend Φ+to a larger domain, D+ κ,δ ∪D+ κ,ext, defined below in (5.21) (see Figure 3). Once extended, the parametrizations of the unstable and stable manifolds will be defined in a common domain containing an interval of R. We will compute the difference of the manifolds in this common domain in Section 9. We remark that we have not been able to find the extension of Φ+in a single step because the Hamilton-Jacobi equation (4.3) is not defined at u= 0. The reason lies in the fact that the method we have used to find Φ+requires computing some integrals along straight lines with some slope in the complex domain where the variable ulives. These straight lines cannot go through 0and hence the current method does not allow us to extend Φ+beyond u= 0 directly. In this section we will find an extension of Φ+to D+ κ,ext by choosing another type of parametrization of the invariant manifold, and then we will go back to the original type of parametrization. However, this extension will not be defined at u= 0. 5.1 From Hamilton-Jacobi parametrization to flow parametrization Taking into account (4.1) and (4.2), the solution Φ+= Φ0+L+ out + Φ+ 2of the Hamilton-Jacobi equation (4.3) obtained in Section 4provides the parametrization of the unstable manifold     q p θ J    = Γ+(u, θ) =     qh(u) 1 ph(u)∂uΦ+(u, θ) θ ∂θΦ+(u, θ)    = Γ0(u, θ) + Γ1(u, θ) + Γ2(u, θ),(5.1) where Γ0(u, θ) =     qh(u) 1 ph(u)∂uΦ0(u) θ 0    ,Γ1(u, θ) =     0 1 ph(u)∂uL+ out(u, θ) 0 ∂θL+ out(u, θ)    (5.2) and Γ2= Γ+−Γ0−Γ1. We recall that ph(0) = 0. However, Γ0is analytic at u= 0 because ph(u)−1∂uΦ0(u) = ph(u). 20 Let X= (Xq, Xp, Xθ, XJ)⊤be the vector field corresponding to H, in (2.3), using the 2-form in (2.1), and let φtbe its flow. We look for a change of variables (u, θ) = (v+f1(v, ϕ), ϕ +f2(v, ϕ)) such that conjugates Xon the unstable invariant manifold, parametrized by e Γ+(v, ϕ) = Γ+(v+f1(v, ϕ), ϕ +f2(v, ϕ)) (5.3) to the vector field (1, νI0), that is, φt(e Γ+(v, ϕ)) = e Γ+(v+t, ϕ +νI0t) or, equivalently, L(e Γ+) = X◦e Γ+,(5.4) where L(Γ) = ∂vΓ + νI0∂ϕΓis the operator defined in (4.7). An immediate computation shows that (5.3) satisfies (5.4) if and only if f= (f1, f2)is a solution of [∂uΓ+◦(Id + f)](1 + L(f1)) + [∂θΓ+◦(Id + f)](νI0+L(f2)) = X◦Γ+◦(Id + f).(5.5) We emphasize that the above equation has four components. However, the symplectic character of the vector field Xensures that if two of them are satisfied, so are the other two. We choose to solve the equations corresponding to the first and third components of (5.5). Taking into account (5.1), the equality ˙qh=−qhphand the fact that ∂uΦ0=p2 h, we can write these two equations as L(f) = A◦(Id + f),(5.6) where A=A1 A2=p−2 h(∂uL+ out +∂uΦ+ 2) ν(∂θL+ out +∂θΦ+ 2).(5.7) We denote N(f)the right hand side of (5.6). We have that N(f) = A+DAf +R(f),(5.8) where DA denotes the derivative of Aand R(f) = A◦(Id + f)−A−DAf. We will see in a moment that equation (5.6) cannot be solved directly as a fixed point equation because the linear term DAf in (5.8) is too large. We will need to rewrite it in a better suited way. 5.1.1 Preliminaries and technical lemmas to solve equation (5.6) To solve equation (5.6), we consider f= (f1, f2)∈ Xr,s ×Xr+1,s+1, with the norm kfkr,s =kf1kr,s +kf2kr+1,s+1 and the operator Guin (4.13), acting on each component. Also, given a matrix function M=M1,1M1,2 M2,1M2,2 with M1,1, M1,2∈ Xr,s and M2,1, M2,2∈ Xr+1,s+1 we define kMkr,s = max {kM1,1kr,s +kM2,1kr+1,s+1, νI0(kM1,2kr,s +kM2,2kr+1,s+1)}.(5.9) It follows immediately from Lemma 4.2 that, if r, s ∈Rand ˜r, ˜s≥0, kMkr,s ≤KνI0 κ˜s kMkr+˜r,s+˜s(5.10) and, for any r, r′, s, s′∈R, kMfkr+r′,s+s′≤ kMkr,skfkr′,s′.(5.11) 21 Indeed, inequality (5.10) follows immediately from (1) of Lemma 4.2. As for (5.11), using (2) of Lemma 4.2, we have that k(Mf)1kr+r′,s+s′+k(Mf)2kr+r′+1,s+s′+1 =kM1,1f1+M1,2f2kr+r′,s+s′+kM2,1f1+M2,2f2kr+r′+1,s+s′+1 ≤ kM1,1f1kr+r′,s+s′+kM1,2f2kr+r′,s+s′+kM2,1f1kr+r′+1,s+s′+1 +kM2,2f2kr+r′+1,s+s′+1 ≤(kM1,1kr,s +kM2,1kr+1,s+1)kf1kr′,s′+KνI0 κ(kM1,2kr,s +kM2,2kr+1,s+1)kf2kr′+1,s′+1 ≤ kMkr,skfkr′,s′, if κ≥K. We will look for the solution of equation (5.6) in a domain slightly smaller than D+ κ,δ ×Tσ. The norms we use depend on the choice of the domains. In particular, they depend on κ,δand σ. Below, we will increase κto ˜κand decrease δto ˜ δ. In order to have the distance from the points of the boundary of D+ ˜κ,˜ δto D+ κ,δ constant, when taking ˜κ > κ we will take δ−˜ δ=˜κ−κ νI0 cos β1 cos β2, sometimes without explicit mention of it. To clarify the exposition, till the end of the section, we include σand κas subscripts in the norms, but not δ, since we will understand that when changing the domain the previous rule applies. In this way kfkr,s,κ,σ will denote the norm kfkr,s for fdefined either in D+ κ,δ or in D+ κ,δ ×Tσ, depending on the setting. We remark that if ˜κ > κ (with ˜κ(νI0)−1<˜ δ) and 0<˜σ < σ, then D+ ˜κ,˜ δ×T˜σ⊂ D+ κ,δ ×Tσ. Analogously, we will denote the spaces by Xr,s,κ,σ to clarify their dependence on the domain. The reduction of domain will only be done a finite number of times. Lemma 5.1. For all ˜κ > κ and 0<˜σ < σ, there exists C > 0such that if B∈ Xr,s,κ,σ, then, for all m, n ∈N,∂m u∂n θB∈ Xr,s,˜κ,˜σwith k∂m uBkr,s,˜κ,˜σ≤C˜κ κs(νI0)mm! (˜κ−κ)m 1 (cos β1)mkBkr,s,κ,σ, m ≥0, k∂m u∂n θBkr,s,˜κ,˜σ≤C˜κ κs(νI0)mm!n! (˜κ−κ)m(σ−˜σ)n+1 1 (cos β1)mkBkr,s,κ,σ, m ≥0, n ≥1. When applying the previous lemma, we will choose ˜κ=κ+κ0, where κ0>0is fixed. In this way, ˜κ/κ = 1 + κ0/κ < 2, if κis large enough. Proof of Lemma 5.1.We note that (∂m u∂n θB)[k]= (ik)n∂m uB[k]. By Cauchy formula, ∂m uB[k](u) = m! 2πi Zγu B[k](z) (z−u)m+1 dz, taking γuto be the circle of radius (˜κ−κ)(νI0)−1cos β1, we immediately have, for some C1>0, k∂m uB[k]kr,s,˜κ,˜σ≤C1˜κ κsm!(νI0)m (˜κ−κ)m 1 (cos β1)mkB[k]kr,s,κ,σ. Hence, since kB[k]kr,s,κ,σ ≤ kBkr,s,κ,σe−|k|σ, we have that k∂m u∂n θB[k]kr,s,˜κ,˜σ≤C1˜κ κsm!|k|n(νI0)m (˜κ−κ)m 1 (cos β1)me−|k|σkBkr,s,κ,σ. Therefore, k∂m u∂n θBkr,s,˜κ,˜σ=X k∈Zk∂m u∂n θB[k]kr,s,˜κ,˜σe|k|˜σ ≤C1 (cos β1)m˜κ κsm!(νI0)m (˜κ−κ)mkBkr,s,κ,σ X k∈Z|k|ne−|k|(σ−˜σ) ≤C1C2 (cos β1)m˜κ κsm!n!(νI0)m (˜κ−κ)m(σ−˜σ)n+1 kBkr,s,κ,σ, where we have used that Pk≥0kne−bk ≤C2n!/bn+1, for b > 0and n≥1, for some C2. 22 Lemma 5.2. Let Cbe the constant given by Lemma 5.1 and r′≥0and s′>0. Given K > 0and δ > 0, for any 0<˜σ < σ, there exist κ0>0such that for any ˜κ > κ +δ > κ ≥κ0, if B∈ Xr,s,κ,σ and f∈ Xr′,s′,˜κ,˜σ×Xr′+1,s′+1,˜κ,˜σwith kfkr′,s′,˜κ,˜σ≤K/(νI0)s′+1, then B◦(Id + f)∈ Xr,s,˜κ,˜σand kB◦(Id + f)kr,s,˜κ,˜σ≤2C σ−˜σ˜κ κs kBkr,s,κ,σ. Proof. First of all, by (1) of Lemma 4.2, we have that kf1k0,0,˜κ,˜σ≤K(νI0)s′ ˜κs′kf1kr′,s′,˜κ,˜σ≤K ˜κs′νI0 , kf2k0,0,˜κ,˜σ≤K(νI0)s′+1 ˜κs′+1 kf2kr′+1,s′+1,˜κ,˜σ≤K ˜κs′+1 . Then, expanding Bin Taylor series and using Lemma 5.1, we have that, writing α1= cos β1, kB◦(Id + f)kr,s,˜κ,˜σ≤X j≥0 1 j! j X ℓ=0 j ℓk∂j−ℓ u∂ℓ θBfj−ℓ 1fℓ 2kr,s,˜κ,˜σ ≤X j≥0 j X ℓ=0 1 (j−ℓ)!ℓ!k∂j−ℓ u∂ℓ θBkr,s,˜κ,˜σkf1kj−ℓ 0,0,˜κ,˜σkf2kℓ 0,0,˜κ,˜σ ≤C σ−˜σ˜κ κsX j≥0 j X ℓ=0 K ˜κs′νI0j−ℓK ˜κs′+1 ℓ(νI0)j−ℓ (α1(˜κ−κ))j−ℓ(σ−˜σ)ℓkBkr,s,κ,σ ≤C σ−˜σ˜κ κsX j≥0 Kj j X ℓ=0 1 (˜κs′(α1(˜κ−κ))j−ℓ 1 (˜κs′+1(σ−˜σ))ℓkBkr,s,κ,σ ≤C σ−˜σ˜κ κsX j≥0K1 ˜κs′α1(˜κ−κ)+1 ˜κs′+1(σ−˜σ)j kBkr,s,κ,σ ≤2C σ−˜σ˜κ κs kBkr,s,κ,σ, where we have chosen κ0such that K ˜κs′1 α1(˜κ−κ)+1 ˜κ(σ−˜σ)≤1 2. 5.1.2 Solution of equation (5.6) Let Gube the operator defined in (4.13). We formally define f0=Gu(A), F0=Gu(DA).(5.12) Lemma 5.3. Let Abe the function defined in (5.7). Let ˜κ > κ and 0<˜σ < σ, with κbig. Then, there exists K > 0such that (1) kAk3,1,κ,σ ≤K(νI0)−1, (2) kf0k3,1,κ,σ ≤K(νI0)−2, (3) kDAk3,1,˜κ,˜σ≤K, (4) kF0k3,1,˜κ,˜σ≤K(νI0)−1. Proof. We write A=A0+A1, where A0=p−2 h∂uL+ out ν∂θL+ out , A1=p−2 h∂uΦ+ 2 ν∂θΦ+ 2. 23 Taking into account that p−2 h∈ X−2,−2,κ,σ and the properties of L+ out in Proposition 4.4, we have that kA0k3,1,κ,σ ≤K(νI0)−1and hA0i= 0. Also, by the properties of Φ+ 2in Proposition 4.5, we have that kA1k4,2,κ,σ ≤K(νI0)−2. Hence, kA1k3,1,κ,σ ≤Kκ−1(νI0)−1. This proves (1). We bound kGu(A0)k3,1,κ,σ and kGu(A1)k3,1,κ,σ separately. Since hA0i= 0, by (1) of Lemma 4.3, kGu(A0)k3,1,κ,σ ≤K νI0kA0k3,1,κ,σ ≤K (νI0)2. Using again that p−2 h∈ X−2,−2, the properties of Φ+ 2in Proposition 4.5 imply that kA1k4,2,κ,σ ≤ K(νI0)−2. Then, by (2) of Lemma 4.3, kGu(A1)k3,1,κ,σ ≤KkA1k4,2,κ,σ ≤K (νI0)2, from which (2) follows. We write DA =DA0+DA1. From the previous bounds for A0and A1, applying Lemma 5.1 and the definition of the matrix norm (5.9), we have that, in the reduced domain, kDA0k3,1,˜κ,˜σ≤K, kDA1k4,2,˜κ,˜σ≤K κνI0 . Here the constants depend on ˜κ−κand σ−˜σ. This proves (3). We finally prove (4). Since hA0i= 0, we have that hDA0i= 0. By (1) of Lemma 4.3, kGu(DA0)k3,1,˜κ,˜σ≤K νI0 . Moreover, by (2) of Lemma 4.3, kGu(DA1)k3,1,˜κ,˜σ≤KkDA1k4,2,˜κ,˜σ≤K νI0 . Since F0=Gu(DA0) + Gu(DA1), the claim follows. We introduce ˜ fby f=f0+ (Id + F0)˜ f. Observe that, by (4) of Lemma 5.3 and (1) of Lemma 4.2, kF0k0,0,˜κ,˜σ≤KνI0 κkF0k3,1,˜κ,˜σ≤K κ. Hence, if κis large enough, Id+F0is invertible and k(Id+F0)−1k0,0,κ,σ ≤2. Using that, by (5.12), L(f0) = Aand L(F0) = DA, we rewrite equation (5.6) as L(˜ f) = e N(˜ f),(5.13) where e N(˜ f) = (Id + F0)−1DA f0+ (Id + F0)−1DA F0˜ f+ (Id + F0)−1Rf0+ (Id + F0)˜ f and Rwas introduced in (5.8). Using the operator Gu, we rewrite equation (5.13) as the fixed point equation ˜ f=Gu◦e N(˜ f).(5.14) Proposition 5.4. For all ˆκ > ˜κ > κ and 0<ˆσ < ˜σ < σ, with κbig enough, equation (5.14)has a solution ˜ f+, defined in D+ ˆκ,ˆ δ×Tˆσ, and ˜ f+∈ X3,1,ˆκ,ˆσ× X4,2,ˆκ,ˆσwith k˜ f+k3,1,ˆκ,ˆσ≤C(νI0)−2. Consequently, e Γ+= Γ+◦(Id+f0+(I+F0)˜ f+)satisfies the invariance equation (5.4)in D+ ˆκ,δ ×Tˆσ. 24 Proof of Proposition 5.4.We claim that there exists K1>0such that kGu◦e N(0)k3,1,ˆκ,ˆσ≤K1 (νI0)2. Indeed, first we notice that e N(0) = (Id+F0)−1(DA f0+R(f0)) = (Id+F0)−1DA f0+Z1 0 (1 −s)D2A◦(Id + sf0)ds (f0)⊗2 = (Id + F0)−1DA f0+Z1 0 (1 −s)f0 1∂uDA ◦(Id + sf0)f0+f0 2∂θDA ◦(Id + sf0)f0ds. Next, we note that kDA f0k4,2,ˆκ,ˆσ≤ kDA f0k6,2,ˆκ,ˆσ≤ kDAk3,1,ˆκ,ˆσkf0k3,1,ˆκ,ˆσ≤K (νI0)2. By (3) of Lemma 5.3 and Lemma 5.1, for ˆκ > ˜κand 0<ˆσ < ˜σ, we have k∂uDAk3,1,ˆκ,ˆσ≤KνI0kDAk3,1,˜κ,˜σ≤KνI0, k∂θDAk3,1,ˆκ,ˆσ≤KkDAk3,1,˜κ,˜σ≤K. (5.15) Moreover, by (1) of Lemma 4.2 and (2) of Lemma 5.3, kf0 1∂uDA ◦(Id + sf0)f0k4,2,ˆκ,ˆσ≤KνI0 κkf0 1∂uDA ◦(Id + sf0)f0k9,3,ˆκ,ˆσ≤K κ(νI0)2 and, since f0 2∈ X4,2,ˆκ,ˆσ, kf0 2∂θDA ◦(Id + sf0)f0k4,2,ˆκ,ˆσ≤KνI0 κ2 kf0 2∂vDA ◦(Id + sf0)f0k10,4,ˆκ,ˆσ≤K κ2(νI0)2. Hence, kGu◦e N(0)k3,1,ˆκ,ˆσ≤Kke N(0)k4,2,ˆκ,ˆσ≤K1 κ(νI0)2. Let C= 2K1. Let f, f′∈ X3,1,ˆκ,ˆσ×X4,2,ˆκ,ˆσwith kfk3,1,ˆκ,ˆσ,kf′k3,1,ˆκ,ˆσ≤e C. We have that e N(f)−e N(f′) = (I+F0)−1(M1+M2) (f−f′),(5.16) where M1=DA F0, M2(f, f′) = Z1 0Z1 0 D2A◦(Id + tus(f, f′)) dt us(f, f′)ds (I+F0) with us(f, f′) = f0+ (Id + F0)f′+s(I+F0)(f−f′). Then, using (1) of Lemma 4.2 and (3) and (4) of Lemma 5.3, we have that kM1k1,1,ˆκ,ˆσ≤KνI0 κkDAF 0k6,2,ˆκ,ˆσ≤KνI0 κkDAk3,1,ˆκ,ˆσkF0k3,1,ˆκ,ˆσ≤K κ.(5.17) Observe that, by (2) of Lemma 5.3 and the hypotheses on f,f′, kus(f, f′)k3,1,ˆκ,ˆσ≤K(νI0)−2. Hence, also using (5.15) kus,1∂uDA ◦(Id + tus)k1,1,ˆκ,ˆσ≤KνI0 κkus,1∂uDA ◦(Id + tus)k6,2,ˆκ,ˆσ ≤KνI0 κkus,1k3,1,ˆκ,ˆσk∂uDA ◦(Id + tus)k3,1,ˆκ,ˆσ≤K κ 25 where Lin(T2) = ∂vT2+∂θT2, Fin(T2) = −ν 2(∂θL+ in)2+ 2v2(∂vL+ in)2+ν∂θL+ in∂θT2+ν 2(∂θT2)2+ 4v2∂vL+ in∂vT2+ 2v2(∂vT2)2, where we have used that Lin(L+ in) = −E(T0). 6.1 Spaces and technical lemmas To deal with equation (6.9) we have to introduce some function spaces and provide basic properties of the operator Lin. We consider the domain D+ κ,in ={v∈C| ℑv < −(tan β1)ℜv−κ}∪{v∈C| ℑv > (tan β1)ℜv+κ},(6.10) with κ > 1. See Figure 5. We notice that the image of D+ κ,δ ∪ D+ κ,ext by the transformation u=i+ (νI0)−1vis contained in D+ κ,in. Figure 5: The domain D+ κ,in defined in (6.10), shaded in gray. To solve equation (6.9), for r∈R, we introduce the Banach space of 2π-periodic in θ, analytic functions Yr={R:D+ κ,in ×Tσ→C| kRkr<∞},(6.11) where, taking into account the Fourier expansion of R(v, θ) = Pk∈ZR[k](v)eikθ , kRkr=X k∈ZkR[k]kre|k|σ,(6.12) and, for an analytic function f:D+ κ,in →C, kfkr= sup v∈D+ κ,in |vrf(v)|. Lemma 6.1. Let r, r1, r2∈R. (1) If R∈ Yr+s,s≥0, then R∈ Yrand there exists K > 0such that kRkr≤K1 κskRkr+s. 32 (2) If R1∈ Yr1and R2∈ Yr2, then R1R2∈ Yr1+r2and kR1R2kr1+r2≤ kR1kr1kR2kr2. We also introduce the Banach space e Yr={R∈ Yr|∂vR, ∂θR∈ Xr+1,TRUr<∞},(6.13) with the norm TRUr=kRkr+k∂vRkr+1 +k∂θRkr+1.(6.14) We will use the next technical lemma several times. It is analogous to Lemma 4.3. Lemma 6.2. Let Gu in be the operator defined in (6.7). There exists K > 0such that (1) If R∈ Yrwith r > 0and hRi= 0, then Gu in(R)∈ Yrand kGu in(R)kr≤KkRkr. (2) If R∈ Yrwith r > 1, then Gu in(R)∈ Yr−1and kGu in(R)kr−1≤KkRkr. (3) If R∈ Yrwith r > 0, then ∂vGu in(R), ∂θGu in(R)∈ Yrand k∂vGu in(R)kr≤KkRkr,k∂θGu in(R)kr≤KkRkr. As a consequence, if R∈ Yr, with r > 1,Gu in(R)∈e Yr−1and TGu in(R)Ur−1≤KkRkr. The constant Konly depends on rand the constants involved in the definition of D+ κ,in. 6.2 Fixed point equation We consider the equation T2=Gu in ◦Fin(T2).(6.15) We decompose Fin(T2) = −(F0+F1+F2+F3+F4),(6.16) with F0(v, θ) = ν 2(∂θL+ in(v, θ))2+ 2v2(∂vL+ in(v, θ))2, F1(v, θ) = ν∂θL+ in(v, θ)∂θT2(v, θ), F2(v, θ) = ν 2(∂θT2(v, θ))2, F3(v, θ) = 4v2∂vL+ in(v, θ)∂vT2(v, θ), F4(v, θ) = 2v2(∂vT2(v, θ))2. If T2is a solution of the fixed point equation (6.15) it is also a solution of (6.9). In view of (6.8), since V[0] = 0, we have that (L+ in)[0] = 0. The following proposition is analogous to Proposition 4.4. Its proof follows exactly the same lines. Proposition 6.3. We have L+ in, ∂θL+ in ∈ Y2and ∂vL+ in ∈ Y3. Moreover, there exists K > 0, independent of κ, such that kL+ ink2,k∂θL+ ink2,k∂vL+ ink3≤K. Motivated by Proposition 6.3, we introduce the constant ΘV=kL+ ink2 2+k∂θL+ ink2 2+k∂vL+ ink2 31/2.(6.17) Remark 6.4. If one replaces riby ε˜riin the definition of Vin (1.2), then ΘV=O(ε). 33 Proposition 6.5. There exist K1>0and κ0such that, if κ > κ0, the operator Gu in ◦Fin :BK1Θ2 V⊂ e Y3→ BK1Θ2 Vis well defined and a contraction. Hence, equation (6.15)has a solution T+ 2∈e Y3 satisfying TT+ 2U3≤K1Θ2 V.(6.18) Proof. We start by bounding F0. By Proposition 6.3, kF0k4≤ν 2(∂θL+ in)24+ 2kv2(∂vL+ in)2k4≤ν 2k∂θL+ ink2 2+ 2kv2k−2k∂vL+ ink2 3< K2Θ2 V,(6.19) for some K2>0. We take K1= 2kGu inkK2and K∗=K1Θ2 V. In this proof kGu inkstands for the operator norm of Gu in :Y4→e Y3. Let T2∈ BK∗⊂e Y3. We claim that Fin(T2)∈ Y4and kFin(T2)k4≤ kF0k4+Kνk∂θL+ ink2 κ2+KνK∗ 2κ4+K4k∂vL+ ink3 κ+K2K∗ κ2K∗.(6.20) Indeed, next we deal with Fj,j= 1,...,4. By Proposition 6.3 and Lemma 6.1 kF1k4=kν∂θL+ in∂θT2k4=νk∂θL+ ink0k∂θT2k4≤Kνk∂θL+ ink2 κ2TT2U3≤Kνk∂θL+ ink2K∗ κ2, since k∂θT2k4≤TT2U3≤K∗. Again by Lemma 6.1, kF2k4=ν 2(∂θT2(v, θ))24≤ν 2k∂θT2k0k∂θT2k4≤Kν 2κ4k∂θT2k4k∂θT2k4≤Kν(K∗)2 2κ4. Now, by Proposition 6.3, Lemma 6.1 and the fact that k∂vT2k4≤TT2U3≤K∗, kF3k4=4v2∂vL+ in∂vT24≤4v2k−2k∂vL+ ink2k∂vT24≤K4k∂vL+ ink3 κK∗. Finally, kF4k4=2v2(∂vT2(v, θ))24≤2v2∂vT2(v, θ)0k∂vT2(v, θ)k4 ≤ k2v2k−2k∂vT2(v, θ)k2k∂vT2(v, θ)k4≤K2 κ2k∂vT2(v, θ)k4k∂vT2(v, θ)k4≤K2(K∗)2 κ2. This proves (6.20). Now we choose κ0>1such that Kνk∂θL+ ink2 κ2 0 +νK∗ 2κ4 0 +4k∂vL+ ink3 κ0 +2K∗ κ2 0K∗< K2Θ2 V. Hence, for any κ > κ0, if T2∈ BK∗, by the last claim of Lemma 6.2, TGu in ◦Fin(T2)U3≤ kGu inkkFin(T2)k4≤2kGu inkK2Θ2 V=K∗. Next we check that Gu in ◦Fin is a contraction on BK∗. We claim that there exists K3>0such that, if T2, T ′ 2∈ BK∗, kFin(T2)−Fin(T′ 2)k4≤K3 κTT2−T′ 2U3.(6.21) To prove this claim, we write Fin(T2)−Fin(T′ 2) = − 4 X i=1 (Fi(T2)−Fi(T′ 2)) , where F1(R) = ν∂θL+ in∂θR, F2(R) = 1 2ν(∂θR)2, F3(R)(v, θ) = 4v2∂vL+ in(v, θ)∂vR(v, θ), F4(R)(r, θ) = 2v2(∂vR(v, θ))2. 34 We bound each difference separately. First, by Proposition 6.3 and Lemma 6.1, we have that kF1(T2)−F1(T′ 2)k4≤ kν∂θL+ ink0k∂θT2−∂θT′ 2k4≤Kνk∂θL+ ink2 κ2TT2−T′ 2U3. Second, kF2(T2)−F2(T′ 2)k4=ν 2k(∂θT2+∂θT′ 2)(∂θT2−∂θT′ 2)k4 ≤ν 2k(∂θT2+∂θT′ 2)k0k∂θT2−∂θT′ 2k4 ≤Kν 2 1 κ4k∂θT2+∂θT′ 2k4k∂θT2−∂θT′ 2k4 ≤KK∗ν κ4TT2−T′ 2U3. Third, using again Proposition 6.3 and Lemma 6.1, kF3(T2)−F3(T′ 2)k4≤4v2∂vL+ in0k∂vT2−∂vT′ 2k4≤4∂vL+ in3 κTT2−T′ 2U3. And fourth, kF4(T2)−F4(T′ 2)k4≤ k2v2(∂vT2+∂vT′ 2)k0k∂vT2−∂vT′ 2k4 ≤ k2v2k−2k∂vT2+∂vT′ 2k2k∂vT2−∂vT′ 2k4 ≤K2 κ2k∂vT2+∂vT′ 2k4k∂vT2−∂vT′ 2k4 ≤4KK∗ κ2TT2−T′ 2U3. Inequality (6.21) follows from combining these four steps, since κ > κ0>1and κ0is big. Finally, we check that the operator Gu in ◦Fin is a contraction in BK∗. Indeed, if T2, T ′ 2∈ BK∗, using the last statement of Lemma 6.2 and (6.21), TGu in ◦Fin(T2)−Gu in ◦Fin(T′ 2)U3≤ kGu inkkFin(T2)−Fin(T′ 2)k4≤ kGu inkK3 κTT2−T′ 2U3. Hence, by the standard argument, taking κ0large enough, Gu in ◦Fin sends BK∗into itself and has a unique fixed point T+ 2in this ball. The bound (6.18) follows from the definition of K∗. Corollary 6.6. The inner equation (6.2)admits a solution T+=T0+L+ in +T+ 2, where T0is given by (6.3),L+ in is defined in (6.8)and T+ 2is given by Proposition 6.5. 7 Approximation of the manifold in the inner domain Let T+=T0+L+ in +T+ 2be the solution of the inner equation (6.2) in the domain D+ κ,in given by Corollary 6.6. From Proposition 4.5, we have that the solution Φ+(u, θ)of the Hamilton-Jacobi equation (4.3) and its extension to the domain D+ κ,δ ∪D+ κ,ext \B∗ ρ(where B∗ ρis a fixed small ball centered at the origin), with vanishing contour conditions at ℜu=−∞, is given by Φ+= Φ0+L+ out + Φ+ 2, where Φ0was introduced in (4.5), L+ out in (4.14) and Φ+ 2∈e X5,3, hence Φ+ 2(u, θ)∼(νI0)−2 (u−i)3. The approximation of Φ+by Φ0+L+ out is not good enough if u−i∼(νI0)−1, although Proposition 5.8 ensures that the manifolds can be extended to D+ κ,δ ∪ D+ κ,ext \B∗ ρ. In order to obtain a better approximation of Φ+, we compare it with the solution of the inner equation (6.2), T+=T0+L+ in +T+ 2, given by Corollary 6.6. 35 We recall that the function S+(v, θ) = (νI0)−1Φ+(i+ (νI0)−1v, θ) is a solution of the Hamilton-Jacobi equation (6.1) in the transformed domain. Actually, we want to compare S+with the solution T+of the inner equation. Both functions are already determined. We write S+=T0+L+ in +T+ 2+S+ 2 and derive bounds and properties of S+ 2from the equation it satisfies. Using that T+=T0+L+ in +T+ 2is a solution of the inner equation (6.2), we have that S+ 2is a solution of Lin(S2) = b Fin,a(S2),(7.1) where Lin(S2) = ∂vS2+∂θS2was defined in (6.6) and b Fin,a(S2) = −e E+ν∂θT+∂θS2+1 2ν(∂θS2)2+ (2A1∂vT+−1)∂vS2+A1(∂vS2)2(7.2) with A1(v) = 1 2 v2(2i+ (νI0)−1v)2 (i+ (νI0)−1v)2, A2(v, θ) = −1 2 (i+ (νI0)−1v)2 v2(2i+ (νI0)−1v)2+1 2 1 v2(2i+ (νI0)−1v)2V(θ), e E(v, θ) = (A1(v)−2v2)(∂vT+ 0)2+A2(v, θ) + 1 8v2+1 8v2V(θ). (7.3) Let β1>0be the angle involved in the definition of D+ κ,δ (see (4.12)). Fix β3and β4satisfying 0< β3< β4< β1. For α∈(0,1) and κ > 1, we define the domain D+ κ,α ={v∈C| −(νI0)α−tan β4ℜv < ℑv < min{−κ−tan β1ℜu, −κ−tan β3ℜu}}.(7.4) See Figure 6. Observe that d(D+ κ,α,0) = κcos β3and |v| ≤ K(νI0)αfor v∈ D+ κ,α. Figure 6: The domain D+ κ,α defined in (7.4), shaded in gray. 36 In order to solve equation (7.1), we consider the following right inverses, b Guand b Gu s, of Lin. First, b Guis determined by the expression of the Fourier coefficients of the image b Gu(S2): b Gu(S2)[k](v) = Zv v+ 0 eik(t−v)S[k] 2(t)dt if k≥0, b Gu(S2)[k](v) = Zv v− 0 eik(t−v)S[k] 2(t)dt if k < 0, (7.5) where v− 0=(νI0)α−κ tan β1−tan β4 +−tan β4 tan β1−tan β4 + 1(νI0)α+tan β4 tan β1−tan β4 κi, v+ 0=−(νI0)α+κ tan β4−tan β3 + tan β4 tan β4−tan β3−1(νI0)α−tan β4 tan β4−tan β3 κi, with α∈(0,1) and 0< β3< β4< β1< π. See Figure 6. The most general right inverse of Lin has the form b Gu s(S2) = S2,0+b Gu(S2),(7.6) with S2,0∈ker Lin. With this operator we consider the fixed point equation S2=b Gu s◦b Fin,a(S2)(7.7) and we choose S2,0so that the solution S+ 2of (7.7) is the analytical continuation of S+−T0− L+ in −T+ 2to the domain D+ κ,α. Note again that if S+ 2is a fixed point of b Gu s◦b Fin,a, it is also a solution of (7.1). Since S2,0∈ker Lin, S2,0(v, θ) = X k∈Z S[k] 2,0(v)eikθ =X k∈Z g[k]eik(θ−v), which defines the coefficients g[k]. If, moreover, S2is a solution of (7.7), we have S2(v, θ) = S2,0(v, θ) + b Gu◦b Fin,a(S2)(v, θ). For k≥0, the Fourier coefficients satisfy S[k] 2(v) = S[k] 2,0(v) + Zv v+ 0 eik(t−v)(b Fin,a(S2))[k](t)dt and, taking into account that, at the points (v+ 0, θ)of the boundary, S2(v, θ) = S+(v, θ)−T0(v, θ)−L+ in(v, θ)−T+ 2(v, θ), evaluating at v=v+ 0, we get g[k]=S[k] 2,0(v+ 0)eikv+ 0= (S+−T0−L+ in −T+ 2)[k](v+ 0)eikv+ 0 and finally, S[k] 2,0(v) = g[k]e−ikv = (S+−T0−L+ in −T+ 2)[k](v+ 0)e−ik(v−v+ 0).(7.8) Analogously, we obtain that for k < 0 S[k] 2,0(v) = g[k]e−ikv = (S+−T0−L+ in −T+ 2)[k](v− 0)e−ik(v−v− 0).(7.9) Therefore, we choose S2,0(v, θ) = X k∈Z S[k] 2,0(v)eikθ (7.10) with the coefficients (7.8) and (7.9). 37 7.1 Spaces and technical lemmas To solve equation (7.7), for r∈R, we introduce the Banach space of analytic, 2π-periodic in θ functions Zr={R:D+ κ,α ×Tσ→C| kRkr<∞}, with the norm k·krdefined as follows. Using that R(v, θ) = Pk∈ZR[k](v)eikθ, kRkr=X k∈ZkR[k]kre|k|σ, where, for an analytic function f:D+ κ,α →C, kfkr= sup v∈D+ κ,α |vrf(v)|. The following lemma is very similar to Lemma 6.1. Lemma 7.1. Let r, r1, r2∈R. There exists K > 0such that the following holds. (1) If R∈ Zr+swith s≥0, then R∈ Zrand kRkr≤K1 κskRkr+s. (2) If R1∈ Zr1and R2∈ Zr2, then R1R2∈ Zr1+r2and kR1R2kr1+r2≤ kR1kr1kR2kr2. We also introduce e Zr={R∈ Zr|∂vR, ∂θR∈ Zr+1,TRUr<∞}, with the norm TRUr=kRkr+k∂vRkr+1 +k∂θRkr+1. The following result is completely analogous to Lemma 6.2. The only difference is the domain of the functions of the space. Lemma 7.2. Let b Gube the operator defined in (7.5). There exists K > 0such that (1) If R∈ Zr, with r > 0, and R[0] = 0, kb Gu(R)kr≤KkRkr. (2) If R∈ Zrwith r > 1,b Gu(R)∈ Zr−1and kb Gu(R)kr−1≤KkRkr. (3) If R∈ Zrwith r > 0, then ∂vb Gu(R), ∂θb Gu(R)∈ Zrand k∂vb Gu(R)kr≤KkRkr,k∂θb Gu(R)kr≤KkRkr, As a consequence, if R∈ Zr, with r > 1,b Gu(R)∈e Zr−1and Tb Gu(R)Ur−1≤KkRkr. The constant Konly depends on rand the constants involved in the definition of D+ κ,α. 38 7.2 The fixed point equation Before solving the fixed point equation (7.7) we deal with some preliminary estimates. Lemma 7.3. Let A1and e Ebe the functions introduced in (7.3). There exists K > 0such that, in D+ κ,α, we have (1) e E ∈ Z2with ke Ek2≤K(νI0)α−1. (2) A1∈ Z−2with kA1k−2≤K. (3) 2A1∂vT+−1∈ Z0with k2A1∂vT+−1k0≤K(νI0)α−1+Kκ−1. (4) ∂θT+∈ Z2with k∂θT+k2≤K. Proof. We recall that in the domain D+ κ,α,K1κ < |v|< K2(νI0)α, for some K1, K2>0. We start by proving (1). We have that A1(v)−2v2=1 2 v2(2i+ (νI0)−1v)2 (i+ (νI0)−1v)2−2v2= 2v2(νI0)−1v(i+3 4(νI0)−1v) (1 −i(νI0)−1v)2,(7.11) which implies that kA1(v)−2v2k−2≤K(νI0)α−1. Recall that 0< α < 1. We claim that there exists a constant Ksuch that k∂vT+k2≤K. Indeed, since T+=T0+L+ in+ T+ 2, by Propositions 6.3 and 6.5,k∂vL+ ink3≤Kand TT+ 2U3≤K. Hence, by (1) of Lemma 7.1, k∂vL+ ink2≤Kκ−1and k∂vT+ 2k2≤Kκ−2. Since ∂vT0= 1/(4v2), the claim follows immediately from the previous bounds. Then, by (2) of Lemma 7.1, (A1(v)−2v2)(∂vT+)22≤ kA1(v)−2v2k−2k∂vT+k2 2≤K3(νI0)α−1. By the definition of A2in (7.3), we have that A2(v, θ) + 1 8v2+1 8v2V(θ) = −1 2 (i+ (νI0)−1v)2 v2(2i+ (νI0)−1v)2+1 8v2+1 2 1 v2(2i+ (νI0)−1v)2+1 8v2V(θ) =1 v2O((νI0)−1v) which implies A2(v, θ) + 1 8v2+1 8v2V(θ)2≤K(νI0)α−1. Hence, (1) follows. (2) is an immediate consequence of (7.11). Now we deal with (3). We have that 2A1∂vT+−1 = 2A1∂vT0−1+2A1∂v(L+ in +T+ 2) = −1 4 (νI0)−1v(4i+ 3(νI0)−1v) (i+ (νI0)−1v)2+2A1∂v(L+ in +T+ 2) which implies that k2A1∂vT+−1k0≤ k2A1∂vT0−1k0+k2A1∂v(L+ in +T+ 2)k0 ≤K(νI0)α−1+k2A1k−2k∂v(L+ in +T+ 2)k2 ≤K(νI0)α−1+K κk∂v(L+ in +T+ 2)k3. (4) Follows directly from Propositions 6.3 and 6.5. Proposition 7.4. The function S2,0defined in (7.10)satisfies S2,0∈ Z0and kS[k] 2,0k1≤Kmax{(νI0)−2α,(νI0)−1+αlog(νI0)}e−|k|σ, k 6= 0. Consequently, in D+ κ,α ×Tσ′with 0< σ′< σ, kS2,0k1≤Kmax{(νI0)−2α,(νI0)−1+αlog(νI0)}. 39 Proof. We have that (S+−T0−L+ in −T+ 2)(v, θ) = (νI0)−1Φ0(i+ (νI0)−1v)−T0(v) + (νI0)−1L+ out(i+ (νI0)−1v, θ)−L+ in(v, θ) + (νI0)−1Φ+ 2(i+ (νI0)−1v, θ)−T+ 2(v, θ), where Φ0was introduced in (4.5), T0in (6.3), L+ out in (4.14), L+ in in (6.8), Φ+ 2in Proposition 4.5 and T+ 2in Proposition 6.5. First, an explicit computation shows that, in the domain under consideration, (νI0)−1Φ0(i+ (νI0)−1v)−T0(v) = i 4(νI0)−1log((νI0)−1v) + (νI0)−1O((νI0)−1v). Next, also an explicit computation shows that (νI0)−1(L+ out)[k](i+ (νI0)−1v)−(L+ in)[k](v) = −V[k]Zv −∞ 4(νI0)−1i+ (νI0)−2s 8s(2 −(νI0)−1is)2eik(s−v)ds =O(νI0)−1 v(1 + O((νI0)−1v))V[k]. Moreover, since Φ+ 2∈e X5,3with kΦ+ 2k5,3≤K(νI0)−2, |(νI0)−1(Φ+ 2)[k](i+ (νI0)−1v)| ≤ K|v|−3e−|k|σ. Finally, since, by Proposition 6.5,T+ 2∈e Y3with kT+ 2k3≤K, |(T+ 2)[k](v)| ≤ K|v|−3e−|k|σ. Since |v± 0|=O((νI0)α), the bounds of the Fourier coefficients follow. To get the bound in the k·k1 norm we have to restrict the domain to D+ κ,α ×Tσ′with 0< σ′< σ < σ0, where σ0was introduced in (2.9). We consider the fixed point equation S2=b Gu s◦b Fin,a(S2),(7.12) where b Gu sand b Fin,a were defined in (7.6) and (7.2), respectively. Proposition 7.5. Let α∈(0,1). Then, equation (7.12)has a solution S+ 2∈e Z1in D+ κ,α ×Tσ′ with kS+ 2k1≤K(νI0)−2α+K(νI0)−1+αlog(νI0). Proof. We first check that there exists K∗>0such that BK∗⊂e Z1satisfies b Gu s◦b Fin,a(BK∗)⊂ BK∗. To do so, given S2∈ BK∗, we write b Fin,a(S2) = F0+F1+F2+F3+F4, with F0=e E, F1=ν∂θT+∂θS2, F2=1 2ν(∂θS2)2, F3= (2A1∂vT+−1)∂vS2, F4=A1(∂vS2)2, where e E,A1and A2were introduced in (7.3). We claim that Fi∈ Z2,i= 0,...,4, and there exists K > 0such that kF0k2≤K(νI0)α−1,(7.13) kF1k2≤Kκ−2K∗,(7.14) kF2k2≤Kκ−2(K∗)2,(7.15) kF3k2≤K(νI0)α−1+κ−1K∗,(7.16) kF4k2≤K(K∗)2.(7.17) 40 Bound (7.13) is simply (1) of Lemma 7.3. Since k∂θT+k2≤K, by (1) of Lemma 7.1, we have that kν∂θT+∂θS2k2≤νk∂θT+k1k∂θS2k1≤ν κ2k∂θT+k2k∂θS2k2≤Kκ−2K∗, which proves (7.14). Bound (7.15) follows analogously from  1 2ν(∂θS2)22≤ν 2k∂θS2k0k∂θS2k2≤ν 2 1 κ2(K∗)2. By (3) of Lemma 7.3, bound (7.16) follows from k(2A1∂vT+−1)∂vS2k2≤ k2A1∂vT+−1k0k∂vS2k2≤K(νI0)α−1+κ−1K∗. Bound (7.17) follows from (2) of Lemma 7.3 and kA1(∂vS2)2k2≤ kA1k−2k∂vS2k2 2≤K(K∗)2. In particular, from (7.13), Proposition 7.4 and the last claim of Lemma 7.2, since b Fin,a(0) = F0, we deduce that Tb Gu s◦b Fin,a(0)U1=TS2,0+b Gu◦b Fin,a(0)U1≤TS2,0U1+Tb Gu◦b Fin,a(0)U1 ≤Kmax{(νI0)−2α,(νI0)−1+αlog(νI0)}+KkF0k2≤K1max{(νI0)−2α,(νI0)−1+αlog(νI0)}, for some K1>0. By the last claim of Lemma 7.2, the same type of computations imply that, if S2, S′ 2∈ BK∗, Tb Gu s◦b Fin,a(S2)−b Gu s◦b Fin,a(S′ 2)U1≤Kkb Fin,a(S2)−b Fin,a(S′ 2)k2 ≤Kκ−1+ (νI0)α−1+K∗TS2−S′ 2U1. Then, taking K∗= 2Tb Gu s◦b Fin,a(0)U1= 2K1max{(νI0)−2α,(νI0)−1+αlog(νI0)}, the claim follows with the usual argument, taking κand νI0large enough so that Kκ−1+ (νI0)α−1+K∗<1/2. 8 Difference between solutions of the inner equation In this section we compute the difference between the functions T±=T0+L± in +T± 2, with T+is given in D+ κ,in ×Tσwhere D+ κ,in is defined in (6.10) and T0(v) = −1/(4v)is introduced in (6.3), L+ in is defined in (6.8) and T+ 2is given by Proposition 6.5. Moreover, the function T−is defined by T−(v, θ) = −T+(−¯v, −¯ θ) = T0(v)−L+ in(−¯v, −¯ θ)−T+ 2(−¯v, −¯ θ)(8.1) in the domain (−D+ κ,in)×Tσ. We also define L− in(v, θ) = −L+ in(−¯v, −¯ θ)and T− 2(v, θ) = −T+ 2(−¯v, −¯ θ).(8.2) The following lemma is an immediate computation. It follows from the fact that Vrestricted to Ris an even function, i.e. only depends on cosinus. We recall that E(T0)(v, θ) = −V(θ)/8v2. Lemma 8.1. We have that L− in(v, θ) = Z∞ vE(T0)(s, θ +s−v)ds =−X k∈Z V[k] 8eik(θ−v)Z∞ v 1 s2eiks ds. It satisfies the analogous bounds of L+ in in Proposition 6.3. 41 Figure 8: The domain Dκdefined in (9.3), shaded in gray. Compare with Figure 4. 9.1 Straightening the linear operator Since Φ±are solutions of (4.3), e ∆out satisfies e Le ∆out = 0,(9.6) where e Le ∆ = 1 + B 1 + A∂ue ∆ + νI0∂θe ∆ and A=ν 2νI0∂θΦ++∂θΦ−, B=1 2 (1 + u2)2 u2∂uΦ++∂uΦ−−1. (9.7) We look for a change of variables of the form u=v+X(v, θ)such that ∆out(v, θ) = e ∆out(v+X(v, θ), θ) satisfies L∆out = 0, where L∆ = ∂u∆ + νI0∂θ∆was introduced in (4.7). Notice that, provided v7→ v+X(v, θ)is invertible, e ∆out satisfies (9.6) if and only if ∆satisfies 1 1 + ∂vX 1 + B 1 + A|(u,θ)=(v+X(v,θ),θ)−νI0∂θX!∂v∆ + νI0∂θ∆ = 0. Hence, we need to impose that Xsatisfies LX=B−A 1 + A|(u,θ)=(v+X(v,θ),θ) .(9.8) To solve this equation, we consider the inverse Gout,d of the operator Ldefined by the Fourier coefficients of the image Gout,d(X): Gout,d(X)[k](u) = Zu u+ 0 eikνI0(s−u)X[k](s)ds, k > 0, Gout,d(X)[0](u) = Zu u0 X[0](s)ds, Gout,d(X)[k](u) = Zu u− 0 eikνI0(s−u)X[k](s)ds, k < 0, (9.9) 48 where u+ 0,u− 0=u+ 0and u0∈Rare topmost, bottommost and real leftmost points in Dκ, respectively. See Figure 8. Remark 9.1. The choice of the points u± 0and u0implies that, if X(u, θ)is real analytic, so is Gout,d(X). To solve equation (9.8), for r∈R, we introduce the Banach space of analytic, 2π-periodic in θ functions Xs={X:Dκ×Tσ→C| kXks<∞}, with the norm kXks=X k∈ZkX[k]kse|k|σ, where, for an analytic function f:Dκ→C, kfks= sup u∈Dκ|(1 + u2)sf(u)|. Next, we state two technical lemmas completely analogous to Lemmas 4.2 and 4.3. Lemma 9.2. Let s1, s2∈R. (1) If X∈ Xs1+s2and s2≥0, then X∈ Xs1and there exists K > 0, independent of s1, s2such that kXks1≤KνI0 κs2 kXks1+s2. (2) If X1∈ Xs1and X2∈ Xs2, then X1X2∈ Xs1+s2and kX1X2ks1+s2≤ kX1ks1kX2ks2. Lemma 9.3. Let s∈Rand Gout,d be the operator defined by (9.9). (1) If X∈ Xswith s≥0, then Gout,d(X)∈ Xsand kGout,d(X)ks≤KkXks. If, furthermore, hXi= 0, kGout,d(X)ks≤K(νI0)−1kXks. (2) If X∈ Xswith s > 1, then Gout,d(X)∈ Xs−1and kGout,d(X)ks−1≤KkXks. (3) If X∈ Xswith s > 0, then ∂uGout,d(X), ∂θGout,d(X)∈ Xsand k∂uGout,d(X)ks≤KkXks,k∂θGout,d(X)ks≤K(νI0)−1kXks. Now, we summarize the properties of Aand Bwe need. Lemma 9.4. Let Aand Bbe the functions introduced in (9.7). There exists K > 0such that (1) A∈ X2with kAk2≤K(νI0)−2. (2) B=B1+B2where B1∈ X1,hB1i= 0,kB1k1≤K(νI0)−1and B2∈ X2with kB2k2≤ K(νI0)−2. Consequently, B∈ X1with kBk1≤K(νI0)−1. Proof. To prove (1), we use the definitions in (9.1) and (9.7). From Proposition 4.4 we have k∂θL+ outk2≤K(νI0)−1and, from Proposition 4.5 and (1) of Lemma 9.2, k∂θΦ+ 2k2≤νI0 κ2 k∂θΦ+ 2k4≤νI0 κ2 (νI0)−1TΦ+ 2U3≤Kκ−2(νI0)−1. 49 Since the bounds of L− out and Φ− 2are the same as the ones of L+ out and Φ+ 2, respectively, and ∂θΦ0= 0, we have kAk2=ν 2νI0∂θΦ++∂θΦ−2≤ν νI0k∂θL+ outk2+k∂θΦ+ 2k2≤K(νI0)−2. Now we prove (2). Using the definition of Φ0in (4.5) we can check that B=B1+B2, where B1(u, θ) = 1 2 (1 + u2)2 u2(∂uL+ out(u, θ) + ∂uL− out(u, θ)), B2(u, θ) = 1 2 (1 + u2)2 u2(∂uΦ+ 2(u, θ) + ∂uΦ− 2(u, θ)). By Proposition 4.4,k∂uL+ outk3≤K(νI0)−1and since k(1+u2)2/u2k−2=O(1), by (2) of Lemma 9.2, kB1k1≤ 1 2 (1 + u2)2 u2−2k∂uL+ out +∂uL− outk3≤K(νI0)−1. Since hL+ outi= 0 and (1 + u2)2/u2does not depend on θ,hB1i= 0. Finally, by Proposition 4.5,k∂uΦ+ 2k4≤TΦ+ 2U3≤K(νI0)−2. Then, kB2k2≤ 1 2 (1 + u2)2 u2−2k∂uΦ+ 2+∂uΦ− 2k4≤K(νI0)−2. Using the operator Gout,d, we consider the fixed point equation X=Gout,d ◦Fout,d(X),(9.10) where Fout,d(X) = B−A 1 + A|(u,θ)=(v+X(v,θ),θ) .(9.11) It is clear that if Xsatisfies (9.10), then Xis a solution of (9.8). Let X0=Gout,d ◦Fout,d(0). Proposition 9.5. X0∈ X1and kX0k1≤K(νI0)−2. Furthermore, X0is real analytic. Proof. We have that Fout,d(0) = B−A 1+A=B1+B2−A(B1+1) 1+A. By Lemma 9.4,kB1k1≤K(νI0)−1and hB1i= 0. Then, by (1) of Lemma 9.3,kGout,d(B1)k1≤ K(νI0)−1kB1k1≤K(νI0)−2. Also, by Lemma 9.4,kB2k2≤K(νI0)−2. Then, the claim follows from the bound  B2−A(B1+ 1) 1 + A2≤ kB2k2k(1 + A)−1k0+kAk2kB1+ 1k0k(1 + A)−1k0 and (2) of Lemma 9.3. The following lemma is analogous to Lemma 8.3. The only difference is the geometry of the domain. Actually Dκdepends on κ, δ, β1and β2. Having fixed some κ0,δ0,β1and β2with β2> β1 we will write b Dκto denote a family of domains with κ > κ0and δ < δ0and with the same angles β1, β2such that if ˜κ > κ > κ0the distance from b D˜κto the boundary of b Dκis (˜κ−κ)(νI0)−1cos β1. This implies determining ˜ δ, depending on ˜κso that (δ−˜ δ) cos β2= (˜κ−κ)(νI0)−1cos β1. Note that a change in ˜κof order one produces a change in ˜ δof order (νI0)−1. Lemma 9.6. Let C∈ Xsin b Dκ×Tσand X∈ Xtin b D˜κ×Tσwith s, t ≥1and ˜κ > κ. We define CX(v, θ) = C(v+X(v, θ), θ),(v, θ)∈b D˜κ×Tσ. Then, there exists κ0such that for any ˜κ > κ > κ0there exist K0, K1>0, depending on ˜κ, κ but independent on νI0, such that the following holds. 50 (1) ∂j vC∈ Xs+jin b D˜κ×Tσand k∂j vCks+j≤K1j!Kj 0kCks,j≥0. (2) If X∈ Xtin b D˜κ×Tσ, then CX∈Xsin b D˜κ×Tσand kCXks≤ kCks K 1−K0(νI0˜κ−1)tkXkt . (3) If X, X′∈ Xtin b D˜κ×Tσ, with kXkt,kX′kt≤K, kCX−CX′ks+1 ≤K(νI0˜κ−1)tkCkskX−X′kt. Proof. (1) is an immediate consequence of Cauchy estimates in the reduced domain. Using (1) of Lemma 9.2, (2) follows from kCX(v, θ)ks=X j≥0 1 j!∂j vC(v, θ)Xj(v, θ)s ≤X j≥0 1 j!k∂j vC(v, θ)kskXkj 0 ≤X j≥0 K j!νI0 ˜κtj k∂j vC(v, θ)ks+jkXkj t≤KK1X j≥0 νI0 ˜κt K0kXkt!j kCks =kCks KK1 1−νI0 ˜κtK0kXkt . (3) follows from C(v+X(v, θ), θ)−C(v+X′(v, θ), θ) =Z1 0 ∂vC(v+X′(v, θ) + s(X(v, θ)−X′(v, θ)), θ)ds (X−X′), (1) and (2) and the fact that kXk0≤K(νI0/˜κ)tkXkt. To solve equation (9.10), we introduce e Xby setting X=X0+e X. Then, Xis a solution of (9.10) if and only if e Xsatisfies e X=Gout,d ◦e Fout,d(e X),(9.12) where e Fout,d(e X) = Fout,d(X0+e X)−Fout,d(0). Proposition 9.7. There exists κ0such that, for any κ > κ0, equation (9.12)has a unique solution e X1∈ X1with ke X1k1≤K(νI0)−2/κ2. As a consequence, X=X0+e X1∈ X1is a solution of (9.10), kXk1≤K(νI0)−2and it is real analytic. Proof. We first remark that, by Lemma 9.2 kAk0≤K(νI0)2kAk2/κ2,kAk1≤KνI0kAk2/κ, and taking into account Lemma 9.4,(B−A)/(1 + A)∈ X1with k(B−A)/(1 + A)k1≤ kB−Ak1k(1 + A)−1k0≤K(νI0)−1. Since X0∈ X1in b Dκwith kX0k1≤K(νI0)−2, by (3) of Lemma 9.6, taking X′= 0,s= 1, t= 1 and ˜κ > κ we have that e Fout,d(0) ∈ X2in b D˜κwith ke Fout,d(0)k2≤KνI0 ˜κ B−A 1 + A1kX0k1≤Kκ−1(νI0)−2. Hence, by (2) of Lemma 9.3,kGout,d ◦e Fout,d(0)k1≤ kGout,dkke Fout,d(0)k2≤K(νI0)−2/˜κ, where here kGout,dkis the norm of the operator Gout,d :X2→ X1. Let K∗= 2kGout,d ◦e Fout,d(0)k1. With the same argument, if X, X′∈ X1with kXk1,kX′k1≤K∗, ke Fout,d(X)−e Fout,d(X′)k2≤KνI0 ˜κ B−A 1 + A1kX−X′k1≤K ˜κkX−X′k1in b D˜κ, 51 for some K > 0, independent of νI0and ˜κ. Finally, using again (2) of Lemma 9.3, kGout,d ◦e Fout,d(X)−Gout,d ◦e Fout,d(X′)k1 ≤ kGout,dkke Fout,d(X)−e Fout,d(X′)k2≤kGout,dkK ˜κkX−X′k1, which proves that Gout,d ◦e Fout,d is a contraction in b D˜κif ˜κis large enough. Now we can check that Gout,d ◦e Fout,d sends BK∗⊂ X1into itself and therefore Gout,d ◦e Fout,d has a unique fixed point in that ball. The real analyticity claim follows from the definitions of Fout,d and e Fout,d and Remark 9.1. Proposition 9.8. Let Xbe the function, in the domain ˆ Dκ×Tσ, given by Proposition 9.7. Then, the map Θ(v, θ) = v+X(v, θ) θ is a well defined change of variables in ˆ Dκ×Tσ. Its inverse, Θ−1(u, θ) = u+Y(u, θ) θ=u−X(u, θ) + b X(u, θ) θ, is well defined and real analytic in ˆ D˜κ×Tσ, for some ˜κ > κ, with Y∈ X1and kYk1≤K(νI0)−2. Moreover Y=−X+b Xwith b X∈ X1and kb Xk1≤κ−1(νI0)−3in ˆ D˜κ×Tσ. Proof. The function Yis the solution of the fixed point equation Y=P(Y), where P(Y)(u, θ) = −X(u+Y(u, θ), θ). Clearly, P(0) = −X∈ X1, with kP(0)k1=kXk1≤K(νI0)−2. We define K∗= 2K1kXk1. Then, by (2) of Lemma 9.6, if Y∈ BK∗⊂ X1,kP(Y)k1≤K(νI0)−2. Since, by Proposition 9.7,X∈ X1, using (3) of Lemma 9.6,Prestricted to BK∗is Lipschitz with Lipschitz constant K˜κ−1νI0kXk1≤K˜κ−1(νI0)−1. Hence, it is a contraction if ˜κ−1(νI0)−1is small enough. Now, one easily checks that P(BK∗)⊂ BK∗. Let Y0be the unique fixed point of Pin BK∗. Then, writing Y0=−X+b X, kb Xk1=kY0+Xk1=kP(Y0)−P(0)k1≤K1 κνI0kY0k1≤K1 κ(νI0)3. 9.2 The exponentially small formula for Φ+−Φ− In this section we finally obtain the formula for the difference e ∆out = Φ+−Φ−for real values of uand θ. First, we introduce Υ+(u, θ) = νI0∆in(νI0(u+Y(u, θ)−i), θ), where the function Yis given by Proposition 9.8.Υ+is defined in {u∈C| ℑu < 1−κ(νI0)−1+ min{tan β1ℜu, −tan β1ℜu}} which contains the domain Dκ. We also define Υ−(u, θ) = Υ+(u, θ)(9.13) and Υ=Υ++ Υ−.(9.14) This definition implies that Υis real analytic. We recall that, since Lin∆in = 0 (see Theorem 8.6), we have ∆in(v, θ) = X k≥1 fkeik(θ−v), 52 where the coefficients fkdo not depend on νI0. Then, introducing Υk=νI0fke−kνI0, k ≥1,(9.15) we have Υ+(u, θ) = X k≥1 Υkeik(θ−νI0u−νI0Y(u,θ)) and Υ−(u, θ) = X k≤−1 Υ−keik(θ−νI0u−νI0Y(u,θ)), since Yis real analytic. The function Υwill be the desired first order of Φ+−Φ−. In order to prove this fact, we introduce E(u, θ) = e ∆out(u, θ)−Υ(u, θ) defined in Dκ. Proposition 9.9. Let N∈Nbe fixed. There exists K > 0and s > 0such that, for all (u, θ)∈ Dslog(νI0)×Tσsuch that νI0(u−i)∈ D+ slog(νI0),α ∩−D+ slog(νI0),α,0≤j+k≤N, |∂j u∂k θE(u, θ)| ≤ K(νI0)j+1 max{(νI0)−2α,(νI0)−1+αlog(νI0),(νI0)−s} log(νI0), where the domain Dκwas introduced in (9.3)and D+ κ,α in (7.4). Proof. Since Eis real analytic, it is enough to bound it for ℑu≥0. We write E=E1+E2+E3, where E1(u, θ) = Φ+(u, θ)−Φ−(u, θ)−νI0T+(νI0(u−i), θ)−T−(νI0(u−i), θ), the function T+was given by Corollary 6.6 and T−was introduced in (8.1), E2(u, θ) = νI0T+(νI0(u−i), θ)−T−(νI0(u−i), θ)−Υ+(u, θ) and E3(u, θ) = −Υ−(u, θ). We bound each term separately. We start with E1. We claim that |E1(u, θ)| ≤ KνI0 max{(νI0)−2α,(νI0)−1+αlog(νI0)} slog(νI0).(9.16) Indeed, for (u, θ)∈ Dslog(νI0)×Tσsuch that νI0(u−i)∈ D+ slog(νI0),α ∩−D+ slog(νI0),α, by Proposition 7.5, |E1(u, θ)| ≤ 2νI0kS+ 2k0≤KνI0 slog(νI0)kS+ 2k1≤KνI0 max{(νI0)−2α,(νI0)−1+αlog(νI0)} slog(νI0). To bound E2we observe that, by Theorem 8.6 and Propositions 9.5 and 9.8, E2(u, θ) = νI0∆in(νI0(u−i) + Z(νI0(u−i), θ), θ)−∆in(νI0(u−i+Y(u, θ)), θ). Also, taking into account that kZk1≤Kand kYk1=K(νI0)−2, we have that |Z(νI0(u−i), θ)| ≤ K νI0|u−i|, |νI0Y(u, θ)| ≤ K νI0|u−i|. 53 Hence, by the mean value theorem, for (u, θ)∈ Dslog(νI0)×Tσwe have that |E2(u, θ)| ≤ KνI0 s(νI0)slog(νI0)4. Finally, to bound E3we first observe that, for (u, θ)∈ Dslog(νI0)×Tσwith ℑu≥0, since Yis real analytic, which implies that ℑY(ℜu, ℜθ) = 0, and kYk1=O((νI0)−2), ℑ(u+Y(u, θ)) ≥(1 −k∂uYk0)ℑu≥(1 −K(νI0)2/(slog(νI0))2kYk1)ℑu ≥(1 −K/(slog(νI0))2)ℑu≥0, where we have used Cauchy estimates to relate ∂uYand Y, and we have slightly reduced the domain by considering a bigger value of s. Hence, from the definition of Υ−in (9.13) and (9.15), for (u, θ)∈ Dslog(νI0)×Tσwith ℑu > 0, |E3(u, θ)| ≤ X k≤−1 νI0|f−k|ek(νI0−σ)ek(νI0ℑu(1+O((νI0)−1))) ≤X k≤−1 νI0|f−k|ek(νI0−σ)≤KνI0e−(νI0−σ). The bounds of the statement follow applying standard Cauchy estimates and slightly reducing the domain. Finally, we need the following elementary lemma. Lemma 9.10. Let Ψ : Dslog(νI0)×Tσ→Cbe an analytic function. We write Ψ(u, θ) = Pk∈ZΨ[k](u)eikθ. Assume (i) LΨ=0. (ii) There exists M > 0such that |Ψ[k](±(i−is log(νI0)/(νI0)))| ≤ M. Then, Ψ(u, θ) = Pk∈ZΛkeik(θ−νI0u)and |Λ±k| ≤ M(νI0)kse−kνI0, k ≥1. Proof. Since LΨ = 0, there exists a 2π-periodic function Λ(φ) = Pk∈ZΛkeikφ such that Ψ(u, θ) = Λ(θ−νI0u). Since Ψis also periodic with respect to θ, we have that Ψ(u, θ) = Pk∈ZΨ[k](u)eikθ, that is Ψ[k](u) = Λke−ikνI0u. The claim follows evaluating the above equality at u=i−is log(νI0)/(νI0), for k≥1, and at u=−i+is log(νI0)/(νI0), for k≤ −1. Theorem 9.11. Fix αsuch that min{2α, 1−α}>0. Take 0< s < min{2α, 1−α}and fix N∈Nas in Proposition 9.9. There exists Λ0∈Rsuch that, for all (u, θ)∈(Dslog(νI0)∩R)×Tσ, 0≤j+k≤N. Let ∂j u∂k θ(e ∆out(u, θ)−Λ0) = ∂j u∂k θΥ(u+Y(u, θ), θ) + OνI0e−νI0 log(νI0) =νI0e−νI02f1∂j u∂k θ(cos(θ−νI0u)) + O1 log(νI0), where f1is given in Theorem 8.6. Furthermore, if one replaces the riin the definition of the coefficients riof Vin (1.2)by εri, we have that f1=επr1 4+O(ε2).(9.17) 54 Proof. By Proposition 9.7 and Theorem 8.6, we have that e E(u, θ) = e ∆out(u+X(u, θ), θ)−Υ(u+X(u, θ), θ) satisfies Le E= 0. Since kXk1≤K(νI0)−2,e Esatisfies the same bounds as Egiven by Proposition 9.9. Writing e E(u, θ) = Pk∈Ze Ekeik(θ−νI0u), by Lemma 9.10 we have that |e E±1| ≤ KνI0 log(νI0)e−νI0, |e E±k| ≤ KνI0 log(νI0)e−kνI0(1−slog(νI0))/(νI0), k ≥2. Hence, for (u, θ)∈(Dslog(νI0)∩R)×Tσ, |e E(u, θ)−e E0| ≤ KνI0 log(νI0)e−νI0.(9.18) Since kYk1≤K(νI0)−2, we have that |Y(u, θ)| ≤ K(νI0)−2for u∈R. Then, from the definition of Υin (9.14) and using that f1∈R, we have that, for (u, θ)∈(Dslog(νI0)∩R)×Tσ, Υ(u, θ) = νI0 X k≥1 fke−kνI0eik(θ−νI0(u+Y(u,θ))) +X k≤−1 f−kekνI0eik(θ−νI0(u+Y(u,θ)))  =νI0e−νI02f1cos(θ−νI0u) + O((νI0)−1). Hence, using again kYk1≤K(νI0)−2,Ealso satisfies (9.18), for (u, θ)∈(Dslog(νI0)∩R)×Tσ, and, from e ∆out = Υ + E, e ∆out(u, θ)−Λ0=νI0e−νI02f1cos(θ−νI0u) + O((log(νI0))−1), where Λ0=e E0. The last claim follows immediately from (9.15) and Theorem 8.6. 10 Acknowledgements F. B has been partially supported by the grant PID2021-122711NB-C’21, E.F. has been partially supported by the grant PID2021-125535NB-I00, and P.M. has been partially supported by the grant PID2021-123968NB-I00, funded by the Spanish State Research Agency through the programs MCIN/AEI/10.13039/501100011033 and “ERDF A way of making Europe”. Also, E.F. and P.M. authors have been partially supported by the Spanish State Research Agency, through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M). A Proof of Theorem 3.2 We first introduce a new time in system (3.3) so that the origin becomes a true saddle. Since the solutions of (3.3) with initial condition (u0, v0, t0)∈Vρwith u0, v0≥0and u0+v0>0satisfy u(t) + v(t)≥0while they belong to Vρ, we define the new time ssuch that dt/ds = (u+v)−1. Equation (3.3) becomes u′=u(1 + O1(u, v)), v′=−v(1 + O1(u, v)), t′= (u+v)−1, (A.1) where ′denotes d/ds. The O1(u, v)terms depend on sand are uniformly bounded in terms of (u, v)in Vρ. Given w0= (u0, v0, t0)∈Vρ, we define sw0= sup {s > 0|w(˜s)∈Vρ,∀˜s∈[0, s)},(A.2) where wis the solution of (A.1) with initial condition w0. Next lemma implies (1) of Theorem 3.2. Its proof is postponed to Appendix B. 55 Lemma A.1. There exist ρ∈(0,1) and C > 0, satisfying Cρ < 7/8, such that the solution w= (u, v, t)of (3.3)with initial condition w0= (u0, v0, t0)∈Vρwith u0, v0>0satisfies log ρ u01 1+Cρ !≤sw0≤log ρ u01 1−Cρ !. Moreover, for any 0< a ≤ρand 0< δ < a/2, the “Poincaré map" Ψ : Σ1 a,δ →Σ0 a,δ1−Ca , where the sets Σ0 a,δ and Σ1 a,δ are defined in (3.4), is well defined and, if w= (u0, a, t0)∈Σ1 a,δ and Ψ(w) = (a, v1, t1), then u1+Ca 0≤v1≤u1−Ca 0, e C1u−(1−Ca)/2 0≤t1−t0≤e C2u−(1+Ca)/2 0 for some constants e C1,e C2>0depending only on ρ. Let w= (u, v, t)be a solution of (3.3) with initial condition w0∈Vρ. We introduce τ=v/u (A.3) and we will write Oi=Oi(u, v). We have that 0< τ(s)<∞, for all ssuch that w∈Vρ. It is immediate from (A.1) that dτ ds =−(2 + O1)τ. (A.4) The variational equations around a solution of system (3.3) are   ˙ U ˙ V ˙ T = 2u+v+O2u(1 + O1)uO2 −v(1 + O1)−u−2v+O2vO2 0 0 0   U V T .(A.5) To prove (2) of Theorem 3.2, we will study the behavior of the solutions of (A.5) with initial condition U=U06= 0 along solutions of (3.3) with initial condition w0= (u0, v0, t0)∈Vρ∩{u, v > 0}for v0=asmall but fixed and u0arbitrarily small. Equations (A.5) become, in the time sin which the equations in (A.1) are written, and using τin (A.3),  U′ V′ T′ =  2+τ+O1 1+τ 1+O1 1+τuO1 −(1+O1)τ 1+τ−1+2τ+O1 1+τvO1 0 0 0   U V T .(A.6) Proposition A.2. There exists α∗, with 0< α∗<5/12 such that for any ρ > 0small enough, any w= (u, v, t), solution of (A.1)with w|s=0 =w0= (u0, v0, t0)∈Vρand any α∗ 0∈[0, α∗], there exists α: [0, sw0]→R,C∞, where sw0was defined in (A.2), with α(0) = α∗ 0, such that, introducing the new variable e V=V+αU, equation (A.6)becomes   U′ e V′ T′ =  2+τ+O1 1+τ−α1+O1 1+τ 1+O1 1+τuO1 0−1+2τ+O1 1+τ+α1+O1 1+τvO1 0 0 0    U e V T .(A.7) Furthermore, for s∈(0, sw0], 0< α(s)<55 128 τ(s) 1 + τ(s).(A.8) 56 Proof. Given αand e V=V+αU, since τ > 0, the equation for e Vis e V′=−(1 + O1)τ 1 + τ+α′+ (3 + O1)α−α21 + O1 1 + τU(A.9) +−1 + 2τ+O1 1 + τ+α1 + O1 1 + τe V+ (v+αu)TO1.(A.10) The claim will follow finding an appropriate solution of α′=ν0+ν1α+ν2α2,(A.11) where ν0=(1 + O1)τ 1 + τ, ν1(s) = −3 + O1, ν2=1 + O1 1 + τ. Let f(w, α) = ν0+ν1α+ν2α2be the right hand side of (A.11), where we have omitted the dependence of νi,i= 1,2,3, on wand s. We introduce α0and α1, the nullclines of (A.11), by f(w, α) = ν2(α−α0(τ))(α−α1(τ)), and R, where α0(τ) = −ν1 2ν2 1−1−4ν0ν2 ν2 11/2! =3 2+O1 1 + τ−(1 + τ)2−4 9+O1τ1/2! =3 2+O11 + τ−pR(τ). To complete the proof of Proposition A.2, we need the following two auxiliary lemmas. Lemma A.3. The function α0has the following properties. For (u, v)∈Va(that is, 0< τ < ∞), (1) 2√2/3 + O1≤pR(τ)/(1 + τ)<1, (2) limτ→∞ α0(τ) = 1/3 + O1, (3) limτ→0α0(τ)/τ = 1/3 + O1, (4) d dsα0=−(1 + O1)√R−τ+ 1 + O1 √Rα0, (5) −(2 + O1) √Rα0≤d dsα0≤ −(16/9 + O1) √Rα0 (6) limτ→0(dα0/ds)/α0=−2 + O1. Furthermore, 0< α0(τ)<11 32 τ 1 + τ.(A.12) Proof. Items (1) to (6) are proven in [GK12]. The bound (A.12) follows from a direct computation. Next lemma provides solutions of (A.11) close to the nullcline α0. The proofs of the next two lemmas are given in Appendix B. Lemma A.4. For any 0< ρ < 1small enough, the following is true. For any solution w= (u, v, t) of (3.3)with initial condition w0∈Vρ×T, if αis a solution of (A.11)with 0≤α(s0)≤ 5α0(τ(s0))/4for some 0< s0< sw0, then 0< α(s)<5α0(τ(s))/4for all s∈[s0, sw0]. 57