EXPONENTIALLY SMALL SPLITTING OF SEPARATRICES IN THE PERTURBED MCMILLAN MAP

The McMillan map is a one-parameter family of integrable symplectic maps of the plane, for which the origin is a hyperbolic fixed point with a homoclinic loop, with small Lyapunov exponent when the parameter is small. We consider a perturbation of the McMillan map for which we show that the loop breaks in two invariant curves which are exponentially close one to the other and which intersect transversely along two primary homoclinic orbits. We compute the asymptotic expansion of several quantities related to the splitting, namely the Lazutkin invariant and the area of the lobe between two consecutive primary homoclinic points. Complex matching techniques are in the core of this work. The coefficients involved in the expansion have a resurgent origin, as shown in [14]. 1. Preliminaries and main results. 1.

(Communicated by Amadeu Delshams) Abstract. The McMillan map is a one-parameter family of integrable symplectic maps of the plane, for which the origin is a hyperbolic fixed point with a homoclinic loop, with small Lyapunov exponent when the parameter is small. We consider a perturbation of the McMillan map for which we show that the loop breaks in two invariant curves which are exponentially close one to the other and which intersect transversely along two primary homoclinic orbits. We compute the asymptotic expansion of several quantities related to the splitting, namely the Lazutkin invariant and the area of the lobe between two consecutive primary homoclinic points. Complex matching techniques are in the core of this work. The coefficients involved in the expansion have a resurgent origin, as shown in [14].

Preliminaries and main results.
1.1. Introduction. This article and its companion [14] are devoted to the study of the exponentially small splitting of separatrices in a particular family of maps of the plane: a two-parameter family of analytic symplectic maps, which contains a one-parameter subfamily composed of integrable maps known as the McMillan map. The McMillan map was introduced in [15] in connection with the modelization of particle accelerator dynamics; it has a hyperbolic fixed point at the origin, for which there is a homoclinic loop. We prove that, generically, for the perturbed McMillan map (i.e. for our two-parameter family) the homoclinic connection is destroyed: it splits in two invariant curves (stable and unstable manifolds of the hyperbolic fixed point) which intersect transversely. We obtain an asymptotic formula for the area of the lobe delimited by the two curves between two consecutive intersection points and for the Lazutkin invariant, a quantity related to the angle of intersection, introduced in [10] and commonly used in the literature about splitting. Our results generalize and improve those of [4].
In the problem considered, the two parameters play very different roles. One of them, which we will call ε, is a regular parameter. It measures the size of the perturbation (the integrable McMillan map corresponds to ε = 0), and all the quantities and geometric objects under consideration will depend analytically on it; this parameter will not be assumed to be small. The other parameter, h, is precisely the Lyapunov exponent of the origin for the McMillan map. Hence, when this parameter tends to zero, the origin is a weakly hyperbolic fixed point; as a consequence, a well-known result in [6] shows that the splitting of the curves must be exponentially small with respect to h.
The problem of exponentially small splitting has been addressed by several authors (e.g. [19,5,1,23,13,16,2]), because of its relevance for the non-integrability of Hamiltonian systems (see [24] for its relation with Poincaré's mistake in his 1889 memoir) and for the Arnold diffusion mechanism in the case of at least three degrees of freedom. The problem was studied in detail mainly for flows, but there are relatively few works dealing with symplectic maps. The famous Lazutkin paper of 1984 (see [12] for the English translation) was the first work concerning the exponentially small separatrix splitting for a one-parameter family of maps, namely the standard map. Although important ideas where already present in that work, the complete proof of the results did not appear till fifteen years later, in [9]. Some asymptotic computations related with the problem of the exponentially small splitting of the standard map were done in [11,21] and for the Hénon map in [22].
The two-parameter family of maps considered in the present paper is essentially the same as in the article [4]. That article provided a rigorous asymptotic formula for the separatrix splitting in the case where the regular perturbation parameter ε is small enough with respect to the singular parameter h, validating the prediction of the Melnikov formula adapted for maps given by [3] (the possibility of taking ε = O(h p ) with p > 0 is an advantage of the presence of two parameters which has no analogue in a one-parameter family like the standard map). We shall remove the smallness assumption on ε, thus reaching a situation which displays the same complexity as the standard map. We shall see that in the non-perturbative case the Melnikov formula does not predict the correct size of the splitting, whereas it does when ε and h are small but independent. Furthermore, the formula we obtain provides the full asymptotic expansion in h of the first exponentially small term in the splitting.
We now give a brief description of our method and its innovative features. Our study splits in two parts, corresponding to "outer" and "inner" domains; we found it convenient to devote a separate article [14] to the inner part.
As in [12,9,4], the detection of the exponentially small splitting relies on considering suitable parametrizations of the invariant curves. These parametrizations will be analytic in a complex strip whose size is limited by the singularities of the unperturbed homoclinic orbit. When the perturbation parameter ε is small with respect to h, the manifolds are well approximated by the unperturbed homoclinic even off the real line, as in [4]. However, when ε is of order one, we need to deal with different approximations of the parametrizations of the invariant curves in different zones of the complex plane; the leading terms in the asymptotic expansion will be found as solutions of the so-called "inner equation". This equation needs its own study, using Borel resummation techniques and resurgence theory, and this is done in [14]. (A study of an inner equation of the same kind but for the Hénon map can be found in [7].) "Complex matching" techniques are then needed to conclude.
In order to have access to the whole asymptotic expansion with respect to h in the first exponentially small term of the formula of the splitting, we need to study not only the "first inner equation" but all the "inner equations" involved in the problem, related to higher order powers in h. This entails the use of resurgence theory in equations with parameters in [14] and matching procedures at any order in the present article.
One of the main differences between our work and the previous ones is the fact that we do not use "complex flow box variables" to obtain a good "splitting function" which measures the distance between both manifolds. Instead, we provide a formula for the difference of the parametrizations of the manifolds directly in the original variables of the problem-see formula (25) below. The key idea, that was already used in [18] in the case of flows, is to exploit a linear difference equation which is satisfied by this difference and for which a basis of solutions can be described precisely enough; the difference has to be a linear combination of the basis solutions with h-periodic coefficients and one can then resort to a classical lemma about periodic functions of a complex variable (Lemma 3.2) to obtain exponentially small bounds on the real line from larger bounds in a complex strip. where h > 0 is a parameter. It is a symplectic transformation of R 2 (for the standard structure dx ∧ dy), which is integrable in the sense that it admits the following polynomial first integral: H 0 (x, y) = x 2 − 2(cosh h)xy + y 2 + x 2 y 2 .
The origin is a hyperbolic fixed point, with characteristic exponents ±h. Its stable and unstable manifolds coincide: the level curve { H 0 = 0 } is formed of the origin and two homoclinic loops, one of which lies in the first quadrant and is explicitly given by W 0 = {z 0 (t), t ∈ R}, with z 0 (t) = ξ 0 (t − h/2), ξ 0 (t + h/2) and in such a way that F h,0 z 0 (t) = z 0 (t + h). We usually shall not write explicitly the dependence of ξ 0 on h. We shall refer to W 0 as "the unperturbed separatrix"; the other loop is obtained by symmetry with respect to the origin-see Figure 1.
Observe that for small h the homoclinic loops are small: z 0 (t) is O(h) uniformly in t. See [20] and [4] for more on the McMillan map.
From now on, we shall use the notation f (y) = 2y 1 + y 2 , µ = cosh h. The perturbation of the McMillan map that we consider is F h,ε : (x, y) → (x * , y * ) x * = y where the "perturbative potential" is an even analytic function, which is defined in a neighborhood of 0 and supposed to be O(y 4 ), and ε ∈ R is a new parameter (not necessarily small). The maps F h,ε are defined in a neighborhood of the origin and symplectic. The only difference with [4] is that we do not assumeṼ to be entire. SinceṼ (y) = O(y 3 ), the origin is still a hyperbolic fixed point with characteristic exponents ±h; its stable and unstable manifolds are curves which have no reason to coincide any longer. The aim of this paper is precisely to show that, generically, for nonzero ε and small h the stable and unstable curves intersect transversely, and to measure the way they depart one from the other; the homoclinic loops are broken, this is the so-called "separatrix splitting" phenomenon-see Figure 2. As is well-known, the existence of a transversal homoclinic intersection has dramatic dynamical consequences, even though the phenomenon is exponentially small.
We shall focus on the part W s h,ε , resp. W u h,ε , of the stable curve, resp. unstable curve, which lies in the first quadrant. Anyway, since the function µf + εṼ is odd, the dynamics of F h,ε is symmetric with respect to the origin. The analysis will be simplified by another kind of symmetry: the map F h,ε and its inverse F −1 h,ε are conjugate by the involution R : (x, y) → (y, x) (the map is "reversible"); this implies that W s h,ε = R(W u h,ε ).
Moreover, at least for small |ε|, both curves intersect the symmetry line ∆ R = {x = y} because they are close to the unperturbed separatrix W 0 and, by (4), a point of W u h,ε ∩ ∆ R is necessarily a homoclinic point (i.e. it also belongs to W s h,ε ).

Main theorem, geometrical version.
The article [4] shows that, whenṼ is entire and ε = o h 6 /| ln h| , there are generically two primary homoclinic orbits in the first quadrant for small h (one of which has a point on ∆ R ), and it yields an estimate of the lobe area enclosed by W u h,ε and W s h,ε between two successive intersection points (this area is invariant under the dynamics of F h,ε ). We shall see that the same result holds generically in our case with independent parameters ε and h (we shall assume h small but remove the smallness assumption on |ε|).
We shall estimate the algebraic lobe area A (with the same convention for its sign as in [4]-see below) and another quantity: the Lazutkin homoclinic invariant ω [10], the definition of which we now recall.
It is known that there must exist a "natural parametrization" for W u h,ε , i.e. this curve can be injectively parametrized by a solution t → z u (t) of (see e.g. [4], p. 328, or [10], and also Proposition 1.4 below). We shall see that there exists t * such that z u (t * ) ∈ ∆ R . We can assume that this occurs for t * = 0 (by shifting the parametrization if necessary: t → z u (t + t * ) is also solution of (5)).
Using reversibility and defining we then get a natural parametrization of W s h,ε and z s (0) = z u (0) is a homoclinic point. In this situation, the Lazutkin homoclinic invariant is ω = det(ż s (0),ż u (0)).
This is an intrinsic quantity, related to the splitting angle.
Here is the convention for the definition of the algebraic lobe area A: if the intersection of the curves is transversal, i.e. if ω = 0, the preservation of orientation by F h,ε implies that there must exist another homoclinic point between z u (0) = z s (0) and its image z u (h) = z s (h); we say that there are only two primary homoclinic orbits if there is only one such other point, say z s (t * ) = z u (t * * ) with 0 < t * , t * * < h; we then have 1 z s (t * ) = z u (h − t * ) (8) and we call A the area enclosed by the simple loop made of the path t ∈ [0, t * ] → z s (t) followed by t ∈ [t * , h] → z u (h − t), counted positively if and only if this loop is traveled anticlockwise (as on Figure 2).
There exist constants h 0 , c > 0 and real analytic functions B + k (ε), k ∈ N, holomorphic for complex ε of modulus < 1/|2V 2 |, such that -If 0 < h < h 0 and −ε 0 < ε < ε 0 , then W s h,ε and W u h,ε have an intersection point on the half-line {x = y > 0} at which the Lazutkin homoclinic invariant admits the following asymptotic expansion with respect to h: where α is the positive constant defined by -If moreover then the aforementioned intersection is transversal, there are only two primary homoclinic orbits in the first quadrant and the lobe area admits the following asymptotic expansion with respect to h: Theorem 1.1 will be proved in Section 1.5, as a consequence of Theorem 1.5 below.
Remark 1.2. In fact, we shall see in Section 1.5 that condition (13) can be replaced by a more technical but also more general one: there are constants c 0 , c 1 , . . . such that the result holds as soon as there exists an integer N 0 such that (still with 0 < h < h 0 and −ε 0 < ε < ε 0 ). Thus in principle, by an appropriate choice of N 0 , one may increase the range of validity of the result. In particular, if condition (13) fails because B + 0 (ε) happens to be zero, one can still try condition (15) with N 0 = 1, and so on. However, notice that we have little information on the numbers B + k (ε) (apart from the value of B + 0 at ε = 0-see Remark 1.3).
In (11) and (14), the symbol "∼" means that the series in the right hand sides are asymptotic to the left hand sides in the classical sense, i.e. truncating the series at order N provides an expression for the left hand side with an error that is of the order of the first neglected term within the range h ∈ (0, h 0 ) uniformly with respect to ε ∈ (−ε 0 , ε 0 ), with the restriction (13) or (15) in the case of A. However, the series in the right hand sides need not be convergent. In fact, numerical studies in [8] indicate that these series are Gevrey-1, i.e. that there exist constants C, M > 0 such that the coefficient B + k (ε) of h 2k is bounded by CM k (2k)!. The function V defined by (9) is an entire function (because of the Cauchy estimates for the Taylor coefficients ofṼ at the origin); it is the Borel transform ofṼ with respect to 1/y (see [14] for more on the Borel transform). Remark 1.3. Suppose V (2π) = 0 (which is true for genericṼ ). Then there exists ε 1 < ε 0 such that B + 0 (ε) = 0 for |ε| ≤ ε 1 ; thus condition (13) is fulfilled for −ε 1 < ε < ε 1 and 0 < h < h 1 with a value of h 1 independent of ε. This is thus an improvement of the range of validity of the result obtained in [4]: the Melnikov is valid for ε and h small and independent-one can relax the assumption ε = o h 6 /| ln h| . But our result is at the same time an extension to the case when ε is not small; then the Melnikov approximation is no longer correct: one must use the coefficient α −2 B + 0 (ε)ε instead of 4π 2 V (2π)ε. Another improvement is the fact that Theorem 1.1 provides the full asymptotic expansion, involving the new coefficients B + k (ε), k ≥ 1, for the Lazutkin invariant and the lobe area.
Furthermore, the appearance of the Borel transform V in the Melnikov approximation will receive a very natural explanation in our proof; this proof indeed relies on the Borel-Laplace summation process, which is at the basis of resurgence theory, and it attributes to the coefficients B + k (ε) a resurgent origin. The reader is referred to Section 2.7 and [14]. 1.4. Rephrasing in terms of solutions of a second-order difference equation. Analytic version of the theorem. To study the stable and unstable curves, we shall use natural parametrizations as alluded above, i.e.
, with z u and z s particular solutions of the system of first-order difference equations The property x * = y in (3) implies that t → z(t) is solution of (16) if and only if it can be written with t → ξ(t) solution of the second-order difference equation For instance, for the McMillan map (ε = 0), the function ξ 0 defined in (1) satisfies Finding a parametrization z u of W u h,ε which satisfies (5) is thus equivalent to finding a solution of (17) which satisfies lim t→−∞ ξ u (t) = 0 and ξ u (t) > 0 for −t large enough (19) and writing z u (t) = (ξ u (t − h/2), ξ u (t + h/2)) (the positivity condition in (19) is meant to distinguish the part of the unstable curve which starts in the first quadrant; the symmetry of this curve with respect to the origin is reflected in the fact that −ξ u is solution of (17) if ξ u is).
Proposition 1.4. For any h > 0 and ε ∈ R, there exists a solutionξ u of equation (17) which satisfies the boundary condition (19) and which is real-analytic and 2πi-periodic in a half-plane {Re t < −T * }, with a constant T * > 0 (which depends on h and ε). Moreover, such a solutionξ u (t) is unique up to a translatioñ ξ u (t) →ξ u (t − τ ) with arbitrary τ ∈ R (which may depend on h and ε).
Proof. With the change of variable ζ = e t , this corresponds to seeking a solution ζ → Z(ζ) of the equation Z(e h ζ) = F h,ε Z(ζ) , the components of which are holomorphic real-analytic near ζ = 0 and positive for small ζ > 0, with Z(0) = 0 (indeed, the 2πi-periodicity, the holomorphy in a half-plane and (19) imply the existence of a convergent Fourier expansion n≥1 e nt Z n ). It is easy to see that this problem has a solution which is unique up to rescaling Z(ζ) → Z(cζ) with arbitrary c > 0 (this is the analytic version of the stable manifold theorem for F −1 h,ε ; it is sufficient to look at the equations obtained by expanding a solution in the form Z(ζ) = n≥1 ζ n Z n , one finds Z 1 proportional to (1, e h ) with an arbitrary positive proportionality factor, the other terms are determined inductively and easy to bound). 2 From now on, we denote byξ u (t) one of the solutions given by Proposition 1.4. We shall see that it has an analytic continuation to any real interval (−∞, T ], provided h is small enough, 2 and choose τ ∈ R (depending on h and ε) so that satisfies the condition Equation (21) corresponds to the condition z u (0) ∈ ∆ R which was introduced at the beginning of Section 1.3. The reversibility property of F h,ε is reflected in the fact that t → ξ(−t) is solution of (17) whenever t → ξ(t) is. Once ξ u is found, the formula ξ s (t) = ξ u (−t) defines a solution ξ s of (17) which satisfies the boundary conditions (21) and lim t→+∞ ξ s (t) = 0 and ξ s (t) > 0 for t large enough, hence z s (t) = (ξ s (t − h/2), ξ s (t + h/2)) is a natural parametrization of W s h,ε which intersects W u h,ε at t = 0. The splitting problem is thus reduced to studying the difference Theorem 1.5 (Main Theorem, analytical version). Let ε 0 < 1/|2V 2 | and T > 0. There exist h 0 , C 0 > 0 such that, for any h and ε ∈ R with 0 < h < h 0 and |ε| < ε 0 , there exists a unique τ ∈ R such that ξ u (t) =ξ u (t − τ ) extends analytically to (−∞, T ], satisfies (21) and Moreover, there exists a sequence (ξ N,out ) N ≥0 of even real-analytic functions defined on R, with ξ 0,out = ξ 0 , and constants C N > 0 such that, for any N ≥ 0, where α is the constant defined in (12).

Consider the function
2 No smallness condition on h is needed for this whenṼ is defined on the whole of the real axis: the function F = µf + εṼ is then defined on R and the definition ofξ u can be propagated from (−∞, −T * ) to (−∞, −T * + h) and then to any interval (−∞, −T * + nh), n ≥ 1, by rewriting equation (17) asξ u (t) = F ξu (t − h) −ξ u (t − 2h). In the general case, the smallness of h ensures thatξ u (t) remains in the domain of definition ofṼ for t ∈ (−∞, T ] when using the same argument. There exist real analytic functions c 1 , c 2 , ν 1 , ν 2 defined in [−T, T ] such that and • c 1 and c 2 are h-periodic and where with real-analytic functions B + k , holomorphic for |ε| < ε 0 , satisfying (10), • ν 1 and ν 2 satisfy The proof of Theorem 1.5 will start in Section 2. Observe that, in view of (23), the defect of evenness measured by D(t) has to be O(h n ) for any n; in fact it is exponentially small, as shown by the exact formula (25) and the information on c 1 , c 2 , ν 1 , ν 2 provided in (26)-(31), and α −1 c N 2 (t) will account for the dominant part of the splitting phenomenon.
The functions ν 1 and ν 2 will be obtained as particular solutions of a certain linear second-order difference equation. In the theory of linear difference equations, the determinant is called discrete Wronskian (or Casoratian), and it is constant for a pair of solutions of the kind of equations we are interested in-see Section 4.
1.5. Deduction of Theorem 1.1 from Theorem 1.5. Let h ∈ (0, h 0 ), ε ∈ (−ε 0 , ε 0 ) and ξ u (t) be as in Theorem 1.5 and set ξ s (t) = ξ u (−t). We denote by z u = (x u , y u ) and z s = (x s , y s ) the natural parametrizations of the unstable and stable invariant manifolds defined by In view of (7) the Lazutkin invariant at the homoclinic point z u (0) = z s (0) can be written On the other hand, since ξ u − ξ s = D, (33) yields whence Lemma 1.6. For any N ∈ N, with a function E N (depending on N, h, ε) such that where the notation g = O(f ) means that there exists a constant C N > 0, that may depend on N but it is independent of h and ε, such that |g| ≤ C N |f | on the considered interval.
Proof. For any N ∈ N, writing the estimates (23) and (30) at N + 1, we have We thus define E = W h (ξ u − α −1 ν 1 , D) and, using (25), (29) and the h-periodicity of c 1 and c 2 , we get We have as a consequence of the bounds d j c1 dt j = O h −4−j e −π 2 /h (as stated in (26)), d j ν1 dt j = O(εh) (which follows from (1) and (30) (27) and (28) with N = 0) and d j ν2 dt j = O(h −2 ) (as stated in (31)). Together with (38), this implies and the conclusion follows. The asymptotic expansion (11) for ω follows from (35) and Lemma 1.6, since we can write ω = α −2ċN 2 (−h/2) +Ė N (−h/2) and (28) shows thatċ . We now assume that there exists N 0 such that (15) holds, with a constant c N0 that we shall specify later, and we proceed to show that there is only one primary homoclinic orbit other than the orbit of z u (0) = z s (0) and compute the lobe area. To this end, we shall use a linear change of variables, so as to make the manifolds appear as graphs over the first coordinate, and a reparametrization of W s h,ε . Figure 1 suggests the linear symplectic change of variables We definez u = (x u ,ỹ u ) andz s = (x s ,ỹ s ) by means of the above relations. By (33) and (1), at first order in ε one finds .
(42) Sinceẋ u − α −1ẋu |ε=0 andẋ s − α −1ẋs |ε=0 are O(εh 3 ) (because of (23) with N = 0), we can find K > 1 and t 0 > 0 independent of h and ε such that, for t ∈ [−t 0 , t 0 ], In particular,x u andx s are invertible in [−t 0 , t 0 ] and the manifoldsz u ,z s are graphs over thex variable. Moreover, setting t 1 = t 0 /K 2 , is well defined in (−t 1 , t 1 ) and a piece of W s h,ε can be reparametrized as Observe thatx u (0) =x s (0) = 0, thus φ(0) = 0 (and more generally φ(kh) = kh for k ∈ Z, |kh| < t 1 , sincex u andx s coincide on hZ). Homoclinic points correspond to solutions of the equatioñ We know that any t ∈ hZ ∩ (−t 1 , t 1 ) is solution of this equation, and we need to prove that (45) admits only one solution in the interval (0, h). If this is the case and if we denote by t * the unique solution of (45) in (0, h), then there will be exactly two primary homoclinic orbits, the orbits of z u (0) = z s (0) and z u (t * ) = z s φ(t * ) , and according to the definition of Section 1.3 the lobe area will be given by (because the change of variables (41) preserves algebraic area; notice that we'll have φ(t * ) = h − t * as a consequence of the computation of Section 1.3). Let us study equation (45) or, equivalently, the equation ∆(t) = 0.
Lemma 1.7. For any N ∈ N, there exist a positive constant C N (independent of h and ε) and a function F N (depending on N, h, ε) such that Proof. We first compute ψ = φ − Id in terms of the functions (the latter function is exponentially small, according to (39)). By Taylor's formula, Thus, again by Taylor's formula, Now, (ỹ u −ỹ s )ẋ u −ẏ sD = det(ż u ,z u −z s ) = det(ż u , z u − z s ) because (41) is symplectic and, by (34) and Lemma 1.6, for any N ∈ N the value at a point t of this determinant coincides with the value of The term E N (t − h/2) and its derivative are controlled by (37). We are thus left with the question of estimating G and its derivative; the result will follow from To derive (50), we first bound ψ, χ and their derivatives. By Thus, (50) is a consequence of the definition of G and of the bound d jỹs In view of (28), Lemma 1.7 yields, for any N ∈ N, (51) Moreover, ∆(0) = ∆(h) = 0 (in fact, ∆ vanishes on all integer multiples of h). By choosing appropriately N , we shall be in a position to apply next lemma, whose proof is an exercise.
Then ∆ has a unique zero in (0, h); this zero t * satisfies |t * − h 2 | < h 8 . Assuming that condition (15) holds for a certain integer N 0 with the constant c N0 defined as we get |b N0 | > 2C N0 |ε|e −π 2 /h h 2N0+1 (because α 2 < 2) and we can apply Lemma 1.8 to (51) with N = N 0 : inequalities (48) guarantee the existence and uniqueness of a zero of ∆ in (0, h). Now, for any N ∈ N, the zero t * ∈ ( 3h 8 , 5h 8 ) of ∆, which depends on h and ε but not on N , satisfies . The lobe area is thus given by and (14) follows, since the value of the first integral is precisely 1 π b N h, the second integral has absolute value < |b N (t * − h 2 )| = O εh 2N +2 e −π 2 /h and the third integral is O εh 2N +2 e −π 2 /h . 1.6. Description of the proof of the analytic Theorem 1.5. The rest of the paper is devoted to the proof of the Analytic Theorem 1.5. Here we give an informal description of the proof, pointing out the main steps.
The lengthiest and most cumbersome part consists in proving the existence of a suitable solution of the invariance equation (17) satisfying boundary conditions (19) and (21), ξ u , and obtaining a meaningful asymptotic formula for the difference between ξ u (t) and ξ s (t) = ξ u (−t). This is accomplished in several steps, which are listed in the form of theorems, in Section 2. The proof of those theorems, for the sake of clarity, is postponed to subsequent sections and [14].
The scheme of this first part of the proof is the following. First of all, in Proposition 2.1 we perform a scaling which allows to assume that the perturbationṼ is of order 5 instead of 3. This amounts for the constant α in the formulas of Theorems 1.1 and 1.5.
Next, in Section 2.2, we introduce the different domains where we shall work. It is clear that, in order to measure the area between the unstable and stable manifolds, the domains where their natural parametrizations are defined need to have a large enough intersection. On the other hand, the arguments to obtain an exponentially small term in the asymptotic formula rely on finding these natural parametrizations in the widest possible complex strip in t in which these parametrizations are holomorphic. The width of this strip is limited by the functions that appear in the approximations we use. Since the first term in these approximations will be the function ξ 0 defined by (1), whose singularities closest to the real line are located at ±iπ/2, the largest strip we shall be able to deal with is {|Im t| < π/2}.
We divide the domain in which we need to find ξ u in two parts, the outer domain and the inner domain (see Sections 2.2 and 2.8). The outer domain comprises points up to a distance δ of iπ/2, with some δ > h. The inner domain contains the points at a distance between δ and h of iπ/2. (We will choose δ = √ h in the end.) The good approximations of ξ u in the outer domain will be given by its asymptotic expansion in powers of h. We find it indirectly by first expanding in an auxiliary parameter in Section 2.4 (see Proposition 2.2) and expanding in h each coefficient of this auxiliary series, in Section 2.5, by means of the Euler-MacLaurin formula.
It turns out that the asymptotic series in h for ξ u is the same as the one for ξ s , which implies that the difference between the invariant manifolds is smaller than any power of h (see Corollary 2.9). However, these approximations are not longer accurate at points close to iπ/2. To study the behavior of ξ u there, we need to use different approximations.
The formal approach, in Section 2.6, consists in introducing a new variable t = iπ/2 + hz, and expand again in h and z, reordering the series obtained in the outer part. This procedure yields a new formal series whereφ j (z) are well-defined formal power series in z. The tool we use to give rigor to these formal expansions is the so-called resurgence theory. After the introduction of the new variable z, suggested by the above expansions, we expand the invariance equation (17) in powers of h to obtain a family of inner equations. In Section 2.7 we claim the existence of two families of solutions of the full hierarchy of equations, with prescribed expansions in z,φ j , one corresponding to ξ u and the other to ξ s , and an asymptotic formula for their difference (see Theorem 2.17). This study relies on very different techniques than those used here, and the proofs of the results we quote here are given in [14].
Once we have the solutions of the inner equations, in Section 2.9 we find the continuation of the function ξ u up to points with Im t = π/2 − h by matching the outer and inner series (see Theorem 2.18).
At this point, we shall have obtained two different approximations of ξ u and ξ s . The outer one will be good enough in the outer region, but without enough precision in the inner region to capture the exponentially small phenomena we want to study. The inner one will be more accurate; moreover, in the inner region, we shall have refined information on the difference between ξ u and ξ s at our disposal.
In parallel to this work, we will claim in Theorems 2.4 and 2.20 the existence and list some properties of an appropriate set of solutions of equation (69), which is the linearization of the invariance equation (17) around ξ u .
In Section 3 we prove Theorem 1.5. We use the results of Section 2 to obtain the asymptotic formula for D = ξ u − ξ s on the real line. Instead of introducing flow box coordinates as in [4,9], in Section 3.1 we take advantage of the fact that D satisfies a linear homogeneous second order difference equation We find a suitable set of fundamental solutions of this equation, {ν 1 , ν 2 }, using the fact that it is close to equation (69), and we estimate D(t) using that D(t) = c 1 (t)ν 1 (t) + c 2 (t)ν 2 (t), where c 1 and c 2 are h-periodic functions. We will use the already known asymptotic formula for D to obtain an asymptotic expression of the functions c 1 , c 2 . Finally, since c 1 and c 2 are analytic and periodic, we will bound their Fourier coefficients to obtain the desired formula.
The proof of several technical results is placed after Section 3.

2.
Approximation of the manifolds. In this section we find a particular solution ξ u of equation (17) satisfying boundary conditions (19) and (21), as well as three different approximations of this function. The first two are related to the asymptotic expansion of ξ u in powers of h and give arbitrarily good approximations of ξ u at points far from iπ/2, but they fail when t is O(h) close to iπ/2. The third approximation, which formally appears from a suitable reordering of the asymptotic expansion in h of ξ u , will provide the necessary approximation at points t close to iπ/2.

2.1.
Rescaling. We first perform a scaling in order to make the perturbative terms in ε of order five in ξ instead of order three. A straightforward computation yields the following (we use the notation V for ∂ ∂y V ): (12) and Then there exist h 0 , y 0 , C > 0 such that V extends holomorphically to the function V is odd with respect to y and even with respect to h, for all (y, h, ε) ∈ B, and the changeξ = αξ transforms equation (17) intõ Hereafter, we shall write again ξ instead ofξ.  (19) and (21) in some large complex domain, D u , which contains the real interval (−∞, T ]. By reversibility, this will yield a solution ξ s (t) = ξ u (−t) of the equation (54) with boundary conditions (22) and (21) The domain D u for ξ u splits in two domains in which ξ u will have different approximations. The outer domain D u,out δ depends on a parameter δ ∈ (0, π/2) and is depicted on Figure 3. It is defined as follows: The inner domain D u,in h = D u,in h (R) will be rigorously defined in Section 2.8 for any R > 0 such that Rh < δ (this domain will also depend on δ).
In the end, δ will be chosen √ h and R will be chosen after Theorem 2.14.
2.3. Unperturbed linearized invariance equation. It will be crucial for us to control the solutions of the linearization of equation (17) around ξ u . See Section 4 for basic techniques to deal with linear second order difference equations. For ε = 0, the linearization of (18) around ξ 0 is A fundamental set of solutions of this equation is {η 1 , η c 2 } for any c ∈ C, with where independently of c, where this Wronskian is defined according to (32).
We will be particularly interested in η 0 2 , since it is real analytic, but also in because η iπ/2 2 has better bounds around iπ/2.

Outer approximation.
Here we deal with the approximation of ξ u in D u,out We introduce a new parameter ε and replace F by We shall find a solution of the new equation and restore the relation ε = ε at the end. From now on, we will not write explicitly the dependence on h, ε and ε.
Looking for a solution of (61) of the form ξ = k≥0 ε k ξ k , we get equation (18) for the first term and an inductive system of equations for the coefficients ξ k , k ≥ 1: with f k depending only on ξ 0 , . . . , ξ k−1 : Let us use the notation D(0, ρ) = {z ∈ C | |z| < ρ}.
Proposition 2.2. Consider the sequence of equations given by (18) and (62), for k ≥ 1. Let ε 0 < 1/|2V 2 |. There exists h 0 > 0, such that for any h ∈ (0, h 0 ) and δ ∈ (h, π/2), there exists a unique sequence of real analytic functions (ξ u k ) k≥0 defined for (t, ε) ∈ D u,out δ × D(0, ε 0 ) and iπ-antiperiodic in t such that (1), and, for each k ≥ 0, there exists C k > 0, independent of h and δ, such that, for any (t, ε) ∈ D u,out The proof of this proposition can be found in Sections 5.3 and 5.4. Now, we put ε = ε and define the first outer approximation of order N as solution of (54), verifying boundary conditions (19), (21) and The proof of this theorem is placed in Sections 5.5-5.7. At this point, we can define which will provide a parametrization of the invariant stable manifold, being solution of the invariance equation (54) and satisfying the boundary conditions (22) and (21). The linearization of the invariance equation (54) around ξ u is the equation In the forthcoming arguments, we will need the two systems of fundamental solutions of this equation provided by the following , solutions of (69), satisfying the following properties: where ξ u,N is given in (66), and the functions η 0 2 , η iπ/2 2 defined by (56) and (57). Then, there exist h N , C, C N > 0, independent of h and δ, such If moreover Im t ≥ 0, then The proof of this theorem is placed in Sections 5.6-5.7.

2.5.
Outer expansion of ξ u and ξ s . In this section and the next ones we compute the asymptotic expansion in h of the function ξ u,N (t, h, ε) of (66). In view of (67) this provides an asymptotic expansion for ξ u (t, h, ε) up to order 2N + 1. It turns out that the coefficients of this asymptotic expansion are even functions of t, thus the approximation properties are equally valid for the stable solution ξ s (t, h, ε).
We will construct a finite sequence of functions (ξ N k ) k=0,...,N holomorphic in which will contain the asymptotic expansion of the functions ξ u k (t, h, ε) of Proposition 2.2 up to order h 2N +1 . Even though ξ N k (t, h, ε) will have an infinite expansion in powers of h which depends on N , the terms of degree ≤ 2N + 1 will not depend on N (provided 0 ≤ k ≤ N ). Proposition 2.6. Let ε 0 , h 0 as in Theorem 2.3. For any N ≥ 0, there exist a constant C N > 0 and a sequence of real analytic functions ξ N k (t, h, ε) k=0,...,N such that , even and iπ-antiperiodic with respect to t, odd with respect to h and satisfies The proof of this proposition is placed in Section 6, where explicit expressions are given for the functions ξ N k (see formulas (190) and (191)). They are obtained by solving approximately the sequence of equations (18) and (62). Corollary 2.7. Defining the outer expansion as we have that Proof. The first part is obtained by plugging inequalities (76) , using the condition |t ± iπ/2| ≥ δ > ρ N h to control the negative powers of | cosh t|. The second part is an immediate consequence of inequalities (71), (78) and Cauchy estimates. 2 Remark 2.8. The first statements of Theorem 1.5 about ξ u are proved, as a consequence of Theorem 2.3 and Corollary 2.7, taking into account the scaling by α performed in Section 2.1. Inequality (23) follows from (67) and (78) (the passage from 0 < h < h N to 0 < h < h 0 is innocuous, since the ratio of the left-hand side of (23) with |ε|h 2N +3 is bounded for h ∈ [h N , h 0 ] and ε ∈ D(0, ε 0 )).
We remark that in the first statement of Proposition 2.6 the parameter h is complex, while inequality (78) only makes sense if h is real, since the functions ξ u k which are involved in ξ u,N are only defined for real and positive h.
In fact, the function ξ N,out (t, h, ε) in (77) collects all the terms up to order 2N +1 of the asymptotic expansion in h of the first outer approximation ξ u,N . Moreover, by Theorem 2.3, ξ N,out (t, h, ε) also contains all the terms up to order 2N + 1 of the asymptotic expansion in h of the function ξ u . Since ξ N,out is even in t and ξ s is defined through ξ s (t, h, ε) = ξ u (−t, h, ε), we have that the asymptotic expansions of ξ u and ξ s coincide up to order 2N + 1. As N can be any natural number, ξ u and ξ s have the same asymptotic expansion in powers of h.
, the difference between ξ u and ξ s can be bounded as Proof. This is an immediate consequence of inequalities (67), (78), the fact that ξ N,out is even with respect to t and that ξ s (t, h, ε) = ξ u (−t, h, ε). 2 In the analytic context, this contact beyond all orders is related to exponentially small phenomena. In order to compute an asymptotic formula of the difference between ξ u and ξ s we will need to have good approximations of the two functions up to distance O(h ln(1/h)) of ±iπ/2.
2.6. Asymptotic expansions and inner equations. The functions ξ N k (t, h, ε) are holomorphic in the domain U out ε0,h0 and, with respect to t, iπ-antiperiodic and even. We shall expand them in Taylor series with respect to h around 0, and then in Laurent series with respect to t around iπ 2 . Proposition 2.10. Let N ≥ 0. Then the functions (ξ N k ) k=0...N given in Proposition 2.6 verify: with real analytic functions Ξ N k,m (t, ε), even, iπ-antiperiodic and meromorphic in t ∈ C, with poles located in iπ 2 + iπZ. Moreover: the coefficients a N k,m, being holomorphic in ε ∈ D(0, ε 0 ) and purely imaginary whenever ε is real.
For example, since (82). The proof of Proposition 2.10 is given in Section 7.1. By (83) we can define, for each k, m ≥ 0, the meromorphic function which, in view of (76), turns out to be a coefficient of the asymptotic expansion of ξ u k (t, h, ε) with respect to h: Corollary 2.11. For each k ≥ 0 and δ ∈ (0, π 2 ), the function ξ u k of Proposition 2.2 admits the asymptotic expansion where the coefficients Ξ k,m are defined by (85) and the asymptotic property is uniform with respect to (t, ε) ∈ D u,out ) uniformly in t and ε (using the fact that e Re t and 1/ cosh t are bounded in D u,out δ ). On the other hand, Property (84) shows that, for each k, n ≥ 0, the sequence of Laurent series ] the Laurent expansion around ∞ of the meromorphic function φ N k,n , we observe that for each p ∈ Z the coefficient of z −p in this formal series does not depend on N provided N is large enough; in fact, only odd powers are needed, and if p = 2(m + k − n) + 1 we get a well-defined coefficient as soon as N > m; the formal limit 3 can thus be defined as and it is characterized by the fact that These formal seriesφ k,n (z, ε) need not be convergent for any value of z. The analysis of their divergence will be performed through resurgence theory (see next section). We can also setφ Indeed, this series of formal series makes sense since the coefficient of each power z −p is made up of finitely many terms only (because the valuation ofφ k,n (z) increases with k). More concretely, from (87), which thus depend holomorphically on ε in D(0, ε 0 ). For instance, with coefficients which are purely imaginary when ε ∈ R. The variable z is called "inner variable" and the seriesφ k,n (z, ε) orφ n (z, ε) are the "inner expansions". We now introduce the "inner equations" inherited from the invariance equation We have (expanding µ = cosh h and V (y, h) in powers of h). Setting now and inserting this expansion into (91), we obtain a sequence of equations; the first one is non-linear: and is called the first inner equation, while the subsequent ones read: where the right-hand sides are determined inductively: summing over n 0 ≥ 0, r ≥ 1, n 0 +r ≥ 2, 1 ≤ n 1 , . . . , n r ≤ n−1, n 0 +n 1 +· · ·+n r = n. In Thus, the nth of these secondary inner equations (95) is linear non-homogeneous in the nth unknown φ n , with a right-hand side determined by φ 0 , . . . , φ n−1 .
The first inner equation (94) makes sense in the differential ring because the above series of formal series is formally convergent (the valuations increase).
The proof of this proposition is given in Section 7.2.
Remark 2.13. These formal series are not the only odd formal solutions: it turns out (see [14]) thatφ 0 and −φ 0 are the only odd formal solutions of the first inner equation, and that, for each of these choices and for any sequence of complex numbers (b n ) n≥1 , there is a sequence of odd formal solutions (φ n ) n≥1 such that b n is the coefficient of z 3 in φ n (z) (by (87), the coefficient of z 3 inφ 1 (z) is zero).

2.7.
Solutions of the inner equations. The present section is devoted to statements about the inner equations, the proofs of which rely onÉcalle's resurgence theory and are given in the article [14]. Roughly speaking, the approach of [14] consists in checking the Borel summability of the formal seriesφ n (z) and, more than this, studying the analytic continuation of their Borel transforms φ n (ζ) and analyzing their singularities by means of the so-called alien calculus. The presence of singularities in the ζ-plane implies that theφ n (z)'s do not converge for any value of z, because of a "factorial" divergence (the modulus of the coefficient of z −p is larger than M p+1 p! for some M > 0). In particular, theφ n (z)'s are Gevrey-1 series and the asymptotic expansion properties satisfied by their Borel sums φ u n (z) are of Gevrey-1 type. In the present section, we content ourselves with asymptotic expansions in the ordinary (Poincaré) sense and extract from [14] the minimum information which is needed for the proof of Theorem 1.5.
The angle β ∈ 0, π 2 was fixed in the definition of D u,out δ in Section 2.2.
Theorem 2.14. Consider the sequence of odd formal solutions (φ n (z, ε)) n≥0 of the inner equations defined by (87). Then there exist an increasing sequence of numbers R n ≥ 1 and a unique sequence of functions φ u n (z, ε) which are holomorphic which satisfy the system of inner equations (94)-(95) for n ≥ 0 and for any 15. It is shown in [14] how to obtain the function φ u n fromφ n by Borel Laplace summation around the direction of R − : by (88), we can writẽ and it turns out that the Borel transform φ n (ζ, ε) = p≥0 B n+p,n (ε)ζ 2p /(2p)! defines a holomorphic function for ζ near 0 which extends analytically to the halfplanes {Re ζ < 0} and {Re ζ > 0}, and that one can define with θ ∈ (π/2 + β, 3π/2 − β) chosen according to arg z. Since, by Proposition 2.12, B ,n (ε) = −B ,n (ε) for any , n, this entails Later, in Section 2.8, we shall introduce the inner domain D u,in Figure 4). The symmetries of the problem imply that the formulas define a sequence of solutions which are holomorphic in D s in (R n ) × D(0, ε 0 ). We also define, for any N = 0, the functions which are holomorphic in the domains D u,s Since the formal solutions we started with are odd in z, the functions φ s n (z, ε) will have the same asymptotic expansionsφ n (z, ε), but one should not believe that they coincide with the functions φ u n (z, ε) (i.e. that the φ u n 's are odd in z). On the contrary, there is a discrepancy, exponentially small with respect to z and thus invisible from the viewpoint of the usual asymptotic expansion theory, which resurgence theory will allow us to analyze.
, which has two connected components; we shall study it in the lower one, which we denote D in (R N ) (see Figure 4). To state our main result about this difference, we first introduce the solutions of the linearization of the first inner equation (94).
Proposition 2.16. The linear difference equation Moreover, the linear difference equation is the Borel sum ofφ 0 (z, ε) described in Theorem 2.14, admits solutions ψ u 1 (z, ε) and ψ u 2 (z, ε) which are holomorphic in D u in (R 0 ) × D(0, ε 0 ) and satisfy W 1 (ψ u 1 , ψ u 2 ) = 1 and, for j = 1, 2, Sinceφ 0 (z, ε) satisfies (94), it is obvious that its derivativeψ 1 (z, ε) is a solution of the linearized equation (101). The reader is referred to Section 4 for the theory of linear second-order difference equations, in particular for the construction of an independent solutionψ 2 (z, ε) and for the properties of the Wronskian .
where the functions φ u n are given by Theorem 2.14, c) for z ∈ D u in (R n ) and for n ≥ 0, 2) There exist complex numbers A + n and B + n , n ≥ 0, which depend holomorphically on ε ∈ D(0, ε 0 ), such that (with the entire function V (ζ) introduced in (9)) and, for any N ≥ 0, the function satisfies the following: if one denotes by D in (R N ) the lower connected component of D s in (R N ) ∩ D u in (R N ), then, for any θ ∈ (0, 1), the difference of the functions φ u,N and φ s,N defined by (100) can be written The proofs of Theorems 2.14 and 2.17 and of Proposition 2.16 are in [14]. The constants B + n are the ones appearing in Theorems 1.1 and 1.5. The constants A + n and B + n have a resurgent origin: the Borel transforms of the formal seriesφ n (z, ε) give rise to holomorphic functions in the ζ-plane which extend analytically to the universal cover of C \ 2πiZ, with singularities at 2πi which account for the main part of the asymptotic expansion of φ s n (z, ε)−φ u n (z, ε) when z lies in D in (R n ), while the singularities at −2πi are related with the asymptotic expansion in the upper part of D s in (R n ) ∩ D u in (R n ) (the singularities at 2πim with |m| ≥ 2 correspond to exponentially small corrections of higher orders).
2.8. Inner domain. We now define the inner domain, a region in the complex plane closer to the singularities ±iπ/2 of ξ 0 (t, h) than the outer domain D u,out δ , where the functions ξ u (t, h, ε) and ξ s (t, h, ε) will be well approximated by making z = (t − iπ/2)/h in the functions φ u,N (z, h, ε) and φ s,N (z, h, ε) defined in (100). Given R > 0, for any δ ∈ (Rh, π 2 ), we set Observe that this domain is symmetric with respect to the real axis and that, if R ≥ R N and t ∈ D u,in h (R), then t−iπ/2 h ∈ D in (R N ).

2.9.
Matching of the outer and inner approximations. In this section we use the information obtained from the study of the first inner equation (94) and the full hierarchy of equations (95) to improve our knowledge of the functions ξ u and ξ s given by Theorem 2.3 and formula (68). This will be achieved by matching the approximation of these functions found in the outer domain D u,out δ with the appropriate approximations in the inner domain D u,in h (R). Theorem 2.18. Let ε 0 < 1/|2V 2 | and consider the function ξ u of Theorem 2.3, solution of (54) verifying boundary conditions (19) and (21). For any N ≥ 0, besides the constants ρ N and R N introduced in Theorems 2.3 and 2.14, there exist constants h N , κ N , C N > 0 such that, if |ε| < ε 0 , 0 < h < h N , max{ρ N , 2R N }h < δ < π/2 and then ξ u admits an analytic continuation to and, for t ∈ D u,in h (2R N ), Im t > 0, where φ u,N is the function introduced in (100).
The proof of this Theorem is placed in Section 8. Observe that if we choose δ = h 1/2 (which is licit, if h is small enough), then A N δ = 2h N +1/2 . Corollary 2.19. With the same hypotheses as in Theorem 2.18, the function ξ s defined in (68) satisfies if Im t ≥ 0, then Im (−t) ≥ 0 and, by Theorem 2.18, the modulus is bounded by We will need the extension to the inner domain of the functions η u 1 , η u,0 (with the notation of Theorem 2.17).
The proof of Theorem 2.20 is given in Section 8.
Corollary 2.21. There exists C > 0 such that, for t ∈ D u,out Proof. In  3. Proof of the analytic Theorem 1.5. We introduce which is the function defined in (24), scaled by α (see Proposition 2.1). By Theorem 2.3, it is real analytic, holomorphic in where D s (R) = −D u (R) and D u (R) were introduced in (110). Since ξ u and ξ s satisfy the invariance equation (54), we get the linear second order difference equation where If ν 1 and ν 2 are two solutions of (119) such that W h (ν 1 , ν 2 ) = 1, then with c 1 = W h (D, ν 2 ) and c 2 = W h (ν 1 , D) h-periodic. If moreover ν 1 and ν 2 are real analytic and satisfy certain bounds in R, then the fact that D(t) is bounded in R and Lemma 3.2 below will provide exponentially small bounds for the real analytic h-periodic functions c i (t) and then for D(t) for real t. (119). The definition (120) implies that

Solutions of the linear equation
thus equation (119) is close to equation (69). By Theorems 2.4 and 2.20, we have a fundamental system of real analytic solutions, {η u 1 , η u,0 2 }. We look for the solutions ν 1 , ν 2 of equation (119) as small perturbations of η u 1 and η u,0 2 . However, we will not be able to find them in the whole domain R. We define Moreover, for t ∈ R such that |Im t| ≤ 1, with η u 1 and η u,0 where ψ u,N 1 and ψ u 2 are defined in (115) and A is the constant introduced in (58).
Proof. We expand f in Fourier series, f (t) = k∈Z f k e 2kπit/h . Since f is holomorphic in a strip, we can compute the coefficients of the Fourier series along different horizontal lines {Im t = ρ}: for any ρ ∈ [−r, r], By choosing ρ = r for k ≤ −1, we get |f k | ≤ M e −2π|k|r/h , and this inequality holds as well for k ≥ 1, since f k = f −k . Thus, for any t ∈ R, whence the conclusion follows. 2 From now on we fix θ = 1/2 in Theorem 2.17 and δ = h 1/2 , hence A N δ = 2h N +1/2 . In particular, for any N ≥ 13, there exists h N > 0 such that all the hypotheses of Theorems 2.3-3.1 are satisfied for any 0 < h < h N . The following estimate of D(t) will be used to control the coefficients c i (t) when Im t = π/2 − σ 2π h| ln h|. Lemma 3.3. Let ε 0 < 1/|2V 2 |. For any σ > 13 and N ≥ 13, there exist h N , C N > 0 such that, if 0 < h < h N , |ε| < ε 0 and t ∈ R σ with Im t = π/2 − σ 2π h| ln h|, then where the function D N,inn was defined in Theorem 2.17.

Some notes on linear second order difference equations.
Definition 4.1. Given h > 0, we define the first order difference operator ∆ h by In view of (32), the Wronskian of two functions can be written For f, g defined in a subset U of C, ∆ h f (t) and W h (f, g)(t) are defined for all t ∈ U such that t + h ∈ U . In what follows, we shall consider only two types of domains: (I) Given a function r + : (a, b) ⊂ R → R, we define Observe that the closure of the domain D u,out δ defined in Section 2.2 is the closure of a domain of type (I), while the closure of the domain D u,in h of Section 2.8 is the disjoint union of the closures of two domains of type (II).
Given some complex function g, we will need to solve the equation If g is defined on a domain U u r+ , then the formula defines a solution ∆ −1 h,u g of equation (138) which tends to 0 as Re t → −∞ provided that this series is normally convergent. We can consider ∆ −1 h,u as a right inverse of operator ∆ h when both operators are well defined.
If g : U s r+ → C, a right inverse of ∆ h is given by provided this series is normally convergent. The next three lemmas summarize some elementary results whose proofs we omit (see however Section 2.1 and Appendix A.2 of [14]). Lemma 4.2. Given a domain of type (I), U = U u r+ , resp. U = U s r+ , on which a function G is defined, consider the linear second order difference equation where the unknown u is required to be defined on U u r++h , resp. U s r+−h . Then: 1. For any two solutions u 1 and u 2 , the function W h (u 1 , u 2 ) is h-periodic.

If u 1 is a solution which does not vanish, then
3. For any two solutions u 1 and u 2 such that W h (u 1 , u 2 ) does not vanish, the set of solutions of (141) is {u = c 1 u 1 + c 2 u 2 , c 1 and c 2 h-periodic functions}.
Once the solutions of a homogeneous difference equation are found, it is possible to obtain the solutions of the non-homogeneous one. In the case of an unbounded domain extending to the left, we have the following Lemma 4.3. Let r + : (a, b) → R be a function and consider the corresponding unbounded domain U u r+ of type (I), on which two functions G and H are supposed to be defined. Assume that u 1 , u 2 : U u r++h → C are two solutions of (141) such that W h (u 1 , u 2 ) ≡ 1. Then a solution of the equation is given by if this series is absolutely convergent.
In Section 8, we shall have to deal with bounded domains and to find solutions that satisfy some given initial conditions: Lemma 4.4. Let r − , r + : (a, b) → R be functions and consider the corresponding domain U r−,r+ of type (II), on which two functions G and H are supposed to be defined. Assume that u 1 , u 2 : U r−−h,r++h → C are two solutions of (141) (for t ∈ U r−,r+ ) such that W h (u 1 , u 2 ) ≡ 1 (on U r−−h,r+ ). Then, for any function u * : U r−−h,r−+h → C, the equation admits a unique solution which is defined on U r−−h,r++h and satisfies c 1 , c 2 are the h-periodic functions uniquely determined by Observe that, in this lemma, the existence and uniqueness of the solution u is obvious, since equation (144) can be written so that the values of u * on U r−−h,r−+h uniquely determine the values of u on U r−+h,r−+2h , and then on U r−+2h,r−+3h , and so on until the domain U r−+h,r++h ∩ k≥1 U r−+kh,r−+(k+1)h is covered. We call the domain U r−−h,r−+h a "boundary layer". In fact, the function u h is the unique solution of the homogeneous equation (141) whose restriction to the boundary layer is u * , while u p is the unique solution of the non-homogeneous equation (144) whose restriction to the boundary layer vanishes identically.
Remark 4.5. If G, H and u * are analytic, this does not imply that the solution u is itself analytic: there are possible failures of analyticity (or even discontinuities) on the curves {Re t = r − (Im t) + kh}, k ≥ 1. However the above chain of reasoning shows that if G and H admit a continuation which is holomorphic in a neighborhood of U r−,r+ and if u * admits a continuation which is holomorphic in a neighborhood of the closure of U r−−h,r++h and which satisfies equation (144) in a neighborhood of the curve {Re t = r − (Im t)}, then the solution u admits a continuation which is holomorphic in a neighborhood of U r−−h,r++h .
We shall give more details when using a non-linear variant of this in Section 8.5.

Extended domains.
Our intention is to find analytic functions defined in the domain D u,out δ introduced in Section 2.2. In fact, as announced in Remark 2.5, we Figure 6. The extended outer domain U (β 1 , β 2 , r 1 , r 2 , δ). It is symmetrical with respect the real axis.

The linearized equation.
To prove the existence of a solution ξ u of the invariance equation (54), satisfying the boundary conditions (19) and (21) and the properties of the sequence of approximating functions given by Proposition 2.2, and also to find suitable solutions of the linearization of (54) around ξ u , we need to solve equations of the form L(η) = g, where By Lemmas 4.3 and 4.4, the solutions of L(η) = g can be obtained from a fundamental system of solutions of the homogeneous equation, as the one provided by the next lemma, which we borrow from [4].

5.3.
Banach spaces and technical lemmas. We will say that g 1 = O(g 2 ) in some domain U if there exists some positive constant C, that may depend on β, N and other constants, but does not depend on δ nor h, such that |g 1 | ≤ C|g 2 | in U .

Proof. Part (a), (b) and (c) are straightforward. Part (d) is obtained by means of Cauchy inequalities in sectorial domains. 2
Remark 5.3. In part (d) of the preceding lemma we have not explicitly written the constant O(1) involved because we will use it at most N times, with N arbitrary but fixed. Hence, the resulting derivative will still be defined in D u,out δ .
Following (143) in Lemmas 4.3, with a view to solving equations of the form L(η) = g, we define two linear operatorsG and G bỹ where and The functions η 1 and η 0 2 were defined in (56) and (57) and ∆ −1 h,u in (139). Notice that V k is real analytic and πi-periodic. Since η If g ∈ X l,m is iπ-antiperiodic, then so isG(g).
For k ≥ 1, we proceed by induction and rewrite equations (62) as where L is defined in (147) and f k are given by (63)  Now we prove the step k of the induction process. We assume, by induction, that there exists a unique sequence of functions, (ξ u j ) j=0,...,k−1 , ξ u j ∈ X 1,2j+1 , verifying (i), (ii), (iii), and inequality (160).
Remark 5.7. Notice that, although the functionξ u given by Proposition 5.6 satisfies inequality (67) and the boundary condition (19), it is not the one claimed in Theorem 2.3 because it does not necessarily satisfy the boundary condition (21).
Proof. The preservation of iπ-antiperiodicity follows from the definition of N . We write N as We finally compute the Lipschitz constant of the map N . For η,η ∈ B κ , N (η) − N (η) =   Then we can bound the first integral in (170) by Analogously, we obtain the same bound for the second integral in (170).
Notice that, for any real T , the functionξ u (t−T ) is also a πi-antiperiodic real analytic solution of the invariance equation that satisfies the boundary condition (19). Hence, we look for T such that Proof. Equation (182) is equivalent to where v u =ξ u −ξ u,N . In order to solve equation (183), we consider some fixed small neighborhood of the origin, B, the ball of radius 1/2, for instance. Let p(T ) denote the left hand side of (183), and q(T ) the right hand side. In this way, equation (183) can be written as p(T ) = q(T ), which we will treat as a fixed point equation after inverting p.
On the other hand, |q (t)| is bounded from above in B by O(ε N +1 h 2N +3 ) and q(0) = O(ε N +1 h 2N +4 ). Therefore, the composition p −1 • q is well defined in the ball of radius min{O(1), O(ε N +1 h −2N −1 )} to itself and its derivative (p −1 • q) is bounded by O(ε N +1 h 2N +1 ). Since it is a contraction, it has a unique fixed point, T (h), which is the solution of equation (182).
6.1. Euler-MacLaurin Formula and first order difference operators. The Euler-MacLaurin summation formula states that, given a C ∞ function g : where B 2j are the Bernoulli numbers andB 2N (x) are periodic functions related to the Bernoulli polynomials (see, for instance, [17]). We apply the Euler-MacLaurin formula to obtain integral expressions of the operators ∆ −1 h,u and ∆ −1 h,s , defined in (139) and (140), respectively. Lemma 6.1. If φ is a C ∞ function with exponential decay at −∞, then, for any N ≥ 1, If φ is a C ∞ function with exponential decay at ∞, then, for any N ≥ 1, If φ is an even C ∞ function with exponential decay at −∞, then, for any N ≥ 1, The proof is a straightforward computation.
6.2. Introducing the asymptotic expansion. We will obtain the asymptotic expansion of the outer approximation of order N , ξ u,N = N k=0 ε k ξ u k , computing the asymptotic expansion of the functions ξ u k . We recall that these functions were constructed in the following way: 1. ξ u 0 = ξ 0 is the function defined in (1), 2. for k ≥ 1, where f u k is obtained by substituting recursively ξ u 0 , . . . , ξ u k−1 into (63), for k = 1, and into (64), for k ≥ 2, and G, defined in (156), is a right inverse of the operator L given by (147). We remark that we can write operator G in formula (188), using the operator ∆ −1 h,u defined in (139), as In order to obtain the asymptotic expansion in h of ξ u k , we modify the above scheme by substituting G by its asymptotic expansion in h up to order h 2N , provided by the Euler-MacLaurin formula.
More concretely, for a fixed N ≥ 0, we define the sequence of functions ξ N 0 , . . . , ξ N N as follows: where f N k is obtained by substituting recursively ξ N 0 , . . . , ξ N k−1 into (63), for k = 1, and into (64), for k ≥ 2, and T N is defined by T N is obtained formally replacing in (189) the operator ∆ −1 h,u by formula (185), computing ∆ −1 h,u (η 0 2 g)(h/2) with formula (187) and dropping the error terms. We remark also that the operator T N , unlike the operator G in (189), is defined for complex h. In Lemma 6.3 we will give a precise description of the dependence of T N with respect to h.
6.3. The operator T N and its approximating properties. In this section we will prove that the operator T N is well defined between suitable Banach spaces and that it is indeed a good approximation of the operator G used in the recurrence (188) that defines the functions ξ u k . We introduce, for l, m ∈ R, the spaces X e l,m (β 1 , β 2 , r 1 , r 2 , δ) = {ξ ∈ X l,m (β 1 , β 2 , r 1 , r 2 , δ) | ξ is even}.
They are Banach spaces with the norm defined in X l,m (see (152) and (153)). It is clear that X e l,m ⊂ X l,m , with the same norm. Analogously to the function V k (t) introduced in (155), we define We remark that, like V k (t), the map t → V (t, s) is iπ-periodic.
Using this function V and performing the change of variables x = t + s in (191), the operator T N can be written as which is the equivalent expression to G in (156) with integrals instead of sums and the correcting term given by the Euler-MacLaurin formula (see also (154)).
Proof. It is clear that if g is real with respect to t, so is T N (g). On the other hand, since η 1 is odd and η 0 2 is even, we deduce from (191) that T N preserves evenness with respect to t.
Since the function t → V (t, s) is πi-periodic and η 1 is πi-antiperiodic, from (194) we have that T N preserves πi-antiperiodicity. Now we check that T N is a well defined operator from X e l,m (β 1 , β 2 , r 1 , r 2 , δ) to X e 1,m−2 (β 1 ,β 2 ,r 1 ,r 2 , δ), with norm bounded by O(h −2 ). We introduce I N and J N , with T N = I N + J N , where I N is the first line of the right hand side in (194), that is, is the integral part of T N , while J N is the second line of the right hand side of (194).
We claim that both I N and J N are bounded and linear from X e l,m (β 1 , β 2 , r 1 , r 2 , δ) to X e 1,m−2 (β 1 ,β 2 ,r 1 ,r 2 , δ) with norm bounded by O(h −2 ). We prove this claim for each operator separately.
We deal with first with I N . Since real analyticity is preserved, we only need to consider t with Im t ≥ 0. The claim follows using the arguments in the proof of Lemma 5.4 to obtain, if Re t ≤ −1, which proves the claim for I N .
Proof. Since ξ N 0 = ξ 0 is the function defined in (1), the claim is trivial for k = 0. We prove the claim for k ≥ 1 by induction.
It is clearly analytic in U out ε0,h0 . Then, by Lemma 6.3 < r 2 and ξ N j 1,2j+1 = O(h 2j+1 ), and analytic in U out ε0,h0 and odd with respect to h. 2 Lemma 6.6. Under the hypothesis of Proposition 2.6, the functions (ξ u k ) 0≤k≤N and Proof. We use induction. The case k = 0 is trivial, since ξ u 0 = ξ N 0 = ξ 0 . Now we consider the case k = 1. By recurrences (188) and (190), we have that Hence, by Lemma 6.4 with l = m = 5, , which proves the claim for k=1.

Asymptotic expansion in the inner variable.
Here we give the proofs of Propositions 2.10 and 2.12, which are closely related.
7.1. Proof of Proposition 2.10. We recall that, by Proposition 2.6, the functions (ξ N k ) k=0,...,N are analytic in U out ε0,h0 with respect to (t, h, ε), even and πi-antiperiodic with respect to t and odd with respect to h. We expand ξ N k in powers of h where γ H is the positively oriented circumference of radius H around h = 0, with 0 < H < h 0 . By the definition of U out ε0,h0 in (74), the coefficients χ N k,j can be computed by formula (202) for any t such that H < |t − iπ/2| < π. It is clear that they are iπ-antiperiodic and even with respect to t. Moreover, since they do not depend on h, their only singularity in {t | |t − iπ/2| < π} is iπ/2. By Lemma 6.5, for any t with H < |t − iπ/2| < π, 0 < H < h 0 , Hence, if 0 ≤ j < k, χ N k,j ≡ 0. With the same argument, if k ≤ j, for 0 < H < |t − iπ/2| < 2H, we have that (1), which implies that χ N k,j has a pole of order at most 2j + 1. Defining Ξ N k,m = χ N k,k+m , we have proven the first part of Proposition 2.10. Formula (82) follows from the Laurent expansion of Ξ N k,m , making the change t = iπ/2 + hz, and reordering the absolutely convergent series We finally check claim (83). We fix N < N . Assume that there exists m such that m ≤ N − 1 and Ξ N k,m − Ξ N k,m = 0 and take m minimal. In particular, there exist t, C ∈ R such that, for 0 < h < h 0 , But, by Lemma 6.6, we have that ). Hence, Ξ N k,m = Ξ N k,m , for all 0 ≤ m ≤ N − 1. (84) follows from substituting t by iπ/2 + hz and reordering the series. The fact that the coefficients a N k,m,l are purely imaginary whenever ε is real is an immediate consequence of iπ-antiperiodicity and real analyticity.  (77) is analytic in U out ε0,h0 , even and πi-antiperiodic with respect to t, odd with respect to h and ξ N,out where F was introduced in (59).
We first remark that, since ξ N,out is analytic in U out ε0,h0 , E N is analytic iñ (204) Moreover, since ξ N,out is even and iπ-antiperiodic with respect to t and F(y, h, ε) is odd with respect to y and even with respect to h, E N is even and iπ-antiperiodic with respect to t, and odd with respect to h. Hence where γ H is the positively oriented circumference of radius H around h = 0 and 0 < H < h 0 .
Using that the function ξ u , given by Theorem 2.3, is a solution of the invariance equation (54) and inequalities (67) and (78), we have that, for 0 < h < h 0 and t ∈ D u,out δ (skipping the dependence on ε),  (1), from which we can deduce that E N k has a pole of order at most 2k + 1 at iπ/2. Therefore, we can write E N in the form Hence, by substituting t = iπ/2 + hz into (206), we have that Finally, the formal seriesφ n introduced in (87) are the formal limit of the functions φ N k,n given by Proposition 2.10. "Formal limit" means that each coefficient of the z-expansion of φ N n is a finite sum of holomorphic functions in ε. The following pages until Section 8.5 are devoted to proving this proposition and, finally, incorporating the missing factor |ε| in the right-hand side of (211), so as to complete the proof of Theorem 2.18. The proof of Theorem 2.20 will be addressed using similar tools in Sections 8. 6-8.7.
In view of Section 4, we introduce a domain which is larger thanD * in : This is a domain of type (II), which we can write as with certain piecewise analytic functions r ± . We also introducẽ The domain will play the role of a boundary layer: since U ⊂D out ∩D in ⊂ D u in (R N ), the function φ u is holomorphic in a neighborhood of the closure of U = U r−−2,r− , while φ u,N is holomorphic in a neighborhood ofD in = U r−−2,r+ , and the difference equation (91) will provide the continuation of φ u inD in (see Figures 7 and 8).
The starting point for the matching method is Lemma 8.2. Let ψ * denote the restriction of the function φ u −φ u,N to the boundary layer U . Then Moreover, ψ * admits a continuation which is holomorphic in a neighborhood of the closure of U and which satisfies the nonlinear difference equation in a neighborhood of the curve z ∈ C | Re z = r − (Im z) − 1 .
Notice that equation (216) makes sense for z ∈ U r−−1,r+−1 provided that the unknown function ψ has a sufficiently small modulus in this domain (so that F • (φ u,N + ψ) be defined) and is defined inD in = U r−−2,r+ . We will prove that there is a unique such solution whose restriction to U is ψ * ; this function will necessarily be analytic and it will provide the continuation of Ψ. Proof of Lemma 8.2. Equation (216) is just a rephrasing with the unknown ψ = φ − φ u,N of the invariance equation (91), which is indeed satisfied by φ u .
We shall have to solve equations of the form L in (ψ) = Φ, with a given function Φ defined inD in . As for the homogeneous equation L in (ψ) = 0 inD in , we already Then, by Lemma 8.5 with m = −2N + 1 < 3, we get G in (G in,N ) ∈ Y 2 and G in (G in,N ) Lemma 8.7. The operator in defined in (229) induces a bounded linear map from Y 2 to Y 2 with in = O(h 2 ). As a consequence, G in • in induces a bounded linear operator of Y 2 with G in • in = O(δ 2 ).
Knowing that • the function φ u,N is analytic in U r−−2,r+ , • the function F(y) is analytic for |y| < y 0 , • the function ψ * admits a continuation which is analytic in a neighborhood of the closure of U r−−2,r− and satisfies equation (239) for z in a neighborhood of the curve {Re z = r − (Im z)} (as claimed in Lemma 8.2), we deduce that ψ admits a continuation which is analytic in a neighborhood of U r−−2,r+ . Indeed, denoting by S(ψ) the right-hand side of (239), we can argue by induction and suppose that, for a k ∈ N, the restriction ψ |U r − −2,r − +k ∩Ur − −2,r + admits a continuation ψ k which is analytic in an open set W k ⊃ r − (Im z) − 2 ≤ Re z ≤ min{r − (Im z) + k, r + (Im z)} and satisfies ψ k = S(ψ k ) in an open set V k ⊃ r − (Im z) ≤ Re z ≤ min{r − (Im z) + k, r + (Im z)} .
Now, for z ∈D in , we use R N ≤ |z| ≤ δ/h and get We now need to study the operator H in,1 , which is defined with the help of M h .
Then, equation (249) can be written as Our purpose is to solve equation (255) as a fixed point equation. In order to do so, we need to define a right inverse of L u in some suitable spaces. Hence, we introduce the spaces Z µ = {u : R σ → C real analytic, such that u µ < ∞}, where u µ = sup t∈Rσ |u(t) cosh µ t|.
They are Banach spaces. Moreover, Lemma 9.2. Let µ > −2. Then the operator L u admits a bounded right inverse G u : Z µ → Z µ+4 such that G u ≤ 1 h 2 | ln h|. We postpone the proof of this lemma to the end of this section.