This is a preprint of: “Differentiable invariant manifolds of nilpotent parabolic points”, Clara Cufı́- Cabré, Ernest Fontich, Discrete Contin. Dyn. Syst., vol. 41(10), 4667–4704, 2021. DOI: [10.3934/dcds.2021053] Differentiable invariant manifolds of nilpotent parabolic points Clara Cufí-Cabré Departament de Matemàtiques Universitat Autònoma de Barcelona (UAB) Barcelona Graduate School of Mathematics (BGSMath) 08193 Bellaterra, Barcelona, Spain clara@mat.uab.cat Ernest Fontich Departament de Matemàtiques i Informàtica Universitat de Barcelona (UB) Barcelona Graduate School of Mathematics (BGSMath) Gran Via 585. 08007 Barcelona, Spain fontich@ub.edu Abstract We consider a map F of class C r with a fixed point of parabolic type whose differ- ential is not diagonalizable and we study the existence and regularity of the invariant manifolds associated with the fixed point using the parameterization method. Con- cretely, we show that under suitable conditions on the coefficients of F , there exist invariant curves of class C r away from the fixed point, and that they are analytic when F is analytic. The differentiability result is obtained as an application of the fiber contraction theorem. We also provide an algorithm to compute an approximation of a parameterization of the invariant curves and a normal form of the restricted dynamics of F on them. 2020 Mathematics Subject Classification: Primary: 37D10; Secondary: 37C25. Key words and phrases: Parabolic point, invariant manifold, parameterization method. 1 Introduction Invariant manifolds play a central role in the study of dynamical systems. There is a huge amount of literature devoted to study them in many different settings. In this paper we deal with the invariant manifolds of a type of parabolic fixed points in dimension two. Parabolic points appear generically in two-parameter families of planar maps or in one- parameter ones in the case of area-preserving maps. In particular they appear when a family of maps undergoes a Bogdanov-Takens bifurcation [6, 27]. 1 In some problems in Celestial Mechanics it is useful to consider parabolic points or parabolic orbits at infinity in order to use their invariant manifolds (provided they exist) to study features of the dynamics in the finite phase space. The local study in a neighborhood of such points is done by means of a change of variables which sends the infinity to a finite part of the space [22]. Also, the periodic orbits become fixed points of appropriate (families of) Poincaré maps. In such cases the fixed points are parabolic for all values of the parameters of the family and may have invariant manifolds. These manifolds have been used to prove the existence of oscillatory motions in the Sitnikov problem [25, 23] and the restricted planar three-body problem [21, 16, 17] using the transversal intersection of invariant manifolds of parabolic points and symbolic dynamics. Parabolic manifolds also appear in the Manev problem [11]. Parabolic periodic orbits at infinity have been found in Hamiltonian systems related to the study of the scattering of He atoms off Cu surfaces with some corrugation [15]. These manifolds also play a significant role in the study of certain systems [20, 14]. In this paper we consider two-dimensional maps having a parabolic fixed point whose linear- ization does not diagonalize, concretly we assume it has a double eigenvalue equal to 1. By simple changes such maps can be brought to the form   x + cy + f1 (x, y) F (x, y) =  , (1.1) y + f2 (x, y) with c > 0, f1 (0, 0) = f2 (0, 0) = 0 and Df1 (0, 0) = Df2 (0, 0) = 0. The origin has a center manifold of dimension two, however, inside this manifold there may exist curves that behave topologically as stable or unstable curves. This class of maps was considered in [12] and the existence of analytic curves was proved. Concretely the (local) sets considered there and the ones we deal with are Wρs+ = {(x, y) | F n (x, y) ∈ (0, ρ) × (−ρ, ρ), ∀ n ≥ 0, lim F n (x, y) = 0} n→∞ and Wρu+ = {(x, y) | F −n (x, y) ∈ (0, ρ) × (−ρ, ρ), ∀ n ≥ 0, lim F −n (x, y) = 0}. n→∞ The main result of [12] concerns analytic stable invariant curves in the domain {(x, y) ∈ R2 | x ≥ 0, y ≤ 0} under some appropriate conditions on the higher order terms. Then, the existence of both stable and unstable curves in neighborhoods of the origin are deduced from the main result by using the symmetries (x, y) 7→ (−x, y), (x, y) 7→ (x, −y) and (x, y) 7→ (−x, −y) and the inverse map F −1 . Moreover, a detailed study of the local dynamics provide the uniqueness of such curves in the category of C k maps where k is the minimum regularity for having a Taylor expansion providing the relevant nonlinear terms [12]. In this paper we study the existence and regularity of stable curves in the domain {(x, y) ∈ R2 | x ≥ 0, y ≤ 0} using the parameterization method. In the analytic case we recover the existence results of [12] but we also provide approximations of the curves up to an arbitrarily high order. We consider three cases of maps of the form (1.1), already introduced in [12], which depend in some sense on the dominant part of the nonlinear terms. The study depends on each case. Moreover, we consider the differentiable case with the same method and we 2 obtain that the invariant manifolds of F are of the same regularity as F provided some minimum regularity holds. Contrary to other works we do not use the Poincaré normal form for the map, but a simple and easy-to-compute reduced form. This class of maps, assuming the fixed point is not isolated, was studied in [10] motivated by the study of collisions in two-body problems with central force potential satisfying certain asympotic properties at the origin. A special case of this family not previously covered is studied in [19]. These papers use an adapted form of the method of McGehee for parabolic points without nilpotent part [22]. McGehee’s method consists of looking for a sector-like domain S, with the fixed point in the vertex, such that the points whose positive iterates remain on S form a graph of some function ϕ. To prove analyticity, it considers the com- plexified map and uses Rouché’s theorem to obtain the uniqueness of ϕ(x) in terms of x, for x in a complex extension S of S, so that then one can apply the implicit function theorem to obtain the analyticity of ϕ(x) for x ∈ S. Again for maps of the form (1.1), using different tools, some regularity results are obtained in [29]. In that paper, the authors deal with what we denote by case 1 for C ∞ maps and obtain the existence of a stable manifold Wρs+ as the graph of some function ϕ by solving a fixed point equation equivalent to the invariance of the graph of ϕ. This equation is considered for functions ϕ in a suitable subset of the space of functions of class C [(k+1)/2] , where [·] denotes integer part, and it is solved applying the Schauder fixed point theorem. Hence, they obtain invariant manifolds of class C [(k+1)/2] . Instead, in this paper, we use the parameterization method (see Section 2.2) and we obtain, away from the fixed point, analytic invariant manifolds for analytic maps and C r invariant manifolds for C r maps, provided r is larger than some quantity that depends on the nonlinear terms of the map. One-dimensional manifolds of fixed points with linear part equal to the identity are studied in [2] using the parameterization method. Higher-dimensional manifolds in the same setting are considered in [1] using a generalized version of the method of McGehee, and in [4, 5] using the parameterization method, where applications to Celestial Mechanics are given. The Gevrey character of one-dimensional manifolds is studied in [3]. The main results of this paper are Theorems 2.1 and 2.3, concerning the existence of analytic invariant curves of a map F of the form (1.1), and Theorems 2.7 and 2.10, concerning the existence of differentiable invariant curves. In Section 2 we present the parameterization method and the main results of the paper. The results are stated for the stable curves. In Section 2.4 we show that completely analogous results hold true for the unstable ones. In Section 3 we provide an algorithm to obtain parameterizations of approximations of the invariant curves of F , and we provide the existence of such curves in Sections 4, for the analytic case, and in Section 5, for the differentiable case. The proofs of the technical results used along the paper are deferred to Section 6. The paper finishes with a conclusions section where we summarize the results of the paper. 3 2 Statement of the main results 2.1 Reduction of the maps to a simple form In this paper we consider C r , r ≥ 3, or analytic maps F : U ⊂ R2 → R2 , where U is a neighborhood of (0, 0), of the form   x + c y + f1 (x, y) F (x, y) =  , (2.1) y + f2 (x, y) with c > 0 and with f1 (x, y), f2 (x, y) = O(k(x, y)k2 ). Via the C r change of variables given by x̃ = x, ỹ = y + 1c f1 (x, y), F can be written in the form   x + cy F (x, y) =  , y + f (x, y) with f (x, y) = O(k(x, y)k2 ) having the same regularity as F . In the C r case we denote by P (x, y) the Taylor polynomial of degree r of f (x, y). We write P (x, y) in the form P (x, y) = p(x) + yq(x) + u(x, y), where we have collected all the terms independent of y in p(x), the terms that are linear in y in yq(x) and all remaining terms in u(x, y). Note that all terms in u(x, y) have the factor y 2 . More precisely, we write p(x) = xk (ak + · · · + ar xr−k ) and q(x) = xl−1 (bl + · · · + br xr−l ), with 2 ≤ k, l ≤ r. Therefore we have f (x, y) = P (x, y) + g(x, y) with g(x, y) = o(k(x, y)k)r . Also, note that one can always assume that c > 0. If this is not the case, then it can be attained via the linear transformation given by L(x, y) = (x, −y), taking the conjugate map F̃ = L−1 ◦ F ◦ L. Notice however that L sends the lower semi plane to the upper one. Hence, any map F of the form (2.1) can be written in the form   x + cy F̄ (x, y) =  , (2.2) y + p(x) + yq(x) + u(x, y) + g(x, y) with c > 0. In the analytic case we have the same form with g(x, y) analytic. In general we will not write the dependence of p, q, u and g on r. Throughout the paper we will refer to (2.2) as the reduced form of F and we will use the same notation F . We will deal with maps of the form (2.2). We remark that in contrast with other references [12, 29] in which they work with normal forms of F à la Poincaré, we work with the reduced form obtained with a simple change of variables. This is an important advantage when one has to perform effective computations. Following [12], we shall consider three cases depending on the indices k and l: • Case 1: k < 2l − 1 and ak 6= 0, • Case 2: k = 2l − 1 and ak , bl 6= 0, 4 • Case 3: k > 2l − 1 and bl 6= 0. In order to deal, whenever possible, with several cases at the same time we associate to F the integers N and s: N = k in case 1 and N = l in cases 2 and 3; s = 2r in case 1 and s = r in cases 2, 3. Notice that the generic case is case 1 with k = 2. Next we make a comment concerning notation. The superindices x and y on the symbol of a function or an operator that takes values in R2 will denote the first and second components of its image, respectively. In R2 and C2 we will use the norm given by k(x, y)k = max {|x|, |y|}. Throughout the paper, M and ρ0 will denote positive constants, and they do not take neces- sarily the same value at different places. 2.2 The parameterization method To study the stable curves of F we will use the parameterization method (see [7], [8], [9], [18]). It consists in looking for the curves as images of parameterizations, K, together with a representation of the dynamics of the map restricted to them, R, satisfying the invariance equation, F ◦ K = K ◦ R. (2.3) This is a functional equation that has to be adapted to the setting of the problem at hand. Clearly, we need the range of R to be contained in the domain of K. It follows immediately from (2.3) that the range of K is invariant. Essentially, K is a (semi)conjugation of the map restricted to the range of K to R. Equation (2.3) has to be solved in a suitable space of functions. Usually it is convenient to have good approximations of K and R and look for a (small) correction of K, in some sense, while maintaining R fixed. Assuming differentiability and taking derivatives in (2.3) we get DF ◦ K · DK = DK ◦ R · DR which says that the range of DK has to be invariant by DF . In our setting we look for K = (K x , K y ) : [0, ρ) → R2 such that K(0) = (0, 0) and DK(t) satisfies DK y (t)/DK x (t) → 0 as t → 0. We already know that in the parabolic case, in general, there is a loss of regularity of the invariant curves at the origin with respect to the regularity of the map [12], [4], [5]. Then we can not assume a priori a Taylor expansion of high degree of the curve at t = 0. However, we can obtain formal polynomial approximations, Kn and Rn , of K and R, satisfying (2.3) up to a certain order that depends on the degree of differentiability of F . Our results will then provide that these expressions are indeed approximations of true invariant curves, whose existence is rigorously established. On the other hand we can suppose that we have approximations, obtained in some way, that satisfy some conditions and obtain that there are true invariant curves closeby. 2.3 Main results First we state the main results concerning the existence of analytic stable invariant manifolds of analytic maps of the form (2.2). Since an analytic map of the form (2.1) is analytically conjugated to a map of the form (2.2), the results of the next theorems provide invariant manifolds for (2.1). 5 Theorem 2.1. Let F : U ⊂ R2 → R2 be an analytic map in a neighborhood U of (0, 0) of the form (2.2). Assume the following hypotheses according to the different cases: (case 1) ak > 0, (case 2) ak > 0, bl 6= 0, (case 3) bl < 0. Then, there exists a C 1 map K : [0, ρ) → R2 , analytic in (0, ρ), such that   (t2 , Kk+1 y tk+1 ) + (O(t3 ), O(tk+2 )) case 1, K(t) = (2.4)  (t, Kly tl ) + (O(t2 ), O(tl+1 )) cases 2, 3, q √ 2ak b − b2l +4 c ak l with Kk+1 y = − c(k+1) for case 1, Kly = l 2cl for case 2 and Kly = bcll for case 3, and a polynomial R of the form R(t) = t + RN tN + R2N −1 t2N −1 , with Rk = 2c Kk+1y for case 1 and Rl = cKl for cases 2, 3, such that y F (K(t)) = K(R(t)), t ∈ [0, ρ). Remark 2.2. This theorem provides a local stable manifold parameterized by K : [0, ρ) → R2 with ρ small. The proof does not give an explicit estimate for the value of ρ. However, we can extend the domain of K by using the formula K(t) = F −j K(Rj (t)), j ≥ 1, while the iterates of the inverse map F −1 exist (note that R is a weak contraction). In particular, if the map F −1 is globally defined, as it happens for example for the Hénon map, one can extend the domain of K to [0, ∞). This observation also applies for the next theorems 2.3, 2.7 and 2.10. In the analytic case the domain of K can be extended to an open domain of C that contains (0, ρ). Next theorem is an a posteriori version of Theorem 2.1 which, given an analytic approxima- tion, in a certain sense, of the solutions K and R of the conjugation equation F ◦ K = K ◦ R, provides exact solutions of the equation, close to the approximations. Theorem 2.3. Let F : U ⊂ R2 → R2 be as in Theorem 2.1 and let K̂ : (−ρ, ρ) → R2 and R̂ = (−ρ, ρ) → R be analytic maps satisfying   (t2 , K̂k+1 y tk+1 ) + (O(t3 ), O(tk+2 )) case 1, K̂(t) =  (t, K̂ly tl ) + (O(t2 ), O(tl+1 )) cases 2, 3, and R̂(t) = t + R̂N tN + O(tN +1 ), R̂N < 0, such that F (K̂(t)) − K̂(R̂(t)) = (O(tn+N ), O(tn+2N −1 )), (2.5) for some n ≥ 2 in case 1 or n ≥ 1 in cases 2, 3. Then, there exists a C 1 map K : [0, ρ) → R2 , analytic in (0, ρ), and an analytic map R : (−ρ, ρ) → R such that F (K(t)) = K(R(t)), t ∈ [0, ρ) 6 and K(t) − K̂(t) = (O(tn+1 ), O(tn+N )),   O(t2k−1 ) if n ≤ k R(t) − R̂(t) = case 1,  0 if n > k   O(t2l−1 ) if n ≤ l − 1 R(t) − R̂(t) = cases 2, 3.  0 if n > l − 1 Remark 2.4. In case 1, condition (2.5) with n = 2 implies the following relations s 2ak c y y K̂k+1 =± , R̂k = K̂k+1 . c(k + 1) 2 In cases 2 and 3 the condition (2.5) with n = 1 implies   ak + bl K̂ly = lR̂l K̂ly case 2, R̂l = cK̂k+1 y ,  bl = lR̂l case 3. Remark 2.5. Theorem 2.3 provides the existence of a stable manifold assuming it has been previously approximated but the theorem is independent of the way such an approximation has been obtained. Propositions 3.1, 3.4 and 3.5 (in Section 3) provide an algorithm to obtain polynomial maps Kn and Rn that satisfy condition (2.5) of Theorem 2.3 for any n. Remark 2.6. The form of the map R given in the statement of Theorem 2.1 is the normal form of the dynamics of a one-dimensional system in a neighborhood of a parabolic point (see [26, 28]). The following are the main results concerning the existence and regularity of stable invariant manifolds of C r maps of the form (2.2). As in the analytic case, the results provide also the existence of invariant manifolds for maps of the form (2.1). Theorem 2.7. Let F : U ⊂ R2 → R2 be a C r map in a neighborhood U of (0, 0) of the form (2.2) with r ≥ 3. Assume the following hypotheses according to the different cases: • (case 1) ak > 0 and r ≥ 32 k, • (case 2) ak > 0, bl 6= 0, r > k and   β  c k ak  2l β max 2l(l − 1) + 2 β , < 1, (r − 2l + 2)(r − l + 1) bl r−l+1 where β = √2l2|bl | . |bl − bl +4 c ak l| • (case 3) bl < 0, r > 2l − 1 and l(l−1) (r−2l+2)(r−l+1) < 1. 7 Then, there exists a C 1 map H √ : [0, ρ) → R2 , H ∈ C r (0, ρ), of the form (2.4), with Hk+1 y = q bl − b2l +4 c ak l 2ak − c(k+1) for case 1, Hl = y 2cl for case 2 and Hl = cl for case 3, and a polynomial y bl R of the form R(t) = t + RN t + R2N −1 t2N −1 , with Rk = 2c Hk+1 N y for case 1 and Rl = cHly for cases 2, 3, such that F (H(t)) = H(R(t)), t ∈ [0, ρ). If the map F is C ∞ then the parameterization H is C ∞ in (0, ρ). Remark 2.8. The assumptions ak > 0 and k ≤ r for cases 1 and 2 and bl < 0 and l ≤ r for case 3 are necessary conditions for the existence of a formal, locally unique stable invariant curve of F asymptotic to (0, 0). The other hypotheses of the theorem are nondegeneracy conditions on the reduced form of F , sufficient to ensure the existence of a stable invariant curve of class C r asymptotic to (0, 0). We do not claim that these conditions on r are sharp. Remark 2.9. For case 2, the condition on the coefficients of F is always satisfied provided that r is sufficiently larger than l. Another sufficient condition for it to be satisfied is that β is small enough. The smallness of the coefficient β is a measure of how fast the dynamics on the associated invariant manifold is. For case 3, a sufficient nondegeneracy condition for the stable manifold to exist is given by r ≥ 34 (2l − 1). Notice that the assumption r ≥ 2l − 1 is necessary for the constructions we will do. We also provide an a posteriori version of Theorem 2.7. Theorem 2.10. Let F : U ⊂ R2 → R2 be a map satisfying the hypotheses of Theorem 2.7 and let K̂ : (−ρ, ρ) → R2 and R̂ = (−ρ, ρ) → R be analytic maps satisfying   (t2 , K̂k+1 y tk+1 ) + (O(t3 ), O(tk+2 )) case 1, K̂(t) =  (t, K̂ly tl ) + (O(t ), O(t 2 l+1 )) cases 2, 3, and R̂(t) = t + R̂N tN + O(tN +1 ), R̂N < 0, such that F (K̂(t)) − K̂(R̂(t)) = (O(tn+N ), O(tn+2N −1 )), for some n ≥ 2 in case 1 or n ≥ 1 in cases 2, 3. Then, there exists a C 1 map H : [0, ρ) → R2 , H ∈ C r (0, ρ), and an analytic map R : (−ρ, ρ) → R such that F (H(t)) = H(R(t)), t ∈ [0, ρ) and H(t) − K̂(t) = (O(tm ), O(tm+N −1 )), where m = min {n + 1, 2r − 2k + 2} (case 1) and m = min {n + 1, r − 2l + 2} (cases 2, 3), and   O(t2k−1 ) if n ≤ k R(t) − R̂(t) = case 1,  0 if n > k   O(t2l−1 ) if n ≤ l − 1 R(t) − R̂(t) =  cases 2, 3. 0 if n > l − 1 8 The structure of the proof is analogous to the one of Theorem 2.3 and uses the constructions of the approximations in the proofs of Theorems 2.1 and 2.7. It will be omitted. As mentioned, using the conjugations (x, y) 7→ (±x, ±y) and F −1 we can obtain the local phase portraits and the location of the local invariant manifolds of F depending on the studied cases (see [12]). Remark 2.11. The invariant manifolds obtained in Theorems 2.1, 2.3, 2.7 and 2.10 are unique. For that we refer to Theorem 4.1 of [12], where it is proved that if the map F is C k , in all the considered cases the local stable set Wρs+ is a graph and therefore is unique. This is proved by checking that both the iterates of the points that are above and the ones that are below the invariant curve cannot converge to the fixed point by a detailed study of the behaviour of the iterates. However, the parameterizations are not unique because if K and and R satisfy F ◦ K = K ◦ R, then for any invertible map β : [0, ρ] → R, the maps K̃ = K ◦ β and R̃ = β −1 ◦ R ◦ β satisfy F ◦ K̃ = K̃ ◦ R̃. 2.4 Unstable manifolds Assuming F satisfies the hypotheses of Theorem 2.1 if F is analytic, or the ones of Theorem 2.7 if F is differentiable, in cases 1 and 2, the results for the unstable manifolds are obtained from the stated theorems without having to compute the inverse map F −1 . Only in case 2 for differentiable maps one has to check a technical condition as explained below. For case 3, if one assumes bl > 0 instead, then an analogous result is obtained for the existence of an unstable manifold of F . Next, we show that the expansions of the parameterizations of the unstable curves obtained in Section 3 are approximations of true invariant curves, as it happens for the stable ones. Assume we have a map of the form (2.2). Then, by Propositions 3.1, 3.4 or 3.5 we have approximations Kn and Rn such that Gn (t) = F (Kn (t)) − Kn (Rn (t)) = (O(tn+N ), O(tn+2N −1 )), (2.6) with Rn (t) = t + RN tN + O(tN +1 ) and RN > 0, which means that 0 is a repellor for Rn . Also, Rn is invertible and we have N +1 n (t) = t − RN t + O(t R−1 ), N and     x x − cy + cak (x − cy)k + cbl y(x − cy)l−1 + O(xk+1 ) + O(yxl ) F −1   = . y y − ak (x − cy)k − bl y(x − cy)l−1 + O(xk+1 ) + O(yxl ) Then, composing by F −1 and R−1 n in (2.6) we obtain F −1 (Kn (t)) − Kn (R−1 n (t)) = (O(t n+N ), O(tn+2N −1 )). Moreover, there exists a change of variables of the form C(x, y) = (x, −y) + O(k(x, y)kN ) that transforms F −1 into its reduced form G := C −1 ◦ F −1 ◦ C, and then G reads     x x + cy G  =  . y y + ak xk − bl yxl−1 + O(xk+1 ) + O(yxl ) 9 We also have G(C −1 (Kn (t))) − C −1 (Kn (R−1 n (t))) = (O(t n+N ), O(tn+2N −1 )). Thus, if F is in case 1 with ak > 0 then G is also in case 1 with the same coefficient ak positive. Also, if F is in case 2 with ak > 0 and bl 6= 0 then G is also in case 2 with the corresponding coefficients ak positive and bl different from 0. If F is in case 3 with bl > 0 then G is also in case 3 and the coefficient of yxl−1 is given by −bl . Therefore, by Theorem 2.3 there exist a map K : [0, ρ) → R2 , analytic in (0, ρ) and an analytic map R : (−ρ, ρ) → R such that G ◦ K = K ◦ R, with K(t) − C −1 Kn (t) = (O(tn+1 ), O(tn+N )), (2.7)   O(t2k−1 ) if n ≤ k n (t) =  R(t) − R−1 case 1, 0 if n > k   O(t2l−1 ) if n ≤ l − 1 n (t) = R(t) − R−1 cases 2, 3.  0 if n > l − 1 Hence, we have F −1 ◦ C ◦ K = C ◦ K ◦ R, which means that C ◦ K is a parameterization of an unstable manifold of F . Moreover, from (2.7) and the form of C, we have C(K(t)) − Kn (t) = (O(tn+1 ), O(tn+N )), and therefore Kn is an approximation of a parameterization of such unstable manifold. In the C r case one has to apply Theorem 2.10. If F satisfies the conditions of case 1, G also does. The same happens for case 3 if we assume bl > 0 instead of bl < 0. If F satisfies the conditions of case 2, since the coefficient bl of F becomes −bl for G, one has to check the condition involving the maximum taking now β as β = √2l |b2 l | . Then, for cases 1 and |−bl − bl +4 c ak l| 3 or for case 2 when that condition holds, we conclude as we have explained for the analytic case. 3 Formal polynomial approximation of the parameter- izations of the curves In this section we consider C r maps F of the form (2.2) and we provide algorithms, depending on the case, to obtain two polynomial maps, Kn and Rn , that are approximations of solutions K and R of the invariance equation F ◦ K = K ◦ R. (3.1) Because of the nature of the problem, the two components of Kn will have a different order and different degrees. The index n has to be seen as an induction index. Higher values of n mean better approximation. The obtained approximations correspond to formal invariant curves. They correspond to stable curves when the coefficient Rk (case 1) or Rl (cases 2, 3) of Rn are negative (see below). When those coefficients are positive they correspond to unstable curves. 10 Proposition 3.1 (Case 1). Let F be a C r map of the form (2.2) with 2 ≤ k ≤ r. Assume that k < 2l − 1 and ak > 0. Then, for all 2 ≤ n ≤ 2(r − k + 1), there exist two pairs of polynomial maps, Kn and Rn , of the form   t2 + · · · + Knx tn Kn (t) =  y k+1  Kk+1 t + · · · + Kn+k−1 y tn+k−1 and  t + R tk if 2 ≤ n ≤ k, Rn (t) = k t + Rk tk + R2k−1 t 2k−1 if n ≥ k + 1, such that Gn (t) := F (Kn (t)) − Kn (Rn (t)) = (O(tn+k ), O(tn+2k−1 )). (3.2) For the first pair we have s s 2 ak c ak c y y Kk+1 =− , Rk = − = Kk+1 , c (k + 1) 2(k + 1) 2 and for the second one s s 2 ak c ak c y y Kk+1 = , Rk = = Kk+1 . c (k + 1) 2(k + 1) 2 If F is C ∞ or analytic, one can compute the polynomial approximation Kn up to any order. Remark 3.2. The algorithm described in the proof of this (and the next) propositions can be implemented in a computer program to calculate R and the expansion of Kn . Notation 3.3. Along the proof, given a C r one-variable map f , we will denote [f ]n , 0 ≤ n ≤ r, the coefficient of the term of order n of the jet of f at 0. Proof. We will see that we can determine Kn and Rn iteratively. For n = 2, we claim that there exist polynomial maps K2 (t) = (t2 , Kk+1 y tk+1 ) and R2 (t) = t + Rk tk , such that G2 (t) = F (K2 (t)) − K2 (R2 (t)) = (O(tk+2 ), O(t2k+1 )). Indeed, from the expansion of G2 we have   t2 + c Kk+1 y tk+1 − t2 − 2Rk tk+1 + O(t2k ) G2 (t) =  y k+1 , Kk+1 t + ak t2k − Kk+1 y tk+1 − (k + 1)Kk+1 y Rk t2k + O(t2k+1 ) so, if the conditions y c Kk+1 − 2Rk = 0, ak − (k + 1)Kk+1 y Rk = 0, are satisfied, then we clearly have G2 (t) = (O(t2+k ), O(t2k+1 )), and we obtain the values of y Kk+1 and Rk given in the statement. 11 Now we assume that we have already obtained maps Kn and Rn , 2 ≤ n < 2(r − k + 1) such that (3.2) holds true, and we look for   x Kn+1 tn+1 Kn+1 (t) = Kn (t) +  y , Rn+1 (t) = Rn (t) + Rn+k−1 tn+k−1 , Kn+k tn+k such that Gn+1 (t) = (O(tn+k+1 ), O(tn+2k )). Using Taylor’s theorem, we write Gn+1 (t) = F (Kn (t) + (Kn+1 x y tn+1 , Kn+k tn+k )) − (Kn (t) + (Kn+1 x y tn+1 , Kn+k tn+k )) ◦ (Rn (t) + Rn+k−1 tn+k−1 ) = Gn (t) + DF (Kn (t)) · (Kn+1 x y tn+1 , Kn+k tn+k ) − (Kn+1 x y tn+1 , Kn+k tn+k ) ◦ (Rn (t) + Rn+k−1 tn+k−1 ) Z 1 + (1 − s)D2 F (Kn (t) + s(Kn+1 x y tn+1 , Kn+k tn+k )) ds (Kn+1 x y tn+1 , Kn+k tn+k )⊗2 0 − DKn (Rn (t))Rn+k−1 tn+k−1 Z 1 − (1 − s)D2 Kn (Rn (t) + s Rn+k−1 tn+k−1 ) ds (Rn+k−1 tn+k−1 )2 . 0 Performing the computations in the previous expression we have Gn+1 (t) = Gn (t)   [c Kn+k y − (n + 1)Rk Kn+1 x − 2 Rn+k−1 ] tn+k + O(tn+k+1 ) (3.3) + . [k ak Kn+1 x − (n + y k) Rk Kn+k − (k + 1)Kk+1 y Rn+k−1 ] tn+2k−1 + O(tn+2k ) Since, by the induction hypothesis, Gn (t) = (O(tn+k ), O(tn+2k−1 )), to complete the induction step we need to make [Gn+1 x ]n+k and [Gn+1 y ]n+2k−1 vanish. From (3.3) we have [Gn+1 x ]n+k = [Gnx ]n+k + c Kn+k y − (n + 1)Rk Kn+1 x − 2 Rn+k−1 , [Gn+1 ]n+2k−1 = [Gn ]n+2k−1 + k ak Kn+1 − (n + k) Rk Kn+k y y x y − (k + 1)Kk+1 y Rn+k−1 . Thus, the conditions [Gn+1 x ]n+k = [Gn+1 y ]n+2k−1 = 0 are equivalent to      −(n + 1)Rk c x Kn+1 −[Gnx ]n+k + 2 Rn+k−1    = . (3.4) k ak −(n + k) Rk y Kn+k −[Gny ]n+2k−1 + (k + 1) Kk+1 y Rn+k−1 If n 6= k the matrix in the left hand side of (3.4) is invertible, so we can take Rn+k−1 = 0 and then obtain Kn+1 x and Kn+k y in a unique way. When n = k, the determinant of the matrix is zero. Then, choosing 2k Rk [Gnx ]2k + c [Gny ]3k−2 R2k−1 = , 2 (3k + 1) Rk system (3.4) has solutions. In this case, however, Kk+1 x and K2k y are not uniquely determined. 12 Proposition 3.4 (Case 2). Let F be a C r map of the form (2.2), with r ≥ k ≥ 2. We b2 assume k = 2l − 1, ak 6= 0, bl 6= 0 and ak > − 4cll . If ak < 0 we assume also ak 6= − 2l+1 b2 . 3l−1 l Then, for all 1 ≤ n ≤ r − 2l + 2 = r − k + 1, there exists two pairs of polynomial functions Kn and Rn of the form   t + · · · + Knx tn Kn (t) =  y l  (3.5) Kl t + · · · + Kn+l−1 y tn+l−1 and  t + R tl if 1 ≤ n ≤ l − 1, Rn (t) = (3.6) l t + Rl t l + R2l−1 t 2l−1 if n ≥ l, such that Gn (t) := F (Kn (t)) − Kn (Rn (t)) = (O(tn+l ), O(tn+2l−1 )). For the first pair we have q q bl − b2l + 4 c ak l bl − b2l + 4 c ak l Kly = , Rl = = cKly , 2cl 2l and for the second one q q bl + b2l + 4 c ak l bl + b2l + 4 c ak l Kly = , Rl = = cKly . 2cl 2l 2l+1 2 If ak = − 3l−1 bl and bl < 0 we can compute the first pair up to n = l − 1 and the second 2l+1 2 pair for any n ≤ r − 2l + 2. If ak = − 3l−1 bl and bl > 0 we can compute the first pair up to n ≤ r − 2l + 2 and the second pair up to n = l − 1. If F is C ∞ or analytic, one can compute the polynomial approximations Kn up to any order, 2l+1 2 except when ak = − 3l−1 bl . Proposition 3.5 (Case 3). Let F be a C r map of the form (2.2), with r ≥ l ≥ 2. Assume k > 2l − 1, bl 6= 0 Then, for all 1 ≤ n ≤ r − 2l + 2, there exist a pair of polynomial functions Kn and Rn of the form (3.5) and (3.6) respectively, such that Gn (t) := F (Kn (t)) − Kn (Rn (t)) = (O(tn+l ), O(tn+2l−1 )). We have bl bl Kly = , Rl = = cKly . cl l If we further assume that k ≤ r and ak 6= 0, then for 1 ≤ n ≤ r − (k − l)l − 2l + 1 there exists another pair Kn and Rn with   t + · · · + Knx tn Kn (t) =  y  Kk−l+1 tk−l+1 + · · · + Kn+k−l y tn+k−l 13 and  t + R k−l+1 if 2 ≤ n ≤ k − l, k−l+1 t Rn (t) = t + Rk−l+1 tk−l+1 + R2(k−l)+1 t2(k−l)+1 if n ≥ k − l + 1, such that Gn (t) := F (Kn (t)) − Kn (Rn (t)) = (O(tn+k−l+1 ), O(tn+k )). We have ak y Kk−l+1 =− , Rk−l+1 = cKk−l+1 y . bl If F is C ∞ or analytic, one can compute the polynomial approximations Kn up to any order. The proofs are analogous to the one of Proposition 3.1. 4 The analytic case This section is devoted to prove Theorems 2.1 and 2.3. Following the parameterization method, given a map F of the form (2.2), first we consider polynomial approximations Kn : R → R2 and Rn : R → R of solutions of equation (3.1) obtained in Section 3 up to a high enough order, to be determined in the proof. Then, keeping R = Rn fixed, we look for a correction ∆ : [0, ρ) → R2 , for some ρ > 0, of Kn , analytic on (0, ρ), such that the pair K = Kn + ∆, R = Rn satisfies the invariance condition F ◦ (Kn + ∆) − (Kn + ∆) ◦ R = 0. (4.1) The proof of Theorem 2.1 is organized as follows. First, taking into account the structure of F we rewrite equation (4.1) to separate the dominant linear part with respect to ∆ and the remaining terms. This motivates the introduction of two families of operators, Sn, R and Nn, F , and the spaces where these operators will act on. We provide the properties of these operators in Lemmas 4.6 and 4.7. Finally, we rewrite the equation for ∆ as the fixed point equation ∆ = Tn, F (∆), where Tn, F = Sn, −1 R ◦ Nn, F and we apply the Banach fixed point theorem to get the solution. The properties of the operators Tn,F are deduced in Lemma 4.10. At the end of the section we prove Theorem 2.3. 4.1 The functional equation Let F : U ⊂ R2 → R2 be an analytic map in a neighborhood U of (0, 0), satisfying the hypotheses of Theorem 2.1,     x + cy 0 F (x, y) =  + , y p(x) + y q(x) + u(x, y) + g(x, y) 14 where c > 0, p, q and u are the polynomials introduced in Section 2.1 and g(x, y) is an analytic function. We take p, q and u of degree at least k in case 1 and degree at least 2l − 1 in cases 2 and 3. Then we have g(x, y) = O(k(x, y)kk+1 ) for case 1 and g(x, y) = O(k(x, y)k2l ) for cases 2 and 3. We denote v(x, y) = u(x, y) + g(x, y). From Propositions 3.1, 3.4 and 3.5 we take n, with n ≥ k + 1 in case 1 and n ≥ l is cases 2 and 3, and we have that there exist polynomials Kn and R = Rn such that En (t) = (O(tn+N ), O(tn+2N −1 )), (4.2) where En = F ◦ Kn − Kn ◦ R. Since we are looking for the stable manifold we will take the approximations corresponding to R = Rn with the coefficient RN < 0. Hence, we look for ρ > 0 and a map K = Kn + ∆ : [0, ρ) → R2 , analytic on (0, ρ) satisfying (4.1), where Kn and R are the mentioned maps that satisfy (4.2). Moreover, we will ask ∆ to satisfy ∆ = (∆x , ∆y ) = (O(tn ), O(tn+N −1 )). Using (4.2) we can rewrite (4.1) as ∆x ◦ R − ∆x = c ∆y + Enx , ∆y ◦ R − ∆y = p ◦ (Knx + ∆x ) − p ◦ Knx + Kny · (q ◦ (Knx + ∆x ) − q ◦ Knx ) (4.3) + ∆y · q ◦ (Knx + ∆x ) + v ◦ (Kn + ∆) − v ◦ Kn + Eny . 4.2 Function spaces, the operators Sn, N and Nn, N and their prop- erties Next we introduce notation, suitable function spaces, and some operators. Definition 4.1. Given β, ρ > 0 such that ρ < 1 and β < π, let S be the sector n β o S = S(β, ρ) = z ∈ C | | arg(z)| < , 0 < |z| < ρ . 2 Given a sector S = S(β, ρ) let Xn , for n ∈ N, be the Banach space given by |f (z)| Xn = {f : S → C | f ∈ Hol(S), f ((0, ρ)) ⊂ R, kf kn := sup < ∞}, z∈S |z|n where Hol(S) denotes the space of holomorphic functions on S. Note that when n ≥ 1 the functions f in Xn can be continuously extended to z = 0 with f (0) = 0 and, if moreover, n ≥ 2, the derivative of f can be continuously extended to z = 0 with f 0 (0) = 0. Note also that Xn+1 ⊂ Xn , for all n ∈ N, and that if f ∈ Xn+1 , then kf kn ≤ kf kn+1 . Moreover if f ∈ Xm , g ∈ Xn , then f g ∈ Xm+n and kf gkm+n ≤ kf km kgkn . Given n, m ∈ N we denote Xm, n := Xm × Xn the product spaces, endowed with the product norm kf km, n = max {kf x km , kf y kn }, f = (f x , f y ) ∈ Xm, n . 15 Given n ≥ 1, N ≥ 2, we define the space Σn, N = Xn, n+N −1 , endowed with the product norm. Also, given α > 0, we define the closed ball Σαn, N = {f ∈ Σn, N | kf kΣn, N ≤ α}. For the sake of simplicity, we will omit the parameters ρ and β in the notation of the spaces Σn, N and the balls Σαn, N . Now let F be as in Theorem 2.1, and Kn and R = Rn be the polynomials provided in Section 3 satisfying (4.2) with n ≥ k + 1 in case 1 and n ≥ l in cases 2, 3. Since F is analytic in U , it has a holomorphic extension to some neighborhood W of (0, 0) in C2 . Let d > 0 be the radius of a ball in C2 contained in the domain where F is holomorphic. Also, Kn and R are defined on any complex sector S(β, ρ). Then it is possible to set equation (4.3) in a space of holomorphic functions defined in a sector S(β, ρ), and look for ∆ being an analytic function of a complex variable that takes real values when restricted to the real line. To solve equation (4.3), we will consider n big enough and we will look for a solution, ∆, in a closed ball of the space Σn, N . In order for the compositions in (4.3) to make sense we need to ensure the range of Kn + ∆ to be contained in the domain where F is analytic. We take n o 1 d α = min , 2 2 . In this way, since Kn (0) = (0, 0), taking ρK ∈ (0, 1) such that supz∈S(β, ρK ) kKn (z)k < d/2 and ρ ≤ ρK , if ∆ : S(β, ρ) → C2 belongs to the ball of radius α of Xn, m , with n, m ≥ 0, we have d sup k∆(z)k = sup max{ |∆x (z)|, |∆y (z)|} ≤ max { d2 ρn , d ρm } < . z∈S(β, ρ) z∈S(β, ρ) 2 2 Therefore, under the previous conditions, if ρ ≤ ρK and ∆ ∈ Σαn, N then kKn (z) + ∆(z)k < d and the composition F ◦ (Kn + ∆) is well defined. Next we introduce two families of operators that will be used to deal with (4.3). The definition of such operators is motivated by the equation itself. First, we state the following auxiliary result (see [3]), Lemma 4.2. Let R : S(β, ρ) → C be a holomorphic function of the form R(z) = z + RN z N + O(|z|N +1 ), with RN < 0. Assume that 0 < β < Nπ−1 . Then, for any ν ∈ (0, (N − 1)|RN | cos λ), with λ = β N2−1 , there exists ρ > 0 small enough such that |z| |Rj (z)| ≤ , ∀ j ∈ N, ∀ z ∈ S(β, ρ), (1 + j ν |z|N −1 )1/N −1 where Rj refers to the j-th iterate of the map R. In addition, R maps S(β, ρ) into itself. 16 Then, if f is defined in S(β, ρ), with suitable values of the parameters β, ρ, and R satisfies the conditions of the lemma, the composition f ◦ R is well defined. Definition 4.3. Given n ≥ 1, N ≥ 2 and a polynomial R(z) = z + RN z N + O(|z|N +1 ) satisfying the hypotheses of Lemma 4.2, let Sn, R : Σn, N → Σn, N be the linear operator defined component-wise as Sn, R = (Sn, x R , Sn, R ), with y R f = Sn, R f = f ◦ R − f. x y Sn, Remark 4.4. Notice that although both components of Sn, R are formally identical they act on spaces of holomorphic functions of different orders. Definition 4.5. Let F be the holomorphic extension of an analytic map of the form (2.2) satisfying the hypotheses of Theorem 2.1. For n ∈ N, we introduce Nn, F = (Nn,x F , Nn,y F ) : Σαn, N → Xn+N −1, n+2N −2 , by Nn,x F (f ) = c f y + Enx , Nn,y F (f ) = p ◦ (Knx + f x ) − p ◦ Knx + Kny · (q ◦ (Knx + f x ) − q ◦ Knx ) + f y · q ◦ (Knx + f x ) + v ◦ (Kn + f ) − v ◦ Kn + Eny . By the properties of R and the choice of α, the operators Sn, R and Nn, F are well defined and Sn, R is linear and bounded. Using these operators, equations (4.3) can be written as Sn, R ∆ = Nn, F (∆). The following lemma states that the operators Sn, R have a bounded right inverse and provide a bound for the norm kSn, −1 R k. Lemma 4.6. Given N ≥ 2 and n ≥ 1, the operator Sn, R : Σn, N → Σn, N , has a bounded right inverse R : Xn+N −1, n+2N −2 → Σn, N = Xn, n+N −1 , −1 Sn, given by ∞ X Rη = − (4.4) −1 Sn, η ◦ Rj , η ∈ Xn+N −1, n+2N −2 . j=0 Moreover, for any fixed ν ∈ (0, (N − 1)|RN |), there exists ρ > 0 such that, taking S(β, ρ) with β < Nπ−1 as the domain of the functions of Xn+N −1, n+2N −2 , we have the operator norm bounds x k(Sn, R) k ≤ ρ −1 N −1 + ν1 Nn−1 , y k(Sn, R) k ≤ ρ −1 N −1 + ν1 n+N N −1 −1 . The operators Nn, F are Lipschitz and we provide bounds for their Lipschitz constants. Lemma 4.7. For each n ≥ 3, there exists a constant, Mn > 0, for which the operator Nn, F satisfies Lip Nn,x F = c, 17 and Lip Nn,y F ≤ k |ak | + Mn ρ, (case 1), Lip Nn,y F ≤ max{((l − 1) |Kly bl | + k |ak |) + Mn ρ, |bl | + Mn ρ}, (case 2), Lip Nn,y F ≤ max{(l − 1) |Kly bl | + Mn ρ, |bl | + Mn ρ}, (case 3), where ρ is the radius of the sector S(β, ρ) where the functions of Σαn, N are defined. Now, we define the third family of operators, Tn, F . Definition 4.8. Let F be the holomorphic extension of an analytic map of the form (2.2) satisfying the hypotheses of Theorem 2.1. Given n ≥ 3 we define Tn, F : Σαn, N → Σn, N by Tn, F = Sn, −1 R ◦ Nn, F . Remark 4.9. Note that given a map F , to define the previous operators we always take together the associated triple (F, Kn , R) satisfying F ◦Kn −Kn ◦R = En . Then, the operators Sn, R , Nn, F and Tn, F are associated not only with the map F itself but to the approximation of a particular invariant manifold of F . Lemma 4.10. Given an analytic map F satisfying the hypotheses of Theorem 2.1, there exist n0 > 0 and ρ0 > 0 such that if ρ < ρ0 , then, for every n ≥ n0 , we have Tn, F (Σαn, N ) ⊆ Σαn, N and Tn, F is a contraction operator in that ball. The proofs of the previous three lemmas are deferred to Section 6. 4.3 Proofs of Theorems 2.1 and 2.3 Now we are ready to give the proofs of Theorems 2.1 and 2.3. Proof of Theorem 2.1. First we consider the holomorphic extension of F to a neighborhood of the origin which contains a ball of radius d > 0 in C2 and let α = min {1/2, d/2}. Let Kn and R(t) = Rn (t) = t + RN tN + R2N −1 t2N −1 be the polynomials given by Propositions 3.1, 3.4 or 3.5 , with n ≥ k + 1 or n ≥ l respectively, satisfying En (t) = F ◦ Kn (t) − Kn ◦ Rn (t) = (O(tn+N ), O(tn+2N −1 )). We also assume that n > n0 , where n0 is the integer provided by Lemma 4.10. We rewrite F ◦ (Kn + ∆) − (Kn + ∆) ◦ R = 0 in the form (4.3), or using the previously defined operators, Sn, R ∆ = Nn, F (∆). By Lemma 4.6, if ρ is small, Sn,R has a right inverse and we can rewrite the equation as ∆ = Tn, F (∆). 18 By Lemma 4.10 we have that Tn, F maps Σαn, N into itself and is a contraction. Then it has a unique fixed point, ∆∞ ∈ Σαn, N . Note that this solution is unique once Kn is fixed. Finally K = Kn + ∆∞ satisfies the conditions in the statement. The C 1 character of K at the origin follows from the order condition of K at 0. Proof of Theorem 2.3. We write the proof for case 1, the other cases being almost identical except for some adjustments in the indices of the coefficients of Rn . Let n0 be the integer provided by Lemma 4.10. If the value of n given in the statement is such that n < n0 , P 0 first we look for a better approximation Kn0 of the form Kn0 (t) = K̂(t) + nj=n+1 K̂ j (t) with K̂ j (t) = (K̂jx tj , K̂j+k−1 y tj+k−1 ) and  R̂(t) if n ≥ k + 1, Rn0 (t) = R̂(t) + R̂2k−1 t 2k−1 if n ≤ k. The coefficients K̂jx , K̂j+k−1 y and R̂2k−1 are obtained imposing the condition F ◦ Kn0 (t) − Kn0 ◦ Rn0 (t) = (O(tn0 +k ), O(tn0 +2k−1 )). Proceeding as in Proposition 3.1, we obtain K̂ j iteratively. We denote Kj (t) = K̂(t) + Pj m=n+1 K̂ (t) and Rj (t) = R̂(t) + R̃j (t), where R̃j (t) = δj,k+1 R̂2k−1 t . In the iterative m 2k−1 step we have F ◦ Kj (t) − Kj ◦ Rj (t) = (O(tj+k ), O(tj+2k−1 )). Then, F (Kj (t) + K̂ j+1 (t))−(Kj + K̂ j+1 ) ◦ (R̂(t) + R̃j (t)) =F (Kj (t)) − Kj (R̂(t)) + DF (Kj (t))K̂ j+1 (t) − K̂ j+1 (R̂(t) + R̃j (t)) Z 1 + (1 − s)D2 F (Kj (t) + sK̂ j+1 (t))(K̂ j+1 (t))⊗2 ds 0 − DKj (R̂(t))R̃j (t) Z 1 − (1 − s)D2 Kj (R̂(t) + sR̃j (t))(R̃j (t))2 ds. 0 The condition F ◦ Kj+1 (t) − Kj+1 ◦ Rj+1 (t) = (O(tj+k+1 ), O(tj+2k )) leads to the same equation (3.4) as in Proposition 3.1 which we solve in the same way. From this point we can proceed as in the proof of Theorem 2.1 and look for ∆ ∈ Xn0 , n0 +k−1 such that the pair K = Kn0 + ∆, R = Rn0 satisfies F ◦ K = K ◦ R. We have that K(t) − K̂(t) = Kn0 (t) − K̂(t) + ∆(t) = (O(tn+1 ), O(tn+k )) + (O(tn0 ), O(tn0 +k−1 )), with n < n0 . If n ≥ n0 we look for K∗ (t) = K̂(t) + K̂ n+1 (t) with K̂ n+1 (t) = (K̂n+1 x y tn+1 , K̂n+k tn+k ) 19 and  R̂(t) if n ≥ k + 1, R∗n (t) = R̂(t) + R̂2k−1 t2k−1 if n ≤ k. We determine K̂n+1 x , K̂n+k y so that F ◦ K∗ (t) − K∗ ◦ R∗ (t) = (O(tn+k+1 ), O(tn+2k )) as in the previous case and we look for ∆ ∈ Xn+1, n+k such that the pair K = K∗ + ∆, R = R∗ satisfies F ◦ K = K ◦ R. As before we obtain K(t) − K̂(t) = (O(tn+1 ), O(tn+k )). Again, the C 1 character of K at 0 follows form the order condition of K. 5 The differentiable case This section is devoted to prove Theorem 2.7 for     x + cy 0 F (x, y) =  + . y p(x) + y q(x) + u(x, y) + g(x, y) As in Section 4 we use the parameterization method. To get the initial approximation we first consider the Taylor polynomial of F or degree r which we denote by F ≤ and reads     x + cy 0 F ≤ (x, y) =  + . y p(x) + y q(x) + u(x, y) Since F ≤ is analytic, Theorem 2.1 provides a C 1 map K : [0, ρ) → R, analytic on (0, ρ) and a polynomial, R, such that F≤ ◦ K − K ◦ R = 0 on [0, ρ). (5.1) Then, we look for ρ > 0 and a C r function, H = K + ∆ : (0, ρ) → R2 , such that F ◦ (K + ∆) − (K + ∆) ◦ R = 0, (5.2) In Section 5.1, we establish a functional equation for ∆ obtained from (5.2) which will be the object of our study. In Section 5.2 we describe the function spaces where we will set such an equation and the operators SL, R and NL, F together with their properties (Lemmas 5.6 and 5.7). Notice that although the notation of the operators is similar to the one of the operators in Section 4, both pair of families of operators are different. In Section 5.3 we recall the fiber contraction theorem and we also introduce the family of operators TL, F given by TL, F = SL, −1 R ◦NL, F and we describe its properties in Lemmas 5.9 and 5.10. Finally, in Section 5.4 we prove the existence of a solution of the functional equation and we conclude the proof of Theorem 2.7. 20 5.1 The functional equation Let F : U ⊂ R2 → R2 be a C r map of the form (2.2) satisfying the hypotheses of Theorem 2.7. Along the section, once having taken a C r map F of the form (2.2), the maps K and R will always refer to the analytic solutions of F ≤ ◦ K − K ◦ R = 0, on some interval [0, ρ) given by Theorem 2.1. Using (5.1) and the previous notation, condition (5.2) can be rewritten as ∆x ◦ R − ∆x = c ∆y , ∆y ◦ R − ∆y = p ◦ (K x + ∆x ) − p ◦ K x + K y · (q ◦ (K x + ∆x ) − q ◦ K x ) (5.3) + ∆y · q ◦ (K x + ∆x ) + u ◦ (K + ∆) − u ◦ K + g ◦ (K + ∆). Clearly, a continuous function ∆ satisfies (5.2) if and only if it satisfies (5.3). Since we want to prove differentiablity of ∆, next we derive r equations for the derivatives of ∆ by formally differentiating equation (5.3). In our approach we will look for continuous solutions of these equations. After having differentiated (5.3) L times, 1 ≤ L ≤ r, we obtain DL ∆x ◦ R (DR)L − DL ∆x = c DL ∆y + JL,x N (∆, . . . , DL−1 ∆), DL ∆y ◦ R (DR)L − DL ∆y = p0 ◦ (K x + ∆x ) DL ∆x + (K y + ∆y ) q 0 ◦ (K x + ∆x ) DL ∆x (5.4) + q ◦ (K + ∆ ) D ∆ + (Du + Dg) ◦ (K + ∆) · D ∆ x x L y L + JL,y N (∆, . . . , DL−1 ∆), where JL,x F and JL,y F are given by JL,x F (f0 , . . . fL−1 ) = ΛxL, R (f0x , . . . fL−1 x ), (5.5) JL,y F (f0 , . . . fL−1 ) = ΛyL, R (f0y , . . . fL−1 y ) + ΩL, F (f0 , . . . , fL−1 ), and ΛiL, R , i = x, y, by Λi1, R (f0i ) = 0, Λi2, R (f0i , f1i ) = −f1i ◦ R D2 R, (5.6) ΛiL, R (f0i , . . . , fL−1 i ) = D[ΛiL−1, R (f0i , . . . , fL−2 i )] − (L − 1) fL−1 i ◦ R (DR)L−2 D2 R, L ∈ {3, . . . , r}, where in the expansion of the derivative D[ΛiL−1, R (f0i , . . . , fL−2 i )] we substitute Dfi by fi+1 . Note that ΛL, R does not depend on f0 . Moreover, ΩL, F is given by i Ω1, F (f0 ) = DK x (p0 ◦ (K x + f0x ) − p0 ◦ K x ) + DK y · (q ◦ (K x + f0x ) − q ◦ K x ) + K y · DK x (q 0 ◦ (K x + f0x ) − q 0 ◦ K x ) + f0y · DK x q 0 ◦ (K x + f0x ) + (Du ◦ (K + f0 ) − Du ◦ K)DK + Dg ◦ (K + f0 )DK, (5.7) ΩL, F (f0 , . . . , fL−1 ) = D[ΩL−1, F (f0 , . . . , fL−2 )] + D[p0 ◦ (K x + f0x )]fL−1 x + D[(K y + f0y )q 0 ◦ (K x + f0x )]fLx + D[q ◦ (K x + f0x )]fL−1y + D[(Du + Dg) ◦ (K + f0 )] · fL−1 , L ∈ {2, . . . , r}. 21 Note that ΛL, R (f0 , . . . , fL−1 ) comes from the differentiation on the left hand side of (5.3) and ΩL, F (f0 , . . . , fL−1 ) comes from the differentiation on the right hand side of the second equation of (5.3). Expanding the derivatives in (5.6) and (5.7) and changing Dfi by fi+1 we obtain expressions that have to be understood as operators acting on (f0 , . . . , fL−1 ), considering the fj ’s as independent variables. It is important to note that ΛiL,R and ΩiL,F , i = x, y, depend in a polynomial way on fj for j ≥ 1, but not on f0 . 5.2 Function spaces, the operators SL, R and NL, F and their prop- erties We introduce next the notation and the function spaces that we will use to study the func- tional equations (5.3) and (5.4). Definition 5.1. Given 0 < ρ < 1, let Yn , for n ∈ Z, be the Banach space given by |f (t)| Yn = {f : (0, ρ) → R | f ∈ C 0 (0, ρ), kf kn := sup < ∞}, (0, ρ) |t|n where C 0 (0, ρ) denotes the space of continuous functions on (0, ρ). Note that when n ≥ 1 the functions f in Yn can be continuously extended to t = 0 with f (0) = 0 and, if moreover, n ≥ 2, the derivative of f can be continuously extended to t = 0 with f 0 (0) = 0. For n < 0 the functions contained in Yn may be unbounded in a neighborhood of 0. Note also that Yn+1 ⊂ Yn , for all n ∈ Z. If f ∈ Ym , g ∈ Yn , then f g ∈ Ym+n and kf gkm+n ≤ kf km kgkn . If f ∈ Yn+1 , then kf kn ≤ kf kn+1 . Given n, m ∈ Z we denote Ym, n := Ym × Yn the product space, endowed with the product norm kf km, n = max {kf x km , kf y kn }, f = (f x , f y ) ∈ Ym × Yn . Given s, r, N positive integer numbers and L ∈ {0, . . . , r}, we define the spaces L Y ΣL, N = (Ys−2N +2−j, s−N +1−j ), 0≤L≤r j=0 and DΣL−1, N = Ys−2N +2−L, s−N +1−L , 1≤L≤r both endowed with the product norm. Clearly, we have ΣL, N = ΣL−1, N × DΣL−1, N , and Q ΣL, N = Σ0, N × Li=1 DΣi−1, N , for 1 ≤ L ≤ r. For notational convenience we also write DΣ−1, N = Σ0, N . Also, let αi > 0, 1 ≤ i ≤ r. Given L we write α = (α0 , . . . , αL ). We define the closed balls Σα0,0N = {f ∈ Σ0, N | kf kΣ0, N ≤ α0 }, DΣαi−1, N = {f ∈ DΣi−1, N | kf kDΣi−1, N ≤ αi }, i ∈ {1, . . . , r}, i 22 and the products of balls L Y ΣαL, N = Σα0,0N × DΣαi−1, i N, L ∈ {1, . . . , r}, i=1 For notational convenience we will write Σα0, N = Σα0,0N . An element of ΣL, N will be denoted by (f0 , . . . , fL ), with f0 = (f0x , f0y ) ∈ Σ0, N , and fi = (fix , fiy ) ∈ DΣi−1, N , for i = 1, . . . , L. For the sake of simplicity we do not write the dependence with respect to r, s and ρ in the notation of the previous objects. To solve the functional equation (5.2), we look for a solution, f0 , of (5.3) contained in a closed ball Σα0,0N , and for a solution, (f1 , . . . , fL ), of (5.4) in a product ΣαL, N , for each L ∈ {1, . . . , r}. In order for the compositions in (5.4) to be meaningful we have to deal with in n f0 o 1 d a ball of sufficiently small radius. Arguing as in the analytic case we take α0 = min 2 , 2 , where d is the radius of a ball contained in the domain where F is C r . The values of the radii αi , 1 ≤ i ≤ r, will be determined later (see proof of Lemma 5.10). In the differentiable case we consider analogous operators as in the analytical case but now we need a family of them, depending on L, to deal with the equations (5.4) for the derivatives of ∆. Their definitions are determined by the structure of such equations. First, we state two auxiliary results about the iterates of R and their derivatives. Lemma 5.2. Let R : [0, ρ) → R be a differentiable map of the form R(t) = t + RN tN + O(|t|N +1 ), with RN < 0. Then, for any ν, µ such that 0 < ν < (N − 1)|RN | < µ, there exists ρ > 0 such that t t < Rj (t) < , ∀ j ≥ 1, ∀ t ∈ (0, ρ). (5.8) (1 + j µ t N −1 ) 1/N −1 (1 + j ν t −1 )1/N −1 N As a consequence, R maps (0, ρ) into itself. If R were a polynomial the upper bound in Lemma 5.2 would be an immediate corollary of Lemma 4.2. Proof. Let λ > 0 and ϕλ (t) = (1+λ tN −1 t )1/N −1 for t ≥ 0. A computation shows that dtd ϕλ (t) = 1 (1+λ tN −1 )N/N −1 > 0 and hence ϕλ is increasing. We prove (5.8) by induction. When j = 1, it is easy to see that there exists ρ > 0 such that t t ϕµ (t) = < R(t) < = ϕν (t), ∀ t ∈ (0, ρ). (1 + µ tN −1 ) 1/N −1 (1 + ν t −1 )1/N −1 N Assuming (5.8) for j ≥ 1,   t R j+1 (t) = R(R (t)) < ϕν (R (t)) < ϕν j j (1 + j ν tN −1 )1/N −1 t = (1 + (j + 1) ν tN −1 )1/N −1 in the same interval (0, ρ). The lower bound is obtained in a completely analogous way using ϕµ . 23 Lemma 5.3. Let R : [0, ρ) → R be a differentiable map of the form R(t) = t + RN tN + O(|t|N +1 ), with RN < 0, such that DR(t) = 1 + N RN tN −1 + O(|t|N ). For any ν, µ such that 0 < ν < (N − 1)|RN | < µ, let κ = ν/µ. Then, there exists ρ > 0 such that 1 DRj (t) ≤ , ∀ j ∈ N, ∀ t ∈ (0, ρ). (5.9) (1 + j µ tN −1 )κN/N −1 Proof. Since N |RN | > ν NN−1 , by the form of the derivative DR, there exists ρ > 0 such that νN N −1 0 < DR(t) < 1 − t , ∀ t ∈ (0, ρ). N −1 Using the chain rule DRj (t) = Πm=0 j−1 DR(Rm (t)) and the lower bound in (5.8) we can write j−1 X j−1 X   νN DR (t) = exp j log DR(R (t)) ≤ exp m log 1 − (Rm (t))N −1 m=0 m=0 N − 1     −νN j−1 X −νN j−1 X tN −1 ≤ exp  (Rm (t))N −1  ≤ exp   N − 1 m=0 N − 1 m=0 (1 + mµtN −1 ) ! ! −νN Z j tN −1 −νN Z jµtN −1 1 ≤ exp ds = exp dξ N − 1 0 (1 + sµtN −1 ) µ(N − 1) 0 1+ξ   −κN 1 = exp log(1 + jµt N −1 ) = . N −1 (1 + jµt −1 )κN/N −1 N From now on we assume R is as in the previous lemmas and ρ satisfies the conclusions of them, in particular, R(0, ρ) ⊂ (0, ρ). Definition 5.4. Given L ∈ {0, . . . , r}, let SL,R : DΣL−1,N → DΣL−1,N be the linear operator defined component-wise as SL,R = (SL,R x y , SL,R ), with x SL,R f = SL,R y f = f ◦ R (DR)L − f. Notice that although both components are formally identical, they act on different domains. Definition 5.5. Given a map F of class C r satisfying the hypotheses of Theorem 2.7, let N0, F : Σα0, N → Ys−N +1, s be the operator given by x N0,F (f0 ) = c f0y , y N0,F (f0 ) = p ◦ (K x + f0x ) − p ◦ K x + K y · [q ◦ (K x + f0x ) − q ◦ K x ] + f0y · q ◦ (K x + f0x ) + u ◦ (K + f0 ) − u ◦ K + g ◦ (K + f0 ), and let NL, F : ΣαL, N → Ys−N +1−L, s−L , L ∈ {1, . . . , r}, be the operator given by x NL,F (f0 , . . . , fL ) = c fLy + JL,N x (f0 , . . . , fL−1 ), y NL,F (f0 , . . . , fL ) = p0 ◦ (K x + f0x ) · fLx + (K y + f0y ) · q 0 ◦ (K x + f0x ) fLx + q ◦ (K x + f0x ) · fLy + (Du + Dg) ◦ (K + f0 ) · fL + JL,y N (f0 , . . . , fL−1 ), where JL,N are already introduced in (5.5), (5.6) and (5.7). 24 From the definition of the operators SL,R and NL,F , the recursive expressions of ΛL,R and ΩL,F obtained in (5.6) and (5.7) and the choice of α0 it is clear that the operators SL,R and NL,F are well defined and that SL,R is linear and bounded. Note that with the operators introduced above, equations (5.3) and (5.4) can be written now as SL, R DL ∆ = NL, F (∆, . . . , DL ∆), (∆, . . . , DL ∆) ∈ ΣαL, N , for each L ∈ {0, . . . , r} and α0 as fixed previously and some αi > 0, 1 ≤ i ≤ L. In the following lemmas we prove that each of the operators SL, R has a bounded right inverse and we provide a bound for the norm kSL, R k. We also show that each of the operators NL, F −1 is Lipschitz with respect to the last variable and we provide a uniform bound for the Lipschitz constant for the family NL,F , L ∈ {0, . . . , r}. For the proofs of Lemmas 5.6 and 5.7, see Section 6. Lemma 5.6. Let 0 ≤ L ≤ r. Assume r > k in case 1 and r > 2l − 1 in cases 2 and 3. Then, given 0 < ν < (N − 1)|RN | < µ such that κ = ν/µ satisfies κ > 1/N , there exists ρ > 0 small enough such that, taking (0, ρ) as the domain of the functions of Ys−N +1−L, s−L , the operator SL, R : DΣL−1, N → DΣL−1, N has a bounded right inverse, R : Ys−N +1−L, s−L → DΣL−1, N = Ys−2N +2−L, s−N +1−L , −1 SL, given by ∞ X −1 SL,R η=− η ◦ Rj (DRj )L , η ∈ Ys−N +1−L, s−L , (5.10) j=0 and we have the operator norm bound 1 R) k ≤ ρ + x −1 N −1 N −1 k(SL, ν s−2N +2+L(κN −1) , 1 R) k ≤ ρ + y −1 N −1 N −1 k(SL, ν s−N +1+L(κN −1) . Lemma 5.7. Let 0 ≤ L ≤ r. Assume r > k in case 1 and r > 2l − 1 in cases 2 and 3. There exists a constant, M > 0, for which the family of operators NL, F satisfy, for each L ∈ {0, . . . , r}, Lip NL,x F (f0 , . . . , fL−1 , ·) = c, and Lip NL, y F (f0 , . . . , fL−1 , ·) ≤ k |ak | + M ρ, (case 1), Lip y NL, F (f0 , . . . , fL−1 , ·) ≤ max{((l − 1) |Kly bl | + k |ak |) + M ρ, |bl | + M ρ}, (case 2), Lip NL, y F (f0 , . . . , fL−1 , ·) ≤ max{(l − 1) |Kly bl | + M ρ, |bl | + M ρ}, (case 3), where (0, ρ) is the domain of the functions of ΣαL, N . Note that the bound we have found for Lip NL, F (f0 , . . . , fL−1 , ·) does not depend on L, and the obtained bounds for k(S0,x R )−1 k and k(S0,y R )−1 k do not depend on κ. 25 5.3 Main lemmas and the fiber contraction theorem From SL, R and NL, F introduced in Section 5.2, we can define the operators TL,F and TL,×F . Definition 5.8. Given a map F of class C r satisfying the hypotheses of Theorem 2.7, let TL, F : ΣαL, N → DΣL−1, N be the operator given by TL, F = SL, −1 R ◦ NL, F , L ∈ {0, . . . , r}, and let TL,F × : ΣαL, N −→ ΣL, N be the operator given by TL,×F = (T0, F , . . . , TL, F ), L ∈ {1, . . . , r}. In the following results we show that, under appropriate conditions, the operators TL, F have some properties strongly related to the hypotheses of the fiber contraction theorem. Lemma 5.9. Let F be a C r map satisfying the hypotheses of Theorem 2.7, αi > 0, 1 ≤ i ≤ r, and α = (α0 , . . . , αL ), 0 ≤ L ≤ r. Then, for every L ∈ {0, . . . , r − 1}, the operator TL, F : ΣαL, N → DΣL−1, N is Lipschitz on ΣαL, N with respect to (f0 , . . . , fL−1 ), with Lipschitz constant independent of fL . (1) (2) (1) Moreover, the operator Tr, F : Σαr, N → DΣr−1, N can be decomposed as Tr, F + Tr, F , where Tr, F is Lipschitz on Σαr, N with respect to (f0 , . . . , fr−1 ), with Lipschitz constant independent of fr and   (2) Tr, F = 0, (Sr,y R )−1 ◦ (Dr g ◦ (K + f0 )(DK + f1 )r ) , which is continuous with respect to (f0 , f1 ). Next we introduce a convenient rescaling. Given γ > 0, let Tγ (x, y) = (x, γ y). (5.11) We define F̃ = Tγ−1 ◦ F ◦ Tγ . If K and R are analytic maps associated to F , then the corresponding analytic maps associated to F̃ will be given by K̃ = Tγ−1 ◦ K and R̃ = R. Concretely, the parameterizations of F̃ and K̃ with respect to the coefficients of F and K will be given by     x + γcy 0 F̃ (x, y) =  + , y γ −1 ak xk + bl y xl−1 + · · · and   t2 + · · · K̃(t) =  , for case 1, y γ −1 Kk+1 tk+1 + · · ·   t + ··· K̃(t) =  , for cases 2 and 3. γ −1 Kly tl + · · · 26 Lemma 5.10. Given a C r map F satisfying the hypotheses of Theorem 2.7, there exist ρ0 > 0 and a linear transformation Tγ as in (5.11) such that if ρ < ρ0 , then the operator TL, F̃ : ΣαL, N → DΣL−1, N associated to F̃ = Tγ−1 ◦F ◦Tγ , for L ∈ {0, . . . , r}, is contractive with respect to the variable fL ∈ DΣαL−1, N . Moreover, for a proper choice of α = (α0 , . . . , αL ), TL, F̃ maps ΣαL, N into DΣαL−1, L N , for each L ∈ {0, . . . r}. For the proofs of Lemmas 5.9 and 5.10, see Section 6. Remark 5.11. The value α0 denoting the radius of the ball Σα0,0N , obtained previously, is forced by the definition of N0, F (and thus, of T0, N ). Indeed, since we will look for the invariant curves of F as parameterizations of Σα0,0N , their image must be contained in the domain where F is C r . This is not the case for the derivatives of the invariant curves, for which we do not need to put a bound on them to have the operators well defined. Also, the definition of TL, F , for L ∈ {1, . . . , r} does not force any restriction to the size of the arguments f1 , . . . , fL since the dependence with respect to these variables is polynomial. The values α1 , . . . , αr obtained in Lemma 5.10 provide then upper bounds for the norms of the derivatives of the invariant curves of F . Finally, for the convenience of the reader, we recall the fiber contraction theorem [24] which will be used in the proof of Theorem 2.7. We use a version of it stated in [13]. Theorem 5.12 (Fiber contraction theorem). Let Σ and DΣ be metric spaces, DΣ complete, and Γ : Σ × DΣ → Σ × DΣ a map of the form Γ(γ, ϕ) = (G(γ), H(γ, ϕ)). Assume that (a) G has an attracting fixed point, γ∞ ∈ Σ, (b) H is contractive with respect to the second variable, ie, for all γ ∈ Σ, Lip H(γ, ·) < 1. Let ϕ∞ ∈ DΣ be the fixed point of H(γ∞ , ·). (c) H is continuous with respect to γ at (γ∞ , ϕ∞ ). Then, (γ∞ , ϕ∞ ) is an attracting fixed point of Γ. 5.4 Proof of Theorem 2.7. We give next the proof of Theorem 2.7, where we use the setting and the results obtained along the previous sections. Proof of Theorem 2.7. Let F be as in the statement and Tγ , γ > 0, be defined by (5.11). It is clear that given maps H and R, the triple (F, H, R) satisfies F ◦ H = H ◦ R if and only if (F̃ , H̃, R̃) satisfies F̃ ◦ H̃ = H̃ ◦ R̃, where F̃ = Tγ−1 ◦ F ◦ Tγ , H̃ = Tγ−1 ◦ H and R̃ = R. Clearly F and F̃ belong to the same case 1, 2 or 3 of the reduced form (2.2). 27 To prove the theorem, we shall look for ρ > 0 and a function H : (0, ρ) → R2 , with H(0) = 0 and H ∈ C r (0, ρ), and a map of the form R(t) = t + RN tN + R2N −1 t2N −1 , with RN < 0, such that F ◦ H = H ◦ R, (5.12) with N = k for case 1 of (2.2) and N = l for cases 2 and 3. We take the value γ > 0 associated with F provided in Lemma 5.10, and we set F̃ = Tγ−1 ◦ F ◦ Tγ . Let F̃ ≤ be the Taylor polynomial of F̃ of degree r at the origin. Then it is a polynomial of the form     x + γ cy 0 F̃ ≤ (x, y) =  + . y γ −1 ak xk + bl y xl−1 + h.o.t. Since we assumed ak > 0 for cases 1 and 2 and bl < 0 for case 3, then by Theorem 2.1, there exists, for each case, an analytic map K̃ and a polynomial R of the form R(t) = t + RN tN + R2N −1 t2N −1 , with RN < 0, satisfying F̃ ≤ ◦ K̃ − K̃ ◦ R = 0. Given such maps K̃ and R, we look for ρ > 0 and a function ∆ : (0, ρ) → R2 , ∆ ∈ C r (0, ρ), such that F̃ ◦ (K̃ + ∆) − (K̃ + ∆) ◦ R = 0. (5.13) To do so, we consider the set n of or equations described in (5.3) and (5.4). We take α = (α0 , . . . , αr ) with α0 = min 12 , d2 , where d is the radius of a centered ball in R2 contained in the domain where F̃ is of class C r , and α1 , . . . , αr given in Lemma 5.10. We also take the value ρ > 0 associated to F̃ provided in Lemma 5.10. Given such values of ρ and α, we take the function spaces ΣαL, N , for L ∈ {0, . . . , r}, with domain (0, ρ) ⊂ R. With the operators introduced in Definition 5.8, equation (5.3) can be written as f0 = T0, F̃ (f0 ), f0 ∈ Σα0, N , (5.14) and each of the equations (5.4) can be written as fL = TL, F̃ (f0 , . . . , fL ), (f0 , . . . , fL ) ∈ ΣαL, N , for L ∈ {1, . . . , L}, or equivalently, all of them together as a unique equation, (f0 , . . . , fr ) = Tr,×F̃ (f0 , . . . , fr ), (f0 , . . . , fr ) ∈ Σαr, N . (5.15) By Lemma 5.10 and the Banach fixed point theorem, T0, F̃ has an unique attracting fixed point, f0∞ ∈ Σα0,N , which is a solution of equation (5.14) and which ensures that there exists a continuous solution, ∆∞ , of (5.13). We will see next that in fact the solution f0∞ of (5.14) is a function of class C r . We will proceed by induction. First we prove that f0∞ is C 1 . 28 Let us pick a C 1 function f00 ∈ Σα0,0N such that f10 := Df00 belongs to DΣα0,1N . For simplicity we take f00 = 0. Then we take the sequence (f0j , f1j ) = (T1,×F̃ )j (f00 , f10 ). From the definition of the operator T1, F̃ , we have D(T0, F̃ (f00 )) = T1, F̃ (f00 , f10 ). (5.16) Applying (5.16) inductively we have that f1j = Df0j , for all j. Also, since f00 is C 1 and f10 = Df00 , all the iterates f0j = (T0, F̃ )j (f00 ) are C 1 , and as we have said the sequence converges in Σα0,0N to f0∞ . Again, by Lemma 5.10, the operator T1, F̃ : Σα0,N × DΣα0,N 1 → DΣα0,1N is contractive with respect to the variable f1 ∈ DΣα0, N . Thus, T1, F̃ (f0∞ , ·) has a unique attracting fixed point, f1∞ ∈ DΣα0,N . Moreover, by Lemma 5.9, T1, F̃ is continuous with respect to f0 at any point (f0 , f1 ) ∈ Σα1, N . Hence, by the fiber contraction theorem, (f0∞ , f1∞ ) ∈ Σα1, N is an attracting fixed point of T1,×F̃ , which means that the sequence f1i = Df0j converges in DΣ0, N . That is, f1i converges uniformly in C 0 (0, ρ) and therefore we have f1∞ = Df0∞ and thus, f0∞ ∈ C 1 (0, ρ). Now, for every L ∈ {2, . . . , r}, we assume that there exists a unique attracting fixed point of TL−1, × F̃ , given by (f0∞ , . . . , fL−1 ∞ ) ∈ ΣαL−1, N , such that f0∞ ∈ C L−1 (0, ρ) and f1∞ = Df0∞ , . . . , fL−1 ∞ = DL−1 f0∞ . We will see next that in fact f0∞ is of class C L . Let us pick again the function f00 = 0 ∈ C L (0, ρ), and let us take also f10 := Df00 , . . . , fL0 := DL f00 . Then we have (f00 , . . . fL−1 0 ) ∈ ΣαL−1, N and fL0 ∈ DΣαL−1, L N. From the definition of the operator TL, F̃ , we have D(TL−1, F̃ (f00 , . . . , fL−1 0 )) = TL, F̃ (f00 , . . . , fL0 ). (5.17) Then let (f0j , . . . , fLj ) = (TL,×F̃ )j (f00 , . . . , fL0 ). Applying (5.17) inductively we have f1j = Df0j , . . . , fLj = DL f0j , for all j, and then the iterates (f0j , . . . , fL−1 j ) = (TL−1, × N ) (f0 , . . . , fL−1 ) j 0 0 are such that fm ∈ C j L−m , for m ∈ {0, . . . , L− 1}. By the induction hypothesis, the sequence (f0 , . . . , fL−1 ) converges in ΣL−1, N to the solution (f0∞ , . . . , fL−1 j j ∞ ) and f1∞ = Df0∞ , . . . , fL−1 ∞ = DL−1 f0∞ . Also, applying Lemmas 5.9 and 5.10 and the fiber contraction theorem, the sequence fLj = DL f0j converges in DΣL−1, N . That is, fLj converges uniformly in C 0 (0, ρ) and therefore we have fL∞ = DL f0∞ and thus, f0∞ ∈ C L (0, ρ). In conclusion f0∞ ∈ C r (0, ρ). Finally, the C r map H̃ = K̃ + ∆ with ∆ = f0∞ parameterizes the stable manifold of F̃ and therefore it is C r . When F is C ∞ , to see that the stable manifold is C ∞ we take r1 satisfying the hypotheses of the theorem and r2 > r1 . The previous proof provides H1 = Kr1 + ∆1 and H2 = Kr2 + ∆2 29 defined in (0, ρ1 ) and (0, ρ2 ) and of class C r1 and C r2 respectively that parameterize stable manifolds W1 and W2 . Theorem 4.1 of [12], which is proved by geometric methods, provides the uniqueness of the stable manifold in this setting. If ρ2 < ρ1 , since we deal with stable manifolds we can extend W2 iterating by F −1 to recover W1 . Then W1 is C r2 for all r2 > r1 . 6 Proofs of the technical results We will give detailed proofs for Lemmas 5.6, 5.7, 5.9 and 5.10, which correspond to the differentiable case. Lemmas 4.6, 4.7 and 4.10 are simplified complex versions of Lemmas 5.6, 5.7 and 5.10 respectively. 6.1 Properties of the operators SL, R and NL, F Proof of Lemma 5.6. A simple computation shows that the expression (5.10) of SL, R formally satisfies SL, R ◦ (SL, R )−1 η = η, for η ∈ Ys−N +1−L, s−L . We give the details of the proof for the second component SL, y R : Ys−N +1−L → Ys−N +1−L of the operator SL, R , the details for SL, x R : Y s−2N +2−L → Y s−2N +2−L being completely analog- ous. The results for SL, R follow immediately because the components of the operator are uncoupled. We take κ > 1/N and µ, ν such that 0 < ν < (N − 1)|RN | < µ and ν/µ = κ. By Lemmas 5.2 and 5.3 there exists ρ > 0 such that R maps the interval (0, ρ) into itself and the bounds (5.8) and (5.9) hold. Then, given η ∈ Ys−L |(η ◦ Rj (DRj )L )(t)| ≤ kηks−L |Rj (t)|s−L |DRj (t)|L ts−L 1 ≤ kηks−L s−L κN L (1 + j ν tN −1 ) N −1 (1 + j µ tN −1 ) N −1 1 ≤ M kηks−L s+L(κN −1) , ∀ t ∈ (0, ρ), j N −1 hence, since s ≥ r ≥ N > N − 1, (5.10) converges uniformly on (0, ρ) by the Weierstrass P M -test. Thus, (SL, y R) −1 η=− ∞ j=0 η ◦ R (DR ) is continuous on (0, ρ). j j L Now, we prove that (SL, y R) −1 is a bounded operator from Ys−L to Ys−N +1−L and we obtain a bound for its norm. Again, having chosen κ = ν/µ, from Lemmas 5.2 and 5.3 one has, 1 ∞ X R) ηks−N +1−L ≤ sup |η(Rj (t))(DRj (t))L | y −1 k(SL, t∈(0,ρ) ts−N +1−L j=0 1 ∞ X ts−L 1 ≤ kηks−L sup , ts−N +1−L s−L κN L t∈(0,ρ) j=0 (1 + jνtN −1 ) N −1 (1 + jµtN −1 ) N −1 30 and, bounding the sum by an appropriate integral, we obtain the bound 1 ∞ X ts−L 1 ts−N +1−L s−L κN L j=0 (1 + jνtN −1 ) N −1 (1 + jµtN −1 ) N −1 Z ∞ ! 1 ≤t N −1 1+ dx (1 + xνtN −1 ) N −1 s−L+κN L 0 1 N −1 = tN −1 + . ν s − N + 1 + L(κN − 1) Therefore, we get R) y −1 k(SL, ηks−N +1−L ! 1 N −1 ≤ kηks−L sup tN −1 + , η ∈ Xs−L , t∈(0,ρ) ν s − N + 1 + L(κN − 1) which shows that (SL, y R) −1 : Ys−L → Ys−N +1−L is bounded and 1 N −1 y k(SL,R )−1 k ≤ ρN −1 + . ν s − N + 1 + L(κN − 1) In the same way, (SL, x R) −1 : Ys−N +1−L → Ys−2N +2−L is bounded and 1 N −1 R) k ≤ ρ + x −1 N −1 k(SL, . ν s − 2N + 2 + L(κN − 1) Proof of Lemma 4.6. The operators Sn, y R do not contain the term (DR) so that the proof L is similar to the one of Lemma 5.6 with L = 0. However, the domain of the functions in the spaces Xn is the complex sector S(β, ρ), and therefore in this case we have to apply Lemma 4.2. Notice that in this case we do not need lower bounds for Rj (z). Proof of Lemma 5.7. To distinguish the roles of the variables (f0 , . . . , fL−1 ) and fL we will denote the latter by hL . The statement concerning the component NL,F x is clear by the definition of NL,F . For NL,N y we first deal with the case L = 0. Since g(x, y) = o(k(x, y)kr ) and g ∈ C r we have Di g(x, y) = o(k(x, y)kr−1 ), i = 1, 2. For every h0 , h̃0 ∈ Σα0,N , from the definition of the operator N0,F y , one can write N0,y F (h0 ) − N0,y F (h̃0 ) Z 1 = p0 ◦ (K x + h̃x0 + s(hx0 − h̃x0 )) ds 0 Z 1 + (K y + hy0 ) q 0 ◦ (K x + h̃x0 + s(hx0 − h̃x0 )) ds 0 Z 1  + (D1 u + D1 g) ◦ (K + h̃0 + s(h0 − h̃0 )) ds (hx0 − h̃x0 ) 0  Z 1  + q ◦ (K x + h̃x0 ) + (D2 u + D2 g) ◦ (K + h̃0 + s(h0 − h̃0 )) ds (hy0 − h̃y0 ). 0 31 Let us denote, for s ∈ [0, 1] ξs = ξs (h0 , h̃0 ) = K + h̃0 + s(h0 − h̃0 ), Z 1 Z 1 Z 1 ϕ = ϕ(h0 , h̃0 ) = 0 p ◦ ξsx ds + (K +y hy0 ) 0 q ◦ ξsx ds + (D1 u + D1 g) ◦ ξs ds, 0 0 0 Z 1 ψ = ψ(h0 , h̃0 ) = q ◦ (K x + h̃x0 ) + (D2 u + D2 g) ◦ ξs ds, 0 so that we have y kN0,F (h0 ) − N0,F y (h̃0 )ks ≤ kϕ(h0 , h̃0 )(hx0 − h̃x0 )ks + kψ(h0 , h̃0 )(hy0 − h̃y0 )ks . (6.1) For case 1 we have K ∈ Y2, k+1 and, since s = 2r and r > k, then for every h0 , h̃0 ∈ Σα0, k we have (h0 , h̃0 ) ∈ Y4, k+2 . Thus we can bound the norm 1 kξsx k2 = sup |K x (t) + h̃x0 (t) + s(hx0 (t) − h̃x0 (t))| ≤ 1 + M ρ, t∈(0,ρ) t2 for all s ∈ [0, 1]. Moreover, checking the orders of ϕ and ψ, taking into account the properties of p, q, u and g, we have ϕ ∈ Y2k−2 , ψ ∈ Yk ⊂ Yk−1 , ∀ h0 , h̃0 ∈ Σα0, k . More precisely, we can bound kϕk2k−2 ≤ sup (kp0 ◦ ξsx k2k−2 + k(K y + hy0 ) q 0 ◦ ξsx + D1 g ◦ ξs + D1 u ◦ ξs k2k−2 ) s∈[0,1] 1 (6.2) ≤ sup sup (k |ak ||ξsx (t)|k−1 + M t2k−1 ) s∈[0,1] t∈(0,ρ) t2k−2 ≤ k|ak | + M ρ, kψkk−1 ≤ M ρ, (6.3) for all h0 , h̃0 ∈ Σα0,k0 . Then, from (6.1) we have y kN0,F (h0 ) − N0,F y (h̃0 )ks ≤kϕk2k−2 khx0 − h̃x0 ks−2k+2 + kψkk−1 khy0 − h̃y0 ks−k+1 ≤(k|ak | + M ρ)khx0 − h̃x0 ks−2k+2 + ρ M khy0 − h̃y0 ks−k+1 , which proves that Lip N0,F y ≤ k |ak | + M ρ, for case 1. For cases 2 and 3 the bounds for Lip N0,y F are obtained in an analogous way. In these cases we have K ∈ Y1, l and we obtain ξs ∈ Y2, l+1 . Take h0 , h̃0 ∈ Σα0,l . Since r > 2l − 1, ϕ ∈ Y2l−2 , ψ ∈ Yl−1 , with the following bounds for their norms, kϕk2l−2 ≤ k |ak | + (l − 1)|Kly bl | + M ρ, kψkl−1 ≤ |bl | + M ρ, (6.4) 32 in case 2 and kϕk2l−2 ≤ (l − 1)|Kly bl | + M ρ, kψkl−1 ≤ |bl | + M ρ, (6.5) in case 3. The proof for L ≥ 1 is similar. Given f0 , . . . , fL−1 and hL , h̃L ∈ DΣL−1, N , from the definition of NL, y N , we have F (f0 , . . . , fL−1 , hL ) − NL, F (f0 , . . . , fL−1 , h̃L ) y y NL,  = p0 ◦ (K x + f0x ) + (K y + f0y ) q 0 ◦ (K x + f0x )  + (D1 u + D1 g) ◦ (K + f0 ) (hxL − h̃xL )   + q ◦ (K x + f0x ) + (D2 u + D2 g) ◦ (K + f0 ) (hyL − h̃yL ). Given f0 ∈ Σα0,N , we denote ϕ̃ = ϕ̃(f0 ) = p0 ◦ (K x + f0x ) + (K y + f0y ) q 0 ◦ (K x + f0x ) + (D1 u + D1 g) ◦ (K + f0 ), ψ̃ = ψ̃(f0 ) = q ◦ (K x + f0x ) + (D2 u + D2 g) ◦ (K + f0 ), so that we can write y kNL,F (f0 , . . . , fL−1 , hL ) − NL,F y (f0 , . . . , fL−1 , h̃L )ks ≤ kϕ̃(f0 )(hxL − h̃xL )ks + kψ̃(f0 )(hyL − h̃yL )ks . The orders of ϕ̃ and ψ̃ are the same as the ones of the corresponding ϕ and ψ when L = 0, respectively, for each of the cases 1, 2 and 3. That is, ϕ̃ ∈ Y2k−2 , ψ̃ ∈ Yk ⊂ Yk−1 , for case 1 and ϕ̃ ∈ Y2l−2 , ψ̃ ∈ Yl−1 , for cases 2 and 3. As in the case L = 0, for each f0 ∈ Σα0,N 0 , the order of K + f0 is the same as the one of K. Therefore we get the same bounds for the norms of ϕ̃ and ψ̃, namely those obtained in (6.2) - (6.5), and finally the bounds in the statement. Proof of Lemma 4.7. The proof is completely analogous to the proof of Lemma 5.7 in the case L = 0, the only difference being that here the functions in the spaces Xn are defined in sectors S(β, ρ) instead of the interval (0, ρ). 6.2 Proofs of Lemmas 5.9 and 5.10 Proof of Lemma 5.9. As before, to distinguish the roles of the variables fL and (f0 , . . . , fL−1 ) we will denote the former by hL . Since TL, F = SL, −1 R ◦ NL,F and SL,R is linear and bounded, −1 along the proof we will deal only with NL, F . Given a function hL ∈ DΣαL−1,N L we decompose NL,F (f0 , . . . , fL−1 , hL ) = AhL , F (f0 ) + JL,F (f0 , . . . , fL−1 ), 33 where AhL , F := (AxhL , F , AyhL , F ) : Σα0,N → Ys−N +1−L,s−L is the auxiliary operator AxhL , F (f0 ) = c hyL , AxhL , F (f0 ) = p0 ◦ (K x + f0x ) · hxL + (K y + f0y ) · q 0 ◦ (K x + f0x ) hxL + q ◦ (K x + f0x ) · hyL + (Du + Dg) ◦ (K + f0 ) · hL , and we will work on AhL , F and JL,F separately. Clearly AxhL , F is uniformly Lipschitz on Σα0, N . To deal with AyhL , F , let f0 , f˜0 ∈ Σα0, N . Then AyhL , F (f0 ) − AyhL , F (f˜0 ) = ϕhL (f0 , f˜0 )(f0x − f˜0x ) + ψhL (f0 , f˜0 )(f0y − f˜0y ) + θ(f0 , f˜0 )(f0 − f˜0 ) · hL , with Z 1 ϕhL = ϕhL (f0 , f˜0 ) = hxL p00 ◦ (K x + f˜0x + s(f0x − f˜0x )) ds 0 Z 1 + hyL q 0 ◦ (K x + f˜0x + s(f0x − f˜0x )) ds 0 Z 1 + hxL (K y + f0y ) q 00 ◦ (K x + f˜0x + s(f0x − f˜0x )) ds, 0 ψhL = ψhL (f0 , f˜0 ) = hxL q 0 ◦ (K x + f˜0x ), Z 1 θ = θ(f0 , f˜0 ) = (D2 u + D2 g) ◦ (K + f˜0 + s(f0 − f˜0 )) ds. 0 First we deal with case 1. By similar arguments as in Lemma 5.7 we have ϕhL (f0 , f˜0 ) ∈ Y2r−2−L ⊆ Y2k−2−L , ψhL (f0 , f˜0 ) ∈ Yr−1−L ⊆ Yk−1−L . All the entries of the matrix θ(f0 , f˜0 ) belong to Y0 . Also, it is clear that the quantities kϕhL (f0 , f˜0 )k2k−2−L , kψ (f0 , f˜0 )kk−1−L and the k · kY0 -norm of the entries of θ (f0 , f˜0 ) are uniformly bounded for f0 , f˜0 ∈ Σα0,N , the norm depending on α0 in the form ρm α0 for some m > 0 and depending linearly on αL . Then, since hL is fixed, we get kAyhL , F (f0 ) − AyhL , F (f˜0 )k2r−L ≤ kϕhL (f0 , f˜0 )k2k−2−L kf0x − f˜0x k2r−2k+2 + kψh (f0 , f˜0 )kk−1−L kf0y − f˜0y k2r−k+1 L + M khL kDΣL−1,k kf0 − f˜0 kΣ0,k ≤ M αL kf0 − f˜0 kΣ . 0,k Similarly we also obtain kAyhL , F (f0 ) − AyhL , F (f˜0 )k2r−L ≤ M αL kf0 − f˜0 kΣ0,k for cases 2 and 3, where in these cases we have ϕhL ∈ Yr−L−1 , ψhL ∈ Yr−l−L and the entries of θ belong to Y0 . This proves that AhL , F is uniformly Lipschitz on Σα0, N . Next we deal with JL, F . Recall that we have, for every L ∈ {1, . . . , r}, JL,x F (f0 , . . . , fL−1 ) = ΛxL,R (f0x , . . . , fL−1 x ), y JL,F (f0 , . . . , fL−1 ) = ΛyL, R (f0y , . . . , fL−1 y ) + ΩL, F (f0 , . . . , fL−1 ), 34 where ΛxL,R and ΛyL, R are given recursively in (5.6) and ΩL, F is given recursively in (5.7). From (5.6), Λi1,R = 0 and, for L ≥ 2, one can check by induction that L−1 X ΛiL, R (f1i , . . . , fL−1 i )= PL, j fji ◦ R, i = x, y, (6.6) j=1 where each function PL, j is a polynomial on the variable t. Indeed, P2, 1 (t) = −D2 R(t) ∈ YN −2 . Assuming (6.6) and applying the recurrence (5.6) we have L−1 X L−1 X ΛiL+1, R = 0 PL, j fji ◦R+ i PL, j fj+1 ◦ R DR − L fLi ◦ R (DR)L−1 D2 R j=1 j=1 L−1 X   = PL, 0 1 f1 ◦ R + i 0 PL, j + PL, j−1 DR fj ◦ R i j=2   + PL, L−1 DR − L(DR)L−1 D2 R fLi ◦ R. We also have the recurrences PL+1, 1 (t) = PL, 0 1 (t), PL+1, j (t) = PL, 0 j + PL, j−1 DR, 2 ≤ j ≤ L − 1, PL+1, L (t) = PL, L−1 DR − L(DR)L−1 D2 R, and then we also deduce by induction that PL, j = YN +j−1−L . From this, it is clear that ΛL,R = (ΛxL,R , ΛyL,R ) : ΣαL−1,N → Ys−N +1−L,s−L is linear and bounded, so it is uniformly Lipschitz in ΣαL−1, N . Also, from (5.7), one can see that ΩL, F is a polynomial operator on the variables f1 , . . . , fL−1 having coefficients depending on f0 . When L = 1, Ω1, F (f0 ) − Ω1, F (f˜0 ) Z 1 =DK x (p00 ◦ (K x + f˜0x + s(f0x − f˜0x )) ds (f0x − f˜0x ) 0 Z 1 + DK y (q 0 ◦ (K x + f˜0x + s(f0x − f˜0x )) ds (f0x − f˜0x ) 0 Z 1 + (K + f˜0y ) DK x y q 00 ◦ (K x + f˜0x + s(f0x − f˜0x )) ds (f0x − f˜0x ) 0 + (f0y − f˜0y ) DK x q 0 ◦ (K x + f0x ) Z 1 + DK x (D2 u + D2 g) ◦ (K + f˜0 + s(f0 − f˜0 )) ds (f0 − f˜0 ) 0 and hence there exists M > 0 depending on F and α0 such that kΩ1, F (f0 ) − Ω1, F (f0 )ks−1 ≤ M kf0 − f˜0 kΣ0,N . 35 (1) (2) For L > 1, we decompose ΩL, F = ΩL, F + ΩL, F , where (2) (2) ΩL, F = ΩL, F (f0 , f1 ) = DL g ◦ (K + f0 )(DK + f1 )L , and ΩL, F = ΩL, F − ΩL, F . The difference ΩL, F (f0 , . . . , fL−1 ) − ΩL, F (f˜0 , . . . , f˜L−1 ) is a sum (1) (2) (1) (1) of terms of the form cL (f0 , f˜0 )Π, where Π is a product of factors among fjx,y , f˜jx,y and fjx,y − f˜jx,y and such that cL (f0 , f˜0 )Π ∈ Ys−L . From (5.7) we estimate ΩL, F (f0 , . . . , fL−1 ) − (1) ΩL, F (f˜0 , . . . , f˜L−1 ) iteratively, where a part of it comes from (1) D[ΩL−1, F (f0 , . . . , fL−2 ) − ΩL−1, F (f˜0 , . . . , f˜L−2 )]. (1) (1) When one differenciates formally the terms cL−1 (f0 )Π, the new terms cL−1 (f0 , f˜0 )0 Π and cL−1 (f0 , f˜0 )Π0 appear. The factors of each function cL (f0 , f˜0 ) are derivatives of K i , fji , f˜ji , 01 (Q1 (K i + f˜0i + s(f0i − R f˜0i )) ds (f0x − f˜0x ) and (Q2 (K i + f0i ), where Q1 , Q2 are polynomials (derivatives of p, q or u), and the derivative of Z 1 Dm g ◦ (K + f˜0 + s(f0 − f˜0 )) ds (f0 − f˜0 ), m ≤ L − 1. (6.7) 0 When taking a derivative, each term generates several terms, each one having bigger order, (2) the same order or the same order minus one unit. The term ΩL, F is Lipschitz when L < r. When L = r, it is continuous (in the given topology) since Dr g isuniformly continuous in closed balls. On the other hand, when taking a derivative to Π we obtain terms which have the same factors except one which is transformed to its derivative, that is, fjx,y is transformed to fj+1 x,y or fjx,y − f˜jx,y is transformed to fj+1 x,y − f˜j+1 x,y . In any case the order decreases by one unit so we have that their k · ks−L -norm is bounded by ML k(f0 , . . . , fL−1 ) − (f˜0 , . . . f˜L−1 )kΣL,N , where the constant ML depends on α0 , . . . , αL and F but not on the (fji )0 s. Proof of Lemma 5.10. By its definition, the operator TL,F satisfies Lip TL, F (f0 , . . . , fl−1 , ·) ≤ max{k(SL,R x )−1 k Lip NL, x F (f0 , . . . , fL−1 , ·), (6.8) y k(SL,R )−1 k Lip NL,F y (f0 , . . . , fL−1 , ·)}. From the estimates obtained in Lemmas 5.6 and 5.7 we have that the bounds of Lip NL, F (f0 , . . . , fL−1 , ·) do not depend on L, and taking κ < 1 close to 1 the obtained bounds for R k decrease as L increases, so that it holds −1 kSL, Lip TL, F (f0 , . . . , fL−1 , ·) ≤ Lip T0,F (f0 , . . . , fL−1 , ·), ∀ L ∈ {0, . . . , r}. Actually, this inequality is for the obtained bounds for the Lipschitz constants of the family {TL, F }L . Note also that Lip T0, F (f0 , . . . , fL−1 , ·) does not depend on κ. To prove the first part of the lemma we will find an appropriate map Tγ given in (5.11) (that is, an appropriate value for γ) such that if the coefficients of F satisfy the hypotheses of 36 Theorem 2.7, then the corresponding operator TL, F̃ associated to F̃ = Tγ−1 ◦ F ◦ Tγ satisfies Lip TL, F̃ (f0 , . . . , fL−1 , ·) < 1. We start by considering case 1. From (6.8) and the estimates obtained in Lemmas 5.6 and 5.7, given ν ∈ (0, (k − 1)|Rk |) there is ρ̃0 such that for ρ < ρ̃0 we have the bound n 1 k−1  Lip TL, F̃ (f0 , . . . , fL−1 , ·) ≤ max ρk−1 + γ |c|, ν 2r − 2k + 2  1 k − 1  −1 o ρk−1 + (γ k ak + M ρ) . ν 2r − k + 1 Clearly, the condition n |c| 1 k ak 1 o max γ , γ −1 < 1, (6.9) |Rk | 2r − 2k + 2 |Rk | 2r − k + 1 is sufficient to ensure that there exists 0 < ρ0 < ρ̃0 such that Lip TL,F̃ (f0 , . . . , fl−1 , ·) < 1 for ρ < ρ0 , since keeping κ fixed one can choose a value for ν close enough to (k − 1)|Rk |. q k ak 2r−2k+2 Then, taking γ = c 2r−k+1 , condition (6.9) is given by 2k(k + 1) < 1, (2r − 2k + 2)(2r − k + 1) which holds for any k ≥ 2 and r ≥ 23 k. Hence, if r ≥ 32 k, the operator TL, F̃ associated to F̃ = Tγ−1 ◦ F ◦ Tγ for the chosen value of γ satisfies Lip TL, F̃ (f0 , . . . , fL−1 , ·) < 1, for every L ∈ {0, . . . , r}, provided that ρ < ρ0 . For cases 2 and 3 of the reduced form of F the result follows in a similar way choosing an appropriate value for the parameter γ. For case 2 we have, from (6.8) and the estimates obtained in Lemmas 5.6 and 5.7, that the condition n |c| 1 (l − 1) |Kly bl | + k ak 1 |bl | 1 o max γ , γ −1 , < 1, (6.10) |Rl | r − 2l + 2 |Rl | r − l + 1 |Rl | r − l + 1 is sufficient to ensure that there exists ρ0 > 0 such that Lip TL,F (f0 , . . . , fl−1 , ·) < 1 for ρ < ρ0 . r (l−1) |Kly bl |+k ak r−2l+2 Then, taking γ = c r−l+1 , condition (6.10) is given by n β  c k ak  β o max (l − 1) + 2 β , < 1, (r − 2l + 2)(r − l + 1) bl r−l+1 where β = √2l2|bl | , which is the condition for F assumed for case 2. |bl − bl +4 c ak l| For case 3 we have, again from (6.8) and the estimates obtained in Lemmas 5.6 and 5.7, that the condition n |c| 1 (l − 1) |Kly bl | 1 |bl | 1 o max γ , γ −1 , < 1, (6.11) |Rl | r − 2l + 2 |Rl | r − l + 1 |Rl | r − l + 1 37 is sufficient to ensure that there exists ρ0 > 0 such that Lip TL,F (f0 , . . . , fl−1 , ·) < 1 for ρ < ρ0 . r (l−1)(r−2l+2) Taking γ = |bl | |c| l(r−l+1) , condition (6.11) is given by n l(l − 1) l o max , < 1, (r − 2l + 2)(r − l + 1) r − l + 1 that is, l(l − 1) < 1, (r − 2l + 2)(r − l + 1) which is the condition for F assumed for case 3. Finally we prove that given a map F satisfying the hypotheses of Theorem 2.7 such that the associated operators TL, F satisfy Lip TL, F (f0 , . . . , fL−1 , ·) < 1 for ρ < ρ0 , one can find a new ρ0 , maybe smaller than the previous one, and a choice for the values α1 , . . . , αr such that, if ρ < ρ0 , then TL, F maps ΣαL, N into DΣαL−1, L N , for every L ∈ {0, . . . , r}. For later use, we estimate kTL, F (0, . . . , 0)kDΣL−1, N . From Definition 5.5 of NL, F and the definition of JL, F in (5.5) we have NL, F (0, . . . , 0) = JL, F (0, . . . , 0) = (0, DL (g ◦ K)). Moreover DL (g ◦ K)(t) = o(|t|s−L ). Therefore, for every ε > 0, there is ρ0 > 0 such that if ρ < ρ0 , then kTL, F (0, . . . , 0)kDΣL−1,N ≤ k(SL, y R ) k kNL, F (0, . . . , 0)ks−N +1−L, s−L −1 y |DL (g ◦ K)(t)| (6.12) R ) k sup R ) k ε. y −1 y −1 ≤ k(SL, ≤ k(SL, t∈(0, ρ) ts−L Next we proceed by induction. For L = 0, we have, for all f0 ∈ Σα0,N 0 , kT0, F (f0 )kΣ0, N ≤ kT0, F (f0 ) − T0,F (0)kΣ0, N + kT0, F (0)kΣ0, N ≤ α0 Lip T0, F + kT0, F (0)kΣ0, N . We need to see then that there exists ρ0 > 0 such that kT0, F (f0 )kΣ0, N ≤ α0 provided that ρ < ρ0 . Clearly this holds from the estimate obtained in (6.12) since we have Lip T0, F < 1, and then one can take ρ0 such that α0 Lip T0, F + kT0,F (0)kΣ0, N ≤ α0 for ρ < ρ0 . Hence we have T0, F (Σα0,0N ) ⊆ Σα0,0N . Now, we take ρ1 < ρ0 and we denote by εL the quantity εL = kTL, F (0, . . . , 0)kDΣL−1, N , L ∈ {1, . . . , r}, taking as the domain of the functions of ΣαL, N the interval (0, ρ1 ). Continuing with the induction procedure, for each L ∈ {1, . . . , r}, we decompose kTL,F (f0 , . . . , fL )kDΣL−1,N ≤ kTL, F (f0 , . . . , fL ) − TL,F (f0 , . . . , fL−1 , 0)kDΣL−1,N + kTL,F (f0 , . . . , fL−1 , 0) − TL,F (0, . . . , 0)kDΣL−1,N (6.13) + kTL,F (0, . . . , 0)kDΣL−1,N . 38 Also, from the definitions of TL, F and NL, F we have TL,F (f0 , . . . , fL−1 , 0) = SL,R −1 ◦ NL,F (f0 , . . . , fL−1 , 0) = SL,R −1 ◦ JL,F (f0 , . . . , fL−1 ). Now we have to consider separately the cases L < r and L = r. For L < r we have, from Lemma 5.9, that TL,F (f0 , . . . , fL ) is uniformly Lipschitz with respect to (f0 , . . . , fL−1 ) in ΣαL,N , and in particular, Lip TL,F (·, 0) = Lip (SL,R −1 ◦ JL, F ). Therefore, from (6.13) we have kTL, F (f0 , . . . , fL )kDΣL−1, N ≤ Lip TL, F (f0 , . . . , fL−1 , ·) kfL kDΣL−1, N + Lip (SL, −1 R ◦ JL, F ) k(f0 , . . . , fL−1 )kΣL−1, N + kTL, F (0, . . . , 0)kDΣL−1, N (6.14) ≤ αL Lip TL, F (f0 , . . . , fL−1 , ·) + max {α0 , . . . , αL−1 } Lip (SL, −1 R ◦ JL, F ) + εL . Then we can choose a value for the radius αL of DΣαL−1, L N to ensure that TL, F maps ΣL, N α into DΣαL−1, L N . Since we have Lip TL,F (f0 , . . . , fL−1 , ·) < 1, then taking εL + Lip (SL, −1 R ◦ JL,F ) max {α0 , . . . , αL−1 } αL = , 1 − Lip TL, F (f0 , . . . , fL−1 , ·) we have, applying (6.14), kTL, F (f0 , . . . , fL )kDΣL−1, N ≤ αL , for each (f0 , . . . , fL ) ∈ ΣαL, N , as we wanted to see. For L = r we proceed in an analogous way, except for the fact that we use the decompos- (1) (2) (1) ition Tr, F + Tr, F given in Lemma 5.9. Since Tr, F is Lipschitz with respect to (f0 , . . . , fr ), its contribution is as in the cases L < r. As we also have Tr, F (f0 , . . . , fL−1 , 0) = Sr,−1R ◦ (i) (i) Jr, F (f0 , . . . , fr−1 ) and Sr,−1R is linear, we can denote Tr, F (f0 , . . . , fr−1 , 0) = Sr,−1R ◦Jr,F (f0 , . . . , fr−1 ), (2) for i = 1, 2, with Jr, F (f0 , . . . , fr−1 ) = (0, Dr g ◦ (K + f0 )(DK + f1 )r ). We proceed as in (6.13), but now for the second term of the sum we have, applying Lemma 5.9, kTr, F (f0 , . . . , fr−1 , 0)−Tr, F (0, . . . , 0)kDΣr−1, N (1) ≤ Lip (Sr,−1R ◦ Jr, F ) k(f0 , . . . , fr−1 )kΣr−1, N + k(Sr,y R )−1 kkDr g ◦ (K + f0 )(DK + f1 )r ks−r . To bound the quantity kDr g ◦ (K + f0 )(DK + f1 )r ks−r , note that we have Dr g(x, y) = o(k(x, y)k0 ). For case 1 of the reduced form of F we have (DK + f1 )r ∈ Yr and thus, for every ε > 0 there is ρ0 such that if ρ < ρ0 , then 1 r kDr g ◦ (K + f0 )(DK + f1 )r kr = sup r |D g ◦ (K + f0 )(t)(DK + f1 )r (t)| < ε. t∈(0,ρ) t 39 Similarly, for cases 2 and 3 we have (DK + f1 )r ∈ Y0 and kDr g ◦ (K + f0 )(DK + f1 )r k0 = sup |Dr g ◦ (K + f0 )(t)(DK + f1 )r (t)| < ε. t∈(0,ρ) Then, for the chosen radius ρ1 we denote ε̂ = kDr g ◦ (K + f0 )(DK + f1 )r ks−r and similarly as in (6.14) we have kTr, F (f0 , . . . , fr )kDΣr−1, N (1) ≤ αr Lip Tr, F (f0 , . . . , fr−1 , ·) + max {α0 , . . . , αr−1 } Lip (Sr, R ◦ Jr, F ) + εr + ε̂, and therefore the statement of the lemma follows choosing (1) ε̂ + εr + Lip (Sr,−1R ◦ Jr,F ) max {α0 , . . . , αr−1 } αr = . 1 − Lip Tr, F (f0 , . . . , fr−1 , ·) Proof of Lemma 4.10. The proof is a simplified version of the one of Lemma 5.10, here the functions in the spaces Xn being defined in sectors S(β, ρ) instead of the interval (0, ρ). To prove that Tn, N is contractive for n > n0 we proceed as in Lemma 5.10 for L = 0, but here the index n appears in the denominator of the bound obtained for Lip Tn, N , proving that the operator is contractive for n large enough. The second part of the lemma if proved also as in Lemma 5.10 for L = 0. In this case we have Tn, N (0) = Sn, −1 N En , and thus kTn, N (0)kΣn, N < ε for ρ sufficiently small. 7 Conclusions In this paper, we have considered two-dimensional maps with a parabolic fixed point with non- diagonalizable linear part in both the analytic and differentiable cases. We have considered three different cases depending on the nonlinear terms (the generic maps are contained in case 1). In the analytic case, we have proved the existence of an analytic one-dimensional invariant manifold (away from the fixed point) under suitable conditions on some of the coefficients of the nonlinear terms of the map. The existence of an analytic manifold in such case was already proved in [12] using a variation of McGehee’s method. However, here we have used the parameterization method, which provides approximations of the manifolds up to any order, and we have also presented an a posteriori result. In the C r case, first we have used our results for analytic maps applied to the Taylor poly- nomial of degree r of the map. In this way we have obtained an analytic invariant manifold which is used as an approximation to apply the parameterization method to the original map. Moreover, we have applied the fiber contraction theorem to obtain the differentiability result. Concretely, we have proved that if the regularity of the map is bigger than some (easily computable) value, then there exists an invariant manifold of the same regularity, away from the fixed point. 40 This is in contrast with the results in [29], where in our case 1, it is proved that the manifolds are of class C [(k+1)/2] , where k is as in Section 2.1, even if they use the Poincaré normal form instead of our reduced form. In that case, the obtained regularity also holds at the fixed point. Throughout the paper, the results are stated for stable curves. However, we showed that the same results hold true for unstable curves and that they can be obtained directly from the stated results, without having to invert explicitly the given map. As in the analytic case, for the C r case we provided approximations of the invariant manifolds up to an order that depends on the regularity of the map. 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