The parameterization method for invariant manifolds II: regularity with respect to parameters

THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS II: REGULARITY WITH RESPECT TO PARAMETERS XAVIER CABR´ E, ERNEST FONTICH, AND RAFAEL DE LA LLAVE Abstract. We study the regularity with respect to parameters of the invariant manifolds associated to non-resonant subspaces obtained in the previous article [CFdlL00]. 1. Introduction and statements of results This article is devoted to establish the regularity on parameters version of the results obtained in [CFdlL00] on existence and uniqueness of invariant manifolds associated to non-resonant subspaces. We refer to [CFdlL00] for the motivation, properties, notation, and references concerning such manifolds. In the theory of dynamical systems, maps or equations usually appear depending on one or several parameters. It is then important to know the smooth dependence of dynamical objects with respect to the parameters. Moreover, results on regularity with respect to parameters are often needed when carrying out perturbation theories or transversality arguments. The main results of this paper are Theorems 1.1 and 1.2, which are the parameter versions of Theorems 1.1 and 1.2 of [CFdlL00], respectively. Throughout this paper we represent functions Gdefined in subsets of Banach spaces Λ0×X1, or of Λ0×X, by Gλ(x) = G(λ, x). Therefore, we write G,Gλor Gλ(x) according to the context. In particular, in formulas involving composition in the xvariable, where λacts as a parameter, we write Gλ◦Hλfor G(λ, H(λ, ·)). Theorem 1.1. Let Xbe a real or complex Banach space, Uan open set of X,0∈ U, let Λbe an open subset of a Banach space Λ0, and let F: Λ×U⊂Λ0×X→X, that we will write as Fλ(x). Assume that Fλ(0) = 0 and F∈Cr(Λ ×U, X)for some r∈N∪ {∞, ω}, i.e., Fis jointly Crin its two arguments. Let Aλ=DxFλ(0),Nλ(x) = Fλ(x)−Aλx, and X=X1⊕X2be a direct sum decomposition into closed subspaces independent of λ. Denote by π1, π2the corresponding projections. Assume that, uniformly for all λ∈Λ : 0) Fλis a local diffeomorphism. In particular, Aλis invertible. Date: Nov 27, 2002. 1 2 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE 1) The space X1is invariant under Aλ. That is, AλX1⊂X1. Let A1,λ =π1Aλ|X1,A2,λ =π2Aλ|X2and Bλ=π1Aλ|X2. Hence, we have Aλ=A1,λ Bλ 0A2,λ with respect to the above decomposition. Assume: 2) Spec(A1,λ)⊂ {z∈C| |z|<1}. 3) 0 /∈Spec(A2,λ). Let L≥1be an integer independent of λsuch that Spec(A1,λ)L+1 Spec(A−1 λ)⊂ {z∈C| |z|<1},(1) and assume that: 4) Spec(A1,λ)i∩Spec(A2,λ) = ∅for every integer iwith 2≤i≤L(in case that L≥2). 5) L+ 1 ≤r. Then, we can find a map Rλ:X1→X1which is a polynomial in xof degree not larger than Lwith Rλ(0) = 0 , DxRλ(0) = A1,λ ,(2) and a map Kλ:U1⊂X1→X, where U1is an open neighborhood of 0, such that Fλ◦Kλ=Kλ◦Rλ(3) holds in U1, and Kλ(0) = 0 ,(4) π1DxKλ(0) = Id , π2DxKλ(0) = 0 .(5) Moreover, Kis Cr−L−1and Ris Cr−Ljointly in their two arguments. Note that (2)-(5) guarantee that Kλ(U1) is an invariant manifold under Fλ, tangent to X1at the origin. We also recall that in [CFdlL00] we already proved the existence, for every fixed λ, of the polynomial Rλand of the map Kλ. We also established that Kλis Crin the xvariable. Remark 1. When r=∞(that is, Fis C∞with respect to both variables), Theorem 1.1 gives that the invariant manifold is also C∞with respect to both variables. We will prove this result using the finite differentiability result of Theorem 1.1. The detailed proof is given in next section, following Lemma 2.2. If Fis in the class Cωof analytic maps with respect to both variables, then the invariant manifold is also analytic with respect to both variables. The proof in this case is much simpler than the finite differentiability result of Theorem 1.1. It can be obtained from an application of the implicit function theorem; see Remark 3. THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS II 3 Remark 2. In 1) of the previous theorem, the assumption that the invariant subspace X1of Aλdoes not depend on λis not a serious restriction. If the invariant subspace X1,λ depends smoothly on λ, one can make a change of coordinates in such a way that X1,λ becomes independent of λ, and then one can apply Theorem 1.1 to the transformed family of maps. Note that in case that X1,λ ⊕X2,λ is a spectral decomposition, then the smooth dependence of Xi,λ on λis a consequence of the spectral theorem. As in the corresponding theorems of [CFdlL00], the following result is stronger than Theorem 1.1, in the sense that hypothesis (6) below is weaker than (1). The differences in the conclusions of both results are the differentiability of R and that, in the following theorem, Rλis no longer a polynomial in x. Theorem 1.2. Assume hypotheses 0) - 5) of Theorem 1.1 hold uniformly for all λ∈Λ, except that (1) is replaced by Spec(A1,λ)L+1 Spec(A−1 2,λ)⊂ {z∈C| |z|<1}, L ≥1.(6) Then, we can find maps Kλand Rλwhich are Cr−L−1jointly in their two arguments, satisfy (3), Kλ(0) = 0, π1Kλ= Id, π2DxKλ(0) = 0 , and Rλ(0) = 0, DxRλ(0) = A1,λ. We refer to [CFdlL00] for uniqueness statements and for further properties of the maps Kλand Rλin the two previous theorems. Theorems 1.1 and 1.2 will follow from sharper regularity results, Theorems 2.1 and 3.1 below, that provide optimal regularity in a certain class –given by Definition 1.3– of differentiable maps of two Banach space variables. We will prove that, in such a class, the parameterization Khas the same regularity as the map F(that is, no derivative is lost). Since these classes include Crmaps, we will deduce the two previous theorems. Recall that the starting point of our proofs consists of finding maps Rλand K≤ λwhich are polynomials in xof degree L, and satisfy the functional equation Fλ◦Kλ=Kλ◦Rλup to order Lat the origin. We then write K=K≤ λ+K>=K≤ λ+K> λ and we study the functional equation (3) for K>. Remark 3. If we try to use the implicit function theorem to solve (3) for K>, Fλ◦(K≤ λ+K>) = (K≤ λ+K>)◦Rλ, we are led to consider the operator defined by P(λ, K>) = Fλ◦(K≤ λ+K>)−(K≤ λ+K>)◦Rλ.(7) By [CFdlL00] we already know that under the hypotheses of Theorem 1.1, for each λ0there exists K> λ0such that P(λ0, K> λ0) = 0. 4 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE Here we are interested in the regularity of K> λwith respect to λ. The following technical difficulty appears. The map (λ, K>)7→ K>◦Rλ, where K>=K>(x) is a function of xalone, can not be differentiable from Λ ×Cr(X1) to Cr(X1) for any r∈N. Indeed, the derivative of this map with respect to λ should be K>7→ DK>◦RλDλRλ, which maps Crinto Cr−1, but not into Cr. Therefore, it is not possible to apply the implicit function theorem in a straightforward way. The situation changes if it happens that Rdoes not depend on λ. For instance, this is the case when A1,λ =A1does not depend on λ(that is, only the nonlinear terms depend on λ) and the non-resonance assumptions Spec(A1)i∩Spec(A1) = ∅for every integer iwith 2 ≤i≤L , of Theorem 1.1 in [CFdlL00], are satisfied. In such case, Rcan be taken to be equal to A1(in particular, to be independent of λ), and one can obtain regularity results very quickly using the implicit function theorem. However, the results obtained applying the implicit function theorem to Pin (7) are always one derivative short from optimal. The use of the implicit function theorem is presented in an expository way in one of the sections of [CFdlL02]. If Fis analytic, then the map Pin (7) is differentiable in the appropriate spaces of analytic functions for K>considered in [CFdlL00]. This holds in the general case when Rdepends on λ. From this, one can deduce the analytic dependence on λof the solution K>of P= 0; see [CFdlL02] for more details. The method of proof of Theorem 1.1 follows closely the one of optimal regularity in Theorem 1.1 of [CFdlL00]. That is, we study the functional equation for K>as a fixed point problem, and we show that the corresponding operator Tis a contraction in a suitable class of differentiable functions. In this way, we obtain a map K>which has almost optimal regularity with respect to the variables (λ, x). Then, studying an equation satisfied by the derivative DxK>, we get the optimal regularity for K>stated in Theorem 1.1 (see Section 2). This kind of proof does not work for Theorem 1.2, since the parameter version of the fixed point operator Mcorresponding to Theorem 1.2 is not a contraction in the usual norms of spaces of differentiable functions. The proof of Theorem 1.2 follows a different route. We inductively prove that the derivatives in the parameter and in the variable xexist and are continuous and bounded, provided that derivatives of “lower order” exist and are continuous and bounded. In the beginning of Section 3 we describe the proof and then, in the same section, we present all its details. In both Theorems 1.1 and 1.2, the precise definition of derivatives of lower order is somewhat intricate (they are derivatives in two variables, so the ordering has to involve two indices) and will be motivated by the induction proof and the structure of the operators, which involve composition in the xvariable. That is, we will define a suitable order on the sets of indices of derivatives for functions of two variables, in such a way that we can use the functional equation to express THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS II 5 derivatives of a certain index in terms of derivatives with indices which are smaller in the indicated order. We now present the classes of differentiable functions that we will use. We consider functions f: Λ ×U→Y, where Λ and Uare open sets in two Banach spaces, and Yis another Banach space. We say that fis Cl,m when Di λDj xf(λ, x) exist, are continuous and bounded for 0 ≤i≤l, 0 ≤j≤m. We consider the space of such functions endowed with the topology given by the supremum of all the derivatives above. We say that fis jointly Crwhen it is a Crmapping from Λ ×U→Ywith bounded derivatives up to order r. This is equivalent to the existence, continuity and boundedness of Di λDj xf(λ, x) for 0 ≤i+j≤r. More generally, if Σ ⊂(Z+)2is such that (i, j)∈Σ and ˜ı≤i, ˜≤jimplies (˜ı, ˜)∈Σ, we denote by CΣthe set of functions ffor which Di λDj xfexists, is continuous and bounded for every (i, j)∈Σ. We consider CΣendowed with the norm ||f||CΣ(B)= sup (i,j)∈Σ,(λ,x)∈B |Di λDj xf(λ, x)|, which makes it a Banach space. When the set Bis understood from the context, we will suppress it from the notation. It is clear that if Σ0⊂Σ then CΣ⊂CΣ0. In particular, Cl+r⊂Cl,r, Cr,r ⊂Cr,Cl+1,r ⊂Cl,r and Cl,r+1 ⊂Cl,r. Moreover, except for trivial cases, these inclusions are strict. Now we turn to the definition of derivatives of “lower order”. The operators Tand Minvolve compositions. If fλand gλare families of maps, when taking derivatives of order iof the composition hλ=fλ◦gλ=fλ(λ, gλ(λ, ·)) with respect to parameters, we are forced to take the same number iof derivatives of fλwith respect to the variable x, and not only with respect to the parameter. Quantitative versions of this fact are made precise in Lemmas 2.3 and 3.6 below. The previous consideration leads naturally to the following class of differentiable functions. It will be well adapted to the study of regularity with respect to parameters. Definition 1.3. Given (i, j)∈(Z+)2, we define Σi,j ={(a, b)∈(Z+)2|a+b≤i+jand a≤i}. We say that the derivative Da λDb xis of lower order than the derivative Di λDj xif (a, b)∈Σi,j. It is easy to verify that being of “lower order than” is a partial order relation. Note also that CΣr,0=Cr. 2. Proof of Theorem 1.1 The following is the main result of this section. It is formulated in the classes CΣi,j which generalize the Crclass. 6 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE We assume that j≥L+1 and that Fhas derivatives Da λDb xFfor each (a, b) in the set Σi,j. We establish that the parameterization Kgiven by Theorem 1.1 of [CFdlL00] has derivatives Da λDb xKfor each (a, b) in the same set Σi,j . We point out that here we do not assume Fto be jointly Crin its two arguments. We will see that Theorem 1.1 follows easily from this result. Theorem 2.1. Let Λ0and Xbe Banach spaces, Λan open set of Λ0, and Uan open set of Xwith 0∈U. Let F: Λ ×U−→ Xbe a map such that: 1) For some (i, j)∈(Z+)2, F∈CΣi,j , where Σi,j is the set introduced in Definition 1.3, and CΣi,j denotes the set of functions which have all the derivatives of orders in Σi,j defined, continuous and bounded. 2) Conditions 0)–4) of Theorem 1.1 are satisfied. 3) L+ 1 ≤j. Then, we can find maps K∈CΣi,j and R∈Ci,∞⊂CΣi,j satisfying all the properties stated in Theorem 1.1. To prove the previous result, and also Theorem 1.1, we start by studying the regularity with respect to λof the polynomial maps K≤ λand Rλ. This is a simple task. Indeed, we follow the proof of Lemma 3.1 in [CFdlL00]. We find polynomials K≤=PL m=1 Kmand R=PL m=1 Rm, with Kmand Rmsatisfying (see [CFdlL00] for the notation) LA1,λ − Lm,A1,λ,IdK1 m−Rm=−Γ1 m−BK2 m,(8) LA2,λ − Lm,A1,λ,IdK2 m=−Γ2 m,(9) making the same choice (for all the values of λ) on which terms of equations (8) and (9) are eliminated. In this way we get that, under the assumptions of Theorem 2.1, K≤ λand Rλare polynomials in xof degree L, and of class Ciwith respect to λ. To justify the previous statement, note that the mappings that to A1,λ,A2,λ associate Lm,A1,λ,A2,λ are analytic (in fact polynomial). Therefore, if A1,λ,A2,λ are Ciin λ, then LA1,λ ,LA2,λ ,Lm,A1,λ,A−1 2,λ and Lm,A1,λ,A−1 1,λ are also Cias maps from Λ to L(Mm,Mm). Therefore, if the right hand sides of (8) and (9) are Ciin λ, then K1 m,K2 mand Rmare also Ciin λ. For further reference, we summarize the previous argument in the following: Lemma 2.2. Assume that F∈CΣi,j for some j≥L, and that hypotheses 0), 1), 3) and 4) of Theorem 1.1 are satisfied uniformly in λ. Then, THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS II 7 a) We can find polynomials in x,K≤ λ=PL m=1 Kmand Rλ=PL m=1 Rm, of degree not larger than L, where Kmand Rmare homogeneous polynomials of degree m, satisfying Fλ◦K≤ λ(x) = K≤ λ◦Rλ(x) + o(|x|L), and (2),(4),(5), i.e., Rλ(0) = 0,DxRλ(0) = A1,λ,K≤ λ(0) = 0, and DxK≤ λ(0) = (Id,0). b) If we further assume that Spec(A1,λ)j∩Spec(A1,λ) = ∅for some jwith 2≤j≤L , (10) then we can choose Rj= 0. c) The maps K≤and Rbelong to Ci,∞. That is, K≤ λand Rλare polynomials in xof class Ciwith respect to λ. For every given λ∈Λ, the parameterization K> λgiven by Theorem 1.1 of [CFdlL00] is the fixed point of the operator Tλdefined by TλK> λ≡ S−1 λ˜ Hλ= ∞ X k=0 A−(k+1) λ˜ Hλ◦Rk λ,(11) where, for K> λgiven, ˜ Hλis defined by ˜ Hλ=−Nλ◦(K≤ λ+K> λ)−AλK≤ λ+K≤ λ◦Rλ,(12) and K≤ λand Rλare the polynomials of degree Lgiven by Lemma 2.2. To prove Theorem 2.1, we scale Fwith respect to both variables. More precisely, given λ0∈Λ, we consider the family ˜ Fλ(x) = δ−1Fλ0+δλ(δx),(13) where δ > 0. The same expression (13) holds with Fλreplaced by Nλ, the nonlinear part of the map Fλ. Recall that Nλ(0) = DxNλ(0) = 0 for every λ∈Λ. Hence we also have DλNλ(0) = 0 for every λ∈Λ. In particular, N,DxN and DλNall vanish at the point (λ0,0). Therefore, using Taylor’s theorem in the expression (13) for ˜ N, taking δsmall (and renaming the corresponding ˜ Nby N), we can assume that kNkCΣi,j (Λ×B1),kK≤−Id kCi,j (Λ×B1),and kR−A1,λkCi,j (Λ×B1)(14) are as small as we need. Proof of Theorem 1.1. We first consider the case r∈N. We assume that F is Crjointly in its two arguments, and that r≥L+ 1. Since F∈Cr⊂CΣr−L,L , Lemma 2.2 applied with i=r−Land j=Lgives that R∈Cr−L,∞, and hence Ris Cr−Ljointly in its two variables. Regarding the parameterization K, note that F∈Cr⊂CΣr−L−1,L+1 . Applying Theorem 2.1, we conclude that K∈CΣr−L−1,L+1 ⊂Cr−L−1. The case when r=∞follows from the finite differentiability case. The argument is very similar to the corresponding one in [CFdlL00]. The key point 8 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE is to show that when the function Fis C∞, all the finite differentiable solutions obtained by applying Theorem 2.1 agree. This is not completely obvious since the proof of Theorem 2.1 depends on scaling the initial problem and the scaling depends on the degree of differentiability required – recall (13) and condition (14). So that, even if a direct application of Theorem 2.1 provides uniqueness in a small neighborhood for each finite regularity – they are based on a contraction mapping principle – we will need to discuss the possibility that the neighborhoods shrink to just the origin as the regularity increases. Indeed, we recall that this phenomenon happens for center manifolds. There are several observations that allow us to obtain the desired uniqueness and, therefore, the case r=∞from the finite differentiable cases. First, we note that, once we fix r, all the non-resonance conditions up to order rhold uniformly for λ∈Λ. In particular, once we fix the procedure to choose the K≤and the R, we can assume that the K≤and Rare uniformly regular and that, since they are polynomials of degree Ldetermined by the jets, they are independent of rprovided that r > L. We can also assume that Rλis contraction in U1uniformly in λ∈Λ. Second, we observe that, as in the proof of the corresponding theorem in [CFdlL00], if we fix λ, the functional equation (3), Fλ◦Kλ=Kλ◦Rλ, leads to Kλ=F−j λ◦Kλ◦Rj λfor every j≥1.(15) Therefore, since Rλis a uniform contraction, equation (15) (used with jlarge) allows to recover Kλin U1from Kλrestricted to any smaller neighborhood of the origin in X1. In particular, if we fix λ, two Kλthat solve the equation (3) and agree in a open neighborhood of the origin in X1, will also agree in U1. Now, for λ0∈Λ, we consider the scaled family (13). By choosing δsmall enough – depending on r– we can ensure that the smallness conditions (14) are satisfied. Therefore, we obtain a Kdefined on Bρ(λ0)×Bµ(0). The positive numbers ρand µcould depend on r. We will denote them as ρ(r), µ(r) to emphasize this fact. If we now consider the solution for another r0> r, we know that they have to agree on Bρ(r0)(λ0)×Bµ(r0)(0). Now, as we mentioned above, using equation (15) we obtain that they agree on Bρ(r0)(λ0)×U1. Since λ0is arbitrary, we obtain that any solution that is Crin Λ ×U1must be Cr0as well. This finishes the proof of the r=∞case.  Remark 4. Note that using equation (3) to improve the domain of uniqueness of the parameterization Kis the step that is missing in the case of center manifolds. Notice also that the uniqueness holds only once we have specified the algorithm to obtain the K≤and the R. In this case, the uniqueness and the differentiablity is clear since K≤and Rare polynomials that are obtained from the L-jet of Fby applying a finite number of operations that clearly yield smoothness. THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS II 9 The scheme of the proof of Theorem 2.1 is the same as in the proofs of existence and sharp regularity of Kin [CFdlL00]. The main difference is that here we will work in the following spaces of differentiable functions of two Banach space variables. For s≥l, let kHkΓi,s,l := max nmax (a,b)∈Σi,s kDa λDb xHkC0(Λ×B1), max 0≤a≤isup (λ,x)∈Λ×B1|Da λDl xHλ(x)|/|x|o(16) and Γi,s,l ={H: Λ ×B1⊂Λ0×X1−→ Y|H∈CΣi,s (Λ ×B1), Da λDb xHλ(0) = 0 if 0 ≤a≤iand 0 ≤b≤l, kHkΓi,s,l <∞} .(17) The norm k · kΓi,s,l makes Γi,s,l a Banach space. Note also that the terms sup(λ,x)∈Λ×B1|Da λDl xHλ(x)|/|x|included in the definition of the norm k·kΓi,s,l are relevant only when s=l, in the sense that we could omit them when s > l and still get an equivalent norm (since Da λDl xHλ(0) = 0). Note that if H∈Γi,s,l and (a, b)∈Σi,s, using Taylor’s theorem and the definition of k·kΓi,j,l , we have |Da λDb xHλ(x)| ≤ 1 (l−b+ 1)+!kHkΓi,s,l |x|(l−b+1)+,(λ, x)∈Λ×B1. To describe in some detail the structure of the expression for the derivatives of the composition in terms of the derivatives of the functions, we introduce the following auxiliary sets of indices, closely related to Σi,j: Σ∗ i,0={(a, b)∈(Z+)2|a+b≤i, b ≥1}∪{(i, 0)}, Σ∗ i,j ={(a, b)∈(Z+)2|a+b≤i+j, a ≤i, b ≥1}if j≥1,(18) and ˜ Σi,0={(a, b)∈(Z+)2|a+b≤i}, ˜ Σi,j = Σ∗ i,j if j≥1.(19) 2.1. Solution of the fixed point problem. The following three lemmas will prove the existence of a solution K>with almost optimal regularity, more precisely a solution K>∈CΣi,j−1. Lemma 2.3. With the notation above, assuming that H∈CΣi,j ,R∈Ci,j and Aλ∈Ci,∞, we have that Di λDj xhA−(k+1) λHλ◦Rk λi is a polynomial expression that involves only derivatives Dm λA−(k+1) λwith indices 0≤m≤i, derivatives Da λDb xHλevaluated at Rk λwith (a, b)∈˜ Σi,j, and derivatives Dα λDβ xRk λwith 0≤α≤i,0≤β≤j, except for (α, β) = (0,0). 16 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE The next lemma corresponds to Lemma 2.2. It follows from the arguments explained before the statement of Lemma 2.2. Lemma 3.2. Assume that F∈CΣi,j for some j≥L, and that hypotheses 0), 1), 3) and 4) of Theorem 1.2 are satisfied uniformly in λ. Then, for every λ there exists a unique polynomial w≤ λof degree Lsuch that w≤ λ(x) = N(w≤ λ)(x) + o(|x|L),(37) w≤ λ(0) = 0 , Dxw≤ λ(0) = 0 .(38) Moreover, w≤∈Ci,∞and, with a suitable scaling of Fλ, we can get w≤ λas small as needed. Regarding the operator Nin (37), we recall (see [CFdlL00]) that, in the present setting, the functional equation F◦K=K◦Rhas become w=N(w), where (N(w))λ=A−1 2,λ wλ◦ψwλ−A−1 2,λ N2,λ ◦(Id, wλ) (39) and ψwλ=A1,λ +N1,λ ◦(Id, wλ) + Bλwλ; we will write ψλfor ψwλ, to simplify notation. We also write wλ=w≤ λ+w> λ, where w≤ λis the polynomial in xgiven by Lemma 3.2, and then equation (39) becomes w>=M(w>), where, consistent with our notation of denoting the parameter as a subindex, we will write (M(w>))λ= (N(w≤+w>))λ−w≤ λ=Nλ(w≤ λ+w> λ)−w≤ λ.(40) The starting point of the proof of Theorem 3.1 is the following result from the previous paper [CFdlL00]. Under the hypotheses of Theorem 3.1, we have that for every λ,Fλis a Ci+jfunction of x. Hence, from the results of [CFdlL00], we know that the sequence of iterates w> n=Mn(0) (which starts with w> 0= 0) converges uniformly in B1, together with all its derivatives in xup to order i+j−1, to the solution w> ∞produced in that paper. In particular, since all estimates are uniform in λ, we have that w> nand Dxw> n converge uniformly in Λ ×B1to w> ∞and Dxw> ∞, respectively. The proof of Theorem 3.1 proceeds by expressing, for every function w>, the derivative Di λDj xM(w>) as Di λDj xM(w>) = A(w>)Di λDj xw>+B(w>), where A(w>) is a linear operator, and B(w>) is an expression that involves only derivatives of lower order. We will show that An:= A(w> n) is a contractive operator. From this it will follow that, when Dk λDl xMn(0) converge uniformly on compact sets as n→ ∞ for all (k, l) of lower order than (i, j), then Di λDj xMn(0) also converges uniformly on compact sets. THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS II 17 From the finite increment formula for the derivatives of the form Di−1 λDj xw> n or Di λDj−1 xw> n, we will recover the finite increment formula for their limits as n→ ∞. This will give us that the limit of Di λDj xw> nas ntends to infinity is indeed a true derivative. We will encounter the difficulty that in infinite dimensions, continuity on compact sets does not imply continuity and boundedness on a ball, and we will need to use an extra argument. We will show that the derivative which we have obtained as a limit satisfies a certain functional equation. By studying this equation, it will be possible to show that, when the derivatives of lower order are continuous and bounded, then the limit of Di λDj xMn(0) is also continuous and bounded. Next we start with the precise results that will lead to Theorem 3.1. Through this section, we use the notation D1=Dx1and D2=Dx2for derivatives with respect to the variables in X1and in X2, respectively. Proposition 3.3. Assume that Fand w>belong to CΣi,j . Then, the following formula holds: [Di λDj xM(w>)]λ =A−1 2,λDi λDj xw> λ◦ψλ(Dxψλ)⊗j +A−1 2,λhDxwλ◦ψλ(D2N1,λ ◦(Id, wλ) + Bλ)−D2N2,λ ◦(Id, wλ)iDi λDj xw> λ +Vi,j(w> λ),(41) where Vi,j is a multilinear operator in Da λDb xw>with (a, b)∈Σi,j −{(i, j)}, whose coefficients are derivatives of Nof orders in Σi,j, or derivatives of A1,λ or Bλ with respect to λof orders in {0, . . . , i}. This proposition is an immediate consequence of the following three lemmas, which provide a quite detailed structure of the derivatives of the composition. As in the previous section, here we see that the rules for computing higher order derivatives with respect to parameters of a composition suggest the introduction of special sets of indices. We introduce two more auxiliary sets of indices for i∈N, j ≥1: Σ∗∗ i,0={(a, b, 0) ∈(Z+)3|a+b≤i, b ≥1}∪{(i, 0,0)}, Σ∗∗ i,j ={(a, b, c)∈(Z+)3|a+b+c≤i+j, a ≤i, c ≤j, b ≥1}. In the rest of the section, Cmeans a generic constant which depends only on the map F, and which is independent of the functions wand w>. Hence, the same letter may denote different constants in different places. Lemma 3.4. Under the hypotheses of Proposition 3.3, we have that M(w>)∈ CΣi,j and that Di λDj x[M(w>)] 18 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE is an expression of the form A−1 2,λ X (a,b)∈Σ∗ i,j X i1+···+ib=i−a, j1+···+jb=j CDa λDb xwλ◦ψλDi1 λDj1 xψλ· · · Dib λDjb xψλ +C(DλA−1 2,λ)Di−1 λDj x[wλ◦ψλ] + · · · +C(Di λA−1 2,λ)Dj x[wλ◦ψλ] −A−1 2,λ X (a,b,c)∈Σ∗∗ i,j X i1+···+ib=i−a, j1+···+jb=j−c CDa λDc 1Db 2N2,λ ◦(Id, wλ) ·Di1 λDj1 xwλ· · · Dib λDjb xwλ −C(DλA−1 2,λ)Di−1 λDj x[N2,λ ◦(Id, wλ)] − · · · − C(Di λA−1 2,λ)Dj x[N2,λ ◦(Id, wλ)] −Di λDj xw≤ λ. This lemma follows directly from the next two lemmas. Lemma 3.5. Let w∈CΣi,j and ψ∈Ci,j . Then the derivatives of wλ◦ψλ(x) = w(λ, ψ(λ, x)) have the form Di λDj x[wλ◦ψλ] =X (a,b)∈Σ∗ i,j X i1+···+ib=i−a, j1+···+jb=j CDa λDb xwλ◦ψλ·Di1 λDj1 xψλ· · · Dib λDjb xψλ, where Cis a combinatorial coefficient which depends on a, b,i1, . . . , ib,j1, . . . , jb, and Σ∗ i,j is defined by (18). Proof. The proof of this lemma is contained in the proof of Lemma 2.3 when one identifies wand ψwith Hand Rkof that lemma, respectively.  Lemma 3.6. Let N∈CΣi0,j0and w∈CΣi0,j0. Then, we have that Nλ◦ (Id, wλ)∈CΣi0,j0, and for (i, j)∈Σi0,j0 Di λDj x[Nλ◦(Id, wλ)] (42) =X (a,b,c)∈Σ∗∗ i,j X i1,j1,...,ib,jb CDa λDc 1Db 2Nλ◦(Id, wλ)Di1 λDj1 xwλ· · · Dib λDjb xwλ, where i1+· · · +ib=i−a,j1+· · · +jb=j−c, and Cis a coefficient depending on the indices. The only term in the sum (42) that contains Di λDj xwλis D2Nλ◦ (Id, wλ)Di λDj xwλ. Proof. We begin by computing the ith derivative of Nλ◦(Id, wλ) with respect to λ, which we denote by Ti=Di λ[Nλ◦(Id, wλ)]. We have Di λhNλ◦(Id, wλ)i=X (a,b)∈Σ∗ i,0XCDa λDb 2Nλ◦(Id, wλ)Di1 λwλ· · · Dib λwλ, THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS II 19 where the second sum is taken for i1+· · ·+ib=i−a, and Cis a coefficient which depends on a, b, i1, . . . , ib. This follows from Lemma 3.5 identifying Nλ(x, .) with wλ, and wλwith ψλ. Note that Dj xTi=Di λDj x[Nλ◦(Id, wλ)]. The rest of the proof is by induction in j. When j= 0, formula (42) holds. The induction step incrementing j corresponds to taking one more derivative with respect to x. Assuming that (42) is true for j, Dj+1 xTi=X (a,b,c)∈Σ∗∗ i,j X i1,j1,...,ib,jb ChDa λDc+1 1Db 2Nλ◦(Id, wλ) +Da λDc 1Db+1 2Nλ◦(Id, wλ)Dxwλi·[Di1 λDj1 xwλ· · · Dib λDjb xwλ] +X (a,b,c)∈Σ∗∗ i,j X i1,j1,...,ib,jb CDa λDc 1Db 2Nλ◦(Id, wλ)· Dx[Di1 λDj1 xwλ· · · Dib λDjb xwλ]. From the previous formula we see that taking one more derivative has the effect that each term labeled with indices (a, b, c) generates three new terms labeled with indices (a, b, c), (a, b+1, c) and (a, b, c+1), except for the term with b= 0 which only appears when j= 0 and it must have indices (i, 0,0). Such term only generates two terms with indices (i+ 1,0,0) and (i, 1,0). For j > 0 we decompose Σ∗∗ i,j =˜ Σ0 i,j ∪ · · · ∪ ˜ Σm i,j ∪ · · · ∪ ˜ Σi i,j, where ˜ Σm i,j ={(m, b, c)|b+c≤i+j−m, c ≤j, b ≥1}. In this way, it is easy to see that the symbolic process (a, b, c)7→ (a, b, c)+(a, b + 1, c)+(a, b, c + 1) is exhaustive from ˜ Σm i,j to ˜ Σm i,j+1. For the case j= 0 the same argument works, but one has to consider the term (i, 0,0) separately. We note that the highest derivative of w, namely Di λDj xw, appears multiplied by D2N(this means a=c= 0, b= 1).  As we already mentioned, Lemma 3.4 follows directly from Lemmas 3.5 and 3.6, while Proposition 3.3 is an immediate consequence of Lemmas 3.4, 3.5 and 3.6. Now we show inductively the existence of derivatives assuming that all the lower order ones are defined. Given a subset S⊂Λ×B1, we introduce the space Γ∗ i,j(S) ={v∈C0(S, Li(Λ0;Lj(X1;X2))) |sup (λ,x)∈S, x6=0 |vλ(x)|/|x|(L−j+1)+<∞} 20 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE with kvkΓ∗ i,j (S)= sup(λ,x)∈S, x6=0 |vλ(x)|/|x|(L−j+1)+. Given a set of indices Σ, we also define the space Γ(Σ, S) ={w∈CΣ(S)|sup (λ,x)∈S, x6=0 |Da λDb xwλ(x)|/|x|(L−b+1)+<∞,∀(a, b)∈Σ} with kwkΓ(Σ,S)= max(a,b)∈Σsup(λ,x)∈S, x6=0 |Da λDb xwλ(x)|/|x|(L−b+1)+. Proposition 3.7. Assume that F∈CΣi0,j0(Λ ×B1), with j0≥L+ 1, and that Nhas small enough CΣi0,j0norm. Let (i, j)∈Σi0,j0, and consider Σ0 i,j := Σi,j − {(i, j)}. Let w> nbe the sequence of families of functions defined by w> n=Mn(0).(43) Assume that: 1) w> ∞∈CΣ0 i,j (Λ ×B1). 2) For every compact set G⊂Λ×B1,w> nconverges to w> ∞in Γ(Σ0 i,j, G). Then, a)w> ∞∈CΣi,j (Λ ×B1). b)For every compact set G⊂Λ×B1,Di λDj xw> nconverges to Di λDj λw> ∞in Γ∗ i,j(G), and therefore w> nconverges to w> ∞in Γ(Σi,j, G). At the end of this section we will see that Theorem 3.1 follows easily from the previous proposition. To prove Proposition 3.7, we first introduce some notation and we establish some preliminary lemmas. First we remark that, since j0≥1, if F∈CΣi0,j0then N=F−DxF(0) ∈ CΣi0,j0. We write (43) in inductive form, w> n+1 =M(w> n), n ∈Z+, w> 0= 0 .(44) We introduce the notation wn=w≤+w> n and ψλ,n(x) = A1,λx+N1,λ(x, wλ,n(x)) + Bλwλ,n(x), n ∈Z+∪ {∞}.(45) Differentiating in (44), we have Di λDj xw> λ,n+1 =Di λDj x[M(w> λ,n)] .(46) By Proposition 3.3 and Lemmas 3.4 to 3.6, we can expand the right hand side of (46) as a sum of derivatives of different orders. We define a linear operator Anin such a way that AnDi λDj xw> ncontains the terms in such expansion with Di λDj xw> n. We take Bnas the sum of all the remaining terms of the expansion. We allow nto be ∞, meaning the limit. THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS II 21 More explicitly, for n∈Z+∪ {∞}, we define [Anv]λ=A−1 2,λvλ◦ψλ,n(Dxψλ,n)⊗j(47) +A−1 2,λhDxwλ,n ◦ψλ,n (D2N1,λ ◦(Id, wλ,n) + Bλ)−D2N2,λ ◦(Id, wλ,n)ivλ and Bn=Di λDj x[M(w> n)] − AnDi λDj xw> n. We have that Bnis the multilinear expression in the derivatives Da λDb xw> nfor (a, b)∈Σ0 i,j, (a, b)6= (0,0), which contains all terms in the expansion of the right hand side of (42) except for the two only ones which contain Di λDj xw> n. Note that Anand Bnactually depend on iand j, but for typographical reasons we suppress these indices from the notation. To study the convergence of the sequence Di λDj xw> non compact sets, it will be useful to associate to each compact set G⊂Λ×B1, the larger compact set G∗=[ n∈N∪{∞}, l∈Z+ {(λ, ψl λ,n(x)) |(λ, x)∈G} ⊂ Λ×B1. We have that G∗is compact since ψλ,n →ψλ,∞as n→ ∞, and since ψλ,n are uniform contractions. The set G∗has the important property of being invariant by (πλ, ψλ,n) for every n∈N∪ {∞}, where πλ(λ, x) = λ. Lemma 3.8. Assume the hypotheses of Proposition 3.7. Let n∈Z+∪ {∞}. Then, a)Given a compact set G⊂Λ×B1,An: Γ∗ i,j(G∗)−→ Γ∗ i,j(G∗)is a well defined linear bounded operator and, denoting by Y=L(Γ∗ i,j(G∗),Γ∗ i,j(G∗)), its norm satisfies kAnkY≤γ < 1, for some constant γindependent of n and of the compact set G. b)Bn∈Γ∗ i,j(G∗)and kBnkΓ∗ i,j (G∗)≤Mfor some constant M > 0independent of n(but possibly dependent on G). c)For n∈Nand (λ, x)∈Λ×B1,|Di λDj xw> λ,n(x)| ≤ Mi,j|x|(L−j+1)+for some constant Mi,j independent of n. d)Bnconverges uniformly on compact sets to B∞. Proof. As we mentioned in the beginning of this section, we already know that w> nconverges to w> ∞in Γ(Σ0,1,Λ×B1). Using also hypotheses 1) and 2) of Proposition 3.7 together with expressions (45) and (42), we deduce that, for n∈Z+∪ {∞} and (λ, x)∈Λ×B1, |ψλ,n(x)| ≤ (kA1,λk+ε)|x|,(48) |Dxψλ,n(x)| ≤ kA1,λk+ε, (49) with εas small as needed. We also get that, for every compact subset of Λ ×B1, |Da λψλ,n(x)| ≤ C|x|, a ≤iif j≥1,or a≤i−1 if j= 0,(50) where Cis a constant independent of n(but depending on the compact set). 22 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE a) Anis a linear operator. We estimate its norm: kAnvkΓ∗ i,j (G∗) = sup (λ,x)∈G∗ 1 |x|(L−j+1)+kA−1 2,λkn|vλ(ψλ,n(x))| |Dxψλ,n(x)|j +h|Dxwλ,n(ψλ,n(x))| |D2N1,λ(x, wλ,n(x)) + Bλ| +|D2N2,λ(x, wλ,n(x))|i|vλ(x)|o ≤sup (λ,x)∈G∗ kA−1 2,λknkvkΓ∗ i,j (G∗) |ψλ,n(x)|(L−j+1)+ |x|(L−j+1)+|Dxψλ,n(x)|j +h|Dxwλ,n(ψλ,n(x))| |D2N1,λ(x, wλ,n(x)) + Bλ| +|D2N2,λ(x, wλ,n(x))|ikvkΓ∗ i,j (G∗)o ≤ kA−1 2,λkhkA1,λk+εL+1 +εikvkΓ∗ i,j (G∗) by (48), (49) and the fact that kNkC1+kBλkis as small as we want. b) By the previous lemmas, we know that if F∈CΣi0,j0then N∈CΣi0,j0; if w∈CΣi,j then Nλ◦(Id, wλ)∈CΣi,j ; and if w>∈CΣi,j then M(w>)∈CΣi,j . From the way that Nwas constructed, we have [D` xM(0)]λ(0) = D` x[(N(w≤)λ−w≤ λ](0) = 0,0≤`≤L . Then [Da λDb xM(0)]λ(0) = 0,0≤a≤i, 0≤b≤L , and therefore M(0) ∈Γ(Σi,j,Λ×B1).(51) To prove that |Bn(λ, x)| ≤ M|x|(L−j+1)+for (λ, x)∈G∗, we write Bnas: Di λDj xM(0) + Di λDj xhM(w> n)− M(0)i− AnDi λDj xw> n. Next, we study the terms in the formula for Di λDj xhM(w> n)− M(0)i given by Lemma 3.4, and we see that all of them contain the factor |x|(L−j+1)+ and are bounded uniformly in n. Since M(0) = M(w> 0), we study the more general expression Di λDj xhM(w> n)− M(w> m)i− AnDi λDj xw> n+AmDi λDj xw> m,(52) to be able to use the conclusions in the case m=∞. THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS II 23 Every term that appears in the decomposition of (52) given by Lemma 3.4 is the product of a bounded quantity times a factor of one of the following forms: T1,λ =hDa λDc 1Db 2N2,λ ◦(Id, wλ,n) −Da λDc 1Db 2N2,λ ◦(Id, wλ,m)iDi1 λDj1 xwλ,m · · · Dib λDjb xwλ,m (53) with (a, b, c)∈Σ∗∗ i,j and i1+· · ·+ib=i−k−a, 0 ≤k≤i−a,j1+· · ·+jb=j−c; T2,λ =Da λDc 1Db 2N2,λ ◦(Id, wλ,n)hDi1 λDj1 xwλ,n · · · Dib λDjb xwλ,n −Di1 λDj1 xwλ,m · · · Dib λDjb xwλ,mi(54) with (a, b, c)∈Σ∗∗ i,j and i1+· · ·+ib=i−k−a, 0 ≤k≤i−a,j1+· · ·+jb=j−c; T3,λ(x) = D˜ı λD˜ xψλ,n(x)−D˜ı λD˜ xψλ,m(x) (55) with ˜ı≤i, ˜≤j, (˜ı, ˜)6= (i, j); T4,λ =hDa λDb xwλ,n ◦ψλ,n −Da λDb xwλ,m ◦ψλ,miDi1 λDj1 xψλ,m · · · Dib λDjb xψλ,m (56) with (a, b)∈Σ0 i,j,i1+· · · +ib=i−k−a, 0 ≤k≤i−a, and j1+· · · +jb=j; T5,λ =Da λDb xw> λ,n ◦ψλ,n hDi1 λDj1 xψλ,n · · · Dib λDjb xψλ,n −Di1 λDj1 xψλ,m · · · Dib λDjb xψλ,mi(57) with (a, b)∈Σ0 i,j,i1+· · · +ib=i−k−a, 0 ≤k≤i−a, and j1+· · · +jb=j. Before finishing the proof of Lemma 3.8, we state and prove the following lemma, which estimates the terms Tlabove. Lemma 3.9. Assume the hypotheses of Lemma 3.8. Then, for every compact subset Gof Λ×B1there exists a constant Cindependent on nsuch that, for x∈Gwe have: |T1,λ(x)| ≤ C|x|L+1, |T2,λ(x)| ≤ C|x|(L−j+1)+, |T3,λ(x)| ≤ C|x|(L−˜+1)+, |T4,λ(x)| ≤ C|x|(L−j+1)+in case that m= 0,and |T5,λ(x)| ≤ C|x|(L−j+1)+. Proof. For (53) we distinguish two cases. When a+b+c < i0+j0, then we use the mean value theorem and we bound T1by kDa λDc 1Db+1 2N2,λkC0|wλ,n(x)−wλ,m(x)| |Di1 λDj1 xwλ,m(x)· · · Dib λDjb xwλ,m(x)|, 24 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE and we recall that |wλ,n(x)−wλ,m(x)|=|w> λ,n(x)−w> λ,m(x)| ≤ kw> n−w> mkΓ(Σ0 i,j ,G∗)|x|L+1. In the second case when a+b+c=i0+j0, since a+b+c≤i+jthen j≥i0+j0−i≥j0≥L+1 and thus (L−j+1)+= 0. Hence it is enough to see that T1is bounded, which follows immediately from hypothesis 2) of Proposition 3.7. To bound T2, we decompose the differences of (54) in telescopic form. Each difference will have a factor of the form |Dil λDjl xwλ,n(x)−Dil λDjl xwλ,m(x)| ≤ kwn−wmkΓ(Σ0 i,j ,G∗)|x|(L−jl+1)+,(58) with jl≤j. The factor (58) is bounded by C|x|(L−j+1)+because jl≤j. For T3, we write D˜ı λD˜ xψλ,n −D˜ı λD˜ xψλ,m =X (a,b,c)∈Σ∗∗ ˜ı,˜X i1+···+ib=˜ı−a, j1+···+jb=˜−c ChDa λDc 1Db 2N1,λ ◦(Id, wλ,n)Di1 λDj1 xwλ,n · · · Dib λDjb xwλ,n −Da λDc 1Db 2N1,λ ◦(Id, wλ,m)Di1 λDj1 xwλ,m · · · Dib λDjb xwλ,mi + ˜ı X l=0 CDl λBλhD˜ı−l λD˜ xwλ,n −D˜ı−l λD˜ xwλ,mi. We deduce the bound for T3claimed in Lemma 3.9 by applying the bounds for T1and T2with i= ˜ıand j= ˜(with N2replaced by N1). To establish the bound for T4, we take into account that m= 0 and hence w0=w≤ 0. Note that w0is a polynomial and, therefore, it is sufficient to bound the first factor in the definition of T4. For this we add and substract the term Da λDb xwλ,0◦ψλ,n(x), and we use that wλ,n −wλ,0=wλ,n −w≤ λ=w> λ,n. We obtain the bound |Da λDb xw> λ,n ◦ψλ,n(x)|+|Da λDb xwλ,0◦ψλ,n(x)−Da λDb xwλ,0◦ψλ,0(x)| ≤ kDa λDb xw> nkΓ(Σ0 i,j ,G∗)|ψλ,n(x)|(L−b+1)+ +kDa λDb+1 xw0kC0|ψλ,n(x)−ψλ,0(x)| ≤(kA1,λk+ε)(L−b+1)+kDa λDb xw> nkΓ(Σ0 i,j ,G∗)|x|(L−b+1)++C|x|L+1, where we have used the previous bound for T3. Moreover, the second factor of T4,Di1 λDj1 xψm· · · Dib λDjb xψm, is such that j1+· · · +jb=jand hence it must have at least (b−j)+derivatives with index jl= 0. Therefore by (49) this factor is bounded by C|x|(b−j)+. For T5, decomposing the difference Di1 λDj1 xψn· · · Dib λDjb xψn−Di1 λDj1 xψm· · · Dib λDjb xψm THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS II 25 in a telescopic sum, each term will have a factor |Dil λDjl xψλ,n(x)−Dil λDjl xψλ,m(x)| which is bounded by C|x|(L−jl+1)+≤C|x|(L−j+1)+by the bounds for T3. Proof of Lemma 3.8 (continued). Using the estimates of Lemma 3.9 and (51), one easily checks that Bn∈Γ∗ i,j(G∗) and that there exists a constant M independent on nsuch that |Bn(λ, x)| ≤ M|x|(L−j+1)+. The same estimates work in the case n=∞. To establish c) of Lemma 3.8, from the definition of Anand Bnwe write Di λDj xw> n+1 =AnDi λDj xw> n+Bn. Then kDi λDj xw> n+1kΓ∗ i,j (G∗)≤ kAnkYkDi λDj xw> nkΓ∗ i,j (G∗)+kBnkΓ∗ i,j (G∗). Iterating this inequality, using that kAnkand kBnkare uniformly bounded, and kAnk ≤ γ < 1, we conclude that c) holds with Mi,j =M/(1 −γ). Next we prove d) of Lemma 3.8. Given a compact set Gof Λ ×B1, we need to prove that Bn→ B∞converges uniformly on G. For this, we consider the differences (52) with m=∞in the larger compact set G∗. Examining the terms T1to T5in the analogous way as in Lemma 3.9, we see that all terms are bounded by factors which are estimated uniformly in n, times one of the following terms: kw> n−w> mkΓ(Σ0 i,j ,G∗)(59) |Da λDc 1Db 2Nj◦(x, wλ,n(x)) −Da λDc 1Db 2Nj◦(x, wλ,∞(x))|(60) kDa λDb xwλ,∞◦ψλ,n −Da λDb xwλ,∞◦ψλ,∞k(61) and kDa λDb xwλ,n ◦ψλ,n −Da λDb xwλ,∞◦ψλ,nk.(62) The terms (60), (61) and (62) go to zero since Da λDc 1Db 2Njand Da λDb xw∞are continuous, and since the sets {(λ, x, wλ,∞(x)) |(λ, x)∈G∗}and {(λ, ψλ,m(x)) | (λ, x)∈G∗, m ∈Z+∪ {∞}} are compact.  Proof of Proposition 3.7. If we denote vn=Di λDj xw> n, using (46) and the definitions of w> n,Anand Bn, we have v0= 0, vn+1 =Anvn+Bn, n ≥0. Hence we can write vn+1 =Bn+AnBn−1+· · · +AnAn−1· · · A1B0.