Borel summability and Lindstedt series

Transcription

1 Bore summabiity and Lindstedt series O. Costin 1, G. Gaavotti 2, G. Gentie 3 and A. Giuiani 4 1 Department of Mathematics, The Ohio State University, Coumbus, Ohio USA 2 Dipartimento di Fisica, Università di Roma La Sapienza, Roma I Itay 3 Dipartimento di Matematica, Università di Roma Tre, Roma I Itay 4 Department of Physics, Princeton University, Princeton, New Jersey USA Abstract. Resonant motions of integrabe systems subject to perturbations may continue to exist and to cover surfaces with parametric equations admitting a forma power expansion in the strength of the perturbation. Such series may be, sometimes, summed via suitabe sum rues defining C functions of the perturbation strength: here we find sufficient conditions for the Bore summabiity of their sums in the case of two-dimensiona rotation vectors with Diophantine exponent τ = 1 (e.g. with ratio of the two independent frequencies equa to the goden mean). 1. Introduction In the paradigmatic setting of KAM theory, one considers unperturbed motions ϕ ϕ+ω 0 (I)t on the torus T d, d 2, driven by a Hamitonian H = H 0 (I), where I R d are the actions conjugated to ϕ and ω 0 (I) = I H 0 (I). Standard anaytic KAM theorem considers the perturbed Hamitonian H ε = H 0 (I) + εf(ϕ,i), with f anaytic, and a frequency vector ω 0 which is Diophantine with constants C 0 and τ (i.e. ω 0 ν C 0 ν τ ν Z d, ν 0) and which is among the frequencies of the unperturbed system: ω 0 = ω 0 (I 0 ) for some I 0. Suppose H 0 (I) = I 2 /2 and f(ϕ,i) = f(ϕ) for simpicity, that is assume that H 0 is quadratic and the perturbation depends ony on the ange variabes. Then for ε sma enough the unperturbed motion ϕ ϕ + ω 0 t can be anayticay continued into a motion of the perturbed system, in the sense that there is an ε anaytic function h ε : T d T d, reducing to the identity as ε 0, such that ψ+h ε (ψ) soves the Hamiton equations for H ε, i.e. ϕ + ε ϕ f(ϕ) = 0, if ψ is repaced by ψ + ωt, for any choice of the initia data ψ. The function h ε (caed the conjugation) can be constructed as a power series in ε (Lindstedt series) and for ε sma convergence can be proved, expoiting canceations and summation methods typica of quantum fied theory, [GBG]. We can ca this the maxima KAM theorem, as it deas with invariant tori of maxima dimension. The same methods aow us to study existence of perturbed resonant quasi periodic motions in quasi integrabe Hamitonian systems. By resonant here we mean that ω 0 satisfies 1 s d 1 rationa reations, i.e. it can be reduced to a vector (ω,0), ω R d s, with rationay independent components, via a canonica transformation acting as a inear integer coefficients map of the anges. In this representation we sha write ϕ = (α, β) denoting by α the fast variabes rotating with anguar veocity ω and by β the fixed unperturbed anges ( sow variabes ). The study of resonant quasi periodic motions is mathematicay a natura extension of the maxima KAM case and physicay it arises in severa stabiity probems. We mention here ceestia 2/aprie/2009; 18:58 1

2 mechanics, where the phenomenon of resonance ocking between rotation and orbita periods of sateites is a simpe exampe (as in this case the resonant torus is one dimensiona, i.e. it describes a periodic motion). In genera resonant motions arise in presence of sma friction: the most unstabe motions are the maximay quasi periodic ones (on KAM tori). In presence of friction the maximay quasi periodic motions coapse into resonant motions with one frequency rationay reated to the others, then on a onger time scae one more frequency gets ocked to the others and the motion takes pace on an invariant torus with dimension ower than the maxima by 2, and so on. Periodic motions are the east dissipative, and eventuay the motion becomes maximay resonant, i.e. periodic. This is a scenario among others possibe, and its study in particuar cases seems to require a good understanding of the properties of the resonant motions of any dimension. The first mathematica resut is that aso in the resonant case, if ω is a (d s) dimensiona Diophantine vector, a conjugation h ε can be constructed by summations of the Lindstedt series; however the conjugation h ε that one is abe to construct is not anaytic, but, at best, ony C in ε: in genera it is defined ony on a arge measure Cantor set E of ε s, E [ ε 0, ε 0 ] for some positive ε 0, so that C has to be meant in the sense of Withney. It is commony beieved that a conjugation which is anaytic in a domain incuding the origin does not exist in the resonant case. Moreover, contrary to what happens in the maxima case, not a resonant unperturbed motions with a given rotation vector ω appear to survive under perturbation, but ony a discrete number of them. This is not due to technica imitations of the method, and it has the physica meaning that ony points β 0 which are equiibria for the effective potentia (2π) s d dαf(α, β) can remain in average at rest in presence of the perturbation. The foowing natura (informa) question then arises: where do the unperturbed motions corresponding to initia data (α, β), β β 0, disappear when we switch on the perturbation? An intriguing scenario is that the tori that seem to disappear in the construction of h ε actuay condense into a continuum of highy degenerate tori near the ones corresponding to the equiibria β 0. It is therefore interesting to study uniqueness, reguarity and possibe degeneracies of the perturbed tori constructed by the Lindstedt series methods (or by aternative methods, such as cassica Newton s iteration scheme). In the present paper we investigate hyperboic (see beow) perturbed tori with two-dimensiona rotation vectors for a cass of anaytic quasi integrabe Hamitonians. Informay, our main resut is that the hyperboic tori, which survive to the switching of the perturbation and are described by a function h ε that we construct expicity, are independent on the procedure used to construct them iterativey (which is not obvious, due to ack of anayticity). Moreover, if η = ε, the conjugation is Bore summabe in η for ε > 0. The conjugation constructed here is the unique possibe for our probem within the cass of Bore summabe functions. Of course this does not excude existence of other ess reguar quasi periodic motions, not even existence of other quasi periodic soutions which admit the same forma power expansion as h ε. In order to make our resuts more precise, we first introduce the mode and summarize the resuts about existence and properties of ower dimensiona perturbed tori as we need in the foowing. Then we briefy reca the definition and some key properties of Bore summabe functions, and finay we state more technicay our main resut The mode. Consider a Hamitonian system H = 1 2 A B2 + η 2 f(α, β), (1.1) with I = (A,B) R r R s the action coordinates and ϕ = (α, β) T r T s the conjugated ange 2/aprie/2009; 18:58 2

3 coordinates. Let A 0 = ω,b = 0 be an unperturbed resonance with ω ν C 0 ν τ for ν Z r, ν 0 and for some C 0, τ > 0, and et ((A 0,0), (ψ + ω t, β 0 )) be the corresponding unperturbed resonant motions with initia anges α = ψ, β = β 0. The foowing resut hods. Proposition 1. [GG1] [GG2] Let β 0 be a non degenerate maximum of the function f 0 (β) (2π) r dα f(α, β), i.e. β f 0 (β 0 ) = 0 and β 2f 0(β 0 ) < 0, and et ω R r be a Diophantine vector of constants C 0, τ, i.e. ω ν C 0 ν τ for a ν Z r, ν 0. Then for ε > 0, setting η = ε, there exists a function ψ h(ψ, η) = (a(ψ),b(ψ)), vanishing as η 0 and with the foowing properties. (i) The functions t ϕ(t) = (α(t), β(t)) = (ψ + ω t, β 0 ) + h(ψ + ω t, η) satisfy the equation of motion ϕ = ε ϕ f(ϕ) for any choice of ψ T r. (ii) The function h is defined for (ψ, η) T r [0, η 0 ] and can be anayticay continued to an hoomorphic function in the domain D = {η C : Re η 1 > η0 1 }. (iii) (The anaytic continuation of) h is C in η at the origin aong any path contained in D and its Tayor coefficients at the origin satisfy, for suitabe positive constants C and D, the bounds 1 k! k η h(ψ, 0) < DCk k! τ, where τ is the Diophantine exponent of ω. Remarks. (1) The domain D is an open disk in C 2, centered in η 0 /2 and of radius η 0 /2 (hence tangent to the imaginary axis at the origin). (2) A function h was constructed in [GG1] by a perturbative expansion in power series in ε = η 2 (Lindstedt series) and by expoiting mutiscae decomposition, canceations and summations in order to contro convergence of the series. Eventuay h is expressed as a new series which is not a power series in ε = η 2 and which is hoomorphic in D. The procedure eading to the convergent resummed series from the initia forma power series in ε = η 2 reies on a number of arbitrary choices (which wi be made expicit in next section) and a priori is not cear that the resut is actuay independent on such choices. Another (in principe) function h was constructed in [GG2], with a method which appies in more genera cases (see item (4) beow): we sha see that the two functions in fact coincide. (3) The function h(ψ, η) can be regarded as a function of ε = η 2, as in [GG1] and [GG2]. As such the bound in item (iii) woud be modified into 1 k! k εh(ψ, 0) < DC k (2k)! τ, and the anayticity domain in item (ii) woud be as described in [GG1], Fig. 1. We aso note here that the exponent 2τ + 1 in [GG1], Eq. (5.29), was not correct (without consequences as the vaue of the exponent was just quoted and not expoited), as the right one is 3τ: of course for τ = 1 (that is the case we consider in this paper) the two vaues coincide. (4) In [GG2] a simiar statement was proved for β 0 a non degenerate equiibrium point of the function f 0 (β), i.e. β 0 not necessariy a maximum. If β 0 is a maximum the corresponding torus is caed hyperboic, if β 0 is a minimum it is caed eiptic. In the eiptic case for ε rea the domain of definition of h on the rea ine is T r E, where E [0, ε 0 ] is a set with open dense compement in [0, ε 0 ] but with 0 as a density point in the sense of Lebesgue. The reason for stating Proposition 1 as above is that in the present paper we sha restrict our anaysis to the hyperboic case. (5) If τ = 1, i.e. if r = 2 and ω is quadraticay irrationa, than the bound on the coefficients of the power expansion of h in η at the origin is h (k) (ψ) < DC k k!. Hence in this case it is natura to ask for Bore summabiity of h. (6) Even if we consider ony hyperboic tori, in the foowing we sha use the method introduced 2/aprie/2009; 18:58 3

4 in [GG2], because it is more genera and it is that one shoud ook at if one tried to extend the anaysis to the case of eiptic tori. The main difference between the forthcoming anaysis and that of [GG2] is the use of a sharp mutiscae decomposition, instead of a smooth one, as it aows further simpifications in the case of hyperboic tori. We sha come back to this ater Bore transforms. Let F(η) be a function of η which is anaytic in a disk centered at ( 1 2ρ 0, 0) 1 and radius 2ρ 0 (i.e. centered on the positive rea axis and tangent, at the origin, to the imaginary axis), and which vanishes as η 0 as η q for some q > 1. Then one can consider the inverse Lapace transform of the function z F(z 1 ), defined for p rea and positive and ρ > ρ 0 by L 1 F(p) = ρ+i ρ i e z p F ( 1) dz z 2πi. (1.2) If F admits a Tayor series at the origin in the form F(η) k=2 F kη k then the Tayor series for L 1 F at the origin is L 1 p k 1 F(p) F k (k 1)!. (1.3) k=2 If the series in (1.3) is convergent then the sum of the series coincides with L 1 F(p) for p > 0 rea. Of course the series expansion (1.3) provides an expression suitabe for studying anaytic continuation of L 1 F(p) outside the rea positive axis. Note that (1.3) makes sense even in the case the series starts from k = 1. Then given any forma power series F(η) k=1 F kη k we sha define F B (p) = k=1 F k pk 1 (k 1)! as the Bore transform of F(η), whenever the sum defining it is convergent. It is remarkabe that in some cases the map F F B is invertibe. If this is the case one says that the Tayor series of F is Bore summabe and we aso say that the function F is Bore summabe: this is made precise as foows. Definition 1. Let a function η F(η) be such that (i) it is anaytic in a disk centered at ( 1 1 2ρ 0, 0) and radius 2ρ 0, and admits an asymptotic Tayor series at the origin where it vanishes, (ii) its Tayor series at the origin admits a Bore transform F B (p) which is anaytic for p in a neighborhood of the positive rea axis, and on the positive rea axis grows at most exponentiay as p +, (iii) it can be expressed, for η > 0 sma enough, as F(η) = + Then we ca F Bore summabe (at the origin). 0 e p/η F B (p)dp. (1.4) Remarks. (1) If F is Bore summabe, then one says that F is equa to the Bore sum of its own Tayor series. (2) For instance one checks that a function F hoomorphic at the origin and vanishing at the origin is Bore summabe; its Bore transform is entire. (3) The function F(η) = k=1 2 k η 1+kη is not anaytic at the origin but it is Bore summabe. (4) If F, G are Bore summabe then aso FG is Bore summabe and (FG) B (p) = p 0 F B(p )G B (p p )dp (F B G B )(p) on the common anayticity domain of F B and G B, with the integra which 2/aprie/2009; 18:58 4

5 can be computed aong any path from 0 to p in the common anayticity domain. Of course by definition F B G B F B G B, where in the r.h.s. the convoution is aong any path from 0 to p in the common anayticity domain of F B and G B (note however that now the convoution F B G B depends on the path). (5) The Bore transform of η k is p k 1 /(k 1)!. The Bore transform of η/(1 αη) is e αp. If F B (p) = e αp p k1 /k 1! and G B (p) = e αp p k2 /k 2!, then F B G B (p) = e αp p k1+k2+1 /(k 1 + k 2 + 1)! (6) More generay if F i, i = 1, 2, have Bore transforms F i B bounded in a sector around the rea axis and centered at the origin by F i B (p) Ce αi p +βi Im p p ki /k i!, with α i and β i rea, then F 1 B F 2 B (p) C 2 e α p +β Im p p k1+k2+1 /(k 1 + k 2 + 1)! where α = max α i and β = max β i. We sha make use of this bound repeatedy beow Main resuts. We are now ready to state more precisey our main resuts. Proposition 2. Let us consider Hamitonian (1.1) with r = 2 and f(α, β) an anaytic function of its arguments. Let β 0 T s satisfy the assumptions of Proposition 1 and ω R 2 be a Diophantine vector with constants C 0 > 0 and τ = 1, i.e. ω ν C 0 ν 1 for a ν Z 2, ν 0. Then there exists a unique Bore summabe function h(ψ, η) satisfying properties (i) (iii) of Proposition 1. Note that, once existence of a Bore summabe function h(ψ, η) satisfying properties (i) (iii) of Proposition 1 is obtained, the uniqueness in the cass of Bore summabe functions is obvious, by the very definition of Bore summabiity. In fact a such functions have a Bore transform h B (ψ, p) that is p anaytic in an open domain encosing R + (hence aso in a neighborood of the origin), where they coincide (because they a have the same expansion at the origin), then they a coincide everywhere. The proof of Proposition 2 wi proceed by showing Bore summabiity of (one of) the function(s) h(ψ, η) constructed in [GG2]. In particuar a coroary of the proof is that h(ψ, η) constructed in [GG2] is independent of the arbitrary choices mentioned in Remark (2) after Proposition 1. Our proof of Bore summabiity of h does not use Nevaninna s theorem, [Ne] [So]. In fact we faied in checking that h satisfies the hypothesis of the theorem. Our strategy goes as foows. We introduce a sequence of approximants h (N) to h, that is naturay induced by the mutiscae construction of [GG2]. We expicity check that h (1) is Bore summabe and that its Bore transform is a neighborood B of R+ (not shrinking to 0 as N ) and that h (N) B grows very fast at infinity in B (in genera faster than exponentia). However the resuts of Proposition 1 impy that the growth of h (N) B on the is entire. Then we show inductivey that the anayticity domain of h (N) B positive rea ine is uniformy bounded by an exponentia. Then Bore summabiity of h foows by performing the imit N and using uniform bounds that we sha derive on the approximants and on their Bore transforms. In the next section we wi reca the structure and the properties of the resummed series obtained in [GG2], defining the function h(ψ, η) of Proposition 1. In Section 3 we define the sequence h (N) of approximants and we show that the inverse Lapace transform of h (N) is uniformy bounded by an exponentia on the positive rea ine. In Section 4 we prove Bore summabiity of h(ψ, η) in the easier case in which the perturbation f(α, β) in (1.1) is a trigonometric poynomia in α. In Appendix A1 we discuss how to extend the method to cover the genera anaytic case. Finay, in Appendix A2 we show that the same resut appies to the function h constructed in [GG1]: this aows us to identify the functions constructed with the two methods of [GG1] and [GG2], since they are both Bore summabe and admit the same forma expansion at the origin. Note that the conjugation functions constructed in [GG1] and [GG2] admit the same forma expansion at the 2/aprie/2009; 18:58 5

6 origin simpy because they were obtained by two different resummations schemes (both consisting of a sequence of forma agebraic identities) of the same forma Tayor series. 2. Lindstedt series Denote by a(ψ),b(ψ) the α, β components of h, respectivey. In [GG2] an agorithm is described to construct order by order in η 2 the soution to the homoogic equation { (ω ψ ) 2 a(ψ) = η 2 α f(ψ + a(ψ), β 0 + b(ψ)), (ω ψ ) 2 b(ψ) = η 2 β f(ψ + a(ψ), β 0 + b(ψ)). (2.1) The resuting series, caed the Lindstedt series, is widey beieved to be divergent. A summation procedure has been found which coects its terms into famiies unti a convergent series is obtained. The resummed series (no onger a power series) can be described in terms of suitaby decorated tree graphs, i.e. h can be expressed as a sum of vaues of tree graphs: h γ,ν (η) = θ Θ ν,γ Va(θ), (2.2) where h ν is the ν th coefficient in the Fourier series for h, and γ = {1,..., 2 + s} abes the component of the vector h ν (reca that 2+s is the number of degrees of freedom of our Hamitonian, 2 being the number of fast variabes α and s being the number of sow variabes β). Θ ν,γ is the set of decorated trees contributing to h γ,ν (η) and, given θ Θ ν,γ, Va(θ) is its vaue, both sti to be defined. We now describe the rues to construct the tree graphs and to compute their vaue. We sha need the expicit structure in the proof of Bore summabiity in next sections, and this is why we are reviewing it here. Given the rues beow one can formay check that the sum (2.2) is a soution to the Hamiton equations, see [GG2]. A few differences (in fact simpifications) arise here with respect to [GG2], and we provide some detais with the aim of making the discussion sef-consistent. Essentiay, the changes consist of: (i) shifting the order of factors in products appearing in the definition of the vaues Va(θ) to an order that makes it easier to organize the recursive evauation of severa Bore transforms; see remarks foowing (2.9), and (ii) using a sharp mutiscae decomposition; see item (f) beow. Consider a tree graph (or simpy tree) θ with k nodes v 1,...,v k and one root r, which is not 2/aprie/2009; 18:58 6

7 considered a node; the tree ines are oriented towards the root (see Fig.1). r v γ η γ v ν v0 0 ν=ν 0 v 0 η ν v1 v 1 η v 5 v 3 v 6 v 7 v 2 v 4 v 8 v 9 v 10 Figure 1. A tree θ with 12 nodes; one has p v0 = 2, p v1 = 2, p v2 = 3, p v3 = 2, p v4 = 2. The ength of the ines shoud be the same but it is drawn of arbitrary size. The separated ine iustrates the way to think of the abe η = (γ, γ). v 11 The ine entering the root is caed the root ine. We denote by V (θ) and Λ(θ) the set of nodes and the set of ines in θ, respectivey. (a) On each node v a abe ν v Z 2, caed the mode abe, is appended. (b) To each ine a pair of abes η = (γ, γ) is attached. γ and γ are caed the eft or right component abes, respectivey: γ (1,...,2 + s) is associated with the eft endpoint of and γ (1,..., 2+s) with the right endpoint (in the orientation toward the root, see Fig.1). The abe γ associated with the root ine wi be denoted by γ(θ). (c) Each node v wi have p v 0 entering ines 1,..., pv. Hence with the node v we can associate the eft component abes γ v1,..., γ vp v of the entering ines, and an extra abe γ v0 = γ attached to the right endpoint of the ine exiting from v. Thus a tensor γv0γ v1 γ vpv f νv (β 0 ) can be associated with each node v, with γ denoting the derivative with respect to β γ if γ > 2 and mutipication by iν γ if γ 2. (d) A momentum ν is associated with each ine = v v oriented from v to v : this is a vector in Z 2 defined as ν = w v ν w. The root momentum, that is the momentum through the root ine, wi be denoted by ν(θ). (e) A number abe k {1,..., Λ(θ) } is associated with each ine, with Λ(θ) {k } = {1,..., Λ(θ) }. The number abe is used for combinatoria purposes: two trees differing ony because of the number abes are sti considered distinct. (f) Each ine aso carries a scae abe n = 1, 0, 1, 2,...: this is a number which determines the size of the sma divisor ω ν, in terms of an exponentiay decreasing sequence {γ p } p=0 of positive numbers that we sha introduce in a moment. If ν = 0 then n = 1. If ω ν γ 0 then n = 0, and we say that the ine (or ese ω ν ) is on scae 0. If γ p ω ν < γ p 1 for some p then n = p, and we say that the ine (or ese ω ν ) is on scae p. The sequence {γ p } p=0 is such that γ p C 0 [2 p 2, 2 p 1 ) for a p 0 and, furthermore, ω ν not ony stays bounded beow by C 0 ν 1 (because of the Diophantine condition) but it stays aso far from the vaues γ p for ν not too arge, i.e. for ν at most of order 2 p ; cf. [GG] for a proof of the existence of the 2/aprie/2009; 18:58 7

8 sequence (without further assumptions on ω). Precisey, (1) ω ν C 0 ν 1, 0 ν Z 2, (2) min ω ν γp > C0 2 n if n 0, 0 < ν 2 (n 3). 0 p n Note that the definition of scae of a ine depends on the arbitrary choice of the sequence {γ p }: we coud as we used a sequence scaing as γ p γ p, with γ any number > 1 instead of γ = 2; or we coud have used a smooth cutoff function (as in [GG2]) repacing the sharp cutoff function 11(γ p ω ν < γ p 1 ) impied in the definition above. (g) The scae abes aow us to define hierarchicay ordered custers. A custer T of scae n is a maxima connected set of ines on scae n, with n n, containing at east one ine on scae n. The ines which are connected to a ine of T but do not beong to T are caed the externa ines of T: according to their orientations, one of them wi be caed the exiting ine of T, whie a the others wi be the entering ines of T. A the externa ines are on scaes n with n > n. The set of ines of T, caed the interna ines of T, wi be denoted by Λ(T) and the set of nodes of T wi be denoted by V (T). (h) Not a arrangements of the abes are permitted. The aowed trees wi have no nodes with 0 momentum and with ony one entering ine and the exiting ine aso carrying 0 momentum. We aso discard trees which contain custers with ony one entering ine and one exiting ine with equa momentum and with no ine with 0 momentum on the path joining the entering and exiting ines ( sef energy custers or resonances ). (2.3) Remark. One can verify that chains of sef energy custers can actuay appear in the initia forma Lindstedt series. One of the main points in [GG1] and [GG2] is to show that if one modifies the series by descarding a chains of sef energy custers, then the resuting series is convergent (a form of Bryuno s emma that appears in KAM theory). In both [GG1] and [GG2] it is shown that, in order to dea with chains of sef energy diagrams one can iterativey resum them into the propagators (i.e. the factors associated with the tree ines in the vaue of a tree, see beow for a definition), that wi then turn out to be different from those appearing in the naive forma Lindstedt series (which are simpy (ω ν) 2 ). Such resummation is the anaogue of Dyson s equations in quantum fied theory and the iterativey modifed propagator has been, therefore, caed the dressed propagator. Here there is further freedom in the choice of the sef energy custers. The idea is that the sef energy custers must incude the diverging contributions affecting the initia forma power series. But if we change the definition of sef energy custers by adding to the cass a new cass of non-diverging custers, the construction can be shown to go through as we. We find convenient the specific choice above but this is of course not necessary. This is the second arbitrary choice we do in the iterative construction of the resummed series. It can fue doubts about the uniqueness of the resut which can ony be dismissed by further arguments (ike the Bore summabiity that we are proving). The set of a aowed trees with abes γ(θ) = γ and ν(θ) = ν is denoted by Θ νγ (this is the set appearing in (2.2)). The abes described above are used to define the vaue Va(θ) of a (decorated) tree θ Θ νγ : this is a number obtained by mutipying the foowing factors: (1) a factor F v = γv0γ v1 γ vpv f ν v (β 0 ), caed the node factor, per each node v; (2) a factor g [n ] = g [n ] γ,γ (ω ν ; η), caed the propagator, per each ine of scae n, momentum ν and component abes γ, γ, see items (I) (V) beow for a definition. 2/aprie/2009; 18:58 8

9 The vaue is then defined as Va(θ) = 1 ( )( F v g [n ) ], (2.4) Λ(θ)! v V (θ) Λ(θ) where it shoud be noted that a abes γ (of the tensors F v and of the matrices g [n ] ) appear repeated twice because they appear in the propagators as we as in the tensors associated with the nodes, with the exception of the abe γ associated with the eft endpoint of the ine ending in the root (as the root is not a node and therefore there is no tensor associated with it). Adopting the convention of summation over repeated component abes Va(θ) depends on the root abe γ(θ) = γ = 1,...,2 + s so that it defines a vector in C 2+s. The recursive definition of the propagators is such that the series in (2.2) is convergent and gives the ν-th Fourier component of the function h(ψ, η) in Sect. 1. The definition of propagators we adopt here is the same introduced in [GG2]: the definition in [GG1] is sighty different (see Appendix A2), but it has the drawback that it is specific for hyperboic resonances, whie the definition in [GG2] can be (expected to be) extended aso to the theory of eiptic resonances and, therefore, might turn out to be usefu in view of possibe extensions of the main resuts of this work to eiptic resonances. (I) For n = 1 the propagator of the ine is defined as the bock matrix ( ) g [ 1] def 0 0 = 0 ( β 2f 0(β 0 )) 1. (2.5) (II) For n = 0, if the ine carries a momentum ν and if x def = ω ν, the propagator is the matrix ( β 2f 0(β 0 ) (III) For n > 0 the propagator is the matrix with M 0 def = g [0] = g [0] η 2 (x; η) = x 2 + η 2, (2.6) M 0 ). By the assumptions of Proposition 2 one has M 0 0. g [n] = g [n] (x; η) = η 2 x 2 + M [ n] (x; η). (2.7) with M [ n] (x; η) = M [0] (x; η) + M [1] (x; η) M [n] (x; η), where M [0] (x; η) = η 2 M 0, whereas M [j] (x; η), j 1, are matrices, caed sef energy matrices, whose expansion in η starts at order η 4 and are defined as described in the next two items. (IV) Let T be a sef energy custer on scae n (see item (h) above) and et us define the matrix 1 V T (ω ν; η) as ( V T (ω ν; η) = η2 )( ) F v g [n ], (2.8) Λ(T)! v V (T) Λ(T) 1 This is a matrix because the sef energy custer inherits the abes γ, γ attached to the eft of the entering ine and to the right of the exiting ine. 2/aprie/2009; 18:58 9

10 where, necessariy, n n for a Λ(T). The matrix (2.8) wi be caed the sef energy vaue of T. The set of the sef energy custers with vaue proportiona to η 2k, hence with k 1 interna ines with n 0, and with maximum scae abe n wi be denoted Sk,n R. (V) The sef energy matrices M [n] (x; η), n 1, are defined recursivey for x γ n 1 (i.e. for x on scae n) as M [n] (x; η) = V T (x; η), (2.9) k=2 T S R k,n 1 where the sef energy vaues are evauated by means of the propagators on scaes p, with p = 1, 0, 1,..., n 1. Remarks. (1) With respect to [GG2] the second argument of the propagators (and of the sef energy vaues and matrices) has been denoted η instead of ε; we reca that the variabe η 2 appearing here is the same as the variabe ε appearing in [GG1] and [GG2]. We make this choice because it is natura to study Bore summabiity in η and not in ε = η 2. (2) The association of the factors η 2 with the ines themseves rather than with the nodes (as in [GG1] and [GG2]) wi be more convenient when considering the Bore transforms of the invoved quantities. (3) The mutiscae decomposition used in [GG2] may ook quite different from the one we are using here, but this is not reay so. First, even though the decomposition in [GG2] was based on the propagator divisors [n] (x; ε) = min j x 2 λ [n] j (ε), where the sef energies λ [n] j (ε) were defined recursivey in terms of the sef energy matrices, in the case of hyperboic tori one has identicay [n] (x; ε) = x 2 : indeed a sef energies which are non-zero are stricty negative. Then the ony rea difference is that here we are using a sharp decomposition instead of a smooth one, but the atter is not a reevant difference. In fact the choice of the sequence {γ p } p=0 impies that the ines appearing in the groups of graphs that wi be coected together to exhibit the necessary canceations have currents ν such that ν ω stays reativey far from the extremes of the intervas [γ p+1, γ p ] that define the scae abes, and this aows us to use a sharp mutiscae decompositions instead of the smooth one used in [GG2]. In other words this change with respect to [GG2] is done ony to avoid introducing partitions of unity by smooth functions and the reated discussions. Therefore the expression (2.2) makes sense and in fact the function h mentioned in Proposition 1 is exacty the Fourier sum of the r.h.s. of (2.2). In particuar in [GG2] it was proved that the Fourier sum of the r.h.s. of (2.2) satisfies the properties (i) (iii) in Proposition Integra representation of the resummed Lindstedt series Given the definitions of Sect. 2 consider the function h (N) defined in the same way as h but restricting the sum in (2.2) to the trees containing ony ines of scae n N. The functions h (N) have the same convergence and anayticity properties of the functions h and the same bounds on the Tayor coefficients at the origin. Moreover h (N) h: this is N a consequence of the intermediate steps in the proof of the above proposition in [GG2], as the strategy of the proof is to define h (N) making sure that the convergence and anayticity properties are uniform in N. In fact h (N) is even anaytic in η near the origin for η η N (but η N N 0). Therefore the functions h (N) are triviay Bore summabe, and have an entire Bore transform, but the growth at p + of their Bore transforms is N dependent whie, to show Bore summa- 2/aprie/2009; 18:58 10

11 biity of h, uniform estimates are needed. This section is devoted to a first attempt at such bounds which uses minimay the informations on the resummed series that can be gathered from [GG2], i.e. the convergence properties just mentioned. The Bore transform of the functions h (N) (ψ, η) is an entire function that can be written for p rea and positive as (h (N) ) B (ψ, p) = L 1 h (N) (ψ, p) = η 1 N +i η 1 N i e pz h (N) ( ψ, 1 z ) dz 2πi, p R+, (3.1) where η N is the convergence radius of h (N) (ψ, η). The key remark is that we aso know that by property (ii) in Proposition 1 (actuay by the same property for h (N) that foows from the construction in [GG2]) the function h (N) (ψ, 1 z ) is anaytic for z > 2η 1 0, so that the integra in (3.1) can be shifted to a contour on the vertica ine with abscissa ρ > 2η0 1, i.e. with N-independent abscissa. Therefore (h (N) ) B (ψ, p) = ρ+i ρ i e pz h (N)( ψ, 1 z ) dz 2πi, p R+, (3.2) and, for a N, the function h (N) (ψ, 1 1 z ) is uniformy bounded by O( z ) on the integration contour, 2 because, by property (iii) in Proposition 1, h is twice differentiabe at the origin aong any path contained in D (in particuar aong the circuar path Reη 1 = ρ). Hence the atter boundedness property of h (N) (ψ, 1 z ) and (3.2) impy the bound, for p > 0 and for a suitabe constant C, (h (N) ) B (ψ, p) C e ρp, p R +, (3.3) and for a ρ > 2η0 1. The existence of the imit im N h (N) (ψ, 1 z ) = h(ψ, 1 z ) for 1 z sma (by [GG2]) impies existence of the imit F(ψ, p) as N of (h (N) ) B (ψ, p) for p R + and F(ψ, p) satisfies the bound (3.3) on R +. Hence the functions h(ψ, η) can be expressed, for 0 η < η 0, as h(ψ, η) = 0 e p/η ( im N (h(n) ) B (ψ, p) ) dp = 0 e p/η F(ψ, p)dp, (3.4) which provides us with an integra representation of the resummed series and shows that the resummation (2.2) generates a Bore sum of the forma Lindstedt series provided F(ψ, p) can be shown to be anaytic in a neighborhood of the positive axis, as required by the very definition of Bore summabiity, see property (ii) in Definition 1 of Section 1.2. Note that, because of property (iii) with τ = 1 in the statement of Proposition 1, the functions (h (N) ) B (ψ, p) are anaytic in an N uniform neighborhood of the origin, and so is F(ψ, p). We are then eft with showing that F(ψ, p) can be anayticay extended to a neighborhood of the positive axis. In order to prove this we wi show that the approximants (h (N) ) B (ψ, p) are anaytic in an N uniform strip around the positive axis and that there they admit N uniform bounds: by Vitai s convergence theorem we wi then concude that the imit F(ψ, p) of the approximants is anaytic in the same strip and it satisfies the same bounds in its anayticity domain. 4. Bore summabiity By the remark at the end of Sect. 3 Bore summabiity of h wi be estabished once the natura candidate for its Bore transform, namey the function F in (3.4), is shown to be anaytic in a region 2/aprie/2009; 18:58 11

12 containing the positive rea axis. Here we wi prove that the anayticity domain of F(ψ, p) (which is not trivia, since by construction it contains a neighborhood of the origin) can be extended to a strip of finite width around the rea axis. The proof of this caim wi be based on an inductive assumption on (g [n] ) B (x; p) formuated by introducing the matrix g [n] η (x; η) = 2 x 2 +M [ n] (x;η) for a x < γ n 1. Note that if χ n (x) def = 11(γ n x < γ n 1 ) is the indicator of the scae of x, the propagator g [n] (x; η) is given by g [n] (x; η) = χ n (x) g [n] (x; η). Furthermore the matrices g [n] (x; η) satisfy the recursive equations ( g [n] (x; η) ) 1 = ( g [n 1] (x; η) ) 1 + η 2 M [n] (x; η), x < γ n 1. (4.1) We suppose, inductivey, that for κ 0 = M 0 and Im p < σ for a suitabe σ one has ( g [n] ) B (x; p) K 0 p x 2 e(cn+c n x 1/2 ) p +κ 0 Im p x 1, x < γ n 1, n 0, (4.2) where the matrix norm is M = max j i M ij and the growth of the coefficients c n, c n wi be specified beow. Note that ( g [0] ) B (x; p) = 1 sin p M 0 x 2 x 2, (4.3) M0 x 2 so at the first step (4.2) is vaid with K 0 = s (where s is the dimension of the non trivia bock in M 0 ) and c 0 = c 0 = 0. The constant K 0 comes from our choice of the matrix norm M = max j i M ij : with this choice for any d d matrix we have d 1/2 M 2 M d 1/2 M 2 where 2 is the spectra norm, so that in particuar sinm/m, cosm d 1/2. Assuming the inductive assumption (4.2) to be vaid for n N 1, we remark that this impies a bound for on ( g [N] ) B (x; p) via the expansion ( g [N] ) B (x; p) = ( g [N 1] (x; η) m=0 ( ) m ) η 2 M [N] (x; η) g [N 1] (x; η). (4.4) B Taking the Bore transform and performing a convoutions aong a straight ine from 0 to p we get ( g [N] ) B (x; p) ( g [N 1] ) B (x; p) k=0 [ ( η 2 M [N]) ] k B (x; p) ( g [N 1] ) B (x; p) (4.5) m def with f = f... f m times. Now ( g [N 1] ) B (x; p) is bounded using the inductive assumption (4.2), whie ( η 2 M [N]) B (x; p) is estimated via (2.9), that is ( η 2 M [N]) (x; p) B k=2 T S R k,n 1 1 ( Λ(T)! Λ(T) n 0 K 0 p x 2 )( e (cn +c n γ 1/2 n ) p +κ 0 Im p γn 1 where x = ω ν, the is a convoution product and F v = Λ(T) n = 1 )( ) g [ 1] F v v V (T) (4.6) max (F v ) γv0,γ. (4.7) γ v1,γ v2,...,γ v1 γ v2...γ vpv vpv γ v0 2/aprie/2009; 18:58 12

13 Hence using the bound Λ(T) n 0 ( p e (c n +c n γ 1/2 n ) p +κ 0 Im p γ ) 1 n p 2k 3 (2k 3)! e(cn 1+c N 1 γ 1/2 ) p +κ0 Im p γ 1 N 1 N 1, (4.8) and using the estimates in [GG2] to contro the sum over the sef energy custers, the bound becomes ( η 2 M [N]) (x; p) B 2k p 2k 3 1+c N Γ (2k 3)! e(cn 1 γ 1/2 ) p +κ0 Im p γ 1 N 1 N 1 e 2κ2 N D 0 p e dn p (4.9) e κ2n, k=2 where Γ and κ are suitabe constants derived in Appendix A3 of [GG2], D 0 = Γ 4, d N = Γ+c N 1 + c N 1 γ 1/2 N 1 and, using that γ 1 N 1 4C 1 0 2N, we chose Im p so sma that 4κ 0 C0 1 Im p κ. Remark. The step eading from (4.6) and (4.8) to (4.9) is non-trivia and the possibiity of bounding the sma divisors x 2 and the sum over sef energy custers, after defining the propagators as above, is the main technica aspect of the work in [GG2]. We take the existence of Γ from [GG2]. We do not repeat here the anaysis performed in Sects. 5 and 6 (and the corresponding Appendices) of [GG2]: the constant Γ 2 has been caed ε 1 in Theorem 1 of [GG2]. Using (4.2) and (4.6) in (4.5) we can get a bound on ( g [N] ) B (x; p): ( g [N]) ( B (x; p) p K 0 k=0 [( D 0 p e dn p e κ2n) 1+c x 2 e(cn N 1 x 1/2 ) p +κ 0 Im p x 1) ( K 0 p x 2 e(cn 1+c N 1 x 1/2 ) p +κ 0 Im p x 1 )] k, (4.10) and, since p ( p p ) k = p 4k+1 (4k+1)! for k 0, the k-th term in the sum is bounded by (K 0 D 0 e κ2n ) k p p 4k K 0 (4k + 1)! x 2 e(dn+cn 1 x 1/2 ) p +κ 0 Im p x 1. (4.11) x2k Summing (4.11) over k 0 and comparing the resut with the inductive assumption (4.2), we see that we can take c N = d N = Γ + c N 1 + c N 1 γ 1/2 N 1 and c N = c N 1 + K 0D 0 e κ2n. Soving the iterative equations for c N, c N, we see that, for x on scae n, ( g[n] ) B (x; p) can be bounded as ( g [n] ) B (x; p) K 0 p x 2 ec2 n/2 p +κ 1 Im p 2 n, γ n x < γ n 1, n 0, (4.12) for some constants c, κ 1. Pugging this bound into the expansion for h (N) B (ψ, p), denoting by N(θ) the maxima scae in θ and choosing Im p sma enough, we finay get (h (N) ) B (ψ, p) k=1 ν n 0 0 ( Λ(T) n = 1 k=1 n 0 0 θ Θν,γ N(θ)=n 0 g [ 1] 1 ( Λ(θ)! Λ(θ) n 0 )( v V (T) ) F v K ) 0 x 2 p e c2n /2 p +κ 1 Im p 2 n 2k p 2k 1 n 0 Γ /2 (2k 1)! ec2 p +κ 1 Im p 2 n 0e 2κ2n 0 Γ p e c p 2, (4.13) 2/aprie/2009; 18:58 13

14 for some new constants Γ, c and where we chose Im p < σ, σ = κ/κ 1. The concusion of previous discussion is that the functions (h (N) ) B (ψ, p), which we aready knew to be anaytic in a neighborhood of the origin, can be anayticay continued to a strip of width 2σ around the rea positive axis, where they admit the N uniform bounds (4.13). By Vitai s convergence theorem, the imit F(ψ, p) of (h (N) ) B (ψ, p) as N is an anaytic function in the same domain satisfying the same bound F(ψ, p) Γ p e c p 2, for Im p σ. (4.14) As remarked after (3.3), a consequence of the resuts of [GG2] is that F(ψ, p) satifies the bounds (3.3) p R +. Then, by the very definition of Bore summabiity, using the representation (3.4) and the anayticity of F(ψ, p) proved in this section, we find that h(ψ, η) is Bore summabe (in η). 5. Concuding remarks (1) It is interesting to stress that the recursive bounds on the Bore transform of the propagators in (4.5) have been derived without making use of the canceations that payed such an essentia roe in the theory in [GG2] (and [GG1]) and by undoing at each step the resummations which ed to the construction of h and to the proof of Proposition 1, see (4.4) above. However the properties of h and the resut of Proposition 1 have payed a key roe in the derivation of (3.3) and (3.4). Without the uniform bounds (in N) on the convergence radii in p of the series expressing (h (N) ) B, which depend on the canceations and on the resummations, the bounds in Sect. 4 or, in the non-trigonometric case, of Appendix A1, woud remain the same but they woud be useess for our purposes of estabishing (3.4) and the Bore summabiity. (2) It has been remarked above that the resummation procedure is based on severa arbitrary a priori choices which therefore may ead to the existence of severa soutions of h with the same asymptotic series at ε = 0. A the choices in [GG2], as we as that used in [GG1] (see Appendix A2), ead however to a Bore summabe series: this proves that a soutions coincide and the resuts of the resummations are independent of the particuar choices provided the Diophantine constant τ is τ = 1 (hence r = 2 and ω is a Diophantine vector with τ = 1, e.g. ω 1 /ω 2 is a quadratic irrationa). (3) If τ > 1, in particuar if r > 2, the probem of Bore summabiity and of independence of the resut from the summation method remains open. (4) The existence or non-existence of soutions h which are C at the origin and give soutions to the equations of motion but which are not Bore summabe is aso an open probem. Note that even in the case of non-resonant motions, i.e. in the case of maxima tori, the probem of the uniqueness at fixed ω is not trivia, and aso the recent resuts in [BT] do not excude the possibiity of other quasi-periodic motions besides those constructed through the KAM agorithm. (5) The case ε < 0 with β 0 a minimum point for f 0 (β), i.e. of eiptic motions is quite different. We have used on purpose the resummation technique of [GG2] which works for hyperboic as we as for eiptic resonances, but the present resuts sti ony appy to the hyperboic case and it is not cear whether the above techniques can be extended to prove Bore summabiity of the parametric equations for h in eiptic cases. (6) In the case f is a trigonometric poynomia the bounds of previous section can be improved, the resut being that F(ψ, p) is an entire function, satisfying F(ψ, p) Γ 2 p e Γ p ec p in the whoe 2/aprie/2009; 18:58 14

15 compex pane. A proof of this caim is given in Appendix A1. Appendix A1. Poynomia perturbations In this Appendix we want to prove that, in the case f is a trigonometric poynomia, the bounds in Section 4 can be improved to show that F(ψ, p) is an entire function of p, with expicit bounds on its grow at infinity. Assume, inductivey, that x = ω ν the functions (M [ k] ) B (x; p) are entire functions of p and ( M [ k]) B (x; p) Γ p e d k p, for a x on scae k, with k k 0, (A1.1) for a p compex. Note that (M [0] ) B (p) = pm 0 (so that the inductive assumption in (A1.1) is vaid at the first step k = 0 with d 0 = 0 if Γ M 0 ) and (g [0] ) B (x; p) is given by (4.3): in particuar (g [0] ) B (x; p) p x e p c0, x γ 2 0 with c 0 = M 0 /γ 0. Supposing the inductive assumption to be vaid for k = 1,...,N 1 we remark that this impies a bound on (g [k] ) B (x; p) for k = 1,...,N 1 via the expansion ( η (g [k] 2 ) B (x; p) = x 2 m=0 ( M [ k] (x; ) x 2 ) m ) B. (A1.2) Taking the Bore transform and performing a convoutions aong a straight ine from 0 to p, for x on scae 1 k N 1, we get (g [k] ) B (x; p) 1 x 2 p m=0 x 2 ed k p +Γ 1/2 γ p (Γ p ed k p ) m x 2m 1 k p x 2 p p x 2 ec k p Then (M [ N] M [0] ) B (x; p) is estimated via (2.9), that is (M [ N] M [0] ) B (x; p) k=2 T N 1 j=0 SR k,j 1 Λ(T)! p ( Λ(T) n 0 m=0 1 p e p )( cn x 2 Λ(T) n = 1 e d k p Γm p 2m (2m)! 1 x 2m. (A1.3) )( ) g [ 1] F v. v V (T) (A1.4) ( ) Bounding p p e cn p by e cn 1 p p 2k 1 /(2k 1)! and using the estimates in [GG2] to contro the sum over the sef energy custers, the bound becomes (M [ N] ) B (x; p) Γ p + k=2 Γ 2k p 2k 1 (2k 1)! ecn 1 p Γ p e dn p, (A1.5) where Γ is a suitabe constant derived in [GG1], see Remark foowing (4.9). Thus the inductive assumption hods for a N, the constants c N, d N can be taken c2 N for some c, and for a x on scae N one has (g [N] ) B (x; p) p x 2 e2n c p. (A1.6) 2/aprie/2009; 18:58 15

16 This eads to a bound on (h (N) ) B (ψ, p), via (2.4) and (2.2): (h (N) ) B (ψ, p) 1 ( Λ(θ)! ν Θ ν,γ k=1 k=1 Λ(θ) n 0 1 p e p )( cn x 2 2k p 2k 1 Γ ec2nmax p (2k 1)! k=1 Λ(T) n = 1 )( g [ 1] v V (T) ) F v 2k p 2k 1 k Γ p (2k 1)! ec Γ 2 p p e Γ p ec, (A1.7) because the maximum scae n max of the ines of a graph with k ines can be at most og 2 (bk) by our assumption that f is a trigonometric poynomia: in fact the maximum momentum on a ine can be ν b 0 k (for f a trigonometric poynomia), so that the smaest x can be C0 b if C 0k 0 and τ = 1 are the Diophantine constants, hence the scae of x can be at most og 2 (b 0 k/4) and c2 nmax c k for a suitabe c. Therefore the functions (h (N) ) B (ψ; p) are entire and bounded by (h (N) ) B (ψ; p) Γ 2 p e Γ p ec p (A1.8) independenty of N. By the same argument at the end of Section 4 (i.e. by an appication of Vitai s convergence theorem) we infer that the imit F(ψ; p) of (h (N) ) B (ψ; p) as N is entire and satisfies the same bound (A1.8). By the definition of Bore summabiity and the resuts of Section 3, h(ψ, η) is Bore summabe (in η). Appendix A2. Comparison with the method of [GG1] In this Appendix we briefy discuss how the function h constructed in [GG1] can be identified with the Bore summabe function of Proposition 1. By the uniqueness in the cass of Bore summabe functions, it is enough to prove that aso the function h of [GG1] is Bore summabe. We begin by reviewing the differences of the construction envisaged in [GG1] with respect to that of [GG2]. Trees, abes and custers are defined in the same way as in items (a) to (h) of Section 2. What changes is the definition of the propagators, which is iterative. By writing x = ω ν, ν 0, we set g [0] = 1/x 2, and, for k 1, g [k] η 2 = x 2 + M [k] (x; η). (A2.1) with M [k] (x; η) defined as M [k] (x; η) = T V T (x; η), (A2.2) where the sum is restricted to the sef-energy custers T with scae n T n + 3, where n is such that γ n 1 x < γ n, and the sef-energy vaue is given by ( V T (ω ν; η) = η2 Λ(T)! v V (T) F v )( Λ(T) ) g [k 1]. (A2.3) Note that k abes the iterative step and, in principe, it has no reation with the scae n of x. However the matrices M [k] (x; η) are obtained from resummations of sef energy custers with 2/aprie/2009; 18:58 16

17 height up to k = n (for definitions and detais we refer to [GG1], where the sef energy custers are caed sef energy graphs). Hence they stop fowing at k = n if x is on scae n: this means that M [k] (x; η) = M [n] (x; η) as soon as k n. Then h (N) wi be expressed in terms of trees as in (2.2), if Va(θ) is defined as in (2.4), with g [n ] repaced with g [N], and h (N) is obtained as the imit of h (N) as N. Let us consider for simpicity the case of poynomia perturbations. Then we can proceed as in Sec. 4, and prove by induction the bound ( M [k]) B (x; p) Γ p e d k p, for a x, (A2.4) for a p compex. Supposing inductivey the bound (A2.4) we obtain that the Bore transform of g [k] can be bounded as (g [k] ) B (x; p) p x 2 ec k p. (A2.5) This is trivia for k = 0, as (g [0] ) B (x; p) = p/x 2, whie it foows from (A2.4) for n 1 by the inductive hypothesis (and it can be proved as the anaogous bound in Sec. 4). Therefore we can write M [N] (x; η) according to (A2.2), and its Bore transform (M [N] ) B (x; p) can be computed and bounded as done in (A1.4) and (A1.6), so that at the end the bound (A2.4) is obtained for k = N. In particuar the bounds (A2.5) on the propagators impy that h (N) admits the same bound (A1.8) found in Sec. 4. Therefore we can take the imit for N, and Bore summabiity for h foows. References [BT] H. Broer, F. Takens, Unicity of KAM tori, Preprint, Groningen, [GBG]G. Gaavotti, F. Bonetto, G. Gentie, Aspects of ergodic, quaitative and statistica theory of motion, Texts and Monographs in Physics, Springer, Berin, [GG] G. Gaavotti, G. Gentie, Majorant series convergence for twistess KAM tori, Ergodic Theory and Dynamica Systems 15 (1995), [GG1] G. Gaavotti, G. Gentie, Hyperboic ow-dimensiona invariant tori and summations of divergent series, Communications in Mathematica Physics 227 (2002), no. 3, [GG2] G. Gaavotti, G. Gentie, Degenerate eiptic resonances, Communications in Mathematica Physics 257 (2005), [JLZ] À. Jorba, R. de a Lave, M. Zou, Lindstedt series for ower-dimensiona tori, Hamitonian systems with three or more degrees of freedom (S Agaró, 1995), , NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 533, Kuwer Acad. Pub., Dordrecht, [Ne] [So] F. Nevaninna, Zur Theorie der Asymptotischen Potenzreihen, Annaes Academiae Scientiarum Fennicae. Series A I. Mathematica 12 (1916), A.D. Soka, An improvement of Watson s theorem on Bore summabiity, Journa of Mathemnatica Physics 21 (1980), no. 2, /aprie/2009; 18:58 17