DISCRETE AND CONTINUOUS doi:10.3934/dcdss.010.3.545 DYNAMICAL SYSTEMS SERIES S Volume 3, Number 4, December 010 pp. 545 578 PROPERLY DEGENERATE KAM THEORY FOLLOWING V. I. ARNOLD Luigi Chierchia Dipartimento di Matematica Università Roma Tre Largo S. L. Murialdo 1, I-00146 Roma, Italy Gabriella Pinzari Dipartimento di Matematica ed Applicazioni R. Caccioppoli Università di Napoli Federico II Monte Sant Angelo Via Cinthia I-8016 Napoli Italy This paper is dedicated to the memory of Professor Nikolaĭ Nekhoroshev Abstract. Arnold s Fundamental Theorem on properly degenerate systems [3, Chapter IV] is revisited and improved with particular attention to the relation between the perturbative parameters and to the measure of the Kolmogorov set. Relations with the planetary many body problem are shortly discussed. 1. Introduction and results. A problem that one often encounters in applications of KAM theory is related to the presence of degeneracies. An important example which actually motivated the birth of KAM theory is the problem of finding a positive measure set in phase space corresponding to quasi periodic motions in the planetary 1 + n body problem i.e., 1 + n point masses interacting only under a gravitational potential modeling a system formed by a star and n planets. In this case the integrable limit i.e., the n uncoupled two body systems formed by the star and one planet does not depend upon a full set of action variables proper degeneracy and therefore typical non degeneracy conditions such as Kolmogorov s non degeneracy or Arnold s iso energetical non degeneracy are strongly violated. To deal with properly degenerate systems V.I. Arnold developed in [3] a new KAM technique, which is summarized in what he called the Fundamental Theorem [3, Chapter IV]. Arnold then applied the Fundamental Theorem to the planar, planetary, nearly circular three body problem n = proving for the first time relatively bounded motions for a positive set of initial data. A full proof of this result in the general spatial many body problem turned out to be more difficult than expected. After an extension to the spatial three body case [15], a first complete proof was published only in 004 [10], where a different 000 Mathematics Subject Classification. Primary: 70H08, 70K43, 34C7, 34C9, 34D10; Secondary: 70H1, 70K45, 70F15, 70F10. Key words and phrases. KAM theory, Kolmogorov set, many body problem, small divisors, invariant tori, degeneracies. Partially supported by European Research Council under F P 7 project New connections between dynamical systems and Hamiltonian PDEs with small divisors phenomena. 545
546 LUIGI CHIERCHIA AND GABRIELLA PINZARI smooth KAM technique due to M.R. Herman was used; for a real analytic proof, see [6]. In this paper we revisit and extend Arnold s Fundamental theorem so as to weaken its hypotheses and to improve the measure estimates on the Kolmogorov set i.e., the union of maximal invariant quasi-periodic tori. In properly degenerate KAM theory it is not enough to make non degeneracy assumptions on the unperturbed limit as in standard KAM theory. To describe a typical setting, let us consider a Hamiltonian function of the form HI, ϕ, p, q; µ := H 0 I + µp I, ϕ, p, q; µ, 1 where 1 I, ϕ V T n1 R n1 T n1 and p, q B R n are standard symplectic variables; here V is an open, connected set in R n1 and B is a n ball around the origin; n, where n := n 1 + n is the dimension of the phase space P := V T n1 B, which is endowed with the standard symplectic two form n 1 n di dϕ + dp dq = di j dϕ j + dp j dq j. j=1 The Hamiltonian H is assumed to be real analytic. When the perturbative parameter µ is set to be zero in the planetary case µ measures the ratio between the masses of the planets and that of the star the system is integrable but depends only on n 1 < n action variables. A typical further assumption is that the averaged or secular perturbation, P av p, q; I, µ := T n 1 j=1 P I, ϕ, p, q; µ dϕ, 3 n1 π has an elliptic equilibrium in the origin with respect to the variables p, q. Under suitable assumptions on the first and/or second order Birkhoff invariants see [11] for general information one can guarantee the existence of maximal KAM tori near the secular tori I} T n1 Tη n, 4 where η = η 1,..., η n, Tη n denotes a n dimensional torus given by the product of n circles of radii η j > 0 and ɛ := max η j is small. More precisely, Arnold makes the following assumptions 3 : A1 I V I H 0 is a diffeomorphism; n A P av p, q; I = P 0 I+ Ω i Ir i + 1 i=1 n i,j=1 β ij Ir i r j + n i,j,k=1 λ ijk Ir i r j r k +o 6 where r i := p i + q i and o 6 / p, q 6 0 as p, q 0; A3 the matrix of the second order Birkhoff invariants is not singular, i.e., det βi const > 0 for all I V. 1 T n denotes the standard n dimensional flat torus R n /πz n. An interesting point is what is the relation between ɛ and µ, especially in view of physical applications in the planetary case ɛ measures the eccentricities and relative inclinations of the star planet motions: this matter will be further discussed in the following. 3 From now on we drop the dependence on µ of the perturbation, assuming that such dependence is smooth enough, say C 1, and that the norms are uniform in µ.
PROPERLY DEGENERATE KAM THEORY 547 We can now state Arnold s Fundamental Theorem. Denote by B ɛ = Bɛ n R n : y < ɛ} the n ball of radius ɛ and let = y P ɛ := V T n1 B ɛ 5 and recall the definitions of H and the phase space P in, respectively, 1 and. Theorem 1.1. Arnold s Fundamental Theorem [3, p. 143] Let H be real analytic on P and assume A1 A3. Then, there exists ɛ > 0 such that, for 0 < ɛ < ɛ, 0 < µ < ɛ 8, 6 one can find a set K P ɛ P formed by the union of H invariant n dimensional tori close to the secular tori in 4, on which the H motion is analytically conjugated to linear Diophantine 4 quasi periodic motions. The set K is of positive Liouville Lebesgue measure and satisfies meas K > 1 const ɛ a meas P ɛ, where a := 1/8n + 4. 7 Remark 1. By Birkhoff s theory compare Proposition below, the expansion in A for P av may be achieved if one assumes that p, q P av p, q; I has an elliptic equilibrium in p = q = 0 and the first order Birkhoff invariants Ω i are non resonant up to order 6, ie 5, n Ω j Ik j const > 0 I V, 0 < k 6, k Z n. 8 j=1 In this paper we relax condition 6 and replace assumption A with either A p, q P av p, q; I has an elliptic equilibrium in the origin p = q = 0 and the first order Birkhoff invariants are non resonant up to order four, i.e., they verify 8 with 6 replaced by 4. or n A P av p, q; I = P 0 I + Ω i Ir i + 1 i=1 and o 4 / p, q 4 0 as p, q 0. We shall prove the following two theorems. n i,j=1 β ij Ir i r j + o 4 with r i := p i + q i Theorem 1.. Let H be real analytic on P and assume A1, A and A3. Then, there exist positive numbers ɛ, C and b such that, for 1 0 < ɛ < ɛ, µ < C log ɛ 1, 9 b one can find a set K P formed by the union of H invariant n dimensional tori, on which the H motion is analytically conjugated to linear Diophantine quasi periodic motions. The set K is of positive Liouville Lebesgue measure and satisfies µ meas P ɛ > meas K > 1 C log ɛ 1 b + ɛ meas P ɛ. 10 Next theorem needs stronger hypotheses on µ but there are no conditions on the first order Birkhoff invariants. 4 I.e., the flow is conjugated to the Kronecker flow θ T n θ + ω t T n, with ω = ω 1, ω satisfying 19 below. 5 Here and below, for integer vectors k Z m, k := k 1 = m j=1 k j. See also notation in Chapter.
548 LUIGI CHIERCHIA AND GABRIELLA PINZARI Theorem 1.3. Let H be real analytic on P and assume A1, A and A3. Then, there exist positive numbers ɛ, C and b such that, for 0 < ɛ < ɛ, 0 < µ < ɛ 6 log ɛ 1, 11 b one can find a set K P ɛ formed by the union of H invariant n dimensional tori, on which the H motion is analytically conjugated to linear Diophantine quasi periodic motions. The set K is of positive Liouville Lebesgue measure and satisfies meas P ɛ > meas K > 1 C ɛ meas P ɛ. 1 Let us make a few remarks. i Under assumption A, near p = 0 = q, the dynamics is approximated by the dynamics governed by the integrable secular averaged and truncated Hamiltonian n H sec := H 0 I + µ P 0 I + Ω i I p i + q i. 13 The phase space P is foliated by n dimensional H sec invariant tori as in 4 with 0 < ɛ < ɛ, where ɛ denotes the radius of the ball B in. Indeed, in this case Tη n are simply given by p j + q j = η j, j n } with η j ɛ. In the perturbed case the fate of the secular tori may be different according to the relation between ɛ and µ. In fact what happens is that, if µ < ɛ α, with α > 1 in particular, if 6 or 11 holds, then K P ɛ as in Arnold s Theorem, but if µ > ɛ α then, is general K is not contained in P ɛ and the persistent tori may be not so close to the secular tori I} T n1 p j + q j = η j, j n } but rather they are close to the translated tori I} T n1 p j p 0 j + q j qj 0 = η j, j n } where p 0 j, q0 j = p 0 j I; µ, q0 j I; µ are the coordinates of a new equilibrium, which depend upon the full averaged system and which may be logarithmically distant from the origin as far as 1/ log ɛ 1. In any case, the set K fills almost completely a region diffeomorphic to and of equal measure of P ɛ. A precise geometrical description of the Kolmogorov set K is given in Step 6 of 3. ii As mentioned above, in the planetary problem, µ measures the mass ratio between the planets and the star, while ɛ is related to the eccentricities and inclinations of the instantaneous two body systems planet star. Condition 9 is much weaker than Arnold s condition 6 and allows, at least in principle, applications to a wider class of planetary systems. Clearly, in order to apply properly degenerate KAM theory to a concrete system such as the outer Solar system 6 one should also estimate ɛ in 9, which would be quite a technical achievement 7. i=1 6 In the outer Solar System Sun, Jupiter, Saturn, Uranus and Neptune µ is of order 10 3 and the largest eccentricity is of 0.05 Saturn. 7 For partial results in this direction, see [5] and [1].
PROPERLY DEGENERATE KAM THEORY 549 iii Arnold declared [3, end of p. 14] that he made no attempt to achieve elegance or precision in evaluating constants adding that the reader can easily strengthen the results. However, the authors are not aware of improvements on Arnold s results in the full torsion case, compare next item and especially on the issue of giving possibly sharp estimates on the measure of the Kolmogorov set arising in properly degenerate systems. At this respect it would be interesting to know whether estimate 10 could be improved or not. iv Relaxing 8, i.e., bringing to four the order of non resonance to be checked, has an interesting application in the case of the 1+n body problem. In fact, Herman and Féjoz showed [10] that, in the spatial case n 1 = n and n = n, the only linear relations satisfied by the first order Birkhoff invariants Ω j are up to rearranging indices: Ω n = 0, n 1 j=1 Ω j I = 0. 14 The first relation is due to rotation invariance of the system, while the second relation is usually called Herman resonance 8. Now, since in the spatial case, Herman resonance is of order n 1, one sees that for n 3 it is not relevant for A but it is for 8. Actually, at this respect, Theorem 1.3 might be even more useful since it involves no assumption on the Ω j so that in possible application to the spatial 1 + n body problem, Herman resonance plays no rôle. v The properly degenerate KAM theory developed in [10] for the C case and in [6] for the analytic case, being based on weaker non degeneracy assumptions, is different from Arnold s theory. Roughly speaking, while Arnold s approach is ultimately based on Kolmogorov s non degeneracy condition full torsion in a two scale setting, the approach followed in [10, 6] which might be called weak properly degenerate KAM theory is based on the torsion of the frequency map, exploiting conditions studied by Arnold himself, Margulis, Pyartli, Parasyuk, Bakhtin and especially Rüssmann [16]; for a review, see [17]. Indeed, for Arnold s properly degenerate theory one has to check that the matrix of the second order Birkhoff invariants is not singular condition A3 above, while for the weak properly degeneracy theory it is enough to check a generic property involving only the first order Birkhoff invariants: Conditions A and A3 are replaced by the requirement that the re scaled frequency map I V ˆωI := H 0 I, ΩI is non planar, i.e., ˆωV does not lie in any n 1 dimensional linear subspace of R n. Incidentally, the presence of the resonances 14 makes difficult a direct application of weak properly degenerate KAM theory to the spatial 1 + n body problem in standard Poincaré variables 9. Explicit measure estimates on the set of persistent tori in the context of weak properly degenerate KAM theory are not readily available 10. 8 Compare also [1]. 9 Application of the properly degenerate KAM theory developed in the present paper using Deprit variables [7] will be matter of a future paper by the authors. 10 In fact, although Pyartli s theorem on the measure of Diophantine points on a non planar curve is quantitative compare [10, Théorème 55], explicit measure estimates of the Kolmogorov set in the N body problem, following the strategy in [10], do not appear completely obvious.
550 LUIGI CHIERCHIA AND GABRIELLA PINZARI vi Let us briefly and informally recall Arnold s scheme of proof. First, by classical averaging theory see, e.g., [] the Hamiltonian 1 is conjugated to a Hamiltonian H satisfying, for any small 11 σ > 0, H = H 0 + µp av + Oµ σ 15 where P av is as in A. Denoting P av [6] the truncation in p, q at order 6 of P av, one sees that 15 can be rewritten as H = H 0 + µp [6] av + Oµɛ 7 + Oµ σ 16 if p, q < ɛ. In turn, 16 is of the form H = H 0 + µp [6] av + Oµɛ 7 if 6 holds. At this point, a two time scale KAM theorem can be applied. The scheme of proof of Theorem 1.3 is similar, but we use more accurate estimates based on the averaging theory described in.1 below and, especially, on the two scale KAM theorem described in.3 below; this last result, in particular, is not available in literature and we include its proof in Appendix B. To relax significantly the relation between µ and ɛ, the above strategy has to be modified. The scheme to prove Theorem 1. is the following: step 1: averaging over the fast angles ϕ s; step : determination of the elliptic equilibrium for the secular system ; step 3: symplectic diagonalization of the secular system; step 4: Birkhoff normal form of the secular part; step 5: global action angle variables for the full system; step 6: construction of the Kolmogorov set via an application of a two scale KAM theorem and estimate of its measure. Properly degenerate systems present naturally two different scales: a scale of order one related to the unperturbed system the typical velocity of the fast angles ϕ s and a scale of order µ typical size of the secular frequencies related to the strength of the perturbation. Furthermore, a third scale appears naturally, namely, the distance from the elliptic equilibrium in the p, q variables. We now give a more technical and detailed statement, from which Theorem 1. follows at once. Theorem 1.4. Under the same notations of Theorem 1. and assumptions A1, A and A3, let τ > n 1 and 1 τ > n := n 1 + n, with n 1, n positive integers. Then, there exist ɛ < 1, γ, C > 1 such that, if 9 holds and if, γ 1, are taken so as to satisfy µ γ 1 and γ max µlog ɛ 1 τ+1, γ ɛ 5/ < γ 1 < γ γ ɛ 5/ log ɛ 5 /γ 1 1 τ +1 < < γ ɛ, 3 µɛlog ɛ 1 τ+1, ɛ log ɛ 1 τ+1 } < < γ 11 The appearance of the exponents σ rather than the more natural exponent is due to the presence of small divisors. 1 At contrast with classical KAM theory, where the Diophantine constant can be taken greater than n 1, here one needs τ > n in [3] it is taken n + 1: this is due to the asymmetry of the frequency domain having n 1 dimensions of order one and n n 1 = n dimensions small with the perturbative parameters. 17
PROPERLY DEGENERATE KAM THEORY 551 then, one can find a set K P formed by the union of H invariant n dimensional tori close to the secular tori in 4, on which the H motion is analytically conjugated to linear Diophantine quasi periodic motions. The set K is of positive measure and satisfies meas P ɛ > meas K > [1 C + γ 1 + ɛ + ɛn/] meas P ɛ. 18 Furthermore, the flow on each H invariant torus in K is analytically conjugated to a translation ψ T n ψ+ωt T n with Diophantine vector ω = ω 1, ω R n1 R n satisfying, for each k = k 1, k Z n1 Z n \0}, ω 1 k 1 + ω k γ 1 k τ if k 1 0 ; µ k τ if k 1 = 0, k 0. To obtain Theorem 1. from Theorem 1.4 one can choose 19 = γ max µlog ɛ 1 τ+1, ɛ /3 log ɛ 1 τ+1}, γ 1 = = γ ɛ 5/ ; 0 then 10 follows easily 13, with b = τ + 1. The proof of Theorem 1.3, as already mentioned, is simpler and it will be shortly given in 4.. Tools: Averaging, Birkhoff normal form and two scale KAM. First of all we fix some notation, which will be used throughout the paper. in R n1 we fix the 1 norm: I := I 1 := I i ; 1 i n 1 in T n1 we fix the sup metric : ϕ := ϕ := max ϕ i mod π; 1 i n 1 in R n we fix the sup norm : p := p := max p i, q := q 1 i n := max q i ; 1 i n for matrices we use the sup norm : β := β := max i,j β ij ; if A R ni, or A T n1, and r > 0, we denote by A r := x A z C ni : } z x < r the complex r neighborhood of A according to the prefixed norms/metrics above; if f is real analytic on a complex domain of the form U v T m s with U R d, we denote by f Uv T m, or, simply, f s v,s its sup Fourier norm : f v,s := sup f k u e k s, k := k i u U v k Z m 1 i m where f k u denotes the k th Fourier coefficient of f = k Z m f k ue ik ϕ ; 13 First, let us check that 17 holds. From 9 it follows that < γ provided C > γ and ɛ is small enough. The lower bound on is checked by considering the cases ɛ µ and µ < ɛ separately. The bounds on γ 1 are obvious.the bounds on are true for γ big enough. Thus, 17 is checked. Finally, 0 and 18 imply easily 10.
55 LUIGI CHIERCHIA AND GABRIELLA PINZARI if f is as in the previous item, K > 0 and Λ is a sub-lattice of Z m, T K f and Π Λ f denote, respectively, the K truncation and the Λ projection of f: T K f := f k ue ik ϕ ; k K f k ue ik ϕ, Π Λ f := k Λ if f : A R d R n is a Lipschitz function and ρ > 0 a weight, we denote its ρ Lipschitz norm by f Lip ρ,a := ρ 1 sup A fi fi f + Lf, Lf := sup I I A I I. 1 D γ1,γ,τ R n1+n denotes the set of Diophantine γ 1, γ, τ numbers, i.e., the set of vectors ω R n1+n satisfying for any k = k 1, k Z n1+n \0}, inequality 19 with τ = τ, = γ and µ = 1. When γ 1 = γ = γ, we obtain the usual Diophantine set D γ,τ..1. Averaging theory. The first step of the proof of Theorem 1.4 and hence of Theorem 1. is based upon averaging theory. We shall follow the presentation given in [4, Appendix A], which in turn is based upon [14]. Proposition 1 Averaging theory. Let K, s and s be positive numbers such that Ks 6 and let α 1 α > 0; let A B B R l1 R l R m R m, and v = r, r p, r q a triple of positive numbers. Let H := hi + fi, ϕ, p, q be a realanalytic Hamiltonian on W v, s+s := A r B rp B r q T l1+l s+s. Finally, let Λ be a possibly trivial sub lattice of Z l1+l and let ω = ω 1, ω denote the gradient I1 h, I h R l1+l. Let k = k 1, k Z l1 Z l and assume that α1, if k ω k 1 0 I A α, if k 1 = 0 r, k = k 1, k / Λ, k K E := f v, s+s < α d, where d := minrs, r 7 p r q }, c m := c m Ks Then, there exists a real-analytic, symplectic transformation such that e1 + em.3 Ψ : I, ϕ, p, q W v/, s+s/6 I, ϕ, p, q W v, s+s 4 with g in normal form and f small: g = k Λ H := H Ψ = h + g + f, g k I, p, q e ik ϕ, g Π Λ T Kf v/, s+s/6 1 11 7 c m E E α d 4, f v/, s+s/6 e Ks/6 9 c m E e Ks/6 E. 5 α d Moreover, denoting by z = zi, ϕ, p, q, the projection of ΨI, ϕ, p, q onto the z variables z = I 1, I, ϕ, p or q one has maxα 1 s I 1 I 1, α s I I, α r ϕ ϕ, α r q p p, α r p q q } 9E. 6 This Proposition is essentially Proposition A.1 of [4] with two slight improvements. The first improvement is trivial and concerns the introduction of the parameter s so as to separate the rôle of the analyticity loss in the angle variables from the initial angle domain. Such variation is important, for example, in applying Proposition 1 infinitely many times.
PROPERLY DEGENERATE KAM THEORY 553 The second improvement is a bit more delicate and we use it in the proof of Proposition 3 below. It concerns the separation of two scales in the frequencies ω = I h. Proposition 1 holds also for l 1 0, l = 0 i.e., there is only one action scale, in which case α := α 1 = α, and in the case m = 0 i.e., there are no p, q variables, in which case one can take d = rs, c m = c 0 = e/. In the following, Proposition 1 will be applied twice: in step 1 of 3 with l 1 = n 1, l = 0, m = n and in Appendix B with m = 0. Proposition 1 is proved in Appendix A... Birkhoff normal form. We now recall a fundamental result due to Birkhoff on normal forms. We follow [11]. Proposition Birkhoff normal form. Let α > 0, s 3; let Ω = Ω 1,, Ω m R m be non resonant of order s, i.e., and let z = p, q Bɛ m 0 of the form m Hz = Ω k α > 0, k Z m with 0 < k s 7 = z : z < ɛ 0 } R m Hz be a real analytic function i=1 Ω i r i + O z 3 where r i := p i + q i. 8 Then, there exists 0 < ɛ ɛ 0 and a real analytic and symplectic 14 transformation φ : z = p, q B m ɛ z + ẑ z B m ɛ 0 which puts H into Birkhoff normal form up to order s, i.e. 15, H := H φ = m i=1 [s/] Ω i r i + Q j r + O z s+1 9 where, for j [s/], the Q j s are homogeneous polynomials of degree j in r = r 1,, r m with r j := p j + q j j=. The polynomials Q j do not depend on φ. Following the proof of this classical result as presented in [11] one can easily achieve the following useful amplifications. 1. The construction of the transformation φ is iterative and can be described as follows. There exist positive numbers ɛ := ɛ s < ɛ s 3 < < ɛ 0, and a symplectic transformations φ i such that φ = ˆφ s := φ 1 φ s, that H ˆφ i is in Birkhoff normal form up to order i + and φ i : sup B m ɛ i z = p, q B m ɛ i ẑ i 1 c i 1 m i 1 α ɛ i 1 i+1, z + ẑ i 1 z B m ɛ i 1, 1 i s, where c i 1 depend only on the dimension m and m i 1 are defined as follows. For i 1 = 0, let P 0 the homogeneous polynomial of degree 3 for which Hz 14 With respect to the standard form dp dq = 15 [x] denotes the integer part of x. 1 i m dp i dq i.
554 LUIGI CHIERCHIA AND GABRIELLA PINZARI m j=1 Ω jr j = P 0 + O z 4, while, for i 1 1, let P i 1 the homogeneous polynomial of degree i + for which Hz ˆφ i 1 m j=1 [i+1/] Ω j r j Q j r = P i 1 + O z i+3. j= Write P i 1 = α + β =i+ c m α,β j=1 p j + iq j αj p j iq j βj, where i:= 1. Then, m i 1 := max c α,β. 30 α,β:α β. Proposition can be easily extended to the case of a real analytic function m Hz; I = Ω i I r i + O z 3, i=1 which also depends on suitable action variables I. More precisely, if A is an open subset of R n, ρ 0, s 0, ɛ 0 are positive numbers, I, ϕ and z = p, q, with I, ϕ, z A ρ0 T n σ 0 Bɛ m 0, are conjugate couples of symplectic variables with respect to the standard form di dϕ + dp dq and Ω = Ω 1,, Ω m is a suitable real analytic function defined on A ρ0 verifying 7 on A ρ0, then, one can prove that for suitable 0 < ɛ = ɛ s < < ɛ 0, 0 < σ = σ s < < σ 0, 0 < ρ = ρ s < < ρ 0, c i, there exist s real analytic, symplectic transformations which we still denote φ i, φi : A ρi T n σ i Bɛ m i A ρi 1 T n σ i 1 B ɛ m i 1 φ i Ĩ, ϕ, z = Ĩ, ϕ + ˆϕ i 1 z; Ĩ, z + ẑ i 1 z; Ĩ such that 9 holds with φ = ˆφ s = φ 1 φ s, Ω i = Ω i I and suitable homogeneous polynomials Q j r; I of degree j in r = r 1,, r m whose coefficients are analytic functions on A ρ. At each step, the functions z; Ĩ ẑ i 1 z; Ĩ, z; Ĩ ˆϕ i 1 z; Ĩ verify B m ɛ i B m ɛ i m i 1 sup ẑ i 1 c i 1 A ρi α ɛ i 1 i+1, m i 1 sup ˆϕ i 1 c i 1 ɛ i 1 i+ 31 A ρi αρ 0 where, if, for any fixed I A ρi 1, m i 1 I are defined as in 30 with c α,β = c α,β I, then, m i 1 = sup A ρi 1 m i 1 I..3. Two scale KAM theory. The invariant tori of Theorem 1.3 and 1.4 will be obtained as an application of a KAM Theorem, adapted to two different frequency scales, which is described in the following Proposition 3 Two scale KAM Theorem. Let n 1, n N, n := n 1 + n, τ > n, γ 1 γ > 0, 0 < 4s s < 1, ρ > 0, D R n1 R n, A := D ρ, and let HJ, ψ = hj + fj, ψ be real analytic on A T n s+s. Assume that ω 0 := h is a diffeomorphism of A with non singular Hessian matrix U := h and let Û denote the n n 1 submatrix of U, i.e., the matrix with entries Ûij = U ij, for n 1 + 1 i n, 1 j n. Let M sup U, A ˆM sup Û, A M sup U 1, E f ρ, s+s ; A
PROPERLY DEGENERATE KAM THEORY 555 define ĉ := max 8 4 τ +1 } n, 6 EM 1 L where log + a := max1, log a} K := 6 s log + γ1 γ 1 ˆρ := min 3MK τ +1, γ 3 L := max M, M 1, ˆM 1} Ê := EL ˆρ ; }, ρ ˆMK τ +1 finally, let M1, M upper bounds on the norms of the sub-matrices n 1 n, n n of U 1 of the first n 1, last n rows 16. Assume the perturbation f so small that the following KAM condition holds ĉê < 1. 3 Then, for any ω Ω := ω 0 D D γ1,γ,τ, one can find a unique real analytic embedding φ ω : ϑ T n vϑ; ω, ϑ + uϑ; ω Re D r T n 33 where r := 0nÊ ˆρ such that T ω := φ ω T n is a real analytic n dimensional H invariant torus, on which the H flow is analytically conjugated to ϑ ϑ + ω t. Furthermore, the map ϑ; ω φ ω ϑ is Lipschitz and one to one and the invariant set K := T ω satisfies the following measure estimate ω Ω meas Re D r T n \ K c n meas D \ D γ1,γ,τ T n + meas Re D r \ D T n 34, where D γ1,γ,τ denotes the ω 0 pre-image of D γ1,γ,τ in D and c n can be taken to be c n = 1 + 1 + 8 nên. Finally, on T n Ω, the following uniform estimates hold v 1 ; ω I1 0 M1 ω 10n M + ˆM Ê ˆρ M v ; ω I 0 M ω 10n M + ˆM Ê ˆρ, M u ; ω Ê s 35 where v i denotes the projection of v R n1 R n over R ni and I 0 ω = I 0 1 ω, I 0 ω D is the ω 0 pre image of ω Ω. This result is proved in Appendix B. 16 I.e., Mi sup T i, i = 1,, if U 1 T1 = D ρ T.
556 LUIGI CHIERCHIA AND GABRIELLA PINZARI 3. Proof of Theorem 1.4. In this section we prove Theorem 1.4 and hence Theorem 1.; compare the remark following the formulation of Theorem 1.4 in 1. In what follows, C denotes suitably positive constants greater than one independent of ɛ and µ,, γ 1, but which may depend on n 1, n, H 0, s 0, etc. Without loss of generality, we may assume that H has an analytic extension to a domain P ρ0,ɛ 0,s 0 := V ρ0 T n1 s 0 B ɛ0 with s 0 < 1 and with ω 0 := H 0 a diffeomorphism of V ρ0. We can also assume that the perturbation P has sup Fourier norm P ρ0,ɛ 0,s 0 1 up to change the definition of µ. Preliminary step. In view of A on p. 547, we can assume that the quadratic part of P av p, q; I is in standard form P 0 I + Ω i I r i + o, where Ω i I 1 i n are the first order Birkhoff invariants; compare [18]. Furthermore again by A, since the Ω i I are non resonant up to the order four, by Birkhoff theory compare. above, one can find a symplectic transformation, O p, q close to the identity, which transforms the original Hamiltonian into 17 1, with P av as in the standard form in A. Step 1 Averaging over the fast angles ϕ s Let 0 < ɛ < e 1/5. The first step consists in removing, in H, the dependence on ϕ up to high orders namely, up to Oµ ɛ 5. To do this, we use Averaging theory Proposition 1 above, with l 1 = n 1, l = 0, m = n h = H 0, g 0, f = µp, B = B = 0}, r p = r q = ɛ 0, s = s 0, s = 0, Λ = 0}, K such that e Ks 0/6 := ɛ 5 i.e., K = 30 s 0 log ɛ 1, 36 A = D, r = ρ, where D, ρ are defined as follows. Let τ > n 1, M := maxi,j sup Vρ0 ij H 0I, γ max1, 5 30/s 0 τ+1 c n M}, γ µlog ɛ 1 τ+1. Then, take D := ω 0 1 } D,τ V and ρ := min M K, ρ τ+1 0, 37 where D,τ R n1 is defined just before.1. From the Diophantine inequality it follows that ω 0, so that γ µlog ɛ 1 τ+1 ω 0 ρ0. By the choice of D, the following standard measure estimate holds meas V \ D C meas V 38 where C depends on the C 1 norm of H 0. By the previous choices, when I D ρ, the unperturbed frequency map ω 0 = H 0 verifies, with α 1 = α = ᾱ := in fact: inf J D ρ, 0< k K ω 0 J k inf ω 0 I k I D, 0< k K sup ω 0 J ω 0 I k J D ρ,i D, 0< k K K τ, K ρ K M. 39 τ K τ 17 By abuse of notations, we use the same name for the variables, but, strictly speaking, they differ from the original variables by a quantity of O p, q in p, q, O p, q 3 in ϕ the actions I are the same.
PROPERLY DEGENERATE KAM THEORY 557 The smallness condition 3 is easily checked, provided E = µ is chosen small enough, because the choice of and γ implies } µ 7 c n Ks0 1 max ᾱd, C µ < 1. Condition Ks 0 6 is trivially satisfied. Thus, by Proposition 1, we find a real analytic symplectomorphism φ : Ī, ϕ, p, q W v, s I, ϕ, p, q W v0,s 0 v := v 0 / := ρ/, ɛ 0 /, s := s 0 /6 where W v0,s 0 := D ρ0 T n1 s 0 B ɛ0 v 0 = ρ 0, ɛ 0, and, by the choice of K in 36, H is transformed into 18 HĪ, ϕ, p, q = H φī, ϕ, p, q By 5, P v, s C and In view of 6, the transformation φ verifies I Ī, p p, q q C µlog ɛ 1 τ = H 0 Ī + µ NĪ, p, q + µe Ks/6 P Ī, ϕ, p, q = H 0 Ī + µ NĪ, p, q + µɛ5 P Ī, ϕ, p, q. 40 sup N P av C µlog ɛ 1 τ+1 D ρ/. 41, ϕ ϕ C µlog ɛ 1 τ+1. 4 Remark. The right hand sides of 41 and 4 can be made small as we please, provided µ and ɛ are small and is chosen suitably. The precise choice will be discussed below. Step Determination of the elliptic equilibrium for the secular system Since P av has a 4 non resonant and non degenerate elliptic equilibrium point at 0 and, in view of 41, N Pav is of order µlog ɛ 1 τ+1, using the Implicit Function Theorem and standard Cauchy estimates 19, for small values of this parameter, for any fixed Ī D ρ/, N also has a µlog ɛ 1 τ+1 close to 0 elliptic equilibrium point, which we call p 0 I, q 0 I. We can thus assume that p 0 I, q 0 I < ɛ 0 /4 for any I and consider a small neighborhood of radius 0 < ɛ < ɛ 0 /4 around p 0 I, q 0 I. We let φ : Ĩ, ϕ, p, q W ṽ, s Ī, ϕ, p, q W v, s ṽ := ρ/4, ɛ, s := s 0 /1 be the transformation having as generating function sĩ, p, ϕ, q = Ĩ ϕ + p + p 0 Ĩ q q 0 Ĩ, which acts as the identity on the Ĩ variables, while shifts the equilibrium point into the origin and suitably lifts the angles ϕ accordingly to Ī = Ĩ, p = p0 Ĩ + p, q = q0 Ĩ + q, ϕ = ϕ Ĩ p + p 0 Ĩ q q 0 Ĩ. 18 For simplicity of notation, we do not write explicitly the dependence on µ, ɛ,, that is, we write HĪ, ϕ, p, q, etc., in place of HĪ, ϕ, p, q; µ, ɛ,, etc. 19 See, e.g., [4, Lemma A.1].
558 LUIGI CHIERCHIA AND GABRIELLA PINZARI The transformation φ is close to the identity, since Ī = Ĩ and p p, q q C µlog ɛ 1 τ+1, ɛ log ɛ 1 τ+1 ϕ ϕ C max, µɛlog ɛ 1 3τ+ } 3. 43 Let us check, for example, the bound on ϕ ϕ as the other ones are immediate: If Dµ, ɛ, := Ĩ, p, q : Ĩ D ρɛ/4, p, q q 0 Ĩ B ɛ}, then, by Cauchy estimates, ϕĩ, p, q ϕ = Ĩ p + p 0 Ĩ q q 0 Ĩ q=q0+ q sup Ĩ p + p 0 Ĩ q q 0 Ĩ Dµ,ɛ, C sup Dµ,ɛ, p + p 0 Ĩ ρɛ/4 C ɛ + µlog ɛ 1 τ+1 ɛ log ɛ 1 τ+1 ɛ log ɛ 1 τ+1 C max, By construction, the transformation φ puts H into the form q q 0 Ĩ µɛlog ɛ 1 3τ+3 } 3. H := H φ = H 0 Ĩ + µñĩ, p, q + µɛ5 P Ĩ, ϕ, p, q, with Ñ := N φ, P := P φ. Observe that P ṽ, s C and Ñ has a 4 non resonant and non degenerate elliptic equilibrium point into the origin of the p, q coordinates. Step 3 Symplectic diagonalization of the secular system The standard diagonal form 8 can be achieved by a symplectic diagonalization as in [18]. In fact, by [18], one can find a symplectic map ˆφ : Î, ˆϕ, ˆp, ˆq Wˆv,ŝ Ĩ, ϕ, p, q W ṽ, s ˆv := ρ/8, ɛ/, ŝ := s 0 /4 which acts as the identity on the Î variables, is linear in the variables ˆp, ˆq and close to the identity in the sense p ˆp, q ˆq C µɛlog ɛ 1 τ+1, ϕ ˆϕ C µɛ log ɛ 1 3τ+ 3. 44 Such estimates are a consequence of the assumptions on P av compare the preliminary step above, the estimate Ñ = P av + O µɛ log ɛ 1 τ+1, for which Ñ is O µɛ log ɛ 1 τ+1 close to be diagonal and Cauchy estimates 0. Moreover, one 0 The generating function of this transformation is µɛ log ɛ 1 τ+1 close to the generating function Î ϕ + ˆp q of the identity map. Taking the derivatives and using Cauchy estimates, with a loss of analyticity Cɛ in ˆp, ˆq, and logɛ 1 τ+1 in the ˆϕ, we find 44.
PROPERLY DEGENERATE KAM THEORY 559 has that 1 ˆN Î, ˆp, ˆq := Ñ ˆφÎ, ˆp, ˆq = P 0 Î + ˆΩÎ ˆr + ˆR where ˆΩ Ω, ˆR C µɛlog ɛ 1 τ+1 45 and ˆR having a zero of order 3 for ˆp, ˆq = 0 and that ˆφ transforms H into Ĥ := H ˆφ = H 0 Î + µ ˆNÎ, ˆp, ˆq + µɛ5 ˆP Î, ˆϕ, ˆp, ˆq, ˆP := P ˆφ. Step 4 Birkhoff normal form of the secular part By Proposition and subsequent remark there exists a Birkhoff transformation ˇφ : Ǐ, ˇϕ, ˇp, ˇq Wˇv,š Î, ˆϕ, ˆp, ˆq Wˆv,ŝ ˇv := ρ/16, ɛ/4, š := s 0 /48 which acts as the identity on the Ǐ variables, is close to the identity as compare 31: ˆp ˇp, ˆq ˇq C µɛ log ɛ 1 τ+1, ˆϕ ˇϕ C µɛ3 log ɛ 1 3τ+ 3. 46 The previous estimates follow from the fact that, in 31, the coefficients c α,β of the non normal part of can be upper bounded by m 1 := µlog ɛ 1 τ+1, uniformly in I; α can be taken of order 1 in ɛ and µ; ρ 0 of O log ɛ 1. Furthermore, ˇφ puts ˆN τ+1 into Birkhoff normal form up to order 4, hence, transforms Ĥ into the form Ȟ := Ĥ ˇφ := H 0 Ǐ + µ P0 Ǐ + ˆΩǏ ř + 1 ř ˇβǏř + O ˇp, ˇq 5 +µɛ 5 ˆP ˇφ =: H 0 Ǐ + µňǐ, ř + µɛ5 ˇP Ǐ, ˇϕ, ˇp, ˇq, where ŇǏ, ř = P 0 Ǐ + ˆΩǏ ř + 1 ř ˇβǏř, ř i = ˇp i + ˇq i, ˇP ˇv,š C. Step 5 Global action angle variables for the full system We finally introduce a set of action angle variables using symplectic polar coordinates. Fix the real n dimensional annulus } Aɛ := J R n : č 1 ɛ 5/ < J i < č ɛ, 1 i n 47 where č 1 will be fixed later on so as to maximize the measure of preserved tori and ɛ small enough with respect to 1/č 1, while č is a constant depending only on the dimensions. Let D := D Aɛ, ρ := minč 1 ɛ 5/ /, ρ/16}, s := ŝ = s 0 /48, 48 where D is the set in 37. On D ρ T n s, let φ : J, ψ = J 1, J, ψ 1, ψ Ǐ, ˇϕ, ˇp, ˇq be defined by J 1 = Ǐ, ψ 1 = ˇϕ, ˇp i = J i cos ψ i, ˇq i = J i sin ψ i 1 i n. 49 1 ˆr i := ˆp i + ˆq i /. This is needed to avoid the singularity introduced by the polar coordinates. Notice that J i ɛ compared to p, q ɛ.
560 LUIGI CHIERCHIA AND GABRIELLA PINZARI For ɛ small enough, ˇp, ˇq B ɛ/4. The transformation φ puts Ȟ into the form HJ, ψ := Ȟ φ = H 0 J 1 + µ NJ + µɛ 5 P J, ψ, N := Ň φ = P 0 J 1 + ˆΩJ 1 J + 1 J ˇβJ 1 J, where P := ˆP φ From the above construction there follows that the transformation φ := φ φ ˆφ ˇφ φ : J, ψ I, ϕ, p, q 50 is well defined 3 and verifies I J 1 C µlog ɛ 1 τ ϕ ψ 1 C max µlog ɛ 1 τ+1, p i p 0 i J i cos ψ i C max q i q 0 i J i sin ψ i C max } ɛ log ɛ 1 τ+1 µɛlog ɛ, 1 3τ+3 3 } µlog ɛ 1 τ µɛlog ɛ, 1 τ+1 } µlog ɛ 1 τ µɛlog ɛ, 1 τ+1 Step 6 Construction of the Kolmogorov set and estimate of its measure Fix γ 1 and γ = µ, with γ 1, satisfying µ γ 1 and 17. We apply now the two scale KAM Theorem Proposition 3, with compare Step 5 above H = H, h = H 0 J 1 + µ NJ, f = µɛ 5 P J, ψ, D = D, ρ = ρ and s = s/5, s = 4 s/5. It is easy to check that, for small values of 4 µlog ɛ 1 τ+1, the frequency map ω µ := H 0 J 1 +µ NJ is a diffeomorphism of D ρ, with non singular hessian matrix H 0 J 1 + µ NJ. Then, we see that for a suitable constant C we can take M, ˆM,, M in Proposition 3 as follows: Then, M = C, ˆM = Cµ, M = Cµ 1, E = Cµɛ 5, M1 = C, M = Cµ 1. and recall also 37 ˆρ c min Finally, L Cµ 1, K C log ɛ 5 /γ 1 1 γ 1 log ɛ 5 /γ 1 1 τ+1, log ɛ 5 /γ 1 1 τ+1, ĉê C max ɛ 5 log γ1 ɛ 5 τ+1 max 1 γ 1 51 } log ɛ 1 τ+1, č 1ɛ 5/, ρ 0. 5, 1 }, ɛ5 log ɛ 1 τ+1, with a constant C not involving č 1. Then, from 17 it follows that 1, č 1 ɛ 5 ρ 0 }, 53 1 ĉê < C max, 1 γ č, ɛ 5} < 1 54 1 3 If is chosen as to satisfy the first inequality in 17, then, the right hand sides of 41, 4, 43, 44, 45 and 51 can all be bounded by 1/γ. Choosing γ big enough, the quantities involved are small as we please. 4 Such inequality is implied by γ µ logɛ 1 τ+1 <, which appears in 17.
PROPERLY DEGENERATE KAM THEORY 561 provided γ, č 1 > C and ɛ 5 < C 1. Finally, since the KAM condition ĉê < 1 is met, Proposition 3 holds in this case. In particular, for any ω in the set Ω := D γ1,µ,τ ω µ D, we find a real analytic embedding φ ω : T n T ω := φ ω T n Re D r T n with r Ĉ ˆρ C such that, on T ω, the H flow analytically conjugated to ϑ ϑ + ω t. We set T ω := φt ω, where φ is the symplectic transformation defined in 50. Using 51 and 35, the parametric equations of T ω may be written as I = J1 0 + J 1 with J µlog ɛ 1 τ 1 C + µˆρ p i p 0 i + q i qi 0 = Ji 0 + J i with č 1 ɛ 5/ < Ji 0 < č ɛ, and J µ log ɛ 1 τ i C + µ ɛ log ɛ 1 4τ+ 4 + ˆρ, where J1 0, J 0 is the ω µ pre image of ω, ˆρ is much smaller than č 1 ɛ 5/ compare 5; finally, by 51, p 0, q 0 C µlog ɛ 1 τ+1. It remains to estimate the measure of the Kolmogorov set K := φk = T ω, where K := T ω. ω Ω ω Ω namely, 18. Let Dγ 1,µ,τ := ωµ 1 D γ1,µ,τ D, where D is the set in 48, with D as in 37 and D γ1,µ,τ is defined just before.1. Then, by 34 and because φ is volume preserving, we have meas K = meas K meas Re D r T n 55 C meas D \ Dγ 1,µ,τ T n + meas Re D r \ D T n. Now, let V := V B n ĉ ɛ, where B n č ɛ denotes the open set J i < č ɛ }. Observe that D V; define P ɛ := V T n1 p i + q i < ɛ } compare 5. Then, by the estimate 38 and the definition 47 of Aɛ, meas D r T n meas D T n = meas D meas Aɛ meas T n 1 C Cɛ n/ meas V T n Similarly, denoting for short B := B n č ɛ, one has that 5 = 1 C Cɛ n/ meas P č ɛ 56 meas Re V r \ V C meas V, meas B r \ B meas B C \ B C ɛ meas B meas V \ D C + ɛ n/ meas V. 5 Recall that r < C.
56 LUIGI CHIERCHIA AND GABRIELLA PINZARI Thus, meas Re D r \ D T n meas Re V r \ D T n 57 meas Re V r \ V T n + meas V \ D T n C ɛ + + ɛn/ meas V T n = C ɛ + + ɛn/ meas P č. ɛ Finally, the frequency map ω µ := J1 H 0 + µ N, µ J N is a diffeomorphism of n a ρ-neighborhood of D B č ɛ. Note that ω µ as a function of J 1 is defined on D ρ and as a function of J is a polynomial; notice also that B n č ɛ is just the full closed ball around the annulus Aɛ compare 47. Then, the measure of the set D \ Dγ 1,µ,τ does not exceed the measure of the γ 1, µ resonant set for ω µ in the set D B n č ɛ. Such set of resonant points may be estimated by the following technical Lemma, whose proof is deferred to Appendix C. Lemma 3.1. Let n 1, n N, τ > n := n 1 + n, γ 1, γ > 0, 0 < ˆr < 1, D be a compact set. Let n ω = ω 1, ω : D B ˆr Ω R n1 R n be a function which can be extended to a diffeomorphism on an open neighborhood of D B n ˆr, with ω of the form ω I 1, I = ω 0 I 1 + βi 1 I where I 1 βi 1 is a n n matrix, non singular on D. Let R 1 > max ω 1, a > max β, cn, τ := 1 D B n D k τ, ˆr 0 k Z n and denote Then, where R γ1,γ,τ := I = I 1, I D } B n ˆr : ωi / D γ1,γ,τ. γ meas R γ1,γ,τ c 1 γ 1 + c ˆr c 1 := max D B n ˆr c := max D meas D B n ˆr ω 1 n Rn1 1 1 an cn, τ p meas D β 1 n a n 1 cn, τ for a suitable integer p depending on D and ω 1. 58 By Lemma 3.1 with γ 1 as in Step 6, D as in 37 and ˆr = č ɛ, ω = ω resc := J1 H 0 + µ N, J N, γ =, a = R 1 = C, we see that meas D \ D γ 1,µ,τ T n meas R γ1,,τ T n maxc 1, c č }γ 1 + ɛ meas D Bče Tn maxc 1, c č }γ 1 + ɛ meas P č ɛ 59
PROPERLY DEGENERATE KAM THEORY 563 with c i independent of ɛ and µ. Then, in view of 55 59, 18 follows, with ɛ replaced by č ɛ. The proof of Theorem 1.4 is finished. 4. Proof of Theorem 1.3. Since most of the arguments are similar but simpler than the ones used in the proof of Theorem 1., we will skip most technical details. We can write P av p, q; I = NI, r + P av p, q; I, where NI, r := P 0 I + and, for a suitable C > 0, Ω i Ir i + 1 β ij Ir i r j 1 i n 1 i,j n sup B n ɛ V ρ0 P av Cɛ 5, 0 < ɛ < ɛ 0. 60 Step 1 Fix τ > n 1, 0 < ɛ < ɛ 0, and ɛ 6 30 τ+1 µlog ɛ 1 µ < logɛ 1 τ+1, τ+1 s 0 ɛ 5/. 61 In place of Proposition 1, we use Lemma A.1 below, where we take r p = r q = ɛ 0, ρ = ρ with ρ as in 37, ρ p = ρ q = ɛ 0 /4, σ = s 0 /6 and the remaining quantities as in Step 1 of the proof of Theorem 1.4, namely, l 1 = n 1, l = 0, m = n h = H 0, g 0, K τ, f = µp, A = D, r = ρ, as in 37 B = B = 0}, s = s 0, α 1 = α = ᾱ = where K as in 36 and Λ = 0}. With such choice, the check of the non resonance assumption 64 for ω 0 = H 0 in D ρ is the same as in Step 1 of the proof of Theorem 1.4 and the smallness condition 65 is implied by 61. Then, there exists a symplectic transformation φ such that H φ = H 0 + g + + f + as in Lemma A.1. Since g + coincides with µp av, on the domain Wṽ, s recall the definition of W v,s just above the 40, where ṽ = ρ/, ɛ 0 / and s = s 0 /3, we find HĪ, ϕ, p, q := H φī, ϕ, p, q 6 = H 0 Ī + µp av p, q; Ī + P Ī, ϕ, p, q = H 0 Ī + µnī, r + µ P av p, q; Ī + P Ī, ϕ, p, q. By 69 below, the transformation φ satisfies the estimates 4. Furthermore, by 61, the choice of K in 36 and 68 below, the function P in 6 satisfies P v, s Cµ τ+1 µ K max, µ K τ e Ks 0/6 } Cµɛ 5. 63 By such estimate and 60, the perturbation P := µ P av + P, on the smaller domain W v, s, where v = ρ/, ɛ/, s = s, is bounded by Cµɛ 5. Step and conclusion At this point, we proceed as in Steps 5 and 6 of Theorem 1.4, with W v, s, N and P replacing, respectively, W v, s, N and µɛ 5 P. Now, choose γ big enough so that the KAM condition 53, 54 is satisfied, and fix γ, satisfying µ γ 1 and last two lines in 17. Then, we can find a set of invariant tori K D r T n1 č 1 ɛ 5/ < p i + q i < č ɛ, i } r P č ɛ r
564 LUIGI CHIERCHIA AND GABRIELLA PINZARI with r < C satisfying the measure estimate meas P č \ K ɛ meas P č ɛ r \ K C + γ 1 + ɛ + ɛn/ meas P č ɛ. Finally, taking, as in the proof of Theorem 1., γ 1, as in 0, and choosing as the value in the right hand side of 61, the theorem is proved with K := K Pčɛ and č ɛ replacing ɛ. Appendix A. Averaging theory Proposition 1. In this appendix we generalize Proposition A.1 in [4] to a two frequency scale, as needed in Appendix B below. Proposition A.1 in [4] is based on the application of an iterative lemma. The following lemma is the easy generalization of the iterative lemma Lemma A.5 in [4] suitable for our purpose. Lemma A.1. Let 0 < α α 1, l = l 1 + l with l i N and let Λ be a sub-lattice of Z l. Let g = k Λ g k ue ik ϕ and Hu, ϕ = hi + gu, ϕ + fu, ϕ be real analytic on W v,s := A r B rp B r q T l s, where A B B R l1 R l R m R m and v = r, r p, r q. Let ρ < r/, ρ p < r p /, ρ q < r q /, σ < s/, ν := ρ, ρ p, ρ q. Suppose that k = k 1, k Z l1 Z l and that α1 if k ωi k 1 0 k Z l1 Z l, k / Λ, k α if k 1 = 0 K, I A r. 64 Assume also that the following smallness condition holds: f v,s < α δ c m, where δ := minρσ, ρ p ρ q }, c m := e1 + em/. 65 Then,there exists a real analytic symplectic transformation such that φ : I, ϕ, p, q W v ν,s σ I, ϕ, p, q W v,σ, 66 H + := H φ = h + g + + f +, 67 with g + g = Π Λ T Kf and f + v ν,s σ 1 c 1 [ ] m α δ f cm v,s α δ f v,s + g, H φ } v,s + e Kσ f v,s. 68 Furthermore, the following uniform bounds hold: } max α 1 σ I 1 I 1, α σ I I, α ρ ϕ ϕ, α ρ q p p, α ρ p q q f v,s. Proof. Assumptions 64, 65 allow to apply the iterative lemma [4, Lemma A.5], with n = l, D = A, E = B, F = B, K = K, α = α, so as to find an analytic transformation Φ := φ verifying 66 68 and the bounds on I I, p p, q q into 69. In order to prove the bound on I 1 I 1, we recall that such transformation is obtained 6 as the time one map associated to the Hamiltonian 69 6 Compare [4, A.14].
PROPERLY DEGENERATE KAM THEORY 565 flow of Then, one can split H φ as with H φ u, ϕ = H 1 φ H φ u, ϕ = k K, k Λ f k u ik ωi eik ϕ. 70 u, ϕ + H φ u, ϕ, where ϕ = ϕ 1, ϕ T l1 T l H 1 φ u, ϕ := H φ u, ϕ := k K, k Λ,k 1 0 k K, 0,k Λ Using 64, one finds that H 1 φ, H φ verify H 1 φ v,s f v,s, H φ α 1 f k u ik ωi eik ϕ, f k u ik ω I eik ϕ. v,s f v,s α. 71 Since H φ is independent on ϕ 1, from the generating equations of φ, equation 71 and Cauchy estimates, the bounds for I 1 I 1 69 follow. Finally, when f does not depend on p, q one can simply take m = 0. We now may proceed to sketch the proof of the Averaging theory in, i.e., Proposition 1. By the same considerations of footnote 0 of [4], we can limit ourselves to the case e Ks/6 3c m E/α d. As in [4], we apply once Lemma A.1 with ν = ν 0 := v/8; σ = σ 0 := s/6, thus, constructing a transformation Φ 0 : W1 := W v1, s+s 1 W v, s+s, with v 1 = 3/4v, s 1 = /3s which transforms H = h + f into H 1 = h + g 0 + f 1. Similarly to A.19 of [4], it follows that f 1 W1 E 6. By 69, one can replace A.0 of [4] with α 1 s I 1 1 I 1, α s I 1 I, α r q p 1 p, α r p q 1 q, α r ϕ 1 ϕ 8E. Next, one proceeds as in A1 A.6 of [4], with W i := W vi, s+s i replacing W i, α replacing α, E i := f i Wi replacing ɛ i, in order to prove 5. Finally, 6 follows by the same telescopic argument as in 7 [4], except for taking into account, as done above, the double scale 69 of the α i s. Appendix B. Two scale KAM theory proof of Proposition 3. The proof of Proposition 3 is based on the following iterative lemma. For the purpose of this proof, we replace τ with τ. Lemma B.1 Iterative Lemma. Under the same assumptions and notations of Proposition 3, for any 1 j N, there exists a domain D j R n, two positive numbers ρ j, s j and a real analytic and symplectic transformation Φ j on W j := D j ρj T n s+s j which conjugates H 0 := H to H j = H 0 Φ j = h j + f j and such that the following holds. Letting, for j = 0, D 0 := ω 1 0 D γ 1,γ,τ D, s 0 := s, ρ 0 := ρ, M 0 := M, ˆM0 := M, M0 := M, L 0 := L, E 0 := E, K 0 := K, 7 Compare [4]: last formula before Appendix B.
566 LUIGI CHIERCHIA AND GABRIELLA PINZARI ˆρ 0 := ˆρ, Ê 0 := Ê and, for j 1, s j := s j 1 /1, Mj := M j 1, ˆMj := ˆM j 1, L j := L j 1, ρ j := ˆρ j 1 /16, then E j := E j 1 L j 1Mj 1, K γ j := 6 Ej L j M 1 j log 1 s + 7 j γ 1 γ 1 ˆρ j := min 3M j K τ+1, j γ 3 ˆM j K τ+1 j, ρ j }, Ê j = E jl j ˆρ j 73 i D j D j 1 ˆρj 1/16; the frequency map ω j := h j is a diffeomorphism of D j ρj such that ω j D j = ω j 1 D j 1 ; the map verifies î j = î j1, î j : I D j 1 ω 1 j ω j 1 I D j sup î j1 id 5n M 1 D j 1 M Êj 1 ˆρ j 1 5nÊj 1 ˆρ j 1, sup î j id 5n M D j 1 M Êj 1 ˆρ j 1 5nÊj 1 ˆρ j 1 74 Lî j id 6 nêj 1 75 ii the perturbation f j has sup Fourier norm on W j f j Wj E j 76 iii the real analytic symplectomorphism Φ j is obtained as Φ j = Ψ 1 Ψ j, where verifies Ψ k : I k, ϕ k W k I k 1, ϕ k 1 W k 1 sup I k 1,1 I k, ϕ k I k,1 3 ˆM D k ρk T n s+s 4 M Êk 1 ˆρ k 1 k sup D k ρk T n s+s k I k 1, I k, ϕ k I k, 3 4Êk 1 ˆρ k 1 sup ϕ k 1 I k, ϕ k ϕ k 3 D k ρk T n s+s 4Êk 1s k 1 77 k and the rescaled dimensionless map ˇΨ k id := 1ˆρ,s Ψ k 1 1 ˆρ,s id, has Lipschitz constant on Ďk ρk /ρ Ťn s+s k /s L ˇΨ 4 τ+1 k 1 k id 4n Êk 1 78 6 where id denotes the identity map, 1 d denotes the d d identity matrix, 1 ρ,σ := ρ 1 1 n, σ 1 1 n, Ďk := ˆρ 1 D k, Ť := R/π/sZ; iv for any j 1, Êj < Ê j 1.
PROPERLY DEGENERATE KAM THEORY 567 Remark 3. Lemma B.1 generalizes the inductive theorem of [3, p. 144]. In [3], the quantities E, γ 1, γ are estimated as µɛ 7, ɛ +a, µɛ +a, respectively 8. Indeed, our approach allows to have E µɛ 5 and hence, essentially, to replace assumption A with A, taking for γ 1, γ the smallest possible values compatibly with convergence, namely, γ 1 ɛ 5/, γ µɛ 5/ compare 0 above. Such smaller choice of γ 1, γ with respect to [3] is important in order to improve the density of the invariant set as in 18. Proof. The proof is based on Proposition 1 with m = 0, l 1 = n 1, l = n, B = B =. Notice that, by assumption and the choice of D 0, the following inequalities hold, for j 1 = 0 ĉêj 1 < 1 79 L j 1 max } Mj 1, M 1 1 j 1, ˆM j 1 80 ωd j 1 D γ1,γ,τ. 81 Let us assume inductively that, when j 1 0 79, 80 and 81 hold and that, for j 1 1, the Lemma holds with Φ j 1 = Ψ 1 Ψ j 1. In order to describe the j th step, for simplicity, we write ρ, ˆρ, s, M, ˆM, M, L, K, E, Ê, D, H, h, f, ω = ω 1, ω for ρ j 1, ˆρ j 1, s j 1,etc., and ρ +, ˆρ +, s +, etc., for ρ j, ˆρ j, s j, etc., the corresponding initial quantities will be called, as in the statement, ρ 0, ˆρ 0, s 0, etc.,. By 81 and the choice of ˆρ equation 73, when 0 < k K and I Dˆρ the following non resonance inequalities hold which are checked as in 39 above γ 1 3K τ =: α 1 when k 1 0 ; ω 1 I k 1 + ω I k 8 γ 3K τ =: α when k 1 = 0 & k 0. The inequality Ks 6 is trivial by definition of K see 7 and also the smallness condition 3 is easily met, since in this case d = ˆρs and hence 7 c 0 K α ˆρ f D ˆρ T n s+s 6 c 0 E ˆM ˆρ ĉe < 1 having used α ˆMK ˆρ, L ˆM 1, 6 c 0 < ĉ and 79. Thus, by Proposition 1 with Λ = 0}, h = h,, g 0, f = f, Wˆρ,s = Dˆρ T n s+s, H may be conjugated to where, by 5 and the choice of K, f + D ˆρ/ T n s+s/6 H + := H Ψ + = h + + f + e Ks/6 E ELM The conjugation is realized by an analytic transformation γ 1 E = E +. 83 Ψ + : I +, ϕ + Dˆρ/ T n s+s/6 I, ϕ Dˆρ T n s+s. 8 With a as in 7 above.
568 LUIGI CHIERCHIA AND GABRIELLA PINZARI Furthermore, in view of 6 with α 1, α as in 8, of α 1 LK ˆρM/ ˆM, of α LK ˆρ, of Ks 6 and of L M 1, ˆM 1, we have Similarly, sup D ˆρ/ T n s+s/6 I 1 I +, ϕ + I +,1 3 ˆM Ê ˆρ. 84 4 M sup I I +, ϕ + I +, 3 ˆρ, sup ϕi +, ϕ + ϕ + 3 D ˆρ/ T n s+s/6 4Ê D ˆρ/ T n s+s/6 4Ês. 85 Lemma B.. The new frequency map ω + := h + is injective on Dˆρ/8 and maps Dˆρ/16 over ωd. The map î + = î +1, î + := ω 1 + ω D which assigns to a point I 0 D the ω + pre-image of ωi 0 in Dˆρ/16 satisfies sup î +1 id 5n M 1 E D ˆρ 5n ME ˆρ, sup î + id 5n M E D ˆρ 5n ME ˆρ Lî + id 6 n ME ˆρ. 86 Finally, the Jacobian matrix U + := h + is non singular on Dˆρ/8 and the following bounds hold M + := M sup U +, ˆM+ := ˆM sup Û+, 87 I D ˆρ/8 I D ˆρ/8 M + := M sup U+ 1, I D ˆρ/8 Mi+ := M i sup T i+, I D ˆρ/8 i = 1,. where U+ 1 T+1 =:. T + Postponing for the moment the proof of this Lemma, we let ρ + := ˆρ/16, s + := s/1, and D + := î + D. By Lemma B., D + is a subset of Dˆρ/16 and hence D + ρ+ Dˆρ/8. 88 At this point, 76 follows from 83 and 77 from 84, 85. We are now ready to prove that Ê+ = E+L+ ˆρ Ê. Since + s + = s E+ L + M 1 + x ELM 1 and x + := 1 γ = where x := 89 1 8 γ 1 we have K + = 6 1 log x + = 1 s + s Thus, and γ 1 ˆρ + = min 3M + K+ τ+1, Ê + = E +L + ˆρ + E LM γ 1 3 log 36 log log x = 4K < 4K. s s γ 3 ˆM + K τ+1 +, ρ + = ˆρ 16 L ˆρ 44τ+1 τ+1 E LM = 84 Ê γ 1 } ˆρ 4 τ+1 90,
PROPERLY DEGENERATE KAM THEORY 569 Now, using, in the last inequality, the bound E LM γ 1 = 1 9K τ+1 since K 6/s we find having used s 1/4. EL [γ 1 /3MK τ+1 ] 1 EL 9 6/s τ+1 ˆρ = 1 s 9 6 τ+1 Ê Ê + 8 9 4sτ+1 Ê < Ê 91 The estimate in 78 is a consequence of 84, 85, 88, 89, 90 and Cauchy estimates: L ˇΨ j id n sup D jρ j T n s+s j D ˇΨ id 3 n 4Êj 1 maxˆρ j 1 /ρ 0, s/s 0 } min3ˆρ j 1 /8ˆρ 0, s j 1 /1s 0 } n 3/41/1j 1 4 τ+1 j 1 j 1 Ê j 1 = 4n Êj 1. 1 6 3/8 4 τ+1 Equations 74, 75 follow from 86, 87 and the inequality L M at once. Finally, by the bounds 87, we see that L + = L is an upper bound for M +, M 1 ˆM 1 + ; 91 easily implies ĉe + < 1 and ω + D + = ωd D γ1,γ,τ by definition of D +. Take then Φ j := Φ j 1 Ψ j, where Ψ j = Ψ +, Φ 0 = id. Having also checked inequalities 79, 80, 81 after the j th step, Lemma B.1 is proved. Proof of Lemma B. Since h + = h + g and, by 5, +, sup g sup g f + sup f 5 D ˆρ/ D ˆρ/ D ˆρ/ 4 E, 9 where f denotes the average of f, by Cauchy inequality, hence, sup g = sup g sup D g 3 ˆρ/8 E D ˆρ/4 D ˆρ/4 ˆρ/8 ˆρ/8 Ḙ 7 ρ ; 93 sup U + = sup h + = sup h + g D ˆρ/4 D ˆρ/4 D ˆρ/4 sup h + sup g M + 7 Ḙ D ˆρ/4 D ˆρ/4 ρ M. The proof of sup D ˆρ/4 Û+ ˆM is similar. Using sup g h 1 sup g sup U 1 7 E M D ˆρ/4 D ˆρ/4 D ˆρ/4 ˆρ 1, 94 and this implies 1 + g h 1 1, so, M +1 M 1, M+ M, M+ M. 95 Injectivity of ω + = ω + g on Dˆρ/8 follows from the non singularity of h + over Dˆρ/4 and the observation that two points I +, I + Dˆρ/8 with the same image would