Quasi-Töplitz Functions in KAM Theorem

arXiv:1102.1066v2 [math.AP] 8 Feb 2011

Quasi-T¨oplitz Functions in KAM Theorem Xindong Xu, M. Procesi Dipartimento di Matematica e Applicazioni “Renato Caccioppoli” Universit` a degli Studi di Napoli ”Federico II” 80126, Italy Email: [email protected] , [email protected]

Abstract We define and describe the class of Quasi-T¨ oplitz functions. We then prove an abstract KAM theorem where the perturbation is in this class. We apply this theorem to a Non-Linear-Scr¨odinger equation on the torus Td , thus proving existence and stability of quasi–periodic solutions and recovering the results of [10]. With respect to that paper we consider only the NLS which preserves the total Momentum and exploit this conserved quantity in order to simplify our treatment.

1

Introduction

In this paper, we study a model NLS with external parameters on the torus Td . The purpose is to apply KAM theory and prove existence and stability of quasi–periodic solutions. We focus on the equation iut + △u + Mξ u + f (|u|2 )u = 0,

x ∈ Td , t ∈ R,

(1.1)

where f (y) is a real analytic function with f (0) = 0, while Mξ is a Fourier multiplier, namely a linear operator which commutes with the Laplacian and whose role is to introduce b parameters in order to guarantee that equation (1.1) linearized at u = 0 admits q a quasi–periodic solution with b 1 ihn,xi e , n ∈ Zd 4π 2 {n(1) , · · · , n(b) } with n(i)

frequencies. More precisely, let φn (x) =

be the standard

Fourier basis, we choose a finite set ∈ Zd and define Mξ so that the eigenvalues of the operator △ + Mξ are  ωj = |n(j) |2 + ξj , 1≤j≤b (1.2) Ωn = |n|2 , n∈ / {n(1) , · · · , n(b) } Key words: Schr¨ odinger equation, KAM Tori, Quasi T¨ oplitz Supported by the European Research Council under FP7, project ”Hamiltonian PDEs

1

Equation (1.1) is a well known model for the natural NLS, in which the Fourier multiplier is substituted by a multiplicative potential V . Existence and stability of quasi–periodic solutions via a KAM algorithm was proved in [10] for the more general case where f (y) is substituted with f (y, x), x ∈ Td . With respect to that paper Rwe strongly exploit the fact that equation has ¯∇u as an integral of motion, this induces the total momentum M = Td u some significant simplifications which we think are interesting. Our dynamic result is Theorem 1 There exists a positive-measure Cantor set C such that for any ξ = (ξ1 , · · · , ξb ) ∈ C, the above nonlinear Schr¨ odinger equation (1.1) admits a linearly stable small–amplitude quasi-periodic solution. Before giving a more detailed comparison let us make a brief excursus on the literature on quasi–periodic solutions for PDEs on Td and on the general strategy of a KAM algorithm. The existence of quasi–periodic solutions for equation (1.1) (as well as for the non–linear wave equation) was first proved by Bourgain, see [3] and [4], by applying a combination of Lyapunov-Schmidt reduction and Nash–Moser generalized implicit function theorem, in order to solve the small divisor problem. This method is very flexible and may be effectively applied in various contexts, for instance in the case where f (y) has only finite regularity, see [6] and [7]. As a drawback this method only establishes existence of the solutions but not the existence of a stable normal form close to them. In order to achieve this stronger result it is natural to extend to (1.1) on Td by now classical KAM techniques, which were developed to study equation (1.1) with Dirichlet boundary conditions on the segment [0, π]. A fundamental hypothesis in the aforementioned algorithms is that the eigenvalues Ωn are simple, and this is clearly not satisfied already in the case of equation (1.1) on T1 , where the eigenvalues are double. We mention that this hypothesis was weakened for the non–linear wave equation in [8] by only requiring that the eigenvalues have finite and uniformly bounded multiplicity. Their method however does not extend trivially to the NLS on T1 and surely may not be applied to the NLS in higher dimension, where the multiplicity of Ωn is of (d−1)/2 order Ωn . The first result on KAM theory on the torus Td was given in [12] for the non–local NLS: iut + △u + Mξ u + f (|Ψs (u)|2 )Ψs (u) = 0,

x ∈ Td , t ∈ R,

where Ψs is a linear operator, diagonal in the Fourier basis and such that Ψs (φn ) = |n|−2s φn for some s > 0. The key points of that paper are: 1. 2

the use of the conservation of the total momentum to avoid the problems arising from the multiplicity of the Ωn and 2. the fact that the presence of the non–local operator Ψs simplifies the proof of the Melnikov non–resonance conditions throughout the KAM algorithm. Let us briefly describe the general strategy in the KAM algorithm for equation (1.1). P qn φn (x) and inWe expand the solution in Fourier series as u = d n∈Z p troduce standard action-angle coordinate: qnj = Ij eiθj , j = 1, · · · , b; qn = zn , n 6= {n(1) , · · · , n(b) }. We get X X H= ω j Ij + Ωn |zn |2 +P (I, θ, z, z¯), Zd1 := Zd \{n1 , . . . , nb }. (1.3) 1≤j≤b

n∈Zd1

It is easily seen that H andP hence P preserve the total momentum (see 2.5 below) moreover P (and (Ωm − |m|2 )zm z¯m ) are T¨ oplitz/anti-T¨ oplitz σ,σ′ (σm + σ ′ n, z, z σ ∂ σ′ P = H ¯ ). functions, namely the Hessian matrix ∂zm zn Informally speaking the KAM algorithm consists in constructing a convergent sequence of symplectic transformations such that X (ν) X 2 Φν H = Hν = ωj (ξ)Ij + ¯), (1.4) Ω(ν) n (ξ)|qn | + Pν (ξ, I, θ, z, z 1≤j≤b

n∈Zd1

where Pν → 0 in some appropriate norm. The symplectic transformation is well defined for all ξ which satisfy the Melnikov non–resonance conditions: ′

′

|hω (ν ) , ki + Ω(ν ) · l| ≥ γν ′ (1 + |k|)−τ , d

(1.5)

for all ν ′ ≤ ν and ∀k ∈ Zb , l ∈ ZZ1 such that (k, l) 6= (0, 0) , |l| ≤ 2. Here τ is an appropriate constant, while γν is a sequence of positive numbers. (ν ′ ) (ν ′ ) With this conditions in mind it is clear that a degeneracy Ωn = Ωm poses problems since the left hand side in (1.5) is identically zero for h = 0, l = em − en (em with m ∈ Zd1 is the standard basis vector). To avoid this problem we use the fact that all the Hν have M as constant of motion. This in turn implies that some of the Fourier coefficients of Pν are identically zero so (1.5) need to be imposed only on those k, l such that Pbthat the conditions P ml n k + d m = 0. Then, in our example, k = 0 automatically i i i=1 m∈Z1 implies n = m. This is the key argument used in [12]. However, once that one has proved that the left hand side of (1.5) is never identically zero, one still has to show that the quantitative bounds of (1.5) may be imposed on some positive measure set of parameters ξ. This is an easy task when 3

|l| = 0, 1 or l = em +en but may pose serious problems in the case l = em −en where the non–resonance condition is ′

′

′

(ν ) (1.6) |hω (ν ) , ki + Ωm − Ωn(ν ) | ≥ γν ′ (1 + |k|)−τ , ∀k ∈ Zb , n, m ∈ Zd1 P where n − m = bi=1 ni ki . Indeed in this case for every fixed value of k one should in principle impose infinitely many conditions, since the momentum conservation only fixes n − m. In [12], the presence of Ψs implies that (ν) ε s τ the variation of Ω(ν) is negligible. Ωm − |m|2 ≈ |m| s so that if |m| > c|k| This implies in turn that one has to impose only finitely many conditions for each k. In the case of equation (1.1) however s = 0, so that this argument may not be applied. To show that it is still true that one may impose the non resonance conditions by verifying only a finite number of bounds for each k (ν) one needs some control on Ωm − |m|2 , for |m| large, throughout the KAM (ν) algorithm. The ideal setting is when Ωm − |m|2 is m–independent. This holds true for the first step of the KAM algorithm due to the fact that P is a T¨ oplitz function. However it is easily seen that already P1 is not a T¨ oplitz function. Our strategy is to define a class of functions, the quasi–T¨ oplitz functions, and show that all the Pν belong to this set. Informally speaking a quasi–T¨ oplitz function is a function whose Hessian restricted to affine subspaces defined by equations with integer coefficients is well approximated by a T¨ oplitz matrix. To explain the use of this property we give a more detailed description of the control that we have on the variation of the normal frequencies. Let τ0 be a parameter such that

|hω, ki − h| > γ0 (1 + |k|)−τ0 ,

(k, h) 6= (0, 0)

may hold for a positive measure set of ξ. We introduce a parameter τ > τ0 , fixed in formula (2.6). (ν−1) The correction to the normal frequency Ωm is given (at step ν) by the diagonal terms in h∂zm ∂z¯m Pν−1 |z=¯z=0 i. The quasi-T¨ oplitz property (see Definition 2.6 and Formula (4.10)) states that for all K large enough (K ≥ Kν , the ultraviolet cut–off at step ν) and for all |m| > K τ there exists a parameter τ0 < τ1 < τ /4d such that ˜ ¯ m K −4dτ1 , Ωm = |m|2 + Ω[m] +Ω ˜ assumes at most K 3dτ1 different values, while Ω ¯ m is where the function Ω ˜ on m is determined by the notion of standard cut, bounded. The value of Ω see Definition 2.3 and Lemma 2.2, which assigns to each point |m| > K τ a 4

˜ is constant. We portion of an affine subspace (depending on K) on which Ω 3dτ 1 show that such subspaces are finite (bounded by K ). Finally we show that one may verify all the conditions (1.6) by imposing them only for one point for each affine subspace. Eliasson and Kuksin in [10] consider an NLS equation which does not have M as a constant of motion. This implies that some of the Melnikov non–resonance conditions (1.6) may not be imposed, and terms like P0,0,n,m with |n| = |m| are kept, at each step of the KAM algorithm they thus obtain a more complicated normal form, which then must be reduced to the standard one through a linear symplectic change of variables. To deal with the variation of the normal frequencies they introduce a property, T¨ oplitz-Lipschitz, which can be preserved by KAM iteration. In the definition of T¨ oplitz-Lipschitz, if M is the Hessian matrix corresponding to an analytic function, they require that for any a, b, c ∈ Z d , the limit a±tc M (±, c) := limt→∞ Mb+tc exists; and the limit matrix M (±, c) still required to satisfy this condition in a lower dimension space Z d−1 . The speed tends to the limit is controlled by 1t . Whats more, except a finite set, they cover any neighborhood {|a − b| ≤ N } ⊂ Z d × Z d of diagonal by finite Lipschitz domain. This reduces the infinitely many second Melnikov conditions to a finite number. In [13] an understanding of this property in T2 is given.

2 2.1

Relevant notations and definitions Function spaces and norms

We start by introducing some notations. We fix b vectors {n(1) , · · · , n(b) } in Zd called the tangential sites. We denote by Zd1 := Zd \ {n(1) , · · · , n(b) } the complement, called the normal sites. Let z = (· · · , zn , · · ·)n∈Zd , and its 1 complex conjugate z¯ = (· · · , z¯n , · · ·)n∈Zd . We introduce the weighted norm 1 X kzkρ = |zn |2 e2|n|ρ |n|d+1 , n∈Zd1

q

where |n| = n21 + n22 + · · · + n2d , n = (n1 , n2 , · · · , nd ) and ρ > 0. We denote by ℓρ the Hilbert space of lists {wj = (zj , z¯j )}j∈Zd with kzkρ < ∞. 1 We consider the real torus Tb := Rb /Zb as naturally contained in the space Cb /Zb × ℓρ as the subset where I = z = z¯ = 0. We then consider in this space the neighborhood of Tb : D(r, s) := {(θ, I, z, z¯) : |Im θ| < s, |I| < r 2 , kzkρ < r, k¯ z kρ < r}, 5

where | · | denotes the sup-norm of complex vectors. Denote by O an open and bounded parameter set in Rb and let D = maxξ,η∈O |ξ − η|. We consider functions F (I, θ, z; ξ) : D(r, s) × O → C analytic in I, θ, z 1 in ξ. We expand in Taylor–Fourier series as: and of class CW X F (θ, I, z, z¯; ξ) = Flkαβ (ξ)I l eihk,θiz α z¯β , (2.1) l,k,α,β

1 (in the sense of Whitney), where the coefficients Flkαβ (ξ) are of class CW the vectors α ≡ (· · · , αn , · · ·)n∈Zd , β ≡ (· · · , βn , · · ·)n∈Zd have finitely many 1 1 Q non-zero components αn , βn ∈ N, z α z¯β denotes n znαn z¯nβn and finally h·, ·i is the standard inner product in Cb . If S is a set of monomials in Ij , eiθj , zm , z¯n , we define the projection operator ΠS which to a given analytic function F associates the part of the series only relative to the monomials in S. The analiticity of the functions implies total convergence of the TaylorFourier series with respect to the following weighted norm of F : X kF kr,s = kF kD(r,s),O ≡ sup |Fklαβ |O r 2|l| e|k|s |z α ||¯ z β |, (2.2) kzkρ j, with ℓ obtaining v ∈ BK 1 j h a new presentation. Again we deduce by minimality in the lexicographical order, that |(v, x)| ≥ |ph | ≥ |pj |. Given an affine subspace A := {x |vi · x = pi ,

i = 1, . . . , ℓ} by the

K

notation A→[vi ; pi ]ℓ we mean that the given presentation is K optimal. Remark 2.2 For fixed K, ℓ, p the number of affine spaces in HK of codimension ℓ and such that |pℓ | = p is bounded by (CK)ℓd (2p)ℓ−1 .

2.2.2

Cuts

We are particularly interested in K-optimal presentations of points. Given K a point m we write m→[vi ; pi ] dropping the index ℓ which for a point is always ℓ = d. We need to analyze certain cuts, by this we mean a pair (ℓ, τ1 ) where 0 ≤ ℓ ≤ d and τ1 a suitable positive number. Recall τ = (4d)(d+1) (τ0 + 1), τ0 > max(d + b, 12). Set by convention p0 = 0 and pd+1 = ∞. Definition 2.3 We say that, a pair (ℓ, τ1 ) where 0 ≤ ℓ ≤ d and τ0 ≤ K

τ1 ≤ τ /4d is a compatible cut for the point m→[vi ; pi ], with parameters 1 τ1 4dτ1 . 2 ≤ λ, µ ≤ 4, if ℓ is such that |pℓ | < µK , |pℓ+1 | > λK 8

Notice that once we have fixed K, τ1 , λ, µ, for any given m ∈ Zd1 there is at most one choice of ℓ such that m has a (ℓ, τ1 ) cut with parameters λ, µ. K

Of course, if (ℓ, τ1 ) is a compatible cut for the point m→[vi ; pi ], with parameters 12 ≤ λ′ , µ′ ≤ 4 it is also so for parameters λ, µ with λ ≤ λ′ , µ′ ≤ µ. In particular we are interested in the extremal case where µ = 1/2, λ = 4, K

in this case we say that (ℓ, τ1 ) is a standard cut for the point m→[vi ; pi ]. The following construction will be useful: we divide 1 1 Si Si+1 [4K 4dτ0 , K τ /4d ) = ∪d−1 ] ∪ [K sd , K τ /4d ) i=1 [K , K 2 2 by setting K S1 = 4K 4dτ0 and defining recursively K Si+1 = 4 · 24d K 4dSi ,

i = 1, . . . d − 1.

By definition we get Pd−1

2K Sd = 8

i=0

(4d)i

dτ 0

K (4d)

d−1 +(4d)d τ 0

≤ K d(4d)

≤ K τ /4d

since K > 8 and τ ≥ (4d)d+1 (τ0 + 1) . Lemma 2.2 Each point m ∈ Zd1 has a standard compatible cut (ℓ, τ1 ) for some 0 ≤ ℓ ≤ d and τ0 ≤ τ1 ≤ τ /4d. If |m| ≥ K τ , then ℓ < d. K

Proof: Let m→[vi ; pi ]. If |pd | < 12 K τ /4d then we set ℓ = d and τ1 = τ /4d. If |p1 | > 4K 4dτ0 then we set ℓ = 0 and τ1 = τ0 . Otherwise if |p1 | < 4K 4dτ0 and |pd | > 12 K τ /4d then at least one of the d − 1 intervals (K Si , K Si+1 ) with i = 1, . . . , d − 1 does not contain any element of the ordered list {|p2 |, . . . , |pd−1 |}. The parameters (ℓ, τ1 ) are fixed by setting 1 τ1 = K S¯ı where ¯ı is the smallest index such that the interval (K Si , K Si+1 ) 2K does not contain any points of the list {|p2 |, . . . , |pd−1 |}; finally we choose ℓ accordingly so that |pℓ | < K S¯ı = 12 K τ1 and |pℓ+1 | > K S¯ı+1 = 4K 4dτ1 . τ If |pd | ≤ K 4d , by Cramers rule |m| = |V −1 p| < 4K τ /4d Kd−1 < K τ . Remark 2.3 The purpose of defining a cut (ℓ, τ1 ) is to separate the numbers pi into small and large. The number τ1 gives a quantitative meaning to this statement. Let [vi ; pi ]ℓ ∈ HK be a K–optimal presentation of an affine space of τ codimension 1 ≤ ℓ < d with |pℓ | ≤ 4K 4d .

9

Definition 2.4 The set: n

x ∈ [vi , pi ]ℓ |

[vi ; pi ]gℓ :=

(2.8)

a |x| > 4K τ , |(v, x)| > 4 max(K 4dτ0 , 24d |pℓ |4d ), ∀v ∈ BK \ hvi iℓ

o

will be called the K−good portion of the subspace [vi ; pi ]ℓ . Remark 2.4 For all 12 ≤ λ, µ ≤ 4 and for each point m ∈ [vi ; pi ]gℓ there exists τ1 such that m has [vi ; pi ]ℓ as a (τ1 , ℓ) cut with parameters (λ, µ). Indeed it is sufficient to choose τ1 = τ0 if K0 ≤ |pℓ and µK1τ = |pℓ | otherwise. Lemma 2.3 Consider m, r ∈ Zd1 such that |m + αr| < 5K 3 with α = ±1. K

K

Let m→[vi ; pi ] and r →[wi ; qi ]. Suppose that (ℓ, τ1 ) is a compatible cut for m with the parameters µ′ , λ′ . Then: (i) (ℓ, τ1 ) is a compatible cut for the point r, for all the parameters 12 ≤ λ < λ′ , µ′ < µ ≤ 4 such that (µ − µ′ )K τ1 , (λ′ − λ)K 4dτ1 > 5K 4 . (ii) If these conditions are satisfied we have hw1 , . . . , wℓ i = hv1 , . . . , vℓ i. Proof: (i) Assume that (µ − µ′ )K τ1 , (λ′ − λ)K 4dτ1 > 5K 4 . Write α(vi , r) = (vi , m + αr) − pi . For i ≤ ℓ: |(vi , r)| ≤ |pi | + |vi ||m + αr| < µ′ K τ1 + 5K 4 ≤ µK τ1 .

(2.9)

a \ From Formula (2.7) by the definition of K–optimal, for all v ∈ BK hv1 , . . . , vℓ i one has

|(v, r)| = |(v, m)−(v, αr+m)| ≥ |pℓ+1 |−|v||m+αr| ≥ λ′ K 4dτ1 −5K 4 ≥ λK 4dτ1 . (2.10) This proves (i). (ii) By induction on i we show that |qi | < µK τ1 and wi ∈ hv1 , . . . , vℓ i a \ hv , . . . , v i we have |(v, m)| ≥ |p for all i ≤ ℓ. By (2.10) if v ∈ BK 1 ℓ ℓ+1 | > 4dτ 1 λK . For 0 ≤ i < ℓ, suppose hw1 , . . . , wi i ⊂ hv1 , . . . , vℓ i. Since vi are independent, there exist h ≤ ℓ such that vh ∈ / hw1 , . . . , wi i. However by (2.9) |qi+1 | ≤ |(vh , r)| ≤ µK τ1 . Again, by (2.10), this also implies wi+1 ∈ hv1 , . . . , vℓ i. Since the wi are linearly independent (and so are the vi ), clearly hv1 , . . . , vℓ i = a \ hv , . . . , v i for s > j. This implies that |q hw1 , . . . , wℓ i, so ws ∈ BK 1 j+1 | > ℓ 4dτ 1 λK . 10

Remark 2.5 With the above lemma we are stating that if m has a (τ1 , ℓ) cut with parameters λ′ , µ′ then, for all choices of λ < λ′ , µ′ < µ, there exists a neighborhood B of m such that all points r ∈ N have a (τ1 , ℓ) cut with parameters λ, µ. The radius of B is determined by the difference in K K the parameters. In the same way if both m→[vi ; pi ] and r →[wi ; qi ] with |m + σr| < 5K 3 , have a (τ1 , ℓ) cut with parameters λ, µ then hvi iℓ = hwi iℓ .

2.3

Quasi–T¨ oplitz functions

Definition 2.5 Given K, λ, µ, τ1 such that 1/2 < λ, µ < 4, τ0 ≤ τ1 ≤ τ /4d and 4K 3 < 12 K τ we say that a monomial σ σ ei(k,θ) I l z α z¯β zm zn

′

is (K, λ, µ, τ1 )–bilinear with the cut [vi ; pi ]ℓ if it satisfies momentum conservation (2.5) i.e. σm + σ ′ n = −π(k, α, β), X |k| < K , min(|n|, |m|) > λK τ , |j|(αj + βj ) < µK 3 . (2.11) j

and moreover there exists 0 ≤ ℓ < d such that both n, m have a (ℓ, τ1 ) K

K

cut with parameters λ, µ. By convention if m→[vi ; pi ] and n→[wi ; qi ] we suppose that (p1 , . . . , pℓ , v1 , . . . , vℓ )  (q1 , . . . , qℓ , w1 , . . . , wℓ ) and denote the cut by [vi ; pi ]ℓ . Notice that by Lemma 2.3 (ii) the cut [wi ; qi ]ℓ is completely fixed by [vi ; pi ]ℓ and σm + σ ′ n. In Ar,s we consider the subspace of (K, λ, µ, τ1 )–bilinear functions and call Π(K,λ,µ,τ1 ) the projection onto this subspace. Having chosen 1/2, 4 as bounds for the parameters λ, µ we will call low σ momentum variables, denoted by wL and spanning the space ℓL ρ , the zj such 3 that |j| < 4K . Similarly we call high momentum variables, denoted by wH σ τ and spanning the space ℓH ρ , the zj such that |j| > K /2. Notice that the low and high variables are separated. d

K

Z1 b d For all σ, σ ′ = ±1, k ∈ Z K with P, h ∈ Z1 α, β ∈ N and3 A→[wi ; qi ]ℓ ∈ Hτ /4d |k| < K, h = −π(k, α, β), j∈Zd |j|(αj + βj ) < µK and |qj | < 4K , let ′

1

σ,σ (h, [wi ; qi ]ℓ ; I) be an analytic function of I for |I| < r 2 . We construct gk,α,β K

T¨ oplitz (K, λ, µ) bilinear functions with a τ1 cut on the subspace A→[vi ; pi ]ℓ

11

by setting: g(A) :=

∗ X

′

′

σ σ zn , gσ,σ (σm + σ ′ n, [vi ; pi ]ℓ )eihk,θi z α z¯β zm

n,m,σ,σ′ ∗ P where the sum means the restriction to the (K, λ, µ, τ1 ) monomials with a (ℓ, τ1 ) cut given by [vi ; pi ]ℓ (note that with our convention this means that m ∈ A moreover each point m is in at most one A). Finally we define F(τ1 , K) as the space of functions X g= g(A). A∈HK K A→[vi ;pi ]ℓ | |pℓ | 5K4 . 2

T ˜ zikT kXP kD(r,s),O < ∞ , khΩz, 0 is small enough, ε = ε(γ, b, d, L, M, K, λ, µ) is a positive constant. If T kXP kD(r,s),O ≤ ε, then there exists a Cantor set Oγ ⊂ O with meas(O\Oγ ) = 1 in ξ) O(γ) and two maps (analytic in θ and CW Ψ : Tb × Oγ → D(r, s),

ω ˜ : Oγ → Rb ,

where Ψ is γε2 -close to the trivial embedding Ψ0 : Tb × O → Tb × {0, 0, 0} and ω ˜ is ε-close to the unperturbed frequency ω, such that for any ξ ∈ Oγ and θ ∈ Tb , the curve t → Ψ(θ + ω ˜ (ξ)t, ξ) is a linearly stable quasi-periodic solution of the Hamiltonian system governed by H = N + P . 13

4

KAM step

Theorem 2 is proved by an iterative procedure. We produce a sequence of hamiltonians Hν = Nν + Pν and a sequence of symplectic transformations XF1 ν−1 Hν−1 := Hν , well defined on a domain D(rν , sν ) × Oν . At each step, the perturbation becomes smaller at cost of reducing the analyticity and parameter domain. More precisely, the perturbation should satisfy kXPν+1 kD(rν+1,sν+1 ),Oν ≤ εκν , κ > 1. The sequence rν → 0 while sν → s/4 and Oν → O∞ . For simplicity of notation, we denote the quantities in the ν-th step without subscript, i.e. Oν = O, ω ν = ω and so on. The quantities in the (ν + 1)th step are denoted with subscript ” + ”. Most of the KAM procedure is completely standard, see [12] for proofs. The new part ˜ z¯i is kept by is: 1. to show that Quasi T¨ oplitz property (A4) for P and hΩz, KAM iteration and 2. prove the measure estimate using the Quasi T¨ oplitz property. For simple, below we always use C (could be different) denote the constant independent on iteration. One step Given Hamiltonian (3.1) well defined in D(r, s)×O satisfies ˜ z¯i are Quasi T¨ (A1 − A4). P and hΩz, oplitz with parameter (K, λ, µ). And we have T ˜ n | 1 ≤ L, khΩz, ˜ z¯ikT |ω|C 1 , |∇ω −1 |O ≤ M, |Ω C D(r,s),O ≤ L, kXP kD(r,s),O ≤ ε. W

W

our aim is to construct: (1) a open set O+ ⊂ O of positive measure, (2) a 1-parameter group of symplectic transformations ΦtF , well defined for all ξ ∈ O+ , t ≤ 1 , such that Φ1F H := H+ = N+ +P+ still satisfies (A1)−(A4) in ˜ + z, z¯i are Quasi T¨ domain D(r+ , s+ ). P+ and hΩ oplitz with new parameter (K+ , λ+ , µ+ ). And we have −1 ˜ + | 1 ; khΩ ˜ + z, z¯ikT |O ≤ M+ , |Ω |ω+ |C 1 , |∇ω+ n C D(r+ ,s+ ),O+ ≤ L+ W

W

kXP+ kTD(r+ ,s+ ),O+

≤ ε+ = εκ .

Let us define R :=

X

i(k,θ) p α β

Pk,p,α,β e

I z z¯ ,

hRi :=

b X i=1

k,2|p|+|α|+|β|≤2

P0,ei ,0,0 Ii +

X

P0,0,ej ,ej |zj |2

j∈Zd1

The generating function of our symplectic transformation, denoted by F , solves the “homological equation”: {N, F } = Π≤K R − hRi 14

(4.1)

where Π≤K is the projection which collects all terms in R with |k| ≤ K and ˜ It’s well known (and K is fixed to be the quasi-T¨ oplitz parameter of P, Ω. immediate) that F is uniquely defined by homological equation for those ξ such that hω(ξ), k, i + Ω(ξ) · l 6= 0. To have quantitative bounds, we restrict to a set O+ where (see Lemma 4.1): |hω(ξ), k, i + Ω(ξ) · l| ≥ γK−τF ,

|k| ≤ K, |l| ≤ 2, (k, l) 6= 0,

(4.2)

d

where k ∈ Zb , l ∈ ZZ1 and (k, l = α − β) satisfy momentum conservation (2.5); τF is a fixed parameter. Then H in the new variables is: H+ := e{F,·} H = N+ + P+ where N+ = N + hRi and P+ = e{F,·} H − N+ .

4.1

The set O+

Definition 4.1 O+ is defined to be the open subset of O such that: i) For all |k| < K, h ∈ Z, (h, k) 6= (0, 0). |hω, ki + h| > 2γK−τ0 .

(4.3)

ii) For all |k| < K, l ∈ Z∞ , |l| = 1. |hω, ki + Ω · l| > 2γK−τ0 .

(4.4)

iii) For all |k| < K, |l| = 2; l 6= em − en or l = em − en and max(|m|, |n|) ≤ 8Kτ . |hω, ki + Ω · l| > 2γK−2dτ . (4.5) iv) For all K with K ≤ K ≤ 2Kτ /τ0 , for all affine spaces [vi , pi ]ℓ in HK with |pℓ | < K τ /4d we choose a point mg ∈ [vi ; pi ]gℓ . For all such mg and for all k such that |k| ≤ K, we require: | ± hω, ki + Ωmg − Ωng | > 2γ min(K −2dτ0 , 2−4d |pℓ |−2d ),

(4.6)

where ng = mg + π(k). By assumption O+ is open, and we have d

Lemma 4.1 For all ξ ∈ O+ , for all k ∈ Zb , |k| ≤ K and l ∈ ZZ1 , |l| ≤ 2 which satisfy momentum conservation, we have |hω, ki + l · Ω| ≥ γKτF , 15

τF = 2dτ.

(4.7)

Proof: The cases with |l| = 0, 1 follow trivially since τF is large with respect to τ0 ; same for l = em +en and l = em −en with max(|m|, |n|) < 8Kτ . For the remaining cases we proceed in two steps: first we fix k, K = K and one subspace [vi ; pi ]ℓ , we consider (4.6) with this choice of k, [vi ; pi ]ℓ . We show that this inequality implies that (4.7) holds for all l = em − en such that m ∈ [vi ; pi ]gℓ and n = m + π(k). We prove this fact by using the ˜ Finally we show that every point m with hypothesis that (A4) holds for Ω. τ |m| > 4K must belong to some [vi ; pi ]gℓ . Let m be any point in [vi ; pi ]gℓ . Let us first notice that hω, ki + |m|2 − |n|2 = hω, ki + |π(k)|2 − 2hπ(k), mi,

(4.8)

hence (4.7) with l = em − en is surely satisfied if |(π(k), m)| ≥ 2K3 because in that case (4.8) is greater than 2K3 − C12 K2 − |ω|K > K3 provided that K is large with respect to C1 and |ω . a is in hv i , If on the other hand |(π(k), m)| < 2K3 , then π(k) ∈ BK i ℓ otherwise we would have |(π(k), m)| > 12 K4dτ0 by definition of [vi ; pi ]gℓ and recalling that K4dτ0 > 4K3 by hypothesis. Thus for all m ∈ [vi ; pi ]gℓ either (4.7) is trivially satisfied or |m|2 − |n|2 = |π(k)|2 − 2hπ(k), mi = |π(k)|2 − 2hπ(k), mg i, recall that mg is one fixed point in [vi ; pi ]gℓ on which we have imposed the non–resonance conditions (4.6). We have seen in Remark 2.5 that all m ∈ [vi ; pi ]gℓ have a (ℓ, τ1 ) cut with parameters (4, 12 ). P ˜ 2 oplitz hence, by definition: We know that m Ω m |zm | is quasi-T¨ X ˜ m |zm |2 = Ω (4.9) Π(K,λ,µ,τ1 ) m

X

[vi ;pi ]ℓ ∈H(K) |pℓ |≤µKτ1

where

∗ P m

∗ X

˜ i ; pi ]ℓ )|zm |2 + K −4dτ1 Ω([v

m

∗ X

¯ m |zm |2 , Ω

m

is the restriction to those m with |m| > λKτ and have (ℓ, τ1 ) cut

with parameters (λ, µ) given by [vi ; pi ]ℓ . All m ∈ [vi ; pi ]gℓ satisfy such conditions with K = K, hence: ˜ m − Ω([v ˜ i ; pi ]ℓ )| < LK−4dτ1 , |Ω 16

in particular this relation holds for mg . K We proceed in the same way for n = m+π(k). Let n→[wi ; qi ], By hypothesis (µ − 12 )K1τ , (4 − λ)K4dτ1 > 5K4 and we apply Lemma 2.3 with λ, µ, λ′ = 4, µ′ = 12 . we produce a (τ1 , ℓ) cut [wi ; qi ]ℓ for n with parameters λ, µ. Now, by Lemma 2.3 ii), [wi ; qi ]ℓ is completely fixed by [vi ; pi ]ℓ and k. We have ˜n − Ω ˜ ([wi ; qi ]ℓ )| < LK−4dτ1 , |Ω and this relation holds also for ng = mg + π(k). This implies that ˜m − Ω ˜n − Ω ˜ mg + Ω ˜ ng | ≤ 4LK−4dτ1 , |Ω where by definition of τ1 , Kτ1 = max(Kτ0 , 2|pℓ |) and hence: |hω, ki + Ωm − Ωn | ≥ |hω, ki + Ωmg − Ωng | − 4LK−4dτ1 ≥ γ min(K−4dτ0 , 2−4d |pℓ |−4d ) ≥ γK−τ .

(4.10)

We know that each point m ∈ Zd1 admits a “standard cut” based on its K– K

τ

optimal presentation m→[vi ; pi ] (see Definition 2.3). If we have |pd | < 4K 4d , then by Cramers rule we have |m| = |V −1 p| < 4Kτ /4d Kd−1 < Kτ and hence max(|m|, |m + π(k)|) < 2Kτ . So the measure estimates for the points m which fall in this case are covered by (4.5). If we have |p1 | > K4dτ0 then | ± hω, ki + Ωm − Ωn | > | ± hω, ki + |m|2 − |n|2 | − 2LK−4dτ0 > γKτ0 . τ such that |pℓ | < 21 Kτ1 Otherwise, there exist 1 ≤ j < d, and τ0 < τ1 < 4d and |pℓ+1 | > 4K4dτ1 > 4 · max(K4dτ0 , 24d |pℓ |4d ). Thus m ∈ [vi ; pi ]gℓ . We have shown that conditions ii)-iv) in O+ imply (4.7).

Remark 4.1 This lemma essentially saying that by improving only one non resonant condition (4.6), we impose all the conditions (4.7) with l = em − en such that m ∈ [vi ; pi ]gj and n = m + π(k). Remark 4.2 Notice that up to now we only use (4.6) with K = K. Indeed the other non–resonance conditions are only required in order to show that the quasi–T¨ oplitz property is preserved in solving the homological equation. Lemma 4.2 The set O+ is open and has |O \ O+ | ≤ CγK−τ0 +b+d/2 .

17

For the measure estimates we define  Rτk,l := ξ ∈ O| |hω, ki + Ω · l| < γK−τ , Lemma 4.3 For all (k, l) 6= (0, 0) |k| ≤ K and |l| ≤ 2, which satisfy momentum conservation, one has |Rτk,l | ≤ CγK−τ . Proof: By assumption O is contained in some open set of diameter D. Choose a to be a vector such that hk, ai = |k|, we have |∂t (hk, ω(ξ + ta)i + Ω · l)| ≥ M (|k| − M L) ≥ lead to Z

Rτk,l

dξ ≤ 2M

−1

γK

−τ

Z

ξ+tRa∩Rτk,l

dt

Z

M . 2

dξ2 . . . dξb ≤ 2M −1 D b−1 γK−τ

Proof of Lemma 4.2: Proof: The first statement is trivial, indeed i)-iv) are a finite number of inequalities; notice that in iv) for each [vi , pi ]gℓ and k we impose only one condition by choosing one couple mg , ng . Finally by Remark 2.2 there are a finite number of [vi , pi ]gℓ . Let us prove the measure estimates; to impose (4.3) with h = 0 we have to remove 0 | ∪|k|≤K Rτk,0 | ≤ CγK−τ0 +b , the case h ∈ Z is exactly the same. In (4.4), by momentum conservation l = ±em implies that ±m = −π(k). Hence to impose (4.4) we have to remove: | ∪k≤K ∪

l=±em , ±m=−π(k)

0 Rτk,l | ≤ CγK−τ0 +b .

If l = ±(em + en ) we notice that the condition ˜m + Ω ˜ n| < | ± hω, ki + |m|2 + |n|2 + Ω

1 2

implies | ± hω, ki + |m|2 + |n|2 | < 1 and hence |m|2 + |n|2 < 2|ω|K: | ∪k≤K ∪

l=±(em +en ) , √ m+n=−∓π(k) ,|m|≤C(b) K

18

0 Rτk,l | ≤ CγK−τ0 +b+d/2 ,

In conclusion one gets (4.3) and (4.4) with τ0 > b + d/2 and l 6= ±(em − en ) by removing an open set of measure CγK−τ0 +b+d/2 . One trivially has −dτ +b , | ∪k≤K ∪ l=±(em −en ) ,m−n=∓π(k) , R2dτ k,l | ≤ CγK max(|m|,|n|)≤8Kτ

so we have (4.5) by removing an open set of measure CγK−dτ +b . To deal with the last case, for all natural K such that K ≤ K ≤ 2Kτ /τ0 , for all affine subspaces [vi ; pi ]ℓ and for all |k| ≤ K we set −4dτ0 −4d g g RK , 2 |pℓ |−4d )}(4.11) k,[vi ;pi ]g := {ξ | |hω, ki + Ωm − Ωn | < 2γ min(K ℓ

Following Lemma 4.3, |RK | < Cγ min(K −4dτ0 , 2−4d |pℓ |−4d ). By k,[vi ;pi ]gℓ Remark 2.2 we have: | ∪K≤K≤Kτ /τ0 ∪ℓ=0,···,d−1 ∪ 1 K τ0 ≤|p 2

≤ Cγ

d−1 XX

X

τ ℓ |≤4K 4d

∪ [vi ;pi ]gℓ RK k,[vi ;pi ]g | |k| 1 K τ0 2

so that we have (4.6) by removing an open set of measure CγK−dτ0 +b . Our Lemma is proved automatically.

4.2

Quasi-Toplitz property

˜ + |z|2 are quasi-T¨ Proposition 1 The functions P+ , Ω oplitz with parameters (K+ , λ+ , µ+ ) such that: p 4dτ0 −1 4 4K+ < (µ − µ+ )(K+ )3/2 , 4µ+ K+ < (λ+ − λ)K+ .

The key of our strategy is based on the following two propositions which are proved in the appendix. Proposition 2 For all K ≥ K, k ∈ Zb with |k| < K and for all |m|, |n| ≥ K

K

λK τ such that m−n = −π(k), m→[vi ; pi ], n→[wi ; qi ] and m, n have a (ℓ, τ1 ) cut with parameters λ, µ for some choice of ℓ, τ1 one has ˜ i ; pi ]ℓ )−Ω([w ˜ |hω, ki+|m|2 −|n|2 +Ω([v i ; qi ]ℓ )| = ˜ i ; pi ]ℓ ) − Ω([w ˜ |hω, ki + |π(k)|2 − 2hπ(k), mi + Ω([v i ; qi ]ℓ )| ≥  γK−2dτ τ1 /τ0 , π(k) ∈ hvi iℓ , 1 4dτ1 , otherwise 2K ˜ i ; pi ]ℓ ) and Ω ˜ ([wi ; qi ]ℓ ) are defined by Formula (4.9). where Ω([v 19

Proposition 3 For ξ ∈ O+ , the solution of the homological equation F is quasi-T¨ oplitz for parameters (K, λ, µ), moreover one has the bound kXF kTr,s ≤ Cγ −2 K2τ

2 /τ 0

kXP kTr,s ,

(4.12)

where C is some constant. Analytic quasi-T¨ oplitz functions are closed under Poisson bracket. More precisely: Proposition 4 Given f (1) , f (2) ∈ Ar,s , quasi-T¨ oplitz with parameters (K, λ, µ) oplitz for all parameters (K′ , λ′ , µ′ ) we have that {f (1) , f (2) } ∈ Ar′ ,s′ , is quasi-T¨ such that K′ , λ′ , µ′ , r ′ , s′ satisfy: 3

3

K′ < (µ − µ′ )K′ , 2µ′ K′ < (λ′ − λ)K′ ′

′

4dτ0 −1

,

τ

e−(s−s )K K′ < 1.

(4.13)

kX{f (1) ,f (2) } kTr′ ,s′ ≤ Cδ−1 kXf (1) kTr,s kXf (2) kTr,s

(4.14)

We have the bounds

where δ = min(s − s′ , 1 −

r′ r)

(ii) Given f (1) , f (2) as in item (i), with kXf (1) kTr,s δ−1 ≪ 1, the func(1)

oplitz in D(r ′ , s′ ) with tion f (2) ◦ φtf (1) := et{f ,·} f (2) , for t ≤ 1, is quasi-T¨ parameters (K′ , λ′ , µ′ ) for K′ ≤

5 5.1

p

(µ − µ′ )K′

3/2

,

4

2µ′ K′ < (λ′ − λ)K′

4dτ0 −1

.

(4.15)

Estimate and KAM Iteration Estimate on the coordinate transformation

We estimate XF and φ1F where F is given by (4.1). Lemma 5.1 Let Di = D( 4i r, s+ + 4i (s − s+ )), 0 < i ≤ 4. Then kXF kD3 ,O+ ≤ cγ −2 K4dτ ε ,

kXF kTD3 ,O+ ≤ Cγ −2 K2τ

20

2 /τ 0

ε

(5.1)

1

Lemma 5.2 Let η = ε 3 , Diη = D( 4i ηr, s+ + 4i (s − s+ )), 0 < i ≤ 4. If 3 2 ε ≪ ( 12 γ 2 K−2τ /τ0 ) 2 , we then have φtF : D2η → D3η ,

−1 ≤ t ≤ 1,

(5.2)

Moreover, kDφtF − IdkD1η ≤ Cγ −2 K4dτ ε ,

kDφtF − IdkTD1η ≤ Cγ −2 K2τ

2 /τ 0

ε

(5.3)

The first bound in the two lemmas is completely standard and corresponds respectively to Lemma 4.2 and 4.3 in [12], we refer to that paper for the proof. Let us fix K+ so that (4.15) holds. Let us prove the quasi–T¨ oplitz bound in Lemma 5.1. This follows by Formula (4.12) of Proposition 1 with K′ = K + 34 (K+ − K), λ′ = λ + 43 (λ+ − λ) and µ′ = µ − 34 (µ+ − µ):

5.2

Estimate of the new perturbation

The symplectic map φ1F defined above transforms H into H+ = N+ + P+ , where N+ = N + hRi and Z 1 Z 1 t {Π≤K R, F } ◦ φtF dt + (P − Π≤K R) ◦ φ1F (1 − t){{N, F }, F } ◦ φF dt + P+ = 0 0 Z 1 {R(t), F } ◦ φtF dt + (P − Π≤K R) ◦ φ1F , (5.4) = 0

with R(t) = (1 − t)(N+ − N ) + tΠ≤K R. Hence Z 1 (φtF )∗ X{R(t),F } dt + (φ1F )∗ X(P −Π≤K R) . XP+ = 0

Lemma 5.3 The new perturbation P+ satisfies the estimate kXP+ kD(r+ ,s+ ) ≤ Cγ −2 K4dτ ε4/3 . Proof: According to Lemma 5.2, kDφtF − IdkD1η ≤ cγ −2 K4dτ ε,

−1 ≤ t ≤ 1,

thus kDφtF kD1η ≤ 1 + kDφtF − IdkD1η ≤ 2,

−1 ≤ t ≤ 1.

kX{R(t),F } kD2η ≤ Cγ −2 K4dτ η −2 ε2 , 21

and kX(P −Π≤K R) kD2η ≤ Cηε, we have kXP+ kD(r+ ,s+ ) ≤ Cηε + C(γ −2 K4dτ )η −2 ε2 ≤ Cγ −2 K4dτ ε4/3 .

We need to show that P+ is quasi–T¨ oplitz and estimate its T¨ oplitz norm. We notice that R(t) and P −Π≤K R in (5.4) are quasi–T¨ oplitz, by hypothesis (A4). Then, by Proposition 4 ii), we have that R(t) ◦ φtF = e{F,·} R(t) and (P − Π≤K R) ◦ φtF are quasi–T¨ oplitz as well. We repeat the reasoning of Lemma 5.3 only with the T¨ oplitz norm. We have the following 2

Lemma 5.4 We set ε+ := Cγ −2 K2τ /τ0 ε4/3 for an appropriate constant C, we have kXP+ kTD(r+ ,s+ ) ≤ ε+ .

5.3

Iteration lemma

To make KAM machine work fluently, a sequence of iteration is given: For any given s, ε, r, γ and for all ν ≥ 1, we define the following sequences sν = s(1 −

ν+1 X

2−i ),

i=2

2τ 2 /τ

4

3 , εν = cγ −2 Kν−1 0 εν−1

γν = γ(1 −

ν+1 X

2−i ),

(5.5) 1

, ην = εν3

i=2

Mν = Mν−1 + εν−1 ,

Lν = Lν−1 + εν−1 , ν−1

Y 1 1 rν = ην−1 rν−1 = 2−2ν ( εi ) 3 r0 , 4 i=0

µν = µ −

ν X

(χ)−i ,

λν = λ +

ν X i=1

i=1

ρν = ρ(1 −

ν+1 X i=2

22

2−i ),

(χ)−i

Kν = c(ρν−1 − ρν )−1 ln ε−1 ν , where c, 1 < χ < 43 is a constant, and the parameters r0 , ε0 , γ0 , L0 , s0 and K0 are defined to be r, ε, γ, L, s and bounded by ln ε−1 respectively. We iterate the KAM step, and proceed by induction. Lemma 5.5 Suppose at the ν–step of KAM iteration, hamiltonian Hν = Nν + Pν , is well defined ν , sν )×Oν , where Nν is usual ”integrable normal form”, P ˜ ν in D(r 1 smooth Pν and Ωn |zn |2 satisfy (A4) for (Kν , λν , µν ), ων and Ωνn are CW ˜ ν | 1 ≤ Lν , |ων |C 1 , |∇ων−1 |O ≤ Mν , |Ω n C W

W

kXPν kTD(rν ,sν ),Oν ≤ εν .

|Ωνn − Ωnν−1 |Oν ≤ εν−1 ;

˜ ν z, z¯ikT khΩ D(rν ,sν ),Oν ≤ Lν

Then there exists a symplectic and Quasi-T¨ oplitz change of variables for parameter (Kν+1 , λν , µν ), Φν : D(rν+1 , sν+1 ) × Oν+1 → D(rν , sν ),

(5.6)

d

where |Oν+1 \Oν | ⋖ γK−τ0 +b+ 2 , such that on D(rν+1 , sν+1 ) × Oν+1 we have Hν+1 = Hν ◦Φν = eν+1 +Nν+1 +Pν+1 = eν+1 +hων+1 , Ii+hΩν+1 z, z¯i+Pν+1 , P ν . = Ωνn + P0,0,e lP0,l,0,0 , Ων+1 with ων+1 = ων + n n ,en |l|=1

P ν+1 ˜ n |zn |2 satisfy (A4) Nν+1 is ”integrable normal form”. Pν+1 and Ω 1 smooth are CW for parameters (Kν+1 , λν+1 , µν+1 ). Functions ων+1 and Ων+1 n −1 ˜ ν+1 | 1 ≤ Lν+1 , |ων+1 |C 1 , |∇ων+1 |O ≤ Mν+1 , |Ω n C W

W

kXPν+1 kTD(rν+1 ,sν+1 ),Oν+1 ≤ εν+1 ,

|Ων+1 − Ωνn |Oν+1 ≤ εν ; n

˜ ν+1 z, z¯ikT khΩ D(rν+1 ,sν+1 ),Oν+1 ≤ Lν+1

˜ ν+1 z, zi satisfies Quasi• By Proposition 1, new perturbation Pν+1 and hΩ T¨ oplitz property for parameters (Kν+1 , λν+1 , µν+1 ). As we can see, when we require τ > τ0 > 12, ν ∀K ≥ Kν+1 = c(ρν−1 − ρν )−1 ln ε−1 ν > K0 2

implies inequality p 2K ≤ (µν − µν+1 )K 3/2 ,

4µ′ K 4 < (λν+1 − λν )K 4dτ0 −1 .

• Poisson bracket preserve momentum conservation or result from Lemma 4.4 in [12] lead to Pν+1 satisfies momentum conservation. 23

5.4

Convergence

Suppose that the assumptions of Theorem 2 are satisfied. Recall ε0 = ε, r0 = r, γ0 = γ, s0 = s, M0 = M, L0 = L, N0 = N, P0 = P, O is a compact set. The assumptions of the iteration lemma are satisfied when ν = 0 if ε0 , γ0 are sufficiently small. Inductively, we obtain sequences: Oν+1 ⊂ Oν , Ψν = Φ0 ◦ Φ1 ◦ · · · ◦ Φν : D(rν+1 , sν+1 ) × Oν+1 → D(r0 , s0 ), ν ≥ 0, H ◦ Ψν = Hν+1 = Nν+1 + Pν+1 . ˜ = ∩∞ Oν , since at ν step the parameter we excluded is bounded Let O ν=0 −τ0 +b+d/2 by CγKν , the total measure we excluded with infinity step of KAM ˜ is a nonempty set, actually it iteration is bounded by γ which guarantee O has positive measure. As in [19, 20], with Lemma 5.2, Nν , Ψν , DΨν , ων converge uniformly on ˜ with D(0, 2s ) × O N∞ = e∞ + hω∞ , Ii +

X

Ω∞ ¯n . n zn z

n

2τ 2

4

τ0 3 cγ 2 Kν−1 εν−1

ρν )−1 ln ε−1 ν ,

we have εν = Since Kν = c(ρν−1 − → 0 once ε is sufficiently small. And with this we have ω∞ is slightly different from ω. Let φtH be the flow of XH . Since H ◦ Ψν = Hν+1 , there is φtH ◦ Ψν = Ψν ◦ φtHν+1 .

(5.7)

The uniform convergence of Ψν , DΨν , ων and XHν implies that the limits ˜ we get can be taken on both sides of (5.7). Hence, on D(0, 2s ) × O φtH ◦ Ψ∞ = Ψ∞ ◦ φtH∞ and

(5.8)

s ˜ → D(r, s) × O. Ψ∞ : D(0, ) × O 2

˜ Ψ∞ (Tb ×{ξ}) is an embedded torus which is invariant for From 5.8, for ξ ∈ O, ˜ The normal behavior the original perturbed Hamiltonian system at ξ ∈ O. of this invariant tori is governed by normal frequency Ω∞ .

24

A

Appendix

Proof of Proposition 2: By hypothesis min(|m|, |n|) ≥ λK τ , |qℓ |, |pℓ | ≤ µK τ1 ,

K

m→[vi ; pi ] ,

|qℓ+1 |, |pℓ+1 | ≥ λK 4dτ1 ,

K

n→[wi ; qi ] , [vi ; pi ]ℓ ≺ [wi ; qi ]ℓ

(A.1)

By definition of quasi–T¨ oplitz (see formula (4.9)), one has: ˜ m − Ω([v ˜ i ; pi ]ℓ |, |Ω ˜ m − Ω([v ˜ i ; pi ]ℓ | ≤ LK −4dτ1 |Ω

(A.2)

Recall that m − n = −π(k), so one has |m|2 − |n|2 = hm + n, m − ni = |π(k)|2 − 2hπ(k), mi. If π(k) ∈ / hvi iℓ then |hπ(k), mi| > K 4dτ1 > K3 and the denominator is not small: 1 4dτ1 ˜ i ; pi ]ℓ ) − Ω([w ˜ |hω, ki + |m|2 − |n|2 + Ω([v , i ; qi ]ℓ )| > K 2 ˜ i ; pi ]ℓ )|,|Ω ˜ ([wi ; qi ]ℓ )| ≤ L. since (again by definition of quasi–T¨ oplitz) |Ω([v If π(k) ∈ hvi iℓ then the value of hπ(k), mi is fixed for all m ∈ [vi ; pi ]ℓ . K

We know that m→[vi′ ; p′i ] has a standard cut, so that m ∈ [vi′ ; p′i ]gℓ¯ for ¯ If 24d Kτ < K τ0 then some ℓ. ˜ ([vi ; pi ]ℓ ) − Ω([w ˜ |hω, ki + |π(k)|2 − 2hπ(k), mi + Ω i ; qi ]ℓ )| (A.2)

≥ |hω, ki + Ωm − Ωn | − 2LK −4dτ1 (4.10) γ ≥ γ min(K−4dτ0 , 2−4d |p′ℓ¯|−4d ) − 2L|K|−4dτ1 ≥ min(K−4dτ0 , |p′ℓ¯|−4d ), 2 since |p′ℓ¯| < 4Kτ /4d by the definition of standard cut. If on the other hand we have 24d Kτ > K τ0 we proceed as follows. We have seen that we may restrict to the case π(k) ∈ hvi ij , where |m|2 − |n|2 = |π(k)|2 − 2hπ(k), mi = |π(k)|2 − 2(π(k), mg ), where (notice that K < 2Kτ /τ0 ), mg := mg (K) is the point in [vi ; pi ]gℓ chosen for the measure estimates (4.6). 25

We notice that mg , ng satisfy the conditions (A.1), so we apply (A.2) to m, n, mg , ng . We have ˜ i ; pi ]ℓ ) − Ω([w ˜ |hω, ki + |π(k)|2 − 2hπ(k), mi + Ω([v i ; qi ]ℓ )| ≥ |hω, ki + Ωmg − Ωng | − 2LK −4dτ1 ≥ 2γ min(K −4dτ0 , 2−2d |pℓ |−4d ) − 2LK −4dτ1 ≥ γ min(K −4dτ0 , 2−2d |pℓ |−4d ) ≥ γK

−4dτ1 τ τ0

since by definition |pj | < µ′ K τ1 < 4K τ1 , K ≤ 2Kτ /τ0 and 2 · 84d L < γ. Proof of Proposition 3: The quasi–T¨ oplitz property is a condition on the (K, λ, µ)–bilinear part of F , where F is at most quadratic. Hence we only need to consider the quadratic terms: X Π(K,λ,µ) F = ei(k,x)(Fk,0,em ,en zm z¯n + Fk,0,em +en ,0 zm zn ) + c.c |k|≤K , max(|n|,|m|)>λ′ K τ

with Fk,0,em ,en =

Pk,0,em ,en , hk, ωi + Ωm − Ωn

Fk,0,em +en ,0 =

Pk,0,em +en ,0 . (A.3) hω, ki + Ωm + Ωn

By hypothesis min(|m|, |n|) > λK τ so in the case of Fk,0,em+en ,0 one has |Fk,0,em +en ,0 | = since

|Pk,0,em +en ,0 | ˜m + Ω ˜n hk, ωi + |m|2 + |n|2 + Ω

≤ |Pk,0,em +en ,0 |K −τ ,

˜m + Ω ˜ n > 2K τ − cK − 2L. hk, ωi + |m|2 + |n|2 + Ω

We proceed in the same way for ∂ξ Fk,0,em +en ,0 . This means that Fk,0,em +en ,0 is quasi-T¨ oplitz with the “T¨ oplitz approximation” equal to zero. K Let m→[vi ; pi ] have a cut (ℓ, τ1 ). We wish to show that Fk,0,em,en = Fk (m − n, [vi ; pi ]ℓ ) + K −4dτ1 F¯k,0,em ,en , here Fk is the k Fourier coefficient of the T¨ oplitz approximation F. By hypothesis we have conditions (A.1) and hv1 , . . . , vℓ i = hw1 , . . . , wℓ i. This in turn implies that the subspace [wi , qi ]ℓ is obtained from [vi , pi ]ℓ by translation by m − n = −π(k). If π(k) ∈ / hvi iℓ then the denominator in the first of (A.3) is 1 4dτ1 ˜ i ; pi ]ℓ )−Ω([w ˜ |hk, ωi+Ωm −Ωn | > |hk, ωi+|m|2 −|n|2 +Ω([v i ; qi ]ℓ )|−L > K 4 26

and we may again set Fk = 0. Otherwise we set

Fk (m−n, [vi , pi ]ℓ ) =

hω, ki +

|π(k)|2

Pk (m − n, [vi , pi ]ℓ ) . ˜ i ; pi ]ℓ ) − Ω([w ˜ − 2hπ(k), mi + Ω([v i ; qi ]ℓ )

We notice that hπ(k), mi depends only on the subspace [vi , pi ]ℓ and on π(k). Finally we apply Proposition 2 to bound the denominator. To bound the derivatives in ξ of F we proceed in the same way, only the denominators may appear to the power two. In conclusion: kXF kTr,s ≤ CK

2τ 2 τ0

kXP kTr,s

Before we give a proof to proposition 4, for simple and better to understand, some notation and technical Lemma are given. Let us set up some notation. We divide the Poisson bracket in four terms: {·, ·} = {·, ·}I,θ +{·, ·}L +{·, ·}H +{·, ·}R where the apices identify the variables in which we are performing the derivatives (the apex R summarizes the derivatives in all the wi which are neither low nor high momentum). We call a monomial ei(k,θ) I l z α z¯β P 3 1. of (K, µ)-low momentum if |k| < K and j |j|(αj + βj ) < µK . Denote by ΠL K,µ the projection on this subspace. 2. of K-high frequency if |k| ≥ K. Denote ΠU K the projection on this subspace. Recall projection symbol ΠK,λ,µ,τ1 is given in definition 2.5. A function f U then may be uniquely represented as f = ΠK,λ,µ,τ1 f + ΠL K,µ f + ΠK f + ΠR f where ΠR f is by definition the projection on those monomials which are neither (K, λ, µ, τ1 ) bilinear nor of (K, µ)-low momentum nor of K-high frequency. A technical lemma is given below. Lemma A.1 The following splitting formula holds:  ΠK,λ′ ,µ′ ,τ1 {f (1) , f (2) } = ΠK,λ′ ,µ′ ,τ1 {ΠK,λ,µ,τ1 f (1) , ΠK,λ,µ,τ1 f (2) }H + (A.4) (2) I,θ (2) L (1) (2) {ΠK,λ,µ,τ1 f (1) , ΠL } + {ΠK,λ,µ,τ1 f (1) , ΠL } + {ΠU ,f } K,2µ f K,2µ f Kf  (1) (2) I,θ L (1) (2) L (1) U (2) {ΠL f , Π f } + {Π f , Π f } + {f , Π f } K,λ,µ,τ K,λ,µ,τ K,2µ K,2µ K 1 1

27

Proof: We perform a case analysis: we replace each f (i) with a single monomial to show which terms may contribute non trivially to the projection ΠK,λ′ ,µ′ ,τ1 {f (1) , f (2) }. Consider the expression ΠK,λ′ ,µ′ ,τ1 {ei(k

(1) ,θ)

(1)

(1)

(1)

I l z α z¯β , ei(k

(2) ,θ)

(2)

(2)

(2)

I l z α z¯β }.

If one or both of the |k(i) | > K then one or both monomials are of high frequency and we obtain the last term in the second and third line of (A.4). Suppose now that |k(1) |, |k(2) | < K we wish to understand under which conditions on the α(i) , β (i) this expression is not zero. By direct inspection, one of the following situations (apart from a trivial permutation of the indexes 1, 2) must hold: (1)

1. one has z α z¯β

(1)

(1)

¯(1)

(2)

σ z σ1 and z α z ¯β = z α¯ z¯β zm j

(2)

¯(2)

(2)

′

= z α¯ z¯β znσ zj−σ1 , (1)

¯(1)

(2)

¯(2)

where min(|m|, |n|) ≥ λ′ K τ have a (ℓ, τ1 ) cut for some ℓ and z α¯ z¯β z α¯ z¯β is of (K, µ′ )–low momentum. The derivative in the Poisson bracket is on wj . (1)

(1)

(1)

(1)

(1)

¯(1)

′

(1)

¯(1)

′

(2)

(2)

(2)

¯(2)

σ z σ and z α z ¯β = z α¯ z¯β , where 2. one has z α z¯β = z α¯ z¯β zm n (2) ¯(2) (1) ¯(1) min(|m|, |n|) ≥ λ′ K τ have a (ℓ, τ1 ) cut for some ℓ and z α¯ z¯β z α¯ z¯β is of (K, µ′ )–low momentum. The derivative in the Poisson bracket is on I, θ.

3. one has z α z¯β

(2)

σ z σ z σ1 and z α z ¯β = z α¯ z¯β zm n j

(2)

(2)

¯(2)

= z α¯ z¯β zj−σ1 (1)

¯(1)

(2)

¯(2)

where min(|m|, |n|) ≥ λ′ K τ have a (ℓ, τ1 ) cut for some ℓ and z α¯ z¯β z α¯ z¯β is of (K, µ′ )–low momentum. The derivative in the Poisson bracket is on wj . (1)

(1)

(1)

¯(1)

(2)

(2)

(2)

¯(2)

′

σ and z α z ¯β = z α¯ z¯β znσ where 4. one has z α z¯β = z α¯ z¯β zm (2) ¯(2) (1) ¯(1) min(|m|, |n|) ≥ λ′ K τ have a (ℓ, τ1 ) cut for some ℓ and z α¯ z¯β z α¯ z¯β is of (K, µ′ )–low momentum. The derivative in the Poisson bracket is on I, θ.

In case 1. we apply momentum conservation to both monomials and obtain σ1 j = −σm − π(k(1) , α ¯ (1) , β¯(1) ) = σ ′ n + π(k(2) , α ¯ (2) , β¯(2) ). Recall that X X (2) (i) (1) (i) (1) ¯l + β¯(2)l ) ≤ µ′ K 3 −→ |l|(¯ αl + β¯l ) ≤ µ′ K τ |l|(¯ αl + β¯l + α l∈Zd1

l∈Zd1

28

and by hypothesis |k(i) | ≤ K, this implies that |j| > λ′ K τ − µ′ K 3 − CK > λK τ for K > K′ respecting (4.13) (here C is a constant so that |π(k)| ≤ C|k|). Hence min(|m|, |n|, |j|) > λK τ . By momentum conservation |σm + σ1 j|, | − σ1 j + σ ′ n| ≤ CK + µ′ K 3 ≤ 5K 3 ; by hypothesis n, m have a (ℓ, τ1 ) K

(i)

(i)

(i)

cut. By Lemma 2.3 also j →[wi ; qi ] has a (ℓ, τ1 ) cut. Then ei(k ,θ) z α z¯β are by definition (K, λ, µ, τ1 ) bilinear. The derivative in the Poisson bracket is on j which is a high momentum variable. As m, n run over all possible vectors in Zd1 with min(|m|, |n|) ≥ λ′ K, we obtain the first term in formula (A.4). (1) (1) (1) In case 2. following the same argument ei(k ,θ) z α z¯β is (K, λ′ , µ′ , τ1 ) (2) (2) (2) bilinear and ei(k ,θ) z α z¯β is (K, µ′ ) low momentum. We obtain the second contribution in formula (A.4). In case 3. we apply momentum conservation to the second monomial and obtain −σ1 j = −π(k(2) , α ¯ (2) , β¯(2) ). This implies that X X (1) (1) (1) (1) |j| + |l|(¯ αl + β¯l ) ≤ |π(k(2) , α ¯(2) , β¯(2) | + |l|(¯ αl + β¯l ) ≤ l∈Zd1

CK +

l∈Zd1

X

(1)

|l|(¯ αl

(1)

+ β¯l

(2)

+α ¯l

(2)

+ β¯l ) ≤ µ′ K 3 + CK ≤ µK 3

l∈Zd1 (1)

(1)

(1)

if K > K′ with K′ satisfying (4.13). Then ei(k ,θ) z α z¯β is, by definition, (2) (2) (2) (K, λ, µ, τ1 ) bilinear and ei(k ,θ) z α z¯β is (K, 2µ) low momentum. The derivative in the Poisson bracket is on j which is a low momentum variable. We obtain the third contribution in formula (A.4). In case 4. we apply momentum conservation to both monomials, we get min(|σm|, |σ ′ n|) ≤ max(| − π(k(i) , α ¯(i) , β¯(i) )| ≤ CK + µ′ K 3 , i=1,2

which is in contradiction to the hypothesis min(|m|, |n|) ≥ λ′ K τ . Hence case 4. does not give any contribution. The third line in formula (A.4) is dealt just as the second line by exchanging the indexes 1, 2. In order to show that {f (1) , f (2) } is quasi–T¨ oplitz, for all K > K′ and τ1 we have to provide a decomposition ΠK,λ′ ,µ′ ,τ1 {f (1) , f (2) } = F (1,2) + K −4dτ1 f¯(1,2) so that F (1,2) ∈ F(K, τ1 ) and kXF (1,2) kr′ ,s′ , kXf¯(1,2) kr′ ,s′ < δ−1 CkXf (1) kTr,s kXf (1) kTr,s . 29

(A.5)

for some constant C. We substitute in formula (A.4) ΠK,λ′ ,µ′ ,τ1 f (i) = F (i) + K −4dτ1 f¯(i) , with (i) F ∈ F(τ1 , K). Lemma A.2 Consider the function   (2) (I,θ)+L L (1) (2) (I,θ)+L f } + {Π f , F } F (1,2) = ΠK,λ′ ,µ′ ,τ1 {F (1) , F (2) }H + {F (1) , ΠL K,2µ K,2µ where we have denoted {·, ·}(I,θ)+L = {·, ·}(I,θ) + {·, ·}L . (i) One has F (1,2) ∈ F(τ1 , K). (ii) Setting f¯(1,2) = K 4dτ1 (ΠK,λ′ ,µ′ ,τ1 {f (1) , f (2) } − F (1,2) ) one has that the bounds (A.5) hold. Proof: To prove the first statement it is useful to write F (i) =

X

[vi ;pi ]ℓ ∈HK |pℓ | λ′ K τ we have that the condition |r| > λK τ is automatically fulfilled. By Lemma 2.3 r, n, m all have K

K

K

a (ℓ, τ1 ) cut with parameters (λ, µ). We set m→[vi ; pi ], n→[vi′ ; p′i ], r →[wi ; qi ] and suppose without loss of generality that (p1 , . . . , pℓ , v1 , . . . , vℓ )  (q1 , . . . , qℓ , w1 , . . . , wℓ )  (p′1 , . . . , p′ℓ , v1′ , . . . , vℓ′ ). 30

Again by Lemma 2.3 hvi iℓ = hvi′ iℓ = hwi iℓ , moreover [wi ; qi ]ℓ is completely fixed by [vi ; pi ]ℓ , σ, σ1 and by σm + σ1 r := h. Then we may change variables in the sum over r in (A.6): X X ′ −σ1 [F (1) ]σ,σ1 (h, [vi ; pi ]ℓ )[F (2) ]−σ1 ,σ (σm+σ ′ n−h; [wi ; qi ]ℓ ), σ1 =±1

h :|h| K+ For given K we bound all the terms in e{F,·} G containing N > K Poisson brackets by K −τ by using the analyticity of the nested Poisson bracket in D(r+ , s+ ) and the denominator N !. We then apply (A.7) with N = K, we get the restriction (4.15).

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