Moser’s Theorem for lower dimensional tori∗ Luigi Chierchia
Dingbian Qian
Dipartimento di Matematica
Department of Mathematics
Universit` a “Roma Tre”
Suzhou University
Largo S. L. Murialdo 1, 00146 Roma (Italy)
Suzhou 215006, P. R. China
(
[email protected])
(
[email protected])
May 17, 2002
Abstract `
J.K. Moser’s C –version of Kolmogorov’s theorem on the persistence of maximal quasi– periodic solutions for nearly–integrable Hamiltonian system is extended to the persistence of non–maximal quasi–periodic solutions corresponding to lower–dimensional elliptic tori of any dimension n between one and the number of degrees of freedom. The theorem is proved for Hamiltonian functions of class C ` for any ` > 6n + 5 and the quasi–periodic solutions are proved to be of class C p for 2 ≤ p < 2+a` where a` > 0 (and tends to infinity when ` → ∞). Keywords: nearly–integrable Hamiltonian systems, KAM theory, Kolmogorov’s Theorem, smooth invariant tori, lower dimensional tori, small divisors, fast convergent methods, smoothing techniques. MSC numbers: 70H08, 37J40, 34C27
Contents 1 Introduction and results
2
2 Preliminaries 2.1 Analytic approximants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Newton (KAM) scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Complex variables and norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 6 8 10
3 Proof of the main Theorem 3.1 The linearized equation . . . . . . . . . . . . 3.2 Iteration step . . . . . . . . . . . . . . . . . . 3.3 Convergence . . . . . . . . . . . . . . . . . . . 3.4 Measure estimates (multiplicity of solutions) .
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Supported by MURST Variational Methods and Nonlinear Differential Equations. Dingbian Qian was supported by the “Distinguished Visiting Scholar Program” of China and by the National Natural Science Foundation of China (No.10071055).
i
1
Introduction and results
1.1 J.K. Moser’s main contribution to the so–called KAM theory was to extend Kolmogorov’s invariant–tori–theorem ([9]) to smooth category. Kolmogorov’s celebrated theorem deals, as well known, with the persistence under small, real–analytic perturbations of maximal quasi–periodic solutions (associated to maximal invariant tori) for nearly– integrable Hamiltonian systems. The basic technical tool exploited by Moser in his extension was closely related to ideas of J.F. Nash ([17]) and consisted in using a Newton (quadratic) iteration method, re–inserting at each step enough regularity into the problem so as to beat (together with the so–called “small divisor problem”, already overcome by Kolmogorov and Arnold) the loss of regularity due to the inversion of certain (non– elliptic) differential operators. In the original work of Moser ([14]), which was dealing with twist area–preserving maps (corresponding to the Hamiltonian system case in “one and a half” degrees of freedom), the perturbation was assumed to be C 333 . The regularity assumption (in the twist map case) was later brought down to five by R¨ ussmann [22]; for the Hamiltonian case we refer to [16], [29], and, especially, [19], where Kolmogorov’s theorem is proved under the hypothesis that the perturbation is C ` with ` > 2d, d being the number of degrees of freedom. We recall also that M. Herman ([8]) gave a counterexample in the twist map case with ` = 3 − ε, ε > 0 (corresponding to ` = 4 − ε in the Hamiltonian case with two degrees of freedom). 1.2 Right after KAM theory for maximal tori was established, it appeared clear that an important direction of further investigations was that of the existence of lower dimensional quasi–periodic solutions corresponding to lower dimensional invariant tori, i.e., tori of dimension1 n < d (as above, d stands for the number of degrees of freedom). In 1965 V.K. Melnikov stated a precise result concerning the persistence of stable (or “elliptic”) lower dimensional tori in [13]; the hypotheses of such result are, now, commonly referred to as “Melnikov conditions”. However, a proof of Melnikov’s theorem was given only later by Moser [15] for the case n = d − 1 and, in the general case, by H. Eliasson in [6] and, independently, by S.B. Kuksin [10]; see also [20]. The unstable (or “hyperbolic”) case (i.e., the case for which the lower dimensionale tori are linearly unstable and lie in the intersection of stable and unstable Lagrangian manifolds) is simpler2 and a complete perturbation theory was worked out in [15], [7] and [29]. Various technical progresses have been recently performed in, e.g., [21], [2], [28], [27], [25]. Incidentally we mention that lower dimensional quasi–periodic solutions are particularly relevant in connection with extensions to PDE’s; see, e.g., [5], [11], [12], [21], [3] and references therein. 1.3 All the above mentioned results concerning the extension of Kolmogorov’s theorem to lower dimensional tori deal only with the real–analytic case. It is the purpose 1
Equilibria and periodic orbits, corresponding, respectively, to n = 0 and n = 1, are the simplest examples; in such cases there are no small–divisor problems and existence was already established by Poincar´e by means of the standard Implicit Function Theorem: see [18], Volume I, chapter III. 2 On a technical level: the normal frequencies to the torus do not resonate with the inner (or “proper”) frequencies associated to the quasi–periodic motion.
2
of this paper to extend Moser’s theorem to lower dimensional quasi–periodic solutions proving, under suitable generic assumptions, the persistence and the regularity of lower n–dimensional elliptic tori (corresponding to lower dimensional quasi–periodic solutions) for C ` perturbations of nearly–integrable systems with ` > 6n + 5. Before stating in a more precise way our results, let us mention that it was already remarked by S.M. Graff in3 [7] that combining “soft” tools of invariant manifold theory (based on the standard Implicit Function Theorem) and KAM theory for maximal tori one can conclude that lower dimensional unstable tori persist under small perturbations (but regularity of the continued manifolds may be, in general, quite low). As well known, however, such “partially hyperbolic techniques” do not carry over to the elliptic situation. 1.4 We proceed, now, to formulate the main result proved in this paper. Consider a (smooth) Hamiltonian system with n + m degrees of freedom, governed by a Hamiltonian function of the form H(x, y, u, v; ξ) ≡ N (y, u, v; ξ) + P (x, y, u, v; ξ) ,
(1.1)
where (x, y) ∈ Tn × Rn and (u, v) ∈ R2m are pairs of standard symplectic coordinates4 and ξ is a real parameter running over a compact set Π ⊂ Rn of positive Lebesgue measure5 ; N is in “normal (integrable) form”, meaning, here, that it is assumed to have the form n m X 1X N =e+ ωj (ξ)yj + Ωj (ξ)(u2j + vj2 ), (1.2) 2 j=1 j=1 (e being a constant) and P is a small perturbation. The motions generated by N decouple in a Kronecker flow x ∈ Tn → x + ω(ξ)t times the motion of m (decoupled) harmonic oscillators with characteristic frequences Ωj (ξ) (sometimes refererred to as normal frequencies); in particular, the n–parameter family (parameterized by ξ) of n dimensional tori T0n (ξ) ≡ Tn × {y = 0} × {u = v = 0} , ξ∈Π, are linearly stable (elliptic) invariant tori of dimension n carrying quasi–periodic motions with frequency ω(ξ) ∈ Rn . 3
Compare point b of the introduction in [7], page 6. Graff’s remark has been recently re–considered by Huang, D. and Liu, Z.: On the persistence of lower dimensional invariant hyperbolic tori for smooth Hamiltonian systems, Nonlinearity, 13 (2000), 189-202. 4 Hence the equation of motion are x˙ = Hy ,
y˙ = −Hx ,
u˙ = Hv ,
v˙ = −Hu ;
where Hy := (Hy1 , ..., Hyn ), etc.; Tn := Rn /(2π Zn ). 5 Typically, ξ may indicate an initial datum y0 and y the distance from such point or (equivalently, if the system is non–degenerate in the classical Kolmogorov sense) ξ → ω(ξ) might be simply the identity, which amounts to consider the unperturbed frequencies as parameter.
3
Theorem 1.1 Let ` > 6n + 5 and let H in (1.1) be C ` in a neighborhood of Tn × {y = 0} × {u = v = 0} and (uniformly) Lipschitz continuos in6 {ξ ∈ Π}. Assume that Ωi (ξ) > 0 ,
Ωi (ξ) 6= Ωj (ξ) ,
∀ ξ ∈ Π , ∀ i 6= j .
(1.3)
Assume, also, that ξ ∈ Π → ω(ξ) ∈ Rn is a Lipschitz homeomorphism of Π onto its image and that7 ∀ l ∈ Zm : |l| ≤ 2 . (1.4) Then, if the gradient of P , together with its Lipschitz semi–norm in ξ, is small enough, there exists a family of n–dimensional, C p –embedded (p > 2), linearly stable H–invariant tori T n (ξ) parameterized by (and Lipschitz continuous in) ξ ∈ Π∞ , Π∞ being a subset of Π of positive Lebesgue measure. Furthermore, on T n (ξ) the H–flow is C p –conjugated to the Kronecker flow x → x + ω∞ (ξ)t where ω∞ is a Lipschitz homeomorphysm of Π∞ close to ω; for all ξ ∈ Π∞ , ω∞ (ξ) is a “Diophantine vector”. meas{ξ ∈ Π : hω(ξ), ki + hΩ(ξ), li = 0} = 0 ,
∀ k ∈ Zn \{0} ,
1.5 Let us collect, here, a few remarks on the above statements. 1.5.1 Conditions (1.3)–(1.4) are a generalized version ([21]) of Melnikov’s conditions and represent a rather weak indepence requirement between ω and Ω (obviously satisfied if, for example, Ω is independent of ξ). Notice that, if ω and Ω are C 1 , (1.4) is satisfied whenever8 (taking ω as independent variable) ∂ω hΩ, li 6= k ,
∀ k ∈ Zn \{0} , ∀ l ∈ Zm : |l| ≤ 2 .
(1.5)
in which case the level sets {ω : hk, ωi + hl, Ω(ξ(ω))i = 0} are (n − 1)–dimensional C 1 hypersurfaces (and hence of vanishing n–dimensional measure). 1.5.2 Condition (1.3) requires the normal frequencies to be bounded away from zero and to be “simple”. Recently, in the KAM method of [28], the simplicity of the normal frequencies has been relaxed allowing, in [4], to establish the existence (and the linear stability) of quasi–periodic solutions for the one–dimensional wave equation with periodic boundary conditions. It is conceivable (but not obvious) that methods taken from [28] might lead to remove the second condition in (1.3). 1.5.3 The tori T n (ξ) are a C p –embedding of the standard flat n–torus Tn into the 2(n + m)–dimensional phase space. In fact, the embedding is C p –close to the identity for any p < 2 + a(` − 2) where a is a positive constant depending on ` and tending to 6
A function g is uniformly Lipschitz continuous on Π if |g|Lip = sup
ξ6=ξ0 ξ,ξ0 ∈Π
|g(ξ) − g(ξ 0 )| < ∞; usually, |ξ − ξ 0 |
we shall not indicate explicitely the domain Π in the notations (since it will be clear from context). 7 Here, “meas” denotes P Lebesgue measure; h·, ·i denotes the standard inner product; for integer vectors l = (l1 , ..., lm ), |l| = i |li |. Obviously, ω = (ω1 , ..., ωn ) and Ω = (Ω1 , ..., Ωn ); later, however, Ω will also be identified with the diagonal matrix diag (Ω1 , ..., Ωn ). 8 Actually, it is sufficient to require (1.5) for a finite number of vectors k; compare (3.99) below.
4
1/8 when ` tends to infinity9 . In particular, if P is C ∞ , so are the tori T n (ξ) and the associated quasi–periodic solutions. 1.5.4 The invariant tori T n (ξ), ξ ∈ Π∞ , correspond to non–maximal quasi–periodic solutions with n rationally independent (uniformly Diophantine) frequencies ω∞1 ,...,ω∞n . “Uniformly Diophantine” means that there exist constants γ∞ > 0, τ > n − 1 such that γ∞ , ∀ k ∈ Zn \{0} , ∀ ξ ∈ Π∞ . (1.6) |hω∞ (ξ), ki| ≥ 1 + |k|τ In fact, a slightly stronger Diophantine property holds, since (1.6) holds also replacing hω∞ (ξ), ki with hω∞ (ξ), ki + λ, where λ ≡ λ(ξ) denotes “T n (ξ)–normal frequencies” or differences of such normal frequencies. 1.5.5 A detailed and quantitative version of Theorem 1.1 is given in Proposition 3.1 (convergence of the KAM iteration) and in Proposition 3.2 (measure estimates on Π ∞ ) below. 1.5.6 The “smoothing technique” we shall use is due to Jackson, Moser and Zehnder (compare [26]) and it is rather different from the original strategy introduced by Nash and used by Moser in the context of dynamical systems. The Jackson–Moser–Zehnder technique is based on approximating the C ` perturbation P by real–analytic functions on smaller and smaller complex neighborhoods, solving linearized (analytic) equation to a better and better degree (keeping careful quantitative track of the procedure) and recovering in the limit a smooth (at least C 2 in our case) solution. Roughly speaking, for ν ≥ 1, we pick a suitable (super–exponentially) fast decaying sequence of positive numbers σν ↓ 0 and we fix (using a Lemma of Jackson–Moser–Zehnder) a sequence of functions P (ν) , which are real–anaytic in complex strips of size σν so that P (1) is “close” to P and so that |P (ν+1) − P (ν) | ∼ σν` . We then solve, iteratively, the “homological” equation (N + P (ν) ) ◦ Φν = Nν+1 + Pν+1 , where Φν is symplectic, Nν+1 is in normal form and |Pν+1 | ∼ |Pν |κ for a suitable (fixed) 1 < κ < 4/3. Since P (ν) → P in the C p –norm one can recover, in the limit ν → ∞, C p –quasi–periodic solutions for the original system. For a somewhat more precise – but still informal – presentation of the method, we refer to § 2.2 below. We point out that we do not use directly an analytic theorem (as done, for instance, in [26]), nor an analytic theorem can be immediately extracted from our approach. 1.5.7 The assumption ` > 6n + 5 is certainly not optimal. It would be interesting to find the optimal value: for example, is it true that Theorem 1.1 holds provided ` > 2n (as in the maximal case)? 1.5.8 Part of the proof relies on analytic tools elaborated in [21] and we, therefore, follow quite closely the notations introduced in [21]. Another reason for using notations borrowed from [21] is that it might facilitate the extension of our results to infinite (m = ∞) dimension. However, we restrain to do so here since we believe that such an extension makes sense only if applied to a real infinite dimensional problem, such as, for example, some “relevant” nonlinear PDE. 9
Fix τ > n − 1, let `∗ = 11 + 6τ and let θ > 0 be such that (3.72) below.
5
(1+θ)2 1−3θ
=
` −2 `∗ −2 ;
then a =
2 θ 3 (1+θ)2 ;
compare
1.6 The (normal) form (1.2) of the integrable piece N is, by now, rather standard in the present context (compare, e.g., [21], [27]). However, we mention briefly how more classical situations may be included in the present formulation. As an example, consider a Hamiltonian h(ϕ, I, q, p; ε) = h0 (I, q, p) + εh1 (ϕ, I, q, p; ε) , where (ϕ, I) and (q, p) are pairs of standard symplectic coordinates with ϕ ∈ Tn , I ∈ B1 (0) ⊂ Rn and (p, q) in a small neighborhood of the origin in R2m . Assume that h0 ∈ C `+3 , that h1 ∈ C ` . Fix a point I0 , say I0 = 0, and assume that (q, p) = (0, 0) is a (linearly) stable equilibrium for (q, p) → h(0, q, p). If such an equilibrium is non– 2 degenerate (i.e., if the Hessian matrix ∂(q,p) h0 (0, 0, 0) is invertible), then (up to symplectic change of coordinates) we may assume that (q, p) = (0, 0) is a non–degenerate, stable equilibrium for (q, p) → h0 (I, q, p) for any I ∈ Bρ (0) for some ρ > 0. Assume, further, 2 that the eigenvalues of Jm ∂(q,p) h0 (0, 0, 0), (Jm being the unit standard (m×m)–symplectic matrix), are purely imaginary (“linear stability”) and simple and are given by ±iΩj with Ωj > 0 and j = 1, ..., m. Finally, assume that also the Hessian matrix ∂I2 h0 (0, 0, 0) is invertible (this assumption corresponds to the classical KAM non–degeneracy condition). Then, expanding h0 in a neighborhood of (ξ, 0, 0) ≡ (I0 , 0, 0), (ξ ∈ Bρ/2 ), by a classical result of Weierstrass on the diagonalization of quadratic symplectic forms, one sees that the Hamiltonian h0 + εh1 can be cast in the form (1.1)–(1.2) with ξ ≡ I0 , e ≡ h0 (ξ, 0, 0), ω ≡ ∂I h0 (ξ, 0, 0), y ≡ (I − ξ) ≡ (I − I0 ), (u, v) ≡ (q, p), with a C ` perturbation function P such that P ≡ O(|y|2) + O(|y||(u, v)|) + O(|(u, v)|3) + O(ε) . (1.7) Notice that, because of the simplicity of the eigenvalues, the dependence of Ωj upon ξ (possibly reducing ρ) is of class C `+1 . From Theorem 1.1 (or, more precisely, from its quantitative version given in Propositions 3.1 and 3.2 below), it follows that if we 1 choose r ≡ ε 3 and ε is small enough, then, generically, for any ξ in a Cantor subset of Bρ/2 of density O(1 − εa ), (for some 0 < a < 1), the unperturbed n–dimensional tori y = 0 = u = v, x ∈ Tn may be continued into C p h–invariant tori; for more details, see Remark 3.1 below.
2 2.1
Preliminaries Analytic approximants
We start by recalling a well known and fundamental approximation result. Lemma 2.1 (Jackson, Moser, Zehnder) Let f ∈ C p (Rk ) for some p > 0 with finite C p norm10 over Rk . Let φ be a radial–symmetric, C ∞ function, having as support the older norm of If p is not integer, the C p norm |f |C p denotes the C [p] norm of f plus the (p − [p])–H¨ the derivatives of order [p] ([p] denoting, as usual, the integer part of p). 10
6
closure of the unit ball centered at the origin, where φ is completely flat and takes value 1; let K = φˆ be its Fourier transform and for all σ > 0 define fσ (x) ≡ Kσ ∗ f (x) ≡ σ
−n
Z
Rk
K
x − y
σ
f (y)dy .
(2.1)
Then, there exist a constant c ≥ 1 depending only on p and k such that the following holds. For any σ > 0, the function fσ (x) is a real–analytic function on Ck such that, if ∆kσ denotes the k–dimensional complex strip of width σ ∆kσ ≡ {x ∈ Ck : | Im xj | ≤ σ , ∀ j} ,
(2.2)
then, for all α ∈ Nk such that |α| ≤ p, one has11
sup ∂ α fσ (x) −
x∈∆kσ
and, for all 0 ≤ s ≤ σ,
∂ β+α f ( Re x) (i Im x)β ≤ c |f |C p σ p−|α| , β! |β|≤p−|α| X
sup |∂ α fσ − ∂ α fs | ≤ c|f |C p σ p−|α| .
(2.3)
(2.4)
x∈∆ks
Moreover, the H¨ older norms of fσ satisfy, for all 0 ≤ q ≤ p ≤ r, |fσ − f |C q ≤ c |f |C p σ p−q ,
|fσ |C r ≤ c
|f |C p . σ r−p
(2.5)
The function fσ preserves periodicity (i.e., if f is T –periodic in any of its variable xj , so is fσ ). Finally, if f depends on some parameter ξ ∈ Π ⊂ Rn and if the Lipschitz semi–norm of f and its x–derivatives are uniformly bounded by |f |Lip C ` , then all the above estimates hold with | · | replaced by | · |Lip . Remark 2.1 (i) As pointed out in [26], (2.5) yields easily the following classical bounds, valid for any12 0 ≤ r ≤ p ≤ q: q−r q−p |f |C |f |p−r (convexity estimates) p ≤ c |f |C r Cq , |f g|C p ≤ c (|f |C p |g|C 0 + |f |C 0 |g|C p ) .
(2.6) (2.7)
(ii) The proof of this lemma (including the statement on dependence upon parameters) consists in a direct check (based on standard tools from calculus and complex analysis); for details see [26] and references therein. ∂ α1 +···+αk f αk . 1 ∂xα 1 · · · ∂xk 12 Clearly, in (2.6) the constant c depends on r, p, q, while in (2.7) the constant c depend only on p. 11
“∂ α f ” means
7
In order to apply the lemma so as to construct a sequence {P (ν) } of real–analytic approximants of the perturbation P we first extend P to R2(n+m) (recall that P needs only be defined in a neighborhood of Tn × {y = 0} × {u = v = 0}): it is clear that if P is defined on Tn × Bd1 ,d2 ≡ Tn × {|y| < d1 } × {|u| < d2 , |v| < d2 }, then one can easily construct a C ` –extension Pext of P |Tn ×Bd1 /2,d2 /2 onto R2(n+m) , (mantaining periodicity in the first n variables and sharing the same properties of P with respect to the parameter ξ), and so that13 |Pext |C ` (R2(n+m) ) ≤ a |P |C ` (Tn ×Bd1 ,d2 ) where a is a suitable positive constant depending only on ` and di . Notational Remark 2.1 From now on we shall replace P by such an extension Pext , which, with abuse of notation, we shall again denote P . Now, given a decreasing sequence (to be fixed later) σν ↓ 0, ν ≥ 1, we define the real– analytic approximant P (ν) as14 P (ν) ≡ P2σν ≡ K2σν ∗ P ;
(2.8)
(compare Lemma 2.1). Notational Remark 2.2 From now on, we shall denote the 2(n+m) dimensional complex strip of width σ, ∆2(n+m) , simply ∆σ . σ
2.2
Newton (KAM) scheme
The proof of Theorem 1.1 is based upon the following KAM iteration scheme. We shall construct, inductively, real–analytic symplectic transformations Φν , ν ≥ 1, so that (N + P (ν) ) ◦ Φν = Nν+1 + Pν+1 ,
(2.9)
where the sequence of Nν ’s is in “normal form”, Nν (y, u, v; ξ) ≡ eν (ξ) +
n X
ωνj (ξ)yj +
j=1
m 1X Ωνj (ξ)(u2j + vj2 ) , 2 j=1
(2.10)
and the sequence of real–analytic functions Pν represent real–analytic perturbations of smaller and smaller size: kPν+1 k ∼ kPν kκ , (2.11) where k · k denote analytic norms in suitable smaller and smaller ν–dependent complex neighborhoods of Tn × {y = 0, u = 0, v = 0} (to be specified below) and 1 < κ < 2 is a fixed number. The parameter ξ appearing in (2.10) will vary in smaller and smaller 13
In fact, one can take Pext = ψ · P , ψ being a function of y, u, v having value 1 on Bd1 /2,d2 /2 and vanishing outside Bd1 ,d2 . 14 The (irrelevant) presence of the factor 2 will be explained in § 2.3.
8
compact sets Πν ⊂ Πν−1 ⊂ · · · ⊂ Π1 ⊂ Π, the choice of which will depend upon the solution of the associated small divisor problem. Under suitable conditions, the (Cantor) set Π∞ ≡ ∩ Πν will be shown to have positive Lebesgue measure. We will also show that the sup–norm of P (1) on the 2(n + m)–dimensional strip of width σ1 is small with the C 1 norm of P , provided σ1 is small enough; the smallness of P (1) will enable us to turn on the iteration procedure. The symplectic map Φν will have the form Φν = Φν−1 ◦ φν = φ1 ◦ · · · ◦ φν . Thus, by induction (for ν ≥ 2), (2.9), takes the form (Nν + Pν + (P (ν) − P (ν−1) ) ◦ Φν−1 ) ◦ φν = Nν+1 + Pν+1 .
(2.12)
From (2.4) it follows that the difference P (ν) − P (ν−1) satisfies ` . sup |P (ν) − P (ν−1) | ≤ 2`+1 c|P |C ` σν−1
(2.13)
∆2σν
Therefore, if we choose σν ∼ kPν kq , with a small positive q > 0 (taking into account the iterative relation (2.11) and that ` will be large enough), we see that the term kP (ν) − P (ν−1) k can be bounded by kPν k and equation (2.9) may be rewritten as (Nν + Pν0 ) ◦ φν = Nν+1 + Pν+1 ,
(2.14)
Pν0 ≡ Pν + (P (ν) − P (ν−1) ) ◦ Φν−1 ,
(2.15)
where and, by the above discussion, kPν0 k ∼ kPν k .
(2.16)
Equation (2.14) fits now in more standard KAM approaches and, in fact, the techniques used in, e.g., [21], allow to equip this scheme with the necessary estimates. It is important to notice, however, that in order for this approach to work the map Φν has to verify suitable compatibility relations with respect to the analyticity domains (compare the inductive relation (2.12)). More precisely, if Dν denotes the analyticity domain of Pν (which will be a suitable complex neighborhood of Tn × {y = 0} × {u = 0 = v}), we will show that φν : Dν+1 → Dν , (∀ ν ≥ 1) ,
Φν−1 : Dν → ∆σν , (∀ ν ≥ 2) .
(2.17)
Once all the above has been established, the convergence argument is routine: from the definition of P (ν) it follows that P (ν) tends to P in, say, C `−1 norm; the sequence
9
of diffeomorphysms x → Φν (x, 0, 0, 0; ξ) will be shown to converge in C p –norm (for a suitable p ≥ 2) to a C p diffeomorphysm x → ψ(x; ξ), which is Lipschitz continuous in ξ. Therefore, from (2.9), the (fast) convergence of Nν to N∞ = e∞ (ξ) + hω∞ (ξ), yi + 1 P Ω∞j (ξ)(u2j + vj2 ) (and from the fact that the size of the analyticity radii measuring 2 Dν goes to zero much slower than the size of kPν k) it follows that
T n (ξ) ≡ ψ Tn ; ξ ,
ξ ∈ Π∞ ≡
\
Πν ,
(2.18)
ν
is an invariant torus for N +P and that, on such a torus, the flow is C p –conjugated to the Kronecker flow x → x + ω∞ t, ω∞ being a “Diophantine vector” (see below). Furthermore (because of the form approximated normal forms Nν ) the tori T n (ξ) are linearly stable. On the small divisor problem The small divisors arising in solving the linearized equation associated to (2.14) are of the form hων (ξ), ki + hΩν (ξ), li , (2.19) where the Fourier/Taylor indices k and l verify the constraints n
(k, l) ∈ ZKν ≡ (k, l) ∈ Zn+m \{0} ,
|k| ≤ Kν ,
o
|l| ≤ 2 ,
(2.20)
for a suitable “cut–off” Kν ↑ ∞. The limitation on l comes from the fact that, choosing the neighborhood of the y, u and v origin in a suitable way, one may consider only lower order terms in y and (u, v),“lower” meaning, here, terms up to order 1 in y and up to order 2 in (u, v). The limitation on k is reminiscent of the Fourier “cut–off” introduced originally by Arnold ([1]); the difference being that, while in Arnold’s proof one can take the cut–off Kν to be proportional to the logarithm of the inverse of the size of the perturbation kPν k, here we have to take it to be proportional to a (small) inverse power of the size of the perturbation kPν k, making the treatment of the convergence of the algorithm somewhat more delicate. The sets Πν are, basically, subsets of Π where the small divisors (2.19) obey a Diophantine condition of the type |hων (ξ), ki + hΩν (ξ), li| ≥
γν , 1 + |k|τ
∀ (k, l) ∈ ZKν ,
∀ ξ ∈ Πν ,
(2.21)
where γν is a decreasing sequence bounded away from zero and τ > n − 1. The non– degeneracy assumptions on ω and Ω will guarantee that, at the end, the set Π∞ is non–empty, and, in fact, is of positive Lebesgue measure (compare Proposition 3.2). Finally, the map ξ ∈ Π∞ → ω∞ (ξ) is easily seen to be a Lipschitz homeomorphism so that, in particular, to different ξ correspond different tori T n (ξ).
2.3
Complex variables and norms
To treat the linearized equation, it is convenient to introduce complex variables in a neighborhood of u = v = 0. Consider the following linear change of variable (u, v) ∈
10
C2m → (z, z) ∈ C2m : 1 z = √ (u + iv), 2
1 z = √ (u − iv) , 2
(2.22)
1 u = √ (z + z) , 2
1 v = √ (z − z) . i 2
(2.23)
and its inverse map15
This map is not symplectic; however the Poisson bracket, the symplectic form and Hamilton equations transform in a simple way: if (as above) (x, y) and (u, v) are couple of conjugate symplectic variables and if f and g are functions of (x, y, u, v) then, with the obvious meaning of the symbols16 , {f, g} ≡ {f, g}x,y,u,v = {f, g}x,y + {f, g}u,v = {f, g}x,y − i{f˜, g˜}z,z ≡ {f, g}e .
(2.24)
The symplectic form dx ∧ dy + du ∧ dv reads dx ∧ dy − idz ∧ dz and the Hamiltonian vector field Xf ≡ (fy , −fx , fv , −fu ) , (2.25) is trasformed into17
f ≡ (f˜ , −f˜ , −if˜ , if˜ ) . X z y x z fe
(2.26)
f = e + hω(ξ), yi + hΩ(ξ) z, zi , N
(2.27)
In the variables (x, y, z, z) the function N takes the form
where we identify the vector Ω = (Ω1 , ..., Ωm ) with the diagonal matrix diag (Ω1 , ..., Ωm ), f and an analytic function still denoted Ω. The Poisson bracket between N f (x, y, z, z) =
X
fkqq (y)eihk,xi z q z q ,
k∈Zn q,q∈Nm
is given by f, f }e = −i {N
X
k∈Zn q,q∈Nm
hω, ki + hΩ, q − qi fkqq (y)eihk,xi z q z q .
15
(2.28)
Beware that, as standard in this context, z does not denote the complex conjugate of z; rather, z and z denote a set of 2m independent variables. Of course, when u and v are restricted to the real space then, indeed, z and z are complex conjugate. This change of variables is standard, for example, in the theory of Birkhoff normal forms. P 16 {f, g}x,y = j fxj gyj − fyj gxj , etc.; f˜(x, y, z, z) = f x, y, √12 (z + z), i√1 2 (z − z) , etc. 17 In other words, the Hamilton equation for f (x, y, u, v) are equivalent to the “Hamilton equation” for f˜ given by x˙ = f˜y , y˙ = −f˜x , z˙ = −if˜z , z˙ = if˜z .
11
Let us now fix the norms we shall work with. In CN we shall use maximum norm: if a ∈ CN , |a| ≡ maxi |ai |; for Fourier indices k ∈ ZN or Taylor indices k ∈ NN , |k| denotes, P as usual, i |ki |. As norms on matrices we take the standard operator norm (with respect to the above maximum norms). Following [21], Hamiltonian functions will be measured by the following weighted sup–norm. For s, r > 0, let D(s, r) = { (x, y, z, z) ∈ C2(n+m) : | Im x| < s, |y| < r 2 , |z| < r, |z| < r } .
(2.29)
We then let kXf kr ≡ |fy | +
1 1 1 |fx | + |fz | + |fz | , 2 r r r
and
kXf kr,
D(s,r)
≡ sup kXf kr . D(s,r)
The Lipschitz semi-norm with respect to the parameter ξ ∈ Π (or in subsets of Π, which will be clear from context) is defined analogously18 : Lip kXf kLip + r ≡ |fy |
1 1 1 |fx |Lip + |fz |Lip + |fz |Lip , 2 r r r
Lip and kXf kLip . r, D(s,r) ≡ sup kXf kr D(s,r)
Notational Remark 2.3 In the following we shall use the notation “a ≤ const b” meaning “there exists a constant c depending only on τ , n and ` such that a ≤ cb” (obviously in such estimates, the constants c’s will be, in general, different one from another). The notation k · k∗ stands for either k · k or k · kLip . Since | Im z|, | Im z| ≤ σν
=⇒
we see that the functions
| Im u|, | Im v| ≤
√
2σν ,
z+z z−z Pe (ν) (x, y, z, z; ξ) ≡ P (ν) x, y, √ , √ ; ξ 2 i 2
(2.30)
are analytic and bounded on ∆σν . In fact, for any |α| ≤ `, one finds immediately sup |∂ α Pe (ν) |∗ ≤ const sup |∂ α P (ν) |∗ ,
(2.31)
∆2σν
∆σ ν
so that, from (2.13), there follows
`−|α|
sup |∂ α (Pe (ν) − Pe (ν−1) )|∗ ≤ const |P |∗C ` σν−1 , ∆σ ν
∀ |α| ≤ ` .
(2.32)
Notational Remark 2.4 The KAM algorithm outlined in the previous subsection will be described in terms of the (x, y, z, z) variables but for ease of notation we shall drop systematically the tilde from functions, vector fields and Poisson brackets, keeping in mind the actual meaning just discussed. Obviously, in the convergence argument sketched at the end of the previous subsection one has to go back to the (x, y, u, v) variables (since the original perturbation function P is only defined for real arguments). We shall not come back on this (mathematically) trivial point, hoping that the notation will cause no confusion. 18
Recall footnote 6.
12
3
Proof of the main Theorem
3.1
The linearized equation
In this section, following quite closely [21], we shall describe the linearized equation associated to the KAM scheme presented in § 2.2 (compare, in particular, (2.14)), and how to solve it. Let
P1 ≡ P10 ≡ Pe (1) .
(3.1)
0 < r ν < s ν < σν < 1 ,
(3.3)
and assume that, for ν ≥ 1, Pν and Pν0 have vector fields real–analytic and bounded in a domain Dν ≡ D(rν , sν ) ⊂ ∆σν (3.2) for suitable numbers
to be specified later. Notational Remark 3.1 In this section we shall drop the index ν and replace the index “ν + 1” by the index “+”. Therefore, N , P , P 0 , φ, r,... stand for Nν , Pν , Pν0 , φν , 0 rν ... while N+ , P+ , P+0 , φ+ , r+ ,... stand for Nν+1 , Pν+1 , Pν+1 , φν+1 , rν+1 ,... The symplectic map φ(= φν ) will be taken to be the time–one map of a Hamiltonian flow XFt associated to a Hamiltonian function F (with kXF k ∼ kXP k ∼ kXP 0 k). In such a case, the left hand side of (2.14) takes the form: (N + P 0 ) ◦ XF1 = N + ({N, F } + P 0 ) + O2 ,
(3.4)
where O2 denotes (loosely) terms of order two in F . Therefore, the “linearized equation” to be solved for F has the form c+ O , {N, F } + P 0 = N 2
(3.5)
c denotes a term in “normal form19 ” (i.e., having the same form of N ). Since where N one is interested in solving (3.5) in a small neighborhood of {y = 0, z = z = 0}, one can truncate the Taylor expansion of P 0 up to order one in y and up to order two in (z, z). Also, in order to control the small divisors (for a “large” set of parameter), as in [1], one can truncate the Fourier expansion up to order K. Thus the equation to be solved becomes: c, {N, F } + R = N (3.6)
where
R=
X
0 Pklqq eihk,xi y l z q z q ,
(3.7)
2|l|+|q+q|≤2 |k|≤K
19
Clearly, the equation {N, F } + P = O2 might not have a solution since P , in general, will not belong to the range of the operator {N, ·}.
13
(recall that the Fourier–Taylor coefficients of P 0 are Lipschitz–continuous functions of ξ). Thus, R is a second degree polynomial in (z, z) (and first degree polynomial in y) having the form: R ≡ R0 + R1 + R2 ≡ R0 (x, y) + R1 (x, z, z) + R2 (x, z, z) ,
(3.8)
where (without indicating explicitely the Lipschitz continuous dependence upon ξ) R0 ≡ R000 (x) + hR001 (x), yi , R1 ≡ hR10 (x), zi + hR01 (x), zi , R2 ≡ hR20 (x)z, zi + hR11 (x)z, zi + hR02 (x)z, zi .
(3.9)
We notice (for later reference) that from such definitions there follows P 0 = R + O(|y|2) + O(|z| |y|) + O(|z|3 |) ,
(3.10)
so that R000 = P 0 (x, 0, 0, 0) , R001 = ∂y P 0 (x, 0, 0, 0) , R10 = ∂z P 0 (x, 0, 0, 0) , R01 = ∂z P 0 (x, 0, 0, 0) , 1 R20 = ∂z2 P 0 (x, 0, 0, 0) , R11 = ∂z ∂z P 0 (x, 0, 0, 0) , 2
(3.11) 1 R02 = ∂z2 P 0 (x, 0, 0, 0) . 2
The projection of R onto the kernel of {N, ·} (sometimes referred to as the “mean value of R”) is given by [R]
=
X
0 0 + P0lqq y l z q z q = P0000
= ≡
0 P0l00 yl +
|l|=1
|l|+|q|≤1
R0000
X
hR0001 , yi
hR011 z, zi
X
0 P00qq zq zq
|q|=1
+ + b eˆ + hˆ ω , yi + hΩz, zi .
(3.12)
Therefore, [R] is in normal form and we can set c ≡ [R] . N
(3.13)
At this point, recalling (2.28), we can easily solve (3.6): F =
X
Fklqq eihk,xi y l z q z q ,
with Fklqq =
2|l|+|q+q|≤2 |k|≤K (k,q−q)6=(0,0)
−iRklqq . hω, ki + hΩ, q − qi
(3.14)
Obviously, F is real for real argument. c and F , one can rewrite (3.4) as Having thus defined R, N
(N + P 0 ) ◦ XF1 = N+ + P+ ,
(3.15)
with c, N+ ≡ N + N
P+ ≡
Z
1 0
c + tR, F } ◦ X t dt + (P 0 − R) ◦ X 1 . {(1 − t)N F F
14
(3.16)
3.2
Iteration step
In this section we describe the estimates associated to one step of the KAM iteration described in § 2.2. We start by discussing estimates associated to the solution Fν ≡ F given in (3.14) (we re–insert the dependence upon the iteration step ν). Assume the Diophantine condition (2.21) and assume that |ων |Lip + |Ων |Lip ≤ Mν ,
|ων−1 |Lip ≤ Lν ,
(3.17)
for some positive numbers Lν , Mν such that Lν Mν ≥ 1 (the Lipschitz semi–norms are taken, respectively, on Πν and on ων (Πν )). Then, by classical KAM estimating techniques – mainly based on Cauchy estimates20 – one finds the following bounds; for details, compare with § 2 and, in particular with Lemma 1 and Lemma 2 of [21]: ∗ kXN cν krν ,
kXFν krν ,
D(sν ,rν ) D(sν /2,rν )
kXFν kLip rν , D(sν /2,rν )
≤ const kXRν k∗rν , D(sν ,rν ) , B sν ≤ const kXRν krν , D(sν ,rν ) , γν B sν M kXRν krν , ≤ const kXRν kLip rν , D(sν ,rν ) + γν γν
D(sν ,rν )
,(3.18)
where, changing slightly the notation with respect to [21] and using R¨ ussmann’s subtle arguments to give optimal estimates of small divisor series (see [23] and [24]), B sν ≡ γ ν
2
v u u u t
X
(k,l)∈Zn+m \{0} |l|≤2
|k|2 e−2|k|sν ≤ const sν −τ1 , 4 |hων , ki + hΩν , li|
τ1 ≡ 2τ + 1 . (3.19)
As in [21], we observe that, setting k · kλr ≡ k · kr + λk · kLip , r
(3.20)
the second and the third inequality in (3.18) are equivalent to the inequality kXFν kλrν ,
D(sν /2,rν )
≤ const
B sν kXRν kλrν , γν
D(sν ,rν )
,
∀0≤λ≤
γν . Mν
Thus, in view of (3.19), (3.18) may me rewritten more compactly as ∗ kXN cν krν ,
kXFν kλrν ,
D(sν ,rν ) D(sν /2,rν )
≤ const kXRν k∗rν , D(sν ,rν ) , 1 ≤ const kXRν kλrν , D(sν ,rν ) , τ 1 γν sν
20
(3.21)
∀0≤λ≤
γν .(3.22) Mν
Cauchy estimates give a bound of derivatives of an analytic function on complex domains in terms of the maximum norm of the function in larger domains: if f is analytic on a domain D ⊂ Ck , then supD−δ |∂ α f | ≤ α! δ −|α| supD |f |, where D−δ denotes the set of δ–inner points of D (i.e., those points x for which a ball of center x and radius δ is contained in D). For a generalized version, see, e.g., Lemma A4, pag 147, of [21].
15
To carry on the KAM step we shall make inductive hypotheses that will be checked in the next section, where the convergence of the KAM algorithm is discussed. We assume that Pν and Pν0 are such that γν εν , kXPν krν , D(sν ,rν ) + kXPν kLip rν , D(sν ,rν ) ≤ Mν 2 γν γν sν τ2 ην2 kXPν0 krν , D(sν ,rν ) + kXPν0 kLip ≤ ε ≤ , ν rν , D(sν ,rν ) Mν c0 where: τ2 ≡ τ 1 + 2 ,
(3.23) (3.24) (3.25)
c0 > 1 is a suitable constant depending only on n and τ , 0 < ην < 1/16 will be a small number (to be fixed later). The role of ην will be that of rescaling the y and z, z–neighborhood of the origin so that terms of order two in y or three in (z, z) may be “disregarded”(compare with (3.49) below). In the following estimates we shall make repeated use of Cauchy estimates on smaller domains that we shall denotes here, for short, Dνj ≡ D(2−j sν , 2−j rν ), j = 1, 2, 3, 4. Indeed, we shall take rν sν , rν+1 ≡ ην rν < , σν+1 = 2sν+1 . (3.26) sν+1 ≤ 16 16 Estimates on the symplectic transformation φν ≡ XF1ν . Observing that the gradient of Rν (appearing in the definition of the norm of XRν ) is defined in terms of derivatives of Pν0 , one gets (by Cauchy21 ) kXRν k∗rν ,
Dν1
≤ const kXPν0 k∗rν ,
D(sν ,rν )
.
(3.27)
Recalling (3.22), we find kXFν kλrν ,
Dν1
≤ const
1 kXPν0 kλrν , τ 1 γν sν
D(sν ,rν )
.
(3.28)
Then, by (3.28) and by the assumption (3.23), kXFν krν ,
Dν1
≤ const s2ν ην2 .
(3.29)
Thus, XFt ν : Dν2 → Dν1 for all −1 ≤ t ≤ 1 and, by a standard ODE result22 , we get kXFt ν − idkrν ,
Dν2
≤ const s2ν ην2 .
(3.30)
21
0 Recall (3.8), (3.9) and (3.11). Then, |Rν,y | ≤ |Pν,y | on D(sν , rν ). Next, Rν,x is the second order 0 in (z, z) truncation of Pν,x - and the estimates on D(sν , rν /2) follows from Cauchy estimates on (z, z)– coefficients. Notice that, in fact, there is no need for such estimates of reducing the x–domain. 22 I.e., essentially Gronwall lemma, which, to fit our purposes may be reformulated as follows: Let V be an open domain in a real Banach space E with norm k · k, Π a subset of another real Banach space, and X : V × Π → E a parameter–dependent vector field on V , which is C 1 on V and Lipschitz on Π. Let φt be its flow. Suppose there is a subdomain U ⊂ V such that φt : U × Π → V for −1 ≤ t ≤ 1.Then
kφt − idkU ≤ kXkV ,
Lip kφt kLip U ≤ exp(k∂XkV )kXkV ,
for −1 ≤ t ≤ 1, where all norms are understood to be taken also over Π. Notice that kφ t − idkLip U = t Lip kφ kU . For a (standard) proof, see [21] p. 147.
16
To estimate the derivatives of XFν , we recall that, because of the particular structure of Fν , the x–component of XFν is independent of y, u, v, while the (z, z)–components are independent of y. Thus, by Cauchy estimates (and recalling that rν < sν < 1), we get |∂XFν |Dν2 ≤ const s−1 ν kXFν krν ,Dν1 .
(3.31)
By the above cited standard ODE result, (3.29) and (3.31) we obtain Lip t 2 2 |XFt ν |Lip Dν3 = |XFν − id|Dν3 ≤ const sν ην .
(3.32)
Moreover, by Cauchy estimates, for any −1 ≤ t ≤ 1 and for p = 1, 2, p t ∗ 2−p 2 |∂ p XFt ν |Lip Dν4 , |∂ (XFν − id)|Dν4 ≤ const sν η .
(3.33)
Assuming
1 , (3.34) 8 (a fact which shall be verified in next section), (3.33) implies that the Jacobian matrix of XFt ν , ∂XFt ν , is invertible and close to the identity23 : const ην2 <
|(∂XFt ν )−1 − I|Dν4 ≤ const sν ην2 ,
∀ −1≤t≤1 .
(3.35)
Also, from such relation, (3.33) and (3.34), one obtains the following bound on the Lipschitz semi–norm: 2 −2 2 |(∂XFt ν )−1 |Lip Dν4 ≤ (1 − const sν ην ) (1 + const sν ην ) ≤ 2 .
(3.36)
As already observed above (after (3.29)), φ ≡ XF1ν : Dν2 → Dν1 and, therefore (compare (2.17) and (3.26)), φν : Dν+1 → Dν . (3.37) We, now, make the following inductive assumption (which shall be easily verified in the next section): (3.38) |∂Φν |∗Dν+1 ≤ 2 . From this assumption it follows immediately that
Φν (D(sν+1 , rν+1 )) ⊂ ∆σν+1 ,
(3.39)
completing the proof of (2.17): Suppose, in fact, that w = Φν (ς), ς ∈ D(sν+1 , rν+1 ). Since Φν is real for real argument24 , we have | Im w| = | Im Φν (ς)| = | Im Φν (ς) − Im Φν ( Re ς)| ≤ |Φν (ς) − Φν ( Re ς)| ≤ |∂Φν |D(sν+1 ,rν+1 ) | Im ς| ≤ 2| Im ς|.
P∞ Use Neumann identity A−1 − I = j=1 (I − A)j valid for any matrix A such that |I − A| < 1, (| · | being any operator norm). 24 Φν is composition of φν ’s = XF1ν ’s and Fν is real for real argument (recall (3.14) and the remark after it). 23
17
Estimates on ων+1 , Ων+1 . Recalling (3.13), (3.12) and (3.11), by Cauchy estimates, one finds |ˆ eν |∗ ≤ const rν2 kXPν0 k∗rν ,D(sν ,rν ) ≤ const kXPν0 k∗rν ,D(sν ,rν ) , b ν |∗ ≤ const rν kXPν0 k∗rν ,D(sν ,rν ) ≤ const kXPν0 k∗rν ,D(sν ,rν ) , |ω
(3.40)
b |∗ ≤ const kX 0 k∗ |Ω ν Pν rν ,D(sν ,rν ) .
Definition of Πν+1 and small divisor estimates. Recall that on Πν the small divisor bound (2.21) holds and define [
Πν+1 ≡ Πν \ where
(k,l)∈Zn+m \{0} |l|≤2 , |k|>Kν
Rνkl (γν ) ,
n
Rνkl (γν ) ≡ ξ ∈ Πν : |hων (ξ), ki + hΩν (ξ), li| <
(3.41)
γν o . 1 + |k|τ
(3.42)
For a given Kν+1 > Kν (to be specified later), let γν+1 be such that25 γν+1
τ +1 εν Kν+1 . ≤ γν 1 − const γν
(3.43)
Then, for ξ ∈ Πν+1 the small divisor bound (2.21) with ν replaced by (ν + 1) holds: by (2.21), the definition of Πν+1 , (3.40) and (3.43), for all (k, l) ∈ Zn+m \{0} such that |l| ≤ 2 and |k| ≤ Kν+1 , one has !
b , li| b ν , ki| + |hΩ |hω ν |hων+1 (ξ), ki + hΩν+1 (ξ), li| ≥ |hων (ξ), ki + hΩν (ξ), li| 1 − |hων , ki| + |hΩν , li| τ +1 εν Kν+1 γν 1 − const ≥ τ 1 + |k| γν γν+1 ≥ . (3.44) 1 + |k|τ 0 Estimates on Pν+1 and Pν+1 . Recall the definition of the new “perturbation function” Pν+1 given in (3.16). Let us first discuss the term (Pν0 − Rν ) ◦ φν and, in particular, the norm of the “tail” Qν ≡ Pν0 − Rν on a domain slightly larger than Dν+1 , namely, D(sν /2, 4rν+1 ) (recall (3.26)). First observe that Qν has the form
Qν ≡ Pν0 − Rν =
X
Pν0 ,lqq (x)y l z q z q +
X
|k|>K 2|l|+|q+q|≤2
2|l|+|q+q|>2
Pν0 ,klqq eihk,xi y l z q z q ≡ Q1ν + Q2ν . (3.45)
25 Clearly (recall (3.23)), εν is an upper bound on 2 kXPν krν ,
18
Lip γν D(sν ,rν ) + M ν kXPν krν , D(sν ,rν ) .
Taking into account the dependence on rν of the norm k · krν , one sees easily that26 kXQ1ν k∗ην rν ,
D(sν ,4ην rν )
≤ const ην kXPν k∗rν ,
D(sν ,rν )
.
(3.46)
The estimate for Q2ν brings in the dependence upon Kν (as in [1]) and one finds |∂Q2ν |∗ην rν , D(sν /2,4ην rν )
kXPν k∗rν , D(sν ,rν ) e−(Kν sν )/4 ≤ const . ην2 snν
(3.47)
Thus, assuming
c1 log(ην sν )−1 , sν with a suitable c1 ≡ c1 (n), from (3.46) and (3.47) there follows
(3.48)
Kν ≥
kXPν0 −Rν k∗ην rν ,
D(sν /2,4ην rν )
≤ const ην kXPν0 k∗rν ,
D(sν ,r)ν
.
(3.49)
Now, it is a general fact that, for any functions f and g and for any symplectic map φ, the following relations hold27 : X{f,g} = [Xf , Xg ] ≡ Jf 00 Xg − Jg 00 Xf ,
Xf ◦φ = φ∗ Xf ≡ (∂φ)−1 Xf ◦ φ .
(3.50)
At this point one has all the ingredients to estimate kXPν+1 krν+1 ,Dν+1 , arriving to the bound28 ! γν 1 . (kXPν kλrν )2 + ην kXPν kλrν , ∀ 0 ≤ λ ≤ kXPν+1 kλrν+1 ,Dν+1 ≤ const τ 2 2 γν sν ην Mν (3.51)
3.3
Convergence
In this section we iterate the KAM algorithm presented above and show its convergence. Let us introduce the following recursive parameters for ν ≥ 1. Let 1 < κ < 2, 0 < q < 1 be suitable constants (to be chosen later); let c2 ≡ c2 (τ, n, `) be a positive large enough constant29 . Then, for some 0 < ε1 1 and r1 ≥ εκ1 (to be specified later), we set γ Mν ≡ M (2 − 2−ν+1 ) , Lν ≡ L(2 − 2−ν+1 ) , γν ≡ (1 + 2−ν+1 ) , 2 κ κ−1 σν c2 ε ε εν+1 ≡ 1/3ν , σν ≡ εqν , sν ≡ , ην ≡ ν1/3 , 2 γν γν ν κ rν+1 ≡ ην rν , K ≡ c2 log ε−1 . (3.52) Kν ≡ K 1 , σν Observe that: M ≡ M1 ≤ Mν ↑ 2M ,
L ≡ L1 ≤ Lν ↑ 2L ,
1 > γ ≡ γ 1 ≥ γν ↓ γ∞ ≡
2−(2|l|+|q+q|)
26
γ . 2
(3.53)
2−(2|l|+|q+q|)
For | Im x| < sν , use |∂x Pν ,lqq | ≤ kXPν0 krν , D(sν ,rν ) rν ≤ 2kXPν krν , D(sν ,rν ) rν . 27 J denotes the standard symplectic matrix and f 00 the Hessian of f . 28 For full details, see [21], pag. 130–132. 29 In particular, one can take c2 = 16c where c denotes here the largest among all constants “ const ” appearing in the preceding sections.
19
Notational Remark 3.2 In this section the constant ci will denote suitable constants depending on τ , n, κ, q and `. We shall need some simple relations among the above parameters: Lemma 3.1 For any ν ≥ 1 (Aε1 )κ εν ≤ A
ν−1
,
A≡
c3 , γ a1
(3.54)
1
1
1 > 1 and c3 ≡ (2 3 c2 ) κ−1 . Furthermore, if ε1 is small enough, i.e., if, for with a1 ≡ (κ−1) a suitable c4 ≥ c3 ,
c4
ε1 <1, γ a2
then, for any ν ≥ 1,
n
κ 1 o , , 3(κ − 1)2 1 − qτ2
a2 ≡ max a1 ,
1 εν+1 < 1 . εν 16 q 2ν+1
rν ≥ εκν ,
(3.55)
(3.56)
Proof From (3.52) and (3.53), it follows that 1
εν+1
2 3 c2 κ εν . ≤ γ
(3.57) 1
1
Iterating such relation one gets (3.54) with c3 ≡ (2 3 c2 ) κ−1 . As for (3.56), observe that from the definitions (3.52) there follows εν+1 = c2 ην εν = cν2
ν Y
ηj ε1 ,
j=1
rν+1 =
ν Y
ηj r1 ,
j=1
hence rν+1 = εν+1
1 r1 , cν2 ε1
(3.58)
(3.59)
and the first relation in (3.56) is seen to be equivalent to ν−1
εν c2κ−1 ≤
r 1
1 κ−1
ε1
,
(3.60)
1
which, since εr11 κ−1 ≥ ε1 , follows from (3.54) and (3.55). From (3.54), choosing c4 big 1 ), there follows enough (and since a2 > 3(κ−1) 2ν+1
εν+1 εκ−1 = 2ν+1 c2 ν 1 εν γν3 ≤
1 16
1 q(κ−1)
c4 γ
1 3(κ−1)
20
ε1
κ−1
≤
1 1
16 q
.
Next Proposition is a detailed version of the main Theorem 1.1 apart from the claim concerning the measure of Π∞ , which shall be discussed in the next section. To state such Proposition we need some definitions. Given γ and M we introduce two numbers, β and δ, measuring the regularity and certain geometric properties of the perturbation P . Let β > 0 be such that n
max 1 ,
o γ γ , |P |C ` , |P |Lip ≤β . ` C M M
(3.61)
Now, let o
n
A1 ≡ α : |α| = 1 and |∂ α P |C 0 6= 0 ,
n
o
Lip α ALip 1 ≡ α : |α| = 1 and |∂ P |C 0 6= 0 ; (3.62)
let, then, δ > 0 be such that δ ≡ inf
(
inf α∈A1 |∂ α P |Lip inf α∈A1 |∂ α P |C 0 C0 , α α supα∈A1 |∂ P |C 0 supα∈A1 |∂ P |Lip C0
)
.
(3.63)
Finally, let n
o
DR1 ≡ (x, y, u, v) ∈ R2(n+m) : |y| < R12 , |u| < R1 , |v| < R1 , and define
ε0 ≡ kXP kR1 ,DR1 +
γ kXP kLip R1 ,DR1 , M
εˆ0 ≡ |XP |DR1 +
R1 ≡ 2r1 ,
γ |XP |Lip DR1 . M
(3.64)
(3.65)
Proposition 3.1 Let ` > `∗ ≡ 2 + 3τ2 = 11 + 6τ and let θ ≡ θ(`, τ ) > 0 such that30
and define q≡
(1 + θ)2 ` −2 = , 1 − 3θ `∗ − 2
(3.66)
1 − 3θ , τ2
(3.67)
κ≡1+θ .
Let31 ω1 ≡ ω, Ω1 ≡ Ω, L1 ≡ L, M1 ≡ M , γ1 ≡ γ. Assume (3.17) for ν = 1, let Π1 such that (2.21) holds for ν = 1. There exist a constant c5 > c4 > 1, depending on τ , n and ` and constants C1 , C2 > 1, depending upon τ , n, `, (LM ), γ, β and δ, such that, if ε1 ≡ c 5 β ε0 ,
C1 ε0 ≤ 1 ,
1 ) 1/(2+ κ
C2 εˆ0
≤ r1 ≤ 1 ,
(3.68)
then the following holds. Let Mν , Lν , γν , εν , rν , sν , σν , Kν be as in (3.52) with ε1 and r1 as in (3.68); let Dν be as in (3.2); let P (ν) be as in32 (2.30); let P1 be as in (3.1). Then, for ν ≥ 1, one can iteratively construct, as described in § 3.1, a sequence of real–analytic Whence, θ ∈ (0, 13 ). Beware, instead, that Π1 6= Π and P 6= P1 , K1 6= K. 32 Recall the Notational Remark 2.4. 30
31
21
symplectic transformations φν (and Φν ≡ φ1 ◦ · · · ◦ φν ) satisfying (2.17), and a sequence of functions Nν , Pν , Pν0 real–analytic on Dν satisfying (2.14). The functions indexed by ν are Lipschitz continuous in ξ ∈ Πν , where Πν is iteratively defined in (3.41). The following conditions hold for any33 ν: |ων |Lip + |Ων |Lip ≤ Mν ,
|ων−1 |Lip ≤ Lν ,
kXPν krν ,
Dν
+
εν γν kXPν kLip , (3.69) r ν , Dν ≤ Mν 2
γν 2γν sτν2 ην2 Lip 0 kX krν , Dν + , (3.70) kXPν krν , Dν ≤ εν ≤ Mν c0 as well as conditions (3.3), (3.26), (3.34), (3.38), (3.43) and (3.48). Furthermore, e ν (e1 ≡ 0), ων and Ων converge (super–exponentially fast) to functions e∞ , ω∞ and Ω∞ , which are Lipschitz continuous on Π∞ ≡ ∩ Πν and obey the bounds Pν0
|ω∞ |Lip + |Ω∞ |Lip ≤ 2M ,
−1 Lip |ω∞ | ≤ 2L .
For any 2 ≤ p < p∗ ≡ 2 + a(` − 2) ,
a≡
(3.71)
θ 2 , 3 (1 + θ)2
(3.72)
the diffeomorhysms x ∈ Tn → Φν (x, 0, 0, 0; ξ) converge in C p –norm to a C p –diffeomorphysm ψ(x; ξ), which is Lipschitz continuous in ξ ∈ Π∞ . In fact, for a suitable c6 > 1: |ψ(x; ξ) − x|C p ≤
c6
2θ
ε1
2
γ3
p∗ −p p∗ −2
2(q+θ)
,
∀ ξ ∈ Π∞ ;
|ψ|Lip ≤ c2
ε1
2
γ3
.
(3.73)
Finally, the tori T n (ξ) defined in (2.18) are invariant tori for N + P and, on such tori, the flow is C p –conjugated to the Kronecker flow x → x + ω∞ t where ω∞ verifies the Diophantine relation |hω∞ (ξ), ki + hΩ∞ (ξ), li| ≥
γ , 2(1 + |k|τ )
∀ (k, l) ∈ Zn+m \{0} , |l| ≤ 2 ,
∀ ξ ∈ Π∞ . (3.74)
Proof As a first step, let us check that the relation between εˆ0 and r1 in (3.68), namely 1/(2+ κ1 )
C2 εˆ0
≤ r1
(3.75)
implies that34 :
γ (3.76) kXPe1 kLip r 1 , D1 ≤ ε 1 . M Notice that, by the definition of norms and complex variables (2.22), it follows that kXPe1 kr1 ,
33
D1
kXPe1 kr1 ,
+
D1
≤ 2kXP1 kR1 ,D(R1 ,s1 ) ,
(3.77)
Recall (3.23). Only for the purpose of this check we re–introduce tildas to distinguish between functions of (x, y, z, z) and functions of (x, y, u, v); recall the Notational Remark 2.4. 34
22
so that, in the following argument, we may use directly the (x, y, u, v) variables. Introduce, also, for the purpose of this check, the short–hand notation “| · |• ” to denote either “| · |” or “(γ/M )| · |Lip” and observe that from the definitions of δ ((3.63)) and εˆ0 ((3.65)), it follows that ∀ α ∈ A1 . (3.78) δ εˆ0 ≤ |∂ α P |•C 0 , Observe, also, that, if 1
C2 ≥ const
β2 1 2q¯
δ
q¯ ≡ q(` − 1) ≡ 3(1 + θ)2
,
`−1 `−2
(3.79)
(for a suitable const ), then, since (as it easy to check) 1 q¯ − 1 > 2¯ q 2+
1 κ
,
equation (3.75) yields
β q¯ q¯−1 εˆ ≤ r12¯q . (3.80) δ 0 Thus, taking into account the weight of the norm k · kR1 appearing in the definition of ε0 , recalling the definitions of σ1 = εq1 , ε1 , q¯, (3.80) and (3.78), we find const
σ1`−1 ≡ εq1¯ = const β q¯εq0¯ ≤ const β q¯
εˆq0¯ ˆ0 δ ≤ const |∂ α P |•C 0 , 2¯ q ≤ const ε r1
∀ α ∈ A1 .
(3.81) Now, if ζ ≡ (x, y, u, v) ∈ ∆σ1 , by Lemma 2.1, the definition of β in (3.61), the convexity estimates (2.6) and (3.81) we find, for any α ∈ A1 , α
|∂ P1 (ζ)|
•
≤
α ∂ P1 (ζ) −
≤
cβσ1`−1
+
+c
• ∂ β+α P ( Re ζ) (i Im ζ)β β! |β|≤`−1 X
• ∂ β+α P ( Re ζ) (i Im ζ)β β! |β|≤`−1 X
`−1 X
m=0
|∂ α P |•C m σ1m
≤ cβσ1`−1 + const `−1 X
`−1 X
m=0
|∂ α P |•C 0
≤
const β
≤
const β
≤
const β |∂ α P |•C 0 .
m=0 `−1 X m=0
|∂ α P |•C 0
23
|∂ α P |•C 0
`−1−m `−1
`−1−m
|∂ α P |•C `−1
σ1m
`−1−m `−1
`−1
|∂ α P |•C 0
m `−1
m `−1
σ1m
From this relation, (3.77) and the definition of ε1 , we find immediately35 kXPe1 kr1 ,
D1
≤ const βε0 ≡ ε1 .
(3.82)
Recall (compare sentence before (3.52)) that we have to check that r1 ≥ εκ1 or, equiva− κ1
lently, ε1 r1
≤ 1: in fact, from the definition of ε1 and from (3.68), there follows 1
ε1 1 κ
r1
εˆ0
≡ const β
2+ 1 r1 κ
1
1 1 2+ κ
β 2+ κ εˆ0 = const r1
provided
1
≤1,
1
C2 ≥ const β 2+ κ .
(3.83)
To proceed, it is convenient to reformulate the smallness condition, C1 ε0 ≤ 1, on ε0 (which will not appear any more in the sequel) in terms of ε1 . It is easily seen that C1 ε0 ≤ 1 implies that36 1
c7 β θ (LM )
n
2(τ +1) ε1 (log ε−1 1 ) <1, γ a3
a3 ≡ max a2 ,
2 3κ − 1 o , , 3θ κ(κ − 1)
(3.84)
for a suitable c7 > c5 . Notice that (3.84), in turn, implies (3.55). Next, the inequality ε1 ≤ is equivalent to
2 τ2 c0
γsτ12 η12 c0
1−qτ2 −2(κ−1)
2 τ2 c0
εθ1 ≤ 1 , 2 γ γ3 which follows from the smallnes condition (3.84) (and the fact that a3 ≥ 2/(3θ)). Thus (3.23) and (3.24) are satisfied for ν = 1 and the KAM iterative procedure, discussed in the previous sections, can bu turned on. 2 3
ε1
≡
We shall, now, proceed to check all iterative conditions claimed in the thesis of the Proposition. (3.3): First notice that (3.3) (the only non trivial part of which is rν < sν ) for ν = 1 holds because κ > 1 > q; to check (3.3) for ν > 1 use (3.59). (3.26) and (3.34): sν+1 ≤ sν /16 is equivalent to εν+1 ≤ εν /16, which is implied by (3.56). Also, from the definition of εν+1 , ην and (3.56) it follows that 1
1 γν3 εν+1 < ν+1 , ην = c2 εν 2 c2 35
(3.85)
From the definition of P1 (compare (2.1)) it follows that if |∂ α P |∗C 0 = 0, for some α, then also |∂ P1 |∗∆σ = 0. 1 1 36 For example, one can take C1 > C¯1 , where C¯1 > const c5 β 1+ θ LM γ a3 and C1 ≥ exp(2(τ + 1)) solves 1 ¯ 2 log C1 = (C1 /C1 ) (τ + 1). α
24
which implies (3.26) and (3.34) because of the definition of c2 . (3.38) is consequence of37 (3.33) and (3.85): |∂Φν |Dν+1 = |(∂φ1 ◦ φ2 ◦ · · · ◦ φν ) (∂φ2 ◦ φ3 ◦ · · · ◦ φν ) · · · (∂φν )|Dν+1 ν ν Y Y 1 ≤ (1 + const sj ηj2 ) < 1 + j+1 < 2 . 2 j=1 j=1 Similarly one obtains38 2
|∂ Φν |Dν+1 =
ν X (∂ 2 φj j=1
×
Y
i6=j
≤ 4ν .
◦ φj+1 ◦ · · · ◦ φν )(∂φj+1 ◦ φj+2 ◦ · · · ◦ φν ) · · · (∂φν )
(∂φi ◦ φi+1 ◦ · · · ◦ φν )
Dν+1
(3.86)
1 Now, assume, by induction up to j = ν − 1, that |∂Φj |Lip Dj+1 ≤ 1 + αj ≡ 2 − 2j (which, for j = 1 is certainly true, in view of (3.33), since Φ1 ≡ φ1 ). Then (shortening, here, “ const ” with “c”, using again (3.33), the smallness of ην , (3.85) and (3.86)),
|∂Φν (·, ξ 0 ) − ∂Φν (·, ξ)| |∂Φν−1 (φν (·, ξ 0 ), ξ 0 )∂φν (·, ξ 0 ) − ∂Φν−1 (φν (·, ξ), ξ)∂φν (·, ξ)| = |ξ 0 − ξ| |ξ − ξ 0 | |∂Φν−1 (φν (·, ξ 0 ), ξ 0 ) − ∂Φν−1 (φν (·, ξ 0 ), ξ)| |∂φν (·, ξ 0 )| ≤ |ξ 0 − ξ| |∂Φν−1 (φν (·, ξ 0 ), ξ) − ∂Φν−1 (φν (·, ξ), ξ)| + |∂φν (·, ξ 0 )| |ξ 0 − ξ| |∂φν (·, ξ 0 ) − ∂φν (·, ξ)| +|∂Φν−1 (φν (·, ξ), ξ)| |ξ 0 − ξ| ≤ (1 + αν−1 )(1 + csν ην2 ) + |∂ 2 Φν−1 | |φν |Lip |∂φν | + 2c sν ην2 ≤ (1 + αν−1 )(1 + csν ην2 ) + 4νc2 s2ν ην2 (1 + csν ην2 ) + 2csν ην2 ≤ 1 + αν ; last inequality follows easily by the smallness of ην . (2.17): recall from §3.3 that (3.37) holds because of (3.34) and that (3.39) is consequence of (3.38) (and recall also that (2.17) is (3.37) plus (3.39)). (3.43): Since
γν+1 γν
=1−
1 , 2+2ν
(3.43) is implied by const
37 38
τ +1 εν Kν+1 1 ≤ ν+1 , γν 2
Recall that φν = XF1ν and that from (3.33) it follows that |∂φν |Dν+1 ≤ 1 + const sν ην2 . Use that from (3.33) with p = 2 there follows that |∂ 2 Φj |Dj+1 ≤ const ηj2 .
25
(3.87)
which is seen to hold because of the definition of Kν , (3.54), the fact that 1 − q(τ + 1)κ > 1/2 (recall the definition of q and κ in Lemma 3.1) and (3.84). ν
(3.48) follows from the definition of Kν , the fact that εν+1 ≥ εκν ≥ εκ1 and the explicit definition of K (used only here), K = c2 log ε−1 1 . Second inequality in (3.70): Using the definitions of sν , ην and the fact that 1−qτ2 −2κ = θ, one sees that the claim follows from const
εν 1
γ3
<1,
which in turn (using (3.54) and the fact that a3 ≥
1 ) 3θ
is implied by (3.84).
(3.69) for ν > 1 is proven by induction: Assume it holds up to ν. Then observing that Mν+1 − Mν = M/2ν , using the bounds (3.40), the fact that kXPν0 kLip rν ,Dν ≤ Mν εν /γν (see (3.70)) and the fact that a3 ≥ (3κ − 1)/(κ(κ − 1)), the first of (3.69) is seen to follow from (3.84).To check the second inequality in (3.69), observe that −1 Lip |ων+1 | ≤
Lν Lν ≤ . b ν |Lip 1 − L ν |ω 1 − const Lν Mν εν /γν
Thus, the claim follows from the smallness assumption (3.84) (it is only here that the presence of the term (LM ) in (3.84) is used), since a3 ≥ (3κ − 1)/(κ(κ − 1)) > a1 + κ1 = (2κ − 1)/(κ(κ − 1)). We turn to the third relation in (3.70). By (3.51) and using the fact that 2 − qτ2 − 2(κ − 1) = κ one sees that the claim follows from the definition of εν+1 . First inequality in (3.70) (for ν ≥ 2): For the purpose of this check call Pbν ≡ (P (ν) − P (ν−1) ) ◦ Φν−1 .
In view of the already verified bound (3.69), the claim is implied by kXPbν krν ,Dν +
γν εν kXPbν kLip . rν ,Dν ≤ Mν 2
(3.88)
Observe, as above, that, by definition of Hamiltonian vector field and of our weighted norms, kXPbν k∗rν ,Dν ≤ rν−2 |∂ Pbν |∗Dν . Now, by (3.38) and (2.31), (on the proper domains), `−1 |∂ Pbν | ≤ |∂(P ν − P ν−1) )| |∂Φν−1 | ≤ const |P |C ` σν−1 .
(3.89)
To bound the Lipschitz part, first observe that, by the chain rule, by (3.32), the fact that q + κ > 1 and (3.84), |Φν |Lip = |Φν−1 (φν , ξ)|Lip ≤ |∂Φν−1 | |φν |Lip + |Φν−1 |Lip ≤ |Φν−1 |Lip + const s2ν ην2 ≤ |Φ1 |Lip + const
ν−1 X j=2
s2ν ην2 ≤ const
26
ν−1 X j=1
2(q+κ)
s2ν ην2 ≤ const ε1
≤ ε1 .
(3.90)
Now, by the chain rule, (3.38), (2.31), (3.90), (on the proper domains), |∂ Pbν |Lip = |∂(P ν − P ν−1) ) · ∂Φν−1 |Lip
Lip
≤ ∂(P (ν) − P (ν−1) ) ◦ Φν−1 ≤
≤ ≤
+ ∂(P (ν)
|∂Φν−1 |
− P (ν−1) ) ◦ Φν−1 |∂Φν−1 |Lip
Lip
`−1 const |P |C ` σν−1 | + ∂(P (ν) − P (ν−1) ) ◦ Φν−1
`−1 const |P |C ` σν−1 + |∂ 2 (P (ν) − P (ν−1) )| |Φν−1 |Lip + |∂(P (ν) − P (ν−1) )|Lip
`−1 `−2 const (|P |C ` + |P |Lip C ` )σν−1 + |P |C ` σν−1 ε1 .
(3.91)
Putting together (3.89) and (3.91), and using (3.84), the first inequality in (3.56), the relation εκν−1 ≤ εν and the fact that q(`−2) − 2κ = (1 + θ) > 1, one gets κ
which is (3.88).
kXPbν krν ,Dν +
`−2 γν εν q σν−1 kXPbν kLip ≤ (βε ) , ≤ ε1+θ < 1 rν ,Dν ν 2 Mν rν 2
(3.92)
The convergence of39 eν , ων and Ων to e∞ , ω∞ and Ω∞ is, at this point, proved, as well as the bounds (3.71), which follows at once from (3.69). P
First estimate in (3.73): Write Φν = φ1 + νj=2 (Φj −Φj−1 ) and introduce, here, the short– hand notation Φ0j (x; ξ) ≡ Φj (x, 0, 0, 0; ξ) and φ0j (x; ξ) ≡ φj (x, 0, 0, 0; ξ) so that ψ(x; ξ) = limν→∞ Φ0ν (x; ξ). Notice that, for | Im x| ≤ sj , by (3.38) and (3.30), one has |Φj−1 (φ0j (x; ξ)) − Φ0j−1 (x; ξ)| ≤
sup |∂Φj−1 | |φ0j (x) − x| ≤ const σj2 ηj2 .
| Im x|≤sj
Then, for any x ∈ Tn and ξ ∈ Π∞ , for any α ∈ Nn with |α| ≤ p, by Cauchy estimates, by the definitions of sj , ηj , q and40 κ, we hav α ∂x ψ(x; ξ) − x
≤ ∂xα φ1 (x; ξ) − x + ≤ =
const
const 2
γ3
∞ X
2−|α| 2 ηj
sj
j=1 ∞ X
≤
2θ+q(2−p)
εj
j=1
∞ X α ∂x Φj−1 (φ0j (x; ξ)) − Φ0j−1 (x; ξ)
j=2
∞ const X 2 3
=
q(2−p)+2(κ−1)
εj
γ j=1 ∞ const X 2
γ3
j=1
2θ
εj
p∗ −p p∗ −2
≤
const 2
γ3
2θ
ε1
p∗ −p p∗ −2
.
For the bound on the Lipschitz semi–norm just take the limit in (3.90). Finally, the Diophantine relation (3.74) is obtained as the limiting case of (2.21). 39
Observe that eν obey the same bound of ω bν so that its convergence follows from the above discussion; in any case e∞ has no dynamical relevance. θ 40 ∗ −p Note: a(` − 2) = 2τ2 1−3θ = 2 qθ , 2θ + q(2 − p) = 2θ pp−2 .
27
3.4
Measure estimates (multiplicity of solutions)
In this section, assuming the notations and hypotheses of Proposition 3.1, we shall prove and make quantitative the claims in Theorem 1.1 concerning the measure of Π∞ , hence establishing multiplicity results for the lower–dimensional quasi–periodic solutions found in Proposition 3.1. Following [21], we note that if |k| is large, then the discarted “resonant set” Rνkl (γν ) defined in (3.42) is small41 : Lemma 3.2 If |k| ≥ K0 ≡ 16LM , then, for any ν ≥ 1 and any |l| ≤ 2, meas(Rνkl (γν )) ≤
λ , |k|τ +1
λ ≡ const (LM )n
γ ( diam Π)n−1 . M
(3.93)
This lemma is essentially Lemma 4, page 136, in [21], to which we refer for the simple proof42 . Proposition 3.2 Assume that ε1 satisfies also ε1 (LM )a < 1 , and that
a ≡ max
n1
θ
,
o 1 , q(τ − n + 1)
n
o
|Ωi (ξ)| , |Ωi (ξ) − Ωj (ξ)| . 0 < γ < min ξ∈Π i6=j
Then, meas Π∞ ≥ meas Π0 − const
γ (LM diam Π)n−1 , M
(3.94)
(3.95)
(3.96)
where n
Π0 ≡ Π0 (γ) ≡ |hω(ξ), ki + hΩ(ξ), li| ≥ Furthermore,
o γ , ∀ 0 < |k| ≤ K , |l| ≤ 2 . 0 1 + |k|τ
lim meas Π\Π0 (γ) = 0 , γ↓0
41
(3.97)
(3.98)
Recall that τ > n − 1. Just for completeness we sketch here an alternative argument: ων and Ων are Lipschitz in ξ and in fact ων is a Lipschitz diffeomorphysm. Thus, such function have derivatives in L1 and the standard formula for the change of variables in integrations holds. Using ω = ων (ξ) as independent variable, up to a suitable k–dependent rotation, we see that it is enough to estimate sets of the form {ω ∈ ω ν (Πν ) : |ω1 +gk (ω)| < γk /|k|τ +1 } where gk is a Lipschitz function that because of the assumption on |k| is smaller than, say, 1/2. Now, make a further change of variables setting ω10 = ω1 + gk (ω), ω20 = ω2 ,...,ωn0 = ωn , etc. 42
28
showing that meas Π∞ > 0 provided γ is small enough. Finally, if ω and Ω are C 1 (Π) and if (taking ω as independent variable43 ) µ≡
min
0<|k|≤K0 ,|l|≤2 ω∈Skl
then
Skl ≡ ω ∈ ω(Π) : hω, ki + hΩ(ω), li = 0 .
γ (LM diam Π)n−1 . Mµ
∂hΩ, li |k|−1 k + >0, ∂ω
n
meas Π\Π0 (γ) ≤ const
o
(3.99) (3.100)
Remark 3.1 Recall point 1.6 in § 1 and especially (1.7) and let r ≡ r1 . Notice that, 1 in such a case, εˆ0 ∼ r13 + ε and ε0 ∼ r1 + rε1 . Thus, choosing r ≡ r1 ≡ ε 3 , we see that 1 ε0 ∼ ε1 ∼ ε 3 and that the hypotheses (3.68) and (3.94) are satisfied and the claim in 1.6 follows; “generiticy” refers to conditions (1.3)–(1.4). Proof Notice that by definition of Kν in (3.52) and (3.94), there follows that Kν ≥ K1 > K0 ≡ 16(LM ). Thus, by Lemma 3.2 and the definition of Πν+1 , meas (Πν+1 ) ≥
X
meas (Πν ) −
meas (Rνkl (γν ))
|l|≤2 |k|>Kν
X
|k|−τ +1
≥
meas (Πν ) − const λ
|k|>Kν
≥
meas (Πν ) − const λ
Kντ −n+1
1
.
Iterating this relation, using the definition of Kν and (3.94), we get meas (Πν+1 ) ≥
≥
q(τ −n+1)
meas (Π1 ) − const λε1 γ meas (Π1 ) − const (LM diam Π)n−1 , M
which implies meas (Π∞ ) ≥ meas (Π1 ) − const
γ (LM diam Π)n−1 . M
(3.101)
From (3.95) it follows that Π1 = Π 0 \
[
K0 <|k|≤K1 |l|≤2
R1kl (γ) ,
and we see, again by Lemma 3.2, that meas (Π1 ) ≥ meas (Π0 ) − const λ(LM )−1 , 43
I.e., Ω(ω) is, by definition, Ω(ξ(ω)) where ω → ξ(ω) is the C 1 inverse function of ξ → ω(ξ).
29
which, together with (3.101), implies (3.96). The claim in (3.98) follows immediately from the compactness of Π, assumption (1.4) and the “monotonicity” of the sets Rνkl (γ) in γ (i.e., Rνkl (γ) ⊂ Rνkl (γ 0 ) if γ < γ 0 ). The claim in (3.100) follows easily by noting that (3.99) implies that Skl are C 1 hyper– surfaces in ω(Π) and observing that µ is a lower bound on the norm of the gradient of the function hω, ki + hΩ(ω), li.
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