PERSISTENCE OF LOWER DIMENSIONAL TORI OF GENERAL TYPES IN HAMILTONIAN SYSTEMS

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1 PERSISTENCE OF LOWER DIMENSIONAL TORI OF GENERAL TYPES IN HAMILTONIAN SYSTEMS YONG LI AND YINGFEI YI Abstract. The work is a generalization to [4] in which we study the persistence of lower dimensional tori of general type in Hamiltonian systems of general normal forms. By introducing a modified linear KAM iterative scheme to deal with small divisors, we shall prove a persistence result, under a Melnikov type of non-resonance condition, which particularly allows multiple and degenerate normal frequencies of the unperturbed lower dimensional tori. 1. Introduction The present work concerns the study of the persistence of lower dimensional tori in perturbative, partially integrable Hamiltonian systems of the following normal form: (1.1) H = e(ω) + ω, y + 1 ( ) ( ) y y 2, M(ω) + h(y, z, ω) + P (x, y, z, ω), z z where (x, y, z) T n R n R 2m, ω is a parameter in a bounded closed region O R n, M(ω) is a (n + 2m) (n + 2m) real symmetric for each ω O, h = O( (y, z) 3 ), P is the perturbation whose smallness will be specified later, and both h and P are real analytic in (x, y, z) for (x, y, z) lying in a complex neighborhood D(r, s) = {(x, y, z) : Im x < r, y < s, z < s} of T n {} {}. In the above, all ω dependences are of class C l for some l 4m 2, and the derivatives of h and P up to order l are uniformly bounded on their domain of definitions. With the symplectic form n m dx i dy i + dz j dz m+j, i=1 j=1 the equation of motion associated to (1.1) reads ẋ = ω + M 11 y + M 12 z + h y + P y, ẏ = P x, ż = JM 22 z + JM 21 y + J h z + J P z, 2 Mathematics Subject Classification. 37J4. Key words and phrases. Hamiltonian systems, invariant tori, KAM theory, Melnikov problem, persistence. The first author was partially supported by NSFC grant , National 973 Project of China: Nonlinearity, the outstanding young s project of Ministry of Education of China, and National outstanding young s award of China. The second author was partially supported by NSF grants DMS and DMS

2 2 Y. LI AND Y. YI where M 11, M 12, M 21, M 22 denote the n n, n 2m, 2m n, 2m 2m blocks of M = M(ω) respectively, and J is the standard 2m 2m symplectic matrix. Clearly, as ω varies in O, the unperturbed system associated to (1.1) admits a family of invariant n-tori T ω = T n {} {} which are parameterized by the toral frequency ω O. Hamiltonian (1.1) can be viewed as a canonical form in studying the persistence of lower dimensional tori in Hamiltonian systems. Under suitable symplectic coordinates, typical Hamiltonian systems in the vicinity of a family of lower dimensional tori (e.g., those near an equilibrium point) may be reduced to a system of form (1.1) in which there is a natural dependence of M on the tori, or a natural coupling of M with the toral frequencies ω (see Section 6 for more details). Along the line of the classical KAM theorem (Kolmogorov [17], Arnold [1], Moser [28]), the persistence problem of lower dimensional tori for (1.1) is to find appropriate non-degenerate and non-resonance conditions on M and ω, respectively, so that the majority (with respect to the Lebesgue measure on O) of these tori, associated to ω O, will persist after small perturbations. Such persistence problem has been studied in various normally non-degenerate cases (i.e. M 22 is non-singular over O). A persistence theorem of elliptic tori was formulated by Melnikov ([25, 26]) for Hamiltonian systems of the form (1.1) in which M 11 (ω), M 12 (ω), M 21 (ω) and M 22 (ω) is a diagonal matrix satisfying the so-called Melnikov s second non-resonance condition, i.e., the set (1.2) {ω O : k, ω + l, Λ, for all (k, l) (Z n Z 2m ) \ {}, l 2} admits full Lebesgue measure relative to O, where Λ = Λ(ω) is the column vector formed by the eigenvalues (diagonal) of M 22. The Melnikov s theorem was first proved by Eliasson in [11]. Refinements of the result with generalizations to infinite dimension were made by Kuksin ([19, 2]) and Pöschel ([32, 33]). We note that, with the Melnikov s second non-resonance condition, all normal frequencies have to be simple. Recently, Bourgain ([2, 3]) obtained a sharp persistence result (in both finite and infinite dimensions) of elliptic tori in the Melnikov s setting only under the Melnikov s first non-resonance condition (i.e., l 1 in (1.2)), which therefore allows the multiplicity of normal frequencies. Following a pioneer work of Moser ([27]), there were also studies on the persistence of normally non-degenerate, lower dimensional tori of general types. For example, Broer, Huitema and Takens ([5]), Broer, Huitema and Sevryuk ([6, 7]), Sevryuk ([34]) studied cases that M 11 (ω), M 12 (ω), M 21 (ω) and all eigenvalues of JM 22 (ω) are simple (hence JM 22 (ω) can be smoothly and symplectically diagonalized) and satisfy a Melnikov s second non-resonance condition. Jorba and Villanueva ([16]) considered a case with generic, non-degenerate perturbation that, M 12 (ω), M 21 (ω), M 11 (ω) is non-singular over O, and JM 22 (ω) is diagonal with distinct eigenvalues ±λ i (ω), i = 1, 2,, m, satisfying a similar Melnikov second non-resonance condition. Recently, You ([4]) considered the same persistence problem for (1.1) with M 11 (ω), M 12 (ω), M 21 (ω), but under the following weaker Melnikov second non-resonance condition on the eigenvalues λ i, i = 1, 2,, 2m, of JM 22 : (1.3) 1 k, ω λi (ω) λ j (ω), for all k Z n \ {}, 1 i, j 2m, which in particular allows the multiplicity of these eigenvalues.

3 HAMILTONIAN SYSTEMS 3 The persistence problem for hyperbolic tori was first considered in the work of Moser ([27]) and later studied by Graff ([12]) and Zehnder ([41]) for a fixed Diophantine toral frequency. Hyperbolic case with degeneracy and varying frequencies is recently considered by the authors in [22]. The aim of the present study is to generalize the result of You ([4]) to allow normal degeneracy (i.e., the singularity of M 22 (ω)) of the unperturbed lower dimensional tori. We shall show under the same weaker Melnikov second non-resonance condition (1.3) that the majority of unperturbed lower dimensional tori in (1.1) will persist if either M 22 (ω) or M(ω) is everywhere non-singular. In the latter case, the normal degeneracy of the lower dimensional tori can be made to an extreme. For example, one can simply take M 22 if M(ω) is everywhere non-singular. The proof of our results uses the classical KAM procedure. However, to be able to handle a general normal form like (1.1) especially when normal degeneracy occurs, a modified KAM linear iterative scheme will be introduced in order to construct the desired symplectic transformations. An essential point of this scheme is to treat the tangential variable y and the normal variable z in a same scale rather than the traditional way of treating z as a much smaller variable than y. Having done so, the present problem is in spirit closely related to the KAM problem in Hamiltonian systems with more action than angle variables (we refer the readers to [24, 3] and references therein for KAM theory in Hamiltonian or Poisson-Hamilton systems with a distinct number of action and angle variables). The persistence problem of lower dimensional tori can also be considered in the resonance zone of a nearly integrable Hamiltonian system in which the lower dimensional tori are natrually split from a family of unperturbed, full dimensional, resonant tori. However, the persistence problem of this nature depends on the Poincaré non-degenerate condition rather than conditions of Melnikov type. In this case the normal form near the lower dimensional tori, derived from splitting of the resonance, is similar to (1.1) but with M being in the same scale as the perturbation (see [9, 21, 23, 37] for details). The paper is organized as follows. In Section 2, we describe our main result followed by some comments and remarks. We use the modified KAM linear scheme in Section 3 to construct the symplectic transformation for one KAM cycle and give estimates to the transformation and the transformed Hamiltonian. In Section 4, an iteration lemma will be proved to ensure the validity of all KAM steps. We shall complete the proof of our result in Section 5 by showing the convergence of iterative sequences and estimating the measure of the limiting frequency set. Some applications of our results will be discussed in Section 6 for quasi-periodic motions of Hamiltonian systems near equilibria, in the spirit of the Lyapunov center theorem. Unless specified otherwise, we shall use the same symbol to denote an equivalent vector norm and its induced matrix norm, absolute value of functions, and measure of sets etc., and use D to denote the sup-norm of functions on a domain D. Also, for any two complex column vectors ξ, ζ of same dimension, ξ, ζ always stands for ξ ζ, i.e., the transpose of ξ times ζ. For the sake of briefness, we shall not specify smoothness orders for functions having obvious orders of smoothness indicated by their derivatives taken. We would like to thank the referee for valuable comments and suggestions which led to a significant improvement of the paper. This work was partially done when

4 4 Y. LI AND Y. YI the first author was visiting the Center of Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, and the second author was visiting the National University of Singapore, in We would also like to thank both Institutions for support. 2. Statement of Main Results Consider (1.1) and let λ 1 (ω),, λ 2m (ω) be eigenvalues of JM 22 (ω). We assume the weak form of the Melnikov second non-resonance condition (1.3), i.e., NR) The set {ω O : 1 k, ω λ i (ω) λ j (ω), for all k Z n \ {}, 1 i, j 2m} admits full Lebesgue measure relative to O. For γ >, let Ôγ denote the Diophantine set {ω O : k, ω > γ k τ, for all k Zn \ {}}, n where k = k i and τ > n 1 is fixed. Our main results are stated as follows. i=1 Theorem 1. Suppose NR) and that either M(ω) or M 22 (ω) is non-singular on O. For given r, s >, a positive integer l and σ (, 1), if there is a sufficient small µ = µ(r, s, l, σ ) > such that (2.1) l ω i x j (y,z) P γ3(1+σ)a µ for all (x, y, z) D(r, s), ω O, and (l, i, j) Z n + Zn + Zn+2m + with l + i + j l, where a = 4m 2 (l + 1), then there exist Cantor-like sets O γ, with O \ O γ as γ, on which the following holds. 1) If M 22 (ω) is non-singular on O, then there exists a C l 1 Whitney smooth family of real analytic, symplectic transformations Ψ ω : D( r 2, s 2 ) D(r, s), ω O γ, which is C l uniformly close to the identity, such that (2.2) H Ψ ω = e (ω)+ Ω (ω), y + 1 ( ) ( ) y y 2, M (ω) +h (y, z, ω)+p (x, y, z, ω), z z where e, Ω, M are C l 1 Whitney smooth with l ω e l ω e O γ, l ω (Ω id) Oγ, l ω M l ω M O γ = O(γ a µ 1 4 ), l l 1, h (y, z, ω) = O( (y, z) 3 ), h and P are real analytic in their associated phase variables and C l 1 Whitney smooth in the parameter ω, and j y k z P (y,z)=(,) = for all x T n, ω O γ, j Z+, n k Z+ 2m with j + k 2. Moreover, Ω (O γ ) Ôγ. Thus, for each ω O γ, the unperturbed torus T ω persists and gives rise to a slightly deformed, analytic, Floquet, quasi-periodic, invariant torus of the perturbed system with the Diophantine toral frequency Ω (ω). These perturbed tori also form a C l 1 Whitney smooth family.

5 HAMILTONIAN SYSTEMS 5 2) If M(ω) is non-singular on O, then 1) holds with Ω = id, i.e., for each ω O γ Ôγ, the perturbed torus also preserves the unperturbed (Diophantine) toral frequency ω. Theorem 2. Suppose that the Hamiltonian (1.1) is also real analytic in parameter ω. Then the conclusions of Theorem 1 hold for a = 4m 2 and any l > 1, provided that the condition (2.1) is replaced by (2.3) P γ 3(1+σ)a µ, (x, y, z) D(r, s), ω O. Remark: 1) Part 1) of Theorem 1 becomes the result of You ([4]) when M 11, M 12, M 21, h. There are certainly cases in application that some of these terms are non-zero and are not able to be eliminated via a smooth family of canonical transformations or via rescaling (see Section 6). In the case that M 22 is everywhere non-singular, our results hold for arbitrary smooth matrices M 11, M 12 = M21 and apply to non-degenerate lower invariant tori of all types. Similar to [4], the results also allow multiple eigenvalues of JM 22 (or M 22 ), for example, when k, i can be equal to j in NR). 2) Part 2) of Theorem 1 is of a particular interest in the presence of normal degeneracy. As a simple example, we let M(ω) be any smooth, non-singular, real symmetric, (n + 2m) (n + 2m) matrix on O such that M 22 (ω) (in particular, all normal eigenvalues equal to ), say, M = Then the non-resonance condition NR) holds automatically, which therefore assures the persistence of n-tori when the perturbation is sufficiently small. With respect to normal degeneracy, an essential point of the result in part 2) of Theorem 1 is that possible normal degeneracy can be compensated by the overall non-degeneracy of M(ω). Unless there are further non-degenerate constraints in the perturbation, such non-degeneracy is generally needed in the presence of normal degeneracy to eliminate first order resonant terms in the Hamiltonian, for otherwise, there would not be any persistent n-torus. For example, in (1.1) if M 12 = M21, M 22, h, P = ε ξ, z, where ξ is a non-zero 2m vector, then for any matrix M 11 (ω), no invariant n-torus can exist when ε. 3) Like the classical KAM case, Theorem 1 concludes that perturbed tori are actually Floquet (i.e., the quadratic terms in their normal forms (2.2) are independent of angle variables) - a useful structure characterizing the normal spectra of the perturbed tori which may be used in studying problems such as bifurcation from tori, stability and integrability etc. (see [7]). The non-resonance condition NR) seems to be a best possible condition in general for the persistence of Floquet lower dimensional tori in Hamiltonians of form (1.1)..

6 6 Y. LI AND Y. YI (2.4) A parallel first non-resonance condition of Melnikov-Bourgain type with respect to the general setting (1.1) can be described as follows: on a frequency set of full measure, 1 k, ω λi (ω), for all k Z n \ {}, 1 i 2m, where λ i s are as in NR). Thus, the condition NR), when restricted to the elliptic case, is weaker than the Melnikov s second condition but is stronger than the first condition. At this point, it is not clear to us whether it is possible to generalize the Bourgain s theorem and our results to obtain a persistence result for general type of non-degenerate, lower dimensional tori in Hamiltonian system of general form (1.1), by only imposing the first non-resonance condition (2.4). Even if it does, we suspect that the perturbed tori under the first non-resonance condition (2.4) would in general be non-floquet. This is because, as far as KAM type of symplectic transformations are concerned, the first non-resonance condition (2.4) is not sufficient for the elimination of the angular dependence among quadratic order perturbative terms of a Hamiltonian system. Some exciting developments of the quasi-periodic Floquet theory are made recently (e.g., [1, 13, 15, 18, 29, 38]) concerning the reducibility of certain linear systems with quasi-periodic coefficients including some cases of quasi-periodic linear Hamiltonian systems. With these developments, one might obtain the Floquet tori alternatively by first obtaining the invariant tori then reducing their normal parts to Floquet forms. However, since the quasi-periodic Floquet theory gives no particular information on the reducibility near a given invariant torus, we feel that the weak second Melnikov condition NR) is still needed for the reducibility near all invariant tori corresponding to the set O γ. Besides, the Hamiltonian system near each of these perturbed tori will not be quasi-periodically forced in general. 4) Part 2) of Theorem 1 also gives a general result on the preservations of toral frequencies, which has been known mainly in the classical KAM case and hyperbolic cases (see [12, 41]). However, it does not mean the persistence of any fixed lower dimensional torus satisfying the Melnikov condition NR). Nevertheless, unless there are further non-degenerate constraints in the perturbation, the non-degeneracy of M assumed in part 2) of Theorem 1 is a sharp condition for the preservation of toral frequencies in Hamiltonian systems of form (1.1) in the sense stated in the theorem. Indeed consider (1.1) and let M = M(ω) be any (n + 2m) (n + 2m) real symmetric matrix depending smoothly on ω and satisfying NR). Take h, P = ε( y, y + z, z ), where ε is sufficiently small and y = y (ω) R n, z = z (ω) R 2m. Then the equation of motions associated to the Hamiltonian becomes ẋ = ω + M 11 y + M 12 z + εy, ẏ =, ż = JM 22 z + JM 21 y + εjz. Suppose that the Hamiltonian admits a family of invariant n-tori T ε,ω which preserves the toral frequencies ω of the corresponding unperturbed tori. Let (x(t), y(t), z(t)) be an orbit on a perturbed torus T ε,ω. Then y is a constant

7 HAMILTONIAN SYSTEMS 7 and z must be a quasi-periodic function with basic frequencies chosen from components of ω. Since NR) implies the condition (2.4) (see Lemma 5.1 in Section 5), i.e, none of the eigenvalues of JM 22 can rationally depend on the basic frequencies, it is easy to see that z must be a constant solution of the last equation above, i.e., (2.5) M 22 z + M 21 y + εz =. Substituting the constant solutions y, z into the x-equation above, we see that the frequencies of x will be drifted unless (2.6) M 11 y + M 12 z + εy =. Now, if M is singular on a positive measure set Ô O, then by choosing (y, z ) as a basis vector of kerm(ω) (which varies smoothly in ω) we see that the system {(2.5), (2.6)} admits no solutions on Ô for any ε, which leads to a contradiction. In other word, we have shown that, given any (n + 2m) (n + 2m) real symmetric matrix M depending smoothly on ω and satisfying NR), if M is singular on a positive measure set, then there exists an arbitrary small perturbation which preserves no toral frequency lying in a positive measure set. 3. KAM Step We shall only prove Theorem 1 in the sequel. The proof of Theorem 2 can be carried out by similar arguments along with the standard Cauchy estimate. Due to the presence of small divisors, one cannot remove all x-dependent terms in a finite number of steps. An essential idea of the KAM theory is to construct a symplectic transformation, consisting of infinitely many successive steps (referred to as KAM steps) of iterations, so that the x-dependent terms are pushed into higher order perturbations after each step. As all KAM steps can be carried out inductively, below, we only show detailed constructions of the symplectic transformation for one KAM cycle (i.e., from one KAM step to the successive one). Initially, we set N = N, e = e, Ω = id, M = M, M22 = M 22, h = h, P = P, O = O, r = r, β = s, s = γ (1+σ)a µ 1 4, µ = γ σ µ 1 2, γ = γ. Without loss of generality, we assume that < r, β, µ, γ 1, s β and write µ = µ. For j Z n+2m +, define a j = a(1 sgn( j )sgn( j 1)sgn( j 2)) = 2, j =, κ j = 1 sgn( j 1) = 1, j = 1,, j 2, d j = 1 λ sgn( j )sgn( j 1)sgn( j 2) = where λ (, 1) is fixed. Then by (2.1), { a, j =, 1, 2,, j 3, { 1, j =, 1, 2, 1 λ, j 3, l ω i x j (y,z) P D(r,s ) O γ aj sκj µdj, l + i + j l.

8 8 Y. LI AND Y. YI Suppose that after a νth KAM step, we arrive at a real analytic, parameterdependent Hamiltonian (3.1) H = H ν = N + P, N = N ν = e + Ω(ω), y + 1 ( ) ( ) y y 2, M(ω) + h(y, z, ω), z z where (x, y, z) D = D ν = D(r, s), r = r ν r, s = s ν s, ω O = O ν O, e(ω) = e ν (ω), Ω(ω) = Ω ν (ω) are smooth on O, M(ω) = M ν (ω) is real symmetric and smooth on O, h = h ν (y, z, ω) = O( (y, z) 3 ), h and P = P ν (x, y, z, ω) are real analytic in (y, z) D = D(s) = {(y, z) : y < s, z < s} and in (x, y, z) D respectively and smooth in ω O, and moreover, (3.2) l ω i x j (y,z) P D O γ aj s κj µ dj, l + i + j l, for some µ = µ ν >, γ = γ ν >. We wish to construct a symplectic transformation Φ + = Φ ν+1, which, in smaller frequency and phase domains, should carry the above Hamiltonian into the next KAM cycle. More precisely, the transformed Hamiltonian should be of a form similar to (3.1) with a smaller perturbation term satisfying an inequality similar to (3.2). Thereafter, quantities (domains, normal form, perturbation, etc.) in the next KAM cycle will be simply indexed by + (=ν + 1), and we shall not specify the dependence of M, M +, P, P + etc. on their arguments. Also, all constants c 1 c 5 below are positive and independent of the iteration process. For simplicity, we also use c to denote any intermediate positive constant which is independent of the iteration process. Let b, σ, d, δ be positive constants such that Define δ = 1 d, σ (b + σ)(4b + 5σ) >, δ(1 + b + σ) > 1, σ σ b + σ (1 + σ )a >, (1 + σ )(1 + b + σ)(1 (b + 2σ)) > 1. r + = δr + d(1 δ2 2 )r, s + = s 1+b+σ, β + = β 2 + β 4, γ + = γ 2 + γ 4, K + = ([log 1 s ] + 1)3, D + = D(β + ), D + = D(r +, s + ), D + = D(r (r r +), β + ), D i = D(r + + i 1 8 (r r +), is + ), i = 1, 2,, 8,

9 HAMILTONIAN SYSTEMS 9 D(λ) = D(r (r r +), λ), Γ(λ) = e r (1 δ)δ2 16 k χ e λ < k K + where λ >, χ = 2(l + 1)(2m 2 + 1)τ, and τ > n 1 is fixed Outline of the construction. The transformation Φ + will be constructed in two steps. The first step is to average out terms in P up to quadratic order. First order resonant terms will then be removed in the second step by a translation of coordinate. We first consider the averaging process. Let R be the truncation of the Taylor-Fourier series of P up to quadratic order, i.e., R = p kıj y ı z j e 1 k,x (3.3) = k K +, ı + j <3 k K + (P k + P k1, y + P k1, z + y, P k2 y + z, P k11 y + z, P k2 z )e 1 k,x, where K + is the truncation order in x to be specified later. We shall seek for an averaging transformation as the time 1-map φ 1 F of the flow φ t F generated by a Hamiltonian F of the form (3.4) F = (F k + F k1, y + F k1, z < k K + + y, F k2 y + z, F k11 y + z, F k2 z )e 1 k,x, where F kij, i + j 2, are ω-dependent vectors or matrices of obvious dimensions. We wish to determine F through the following linear homological equation: (3.5) {N, F } + R [R] Q =, where [R] = R(x, )dx is the average of R with respect to x T n and Q consists T n of all terms in {N, F } of size O(s 3 µ) including those of order O(y ı z j ) for ı + j 3. We note that, by taking the term Q into account, (3.5) modifies the usual linear homological equation adopted in standard KAM linear schemes. At the moment, let us suppose that (3.5) is solvable on a suitable frequency domain. Then H φ 1 F = (N + R) φ 1 F + (P R) φ 1 F = N + [R] + ({N, F } + R [R] Q) = N + [R] + {R, F } φ t F dt + (P R) φ1 F + Q 1 1 8, {R t, F } φ t F dt + (P R) φ1 F + Q, (1 t){{n, F }, F } φ t F dt

10 1 Y. LI AND Y. YI where (3.6) R t = (1 t){n, F } + R = (1 t)(q + [R] R) + R. (3.7) then Therefore, if we let N + = N + [R], P + = 1 {R t, F } φ t F dt + (P R) φ 1 F + Q, H φ 1 F = N + + P +. This completes the averaging process. Write M into blocks: ( ) M11 M M = 12, M 21 M 22 where M 11, M 12, M 21, M 22 are n n, n 2m, 2m n, 2m 2m blocks of M respectively. As for the second step, we need to eliminate the first order resonant terms in [R]. Assume the following condition: H1) If M 22 (M resp.) is non-singular on O, then so is M 22 (M resp.), and moreover, Let (y, z ) satisfy M 1 22 O 2 (M 22 ) 1 O ( M 1 O 2 (M ) 1 O resp.). (3.8) (M(ω) + 2 (y,z) h(y, z, ω)) ( y z ) + (y,z) h(y, z, ω) = if M is non-singular on O, or ( ) y =, (diag(o, M 22 (ω)) + (y,z) 2 h(y y (3.9), z, ω)) z ( + (y,z) h(y, z, ω) = ( P1 P 1 P 1 if M 22 is non-singular on O. By the implicit function theorem, (y, z ) = (y (ω), z (ω)) exist and depend smoothly on ω. Let e + = e + P + h(y, z, ω) + Ω + (ω), y + P 1, z + 1 ( ) ( )( ) 2 y P2 P, 11 y (3.1) z P11, P 2 z (3.11) Ω + = Ω + P, ( ) M + P2 P = M P11, P 2 h + = h(y + y, z + z, ω) h(y, z, ω) (y,z) h(y, z, ω), 1 ( ) ( ) y y (3.12) 2, (y,z) 2 z h(y, z, ω), z ) ) ( ) y z

11 HAMILTONIAN SYSTEMS 11 where P = { P1, if M 22 is non-singular on O,, if M is non-singular on O. We consider the translation φ : x x, ( ) y z ( ) y + z ( y z ). Then Φ + = φ 1 F φ will transform H into the Hamiltonian H + in the next KAM cycle, i.e., H + = H Φ + = N + + P +, where (3.13) (3.14) )( y ( ) ( ) N + = N y P2 P + φ 2, 11 z P11 P 2 z = e + + Ω + (ω), y + 1 ( ) ( ) y y 2, M + + h + (y, z, ω), z z ( ) ( ) ( ) P + = P y P2 P + φ + 2, 11 y z P11. P 2 z 3.2. Solving the linear homological equation. In (3.5), we let (3.15) Q = 1 < k K + k, M 11 y + M 12 z + h y (F k + F k1, y + F k1, z + y, F k2 y + z, F k11 y + z, F k2 z )e 1 k,x h z, J(F k1 + F k11 y + F k2 z + Fk2 z) e 1 k,x. < k K + Then substitutions of (3.3), (3.4), (3.15) into (3.5) yield 1 k, Ω (Fk + F k1, y + F k1, z (3.16) < k K + + y, F k2 y + z, F k11 y + z, F k2 z )e 1 k,x + M 21 y + M 22 z, J(F k1 + F k11 y + F k2 z + Fk2 z) e 1 k,x < k K + = (P k + P k1, y + P k1, z < k K + + y, P k2 y + z, P k11 y + z, P k2 z )e 1 k,x.

12 12 Y. LI AND Y. YI By comparing the coefficients of (3.16), we obtain the following linear equations for all < k K + : (3.17) 1 k, Ω Fk = P k, (3.18) 1 k, Ω Fk1 + M 12 JF k1 = P k1, (3.19) ( 1 k, Ω I 2m M 22 J)F k1 = P k1, (3.2) 1 k, Ω Fk (M 12JF k11 F k11jm 21 ) = P k2, (3.21) (3.22) ( 1 k, Ω M 22 J)F k11 + (Fk2 + F k2 )JM 21 = P k11, 1 k, Ω Fk2 M 22 JF k2 + F k2 JM 22 = P k2. For any given matrix A = (a ij ) pq, let σ(a) denote the pq-vector which lines up the row vectors of A from left to the right, i..e., σ(a) = (a 11 a 1q a p1 a pq ). Then for any matrices A, B, C such that ABC is well defined, we have (3.23) σ(abc) = (A C )σ(b), where denotes the usual tensor product of matrices, i.e., (3.24) U V = (u ij V ), U = (u ij ) for any two matrices U, V. For < k K +, define F k = σ(f k2 ), P k = σ(p k2 ), L k = 1 k, Ω, L 1k = 1 k, Ω I 2m M 22 J, L 2k = 1 k, Ω I 4m 2 (M 22 J) I 2m I 2m (M 22 J). Using (3.23), (3.24) and the above notions, the linear equations (3.17)-(3.22) can be rewritten equivalently as (3.25) (3.26) (3.27) L k F k = P k, L 1k F k1 = P k1, L k F k1 = P k1 M 12 JF k1, (3.28) L k F k2 = P k (F k11jm 21 M 12JF k11 ), (3.29) (3.3) L 1k F k11 = P k11 (F k2 + F k2 )JM 21, L 2k F k = P k, < k K +, which are clearly solvable as long as all L k, L 1k, L 2k are invertible. Consider the set O + = O(K + ) = {ω O : L k > γ k τ, detl 1k > γ2m k 2mτ, (3.31) detl 2k > γ4m2 k 4m2 τ, for all < k K +}.

13 HAMILTONIAN SYSTEMS 13 Then, on O +, equations (3.25)-(3.3) can be solved uniquely to obtain solutions F kij, depending smoothly on ω O + and satisfying F kij = F kij, for all < k K +, i + j 2, which uniquely determine the Hamiltonian F in (3.4) Estimate on the truncation R. Lemma 3.1. Assume that H2) s + s 16 ; H3) λ n+l e λ r r + 16 dλ s. K + Then there is a constant c 1 such that for all l + i + j l, ω O, Proof. Denote where l ω i x j (y,z) (P R) D 8 c 1 γ aj (s κj+1 + s κj 3 s 3 +)µ dj. I = II = = 1 k,x, p kıj y ı z j e k >K + p kıj y ı z j e 1 k,x k K +, ı + j 3 (p,q) y p z q k K +, ı + j 3 is the obvious anti-derivative of (p,q) y p z q P R = I + II. p kıj e 1 k,x y ı z j dydz, for p + q = 3. Then, Since, by H2), D 8 D(s), and by Cauchy s estimate and (3.2), p kıj y ı z j ω l j (y,z) P D(r,s)e k r l ω j (y,z) ı Z n +,j Z2m + γ aj s κj µ dj e k r, y, z < s, for all k and l + j l, H3) implies that ω l x i j (y,z) I D(s) γ aj k i s κj µ dj e k r e k (r++ 7 k K + It follows that γ aj s κj µ dj η n+l e η r r + 8 η=k + γ aj s κj µ dj λ n+l e λ r r + K + 16 dλ γ aj s κj+1 µ dj, l + i + j l. 8 (r r+)) l ω i x j (y,z) (P I) D(s) l ω i x j (y,z) P D(r,s) + l ω i x j (y,z) I D(s) 2γ aj s κj µ dj, l + i + j l.

14 14 Y. LI AND Y. YI By performing Cauchy s estimate of ω l i x j (y,z)(p I) on D(s), we have Thus, ω l x i j (y,z) II D 8 ω l x i j (y,z) ( 2 (p,q) (p,q) y p z q k K +, ı + j 3 y p z q l ω i x j (y,z) (P I) dy dz D 8 ) 3 1 γ aj s κj µ dj dy dz D8 s 8s + cγ aj µ dj s κj 3 s 3 +, l + i + j l. p kıj y ı z j e 1 k,x dy dz D8 l ω i x j (y,z) (P R) D 8 cγ aj (s κj+1 + s κj 3 s 3 + )µdj, l + i + j l Estimate on the transformation Φ +. Lemma 3.2. Assume H3) and also that H4) l ωm l ωm O, l ω(ω id) O µ 1 4, l l. If F is the Hamiltonian defined in Sections 3.1, 3.2, then there is a constant c 2 such that (3.32) (3.33) l ω i x j (y,z) F D(s) O + c 2 s κj µγ(r r + ), l ω i x j (y,z) F D(β) O + c 2 µγ(r r + ) for all < k K +, l l, i + j l + 1. Proof. Let q =, 1, 2, < k K +. By H4), (3.34) l ωl qk O+ c k, l l, and by (3.31), i.e., k 1 O + = k, Ω(ω) O + c k τ γ, L 1 L 1 1k O + = adjl 1k detl 1k O+ c k 2mτ+2m 1 γ 2m, L 1 2k O + = adjl 2k O+ c k 4m 2 τ+4m 2 1, detl 2k γ 4m2 (3.35) L 1 qk O + c k (2m) q τ+(2m) q 1. γ (2m)q Using (3.34), (3.35) and applying the identity l ω l L 1 qk = l =1 ( ) l l ( l l ω L 1 qk l ω L qk )L 1 qk

15 HAMILTONIAN SYSTEMS 15 inductively, it is easy to see that +( l +1)(2m) q τ (3.36) ω l L 1 qk c k l, l l γ ( l +1)(2m) q. By Cauchy s estimate, we also have (3.37) l ωp kıj O l ωp D(r,s) O s (ı+j) e k r γ a s 2 ı j µe k r, ı + j 2. It now follows from (3.26)-(3.3), (3.36), (3.37) that l ωf kıj O+ c k l +( l +1)4m2 τ γ ( l +1)4m2 γ a s 2 ı j µe k r c k l +( l +1)4m2τ s 2 ı j µe k r, l l, ı + j 2. Since F is of quadratic orders in y, z, we have, on D(s) O +, that l ω i x j (y,z) F c < k K + k i ( l ω F k + l ω F k1 s 1 sgn( j ) + l ω F k1 s 1 sgn( j ) + l ω F k2 s 1 sgn( j 1) + l ω F k11 s 1 sgn( j 1) + ωf l k2 s 1 sgn( j 1) )e k (r (r r+)) cs κj r r + µ 8 cs κj µγ(r r + ), l l, i + j l + 1. < k K + k χ e k This proves (3.32). The proof of (3.33) is similar. Lemma 3.3. Assume H1)-H4) and also that H5) c 2 µγ(r r + ) < 1 8 (r r +), H6) c 2 sµγ(r r + ) < s +, H7) c 2 µγ(r r + ) + c 2 µ < β β +. 1) Let φ t F be the flow generated by F. Then for all t 1 φ t F : D 3 D 4, φ : D 1 D 3 are well defined, real analytic and depend smoothly on ω O +. 2) Let Φ + = φ 1 F φ. Then for all ω O +, (3.38) Φ + : D + D, D + D(r, β). 3) There is a constant c 3 such that for all t 1, l l, c 3 sµγ(r r + ), i + j =, l 1, (3.39) ω l x i j (y,z) φt F D3 O + c 3 µγ(r r + ), 2 l + i + j l + 1, c 3, otherwise, (3.4) l ω p ξ (Φ + id) D+ O + c 3 µγ(r r + ), p l + 1, (3.41) ωφ l D+ O + c 3 γ a sµ, where ξ = (x, y, z).

16 16 Y. LI AND Y. YI Proof. Let φ t F 1, φt F 2, φt F 3 be components of φt F note that t (3.42) φ t F = id + X F φ λ F dλ, in x, y, z planes respectively. We where X F = (F y, F x, JF z ) denotes the vector field generated by F. For any (x, y, z) D 3, let t = sup{t [, 1] : φ t F (x, y, z) D 4}. Since, by H2), D 4 D(s), it follows from H5)-H7) and (3.32) that t φ t F 1 (x, y, z) = x + F y φ λ F dλ x + F y D(s) r (r r +) + c 2 sµγ(r r + ) < r (r r +), t φ t F 2 (x, y, z) = y + F x φ λ F dλ y + F x D(s) 3s + + c 2 s 2 µγ(r r + ) < 4s +, t φ t F 3 (x, y, z) = z + JF z φ λ F dλ z + F z D(s) 3s + + c 2 s 2 µγ(r r + ) < 4s +, i.e., φ t F (x, y, z) D 4 for all t t. Thus, t = 1 and φ t F (x, y, z) D 4, t 1. Next, we note by H3), H4) and (3.37) that l ωz O+, l ωy O+ c 2 γ a sµ, l l, where y, z are as in (3.8). It follows from H2) that φ : D 1 D 3 is well defined and satisfies (3.41). Thus, Φ + : D + D is well defined. Using H5), H7), (3.33) and a similar argument as above, one sees that φ t F : ˆD + D(r, β) is well defined for all t 1, where ˆD + = D(r (r r +), β + + c 2 µ). Thus, Φ + : D + D(r, β) is also well defined. As the proofs for (3.39) and (3.4) are similar, we only consider (3.4). Observe that Φ + id = (φ 1 F id) φ + y z Let ω O +, t 1. By (3.41), to prove (3.4), it suffices to show that Note that. l ω p ξ (φt F id) ˆD+ cµγ(r r + ), p l + 1, l l. (3.43) D l X F D O + c D l +1 ξ F D O +, l l. Using (3.42) and (3.33), we immediately have Differentiating (3.42) yields D ξ φ t F = I 2n+2m + t φ t F id ˆD+ cµγ(r r + ). (DX F )D ξ φ λ F dλ = I 2n+2m + t J(D 2 ξ F )D ξφ λ F dλ.

17 HAMILTONIAN SYSTEMS 17 It follows from (3.33) and the Gronwall s inequality that D ξ φ t F I 2n+2m ˆD+ t t D 2 ξ F D(β) D ξ φ λ F I 2n+2m ˆD+ dλ + D 2 ξ F D(β) D 2 ξ F D(β) D ξ φ λ F I 2n+2m ˆD+ dλ + c 2 µγ(r r + ) cµγ(r r + ). Similarly, by using (3.33), the Gronwall s inequality and (3.43) inductively, we have ω l p ξ φt F ˆD+ O + cµγ(r r + ), 2 p + l l + 1, l l, or, p =, l Estimate on the new Hamiltonian. We now estimate the new Hamiltonian H + = H Φ + = N + + P +, where N +, P + are as in (3.13), (3.14). Lemma 3.4. There is a constant c 4 such that for all l l, ωω l + ωω l O+ c 4 γ a sµ, ω l e + ω l e O + c 4 γ a s 2 µ, ωm l + ωm l O+ c 4 γ a µ, ω l j (y,z) (h + h) D+ O + c 4 γ a µ, j l. Proof. The lemmas follows easily from (3.1)-(3.12) and (3.37). Let (3.44) j = (γ a s κj + s 2 sgn( j ) sgn( j 2) )µ 2 Γ 3 (r r + ) + s 1 sgn( j ) + s 2 sgn( j )sgn( j 1) sgn( j )sgn( j 1)sgn( j 2) µγ(r r + ) + γ aj (s κj+1 + s κj 3 s 3 +)µ dj. Lemma 3.5. Assume H1)-H6). Then there is a constant c 5 such that Thus, if l ω i x j (y,z) P + D+ O + c 5 j, l + i + j l. H8) c 5 j γ aj + sκj + µdj +, then l ω i x j (y,z) P + D+ O+ γ aj + sκj + µdj +, l + i + j l. Proof. Let l + i + j l. Similar to the proof of (3.32), we have, by H4), (3.15), (3.38) and Lemma 3.4 that (3.45) (3.46) l ω i x j (y,z) Q D 3 O + cs 1 sgn( j ) + s 2 sgn( j )sgn( j 1) sgn( j )sgn( j 1)sgn( j 2) µγ(r r + ), l ω i x j (y,z) Q D(s) O + cs 2 sgn( j ) sgn( j 2) µγ(r r + ). Using the expression of R in (3.3), we also have (3.47) l ω i x j (y,z) [R] D(s) O + l ω i x j (y,z) R D(s) O cγ a s κj µγ(r r + ).

18 18 Y. LI AND Y. YI It follows from (3.32) and (3.47) that ( l ω i x j (y,z) {[R], F } + l ω i x j (y,z) {R, F } ) D(s) O + cγ a s κj µ 2 Γ 2 (r r + ), and from (3.32) and (3.46) that l ω i x j (y,z) {Q, F } D(s) O + cs 2 sgn( j ) sgn( j 2) µ 2 Γ 2 (r r + ). Hence, by (3.39), we have on D 2 O + that l ω i x j (y,z) (3.48) 1 1 {R t, F } φ t F dt = {(1 t)(q + [R] R) + R, F } φ t F dt c(γ a s κj + s 2 sgn( j ) sgn( j 2) )µ 2 Γ 3 (r r + ). Combining (3.45), (3.48) and Lemma 3.1 with (3.7), we have l ω i x j (y,z) P + D2 O + c((γ a s κj + s 2 sgn( j ) sgn( j 2) )µ 2 Γ 3 (r r + ) +s 1 sgn( j ) + s 2 sgn( j )sgn( j 1) sgn( j )sgn( j 1)sgn( j 2) µγ(r r + ) +γ aj (s κj+1 + s κj 3 s 3 + )µdj ), which, together with (3.45) and (3.41), implies that l ω i x j (y,z) P + D+ O + c j Estimate on the new frequency domain. Define ω O, k Z n \ {}. Lemma 3.6. If L + k = 1 k, Ω +, L + 1k = 1 k, Ω + I 2m M + 22 J, L + 2k = 1 k, Ω + I 4m 2 (M + 22 J) I 2m I 2m (M + 22 J), H9) 3c 4 µk 8m2 τ+8m 2 + < min{ γ γ+ then for all < k K +, ω O +, γ, γ2m γ 2m + γ 2m 2, γ4m γ 4m 2 + γ 4m2 L + k > γ + k τ, detl+ 1k > γ2m + k 2mτ, detl+ 2k > γ4m + k. 4m2 τ Proof. Let < k K +, ω O +. We note by (3.31) that L k > γ k τ, detl 1k > γ2m k 2mτ, detl 2k > γ4m 2 k. 4m2 τ It follows from Lemma 3.4 and H9) that L + k > L k c 4 µk + > γ + k τ, detl + 1k > detl 1k 2c 4 νk 2m detl + 2k > detl 2k 3c 4 νk 4m2 }, + > γ2m + k 2mτ, + > γ4m 2 + k 4m2 τ. 2

19 HAMILTONIAN SYSTEMS Iteration Lemma In this section, we shall prove an iteration lemma which guarantees the inductive construction of the transformations in all KAM steps. Let r, s, µ, γ, O, H, N, e, Ω, M, M 22, h, P be as defined at the beginning of Section 3 and define D = D(β ), D = D(r, β ), D = D(r, s ), K =, Φ = id. For any ν =, 1, we index all index-free quantities in Section 3 by ν and index all + -indexed quantities in Section 3 by ν +1. This yields the following sequences: for ν = 1, 2,. In particular, r ν, s ν, µ ν, K ν, O ν, D ν, D ν, Dν, H ν, N ν, e ν, Ω ν, M ν, M ν 22, Lν k, Lν 1k, Lν 2k, h ν, P ν, Φ ν H ν = N ν + P ν, N ν = e ν + Ω ν, y + 1 ( ) ( ) y y 2, M ν + h ν (y, z, ω), z z ( ) M M ν ν = 11 M12 ν (M12 ν ) M22 ν, L ν k = 1 k, Ω ν, L ν 1k = 1 k, Ω ν I 2m M ν 22J, L ν 2k = 1 k, Ω ν I 4m 2 (M ν 22J) I 2m I 2m (M ν 22J), r ν = r (1 1 ν 2 (1 δ) δ i+1 ), s ν = s 1+b+σ ν 1, β ν = β (1 ν i=1 1 ), 2i+1 i=1 µ ν = c s σ ν 1µ ν 1, c = max{1, c 1,, c 5 }, ν 1 γ ν = γ (1 ), 2i+1 K ν = ([log i=1 1 s ν 1 ] + 1) 3, O ν = {ω O ν 1 : L ν 1 k > γ ν 1 k τ, detlν 1 1k detl ν 1 2k > γ4m 2 ν 1 k, < k K 4m2 τ ν}, D ν = D(β ν ), D ν = D(r ν, s ν ), D ν = D(r ν (r ν 1 r ν ), β ν ). We note that c only depends on r, β, l. > γ2m ν 1 k 2mτ,

20 2 Y. LI AND Y. YI Lemma 4.1 (Iteration Lemma). If (2.1) holds for sufficiently small µ = µ(r, β, l ), then the KAM steps described in Section 3 are valid for all ν =, 1,, and the following holds for all ν = 1, 2,. 1) e ν = e ν (ω), Ω ν = Ω ν (ω) are smooth on O ν, M ν = M ν (ω) is symmetric and smooth on O ν, h ν = h ν (y, z, ω) = O( (y, z) 3 ) and P ν = P ν (x, y, z, ω) are real analytic in (y, z) D ν and in (x, y, z) D ν respectively and smooth in ω O ν. Moreover, for all l l, (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) (4.9) ωe l ν ωe l ν 1 Oν γ2(1+σ)a+a µ ν, ω l e ν ω l e Oν γ 2(1+σ)a+a µ 1 4, l ω Ω ν l ω Ω ν 1 Oν γ(1+σ)a+a µ ν, l ω(ω ν id) Oν γ (1+σ)a+a µ 1 4, l ω M ν l ω M ν 1 Oν γa µ ν, l ωm ν l ωm Oν γ a µ 1 4, ω l j (y,z) (h ν h ν 1 ) Dν O ν γa µ ν, j l, ω l j (y,z) (h ν h ) Dν O ν γ a µ 1 4, j l, l ω i x j (y,z) P ν Dν O ν γ aj ν sκj ν µdj ν, l + i l, j l. 2) If M is non-singular on O, then Ω ν (ω) ω. 3) Φ ν : D ν O ν D ν 1, D ν O ν D ν 1 is symplectic for each ω O ν, real analytic in (x, y, z) D ν and smooth in ω O ν, and Moreover, H ν = H ν 1 Φ ν = N ν + P ν. (4.1) l ω p ξ (Φ ν id) Dν O ν µ 1 4 4) where ξ = (x, y, z). O ν = {ω O ν 1 : L ν 1 k > γ ν 1 2 ν, p l + 1, l l, k τ, detlν 1 1k > γ2m ν 1 k 2mτ, detl ν 1 2k > γ4m 2 ν 1 k, for all K 4m2 τ ν 1 < k K ν }. Proof. We need to verify the conditions H1)-H9) in Section 3 for all ν =, 1,. Note that (4.11) (4.12) (4.13) µ ν = (c ) ν µ s σ b+σ ((1+b+σ)ν 1), s ν = s (1+b+σ)ν, s = γ (1+σ)a µ 1 4.

21 By (4.12), we see that if µ < ( 1 16 ) 4 b+σ, then i.e., H2) holds. To verify H3), we denote HAMILTONIAN SYSTEMS 21 s ν+1 s b+σ s ν s ν 16, E ν = r ν r ν+1 = r (1 δ)δ ν+2. Since δ(1 + b + σ) > 1, we see that if µ is small, then E ν 2 log 1 s ν = 1 32 r (1 δ)δ 2 (δ(1 + b + σ)) ν log s (4.14) > 1 32 r (1 δ)δ 2 log s > 1. It follows from (4.12), (4.13), (4.14) that log(n + l + 1)! + 3(n + l ) log([log 1 s ν ] + 1) E ν ([log 1 s ν ] + 1) 3 (n + l ) log E ν log(n + l + 1)! + 3(n + l ) log(log 1 s ν + 2) (log 1 s ν ) 2 + (n + l ) log 1 s ν log 1 s ν, provided that µ (hence s ) is sufficiently small. Thus, K ν+1 λ n+l e λeν dλ (n + l + 1)! K n+l ν+1 Eν n+l i.e., H3) holds. Denote η = n + 2(l + 1)(2m 2 [τ] + 2m 2 + 1) and fix an < ε 1 such that e Kν+1Eν s ν, (4.15) (1 + σ )(1 + b + σ)(1 b 2σ 2ε) > 1, σ b + σ > (4b + 5σ + 3ε) + 3ε, λ (1 λ )σ 2ε >. We let µ (hence s ) be sufficiently small so that s ε ( ) 4η ( ) ν 32 s ε(1+b+σ)ν 4η ( ) 32 s Eν 4η = r (1 δ)δ 2 δ 4ην r (1 δ)δ 2 s ε εb ν δ 4η ( ) 4η (4.16) 32 r (1 δ)δ 2 s ε 1 (η!) 4 e. 8E Using (4.16) and the fact that 1 Γ ν = Γ(r ν r ν+1 ) = Γ( 1 2 r δ ν+2 (1 δ)) e 2E we have (4.17) s ε ν Γ4 ν (η!)4 e 8E s ε ν Since σ Eν 4η 1. σ b + σ (1 + σ )a >, 1 λ η e λeν dλ η!e2e Eν η,

22 22 Y. LI AND Y. YI it follows that if µ is small, then ( ) c µ ν Γ ν s c µ ν sν 2ε ε (4.18) ν E ν Eν 3η (s ε ν Γ3 ν ) c µ ν sν 2ε = c µ s σ b+σ c ν s (1+b+σ)ν ( σ b+σ 2ε) c µ s σ b+σ (c s σ 2ε(b+σ) ) ν c µ s σ b+σ c µ 1 4 < 1, i.e., H5) holds. (4.11)-(4.13), (4.15) and ν 1 yield that (4.19) µ ν s 4b+5σ+2ε ν = (c ) ν µ s σ b+σ ((1+b+σ)ν 1) s (1+b+σ)ν (4b+5σ+2ε) = s σ b+σ µ (c ) ν s ( σ s σ b+σ µ (c s ε )ν µ 1 4, b+σ (4b+5σ+2ε))(1+b+σ)ν provided that µ is small so that c s ε 1. This together with (4.17) implies that c s ν µ ν Γ ν s ν+1 c µ ν s b+σ+ε ν c µ 1 4 < 1, i.e., H6) holds. H7) is obvious as µ small. To verify H8), we let j = ν j be the sequence defined in (3.44) for the νth KAM step. Then, c 1 = c ((γν a s2 ν + s3 ν )µ2 ν Γ3 ν + s ν+1s 2 ν µ νγ ν + γν a (s3 ν + s 1 ν s3 ν+1 )µ ν), c 2 = c ((γν a s ν + s 2 ν )µ2 ν Γ3 ν + s2 ν µ νγ ν + γν a (s2 ν + s 2 ν s3 ν+1 )µ ν), c 3 = c ((γν a + s ν)µ 2 ν Γ3 ν + s νµ ν Γ ν + γν a (s ν + s 3 ν s3 ν+1 )µ ν), c j = c ((γν a + 1)µ2 ν Γ3 ν + µ νγ ν + γν a (s ν + s 3 ν s3 ν+1 )µ1 λ ν ), j 3. Using (4.17), (4.19), we have c 1 Γ ν γ a ν+1 s2 ν+1 µ ν+1 (2s 2(b+σ) ν + γ a s1 b 2σ 2ε ν + s 1 2b 3σ ν + s b ν)γ ν, c 2 Γ ν γν+1 a s (2s 3(b+σ) ν + γ a ν+1µ s1 b 2σ 2ε ν + sν 1 b 2σ + s 2b+σ ν )Γ ν, ν+1 c 3 Γ ν γν+1 a µ (2s 4(b+σ) ν + γ a s1 b σ 2ε ν + sν 1 σ + sν 3b+2σ )Γ ν, ν+1 c j 2s 4(b+σ) ν + sν λ (1 λ)σ 2ε + sν 1 σ + sν 3b+2σ, j 3. µ ν+1 Since, by (4.15) and ν 1, s 1 b 2σ 2ε ν s (1+b+σ)(1 b 2σ 2ε) Y = γ (1+σ)a(1+b+σ)(1 b 2σ 2ε) µ 1 b 2σ 2ε 4 γ a µ 1 b 2σ 2ε 4, it is clear that one can make µ small such that γ a s1 b σ 2ε ν γ a s1 b 2σ 2ε ν < 1 4.

23 HAMILTONIAN SYSTEMS 23 By (4.12), (4.13), all other terms in the above can also be made smaller than 1 4 by assuming µ small. This verifies H8). As µ sufficiently small, we have by (4.19) that 3c µ ν K 8m2 τ+8m 2 ν+1 < This verifies H9). Since by (4.17), (4.19), < we can make µ small so that (4.2) 3c ν ([log 1 2 (ν+2)(4m2 ) s4b+5σ+2ε 1 2 < 1 (ν+2)(4m2 ) 2 < 1 (ν+2)(4m) 2 ν+2. µ ν Γ ν µ 1 4 (c s ε )ν (s ε ν Γ ν)s 4b+5σ ν µ 1 4 (c s ε )ν, c µ ν Γ ν µ ν+1, ] + 1) 3(4m 2 τ+4m 2 ) µ ν c γ a µ ν Γ ν γa µ 1 4 (4.21) 2 ν+1 for all ν. We verify H1), H4) by induction. As H1), H4) trivially hold for ν =, the KAM step described in Section 3 is valid for ν =. We now assume that for some positive integer ν H1), H4) hold for all ν =, 1,, ν, i.e., the KAM steps are valid for all ν = 1, 2,, ν. Applying Lemma 3.4 and (4.21) for all ν =, 1,, ν, we have (4.22) ω l M ν ω l M 22 O ν +1 ω l M ν +1 ω l M Oν +1 l ω(ω ν +1 id) Oν +1 ν i= ν i= l ω (M i+1 M i ) Oi+1 l ωω i+1 l ωω i Oi+1 γ (1+σ)a+a µ 1 4, ν i= ν i= γ a µ i+1 γa µ 1 4, γ (1+σ)a+a µ i+1 i.e., H4) holds for ν = ν + 1. Denote ˆM ν +1 = M ν +1, M ν +1 22, ˆM = M, M 22, respectively. If µ is small so that s < 1 2 ( ˆM ) 1 O, then by (4.22) and the invertibility of ˆM on O, we see that ˆM ν +1 is non-singular on O ν +1, and ( ˆM ν +1) 1 Oν +1 ( ˆM ) 1 O 1 ˆM ν +1 ˆM Oν +1 ( ˆM ) 1 O 2 ( ˆM ) 1 O. Hence H1) holds for ν = ν + 1. Therefore, all H1)-H9) hold and the KAM steps described in Section 3 are valid for all ν. By performing the KAM steps described in Section 3 inductively, we then obtain the desired sequences stated in the lemma. Now, 2) clearly follows from (3.11), and (4.1), (4.3), (4.5), (4.9), (4.1) follow from Lemmas and (4.2), (4.21) accordingly. By the same argument as in (4.22), we also obtain (4.2), (4.4), (4.6) and (4.8) from (4.1), (4.3), (4.5) and (4.7) respectively. Note that 4) automatically holds for ν = 1. We now let ν > 1. By Lemma 3.6, it is clear that O ν 1 = {ω O ν 1 : L ν 1 k > γ ν 1 k τ, detlν 1 1k > γ2m ν 1 k 2mτ,

24 24 Y. LI AND Y. YI Denote Then (4.23) detl ν 1 2k > γ4m 2 ν 1 k, 4m2 τ for all < k K ν 1}. Ô ν = {ω O ν 1 : k, ω > γ ν 1 k τ, detlν 1 1k > γ2m ν 1 k 2mτ, detl ν 1 2k > γ4m 2 ν 1 k, for all K 4m2 τ ν 1 < k K ν }. O ν = {ω O ν 1 : k, ω > γ ν 1 k τ, detlν 1 1k > γ2m ν 1 k 2mτ, detl ν 1 2k > γ4m 2 ν 1 k, for all < k K 4m2 τ ν} = O ν 1 Ôν = Ôν. The lemma is now complete. 5. Proof of the Theorem 5.1. Convergence. By assuming µ = µ(r, s, l ) small, one can apply Lemma 4.1 inductively to obtain the following sequences: for ν =, 1,. Let Ψ ν = Φ Φ 1 Φ ν : D ν O ν D, H Ψ ν = H ν = N ν + P ν, N ν = e ν + Ω ν, y + 1 ( ) ( ) y y 2, M ν (ω) + h ν (y, z, ω), z z O = O ν. First, we show the uniform convergence of Ψ ν on D( r 2, β 2 ) O. Note that ν (5.1) Ψ ν = Ψ + (Ψ i Ψ i 1 ), where, for each i = 1, 2,, (5.2) (5.3) ν= i=1 Ψ i Ψ i 1 = Φ Φ i Φ Φ i 1 = 1 Since, by (4.1), on D( r 2, β 2 ) O, D(Φ Φ i 1 )(id + θ(φ i id)) D(Φ Φ i 1 )(id + θ(φ i id))dθ(φ i id). DΦ (Φ 1 Φ i 1 )(id + θ(φ i id))) DΦ i 1 (id + θ(φ i id)) µ 1 (1 + µ )(1 + 2 ) (1 + µ i 1 ) 1 e i 1 e 2,

25 HAMILTONIAN SYSTEMS 25 we have Ψ i Ψ i 1 r D( 2, β 2 ) O e 2 Φ i id r D( 2, β 2 ) O e 2 µ i, for all i = 1, 2,. Thus, Ψ ν converges uniformly on D( r 2, β 2 ) O. We denote its limit by Ψ. Then (5.4) Ψ = Ψ + (Ψ i Ψ i 1 ) = id + (Ψ i Ψ i 1 ). i= It follows that Ψ is real analytic in ξ = (x, y, z) and uniformly close to the identity. In fact, using a similar argument, Ψ can be shown to be C l uniformly close to the identity. To show the Whitney smoothness of Ψ in ω, one can apply the standard Whitney extension theorem (see [31, 36]) to uniformly extend all Φ ν, with respect to ω, to functions on D ν O of class C l σ in ω O for some fixed < σ < 1, whose Hölder norms satisfy the same estimate (4.1), up to multiplication of a constant. This results in a sequence of extended transformations Ψ ν defined on D ν O. Similar to (5.2), (5.3), one can use (4.1) to obtain the estimates i= ω l Ψ ν ω l Ψ ν 1 r D( 2, β 2 ) O c µ 1 4 for all ν = 1, 2,, which shows the uniform convergence of ωψ l ν, on D( r 2, β 2 ) O. Let Ψ be defined as in (5.4) in terms of the extended transformations Ψ ν. Then a similar argument shows that ωψ l ν are equally (Hölder) continuous, and ω l Ψ ν ω l Ψ, 1 l l 1, uniformly on D( r 2, β 2 ) O. Thus, the original limit Ψ is C l 1 Whitney smooth in ω O. Next, we show the convergence of the Hamiltonians H ν. By Lemma 4.1 1), e ν, Ω ν, M ν converge uniformly on O and h ν converges uniformly on D( β 2 ) O, as ν. We denote their limits by e, Ω, M, h respectively. Clearly, h = O( (y, z) 3 ). By a similar application of the Whitney extension theorem, one can uniformly extend all P ν, with respect to ω, to functions on D ν O of class C l σ in ω O, whose Hölder norms satisfy the same estimate (4.9), up to multiplication of a constant. For such extended functions P ν, we then use the same formula (3.12) to define smooth extensions of M ν on O. Then the estimates in (4.5), (4.6), up to multiplication of a constant, are still valid for all extended matrices M ν on O, which implies the uniform convergence of ωm l ν on O for all 1 l l 1. Using the Hölder norms of the extended functions P ν one further shows the Hölder continuities of ω l M ν which implies that the limit of M ν on O is of class C l 1. Thus, M is C l 1 Whitney smooth on O, and l ω M l ω M O = O(γ a µ 1 4 ), l l 1, in the sense of Whitney (see [31]). Similarly, e (ω), Ω (ω) are C l 1 Whitney smooth on O, h (y, z, ω) is real analytic in (y, z) D( β 2 ) and Cl 1 Whitney smooth in ω O, and l ωe l ωe O = O(γ 2(1+σ)a+a µ 1 4 ), 2 ν l ω (Ω id) O = O(γ (1+σ)a+a µ 1 4 ), l ω j (y,z) h l ω j (y,z) h D( β 2 ) O = O(γ a µ 1 4 ),

26 26 Y. LI AND Y. YI for all j l, l l 1. It follows that, on D( r 2, β 2 ) O, N ν and all their possible derivatives converge uniformly to N = e (ω) + Ω, y + 1 ( ) ( ) y y 2, M (ω) + h (y, z, ω) z z and its corresponding derivatives, in the sense of Whitney. By the definition of O, it is easy to see that Ω (O ) Ôγ. Let P = H Ψ N. Using the uniform convergence of N ν, Ψ ν to N, Ψ, respectively, we see that, for all l l 1, ω l P ν ω l P uniformly on D( r 2, β 2 ) O in the sense of Whitney, and P is real analytic in (x, y, z) D( r 2, s 2 ) and Cl 1 Whitney smooth in ω O. For any ν Z +, ω O, j Z+ n, k Z2m + with j + k 2, by applying the inequality P ν Dν γ a s 2 ν µ ν and performing Cauchy s estimate of P ν on D(r ν, 1 2 s ν), we see that j y k z P ν 2 j+2 γ a ν µ ν, for all j + k 2 and ν = 1, 2,. By (4.11), it is easy to see that the right hand side of the above converges to as ν. Hence, on D( r 2, ) O, j y k z P (y,z)= = for all x T n, ω O, j Z+ n, k Z2m + with j + k 2. It follows that for each ω O, T ω = T n {} {} is an analytic, quasi-periodic, invariant n-torus associated to the Hamiltonian H = H Ψ with the Diophantine toral frequency Ω (ω), which corresponds to a perturbed invariant n-torus of (1.1) with the same properties. Moreover, these perturbed tori form a C l 1 Whitney smooth family. In the case that M is non-singular on O, it follows from Lemma 4.1 2) that Ω (ω) ω, i.e., toral frequencies are also preserved in this case Measure estimates. We wish to show that (5.5) O \O, as γ. The proof of (5.5) will be based on the following two lemmas. The first lemma shows the connection between the non-resonance condition NR) with the small divisor condition involved in (3.31) for ν =. The second lemma deals with measure estimates. Lemma 5.1. Let λ j (ω), j = 1, 2,, 2m, be eigenvalues of JM22 (ω). Then the following holds. 1) For all k Z n, detl 1k = detl 2k = 2m i=1 2m i,j=1 ( 1 k, ω λ i ), ( 1 k, Ω λ i λ j ).

27 HAMILTONIAN SYSTEMS 27 2) The set {ω O : k, ω, detl 1k, detl 2k, for all k Z n \ {}} admits full Lebesgue measure relative to O. Proof. 1) Let E be the Jordan canonical form of M22 J and let T be the non-singular matrix such that T 1 (M22J)T = E. Then detl 1k = det( 1 k, ω I 2m M 22J) = det( 1 k, ω I 2m E) = 2m i=1 ( 1 k, ω λ i ). Since, for any square matrices A, B, C, D of the same dimension, we have (A B)(C D) = (AC BD), (T 1 T 1 )((M 22 J) I 2m + I 2m (M 22 J))(T T ) = E I 2m + I 2m E. It follows that detl 2k = det( 1 k, Ω I 4m 2 M 22J I 2m I 2m (M 22J)) = det( 1 k, Ω I 4m 2 E I 2m I 2m E) = 2m i,j=1 ( 1 k, Ω λ i λ j ). The proof of 1) is now complete since JM22, M 22 J have the same eigenvalues. 2) Denote O 1 = {ω O : 1 k, ω λ i for all k Z n \ {}, 1 i 2m}, O 2 = {ω O : 1 k, ω λ i λ j for all k Z n \ {}, 1 i, j 2m}. For any ω O 2, k Z n \{}, 1 i 2m, we have that 1 2k, ω λ i λ i, i.e., ω O 1. Thus, O 2 O 1, and 2) easily follows from NR) and 1). Lemma 5.2. Suppose that g C p (Ī), p 2, where I R1 is a finite interval. Let I h = {x I : g(x) h}, h >. If, on I, g (p) (x) c > for some constant c, then I h c h 1 p, where c = p c. Proof. See [39], Lemma 2.1. Let R ν+1 k (γ) = {ω O ν : L ν k γ k τ, or detlν 1k We note by Lemma 4.1 4) that O ν+1 = O ν \ for all ν =, 1,, which implies that O \O = K ν< k K ν+1 R ν+1 R ν+1 ν= K ν< k K ν+1 γ2m k 2mτ, or detlν 2k k (γ ν ), k (γ ν ). γ4m 2 k }. 4m2 τ

28 28 Y. LI AND Y. YI Thus, the measure estimate (5.5) amounts to the estimate of R ν+1 k (γ) for all ν and k. Given k = (k 1, k 2,, k n ) Z n \ {}, ν =, 1,. Without loss of generality, we assume that k 1 = max{ k i }. For any (ω 2,, ω n ), we consider the sets I = {ω 1 : ω = (ω 1, ω 2,, ω n ) O ν }, and S 1 = {ω 1 I : ω = (ω 1, ω 2,, ω n ), g(ω) where g(ω) = det(l ν 2k (ω)). By (4.6), we have that, on I, A k = 4m2 ω1 4m2 γ4m2 ν k 4m2 τ }, g(ω) = k 1 4m2 ((4m 2 1 )! + O( k + 1 ) + O(µ 1 4 )), where O( 1 k +1 ), O(µ 1 4 ) are independent of ν, ω, l. Thus, there is a positive integer n such that A k k 1 1, provided that k n and µ is small. Hence by Lemma 5.2, (5.6) S 1 (4m 2 + 3) γ ν k τ, provided that k n. It follows from (5.6) and Fubini s theorem that if k n, then {ω O ν : g(ω) γ4m2 ν k } c S 4m2 τ 1 c γ k τ. Similarly, by making n larger if necessary, we have that {ω O ν : L ν k γ ν k τ } c γ k τ, {ω O ν : detl ν 1k γ2m ν γ } c k 2mτ k τ for all k n. As all constants above are independent of k, ν, we conclude that (5.7) R ν+1 k (γ ν ) c γ k τ for all ν and all k n. Let ν be such that K ν n as ν ν. Then (5.8) 1 k (γ ν ) cγ k τ = O(γ). ν=ν ν=ν K ν< k K ν+1 K ν< k K ν+1 R ν+1 We now estimate R ν+1 k (γ ν ) for < k K ν, ν ν. Since, by Lemma 4.1 1), Ω ν id Oν, M ν M Oν = O(γ 2a ), one can make γ small such that R ν+1 k (γ ν ) is contained in the set {ω O : k, ω 2γ k τ, or detl 1k 2γ2m k 2mτ, or detl 2k 2γ4m 2 k } 4m2 τ for all < k K ν, ν ν. It follows from NR) and Lemma 5.1 that uniformly for all < k K ν, ν ν. R ν+1 k (γ), as γ,