Persistence of invariant tori on sub-manifolds in Hamiltonian systems Shui-Nee Chow School of Mathematics, Georgia Institute of Technology, Atlanta, GA 3033, USA Yong Li Department of Mathematics, Jilin University, Changchun 3003, PRC Yingfei Yi School of Mathematics, Georgia Institute of Technology, Atlanta, GA 3033, USA Abstract Generalizing the degenerate KAM theorem under the Rüssmann non-degeneracy and the isoenergetic KAM theorem, we employ a quasi-linear iterative scheme to study the persistence and frequency preservation of invariant tori on a smooth sub-manifold for a real analytic, nearly integrable Hamiltonian system. Under a nondegenerate condition of Rüssmann type on the sub-manifold, we shall show the following: a) the majority of the unperturbed tori on the sub-manifold will persist; b) the perturbed toral frequencies can be partially preserved according to the maximal degeneracy of the Hessian of the unperturbed system and be fully preserved if the Hessian is nondegenerate; c) the Hamiltonian admits normal forms near the perturbed tori of arbitrarily prescribed high order. Under a sub-isoenergetic nondegenerate condition on an energy surface, we shall show that the majority of unperturbed tori give rise to invariant tori of the perturbed system of the same energy which preserve the ratio of certain components of the respective frequencies. Keywords. Hamiltonian system, KAM theory, persistence on sub-manifolds. AMS(MOS). Mathematics Subject Classification. 58F05, 58F7, 58F30. Introduction We consider an analytic family of real analytic Hamiltonian systems of the following actionangle form H = N(y) + εp (x, y, ε), (.) where (x, y) lies in a complex neighborhood {(x, y) : Imx < r, dist(y, G) < β} of T d G, G R d (d > ) is a bounded closed region, and ε is a small parameter. Partially supported by NSFC grant 99704, National 973 key Project: Nonlinearity in China and the outstanding youth project of Ministry of Education of China. Partially supported by NSF grant DMS980358. This work is partially done when the second and third authors were visiting the National University of Singapore.
With the symplectic form n dx i dy i, i= the associated unperturbed motion of (.) is simply described by the equation { ẋ = ω(y), ẏ = 0, where ω(y) = N y (y). Thus, for ε = 0, the phase space G T d is foliated into invariant tori T y = {y} T d with the frequency vectors ω(y), y G. Under the Kolmogorov nondegenerate condition, i.e., K) the Hessian A(y) N (y) is nonsingular for all y G, y the classical KAM theorem (see Kolmogorov [7], Arnold [], Moser [0]) says that the majority of the invariant d-tori will persist as ε sufficiently small. More precisely, there is a family of Cantor sets G ε G, with G\G ε 0, as ε 0, such that for each y G ε, the torus T y persists and gives rise to a slightly deformed, analytic, quasi-periodic, invariant torus T ε y of the perturbed system. Moreover, for each y G ε the perturbed torus T ε y preserves the frequency ω(y) of the corresponding unperturbed torus T y. Recently, a fair amount of attention was given to the persistence of a fixed Diophantine torus with the preservation of the toral frequency, see Benettin et. al. [4] for a KAM approach, Eliasson [], Gallavotti [3], Chierchia and Falcolini [0] for a direct method using Lindstedt series, and Gallavotti, Gentile and Mastropietro [4] and Bricmont, Gawedzki, and Kupiainen [5] for using renormalization groups techniques. Important generalizations to the classical KAM theorem were also made for various degenerate cases (i.e., when the Hessian A(y) becomes singular), see Bruno ([8]), Cheng and Sun ([9]), Rüssmann ([4]), Xu, You and Qiu ([7]), Sevryuk ([5]) and references therein. The persistence of KAM tori has been shown under various partially nondegenerate conditions. The weakest such condition was given by Rüssmann in [4] which says that the frequencies {ω(y) : y G} should not lie in any hyperplane of R d. In the real analytic case, it is shown in [7] that the Rüssmann condition is equivalent to that ω R) max y G rank{ α : α d } = d. yα The matrix in the above is formed by d dimensional column vectors of all the partial derivatives of ω(y) of orders up to d. In this paper, instead of the persistence of invariant tori in the whole domain of the action variable, we shall study the persistence problem on a given smooth sub-manifold M in the action space G, e.g., a curve or a surface, which is either closed or with boundary. Clearly, such persistence will depend on both the non-degeneracy of the unperturbed system and the differential structure of the sub-manifold. By using a quasi-linear iterative scheme introduced in [8], we shall show the following results for (.) as ε sufficiently small:
) The majority of the unperturbed tori {T y : y M} will persist under a nondegenerate condition of Rüssmann type on M. ) The maximal number of the preserved frequency components of a perturbed torus is characterized by the maximal rank of the Hessian matrices {A(y) : y M}. 3) If A(y) is nonsingular on M, i.e., if the Kolmogorov nondegenerate condition is satisfied on M, then all Diophantine tori of the unperturbed system on M persist with unchanged toral frequencies. 4) If the unperturbed system admits a sub-isoenergetic non-degeneracy on an energy surface, then the majority of the unperturbed tori on the energy surface will persist and give rise to perturbed tori of the same energy, whose frequency ratios of the respective nondegenerate components are preserved. These results generalize both the KAM theorem under the Rüssmann non-degeneracy and the isoenergetic KAM theorem ([],[3],[6]). Similar to the isoenergetic case, one interesting phenomenon is that the Kolmogorov and the Rüssmann non-degeneracy can be independent conditions on a sub-manifold of G, i.e., one can have the Kolmogorov but not the Rüssmann non-degeneracy on a sub-manifold and vice versa. In contrast to the KAM theory on the entire region G, the validity of the Kolmogorov nondegenerate condition on a sub-manifold does not automatically guarantee the existence of a Diophantine torus on the manifold (hence the persistence of any torus), unless the Rüssmann nondegenerate condition is also satisfied on the manifold. It should be noted the Rüssmann nondegenerate conditions on the whole domain and on a sub-manifold are also independent conditions. Hence the persistence on a particular sub-manifold does not follow from the Rüssmann non-degeneracy on the entire domain (see the examples in Section for detail). The quasi-linear scheme we employed follows the standard KAM iterative procedure but involves solving a system of quasi-linear equations at each KAM step instead of linear ones. This has the advantage of eliminating any prescribed number of high order angulardependent terms in just one iteration, resulting in a normal form in the vicinity of a perturbed torus of arbitrarily high order. The paper is organized as follows. In Section, we state our results with respect to both (.) and a parameterized Hamiltonian system, along with some discussion and examples. The quasi-linear iterative scheme will be described in Section 3 for one KAM cycle. We complete the proof of our results in Section 4 by deriving an iteration lemma and giving measure estimates. Acknowledgment. The authors would like to thank the referees and editors for their valuable suggestions and comments which leads to a significant improvement of the paper. Main results Below, unless specified otherwise, we shall use the same symbol to denote an equivalent (finite dimensional) vector norm and its induced matrix norm, absolute value of functions, and measure of sets etc., and use D to denote the supremum norm of functions on a domain D. Also, for any two complex column vectors ξ, ζ of the same dimension, ξ, ζ 3
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. Moreover, all Hamiltonians in the sequel are associated to the standard symplectic structure. We first consider the following parameter-dependent, real analytic Hamiltonian system H = e(λ) + ω(λ), y + h(y, λ) + P (x, y, λ), (.) where (x, y) lies in a complex neighborhood D(r, s) = {(x, y) : Imx < r, y < s} of T d {0} T d R d, λ is a parameter lying in a bounded closed region Λ R d 0, h(y, λ) = O( y ). In the above, all λ dependence are of class C l 0 for some l 0 d. Write h = y, A(λ)y + ĥ(y, λ), where A(λ) is real symmetric for each λ Λ and ĥ(y, λ) = O( y 3 ). following conditions: We assume the A) rank{ α ω : α d } = d for all λ Λ. λα A) ranka(λ) n on Λ, and, there is a smoothly varying, nonsingular, n n principal minor A(λ) of A(λ). Remark. ) In the case that ω is real analytic and Λ is connected, the condition A) can be replaced by max λ Λ rank{ α ω λ α : α d } = d, which becomes the Rüssmann condition R) when d 0 = d. Indeed, as pointed out in [7] for the case d 0 = d, the Rüssmann condition implies that A) holds on an open subset Λ 0 Λ with Λ \ Λ 0 = 0. ) The condition A) is more or less automatic in the sense that if n = max ranka(λ) = λ Λ A(λ 0 ), then due to the symmetry of A(λ) there is an n n principal minor of A(λ) which is smoothly varying and nonsingular in a neighborhood of λ 0. Denote i, i,, i n as the row indices (in the natural order) of A(λ) in A(λ), and set d = max{d 0, d}. Our main result is as follows. Theorem A. Consider (.) and let m be a given integer. ) Assume A),A) and let τ > d(d ) be fixed. Then there exists a µ = µ(r, s, m, l 0, τ) > 0 sufficiently small such that if l λp D(r,s) Λ γ m+l 0+5 s m µ, l l 0, (.) then there exist Cantor sets Λ γ Λ with Λ \ Λ γ = O(γ d ) and a C l 0 Whitney smooth family of C m symplectic transformations Ψ λ : D( r, s ) D(r, s), λ Λ γ, 4
which is real analytic in x and C m uniformly close to the identity such that H Ψ λ (x, y) = e (λ) + ω (λ), y + h (y, λ) + P (x, y, λ), (.3) where, for all λ Λ γ and (x, y) D( r, s ), h (y, λ) = O( y ), P (x, y, λ) = O( y m+ ), l λ (e e) = O(γ m+l 0+4 µ), l l 0, l λ(ω ω) = O(γ m+l 0+4 µ), l l 0, l λ j y(h h) = O(γ m+l 0+4 µ ), l l0, j m, and moreover, k, ω (λ) > γ k τ, for all k Zd \ {0}, (ω (λ)) iq (ω(λ)) iq, for all q n. Thus, for each λ Λ γ, the unperturbed torus T λ = T d {0} associated to the toral frequency ω(λ) persists and gives rise to an analytic, Diophantine, invariant torus of the perturbed system with the toral frequency ω (λ) which preserves the frequency components ω i (λ),, ω in (λ) of the unperturbed toral frequency ω(λ). Moreover, these perturbed tori form a C l 0 Whitney smooth family. ) Assume that A(λ) is nonsingular on Λ and let τ > d be fixed. Then there exists a µ = µ(r, s, m, l 0, τ) > 0 sufficiently small such that if (.) holds, then each Diophantine torus T λ = T d {0}, λ Λ, whose toral frequency ω(λ) having the Diophantine type (γ, τ), will persist, with the normal form (.3), and gives rise to an analytic, Diophantine, invariant perturbed torus with the same toral frequency. In the above theorem, d 0 can be any positive integer. The case d 0 > d will typically occur when the nondegenerate condition A) fails with respect to the original parameters of a Hamiltonian system and extra deformation parameters need to be added so that a joint nondegenerate condition of type A) can hold with respect to the extended parameters. When d 0 d, the theorem has a direct application to nearly integrable Hamiltonian systems of form (.) with respect to the persistence of invariant tori on a sub-manifold of G. Consider (.) and let M be a d 0 ( d) dimensional, C l 0 (l 0 d) sub-manifold of G which is either closed or with boundary. Denote ω(y) = N y (y), A(y) = N (y), y G. y We assume the following conditions: A) For any coordinate chart (φ, U) of M, rank{ α (ω φ ) λ α λ φ(u) R d 0. : α d } = d for all 5
A) ranka(y) n on M, and, there is a smoothly varying, nonsingular, n n principal minor A(y) of A(y) on M. Corollary. Consider (.). Let m be given and r, β be as in (.). ) Assume A),A) and let τ > d(d ) be fixed. Then there is an ε 0 = ε 0 (r, β, l 0, m, M, τ) > 0 and a family of Cantor sets M ε M, 0 < ε ε 0, with (d )(m+l M \ M ε = O(ε 0 +5) ), where d = max{d 0, d}, such that for each y M ε, the unperturbed torus T y persists and gives rise to an analytic, Diophantine, invariant torus of the perturbed system whose toral frequency ω ε (y) satisfies k, ω ε (y) > γ k τ, for all k Zd \ {0}, (ω ε (y)) iq = (ω(y)) iq, for all q n, (m+l where 0 < γ ε 0 +5), i,, i n are the row indices (in the natural order) of A(y) located in A(y). Moreover, these perturbed tori form a Whitney smooth family. ) Assume that A(y) is non-singular on M and let τ > d be fixed. Then each Diophantine torus T y, y M, whose toral frequency ω(y) having the Diophantine type (γ, τ) for some 0 < γ ε (m+l 0 +5), will persist and gives rise to an analytic, Diophantine, invariant perturbed torus with the same toral frequency. 3) Let y 0 M ε in ) or ω(y 0 ) be Diophantine in ). Then (.) admits the following normal form on D y0 ( r, β ) = {(x, y) : Imx < r, y y 0 < β}: H y0 (x, y) = e (y 0 ) + ω (y 0 ), y y 0 + h (y, y 0 ) + P (x, y, y 0 ), (.4) where ω (y 0 ) is the toral frequency of the perturbed torus associated to y 0 (hence equals ω(y 0 ) in the case )), h (y, y 0 ) = O( y y 0 ), P (x, y, y 0 ) = O( y y 0 m+ ), which satisfy similar properties as described in part ) of Theorem A with µ = ε m+l 0 +5, γ = ε (m+l 0 +5). Remark. ) Under the Kolmogorov or isoenergetic non-degeneracy, the arbitrarily high order normal forms of type (.4) around a perturbed Diophantine torus plays the role of the classical Birkhoff normal forms and the existence of such has implications on the measure of the set of invariant tori around the torus (see [],[],[] and references therein). In particular, a more or less straightforward application of Theorem 4 and its proof in [] to the normal form (.4) gives rise to an exponential measure estimate of the set of invariant tori around a perturbed Diophantine torus of (.). More precisely, if y 0 is as in part 3) of the Corollary, B ρ is a ball in R d centered at y 0 with sufficiently small radius ρ, and T ρ T d B ρ is the set of points lying in invariant d-tori of (.4), then (T d B ρ ) \ T ρ = O(e m 8 ). Furthermore, corresponding to the original Hamiltonian (.), one can choose m = [(cγ/ρ) τ+ ] for some constant c to conclude that the Lebesgue 6
measure of the set of points in T d B ρ which do not lie in any invariant d-torus of (.) is of the order of O(exp{ (cγ/6ρ) τ+ }) (see [], page 93). Given the above, it would be interesting to know whether the normal form (.4) can also lead to a similar measure estimate under the Rüssmann non-degeneracy, or more generally the condition A). In the later, relative measure estimates of a similar nature on a sub-manifold should be considered. ) Both Theorem A and the Corollary trivially hold when d =. In this case, one can simply take τ > 0, d = (see [9] for more discussions). To illustrate the significance and application of the Corollary, we now consider (.) with d = and assume that N(y) has the form N(y) = h (y ) + h (y ). Particular examples of N(y) to be considered are N (y) = y + y, N (y) = y + 3 y3, N 3 (y) = y + y. It is clear that the Hessian matrices associated to N, N, N 3 read ( ) ( ) 0 0 0 A (y) =, A 0 (y) =, A 0 y 3 (y) = ( 0 0 respectively. Below, we discuss the application of the Corollary to the Hamiltonians above on three type of analytic plan curves: line segment, parabola, and circle. Since d 0 = in these cases, the non-degenerate condition A) will give rise to certain twist conditions on these curves. Example (Line segment). Consider the line segment: M : y (λ) = a λ, y (λ) = a λ, λ [, ], where (a, a ) is a non-zero vector. Then it is easy to see that A) is equivalent to the twist condition h h h h a y y a y y 0 (.5) on M. For N, (.5) becomes a 0. Thus, if a is non-zero and a is arbitrarily given, then it follows from part ) of the Corollary and the expression of A that the majority of -tori on M will persist with unchanged second components of toral frequencies. Since A is singular, part ) of the Corollary is not applicable. For N, (.5) becomes a a 0. Thus, if both a and a are non-zero, then part ) of the Corollary and the fact that ranka imply that all Diophantine -tori on M will 7 )
persist with unchanged toral frequencies. In fact, since A is non-singular on M if a 0, the same conclusion also follows from part ) of the Corollary (a 0 is still required since Diophantine tori are considered). We note that only due to the application of part ) of the Corollary one knows that the set of Diophantine -tori on M is a nearly full measure set (hence non-empty). For N 3, (.5) will never be satisfied with any choice of a, a. Hence part ) of the Corollary is not applicable. But since A 3 is always non-singular on M, one can apply part ) of the Corollary to conclude that all Diophantine -tori on M will persist with unchanged toral frequencies. However, we note in this special situation that a toral frequency of N 3 on M is Diophantine if only if (a, a ) is. Hence the above conclusion holds only if (a, a ) is Diophantine, in which case the persistence of all -tori on M follows. Example (Parabola). Consider the following parabola: M : y (λ) = a λ, y (λ) = a λ, λ [, ], where (a, a ) is a nonzero vector. Then it is easy to see that A) is equivalent to the twist condition a λ h h h h y y a y y 0 (.6) on M. For N, (.6) becomes a 0. Hence the same conclusion for N in Example is valid. For N, (.6) becomes a a 0. Thus, the same conclusion for N in Example is valid. For N 3, (.6) also becomes a a 0. Still, A 3 is always non-singular on M. Hence, if both a and a are non-zero, then one can apply either parts ) ) of the Corollary to conclude the persistence and the preservation of toral frequencies of all Diophantine -tori on M. But, again, it is because of the application of part ) of the Corollary that one can actually conclude the existence of Diophantine -tori on M in the case a a 0. Example 3 (Circle). Consider the unit circle: M 3 : y (λ) = cos πλ, y (λ) = sin πλ, λ [0, ]. Then it is easy to see that A) is equivalent to the twist condition h h h h y y y + y y y 0 (.7) on M 3. For N, (.7) becomes cos πλ 0, i.e., A) is satisfied on M 3 except two points (0, ), (0, ). In view of Remark. ), one can still apply part ) of the Corollary to conclude the persistence of the majority of invariant -tori on M 3 and the preservation of their second frequency components. For N, (.7) becomes sin πλ 0, i.e., A) is satisfied on M 3 except two points (, 0), (, 0). Since the lower right minor of A also vanishes at these points, the 8
application of part ) of the Corollary (based on Remark. )) will only guarantee the persistence of the majority of invariant -tori on M 3 and the preservation of their first frequency components. In order to apply part ) of the Corollary to obtain the persistence of all Diophantine -tori with unchanged toral frequencies, one needs to restrict to a closed portion of M 3 which does not contain these two points. For N 3, (.7) always holds and A 3 is always non-singular on M 3. One can use both parts of the Corollary to conclude the persistence and the frequency preservation of all Diophantine -tori on M 3. We now consider the case that M is a fixed energy surface {N(y) = E} in G. Then the usual isoenergetic non-degeneracy on M implies the Rüssmann non-degeneracy A) on M. Indeed, if (φ, U) is any coordinate chart on M, then one clearly has ( N φ ) y λ 0 on φ(u). It follows from the isoenergetic non-degeneracy that ) ( N φ d = rank y λ 0 0 on φ(u), hence rank = rank = rank ( N y ( N y ) 0 N y ( N y ( N y φ λ φ ) φ λ N y λ 0 ) N y 0 0 ( N φ y λ, N ) ( ) N = rank y y, N φ y = d λ on φ(u). Thus, our Corollary asserts that the perturbed system admits invariant tori conjugating to certain unperturbed ones on the energy surface M, and moreover, the perturbed toral frequencies are preserved if the Kolmogorov non-degeneracy also holds on M. However, it should be noted that such perturbed tori do not lie on the same energy level E in general, simply because the perturbed tori on the same energy level are generally equivalent (not necessary conjugated) to the unperturbed ones and only the preservation of frequency ratios can be expected (see [6]). To generalize the standard isoenergetic KAM theorem, it turns out that an additional sub-isoenergetic nondegenerate condition is needed besides the Rüssmann non-degeneracy on an energy surface. More precisely, let M be a sufficiently smooth, relatively open, bounded subset of {N(y) = E}. We assume A) on M and also the following subisoenergetic non-degeneracy: A) There is a smoothly varying n n principal minor A(y) of A(y) on M such that ( ) A(y) ω det (y) ω (y) 0 0 on M, where ω (y) = N y (y), y = (y i,, y in ), and i,, i n denote the row indices of A(y) in A(y). 9 )
Theorem B. Consider (.). Let m be given, r, β be as in (.), and M be a sufficiently smooth, relatively open, bounded subset of {N(y) = E}. ) Assume A) on M and let τ > d(d ) be fixed. Then there is an ε 0 = ε 0 (r, β, l 0, m, M, τ) > 0 and a family of Cantor sets M ε M, 0 < ε ε 0, with M \ M ε = O(ε (d )(m+l 0 +5) ), where d = max{d 0, d}, such that for each y M ε, the unperturbed torus T y persists and gives rise to an analytic, Diophantine, invariant torus T ε,y of the perturbed system on the energy surface {H(x, y) = E}, whose toral frequency ω ε (y) satisfies k, ω ε (y) > γ k τ for all k Z d \ {0}. Moreover, these perturbed tori form a local Whitney smooth family. ) If A) also holds on M, then each perturbed torus T ε,y preserves the ratio of the i, i,, i n components of its toral frequency ω ε (y), i.e., [ω ε,i (y) : : ω ε,in (y)] = [ω i (y) : : ω in (y)], where ω ε,ij (y) and ω ij (y) are the i j -th components of ω ε (y) and ω(y) = N y (y) respectively, for j =,,, n. 3) For y 0 M ε, (.) admits the same normal form as in part 3) of the Corollary. In the case that A(y) N(y) y (y), the condition A) coincides with the isoenergetic non-degeneracy which also implies the Rüssmann condition A) on the energy surface. Hence, part ) of the Theorem B generalizes the standard isoenergetic KAM theorem. 3 KAM step In this section, we describe the quasi-linear iterative scheme for the Hamiltonian (.) in one KAM cycle, say, from a νth KAM step to the (ν + )th-step. For simplicity, we set l 0 = d. Consider (.) and define r 0 = r, γ 0 = 4γ, β 0 = s, Λ 0 = Λ, H 0 = H, e 0 = e, A 0 = A, A 0 = A, h 0 = h, ĥ0 = ĥ, P 0 = P, N 0 = e 0 (λ) + ω 0 (λ), y + h 0. Without loss of generality, we assume that 0 < r 0, β 0, γ 0 and A 0 is the ordered n n principal minor of A 0. By monotonicity, we define µ 0, s 0 implicitly through the following equations µ = s 0 = 4 d+5 µ 0 (M + ) m K mτ, β 0 γ 0 6(M + )K τ, (3.) 0
where M = max l d, j m+5, y β 0,λ Λ 0 l λ j yh 0 (y, λ), K = ([log µ 0 ] + ) 3η, η is a fixed positive integer such that ( + σ) η > for σ = (m+). It is clear that µ 0 is small if and only if µ is, and, µ 0 = o(µ ɛ ) (3.) for any 0 < ɛ <. By making µ small, we assume without loss of generality that Hence s 0 < min{β 0, γ 0 }. Since (.) becomes 6(M + )K τ >. ( ) µ m = 4 d+m+5 s0, (3.3) µ 0 β 0 γ l λp 0 D(r0,s 0 ) γ d+m+5 0 s m 0 µ 0, l d. (3.4) Now, suppose that after a νth-step, we have arrived at the following real analytic Hamiltonian: H = N + P, (3.5) N = e(λ) + ω(λ), y + h(y, λ) which is defined on a phase domain D(r, s) and depends smoothly on λ Λ, where Λ Λ 0, h = y, A(λ)y + ĥ, ĥ = ĥ(y, λ) = O( y 3 ). In addition, suppose that the n n ordered principal minor A of A is non-singular on Λ, and, P = P (x, y, λ) satisfies l λ P D(r,s) γ d+m+5 s m µ, l d (3.6) for some 0 < µ µ 0, 0 < γ γ 0. By considering both averaging and translation, we shall find a symplectic transformation Φ +, which, on a small phase domain D(r +, s + ) and a smaller parameter domain Λ +, transforms the Hamiltonian (3.5) into the Hamiltonian of the next KAM cycle (the (ν + )th-step), i.e., H + = H Φ + = N + + P +, where N +, P + enjoy similar properties as N, P respectively on D(r +, s + ) Λ +. For simplicity, we shall omit index for all quantities of the present KAM step (the νthstep) and index all quantities (Hamiltonian, normal form, perturbation, transformation, and domains, etc) in the next KAM step (the (ν + )-th step) by +. All constants c c 7 below are positive and independent of the iteration process, and, we shall also use c to denote any intermediate positive constant which is independent of the iteration process. To simplify the notations, we shall suspend the λ dependence in most terms of this section.
Define 3. Truncation r + = r + r 0 4, s + = 8 αs, α = µσ = µ m+, β + = β + β 0 4, γ + = γ + γ 0 4, K + = ([log µ ] + )3η, D i 8 α = D(r + + i 8 (r r +), i αs), i =,,, 8, 8 D(ξ) = {y C d : y < ξ}, ξ > 0, ˆD(ξ) = D(r + + 7 8 (r r +), ξ), ξ > 0, D + = D 8 α = D(r +, s + ), D + = D(r + + 3 4 (r r +), β + ), Λ + = {λ Λ : k, ω(λ) > γ k τ, for all 0 < k K +}, r r + Γ(r r + ) = 8. 0< k K + k (d+m+6)τ+d+m+6 e k Consider the Taylor-Fourier series of P : P = p kj y j e k,x k Z d,j Z d + and let R be the truncation of P of the form Lemma 3. Assume that H) K + t d+m e t r r + 6 dt µ. R = k K +, j m p kj y j e k,x. (3.7) Then there is a constant c such that for all j Z+ d, l d, λ Λ, l λ(p R) D 78 α l λ R D 78 α c γ d+m+5 s m µ, c γ d+m+5 s m µ.
Proof: Without loss of generality, we let µ 0 8. Hence α. Let I = p kj y j e k,x, II = k >K +,j Z+ d k K +, j m+ p kj y j e k,x. Then P R = I + II. By using the standard Cauchy estimate, we have l λi ˆD(s) It follows that For q = m+, let k >K + l λp D(r,s) e k γ d+m+5 s m µ of l λ (P I) on ˆD(s) yields K + t d+m e t r r + 8 γ d+m+5 s m µ r r + 6 dt γ d+m+5 s m µ. κ=k + κ d+m e κ l λ(p I) ˆD(s) l λp D(r,s) + l λi ˆD(s) γ d+m+5 s m µ. l λii D 78 α be the obvious anti-derivative of = l λ c s m+ r r + 8 q. Then the Cauchy estimate yq q y q p kj e k,x y j dy D 78 α k K +, j m+ λ l (P I R) ˆD(s) dy D 78 α cγ d+m+5 s m µ s m+ dy D 78 α c(αs) m+ γ d+m+5 µ s = cγd+m+5 s m µ. Thus, and therefore, l λ(p R) D 78 α cγd+m+5 s m µ, (3.8) l λ R D 78 α P R D 78 α + P D(r,s) cγ d+m+5 s m µ. 3. Averaging and quasi-linear equations As usual, we shall construct the averaging transformation as the time -map φ F of the flow generated by a Hamiltonian F. Let F have the following form: F = f kj y j e k,x, (3.9) 0< k K +, j m 3
where f kj are (matrix valued) functions of y. Let [R] = R(x, )dx be the average of the truncation R defined in (3.7). Substituting F into the equation T n {N, F } + R [R] = 0 (3.0) yields k, ω(λ) + y h f kj y j e k,x = 0< k K +, j m 0< k K +, j m p kj y j e k,x. By equating the coefficients above, we then obtain the following quasi-linear equations: k, ω(λ) + y h f kj = p kj, j m, 0 < k K +. (3.) Lemma 3. Assume that H) max l d, j m+5 l λ j y h l λ j y h 0 D(s) Λ+ µ 0, H3) s < γ γ + (M +)K τ+ +. Then the quasi-linear equations (3.) can be uniquely solved on D(s) Λ + to obtain a family of functions f kj which are analytic in y, smooth in λ, and satisfy the following properties: f kj (ȳ, λ) = f kj (y, λ), (3.) l λ j yf kj D(s) Λ+ c k ( l + j +)τ+ l + j + γ d+m+4 l j s m j e k r (3.3) for all j m, 0 < k K +, l d, j m + 4, where c is a constant. Proof: It follows that Hence Let (y, λ) D(s) Λ +. By H), H3), is non-vanishing on Λ +, and, y h(y) (M + ) y < (M + )s < k, ω(λ) + y h(y) > γ k τ γ k τ = L k = k, ω(λ) + y h(y) f kj = f kj (y, λ) = L k p kj γ k τ+. γ k τ. for all (y, λ) D(s) Λ +, 0 < k K +, j m, from which (3.) clearly follows. Let 0 < k K +. We note by the Cauchy estimate that l λp kj Λ+ l λ j yp D(r,s) Λ+ e k r γ d+m+5 s m j µe k r, l d, j m, (3.4) 4
and by H) that l λ j y L k Therefore, D(s) Λ + c k j + L k j + c k ( l + j +)τ+ l + j + γ l + j +, l d, j m + 4. λ l j y f kj D(s) Λ+ c k ( l + j +)τ+ l + j + γ l + j + γ d+m+5 s m j e k r = c k ( l + j +)τ+ l + j + γ d+m+4 l j s m j e k r, l d, j m + 4. Let F be the Hamiltonian (3.9) with coefficients given by Lemma 3.. If φ t F denotes the flow generalized by F, then where with N + = N + [R], P + = This completes the averaging process. 0 H φ F = N + + P +, {R t, F } φ t F dt + (P R) φ F, (3.5) R t = ( t)[r] + tr. 3.3 Translation and partial non-degeneracy Let Y, P 0 be the vectors formed by the first n components of y, p 0 respectively and denote Ĥ(Y ) = ĥ(( Y 0 ) ). Then by the implicit function theorem, the equation AY + Y Ĥ(Y ) = P 0 (3.6) admits a unique solution Y on D(s) which also smoothly depends on λ. Define ( ) Y y =. 0 By (3.6), we clearly have and let Consider the translation Ay + y ĥ(y ) = ( ) P0 φ : x x, y y + y Φ + = φ F φ. 0. (3.7) 5
Then where H Φ + = N + + P +, N + = N + φ ψ = e + + ω +, y + h + (y), P + = P + φ + ψ, (3.8) e + = e + ω, y + y, Ay + ĥ(y ) + [R](y ), (3.9) ) ω + = ω + p 0 ( P0 0, (3.0) h + (y) = y, A +y + ĥ+(y), (3.) A + = A + yĥ(y ) + y[r](y ), (3.) ĥ + (y) = ĥ(y + y ) ĥ(y ) y ĥ(y ), y y, yĥ(y )y (3.3) + [R](y + y ) [R](y ) y [R](y ), y y, y[r](y )y, ( ) ψ = y [R](y j ), y p 0, y = p 0j y j j y. (3.4) j m, j j m, j = 3.4 Estimate on N + Lemma 3.3 Assume H), H3). Then there is a constant c 3 such that the following holds for all l d: l λ y Λ+ c 3 γ d+m+5 s m µ; (3.5) l λe + l λe Λ+ c 3 γ d+m+5 s m µ; (3.6) λω l + λω l Λ+ c 3 γ d+m+5 s m µ; (3.7) { λ l j y h + λ l j y h c3 γ D(s + ) Λ + d+m+5 s m j µ, j m; c 3 γ d+m+5 (3.8) µ, j > m. Proof: Denote M = max λ Λ0 A 0 (λ) +. By (3.), we can make µ 0 small, say µ 0 < 8M (M +), such that M (M + )s 0 < 4. Let λ Λ +. To prove (3.5), we denote Then by (3.6), B(y) = A + ( 0 yĥ(θy)dθ)y. B(Y )Y = P 0. (3.9) Since, by H), A A 0 Λ µ 0 and yĥ D(s) (M + )s, we have that A 0 B(Y ) A A 0 + B(Y ) A 0 µ 0 + (M + )s µ 0 + (M + )s 0 M. 6
It follows that B(Y ) is non-singular and Hence B (Y ) A 0 A 0 B(Y ) A 0 M. y = Y M P 0 = M p 0 M y P D(s) M γ d+m+5 s m µ. (3.30) Differentiating (3.9) with respect to λ yields Therefore, B(Y ) λ Y + y B(Y )( λ Y )Y + λ B(Y )Y = λ P 0. λ Y 4M (M + )γ d+m+5 s m µ λ Y + M c Y + M c λ y P D(s). The estimate (3.5) now follows from (3.6), (3.30) and induction. Using (3.9) ((3.0) respectively), (3.6) ((3.7) respectively) easily follows from H), (3.5) and (3.4). Also, it follows from (3.) that Note by (3.5) that We have that Therefore, ĥ + = l λ A + l λ A Λ + c 3 γ d+m+5 s m µ. (3.3) j 3 = ĥ + ĥ = j 3 j 3 j! j yĥ(y )y j + j! j yĥ(y )y j + 3 j m 3 j ı m j! ( j yĥ(y ) j yĥ(0))yj + l λ j yĥ+ l λ j yĥ D(s) Λ + Combining the above with (3.3), we obtain (3.8). j! j y[r](y )y j ( ) ı p 0ı (y ) ı j y j. j 3 j ı m ( ) ı p 0ı (y ) ı j y j. j { c3 γ d+m+5 s m j µ, j m; c 3 γ d+m+5 µ, j > m. 3.5 Estimate on Φ + Let F be as in (3.9) with coefficients given by Lemma 3.. By (3.), F is real analytic in (x, y) D(r, s). Lemma 3.4 Assume H), H3). Then the following holds. 7
) There is a constant c 4 such that for all l d, i m + 4, { λ l x i yf j c4 γ ˆD(s) Λ + d+m+4 l j s m j µγ(r r + ), j m; c 4 γ d+m+4 l j µγ(r r + ), m < j m + 4. ) F, y can be extended to functions of Hölder class C m+3,d +σ 0 ( ˆD(β 0 ) Λ 0 ), C d +σ 0 (Λ 0 ), respectively, where 0 < σ 0 < is fixed. Moreover, there is a constant c 5 such that where Proof: F C m+3,d +σ 0 ( ˆD(β0 ) Λ 0 ) c 5 µγ(r r + ), y C d +σ 0 (Λ0 ) c 5 µγ(r r + ). By (3.9), (3.3), we have λ l x i yf j c k i y( j λf l kj y j ) e k (r++ 7 8 (r r +)) j m,0< k K + c k ( l + j +)τ+ l + i + j + γ d+m+4 l j s a( j ) µe k 0< k K + cγ d+m+4 l j s a( j ) µγ(r r + ), a( j ) = { m j, if j m, 0, if m < j m + 4. This proves ). ) follows from the standard Whitney extension theorem (see [], [6]). Lemma 3.5 In addition to H), H3), assume that H4) c 4 s m µγ(r r + ) < 8 (r r +); H5) c 4 s m µγ(r r + ) < 8 αs; H6) c 3 s m µ < 8 αs. Then for all 0 t, φ t F : D 4 α D α, (3.3) φ : D 8 α D 4 α (3.33) are well defined, real analytic and depend smoothly on λ Λ +. Proof: (3.33) follows immediately from Lemma 3.3 and H6). To show (3.3), we write φ t F = (φt, φt ), where φ t, φt are components of φt F in the directions x, y respectively. Let (x, y) D 4 α and let t = Sup{t [0, ] : φ t F (x, y) D α}. Then for any 0 t t, φ t F (x, y) x t φ t F (x, y) y 0 t 0 r r + 8 F y φ u F D α du F y ˆD(s) c 4 s m µγ(r r + ) < 6 (r r +), F x φ u F Dα du F x ˆD(s) c 4 s m µγ(r r + ) < 6 αs. It follows that φ t F (x, y) < r + + 3 8 (r r +), φ t F (x, y) < 3 8 αs, i.e., φt F (x, y) D α D α. Thus, t = and (3.3) holds. 8
The above lemma implies that Φ + : D + D α is well defined, symplectic and real analytic for all λ Λ +. We now consider Φ + on the domain D +. Lemma 3.6 Assume H), H3) and also the following: H7) c 5 µγ(r r + ) < 8 (r r +); H8) c 5 µγ(r r + ) + c 3 δµ < β β +. Let F, y be the extended functions defined in Lemma 3.4 ). Then Φ + = φ F φ : ˆD + D(r, β) (3.34) is of class C m+ and also depends C d +σ 0 smoothly on λ Λ 0, where σ 0 is as in Lemma 3.4 ). Moreover, there is a constant c 6 such that Φ + id C m+,d +σ 0 ( D+ Λ 0 ) c 6µΓ(r r + ). (3.35) Proof: By a similar argument as in Lemma 3.5, it is easy to see that Φ + maps ˆD + into D(r, β) for all λ Λ 0. Let X F = (F y, F x ) be the vector field generated by F. We note that φ t F = id + t 0 X F φ u F du, 0 t, X F C m+,d +σ 0 ( ˆD(β0 ) Λ 0 ) c F C m+3,d +σ 0 ( ˆD(β 0 ) Λ 0 ). By applying Lemma 3.4 ) and the Gronwall inequality inductively, we have that, on D + Λ 0, φ t id, y φ t F I n, j yφ t F cµγ(r r + ), j m +, 0 t. (3.36) The lemma now follows from Lemma 3.4 ) and the identity ( ) Φ + id = (φ 0 F id) φ + y. (3.37) 3.6 Frequency property Lemma 3.7 Assume H),H3),H6). Then for all λ Λ + and 0 < k K +. Proof: By H3), H6), we have It follows from Lemma 3.3 that k, ω + (λ) > γ + k τ, c 3 s m+ µk τ+ + < γ γ +. as desired. k, ω + (λ) k, ω(λ) c 3 γ 0 s m µk + γ k τ c 3γ 0 s m µk + > γ + k τ, 9
3.7 Estimate on P + Lemma 3.8 Assume H)-H6). Then, there is a constant c 7 such that, on D + Λ +, λ l P + c 7 γ d+m+5 s m µ (Γ (r r + ) + ), l d. (3.38) Proof: By Lemmas 3., 3.4 ) and (3.36), we see that, for all l d, 0 t, λ l {R t, F } φ t F D 4 α Λ + cγd+m+5 s m µ Γ (r r + ), λ l (P R) φ F D 4 α Λ + cγd+m+5 s m µ Γ(r r + ). Hence, by (3.5), λ l P + D 4 α Λ + cγd+m+5 s m µ (Γ (r r + ) + ), l d. Since, by (3.4), l λ p 0j Λ+ cγ d+m+5 s m µ, l d, it follows from (3.4), (3.5) that By (3.5), we also have l λ ψ D + Λ + γ d+m+5 s m j µ, l d. l λφ D+ Λ + cγ d+m+5 s m µ, l d. The lemma now follows from (3.8) and the above estimates. Let c 0 = max{, c,, c 7 } and define If we assume that H9) µ σ (Γ (r r + ) + ) γd+m+5 +, γ d+m+5 then, on D + Λ +, µ + = 8 m c 0 µ +σ. l λ P + 8 m c 0 s m + µ+σ µ σ m m+ (µ σ γ d+m+5 (Γ (r r + ) + )) γ d+m+5 + s m + µ +, l d. This completes one cycle of KAM steps. 4 Proof of Main Results 4. Iteration Lemma Consider (.) and let r 0, s 0, γ 0, β 0, µ 0, Λ 0, H 0, N 0, e 0, ω 0, h 0, A 0, ĥ0, P 0 be given at the beginning of Section 3 and let ˆD 0 = D(r 0, β 0 ), K 0 = 0. We define the following sequences 0
inductively for all ν =,, : r ν = r 0 ( ν i= s ν = 8 α ν s ν, α ν = µ σ ν = µ µ ν = 8 m c 0 µ +σ β ν = β 0 ( γ ν = γ 0 ( ), i+ m+ ν, ν, ν i= ν i= ), i+ ), i+ K ν = ([log( )] + ) 3η, µ ν Λ ν = {λ Λ ν : k, ω ν (λ) > γ ν k τ for all 0 < k K ν }, D ν = D(r ν, s ν ), D ν = D(r ν + 3 4 (r ν r ν ), β ν ). Lemma 4. If (3.4) holds for a sufficiently small µ 0 = µ 0 (r 0, β 0, m, d, τ), or equivalently, µ = µ(r, s, m, d, τ), then the KAM step described in Section 3 is valid for all ν = 0,,, resulting in sequences: ν =,,, with the following properties. Λ ν, H ν, N ν, e ν, ω ν, h ν, A ν, ĥν, P ν, Φ ν, ) Φ ν : ˆD ν Λ 0 ˆD ν, D ν Λ ν D ν is symplectic for each λ Λ 0 or Λ ν, and is of class C m+,d +σ 0, C α,d, respectively, where α stands for real analyticity and 0 < σ 0 < is fixed, and, Moreover, on ˆD ν Λ ν, Φ ν id C m+,d +σ 0 ( ˆDν Λ 0 ) µ ν. (4.) H ν = H ν Φ ν = N ν + P ν, where H ν = N ν + P ν, N ν = e ν + ω ν, y + h ν, h ν = y, A νy + ĥν, A ν is real symmetric with its n n ordered principal minor A ν being non-singular on Λ ν, ĥν = O( y 3 ).
) (ω ν (λ)) q = (ω ν (λ)) q for all q =,,, n and λ Λ ν. 3) For all l d, λe l ν λe l ν Λν γ0 d+m+4 µ ν, (4.) λe l ν λe l 0 Λν γ0 d+m+4 µ, (4.3) λ l ω ν λ l ω ν Λν γ0 d+m+4 µ ν, (4.4) λ l ω ν λ l ω 0 Λν γ0 d+m+4 µ, (4.5) l λ j yh ν l λ j yh ν D(sν) Λ ν γ d+m+4 0 µ, j m +, (4.6) ν l λ j y h ν l D(s ν) λ j y h 0 Λν γ d+m+4 0 µ, j m +, (4.7) l λ P ν Dν Λ ν γ d+m+5 ν s m ν µ ν. (4.8) 4) Λ ν = {λ Λ ν : k, ω ν (λ) > γ ν k τ for all K ν < k K ν }. Proof: The proof amounts to the verification of H)-H9) for all ν. For simplicity, we let r 0 = β 0 =. First, it is obvious from (3.) that H3) holds for ν = 0. By choosing µ 0 small, we also see that H), H4)-H9) hold for ν = 0 and H6) holds for all ν. By the definition of µ ν, we have that Then µ ν = (8 m c 0 ) (+σ)ν µ (+σ)ν 0. (4.9) Let ζ be fixed and µ 0 be sufficiently small such that ( ) σ µ 0 < 8 m <. (4.0) c 0 ζ Denote We note that Since Γ ν µ = 8 m c 0 µ +σ 0 < ζ µ 0 <, µ = 8 m c 0 µ +σ < ζ µ < ζ µ 0, µ ν = 8 m c 0 µ +σ ν < < ζ ν µ 0. (4.) Γ ν = Γ(r ν r ν+ ). r ν r ν+ = ν+ = β ν β ν+ β 0. (4.) λ (d+m+6)τ+d+m+6 e λ ν+6 dλ ([(d + m + 6)τ] + d + m + 7)! (ν+6)((d+m+6)τ+d+m+6),
it is clear that if ζ is sufficiently large, then µ σ ν Γ i ν < µ σ ν(γ i ν + ) < γd+m+5 ν+ In particular, H9) holds for all ν, and, µ ν Γ ν µ σ γν d+m+5 ν µ σ 0, i =,. (4.3). (4.4) ζ ( σ)ν By (4.) and (4.4), it is easy to see that if ζ is sufficiently large and µ 0 is sufficiently small, then H4), H5), H7), H8) hold for all ν. Since t ν+3 dt (d + m + )! (ν+6)(d+m) Kν+e n K ν+ K ν+ t d+m e ν+, it follows from (4.9) and the inequality ( + σ) η > that H) holds for all ν 0 as µ 0 small. For the verification of H3), we observe by (4.) that as µ 0 small. Then 4 (M + )µ σ ν Kτ+ ν+ < ν+, (M + )s ν K τ+ ν+ s ν 4 (M + )µ σ ν K τ+ ν+ s 0 ν+ < γ 0 ν+ < γ ν γ ν+, which verifies H3) for all ν. Let ζ σ in (4.0), (4.). We have by (3.)-(3.3) that if µ 0 is sufficiently small, then the following holds for all ν : c 0 µ ν µ 0 ν µ ν, (4.5) c 0 µ ν Γ ν µ σ 0 c 0 s m ν ν µ ν, (4.6) µ ν µ+σ(m ) 0 s0 m ν+3 (µ 0s m 0 ) µ σ(m ) 0 ν µ 8s 0 ν. (4.7) The verification of H) follows from (4.5) and an inductive application of (3.8) for all ν = 0,,. Above all, the KAM steps described in Section 3 are valid for all ν, which gives the desired sequences stated in the lemma. Now, ) follows from Lemma 3.7, ) follows from (3.0) and induction, (4.), (4.4), (4.6) follow from (4.5), (4.7) and Lemma 3.3, and (4.8) follows from Lemma 3.8 and H9). By adding up (4.), (4.4), (4.6) for all ν =,,, we also obtain (4.3), (4.5), (4.7) respectively. 4) clearly holds for ν = 0. We now assume that ν > 0. Then by Lemma 3.6, Λ ν = {λ λ ν : k, ω ν (λ) > γ ν k τ, 0 < k K ν}. 3
Hence Λ ν+ = {λ Λ ν : k, ω ν (λ) > γ ν k τ, 0 < k K ν+} The lemma is now complete. = {λ Λ ν : k, ω ν (λ) > γ ν k τ, 0 < k K ν} {y 0 Λ ν : k, ω ν (λ) > γ ν k τ, K ν < k K ν+ } = Λ ν {y 0 Λ ν : k, ω ν (λ) > γ ν k τ, K ν < k K ν+ } = {y 0 Λ ν : k, ω ν (λ) > γ ν k τ, K ν < k K ν+ }. 4. Convergence Let Then Ψ ν : D ν Λ 0 D 0, and, Ψ ν = Φ Φ Φ ν, ν =,,. H 0 Ψ ν = H ν = N ν + P ν, N ν = e ν + ω ν (λ), y + h ν (y, λ), ν = 0,,, where Ψ 0 = id. Using (4.) and the identity ν Ψ ν = id + (Ψ i Ψ i ), i= it is easy to see that Ψ ν converges in C m+,d +σ 0 (D( r 0, β 0 ) Λ 0 ) norm to a function Ψ C m,d (D( r 0, β 0 ) Λ 0 ) such that Ψ λ = Ψ (, λ), λ Λ 0, are symplectic and C m uniformly close to the identity. Let Λ = Λ ν. ν 0 Then {Ψ λ : λ Λ } is a C d Whitney smooth family of analytic symplectic transformations on D( r 0, s 0 ). By Lemma 4., it is also clear that e ν, ω ν converge uniformly on Λ and h ν converge uniformly on D( s 0 ) Λ. Denote e, ω, h as the limit of e ν, ω ν, h ν respectively. Then, on D( s 0 ) Λ, N ν converge uniformly to N = e + ω (λ), y + h (y, λ). Hence, on D( r 0, s 0 ) Λ, converge uniformly to P ν = H 0 Ψ ν N ν, P = H 0 Ψ N. 4
Since P ν is real analytic on D ν and the Cauchy estimate yields that P ν Dν γ d+m+5 ν s m ν µ ν, j yp ν D(rν+m, sν) ( m r 0 ) m mν+ m + γ d+m+5 ν µ ν, j m. By (4.9), the right hand side of the above converges to 0 as ν, provided that µ (hence µ 0 ) is sufficiently small. Thus, on D( r 0, 0) Λ, j yp = 0, j m. Hence for each λ Λ, T d {0} is an analytic invariant torus of H with the toral frequency ω (λ), which, by definition of Λ ν and Lemma 4. ), satisfies γ k, ω (λ) > k τ, for all k Zd \ {0}, (ω (λ)) q (ω 0 (λ)) q, for all q n. Following the Whitney extension of Ψ ν s, all e ν, ω ν, h ν, P ν, ν = 0,,, admit uniform C d +σ 0 extensions in λ Λ 0 with derivatives in λ up to order d satisfying the same estimates (4.)-(4.8). Thus, e, ω, h, P are C d Whitney smooth in λ Λ, and, the derivatives of (e e 0 ), (ω ω 0 ), (h h 0 ) satisfy similar estimates as in (4.3),(4.5),(4.7). Consequently, the perturbed tori form a C d Whitney smooth family on Λ. 4.3 Measure estimate Lemma 4. Let Λ R d, d >, be a bounded closed region and let g : Λ R d be such that rank{ α g : α d } = d. λα Then for a fixed τ > d(d ) {λ Λ : g(λ), k γ ( ) k τ } γ c(λ, d, τ) d, k Z d k τ+ \ {0}, γ > 0. Proof: See Theorem B in [7]. We note that the constant c above does not depend on g but rather on a lower bound of the derivatives of g up to order d. The following measure estimate is adopted from [9]. We consider the following three cases. Case : d 0 = d. Let R ν+ k ˆR ν+ k = {λ Λ ν : k, ω ν (λ) γ ν k τ }, k Rd \ {0}, = {λ Λ 0 : k, ω ν (λ) γ ν k τ }, k Rd \ {0}, 5
for all ν = 0,,. Then by Lemma 4. 4), and, Λ ν+ = Λ ν \ Λ 0 \ Λ = Rk ν+, K ν< k K ν+ Rk ν+. ν=0 K ν< k K ν+ Since (4.5) is also satisfied by the extended toral frequencies ω ν on Λ 0, A) implies that if µ is sufficiently small, then rank{ α ω ν λ α : α d } = d for all λ Λ 0, ν = 0,,. It follows from Lemma 4. that R ν+ k ( ) ν+ γ d ˆR k c, k τ+ for all k Z d \ {0} and ν = 0,,, where c is a constant independent of ν. Hence Λ 0 \ Λ Rk ν+ cγ d ν=0 K ν< k K ν+ ν=0 = O(γ d ) = O(γ d ), as desired. Case : d 0 < d. Let Λ = [, ] d d 0 Then it is clear that and define Λ = Λ 0 Λ, Λ = Λ Λ, λ = (λ, λ), λ Λ, ω ν ( λ) = ω ν (λ), rank{ α ω ν λ α ν = 0,,, λ Λ. : α d } = d K ν< k K ν+ k τ+ d on Λ for all ν = 0,,, as µ sufficiently small. Similar to Case, we have that By Fubini s theorem, Λ \ Λ = O(γ d ). Λ 0 \ Λ = O(γ d ) = O(γ d ), as desired. Case 3: d 0 > d. For any λ Λ 0, A) implies that there exist indexes α i {α Z d 0 + : α d }, i = 0,,, d, 6
such that rank{ αi ω λ αi (λ) : i = 0,,, d } = d. Since rank{ ω λ (λ)} d, there are λ i, λ i,, λ id0 d Then Define such that ω (λ) / { αi ω (λ) : i = 0,,, d }, j =,,, d 0 d. λ ij λ αi Ω(λ) = (λ i, λ i,, λ id0 d ), λ Λ 0, ω ν (λ) = (ω ν (λ), Ω(λ)), ν = 0,,, λ Λ 0, R ν+ k = {λ Λ ν : k, ω ν (λ) γ ν k τ }, k Zd 0 \ {0}, ν = 0,,, Λ ν+ = Λ ν \ k, ν = 0,,, Λ = ν 0 Λ ν. K ν< k K ν+ Rν+ rank{ αi ω ν (λ) : i = 0,,, d ; ω ν (λ) : j =,, d 0 d} = d 0 λ αi λ ij on Λ 0 for all ν = 0,,. It follows that rank{ α ω ν λ α : α d 0 } = d 0 on Λ 0 for all ν = 0,,. Similar to Case ), we have that Since Λ Λ, Λ \ Λ = O(γ d 0 ) = O(γ d ). Λ 0 \ Λ Λ 0 \ Λ = O(γ d ), as desired. This proves part ) of Theorem A. Given the convergence in Section 4., part ) of Theorem A clearly follows from Lemma 4. ). 4.4 Proof of Corollary Without loss of generality, we assume that M admits a global coordinate, i.e., there is a bounded closed region Λ R d 0 and a C l 0 diffeomorphism y : Λ M such that M = y(λ). Let λ Λ and consider the transformation Then (.) gives rise to y y + y(λ). H(x, y, λ, ε) = e(λ) + ω(λ), y + h(y, λ) + P (x, y, λ, ε), 7
where e(λ) = N(y(λ)), ω(λ) = N y (y(λ)), h(y, λ) = y, A(λ)y + ĥ(y, λ), A(λ) = N y (y(λ)), ĥ(y, λ) = O( y 3 ), P (x, y, λ, ε) = εp (x, y + y(λ), ε). Let r be fixed and take s = ε m+l 0 +5, γ = ε (m+l 0 +5), µ = ε m+l 0 +5. Then (.) holds and the Corollary follows immediately from the theorem as ε sufficiently small. 4.5 Proof of Theorem B By choosing λ, Λ as in the Section 4.4 above with the present M, the proof of Theorem B essentially follows from that of Theorem A, except that the translation φ : x x, y y + y in Section 3.3 should be defined for the purpose of eliminating the energy drift at each KAM step. In the case of part ) of Theorem B, y is defined so that e + = e = E. Hence, instead of (3.7), we consider the equation ω, y + y, Ay + ĥ(y ) + [R](y ) = 0, which, by the implicit function theorem, clearly admits a local smooth solution y on M. In the case of part ) of Theorem B, y is defined so that e + = e = E, and, [ω +,i : : ω +,in ] = [ω i : : ω in ]. Hence, instead of (3.7), we consider the equations ( ) ĥ A + (y i,, y in ) (y ) (yi,, yi n ) t (ω i : : ω in ) = (p 0,i,, p 0,in ), (ω i,, ω in ), (y i,, y i n ) + y, Ay + ĥ(y ) + [R](y ) = 0, which, by the sub-isoenergetic nondegenerate condition A) and the implicit function theorem, admits a local smooth solution (y, t ), y M, t R, such that yj = 0 if j / {i,, i n }. 8
Let φ F be as in Section 3 and φ be as in the above. Then under the symplectic transformation Φ + = φ F φ, the new Hamiltonian reads H Φ + = N + + P +, N + = N + φ ψ = E + ω +, y + h + (y), P + = P + φ + ψ, where, with respect to y defined above, ω + = ω + p 0 + Ay + y ĥ(y ), and, h + (y), A +, ĥ+(y), ψ have the same forms as in (3.)-(3.4). Thus, with estimates on the present y similar to those in Sections 3.4, 3.5, the remaining proof of Theorem A is valid. Part 3) of Theorem B is a special case of part 3) of the Corollary. References [] V.I. Arnold, Proof of A. N. Kolmogorov s theorem on the preservation of quasi periodic motions under small perturbations of the Hamiltonian, Usp. Math. USSR 8 (963), 3-40 [] V.I. Arnold, Mathematical methods of classical mechanics, Springer-Verlag, New York, 980 [3] V.I. Arnold and A. Avez, Probl `mes ergodiques de la mécanique classique, Gauthier- Villars, Paris, 967 [4] G. Benettin, L. Galgani, A Giorgilli and J. Strelcyn, A proof of Kolmogorov s theorem on invariant tori using canonical transformations defined by the Lie method, Nuovo. Cimento B. 79 (984), 0-3 [5] J. Bricmont, K. Gawedzki and A. Kupiainen, KAM theorem and quantum field theory, Comm. Math. Phys. 0 (999), 699-77 [6] H. Broer and G. Huitema, A proof of the isoenergetic KAM-theorem from the ordinary one, J. Diff. Eqs. 90 (99), 5-60 [7] H. Broer, G. Huitema and M. Sevryuk, Quasi-periodic motions in families of dynamical systems, Lecture Notes in Math., 645, Springer-Verlag, 996 [8] A. D. Bruno, Analytic form of differential equations, Trans. Moscow Math. Soc. 5 (97), 3-88 [9] C-Q. Cheng and Y-S. Sun, Existence of KAM tori in degenerate Hamiltonian systems, J. Diff. Eqs. 4 (994), 88-335 9
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