Eeective Stability and Kam Theory

The two main stability results for nearly-integrable Hamiltonian systems are revisited: Nekhoroshev theorem, concerning exponential lower bounds for the stability time (effective stability), and KAM theorem, concerning the preservation of a majority of the nonresonant invariant tori (perpetual stability). To stress the relationship between both theorems, a common approach is given to their proof, consisting of bringing the system to a normal form constructed through the Lie series method. The estimates obtained for the size of the remainder rely on bounds of the associated vector elds, allowing to get the \optimal" stability exponent in Nekhoroshev theorem for quasiconvex systems. On the other hand, a direct and complete proof of the isoenergetic KAM theorem is obtained. Moreover, a modi cation of the proof leads to the notion of nearly-invariant torus, which constitutes a bridge between KAM and Nekhoroshev theorems. 1 INTRODUCTION 1 1 Introduction We consider a nearly-integrable Hamiltonian written in action{angle variables: H( ; I) = h(I) + f( ; I); (1) where = ( 1; : : : ; n) 2 Tn and I = (I1; : : : ; In) 2 G Rn are, respectively, the angular and action variables, and f is a small perturbation, of size ", of the integrable Hamiltonian h. It is well-known that the dynamics associated to the unperturbed Hamiltonian h is very simple: the action I(t) remains constant for all motions. Then, all n-dimensional tori I = const. in phase space Tn G are invariant. The ow on each torus is linear, with frequency vector !(I) = gradh(I). In general, for the perturbed system associated to (1), the dynamics can be very complicated. It is thought that there are unstable motions, and that Arnold di usion takes place. Concerning stability, the main results are provided by Nekhoroshev and KAM (Kolmogorov{Arnold{Moser) theorems. Nekhoroshev theorem, which was rst proved in [24], leads to the concept of e ective stability. Roughly speaking, it states that an estimate of the type jI(t) I(0)j < R0 "b for jtj T0 exp "0 " a ; holds for all initial conditions ( (0); I(0)) 2 Tn G, provided steepness conditions are ful lled by h. The stability exponents a and b are positive constants. For the case of a perturbation of a quasiconvex Hamiltonian (the simplest kind of steepness), these exponents have been successively improved along several papers. Thus, the exponent a = 2=(n2 + n) was found in [2]; and a < 1=(2n+ 1) in [15]. Finally, the exponents a = b = 1 2n ; are stated in [17], [27]. It has been conjectured [6] that the exponent a = 1=2n is optimal. Estimates of an analogous type can be obtained for the case of a perturbation of a system of harmonic oscillators: H( ; I) = ! I + f( ; I), where ! is now a constant vector satisfying a Diophantine condition: jk !j jkj 1 8k 2 Zn n f0g; (2) for some n 1 and > 0 (it is well-known that, if > n 1, the set of vectors ! satisfying this condition for a given > 0 has relative measure 1 O( ) in Rn). We use the notation jkj1 = Pnj=1 jkjj for k = (k1; : : : ; kn). We say a vector ! satisfying (2) to be ; -Diophantine. In this case, the optimal stability exponent seems to be a = 1=( +1). This exponent has been obtained in [11], [10] and [7]. KAM theorem states, under a suitable nondegeneracy condition, that most of the n-dimensional invariant tori are preserved with some deformation in the perturbed system (1) if the size " of the perturbation is small enough. More precisely, this preservation is guaranteed for tori that have frequency vector !(I) = gradh(I) satisfying a Diophantine condition. In this way, one gets perpetual stability, but only for initial conditions on 1 INTRODUCTION 2 a Cantorian set, which does not contain any open set although its measure is large. In fact, it was rst stated by Kolmogorov [13], for analytic Hamiltonians, the preservation of one given torus, suitably chosen. Afterwards, Arnold proved in [1] (see also [26]) the existence of a large family of invariant tori and estimated the measure of the complement of the invariant set. An analogous theorem for area-preserving maps of the plane was proved by Moser [20], without the hypothesis of analiticity. Concerning the nondegeneracy condition required for the validity of KAM theorem, two sorts of conditions are usually imposed on the unperturbed frequency map ! = grad h, namely the (standard) nondegeneracy and the isoenergetic nondegeneracy (see de nitions (31{32) in section 4.1). There are slight di erences between the statements of KAM theorem under both nondegeneracies. Indeed, in the standard case every preserved invariant torus keeps its frequency vector in the perturbation. In the isoenergetic case, the frequency vector is not usually kept but, nevertheless, every invariant torus keeps its frequency ratios and its energy and, moreover, on each xed energy hypersurface most of the invariant tori are preserved. A well-known consequence is that, for two degrees of freedom (n = 2), it follows from the isoenergetic nondegeneracy the stability of the perturbed system. The usual proofs of Nekhoroshev and KAM theorems do not allow to stress the close relationship existing between the di erent types of stability provided by these theorems. Actually, no use is made of the existence of the KAM tori in the proof of Nekhoroshev theorem, which gives a uniform stability time for all trajectories in phase space. These trajectories include the ones lying in KAM tori, which are the most numerous, and clearly have an in nite stability time. But one can also expect that, for a trajectory starting near a KAM torus, the stability time is much larger than the one predicted by Nekhoroshev theorem. Results concerning this \stickiness" of KAM tori, with Nekhoroshev-like estimates, have recently been obtained in [25], [19]. In this paper we are concerned about a uni ed approach to Nekhoroshev and KAM theorems, already announced in [8]. After a preliminary part where the common method is set up, we give quantitative proofs of Nekhoroshev theorem under the assumption of quasiconvexity (theorem D in section 3.5) and the isoenergetic KAM theorem (theorem E in section 4.4). We notice that our approach to the isoenergetic theorem is direct, unlike the usual proofs where it is deduced from the standard KAM theorem (see, for example, [4]) or making use of the associated Poincar e map (see [21]). Moreover, under the same hypothesis of KAM theorem, we get a Nekhoroshev-like stability result (theorem F in section 4.5) which is slighlty di erent from the ones of [25] and [19]. The result we prove considers the invariant tori of the unperturbed system such that their associated frequency vector satis es appoximately, up to a given precision r, a Diophantine condition. In the perturbed system, these tori survive in the form of nearly-invariant tori, i.e., the trajectories starting on such a torus remain near to it up to a stability time which is exponentially long in 1=r. This result is similar to the one of [18] where, however, the estimates are expressed in terms of the stability time, which is previously xed. On the other hand, in [25] and [19] the stability estimates are expressed in terms of the distance to a given KAM torus, meaning in this way the \stickiness" of KAM tori. The stability time is then exponentially long in the inverse of this distance (or even \superexponentially" long for quasiconvex Hamiltonians [19]). We do not prove that 1 INTRODUCTION 3 KAM tori are \sticky" but we believe that our result is more useful for practical purposes, since our estimates for nearly-invariant tori do not require the existence of a KAM torus nearby. The method we follow for the proof of Nekhoroshev and KAM theorems is standard in classical perturbation theory. It consists of seeking for a suitable canonical transformation , bringing our Hamiltonian H into a normal form H = H , asked to depend on less angles, none if possible. The transformation is constructed iteratively as a product of successive canonical transformations (1); (2); : : :, near to the identity, which provide a sequence of Hamiltonians H(1); H(2); : : : coming nearer and nearer to the normal form. We construct the successive canonical transformations with the help of the well-known Lie series formalism, which we describe in section 2.1. This is a very suitable procedure for practical applications, since it allows to carry out explicit computations in concrete examples. The procedure can be directly implemented in computers, since we only use harmonics of nite order. It is well-known that an obstruction for the construction of the normal form is found on the resonances or near them. A resonant manifold is characterized by a given module M Zn: SM := fI 2 G : k !(I) = 0 8k 2 Mg: The obstruction comes from the presence of the small divisors k !(I), with k 2 M, which can be zero or too small. It is because of the presence of the small divisors that one considers, near the resonance SM, a resonant normal form, which accepts dependence on combinations of angles k , with k 2 M. The union of all resonances is dense in the set of frequencies, but one only needs to consider resonances up to a given suitable nite order: jkj1 K, since it turns out that the e ect of higher-order resonances is exponentially small in K. Thus, we say a function g( ; I) to be in normal form with respect to M of degree K if its Fourier series expansion in the angular variables is restricted to the form g( ; I) = X k2M jkj1 K gk(I) ei k : We express this by writing g 2 R(M; K). Note that a fuction is in normal form with respect to the trivial module M = 0 if it does not depend on the angular variables. In the rst part of this paper, we restrict ourselves to a subset G G, where the frequency vector !(I) is allowed to satisfy resonance relations corresponding to a xed module M, but a neighborhood of all other resonances of order less or equal than K are excluded. For such a set we say that !(G) is nonresonant modulo M up to order K (see de nition (16) in section 2.3). On this set G we make successive reductions to the type of normal form de ned above: the harmonics corresponding to integer vectors satisfying k = 2 M, jkj1 K, become smaller and smaller in the successive Hamiltonians, whereas the harmonics corresponding to k 2 M or jkj1 > K have to be kept because in the set G the small divisors k !(I) associated to these harmonics are not avoided. In this way the nal Hamiltonian H = H can be written as H ( ; I) = h (I) + Z ( ; I) +R ( ; I) ; where Z 2 R(M; K) and the remainder R is exponentially small in K. 1 INTRODUCTION 4 As in [24] and [2], the proof of Nekhoroshev theorem is divided two parts, usually named analytic and geometric ones. The analytic part concerns the iterative process and the estimates for the successive remainders. In the geometric part the whole action space G is covered by a family of sets associated to every module in order to get stability estimates for the trajectories corresponding to all initial conditions. A similar distinction may also be carried out in the proof of KAM theorem. In this case the geometric part concerns the estimates for the measure of the complement of the invariant set. For given M and K, we consider a subset G G where the nonresonance condition quoted above is satis ed. In section 2.1, we show how the Lie series method is applied to the construction of the iterative process, which is nite for the proof of Nekhoroshev theorem and in nite for KAM theorem. In the case of KAM theorem we always take M = 0. The Iterative Lemma (theorem A in section 2.3), which is common to both proofs, provides estimates for one given step of this process. We make use of a vector eld norm, introduced in section 2.2, which allows us to optimize the estimates of the Iterative Lemma with respect to other related papers. From successive application of the Iterative Lemma, with a xed K, and carrying out an appropiate number of iterative steps, we get the Normal Form Theorem (theorem B in section 3.1), where the estimate obtained for the remainder is exponentially small in K, and hence H is an approximate normal form, speci c for the set G. This theorem completes the analytic part of the proof of Nekhoroshev theorem. We point out that this approach is carried out along the lines of P oschel's proof [27] of Nekhoroshev theorem, but our proof is somewhat simpler because the Iterative Lemma has been optimized. From the Normal Form Theorem, one can deduce stability estimates for the trajectories starting in Tn G, which hold up to an exponentially long time. The estimates for nonresonant (M = 0) and resonant (M 6= 0) regions are given in sections 3.2 and 3.3, respectively. In the resonant case we impose a quasiconvexity condition. Following [27], in the geometric part of the proof, the whole domain G is covered by a family of sets G = GM associated to the di erent modules M Zn, with suitably chosen parameters (section 3.4). One sees that it su ces to consider K-modules (a module M Zn is said to be a K-module if it is generated by vectors of order less or equal than K). To complete the proof of Nekhoroshev theorem (section 3.5), with optimal exponents, we choose K as a suitable function of " and apply the stability estimates to each set GM. In this way, we obtain estimates for all trajectories starting in Tn G. As an additional application of the Normal Form Theorem, we also consider a perturbation of a system of n harmonic oscillators with a Diophantine frequency vector. The nonresonant estimates of the case M = 0 give rise to e ective stability in such a system (theorem C in section 3.2). Our approach to KAM theorem is parallel, in its main lines, to the Arnold's one [1]. We rst prove the Inductive Lemma (proposition 11 in section 4.3), which concerns the estimates given by the Iterative Lemma, with M = 0, for one given step of the iterative process. In this case, it does not su ce to bring our Hamiltonian H to an approximate normal form with an exponentially small remainder. It is necessary to perform an in nite number of iterations, with orders K1; K2; : : : increasing to in nity. Then, the resonances up to higher and higher orders are removed from the domain along the successive iterative steps. In this way, the remainders tend quickly to zero and the nal Hamiltonian becomes integrable: H ( ; I) = h (I). Therefore, the domain where the transformation holds is 2 THE COMMON PART 5 lled with n-dimensional invariant tori with linear ow, but it shrinks to a Cantorian set corresponding to Diophantine frequencies. To nish the proof of KAM theorem, the measure of the invariant set can be estimated assuming that a suitable nondegeneracy condition is ful lled by the unperturbed system. In section 4.4, we give this direct proof of KAM theorem under the hypothesis that the unperturbed frequency map ! is isoenergetically nondegenerate. We point out that the same scheme would be useful for the standard nondegeneracy. An explanation of both nondegeneracy conditions and the technical di culties arising in the isoenergetic case is given in section 4.1 (quantitative lemmas are provided in section 4.2). This common approach to both nondegeneracy conditions can be seen as a rst step towards the proof of KAM theorem under higher-order nondegeneracy conditions. We recall that a very general condition has been announced by R ussmann [28]. See [29], [5], [30] for very recent results along this line. Finally, we see in section 4.5 that, inside the same iterative scheme used for KAM theorem but stopping it at an appropiate step, instead of carrying it to the limit, we nd that Nekhoroshev-like estimates hold for the trajectories starting in the domain at this step. This domain is then lled with nearly-invariant tori (theorem F). This result is quantitatively very close to KAM theorem. Qualitatively, the perpetual stability of KAM tori is sacri ced but, on the other hand, the domain where the result holds contains inner points, and hence it is not a Cantorian set. It is worth reminding that KAM theorem is meaningless from a practical point of view despite its theoretical importance. This is due to the fact that, from an approximation of a concrete frequency vector, one cannot decide whether this vector is Diophantine or not. The notion of nearly-invariant torus may be understood as an attempt to compensate this de ciency. Acknowledgements The authors would like to thank G. Benettin, A. I. Neishtadt, J. P oschel, C. Sim o and D. V. Treschev for their very useful comments, suggestions and general interest. 2 The common part 2.1 Normal forms via the Lie series method We describe in this section the iterative process leading our Hamiltonian H( ; I) = h(I) + f( ; I) to normal form. This setup provides a common environment for the proofs of Nekhoroshev and KAM theorems. According to the approach described in the introduction, we restrict our Hamiltonian H to a subset G G, where it is assumed that the frequency set !(G) is nonresonant moduloM up to order K for givenM and K. For notational convenience, we consider the starting Hamiltonian H written, on the set G, in the form H( ; I) = h(I) + Z( ; I) +R( ; I); (3) with Z 2 R(M; K). For instance, we may choose Z = 0 and R = f . However, if the starting Hamiltonian is already near to the normal form, we may write it in the form (3), with a small R with respect to Z, and seek for a better approximation to the normal form. 2 THE COMMON PART 6 The transformation leading to normal form is constructed as a product of canonical transformations (1); (2); : : : We put (q) = (1) (q). At the step q, the transformed Hamiltonian is written in the form H(q) = H (q) = h+ Z(q) +R(q); with Z(q) 2 R(M; K). Obviously we start with Z(0) = Z and R(0) = R. Now, to describe a concrete iterative step, we write H, Z, R, ~ Z, ~ R, , instead of H(q 1), Z(q 1), R(q 1), Z(q), R(q), (q), respectively. Following the Lie series method, as in [7], we construct as the ow at time 1 associated to a generating Hamiltonian W to be determined. More precisely, if t denotes the ow at time t of an autonomous Hamiltonian W , it is known from the Hamiltonian theory that, for any function f , the derivative of f t with respect to t can be expressed in terms of the Poisson bracket of f and W : d dt(f t) = ff;Wg t: So, assuming analyticity in t and taking the Taylor expansion, one has the Lie series f t = 1 X m=0 tm m! L m W f; where we denote L 0 Wf = f and L m W f = nL m 1 W f;Wo for m 1. For the remainders of the Lie series, we use the notation rm(f;W; t) := f t m 1 Xl=0 tl l! L l Wf = 1 X l=m tl l! L l Wf (4) for m 0. With this notation, we have for the transformed Hamiltonian the following expression: H = h+ Z +R + fh;Wg+ r2(h;W; 1) + r1(Z +R;W; 1): (5) We want ~ R to be smaller than R in order to get H closer to normal form than H. Consequently W should be chosen in such a way that R + fh;Wg be in normal form. As a matter of fact, this can only be guaranteed up to order K because the nonresonance condition on !(G) does not avoid the small divisors corresponding to higher orders. Thus, we seek for Z 2 R(M; K) and W solving the linear functional equation fW;hg+ Z = R K ; (6) where we write R K( ; I) = X jkj1 KRk(I) ei k . The resolution of equation (6) is standard. In terms of Fourier coe cients, we have the solution Wk(I) = Rk(I) i k !(I) ; Zk(I) = 0; for k 2 Zn nM, jkj1 K; Wk(I) = 0; Zk(I) = Rk(I); for k 2 M, jkj1 K; Wk(I) = 0; Zk(I) = 0; for jkj1 > K: (7) 2 THE COMMON PART 7 This is the only solution of equation (6) if we require W to have no resonant terms with respect toM and to be of degree K. We denote by NR(M; K) the set of the functions satisfying these requirements. If h and R are real functions, we see from (7) that Z and W are also real. The new Hamiltonian can be put as H = h+ ~ Z + ~ R; with ~ Z = Z + Z 2 R(M; K); (8) ~ R = R>K + r2(h;W; 1) + r1(Z +R;W; 1); (9) where we write R>K = R R K. If h, Z and R are real, then the transformation preserves real domains, and the new Hamiltonian is also real. Recall that the algorithm explicited in (7{9) is just one step of the iterative process. It describes how to get H(q) = H(q 1) (q). Roughly speaking, this procedure can be considered as linear if we ignore the term R(q 1) >K . Indeed, if Z(q 1) = O("), R(q 1) = O("q), we see from equations (8{9) that R(q) = O("q+1), since the generating Hamiltonian for (q) is taken of the same order as R(q 1). We use this procedure for the proof of Nekhoroshev theorem (section 3.5). The term R(q 1) >K is exponentially small in K. So its in uence can be overcome by choosing K large enough. We remark that the canonical transformation (q) could also be constructed by means of a quadratic procedure: if R(q 1) = O "2q 1 , then R(q) = O "2q . We can attain this aim by taking another term of the Lie series in (5), which gives rise to an alternative algorithm for the reduction to normal form. However, for an arbitrary module M, the linear equation substituting (6) is not easily resoluble, and an approximate solution does not seem to improve the estimates of Nekhoroshev theorem. In the case of KAM theorem (section 4), whereM = 0, the linear procedure described in (7{9) is almost quadratic, provided we take Z(q 1) = 0 at each step (the procedure can never be strictly quadratic because of the presence of small divisors). This forces us to a little change in the algorithm: we include Z(q 1) in the integrable part in order to have Z(q) = 0 for the following step. In this way, the integrable part changes at every step: we begin the step q with H(q 1)( ; I) = h(q 1)(I) + R(q 1)( ; I), and the new Hamiltonian can be written as H(q)( ; I) = h(q)(I) +R(q)( ; I); (10) where, by (7), h(q) = h(q 1) + Z(q 1) = h(q 1) +R(q 1) 0 (11) (note that the function R(q 1) 0 (I) is the angular average of R(q 1)( ; I)). The new remainder R(q) is obtained like in (9), which then gives a fastly convergent procedure if the term R(q 1) >K is ignored. To solve the di culty caused by this term, we take increasing orders Kq, tending to in nity for q !1. We can then see the convergence to zero of the remainders R(q), and hence the existence of invariant tori. However, resonances of successive higher orders have to be removed along the procedure and hence the nal domain is reduced to a Cantorian set. 2 THE COMMON PART 8 Another remark is that in section 4.4 we prove KAM theorem without showing explicitly that the remainders R(q) converge in a fast way, but linear. Nevertheless, we use the almost quadratic convergence of the remainders to show the existence of nearly-invariant tori, with exponential estimates (see section 4.5). 2.2 A norm for Hamiltonian vector elds In order to obtain rigorous estimates for the successive remainders, we need to de ne norms for the functions taking part in the iterative process introduced in section 2.1. An important remark is that a Hamiltonian function H does not take part directly in the Hamiltonian equations, but rather its derivative DH = @H @ ; @H @I ! = @H @ 1 ; : : : ; @H @ n ; @H @I1 ; : : : ; @H @In! : Then, to obtain the stability estimates leading to the proof of Nekhoroshev and KAM theorems, we do not need to obtain estimates for the successive remainders provided by (9), but estimates for the derivatives of these remainders su ce. Looking carefully at equations (8{9), one realizes that it is possible to bound the derivatives D ~ Z and D ~ R from the derivatives DZ and DR, since the Lie remainders r1, r2 have been de ned in (4) in terms of Poisson brackets. So it would be a nice tool to work with a suitable vector eld norm for the derivatives, which would avoid unnecessary uses of the Cauchy inequalities in estimating derivatives. This idea was suggested to us by A. I. Neishtadt, although it goes back to [10], where estimates for the Lie series method for not necessarily Hamiltonian vector elds are fully developed. However, we cannot avoid all uses of the Cauchy inequalities, since the remainders r1, r2 in (9) have to be di erentiated in order to estimate D ~ R. Moreover, a di erentiation has to be done before starting the rst iterative step. Thus, we work with analytic functions on complex neighborhoods of the domain Tn G. Given = ( 1; 2) 0 (i.e. j 0, j = 1; 2), we rst introduce the sets: W 1(Tn) := f : Re 2 Tn; jIm j1 1g; V 2(G) := fI 2 Cn : jI I 0j 2 for some I 0 2 Gg; where j j1 and j j = j j2 denote, respectively, the maximum norm and the Euclidean norm for vectors. We then de ne: D (G) :=W 1(Tn) V 2(G): Several kinds of norms are used along this paper. First, we consider functions of the n action variables. Given a (real or complex) function f(I), de ned on a complex neighborhood V (G), 0, we introduce the supremum norm: jf jG; := sup I2V (G) jf(I)j ; jf jG := jf jG;0 : In this way, the subscript is removed from the notation if = 0. This remark applies throughout this section. 2 THE COMMON PART 9 In an analogous way, we consider the supremum norm for vector-valued functions, i.e. vector elds. Given F : V (G) ! Cn and 1 p 1, we de ne jF jG; ;p := sup I2V (G) jF (I)jp ; jF jG; := jF jG; ;2 : In this de nition, j jp means the p-norm for vectors in Cn, i.e.: jvjp = Pnj=1 jvjjp 1=p for 1 p <1, and jvj1 = max1 j n jvjj. Note that we remove the subscript p to mean the Euclidean norm (p = 2). We also de ne the supremum norm for matrix-valued fuctions or even tensor-valued functions (e.g. successive total derivatives of a function). The de nition is analogous, taking for matrices and tensors the norm induced by the Euclidean norm for vectors (we only consider the case p = 2). Next we consider functions of the action{angle variables. For a given complex function f( ; I) (2 -periodic in ) de ned on the neighborhood D (G), = ( 1; 2) 0, we may consider its supremum norm: jf jG; := sup ( ;I)2D (G) jf( ; I)j : (12) But if f is analytic on (a neighborhood of) the set D (G), we may de ne an exponentially weighted norm in terms of the Fourier series of f . Writing f( ; I) = X k2Zn fk(I) ei k , we introduce kfkG; := X k2Zn jfkjG; 2 ejkj1 1 : (13) Note that jf jG; kfkG; : This exponentially weighted norm, analogous to the one used in [27], allows to carry out easily a separate control of harmonics in estimating the solution of the linear functional equation (6), in proposition 4 of section 2.3. This would be more di cult by using the supremum norm. Exactly in the same way as before we may extend the de nitions of the norms (12{13) to the case of vector-valued functions. Given F : D (G) ! Cn and 1 p 1, and writing F ( ; I) = X k2Zn Fk(I) ei k , where Fk : V 2(G) ! Cn, we de ne kFkG; ;p := X k2Zn jFkjG; 2;p ejkj1 1; kFkG; := kFkG; ;2 : Let us recall the Cauchy inequalities for the -derivatives and the I-derivatives (see also [27]). Given f analytic on D (G), for 0 < < (i.e. 0 < j < j, j = 1; 2) one has @f @ G;( 1 1; 2);1 1 e 1 kfkG; ; @f @I G;( 1; 2 2);1 1 2 kfkG; : To have a more compact writing and to avoid to carry out separate estimates for the -derivatives and the I-derivatives along the iterative process, we introduce for Df = @f @ ; @f @I the vector eld norm kDfkG; ;c := max0@ @f @ G; ;1 ; c @f @I G; ;11A ; (14) 2 THE COMMON PART 10 where c > 0 is a parameter to be xed in subsequent sections. This parameter (having the physical dimension of the action variables) is introduced in order to compensate the di erence between the Cauchy inequalities for -derivatives and I-derivatives. Lemma 1 Let f , g be analytic functions on D (G). For 0 < = ( 1; 2) < and c > 0 given, let us denote ̂c := min(c 1; 2): Then, a) kDfkG; ;c ĉ c kfkG; : b) kff; ggkG; 2c kDfkG; ;c kDgkG; ;c : c) kD (f>K)kG;( 1 1; 2);c e K 1 kDfkG; ;c : The proof of the properties contained in this lemma is very simple. In subsequent sections, we shall see that an appropiate choice for the parameter c makes possible to obtain better estimates, even in case of very di erent 1, 2. More notation is required. At every step of the iterative process described in section 2.1, the canonical transformation leading our Hamiltonian to normal form is constructed as a ow associated to the generating Hamiltonian W de ned in (7). To know how near to the identity map this canonical transformation is, we need to de ne a norm for maps like id. This map is de ned in D (G), and we may consider it taking values in C2n. First, we take the parameter c > 0 of de nition (14) and, for a 2n-vector x = ( ; I), we introduce its \c-norm" as jxjc := max (c j j1 ; jIj) : Then, for a map : D (G) ! C2n, we de ne the norms: j jG; ;c := sup x2D (G) j (x)jc ; jD jG; ;c := sup x2D (G) jD (x)jc ; where, for the second de nition, the matrix c-norm is the one induced by the c-norm for vectors. With these notations, it is easy to prove the following property: if is analytic on D (G), then jD jG; ;c j jG; ;c ̂c : (15) In the following lemma, the e ect of the ow associated to a generating Hamiltonian is estimated in terms of the norms introduced above. Moreover, a bound for the remainder of a Lie series is found. The proof is given in section 5. Lemma 2 Let W be an analytic function on D (G), > 0, and t its associated Hamiltonian ow at time t, t 0. Let = ( 1; 2) > 0 and c > 0 given. Assume that kDWkG; ;c ̂c. Then, t maps D t (G) into D (G) and one has: a) j t idjG; t ;c t kDWkG; ;c : 2 THE COMMON PART 11 b) t(D 0(G)) D 0 t (G) for 0 t . c) Assuming that kDWkG; ;c < ̂c=2e, for any given function f , analytic on D (G), and for any integer m 0, the following bound holds: krm(f;W; t)kG; t 24 1 Xl=0 1 l+m m 2e kDWkG; ;c ̂c !l35 tm m! kLm WfkG; = m 2e kDWkG; ;c ̂c ! tm kLm WfkG; ; where, for 0 x < 1, we de ne m(x) := 1 Xl=0 l! (l +m)! xl: 2.3 The Iterative Lemma Now we are going to obtain estimates for one step of the procedure of section 2.1, with the help of the norm introduced in section 2.2. We consider the Hamiltonian (3), real analytic on D (G), with Z 2 R(M; K), and we restrict it to a subset G G such that !(G) is nonresonant moduloM up to order K (see the introduction). We rst introduce, following [27], a quantitative version of this nonresonance condition. Given a module M, an integer K and > 0, a subset F of the n-dimensional frequency space is said to be ;K-nonresonant modulo M if jk vj 8k 2 Zn nM; jkj1 K; 8v 2 F: (16) We begin by seeing that this nonresonance condition on the set !(G) can be extended to a complex neighborhood of small enough radius 2. Lemma 3 Let h(I) be a real analytic function on V 2(G), let ! = gradh, and assume that !(G) is ;K-nonresonant modulo M. Assume that @2h @I2 G; 2 M . If 2 2MK ; (17) then ! (V 2(G)) is 2 ; K-nonresonant modulo M. The proof is a simple application of the mean value theorem. We point out that, as shown in section 3.5, condition (17) on 2 imposes an important restriction on the domain. An exception is the very special case of a system of harmonic oscillators, where M = 0 (see section 3.2). The next result provides estimates for the functions Z and W solving the linear functional equation (6). 2 THE COMMON PART 12 Proposition 4 Let h(I), Z( ; I), R( ; I) be real analytic functions on D (G), let ! = grad h, and assume that !(G) is ;K-nonresonant modulo M, and that Z 2 R(M; K). Assume that @2h @I2 G; 2 M , and 2 2MK : Let c > 0 given. Then the functions Z 2 R(M; K) and W 2 NR(M; K) given in (7), which solve the linear equation (6), are both real analytic on D (G), and the following bounds hold: kD( Z)kG; ;c kDRkG; ;c ; kD(R Z)kG; ;c kDRkG; ;c ; kDWkG; ;c 2A kDRkG; ;c ; where we de ne A := 1 + 2Mc : (18) Proof We obtain the estimates from the explicit solution given in (7), in terms of Fourier coe cients. The two rst ones are clear, since Z and R Z are obtained from R just removing the appropiate Fourier harmonics. To estimate DW , we bound @W @ and @W @I . Using lemma 3, it is easy to see that @W @ G; ;1 2 @R @ G; ;1 : Next we write, for k 2 Zn nM; jkj1 K, @Wk @I = @Rk @I i k !(I) Rk(I) @ @I (i k !(I)) (i k !(I))2 = @Rk @I i k !(I) + h@R @ ik @! @I (k !(I))2 ; where we have used that h@R @ ik = i Rk(I) k (di erentiating the Fourier expansion of R). From lemma 3, we obtain @Wk @I G; 2;1 2 @Rk @I G; 2;1 + 4M 2 "@R @ #k G; 2 : Thus, @W @I G; ;1 2 @R @I G; ;1 + 4M 2 @R @ G; and nally kDWkG; ;c 2 + 4Mc 2 kDRkG; ;c = 2A kDRkG; ;c : 2 Remarks 1. These estimates do not involve a reduction of the domainD (G). This becomes more di cult if we use a norm that does not take into account the explicit expansion in Fourier series (for example, the supremum norm). One exception is the case dimM = n 1, i.e. near periodic orbits, where integral expressions for the solution of equation (6) are available (see [15] and also [14]). 2 THE COMMON PART 13 2. The value of A could be big (of the order of 1= ). Therefore, it would be an obstruction to the obtainment of the optimal exponent, unless we chose c small. But we shall see in the subsequent sections that our choice of c allows to bound A by a constant not depending on . Theorem A (Iterative Lemma) Let H( ; I) = h(I) +Z( ; I) +R( ; I) real analytic on D (G), let ! = gradh, and assume that !(G) is ;K-nonresonant modulo M, and that Z 2 R(M; K). Assume that @2h @I2 G; 2 M . Let < and c > 0 given, and let A de ned as in (18). Assume: 2 2MK ; kDRkG; ;c ̂c 74A : (19) Then, there exists a real analytic canonical transformation : D 2 (G) ! D (G) such that H = h+ ~ Z + ~ R, with ~ Z 2 R(M; K), and one has: a) D ~ Z G; ;c kDZkG; ;c + kDRkG; ;c : b) D ~ R G; ;c e K 1 kDRkG; ;c + 14A ̂c kDZkG; ;c + kDRkG; ;c kDRkG; ;c : c) j idjG; 2 ;c 2A kDRkG; ;c : d) (D 0(G)) D 0 2 (G) for 0 2 : Proof We take Z, W and as constructed in section 2.1. Then the bounds of proposition 4 for D( Z), D(R Z) and DW hold. In particular, kDWkG; ;c 2A kDRkG; ;c ̂c 37 < ̂c 4e ; and therefore lemma 2 applies, with t = 1 and with =2 instead of . We obtain : D 2 (G) ! D (G) and expressions (8{9) hold for the transformed Hamiltonian. From (8) and proposition 4, we easily get estimate (a). On the other hand, from (9) and parts (a) and (c) of lemma 1, D ~ R G; ;c e K 1 kDRkG; ;c + 2ĉ c kr2(h;W; 1)kG; 2 + kr1(Z +R;W; 1)kG; 2 : From part (c) of lemma 2, kr2(h;W; 1)kG; 2 2 4e kDWkG; ;c ̂c ! kffh;Wg ;WgkG; ; kr1(Z +R;W; 1)kG; 2 1 4e kDWkG; ;c ̂c ! kfZ +R;WgkG; : 2 THE COMMON PART 14 We estimate the Poisson brackets using part (b) of lemma 1: kfZ +R;WgkG; 2c kDZkG; ;c + kDRkG; ;c kDWkG; ;c ; kffh;Wg ;WgkG; = kf Z R K ;WgkG; 2c kDRkG; ;c kDWkG; ;c where, in the second estimate, we have used proposition 4 to ensure that kD( Z R K)kG; ;c kD( Z R)kG; ;c kDRkG; ;c : For 0 < x < 1, one has 1(x) = ln(1 x) x ; 2(x) = x + (1 x) ln(1 x) x2 : Using that these functions are increasing and evaluating them at x = 4e=37, we obtain kr2(h;W; 1)kG; 2 + kr1(Z +R;W; 1)kG; 2 2c 2 4e 37 + 1 4e 37 kDZkG; ;c + kDRkG; ;c kDWkG; ;c 7A c kDZkG; ;c + kDRkG; ;c kDRkG; ;c : (20) By putting all of these estimates together, we get (b). Finally, we deduce from lemma 2 (with =2 instead of ) the statements (c) and (d), concerning the distance from to the identity. 2 Remarks 1. The Iterative Lemma provides a description for one step of the transformation to normal form constructed in section 2.1. The improvement of this result, with respect to related papers (for instance, [27]), is the main contribution of the vector eld norm (14). It avoids a subsequent application of the Cauchy inequalities, which would cause an extra division by ̂c in estimate (b). 2. In the statement of the Iterative Lemma the value of the parameter c is still free. From now onwards, we shall take c = 2 1 ; and hence ̂c = 2. This choice of c seems to be the best because it leads to the smallest possible value for the quotient kDRkG; ;c ̂c ; appearing implicitly in condition (19) and estimate (b). 3 NEKHOROSHEV ESTIMATES AND RELATED RESULTS 15 3 Nekhoroshev estimates and related results 3.1 Estimates for the normal form Now, starting with H( ; I) = h(I) + Z( ; I) + R( ; I) on D (G), we apply Q times the Iterative Lemma and obtain an estimate for the remainder. By choosing Q = Q(K) adequately, we get an exponentially small remainder in the next theorem. Theorem B (Normal Form Theorem) Let H( ; I) = h(I) + Z( ; I) + R( ; I) real analytic on D (G), let ! = gradh, and assume that !(G) is ;K-nonresonant modulo M, and that Z 2 R(M; K). Assume that @2h @I2 G; 2 M . Let < given, c = 2= 1, and let A the constant de ned in (18). Assume: 2 2MK ; kDZkG; ;c + kDRkG; ;c 2 61AK 1 : (21) Then, there exists a real analytic canonical transformation : D (G) ! D (G) such that H = h + Z +R , with Z 2 R(M; K), and one has: a) kDZ kG; ;c + kDR kG; ;c kDZkG; ;c + 2 kDRkG; ;c : b) kDR kG; ;c 3 e K 1 2 kDRkG; ;c : c) j idjG; ;c 4A kDRkG; ;c : d) (D 0(G)) D 0 2 (G) for 0 . Proof Let Q 1 be an integer to be chosen below, and let us introduce the sequence (q) = q Q ; 0 q Q: We take (q) = =Q for every 1 q Q. Next we shall construct a sequence of real analytic canonical transformations (q) : D (q)(G) ! D (q 1)(G), 1 q Q. Denoting (q) = (1) (q), the successive transformed Hamiltonians will be written in the form H(q) = H (q) = h+Z(q) +R(q), with Z(q) 2 R(M; K). Moreover, we are going to show that, if K 1 2Q; (22) the following statements are true for 0 q Q: 1q) DZ(q) G; (q);c kDZkG; ;c + q 1 X s=0 DR(s) G; (s);c : 2q) DR(q) G; (q);c 1 eq kDRkG; ;c : 3 NEKHOROSHEV ESTIMATES AND RELATED RESULTS 16 We proceed by induction. The results are obviously true for q = 0. For 1 q Q, note that, by (2q 1) and condition (21), DR(q 1) G; (q 1);c 1 eq 1 kDRkG; ;c 2 61AK 1 2 122AQ and hence the Iterative Lemma applies, with =Q instead of , and we obtain the canonical transformation (q). We immediately get (1q). The bound (2q) comes from the following estimate: DR(q) G; (q);c e K 1 Q DR(q 1) G; (q 1);c + 14AQ 2 DZ(q 1) G; (q 1);c + DR(q 1) G; (q 1);c DR(q 1) G; (q 1);c 1 e2 + 28AQ 2 kDZkG; ;c + kDRkG; ;c DR(q 1) G; (q 1);c 1 e2 + 28 122 DR(q 1) G; (q 1);c 1e DR(q 1) G; (q 1);c : Now, let us assume that K 1 2 (if K 1 < 2, all results are obvious if we take as the identity map). Then, we may choose Q = Q(K) as the maximum integer satisfying (22), i.e. Q = hK 1 2 i. Denoting = (Q), Z = Z(Q), R = R(Q), we have H = h+ Z +R . Then, part (a) comes from (1Q). For part (b), we use (2Q): kDR kG; ;c 1 eQ kDRkG; ;c 1 eK 1 2 1 kDRkG; ;c 3 e K 1 2 kDRkG; ;c : The proof of (c) is very simple from the analogous bound in the Iterative Lemma and the inequalities (2q): (Q) id G; ;c Q X q=1 (q) id G; (q);c Q X q=1 2A DR(q 1) G; (q 1);c 4A kDRkG; ;c : Finally, to get (d) it su ces to prove that, for 0 q Q, (q) (D 0(G)) D 0 q 2Q (G) if 0 q Q : Indeed, this inclusion is obvious for q = 0. By induction, we assume it for q 1: (q 1) (D 00(G)) D 00 (q 1) 2Q (G) if 00 (q 1) Q : Then, taking 00 = 0 2Q in this inclusion and applying part (d) of the Iterative Lemma, we get: (q) (D 0(G)) = (q 1) (q) (D 0(G)) (q 1) D 0 2Q (G) D 0 q 2Q (G): 2 3 NEKHOROSHEV ESTIMATES AND RELATED RESULTS 17 Remark This result is essentially equivalent to the analogous one in [27], and seems to be \optimal" in the sense that the exponent for K in the second condition of (21) is 1. The di erence is that our proof is much simpler because the Iterative Lemma is also optimal, whereas the proof appearing in [27] relies in a very careful choice of the size of the successive reductions of the domain (see also [23]). 3.2 Nonresonant stability estimates: application to the harmonic oscillators case From theorem B, one can obtain estimates for the variation of the action variables on the set G where this theorem is applied. This is very simple in a nonresonant region (M = 0), where no extra geometric condition on the unperturbed Hamiltonian h is required. Lemma 5 Let H( ; I) = h(I) + Z(I) +R( ; I) real analytic on D (G), let ! = gradh, and assume that !(G) is ;K-nonresonant modulo 0. Assume that @2h @I2 G; 2 M . Let c = 2= 1, and assume: 2 2MK ; kDZkG; ;c + kDRkG; ;c 2 122K 1 : (23) Then, for every trajectory ( (t); I(t)) of H, with ( (0); I(0)) 2 Tn G, one has jI(t) I(0)j 24 kDRkG; ;c for jtj 2 eK 1 6 : (24) The proof is deferred to section 5. Now, as a simple application, we consider the case h(I) = ! I, i.e. H is a perturbation of a system of n harmonic oscillators. The frequency vector ! 2 Rn is assumed to be ; -Diophantine (see (2)), for n 1 and > 0 given. This case, where no geometric part is required, is also considered in [12], [11], [10], [27] and [7]. We obtain, like in the last four of the quoted papers, the \optimal" stability exponent a = 1=( + 1). We remark that, since M = 0 in this case, condition (17) does not impose any restriction on 2. Theorem C Let H( ; I) = ! I + f( ; I) real analytic on D (G), and assume that the vector ! is ; -Diophantine for some n 1 and > 0. Assume: " := kfkG; "0 := 2 244 : Then, for every trajectory ( (t); I(t)) of H, with ( (0); I(0)) 2 Tn G, one has jI(t) I(0)j 2 5 1 " "0 1=( +1) for jtj 2 exp( 1 24 "0 " 1=( +1)). Proof Let c = 2= 1. We notice that we may take M = 0 in lemma 5. For a xed K to be chosen, the set f!g is clearly K ; K-nonresonant modulo 0. We are going to apply lemma 5 with Z = 0, R = f , and =2 instead of . Since kDfkG; 2 ;c 2" 1 ; 3 NEKHOROSHEV ESTIMATES AND RELATED RESULTS 18 the second condition of (23) is satis ed for " 2 244K +1 : We then choose K = "0 " 1=( +1) and obtain: jI(t) I(0)j 24K 2" 1 2 5 1 " "0 1=( +1) : Concerning the stability time, it is easily obtained from the one of lemma 5 if we take into account that, since " "0, we have K 1 2 "0 " 1=( +1). 2 Remarks 1. One may notice that our results in the case of a perturbation of a system of harmonic oscillators are slightly worse than the ones obtained in [10]. Indeed, the stability exponent a = 1=( +1) is the same but we have obtained b = 1=( +1) instead of b = 1. This di erence comes from a di erent performance of the iterations leading to normal form. Indeed, in [10] the linear functional equation (6) is solved without cutting the Fourier expansions at order K (but making a reduction of the domain). This approach makes the estimates of proposition 4 better, but it is limited to the nonresonant case (M = 0). Although our approach leads to worse estimates, it avoids dealing with in nitely many small divisors, and also allows to treat the resonant case. 2. Even in the harmonic oscillators case, our approach looks more signi cative from a practical point of view. Indeed, if we consider " xed (i.e. a concrete Hamiltonian), then the result of theorem C still holds if Diophantine condition (2) is required just for jkj1 "0 " 1=( +1). For instance, if the frequency ! is known only up to a nite precision then it has no sense to check the Diophantine condition farther than a certain nite order, but our estimates could also be applied. 3.3 Resonant stability estimates Now we restrict ourselves to a neighborhood of the resonance associated to a given module M Zn, and afterwards the whole domain G will be divided in resonant and nonresonant regions corresponding to the di erent modules. A set of frequencies F Rn is said to be -close to M-resonances if jv Mvj 8I 2 F; where M denotes the orthogonal projection onto the space of exactM-resonant frequencies M? = fv 2 Rn : k v = 0 8k 2 Mg : 3 NEKHOROSHEV ESTIMATES AND RELATED RESULTS 19 To obtain stability estimates for the trajectories with initial condition in a set G such that !(G) is close to a resonance, we need to impose some geometric condition on the unperturbed Hamiltonian h. In the original Nekhoroshev's proof [24], a general steepness condition was imposed. But the main geometric ideas of the proof are contained in the simpler quasiconvex case, considered for instance in [2] (convex case), [15] and [27]. Following [15], we say the function h(I) to be m-quasiconvex on a set U if @2h @I2 (I) (v; v) m jvj2 8v 2 h!(I)i? ; 8I 2 U (this de nition is slightly di erent from the one given in [27]). One remarks that the level hypersurfaces of h are convex if h is m-quasiconvex. Moreover, the quasiconvexity implies that, for every module M, the resonant manifold SM and the vector subspace generated by M are always transversal. Under this condition, the next lemma (called Resonant Stability Lemma in [27]) provides stability estimates on a region G G such that !(G) is assumed to be close to the resonance associated to a given module M and satisfying a nonresonant condition modulo M. Our proof is standard. It follows [15] and [27] in the main ideas, which go back (for convex systems) to [3]. The basic point is that, for a Hamiltonian in normal form with respect to M with an exponentially small remainder, the speed of variation of the action variables along the M?-direction is exponentially small. On the other hand, the quasiconvexity condition forces the energy hypersurface of h passing through a point of the resonance SM to have a contact of order two with the M-direction. Then, by energy conservation, one may bound the variation of the actions along the M-direction, giving rise to the stability estimate. It has to be noticed that this approach di ers from the one of [2], where a di erent feature of (quasi)convex Hamiltonians is used: the transversality between the resonant manifold SM and the M-direction. We de ne a real neighborhood of the domain G as U 2(G) := fI 2 Rn : jI I 0j 2 for some I 0 2 Gg = V 2(G) \Rn: (25) Lemma 6 Let H( ; I) = h(I) + Z( ; I) + R( ; I) real analytic on D (G), and let ! = grad h. For a given module M 6= Zn, assume that !(G) is -close to M-resonances, and ;K-nonresonant moduloM, with K 1, and assume also that Z 2 R(M; K). Assume that @2h @I2 G; 2 M; j!jG L; and that h is m-quasiconvex on U 2(G). Let c = 2= 1, and assume: 2 m 48M2K ; m 2 60 ; kDZkG; ;c + kDRkG; ;c m 2 2 350 ; (26) where we write := min 1; 1 pn 1!. Then, for every trajectory ( (t); I(t)) of H, with ( (0); I(0)) 2 Tn G, one has jI(t) I(0)j 2 for jtj m 2 2 74L kDRkG; ;c emK 1 6M . (27) 3 NEKHOROSHEV ESTIMATES AND RELATED RESULTS 20 We give the proof of this result in section 5. Remark If R = 0, this lemma implies that, if the quasiconvexity condition is ful lled, all trajectories starting in Tn G have perpetual stability. Nevertheless, one could deduce this fact in a more direct way, because in this case the Hamiltonian H = h+Z is already in normal form with respect to M. 3.4 Geometry of resonances Next we return to the Hamiltonian (1), which we assume real analytic on D (G). The stability estimates obtained in lemmas 5 and 6 only apply to the trajectories starting in a subset G G where the frequencies are assumed to be close to the resonance characterized by a xed moduleM Zn and satisfying a nonresonance condition moduloM. In order to obtain stability estimates for all trajectories starting inTn G, the whole action domain G is covered by a family of sets GM, called resonant and nonresonant blocks (for M 6= 0 and M = 0, respectively). For each moduleM, the frequencies on the block GM have to be close to M-resonances, and to satisfy the nonresonance condition (16) up to a xed order K. A construction of such a covering is carried out in [24] and [2]. The quantitative aspect was improved in [27]. The Geometric Lemma stated below has been taken from [27] with no changes. Actually one may work in frequency space. We obtain for this space a covering fBMg, which can be pulled back by the frequency map ! to a covering fGMg for G. Before stating the Geometric Lemma we recall some concepts and terminology introduced in [27]. For each module M Zn, we consider the space of M-resonant frequencies M?. Note that there are a lot of modules giving rise to the same resonant space. Obviously we only have to consider the maximal one. A module M Zn is said to be maximal if it is not properly contained in any other module of the same dimension. See appendix 3 of the book [16] for an explicit characterization of the maximal modules in Zn. Given a maximal d-dimensional moduleM Zn, the set BM is constructed by taking a neighborhood of the space M? and removing from it a neighborhood of the resonant spaces associated to the (d + 1)-dimensional modules. The set constructed in this way would not contain any open set. However, one remarks that, to satisfy the nonresonance condition (16) up to order K, it su ces to consider K-modules. A module M Zn is said to be a K-module if it is generated by vectors of order less or equal than K. To make these ideas quantitative, one requires the notion of volume of a module. For a d-dimensional module M Zn, 1 d n, let C be the (n d)-matrix obtained by choosing a basis of M and putting its vectors as columns. The volume of M is then de ned as jMj := qdet(C>C); i.e. the d-dimensional volume of the parallelepiped spanned by the vectors of the basis. The choice of the basis does not have in uence in this de nition. Let d > 0, for 1 d n, be xed parameters. For each maximal d-dimensional 3 NEKHOROSHEV ESTIMATES AND RELATED RESULTS 21 K-module M, one introduces M := d jMj ; and the resonant zone associated toM is de ned as a neighborhood of radius M around M?. Recalling that M denotes the orthogonal projection onto M?, one de nes AM := fv 2 Rn : jv Mvj < Mg : Then, the resonant block associated to M is de ned as BM := AM n A d+1; where A l , 1 l n, stands for the union of all resonant zones corresponding to maximal l-dimensionalK-modules, and A n+1 = ;. Note that every resonant block BM is M-close to M-resonances. For the trivial module one de nes the nonresonant block: B0 := Rn n A 1: It is easy to see that the whole frequency space is covered by the blocks BM. Lemma 7 (Geometric Lemma) Let us x K 1, E > 0 and F E+p2. Assume: d+1 d FK for 1 d < n. Then, the blocks BM de ned above are M; K-nonresonant modulo M, with M := EK M for M 6= 0, 0 := 1: For the proof, see [27]. The desired covering for G is then obtained from this lemma by putting GM = ! 1 (BM) for each maximal K-module M, except for the ones giving rise to an empty set. 3.5 Global e ective stability Our main result on e ective stability concerns estimates holding for all motions in phase space. Like in [27], these estimates are obtained by considering the covering supplied by lemma 7 with a xed order K (which is chosen as a suitable function of the size " of perturbation) and then applying the stability estimates to each block of the covering. Theorem D (Nekhoroshev Theorem) Let H( ; I) = h(I) + f( ; I) real analytic on D (G), let ! = gradh, and assume that @2h @I2 G; 2 M; j!jG L: 3 NEKHOROSHEV ESTIMATES AND RELATED RESULTS 22 Assume also that h is m-quasiconvex on U 2(G). Let > 0 given, and assume: 23M2 2 m ; " := kfkG; "0 := m4n 1̂ 2 224n 2M4n ; (28) where we write ̂ := min 1; 2 pn!. Then, for every trajectory ( (t); I(t)) of H, with ( (0); I(0)) 2 Tn G and satisfying j!(I(0))j , one has jI(t) I(0)j 2 " "0 1=2n for jtj 4 L exp(m 1 24M "0 " 1=2n). (29) Proof Fix K 1 to be chosen below. Let F = 2882M2 m2 ; E = F p2: For 1 d n, we put: d = (FK)n d : Then, lemma 7 provides a covering fGMg of G, with GM = ! 1 (BM), and its parameters are M = jMj (FK)n d ; M = E jMj F n dKn d 1 for every maximal d-dimensional K-moduleM, 1 d n, and 0 = (FK)n 1 for the trivial module. We also put 0 = 0. We are going to apply lemma 6 with Z = 0 and R = f on all blocks, except on the one corresponding to M = Zn. In this way the estimates hold for all initial conditions satisfying j!(I(0))j Zn = . Unlike the case of theorem C (harmonic oscillators), the smallness condition (17) on 2 makes us restrict the domain. For every M we take (M) = (M) 1 ; (M) 2 , with (M) 1 = 1 2 ; (M) 2 = m M 48M2K ; cM = (M) 2 (M) 1 : (30) For every d-dimensional K-module M, 0 d n 1, one has 60 mF n dKn (M) 2 61 m(FK)n d 2 2 ; where we used that 1 jMj Kd. We have kDfkGM; (M);cM " (M) 1 2" 1 : 3 NEKHOROSHEV ESTIMATES AND RELATED RESULTS 23 To apply lemma 6 on GM, we must verify the three inequalities of (26). The two rst ones are easily veri ed and the last one is ful lled for allM if 2" 1 2"0 1K2n m 350 60 m(FK)n!2 m 350 (M) 2 2 ; with = min 1; 2 pn 1!. Thus, we choose K = "0 " 1=2n . For everyM, the stability radius and the stability time for the trajectories starting in Tn GM are obtained from lemma 6: (M) 2 61 mFK 122 mF " "0 1=2n 2 " "0 1=2n ; m (M) 2 2 74L 2" 1 emK 1 12M 4 L exp(m 1 24M "0 " 1=2n) : 2 Remarks 1. These estimates would have been a little better if, on the nonresonant block G0, we had used lemma 5 instead of lemma 6. But the stability exponents obtained would have been the same, so we used lemma 6 on all blocks for sake of simplicity. 2. We also point out that, actually, condition (28) on " is not essential. It can be removed with some additional e ort, but we omit the details. Note, however, that for a large " Nekhoroshev estimate (29) is meaningless. The same remark holds for theorem C. In the proof of theorem D, we have obtained the stability exponents a = b = 1 2n by carrying out the estimates on every block GM and always taking the worst possible case. The key point is to nd greater and lower bounds for (M) 2 , valid for all modules M. However, the stability exponents can be improved by means of a particular analysis, if one is only interested in a given region. In particular, one remarks the case of the nonresonant block G0. This case corresponds to the smallest (0) 2 , which gives rise to the smallest stability radius. It is not hard to see that, as stated in theorem 2 of [27] and theorem 3 of [7], the stability exponents obtained for this case are a = 1 2n ; b = 12 : However, if lemma 5 were used to obtain the stability estimate, one may check that the exponents would be a = 1 2n ; b = n+ 1 2n : 4 KAM THEOREM AND NEARLY-INVARIANT TORI 24 It is also interesting to consider, for a xed moduleM0, a neighborhood of the resonant manifold SM0 . This set can be covered by the blocks GM associated to the modules M containing M0. If we restrict ourselves to these modules, the lower bound for (M) 2 is greater than the one obtained in theorem D. This allows to choose K greater, and leads to the exponents: a = b = 1 2 0 ; where 0 is the codimension ofM0. A precise statement of this result is given in theorem 3 of [27]. 4 KAM theorem and nearly-invariant tori 4.1 Nondegeneracy conditions Now our aim is to prove that, for a nearly-integrable Hamiltonian H( ; I) = h(I)+f( ; I), analytic on D (G), most orbits lie in n-dimensional invariant tori if the perturbation f is small enough. To reach this result, a suitable nondegeneracy condition has to be ful lled by unperturbed system. In the usual statements of KAM theorem, two sorts of nondegeneracy conditions are imposed on ! = gradh. These are the (standard) nondegeneracy and the isoenergetic nondegeneracy. The frequency map ! is said to be nondegenerate if det @! @I (I)! 6= 0 8I 2 G; (31) and isoenergetically nondegenerate if det @! @I (I) !(I) !(I)> 0 ! 6= 0 8I 2 G: (32) An equivalent formulation for the isoenergetic nondegeneracy is to require that ! is nonvanishing on G and @! @I (I) v + !(I) 6= 0 8v 2 h!(I)i? n f0g; 8 2 R; 8I 2 G; (33) In action space, condition (33) can be interpreted as transversality, at every point, between any energy level ME = h 1(E) and the hypersurfaces !(I) v = 0 (which include the resonant ones). The interpretation in frequency space is that the image ! (ME) of any energy level and the subspace h!(I)i are always transversal. It is easy to construct examples showing that conditions (31) and (32) are independent: h(I1; I2) = ln I2 I1 ; h(I1; I2) = 1 2I 2 1 + I2 : (34) The rst one is only nondegenerate on its whole domain, and the second one is only isoenergetically nondegenerate. We give in the next sections a direct and quantitative proof of KAM theorem under the assumption of isoenergetic nondegeneracy, although the same approach holds for the 4 KAM THEOREM AND NEARLY-INVARIANT TORI 25 standard nondegeneracy. Our setup di ers from the one which can be found in [9], [4], where the isoenergetic KAM theorem is proved from the standard one. We rst remind that the standard version of KAM theorem (under the standard condition (31)) states that, given > n 1 and > 0 previously xed, and assuming for the size of the perturbation f a smallness condition of the type " = O 2 ; (35) then every invariant torus of the unperturbed Hamiltonian h having ; -Diophantine frequency (i.e. satisfying (2)) is preserved in the perturbed system with the same frequency vector. Moreover, the measure of the complement of the set lled with the invariant tori is O( ). In proving this statement it is crucial to use that, under condition (31), the map ! is a local di eomorphism and therefore the unperturbed invariant tori can be locally parametrized by their frequency vector. The preservation of the invariant tori with the same frequency vector can be false under the isoenergetic condition (32). Indeed, it su ces to consider h(I) as in the second example of (34), where the frequency map !(I1; I2) = (I1; 1) maps the whole plane into a straight line, and f( ; I) = " h(I) as a perturbation. Nevertheless, it is known that in the isoenergetic case the unperturbed invariant tori can be locally parametrized on each energy level ME by their frequency ratios. More precisely, if we assume, with no loss of generality, that the component !n does not vanish on G, then the isoenergetic condition is equivalent to requiring that the map (I) := !(I) !n(I) ; h(I)! = !1(I) !n(I) ; : : : ; !n 1(I) !n(I) ; h(I)! (36) is a local di eomorphism on G. We use, in this section and in the subsequent ones, the notation v = (v1; : : : ; vn 1) for v = (v1; : : : ; vn 1; vn). Note that, including the last component h(I) in the de nition of , we avoid to consider each energy level separately. Using the nondegeneracy of the map , we are able to state that, if a smallness condition on " like (35) is ful lled, then for every ; -Diophantine torus of the unperturbed Hamiltonian there exists an invariant torus of the perturbation, with the same frequency ratios (though the frequency itself can vary) and the same energy. Moreover, like in the standard case, we get that the measure of the complement of the invariant set can be estimated as O( ). One deduces that in the isoenergetic case most of the invariant tori on any energy level are preserved under the perturbation. Indeed, since !n(I) 6= 0 for I 2 G, the frequency vector !(I) associated to a given torus is Diophantine if the vector !(I) !n(I) ; 1 is also Diophantine. This occurs for most of the unperturbed tori on a xed energy level, since these tori can always be parametrized by their frequency ratios. It is worth noting that the isoenergetic version of KAM theorem is more signi cative from the point of view of stability, because in this case the existence of a large family of invariant tori is ensured on every xed energy level. For two degrees of freedom (n = 2), it follows the stability of the system (in the sense that all motions are bounded), since a given energy level is always separated by the invariant tori. Under the standard nondegeneracy condition, one cannot deduce from KAM theorem that on a given energy level any invariant tori is preserved. On the other hand, for more than two degrees of freedom, stability cannot be guaranteed under any nondegeneracy condition but in the isoenergetic case the preserved tori seem to be stronger barriers to Arnold di usion. 4 KAM THEOREM AND NEARLY-INVARIANT TORI 26 Another remark is that the isoenergetic KAM theorem can eventually be applied to periodic nonautonomous Hamiltonians, by taking the time variable as an additional angular variable. Let us make an outline of the method we use for the proof of KAM theorem and the main technical problems found. As established in section 2.1, we use an iterative procedure leading to successive Hamiltonians of the form H(q) = h(q) + R(q), with the integrable part h(q) changing from every step to the next one. The domain is restricted by removing at every step resonant strips up to successive orders Kq increasing to in nity. A rst problem is that we need to guarantee the nondegeneracy condition for the frequency map !(q) = gradh(q) at every iteration. Hence the analytic part and the geometric part cannot be separated and must be carried out simultaneously along the proof. Another technical problem is that, to move the bounds of the measure of the resonant zones from frequency space to phase space, we need to estimate from below the Jacobian determinant of a suitable di eomorphism. In the standard case, the frequency map ! itself is a local di eomorphism, and one can see that the successive maps !(q) are still di eomorphisms on their domains. One can assume the starting map ! to be one-toone (restricting the domain if necessary). To guarantee the perturbed maps !(q) to be also one-to-one, their domains still have to be slightly restricted after having removed resonances. The isoenergetic case is, in this aspect, more cumbersome. As showed above, the frequency map ! is not necessarily a local di eomorphism, but we may use the map introduced in (36), which we shall assume to be one-to-one on G. Then, like one would do in the standard case, successive perturbations (q) of this map will be guaranteed to be also one-to-one provided their domain is slightly restricted. We notice that the resonant strips to be removed along the successive iterations are better expressed in the image space !(G) or (G). Indeed, the strips can then be taken as linear and are thus easier to handle. To be more precise, in the isoenergetic case we remove from (G) resonant strips of the form (k; ) := nJ 2 Rn : k J + kn < o ; (37) with > 0. This is very appropiate for the isoenergetic case since maps every resonant zone jk !(I)j < into such a linear strip. The only exception is the case k = (0; : : : ; 0; 1), since (k; ) is then empty if < 1, but this integer vector corresponds to the resonance !n(I) = 0, which has previously been excluded from the domain. We also remark that it is easy to bound the measure of a linear strip like (37), and that this bound can be pulled back to action space by estimating from below the Jacobian determinant of the di eomorphism. It is possible to construct a common environment for both the standard and the isoenergetic nondegeneracy conditions. Indeed, let us consider the condition: @! @I (I) v 6= 0 8v 2 h!(I)i? n f0g; 8I 2 G; (38) which means that the restriction of ! to each energy level ME is a local di eomorphism. It is not hard to check that this condition is equivalent to that, at every point I 2 G, the standard or the isoenergetic condition holds. 4 KAM THEOREM AND NEARLY-INVARIANT TORI 27 It is known that the measure of the set of vectors which do not satisfy Diophantine condition (2), with given > n 1 and , is O( ). Note also that the nondegeneracy condition (38) implies that every resonance k !(I) = 0, with k 6= 0, is a regular hypersurface. Then, the measure of the complement of the invariant set is also O( ), since this estimate is obtained by pulling back to action space the measure of every resonant strip removed along the iterative process. This is the idea of the measure estimates for both nondegeneracy conditions (31) and (32), and the only reason for carrying out the proofs separately is the technical problem concerning the appropiate di eomorphisms described above. A higher-order nondegeneracy condition has been announced by R ussmann [28]: the Taylor expansion of ! at a point I is required to contain n linearly independent coe cients. As a matter of fact, it is easy to see (using that @! @I (I) is a symmetric matrix) that condition (38) is equivalent to imposing that n of the vectors !(I); @! @I1 (I); : : : ; @! @In (I) are linearly independent, namely R ussmann's condition at order 1. As said in [28], KAM theorem remains true even under the most general R ussmann's condition. However, the estimate one would get for the measure of the complement of the invariant set would then be larger: O( b), with b < 1. Very recent results along these lines can be found in [29], [5] and [30]. 4.2 Isoenergetic nondegeneracy: quantitative results We give in this section several quantitative results related to the isoenergetic nondegeneracy. All proofs are technical and we postpone them to section 5. We need to work with a quantitative version of the isoenergetic nondegeneracy in its form (33). If h(I) is de ned on a set G Rn, we say the frequency map ! = grad h to be -isoenergetically nondegenerate if ! does not vanish on G and @! @I (I) v + !(I) jvj 8v 2 h!(I)i? ; 8 2 R; 8I 2 G: Moreover, we modify slightly the map de ned in (36). For a xed constant a > 0, we de ne !;h;a(I) := !(I) !n(I) ; a h(I)! ; I 2 G: (39) The constant a is introduced for quantitative reasons. It has the appropiate dimensions to make the components of the map (39) dimensionally coherent. But its main motivation is that the estimates given in the next lemma are better with a good choice of a. Lemma 8 Let h be a real function of class C3 on G Rn, and ! = gradh. Assume the bounds: @2h @I2 G M; @3h @I3 G M 0; j!jG L and j!n(I)j l 8I 2 G: Assume also that ! is -isoenergetically nondegenerate on G. Let a 2M=l2 be a xed constant, and denote = !;h;a : One has: 4 KAM THEOREM AND NEARLY-INVARIANT TORI 28 a) @ @I G 2La. b) @ @I (I) v 2L jvj 8v 2 Rn; 8I 2 G. c) det @ @I (I)! n 1a Ln 2 8I 2 G. d) @2 @I2 G M 0 2M + 3Ml !La. Next we state how a slight variation of the frequency map a ects the constant of the isoenergetic nondegeneracy condition. This result may be expressed in terms of vectors and matrices. Lemma 9 Let !; ~ ! 2 Rn, and let A; ~ A be (n n)-matrices. Let " = j~ ! !j, "0 = ~ A A , and de ne l = min (j!j ; j~ !j), M = max jAj ; ~ A . For some > 0, assume that jAv + !j jvj 8v 2 h!i? ; 8 2 R: Then, ~ Av + ~ ! 4M" l "0 jvj 8v 2 h~ !i? ; 8 2 R: Finally, we see that a small perturbation of a one-to-one map is still one-to-one provided the domain is slightly restricted. Previously, for b 0 we de ne the set G b := fI 2 G : Ub(I) Gg ; where Ub(I) means the closed ball of radius b centered at I, according to (25). Lemma 10 Let G Rn a compact, and let ; ~ : G ! Rn maps of class C2, with ~ G ". Assume that is one-to-one on G, and let F = (G). Assume the bounds: @ @I G M; @ ~ @I G ~ M; @2 ~ @I2 G ~ M 0; @ @I (I) v m jvj ; @ ~ @I (I) v ~ m jvj 8v 2 Rn; 8I 2 G; with 0 < ~ m < m, ~ M > M . Assume also that " ~ m2 4 ~ M 0 : (40) Then, given a subset ~ F F 4M" ~ m and writing ~ G = ~ 1 ~ F , the map ~ is one-toone from ~ G to ~ F , and one has the inclusions ~ G G 2" ~ m ; ~ G ~ F ": 4 KAM THEOREM AND NEARLY-INVARIANT TORI 29 Moreover, the following estimate holds: ~ 1 1 ~ F " m : 4.3 Analytic and geometric estimates for one step We provide in proposition 11 below quantitative estimates for one concrete step of the iterative process described in section 2.1 for the isoenergetic KAM theorem. A parallel result for the standard theorem is called Inductive Lemma in [1]. Let us outline what we call the \analytic part". If the starting Hamiltonian in a step of the iterative process is written in the form H( ; I) = h(I) + R( ; I), we get, from the Iterative Lemma (section 2.3), estimates for the new Hamiltonian ~ H( ; I) = ~ h(I) + ~ R( ; I). Assuming the starting frequencies !(I) to be Diophantine up to a given order K, we ensure the Iterative Lemma to apply with M = 0. To be more precise, the Diophantine condition up to order K is slightly modi ed (see condition (41) below) in view of the resonant strips introduced in (37). In this way, we obtain the estimates of parts (a{e) of proposition 11, which could be stated without the hypothesis of isoenergetic nondegeneracy. On the other hand, some \geometric part" is required. Indeed, assuming that ! is isoenergetically nondegenerate on the domain G, we need to guarantee that the new frequency map ~ ! = grad ~ h is also isoenergetically nondegenerate, with a new parameter, in order to allow further iterations. This is established in part (f) of proposition 11. Moreover, if the map !;h;a introduced in (39) is one-to-one, we see in part (g) that the new map ~ !;~ h;a is also one-to-one on a new domain ~ G G. Proposition 11 (Inductive Lemma) Let G Rn a compact, and H( ; I) = h(I) + R( ; I) real analytic on D (G). Let ! = gradh, and assume the bounds: @2h @I2 G; 2 M; j!jG L and j!n(I)j l 8I 2 G: Assume also that ! is -isoenergetically nondegenerate on G. Let ~ M > M , ~ L > L, ~l < l and ~ < given. For a xed constant a 2 ~ M=~l2, assume that the map = !;h;a is one-to-one on G, and let F = (G). For > 0, 0 < 1 and K given, K an integer, assume the nonresonance condition: F \ k; jkj 1 ! = ; 8k = k; kn 2 Zn; jkj1 K; k 6= 0: (41) Let < given, c = 2= 1 and A = 1 + 2McK l : Let " := kDRkG; ;c ; := jR0jG; 2 and := @R0 @I G; 2 (where R0(I) is the angular average of R( ; I)), and assume: 2 ~l 2 ~ MK +1 ; (42) 4 KAM THEOREM AND NEARLY-INVARIANT TORI 30 " l 2 74AK ; min ~ M M 2 ; ~ L L ; l ~l ; ( ~ ) 2 3 ! ; (43) 0 := L 2 ~ M + ~ 2( 2 2) 32~ L3a2 : (44) Then, there exists a real analytic canonical transformation : D 2 (G) ! D (G) and a descomposition H = ~ h(I) + ~ R( ; I) such that, writing ~ ! = grad ~ h and ~ = ~ !;~ h;a ; one has: a) j~ ! !jG; 2 = , ~ h h G; 2 = . b) ~ " := D ~ R G; ;c e K 1 "+ 14AK l 2 "2. c) ~ := ~ R0 G; 2 2 2 7AK c l "2. d) j idjG; 2 ;c 2AK l " : e) @2~ h @I2 G; 2 2 ~ M , j~ !jG ~ L and j~ !n(I)j ~l 8I 2 G. f) ~ ! is ~ -isoenergetically nondegenerate on G. g) Given a subset ~ F F 16L~ La2 0 ~ and writing ~ G = ~ 1 ~ F , the map ~ is one-to-one from ~ G to ~ F , and one has the inclusions ~ G G 4~ La 0 ~ ; ~ G ~ F a 0: Moreover, the following estimates hold: ~ G a 0; ~ 1 1 ~ F 2La 0 : Proof By condition (41) we have, for every I 2 G and 0 < jkj1 K, k 6= 0, jk !(I)j jkj 1 j!n(I)j l K : This estimate holds even if k = 0 since j!n(I)j l and 1. Then, the set !(G) is l K ; K-nonresonant modulo 0. This fact and conditions (42) and (43) on 2 and " allow to apply the Iterative Lemma with Z = 0, M = 0 and = l K : We obtain the canonical transformation and, according to (10{11), the new Hamiltonian may be written as H = ~ h + ~ R, where ~ h = h + ~ Z, and we have ~ Z = R0. One sees the estimates of (a) using that j~ ! !jG; 2 = @R0 @I G; 2 = : (45) 4 KAM THEOREM AND NEARLY-INVARIANT TORI 31 The estimates of (b) and (d) are provided directly by the Iterative Lemma. The bound (c) comes from the inequality (20) in the Iterative Lemma. The estimates of part (e) come from (45), condition (43) on and the Cauchy inequality @2~ h @I2 @2h @I2 G; 2 2 2 : We prove (f) using lemma 9, which tells us that ~ ! is 0-isoenergetically nondegenerate on G, where 0 = 4 ~ M~l j~ ! !jG @2~ h @I2 @2h @I2 G 4 ~ M~l 2 3 2 ~ : We have used (45), condition (43) on , the inequality @2~ h @I2 @2h @I2 G 2 and the inequality 2 ~l=2 ~ M , deduced from (42). Before going on, we remark that the bounds on the derivatives of given in lemma 8 are also valid for ~ , on G, provided one replaces M , L, l, , by ~ M , ~ L, ~l, ~ , respectively. We do not need to change the constant a because we are assuming that a 2 ~ M=~l2. We shall now apply lemma 10 but we have to take, instead of the parameters of that lemma M , m, ~ M , ~ m, the values obtained applying lemma 8 to and to ~ . We rst nd the value which will replace " in lemma 10, i.e. an estimate for ~ G. We have ~ (I) (I) j~ !n(I) !n(I)j j!(I)j+ ~ !(I) !(I) j!n(I)j j!n(I)j j~ !n(I)j j~ !(I) !(I)j j!(I)j j!n(I)j j~ !n(I)j L l~l ; ~ n(I) n(I) a ~ h(I) h(I) a : Therefore, ~ G vuut L l~l !2 + (a )2 a 0; where the condition on a has been used. Then, lemma 10 applies if the next inequality (which replaces (40) of that lemma) is ful lled: a 0 ~ 2~ L 2 4 ~ M 0 2 ~ M + 3 ~ M~l ~ La ; (46) where we have taken into account the inequality @3~ h @I3 G ~ M 0 := ~ M 2 2 : It is easy to check that the inequality (46) is guaranteed by condition (44) on 0. Then, lemma 10 gives part (g). 2 4 KAM THEOREM AND NEARLY-INVARIANT TORI 32 4.4 Invariant tori In order to estimate the measure of the resonant strips which we remove along the successive steps, a reasonable condition has to be imposed on the domain. Given F Rn and D > 0 we say F to be a D-set if, for any 0 b1 < b2, mes [(F b1) n (F b2)] D(b2 b1): We remark that the constant D is a rough upper bound of the \area" of the boundary of F , which is forced to be nite. The next technical lemma provides the necessary estimates for the measure when resonant strips of the type introduced in (37) are removed from F . We point out that it su ces to remove resonant strips corresponding to integer vectors k = k; kn 2 Zn with k 6= 0, since it is assumed that !n(I) 6= 0 throughout the domain. The proof of this result is deferred to section 5. Lemma 12 Let F Rn a D-set. For d 0, > 0, 0 and K 0 given, K an integer, let us denote F (d; ;K) := (F d) n [ jkj1 K k 6=0 k; jkj 1 ! : One has: a) Given d0 d, 0 and K 0 K, mes [F (d; ;K) n F (d0; 0; K 0)] D(d0 d) + 2(diamF )n 10BB@ X jkj1 K k 6=0 0 jkj 1 k + X K n 1, > 0 and 0 < < 1 given, and assume: " := kfkG; 2l6 2̂2 +2 24 +32L6M3 2; min 8LM 2 l̂ +1 ; l! ; (47) where we write ̂ := min 1 12( + 2) ; 1!. De ne the set b G = b G := (I 2 G 2 : !(I) is ; -Diophantine) : Then, there exists a real continuous map T : W 1 4 (Tn) b G ! D (G), analytic with respect to the angular variables, such that: a) For every I 2 b G, the set T (Tn fIg) is an invariant torus of H, its frequency vector is colinear to !(I) and its energy is h(I). b) Writing T ( ; I) = ( + T ( ; I); I + TI( ; I)) ; one has the estimates jT jb G;( 1 4 ;0);1 22 +15L2M 2l2̂2 +1 " 2 ; jTI jb G;( 1 4 ;0) 2 +16L3M l3 ̂ +1 " : c) mes h(Tn G) n T Tn b G i C , where C is a (very complicated) constant depending on n, , diamF , D, ̂, M , L, l, . Proof A. Choice of the parameters. Since we make iterative use of proposition 11, we rst introduce appropiate sequences of parameters to replace the constants of that proposition. For q 0, we de ne Mq = 2 1 2q M; Lq = 2 1 2q L; lq = 1 + 1 2q l 2 ; q = 1 + 1 2q 2 : 4 KAM THEOREM AND NEARLY-INVARIANT TORI 34 Note that Mq, Lq increase from M , l to 2M , 2L, respectively, for q !1, and that lq, q decrease from l, to l=2, =2. We also put K0 = 0; Kq = K 2q 1; q 1; where K 1 is an integer to be xed below. Moreover, we de ne := L 1 and, for q 0, we put (q) = (q) 1 ; (q) 2 , with (q) 1 = 1 + 1 2 q 1 4 ; (q) 2 = l 32MK +1 q+1 : We notice that (q) 1 decreases from 1=2 to 1=4, and that (q) 2 decreases to 0. We also write (q) 1 = (q 1) 1 (q) 1 ; (q) 2 = (q 1) 2 (q) 2 ; cq = (q) 2 (q) 1 : Taking into account that the inequalities 2 1 1 2 and 1 1 2 +1 1 2 ; hold for 0 < < 1 and > 0 respectively, it is easy to see that, for every q 1, 1 8 2 (q 1) (q) 1 1 4 2 (q 1) ; (q) 2 l 64MK +1 q ; (48) l 2 (q 1) 16MK +1 q 1 cq l 2 (q 1) 4MK +1 q 1 : (49) Finally, we de ne q = 1 1 2 q ; 0 q = q + q+1 2 ; which are both increasing with limit . It is also easy to check that 0 q =4 for every q 0. We choose K as the minimum integer such that K̂ 1. Then, one has K 2=̂. Using this inequality, our choice = =L and the inequality ̂ 1, we deduce from conditions (47) the inequalites " min 3l2 1 2 22 +20MK2 +1 ; 2l6 2 2 2 +30L4M3K2 +2! ; 8MK +1 2 l : (50) B. Induction. Starting with G0 = G, we shall now construct a decreasing sequence of compacts Gq G and a sequence of real analytic canonical transformations (q) : D (q)(Gq) ! D (q 1)(Gq 1), q 1. Denoting (q) = (1) (q), the transformed Hamiltonians will be written in the form H(q) = H (q) = h(q)(I)+R(q)( ; I). Moreover, we write !(q) = gradh(q) and (q) = !(q);h(q);a. We are going to show that the following statements hold for every q 0: 1q) "q := DR(q) Gq ; (q);cq+1 8" 1 2(2 +2)q : 4 KAM THEOREM AND NEARLY-INVARIANT TORI 35 2q) q := R(q) 0 Gq ; (q) 2 " 2(2 +3)q ; q := @R(q) 0 @I Gq; (q) 2 26MK +1 " l 2( +2)q : 3q) @2h(q) @I2 Gq ; (q) 2 Mq; !(q) Gq Lq and !(q)(I) lq 8I 2 Gq. 4q) !(q) is q-isoenergetically nondegenerate on Gq. 5q) (q) is one-to-one on Gq, and (q)(Gq) = Fq, where we de ne Fq := (F q) n [ jkj1 Kq k 6=0 k; q jkj 1 ! : (51) We proceed by induction. For q = 0, we choose G0 = G, h(0) = h, R(0) = f . Note that (0) 1 = 1 2 ; (0) 2 = l 32MK +1 2 2 by (50). Then, "0 = kDfkG; (0);c1 " (1) 1 8" 1 ; (52) namely (10). The rst estimate of (20) is obvious, and the second one comes from the Cauchy inequality 0 2 2 R(0) 0 G; 2 2" 2 " (0) 2 : The remaining statements (30{50) are clear. For q 1, we assume the statements true for q 1 and we prove them for q. We attain this aim by applying proposition 11 to H(q 1) = h(q 1) +R(q 1), with Kq instead of K. However, in order to ful l condition (41), we must replace the domains Gq 1 and Fq 1 by suitable subsets where the resonances from order Kq 1+1 to order Kq have been removed. More precisely, we de ne F 0 q 1 := (F q 1) n [ jkj1 Kq k 6=0 k; 0 q 1 jkj 1 ! ; G0q 1 := (q 1) 1 F 0 q 1 : (53) The nonresonance condition (41) is then ful lled by F 0 q 1, 0 q 1 and Kq instead of F , and K, respectively (taking 0 q 1 instead of q 1, we avoid to apply proposition 11 with 0 = 0 for q = 1). The remaining parameters taking part in proposition 11 are Mq 1, Lq 1, lq 1, q 1, (q 1), (q), cq, Mq, Lq, lq, q, replacing M , L, l, , , , c, ~ M , ~ L, ~l, ~ , respectively, and a = 16M l2 2Mq l 2 q : It is clear that condition (42) on (q 1) 2 is satis ed with our choice of the parameters. Concerning condition (43) on "q 1, we rst notice that, by (49) and our choice of K, Aq := 1 + 2Mq 1cqK q lq 1 0 q 1 1 + 32M l l 4MK 1 = 1 + 8 K 1 2: 4 KAM THEOREM AND NEARLY-INVARIANT TORI 36 Then, to see that condition (43) is ful lled we have to check the inequality "q 1 lq 1 0 q 1 (q) 2 148K q : Indeed, recalling the de nitions of the parameters and applying (1q 1) and (48), it su ces to check that 8" 1 l 211K l 64MK +1 ; which can be deduced from (50). Let us verify the second condition of (43), namely q 1 min0@(Mq Mq 1) (q) 2 ; Lq Lq 1 ; lq 1 lq ; ( q 1 q) (q 1) 2 3 1A : By (2q 1) and (48), this condition holds, since we may deduce from (50) the inequality 26MK +1 " l min M2 l 64MK +1 ; L2 ; l 4 ; 12 l 32MK +1! : Finally, we have to check condition (44): 0 q 1 := Lq 1 q 1 2Mq + q 1 2 q (q) 2 32L 3 q a2 : (54) By (2q 1), we have the estimate 0 q 1 L 2M 26MK +1 " l 2( +2)(q 1) + " 2(2 +3)(q 1) 26LK +1 " l 2( +2)(q 1) : (55) Taking into account the value of a, the inequality (54) holds, since 26LK +1 " l l4 2 218L3M2 l 2 +6MK +1 ; as deduced from (50). Applying proposition 11 with the parameters considered above, we obtain a canonical transformation (q) and a new Hamiltonian H(q) = h(q) + R(q). The new domain Gq G0q 1 is chosen below. First, we prove (1q{4q). By the bound (b) of proposition 11 and the inequality cq+1 cq, "q DR(q) Gq ; (q);cq e Kq (q) 1 "q 1 + 14AqK q lq 1 0 q 1 (q) 2 " 2 q 1 : (56) Let us bound the terms of this sum. By (48) and the inequality K̂ 1, Kq (q) 1 K 1 8 2(1 )(q 1) (2 + 3) ln 2; (57) and therefore e Kq (q) 1 1 22 +3 : 4 KAM THEOREM AND NEARLY-INVARIANT TORI 37 Moreover, applying (48), (1q 1) and (50), 14AqK q lq 1 0 q 1 (q) 2 "q 1 28K q l 64MK +1 q l 8" 1 2(2 +2)(q 1) 1 22 +3 2q 1 : (58) Then, we deduce from (56) that "q "q 1 22 +2 ; which gives (1q). For (2q), we use part (c) of proposition 11. Writing (q) 2 = (q 1) 2 (q) 2 =2, we have: q R(q) 0 Gq ; (q) 2 7AqK q cqlq 1 0 q 1 " 2 q 1 (q) 1 "q 1 2 1 22 +3 2q 1 " 2(2 +3)q ; (59) where we have used the inequalities (58), (48) and (1q 1). We obtain the other estimate of (2q) using the Cauchy inequality and (48): q 2 (q) 2 R(q) 0 Gq ; (q) 2 2 (q) 2 " 2(2 +3)q : The statements (3q{4q) are clear from proposition 11. For (5q), we apply part (g) of proposition 11 to the subset Fq. We have to check that Fq F 0 q 1 16Lq 1Lqa2 0 q 1 q : (60) De ning dq := q q 1 2K +1 q and looking at (53) and (51), we have F 0 q 1 dq (F ( q 1 + dq)) n [ jkj1 Kq k 6=0 k ; 0 q 1 jkj 1 + k dq! Fq ; (61) where we have used the inequalities q 1 + dq q ; 0 q 1 jkj 1 + k dq q jkj 1 : Moreover, applying estimate (55) for 0 q 1 and (50), 16Lq 1Lqa2 0 q 1 q 215L2M2 l4 26LK +1 " l 2( +2)(q 1) 4 2 (q 1)K +1 q dq : This bound and the inclusion (61) imply (60). Hence, part (g) of proposition 11 says that (q) is one-to-one on Gq = (q) 1 (Fq). This gives (5q), and the induction is completed. 4 KAM THEOREM AND NEARLY-INVARIANT TORI 38 C. Convergence of the di eomorphisms. We now see the convergence of the successive maps (q) : Gq ! Fq : Applying part (g) of proposition 11 again, we obtain for q 1 the estimates (q) (q 1) Gq a 0 q 1 ; (q) 1 (q 1) 1 Fq 2Lq 1a 0 q 1 q 1 : (62) Therefore, by the bound (55) on 0 q 1, the sequences (q) and (q) 1 converge respectively to maps which we name and , de ned on the sets G := \ q 0Gq; F := \ q 0Fq = (F ) n [ k2Zn k 6=0 k; jkj 1 ! : (63) Note that the compacity of F has been used to establish the second equality. From (62) we deduce: (q) G Xs q a 0 s ; (q) 1 F X s q 2Lsa 0 s s : (64) Note also that, for every q, Gq Gq 1 4Lqa 0 q 1 q ; Fq Fq 1 16Lq 1Lqa2 0 q 1 q : Iterating these inclusions we deduce the following two ones: G Gq X s q 4Ls+1a 0 s s+1 ; F Fq X s q 16LsLs+1a2 0 s s+1 ; (65) to be used below. We are now going to see that is one-to-one on G and that (G ) = F . Given I 2 G , we have (q)(I) 2 Fq for every q. Hence (I) 2 F , and we deduce that (G ) F . Analogously, we have (F ) G . Moreover, we prove that ( (I)) = I for every I 2 G . Indeed, for every q, j ( (I)) Ij ( (I)) (q) 1 ( (I)) + (q) 1 ( (I)) (q) 1 (q)(I) (q) 1 F + 2Lq q (q) G : (66) For the estimate of the second term, we have used the following two facts: on one hand, part (b) of lemma 8 gives the bound @ (q) @I (I 0) v q 2Lq jvj 8v 2 Rn; 8I 0 2 Gq ; on the other hand, by (64{65) the segment joining (q)(I) and (I) is contained in Fq. Then, since the bound obtained in (66) tends to zero for q ! 1, we deduce that ( (I)) = I, and therefore is one-to-one. A symmetric argument allows to prove 4 KAM THEOREM AND NEARLY-INVARIANT TORI 39 that ( (J)) = J for every J 2 F . This implies that (G ) F . Thus, the map is one-to-one and (G ) = F . We also see, from proposition 11, that !(q) !(q 1) Gq ; (q 1) 2 q 1; h(q) h(q 1) Gq; (q 1) 2 q 1: This implies that the sequences !(q) and h(q) converge to some continuous maps ! and h , respectively. Note also that, for I 2 G , (I) = ! (I) ! n(I) ; a h (I)! : (67) From (2q) we deduce, for every q 0, the following bound to be used later: ! !(q) G X s q s 27MK +1 " l 2( +2)q : (68) D. Convergence of the canonical transformations. Next we estimate how near to the identity map the transformations (q) are. Applying part (d) of proposition 11 and using (1q 1), (48) and (50), one deduces that, for every q 1, (q) id Gq ; (q);cq 2AqK q lq 1 0 q 1 "q 1 25K q l 8" 1 2(2 +2)(q 1) = 28K " 2l 1 2( +2)(q 1) (q) 2 8 2q 1 ; (69) where we write (q) = (q 1) (q) 2 =2. Then, applying property (15) of section 2.2, D (q) Id Gq; (q);cq 2 (q) 2 (q) id Gq ; (q);cq 1 4 2q 1 : Let x, y such that the segment joining them is contained D (q)(Gq). The mean value theorem gives the bound: (q)(x) (q)(y) cq D (q) Gq ; (q);cq jx yjcq : By (69), (q)(x) x cq (q) 2 ; (q)(y) y cq (q) 2 : Then, since (q) + (q) = (q 1), it turns out that the segment joining (q)(x) and (q)(y) is contained in D (q 1)(Gq 1). Therefore, (q 1) (q)(x) (q 1) (q)(y) cq 1 D (q 1) Gq 1; (q 1);cq 1 (q)(x) (q)(y) cq 1 2 +1 D (q 1) Gq 1; (q 1);cq 1 (q)(x) (q)(y) cq ; 4 KAM THEOREM AND NEARLY-INVARIANT TORI 40 where we have used that cq 1=cq = 2 +1 . Iterating this argument and putting the successive bounds obtained together, we arrive at the estimate (q)(x) (q)(y) c1 2( +1 )(q 1) D (1) G1; (1);c1 D (2) G2; (2);c2 D (q) Gq ; (q);cq jx yjcq 2( +1 )(q 1) 1 + 1 4 1 + 1 4 2 1 + 1 4 2q 1 jx yjcq 2( +1 )(q 1) e1=2 jx yjcq 2( +1 )(q 1) 2 jx yjcq ; (70) which holds for q 1, and for every x, y such that the segment joining them is contained in D (q)(Gq). Now, given q 2 and x 2 D (q)(Gq), we put y = (q)(x) and apply (70) with q 1 instead of q. We obtain: (q)(x) (q 1)(x) c1 = (q 1) (q)(x) (q 1)(x) c1 2( +1 )(q 2) 2 (q)(x) x cq 1 2( +1 )(q 1) 2 (q)(x) x cq 29K " 2l 1 2(1+ )(q 1) ;(71) where (69) has been used. This estimate holds for q 2, but one readily sees from (69) that it is also true for q = 1 (we put (0) = id). Clearly, estimate (71) implies that the sequence of transformations (q) converges to a map : D( 1 4 ;0) (G ) =W 1 4 (Tn) G ! D (G) and we deduce, for every q 0, the estimate (q) G ;( 1 4 ;0);c1 210K " 2l 1 2(1+ )q : (72) Moreover, by carrying to the limit the equation H (q) = h(q) + R(q), we see that H = h (I) on D( 1 4 ;0) (G ). E. Stability estimates. Next we see that, for q ! 1, the motions associated to the transformed Hamiltonian H(q) = h(q) + R(q) and the quasiperiodic motions of h(q) become closer and closer. Let us denote x(q)(t) = (q)(t); I(q)(t) the trajectory of H(q) corresponding to a given initial condition x(q)(0) = x 0 = ( 0; I 0 ) 2 Tn Gq, and let x̂(q)(t) := ̂(q)(t); I 0 = 0 + !(q) (I 0 ) t; I 0 the corresponding trajectory of the integrable part h(q). It is clear that x̂(q)(t) is de ned for all t 2 R. Like in lemma 5, let us denoteTq := inf nt > 0 : I(q)(t) I 0 > (q+1) 2 or (q)(t) ̂(q)(t) 1 > (q+1) 1 o : (73) Clearly, x(q)(t) is de ned and belongs to D (q)(Gq) for 0 t Tq (we remark that our use of (q+1) instead of (q) is just due to the fact that we shall take some advantage of the \cq+1-norm"). From the Hamiltonian equations associated to H(q) _ I(q)(t) = @R(q) @ x(q)(t) ; _ (q)(t) = !(q) I(q)(t) + @R(q) @I x(q)(t) ; 4 KAM THEOREM AND NEARLY-INVARIANT TORI 41 we get the bounds: _ I(q)(t) @R(q) @ Gq; (q) "q ; (74) _ (q)(t) !(q) (I 0 ) 1 Mq I(q)(t) I 0 + @R(q) @I Gq; (q);1 2M I(q)(t) I 0 + "q cq+1 2M (q+1) 2 + "q cq+1 3M (q+1) 2 : (75) In the second bound, we have used the inequality "q cq+1 M (q+1) 2 ; (76) which comes from the bounds (1q) and (48{50). Thus, since one of the inequalities de ning (73) is an equality for t = Tq, we have (q+1) 2 = I(q) (Tq) I 0 Tq"q or (q+1) 1 = (q) (Tq) ̂(q) (Tq) 1 Tq 3M (q+1) 2 : (77) Therefore, Tq min0@ (q+1) 2"q ; (q+1) 1 3M (q+1) 2 1A T 0 q := 1 3Mcq+1 ; (78) where the inequality (76) has been used again. This implies: x(q)(t) x̂(q)(t) cq+1 (q+1) 2 for jtj T 0 q : (79) Since H(q) = H (q) and (q) is canonical, it turns out that (q) x(q)(t) is a trajectory of H, de ned for jtj T 0 q. For large values of q, this trajectory remains near the torus (q) (Tn fI 0g). Note that T 0 q tends to in nity for q !1. F. Invariant tori. Assume now that x 0 2 Tn G , and write x (t) = ( 0 + ! (I 0 ) t; I 0 ), t 2 R. Note that x̂(q)(t) x (t) cq+1 cq+1 !(q) (I 0 ) ! (I 0 ) 1 jtj cq+1 !(q) ! G ;1 T 00 q (q+1) 2 for jtj T 00 q := (q+1) 1 j!(q) ! jG ;1 ; which also tends to in nity, by (68). Then, x(q)(t) x (t) cq+1 2 (q+1) 2 for jtj T 000 q := min T 0 q; T 00 q . 4 KAM THEOREM AND NEARLY-INVARIANT TORI 42 Next, we see that the trajectory (q) x(q)(t) is very close to (x (t)) for large values of q. Indeed, for jtj T 000 q , (q) x(q)(t) (x (t)) c1 (q) x(q)(t) (q) (x (t)) c1 + (q) (x (t)) (x (t)) c1 2( +1 )(q 1) 4 (q+1) 2 + (q) G ;( 1 4 ;0);c1 = 4c1 (q+1) 1 + (q) G ;( 1 4 ;0);c1 ; (80) where we have applied (70). The bound obtained in (80) tends to zero, by (72). Then, for every xed t, we see that (q) x(q)(t) exists for q large enough, and its limit is (x (t)). This fact and the continuity of the ow of H imply that (x (t)) is also a trajectory of H, which is de ned for all t 2 R. This holds for every initial condition x 0 = ( 0; I 0 ) 2 Tn G . Hence (Tn fI 0g) is an invariant torus of H, with frequency vector ! (I 0 ). Moreover, the energy on the torus is H ( ( 0; I 0 )) = h (I 0 ). The preserved invariant tori are thus parametrized by the transformed actions I 0 2 G . We are now going to parametrize the preserved tori by their original actions. First, let us see that b G F (the Diophantine set F has been introduced in (63)). Indeed, using part (b) of lemma 8 and the fact that = =L, we see that G 2 F . On the other hand, given I 2 b G, we deduce from the Diophantine condition ful lled by !(I) that k (I) + kn = j!n(I)j jkj 1 jkj 1 and therefore (I) = 2 k; jkj 1 for every k 6= 0 (we point out that this estimate motivated our choice = =L). Hence b G F and, since : G ! F is one-to-one, we can take for the set of invariant tori a parameter I0 2 b G (note that some of the invariant tori are thus neglected). We de ne, for ( 0; I0) 2 W 1 4 (Tn) b G, T ( 0; I0) = ( 0; I 0 ) ; where I 0 = ( ) 1 ( (I0)) 2 G . One then obtains part (a): the set T (Tn fI0g) is an invariant torus of H, with frequency ! (I 0 ) and energy h (I 0 ). Since (I 0 ) = (I0), we deduce from (67) that ! (I 0 ) is colinear to !(I0) and that h (I 0 ) = h(I0). For (b), let us write, for ( 0; I 0 ) 2 W 1 4 (Tn) G , ( 0; I 0 ) = 0 + ( 0; I 0 ) ; I 0 + I ( 0; I 0 ) : Then, for ( 0; I0) 2 W 1 4 (Tn) b G, T ( 0; I0) = ( 0; I 0 ) ; TI( 0; I0) = I ( 0; I 0 ) + I0 I 0 : Using (72) and (49), we get the following estimates: ( 0; I 0 ) 1 1 c1 j idjG ;( 1 4 ;0);c1 214MK2 +1 " 2l2 2 ; j I ( 0; I 0 )j j idjG ;( 1 4 ;0);c1 210K " 2l 1 : 4 KAM THEOREM AND NEARLY-INVARIANT TORI 43 By (64) and (55), jI 0 I0j ( ) 1 1 F X s 0 2Lsa 0 s s 8La X s 0 0 s 27LM l2 27LK +1 " l = 214L2MK +1 " l3 : By putting these bounds together, applying the inequalities ̂ 1 and K 2=̂ and writing the estimate in terms of instead of , we get part (b). G. Estimate of the measure. Finally, we carry out the estimate of part (c). Writing b G = ( ) 1 b G , the invariant tori found ll the set T Tn b G = Tn b G . Since the transformations (q) are canonical, mes h (q) Tn b G i = mes Tn b G = (2 )n mes b G : Using estimate (72) and the compacity of Tn b G , we get the inequality mes h Tn b G i (2 )n mes b G : Then, to estimate the measure of the complement of the invariant set, it su ces to bound the measure of G n b G . First we construct an auxiliary set, included in b G , such that the estimates become easier on it. Let ~ = 64LM l2 ; ~ q = 1 1 2 q ~ ; for q 0, and note that ~ . We de ne the sets ~ Fq = F ~ q n [ jkj1 Kq k 6=0 k; ~ q jkj 1 ! ; ~ Gq = (q) 1 ~ Fq ; and ~ F = \ q 0 ~ Fq = F ~ n [ k2Zn k 6=0 k; ~ jkj 1 ! ; ~ G = \ q 0 ~ Gq: From the fact that (q) ~ Gq = ~ Fq for every q, we deduce ~ G = ~ F . Let us check that b G ~ F . Indeed, from part (a) of lemma 8, we get that G 2 F ~ . Moreover, given J = (I) 2 ~ F , for every k 2 Zn with k 6= 0 we have k J + kn ~ jkj 1 and hence jk !(I)j ~ j!n(I)j jkj 1 l ~ jkj 1 jkj 1 : 4 KAM THEOREM AND NEARLY-INVARIANT TORI 44 For k = 0, it is clear that the Diophantine estimate is also ful lled since j!n(I)j l . Hence b G ~ F and therefore b G ~ G . Then, mes G n b G 1 X q=1mes ~ Gq 1 n ~ Gq : For q 1 we have the estimate mes ~ Gq 1 n ~ Gq 22n 7l2Ln 2 n 1M mes ~ Fq 1 n ~ Fq a 0 q 1 : It has been used that (q 1) ~ Gq 1 = ~ Fq 1 and that (q 1) ~ Gq ~ Fq a 0 q 1. This inclusion comes from part (g) of proposition 11. Another point we have used is the bound det @ (q 1) @I (I)! n 1 q 1 a Ln 2 q 1 n 1M 22n 7l2Ln 2 ; given by part (c) of lemma 8. In accordance to the notation of lemma 12, we have ~ Fq 1 = F ~ q 1; ~ q 1; Kq 1 , ~ Fq = F ~ q; ~ q; Kq . Applying that lemma, mes ~ Fq 1 n ~ Fq D ~ q ~ q 1 + 2(diamF )n 10BBB@ X jkj1 Kq 1 k 6=0 ~ q ~ q 1 jkj 1 k + X Kq 1 n 1 in checking that the three series taking part in the right hand side of (81) are convergent. Indeed, for the rst series, X k2Zn k 6=0 1 jkj 1 k X k2Zn 1 k 6=0 X kn2Z pn k 1 + jknj k 1 pn 2n 1 1 X j=1 X kn2Z jn 3 (j + jknj) ; where we have used that the number of vectors k 2 Zn 1 with k 1 = j 1 can be bounded by 2n 1jn 2. The series indexed by kn can be bounded by comparing it with an integral:X kn2Z 1 (j + jknj) 1 j + 2 Z 1 0 dx (j + x) = 1 j + 2 ( 1)j 1 + 1 1 1 j 1 : 4 KAM THEOREM AND NEARLY-INVARIANT TORI 45 We have used that > 1 since n 2. Then, X k2Zn k 6=0 1 jkj 1 k pn 2n 1( + 1) 1 1 X j=1 1 j n+2 ; which converges by the condition > n 1. It is easy to check that the second and the third series of (81) converge, using the bound a 0 q 1 l3 2 2 +20L3M2K +1 2( +2)(q 1) ; which comes from estimates (55) and (50). Writing all bounds in terms of instead of ~ or , we get from (81) a bound of the type mes G n b G C 0 ; where C 0 is a constant depending on n, , diamF , D, K, M , L, l, . We then get estimate (c), with C = (2 )nC 0. This constant may be explicited if desired. 2 Remarks 1. All of the sequences introduced at the beginning of the proof have linear convergence. Of course, alternative choices for those sequences are possible provided the restrictions imposed by proposition 11 are ful lled. 2. The reason of our choice of Kq and (q) 1 will be transparent in the next section, where we see that, for a small , the remainders decrease in an almost quadratic way. 3. The estimates of part (b) on the deformation of the perturbed invariant tori from the unperturbed ones are essentially the same of [22]. 4.5 Fast convergence and nearly-invariant tori We have established in the previous section the linear convergence to zero of the sizes of the successive remainders. This kind of convergence is enough (and very suitable) for the proof of the existence of invariant tori. But in the current section we show that the remainders actually decrease much faster, and we take advantage on this fact. Indeed, by stopping the iterative process at an appropriate step, we deduce that the domain obtained is full of nearly-invariant tori, i.e. Nekhoroshev-like estimates hold for the trajectories starting on these tori. One may look our theorem F on e ective stability as an attempt to make KAM and Nekhoroshev theorems closer. Indeed, we provide Nekhoroshev-like estimates which are very near, from a quantitative point of view, to KAM theorem. From the practical point of view, this result is more signi cative than KAM theorem. Indeed, if the coordinates of a given unperturbed invariant torus are known just approximately, up to a precision r, it is not possible to decide whether the frequency associated to this torus is Diophantine 4 KAM THEOREM AND NEARLY-INVARIANT TORI 46 or not, and therefore one cannot deduce that this torus survives in the perturbation. In fact, in checking the Diophantine condition it has no sense to go farther than a nite order K(r) (tending to in nity as r ! 0). However, this nite test is enough to ensure that the torus is still included in the domain at an appropiate step of the iterative process and that this torus survives in the perturbation at least in the form of a nearly-invariant torus: a trajectory starting on this torus remains near to it up to a stability time which is exponentially long in 1=r. This result is similar to the one of [18], which does not however worry about optimal estimates. Moreover, in that paper the stability estimates are expressed in terms of the stability time, previously xed, instead of r. Another related result is obtained in [25] and [19], where it is shown that KAM tori are \sticky": estimates are given for the time to move away from a xed KAM torus. The estimates are exponential in [25] and \superexponential" (but only for quasiconvex systems) in [19]. This result requires the transformation to normal form to hold in a full neighborhood of the given KAM torus, which is achieved in the quoted papers by carrying out the Kolmogorov's approach to KAM theorem instead of the Arnold's one. But our result seems in practice more useful since the existence of a KAM torus is not used for the estimates. We notice that theorem E gives a large family of invariant tori if is small. But for large values of one cannot guarantee the preservation of any invariant tori even if (47) is satis ed, since the set b G may be empty. Actually, there is a maximum value 0 such that b G is empty for > 0 (in the case n = 2, the set b G 0 corresponds to the noble frequencies). Nevertheless, in theorem F the nearly-invariant tori are parametrized by a set G(r) (de ned below) containing b G properly. Then, for some interval of values > 0 we may still ensure the existence of nearly-invariant tori. Theorem F Consider notations and hypothesis as in theorem E and assume also that " 2 l2̂2 +2 22 +17L2M 2; (82) where we de ne := min s 0 e(2 21 ) 2(1 )s 2(2 +1)s > 0. Let 0 < r r0 := ̂ +1 2 +2M given, and write b G(r) = b G(r) := (I 2 G 2 : jk !(I)j jkj 1 8k 2 Zn; 0 < jkj1 K(r)) ; G(r) = G(r) := Ur b G(r) ; where K(r) := 2̂ r0 r 1=( +1+ ) : Then, there exist an analytic map A(r) : G(r) ! G and a real analytic canonical transformation (r) : D 1 4 ; l r 2 +5L A(r) G(r) ! D (G) such that, writing T (r) = 4 KAM THEOREM AND NEARLY-INVARIANT TORI 47 (r) id A(r) , any torus T (r) (Tn fI0g), with I0 2 G(r) has the property that, for every trajectory ( (t); I(t)) = (r) ̂(t); Î(t) of H with ( (0); I(0)) belonging to this torus, one has: Î(t) A(r)(I0) s 2" M exp( 12 r0 r (1 )=( +1+ )) ; (83) ̂(t) ̂(0) + (r)(I0)t 1 1 4 r r0 =( +1+ ) ; (84) for jtj 1 51pM " r r0 =(2 +2) exp(1 2 r0 r (1 )=( +1+ )) ; (85) where the vector (r)(I0) is colinear to !(I0). Moreover, if E denotes the energy of the trajectory ( (t); I(t)), jE h(I0)j 22n+1" ̂ exp( r0 r (1 )=( +1+ )) : Proof We go again into the iterative process of the proof of theorem E. Improving the argument given there for the linear estimate (1q), we are going to see that the successive sizes of the remainders admit an almost quadratic estimate. We rst notice that, choosing K as in the proof of theorem E, and taking into account the de nition = =L and the inequalities ̂ 2=K and ̂ 1=16, we deduce from (82) the inequality " 3 l2 1 2 220MK2 +1 : (86) We next prove that, for every q 0, "q 32" 1 e 2(1 )q : (87) Indeed, this is true for q = 0 by (52). Given q 1 and assuming the estimate true for q 1, we are going to establish it for q. From (57) and the inequality K̂ 1, we deduce Kq (q) 1 K 1 8 2(1 )(q 1) ln 2 + 2(1 )(q 1); and therefore e Kq (q) 1 1 2 e 2(1 )(q 1) : Moreover, applying the inequalities (48) and (86), the de nition of and the hypothesis that (87) holds for "q 1, 14AqK q lq 1 0 q 1 (q) 2 "q 1 28K q l 64MK +1 q l 32" 1 e 2(1 )(q 1) 1 2 e (21 1) 2(1 )(q 1) : (88) 4 KAM THEOREM AND NEARLY-INVARIANT TORI 48 Then, from (56) we deduce: "q 12 e 2(1 )(q 1) + 1 2 e (21 1) 2(1 )(q 1) "q 1 e (21 1) 2(1 )(q 1) "q 1 ; which gives estimate (87) for "q. The stability time given in (79) can also be improved. Recall that we denote x(q)(t) = (q)(t); I(q)(t) the trajectory of H(q) such that x(q)(0) = ( 0; I 0 ) 2 Tn Gq. We deduce from the inequality (74) that, for 0 t Tq, I(q)(t) I 0 Tq"q (Tq was introduced in (73)). Then, we may replace (75) by the next alternative bound: _ (q)(t) !(q) (I 0 ) 1 2MTq"q + "q cq+1 5MTq"q; where (78) has been used. Then, the inequality (77) becomes (q+1) 1 T 2 q 5M"q; and we obtain: Tq min0B@ (q+1) 2"q ; vuut (q+1) 1 5M"q1CA = ~ Tq := vuut (q+1) 1 5M"q ; where we have used (76) to see that the minimum is given by the second term. Hence, for every initial condition ( 0; I 0 ) 2 Tn Gq, the corresponding trajectory of H(q) satis es: I(q)(t) I 0 ~ Tq"q = vuut (q+1) 1 "q 5M ; (q)(t) 0 + !(q) (I 0 ) t 1 (q+1) 1 ; (89) for jtj ~ Tq : (90) This stability time is much better than the one of (79), because of the quadratic behaviour of "q. This estimate says that Tn fI 0g is a nearly-invariant torus of H(q) for every I 0 2 Gq, since every trajectory starting at a point on this torus remains near a quasiperiodic motion with frequency vector !(q) (I 0 ) for a long time. Then, the torus (q) (Tn fI 0g) is also nearly-invariant for the ow of H. We are now going to choose q = q(r) 1, as large as possible, such that Fq G(r) . Given I0 2 G(r), there exists I 0 0 2 b G(r) such that jI 0 0 I0j r. We get I0 2 G 2 r and hence (I0) 2 F r 2L (91) by part (b) of lemma 8. On the other hand, for 0 < jkj1 K(r), jk !(I0)j jkj 1 jkj Mr 4 KAM THEOREM AND NEARLY-INVARIANT TORI 49 and therefore k (I0) + kn 1 j!n(I0)j jkj 1 jkj Mr! jkj 1 jkj Mr L : (92) We deduce from (91{92) that J0 2 Fq provided the following inequalities are ful lled: K(r) Kq ; r L M q K +1 q : Noting that q = =2 q and reminding that K 2=̂, we see that it su ces to choose q such that the inequality 2( +1+ )(q 1) r0 r (93) holds. Hence, we choose q 1 as the maximum integer such that this happens. We then have also 2( +1+ )q r0 r : (94) With this choice of q, we have G(r) Fq. We take A(r) := (q) 1 , and note that A(r) G(r) Gq. The transformation (r) := (q) is de ned on D (q)(Gq), and we have the inequalities (q) 1 1=4 and (q) 2 = l 32MK +1 2( +1)q l̂ +1 2 +6M 2( +1)q = lr0 24L 2( +1)q l r 2 +5L ; where we have used the inequality (93). This gives, in function of r, the complex domain where (r) is de ned. For ( 0; I0) 2 Tn G(r), we put T (r)( 0; I0) = (q) ( 0; I 0 ) ; where I 0 = A(r)(I0) 2 Gq. If ( (t); I(t)) is a trajectory of H starting on the torus T (r) (Tn fI0g), then ̂(t); Î(t) is a trajectory of H(q), with Î(0) = I 0 . We can thus apply the stability estimate of (89{90). Using the inequalities (87), (48) and (93{94), we get the bounds "q 32" 1 exp( r0 r (1 )=( +1+ )) ; (q+1) 1 1 4 2 q 1 4 r r0 =( +1+ ) ; (q+1) 1 1 16 2 (q 1) 1 16 r r0 =( +1+ ) : Including these bounds in (89{90), we get (83{85). Note that we put (r)(I0) = !(q) (I 0 ), which is colinear to !(I0) since (q) (I 0 ) = (I0). Finally, the energy of the trajectory ( (t); I(t)) is E = H( (0); I(0)) = H(q) ̂(0); I 0 = h(I0) +R(q) ̂(0); I 0 4 KAM THEOREM AND NEARLY-INVARIANT TORI 50 since h(q) (I 0 ) = h(I0). Then, jE h(I0)j R(q) ̂(0); I 0 R(q) 0 Gq + @R(q) @ Gq n q + n"q : To get a quadratic estimate for q, we proceed as in (59) but we now use (88), (48) and (87): q 7AqK q cqlq 1 0 q 1 " 2 q 1 (q) 1 "q 1 2 1 2 e (21 1) 2(1 )(q 1) 2" e 2(1 )q : Then,jE h(I0)j 2 + 32 n 1 ! " e 2(1 )q 22n+1" ̂ exp( r0 r (1 )=( +1+ )) : 2 Remarks 1. To give an idea of how this theorem should be applied in practice, assume that I0 is the action of a given unperturbed torus, for which we just know an approximation I 0 0, with jI 0 0 I0j r. If I 0 0 2 b G(r) , i.e. the frequency ratios of ! (I 0 0) satisfy the Diophantine condition (2) up to order K(r), then the invariant torus corresponding to the action I0 survives as a nearly-invariant torus. 2. We have omitted statements like parts (b) and (c) of theorem E because they would be exactly the same, with T (r) instead of T . 3. If the size " of the perturbation is xed, we could take r small and the stability estimate given in (83{85) is then much better than the one provided by Nekhoroshev theorem. This is due to the fact that the estimate has been expressed in the transformed coordinates ̂; Î provided by the canonical transformation (r). These coordinates are better because the nearly-invariant tori are given by equations Î = const. In Nekhoroshev theorem, the stability estimate includes the coordinate change because it is expressed in the original coordinates. 4. The comparison between the estimates (83{84) for action and angular variables shows that the separation from a given torus remains much smaller than the separation from a linear ow inside this torus. 5. The stability exponent given in (85) is larger (as near to 1=( + 1) as wanted) if we choose the parameter near to zero. 6. Roughly speaking, the set G(r) parametrizing the nearly-invariant tori has a very large boundary for r small. However, the \area" of this boundary is nite, which means that the set G(r) is not as strange as the Cantorian set b G provided by KAM theorem. An alternative way to express this fact is used in [18]: the set of nearly-invariant tori contains balls of a suitable radius, and hence it contains inner points. 5 APPENDIX: PROOFS OF THE TECHNICAL LEMMAS 51 We can also get \superexponential" stability estimates, like in [19], by means of an alternative approach. However, we need to assume that the unperturbed Hamiltonian h is quasiconvex. Applying the iterative process of KAM theorem, our starting Hamiltonian H is transformed after q steps into H(q) = h(q)+R(q). If h is m-quasiconvex with m > 0 and we assume " = O(m), then h(q) is also quasiconvex and Nekhoroshev theorem may be applied to H(q). In this way, we can obtain for every trajectory (q)(t); I(q)(t) of H(q), with initial condition in Tn Gq, a stability estimate of the type I(q)(t) I(q)(0) R for jtj T , with R " 1=2n q ; T exp8<: 1 "q!1=2n9=; : Choosing q = q(r) as in (93{94), we get R " exp 1r c 1=2n ; T exp( 1" exp 1r c 1=2n) ; where we have put c = (1 )=( +1+ ). The stability radius R and the stability time T substitute the ones obtained in (83) and (85), respectively. 5 Appendix: proofs of the technical lemmas Proof of lemma 2 Given ( 0; I0) 2 D t (G), let ( (s); I(s)) = s ( 0; I0). First, we prove that j (s) 0j1 t @W @I G; ;1 ; jI(s) I0j t @W @ G; ;1 ; (95) for 0 s t. Let s0 be the supremum of the s 0 satisfying both inequalities in (95). Clearly s0 > 0, and one of these inequalities is an equality for s = s0. On the other hand, we have ( (s); I(s)) 2 D (G) for 0 s s0. From the mean value theorem, j (s0) 0j1 s0 sup 0 s s0 @W @I ( (s); I(s)) 1 s0 @W @I G; ;1 ; (96) jI(s0) I0j s0 sup 0 s s0 @W @ ( (s); I(s)) s0 @W @ G; s0 @W @ G; ;1 : (97) Thus, s0 t and (95) is true. This implies that t is de ned in D t (G) and that the bound (a) holds. We can deduce the inclusion (b) from the fact that t is the ow at time t of W . To see (c), note that f t is de ned in D t (G). Since W is analytic, f t is also analytic in t, and hence the Lie series expansion (4) for rm(f;W; t) holds. Given 5 APPENDIX: PROOFS OF THE TECHNICAL LEMMAS 52 l m+ 1, let = =(l m). For j = m+ 1; : : : ; l, we have L j Wf G; (j m)t 2c D L j 1 W f G; (j m)t ; c kDWkG; ;c 2 t̂c L j 1 W f G; (j 1 m)t kDWkG; ;c : Thus, L l Wf G; t 2 kDWkG; ;c t̂c !l m kLm WfkG; (l m)! 2e kDWkG; ;c t̂c !l m kLm WfkG; ; where we have used that kk ek k! for k 1. In this way, the bound that we obtain for krm(f;W; t)kG; t is 1 X l=m tl l! L l Wf G; t 24 1 X l=m (l m)! l! 2e kDWkG; ;c ̂c !l m35 tm kLm WfkG; ; this series being convergent for kDWkG; ;c < ̂c=2e. 2 Remark The bounds (96{97) are based on the special structure of Hamiltonian equations. Our choice of the 1-norm for the angular variables was motivated by (96). Concerning the action variables, the best choice would be, according to (97), the 1-norm, but our use of Euclidean geometry in the geometric parts of Nekhoroshev and KAM theorems made us choose the 2-norm. Proof of lemma 5 Assume rst K 1 1. Let = =3. From (18), we see that A = 1 + 2M 2 1 1 + 1 K 1 2: Then, theorem B provides a canonical transformation : D 2 3 (G) ! D (G) such that H = h+ Z +R , with Z = Z (I), and kDR kG; 2 3 ;c 3 e K 1 6 kDRkG; ;c : Let us denote = 8 kDRkG; ;c : By estimate (c) of theorem B and the second condition of (23), j idjG; 2 3 ;c 2 15K 1 2 15 : Moreover, by the inclusion (d) of theorem B, we have D 2 3 (G) D 2 (G) Tn G and therefore we can write ( (0); I(0)) = ( 0; I 0 ) and, since preserves real domains, 5 APPENDIX: PROOFS OF THE TECHNICAL LEMMAS 53 ( 0; I 0 ) 2 Tn V 2 15 (G). Let ( (t); I (t)) be the trajectory of H with ( (0); I (0)) = ( 0; I 0 ). Let T = inf ft > 0 : jI (t) I (0)j > g (the procedure for negative times is exactly the same). For 0 t T , we obtain ( (t); I (t)) 2 D 2 3 (G). Since takes the motions of H into motions of H, we get that ( (t); I(t)) = ( (t); I (t)) is also de ned for 0 t T , and obtain the estimate jI(t) I(0)j jI(t) I (t)j+ jI (t) I (0)j+ jI (0) I(0)j 3 : Next, we proceed to obtain a lower bound for T . Let I = I (T ) I (0). Clearly, j I j = : On the other hand, by using the form of the Hamiltonian equations and the fact that the normal form Z only depends on the action variables, we obtain j I j T @R @ G; 2 3 T kDR kG; 2 3 ;c T 3 e K 1 6 kDRkG; ;c ; and it follows that T 3 kDRkG; ;c eK 1 6 2 eK 1 6 : For K 1 < 1, one obtains, by working in original coordinates, jI(t) I(0)j 24 kDRkG; ;c for jtj 24 ; and it is then easy to see that this stability time is longer than the one proclaimed in (24). 2 Proof of lemma 6 First we assume K 1 M=m. Let = =3. Like in the proof of lemma 5, we have A 2. Note that kDZkG; ;c + kDRkG; ;c m 2 2 350 1 2 122K 1 : (98) Then, theorem B provides a canonical transformation : D 2 3 (G) ! D (G) such that H = h+ Z +R , with Z 2 R(M; K), and kDZ kG; 2 3 ;c + kDR kG; 2 3 ;c kDZkG; ;c + 2 kDRkG; ;c ; kDR kG; 2 3 ;c 3 e K 1 6 kDRkG; ;c : Moreover, by estimate (c) of theorem B and the inequality (98), j idjG; 2 3 ;c 8 kDRkG; ;c 2 15K 1 m 2 15M : Like in lemma 5, we can write ( (0); I(0)) = ( 0; I 0 ), with ( 0; I 0 ) 2 Tn Vm 2 15M (G). Let ( (t); I (t)) be the trajectory of H with ( (0); I (0)) = ( 0; I 0 ). Let T = inf t > 0 : jI (t) I (0)j > 2 2 : 5 APPENDIX: PROOFS OF THE TECHNICAL LEMMAS 54 For 0 t T , we have ( (t); I (t)) 2 D 2 3 (G). We then obtain the estimate jI(t) I(0)j jI(t) I (t)j+ jI (t) I (0)j+ jI (0) I(0)j m 2 15M + 2 2 + m 2 15M 2: We introduce the notations = (T ) (0); I = I (T ) I (0); and, for a function f ( ; I), f = f ( (T ); I (T )) f ( (0); I (0)) : The de nition of T clearly implies that j I j = 2=2. We notice that, since Z is in normal form with respect toM, it does not contribute to I in any direction lying inM?. More precisely, let P denote the orthogonal projection onto the one-dimensional space h M!(I(0))i. By the speci c form of the Hamiltonian equations, we have P I = Z T 0 P @R @ ( (t); I (t))! dt and it follows that the M!(I(0))-component of the vector I is small up to an exponentially long time: jP I j T @R @ G; 2 3 T kDR kG; 2 3 ;c T 3 e K 1 6 kDRkG; ;c : (99) To bound the whole vector I we use the quasiconvexity condition on h. By Taylor formula, one has h = !(I 0) I + Z 1 0 (1 s) @2h @I2 (I 0 + s I ) ( I ; I ) ds: (100) We notice that, since preserves real domains, I 0 + s I 2 U 2(G) for 0 s 1. For a xed s, we write I = Ps I +Qs I , where Ps and Qs denote the orthogonal projections onto h! (I 0 + s I )i and h! (I 0 + s I )i?, respectively. Thus, applying the quasiconvexity condition at the point I 0 + s I (and this is the only time that we use it), we obtain @2h @I2 (I 0 + s I ) (Qs I ; Qs I ) m jQs I j 2 : We deduce that @2h @I2 (I 0 + s I ) ( I ; I ) m jQs I j2 2M jPs I j jQs I j M jPs I j2 = m j I j2 (m jPs I j+ 2M jQs I j+M jPs I j) jPs I j m j I j2 4M j I j jPs I j : 5 APPENDIX: PROOFS OF THE TECHNICAL LEMMAS 55 From formula (100) we obtain m2 j I j2 j hj + j! (I 0 ) I j+ 4M j I j Z 1 0 (1 s) jPs I j ds; and hence m 2 2 8 j hj + j! (I 0 ) I j+ 2M 2 Z 1 0 (1 s) jPs I j ds: (101) Next we bound the terms appearing in the right hand side of (101). To bound jPs I j we use that the vector ! (I 0 + s I ) is near to M!(I(0)). More precisely, we apply the following property: given v; v0 2 Rn, if P(v) and P(v0) denote the orthogonal projections onto the one-dimensional spaces hvi and hv0i, respectively, then for every vector u 2 Rn one has: P(v)u P(v0)u 4 jv v0j jvj juj : First we notice that, from lemma 3 and the fact that M 6= Zn and K 1, we deduce j!(I)j =2 for I 2 U 2(G). Then, applying the property, one has: jPs I P I j 4 j! (I 0 + s I )j j! (I 0 + s I ) M!(I(0))j j I j 4 2 j! (I 0 + s I ) M!(I(0))j : Using that !(G) is -close to M-resonances, we get j! (I 0 + s I ) M!(I(0))j M jI 0 + s I I(0)j+ j!(I(0)) M!(I(0))j m 15 + sM2 2 + m 12 + sM2 2: (102) Thus, jPs I j jP I j+ jPs I P I j jP I j+ 4 2 2 m 12 + sM2 and we obtain, using the rst condition of (26), Z 1 0 (1 s) jPs I j ds 1 2 jP I j+ 4 2 2 m 24 + M12 12 jP I j+ m 2 96M : To bound j! (I 0 ) I j, we put s = 0 in (102): j! (I 0 ) I j j M!(I(0))j jP I j+ j! (I 0 ) M!(I(0))j j I j L jP I j+ m 2 2 24 : Finally, by energy conservation, h = (Z +R ) = Z 1 0 @ (Z +R ) @ ( 0 + s ; I 0 + s I ) + @ (Z +R ) @I ( 0 + s ; I 0 + s I ) I ! ds 5 APPENDIX: PROOFS OF THE TECHNICAL LEMMAS 56 and we deduce j hj @ (Z +R ) @ G; 2 3 ;1 j j1 + @ (Z +R ) @I G; 2 3 j I j + pn c 2 2 ! kD (Z +R )kG; 2 3 ;c 2 +pn 1 kDZkG; ;c + kDRkG; ;c (2 + 1)m 2 2 350 m 2 2 48 : We insert all of these estimates in (101) and obtain: m 2 2 8 1 48 + 1 24 + 1 48 m 2 2 + (L+M 2) jP I j m 2 2 12 + 49L 48 jP I j ; where we used that M 2 L=48 since L. By estimate (99), it follows that T m 2 2 74L kDRkG; ;c eK 1 6 m 2 2 74L kDRkG; ;c emK 1 6M : For K 1 < M=m, one may work in original coordinates. Then, one obtains: jI(t) I(0)j 2 for jtj 2 kDRkG; ;c : It is not hard to check that this stability time is longer than the one proclaimed in (27). 2 Proof of lemma 8 A simple computation gives, for I 2 G and v 2 Rn, @ @I (I) v = @ @I (I) v ; @ n @I (I) v! = 1 !n(I) @! @I (I) v @!n @I (I) v !n(I) !(I)! ; a !(I) v! : We have the bound @ @I (I) v 1 !n(I)2 @! @I (I) v j!n(I)j+ @!n @I (I) v j!(I)j! 1 !n(I)2 @! @I (I) v j!(I)j M j!(I)j l2 jvj (103) and we then get estimate (a): @ @I G s ML l2 2 + (aL)2 2La: (104) To prove parts (b) and (c) we use the isoenergetic condition. For any v 2 h!(I)i?, we have @ @I (I) v = 1 !n(I) @! @I (I) v @!n @I (I) v !n(I) !(I)! = 1 !n(I) @! @I (I) v @!n @I (I) v !n(I) !(I)! j!n(I)j jvj ; (105) 5 APPENDIX: PROOFS OF THE TECHNICAL LEMMAS 57 where we have used the fact that the vector @! @I (I) v @!n @I (I) v !n(I) !(I) has its n-th component vanishing. Moreover, the isoenergetic nondegeneracy has been used to bound the size of this vector from below. Now, consider an orthonormal basis e1; : : : ; en ofRn with the rst n 1 vectors belonging to h!(I)i? and the last one belonging to h!(I)i. Let P = P ; Pn be the (n n)-matrix having these vectors as columns. Let us write: @ @I (I)P = 0@ @ @I (I)P @ @I (I)Pn @ n @I (I)P @ n @I (I)Pn 1A = A b 0 bn ! : (106) It follows directly from (105) that jAvj j!n(I)j jvj 8v 2 Rn 1: Note also that b M j!(I)j =l2 by (103). Moreover, bn = a !(I) en and hence jbnj = a j!(I)j. Then, computing the inverse of the matrix (106) and carrying out a rough bound on its norm, @ @I (I)! 1 0@ A 1 1 bnA 1b 0 1 bn 1A A 1 + 1 jbnj A 1 b + 1 jbnj j!n(I)j + 1 a j!(I)j j!n(I)j M j!(I)j l2 + 1 a j!(I)j L + LM l2 a + 1 al L 1 + 2M l2a : (107) This estimate implies (b), by our condition on a. To obtain the lower bound (c) for the determinant, we take into account the expression (106) again: det @ @I (I)! j!n(I)j!n 1 a j!(I)j n 1a Ln 2 : Finally, we prove (d). For I 2 G and u; v 2 Rn, @2 @I2 (I) (u; v) = @2 @I2 (I) (u; v); @2 n @I2 (I) (u; v)! = 0@ @2! @I2 (I) (u; v) !n(I) @!n @I (I) u @! @I (I) v + @!n @I (I) v @! @I (I) u !n(I)2 @2!n @I2 (I) (u; v) !(I) !n(I)2 + 2 @!n @I (I) u @!n @I (I) v !(I) !n(I)3 ; a @2h @I2 (I) (u; v)1A : We can bound this expression like in (103{104): @2 @I2 (I) (u; v) 1 !n(I)2 @2! @I2 (I) (u; v) j!(I)j+ @! @I (I) u @! @I (I) v ! 5 APPENDIX: PROOFS OF THE TECHNICAL LEMMAS 58 + 2 j!n(I)j3 @!n @I (I) u @!n @I (I) v j!(I)j M 0L l2 + 3M2L l3 ! juj jvj ; and we deduce estimate (d): @2 @I2 G vuut M 0L l2 + 3M2L l3 !2 + (aM)2 M 0 2M + 3Ml !La: 2 Remark If the condition a 2M=l2 is removed, then estimate (b) has to be substituted by (107), which is actually worse if a is taken too small. Proof of lemma 9 Clearly, it su ces to check the result for the vectors v 2 h~ !i? such that jvj = 1. One has j! vj " for these vectors. Writing v = v1 + v2, with v1 2 h!i? and v2 2 h!i, one deduces that jv2j " j!j ; jv1j 1 " j!j : By the hypothesis, jAv + !j jAv1 + !j jAv2j jv1j jAj jv2j 1 " j!j! jAj " j!j 2 jAj " j!j : Then, if we assume that j ~ !j 2 ~ A , we obtain ~ Av + ~ ! jAv + !j ~ A A v j (~ ! !)j 2 jAj " j!j "0 2 ~ A j~ !j " 4M" l "0: In the case j ~ !j > 2 ~ A , the proof is much easier: ~ Av + ~ ! j ~ !j ~ Av ~ A jAj "0 "0: 2 Proof of lemma 10 For a xed J 2 F 4M" ~ m ; we are going to prove that there exists a unique point I 2 G solving ~ (I ) = J . This aim is attained with the help of a modi ed Newton algorithm. Let us consider I(0) = 1(J) 2 G 4" ~ m 5 APPENDIX: PROOFS OF THE TECHNICAL LEMMAS59as a rst approximation, and we have to see that the map(I) = I @ ~@I I(0) ! 1 ~(I) Jhas a unique xed point in G. We rst compute the derivative of this map:@@I (I) = Id @ ~@I I(0) ! 1 @ ~@I (I) = @ ~@I I(0) ! 1 @ ~@I I(0) @ ~@I (I)! ;and therefore@@I (I) ~M 0~m I I(0)if I I(0) 4"~m ;(108)because the segment joining I(0) and I is fully contained in G. Starting at I(0), we considerthe sequence de ned by I(k) = I(k 1) , k 1. We check by induction thatI(k) I(k 1)"2k 1 ~mand I(k) 2 G for every k 1. Indeed, this is true for k = 1. For k > 1, the inductionhypothesis implies that the distance from I(0) to I(k 1) or I(k 2) is less than 2"= ~m. Thesame holds for every point in the segment joining I(k 1) and I(k 2). Then, using (108)and (40) we obtainI(k) I(k 1) = I(k 1)I(k 2)~M 0~m 2"~m I(k 1) I(k 2) 12 I(k 1) I(k 2) :Thus, the sequence I(k) converges, for k ! 1, to a xed point of which we name I .This point satis esI I(0) 2"~m(109)and thereforeI 2 G 2"~m :(110)Then,(I ) 2 F 2"m~m F 2":(111)The point I is the unique xed point of . Indeed, assuming that there is another xedpoint I 6= I , we have ~(I ) = ~(I ) and hence j (I ) (I )j 2". Then, wehave jI I j 2"=m because the whole segment joining (I ) and (I ) is containedin F , by (111). We deduce from (109) that the distance from I(0) to every point in thesegment joining I and I is less or equal than 2"~m + 2"m < 4"~m : Applying (108), we get acontradiction:jI I j = j (I ) (I )j < ~M 0~m 4"~m jI I j jI I j :Given a subset ~F F 4M"~m ; the map ~is one-to-one on ~G = ~1 ~F . Moreover,one has ~G G 2"~m by (110). For the proof of the other inclusion, note that ~G n ~G \ REFERENCES60~F = ;. Then, since ~G ", we have G n ~G \ ~F " = ;, and we deducethat ~G ~F ".Finally, we check that ~11 ~F "=m. For a xed J 2 ~F , let I =1(J),~I = ~1 (J). We have ~I(I) = ~I ~~I ";and therefore the segment joining (I) and ~I is contained in F , since (I) 2 ~FF ". Hence, we obtain ~I I "=m.2Proof of lemma 12 The estimate of part (a) comes from the inclusionF (d; ;K) n F (d0; 0; K 0)((F d) n (F d0))[ [jkj1 Kk 6=0 (F d) \k; 0jkj 1 ! n k; jkj 1 !!![ [Kk6=0 (F d) \ k; 0jkj 1 !!and the fact that, for 00 and k 6= 0,mes [(F d) \ ( (k; 0) n (k; ))] (diamF )n 1 2( 0 )k :(112)Concerning part (b), we rst remark that, for b 0,F (d; ;K) b (F (d+ b)) n [jkj1 Kk6=0k ; jkj 1 + k b! :Then, proceeding like in part (a) and applying (112) again,mes [(F (d; ;K) n (F (d; ;K) b)] Db+ Xjkj1 Kk 6=0 (diamF )n 1 2b :From the fact that the number of integer vectors k 2 Zn n f0g with jkj1 K can beestimated by 2nKn, one gets part (b). It is worth noting that this estimate expresses the\growth" of the boundary of the domain when the resonances are removed.2References[1] V. I. Arnold. Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodicmotions under small perturbations of the Hamiltonian. Russian Math. Surveys 18 (1963),9{36. REFERENCES61[2] G. 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