On the normal behaviour of partially elliptic lower dimensional tori of hamiltonian systems
Abstract
The purpose of this paper is to study the dynamics near a reducible lower dimensional invariant tori of a finite-dimensional autonomous Hamiltonian system with $\ell$ degrees of freedom. We will focus in the case in which the torus has (some) elliptic directions. First, let us assume that the torus is totally elliptic. In this case, it is shown that the diffusion time (the time to move away from the torus) is exponentially big with the initial distance to the torus. The result is valid, in particular, when the torus is of maximal dimension and when it is of dimension 0 (elliptic point). In the maximal dimension case, our results coincide with previous ones. In the zero dimension case, our results improve the existing bounds in the literature. Let us assume now that the torus (of dimension $r$, $0\le r<\ell$) is partially elliptic (let us call $m_e$ to the number of these directions). In this case we show that, given a fixed number of elliptic directions (let us call $m_1\le m_e$ to this number), there exist a Cantor family of invariant tori of dimension $r+m_1$, that generalize the linear oscillations corresponding to these elliptic directions. Moreover, the Lebesgue measure of the complementary of this Cantor set (in the frequency space $\RR^{r+m_1}$) is proven to be exponentially small with the distance to the initial torus. This is a sort of ``Cantorian central manifold'' theorem, in which the central manifold is completely filled up by invariant tori and it is uniquely defined. The proof of these results is based on the construction of suitable normal forms around the initial torus.
Full text
On the Normal Behaviour of Partially Elliptic Lower Dimensional Tori of Hamiltonian Systems Angel Jorba and Jordi Villanueva Octob er 16th, 1996 Departamentde Matematica Aplicada I Universitat Politecnica de Catalunya Diagonal 647, 08028 Barcelona, Spain. E-mails: [email protected] , [email protected] Abstract The purp ose of this pap er is to study the dynamics near a reducible lower dimensional invariant tori of a nite-dimensional autonomous Hamiltonian system with ` degrees of freedom. We will fo cus in the case in which the torus has (some) elliptic directions. First, let us assume that the torus is totally elliptic. In this case, it is shown that the diusion time (the time to moveaway from the torus) is exp onentially big with the initial distance to the torus. The result is valid, in particular, when the torus is of maximal dimension and when it is of dimension 0 (elliptic point). In the maximal dimension case, our results coincide with previous ones. In the zero dimension case, our results improve the existing b ounds in the literature. Let us assume now that the torus (of dimension r , 0 r < ` ) is partially elliptic (let us call m e to the numb er of these directions). In this case we show that, given a xed number of elliptic directions (let us call m 1 m e to this numb er), there exist a Cantor family of invariant tori of dimension r + m 1 , that generalize the linear oscillations corresp onding to these elliptic directions. Moreover, the Leb esgue measure of the complementary of this Cantor set (in the frequency space R r + m 1 ) is proven to be exp onentially small with the distance to the initial torus. This is asort of \Cantorian central manifold" theorem, in which the central manifold is completely lled up byinvariant tori and it is uniquely dened. The pro of of these results is based on the construction of suitable normal forms around the initial torus.
2 Contents 1Intro duction 3 2 Summary 4 2.1 Notation and formulation of the problem . . . . . . . . . . . . . . . . . . . 4 2.1.1 Reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Linear normal b ehaviour of the torus . . . . . . . . . . . . . . . . . 5 2.1.3 Seminormal form: formal description . . . . . . . . . . . . . . . . . 6 2.2 Results and main ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Seminormal form: bounds on the remainder . . . . . . . . . . . . . 8 2.2.2 Elliptic tori are very sticky . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.3 Cantor families of invariant tori . . . . . . . . . . . . . . . . . . . . 9 3 Normal form and eective stability 11 3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Bounding the remainder of the normal form . . . . . . . . . . . . . . . . . 12 3.3 Eective stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4 Estimates on the families of lower dimensional tori 23 4.1 Nondegeneracy conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.1.1 Nondegeneracy of the intrinsic frequencies . . . . . . . . . . . . . . 24 4.1.2 Nondegeneracy of the normal frequencies . . . . . . . . . . . . . . . 26 4.2 Main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.3 Pro of of Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.3.2 The iterativescheme . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.3.3 Convergence of the iterativescheme . . . . . . . . . . . . . . . . . . 37 4.3.4 Bounds on the measure . . . . . . . . . . . . . . . . . . . . . . . . . 38 5 Basic lemmas 40 6 Acknowledgements 44 References 44
A. Jorba and J. Villanueva 3 1Intro duction The study of the solutions close to an invariant ob ject is a classical sub ject in Dynamical Systems. Here we will address the problem of describing the phase space near an invariant torus of a Hamiltonian system. To x the notation, let us call H to a real analytic Hamiltonian with ` degrees of freedom, and let us assume it has an invariant r -dimensional torus, 0 r ` . Note that we are including the two limit cases, that is, when it is an equilibrium p ointand when it is a maximal dimensional torus. To start the discussion, let us assume that the torus has some elliptic directions, this is, that the linearized normal ow contains some harmonic oscillators. A natural question is if these oscillations p ersist when the nonlinear part of the Hamiltonian is added. If the torus is totally elliptic, another natural problem is the (nonlinear) stability around this torus. There are known answers to these questions in some concrete cases. If r = 0 (the torus is an equilibrium p oint) and it is totally elliptic, KAM theory says that there is plentyof maximal dimension invariant tori around the point (see 6]): the complementary of the set of invariant tori has measure exp onentially small with the distance to the point. It is well known that if ` =2, the maximal dimensional tori split the energy levels H = h in disconnected comp onents. This is the basis to prove the nonlinear stability of the point. Unfortunately, if ` > 2, the invariant tori do not separate the energy levels. In this case it is generally b elieved that some diusion can take place in the phase space (see 2]). Nevertheless, it is still p ossible to give lower bounds on the diusion time, that are exp onentially big with the distance to the point (they follow immediately from 6]). If r = ` (the torus has maximal dimension) we can not sp eak ab out normal b ehaviour since there are not \available directions". The nonlinear stability has b een studied in 14] and 13] (among others), where it is shown that the diusion time is also b ounded by an exp onentially big (with the distance to the initial torus) quantity. In 13] it is also related this exp onentially big stability time with the density of invariant maximal dimensional tori around the initial one, by showing that the total measure of the gaps between the invariant tori nearby is not bigger than an exp onentially small quantity with resp ect to the distance to the initial one. In fact, in 13] it is proved that under an extra steepness condition the diusion time is, at least, sup erexp onential. This condition corresp onds to the classical quasi-convexity hyp othesis used to obtain \global" and exp onentially big stability time of a p erturb ed integrable Hamiltonian system with resp ect to the size of the p erturbation (see 5] and references therein). In this work we will consider these problems, without any steepness condition, for a lower dimensional torus. The two limit cases mentioned ab ove are included, and the results obtained can b e summarized as follows: for a totally elliptic torus, wehave obtained lower bounds for the diusion time. They agree with the bounds of 13] in the case r = ` but, for the case r =0, they are b etter than the ones directly derived from 6]. Moreover, we showthe existence of quasip erio dic solutions that generalize the linear oscillations of the normal ow to the complete system. If the torus has normal b ehaviour of the kind \some centres" \some saddles" we obtain, for any combination of centres, a Cantor family of invant tori around the initial one, by adding to the initial set of frequencies new ones that come from the nonlinear oscillations asso ciated to the chosen centres. Those invarianttori have the same normal b ehaviour as the initial one (of course, skipping the centres that give rise to the family). This result is a sort of \Cantorian central manifold" theorem,
4 Normal Behaviour of Lower Dimensional Tori in which we obtain an invariant manifold parametrized on aCantor set and completely lled up byinvarianttori. We note that we obtain a Cantorian central \submanifold" for eachcombination of centres, and that it is uniquely dened. The pro ofs are based on the construction of suitable normal forms. The estimates on the difussion time are obtained b ounding the remainder of this normal form, while the existence of families of lower dimensional tori is proved by applying aKAM scheme to this remainder. The pap er has been organized in the following way: Section 2 summarizes the main ideas and results contained in the work. Section 3 contains the details concerning the normal form and the bounds on the diusion time. Section 4 is devoted to the existence of families of tori near the initial one and, nally, in Section 5, we have included some basic lemmas used along the pap er. 2 Summary Here we have included a technical description of the problem, the metho dology used in the pro ofs and the results obtained. Wehave ommitted the technical details of the pro ofs in order to simplify the reading. 2.1 Notation and formulation of the problem Let H b e a Hamiltonian system of ` degrees of freedom dened on R 2 ` ,having an invariant r -dimensional isotropic torus (that is, the canonical 2-form of R 2 ` restricted to the tangent bundle of the torus vanish), 0 r ` ,with a quasip erio dic ow given by the vector of basic frequencies ^ ! (0) 2 R r . We assume, from the isotropic character of the torus, that we can intro duce (with a canonical change of co ordinates) r angular variables ^ describing the initial torus. Hence, the Hamiltonian in these co ordinates takes the form H ( ^ x ^ I y )= ^ ! (0) > ^ I + 1 2 z > B ( ^ ) z + H 1 ( ^ x ^ I y ) where z > = ( x > y > ). Here, x , y are m -dimensional real vectors, and ^ , ^ I b elong to R r , r + m = ` . Of course, ^ , x are the p ositions and ^ I , y the resp ective conjugate momenta. As ^ is an angular variable, we assume that H depends onitina2 -p erio dic way. Moreover, we will use u > v to denote the scalar pro duct of two vectors. We also supp ose that the Hamiltonian H can be extended to a real analytic function dened on the set D rm ( 0 R 0 ) given by D rm ( 0 R 0 )= f ( ^ x ^ I y ) 2 C r C m C r C m : j Im ^ j 0 j z j R 0 j ^ I j R 2 0 g (1) where j : j denotes the innity norm of a complex vector (we will use the same notation for the matrix norm induced). The dierent scaling for the variables z and ^ I in D rm ( 0 R 0 ) is motivated by the denition of degree for a monomial of the Taylor expansion (with resp ect to z and ^ I , see (10)) used along the pap er: deg h ls ( ^ ) z l ^ I s = j l j 1 +2 j s j 1 (2)
A. Jorba and J. Villanueva 5 with l 2 N 2 m , s 2 N r , and where j k j 1 is dened as P j j k j j . The reason for counting twice the exp onent s will b e clear later (it is motivated, basically,by the prop erties of the Poisson bracket). We assume the initial invariant torus is given by z =0and ^ I =0. Hence, we can take B ( ^ ) as a symmetric 2 m -dimensional matrix, with real co ecients that dep end on ^ in analytic and 2 -p erio dic way. Moreover, the Taylor expansion of H 1 around z =0, ^ I =0 b egins with terms of degree at least three. 2.1.1 Reducibility We will assume that the normal variational ow around this torus (given by the matrix J m B (^ ! (0) t ), where J m is the canonical 2-form of C 2 m ) can be reduced to constant co e- cients with a real linear change of variables that dep ends quasip erio dically on ^ , having ^ ! (0) as a vector of basic frequencies (quasip erio dic Flo quet reduction). 1 The hyp othesis do es not seem to be very restrictive in our context, since all the partially elliptic tori obtained by KAM techniques have reducible normal ow (see, for instance, 7], 15], 8], 11]). This prop erty allows to construct a canonical change of co ordinates that transforms the matrix B ( ^ ) to constant co ecients. Hence, we will assume that B is a real symmetric matrix, indep endent from ^ ,and that the initial Hamiltonian in those Flo quet variables lo oks like: H ( ^ x ^ I y )= ^ ! (0) > ^ I + 1 2 z > B z + H 2 ( ^ x ^ I y ) (3) where H 2 b egins with terms of degree at least three. 2.1.2 Linear normal behaviour of the torus We also assume that the matrix J m B has dierenteigenvalues, given by the complex vector 2 C 2 m , that takes the form > =( 1 ::: m ; 1 ::: ; m ) (this structure comes from the canonical character of the system). We note that in this case, dierent eigenvalues also means nonzero eigenvalues. We will refer to those eigenvalues as the normal eigenvalues of the torus. We remark that if j = i (with 2 R nf 0 g and i = p ; 1) is an eigenvalue, then j + m = ; i . The vectors of R 2 m that are combination of eigenvectors corresp onding to (couples of ) eigenvalues of this form are called the elliptic directions of the torus. The study of the b ehaviour of the initial torus in those directions is the main issue in this pap er. Moreover, there may be other eigenvalues with real part dierent from zero, that dene the hyp erb olic directions of the torus. They can be group ed in one of these two following forms: 1. if j = 2 R nf 0 g , then j + m = ; , 2. if j = + i (with 2 R nf 0 g ), then, from the real character of the matrix B , we can take j +1 = ; i ,and hence, j + m = ; ; i and j + m +1 = ; + i . The imaginary parts of the eigenvalues are usually called normal frequencies of the torus. For reasons that will b e clear later, it is very convenient to put the matrix J m B in diagonal form. This is p ossible with a complex canonical change of basis, that transforms the 1 If the torus is reducible except by an small remainder, it is still p ossible to derive similar results (by adding a perturbative parameter). See 9] and 11] for the main ideas and related results.
6 Normal Behaviour of Lower Dimensional Tori initial real Hamiltonian system into a complex one. Thus, the complexied Hamiltonian has some symmetries b ecause it comes from a real one. As this symmetries are preserved by the transformations used along the pro ofs, the nal Hamiltonian can be realied. In fact, complexication is not necessary, but it simplies the pro ofs. Nevertheless, in the pro ofs we have not written explicitly the preservation of those symmetries. This is because the details are very tedious and cumb ersome and, on the other hand, the interested reader should not have problems in writting them (it is a very standard metho dology). For further uses, we denote by Z > =( X > Y > ) those complex (canonical) variables, and by B the complex symmetric matrix such that J m B =diag( ). 2.1.3 Seminormal form: formal description Now we take a subbundle of R 2 m , G R 2 m ,invariant by the action of the matrix J m B , and such that it only contains eigenvectors of elliptic typ e. We put 2 m 1 = dim( G ) (we recall that this dimension is always even) and we call ~ ! (0) 2 R m 1 to the vector of normal frequencies asso ciated to this subbundle. As G will b e xed along the pap er, we intro duce some notation related to it. First, we assume that the rst m 1 eigenvalues of are the ones asso ciated to G , that is, j = i ~ ! (0) j , j = 1 :::m 1 . We also denote by ^ 2 C 2( m ; m 1 ) the vector obtained skipping from the 2 m 1 eigenvalues asso ciated to G . This intro duces in a natural way the decomp osition X > = ( ~ X > ^ X > ), Y > = ( ~ Y > ^ Y > ), obtained taking apart the rst m 1 comp onents from the last m ; m 1 .Moreover, we dene ~ Z > = ( ~ X > ~ Y > ) and ^ Z > = ( ^ X > ^ Y > ). A similar notation can be used for any vector l 2 N 2 m , splitting l > = ( l > X l > Y ), where l X and l Y are the exp onents of X and Y in the monomial Z l ( Z l = X l X Y l Y ). Then, we intro duce ! (0) 2 R r + m 1 as ! (0) > =(^ ! (0) > ~ ! (0) > ), and we ask for a Diophantine condition of the following form, 2 j ik > ! (0) + l > ^ j j k j 1 k 2 Z r + m 1 nf 0 g l 2 N 2( m ; m 1 ) 0 j l j 1 2 (4) b eing > 0 and >r + m 1 . This nonresonance condition allows to construct (formally) a seminormal form related to the chosen G .If we express the Hamiltonian in terms of the variables Z , this seminormal form is done by removing from H the monomials of the following form (see (11) for the notations): h lsk exp ( ik > ^ ) Z l ^ I s l 2 N 2 m s 2 N r k 2 Z r j k j 1 + j l X ; l Y j 1 6 =0 j ^ l j 1 2 (5) where ^ l is the part of l that corresp onds to ^ . After this normal form pro cess, using the preservation of the symmetries that come from the complexication, we can rewrite this (formal) seminormal form, in terms of suitable real variables, in the following form: H ( ^ x ^ I y )= ! (0) > I + 1 2 ^ z > ^ B ^ z + F ( I )+ 1 2 ^ z > Q ( I )^ z + O 3 (^ z ) (6) where, for simplicity,wedonotchange the name of the Hamiltonian, and where we extend the decomp osition intro duced ab ove to the variables ( x y ). Here, the matrix ^ B is areal 2 The Diophantine condition can b e relaxed when j l j 1 =2and l > ^ only involves hyp erb olic eigenvalues. In this case, the results are proved using a combined method based on a xed p oint scheme for the hyp erb olic directions and a Newton metho d for the remaining ones. This technique allows to have multiple hyperb olic eigenvalues.
A. Jorba and J. Villanueva 7 symmetric matrix obtained by pro jecting B on the directions given by the eigenvalues corresp onding to the eigenvectors ^ . I is a compact notation for I > =( ^ I > ~ I > ), where the actions ~ I can b e taken as ~ I j = 1 2 ( x 2 j + y 2 j ), j =1 :::m 1 ,ifwecho ose the real normal form variables ( x y ) asso ciated to the considered elliptic directions in adequate (and standard) way (see (44) in the pro of of Theorem 2). Of course, F = O 2 ( I )and Q = O 1 ( I ). Now, we pro ceed to describ e the normal b ehaviour of the torus derived from this seminormal form. It is not dicult to check that wehave the following (formal) quasip erio dic solutions for the canonical equations of (6): ^ ( t ) = ^ ! (0) + @ F @ ^ I ( I (0)) ! t + ^ (0) ^ I ( t ) = ^ I (0) ~ x j ( t ) = q 2 ~ I j (0) sin ~ ! (0) + @ F @ ~ I j ( I (0)) ! t + ~ j (0) ! (7) ~ y j ( t ) = q 2 ~ I j (0) cos ~ ! (0) + @ F @ ~ I j ( I (0)) ! t + ~ j (0) ! ^ z ( t ) = 0 : That is, we obtain a 2( r + m 1 )-dimensional invariant manifold (^ z = 0) foliated by a continuous ( r + m 1 )-dimensional family of ( r + m 1 )-dimensional invariant reducible tori, parametrized by I (0). The selection of the parameter I (0) is natural, as I 1 :::I r + m 1 are rst integrals of the Hamiltonian (6) restricted to the invariantmanifold ^ z =0. We remark that the tori of the family collapse to lower dimensional ones when any of the ~ I j (0) b ecome zero. In particular, if we take I (0) = 0 we recover the initial r -dimensional one. In fact, for every 0 m 2 m 1 wehave, for this seminormal form, m 1 m 2 dierent( r + m 2 )- dimensional families of ( r + m 2 )-dimensional invariant tori. They are asso ciated to every invariant real subbundle contained in G . The skeleton of these families comes from the natural r -dimensional family of r -dimensional tori containing the initial one. This family is asso ciated to the neutral directions of the torus (the neutral directions are conjugated to the tangent ones), an it is obtained taking G = 0 in our notation. Moreover, we also remark that in (7) we only have real tori when all the ~ I j 0. This comes directly from the denition of ~ I as a function of the real normal form variables. To explain this fact let us give the classical example of a 1-dimensional p endulum near the elliptic equilibrium point, x + sin( x )=0. The linear (normal) frequency at the equilibrium p ointis1. Moving the energy level in the real phase space we obtain p erio dic orbits with frequency smaller than 1. If one wants p erio dic orbits with frequency bigger than 1, one is forced to extend the phase space from R 2 to C 2 , keeping the time in R .This same phenomenon happ ens when we study the normal elliptic directions of a torus. It is imp ortantto note that, for us, a s -dimensional complex torus is a map from T s to C 2 ` . Hence, we will use the word \dimension" to refer to the real dimension. 2.2 Results and main ideas A basic result in this pap er is the quantitative version of the seminormal form, if we only kill the monomials like (5) up to some nite order. From the estimates on this seminormal form, we deduce (under certain nondegeneracy conditions) that the normal
8 Normal Behaviour of Lower Dimensional Tori behaviour of the initial torus describ ed in Section 2.1.3 is \correct" in the sense of the classical KAM ideas: the \ma jority" of these tori really exist (but slightly deformed) in the initial Hamiltonian system. Moreover, we also deduce the long time eective stability of any real tra jectory close to a totally elliptic torus. In the following sections we present the explicit description of those results, and we explain the main ideas used in the pro ofs. 2.2.1 Seminormal form: b ounds on the remainder We start with the Hamiltonian (3), where the normal owis reduced to constant co e- cients. Then, we p erform a nite number of (semi)normal form steps, by using suitable canonical transformations that remove the monomials (5) up to a nite degree. This allows to showthe convergence of the pro cess on the set D rm ( 1 R ), where 1 is indep endent from R and R is small enough. By selecting the order up to which the seminormal form is done as a suitable function of R , it is p ossible to obtain a remainder for the seminormal form which is exp onentially small with R .This is contained in Theorem 1. 2.2.2 Elliptic tori are very sticky Nowlet us assume that the inital torus has all the normal directions of elliptic typ e. In this case we can take G = R 2 m ,thewhole set of normal directions. Then, using the normal form explained ab ove, one can write the initial Hamiltonian as an integrable one plus an exp onentially small p erturbation. Hence, it is very natural to obtain exp onentially big estimates for the diusion time: the time needed for a real tra jectory to go away from the set D rm ( R ) (for a precise denition of \going away" see Theorem 2) is bigger than T ( R )= const: exp 0 @ const: 1 R 2 +1 1 A (8) b eing the constants on the denition of T ( R ) indep endent from R .As usual, we call the exp onent 2 +1 the stability exp onent. Let us compare this result with previous ones. In the case in which the initial torus is of maximal dimension, note that the normal variables z > =( x > y > ) are missing everywhere. So, the set D rm ( R ) (see (1)) reads D ` 0 ( R )= f ( ^ ^ I ) 2 C ` C ` : j Im ^ j j ^ I j R 2 g : To compare with 13] we must redene R 2 as R ,in order to have the same units. Then, the stability exp onent in (8) coincides with 13]. If the initial torus is an equilibrium p oint, the variables ^ and ^ I are the ones that are missing. Hence, D rm ( R ) b ecomes D 0 ` (0 R )= f ( x y ) 2 C ` C ` : j ( x y ) j R g : Hence, no rescaling is necessary to compare the diusion time of (8) with the one derived from 6]: the improvement is that the exp onent 1 +1 in 6] here b ecomes 2 +1 . We note that this improvement is not only on the diusion time, but also on the measure of the destroyed tori (see Remark 10).
A. Jorba and J. Villanueva 9 2.2.3 Cantor families of invarianttori It is clear from Section 2.1.3 that computing the seminormal formal form asso ciated to G around the initial torus, up to nite order, and skipping the non-integrable remainder, those elliptic directions dene a unique ( r + m 1 )-dimensional family of ( r + m 1 )-dimensional tori around the initial r -dimensional one. When we approach the inital torus, the intrinsic frequencies of the tori of the family can b e selected such that they tend to ! (0) . In this case we will show that when we add the remainder of the seminormal form most of these tori still p ersist in the complete system H ,having also reducible normal ow. The normal eigenvalues of these tori are close to the eigenvalues ^ j (that are the ones not related with G ). Of course, due to the dierent small divisors involved in the problem, we can not prove the p ersistence of all the invariant tori predicted by the normal form. The hyp otheses needed are usual in KAM metho ds. The rst one is a non-resonance condition involving the frequencies ^ ! (0) and the normal ones ,that dep ends on the concrete selection of G and it is explicitly given in (4). The second hyp othesis is a nondegeneracy condition, asking that all the frequencies vary with the actions. Note that, in general, we have more frequencies ( r + m ) than actions ( r + m 1 , m 1 m ). This intro duces the classical lack-of-parameters problem when working with lower dimensional torus, that needs a sp ecial treatement (for related results, see 4], 16] and 11]). The idea that wehave used here is to cho ose a suitable ( r + m 1 )-dimensional set of parameters, and to ask for the existence of lower dimensional tori asso ciated to some of the values of these parameters. Here, the natural parameter is the vector of intrinsic frequencies ! 2 R r + m 1 of the invariant tori. To use this parametrization we need a typical nondegeneracy condition on the frequency map from I to ! ,this is, that this map be a (lo cal) dieomorphism around I = 0. This condition can be explicitly formulated computing the normal form of Section 2.1.3 up to degree 4 and it is given in (50). The control of the remaining m ; m 1 normal frequencies (normal to the ( r + m 1 )-dimensional family of tori) is more dicult, since there are no free parameters to control them. Note that those frequencies are functions of the intrinsic ones. Then, the idea is to eliminate all the frequencies for which the Diophantine conditions needed to construct invariant tori are not satised. This will lead us to eliminate values of ! to: a) control the intrinsic frequencies ! and b) control the normal ones as a function of the intrinsic ones. Tocontrol the measure of the set of intrisic frequencies for which the asso ciated normal ones are close to resonance, we use the same kind of metho d of 11]: we ask for a extra set of nondegeneracy conditions for the dep endence of these normal frequencies with resp ect to the intrinsic ones. Those conditions are given in (54). They have already b een considered in 12] and 7]. With the formulation given ab ove, the result is that the measure of the complementary of the preserved tori is exp onentially small: we intro duce U ( A )= n ! 2 R r + m 1 : j ! ; ! (0) j A o A> 0 (9) and let us dene A ( A ) as the set of frequencies of U ( A )for which we have reducible invariant tori. Then, if A is small enough, we have mes( U ( A ) nA ( A )) mes( U ( A )) const: exp 0 @ ; const: 1 A 1 +1 1 A
16 Normal Behaviour of Lower Dimensional Tori Then, there exists a constant $ , depending only on r , m , , 0 , ^ N 4 , ^ N , ^ S , ^ T 3 and ^ T ,such that the fol lowing bounds hold for the transformed Hamiltonian H ! G 1 , j N (1) ; N j 4 R 4 $ ^ S R p +1 +2 + R 2 p ; 1 2( +2) ! j S (1) 1 j 4 R 4 $ ^ SR p +1 R 2 +1 + R 4 +2 + R p ; 3 +2 + R p ; 2 2( +2) ! j S (1) 2 j 4 R 4 $ ^ SR p R 2 +1 + R 3 +2 + R p ; 1 2( +2) ! j T (1) ; T j 4 R 4 $ ^ S R p +1 +2 + R 2 p ; 1 2( +2) ! : Remark 4 (A very important one) If p is big enough and > R ,the dominant term in the bounds of S (1) 1 and S (1) 2 is given by the factor R 2 = +1 . This wil l be the factor of decreasing of those terms during the normal form process and it al lows to take of order R 2 = ( +1) , that wil l produce the exponent 2 = ( +1) in (36) .As we have 2 = ( +1) < 1 , we can deduce that an adequate selection for p is p =8 .This al lows to keep bounds like (35) during al l the iterative process. If we start with a \raw" Hamiltonian (without any previous step of normal form) the decreasing factor obtained is of order R= +1 , that forces us to select of order R 1 = ( +1) . This produces a worse exponent 1 = ( +1) in (36) . For instance, let us assume that the normal form has been done around an el liptic equilibrium point. Here the important issue is to note that the bounds obtained when kil ling degree 3 are much worse than the bounds obtained for the other degrees (this has been observed numerical ly in 17] ). Hence, to apply the same bounds to al l the degrees results in poor estimates. Remark 5 The exponent 2 = ( +1) in Remark 4 can be improved in some very degenerate cases. For instance, let us consider a total ly el liptic torus, and we take G = R 2 m . Let q be the lowest degree of the monomials of N corresponding to the (formal) normal form of H around the torus (of course, q 4 ). Then, can be taken of order R ( q ; 2) = ( +1) ,that produces the exponent ( q ; 2) = ( +1) in (36) . Pro of: During this pro of we will use dierent constants $ j , j 0, that will dep end only on the same parameters as the nal constant $ of the statement of the lemma. First, from the b ound (24) of Lemma 1, we have that j G 1 j 1 R 1 $ 0 ^ SR p +1 j G 2 j 1 R 1 $ 0 ^ SR p j G j 1 R 1 $ 0 ^ SR p where, as in Lemma 1, j = ; j and R j = R exp ( ; j ). Then, to obtain the b ounds for the dierent terms of the transformed Hamiltonian, we only need to b ound the Poisson brackets that app ear in (29){(33). To obtain precise estimates, we will lo ok carefully into the critical b ounds of the dierent partial derivatives involved, that is, the ones asso ciated to N 4 and T 3 . So, we estimate, separately, the contribution of N 4 , N , T 3 and T , taking into account that N do es not dep end on ^ , N 4 is a p olynomial of degree 4, and T 3 only contains terms of degree 3. Moreover, to b ound e S 2 ( f T G 1 g )we note that (from the denition of S 1 and S 2 )
A. Jorba and J. Villanueva 17 it only contains terms corresp onding to @ @ ^ Z , and not to @ @ ^ or @ @ ^ I .Thus, using the b ounds on the Poisson bracket provided by Lemma 6 (see Remark 11 for the case in which one of the terms has nite degree), we have j e S 2 ( f T G 1 g ) j 2 R 2 $ 1 ^ SR p R 2 +1 + R 3 +2 ! jf T G gj 2 R 2 $ 2 ^ S R p +1 +2 jf N G 1 gj 2 R 2 $ 3 ^ SR p +1 R 2 +1 + R 4 +2 ! jf N G gj 2 R 2 $ 4 ^ SR p R 2 +1 + R 4 +2 ! jf S G gj 2 R 2 $ 5 ^ S R 2 p ; 2 +2 jff T G g G gj 3 R 3 $ 6 ^ S R 2 p ; 1 2( +2) jff N G g G gj 3 R 3 $ 7 ^ S R 2 p 2 +3 + R 2 p +2 2( +2) ! and nally j H j 4 R 4 $ 8 ^ S R 2 p ; 2 +2 + R 2 p ; 1 2( +2) ! : From that, with a suitable denition of $ as a function of $ 0 {$ 8 , the b ounds of the statement of the lemma are clear, if we recall that we have taken p 6. Now, we are in conditions to formulate a quantitative result ab out \partial reduction to seminormal form" of the initial Hamiltonian. For this purp ose, we consider the Hamiltonian H of (3), written as in (17) in terms of the Z variables. We assume that H is dened on D rm ( 0 R 0 ), for some 0 < 0 < 1 and 0 <R 0 < 1, with the following b ounds: j N j 0 R ^ NR 4 , j S j 0 R ^ SR 3 and j T j 0 R ^ TR 3 ,for any 0 < R R 0 , b eing ^ N , ^ S and ^ T , p ositive constants (indep endent from R ). Then, we prove the following result: Theorem 1 We consider the Hamiltonian H of (17) ,with the hypotheses previously described. We suppose that there exists 0 > 0 and >r + m 1 such that j ik > ^ ! (0) + l > j 0 ( j k j 1 + j l x ; l y j 1 ) 8 ( l s ) 2S 8 k 2 Z r with j l x ; l y j 1 + j k j 1 6 =0 : Then, for any R> 0 smal l enough (this condition on R depends only on r , m , , 0 , 0 , R 0 , ^ N , ^ S and ^ T ), there exists an analytical canonical transformation ! R such that 1. ! R ; Id and (! R ) ; 1 ; Id are 2 -periodic on ^ . 2. ! R : D rm (3 0 = 4 R exp ( ; 0 = 4)) ;!D rm ( 0 R ) and (! R ) ; 1 : D rm (11 0 = 16 R exp ( ; 5 0 = 16)) ;!D rm ( 0 R ) :
18 Normal Behaviour of Lower Dimensional Tori 3. If we take ( ^ X ^ I Y ) 2D rm (3 0 = 4 R exp ( ; 0 = 4)) and we dene ( ^ X ^ I Y )= ! R ( ^ X ^ I Y ) , then j ^ ; ^ j 0 = 16 , j Z ; Z j R 0 exp ( ; 1 = 2) = 32 , j ^ I ; ^ I j R 2 0 exp ( ; 1) = 16 .Moreover, the same bounds hold for (! R ) ; 1 if ( ^ X ^ I Y ) 2 D rm (11 0 = 16 R exp ( ; 5 0 = 16)) . 4. ! R transforms H R := H ! R = ^ ! (0) > ^ I + 1 2 Z > B Z + N R + S R + T R decomposition analogous to (17) ,with the bounds: j N R ; N 4 j 3 0 = 4 R exp ( ; 0 = 4) const:R 6 , j T R ; T 3 j 3 0 = 4 R exp ( ; 0 = 4) const:R 4 , where N 4 and T 3 were introducedin (34) , and can becomputed with a normal form with respect to S up to degree 4 ,and j S R j 3 0 = 4 R exp ( ; 0 = 4) const: exp 0 @ ; const: 1 R 2 +1 1 A R 8 (36) being the constats that appear in the bounds of N R , T R and S R , positive and independent from R .Moreover, for any R for which the result holds, H R is in normal form with respect to S , at least up to degree 8 . Remark 6 The dependence of ! R on R is not continuous but piecewise analytic. Remark 7 From the bounds providedbyLemma 2 for the iterative normal form procedure described in Lemma 1 , this exponential ly smal l bound seems to be the best that one can obtains by using this linearly convergent scheme. Pro of: The pro of is done simultaneously for any0 <R R 0 . The b ounds where R is not written explicitly are indep endent from R . All these bounds and the dierent conditions on the smallness of R will dep end only on the xed parameters of the statement. The main idea of this pro of is to use Lemma 1 recursively,and to iterate the b ounds provided by Lemma 2 for p = 8 (see Remark 4). Hence, to use this lemma, we need to put the initial Hamiltonian in normal form with resp ect to S ,up to degree at least 8. For this purp ose, we construct recursively the generating functions G (0) , G (1) , ::: , G (5) ,provided by Lemma 1. Putting H (0) = H , we can dene H ( n +1) := H ( n ) ! G ( n ) 1 = H ( n ) + f H ( n ) G ( n ) g + 1 2! ff H ( n ) G ( n ) g G ( n ) g + (37) for n = 0 ::: 5. Let us consider rst the expression (37) as a formal transformation. From the prop erty (14) for the Poisson bracket, and from the way in whichthe dierent G ( j ) are selected in Lemma 1 (see Remark 2), we can ensure that the non-resonant terms asso ciated to S that remain in H (6) are of degree at least 9. Toshow that this construction is not only formal, we are going to provethewell dened character of the transformations ! G ( n ) 1 , n =0 ::: 5, and to b ound H ( n ) , n =1 ::: 6. For this purp ose, we expand H ( n ) as in (17), but adding the sup erscript \( n )" to N , S and T . We dene 0 = 0 192 ,tointro duce (0) = 0 , R (0) = R , and ( n ) = ( n ; 1) ; 4 0 , R ( n ) = R ( n ; 1) exp ( ; 4 0 ), n =1 ::: 6. Then, we are going to show that taking 0 in Lemma 1, we have for n =0 ::: 6, that, if R is small enough, j N ( n ) j ( n ) R ( n ) ^ N ( n ) ( R ( n ) ) 4 j S ( n ) j ( n ) R ( n ) ^ S ( n ) ( R ( n ) ) n +3 j T ( n ) j ( n ) R ( n ) ^ T ( n ) ( R ( n ) ) 3 : (38)
A. Jorba and J. Villanueva 19 This is proved by (nite) induction: assuming that (38) holds for some n (0 n 5) and using that, if R is suciently small, " ^ S ( n ) ( R ( n ) ) n +1 +2 0 1 (39) (" is provided by Lemma 1), we have ! G ( n ) 1 ! G ( n ) ; 1 : D rm ( ( n +1) R ( n +1) ) ;!D rm ( ( n ) ; 3 0 R ( n ) exp ( ; 3 0 )) : (40) Then, the fact that the successive steps increase at least by one the degree of the normal form with resp ect to S , makes evident the estimates of (38) for n +1. For more details one can rewrite, with minor changes, the pro of of Lemma 2, using (38) instead of (35). Here, the dierent R -indep endent constants ^ N ( n ) , ^ T ( n ) and ^ S ( n ) ,dened recursively for n = 0 ::: 6, dep end only on the same parameters involved in the formulation of the Theorem. We remark that condition (39) for n =0 ::: 5, imp oses only anite number of restrictions on R .Let R 0 the biggest value of R for whichthey hold. The next step is to continue with the iterative normal form pro cess, but using Lemma 2 (with p = 8) to bound H ( n ) , 6 n L +1 ( L will be determined b elow). This will be done in an inductiveway,showing that b ounds like (35) hold for each H ( n ) , n 6. Hence, we add in (35) the sup erscript \( n )" to S , S 1 , S 2 , N and T ,andwe replace ^ S , ^ N and ^ T by ^ S ( n ) , ^ N ( n ) and ^ T ( n ) . All these b ounds have b een taken on D rm ( ( n ) R ( n ) ), for some ( n ) , R ( n ) that will be determined below. Initially, for n =6, we can take for instance ^ N (6) 4 = ^ N (6) and ^ N (6) = ^ N (6) = ( R 0 ) 2 . The denition of the other sup er-(6) constants can be done similarly. Before continuing the iterative pro cedure, we remark that, as the following steps only afect high order terms, N 4 and T 3 remain invariant during all the normal form pro cedure. Then, Remark 4 suggests the denition ( R )=( AR ) 2 = ( +1) , where A 1 will be determined later (indep endently from R ). From this value of we dene, recursively, ( n +1) = ( n ) ; 4 , R ( n +1) = R ( n ) exp ( ; 4 ), for n 6. To preserve the p ositiveness of ( n ) , we restrict n L ( R ), being L ( R )the greatest integer for which we have 4( L ; 5) 0 = 8. This implies the following restriction on L : L 5+ 0 32 1 AR 2 +1 : (41) Hence, we take as L the integer part of (41). This implies R exp ( ; 0 = 4) R ( n ) R if 6 n L +1. To apply Lemma 2, we assume that for the current Hamiltonian H ( n ) , 6 n L , we have ^ S ( n ) ^ S (6) , ^ N ( n ) ^ N and ^ T ( n ) ^ T ,for some ^ N and ^ T to b e precised later (those b ounds are necessary to dene $ in Lemma 2, indep endent from n ). If for the current value of n we have " ^ S ( n ) ( R ( n ) ) 6 +2 1 (42) then, the canonical transformation ! G ( n ) 1 given by Lemma 1 acts like (40), replacing 0 by . Therefore, using Lemma 2, and recalling that 2 +1 < 1, A 1and R ( n ) < R < 1, one obtains the following b ounds for the transformed Hamiltonian: j N ( n +1) ; N ( n ) j ( n +1) R ( n +1) $ ^ S ( n ) ( R ( n ) ) 9 A 2 R 2 + ( R ( n ) ) 15 2 A 4 R 4 ! 2$ ^ S ( n ) A 2 ( R ( n ) ) 6
20 Normal Behaviour of Lower Dimensional Tori j S ( n +1) 1 j ( n +1) R ( n +1) $ ^ S ( n ) ( R ( n ) ) 9 ( R ( n ) ) 2 A 2 R 2 + ( R ( n ) ) 4 A 2 R 2 + ( R ( n ) ) 5 A 2 R 2 + ( R ( n ) ) 6 2 A 4 R 4 ! 4$ ^ S ( n ) A 2 ( R ( n ) ) 9 j S ( n +1) 2 j ( n +1) R ( n +1) $ ^ S ( n ) ( R ( n ) ) 8 ( R ( n ) ) 2 A 2 R 2 + ( R ( n ) ) 3 A 2 R 2 + ( R ( n ) ) 7 2 A 4 R 4 ! 3$ ^ S ( n ) A 2 ( R ( n ) ) 8 j T ( n +1) ; T ( n ) j ( n +1) R ( n +1) $ ^ S ( n ) ( R ( n ) ) 9 A 2 R 2 + ( R ( n ) ) 15 2 A 4 R 4 ! 2$ ^ S ( n ) A 2 ( R ( n ) ) 4 : We take A = max n 1 q 8$ exp (1) o ,and then, recalling that R ( n +1) = R ( n ) exp ( ; 4 ), we can dene inductively ( n 6), ^ S ( n +1) = (exp (4 )) 9 exp (1) ^ S ( n ) ^ N ( n +1) =(exp (4 )) 9 ^ N ( n ) + 1 exp (1) ^ S ( n ) ! ^ T ( n +1) =(exp (4 )) 9 ^ T ( n ) + 1 exp (1) ^ S ( n ) ! : Assuming R small enough such that 1 = 72, we obtain ^ S ( n ) = ^ S (6) exp ((6 ; n )(1 ; 36 )) ^ S (6) exp ((6 ; n ) = 2) (43) ^ N ( n ) exp (36 ( n ; 6)) ^ N (6) + ^ S (6) 1 (exp (1) ; 1) ! ^ T ( n ) exp (36 ( n ; 6)) ^ T (6) + ^ S (6) 1 (exp (1) ; 1) ! : As we are only intrerested in those b ounds for n L + 1, from the restriction on L in (41) we can easily intro duce n -indep endent bounds ^ N and ^ T for ^ N ( n ) and ^ T ( n ) . Now, assuming that all the steps are well dened, if one puts n L ( R ) + 1 in (43), we obtain the exp onentially small b ound of the statement for ^ S ( L +1) . To justify that we can reach this value, we note that (42) holds for all the previous n ,ifwe restrict R with " ^ S (6) R 3 1. Then, to prove the Theorem, we only have to intro duce ! R = ! G (0) 1 ::: ! G ( L ) 1 , and hence, (! R ) ; 1 = ! G ( L ) ; 1 ::: ! G (0) ; 1 .If those transformations act as it has b een said in the statement, the pro of is nished. First, and from the domains of denition of the dierent canonical transformations ! G ( n ) 1 (see (40), replacing 0 by if n 6), we deduce that ! R is dened on the domain given in the statement. Moreover, from the b ounds for the dierentcomponents of ! G ( n ) 1 ; Id given by Lemma 1, and remarking that 6 0 +( L ; 5) 0 = 16, the nal bounds for ! R ; Id follow immediatly. We consider now(! R ) ; 1 . In this case, and using the same arguments on ! G ( n ) ; 1 ,one can check that if we dene n =11 0 = 16 + n 0 , R n = R exp ( ; 5 0 = 16 + n 0 ) for n = 0 ::: 6, and n =11 0 = 16 + 6 0 +( n ; 6) , R n = R exp ( ; 5 0 = 16 + 6 0 +( n ; 6) )for n =6 :::L +1, then we have ! G ( n ) ; 1 : D rm ( n R n ) ;!D rm ( n +1 R n +1 ) for 0 n L . The pro of of this fact can b e done by combining the b ounds on ! G ( n ) ; 1 ; Id with the inequality (27). Moreover, this allows to estimate (! R ) ; 1 ; Id as it has b een done with the case of ! R .
A. Jorba and J. Villanueva 21 3.3 Eective stability An immediate consequence of Theorem 1 is that we can b ound the diusion sp eed around a linearly stable torus of a Hamiltonian system. In this case, we take G = R 2 m , and hence, S = N . Then, we apply Theorem 1, without taking into account the term T of the decomp osition (17). In fact, in this case one can rewrite the pro ofs of Lemmas 1, 2, and Theorem 1, in a simpler form (although the actual formulation also holds in this particular case), to obtain exp onentially small b ounds for the remainder S R . Theorem 2 We consider the real analytic Hamiltonian (3) dened on D rm ( 0 R 0 ) for some 0 < 0 < 1 and R 0 > 0 . We also assume that al l the eigenvalues of J m B are of el liptic type, and that there exists 0 > 0 and >` such that j ik > ^ ! (0) + l > j 0 ( j k j 1 + j l x ; l y j 1 ) 8 l 2 N 2 m 8 k 2 Z r with j l x ; l y j 1 + j k j 1 6 =0 : Let R 2 (0 R 0 ) , and let us take real initial conditions at t = 0 contained in D rm (0 R ) . Then, we can dene > 2 such that, if R is smal l enough, the corresponding trajectories belong to D rm (0 R ) for any time 0 t T ( R ) ,with T ( R )= const: exp 0 @ const: 1 R 2 +1 1 A being the constants in the denition of T ( R ) independent from R . Remark 8 In the proof, and only for technical reasons, depends on 0 . Nevertheless, we can take as close as we want to 2 (by taking an initial 0 smal l enough, see the proof for details), but this implies a reduction on the set of al lowed R , and on the constants of the stability time. The reason that forces to take > 2 is the norm used for the normal variables. If one takes the Euclidean norm instead of the supremum norm, the condition > 2 is replaced by > 1 . Pro of: In order to simplify the pro of, we assume that the initial real variables ( x y ) of (3) corresp ond to the ones that put B in canonical real form, that is, z > B z = m X j =1 j ( x 2 j + y 2 j ) with j = i j , j =1 :::m .Moreover, we assume that R 0 < 1. We intro duce ( X Y )to denote the complexied variables x j = X j + iY j p 2 y j = iX j + Y j p 2 j =1 :::m (44) that put the matrix J m B in the diagonal form J m B .Then, we can write the Hamiltonian in these variables as H = ^ ! (0) > ^ I + 1 2 Z > B Z + N ( X ^ I Y )+ S ( ^ X ^ I Y )
22 Normal Behaviour of Lower Dimensional Tori where N can b e rewritten as a function of I , I > =( ^ I > ~ I > ), with ~ I j = iX j Y j = 1 2 ( x 2 j + y 2 j ), and S veries N ( S )=0. This corresp onds to the decomp osition (17) if one puts S = N . H is dened on D rm ( 0 R 0 = p 2), with bounds of the following form: j N j 0 R ^ NR 4 and j S j 0 R ^ SR 3 ,for any 0 < R R 0 = p 2. Now, we apply Theorem 1 and we obtain, for any R small enough, a canonical change ! R such that in the new co ordinate system ( ^ X ^ I Y )=! R ( ^ R X R ^ I R Y R ), we have H R := H ! R = ^ ! (0) > ^ I R + 1 2 ( Z R ) > B Z R + N R ( X R ^ I R Y R )+ S R ( ^ R X R ^ I R Y R ) b eing N R a function of ( I R ) > = (( ^ I R ) > ( ~ I R ) > ), with ~ I R j = iX R j Y R j . H R is dened on D rm (3 0 = 4 R exp ( ; 0 = 4)), with j S R j 3 0 = 4 R exp ( ; 0 = 4) const: exp 0 @ ; const: 1 R 2 +1 1 A R 8 := M ( R ) : The canonical equations for ( X R ^ I R Y R )are _ X R j = @ H R @Y R j _ Y R j = ; @ H R @X R j j =1 :::m _ I R j = @ H R @ ^ R j = @S R @ ^ R j j =1 :::r: (45) From this, one obtains (using that N R is in fact only a function of I , and recalling ` = r + m ), _ I R j = i @ H R @Y R j Y R j ; i @ H R @X R j X R j = i @S R @Y R j Y R j ; i @S R @X R j X R j j = r +1 :::`: Weput I R j ( ^ R X R ^ I R Y R ) for the expressions on the right-hand side of _ I R j , j =1 :::` . We use Lemma 5 to b ound these expressions. Then, for j =1 :::r one has jI R j j 0 R exp ( ; 0 = 2) 4 M ( R ) 3 0 exp (1) : (46) If we combine Lemma 5 with the inequality (27), one obtains for j = r +1 :::` that, jI R j j 0 R exp ( ; 0 = 2) 2 M ( R ) exp ( ; 0 = 2) exp ( ; 0 = 4)(1 ; exp ( ; 0 = 4)) 16 M ( R ) exp ( ; 0 = 4) 0 : (47) Tocontinue the pro of, we put ( x R y R )forthevariables that come from the \realication" of ( X R Y R ), that is, X R j =( x R j ; iy R j ) = p 2, Y R j =( y R j ; ix R j ) = p 2. In fact, as ! R preserves the symmetries of H (due to the complexication of a real Hamiltonian, see Section 2.1.2), wehave that the Hamiltonian in the variables ( x R y R ) is real analytic. Towork with those dierent representations of the variables, we give the following remarks: ( i ) the set of real variables ( x j y j )such that j x j j j y j j A is contained in the set of complex ( X j Y j ) such that j X j j j Y j j A , ( ii )the set of complex ( X j Y j )suchthat j X j j j Y j j A , is contained in the complex set for ( x j y j ) such that j x j j j y j j p 2 A (this prop erty has been used to say that H is dened in D rm ( 0 R 0 = p 2)), ( iii ) the set of real ( x j y j ) such that I j A 2 ,
A. Jorba and J. Villanueva 23 is contained in the set of real ( x j y j )such that j x j j j y j j p 2 A . Those remarks are used when working with these dierent kind of variables, and one wants to control the size of the corresp onding domains, when we change the variable representation. Now, we take real values for ( ^ R x R ^ I R y R ) as initial conditions at t = 0. To prove the lower bound for the stability time, we consider a xed 0 < < 1, and we restrict to initial conditions such that, when expressed in terms of ( ^ R X R ^ I R Y R ), they b elong to D rm (0 R exp ( ; 0 = 2) = p 2). Then, we have that the corresp onding initial actions I R are b ounded by j I R j (0) j R 2 2 exp ( ; 0 ) = 2, j = 1 :::` . Using this, we deduce from the b ounds (46) and (47) that, for the tra jectories of the Hamiltonian equations (45), we have j I R j ( t ) j R 2 exp ( ; 0 ) = 2 for 0 t T ( R ), where we can take T ( R )= R 2 0 exp ( ; 3 0 = 4)(1 ; 2 ) 32 M ( R ) : This bound comes from (47), that is the worst case. This is the expression for the stability time of the statement of the Theorem. To use the bounds (46) and (47) for I R j , we need that these tra jectories expressed in terms of ( ^ R X R ^ I R Y R ) b elong to D rm (0 R exp ( ; 0 = 2)) up to time T ( R ). As we have j I R j ( t ) j R 2 exp ( ; 0 ) = 2, this follows from remarks ( iii ) and ( i ). From that we deduce, using the b ounds for (! R ) ; 1 ; Id provided by Theorem 1 and remark ( ii ), that the corresp onding real tra jectories in terms of ( ^ x ^ I y ) are contained in D rm (0 R 1 ), being R 1 dened by R 1 =max 8 < : p 2 R exp ; 0 2 + R 0 exp ( ; 1 = 2) 32 ! s R 2 exp ( ; 0 )+ R 2 0 exp ( ; 1) 16 9 = : Then, if we give for ( ^ x ^ I y ) a real set of points such that, expressed in terms of ( ^ X ^ I Y ), they b elong to the domain (! R ) ; 1 ( D rm (0 R exp ( ; 0 = 2) = p 2)), then, the tra jectories of H with initial conditions in this set remain in D rm (0 R 1 ) for a time span T ( R ). With similar arguments as the ones used to dene R 1 (using now remark ( ii )), one can check that this domain can be taken as D rm (0 R 2 ), b eing R 2 dened by R 2 =min 8 < : R exp ( ; 0 = 2) p 2 ; R 0 exp ( ; 1 = 2) 32 s R 2 2 exp ( ; 0 ) 2 ; R 2 0 exp ( ; 1) 16 9 = : If one considers the ow ! H t dened from D rm (0 R 2 ) \ R 2 ` to D rm (0 R 1 ) \ R 2 ` , for 0 t T ( R ), then, putting R R 2 in the statement, and taking an R -indep endent value of close enough to1such that R 2 > 0, we can dene = R 1 =R 2 . 4 Estimates on the families of lower dimensional tori Let us consider the real analytic reduced Hamiltonian H of (3) and a xed subbundle G of elliptic directions of J m B . In Theorem 1 wehave proved that, under standard Diophantine conditions, one can put H in normal form with resp ect to the set S (see (15) and (16) for the denition), with an exp onentially small remainder. If we write this seminormal form in terms of the complexied variables Z ,and without changing the name of the Hamiltonian, one has H = ! (0) > I + 1 2 ^ Z > ^ B ^ Z + F ( I )+ 1 2 ^ Z > Q ( I ) ^ Z + T ( ^ X ^ I Y )+ R ( ^ X ^ I Y ) : (48)
24 Normal Behaviour of Lower Dimensional Tori To explain the notation used, let us recall that the dierent resonant terms dep end only on ^ I and on the pro ducts X j Y j , j =1 :::m , but, from the structure of S , not all the p ossible combinations of those monomials takeplacein M ( S ). Then, weintro duce I > =( ^ I > ~ I > ), with ~ I j = iX j Y j , j =1 :::m 1 ,andwiththis denition (48) can b e describ ed as follows: the symmetric matrix ^ B is dened from B skipping the 2 m 1 eigenvalues asso ciated to G , J m ; m 1 ^ B = diag( ^ ). F and Q corresp ond to the normal form with resp ect to S ,with the expansion of F starting at second order with resp ect to I ,and with Q (0) =0. It is not dicult to check that by cho osing the variables ~ Z in suitable form (as it has b een done in the pro of of Theorem 2), F is real analytic. Moreover, Q is a symmetric matrix such that J m ; m 1 Q is diagonal, T2M ( NnS )(so T O 3 ( ^ Z )) and R2 f M ( S ). We assume that this normal form has b een done for a given (and small enough) R ,as in the formulation of Theorem 1. We only consider the R -dep endence when we give the bounds of the dierent terms of (48). To obtain these b ounds, let us dene 1 =3 0 = 4, where we recall that 0 is the width of the strip of analiticity, with resp ect to ^ , for the initial Hamiltonian. Then, Theorem 1 implies that, for any R small enough, we have jF j 0 R ^ F R 4 jF 3 j 0 R ^ F 3 R 6 jQj 0 R ^ Q R 2 jQ 2 j 0 R ^ Q 2 R 4 jT j 1 R ^ T R 3 jRj 1 R const: exp ; const: 1 R 2 +1 R 8 (49) To derive these bounds on D rm (0 R ), we have considered the functions that dep end on ~ I as functions of ~ Z . Here, we have split F = F 2 + F 3 and Q = Q 1 + Q 2 . F 2 and the comp onents of Q 1 are p olynomials on I of degrees 2 and 1 resp ectively. F 3 and Q 2 contain the remaining terms. We note that the denition of F 2 and Q 1 do es not dep end on the order of the seminormal form. This seminormal form has b een formally explained in Section 2.1.3, and we will use the notation related to (7) to represent the normal form tori. The main purp ose of this section is to study the p ersistence of those tori when we add the remainder R . We note that, as jRj is exp onentially small with R ,we can exp ect that the tori of (7) will survive, except the ones corresp onding to a set of parameters ( I (0)) of exp onentially small measure with resp ect to R . We will showthat this assertion holds, assuming certain standard nondegeneracy conditions on this family of tori, that have b een explained in Section 2.2.3 (conditions that, as we will see, can b e checked by computing a normal form up to degree 4, that is, from F 2 and Q 1 ). As it is a more natural parameter, the results will b e formulated in terms of frequencies instead of actions. 4.1 Nondegeneracy conditions Before the rigorous formulation of the results, let us give in explicit form these nondegeneracy conditions. 4.1.1 Nondegeneracy of the intrinsic frequencies The rst one is a standard nondegeneracy condition on the dep endence of the frequencies with resp ect to the actions: we require det C 6 =0 C = @ 2 F 2 @I 2 (0) : (50)
A. Jorba and J. Villanueva 25 This allows to parametrize the tori of the family by their vector of intrinsic frequencies (instead of I (0)). Of course, wehave to b e close enough to the initial r -dimensional torus. This assertion is justied by the following lemma: Lemma 3 Let us assume that det C 6 =0 . Then, if R is smal l enough, there exists a real analytic vectorial function I ( ! ) ,dened on the set ! 2 C r + m 1 : j ! ; ! (0) j 1 8 ( jC ; 1 j ) ; 1 R 2 (51) such that @ F @I ( I ( ! )) = ! ; ! (0) with I ( ! (0) )=0 . Moreover, we have jI ( ! ) j 1 4 R 2 for any ! in the set (51) ,and if ! (1) , ! (2) belong in (51) , then jI ( ! (1) ) ;I ( ! (2) ) j 2 jC ; 1 jj ! (1) ; ! (2) j : Of course, we are stil l using the notation of Section 4 . Pro of: We have F ( I ) = 1 2 I > C I + F 3 . Then, we take a xed ! in the set (51), and we want to solve the equation: I ( ! )= C ; 1 ! ; ! (0) ; @ F 3 @I ( I ( ! )) ! : (52) Putting the sup erscripts \( k + 1)" and \( k )" to I ( ! ) in (52), we can consider this expression as an iterative pro cedure, using I (0) ( ! ) = 0 as the seed. If we assume jI ( k ) ( ! ) j 1 4 R 2 , then, using Cauchy inequalities, we have for R small enough, jI ( k +1) ( ! ) j jC ; 1 j 1 8 ( jC ; 1 j ) ; 1 R 2 + ^ F 3 R 6 3 4 R 2 ! 1 4 R 2 where we have used the b ounds of (49) for F 3 , remarking that jF 3 j 0 R is abound for the supremum norm of F 3 ( I ) if j I j R 2 . Moreover, to ensure convergence, we remark that using the main value theorem one has, jI ( k +1) ( ! ) ;I ( k ) ( ! ) j ( r + m 1 ) ^ F 3 R 6 3 8 2 R 4 jI ( k ) ( ! ) ;I ( k ; 1) ( ! ) j 1 2 jI ( k ) ( ! ) ;I ( k ; 1) ( ! ) j if R is small enough. Clearly, the limit function is analytic with resp ect to ! ,and from the real analytic character of F , I is in fact real analytic. Taking ! (1) , ! (2) in the set (51), one has I ( ! (1) ) ;I ( ! (2) )= C ; 1 ( ! (1) ; ! (2) )+ C ; 1 @ F 3 @I ( I ( ! (2) )) ; @ F 3 @I ( I ( ! (1) )) ! and with the same arguments previously used, we obtain for R small enough jI ( ! (1) ) ;I ( ! (2) ) j 2 jC ; 1 jj ! (1) ; ! (2) j :
32 Normal Behaviour of Lower Dimensional Tori 4.3.2 The iterative scheme Now, we can describ e the iterative pro cedure used to construct invariant ( r + m 1 )- dimensional tori. This pro cess is given by a sequence of canonical changes of variables, constructed as the time one ow of a suitable generating function S ! .The changes are constructed to kill the terms that obstructs the existence of an invariant reduced torus with vector of basic frequencies given by ! . As usual (to overcome the eect of the small divisors), the changes are chosen to pro duce a quadratically convergent scheme, instead of the linear one of Lemma 1. First, we describ e a generic step of this iterative pro cess. For this purp ose, we expand the Hamiltonian H (0) in the following form H (0) = a ( )+ b ( ) > ^ Z + c ( ) > I + 1 2 ^ Z > B ( ) ^ Z + I > E ( ) ^ Z + 1 2 I > C ( ) I +)( ^ X I ^ Y ) (67) where we do not write explicitly the ! -dep endence and where wehave skipp ed the sup erscript \(0)" in the dierent parts of the Hamiltonian. From this expansion, weintro duce the following notations: H (0) ] ( ^ Z ^ Z ) = B , H (0) ] ( I ^ Z ) = E and <H (0) > = H (0) ; ). From the bounds on the terms of the decomp osition (60), we have that ~ a , b , c ; ! , B ; ^ B (0) , C ;C (0) and E are all O ( ^ H (0) ). Note that if we are able to kill the terms ~ a , b and c ; ! , we will obtain an invariant torus with intrinsic frequency ! .Nevertheless, as we want to have simple equations at every step of the iterativescheme (this is, linear equations with constant co ecients), we are forced to kill something more. Then, we ask the nal torus to have reducible normal ow given by a diagonal matrix. This is, we want that the new matrix B veries B = J m ; m 1 ( B ) where, for a (2 s )-dimensional matrix A ( ) dep ending 2 -p erio dically on , we dene J s ( A )= ; J s dp( J s A ). Here, dp( A ) denotes the diagonal matrix obtained taking the diagonal entries of A . Moreover, we have to eliminate E to uncouple the \neutral" and the normal directions of the torus up to rst order. Thus, for each step of the iterative pro cess, we use a canonical change of variables, given by a generating function of the form S ( ^ X I ^ Y )= > + d ( )+ e ( ) > ^ Z + f ( ) > I + 1 2 ^ Z > G ( ) ^ Z + I > F ( ) ^ Z where 2 C r + m 1 , d =0, f = 0 and G is a symmetric matrix, with J m ; m 1 ( G )=0. The transformed Hamiltonian is H (1) = H (0) ! S 1 . We expand H (1) in the same way as H (0) in (67), keeping the same name for the new variables, but adding the sup erscript \(1)" to a , b , c , B , C , E and ). Then, we ask ~ a (1) =0, b (1) = 0, c (1) ; ! = 0, E (1) = 0 and B (1) = J m ; m 1 ( B (1) ). We will show that this can be achieved up to rst order in the size of ^ H (0) . For this purp ose, we write those conditions in terms of the initial Hamiltonian and the generating function, and then, we obtain the following equations: ( eq 1 ) ~ a ; @d @ ! =0, ( eq 2 ) b ; @e @ ! + ^ B (0) J m ; m 1 e =0, ( eq 3 ) c ; ! ; @f @ ! ;C (0) + @d @ > =0, ( eq 4 ) B ;J m ; m 1 ( B ) ; @G @ ! + ^ B (0) J m ; m 1 G ; GJ m ; m 1 ^ B (0) =0,
A. Jorba and J. Villanueva 33 ( eq 5 ) E ; @F @ ! ; FJ m ; m 1 ^ B (0) =0, b eing B = B ; 2 4 @H (0) @I 0 @ + @d @ ! > 1 A ; @H (0) @ ^ Z J m ; m 1 e 3 5 ( ^ Z ^ Z ) E = E ;C (0) @e @ ! > ; 2 4 @H (0) @I 0 @ + @d @ ! > 1 A ; @H (0) @ ^ Z J m ; m 1 e 3 5 ( I ^ Z ) : To solve those homological equations, we expand them in Fourier series and we equate the corresp onding co ecients, obtaining the formal solutions. The next step is to derive bounds on those solutions. As we will use these bounds in iterative form, we want to make clear which expressions change from one step to another, and which ones can be b ounded indep endently from the step. For this purp ose, we take xed p ositive constants m , ^ m , ~ m , 2 , 1 , ^ , ~ dened as twice the corresp onding inital values m (0) , ^ m (0) , ~ m (0) , (0) 2 , (0) 1 , ^ (0) , ~ (0) and a xed 1 , 0 < 1 < (0) 1 .In what follows, ^ N will denote an expression dep ending only on m , ^ m , 1 , 2 , ^ ,the dierent dimensions r , m , m 1 , plus and 0 . ^ N will be redened during the description of the iterative scheme to meet a nite number of conditions. The idea is to p erform the bounds on the iterative scheme putting the sup erscript \(0)" on the terms that change at every iteration. Hence, we write the b ounds on ^ H (0) as k ^ H (0) k E (0) (0) R (0) M (0) and L E (0) (0) R (0) f ^ H (0) g L (0) , with M (0) ( R ) M ( R ) and L (0) ( R ) ( M ( R )) 1 ; . Hence, using Lemma 5, k a ; (0) k E (0) (0) M (0) k E k E (0) (0) 2( m ; m 1 ) M (0) ( R (0) ) 3 k c ; ! k E (0) (0) M (0) ( R (0) ) 2 k B ; ^ B (0) k E (0) (0) (2( m ; m 1 )+1) M (0) ( R (0) ) 2 k b k E (0) (0) M (0) R (0) k C ;C (0) k E (0) (0) (2( r + m 1 )+1) M (0) ( R (0) ) 4 k ) k E (0) (0) R (0) ^ (0) + M (0) : (68) Moreover, we can use Lemma 11 to deduce that the same b ounds hold for their Lipschitz constants on E (0) , replacing M (0) by L (0) ,and ^ (0) by ~ (0) . Then, to prove the convergence of the expansion of S ,we need some kind of control on the dierent small divisors involved. For this purp ose, we restrict the parameter ! to the subset E (1) ( R ) E (0) ( R )for which the following Diophantine estimates hold: we say that ! 2E (1) , if ! 2E (0) ,and j ik > ! + l > ^ (0) ( ! ) j (0) ( R ) j k j 1 k 2 Z r + m 1 nf 0 g l 2 N 2( m ; m 1 ) 0 < j l j 1 2 (69) for certain (0) > 0. We exp ect the measure of E (0) nE (1) to be of order (0) and, hence, as we wantto have exp onentially small b ounds for this measure, we take (0) ( M (0) ) . Then, we pro ceed to b ound the solutions of the dierent homological equations. For this purp ose, we use Lemma 4. More precisely, we dene (0) =( M (0) ) ,and we take (0) as a value for to use the dierent estimates provided by this lemma. In order to simplify the pro ofs, we assume (0) ; N (0) 0 = 4, where N 2 N will b e a xed integer that will be determined before the description of the iterative scheme. Moreover, we also assume that ( M (0) ) R (0) 1. Then, one can solve ( eq 1 ) ; ( eq 5 )asfollows:
34 Normal Behaviour of Lower Dimensional Tori ( eq 1 ) For d ,we have d ( )= X k 2 Z r + m 1 nf 0 g a k ik > ! exp( ik > ) that implies, k d k E (1) (0) ; (0) (0) exp (1) ! k ~ a k E (1) (0) (0) ^ N ( M (0) ) 1 ; ; : ( eq 2 ) For any j ,1 j 2( m ; m 1 ), we have e j ( )= X k 2 Z r + m 1 b jk ik > ! + ^ (0) j exp( ik > ) and hence, k e k E (1) (0) ; (0) 2 1 + (0) exp (1) ! 1 (0) ! k b k E (1) (0) ^ N ( M (0) ) 1 ; 2 ; : ( eq 3 ) Taking average with resp ect to , we obtain =( C (0) ) ; 1 0 @ c ; ! ; C (0) @d @ ! > 1 A : Thus, k k E (1) = k ( C (0) ) ; 1 C (0) k E (1) k ( C (0) ) ; 1 k E (1) k C (0) k E (1) m 0 B @ k c ; ! k E (1) 0 + C (0) @d @ ! > E (1) 0 1 C A m k c ; ! k E (1) (0) + ^ m k d k E (1) (0) ; (0) ( (0) ; (0) ) exp (1) ! ^ N ( M (0) ) 1 ; ; : Tosolve the equation for f , we dene c =~ c ; ~ C (0) ;C (0) @d @ ! > + C (0) @d @ ! > and then, for any 1 j r + m 1 , we have f j ( )= X k 2 Z r + m 1 nf 0 g c jk ik > ! exp( ik > ) : To b ound f , rst we have that k c k E (1) (0) ; 2 (0) k ~ c k E (1) (0) + kC (0) k E (1) (0) k k E (1) + k d k E (1) (0) ; (0) (0) exp (1) ! ^ N ( M (0) ) 1 ; 2 ; and from here k f k E (1) (0) ; 3 (0) (0) exp (1) ! k c k E (1) (0) ; 2 (0) (0) ^ N ( M (0) ) 1 ; 3 ; 2 :
A. Jorba and J. Villanueva 35 ( eq 4 ) We dene B = B ;J m ; m 1 ( B ), and then, if G = ( G jl ), 1 j l 2( m ; m 1 ), we have G jl ( )= X k 2 Z r + m 1 B jlk ik > ! + ^ (0) j + ^ (0) l exp( ik > ) : In this sum we have to avoid the indices ( j l k )for which j j ; l j = m ; m 1 and k =0. In these cases wehave trivial zero divisors, but also the co ecient B jl 0 is 0. Moreover, we remark that the matrix G is symmetric. Then, to bound G , we have to b ound B . First, we have k B ; ^ B (0) k E (1) (0) ; 2 (0) k B ; ^ B (0) k E (1) (0) ; 2 (0) + +(2( m ; m 1 )+1)( r + m 1 ) k H (0) k E (1) (0) R (0) ( R (0) ) 4 k k E (1) + k d k E (1) (0) ; (0) (0) exp (1) ! + +24( m ; m 1 ) 2 k H (0) k E (1) (0) R (0) ( R (0) ) 3 k e k E (1) (0) ; (0) ^ N ( M (0) ) 1 ; 6 ; and from the denition of B and the norm used, the same bound holds for B . Then, k G k E (1) (0) ; 3 (0) 1 1 + (0) exp (1) ! 1 (0) ! 2( m ; m 1 ) k B k E (1) (0) ; 2 (0) ^ N ( M (0) ) 1 ; 7 ; 2 : ( eq 5 )The dierent comp onents of F are given by F jl ( )= X k 2 Z r + m 1 E jlk ik > ! + ^ (0) l exp( ik > ) for j =1 :::r + m 1 and l =1 ::: 2( m ; m 1 ). Thus, k E k E (1) (0) ; 2 (0) k E k E (1) (0) +2( m ; m 1 ) kC (0) k E (1) (0) k e k E (1) (0) ; (0) (0) exp (1) + +4( m ; m 1 )( r + m 1 ) k H (0) k E (1) (0) R (0) ( R (0) ) 5 k k E (1) + k d k E (1) (0) ; (0) (0) exp (1) ! + +8( m ; m 1 ) 2 k H (0) k E (1) (0) R (0) ( R (0) ) 4 k e k E (1) (0) ; (0) ^ N ( M (0) ) 1 ; 7 ; and, hence, k F k E (1) (0) ; 3 (0) 2 1 + (0) exp (1) ! 1 (0) ! 2( m ; m 1 ) k E k E (1) (0) ; 2 (0) ^ N ( M (0) ) 1 ; 8 ; 2 : We use these estimates to b ound the transformed Hamiltonian H (1) . For this purp ose, we dene H (0) := f H (0) S g = H (0) 1 + H (0) 2 ,with H (0) 1 = ! > I + 1 2 ^ Z > ^ B (0) ^ Z + 1 2 I > C (0) I + H (0) S
36 Normal Behaviour of Lower Dimensional Tori and H (0) 2 = f ^ H (0) S g . Note that we are splitting the contributions that are O 1 ( ^ H (0) ) and O 2 ( ^ H (0) ). Then, by construction of S ,one has H (0) + H (0) 1 = (1) + ! > I + 1 2 ^ Z > ^ B (1) ^ Z + 1 2 I > C (1) ( ) I + H (1) with ^ B (1) = J m ; m 1 ( ^ B (1) ) and <H (1) > =0. Hence, H (1) takes the same form as H (0) in (60) if we dene ^ H (1) = H (0) ! S 1 ; H (0) ; H (0) 1 = Z 1 0 H (0) 2 +(1 ; t ) f H (0) 1 S g ! S t dt: (70) To b ound the dierent terms of H (1) , we use Lemma 6 to b ound the Poisson brackets involved in the previous expressions: k H (0) 1 k E (1) (0) ; 4 (0) R (0) exp ( ; (0) ) ^ N ( M (0) ) 1 ; 12 ; 2 kf H (0) 1 S gk E (1) (0) ; 5 (0) R (0) exp ( ; 2 (0) ) ^ N ( M (0) ) 2 ; 24 ; 4 k H (0) 2 k E (1) (0) ; 4 (0) R (0) exp ( ; (0) ) ^ N ( M (0) ) 2 ; 12 ; 2 : Hence, to b ound ^ H (1) one only needs to control the eect of ! S t . To this end, we remark that from the b ounds on the solutions of ( eq 1 ) ; ( eq 5 ), one has kr S k E (1) (0) ; 4 (0) R (0) ^ N ( M (0) ) 1 ; 9 ; 2 (71) where r S is taken with resp ect to ( ^ X I ^ Y ). If we assume that kr S k E (1) (0) ; 4 (0) R (0) ( R (0) ) 2 (0) exp ( ; 1) = 2 (72) then, ! S t is well dened from D r + m 1 m ; m 1 ( (0) ; 5 (0) R (0) exp ( ; (0) )) to D r + m 1 m ; m 1 ( (0) ; 4 (0) R (0) ), for any ; 1 t 1, and for any ! 2 E (1) (this follows from Lemma 9 and (27)). More precisely, we have that k ! S t ; Id k E (1) (0) ; 5 (0) R (0) exp ( ; (0) ) kr S k E (1) (0) ; 4 (0) R (0) (73) for any ; 1 t 1. From (71) wehave that (72) holds if ^ N ( M (0) ) 1 ; 12 ; 2 1, condition that will follow immediately from the inductive restrictions. Applying the b ounds (71), (71) and (73) to (70) and using Lemma 7, we deduce k ^ H (1) k E (1) (0) ; 6 (0) R (0) exp ( ; 3 (0) ) ^ N ( M (0) ) 2 ; 24 ; 4 : (74) Moreover, the b ound on H (0) 1 pro duces k (1) ; (0) k E (1) ^ N ( M (0) ) 1 ; 12 ; 2 k ^ B (1) ; ^ B (0) k E (1) ^ N ( M (0) ) 1 ; 14 ; 2 kC (1) ;C (0) k E (1) (0) ; 4 (0) ^ N ( M (0) ) 1 ; 16 ; 2 k H (1) ; H (0) k E (1) (0) ; 4 (0) R (0) exp ( ; (0) ) ^ N ( M (0) ) 1 ; 12 ; 2 : (75) We take N 6, and we dene (1) = (0) ; N (0) ,and R (1) = R (0) exp ( ; ( N ; 3) (0) ). Then, it is not dicult to rewrite the bounds on H (1) as the ones on H (0) , but now on D r + m 1 m ; m 1 ( (1) R (1) ). To iterate this scheme, we only need to check that the b ounds assumed on H (0) to dene ^ N still hold on H (1) . This is done in the next section.
A. Jorba and J. Villanueva 37 4.3.3 Convergence of the iterative scheme Lo oking at the b ounds of the previous section, we take > 0 small enough such that, for s =2(1 ; 16 ; 2 ), wehave s> 1. Then, assuming ^ N 1, wedene M (1) =( ^ NM (0) ) s (note that this is a b ound for the norm of ^ H (1) in (74)). If the hyp otheses needed to iterate hold, we obtain recursively M ( n ) = ( ^ NM (0) ) s n , and hence, for R small enough, we have lim n !1 M ( n ) = 0. Let us dene E ( R )as the set of parameters ! for which all the steps are well dened. We assume that, for any ! 2 E ( R ), the comp osition of canonical transformations ! = ! S (0) 1 ! S (1) 1 ::: (b eing S ( n ) the generating function used at the n -step of the iterative pro cedure) is convergent. Then, the limit Hamiltonian H = H (0) ! takes the form: H = ( ! )+ ! > I + 1 2 ^ Z > ^ B ( ! ) ^ Z + 1 2 I > C ( ! ) I + H ( ^ X I ^ Y! ) with < H > =0. This is, we obtain for any ! 2 E a Hamiltonian with an ( r + m 1 )- dimensional reducible torus, with linear quasip erio dic ow given by ! . Let us prove that the inductive bounds hold. First, we check that we can dene, recursively, constants m ( n ) , ^ m ( n ) , ( n ) 1 , ( n ) 2 and ^ ( n ) , replacing the initial sup er-\(0)" ones, such that they are also b ounded by m , ^ m , 1 , 2 and ^ , resp ectively. Toprove that, we note that the expressions in the right-hand side of (75) can b e b ounded by( ^ NM (0) ) s= 2 (we remark that the same b ound holds for (71)). Hence, iterating this bounds, we only need to use that the sum X n 0 ^ NM (0) s n +1 2 (76) is convergent for R small enough (and in fact, that it go es to zero when R do es), to justify these n -indep endent b ounds. The same arguments can be used to prove that k k E < + 1 . Here, we only check the b ound m ( n ) m , b ecause is the only one that do es not follow directly: note that one can dene m (1) = m (0) 1 ; m (0) ( ^ NM (0) ) s= 2 and then, taking R small enough, wehave m (1) m . Hence, iterating this denition and assuming m ( n ) m by induction, we have m ( n ) m (0) n ; 1 Y j =0 1 1 ; m ( ^ NM (0) ) s n +1 = 2 : Under this inductive hyp otesis, one can b ound m ( n ) by an innite pro duct that it is convergent b ecause (76) do es. From here, the b ound m ( n ) m follows immediately for R small enough. Finally, with the inductive denitions ( n +1) = ( n ) ; N ( n ) and R ( n +1) = R ( n ) exp ( ; ( N ; 3) ( n ) ), n 0, we need to checkthat ( n ) 0 = 4and R ( n ) M ( n ) . We remark that, as we take ( n ) =( M ( n ) ) , we have, X n 0 ( n ) ( M (0) ) + X n 1 ( ^ NM (0) ) s n 2( M (0) ) (77) at least for R small enough. Then, as N will be a xed numb er, the b ound on ( n ) is clear, taking R small enough. Moreover, wealsohave R ( n ) R (0) exp ( ; 0 = 4) >R (0) = 2=
38 Normal Behaviour of Lower Dimensional Tori M (0) M ( n ) . To justify this last inequality, we only need to take R small enough such that M (1) M (0) . Under this assumption, the sequence f M ( n ) g n 0 is clearly decreasing. Finally, to prove the well dened character of the limit Hamiltonian, it only remains to checkthe convergence of ! . To do that we write, for simplicity, ! ( n ) = ! S ( n ) 1 and we dene ( ! ( n ) = ! (0) ::: ! ( n ) ,for n 0. We also put 0 n = ( n ) ; 0 = 8 and R 0 n = R ( n ) exp ( ; 0 = 8), n 1. Then, using in inductive form the b ounds (73), (71) and (72), it is not dicult to check that from Lemma 8 we have k ( ! ( n +1) ; ( ! ( n ) k E 0 n +2 R 0 n +2 (1 + ^ "( ^ NM (0) ) s 2 ; 2 ) k ! (1) ::: ! ( n +1) ; ! (1) ::: ! ( n ) k E 0 n +2 R 0 n +2 where ^ "only dep ends on r , m , m 1 , 0 and ^ N . Iterating this b ound and taking small enough, one obtains for R small enough k ( ! ( n +1) ; ( ! ( n ) k E 0 n +2 R 0 n +2 n Y j =0 (1 + ^ "( ^ NM (0) ) s j +1 2 ; 2 )( ^ NM (0) ) s n +2 2 2( ^ NM (0) ) s n +2 2 where we have used again the convergent character of the sum (76). From this b ound, it is clear that if p>q 0, then k ( ! ( p ) ; ( ! ( q ) k E 0 = 8 R (0) exp ( ; 3 0 = 8) X j q 2( ^ NM (0) ) s n +2 2 boundthatgoestozero as p q ! + 1 .This allows to check that the limit canonical transformation ! go es from D r + m 1 m ; m 1 ( 0 = 8 R (0) exp ( ; 3 0 = 8)) to D r + m 1 m ; m 1 ( (0) R (0) ). 4.3.4 Bounds on the measure Then, wehave shown the existence of real invariant reducible tori for a set of parameters ! 2E .It only remains to b ound the measure of E or, equivalently,the measure of the complementary set. To do that, we start recalling how E is constructed. Iterating the denition of E (1) from E (0) ,we dene E ( n +1) from E ( n ) in the same way as it has b een done in (69), replacing (0) ( M (0) ) by ( n ) ( M ( n ) ) . Then, we have E = \ n 1 E ( n ) . This is, E is constructed by taking out, in recursive form, the set of parameters ! for which the Diophantine conditions (69), formulated on the eigenvalues of the previous step and dep ending on the size of the remaining p erturbative terms, do not hold. Then, the set of removed parameters can b e obtained as union of sets for which one of those conditions is not satised at some step of the iterative pro cess. To estimate the size of the removed sets, we will use a Lipschitz condition with resp ect to ! for the dierent eigenvalues ^ ( n ) j of B ( n ) ,for n 0. To this end, we will prove that this kind of regularity holds for the successive transformed Hamiltonians. As this condition holds for the initial one, we have to check, by induction, that the canonical transformations used preserve this kind of dep endence. The key point is to b ound the Lipschitz constants of the dierent solutions of ( eq 1 ) ; ( eq 5 ). To do it, we recall that we have b ounds likethe ones of (68) for the Lipschitz constants of the dierent terms of the decomp osition (60) of H (0) .Then, we only have to prove that those bounds for the Lipschitz constants, can be iterated in the same way as the b ounds on the norms. To see that, we can use the dierent results given in item ( a )of Lemma 11 to b ound the
A. Jorba and J. Villanueva 39 Lipschitz constants of the solutions of ( eq 1 ) ; ( eq 5 ). We remark that, for the denominators that app ear solving these equations, we have L E (0) f ik > ! + l > ^ (0) gj k j 1 + (0) 1 j l j 1 : Then, combining Lemma 11 with standard inequalities to b ound the Lipschitz constants of sums and pro ducts, it is not dicult to check that one can iterate b ounds of the following form: L E (1) (1) R (1) f ^ H (1) g ~ N ( M (0) ) 2 s 1 L E (1) (1) R (1) f ^ B (1) ; ^ B (0) g ~ N ( M (0) ) s 1 L E (1) (1) R (1) fC (1) ;C (0) g ~ N ( M (0) ) s 1 L E (1) (1) R (1) f H (1) ; H (0) g ~ N ( M (0) ) s 1 that are analogous to the ones of (74) and (75). ~ N 1 dep ends on the same parameters as ^ N , plus ~ m , ~ and 1 . Moreover, taking small enough, we have 2 s 1 > 1. Here, the selection of N (used to dene (1) and R (1) ) is done dep ending on the numb er of times that we need to use Cauchy estimates to bound the dierent norms and Lipschitz constants. Iterating those expressions, it is not dicult to check (by induction) that we can dene inductively ~ m ( n ) , ~ ( n ) and ( n ) 1 for whichthe assumed n -indep endent bounds hold. The deduction of those Lipschitz b ounds is tedious but it only involves simple inequalities. Full details in a very similar context can be found in 10] or 11]. Let us particularize those b ounds on the eigenvalues of B ( n ) .If we expand ^ ( n ) j , j =1 ::: 2( m ; m 1 ), n 0, as in (66), replacing only the sup erscript \(0)"by\( n )", we have that L E ( n ) f ( ( n ) j g ( NR , b eing ( N a p ositive constant indep endent from R , j and n . To justify this assertion, we note that it holds for n =0, and that the contributions that come from the next steps are exp onentially small with R . Those b ounds on the Lipschitz constants of ( n ) j plus the nondegeneracy conditions (66) are the key to control the measure of E ( n ) nE ( n +1) . We consider the decomp osition E ( n ) nE ( n +1) = l 2 Z 2( m ; m 1 ) 0 < j l j 1 2 l ^ X 6 = l ^ Y k 2 Z r + m 1 nf 0 g R ( n ) lk with R ( n ) lk ( R )= ( ! 2E ( n ) ( R ): j ik > ! + l > ^ ( n ) ( ! ) j < ( n ) ( R ) j k j 1 ) : To estimate the measure of R ( n ) lk ,we take ! (1) and ! (2) in this set and then, we have j ik > ( ! (1) ; ! (2) )+ l > ( ^ ( n ) ( ! (1) ) ; ^ ( n ) ( ! (2) )) j < 2 ( n ) j k j 1 : Let us start with the case j l j 1 = 1. Then, l > ^ ( n ) = ^ ( n ) j for some j = 1 ::: 2( m ; m 1 ). Hence, the previous expression can be rewritten as j i ( k + v j ) > ( ! (1) ; ! (2) )+ ( ( n ) j ( ! (1) ) ; ( ( n ) j ( ! (2) ) j < 2 ( n ) j k j 1 :
40 Normal Behaviour of Lower Dimensional Tori Assuming that ! (1) ; ! (2) is parallel to k +Re( v j ), we have j ! (1) ; ! (2) j 2 = j ( k + Re( v j )) > ( ! (1) ; ! (2) ) j j k +Re( v j ) j 2 j ( k + v j ) > ( ! (1) ; ! (2) ) j j k + Re( v j ) j 2 1 j k + Re( v j ) j 2 j ( ( n ) j ( ! (1) ) ; ( ( n ) j ( ! (2) ) j + 2 ( n ) j k j 1 ! 1 j k + Re( v j ) j 2 ( NR j ! (1) ; ! (2) j + 2 ( n ) j k j 1 ! : b eing j : j 2 the Euclidean norm of a real vector. Using that Re( v j ) 6 = 0 (see (66)), we obtain that there exists apositive constant $ 1 , indep endent from j , k and n ,such that j ! (1) ; ! (2) j 2 $ 1 ( n ) j k j 1 for R small enough. In fact, this bound can be extended to the case j l j 1 = 2, l x 6 = l y , using that Re( v j 1 j 2 ) 6 =0 if j 1 6 = j 2 .This is abound for the width of a section of R ( n ) lk by a line in the direction k + Re( v j ). Then, the measure of R ( n ) lk can be bounded by mes( R lk ) $ 1 ( n ) j k j 1 p r + m 1 1 4 ( jC ; 1 j ) ; 1 R 2 r + m 1 ; 1 where 2 p r + m 1 1 8 ( jC ; 1 j ) ; 1 R 2 is a bound for the diameter of E (0) ( R ). Then, we have mes( E ( n ) nE ( n +1) ) $ 2 R 2( r + m 1 ; 1) ( n ) X k 2 Z r + m 1 nf 0 g 1 j k j 1 where $ 2 do es not dep end on n and R . Using that # f k 2 Z r + m 1 : j k j 1 = j g 2( r + m 1 ) j r + m 1 ; 1 and that >r + m 1 we obtain mes( E ( n ) nE ( n +1) ) $ 2 R 2( r + m 1 ; 1) ( n ) X j 1 2( r + m 1 ) j r + m 1 ; 1 ; $ 3 R 2( r + m 1 ; 1) ( n ) b eing $ 3 also indep endent from n and R . As ( n ) =( M ( n ) ) ,we deduce, using (77), that for R 1 small enough, mes( E (0) nE ) $ 3 R 2( r + m 1 ; 1) 0 @ ( M (0) ) + X n 1 ( ^ NM (0) ) s n 1 A 2$ 3 ( M (0) ) : Taking into account the b ound on the measure of W 1 8 ( jC ; 1 j ) ; 1 R 2 nE (0) (wehave shown, from (61), that it is of order ( M (0) ) 2 ), one obtains the exp onentially small b ounds on the measure of destroyed tori. To nish the pro of, we dene A as 0 <R R E ( R ), where R is the maximum value of R for which the iterativescheme converges. 5 Basic lemmas In this section, we give some basics results used to bound the norms (12) and (13) and the related Lipschitz constants, as well as the expressions and transformations involved in the dierent pro ofs. Similar lemmas app ear in 11].
A. Jorba and J. Villanueva 41 Lemma 4 Let f ( ) and g ( ) be analytic functions of r complex arguments dened on a strip of width > 0 , 2 -periodic on , and taking values in C . Let us denote by f k the Fourier coecients of f , f = P k 2 Z r f k exp ( ik > ) . Then, we have: ( i ) j f k jj f j exp ( ;j k j 1 ) . ( ii ) j fg j j f j j g j . ( iii ) For every 0 < < , @f @ j ; j f j exp (1) j =1 :::r: ( iv ) Let f d k g k 2 Z r nf 0 g C , with j d k j j k j 1 , for some > 0 and 0 . If we assume that f =0 , then, for any 0 < < , we have that the function g dened as g ( )= X k 2 Z r nf 0 g f k d k exp ( ik > ) satises the bound j g j ; exp(1) ! j f j : All these bounds can be extended to the case in which f and g take values in C n 1 or M n 1 n 2 ( C ) . Pro of: Items (i) and (ii) are easily veried. Pro ofs of (iii) and (iv) follows immediately using (23). Lemma 5 Let f ( x I y ) and g ( x I y ) be analytic functions on D rm ( R ) , and 2 - periodic on . Then, ( i ) If f = P ( ls ) 2 N 2 m N r f ls ( ) z l ^ I s ,we have j f ls j j f j R R j l j 1 +2 j s j 1 . ( ii ) j fg j R j f j R j g j R . ( iii ) For every 0 < < and 0 << 1 ,we have for j =1 :::r and k =1 ::: 2 m : @f @ j ; R j f j R exp(1) @f @I j R j f j R (1 ; 2 ) R 2 @f @z k R j f j R (1 ; ) R : As in Lemma 4 , al l the bounds hold if f and g take values in C n 1 or M n 1 n 2 ( C ) . Pro of: The pro of of (i) and (ii) is straightforward. (iii) is proved using item (iii) of Lemma 4 and applying Cauchy estimates to the function P ( ls ) 2 N 2 m N r j f ls j z l ^ I s . Lemma 6 Let us consider f ( x I y ) and g ( x I y ) complex-valued functions, such that f and r g are analytic functions dened on D rm ( R ) , 2 -periodic on . Then, for every 0 < < and 0 << 1 , we have: jf f g gj ; R r j f j R exp (1) @g @I ; R + r j f j R R 2 (1 ; 2 ) @g @ ; R + 2 m j f j R R (1 ; ) @g @z ; R :