On the persistence of lower dimensional invariant tori under quasiperiodic perturbations

On thePers i stence of Lower Dimens ional Invar iantTor i under Quas ip er io dic Perturbations  Angel Jorba  and Jordi Villanueva y DepartamentdeMatematica AplicadaI Univers itatPolitecnica deCatalunya Diagonal 647, 08028 Barcelona, Spain. Ab stract In thi s workweconsider timedep endent quasiperiodic perturbations of autonomous Hamiltonian systems. We fo cus on the eect thatthi s kindofperturbations has on lower dimens ional invar ianttor i. Our re sultsshowthat, under standard conditions of analyticity, nondegeneracy and nonre sonance, most of these tor i survive, adding the f requencie s of theperturbation totheones they already have. The pap er also contains e stimates on the amount of survivingtor i. Theworst situation happ ens when the initial tor i are normally elliptic. In thi s cas e, a torus (identie d bythevector of intr ins ic f requencie s) can b e continue d with respect to aperturbativeparameter " 2 0 " 0 ], except for a s et of " of measure exp onentially small with " 0 .Incasethat " i s xe d (and suciently small), we provethe exi stence of invar ianttor i for every vector of f requencie s clos e totheoneof the initial torus, except for a s et of f requencie s of measure exp onentially small withthedistance to theunp erturb e d torus. As a particular cas e, if theperturbation i s autonomous, these re sults also givethesamekindofestimates on themeasure of destroyed tor i. Finally,these results are applie d tosome problems of cele stial mechanics, in order tohelp in thede scr iptionofthephas e space of some concretemodels.  [email protected] y [email protected]s 2 Contents 1 Intro duction 3 2 Main ideas 5 2.1 Re ducibility :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: ::: :: 6 2.2 Normal form aroundthe initial torus : :: :: ::: :: :: :: :: ::: :: 7 2.3 Theiterativescheme : :: ::: :: :: :: :: ::: :: :: :: :: ::: :: 9 2.4 Estimate s on theme asure of pre s erved tor i :: ::: :: :: :: :: ::: :: 12 2.5 Other parameters: f amilie s of lower dimens ional tor i :: :: :: :: ::: :: 13 3 Statementof theresults 14 3.1 Remarks :: :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: ::: :: 15 4 Applications 15 4.1 The bicircular mo del near L 4  5 : :: :: :: :: ::: :: :: :: :: ::: :: 15 4.1.1 Extens ions : :: :: ::: :: :: :: :: ::: :: :: :: :: ::: :: 17 4.2 Halo orbits : :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: ::: :: 17 5 Pro ofs 18 5.1 Notations : :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: ::: :: 18 5.1.1 Norms and Lip schitz constants : :: :: ::: :: :: :: :: ::: :: 18 5.1.2 Canonical transformations : :: :: :: ::: :: :: :: :: ::: :: 19 5.2 Bas ic lemmas : :: :: :: ::: :: :: :: :: ::: :: :: :: :: ::: :: 20 5.2.1 Lemmas on norms andLipschitz constants :::::::::::::: 20 5.2.2 Lemmas on canonical transformations : ::: :: :: :: :: ::: :: 24 5.2.3 Convergence lemma ::: :: :: :: :: ::: :: :: :: :: ::: :: 25 5.2.4 Lemmas on the controlof themeasure ::: :: :: :: :: ::: :: 25 5.3 Iterativelemma :: :: :: ::: :: :: :: :: ::: :: :: :: :: ::: :: 29 5.4 Pro of of thetheorem : :: ::: :: :: :: :: ::: :: :: :: :: ::: :: 37 5.4.1 Linear schemewith respect to ":: :: ::: :: :: :: :: ::: :: 37 5.4.2 Intro duction of thevector of f requencie s as a parameter :: ::: :: 38 5.4.3 Inductive part : :: ::: :: :: :: :: ::: :: :: :: :: ::: :: 39 5.4.4 Convergence of thechange s of var iable s ::: :: :: :: :: ::: :: 41 5.4.5 Controlofthemeasure : :: :: :: :: ::: :: :: :: :: ::: :: 43 6 Acknowle dgements 44 Reference s 44 A. Jorba and J. Villanueva 3 1 Intro duction Let H b e an autonomous Hamiltonian system with ` degree s of f ree dom, havingthe or igin as an elliptic equilibr iumpoint. If wetakethelinearization atthi s p oint as a rst approximation tothedynamics, we s ee thatallthesolutions are quasiperiodic and can be de scr ib e d as the pro duct of ` line ar o scillators. Thesolutions of e ach o scillator can b e parametr ize d bytheamplitudeof the orbits. When the nonline ar part i s added, each o scillator b ecomes a oneparametr ic f amily of p er io dic orbits (usually calle d Lyap ounov orbits), thatcan bestill parametr ize d bythe amplitude, at le ast near the or igin (s ee 31]). Gener ically,the f requency of the s e orbits var ie s withthe ir amplitude. The eect thatthe nonlinear part of the Hamiltonian has on the quasiperiodic solutions i s more complex. Withoutgoingintothedetails, KAM theorem states thatunder gener ic conditions of nonre sonance on the f requencie s of the o scillators andgener ic conditions of nondegeneracy on the nonline ar part of the Hamiltonian, mo st of these solutions still survive. The ir f requencie s nowvary withtheamplitude, andtheme asure of thede stroyed tor i i s exp onentially small withthedistance tothe or igin (s ee 7]). Usually these results are proved putting rst theHamiltonian intothe more general form H = H 0 ( I )+ H 1 (  I )  I =( I 1 :::I ` )   =(  1 ::: ` )  where H 1 is small near the or igin. Thi s can b e achieve d, for instance, applyingsomestep s of theprocess toputtheHamiltonian in (Birkho ) normal form. If weneglect H 1 ,each quas ip er io dic solution takes place on a torus I = I  with f requencie s given by r H 0 ( I  ). Here the que stion i s if these invar ianttor i are pre s erved when the p erturbingterm H 1 is added. The usual hyp othesis are, essentially,two: 1. Nonre sonance. The f requencie s of thetorus must sati sfy a Diophantinecondition: j k > r H 0 ( I  ) j  j k j  1 >` ; 1  where k > r H 0 ( I  )denotes the scalar pro duct of k withthe gradientof H 0 ,and j k j 1 = j k 1 j +  + j k ` j . 2. Nondegeneracy.The f requencie s must dep endontheactions: det  @ 2 H 0 @I 2 ( I  ) ! 6 =0 : Thenece s s ityof therst hyp othesis comes from thefactthat, dur ingtheproof, we obtain the divi sors k > r H 0 ( I  ). Hence, if they are too smallitisnot possible to provethe convergence of the s er ie s thatapp e ar in the pro of (s ee 1] for thedetails). An intere stingcase is when k > r H 0 ( I  )isexactly zero, for some k . Thi s implie s that, as the f requencie s are rationally dep endent, theowonthetorus I = I  i s not dens e. More preci s ely,if onehas ` i indep endent f requencie s, thetorus I = I  contains an ( ` ; ` i )-f amily of ` i -dimens ional invar ianttor i, andeachof these tor i i s dens ely lle d up bytheow. Here thenatural problem i s also tostudy the p ers i stence of these lower dimens ional invar iant tor i when the nonintegrable part H 1 is taken intoaccount. Gener ically, someof these tor i 4 Pers i stence of Lower Dimens ional Tor i survivebutthe ir normal b ehaviour can b e e ither elliptic or hyp erb olic (s ee 35], 25 ], 11 ], 25] and 19]). Theinvar iantmanifolds as so ciated tothehyp erb olic directions of these tor i (usually calle d \whi skers") s eem tobetheskeleton that organize s the dius ion (s ee 2]). Moreover, there are other f amilie s of lower dimens ional tor i thatcomefrom the Hamiltonian in normal form H 0 .They can b e obtaine d combining someof the elliptic directions as so ciated tothe xe d p oint, that i s, they comefromthe pro duct of someof the o scillators of the linearization. These tor i are gener ically nonre sonant, andsomeof them also survive when we add the nonitegrable part H 1 (s ee 9]). They are thegeneralitzation of the p er io dic Lyap ounov orbitsto higher dimens ional tor i andhence, we will call them Lyap ounov tor i. In thi s pap er we will fo cus on every kindof nondegeneratelowdimens ional torus, in the s ens e thatits normal b ehaviour only contains elliptic or hyp erb olic directions butnot degenerateone s (s ee 17] and 36] for re sultsinthehyp erb olic cas e and 27], 9] and28] for previous re sultsinthe general cas e). Thi s implie s thatthetorus i s not contained in a (re sonant) higher dimens ional invar ianttorus. We will develop a p erturbation theory for these tor i, fo cus s ingonthe cas e in whichthe p erturbation also dep ends on time in a quasiperiodic way.The Hamiltonian i s of the form H (   x I  y )= ~ ! (0) > ~ I + H 0 ( ^  x ^ I y )+ H 1 ( ^  ~  x ^ I y )  (1) with re sp ect tothesymplectic form d ^  ^ d ^ I + d ~  ^ d ~ I + dx ^ dy . Here, ^  are theangular var iable s thatde scr ib e an initial r -dimens ional torus of H 0 , x and y are thenormal directions tothetorus, ~  are theangular var iable s thatdenotes thetime, ~ I are the corre sp onding momenta(thathas only b een added toputtheHamiltonian in autonomous form) and~ ! (0) is the f requency as so ciated totime. These kind of Hamiltonians app e ar in s everal problems of cele stial mechanics: for instance, tostudy thedynamics of a small particle (an asteroid or spacecraft) near the equilateral libration p oints (34]) of theEarth{Mo on system, one can takethe Earth{Mo on system as a re str icted three b o dy problem (that can b e wr itten as an autonomous Hamiltonian) plus p erturbations comingfor therealmotion of EarthandMoonandthe pre s ence of theSun. As the s e p erturbations can b e very well approximated by quas ip er io dic functions (at le ast for mo deratetime spans), it i s usual to do so. Hence, oneends up with an autonomous mo del p erturb e d witha function thatdep ends on timeinaquasiperiodic way. Details on these models andtheir applications can b e found in 8], 12], 14 ] and 16 ]. For more theoretical re sults, s ee 21], 23] and20]. Theproblem of the pre s ervation of maximal dimens ion tor i of Hamiltonians like (1) has alre ady b een cons idere d in 23 ]. There it i s proved thatmost(inthe usual me asure s ens e) of thetor i of theunp erturb e d system survivetothe p erturbation, but addingthe p erturbing f requencie s totheones they alre ady have. Here wewillconsider the problem of the pre s ervation of lower dimens ional invar ianttor i, under thesamekindof perturbations. We will showthat, under somehyp othe s i s of nondegeneracy and nonre sonance (tobe precised later) someof the(lower dimens ional) tor i are not destroyed butonlydeformed bythe p erturbation, addingtheperturbing f requencie s totheones they previously had. Oneofthemain contr ibutions of thi s pap er are theestimates on theme asure of the de stroyed tor i. Wehavetaken twoapproaches tothat p oint. In the rst onewestudy the persistence of a s ingle invar ianttorus of the initial Hamiltonian, under a quas ip er io dic time-dep endent p erturbation, us ing as a parameter thesize( " )ofthi s p erturbation. Our re sultsshowthatthi s torus can b e continue d for a Cantor s et of value s of " , addingthe A. Jorba and J. Villanueva 5 p erturbing f requencie s totheone s it alre ady have. Moreover, if " 2 0 " 0 ], theme asure of the complementary of that Cantor s et i s exp onentially small with " 0 .Iftheperturbation is autonomous thi s re sult i s alre ady contained in 19] but for 4-D symplectic map s. The s econdapproachistoxthesizeofthepertubation to a given (andsmall enough) value. Then it i s p o s s ible thatthelatter re sult can not b e applie d b ecaus e " can b e in thecomplementary of theabove-dened Cantor s et. In thi s cas e, it i s still possible to provetheexistence of invar ianttor i with f requencie s theones of theperturbation plus f requencie s clo s e totheone s of theunp erturb e d torus. These tor i are a Cantor f amily parametr ize d (for instance) bythe f requencie s of theunp erturbed problem. Again, the me asure of the complementary of thi s Cantor setisexponentially small withthedistance tothe f requencie s of theinitial torus. It i s intere stingto notethe implications of thi s last as s ertion when theperturbation is autonomous andthe s ize of the p erturbation i s xe d: in thi s cas e we are proving, for the p erturb e d Hamiltonian, theexistence of a Cantor f amily of invar ianttor i near the initial one (s ee 9] and 33 ]). Moreover, theme asure of the complementary of thi s s et i s exp onentially small withthedistance totheinitial tor i. The mo st dicultcaseiswhen the normal b ehaviour of thetorus contain some elliptic directions, b ecaus e the(small) divi sors obtaine d contain combinations of theintr ins ic f requencie s withthe normal ones. As we will s ee, it i s not diculttocontrolthevalue of theintr ins ic f requencie s butthen wehaveno control (in pr inciple) on the corre sp onding normal ones. Thi s i s equivalenttosaythatwe can not s elect a torus withgiven b oth intr ins ic and normal f requencie s, b ecaus e there are not enough available parameters (s ee 27], 4] and 33]). Themain tr ickin the pro ofs i s toassumethatthe normal f requencie s moveasafunction of " (then weder ivetheexistence of thetorus for a Cantor s et of " )or as afunction of theintr ins ic f requencie s (then weobtain the exi stence of theabove-mentioned f amily of tor i, clo s e tothe initial one). When the initial torus i s normally hyp erb olic wedo not nee d to controlthe e igenvalue s in the normal direction and, hence, we do not havetodeal withthelack of parameters. Of cours e, in thi s cas e theresults are muchbetter andthe pro ofs can b e s een as s implications of theone s contained here. Hence, thiscaseisnot explicity cons idere d. Finally,wehave also included example s where theapplication of the s e re sultshelp s to understandthedynamics of concrete problems. The pap er has b een organize d as follows: s ection 2 contains themain ideas used toder ivethe s e re sults. Section 3 contains the r igorous statementof the re sults. The applications of the s e re sultsto some concreteproblems can b e foundinsection 4 and, nally, s ection 5 contains thetechnical details of the pro ofs. 2 Main ideas Let H beaHamiltonian system of ` degree s of f ree dom in C 2 ` havingan invar iant r - dimens ional torus, 0  r  ` ,with a quas ip er io dic owgiven bythevector of bas ic f requencie s ^ ! (0) 2 R r . Let us cons ider the (p erturb e d) Hamiltonian system H = H + " ^ H . As it has b een mentione d b efore, we do not re str ict ours elves tothecase of autonomous p erturbations, butwe will as sumethat ^ H dep ends on time in a quasiperiodic way,with vector of bas ic f requencie s given by~ ! (0) 2 R s . 6 Pers i stence of Lower Dimens ional Tor i 2.1 Re ducibility Let us cons ider thevar iational owaroundoneof the quasiperiodic orbitsofthe initial r -dimens ional invar ianttorus of H .Thevar iational equations are a linear system with quas ip er io dic timedep endence, withvector of bas ic f requencie s ^ ! (0) .When thetorus i s a p er io dic orbit, thewell know Flo quet theorem states thatwecan reduce thi s p er io dic system to constant co ecientswitha line ar p er io dic change of var iable s (withthe same period of the system). Thi s change can b e s elected to b e canonical if the equations are Hamiltonian. So, thereduce d matr ix has a pair of zero e igenvalue s (as so ciated tothe tangentvector tothe p er io dic orbit) plus e igenvalue s thatde scr ib e s the line ar normal behaviour aroundthetorus. We will as sumethatthese eigenvalue s are all dierent(thi s condition implie s, f rom the canonical character of the system, thatthey are also non-zero). Thi s implie s thatthe p er io dic orbit i s not containe d in a (re sonant) higher dimens ional torus. Usually,theimaginary partsofthese eigenvalue s are calle d normal f requencie s, and ^ ! (0) i s calle d thevector of intr ins ic f requencie s of thetorus. The quasiperiodic case ( r> 1) i s more complex, b ecaus e we can not guarantee in general thereducibilityto constantcoecientsof thevar iational equations witha linear quas ip er io dic change of var iable s withthe same bas ic f requencie s as theinitial system. The que stion of re ducibility of linear quasiperiodic systems (prove d in some cas e s, s ee 18], 5], 10], 21], 23], 19 ], and 24 ], amongothers) remains open in the general cas e. However, we can say thatifthi s re duction i s p o s s ible, wehave2 r zero e igenvalue s (related tothe r tangentvectors tothetorus). Here we will as sumethatsuchreduction is possible for theinitial torus. Wewantto remarkthatif thi s initial torus come s f rom an autonomous p erturbation of a re sonanttorus of an integrable Hamiltonian, thi s hyp othesis is not very strong. To justify thi s as s ertion, wemention thefollowing f act: let us wr itethe Hamiltonian as H = H ( I )+ " ^ H (  I ), and let T 0 bealowdimens ional invar ianttor i of theintegrable Hamiltonian H ( I )that survives tothe p erturbation " ^ H (  I ). Then, under gener ic hyp othesis of nondegeneracy and nonre sonance, thi s lowdimens ional torus exi stsanditsnormal ow i s also re ducible for a Cantorsetofvalue s of " .TheLeb e sgue me asure of the complementary of thi s s et in 0 " 0 ] i s exp onentially small with " 0 . Thi s f act i s prove d for symplectic dieomorphi sms of R 4 in 19 ], butitisimmediatetoextendtoother cas e s. Moreover, let us as sumethatwecan intro duce (with a canonical change of co ordinates) r angular var iable s ^  de scr ibingthe initial torus. Hence, the Hamiltonian takes theform H ( ^  x ^ I y )= ^ ! (0) > ^ I + 1 2 z > B z + H  ( ^  x ^ I y )  where z > =( x > y > ), being z , ^  and ^ I complex vectors, x and y elementsof C r and ^  and ^ I elementsof C s ,with r + m = ` . Here, ^  and x are thepositions and ^ I and y are the conjugatemomenta. In thi s notation B i s a symmetric 2 m -dimens ional matr ix (with complex co ecients). Moreover, H  i s an analytic function (with re sp ect to all its arguments) with2  -p er io dic dep endence on ^  . More concretely,wewillassumethatitis analytic on a ne ighbourhood of z =0, ^ I =0, and on a complex str ip of p o s itivewidth  for thevar iable ^  ,that i s, if j Im ^  j j  ,forall j =1 :::r .Then, if weassumethat H has an invar iant r -dimens ional torus withvector of bas ic f requencie s ^ ! (0) ,given by ^ I =0 and z =0, thi s implie s thattheTaylor expans ion of H  must b egin withterms of s econd order in thevar iable s ^ I and z . If wehavethatthenormal var iational owaroundthi s A. Jorba and J. Villanueva 7 torus can b e re duce d toconstantcoecients, we can as sumethatthe quadratic terms of H  in the z var iable s vani sh. Hence, thenormal var iational equations are given bythe matr ix J m B ,where J m is the canonical 2-form of C 2 m .Wealso assumethatthematr ix J m B i s in diagonal form with dierenteigenvalue s  > =(  1 ::: m  ;  1 ::: ;  m ). Let us give someremarks on the s e co ordinate s. First, notethatwehaveassumed that the initial torus i s i sotropic (thi s i s, the canonical 2-form of C 2 ` re str icted tothetangent bundle of thetorus vani shes everywhere). This fact (thatisalways true for a p er io dic orbit) i s not a strongassumption for a torus, b ecaus e all thetor i obtained byapplying KAM technique s tonear-integrable Hamiltonian systems are i sotropic. Another p ointworthto commentis therealorcomplex character of thematr ix B . In thi s pap er wework, in pr inciple, withcomplex analytic Hamiltonian systems, butthe mo st intere sting cas e happens when wedeal withrealanalytic ones, andwhen the initial torus i s also re al. In thi s cas e, to guarantee thattheperturbativescheme pre s erves the re al character of thetor i, wewantthatthe initial re duce d matr ix B come s f rom a re al matr ix. We notethatthisisequivalenttoassumethatif  is an eigenvalue of J m B ,then   i s also an e igenvalue. Thi s as sumption i s not true in general for every re ducible torus of arealanalytic Hamiltonian system, butitholds for mo st of thetor i one can obtain near an initial torus withanormal owreducible over R .Notethatinthe cas e of a p er io dic orbit one can always as sumethat B i s re al (doublingtheperiod ifnece s sary). The f act that B i s re al guarantee s that all thetor i obtaine d are also re al. Toseeit,we notethat we can us e the same pro of butputting J m B in re al normal form inste ad of diagonal form, andthi s makes thatall thestep s of the pro of are also re al. However, thetechnical details in thi s cas e are a little more te dious and, hence, wehave prefere d toworkwith a diagonal J m B . 2.2 Normal form aroundtheinitial torus The rst step i s to re arrange the initial Hamiltonian H (0) := H in a suitable form toapply an inductive pro ce dure. In whatfollows, we will denedegree of a monomial z l ^ I j as j l j 1 +2 j j j 1 .Thisdenition i s motivated below. Let us expand H (0)  in p ower series with re sp ect to z and ^ I around the or igin: H (0)  = X d  2 H (0) d  where H (0) d are homogeneous p olynomials of degree d ,thatis, H (0) d = X l 2 N 2 m j 2 N r j l j 1 +2 j j j 1 = d h (0) lj ( ^  ) z l ^ I j : We also expandthe (p er io dic) co ecientsin Fourier series: h (0) lj ( ^  )= X k 2 Z r h (0) ljk exp ( ik > ^  )  (2) being i = p ; 1. Thedenition of degree for a monomial z l ^ I j countingtwice thecontr ibution of thevar iable ^ I i s motivated bythedenition of thePoi s son bracket of twofunctions 8 Pers i stence of Lower Dimens ional Tor i dep endingon ( ^  x ^ I y ): f f g g = @f @ ^   @g @ ^ I ! > ; @f @ ^ I  @g @ ^  ! > + @f @z J m  @g @z ! > : Notethat, if f i s an homogeneous p olynomial of degree d 1 and g is an homogeneous polynomial of degree d 2 then f f g g is an homogeneous p olynomial of degree d 1 + d 2 ; 2. Thi s propertyshows thatifwe try to construct canonical change s us ingthe Lie series method, theadequateformtoput H (0) in normal form i s to remove in an incre as ing order theterms of degree 3, 4, ::: ,witha suitable generatingfunction. Tointro duce someof the parameters (s ee s ection 2.5), it i s very convenientthatthe initial Hamiltonian has thefollowingprop ertie s: P1 The co ecientsof the monomials ( z ^ I )(degree 3) and( z ^ I ^ I )(degree 5) are zero. P2 The co ecientsof the monomials ( z z ^ I )(degree 4) and( ^ I ^ I )(degree 4) do not dep endon ^  and, in the cas e of ( z z ^ I ), they vani sh except for the co ecientsof the tr ivial re sonantterms. Here, wehaveused thefollowingnotation: for instance, bytheterms of order ( z z ^ I )we denotethe monomials z l ^ I j , with j l j 1 =2 and j j j 1 =1, withthe corre sp onding co ecients. We will apply three step s of a normal form pro ce dure in order toachievethese conditions. Eachstep i s doneusing a generatingfunction of thefollowingtyp e: S ( n ) ( ^  x ^ I y )= X l 2 N 2 m j 2 N r j l j 1 +2 j j j 1 = n s ( n ) lj ( ^  ) z l ^ I j  for n =3, 4 and5. Then, if weput S ( n ) for theowattimeoneoftheHamiltonian system as so ciated to S ( n ) ,we transform the initial Hamiltonian into H ( n ; 2) = H ( n ; 3)   S ( n ) = = H ( n ; 3) + fH ( n ; 3) S ( n ) g + 1 2! ffH ( n ; 3) S ( n ) g S ( n ) g + O n +1 = = ^ ! (0) > ^ I + 1 2 z > B z + H ( n ; 3)  + f ^ ! (0) > ^ I + 1 2 z > B z S ( n ) g + O n +1  for n =3  4  5. In e achstep, wetake S ( n ) suchthat H ( n ; 3) n + f ^ ! (0) > ^ I + 1 2 z > B z S ( n ) g satises conditions P1 and P2 for the monomials of degree n ( n =3  4  5). Tocompute S ( n ) we expand H ( n ; 3) n andwend (formally) an expans ion for S ( n ) : s ( n ) ljk = h ( n ; 3) ljk ik > ^ ! (0) + l >   where theindice s havethe sameme aningasin(2). Ifwesplit l =( l x l y )( z l = x l x y l y ), the exactly re sonantterms corre sp ondto k =0and l x = l y (we recall that  > = (  1 ::: m  ;  1 ::: ;  m )). Hence, it would be possible to formally compute a normal form dep endingonlyon ^ I andthe pro ducts x j y j , j =1 :::m .Asithas b een mentioned b efore, our purpose is much more mo de st. Tokillthe monomials mentioned above (in A. Jorba and J. Villanueva 9 conditions P1 and P2 ) withaconvergentchange of var iable s, onenee ds a condition on thesmallness of j ik > ^ ! (0) + l >  j , k 2 Z r nf 0 g , l 2 N 2 m and j l j 1  2. Wehave used the usual one j ik > ^ ! (0) + l >  j  0 j k j  1  thatwewillassumetrue in thestatementof the re sults. Wenotice thatwiththe s e conditions we can construct convergent expre s s ions for the dierentgeneratingfunctions S ( n ) , n =3  4  5, toachieveconditions P1 and P2 .We can call thi s pro ce s s a s eminormal form construction. Then, thenal form for the Hamiltonian i s H =^ ! (0) > ^ I + 1 2 z > B z + 1 2 ^ I > C ^ I + H  ( ^  x ^ I y )  (3) for which conditions P1 and P2 holds. Here C is a symmetr ic constantmatr ix andwe will as sumedet C 6 =0 (thi s i s oneof thenondegeneracy hyp othesis). Now let us intro duce the quasiperiodic time-dep endent p erturbation. Tosimplify thenotation, wewritethi s p erturbation in the normal form var iable s, andweadd thi s p erturbation to (3). We call H tothenew Hamiltonian: H (   x I  y  " )= ! (0) > I + 1 2 z > B z + 1 2 ^ I > C ^ I + H  ( ^  x ^ I y )+ " ^ H (  x ^ I y " )  (4) foraxed ! (0) > =(^ ! (0) >  ~ ! (0) > ), ! (0) 2 R r + s ,where  > =( ^  >  ~  > ), I > =( ^ I >  ~ I > )and z > =( x > y > ), being ~  , ~ I ( s -dimens ional complex vectors) thenew p o s itions and momenta added toputinautonomous form the quasiperiodic perturbation. Hence, H is 2  -p er io dic in  . Moreover, " is a small p o s itive parameter. This is the Hamiltonian thatwe cons ider in the formulation of theresults. 2.3 Theiterativescheme Before the explicit formulation of theresults, let us de scr ib e a gener ic step of theiterative metho d us e d in the pro of. So, let us cons ider a Hamiltonian of theform: H (   x I  y )= ! (0) > I + 1 2 z > B z + 1 2 ^ I > C (  ) ^ I + H  (  x ^ I y )+ " ^ H (  x ^ I y )  (5) withthe samenotations of (4), where weassumethat skippingtheterm " ^ H ,wehavethat z =0, ^ I =0 is a reducible ( r + s )-dimens ional torus withvector of bas ic f requencie s ! (0) , suchthatthevar iational normal owisgiven by J m B =diag(  1 ::: m  ;  1 ::: ;  m ), andthatdet  C 6 =0, where  C me ans theaverage of C with re sp ect toitsangular var iable s (although initially C do e s not dep endon  ,dur ingtheiterativescheme it will). Moreover, we suppose thatin H  theterms of order ( ^ I z )vani sh (thatis, we suppose thatthe \central" and \normal" directions of theunp erturb e d torus havebeen uncoupledupto rst order). Here weonlyusethe parameter " toshowthattheperturbation " ^ H is of O ( " ). We expand ^ H in p ower s er ie s around ^ I =0, z =0andweadd these terms tothe previous expans ion of theunp erturb e d Hamiltonian. Thi s makes thattheinitial torus i s not longer invar iant. Hence, the expre s s ion of the Hamiltonian must b e (withoutwritting explicitythedep endence on " ): H (   x I  y )= ~ ! (0) > ~ I + H  (  x ^ I y )  (6) 16 Pers i stence of Lower Dimens ional Tor i as sumed that EarthandMoon revolve in a circular orbits aroundtheir centre of mas s e s, andthatthi s centre of masses move s in circular orbit aroundtheSun. Usually,inorder to s implify the equations, theunitsoflenght, timeandmas s are chosen suchthatthe angular velo cityof rotation of Earthand Mo on (aroundthe ir centre of mas s e s), thesum of mas s e s of EarthandMoon andthegravitational constant are all equal toone. With the s e normalize d units, the Earth{Mo on di stance i s also one. Thesystem of reference i s dene d as follows: the or igin i s taken atthe centre of mas s of theEarth{Mo on system, the X axi s i s given bythelinethatgoesfromMoon to Earth, the Z axi s has the direction of theangular momentumof EarthandMoon andthe Y axis is taken suchthatthesystem i s orthogonal and positive-or iented. Notethat, in thi s (non-inertial) f rame, calle d syno dic system, EarthandMoon have xe d p o s itions andtheSunisrotating aroundthe barycentre of the Earth-Mo on system. If wedenemomenta P X = _ X ; Y , P Y = _ Y + X and P Z = _ Z , in the s e co ordinates, themotion of a innitessimal particle movingunder thegravitational attraction of Earth, Mo on andSunisgiven bythe Hamiltonian H = 1 2 ( P 2 X + P 2 Y + P 2 Z )+ YP X ; XP Y ; 1 ;  r PE ;  r PM ; m s r PS ; m s a 2 s ( Y sin  ; X co s  )  where  = w S t , being w S themean angular velo cityof theSun in syno dic co ordinates,  themas s parameter for the Earth{Mo on system, a s thesemima jor axi s of theSun, m s the Sunmas s, and r PE , r PM , r PS are dened in thefollowingform: r 2 PE = ( X ;  ) 2 + Y 2 + Z 2  r 2 PM = ( X ;  +1) 2 + Y 2 + Z 2  r 2 PS = ( X ; X s ) 2 +( Y ; Y s ) 2 + Z 2  where X s = a s co s  and Y s = ; a s sin  . Notethatone can lo ok atthi s mo del as a time-p er io dic p erturbation of an autonomous system, the Re str icted Three Body Problem (usually calle d RTBP, s ee 34 ] for denition and bas ic prop ertie s). Hence, the Hamiltonian i s of theform H = H 0 ( x y )+ "H 1 ( x y  t )  where " i s a parameter suchthat " = 0 corre sp onds totheunp erturbed RTBP and " =1 tothe bicircular mo del withtheactual value s for the p erturbation. Notethatthe bicircular mo del i s not dynamically cons i stent, b ecaus e the motion of Earth, Mo on andSun does not follow a true orbit of thesystem (we are not takinginto accounttheinteraction b etween theSunandtheEarth{Mo on system). Neverthele s s, numer ical s imulation shows that, in some regions of thephas e space, thi s mo del gives the same qualitativebehaviour as therealsystem andthi s makes it worthtostudy (s ee 32 ]). We are goingto fo cus in the dynamics near the equilateral p oints L 4  5 of theEarth{ Mo on system. These points are linearly stable for theunp erturb e d problem ( " = 0), so we can as so ciatethree f amilie s of p er io dic (Lyap ounov) orbitstothem: theshort p er io d f amily,thelong p er io d f amily andthevertical f amily of p er io dic orbits. Clas s ical re sults aboutthe s e f amilie s can b e foundin34]. When the p erturbation i s added the p oints L 4  5 b ecome(stable) p er io dic orbitswith the same p er io d as the p erturbation. The s e orbits b ecomeunstable for theactual value of theperturbation ( " =1 in thenotation above). In thi s last cas e, numer ical s imulation A. Jorba and J. Villanueva 17 shows the exi stence of a region of stabilitynot very clo s e totheorbit andoutsideofthe planeofmotion of Earthand Mo on. Thi s region s eems tobecentere d aroundsomeof the (Lyap ounov) periodic orbitsof thevertical f amily. See 16] or 32] for more details. Let us cons ider thedynamics near L 4  5 for " small. In thi s cas e, the equilibr ium p oint has b een replace d bya small p er io dic orbit. Our re sults imply thatthethree f amilie s of Lyap ounov p er io dic orbitsbecomethree cantor ian f amilie s of 2-D invar ianttor i, adding the p erturbing f requency totheoneof the p er io dic orbit. Moreover, theLyap ounovtor i (the2-D invar ianttor i of theunp erturb e d problem that are obtained by \pro duct" of two f amilie s of p er io dic orbits) b ecome3-D invar ianttor i, provided they are nonre sonantwith the p erturbation. Finally,themaximal dimens ion (3-D) invar ianttor i of theunp erturb e d problem b ecome 4-D tor i, addingthe f requency of theSuntotheones they alre ady had (thi s last re sult i s alre ady contained in 23]). Now let us cons ider " =1. Thi s value of " is too big toapply these results. In particular, " i s big enough tocaus e a change of stabilityin the p er io dic orbit that replace s the equilibr ium p oint. Hence, if onewantstoapply theresultsofthi s pap er tothi s case,itisnece s sary tostart byputtingthe Hamiltonian in a suitable form. Tode scr ib e thedynamics near theunstable periodic orbit that replace s the equilibr iumpoint, we can p erform somestep s of a normal form pro ce dure towritethe Hamiltonianasanautonomous (andintegrable) Hamiltonian plus a small timedep endent p er io dic p erturbation (s ee 16 ], 22] or 32] for more details aboutthese kindof computations). Then, if we are clo s e enough tothe p er io dic orbit, Theorem 1 applie s andwehaveinvar ianttor i of dimens ions 1, 2 and3. They are in the \central" directions of the p er io dic orbit. Theapplication tothestable region thatisinthevertical direction i s more dicult. A possiblityis to compute(numer ically) an approximation to a 2-D invar ianttorus of thevertical f amily (notethatits exi stence has not alre ady b een prove d r igorously) and to p erform somestep s of a normal form pro ce dure, in order towritethe problem as an integrable autonomous Hamiltonian plus a timedep endent p er io dic p erturbation. Then, if the (approximate) torus satises theequations within a small enough error, it should be possible toshowthe exi stence of a torus nearby,andtoestabli sh thatitisstable and surrounded byinvar ianttor i of dimens ions 1 to4. Numer ical exp er iments sugge st (s ee 16] or 32]) thatthi s i s whathapp ens in thi s cas e. 4.1.1 Extens ions In f act, the bicircular mo del i s only the rst step in thestudyofthedynamics near the libration p ointsofEarth-Mo on system. One can construct b etter mo dels takinginto accountthe non-circular motion of Earthand Mo on (s ee 8], 14 ], 16]). Our re sults can be applie d tothese models in the sameway it has b een doneinthe bicircular cas e. The main dierence i s thatnowthe equilibr iumpointisreplace d by a quas ip er io dic solution that, due tothe re sonance s, do e s not exi st for all value s of " but only for a Cantor s et of them (s ee 23]). 4.2 Halo orbits Let us cons ider the EarthandSunasaRTBP,and let us fo cus in thedynamics near the equilibr ium p ointthat itisinbetween (the so calle d L 1 point). It i s well known the exi stence of a f amily of p er io dic orbits (calle d Halo orbits, s ee 29 ]) suchthat, when one 18 Pers i stence of Lower Dimens ional Tor i lo oks atthem f rom the Earth, they s eem tode scr ib e an halo aroundthesolar di sc. These orbits areavery suitable place toput a spacecraft tostudy theSun: f rom thatplace, the Sunisalways vi s ible and itisalways p o s s ible tosenddatabackto Earth(becaus e the prob e do e s not cro s s thesolar di sc, otherwi s e the noi s e coming f rom theSunwould make communications imp o s s ible). These orbitshavebeenused by missions ISEE-C (from 1978 to 1982) and SOHO (launche d in 1995). In theRTBP, Halo orbitsareaone parameter f amily of p er io dic orbitswith a normal behaviour of thetyp e centre  saddle. Unfortunately,theRTBP i s too simple to pro duce good approximations tothe dynamics. If onewantstohavea cheap station keepingit is nece s sary to computethe nominal orbit withavery accuratemodel (s ee 13 ], 14], 15] and 16 ]). The usual analytic mo dels for thi s problem are wr itten as an autonomous Hamiltonian (theRTBP) plus the eect coming f rom therealmotion of EarthandMoon,the eect of Venus, etc. All the s e eects can b e mo delle d very accurately us ing quasiperiodic functions thatdep endon time in a quas ip er io dic way. Hence, weendup withanautonomous Hamiltonian plus a quasiperiodic timedep endent p erturbation with r> 0 f requencie s. As usual, we add a parameter " in f rontofthi s p erturbation. Then, Theorem 1 implie s that, if " is small enough, theHalo orbits b ecome a cantor ian f amily of ( r + 1)-D invar ianttor i. The normal b ehaviourofthese tor i i s also of thetyp e centre  saddle. Tostudy the cas e " =1 we refer totheremarks for thecaseof the bicircular problem. 5 Pro ofs Thi s s ection contains the pro of of Theorem 1. It has b een split in s everal partsto s implify the re ading. Section 5.1 intro duce s the bas ic notation us e d alongthe pro of. In s ection 5.2 wegivethe bas ic lemmas nee ded dur ingthe pro of. Section 5.3 give s quantitativeestimates on onestep of theiterativeschemeand s ection 5.4 contains thetechnical details of the pro of. 5.1 Notations Here weintro duce someof the notations us e d toprovethe dierentresults. 5.1.1 Norms andLipschitz constants As usual wedenoteby j v j theabsolutevalue of v 2 C ,andweusethe samenotation to refer tothe(maximum)vector ial or matr ix norm on C n or M n 1 n 2 ( C ). Let us denoteby f an analytic function dene d on a complex str ip of width > 0, having r argumentsand being2  -p er io dic in all of them. The range of thi s function can be in C , C n or M n 1 n 2 ( C ). If wewriteitsFour ier expans ion as f (  )= X k 2 Z r f k exp ( ik >  )  we can intro duce the norm j f j  = X k 2 Z r j f k j exp ( j k j 1  ) : A. Jorba and J. Villanueva 19 Let f (  q )be a 2  -p er io dic function on  ,andanalytic on thedomain U rm R = f (  q ) 2 C r  C m : j Im  j  j q j R g : If wewriteitsTaylor expans ion around q =0 as: f (  q )= X l 2 N m f l (  ) q l  then, f rom thi s expans ion wedenethenorm: j f j R = X l 2 N m j f l j  R j l j 1 : If f takes value s in C ,weput r f todenotethe gradientof f withrespectto(  q ). Now, weintro duce thekindof Lipschitz dep endence cons idered. Assumethat f ( ' )is afunction dened for ' 2E , E R j for some j ,andwithvalue s in C , C n or M n 1 n 2 ( C ). We call f a Lip schitz function with respect to ' on theset E if: L E f f g = sup ' 1 ' 2 2E ' 1 6 = ' 2 j f ( ' 2 ) ; f ( ' 1 ) j j ' 2 ; ' 1 j < + 1 : Thevalue L E f f g i s calle d the Lip schitz constantof f on E .For these kindof functions wedene k f k E = sup ' 2E j f ( ' ) j . Similarly,if f (  ' )isa2  -p er io dic analytic function on  for every ' 2E ,wedenote: L E  f f g =sup ' 1 ' 2 2E ' 1 6 = ' 2 j f ( : ' 2 ) ; f ( : ' 1 ) j  j ' 2 ; ' 1 j : In the sameway wecanintro duce L E R f f g ,ifweworkwith f (  q  ' )andthe norm j : j R . We can also extend k : k E toboth cases todene k : k E  and k : k E R . 5.1.2 Canonical transformations Thechange s of var iable s are p erformed bymeans of a Lie series method, withasuitable generatingfunction. For thesakeof clarity,wewillusehere thesamenotations for the dierentvar iablesasinthe formulation of theresults. Wewanttokeep the quasiperiodic timedep endence (after e ach transformation) withthesamevector of bas ic f requencie s ~ ! (0) as the initial one. This is achieved when the generatingfunction do e s not dep endon ~ I . Let us cons iderageneratingfunction S (  x ^ I y )suchthat r S dep ends analytically on (  x ^ I y )anditis2  -p er io dic in  .Theequations related tothe Hamiltonian function S are _ ^  =  @S @ ^ I ! >  _ ~  =  @S @ ~ I ! > =0  _ ^ I = ;  @S @ ^  ! >  _ ~ I = ;  @S @ ~  ! >  _ z = J m  @S @z ! > : Wedenoteby S t (  x I  y )theowattime t of S withinitial conditions (   x I  y )when t =0. We notethat S t i s (for a xe d t) a canonical change of var iable s that actsin 20 Pers i stence of Lower Dimens ional Tor i a tr ivial way on ~  .If weput(  ( t ) x ( t ) I ( t ) y ( t )) =  S t (  (0) x (0) I (0) y (0)), we can expre s s thechange as : ^  ( t )= ^  (0) + Z t 0  @S @ ^ I (  (  ) x (  )  ^ I (  ) y (  )) ! > d  I ( t )= I (0) ; Z t 0  @S @ (  (  ) x (  )  ^ I (  ) y (  )) ! > d  z ( t )= z (0) + J m Z t 0  @S @z (  (  ) x (  )  ^ I (  ) y (  )) ! > d  and ~  ( t )= ~  (0). Wenotethatthefunction  S t ; Id do e s not dep endon theauxiliar var iable s ~ I .Then, weput  (0) =  , ^ I (0) = ^ I and z (0) = z tointro duce the transformations ^  S t and ^ # S t ,dened as ^  S t (  x ^ I y )=(  ( t ) x ( t )  ^ I ( t ) y ( t )) and ^ # S t = ^  S t ; Id .Itisnot diculttocheckthat ^ # S t (  x ^ I y ) i s (for a xe d t )2  -p er io dic in  . If we cons ider the Hamiltonian function H of (6), andweput H  = f H  S g; @S @ ~  ~ ! (0)  (15)  S t transforms the Hamiltonian H into H   S t (   x I  y )= ~ ! (0) > ~ I + H  (  x ^ I y )+ tH  (  x ^ I y )+ $ t ( H  S )(  x ^ I y )  where $ t ( H  S )= X j  2 t j j ! L j ; 1 S ( H  )  (16) with L 0 S ( H  )= H  and L j S ( H  )= f L j ; 1 S ( H  ) S g ,for j  1. Weremarkthatifwe transform a Hamiltonian function H bythe canonical change of var iable s  S t ,we only nee d tocontrolthe transformation ^ # S t andto s ee thatthenew Hamiltonian, H   S t ,iswell dened on a suitable domain. Finally,as thechange of var iable s i s s electe d as theowattimeone of a Hamiltonian S ,inwhatfollows we will omit thesub scr ipt t andwe will as sumethatitmeans t =1. 5.2 Bas ic lemmas 5.2.1 Lemmas on norms and Lip schitz constants In thi s s ection we give somebounds us e d when workingwiththe norms andLipschitz constantsintro duce d in s ection 5.1.1. Wefollowhere the samenotations of s ection 5.1.1 for the dierentanalytic functions us e d in the lemmas. Lemma1 Let f (  ) and g (  ) be analytic functions on a strip of width > 0 , 2  -periodic in  and taking values in C . Let us denote by f k the Fourier coecients of f , f (  )= P k 2 Z r f k exp ( ik >  ) . Then we have: ( i ) j f k jj f j  exp ( ;j k j 1  ) . ( ii ) j fg j  j f j  j g j  . A. Jorba and J. Villanueva 21 ( iii ) For every 0 < 0 <      @f @ j       ;  0  j f j   0 exp(1) j =1 :::r: ( iv ) Let f d k g k 2 Z r nf 0 g  C , with the fol lowing bounds: j d k j  j k j  1 exp ( ;  j k j 1 )  for some > 0 ,   0 , 0  < . If we assume that  f =0 , then the function g denedas g (  )= X k 2 Z r nf 0 g f k d k exp ( ik >  )  satises the bound j g j  ;  0    (  0 ;  )exp(1) !  j f j    for every  0 2 ]   . Al l these bounds can be extended to the case when f and g take values in C n or M n 1 n 2 ( C ) . Of course, in the matrix case, in ( ii ) it is necessary that the product fg be wel l dened. Pro of: Items ( i )and( ii ) are e as ily ver ie d. Pro ofs of ( iii )and( iv ) are e s s entially containe d in 23 ], butworkingwiththe supremum norm. Lemma2 Let f (  q ) and g (  q ) be analytic functions on a domain U rm R and 2  -periodic in  . Then we have: ( i ) If we expand f (  q )= P l 2 N m f l (  ) q l , then j f l j   j f j R R j l j 1 . ( ii ) j fg j R j f j R j g j R . ( iii ) For every 0 < 0 < and 0 <R 0 <R , we have:      @f @ j       ;  0 R  j f j R  0 exp(1) j =1 :::r and      @f @q j      R ; R 0  j f j R R 0 j =1 :::m: As in lemma 1, al l the bounds hold if f and g take values in C n or M n 1 n 2 ( C ) . Pro of: Items ( i )and( ii ) are straightforward. The rst part of ( iii ) i s a cons equence of lemma1. The s econdpartisobtained applyingstandard Cauchyestimates tothefunction F ( q )= P l 2 N m j f l j  q l . 22 Pers i stence of Lower Dimens ional Tor i Lemma3 Let us take 0 < 0 < and 0 <R 0 <R ,andlet usconsider analytic functions %(  q ) (with values in C r )and X (  q ) (with values in C m ), both 2  -periodic on  ,and such that j % j  0 R 0   ;  0 and j X j  0 R 0  R ; R 0 .Let f (  q ) be a given ( 2  -periodic on  ) analytic function. If we dene: F (  q )= f (  +%(  q ) q + X (  q ))  then, j F j  0 R 0 j f j R . Pro of: Expanding f in Taylor s er ie s (as ( i )inlemma2) oneobtains the expans ion of F asafunction of % and X .Then thebound i s a cons equence of ( ii )in 2. Lemma4 Let us consider % ( j ) and X ( j ) , j =1  2 , with the same conditions as % and X lemma 3, but with the fol lowing bounds: j % ( j ) j  0 R 0   ;  0 ;  and j X ( j ) j  0 R 0  R ; R 0 ;  , with 0 < < ;  0 and 0 <<R ; R 0 . Then, if we dene F ( j ) (  q )= f (  +% ( j ) (  q ) q + X ( j ) (  q )) j =1  2  one has j F (1) ; F (2) j  0 R 0   j % (1) ; % (2) j  0 R 0 exp (1)  + m j X (1) ; X (2) j  0 R 0  ! j f j R : Pro of: Wecan use here thesameide as as in lemma 3, combine d withtheones used to prove lemmas 1 and2. Nowwe givesome bas ic re sults related totheLipschitz dep endence s intro duce d in s ection 5.1.1. For that purpose, weworkwith a parameter ' on theset E R j , for some j  1. Lemma5 We consider Lipschitz functions f ( ' ) and g ( ' ) dened for ' 2E with values in C , then: ( i ) L E f f + g gL E f f g + L E f g g . ( ii ) L E f fg gk f k E L E f g g + k g k E L E f f g . ( iii ) L E f f=g gk 1 g k E L E f f g + k f k E k 1 g k 2 E L E f g g ,if g does not vanish. Moreover, ( i ) holds if f and g take values in C n or M n 1 n 2 ( C ) , and ( ii ) also holds when f and g are matrix-valued functions (such that the matrix product fg is wel l dened). Pro of: It i s straightforward. Lemma6 We assume that B ( ' ) is denedfor ' 2E with values in M nn ( C ) , and that B ; 1 exist for al l ' . Then L E f B ; 1 g k B ; 1 k 2 E L E f B g : Pro of: It i s straightforward. Remark1 In lemmas 5 and 6, we obtain analogous results if we work with functions of the form f (  ' ) or f (  q  ' ) , denedfor ' 2E and analytical with respect to the variables (  q ) and the norms j : j  , j : j R . A. Jorba and J. Villanueva 23 Lemma7 We assume that f (  ' ) is, for every ' 2E , an analytic 2  -periodic function in  on a strip of width > 0 , with Lipschitz dependence with respect to ' .Let us expand f (  ' )= P k 2 Z r f k ( ' )exp( ik >  ) . Then, we have: ( i ) L E f f k gL E  f f g exp ( ;j k j 1  ) . ( ii ) For every 0 < 0 < L E  ;  0 ( @f @ j )  L E  f f g  0 exp (1) j =1 :::r: ( iii ) Let f d k ( ' ) g k 2 Z r nf 0 g be a set of complex-valued functions dened for ' 2E , with the fol lowing bounds: j d k ( ' ) j  j k j  1 exp ( ;  j k j 1 )  and L E f d k g A + B j k j 1  for some > 0 ,   0 , 0  2 < , A  0 and B  0 . As in lemma 1 we assume  f =0 for every ' 2E .If g (  ' )= X k 2 Z r nf 0 g f k ( ' ) d k ( ' ) exp ( ik >  )  then, for every  0 , 2 < 0 < , we have: L E  ;  0 f g g    (  0 ;  )exp(1) !  L E  f f g  +  2  +1 (  0 ; 2  ) exp(1) ! 2  +1 k f k E   2 B + +  2  (  0 ; 2  ) exp(1) ! 2  k f k E   2 A: Pro of: It i s analogous tolemma1, usingalso the re sultsoflemma5. Lemma8 We assume that f (  q  ' ) is, for every ' 2E , an analytic function on U rm R and 2  -periodic in  . Then we have: ( i ) If we write f (  q  ' )= P l 2 N m f l (  ' ) q l , then L E  f f l g L E R f f g R j l j 1 . ( ii ) For every 0 < 0 < and 0 <R 0 <R , we have: L E  ;  0 R ( @f @ j )  L E R f f g  0 exp (1) j =1 :::r and L E R ; R 0 ( @f @q j )  L E R f f g R 0 j =1 :::m: Pro of: As in lemma7, butusingnowthe sameide as as in lemma2. 24 Pers i stence of Lower Dimens ional Tor i 5.2.2 Lemmas on canonical transformations We givehere some lemmas thatwewillusetoworkwiththe canonical transformations thatwehaveintro duce d in s ection 5.1.2. Thepurpose istoboundthechange s as well as the transforme d Hamiltonian. We also takeinto accountthe p o s s ibilitythatthegenerating function dep ends on a parameter ' 2E in a Lip schitz way. To s implify thenotations in the lemmas of thi s s ection, wedene &  0 R 0 = r  0 exp (1) + r +2 m R 0  (17) andwe will us e (withoutexplicit mention) thenotations intro duce d in s ection 5.1. The pro ofs of lemmas 9, 10, 11 and 12 can b e obtaine d f rom thebounds of lemmas of s ection 5.2.1. The pro of of lemma10 is essentially containe d in 6]. The pro of of 12 i s s imilar. The pro of of lemma 13 can also b e foundin6],whereitisproved workingwith the supremum norm. In our cas e the pro of i s analogous f rom the explicit expre s s ions for the transformation ^  S given in s ection 5.1.2, us ingthe re sult of lemma3 toboundthe comp o s itions. Lemma9 Let us consider f (  x ^ I y ) and g (  x ^ I y ) complex-valued functions such that f and r g are analytic functions denedon U r + sr +2 m R , 2  -periodic on  . Then, for every 0 < 0 < and 0 <R 0 <R , we have: jf f g gj  ;  0 R ; R 0  &  0 R 0 jr g j R j f j R : Lemma10 With the same hypothesis of lemma 9 we have, for the expression $( f g ) introducedin (16) , j $( f g ) j  ;  0 R ; R 0  X j  1 1 j +1 (&  0 R 0 exp (1) jr g j R ) j j f j R : Lemma11 Assume that the complex-valued functions f (  x ^ I y ' ) and g (  x ^ I y ' ) verify that, for every ' 2E , f and r g are analytic functions on U r + sr +2 m R , 2  -periodic on  , with Lipschitz dependenceon ' . Then, if k f k E R  F 1 , kr g k E R  F 2 , L E R f f g L 1 and L E R fr g g L 2 , we have that, for every 0 < 0 < and 0 <R 0 <R , L E  ;  0 R ; R 0 ff f g gg  &  0 R 0 ( F 1 L 2 + F 2 L 1 ) : Lemma12 With the same hypothesis of lemma 11, we have: L E  ;  0 R ; R 0 f $( f g ) g X j  1  1 j +1 (&  0 R 0 exp (1)) j F j ; 1 2 ( jL 2 F 1 + L 1 F 2 ) ! : Lemma13 We assume that the generating function S (  x ^ I y ) of section 5.1.2 veri- es that r S is analytic on U r + sr +2 m R , 2  -periodic in  , with jr S j R   , where < min f  R g . Then, with the notations of section 5.1.2, we have: ( i ) j ^ # S j  ; R ;  jr S j R . ( ii ) ^  S : U r + s 2 m + r  ; R ;  ;!U r + s 2 m + r R . A. Jorba and J. Villanueva 25 5.2.3 Convergence lemma We will us e thefollowing lemmadur ingthe pro of of Theorem 1, to relatethebounds on the Hamiltonian after n step s of theiterativeschemeas a function of b ounds for the initial Hamiltonian. Lemma14 Let f K n g n  1 beasequenceofpositive numbers with K n +1  an b K 2 n exp ( % n c ) if n  1 ,being a> 0 , b  0 , c> 0 and 1 <%< 2 . Then: K n +1  1 a   5 3  b aK 1 exp  c% 2 ; % !! 2 n : Pro of: The pro of i s a direct combination of re sults containe d in 21] and 23 ]. 5.2.4 Lemmas on the controlof theme asure In thefollowing lemmas, weconsider a xe d ! (0) > =(^ ! (0) >  ~ ! (0) > ), with^ ! (0) 2 R r and ~ ! (0) 2 R s .Let  ( ' )beafunction dened on E R r +1 withrange in C ,where ' > = (^ ! > " ), with^ ! 2 R r and " 2 R .Weassumethat  takes theform:  ( ' )=  0 + iu" + iv > (^ ! ; ^ ! (0) )+ ~  ( ' )  where  0 , u 2 C , v 2 C r and, if wedenoteby  E =  E (  # ):= n ' 2E : j ' ; ' (0) j  # o , ' (0) > =(^ ! (0) >  0), then wehavethat L  E f ~  g L  # for certain L  0, for all 0   #  # 0 . We also as sumethat j  ( ' ) ;  0 j M j ' ; ' (0) j for all ' 2  E ( # 0 ). Weremarkthatthe Lip schitz b ound for ~  formulate d on a suciently smo othfunction, means that ~  is of O 2 ( ' ; ' (0) ). Now, wetake > 0, >r + s ; 1and0 <  1todenefrom  and E thefollowing \re sonant" s ets: R ( " 0 R 0 ) = n ^ ! 2 R r : j ^ ! ; ^ ! (0) j R 0  (^ ! > " 0 ) > = ' 2E and 9 k 2 Z r + s nf 0 g suchthat j ik > ! +  ( ' ) j <  j k j  1 exp ( ;  j k j 1 ) )  for every " 0  0and R 0  0, and A ( " 0  ^ ! ) = n " 2 0 " 0 ]: (^ ! > " ) > = ' 2E and 9 k 2 Z r + s nf 0 g suchthat j ik > ! +  ( ' ) j <  j k j  1 exp ( ;  j k j 1 ) )  for every ^ ! 2 R r and " 0 > 0, where in b oth cas e s ! 2 R r + s is dened from ' > =(^ ! > " ) as ! > =(^ ! >  ~ ! (0) > ). Notethatthese setsdep endon  and  . As the purpose of thi s s ection i s todeal withtheme asure of these resonantsets, we will always as sumewe are in theworst cas e: Re  0 =0. When thi s i s not true (thi s i s, when there are no re sonance s) it i s not dicultto s ee thatthesets R and A are empty if we are clo s e enough to ' (0) (thevalue of the parameter for theunp erturb e d system). Wewantto remarkthatwe are not makinganyassumption on thevalue s Im u andIm v . Accordingtothe s ize of the re sonantsetstheworst cas e happ ens when Im u =0 and/or Im v = 0. Hence, the pro of will b e valid in thi s cas e, although it is possible to improve themeasure estimates assumingthatIm u 6 =0 andIm v 6 =0. 32 Pers i stence of Lower Dimens ional Tor i ( eq 2 )Wehave for e j e j  ;    2   1 +   (  ;  ) exp (1) !  1  ! j b j   for all >> . Cons equently: j e j  ;   ^ N M (  ;  )   : ( eq 3 ) First webound  : j  j = j (  C ) ; 1  C  jj (  C ) ; 1 jj  C  j  m         c ; ^ ! ;C  @d @ ^  ! >        ;     m  0 B @ j c ; ^ ! j  +       C  @d @ ^  ! >        ;  1 C A   m   ^ NM + jC j  2 j d j  ; = 2  exp (1) !  where   > 2  . Hence, j  j ^ N M (  ; 2  )  +1   for all   > 2  .Then, for c  wehave: j c  j  ;         ~ c ; ~ C  ;C  @d @ ^  ! > + C  @d @ ^  ! >        ;    j ~ c j  + jC j  0 B @ j  j +        @d @ ^  ! >        ;  1 C A   j c ; ^ ! j  +^ m   j  j + 2 j d j  ; = 2  exp (1) !  ^ N M (  ; 2  )  +1  : Hence, if >> 3  , j f j  ;    3  (  ; 3  ) exp (1) !  j c  j  ; 2 = 3   ^ N M (  ; 3  ) 2  +1  2 : ( eq 4 )From thedenition of B  given in (8), wehave j B  ;Bj  ;   j B ;Bj  ;  + +        2 4 @H  @ ^ I 0 @  +  @d @ ^  ! > 1 A 3 5 ( zz )         ;  +       " @H  @z J m e # ( zz )        ;    ^ NM +(2 m +1) r j H  j R ( R  ) 3  j  j + 2 j d j  ; = 2  exp (1) ! +24 m 2 j H  j R ( R  ) 3 j e j  ;   andthen j B  ;Bj  ;   ^ N M (  ; 2  )  +1   A. Jorba and J. Villanueva 33 if >> 2  ,andthesameboundholds for j B  j  ;  (s ee (10)). Lemma1 allows to bound j G j  ;    1   1 +  3  (  ; 3  )exp(1) !  1  ! 2 m j B  j  ; 2 = 3  with >> 3  . Hence, j G j  ;   ^ N M (  ; 3  ) 2  +1  2 : ( eq 5 )If >> 2  ,wehave for E  dene d in (9): j E  j  ;   j E j  ;  +       C  @e @ ^  ! >        ;  +        2 4 @H  @ ^ I 0 @  +  @d @ ^  ! > 1 A 3 5 ( ^ Iz )         ;  + +       " @H  @z J m e # ( ^ Iz )        ;   ^ NM + jC j  2 m 2 j e j  ; = 2  exp (1) + +4 mr j H  j R ( R  ) 3  j  j + 2 j d j  ; = 2  exp (1) ! +8 m 2 j H  j R ( R  ) 3 j e j  ;  : Then, j E  j  ;   ^ N M (  ; 2  )  +1  : Now, if >> 3  , j F j  ;   2 m  2   1 +  3  (  ; 3  ) exp (1) !  1  ! j E  j  ; 2 = 3  that implie s j F j  ;   ^ N M (  ; 3  ) 2  +1  2 : Now, we repeatthesame pro ce s s toboundthe Lip schitz constantsforthesolutions of the s e equations. For that purpose, wewillalso nee d theresultsoflemmas 7 and8 towork withthe dierent Lip schitz dep endence s. Weremarkthat, for the dierentdenominators, we can b ound: L E f ik > ! + l >  gj k j 1 +   2 2 j l j 1  for every k 2 Z r + s , l 2 N 2 m , j l j 1  2. Moreover, wewillalso use thehyp othesis M  L to s implify thebounds. Then wehave: ( eq 1 )Wenee d totakeinto accountthe ' dep endence for all thefunctions, andsofor d we have d (  ' )= X k 2 Z r + s nf 0 g a k ( ' ) ik > ! exp( ik >  ) : Then, us ing lemma7 and L E  f ~ a g L E  f a ;  g ,oneobtains L E  ;  f d g    (  ;  ) exp (1) !  L E  f ~ a g  +  2  +1 (  ; 2  ) exp (1) ! 2  +1 k ~ a k E   2   ^ N L (  ; 2  ) 2  +1  2  34 Pers i stence of Lower Dimens ional Tor i for every >> 2  . ( eq 2 ) L E  ;  f e g    (  ;  ) exp (1) !  L E  f b g  +  2  +1 (  ; 2  )exp(1) ! 2  +1 k b k E   2 + +  2  (  ; 2  ) exp (1) ! 2  k b k E   2   2 2 + 2   1 L E  f b g + 4 (   1 ) 2 k b k E    2 2   ^ N L (  ; 2  ) 2  +1  2  if >> 2  . ( eq 3 )If   > 3  ,wehave: L E f  g  L E n (  C ) ; 1 o        c ; ^ ! ; C  @d @ ^  ! >       E  ;  + + k (  C ) ; 1 k E L E  ;  8 < :  c ; ^ ! ; C  @d @ ^  ! > 9 =   ^ N L (  ; 3  ) 2  +2  2  where wehaveusedthat, f rom lemma6, L E f (  C ) ; 1 gk (  C ) ; 1 k 2 E L E f  Cg  ( m  ) 2 L E  fC g  and also that L E  ;  8 < :  @d @ ^  ! > 9 =   3  exp (1) L E  ; 2 = 3 f d g  and L E  ;  f  c ; ^ ! gL E  f c ; ^ ! g : Then, if >> 3  ,usingthat L E  ;  f ~ c g L E  ;  f c ; ^ ! g  onehas L E  ;  f c  gL E  ;  f ~ c g + L E  ;  n ~ C  o + L E  ;  8 < : C  @d @ ^  ! > 9 =   ^ N L (  ; 3  ) 2  +2  2 : Hence, L E  ;  f f g   3  (  ; 3  ) exp (1) !  L E  ; 2 = 3 f c  g  + +  2(2  +1) (  ; 6  ) exp (1) ! 2  +1 k c  k E  ; 2 = 3  2  ^ N L (  ; 6  ) 3  +2  3  if >> 6  . ( eq 4 )We rst b ound: L E  ;  f B  ;Bg  L E  ;  f B ;Bg + L E  ;  8 > < > : 2 4 @H  @ ^ I 0 @  +  @d @ ^  ! > 1 A 3 5 ( zz ) 9 > = >  + + L E  ;  8 < : " @H  @z J m e # ( zz ) 9 =   ^ N L (  ; 3  ) 2  +2  2  A. Jorba and J. Villanueva 35 if >> 3  ,andthesameboundholds for L E  ;  f B  g . Thi s implie s L E  ;  f G g  (2 m ; 1) 1   1 L E  ;  f B  g +(2 m ; 1) 1 (   1 ) 2 k B  k E  ;    2 + +2 m  3  (  ; 3  ) exp (1) !  L E  ; 2 = 3 f B  g  + +2 m  3(2  +1) (  ; 6  ) exp (1) ! 2  +1 k B  k E  ; 2 = 3  2 + +2 m  6  (  ; 6  ) exp (1) ! 2  k B  k E  ; 2 = 3  2   2   ^ N L (  ; 6  ) 3  +2  3  if >> 6  . ( eq 5 )From thedenition of E  , L E  ;  f E  g  L E  ;  f E g + L E  ;  8 > < > : 2 4 @H  @ ^ I 0 @  +  @d @ ^  ! > 1 A 3 5 ( ^ Iz ) 9 > = >  + + L E  ;  8 < : C  @e @ ^  ! > 9 =  + L E  ;  8 < : " @H  @z J m e # ( ^ Iz ) 9 =    ^ N L (  ; 4  ) 2  +2  2  if >> 4  . Hence, if now >> 6  ,we can b ound: L E  ;  f F g  2 m 2   1 L E  ;  f E  g +2 m 4 (   1 ) 2 k E  k E  ;    2 2 + +2 m  3  (  ; 3  ) exp (1) !  L E  ; 2 = 3 f E  g  + +2 m  3(2  +1) (  ; 6  ) exp (1) ! 2  +1 k E  k E  ; 2 = 3  2 + +2 m  6  (  ; 6  ) exp (1) ! 2  k E  k E  ; 2 = 3  2   2 2  +  ^ N L (  ; 6  ) 3  +2  3 : Before b oundingthe transformed Hamiltonian, let us checkthatthechange given by the generatingfunction S is well dene d. First, wehavethat: kr S k E  ; R  ^ N M (  ; 4  ) 2  +2  2  (28) andthat L E  ; R fr S g ^ N L (  ; 7  ) 3  +3  3  36 Pers i stence of Lower Dimens ional Tor i provided that >> 7  .Ifwe s elect  =8  ,andifweconsider (28), wehavea bound of thetyp e: kr S k E  ; 8 R ; 8 ^   ^ N M  2  +2  2 : Before continuing, let us ask tothequantity  r +( r +2 m )exp(1) max f 1  ~  0 g  ^ N M  2  +3  2  (29) tobebounded by1 = 2(thiswillbeused in(30) and (31)). Wecan bound expre s s ion (29) by%:= ^ N M  2  +3  2 witharedenition of ^ N ,andhence, thi s condition (on the s ize of M ) can b e re duce d to%  1 = 2. Wenotethat ^ N only dep ends on theabsolute constants given in thehyp othesis of the lemma, andthi s i s, in f act, theassumptiononthesizeof%that app e ars in those hyp othesis. In whatfollows, wewillredenethevalue of ^ N in order to meet a few more conditions, butthi s re denition will not change thefact thatthenal ^ N i s an absolute constant. From thislastboundoneobtains, kr S k E  ; 8 R ; 8 ^   %  max f 1  ~  0 g  min f = ~  0 g min f  ^  g  (30) and &  ^  exp (1) kr S k E  ; 8 R ; 8 ^   %  (31) where weusethedenition of &  ^  given in (17). From (30) andlemma13 wehavethat S is well dene d (for every ' 2E ), according to (21). From (31) andlemma10 wecan boundthe expre s s ion of $( H  S )thatappear in the transforme d Hamiltonian, k $( H  S ) k E  (1) R (1)  0 @ X j  1 1 j +1  1 2  j ; 1 1 A % k H  k E  ; 8 R ; 8 ^  : and, s imilarly, for theLipschitz constantwe can us e lemma12 to pro duce L E  (1) R (1) f $( H  S ) g  X j  1  1 j +1  &  ^  exp (1)  j ^ F j ; 1 2 ( j ^ L 2 ^ F 1 + ^ L 1 ^ F 2 ) !  with ^ F 1 = k H  k E  ; 8 R ; 8 ^  , ^ F 2 = kr S k E  ; 8 R ; 8 ^  , ^ L 1 = L E  ; 8 R ; 8 ^  f H  g and ^ L 2 = L E  ; 8 R ; 8 ^  fr S g .Then L E  (1) R (1) f $( H  S ) g  0 @ X j  1 j j +1  1 2  j ; 1 1 A &  ^  exp (1) ^ L 2 ^ F 1 + + 0 @ X j  1 1 j +1  1 2  j ; 1 1 A &  ^  exp (1) ^ L 1 ^ F 2 : Withtho s e expre s s ions, tobound$( H  S )is reduce d tobound H  ,withthe only remarkthatthesums P j  1 1 j +1  j ; 1 = ; ln (1 ;  )+   2 and P j  1 j j +1  j ; 1 = (1 ;  )ln(1 ;  )+   2 (1 ;  ) are well dene d for  =1 = 2. A. Jorba and J. Villanueva 37 Now, we can b oundthe transformed Hamiltonian. From thebounds thatcome f rom thesolutions of ( eq 1 ) ; ( eq 5 )wehave: k H  1 k E  ; R ;   ^ N M (  ; 4  ) 2  +3  2 max f 1 = g  and L E  ; R ;  f H  1 g ^ N L (  ; 7  ) 3  +4  3 max f 1  = g : Toobtain these bounds, weuse theexplicit expre s s ion of H  1 given in (23), and lemmas 1, 2, 7 and8toboundthe dierentpartial der ivatives. Weremarkthathere wenee d to us e that j ! j   for any ' 2E .Moreover, f rom theboundfor thePoi s son brackets given in lemmas 9 and11 wehave, for H  2 , k H  2 k E  ; R ;   ^ N M 2 (  ; 4  ) 2  +3  2 max f 1 = g  and L E  ; R ;  f H  2 g ^ N LM (  ; 7  ) 3  +4  3 max f 1  = g : Thetechnique s thatweusetocontrolthereduction in the dierentdomains when weuse Cauchyestimate s, are analogous totheones used in all the previous b ounds. Hence, it i s cle ar thatwe can e stimate H  withan analogous b ounds as theone s for H  1 . Finally,using all those bounds and f rom theexplicit expre s s ions of  (1) , B (1) , C (1) , H (1) 1 and ^ H (1) in (24){(27) it i s not diculttoobtain thenal ^ N suchthatallthebounds in thestatementsof the lemmahold. 5.4 Pro of of thetheorem Wesplit the pro of of thetheorem in s everal parts: in the rst oneweuseonestep of theiterativemethod de scr ib e d in s ection 2.3 as a linear schemetoreduce thesizeof the p erturbation. Then, weintro duce ^ ! as a new parameter todescribe the f amily of lower dimens ional tor i near the initial one. Thenext step i s toapply thebounds of theiterative scheme given by lemma 18, andwe provetheconvergence of thi s scheme for a suitable s et of parameters. Finally,weobtain the dierentestimate s on theme asure of thi s s et. 5.4.1 Linear schemewith re sp ect to " We cons ider the initial Hamiltonian given in the formulation of Theorem 1, andweapply onestep of theiterativemethod de scr ib e d in s ection 2.3. Weremarkthatfromthe Diophantinebounds in thestatementsof thetheorem, we can guarantee thatthi s step i s possible for small enough value s of " ,andthatitkeep s theinitial C 2 dierentiabilitywith re sp ect to " on the transforme d Hamiltonian. Weput H (0) for thi s Hamiltonian that, if we skip the constantterm, lo oks like: H (0) = ! (0) > I + 1 2 z > B (0) ( " ) z + 1 2 ^ I > C (0) (  " ) ^ I + H (0)  (  x ^ I y " )+ " 2 ^ H (0) (  x ^ I y " )  (32) withthe same kindof analytic prop ertie s with re sp ect to(  x ^ I y )astheinitial one, in anew domain thatisindep endenton " (small enough). Weremarkthatthenew matr ice s 38 Pers i stence of Lower Dimens ional Tor i B (0) and C (0) dep endon " ,andthat C (0) dep ends also on  .Moreover, for H  wedo not havethe s emi-normal form conditions given in P1 and P2 . As thi s step come s f rom a p erturbative (line ar) method, wehavethat B (0) ;B , C (0) ;C and H (0)  ; H  are of O ( " ). Our aim i s to rep e atthe sameiterativescheme. We remarkthatinthenext step and in theones thatfollows, we can not guarantee go o d Diophantinepropertie s for thenew e igenvalue s of J m B (0) becaus e thi s matr ix change s ateachstep of theprocess. This isthe re ason that force s us to us e parameters tocontrolthese eigenvalue s. So, we can only work in the s et of parameters for which certain Diophantinebounds hold. But b efore that, we wanttointro duce a new parameter. 5.4.2 Intro duction of thevector of f requencie s as a parameter Here weaddanew parameter tointro duce a f amily of Hamiltonians H (0) .We cons ider value s of ^ ! 2 R r close to^ ! (0) ,and for anyof these value s we p erform thechange given in (14). So, followingthe notation intro duce d in s ection 5.2.4, weput ' > =(^ ! > " )andthen wewritethi s f amily of Hamiltonians in thefollowing form: H (1) (   x I  y  ' ) = ~ ! (0) > ~ I +^ ! (0) > ( ^ I + C ; 1 (^ ! ; ^ ! (0) )) + 1 2 z > B (0) ( " ) z + + 1 2 ( ^ I + C ; 1 (^ ! ; ^ ! (0) )) > C (0) (  " )( ^ I + C ; 1 (^ ! ; ^ ! (0) )) + + H (0)  (  x ^ I + C ; 1 (^ ! ; ^ ! (0) ) y" )+ + " 2 H (0) (  x ^ I + C ; 1 (^ ! ; ^ ! (0) ) y" )  that can b e expanded as H (1) =  (1) ( ' )+ ! > I + 1 2 z > B (1) ( ' ) z + 1 2 ^ I > C (1) (  ' ) ^ I + H (1)  (  x ^ I y ' )+ ^ H (1) (  x ^ I y ' )  (we recall ! > =(^ ! >  ~ ! (0) > )) withanalogous prop ertie s for the dierentterms as in (32), where we can take ^ H (1) of O 2 ( ' ; ' (0) ), ' (0) > =(^ ! (0) >  0). Thi s comes from the s eminormal form structure thatwehavefor H  ,andthefactthat H (0)  is " -clo s e to H  .We also remarkthatwehave dierentiable dep endence of thi s Hamiltonian with re sp ect to ' (in f act it i s analytic with respect to^ ! ), but s ince wewillworkinthefollowingstep s on Cantor s ets, we can not keep thi s kindofdep endence. So, wereplace the dierentiable dep endence by a Lip schitz one, in the s ens e given in s ection 5.1.1. To quantify all these f acts, wetake0 <  1, 0 <R  1and0 <# 1  1, suchthatifweput  (1) =  and R (1) = R ,then wehaveanalogous b ounds as theones de scr ib e d in s ection 5.3 for (19), given by  (1) , R (1) ,and some positive constants  (1) 1 ,  (1) 2 ,  (1) 2 , m (1) ,^ m (1) ,~ m (1) ,^  (1) and ~  (1) on theset E (1) = f ' 2 R r +1 : j ' ; ' (0) j # 1 g ,with re sp ect tothe\unp erturb e d part". For the p erturbation ^ H (1) ,ifweworkwithsetsoftheform  E (1) =  E (1) (  # ):= n ' 2E (1) : j ' ; ' (0) j  # o ,for all0   #  # 1 ,we can replace M and L by N 1  # 2 and N 1  # , for some N 1 > 0. Tosimplify thefollowingbounds weassume, withoutlossof generality,that N 1  1. Finally,we ni sh thi s part with an explicit formulation of thenondegeneracy hyp othesis of the normal e igenvalue s with re sp ect tothe parameters. Let us cons ider B (1) . By construction, wehavethat J m B (1) i s a diagonal matr ix withthe same kind of e igenvalue s A. Jorba and J. Villanueva 39 as thematr ix J m B thatapp e ars in thestatementof thetheorem. Then, us ingthe C 2 dierentiability with re sp ect to ' ,we can wr iteits e igenvalue s as:  (1) j ( ' )=  j + iu j " + iv > j (^ ! ; ^ ! (0) )+ ~  (1) j ( ' )  (33) for j =1 ::: 2 m ,with u j 2 C and v j 2 C r ,andwhere theLipschitz constantof ~  (1) j on  E (1) is of O (  # ). Then, those gener ic nondegeneracy conditions are: NDC For any j suchthatRe  j =0, wehave u j 6 =0 and Re ( v j ) = 2 Z r . Moreover, if we dene u jl = u j ; u l and v jl = v j ; v l ,wehavethese sameconditions for u jl and v jl for any j 6 = l suchthatRe(  j ;  l )= 0. Notethatwehave used the C 2 dep endence on ' to ensure thattheLipschitz constantof ~  (1) j on  E (1) is O (  # ). If thedep endence i s C 1 we can only say thatthi s constantis o (  # ). Nevertheless, it is still possible in thi s cas e toder ivethe sameresultsasinthe C 2 cas e, butthedetails are more te dious. Thenondegeneracy conditions with re sp ect to " are thesameone s us e d in 23] tostudy the quas ip er io dic p erturbations of elliptic xe d p oints, andthenondegeneracy conditions with respect tothe^ ! -dep endence are analogous totheones app e are d in 26] and 9], but in those cases they were formulated for an unp erturb e d system havingan r -dimens ional analytic f amily of r -dimens ional re ducible elliptic tor i. 5.4.3 Inductive part Wewanttoapply here theiterative lemmain an inductive form. For this purpose, we dene  n = 6  2 n 2 for every n  1, andwenotethat P n  1  n =1. From thi s denition, we put  n =  n  18 , ^  n =  n R 18 andweintro duce  ( n +1) =  ( n ) ; 9  n and R ( n +1) = R ( n ) ; 9 ^  n for every n  1. We also cons ider a xe d 1 <%< 2, todene  n =exp( ; % n )  0 . We suppose that, atstep n ,wehave a Hamiltonian H ( n ) like H (1) dened for ' in a set E ( n ) E (1) , withanalogous b ounds as H (1) , replacingthe sup erscr ipt (1) by( n )inthe unp erturb e d part, andwithbounds for the p erturbation given by  M n = M n (  # )= N n  # 2 n and  L n = L n (  # )= N n  # 2 n ; 1 ,inevery s et of theform  E ( n ) (  # ), for all 0   #  # 1 ,being N n indep endenton  # .Wewillshowthatthi s i s p o s s ible if # 1 is small enough, withconditions on # 1 that are indep endenton theactual step. Atthi s p oint, wedenethenew s et E ( n +1) of go o d parameters f rom E ( n ) lo okingat thenew Diophantine conditions. Wehavethat ' 2E ( n +1) if ' 2E ( n ) andthefollowing conditions hold: j k > ! + l >  ( n ) ( ' ) j  n j k j  1 exp ( ;  n j k j 1 )  (34) for all k 2 Z r + s nf 0 g , l 2 N 2 m , j l j 1  2. Now, weuse theiterative lemmafor ' 2E ( n +1) .Firstwe remarkthatatevery step wehave  ( n )   , R= 2  R ( n )  R ,  n = ^  n = =R ,  n   0 and, as # 1  1, wehavethat for every ' 2E (1) , j ! j max fj ~ ! (0) j  j ^ ! (0) j +1 g . Moreover, weassumethatwe can b ound  (1) 1 = 2   ( n ) 1 ,  ( n ) 2  2  (1) 2 ,  ( n ) 2  2  (1) 2 , m ( n )  2 m (1) ,^ m ( n )  2^ m (1) ,~ m ( n )  2~ m (1) , ^  ( n )  2^  (1) ~  ( n )  2~  (1) and N n # 2 n ; 2 1  N 1 .Weremarkthatallthose bounds hold for n =1. Then we cons ider the constant ^ N ,given bytheiterativelemma, corre sp ondingto these bounds. 40 Pers i stence of Lower Dimens ional Tor i If weassumethatinthe actual step wehavefor % n := ^ N N n # 2 n 1  2  +3 n  2 n ,% n  1 = 2, then we can apply theiterativelemmatoobtain thegeneratingfunction S ( n ) (  x ^ I y ' ), with kr S ( n ) k E ( n +1)  ( n ) ; 8  n R ( n ) ; 8 ^  n  min f  n  ^  n g .So,inthi s cas e wehave for  S ( n ) ^  S ( n ) : U r + s 2 m + r  ( n +1) R ( n +1) ;!U r + s 2 m + r  ( n ) ; 8  n R ( n ) ; 8 ^  n : Thenext step i s toboundthe transforme d Hamiltonian H ( n +1) = H ( n )   S ( n ) .Wework in a s et of theform  E ( n +1) , for all 0 <  #  # 1 .From thebounds of theiterative lemma, andtheexplicit expre s s ions of  n ,  n and  n ,wecan deduce thatthere exi sts ~ N (we can as sume ~ N  1) dep endingon thesame constantsas ^ N ,suchthat kr S ( n ) k  E ( n +1)  ( n ) ; 8  n R ( n ) ; 8 ^  n  ~ Nn 4+4  (exp ( % n )) 2 N n  # 2 n  k  ( n +1) ;  ( n ) k  E ( n +1)  ~ Nn 2+2  exp ( % n ) N n  # 2 n  kB ( n +1) ;B ( n ) k  E ( n +1)  ~ Nn 2+2  exp ( % n ) N n  # 2 n  L  E ( n +1) fB ( n +1) ;B ( n ) g  ~ Nn 4+4  (exp ( % n )) 2 N n  # 2 n ; 1  kC ( n +1) ;C ( n ) k  E ( n +1)  ( n +1)  ~ Nn 4+4  (exp ( % n )) 2 N n  # 2 n  L  E ( n +1)  ( n +1) fC ( n +1) ;C ( n ) g  ~ Nn 6+6  (exp ( % n )) 3 N n  # 2 n ; 1  k H ( n +1)  ; H ( n )  k  E ( n +1)  ( n +1) R ( n +1)  ~ Nn 6+4  (exp ( % n )) 2 N n  # 2 n  L  E ( n +1)  ( n +1) R ( n +1) f H ( n +1)  ; H ( n )  g  ~ Nn 8+6  (exp ( % n )) 3 N n  # 2 n ; 1  k ^ H ( n +1) k  E ( n +1)  ( n +1) R ( n +1)  ~ Nn 12+8  (exp ( % n )) 4 N 2 n  # 2 n +1  L  E ( n +1)  ( n +1) R ( n +1) f ^ H (1) g  ~ Nn 14+10  (exp ( % n )) 5 N 2 n  # 2 n +1 ; 1 : Moreover, weassumethatwecanbound% n  ~ Nn 6+4  (exp ( % n )) 2 N n # 2 n 1 , withthe same constant ~ N . Then, weuse allthe s e expre s s ions as a motivation todene N n +1 = ~ Nn 14+10  (exp ( % n )) 5 N 2 n , for n  1. Toboundhowfast N n +1 grows with n and N 1 we us e lemma 14: N n  1 ~ N   5 3  14+10  ~ NN 1 exp  5 % 2 ; % !! 2 n ; 1  if n  1. If we also dene ~ N n +1 = ~ Nn 8+6  (exp ( % n )) 5 N n ,for n  1, we cle arly have, us ing that N 1  1and ~ N  1, that ~ N n  N n for n  2. Now, wehaveto justify thatwecan use theiterative lemmain thi s inductive form when n  2. Tothi s endwenee d to s ee thatthebounds thatwehaveassumed atthestep n (todene ^ N andtousetheiterativelemma) hold atevery step if # 1 is small enough. So, we notethatif # 1 is small enough, thefollowingsum: X n  1 N n +1 # 2 n ; 2 1  (35) is bounded by ^ N  thatdep ends on % andthesameconstantsas ^ N . Thi s b oundisnot dicultto obtain if welook athowfast ~ N n grows. Moreover thesameideas can be used to provethat N n # 2 n ; 2 1  N 1 ,if n  1and # 1 is small enough. Then, we can dene  ( n +1) 1 =  ( n ) 1 ; 2 N n +1 # 2 n 1 ,  ( n +1) 2 =  ( n ) 2 +2 N n +1 # 2 n 1 ,  ( n +1) 2 =  ( n ) 2 + N n +1 # 2 n ; 1 1 ,^ m ( n +1) =^ m ( n ) + N n +1 # 2 n 1 ,~ m ( n +1) =~ m ( n ) + N n +1 # 2 n ; 1 1 ,^  ( n +1) =^  ( n ) + N n +1 # 2 n 1 and~  ( n +1) =~  ( n ) + N n +1 # 2 n ; 1 1 ,thatfromtheconvergence of (35) allows toapply another A. Jorba and J. Villanueva 41 step of theiterativescheme, at le ast for suciently small value s of # 1 . Moreover, it i s cle ar that% n  N n +1 # 2 n 1  ^ N  # 2 1  1 = 2taken # 1 small enough. Then, it only re amains tobound  m ( n +1) .For that purpose we rst cons ider thebound k  C ( n +1) ;  C ( n ) k E ( n +1)  ( n +1)  N n +1 # 2 n 1 , andthen, if wework witha xedvalue of ' 2  E ( n +1) ( # 1 ), wehavefor any W 2 C r : j  C ( n +1) W jj  C ( n ) W j;      C ( n +1) ;  C ( n )  W        m ( n )  ; 1 ; N n +1 # 2 n 1  j W j : We notethat, f rom the equivalence j   C ( n )  ; 1 j  m ( n ) ()j  C ( n ) W j   m ( n )  ; 1 j W j ,for any W 2 C r ,we can take m ( n +1) =  m ( n ) 1 ;  m ( n ) N n +1 # 2 n 1 ,provided that m ( n ) N n +1 # 2 n 1 < 1. Then, us ingthis expression wecanseethat m ( n )  2 m (1) for any n  1, if # 1 is small enough: if weassumethatitholds for n ,when we compute m ( n +1) wehavethat  m ( n ) N n +1 # 2 n 1  2 m (1) N n +1 # 2 n 1  1 2  if # 1 is small enough. Moreover, wehavebyinduction that  m ( n +1)   m (1) n Y j =1 1 1 ; 2 m (1) N j +1 # 2 j 1 : So, it i s cle ar that, if # 1 is small enough, X j  1 2 m (1) N j +1 # 2 j 1  2 m (1) ^ N  # 2 1  1 2 ln (2)  andhence, if wenotethatwhen 0  X  1 = 2, ln  1 1 ; X  =ln  1+ X 1 ; X   X 1 ; X  2 X we can b oundln   m ( n +1)   ln (  m (1) ) + ln (2), thatproves  m ( n +1)  2 m (1) . 5.4.4 Convergence of thechangesofvar iable s Now, we are goingto provetheconvergence of the composition of change s of var iable s. Let E  = \ n  1 E ( n ) be the s et of ' where everythingiswell dene d for all thestep s. We cons ider a xe d ' 2E  ,but in f act, the re sultswillhold in thewhole s et E  provided that # 1 is small enough. Weput )  ( n ) = ^  (1)  :::  ^  ( n ) for n  1, thatgoes from U r + s 2 m + r  ( n +1) R ( n +1) to U r + s 2 m + r R , where ^  ( n ) means ^  S ( n ) .Then, if p>q  1, wehave )  ( p ) ; )  ( q ) = p ; 1 X j = q  )  ( j +1) ; )  ( j )  : Tobound )  ( j +1) ; )  ( j ) ,wedene  0 j =  ( j ) ; = 4and R 0 j = R ( j ) ; R= 4, andweput ^ & R = 1 exp (1)  + r +2 m R .Now, let us s ee that j )  ( j +1) ; )  ( j ) j  0 j +2 R 0 j +2 = j ^  (1)  :::  ^  ( j +1) ; ^  (1)  :::  ^  ( j ) j  0 j +2 R 0 j +2    1+4 ^ & R j ^ # (1) j  (2) R (2)  j ^  (2)  :::  ^  ( j +1) ; ^  (2)  :::  ^  ( j ) j  0 j +2 R 0 j +2  (36)