Exponentially small splitting and Arnold di(cid:11)usion for multiple time s ale systems Mi hela Pro esi Abstra t We onsider the lass of Hamiltonians: 1Pn(cid:0)1 2 1 2 p2 2 Pn 2 j=1 Ij + 2"In+ 2 +"[( osq(cid:0)1)(cid:0)b ( os2q(cid:0)1)℄+"(cid:22)f(q) i=1sin( i); 1 iq where 0 (cid:20) b < 2, and the perturbing fun tion f(q) is a rational fun tion of e . We prove upper and lower bounds on the splitting for su h lass of systems, in regions of the phase spa e hara terizedbyonefastfrequen y. FinallyusinganappropriateNormalFormtheoremweprove theexisten e of hains of hetero lini interse tions. Contents 1 Presentation of the model and main Theorems. 1 2 Perturbative onstru tion of the homo lini traje tories 5 t 2.1 Whisker al ulus, the \primitive" = . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 The re ursive equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Proofs of the Theorems 9 3.1 The formal linear equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Lower bounds on the Melnikov term . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Hetero lini interse tion for systems with one fast frequen y . . . . . . . . . . . . . . . 13 4 Tree representation 16 4.1 De(cid:12)nitions of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Admissible trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.3 Values of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.4 Tree identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.4.1 Mark adding fun tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.4.2 Fruit adding fun tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.4.3 Changing the (cid:12)rst node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.5 Upper bounds on the values of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 A Appendix 31 A.1 Proof of proposition 4.16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 A.2 Normal form theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 A.3 Proof of Lemma 4.23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1 Presentation of the model and main Theorems. Thegeneralsettingofthispaperistheproblemofhomo lini splittingandArnol'ddi(cid:11)usionina-priori stable systems with three or more relevant time s ales. The general strategy is the one proposed in [A2℄and[CG℄andinparti ulartheappli ationtoa-prioristablesystemsproposedin[G1℄andfurther 2 1 Presentation of the model and main Theorems. developed in [GGM1℄. More pre isely we onsider a lass of lose to integrable n degrees of freedom Hamiltonian systems for whi h one an prove the existen e of n(cid:0)1 dimensional unstable KAM tori togetherwith their stable and unstable manifolds. We use a perturbative diagrammati onstru tion (proposedanddevelopedin[C℄,[G1℄and[GGM1℄)toproveupperboundsontheanglesofinterse tion ofthestableandunstablemanifoldsofaKAMtorus(homo lini splitting). Su hboundsaregenerally exponentially small in the perturbation parameter and depend on the hosen torus and in parti ular onthenumberoffastdegreesoffreedom. Forsystemswithonefastdegreeoffreedomweproveaswell lowerbounds onthe homo lini splitting through the me hanismof Melnikovdominan e. Finally for su hsystemsweprovetheexisten eof\long" hainsofhetero lini interse tions;namelyweprodu e a list of unstable KAM tori T1:::;Th su h that T1;Th are at distan es of order one in the a tion variables and the unstable manifold of ea h Ti interse ts the stable manifold of Ti+1. This paper is a generalization of the results of [G1℄,[GGM1℄,[GGM2℄, therefore in proving our laims we will rely heavily on intermediate results proved in the latter papers whi h we will not prove again. Consider the lass of Hamiltonians: 1nX(cid:0)1I~2+ 1"I~n2+ p~2 +"[( osq~(cid:0)1)(cid:0) 1(cid:0) 2( os2q~(cid:0)1)℄+"(cid:22)f(q~)Xn sin ~i; (1.1) 2 2 2 4 j=1 i=1 where the pairs I~ 2 Rn; ~ 2 Tn and p~ 2 R;q~ 2 T are onjugate a tion-angle oordinates, 0 < (cid:20) 1,f(q~) is odd and analyti on the torus and (cid:22)," are small parameters. We will onsider them P independent and then prove that one an prove Arnold Di(cid:11)usion for (cid:22)(cid:20)" , for an appropriate P. This lass of Hamiltonians is a model for a near to integrable system lose to a simple resonan e where the dependen e on the hyperboli variables is not through the standard pendulum, but still maintainsvariousqualitativepropertiesofthependulum. Namelywehavea\generalizedpendulum": 2 2 p~ 1(cid:0) +"[( osq~(cid:0)1)(cid:0) ( os2q~(cid:0)1)℄ 2 4 p whi h has an unstable (cid:12)xed point in p~=q~=0 with Lyapunov exponent (cid:21)= ". Generally one re-s ales the time and a tion variables so that the Lyapunov exponent is one: I(t) = I~( ppt"); (t)= ~( pt ); p(t) = p~( ppt"); q(t)=q~( pt ): (1.2) " " " " Su h res aling sends Hamiltonian 1.1 in: (I;A(")I) p2 1 1(cid:0) 2 Xn 2 + 2 + 2[( osq(cid:0)1)(cid:0) 4 ( os2q(cid:0)1)℄+(cid:22)f(q) sin( i) (1.3) i=1 where A(") is the diagonal matrix with eigenvalues ai = 1 for i = 1;:::;n(cid:0)1 and an = ". So from nowonwewillworkonHamiltonian1.3andturnba ktoHamiltonian1.1onlytoprovetheexisten e of hetero lini hains. The system (1.3) is integrable for (cid:22) = 0. It represents a list of n un oupled rotators and a generalized pendulum (depending on the parameter ). We will denote the frequen y of the rotators (whi h determines the initial data I(0)) by ! so that: (cid:0)1 I(t)=I(0)=A !; (t)= (0)+!t: The initial data are hosen in an appropriate domain (physi ally interesting in the variables I~) so thatthereareatleastthree hara teristi ordersofmagnitudeforthe frequen iesof theunperturbed system. 3 De(cid:12)nition 1.1. In frequen y spa e we (cid:12)rst onsider the ellipsoid: (cid:8) Xn (cid:9) n 2 (cid:6):= x2R : xi=ai =2E i=1 1 where E is an order one onstant E (cid:24)O"(1). For notational onvenien e we split the frequen y ! in two ve torial omponents: ! =(p!1";"(cid:11)!2) with m n(cid:0)m 1 !1 2R , !2 2R , and 0(cid:20)(cid:11)(cid:20) 2. Finally, given two suitable order one onstants R;r (cid:24)O"(1), we onsider the region: p p p n (cid:11) (cid:11) (cid:10)(cid:17)f!2R : "! 2(cid:6) ;r <j!1;ij<R and r <j!2j<R ; " j!2;ij(cid:21) "; " j!2;n(cid:0)mj(cid:24) "g: We have hosen the generalized pendulum so that its dynami s on the separatrix is parti ularly 2 simple , namely: 1 iq(t) sinh((cid:6)t)+i q(t)=2ar otg( sinh((cid:6)t)); e = : (1.4) sinh((cid:6)t)(cid:0)i There areat least three hara teristi time s alesO"("(cid:0)21), O"("(cid:11)), O"(p") ( oming from the degen- erate variable In) and 1 whi h is the Lyapunov exponent of the unperturbed pendulum. m Wewill all 1;(cid:1)(cid:1)(cid:1) ; m thefastvariablesandwewillsometimesdenotethemas F 2T . Conversely n(cid:0)m we will all m+1;(cid:1)(cid:1)(cid:1) ; n the slow variables S 2T . Theperturbingfun tionisatrigonometri polynomialofdegreeoneintherotators andarational 3 iq fun tion in e . We have de oupled the dependen e of and q only to simplify the omputations. n For ea h ! 2R the unperturbed system has an unstable (cid:12)xed torus : (cid:0)1 p(t)=q(t)=0; I(t)=I(0)=A !; (t)= (0)+!t: Thestableandunstablemanifoldsofsu htori oin ideand anbeexpressedasgraphsontheangles. 1 De(cid:12)nition 1.2. Given any (cid:13) 2R, "<(cid:13) (cid:20)O("2) and a (cid:12)xed (cid:28) >n(cid:0)1, we de(cid:12)ne the set n o (cid:13) n (cid:10)(cid:13) (cid:17) ! 2(cid:10):j!(cid:1)lj> jlj(cid:28) 8l2Z =f0g of (cid:13);(cid:28) Diophantine ve tors in (cid:10). Now we onsider (cid:3) 1 1 (cid:10)(cid:13) (cid:17)(cid:10)(cid:13) (cid:2)((cid:0) ; ) 2 2 (cid:3) and for all (!;(cid:26))2(cid:10)(cid:13) we set !(cid:26) =(1+(cid:26))!. (cid:3) n (cid:13) For all (!;(cid:26))2(cid:10)(cid:13) and for all l2Z =f0g j!(cid:26)(cid:1)lj> 2jlj(cid:28). !2(cid:10)(cid:13) implies that !1 and !2 are Diophantine as well; we will all (cid:28)F and (cid:28)S their exponents. 2 KAM like theorems (see [CG℄,[C℄) imply that there exists (cid:22)0(";(cid:13)) (cid:24) " su h that if j(cid:22)j (cid:20) (cid:22)0 (cid:3) and if (!;(cid:26))2(cid:10)(cid:13), there exists one and only one n-dimensional H(cid:22)-invariant unstable torus T(cid:22)(!;(cid:26)) n whose Hamiltonian (cid:13)ow is analyti ally onjugated to the (cid:13)ow T 3 # !#+!(cid:26)t. Moreover one an (cid:6) parameterizethestableandunstablemanifoldsofT(cid:22)(!;(cid:26))byfun tionsI (!;';q;(cid:22)), analyti inthe n 3 3 last three arguments, with ';q 2T (cid:2)[(cid:0) (cid:25); (cid:25)℄. Namely given 2 2 (cid:0) (cid:1) (cid:6) (cid:6) (cid:6) z (!;';q;(cid:22))= I (!;';q;(cid:22));p (!;';q;(cid:22));';q ; 1nowandinthefollowingwewillsaya(")(cid:24)O"(f("))if lim"!0+ fa(("")) =L6=0. 2The motion on the separatrix an be easily obtained by dire t omputation; the mainfeature is that the motion ontheseparatrixissu hthateiq(t) isarationalfun tionofet. Hereweare onsideringthesimplest lassofexamples, whi h ontainsthestandardpendulum =1. 3A tuallyitissuÆ ientthatthesingularityoff( (t);q(t)),whi hisnearesttotherealaxisispolarandisolated. 4 1 Presentation of the model and main Theorems. 4 where the pendulum a tion is derived by energy onservation, the traje tory : (cid:26) t + (cid:8)Hz (!;';q;(cid:22)) if t>0 z(!;';q;(cid:22);t)= t (cid:0) (cid:8)Hz (!;';q;(cid:22)) if t<0 tends exponentially to a quasi-periodi fun tion of frequen y !. 5 Remark 1.3. We have introdu ed the variable (cid:26) in order to (cid:12)x the energy of the perturbed system , namely given a list of !i 2(cid:10)(cid:13) one an (cid:12)nd (cid:26)(!i;(cid:22)) su h that all the orresponding whiskered tori are on the same energy surfa e, see for instan e [C℄. De(cid:12)nition 1.4. We will study the di(cid:11)eren e between the stable and unstable manifolds on an hyper- plane transverse to the (cid:13)ow (a Poin ar(cid:19)e se tion), we hoose the hyper-plane q =(cid:25) and onsequently drop the dependen e on q. We all 0 1 (cid:0) (cid:0) + (cid:1) Gj(';!)= aj Ij(';!;0 )(cid:0)Ij(';!;0 ) 2 0 the splitting ve tor and prove that Gj('=0;!)=0. A measure of the transversality is 0 0 (cid:1)ij ='jGi(')j'=0 alled splitting matrix. We will prove the following theorems: 0 6 Theorem 1. The splitting matrix (cid:1) satis(cid:12)es the formal power series relation : 0 0 (cid:1) (cid:24)AD B 0 where A;B are lose to identity matri es and D is the \holomorphi part" of the splitting matrix; namely its entries are expressed as integrals over R of analyti fun tions. Moreover the formal power 7 series involved are all asymptoti This statement was posed as a onje ture in [G2℄ Paragraph3. Corollary 1.5. The pre eding Theorem implies that Hamiltonian (1.3), in regions of the a tion vari- ables orrespondingtom6=0fasttimes ales,hasexponentiallysmallupperboundsonthedeterminant of the splitting matrix: jdet(cid:1)0j(cid:20)Ce(cid:0)" b ; withb= 1 ; 2m provided that (cid:22)<"1+2mn . Noti e that Theorem 1 an be proved for mu h more general systems than model (1.3). Theorem2. Consider Hamiltonian 1.3 inregions of thea tion variables orresponding tom=1fast variables and for perturbing fun tions f(q) su h that the pole f(q(t)) losest to the imaginary axis, say t(cid:22), is su h that jIm t(cid:22)j=d(cid:20)ar sin . Setting (cid:22)(cid:20)"P with P =p=2+8+4n where p is the degree of the pole of f(q(t)) in t(cid:22)we prove that: C1"(cid:0)p1e(cid:0)djp!"1j (cid:20)jdet(cid:1)0j(cid:20)C2"(cid:0)p2e(cid:0)djp!"1j where C1;C2;p1;p2 are appropriate order one onstants. 4(cid:8)tH istheevolutionattimetoftheHamiltonian(cid:13)ow1.3. 5The(cid:12)nalgoalisto(cid:12)ndhetero lini interse tionsonthe(cid:12)xedenergysurfa e,andso\Arnolddi(cid:11)usion",butinthe followingse tionswewilldis ussonlyhomo lini interse tionsandsowewilldroptheparameter(cid:26) 6wedenoteformalpowerseriesidentitieswiththesymbolA(cid:24)B P 7Aformalpower series (cid:22)nan(") isasymptoti iffor allq>0 there exists Q>0 su h that foralln(cid:20)"(cid:0)q then an(")(cid:20)"(cid:0)Qn. 5 Corollary 1.6. Under the onditions of Theorem 2 the Hamiltonian (1.1) has hetero lini hains, 1 N 8 namely a set of N (cid:21) 1 traje tories z (t);:::;z (t) together with N + 1 di(cid:11)erent minimal sets T0;:::;TN su h that for all 1(cid:20)i(cid:20)N i i lim dist (z (t);Ti(cid:0)1)=0= lim dist (z (t);Ti): t!(cid:0)1 t!1 Moreover one an onstru t su h hains between tori T(!a;(cid:22)), T(!b;(cid:22)) su h that !a;!b 2 (cid:10)(cid:22) (cid:26) (cid:10)(cid:13) and j"(cid:0)12(!na (cid:0)!nb)j(cid:24)O"(1): Thete hniquesusedforprovingtheTheoremsarethoseproposedin[G1℄anddevelopedin[GGM1℄ for partially iso hronous three time s ale systems with three degrees of freedom. In this paper, par- ti ularattentionisgiventotheformalizationofthetreeexpansionsandofthe\Dysonequation"and relative an ellations proposed in[GGM1℄. This enables us to extend Theorem 1 to systems with n degrees of freedom and at least two time s ales; moreover the proof is de(cid:12)nitely simpli(cid:12)ed and quite ompa t. Inthisarti lewehave onsidered ompletelyaniso hronoussystemsonlyto(cid:12)xanexample; generalizingto partially(or totally, thus re overingthe results of [BB1℄) iso hronoussystems is om- pletely trivial. Indeed Theorem 1 and hen e Corollary1.5 an be provedfor very generalsystems, as we will show in a forth oming paper. Moreover we have generalized the lass of perturbing fun tions and the \ pendulum"(the literature onsiders only trigonometri polynomials and the standard pendulum); the latter generalizationsare quite te hni al but nevertheless non trivial and interesting, we think, as the te hniques we propose areeasilygeneralizableandgivea learpi ture ofthe limits of provingArnolddi(cid:11)usion viaMelnikov dominan e. 2 Perturbative onstru tion of the homo lini traje tories One an use perturbation theory to (cid:12)nd the (analyti for (cid:22)(cid:20)(cid:22)0) traje tories on the S/U manifolds 9 of Hamiltonian (1.3) X k k z(';!;t)= ((cid:22)) z (';!;t): k Namely we insert the expansion in (cid:22) in the Hamilton equations of system (1.3): I_j =(cid:0)((cid:22)) os jf(q); _j =ajIj; 1 2 Xn df (2.1) p_ = 2 sinq(1(cid:0)(1(cid:0) ) osq)(cid:0)((cid:22)) sin idq(q); q_=p; i=1 (cid:6) (cid:6) and (cid:12)nd initial data I(!;';(cid:22);0 ) (and onsequently p(!;';(cid:22);0 )) su h that the solution of (2.1) tends exponentially to a quasi-periodi fun tion of frequen y !. Inserting in the Hamilton equations the onvergentpower series representation: P1 k k P1 k k I(t;';(cid:22))= k=0((cid:22)) I (t;'); (t;';(cid:22))= k=0((cid:22)) (t;'); P1 k k 0 P1 k k p(t;';(cid:22))= k=0((cid:22)) p (t;'); q(t;';(cid:22))=q (t)+ k=1((cid:22)) 0(t;') 8A losed subset of the phase spa e is alled minimal(with respe t to a Hamiltonian (cid:13)ow (cid:30)th) if it is non-empty, invariant for (cid:8)th and ontains a dense orbit. In our ase the minimal sets will be unstable tori T(I) with !(I) Diophantine. 9Noti ethattheapexkonthefun tionsI; representstheorderintheexpansionin(cid:22)NOTanexponent. Toavoid onfusion,whenweneedtoexponentiate wealwayssettheargumentinparentheses. 6 2 Perturbative onstru tion of the homo lini traje tories 10 we obtain, for k >0, the hierar hy of linear non-homogeneousequations : I_jk =Fjk(f ihgi=0;:::;n); _jk =ajIjk; forj =1;:::;n; h<k p_k = 12(cid:16) os(q0(t))(cid:0)(1(cid:0) 2) os(2q0(t))(cid:17) 0k+F0k(cid:0)f ihgi=h0<;::k:;n(cid:1); _0k =pk; (2.2) k 1 dk where the fun tions Fi are de(cid:12)ned as follows. Set: [(cid:1)℄k = k!d(cid:22)k((cid:1))j(cid:22)=0; we have: h kX(cid:0)1 i h kX(cid:0)1 i Fjk(t)=(cid:0)  jf1( ((cid:22))h ~h(t)) (cid:0)Æj0  0f0( ((cid:22))h 0h(t)) ; j =0;:::;n k(cid:0)1 k h=1 h=1 where ~h(t) is the ve tor 0h(t);:::; nh(t), f1( ~)=Xn sin if( 0); f0( 0)= 12(cid:0)( os 0(cid:0)1)+ 1(cid:0)2 2 sin2 0(cid:1); i=1 (cid:12)nally Æji denotes the Krone kerdelta. For k =0 we obtain the unperturbed homo lini traje tory: 0 (cid:0)1 0 0 z (t)=(A !;p (t);'+!t;q (t))); 0 0 (q (t);p (t)) is the lower bran h of the pendulum separatrix starting at q = (cid:25) written in Equation (1.4). For k >0 we have a linear non-homogeneous ODE that we an solve by variation of onstants. The fundamental solution of the linearized pendulum equation is given by: (cid:12) (cid:12) W(t)=(cid:12)(cid:12)(cid:12)ww_00 xx_0000(cid:12)(cid:12)(cid:12) ; w0 = 21(cid:27)(t)x10 where (cid:27)(t)= sign(t) 0 2 osh(t) 1 (cid:27)(t)x00 (cid:16) (cid:0) 2(cid:1) (cid:0) 2(cid:1)2 (cid:17) x0 = 2+sinh(t)2 ; x0 = 2 4 2 (cid:0)3+4 t+sinh(2t)+4 (cid:0)1+ tanh(t) : (2.3) It is easily seen (see [G1℄ or [C℄) that one an hoose an appropriate \primitive" in the right hand side of the (cid:12)rst olumn of equations (2.2) so that the solutions are exponentially quasi-periodi . t 2.1 Whisker al ulus, the \primitive" = Let us (cid:12)rst de(cid:12)ne the fun tion spa es on whi h we work, all the de(cid:12)nitions and statements of this Subse tion and of the following one are proposed and explained in detail in [G1℄, we are simply reformulating them to suit our needs. De(cid:12)nition 2.1. (i) H is the ve tor spa e (on C) generated by monomials of the form: j ajtj h i('+!t)(cid:1)(cid:23) n m = (cid:27)(t) x e whereh2Z; (cid:23) 2Z ; j 2N; j! (cid:0)jtj x=e ; a=0;1; (cid:27)(t)= sign(t): (2.4) (ii) Given two positive onstants b and d, H(b;d) is the subset of fun tions f(t) analyti on the real axis in t6=0 that admit, separately for t>0 and t<0, a (unique) representation: Xk jtjj (cid:27)(t) f(t)= Mj (x;'+!t); (2.5) j! j=0 (cid:27)(t) (cid:27)(t) with Mj (x;') trigonometri polynomials in ' and the fun tion Mk not identi ally zero . 10whenitisnotstri tlyne essarywewillomitthepre(cid:12)xedinitialdataoftheangles'= 1(0);(cid:1)(cid:1)(cid:1); n(0); 0(0)=(cid:25) t 2.1 Whisker al ulus, the \primitive" = 7 (cid:27)(t) The Fourier oeÆ ients Mj(cid:23) (x) are all holomor- phi in the x-plane in a region eb (cid:0)b f0<jxj<e g[fjargxj<dg d and have possible polar singularities at x=0. k is alled the t degree of f. In Figure1 we haverepresenteda possible domain of analyti ity for the M(cid:23)j Noti e that H is on- tained in all the spa es H(b;d); Figure 1: moreover if jtj > b, f(t) an be represented as an absolutely onvergent series of monomials of the type m, separately for t > b and t < (cid:0)b. One an easily he k that the fun tional that a ts on monomials m of the form (2.4) as: ( a+1 h i( +!t)(cid:1)(cid:23)Pj jtjj(cid:0)p t (cid:0)(cid:27) x e p=0 (j(cid:0)p)!(h(cid:0)i(cid:27)!(cid:1)(cid:23))p+1 if jhj+j(cid:23)j6=0 = (m)= (cid:27)a+1jtjj+1 (2.6) (cid:0) ifjhj+j(cid:23)j=0 (j+1)! is a primitive of m. t We an extend = , with jtj > b, to a primitive on fun tions f 2 H(b;d) by expanding f in the monomials m ( we obtain absolutely onvergentseries) and applying (2.6). Then if jtj(cid:20)b we set Zt t 2(cid:27)(t)b = (cid:17)= + ; (2.7) 2(cid:27)(t)b obviously the hoi e of 2b is arbitrary and this is still the same primitive of f. t In H(b;d) we an extend = to omplex values of t su h that t2C(b;d) where: C(b;d):=ft2C :j Im tj(cid:20)d; j Re tj(cid:20)bg[ft2C :j Im tj(cid:20)2(cid:25); j Re tj>bg; is the domain in Figure 1 in the t variables. t An equivalent (and quite useful) de(cid:12)nition of = is I Zt t du (cid:0)(cid:27)((cid:28))u(cid:28) = f = e f((cid:28))d(cid:28); (2.8) 2i(cid:25)u (cid:27)(t)1+is where(cid:27)(t)= sign(Ret), t=t1+is, with t1;s2R and the integralisperformed on the line Im(cid:28) =s; (cid:12)nally the integrals in u have to be onsidered to be the analyti ontinuation on u from u positive and large. This de(cid:12)nition is learly ompatible with the formal de(cid:12)nition given above and one easily sees that t H(b;d) is losed under the appli ation of = . De(cid:12)nition 2.2. H0(b;d) is the subspa e of H(b;d) of fun tions that an be extended to analyti fun tions in C(b;d). Noti e that f is in H0(b;d) if it is in H(b;d) and f(t) is analyti at t=0. Remark 2.3. If f 2 H0(b;d) then generally =f 2= H0(b;d) and has a dis ontinuity in t = 0. For instan e if f 2L1 is positive, then: Z1 0(cid:0) 0+ =(f):=(= (cid:0)= )f = f 6=0: (cid:0)1 8 2 Perturbative onstru tion of the homo lini traje tories 0(cid:0) 0+ We an onstru t operators whi h preserve H0(b;d); let === (cid:0)= and (cid:26) (cid:26) t t t = if t(cid:21)0 t = if t(cid:20)0 =+ = t =(cid:0) = t = (cid:0)= if t<0; = += if t>0: The operator 1 X t t 1 =(cid:26) == (cid:0) (cid:27)(t)= 2 2 (cid:26)=(cid:6)1 preserves the analyti ity. Now let us ite two important properties of H0(b;d), proved in [G1℄. Lemma 2.4. In H0(b;d) we have the following shift of ontour formulas: 8f 2H0(b;d) and for all d>s2R (i) =f((cid:28))==f((cid:28) +is); I Zt X t+is dR X (cid:0)R(cid:27)((cid:28))((cid:28)+is) (ii) =(cid:26) f((cid:28))= e f((cid:28) +is)d(cid:28): 2i(cid:25)R (cid:26)=(cid:6)1 (cid:26)=(cid:6)1(cid:26)1 2.2 The re ursive equations 1 0 One an easily verify that f ( 0(t);q0(t)) and f (q0(t)) are in H0(a;d) (and bounded at in(cid:12)nity) for some \optimal" values a;d orresponding respe tively to the maximal distan e from the imaginary axis and the minimal distan e form the real axis of the poles of su h fun tions. One an prove by indu tion, see [G1℄ or [C℄ for the details, that the solutions of equations (2.2) tend to quasi-periodi fun tions provided that the initial data are hosen to be: k (cid:6) X k 0(cid:6) k (cid:6) X k 0(cid:6) 0 k Ij(';!;0 )= (cid:22) = Fj ; p(';!;0 )= (cid:22) = x0F0: k k k Moreover one an prove that Fj (';!;t) has no onstant omponent. Consequently it is onvenient t to express the traje toriesin terms of the \primitives" = in the form (a0 =1): k k k t k 0 1k 1 0k ((cid:22)) j(';t)=((cid:22)) ajQjFj +xjGj +xjGj 0 1 i where xj =1, xj =jtj for j 6=0 while the x0 are de(cid:12)ned in equation 2.3, t 1 t t h(cid:0) 0 1 1 0 (cid:1) i ik k1 i k Qj[f℄= (=++=(cid:0)) xj(t)(cid:27)((cid:28))xj((cid:28))(cid:0)xj(t)(cid:27)(t)xj((cid:28)) f((cid:28)) ; Gj =((cid:22)) aj=xjFj: 2 2 For the proofs of these assertionssee [G1℄ or [C℄. Noti e that by our de(cid:12)nitions: X (cid:0) + (cid:0)1 0k (cid:0)1 0 0 (cid:0) + Ij(';0 )(cid:0)Ij(';0 )=2aj Gj (cid:17)2aj Gj ; 2G0 =p(';0 )(cid:0)p(';0 ) k We de(cid:12)ne the formal power series X lk l Gj (')(cid:17)Gj(');j =0;:::;n; l=0;1 k 0 Noti e that by the KAM theorem the Gj are onvergentseries. 9 Remark 2.5. (i) We will often use formal power series and in parti ular formal power series iden- tities, namely identities whi h hold only at ea h order k in the series expansion in (cid:22); we will mark su h identities with the symbol A(cid:24)B. In se tion 4.5 we will prove that the formal power series we use are \ asymptoti ". Asa de(cid:12)nition P n of asymptoti power series we will assume that a formal power series (cid:22) an(") is asymptoti if for (cid:0)q (cid:0)Qn all q >0 there exists Q>0 su h that, for all n(cid:20)" , an(")(cid:20)" .This implies that we an ontrol (cid:0)q Q the (cid:12)rst " terms provided that (cid:22)<" . (ii) It should be stressed that we do not need to prove onvergen e for all the asymptoti power series involved in a given identity to obtain information on those series whi h are known to be onvergent (by the KAM theorem). The following Proposition ontains some important properties of the operators Qj all proved in [G1℄. Proposition 2.6 (Chier hia). (i)The operators Qj are \symmetri " on H(a;d): =(fQjg)==(gQjf): t (ii) H0(a;d) is losed under the appli ation of Qj. (iii) The operators Qj preserve parities and if f 2H0(a;d) is odd then =f =0 (iv)IfF;G2H(a;d)aresu hthattheproje tion onpolynomials, (cid:25)PF(cid:1)G, hasno onstant omponent, then: 0(cid:27) (cid:27) (cid:27) 0(cid:27) = G((cid:28))(cid:28)F((cid:28))=F(0 )G(0 )(cid:0)= F((cid:28))(cid:28)G((cid:28)) Proposition 2.6(iii) immediately implies the following (again proved in [G1℄) ik Corollary2.7. Forallk 2N,j =0;:::;ni=0;1thefun tionGj (')iszerofor'=0. Inparti ular the splitting ve tor is zero for '=0 and the system has an homo lini point. i1 Proof. We pro eed by indu tion; by Proposition 2.6(iii) Gj (' = 0) = 0 as it is the integral of an 1 odd analyti fun tion. Consequently j('=0;t) is both odd and in H0(a;d). Now we suppose that ih h Gj ('=0)=0 and j('=0;t) is odd and in H0(a;d) for all h<k and j =0;:::;n. The fun tion P k Æ h h Fj isanoddanalyti fun tionoftheangles i ( jf ) omputedat = h<k((cid:22)) j('=0;t)whi h ik k is again odd and in H0(a;d). We an apply Proposition 2.6(iii) so Gj (' = 0)=0 and j(' =0;t) is both odd and in H0(a;d). 3 Proofs of the Theorems We de(cid:12)ne the formal power series: a a a a (cid:1)i;j ='iGj ; forj =1;:::;n; Æi ='iG0; fora=0;1: Noti e that su h series are known a priori to be onvergentonly for a=0. Lemma 3.1. The stable and unstable manifolds are on the same energy surfa e so that: Xn 0 + (cid:0) 0 + (cid:0) Gj(')(Ij(';0 )+Ij(';0 ))=(cid:0)G0(')(p(';0 )+p(';0 )) (3.1) j=1 this relation implies that at the homo lini point '=0: I~('=0;0+)(cid:1)0 =(cid:0)Æ0p('=0;0+): Proof. The equation 3.1 are simply the energy onservation at time t=0: + + 2 + (cid:0) (cid:0) 2 (cid:0) (I(';0 );AI(';0 ))+p (';0 )=(I(';0 );AI(';0 ))+p (';0 ); thepotentialpartoftheHamiltonian an elsastheperturbationdependsonlyontheangles. Finally 0 we di(cid:11)erentiate in ' and ompute at the homo lini point where Gj =0 by Corollary2.7. 10 3 Proofs of the Theorems 3.1 The formal linear equation i In the re ursive onstru tion of j, and onsequently of Gj, we have distinguished three \blo ks" : 0 0k 1 1k k t k (0) xjGj ; (1) xjGj ; (2) ((cid:22)) ajQj(Fj ); (3.2) ih k 0k astheGj anbebroughtoutoftheintegralwe ansaythat j andGj (j =1;:::;n)arepolynomials rh in the Gl with l = 0;:::;n, h = 1;:::;k(cid:0)1, r = 0;1. This an be seen as a formal power series identity: X Xn 0 0 [r℄ r [r℄ r Gj(')(cid:24)Jj(')+ ( Njl (')Gl(')+nj G0)+ quadrati terms+(cid:1)(cid:1)(cid:1) [r℄=jr(cid:0)1j r=0;1 l=1 Following [GGM1℄ we di(cid:11)erentiate this relation in the parameter ' and evaluate it on the homo- i 0 lini point where Gj (cid:24)0, this leads to a linear formal identity for (cid:1) : 0 0 1 0 0 1 1 0 0 1 (cid:1) (cid:24)D +N (cid:1) +N (cid:1) +n Æ +n Æ (3.3) 0 0 where Dij ='jJij'=0. i i Noti e that we do not have an expli it expression for the matri es N and n although we have a re ursive algorithm for the oeÆ ients of the series expansion; we will use trees to (cid:12)nd su h expli it 0 expressions. We an noti e however that D is the holomorphi part of the splitting matrix, namely it is obtained by using only the holomorphi blo k (2) of (3.2) in the onstru tion of the homo lini traje tory. We insert the energy onservation relations in equation (3.3): (cid:16) (cid:17) 1(cid:0)N1+ 1 n1I~('=0;0+) (cid:1)0 (cid:24)D0+N0(cid:1)1+n0Æ1; (3.4) p('=0;0+) The tree representation of the traje tories leads to the following Propositions all proved in the next Se tions: Proposition 3.2. The following formal power series relations hold: 0 0 0 0 (i)D (cid:24)N ; (ii)n (cid:24) D !; 2 0 0 0 (iii) D0 is the Hessian of a fun tion S at the homo lini point: Dij ='i'jS (')j'=0. This Proposition generalizes to Hamiltonian 1.3 similar results of [GGM1℄. Relations (i) and (ii) inserted in (3.4) dire tly imply that: (1(cid:0)N1+ 1 n1I~('=0;0+))(cid:1)0 (cid:24)D0(1+(cid:1)1+ 1!Æ1): (3.5) p('=0;0+) 2 Proposition 3.3. One an use the tree representation to (cid:12)nd appropriate bounds on the order k k terms of the series expansion of the formal power series of Equation (3.5). If we denote by M the order k term of the (cid:22) expansion of a formal power series M, we have: 1k k + k + 1k 1k 1 (cid:0)1 k max(jN j;jp ('=0;0 )j;jIj('=0;0 )j;jÆ j;j(cid:1) j)(cid:20)(k!) (C" ) : 0k Moreover the Fourier oeÆ ients of the fun tion S : X S0k(')= ei'(cid:1)(cid:23)S^0k((cid:23)); (cid:23)2Zn:j(cid:23)j(cid:20)k respe t the inequalities (cid:26) p+7 jS^0k((cid:23))j(cid:20) (k!) 1(C"(cid:0) 2 )ke(cid:0)j!(cid:1)(cid:23)jd ; (k!) 1Ck"(cid:0)ke(cid:0)j!(cid:1)(cid:23)j where C; <d are appropriate order one onstants, 1 =4(cid:28)+4 ((cid:28) is the Diophantine exponent of !) 0 (cid:12)nally p is the degree of the pole nearest to the real axis of f(q (t)).