JIANSHENG GENG AND YINGFEI YI

A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS JIANSHENG GENG AND YINGFEI YI Abstract. We cosider Hamitoia etworks of og raged ad weaky couped osciators with variabe frequecies. By derivig a abstract ifiite dimesioa KAM type of theorem we show that for ay give positive iteger N ad a fixed positive measure set O of N variabe frequecies there is a subset O O of positive measure such that each ω O correspods to a sma ampitude quasi-periodic breather (i.e. a soutio which is quasi-periodic i time ad expoetiay ocaized i space) of the Hamitoia etwork with N-frequecies which are sighty deformed from ω. 1. Itroductio ad Mai Resut Associated with the sympectic structure p q we cosider Hamitoia etworks defied by rea aaytic Hamitoias of the form (1.1) H = Z ( p V (q )) W ({q }) where V s are the o-site potetias satisfyig V () = V () = ad V () β β > ad W is a coupig potetia. Hamitoia etworks have bee used i soid state physics i describig the vibratio of partices (atoms) i a attice (see [1 11 ]) ad aso used to mode DNA chais (see [1 14 34]). They aso arise aturay as spatia discretizatio of Hamitoia PDEs such as oiear wave equatios. Amog the soutios of a Hamitoia etwork of particuar physica iterests are the socaed breathers or quasi-periodic breathers which are sef-ocaized time periodic or quasi-periodic soutios whose ampitudes decay at east expoetiay as. Breathers or quasi-periodic breathers are ofte referred to as dyamica soitos or itrisic ocaized modes i physics ad they have bee argey foud via umerics i may physica modes (see [1 9] ad refereces therei). The existece of breathers i Hamitoia etworks associated with Hamitoias (1.1) was rigorousy aayzed whe β β by Aubry [1 ] Mackay Aubry [5] for the iter-partice earest eighbor coupig potetia W ({q }) = (q 1 q ) ad by Bambusi [3] for the og rage coupig potetia W ({q }) = 1 m α (q q m ) α > 1. m Like i [5] breathers i the earest eighbor coupig case ca be studied ear a fixed periodic orbit of the ucouped Hamitoia by certai cotiuatio or perturbatio argumets provided that the coupigs are weak ad o sma divisors eed to be cosidered i such perturbatio probems. These perturbatio techiques are aso appicabe i fidig quasi-periodic breathers with 1991 Mathematics Subject Cassificatio. Primary 37K6 37K55. Key words ad phrases. Couped osciators Hamitoia etworks og raged coupig KAM theory quasiperiodic breathers. The secod author is Partiay supported by NSF grat DMS4119. 1

JIANSHENG GENG AND YINGFEI YI two or three frequecies for certai modes with symmetries (see Bambusi Vea [4] Johasso Aubry [19]). Usig a modified KAM techique the existece of quasi-periodic breathers with ay fiite umber of frequecies was recety show by Yua [33] for the higher order earest-eighbor coupig potetia (1.) W ({q }) = (q 1 q ) 3. Amost periodic breathers with ifiitey may frequecies have aso bee ivestigated. Associated with the potetia (1.) Fröhich-Specer-Waye [15] cosidered the case whe the frequecies are o-egative radom variabes with smooth distributio of fast decay at ifiity ad showed that there is a set Ω R with positive probabiity measure such that each ω Ω correspods to a amost periodic breather with ifiite may frequecies (see aso Pösche [8] for more geera spatia structures). I this paper we wi study the existece of quasi-periodic breathers for the Hamitoia (1.1) with the foowig higher order og-raged coupig potetia (1.3) W ({q }) = 1 e m α (q q m ) 3 α 1 3 or equivaety the Hamitoia etwork m (1.4) d q dt V (q ) = m Z e m α (q q m ). For a give iteger N > 1 we specify N itegers {i 1 i N } ad et Z 1 = Z \ {i 1 i N }. We treat ω = (β i1 β in ) as parameters i a bouded cosed regio O i R N ad assume the foowig spectra gap coditio: SG) There exist 1 d < ad > such that where Λ = 1 are sets satisfyig ad We wi show the foowig resut. {β } Z1 = =1Λ #(Λ ) d for a β β m for a β Λ β m Λ j j. Theorem A. Assume SG) with sufficiety sma. The there exists a Cator set O O with meas(o \ O ) = O() such that for ay ω O the Hamitoia etwork (1.4) associated with ω admits a sma ampitude ieary stabe quasi-periodic breather q(t) = ({q (t)}) of N-frequecy ω which is cose to ω ad moreover q e. The coditio SG) ceary hods whe β = Z 1. Comparig with the cases of oiear wave equatios [1 17 6 7] i which β Z the vaidity of Theorem A cruciay depeds o the coupig potetia or perturbatio (1.3) which admits a weaker reguarity but a higher order perturbatio. For the case of earest-eighborig couped Hamitoia etworks breathers were show to be super-expoetiay ocaized i space ([33]). This is due to the at most iear growth of the orma compoets i the orma form associated with the short-raged coupig potetia (1.). Our resut oy asserts the expoetia ocaizatio of quasi-periodic breathers due to the expoetia growth of the orma compoets i the orma form associated with the expoetiay weighted og-raged coupig potetia (1.3). If the og-raged coupig potetia W ({q }) = 1 1 3 m α (q q m ) 3 α > 1 m

A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS 3 is cosidered istead the the orma compoets i the associated orma form wi have a superexpoetia growth ad our method wi equay appicabe to yied quasi-periodic breathers which 1 are ocaized ike i space. α Theorem A wi be proved by usig KAM (Komogorov-Arod-Moser) method. I fact we wi preset a abstract ifiite dimesioa KAM type of theorem from which Theorem A wi foow. Such a ifiite dimesioa KAM theorem differs sigificaty from those for Hamitoia PDEs ike oiear Schödiger wave beam ad KdV equatios studied by may authors usig either KAM or CWB (Craig-Waye-Bourgai) method (see [5 6 7 8 9 1 13 16 17 18 1 3 6 7 3] ad refereces therei). This is maiy due to the fact that whe the orma frequecies of a Hamitoia etwork have iear growth the perturbatio (1.3) admits weaker reguarity tha those of Hamitoia PDEs uder KAM or Newto iteratios. Simiar to the short-raged coupig cases cosidered i [5 33] it is aso importat to study the existece of quasi-periodic breathers for Hamitoia etworks with og-raged coupig potetias ad costat frequecies β β Z i.e. those formed by weaky couped idetica osciators. As the KAM iteratio mechaism ad measure estimates for the costat frequecy case sigificaty differ from the variabe oes to be studied i this paper we wi cosider the costat frequecy case i a separate work. The paper is orgaized as foows. I Sectio we state a abstract ifiite dimesioa KAM theorem ad prove Theorem A as a coroary. Sectios 3 ad 4 are devoted to the proof of the abstract KAM theorem. More precisey i Sectio 3 we give detaied costructio of the KAM iteratio for oe KAM step. We compete the proof of the abstract ifiite dimesioa KAM theorem i Sectio 4 by showig a iteratio emma covergece ad measure estimate. Some techica emmas are provided i the Appedix.. A Abstract KAM Theorem I this sectio we wi formuate a abstract KAM theorem which ca be appied to the Hamitoia etworks of og-raged ad weaky couped osciators with variabe frequecies. Theorem A wi be proved by usig the abstract KAM theorem ad orma form reductios..1. The abstract theorem. We begi with some otatios. Let itegers N > 1 d 1 ad rea umbers r s > be give. We cosider the compex eighborhood D(r s) of T N {} {} {} T N R N 1 1 defied by D(r s) = {(θ I w w) : Imθ < r I < s w < s w < s} where deote the sup-orm of compex vectors ad deote the 1 orm. Aso et O be a positive (Lebesgue) measure set i R N. Let F (θ I w w) be a rea aaytic fuctio o D(r s) which depeds o a parameter ξ O C d -Whitey smoothy (i.e. C d i the sese of Whitey). We expad F ito the Tayor-Fourier series with respect to θ I w w: F (θ I w w) = αβ F αβ w α w β where α ( α ) β ( β ) α β N are muti-idices with fiitey may o-vaishig compoets ad F αβ = F kαβ (ξ)i e i kθ. k Z N N N We defie the weighted orm of F by F D(rs)O sup w <s w <s F αβ w α w β αβ

4 JIANSHENG GENG AND YINGFEI YI where F αβ F kαβ O s e k r F kαβ O sup{ max ξ O m d m ξ F kαβ }. k I the above ad aso for the rest of the paper derivatives i ξ O are take i the sese of Whitey. For a vector-vaued fuctio G : D(r s) O C < we simpy defie its weighed orm by G D(rs)O G i D(rs)O. For the Hamitoia vector fied i=1 X F = (F I F θ {if w } { if w }) associated with a Hamitoia fuctio F o D(r s) O we defie its weighted orm by X F D(rs)O F I D(rs)O 1 s F θ D(rs)O 1 s ( F w D(rs)O F w D(rs)O ). Associated with the sympectic structure di dθ i Z dw d w we cosider the foowig famiy of rea aaytic parameterized Hamitoias (.1) H = N P N = ω(ξ) I Z Ω w w P = P (θ I w w ξ) where (I θ w w) D(r s) ξ O Ω s are positive ad idepedet of ξ ad a ξ-depedece are of cass C d i the sese of Whitey. It is cear that whe P = the uperturbed Hamitoias N are competey itegrabe admittig a famiy of quasi-periodic soutios (θ ωt ) correspodig to ivariat N-tori i the phase space. To study the persistece of some of these N-tori we eed the foowig assumptios o ω(ξ) Ω ad the perturbatio P : (A1) No-degeeracy of tagetia frequecies: There is a costat δ > such that det ( ω ) δ. ξ (A) Gap coditios of orma frequecies: There exist sufficiety sma > ad sets Λ = 1 such that {Ω } Z = =1Λ #(Λ ) d for a Ω Ω m if Ω Λ Ω m Λ j j.

A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS 5 (A3) Decay property of the perturbatio: P = P Ṕ `P where P = P (θ I w w ξ) Ṕ = Ṕ (θ I w w ξ) `P = `P (θ I w w ξ) are such that (.) (.3) (.4) P = P (θ I ξ) Z αβ 1 Ṕ = Ṕ m (ξ)w α m Z m αβαmβm 1 αβαmβm 3 `P = Z O( w 3 ). P (θ I ξ)w α Our abstract KAM theorem states as the foowig. w β P (θ I ξ) e ; w β wm αm w m βm Ṕm(ξ) e m ; Theorem B. Cosider the Hamitoia (.1) ad assume (A1)-(A3). For a fixed > sufficiety sma there exists a positive costat ε = ε(o d δ N r s) such that if X P D(rs)O < ε the the foowig hods. There exist Cator sets O O with meas(o \ O ) = O() ad maps Ψ : T N O D(r s) ω : O R N which are rea aaytic i θ ad C d -Whitey smooth i ξ with Ψ Ψ D( r )O ad ω ω O as where Ψ is the trivia embeddig: T N O T N { } such that each ξ O ad θ T N correspods to a ieary stabe N-frequecy quasi-periodic soutio Ψ(θ ω(ξ)t ξ) = (θ ω(ξ)t I(t) {w (t)}) of the Hamitoia (.1). Moreover w e. Sice our perturbatio has a weaker reguarity the frequecies of these ivariat tori are i geera o-resoat istead of Diophatie. Comparig with resuts o quasi-periodic soutios for Hamitoia PDEs (see e.g. [5 6 7 8 9 1 13 16 17 18 1 3 6 7 3]) the above theorem reaxes the iear or super-iear growth coditios o the orma frequecies Ω. Ideed it is easy to see that the gap coditio (A) above is weaker tha the iear or sup-iear growth coditios o the orma frequecies. The assumptio (A3) is ew but atura for etworks of og raged ad weaky couped osciators. It is ot cear whether a Lyapuov ceter theorem is possibe for a Hamitoia etwork whose orma frequecies satisfy the gap coditio (A). At east the above theorem assert a quasiperiodic type of Lyapuov ceter resut i the sese of measure... Proof of Theorem A. Reca that the Hamitoia etworks of og raged weaky couped osciators cosidered i Theorem A is described by the Hamitoia H = [ p V (q )] 1 e m α (q q m ) 3 α 1 3 Z m which i terms of the Tayor expasio at q = ca be equivaety rewritte as H = [ p β q ] 1 e m α (q q m ) 3 O( q 3 ). 3 Z m Z Let ε > be sufficiety sma. With the re-scaigs p q εp εq the re-scaed Hamitoia reads ε H(εp εq) = [ p β q ] ε e m α (q q m ) 3 ε O( q 3 ). 3 Z m Z Let N {i 1 i N } ad Z 1 = Z \ {i 1 i N } be as i Theorem A. For a give vaue a = (a 1 a N ) R N we itroduce the stadard actio-age-orma variabes (I θ w w) =

6 JIANSHENG GENG AND YINGFEI YI (I θ {w } Z1 { w } Z1 ) R N T N 1 i.e. p ij = β ij Ij a j cos θ j q ij = 1 β ij Ij a j si θ j 1 j N β (w w ) p = q = w w i Z 1. β Let ξ = (ξ 1 ξ N ) = (β i1 β in ). The i terms of the actio-age-orma variabes the above Hamitoia becomes (.5) H = N P = ω(ξ) I Z 1 Ω w w P (θ I w w ξ) where ad satisfyig P = P (θ I ξ) Z1 αβ 1 Ṕ = Ṕ m (ξ)w α m Z 1 m αβαmβm 1 αβαmβm 3 `P = Z1 O( w 3 ). ω(ξ) = (ω 1 (ξ) ω N (ξ)) = (ξ 1 ξ N ) Ω = β Z 1 P = P Ṕ `P P (θ I ξ)w α w β P (θ I ξ) e α e ; w β wm αm w m βm Ṕm(ξ) e m α e m ; It is aso easy to see that we ca choose appropriate r s > such that X P D(rs)O < ε. Hece the Hamitoia (.5) satisfies a coditios of Theorem B from which Theorem A foows. 3. KAM Step I what foows we wi perform KAM iteratios to (.1) which ivoves ifiite may successive steps caed KAM steps of iteratios to eimiate ower order θ-depedet terms i P. Each KAM step wi make the perturbatio smaer tha the previous oe at a cost of excudig a sma measure set of parameters. At the ed the KAM iteratios wi be show to coverge ad the measure of the tota excudig set wi be show to be sma. To begi with the KAM iteratio we set Ω = Ω Z 1 r = r s = s. 3.1. Norma form. We first covert the Hamitoia (.1) ito a more coveiet form i order to perform the KAM iteratio. Let ε ε 5 4 ad choose a K such that K ε. Accordig to the forms of (.3) (.4) i the Assumptio (A3) we ca make s smaer if ecessary such that XṔ `P D(rs ) O ε. We ow treat the term P. Accordig to the form of (.) ad the defiitio of the orm we have P = P (θ I ξ) P (θ I ξ)w α w β where = k αβ 1 P k I e i kθ k αβ 1 P k e k r P kαβ I e i kθ w α w β P kαβ e k r e.

A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS 7 Thus we ca make r s smaer such that Let X k >K 1 P k I e i kθ >K or k >K αβ 1 R = k K 1 P k I e i kθ P kαβ e i kθ w α w β O( I )O( w 3 ) ε. K k K 1 αβ P kαβ e i kθ w α w β. We first costruct a sympectic trasformatio Φ = Φ 1 F defied as the time-1 map of the Hamitoia fow associated to a Hamitoia F of the form F = F k I e i kθ (F k1 w F k1 w )e i kθ < k K 1 k K K (F k w w F k w w )e i kθ k K K F k11 w w e i kθ < k K K such that a o-resoace terms P k I e i kθ kα < k K 1 P β e i kθ w α w β < k K α β are eimiated ad terms P I 11 1 P w w K are added to the orma form part of the ew Hamitoia. More precisey et the coefficiets of F satisfy the homoogica equatio (3.1) {N F } R = P I w w. 1 K P 11 It is easy to see that the homoogica equatio (3.1) is sovabe o the parameter set k ω K < k K τ O = ξ O : k ω Ω K k K τ K k ω Ω k K K. Hece we obtai the trasformatio Φ such that K τ H = H Φ = N P Ṕ `P N = e ω (ξ) I P 11 where ω = ω P ( = 1) Ω = Ω ad P = P (θ I w ( K) w ( K) ξ) satisfies K Ω w w >K αβ 1 >K Ω w w P (θ I w m( m K) w m( m K) ξ)w α P (θ I w m( m K) w m( m K) ξ) e ( K). I the above the first ad the secod term of P come from P Φ ad P Φ Ṕ Φ respectivey. The decay property of P foows from the fact that Φ depeds oy o I θ ad w m w m for m K. w β

8 JIANSHENG GENG AND YINGFEI YI Next write = >K αβ 1 >5K αβ 1 K < 5K αβ 1 P (θ I w m( m K) w m( m K) ξ)w α P (θ I w m( m K) w m( m K) ξ)w α P (θ I w m( m K) w m( m K) ξ)w α O the domai D(r s ) O it is easy to see that the orm of the vector fied associated with the first term above is bouded by ε. To hade the secod term above we ote that due to the gap coditio (A) of orma frequecies whe Ω Ω m beog to the same Λ terms of the form P mw m w P mw w m are ot abe to be eimiated via sovig a homoogica equatio. Hece they eed to be icuded i the orma form part of the Hamitoia. More precisey accordig to the assumptio (A) we et L be the smaest positive iteger such that {Ω } 5K ie i Λ 1 Λ L. Let K = 5K. Ceary L K. Let ad et R = k K 1 F = P ki e i kθ satisfy the homoogica equatio (3.) {N F } R = m K K k K 1 αmβ w β w β w β. P m kαmβ e i kθ (wm αm w β w m αm w β ) f k e i kθ I (f k1 w f k1 w )e i kθ < k K 1 k K K (fm k w w m fm k11 w w m fm k w w m )e i kθ k K m K K (f k w w f k w w )e i kθ k K K < K f k11 w w e i kθ < k K K < K 1 P I m K K P 11 m w w m It is cear that the equatio (3.) is sovabe o the domai k ω Kτ < k K K < K P 11 w w. O = ξ O : k ω Ω Kτ k K K k ω Ω Kτ k K K < K k ω Ω m ± Ω Kτ k K m K k ω Ω m ± Ω Kτ k K m K K < K k ω Ω Kτ k K K < K. Cosider the sympectic trasformatio Φ = Φ 1 F. The H = H Φ = N P N = L e ω (ξ) I A z z Ω w w =1 > K

A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS 9 where L =1 with e = e P ω = ω P ( = 1) A z z = [ K Ω w w m K K ( Ω = Ω 11 P P = P Ṕ `P P = P = P (θ I w ( K) w ( K ξ) ) def = P (θ I z z ξ) > K αβ 1 m αβαmβm 1 αβαmβm 3 11 11 P m w m w P m w w m )] > K αβ 1 P (θ I z z ξ)w α P (θ I w m( m K) w m( m K ) ξ)wα w β P (θ I z z ξ) e ( K ) > K Ṕ = Ṕ = Ṕm(ξ)w α w β wm αm w m βm Ṕ m(ξ) e m `P = `P = Z O( w 3 ) z = ( w ) Ω Λ K z = (z 1 z L ) diam(a ) d ad X P D(r s )O ε 5 4 def = ε. Suppose that after a νth KAM step we arrive at a Hamitoia H H ν = N P = N P Ṕ `P L N = N ν = ω(ξ) I A z z Ω w w =1 P = P ν = P (θ I z ν z ν ξ) > K αβ 1 = P ν (θ I z ν z ν ξ) > Kν αβ 1 > K P (θ I z z ξ)w α w β P ν (θ I z ν z ν ξ)w α defied o a domai D(r s) O = D(r ν s ν ) O ν where K = K ν is a positive costat L = L ν is the smaest positive iteger such that {Ω } K ie i Λ 1 Λ L z = z ν = ( w ) Ω Λ K z = z ν = ( w ) Ω Λ K z = z ν = (z 1 z L ) z = z ν = ( z 1 z L ) P = P ν for some ε = ε ν ad P (θ I z z ξ) D(rs)O e ( K) > K. It is cear that L K. We wi costruct a sympectic trasformatio Φ = Φ ν which i smaer frequecy ad phase domais carries the above Hamitoia ito the ext KAM cyce. Beow a costats c 1 c 1 w β w β

1 JIANSHENG GENG AND YINGFEI YI are positive ad idepedet of the iteratio process. The tesor product (or direct product) of two m k matrices A = (a ij ) B is a (mk) () matrix defied by A B = (a ij B) = a 11B a 1 B. a m1 B a m B We aso use to deote the operator matrix orm i.e. for a matrix M M = sup y =1 My. Let K = 7 K 4 1 4 K. I this KAM step Ω with K < K wi be added to the ew orma matrix A accordig to the assumptio (A). I order to have a compact formuatio whe sovig a homoogica equatio we rewrite N as L N = e ω(ξ) I A z z =1 K< K Ω w w > K Ω w w L def = e ω(ξ) I Ãz z =1 > K Ω w w where dim(ã) d L is the smaest positive iteger such that {Ω } K ie i Λ 1 Λ L (hece L K ) z ( ) A Ã = Ω Ω Λ K < K = ( w ) Ω Λ K z = ( w ) Ω Λ K z = (z 1 z L ) z = ( z 1 z L ). 3.. Trucatio. We first expad P ito the Tayor-Fourier series P = P kαβ e i kθ I z α z β P kαβ e i kθ I z α z β w α kαβ kαβ > Kαβ 1 where k Z N N N ad the muti-idex α (resp. β) rus over the set α (α 1 α α L ) for α = ( α m ) Ωm Λ α m N (resp. β (β 1 β β L ) for β = ( β m ) Ωm Λ m K m K β m N). Let R be the foowig trucatio of P : R(θ I z z w w) = k K L ( k K j L ( k K 1 P k e i kθ I k1 k1 P z P z )e i kθ k P j z z j k K L K< K ( k K K< K ( where K = 3 4 K 1 4 K. k11 P j z z j k P z w k K K< K ( k P j z z j )e i kθ k11 P z w k11 P w z k k11 k P w w P w w P w w )e i kθ w β k1 k1 P w P w )e i kθ k P z w )e i kθ

A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS 11 Remark 3.1. Due to decay property terms i the Tayor-Fourier expasio of P correspodig to k > K or > K are sma eough to be postpoed to the ext KAM step. I additio due to the decay property of P ad the fact that Ṕ starts from third order terms there are o coupig terms of the form m K< m K w w m i R. If Ṕ starts from secod order terms the coupigs betwee differet osciators are so fast that we have to cosider a orma frequecies - a case we are ot abe to hade with the method i this paper. Accordig to the orma form N ad the assumptio (A) we may rewrite R as R(θ I z z ) = R R 1 R = P k e i kθ I where It is cear that R k j ( R k1 k K L ( Rj k k K j L k K 1 ( P R k1 k1 = z Rk1 z )ei kθ z z j Rk11 j z z j Rk j z z j )ei kθ P k1 ( P R k1 k1 = R k j = R k11 j = R k j = = (Rj k ) Rj k11 ( P k j P k ( P k11 j P k11 ( P k j P k P k1 ) ) P j k P k P j k11 P k11 P j k P k K< K K< K ) ) ) K< K K< K K< K. = (Rj k11 ) ad Rj k = (Rj k ). Remark 3.. Note that R is depedet o the former K orma compoets ad idepedet of the atter orma compoets. As a resut i the KAM step we do ot eed to cosider sma divisors k ω Ω k ω ± Ω ± Ω m > K or m > K. This eabes us to oy cosider fiite sma divisor coditios at each KAM step. Cosequety the excuded measure is sufficiety sma at each step. As K wi icrease aog KAM iteratios we utimatey hade a sma divisor coditios. Rewrite H as H = N R (P R). By the defiitio of orms we immediatey have Note that P R = k >K Pk (I z z ξ)e i kθ X R D(rs)O X P D(rs)O ε. > K α β 1 P (θ I z z ξ)w α w β h.o.t. where h.o.t. deotes the terms of the form O( I I w w 3 ). Let r = r r 4 η = ε 1 4. Usig the facts P (θ I z z ξ) e ( K) P k (I z z ξ) e k r we have that if C1) e K r r ε 5 4

1 JIANSHENG GENG AND YINGFEI YI the (3.3) X P R D(r r r e k r r e ( K) 5 h.o.t. c ηs)o 1 ε 4. k >K > K 3.3. The homoogica equatio. We ow ook for a Hamitoia F defied i a domai D = D(r s ) such that the time-1 map Φ = Φ 1 F of the Hamitoia vector fied X F defies a map from D to D ad trasforms H ito H i the ext KAM cyce. Let F have the form F (θ I z z ) = F F 1 F = F k e i kθ I (3.4) = < k K 1 (f k1 k K K (fm k w w m fm k w w m )e i kθ k K m K fm k11 w w m e i kθ k K m K < k K 1 F k e i kθ I ( Fj k z z j k K j L k K j L k j = ad satisfy the homoogica equatio (3.5) Fj k11 z z j ei kθ {N F } R = P ω I ( F k1 k K L w f k1 z k Fj z z j )ei kθ L =1 R 11 z z w )e i kθ k1 F z )ei kθ where P ω = I dθ z = z =I=. I the rest of the sub-sectio k is aways bouded by K m are aways bouded by K ad j are aways bouded by L. Lemma 3.1. Equatio (3.5) is equivaet to (3.6) k ω F k = i P k k 1 k1 ( k ω I Ã)F = ir k1 ( k ω I = ir k1 ( k ω I Ã)F k1 Ãj)F k j k11 ( k ω I Ãj)Fj k ( k ω I Ãj)Fj F k j à = ir k j Fj k11 à = irj k11 k j Fj k à = irj k. Proof. It is cear that (3.5) is equivaet to the foowig equatios {N F } R = P ω I (3.7) {N F 1 } R 1 = {N F } R = L =1 R 11 z z.

A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS 13 By comparig the coefficiets the first equatio i (3.7) is obviousy equivaet to the first equatio i (3.6). Sice ad {N F 1 } = i k i k = i k i k ( k ω F k1 ( k ω F k1 ( k ω I {N F } = i ( k ω Fj k z z j kj i ( k ω Fj k11 z z j k j = i ( k ω Fj k z z j kj z Ãz F k1 )e i kθ z à z F k1 )e i kθ k1 Ã)F z ei kθ k1 ( k ω I Ã)F z ei kθ k Fj z Ãjz j Ãz k11 Fj z Ãj z j Ãz k Fj z Ãj z j à z k (Fj ) T z j )ei kθ k11 (Fj ) T z j )ei kθ k (Fj ) T z j )ei kθ = i ( k ω Fj k z z j kj i ( k ω Fj k11 z z j k j = i ( k ω Fj k z z j kj = i ( k ω Fj k kj i k j = k ÃjFj ( k ω F k11 j i ( k ω Fj k kj k ÃjFj (ÃjF k j (ÃjF k11 j (ÃjF k j Fj k à )z z j )ei kθ Fj k11 à )z z j )ei kθ Fj k à ) z z j )ei kθ Fj k à )z z j ei kθ k11 ÃjFj Fj k11 à )z z j ei kθ Fj k à ) z z j ei kθ we see from the secod ad the third equatio i (3.7) that F k1 the respective equatios i (3.6). Let O = ξ O : k ω 1 ( K τ ) d ( K τ ) d ( k ω I Ã) 1 < k K ( k ω I à I I Ãj) 1 ( k ω I à I I Ãj) 1 F k1 k K L ( K τ ) d ( K τ ) d F k j Fj k11 Fj k k K j L k j k K j L satisfy The first three equatios i (3.6) ca be immediatey soved o O. The sovabiity of the remaiig equatios i (3.6) foows from the foowig eemetary agebraic resut from matrix theory..

14 JIANSHENG GENG AND YINGFEI YI Lemma 3.. Let A B C be m m m matrices respectivey ad et X be a m ukow matrix. The matrix equatio (3.8) AX XB = C is sovabe if ad oy if I m A B I is osiguar. Moreover X (I m A B I ) 1 C. Proof. See [4 3]. By takig the traspose of the fourth equatio i (3.6) oe sees that (Fj k ) satisfies the same equatio as Fj k. We have by the uiqueess of soutio that Fj k = (Fj k ). Simiary Fj k11 = (Fj k11 ) ad Fj k = (Fj k ). Hece the Hamitoia F is uiquey determied o O. We proceed to estimate X F ad Φ 1 F.. Lemma 3.3. Let D i = D(r i 4 (r r ) i 4s) < i 4. If C) K τd4 ε 1 4 the there is a costat c > such that X F D3O c d4 (r r ) N ε 3 4. Proof. By the defiitio of O Lemma 3.1 Lemma 3. ad Lemma 5.5 Lemma 5.6 i the Appedix we have It foows that F k O k ω d P k O d4 K τd4 P k O k ; F k1 O d4 K τd4 R k1 O ; F k1 O d4 K τd4 R k1 O ; F k j O d4 K τd4 R k j O ; F k11 j O d4 K τd4 R k11 j O k j ; F k j O d4 K τd4 R k j O. 1 s F θ D3O 1 s ( k k k 1 kj kj F k s k e k (r 1 4 (r r)) ( F k1 z ) k e k (r 1 4 (r r)) ( F k1 z ) k e k (r 1 4 (r r)) Fj k z z j k e k (r 1 4 (r r)) k j = Fj k11 z z j k e k (r 1 4 (r r)) F k j z z j k e k (r 1 4 (r r)) )

A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS 15 k k K d4 τd4 s ( P k s k e k (r 1 4 (r r)) k 1 kj kj ( R k1 z ) k e k (r 1 4 (r r)) ( R k1 z ) k e k (r 1 4 (r r)) Rj k z z j k e k (r 1 4 (r r)) k j = Rj k11 z z j k e k (r 1 4 (r r)) R k j z z j k e k (r 1 4 (r r)) ) Simiary c 3 d4 (r r ) N K τd4 X R c 3 d4 (r r ) N ε 3 4. F I D3O = F k e k (r 1 4 (r r)) c 4 d4 (r r ) N ε 3 4. =1 X F1 D3O 1 s ( F 1w F 1 w ) 1 s ( F 1z F 1 z ) X F D3O 1 s ( c 5 d4 (r r ) N K τd4 X R1 c 5 d4 (r r ) N ε 3 4. F w F w ) 1 s ( F z c 6 d4 (r r ) N K τd4 X R c 6 d4 (r r ) N ε 3 4. The proof is ow competed by addig the estimates above together. Let D iη = D(r i 4 (r r ) i 4ηs) < i 4. Lemma 3.4. If C3) c d4 (r r ) N ε 1 < 1 the (3.9) Φ t F : D η D 3η 1 t 1 ad moreover (3.1) DΦ t F I D1η < c 7 d4 (r r ) N ε 3 4. Proof. Let i α β F z ) D m F DO = max{ θ i I (z ) α ( z ) β F DO i α β = m }. We ote that F is a poyomia of order 1 i I ad of order i z z. It foows from Lemma 3.3 ad the Cauchy iequaity that for ay m. Usig the itegra equatio D m F DO < c 8 d4 (r r ) N ε 3 4 Φ t F = id t X F Φ s F ds

16 JIANSHENG GENG AND YINGFEI YI ad Lemma 3.3 we easiy see that Φ t F : D η D 3η DΦ t F = Id t (DX F )DΦ s F ds = Id 1 t 1. Sice t J(D F )DΦ s F ds where J deotes the stadard sympectic matrix. Let c 7 = c 8. It foows that DΦ t F I D F c 7 d4 (r r ) N ε 3 4. 3.4. The ew Hamitoia. Let Φ = Φ 1 F s = 1 8 ηs D = D(r s ) ad where N = e ω I P = P Ṕ `P e = e P ω = ω P ( = 1) L =1 A z z A = Ã R 11 L z = ( w ) K z = ( w ) K z = (z 1 z L ) z = ( z 1 z L ) P = (1 t){{n F } F } Φ t F dt > K Ω w w {R F } Φ t F dt ( P R) Φ 1 F The Φ : D O D ad by the secod order Tayor formua H H Φ = (N R) Φ (P R) Φ = N {N F } R (1 t){{n F } F } Φ t F dt {R F } Φ t F dt ( P R) Φ 1 F (Ṕ `P ) Φ 1 F = N {N F } R P Ṕ `P = N P {N F } R P ω I = N P. L =1 R 11 z z Beow we show that the ew Hamitoia H ejoys simiar properties as H. By the Assumptios of P we have that there is a costat c 9 > such that ω ω O c 9 ε A Ã O c 9 ε. Deote R(t) = (1 t)(n N) tr. We ca rewrite P as Hece P = By Lemma 3.4 if = (1 t){{n F } F } Φ t F dt {R(t) F } Φ t F dt (P R) Φ 1 F. X P = C4) c 7 d4 (r r ) N ε 3 4 1 {R F } Φ t F dt (P R) Φ 1 F (Φ t F ) X {R(t)F } dt (Φ 1 F ) X (P R). {Ṕ `P F } Φ t F dt.

A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS 17 the DΦ t F D1η 1 DΦ t F I D1η 1 t 1. By Lemma 5.4 ad (3.3) we aso have X {R(t)F } Dη c 1 d4 (r r ) N η ε 7 4 X (P R) Dη c 1 ε 5 4. Let c = max{c 1 c 1 c 11 c 1 } where c 11 c 1 wi be defied ater ad et The ε = 4c d4 (r r ) N ε 5 4. X P DO c 1 ε 5 4 c1 d4 (r r ) N ε 5 4 ε. We ow exam the decay property of P. More precisey write P = P (θ I z z ξ) P (θ I z z ξ)w α We show that > K α β 1 P (θ I z z ξ) DO e ( K ) > K. Sice F oy ivoves the orma compoets w w for K so does {N F }. Hece (1 t){{n F } F } Φt F dt oy ivoves the orma compoets w w for K. Reca w β. that `P = O( w 3 ). Hece { `P F } oy ivoves the orma compoets w w for K so does { `P F } Φ t F dt. Sice R is a trucatio of P we oy eed to cosider the terms ( P R) ad { P Ṕ F } Φ t F dt. Reca that P = P (θ I z z ξ) m αβαmβm 1 αβαmβm 3 > Kα β 1 P (θ I z z ξ)w α P (θ I z z ξ) D(rs)O e ( K). Ṕ = Ṕm(ξ)w α w β wm αm w m βm Ṕ m(ξ) D(rs)O e m. Sice R oy ivoves the orma compoets w w for K the terms correspodig to the orma compoets w w for > K i P R are just those correspodig for > K i P for which we have the decay property P (θ I z z ξ) D(rs )O e ( K) e ( K ). To prove the decay estimates of { P Ṕ F } Φ t F dt we oy eed to cosider the terms correspodig to the orma compoets w w for > K. Sice F is idepedet of orma compoets w w for > K so is { P (θ I z z ξ) F } Φ t F dt. Simiary { m m K αβαmβm 1 αβαmβm 3 Ṕ m(ξ)w α w β w β wm αm w m βm F } Φ t F dt

18 JIANSHENG GENG AND YINGFEI YI is idepedet of the orma compoets w w for > K. Thus it remais to cosider terms { P (θ I z z ξ)w α w β F } Φ t F dt (3.11) ad (3.1) Let = = = = { > Kα β 1 > Kα β 1 > Kα β 1 ( m > K m K αβαmβm 1 αβαmβm 3 > K { ( > K m K αβαmβm 1 αβαmβm 3 { m K αβαmβm 1 αβαmβm 3 { P (θ I z z ξ) F } Φ t F w α { P (θ I z z ξ) F } Φ t F dt)w α Ṕ m(ξ)w α P = P (θ I z z ξ) Ṕ m(ξ)w αm m Ṕ m(ξ)w αm m w β dt w β w β wm αm w m βm F } Φ t F dt m K αβαmβm 1 αβαmβm 3 We combie (3.11) ad (3.1) to cosider decay property of ( { P F } Φ t F dt)w α > Kα β 1 w m βm F } Φ t F w α w β dt w m βm F } Φ t F dt)w α w β. Ṕ m(ξ). By reaxig decay properties of e ( K) e m to e ( K ) we have by Lemma 5.3 that w β. { P F } D(r σ 1 s) c 11 d4 (r r ) N σ 1 s ε 3 4 e ( K ). It foows from Cauchy estimate that Hece by Lemma 3.4 if X { PF } D(r σ 1 4 s) c 1 d4 (r r ) N σ s 4 ε 3 4 e ( K ). C5) c 11 d4 (r r ) N η ε 3 4 1 C6) c 1 c ( d4 (r r ) N η ε 3 4 ) 1 the { P F } Φ t F dt D(rs ) { P F } Φ t F D(rs ) { P F } D(rs ) { P F } Φ t F { P F } D(rs ) { P F } D(rs ) X { PF } D η Φ t F id D1η c 11 d4 (r r ) N η ε 3 4 e ( K ) c 1 c ( d4 (r r ) N η ε 3 4 ) e ( K ) e ( K ).

A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS 19 This competes oe step of KAM iteratios. 4. Proof of Theorem B Let r s ε K K O H be give at the begiig of Sectio 3. For each ν = 1 we abe a idex-free quatities by ν ad abe a -idexed quatities by ν 1. This defies for a ν = 1 the foowig sequeces: ν1 r ν = r (1 i ) i= ε ν = 4c d4 (r ν 1 r ν ) N ε 5 4 ν 1 s ν = 1 ν 1 8 η ν 1s ν 1 = 3ν ( ε i ) 1 4 s η ν = ε 1 4 ν K ν = 4K ν 1 K ν = K ν 1 K ν D ν = D(r ν s ν ) i= D ν = D(r ν1 1 4 (r ν r ν1 ) 1 4 η νs ν ) H ν = N ν P ν L ν N ν = e ν ω ν (ξ) I A ν z ν z ν Ω w w =1 > K ν O ν = ξ O ν 1 : k ω ν 1 1 ( K τ ν ) d ( k ω ν I Ãν 1 ) 1 ( k ω ν 1 I Ãν 1 ( k ω ν 1 I Ãν 1 < k K ν ( K τ ν ) d I I Ãν 1 j ) 1 I I Ãν 1 j ) 1 k K ν L ν K ν ( K τ ν ( K τ ν ) d ) d k K ν j L ν k j = k K ν j L ν where L ν is the smaest positive iteger such that {Ω } ie i Λ Kν 1 Λ Lν ad ( ) Ã ν 1 A ν 1 = Ω Ω Λ Kν 1 < K ν. 4.1. Iteratio Lemma. The precedig aaysis may be summarized as foows. Lemma 4.1. If ε is sufficiety sma the the foowig hods for a ν = 1. a) H ν is rea aaytic o D ν O ν L ν1 N ν = e ν ω ν (ξ) I Ãν z ν1 z ν1 Ω w w =1 > K ν1 P ν = P ν Ṕ `P

JIANSHENG GENG AND YINGFEI YI ad moreover with ω ν1 ω ν Oν c ε ν Ãν1 Ãν Oν c ε ν X P ν DνO ν ε ν P ν = P ν (θ I z ν z ν ξ) Ṕ = `P = m αβαmβm 1 αβαmβm 3 O( w 3 ) > K να β 1 Ṕ m(ξ)w α b) There is a sympectic trasformatio such that w β wm αm w m βm P ν (θ I z ν z ν ξ)w α P ν (θ I z ν z ν ξ) DνO ν e ( K ν) Ṕ m(ξ) DνO ν e m. Φ ν : D ν O ν1 D ν H ν1 = H ν Φ ν. Proof. It is sufficiet to verify the coditios C1) C6) for a ν = 1 which are easiy see to foow from the foowig coditios D1) r ν r ν1 1 5 K ν1 1 1 ε 4 ν ε 4τd ν 4 D) c d4 (r ν r ν1 ) N ε 1 4 ν 1 for a ν = 1. We first et ε (hece ε ) be sufficiety sma such that where r 5N ε < mi{ 5d4 1 9N6 c 5 Ψ(r ) δ } Ψ(r ) = [(r i 1 r i ) 5N ] ( 4 5 )i i=1 which is easiy see to be we-defied. The c d4 (r r 1 ) N ε 1 4 1 i.e. D) hods for ν =. Reca that 8 1 K r ε 5 1 4 ε 1 4τd 4. w β We see that D1) aso hods for ν =. Now for ay ν 1 we have by iductio that c d4 (r ν r ν1 ) N ε 1 4 ν = c d4 (r ν r ν1 ) N (4c d4 (r ν 1 r ν ) N ε 5 4 ν 1 ) 1 4 ( 4N c 5 5d4 (r ν 1 r ν ) 5N ε 5 4 ν 1 ) 1 4 ( 4N c 5 5d4 Ψ(r )ε ) 1 4 ( 5 4 )ν ( r5n 5N4 ) 1 4 ( 5 4 ) ν 1

A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS 1 ad i.e. D1) ad D) hod true. 1 ν3 r ν r ν1 ε 5 4 ν r = K ν1 1 ε ( 6 5 )ν 1 4τd 4 1 ε 1 4τd 4 ν 1 ε ( 5 4 )ν 3ν r 1 ε 4 ν K 4.. Covergece. Let Ψ ν = Φ Φ 1 Φ ν 1 ν = 1. Iductivey we have that Ψ ν : D ν O ν1 D ad H Ψ ν = H ν = N ν P ν for a ν = 1. Let Õ = ν=o ν. Appyig Lemma 4.1 ad stadard argumets (e.g. [ 7]) we cocude that H ν e ν N ν P ν Ψ ν DΨ ν ω ν coverge uiformy o D( 1 r ) Õ say to H e N P Ψ DΨ ω respectivey. It is cear that N = e ω I A z z. Sice ε ν = 4c d4 (r ν 1 r ν ) N ε 5 4 ν 1 (4c d4 Ψ(r )ε ) ( 5 4 )ν we have by Lemma 4.1 that X P D( 1 r) Õ. Let Φ t H deote the fow of ay Hamitoia vector fied X H. Sice H Ψ ν = H ν we have that (4.1) Φ t H Ψ ν = Ψ ν Φ t H ν. The uiform covergece of Ψ ν DΨ ν X Hν impy that oe ca pass the imit i the above to cocude that Φ t H Ψ = Ψ Φ t H o D( 1 r ) Õ. It foows that Φ t H (Ψ (T N {ξ})) = Ψ Φ t N (T N {ξ}) = Ψ (T N {ξ}) for a ξ Õ. Hece Ψ (T N {ξ}) is a embedded ivariat torus of the origia perturbed Hamitoia system at ξ Õ. The frequecies ω (ξ) associated with Ψ (T N {ξ}) are sighty deformed from the uperturbed oes ω(ξ) ad the orma behaviors of the ivariat tori Ψ (T N {ξ}) are govered by their respective orma frequecy matrices A. 4.3. Measure estimates. For each ν = 1 et { } R ν k() = ξ O ν 1 : k ω ν 1 1 > ( Kτ ν )d { ad R ν k() = =1 ξ O ν 1 : ( k ω ν 1 I Ãν 1 ) 1 > ( Kτ ν )d R ν kj () = {ξ O ν 1 : ( k ω ν 1 I Ãν 1 R ν kj () = {ξ O ν 1 : ( k ω ν 1 I Ãν 1 The O ν O ν 1 \(( R ν k()) ( R ν k()) ( k K ν k K ν K ν } I I Ãν 1 j ) 1 > ( Kτ ν )d } I I Ãν 1 j ) 1 > ( Kτ ν )d } k j =. k K νj K ν R ν± kj ()))

JIANSHENG GENG AND YINGFEI YI for a ν = 1. Cosider the resoat sets R ν = ( R ν k()) ( R ν k()) ( k K ν k K ν K ν It is cear that O \ Õ ν 1 R ν. Lemma 4.. There is a costat C 1 > such that for a k K ν j K ν ad ν = 1. meas(r ν k() R ν k() R ν± kj ()) C 1 k K νj K ν R ν± K τ 1 ν kj ()). Proof. The proof foows from argumets i [31]. For simpicity we oy estimate the measures of R ν kj (). Measure estimates for Rν k Rν k ad Rν kj () ca be obtaied simiary. We ote that (4.) k ω ν (ξ) I Ãν 1 I I Ãν 1 j = k ω ν 1 (ξ) I diag(ω 1 Ω m1 Ω r Ω mr )W (ξ) where Ω 1 Ω m1 Ω r Ω mr are uperturbed frequecies idepedet of parameters ad dim(ãν 1 I) = r d ad W (ξ) Oν 1 ε. I the case k = ad j sice by assumptio (A) Ω j Ω mj > j = 1 r we have by the stadard Neuma series expasio that Ãν 1 ε 1 i.e. R ν j () =. I the case k we have by Lemma 5.7 that R ν kj {ξ O ν 1 : det( k ω ν 1 (ξ) I Ãν 1 I I Ãν 1 Deote g(ξ) = det( k ω ν 1 (ξ) I Ãν 1 The it foows from (4.) that r g(ξ) = ω ν 1 (ξ) Ω i Ω mi ) i=1( k α where a α Oν 1 r i=1 α i r 1. Due to the choice of ε we have that I I Ãν 1 j ) 1 < < ( Kτ ν1 ) d as j ) < ( I I Ãν 1 j ). K τ 1 ν a α ( k ω ν 1 (ξ) Ω Ω m ) α ) d }. ε the muti-idex α rus over the set α = (α 1 α r ) α i = or 1 ad r ξ g(ξ) Hece by Lemma 5.8 r k ξ ω ν (ξ) ε k r 1 (δ ε ) k r ε k r 1 δ k r. i=1 meas(r ν kj ) C 1[( K τ 1 ν ) d ] 1 r C1 Kν τ 1. Lemma 4.3. meas(o \ Õ) meas( ν 1 R ν ) = O(). Proof. By Lemma 4. there exists a costat C > such that meas( kj ) k K νj K ν R ν± k K νj K ν C 1 4C 1 k K ν K τ 3 ν K τ 1 ν C K τ b 3 ν.

A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS 3 Simiary there are costats C 3 > C 4 > such that meas( R ν k() C 3 k K ν ad Let τ b 4. We have that meas( k K ν K ν R ν k()) C 4 meas(o \ Õ) meas( ν 1 R ν ) = meas[ (( R ν k) ( ν 1 k K ν = O( ) = O(). K ν1 ν K τ b 3 ν k K ν K ν R ν k) K τ b 3 ν (. R ν± kj ))] k K νj K ν This competes the measure estimate. 5. Appedix Lemma 5.1. F G D(rs)O F D(rs)O G D(rs)O. Proof. Sice (F G) kαβ = k α β F k k α α β β G k α β we have F G D(rs)O = sup (F G) kαβ s w α w β e k r w <s w <s kαβ F k k α α β β G k α β s w α w β e k r kαβ k α β sup w <s w <s F D(rs)O G D(rs)O. Lemma 5.. (Geeraized Cauchy iequaities) F θ D(r σs) 1 σ F D(rs) F I D(r 1 s) 4 s F D(rs) F w D(r 1 s) s F D(rs) Proof. See [7]. F w D(r 1 s) s F D(rs). Let { } deote the Poisso bracket of smooth fuctios: {F G} = F I G θ F θ G I i ( F w G F w w G w ). Lemma 5.3. There exists a costat c > such that if the F D(rs) < e G D(rs) < ε {F G} D(r σ 1 s) < cσ 1 s F D(rs) G D(rs) cσ 1 s εe.

4 JIANSHENG GENG AND YINGFEI YI Proof. By Lemmas 5.1 5. F I G θ D(r σ 1 s) < 4σ 1 s F G F θ G I D(r σ 1 s) < cσ 1 s F G m F wm G wm D(r 1 s) m F wm D(r 1 s) G wm D(r 1 s) F w D(r 1 s) G w D(r 1 s) 4s F G m F wm G wm D(r 1 s) m F wm D(r 1 s) G wm D(r 1 s) It foows that F w D(r 1 s) G w D(r 1 s) 4s F G. {F G} D(r σ 1 s) < cσ 1 s F D(rs) G D(rs) cσ 1 s εe. Lemma 5.4. There exists a costat c > such that if for some ε ε > the X F D(rs) < ε X G D(rs) < ε X {FG} D(r σηs) < cσ 1 η ε ε for ay < σ < r ad < η 1. I particuar if η ε 1 4 ε ε ε ε 3 4 the Proof. See [16]. X {FG} D(r σηs) ε 5 4. The foowig emmas ca be foud i the Appedix of [1 3]. Lemma 5.5. Let O be a compact set i R N for which sma divisor coditios hod. Suppose that ω(ξ) are C d Whitey-smooth fuctios i ξ O with derivative bouded by L ad f(ξ) are C d Whitey-smooth fuctios i ξ O with CW d orm bouded by L. The g(ξ) f(ξ) k ω(ξ) is C d Whitey-smooth i O with g O < c d4 K τd4 L. Lemma 5.6. Let O be a compact set i R N for which sma divisor coditios hod. Suppose that A (ξ) R (ξ) are respectivey C d Whitey-smooth matrices ad vectors ad ω(ξ) is a C d Whitey-smooth fuctio with derivatives bouded by L. The F (ξ) = M 1 R (ξ) is C d Whitey-smooth with F O c d4 K τd4 L where M stads for either k ω I Ã or k ω I ± Ã I ± I Ãj.

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