Chaotic scattering of He atoms off a Cu surface with corrugated Morse potential

a Xi :2401.05795 1 [ma h.DS] 11 Jan 2024 Chao ic sca e ing o He a oms o a Cu su ace wi h co uga ed Mo se po en ial Flo en ino Bo ondo∗ , E nes Fon ich† , and Pau Ma ín‡ . Janua y 12, 2024 Abs ac We conside a Hamil onian sys em ha models he sca e ing o helium a oms o a coppe su ace. The in e ac ion be ween he He and he Cu a oms is desc ibed by a co uga ed Mo se po en ial. Using co uga ion coe icien s alues in he po en ial ob ained by i ing o expe imen al alues, we p o e ha , p o ided some coe icien o an auxilia y unc ion is di e en om 0, he e a e egions o he phase space, co esponding o su icien ly la ge ene gy o he inciden a om, whe e he sca e ing is chao ic. Fu he mo e, we p o e ha he sys em has oscilla o y mo ions. Keywo ds: chao ic sca e ing, Hamil onian sys ems, oscilla o y o bi s, exponen ially small spli ing, inne equa ion. Con en s 1 In oduc ion 2 1.1 The He-Cu sca e ing p oblem. Main s a emen . . . . . . . . . . . . . . . . . . . . 2 1.2 Exponen ially small spli ing o in a ian mani olds . . . . . . . . . . . . . . . . . . 4 1.3 S uc u e o he pape .................................. 4 2 Hamil onian o mula ion o he He-Cu sca e ing p oblem and spli ing o sepa a ices 5 2.1 McGehee-like coo dina es and as dynamics ...................... 5 2.2 Dynamics o H0...................................... 6 2.3 The Melniko po en ial ................................. 8 2.4 Dynamics o he ull sys em. Spli ing o he in a ian mani olds . . . . . . . . . . 9 3 Chao ic dynamics 9 3.1 Poinca é-Ca an educ ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Local coo dina es a ound q=p= 0 and a pa abolic λ-lemma . . . . . . . . . . . . 10 3.3 The local and he global maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.4 Symbolic dynamics. P oo o Theo em 1.1 . . . . . . . . . . . . . . . . . . . . . . . 12 4 Hamil on-Jacobi equa ion 14 4.1 No a ion .......................................... 14 4.2 De i ing he Hamil on-Jacobi equa ion . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.3 De ini ions and echnical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.4 Solu ion o he Hamil on-Jacobi equa ion (4.3) . . . . . . . . . . . . . . . . . . . . 18 ∗ .b[email p o ec ed], Depa amen o de Química, Uni e sidad Au ónoma de Mad id. † on [email protected], Depa amen de Ma emà iques i In o mà ica, Uni e si a de Ba celona (UB), and Cen e de Rece ca Ma emà ica (CRM). ‡[email p o ec ed], Depa amen de Ma emà iques, Uni e si a Poli ècnica de Ca alunya (UPC), and Cen e de Rece ca Ma emà ica (CRM) 1 5 Fi s ex ension o he in a ian mani old 20 5.1 F om Hamil on-Jacobi pa ame iza ion o low pa ame iza ion . . . . . . . . . . . 20 5.1.1 P elimina ies and echnical lemmas o sol e equa ion (5.6) . . . . . . . . . . 21 5.1.2 Solu ion o equa ion (5.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Ex ension o he low pa ame iza ion . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.3 F om low pa ame iza ion o Hamil on-Jacobi . . . . . . . . . . . . . . . . . . . . 29 6 The inne equa ion 31 6.1 Spaces and echnical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.2 Fixed poin equa ion ................................... 33 7 App oxima ion o he mani old in he inne domain 35 7.1 Spaces and echnical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7.2 The ixed poin equa ion ................................. 39 8 Di e ence be ween solu ions o he inne equa ion 41 9 Di e ence o he solu ions o he Hamil on-Jacobi equa ion 47 9.1 S aigh ening he linea ope a o . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 9.2 The exponen ially small o mula o Φ+−Φ−. . . . . . . . . . . . . . . . . . . . . 52 10 Acknowledgemen s 55 A P oo o Theo em 3.2 55 B P oo s o claims in Appendix A 59 B.1 P oo o Lemma A.1 ................................... 59 B.2 P oo o Lemma A.4 ................................... 60 B.3 P oo o Lemma A.5 ................................... 61 1 In oduc ion 1.1 The He-Cu sca e ing p oblem. Main s a emen We conside he mo ion o a helium a om bouncing o a coppe su ace. The p oblem a ises om expe imen al echniques, whe e he sca e ing o noble gas a oms a e collisions wi h a su ace is used o cha ac e ize su ace s uc u es in a non-des uc i e way [Hul92]. The co uga ion o he su ace is modeled by he e ec i e po en ial seen by he a oms as hey come nea he su ace. Co uga ion depends on he inciden ene gy: he highe he ene gy o he He a om, he close will ge o he su ace. Nume ical e idence o he exis ence o chao ic sca e ing was gi en in [GBMA97,BGB+99], whe e i was ela ed o some in a ian mani olds in he sys em. The pu pose o he p esen pape is o p o ide an ac ual igo ous p oo o such chao ic beha io , hus es ablishing a link be ween expe imen s and heo y on i m g ounds. We simpli y he p oblem assuming ha he He a om mo ion akes place on a plane; ac ually he ou -o -plane sca e ing is ela i ely small o he alues o he ene gies and inciden angles conside ed in he expe imen s. We deno e by (x, z)∈R2 he posi ion o he He a om, whe e xand za e he ho izon al and e ical displacemen s, espec i ely. Le (px, pz)be he conjuga e momen a. We model he in e ac ion o he He a om wi h he coppe su ace by a co uga ed Mo se po en ial, whe e he co uga ion ep esen s he p esence o he coppe a oms in he su ace. Mo e conc e ely, we conside he Hamil onian HCM(x, z, px, pz) = 1 2m(p2 x+p2 z) + VM(z) + VC2πx a, z,(1.1) whe e VM(z) = De−αz(e−αz −2), VC(θ, z) = De−2αz V(θ), V(θ) = X n≥1 ( ncos(nθ) + snsin(nθ)) . (1.2) 2 The coe icien s nand sna e de e mined expe imen ally. Thei alues a e 1= 0.06, 2= 0.008, n= 0, n ≥3, sn= 0, n ≥1, D= 6.35 meV, a= 3.6Å, α= 1.05 Å−1. See [GBMA97]. In pa icula , 16= 0. In he p esen pape , ou only equi emen on Vis ha i is an analy ic e en unc ion, ha is, sn= 0, o all n. We emphasize ha we do no equi e V o be a igonome ic polynomial. E enness is no an impo an equi emen and he esul s will hold wi h he same echniques o non-e en V, bu he added symme y will simpli y some o he de ails, pa icula ly he nume ical compu a ions pe o med in he p oblem ha a e ca ied ou in [BBF+23]. Since θ= 2πx/a appea s in he equa ion h ough V, we will ake θ∈T=R/2πZ. The pu pose o his pape is wo old. On he one hand, we wan o p o e he p esence o chaos in some pa s o he phase space o (1.1). He e, he no ion o chaos is he one in oduced by Smale in [Sma65] and i is based on he p esence o a Smale ho seshoe and a conjuga ion wi h he shi on a space o sequences o symbols. I is wo h o ema k ha he se whe e chaos akes place is a hype bolic se . We ecall he de ini ion o symbolic dynamics wi h an in ini e numbe o symbols, as in oduced by Mose in [Mos73]. Le S={s= (...,s−1, s0, s1,...)|si∈N}be he space o wo sided sequences o in ini e symbols, wi h he opology induced by he neighbo hood basis o s∗= (...,s∗ −1, s∗ 0, s∗ 1,...), Ij(s∗) = {s∈S|sk=s∗ k,|k|< j}, s∗∈S. I is well known ha he shi σ:S→Sde ined by σ(s)i=si+1 is a homeomo phism. The shi σis ansi i e and he se o i s pe iodic o bi s is dense in S. I has sensi i e dependence on ini ial condi ions. I is one o he pa adigms o chaos. The second goal o he pape is o p o e he exis ence o oscilla o y o bi s, ha is, solu ions (x( ), z( )) o (1.1) wi h he p ope y ha lim sup z( ) = ∞and lim in z( )<∞, ha is, solu ions ha go highe and highe bu always go back again o a ini e dis ance o he Cu su ace. The claims in he p esen pape ollow he scheme p oposed by Mose in [Mos73], which was also used in he es ic ed plana h ee body p oblem in [LS80]. In his las pape , an impo - an echnicali y, no p esen in Mose ’s wo k, appea ed: he exponen ially small beha io o he di e ence be ween he mani olds which gi e ise o he in e es ing dynamics. This p oblem was la e o e come in [GMS16]. This exponen ially small beha io also appea s in he model unde conside a ion he e and dealing wi h i will ep esen an impo an pa o ou wo k. Ou main heo em is he ollowing. Theo em 1.1. The e exis s a unc ion 1o he coe icien s 1, 2,... such ha i 16= 0, hen, o any hla ge enough, he e exis s a sec ion Σ⊂ {HCM =h}o he ec o ield associa ed o HCM and a subse I ⊂ Σsuch ha he Poinca é map Ψ : I → I is a homeomo phism and is conjuga e o he shi σo in ini e symbols. The se Iis a hype bolic se o Ψ. Fu he mo e, o any hla ge enough, he Hamil onian HCM in (1.1)possesses oscilla o y o bi s. Rema k 1.2. The unc ion 1appea s in a p oblem independen o he ene gy h. This equa ion is usually known in he li e a u e as inne equa ion. The nume ical e idence in [BBF+23] s ongly sugges s ha o he expe imen al alues men ioned abo e, 1is non-ze o. Fu he mo e, we p o e ha i one eplaces iby ε˜ iin he de ini ion o V, in (1.2), ha is, i we conside he e m VCas a small pe u ba ion, hen 1=επ 4˜ 1+O(ε2). The es o he pape is de o ed o p o e Theo em 1.1. In pa icula , i will be an immedia e consequence o Theo em 3.5. We would like o ema k ha he hype bolic se , as is he case o he Si niko p oblem conside ed in [Mos73] o he es ic ed plana h ee body p oblem in [LS80,GMS16], is ela ed o he in a ian mani olds o a ce ain pe iodic o bi a z=∞. In ou case, since he equa ions o mo ion a e ˙x=1 mpx, ˙z=1 mpz, ˙px=−2π aDe−2αzV′2πx a, ˙pz=−2Dα e−αz 1−e−αz 1 + V2πx a, (1.3) 3 we ha e ha he se {(x, z, px, pz)∈R/aZ×R×R2|z=∞, pz= 0, px= (2mh)1/2}(1.4) is a pe iodic o bi o (1.3) a in ini y in he ene gy le el h. The p oo o Theo em 1.1 consis s in checking ha his pe iodic o bi possesses in a ian s able and uns able mani olds, ha hese in a ian mani olds in e sec ans e sally i his la ge enough and hen p o e a sui able λ-lemma ha ensu es ha his ans e sal in e sec ion gi es ise o he s anda d isola ing blocks wi h cone condi ions. This las pa is due o he ac ha he pe iodic o bi (1.4) is nei he hype bolic no ellip ic, bu degene a e. Howe e , al hough degene a e, i possesses s able and uns able in a ian mani olds. The s udy o he in a ian mani olds o hese ype o degene a e objec s goes back o [McG73]. See also [BFdlLM07], whe e he pa ame iza ion me hod is used, and he subsequen wo ks, [BFM20a,BFM20b,BFM17,BFM20c]. In his case, when he pe iodic o bi is degene a e, he s anda d λ-lemma does no hold. In pa icula , i is no ue ha he o wa d images o a mani old in e sec ing ans e sally he s able mani old o he o bi accumula e o he whole uns able mani old. Hence, a di e en a gumen is needed o con ol he passage close o he pe iodic o bi . This was al eady known by Mose in [Mos73]. Howe e , his p oo does no di ec ly apply o he p esen case, because he deg ee o degene acy o ou case is di e en om he one o he Si niko p oblem. We p esen ano he p oo , based in he ideas in [GMPS22]. 1.2 Exponen ially small spli ing o in a ian mani olds One o he main di icul ies o he p esen wo k is o es ablish ha he angle o in e sec ion be ween he s able and uns able in a ian mani olds o ce ain pe iodic o bi s is non-ze o. Indeed, since ou goal is o deal wi h he physical p oblem, in which he co uga ion, modelled by he unc ion V in (1.2), is ixed, he only pa ame e we will ha e o deal wi h he p oblem will be he ene gy o he sys em. We will see ha he angle o in e sec ion is in ac exponen ially small in he ene gy hence p ecluding he use o he s anda d Melniko heo y. S a ing wi h he seminal pape o Lazu kin [Laz84] (see he English ansla ion in [Laz03]), he e is a long lis o wo ks in he li e a u e conce ning he exponen ially small spli ing o sepa a- ices. In pa icula , he me hod in oduced in [Sau01] has been essen ial in pos e io de elopmen s o he ield. See [BFGS12] and he e e ences he ein. Ou app oach he e is simila o he he one es ablished in [BFGS12] and [Bal06]. I is impo an o ema k ha he esul s in hese las wo pape s do no apply o ou se ing. As a ma e o ac , i one ies o w i e Hamil onian (1.1), which has 2 deg ees o eedom, as a 11 2deg ees o eedom one using he Poinca é-Ca an e- duc ion, he educed Hamil onian does no sa is y some o he hypo heses in [BFGS12,Bal06]. Pa icula ly, when w i en as an in eg able Hamil onian plus a pe u ba ion, he pe u ba ion is no polynomial and has b anching singula i ies when ex ended o he complex domain. This is ue e en in he McGehee a iables ha a e in oduced in Sec ion 2. Since we a e no assuming he pe u ba ion o be small, we need o s udy he inne equa ion associa ed o he p oblem in o de o cap u e he leading e m o he exponen ially small beha io o he spli ing. This inne equa ion was used by Gel eich in [Gel99] o p o e he spli ing o sepa ices in he Chi iko s anda d map. See also [MSS11a,MSS11b], whe e he McMillan map was s udied by using esu gence heo y. The s udy o he inne equa ion o he Hénon map was pe o med in [GS01]. The exponen ially small spli ing o in a ian mani olds also appea s in o he physical p oblems, since i is ela ed o he exis ence o some as equencies on he sys em. In pa icula , i has been deal wi h success ully in some p oblem o celes ial mechanics [GMS16,GMPS22,GPS23]. In hese las cases, howe e , he Melniko unc ion p edic s co ec ly he i s o de o he spli ing. 1.3 S uc u e o he pape The s uc u e o he pape is as ollows. In Sec ion 2we w i e he sys em in sui able McGehee coo dina es, desc ibe he geome ic beha iou o i and claim he main heo em o he spli ing o sepa a ices. In Sec ion 3we apply he spli ing heo em o deduce he exis ence o chao ic dynamics o he sys em. To do so, we need o cope wi h he p oblem ha he pe iodic o bi s unde conside a ion a e no hype bolic bu degene a e, which implies ha he s anda d lambda lemma does no apply. 4 He e we use a e sion o he pa abolic lambda lemma in [GMPS22] o ob ain he conjuga ion o he Be noulli shi . Sec ions 4 o 9deal wi h he ac ual spli ing o he in a ian mani olds o he pe iodic o bi a in ini y. In Sec ions 4and 5we ob ain sui able app oxima ions o he in a ian mani olds by sol ing an app op ia e Hamil on-Jacobi equa ion. These app oxima ions a e alid in a ce ain complex domain. Howe e , we a e no able o ob ain enough in o ma ion o hei di e ence. To do so, we in oduce he inne equa ion in Sec ion 6, om which we ob ain wo solu ions. In Sec ion 7, we check ha hese solu ions a e also good app oxima ions o he in a ian mani olds in some egion o hei complex domain o de ini ion. In Sec ion 8we ob ain an exponen ially small o mula o he di e ence o solu ions o he inne equa ion which, in u n, we use in Sec ion 9 o ob ain he exponen ially small o mula o he di e ence o he in a ian mani olds. We ha e le o Appendix A he p oo o he pa abolic lambda lemma (Theo em 3.2). Ap- pendix B.1 con ains he p oo s o some echnical lemmas used in he p oo o Theo em 3.2. 2 Hamil onian o mula ion o he He-Cu sca e ing p oblem and spli ing o sepa a ices 2.1 McGehee-like coo dina es and as dynamics We a e in e es ed in mo ions o which he He pa icle a i es o z=∞wi h ze o momen um. Fo his eason, we in oduce he McGehee-like coo dina es    Aq2=e−αz,a 2πθ=x, Bp =pz, CI =px, whe e A= 2, B =aα 4πC, C2= 2mD 8π aα2 . Wi hou loss o gene ali y, we ake C > 0. This change ans o ms he s anda d 2- o m dx ∧dpx+ dz ∧dpzin o he b-symplec ic o m (aC/2π)ω, whe e ω=dθ ∧dI −1 qdq ∧dp (2.1) and he Hamil onian unc ion HCM in (1.1) becomes 8D·H, whe e H(q, p, θ, I) = 1 2(νI2+p2)−1 2q2+1 2q4+1 2q4V(θ)(2.2) and ν=4π aα2 . Fo he expe imen al alues o he He-Cu p oblem, he alue o νis ν= 11.051879175935... Rescaling ime, he dynamics o HCM is equi alen o he one gene a ed by Hwi h espec o he o m ω. To s udy he beha io o he sys em o la ge alues o I0, we in oduce he new a iable J by I=I0+Jin (2.2) wi h I0≫1. This change p ese es he 2- o m ω. We will deno e he Hamil onian in hese new a iables wi h he same le e , namely, H(q, p, θ, J) = H0(q, p, J) + H1(q, p, θ, J),(2.3) whe e H0(q, p, J) = 1 2(ν(I0+J)2+p2)−1 2q2+1 2q4, H1(q, p, θ, J) = 1 2q4V(θ). (2.4) 5 Since H0does no depend on θ, i is in eg able. We ema k ha Hlooks like a pe iodically pe u bed Du ing equa ion. Howe e , since he ec o ield gene a ed by His ob ained h ough he non-s anda d 2- o m ω, he sys em is no equi alen o he Du ing equa ion. Indeed, he equa ions o mo ion gene a ed by Ha e ˙q=−q∂H ∂p =−qp, ˙ θ=∂H ∂I =νI0+νJ, ˙p=−q−∂H ∂q =−q2+ 2q4+ 2q4V(θ),˙ J=−∂H ∂θ =−q4 2V′(θ), (2.5) which is no a Du ing oscilla o due o he p esence o he ac o qin he (q, p)componen s o he ec o ield. We conside , a he ene gy le el νI2 0/2o Hin (2.3), he se ΛνI0={q=p=J= 0, θ ∈T}. F om (2.5), i is in a ian and he dynamics on ΛνI0is gi en by θ=θ0+νI0 , ha is, i is pe iodic wi h equency νI0/(2π), which is as when νI0is la ge. Tha is, a he ene gy le el νI2 0/2, o la ge alues o νI0,Hcan be seen as a as pe iodic pe u ba ion o he in eg able Hamil onian H0. We emphasize ha he pe u ba ion H1is no small since we a e assuming ha 1is ixed and we will conside mo ions in which qwill each size o O(1). The p oo o Theo em 1.1 consis s in p o ing ha each o hese pe iodic o bi s possesses in a ian mani olds and ha hey in e sec ans e sally. F om his in e sec ion we will deduce he exis ence o a ho seshoe wi h in ini ely many symbols. Howe e , since he sys em is a as pe iodic pe u ba ion o an in eg able sys em, i is well known ha he angle o in e sec ion o he mani olds will be smalle han any powe o he in e se o he equency which makes he ques ion o compu ing i a beyond all o de s phenomenon. Rema k 2.1. Al hough H1is no small, i is well known ha he ac ha depends on a as angle implies ha i s con ibu ion a e ages ou up o an exponen ially small emainde . Indeed, we can apply a s ep o a e aging and s ill ob ain an explici exp ession o he emainde , which is o size O((νI0)−1). I is immedia e o check ha he change θ= Θ, J =K+A′(Θ)Q4, q =Q, p =P+ 4A(Θ)Q4, p ese es he 2- o m ωin (2.1). In hese new a iables, Hbecomes e H(Q, P, Θ, K) =1 2(ν(I0+K−A′(Θ)Q4)2+ (P+ 4A(Θ)Q4)2)−1 2Q2+1 2Q4+1 2Q4V(Θ) =1 2(ν(I0+K)2+P2)−1 2Q2+1 2Q4+1 2V(Θ) −νI0A′(Θ)Q4 −νA′(Θ)KQ4+ 4A(Θ)PQ4+ 8A(Θ)2Q8+1 2νA′(Θ)2Q8. Taking Asuch ha A′(Θ) = −V(θ)/(2νI0)and in oducing W(Θ) = RΘV(ϑ)dϑ, we ob ain e H(Q, P, Θ, K) = H0(Q, P, K) + e H1(Q, P, Θ, K), whe e e H1(Q, P, Θ, K) = 1 2νI0−νV (Θ)KQ4+ 4W(Θ)PQ4+1 4I0 V(Θ)2Q8+4 νI0 W(Θ)2Q8. The compu a ions o he Melniko po en ial in Sec ion 2.3 below lead o a o mula o he same o de . We will no ollow his app oach. On he con a y, we will wo k di ec ly wi h he Hamil onian H=H0+H1in (2.3). 2.2 Dynamics o H0 In iew o Rema k 2.1, al hough H1is no small, we can see H0as a e e ence sys em o H. He e we desc ibe i s dynamics. Fi s we obse e ha , since H0does no depend on θ,Jis a conse ed quan i y. Hence, H0is in eg able. Using he 2- o m ωin (2.1), he equa ions o mo ion o H0a e ˙q=−qp, ˙ θ=νI0+νJ, ˙p=−q2+ 2q4,˙ J= 0.(2.6) 6 The se ΛνI0={q=p=J= 0, θ ∈T} is a pe iodic o bi in {H0=νI2 0/2}. F om (2.6), as we ha e al eady men ioned, i is clea ha he es ic ion o ΛνI0 o {H0=νI2 0/2, J = 0}is no hype bolic. Howe e , i possesses in a ian s able and uns able mani olds, ha gene a e wo homoclinic loops, gi en by W0 νI0=(q, p, θ, J)|J= 0, θ ∈T, H0(q, p, 0) = 1 2νI2 0. See Figu e 1. We will ocus on he loop wi h q≥0. The igu e is sligh ly misleading: o any J, he se {q= 0, p =p0}is a pe iodic o bi . In pa icula , {q= 0}is in a ian , e en o he ull Hamil onian H. This means ha , in ac , in he (q, p)-plane, (q, p) = (0,0) is no a opological saddle because, in pa icula , is no an isola ed equilib ium poin because q= 0 is a line o equilib ia. As a consequence, he s anda d λ-lemma is no longe ue and ano he s a emen will be necessa y. The needed esul is Theo em 3.2 below. We will be in e es ed in he pa o he phase space whe e q > 0. Figu e 1: The p ojec ion o he in a ian mani olds W0 νI0on o he (q, p)-plane. Fo each p=p0, {(0, p0, θ, 0) |θ∈T}is a pe iodic o bi . The ollowing lemma, whose p oo is a s aigh o wa d compu a ion, p o ides he ime pa ame- iza ion o he igh hand side homoclinic o bi . Lemma 2.2. The ime pa ame iza ion o he homoclinic o bi o (0,0) o (H0)|J=0 ha passes h ough (1,0) a = 0 is gi en by qh( ) = 1 √1 + 2, ph( ) = −˙qh( ) qh( )= 1 + 2. As a consequence, Γ0(u, θ) =     qh(u) ph(u) θ 0    (2.7) pa ame izes W0 νI0. Deno ing φ0 he low gene a ed by H0, his pa ame iza ion sa is ies φ0 ◦ Γ0(u, θ) = Γ0(u+ , θ +νI0 ). 7 2.3 The Melniko po en ial As usual, we de ine he Melniko po en ial associa ed o he Hamil onian Has L(u, θ) = −Z∞ −∞ H1(qh(u+ ), ph(u+ ), θ +νI0 , 0) d . (2.8) Expanding Vin Fou ie se ies, V(θ) = X k∈Z V[k]eikθ, by (2.4), we can w i e H1(q, p, θ, 0) = q4 2V(θ) = q4 2X k∈Z V[k]eikθ. Since Vis eal analy ic, e en and has ze o a e age, we ha e ha V[k]∈R, V [0] = 0, V [k]=V[−k], k ∈Z {0} and he e exis s σ0>0and K > 0such ha , o all k∈Z, |V[k]| ≤ Ke−|k|σ0.(2.9) Hence, L(u, θ) = −Z∞ −∞ H1(qh( ), ph( ), θ −νI0u+νI0 , 0) d =−X k∈Z eik(θ−νI0u)Z∞ −∞ q4 h( ) 2V[k]eikνI0 d . Tha is, L(u, θ) = X k∈Z L[k](νI0)eik(θ−νI0u), whe e L[k](νI0) = −1 2V[k]Z∞ −∞ q4 h( )eikνI0 d . P oposi ion 2.3. The Melniko po en ial Lsa is ies L[k](νI0) = −πνI0V[k] 4e−|k|νI0|k|+1 νI0, k ∈Z. In pa icula , since V[0] = 0 hen L[0](νI0) = 0. P oo . Since, in iew o Lemma 2.2,q4 his a me omo phic unc ion ha has poles o o de 2a ±i, he claim ollows om a s aigh o wa d esidue compu a ion. Co olla y 2.4. I 16= 0 in (1.2), hen L(u, θ) = −π 4νI0 1e−νI01 + 1 νI0cos(θ−νI0u) + O(e−νI0). I he coe icien s iin he de ini ion o Vin (1.2) sa is y i=ε˜ iand ˜ 16= 0, he s anda d Melniko heo y implies ha he dis ance be ween he uns able and s able in a ian mani olds o ΛνI0is gi en by he de i a i es o εL plus an e o o size O(ε2). Howe e , since Lhas size O(e−νI0), his means ha a he ene gy le el νI2 0/2,εL gi es he leading o de o he dis ance only i ε < O(e−νI0), ha is, o a ixed alue o he ene gy, he in a ian mani olds o in ini y spli only i he co uga ion is exponen ially small in he ene gy. We will see ha εand νI0can be aken as independen pa ame e s while he exponen ially small o mula o he dis ance will emain alid. The physical implica ions a e impo an : we will see ha , o a gi en co uga ion, i ce ain coe icien is di e en om 0( ha depends on he coe icien s o he co uga ion and is gene ically non-ze o), o any la ge enough ene gy, he in a ian mani olds o in ini y spli . Mo eo e , his coe icien depends on i s o de on L[1] i εis small. I is impo an o ema k ha i 1= 0, he Melniko unc ion is o o de e−2νI0. Howe e , in his case, he ue beha io , gi en by o mula (2.11) in Theo em 2.5, below, can be o o de e−νI0, i ce ain coe icien 1is di e en om 0. See [BMS23]. 8 2.4 Dynamics o he ull sys em. Spli ing o he in a ian mani olds The pe iodic o bi ΛνI0={q=p=J= 0, θ ∈T}o H0 emains when we conside he ull sys em Hin (2.3), wi h he same equency νI0. Al hough i is no hype bolic, we will p o e ha i has in a ian uns able and s able mani olds, which, i ce ain non-degene acy condi ion holds, no longe coincide. Nex heo em summa izes he claim. I is he i s s ep in he p oo o Theo em 1.1. Theo em 2.5. I νI0is la ge enough, he pe iodic o bi ΛνI0={q=p=J= 0, θ ∈T}possesses in a ian uns able and s able mani olds, W± νI0. Mo eo e , o any 0< u0< u1, he e exis analy ic unc ions Φ±: [u0, u1]×T→Rsuch ha Γ±(u, θ) =     qh(u) ph(u)−1∂uΦ±(u, θ) θ ∂θΦ±(u, θ)    (2.10) a e pa ame iza ions o pieces o W± νI0sa is ying (1) Γ±= Γ0+O((νI0)−1), whe e Γ0is gi en in (2.7), (2) he e exis s 1,Λ0∈Rsuch ha , o any j, k ≥0,0≤j+k≤3, ∂j u∂k θ(Φ+(u, θ)−Φ−(u, θ)−Λ0) = (−1)j(νI0)j+1e−νI02 1∂j+k αcos α|α=(θ−νI0u)+O1 log(νI0).(2.11) The coe icien s 1and Λ0do no depend on νI0. They only depend on he coe icien s io he unc ion Vin H1. I one assumes ha i=ε˜ i, i.e. he co uga ion e m H1is a small pe u ba ion o H0, hen 1=π 2˜ 1ε+O(ε2).(2.12) Rema k 2.6. Fo he expe imen al alues o Vco esponding o he in e ac ion be ween helium and coppe a oms, nume ical expe imen s in [BBF+23] sugges ha 16= 0. Howe e , i i=ε˜ i, we will see ha 1is an analy ic unc ion o ε. Then, (2.12)implies ha , i ˜ 16= 0, 1can only anish o a disc e e numbe o alues o ε. Co olla y 2.7. I νI0is la ge enough, he in a ian mani olds W± νI0o ΛνI0in e sec ans e sely along wo p ima y homoclinic o bi s. P oo . Since W± νI0a e con ained in he le el νI2 0/2o H, he exp ession o Γ±in (2.10) implies ha hei in e sec ions a e gi en by ∂uΦ+(u, θ)−∂uΦ−(u, θ) = 0. Then, (2.11) allows us o apply he s anda d implici unc ion heo em o ob ain ha he p ima y homoclinic o bi s a e gi en by θ−νI0u+O((νI0)−1) = kπ. Theo em 2.5 p o ides an exponen ially small o mula o he di e ence o he in a ian mani olds W± νI0, wi h he ene gy o he sys em as pa ame e . We will ind sui able app oxima ions o he in a ian mani olds o ΛνI0which ex end o complex alues o hei a iables. The leading e m o hese app oxima ions, o bo h he s able and uns able mani olds, will be he unpe u bed sepa a ix Γ0. This is done in Sec ions 4and 5. In many cases, his is enough o ob ain he exponen ially small o mula, whose leading e m is gi en by he Melniko po en ial (2.8). He e, howe e , in o de o cap u e he leading e m o he exponen ially small di e ence be ween he mani olds, we need o eso o wha is o en known as he inne equa ion. This is done in Sec ion 6 and 8. In Sec ion 7 he mani olds a e compa ed wi h he solu ions o he inne equa ion and, inally, all he in o ma ion is ga he ed in Sec ion 9, comple ing he p oo o Theo em 2.5. 3 Chao ic dynamics He e we de ine a e u n map on a sui able sec ion such ha i possesses an in a ian hype bolic se and when es ic ed o his se , i is opologically conjuga e o he shi o in ini e symbols. The cons uc ion is analogous o he one o Mose in [Mos73]. 9 equa ion (4.6) becomes L(Φ1) = F(Φ1).(4.9) We ema k ha F(0)(u, θ) = −H1(qh(u), θ) = −1 2 1 (1 + u2)2V(θ).(4.10) To sol e (4.9), we will ew i e i as a ixed poin equa ion by means o a sui able igh in e se Go L, o be de ined la e on, ha is, Φ1=G ◦F(Φ1).(4.11) We will de o e he es o he sec ion o p o e he exis ence o a solu ion o (4.6) wi h he bounda y condi ions as u→ −∞ in (4.4). 4.3 De ini ions and echnical lemmas Le 0< β1< β2< π/2be ixed. Fo κ > 1and δ∈(0,1/2), we conside he complex domain D+ κ,δ ={u∈C| |ℑu|<− an β1ℜu+ 1 −κ(νI0)−1,|ℑu|> an β2ℜu+ 1 −δ.(4.12) We will be in e es ed in he case whe e I0and κa e big bu sa is y κ(νI0)−1< δ. Obse e ha , i κ(νI0)−1< δ, hen d(D+ κ,δ,±i) = Cκ(νI0)−1wi h C= cos β1. See Figu e 2. Figu e 2: The domain D+ κ,δ de ined in (4.12). To sol e equa ion (4.9), o , s ∈Rwe in oduce he Banach space o eal analy ic, 2π-pe iodic in θ, unc ions X ,s ={Ψ : D+ κ,δ ×Tσ→C|Ψ eal analy ic,kΨk ,s <∞} and, aking in o accoun he Fou ie expansion Ψ(u, θ) = Pk∈ZΨ[k](u)eikθ, he no m kΨk ,s =X k∈ZkΨ[k]k ,se|k|σ, whe e, o an analy ic unc ion :D+ κ,δ →C, k k ,s = max   sup u∈D+ κ,δ,ℜu≤−u0|u (u)|,sup u∈D+ κ,δ,ℜu>−u0|(1 + u2)s (u)|  , whe e u0∈Ris chosen such ha u0≥max{1,(1 −δ)/ an β2}. 16 Lemma 4.2. Le , 1, 2, s, s1, s2∈R. We ha e (1) I Ψ∈ X + 1,s+ 2wi h 1, 2≥0 hen Ψ∈ X ,s and kΨk ,s ≤KνI0 κ 2 kΨk + 1,s+ 2, whe e K > 0is independen o νI0and κ. (2) I Ψ1∈ X 1,s1and Ψ2∈ X 2,s2 hen Ψ1Ψ2∈ X 1+ 2,s1+s2and kΨ1Ψ2k 1+ 2,s1+s2≤ kΨ1k 1,s1kΨ2k 2,s2. The p oo o he p e ious lemma is s aigh o wa d om he ideas in [Sau01]. We also in oduce e X ,s ={Ψ∈ X ,s |∂uΨ∈ X +1,s+1, ∂θΨ∈ X +1,s+1,TΨU ,s <∞}, whe e TΨU ,s =kΨk ,s +k∂uΨk +1,s+1 +νI0k∂θΨk +1,s+1. Gi en Ψ∈ X ,s, wi h > 0,s≥0, we o mally de ine Gu(Ψ)(u, θ) = Z0 −∞ Ψ(u+ξ, θ +νI0ξ)dξ =Zu −∞ Ψ(ξ, θ +νI0(ξ−u)) dξ. (4.13) Clea ly, when Gu(Ψ) is well de ined, i Ψis eal analy ic, so is Gu(Ψ). The ope a o Gu o mally sa is ies L◦Gu(Ψ) = Ψ, whe e Lis he di e en ial ope a o in oduced in (4.7). The nex echnical lemma will be he main ool o ind he desi ed solu ions o equa ion (4.11). I s p oo is an immedia e a ia ion o he a gumen s in [GOS10]. Lemma 4.3. Le Gube he ope a o de ined in (4.13). (1) I Ψ∈ X ,s, wi h > 1,s≥0, hen Gu(Ψ) ∈ X −1,s and kGu(Ψ)k −1,s ≤KkΨk ,s. I , s > 0and hΨi= 0, hen Gu(Ψ) ∈ X ,s and kGu(Ψ)k ,s ≤K(νI0)−1kΨk ,s. (2) I Ψ∈ X ,s, wi h , s > 1,Gu(Ψ) ∈ X −1,s−1and kGu(Ψ)k −1,s−1≤KkΨk ,s. (3) I Ψ∈ X ,s, wi h ≥1,s > 0,∂uGu(Ψ), ∂θGu(Ψ) ∈ X ,s and k∂uGu(Ψ)k ,s ≤KkΨk ,s,k∂θGu(Ψ)k ,s ≤K(νI0)−1kΨk ,s. As a consequence, i Ψ∈ X ,s, wi h , s > 1,Gu(Ψ) ∈e X −1,s−1and TGu(Ψ)U −1,s−1≤KkΨk ,s. The cons an Konly depends on , s and he cons an s in ol ed in he de ini ion o D+ κ,δ bu i is independen o νI0and κ. 17 4.4 Solu ion o he Hamil on-Jacobi equa ion (4.3) Le Fand Gube he ope a o s de ined by (4.8) and (4.13), espec i ely. We o mally in oduce L+ ou (u, θ) = Gu◦F(0)(u, θ) = −Zu −∞ H1(qh(s), θ +νI0(s−u)) ds =−Zu −∞ 1 2(1 + s2)2V(θ−νI0(s−u)) ds. (4.14) Fo la e use we also in oduce L− ou (u, θ) = −L+ ou (−u, −θ) = Z∞ u 1 2(1 + s2)2V(θ+νI0(s−u)) ds and Lou (u, θ) = L+ ou (u, θ)−L− ou (u, θ) = −Z∞ −∞ 1 2(1 + s2)2V(θ+νI0(s−u)) ds. Obse e ha Lou =L, whe e Lis he Melniko po en ial in oduced in (2.8). P oposi ion 4.4. The unc ion L+ ou ∈ X4,2in D+ κ,δ and sa is ies kL+ ou k4,2,k∂θL+ ou k4,2,k∂uL+ ou k5,3≤K νI0 , o some K > 0, independen o νI0and κ. Fu he mo e, hL+ ou i= 0. P oo . We ecall ha F(0)(u, θ) = −H1(qh(u), θ) = −1 2q4 h(u)V(θ) = −1 2 1 (1 + u2)2V(θ), whe e H1was in oduced in (2.4) and he unc ion Vin (1.2). Clea ly, F(0) ∈ X4,2and hF(0)i= 0. The bounds on kL+ ou k4,2and on k∂θL+ ou k4,2 ollow om (1) and (3) o Lemma 4.3, espec i ely. To ob ain he bound on k∂uL+ ou k5,3we obse e ha , by in eg a ion by pa s, i k6= 0, ∂uZu −∞ 1 2 1 (1 + s2)2eikνI0(s−u)eikθ ds =−2Zu −∞ s (1 + s2)3eikνI0(s−u)eikθ ds. Hence, de ining g(u, θ) = −2u (1 + u2)3V(θ), we ha e ha ∂uL+ ou =Gu(g). Since hgi= 0 and g∈ X5,3, using again (1) o Lemma 4.3, k∂uL+ ou k5,3=kGu(g)k5,3≤K(νI0)−1kgk5,3. The las claim ollows om he ac ha hF(0)i= 0. We use L+ ou o ew i e equa ion (4.11) as a new ixed poin equa ion, wi h be e con ol o i s solu ion. To do so, we in oduce Φ2by Φ1=L+ ou + Φ2. Then, Φ1is a solu ion o (4.11) i and only i Φ2sa is ies Φ2=Gu◦e F(Φ2), whe e e F(Φ2) = F(L+ ou + Φ2)−F(0).(4.15) P oposi ion 4.5. The e exis s K∗>0such ha , i κis big enough, he ope a o Gu◦e F: BK∗(νI0)−2⊂e X5,3→ BK∗(νI0)−2is well de ined and a con ac ion. Le Φ+ 2be i s ixed poin . As a consequence, since Φ+ 2is he ixed poin o Gu◦e F, hen Φ+ 1=L+ ou + Φ+ 2is a solu ion o equa ion (4.9). 18 P oo . We i s claim ha e F(0) ∈ X6,4and ke F(0)k6,4≤K1(νI0)−2, o some K1independen o νI0. Indeed, by he de ini ions o Fin (4.8) and e Fin (4.15), using Lemma 4.2, P oposi ion 4.4, he ac ha p−2 h∈ X−2,−2and k∂θL+ ou k2,2≤ k∂θL+ ou k4,2≤K νI0 , we ha e ha ke F(0)k6,4= 1 2p2 h (∂uL+ ou )2+ν 2(∂θL+ ou )26,4 ≤ 1 2p2 h ∂uL+ ou 1,1k∂uL+ ou k5,3+ν 2k∂θL+ ou k2,2k∂θL+ ou k4,2 ≤K 2kp−2 hk−2,−2k∂uL+ ou k3,3 1 νI0 +νK 2k∂θL+ ou k2,2 1 νI0 ≤K2 2kp−2 hk−2,−2 1 (νI0)2+νK2 2(νI0)2 ≤K1 (νI0)2. Hence, by he las claim o Lemma 4.3,TGu◦e F(0)U5,3≤Kke F(0)k6,4≤KK1 (νI0)2<∞. We ake K∗= 2KK1. Le Ψ∈ BK∗(νI0)−2⊂e X5,3. In iew o (4.8) and (4.15), we w i e e F(Ψ) = e F(0) + e F(Ψ) −e F(0) = F0−F1−F2, whe e F0=e F(0), F1=1 2p2 h(∂uL+ ou +∂uΨ)2−(∂uL+ ou )2, F2=ν 2(∂θL+ ou +∂θΨ)2−(∂θL+ ou )2. We s a wi h F1. Using ha kp−2 hk−2,−2<∞,k∂uL+ ou k5,3≤K(νI0)−1,Ψ∈ BK∗(νI0)−2⊂ e X5,3and Lemma 4.2, we ha e ha kp−2 h(2∂uL+ ou +∂uΨ)k0,0≤ kp−2 hk−2,−2k2∂uL+ ou +∂uΨk2,2 ≤K2νI0 κk∂uL+ ou k5,3+(νI0)2 κ2k∂uΨk6,4≤K1 κ+K∗ κ2(4.16) o some cons an s Kindependen o νI0and κ. Then, using again Lemma 4.2, kF1k6,4=1 2kp−2 h(2∂uL+ ou +∂uΨ)∂uΨk6,4 ≤1 2kp−2 h(2∂uL+ ou +∂uΨ)k0,0k∂uΨk6,4 ≤K 21 κ+K∗ κ2K∗ (νI0)2. Now we deal wi h F2. We obse e ha , since kL+ ou k4,2=K(νI0)−1and Ψ∈ BK∗(νI0)−2⊂e X5,3, by Lemma 4.2, k2∂θL+ ou +∂θΨk0,0≤2K(νI0)2 κ2k∂θL+ ou k4,2+K(νI0)4 κ4k∂θΨk6,4≤KνI0 κ2+K∗νI0 κ4.(4.17) Then, kF2k6,4≤ν 2k2∂θL+ ou +∂θΨk0,0k∂θΨk6,4≤ν 2K1 + K∗ κ21 κ2(νI0)2. 19 Then, e F(Ψ) ∈ X6,4and, by Lemma (4.3), aking κla ge enough, Gu◦e F(Ψ) ∈ BK∗(νI0)−2⊂e X5,3. Now we check ha e Fis Lipschi z wi h Lip e F ≤ Kκ−1. Le Ψ,Ψ′∈ BK∗(νI0)−2⊂e X5,3. Obse e ha kp−2 h(2∂uL+ ou +∂uΨ + ∂uΨ′)k0,0and k2∂θL+ ou +∂θΨ + ∂θΨ′k0,0a e bounded as in (4.16) and (4.17). Then, ke F(Ψ) −e F(Ψ′)k6,4≤1 2kp−2 h(2∂uL+ ou +∂uΨ + ∂uΨ′)k0,0k∂uΨ−∂uΨ′k6,4 +νk2∂θL+ ou +∂θΨ + ∂θΨ′k0,0k∂θΨ−∂θΨ′k6,4 ≤K1 κ+2K∗ κ2TΨ−Ψ′U5,3+νK νI0 κ2+ 2K∗νI0 κ4(νI0)−1TΨ−Ψ′U5,3 ≤K κTΨ−Ψ′U5,3, o some cons an K2>0. Then, by he las claim o Lemma 4.3, TGu◦e F(Ψ) −Gu◦e F(Ψ′)U5,3≤Kke F(Ψ) −e F(Ψ′)k6,4≤K κTΨ−Ψ′U5,3. The claim ollows aking κla ge enough. Then, we easily check ha Gu◦e Fsends he ball BK∗(νI0)−2 in o i sel and has a unique ixed poin he e. 5 Fi s ex ension o he in a ian mani old In Sec ion 4we ha e ound he ex ension o he local uns able in a ian mani old o he complex domain D+ κ,δ ×Tσ(see Figu e 2) by means o he unc ion Φ+= Φ0+L+ ou + Φ+ 2, whe e Φ0is in oduced in (4.5), L+ ou in (4.14) and Φ+ 2in P oposi ion 4.5. Hence, as commen ed in Rema k 4.1, he pa ame iza ion o he s able mani old is gi en by Φ−(u, θ) = −Φ+(−u, −θ). I is de ined in D− κ,δ ×Tσ, whe e D− κ,δ =−D+ κ,δ. This ex ension o Φ+is no enough o ou pu poses, because D+ κ,δ ∩ D− κ,δ ∩R=∅and we canno compu e he di e ence be ween he mani olds in he eals. In his sec ion we will ex end Φ+ o a la ge domain, D+ κ,δ ∪D+ κ,ex , de ined below in (5.21) (see Figu e 3). Once ex ended, he pa ame iza ions o he uns able and s able mani olds will be de ined in a common domain con aining an in e al o R. We will compu e he di e ence o he mani olds in his common domain in Sec ion 9. We ema k ha we ha e no been able o ind he ex ension o Φ+in a single s ep because he Hamil on-Jacobi equa ion (4.3) is no de ined a u= 0. The eason lies in he ac ha he me hod we ha e used o ind Φ+ equi es compu ing some in eg als along s aigh lines wi h some slope in he complex domain whe e he a iable uli es. These s aigh lines canno go h ough 0and hence he cu en me hod does no allow us o ex end Φ+beyond u= 0 di ec ly. In his sec ion we will ind an ex ension o Φ+ o D+ κ,ex by choosing ano he ype o pa ame iza ion o he in a ian mani old, and hen we will go back o he o iginal ype o pa ame iza ion. Howe e , his ex ension will no be de ined a u= 0. 5.1 F om Hamil on-Jacobi pa ame iza ion o low pa ame iza ion Taking in o accoun (4.1) and (4.2), he solu ion Φ+= Φ0+L+ ou + Φ+ 2o he Hamil on-Jacobi equa ion (4.3) ob ained in Sec ion 4p o ides he pa ame iza ion o he uns able mani old     q p θ J    = Γ+(u, θ) =     qh(u) 1 ph(u)∂uΦ+(u, θ) θ ∂θΦ+(u, θ)    = Γ0(u, θ) + Γ1(u, θ) + Γ2(u, θ),(5.1) whe e Γ0(u, θ) =     qh(u) 1 ph(u)∂uΦ0(u) θ 0    ,Γ1(u, θ) =     0 1 ph(u)∂uL+ ou (u, θ) 0 ∂θL+ ou (u, θ)    (5.2) and Γ2= Γ+−Γ0−Γ1. We ecall ha ph(0) = 0. Howe e , Γ0is analy ic a u= 0 because ph(u)−1∂uΦ0(u) = ph(u). 20 Le X= (Xq, Xp, Xθ, XJ)⊤be he ec o ield co esponding o H, in (2.3), using he 2- o m in (2.1), and le φ be i s low. We look o a change o a iables (u, θ) = ( + 1( , ϕ), ϕ + 2( , ϕ)) such ha conjuga es Xon he uns able in a ian mani old, pa ame ized by e Γ+( , ϕ) = Γ+( + 1( , ϕ), ϕ + 2( , ϕ)) (5.3) o he ec o ield (1, νI0), ha is, φ (e Γ+( , ϕ)) = e Γ+( + , ϕ +νI0 ) o , equi alen ly, L(e Γ+) = X◦e Γ+,(5.4) whe e L(Γ) = ∂ Γ + νI0∂ϕΓis he ope a o de ined in (4.7). An immedia e compu a ion shows ha (5.3) sa is ies (5.4) i and only i = ( 1, 2)is a solu ion o [∂uΓ+◦(Id + )](1 + L( 1)) + [∂θΓ+◦(Id + )](νI0+L( 2)) = X◦Γ+◦(Id + ).(5.5) We emphasize ha he abo e equa ion has ou componen s. Howe e , he symplec ic cha ac e o he ec o ield Xensu es ha i wo o hem a e sa is ied, so a e he o he wo. We choose o sol e he equa ions co esponding o he i s and hi d componen s o (5.5). Taking in o accoun (5.1), he equali y ˙qh=−qhphand he ac ha ∂uΦ0=p2 h, we can w i e hese wo equa ions as L( ) = A◦(Id + ),(5.6) whe e A=A1 A2=p−2 h(∂uL+ ou +∂uΦ+ 2) ν(∂θL+ ou +∂θΦ+ 2).(5.7) We deno e N( ) he igh hand side o (5.6). We ha e ha N( ) = A+DA +R( ),(5.8) whe e DA deno es he de i a i e o Aand R( ) = A◦(Id + )−A−DA . We will see in a momen ha equa ion (5.6) canno be sol ed di ec ly as a ixed poin equa ion because he linea e m DA in (5.8) is oo la ge. We will need o ew i e i in a be e sui ed way. 5.1.1 P elimina ies and echnical lemmas o sol e equa ion (5.6) To sol e equa ion (5.6), we conside = ( 1, 2)∈ X ,s ×X +1,s+1, wi h he no m k k ,s =k 1k ,s +k 2k +1,s+1 and he ope a o Guin (4.13), ac ing on each componen . Also, gi en a ma ix unc ion M=M1,1M1,2 M2,1M2,2 wi h M1,1, M1,2∈ X ,s and M2,1, M2,2∈ X +1,s+1 we de ine kMk ,s = max {kM1,1k ,s +kM2,1k +1,s+1, νI0(kM1,2k ,s +kM2,2k +1,s+1)}.(5.9) I ollows immedia ely om Lemma 4.2 ha , i , s ∈Rand ˜ , ˜s≥0, kMk ,s ≤KνI0 κ˜s kMk +˜ ,s+˜s(5.10) and, o any , ′, s, s′∈R, kM k + ′,s+s′≤ kMk ,sk k ′,s′.(5.11) 21 Indeed, inequali y (5.10) ollows immedia ely om (1) o Lemma 4.2. As o (5.11), using (2) o Lemma 4.2, we ha e ha k(M )1k + ′,s+s′+k(M )2k + ′+1,s+s′+1 =kM1,1 1+M1,2 2k + ′,s+s′+kM2,1 1+M2,2 2k + ′+1,s+s′+1 ≤ kM1,1 1k + ′,s+s′+kM1,2 2k + ′,s+s′+kM2,1 1k + ′+1,s+s′+1 +kM2,2 2k + ′+1,s+s′+1 ≤(kM1,1k ,s +kM2,1k +1,s+1)k 1k ′,s′+KνI0 κ(kM1,2k ,s +kM2,2k +1,s+1)k 2k ′+1,s′+1 ≤ kMk ,sk k ′,s′, i κ≥K. We will look o he solu ion o equa ion (5.6) in a domain sligh ly smalle han D+ κ,δ ×Tσ. The no ms we use depend on he choice o he domains. In pa icula , hey depend on κ,δand σ. Below, we will inc ease κ o ˜κand dec ease δ o ˜ δ. In o de o ha e he dis ance om he poin s o he bounda y o D+ ˜κ,˜ δ o D+ κ,δ cons an , when aking ˜κ > κ we will ake δ−˜ δ=˜κ−κ νI0 cos β1 cos β2, some imes wi hou explici men ion o i . To cla i y he exposi ion, ill he end o he sec ion, we include σand κas subsc ip s in he no ms, bu no δ, since we will unde s and ha when changing he domain he p e ious ule applies. In his way k k ,s,κ,σ will deno e he no m k k ,s o de ined ei he in D+ κ,δ o in D+ κ,δ ×Tσ, depending on he se ing. We ema k ha i ˜κ > κ (wi h ˜κ(νI0)−1<˜ δ) and 0<˜σ < σ, hen D+ ˜κ,˜ δ×T˜σ⊂ D+ κ,δ ×Tσ. Analogously, we will deno e he spaces by X ,s,κ,σ o cla i y hei dependence on he domain. The educ ion o domain will only be done a ini e numbe o imes. Lemma 5.1. Fo all ˜κ > κ and 0<˜σ < σ, he e exis s C > 0such ha i B∈ X ,s,κ,σ, hen, o all m, n ∈N,∂m u∂n θB∈ X ,s,˜κ,˜σwi h k∂m uBk ,s,˜κ,˜σ≤C˜κ κs(νI0)mm! (˜κ−κ)m 1 (cos β1)mkBk ,s,κ,σ, m ≥0, k∂m u∂n θBk ,s,˜κ,˜σ≤C˜κ κs(νI0)mm!n! (˜κ−κ)m(σ−˜σ)n+1 1 (cos β1)mkBk ,s,κ,σ, m ≥0, n ≥1. When applying he p e ious lemma, we will choose ˜κ=κ+κ0, whe e κ0>0is ixed. In his way, ˜κ/κ = 1 + κ0/κ < 2, i κis la ge enough. P oo o Lemma 5.1.We no e ha (∂m u∂n θB)[k]= (ik)n∂m uB[k]. By Cauchy o mula, ∂m uB[k](u) = m! 2πi Zγu B[k](z) (z−u)m+1 dz, aking γu o be he ci cle o adius (˜κ−κ)(νI0)−1cos β1, we immedia ely ha e, o some C1>0, k∂m uB[k]k ,s,˜κ,˜σ≤C1˜κ κsm!(νI0)m (˜κ−κ)m 1 (cos β1)mkB[k]k ,s,κ,σ. Hence, since kB[k]k ,s,κ,σ ≤ kBk ,s,κ,σe−|k|σ, we ha e ha k∂m u∂n θB[k]k ,s,˜κ,˜σ≤C1˜κ κsm!|k|n(νI0)m (˜κ−κ)m 1 (cos β1)me−|k|σkBk ,s,κ,σ. The e o e, k∂m u∂n θBk ,s,˜κ,˜σ=X k∈Zk∂m u∂n θB[k]k ,s,˜κ,˜σe|k|˜σ ≤C1 (cos β1)m˜κ κsm!(νI0)m (˜κ−κ)mkBk ,s,κ,σ X k∈Z|k|ne−|k|(σ−˜σ) ≤C1C2 (cos β1)m˜κ κsm!n!(νI0)m (˜κ−κ)m(σ−˜σ)n+1 kBk ,s,κ,σ, whe e we ha e used ha Pk≥0kne−bk ≤C2n!/bn+1, o b > 0and n≥1, o some C2. 22 Lemma 5.2. Le Cbe he cons an gi en by Lemma 5.1 and ′≥0and s′>0. Gi en K > 0and δ > 0, o any 0<˜σ < σ, he e exis κ0>0such ha o any ˜κ > κ +δ > κ ≥κ0, i B∈ X ,s,κ,σ and ∈ X ′,s′,˜κ,˜σ×X ′+1,s′+1,˜κ,˜σwi h k k ′,s′,˜κ,˜σ≤K/(νI0)s′+1, hen B◦(Id + )∈ X ,s,˜κ,˜σand kB◦(Id + )k ,s,˜κ,˜σ≤2C σ−˜σ˜κ κs kBk ,s,κ,σ. P oo . Fi s o all, by (1) o Lemma 4.2, we ha e ha k 1k0,0,˜κ,˜σ≤K(νI0)s′ ˜κs′k 1k ′,s′,˜κ,˜σ≤K ˜κs′νI0 , k 2k0,0,˜κ,˜σ≤K(νI0)s′+1 ˜κs′+1 k 2k ′+1,s′+1,˜κ,˜σ≤K ˜κs′+1 . Then, expanding Bin Taylo se ies and using Lemma 5.1, we ha e ha , w i ing α1= cos β1, kB◦(Id + )k ,s,˜κ,˜σ≤X j≥0 1 j! j X ℓ=0 j ℓk∂j−ℓ u∂ℓ θB j−ℓ 1 ℓ 2k ,s,˜κ,˜σ ≤X j≥0 j X ℓ=0 1 (j−ℓ)!ℓ!k∂j−ℓ u∂ℓ θBk ,s,˜κ,˜σk 1kj−ℓ 0,0,˜κ,˜σk 2kℓ 0,0,˜κ,˜σ ≤C σ−˜σ˜κ κsX j≥0 j X ℓ=0 K ˜κs′νI0j−ℓK ˜κs′+1 ℓ(νI0)j−ℓ (α1(˜κ−κ))j−ℓ(σ−˜σ)ℓkBk ,s,κ,σ ≤C σ−˜σ˜κ κsX j≥0 Kj j X ℓ=0 1 (˜κs′(α1(˜κ−κ))j−ℓ 1 (˜κs′+1(σ−˜σ))ℓkBk ,s,κ,σ ≤C σ−˜σ˜κ κsX j≥0K1 ˜κs′α1(˜κ−κ)+1 ˜κs′+1(σ−˜σ)j kBk ,s,κ,σ ≤2C σ−˜σ˜κ κs kBk ,s,κ,σ, whe e we ha e chosen κ0such ha K ˜κs′1 α1(˜κ−κ)+1 ˜κ(σ−˜σ)≤1 2. 5.1.2 Solu ion o equa ion (5.6) Le Gube he ope a o de ined in (4.13). We o mally de ine 0=Gu(A), F0=Gu(DA).(5.12) Lemma 5.3. Le Abe he unc ion de ined in (5.7). Le ˜κ > κ and 0<˜σ < σ, wi h κbig. Then, he e exis s K > 0such ha (1) kAk3,1,κ,σ ≤K(νI0)−1, (2) k 0k3,1,κ,σ ≤K(νI0)−2, (3) kDAk3,1,˜κ,˜σ≤K, (4) kF0k3,1,˜κ,˜σ≤K(νI0)−1. P oo . We w i e A=A0+A1, whe e A0=p−2 h∂uL+ ou ν∂θL+ ou , A1=p−2 h∂uΦ+ 2 ν∂θΦ+ 2. 23 Taking in o accoun ha p−2 h∈ X−2,−2,κ,σ and he p ope ies o L+ ou in P oposi ion 4.4, we ha e ha kA0k3,1,κ,σ ≤K(νI0)−1and hA0i= 0. Also, by he p ope ies o Φ+ 2in P oposi ion 4.5, we ha e ha kA1k4,2,κ,σ ≤K(νI0)−2. Hence, kA1k3,1,κ,σ ≤Kκ−1(νI0)−1. This p o es (1). We bound kGu(A0)k3,1,κ,σ and kGu(A1)k3,1,κ,σ sepa a ely. Since hA0i= 0, by (1) o Lemma 4.3, kGu(A0)k3,1,κ,σ ≤K νI0kA0k3,1,κ,σ ≤K (νI0)2. Using again ha p−2 h∈ X−2,−2, he p ope ies o Φ+ 2in P oposi ion 4.5 imply ha kA1k4,2,κ,σ ≤ K(νI0)−2. Then, by (2) o Lemma 4.3, kGu(A1)k3,1,κ,σ ≤KkA1k4,2,κ,σ ≤K (νI0)2, om which (2) ollows. We w i e DA =DA0+DA1. F om he p e ious bounds o A0and A1, applying Lemma 5.1 and he de ini ion o he ma ix no m (5.9), we ha e ha , in he educed domain, kDA0k3,1,˜κ,˜σ≤K, kDA1k4,2,˜κ,˜σ≤K κνI0 . He e he cons an s depend on ˜κ−κand σ−˜σ. This p o es (3). We inally p o e (4). Since hA0i= 0, we ha e ha hDA0i= 0. By (1) o Lemma 4.3, kGu(DA0)k3,1,˜κ,˜σ≤K νI0 . Mo eo e , by (2) o Lemma 4.3, kGu(DA1)k3,1,˜κ,˜σ≤KkDA1k4,2,˜κ,˜σ≤K νI0 . Since F0=Gu(DA0) + Gu(DA1), he claim ollows. We in oduce ˜ by = 0+ (Id + F0)˜ . Obse e ha , by (4) o Lemma 5.3 and (1) o Lemma 4.2, kF0k0,0,˜κ,˜σ≤KνI0 κkF0k3,1,˜κ,˜σ≤K κ. Hence, i κis la ge enough, Id+F0is in e ible and k(Id+F0)−1k0,0,κ,σ ≤2. Using ha , by (5.12), L( 0) = Aand L(F0) = DA, we ew i e equa ion (5.6) as L(˜ ) = e N(˜ ),(5.13) whe e e N(˜ ) = (Id + F0)−1DA 0+ (Id + F0)−1DA F0˜ + (Id + F0)−1R 0+ (Id + F0)˜  and Rwas in oduced in (5.8). Using he ope a o Gu, we ew i e equa ion (5.13) as he ixed poin equa ion ˜ =Gu◦e N(˜ ).(5.14) P oposi ion 5.4. Fo all ˆκ > ˜κ > κ and 0<ˆσ < ˜σ < σ, wi h κbig enough, equa ion (5.14)has a solu ion ˜ +, de ined in D+ ˆκ,ˆ δ×Tˆσ, and ˜ +∈ X3,1,ˆκ,ˆσ× X4,2,ˆκ,ˆσwi h k˜ +k3,1,ˆκ,ˆσ≤C(νI0)−2. Consequen ly, e Γ+= Γ+◦(Id+ 0+(I+F0)˜ +)sa is ies he in a iance equa ion (5.4)in D+ ˆκ,δ ×Tˆσ. 24 P oo o P oposi ion 5.4.We claim ha he e exis s K1>0such ha kGu◦e N(0)k3,1,ˆκ,ˆσ≤K1 (νI0)2. Indeed, i s we no ice ha e N(0) = (Id+F0)−1(DA 0+R( 0)) = (Id+F0)−1DA 0+Z1 0 (1 −s)D2A◦(Id + s 0)ds ( 0)⊗2 = (Id + F0)−1DA 0+Z1 0 (1 −s) 0 1∂uDA ◦(Id + s 0) 0+ 0 2∂θDA ◦(Id + s 0) 0ds. Nex , we no e ha kDA 0k4,2,ˆκ,ˆσ≤ kDA 0k6,2,ˆκ,ˆσ≤ kDAk3,1,ˆκ,ˆσk 0k3,1,ˆκ,ˆσ≤K (νI0)2. By (3) o Lemma 5.3 and Lemma 5.1, o ˆκ > ˜κand 0<ˆσ < ˜σ, we ha e k∂uDAk3,1,ˆκ,ˆσ≤KνI0kDAk3,1,˜κ,˜σ≤KνI0, k∂θDAk3,1,ˆκ,ˆσ≤KkDAk3,1,˜κ,˜σ≤K. (5.15) Mo eo e , by (1) o Lemma 4.2 and (2) o Lemma 5.3, k 0 1∂uDA ◦(Id + s 0) 0k4,2,ˆκ,ˆσ≤KνI0 κk 0 1∂uDA ◦(Id + s 0) 0k9,3,ˆκ,ˆσ≤K κ(νI0)2 and, since 0 2∈ X4,2,ˆκ,ˆσ, k 0 2∂θDA ◦(Id + s 0) 0k4,2,ˆκ,ˆσ≤KνI0 κ2 k 0 2∂ DA ◦(Id + s 0) 0k10,4,ˆκ,ˆσ≤K κ2(νI0)2. Hence, kGu◦e N(0)k3,1,ˆκ,ˆσ≤Kke N(0)k4,2,ˆκ,ˆσ≤K1 κ(νI0)2. Le C= 2K1. Le , ′∈ X3,1,ˆκ,ˆσ×X4,2,ˆκ,ˆσwi h k k3,1,ˆκ,ˆσ,k ′k3,1,ˆκ,ˆσ≤e C. We ha e ha e N( )−e N( ′) = (I+F0)−1(M1+M2) ( − ′),(5.16) whe e M1=DA F0, M2( , ′) = Z1 0Z1 0 D2A◦(Id + us( , ′)) d us( , ′)ds (I+F0) wi h us( , ′) = 0+ (Id + F0) ′+s(I+F0)( − ′). Then, using (1) o Lemma 4.2 and (3) and (4) o Lemma 5.3, we ha e ha kM1k1,1,ˆκ,ˆσ≤KνI0 κkDAF 0k6,2,ˆκ,ˆσ≤KνI0 κkDAk3,1,ˆκ,ˆσkF0k3,1,ˆκ,ˆσ≤K κ.(5.17) Obse e ha , by (2) o Lemma 5.3 and he hypo heses on , ′, kus( , ′)k3,1,ˆκ,ˆσ≤K(νI0)−2. Hence, also using (5.15) kus,1∂uDA ◦(Id + us)k1,1,ˆκ,ˆσ≤KνI0 κkus,1∂uDA ◦(Id + us)k6,2,ˆκ,ˆσ ≤KνI0 κkus,1k3,1,ˆκ,ˆσk∂uDA ◦(Id + us)k3,1,ˆκ,ˆσ≤K κ 25 whe e Lin(T2) = ∂ T2+∂θT2, Fin(T2) = −ν 2(∂θL+ in)2+ 2 2(∂ L+ in)2+ν∂θL+ in∂θT2+ν 2(∂θT2)2+ 4 2∂ L+ in∂ T2+ 2 2(∂ T2)2, whe e we ha e used ha Lin(L+ in) = −E(T0). 6.1 Spaces and echnical lemmas To deal wi h equa ion (6.9) we ha e o in oduce some unc ion spaces and p o ide basic p ope ies o he ope a o Lin. We conside he domain D+ κ,in ={ ∈C| ℑ < −( an β1)ℜ −κ}∪{ ∈C| ℑ > ( an β1)ℜ +κ},(6.10) wi h κ > 1. See Figu e 5. We no ice ha he image o D+ κ,δ ∪ D+ κ,ex by he ans o ma ion u=i+ (νI0)−1 is con ained in D+ κ,in. Figu e 5: The domain D+ κ,in de ined in (6.10), shaded in g ay. To sol e equa ion (6.9), o ∈R, we in oduce he Banach space o 2π-pe iodic in θ, analy ic unc ions Y ={R:D+ κ,in ×Tσ→C| kRk <∞},(6.11) whe e, aking in o accoun he Fou ie expansion o R( , θ) = Pk∈ZR[k]( )eikθ , kRk =X k∈ZkR[k]k e|k|σ,(6.12) and, o an analy ic unc ion :D+ κ,in →C, k k = sup ∈D+ κ,in | ( )|. Lemma 6.1. Le , 1, 2∈R. (1) I R∈ Y +s,s≥0, hen R∈ Y and he e exis s K > 0such ha kRk ≤K1 κskRk +s. 32 (2) I R1∈ Y 1and R2∈ Y 2, hen R1R2∈ Y 1+ 2and kR1R2k 1+ 2≤ kR1k 1kR2k 2. We also in oduce he Banach space e Y ={R∈ Y |∂ R, ∂θR∈ X +1,TRU <∞},(6.13) wi h he no m TRU =kRk +k∂ Rk +1 +k∂θRk +1.(6.14) We will use he nex echnical lemma se e al imes. I is analogous o Lemma 4.3. Lemma 6.2. Le Gu in be he ope a o de ined in (6.7). The e exis s K > 0such ha (1) I R∈ Y wi h > 0and hRi= 0, hen Gu in(R)∈ Y and kGu in(R)k ≤KkRk . (2) I R∈ Y wi h > 1, hen Gu in(R)∈ Y −1and kGu in(R)k −1≤KkRk . (3) I R∈ Y wi h > 0, hen ∂ Gu in(R), ∂θGu in(R)∈ Y and k∂ Gu in(R)k ≤KkRk ,k∂θGu in(R)k ≤KkRk . As a consequence, i R∈ Y , wi h > 1,Gu in(R)∈e Y −1and TGu in(R)U −1≤KkRk . The cons an Konly depends on and he cons an s in ol ed in he de ini ion o D+ κ,in. 6.2 Fixed poin equa ion We conside he equa ion T2=Gu in ◦Fin(T2).(6.15) We decompose Fin(T2) = −(F0+F1+F2+F3+F4),(6.16) wi h F0( , θ) = ν 2(∂θL+ in( , θ))2+ 2 2(∂ L+ in( , θ))2, F1( , θ) = ν∂θL+ in( , θ)∂θT2( , θ), F2( , θ) = ν 2(∂θT2( , θ))2, F3( , θ) = 4 2∂ L+ in( , θ)∂ T2( , θ), F4( , θ) = 2 2(∂ T2( , θ))2. I T2is a solu ion o he ixed poin equa ion (6.15) i is also a solu ion o (6.9). In iew o (6.8), since V[0] = 0, we ha e ha (L+ in)[0] = 0. The ollowing p oposi ion is analogous o P oposi ion 4.4. I s p oo ollows exac ly he same lines. P oposi ion 6.3. We ha e L+ in, ∂θL+ in ∈ Y2and ∂ L+ in ∈ Y3. Mo eo e , he e exis s K > 0, independen o κ, such ha kL+ ink2,k∂θL+ ink2,k∂ L+ ink3≤K. Mo i a ed by P oposi ion 6.3, we in oduce he cons an ΘV=kL+ ink2 2+k∂θL+ ink2 2+k∂ L+ ink2 31/2.(6.17) Rema k 6.4. I one eplaces iby ε˜ iin he de ini ion o Vin (1.2), hen ΘV=O(ε). 33 P oposi ion 6.5. The e exis K1>0and κ0such ha , i κ > κ0, he ope a o Gu in ◦Fin :BK1Θ2 V⊂ e Y3→ BK1Θ2 Vis well de ined and a con ac ion. Hence, equa ion (6.15)has a solu ion T+ 2∈e Y3 sa is ying TT+ 2U3≤K1Θ2 V.(6.18) P oo . We s a by bounding F0. By P oposi ion 6.3, kF0k4≤ν 2(∂θL+ in)24+ 2k 2(∂ L+ in)2k4≤ν 2k∂θL+ ink2 2+ 2k 2k−2k∂ L+ ink2 3< K2Θ2 V,(6.19) o some K2>0. We ake K1= 2kGu inkK2and K∗=K1Θ2 V. In his p oo kGu inks ands o he ope a o no m o Gu in :Y4→e Y3. Le T2∈ BK∗⊂e Y3. We claim ha Fin(T2)∈ Y4and kFin(T2)k4≤ kF0k4+Kνk∂θL+ ink2 κ2+KνK∗ 2κ4+K4k∂ L+ ink3 κ+K2K∗ κ2K∗.(6.20) Indeed, nex we deal wi h Fj,j= 1,...,4. By P oposi ion 6.3 and Lemma 6.1 kF1k4=kν∂θL+ in∂θT2k4=νk∂θL+ ink0k∂θT2k4≤Kνk∂θL+ ink2 κ2TT2U3≤Kνk∂θL+ ink2K∗ κ2, since k∂θT2k4≤TT2U3≤K∗. Again by Lemma 6.1, kF2k4=ν 2(∂θT2( , θ))24≤ν 2k∂θT2k0k∂θT2k4≤Kν 2κ4k∂θT2k4k∂θT2k4≤Kν(K∗)2 2κ4. Now, by P oposi ion 6.3, Lemma 6.1 and he ac ha k∂ T2k4≤TT2U3≤K∗, kF3k4=4 2∂ L+ in∂ T24≤4 2k−2k∂ L+ ink2k∂ T24≤K4k∂ L+ ink3 κK∗. Finally, kF4k4=2 2(∂ T2( , θ))24≤2 2∂ T2( , θ)0k∂ T2( , θ)k4 ≤ k2 2k−2k∂ T2( , θ)k2k∂ T2( , θ)k4≤K2 κ2k∂ T2( , θ)k4k∂ T2( , θ)k4≤K2(K∗)2 κ2. This p o es (6.20). Now we choose κ0>1such ha Kνk∂θL+ ink2 κ2 0 +νK∗ 2κ4 0 +4k∂ L+ ink3 κ0 +2K∗ κ2 0K∗< K2Θ2 V. Hence, o any κ > κ0, i T2∈ BK∗, by he las claim o Lemma 6.2, TGu in ◦Fin(T2)U3≤ kGu inkkFin(T2)k4≤2kGu inkK2Θ2 V=K∗. Nex we check ha Gu in ◦Fin is a con ac ion on BK∗. We claim ha he e exis s K3>0such ha , i T2, T ′ 2∈ BK∗, kFin(T2)−Fin(T′ 2)k4≤K3 κTT2−T′ 2U3.(6.21) To p o e his claim, we w i e Fin(T2)−Fin(T′ 2) = − 4 X i=1 (Fi(T2)−Fi(T′ 2)) , whe e F1(R) = ν∂θL+ in∂θR, F2(R) = 1 2ν(∂θR)2, F3(R)( , θ) = 4 2∂ L+ in( , θ)∂ R( , θ), F4(R)( , θ) = 2 2(∂ R( , θ))2. 34 We bound each di e ence sepa a ely. Fi s , by P oposi ion 6.3 and Lemma 6.1, we ha e ha kF1(T2)−F1(T′ 2)k4≤ kν∂θL+ ink0k∂θT2−∂θT′ 2k4≤Kνk∂θL+ ink2 κ2TT2−T′ 2U3. Second, kF2(T2)−F2(T′ 2)k4=ν 2k(∂θT2+∂θT′ 2)(∂θT2−∂θT′ 2)k4 ≤ν 2k(∂θT2+∂θT′ 2)k0k∂θT2−∂θT′ 2k4 ≤Kν 2 1 κ4k∂θT2+∂θT′ 2k4k∂θT2−∂θT′ 2k4 ≤KK∗ν κ4TT2−T′ 2U3. Thi d, using again P oposi ion 6.3 and Lemma 6.1, kF3(T2)−F3(T′ 2)k4≤4 2∂ L+ in0k∂ T2−∂ T′ 2k4≤4∂ L+ in3 κTT2−T′ 2U3. And ou h, kF4(T2)−F4(T′ 2)k4≤ k2 2(∂ T2+∂ T′ 2)k0k∂ T2−∂ T′ 2k4 ≤ k2 2k−2k∂ T2+∂ T′ 2k2k∂ T2−∂ T′ 2k4 ≤K2 κ2k∂ T2+∂ T′ 2k4k∂ T2−∂ T′ 2k4 ≤4KK∗ κ2TT2−T′ 2U3. Inequali y (6.21) ollows om combining hese ou s eps, since κ > κ0>1and κ0is big. Finally, we check ha he ope a o Gu in ◦Fin is a con ac ion in BK∗. Indeed, i T2, T ′ 2∈ BK∗, using he las s a emen o Lemma 6.2 and (6.21), TGu in ◦Fin(T2)−Gu in ◦Fin(T′ 2)U3≤ kGu inkkFin(T2)−Fin(T′ 2)k4≤ kGu inkK3 κTT2−T′ 2U3. Hence, by he s anda d a gumen , aking κ0la ge enough, Gu in ◦Fin sends BK∗in o i sel and has a unique ixed poin T+ 2in his ball. The bound (6.18) ollows om he de ini ion o K∗. Co olla y 6.6. The inne equa ion (6.2)admi s a solu ion T+=T0+L+ in +T+ 2, whe e T0is gi en by (6.3),L+ in is de ined in (6.8)and T+ 2is gi en by P oposi ion 6.5. 7 App oxima ion o he mani old in he inne domain Le T+=T0+L+ in +T+ 2be he solu ion o he inne equa ion (6.2) in he domain D+ κ,in gi en by Co olla y 6.6. F om P oposi ion 4.5, we ha e ha he solu ion Φ+(u, θ)o he Hamil on-Jacobi equa ion (4.3) and i s ex ension o he domain D+ κ,δ ∪D+ κ,ex B∗ ρ(whe e B∗ ρis a ixed small ball cen e ed a he o igin), wi h anishing con ou condi ions a ℜu=−∞, is gi en by Φ+= Φ0+L+ ou + Φ+ 2, whe e Φ0was in oduced in (4.5), L+ ou in (4.14) and Φ+ 2∈e X5,3, hence Φ+ 2(u, θ)∼(νI0)−2 (u−i)3. The app oxima ion o Φ+by Φ0+L+ ou is no good enough i u−i∼(νI0)−1, al hough P opo- si ion 5.8 ensu es ha he mani olds can be ex ended o D+ κ,δ ∪ D+ κ,ex B∗ ρ. In o de o ob- ain a be e app oxima ion o Φ+, we compa e i wi h he solu ion o he inne equa ion (6.2), T+=T0+L+ in +T+ 2, gi en by Co olla y 6.6. 35 We ecall ha he unc ion S+( , θ) = (νI0)−1Φ+(i+ (νI0)−1 , θ) is a solu ion o he Hamil on-Jacobi equa ion (6.1) in he ans o med domain. Ac ually, we wan o compa e S+wi h he solu ion T+o he inne equa ion. Bo h unc ions a e al eady de e mined. We w i e S+=T0+L+ in +T+ 2+S+ 2 and de i e bounds and p ope ies o S+ 2 om he equa ion i sa is ies. Using ha T+=T0+L+ in +T+ 2is a solu ion o he inne equa ion (6.2), we ha e ha S+ 2is a solu ion o Lin(S2) = b Fin,a(S2),(7.1) whe e Lin(S2) = ∂ S2+∂θS2was de ined in (6.6) and b Fin,a(S2) = −e E+ν∂θT+∂θS2+1 2ν(∂θS2)2+ (2A1∂ T+−1)∂ S2+A1(∂ S2)2(7.2) wi h A1( ) = 1 2 2(2i+ (νI0)−1 )2 (i+ (νI0)−1 )2, A2( , θ) = −1 2 (i+ (νI0)−1 )2 2(2i+ (νI0)−1 )2+1 2 1 2(2i+ (νI0)−1 )2V(θ), e E( , θ) = (A1( )−2 2)(∂ T+ 0)2+A2( , θ) + 1 8 2+1 8 2V(θ). (7.3) Le β1>0be he angle in ol ed in he de ini ion o D+ κ,δ (see (4.12)). Fix β3and β4sa is ying 0< β3< β4< β1. Fo α∈(0,1) and κ > 1, we de ine he domain D+ κ,α ={ ∈C| −(νI0)α− an β4ℜ < ℑ < min{−κ− an β1ℜu, −κ− an β3ℜu}}.(7.4) See Figu e 6. Obse e ha d(D+ κ,α,0) = κcos β3and | | ≤ K(νI0)α o ∈ D+ κ,α. Figu e 6: The domain D+ κ,α de ined in (7.4), shaded in g ay. 36 In o de o sol e equa ion (7.1), we conside he ollowing igh in e ses, b Guand b Gu s, o Lin. Fi s , b Guis de e mined by he exp ession o he Fou ie coe icien s o he image b Gu(S2): b Gu(S2)[k]( ) = Z + 0 eik( − )S[k] 2( )d i k≥0, b Gu(S2)[k]( ) = Z − 0 eik( − )S[k] 2( )d i k < 0, (7.5) whe e − 0=(νI0)α−κ an β1− an β4 +− an β4 an β1− an β4 + 1(νI0)α+ an β4 an β1− an β4 κi, + 0=−(νI0)α+κ an β4− an β3 + an β4 an β4− an β3−1(νI0)α− an β4 an β4− an β3 κi, wi h α∈(0,1) and 0< β3< β4< β1< π. See Figu e 6. The mos gene al igh in e se o Lin has he o m b Gu s(S2) = S2,0+b Gu(S2),(7.6) wi h S2,0∈ke Lin. Wi h his ope a o we conside he ixed poin equa ion S2=b Gu s◦b Fin,a(S2)(7.7) and we choose S2,0so ha he solu ion S+ 2o (7.7) is he analy ical con inua ion o S+−T0− L+ in −T+ 2 o he domain D+ κ,α. No e again ha i S+ 2is a ixed poin o b Gu s◦b Fin,a, i is also a solu ion o (7.1). Since S2,0∈ke Lin, S2,0( , θ) = X k∈Z S[k] 2,0( )eikθ =X k∈Z g[k]eik(θ− ), which de ines he coe icien s g[k]. I , mo eo e , S2is a solu ion o (7.7), we ha e S2( , θ) = S2,0( , θ) + b Gu◦b Fin,a(S2)( , θ). Fo k≥0, he Fou ie coe icien s sa is y S[k] 2( ) = S[k] 2,0( ) + Z + 0 eik( − )(b Fin,a(S2))[k]( )d and, aking in o accoun ha , a he poin s ( + 0, θ)o he bounda y, S2( , θ) = S+( , θ)−T0( , θ)−L+ in( , θ)−T+ 2( , θ), e alua ing a = + 0, we ge g[k]=S[k] 2,0( + 0)eik + 0= (S+−T0−L+ in −T+ 2)[k]( + 0)eik + 0 and inally, S[k] 2,0( ) = g[k]e−ik = (S+−T0−L+ in −T+ 2)[k]( + 0)e−ik( − + 0).(7.8) Analogously, we ob ain ha o k < 0 S[k] 2,0( ) = g[k]e−ik = (S+−T0−L+ in −T+ 2)[k]( − 0)e−ik( − − 0).(7.9) The e o e, we choose S2,0( , θ) = X k∈Z S[k] 2,0( )eikθ (7.10) wi h he coe icien s (7.8) and (7.9). 37 7.1 Spaces and echnical lemmas To sol e equa ion (7.7), o ∈R, we in oduce he Banach space o analy ic, 2π-pe iodic in θ unc ions Z ={R:D+ κ,α ×Tσ→C| kRk <∞}, wi h he no m k·k de ined as ollows. Using ha R( , θ) = Pk∈ZR[k]( )eikθ, kRk =X k∈ZkR[k]k e|k|σ, whe e, o an analy ic unc ion :D+ κ,α →C, k k = sup ∈D+ κ,α | ( )|. The ollowing lemma is e y simila o Lemma 6.1. Lemma 7.1. Le , 1, 2∈R. The e exis s K > 0such ha he ollowing holds. (1) I R∈ Z +swi h s≥0, hen R∈ Z and kRk ≤K1 κskRk +s. (2) I R1∈ Z 1and R2∈ Z 2, hen R1R2∈ Z 1+ 2and kR1R2k 1+ 2≤ kR1k 1kR2k 2. We also in oduce e Z ={R∈ Z |∂ R, ∂θR∈ Z +1,TRU <∞}, wi h he no m TRU =kRk +k∂ Rk +1 +k∂θRk +1. The ollowing esul is comple ely analogous o Lemma 6.2. The only di e ence is he domain o he unc ions o he space. Lemma 7.2. Le b Gube he ope a o de ined in (7.5). The e exis s K > 0such ha (1) I R∈ Z , wi h > 0, and R[0] = 0, kb Gu(R)k ≤KkRk . (2) I R∈ Z wi h > 1,b Gu(R)∈ Z −1and kb Gu(R)k −1≤KkRk . (3) I R∈ Z wi h > 0, hen ∂ b Gu(R), ∂θb Gu(R)∈ Z and k∂ b Gu(R)k ≤KkRk ,k∂θb Gu(R)k ≤KkRk , As a consequence, i R∈ Z , wi h > 1,b Gu(R)∈e Z −1and Tb Gu(R)U −1≤KkRk . The cons an Konly depends on and he cons an s in ol ed in he de ini ion o D+ κ,α. 38 7.2 The ixed poin equa ion Be o e sol ing he ixed poin equa ion (7.7) we deal wi h some p elimina y es ima es. Lemma 7.3. Le A1and e Ebe he unc ions in oduced in (7.3). The e exis s K > 0such ha , in D+ κ,α, we ha e (1) e E ∈ Z2wi h ke Ek2≤K(νI0)α−1. (2) A1∈ Z−2wi h kA1k−2≤K. (3) 2A1∂ T+−1∈ Z0wi h k2A1∂ T+−1k0≤K(νI0)α−1+Kκ−1. (4) ∂θT+∈ Z2wi h k∂θT+k2≤K. P oo . We ecall ha in he domain D+ κ,α,K1κ < | |< K2(νI0)α, o some K1, K2>0. We s a by p o ing (1). We ha e ha A1( )−2 2=1 2 2(2i+ (νI0)−1 )2 (i+ (νI0)−1 )2−2 2= 2 2(νI0)−1 (i+3 4(νI0)−1 ) (1 −i(νI0)−1 )2,(7.11) which implies ha kA1( )−2 2k−2≤K(νI0)α−1. Recall ha 0< α < 1. We claim ha he e exis s a cons an Ksuch ha k∂ T+k2≤K. Indeed, since T+=T0+L+ in+ T+ 2, by P oposi ions 6.3 and 6.5,k∂ L+ ink3≤Kand TT+ 2U3≤K. Hence, by (1) o Lemma 7.1, k∂ L+ ink2≤Kκ−1and k∂ T+ 2k2≤Kκ−2. Since ∂ T0= 1/(4 2), he claim ollows immedia ely om he p e ious bounds. Then, by (2) o Lemma 7.1, (A1( )−2 2)(∂ T+)22≤ kA1( )−2 2k−2k∂ T+k2 2≤K3(νI0)α−1. By he de ini ion o A2in (7.3), we ha e ha A2( , θ) + 1 8 2+1 8 2V(θ) = −1 2 (i+ (νI0)−1 )2 2(2i+ (νI0)−1 )2+1 8 2+1 2 1 2(2i+ (νI0)−1 )2+1 8 2V(θ) =1 2O((νI0)−1 ) which implies A2( , θ) + 1 8 2+1 8 2V(θ)2≤K(νI0)α−1. Hence, (1) ollows. (2) is an immedia e consequence o (7.11). Now we deal wi h (3). We ha e ha 2A1∂ T+−1 = 2A1∂ T0−1+2A1∂ (L+ in +T+ 2) = −1 4 (νI0)−1 (4i+ 3(νI0)−1 ) (i+ (νI0)−1 )2+2A1∂ (L+ in +T+ 2) which implies ha k2A1∂ T+−1k0≤ k2A1∂ T0−1k0+k2A1∂ (L+ in +T+ 2)k0 ≤K(νI0)α−1+k2A1k−2k∂ (L+ in +T+ 2)k2 ≤K(νI0)α−1+K κk∂ (L+ in +T+ 2)k3. (4) Follows di ec ly om P oposi ions 6.3 and 6.5. P oposi ion 7.4. The unc ion S2,0de ined in (7.10)sa is ies S2,0∈ Z0and kS[k] 2,0k1≤Kmax{(νI0)−2α,(νI0)−1+αlog(νI0)}e−|k|σ, k 6= 0. Consequen ly, in D+ κ,α ×Tσ′wi h 0< σ′< σ, kS2,0k1≤Kmax{(νI0)−2α,(νI0)−1+αlog(νI0)}. 39 P oo . We ha e ha (S+−T0−L+ in −T+ 2)( , θ) = (νI0)−1Φ0(i+ (νI0)−1 )−T0( ) + (νI0)−1L+ ou (i+ (νI0)−1 , θ)−L+ in( , θ) + (νI0)−1Φ+ 2(i+ (νI0)−1 , θ)−T+ 2( , θ), whe e Φ0was in oduced in (4.5), T0in (6.3), L+ ou in (4.14), L+ in in (6.8), Φ+ 2in P oposi ion 4.5 and T+ 2in P oposi ion 6.5. Fi s , an explici compu a ion shows ha , in he domain unde conside a ion, (νI0)−1Φ0(i+ (νI0)−1 )−T0( ) = i 4(νI0)−1log((νI0)−1 ) + (νI0)−1O((νI0)−1 ). Nex , also an explici compu a ion shows ha (νI0)−1(L+ ou )[k](i+ (νI0)−1 )−(L+ in)[k]( ) = −V[k]Z −∞ 4(νI0)−1i+ (νI0)−2s 8s(2 −(νI0)−1is)2eik(s− )ds =O(νI0)−1 (1 + O((νI0)−1 ))V[k]. Mo eo e , since Φ+ 2∈e X5,3wi h kΦ+ 2k5,3≤K(νI0)−2, |(νI0)−1(Φ+ 2)[k](i+ (νI0)−1 )| ≤ K| |−3e−|k|σ. Finally, since, by P oposi ion 6.5,T+ 2∈e Y3wi h kT+ 2k3≤K, |(T+ 2)[k]( )| ≤ K| |−3e−|k|σ. Since | ± 0|=O((νI0)α), he bounds o he Fou ie coe icien s ollow. To ge he bound in he k·k1 no m we ha e o es ic he domain o D+ κ,α ×Tσ′wi h 0< σ′< σ < σ0, whe e σ0was in oduced in (2.9). We conside he ixed poin equa ion S2=b Gu s◦b Fin,a(S2),(7.12) whe e b Gu sand b Fin,a we e de ined in (7.6) and (7.2), espec i ely. P oposi ion 7.5. Le α∈(0,1). Then, equa ion (7.12)has a solu ion S+ 2∈e Z1in D+ κ,α ×Tσ′ wi h kS+ 2k1≤K(νI0)−2α+K(νI0)−1+αlog(νI0). P oo . We i s check ha he e exis s K∗>0such ha BK∗⊂e Z1sa is ies b Gu s◦b Fin,a(BK∗)⊂ BK∗. To do so, gi en S2∈ BK∗, we w i e b Fin,a(S2) = F0+F1+F2+F3+F4, wi h F0=e E, F1=ν∂θT+∂θS2, F2=1 2ν(∂θS2)2, F3= (2A1∂ T+−1)∂ S2, F4=A1(∂ S2)2, whe e e E,A1and A2we e in oduced in (7.3). We claim ha Fi∈ Z2,i= 0,...,4, and he e exis s K > 0such ha kF0k2≤K(νI0)α−1,(7.13) kF1k2≤Kκ−2K∗,(7.14) kF2k2≤Kκ−2(K∗)2,(7.15) kF3k2≤K(νI0)α−1+κ−1K∗,(7.16) kF4k2≤K(K∗)2.(7.17) 40 Bound (7.13) is simply (1) o Lemma 7.3. Since k∂θT+k2≤K, by (1) o Lemma 7.1, we ha e ha kν∂θT+∂θS2k2≤νk∂θT+k1k∂θS2k1≤ν κ2k∂θT+k2k∂θS2k2≤Kκ−2K∗, which p o es (7.14). Bound (7.15) ollows analogously om  1 2ν(∂θS2)22≤ν 2k∂θS2k0k∂θS2k2≤ν 2 1 κ2(K∗)2. By (3) o Lemma 7.3, bound (7.16) ollows om k(2A1∂ T+−1)∂ S2k2≤ k2A1∂ T+−1k0k∂ S2k2≤K(νI0)α−1+κ−1K∗. Bound (7.17) ollows om (2) o Lemma 7.3 and kA1(∂ S2)2k2≤ kA1k−2k∂ S2k2 2≤K(K∗)2. In pa icula , om (7.13), P oposi ion 7.4 and he las claim o Lemma 7.2, since b Fin,a(0) = F0, we deduce ha Tb Gu s◦b Fin,a(0)U1=TS2,0+b Gu◦b Fin,a(0)U1≤TS2,0U1+Tb Gu◦b Fin,a(0)U1 ≤Kmax{(νI0)−2α,(νI0)−1+αlog(νI0)}+KkF0k2≤K1max{(νI0)−2α,(νI0)−1+αlog(νI0)}, o some K1>0. By he las claim o Lemma 7.2, he same ype o compu a ions imply ha , i S2, S′ 2∈ BK∗, Tb Gu s◦b Fin,a(S2)−b Gu s◦b Fin,a(S′ 2)U1≤Kkb Fin,a(S2)−b Fin,a(S′ 2)k2 ≤Kκ−1+ (νI0)α−1+K∗TS2−S′ 2U1. Then, aking K∗= 2Tb Gu s◦b Fin,a(0)U1= 2K1max{(νI0)−2α,(νI0)−1+αlog(νI0)}, he claim ollows wi h he usual a gumen , aking κand νI0la ge enough so ha Kκ−1+ (νI0)α−1+K∗<1/2. 8 Di e ence be ween solu ions o he inne equa ion In his sec ion we compu e he di e ence be ween he unc ions T±=T0+L± in +T± 2, wi h T+is gi en in D+ κ,in ×Tσwhe e D+ κ,in is de ined in (6.10) and T0( ) = −1/(4 )is in oduced in (6.3), L+ in is de ined in (6.8) and T+ 2is gi en by P oposi ion 6.5. Mo eo e , he unc ion T−is de ined by T−( , θ) = −T+(−¯ , −¯ θ) = T0( )−L+ in(−¯ , −¯ θ)−T+ 2(−¯ , −¯ θ)(8.1) in he domain (−D+ κ,in)×Tσ. We also de ine L− in( , θ) = −L+ in(−¯ , −¯ θ)and T− 2( , θ) = −T+ 2(−¯ , −¯ θ).(8.2) The ollowing lemma is an immedia e compu a ion. I ollows om he ac ha V es ic ed o Ris an e en unc ion, i.e. only depends on cosinus. We ecall ha E(T0)( , θ) = −V(θ)/8 2. Lemma 8.1. We ha e ha L− in( , θ) = Z∞ E(T0)(s, θ +s− )ds =−X k∈Z V[k] 8eik(θ− )Z∞ 1 s2eiks ds. I sa is ies he analogous bounds o L+ in in P oposi ion 6.3. 41 Figu e 8: The domain Dκde ined in (9.3), shaded in g ay. Compa e wi h Figu e 4. 9.1 S aigh ening he linea ope a o Since Φ±a e solu ions o (4.3), e ∆ou sa is ies e Le ∆ou = 0,(9.6) whe e e Le ∆ = 1 + B 1 + A∂ue ∆ + νI0∂θe ∆ and A=ν 2νI0∂θΦ++∂θΦ−, B=1 2 (1 + u2)2 u2∂uΦ++∂uΦ−−1. (9.7) We look o a change o a iables o he o m u= +X( , θ)such ha ∆ou ( , θ) = e ∆ou ( +X( , θ), θ) sa is ies L∆ou = 0, whe e L∆ = ∂u∆ + νI0∂θ∆was in oduced in (4.7). No ice ha , p o ided 7→ +X( , θ)is in e ible, e ∆ou sa is ies (9.6) i and only i ∆sa is ies 1 1 + ∂ X 1 + B 1 + A|(u,θ)=( +X( ,θ),θ)−νI0∂θX!∂ ∆ + νI0∂θ∆ = 0. Hence, we need o impose ha Xsa is ies LX=B−A 1 + A|(u,θ)=( +X( ,θ),θ) .(9.8) To sol e his equa ion, we conside he in e se Gou ,d o he ope a o Lde ined by he Fou ie coe icien s o he image Gou ,d(X): Gou ,d(X)[k](u) = Zu u+ 0 eikνI0(s−u)X[k](s)ds, k > 0, Gou ,d(X)[0](u) = Zu u0 X[0](s)ds, Gou ,d(X)[k](u) = Zu u− 0 eikνI0(s−u)X[k](s)ds, k < 0, (9.9) 48 whe e u+ 0,u− 0=u+ 0and u0∈Ra e opmos , bo ommos and eal le mos poin s in Dκ, espec- i ely. See Figu e 8. Rema k 9.1. The choice o he poin s u± 0and u0implies ha , i X(u, θ)is eal analy ic, so is Gou ,d(X). To sol e equa ion (9.8), o ∈R, we in oduce he Banach space o analy ic, 2π-pe iodic in θ unc ions Xs={X:Dκ×Tσ→C| kXks<∞}, wi h he no m kXks=X k∈ZkX[k]kse|k|σ, whe e, o an analy ic unc ion :Dκ→C, k ks= sup u∈Dκ|(1 + u2)s (u)|. Nex , we s a e wo echnical lemmas comple ely analogous o Lemmas 4.2 and 4.3. Lemma 9.2. Le s1, s2∈R. (1) I X∈ Xs1+s2and s2≥0, hen X∈ Xs1and he e exis s K > 0, independen o s1, s2such ha kXks1≤KνI0 κs2 kXks1+s2. (2) I X1∈ Xs1and X2∈ Xs2, hen X1X2∈ Xs1+s2and kX1X2ks1+s2≤ kX1ks1kX2ks2. Lemma 9.3. Le s∈Rand Gou ,d be he ope a o de ined by (9.9). (1) I X∈ Xswi h s≥0, hen Gou ,d(X)∈ Xsand kGou ,d(X)ks≤KkXks. I , u he mo e, hXi= 0, kGou ,d(X)ks≤K(νI0)−1kXks. (2) I X∈ Xswi h s > 1, hen Gou ,d(X)∈ Xs−1and kGou ,d(X)ks−1≤KkXks. (3) I X∈ Xswi h s > 0, hen ∂uGou ,d(X), ∂θGou ,d(X)∈ Xsand k∂uGou ,d(X)ks≤KkXks,k∂θGou ,d(X)ks≤K(νI0)−1kXks. Now, we summa ize he p ope ies o Aand Bwe need. Lemma 9.4. Le Aand Bbe he unc ions in oduced in (9.7). The e exis s K > 0such ha (1) A∈ X2wi h kAk2≤K(νI0)−2. (2) B=B1+B2whe e B1∈ X1,hB1i= 0,kB1k1≤K(νI0)−1and B2∈ X2wi h kB2k2≤ K(νI0)−2. Consequen ly, B∈ X1wi h kBk1≤K(νI0)−1. P oo . To p o e (1), we use he de ini ions in (9.1) and (9.7). F om P oposi ion 4.4 we ha e k∂θL+ ou k2≤K(νI0)−1and, om P oposi ion 4.5 and (1) o Lemma 9.2, k∂θΦ+ 2k2≤νI0 κ2 k∂θΦ+ 2k4≤νI0 κ2 (νI0)−1TΦ+ 2U3≤Kκ−2(νI0)−1. 49 Since he bounds o L− ou and Φ− 2a e he same as he ones o L+ ou and Φ+ 2, espec i ely, and ∂θΦ0= 0, we ha e kAk2=ν 2νI0∂θΦ++∂θΦ−2≤ν νI0k∂θL+ ou k2+k∂θΦ+ 2k2≤K(νI0)−2. Now we p o e (2). Using he de ini ion o Φ0in (4.5) we can check ha B=B1+B2, whe e B1(u, θ) = 1 2 (1 + u2)2 u2(∂uL+ ou (u, θ) + ∂uL− ou (u, θ)), B2(u, θ) = 1 2 (1 + u2)2 u2(∂uΦ+ 2(u, θ) + ∂uΦ− 2(u, θ)). By P oposi ion 4.4,k∂uL+ ou k3≤K(νI0)−1and since k(1+u2)2/u2k−2=O(1), by (2) o Lemma 9.2, kB1k1≤ 1 2 (1 + u2)2 u2−2k∂uL+ ou +∂uL− ou k3≤K(νI0)−1. Since hL+ ou i= 0 and (1 + u2)2/u2does no depend on θ,hB1i= 0. Finally, by P oposi ion 4.5,k∂uΦ+ 2k4≤TΦ+ 2U3≤K(νI0)−2. Then, kB2k2≤ 1 2 (1 + u2)2 u2−2k∂uΦ+ 2+∂uΦ− 2k4≤K(νI0)−2. Using he ope a o Gou ,d, we conside he ixed poin equa ion X=Gou ,d ◦Fou ,d(X),(9.10) whe e Fou ,d(X) = B−A 1 + A|(u,θ)=( +X( ,θ),θ) .(9.11) I is clea ha i Xsa is ies (9.10), hen Xis a solu ion o (9.8). Le X0=Gou ,d ◦Fou ,d(0). P oposi ion 9.5. X0∈ X1and kX0k1≤K(νI0)−2. Fu he mo e, X0is eal analy ic. P oo . We ha e ha Fou ,d(0) = B−A 1+A=B1+B2−A(B1+1) 1+A. By Lemma 9.4,kB1k1≤K(νI0)−1and hB1i= 0. Then, by (1) o Lemma 9.3,kGou ,d(B1)k1≤ K(νI0)−1kB1k1≤K(νI0)−2. Also, by Lemma 9.4,kB2k2≤K(νI0)−2. Then, he claim ollows om he bound  B2−A(B1+ 1) 1 + A2≤ kB2k2k(1 + A)−1k0+kAk2kB1+ 1k0k(1 + A)−1k0 and (2) o Lemma 9.3. The ollowing lemma is analogous o Lemma 8.3. The only di e ence is he geome y o he domain. Ac ually Dκdepends on κ, δ, β1and β2. Ha ing ixed some κ0,δ0,β1and β2wi h β2> β1 we will w i e b Dκ o deno e a amily o domains wi h κ > κ0and δ < δ0and wi h he same angles β1, β2such ha i ˜κ > κ > κ0 he dis ance om b D˜κ o he bounda y o b Dκis (˜κ−κ)(νI0)−1cos β1. This implies de e mining ˜ δ, depending on ˜κso ha (δ−˜ δ) cos β2= (˜κ−κ)(νI0)−1cos β1. No e ha a change in ˜κo o de one p oduces a change in ˜ δo o de (νI0)−1. Lemma 9.6. Le C∈ Xsin b Dκ×Tσand X∈ X in b D˜κ×Tσwi h s, ≥1and ˜κ > κ. We de ine CX( , θ) = C( +X( , θ), θ),( , θ)∈b D˜κ×Tσ. Then, he e exis s κ0such ha o any ˜κ > κ > κ0 he e exis K0, K1>0, depending on ˜κ, κ bu independen on νI0, such ha he ollowing holds. 50 (1) ∂j C∈ Xs+jin b D˜κ×Tσand k∂j Cks+j≤K1j!Kj 0kCks,j≥0. (2) I X∈ X in b D˜κ×Tσ, hen CX∈Xsin b D˜κ×Tσand kCXks≤ kCks K 1−K0(νI0˜κ−1) kXk . (3) I X, X′∈ X in b D˜κ×Tσ, wi h kXk ,kX′k ≤K, kCX−CX′ks+1 ≤K(νI0˜κ−1) kCkskX−X′k . P oo . (1) is an immedia e consequence o Cauchy es ima es in he educed domain. Using (1) o Lemma 9.2, (2) ollows om kCX( , θ)ks=X j≥0 1 j!∂j C( , θ)Xj( , θ)s ≤X j≥0 1 j!k∂j C( , θ)kskXkj 0 ≤X j≥0 K j!νI0 ˜κ j k∂j C( , θ)ks+jkXkj ≤KK1X j≥0 νI0 ˜κ K0kXk !j kCks =kCks KK1 1−νI0 ˜κ K0kXk . (3) ollows om C( +X( , θ), θ)−C( +X′( , θ), θ) =Z1 0 ∂ C( +X′( , θ) + s(X( , θ)−X′( , θ)), θ)ds (X−X′), (1) and (2) and he ac ha kXk0≤K(νI0/˜κ) kXk . To sol e equa ion (9.10), we in oduce e Xby se ing X=X0+e X. Then, Xis a solu ion o (9.10) i and only i e Xsa is ies e X=Gou ,d ◦e Fou ,d(e X),(9.12) whe e e Fou ,d(e X) = Fou ,d(X0+e X)−Fou ,d(0). P oposi ion 9.7. The e exis s κ0such ha , o any κ > κ0, equa ion (9.12)has a unique solu ion e X1∈ X1wi h ke X1k1≤K(νI0)−2/κ2. As a consequence, X=X0+e X1∈ X1is a solu ion o (9.10), kXk1≤K(νI0)−2and i is eal analy ic. P oo . We i s ema k ha , by Lemma 9.2 kAk0≤K(νI0)2kAk2/κ2,kAk1≤KνI0kAk2/κ, and aking in o accoun Lemma 9.4,(B−A)/(1 + A)∈ X1wi h k(B−A)/(1 + A)k1≤ kB−Ak1k(1 + A)−1k0≤K(νI0)−1. Since X0∈ X1in b Dκwi h kX0k1≤K(νI0)−2, by (3) o Lemma 9.6, aking X′= 0,s= 1, = 1 and ˜κ > κ we ha e ha e Fou ,d(0) ∈ X2in b D˜κwi h ke Fou ,d(0)k2≤KνI0 ˜κ B−A 1 + A1kX0k1≤Kκ−1(νI0)−2. Hence, by (2) o Lemma 9.3,kGou ,d ◦e Fou ,d(0)k1≤ kGou ,dkke Fou ,d(0)k2≤K(νI0)−2/˜κ, whe e he e kGou ,dkis he no m o he ope a o Gou ,d :X2→ X1. Le K∗= 2kGou ,d ◦e Fou ,d(0)k1. Wi h he same a gumen , i X, X′∈ X1wi h kXk1,kX′k1≤K∗, ke Fou ,d(X)−e Fou ,d(X′)k2≤KνI0 ˜κ B−A 1 + A1kX−X′k1≤K ˜κkX−X′k1in b D˜κ, 51 o some K > 0, independen o νI0and ˜κ. Finally, using again (2) o Lemma 9.3, kGou ,d ◦e Fou ,d(X)−Gou ,d ◦e Fou ,d(X′)k1 ≤ kGou ,dkke Fou ,d(X)−e Fou ,d(X′)k2≤kGou ,dkK ˜κkX−X′k1, which p o es ha Gou ,d ◦e Fou ,d is a con ac ion in b D˜κi ˜κis la ge enough. Now we can check ha Gou ,d ◦e Fou ,d sends BK∗⊂ X1in o i sel and he e o e Gou ,d ◦e Fou ,d has a unique ixed poin in ha ball. The eal analy ici y claim ollows om he de ini ions o Fou ,d and e Fou ,d and Rema k 9.1. P oposi ion 9.8. Le Xbe he unc ion, in he domain ˆ Dκ×Tσ, gi en by P oposi ion 9.7. Then, he map Θ( , θ) =  +X( , θ) θ is a well de ined change o a iables in ˆ Dκ×Tσ. I s in e se, Θ−1(u, θ) = u+Y(u, θ) θ=u−X(u, θ) + b X(u, θ) θ, is well de ined and eal analy ic in ˆ D˜κ×Tσ, o some ˜κ > κ, wi h Y∈ X1and kYk1≤K(νI0)−2. Mo eo e Y=−X+b Xwi h b X∈ X1and kb Xk1≤κ−1(νI0)−3in ˆ D˜κ×Tσ. P oo . The unc ion Yis he solu ion o he ixed poin equa ion Y=P(Y), whe e P(Y)(u, θ) = −X(u+Y(u, θ), θ). Clea ly, P(0) = −X∈ X1, wi h kP(0)k1=kXk1≤K(νI0)−2. We de ine K∗= 2K1kXk1. Then, by (2) o Lemma 9.6, i Y∈ BK∗⊂ X1,kP(Y)k1≤K(νI0)−2. Since, by P oposi- ion 9.7,X∈ X1, using (3) o Lemma 9.6,P es ic ed o BK∗is Lipschi z wi h Lipschi z cons an K˜κ−1νI0kXk1≤K˜κ−1(νI0)−1. Hence, i is a con ac ion i ˜κ−1(νI0)−1is small enough. Now, one easily checks ha P(BK∗)⊂ BK∗. Le Y0be he unique ixed poin o Pin BK∗. Then, w i ing Y0=−X+b X, kb Xk1=kY0+Xk1=kP(Y0)−P(0)k1≤K1 κνI0kY0k1≤K1 κ(νI0)3. 9.2 The exponen ially small o mula o Φ+−Φ− In his sec ion we inally ob ain he o mula o he di e ence e ∆ou = Φ+−Φ− o eal alues o uand θ. Fi s , we in oduce Υ+(u, θ) = νI0∆in(νI0(u+Y(u, θ)−i), θ), whe e he unc ion Yis gi en by P oposi ion 9.8.Υ+is de ined in {u∈C| ℑu < 1−κ(νI0)−1+ min{ an β1ℜu, − an β1ℜu}} which con ains he domain Dκ. We also de ine Υ−(u, θ) = Υ+(u, θ)(9.13) and Υ=Υ++ Υ−.(9.14) This de ini ion implies ha Υis eal analy ic. We ecall ha , since Lin∆in = 0 (see Theo em 8.6), we ha e ∆in( , θ) = X k≥1 keik(θ− ), 52 whe e he coe icien s kdo no depend on νI0. Then, in oducing Υk=νI0 ke−kνI0, k ≥1,(9.15) we ha e Υ+(u, θ) = X k≥1 Υkeik(θ−νI0u−νI0Y(u,θ)) and Υ−(u, θ) = X k≤−1 Υ−keik(θ−νI0u−νI0Y(u,θ)), since Yis eal analy ic. The unc ion Υwill be he desi ed i s o de o Φ+−Φ−. In o de o p o e his ac , we in oduce E(u, θ) = e ∆ou (u, θ)−Υ(u, θ) de ined in Dκ. P oposi ion 9.9. Le N∈Nbe ixed. The e exis s K > 0and s > 0such ha , o all (u, θ)∈ Dslog(νI0)×Tσsuch ha νI0(u−i)∈ D+ slog(νI0),α ∩−D+ slog(νI0),α,0≤j+k≤N, |∂j u∂k θE(u, θ)| ≤ K(νI0)j+1 max{(νI0)−2α,(νI0)−1+αlog(νI0),(νI0)−s} log(νI0), whe e he domain Dκwas in oduced in (9.3)and D+ κ,α in (7.4). P oo . Since Eis eal analy ic, i is enough o bound i o ℑu≥0. We w i e E=E1+E2+E3, whe e E1(u, θ) = Φ+(u, θ)−Φ−(u, θ)−νI0T+(νI0(u−i), θ)−T−(νI0(u−i), θ), he unc ion T+was gi en by Co olla y 6.6 and T−was in oduced in (8.1), E2(u, θ) = νI0T+(νI0(u−i), θ)−T−(νI0(u−i), θ)−Υ+(u, θ) and E3(u, θ) = −Υ−(u, θ). We bound each e m sepa a ely. We s a wi h E1. We claim ha |E1(u, θ)| ≤ KνI0 max{(νI0)−2α,(νI0)−1+αlog(νI0)} slog(νI0).(9.16) Indeed, o (u, θ)∈ Dslog(νI0)×Tσsuch ha νI0(u−i)∈ D+ slog(νI0),α ∩−D+ slog(νI0),α, by P oposi ion 7.5, |E1(u, θ)| ≤ 2νI0kS+ 2k0≤KνI0 slog(νI0)kS+ 2k1≤KνI0 max{(νI0)−2α,(νI0)−1+αlog(νI0)} slog(νI0). To bound E2we obse e ha , by Theo em 8.6 and P oposi ions 9.5 and 9.8, E2(u, θ) = νI0∆in(νI0(u−i) + Z(νI0(u−i), θ), θ)−∆in(νI0(u−i+Y(u, θ)), θ). Also, aking in o accoun ha kZk1≤Kand kYk1=K(νI0)−2, we ha e ha |Z(νI0(u−i), θ)| ≤ K νI0|u−i|, |νI0Y(u, θ)| ≤ K νI0|u−i|. 53 Hence, by he mean alue heo em, o (u, θ)∈ Dslog(νI0)×Tσwe ha e ha |E2(u, θ)| ≤ KνI0 s(νI0)slog(νI0)4. Finally, o bound E3we i s obse e ha , o (u, θ)∈ Dslog(νI0)×Tσwi h ℑu≥0, since Yis eal analy ic, which implies ha ℑY(ℜu, ℜθ) = 0, and kYk1=O((νI0)−2), ℑ(u+Y(u, θ)) ≥(1 −k∂uYk0)ℑu≥(1 −K(νI0)2/(slog(νI0))2kYk1)ℑu ≥(1 −K/(slog(νI0))2)ℑu≥0, whe e we ha e used Cauchy es ima es o ela e ∂uYand Y, and we ha e sligh ly educed he domain by conside ing a bigge alue o s. Hence, om he de ini ion o Υ−in (9.13) and (9.15), o (u, θ)∈ Dslog(νI0)×Tσwi h ℑu > 0, |E3(u, θ)| ≤ X k≤−1 νI0| −k|ek(νI0−σ)ek(νI0ℑu(1+O((νI0)−1))) ≤X k≤−1 νI0| −k|ek(νI0−σ)≤KνI0e−(νI0−σ). The bounds o he s a emen ollow applying s anda d Cauchy es ima es and sligh ly educing he domain. Finally, we need he ollowing elemen a y lemma. Lemma 9.10. Le Ψ : Dslog(νI0)×Tσ→Cbe an analy ic unc ion. We w i e Ψ(u, θ) = Pk∈ZΨ[k](u)eikθ. Assume (i) LΨ=0. (ii) The e exis s M > 0such ha |Ψ[k](±(i−is log(νI0)/(νI0)))| ≤ M. Then, Ψ(u, θ) = Pk∈ZΛkeik(θ−νI0u)and |Λ±k| ≤ M(νI0)kse−kνI0, k ≥1. P oo . Since LΨ = 0, he e exis s a 2π-pe iodic unc ion Λ(φ) = Pk∈ZΛkeikφ such ha Ψ(u, θ) = Λ(θ−νI0u). Since Ψis also pe iodic wi h espec o θ, we ha e ha Ψ(u, θ) = Pk∈ZΨ[k](u)eikθ, ha is Ψ[k](u) = Λke−ikνI0u. The claim ollows e alua ing he abo e equali y a u=i−is log(νI0)/(νI0), o k≥1, and a u=−i+is log(νI0)/(νI0), o k≤ −1. Theo em 9.11. Fix αsuch ha min{2α, 1−α}>0. Take 0< s < min{2α, 1−α}and ix N∈Nas in P oposi ion 9.9. The e exis s Λ0∈Rsuch ha , o all (u, θ)∈(Dslog(νI0)∩R)×Tσ, 0≤j+k≤N. Le ∂j u∂k θ(e ∆ou (u, θ)−Λ0) = ∂j u∂k θΥ(u+Y(u, θ), θ) + OνI0e−νI0 log(νI0) =νI0e−νI02 1∂j u∂k θ(cos(θ−νI0u)) + O1 log(νI0), whe e 1is gi en in Theo em 8.6. Fu he mo e, i one eplaces he iin he de ini ion o he coe icien s io Vin (1.2)by ε i, we ha e ha 1=επ 1 4+O(ε2).(9.17) 54 P oo . By P oposi ion 9.7 and Theo em 8.6, we ha e ha e E(u, θ) = e ∆ou (u+X(u, θ), θ)−Υ(u+X(u, θ), θ) sa is ies Le E= 0. Since kXk1≤K(νI0)−2,e Esa is ies he same bounds as Egi en by P oposi ion 9.9. W i ing e E(u, θ) = Pk∈Ze Ekeik(θ−νI0u), by Lemma 9.10 we ha e ha |e E±1| ≤ KνI0 log(νI0)e−νI0, |e E±k| ≤ KνI0 log(νI0)e−kνI0(1−slog(νI0))/(νI0), k ≥2. Hence, o (u, θ)∈(Dslog(νI0)∩R)×Tσ, |e E(u, θ)−e E0| ≤ KνI0 log(νI0)e−νI0.(9.18) Since kYk1≤K(νI0)−2, we ha e ha |Y(u, θ)| ≤ K(νI0)−2 o u∈R. Then, om he de ini ion o Υin (9.14) and using ha 1∈R, we ha e ha , o (u, θ)∈(Dslog(νI0)∩R)×Tσ, Υ(u, θ) = νI0 X k≥1 ke−kνI0eik(θ−νI0(u+Y(u,θ))) +X k≤−1 −kekνI0eik(θ−νI0(u+Y(u,θ)))  =νI0e−νI02 1cos(θ−νI0u) + O((νI0)−1). Hence, using again kYk1≤K(νI0)−2,Ealso sa is ies (9.18), o (u, θ)∈(Dslog(νI0)∩R)×Tσ, and, om e ∆ou = Υ + E, e ∆ou (u, θ)−Λ0=νI0e−νI02 1cos(θ−νI0u) + O((log(νI0))−1), whe e Λ0=e E0. The las claim ollows immedia ely om (9.15) and Theo em 8.6. 10 Acknowledgemen s F. B has been pa ially suppo ed by he g an PID2021-122711NB-C’21, E.F. has been pa ially suppo ed by he g an PID2021-125535NB-I00, and P.M. has been pa ially suppo ed by he g an PID2021-123968NB-I00, unded by he Spanish S a e Resea ch Agency h ough he p og ams MCIN/AEI/10.13039/501100011033 and “ERDF A way o making Eu ope”. Also, E.F. and P.M. au ho s ha e been pa ially suppo ed by he Spanish S a e Resea ch Agency, h ough he Se e o Ochoa and Ma ía de Maez u P og am o Cen e s and Uni s o Excel- lence in R&D (CEX2020-001084-M). A P oo o Theo em 3.2 We i s in oduce a new ime in sys em (3.3) so ha he o igin becomes a ue saddle. Since he solu ions o (3.3) wi h ini ial condi ion (u0, 0, 0)∈Vρwi h u0, 0≥0and u0+ 0>0sa is y u( ) + ( )≥0while hey belong o Vρ, we de ine he new ime ssuch ha d /ds = (u+ )−1. Equa ion (3.3) becomes u′=u(1 + O1(u, )), ′=− (1 + O1(u, )), ′= (u+ )−1, (A.1) whe e ′deno es d/ds. The O1(u, ) e ms depend on sand a e uni o mly bounded in e ms o (u, )in Vρ. Gi en w0= (u0, 0, 0)∈Vρ, we de ine sw0= sup {s > 0|w(˜s)∈Vρ,∀˜s∈[0, s)},(A.2) whe e wis he solu ion o (A.1) wi h ini ial condi ion w0. Nex lemma implies (1) o Theo em 3.2. I s p oo is pos poned o Appendix B. 55 Lemma A.1. The e exis ρ∈(0,1) and C > 0, sa is ying Cρ < 7/8, such ha he solu ion w= (u, , )o (3.3)wi h ini ial condi ion w0= (u0, 0, 0)∈Vρwi h u0, 0>0sa is ies log ρ u01 1+Cρ !≤sw0≤log ρ u01 1−Cρ !. Mo eo e , o any 0< a ≤ρand 0< δ < a/2, he “Poinca é map" Ψ : Σ1 a,δ →Σ0 a,δ1−Ca , whe e he se s Σ0 a,δ and Σ1 a,δ a e de ined in (3.4), is well de ined and, i w= (u0, a, 0)∈Σ1 a,δ and Ψ(w) = (a, 1, 1), hen u1+Ca 0≤ 1≤u1−Ca 0, e C1u−(1−Ca)/2 0≤ 1− 0≤e C2u−(1+Ca)/2 0 o some cons an s e C1,e C2>0depending only on ρ. Le w= (u, , )be a solu ion o (3.3) wi h ini ial condi ion w0∈Vρ. We in oduce τ= /u (A.3) and we will w i e Oi=Oi(u, ). We ha e ha 0< τ(s)<∞, o all ssuch ha w∈Vρ. I is immedia e om (A.1) ha dτ ds =−(2 + O1)τ. (A.4) The a ia ional equa ions a ound a solu ion o sys em (3.3) a e   ˙ U ˙ V ˙ T = 2u+ +O2u(1 + O1)uO2 − (1 + O1)−u−2 +O2 O2 0 0 0   U V T .(A.5) To p o e (2) o Theo em 3.2, we will s udy he beha io o he solu ions o (A.5) wi h ini ial condi ion U=U06= 0 along solu ions o (3.3) wi h ini ial condi ion w0= (u0, 0, 0)∈Vρ∩{u, > 0} o 0=asmall bu ixed and u0a bi a ily small. Equa ions (A.5) become, in he ime sin which he equa ions in (A.1) a e w i en, and using τin (A.3),  U′ V′ T′ =  2+τ+O1 1+τ 1+O1 1+τuO1 −(1+O1)τ 1+τ−1+2τ+O1 1+τ O1 0 0 0   U V T .(A.6) P oposi ion A.2. The e exis s α∗, wi h 0< α∗<5/12 such ha o any ρ > 0small enough, any w= (u, , ), solu ion o (A.1)wi h w|s=0 =w0= (u0, 0, 0)∈Vρand any α∗ 0∈[0, α∗], he e exis s α: [0, sw0]→R,C∞, whe e sw0was de ined in (A.2), wi h α(0) = α∗ 0, such ha , in oducing he new a iable e V=V+αU, equa ion (A.6)becomes   U′ e V′ T′ =  2+τ+O1 1+τ−α1+O1 1+τ 1+O1 1+τuO1 0−1+2τ+O1 1+τ+α1+O1 1+τ O1 0 0 0    U e V T .(A.7) Fu he mo e, o s∈(0, sw0], 0< α(s)<55 128 τ(s) 1 + τ(s).(A.8) 56 P oo . Gi en αand e V=V+αU, since τ > 0, he equa ion o e Vis e V′=−(1 + O1)τ 1 + τ+α′+ (3 + O1)α−α21 + O1 1 + τU(A.9) +−1 + 2τ+O1 1 + τ+α1 + O1 1 + τe V+ ( +αu)TO1.(A.10) The claim will ollow inding an app op ia e solu ion o α′=ν0+ν1α+ν2α2,(A.11) whe e ν0=(1 + O1)τ 1 + τ, ν1(s) = −3 + O1, ν2=1 + O1 1 + τ. Le (w, α) = ν0+ν1α+ν2α2be he igh hand side o (A.11), whe e we ha e omi ed he dependence o νi,i= 1,2,3, on wand s. We in oduce α0and α1, he nullclines o (A.11), by (w, α) = ν2(α−α0(τ))(α−α1(τ)), and R, whe e α0(τ) = −ν1 2ν2 1−1−4ν0ν2 ν2 11/2! =3 2+O1 1 + τ−(1 + τ)2−4 9+O1τ1/2! =3 2+O11 + τ−pR(τ). To comple e he p oo o P oposi ion A.2, we need he ollowing wo auxilia y lemmas. Lemma A.3. The unc ion α0has he ollowing p ope ies. Fo (u, )∈Va( ha is, 0< τ < ∞), (1) 2√2/3 + O1≤pR(τ)/(1 + τ)<1, (2) limτ→∞ α0(τ) = 1/3 + O1, (3) limτ→0α0(τ)/τ = 1/3 + O1, (4) d dsα0=−(1 + O1)√R−τ+ 1 + O1 √Rα0, (5) −(2 + O1) √Rα0≤d dsα0≤ −(16/9 + O1) √Rα0 (6) limτ→0(dα0/ds)/α0=−2 + O1. Fu he mo e, 0< α0(τ)<11 32 τ 1 + τ.(A.12) P oo . I ems (1) o (6) a e p o en in [GK12]. The bound (A.12) ollows om a di ec compu a ion. Nex lemma p o ides solu ions o (A.11) close o he nullcline α0. The p oo s o he nex wo lemmas a e gi en in Appendix B. Lemma A.4. Fo any 0< ρ < 1small enough, he ollowing is ue. Fo any solu ion w= (u, , ) o (3.3)wi h ini ial condi ion w0∈Vρ×T, i αis a solu ion o (A.11)wi h 0≤α(s0)≤ 5α0(τ(s0))/4 o some 0< s0< sw0, hen 0< α(s)<5α0(τ(s))/4 o all s∈[s0, sw0]. 57