Holomorphic Normal Form of Nonlinear Perturbations of Nilpotent Vector Fields

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1 ISSN , Regula ad Chaotic Dyamics, 2016, Vol. 21, No. 4, pp c Pleiades Publishig, Ltd., Holomophic Nomal Fom of Noliea Petubatios of Nilpotet Vecto Fields Lauet Stolovitch 1* ad Feek Vestige 2** 1 CNRS, Laboatoie J.-A. Dieudoé U.M.R. 6621, Uivesité de Nice Sophia Atipolis, Pac Valose Nice Cedex 02, Face 2 Royal Obsevatoy of Belgium, Riglaa 3, 1180 Bussels, Belgium Received Mach 08, 2016; accepted Jue 22, 2016 Abstact We coside gems of holomophic vecto fields at a fixed poit havig a ilpotet lieapatatthatpoit,idimesio 3. Based o Belitskii s wok, we kow that such a vecto field is fomally cojugate to a (fomal omal fom. We give a coditio o that omal fom which esues that the omalizig tasfomatio is holomophic at the fixed poit. We shall show that this sufficiet coditio is a ilpotet vesio of Buo s coditio (A. I dimesio 2, o coditio is equied sice, accodig to Stóżya Żo ladek, each such gem is holomophically cojugate to a Takes omal fom. Ou poof is based o Newto s method ad sl 2 (C-epesetatios. MSC2010 umbes: 34M35, 34C20, 37J40, 37F50, 58C15, 34C45 DOI: /S Keywods: local aalytic dyamics, fixed poit, omal fom, Belitskii omal fom, small divisos, Newto method, aalytic ivaiat maifold, complete itegability 1. INTRODUCTION I this aticle, we coside gems of holomophic vecto fields i a eighbohood of a fixed poit i C, 2. We ae iteested i the local classificatio ude the actio of the goup of gems of biholomophisms pesevig the fixed poit, which we may assume to be the oigi. I the sequel, the gem of a vecto field o map efes to a holomophic gem at the oigi. The idea is to simplify, by a chage of coodiates, the system of diffeetial equatios i ode to bette udestad its dyamics. A elemetay istace of such a poblem is the followig: Let A be a matix with complex coefficiets. I ode the udestad the obits {A k z} k Z, z ea the fixed poit 0 of A, it is much easie to tasfom A ito its Joda omal fom J by a chage of coodiates P, A = PJP 1. The oe studies {J k y} k Z, Py = z ad pulls back the ifomatio to the oigial poblem. It was the idea of Poicaé to develop this poit of view fo vecto fields. The difficulty is that the Lie algeba of gems of vecto fields is ifiite-dimesioal. The liea tasfomatios ae eplaced by gems of diffeomophisms fixig the oigi. The special epesetative of the obit of the actio that plays the ole of the Joda omal fom is called a omal fom. A pecise defiitio is give below. This idea has bee extesively developed by Aold ad his school, Buo, Mose, Ecalle, Matiet Ramis, Yoccoz,...i the case whee the liea pat at the fixed poit is a semisimple matix. The issue is that, although cojugacy to a fomal omal fom ca always be obtaied by a suitable fomal tasfomatio, it may ot be possible to each a omal fom by a holomophic tasfomatio at the fixed poit. I this aticle, we ivestigate the omal fom poblem, i dimesio 3, fo holomophic vecto fields with a ilpotet liea pat at the fixed poit (assumed to be the oigi. We give a sufficiet coditio that esues that a gem of a holomophic vecto field ca be holomophically cojugated to a omal fom. * stolo@uice.f ** feek.vestige@oma.be 410

2 HOLOMORPHIC NORMAL FORM OF NONLINEAR PERTURBATIONS Fomal Nomal Foms Followig G. Belitskii [2, 3], E. Lombadi ad L. Stolovitch defied i [22] a otio of omal fom of highe-ode petubatios of a give quasi-homogeeous vecto field S ad the associated otio of geealized esoaces (these ae obstuctios to fidig a fomal powe seies tasfomatio cojugatig the petubatio of S back to S itself. Hee, we shall fist focus o the case whee S is liea, ad the o the case whee S is ilpotet. Let H k be the space of homogeeous vecto fields of degee k. Fo each atual umbe k 2, we coside the cohomological opeato d 0 := [S,. ]:H k H k associated to S. Hee, [S,. ] deotes the Lie backet with S. We defie a space of omal foms of degee k to be a supplemetay space C k to Im d 0 Hk. We ca show that thee exists a fomal tasfomatio that cojugates each petubatio of S of ode 2, X := S + R 2, to a fomal vecto field NF := S + k 2 v k whee v k C k fo each k. WecallNF a omal fom (of style Ĉ = C j, see [27]. Ideed, by iductio o the degee k 2, assume that X is omalized up to ode k 1, that is, X = NF k 1 + R k ad NF k 1 j=2 k 1 C j.letπ Im d0 deote the pojectio oto the age of d 0 alog C k.letus choose U k H k to solve the cohomological equatio d 0 (U k = π Im d0 R k,wheer k deotes the homogeeous polyomial of degee k of R k.the(i π Im d0 R k C k ad we have (id + U k 1 X = NF k 1 +[S, U k ]+R k + h.o.t. = NF k + R >k, whee NF k := NF k 1 +(I π Im d0 R k. Hee, Φ X := DΦ(Φ 1 X(Φ 1 deotes the cojugacy of X by the diffeomophism Φ. It is classical that a omal fom is ot uiquely defied sice oe ca add ay U 0 Ke d 0 Hk to the solutio U k of d 0 (U k = π Im d0 R k. Theefoe, defiig fo ay k 2, a supplemetay space Ṽk to Ke d 0 Hk i H k, we ca estict the solutio U k to belog to Ṽk, fo all k, ad efe to the omal fom so obtaied as the omal fom of X. The coespodig fomal tasfomatio (id + U k (id + U 1 iscalledthe associated omalizig tasfomatio. Of couse, we ae iteested i the case whee these supplemetay spaces ca be computed effectively. Oe way to do so is as follows: H k is povided with a Hemitia scala poduct (see (2.1. Let us defie C k (esp. Ṽ k to be the othogoal complemet to Im(d 0 Hk (esp. to be equal to Im(d 0 H k, that is, H k =Im(d 0 Hk C k. The space C k is the keel of the adjoit d 0 H k of d 0 Hk with espect to the Hemitia scala poduct. Whe S is a semisimple liea vecto field with eigevalues λ 1,...,λ, the space of omal foms is defied i tems of esoace elatios j q jλ j = λ i : the space C k geeated by moomial vecto fields x Q xi with Q =(q 1,...,q N, Q = q q = k 2, 1 i which satisfies a esoace elatio. We ca also choose Ṽk = C k ad obtai the classical Poicaé Dulac omal fom [1]. Whe S is ilpotet, the omal fom spaces ae moe divese ad thee ae diffeet ways of defiig omal foms. I dimesio = 2, F. Takes [39] has show that the space of omal foms associated to S = y x, C k, ca be chose to be the vecto space geeated by x k x ad x k y, k 2. This has bee geealized ecetly by E. Stóżya ad H. Żo ladek [37] to the case 2. Thee is aothe classical way to defie omal foms of petubatios of ilpotet liea vecto fields, by cosideig a sl(2, C-tiple associated to S ad thei epesetatios (see [9, 26]. Fist of all, it is classical (Jacobso Moozov theoem [4, 31] that thee exist liea vecto fields N := S, M ad H of C such that [N,M] =H, [H, N] =2N, [H, M] = 2M. A epesetatio of sl(2, C i a fiite-dimesioal vecto space V is a tiple of edomophisms X, Y, Z such that [X, Y ]=Z, [Z, X] =2X, [Z, Y ]= 2Y. Hee, [X, Y ] deotes the backet of edomophism XY YX. Such a family {X, Y, Z} is called a sl(2, C-tiple. It is classical that V =ImX Ke Y. We apply this to X =[N,.], Y =[M,.]actig o the space of homogeeous polyomial vecto fields: The supplemetay space to the image of d 0 is the defied to be the keel of ad M := [M,.] esticted to the space of homogeeous polyomial vecto fields.

3 412 STOLOVITCH, VERSTRINGE Thee exist also moe ivolved costuctios of fomal omal foms dealig with uiqueess poblems such as i [19] fo petubatio of (quasi-homogeeous vecto fields o i [14, 30], based o spectal sequeces Aalytic Nomal Fom Poblem As we have see, fo each highe-ode petubatio of S, thee exists a fomal tasfomatio (i. e., a fomal chage of coodiates to a fomal omal fom. We ae iteested i the situatio whee ot oly the petubatio is aalytic i a eighbohood of the oigi but also the omalizig tasfomatio to its omal fom (which is thus aalytic too. Hece we ae led to solve ad estimate iteatively the solutio U k of the cohomological equatio d 0 (U k =F k fo a give polyomial F k of degee k i the age of d 0. If S is semisimple, say S = i=1 λ ix i x i, this leads to the so-called small diviso poblem [1, 25, 33]: the om of U k is bouded by the om of F k divided by the smallest of the ozeo umbes of the fom q 1 λ q λ λ j, Q =(q 1,...,q N, Q := q q = k. These umbes may accumulate to the oigi whe the degee k teds to + causig a blow up of the omalizig tasfomatio. These umbes ae called small divisos (although they may ot be small. The faste these umbes accumulate to zeo, the smalle the adius of covegece of the fomal tasfomatio is. Fo istace, C. L. Siegel poved [32] that if thee exists τ 0 such that q 1 λ q λ λ j > fo all Q N, Q 2, the ay highe-ode aalytic C Q τ petubatio of S is aalytically lieaizable. This coditio has bee weakeed to the so-called (ω- coditio by Buo [5] (we shall ot ecall hee its defiitio sice we shall ot use it. Nevetheless, the cotol of these small divisos is ot sufficiet to esue the aalyticity of the omalizig tasfomatio to a (oliea omal fom. Ideed, thee ae well kow situatio fo which, although the small divisos ae bouded away fom the oigi, the omalizig tasfomatio is ayway diveget at the oigi [23]. Fially, A. D. Buo [5] foud a ecessay ad sufficiet coditio o the fomal omal fom coditio (A that esues that if the semisimple liea pat satisfies the diophatie coditio (ω, the ay aalytic petubatio of S that has a fomal omal fom satisfyig (A ca be aalytically omalized. This is a geealizatio of H. Rüssma s Hamiltoia vesio [29]. These woks have bee geealized to seveal commutig vecto fields i seveal diectios [17, 18, 34, 35, 43, 44] i the spiit of J. Vey [40, 41]. I [22], a boade otio of small divisos associated to a liea vecto field S is defied. They ae defied as the squae oots of the ozeo eigevalues of the box opeato Hk := d 0 Hk d 0 H k fo all k>2. Whe S is a ilpotet liea vecto field i dimesio two ad thee, G. Iooss ad E. Lombadi [16] computed the coespodig geealized esoaces as well as the small divisos. Recetly, P. Bockaet ad F. Vestige succeeded i [6] i estimatig the geealized small divisos fo (almost ay ilpotet liea pat. It is show thee that they ae always bouded away fom the oigi. Oe of the emaiig questios was whethe a Takes omal fom (i dimesio 2 could be obtaied systematically by a coveget tasfomatio. It took almost foty yeas to be asweed. I dimesio 2, afte a attempt to solve this questio by X. Gog [11], fially E. Stóżya ad H. Żo ladek [36] showed that ay holomophic oliea petubatio of y x has a gem of biholomophism that cojugate it to a Takes omal fom: (y + f(x, y x + g(x, y y is aalytically cojugate to (y + k(x x + l(x y ;heef,g,k,l ae aalytic gems of ode 2atthe oigi. A moe geometic poof was give i [21]. It was show also by E. Stóżya ad H. Żo ladek that the equivalet theoem i a highe-dimesioal settig is false [37, 38]. Fo the simila poblem with the scala poduct omal fom, we emphasize that i dimesio 2 the small divisos ted to ifiity with the degee of homogeeity. This pheomeo has a egulaizig effect o the solutio of the cojugacy equatio to a omal fom. This coespods somehow to the situatio of the Poicaé domai fo ozeo semisimple liea pat, whee o equiemets ae eeded o the petubatio fo it to have a aalytic tasfomatio to a omal fom.

4 HOLOMORPHIC NORMAL FORM OF NONLINEAR PERTURBATIONS 413 I highe dimesio a example i [16] shows that the small divisos may ot ted to ifiity with the degee of homogeeity: they may accumulate fiite umbes as well as ifiity. The highedimesioal situatio fo the ilpotet case coespods somehow to the Siegel domai : oe eeds to impose coditios o the fomal omal fom i ode to obtai the covegece of such a tasfomatio. The aim of this aticle is to ivestigate, i dimesio 3, the holomophic cojugacy to a omal fom poblem fo a highe-ode aalytic petubatio of the ilpotet liea vecto field. We shall defie a ilpotet vesio of Buo s coditio (A. We shall state ad pove a ilpotet vesio of the sufficiecy pat of Buo s theoem: i dimesio 3, if a aalytic highe petubatio of a liea ilpotet N has its fomal omal fom that satisfies this coditio, the thee exists a aalytic tasfomatio to its omal fom Mai Result I this sectio we fomulate ou mai esult usig the otatios itoduced i the itoductio. We fist itoduce some exta temiology. Let S be a liea vecto field. We deote by ÔS the ig of fomal fist itegals of S, that is, the set of fomal powe seies ˆf fo which the Lie deivative L S ( ˆf = 0 vaishes. A ilpotet matix is said to be egula if its Joda omal fom does ot cotai zeo blocks. I othe wods, a ilpotet edomophism is egula if thee is o ivaiat subspace upo which the opeato acts as a zeo opeato. We shall deote by R 2 agemofa holomophic vecto field (o fuctio, map depedig o the cotext vaishig at ode 2atthe oigi. Theoem 1. Let N be a egula ilpotet liea vecto field of C ad assume that {N,N, [N,N ]} is a sl(2, C-tiple. Let V = N + R 2 be a gem of a holomophic vecto field at the oigi of C. Assume that its fomal omal fom (as defied by the iteatio pocedue descibed i the itoductio has the fom V = N + fn whee f Ô N Ô N. (1.1 The its associated fomal diffeomophism coveges at the oigi. This is the mai esult of ou aticle ad should be udestood as a complete itegability theoem alog the same lies as (ad i fact a cotiuatio of esults by A. D. Buo [5] (see also [23, 43], ad by the fist autho [34, 35]. Ideed, i these aticles, the authos coside a (gem of of a vecto field (o commutig families of vecto fields V = S + R 2 that is a holomophic oliea petubatio of a semisimple liea vecto field S = i=1 λ ix i x i. The mai (easiest theoem of Buo ca be stated as follows: Assume that the omal fom of V is of the fom S + fs + g S whee S := λ i=1 i x i x i ad f,g ÔS, f(0 = g(0 = 0 (this is Buo s coditio (A. If S satisfies Buo s small divisos coditio (ω, the its associated fomal biholomophism is coveget at the oigi ad it cojugates V to a (coveget omal fom. Theoem 1 ca be see as a ilpotet vesio of Buo s theoem. Ideed, S plays the ole of N ad we have ÔS = ÔS Ô S. I ou omal fom (1.1, thee is o equivalet tem to fs sice the atual oe would be fn which is ot a omal fom w..t. to N, i.e.,fn / Ke ad N. Remak 1. Oe of the key poits of [6] is that, if the liea ilpotet vecto field S is egula, the thee exists a liea chage of coodiates L of C such that N := L S, N, the vecto field defied as the adjoit of the diffeetial opeato actig o polyomials of N w..t. the scala poduct used above, ad H =[N,N ] defie a sl(2, C-tiple. Moeove, {ad N, ad N, ad N ad N ad N ad N } is also a sl(2, C-tiple. I the sequel, we shall assume that the liea chage of coodiates L has bee applied ad, hece, that both {N,N, [N,N ]} ad {ad N, ad N, [ad N, ad N ]} ae sl(2, C-tiples. Remak 2. If V = N + R 2 is a gem of the holomophic vecto field at the oigi of C that ca be embedded ito a aalytic sl(2, C-tiple, that is, if thee ae gems Ṽ = N + h.o.t., V = H + h.o.t. of holomophic vecto fields such that {V,Ṽ,V } is (Lie-isomophic to sl(2, C, the V,Ṽ ad V ae simultaeously ad holomophically lieaizable. Ideed, at the fomal level, this follows fom [15], while the aalytic esult follows fom [13, 20]. It should be emphasized that this esult does ot hold i the smooth categoy [8, 13].

5 414 STOLOVITCH, VERSTRINGE Remak 3. If V = N + R 2, a gem of the holomophic vecto field at the oigi of C ca be embedded ito a fomal sl(2, C-tiple, that is, if thee ae fomal vecto fields Ṽ = N + h.o.t., V = H + h.o.t. such that {V,Ṽ,V } is (Lie-isomophic to sl(2, C, the V,Ṽ ad V ae simultaeously fomally lieaizable. Accodig to [6, p. 2223], the small divisos ae bouded away fom the oigi. The, accodig to [22, Theoem 5.8], V is aalytically lieaizable. Thee ae othe esults coceig the poblem of classificatio of petubatios ot of ilpotet vecto fields but athe of quasihomogeous vecto fields with a ilpotet liea pat at the oigi. They ae all i dimesio 2 (see, fo istace, [7, 24] ad ae ot immediately coceed with holomophic cojugacy (i a eighbohood at the oigi to a omal fom Geometic Itepetatio Accodig to Weitzeböck s theoem [42], the ig of fomal fist itegals ÔN of ay ilpotet liea vecto field N is fiitely geeated ove C (see [28, Theoem 6.2.1] fo this fomulatio. The ig of commo fomal fist itegals ÔN ÔN is hece fiitely geeated ove C. Let P 1,...,P be a set of geeatig polyomials. If they ae algebaically depedet, the thee exist polyomials Q 1,...,Q l o C such that Q i (P 1,...,P = 0 fo all i. Let us coside C = {z C Q i (z =0,i=1,...,l} the zeo locus of these polyomials. If the P j s ae algebaically idepedet, the C = C. Coside the map π : C (C, 0, x (P 1 (x,...,p (x. We apply ou mai esult ad assume that the vecto field V has bee cojugated ito a omal fom by a gem of holomophic tasfomatio at the oigi. I these holomophic coodiates, we have V = N + fn, whee f O N O N is a gem of the holomophic fuctio. Let b (C, 0 be a poit i the image of π. The 1. V is taget to the level set of Σ b := π 1 (b, itesected with a eighbohood of the oigi. Ideed, both N ad N ae taget to this level set ad the fuctio f is costat o this level set. 2. the estictio of the vecto field V, V Σb, is the estictio to Σ b of a liea vecto field. 3. the estictio V Σb belogs to the estictio of a sl(2, C-actio 1, amely, {N Σb,N Σ b,h Σb } (compae with Remaks 2 ad Moeove, this holds fo evey fibe (i a eighbohood of the oigi i C withia eighbohood of the oigi i C Im π. As a bypoduct, we also fid that the zeo fibe of π is ivaiat, the estictio to which V is equal to the estictio of the liea ilpotet field N. As illustated i the ext example, this last esult could have bee obtaied ude a much weake assumptio usig [22, Theoem 5.6] combied with the o-small-divisos statemet of [6]. 1 We thak Nguye Tie Zug fo this emak.

6 HOLOMORPHIC NORMAL FORM OF NONLINEAR PERTURBATIONS 415 Example 1. Coside the thee-dimesioal liea vecto field: N := y x + z y. We have Letusset N := x y + y z. h = xz y2 2. It follows that N(h =0=N (h adôn 3 ÔN 3 = C[[h]]. A fomal omal fom of a oliea petubatio of N is a fomal vecto field of the fom ẋ = y + xp 1 (x, h, ẏ = z + yp 1 (x, h+xp 2 (x, h, ż = zp 1 (x, h+yp 2 (x, h+p 3 (x, h, whee the P i s ae fomal powe seies [16]. We apply ou mai esult: Let V = N + R 2 be a gem of a holomophic vecto field that is a oliea petubatio of N. Assume that the omal fom of V is of the fom ˆΦ V = N + ˆf (xz y2 2 N, whee ˆf is a fomal powe seies of oe vaiable (that is, P 1 = P 3 =0,P 2 (x, h = ˆf(h. The ˆΦ defies a gem of holomophic chage of vaiables Φ at the oigi of C 3 ad ˆf defies a gem of the holomophic fuctio f at 0 C such that Φ V = N + f (xz y2 N. 2 I these holomophic coodiates, the vecto field is taget to each fibe } Σ b = {(x, y, z (C 3, 0 xz y2 2 = b i a eighbohood of the oigi, b beig sufficietly small. I paticula, i these ew holomophic coodiates, the vecto field is taget to Σ 0 ad its estictio to it is equal to (the estictio of N. We emphasize that the last poit ca be achieved ude a much weake assumptio: Assume that the fomal omal fom is such that P i (x, 0 = 0, i =1, 2, 3. The, accodig to [22, Theoem 5.6] applied to I =(h, thee ae holomophic coodiates fo which Σ 0 is a ivaiat set of the iitial vecto field X. Futhemoe, its estictio to Σ 0 is equal to N Sketch of the Poof The poof of the mai theoem is doe usig a Newto scheme. We assume that V = NF m + R m+1 is omalized up to ode m =2 k. By assumptio, the patial omal fom NF m is of the fom N + f m N whee f m ÔN ÔN is a polyomial of degee m 1. Thee is a uique polyomial vecto field U of the fom U = d 0 (V, of ode m +1adofdegee 2m such that the diffeomophism (id + U 1 omalizes V up to ode 2m. The vecto field U solves a equatio of the fom J 2m ([NF m,u] = B fo some kow B, i the age of the box opeato. Hee, J 2m deotes the tucatio at degee 2m of the Taylo expasio at the oigi. We the eed to estimate U i tems of B. Fo (1.2

7 416 STOLOVITCH, VERSTRINGE that pupose, we shall fist defie a suitable aalytic om (depedig o some adius, see Sectio 2.2 based o the scala poduct we metioed above (see Sectio 2.1. Followig (3.10, we shall the ewite the pevious equatio as V =(id+q 1 + Q B, whee deotes the estictio of the opeato := d 0 d 0 to its image ad Q 1,Q 2 ae opeatos defied by (3.9 ad (3.8, depedig o f m. Usig all popeties of sl(2, C-tiples (see Sectio 2.4, we shall be able to estimate the omal of Q 1,Q 2 ad 1. As a cosequece, we shall obtai a estimate of V i tems of B ad the a estimate of U i tems of B (see Popositio 4. The the cojugacy of V by (id + U 1 is omalized up to ode 2m: (id + U 1 V = NF 2m + R 2m+1 whee NF 2m := N + f 2m N is a omal fom. Choosig R< i a appopiate way, we shall show that the ew omal fom NF 2m as well as the ew emaide R 2m+1 satisfy the same estimate w..t. the R-om that NF m ad R m+1 satisfy w..t. the -om (Popositio 5. This allows us to do a iductio pocess. The key poit (Lemma 16 is that the sequece of successive adii {R k } coveges to a positive umbe. This is due to the shap estimate of the solutio U we obtai (fo each step. A classical agumet the shows that the sequece of (patial omalizig diffeomophisms coveges to a geuie holomophic diffeomophism that cojugates V to a omal fom Covegece to the Nomal Fom Requies a Coditio We do t kow whethe coditio (1.1 is ecessay fo the covegece of the omalizig tasfomatio to hold (as is the case with Buo s coditio (A i the semisimple case. Howeve, Stóżya ad Żo ladek have show i [38] that the omalizig tasfomatio of the vecto field ẋ = y x 2 + x 3, ẏ = z +2x 4, ( ż =0 to its (geealized Takes omal fom is diveget at the oigi. Although we shall ot ivestigate the dictioay betwee diffeet styles of omal fom, we show that the scala poduct omal fom of ( does t satisfy (1.1. Ideed, we have the followig Lemma 1. If f(x, y, z isafomalfistitegalof(, thef is a fuctio of z oly. Poof. Let L be the vecto field associated to ( ad let N := y x + z y be its liea pat at 0. Let us assume, by iductio, that f = k i=1 f iz i + j k+1 g j whee g j is homogeeous of degee j i all vaiables, adf i s ae complex coefficiets. Hece, we have 0 = L(f =L(g k+1 + g k+2 + h.o.t, so N(g k+1 = 0. Theefoe, g k+1 is a polyomial P k+1 of z ad h = xz y2 2. Let us show that P k+1 = α k+1 z k+1 ad that g k+2 = α k+2 z k+2 + β k+2 z k h. Ideed, the Taylo polyomial of degee k +2ofL(f is N(g k+2 x 2 g k+1 x =0=N(g k+2 x 2 z P k+1 h. Settig z = 0, we obtai y g k+2 x (x, y, 0 = 0. Theefoe, g k+2 = z g k+2 ad N( g k+2 x 2 P k+1 h =0. Let us wite x 2 P k+1 h = αx 2 h m + x 2 zp(z,h fo some polyomial p ad a costat α. Sice x 2 h m Ke N, x 2 P k+1 h ca belog to the age of N oly whe α =0.Thus,x 2 P k+1 h = x 2 zp(z,h, g k+2 = zg ad N(G =x 2 p(z,h. We cotiue this pocess. We fially fid that g k+2 = z k G 2 ad N(G 2 =αx 2 fo some quadatic polyomial G 2.SiceN (x 2 =0,wehaveα =0adthusG 2 is a fist itegal: G 2 = αz 2 + βh. This meas that g k+2 = α k+2 z k+2 + β k+1 z k h. As a cosequece, N(g k+2 = 0, so that P k+1 h =0adP k+1 = α k+1 z k+1.

8 HOLOMORPHIC NORMAL FORM OF NONLINEAR PERTURBATIONS 417 As a cosequece, system ( caot be fomally cojugate to a system satisfyig (1.1. Ideed, if it wee, the its fomal fist itegal would be a push-fowad of a fist itegal of the secod system. Sice (1.1 has h = xz y2 2 as a fist itegal, ( would have a fomal fist itegal of the fom h Φ, fo some fomal diffeomophism Φ, taget to idetity. But such a fuctio caot be a fuctio of the vaiable z oly. 2. BACKGROUND Fo ay oegative itege k, P k (esp. V k deotes the space of homogeeous polyomials (esp. polyomial vecto fields of degee k. We shall also wite P k,m := m j=k P j (esp. P k := j=k P j as well as V k,m := m j=k V j (esp. V k := j=k V j 2.1. Scala Poduct We itoduce fist some otatio. Let α N, we shall wite α! :=α 1!...α!. We defie P δ to be the space of homogeeous polyomials of degee δ i vaiables x 1,...,x. We fist coside the scala poduct o P δ itoduced by E. Fische [10] ad G. Belitskii [2]: Lemma 2. The scala poduct defied o P δ : a α x α, c β x β := a α c α α! B,δ α =δ has the followig popety: the opeato β =δ α =δ is adjoit to the multiplicatio by the x j opeato, that is, < f,g > B,δ =<f,x j g> B,δ+1 ad <g, f > B,δ =<x j g, f > B,δ+1 fo all (f,g P δ+1 P δ. I [22], a vaiat of this scala poduct was itoduced, amely, a α x α, c β x β = α! a α c α δ α!. (2.1 α =δ β =δ Let. δ be its associated om. Sice homogeeous polyomials of diffeet degee ae othogoal to each othe, we ae able to defie the om of fomal powe seies f = k 0 f k whee f k is a homogeeous polyomial of degee k as α =δ f 2 := k 0 f k 2 k. It has some impotat popeties that ae summaized i the followig popositio: Popositio 1 ([22, Popositios ]. Let f,g be fomal powe seies o C.Letuswite f = k 0 f k whee f k is a homogeeous polyomial of degee k. The fg f g, f defies a gem of the holomophic fuctio at the oigi if ad oly if thee exist positive costats M,C such that f k k MC k. This scala poduct iduces a scala poduct o V δ, the space of homogeeous vecto fields of degee δ i vaiables. Such a vecto field V cabewitteasv = k=1 V k x k,wheev k P δ. We the defie V k, W k x k x = V k,w k δ. (2.2 k δ k=1 k=1 k=1

9 418 STOLOVITCH, VERSTRINGE 2.2. The Aalytic Nom Let us defie, fo δ N, c δ := ( Q! 1/2. Q =δ Q! Letf be a fomal powe seies vaishig at the oigi. It ca be witte as a sum of homogeeous polyomials f = δ 1 f δ,wheef δ P δ.we defie, fo 0, f := f δ δ c δ δ, (2.3 δ 1 i this fomula f δ δ is the om associated to the scala poduct (2.1: f δ 2 δ = f k x k 2 δ = f k 2 k! k!. k =δ k =δ I a simila mae, we exted this om to the space of vecto fields usig (2.2. Lemma 3. c δ = δ/2. Poof. This follows immediately by choosig z i =1,1 i i the fomula (see, fo istace, [16, Lemma A.1] (z z δ = Q! Q! zq. Q =δ Lemma 4. Let f,g be fomal powe seies. The fg f g. Poof. We have fg = {fg} k k c k k = f i g j c k k. k 0 k 0 i+j=k k We ecall that {fg} k deotes the homogeeous pat of degee k of fg i the Taylo expasio at the oigi. Sice the scala poduct (2.1 defies a Baach algeba om, we have i+j=k f k ig j i+j=k f i i g j j.sicec k = k/2 = c i c j if i + j = k, wehave fg f i i c i i g j j c j j = f i i c i i g j j c j j. k 0 i+j=k i 0 j 0 Lemma 5. Suppose that u R, the f u R f. Poof. Let f = Q N f Qx Q be the Taylo expasio of f at the oigi. f u R = f Q (u(x Q f Q (u(x Q R δ 1 Q =δ δ 1 R Q =δ f Q (u(x Q R f Q Q δ 1 Q =δ δ 1 Q =δ Q! Q! f Q Q! Q! Q δ 1 Q =δ (Apply Cauchy Schwatz

10 HOLOMORPHIC NORMAL FORM OF NONLINEAR PERTURBATIONS 419 ( f Q 2 Q! 1/2 ( Q! Q! Q! 2 Q 1/2 δ 1 Q =δ Q =δ ( f Q 2 Q! 1/2cδ δ = f δ δ c δ δ Q! δ 1 Q =δ δ 1 = f. Lemma 6. Let f be a polyomial of degee at most m, the f x i m f. Poof. We emak fist that it eadily follows fom Lemma 3 that c δ 1 c δ. We estimate f δ 2 δ 1 Q =δ Q =δ f Q 2 q 2 j Q! q j (δ 1! Q =δ f Q 2 Q! δ! (δq j δ 2 Q =δ f Q 2 Q! δ! f Q 2 Q! δ!. δ!q j (δ 1! Hece we have f δ x i δ 1 δ f δ δ. As a cosequece, we obtai the followig estimates: f m = f δ m x i x i c δ 1 δ 1 δ f δ δ c δ 1 δ 1 δ=1 δ 1 δ=1 m m f δ δ c δ 1 δ m m f δ δ c δ δ δ=1 m f. δ= Nomal Foms Let F = ( i=1 j=1 f i,jx j x i be a liea vecto field o C. It acts by deivatio as a liea map fom P δ to itself. Accodig to Lemma 2, the adjoit of this opeato w..t. Belitskii scala poduct is the deivatio F := ( f i,j x i j=1 i=1. The adjoit w..t. the scala poduct (2.1 is the same. Let Ñ be a ilpotet liea vecto field i C. Let us fist ecall the fudametal esult that liks the sl(2, C-tiple actios ad adjoit opeatos with espect to the scala poduct. Popositio 2 ([6]. Assume that Ñ is egula. The thee exists a liea chage of coodiates L such that N = L Ñ ad its adjoit N geeates a sl(2, C-tiple. Moe pecisely, the diffeetial opeatos X = N, Y = N ad H = N N NN, actig o gems of holomophic fuctios o fomal powe seies, fulfill the sl(2, C-elatios fom fomula (2.5. Futhemoe,N acts o vecto fields of V V δ, δ 2, as d 0,δ (V :=[N,V ], whee [.,.] deotes the Lie backet of vecto fields. The its adjoit d 0,δ w..t. to the scala poduct (2.2 is d 0,δ (V :=[N,V] ad the tiple X = d 0,δ, Y = d 0,δ, H =[d 0,δ,d 0,δ] satisfies the sl(2, C-elatios fom fomula (2.5.

11 420 STOLOVITCH, VERSTRINGE The opeato d 0 is called the cohomological opeato ad we shall defie the box opeato of degee δ to be δ := d 0,δ d 0,δ. (2.4 We ecall the mai fomal omal fom esult: Popositio 3 ([22]. Let X := N + R 2 be a oliea holomophic (o fomal petubatio of the ilpotet liea vecto field N i a eighbohood of the oigi of C. The thee exists a fomal diffeomophism ˆΦ taget to the idetity that cojugates X to a fomal omal fom, that is, ˆΦ X N Ke ad N. Moeove, thee exists a uique omalizig diffeomophism Φ=Id+U such that U has a zeo pojectio o the keel of d 0 =[S,.]. Defiitio 1. The omal fom of X obtaied by cojugacy of the uique omalizig diffeomophism Φ=Id+U such that U has a zeo pojectio o the keel of d 0 =[S,.] will be called the omal fom of X sl(2, C-epesetatios Thoughout this sectio we wok withi the followig settig. We coside a epesetatio {X, Y, H} of sl(2, C actig o a fiite-dimesioal vecto space V, that is, liea opeatos of V ad satisfy the elatios [X, Y ]=H, [H, X] =2X, [H, Y ]= 2Y. (2.5 Such a family {X, Y, H} is also called a sl(2, C-tiple. The mai defiitios ad popositios of this sectio ca be foud i [4, 31]. Defiitio 2. Let {X, Y, H} be a sl(2, C-tiple actig o a fiite-dimesioal vecto space V. A ozeo vecto b such that Xb =0ad Hb = λb is called a pimitive vecto. The coespodig eigevalue λ is called a weight. We use the followig lemma fom sl(2, C-epesetatio theoy: Lemma 7 ([31, IV-3]. Let {X, Y, H} be a sl(2, C-tiple actig o a vecto space V. Letb be a pimitive vecto. Let us defie e m := Y m b m!, v m := Y m b, m 0 ad e 1 := v 1 := 0. The followig popeties hold: 1. H(e m =(λ 2m e m, 1. H(v m =(λ 2m v m, 2. Y(e m =(m +1e m+1, 3. X(e m =(λ m +1e m Y(v m =v m+1, 3. X(v m =m (λ m +1v m 1. Poof. Let x be a abitay vecto of V such that Hx = λx; the HY x =[H, Y ]x + YHx= 2Yx+ λy x =(λ 2 Yx. As a cosequece, we obtai He m =(λ 2m e m ad Y (e m =(m +1e m+1 follows fom the defiitio of e m. Fially we fid that mxe m = XY e m 1 =[X, Y ]e m 1 + YXe m 1 = He m 1 +(λ m +2Ye m 2 = m (λ m +1e m 1. The fomulas fo v m ae immediately deduced fom the fomulas fo e m. Coollay 1. If V is a fiite-dimesioal vecto space, the the weight λ is a itege.

12 = C m Y 1 e 0,Y m 1 f 0 =...= Cm C m 1...C 1 e0,y m f 0 =0. HOLOMORPHIC NORMAL FORM OF NONLINEAR PERTURBATIONS 421 Coollay 2 ([31, Chapte IV, Theoems 2 ad 3]. Let e be a pimitive vecto ad coside the vecto subspace W = Spa{e, Y e, Y 2 e,...} of V; thew is a ieducible sl(2, C-module. Evey sl(2, C-module ca be decomposed ito a diect sum of ieducible sl(2, C-modules. Let us also assume that the vecto space V is povided with a scala poduct.,. ad a associated om.. Assume that the opeatos X, Y ae adjoit w..t. the give scala poduct: fo each v, w V,wehave Xv,w = v, Y w, Yv,w = v, Xw. Lemma 8. The opeato H = XY YX is self-adjoit w..t. the scala poduct. Lemma 9. Let e 0, f 0 be two distict pimitive vectos of V such that e 0,f 0 =0. The the followig popeties hold: 1. Y m e 0,Y e 0 =0fo all m, N, m. 2. Y m e 0,Y f 0 =0fo all m, N. Poof. We show the fist popety by iductio. We have Ye 0,e 0 = e 0,Xe 0 =0. Let >m, the it follows fom Lemma 7 that Y e 0,Y m e 0 = Y 1 e 0, (XY Y m 1 e 0 = Cm Y 1 e 0,Y m 1 e 0 fo a cetai costat C m that depeds o m. Cotiuig this way cocludes the fist popety. We poceed with the secod popety. Remak fist that fo all 0 Y e 0,f 0 = Y 1 e 0,Xf 0 =0. Let ow m Y e 0,Y m f 0 = Y 1 e 0, (XY Y m 1 f 0 Coollay 3. Let {X, Y, H} be a sl(2, C-tiple actig o a fiite-dimesioal Hilbet space V. Assume that the opeatos X, Y ae adjoit w..t. the give scala poduct. The thee exists a othogoal decompositio ito ieducible sl(2, C-submodules of V. Poof. Accodig to Lemma 2, V ca be witte as a diect sum of ieducible submodules W i s. Each of them is geeated by a pimitive elemet b (i which is a eigevecto of H ad belogs to the keel of X. LetṼ be the vecto space spaed by these b(i s. Sice [H, X] =2X, thekeelofx is left-ivaiat by H: ifxf = 0, the X(Hf = 0. I paticula, Ṽ is ivaiat by H. The opeato H is self-adjoit. Let { b (0 Ṽ,..., b (j } be a othoomal basis of eigevectos of H Ṽ.Each b (l is a pimitive elemet ad geeates a ieducible sl(2, C-module W l := Spa{Y m ( b (l m N}. Accodig to Lemma 9, the W l s ae paiwise othogoal. Hece, V = W i = Wi. We explai ow how the om acts o the idividual ieducible epesetatios Spa{Y l (b l N}. We have the followig lemma. Lemma 10. Suppose that b is a pimitive elemet of V ad Hb = λb, the fo each m 0 we have Y m b 2 = m!λ! (λ m! b 2. (2.6 Poof. Accodig to Lemma 7, we have Y m b 2 = Y m b, Y m b = Y m 1 b, XY m b = m (λ m +1 Y m 1 b, Y m 1 b. It follows, by iductio, that Y m b 2 = m!λ! (λ m! b 2.

13 422 STOLOVITCH, VERSTRINGE Followig the poof of Lemma 3, let {b (0,...,b (j } be a basis of othogoal eigevectos of the opeato H Ke(X. Let us defie b m (i := Y m (b i / Y m (b (i fo each i N, 0 i j. Usig the same agumetatio as i the poof of Coollay 3 it follows that {b m (i m N, 0 i j}\{0} (2.7 is a basis of othoomal eigevectos of H. We study the actio of X, Y, H, XY ad YX o this omed basis, as this will be coveiet fo late use. Lemma 11. Let b Ke(X be a eigevecto of H associated with eigevalue λ. Let us set b m := Y m (b/ Y m (b, m 0. The the vectos b m satisfy the followig popeties: H(b m =(λ 2mb m, Y (b m = (m +1(λ mb m+1, X(b m = m(λ m +1b m 1, YX(b m =m(λ m +1b m, (2.8 XY (b m =(m +1(λ mb m. Poof. We use Lemma 2.6 ad compute: H(b m = H(Y m (b Y m (b =(λ 2mb m, Y (b m = Y (Y m (b Y m (b = Y m+1 (b Y m (b = b m+1 Y m+1 (b Y m = (m +1(λ mb m+1, (b X(b m = X(Y m (b m 1 Y m (b = m(λ m +1Y (b Y m (b = m(λ m +1 b m 1 Y m 1 (b Y m = m(λ m +1b m 1, (b YX(b m = m(λ m +1Yb m 1 = m(λ m +1b m, XY (b m = (m +1(λ mxb m+1 =(m +1(λ mb m. It is a classical fact that XY is self-adjoit positive semidefiite such that V =Ke(XY Im(XY =Ke(Y, Im(XY =Im(X, Ke(XY =Ke(Y (the same holds fo YX if we itechage the ole of X ad Y. 3. THE NORMAL FORM PROCEDURE: THE NEWTON METHOD We coside a gem of the holomophic vecto field at the oigi of C, V = N + R 2,whee N is its (ilpotet liea pat ad R 2 = k 2 R k,withr k V k, is a oliea petubatio of N. I this sectio, we ivestigate the cojugacy of V to a omal fom. We assume that {N,N,D := [N,N]} (ad, as a cosequece {d 0,d 0,H := [d 0,d 0]} isasl(2, C-tiple actig o each space of homogeeous polyomials P k (esp. V k,k 2. Hece, we ca apply the etie abstact sl(2, C-theoy developed i Sectio 2.4. Let us assume that, at step m, we stat with a gem of the holomophic vecto field at the oigi of the fom V m = N + f m N + R m+1. I this fomula, NF m := N + f m N is a polyomial omal fom of degee m, R m+1 := B + C whee B V m+1,2m, C V >2m ad f m is a polyomial of degee m 1 which vaishes at the oigi ad which is a ivaiat of both N ad N,thatis, N(f m =N (f m =0. At each step m of the pocedue, we ty to fid a suitable polyomial coodiate tasfomatio Φ 1 =id+u, wheeu P m+1,2m such that the diffeomophism Φ 1 =id+u omalizes V m up to ode 2m. This meas that Φ 1 cojugates V m to V 2m := Φ V m = N + f m N + B + C whee

14 HOLOMORPHIC NORMAL FORM OF NONLINEAR PERTURBATIONS 423 B V m+1,2m is a omal fom ad C V >2m. Let us set R 2m+1 := C. The cojugacy equatio eads: ( (id + DU N + f m N + B + C =(N + f m N + B + C (id + U. The pevious equatio becomes NF m + B + C + DU.(NF m +DU.( B + C = NF m + D(NF m.u + B (3.1 +(B (id + U B+C (id + U + {NF m (id + U NF m D(NF m.u}. O the othe had, we also have C = NF m (id + U NF m +(B + C (id + U B DU.(NF m + B + C. (3.2 Hece, C = 1 0 D(NF m.(y + tu(yu(y dt +(B + C (id + U B DU.(NF m + B + C. (3.3 This will be used to estimate C withi the Newto pocess. We ewite Eq. (3.1 ude the followig fom: B B +[NF m,u]+ C = (B (id + U B+C (id + U + {NF m (id + U NF m DNF m.u} ( DU B + C. (3.4 Sice U V m+1,2m, the Taylo expasio at the oigi of the ight-had side of Eq. (3.4 cotais oly tems of ode stictly lage tha ( 2m at the oigi. Theefoe, we have J 2m B B +[NFm,U] =0, whee J 2m deotes the tucatio at degee 2m of the Taylo expasio at the oigi. We ae led atually to defie the opeato d 0,m+1,2m : V m+1,2m V m+1,2m U J 2m ([NF m,u]. LetussetZ := π Im( B ad U := d 0 (V fosomev V m+1,2m. We wat to solve the iteated cohomological equatio π Im( (J 2m ([N + f m N,d 0(V ] = Z = π Im( J 2m (R m+1. (3.5 The pat that we do ot emove fom the cojugacy Eq. (3.4 is the give by B := π Ke( B π Ke( (J 2m ([N + f m N,d 0(V ]. (3.6 By assumptio, we have that B = g 2m N whee g 2m O N N ON N 2m 1. We defie NF 2m := NF m + B. The mai esult of this sectio is the estimate of the cohomological equatio. is a polyomial of degee Popositio 4. Thee exists a positive costat η, idepedet of m such that if 1/2 1 ad D(NF m N + NF m N <η, fo ay Z π Im( (V m+1,2m, the the uique solutio U = d 0 V V m+1,2m of the iteated cohomological equatio (3.5 π Im( (d 0,m+1,2m (U = Z satisfies U 2md Z. (3.7

15 424 STOLOVITCH, VERSTRINGE The ext sectio is devoted to the poof of this popositio, but we stat with some pepaatios. Recall that = d 0 d 0 is a self-adjoit, positive semidefiite opeato ad that we have the decompositio P m+1,2m =Im(d 0 Ke(d 0 =Im( Ke( togethe with Im(d 0=Im( ad Ke(d 0 =Ke(. As a cosequece, the opeato :Im( Im( :V (V is a ivetible opeato. We defie P 1 : V m+1,2m V m+1,2m ; P 2 : V m+1,2m V m+1,2m V J 2m (f m d 0d 0(V, V J 2m (d 0(V (f m N. (3.8 We also defie Q 1 := 1 π Im( P 1 Im(, Q 2 := 1 π Im( P 2 Im(. (3.9 Hece, we have π Im( J 2m ([N + f m N,d 0(V ] = π Im( ((V +J 2m (f m d 0d 0(V J 2m (d 0(V (f m N ( = π Im( id + Q 1 + Q 2 (V. (3.10 We coside the opeato id + Q 1 + Q 2 as actig o Im(. If it is ivetible, the the solutio of Eq. (3.5 is give by V =(id+q 1 + Q Z. (3.11 We show i the ext sectio that the opeato Q 1 + Q 2 is, i fact, ilpotet ad we give a boud o this opeato. This allows us to ivet ad to estimate the solutio of the equatio sice it leads to α (id + Q 1 + Q 2 1 = (Q 1 + Q 2 l. Although the degee of ilpotecy α depeds o m, we essetially show, ad this is the key step, that (id + Q 1 + Q 2 1 ca be bouded idepedetly of degee m. Such a shap boud is eeded to pove Popositio 4. l= Estimate of the Solutio of the Iteated Cohomological Equatio This sectio is devoted to the poof of Popositio 4. We stat by showig the ilpotecy popeties of seveal opeatos defied by (3.8, (3.9 actig o V m+1,2m. Lemma 12. The opeatos P 1,P 2,Q 1,Q 2,P 1 + P 2,Q 1 + Q 2 ae ilpotet. Moeove, P 1 ad P 2 commute paiwise. Poof. Sice f m vaishes at the oigi, the multiplicatio by f m iceases the ode by 1. Theefoe, the opeato P 1 is ilpotet. The opeato P 2 is ilpotet too. I fact, sice f m O N O N,we have P 2 (P 2 (V = P 2 (d 0(V (f m N =J 2m (d 0(d 0(V (f m N (f m N ( = J 2m N ( d 0(V (f m N (f m N =0. To pove that P 1 ad P 2 commute with each othe is a little bit moe iticate. We fist compute P 1 P 2 (V =J 2m (f m d 0d 0 (d 0(V (f m N = J 2m (f m N N (d 0(V (f m N = J 2m (f m N N N (V (f m N.

16 HOLOMORPHIC NORMAL FORM OF NONLINEAR PERTURBATIONS 425 O the othe had, we have P 2 P 1 (V =J 2m (d 0 (f m d 0d 0(V (f m N =J 2m (f m d 0d 0d 0(V (f m N = J 2m (f m N d 0d 0(V (f m N =J 2m (f m N (N (N (V (f m N. This poves that P 1 ad P 2 ae paiwise commutig. Sice P 1 is ilpotet, thee exists a atual umbe N>1 (depedig o m such that P1 N =0. Sice P 2 2 = 0, we have (P 1 + P 2 N+1 = N+1 ( N+1 k=1 k P k 1 P2 N+1 k =0. Let us show that both P 1 ad P 2 ae uppe tiagula ad is diagoal with espect to a well-chose basis. To this ed it suffices to wite dow the basis descibed i fomula (2.7 odeed by iceasig degee. Sice the opeatos P 1 ad P 2 icease the degee, they ae automatically uppe tiagula i this basis. Futhemoe, it follows immediately fom Lemma 11 that acts diagoally o this basis. Oe immediately deduces fom these facts that the opeatos Q 1, Q 2 ad Q 1 + Q 2 ae uppe tiagula ad hece ilpotet. We ecall that Y = d 0, X = d 0, H =[d 0,d 0]= H defies a sl(2, C-tiple actig o V m+,2m. Followig Coollay 3, we coside a othoomal basis {b (1,...,b (k } of pimitive elemets of V m+1,2m (k depeds o m so that V m+1,2m = 1 i k is a othogoal decompositio ito ieducible sl(2, C-submodules: we have, fo 1 i k, { } W i := Spa b (i, N with b (i = d 0 (b(i 0 N. d 0 (b(i 0, The set B m+1,2m := {b (i 1 i k, N}\{0} (3.12 is a othoomal basis of V m+1,2m as immediately follows fom Lemma 9. The followig lemma gives a boud fo the opeato Q 1. Lemma 13. The opeato Q 1 is bouded by 6 f m. (3.13 Futhemoe, we have Q 1 (V 6 f m V. (3.14 Poof. We ecall the defiitio of the opeato Q 1 (V = 1 π Im( f m d 0d 0 Im( (V. Sice = d 0 d 0 ad sice f m is a ivaiat of both N ad N, the multiplicatio opeato by f m ad the opeato commute paiwise. Theefoe, we have Q 1 (V =f m Q1 (V, with Q 1 (V := 1 π Im( d 0d 0 Im( (V. We split the opeato Q 1 ito two pats: d 0 d 0 ad 1 ad compute thei actios usig the decompositio ito ieducible sl(2, C-modules. I paticula, we compute thei actios o the basis (3.12. We use the theoy developed i Sectio 2.4 fo that pupose. Accodig to Lemma 11 ad usig the othogoal basis B m+1,2m, we obtai fo each j: d 0d 0b (j b (j = X 2 b (j = d 0 d 0 = YXb (j W i = (λ +2( 1 (λ +1b (j 2. = (λ +1b (j.

17 426 STOLOVITCH, VERSTRINGE Hee, λ deotes the weight of the pimitive elemet b (j 0. It follows that Q 1(b (j =0fo0 2 ad fo 3: (λ +2( 1 (λ +1 Q 1 (b (j = b (j 2 ( 2(λ +3. O the othe had, we have (λ +2( 1 (λ ( 2(λ +3 Let V = k j=1 0 V j b (j V ( m+1,2m. Accodig to the pevious computatios ad estimates, we have Q 1 (V 2 k = V j Q 1 (b (j j=1 0 2 k 6 2 V j 2 b (j V 2. j=1 0 Sice Q 1 leaves each space of homogeeous vecto fields ivaiat, we obtai as a cosequece: Q 1 (V m V 2m f m Q1 (V m V 2m This eds the poof of the lemma. 2m f m Q 2m 1 (V k = f m Q 1 (V k k c k k k=m+1 k=m+1 2m 6 f m V k k c k k =6 f m V. k=m+1 We poceed with a estimate o the opeato Q 2. Hee the thigs ted to become a little moe difficult sice the opeato Q 2 is ot actig as a ladde opeato as befoe. We ca, howeve, make use of the followig ivaiat subspace lemma: Lemma 14. The vecto subspace of vecto fields of the fom AN + BN + CH, A, B, C P m+1,2m, is ivaiat ude the actio of d 0,d 0. Poof. It is sufficiet to show that this space is ivaiat by the actio of d 0 ad d 0. We ecall that we coside the sl(2, C-tiple {N,N,H },withh = H =[N,N]. [N,AN + BN + CH ]=N(AN + N(BN + N(CH + B[N,N ]+C[N,H ] =(N(A+2C N + N(BN +(N(C B H (3.15 [N,AN + BN + CH ]=N (AN + N (BN + N (CH + A[N,N]+C[N,H ] = N (AN +(N (B 2C N +(N (C+A H. (3.16 Lemma 15. Let f m = f m x f m x. The opeato Q 2 is bouded by C 0 f m, fo some positive costat C 0, idepedet of m. Moeove, we have ( f m Q 2 (V C 0 x f m x max( N, N, H V. Poof. We ecall the defiitio of the opeato Q 2. Q 2 (V = 1 π Im( d 0(V (f m N. Lemma 14 above shows that the space of vecto fields of the fom AN + BN + CH is ivaiat ude the actio of the Lie algeba geeated by d 0 ad d 0. We compute the actio of o this

18 HOLOMORPHIC NORMAL FORM OF NONLINEAR PERTURBATIONS 427 space ad deduce the actio of 1 fo elemets of the fom B N := d 0 (V (f mn.ofcouse, we shall obtai coditios o B i ode to solve the equatio. We the use this expessio to compute a boud. Usig Eqs. (3.15 ad (3.16, we deduce that (AN + BN + CH = (NN (A+2(N (C+A N +(NN (B 2N(C N +(NN (C+2C + N(A N (B H =: A N + B N + C H. (3.17 Let D := NN be the opeato actig o polyomials. Let us set K := D +2I. I matix otatio, the opeato thus acts as D +2I 0 2N A A 0 D 2N B = B. (3.18 N N D +2I C C We wat compute the ivese of that opeato fo a vecto of the type B N =[N,V](f m N,so we set A = C = 0. Theefoe, we have KA = 2N (C. (3.19 Composig the thid lie by Y = N ad addig it to the secod lie leads to Fially, we have N ( D 2NK 1 N (C =B. (3.20 D(B =B +2N(C. (3.21 We fist decompose V alog the basis B m+1,2m as defied by (3.12: V = i, V (i b (i. Hece, we have [N,V](f m = i, V (i d 0 (b(i (f m. This motivates the followig otatio: Fo each ieducible submodule W i of V m+1,2m associated to the idex 1 i k, letλ be its weight (we omit to wite i, we shall set fo 0 λ: a( :=(λ +1. We shall use the covetio that a( =0adb (i v := d 0(b (i (f m, =0fo 0.Letusalsoset w := b (i (f m. Sice f m is a ivaiat of both N ad N,wehave N(w =[N,b (i ](f = ( +1(λ b (i +1 (f m= ( +1(λ w +1, (3.22 N (w =[N,b (i ](f = (λ +1b (i 1 (f m= (λ +1w 1. (3.23 H(w =N N NN (w =N ( a( +1w +1 N( a(w 1 =(λ 2w. (3.24 Sice the elemets w have distict weights, those which ae ozeo ae liealy idepedet [31, Popositio 1, p. 18]. Futhemoe, we have v =[N,b (i ](f m =N (b (i (f m = N (w = a(w 1. Sice we wat to ivet o elemets which ae othogoal to the keel of d 0,itissufficietto coside B N as a liea combiatio of elemets of the fom d 0 (b(i (f m N, 2. Ideed, we have d 0 (b(i 0 = 0 (by defiitio ad d 0 (d 0 (b(i 1 (f m N =0.Letuswite B = i B (i b (i (f m = i B (i w.

19 428 STOLOVITCH, VERSTRINGE LetussetV = i 1 V (i b (i V m+1,2m.siceb =[N,V](f m,accodigto(3.23, we have B (i = V (i +1 a( +1. (3.25 We ecall that K = D +2I. The opeatos D ad thus K ae both diagoal whe expessed i the basis {w }. Let us explai this i detail ad let us compute thei eigevalues. Ideed, usig Lemma 11 we have Kw =(NN +2I b (i (f m =[(d 0 d 0 +2I b (i ](f m =(a(+2w. It follows that As a cosequece, we have K 1 w = w (a(+2, Dw 1 = a( 1w 1. K 1 N w 1 = K 1 a( 1w 2 = NK 1 N w 1 = a( 1 a( Nw 2 = (D 2NK 1 N a( 1a( 2 w 1 = w 1, a( a(a( 1a( 2 N(D 2NK 1 N w 1 = w. a( Theefoe, accodig to (3.26, if a( 1 a( w 2, a( 1 a( w 1, (3.26 B (i 0 = B (i 1 = B (i 2 =0, (3.27 the Eq. (3.20 has a uique solutio C (i 1 := a( B (i, a(a( 1a( 2 3 (3.28 with C (i 0 = C (i 1 =0.Sicewehave K 1 N w = Eq. (3.19 has a uique solutio give by A (i 1 a( := 2 a( C(i ad A (i 0 = 0. Fially, we have fo 3 B (i w +2C (i 1 N(w 1 = B (i w +2C (i = = ( a( a( w 1, B (i ( 1+ 1 N(w 1 =0, 2, ( 1+ B (i = 1 a( 2 B (i +1, 2 (3.29 a( +1a( 1 +2 a(c (i 1 2(a( a( 1a( 2 w B (i w, B (i w, 3. D 1 (B (i w +2C (i 1 N(w 1 = 1 2(a( a( a( 1a( 2 Theefoe, the uique solutio of (3.21 such that B (i 0 = B (i 1 = B (i 2 =0isgiveby ( 1+ 2(a( a( 1a( 2 B (i, 3. (3.30

20 HOLOMORPHIC NORMAL FORM OF NONLINEAR PERTURBATIONS 429 Sice λ is a positive itege, we have ( λ +1 2 λ a(, 1 λ. 2 Theefoe, the umbes α := β := γ := 2 a( +2 a( +1a( 1, 2 a( +1 (1+ 2(1 + 2 a( 2 a( 1 a( a( + 1(1 + 2 a( 2, 3 a(a( 1, 3 ae uifomly bouded with espect to ad λ. Hece, sup α,sup β,sup γ ae uifomly bouded with espect to m ad i (the label of ieducible submodules of V m+1,2m. Accodig to Eqs. (3.25, (3.29, (3.30 ad (3.28, we deduce that A = α V (i +2 b(i 1 (f m i 2 B = β V (i +1 b(i (f m i 3 C = γ V (i +1 b(i 1 (f m i 3 solve Eq. (3.17 with A = C =0adB = d 0 (V (f m with estictios (3.27. Let X = j=1 X j be a polyomial vecto field. We have X j X sice X = j=1 X j 2. As a cosequece, we have X(f = f X j x j=1 j X j f =1 j X f =1 j X f. Let V V m+1,2m be ay uit vecto field. It ca be witte alog the othoomal basis B m+1,2m as V := (,i V,ib (i,whee (,i V,i 2 = 1. We compute Q 2 (V V +2,i α b (i 1 (f mn i 2 + V +1,i β b (i (f m N i 3 + V +1,i δ b (i 1 (f mh i 3 f m N sup + f m N sup C 0 f m. α V + f m N sup β V δ V Hee C 0 is idepedet of m. We have hece show that Q 2 C 0 f m. Let us pove the secod iequality. We fist have V +2,i α b (i 1 (f mn V +2,i α b (i 1 (f m N. (,i (,i

21 430 STOLOVITCH, VERSTRINGE Let us coside the polyomial vecto field X := (,i V +2,iα b (i 1. Let us show that ( f m X(f m X x f m x. Ideed, accodig to Lemma 4, we have ( f m X(f m max X j j x f m x. We have X j X sice {X j } k {X} k whee {X j } k deotes the homogeeous compoet of degee k of X j. As a cosequece, we have V +2,i α b (i 1 (f mn V +2,i α b (i 1 (,i (f m N (,i ( V +2,i α b (i f m 1 x f m x N. (,i Fo m +1 k 2m, leti k be the set of idex i fo which b (i is homogeeous of degee k (fo all. The we have V +2,i α b (i 2m 1 = V +2,i α b (i (,i 1 c k k. k=m+1 i I k Sice the B m+1,2m is a othoomal basis, we have V +2,i α b (i Theefoe, we have 2m V +2,i α b (i k=m+1 i I k c k k max V+2,iα 2 2 max α2 2m α k=m+1 i I k We obtai simila estimates fo the othe tems. The poof is complete. We ecall the cohomological equatio (3.11 V 2,i. V 2,i c k k =max α V. V =(id+q 1 + Q Z whee Z belogs to π Im( (V m+1,2m. Accodig to the pevious lemmas, we have α(m ( k V 6 f m + C 0 f m max( N, N, H 1 Z, (3.31 k=0 wheewehavewitte ( f m f := x f m x. The aim of the emaide of the poof is to obtai a uppe boud fo the om of the ight-had side of (3.31. By defiitio, we have NF m N = f m N = f m s i,j x j. x i i=1 j=1

22 HOLOMORPHIC NORMAL FORM OF NONLINEAR PERTURBATIONS 431 LetussetL i (x := i=j s i,jx j fo 1 i. Wewitef fo f m ad we decompose f = m 1 k=1 f (k as a sum of homogeeous polyomials f (k of degee k. Wehave m fl i (x 2 = L i (xf,l i (xf = L i (xf (k,l i (xf (k k=1 k=1 m 1 = f (k (L i f (k, s i,j (k +1! k=1 j=1 B m 1 = f (k, s i,j 2 f (k + L i (x (k +1! j=1 j=1 f (k s i,j. B As a cosequece, we have fl i (x 2 = m 1 s i,j 2 f (k 2 B + L i ( f (k 2 B, (k +1! k=1 j=1 whee L i ( (f k= j=1 s i,j f k. Hece, we have fl i = = m 1 k=1 m 1 k=1 L i (xf (k,l i (xf (k 1/2 ck+1 k+1 1 s i,j 2 f (k 2 + L i ( f (k 2 (k +1 j=1 1/2 N s i,j 2 f (k c k k m 1 1/2 k +1 k=1 s i,j 2 j=1 j=1 1/2 f. 1/2 c k+1 k+1 Sice 1/2 ad sice thee exists a i fo which j=1 s i,j 2 0, we fially obtai f m 2 j=1 s i,j 2 NFm N. (3.32 = f m L i We have f ml i of f, thee exists a i such that, if 1/2, the + L i (x f m. Accodig to the pevious estimate applied to f f m 2 j=1 s i,j 2 f m L i f m s i,j. istead As a cosequece, thee exists a positive costat c such that fo all 1 j ad all m 2 f m c ( D(NF m N + NF m N. (3.33 Let us choose η>0 such that (6 + C 0 c max( N 1, N 1, H 1 η <1/2. (3.34

23 432 STOLOVITCH, VERSTRINGE Usig estimates (3.32 ad (3.33 i (3.31, we fid that if 1/2 1ad D(NF m N + NF m N <η, the V 2 1 Z. Accodig to Lemma 6, sice V is a polyomial vecto field of degee 2m ad 1/2, wehave d 0V V DN + DV N md V (3.35 fo some positive costat d, idepedet of m. O the othe had, accodig to (2.8 we have 1 Z Z. As a cosequece, if 1/2 1ad D(NF m N + NF m N <η, the the uique solutio U = d 0V of the cohomological equatio (3.5 satisfies U 2md Z. (3.36 The poof is complete. 4. THE ITERATION PROCEDURE TOWARDS CONVERGENCE We adapt the poof by the fist autho i [34, Sectio 8]. Let 1/2 < 1adη>0beapositive umbe that is small eough so that coditio (3.34 is satisfied. Fo ay itege m 8/η +1 we defie { NF m ( = X V >0 max( X N, D(X N <η 8 }, { m } B m+1 ( = X V >m X < 1. Let m =2 k fo some itege k 1 ad defie ρ = m 2/m ad R = γ k m 4/m, whee γ k =(2md 1/m. Sice m 1/m 1, it is eadily veified that ρ<r< 1fomlage eough, say m m 0. We etu to the omal fom pocedue that is discussed i Sectio 3. Suppose that we have aleady omalized ou vecto field up to ode m. Ou statig poit is a vecto field of the fom NF m + R m+1,wheenf m = N + f m N is the polyomial pat of degee m of the omal fom ad R m+1 is a aalytic gem of ode m + 1 at the oigi. The followig popositio will play the ole of oe step i the Newto pocess: Popositio 5. Assume that NF m NF m (, R m+1 B m+1 (. Suppose that m is lage eough, say m m 0, m 0 idepedet of, the the uique U d 0 (V m+1,2m solutio of (3.5 is such that: 1. Φ:=(id + U 1 is a diffeomophism such that id + U R <ρ, 2. Φ (NF m + R m+1 =N 2m + R 2m+1 is omalized up to ode 2m, 3. NF 2m NF 2m (R, R 2m+1 B 2m+1 (R. Poof. Accodig to the popeties of the om used (see Sectio 2.2 ad the defiitio of the ew emaide (3.3, the poof of the fist poit, secod poit ad of the iequality R 2m+1 B 2m+1 (R of the popositio is idetical to the poof of [34, Popositio 8.0.2]. Ideed, fom Popositio 4 we obtaied the estimate U γ m k. We eed to pove NF 2m NF 2m (R. Its poof slightly diffes fom the equivalet oe i [34, Popositio 8.0.2]. We have to estimate the ew omal fom NF 2m = NF m + B as defied i (3.6. This expessio cotais a Lie backet that we eed to estimate. Accodig to (3.6, we have B = π Ke( B π Ke( (J 2m ([f m N,d 0(V ].