Stability of parabolic points of area preserving analytic diffeomorphisms

Pub . Ma . UAB N° 22 No . 1980 Ac es VII JMHL STABILITY OF PARABOLIC POINTSOF AREA PRESERVING ANALYTIC DIF- FEOMORPHISMS Ca lesSimó Facul a de Ma emi iques Uni e si a .d e Ba celona Abs ac . Theo ems cha ac e izing s able pa abolic poin sa e p o ed . Essen- ially, s abili y is equi alen o he ac ha he gene a ing unc ion o he di e omo phism, akingou he pa which gene a es he iden i y, has a s ic ex emum a he ixedpoin . Wi h hese esul s, he s udy o he s a bili y o ixed poin s o analy ic a ea p ese ing mappings (APM) is ended . Some examples a e included, specially' he case o ellip ic poin s whose el-- gen aluesa e cubic o ou h oo s o uni y . 51 . In oduc ion and esul s . Le T an analy ic APM . The (Lyapuno ) s abili- y o ixed poin s o T is a me hod usually employed _o he s udy o he qua li a i e p ope ies o pe iodic o bi s (P . O .) in hamil onian sys ems wi h wo deg ees o eedom, ia he Poinca é mapping wi h espec o_a su ace ans e sal o he P .O . in he ene gy le el H =h . I he ixed poin , ha we ake as he o igin, is hype bolic, he ines abili y o he linea pa emains when nonlinea e ms a e aken in o accoun . I he ixed poin is ellip ic he s abili y o he linea pa is p ese ed p o ided ha he éigen alues a e no hi d o ou h oo s o uni y and ha sui ablecoe icien s o he Bi kho No mal Fo m (B.N .F .) a e no ze o . When he ixed poin is deg_e ne a ed o pa abolic, i .e ., Spec DT(0)C{±1}, he s abili y is a mo e sub le ques ion . I is no alwaysenough o conside only he lowe deg eenonli- - nea e ms o decide abóu s abili y . Besides he cases A 3 =1 and 14 = 1, di-- icul ies can appea o e e y A, k- h oo o uni y i all he de e mined coe icien s ( he i s [ k 2 2 ] ones) in he B .N .F . a e .ze o . Example s o ines abili y exis o all k [91 . The case o A being a k- oo o uni y is edu- ceo o ne pa abolicone axlng '1' ins eaa o= 'l . wl nou - loss OT :j- : .c alll` we can suppose ha in he pa abolic case he eigen alues a e equal o one . (Take T2 i necessa y . This accoun s also o T o ien a ion e e sing) . In [111 he ollowing esul s a e p o en o he,pa abolic casewhen DT(0) 0 can no be educed o diagonal o m, i .e ., DT(0) =( ;) in a sui able basis : 1 .1 . Lemma . Le T(x,y) = (x + (x,y), x+y+g(x,y)) be an analy íe APM uwí h ,g beg .ínn,¿ng wi h . enme oj degnee a .Leas wo . Then heAe ex,í,a 6 a neah . he .íden, c y polynomía .2 change o6 a~u :ables c such . ha xhe ms6onmed mapp .íng T* =c -1 Tc  .íl6 g .í en by T* (x, y) = (x+F n (x,y)+on+1 , x+y+O n+1 )whexe Fn .c,s a  de- gnee n polynom .í .a e . w hou P¿nean . enme and os e and6 6oA a &eA,íeh wí h zeAme o6 .1' oweA degAee a keas s . 1 .2 . Theonem . In he hypo hee .í s o6 1 .1 l e P n (z) = amzm+0m+1' a ,~ o . Then he o ígín .íes e~e undeA T* (and heAelo e unde T) í66 m íz odd and a m <O . The objec o he communica ion is o gi e a heo em cha ac e izing  he s able pa abolic poin s o he emaining case, i .e ., when DT(0) can be pu in diagonal o m . (Then T is nea he iden i y in a neighbou nhood U o he ixed poin ) . Le (x',y') = T(x,y) a canonical mapping .  I Dy y' is egula  (as happens in ou case) we can de ine an analy ic gene a ing unc ion (see [11) G(x,y') such ha G(x,y') = xy' +G (x,y')  and x'= D  y= D C . Fo he nondianonal yx case o  1 .2 . we ge G(x,y')=-x 2 /2 + Fn (u) du +0 n+2(x,y') . Theo em 1 .2 can 0 be e o mula ed as : s abili y is equi alen o G(x,y') ha ing a s ic ex e mum a he o igin . Tha his cha ac e iza ion is applicable o he diagonal case is s a ed in he main esul : 1 .3 . Theonem . Le Pbe a paAabolc :c 6 .íxed po,íw o6 an analy í .c APM, T,  and G(x,y') =xy' +G(x,y') a geneAa í,ng 6unc c :an 60A T . Then P Zb Lyapuno 6.2e i« Ghay a a h,íc ex xemum a P . ones o McGehee [71 bu only o he conse a i e case . §2 . Ske ch o he p oo . Only he case DT(0) =  1 0 ~0 1/ o 1 .3 . emains o be p o ed . Ins ead o using he ac ha G has a s ic ex emum a he o igin we can equi alen ly conside ha , nea he o igin, he se s G=g wi h Ig1 small and sui able sign, a e closedcu esa ound he o igin . The algo i hm o decide whe he o no G has a s ic ex emum a he o igin using he New on polygon is de e ed o he nex sec ion . 68 As a as ins abili y is conce ned he esul s ob ainedhe e ex end he Le (D 1 be he ime uni low associa ed o he hamil onian sys em wi h ha mil onian G : m 1 (x,y) = (x,y) . We in end o use m1 as an app oxima ion o T in U . By he way, i G, iis he pa ialde i a i e o Gw. . . he i- h a gumen , G i , Gi , k , . . . he second, hi d, . . . pa ial de i a i es, be e app oxima- ) 7 ions o T can be ob ained wi h modi ied hamil onians : H = G - 2 G 1 G 2 + 112 (G11G2 + 4G 12 G 1 G2 + G 22 G~ ) - 6 (G 112G 1 G2 + G 122 G 2 G2 + G 11 G 12 G2 + + G 22G12 G 2+ G11G22G1G2 +3G 2 2 G 1 G 2 )  + . . . We ge inc easingapp oxima ion aking e ms o inc easingo de . Howe e H=G is enough o he p oo . In U - (01  we de ine =G(x,y), a=2n /T( )whe e T( ) is he pe iod o he low o hamil onian G along he closed cu e Y=1G(x,y)= I . He e s ands he ime in e al in going om (x 0 ,0) o (x,y) along Y, wi h x0 >0  (one shows ha Yis s a -shaped w. . . he o igin i I l is small enough) . he ( ,a) a iables one has m 1 ( ,a) = ( ,a+2n/T( )) . A d T( )/d = 0( p), /3<0 . The e o e p essed in he ( ,a) a iables as T( ,a) = ( +¿á , a +2n/T( )+,áa), begin wi h e ms o ela i e high o de . Hence T can be seen as wis [81 and his gua an ees he exis en e o in a ian cu es om he s abili y ollows . 0 1 is a wis . The In compu a ion gi es ini ial map can be whe e A ,Ja a pe u bed whe e I G=g, Ig1 small, does no de ineclosed cu es in U bu G=0 has al b anches h ough he o igin hen we ge ins abili y unde 01 [61 he e o e,unde T . Comple e p oo sappea in [121 . o ex se e-- and, §3 . An algo i hm o decide abou s abili y . Fi s we plo he New onpoly-- gon associa ed o G . A necessa ycondi ion o s abili y is ha all he e icesha e e en coo dina es . Le m+ka, n-k/3, a,PEZ + , g .c .d . (a,P)=1, m+ka k=0 : be poin s in one side o he polygon . The e we ge G= . . .+Y- a k x ,y n-k~3 + . . . . Le (P=  akzk  0 . An addi ional necessa y condi ion is ha all 0 he eal ze os o (p be o e en mul iplici y . I he mul iplici y is ze o his is enough o s abili y . I i is no ze o, h ee cases a e posible, associa ed o each o such ze os :  y=0 (x) ;  x = 0 (y a ) , a > 1 ;  y = 0 (x a ) , a> 1 . The i s and second cases can be educed o he hi d one h ough a o a ion o a elabelling o he axes, espec i ely . The e o e, we can suppose y = mx p/q + . . . ,  p/q >l .  In oducing x =u q , y = mu p +z, we ge a new New on polygon and we p oceed o he analysis o e ms o he o m z = 0(Up),P>q . §4 . Some examples . a) We conside he case ,1 a ou h oo o uni y . A simple map is T(x,y) = _ (-y,x+ (y)) (a de Jonquié e map, no mal o m i Tis a C emona map o p i me deg ee [31 ) . Taking T 4 we can apply 1 .3 and §3 . I begins wi h e ms o deg ee k and k is odd, he o igin is s able . I k is e en and has only one e m (yk a e scaling) he o igin is s able . This is he case o he classical Hénon map [5,101 wi h o a ion angle a =7 /2 (k=2) . The in a ian cu es a e la in c oss shaped . Howe e , highe o de e ms can p oduce ins- abili y . Fo ins ance, T(x,y) = (-y, x+y 2 +ay 3 ) is uns able o a E[-1,0) .See [121 . b) I 11 is a cubic oo o uni y and we es ic ou sel es o T(x,y) _ = R2,c/3c) (x,y -x k ) , whe e Rá is a o a ion o anggle S a ound he o igin, we ge s abili y (uns abili y) i k is odd (e en) . c) Conce ning he es ic ed h ee-body p oblem, he s abili y o 4 o ma sses ,a equal o he c i ical alues o Rou h (see [21) only he alues ~¿2,u3 emain o se le he ques ion . Wi h he help o some leng hly compu a ions he esul s will appea elsewhe e [131 . O he applica ions o he s a bili y o bi u ca ion o bi scan be ound in [41 . Re e ences [11  A nold, V . I ., A ez, A . :"E godic p oblems o classical mechanics", [21 [31 [41 [51 [61 [71 [91 min, 1968 . Dép i , A ., Dép i , A ., As on, J . 72 (1967), 173-179 . Engel, W ., Ma h . Ann . 136 (1958), 319-325 . Gómez, G ., Doc o al Disse a ion, Uni . Au ónoma Ba celona, 1980 . Hénon, M ., Qua .Appl . Ma h . 27 (1969), 291-312 . . Le sche z, S . : "Di e en ial Equa ions : Geome ic heo y",Wiley, McGehee, R ., J . Di e en ial Eq . 14 (1973), 70-88 . Mose , J . : "S able and andommo ion in dynamical sys ems", P ince on Siegel, C .L ., Mose , J .K . : "Lec u es in Celes ial Mechanics", Sp inge , 1971 . [101 Simó, C ., (111 Simó, c ., [121 Simó, C . , [131 Simó, C ., Uni . P ess, 1973 . Ac as V Reunión Ma emá icos Exp esión La ina, Palma 1978, 361-369 . P oceed . In e n . Con . GlobalTheo y o Dynamical Sys ems,Ch_i cago 1979, o appea . o appea . o appea . Benja 1963 .