Attracting domains for semi-attractive transformations of Cp

Publicacions Matemàtiques, Vol 38 (1994), 479-499 . ATTRACTING DOMAINS FO R SEMIATTRACTIVE TRANSFORMATION S C]F C P MONI+QUE HAKI M Abstract Let F be a germ of analytic transformation of (C p , o) . We sa y that F is semi-attractive at the origin, if F { o) has one eigenvalu e equal to 1 and if the other ones are of modulus strictly less than 1 . The main result is : either there exists a curve of fixed points, o r F — Id has multiplicity k and there exists a domain of attractio n with k — 1 petals . We study also the case where F is a globa l isomorphism of C 2 and F — Id has multiplicity k at the origin . This work has been inspired by two papers : one of P . Fatou (1924 ) and the other one of T . Ueda (1986) . 1 . Introductio n Let F be a germ of analytic transformation from (CP, 0) to ( C P , O), Le . , a holomorphic map defined in a neighborhood of the origin in C P whic h leaves the origin o = (o, o) of C P fixed . We are interested in the behavio r of the iterates (Xn,yr j ) = F (n) (x, y) of points (x, y) near the origin . W e implicitly assume that they are defined . We study the situation wher e F~ o }has one eigenvalue equal to 1 and the others are {Á} 2 <<, wit h o < I) 9 1 < 1 . We call semiattractive such trasnformations . We want t o investigate the existence of attracting domains at o in a neighbourhoo d of O . As the partial derivative ¡-Fr(0) = 1 in some coordinate systern , the family {F () } cannot converge to o in a neighborhood of O . So b y attracting domains in a neighborhood of o, we mean open domains D with o E aD such that x as = F (n) (x) converge to o for x E D . When p = = 1, the dynamics of analytic transformations from (C, o) t o (C, o) with eigenvalue 1, i .e ., transformations which can be written wit h convergent power series in x a s F(x) = x l = x(1 + a i x + a 2 x 2 + . . . ) 480  M. H AKI M have been studied by Fatou and Leau . Their theory is quite complet e (see for instance [B]) . In his paper [F] on transformations of (C 2 , 0) Fatou investigates th e case of transformations with eigenvalues 1 and b, with 0 < i b j < 1 . H e proves the existence of a coordinate system (x, y) where F can be writte n xi = f( x , y ) = a l ( y ) x + a2(y)x 2 + . . . yi = g ( x , y ) = by + bi ( y ) x + b 2( y ) x2 + where the a 3 (y), b j ( y ) are holomorphic functions in a neighborhood o f 0 E C such that a l (0) = 1, b 1 (0) = 0, the x -coordinate being chosen i n such a way that {x = 0} is the invariant curve of Poincaré . Then Fato u shows that, if a 2 (0) 0, there exists an attracting domain at O . Th e projection on the x -plane of the dynamics is of the same type as the on e we get in C with the a j 's constant . This case has been studied again b y Ueda [U] with a better reduced form for F, which allows him to give a simple and more complete description of the domain of convergence . When y = = 0 is a curve of fixed points, the transformation F is of typ e yi = y( b + b i (y)x + b 2 (y)x 2 + . . . ) a case considered by Lattès [L] . Fatou shows that the coordinates ca n be chosen in such a way that we get the reduced for m yi = ~ J y1 -- y b+ E b ij x z y i i-I-j 7 1 So a neighborhood of 0 is attracted by the curve of fixed points alon g the trajectories x = constant . Then Fatou asks if there exis other case s for which there is no attracting domain at 0 . He asks what happens fo r instante with a transformation like 1-fxy ~ yl = by + x 2 In this paper, we will see that there is an attracting domain of 0 . W e prove the following result . x SEMI--ATTRACTIVE TRANSFORMATIONS OF C P  48 1 Theorem 1 .1 . Let F be an analytic germ of transformation fro m (C P , O) to (C v, O), w ith eigenvalues in 0 {1, {À} 2 <<}, such that o Ç I aj i < 1, f or 2 Ç j Ç p, then either there exists a curve o f fixed points o r there exists an attracting domain of O . More precisely, let Id be the identity of (C P , 0) . Either there exist s a curve of fixedpoints or F — Id has a finite multiplicity k > 2 . W e show that in the case of multiplicity k ? 2, there exists an attractin g domain D of 0 made of k — 1 attracting petals, i .e . k — 1 disjoint ope n sets {D} positively invariant by F such that 0 E aD j and that every x E D i is attracted by 0 . Conversely, if a point x is attracted by 0, fo r n big enough, xn = F (n) (x) is in one of the Di . Let us recall (see far instance [ the definition of the multiplicity o f a holomorphic transformation ~ from (CP, 0) to (C P , O) such that 0 i s isolated in the fiber (1 .-I({0}) . Let V be a compact neighborhood of 0 such that the restriction  of (I) to V is proper from V to W = ( I : . ( V ) ; let Breli) be the branched locus of  i .e . the set of z E V wher e Det(J(1}(z) } = 0 ; here J( c l)} is the matrix (o-  . Th e 1ÇiÇ 1Ç ' C multiplicity of ~ at 0 is then the number of points in a fiber (19 -1 ({(} ) for ( a point in W which does not belong to o ll(Bre : D) } . We use then th e lemm a Lemma 1 .2 [C, p . 1021 . Let <1) be a holomorphic transformation ~ from (CP, O) to (C P , O) such that o is isolated in its fiber , P -1 ({o}) . Sup - pose that the matrix a has rank p — 1 . Let C be th e 3 2CiCp, 2Ç j C p curve {z i = cp j (z 1 )} 2C j c p defined via the implicit function theorem b y 1 ' 2 = ' D3 = - • • = ~p = 0, then the multiplicity of ~ at o is equal to th e multiplicity at 4 of the function of one variabl e . DI(zI, Ç02(z1), .- , , Pp( z i)) . Proof of Lemma 1 .2 : From the relation s ad),  P , + E (p j (z 1 )  -0, for2 i p az 1  az j ji 2 on C, we get p,  a~ 1  al' . Det(J(~} i c} —  + E cp j (z 1 )  De t az1  -2  `iZj  azj 2CiÇp, 2Ç jÇ p j 482  M . HAKI M As 0 is isolated in its fiber j= 2 + =  1 ~ C  a z 1 ~ (z1) &D 1 a z . , ~ is not identically zero . Hence C is not in BO) . So to count the multiplicity, we can restrict ( to (C), and we have just to count the zeros o f 0 1 (z 1 ) = 1 . 1 (z 1 , cp 2 (z), . . . , (p p ( z 1 )} = (1 for (1 in a neighborhood of o i n C . By Rouché ' s theorem, this multiplicity is given by the order in o o f f .~ 1 . Let us for instance, compute the multiplicity of ~ = F — Id in th e example (1 .3) . According to lemma 1 . 2, we solve the equatio n y = by + x 2 and replace y in the first relation . We ge t x x l =  = x 1 - - 1-#- Is-3b  1 — b The multiplicity of F — Id at O is 4 . By theorem 1 we see that ther e exists an attracting domain at o with three petats . The result is also true if F~ o ) is not invertible . For instance the trans - formation Y] . = y 2 + x 2 where F~ o ) has for eigenvalues 1 et o, F — Id has multiplicity 6 in o . Indeed, solve y = y 2 +x 2 , we get ? . .~ f - x2 + x4 + . . . , replacing y in the first relatio n x l - x = x 2 y - - x 4 = x 6 + 0(x 6 ) . So there is an attracting domain at o with 5 petals . ■ In [U] , Ueda studies the analytic transformations of (C 2 , o) with eigen - values {1, b} at o, such that o < lb l < 1 . He calls them transformation s of type (1, b) . Then he defines a classification «1, b} k} , for k integer , x 3 SEMI-ATTRACTIVE TRANSFORMATIONS OF C'  483 1 < k Ç + oo, on these transformations . In fact, the integer k + 1 fo r type (1, b) k of Ueda is precisely the multiplicity of F — Id . Ueda concentrates his work on the case (1, b)1 . This is the case considered by Fato u when in the expression (1 .1), we have a 2 (0) O . Ueda introduces transformations of the coordinates which give simple reduced forms to stud y the attracting domain, in the case a 2 (0) O . Similar transformation s will be used here . Ueda studies also the case of a global automorphism F in C 2 . Le t F be a global automorphism in c 2 with a fixed point of type (1, b)1 , he proves that, like in the examples given by Fatou and Bieberbac h with eigenvalues of modulus strictly less than 1, the attracting domai n is isomorphic to c 2 . This is, for instance, the case for the attractio n domain of the Hénon transformatio n x l = x(1 + b) — by + x 2 , l yi = x for o <  C 1 . This statement has the following generalization . Theorem 1 .3 . Let F be a global autornorphism in C 2 with a fixe d point p, such that F' ( ' r) has eigenvalues {1, a}, with P■1 < 1, and tha t F —Id has a multiplicity k -}-1 in p . The attracting domain of p has ther i k components and each component isisomorphic to C 2 . Theorem 1 .3 applies for instance to Hénon transformation s where P is a polynomial with a zero of order k + 1 at the origin . 2 . Reduced forms of semi-attractive transformation s Proposition 2 .1 . Let F be a semiattractive germ of transformatio n of (C P , a), with eigenvalues {1, {À} 2 <<}, o Ç < 1, for 2 Ç j p . There exist coordinates (x, y), x E C, y E C P— 1 in which F has the for m { x z = u ( x , y ) = a l( y ) x + a z( y ) x2 + - . . y l = v(x,y) = g(y) + xh(x,y ) where {a( .)}, j = 1, 2, . . . , g( .) and h( ., .) are respectively germs of holomorphic functions from (GP', O) to C, from (G P ', O) to C P—1 and 484  M . HAKI M from (CP, D) to C P - 1 , with a 1 (o) = 1, g(0) = o, h(o, o) = o, and glo} i s triangular with eigenvalues {Àj}2<j<p . Proof : Let E 1 19E2 be the Jordan decomposition of C P in characteristi c subspaces . Here E 1 is associated to the eigenvalue 1 and E 2 to the set o f the eigenvalues{)j}2<j<p . There exists an analytic stable submanifol d X attracted by o and tangent to E 2 (see [R] for a sketch of the proo f and a complete bibliography) . We then just choose coordinates (x, y) , x E C, y E C P -1 , in such a way that X is {x = O} and that matrix F ~o } is triangular . ■ Proposition 2 .2 . Let F be a semi -attractive germ of transformatio n of (C I ',O) . For every integer m there exists coordinates (x, y), x E C , y E C P-1 , in which the transformation has the forr n x1 = x + a 2x 2 + . . . a 7z x m + a riz+l lyJxm+1 + . . . (2 .2) y 1 =g(y) + xh(x, y ) i .e . libe in (2 .1), but with a 1 = 1 and a 2 , . . . , a, n are constants . Remark . This proposition is in [U] in the case of a semiattractiv e invertible germ of (C 2 , o) . The following proof is just a generalization o f it Proof : We start with x i = a i ( y ) x + az(y) x2 + . . . 1 yi = g(y) + xh(x, y ) and we proceed inductively on h . 1) Reduction to a l (y) = 1 . We use the coordínate transformatio n or f s= X/u(Y ) l y = Y with u( .) a germ of analytic function from (CP', 0) to C such tha t u(0) = 1, to be chosen . We avan t X i = u ( y i) x i = u(g(y) + xh(x, y ))[ a i( y ) x + a 2 (y)x 2 + . . . ] = u(g(Y) + . ) [a l (Y)X/u(Y) + . . . ] al(Y)u(g( Y )) x + o(x 2 ) x + 0(X 2 ) . u(Y) SEMI-ATTRACTIVE TRANSFORMATIONS OF C p  48 5 So we must choose u such tha t u ( Y ) = ai(Y)u(9(Y) ) u(g(Y)) = a i ( 9 ( Y ) u ( 9 (2) ( Y )) u(g( n ) ( y)) = a l (g( n ) (y))u(g(n+l) (y)) . This gives for u the unique solutio n u(Y) = H a l ( g (n) (Y)) . Since a l (0) = 1 and since there exists a, 0 < a < 1, such that fo r small enough, one has Ilg(y)II < ailyll, so lthe infinit y product aboye is convergent in a neighborhood of 0 . 2) Suppose that for m > 2, with some coordinates (x, y), F takes th e form { x l = x + a 2 x 2 + . . .  + a rn (y)x' + . . . g(y) + xh(x, y ) with the a j 's constant for 1 < j < m – 1 . We then use a coordínat e transformatio n X = x + v(y)x'  x = X – v(Y) .K m + . i Yy o r _  i y _ Y with v(y) a holomorphic function in a neighborhood of 0 in C p-1 suc h that v(0) = 0, v to be chosen . We ge t X i = x i + v(y i )x i = x + a 2 x 2 + . . . ¢,,,, - ixi`-i + a ,,,,(y ) xm + v ( 9 ( y )) x ' +O ( x m+l ) = X –v(Y)X m +CLZX 2 + . .+a ni (y)X m + v(g(y))X' + O(X' +1 ) . So we need tha t v(Y) – v(g(Y)) = a,(y) – a m (0) , v(g(Y)) – v(g 2 (Y)) = a m (g(y)) – a m (0 ) v ( 9 n ( Y )) – 1J(9n+1(Y)) = a m ( 9 n (y)) – a m( 0 ) • The unique solution is the n v(y) = E {a,n(gn(y}} — a, n (0)} . n= 0 The series converges in a neighborhood of 0 because g is contraction an d because a m (y) — am(0) = 0 for y = 0 . ■ 486  M . HAKI M Proposition 2 .3 . Let F be a semi -attractive germ of transformatio n of (CF, O), such that F --- Id is of multiplicity k + 1 in O . Then th e transformation can be written in sorrze coordinates (x, y), x E C, y E CP - 1 Y], = g(y) + xh(x, y ) x i = x(1 + x k + Cx 2k + a2k+i(y)x2k+i + ) (2 .3 ) with C a constant . Prod .. Assume then that the transformation is written in th e form (2 .2) with m ~ k + 1 . We want to evaluate the multiplicity o f F — Id at O . Since (4- . 1 — g ' )(o) is invertible, we can use lemma 1 .2 . Using the implicit function theorem, we can solve locally in y = y(x ) the equation y = yl . The multiplicity is then given by the order at th e origin of xl — x = a 2 x 2 + . . . amx m + am+l (y)x m +1 + . . . Since F — Id is off multiplicity k + 1 in 6, we have a l = • • • = a k = 6 and a k + 1 O . If we use proposition (2 .2) with m = 2k + 1, we ge t x l = x(1 + ak+ l x k + ... + a2k+1X2k + a 2k+2(y) x2k+l + . . .) . As in the case of onevariable, a polynomial transformation in the singl e variable x leads to the required form . The coefficient of x k +l can be a n arbitrary constant not equal to 6, the coefficient of x 2k+1 is then fixe d (see for instance [B, theorem 6 .5 .7, page 1221 . ■ 3 . Existence of attracting domain s and curve of fixed point s We will now prove the . theorem 1 .1 stated in the introduction . Let F be a semi -attractive germ of transformation of (C,O), given as befor e in the form f x l = u ( x , y) = x(1 + a2(y)x + ... ) y 1 = v (x, y) = g(y) + xh(x,y ) with x E C, y E C P—1 and g, h like in proposition(2 .1) . To find th e fixed points of F, one can first solve locally in y = y (x) the relatio n y = v(x, y), thus obtaining an analytic curve y = (p(x) . There exists a curve of fixed points if and only if the relatio n x = u(x, (x) ) is also satisfied . If not, F — Id is of finite multiplicity, and the multiplicit y is given by the order at 0 of x — u(x, cp(x)} . The following corollary is an answer to a question of Fatou [F, pa - ge 131] } who asked if for such F, there could exist a curve of fixed point s through 0 for some iterate F e ' } but not far F . SEMI-ATTRACTIVE TRANSFoRIV .IATIONS OF C P  48 7 Corollary . Let F be a semi -attractive germ of transformation o f (C,O) . There exists a curve of fixed points through Q for an iterat e F (n) of F if and only if there is one for F . Proof : It is an immediate consequence of proposition (2 .3), for if F i s in the form (2 .3), we hav e xn—x ( 1+nx~ + . . . ) , so F (n) — Id has the same multiplicity as F — Id . ■ The proof of theorem 1 .1 is then a consequence of the following propo - sition . Proposition 3 .1 . L et F be a semi -attractive germ of transformatio n of (C,O), of multiplicity k + 1 in o, then there exists an attractin g domain with k petals . Proof : We can suppose that F is in the form given by proposition (2 .3 ) x1 = x(1 + a k x k + a2kx2k + a 2k+1 (y)x2k+1 + . . . ) Y] . = g(y) + xh(x, y ) with a k O . Changing x in some ax one can assurne a k _ — ~ . We ca n then imitate a Fatou's method simplified here by using the reduced for m for F which gives easily the Abel -Fatou invariant functions . Let R and p be positive constants to be adjusted later . The hai f complex -plane P R and the subset Y R ,P of C x C p—1 are defined b y P R = { XE C ; Re X > R} , V R,p - {fix, y) E C X C p - 1 ; X E P R , II y II < P} . Let D R and U R,P be the images of P R and V R,P by the inversion z = + , so we have D--- z E C ; z — 1 C 1 R i  '  2R 2 R UR ,P -- {(z,y) E C x C p-1 y z E D RS IIyII < p} . There are k branches of z 1/k in D R . Let { A Ri }O<j<k -1 be the images o f D R by these determinations . We will show that, for R big enough an d p small enough, the domain s (3 .2) W R,P,j = {(x,y) E CxC p—1 ; x E OR~~ Ilyii < P}, 0 j < k—1, 494 M . HAKI M Define B = Sp (W ) . Let us consider the diagram obtained by restrictio n of the preceeding one to 99 -1 (B), that i s (p-1(B)  " D/(F ) ç0 l  Ç z, c o As follows from proposition (5 .1), we know that C =  (B—n), so tha i n= 0 the restriction E B :B~C * is surjective . For s E B, Ueda defines a holomorphic family of holomorphic functions : ( p — 1 (s) --4 C on the fibers of cp which gives t o D/(F) —> C* a fiber bundle structure with fibers isomorphic to C an d with transition group the additive group of holomorphic functions on C* . This fiber bundle structure is necessarily trivial because H 1 ( C *, O) = O . Lifting this structure to ço : D —} C by E, we get a trivial fiber bundl e structure on D and this gives an isomorphism from D to C 2 .  The definition of the  is obtained by integrating on the fiber s s = Constant a holomorphic differential 1 -form satisfying a functiona l equation, that we have now to define . ■ Definition of a holomorphic family of 1 -forms on W . We will define on W a family of holomorphic differential 1 -forms {w 8 } on the fiber of the Abel Fatou function depending holomorphically on s , of the form w s ( p ) = n( s , y) dy = n( p ) dy , with ri a holomorphic function in B = c p(W ) satisfying the functiona l equation ay , (5 .8)  n( F (P))  (P) = n(P) - We notice that an+1  = (5 .9)  ~ ~ p }  a ( F ( P))—(P) .  ay  y  y So that if the sequence áyy were uniformly converging, its limit woul d be a good candidate for r7 . But this is not the case sinc e a Y n  a~ .~ j, I a~ .~ j . 2  a7 .J . n . ay  ay cy1 . . n-1  I  g =bn~(  b k 1  L  Sh l 1+ b h+ . . .+ bs +0 Y( so y + l )) ' h SEMI-ATTRACTIVE TRANSFORMATIONS OF C P  49 5 So we will replace the sequence ay by a sequence {hQ }n ay wher e {hn,} is a sequence of holomorphic functions such tha t (5 .10)  h n( F ( p )) = h n+l(P) , and {hn  ~~ is uniformly convergent . We can ta p e h n (p) to be of the for m h ra( p ) = 9(S)u(s)u(s1) . . .2l,(sn ) with g and u depending only on s, holomorphic in W such tha t (5 .11)  g(s)u(s) = g(s i ) and /  1 (5 .12)  u(s) = b -1 I 1 + b s + .. . + bb s) -  1 + O( 1 0+ i The condition (5 .11) implies indeed (5 .10) and the condition (5 .12) implies tha t hn(P) ay . = n h=o ( 1+o yHk+ 1 (  )/ = H h=o (1+0 \~ g/k / ) So that rl will be defined by a uniformly convergent infinite product . We have then to show that there exists a function g holomorphic in B such that u(s) = g (s i )/g( s ) satisfies (5 .12) . We define g as a product o f three functions g  •929 3 where g i (s) = b -sk (for any choice of log ó) . In fact, this give s gl(sl) =b - 1 9i( s ) We then choose a function g 2 satisfyin g 92(si)  ) ' +o() . (5 .14)  9z (s)  b s  b Sk i  ■ The existence of g 2 is proved by the following lemm a (5 .13) 496 M . HAKI M Lemma 5 .3 . Por any (c i , c 2 , . . . , c k _ 1 ) E C kl , t here exists a polynomial in 1 5 1  a l  a 2  a ~- 1 s  s  s 2  s - 1 such that h(s) = exp (s k P (s)) satisfie s h(s i)  C1  C2  Ck—1  1 h(s) - 1 + S + SZ + . . . + Sk_1 + O S ~ Proof .• From = s k +1 , we get 1  1 1 1 +~¡ 1 Sl — s k S k+i  ~ S z/c+l J so for a polynomial P 1  1  1 1  , 1 P VSi) P s  k sk+i P (-s) + ~ ( S a p + i 1 From that we deduc e S1PC  - S k PC)- Sk CPC  - PC)) + P S  S P s) ks ( S) +O ( - S k 1 +1 ) k—1 ( 1 = E i=l ) a i+ O ( k ) . We can then compute the a j 's by identifying the expression aboye wit h the polynomial part of degree C k - 1 in the Taylor expansion o f - Log (1 + Çl -f- ~2 + . . . + ~~ i 1 s  s  s - at infinity . ■ Choice of g 3 . It is a consequence of the choice of g l and g 2 that i t exists c such tha t 9i( S i) 9a(si)  b l 1  b k 1  c  1 9i ~ S ) 9z (s )  b 1 + b s + . . . + b s k - 1 + sk + 0 s k+i SEM1-ATTRACTIVE TRANSFORMATIONS OF C P  49 7 So we need a function g 3 such tha t g 3 (s i ) — 1 _ c 0   1 s~ +  8 k-F 1 g 3 (s ) We can take g 3 defined by g 3 (s) = s -kc = exp( - c Lo g s k ) (with th e branch of logarithm which is real on positive numbers) . Indeed, we ge t g 3(si)  e  1 = (i+_c=i_+o ( s~  s c  s k+ 1 Os ) Once w, is defined in W, we just follow the construction of Ueda t o define the ~~ . We recall this construction for reader's convenience . Construction of ~s : cp -1 (s) —> C . We have seen that if p E D, for n big enough, s = F n (p) is in B and the fiber D n Sp -1 (s) contains a disk q p = {y E C ; lly~~ < p} . So w e can fist suppose that W is a set of the form W = B x q p and that w e still have D = ~ J F -n (W) . VLTe first define 'IbS (p) when p is in W and n= o cp(p) = s by integrating the form (p) dy on the fiber s = constant, alon g a path joining po = (s, o) to p = (s, y(p)) in {s} x q p y (P ) ~s ( p ) =  r~(s, y) dy . o From the functional equation (5 .8) verified by r) we ge t y ( F (P) ) ( F (P)) = n(81,y1)) d y 1 f  y ( F (P)}  y ( F (Pa) } 71(s 1 ,y 1 )dy 1 +  ~ 1( s 1 , y ) d y  y(F( P ~ }}  o = y~s (p) + h(s ) where h(s) = f o Y('(P0)) ,r 1 ( s 1 i y) d y is a holomorphic function of s in B . We get in this way a holomorphic function ~ : W —> C defined by IP(p) = O s (p) for s = (())1/k such tha t  (5 .15)  'K p } =  (p) =  (f (p)) - - h ( s ) . The relation (5 .15) allows the extension of the definition of ~ to ço -I (B ) in the following way : let p E (p -1 (B) . For n big enough, we have F n (p) E W and we define ~~p} by the formul a  (5 .10)  ~( p ) _ I P( Fn ( p )) - (h(s) + h(s i ) + . + h(s n ) } where s i = (ço(fa (y))) 1 / k for j = O, 1, . . . , n . It is clear that the definitio n doesn't depend on n and that the function is holomorphic in (p R 1 ( B ) . 498  M . HAKI M Proposition 5 .4 (Ueda) . For all s E B,  : ço -I (s) —> C is a n isomorphism . Proof Let us prove first the injectivity of Let p and p ' E c,o -I (s ) such that (p) _ (p') . According to property (5 .16), one can assum e that p and p ' are in W and that we have with s big and y(p) and y (p' ) small y (P)  y ( P ' ) n(s, y) dy = f  r l ( S , ~ J) dy . 0 The results is just a consequence of the fact that the functio n f o y(P) n(s, y) dy is holomorphic in y(p) and has a derivativa n( s a o) O . To prove the surjectivity, we remark that rl(s, o)_ b - sk s -- k c when s — > oo and as h(s) = Jo y(F( P a)} , y) d y with y (F (po )} = y 1 (s, o) _ bks+k+  o) + . (see the form of F in coordinate s (s,y)) . VLTe have h(s)_77(s, D b k + 1 (o ) )  k s + 1 W e deduce from this that the image O s (W) contains a disk centered a t o with radius Epl b -sk s -kc i for some constant E > O . The relation (5 .16 ) then implies that the image of F -n (W) n ç o -I (s) contains a disk centere d in Cn = h(s) + h( s i ) + • • • + h( s n ) and with radius Rn = Epl b —s n k s n -kcI . An elementary calculation proves that R n et Itend to +oo whil e -4 o, so the union of the disks D l(n, R n ) contained in the image o f R n ~-i (s) is equal to C . It is then easy using the 's to define a structure of locally trivial fibe r bundle on D/(F) —> C*, with fibers isomorphic toC and with structur e group, the group of translations by holomorphic functions in s . Thi s ends the proof of the existence of an isomorphism from D to C 2 . ■ . Reference s [B] BEARDON A . F ., "Iteration of rational functions," Springer, Ne w York, Berlin and Heidelberg, 1991 . [C] CHIRKA E . M ., "ComplexAnalytic sets," vol . 46, Kluwer Academic Publishers, Dadrech, Boston, London, 1989 . Translation i n English . SEMI--ATTRACTIVE TRANSFORMATIONS OF C P  49 9 [F] FATOU P ., Substitutions analytiques et equations fonctionnelles á deux variables,An . Sc . Ec . N. (1924), 67142 . [R] RUELLE D ., "Elements of differentiable dynamics and bifurcatio n theory," Academic Press, Boston, 1989 . [U] UEDA T ., Local structure of analytic transformations of two complex variables I, J . Math . Kyoto Univ . 26(2) (1956), 233-261 . Département de Mathématique s Bat . 42 5 Unité Associée CNR S Université de Paris -Su d 91405 Orsay Cede x FRANC E Rebut el 5 d'Octubre de 1994