ON THE HYPERBOLICITY OF THE FEIGENBAUM FIXED POINT 3 0 0 DANIELSMANIA 2 n a Abstract. WeshowthehyperbolicityoftheFeigenbaumfixedpointusingthe J inflexibility of the Feigenbaum tower, the Man˜e-Sad-Sullivan λ- Lemma and theexistenceofparabolicdomains(petals)forsemi-attractivefixedpoints. 1 1 ] S 1. Introduction and statement of results D . Let g: U → V be a quadratic-like map. This means that g is a ramified h 0 0 t holomorphic covering map of degree two, where U and V are simply connected a domains,U ⋐V. Wealsoassumethatthefilled-inJuliasetofg,K(g):=∩ g−nV, m n is connected. We say that g is renormalizablewith period two if there exist simply [ connectedsubdomainsU ,V sothatg2: U →V isalsoaquadratic-likemapwith 1 1 1 1 1 connected filled-in Julia set. v Two quadratic like maps h: U → V and g: U → V , both with connected h h g g 8 filled-inJuliaset,definesthe samequadratic-likegermifK(h)coincideswithK(g) 1 and h coincides with g in a neighborhood of K(g). If g is renormalizable, then 1 the renormalization of the germ defined by g is the unique quadratic-like germ 1 0 defined by the normalization of any possible induced map g2: U1 → V1 which are 3 quadratic-like maps with connected filled-in Julia set (normalize the germ using 0 an affine conjugacy, setting the critical point at zero and the unique fixed point / h in K(g1) which does not cut K(g1) in two parts, the so-called β fixed point of g1, t to 1). The operator R is called the Feigenbaum renormalization operator. In the a m setting of quadratic-like germs which have real values in the real line, there exists an unique fixed point to the Feigenbaum renormalization operator (proved by D. : v Sullivan: see also [McM96]), denoted f⋆ (it is a open question if this is the unique i fixed point in the set of all quadratic-like germs). X It is a consequence of the so-called a priori bounds [Lyu99] that we can choose r a a simply connected domain U, K(f) ⊂ U, so that, if B(U) denotes the Banach space of the complex analytic functions g, Dg(0)=0, with a continuous extension to U, provided with the sup norm, and B (U) denotes the affine subspace of nor the functions g so that g(1) = 1, then the fixed point f⋆ has a complex analytic extension which belongs to B (U) and there exists N so that the operator RN nor canbe representedas a compactoperatordefined in a smallneighborhoodof f⋆ in 2000 Mathematics Subject Classification. 37F25,37F45, 37E20. Key words and phrases. renormalization,parabolicdomain,petals, Feigenbaum, universality, semi-attractive,hyperbolicity. This work was partially supported by CNPq-Brazil grant 200764/01-2 and University of Toronto. IwouldliketothankIMS-SUNYatStonyBrook,UniversityofTorontoandspeciallyto M.LyubichbymywonderfulstayatStonyBrookandToronto. IamalsoindebittoM.Lyubich bytheusefulcomments. 1 2 DANIELSMANIA B (U). Moreprecisely,thereexistsalargerdomainU˜ ⋑U andacomplexanalytic nor operator R˜: B (f⋆,ǫ) → B (U˜) so that, if i denotes the natural inclusion Bnor(U) nor i: B(U˜)→B(U), then RN =i◦R˜, where the equality holds in the intersection of the domains ofthe operators. To simplify the notation, we willassume thatN =1 and identify R with its complex analytic extension in B (U). nor Two quadratic-like maps g and g are in the same hybrid class if there exists 0 1 a quasiconformal conjugacy φ between them, in a neighborhood of their filled-in Julia sets, so that ∂φ ≡ 0 on K(g ). Note that quadratic-like maps in the hybrid 0 class of f⋆ are infinitely renormalizable. We will provide a new approach to the following result: Theorem 1 (Exponential contraction:[McM96] and [Lyu99]). There exists λ < 1 so that, for every quadratic-like map f which is in the hybrid class of f⋆, there exist n = n (f) and C = C(f) > 0 so that Rnf ∈ B (f⋆,ǫ), for n ≥ n , and 0 0 B(U) 0 |Rn0+nf −f⋆|B(U) ≤Cλn. A major attractive of this new proof is that it is essentially infinitesimal and has a “dynamical flavor”: we will prove that the derivative of the renormalization operator is a contraction in the tangent space of the hybrid class (the contraction ofthe derivative ofthe renormalizationoperatoronthe hybridclasswasprovedby Lyubich [Lyu99], but his proof is not infinitesimal). Moreover, the method seems to be so general as the previous ones: it also applies to the classical renormalization horseshoe [Lyu99] and the Fibonacci renormalization operator [Sm02a], for instance. We will also obtain, as a corollary of McMullen theory of towers [McM96], the local behavior of semi-attractive fixed points [H] and an easy application of the λ-lemma [MSS] that Theorem 2 ([Lan][Lyu99]). The Feigenbaum fixed point is hyperbolic. The reader will observe that we assume the Feigenbaum combinatorics just to simplify the notation: the argument in the proof of Theorem 2 works as well to prove the hyperbolicity of real periodic points of the renormalization horseshoe. 2. Preliminaries 2.1. Parabolicdomainsforsemi-attractivefixedpoints. Consideracomplex Banach space B, and let F: A ⊂ B → B be a complex analytic operator defined in an open set A. Suppose that p ∈ A is a fixed point for F. We say that p is a semi-attractive fixed point for F if • The value 1 is an eigenvalue for DF . p • ThereexistsaBanachsubspaceEs,with(complex)codimensionone,which is invariant by the action of DF and furthermore the spectrum of DF , p p restricted to Es, is contained in {z: |z|≤r}, where r <1. The followingresultwasprovedbyM.Hakim[H]for finite-dimensionalcomplex Banach spaces (Cn), but the proof can be carry out as well for a general complex Banach space: Proposition 2.1 ([H]). Consider a compact complex analytic operator F, defined in an open set of a complex Banach space B. Let p be a semi-attractive fixed point. Then one of the following statements holds: THE FEIGENBAUM FIXED POINT 3 (1) Curve of fixed points: There exists a complex analytic curve of fixed points which contains p. (2) Parabolic domains (Petals): There existsk ≥1 so that, for every ǫ>0 thereexistsaconnectedopensetU,whosediameterissmallerthanǫ,which is forward invariant by the action of F and, moreover, Fnu→ p, for every u∈U, n where the speed of this convergence is subexponential: for each u∈U, there exists C =C(u) so that 1 1 1 ≤|Fnu−p|≤C . C n1/k n1/k An outline of Hakim’s proof can be found in the Appendix. 3. Infinitesimal contraction on the horizontal space Let f: V → V be a quadratic-like map with connected Julia set and with 1 2 an analytical extension to B (U), with K(f) ⊂ U. The horizontal subspace nor (introduced by Lyubich[Lyu99]) of f, denoted Eh, is the subspace of the vectors f v ∈B(U) so that there exists a quasiconformal vector field in the Riemann sphere α satisfying v = α◦f −Df ·α in a neighborhood of K(f), with ∂α ≡ 0 on K(f) and α(0)=α(1)=α(∞)=0. We will not use the following information here, but certainly it will clarify the spirit of our methods: in an appropriated setting, the hybridclass is a complex analytic manifold and the horizontalspace is the tangent space of the hybrid class at f(see [Lyu99]). Lemma3.1([ALdM]). Letf beaquadratic-like mapwithanextensiontoB (U) nor and connected Julia set contained in U. Assume that f does not support invariant line fields in its filled-in Julia set. Let V ⋐U be a domain with smooth boundary so that K(f)⊂V. Then there exist C,ǫ>0 so that, if |f−g| ≤ǫ and g: g−1V → B(U) V is a quadratic-like map with connected Julia set, then, for every v ∈ Eh there g exists a C|v| -quasiconformal vector field α in C so that v = α◦g−Dg·α on B(U) V. WiththeaidofacompactnesscriteriumtoquasiconformalvectorfieldsinC,we have: Corollary 3.2 ([ALdM]). Assume that (f ,v )→ (f ,v ) in B (U)×B(U), n n n ∞ ∞ nor where f : f−1V →V, i∈N∪{∞}, are quadratic-like maps with connected filled-in i i Julia sets K(f ) ⊂ V ⋐ U. Furthermore, assume that v ∈ Eh , for n ∈ N. If f i n fn does not support invariant line fields in K(f), then v ∈ Eh. In particular Eh is ∞ f f closed. If R is the nth iteration of the Feigenbaum renormalization operator and f is close to f⋆ in B(U), denote by β the analytic continuation of the β-fixed point of f the small Julia set associated with the nth renormalization of f⋆ [McM96]. The following result gives a description of the action of the derivative in a horizontal vector v =α◦f −Df ·α in terms of α: Proposition 3.3. Let V be a neighborhood of K(f⋆). Replacing R by an iteration of it, if necessary, the following property holds: If f ∈ B (U) is close enough to nor 4 DANIELSMANIA f⋆ and v =α◦f −Df ·α on V, where v ∈B(U) and α is a quasiconformal vector field in the Riemann sphere, normalized by α(0)=α(1)=α(∞)=0, then (1) DR ·v =r(α)◦Rf −D(Rf)·r(α), f on U, where 1 1 r(α)(z):= α(β z)− α(β )·z. f f β β f f In particular, if f is renormalizable, then DR Eh ⊂Eh . f f Rf Thisresultisconsequenceofasimplycalculationandthecomplexboundstof⋆. Notethat,apartthenormalizationbyalinearvectorfield,r(α)isjustthepullback of the vector field α by a linear map. In particular, if α is a C-quasiconformal vector field, then r(α) is also a C-quasiconformal vector field: this will be a key point in the proof of the infinitesimal contraction of the renormalization operator in the horizontal subspace (Proposition 3.4). Let f⋆: V → V be a quadratic-like representation of the fixed point. The 1 2 Feigenbaumtoweristheindexedfamilyofquadratic-likemapsf⋆: β−iV →β−iV , i f⋆ 1 f⋆ 2 i∈N, defined by f⋆(z):=β−if⋆(βi ·z). i f⋆ f⋆ Proposition 3.1 ([McM96]). The Feigenbaum tower does not support invariant line fields: this means that there is not a measurable line field which is invariant by all (or even an infinite number of) maps in the Feigenbaum tower. Proposition 3.2 ([Su] and[McM96]). Let f be a quadratic-like map which admits a hybrid conjugacy φ with f⋆. Then φn(z) := βf−⋆n·φ(βRn−1f···βf ·z) converges to identity uniformly on compact sets in the complex plane. In particular, there exists n = n (f) so that Rnf ∈ B (f⋆,ǫ), for n > n , and Rnf → f⋆ on 0 0 Bnor(U) 0 n B (U). nor Theorem 1 says that this convergenceis, indeed, exponentially fast. The following proposition has a straightforwardproof: Proposition 3.3. Let R=i◦R˜, where R˜: V ⊂B →B˜is an operator defined in a neighborhood V ofaBanach spaceB,toanotherBanach spaceB˜, andi: B˜→B is a compact lineartransformation. Let S ⊂B×B beasetwith thefollowing properties: (1) Vector bundle structure: If (f,v ) and (f,v )∈S, then (f,α·v +v )∈ 1 2 1 2 S, for every α∈C, (2) Semicontinuity: If (f ,v )→(f,v) and (f ,v )∈S, then (f,v)∈S, n n n n (3) Invariance: If (f,v)∈S then (Rf,DR ·v)∈S, f (4) Compactness: {(R˜f,DR˜ ·v): (f,v) ∈ S,|v| ≤ 1} is a bounded set in f B˜×B˜, (5) Uniform continuity: Denote E := {(f,v): (f,v) ∈ S}. There exists f C >0 so that, for every f and n≥0, |DRn| ≤C, f Ef (6) If (f,v)∈S then |DRn·v|→ 0, f n Then there exist λ < 1 and N ∈ N so that |DRN| ≤ λ, for every f so that f Ef E 6=∅. f Proposition 3.3 is a generalization of the following fact about compact linear operators T: B →B: if Tnv →0, for every v ∈B, then the spectral radius of T is strictly smaller than one. THE FEIGENBAUM FIXED POINT 5 Given ǫ, K >0,and a domainV ⋐U so thatK(f⋆)⊂V, denote by A(ǫ,K,V) the setofmapsf ∈B (U) sothatthere existsa K-quasiconformalmapφ inthe nor complex plane so that φ(V) ⊂ U and φ◦f⋆ = f ◦φ on V; moreover, for n ≥ 0, we have |Rnf −f⋆| ≤ ǫ. Note that A := A(ǫ,K,V) is closed. Furthermore, B(U) replacing R by an iterate, if necessary, we can assume that A is invariant by the actionofR. SelectingKandǫproperly,bythetopologicalconvergence(Proposition 3.2) and Lemma 2.2 in [Lyu02], for every f in the hybrid class of f⋆, there exists N =N(f) so that RNf ∈A. Proposition 3.4 (Infinitesimal contraction: cf. [Lyu99]). There exist λ < 1 and N >0 so that |DRN| ≤λ, for every f ∈A(ǫ,K,V). f Eh f Proof. Consider the set S := {(f,v): f ∈A, v ∈Eh}. It is sufficient to verify the f propertiesinthestatementofProposition3.3. SinceAisclosed,property2follows of Corollary 3.2. Since A is invariant by R, property 3 follows of Proposition 3.3. The compactness property is obvious, if ǫ is small enough. To prove the uniform continuity property, by Propositions 3.1 and 3.3, we have that, for (f,v) ∈ S and n≥1, DRn·v =α ◦RNf −D(RNf)·α on U, with f n n 1 1 α (z):= α(β ...β z)− α(β ...β )z, n n−1 0 n−1 0 β ...β β ...β n−1 0 n−1 0 where β =β and α are K·|v| -quasiconformal vector fields. Note that K i Rif n B(U) doesnotdependson(f,v)∈S orn≥1. BythecompactnessofK-quasiconformal vector fields (recall that α (0) = α (1) = α (∞) = 0), we get |DRn| ≤ C, for n n n f Eh f some C > 0. To prove assumption 6, note that ∂α is an invariant Beltrami field n to the finite tower 1 1 Rnf, Rn−1f(β z),..., f(β ···β z). n−1 n−1 0 β β ···β n−1 n−1 0 But,bythetopologicalconvergence,thesefinitetowersconvergestotheFeigenbaum tower. Hence, if a subsequence α converges to a quasiconformal vector field nk α , then ∂α is an invariant Beltrami field to the Feigenbaum tower (since ∂α ∞ ∞ nk converges to ∂α in the distributional sense), so, by Proposition 3.1, α is a ∞ ∞ conformal vector field in the Riemann sphere. Since α vanishes at three points, ∞ α ≡0. Hence α →0 uniformly oncompactsets in the complex plane, so we get ∞ n Rn·v →0 (Note that |D(Rnf)| is uniformly bounded, for n≥1). (cid:3) f We are going to prove Theorem 1: Let f be a quadratic-like map in the hybrid classoff⋆. Thenthereexistsaquasiconformalmapφ: C→Cwhichisaconjugacy betweentheminaneighborhoodoftheirJuliasets. ConsiderthefollowingBeltrami pathf betweenthetwomaps,inducedbyφ: ifφ ,|t|≤1,istheuniquenormalized t t quasiconformal map so that ∂/∂φ = t·∂/∂φ, then f = φ ◦f ◦φ−1. By the t t t t topological convergence, there exists n0 so that Rn0+nft ∈ A, for n ≥ 0, |t| ≤ 1. An easy calculation shows that dRn0+nft(cid:12) ∈Eh , dt (cid:12)(cid:12)t=t0 Rn0+nft0 for |t |≤1. The infinitesimal contraction finishes the proof. 0 Remark 1. The first step in the above proof on Theorem 1, to prove that α → n 0 (in the proof of Proposition 3.4), must be compared with the proof of Lemma 6 DANIELSMANIA 9.12 in [McM96]. In C. McMullen argument, additional considerations should be done to arrive in exponential contraction; firstly it is proved that quasiconformal deformations (asthequasiconformal vectorfieldαinthedefinitionofthehorizontal vectors) areC1+β-conformal at thecritical point ( Lemma9.12 in [McM96]andthe deepness of the critical point have key roles in this proof), and then it is necessary tointegratethis result. In M. Lyubich argument [Lyu99], firstly it is proved that the hybrid class is a complex analytic manifold and then the topological convergence is converted in exponential contraction via Schwartz’s Lemma. 4. Hyperbolicity of the Feigenbaum fixed point We are going to prove Theorem 2. Firstly we will prove that (2) σ(DR2 )∩S1 ⊂{1}. f⋆ Indeed,ifDR ·v =λv,thenthevectorv˜(z):=v(z)isasolutiontoDR ·v˜=λv˜. f⋆ f⋆ So if λ ∈ S1 \ {−1,1} then codim Eh > 1, which is a contradiction (Eh has codimension one [Lyu99]. The same result can be proven in an easy way using the argument explained in section 12 on [Sm02a]). Indeed, we can prove, using the contractiononthe horizontaldirectionandresultson[Sm02b],whichusesonly elementary methods, that σ(DRf⋆)∩S1 ⊂{1}, but the proof is more involving. Furthermore σ(DR ) is not contained in D (see Lyubich[Lyu99]. We can also f⋆ use the results in [Sm02b] to prove this claim). So either f⋆ is a hyperbolic fixed point (with a onedimensional expanding direction) or it is a semi-attractive fixed point,sincebyProposition3.4thederivativeoftherenormalizationoperatoratthe fixed point is a contraction on the horizontal space, which has codimension one. Assume that f⋆ is semi-attractive and let’s arrive in a contradiction. Indeed, by Proposition 2.1, one of the following statements holds: Casei. ThereexistsaconnectedopensetofmapsU ⊂B (U),whosediameter nor canbe takensmall,whichisforwardinvariantbythe actionofR2 andsothateach mapin U is attractedatasubexponentialspeedto the fixedpointf⋆. Becausethe maps in U are very close to f⋆ and U is forward invariant, all the maps in U are infinitelyrenormalizable(thisargumentiseasy: seeLemma5.8in[Lyu99]). Sotheir filled-inJuliasetshaveemptyinteriorandtheirperiodicpointsarerepelling,hence therearenotbifurcationsofperiodicpointsinU. Considertwomapsg,g˜inU which admitacomplexanalyticpathg: D→U betweenthem(g =g and,forsome|λ|< 0 1,g˜=g ). BecauseDissimplyconnectedandtherearenotbifurcationsofperiodic λ points in U, each periodic point p ∈ K(g ) has an unique analytic continuation 0 h(p,λ), λ ∈ D: this means that h(p,0) = p and h(p,λ) is a periodic point of g . λ So the function h: Per(g )×D → C defines a holomorphic motion on Per(g ) = 0 0 {p: ∃n > 0 s.t. gn(p) = p} (note that h(p,λ) 6= h(q,λ), if p 6= q, since there are 0 not bifurcations of periodic points). Moreover,providedU is small enough, we can select a domain U with a real analytic Jordan curve boundary so that, for every 1 λ∈D,g : g−1U →U isaquadratic-likerepresentation. Wecanalsoeasilydefine λ λ 1 1 aholomorphicmotionh: U \g−1U ×D→Csothath(x,λ)≡x,forx∈C\U and 1 0 0 1 g (h(x,λ)) = h(g (x),λ), for x ∈ ∂g−1U . Since g have connected filled-in Julia λ 0 0 1 λ sets,wecanextendtheholomorphicmotiontoaholomorphicmotionh: C\K(g )× 0 D → C so that g (h(x,λ)) = h(g (x),λ), for x ∈ g−1U \ K(g ). So we have λ 0 0 1 0 definedaholomorphicmotionhontheeverywheredensesetPer(g )∪(C\ K(g )) 0 0 which commutes with the dynamics. By the λ-lemma [MSS], this holomorphic THE FEIGENBAUM FIXED POINT 7 motion extends to the whole Riemann sphere, so all maps g are quasiconformaly λ conjugated. Sincethereisapiecewisecomplexanalyticpathbetweenanytwomaps inU,weconcludethatallmapsinU areinthesamequasiconformalclass. Notethat the aboveconstructiondoes notgiveany upper bound for the quasiconformalityof the conjugacy: the quasiconformality could be large when the Kobayashi distance between g and g˜ on U is large. We claim that, provided U is small enough, it is possible to choose a quasiconformal conjugacy between any two maps in U so that the quasiconformality is uniformlyboundedoutsidetheirfilled-inJuliasets,usingtheargumentintheproof ofLemma2.3in[Lyu02]: inasmallneighborhoodV ⊂B (U)off⋆,itispossible nor to find a domainU so that g: g−1U →U is a quadratic-like restrictionof g (but 1 1 1 note that the Julia sets of these quadratic-like restrictions are not, in general,connected). This define the holomorphic moving fundamental annulus U \g−1U . In 1 1 particular,provided U is small enough, there exists B >0 so that for every g and 0 g whichbelongtoU,thereexistsaB-quasiconformalmappinghbetweenC\g−1U 1 0 1 and C\g−1U so that h≡Id on C\U and g ◦h=h◦g on ∂g−1U . Since the 1 1 1 1 0 0 1 Julia sets of g and g are connected, we can extend h to a B-quasiconformalmap 0 1 h: C\K(g )→C\K(g ) 0 1 whichisaconjugacyong−1U \K(g ). Oncewealreadyknowthatg andg arein 0 1 0 0 1 the same quasiconformal class, h has a quasiconformal extension h to C (this g0,g1 followsasintheproofofLemma1,in[DH,pg. 302]: ifh˜ isaquasiconformalconju- gacybetweeng andg ,thenh˜−1◦hcommuteswithg outsideK(g ),whichimplies 0 1 0 0 that h˜−1◦h extends to a homeomorphism in C which coincides with Id on K(g ). 0 By the Rickmann removability theorem (see the statement in [DH]), this map is a quasiconformal homeomorphism, so h is a quasiconformal homeomorphism). This finishes the proof of the claim. Since all renormalizations of these maps are very close to f⋆, they also satisfies theunbranchedcomplexbounds condition(this isconsequenceofashortlemmain [Lyu97]). Inparticulartherearenotinvariantlinefieldssupportedontheirfilled-in Juliasets[McM94],andhencethequasiconformalityoftheconjugacyh : C→C g0,g1 isuniformlyboundedonthewholecomplexplanebyB. Butf⋆ isaboundarypoint ofU,sothecompactnessofB-quasiconformalmaps(notethattheconjugaciesh g0,g1 satisfies h (0) = 0, h (1) = 1 and h (∞) = ∞) and the non-existence of g0,g1 g0,g1 g0,g1 invariant line fields supported on the filled-in Julia set of f⋆ imply that all these maps are hybrid conjugated with f⋆. But this implies that the subexponential speed of convergence given by Proposition 2.1 is impossible, since by Theorem 1 the maps in the hybrid class of f converges to f⋆ exponentially fast. Case ii. There exists a connected complex analytic curve of fixed points which contains f. We will apply essentially the same argumentused in Case i: Note that in a similar way we can prove that all these fixed points of the operator R2 are polynomial-like maps which are infinitely renormalizable: in particular their filled- inJuliasetshaveemptyinteriorandalltheirperiodicpointsarerepelling. Sothere arenotbifurcationsofperiodicpointsinthiscurveoffixedpoints. Usetheλ-lemma [MSS] to conclude that all these fixed points are quasiconformally conjugated (the argument is as in Case i). Since the fixed point f⋆ does not support invariant line fields in its filled-in Julia set, we conclude that all these fixed points are hybrid 8 DANIELSMANIA conjugated, which is impossible, since iterations of maps in the hybrid class of f⋆ converges to the fixed point f⋆. So we concludedthat f⋆ must be a hyperbolic fixed pointwith codimensionone stable manifold. Appendix: Outline of Hakim’s proof To convince the reader of the existence of parabolic petals for semi-attractive compactoperatorsinBanachspaces,wewillgiveanoutlineofHakim’sproofofthe existenceofparabolicdomains: wedonotclaimanysortoforiginalityforourselves in the following exposition and we refer to the quite clear work [H] for details. We will use the notation introduced in Section 2.1. Consider a complex analytic operatorT with a semi-attractive fixedpoint 0. Assume DT ·v =v, v 6=0. In the 0 followinglines,wewillidentifyB withC×Es bytheisomorphism(x,y)→x·v+y. BytheStableManifoldTheoremforcompactoperators(seeMan˜e[M]),forδ >0 and ǫ>0 small the set Ws ={x: ∃ C s.t. |Tnx|<δ and |Tnx|≤C(1−ǫ)n, for n≥0} δ,ǫ is a codimension one complex analytic manifold. More precisely, there exists a holomorphic function ψ: V → C, where V is a neighborhood of 0 on Es, with Dψ(0)=0, so that Ws ={(ψ(y),y): y ∈V}. δ,ǫ In particular, after the local biholomorphic changes of variables X =x+ψ(y) (3) Y =y itis possible to representT asT: C×Es →C×Es, where T(x,y)=(x′,y′), with x′ =F(x,y)=a (y)x+O (x2) 1 y (4) y′ =G(y)+xh(x,y) where G is a (compact) contraction around 0 and a (0) = 1. After the local 1 biholomorphic change of variables X =v(y)x (5) Y =y where v(y):=Π a (Gi(y)), i≥0 1 we can assume that a ≡1. 1 Note that, for every n, T has the form x′ =F(x,y)=x+ X ai(y)xn+Oy(xn+1) (6) 2≤i≤n y′ =G(y)+xh(x,y) where G is a (compact) contraction around0. We claim that we can assume, after certain biholomorphic changes of variables, that a , a , ···, a do not depend on 2 3 n y. Indeed, assume by induction that T can put in the form x′ =F(x,y)=x+ X a˜ixn+añ+1(y)xn+1+Oy(xn+2) (7) 2≤i≤n y′ =G(y)+xh(x,y) THE FEIGENBAUM FIXED POINT 9 Then after the local change of variables X =x+v(y)xn+1 (8) Y =y where v(y):= (a˜ (Gi(y))−a˜ (0)), T will have the form Pi≥0 n+1 n+1 x′ =F(x,y)=x+ X a˜ixn+añ+1(0)xn+1+añ+2(y)xn+2+Oy(xn+2) (9) 2≤i≤n y′ =G(y)+xh(x,y). Now we are going to introduce the concept of multiplicity of the fixed point 0 for transformationson the form ofEq. (6). By the implicit function theorem, for each transformationin that form there exists a complex analyticcurve y: U ⊂C→Es, with 0∈U, which is the unique solution for the equation y(x)=G(y(x))+xh(x,y(x)). Consider the function q: U →C defined by q(x):=F(x,y(x))−x. The multiplicity of T at 0 is defined as the order of q at 0. Note that the multiplicity of T at 0 is finite if and only if 0 is an isolated fixed point and infinity if and only if q(x) vanishes everywhere and (x,y(x)) is a complex analytic curve of fixed points for T (which contains all the fixed points in a neighborhood of 0). Moreover, if T has the form Eq. (7), with a˜ = ··· = a˜ = 0 and a˜ 6= 0, then 2 n−1 n the multiplicity of T is exactly n. ConsideratransformationT asinEq. (6)andbiholomorphicchangeofvariables W(x,y)=(X,Y) of the type X =x+v(y)xk (10) Y =y wherev isa holomorphicfunction andk>1. ThenW−1◦T ◦W hasalsothe form in Eq. (6). Moreover Proposition 4.1. The multiplicity of W−1◦T◦W at 0 is equal to the multiplicity of T at 0. Proof. (suggestedbyM.Lyubich)Assumethatitisfinite(otherwisetheinvariance is trivial): then 0 is an isolated fixed point. Consider the one- parameter family of change of variables W defined by λ X =x+λv(y)xk (11) Y =y Then W−1◦T ◦W has the form λ λ x′ =F (x,y)=x+O (x2) λ y,λ (12) y′ =G (y)+xh (x,y) λ λ Note that we can choose δ small enough so that for all |λ| ≤ 1, 0 is the unique 0 fixedpointforW−1◦T◦W on{(x,y),|x|≤δ ,|y|≤δ }.Moreover,bytheimplicit λ λ 0 0 10 DANIELSMANIA function theorem and the compactness of {λ: |λ| ≤ 1} there exists a holomorphic function y (x)=y(λ,x), defined on λ {λ: |λ|<1+δ }×{x: |x|<δ } 1 2 so that y (x)=G (y)+xh (x,y (x)) λ λ λ λ Choosing δ ,δ small enough, for each λ the point 0 is the unique solution for the 1 2 equation q (x):=F (x,y (x))−x=0 λ λ λ on {x: |x|≤δ }. By Rouché’s Theorem, ord q does not depends on λ. (cid:3) 1 0 λ AssumethatT hasthe formofEq. (6)andfinite multiplicityn. Afterappropri- ated changes of variables, we can assume that a , ..., a does not depends on 2 2n−1 y. Sincethemultiplicityisinvariantbytheabovechangesofvariables,weconclude that a = ···= a = 0 and a 6= 0. Doing appropriated changes of variables in 2 n−1 n the form of Eq. (10) (indeed, in this case v does not depends on y) and replacing the coordenate x by θx, for some θ 6= 0, if necessary, it is possible to put T in the form 1 x′ =x− xn−1+ax2(n−1)+O (|x|2(n−1)+1) y (13) n−1 y′ =G(y)+xh(x,y). Under the above form, the set 1 1 P ={(x,y): |xn−1− |< and |y|<ρ} R,ρ 2R 2R is a parabolic domain, provided R and ρ are small enough: here Hakim’s proof is very similar to the one-dimensional situation: make the “change of variables” X =xn−1 (14) Y =y and X =1/x (15) Y =y to put T in the form 1 1 x′ =x+1+c +O ( ) x y |x|1+1/(n−1) (16) 1 y =G(y)+O ( ). y |x|1/(n−1) and now the proof is easy. References [AB] L.AhlforsandL.Bers.Riemann’smappingtheoremforvariablemetrics.Ann.ofMath., 72:385–404, 1960. [ALdM] A.Avila,M.LyubichandW.deMelo.Regularorstochasticdynamicsinrealanalytic familiesofunimodalmaps. preprint,2001. [DH] A. Douady and J. Hubbard. On the dynamics of polynomial-likemappings. Ann. Sci. EćoleNorm.Sup(4),18:287–343, 1985. [H] M. Hakim. Attracting domains for semi-attractive transformations of Cp. Publ. Mat. 38,479–499, 1994.

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