Fatou flowers and parabolic curves Marco Abate Dipartimento di Matematica, Universit`a di Pisa Largo Pontecorvo5, 56127 Pisa, Italy. E-mail: [email protected] July 2015 5 1 Abstract. In this survey we collect the main results known up to now (July 2015)regarding 0 possible generalizations to several complex variables of the classical Leau-Fatou flower theorem 2 about holomorphic parabolic dynamics. l u 1. The original Leau-Fatou flower theorem J 4 In this surveywe shallpresentthe knowngeneralizationsofthe classicalLeau-Fatoutheoremdescribing the 1 localholomorphicdynamicsaboutaparabolicpoint. Butletus startwithanumber ofstandarddefinitions. ] Definition 1.1: A local n-dimensional discrete holomorphic dynamical system (in short, a local dy- S namical system) is a holomorphic germ f of self-map of a complex n-dimensional manifold M at a point D p∈M such that f(p)=p; we shall denote by End(M,p) the set of such germs. . h If f, g belongs to End(M,p) their composition g◦f is defined as germ in End(M,p); in particular, we at can consider the sequence {fk}⊂End(M,p) of iterates of f ∈End(M,p), inductively defined by f0 =idM m and fk =f ◦fk−1 for k ≥1. The aim of local discrete dynamics is exactly the study of the behavior of the [ sequence of iterates. 3 Remark1.1: Inpractice,weshallworkwithrepresentatives,thatiswithholomorphicmapsf:U →M, v where U ⊆ M is an open neighborhood of p ∈ U, such that f(p)= p. The fact we are working with germs 6 willbereflectedinthefreedomwehaveintakingU assmallasneeded. Weshallalsomostly(butnotalways) 7 take M =Cn and p=O; indeed a choice of local coordinates ϕ for M centered at p yields an isomorphism 1 ϕ :End(M,p)→End(Cn,O) preserving the composition by setting ϕ (f)=ϕ◦f ◦ϕ−1. 2 ∗ ∗ 0 Definition 1.2: Let f:U → M be a representative of a germ in End(M,p). The stable set K ⊆ U . f 1 of f is the set of points z ∈ U such that fk(z) is defined for all k ∈ N; clearly, p ∈ K . If z ∈ K , the f f 0 set {fk(z)} is the orbit of z; if z ∈ U \K we shall say that z escapes. The stable set depends on the f 5 chosenrepresentative,butits germatpdoesnot;soweshallfreelytalkaboutthe stablesetofanelementof 1 End(M,p). An f-invariant set is a subset P ⊆U such that f(P)⊆P; clearly, the stable set is f-invariant. : v i Definition 1.3: Alocaldynamicalsystemf ∈End(M,p)isparabolic(andsometimesweshallsaythat X p is a parabolic fixed point of f) if df is diagonalizable and all its eigenvalues are roots of unity; is tangent p r to the identity if df =id. We shall denote by End (M,p) the set of localdynamical systems tangent to the a p 1 identity in p. Remark 1.2: If f ∈ End(M,p) is parabolic then a suitable iterate fq is tangent to the identity; for this reason we shall mostly concentrate on germs tangent to the identity. Furthermore, if f ∈ End(M,p) is tangent to the identity then f−1 is a well-defined germ in End(M,p) still tangent to the identity. Definition 1.4: The order ord (f) of a holomorphic function f:M → C at p ∈ M is the order of p vanishingatp,thatisthedegreeofthefirstnon-vanishingtermintheTaylorexpansionoff atp(computed inanysetoflocalcoordinatescenteredatp). Theorderord (F)ofaholomorphicmapF:M →Cn atp∈M p is the minimum order of its components. A germ f ∈ End(Cn,O) can be represented by a n-tuple of convergent power series in n variables; collecting terms of the same degree we obtain the homogeneous expansion. 2010MathematicsSubjectClassification: Primary32H50;37F75, 37F99. PartiallysupportedbyFIRB2012project“GeometriaDifferenzialeeTeoriaGeometricadelleFunzioni” 2 MarcoAbate Definition 1.5: A homogeneous map of degree d ≥ 1 is a map P:Cn → Cn where P is a n-tuple of homogeneous polynomials of degree d in n variables. The homogeneous expansion of a germ tangent to the identity f ∈End1(Cn,O), f 6≡idCn, is the (unique) series expansion f(z)=z+P (z)+P (z)+··· (1.1) ν+1 ν+2 where P is a homogeneousmap of degree k, andP 6≡O. The number ν ≥1 is the order (or,sometimes, k ν+1 multiplicity) ν(f)off atO, andP isthe leadingtermoff. Itis easytocheckthatthe orderisinvariant ν+1 underchangeofcoordinates,andthusitcanbedefinedforanygermtangenttotheidentityf ∈End (M,p); 1 we shall denote by End (M,p) the set of germs tangent to the identity with order at least ν. ν In the rest of this section we shall discuss the 1-dimensional case, where the homogeneous expansion reduces to the usual Taylor expansion f(z)=z+a zν+1+O(zν+2) (1.2) ν+1 with a 6=0. ν+1 Definition1.6: Letf ∈End (C,0)betangenttotheidentitygivenby(1.2). Aunitvectorv ∈S1isan 1 attracting (respectively, repelling) direction for f at 0 if a vν is real and negative (respectively, positive). ν+1 Clearly, there are ν equally spaced attracting directions, separated by ν equally spaced repelling directions. Example 1.1: To understand this definition, let us consider the particular case f(z) = z+azν+1. If v ∈S1 is suchthatavν >0thenforeveryz ∈R+v we havef(z)∈R+v and|f(z)|>|z|;inotherwords,the half-line R+ is f-invariantand repelledfromthe origin. Conversely,if v ∈S1 is suchthat avν <0 then R+v is again f-invariantbut now |f(z)|<|z| if z ∈R+v is small enough; so there is a segment of R+v attracted by the origin. Remark 1.3: If f ∈End (C,0) is given by (1.2) then 1 f−1(z)=z−a zν+1+O(zν+2). ν+1 Inparticular,ifv ∈S1isattracting(respectively,repelling)forf thenitisrepelling(respectively,attracting) for f−1, and conversely. To describe the dynamics of a tangent to the identity germ two more definitions are needed. Definition 1.7: Let v ∈ S1 be an attracting direction for a f ∈ End (C,0) tangent to the identity. 1 The basin centeredat v is the set of points z ∈K \{0} such that fk(z)→0 andfk(z)/|fk(z)|→v (notice f that, up to shrinking the domain of f, we can assume that f(z) 6= 0 for all z ∈ K \{0}). If z belongs to f the basin centered at v, we shall say that the orbit of z tends to 0 tangent to v. A slightly more specialized (but more useful) object is the following: Definition 1.8: Let f ∈ End (C,0) be tangent to the identity. An attracting petal with attracting 1 central direction v ∈ S1 for f is an open simply connected f-invariant set P ⊆ K \{0} with 0 ∈ ∂P such f that a point z ∈ K \{0} belongs to the basin centered at v if and only if its orbit intersects P. In other f words, the orbit of a point tends to 0 tangent to v if and only if it is eventually contained in P. A repelling petal (with repelling central direction) is an attracting petal for the inverse of f. WecannowstatetheoriginalLeau-Fatouflowertheorem,describingthedynamicsofaone-dimensional tangent to the identity germ in a full neighborhood of the origin (see, e.g., [M] for a modern proof): Theorem 1.1: (Leau, 1897 [Le]; Fatou, 1919-20 [F1–3]) Let f ∈ End (C,0) be tangent to the identity of 1 order ν ≥1. Let v+,...,v+ ∈S1 be the ν attracting directions of f at the origin, and v−,...,v− ∈S1 the 1 ν 1 r ν repelling directions. Then: (i) for each attracting (repelling) direction v+ (v−) we can find an attracting (repelling) petal P+ (P−) j j j j such that the union of these 2ν petals together with the origin forms a neighborhood of the origin. Furthermore, the 2ν petals are arranged cyclically so that two petals intersect if and only if the angle between their central directions is π/ν. Fatouflowersandparaboliccurves 3 (ii) K \{0} is the (disjoint) union of the basins centered at the ν attracting directions. f (iii) If B is a basin centered at one of the attracting directions, then there is a function χ:B →C such that χ◦f(z)=χ(z)+1 for all z ∈B. Furthermore, if P is the corresponding petal constructed in part (i), then χ| is a biholomorphism with an open subset of the complex plane containing a right half-plane P — and so f| is holomorphically conjugated to the translation z 7→z+1. P Definition 1.9: The function χ:B →C constructed in Theorem 1.1.(iii) is a Fatou coordinate on the basin B. Remark 1.4: Up to a linear change of variable, we can assume that a = −1 in (1.2), so that the ν+1 attracting directions are the ν-th roots of unity. Given δ >0, the set D ={z ∈C||zν −δ|<δ} (1.3) ν,δ hasexactlyν connectedcomponents(eachonesymmetricwithrespecttoadifferentν-throotofunity),and it turns out that when δ > 0 is small enough these components can be taken as attracting petals for f — even though to cover a neighborhood of the origin one needs slightly larger petals. The components of D ν,δ are distributed as petals in a flower; this is the reason why Theorem 1.1 is called “flower theorem”. So the union of attracting and repelling petals gives a pointed neighborhood of the origin, and the dynamicsoff oneachpetalisconjugatedtoatranslationviaaFatoucoordinate. Therelationshipsbetween different Fatou coordinates is the key to E´calle-Voronin holomorphic classification of parabolic germs (see, e.g., [A4] and references therein for a concise introduction to E´calle-Voronin invariants), which is however outsideofthescopeofthissurvey. Weendthissectionwiththe statementoftheLeau-Fatouflowertheorem for general parabolic germs: Theorem1.2: (Leau,1897[Le];Fatou,1919-20[F1–3]) Letf ∈End(C,0)beoftheformf(z)=λz+O(z2), where λ∈S1 is a primitive rootof the unity of order q. Assume that fq 6≡id. Then there exists µ≥1 such that fq has order qµ, and f acts on the attracting (respectively, repelling) petals of fq as a permutation composed by µ disjoint cycles. Finally, K =K . f fq In the subsequent sections we shall discuss known generalizations of Theorem 1.1 to several variables. 2. E´calle-Hakim theory From now on we shall work in dimension n ≥ 2. So let f ∈ End (Cn,O) be tangent to the identity; we 1 would like to find a multidimensional version of the petals of Theorem 1.1. If f had a non-trivial one-dimensional f-invariantcurve passing through the origin, that is an injective holomorphic map ψ:∆→Cn, where ∆⊂C is a neighborhood of the origin, such that ψ(0)=O, ψ′(0)6=O and f ψ(∆) ⊆ ψ(∆) with f| 6≡ id, we could apply Leau-Fatou flower theorem to f| obtaining a ψ(∆) ψ(∆) one-dimensional Fatou flower for f inside the invariant curve. In particular, if zo ∈ ψ(∆) belongs to an attrac(cid:0)tive pe(cid:1)tal, we would have fk(zo) → O and [fk(zo)] → [ψ′(0)], where [·]:Cn\{O} → Pn−1(C) is the canonical projection. The first observation we can make is that then [ψ′(0)] cannot be any direction in Pn−1(C). Indeed: Proposition 2.1: ([H2]) Let f(z) = z+P (z)+··· ∈ End (Cn,O) be tangent to the identity of order ν+1 1 ν ≥ 1. Assume there is zo ∈ K such that fk(zo) → O and [fk(zo)] → [v] ∈ Pn−1(C). Then P (v) = λv f ν+1 for some λ∈C. Definition 2.1: Let P:Cn → Cn be a homogeneous map. A direction [v] ∈ Pn−1(C) is characteristic for P if P(v) = λv for some λ ∈ C. Furthermore, we shall say that [v] is degenerate if P(v) = O, and non-degenerate otherwise. Remark 2.1: From now on, given f ∈ End (Cn,O) tangent to the identity of order ν ≥ 1, every 1 notion/object/concept introduced for its leading term P will be introduced also for f; for instance, a ν+1 (degenerate/non-degenerate) characteristic direction for P will also be a (degenerate/non-degenerate) ν+1 characteristic direction for f. Remark 2.2: If f ∈End (Cn,O) is given by (1.1), then f−1 ∈End (Cn,O) is given by 1 1 f−1(z)=z−P (z)+··· . ν+1 4 MarcoAbate In particular, f and f−1 have the same (degenerate/non-degenerate)characteristic directions. Remark 2.3: If ψ:∆ → Cn is a one-dimensional curve with ψ(0) = O and ψ′(0) 6= O such that f| ≡id, it is easy to see that [ψ′(0)] must be a degenerate characteristic direction for f. ψ(∆) So if we have an f-invariantone-dimensionalcurve ψ through the originthen [ψ′(0)] must be a charac- teristicdirection. However,ingeneraltheconverseisfalse: therearenon-degeneratecharacteristicdirections which are not tangent to any f-invariant curve passing through the origin. Example 2.1: ([H2]) Let f ∈End(C2,O) be given by z f(z,w)= ,w+z2 , 1+z (cid:18) (cid:19) so that f is tangent to the identity of order 1, and P (z,w) = (−z2,z2). In particular, f has a degenerate 2 characteristic direction [0 : 1] and a non-degenerate characteristic direction [v] = [1 : −1]. The degenerate characteristic direction is tangent to the curve {z = 0}, which is pointwise fixed by f, in accord with Remark 2.3. We claim that no f-invariant curve can be tangent to [v]. Assume,bycontradiction,thatwehaveanf-invariantcurveψ:∆→C2withψ(0)=Oand[ψ′(0)]=[v]. Without loss of generality, we can assume that ψ(ζ) = ζ,u(ζ) with u∈End(C,0). Then the condition of f-invariance becomes f ζ,u(ζ) =u f (ζ,u(ζ) , that is 2 1 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1) ζ u(ζ)+ζ2 =u . (2.1) 1+ζ (cid:18) (cid:19) Put g(ζ) = ζ/(1+ζ), so that gk(ζ) = ζ/(1+kζ); in particular, gk(ζ) → 0 for all ζ ∈ C\{−1 | n ∈ N∗}. n This means that by using (2.1) we can extend u to C\{−1 |n∈N∗} by setting n k−1 u(ζ)=u gk(ζ) − [gj(ζ)]2 j=0 (cid:0) (cid:1) X where k ∈N is chosen so that gk(ζ)∈∆. Analogously, (2.1) implies that for |ζ| small enough one has u g−1(ζ) + g−1(ζ) 2 =u(ζ); so we can use this relation to extend u(cid:0)to all o(cid:1)f C(cid:0), and th(cid:1)en to P1(C), because g−1(∞) = −1. So u is a holomorphic function defined on P1(C), that is a constant; but no constant can satisfy (2.1), contradiction. Remark 2.4: Rib´on [R] has given examples of germs having no holomorphic invariant curves at all. For instance, this is the case for germs of the form f(z,w)=(z+w2,w+z2+λz5) for all λ∈C outside a polar Borel set. The first important theorem we would like to quote is due to E´calle [E] and Hakim [H2], and it says thatwe do alwayshaveaFatouflowertangentto a non-degeneratecharacteristicdirection,evenwhenthere areno invariantcomplex curvescontainingthe originin their relativeinterior. To state it, we needto define what is the correct multidimensional notion of petal. Definition 2.2: Aparaboliccurveforf ∈End (Cn,O)tangenttotheidentity isaninjectiveholomor- 1 phic map ϕ:D →Cn\{O} satisfying the following properties: (a) D is a simply connected domain in C with 0∈∂D; (b) ϕ is continuous at the origin, and ϕ(0)=O; (c) ϕ(D) is f-invariant, and (f| )k →O uniformly on compact subsets as k →+∞. ϕ(D) Furthermore,if [ϕ(ζ)] →[v] in Pn−1(C) as ζ →0 in D, we shall say that the paraboliccurve ϕ is tangentto the direction[v]∈Pn−1(C). Finally, a Fatou flowerwith ν petals tangent to a direction [v] is a holomorphic map Φ:D →C, where D is given by (1.3), such that Φ restricted to any connected component of D ν,δ ν,δ ν,δ is a parabolic curve tangent to [v], a petal of the Fatou flower. If ν is the order of f then we shall talk of a Fatou flower for f without mentioning the number of petals. Then E´calle, using his resurgence theory (see, e.g., [S] for an introduction to E´calle’s resurgence theory inonedimension),andHakim,usingmoreclassicalmethods,haveprovedthefollowingresult(seealso[W]): Fatouflowersandparaboliccurves 5 Theorem 2.2: (E´calle, 1985 [E]; Hakim, 1997 [H2, 3]) Let f ∈ End(Cn,O) be tangent to the identity, and [v] ∈ Pn−1(C) a non-degenerate characteristic direction for f. Then there exists (at least) one Fatou flower tangent to [v]. Furthermore, for every petal ϕ:∆ → Cn of the Fatou flower there exists a injective holomorphic map χ:ϕ(∆)→C such that χ f(z) =χ(z)+1 for all z ∈ϕ(∆). Definition 2.3: The function χ constr(cid:0)ucted(cid:1)in the previous theorem is a Hakim-Fatou coordinate. Remark 2.5: A characteristic direction is a complex direction, not a real one; so it should not be confused with the attracting/repelling directions of Theorem 1.1. All petals of a Fatou flower are tangent to the same characteristic direction, but each petal is tangent to a different real direction inside the same complex (characteristic) direction. In particular, Fatou flowers of f and f−1 are tangent to the same characteristicdirections(seeRemark2.2)butthecorrespondingpetalsaretangenttodifferentrealdirections, as in Theorem 1.1. In particularthere existparabolic curvestangentto [1:−1]for the system ofExample 2.1eventhough there are no invariant curves passing through the origin tangent to that direction. Parabolic curves are one-dimensional objects in an n-dimensional space; it is natural to wonder about the existence of higher dimensional invariant subsets. A sufficient condition for their existence has been given by Hakim; to state it we need to introduce another definition. Definition 2.4: Let[v]∈Pn−1(C)beanon-degeneratecharacteristicdirectionforahomogeneousmap P:Cn →Cnofdegreeν+1≥2;inparticular,[v]isafixedpointforthemeromorphicself-map[P]ofPn−1(C) induced by P. The directors of P in [v] are the eigenvalues α ,...,α ∈C of the linear operator 1 n−1 1 d[P] −id :T Pn−1(C)→T Pn−1(C). ν [v] [v] [v] As usual, if f ∈End (Cn,O) is of(cid:0)the form (1(cid:1).2), then the directors of f in a non-degenerate characteristic 1 direction [v] are the directors of P in [v]. ν+1 Remark 2.6: Definition 2.4 is equivalent to the original definition used by Hakim (see, e.g., [ArR]). Furthermore, in dimension 2 if [v] = [1 : 0] is a non-degenerate characteristic direction of P = (P ,P ) we 1 2 have P (1,0)6=0, P (1,0)=0 and the director is given by 1 2 1 d P (1,ζ)−ζP (1,ζ) 1 ∂P2(1,0) 2 1 = ∂z2 −1 . ν dζ P (1,ζ) ν P (1,0) 1 (cid:12)ζ=0 " 1 # (cid:12) Remark 2.7: Recalling Remark 2.2 one sees tha(cid:12)(cid:12)t a germf ∈End1(Cn,O) tangent to the identity and its inverse f−1 have the same directors at their non-degenerate characteristic directions. Remark2.8: TheproofofTheorem2.2becomessimplerwhennodirectorisoftheform k withk∈N∗; ν furthermore, in this case the parabolic curves enjoy additional properties (in the terminology of [AT1] they are robust; see also [Ro3]). Definition 2.5: A parabolic manifold for a germ f ∈ End (Cn,O) tangent to the identity is an f- 1 invariantcomplexsubmanifoldM ⊂Cn\{O}withO ∈∂M suchthatfk(z)→O forallz ∈M. Aparabolic domainisaparabolicmanifoldofdimensionn. WeshallsaythatM isattachedtothecharacteristicdirection [v]∈Pn−1(C) if furthermore [fk(z)]→[v] for all z ∈M. Then Hakim has proved (see also [ArR] for the details of the proof) the following theorem: Theorem 2.3: (Hakim, 1997 [H3]) Let f ∈ End (Cn,O) be tangent to the identity of order ν ≥ 1. Let 1 [v] ∈ Pn−1(C) be a non-degenerate characteristic direction, with directors α ,...,α ∈ C. Furthermore, 1 n−1 assume that Reα ,...,Reα >0 and Reα ,...,Reα ≤0 for a suitable d≥0. Then: 1 d d+1 n−1 (i) There exist (at least) ν parabolic (d+1)-manifolds M ,...,M of Cn attached to [v]; 1 ν (ii) f| is holomorphically conjugated to the translation τ(w ,w ,...,w )= (w +1,w ,...,w ) defined Mj 0 1 d 0 1 d on a suitable right half-space in Cd+1. Remark2.9: Inparticular,ifallthedirectorsof[v]havepositiverealpart,thereisatleastoneparabolic domain. However,theconditiongivenbyTheorem2.3isnotnecessaryfortheexistenceofparabolicdomains; 6 MarcoAbate see [Ri1], [Us], [AT3] for examples, and [Ro8] for conditions ensuring the existence of a parabolic domain when some directors have positive real part and all the others are equal to zero. Moreover, Lapan [L1] has provedthatifn=2andf hasauniquecharacteristicdirection[v]whichis nondegeneratethenthereexists a parabolic domain attached to [v] even though the director is necessarily 0. Two natural questions now are: how many characteristic directions are there? Does there always exist a non-degenerate characteristic direction? To answer the first question, we need to introduce the notion of multiplicity of a characteristic direction. To do so, notice that [v] = [v : ··· : v ] ∈ Pn−1(C) is a 1 n characteristic direction for the homogeneous map P = (P ,...,P ) if and only if v P (v)−v P (v) = 0 1 n h k k h for all h, k = 1,...,n. In particular, the set of characteristic directions of P is an algebraic subvariety of Pn−1(C). Definition 2.6: If the maximal dimension of the irreducible components of the subvariety of charac- teristic directions of a homogeneous map P:Cn → Cn is k, we shall say that P is k-dicritical; if k = n we shall say that P is dicritical; if k =0 we shall say that P is non-dicritical. Remark 2.10: A homogeneous map P:Cn → Cn of degree d is dicritical if and only if P(z) = p(z)z for some homogenous polynomial p:Cn → C of degree d−1. In particular, the degenerate characteristic directions are the zeroes of the polynomial p. In the non-dicritical case we can count the number of characteristic directions, using a suitable multi- plicity. Definition 2.7: Let [v]=[v :···:v ]∈Pn−1(C) be a characteristic direction of a homogeneous map 1 n P = (P ,...,P ):Cn →Cn. Choose 1 ≤j ≤ n so that v 6= 0. The multiplicity µ ([v]) of [v] is the local 1 n 0 j0 P intersection multiplicity at [v] in Pn−1(C) of the polynomials z P −z P with j 6= j if [v] is an isolated j0 j j j0 0 characteristic direction; it is +∞ if [v] is not isolated. Remark 2.11: The localintersection multiplicity I(p ,...,p ;zo) of a set {p ,...,p } of holomorphic 1 k 1 k functions at a point zo ∈Cn can be defined (see, e.g., [GH]) as I(p ,...,p ;zo)=dimO /(p ,...,p ), 1 k n,zo 1 k whereOn,zo isthelocalringofgermsofholomorphicfunctionsatzo,andthedimensionisasvectorspace. It iseasytocheckthatthedefinitionofmultiplicityofacharacteristicdirectiondoesnotdependontheindexj 0 chosen. Furthermore, since the local intersection multiplicity is invariant under change of coordinates, we can use local charts to compute the local intersection multiplicity on complex manifolds. Remark 2.12: When n=2, the multiplicity of [v]=[1:v ] as characteristicdirection of P =(P ,P ) 2 1 2 is the order of vanishing at t = v of P (1,t)−tP (1,t); analogously, the multiplicity of [0 : 1] is the order 2 2 1 of vanishing at t=0 of P (t,1)−tP (t,1). 1 2 Then we have the following result (see, e.g., [AT1]): Proposition 2.4: Let P:Cn →Cn be a non-dicritical homogeneous map of degree ν+1≥2. Then P has exactly n−1 1 n (ν+1)n−1 = νj ν j+1 j=0(cid:18) (cid:19) (cid:0) (cid:1) X characteristic directions, counted according to their multiplicity. Inparticular,whenn=2thenahomogeneousmapofdegreeν+1eitherisdicritical(andalldirections are characteristic) or has exactly ν +2 characteristic directions. But all of them can be degenerate; an example is the following (but it is easy to build infinitely many others). Example 2.2: Let P(z,w)=(z2w+zw2,zw2). Then the characteristic directions of P are [1:0] and [0:1], both degenerate. Using Remark 2.12, we see that µ ([1:0])=3 and µ ([0:1])=1. P P So we cannot apply Theorem 2.2 to any germ of the form f(z) = z +P(z)+··· when P is given by Example 2.2. However, as soon as the higher order terms are chosen so that the origin is an isolated fixed point then f does have parabolic curves: Fatouflowersandparaboliccurves 7 Theorem 2.5: (Abate,2001[A2]) Letf ∈End (C2,O)betangenttotheidentitysuchthatOisanisolated 1 fixed point. Then f admits at least one Fatou flower tangent to some characteristic direction. In the next sectionwe shallexplainwhy this theoremholds,we shallgivemore generalstatements,and weshallgiveanexample(Example3.1)showingthe necessityofthe hypothesisthatthe originis anisolated fixed point. 3. Blow-ups, indices and Fatou flowers Intheprevioussectionwesawthatforstudyingthedynamicsofagermtangenttotheidentityitisusefulto consider the tangentdirections at the fixed point. A useful way for dealing with tangentdirections consists, roughly speaking, in replacing the fixed point by the projective space of the tangent directions, in such a way that the new space is still a complex manifolds, where the tangent directions at the originalfixed point are now points. We refer to, e.g., [GH] or [A1] for a precise description of this construction; here we shall limit ourselves to explain how to work with it. Definition 3.1: Let M be a complex n-dimensional manifold, and p ∈ M. The blow-up of M of center p is a complex n-dimensional manifold M˜ equipped with a surjective holomorphic map π:M˜ → M such that (i) E = π−1(p) is a compact submanifold of M˜, the exceptional divisor of the blow-up, biholomorphic to P(T M); p (ii) π| :M˜ \E →M \{p} is a biholomorphism. M˜\E Letusdescribetheconstructionfor(M,p)=(Cn,O); usinglocalchartsonecanrepeattheconstruction foranymanifold. Asaset,Cn isthe disjointunionofCn\{O}andE =Pn−1(C);weshalldefine amanifold structureusingcharts. Forj =1,...,nletU′ ={[v :···:v ]∈Pn−1(C)|v 6=0},U′′ ={w∈Cn |w 6=0} j 1 n j j j and U˜ =U′∪U′′ ⊂Cn. Deffine χ :U˜ →Cn by setting j j j j j f v1,...,vj−1,0,vj+1,...,vn if q =[v :···:v ]∈U′, χ (q)= vj vj vj vj 1 n j j (cid:16)w1,...,wj−1,w ,wj+1,...,(cid:17)wn if q =(w ,...,w )∈U′′.  wj wj j wj wj 1 n j (cid:16) (cid:17) We have  [w :···:w :1:w :···:w ] if w =0, χ−1(w)= 1 j−1 j+1 n j j (w w ...,w w ,w ,w w ,...,w w ) if w 6=0, j , j j−1 j j j+1 j n j (cid:26) and it is easy to check that {(U˜ ,χ ),...,(U˜ ,χ )} is an atlas for Cn, with χ ([0 : ··· : 1 : ··· : 0]) = O 1 1 n n j and χ U˜ ∩Pn−1(C) ={w =0}⊂Cn. We can then define the projection π:Cn →Cn in coordinates by j j j setting f (cid:0) (cid:1) π◦χ−1(w)=(w w ,...,w w ,w ,w w ,...,w w f); j 1 j j−1 j j j+1 j n j it is easy to check that π is well-defined, that π−1(O) = Pn−1(C) and that π induces a biholomorphism between Cn\Pn−1(C) and Cn\{O}. Notice furthermore that Cn has a canonical structure of line bundle over Pn−1(C) given by the projection π˜:Cn →Pn−1(C) defined by f f [v] if q =[v]∈Pn−1(C), π˜(q)= f [w :···:w ] if q =w ∈Cn\{O}; (cid:26) 1 n the fiber over [v]∈Pn−1(C) is given by the line Cv ⊂Cn. Two more definitions we shall need later on: Definition 3.2: Let π:M˜ → M be the blow-up of a complex manifold M at p ∈ M. Given a subset S ⊂ M, the full (or total) transform of S is π−1(S), whereas the strict transform of S is the closure in M˜ of π−1(S\{O}). Clearly, the full and the strict transform coincide if p∈/ S; if p∈S then the full transform is the union of the strict transform and the exceptional divisor. Furthermore, if S is a submanifold at p then its strict transform is (S\{p})∪P(T S). p 8 MarcoAbate Definition 3.3: Let f ∈ End(M,p) be a germ such that df is invertible. Choose a representative p (U,f) of the germ such that f is injective in U. Then the blow-up of f is the map f˜:π−1(U)→M˜ defined by [df (v)] if q =[v]∈E =P(T M), f˜(q)= p p f(w) if q =w ∈U \{p}. (cid:26) In this way we get a germ about the exceptional divisor of a holomorphic self-map of the blow-up, given by the differential of f along the exceptional divisor and by f itself elsewhere, satisfying π◦f˜= f ◦π. In particular, K = π−1(K ) = K \{O} ∪E, and to study the dynamics of f˜in a neighborhood of the f˜ f f exceptional divisor is equivalent to studying the dynamics of f in a neighborhood of p. (cid:0) (cid:1) If f ∈End (Cn,O) is tangent to the identity, its blow-up f˜in the chart (U ,χ ) is given by 1 1 1 w if w =0, 1 χ1◦f˜◦χ1−1(w)= f (w ,w w ,...,w w ),f2(w1,w1w2,...,w1wn),...,fn(w1,w1w2,...,w1wn) if w 6=0; ( 1 1 1 2 1 n f1(w1,w1w2,...,w1wn) f1(w1,w1w2,...,w1wn) 1 (cid:16) (cid:17) similarformulasholdinthe othercharts. Inparticular,writingw =(w ,w′)andχ ◦f˜◦χ−1 =(f˜,...,f˜ ), 1 1 1 1 n if f is tangent to the identity of order ν ≥1 and leading term P =(P ,...,P ), we get ν+1 ν+1,1 ν+1,n f˜(w)=w +wν+1P (1,w′)+O(wν+2), 1 1 1 ν+1,1 1 (3.1) f˜(w)=w +wν P (1.w′)−w P (1,w′) +O(wν+1) if j 6=1. (cid:26) j j 1 ν+1,j j ν+1,1 1 (cid:0) (cid:1) It follows immediately that: – if ν ≥2 then f˜is tangent to the identity in all points of the exceptional divisor; – if ν = 1 then f˜ is tangent to the identity in all characteristic directions of f; in other points of the exceptional divisor the eigenvalues of the differential of f˜are all equal to 1 but the differential is not diagonalizable. This means that we can always repeat the previous construction blowing-up f˜at a characteristic direction of f; this will be important in the sequel. As a first application of the blow-up construction, let us use it for describing the dynamics of dicritical maps. Iff ∈End (Cn,O)isdicritical,Theorem2.2yieldsaparaboliccurvetangenttoalldirectionsoutside 1 a hypersurface of Pn−1(C) (notice that all directors are zero), and the same holds for f−1. One can then summarize the situation as follows: Proposition 3.1: ([Br1, 2]) Let f ∈ End (Cn,O) be a dicritical germ tangent to the identity of or- 1 der ν ≥ 1. Write P (z) = p(z)z, and let D = {[v] ∈ Pn−1(C) | p(v) = 0}. Then there are two open sets ν+1 U+, U− ⊂Cn\{O} such that: (i) U+∪U− is a neighborhood of Pn−1(C)\D in the blow-up Cn of O; (ii) the orbit of any z ∈U+ converges to the origin tangent to a direction [v]∈Pn−1(C)\D; (iii) the inverse orbit (that is, the orbit under f−1) of any z ∈fU− converges to the origin tangent to a direction [v]∈Pn−1(C)\D. Coming back to the general situation, when f ∈ End (Cn,O) is tangent to the identity its blow-up f˜ 1 fixes pointwise the exceptional divisor; more precisely, the fixed point set of f˜is the full transform of the fixed point set of f, and in particular f˜has a at least a hypersurface of fixed points. This is a situation important enough to deserve a special notation. Definition 3.4: Let E be a connected (possibly singular) hypersurface in a complex manifold M. We shall denote by End(M,E) the set of germs about E of holomorphic self-maps of M fixing pointwise E. If E is a hypersurface in a complex manifold M, we shall denote by O the sheaf of holomorphic M functions on M, and by I the subsheaf of functions vanishing on E. Given f ∈End(M,E), f 6≡id , take E M p∈E. For every h∈O , the germ h◦f is well-defined, and h◦f −h∈I . Following [ABT1] (see also M,p E,p [ABT2, 3]), we can then introduce a couple of important notions. Fatouflowersandparaboliccurves 9 Definition 3.5: Let E be a connected hypersurface in a complex manifold M. Given f ∈End(M,E), p∈E and h∈O , let ν (h)=max{µ∈N|h◦f −h∈Iµ }. Then the order of contactν of f with the M,p f E,p f identity along E is ν =min{ν (h)|h∈O }; f f M,p it can be shown ([ABT1]) that ν does not depend on p ∈ E. Furthermore, we say that f is tangential if f min{ν (h)|h∈I }>ν for some (and hence any; see again [ABT1]) p∈E. f E,p f Let (z ,...,z ) be localcoordinatesin M centeredat p∈E, andℓ∈I a reduced equationofE atp 1 n E,p (that is, a generator of I ). If (f ,...,f ) is the expression of f in local coordinates, it turns out [ABT1] E,p 1 n that we can write f (z)=z +ℓ(z)νfgo(z) (3.2) j j j for j =1,...,n, where there is a j such that ℓ does not divide go ; furthermore, f is tangential if and only 0 j0 if ν (ℓ)>ν . f f Remark3.1: IfE issmoothatp,wecanchooselocalcoordinatessothatlocallyE isgivenby{z =0}, 1 that is ℓ=z . Then we can write 1 f (z)=z +zνfgo(z) j j 1 j with z not dividing some go; and f is tangential if z divides go, that is if f (z) = z + zνf+1ho(z). 1 j 1 1 1 1 1 1 More generally, if E has a normal crossing at p with 1 ≤ r ≤ n smooth branches, then we can choose local coordinatessothatℓ=z1···zr,sothatfj(z)=zj+(z1···zr)νfgjo(z)withsomegjo0 notdivisiblebyz1···zr; in this case f is tangential if and only if z divides go for j = 1,...,r. In particular, in the terminology of j j [A2] f is tangential if and only if it is nondegenerate and b =1. f Definition 3.6: We say that p ∈ E is a singular point for f ∈ End(M,E) (with respect to E) if go(p)=···=go(p) in (3.2); it turns out [ABT1] that this definition is independent of the localcoordinates. 1 n Furthermore, the pure order (or pure multiplicity) ν (f,E) of f along E at p is o ν (f,E)=min{ord (go),...,ord (go)}. o O 1 O n It is easy to see that the pure order does not depend on the local coordinates;in particular, p is singular for f with respect to E if and only if ν (f,E)≥1. If E is the fixed point set of f at p then we shall talk of the o pure order ν (f) of f at p. o Remark3.2: Whenf istheblow-upofagermf ∈End (Cn,O)tangenttotheidentityoforderν ≥1, o 1 then (3.1) implies that: – f is tangential if and only if f is not dicritical; in particular, in this case being tangential is a generic o condition; – ν =ν if f is not dicritical, and ν =ν+1 if f is dicritical; f o f o – if f is non dicritical, then [v] ∈ Pn−1(C) is singular for f if and only if it is a characteristic direction o of f . o Using the notion of singular points we can generalize Proposition 2.1 as follows: Proposition 3.2: ([ABT1]) Let E ⊂M be a hypersurface in a complex manifold M, and f ∈End(M,E), f 6≡ id , tangential to E. Let p ∈ E be a smooth point of E which is non-singular for f. Then no infinite M orbit of f can stay arbitrarily close to p, that is, there exists a neighborhood U of p such that for all q ∈U either the orbit of q lands on E or fn0(q)∈/ U for some n0 ∈N. In particular,no infinite orbit is converging to p. More generally, we have: Proposition 3.3: ([AT1]) Let f ∈End(Cn,O) be of the form z +z ( r zνh)g (z) for 1≤j ≤r, f (z)= j j h=1 h j (3.3) j z +( r zνh)g (z) for r+1≤j ≤n, (cid:26) j hQ=1 h j for suitable 1 ≤ r < n, with ν1,...,νr ≥Q1 and g1,...,gn ∈ OCn,O. Assume that gj0(O) 6= 0 for some r+1≤j ≤n. Then no infinite orbit can stay arbitrarily close to O. 0 A very easy example of this phenomenon, promised at the end of the previous section, is the following: 10 MarcoAbate Example 3.1: Let f(z,w) = (z,w+z2). Then f is tangent to the identity at the origin; the fixed pointsetis{z =0},andthus O is notanisolatedfixedpoint. We havefk(z,w)=(z,w+kz2); thereforeall orbitsoutsidethe fixedpointsetescapeto infinity,andinparticularnoorbitconvergesto theorigin. Notice thatthisgermhasonlyonecharacteristicdirection,whichisdegenerate(andtangenttothefixedpointset). Moreover, f is tangential with order of contact 2 to its fixed point set, but the origin is not singular. After these generalities, in the rest of this section we specialize to the case n = 2 and to tangential maps(because ofRemark3.2;see anyway[ABT1]forinformationonthe dynamicsofnon-tangentialmaps). Take f ∈End(M,E), where M is a complex surface and E ⊂M is a 1-dimensional curve smooth at p∈E, and assume that f is tangential to E with order of contact ν ≥ 1. Then we can choose local coordinates f centered at p so that we can write f (z)=z +zνf+1ho(z), 1 1 1 1 (3.4) f (z)=z +zνfgo(z), (cid:26) 2 2 1 2 where z does not divide go; notice that ho(0,·)= ∂g1o(0,·), where go =z ho. In particular, O is singular if and only1 if go(O)=0. We2then introduce1the follow∂izn1g definitions: 1 1 1 2 Definition 3.7: Let f ∈End(M,E) be written in the form (3.4). Then: – the multiplicity µ of f along E at p is µ = ord go(0,·) , so that p is a singular point if and only if p p 0 2 µ ≥1; p (cid:0) (cid:1) – the transversalmultiplicity τ of f along E at p is τ =ord ho(0,·) ; p p 0 1 – p is an apparent singularity if 1≤µ ≤τ ; p p (cid:0) (cid:1) – p is a Fuchsian singularity if µ =τ +1; p p – p is an irregular singularity if µ >τ +1; p p – p is a non-degenerate singularity if µ ≥1 but τ =0; p p – p is a degenerate singularity if µ , τ ≥1; p p – the index ι (f,E) of f at p along E is p ho(0,·) ι (f,E)=ν Res 1 ; p f 0go(0,·) 2 – the induced residue Res0(f) of f along E at p is p Res0(f)=−ι (f,E)−µ . p p p It is possible to prove(see [A2, ABT1,AT3]; notice that our index is ν times the residualindex introduced f in [A2]) that these definitions are independent of the local coordinates; see also Remark 4.3. Remark 3.3: Recalling (3.1), we see that if f is obtained as the blow-up of a non-dicritical map f , o and E is the exceptional divisor of the blow-up, then: – the multiplicity of [v] as characteristic direction of f is equal to the multiplicity of f along E at [v]; o – [v] is a degenerate/non-degenerate characteristic direction of f if and only if it is a degenerate/non- o degenerate singularity of f. Furthermore, if we write P (1,w) = ν+1a wk and P (1,w) = ν+1b wk then [1: 0] is a charac- ν+1,1 k=0 k ν+1,2 k=0 k teristic direction if and only if b =0, non-degenerateif and only if moreovera 6=0, and (setting b =0) 0 0 ν+2 P P ho(0,ζ) 1 a + ν+1a ζk 1 = 0 k=1 k . go(0,ζ) ζ (b −a )+ ν+1(b −a )ζk 2 1 0 Pk=1 k+1 k P So if b 6=a we have 1 0 νa (ν−1)a +b µ =1, ι (f,E)= 0 , Res0 (f)= 0 1 ; O [1:0] b −a [1:0] a −b 1 0 0 1