Discrete holomorphic local dynamical systems Marco Abate Dipartimento di Matematica, Universita` di Pisa Largo Pontecorvo 5, 56127 Pisa, Italy E-mail: [email protected] November 2008 1. Introduction Let us begin by defining the main object of study in this survey. Definition 1.1: Let M be a complex manifold, and p M. A (discrete) holomorphic local dynamical ∈ system at p is a holomorphic map f:U M such that f(p) = p, where U M is an open neighbourhood → ⊆ ofp;weshallalsoassumethatf id . WeshalldenotebyEnd(M,p)thesetofholomorphiclocaldynamical U 6≡ systems at p. Remark 1.1: Since we are mainly concerned with the behavior of f nearby p, we shall sometimes replace f by its restrictionto some suitableopen neighbourhood of p. It is possible to formalizethis fact by using germs of maps and germs of sets at p, but for our purposes it will be enough to use a somewhat less formal approach. Remark 1.2: In this survey we shall never have the occasion of discussing continuous holomorphic dynamical systems (i.e., holomorphic foliations). So from now on all dynamical systems in this paper will be discrete, except where explicitly noted otherwise. Totalkaboutthedynamicsofanf End(M,p)weneedtodefinetheiteratesoff. Iff isdefinedonthe ∈ setU,thentheseconditeratef2 =f f isdefinedonU f 1(U)only,whichstillisanopenneighbourhood − ◦ ∩ of p. More generally, the k-th iterate fk = f fk 1 is defined on U f 1(U) f (k 1)(U). This − − − − ◦ ∩ ∩···∩ suggests the next definition: Definition 1.2: Let f End(M,p) be a holomorphic local dynamical system defined on an open set ∈ U M. Then the stable set K of f is f ⊆ 1 K = f k(U). f − k=0 \ In other words, the stable set of f is the set of all points z U such that the orbit fk(z) k N is ∈ { | ∈ } well-defined. If z U K , we shall say that z (or its orbit) escapes from U. f ∈ \ Clearly, p K , and so the stable set is never empty (but it can happen that K = p ; see the next f f ∈ { } section for an example). Thus the first natural question in local holomorphic dynamics is: (Q1) What is the topological structure of K ? f For instance, when does K have non-empty interior? As we shall see in Proposition 4.1, holomorphic local f dynamical systems such that p belongs to the interior of the stable set enjoy special properties. Remark 1.3: Both the definition of stable set and Question 1 (as well as several other definitions and questionsweshallseelateron)aretopologicalincharacter;wemightstatethemforlocaldynamicalsystems whicharecontinuousonly. Asweshallsee,however,theanswerswillstronglydependontheholomorphicity of the dynamical system. Definition 1.3: Given f End(M,p), a set K M is completely f-invariant if f 1(K) = K (this − ∈ ⊆ implies, in particular, that K is f-invariant, that is f(K) K). ⊆ Clearly, the stable set K is completely f-invariant. Therefore the pair (K ,f) is a discrete dynamical f f system in the usual sense, and so the second natural question in local holomorphic dynamics is (Q2) What is the dynamical structure of (K ,f)? f 2 MarcoAbate For instance, what is the asymptotic behavior of the orbits? Do they converge to p, or have they a chaotic behavior? Is there a dense orbit? Do there exist proper f-invariant subsets, that is sets L K such f ⊂ that f(L) L? If they do exist, what is the dynamics on them? ⊆ To answer all these questions, the most efficient way is to replace f by a “dynamically equivalent” but simpler (e.g., linear) map g. In our context, “dynamically equivalent” means “locally conjugated”; and we have at least three kinds of conjugacy to consider. Definition 1.4: Let f :U M and f :U M be two holomorphic local dynamical systems 1 1 1 2 2 2 → → at p M and p M respectively. We shall say that f and f are holomorphically (respectively, 1 1 2 2 1 2 ∈ ∈ topologically) locally conjugated if there are open neighbourhoods W U of p , W U of p , and a 1 1 1 2 2 2 ⊆ ⊆ biholomorphism (respectively, a homeomorphism) ϕ:W W with ϕ(p )=p such that 1 2 1 2 → f1 =ϕ−1◦f2◦ϕ on ϕ−1 W2∩f2−1(W2) =W1∩f1−1(W1). ° ¢ If f :U M and f :U M are locally conjugated, in particular we have 1 1 1 2 2 2 → → ∀k ∈N f1k =ϕ−1◦f2k◦ϕ on ϕ−1 W2∩···∩f2−(k−1)(W2) =W1∩···∩f1−(k−1)(W1), ° ¢ and thus K =ϕ(K ). f2|W2 f1|W1 So the local dynamics of f about p is to all purposes equivalent to the local dynamics of f about p . 1 1 2 2 Remark 1.4: Using local coordinates centered at p M it is easy to show that any holomorphic ∈ local dynamical system at p is holomorphicallylocally conjugated to a holomorphic local dynamical system at O Cn, where n=dimM. ∈ Wheneverwehaveanequivalencerelationinaclassofobjects,thereareclassificationproblems. Sothe third natural question in local holomorphic dynamics is (Q3) Find a (possibly small) class of holomorphic local dynamical systems at O Cn such that every holo- F ∈ morphic local dynamical system f at a point in an n-dimensional complex manifold is holomorphically (respectively,topologically)locallyconjugatedtoa(possibly)uniqueelementof ,calledtheholomorphic F (respectively, topological) normal form of f. Unfortunately,theholomorphicclassificationisoftentoocomplicatedtobepractical;thefamily ofnormal F forms might be uncountable. A possible replacement is looking for invariants instead of normal forms: (Q4) Find a way to associate a (possibly small) class of (possibly computable) objects, called invariants, to any holomorphic local dynamical system f at O Cn so that two holomorphic local dynamical systems ∈ at O can be holomorphically conjugated only if they have the same invariants. The class of invariants is furthermore said complete if two holomorphic local dynamical systems at O are holomorphically conjugated if and only if they have the same invariants. Asremarkedbefore,uptonowallthequestionsweaskedmadesensefortopologicallocaldynamicalsystems; the next one instead makes sense only for holomorphic local dynamical systems. AholomorphiclocaldynamicalsystematO Cn is clearlygivenby anelement ofC0 z1,...,zn n, the ∈ { } space of n-uples of converging power series in z1,...,zn without constant terms. The space C0 z1,...,zn n { } is a subspace of the space C0[[z1,...,zn]]n of n-uples of formal power series without constant terms. An elementΦ C0[[z1,...,zn]]n hasaninverse(withrespecttocomposition)stillbelongingtoC0[[z1,...,zn]]n if and only∈if its linear part is a linear automorphism of Cn. Definition 1.5: We shall say that two holomorphic local dynamical systems f1, f2 C0 z1,...,zn n ∈ { } are formally conjugated if there exists an invertible Φ C0[[z1,...,zn]]n such that f1 = Φ−1 f2 Φ ∈ ◦ ◦ in C0[[z1,...,zn]]n. It is clear that two holomorphically locally conjugated holomorphic local dynamical systems are both formally and topologically locally conjugated too. On the other hand, we shall see examples of holomor- phic local dynamical systems that are topologically locally conjugated without being neither formally nor holomorphicallylocallyconjugated,andexamplesof holomorphiclocal dynamicalsystemsthatare formally Discreteholomorphiclocaldynamicalsystems 3 conjugated without being neither holomorphically nor topologically locally conjugated. So the last natural question in local holomorphic dynamics we shall deal with is (Q5) Find normal forms and invariants with respect to the relation of formal conjugacy for holomorphic local dynamical systems at O Cn. ∈ In this survey we shall present some of the main results known on these questions, starting from the one- dimensionalsituation. ButbeforeenteringthemaincoreofthepaperIwouldliketoheartilythankFranc¸ois Berteloot, Kingshook Biswas, Filippo Bracci, Santiago Diaz-Madrigal, Graziano Gentili, Giorgio Patrizio, Mohamad Pouryayevali, Jasmin Raissy and Francesca Tovena, without whom none of this would have been written. 2. One complex variable: the hyperbolic case Let us then start by discussingholomorphiclocal dynamicalsystems at 0 C. As remarked in the previous ∈ section, such a system is given by a converging power series f without constant term: f(z)=a1z+a2z2+a3z3+ C0 z . ···∈ { } Definition 2.1: The number a =f (0) is the multiplier of f. 1 0 Sincea z is the best linearapproximationof f, it is sensibleto expect thatthe local dynamicsof f will 1 be strongly influenced by the value of a . For this reason we introduce the following definitions: 1 Definition 2.2: Let a1 C be the multiplier of f End(C,0). Then ∈ ∈ – if a <1 we say that the fixed point 0 is attracting; 1 | | – if a =0 we say that the fixed point 0 is superattracting; 1 – if a >1 we say that the fixed point 0 is repelling; 1 | | – if a =0, 1 we say that the fixed point 0 is hyperbolic; 1 | |6 – if a S1 is a root of unity, we say that the fixed point 0 is parabolic (or rationally indifferent); 1 ∈ – if a S1 is not a root of unity, we say that the fixed point 0 is elliptic (or irrationally indifferent). 1 ∈ As we shall see in a minute, the dynamics of one-dimensional holomorphic local dynamical systems with a hyperbolic fixed point is pretty elementary; so we start with this case. Remark 2.1: Notice that if 0 is an attracting fixed point for f End(C,0) with non-zero multiplier, ∈ then it is a repelling fixed point for the inverse map f−1 End(C,0). ∈ Assume first that 0 is attracting for the holomorphic local dynamical system f End(C,0). Then ∈ we can write f(z) = a z +O(z2), with 0 < a < 1; hence we can find a large constant M > 0, a small 1 1 | | constant ε>0 and 0<δ <1 such that if z <ε then | | f(z) (a +Mε)z δ z . (2.1) 1 | |≤ | | | |≤ | | In particular, if ∆ denotes the disk of center 0 and radius ε, we have f(∆ ) ∆ for ε >0 small enough, ε ε ε ⊂ andthestablesetoff is∆ itself(inparticular,aone-dimensionalattractingfixedpointisalwaysstable). |∆ε ε Furthermore, fk(z) δk z 0 | |≤ | |→ as k + , and thus every orbit starting in ∆ is attracted by the origin, which is the reason of the name ε → 1 “attracting” for such a fixed point. Ifinstead0isarepellingfixedpoint,asimilarargument(ortheobservationthat0isattractingforf 1) − shows that for ε > 0 small enough the stable set of f reduces to the origin only: all (non-trivial) orbits |∆ε escape. Itisalsonotdifficulttofindholomorphicandtopologicalnormalformsforone-dimensionalholomorphic local dynamical systems with a hyperbolic fixed point, as shown in the following result, which can be considered as the beginning of the theory of holomorphic dynamical systems: 4 MarcoAbate Theorem 2.1: (Kœnigs, 1884 [Kœ]) Let f End(C,0) be a one-dimensional holomorphic local dynamical ∈ system with a hyperbolic fixed point at the origin, and let a1 C∗ S1 be its multiplier. Then: ∈ \ (i) f is holomorphically (and hence formally) locally conjugated to its linear part g(z) = a z. The conju- 1 gation ϕ is uniquely determined by the condition ϕ(0)=1. 0 (ii) Two such holomorphiclocal dynamicalsystemsareholomorphicallyconjugatedif andonlyif theyhave the same multiplier. (iii) f is topologically locally conjugated to the map g (z) = z/2 if a < 1, and to the map g (z) = 2z < 1 > | | if a >1. 1 | | Proof: Let us assume 0< a <1; if a >1 it will suffice to apply the same argument to f 1. 1 1 − | | | | (i) Choose 0 < δ < 1 such that δ2 < a < δ. Writing f(z) = a z+z2r(z) for a suitable holomorphic 1 1 | | germ r, we can clearly find ε>0 such that a +Mε<δ, where M =max r(z). So we have | 1| z∈∆ε| | f(z) a z M z 2 1 | − |≤ | | and fk(z) δk z | |≤ | | for all z ∆ε and k N. ∈ ∈ Put ϕk =fk/ak1; we claim that the sequence {ϕk} converges to a holomorphic map ϕ:∆ε →C. Indeed we have 1 M M δ2 k ϕ (z) ϕ (z) = f fk(z) a fk(z) fk(z)2 z 2 | k+1 − k | a k+1 − 1 ≤ a k+1| | ≤ a a | | | 1| | 1| | 1|µ| 1|∂ Ø ° ¢ Ø for all z ∈∆ε, and so the telescopic seØries k(ϕk+1−ϕk)Øis uniformly convergent in ∆ε to ϕ−ϕ0. Since ϕ0k(0)=1 for all k ∈N, we havePϕ0(0)=1 and so, up to possibly shrink ε, we can assume that ϕ is a biholomorphism with its image. Moreover, we have fk f(z) fk+1(z) ϕ f(z) = lim =a lim =a ϕ(z), k→+1 °ak1 ¢ 1k→+1 ak1+1 1 that is f =ϕ 1 g ϕ, as°claim¢ed. − ◦ ◦ If √ is another local holomorphic function such that √ (0) = 1 and √ 1 g √ = f, it follows that 0 − ◦ ◦ √ ϕ 1(∏z)=∏√ ϕ 1(z); comparingtheexpansioninpowerseriesofbothsideswefind√ ϕ 1 id, that − − − ◦ ◦ ◦ ≡ is √ ϕ, as claimed. ≡ (ii) Since f = ϕ 1 f ϕ implies f (0) = f (0), the multiplier is invariant under holomorphic local 1 − ◦ 2 ◦ 10 20 conjugation,andsotwoone-dimensionalholomorphiclocaldynamicalsystemswithahyperbolicfixedpoint are holomorphically locally conjugated if and only if they have the same multiplier. (iii) Since a < 1 it is easy to build a topological conjugacy between g and g on ∆ . First choose 1 < ε | | a homeomorphism χ between the annulus a ε z ε and the annulus ε/2 z ε which is the 1 {| | ≤ | | ≤ } { ≤ | | ≤ } identity on the outer circle and given by χ(z)=z/(2a ) on the inner circle. Now extend χ by induction to 1 a homeomorphism between the annuli a kε z a k 1ε and ε/2k z ε/2k 1 by prescribing 1 1 − − {| | ≤| |≤| | } { ≤| |≤ } χ(a z)= 1χ(z). 1 2 Putting finally χ(0) = 0 we then get a homeomorphism χ of ∆ with itself such that g = χ 1 g χ, as ε − < ◦ ◦ required. Remark 2.2: Noticethatg (z)= 1z andg (z)=2z cannotbetopologicallyconjugated,because(for < 2 > instance) K is open whereas K = 0 is not. g< g> { } Remark 2.3: The proof of this theorem is based on two techniques often used in dynamics to build conjugations. The first one is used in part (i). Suppose that we would like to prove that two invertible local dynamical systems f, g End(M,p) are conjugated. Set ϕ =g k fk, so that k − ∈ ◦ ϕ f =g k fk+1 =g ϕ . k − k+1 ◦ ◦ ◦ Therefore if we can prove that ϕ converges to an invertible map ϕ as k + we get ϕ f =g ϕ, and k { } → 1 ◦ ◦ thus f and g are conjugated, as desired. This is exactly the way we proved Theorem 2.1.(i); and we shall see variations of this techniques later on. To describe the second technique we need a definition. Discreteholomorphiclocaldynamicalsystems 5 Definition2.3: Letf:X X beanopencontinuousself-mapofatopologicalspaceX. Afundamental → domain for f is an open subset D X such that ⊂ (i) fh(D) fk(D)=∅ for every h=k N; ∩ 6 ∈ (ii) fk(D)=X; k N (iii) ifS∈z1, z2 D are so that fh(z1)=fk(z2) for some h>k N then h=k+1 and z2 =f(z1) @D. ∈ ∈ ∈ There are other possible definitions of a fundamental domain, but this will work for our aims. Suppose that we would like to prove that two open continuous maps f :X X and f :X X 1 1 1 2 2 2 → → are topologically conjugated. Assume we have fundamental domains D X for f (with j = 1, 2) and a j j j ⊂ homeomorphism χ:D D such that 1 2 → χ f =f χ (2.2) 1 2 ◦ ◦ onD1∩f1−1(D1). Thenwecanextendχtoahomeomorphismχ˜:X1 →X2 conjugatingf1 andf2 bysetting z X χ˜(z)=fk χ(w) , (2.3) ∀ ∈ 1 2 ° ¢ wherek =k(z) Nandw =w(z) D arechosensothatfk(w)=z. Thedefinitionoffundamentaldomain ∈ ∈ 1 and (2.2) implythat χ˜ is well-defined. Clearlyχ˜ f =f χ˜; and using the openness of f and f it is easy 1 2 1 2 ◦ ◦ to check that χ˜ is a homeomorphism. This is the technique we used in the proof of Theorem 2.1.(iii); and we shall use it again later. Thusthedynamicsintheone-dimensionalhyperboliccaseiscompletelyclear. Thesuperattractingcase can be treated similarly. If 0 is a superattracting point for an f End(C,0), we can write ∈ f(z)=a zr+a zr+1+ r r+1 ··· with a =0. r 6 Definition 2.4: The number r 2 is the order (or local degree) of the superattracting point. ≥ Anargumentsimilartotheonedescribedbeforeshowsthatforε>0smallenoughthestablesetoff |∆ε still is all of ∆ , and the orbits converge (faster than in the attracting case) to the origin. Furthermore, we ε can prove the following Theorem 2.2: (Bo¨ttcher, 1904 [Bo¨]) Let f End(C,0) be a one-dimensionalholomorphiclocal dynamical ∈ system with a superattracting fixed point at the origin, and let r 2 be its order. Then: ≥ (i) f isholomorphically(andhenceformally)locallyconjugatedtothemapg(z)=zr,andtheconjugation is unique up to multiplication by an (r 1)-root of unity; − (ii) two such holomorphiclocal dynamicalsystems are holomorphically(or topologically)conjugatedif and only if they have the same order. Proof: First of all, up to a linear conjugation z µz with µr 1 =a we can assume a =1. − r r 7→ Now write f(z)=zrh (z) for a suitable holomorphic germ h with h (0)=1. By induction, it is easy 1 1 1 to see that we can write fk(z)=zrkh (z) for a suitableholomorphicgerm h with h (0)=1. Furthermore, k k k the equalities f fk 1 =fk =fk 1 f yields − − ◦ ◦ hk 1(z)rh1 fk−1(z) =hk(z)=h1(z)rk−1hk 1 f(z) . (2.4) − − ° ¢ ° ¢ Choose 0 < δ < 1. Then we can clearly find 1 > ε > 0 such that Mε < δ, where M = max h (z); we z∈∆ε| 1 | can also assume that h (z)=0 for all z ∆ . Since 1 ε 6 ∈ z ∆ f(z) M z r <δ z r 1 , ε − ∀ ∈ | |≤ | | | | we have f(∆ ) ∆ , as anticipated before. ε ε ⊂ We also remark that (2.4) implies that each h is well-defined and never vanishing on ∆ . So for ev- k ε eryk 1wecanchooseaunique√ holomorphicin∆ suchthat√ (z)rk =h (z)on∆ andwith√ (0)=1. k ε k k ε k ≥ 6 MarcoAbate Set ϕ (z) = z√ (z), so that ϕ (0) = 1 and ϕ (z)rk = f (z) on ∆ ; in particular, formally we have k k 0k k k ε ϕ =g k fk. We claim that the sequence ϕ converges to a holomorphic function ϕ on ∆ . Indeed, we k − k ε ◦ { } have ϕk+1(z) = √k+1(z)rk+1 1/rk+1 = hk+1(z) 1/rk+1 = h fk(z) 1/rk+1 ØØØØ ϕk(z) ØØØØ=ØØØØØ1√+kO(z)rfk+k(1z)ØØØØØ 1/rk+1 =ØØØØ h1k+(z)r1ØØØØ O fk(zØØ)1°=1+¢OØØ 1 , | | rk+1 | | rk+1 µ ∂ Ø ° ¢Ø ° ¢ and so the telescopic producØt (ϕ /ϕ )Øconverges to ϕ/ϕ uniformly in ∆ . k k+1 k 1 ε Since ϕ0k(0)=1 for all k ∈QN, we have ϕ0(0)=1 and so, up to possibly shrink ε, we can assume that ϕ is a biholomorphism with its image. Moreover, we have ϕ f(z) rk =f(z)rk√ f(z) rk =zrkh (z)rkh f(z) =zrk+1h (z)= ϕ (z)r rk , k k 1 k k+1 k+1 and thus ϕ °f =¢[ϕ ]r. Passin°g to t¢he limit we get f =° ϕ 1¢ g ϕ, as claimed.£ § k k+1 − ◦ ◦ ◦ If √ is anotherlocal biholomorphismconjugatingf with g, we must have √ ϕ 1(zr)=√ ϕ 1(z)r for − − ◦ ◦ all z in a neighbourhood of the origin; comparingthe series expansionsat the origin we get √ ϕ 1(z)=az − ◦ with ar 1 =1, and hence √(z)=aϕ(z), as claimed. − Finally, (ii) follows because zr and zs are locally topologically conjugated if and only if r =s (because the order is the number of preimages of points close to the origin). Thereforetheone-dimensionallocaldynamicsaboutahyperbolicorsuperattractingfixedpoint iscom- pletely clear; let us now discuss what happens about a parabolic fixed point. 3. One complex variable: the parabolic case Let f End(C,0) be a (non-linear)holomorphiclocal dynamicalsystem with a parabolic fixed point at the ∈ origin. Then we can write f(z)=e2iπp/qz+a zr+1+a zr+2+ , (3.1) r+1 r+2 ··· with a =0. r+1 6 Definition 3.1: The rational number p/q Q [0,1) is the rotation number of f, and the number ∈ ∩ r+1 2 is the multiplicity of f at the fixed point. If p/q = 0 (that is, if the multiplier is 1), we shall say ≥ that f is tangent to the identity. The first observation is that such a dynamical system is never locally conjugated to its linear part, not even topologically, unless it is of finite order: Proposition3.1: Letf End(C,0)beaholomorphiclocaldynamicalsystemwithmultiplier∏,andassume ∈ that ∏ = e2iπp/q is a primitive root of the unity of order q. Then f is holomorphically (or topologically or formally) locally conjugated to g(z)=∏z if and only if fq id. ≡ Proof: If ϕ 1 f ϕ(z)=e2πip/qz then ϕ 1 fq ϕ=id, and hence fq =id. − − ◦ ◦ ◦ ◦ Conversely, assume that fq id and set ≡ 1q−1fj(z) ϕ(z)= . q ∏j j=0 X Thenit is easy to check thatϕ(0)=1 andϕ f(z)=∏ϕ(z), andso f is holomorphically(andtopologically 0 ◦ and formally) locally conjugated to ∏z. In particular, if f is tangent to the identity then it cannot be locally conjugated to the identity (unless itwastheidentitytobeginwith,whichisnotaveryinterestingcasedynamicallyspeaking). Moreprecisely, the stable set of such an f is never a neighbourhood of the origin. To understand why, let us first consider a map of the form f(z)=z(1+azr) Discreteholomorphiclocaldynamicalsystems 7 for some a=0. Let v S1 C be such that avr is real and positive. Then for any c>0 we have 6 ∈ ⊂ f(cv)=c(1+cravr)v R+v; ∈ moreover, f(cv) > cv . In other words, the half-line R+v is f-invariant and repelled from the origin, that is Kf R+|v = ∅|. C|on|versely, if avr is real and negative then the segment [0, a−1/r]v is f-invariant and ∩ | | attracted by the origin. So K neither is a neighbourhood of the origin nor reduces to 0 . f { } This example suggests the following definition: Definition 3.2: Let f End(C,0) be tangent to the identity of multiplicity r+1 2. Then a unit ∈ ≥ vector v S1 is an attracting (respectively, repelling) direction for f at the origin if a vr is real and r+1 ∈ negative (respectively, positive). Clearly, there are r equally spaced attracting directions, separated by r equally spaced repelling direc- tions: if a = a eiα, then v =eiθ is attracting (respectively, repelling) if and only if r+1 r+1 | | 2k+1 α 2k α θ = π respectively, θ = π . r − r √ r − r! Furthermore, a repelling (attracting) direction for f is attracting (repelling) for f 1, which is defined in a − neighbourhood of the origin. It turns out that to every attracting direction is associated a connected component of K 0 . f \{ } Definition 3.3: Let v S1 be an attracting direction for an f End(C,0) tangent to the identity. ∈ ∈ The basin centered at v is the set of points z K 0 such that fk(z) 0 and fk(z)/fk(z) v (notice f ∈ \{ } → | |→ that, up to shrinking the domain of f, we can assume that f(z) = 0 for all z K 0 ). If z belongs to f 6 ∈ \{ } 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 3.4: An attracting petal centered at an attracting direction v of an f End(C,0) tangent ∈ to the identity is an open simply connected f-invariant set P K 0 such that a point z K 0 f f ⊆ \{ } ∈ \{ } belongs to the basin centered at v if and only if its orbit intersects P. In other words, the orbit of a point tendsto0tangenttov ifandonlyifitiseventuallycontainedinP. Arepellingpetal(centeredatarepelling direction) is an attracting petal for the inverse of f. It turns out that the basins centered at the attractingdirections are exactly the connected components of K 0 , as shown in the Leau-Fatou flower theorem: f \{ } Theorem3.2: (Leau,1897[L];Fatou,1919-20[F1–3]) Letf End(C,0)beaholomorphiclocaldynamical system tangent to the identity with multiplicity r+1 2 at ∈the fixed point. Let v+,...,v+ S1 be the r ≥ 1 r ∈ attracting directions of f at the origin, and v1−,...,vr− ∈S1 the r repelling directions. Then (i) for each attracting (repelling) direction vj± there exists an attracting (repelling) petal Pj±, so that the union of these 2r petals together with the origin forms a neighbourhood of the origin. Furthermore, the 2r petals are arranged ciclically so that two petals intersect if and only if the angle between their central directions is π/r. (ii) K 0 is the (disjoint) union of the basins centered at the r attracting directions. f \{ } (iii) If B is a basin centered at one of the attracting directions, then there is a function ϕ:B C such that → ϕ f(z)=ϕ(z)+1 for all z B. Furthermore, if P is the corresponding petal constructed in part (i), ◦ ∈ then ϕ 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 z+1. P | 7→ Proof: Up to a linear conjugation, we can assume that a = 1, so that the attracting directions are the r+1 − r-th roots of unity. For any δ >0, the set z C zr δ <δ has exactly r connected components, each { ∈ || − | } one symmetric with respect to a different r-th root of unity; it will turn out that, for δ small enough, these connected components are attracting petals of f, even though to get a pointed neighbourhood of the origin we shall need larger petals. 8 MarcoAbate For j =1,...,r let Σj ⊂C∗ denote the sector centered about the attractive direction vj+ and bounded by two consecutive repelling directions, that is 2j 3 2j 1 Σj = z C∗ − π <argz < − π . ∈ r r Ω Ø æ Ø Ø Notice that each Σj contains a unique connecØted component Pj,δ of z C zr δ <δ ; moreover, Pj,δ is { ∈ || − | } tangent at the origin to the sector centered about v of amplitude π/r. j The main technical trick in this proof consists in transfering the setting to a neighbourhood of infinity in the Riemann sphere P1(C). Let √:C∗ C∗ be given by → 1 √(z)= ; rzr itisabiholomorphismbetweenΣj andC∗ R−, withinverse√−1(w)=(rw)−1/r, choosingsuitablyther-th \ root. Furthermore, √(Pj,δ) is the right half-plane Hδ = w C Rew >1/(2rδ) . { ∈ | } When w issolargethat√ 1(w)belongstothedomainofdefinitionoff,thecompositionF =√ f √ 1 − − | | ◦ ◦ makes sense, and we have F(w)=w+1+O(w 1/r). (3.2) − Thus to study the dynamics of f in a neighbourhood of the origin in Σ it suffices to study the dynamics j of F in a neighbourhood of infinity. The first observation is that when Rew is large enough then 1 ReF(w)>Rew+ ; 2 this implies that for δ small enough H is F-invariant (and thus P is f-invariant). Furthermore, by δ j,δ induction one has k w H ReFk(w)>Rew+ , (3.3) δ ∀ ∈ 2 which implies that Fk(w) in H (and fk(z) 0 in P ) as k . δ j,δ →1 → →1 Now we claim that the argument of w =Fk(w) tends to zero. Indeed, (3.2) and (3.3) yield k k 1 wk = w +1+ 1 − O(w−1/r); k k k l l=0 X so Cesaro’s theorem on the averages of a converging sequence implies w k 1, (3.4) k → and thus argw 0 as k . Going back to P , this implies that fk(z)/fk(z) v for every z P . k j,δ j j,δ Since furthermor→e P is c→ent1ered about v+, every orbit converging to 0 tan|gent t|o→v+ must intersec∈t P , j,δ j j j,δ and thus we have proved that P is an attracting petal. j,δ Arguing in the same way with f 1 we get repelling petals; unfortunately, the petals obtained so far − are too small to form a full pointed neighbourhood of the origin. In fact, as remarked before each P j,δ is contained in a sector centered about v of amplitude π/r; therefore the repelling and attracting petals j obtained in this way do not intersect but are tangent to each other. We need larger petals. So our aim is to find an f-invariant subset P+ of Σ containing P and which is tangent at the origin j j j,δ to a sector centered about v+ of amplitudestrictlygreaterthan π/r. To do so, first of all remarkthat there j are R, C >0 such that C F(w) w 1 (3.5) | − − |≤ w 1/r | | Discreteholomorphiclocaldynamicalsystems 9 as soon as w > R. Choose ε (0,1) and select δ > 0 so that 4rδ < R 1 and ε > 2C(4rδ)1/r. Then − | | ∈ w >1/(4rδ) implies | | F(w) w 1 <ε/2. | − − | Set M =(1+ε)/(2rδ) and let ε H˜ε = w C Imw > εRew+Mε Hδ . { ∈ || | − }∪ If w H˜ we have w >1/(2rδ) and hence ε ∈ | | ReF(w)>Rew+1 ε/2 and ImF(w) Imw <ε/2; (3.6) − | − | it is then easy to check that F(H˜ ) H˜ and that every orbit starting in H˜ must eventually enter H . ε ε ε δ Thus P+ =√ 1(H˜ ) is as required, a⊂nd we have proved (i). j − ε To prove (ii) we need a further property of H˜ . If w H˜ , arguing by induction on k 1 using (3.6) ε ε ∈ ≥ we get ε k 1 <ReFk(w) Rew − 2 − ≥ ¥ and kε(1 ε) − < ImFk(w) +εReFk(w) Imw +εRew . 2 | | − | | Thisimpliesthatforeveryw0 H˜εthereexistsak0 1sothatw°ecannothaveFk¢0(w)=w0foranyw H˜ε. Coming back to the z-plane,∈this says that any inv≥erse orbit of f must eventually leave P+. Thus∈every j (forward) orbit of f must eventually leave any repelling petal. So if z K O , where the stable set is f ∈ \{ } computed working in the neighborhood of the origin given by the union of repelling and attracting petals (together with the origin), the orbit of z must eventually land in an attracting petal, and thus z belongs to a basin centered at one of the r attracting directions — and (ii) is proved. To prove (iii), first of all we notice that we have 21+1/rC F (w) 1 (3.7) 0 | − |≤ w 1+1/r | | in H˜ . Indeed, (3.5) says that if w >1/(2rδ) then the function w F(w) w 1 sends the disk of center ε | | 7→ − − w and radius w /2 into the disk of center the origin and radius C/(w /2)1/r; inequality (3.7) then follows | | | | from the Cauchy estimates on the derivative. Now choose w0 Hδ, and set ϕ˜k(w) = Fk(w) Fk(w0). Given w H˜ε, as soon as k N is so large ∈ − ∈ ∈ thatFk(w) H wecanapplyLagrange’stheoremtothesegmentfromFk(w )toFk(w)togetat [0,1] δ 0 k ∈ ∈ such that ϕ˜ (w) F Fk(w) Fk Fk(w ) k+1 1 = − 0 1 = F t Fk(w)+(1 t )Fk(w ) 1 ØØØØ ϕ˜k(w) − ØØØØ ØØØØØ ° Fk(w¢)−2F1+k°1(/wrC0) ¢ − ØØØØØ ØØ 0° kC0 , − k 0 ¢− ØØ ≤ min Re Fk(w),Re Fk(w ) 1+1/r ≤ k1+1/r 0 { | | | |} where we used (3.7) and (3.4), and the constant C is uniform on compact subsets of H˜ (and it can be 0 ε chosen uniform on H ). δ As a consequence, the telescopic product ϕ˜ /ϕ˜ converges uniformly on compact subsets of H˜ k k+1 k ε (anduniformlyonH ),andthusthesequenceϕ˜ converges,uniformlyoncompactsubsets,toaholomorphic δ k function ϕ˜:H˜ε C. Since we have Q → ϕ˜ F(w)=Fk+1(w) Fk(w )=ϕ˜ (w)+F Fk(w ) Fk(w )=ϕ˜ (w)+1+O Fk(w ) 1/r , k 0 k+1 0 0 k+1 0 − ◦ − − | | ° ¢ ° ¢ it follows that ϕ˜ F(w)=ϕ˜(w)+1 ◦ 10 MarcoAbate on H˜ . In particular, ϕ˜ is not constant; being the limit of injective functions, by Hurwitz’s theorem it is ε injective. We now prove that the image of ϕ˜ contains a right half-plane. First of all, we claim that ϕ˜(w) lim =1. (3.8) w + w |w|∈→Hδ1 Indeed, choose η >0. Since the convergence of the telescopic product is uniform on Hδ, we can find k0 N ∈ such that ϕ˜(w) ϕ˜ (w) η − k0 < w w 3 Ø − 0 Ø Ø Ø on Hδ. Furthermore, we have Ø Ø Ø Ø ϕ˜k0(w) 1 = k0+ kj=0−01O(|Fj(w)|−1/r)+w0−Fk0(w0) =O(w 1) − on Hδ; thereforeØØØØwwe−cawn0fi−ndØØØØR>ØØØØØ0 sucPh that w−w0 ØØØØØ | | ϕ˜(w) η 1 < w w − 3 Ø − 0 Ø Ø Ø as soon as w >R in H . Finally, if R is largØe enough weØ also have δ Ø Ø | | ϕ˜(w) ϕ˜(w) ϕ˜(w) w η = < , w w − w w w w 3 Ø − 0 Ø Ø − 0ØØ 0Ø Ø Ø Ø ØØ Ø and (3.8) follows. Ø Ø Ø ØØ Ø Ø Ø Ø ØØ Ø Equality (3.8) clearly implies that (ϕ˜(w) wo)/(w wo) 1 as w + in Hδ for any wo C. But − − → | |→ 1 ∈ thismeansthatifRewo islargeenoughthenthedifferencebetweenthevariationoftheargumentofϕ˜ wo − along a suitably small closed circle around wo and the variation of the argument of w wo along the same − circle will be less than 2π — and thus it will be zero. Then the argument principle implies that ϕ˜ wo and − w wo have the same number of zeroes inside that circle, and thus wo ϕ˜(H ), as required. δ −So setting ϕ=ϕ˜ √, we have defined a function ϕ with the require∈d properties on P+. To extend it to ◦ j the whole basin B it suffices to put ϕ(z)=ϕ fk(z) k , (3.9) − where k N is the first integer such that fk(z) P+°. ¢ ∈ ∈ j Remark 3.1: It is possible to construct petals that cannot be contained in any sector strictly smaller than Σj. To do so we need an F-invariant subset Hˆε of C∗ R− containing H˜ε and containing eventually \ every half-line issuing from the origin (but R−). For M >>1 and C >0 large enough, replace the straight lines bounding H˜ on the left of Rew = M by the curves ε − Clog Rew if r =1, Imw = | | | | C Rew 1 1/r if r >1. Ω | | − Then it is not too difficult to check that the domain Hˆ so obtained is as desired (see [CG]). ε So we have a complete description of the dynamics in the neighbourhood of the origin. Actually, Camacho has pushed this argument even further, obtaining a complete topological classification of one- dimensional holomorphic local dynamical systems tangent to the identity (see also [BH, Theorem 1.7]): Theorem 3.3: (Camacho, 1978 [C]; Shcherbakov, 1982 [S]) Let f End(C,0) be a holomorphic local ∈ dynamical system tangent to the identity with multiplicity r+1 at the fixed point. Then f is topologically locally conjugated to the map g(z)=z zr+1 . − The formal classification is simple too, though different (see, e.g., Milnor [Mi]):