Automorphisms of C 2 with parabolic cylinders
aa r X i v : . [ m a t h . D S ] F e b AUTOMORPHISMS OF C WITH PARABOLIC CYLINDERS
LUKA BOC THALER † , FILIPPO BRACCI †† , AND HAN PETERS Abstract. A parabolic cylinder is an invariant, non-recurrent Fatou component Ω of anautomorphism F of C satisfying: (1) The closure of the ω -limit set of F on Ω containsan isolated fixed point, (2) there exists a univalent map Φ from Ω into C conjugating F to the translation ( z, w ) ( z + 1 , w ), and (3) every limit map of { F ◦ n } on Ω hasone-dimensional image. In this paper we prove the existence of parabolic cylinders foran explicit class of maps, and show that examples in this class can be constructed ascompositions of shears and overshears. Introduction
Main result.
The description of Fatou components plays a central role in our under-standing of holomorphic dynamical systems. In one complex variable the different kindsof Fatou components have been precisely characterized. In the last decade there has beensignificant progress in higher dimensions as well, but many fundamental questions remainunanswered. A primary objective is to describe invariant Fatou components in terms ofthe limit behavior of orbits, their complex geometry, and if possible, to give a normalform for the action of the map on the Fatou component.The purpose of this paper is to shed more light on a specific kind of non-recurrentFatou component, which we will call a parabolic cylinder . Recall that an invariant Fatoucomponent is called non-recurrent if every orbit on the component eventually departsfrom every compact subset of the component.
Definition 1.1.
Let F be an automorphism of C . An invariant non-recurrent Fatoucomponent Ω is called a parabolic cylinder if(1) the closure of the ω -limit set of F on Ω contains an isolated fixed point, Mathematics Subject Classification.
Key words and phrases. holomorphic dynamics; local dynamics; automorphisms; Fatou components. † Supported by the SIR grant “NEWHOLITE - New methods in holomorphic iteration” no.RBSI14CFME and by the research program P1-0291 from ARRS, Republic of Slovenia. †† Partially supported by the MIUR Excellence Department Project awarded to the Department ofMathematics, University of Rome Tor Vergata, CUP E83C18000100006 and PRIN
Real and ComplexManifolds: Topology, Geometry and holomorphic dynamics n.2017JZ2SW5. (2) there exists a univalent map Φ : Ω → C , conjugating F to the translation( z, w ) ( z + 1 , w ) , (3) all limit maps of F on Ω have dimension one.Our main result is the following: Theorem 1.2.
Let F be an automorphism of C of the form (1.1) F ( z, w ) = (cid:0) z + f ( w ) z + O ( z , z w ) , e πiθ w + g ( w ) z + O ( z , z w ) (cid:1) , where θ ∈ R \ Q is Diophantine, f (0) = 0 , and g ( w ) = O ( w ) . Then there exists aparabolic cylinder Ω for F that is biholomorphically equivalent to C . Moreover, everylimit map of F on Ω has image { } × C . The first example of an automorphism of C with a non-recurrent Fatou component onwhich all limit maps have rank 1 was given in [9]. There an explicit map of the form(1.2) G ( z, w ) = ( z + z + O ( z , z w, z w ) , w − z w O ( z , z w, z w )) , was constructed, and it was shown that G exhibits a Fatou component on which the orbitsconverge to the fixed plane { } × C . By composing G with a rotation ( z, w ) ( z, e πiθ w ),one obtains a non-recurrent Fatou component where the orbits converge to the rotating w -axis. Our main result implies that one can also obtain the normal form ( z, w ) ( z +1 , w ).We emphasize that the term g ( z ) w that is allowed to appear in the second coordinateof the maps F vanishes for the map G given in equation (1.2). The term of the form g ( z ) w makes it significantly harder to prove convergence of iterates, and the proof givenin [9] breaks down for the family considered here. In fact, it will be clear that our prooffails when θ = 0. The requirement that θ ∈ R \ Q is Diophantine can likely be relaxed,but we chose it for convenience.1.2. Fatou components in several variables.
Let F be a holomorphic self-map of acomplex manifold X . The Fatou set of F is the set of points p ∈ X for which there existsan open neighborhood U ∋ p such that the sequence of iterates { F ◦ n } form a normal familyon U . The connected components of the Fatou set of F are called Fatou components of F . A Fatou component Ω ⊂ X of F is invariant if F (Ω) = Ω. Following Bedford-Smillie[5], an invariant Fatou component is called recurrent if it contains a recurrent orbit, i.e.an orbit that accumulates at a point in Ω. In a non-recurrent invariant Fatou componentall orbits eventually leave any compact subset.For rational functions in one complex variable there is a complete description of allpossible Fatou components and the dynamics of the map on such components is quitewell understood. In particular, the invariant non-recurrent Fatou components are “Leau-Fatou petals” at a parabolic fixed point. All orbits in such petals converge to the fixed ARABOLIC CYLINDERS 3 point and on such petals the map is conjugated to a translation via the so-called “Fatoucoordinate”.Despite significant recent progress, including the construction of wandering domains[2, 6] and the classification of invariant Fatou components [5, 10], the situation is notnearly as well understood in C .Let F be an automorphism of C . If Ω ⊂ C is a Fatou component of F , we saythat a holomorphic map h : Ω → C ∪ {∞} is a limit map of F on Ω if there exists asequence { F ◦ n k } which converges uniformly on compacta of Ω to h — here, for the sakeof uniformizing notation, we let h ≡ ∞ in case { F ◦ n k } compactly diverges to ∞ .1.3. Fatou components of polynomial automorphisms. If F is a polynomial auto-morphism of C , the Jacobian determinant δ is necessarily constant and different from 0.When | δ | = 1 all Fatou components Ω of F are recurrent, the so-called Siegel domains,and h (Ω) = Ω for any limit map h . This does not complete the description, as it remainsan open question whether Ω must be topologically trivial, see for example [4].In the case | δ | < ⊂ Ω, see [5]. In the latter case, which could be called an attracting cylinder , theaction of f on the invariant set Σ is that of an irrational rotation, and Σ is equivalent toeither the disk or an annulus. Whether an annulus can actually occur is a pressing openquestion.The non-recurrent case has been described in [10], under the additional assumption | δ | < ( f ) . In this case all orbits converge to a parabolic-attracting fixed point, and thecomponent is biholomorphic to C , by a result of Ueda [16], and F is conjugate on Ω toa map ( z, w ) ( z + 1 , w ).1.4. Fatou components of holomorphic automorphisms.
Little is known aboutwhich other phenomena can occur when considering non-polynomial automorphisms of C . An invariant Fatou component is called attracting if all the orbits in the componentconverge to the same (necessarily fixed) point p ∈ C . By [11, 14], a recurrent attractingFatou component Ω is necessarily biholomorphic to C , the spectrum of dF p is containedin the (open) unit disk and F is conjugate to a polynomial triangular map on Ω.In [8] (see also [12] for the construction of multiple “petals”) the authors constructedan attracting non-recurrent Fatou component biholomorphic to C × C ∗ , where the mapis semi-conjugate to a translation over C . It is an open question whether all attractingFatou components in C are conjugate to either C or C × C ∗ .Contrary to the polynomial case, there are known to exist holomorphic automorphismswith non-recurrent Fatou components that are not attracting. Examples of such mapswere given in [9], including the map G mentioned in equation (1.2). Here we provethe existence of these non-recurrent Fatou components for a considerably larger class ofmaps, prove these examples are all biholomorphic to C , and construct Fatou coordinates, L. BOC THALER, F. BRACCI, AND H. PETERS showing that the dynamics is conjugate to a translation. We note that in [13], J. Reppekus,exploiting the example in [8] and blowing-up, shows that there exist parabolic cylindersthat are not biholomorphically equivalent to C × C ∗ . Whether C and C × C ∗ are theonly two possibilities is again an open question.Another natural open question, partially addressed in [9, 10], concerns the uniquenessof limit sets, i.e. whether all limit maps must have the same image.1.5. Outline of the paper.
In section (2) we introduce several coordinate changes,defined only locally near the invariant w -axis. In section (3), Proposition 3.4 we showthat the map F is locally conjugate to a map H of the form H ( u, w ) = (cid:18) u + 1 + Au + O (cid:18) u (cid:19) , λw + O (cid:18) u (cid:19)(cid:19) . This simpler form is exploited in Proposition 3.7 to prove the existence of a non-recurrentFatou component on which the orbits converge to the w -axis. In section (4), Proposi-tion 4.3 we prove the existence of the Fatou components, which implies that the Fatoucomponent is biholomorphically equivalent to C .In section (5) we show that maps F satisfying the constraints in the theorem can ac-tually be constructed as finite compositions of shears and overshears. Recall that suchcompositions form a dense subset of all automorphisms of C in the compact-open topol-ogy [1]. We note that F cannot have constant Jacobian determinant, and therefore cannotbe approximated by compositions of shears only. Acknowledgments.
In a first version of this paper, “parabolic cylinders” were named“non-recurrent Siegel cylinders”. However, the term “parabolic cylinders” seems to bemore appropriate, due to Property (2) in Definition 1.1. We thank Eric Bedford forstimulating discussions about this and other facts related to the paper. We also thank thereferee for very useful comments which improved much the original paper. In particular,for finding a mistake in the original version of Lemma 2.2.2.
Preliminaries
In this section we introduce various maps, together with useful estimates, that willserve in the next sections to prove our main result.Throughout this paper we will use λ = e πiθ where θ ∈ R \ Q is diophantine , i.e. thereexist c, r > | λ n − | ≥ cn − r for every n ≥
1. Such numbers form a densesubset of the unit circle with full measure. Note that if λ is diophantine then λ − is alsodiophantine and satisfies the same estimates.We will be using the following notation. Let u ( x ) and v ( x ) be two functions. By writing u ( x ) = O ( v ( x )) we mean that there exist a constant C > | u ( x ) | ≤ C | v ( x ) | for all x in a neighborhood of the origin where u and v are defined. The notation u ( x ) = o ( v ( x )) as x → a means that u ( x ) /v ( x ) → x → a . In case of a sequence u n of ARABOLIC CYLINDERS 5 complex numbers, the notation u n = O ( v ( n )) has to be understood as | u n | ≤ C | v ( n ) | forall n ∈ N .The following was used in [3], we repeat the proof for convenience of the reader. Lemma 2.1.
There exist constants
C, r > such that for every integer n ≥ and forevery m ≥ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = m λ jn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < Cn r . Proof.
Let N ≥ m . Since λ = e πiθ where θ ∈ R \ Q is diophantine, there exist c, r > | λ n − | ≥ cn − r for all n . This gives the bound (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X j = m λ nj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X j = m λ n ( j +1) − λ nj λ n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ n − N X j = m ( λ n ( j +1) − λ nj ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < (cid:12)(cid:12)(cid:12)(cid:12) λ n − (cid:12)(cid:12)(cid:12)(cid:12) < Cn r , and we are done. (cid:3) Let γ : [0 , ∞ ) → [0 , ∞ ) be an increasing function that satisfies γ (2 x ) ≥ γ ( x ) for all x ≥
0. For
R > δ > K R,δ := { u ∈ C | Re ( u ) > − γ ( δ ) and arg( u − R ) ∈ [ − π/ , π/ } , and U R,δ := { ( u, w ) ∈ C | u ∈ K R,δ and | w | < δ } . To be precise, both the set K R,δ and U R,δ depend on the choice of the function γ , sothat one should more appropriately name them K R,δ,γ and U R,δ,γ . However, in order notto burden notation, and since no confusion should arise, we avoid mentioning γ in thisdefinition.Note that U R ,δ ⊂ U R ,δ when R ≤ R and δ ≤ δ . Lemma 2.2.
Let
R > and δ > . There exist r > and ˜ C > such that for everyinteger n ≥ , m ≥ and u ∈ K R,δ , (2.2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = m λ nj u + j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ˜ Cn r | u + m | . In particular, the series P ∞ j =0 λ nj u + j is converging uniformly on compacta of K R,δ for every n ≥ . L. BOC THALER, F. BRACCI, AND H. PETERS
Proof.
Let N ≥ m . Lemma 2.1 gives (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X j = m λ nj u + j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u + N N X j = m λ nj − N − X j = m (cid:18) u + j + 1 − u + j (cid:19) j X k = m λ nk (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < Cn r | u + N | + Cn r · N − X j = m | ( u + j + 1)( u + j ) | < ˜ Cn r | u + m | , with the constant ˜ C chosen to be independent from N and u , and we are done. Notethat here we have used the fact that for every u ∈ K R,δ and for every m ≥ | u + m | < | u + k | for all k > M := max { γ ( δ ) , m } , hence there is a constant C ′ ( m, γ ( δ ))such that 1min k ≥ m | u + k | = 1min m ≤ k ≤ M | u + k | ≤ C ′ | u + m | for all u ∈ K R,δ . (cid:3) Lemma 2.3.
Let g : C → C be an entire function such that g (0) = g ′ (0) = 0 . Let g ( w ) = P ∞ ℓ =2 d ℓ w ℓ be its expansion at . Then for every δ > , there exists R = R ( δ ) > ,which depends continuously on δ , such that the map Φ( u, w ) := u, w + λ − ∞ X ℓ =2 d ℓ w ℓ ∞ X k =0 λ ( ℓ − k u + k !! is univalent on U R,δ . Moreover Φ( u, w ) = (cid:0) u, w + O (cid:0) u (cid:1)(cid:1) and for every δ > and R > there exists R ′ ≥ R such that Φ is univalent on U R ′ ,δ and Φ( U R ′ ,δ ) ⊂ U R, δ . Also, forevery δ > there exists R ′′ ≥ R ( δ ) such that U R ′′ ,δ/ ⊂ Φ( U R ( δ ) ,δ ) .Proof. Let
R, δ >
0. By Lemma 2.2 the map Φ( u, w ) is a well defined holomorphic mapon { ( u, w ) ∈ C | u ∈ K R,δ } and Φ( u, w ) = (cid:0) u, w + O (cid:0) u (cid:1)(cid:1) . In order to check injectivity,first observe that Φ( u, w ) = Φ( u ′ , w ′ ) implies u = u ′ . Therefore Φ( u, w ) = Φ( u ′ , w ′ ) if andonly if w − w ′ + λ − ∞ X ℓ =2 d ℓ ( w ℓ − w ′ ℓ ) ∞ X k =0 λ ( ℓ − k u + k = 0 . Assuming that w = w ′ we can divide this equation by w − w ′ to obtain1 + λ − ∞ X ℓ =2 d ℓ (cid:18) w ℓ − w ′ ℓ w − w ′ (cid:19) ∞ X k =0 λ ( ℓ − k u + k = 0 . ARABOLIC CYLINDERS 7
Since (cid:12)(cid:12)(cid:12) w ℓ − w ′ ℓ w − w ′ (cid:12)(cid:12)(cid:12) ≤ ℓδ ℓ − , taking into account Lemma 2.2 and that g is entire, we can choose R large enough so that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X ℓ =2 d ℓ (cid:18) w ℓ − w ′ ℓ w − w ′ (cid:19) ∞ X k =0 λ ( ℓ − k u + k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ˜ C | u | ∞ X ℓ =2 | d ℓ | δ ℓ − ℓ ( ℓ − r < U R,δ , and hence Φ( u, w ) = Φ( u ′ , w ′ ) implies ( u, w ) = ( u ′ , w ′ ). This lastinequality also implies that R depends continuously on δ .In order to prove the final statement, let(2.3) h ( u, w ) := λ − ∞ X ℓ =2 d ℓ w ℓ ∞ X k =0 λ ( ℓ − k u + k ! = O (cid:18) u (cid:19) , and let R ′′ ≥ R ( δ ) be such that | h ( u, w ) | < δ for u ∈ K R ′′ , δ and | w | ≤ δ . Let u ∈ K R ′′ , δ and let | w | ≤ δ/
2. Let C := { w ∈ C : | w − w | = δ/ } . By the triangular inequality | w | ≤ δ for all w ∈ C . Thus, for all w ∈ C , | w − w | = δ > | h ( u , w ) | . Hence, by the Rouch´e theorem, the functions w w − w and w w + h ( u , w ) − w have the same number of zeros in { w ∈ C : | w − w | < δ/ } . In particular, there exists w ∈ C such that | w | < δ and w + h ( u , w ) = w . Therefore, ( u , w ) ∈ Φ( U R ( δ ) ,δ ).By the arbitrariness of ( u , w ), this proves that U R ′′ ,δ/ ⊂ Φ( U R ( δ ) ,δ ). Finally it followsfrom (2.3) that for sufficiently large R ′ > R ( δ ) we have | h ( u, w ) | < on U R ′ ,δ , henceΦ( U R ′ ,δ ) ⊂ U R, δ . (cid:3) Lemma 2.4.
Let f : C → C be an entire function such that f (0) = 1 . Let f ( w ) =1 + P ∞ ℓ =1 d ℓ w ℓ be its expansion at . The map Ψ : C → C defined as Ψ( u, w ) := u + ∞ X ℓ =1 d ℓ λ ℓ − w ℓ , w ! is a holomorphic automorphism of C . Furthermore, let C, r > be as in Lemma anddefine (2.4) γ ( δ ) := C ∞ X ℓ =1 | d ℓ | ℓ r δ ℓ . Then γ : [0 , ∞ ) → [0 , ∞ ) is an increasing function satisfying γ (2 δ ) ≥ γ ( δ ) for all δ ≥ .Moreover, for every δ > we have || Ψ( u, w ) − ( u, w ) || < γ ( δ ) for all ( u, w ) ∈ C × { w ∈ L. BOC THALER, F. BRACCI, AND H. PETERS C : | w | ≤ δ } . Finally, using (2.4) in the definition of K R,δ (see (2.1) ), we also have that,for every δ > there exists M δ > such that for all R ≥ M δ Ψ( U R,δ ) ⊂ U R, δ . Proof.
Clearly the series P ∞ ℓ =1 d ℓ w ℓ is absolutely convergent on compacta of C . Recallthat by our assumption λ satisfies the condition | λ n − | > cn − r , for some c, r >
0. Itfollows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X ℓ =1 d ℓ λ ℓ − w ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < C ∞ X ℓ =1 | d ℓ | ℓ r | w | ℓ . From this last inequality we can deduce that the series which appears in the first coordinateof the map Ψ is absolutely convergent on compacta in C . Therefore Ψ is an automorphismof C .From the previous considerations and the definition of γ , it follows immediately that || Ψ( u, w ) − ( u, w ) || < γ ( δ ) for all ( u, w ) ∈ C × { w ∈ C : | w | ≤ δ } .In order to prove the last statement, let us write ( u , w ) = Ψ( u , w ) and observe that w = w . Since ( u , w ) ∈ U R,δ it follows that Re ( u ) > − γ ( δ ) which implies Re ( u ) = Re ( u ) − γ ( δ ) > − γ ( δ ) > − γ (2 δ ) . From here one can easily deduce that there exists M δ >
0, such that for all
R > M δ wehave Ψ( U R,δ ) ⊂ U R, δ . (cid:3) Lemma 2.5.
Let f : C → C be an entire function with f (0) = 0 . Let f ( w ) = P ∞ ℓ =1 d ℓ w ℓ be its expansion at . Then for every δ > , there exists R > such that the map τ ( u, w ) := u − ∞ X k =0 f ( λ k w ) u + k , w ! is univalent on U R,δ . Moreover R depends continuously on δ and τ ( u, w ) = (cid:0) u + O (cid:0) u (cid:1) , w (cid:1) .In particular, for every δ > and R > there exists R ′ ≥ R such that τ is univalenton U R ′ ,δ and τ ( U R ′ ,δ ) ⊂ U R, δ . Also, for every δ > there exists R ′′ ≥ R ( δ ) such that U R ′′ ,δ/ ⊂ τ ( U R ( δ ) ,δ ) .Proof. We first prove that the map is well defined, i.e. , we prove that for every
R, δ > ∞ X k =0 f ( λ k w ) u + k ARABOLIC CYLINDERS 9 converges uniformly on compacta of { ( u, w ) ∈ C | u ∈ K R,δ } . By Lemma 2.2, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k =0 f ( λ k w ) u + k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k =0 ∞ X j =1 λ kj d j w j u + k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j =1 d j w j N X k =0 λ kj u + k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ˜ C | u | ∞ X j =1 | d j || w | j j r . Since f is entire, the last series converges uniformly on compacta, and so does the se-ries (2.5).In order prove injectivity, we first observe that τ ( u, w ) = τ ( u ′ , w ′ ) implies that w = w ′ .If u = u ′ then τ ( u, w ) = τ ( u ′ , w ′ ) if and only if u − u ′ − ∞ X k =0 f ( λ k w ) u + k − f ( λ k w ) u ′ + k = 0 . Dividing this equation by u − u ′ we obtain(2.6) 1 + ∞ X k =0 f ( λ k w )( u + k )( u ′ + k ) = 0 . Given δ >
0, we can find R large enough such that for all u, u ′ ∈ K R,δ we have ∞ X k =0 | ( u + k )( u ′ + k ) | < | w | <δ | f | . Therefore, (2.6) cannot be satisfied in U R,δ , and hence τ is injective in U R,δ .By the previous considerations it follows also that τ ( u, w ) = (cid:0) u + O (cid:0) u (cid:1) , w (cid:1) .The last statement follows by applying Rouch´e’s theorem as in Lemma 2.3. (cid:3) Remark . It is worth noticing that, except for Lemma 2.4 where we need to choose asuitable γ in the definition of the set K R,δ , for the other lemmas, any choice of γ workswell. 3. Non-recurrent Fatou component
Let F be a holomorphic automorphism of C of the form(3.1) F ( z, w ) = (cid:0) z + f ( w ) z + O ( z ) , λw + g ( w ) z + O ( z ) (cid:1) , where f and g are entire functions in C , f (0) = 0 and g ( w ) = O ( w ). Notice that theinverse F − has the same form as F . From now on we assume without loss of generalitythat f (0) = 1, since otherwise we can simply conjugate F with a dilatation in the firstfactor.The aim of this section is to show that F has an invariant non-recurrent Fatou com-ponent Ω with ω -limit set { } × C ⊂ ∂ Ω (Proposition 3.7). To achieve this, we use the maps introduced in the previous section to change coordinates and get estimates allowingto show that F maps sets of the type “a small sector times a disc of radius δ ” (which,in the ( u, w ) coordinates, correspond to the sets U R,δ introduced in the previous section)into sets of the same type. Moreover, we prove that for each orbit of F on such sets, thefirst variable moves toward 0, while the modulo of the second variable stays close to themodulo of the second variable of the starting point of the orbit. The Fatou component Ωis then defined as the Fatou component of F which contains the union of all such (suitablychosen) “small sectors times discs” sets.Let f ( w ) = 1 + P ∞ ℓ =1 d ℓ w ℓ be the expansion of f at 0, and let C > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X ℓ =1 d ℓ λ ℓ − w ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < C ∞ X ℓ =1 | d ℓ | ℓ r | w | ℓ . Define γ ( δ ) := C ∞ X ℓ =1 | d ℓ | ℓ r δ ℓ , and observe that sup | w |≤ δ | f ( w ) − | ≤ γ ( δ ) and γ (2 δ ) > γ ( δ ) for all δ > Assumption:
From now on, without further mentioning it, we use the above function γ in the definition of the sets K R,δ (see (2.1)) and U R,δ .We define the biholomorphic map Θ : C ∗ × C → C ∗ × C as(3.2) Θ( u, w ) := ( − u , w ) . Since F is an automorphism of C which leaves invariant the affine line { } × C , it followsthat the map ˜ F := Θ − ◦ F ◦ Θis a well defined biholomorphic map ˜ F : C ∗ × C → C ∗ × C . A quick computation shows˜ F ( u, w ) = (cid:18) u + f ( w ) + O (cid:18) u (cid:19) , λw − g ( w ) u + O (cid:18) u (cid:19)(cid:19) . Let δ ′ > U R ′ ,δ ′ ⊂ C ∗ × C for all R ′ >
0. Therefore, ˜ F is welldefined and univalent on U R ′ ,δ ′ .Moreover, given R ′′ >
0, we can find R ′ > F ( U R ′ ,δ ′ ) ⊂ U R ′′ , δ ′ . In order tosee this, let ( u , w ) = ˜ F ( u , w ). Since | w | = | w | + O (1 /u ) it is easy to see that forsufficiently large R ′ we obtain | w | < δ ′ . Also, observe that Re ( u ) = Re ( u ) + 1 + Re ( f ( w ) −
1) + Re ( O (cid:18) u (cid:19) ) > − γ ( δ ′ ) + 1 − O (cid:18) u (cid:19) > − γ (2 δ ′ ) , ARABOLIC CYLINDERS 11 where the last inequality holds for all sufficiently large R ′ .Let g ( w ) = P ∞ ℓ =2 b ℓ w ℓ be the expansion of g at 0 and let Φ be as in Lemma 2.3.Fix δ >
0. By Lemma 2.3, there exists R ′′ > − is well defined andunivalent on U R ′′ , δ . By the previous considerations, there exists R ′ ≥ R ′ ( δ ) such that˜ F ( U R ′ , δ ) ⊂ U R ′′ , δ and finally, by Lemma 2.3, there exists R > U R,δ and Φ( U R,δ ) ⊂ U R ′ , δ . Thus, G := Φ − ◦ ˜ F ◦ Φis well defined and univalent on U R,δ . Lemma 3.1.
For ( u, w ) ∈ U R,δ , we have G ( u, w ) = (cid:18) u + f ( w ) + O (cid:18) u (cid:19) , λw + O (cid:18) u (cid:19)(cid:19) . Proof.
Let us write ( u , w ) := G ( u, w ). First observe that( ˜ F ◦ Φ)( u, w ) = u + f ( w ) + O (cid:18) u (cid:19) , λw + ∞ X ℓ =2 b ℓ w ℓ ∞ X k =0 λ ( ℓ − k u + k − u ∞ X ℓ =2 b ℓ w ℓ + O (cid:18) u (cid:19)! . Since Φ − ( u, w ) = u, w − λ − ∞ X ℓ =2 b ℓ w ℓ ∞ X k =0 λ ( ℓ − k u + k + O (cid:18) u (cid:19)! , it follows that u = u + f ( w ) + O (cid:18) u (cid:19) w = λw + ∞ X ℓ =2 b ℓ w ℓ ∞ X k =0 λ ( ℓ − k u + k − u ∞ X ℓ =2 b ℓ w ℓ − λ − ∞ X ℓ =2 b ℓ λ ℓ w ℓ ∞ X k =0 λ ( ℓ − k u + k + f ( w )= λw + ∞ X ℓ =2 b ℓ w ℓ − u + ∞ X k =0 λ ( ℓ − k u + k − ∞ X k =1 λ ( ℓ − k u + k + ( f ( w ) −
1) + O (cid:0) u (cid:1) ! + O (cid:18) u (cid:19) = λw + ∞ X ℓ =2 b ℓ w ℓ ∞ X k =1 λ ( ℓ − k ( f ( w ) −
1) + O (cid:0) u (cid:1) ( u + k ) (cid:0) u + k + ( f ( w ) −
1) + O (cid:0) u (cid:1)(cid:1) + O (cid:18) u (cid:19) = λw + ( f ( w ) − ∞ X ℓ =2 b ℓ w ℓ ∞ X k =1 λ ( ℓ − k ( u + k ) (cid:0) u + k + ( f ( w ) −
1) + O (cid:0) u (cid:1)(cid:1) + O (cid:18) u (cid:19) = λw + O (cid:18) u (cid:19) . The last equality follows from the fact that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k =1 λ ( ℓ − k ( u + k ) (cid:0) u + k + ( f ( w ) −
1) + O (cid:0) u (cid:1)(cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < C ( ℓ − r | u | on U R,δ for some
C >
0, which follows similarly as Lemma 2.2. (cid:3)
Recall that f ( z ) = 1 + P ∞ k =1 d k w k and letΨ( u, w ) = u + ∞ X k =1 d k λ k − w k , w ! be the map defined in Lemma 2.4. Observe that (cid:12)(cid:12)(cid:12)P ∞ k =1 d k λ k − w k (cid:12)(cid:12)(cid:12) ≤ γ ( δ ) for all for all | w | ≤ δ . Therefore, by Lemma 2.4, there exists R ′ > R such that Ψ( U R ′ ,δ/ ) ⊂ U R,δ . Thisimplies that ˜ G := Ψ − ◦ G ◦ Ψis a well defined univalent map on U R ′ ,δ/ .In order to avoid burdening notations, we still denote R ′ by R and δ by δ , so that ˜ G isunivalent on U R,δ . Lemma 3.2.
Given δ > , we can choose R sufficiently large so that for ( u, w ) ∈ U R,δ we have (3.3) ˜ G ( u, w ) = (cid:18) u + 1 + O (cid:18) u (cid:19) , λw + O (cid:18) u (cid:19)(cid:19) . Proof.
Let us write ( u ′ , w ′ ) = ˜ G ( u, w ). By Lemma 2.4 the function q ( w ) := P ∞ k =1 d k λ k − w k is an entire function, therefore we can choose R so large that | q ( w ) | < R for all | w | ≤ δ .It follows that w ′ = λw + O (cid:18) u (cid:19) . In the first coordinate we get u ′ = u + ∞ X k =1 d k λ k − w k + f ( w ) − ∞ X k =1 d k λ k − (cid:18) λw + O (cid:18) u (cid:19)(cid:19) k + O (cid:18) u + q ( w ) (cid:19) + O (cid:18) u (cid:19) . ARABOLIC CYLINDERS 13
Next observe that O (cid:16) u + q ( w ) (cid:17) = O (cid:0) u (cid:1) . Hence, u ′ = u + ∞ X k =1 d k λ k − w k + f ( w ) − ∞ X k =1 d k λ k − (cid:18) λw + O (cid:18) u (cid:19)(cid:19) k + O (cid:18) u (cid:19) = u + 1 + ∞ X k =1 d k λ k − w k + ∞ X k =1 d k w k − ∞ X k =1 d k λ k − (cid:18) λw + O (cid:18) u (cid:19)(cid:19) k + O (cid:18) u (cid:19) = u + 1 + ∞ X k =1 d k λ k − λ k w k − (cid:18) λw + O (cid:18) u (cid:19)(cid:19) k ! + O (cid:18) u (cid:19) = u + 1 + O (cid:18) u (cid:19) , where we used the fact that P ∞ k =1 d k λ k − (cid:16) λ k w k − (cid:0) λw + O (cid:0) u (cid:1)(cid:1) k (cid:17) = O (cid:0) u (cid:1) . (cid:3) Remark . It follows from (3.3) that for any R ′′ > R ′ ≥ R such that˜ G ( U R ′ ,δ ) ⊂ U R ′′ , δ .By previous lemma, we can write˜ G ( u, w ) = (cid:18) u + 1 + h ( w ) u , λw (cid:19) + O (cid:18) u (cid:19) , for some holomorphic function h : { w ∈ C : | w | ≤ δ } → C . Let us denote A := h (0) andlet(3.4) τ ( u, w ) = u − ∞ X k =0 h ( λ k w ) − Au + k , w ! . By Lemma 2.5, there exists R ′′ > τ − is well defined on U R ′′ , δ . By Remark 3.3we can choose R ′ ≥ R such that ˜ G ( U R ′ ,δ ) ⊂ U R ′′ , δ . Finally, we can choose R > τ is well defined and univalent on U R ,δ/ and τ ( U R ,δ/ ) ⊂ U R ′ ,δ .Once again, in order to simplify notation, we will write R instead of R and δ insteadof δ , so that(3.5) H ( u, w ) := τ − ◦ ˜ G ◦ τ is well defined and univalent on U R,δ . Proposition 3.4.
For ( u, w ) ∈ U R,δ , we have H ( u, w ) = (cid:0) u + 1 + Au + O (cid:0) u (cid:1) , λw + O (cid:0) u (cid:1)(cid:1) .Proof. By Lemma 2.5, τ ( u, w ) = ( u + O (cid:0) u (cid:1) , w ). This, together with (3.3), implies˜ G ◦ τ ( u, w ) = u + 1 + h ( w ) u − ∞ X k =0 h ( λ k w ) − Au + k + O (cid:18) u (cid:19) , λw + O (cid:18) u (cid:19)! . Next observe that τ − ( u, w ) = u + ∞ X k =0 h ( λ k w ) − Au + k + O (cid:18) u (cid:19) , w ! . Let us write ( u ′ , w ′ ) = H ( u, w ) and observe that w ′ = λw + O (cid:0) u (cid:1) . Let us write w ′ = λw + α ( u ) where α ( u ) = O (cid:0) u (cid:1) is a holomorphic function (with coefficients dependingon w ). In the first coordinate we get u ′ = u + 1 + h ( w ) u − ∞ X k =0 h ( λ k w ) − Au + k + ∞ X k =0 h ( λ k +1 w + λ k α ( u )) − Au + k + 1 + O (cid:0) u (cid:1) + O (cid:18) u (cid:19) = u + 1 + Au − ∞ X k =0 h ( λ k w ) − Au + k + ∞ X k =1 h ( λ k w + λ k − α ( u )) − Au + k + O (cid:0) u (cid:1) + O (cid:18) u (cid:19) = u + 1 + Au + ∞ X k =1 h ( λ k w + λ k − α ( u )) − h ( λ k w ) u + k + O (cid:0) u (cid:1) − ∞ X k =1 h ( λ k w ) O (cid:0) u (cid:1) ( u + k + O (cid:0) u (cid:1) )( u + k ) + O (cid:18) u (cid:19) . We are going to show that both of the infinite sums in the above expression are of order O (cid:0) u (cid:1) .Clearly (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k =1 h ( λ k w ) O (cid:0) u (cid:1) ( u + k + O (cid:0) u (cid:1) )( u + k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup | w |≤ δ | h ( w ) | · O (cid:18) | u | (cid:19) · ∞ X k =1 | ( u + k + O (cid:0) u (cid:1) )( u + k ) | = O (cid:18) | u | (cid:19) . As before we can write convergent power series: h ( w + λ − α ( u )) − h ( w ) = ∞ X ℓ =1 ∞ X j =0 ( ℓ + j )! λ − ℓ j ! b j + ℓ w j ( α ( u )) ℓ , where h ( w ) = P ∞ j =0 b j w j . Now observe that the same computation as in Lemma 2.2 tellsus that the sum ∞ X k =0 λ k ( j + ℓ ) u + k + O (cid:0) u (cid:1) converges and that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k =0 λ k ( j + ℓ ) u + k + O (cid:0) u (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( j + ℓ ) r | u | . ARABOLIC CYLINDERS 15
Therefore (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k =0 h ( λ k w + λ k − α ( u )) − h ( λ k w ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X ℓ =1 ∞ X j =0 ( ℓ + j )! λ − ℓ j ! b j + ℓ w j ( α ( u )) ℓ ∞ X k =0 λ k ( j + ℓ ) u + k + O (cid:0) u (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = C | u | ∞ X ℓ =1 ∞ X j =0 ( ℓ + j )!( j + ℓ ) r j ! (cid:12)(cid:12) b j + ℓ w j ( α ( u )) ℓ (cid:12)(cid:12) = O (cid:18) u (cid:19) . (cid:3) For given ( u , w ) ∈ U R,δ and integer n ≥ u n , w n ) := H n ( u , w ). Lemma 3.5.
Given δ > there exists T δ ≥ R such that for every T ≥ T δ and ( u , w ) ∈ U T, δ , we have ( u n , w n ) ∈ U T + n ,δ for every n ≥ . Moreover given < ε ≤ δ , there exists T ε ≥ T δ such that | w n − λ n w | < ε for every ( u , w ) ∈ U T ε ,δ/ and for every n ≥ .Proof. Fix 0 < ε ≤ δ/
2. By Proposition 3.4 we can choose T ≥ R and C > | u − u − | < and | w − λw | < C | u | on U T,δ and such that P ∞ n =0 C ( T + n ) < ε . Usinginduction it is easy to see that Re ( u n ) > T + n and | w n − λ n w | < ε . (cid:3) Lemma 3.6.
Let δ > and T ≥ T δ be as in Lemma . For every compact subset K ⊂ U T, δ there exists a constant C > so that for every ( u , w ) ∈ K and every n ≥ we have | u n | ≤ C n and (cid:12)(cid:12)(cid:12)(cid:12) u n − n (cid:12)(cid:12)(cid:12)(cid:12) ≤ C log nn . Proof.
The first inequality follows directly from Re ( u n ) > T + n . As for the secondinequality first observe that(3.6) (cid:12)(cid:12)(cid:12)(cid:12) u n − n (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | u n − n | n . Recall that ( u ′ , w ′ ) := H ( u, w ) = ( u +1+ Au , λw )+ O (1 /u ) where (cid:12)(cid:12) u ′ − u + 1 + Au (cid:12)(cid:12) ≤ C ′ | u | for some C ′ >
0. Using the inequality (3.6) we can now deduce that | u n − n | ≤ (cid:12)(cid:12)(cid:12)(cid:12) u + n + A (cid:18) u + . . . + 1 u n − (cid:19) − n (cid:12)(cid:12)(cid:12)(cid:12) + C ′ C n X k =1 k ≤ | u | + | A | C n X k =1 k + C ′ C n X k =1 k ≤ | u | + C log n + C where C , C >
0. Finally we obtain (cid:12)(cid:12)(cid:12)(cid:12) u n − n (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( | u | + C log n + C ) n = O (cid:18) log nn (cid:19) . (cid:3) Proposition 3.7.
Let F be an automorphism of the form (3.1) . Then F has an invariantnon-recurrent Fatou component Ω with ω -limit set { } × C ⊂ ∂ Ω .Proof. Let δ >
R >
0. Let Θ be the map defined in (3.2). Let V R,δ := Θ( U R,δ ) andobserve that V R,δ = Λ
R,δ × D δ , where Λ R,δ := { z ∈ C | − z ∈ K R,δ } and D δ := { w ∈ C : | w | < δ } .We divide the proof in four steps. First, we prove that for every δ > R ( δ ) > { F ◦ n } is a normal family on V R ( δ ) ,δ/ . Next, we prove that theimage of every limit map of { F ◦ n } on such a set is contained in the affine line { z = 0 } andalways contains a disc of radius δ/
16. Then, we show that every limit map of the Fatoucomponent Ω of F which contains V R ( δ ) ,δ/ (for δ → + ∞ ), has image { } × C . Finally,we show that Ω is non-recurrent. Step 1: For every δ > there exists R ( δ ) > such that { F ◦ n } is a normal family on V R ( δ ) ,δ/ .Let δ >
0. Let τ be the map defined in (3.4) and let Φ be the one defined in Lemma 2.3.Let M δ > T ≥ M δ there exists R ( δ, T ) > τ − ◦ Ψ − ◦ Φ − ◦ Θ − ( V R ( δ,T ) ,δ/ ) ⊂ Θ − ( V T, δ ) = U T, δ and Φ ◦ Ψ ◦ τ ( U T, δ ) ⊂ U T, δ , for every n ≥ ARABOLIC CYLINDERS 17
Therefore, if T δ > T := max { M δ , T δ } and R ( δ ) := R ( δ, T ). Since(3.7) F ◦ n = Θ ◦ Φ ◦ Ψ ◦ τ ◦ H n ◦ τ − ◦ Ψ − ◦ Φ − ◦ Θ − , Lemma 3.5 implies at once that the family { F ◦ n } is normal on V R ( δ ) ,δ/ . Step 2: Let g be a limit map of { F ◦ n } on V R ( δ ) ,δ/ . Then { } × D δ ⊂ g ( V R ( δ ) ,δ/ ) ⊂ { } × C . Let ε = δ/
16. Let T ε be as in Lemma 3.5, and let R ≥ max { M δ , T ε } . Denote by π j : C → C , j = 1 , j -th component, that is, π ( z, w ) = z , π ( z, w ) = w . It follows from Lemma 3.5 that for all ( z, w ) ∈ V R,δ/ and n ≥ | π ◦ Ψ ◦ τ ◦ H n ◦ τ − ◦ Ψ − ◦ Φ − ◦ Θ − ( z, w ) − λπ ◦ Φ − ◦ Θ − ( z, w ) | < δ . Taking into account that, by Lemma 2.3, π ◦ Φ( u, w ) and π ◦ Φ − ( u, w ) are of the form w + O (1 /u ), the previous equation and (3.7) imply that there exist R ′ ≥ R such that(3.8) | π ◦ F ◦ n ( z, w ) − λ n w | < δ z, w ) ∈ V R ′ ,δ/ and n ≥ η >
0, there exists n such that for all n ≥ n ,(3.9) | π ◦ F ◦ n ( z, w ) | ≤ η for all ( z, w ) ∈ V R ′ ,δ/ .Let { n j } be any increasing sequence for which F ◦ n j converges uniformly on compactaof V R ( δ ) ,δ/ to a holomorphic function g and λ n j converges to some µ ∈ ∂ D . It followsfrom (3.9) that g ( z, w ) = (0 , g ( z, w )) for all ( z, w ) ∈ V R ( δ ) ,δ/ , where g : V R ( δ ) ,δ/ → C isholomorphic. Moreover, by (3.8),(3.10) | g ( z, w ) − µw | < δ z, w ) ∈ V R ′ ,δ/ .Fix w ∈ C , | w | < δ/
16. Let z ∈ C be such that − /z ∈ K R ′ , δ . Hence, ( z , w ) ∈ V R ′ ,δ/ . Moreover, { z } × { w ∈ C : | w − w | < δ/ } ⊂ V R ′ ,δ/ and (3.10) holds on | w − w | = δ/
8. Therefore, by Rouch´e’s theorem, µg ( z , w ) − µw and w − µw have thesame number of zeros in { w ∈ C : | w − w | < δ/ } . Since | µw − w | ≤ | w | < δ/
8, itfollows that there exists w , with | w − w | < δ/
16 such that µg ( z , w ) = w . By thearbitrariness of w , it follows that D δ/ ⊂ g ( V R ( δ ) ,δ/ ), which completes step 2. Step 3: There exists an invariant Fatou component Ω such that the image of any limitmap of { F ◦ n | Ω } is { } × C . Let { δ m } be an increasing sequence of positive real numbers which converges to + ∞ .We can choose R ( δ m +1 ) ≥ R ( δ m ) for all m ≥
0. Let V m := V R ( δ m ) ,δ m / , m ≥
0. Hence, (cid:26) ( z, w ) ∈ V m +1 : | w | < δ m Re ( − z ) > − γ (cid:18) δ m (cid:19)(cid:27) ⊂ V m . Therefore V := ∪ m ≥ V m is open and connected. Since { F ◦ n } is a normal family on V m for all m ≥ { F ◦ n } is a normal family on V and, hence, { F ◦ n } is normal on(3.11) V := ∞ [ n =0 F ◦ n ( V ) . By (3.9) and (3.8), F ◦ n ( V ) ∩ V = ∅ for every n ≥
1. Hence V is a F -forward invariant,open, connected set on which { F ◦ n } is normal. Therefore, there exists an invariant Fatoucomponent Ω which contains V .Now, let g be a limit map of { F ◦ n } on Ω. Hence, g | V m is a limit map of { F ◦ n | V m } forall m ≥
0. Then, it follows from Step 2 that g ( V ) = { } × C . Since V is open in V , itfollows as well that g (Ω) = { } × C . Step 4: Ω is non-recurrent .Observe that F ◦ n (0 , w ) = (0 , λ n w ) for all w ∈ C . Equation (3.1) therefore implies that ∂ π ( F ◦ n ) ∂z (0 , w ) = 2 n − X k =0 f ( λ k w ) = 2 n − X k =0 ( f ( λ k w ) −
1) + 2 n. Let us first observe that P n − k =0 ( f ( λ k w ) −
1) is uniformly bounded in w with respect to n . We have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − X k =0 ( f ( λ k w ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − X k =0 ∞ X ℓ =1 d ℓ λ kℓ w ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X ℓ =1 d ℓ w ℓ n − X k =0 λ kℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X ℓ =1 | d ℓ || w ℓ | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − X k =0 λ kℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ∞ X ℓ =1 | d ℓ | ℓ r | w ℓ | . ARABOLIC CYLINDERS 19
Since f ( w ) is an entire function, the last sum converges on uniformly on compacta of C .This implies that lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) ∂ π ( F ◦ n ) ∂z (0 , w ) (cid:12)(cid:12)(cid:12)(cid:12) = ∞ . Therefore, (0 , w ) cannot be contain in any Fatou component of F for all w ∈ C . Thus { } × C ⊂ ∂ Ω, which completes the proof. (cid:3)
Remark . The set V defined in (3.11) depends on the sequence { δ n } and on the choiceof { R ( δ n ) } . In particular, given any η > V of the form(3.11) such that(3.12) sup {| z | : ( z, w ) ∈ V} < η. Indeed, unrolling the definition of V , we see that V = ∪ n ≥ ∪ m ≥ F ◦ n ( V m ) , where V m = V R ( δ m ) ,δ m / , with { δ m } an increasing sequence converging to ∞ and R ( δ m )a suitable increasing sequence of real positive numbers. By construction one can replace R ( δ m ) with any R m ≥ R ( δ m ) so that { R m } is still increasing. If we choose R m sufficientlylarge, by (3.9), | π ( F ◦ n ( z, w )) | < η for all ( z, w ) ∈ F ◦ n ( V m ) for all n ≥ m ≥ V satisfies (3.12).4. Fatou coordinates
Let F be an automorphism of C of the form (3.1). Let Ω be the invariant non-recurrentFatou component of F defined in Proposition 3.7.In this section we prove that there exists a global change of holomorphic coordinates onΩ—the “Fatou coordinates”—which makes Ω biholomorphic to C , and so that F has theform ( z, w ) ( z + 1 , λw ) in this new coordinates. In order to make such a construction,we first show that Ω coincides with the set of points whose orbits pointwise accumulateto { } × C . Then we define “local approximating Fatou coordinates” and show that theydo converge to the Fatou coordinates.Let { δ m } be an increasing sequence of positive real numbers converging to ∞ and let { R m } be an increasing sequence of positive real numbers such that R m ≥ R ( δ m ) (wherethe R ( δ m )’s are defined in Step 1 of the proof of Proposition 3.7). Let V m := V R m ,δ m / and let V := ∪ n ≥ ∪ m ≥ F ◦ n ( V m ) . Let W ι := ∞ [ n =0 ( F − ) ◦ n ( V ) . Note that W ι is an open connected set such that F ( W ι ) = W ι , and, since V ⊂
Ω, itfollows that W ι ⊆ Ω. Lemma 4.1.
Let ( z , w ) ∈ C \ ( { } × C ) be such that lim n →∞ π ( F ◦ n ( z , w )) = 0 and { π ( F ◦ n ( z , w )) } is bounded, where, as before, π j is the projection on the j -th coordi-nate, j = 1 , . Then for every set V as before, there exists n = n ( V , z , w ) such that F ◦ n ( z , w ) ∈ V for all n ≥ n . In particular, Ω = W ι .Proof. If F ◦ n ( z, w ) ∈ V for some n , then F ◦ n ( z, w ) ∈ V for all n ≥ n since F ( V ) ⊂ V by construction.Therefore, we assume by contradiction that F ◦ n ( z, w )
6∈ V for all n ≥ F − has the same form of F , that is, F − ( z, w ) = ( z + ˜ f ( w ) z + O ( z ) , λw + ˜ g ( w ) z + O ( z )) , where ˜ f , ˜ g : C → C are holomorphic, ˜ f (0) = − f (0) = 1) and g ( w ) = O ( w ). Let χ ( z, w ) = ( − z, w ). The automorphism χ ◦ F − ◦ χ has the same formas F − , but the coefficient of z in the first coordinate is 1. Hence, by Proposition 3.7, thereexists an invariant non-recurrent Fatou component Ω − of χ ◦ F − ◦ χ with { } × C ⊂ ∂ Ω − ,and Ω − contains a connected open set ˜ V − of the same form as (3.11). Moreover, byRemark 3.8, we can assume that | z | < | z | for all ( z, w ) ∈ ˜ V − .In particular, χ (Ω − ) is a non-recurrent Fatou component of F − with { } × C on theboundary, contains the open set V − := χ ( ˜ V − ), and | z | < | z | for all ( z, w ) ∈ V − , that is,( z , w )
6∈ V − .By the very definition of V and V − , it follows that V ∪ V − ∪ ( { } × C ) is a neighborhoodof { } × C . Therefore, since lim n →∞ π ( F ◦ n ( z , w )) = 0, { π ( F ◦ n ( z , w )) } is boundedand F ◦ n ( z , w )
6∈ V , the sequence { F ◦ n ( z, w ) } has to be eventually contained in V − .However, since F − ( V − ) ⊂ V − , it follows that the entire orbit { F ◦ n ( z, w ) } n ∈ Z is con-tained in V − , hence ( z , w ) ∈ V − , a contradiction. (cid:3) For natural numbers n + 1 > j ≥
1, we let Q n ( u, w ) = ( u − n − A log n, λ − n w )and let ϕ n = Q n ◦ H n , where the H is defined in (3.5).For n > ϕ n ◦ H = χ n ◦ ϕ n +1 , where χ n ( u, w ) = ( u + 1 + A log(1 + n ) , λw ). Lemma 4.2.
For every δ > there exists S δ > such that the sequence { ϕ n } n ∈ N convergesuniformly on compacta of U S δ ,δ/ to a univalent map ϕ : U S δ ,δ/ → C such that (4.2) ϕ ◦ H = χ ◦ ϕ, ARABOLIC CYLINDERS 21 where χ ( u, w ) = ( u + 1 , λw ) and (4.3) ϕ ( u, w ) = ( u − A log( u ) + o (1) , w + o (1)) as Re ( u ) → ∞ . Moreover, given any increasing sequence { δ m } of positive real numbersconverging to ∞ , ϕ : S m ≥ U S δm ,δ m / → C is univalent.Proof. Fix δ > T δ be given by Lemma 3.5. Let ( u , w ) ∈ U T δ ,δ/ and set( u n , w n ) := H n ( u , w ). By Lemma 3.5 we have ( u n , w n ) ∈ U T δ ,δ for all n . Hence, byProposition 3.4,( u n +1 , w n +1 ) = (cid:18) u n + 1 + Au n + O (cid:18) u n (cid:19) , λw n + O (cid:18) u n (cid:19)(cid:19) , where the bounds in the O ’s are uniform in n . Let us write( u ′ , w ′ ) = ϕ n +1 ( u , w ) − ϕ n ( u , w ) . Observe that w ′ = λ − ( n +1) (cid:18) λw n + O (cid:18) u n (cid:19)(cid:19) − λ − n w n = O (cid:18) u n (cid:19) and u ′ = u n + 1 + Au n + O (cid:18) u n (cid:19) − ( n + 1) − A log( n + 1) − u n + n + A log n = A (cid:18) u n − n (cid:19) + A (cid:18) n − log(1 + 1 n ) (cid:19) + O (cid:18) u n (cid:19) . By Lemma 3.6 we have (cid:12)(cid:12)(cid:12)(cid:12) u n − n (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18) log nn (cid:19) and O (cid:18) | u n | (cid:19) = O (cid:18) n (cid:19) . Next observe that (cid:0) n − log(1 + n ) (cid:1) = O ( n ). Since all bounds are uniform on compactsubsets of U T,δ/ and independent from n it follows that P ∞ n = j ( ϕ n +1 − ϕ n ) convergesabsolutely, hence the sequence { ϕ n } converges uniformly on compacta of U T δ ,δ/ to a map ϕ . Fix ε >
0. By Lemma 3.5, ( u n , w n ) ∈ U T δ + n ,δ/ for every ( u , w ) ∈ U T δ/ ,δ/ . Therefore,by the same lemma, there exists n such that for all n ≥ n and all ( u, w ) ∈ U T δ/ ,δ/ , | ϕ ( u, w ) − ϕ n ( u, w ) | < ε. Since all maps ϕ n are univalent it follows that ϕ is also univalent on U T δ/ ,δ/ . Setting S δ := T δ/ we have the first result. Since ϕ n ( u, w ) = P n − k =0 ( ϕ k +1 ( u, w ) − ϕ k ( u, w )), (4.3) follows immediately from theprevious computations. The functional equation (4.2) follows from (4.1) passing to thelimit. Finally observe that given any increasing sequence { δ m } of positive real numbersconverging to ∞ there exist a sequence of positive real numbers { S δ m } so that ϕ isunivalent on S m ≥ U S δm ,δ m / . (cid:3) Let { δ m } be an increasing sequence of positive real numbers converging to ∞ and let { S δ m } be the sequence given by Lemma 4.2. Let { R m } be an increasing sequence ofpositive real numbers such that R m ≥ max { R ( δ m ) , S δ m } (where, as before, the R ( δ m )’sare defined in Step 1 of the proof of Proposition 3.7). Let V m := V R m ,δ m / .We define P = (Θ ◦ Φ ◦ Ψ ◦ τ ◦ ϕ − ) − . By Lemma 4.2, P is a univalent map defined on S ∞ m =1 V m .By (3.7) and (4.2),(4.4) F = P − ◦ χ ◦ P on S ∞ m =1 V m for all n ≥ Proposition 4.3.
The Fatou component Ω is biholomorphic to C and there exists aunivalent map Q defined on Ω such that (4.5) Q ◦ F = χ ◦ Q where χ ( u, w ) = ( u + 1 , λw ) .Proof. Let V m as before. Let V := ∪ n ≥ ∪ m ≥ F ◦ n ( V m ). By Lemma 4.1, Ω = ∪ ∞ k =0 F − k ( V ).We extend P to a univalent map Q defined on Ω as follows. If ( z, w ) ∈ Ω, there existsa natural number n such that F ◦ n ( z, w ) ∈ V m for some m . Hence, we set Q ( z, w ) = ( χ − ) ◦ n ◦ P ◦ F ◦ n ( z, w ) . By (4.4), this definition is well posed and Q : Ω → C is univalent. The functionalequation (4.5) therefore follows from (4.4).Now we prove that Q (Ω) = C . Let Ω n := ∪ nk =0 ( F − ) ◦ n ( V ). Observe that Q (Ω) = ∪ ∞ n =0 ( χ − ) ◦ n ◦ P ◦ F ◦ n (Ω n ) = ∪ ∞ n =0 ( χ − ) ◦ n ◦ P ( V ) . From the definition of the maps Θ, Φ, Ψ and τ and the set V we can find a sequenceof ρ n → ∞ satisfying ∪ ∞ k =0 { u ∈ C : Re( u ) > ρ k } × D k ⊂ τ − ◦ Ψ − ◦ Φ − ◦ Θ − ( V ) . It follows ϕ ( ∪ ∞ k =0 { u ∈ C : Re( u ) > ρ k } × D k ) ⊂ P ( V ) . ARABOLIC CYLINDERS 23
Therefore ∪ ∞ n =0 ∪ ∞ k =0 ( χ − ) ◦ n ◦ ϕ ( { u ∈ C : Re( u ) > ρ k } × D k ) ⊆ Q (Ω) . Equation (4.3) shows that for every k we can find r k ≥ ρ k such that { u ∈ C : Re( u − r k ) > | Im ( u ) |} × D k/ ⊆ ϕ ( { u ∈ C : Re( u ) > ρ k } × D k ) , hence C = ∪ ∞ n =0 ∪ ∞ k =0 ( χ − ) ◦ n (cid:0) { u ∈ C : Re( u − r k ) > | Im ( u ) |} × D k/ (cid:1) ⊆ Q (Ω) , and we are done. (cid:3) Theorem 1.2 now follows from Proposition 3.7 and Proposition 4.3.5.
An example
Using shears and overshears we can construct an explicit automorphism of the form(3.1). We first define automorphisms F ( z, w ) = ( z, λw + z ) ,F ( z, w ) = ( ze w , w ) ,F ( z, w ) = ( z, w − z ) ,F ( z, w ) = ( ze − w , w ) ,F ( z, w ) = ( z, we z ) , and finally F ( z, w ) := ( F ◦ F ◦ F ◦ F ◦ F )( z, w ) . Quick computation shows that F ( z, w ) = z + e λw z + O ( z ) , λw − z ∞ X k =2 λ k k ! w k + O ( z ) ! and F − ( z, w ) = z − e w z + O ( z ) , λ − w + λ − z ∞ X k =2 k ! w k + O ( z ) ! . References
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L. Boc Thaler: Faculty of Education, University of Ljubljana, SI–1000 Ljubljana,Slovenia.
E-mail address : [email protected] F. Bracci: Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, Via dellaRicerca Scientifica 1, 00133, Roma, Italia.
E-mail address : [email protected] H. Peters: Korteweg de Vries Institute for Mathematics, University of Amsterdam,the Netherlands
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