Matthieu Astorg

Boc Luka

Thaler

Dynamics of skew-products tangent to the identity

come

Introduction

Skew-products are holomorphic self-maps of C 2 of the form P (z, w) = (p(z), q(z, w)).

An important feature of these maps is that they preserve the set of vertical lines in C 2 . This means that we can view the restriction of P n to a line {z} × C as the composition of n entire functions on C, which allows techniques from one-dimensional complex dynamics to be applied. The dynamics of skew-products is therefore in some ways reminiscent of the dynamics of one-variable maps; however, in recent years, several important results have shown that these maps have rich and interesting dynamics, see [20,25,26,31]. For example, in [7], it was shown that there exists polynomial skew-products, i.e. P is a polynomial map, with wandering Fatou components, a dynamical phenomenon that is known not to occur for polynomial maps in one complex dimension. The proof of the main result in that paper involves the adaptation of parabolic implosion to the skewproduct setting (see also [8,10,6] for further results on parabolic implosion in several complex variables). Polynomial skew-products were also used in [13] and [29] to construct robust bifurcations, i.e. open sets contained in the bifurcation locus of the family of endomorphisms of P 2 of given algebraic degree d ≥ 2.

Given a germ of a holomorphic self-map P of C 2 that xes the origin, we say that P is tangent to the identity if it is of the form P = Id + P k (z, w) + O( (z, w) k+1 ), where k ≥ 2 and P k : C 2 → C 2 is a non-trivial homogeneous polynomial map of degree k. The he (rst uthor ws prtilly supported y the ex grnt ptou exEIUEgiRHEHHHPEHIF he seond uthor ws supported y the reserh progrm tIEQHHS from eD epuli of loveniF 1 study of local dynamics of germs tangent to the identity has received signicant attention over the last decades. For general germs of (C 2 , 0) tangent to the identity, a complete description of the dynamics on a full neighborhood of the origin is for now far out of reach. Much eort has been instead devoted to investigating the existence of invariant manifolds or invariant formal curves on which the dynamics converges to the origin (see e.g. [17,1], and more recently [22,21]).

In this paper we investigate the local dynamics of skew-products P which are tangent to the identity and have a non-degenerate second order dierential at the origin. Up to conjugacy by a linear automorphism of C 2 , such maps have the form

P : (z, w) → z -z 2 + O(z 3 ), w + w 2 + bz 2 + O( (z, w) 3 ) ,
and after a second conjugacy by an automorphism of C 2 of the form (z, w) → (z, e Az w + Bz 2 ), we may nally assume that P is of the form P (z, w) = (p(z), q(z, w)) with (1.1) p(z) := z -z 2 + az 3 + O(z 4 ) q(z, w) := w + w 2 + bz 2 + b 0,3 w 3 + b 3,0 z 3 + O( (z, w) 4 ) where a, b, b 0,3 , b 3,0 ∈ C.

Throughout this paper, we will be using the notation q z (w) := q(z, w) (in particular, q 0 = q(0, •)).

A study of the local dynamics of skew-products in the case b = 0 in (1.1) has been undertaken in [31], where a full description of the dynamics on a neighborhood of a parabolic xed point at the origin was achieved. However, most of the diculty and richness of the dynamics (including the phenomenon of parabolic implosion and the existence of wandering domains) comes precisely from this term bz 2 .

In fact, although maps of the form (1.1) are generic among polynomial skew-products which are tangent to the identity (after analytic conjugacy), we will see that they have considerably complicated local dynamics. We see the investigation of those maps (1.1) and the results of this paper as a rst step (generic case) towards the systematic analysis of the local dynamics of all polynomial skew-products which are tangent to the identity.

1.1. Parabolic domains and parabolic implosion. Denition 1.1. Let P be a holomorphic self-map of C 2 with a parabolic xed point at the origin. A parabolic domain of P is a maximal connected domain U ⊂ C 2 such that the origin is contained in the boundary of U and the iterates P n |U converges locally uniformly on U to the origin.

We begin by discussing the existence of parabolic domains for maps of the form (1.1), which depends only on b:

Theorem 1.2. Let P be a map of the form (1.1). Then

(1) If b ∈ ( 1 4 , +∞), the map P has at least two invariant parabolic domains, in which orbits converge non-tangentially to the origin.

(2) If b ∈ C\( 1 4 , +∞), the map P has an invariant parabolic domain, in which each point is attracted to the origin along trajectories tangent to one of its nondegenerate characteristic directions.

The main novelty here lies in the rst statement of this theorem, while the second statement can be deduced from results of Hakim and Vivas. Invariant parabolic domains in which points converge non-tangentially to the origin are also sometimes called spiral domains (see the beginning of Section 2 for a precise denition). Such domains were rst constructed by Rivi in her thesis [27,Proposition 4.4.4]. In [28], Rong gave sucient conditions for the existence of spiral domains for some class of maps tangent to the identity (see [28,Theorem 1.4]). However, his result does not apply to maps of the form (1.1).

From now on we will assume that b > 1 4 , and we introduce the following notations:

(1.2)

c := √ 4b -1 2
, α 0 := e π/c , β 0 := (b 0,3 -a)(α 0 -1).

Observe that since b > 1 4 , we have c > 0 and α 0 > 1. In what follows we will see that in the case b > 1 4 and β 0 ∈ R, there is parabolic implosion, which has many interesting dynamical consequences.

Denition 1.3. Let P be of the form (1.1), and α, σ ∈ C. Its generalized Lavaurs map of phase σ and parameter α is dened as (1.3) L(α, σ; z, w) := (φ o q 0 ) -1 αφ ι q 0 (w) + (1 -α)φ ι p (z) + σ , where φ ι p is the incoming Fatou coordinate of p and φ ι/o q 0 are the incoming and outgoing Fatou coordinates of q 0 respectively.

The denitions and basic properties of Fatou coordinates are recalled in Subsection 3.1; for some more background on Fatou coordinates, Lavaurs maps and horn maps, see e.g. the Appendix of [7]. The generalized Lavaurs map is dened for (z, w) ∈ B p × B q 0 , where B p and B q 0 are basins of a parabolic xed point at the origin for p and q 0 respectively. If α = 1, then the map w → L(α, σ; z, w) does not depend on z and coincides with the classical Lavaurs map of phase σ of the one-variable polynomial q 0 . Moreover, generalized Lavaurs maps satisfy the following functional relation:

(1.4) L(α, σ; p(z), q 0 (w)) = q 0 • L(α, σ; z, w) = L(α, σ + 1; z, w) for all (z, w) ∈ B p × B q 0 . Denition 1.4. Given real numbers α > 1 and β ∈ R we say that a strictly increasing sequence of positive integers (n k ) k≥0 is (α, β)-admissible if and only if its phase sequence (σ k ) k≥0 , dened by σ k := n k+1 -αn k -β ln n k , is bounded. In the case where β = 0, we will simply call such a sequence α-admissible.

Observe that for any α > 1 and β ∈ R, there always exists (α, β)-admissible sequences: it suces to dene inductively n k+1 := αn k + β ln n k and take n 0 ∈ N large enough, where • denotes the oor function. However, describing the phase sequence is in general a dicult problem; for instance, even in the particular case of the 3 2 -admissible sequences

n k+1 = 3
2 n k , the phase sequence is not fully understood (see [12]). An interesting question is the existence of (α, β)-admissible sequences with converging phase sequence, which will be discussed in detail below.

The following is the main technical result of this paper.

Main Theorem. Let P be a map of the form (1.1). Let α 0 , β 0 be as in (1.2), and assume that b > 1 4 and β 0 ∈ R. Let (n k ) k≥0 be an (α 0 , β 0 )-admissible sequence and let (σ k ) k≥0 denote its phase sequence. Then

P n k+1 -n k (p n k (z), w) = (0, L(α 0 , Γ + σ k ; z, w)) + o(1) ( as k → +∞)
with uniform convergence on compacts in B p ×B q 0 and where Γ is a constant depending only on a, b, b 0,3 , b 3,0 (see (4.1) for its explicit expression).

(k, m) ∈ N × N * , we have P k+m (Ω) ∩ P k (Ω) = ∅. A non-wandering Fatou component is a pre-periodic Fatou component.
The rst examples of polynomial maps with wandering Fatou components were introduced in [7] by Bu, Dujardin, Peters, Raissy and the rst author (see also [6]); other examples were constructed by Berger and Biebler in [9], by completely dierent methods, for Hénon maps and polynomial endomorphisms of P 2 . In the opposite direction, Ji gave in [18] and [19] sucient conditions to guarantee the absence of wandering domains near an attracting invariant ber for a skew-product map.

The examples from [7] are polynomial skew-products of the form

(z, w) → p(z), q(w) + π 2 4 z with p(z) = z -z 2 + O(z 3
) and q(w) = w + w 2 + O(w 3 ), and are not tangent to the identity at the origin. One can simplify the investigation of these maps by passing to a nite branched cover y 2 = z. This brings these maps to a form that is tangent to the identity, but with degenerate second order dierential at the origin. In particular, these maps are not of the form (1.1) considered in the present paper, which explain the dierence in dynamical features.

Denition 1.5. We dene the rank of a Fatou component Ω as the maximal rank of dh x , where x ∈ Ω and h ranges over all Fatou limit functions of (P n ) n≥0 on Ω.

Note that for endomorphism of C 2 , any wandering domain either has rank 0 (all Fatou limits are constant) or rank 1. So far, the only known examples of wandering domains in C 2 have rank 0 (that is, the examples constructed in [7], [6] and [9]). In other words, Theorem 1.6 below gives the rst examples of rank 1 wandering domains in complex dimension 2.

Theorem 1.6. Let P be a map of the form (1.1), and assume that there exists an (α 0 , β 0 )-admissible sequence with converging phase sequence. Then P has a wandering domain of rank 1.

We are therefore led to the question: for which values of α and β does such a sequence exist? Before stating an answer, recall the denition of Pisot numbers: Denition 1.7. A real algebraic integer α > 1 is called a Pisot number if all of its Galois conjugates are in the open unit disk in C (in particular, integers ≥ 2 are Pisot numbers).

The next denition might not be standard terminology, but it will be convenient for our purposes: Denition 1.8. We say that α > 1 has the Pisot property if there exist a real number ζ such that ζα k → 0, where • denotes the distance to the nearest integer.

We recall here two classical results from number theory that justify the terminology of "Pisot property":

(Pisot): Let α > 1 be an algebraic number and ζ be a non-zero real number such that ζα k → 0. Then, α is a Pisot number and ζ lies in the eld Q(α).

(Viiayaraghavan): There are only countably many pairs (ζ, α) of real numbers such that ζ = 0, α > 1, and the sequence ({ζα k }) k≥0 has only nitely many limit points. Moreover if (ζ, α) is such a pair where α is an algebraic number, then α is a Pisot number and ζ lies in the eld Q(α). Here {•} denotes the fractional part of the number.

In particular, an algebraic number has the Pisot property if and only if it is a Pisot number. Moreover, it is a long-standing conjecture known as the Pisot-Viiayaraghavan problem that Pisot numbers are the only real numbers with the Pisot property. Denition 1.9. We say that a sequence (σ k ) k≥0 converges to a cycle of period if the subsequence (σ k +j ) k≥0 converges for every 0 ≤ j < .

We can now state an almost sharp diophantine condition on α and β for the existence of an (α, β)-admissible sequence with converging phase: Theorem 1.10. Let α > 1 and β ∈ R. Then

(1) There exists an α-admissible sequence with phase sequence converging to a cycle if and only if α has the Pisot property. Moreover, in that case there exists an α-admissible sequence with phase sequence converging to 0.

(2) (a) If there exists an (α, β)-admissible sequence with phase sequence converging to a periodic cycle, then α has the Pisot property. (b) Conversely, if α has the Pisot property and

β = α-1 ln α k 1 k 2 ,
where k 1 and k 2 are coprime integers with k 2 ≥ 1, then there exists an (α, β)-admissible sequence whose phase sequence converges to a cycle of period k 2 . (non-integer Pisot number) Figure 1. Vertical slices z = constant of quadratic skew-products (1.5) for two dierent values of α 0 . In red, wandering domains; in blue, the two parabolic basins; in shades of grey, the basin of innity. Observe that gure (A) is q 0 -invariant while gure (B) is not.

Note that if the Pisot-Viijayaraghavan conjecture is true, then there exists an αadmissible sequence with converging phase sequence if and only if α is a Pisot number.

It is natural to ask whether the condition of Theorem 1.6 is necessary or not. In the case that there are no (α, β)-admissible sequences whose phase sequence converge to a periodic cycle, it means that any wandering Fatou component whose orbit remains in B p × B q 0 would have to remain bounded under a sequence of non-autonomous compositions of generalized Lavaurs maps with non-periodic sequences of phases. Proving rigorously whether such a thing is possible or not is likely to be very dicult, but it seems reasonable to expect that for generic values α it is not the case.

If we now specialize to the case of degree 2, Theorems 1.6 and 1.10 imply that for any Pisot number α 0 > 1, the map

(1.5) (z, w) → z -z 2 , w + w 2 + 1 4 + π 2 (ln α 0 ) 2 z 2
has a wandering domain of rank 1 (see Figure 1). Those are the rst completely explicit examples of polynomial maps with wandering domains, as well as the rst examples in degree 2 and the rst examples of wandering domains with rank 1.

Recall that two Fatou components Ω 1 and Ω 2 are in the same grand orbit (of Fatou components) for P if there exists n 1 , n 2 ∈ N such that P n 1 (Ω 1 ) = P n 2 (Ω 2 ). One may ask whether for polynomial endomorphisms of P 2 there exists a bound on the number of grand orbits of wandering domains that would depend only on the degree. The following theorem gives a negative answer: Theorem 1.11. Let P be of the form (1.5) and let α 0 > 1 be an integer. Then P has countably many distinct grand orbits of rank 1 wandering domains.

Note that contrary to something like the classical Newhouse phenomenon, we do not use perturbative arguments in the proof of Theorem 1.11, and the maps considered are completely explicit. In fact, more precisely, we construct an injective map from the set of hyperbolic components in a specic family of modied horn maps into the set of grand orbits of wandering Fatou components of P , see Theorem 6.6 and the beginning of Section 6. 1.3. Topological invariants and horn maps. We will now investigate a few consequences of the Main Theorem on the topological classication of skew-products tangent to the identity.

Recall that in dimension one, the topological classication of germs tangent to the identity is just given by the parabolic multiplicity, that is, the order of vanishing of f -Id at the origin. However, the analytic classication of germs tangent to the identity is extremely complicated: by a result proved independantly by Écalle and Voronin ([32],[14]) the so-called horn maps (also called Écalle-Voronin invariant) are complete invariants. These horn maps are themselves two holomorphic germs xing 0 and ∞ respectively; see e.g. the Appendix in [7] for more details.

To our knowledge, no complete topological classication is available for germs tangent to the identity in C 2 . Our results imply that such a classication must in fact also be complicated even in the seemingly simple class of skew-products; in fact, it resembles the analytic classication for one-dimensional parabolic germs. Denition 1.12. Let us dene the lifted horn map of P of phase σ by (1.6)

Hσ (Z, W ) := (Z, α 0 • φ ι q 0 • (φ o q 0 ) -1 (W ) + (1 -α 0 )Z + σ) = (Z, HZ,σ (W ))
The map Hσ satises the functional relation Hσ (Z + 1, W + 1) = Hσ (Z, W ) + (1, 1), so it descends to a map H σ dened on C 2 /Z(1, 1), which we call the horn map of phase σ of P . Remark 1.13. Observe that we have the following semi-conjugation:

(1.7)

(φ o q 0 ) -1 • ( HZ,σ (W )) = L(α 0 , σ; z, (φ o ) -1 (W ))
where Z = φ ι p (z).

The following is the main result of this subsection:

Theorem 1.14. Assume that two maps P 1 and P 2 of the form (1.1) are topologically conjugated on a neighborhood of the origin, and let H i σ denote their respective horn maps. Then there exists σ 1 , σ 2 ∈ C such that H 1 σ 1 and H 2 σ 2 are also topologically conjugated on

C 2 /Z.
The following Proposition will be needed in order to prove Theorem 1.14, but it also has an intrinsic interest:

Proposition 1.15. The real numbers α 0 and β 0 from (1.2) are topological invariants (and therefore so is b).

Finally, using Theorem 1.14, we can obtain: Corollary 1.16. If P 1 and P 2 are topologically conjugated near (0, 0), then the number of critical points of q i in B q i is the same. In particular, there is no k ∈ N such that the local topological conjugacy class of maps of the form (1.1) depend only the k-jet of P at the origin.

In [2] Abate raised the question whether maps of the form

(3 α,β,1 ) : f (z, w) = (z+αz 2 +(1-α)zw, w+βw 2 +(1-β)zw), with α+β = 1 and α, β = 0 are topologically conjugated to each other. Using Proposition 1.15 we can now answer this question negatively. Indeed, observe that for α = 1 and β = 0 this map is conjugate, via a linear automorphism, to a map

(z, w) → z -z 2 , w + w 2 + 1 -β 2 4 z 2
which is of the form (1.1).

1.4. Fatou components with historic behaviour. In [9], Berger and Biebler construct wandering Fatou components Ω for some maps f (which are Hénon maps or endomorphisms of P 2 ) that have historic behaviour, meaning that for any x ∈ Ω, the sequence of empirical measures

e n (x) := 1 n n k=1 δ f k (x)
does not converge.

To our knowledge, these are the only known examples so far of Fatou components for endomorphisms of P k or for Hénon maps with historic behaviour. Note that in the case of the wandering Fatou components constructed in [7] and [6], the sequences (e n ) n∈N converge to the dirac mass centered at the parabolic xed point at the origin.

In dimension 1, it follows easily from the Fatou-Sullivan classication that no Fatou components of a rational map on P 1 can have historic behaviour; and for moderately dissipative Hénon maps, it follows from the classication of Lyubich and Peters [23] that periodic Fatou components cannot have historic behaviour.

Using the Main Theorem of this paper, we give here new, explicit examples of polynomial skew-products that extend to endomorphisms of P 2 which have a Fatou component with historic behaviour: Corollary 1.17. Let P (z, w) = (p(z), q(z, w)) be a polynomial skew-product satisfying the following properties:

(1)

p(z) = z -z 2 + O(z 3 )
(2) P has two dierent xed points tangent to the identity of the form (0, w 1 ) and (0, w 2 ), which both satisfy the conditions that α i ∈ N * and β i = 0, with the same notations as in the Main Theorem and in appropriate local coordinates. Then P has a Fatou component Ω with historic behaviour. More precisely, for any (z, w) ∈ Ω, the sequences (e n (z, w)) n∈N accumulates on

µ 1 := α 1 α 2 -α 2 α 1 α 2 -1 δ (0,w 1 ) + α 2 -1 α 1 α 2 -1 δ (0,w 2 )
and on

µ 2 := α 1 -1 α 1 α 2 -1 δ (0,w 1 ) + α 1 α 2 -α 1 α 1 α 2 -1 δ (0,w 2 ) .
More explicitly, these conditions are given by: (1) p(z) = z -z 2 + O(z 3 ) (2) P has two dierent xed points tangent to the identity of the form (0, w 1 ) and (0, w 2 ), with q 0 (w

i ) = 2 (3) d 3 dz 3 |z=0 p(z) = d 3 dw 3 |w=w i q 0 (w) (4) If b i := 1 2 ∂ 2 ∂z 2 (z,w)=(0,w i ) q(z, w), then b i > 1 4
, and

α i := e 2π √ 4b i -1 ∈ N * .
Example 1.18. With

p(z) := z -z 2 + z 3 + z 7
and q(z, w) := q 0 (w) + a(z) with

q 0 := w + w 2 + w 3 - 35 3 w 4 + 39 2 w 5 -13w 6 + 19 6 w 7 and a(z) := 1 4 + π 2 (ln 2) 2 z 2 (1 -z) 2 ,
the map P satises the conditions above, with w 1 = 0 and w 2 = 1, α i = 2 and β i = 0.

Although we believe that the Fatou component constructed in Corollary 1.17 is wandering, we were not able to prove so. Note however that if it is not the case, then this would be the rst example of an invariant (for some iterate of P ) non-recurrent Fatou component whose limit sets depend on the limit map, which would give an armative answer to [23,Question 30] for the case X = C 2 and X = P 2 .

Structure of the paper. In Section 2, we recall classical properties of parabolic curves and prove Theorem 1.2. In Section 3, we introduce some notations, recall some basic facts concerning Fatou coordinates, and introduce approximate Fatou coordinates. We also prove some important estimates on the error function A which measures how close the dynamics is to a translation in these approximate Fatou coordinates. The Main Theorem is proved in Section 4. Finally, Sections 5, 6, 7, 8 and 9 are devoted to the proofs of Corollary 1.6, Theorem 1.11, Theorem 1.10, Theorem 1.14 and Corollary 1.17 respectively.

Acknowledgements. We thank Marco Mancini for invaluable help in writing the code used to produce Figure 3, and Arnaud Chéritat for helpful discussions.

Parabolic domains

Let P be a holomorphic germ xing the origin which is tangent to the identity of order k ≥ 2, i.e. a map with a homogeneous expansion P = Id + P k + P k+1 + . . . where P k ≡ 0. We say that v ∈ C 2 is a characteristic direction for P if there exists a λ ∈ C so that P k (v) = λv. If λ = 0 then v is said to be non-degenerate otherwise, it is degenerate. The director of a characteristic direction v is an eigenvalue of a linear operator

d(P k ) [v] -Id : T [v] P 1 → T [v] P 1 .
A parabolic curve for P is an injective holomorphic map ϕ : ∆ → C 2 , satisfying the following properties:

(1) ∆ is simply connected domain in C with 0 ∈ ∂∆ (2) ϕ is continuous at the origin and ϕ(0) = (0, 0), (3) ϕ(∆) is invariant under P and P n | ϕ(∆) → (0, 0) uniformly on compact subsets. We say that a parabolic curve is tangent to

[v] ∈ P 1 if [ϕ(ξ)] → [v] as ξ → 0 in ∆.
This implies that for any point given point z in the parabolic curve the orbit (P n (z)) converges to the origin tangentially to v, i.e. [P n (z)] → [v] in P 1 . We now recall the following classical result due to Hakim [16,17]:

Theorem 2.1. Let P : C 2 → C 2 be a holomorphic germ xing the origin which is tangent to the identity of order k ≥ 2. Then for any non-degenerate characteristic direction v there exist (at least) k -1 parabolic curves for P tangent to [v]. Moreover if the real part of the director of a non-degenerate characteristic direction v is strictly positive, then there exists an invariant parabolic domain in which every point is attracted to the origin along a trajectory tangent to v.

From now on let P be a map of the form (1.1) and observe that its characteristic directions are given by the equations

-z 2 = λz w 2 + bz 2 = λw
Then, aside from the trivial parabolic curve z = 0 with non-degenerate characteristic direction (0, 1) , there are two parabolic curves z → (z, ζ ± (z)) which are tangent to the non-degenerate characteristic directions (1, c ± ), where c ± are the roots of (2.1)

u 2 + u + b = 0.
We break Theorem 1.2 into the following two propositions.

Proposition 2.2. If b ∈ C -( 1 4 , ∞)
, then the map P has an invariant parabolic domain, in which each point is attracted to the origin along trajectories tangent to one of its non-degenerate characteristic directions. Proof. We have two cases:

Case 1:

Let b ∈ C -[ 1 4 , ∞).
A straightforward computation shows that directors of P in the directions (0, 1) and (1, c ± ) are -1 2 and -1 -2c ± respectively. Since c ± are the solutions of the equation (2.1) it follows that c ± = -1 2 ± x 2 , where x is the solution of

x 2 = 1 -4b, hence Re(-1 -2c ± ) = ∓Re(x). Observe that for b ∈ C -[ 1 4 , ∞)
we have Re(x) = 0, hence exactly one of the directions (1, c ± ) has a director with a strictly positive real part. By Theorem 2.1 we know that if the real part of the director of a non-degenerate characteristic direction v is strictly positive, then there is an invariant parabolic domain in which each point is attracted to the origin along trajectories tangent to v Case 2: Let b = 1 4 . First, observe that as b → 1 4 , the characteristic directions (1, c ± ) are getting closer to each other, and in the limit they merge to a single characteristic

direction v = (1, -1 2 ).
In the terminology of Abate-Tovena [3], v is an irregular characteristic direction, hence by the result of Vivas [30, Theorem 1.1] there exists an invariant parabolic domain, in which each point is attracted to the origin along trajectories tangent to v.

Proposition 2.3. If b > 1 4 , then the map P always has at least two invariant parabolic domains, where points converge non-tangentially to the origin.

Proof. Let z → (z, ζ ± (z)) be a parabolic curve tangent to a non-degenerate characteristic direction (1, c ± ). Since it is invariant under P , it has to satisfy the equality

q z (ζ ± (z)) = ζ ± (p(z)). A direct computation then gives us ζ ± (z) := c ± z + O(z 2 ).
For z close to the origin, we can dene a change of coordinates ψ ± (z) = (z, w + ζ ± (z)) which conjugates our map P to a map of the form

(2.2) (z, w) → (z -z 2 + O(z 3 ), w + w 2 + 2c ± zw + O(zw 2 , z 2 w, w 3 )), where c ± = -1 2 ± i x
2 is the solution of the equation u 2 + u + b = 0 and x > 0. Note that (1, 0) is now a non-degenerate characteristic direction of this map. For the rest of the proof let us focus only on the case of c + ; in the case of c -, one can follow computations verbatim with an appropriate change of sign.

By making a blow-up w = uz of the map (2.2), we obtain (2.3)

P (z, u) = (z -z 2 + O(z 3 ), u(1 + ixz) + zu 2 + O(z 2 u)).
where O(z 2 u) is holomorphic on some neighbourhood of the origin. Let us dene D(r, r) = {z ∈ C : |z -r| < r} and D r := {(z, u) | |u| < r, z ∈ D(r, r)} and assume that r is suciently small so that p(D(r, r)) ⊂ D(r, r).

Lemma 2.4. There exists a sequence of real numbers 0 < r j < r such that for any

(z 0 , u 0 ) ∈ D := j≥1 {(z, u) | |u| < r j , z ∈ D(r, r j j+1 )} we have P n (z 0 , u 0 ) ∈ D r for all n ≥ 0.
Moreover, the sequence P n (z 0 , u 0 ) is bounded away from the origin. Proof of Lemma 2.4. First observe that for suciently small r > 0, there exists a holomorphic function h(z) such that

P (z, u) = (p(z), q(z, u)) = (z -z 2 + O(z 3 ), ue ixz+z 2 h(z) + zu 2 + O(z 2 u 2 )). Let j ∈ N * and z 0 ∈ K j := D(r, r j j+1 ). Note that we have Re(p n (z 0 )) = 1 n + O ln n n 2
and Im(p n (z 0 )) = O ln n n 2 , with uniform bounds depending only on K j for all n ≥ 1 (see Section 3). Using this, we dene

f n (u) := proj 2 (P (p n-1 (z 0 ), u)) = ue ix n +Θn(z 0 ) + u 2 n + O u 2 ln n n 2
where Θ n = O ln n n 2 depends only on z 0 and is uniformly bounded on K j and the constant in

O u 2 ln n n 2 is uniform on K j × D(0, r).
We need to prove that there exists an 0 < r j < r such that for every u 0 ∈ D(0, r j ) and every z 0 ∈ K j we have (z n , u n ) := P n (z 0 , u 0 ) ∈ D r for all n ≥ 1. In particular we need to prove that |u n | < r for all n ≥ 1.

Observe that u

n = (f n • . . . • f 1 )(u 0 ) for all n ≥ 1 and let U := τ (u) = -1 u . For n ≥ 1 we dene g n (U ) := (τ • f n • τ -1 )(U ) = U e -ix n -Θn + 1 n + O ln n n 2 , ln n U n 2 .
It suces to prove that there exists 0 < r j < r such that for all (z 0 , U 0 ) where

z 0 ∈ K j and |U 0 | > 1 r j we have |g n • . . . • g 1 (U 0 )| > 1 r for all n ≥ 1. Observe that since x is real, there exists Cj > 0 such that C-1 j < e -n-1 k=1 ix k +Θ k < Cj on K j for all n ≥ 1.
By making a non-autonomous change of coordinates

ψ n (U ) = e -n-1 k=1 ix k +Θ k U,
we obtain

G n (U ) = ψ -1 n+1 • g n • ψ n (U ) = U + 1 n e n k=1 ix k +Θ k + O ln n n 2 , ln n U n 2 = U + 1 n e ix ln n+ixγ+h(z 0 ) + O ln n n 2 , ln n U n 2 where h := ∞ k=1 Θ k is a holomorphic function of z 0 .
Here, we have used the fact that n k=1

1 k = γ + ln n + O( 1 n ) and that n k=1 Θ k (z 1 ) = h(z 0 ) + O ln n
n , where the bounds are uniform on K j .

Since x = 0 is real, it follows from Abel's summation formula that there exists a constant C > 0 such that

n k=1 1 k e ix ln k = n k=1 k -(1-ix) < C for all n ≥ 1. This implies that G n • . . . • G 1 (U ) = U + O(1) for all n ≥ 1, where the constant in O(1) depends only on K j . Next observe that g n • . . . • g 1 (U ) = ψ n+1 • G n • . . . • G 1 (U ), hence there exists A j > 0 such that for all |U 0 | > 1
r and all z 0 ∈ K j we have

C-1 j |U 0 | -A j < |g n • . . . • g 1 (U 0 )| < Cj |U 0 | + A j for all n ≥ 1.
From here it immediately follows that there exists an 0 < r j < r such that for every

|U 0 | > 1 r j we have |g n • . . . • g 1 (U 0 )| > 1 r for all n ≥ 1. Moreover for every |U 0 | > 1 r j the sequence g n • . . . • g 1 (U 0 ) is bounded away from innity.
Therefore we have proven that for any (z 0 , u 0 ) ∈ K j × D(0, r j ), we have (z n , u n ) ∈ D r for all n ≥ 0, where the sequence (u n ) n≥0 is bounded away from the origin. This concludes the proof of Lemma 2.4.

Let us resume with the proof of Proposition 2.3. Let Ω := {(z, zu) | (z, u) ∈ D}: it is a connected open set whose boundary contains the origin and such that P (Ω) ∩ Ω = ∅. From the Lemma above, it immediately follows that the iterates P n |Ω converge to the origin locally uniformly on Ω, which is therefore contained in some invariant parabolic domain. It remains to prove that orbits of points converge non-tangentially to the origin in that parabolic domain. Indeed, let (z 0 , w 0 ) ∈ Ω and (z n , w n ) = P n (z 0 , w 0 ) and observe that since z n = 0, for all n ∈ N we have [z n :

w n ] = [1 : wn zn ] = [1 : u n ].
From the proof of Lemma 2.4 we can see that every limit map of the iterates ( P n ) on D is of the form (z, u) → (0, η(z, u)), where η is a non-constant holomorphic function and ∂η ∂u ≡ 0.

Therefore, there is no vector v ∈ C 2 such that the sequence [P n (z, w)] would converge to [v] in P 1 for all (z, w) ∈ Ω.

Fatou coordinates and properties of the error function

3.1. Fatou coordinates. Consider a holomorphic function f (z) = z+a 2 z 2 +a 3 z 3 +O(z 4 )
where a 2 = 0. For r > 0 small enough we dene incoming and outgoing petals

P ι f = {|a 2 z + r| < r} and P o f = {|a 2 z -r| < r}.
The incoming petal P ι f is forward invariant, and all orbits in P ι f converge to 0. Moreover, any orbit which converges to 0 but never lands at 0 must eventually be contained in P ι f . Therefore we can dene the parabolic basin as

B f = f -n (P ι f ).
The outgoing petal P o f is backwards invariant, with backwards orbits converging to 0. On P ι f and P o f one can dene incoming and outgoing Fatou coordinates φ ι f :

P ι f → C and φ o f : P o f → C, solving the functional equations φ ι f • f (z) = φ ι f (z) + 1 and φ o f • f (z) = φ o f (z) + 1,
where φ ι f (P ι f ) contains a right half plane and φ o f (P o f ) contains a left half plane. By the rst functional equation the incoming Fatou coordinates can be uniquely extended to the attracting basin B f . On the other hand, the inverse of φ o f , denoted by (φ o f ) -1 , can be extended to the entire complex plane, still satisfying the functional equation

f • (φ o f ) -1 (Z) = (φ o f ) -1 (Z + 1).
This entire function is then called an outgoing Fatou parametrization. We note that both incoming and outgoing Fatou coordinates are (on the corresponding petals) of the form

φ ι f (z) = - 1 a 2 z -b log - 1 a 2 z + o(1)
and

φ o f (z) = - 1 a 2 z + b log 1 a 2 z + o(1)
where

b := 1 -a 3 a 2 2 .
Finally note that for every z 0 ∈ B f we have

z k := f k (z 0 ) = (φ ι f ) -1 (φ ι f (z 0 ) + k) = - 1 a 2 k + b ln k + φ ι f (z 0 ) + O ln k k -1 = - 1 a 2 1 k - b ln k k 2 - φ ι f (z 0 ) k 2 + O ln 2 k k 3 , hence Re(a 2 z k ) = -1 k + O ln k k 2
and Im(a

2 z k ) = O ln k k 2 .
3.2. The error functions. Here, we introduce and study properties for one of the main objects to appear in our arguments: the functions Ã(z, w), A(z, w) and A 0 (z).

Let P be a skew-product of the form (1.1), and recall that v = (1, c ± ) are two nondegenerate characteristic directions of P , where c ± := -1 2 ± ic. From Hakim's explicit construction [16], we know that there are two parabolic curves z → (z, ζ ± (z)) associated to these directions, which are both graphs over a small petal P ι p . Since parabolic curves are invariant under P , it follows that the functions ζ ± (z) satisfy the following functional equation:

q z (ζ ± (z)) = ζ ± (p(z)).
From here we can easily compute the rst few terms of their (formal) power series expansion:

ζ ± (z) := c ± z + c ± Θ + a 3 + (b -1)b 0,3 2 z 2 + O(z 3 ),
where

Θ := b 0,3 + a 3 -b 0,3 +b 3,0 2b Denition 3.1. Let ψ z (w) := 1 2ic log ζ + (z) -w w -ζ -(z)
where log is the principal branch of logarithm and let

ψ ι/o z (w) := ψ z (w) ± π 2c .
Note that with this choice of branch, ψ z is dened on C\L z , where L z is the real line through

ζ + (z) and ζ -(z) minus the segment [ζ -(z), ζ + (z)]. In particular, ψ ι z and ψ o z are both dened in a disk centered at w = 1 2 (ζ + (z) + ζ -(z)) whose radius is of order z. Denition 3.2. Let (1) A(z, w) := ψ ι/o p(z) • q z (w) -ψ ι/o z (w) -z (2) A 0 (w) := -1 q 0 (w) + 1 w -1
Note that the formula for A(z, w) does not depend on whether the ingoing or outgoing coordinate ψ z is used, and is therefore well dened. Proposition 3.3. We have that:

(1) A 0 (w) = (b 0,3 -1)w + O(w 2 ) is analytic near zero.

(2) There exists r > 0 such that for all z = 0 in a neighborhood of zero, A(z, •) is analytic on the disk D(0, r).

Proof. The item ( 1) is an easy computation. For (2), observe that

A(z, w) = 1 2ic log q z (w) -ζ + (p(z)) q z (w) -ζ -(p(z)) - 1 2ic log w -ζ + (z) w -ζ -(z) -z = 1 2ic log q z (w) -ζ + (p(z)) w -ζ + (z) : q z (w) -ζ -(p(z)) w -ζ -(z) -z = 1 2ic log q z (w) -q z (ζ + (z)) w -ζ + (z) : q z (w) -q z (ζ -(z)) w -ζ -(z) -z.
It follows that A(z, w) has removable singularities at w = ζ ± (z) unless these are critical points. But up to taking r > 0 small enough, D(0, r) contains no critical point of q 0 . Proposition 3.4. We have

A(z, w) = zA 0 (w) + Θ + 1 2 -b 0,3 z 2 + O(z 3 , z 2 w)
where the constants in the O are uniform for (z, w) ∈ C 2 near (0, 0) (with z ∈ P ι p ). Proof. Let w ∈ K be a compact in C * . By a straightforward computation we obtain

1 2ic log w -ζ + (z) w -ζ -(z) = 1 2ic ζ -(z) -ζ + (z) w - (ζ + (z)) 2 -(ζ -(z)) 2 2w 2 + O(z 3 ) = - z w - Θz 2 w + z 2 2w 2 + O(z 3 ).
Using this we can now show that

ψ ι/o p(z) • q z (w) = - p(z) q z (w) - Θ(p(z)) 2 q z (w) + (p(z)) 2 2(q z (w)) 2 + O(z 3 ) = - z -z 2 q 0 (w) - Θz 2 q 0 (w) + z 2 2(q 0 (w)) 2 + O(z 3 ).

This implies that

A(z, w) = zA 0 (w) + Θz 2 1 w - 1 q 0 (w) + z 2 2 1 (q 0 (w)) 2 - 1 w 2 + 2 q 0 (w) + O(z 3 ) = zA 0 (w) + Θz 2 + z 2 2 (1 -2b 0,3 ) + O(z 3 , z 2 w) = zA 0 (w) + Θ + 1 2 -b 0,3 z 2 + O(z 3 , z 2 w).
Here, we used the fact that A(z, w) is analytic, hence all terms of w with the negative power are cancelled. Note that the constant in the

O(z 3 , z 2 w) a priori depends on K ⊂ C * . Let φ z (w) := A(z,w)-zA 0 (w) z 2
and note that by Proposition 3.3 it is holomorphic on D(0, r). We have proved that for all compact K ⊂ C * , for all w ∈ K, and for all small z = 0 with Re(z) > 0, we have |φ z (w)| ≤ C K . By taking K = {|w| = r 2 } we therefore obtain the same estimate |φ z (w)| ≤ C K for all |w| ≤ r 2 because of the maximum modulus principle. This gives the desired uniformity. Denition 3.5. As in [7], let ν ∈ ( 1 2 , 2 3 ) and (1)

r z := |z| 1-ν (2) R z := {W ∈ C : rz 10 < Re(W ) < π c -rz 10 and -1 2 < Im(W ) < 1 2 } Denition 3.6. Let χ z (W ) = W + (b 0,3 -1)R(z, W ), where (3.1) R(z, W ) := cze W F c (W )
and

F c is the primitive on R 0 of W → e -W cot(cW ) vanishing at π 2c . A straightforward computation shows that R(z, W ) is a solution of the linear PDE (3.2) -z ∂R ∂z + ∂R ∂W = cz cot(cW ).
Lemma 3.7. We have

(ψ ι/o z ) -1 (W ) = -cz cot(cW ) - z 2 + O(z 2 cot(cW ), z 2 )
Proof. We have:

(3.3) (ψ ι/o z ) -1 (W ) = ζ + (z) -ζ -(z)e 2icW 1 -e 2icW
and using the fact that ζ ± (z) := (-

1 2 ± ic)z + a+(b-1)b 0,3 -Θ 2 ± icΘ z 2 + O(z 3 ), we nd (3.4) (ψ ι/o z ) -1 (W ) = -cz cot(cW ) - z 2 + O(z 2 cot(cW ), z 2 ).
Lemma 3.8. Assume that ψ ι z (w) ∈ R z , and let

W := ψ ι z (w), W 1 := ψ ι p(z) • q z (w) and z 1 := p(z). Then (3.5) R(z 1 , W 1 ) -R(z, W ) -cz 2 cot(cW ) = O |z| 2+δ
for some δ > 0.

Proof. Let x := (z, W ) and h := (z 1 , W 1 ) -(z, W ). Then by Taylor-Lagrange's formula, we have

(3.6) R(x + h) -R(x) -dR x (h) = 1 0 (1 -t) 2 2 d 2 R x+th (h, h)dt and d 2 R y (h, h) = R zz (y)h 2 1 + 2R zW (y)h 1 h 2 + R W W (y)h 2 2 = 0 + O(z 3 cot(cW )) + O(z 3 cot 2 (cW )) (Here, R zz := ∂ 2 R ∂z 2 , etc.). Since W ∈ R z by assumption, we have z 3 cot 2 (cW ) = O(|z| 1+2ν ) = O(|z| 2+δ ) for some δ > 0. Therefore |R(z 1 , W 1 ) -R(z, W ) -dR x (h)| = O |z| 2+δ .
It now remains to compare dR x (h) and cz 2 cot(cW ). First, note that

h = (-z 2 + O(z 3 ), z + O(zw)) = (-z 2 + O(z 3 ), z + O(z 2 cot(cW )).
Therefore

dR x (h) = R z (x)h 1 + R W (x)h 2 = -z 2 R z (x) + zR W (x) + O(z 3 R W , z 2 cot(cW )R W ) = cz 2 cot(cW ) + O(z 3 cot(cW ), z 3 cot 2 (cW ))
hence we have

|dR x (h) -cz 2 cot(cW )| = O |z| 2+δ .
Denition 3.9. We dene Ã(z, w)

:= χ p(z) • ψ ι p(z) • q z (w) -χ z • ψ ι z (w) -z.
Proposition 3.10 (Almost translation property). There exists δ > 0 (depending only on the choice of ν) such that

| Ã(z, w) -Λz 2 | = O |z| 2+δ for all (z, w) such that ψ ι z (w) ∈ R z , where Λ := Θ + 1 -3b 0,3 2
Proof. Let z 1 := p(z), W := ψ ι z (W ) and W 1 := ψ ι z 1 • q z (w). We have

Ã(z, w) = χ z 1 • ψ ι z 1 • q z (w) -χ z • ψ ι z (w) -z = ψ ι z 1 • q z (w) -ψ ι z (w) -z + (b 0,3 -1)(R(z 1 , W 1 ) -R(z, W )).
By Lemma 3.8

| Ã(z, w) -A(z, w) -cz 2 (b 0,3 -1) cot(cW )| = O |z| 2+δ
On the other hand, by Proposition 3.4 we have

A(z, w) = zA 0 (w) + Θ + 1 2 -b 0,3 z 2 + O(z 2 w, z 3 ) = (b 0,3 -1)zw + Θ + 1 2 -b 0,3 z 2 + O(zw 2 , z 2 w, z 3 )
so using Lemma 3.7:

A(z, w) = (1 -b 0,3 )cz 2 cot(cW ) + z 2 Θ + 1 - 3b 0,3 2 + O zw 2 , z 2 w, z 3 , z 3 cot(cW )
Putting all of these estimates together, we get:

| Ã(z, w) -Λz 2 | = O |zw 2 |, |z 2 w|, |z| 3 , |z| 2+δ
Finally, note that since by assumption ψ z (w) ∈ R z , we have |w| = O (|z| ν ). Moreover, recall that ν > 1 2 , so that:

• |zw 2 | = O |z| 1+2ν • |z 2 w| = O |z| 2+ν
Lemma 3.11. As W → 0 in R 0 , we have

(3.7) F c (W ) = 1 c log(cW ) - 1 c π 2c 0 e -u ln sin(cu)du + o(1)
Similarly, as W → π c in R 0 , we have:

(3.8) F c (W ) = e -π c 1 c log (π -cW ) + + 1 c π c π 2c e -u ln sin(cu)du + o(1) Proof. Recall that F c (W ) = W π 2c
e -u cot(cu)du. An integration by parts gives:

F c (W ) = 1 c e -W log sin(cW ) + 1 c W π 2c
e -u log sin(cu)du from which it follows that as W → 0:

F c (W ) = 1 c log(cW ) + o(1) - 1 c π 2c 0 e -u ln sin(cu)du
and as W → π c :

F c (W ) = e -π c 1 c log(π -cW ) + 1 c π c π 2c
e -u ln sin(cu)du + o(1)

Proof of the main theorem

We begin this section by explaining how the map ψ z , dened in the previous section, transforms the complex plane.

Let D z be the disk of radius

1 2 |ζ + (z) -ζ -(z)| = c|z| + O(z 2 ) centered at 1 2 (ζ + (z) + ζ -(z)).
Let S(z, R) be the union of the two disks of radius R that both contain the points ζ + (z), ζ -(z) on their boundary. The radius R will be a suciently small number, to be xed later. The denition of S(z, R) of course only makes sense when the distance between ζ + (z) and ζ -(z) is less than 2R, which once R is xed will be satised for z suciently small. Our choice of R will depend on the map q 0 , but not on z.

The line L z through ζ + (z) and ζ -(z) cuts the complex plane into the left half plane Figure 2 Key observation: There are positive real constants r 0 , R, s, t, δ such that:

(i) The invariant curves z → (z, ζ ± (z)) are graphs over the disk D(r 0 , r 0 ) ⊂ B p .

(ii) We have

- π 2c + s|z|, π 2c -s|z| × iR ⊂ ψ z (S(z, R)) ⊂ - π 2c + t|z|, π 2c -t|z| × iR
for all z ∈ D(r 0 , r 0 ). (iii) For every compact K ⊂ P ι R there exists 0 < r < r 0 such that

K ⊂ S ι (z, R) ⊂ P ι 2R for all z ∈ D(r , r ), (iv) | Ã(z, w) -Λz 2 | < |z 2+δ | for all (z, w) ∈ D(r 0 , r 0 ) × D(0, 4R) (see Proposition 3.10). (v) the inverse q -1 0 (w) is well dened on D(0, 4R). (vi) q 0 (P ι 2R ) ⊂ P ι 2R and q -1 0 (P o 2R ) ⊂ P o 2R Remark 4.1. Recall that ψ ι z = ψ z + π 2c , therefore (ii) implies ψ ι z (S(z, R)) ⊂ R z for all z ∈ D(r,
r) assuming that r > 0 is suciently small (recall that R z was introduced in Denition 3.5).

We are now ready to start with the proof: Fixing a constant: We now x these constants r 0 , R, s, t, δ and dene

k n := [n ν ] for the constant ν ∈ ( 1 2 , 2
3 ) already dened in Denition 3.5.

Notation: Given a point (z 0 , w 0 ) ∈ B p × B q 0 and an integer n > 0 we will write j := p n+j (z 0 ) and w j := q j • q j-1 • . . . • q 1 (w 0 ).

Fixing a compact: For the rest of this section we x a compact subset K × K ⊂ B p × B q 0 . Let n 0 be suciently large integer so that p kn (K ) ⊂ D(r 0 , r 0 ) for all n > n 0 . By taking even larger n 0 if necessary we may assume that w kn ∈ P ι R and therefore by (iii) above w kn ∈ S ι ( kn , R) for all (z 0 , w 0 ) ∈ K × K and all n ≥ n 0 . Finally we x a point (z 0 , w 0 ) ∈ K × K.

Remark 4.2. Unless otherwise stated, all the constants appearing in estimates depend only on the compact K × K, but not on the point (z 0 , w 0 ) nor the integer n.

4.1. Entering the eggbeater.

Lemma 4.3. We have φ ι q 0 (w kn ) = φ ι q 0 (w 0 ) + k n + o(1) and hence

w kn = -1 kn + O ln n k 2 n .
Proof. For 0 ≤ j ≤ k n we have

φ ι q 0 (w j+1 ) = φ ι q 0 (q 0 (w j ) + b 2 j + O( 3 j )) = φ ι q 0 (w j ) + 1 + O((φ ι q 0 ) (w j ) 2 j ) = φ ι q 0 (w j ) + 1 + O 2 j w 2 j = φ ι q 0 (w j ) + 1 + O k 2 n n 2
Therefore by induction,

φ ι q 0 (w kn ) = φ ι q 0 (w 0 ) + k n + O( k 3 n n 2 ), and k 3 n n 2 = o(1)
by the choice of k n . Final conclusion follows from the fact that φ ι q 0 (w) = -

1 w + (1 -b 0,3 ) log(-w) + o(1).
Lemma 4.4. We have

ψ ι kn (w kn ) = - kn w kn + 2 kn 2w 2 kn + o ( kn ) .
Proof. This follows directly from the computation in the proof of Proposition 3.4. Denition 4.5 (Approximate Fatou coordinate). Let Φ z := χ z • ψ ι z . Lemma 4.6 (Comparison with incoming Fatou coordinates). We have

1 kn Φ kn (w kn ) = φ ι q 0 (w kn ) + k 2 n 2n + (1 -b 0,3 ) ln n + E ι + o(1)
where

E ι := (b 0,3 -1) ln c - π 2c
0 e -u ln sin(cu)du . Proof. Recall that by Lemma 4.3 and Lemma 4.4 we have

w kn = - 1 k n + O ln n k 2 n and ψ ι kn (w kn ) = - kn w kn + 2 kn 2w 2 kn + o( kn ).
Next, we have:

1 kn Φ kn (w kn ) = 1 kn χ kn • ψ ι kn (w kn ) = W kn -c(1 -b 0,3 )e W F c (W ) ,
where

W := ψ ι kn (w kn ) = - 1 w kn + k 2 n 2n -c(1 -b 0,3 )e W F c (W ) + o(1)
and by Lemma 3.11, Putting all together we get

1 kn Φ kn (w kn ) = - 1 w kn + (1 -b 0,3 ) log(-w kn ) + k 2 n 2n + (1 -b 0,3 ) ln n + E ι + o(1) = φ ι q 0 (w kn ) + k 2 n 2n + (1 -b 0,3 ) ln n + E ι + o(1).
4.2. Passing through the eggbeater. Denition 4.7. Let α 0 , β 0 be as in (1.2) and dene M n := (α 0 -1)n + β 0 ln n where • is the oor function. Let n := e π c k n and ρ n := {(α 0 -1)n + β 0 ln n}, where {•} denotes the fractional part. Finally we dene W j := Φ j (w j ) Lemma 4.8. We have

W kn + Mn-n-1 j=kn j + Λ 2 j = π c + G n n + o 1 n
where

G n := -e -π c n + k 2 n 2n + (1 -b 0,3 )e -π c ln n -e -π c ρ n + φ ι q 0 (w 0 ) + C
and

C := (1 -a)e -π c π c + (1 -e -π c ) Θ + 3 2 (1 -b 0,3 ) + (a -1) -φ ι p (z 0 ) + E ι .
Proof. First recall that by Lemma 4.3 and Lemma 4.6 we have:

W kn = 1 n φ ι q 0 (w 0 ) + k n + k 2 n 2n + (1 -b 0,3 ) ln n + E ι + o 1 n .
Next recall that

j = 1 n + j - (1 -a) ln(n + j) + φ ι p (z 0 ) (n + j) 2 + O ln 2 n n 3
and observe that by the Euler-MacLaurin formula we get

Mn-n-1 j=kn j = Mn-n kn j dj + 1 2 ( kn -Mn-n ) + o 1 n = Mn-n kn j dj + 1 2n (1 -e -π/c ) + o 1 n and Mn-n-1 j=kn 2 j = kn -Mn-n + o 1 n = 1 n (1 -e -π c ) + o 1 n .

Furthermore we have

Mnn kn j dj = ln

n + M n -n n + k n + ((1 -a) + φ ι p (z 0 )) 1 n + M n -n - 1 n + k n + (1 -a) ln(n + M n -n ) n + M n -n - ln(n + k n ) n + k n + o 1 n = π c + (b 0,3 -a)(1 -e -π c ) ln n n -e -π c 1 n ρ n - k n n -e -π c n n + 1 n (1 -a) e -π c -1 + 1 n e -π c -1 φ ι p (z 0 ) + 1 n (1 -a)e -π c π c + (1 -a)(e -π c -1) ln n n + o 1 n = π c + (b 0,3 -1)(1 -e -π c ) ln n n - k n n -e -π c n n + 1 n (1 -a + φ ι p (z 0 )) e -π c -1 + (1 -a)e -π c π c -e -π c ρ n + o 1 n .
Putting all together we obtain

W kn + Mn-n-1 j=kn j + Λ 2 j = π c -e -π c n n + (1 -b 0,3 )e -π c ln n n + k 2 n 2n 2 + 1 n φ ι q 0 (w 0 ) + C -e -π c ρ n + o 1 n
Lemma 4.9. For k n ≤ j ≤ M nn , we have W j ∈ R j and

W j = W kn + j-1 i=kn i + Ã( i , w i )
Proof. We prove this by induction on j.

• Initialization: it comes from the fact that W kn = kn n + o( kn n ) (Lemma 4.3 and Lemma 4.4).

• Heredity: it follows immediately from Proposition 3.10 and the computation above.

4.3. Exiting the eggbeater.

Lemma 4.10 (Comparison with outgoing Fatou coordinates). We have w Mn-n ∈ P o R , and

1 Mn-n Φ Mn-n (w Mn-n ) - π c = φ o q 0 (w Mn-n ) + e -π c 2 n 2n + (1 -b 0,3 ) ln n + E o + o(1)
where

E o := (1 -b 0,3 ) π c -ln c -e π c
π/c π/2c e -u ln sin(cu)du . Proof. By Lemma 4.8 and Lemma 4.9 we know that W Mn-N ∈ R Mn-N and that

W Mn-N = π c -e -π c n n + k 2 n 2n 2 + o( 1 n ). Since w Mn-n = -c Mn-cot(cW Mn-n ) + O 1 n
we have w Mn-n ∼ 1 n hence for all suciently large n we have w Mn-n ∈ P o R . By the same computation as in the incoming case, we have

ψ o Mn-n (w Mn-n ) = - Mn-n w Mn-n + 2 Mn-n 2w 2 Mn-n + o( Mn-n ) = Mn-n - 1 w Mn-n + e -π c 2 n 2n + o(1)
Recall that φ o q 0 (w) = -1 w + (1 -b 0,3 ) log(w) + o(1).

Next, we have:

Φ Mn-n (w Mn-n ) = χ Mn-n • ψ ι Mn-n (w Mn-n ) = χ Mn-n ψ o Mn-n (w Mn-n ) + π c = π c + W -c Mn-n (1 -b 0,3 )e W + π c F c W + π c
where W := ψ o Mn-n (w Mn-n ), and by Lemma 3.11 Putting all together we obtain

ce W + π c F c (W + π c ) = log
1 Mn-n Φ Mn-n (w Mn-n ) - π c = = - 1 w Mn-n + (1 -b 0,3 ) log w Mn-n + e -π c 2 n 2n + (1 -b 0,3 ) ln n + E o + o(1) = φ o q 0 (w Mn-n ) + e -π c 2 n 2n + (1 -b 0,3 ) ln n + E o + o(1)
Lemma 4.11. We have

φ o q 0 (w Mn ) = φ o q 0 (w Mn-n ) + n + o(1) Proof. Recall that w Mn-n = O 1 n . For M n -n ≤ j ≤ M n we have φ o q 0 (w j+1 ) = φ o q 0 (q 0 (w j ) + b 2 j + O( 3 j )) = φ o q 0 (w j ) + 1 + O((φ o q 0 ) (w j ) 2 j ) = φ o q 0 (w j ) + 1 + O 2 j w 2 j = φ o q 0 (w j ) + 1 + O 2 n n 2
Therefore by induction,

φ o q 0 (w Mn ) = φ o q 0 (w Mn-n ) + n + O 3 n
n 2 , and

3 n n 2 = o(1) since n ∼ n ν with ν ∈ 1 2 , 2 3 .
4.4. Conclusion. Finally we can prove the Main Theorem. We state here a technical, equivalent formulation: Theorem 4.12. We have

w Mn = (φ o q 0 ) -1 e π c φ ι q 0 (w 0 ) -e π c -1 φ ι p (z 0 ) -ρ n + Γ + o(1)
where

(4.1) Γ := (e π c -1) a -b 0,3 + b 3,0 2b + a + 1 2 (1 -b 0,3 ) + (b 0,3 -1) ln c + (b 0,3 -a) π c + e π c (1 -b 0,3 ) π c 0 e -u ln sin(cu)du
Proof. We have:

φ o q 0 (w Mn ) = φ o q 0 (w Mn-n ) + n + o(1) = 1 Mn-n Φ Mn-n (w Mn-n ) - π c -e -π c 2 n 2n -(1 -b 0,3 ) ln n -E o + n + o(1) = e π c φ ι q 0 (w 0 ) -ρ n + e π c C -E o + o(1).
where the rst equality follows from Lemma 4.11, the second equality follows from Lemma 4.10 and the last equality follows from Lemma 4.8 and Lemma 4.9. Note that in this computation we used the fact that 1 2n (e

π c k 2 n -e -π c 2 n ) = o(1).
Finally recall that

Θ = b 0,3 + a -b 0,3 + b 3,0 2b , E ι = (b 0,3 -1) ln c - π 2c 0 e -u ln sin(cu)du , C = (1 -a)e -π c π c + (1 -e -π c ) Θ + 3 2 (1 -b 0,3 ) + (a -1) -φ ι p (z 0 ) + E ι , E o = (1 -b 0,3 ) π c -ln c -e π c π c π 2c
e -u ln sin(cu)du .

A quick computation now gives

e π c C -E o = -e π c -1 φ ι p (z 0 ) + Γ, hence φ o 0 (w Mn ) = e π c φ ι q 0 (w 0 ) -e π c -1 φ ι p (z 0 ) -ρ n + Γ + o(1).
Remark 4.13. Note that Theorem 4.12 has been proved under the assumption that β 0 ∈ R. Following essentially the same proof in the case where β 0 ∈ C (only replacing the denition of M n and ρ n in Denition 4.7 by M n := (α 0 -1)n + Re(β 0 ) ln n and ρ n := {(α 0 -1)n + Re(β 0 ) ln n} ), one could prove that

w Mn = (φ o q 0 ) -1 e π c φ ι q 0 (w 0 ) -e π c -1 φ ι p (z 0 ) -ρ n + Γ + iIm(b 0,3 -a) ln n + o(1).
It then seems likely that (z n+Mn , w Mn ) belongs to one of the two parabolic domains U ± from Theorem 1.2, which in turn would imply that (z n , w) belongs to the parabolic basin of (0, 0) for all n large enough. This also seems to be supported by numerical experiments. Proof of the Main Theorem from Theorem 4.12. It only remains to rephrase Theorem 4.12 in terms of admissible sequences. Let (n k ) k≥0 be an (α 0 , β 0 )-admissible sequence.

By denition of M n and ρ n , we have

M n k = (α 0 -1)n k + β 0 ln n k ,
and

ρ n k = {(α 0 -1)n k + β 0 ln n k }.
Therefore, by denition of an (α 0 , β 0 )-admissible sequence, there exists a bounded sequence of integers (m k ) k≥0 such that

n k+1 -n k = M n k + m k ,
and the phase sequence of (n k ) k≥0 is given by

σ k = n k+1 -α 0 n k -β 0 ln n k = n k+1 -(M n k + n k + ρ n k ) = m k -ρ n k .
By Theorem 4.12, we have

P Mn k (p n k (z), w) = p n k +Mn k (z), L(α 0 , Γ -ρ n k ; z, w) + o(1)
and therefore, by the functional equation satised by L,

P n k+1 -n k (p n k (z), w) = P Mn k +m k (p n k (z), w) = p n k +Mn k +m k (z), L(α 0 , Γ + m k -ρ n k ; z, w) + o(1) = (p n k+1 (z), L(α 0 , Γ + σ k ; z, w)) + o(1)
which is the desired result.

Wandering domains of rank 1

The aim of this section is to prove Theorem 1.6.

Proof of Theorem 1.6. By our assumption, if (σ k ) k∈N denotes the phase sequence associated to the (α 0 , β 0 )-admissible sequence (n k ) k∈N , then

σ k := n k+1 -α 0 n k -β 0 ln n k k→+∞ ----→
θ, and hence by the Main Theorem we have

P n k+1 -n k (p n k (z), w) k→∞ ---→ (0, L z (w))
where

L z (w) := L(α 0 , Γ + θ; z, w).
Let E(W ) := φ ι q 0 • (φ o q 0 ) -1 (W ) be the lifted horn map of q 0 . The map E is dened on the open set U q 0 := ψ o q 0 -1 (B q 0 ), which has at least two connected components, one containing an upper half-plane and the other containing a lower half-plane. Moreover, it commutes with the translation by 1: for all W ∈ U q 0 , E(W + 1) = E(W ) + 1.

Let us dene σ := Γ + θ, where Γ is the constant from the Main Theorem, and

HZ,σ (W ) := α 0 E(W ) + (1 -α 0 )Z + σ
as in Denition 1.12 Lemma 5.1. There exists a point (z 0 , w 0 ) ∈ B p × B q 0 such that w 0 is a super-attracting xed point of the map L z 0 (w).

Proof. First observe that L z is semi-conjugate to HZ,σ , where Z := φ ι p (z). Indeed, we have L z • (φ o q 0 ) -1 = (φ o q 0 ) -1 • HZ,σ , hence it suces to prove that the map HZ,σ has a super-attracting xed point for appropriate choice of Z ∈ C.

Let W 0 be a critical point of E and observe that since E commutes with the translation by 1, it follows that for every N ∈ N the point W 0 + N is also a critical point of E.

Next, observe that

α 0 E(W 0 + N ) -(W 0 + N ) + σ α 0 -1 = α 0 E(W 0 ) -W 0 + σ α 0 -1 + N,
hence for suciently large N 0 ∈ N there exists z 0 ∈ B p such that

Z 0 := φ ι p (z 0 ) = α 0 E(W 0 + N 0 ) -(W 0 + N 0 ) + σ α 0 -1 .
It is then straightforward to check that W 0 + N 0 is a super-attracting xed point of

HZ 0 ,σ (W ).
Let (z 0 , w 0 ) ∈ B p × B q 0 such that w 0 is a super-attracting xed point of L z 0 (w). Let A := {(z, w) ∈ B p × B q 0 | L z (w) = w}. The analytic set A has pure dimension 1, and since w 0 is a super-attracting xed point of L z 0 (w), the Implicit Function Theorem implies that the point (z 0 , w 0 ) is contained in a regular part of A. Therefore, there exists a small disk ∆ z 0 centered at z 0 and a holomorphic function η : ∆ z 0 → B q 0 that satises η(z 0 ) = w 0 and h(∆ z 0 ) ⊂ A where h(z) := (z, η(z)). Moreover by restricting that disk if necessary, we can assume that |L z (η(z))| < 1 2 on ∆ z 0 .

Lemma 5.2. The map η : ∆ z 0 → C is non-constant.

Proof. Recall that we constructed Z 0 , W 0 ∈ C such that HZ 0 ,σ (W 0 ) = W 0 , and

Z 0 = φ ι p (z 0 ), η(z 0 ) = (φ o q 0 ) -1 (W 0 ).
Again by the Implicit Function Theorem, there exists a holomorphic map η :

∆ Z 0 → C such that η(Z) is a xed point of HZ,σ for all Z ∈ ∆ Z 0 , where ∆ Z 0 is a small disk centered at Z 0 . Moreover, we have η = (φ o q 0 ) -1 • η • φ ι p .
From the expression of HZ,σ , it is not dicult to nd that η (Z 0 ) = 1 -α 0 = 0, therefore η and also η are non-constant.

By the Main Theorem, for each z ∈ ∆ z 0 there exist a disk D z ⊂ B q 0 centered at η(z) and k 0 > 0 such that (5.1) proj 2 (P n k+1 -n k (p n k (z) × D z )) D z for all k ≥ k 0 , where proj 2 : C 2 → C denotes the projection on the second coordinate. Moreover, we can nd a continuously varying family of disks {z} × D z ⊂ B p × B q 0 and a uniform constant k 0 with respect to the parameter z ∈ ∆ z 0 for which (5.1) holds. Let us dene an open set (5.2)

V := z∈∆z 0 {p n k 0 (z)} × D z .
and let U be a connected component of the open set P -n k 0 (V ) containing a point (z 0 , w ) for which P n k 0 (z 0 , w ) = (p n k 0 (z 0 ), w 0 ). Observe that by the Main Theorem, the sequence (P n k ) k≥0 converges uniformly on compacts in U to a holomorphic map ϕ(z, w) := (0, η(z)) where η is as above. Moreover, since P is a skew-product, this implies that the sequence of iterates (P n ) n≥0 is bounded on U and therefore that U is contained in some Fatou component Ω ⊂ C 2 .

Lemma 5.3. The map η extends holomorphically to a map η : proj 1 (Ω) → B q 0 , and there exists a subsequence (P m k ) k≥0 that converges locally uniformly on Ω to the map Φ : Ω → {0} × B q 0 dened by Φ(z, w) = (0, η(z)). Proof. Since (P n k ) is normal on Ω, we know that it has a convergent subsequence, let us denote it by (P m k ). Moreover since P is a skew-product we know that Ω ⊂ B p × C and therefore any limit function of a convergent subsequence of (P n k ) must be of the form Φ(z, w) = (0, κ(z, w)), and κ(z, w) = η(z) for all (z, w) ∈ U . By the identity principle, we therefore have ∂κ ∂w = 0 on Ω, and so κ gives a holomorphic continuation of η on proj 1 (Ω), which we still denote by η. Finally, let us argue that η : proj 1 (Ω) → B q 0 . First, observe that if (z, w) ∈ Ω, then any ω-limit point of (z, w) has bounded orbit under P . This implies that η takes values in K(q 0 ), the lled-in Julia set of q 0 . Moreover, by Lemma 5.2, η is non-constant and therefore open; and by denition, η(∆ z 0 ) ⊂ B q 0 . So η must take values in

B q 0 . Since E(W ) = W -πi(1 -b 0,3 ) + o(1) as |Im(W )| → +∞ (see [7], Appendix), we have (5.3) HZ,σ (W ) = α 0 W + (1 -α 0 )Z + C + o(1) as |Im(W )| → +∞
for some constant C ∈ C. Let Hσ (Z, W ) := (Z, HZ,σ (W )) be the lifted horn map of P , with the notations of the introduction, and recall that it commutes with the map T (Z, W ) = (Z + 1, W + 1). This map is well dened on C × U q 0 . The set of xed points of Hσ can be explicitly written as

(5.4) Fix Hσ :=

α 0 E(W ) -W α 0 -1 + σ α 0 -1 , W | W ∈ U q 0 .
Let us dene ψ(Z, W ) = (Z -W, e 2πiW ) and Ũ := ψ(C × U q 0 ) ⊂ C × C * . Observe that there is a small punctured disk ∆ * such that C × ∆ * ⊂ Ũ and that there exists a holomorphic map

Ψ : Ũ → C × C * such that Ψ • ψ = ψ • Hσ .
(This map Ψ is holomorphically conjugated to the horn map H σ of P , see Denition 1.12). It extends holomorphically over C × {0} with Φ(X, 0) = (α 0 X + α 0 πi(1 -b 0,3 ) -σ, 0). We still denote by Ψ this extended map.

Lemma 5.4. Ω is a wandering domain. Proof. Let Φ(z, w) = (0, η(z)) be the limit function as in lemma above and dene

Λ := proj 1 (Ω) ⊂ B p . Observe that η(Λ) = proj 2 (Φ(Ω)) and that Σ := {(z, η(z)) | z ∈ Λ} is connected.
Let Fix Ψ be the analytic variety of xed points of Ψ and observe that Fix Ψ is closed in the domain of denition of Ψ. Moreover, observe that (5.5)

ψ α 0 E(W ) -W α 0 -1 + σ α 0 -1 , W = α 0 (E(W ) -W ) α 0 -1 + σ α 0 -1 , e 2πiW ,
and hence

(5.6)

ψ α 0 E(W ) -W α 0 -1 + σ α 0 -1 , W Im(W )→+∞ --------→ -α 0 πi(1 -b) + σ α 0 -1 , 0 . Since Fix Ψ is closed, it follows that -α 0 πi(1-b)+σ α 0 -1 , 0 ∈ Fix Ψ . Let B z (w) := α 0 φ ι q 0 (w) + (1 -α 0 )φ ι p (z) + σ. Observe that L z = (φ o q 0 ) -1 • B z , and that if Z := φ ι p (z), then HZ,σ = B z • (φ o q 0 ) -1 .
In other words, B z also semi-conjugates L z and HZ,σ . We let Ξ(z, w) := (φ ι p (z), B z (w)), and let Σ := Ξ(Σ) ⊂ Fix Hσ be the "lift" of Σ. Since Ξ is continuous and Σ is connected, so is Σ .

Let us assume that Ω is not wandering. Up to replacing Ω with P (Ω) we may assume that it is periodic, i.e. P m (Ω) = Ω. Observe that this implies that Σ is forward invariant under the translation T m . Let γ : I → Σ be a smooth curve such that γ(0) = (Z 0 , W 0 ) and γ(1) = (Z 0 + m, W 0 + m) (this is possible since Σ is connected), and such that ψ(γ(I)) is a Jordan curve. Now observe that by (5.4), Fix H is a holomorphic graph above U q 0 and therefore is conformally equivalent to an upper half-plane; and by (5.5) and (5.6), its image under ψ is conformally equivalent to a punctured disk. After the addition of the xed point

-α 0 πi(1-b)+σ α 0 -1
, 0 , we therefore see that Fix Ψ is conformally equivalent to a disk. The curve ψ(γ) is a Jordan curve around -α 0 πi(1-b)+σ α 0 -1 , 0 in that disk. Now let us consider the holomorphic map det dΨ : Fix Ψ → C. We have | det dΨ| < 1 on ψ(γ), since by construction the eigenvalues of dF on γ are 1 and another one which lies in the unit disk. But we have computed that det dΨ -α 0 πi(1-b)+σ α 0 -1 , 0 = α 0 > 1, which contradicts the maximum principle.

This completes the proof of Theorem 1.6. 6. Wandering domains for higher periods 6.1. Simply connected hyperbolic components. In this section we assume that α 0 ∈ N * and q 0 (w) = w + w 2 . We let ĥ denote the classical horn map of q 0 , and recall that (6.1) e 2iπ(1-α 0 )Z+2iπσ ĥ(e 2iπW ) α 0 = e 2iπ HZ,σ (W )

We let h := ĥα 0 and λ := e 2iπ(1-α 0 )Z+2iπσ ∈ C * , and consider the family (h λ ) λ∈C * , dened by h λ := λh. Observe that by the choice of q 0 , the maps h λ have exactly 3 singular values:

(1) 0 and ∞, which are asymptotic values that are also superattracting xed points (2) one free critical value v λ := λv, where v := e 2iπφ ι q 0 (-1 2 ) . In particular, if h λ has an attracting cycle dierent from 0 and ∞, then it must capture v λ . Denition 6.1. A hyperbolic component of period m in the family (h λ ) λ∈C * is a connected component of the set of λ ∈ C * such that h λ has an attracting cycle of period m dierent from 0 and ∞.

Note that by [5, Theorem E], hyperbolic components as they are dened here are also stability components. In order to prove that the Fatou components that we construct are indeed wandering, we will use the following result, which also has intrinsic interest: Theorem 6.2. Hyperbolic components in the family (h λ ) λ∈C * are simply connected.

Before proving Theorem 6.2, we introduce some further notations: Denition 6.3. We let

P m := {(λ, z) ∈ C * × C * : z = h m λ (z)}, and ρ : P m → C be the map dened by ρ(λ, z) = (h m λ ) (z).
Let U be a hyperbolic component of period m and D ⊂ C the unit disk. Then U = proj 1 (Π), where Π is a connected component of ρ-1 (D) and proj 1 : C 2 → C is the projection on the rst coordinate. Since for every λ ∈ C * , h λ has only one free singular value, it may have at most one attracting cycle dierent from 0 and ∞; therefore if (λ, z 1 ) and (λ, z 2 ) are in a same ber of the map proj 1 : Π → U , then z 1 and z 2 must be periodic points of the same attracting cycle. This means that the function ρ : Π → D descends to a well-dened holomorphic function ρ : U → D satisfying ρ = ρ • proj 1 .

Figure 3. Parameter space of (h λ ) λ∈C * . Hyperbolic components are in black. Red and blue correspond to parameters λ for which v λ is captured by 0 or ∞ respectively, and white to λ such that v λ eventually exits the domain of h λ . Observe that for all |λ| large enough, v λ is captured by ∞ (blue). Right: a zoom on a copy of the Mandelbrot set (bottom center of the left gure). Lemma 6.4. Let U 0 := U \ρ -1 ({0}). The map ρ : U 0 → D * is locally invertible.

Proof. We will prove this using a classical surgery argument, originally due to Douady-Hubbard for the case of the quadratic family ([11]). Let λ 0 ∈ U 0 , and let V be a simply connected open subset of D * containing ρ(λ 0 ). Using a standard surgery procedure, we construct for any t ∈ V a quasiconformal homeomorphism g t such that g t • h λ 0 • g -1 t is holomorphic, and g t (z 0 ) is a periodic point of period m and multiplier t. We refer to [6, Proposition 6.7], for the details (see also e.g. [15, Theorem 6.4]).

We let φ : V → Teich(h λ 0 ) be the holomorphic map induced by t → µ t , where µ t is the Beltrami form associated to g t and Teich(h λ 0 ) is the dynamical Teichmüller space of h λ 0 . For a denition of the dynamical Teichmüller space, see [24], [4]. Let V ⊂ U 0 be a simply connected domain containing λ 0 . Since for all λ ∈ V the free critical value v λ remains captured by the attracting cycle, the family (h λ ) λ∈ V is J-stable by [5, Theorem E]. In fact, since there are no non-persistent singular relations for the family (h λ ) λ∈ V , by [24, Theorem 7.4] (stated for rational maps, but whose proof carries over verbatim in this setting), the map h λ 0 is in fact structurally stable on P 1 : there is a second holomorphic family ĝλ of quasiconformal homeomorphisms ĝλ :

P 1 → P 1 such that h λ := ĝλ • h λ 0 • ĝ-1 λ for all λ ∈ V , and ĝλ 0 = Id.
We let φ : V → Teich(h λ 0 ) denote the map induced by λ → μλ , where μλ is the Beltrami form associated to ĝλ . Let ξ := d dλ |λ=λ 0 ĝλ , and observe that since ĝλ (v λ 0 ) = v λ = λv, we have ξ(v λ 0 ) = 0. By [4, Proposition 5], the derivative φ (λ 0 ) is therefore non-zero. Therefore, up to restricting V , we may assume that φ(V ) ⊂ φ( V ) and that there exists a well-dened inverse branch φ-1 : φ(V ) → V . Let c : V → V be the map dened by c := φ-1 • φ. Then c is a holomorphic local inverse of ρ, which maps ρ(λ 0 ) to λ 0 ; the Lemma is proved. Lemma 6.5. The map ρ : U 0 → D * is a covering map of nite degree.

Proof of Lemma 6.5. We start by claiming that Π is relatively compact in P m . Indeed, U = proj 1 (Π) is relatively compact in C * because if |λ| is small (respectively large) enough, v λ is captured by the super-attracting xed point 0 (respectively ∞). Moreover, by [5, Theorem A], the map proj 1 : P m → C * is proper, because the only two asymptotic values in the family (h λ ) λ∈C * are persistently xed. Therefore Π is relatively compact in P m . Since ρ is analytic (hence continuous) on P m , and since the set

Π 0 is a connected component of ρ-1 (D * ), this proves that ρ : Π 0 → D * is proper. Consequently, so is ρ : U 0 → D * .
By Lemma 6.4, the map ρ : U 0 → D * is also locally invertible; therefore it is a nite degree covering map.

Proof of Theorem 6.2. By the lemma above, ρ : U 0 → D * is a nite degree covering map. This implies that there exists λ 0 ∈ U such that U 0 = U \{λ 0 }, and that U 0 is isomorphic to a punctured disk and U to a disk. 6.2. Proof of Theorem 1.11. We state here is a slightly more precise statement of Theorem 1.11: Theorem 6.6. To each hyperbolic component U of the family (h λ ) λ∈C * , we can associate a wandering Fatou component Ω U of P . Moreover, if U 1 = U 2 , then Ω U 1 and Ω U 2 are in dierent grand orbits of P .

Since α 0 is an integer we know already that there exists an α 0 -admissible sequence n k and σ ∈ C, such that P n k+1 -n k (p n k (z), w) → L(α 0 , σ; z, w) uniformly on compacts in B p × B q 0 . Let (λ 0 , x 0 ) ∈ C * × C * be such that x 0 is a super-attracting periodic point of exact period m for h λ 0 . Let (z 0 , w 0 ) ∈ B p × B q 0 be such that e 2iπ(1-α 0 )φ ι p (z 0 ) = λ 0 and e 2iπφ ι q 0 (w 0 ) = x 0 . Then w 0 is an attracting xed point of L(α 0 , σ; z 0 , •). Applying the Main Theorem as in Section 5, we know that (z n , w 0 ) belongs to the Fatou set of P for large enough n, where z n := p n (z). We let n 0 be large enough, and Ω = Ω (zn 0 ,w 0 ) denote the Fatou component containing (z n 0 , w 0 ). We also know (again, by applying inductively the Main Theorem as in Section 5), than there is an increasing sequence of integers (n k ) such that P n k -n 0 |Ω (z, w) → (0, η(z)), where η(z) is an attracting periodic point of period m of L(α 0 , σ; z, •), with η(z n 0 ) = w 0 . Lemma 6.7. The Fatou component Ω (zn 0 ,w 0 ) is wandering.

Proof. The proof is similar to the one in Section 5, and we use some of the same notations. We assume for a contradiction that Ω is not wandering: then P k+ (Ω) = P k (Ω), for some k ∈ N and ∈ N * . Up to replacing Ω by P k (Ω), we may assume k = 0.

There exists some continuous curve joining (z n 0 , w 0 ) and P (z n 0 , w 0 ) inside Ω. Using the convergence of P n k -n 0 to (0, η), we obtain a curve joining η(z n 0 ) and η(z n 0 + ) inside Σ := η(Λ), where Λ := proj 1 (Ω). We let Σ be as in Section 5: Σ is an open subset of Per m ( Hσ ) := {(Z, W ) ∈ C 2 : Hm σ (Z, W )}: then there is a curve in Σ joining (Z 0 , W 0 ) and (Z 0 + , W 0 + ), where Z 0 := φ ι p (z n 0 ) and W 0 := α 0 φ ι q 0 (w 0 ) + (1 -α 0 )Z 0 . Finally, we consider the image of this curve under the map (6.2) e : (Z, W ) → (e 2iπ(1-α 0 )Z , e 2iπW ).

It now becomes a closed loop in Π := e(Σ ), which we denote by γ := (γ 1 , γ 2 ). By construction, the loop γ 2 is non-contractible in C * ; however, it is contained in the hyperbolic component U = proj 1 (Π). But this contradicts Theorem 6.2. Lemma 6.8. If (z i , w i ) (i = 1, 2) are such that λ i = e 2iπ(1-α 0 )φ ι p (z i ) belong to two dierent hyperbolic components U i , then the wandering Fatou components Ω i := Ω (z i,n i ,w i ) are not in the same grand orbit.

Proof. The idea of the proof is similar. Let us consider two wandering Fatou components

Ω i := Ω (p n i (z i ),w i ) constructed above.
Recall that (z i , w i ) are such that w i is a superattracting periodic point of L(α 0 , σ; z i , •), and that (p n i (z i ), w i ) ∈ Ω i . Let us denote by i the periods of w i .

Assume by contraposition that Ω 1 and Ω 2 are in the same grand orbit of Fatou components for P : then there exists m i ∈ N such that P m 1 (Ω 1 ) = P m 2 (Ω 2 ) =: Ω. Moreover, there is an increasing sequence of integers (n k ) k≥0 such that P n k -n i |Ω i converge to the maps (z, w) → (0, η i (z)), where for all z ∈ Λ i := proj 1 (Ω i ), η i (z i ) are periodic points of respective periods i of the maps L z i := L(α 0 , σ; z i , •). (This sequence (n k ) is obtained by taking an α 0 -admissible sequence, and then extracting a subsequence by taking only one term every lcm( 1 , 2 )). By normality, it is easy to see that the multipliers of those xed points cannot be repelling: ρ i (z) := (L i z ) (η i (z)) ∈ D for all z ∈ Λ i . Since nonconstant holomorphic functions are open and ρ i (z i ) = 0 ∈ D, we must therefore in fact have ρ i (z) ∈ D for all z ∈ Λ i .

Next, we claim that there exists ξ : Λ := proj 1 (Ω) → C such that

η i = q N i 0 • ξ • p m i for some N i ∈ N. Indeed, since P n k -n i |Ω i → (0, η i )
on Ω i , there exists functions ξ i : Λ → C such that P n k -n i -m i → (0, ξ i ) on Ω, and

η i = ξ i • p m i .

Assume without loss of generality that N

0 := n 1 + m 1 -(n 2 + m 2 ) ≥ 0. Then (0, ξ 2 ) = lim k P n k -n 2 -m 2 = lim k P n k -n 1 -m 1 +N 0 = P N 0 • (0, ξ 1 ) so that q N 0 0 • ξ 2 = ξ 1 . So we can take ξ := ξ 2 , N 1 := 0 and N 2 := N 0 . Recall now that Ξ(z, w) := (φ ι p (z), α 0 φ ι q 0 (w) + (1 -α 0 )φ ι p (z) + σ), Σ i := {(z, η(z)) : z ∈ Λ i }, and Σ i := Ξ(Σ i ). Let γ = (γ 1 , γ 2 ) : [0, 1] → Ω be a continuous curve joining P m 1 (p n 1 (z 1 ), w 1 ) and P m 2 (p n 2 (z 2 ), w 2 ) in Ω. Let (Z i , W i ) := Ξ(p n i (z i ), w i ).
Note that for all k, ∈ N and (z, w) ∈ B p × B q 0 , Ξ(p k (z), q 0 (w)) -Ξ(z, w) ∈ Z 2 . In particular, γ(t) := Ξ(γ 1 (t), ξ(γ 1 (t))) gives a continuous curve satisfying the following properties:

(1)

γ(0) -Ξ(z 1 , η 1 (z 1 )) ∈ Z 2 (2) γ(1) -Ξ(z 2 , η 2 (z 2 )) ∈ Z 2
(3) for all t ∈ [0, 1], γ(t) ∈ Per ( Hσ ), where := lcm( 1 , 2 ), Hσ (Z, W ) is the lifted horn map dened in (1.6), and

∂ ∂W H Z,σ (γ(t)) ∈ D.
(Property 3 comes from the previous observation that ρ i (z) ∈ D for all z ∈ Λ i .) Finally, we consider a curve (γ 1 , γ2 ) := e • γ, were e is given by equation (6.2). Then γ1 is a continuous curve joining λ 1 and λ 2 , inside a hyperbolic component of period (dividing) for the family (h λ ) λ∈C * , which is a contradiction. Thus Lemma 6.8 and Theorem 6.6 are proved.

Finally, to obtain Theorem 1.11 from Theorem 6.6, we just need to know that there are countably many hyperbolic components in the family (h λ ) λ∈C * . Since we have proved that the multiplier map is a conformal uniformization of any hyperbolic component on the unit disk, it is enough to prove that there are countably many λ ∈ C * such that h λ has a super-attracting periodic point (dierent from 0 or ∞). But this follows from e.g. [[5], Proposition 5.1].

Admissible sequences and Pisot numbers

We will give in this section the proof of Theorem 1.10. Lemma 7.1. For every (α, β)-admissible sequence (n k ) k≥0 there exist a real number ζ > 0 and a bounded sequence of real numbers (d k ) k≥0 such that

n k = ζα k -k β ln α α -1 + d k , ∀k ≥ 0.
Moreover, if we let ρ k := n k+1 -αn k -kβ ln α, then

ρ k = σ k + β ln ζ + o(1)
and

ζ = n 0 + β ln α (α -1) 2 + 1 α ∞ j=0 ρ j α j and d k = - β ln α (α -1) 2 - 1 α ∞ j=0 ρ j+k α j .
Proof. First we study the asymptotic behaviour of the (α, β)-admissible sequences.

Claim 1: For every (α, β)-admissible sequence (n k ) k≥0 there exist constants ζ, C ≥ 0 such that n k -ζα k ≤ Ck for all k > 0.

Proof of Claim 1. Let us dene

u k := n k α k and observe that u k = u k-1 + β α k ln u k-1 + β(k-1) ln α α k + σ k-1
α k where (σ k ) k≥0 denotes the phase sequence of (n k ) k≥0 . Since the sequence of phases is bounded and α > 1 it is easy to see that there exists D > 0 such that |u k | < Dk for all k > 0, hence the sequence u k converges to some positive real number ζ. It follows that

ζ = u 0 + ∞ j=0 (u j+1 -u j ) = n 0 + ∞ j=0 β α j+1 ln u j + jβ ln α α j+1 + σ j α j+1
where the sum converges absolutely. Finally observe that

n k = α k u k = ζα k - 1 α ∞ j=0 β α j ln u j+k + (j + k)β ln α α j + σ j+k α j
and that there exists C > 0 such that

1 α ∞ j=0 β α j ln u j+k + (j + k)β ln α α j + σ j+k α j < Ck, ∀k ≥ 0.
Claim 2: We have

ρ k = σ k + β ln ζ + o(1)
. Proof of Claim 2. Observe that by the previous lemma, we have

ρ k = n k+1 -αn k -kβ ln α = σ k + β ln n k -kβ ln α = σ k + β ln u k = σ k + β ln ζ + o(1)
Claim 3: We have

ζ = n 0 + β ln α (α -1) 2 + 1 α ∞ j=0 ρ j α j , d k = - β ln α (α -1) 2 - 1 α ∞ j=0 ρ j+k α j .
Proof of Claim 3. Recall that ρ k = σ k +β ln u k . From the proof of Claim 1 it now follows that

ζ = n 0 + ∞ j=0 β α j+1 ln u j + jβ ln α α j+1 + σ j α j+1 = n 0 + 1 α ∞ j=0 jβ ln α α j + ρ j α j = n 0 + β ln α (α -1) 2 + 1 α ∞ j=0 ρ j α j and d k = k β ln α α -1 + n k -ζα k = k β ln α α -1 - 1 α ∞ j=0 β α j ln u j+k + (j + k)β ln α α j + σ j+k α j = k β ln α α -1 - 1 α ∞ j=0 (j + k)β ln α α j + ρ j+k α j = - β ln α (α -1) 2 - 1 α ∞ j=0 ρ j+k α j
Corollary 7.8. Let α > 1 and m ∈ N * arbitrary. The following are equivalent:

(1) α has the Pisot property, (2) there exists an α-admissible sequence whose phase sequence converge, (3) there exists an α-admissible sequence whose phase sequence converge to a cycle of exact period m.

Let us mention that for a very special type of α-admissible sequences a similar conclusions were already made by Dubickas, see [12].

Finally the claim (2) of Theorem 1.10 follows from Lemma 7.4, Corollary 7.8 and the following remark.

Remark 7.9. Let (n k ) k≥0 be an (α, β)-admissible sequence and denote θ = β ln α α-1 and m k = n k + kθ . Observe that by Lemma 7.1 we have

n k+1 -αn k -β ln n k = n k+1 -αn k -kβ ln α -β ln ζ + o(1) = m k+1 -αm k + {(k + 1)θ} -α{kθ} -θ -β ln ζ + o(1).
It follows that the phase sequence of (n k ) k≥0 converges to a cycle if and only if the sequence (m k ) k≥0 is α-admissible and the sequence m k+1 -αm k + {(k + 1)θ} -α{kθ} converges to a cycle of the same period as σ(n k ).

Hence if we take (m k ) k≥0 to be an α-admissible the sequence whose phase sequence converges to zero (note that such always exists since α has the Pisot property) and if

θ = k 1 k 2 ∈ Q then clearly the sequence n k := m k -kθ is an (α, β)-admissible sequence whose phase sequence converges to a cycle of period k 2 .
Note that the sequence ({(k + 1)θ} -α{kθ}) k≥0 is uniformly distributed modulo 1 if and only if θ is an irrational number, therefore it is reasonable to consider the following question.

Question 7.10. Let α > 1 have the Pisot property. From the previous remark we already know that θ ∈ Q is a sucient condition for the existence of an α-admissible sequence (n k ) k≥0 such that the sequence n k+1 -αn k + {(k + 1)θ} -α{kθ} converges to a cycle. Is this condition also necessary? 8. Proof of Theorem 1.14 Let P 1 , P 2 be two skew-products that are topologically conjugated in a neighborhood of the origin, that is, there is a homeomorphism h : U → V with h • P 1 = P 2 • h and U, V are open neighborhoods of (0, 0). We will assume without loss of generality that U, V are bounded in C 2 . The map h is of the form h(z, w) = (f(z), g(z, w)), and f conjugates locally p 1 to p 2 : f • p 1 = p 2 • f. We will write g z (w) := g(z, w). We will also denote by L i , α i , β i for i ∈ {1, 2} the quantities appearing in the Main Theorem, and

(n i k ) k∈N two (α i , β i )-admissible sequences dened by n i k+1 := α i n i k + β i ln n i
k , where • is the oor function and n i 0 = n 0 is chosen large enough that both sequences are strictly increasing, and let σ i k denote their phase sequences. We let U 0 := U ∩ {z = 0}, and we assume without loss of generality that U 0 is a disk centered at w = 0.

In what follows we write q i (w) := proj 2 • P i (0, w) for i ∈ {1, 2}.

Lemma 8.1. Let z ∈ B p 1 , w ∈ B q 1 ∩ U 0 , and for any n ∈ N, let z n := p n 1 (z). Then there exists m ∈ N such that for all k large enough and all 0

≤ j ≤ n 1 k+1 -n 1 k -m, P j 1 (z n 1 k , w) ∈ U . Proof.
First, note that since lim n→∞ z n = 0, z j+n k belongs to any arbitrary neighborhood for j ≥ 0 and k large enough. Therefore, if we let w j denote the second component of

P j 1 (z n 1 k , w
), it is enough to prove that for k and m large enough, w j remains in U 0 for all 0

≤ j ≤ n 1 k+1 -n 1 k -m. For 0 ≤ j ≤ t k := (n 1 k ) ν , this follows from Lemma 4.3. For t k ≤ j ≤ n 1 k+1 -n 1 k -α 1 t k , this follows from Lemma 4.9 (recall that ν is a xed constant in ( 1 2 , 2 3 )). Finally, the existence of m > 0 (independant from n 1 k ) such that for all n 1 k+1 -n 1 k -α 1 t k ≤ j ≤ n 1 k+1 -n 1
k -m we have w j ∈ U 0 follows from Lemma 4.10.

Let us now prove Proposition 1.15.

Proposition 8.2. The real numbers α and β are topological invariants, i.e. α 1 = α 2 and β 1 = β 2 .

Proof. Let z ∈ B p ∩ Dom(f) and w ∈ B q 1 ∩ U 0 . By Lemma 8.1, we have (8.1)

h • P j 1 (p n 1 k 1 (z), w) = P j 2 • h(p n 1 k 1 (z), w) for all 0 ≤ j ≤ n 1 k+1 -n 1 k -m.
In particular, both sides of the equation belong to

V . Let M k := (α 2 -1)n 1 k + β 2 ln n 1 k , and let ρ k := {(α 2 -1)n 1 k + β 2 ln n 1 k }. Chose R > 0 large enough that V ⊂ D(0, R) 2 , and choose (z, w) ∈ U so that |L 2 (α 2 , Γ 2 -ρ k ; f(z), g 0 (w))| > R
for arbitrarily large values of k. This is always possible: indeed, let ρ ∈ [0, 1) be an accumulation point of the sequence s k . From the functional equation

L 2 (α 2 , Γ 2 -ρ; p 2 (z), q 2 (w)) = q 2 • L 2 (α 2 , Γ 2 -ρ; z, w),
it follows that (z, w) → L 2 (α 2 , Γ 2 -ρ; z, w) takes arbitrarily large values on U . Then any

(z, w) ∈ U such that |L 2 (α 2 , Γ 2 -ρ; z, w)| > R works.
Next, we observe that it follows from the Theorem 4.12 that

P M k 2 • h(p n 1 k 1 (z), w) = P M k 2 p n 1 k 2 (f(z)), g 0 (w) + o(1) = (0, L 2 (α 2 , Γ 2 -ρ k ; f(z), g 0 (w)) + o(1).
Therefore, by (8.1) and our choice of R, z and w, we must have M k > n 1 k+1 -n 1 k -m for arbitrarily large values of k. Therefore α 2 ≥ α 1 ; but then by symmetry, α 2 = α 1 . Then, using again the fact that M k > n 1 k+1 -n 1 k -m, we nd β 2 ≥ β 1 , and therefore we nally have, again by symmetry, β 1 = β 2 .

We are now ready to prove Theorem 1.14.

Proof of Theorem 1.14. By Proposition 1.15, we have n

1 k = n 2 k =: n k , σ 1 k = σ 2 k =: σ k and α 1 = α 2 =: α,
so by (8.1) and the Main Theorem, we have

h (o(1), L 1 (α, Γ 1 + σ k -m; z, w) + o(1)) = (o(1), L 2 (α, Γ 2 + σ k -m; f(z), g 0 (w)) + o(1)
for all k large enough, z ∈ B p , and w ∈ U 0 ∩ B q 0 . Therefore, for any accumulation point σ of the sequence (σ k ) k≥0 , we have:

(8.2) g 0 (L 1 (α, Γ 1 + σ -m; z, w)) = L 2 (α, Γ 2 + σ -m; f(z), g 0 (w))
Let us write for simplicity L i (z, w) := L i (α, Γ i + σ -m; z, w). Observe that since f and g 0 conjugate p 1 to p 2 and q 1 to q 2 respectively, there exists homeomorphisms f : C → C and g0 : C → C commuting with the translation by 1 such that:

(8.3) g0 • φ o 1 = φ o 2 • g 0 ,

and

(8.4)

f • φ ι p 1 = φ ι p 2 • f,
where φ o i denotes the outgoing Fatou coordinate of q i . For z, w as above, let Z := φ ι p 1 (z) and W := φ ι 1 (w). Let us compute:

g0 • H1 Z,σ 1 (W ) = φ o 2 • g 0 • (φ o 1 ) -1 • H1 Z,σ 1 (W ) (by (1.7)) = φ o 2 • g 0 • L 1 (z, (φ o 1 ) -1 (W )) (by (8.2)) = φ o 2 • L 2 (f(z), g 0 • (φ o 1 ) -1 (W )) = αφ ι 2 • g 0 • (φ o 1 ) -1 (W ) + (1 -α)φ ι p 2 (f(z)) + σ 2 = αφ ι 2 • (φ o 2 ) -1 • g0 (W ) + (1 -α) f(Z) + σ 2 = H2 f(Z),σ 2 (g 0 (W ))
where

σ i = σ + Γ i -m.
Therefore, if we let G(Z, W ) = ( f(Z), g0 (W )) we have proved that

G • H1 σ 1 (Z, W ) = H2 σ 2 • G(Z, W ).
This relation holds for all z ∈ B p 1 and for all w ∈ B q 1 ∩ U 0 ; therefore it holds for all Z ∈ C and all W ∈ C with Re(W ) large enough.

But since the lifted horn maps Hi σ i commute with the translation of vector (1, 1), this conjugation descends to a conjugation of the horn maps on C 2 /Z. Corollary 8.3. If P 1 and P 2 are topologically conjugated near (0, 0), then the number of critical points of q i in B q i is the same. In particular, there is no k ∈ N such that the local topological conjugacy class of maps of the form (1.1) depend only the k-jet of P at the origin.

Proof. For any Z ∈ C, the number of critical values of HZ,σ in {0 < Re W ≤ 1} is exactly equal to the number of critical points of q 0 in B q 0 . The former is clearly preserved under the topological conjugacy G, therefore so is the latter.

For the second assertion, it suces to observe that this number cannot depend on any k-jet of q 0 at w = 0. 9. Proof of Corollary 1.17

Finally, we will prove Corollary 1.17 in this Section. Let L (i) denote the extended Lavaurs maps associated to both parabolic xed points (0, w i ), and let

L (i) z (w) := L(α i , Γ i ; z, w). Let M z := L (2) z • L (1)
z . We denote by B i the parabolic basins of w i for q 0 , so that (z, w) → M z (w) is dened on B p × B 1 . We start by recalling the notion of islands, named after Alhlfors famous Five Islands theorem. Denition 9.1. Let f : U → P 1 be a holomorphic map, where U ⊂ P 1 is a domain. Let D 1 ⊂ P 1 be a Jordan domain. We say that D is an island for Let D k := D 0 + k. It is well-known that φ ι f : B f → C is a branched cover whose critical points are the pre-critical orbits of f in B f ; therefore, by the assumptions on D 0 , D k is simply connected and doesn't contain any critical value of φ ι f , so φ ι f has an island U 0 above D k .

f over D 1 if f : D → D 1 is a conformal isomorphism. Lemma 9.2. Let f (z) = z + z 2 + O(z 3 ) be a
By assumption, U 0 doesn't meet any critical orbits of f , and it is simply connected, so we may dene univalent inverses branches of f -k for all k, and for k large enough, at least one such branch g k of f -k will map U 0 compactly into Ω (by normality and the equidistribution of preimages). Let U k := g k (U 0 ). We then have:

φ ι f (U 0 ) = φ ι f • f k (U k ) = φ ι f (U k ) + k = D 0 + k so that φ ι f (U k ) = D 0 .
The domain U k is the desired island above D 0 . The second item follows immediately from the other well-known fact that

ψ o f : C\(ψ o f ) -1 (P f ) → C\P f
is a covering map, where P f denotes the post-critical set of f . Lemma 9.3. There exists z 0 ∈ B p such that M z 0 has a super-attracting xed point w 0 .

Proof. The diculty is that we cannot apply Montel's theorem, as the domain of M n z shrinks as n → +∞. Instead, we will follow closely the proof of the Shooting Lemma from [5]. Let φ ι i (with i = 1, 2) denote the incoming Fatou coordinates of w i for q 0 , and let ψ i denote the outgoing Fatou parametrizations associated to w i for q 0 . Let Z := φ ι p (z), A i,Z (W ) := α 0 W + (1 -α 0 )Z + Γ i , so that (9.1) This follows from the fact that it takes n 2k+1 -n 2k iterations to "pass through the eggbeater" associated to (0, w 1 ), and n 2k+2 -n 2k to pass through the one associated to (0, w 2 ) (more precisely, this follows from Lemma 4.9). Let (z, w) ∈ V , and let us consider and similarly, using (9.5),

M z = ψ 2 • A 2,Z • φ ι 2 • ψ 1 • A 1,Z • φ ι 1 .
e n 2k = 1 α 2 e n 2k-1 + (1 - 1 α 2
)δ (0,w 2 ) + o(1).

Putting last two equations together, we nd:

(9.6)

e 2k = α 1 α 2 -α 2 α 1 α 2 -1 δ (0,w 1 ) + α 2 -1 α 1 α 2 -1 δ (0,w 2 ) + o(1)
and (9.7)

e 2k+1 = α 1 -1 α 1 α 2 -1 δ (0,w 1 ) + α 1 α 2 -α 1 α 1 α 2 -1 δ (0,w 2 ) + o(1).
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  (a) α 0 = 2 (b) α 0 = 1+ √ 5 2

-

  right half plane H o z . We dene S ι/o (z, R) := S(z, R) ∩ H ι/o z . The map ψ z maps the disk D z to the shaded strip [-π 4c , π 4c ] × iR. The image of S(z, R) is bounded by two vertical lines, intersecting the real line in points of the form ∓ π 2c ± O(z), see Figure 2. Next we dene P ι/o R := D(∓R, R) and observe that S ι/o (z, R) z→0

  ce W F c (W ) = log -kn w kn + ln c -π 2c 0 e -u ln sin(cu)du + o(1) = -log(-w kn ) -ln n + ln c -π 2c 0 e -u ln sin(cu)du + o(1).

  ln sin(cu)du + o(1)

δ

  P j (z,w) = (0, w 2 ).

  e n = e n (z, w)

  rivXPHHIFHVTRW @PHPHAF IHF prizio finhiD Parabolic implosion for endomorphisms of C 2 D tF iurF wthF oF @tiwA 21 @PHIWAD noF IPD QUHW!QUQUF w RHPPUIQ IIF edrien houdy nd tohn rF rurdD Étude dynamique des polynômes complexes. Partie ID uliE tions wthémtiques d9yrsy wthemtil ulitions of yrsyD volF VRD niversité de risEudD héprtement de wthémtiquesD yrsyD IWVRF w UTPRQI IPF ert© urs huiksD On integer sequences generated by linear mapsD qlsgF wthF tF 51 @PHHWAD noF PD PRQ!PSPF w PSHHURU IQF omin hujrdinD Non-density of stability for holomorphic mappings on P k D tF ÉF polytehF wthF 4 @PHIUAD VIQ!VRQF w QTWRHWT IRF ten ÉlleD Les fonctions résurgentes. Tome IIID ulitions wthémtiques d9yrsy wthemtiE l ulitions of yrsyD volF VSD niversité de risEudD héprtement de wthémtiquesD yrsyD IWVSD v9éqution du pont et l lssi(tion nlytique des ojets louxF he ridge eqution nd nlyti lssi(tion of lol ojetsF w VSPPIH ISF xúri pgell nd vind ueenD Stable components in the parameter plane of transcendental functions of nite typeD tF qeomF enlF 31 @PHPIAD noF SD RVIT!RVSSF w RPRRVVU ITF wonique rkimD Attracting domains for semi-attractive transformations of C p D ulF wtF 38 @IWWRAD noF PD RUW!RWWF w IQITTRP IUF D Analytic transformations of (C p , 0) tangent to the identityD huke wthF tF 92 @IWWVAD noF PD RHQ!RPVF w ITIPUQH IVF huho tiD Non-uniform hyperbolicity in polynomial skew productsD riv preprint rivXIWHWFHTHVRY o pper in swx @PHIWAF IWF D Non-wandering Fatou components for strongly attracting polynomial skew productsD tF qeomF enlF 30 @PHPHAD noF ID IPR!ISPF w RHSVSHV PHF wttis tonssonD Dynamics of polynomial skew products on C 2 D wthF ennF 314 @IWWWAD noF QD RHQ!RRUF w IUHRSRQ PIF voren vópezErernnzD tsmin issyD tvier iónD nd pernndo nzEánhezD Stable manifolds of two-dimensional biholomorphisms asymptotic to formal curvesD sntF wthF esF xotF swx @PHPIAD noF IUD IPVRU!IPVVUF w RQHUTUT PPF voren vópezErernnzD tvier iónD pernndo nzEánhezD nd viz ivsD Stable manifolds of biholomorphisms in C n asymptotic to formal curvesD riv preprint rivXPHHPFHUIHP @PHPHAF PQF wikhil vyuih nd rn etersD Classication of invariant Fatou components for dissipative Hénon mapsD qeomF puntF enlF 24 @PHIRAD noF QD VVU!WISF w QPIQVQP PRF gurtis F wwullen nd hennis F ullivnD Quasiconformal homeomorphisms and dynamics. III. The Teichmüller space of a holomorphic dynamical systemD edvF wthF 135 @IWWVAD noF PD QSI!QWSF w ITPHVSH PSF rn eters nd sris wrjn mitD Fatou components of attracting skew-productsD tF qeomF enlF 28 @PHIVAD noF ID VR!IIHF w QURSVSH PTF rn eters nd viz quel ivsD Polynomial skew-products with wandering Fatou-disksD wthF F 283 @PHITAD noF IEPD QRW!QTTF w QRVWHUH PUF wrzi iviD Local behaviour of discrete dynamical systemsD hFhF hesisD niversit di pirenze @IWWVAF PVF peng ongD The non-dicritical order and attracting domains of holomorphic maps tangent to the identityD snterntF tF wthF 25 @PHIRAD noF ID IRSHHHQD IHF w QIVWUTH PWF tohn )inD Blenders near polynomial product maps of C 2 D tF iurF wthF oF @tiwA 23 @PHPIAD noF IID QSSS!QSVWF w RQIHVIP QHF viz ivsD Degenerate characteristic directions for maps tangent to the identityD sndin nivF wthF tF 61 @PHIPAD noF TD PHIW!PHRHF w QIPWIHH QIF D Local dynamics of parabolic skew-productsD ro wthemti 31 @PHPHAD noF TID SQ!UIF QPF ergei wF oroninD Analytic classication of germs of conformal mappings (C, 0) → (C, 0)D punktE sionlF enlF i rilozhenF 15 @IWVIAD noF ID I!IUD WTF w THWUWH

  See Remark 4.13 for a discussion of the case where b > 1 4 and β 0 / ∈ R. This technical Lavaurs-type theorem has a number of consequences about the local dynamics of the maps P , which we will now state. 1.2. Existence of wandering domains and Pisot numbers. The Fatou set is the largest open set in C 2 on which the family of iterates (P n ) n∈N is normal. A Fatou component Ω is a connected component of the Fatou set, and it is called wandering if for every

  polynomial map with a parabolic xed point, and let φ ι f : B f → C and ψ o f : C → C denote its incoming Fatou coordinate and outgoing Fatou parametrization respectively. Then (1) For every Jordan domain D 1 ⊂ C such that (φ ι f ) -1 (D 1 ) doesn't intersect critical orbits of f , for every open set Ω intersecting ∂B f , φ ι f has an island D Ω over D 1 (2) For every Jordan domain D 1 ⊂ C that doesn't intersect the postcritical set of f , ψ o f has an island D over D 1 . Proof. Let D 0 ⊂ C be a Jordan domain, and let Ω be an open set intersecting ∂B f .

This completes the proof of the proposition.

Remark 7.2. Note that by Lemma 7.1, the phase sequence (σ k ) k≥0 of a (α, β)-admissible sequence converges to a cycle if and only if the sequence (ρ k ) k≥0 converges to a cycle of the same period. Hence the phase sequence (σ k ) k≥0 converges to a cycle if and only if the sequence (d k ) k≥0 converges to a cycle of the same period. Corollary 7.3. If (n k ) k≥0 is an α-admissible sequence whose phase sequence converges to zero, then α has the Pisot property.

Proof. Since (n k ) k≥0 is an α-admissible sequence, β = 0 and ρ k = σ k (using the notation from Lemma 7.1). Moreover, since (σ k ) k≥0 converges to zero the same holds for the sequence (d k ) k≥0 , and hence we have ζα k → 0.

Lemma 7.4. Let (n k ) k≥0 be an (α, β)-admissible sequence and (σ k ) k≥0 its phase sequence. Then (σ k ) k≥0 converges to a cycle of period if and only if m k := n k+ -n k is α-admissible sequence and whose phase sequence converges to β ln α. Proof. Observe that

Corollary 7.5. If (n k ) k≥0 is α-admissible with converging phase sequences, then m k := n k+1 -n k is α-admissible and has phase converging to zero. Proof. This follows from the previous lemma with β = 0. Lemma 7.6. If α has the Pisot property, then there exists an α-admissible sequence whose phase sequence converges to 0.

Proof. Since α has the Pisot property there is ζ > 0 such that ζα k → 0. Now dene n k := [ζα k ] and observe that n k+1 -αn k = -ζα k+1 + α ζα k → 0.

Lemma 7.7. Let (n k ), (m k ) be two α-admissible sequences, and let j, j 1 , j 2 ∈ Z. Then:

(1) n k+j is again an α-admissible sequence, and

is strictly increasing, then it is an α-admissible sequence, and

Observe that Corollary 7.3, Corollary 7.5, Lemma 7.6 and Lemma 7.7 imply the following result which settles claim (1) of Theorem 1.10. Let c ∈ B 1 be a critical point for φ ι 1 , and let

and let

x, then this will mean that M z (c) = c, where φ ι p (z) = Z, which will prove the Lemma.

is an open set with non-empty boundary. From the expression of γ, if we x any W 0 ∈ ∂U 0 , we can nd explicitly some

Let D be a disk centered at x such that D contains no critical values of g Z 0 . (This is possible because the set of critical values of g Z is discrete, in fact nite in C/Z, and we may assume that x is not one of them). Let > 0 be small enough that h(D(Z 0 , )) D.

Let

By the denition of and V , we therefore have

and D, V are disks with smooth boundaries. It then follows from the Argument Principle that there exists Z ∈ V satisfying (9.2), and the Lemma is proved.

Proof of Corollary 1.17. We consider an inductive sequence of integers dened by

By the Main Theorem applied twice, we have

with local uniform convergence for (z, w) suciently close to the point (z 0 , w 0 ) given by Lemma 9.3. Since w 0 is a super-attracting xed point for M z 0 , there exists r > 0 such that

2 ), and by continuity there exists η > 0 such that for all z ∈ D(z 0 , η) we have M z (D(w 0 , r)) D(w 0 , r).

Let V be a connected component of P -n 0 (p n 0 (D(z 0 , η)) × D(w 0 , r)). For n 0 large enough and (n k ) satisfying the induction relation above, we have, for any k ∈ N and (z, w) ∈ V : (9.3) P n 2k (z, w) ∈ B p × D(w 0 , r).

In particular, V ⊂ K(P ), and therefore V is contained in the Fatou set of P . Let Ω be the Fatou component of P containing V .

Finally, let us prove that Ω satises the historicity property. Observe that (9.4)