Xavier Buff

Arnaud Ch Éritat

Adrien Douady

QUADRATIC JULIA SETS WITH POSITIVE AREA

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Quadratic Julia sets with positive area

Introduction

Assume P : C → C is a polynomial of degree 2. Its Julia set J(P ) is a compact subset of C with empty interior. Fatou suggested that one should apply to J(P ) the methods of Borel-Lebesgue for the measure of sets.

It is known that the area (Lebesgue measure) of J(P ) is zero in several cases including: [START_REF] Douady | Hubbard Etude dynamique des polynômes complexes I & II[END_REF] or [L1]); • if P is not infinitely renormalizable ( [L3] or [START_REF] Shishikura | Topological, geometric and complex analytic properties of Julia sets[END_REF]);

• if P is hyperbolic; 1 • if P has a parabolic cycle
1 Conjecturally, this is true for a dense and open set of quadratic polynomials. If there were an open set of non-hyperbolic quadratic polynomials, those would have a Julia set of positive area (see [MSS]).

• if P has a (linearizable) indifferent cycle with multiplier e 2iπα such that α = a 0 + 1 a 1 + 1 a 2 + . . . with log a n = O( √ n) ( [PZ]).2

In [L1], Lyubich showed that the postcritical set is a measure-theoretic attractor, which implies that the Julia sets of Misiurewicz and parabolic maps have area zero. In the same note, he also observed that the filled-in Julia set depends upper semi-continuously on the map, and concluded that generic (in the Baire sense) quadratic maps in the boundary of the Mandelbrot set have Julia set of zero area (see also [L2]). Of course, the later result of [L3] and [START_REF] Shishikura | Topological, geometric and complex analytic properties of Julia sets[END_REF] implies this since nonrenormalizable maps are generic in the boundary of the Mandelbrot set.

Recently, we completed a program initiated by Douady with major advances by the second author in [C1]: there exist quadratic polynomials with a Cremer fixed point and a Julia set of positive area. For a presentation of Douady's initial program, the reader is invited to consult [C2]. In this article, we present a slightly different approach (the general ideas are essentially the same).

Theorem 1. There exist quadratic polynomials which have a Cremer fixed point and a Julia set of positive area.

We also have the following two results.

Theorem 2. There exist quadratic polynomials which have a Siegel disk and a Julia set of positive area.

Theorem 3. There exist infinitely satellite renormalizable quadratic polynomials with a Julia set of positive area.

We will give a detailed proof of Theo. 1 and 2. We will only sketch the proof of Theo. 3.

The proofs are based on • McMullen's results [McM] regarding the measurable density of the filled-in Julia set near the boundary of a Siegel disk with bounded type rotation number; • Chéritat's techniques of parabolic explosion [C1] and Yoccoz's renormalization techniques [Yo] to control the shape of Siegel disks; • Inou and Shishikura's results [IS] to control the post-critical sets of perturbations of polynomials having an indifferent fixed point. In [Ya], Yampolsky outlines an alternative to deal with the final piece of the argument by means of the Renormalization Theorem for Siegel disks (also using the Inou-Shishikura's result).

The Cremer case

Let us introduce some notations.

Definition 1. For α ∈ C, we denote by P α the quadratic polynomial

P α : z → e 2iπα z + z 2 .
We denote by K α the filled-in Julia set of P α and by J α its Julia set.

1.1. Strategy of the proof. The main gear is the following Proposition 1. There exists a non empty set S of bounded type irrationals such that: for all α ∈ S and all ε > 0, there exists α ′ ∈ S with

• |α ′ -α| < ε,

• P α ′ has a cycle in D(0, ε) \ {0} and

• area(K α ′ ) ≥ (1 -ε)area(K α ).
The proof of Prop. 1 will occupy sections 1.2 to 1.7.

Remark. Since α ∈ S has bounded type, K α contains a Siegel disk [Si] and thus, has positive area.

Figure 1. Two filled-in Julia sets K α and K α ′ , with α ′ a wellchosen perturbation of α as in Prop. 1. This proposition asserts that if α and α ′ are chosen carefully enough the loss of measure from K α to K α ′ is small.

Remark. We do not know what is the largest set S for which Prop. 1 holds. It might be the set of all bounded type irrationals.

Proposition 2. The function α ∈ C → area(K α ) ∈ [0, +∞[ is upper semicontinuous.

Proof. Assume α n → α. By [D2], for any neighborhood V of K α , we have K αn ⊂ V for n large enough. According to the theory of Lebsegue measure, area(K α ) is the infimum of the area of the open sets containing K α . Thus, area(K α ) ≥ lim sup n→+∞ area(K αn ).

Proof of Theo. 1 assuming Prop. 1. We choose a sequence of real numbers ε n in (0, 1) such that (1ε n ) > 0. We construct inductively a sequence θ n ∈ S such that for all n ≥ 1 Figure 2. A zoom on K α ′ near its linearizable fixed point. The small cycle is highlighted.

• P θn has a cycle in D(0, 1/n) \ {0},

• area(K θn ) ≥ (1ε n )area(K θn-1 ). Every polynomial P θ with θ sufficiently close to θ n has a cycle in D(0, 1/n) \ {0}. By choosing θ n sufficiently close to θ n-1 at each step, we guarantee that

• the sequence (θ n ) is a Cauchy sequence that converges to a limit θ,

• for all n ≥ 1, P θ has a cycle in D(0, 1/n) \ {0}. So, the polynomial P θ has small cycles and thus is a Cremer polynomial. In that case, J θ = K θ . By Prop. 2: area(J θ ) = area(K θ ) ≥ lim sup

n→+∞ area(K θn ) ≥ area(K θ0 ) • n≥1 (1 -ε n ) > 0.
1.2. A stronger version of Prop. 1. For a finite or infinite sequence of integers, we will use the following continued fraction notation:

[a 0 , a 1 , a 2 , . . .] := a 0 + 1 a 1 + 1 a 2 + . . .

.

For α ∈ R, we will denote by ⌊α⌋ the integral part of α.

Definition 2. If N ≥ 1 is an integer, we set

S N := α = [a 0 , a 1 , a 2 , . . .] ∈ R \ Q (a k
) is bounded and a k ≥ N for all k ≥ 1 .

Note that S N +1 ⊂ S N ⊂ • • • ⊂ S 1 and S 1 is the set of bounded type irrationals. If α ∈ S 1 , the polynomial P α has a Siegel disk bounded by a quasicircle containing the critical point (see [D1], [He], [Sw]). In particular, the post-critical set of P α is contained in the boundary of the Siegel disk.

Prop. 1 is an immediate consequence of the following proposition. Then, for all ε > 0, if n is sufficiently large,

• P αn has a cycle in D(0, ε) \ {0} and • area(K αn ) ≥ (1ε)area(K α ).

The rest of section 1 is devoted to the proof of Prop. 3. In the sequel, unless otherwise specified,

• α is an irrational number of bounded type, • p k /q k are the approximants to α given by the continued fraction algorithm and • (α n ) is a sequence converging to α, defined as in Prop. 3.

Note that for k ≤ n, the approximants p k /q k are the same for α and for α n . The polynomial P α (resp. P αn ) has a Siegel disk ∆ (resp. ∆ n ). We let r (resp. r n ) be the conformal radius of ∆ (resp. ∆ n ) at 0 and we let φ : D(0, r) → ∆ (resp. φ n : D(0, r n ) → ∆ n ) be the conformal isomorphism which maps 0 to 0 with derivative 1.

1.3. The control of the cycle. We first recall results of [C1] (see also [START_REF] Buff | Upper Bound for the Size of Quadratic Siegel Disks[END_REF] Props. 1 and 2), which we reformulate as follows.

The first proposition asserts that as θ varies in the disk D(p/q, 1/q 3 ), the polynomial P θ has a cycle of period q which depends holomorphically on q θp/q and coalesces at z = 0 when θ = p/q. Proposition 4. For each rational number p/q (with p and q coprime), there exists a holomorphic function χ : D(0, 1/q 3/q ) → C with the following properties.

(1) χ(0) = 0.

(2) χ ′ (0) = 0.

(3) If δ ∈ D(0, 1/q 3/q ) \ {0}, then χ(δ) = 0. (4) If δ ∈ D(0, 1/q 3/q ) \ {0} and if we set ζ := e 2iπp/q and θ := p q + δ q , then, χ(δ), χ(ζδ), . . . , χ(ζ q-1 δ) forms a cycle of period q of P θ . In particular, ∀δ ∈ D(0, 1/q 3/q ), χ(ζδ) = P θ χ(δ) .

3 The choice of N will be specified in equation 3

4 For example, one can choose An := q qn n . However, we think that the proposition holds for more general sequences (αn) for instance as soon as qn √ An → +∞. This condition guaranties the existence of a small cycle. The condition qn √ log An -→ n→+∞ 1 is used at the end of the proof of Lemma 5.

A function χ : D(0, 1/q 3/q ) → C as in Prop. 4 is called an explosion function at p/q. Such a function is not unique. However, if χ 1 and χ 2 are two explosions functions at p/q, they are related by χ 1 (δ) = χ 2 (e 2iπkp/q δ) for some integer k ∈ Z.

The second proposition studies how the explosion functions behave as p/q ranges in the set of approximants of an irrational number α such that P α has a Siegel disk.

Proposition 5. Assume α ∈ R\Q is an irrational number such that P α has a Siegel disk ∆. Let p k /q k be the approximants to α. Let r be the conformal radius of ∆ at 0 and let φ : D(0, r) → ∆ be the isomorphism which sends 0 to 0 with derivative 1. For k ≥ 1, let χ k be an explosion function at p k /q k and set

λ k := χ ′ k (0). Then, (1) |λ k | -→ k→+∞ r and
(2) the sequence of maps ψ k : δ → χ k (δ/λ k ) converges uniformly on every compact subset of D(0, r) to φ : D(0, r) → ∆.

Corollary 1. Let (α n ) be the sequence defined in Prop. 3. Then, for all ε > 0, if n is sufficiently large, P αn has a cycle in D(0, ε) \ {0}.

Proof. Let χ n be an explosion at p n /q n and let C n be the set of q n -th roots of

α n - p n q n = (-1) n q n (q n A ′ n + q n-1 ) with A ′ n := [A n , N, N, N, . . .].
Since qn A ′ n -→ n→+∞ +∞, for n large enough, the set C n is contained in an arbitrarily small neighborhood of 0 and χ n (C n ) is a cycle of P αn contained in an arbitrarily small neighborhood of 0.

1.4. Perturbed Siegel disks.

Definition 3. If U and X are measurable subsets of C, with 0 < area(U ) < +∞, we use the notation dens U (X) := area(U ∩ X) area(U ) .

In the whole section, α is a Bruno number, p n /q n are its approximants, and χ n : D n := D(0, 1/q 3/qn n ) → C are explosion functions at p n /q n . Proposition 6 (see figure 3). Assume α := [a 0 , a 1 , . . .] and θ := [0, t 1 , . . .] are Brjuno numbers and let p n /q n be the approximants to α. Assume

α n := [a 0 , a 1 , . . . , a n , A n , t 1 , t 2 , . . .] with (A n ) a sequence of positive integers such that (1) lim sup n→+∞ qn log A n ≤ 1. 5
Let ∆ be the Siegel disk of P α and let ∆ ′ n be the Siegel disk of the restriction of P αn to ∆. 6 For any non empty open set U ⊂ ∆,

lim inf n→+∞ dens U (∆ ′ n ) ≥ 1 2 .
5 We think that the condition lim sup qn log An ≤ 1 is not needed. It is used at the end of the proof of Lemma 5. Proof. Set

ε n := α n - p n q n = (-1) n q 2 n (A n + θ) + q n q n-1 ∼ n→+∞ (-1) n q 2 n A n . Note that qn |ε n | ∼ n→+∞ 1 qn √ A n (where the notation u n ∼ v n means u n = v n • (1 + δ n ) with δ n → 0). For ρ < 1, define X n (ρ) := z ∈ C ; z qn z qn -ε n ∈ D(0, s n ) with s n := ρ qn ρ qn + |ε n | .
This domain is star-like with respect to 0 and avoids the q n -th roots of ε n . 7 It is contained but not relatively compact in D(0, ρ). For all non empty open set U contained in D(0, ρ),

lim inf n→+∞ dens U X n (ρ) ≥ 1 2 .
Since the limit values of the sequence (χ n :

D n → C) are isomorphisms χ : D → ∆, Prop.
6 is a corollary of Prop. 7 below.

Proposition 7. Under the same assumptions as in Prop. 6, for all ρ < 1, if n is large enough, the Siegel disk ∆ ′ n contains χ n X n (ρ) .

7 It is the preimage by the map z → z qn of a disk which is not centered at 0, contains 0 but not εn. Proof. We will proceed by contradiction. Assume there exist ρ < 1 and an increasing sequence of integers

n k such that χ n k X n k (ρ) is not contained in ∆ ′ n k .
Extracting a subsequence, we may assume

A 1/qn k n k → A ∈ [1, +∞].
To simplify notations, we will drop the index k.

• Assume A = 1. Then, any compact K ⊂ ∆ is contained in ∆ ′ n for n large enough (for a proof, see for example in [ABC], Prop. 2, the remark following Prop. 2 and Theo. 3). Note that X n (ρ) ⊂ D(0, ρ) and the limit values of the sequence (χ n : D n → C) are isomorphisms χ : D → ∆. It follows that for n large enough,

χ n X n (ρ) ⊂ χ n D(0, ρ) ⊂ χ D(0, √ ρ) ⊂ ∆ ′ n .
This contradicts our assumption.

• Assume A > 1. Without loss of generality, increasing ρ if necessary, we may assume that ρ > 1/A. We will show that for ρ < ρ ′ < 1, if n is large enough, the orbit under iteration of P αn of any point z ∈ χ n X n (ρ) remains in χ n D(0, ρ ′ ) ⊂ ∆. This will show that χ n X n (ρ) ⊂ ∆ ′ n , completing the proof of Prop. 7.

Since the limit values of the sequence χ n : D n → C are isomorphisms χ : D → ∆, there is a sequence r ′ n tending to 1 such that χ n is univalent on D ′ n := D(0, r ′ n ) and the domain of the map

f n := χ n | D ′ n -1 • P αn • χ n | D ′ n
eventually contains any compact subset of D. So, Prop. 7 is a corollary of Prop. 7' below.

Proposition 7'. Assume

0 ≤ 1 A < ρ < ρ ′ < 1.
If n is large enough, the orbit under iteration of f n of any point z ∈ X n (ρ) remains in D(0, ρ ′ ).

The rest of section 1.4, is devoted to the proof of Prop. 7'. There will be several changes of coordinates which are summarized on Figure 5 in order to help the reader (we would like to thank Misha Lyubich for suggesting this picture).

1.4.1. A vector field. Let ε n and f n be defined as previously. To prove Prop. 7', it is not enough to compare the dynamics of f n with the dynamics of a rotation. Instead, we will compare it with the (real) dynamics of the polynomial vector field ξ n which has simple roots exactly at 0 and the q n -th roots of ε n and which has derivative 2πiq n ε n at 0. Then, the time-1 map of ξ n fixes 0 and the q n -th roots of ε n (which are also fixed points of f •qn n ) with multiplier e 2πiqnεn at 0 (which is also the multiplier of f •qn n at 0). Thanks to those properties, there is a good hope that the time-1 map of ξ n very well approximates f •qn n . This vector field is

ξ n = ξ n (z) d dz := 2πiq n z(ε n -z qn ) d dz .
The vector field ξ n is invariant by the rotation z → e 2πi/qn z. It is semiconjugate by z → v = z qn to the vector field

2πiq 2 n v(ε n -v)
d dv which vanishes at 0 and ε n . Let us now consider the further change of coordinates

v → w = v/(v -ε n ) in which the vector field becomes 2πiq 2 n w d dw .
This vector field is tangent to Euclidean circles centered at 0. The boundary of X n (ρ) is mapped to such a Euclidean circle by the map z → w = z qn /(z qnε n ). It follows that the vector field ξ n is tangent to the boundary of X n (ρ) which is therefore invariant be the real dynamics of ξ n .

In addition, the unit disk is invariant by its real flow, and the open set

Ω n := z ∈ C | w = z qn z qn -ε n ∈ D
is invariant by the real flow of the vector field ξ n . The map

z → w = z qn z qn -ε n : Ω n → D
is a ramified covering of degree q n , ramified at 0. Thus, there is an isomorphism

ψ n : Ω n → D such that ψ n (z) qn = z qn z qn -ε n . The change of coordinates Ω n ∋ z → θ = ψ n (z) ∈ D conjugates the vector field ξ n to 2πiq n d dθ .
Finally, let π n : H → Ω n \ {0} (H is the upper half-plane) be the universal covering given by

π n (Z) := ψ -1 n e 2iπqnεnZ . Then, π * n ξ n = d dZ . ε 1/q X(ρ) z ξ Ω ∆ w = u q w = v v -ε ψ v = z q v = e 2πiqεZ π χ n f P αn D w s = ρ q ρ q + ε D η = qε d dθ u 0 ε ε/2 -ρ 2 v H 1 qε Z d dZ τ (ρ)
Figure 5. Several changes of coordinates involved in the proof.

Figure 6. Some real trajectories for the vector field ξ n ; zeroes of the vector field are shown.

Figure 7.

An example of open set Ω n for q n = 3. It is bounded by the black curves. Some trajectories of the vector field ξ n (red in Ω n and green outside).

1.4.2. Working in the coordinate straightening the vector field. For simplicity, we assume from now on that n is even in which case ε n > 0. In the sequel, r ∈ [ρ, 1). Then, X n (ρ) ⊂ X n (r) ⊂ Ω n and the preimage of X n (r) is the half-plane

H n (r) := Z ∈ C ; Im(Z) > τ n (r) with τ n (r) := 1 2πq 2 n ε n log 1 + ε n r qn ∼ n→+∞ 1 2πq 2 n r qn . The map π n : H n (r) → X n (r) \ {0} is a universal covering.
Remark. Note that τ n (r) increases exponentially fast with respect to q n . More precisely,

qn τ n (r) -→ n→+∞ 1 r .
Definition 4. We say that a sequence (B n ) is sub-exponential with respect to q n if lim sup

n→+∞ qn |B n | ≤ 1.
Proposition 8. Assume r < 1. If n is large enough, there exist holomorphic maps

F n : H n (r) → H and G n : H n (r) → H such that • π n semi-conjugates F n to f •qn n and G n to f •qn-1 n : π n • F n = f qn n • π n and π n • G n = f qn-1 n • π n ,
• F n -Id and G n -Id are periodic of period 1/(q n ε n ) and • as Im(Z) → +∞, we have

F n (Z) = Z + 1 + o(1) and G n (Z) = Z -(A n + θ) + o(1).
In addition, the sequences sup Z∈Hn(r)

F n (Z) -Z -1 and sup

Z∈Hn(r) G n (Z) -Z + A n + θ are sub-exponential with respect to q n .
Proof. We will use the following theorem of Jellouli (see [J1] or [J2] Theo. 1) to show that the domains of f •qn n and f

•qn-1 n eventually contain any compact subset of D.

Theorem (Jellouli). Assume P α has a Siegel disk ∆ and let χ : D → ∆ be a linearizing isomorphism. For r < 1, set ∆(r) := χ D(0, r) . Assume

α n ∈ R and b n ∈ N are such that b n • |α n -α| = o(1). 8 For all r ′ 1 < r ′ 2 < 1, if n is sufficiently large, ∆(r ′ 1 ) ⊂ z ∈ ∆(r ′ 2 ) ; ∀j ≤ b n , P •j αn (z) ∈ ∆(r ′ 2 )
. Corollary 2. For all r 1 < r 2 < 1, if n is sufficiently large, then for all z ∈ D(0, r 1 ) and for all j ≤ q n , we have f •j n (z) ∈ D(0, r 2 ). Proof. Choose r ′ 1 and r ′ 2 such that r 1 < r ′ 1 < r ′ 2 < r 2 . Let χ : D → ∆ be a linearizing isomorphism of P α . Set ∆(r ′ 1 ) := χ D(0, r ′ 1 ) and ∆(r ′ 2 ) := χ D(0, r ′ 2 ) . Since limit values of the sequence χ n : D ′ n → C are linearizing isomorphisms χ : D → ∆, for n sufficiently large,

χ n D(0, r 1 ) ⊂ ∆(r ′ 1 ) ⊂ ∆(r ′ 2 ) ⊂ χ n D(0, r 2 ) . It is therefore enough to show that for n large enough, ∆(r ′ 1 ) ⊂ z ∈ ∆(r ′ 2 ) ; ∀j ≤ q n , P •j αn (z) ∈ ∆(r ′ 2 ) . This is Jellouli's theorem with b n = q n since q n |α n -α| ∼ n→+∞ q n p n q n -α = n→+∞ o(1).
8 In fact, Jellouli's theorem is stated for the sequence αn = pn/qn and bn = o(qnq n+1 ) but the adaptation to bn • |αn -α| = o(1) is straightforward.

In particular, for r < 1, if n is large enough, then f •qn n and f

•qn-1 n are defined on X n (r). We will show that if n is large enough, then

∀z ∈ X n (r) \ {0}, f •qn n (z) ∈ Ω n \ {0} and f •qn-1 n (z) ∈ Ω n \ {0}.
We can then lift them via π n so that the following diagrams commute:

H n (r) Fn / / πn H πn X n (r) -{0} f •qn n / / Ω n -{0} and H n (r) Gn / / πn H πn X n (r) -{0} f •q n-1 n / / Ω n -{0}.
The periodicity of F n and G n then follows from

π n Z + 1 q n ε n = π n (Z).
The lifts F n and G n are determined uniquely up to addition of a integer multiple of 1/(q n ε n ). We have

q n α n -p n = q n ε n q n-1 α n -p n-1 = - 1 q n + q n-1 ε n .
So, the lift F n and G n are uniquely determined if we require that

F n (Z) -Z -→ Im(Z)→+∞ 1 and G n (Z) -Z -→ Im(Z)→+∞ - 1 q 2 n ε n + q n-1 q n = -A n -θ.
Lemma 1 below asserts that f •qn n is very close to the identity and bounds the difference.

Lemma 1. There exist a holomorphic function g n , defined on the same set as f •qn n , such that f •qn n (z) = z + ξ n (z) • g n (z). For all r < 1, the sequence sup

D(0,r)
|g n | is sub-exponential with respect to q n . Proof. According to the definition of the map χ n , the map f •qn n fixes 0 and the q n -th roots of ε n . This shows that f •qn n can be written as prescribed. To prove the estimate on the modulus of g n , note that f •qn n takes its values in D and thus, ξ n (z) • g n (z) ≤ 2. Choose a sequence r n ∈ ]0, 1[ tending to 1 so that g n is defined on D(0, r n ). By the maximum modulus principle, if n is large enough so that r n > max(r, 1/A), we have sup

|z|≤r g n (z) ≤ sup |z|≤rn g n (z) ≤ B n := sup |z|=rn 2 ξ n (z)
.

As n → +∞, inf |z|=rn ξ n (z) ∼ 2πq n r 1+qn n and thus qn B n ∼ r n → 1.
Recall that we assume n even, in which case

ε n > 0 and q n-1 • p n q n = - 1 q n mod (1).
Lemma 2 below asserts that f

•qn-1 n is very close to the rotation of angle -1/q n and bounds the difference.

Lemma 2. There exists a holomorphic function h n , defined on the same set as f

•qn-1 n , such that e 2iπ/qn f •qn-1 n (z) = z + ξ n (z) • h n (z).
For all r < 1, the sequence sup

D(0,r)
|h n | is sub-exponential with respect to q n .

Proof. According to the definition of the map χ n , the map f n coincides with the rotation of angle p n /q n on the set of q n -th roots of ε n and q n-1 • (p n /q n ) = -1/q n mod(1). Thus, e 2iπ/qn f

•qn-1 n (z) fixes 0 and the q n -th roots of ε n . This shows that e 2iπ/qn f •qn-1 n can be written as prescribed. The same method as in lemma 1 yields the bound on h n .

Proof of Prop. 8, continued. Now, given r < 1, set

R n := min 1 q n ε n , τ n (r) . Note that qn R n -→ n→+∞ min A, 1 r .
Hence, R n increases exponentially fast with respect to q n . For all n and all Z ∈ H n (r), the map π n is univalent on D(Z, R n ) and takes its values in Ω n \ {0}. By Koebe 1/4-theorem, its image contains a disk centered at z := π n (Z) with radius

π ′ n (Z) • R n 4 = ξ n (z) • R n 4 .
In particular, if the sequence (B n ) is sub-exponential with respect to q n and if n is large enough so that B n ≤ R n /4, we have

∀z ∈ X n (r), D z, ξ n (z) • B n ⊂ Ω n \ {0}.
Therefore, it follows from lemmas 1 and 2 that for all r < 1, if n is large enough, then

∀z ∈ X n (r) \ {0}, f qn n (z) ∈ Ω n \ {0} and f qn-1 n (z) ∈ Ω n \ {0}.
Lemmas 1 and 2 and Koebe distortion theorem applied to π n : D Z, R n → C imply that the sequences sup Z∈Hn(r)

F n (Z) -Z -1 and sup

Z∈Hn(r) G n (Z) -Z + A n + θ
are sub-exponential with respect to q n . This completes the proof of Prop. 8.

We will need the following improved estimate for F n .

Proposition 9. Assume r < 1. There exists a sequence (B n ), sub-exponential with respect to q n , such that for all Z ∈ H n (r),

F n (Z) -Z -1 ≤ B n • |ε n | + ε n -π n (Z) qn .
Proof. Lemma 3 below gives a similar estimate for f •qn n on X n (r). This estimate transfers to the required one by Koebe distortion theorem as in the previous proof.

Lemma 3. There exist a complex number η n and a holomorphic function k n , defined on the same set as f

•qn n , such that f •qn n (z) = z + ξ n (z) • 1 + η n + (ε n -z qn )k n (z) .
For all r < 1, there exists a sequence (B n ), sub-exponential with respect to q n , such that

|η n | ≤ B n • |ε n | and ∀z ∈ D(0, r) k n (z) ≤ B n .
Proof. By lemma 1, we know that

f •qn n (z) = z + ξ n (z) • g n (z) with, B n := sup D(0,r)
|g n | a sub-exponential sequence with respect to q n . The map

f •qn n has the same multiplier at each q n -th roots of ε n . If ω is a q n -th root of ε n , then (f •qn n ) ′ (ω) = 1 -2πiq 2 n ε n g n (ω
). Thus, g n (ω) is independant of the choice of q n -th root and we set

η n := g n (ω) -1. It follows that g n (z) = 1 + η n + (ε n -z qn )k n (z)
as prescribed. Since qn √ ε n → 1/A < r < 1, the q n -th roots of ε n belong to D(0, r) for n large enough. In that case, the bound on g n , taken at any of the q n -th roots of ε n , shows that

|1 + η n | ≤ B n and thus ∀z ∈ D(0, r) (ε n -z qn )k n (z) ≤ 2B n .
As in lemma 1, we have for any sequence r n → 1 and for n large enough:

sup |z|≤r |k n (z)| ≤ B ′ n := 2B n r qn n -ε n and (B ′
n ) is sub-exponential with respect to q n . Looking at z = 0 gives:

1 + η n + ε n k n (0) = g n (0) = (f •qn n ) ′ (0) -1 ξ ′ n (0) = e 2πiqnεn -1 2πiq n ε n .
As n → +∞, the left hand of this equality expands to 1

+ iπq n ε n + o(q n ε n ). Therefore |η n | ≤ ε n |k n (0)| + πq n + o(q n ) . Since |k n (0)| ≤ B ′ n ,
F n (Z) -Z -1 -→ n→+∞ 0 and sup Z∈Hn(r) F ′ n (Z) -1 -→ n→+∞ 0.
Proof. The first is an immediate consequence of Prop. 9. For the second, use the first on H n (r ′ ) with r < r ′ < 1.

1.4.3. Iterating the commuting pair (F n , G n ).

Proposition 10. Assume 1/A < r 1 < r 2 < 1. If n is sufficiently large, the following holds. Given any point Z ∈ H n (r 1 ), there exists a sequence of integers (j ℓ ) ℓ≥0 such that for any integer ℓ ≥ 0 and any integer j ∈ [0, j ℓ ], the point

F •j n • G n • F •j ℓ-1 n • G n • • • • • F •j1 n • G n • F •j0 n (Z)
is well defined and belongs to H n (r 2 ).

Proof. We will need to control iterates of F n for a large number of iterates. We will use the following lemma.

Lemma 4. Assume F : H → C satisfies F (Z) -Z -1 < u Re(Z)
with u : R → ]0, 1/10[ a function such that log u is 1/2-Lipschitz. Let Γ be the graph of an antiderivative of -2u. Then, every Z ∈ H which is above Γ has an image above Γ.

Proof. Let U be the antiderivative whose graph is Γ.

Let Z = X + iY ∈ H. The point Z ′ = X ′ + iY ′ = F (Z) satisfies X ′ ∈ [X + 9 10 , X + 11 10 ]. Since log u is 1/2-Lipschitz ∀x ∈ X, X + 11 10 , log u(x) ≥ log u(X) - 11 20 . Therefore, from X to X ′ , U decreases of at least 2 X ′ X u(x) dx ≥ 2(X ′ -X)e -11/20 u(X) ≥ 18 10 e -11/20 u(X) > u(X) > Y -Y ′ . Γ Z = X + iY Z + 1 u(X) Lemma 5. Assume 1/A < r < r ′ < 1.
If n is sufficiently large, then for all Z ∈ H n (r) there exists an integer j(Z) such that • for all j ≤ j(Z), we have

F •j n • G n (Z) ∈ H n (r ′ ) and • Re F •j(Z) n • G n (Z) > Re(Z).
Proof. Let us first recall that there exists a sequence (B n ), sub-exponential with respect to q n , such that for n large enough, for all Z ∈ H n (r),

G n (Z) -Z + A n + θ ≤ B n . In particular, if n is sufficiently large, Re G n (Z) ≥ Re(Z) -A n -θ -B n and Im G n (Z) ≥ τ n (r) -B n .
We will apply lemma 4 to control the orbit of G n (Z) under iteration of F n . More precisely, we will prove the existence of a function u n such that:

a) F n (Z) -Z -1 ≤ u n Re(Z) , b) for n large enough u n ∈ ]0, 1/10[, c) for n large enough, log u n is 1/2-Lipschitz and d) the sequence C n := Re(Z) Re Gn(Z)
2u n (X) dX is sub-exponential with respect to q n . Since τ n (r)/τ n (r ′ ) grow exponentially with respect to q n , if n is taken sufficiently large, we have

τ n (r) ≥ τ n (r ′ ) + B n + C n + 1 10 .
It then follows from lemma 4 that there is an integer j(Z) such that • for all j ≤ j(Z), we have

F •j n • G n (Z) ∈ H n (r ′ ) and • Re F •j(Z) n • G n (Z) > Re(Z). ≤Bn Cn Gn(Z) Z Hn(r ′ ) Hn(r) F •j(Z) n •Gn(Z)
a) By Prop. 9, there is a sequence (B ′ n ), sub-exponential with respect to q n , such that for all Z ∈ H n (r ′ ),

F n (Z) -Z -1 ≤ B ′ n ε n + ε n -π n (Z) qn . Set T n := 1/(2πq 2 n ε n ) → +∞.
We have (see Figure 5)

π n (Z) qn = ε n 1 -e -iZ/Tn . Using B ′ n ε n + ε n -π n (Z) qn ≤ B ′ n 2ε n + π n (Z) qn we see that for all Z ∈ H n (r ′ ), F n (Z) -Z -1 ≤ B ′ n ε n 2 + 1 1 -e -iZ/Tn ≤ B ′ n ε n 2 + 1 s n e -iRe(Z)/Tn -1 with e Im(Z)/Tn ≥ s n := e τn(r ′ )/Tn = 1 + ε n (r ′ ) qn .
Since 1/A < r ′ , we have ε n /(r ′ ) qn → 0 and thus s n → 1. Thus, for n large enough

1 3 ≤ 1 |s n e -iRe(Z)/Tn -1| ,
and for all Z ∈ H n (r ′ ),

F n (Z) -Z -1 ≤ u n Re(Z) with u n (X) := 7B ′ n ε n |s n e iX/Tn -1| . b) Let us show that for n large enough u n ∈ ]0, 1/10[. Note that ∀X ∈ R, u n (X) ≤ 7B ′ n ε n s n -1 = 7B ′ n (r ′ ) qn -→ n→+∞ 0.
Thus u n tends uniformly to 0 as n → +∞.

c) Let us now check that for n large enough, log u n is 1/2-Lipschitz. Letting s n = cotan(ω/2), we have

log |s n e iX/Tn -1| 2 = log(1 -sin ω cos β) + const
where β = X/T n and the constant stands for something independent of it). The β-derivative of this expression is equal to

tan ω cos ω sin β 1 -sin ω cos β = tan ω 1 - 1 -sin(β + ω) 1 -sin ω cos β ≤ tan ω = 2s n s 2 n -1 . It follows that d dX log u n (X) = 1 2 d dX log |s n e iX/Tn -1| 2 ≤ s n T n (s 2 n -1) ∼ n→+∞ πq 2 n (r ′ ) qn .
Thus, d dX log u n (X) converges uniformly to 0 as n → +∞, and for n large enough,

log u n is 1/2-Lipschitz.
d) Let us finally show that the sequence

C n := Re(Z) Re Gn(Z) 2u n (X) dX is sub-exponential with respect to q n . Let us recall that 2πT n ∼ 1/(q 2 n ε n ) ∼ A n . If n is large enough, Re G n (Z) ≥ Re(Z) -A n -θ n -B n ≥ Re(Z) -4πT n . Since u n is 2πT n -periodic, C n ≤ B ′′ n := Re(Z) Re(Z)-4πTn 2u n (X) dX = 4 πTn -πTn u n (X) dX.
The change of variable θ = X/T n , which yields

B ′′ n = 14B ′ n πq 2 n π -π dθ s 2 n + 1 -2s n cos θ . It follows that B ′′ n ∼ n→+∞ 28B ′ n πq 2 n log 1 s n -1 = 28B ′ n πq 2 n log r ′ qn ε n ∼ n→+∞ 28B ′ n πq 2 n log(r ′ qn A n ).
By assumption (condition (1) in the statement of Prop. 6; this is the only place where it is used), the sequence log A n is sub-exponential with respect to q n . As a consequence, (B ′′ n ), and thus (C n ), is sub-exponential with respect to q n .

Proof of Prop. 10, continued. Remember that we are given r 1 and r 2 with 1/A < r 1 < r 2 < 1 and we want to prove that for n sufficiently large, any point of H n (r 1 ) has an infinite orbit remaining in H n (r 2 ) along a well chosen composition of F n and G n . It is enough to show that this is true for any sequence of points

Z n = X n + iY n ∈ H n (r 1 ).
We will use Douady-Ghys-Yoccoz's renormalization techniques and follow the presentation in [ABC] Section 3.2.

Step 1. Construction of a Riemann surface:

V n . Choose n sufficiently large so that F n is defined in the upper half-plane Z ∈ C ; Im(Z) ≥ τ n (r 2 ) -1/10} with F n (Z) -Z -1 ≤ 1 10 and F ′ n (Z) -1 ≤ 1 10
.9

Set

P n := X n + i τ n (r 2 ) - 1 10 . Let L n := X n + it ; t > Im(P n
) be the vertical half-line starting at P n and passing through Z n (see Figure 8). The union

L n ∪ P n , F n (P n ) ∪ F n (L n ) ∪ {∞} forms a
Jordan curve in the Riemann sphere bounding a region U n such that for Y > Im(P n ), the segment X n + iY, F n (X n + iY ) is contained in U n (see [ABC] Section 3.2). We set U n := U n ∪ L n . If we glue the sides L n and F n (L n ) of U n via F n , we obtain a topological surface V n . We denote by ι n : U n → V n the canonical projection. The space V n is a topological surface with boundary, whose boundary

ι n [P n , F n (P n )] is denoted ∂V n . We set V n = V n \ ∂V n .
Since the gluing map F n is analytic, the surface V n has a canonical analytic structure induced by the one of U n . It is possible to show that V n is quasiconformally homeomorphic, thus isomorphic to H/Z ≃ D * (see [ABC] Section 3.2 for details). Let φ n : V n → D * be an isomorphism. Hence, we have the following composition:

φ n • ι n : U n → D * .
We set

ζ n := φ n • ι n (Z n ) ∈ D.
Step 2. The renormalized map g n . Choose

r 3 ∈ ]r 1 , r 2 [. Set P ′ n := X n + i τ n (r 3 ) + 1 10 .
Let U ′ n be the set of points of U n which are above the segment

P ′ n , F n (P ′ n ) and let V ′ n be the image of U ′ n in V n .
Choose n sufficiently large so that lemma 5 can

Z G n (Z) W U n U ′ n H n (r 2 ) H n (r 3 ) H n (r 1 ) L n F n (L n ) P n Z n F n (P n ) P ′ n F n (P ′ n ) V n ζ n [Z] g n ([Z])
Figure 8. Construction of the Riemann surface V n and the renormalized map g n .

be applied with r = r 3 and r ′ = r 2 . Then, for all Z ∈ U ′ n ⊂ H n (r 3 ), there exists an integer j(Z) such that

W := F •j(Z) n • G n (Z) ∈ U n and ∀j ∈ 0, j(Z) F •j n • G n (Z) ∈ H n (r 2 ). The map Z → W induces a univalent map g n : φ n (V ′ n ) → D * .
10 By the removable singularity theorem, this map extends holomorphically to the origin by g n (0

) = 0. Since F n (Z) = Z + 1 + o(1) and G n (Z) = Z -A n -θ + o(1) as Im(Z) → +∞, we have that g ′ n (0) = e -2iπ(An+θ) = e -2iπθ
(see the Proposition on page 33 in [Yo] for details).

Step 3. The orbit of ζ n .

We will show that the orbit of ζ n under iteration of g n is infinite. For this, let ρ n be the radius of the largest disk centered at 0 and contained in φ n (V ′ n ). We will show that a) ∃c > 0 such that g n has a Siegel disk which contains

D(0, cρ n ) b) |ζ n | = o(ρ n ).
a) The restriction of g n to D(0, ρ n ) is univalent. It fixes 0 with derivative e -2iπθ . Remember that θ is a Brjuno number. It follows (see [Brj] or [Yo] for example) that there is a constant c θ > 0 depending only on θ such that g n has a Siegel disk containing D(0, c θ ρ n ). Indeed, according to Theorem on page 21 in [Yo], there is a constant c > 0 such that for all Brjuno number θ, any univalent map f : D(0, 1) → C which fixes 0 with derivative e 2πiθ has a Siegel disk containing D(0, ce -B(θ) ), where B(θ) is the Brjuno function.

b) Denote by B n the half-strip

B n = {Z ∈ C ; 0 < Re(Z) < 1 and Im(Z) > Im(P n )}
and consider the map H n : B n → U n defined by

H n (Z) = (1 -X) • (X n + iY ) + X • F n (X n + iY ) where Z = X+iY , (X, Y ) ∈ [0, 1]× Im(P n ), +∞ . The map H n sends each segment [iY, iY + 1] to the segment X n + iY, F n (X n + iY ) .
R n := Z ∈ C ; 0 ≤ Re(Z) < 1 and Im(P ′ n ) < Im(Z) < Im(Z n ) . Note that H n (R n ) ⊂ U ′ n and observe that A n := φ n • ι n • H n (R n ) is an annulus contained in φ n (V ′ n ) that surrounds 0 and ζ n . R n H n (R n ) -→ H n Im(Pn) Im(P ′ n ) Im(Zn) Pn P ′ n Zn H/Z V n Z∼Z+1 -→ Z∼Fn(Z) -→ ιn -→ φn ρn ζn 0 An U n B n
The image of R n in H/Z is an annulus of modulus

M n := Im(Z n ) -Im(P ′ n ) ≥ τ n (r 1 ) -τ n (r 3 ) - 1 10 -→ n→+∞ +∞.
11 For a proof that Hn is 5/4-quasiconformal homeomorphism, see for example [ABC] section 3.2 or [START_REF] Shishikura | Bifurcation of parabolic fixed points The Mandelbrot set, theme and variations[END_REF] section 2.5.

Note that H n induces a 5/4-quasiconformal homeomorphism between this annulus and A n . It follows that

modulus(A n ) ≥ 4 5 M n -→ n→+∞ +∞.
Since A n separates 0 and ζ n from ∞ and a point of modulus

ρ n in ∂φ n (V ′ n ), the claim follows: as n → +∞, |ζ n | = o(ρ n ).
Step 4. Controlling the orbit of Z n .

We know that the orbit of ζ n under iteration of g n is infinite. Thus, we have a sequence

ζ n ∈ V ′ n gn -→ ζ 1 n ∈ V ′ n gn -→ ζ 2 n ∈ V ′ n gn -→ • • • .
Now, for each ℓ ≥ 0, we have

ζ ℓ n = φ n • ι n (Z ℓ n ) for some Z ℓ n ∈ U ′ n .
Moreover, by definition of g n , there exists an integer j ℓ such that

Z ℓ+1 n = F •j ℓ n • G n (Z ℓ n ) and ∀j ∈ [0, j ℓ ] F •j n • G n (Z ℓ n ) ∈ H n (r 2 ).
In other words,

ζ ℓ n ∈ V ′ n gn -→ ζ ℓ+1 n ∈ V ′ n corresponds to Z ℓ n ∈ U ′ n Gn -→ • ∈ H n (r 2 ) Fn -→ • ∈ H n (r 2 ) Fn -→ • • • Fn -→ Z ℓ+1 n ∈ U ′ n .
Thus, for n sufficiently large, any point Z n ∈ H n (r 1 ) has an infinite orbit remaining in H n (r 2 ) along a well chosen composition of F n and G n . This completes the proof of Prop. 10. Proof of Prop. 7', continued. Remember that 0 < 1/A < ρ < ρ ′ < 1. Choose r 1 = ρ < r 2 < ρ ′ . By Prop. 10, for n sufficiently large, any point Z ∈ H n (ρ) has an infinite orbit remaining in H n (r 2 ) under a well chosen composition of F n and G n . This means that any point z ∈ X n (ρ) has an infinite orbit remaining in X n (r 2 ) under a well chosen composition of f •qn n and f

•qn-1 n . By Cor. 2, if n is sufficiently large, we know that any point in X n (r 2 ) ⊂ D(0, r 2 ) has its first q n iterates in D(0, ρ ′ ). This shows that any point z ∈ X n (ρ) has an infinite orbit remaining in D(0, ρ ′ ) under iteration of f n , as required.

In other words,

• ∈ H n (r 2 ) Gn -→ • ∈ H n (r 2 ) corresponds to • ∈ X n (r 2 ) f •q n-1 n -→ • ∈ X n (r 2 )
and

• ∈ H n (r 2 ) Fn -→ • ∈ H n (r 2 ) corresponds to • ∈ X n (r 2 ) f •qn n -→ • ∈ X n (r 2 ).
Moreover, for n sufficiently large,

• ∈ X n (r 2 ) f •q n-1 n -→ • ∈ X n (r 2 ) and • ∈ X n (r 2 ) f •qn n -→ • ∈ X n (r 2 )
decompose as

• ∈ X n (r 2 ) ⊂ D(0, r 2 ) fn -→ • ∈ D(0, ρ ′ ) fn -→ • • • fn -→ • ∈ D(0, ρ ′ ) fn -→ • ∈ X n (r 2 ).
This completes the proof of Prop. 7'.

1.5. The control of the post-critical set.

Definition 5. We denote by ∂ the Hausdorff semi-distance:

∂(X, Y ) = sup x∈X d(x, Y ).
Definition 6. We denote by PC(P α ) the post-critical set of P α :

PC(P α ) := k≥1 P •k α (ω α ) with ω α := - e 2iπα 2 .
This section is devoted to the proof of the following proposition. Remember that S N is the set of irrational numbers of bounded type whose continued fractions have entries greater than or equal to N .

Proposition 11. There exists N such that as α ′ ∈ S N → α ∈ S N , we have

∂ PC(P α ′ ), ∆ α → 0, with ∆ α being the Siegel disk of P α .
The corollary we will use later is the following.

Corollary 4. Let (α n ) be the sequence defined in Prop. 3. If n is large enough, the post-critical set of P αn is contained in the δ-neighborhood of the Siegel disk of P α .

The proof of Prop. 11 will rely on some (almost) classical results on Fatou coordinates and perturbed Fatou coordinates. We refer the reader to appendix A and to [START_REF] Shishikura | Bifurcation of parabolic fixed points The Mandelbrot set, theme and variations[END_REF] for more details. The proof will also rely on results of Inou and Shishikura [IS] that we will now recall. 1.5.1. The class of Inou and Shishikura. Consider the cubic polynomial

P (z) = z(1 + z) 2 .
This polynomial has a multiple fixed point at 0, a critical point at -1/3 which is mapped to the critical value at -4/27, and a second critical point at -1 which is mapped to 0. We set R := e 4π and v := -4/27. Let U be the open set defined by

U := P -1 D(0, |v|R) \ ]-∞, -1] ∪ B ,
where B is the connected component of P -1 D(0, |v|/R) which contains -1.

Consider the following class of maps (Inou and Shishikura use the notation F P 2 in [IS]):

IS 0 := f = P • ϕ -1 : U f → C with ϕ : U → U f isomorphism such that ϕ(0) = 0 and ϕ ′ (0) = 1 .
Remark. The set IS 0 is identified with the space of univalent maps in U fixing 0 with derivative 1, which is compact. A sequence of univalent maps (ϕ n :

U → C) satisfying ϕ n (0) = 0 and ϕ ′ n (0) = 1 converges uniformly to ϕ : U → C on every compact subset of U , if and only if the sequence (f n = P • ϕ -1 n ) converges to f = P • ϕ -1 on every compact subset of U f = ϕ(U ).
A map f ∈ IS 0 fixes 0 with multiplier 1. The map f : U f → D 0, |v|R is surjective. It is not a proper map. Inou and Shishikura call it a partial covering.

The map f has a critical point ω f := ϕ f (-1/3) which depends on f and a critical value v := -4/27 which does not depend on f .

f ; f ∈ IS 0 } is a compact subset of C * .
In particular, for all f ∈ IS 0 , c f = 0 and f has a multiple fixed point of multiplicity 2 at 0. If we make the change of variables

z = τ f (w) := - 1 c f w ,
we find F (w) = w + 1 + o(1) near infinity. To lighten notation, we will write f and F for pairs of functions related as above; ω f := φ f (-1/3) and ω F := τ -1 f (ω f ) will denote their critical points. Lemma 6. There exists R 0 such that for all f ∈ IS 0

• F is defined and univalent in a neighborhood of C \ D(0, R 0 ) and

• for all w ∈ C \ D(0, R 0 ), F (w) -w -1 < 1 4 and F ′ (w) -1 < 1 4 .
Proof. This follows from the compactness of IS 0 .

If R 1 > √ 2R 0 , the regions

Ω att := w ∈ C ; Re(w) > R 1 -|Im(w)| and Ω rep := w ∈ C ; Re(w) < -R 1 + |Im(w)| are contained in C \ D(0, R 0 ).
Then, for all f ∈ IS 0 , In addition, there are univalent maps Φ att F : Ω att → C (attracting Fatou coordinate for F ) and Φ rep

F (Ω att ) ⊂ Ω att and F Ω rep ⊃ Ω rep . τ f ←- Ω att,f Ω rep,f Ω att Ω rep 0 R 1 -R 1
F : Ω rep → C (repelling Fatou coordinate for F ) such that Φ att F • F (w) = Φ att F (w) + 1 and Φ rep F • F (w) = Φ rep F (w)
+ 1 when both sides of the equations are defined. The maps Φ att F and Φ rep F are unique up to an additive constant. In addition, as w ∈ Ω att ∩Ω rep tends to infinity, Φ att F -Φ rep F tends to a constant.

Result of Inou-Shishikura (Main theorem 1 part a). For all f ∈ IS 0 , the critical point ω f is attracted to 0.

The following lemma easily follows, using the compactness of the class IS 0 .

Lemma 7. There exists k such that for all f ∈ IS 0 we have

F •k (ω F ) ∈ Ω att .
Proof. By contradiction, suppose that there is a sequence (f n ) ∈ IS 0 such that for k ≤ n we have F •k n (ω Fn ) / ∈ Ω att . By compactness of IS 0 we may assume that the sequence F n converges to F ∞ . But since f ∞ ∈ IS 0 , the orbit of the critical point ω f∞ converges to 0, so for some k we have

F •k ∞ (ω F∞ ) ∈ Ω att . But F •k ∞ (ω F∞ ) = lim n→∞ F •k
n (ω Fn ) and this is a contradiction.

Since the maps Φ att

F and Φ rep F are only defined up to an additive constant, we can normalize Φ att F so that

Φ att F F •k (ω F ) = k. Then, we can normalize Φ rep F so that Φ att F (w) -Φ rep F (w) → 0 when Im(w) → +∞ with w ∈ Ω att ∩ Ω rep . Coming back to the z-coordinate, we define Ω att,f := τ f (Ω att ) and Ω rep,f := τ f (Ω rep ) and we set Φ att,f := Φ att F • τ -1 f and Φ rep,f := Φ rep F • τ -1 f .
The univalent maps Φ att,f : Ω att,f → C and Φ rep,f : Ω rep,f → C are called attracting and repelling Fatou coordinates for f . Note that our normalization of the attracting coordinates is given by Φ

att,f f •k (ω f ) = k.
The following result of Inou and Shishikura asserts that the attracting Fatou coordinate can be extended univalently up to the critical point of f . It easily follows from [IS] Prop. 5.6.

Result of Inou-Shishikura (see figure 11). For all f ∈ IS 0 , there exists an attracting petal P att,f and an extension of the Fatou coordinate, that we will still denote Φ att,f : On the left, we pulled this coloring back by Φ att,f .

P att,f → C, such that • v ∈ P att,f , • Φ att,f (v) = 1, • Φ att,f is
Definition 7 (see figure 12). For f ∈ IS 0 , we set:

V f := z ∈ P att,f ; Im Φ att,f (z) > 0 and 0 < Re Φ att,f (z) < 2
and

W f := z ∈ P att,f ; -2 < Im Φ att,f (z) < 2 and 0 < Re Φ att,f (z) < 2 .
We now come to the key result of Inou and Shishikura. The result stated below easily follows from [IS] Prop. 5.7 and 5.8 and section 5.M. Our domain [IS] to the interior of

V -k f ∪ W -k f below corresponds in
D -k ∪ D ♯ -k ∪ D ′′ -k ∪ D -k+1 ∪ D ♯ -k+1 ∪ D ′ -k+1 . Φ att,f -→ V f W f
D -k ∪ D ′′ -k ∪ D -k+1 ∪ D ′ -k+1 .
Result of Inou-Shishikura (see figure 13). For all f ∈ IS 0 and all k ≥ 0,

• the unique connected component

V -k f of f -k (V f ) which contains 0 in its closure is relatively compact in U f (the domain of f ) and f •k : V -k f → V f is an isomorphism and • the unique connected component W -k f of f -k (W f ) which intersects V -k f is relatively compact in U f and f •k : W -k f → W f is a covering of degree 2 ramified above v. In addition, if k is large enough, then V -k f ∪ W -k f ⊂ Ω rep,f .
The following lemma asserts that if k is large enough, then for all map f ∈ IS 0 , the set

V -k f ∪ W -k f
is contained in a repelling petal of f , i.e. the preimage of a left half-plane by Φ rep,f . Lemma 8 (see figure 14). There is an R 2 > 0 such that for all f ∈ IS 0 , the set

Φ rep,f (Ω rep,f ) contains the half-plane {w ∈ C ; Re w < -R 2 }.
There is an integer k 0 > 0 such that for all k ≥ k 0 , we have

V -k f ∪ W -k f ⊂ z ∈ Ω rep,f ; Re Φ rep,f (z) < -R 2 . Remark. Of course, R 2 can be replaced by any R 3 ≥ R 2 , replacing if necessary k 0 by k 1 := k 0 + ⌊R 3 -R 2 ⌋ + 1. Proof. For all f ∈ IS 0 , Φ f (Ω rep,f )
W F ∪ V F and W -7 F ∪ V -7 F . V f ∪ W f V -k f ∪ W -k f P rep,f P att,f f •k Figure 14. If k is large enough, V -k f ∪ W -k f
is contained in the repelling petal P rep,f . By Inou and Shishikura's result, we know that for all f ∈ IS 0 there is an integer k > 0 such that W -k f is relatively compact in Ω rep,f . It follows from the compactness of IS 0 that there is an integer k 1 > 0 and a constant M , such that for all f ∈ IS 0 , W -k1

f ⊂ Ω rep,f and sup w∈W -k 1 f Re Φ rep,f (w) < M. Set k 0 := k 1 + M + ⌊R 2 ⌋ + 3. Then, sup w∈W -k 0 f Re Φ rep,f (w) < -R 2 -2.
We will show that we then automatically have

(2) V -k0 f ⊂ Ω rep,f and sup w∈V -k 0 f Re Φ rep,f (w) < -R 2 .
It will follow immediately that

∀k ≥ k 0 and ∀w ∈ V -k f ∪ W -k f , Re Φ rep,f (w) < -R 2 ,
which will conclude the proof of the lemma.

In order to get (2), we fix f ∈ IS 0 and consider k ≥ k 0 large enough so that V -k f ⊂ Ω rep,f (this is possible thanks to Inou and Shishikura). Note that sup

w∈W -k f Re Φ rep,f (w) < -R 2 -2 -k + k 0 . Denote by g : V f → V -k f the inverse branch of f •k : V -k f → V f . Set B := w ∈ C ; 0 < Re(w) < 2 and 0 < Im(w) . Note that B = Φ att,f (V f ). Consider the map Ψ : B → C defined by Ψ := Φ rep,f • g • Φ -1
att,f . Since Ψ commutes with translation by 1, so that Ψ(w)w is 1-periodic, the maximum modulus principle yields

sup w∈B Re Ψ(w) -w = sup w∈[0,2] Re Ψ(w) -w . Note that g • Φ -1 att,f [0, 2] ⊂ W -k f and thus sup w∈[0,2] Re Ψ(w) -w < -R 2 -2 -k + k 0 .
Hence, sup

w∈V -k f Re Φ rep,f (w) = sup w∈B Re Ψ(w) < -R 2 -k + k 0 .

It now follows that sup

w∈V

-k 0 f Re Φ rep,f (w) < -R 2 .
This completes the proof of (2) and of lemma 8.

1.5.3. Perturbed Fatou coordinates. For α ∈ R, we denote by IS α the set of maps of the form z → f (e 2iπα z) with f ∈ IS 0 . If A is a subset of R, we denote by IS A the set

IS A := α∈A IS α .
Note that

IS α = f = P • ϕ -1 : U f → C with ϕ : U → U f isomorphism such that ϕ(0) = 0 and ϕ ′ (0) = e -2iπα
and

IS R = f = P • ϕ -1 : U f → C with ϕ : U → U f isomorphism such that ϕ(0) = 0 and ϕ ′ (0) = 1 .
The map f depends on φ in a one-to-one way. Thus we get a one-to-one correspondence between IS R and the set of univalent maps on U fixing 0 with derivative of modulus 1. We put the compact-open topology on this set of univalent maps. This induces a topology on IS R .

Remark. A sequence (f n = P • ϕ -1 n ∈ IS R ) converges to f = P • ϕ -1 ∈ IS R if and only if the sequence (f n ) converges to f on every compact subset of U f = ϕ(U ). If f ∈ IS [0,1[ , we denote by α f ∈ [0, 1[ the rotation number of f at 0, i.e. the real number α f ∈ [0, 1[ such that f ′ (0) = e 2iπα f .
Lemma 9. There exist ε 0 ∈ ]0, 1[ and r > 0 such that for all f ∈ IS [0,ε0[ , the map f has two fixed points in D(0, r) (counting multiplicities), one at z = 0 the other one denoted by σ f . The map σ :

IS [0,ε0[ → D(0, r) defined by f → σ f is continuous.
Proof. According to Inou and Shishikura, maps f ∈ IS 0 have a double fixed point at 0. By compactness of IS 0 , there is an r ′ > 0 such that maps f ∈ IS 0 have only 2 fixed points in D(0, r ′ ). Choose r ∈ ]0, r ′ [. By Rouché's theorem and by compactness of IS 0 , there is an ε 0 > 0 such that maps f ∈ IS [0,ε0[ have exactly two fixed points in D(0, r). The result follows easily.

The following results are consequences of results in [START_REF] Shishikura | Bifurcation of parabolic fixed points The Mandelbrot set, theme and variations[END_REF], the compactness of the class IS 0 and the results of the previous paragraph. Proposition 12 (see figure 15). There are constants K > 0,

ε 1 > 0 and R 3 ≥ R 2 with 1/ε 1 -R 3 > 1,
such that for all f ∈ IS ]0,ε1[ the following holds.

(1) There is a Jordan domain P f ⊂ U f (a perturbed petal) containing v, bounded by two arcs joining 0 to σ f and there is a branch of argument defined on P f such that

sup z∈P f arg(z) -inf z∈P f arg(z) < K.
(2) There is a univalent map Φ f :

P f → C (a perturbed Fatou coordinate) such that • Φ f (v) = 1, • Φ f (P f ) = w ∈ C ; 0 < Re(w) < 1/α f -R 3 , • Im Φ f (z) → +∞ as w → 0 and Im Φ f (z) → -∞ as w → σ f and • when z ∈ P f and Re Φ f (z) < 1/α f -R 3 -1, then f (z) ∈ P f and Φ f • f (z) = Φ f (z) + 1. For f ∈ IS 0 , we set P rep,f := z ∈ Ω rep,f ; Re Φ rep,f (z) < -R 3 .
(3) If (f n ) is a sequence of maps in IS ]0,ε1[ converging to a map f 0 ∈ IS 0 , then • any compact K ⊂ P att,f0 is contained in P fn for n large enough and the sequence (Φ fn ) converges to Φ att,f0 uniformly on K, and • any compact K ⊂ P rep,f0 is contained in P fn for n large enough and the sequence (Φ fn -1 α fn ) converges to Φ rep,f0 uniformly on K. Proof. Thanks to the compactness of the class IS 0 , it is enough to show that if (f n ) is a sequence of maps in IS ]0,1[ converging to a map f 0 ∈ IS 0 , there is a number R 3 ≥ R 2 such that properties (1), ( 2) and (3) hold.

So, assume f n is such a sequence, and for simplicity, write α n , σ n , . . . instead of α fn , σ fn , . . . Let τ n : C → P 1 \ {0, σ n } be the universal covering given by

τ n (w) := σ n 1 -e -2iπαnw so that τ n (w) -→ Im(w)→+∞ 0 and τ n (w) -→ Im(w)→-∞ σ n .
Denote by T n : C → C the translation

T n : w → w - 1 α n .
Recall that f 0 (z) = z + c 0 z 2 + O(z 3 ) with c 0 = 0, and

τ 0 (z) := - 1 c 0 z .
The following observations follow from [START_REF] Shishikura | Bifurcation of parabolic fixed points The Mandelbrot set, theme and variations[END_REF]. We let R 0 and R 1 be the constants introduced in paragraph 1.5.2.

(1) The sequence (τ n ) converges to τ 0 uniformly on every compact subset of C * . (2) If n is sufficiently large, there is a map F n : D n → C, defined and univalent in

D n := C \ k∈Z D(k/α n , R 0 ) which satisfies • f n • τ n = τ n • F n , • F n (w) -w is 1/α n -periodic (or equivalently, F n • T n = T n • F n ), • F n (w) -w → 1 as Im(w) → +∞.
Remark. This lift F n of f n may be defined by

F n (w) := w + 1 2iπα n log f n (z) -σ n f n (z) • z z -σ n with z = τ n (w).
(3) As n tends to +∞, the sequence (F n ) converges to F 0 uniformly on every compact subset of C \ D(0, R 0 ). (4) The set

Ω n := w ∈ C ; Re(w) > R 1 -Im(w) and Re(w) < 1 α n -R 1 + Im(w)
is contained in D n (see figure 16).

Ω n (5) Remember that for all w ∈ C \ D(0, R 0 ),

1 αn -R1 R1 -R1 R1-1 αn D n
F 0 (w) -w -1 < 1 4 and F ′ 0 (w) -1 < 1 4 .
It follows from the convergence of (F n ) to F 0 that if n is sufficiently large, then for all w ∈ Ω n ,

F n (w) -w -1 < 1 4 and F ′ n (w) -1 < 1 4 .
(6) Increasing n if necessary, we may assume that 1/α n > 2R 1 + 2. Then, there is a univalent map Φ n : Ω n → C, called a perturbed Fatou coordinate for

F n , such that Φ n • F n (w) = F n (w) + 1
when w ∈ Ω n and F n (w) ∈ Ω n . This map is unique up to post-composition with a translation. (7) Remember that there is a k such that f

•k 0 (ω 0 ) ∈ Ω att , with ω 0 the critical point of f 0 . For n large enough, f •k n (ω n ) is in τ n (Ω n ), with ω n the critical point of f n . There is a point w n ∈ Ω n such that τ n (w n ) = f •k n (ω n ) with w n -→ n→+∞ τ -1 0 f •k 0 (ω 0 ) . We can normalize Φ n by Φ n (w n ) = k. Then, Φ n -→ n→+∞ Φ att 0
uniformly on every compact subset of Ω att . Due to the normalization Φ att 0 (w) -Φ rep 0 (w) → 0 as Im(w) → +∞ with w ∈ Ω att ∩ Ω rep , we have

T n • Φ n • T -1 n -→ n→+∞ Φ rep 0
uniformly on every compact subset of Ω rep . Coming back to the z-coordinate is not immediate. Indeed, the map τ n is not injective on Ω n and we cannot define a Fatou coordinate for f n on τ n (Ω n ). We will instead restrict to a subset P n ⊂ Ω n whose image by Φ n is a vertical strip and on which τ n is injective. The precise statement is the following. The proof is given in appendix A. It is a consequence of results in [START_REF] Shishikura | Bifurcation of parabolic fixed points The Mandelbrot set, theme and variations[END_REF], but is not stated there.

Lemma 10 (see figure 17). If K > 0 and R ≥ R 2 are sufficiently large, then for n large enough:

• Φ n (Ω n ) contains the vertical strip

U n := w ∈ C ; R < Re(w) < 1/α n -R and • τ n is injective on P n := (Φ n ) -1 (U n ).
• there is a branch of argument defined on τ n (P n ) such that sup

z∈τn(P n ) arg(z) -inf z∈τn(P n ) arg(z) < K. τ n ' Φ n A τ n (P n ) P n U n 0 σ n s s R 1 αn -R Figure 17. The map τ n is injective on P n := (Φ n ) -1 (U n ).
Let M > R be an integer. Note that w ∈ C ; Re(w) > M ⊂ Φ att,0 (Ω att,0 ) and w ∈ C ; Re(w) < -M ⊂ Φ rep,0 (Ω rep,0 ). Set

P ′ 0 := z ∈ Ω att,0 ; Re Φ att,0 (z) > M ∪ z ∈ Ω rep,0 ; Re Φ rep,0 (z) < -M and P ′ n := τ n w ∈ P n ; M < Re Φ n (w) < 1/α n -M .
For any r > 0, if n is sufficiently large so that σ n ∈ D(0, r), then points with large (positive or negative) imaginary part are mapped by τ n into D(0, r). It therefore follows from point (7) above that P ′ n → P ′ 0 as n → +∞. Set In a simply connected neighborhood of P ′ 0 , the function f •M 0 (z)/z does not vanish (and extends by 1 at z = 0). It follows that for n large enough, there are branches of argument of f •M n (z)/z which are uniformly bounded on P n . It is now easy to check that Proposition 12 holds for the maps f n with n large enough. 1.5.4. Renormalization. Recall that for maps f ∈ IS 0 we defined sets V f ⊂ P att,f and W f ⊂ P att,f . We claimed (see lemma 8) that for k ≥ 0 there are components V -k f and W -k f properly mapped by f •k respectively to V f with degree 1 and W f with degree 2. In addition, there is an integer k 0 > 0 such that

∀f ∈ IS 0 , V -k0 f ∪ W -k0 f ⊂ P rep,f .
We will now generalize this to maps f ∈ IS ]0,ε[ with ε sufficiently small. If f ∈ IS ]0,ε1[ , we set

V f := z ∈ P f ; Im Φ f (z) > 0 and 0 < Re Φ f (z) < 2 and W f := z ∈ P f ; -2 < Im Φ f (z) < 2 and 0 < Re Φ f (z) < 2 .
Proposition 13 (see figure 19). There is a number ε 2 > 0 and an integer k 1 ≥ 1 such that for all f ∈ IS ]0,ε2[ and for all integer k ∈ [1, k 1 ],

(1) the unique connected component

V -k f of f -k (V f ) which contains 0 in its closure is relatively compact in U f (the domain of f ) and f •k : V -k f → V f is an isomorphism, (2) the unique connected component W -k f of f -k (W f ) which intersects V -k f is relatively compact in U f and f •k : W -k f → W f is a covering of degree 2 ramified above v. (3) V -k1 f ∪ W -k1 f ⊂ z ∈ P f ; 2 < Re Φ f (z) < 1 α f -R 3 -5 . V f ∪ W f V -k f ∪W -k f P f f •k Figure 19. If k is large enough, V -k f ∪ W -k f is contained in the perturbed petal P f .
Proof. Set k 1 := k 0 + 7. By compactness of IS 0 , there is an ε 2 > 0 such that for all f ∈ IS ]0,ε2[ , properties (1) and ( 2) hold for all integers k ∈ [1, k 1 ], and further,

W -k1 f is contained in z ∈ P f ; 4 < Re Φ f (z) < 1 α f -R 3 -7 . To see that V -k1 f is a subset of z ∈ P f ; 2 < Re Φ f (z) < 1 α f -R 3
-5 , we proceed as in the proof of lemma 8.

We now come to the definition of the renormalization of maps f ∈ IS ]0,ε2[ . Result of Inou-Shishikura (Main theorem 3 and section 5.M).

If f ∈ IS ]0,ε2[ , the map Φ f • f •k1 • Φ -1 f : Φ f V -k1 f ∪ W -k1 f → Φ f V f ∪ W f projects via w → -4 27 e 2iπw to a map R(f ) ∈ IS -1/α f . Definition 8. The map R(f ) is called the renormalization of f .
The polynomial P α does not belong to the class IS α . However, according to [IS], the construction we described also works for polynomials P α with α > 0 sufficiently close to 0. In other words, if α > 0 is sufficiently close to 0, there are perturbed petals and perturbed Fatou coordinates, and there is a renormalization R(P α ) which belongs to IS -1/α . In the sequel, ε 2 > 0 is chosen sufficiently small so that for α ∈ ]0, ε 2 [, a map f which either is a polynomial P α , or belongs to IS α , has a renormalization R(f ) ∈ IS -1/α . 1.5.5. Renormalization tower. Assume 1/N < ε 2 . Denote by Irrat ≥N the set:

Irrat ≥N := α = [a 0 , a 1 , a 2 , . . .] ∈ R \ Q ; a k ≥ N for all k ≥ 1 . Assume α = [a 0 , a 1 , a 2 , . . .] ∈ Irrat ≥N . For j ≥ 0, set α j := [0, a j+1 , a j+2 , . . .].
Note that for all j ≥ 1,

α j+1 = 1 α j - 1 α j .
The requirement α ∈ Irrat ≥N translates into ∀j,

α j ∈]0, 1/N [.
Denote by p j /q j the approximants to α 0 given by the continued fraction algorithm. Now, if either f 0 = P α or f 0 ∈ IS α , we can define inductively an infinite sequence of renormalizations, also called a renormalization tower, by

f j+1 := s • R(f j ) • s -1 , the conjugacy by s : z → z being introduced so that f ′ j (0) = e 2iπαj .

It will be convenient to define

Exp : C → C * w → -4 27 s(e 2iπw ). For j ≥ 0, we define φ j := Exp • Φ fj : P fj → C. The map φ j goes from the j-th level of the renormalization tower to the next level.

We now want to relate the dynamics of maps at different levels of the renormalization tower. For this purpose, we will use the following lemma.

Lemma 11. There is a constant K > 0 such that for all f ∈ IS ]0,ε2[ , there is an inverse branch of Exp which is defined on P f and takes its values in the strip w ∈ C ; 0 < Re(w) < K .

Proof. This is an immediate consequence of Prop. 12 part (1).

From now on, we assume that N is sufficiently large so that

(3) 1 N < ε 2 and 1 N -R 3 > K.
Then, according to lemma 11, for all j ≥ 1, there is an inverse branch ψ j of φ j-1 defined on the perturbed petal P fj with values in P fj-1 (there are several possible choices, we choose any one). Remember that

P fj+1 P fj E Φ fj ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ Exp E ψ j+1
Φ fj (P fj ) = w ∈ C ; 0 < Re(w) < 1/α j -R 3 .
Define P j ⊂ P fj and P ′ j ⊂ P fj by P j := z ∈ P fj ; 0 < Re Φ fj (w) < 1/α j -R 3 -1 and P ′ j := z ∈ P fj ; 1 < Re Φ fj (w) < 1/α j -R 3 . Note that f j maps P j to P ′ j isomorphically. Set Q j := Ψ j (P j ) and Q ′ j := Ψ j (P ′ j ).

Proposition 14. The map Ψ j conjugates f j :

P j → P ′ j to f •qj 0 : Q j → Q ′ j .
In other words, we have the following commutative diagram:

Q j ⊂ Ψ j (P fj ) f •q j 0 / / Q ′ j ⊂ Ψ j (P fj ) P j ⊂ P fj fj / / Ψj O O P ′ j ⊂ P fj . Ψj O O
Proof. We must show that if z j ∈ P j and z ′ j := f j (z j ) ∈ P ′ j , then the points z 0 := Ψ j (z j ) and z ′ 0 := Ψ j (z ′ j ) are related by

z ′ 0 = f •qj 0 (z 0 ). Let us first show that there is an integer k such that z ′ 0 = f •k 0 (z 0 ).
Our proof is based on the following lemma.

Lemma 12. Assume ℓ ≥ 0, w ∈ U f ℓ+1 and w ′ := f ℓ+1 (w). Let z ∈ P f ℓ and z ′ ∈ P f ℓ be such that

Exp • Φ f ℓ (z) = w and Exp • Φ f ℓ (z ′ ) = w ′ .
Then, there is an integer

k ≥ 1 such that z ′ = f •k ℓ (z). Proof. Let z ′ 1 ∈ P f ℓ be the unique point such that Re Φ f ℓ (z ′ 1 ) ∈ ]0, 1] and Exp • Φ f ℓ (z ′ 1 ) = w ′ . By definition of the renormalization f ℓ+1 , there is a point z 1 ∈ V -k1 f ℓ ∪ W -k1 f ℓ such that Exp • Φ f ℓ (z 1 ) = w and f •k1 ℓ (z 1 ) = z ′ 1 . We then have Φ f ℓ (z 1 ) = Φ f ℓ (z) + m 1 and Φ f ℓ (z ′ ) = Φ f ℓ (z ′ 1 ) + m ′ 1 with m 1 ∈ Z and m ′ 1 ∈ N. If m 1 ≥ 0, we have z 1 = f •m1 ℓ (z) and z ′ = f •m ′ 1 ℓ (z ′ 1 ). Since k 1 ≥ 0, we then have z ′ = f •k (z) with k := k 1 + m 1 + m ′ 1 ≥ 1. If m 1 < 0, then z = f •-m1 ℓ (z ′ 1 )
. However, for m ≤ -m 1 , we have f •m ℓ (z ′ 1 ) ∈ P f ℓ , and so, k 1 ≥ -m 1 + 1. Thus, we can write

z ′ 1 = f •m2 ℓ (z) with m 2 := k 1 + m 1 ≥ 1.
In that case,

z ′ = f •k (z) with k := m 2 + m ′ 1 ≥ 1.
It follows by decreasing induction on ℓ from j to 0 that for all z j ∈ P j , there is an integer k ≥ 1 such that z ′ 0 = f •k 0 (z 0 ). We will now show that we have a common integer k, valid for all points z j ∈ P j .

Lemma 13. There is an integer k 0 ≥ 1 such that for all point z j ∈ P j , we have

z ′ 0 = f •k0 0 (z 0 ).
Proof. We will use the connectivity of P j . For k ≥ 1, set

O k := {z ∈ P j ; f •k 0 Ψ j (z) is defined
This is an open set. Set

X k := z ∈ O k ; f •k 0 Ψ j (z) = Ψ j f j (z) }.
Note that for every component

O of O k , either X k ∩ O = O, or X k is discrete in O, in particular countable. Indeed, X k is the set of zeroes of the holomorphic function f •k 0 • Ψ j -Ψ j • f j : O k → C. Since P j = k≥1 X k there is a smallest integer k 0 ≥ 1 such that X k0 is not countable. Then, there is a component O of O k0 such that on O, we have f •k0 0 • Ψ j = Ψ j • f j . Since O is a component of O k0 , we have ∂O ∩ P j ⊂ C \ O k0 . It follows that ∂O ∩ P j ⊂ X 1 ∪ . . . X k0-1
since the remaining X k 's are contained in O k0 . So, ∂O ∩ P j is countable. This is only possible if ∂O ∩ P j = ∅ since in any neighborhood of a point z ∈ C \ O k0 , there are uncountably many points in C \ O k0 . As a consequence, O = P j , which concludes the proof of the lemma.

We must now show that k 0 = q j . Let L j ⊂ P j be the curve defined by

L j := z ∈ P j ; Re Φ fj (z) = 1 .
Set L ′ j := f j (L j ), i.e. the curve L ′ j := z ∈ P j ; Re Φ fj (z) = 2 . Those curves both have an end point at z = 0. They both have tangents at z = 0. Since the linear part of f j at z = 0 is the rotation of angle α j , the angle between L j and L ′ j at z = 0 is α j . It follows that the curves Ψ j (L j ) and Ψ j (L ′ j ) have tangents at z = 0 and the angle between those curves is α

0 α 1 • • • α j . So, the linear part of f •k0 0 at z = 0 is the rotation of angle α 0 α 1 • • • α j . It follows that k 0 = q j . Set D j := V -k1 fj ∪ W -k1 fj , D ′ j := V fj ∪ W fj , C j := Ψ j (D j ) and C ′ j := Ψ j (D ′ j ). Note that f •k1 j maps D j to D ′ j .
Proposition 15. The map Ψ j conjugates the map f •k1

j : D j → D ′ j to the map f •(k1qj +qj-1) 0 : C j → C ′ j .
In other words, we have the following commutative diagram:

C j ⊂ Ψ j (P fj ) f •(k 1 q j +q j-1 ) 0 / / C ′ j ⊂ Ψ j (P fj ) D j ⊂ P fj f •k 1 j / / Ψj O O D ′ j ⊂ P fj . Ψj O O
Proof. The proof is similar to the one of Prop. 14.

1.5.6. Neighborhoods of the postcritical set. We can now see that the post-critical set of maps f ∈ IS α with α ∈ Irrat ≥N is infinite.

Proposition 16 . For all α ∈ Irrat ≥N and all f ∈ IS α , the postcritical set of f is infinite.

Proof. For j ≥ 1, the map f •k1 j : W -k1 fj → W fj is a ramified covering of degree 2, ramified above v. Denote by w j the critical point of this ramified covering. Set w 0 := Ψ j (w j ). According to Prop. 15, we can iterate f 0 at least k 1 q j + q j-1 times at w 0 , w 0 is a critical point of f

•(k1qj +qj-1) 0 and its critical value is Ψ j (v). In particular, Ψ j (v) is a point of the postcritical set of f 0 .

Note that v ∈ P j . According to Prop. 14, we can iterate f 0 at least q j times at Ψ j (v). This shows that we can iterate f 0 at least q j times at v. Since j ≥ 1 is arbitrary, the postcritical set of f 0 is infinite.

For every α ∈ Irrat ≥N , we are going to define a sequence (U j ) of open sets containing the post-critical set of P α . We still use the notations of the previous paragraph. In particular, for j ≥ 1, the j-th renormalization of f 0 := P α has a perturbed petal P fj , a perturbed Fatou coordinate

Φ fj : P fj → w ∈ C ; 0 < Re(w) < 1/α j -R 3 .
The set

D j := V -k1 fj ∪ W -k1 fj ⊂ P fj is mapped by f •k1 j to D ′ j := z ∈ P fj ; 0 < Re Φ fj (z) < 2 and Im Φ fj (z) > -2 .
There is a map Ψ j , univalent on P fj , with values in the dynamical plane of P α ,

conjugating f •k1 j : D j → D ′ j to P •(k1qj +qj-1) α : C j → C ′ j with C j := Ψ j (D j ) and C ′ j := Ψ j (D ′ j ). Definition 9. For α ∈ Irrat ≥N and j ≥ 1 we set U j (α) := qj+1+ℓqj k=0 P •k α (C j )
where ℓ := k 1 -⌊R 3 ⌋ -4 ∈ N.

Figure 21 shows the open set U 1 (α) for an α of bounded type.

Proposition 17. For all α ∈ Irrat ≥N and all j ≥ 1, the post-critical set PC(P α ) is contained in U j (α). Proof. We will show that for all j ≥ 1, there is a point z 0 ∈ C j which is a precritical point of P α , and a sequence of positive integers with t 1 < t 2 < t 2 < . . . such that • for all m ≥ 1, t m+1t m < q j+1 + k 1 -⌊R 3 ⌋ -4)q j and

• P •tm α (z 0 ) ∈ C j .
The proof follows immediately. Denote by ω j+1 the critical point of f j+1 . According to Prop. 16 the orbit of ω j+1 under iteration of f j+1 is infinite. In particular, for all m ≥ 0, f •m j+1 (ω j+1 ) is in the domain U fj+1 of f j+1 . Remember that the map φ j := Exp • Φ fj : D j → U fj+1 is surjective. So, for all m ≥ 0, we can find a point w m ∈ D j such that

φ j (w m ) = f •m j+1 (ω j+1 ). Set z m := Ψ j (w m ) ∈ C j .
Then, z 0 is a precritical point of P α and according to lemma 12, there is an increasing sequence (t m ) such that z m = P •tm α (z 0 ). It is therefore enough to show that for all m ≥ 1, t m+1t m < q j+1 + k 1 -⌊R 3 ⌋ -4)q j . Note that for m ≥ 0,

w m ∈ D j , w ′ m := f •k1 j (w m ) ∈ D ′ j .
By definition of the renormalization f j+1 , we have

φ j (w ′ m ) = f j+1 φ j (w m ) = f •(m+1) j+1 (ω j+1 ) = φ j (w m+1 ).
In addition, since w ′ m ∈ D ′ j and w m+1 ∈ D j ,

0 < Re Φ fj (w ′ m ) < 2 and 2 < Re Φ fj (w m+1 ) < 1 α j -R 3 -5. Thus, Φ fj (w m+1 ) -Φ fj (w ′ m ) is a positive integer ℓ m , w m+1 = f •ℓm j (w ′ m ), and since a j+1 = ⌊1/α j ⌋, ℓ m ≤ a j+1 -⌊R 3 ⌋ -4. Set z ′ m := Ψ j (w ′ m )
. According to Prop. 14 and 15, we have

z ′ m = P •(k1qj +qj-1) α (z m ) and z m+1 = P •ℓmqj α (z ′ m ). Thus, t m+1 -t m = k 1 q j + q j-1 + ℓ m q j ≤ (a j+1 + k 1 -⌊R 3 ⌋ -4)q j + q j-1 .
The result now follows immediately from q j+1 = a j+1 q j + q j-1 .

We will now assume that α ∈ S N , i.e. α ∈ Irrat ≥N is a bounded type irrational number (the coefficients of the continued fraction are bounded). We will use the additional hypothesis that α has bounded type in order to obtain the following result.

Proposition 18. For all α ∈ S N , for all ε > 0, if j is large enough, the set U j (α) is contained in the ε-neighborhood of the Siegel disk ∆ α .

Proof. Consider the renormalization tower associated to f 0 := P α and let us keep the notations we have introduced so far. Set

D ′′ j := f •(aj+1+ℓ) j (D j ). Define N j := a j+1 -⌊R 3 ⌋ -1 < 1 α j -R 3 .
Note that

D ′′ j = z ∈ C ; N j -3 < Re Φ fj (z) < N j -1 and Im(w) > -2 . In particular, D ′′ j ⊂ P fj . Set C ′′ j := Ψ j (D ′′ j )
. According to Prop. 14 and 15,

C ′′ j = P •(qj+1+ℓqj ) α (C j ).
Lemma 14. There exists M such that for all j ≥ 1, the disk D 0, |v|e -2πM is contained in the Siegel disk of f j .

Proof. Let B(α j ) be the Brjuno sum defined by Yoccoz as

B(α j ) := +∞ k=0 α j • • • α j+k-1 log 1 α j+k .
Since α is of bounded type, there is a constant B such that for all j ≥ 1, B(α j ) ≤ B.

The map f j has a univalent inverse branch g j : D 0, |v| → C fixing 0 with derivative e -2iπαj . According to a theorem of Yoccoz [Yo], there is a constant C, which does not depend on j, such that the Siegel disk of g j contains the disk centered at 0 with radius |v|e -2π(B(αj)+C) ≥ |v|e -2π(B+C) .

The lemma is proved with M := B + C.

Let us now show that for any ε > 0, for j large enough, C ′′ j is contained in the ε-neighborhood of ∆ α . Denote by D ′′ j ♯ the set of points in D ′′ j which are mapped by

φ j = Exp • Φ fj in D 0, |v|e -2πM and set D ′′ j ♭ := D ′′ j \ D ′′ j ♯ .
In addition, set

C ′′ j ♯ := Ψ j D ′′ j ♯ and C ′′ j ♭ := Ψ j D ′′ j ♭ .
Points in D 0, |v|e -2πM have an infinite orbit under iteration of f j+1 . It follows that points in D ′′ j ♯ have an infinite orbit under iteration of f j . Thus, the orbit of

points in C ′′ j ♯ remains in U j (α), thus is bounded. As a consequence, C ′′ j ♯ (which is
open) is contained in the Fatou set of P α , and since it contains 0 in its boundary,

C ′′ j ♯ is contained in the Siegel disk of P α .
So, in order to show that

C ′′ j is contained in the ε-neighborhood of ∆ α , it is enough to show that C ′′ j ♭ is contained in the ε-neighborhood of ∆ α . Note that D ′′ j ♭ is the image of the rectangle w ∈ C ; N j -3 < Re(w) < N j -1 and -2 < Im(w) ≤ M by the map Φ -1 fj which is univalent on the strip w ∈ C ; 0 < Re(w) < 1/α j -R 3 . Since 1 < N j -3 < N j < 1/α j -R 3 ,
the modulus of the annulus P fj \ D ′′ j ♭ is bounded from below independently of j. It follows from Koebe's distortion lemma that there is a constant

K such that diam(C ′′ j ♭ ) ≤ K • d(z j , z ′ j ) where z j := Ψ j • Φ -1 fj (N j -3) and z ′ j := Ψ j • Φ -1 fj (N j -2
). According to Prop. 14,

z j = P •(Nj-3)qj α (ω α ) and z ′ j = P •qj α (z j
). The boundary of ∆ α is a Jordan curve, and

P α : ∂∆ α → ∂∆ α is conjugate to the rotation of angle α on R/Z. It follows that diam(C ′′ j ♭ ) ≤ K • max z∈∂∆α P •qj α (z) -z .
Without loss of generality, we may assume that M ≥ 2. If z ∈ U j (α), then there is a k ≤ q j+1 + ℓq j such that P

•k α (z) ∈ C ′′ j . Then, • either P •k α (z) ∈ C ′′ j ♯ in which case z ∈ ∆ α , • or P •k α (z) ∈ C ′′ j ♭ in which case z belongs to the connected component O -k j of P -k α (C ′′ j ♭ ) intersecting ∆ α .
In the second case, O -k j contains two points z -k j and z ′ j -k which are in the boundary of ∆ α and which are respectively mapped to z j and z ′ j by P k α . We have z ′

j -k = P •qj α (z -k j ). Note that since α is of bounded type, there is a constant A such that ∀j ≥ 1 q j+1 + ℓq j ≤ A • q j .
According to lemma 15 below, there is a constant K ′ such that for all j ≥ 1 and all k ≤ q j+1 + ℓq j

diam(O -k j ) ≤ K ′ • z ′ j -k -z -k j ≤ K ′ • max z∈∂∆α P •qj α (z) -z .
So, we see that sup

z∈Uj (α) d(z, ∆ α ) ≤ max(K, K ′ ) • max z∈∂∆α P •qj α (z) -z -→ j→+∞ 0.
This completes the proof of Prop. 18.

Assume α ∈ R \ Q is of bounded type. If z ∈ ∂∆ α , we set r j (z) = P •qj α (z) -z .
Lemma 15. For all α ∈ R \ Q of bounded type, all A ≥ 1 and all K ≥ 1, there exists a K ′ such that the following holds.

If j ≥ 1, if k ≤ A • q j , if z 0 ∈ ∂∆ α , if z k = P •k α (z 0 ) and if O is the connected component of P -k α D(z k , K • r j (z k )) containing z 0 , then diam(O) ≤ K ′ • r j (z 0 ).
Proof. The constants M and m which will be introduced in the proof depend on α, A and K, but they do not depend on j, k or z.

Set D := D z k , K • r j (z k ) and D := D z k , 2K • r j (z k ) .
Since ∂∆ α is a quasicircle and since P α : ∂∆ α → ∂∆ α is conjugate to the rotation of angle α on R/Z, the number of critical values of P •k α in D is bounded by a constant M which only depends on α, A and K.

Let O (respectively O) be the connected component of P -k α (D) (respectively P -k α ( D)) containing z 0 . The degree of P •k α : O → D is bounded by 2 M . On the one hand, it easily follows from the Grötzsch inequality that the modulus of the annulus O \ O is bounded from below by log 2/(2π2 M ) (see for example [ShT] lemma 2.1).

On the other hand, it follows from Schwarz's lemma that the hyperbolic distance in O between z 0 and P •qj α (z 0 ) is greater than the hyperbolic distance in D between z k and P

•qj α (z k ), i.e. a constant m which only depends on α, A and K. Lemma 15 now follows easily from the Koebe distortion lemma.

Note that for each fixed j, the set U j (α) depends continuously on α as long as the first j + 1 approximants remain unchanged. Hence, given α ∈ S N and δ > 0, if α ′ ∈ Irrat ≥N is sufficiently close to α (in particular, the first j entries of the continued fractions of α and α ′ coincide), then U j (α ′ ) is contained in the δ-neighborhood of U j (α). This completes the proof of Prop. 11. 1.6. Lebesgue density near the boundary of a Siegel disk. Definition 10. If α is a Brjuno number and if δ > 0, we denote by ∆ the Siegel disk of P α and by K(δ) the set of points whose orbit under iteration of P α remains at distance less than δ from ∆.

Our proof will be based on the following theorem of Curtis T. McMullen [McM].

Theorem 4 (McMullen). Assume α is a bounded type irrational and δ > 0. Then, every point z ∈ ∂∆ is a Lebesgue density point of K(δ). Proof. We proceed by contradiction. Assume we can find a sequence (z j ) such that • d j := d(z j , ∂∆) → 0 and • ρ j := dens D(zj ,dj) C \ K(δ) → 0. Extracting a subsequence if necessary, we may assume that the sequence (z j ) converges to a point z 0 ∈ ∂∆ and that lim ρ j = ρ > 0.

Set η := ρ/5 and for i ≥ 1, set

X i := w ∈ ∂∆ (∀r ≤ 1/i) dens D(w,r) C \ K(δ) ≤ η .
The sets X i are closed. By McMullen's Theo. 4, X i = ∂∆. By Baire category, one of these sets X i contains an open subset W of ∂∆. Then, for all sequence of points w j ∈ W and all sequence of real number r j converging to 0, we have

(4) lim sup j→+∞ dens D(wj ,rj ) C \ K(δ) ≤ η = ρ 5 .
We claim that there is a map g defined and univalent in a neighborhood U of z 0 , such that

• g(z 0 ) = w 0 ∈ W , • g K(δ) ∩ U = K(δ) ∩ g(U ) and • g(∂∆ ∩ U ) = ∂∆ ∩ g(U ).
Indeed, if z 0 is not precritical, we can find an integer k ≥ 0 such that f •k (z 0 ) ∈ W and we let g be the restriction of f •k to a sufficiently small neighborhood of z 0 . If z 0 is precritical, we can find a point w 0 ∈ W and an integer k ≥ 0 such that f •k (w 0 ) = z 0 and we let g coincide the restriction of the branch of f -k sending z 0 to w 0 , to a sufficiently small neighborhood of z 0 . Let z ′ j ∈ ∂∆ be such that |z jz ′ j | = d j . Then, z ′ j -→ j→+∞ z 0 . Let j be sufficiently large so that z ′ j ∈ U and set w j := g(z ′ j ). On the one hand, w j -→ j→+∞ w 0 . Thus, w j ∈ W for j large enough. On the other hand,

dens D(z ′ j ,2dj ) C \ K(δ) ≥ 1 4 dens D(zj,dj ) C \ K(δ)
and so lim inf

j→+∞ dens D(z ′ j ,2dj) C \ K(δ) ≥ ρ 4 .
Since g is holomorphic at z 0 , lim inf

j→+∞ dens D(wj ,rj) C \ K(δ) ≥ ρ 4 with r j := g ′ (w 0 ) • 2d j -→ j→+∞ 0.
This contradicts (4).

1.7. The proof. We will now prove Prop. 3. We let N be sufficiently large so that the conclusions of Prop. Note that since α is of bounded type, the Julia set J α has zero Lebesgue measure (see [P]). Prop. 6 then easily implies that lim inf area(K αn ) ≥ 1 2 area(K α ).

Everything relies on our ability to promote the coefficient 1/2 to the coefficient 1.

Let us first give an overall idea of the strategy of the proof. Denote by K (resp. K n ) the filled-in Julia set of P α (resp. P αn ) and by ∆ (resp. ∆ n ) its Siegel disk.

The idea of the proof is the following. For all S ≥ 1, one can find a nested sequence of toll belts (W s ) 1≤s≤S

W s := z ∈ C 2δ s < d(z, ∆) < 8δ s with 8δ s+1 < δ s ,
surrounding the Siegel disk ∆ such that for n large enough the following holds.

• The orbit under iteration of P αn of any point in ∆ \ K n must pass through all the toll belts. • Thanks to Corollary 4, the toll belts surround the Siegel disk ∆ n .

• Thanks to Corollary 5 and Proposition 6, under the iterates of P αn , at least 1/2ε of points in the toll belt W s+1 will be captured by the Siegel disk ∆ n without being able to enter the toll belt W s .

• Since the toll belts surround the Siegel disk ∆ n , they are free of the postcritical set of P αn . This gives us Koebe control of points passing through the belt, implying that at most 1/2 + ε of points in ∆ that manage to reach W s+1 under iteration of P αn will manage to reach W s . As a consequence, at most (1/2 + ε) S points in ∆ can have an orbit under iteration of P αn that passes through all the belts and we are done by choosing S large enough.

There are minor boundary effects which complicate slightly the argument and we proceed as follows. For δ > 0, set

V (δ) := z ∈ C d(z, ∆) < δ , K(δ) := z ∈ V (δ) (∀k ≥ 0) P •k α (z) ∈ V (δ) and K n (δ) := z ∈ V (δ) (∀k ≥ 0) P •k αn (z) ∈ V (δ) . Define ρ n : +∞[ → [0, 1] by ρ n (δ) := dens ∆ C \ K n (δ) .
Lemma 16. For all δ > 0, there exist δ ′ > 0 (with δ ′ < δ) and a sequence (c n > 0) converging to 0, such that

ρ n (δ) ≤ 3 4 ρ n (δ ′ ) + c n . 12
This lemma enables us to complete the proof of Prop. 3 as follows. We set ρ(δ) := lim sup n→+∞ ρ n (δ) (≤ 1).

Then, for all δ > 0, there is a δ ′ > 0 such that ρ(δ) ≤ 3 4 ρ(δ ′ ). Since ρ is bounded from above by 1, this implies that ρ identically vanishes. In other words

(5) (∀δ > 0) dens ∆ K n (δ) -→ n→+∞ 1. Since K n (δ) ⊂ K n , we deduce that dens ∆ (K n ) -→ n→+∞ 1.
We know that

• P αn converges locally uniformly to P α ,

• the orbit under iteration of P α of any point in K \ ∂K eventually lands in ∆ and

• P -1 αn (K n ) = K n . It follows that dens K\∂K (K n ) -→ n→+∞ 1.
Since the Julia set ∂K has Lebesgue measure zero, this implies that lim inf area(K n ) ≥ area(K). This completes the proof of Prop. 3 modulo Lemma 16.

Proof of Lemma 16. Let us sum up what we obtained in sections 1.4, 1.5 and 1.6.

(A) For all open set U ⊂ ∆ and all δ > 0, lim inf

n→+∞ dens U K n (δ) ≥ 1 2
. This is an immediate consequence of Prop. 6 in section 1.4. (B) For all δ > 0, if n is sufficiently large, the post-critical set of P αn is contained in V (δ). This is just a restatement of Cor. 4 in section 1.5. (C) For all η > 0 and all δ > 0, there exists δ ′ 0 > 0 such that if δ ′ < δ ′ 0 and if z ∈ V (8δ ′ )\V (2δ ′ ), then dens D(z,δ ′ ) C\K(δ) < η. This is an easy consequence of Cor. 5 in section 1.6.

Step 1. By Koebe distortion theorem, there exists a constant κ such that for all map φ : D := D(a, r) → C which extends univalently to D(a, 3r/2), we have sup

D |φ ′ | ≤ κ inf D |φ ′ |.
We choose η > 0 such that 8πκ 2 η < 1 4 .

Step 2. Fix δ > 0. We claim that there exists δ ′ > 0 such that:

(i) 9δ ′ < δ and (2 + 3κ) • δ ′ < δ, 13 (ii) if d(z, ∆) < 2δ ′ , then d P α (z), ∆ < 8δ ′ and (iii) if z ∈ V (8δ ′ ) \ V (2δ ′ ), then dens D(z,δ ′ ) C \ K(δ) < η.
Indeed, it is well-known and easy to check that for α ∈ R, P ′ α < 4 on K α . As a consequence, if δ ′ > 0 is sufficiently small, then P ′ α < 4 on V (2δ ′ ). It follows that (ii) holds for δ ′ > 0 sufficiently small. Claim (iii) follows from the aforementioned point (C).

From now on, we assume that δ ′ is chosen so that the above claims hold and we set

W := V (8δ ′ ) \ V (2δ ′ ).
Step 3. Set Y ℓ := z ∈ K(δ) P •ℓ α (z) ∈ ∆ . The set of points in K(δ) whose orbits do not intersect ∆, is contained in the Julia set of P α . This set has zero Lebesgue measure. Thus, K(δ) and Y ℓ coincide up to a set of zero Lebesgue measure. The sequence (Y ℓ ) ℓ≥0 is increasing. From now on, we assume that ℓ is sufficiently large so that

∀w ∈ W dens D(w,δ ′ ) (C \ Y ℓ ) < η.
Step 4. Assume φ is univalent on D(w, 3δ ′ /2) with w ∈ W , r is the radius of the largest disk centered at φ(w) and contained in φ D(w, δ ′ ) and Q is a square contained in φ D(w, δ ′ ) with side length at least r/ √ 8. Set D := D(w, δ ′ ). Then,

r ≥ inf D |φ ′ | • δ ′ and thus, area(Q) ≥ inf D |φ ′ | 2 • (δ ′ ) 2 8 .
In addition, sup

D |φ ′ | ≤ κ inf D |φ ′ | and so, dens Q C \ φ(Y ℓ ) ≤ area φ(D \ Y ℓ ) area(Q) ≤ sup D |φ ′ | 2 • π(δ ′ ) 2 • η inf D |φ ′ | 2 • (δ ′ ) 2 /8 ≤ 8πκ 2 η < 1 4 .
As a consequence,

dens Q φ(Y ℓ ) > 3 4 .
Step

dens U ′ (X n ) ≥ λ. 14
Assume f : V → U is a holomorphic map, nowhere locally constant, and (f n :

V n → C) is a sequence of holomorphic maps such that • every compact subset of V is eventually contained in V n and • the sequence (f n ) converges uniformly to f on every compact subset of V . Then, lim inf

n→+∞ m| Xn ≥ λ • m| U =⇒ lim inf n→+∞ m| f -1 n (Xn) ≥ λ • m| V . Step 6. Set Y ℓ n := z ∈ V (δ) (∀j ≤ ℓ) P •j αn (z) ∈ V (δ) and
dens U ′ (Xn) ≥ λ • dens U ′ (U ).
Indeed, assume λ is a limit value of the sequence dens Qn φ n K n (δ) .

Post-composing the maps φ n with affine maps and extracting a subsequence if necessary, we may assume that (w n ) converges to w ∈ W , (φ n ) converges locally uniformly to φ : D(w, 3δ ′ /2) → C, r n converges to the radius r of the largest disk centered at φ(w) and contained in φ D(w, δ ′ ) and Q n converges to a square Q with side length at least r/ √ 8. According to steps 5 and 6, lim inf

n→+∞ m| φn(Kn(δ)) ≥ 1 2 m| φ(Y ℓ ) .
According to step 4, it follows that

λ ≥ 1 2 dens Q φ(Y ℓ ) ≥ 3 8 .
Step 8. From now on, we assume that n is sufficiently large, so that:

(i) ∆ \ K n (δ) ⊂ X n ⊂ ∆ \ K n (δ ′ ) with X n := z ∈ ∆ (∃k) P •k αn (z) ∈ W (this is possible by step 2); (ii) s n < δ ′ with s n := sup z∈∆ d z, K n (δ ′ )
(this is possible since s n -→ n→+∞ 0 in order for the aforementioned point (A) to hold); (iii) the post-critical set of P αn is contained in V (δ ′ /2) (this is possible by the aforementioned point (B)); (iv) if φ is univalent on D(w, 3δ ′ /2) with w ∈ W , if r is the radius of the largest disk centered at φ(w) and contained in φ D(w, δ ′ ) and if Q is a square contained in φ D(w, δ ′ ) with side length at least r/ √ 8, then dens Q φ K n (δ) ≥ 1 4 (this is easily follows from step 7 by contradiction).

Step 9. Assume z 0 ∈ X n . Then, we have

z 0 ∈ X n Pα n → z 1 ∈ V (2δ ′ ) Pα n → • • • Pα n → z k-1 ∈ V (2δ ′ ) Pα n → z k ∈ W
for some integer k > 0. Since the post-critical set of P αn is contained in V (δ ′ /2), for j ≤ k there exists a univalent map φ j : D

:= D(z k , δ ′ ) → C such that • φ j is the inverse branch of P •k-j
αn which maps z k to z j and • φ j extends univalently to D(z k , 3δ ′ /2). In particular, sup

D |φ ′ j | ≤ κ inf D |φ ′ j |.
Let D(z j , r j ) be the largest disk centered at z j and contained in φ j (D) and D(z j , R j ) be the smallest disk centered at z j and containing φ j (D). Note that D is contained in C \ V (δ ′ ) and so, for j ≤ k -1, D(z j , r j ) ⊂ φ j (D) ⊂ C \ K n (δ ′ ). On the one hand, d(z j , ∆) < 2δ ′ and on the other hand, every point of ∆ is at distance at most s n from a point of K n (δ ′ ). It follows that

R j ≤ κr j ≤ κ • (s n + 2δ ′ ).
If w 0 ∈ φ 0 (D) and w j := P •j αn (w 0 ), then for j ≤ k -1,

d(w j , ∆) ≤ d(w j , z j ) + d(z j , ∆) ≤ κ • (s n + 2δ ′ ) + 2δ ′ < (2 + 3κ) • δ ′ < δ and for j = k, d(w k , ∆) ≤ d(w k , z k ) + d(z k , ∆) ≤ 9δ ′ < δ.
In other words, w 0 , w 1 , . . . , w k all belong to V (δ). As a consequence,

φ 0 K n (δ) ⊂ K n (δ).
Step 10. Continuing with the notations of step 9, we denote by Q z0 the largest douadic square containing z 0 and contained in D(z 0 , r 0 ). On the one hand, since z 0 ∈ ∆ and since φ 0 (D) ⊂ C \ K n (δ ′ ), we have r 0 ≤ s n , and so

Q z0 ⊂ D(z 0 , r 0 ) ⊂ V (s n ) \ K n (δ ′ ).
On the other hand, Q z0 has an edge of length greater than r 0 /2 √ 2 and so, according to step 8 point (iv),

dens Qz 0 K n > 1 4 . As a consequence dens Qz 0 C \ K n (δ) < 3 4 . Given two douadic squares Q and Q ′ , either Q ∩ Q ′ = ∅, or Q ⊂ Q ′ or Q ′ ⊂ Q. It follows that area ∆ \ K n (δ) ≤ 3 4 area z∈Xn Q z ≤ 3 4 area V (s n ) \ K n (δ ′ ) ≤ 3 4 area ∆ \ K n (δ ′ ) + 3 4 area V (s n ) \ ∆ = 3 4 area ∆ \ K n (δ ′ ) + c n • area(∆) with c n := 3 4 area V (s n ) \ ∆ area(∆) .
Step 11. Since s n → 0 and since the boundary of ∆ has zero Lebesgue measure,

area V (s n ) \ ∆ -→ n→+∞ 0. Thus, dens ∆ C \ K n (δ) < 3 4 dens ∆ C \ K n (δ ′ ) + c n with c n -→ n→+∞ 0.
This completes the proof of Lemma 16.

The linearizable case

In order to find a quadratic polynomial with a linearizable fixed point and a Julia set of positive area, we need to modify the argument.

Definition 11. If α is a Brjuno number, we denote by ∆ α the Siegel disk of P α and by r α its conformal radius. For ρ ≤ r α , we denote by ∆ α (ρ) the invariant sub-disk with conformal radius ρ and by L α (ρ) the set of points in K α whose orbits do not intersect ∆ α (ρ).

Figure 23. Two sets L α (ρ) and L α ′ (ρ), with α ′ a well-chosen perturbation of α as in Prop. 19. This proposition asserts that if α and α ′ are chosen carefully enough, the loss of measure from L α (ρ) to L α ′ (ρ) is small. We colored white the basin of infinity, the invariant subdisks ∆ α (ρ) and ∆ α ′ (ρ) their preimages; we colored light grey the remaining parts of the Siegel disks and their preimages; we colored dark grey the pixels where the preimages are too small to be drawn.

Proposition 19. There exists a set S of bounded type irrationals such that for all α ∈ S, all ρ < ρ ′ < r α and all ε > 0, there exists α ′ ∈ S with

• |α ′ -α| < ε, • max ρ, (1 -ε)ρ ′ < r α ′ < (1 + ε)ρ ′ and • area L α ′ (ρ) ≥ (1 -ε)area L α (ρ) .
Proof. We let N be sufficiently large so that the conclusions of Prop. 11 and Cor. 4 apply. We will work with S = S N . Assume α ∈ S N and choose a sequence (A n ) such that lim

n→+∞ qn A n = r α ρ ′ . Set α n := [a 0 , a 1 , . . . , a n , A n , N, N, N, . . .]
. This guaranties that r αn -→ n→+∞ ρ ′ (see [ABC]).

Let ∆ be the Siegel disk of P α . Let us use the notations V (δ), K(δ) and K n (δ) introduced in section 1.7. With an abuse of notations, set ∆(ρ) := ∆ α (ρ) and ∆ n (ρ) := ∆ αn (ρ). Set ∆ ′ (ρ) :

= P -1 α ∆(ρ) \ ∆(ρ).
Then, ∆(ρ) and ∆ ′ (ρ) are symmetric with respect to the critical point of P α . The orbit under iteration of P α of a point z / ∈ ∆(ρ) lands in ∆(ρ) if and only if it passes through ∆ ′ (ρ). We have a similar property for ∆ ′ n (ρ) := P -1 αn ∆ n (ρ) \ ∆ n (ρ). We have proved -see equation ( 5) -that

(∀δ > 0) dens ∆ K n (δ) -→ n→+∞ 1.
The sequence of compact sets ∆ n (ρ) converges to ∆(ρ) for the Hausdorff topology on compact subsets of C, because lim r αn > ρ. It immediately follows that for all δ > 0, dens ∆\∆

(ρ) K n (δ) \ ∆ n (ρ) -→ n→+∞ 1.
Choose δ sufficiently small so that V (δ) does not intersect ∆ ′ (ρ). Then, for n large enough V (δ) does not intersect ∆ ′ n (ρ). In that case, the orbit under iteration of P αn of a point in K n (δ) \ ∆ n (ρ) cannot pass through ∆ ′ n (ρ) and so,

K n (δ) \ ∆ n (ρ) ⊂ L αn (ρ). Thus, dens ∆\∆(ρ) L αn (ρ) -→ n→+∞ 1.
The points of L α (ρ) whose orbits do not intersect ∆ \ ∆(ρ) are contained in the union of the Julia set J α and the countably many preimages of ∂∆(ρ). Thus, they form a set of zero Lebesgue measure. It follows that area

L αn (ρ) -→ n→+∞ area L α (ρ) .
Proof of Theo. 2. We start with α 0 ∈ S and set ρ 0 := r α0 . We then choose ρ ∈ ]0, ρ 0 [ and two sequences of real numbers ε n in (0, 1) and ρ n in (0, ρ 0 ) such that (1ε n ) > 0 and ρ n ց ρ > 0. We can construct inductively a Cauchy sequence (α n ∈ S) such that for all n ≥ 1,

• r αn ∈ (ρ n , ρ n-1 ) and

• area L αn (ρ) ≥ (1 -ε n )area L αn-1 (
ρ) . Let α be the limit of the sequence (α n ). The conformal radius of a fixed Siegel disk depends upper semi-continuously on the polynomial (a limit of linearizations linearizes the limit). So, r α ≥ lim r αn = ρ. Also, by choosing α n sufficiently close to α n-1 at each step, we can guaranty that r α ≤ ρ, in which case r α = ρ.

In addition, the sequence of pointed domains ∆ αn (ρ), 0 converges for the Carathéodory topology to (∆ α , 0). In particular, every compact subset of ∆ α is contained in ∆ αn (ρ) for n large enough. Similarly, every compact subset of

C \ K α is contained in C \ K αn for n large enough. It follows that lim sup L αn (ρ) := m n≥m L αn (ρ) ⊂ L α (ρ). Since r α = ρ, ∆ α (ρ) = ∆ α and L α (ρ) = J α . Thus, lim sup L αn (ρ) ⊂ J α and area(J α ) ≥ area(lim sup L αn (ρ)) ≥ area L α0 (ρ) • (1 -ε n ) > 0.

The infinitely renormalizable case

In order to find an infinitely renormalizable quadratic polynomial with a Julia set of positive area, we need a modification based on Sørensen's construction of an infinitely renormalizable quadratic polynomial with a non-locally connected Julia set.

Proposition 20. There exists a set S of bounded type irrationals such that for all α ∈ S and all ε > 0, there exists α ′ ∈ C \ R with

• |α ′ -α| < ε,

• P α ′ has a periodic Siegel disk with period > 1 and rotation number in S and • area(K α ′ ) ≥ (1ε)area(K α ).

Figure 24. Two filled-in Julia sets K α and K α ′ , with α ′ a wellchosen perturbation of α as in Prop. 20. This proposition asserts that if α and α ′ are chosen carefully enough, P α ′ has a periodic Siegel disk and the loss of measure from K α to K α ′ is small. Left: we hatched the fixed Siegel disk. Right: we hatched the cycle of Siegel disks.

Proof. We can choose S = S N with N large enough (in order to be able to apply Inou and Shishikura techniques). The proof essentially goes as in the Cremer case Given α ∈ S, we let p k /q k be its approximants, and we consider the functions of explosion χ k given by Prop. 4. If α ′ belongs to the disk centered at p k /q k with radius 1/q 3 k , the set C k (α ′ ) := χ k q k α kp k /q k is a cycle of P α ′ . Its multiplier is e 2iπθ k (α ′ ) with θ k : D(p k /q k , 1/q 3 k ) → C a nonconstant holomorphic function which vanishes at p k /q k . We consider a sequence (α n ) converging to α so that

• lim sup Proof. Let p be the period of 0 under iteration of Q c0 and let f λ := Q •p λ : U ′ λ → U λ λ∈Λ be a Mandelbrot-like family centered at c 0 .

Let φ c ′ : U c0⊥c ′ → C be hybrid conjugacies. As c ′ → c, the modulus of the annulus U c0⊥c ′ \ U ′ c0⊥c ′ is bounded from below. So, the φ c ′ can be chosen to have a uniformly bounded quasiconformal dilatation. It follows that if c ′ ∈ M → c ∈ M with area(K c ′ ) → area(K c ), we have

area φ -1 c ′ (K c ′ ) -→ c ′ →c area φ -1 c (K c ) .
It follows easily that area(K c0⊥c ′ ) → area(K c0⊥c ) since almost every point in K c0⊥c has an orbit terminating in φ -1 c (K c ). Proof of Theo. 3. If P c has a periodic Siegel disk then c is on the boundary of a hyperbolic component with center c 0 . We denote by Ω c this hyperbolic component and we set M c := c 0 ⊥ M .

We will denote by S the image of S by the map α → e 2iπα /2e 4iπα /4. Then, c ∈ S if and only if P c has a fixed Siegel disk with rotation number in S. Moreover, P c has a periodic Siegel disk with rotation number in S if and only if c = c 0 ⊥ s with c 0 the center of the hyperbolic component containing c in its boundary and s ∈ S.

It follows from Prop. 21 and 22 that if Q c has a periodic Siegel disk with rotation number in S, then for all ε > 0, we can find c ′ ∈ M c \ Ω c such that

• |c ′ -c| < ε,

• P c ′ has a periodic Siegel disk with rotation number in S and • area(K c ′ ) > (1ε)area(K c ). Let us choose a parameter c 0 ∈ S and a sequence of real number ε n in (0, 1) such that (1ε n ) > 0. We can construct inductively a sequence (c n ) such that

• (c n ) is a Cauchy sequence that converges to a parameter c,

• Q cn has a periodic Siegel disk with rotation number in S,

• for n ≥ 1, c n ∈ M cn-1 \ Ω cn-1 and • area(K cn ) > (1ε n )area(K cn-1 ). Then, P c is infinitely renormalizable (it is in the intersection of the nested copies M cn ). Thus, J c = K c = lim K cn . Finally, area(J c ) = area(K c ) ≥ area(K c0 ) • (1ε n ) > 0.

Appendix A. Parabolic implosion and perturbed petals

The notations used in this appendix are those of section 1.5.3. We postponed the proof of the following lemma to this appendix.

Lemma 17. If R > 0 and K > 0 are sufficiently large, then for n large enough:

(1) Φ n (Ω n ) contains the vertical strip

U n := w ∈ C ; R < Re(w) < 1/α n -R ,
(2) τ n is injective on P n := (Φ n ) -1 (U n ) and

(3) there is a branch of argument defined on τ n (P n ) such that Claim 1. Increasing R 1 if necessary, there is a constant C > 0 such that for all n sufficiently large sup

w∈Ω n Φ n (w) -Ψ n (w) < C.
Proof of Claim 1. According to Prop. 2.6.2 in [START_REF] Shishikura | Bifurcation of parabolic fixed points The Mandelbrot set, theme and variations[END_REF], there are constants R and C such that for all sufficiently large n and for all w ∈ Ω n with d(w, ∂Ω n ) ≥ R, we have

(Φ n ) ′ (w) -(Ψ n ) ′ (w) ≤ C 1 d(w, ∂Ω n ) 2 + F ′ n (w) -1 .
We will first show that we can get rid of F ′ n (w) -1 . Set Those functions are 1/α n periodic. On the one hand, as n → +∞,

• the functions G n are uniformly bounded by 1/4 on ∂Ω n and • the sequence (S n ) converges uniformly to w → 1/w 2 on ∂Ω n , and thus, the functions S n are uniformly bounded away from 0 on ∂Ω n .

As a consequence, the functions G n /S n are uniformly bounded on ∂Ω n . On the other hand, as Im(w) → ±∞, G n (w) → 0. Thus, in C/ 1 αn Z, G n has removable singularities at ±i∞ and vanishes at those points. Since in C/ 1 αn Z, S n has simple zeros at ±i∞, the function G n /S n has removable singularities at ±i∞ in C/ 1 αn Z. It follows that from the maximum modulus principle that there is a constant C 1 such that for all sufficiently large n and all w ∈ Ω n , we have

F ′ n (w) -1 ≤ C 1 πα n sin(πα n w) 2 .
Note that there is a constant C 2 > 0 such that ∀w ∈ C, d(w, Z) ≤ C 2 sin(πw) .

Indeed, the quotient d(w, Z) sin(πw) extends continuously to (C/Z) ∪ {±i∞} which is compact. It follows that for all w ∈ Ω n , πα n sin(πα n w)

2 ≤ C 2 2 π 2 |α n | 2 d(α n w, Z) 2 ≤ C 2 2 π 2 d(w, ∂Ω n ) 2 .
Thus, there is a constant C ′ such that for all sufficiently large n and for all w ∈ Ω n with d(w, ∂Ω n ) ≥ R, we have

(Φ n ) ′ (w) -(Ψ n ) ′ (w) ≤ C ′ d(w, ∂Ω n ) 2 .
Taking R ≥ 1 and replacing R 1 by R 1 + √ 2R, this can be rewritten as: there is a constant C such that for all sufficiently large n and for all w ∈ Ω n

(Φ n ) ′ (w) -(Ψ n ) ′ (w) ≤ C ′ 1 + d(w, ∂Ω n ) 2 .
Let us now assume n is sufficiently large, so that

X n := 1 2α n -R 1 > 0.
Then

C ′ |du| 1 + d(u, ∂Ω n ) 2 ≤ +∞ 0 C ′ ds 1 + √ 2(X n + s) 2 + Xn+|y| 0 C ′ dt 1 + √ 2(X n + |y| -t) 2 ≤ 2C ′ .
This completes the proof of Claim 1.

Claim 2. The map Ψ n is univalent on Ω n , Ψ n (Ω n ) contains the vertical strip

V n := w ∈ C ; Re Ψ n (R 1 ) < Re(w) < Re Ψ n (1/α n -R 1 )

and τ n is injective on

Q n := (Ψ n ) -1 (V n ).
Proof of Claim 2. Note that Ψ n is a straightening map for the vector field ζ n :

(Ψ n ) * ζ n = d dw .
Since F n (w)w ∈ D(1, 1/4) on Ω n , the trajectories of the vector field ζ n are curves which enter Ω n through its left boundary and exit Ω n through the right boundary. In particular, no trajectory is periodic. Since two distinct trajectories cannot intersect, the map Ψ n is injective.

Observe that for w ∈ ∂Ω n , arg (Ψ n ) ′ (w) = -arg F n (w)-w ∈arcsin(1/4), arcsin(1/4) ⊂ -π/12, π/12 .

Integrating (Ψ n ) ′ (w) along ∂Ω n , we conclude that This completes the proof of part (3) of lemma 17.

Appendix B. Arithmetic background

Proposition 3 .

 3 If N is sufficiently large then the following holds. 3 Assume α ∈ S N , choose a sequence (A n ) such that qn A n -→ n→+∞ +∞ and qn log A n -→ n→+∞ 1. 4 Set α n := [a 0 , a 1 , . . . , a n , A n , N, N, N, . . .].

Figure 3 .

 3 Figure 3. Illustration of Prop. 6 for α = θ = [0, 1, 1, . . .], n = 7 and A n = 10 10 . We see the Siegel disk ∆ of P α (light grey), the Siegel disk ∆ ′ n of the restriction of P αn to ∆ (dark grey) and the boundary of the Siegel disk of P αn .

Figure 4 .

 4 Figure 4. The boundary of a set X n (ρ).

Figure 9 .

 9 Figure9. A schematic representation of the set U . We colored gray the set of points in U whose image by P is contained in the lower half-plane.

Figure 10 .

 10 Figure 10. Right: the sets Ω att and Ω rep . Left: the set Ω att,f and Ω rep,f for a map f with c f = 1. The sets Ω att and Ω att,f are shaded. The boundaries of the sets Ω rep and Ω rep,f are dashed.

Figure 11 .

 11 Figure11. Left: the attracting petal P att,f of some map f ∈ IS 0 ; the critical point is ω f , the critical value v and 0 is a fixed point. Right: its image by Φ att,f ; we divided the right half plane ]0, +∞[×R into vertical strips of width 1 of alternating color, highlighted the real axis in red, and put a black dot at the point z = 1. On the left, we pulled this coloring back by Φ att,f .

Figure 12 .

 12 Figure12. On the right, we divided ]0, 2[×] -2, +∞[ into 3 regions of different colors. We subdivided each by a vertical line through z = 1. These 6 pieces were then pulled back on the left by Φ att,f , for the same parabolic f ∈ IS 0 as in figure11. The set V f is the union of the green and red regions. The set W f is the union of the red and yellow regions.

Figure 13 .

 13 Figure 13. Left: among the successive preimages of V f and W f by f , those that compose the sets V -k f , W -k f are shown. The colors are preserved by f . Right: preimage of the left part by τ 0 . We hatched W F ∪ V F and W -7F ∪ V -7 F .

Figure 15 .

 15 Figure 15. The perturbed petal P f whose image by the perturbed Fatou coordinate Φ f is the strip 0 < Re(w) < 1/α f -R 3 .

Figure 16 .

 16 Figure 16. The domain D n (grey) is the complement of a union of disks and the hourglass Ω n (drak grey) is contained in D n .

Figure 18 .

 18 Figure 18. Definition of the perturbed Fatou coordinate Φ n . The perturbed petal P n is grey and the set P ′ n is hatched.

Figure 20 .

 20 Figure 20. The branch ψ j+1 maps P fj+1 univalently into P fj . The map Ψ j := ψ 1 • ψ 2 • . . . • ψ j is then defined and univalent on P fj with values in the dynamical plane of the polynomial f 0 .Remember that

Figure 21 .

 21 Figure 21. If f ∈ IS α with α ∈ Irrat ≥N , the set U 1 (f ) contains the postcritical set PC(f ). If α is of bounded type, this post-critical set is dense in the boundary of the Siegel disk of f .

Figure 22 .

 22 Figure 22. If α = ( √ 5-1)/2, the critical point of P α is a Lebesgue density point of the set of points whose orbit remain in D(0, 1). Left: the set of points whose orbit remains in D(0, 1). Right: a zoom near the critical point.

  11 and Cor. 4 apply. Assume α ∈ S N and choose a sequence (A n ) such that qn A n -→ n→+∞ +∞ and qn log A n -→ n→+∞ 1. Set α n := [a 0 , a 1 , . . . , a n , A n , N, N, N, . . .].

  n→+∞ qn |α np n /q n | = 0 and • θ n (α n ) := [A n , N, N, N, . . .] with lim n→+∞ qn A n = +∞ and lim n→+∞ qn log A n = 1.

  As in[START_REF] Shishikura | Bifurcation of parabolic fixed points The Mandelbrot set, theme and variations[END_REF], the argument consists in comparing the Fatou coordinate Φ n to the Fatou coordinate Ψ n of the time one map of the vector field ζ n defined on D n byζ n = ζ n (w) d dw := F n (w)w d dw .In other words, set w n := 1 2α n and let Ψ n : Ω n → C be defined byΨ n (w) = Φ n (w n ) + w wn du F n (u)u .

G

  n (w) := F ′ n (w) -1 and S n (w) := πα n sin(πα n w) 2 .

2π 3 <

 3 arg Ψ n (w) -Ψ n (R 1 ) < 4π 3 on the left boundary of Ω n and thatπ 3 < arg Ψ n (w) -Ψ n (1/α n -R 1 ) < πIt is therefore enough to show that supw∈Ω n ∩H + Φ n (w)w = O w)w -1 is periodic of period 1/α n ,bounded by 1/3 in Ω n and tends to 0 as Im(w) tends to +∞. It follows from the maximum modulus principle that 1F n (w)w n ∩H + ) |e 2iπαnw | • |e 2iπαnw | ≤ Ce -2παnIm(w)for some constant C which does not depend on n.If w := R + x + iy ∈ Ω n ∩ H + , then |x| < y + 1/α n . So sup w∈Ω n ∩H + Ψ n (w)w ≤ Ψ n (R) -R

  6 ∆ ′ n is the largest connected open subset of ∆ containing 0, on which Pα n is conjugate to a rotation. It is contained in the Siegel disk of Pα n

  we get the desired bound on η n .

	Corollary 3. Assume r < 1. Then,
	sup
	Z∈Hn(r)

  An elementary computationshows that H n is a 5/4-quasiconformal homeomorphism between B n and U n . 11 Since H n (iY + 1) = F n H n (iY ) , the quasiconformal homeomorphism H n : B n → U n induces a homeomorphism between the half cylinder H/Z and the Riemann surface V n . This homeomorphism is clearly quasiconformal on the image of B n in H/Z, i.e., outside a straight line. It is therefore quasiconformal in the whole half cylinder (R-analytic curves are removable for quasiconformal homeomorphisms).Let R n be the rectangle

  5. If X ⊂ C is a measurable set, we use the notation m| X for the Lebesgue measure on X, extended by 0 outside X. If U ⊂ C is an open set, (X n ) is a sequence of measurable subsets of C and λ ∈ [0, 1], we say that lim inf n→+∞ m| Xn ≥ λ • m| U if for all non empty open set U ′ relatively compact in U , we have lim inf

	n→+∞

  Assume φ n is univalent on D(w n , 3δ ′ /2) with w n ∈ W , r n is the radius of the largest disk centered at φ n (w n ) and contained in φ n D(w n , δ ′ ) and Q n is a square contained in φ n D(w n , δ ′ ) with side length at least r n / Equivalently, for all non empty open set U ′ ⊂ C with finite area, lim inf

	P •ℓ αn (z) ∈ ∆ . αn (z) ∈ K n (δ), then z ∈ K n (δ). On the other n and P •ℓ On the one hand, if z ∈ Y ℓ hand, every compact subset of Y ℓ is eventually contained in Y ℓ n and the sequence (P •ℓ αn ) converges uniformly to P •ℓ α on every compact subset of Y ℓ . By the afore-
	mentioned point (A), we have
		lim inf n→+∞	m| Kn(δ) ≥	1 2	m| ∆ .
	So, according to step 5,	lim inf n→+∞	m| Kn(δ) ≥	1 2	m| Y ℓ .
	Step 7. √ 8. Then,
		lim inf n→+∞	dens Qn φ n K n (δ) ≥	3 8	.
						n→+∞

14

  , w n := 1 2α n belongs to Ω n . Fix w := w n + x + iy ∈ Ω n . Note that |x| < X n + |y| and d(w, ∂Ω n ) > √ 2 X n + |y| -|x| .

	It follows that
	Φ

n (w) -Ψ n (w) ≤ [wn,wn+iy]∪[

wn+iy,w]

This is true for almost every α ∈ R/Z.

This is possible by Cor. 3 applied with r > r 2 . Indeed, for n large enough, we have that τn(r 2 ) -1/10 > τn(r) and thus Z ∈ C ; Im(Z) ≥ τn(r 2 ) -1/10} ⊂ Hn(r).

The fact that gn : φn(V ′ n ) → D * is continuous and univalent is not completely obvious; see the Proposition on page 33 in[Yo] for details.

The coefficient 3

Acknowledgements

We would like to thank Adrien Douady, John H. Hubbard, Hiroyuki Inou, Curtis T. McMullen, Mitsuhiro Shishikura, Misha Yampolsky and Jean-Christophe Yoccoz whose contributions were decisive in proving these results. We would like to thank Misha Lyubich and the referees for carefully reading our manuscript and suggesting several improvements in the presentation.

We control the shape of the cycle of Siegel disk as in the Cremer case. For all ρ < 1 and all n sufficiently large, the cycle of Siegel disks contains the χ n Y n (ρ) with

For this purpose, we work in the coordinate given by χ n and compare the dynamics of the conjugated map to the flow of a vector field.

We control the post-critical set of P αn via Inou-Shishikura's techniques.

We then control the loss of area as in the Cremer case.

Definition 12. For c ∈ C, we denote by

With an abuse of notations, we denote by K c its filled-in Julia set and by J c its Julia set. We denote by M the Mandelbrot set, i.e. the set of parameters c for which K c is connected.

The previous proposition can be restated as follows.

Proposition 21. Assume P c has a fixed Siegel disk with rotation number in S.

Then, for all ε > 0, there exists c ′ such that • |c ′ -c| < ε,

• P c ′ has a periodic Siegel disk with period > 1 and rotation number in S and

In fact, such a c is on the boundary of the main cardioid of M and the proof we proposed yields a c ′ which is on the boundary of a satellite component of the main cardioid of M .

Using the theory of quadratic-like maps introduced by Douady and Hubbard [START_REF] Douady | On the dynamics of polynomial-like mappings[END_REF], we can transfer this statement to perturbations of quadratic polynomials having periodic Siegel disks. We will use the notions of renormalization and tuning (see for example [Ha]).

If 0 is periodic of period p under iteration of Q c0 , then c 0 is the center of a hyperbolic component Ω of the Mandelbrot set. This component Ω has a root: the parameter c 1 ∈ ∂Ω such that Q c1 has an indifferent cycle with multiplier 1. In addition, there exist

on the right boundary of Ω n . This proves that Ψ n (Ω n ) contains the vertical strip V n .

Assume by contradiction that τ n is not injective on V n . Then, there is an integer k ∈ Z\{0} and a point w ∈ V n such that w+k/α n is in V n . Note that V n is a union of trajectories for the rotated vector field iζ n . As w runs along those trajectories, the imaginary part of w increases from -i∞ to +i∞. In particular, every trajectory intersects R. Since for all w ∈ D n , we have iζ n (w) = iζ n (w + 1/α n ), the trajectory for iζ n passing through w + k/α n is obtained from the trajectory passing through w by translation by k/α n . This is not possible since the intersection of those trajectories with R is contained in

Let us now come to the proof of parts ( 1) and ( 2) of lemma 17. Assume n is sufficiently large, so that sup

Thus, if R is large enough and if n is sufficiently large, then Φ n (Q n ) contains the vertical strip

Since τ n is injective on Q n , this proves parts (1) and (2) of lemma 17. For all w ∈ P n , we have