QUADRATIC JULIA SETS WITH POSITIVE AREA.

Transcription

1 QUADRATIC JULIA SETS WITH POSITIVE AREA. XAVIER BUFF AND ARNAUD CHÉRITAT Abstract. We prove the existence o quadratic polynomials having a Julia set with positive Lebesgue measure. We ind such examples with a Cremer ixed point, with a Siegel disk, or with ininitely many satellite renormalizations. To Adrien Douady Contents Introduction 1 Acknowledgements 2 1. The Cremer case Strategy o the proo A stronger version o Prop The control o the cycle Perturbed Siegel disks The control o the post-critical set Lebesgue density near the boundary o a Siegel disk The proo The linearizable case The ininitely renormalizable case 51 Appendix A. Parabolic implosion and perturbed petals 54 Reerences 58 Introduction Assume P : C C is a polynomial o degree 2. Its Julia set JP ) is a compact subset o C with empty interior. Fatou suggested that one should apply to JP ) the methods o Borel-Lebesgue or the measure o sets. It is known that the area Lebesgue measure) o JP ) is zero in several cases including: i P is hyperbolic; 1 i P has a parabolic cycle [DH1]); i P is not ininitely renormalizable [L] or [Sh1]); 1 Conjecturally, this is true or a dense and open set o quadratic polynomials. I there were an open set o non-hyperbolic quadratic polynomials, those would have a Julia set o positive area see [MSS]). 1

2 2 XAVIER BUFF AND ARNAUD CHÉRITAT i P has a linearizable) indierent cycle with multiplier e 2iπα such that 1 α = a 0 + with log a n = O n) [PZ]). 2 1 a 1 + a Recently, we completed a program initiated by Douady with major advances by the second author in [C]: there exist quadratic polynomials with a Cremer ixed point and a Julia set o positive area. In this article, we present a slightly dierent approach the general ideas are essentially the same). Theorem 1. There exist quadratic polynomials which have a Cremer ixed point and a Julia set o positive area. We also have the ollowing two results. Theorem 2. There exist quadratic polynomials which have a Siegel disk and a Julia set o positive area. Theorem 3. There exist ininitely satellite renormalizable quadratic polynomials with a Julia set o positive area. We will give a detailed proo o Theo. 1 and 2. We will only sketch the proo o Theo. 3. The proos are based on McMullen s results [McM] regarding the measurable density o the illed-in Julia set near the boundary o a Siegel disk with bounded type rotation number; Chéritat s techniques o parabolic explosion [C] and Yoccoz s renormalization techniques [Y] to control the shape o Siegel disks; Inou and Shishikura s results [IS] to control the post-critical sets o perturbations o polynomials having an indierent ixed point. 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. 1. The Cremer case Let us introduce some notations. Deinition 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 illed-in Julia set o P α and by J α its Julia set. 2 This is true or almost every α R/Z.

3 QUADRATIC JULIA SETS WITH POSITIVE AREA Strategy o the proo. The main gear is the ollowing Proposition 1. There exists a non empty set S o bounded type irrationals such that: or all α S and all ε > 0, there exists α S with α α < ε, P α has a cycle in D0, ε) \ {0} and areak α ) 1 ε)areak α ). The proo o 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 illed-in Julia sets K α and K α, with α a wellchosen perturbation o α as in Prop. 1. This proposition asserts that i α and α are chosen careully enough the loss o measure rom K α to K α is small. Remark. We do not know what is the largest set S or which Prop. 1 holds. It might be the set o all bounded type irrationals. Proposition 2 Douady). The unction α C areak α ) [0, + [ is upper semi-continuous. Proo. Assume α n α. By [D2], or any neighborhood V o K α, we have K αn V or n large enough. According to the theory o Lebsegue measure, areak α ) is the inimum o the area the open sets containing K α. Thus, areak α ) lim sup areak αn ). n + Proo o Theo. 1 assuming Prop. 1. We choose a sequence o real numbers ε n in 0, 1) such that 1 ε n ) > 0. We construct inductively a sequence θ n S such that or all n 1 P θn has a cycle in D0, 1/n) \ {0}, areak θn ) 1 ε n )areak θn 1 ). Every polynomial P θ with θ suiciently close to θ n has a cycle in D0, 1/n) \ {0}. By choosing θ n suiciently close to θ n 1 at each step, we guarantee that

4 4 XAVIER BUFF AND ARNAUD CHÉRITAT Figure 2. A zoom on K α small cycle is highlighted. near its linearizable ixed point. The the sequence θ n ) is a Cauchy sequence that converges to a limit θ, or all n 1, P θ has a cycle in D0, 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: areaj θ ) = areak θ ) lim sup areak θn ) areak θ0 ) 1 ε n ) > 0. n + n A stronger version o Prop. 1. For a inite or ininite sequence o integers, we will use the ollowing continued raction notation: 1 [a 0, a 1, a 2,...] := a a 1 + a For α R, we will denote by α the integral part o α. Deinition 2. I N 1 is an integer, we set S N := { α = [a 0, a 1, a 2,...] R \ Q ak ) is bounded and a k N or all k 1 }. Note that S N+1 S N S 1 and S 1 is the set o bounded type irrationals. I 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 o P α is contained in the boundary o the Siegel disk. Prop. 1 is an immediate consequence o the ollowing proposition.

5 QUADRATIC JULIA SETS WITH POSITIVE AREA. 5 Proposition 3. I N is suiciently large then the ollowing holds. 3 Assume α S N, choose a sequence A n ) such that qn An + and qn log An n + n Set α n := [a 0, a 1,..., a n, A n, N, N, N,...]. Then, or all ε > 0, i n is suiciently large, P αn has a cycle in D0, ε) \ {0} and areak αn ) 1 ε)areak α ). The rest o section 1 is devoted to the proo o Prop. 3. In the sequel, unless otherwise speciied, α is an irrational number o bounded type, p k /q k are the approximants to α given by the continued raction algorithm and α n ) is a sequence converging to α, deined as in Prop. 3. Note that or k n, the approximants p k /q k are the same or α and or α n. The polynomial P α resp. P αn ) has a Siegel disk resp. n ). We let r resp. r n ) be the conormal radius o resp. n ) at 0 and we let φ : D0, r) resp. φ n : D0, r n ) n ) be the conormal isomorphism which maps 0 to 0 with derivative The control o the cycle. We irst recall results o [C] see also [BC1] Props. 1 and 2), which we reormulate as ollows. The irst proposition asserts that as θ varies in the disk Dp/q, 1/q 3 ), the polynomial P θ has a cycle o period q which depends holomorphically on q θ p/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 unction χ : D0, 1/q 3/q ) C with the ollowing properties. 1) χ0) = 0. 2) χ 0) 0. 3) I δ D0, 1/q 3/q ) \ {0}, then χδ) 0. 4) I δ D0, 1/q 3/q ) \ {0} and i we set ζ := e 2iπp/q and θ := p q + δq k, then, χδ), χζδ),..., χζ q 1 δ) orms a cycle o period q o P θ. In particular, δ D0, 1/q 3/q ), χζδ) = P θ χδ) ). A unction χ : D0, 1/q 3/q ) C as in Prop. 4 is called an explosion unction at p/q. Such a unction is not unique. However, i χ 1 and χ 2 are two explosions unctions at p/q, they are related by χ 1 δ) = χ 2 e 2iπkp/q δ) or some integer k Z. The second proposition studies how the explosion unctions behave as p/q ranges in the set o approximants o an irrational number α such that P α has a Siegel disk. 3 The choice o N will be speciied in equation 3 4 For example, one can choose An := q qn n. However, we think that the proposition holds or more general sequences α n) or instance as soon as qn A n +.

6 6 XAVIER BUFF AND ARNAUD CHÉRITAT 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 conormal radius o at 0 and let φ : D0, r) be the isomorphism which sends 0 to 0 with derivative 1. For k 1, let χ k be an explosion unction at p k /q k and set λ k := χ k 0). Then, 1) λ k r and k + 2) the sequence o maps ψ k : δ χ k δ/λ k ) converges uniormly on every compact subset o D0, r) to φ : D0, r). Corollary 1. Let α n ) be the sequence deined in Prop. 3. Then, or all ε > 0, i n is suiciently large, P αn has a cycle in D0, ε) \ {0}. Proo. Let χ n be an explosion at p n /q n and let C n be the set o q n -th roots o α n p n 1) n = q n q n q n A n + q n 1) with A n := [A n, N, N, N,...]. Since qn A n n is contained in an arbitrarily n + small neighborhood o 0 and χ n C n ) is a cycle o P αn contained in an arbitrarily small neighborhood o Perturbed Siegel disks. Deinition 3. I U and X are measurable subsets o C, with 0 < areau) < +, we use the notation areau X) dens U X) :=. areau) In the whole section, α is a Bruno number, p n /q n are its approximants, and χ n : D n := D0, 1/q 3/qn n ) C are explosion unctions at p n /q n. Proposition 6 see igure 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 o positive integers such that 1) lim sup n + qn log An 1. 5 Let be the Siegel disk o P α and n the Siegel disk o the restriction o P α n to. 6 For all non empty open set U, lim in dens U n) 1 n + 2. Proo. Set Note that ε n := α n p n 1) n = q n qn 2A. n + θ) + q n q n 1 qn εn n + 1 qn A n. 5 We think that the condition lim sup qn p log A n 1 is not necessary. 6 n is the largest connected open subset o containing 0, on which P αn is conjugate to a rotation. It is contained in the Siegel disk o P αn

7 QUADRATIC JULIA SETS WITH POSITIVE AREA. 7 Figure 3. Illustration o Prop. 6 or α = θ = [0, 1, 1,...], n = 7 and A n = We see the Siegel disk o P α light grey), the Siegel disk n o the restriction o P α n to dark grey) and the boundary o the Siegel disk o P αn. For ρ < 1, deine X n ρ) := { z C ; } z qn D0, s n ) z qn ε n with s n := ρ qn ρ qn + ε n. 0 ρ X n ρ) D Figure 4. The boundary o a set X n ρ).

8 8 XAVIER BUFF AND ARNAUD CHÉRITAT This domain is star-like with respect to 0 and avoids the q n -th roots o ε n. 7 It is contained but not relatively compact in D0, ρ). For all non empty open set U contained in D0, ρ), lim in dens U Xn ρ) ) 1 n + 2. Since the limit values o the sequence χ n : D n C) are isomorphisms χ : D, Prop. 6 is a corollary o Prop. 7 below. Proposition 7. Under the same assumptions as in Prop. 6, or all ρ < 1, i n is large enough, the Siegel disk n contains χ n Xn ρ) ). Proo. We will proceed by contradiction. Assume there exist ρ < 1 and an increasing sequence o integers n k such that χ nk Xnk ρ) ) is not contained in n k. Extracting a subsequence, we may assume A 1/qn k n k A [1, + ]. To simpliy notations, we will drop the index k. Assume A = 1. Then, any compact K is contained in n or n large enough or a proo, see or example in [ABC], Prop. 2, the remark ollowing Prop. 2 and Theo. 3). Note that X n ρ) D0, ρ) and the limit values o the sequence χ n : D n C) are isomorphisms χ : D. It ollows that or n large enough, χ n Xn ρ) ) χ n D0, ρ) ) χ D0, ρ) ) n. This contradicts our assumption. Assume A > 1. Without loss o generality, increasing ρ i necessary, we may assume that ρ > 1/A. We will show that or ρ < ρ < 1, i n is large enough, the orbit under iteration o P αn o any point z χ n Xn ρ) ) remains in χ n D0, ρ ) ). This will show that χ n Xn ρ) ) n, completing the proo o Prop. 7. Since the limit values o 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 := D0, r n ) and the domain o the map n := χ n D n ) 1 Pαn χ n D n eventually contains any compact subset o D. So, Prop. 7 is a corollary o Prop. 7 below. Proposition 7. Assume 0 1 A < ρ < ρ < 1. I n is large enough, the orbit under iteration o n o any point z X n ρ) remains in D0, ρ ). The rest o section 1.4, is devoted to the proo o Prop It is the preimage by the map z z q n o a disk which is not centered at 0, contains 0 but not ε n.

9 QUADRATIC JULIA SETS WITH POSITIVE AREA A vector ield. It is not enough to compare the dynamics o n with the dynamics o a rotation. Instead, we will compare it with the real) dynamics o the polynomial vector ield ξ n = ξ n z) z := 2iπq nzε n z qn ) z. As we shall see later, the time-1 map o ξ n very well approximates n qn the coeicient 2πq n has been chosen so that their derivatives coincide at 0). For simplicity we will assume that n is even in which case ε n > 0. Note that the polynomial vector ield ξ n is tangent to the boundary o X n ρ), which is thereore invariant by the real) dynamics o ξ n. Figure 5. Some real trajectories or the vector ield ξ n ; zeroes o the vector ield are shown. In order to compare the dynamics o n to that o ξ n, we will work in a coordinate that straightens the vector ield ξ n. Let us irst consider the open set { } z qn Ω n := z C ; D z qn ε n which is invariant by the real low o the vector ield ξ n. The map z zqn : Ω n D z qn ε n is a ramiied covering o degree q n, ramiied at 0. Thus, there is an isomorphism ψ n : Ω n D such that ψn z) ) q n z qn =. z qn ε n We note φ n : D Ω n its inverse and π n : H Ω n \ {0} H is the upper hal-plane) the universal covering given by Then, π n Z) := φ n e 2iπq nε nz ). π nξ n = z.

10 10 XAVIER BUFF AND ARNAUD CHÉRITAT Figure 6. An example o open set Ω n or q n = 3. It is bounded by the black curves. Some trajectories o the vector ield ξ n red in Ω n and green outside). For r < 1, we have X n r) Ω n and the preimage o X n r) is the hal-plane H n r) := { Z C ; ImZ) > τ n r) } 1 with τ n r) := 2πqnε 2 log 1 + ε ) n. 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 ast with respect to q n. More precisely, 1 qn τn r) n + r Working in the coordinate straightening the vector ield. Deinition 4. We say that a sequence B n ) is sub-exponential with respect to q n i qn lim sup Bn 1. n + Proposition 8. Assume r < 1. I 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 qn n and G n to qn 1 n : π n F n = qn n π n and π n G n = qn 1 n π n, F n and G n are periodic o period 1/q n ε n ) and as ImZ) +, we have F n Z) = Z o1) and G n Z) = Z A n + θ) + o1). In addition, the sequences sup Z H nr) Fn Z) Z 1 and sup are sub-exponential with respect to q n. Z H nr) Gn Z) Z + A n + θ

11 QUADRATIC JULIA SETS WITH POSITIVE AREA. 11 Proo. We will use the ollowing theorem o Jellouli see [J1] or [J2] Theo. 1) to show that the domains o n qn and n qn 1 eventually contain any compact subset o D. Theorem Jellouli). Assume P α has a Siegel disk and let χ : D be a linearizing isomorphism. For r < 1, set r) := χ D0, r) ). Assume α n R and b n N are such that b n α n α = o1). 8 For all r 1 < r 2 < 1, i n is suiciently large, r 1 ) { z r 2 ) ; j b n, Pα j n z) r 2 )}. Corollary 2. For all r 1 < r 2 < 1, i n is suiciently large, then or all z D0, r 1 ) and or all j q n, we have j n z) D0, r 2 ). Proo. Choose r 1 and r 2 such that r 1 < r 1 < r 2 < r 2. Let χ : D be a linearizing isomorphism o P α. Set r 1 ) := χ D0, r 1 )) and r 2 ) := χ D0, r 2 )). Since limit values o the sequence χ n : D n C are linearizing isomorphisms χ : D, or n suiciently large, χ n D0, r1 ) ) r 1 ) r 2 ) χ n D0, r2 ) ). It is thereore enough to show that or n large enough, r 1) { z r 2) ; j q n, P j α n z) r 2) }. This is Jellouli s theorem with b n = q n since q n α n α q p n n n + α q n = o1). n + In particular, or r < 1, i n is large enough, then qn n and qn 1 n are deined on X n r). In order to lit them via π n as required, it is enough to show that i n is large enough, then z X n r) \ {0}, qn n z) Ω n \ {0} and qn 1 n z) Ω n \ {0}. The periodicity o F n and G n then ollows rom π n Z + 1 ) = π n Z) q n ε n and the behavior as ImZ) + ollows by computing the derivatives o n qn and n qn 1 at 0. Lemma 1 below asserts that n qn is very close to the identity and bounds the dierence. Lemma 1. There exist a holomorphic unction g n, deined on the same set as n qn, such that n qn z) = z + ξ nz) g n z). For all r < 1, the sequence sup D0,r) g n is sub-exponential with respect to q n. 8 In act, Jellouli s theorem is stated or the sequence αn = p n/q n and b n = oq nq n+1 ) but the adaptation to b n α n α = o1) is straightorward.

12 12 XAVIER BUFF AND ARNAUD CHÉRITAT Proo. The map qn n ixes 0 and the q n -th roots o ε n. This shows that qn n can be written as prescribed. To prove the estimate on the modulus o g n, note that n qn 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 deined on D0, r n ). By the maximum modulus principle, i n is large enough so that r n > maxr, 1/A), we have As n +, sup z r gn z) sup gn z) Bn := sup z r n in ξn z) 2πqn rn 1+qn z =r n and thus Recall that we assume n even, in which case Lemma 2 below asserts that n qn 1 bounds the dierence. 2 z =r n ξn z). qn Bn r n 1. ε n > 0 and q n 1 pn q n = 1 q n mod 1). is very close to the rotation o angle 1/q n and Lemma 2. There exists a holomorphic unction h n, deined on the same set as n qn 1, such that e 2iπ/qn n qn 1 z) = z + ξ n z) h n z). For all r < 1, the sequence sup D0,r) h n is sub-exponential with respect to q n. Proo. The map n coincides with the rotation o angle p n /q n on the set o q n -th roots o ε n and q n 1 p n /q n ) = 1/q n mod1). Thus, e 2iπ/qn n qn 1 z) ixes 0 and the q n -th roots o ε n. This shows that e 2iπ/qn n qn 1 can be written as prescribed. The same method as in lemma 1 yields the bound on h n. Proo o Prop. 8, continued. Now, given r < 1, set ) 1 R n := min, τ n r). q n ε n Note that qn Rn A, min 1 ). n + r Hence, R n increases exponentially ast with respect to q n. For all n and all Z H n r), the map π n is univalent on DZ, 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 Rn Z) 4 = ξ nz) Rn 4. In particular, i the sequence B n ) is sub-exponential with respect to q n and i n is large enough so that B n R n /4, we have z X n r), D z, ξ n z) B n ) Ωn \ {0}. Thereore, it ollows rom lemmas 1 and 2 that or all r < 1, i n is large enough, then z X n r) \ {0}, qn n z) Ω n \ {0} and qn 1 n z) Ω n \ {0}.

13 QUADRATIC JULIA SETS WITH POSITIVE AREA. 13 Lemmas 1 and 2 and Koebe distortion theorem applied to π n : D ) Z, R n C imply that the sequences Fn Z) Z 1 and sup Gn Z) Z + A n + θ sup Z H nr) are sub-exponential with respect to q n. This completes the proo o Prop. 8. Z H nr) We will need the ollowing improved estimate or F n. Proposition 9. Assume r < 1. There exists a sequence B n ), sub-exponential with respect to q n, such that or all Z H n r), Fn Z) Z 1 Bn ε n + εn π n Z) qn ). Proo. Lemma 3 below gives a similar estimate or n qn on X n r). This estimate transers to the required one by Koebe distortion theorem as in the previous proo. Lemma 3. There exist a complex number η n and a holomorphic unction k n, deined on the same set as n qn, such that qn n z) = z + ξ nz) 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 D0, r) kn z) Bn. Proo. By lemma 1, we know that with, B n := qn n n qn z) = z + ξ nz) h n z) sup h n a sub-exponential sequence with respect to q n. The map D0,r) has the same multiplier at each q n -th roots o ε n. It ollows that h n z) = 1 + η n + ε n z qn )k n z) as prescribed. Since qn ε n 1/A < r < 1, the bound on h n, taken at any o the q n -th roots o ε n, shows that or n large enough, and thus z D0, r) 1 + η n B n εn z qn )k n z) 2Bn. As in lemma 1, we have or some sequence r n 1 and or n large enough: sup z r k n z) B n := 2B n rn qn ε n and B n ) is sub-exponential with respect to q n. Looking at z = 0 gives: e 2iπqnεn 1 2iπq n ε n = 1 + η n + ε n k n 0). As n +, the let hand o this equality expands to 1 + iπq n ε n + oq n ε n ). Thereore η n ε n kn 0) + πq n + oq n ) ). Since k n 0) B n, we get the desired bound on η n.

14 14 XAVIER BUFF AND ARNAUD CHÉRITAT Corollary 3. Assume r < 1. Then, F n Z) Z 1 sup Z H nr) 0 and sup n + Z H nr) F n Z) 1 0. n + Proo. The irst is an immediate consequence o Prop. 9. For the second, use the irst on H n r ) with r < r < Iterating the commuting pair F n, G n ). Proposition 10. Assume 1/A < r 1 < r 2 < 1. I n is suiciently large, the ollowing holds. Given any point Z H n r 1 ), there exists a sequence o integers j l ) l 0 such that or any integer l 0 and any integer j [0, j l ], the point F j n G n F j l 1 n is well deined and belongs to H n r 2 ). G n F j1 n G n Fn j0 Z) Proo. We will need to control iterates o F n or a large number o iterates. We will use the ollowing lemma. Lemma 4. Assume F : H C satisies F Z) Z 1 < u ReZ) ) with u : R ]0, 1/10[ a unction such that log u is 1/2-Lipschitz. Let Γ be the graph o an antiderivative o 2u. Then, every Z H which is above Γ has an image above Γ. Proo. Let U be the antiderivative whose graph is Γ. Let Z = X + iy H. The point Z = X + iy = F Z) satisies X [X , X ]. Note that [ x X, X + 11 ], log ux) log ux) Thereore, rom X to X, U decreases o at least X X)2e 11/20 ux) e 11/20 ux) > ux) > Y Y. Z = X + iy Z + 1 Γ ux) Lemma 5. Assume 1/A < r < r < 1. I n is suiciently large, then or all Z H n r) there exists an integer jz) such that or all j jz), we have Fn j G n Z) H n r ) and Re Fn jz) G n Z) ) > ReZ).

15 QUADRATIC JULIA SETS WITH POSITIVE AREA. 15 Proo. Let us irst recall that there exists a sequence B n ), sub-exponential with respect to q n, such that or n large enough, or all Z H n r), Gn Z) Z + A n + θ Bn. In particular, i n is suiciently large, Re G n Z) ) ReZ) A n θ B n and Im G n Z) ) τ n r) B n. We will apply lemma 4 to control the orbit o G n Z) under iteration o F n. More precisely, we will prove the existence o a unction u n such that: a) F n Z) Z 1 ) u n ReZ), b) or n large enough u n ]0, 1/10[, c) or n large enough, log u n is 1/2-Lipschitz and d) the sequence C n := to q n. ReZ) I n is taken suiciently large so as to have ) 2u n X) dx is sub-exponential with respect Re G nz) τ n r) τ n r ) + B n + C n , it then ollows rom lemma 4 that there is an integer jz) such that or all j jz), we have Fn j G n Z) H n r ) and Re Fn jz) G n Z) ) > ReZ). B n H nr) G nz) Z C n F jz) n G nz) H nr ) a) By Prop. 9, there is a sequence B n), sub-exponential with respect to q n, such that or all Z H n r ), Fn Z) Z 1 B n εn + εn π n Z) qn ). Set T n := 1/2πq 2 n ε n) +. We have πn Z) ) q n = ε n 1 e iz/tn. Using B n εn + ) εn π n Z) qn B n 2εn + ) πn Z) qn

16 16 XAVIER BUFF AND ARNAUD CHÉRITAT we see that or all Z H n r ), F n Z) Z 1 ) B n ε 1 n e iz/t n ) B nε 1 n 2 + sn e irez)/tn 1 with s n = 1 + ε n r ). qn Since 1/A < r, we have ε n /r ) qn 0 and thus s n 1. Thus, or n large enough and or all Z H n r ), Fn Z) Z 1 un ReZ) ) s n e irez)/tn 1, with u n X) := b) Let us show that or n large enough u n ]0, 1/10[. Note that X R, Thus u n tends uniormly to 0 as n +. 7B nε n s n e ix/tn 1. un X) 7B nε n s n 1 = 7B nr ) qn 0. n + c) Let us now check that or n large enough, log u n is 1/2-Lipschitz. One computes u nx) u n X) = s n sinx/t n ) T n s 2 n + 1 2s n cosx/t n ). This unction reaches its extrema when s 2 n + 1) cosx/t n) 2s n = 0. It ollows that: u n X) u n X) s n T n s 2 n 1) n + πq2 nr ) qn. Thus, log u nx) converges uniormly to 0 as n +, and or n large enough, X log u n is 1/2-Lipschitz. d) Let us inally show that the sequence C n := ReZ) ) 2u n X) dx Re G nz) is sub-exponential with respect to q n. I n is large enough, Re G n Z) ) ReZ) A n θ n B n 4πT n. Thus, ReZ) πtn C n B n := 2u n X) dx = 4 u n X) dx. ReZ) 4πT n πt n The change o 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.

17 QUADRATIC JULIA SETS WITH POSITIVE AREA. 17 It ollows that B n 28B n n + πqn 2 log 1 s n 1 28B n logr A n ). n + πq n By assumption condition 1) in the statement o Prop. 6), 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. Proo o 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 or n suiciently large, any point o H n r 1 ) has an ininite orbit remaining in H n r 2 ) along a well chosen composition o F n and G n. It is enough to show that this is true or any sequence o points Z n = X n + iy n H n r 1 ). We will use Douady-Ghys-Yoccoz s renormalization techniques and ollow the presentation in [ABC] section 3.2. Step 1. Construction o a Riemann surace: V n. Choose n suiciently large so that F n is deined in the upper hal-plane { Z C ; ImZ) τ n r 2 ) 1/10} with Fn Z) Z 1 1 and F 10 n Z) Set P n := X n + i τ n r 2 ) 1 ). 10 Let L n := { X n + it ; t > ImP n ) } be the vertical hal-line starting at P n and passing through Z n. The union L n [ P n, F n P n ) ] F n L n ) { } orms a Jordan curve in the Riemann sphere bounding a region U n such that or Y > ImP n ), the segment [ iy, F n iy ) ] is contained in U n see [ABC]). We set U n := U n L n. I we glue the sides L n and F n L n ) o U n via F n, we obtain a topological surace V n. We denote by ι n : U n V n the canonical projection. The space V n is a topological surace with boundary, whose boundary ι n [Pn, 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 surace V n has a canonical analytic structure induced by the one o U n. It is possible to show that V n is isomorphic to H/Z D see [Y] or details). Let φ n : V n D be an isomorphism. Hence, we have the ollowing composition: We set φ n ι n : U n D. 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 1/10. 9 This is possible by Cor. 3 applied with r > r2. Indeed, or n large enough, τ nr 2 ) > τ nr) +

18 18 XAVIER BUFF AND ARNAUD CHÉRITAT Let U n be the set o points o U n which are above the segment [ P n, F np n )] and let V n be the image o U n in V n. Choose n suiciently large so that lemma 5 can be applied with r = r 3 and r = r 2. Then, or all Z U n H n r 3 ), there exists an integer jz) such that W := F jz) n G n Z) U n and j [ 0, jz) ] F j n G nz) 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 o1) and G n Z) = Z A n θ + o1) as ImZ) +, it is possible to show that again, see [Y] or details). g n 0) = e 2iπAn+θ) = e 2iπθ L n F n L n ) H n r 1 ) Z n U n H n r 3 ) G n Z) Z P n F n P n) W H n r 2 ) P n U n F n P n ) Step 3. The orbit o ζ n. We will show that the orbit o ζ n under iteration o g n is ininite. For this, let ρ n be the radius o 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 D0, Cρ n ) b) ζ n = oρ n ). a) The restriction o g n to D0, ρ n ) is univalent. It ixes 0 with derivative e 2iπθ. Remember that θ is a Brjuno number. It ollows see [Brj] or [Y] or example) that there is a constant C θ > 0 depending only on θ such that g n has a Siegel disk containing D0, C θ ρ n ). b) Denote by B n the hal-strip B n = {Z C ; 0 < ReZ) < 1 and ImZ) > ImP n )} 10 The act that gn : φ nv n) D is continuous and univalent is not completely obvious; see [Y] or details.

19 QUADRATIC JULIA SETS WITH POSITIVE AREA. 19 and consider the map H n : B n U n deined by H n Z) = 1 X) X n + iy ) + X F n X n + iy ) where Z = X+iY, X, Y ) [0, 1] [ ImP n ), + [. The map H n sends each segment [iy, iy + 1] to the segment [ X n + iy, F n X n + iy ) ]. An elementary computation shows that H n is a 5/4-quasiconormal homeomorphism between B n and U n. 11 Since H n iy + 1) = F n Hn iy ) ), the quasiconormal homeomorphism H n : B n U n induces a homeomorphism between the hal cylinder H/Z and the Riemann surace V n. This homeomorphism is clearly quasiconormal on the image o B n in H/Z, i.e., outside a straight line. It is thereore quasiconormal in the whole hal cylinder R-analytic curves are removable or quasiconormal homeomorphisms). Let R n be the rectangle R n := { Z C ; 0 ReZ) < 1 and ImP n) < ImZ) < ImZ 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. Z n P n H n R n ) Z F nz) ιn A n ρ n φn 0 ζ n U n V n P n H n ImZ n) ImP n ) R n Z Z+1 ImP n) B n H/Z The image o R n in H/Z is an annulus o modulus M n := ImZ n ) ImP n ) τ nr 1 ) τ n r 3 ) 1 10 n For a proo that Hn is 5/4-quasiconormal homeomorphism, see or example [ABC] section 3.2 or [Sh2] section 2.5.

20 20 XAVIER BUFF AND ARNAUD CHÉRITAT Note that H n induces a 5/4-quasiconormal homeomorphism between this annulus and A n. It ollows that modulusa n ) 4 5 M n +. n + Since A n separates 0 and ζ n rom and a point o modulus ρ n in φ n V n), the claim ollows: as n +, ζ n = oρ n ). Step 4. Controlling the orbit o Z n. We know that the orbit o ζ n under iteration o g n is ininite. Thus, we have a sequence ζ n V n Now, or each l 0, we have g n ζ 1 n V n g n ζ 2 n V n g n. ζ l n = φ n ι n Z l n) or some Z l n U n. Moreover, by deinition o g n, there exists an integer j l such that Z l+1 n In other words, ζ l n V 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 ). Z l n U n g n ζ l+1 n V n corresponds to G n Hn r 2 ) Fn H n r 2 ) Fn F n Z l+1 n U n. Thus, or n suiciently large, any point Z n H n r 1 ) has an ininite orbit remaining in H n r 2 ) along a well chosen composition o F n and G n. This completes the proo o Prop. 10. Proo o Prop. 7, continued. Remember that 0 < 1/A < ρ < ρ < 1. Choose r 1 = ρ < r 2 < ρ. By Prop. 10, or n suiciently large, any point Z H n ρ) has an ininite orbit remaining in H n r 2 ) under a well chosen composition o F n and G n. This means that any point z X n ρ) has an ininite orbit remaining in X n r 2 ) under a well chosen composition o n qn and n qn 1. By Cor. 2, i n is suiciently large, we know that any point in X n r 2 ) D0, r 2 ) has its irst q n iterates in D0, ρ ). This shows that any point z X n ρ) has an ininite orbit remaining in D0, ρ ) under iteration o n, as required. In other words, and H n r 2 ) Gn H n r 2 ) corresponds to X n r 2 ) q n 1 n X n r 2 ) H n r 2 ) Fn H n r 2 ) corresponds to X n r 2 ) qn n X n r 2 ). Moreover, or n suiciently large, decompose as X n r 2 ) q n 1 n X n r 2 ) and X n r 2 ) qn n X n r 2 ) X n r 2 ) D0, r 2 ) n D0, ρ ) n n D0, ρ ) n X n r 2 ). This completes the proo o Prop. 7.

21 QUADRATIC JULIA SETS WITH POSITIVE AREA The control o the post-critical set. Deinition 5. We denote by the Hausdor semi-distance: X, Y ) = sup dx, Y ). x X Deinition 6. We denote by PCP α ) the post-critical set o P α : PCP α ) := k 1 P k α ω α ) with ω α := e2iπα 2. This section is devoted to the proo o the ollowing proposition. Remember that S N is the set o irrational numbers o bounded type whose continued ractions have entries greater than or equal to N. Proposition 11. There exists N such that as α S N α S N, we have with α the Siegel disk o P α. PCP α ), α ) 0, The corollary we will use later is the ollowing. Corollary 4. Let α n ) be the sequence deined in Prop. 3. I n is large enough, the post-critical set o P αn is contained in the δ-neighborhood o the Siegel disk o P α. The proo o Prop. 11 will rely on some almost) classical results on Fatou coordinates and perturbed Fatou coordinates. We reer the reader to appendix A and to [Sh2] or more details. The proo will also rely on results o Inou and Shishikura [IS] that we will now recall The class o Inou and Shishikura. Consider the cubic polynomial P z) = z1 + z) 2. This polynomial has a multiple ixed 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 deined by U := P 1 D0, v R) ) \ ], 1] B ), where B is the connected component o P 1 D0, v /R) ) which contains 1. Consider the ollowing class o maps Inou and Shishikura use the notation F2 P in [IS]): IS 0 := { = P ϕ 1 : U C with ϕ : U U isomorphism such that ϕ0) = 0 and ϕ 0) = 1 Remark. The set IS 0 is identiied with the space o univalent maps in U ixing 0 with derivative 1, which is compact. A sequence o univalent maps ϕ n : U C) satisying ϕ n 0) = 0 and ϕ n 0) = 1 converges uniormly to ϕ : U C on every compact subset o U, i and only i the sequence n = P ϕ 1 n ) converges to = P ϕ 1 on every compact subset o U = ϕu). }. A map IS 0 ixes 0 with multiplier 1. The map : U D 0, v R ) is surjective. It is not a proper map. Inou and Shishikura call it a partial covering. The map has a critical point ω := ϕ 1/3) which depends on and a critical value v := 4/27 which does not depend on.

22 22 XAVIER BUFF AND ARNAUD CHÉRITAT U P B 1 3 Figure 7. A schematic representation o the set U. We colored gray the set o points in U whose image by P is contained in the lower hal-plane Fatou coordinates. Near z = 0, elements IS 0 have an expansion o the orm z) = z + c z 2 + Oz 3 ). The ollowing result o Inou and Shishikura is an immediate consequence o the Koebe 1/4-theorem. Result o Inou-Shishikura Main theorem 1 part a). The set {c ; IS 0 } is a compact subset o C. In particular, or all IS 0, c 0 and has a multiple ixed point o multiplicity 2 at 0. I we make the change o variables w = τ z) := 1 c z, we ind F w) = w o1) near ininity. To lighten notation, we will write and F or pairs o unctions related as above; ω := φ 1/3) and ω F := τ 1 ω ) will denote their critical points. Lemma 6. There exists R 0 such that or all IS 0 F is deined and univalent in a neighborhood o C \ D0, R 0 ) and or all w C \ D0, R 0 ), F w) w 1 1 < 4 and Proo. This ollows rom the compactness o IS 0. F w) 1 < 1 4. I R 1 > 2R 0, the regions Ω att := { w C ; Rew) > R 1 Imw) } and Ω rep := { w C ; Rew) < R 1 + Imw) } are contained in C \ D0, R 0 ). Then, or all IS 0, F Ω att ) Ω att and F Ω rep) Ω rep.

23 QUADRATIC JULIA SETS WITH POSITIVE AREA. 23 Ω att τ 0 R 1 R 1 Ω att, Ω rep, Ω rep Figure 8. Right: the sets Ω att and Ω rep. Let: the set Ω att, and Ω rep, or a map with c = 1. The sets Ω att and Ω att, are shaded. The boundaries o the sets Ω rep and Ω rep, are dashed. In addition, there are univalent maps Φ att F : Ωatt C attracting Fatou coordinate or F ) and Φ rep F : Ωrep C repelling Fatou coordinate or F ) such that Φ att F F w) = Φ att F w) + 1 and Φ rep F F w) = Φrep F w) + 1 when both sides o the equations are deined. The maps Φ att F and Φrep F are unique up to an additive constant. Result o Inou-Shishikura Main theorem 1 part a). For all IS 0, the critical point ω is attracted to 0. The ollowing lemma easily ollows, using the compactness o the class IS 0. Lemma 7. There exists k such that or all IS 0 we have F k ω F ) Ω att. Proo. By contradiction, suppose that there is a sequence n ) IS 0 such that or k n we have Fn k ω Fn ) / Ω att. By compactness o IS 0 we may assume that the sequence F n converges to F. But since IS 0, the orbit o the critical point ω converges to 0, so or some k we have F kω F ) Ω att. But and this is a contradiction. Since the maps Φ att F can normalize Φ att F k ω F ) = lim F k n n ω Fn ) and Φrep F are only deined up to an additive constant, we F so that Φ att F so that Then, we can normalize Φ rep F Φ att F F k ω F ) ) = k. w) Φrep F w) 0 when Imw) + with w Ωatt Ω rep. Coming back to the z-coordinate, we deine and we set Ω att, := τ Ω att ) and Ω rep, := τ Ω rep ) Φ att, := Φ att F τ 1 and Φ rep, := Φ rep F τ 1.

24 24 XAVIER BUFF AND ARNAUD CHÉRITAT The univalent maps Φ att, : Ω att, C and Φ rep, : Ω rep, C are called attracting and repelling Fatou coordinates or. Note that our normalization o the attracting coordinates is given by Φ att, k ω ) ) = k. The ollowing result o Inou and Shishikura asserts that the attracting Fatou coordinate can be extended univalently up to the critical point o. It easily ollows rom [IS] Prop Result o Inou-Shishikura see igure 9). For all IS 0, there exists an attracting petal P att, and an extension o the Fatou coordinate, that we will still denote Φ att, : P att, C, such that v P att,, Φ att, v) = 1, Φ att, is univalent on P att, and Φ att, P att, ) = { w ; Rew) > 0 }. v 0 Φ att, ω P att, Figure 9. Let: the attracting petal P att, o some map IS 0 ; the critical point is ω, the critical value v and 0 is a ixed point. Right: its image by Φ att, ; we divided the right hal plane ]0, + [ R into vertical strips o width 1 o alternating color, highlighted the real axis in red, and put a black dot at the point z = 1. On the let, we pulled this coloring back by Φ att,. Deinition 7 see igure 10). For IS 0, we set: V := { z P att, ; Im Φ att, z) ) > 0 and 0 < Re Φ att, z) ) < 2 } and W := { z P att, ; 2 < Im Φ att, z) ) < 2 and 0 < Re Φ att, z) ) < 2 }. We now come to the key result o Inou and Shishikura. The result stated below easily ollows rom [IS] Prop. 5.7 and 5.8 and section 5.M. Our domain V k W k below corresponds in [IS] to the interior o D k D k D k D k+1 D k+1 D k+1.

25 QUADRATIC JULIA SETS WITH POSITIVE AREA. 25 Φ att, The set W k Figure 10. On the right, we divided ]0, 2[ ] 2, + [ into 3 regions o dierent colors. We subdivided each by a vertical line through z = 1. These 6 pieces were then pulled back on the let by Φ att,, or the same parabolic IS 0 as in igure 9. itsel corresponds to the interior o D k D k D k+1 D k+1. Result o Inou-Shishikura see igure 11). For all IS 0 and all k 0, the unique connected component V k o k V ) which contains 0 in its closure is relatively compact in U the domain o ) and k : V k V is an isomorphism and the unique connected component W k o k W ) which intersects V k is relatively compact in U and k : W k W is a covering o degree 2 ramiied above v. In addition, i k is large enough, then V k W k Ω rep,. The ollowing lemma asserts that i k is large enough, then or all map IS 0, the set V k W k is contained in a repelling petal o, i.e. the preimage o a let hal-plane by Φ rep,. Lemma 8 see igure 12). There is an R 2 > 0 such that or all IS 0, the set Φ Ω rep, ) contains the hal-plane {w C ; Re w < R 2 }. There is an integer k 0 > 0 such that or all k k 0, we have V k W k { z Ω rep, ; Re Φ rep, z) ) < R 2 }. Remark. O course, R 2 can be replaced by any R 3 R 2, replacing i necessary k 0 by k 1 := k 0 + R 3 R Proo. For all IS 0, Φ Ω rep, ) contains a let hal-plane. The existence o R 2 ollows rom the compactness o IS 0. By Inou and Shishikura s result, we know that or all IS 0 there is an integer k > 0 such that W k is relatively compact in Ω rep,. It ollows rom the

26 26 XAVIER BUFF AND ARNAUD CHÉRITAT τ 0 Ω rep Figure 11. Let: among the successive preimages o V and W by, those that compose the sets V k, W k are shown. The colors are preserved by. Right: preimage o the let part by τ 0. We hatched W F V F and W 7 F V 7 F. k P att, k V W k V W P rep, Figure 12. I k is large enough, V k repelling petal P rep,. W k is contained in the compactness o IS 0 that there is an integer k 1 > 0 and a constant M, such that or all IS 0, W k1 Ω rep, and sup w W k 1 Set k 0 := k 1 + M + R Then, sup w W k 0 Re Φ rep, w) ) < M. Re Φ rep, w) ) < R 2 2.

27 QUADRATIC JULIA SETS WITH POSITIVE AREA. 27 We will show that we then automatically have 2) V k0 Ω rep, and sup Re Φ rep, w) ) < R 2. w V k 0 It will ollow immediately that k k 0 and w V k W k, Re Φ rep, w) ) < R 2, which will conclude the proo o the lemma. In order to get 2), we ix IS 0 and consider k k 0 large enough so that Ω rep, this is possible thanks to Inou and Shishikura). Note that V k sup w W k Denote by g : V V k Re Φ rep, w) ) < R 2 2 k + k 0. the inverse branch o k : V k V. Set B := { w C ; 0 < Rew) < 2 and 0 < Imw) }. Note that B = Φ att, V ). Consider the map Ψ : B C deined by Ψ := Φ rep, g Φ 1 att,. Since Ψ commutes with translation by 1, so that Ψw) w is 1-periodic, the maximum modulus principle yields Note that and thus Hence, sup w V k It now ollows that sup Re Ψw) w ) = sup Re Ψw) w ). w B w [0,2] g Φ 1 ) att, [0, 2] W k sup Re Ψw) w ) < R 2 2 k + k 0. w [0,2] Re Φ rep, w) ) = sup Re Ψw) ) < R 2 k + k 0. w B sup w V k 0 Re Φ rep, w) ) < R 2. This completes the proo o 2) and o lemma Perturbed Fatou coordinates. For α R, we denote by IS α the set o maps o the orm z e 2iπα z) with IS 0. I A is a subset o R, we denote by IS A the set IS A := IS α. Note that and IS α = IS R = { = P ϕ 1 : U C with { = P ϕ 1 : U C with α A ϕ : U U } isomorphism such that ϕ0) = 0 and ϕ 0) = e 2iπα ϕ : U U isomorphism such that ϕ0) = 0 and ϕ 0) = 1 }.

28 28 XAVIER BUFF AND ARNAUD CHÉRITAT The map depends on φ in a one-to-one way. Thus we get a one-to-one correspondence between IS R and the set o univalent maps on U ixing 0 with derivative 1. We put the compact-open topology on this set o univalent maps. This induces a topology on IS R. Remark. A sequence n = P ϕ 1 n IS R) converges to = P ϕ 1 IS R i and only i the sequence n ) converges to on every compact subset o U = ϕu). I IS [0,1[, we denote by α [0, 1[ the rotation number o at 0, i.e. the real number α [0, 1[ such that 0) = e 2iπα. Lemma 9. There exist ε 0 ]0, 1[ and r > 0 such that or all IS [0,ε0[, the map has two ixed points in D0, r) counting multiplicities), one at z = 0 the other one denoted by σ. The map σ : IS [0,ε0[ D0, r) deined by σ is continuous. Proo. According to Inou and Shishikura, maps IS 0 have a double ixed point at 0. By compactness o IS 0, there is an r > 0 such that maps IS 0 have only 2 ixed points in D0, r ). Choose r ]0, r [. By Rouché s theorem and by compactness o IS 0, there is an ε 0 > 0 such that maps IS [0,ε0[ have exactly two ixed points in D0, r). The result ollows easily. The ollowing results are consequences o results in [Sh2], the compactness o the class IS 0 and the results o the previous paragraph. P ω Φ v α R 3 Figure 13. The perturbed petal P whose image by the perturbed Fatou coordinate Φ is the strip { } 0 < Rew) < 1/α R 3. Proposition 12 see igure 13). There are constants K > 0, ε 1 > 0 and R 3 R 2 with 1/ε 1 R 3 > 1, such that or all IS ]0,ε1[ the ollowing holds. 1) There is a Jordan domain P U a perturbed petal) containing v, bounded by two arcs joining 0 to σ and there is a branch o argument deined on P such that sup argz) in argz) < K. z P z P 2) There is a univalent map Φ : P C a perturbed Fatou coordinate) such that Φ v) = 1,

29 QUADRATIC JULIA SETS WITH POSITIVE AREA. 29 Φ P ) = { w C ; 0 < Rew) < 1/α R 3 }, Im Φ z) ) + as w 0 and Im Φ z) ) as w σ and when z P and Re Φ z) ) < 1/α R 3 1, z) P and Φ z) = Φ z) + 1. For IS 0, we set P rep, := { z Ω rep, ; Re Φ rep, z) ) < R 3 }. 3) I n ) is a sequence o maps in IS ]0,ε1[ converging to a map 0 IS 0, then any compact K P att,0 is contained in P n or n large enough and the sequence Φ n ) converges to Φ att,0 uniormly on K, and any compact K P rep,0 is contained in P n or n large enough and the sequence Φ n 1 α n ) converges to Φ rep,0 uniormly on K. Proo. Thanks to the compactness o the class IS 0, it is enough to show that i n ) is a sequence o maps in IS ]0,1[ converging to a map 0 IS 0, there is a number R 3 R 2 such that properties 1), 2) and 3) hold. So, assume n is such a sequence, and or simplicity, write α n, σ n,... instead o α n, σ n,... Let τ n : C P 1 \ {0, σ n } be the universal covering given by σ n τ n w) := 1 e 2iπαnw so that τ n w) 0 and τ nw) σ n. Imw) + Imw) Denote by T n : C C the translation T n : w w 1. α n Recall that 0 z) = z + c 0 z 2 + Oz 3 ) with c 0 0, and τ 0 z) := 1 c 0 z. The ollowing observations ollow rom [Sh2]. We let R 0 and R 1 be the constants introduced in paragraph ) The sequence τ n ) converges to τ 0 uniormly on every compact subset o C. 2) I n is suiciently large, there is a map F n : D n C, deined and univalent in D n := C \ k Z Dk/α n, R 0 ) which satisies 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 Imw) +. Remark. This lit F n o n may be deined by F n w) := w + 1 ) n z) σ n z log 2iπα n n z) z σ n with z = τ n w).

30 30 XAVIER BUFF AND ARNAUD CHÉRITAT 3) As n tends to +, the sequence F n ) converges to F 0 uniormly on every compact subset o C \ D0, R 0 ). 4) The set { Ω n := w C ; Rew) > R 1 Imw) and Rew) < 1 R 1 + Imw) } α n is contained in D n see igure 14). Ω n R 1 1 αn R 1 R 1 1 αn R1 D n Figure 14. The domain D n grey) is the complement o a union o disks and the hourglass Ω n drak grey) is contained in D n. 5) Remember that or all w C \ D0, R 0 ), F0 w) w 1 < 1 4 and F 0 w) 1 < 1 4. It ollows rom the convergence o F n ) to F 0 that i n is suiciently large, then or all w Ω n, Fn w) w 1 < 1 4 and F n w) 1 < ) Increasing n i necessary, we may assume that 1/α n > 2R Then, there is a univalent map Φ n : Ω n C, called a perturbed Fatou coordinate or 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 0 kω 0) Ω att, with ω 0 the critical point o 0. For n large enough, n k ω n ) is in τ n Ω n ). There is a point w n Ω n such that τ n w n ) = n k ω n ) with w n τ 1 n + 0 k 0 ω 0 ) ). We can normalize Φ n by Φ n w n ) = k. Then, Φ n n + Φatt 0

31 QUADRATIC JULIA SETS WITH POSITIVE AREA. 31 uniormly on every compact subset o Ω att. Due to the normalization Φ att 0 w) Φ rep 0 w) 0 as Imw) + with w Ωatt Ω rep, we have T n Φ n T 1 n n + Φrep 0 uniormly on every compact subset o 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 deine a Fatou coordinate or 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 ollowing. The proo is given in appendix A. It is a consequence o results in [Sh2], but is not stated there. Lemma 10 see igure 15). I K > 0 and R R 2 are suiciently large, then or n large enough: Φ n Ω n ) contains the vertical strip U n := { w C ; R < Rew) < 1/α n R } and τ n is injective on P n := Φ n ) 1 U n ). there is a branch o argument deined on τ n P n ) such that sup argz) z τ np n ) in z τ np n ) argz) < K. τ n P n ) 0 σ n τ n P n Φ n U n 1 R αn R Figure 15. The map τ n is injective on P n := Φ n ) 1 U n ). Let M > R be an integer. Note that { w C ; Rew) > M } Φatt,0 Ω att,0 ) and { w C ; Rew) < M } Φrep,0 Ω rep,0 ).

32 32 XAVIER BUFF AND ARNAUD CHÉRITAT Set and P 0 := { z Ω att,0 ; Re Φ att,0 z) ) > M } { z Ω rep,0 ; Re Φ rep,0 z) ) < M } P n := τ n{ w P n ; M < Re Φ n w) ) < 1/α n M }). For any r > 0, i n is suiciently large so that σ n D0, r), then points with large positive or negative) imaginary part are mapped by τ n in D0, r). It thereore ollows rom point 7) above that P n P 0 as n +. Set P 0 := P att,0 { z Ω rep,0 ; Re Φ rep,0 z) ) < 2M }. Note that P 0 is compactly contained in the domain o 0 M and that 0 M : P 0 P 0 is an isomorphism. In addition, or n suiciently large, n M does not have any critical value in P n. It ollows rom Rouché s theorem that or n large enough, the connected component P n o n M P n) which contains 0 in its boundary is relatively compact in the domain o n, and n M : P n P n is an isomorphism. The perturbed Fatou coordinate Φ n : P n C is deined by Φ n z) := Φ n w) M where w P n is chosen so that τ n w) = M n z) P n. P n z n M z) = τ nw) P n w τ n Φ n Φ n Φ n z) := Φ n w) M Φ n w) Figure 16. Deinition o the perturbed Fatou coordinate Φ n. The perturbed petal P n is grey and the set P n is hatched. In a simply connected neighborhood o P 0 M, the unction 0 z)/z does not vanish and extends by 1 at z = 0). It ollows that or n large enough, there are

33 QUADRATIC JULIA SETS WITH POSITIVE AREA. 33 branches o argument o n M z)/z which are uniormly bounded on P n. It is now easy to check that the proposition holds or n with n large enough Renormalization. Recall that or maps IS 0 we deined sets V P att, and W P att,. We claimed see lemma 8) that or k 0 there are components V k and W k properly mapped by k respectively to V with degree 1 and W with degree 2. In addition, there is an integer k 0 > 0 such that IS 0, V k0 W k0 P rep,. We will now generalize this to maps IS ]0,ε[ with ε suiciently small. IS ]0,ε1[, we set V := { z P ; Im Φ z) ) > 0 and 0 < Re Φ z) ) < 2 } and W := { z P ; 2 < Im Φ z) ) < 2 and 0 < Re Φ z) ) < 2 }. Proposition 13 see igure 17). There is a number ε 2 > 0 and an integer k 1 1 such that or all IS ]0,ε2[ and or all integer k [1, k 1 ], 1) the unique connected component V k o k V ) which contains 0 in its closure is relatively compact in U the domain o ) and k : V k V is an isomorphism, 2) the unique connected component W k o k W ) which intersects V k is relatively compact in U and k : W k W is a covering o degree 2 ramiied above v. 3) V k1 W k1 { z P ; 2 < Re Φ z) ) < 1 α R 3 5 }. I k V W V k W k P Figure 17. I k is large enough, V k perturbed petal P. W k is contained in the

34 34 XAVIER BUFF AND ARNAUD CHÉRITAT Proo. Set k 1 := k By compactness o IS 0, there is an ε 2 > 0 such that or all IS ]0,ε2[, properties 1) and 2) hold or all integers k [1, k 1 ], and urther, W k1 is contained in { z P ; 4 < Re Φ z) ) < 1 α R 3 7 }. To see that V k1 is a subset o { z P ; 2 < Re Φ z) ) < 1 α R 3 5 }, we proceed as in the proo o lemma 8. We now come to the deinition o the renormalization o maps IS ]0,ε2[. Result o Inou-Shishikura Main theorem 3 and section 5.M). I IS ]0,ε2[, the map Φ k1 Φ 1 : Φ V k 1 ) ) W k1 Φ V W projects via w 4 27 e2iπw to a map R) IS 1/α. Deinition 8. The map R) is called the renormalization o. The polynomial P α does not belong to the class IS α. However, the construction we described also works or polynomials P α with α > 0 suiciently close to 0. In other words, i α > 0 is suiciently close to 0, there are perturbed petals and perturbed Fatou coordinates, and there is a renormalization RP α ) which belongs to IS 1/α. In the sequel, ε 2 > 0 is chosen suiciently small so that or α ]0, ε 2 [, a map which either is a polynomial P α, or belongs to IS α, has a renormalization R) IS 1/α 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 or all k 1 }. Assume α = [a 0, a 1, a 2,...] Irrat N. For j 0, set Note that or all j 1, α j := [0, a j+1, a j+2,...]. α j+1 = 1 1 α j The requirement α Irrat N translates into j, α j. α j ]0, 1/N[. Denote by p j /q j the approximants to α 0 given by the continued raction algorithm. Now, i either 0 = P α or 0 IS α, we can deine inductively an ininite sequence o renormalizations, also called a renormalization tower, by j+1 := s R j ) s 1, the conjugacy by s : z z being introduced so that j 0) = e2iπαj. It will be convenient to deine Exp : C C w 4 27 se2iπw ). For j 0, we deine φ j := Exp Φ j : P j C. The map φ j goes rom the j-th level o the renormalization tower to the next level. We now want to relate the dynamics o maps at dierent levels o the renormalization tower. For this purpose, we will use the ollowing lemma.

35 QUADRATIC JULIA SETS WITH POSITIVE AREA. 35 Lemma 11. There is a constant K > 0 such that or all IS ]0,ε2[, there is an inverse branch o Exp which is deined on P and takes its values in the strip { w C ; 0 < Rew) < K }. Proo. This is an immediate consequence o Prop. 12 part 1). From now on, we assume that N is suiciently large so that 1 3) N < ε 1 2 and N R 3 > K. Then, according to lemma 11, or all j 1, there is an inverse branch ψ j o φ j 1 deined on the perturbed petal P j with values in P j 1 there are several possible choices, we choose any one). P j ψ j+1 Φ j Exp P j+1 Figure 18. The branch ψ j+1 maps P j+1 univalently into P j. The map Ψ j := ψ 1 ψ 2... ψ j is then deined and univalent on P j with values in the dynamical plane o the polynomial 0. Remember that Deine P j P j and P j P j by Φ j P j ) = { w C ; 0 < Rew) < 1/α j R 3 }. P j := { z P j ; 0 < Re Φ j w) ) < 1/α j R 3 1 } and P j := { z P j ; 1 < Re Φ j w) ) } < 1/α j R 3. Note that j maps P j to P j isomorphically. Set Q j := Ψ j P j ) and Q j := Ψ j P j).

36 36 XAVIER BUFF AND ARNAUD CHÉRITAT Proposition 14. The map Ψ j conjugates j : P j P j In other words, we have the ollowing commutative diagram: Q j Ψ j P j ) qj 0 Q j Ψ jp j ) to qj 0 : Q j Q j. Ψ j Ψ j P j P j P j P j. j Proo. We must show that i z j P j and z j := jz j ) P j, then the points z 0 := Ψ j z j ) and z 0 := Φ jz j ) are related by z 0 = qj 0 z 0 ). Let us irst show that there is an integer k such that z 0 = k 0 z 0). Our proo is based on the ollowing lemma. Lemma 12. Assume l 0, w U l+1 and w := l+1 w). Let z P l and z P l be such that Exp Φ l z) = w and Exp Φ l z ) = w. Then, there is an integer k 1 such that z = l kz). Proo. Let z 1 P l be the unique point such that Re Φ l z 1) ) ]0, 1] and Exp Φ l z 1) = w. By deinition o the renormalization l+1, there is a point z 1 V k1 l that Exp Φ l z 1 ) = w and k1 l z 1 ) = z 1. We then have Φ l z 1 ) = Φ l z) + m 1 and Φ l z ) = Φ l z 1) + m 1 with m 1 Z and m 1 N. I m 1 0, we have Since k 1 0, we then have z 1 = m1 l z) andz = m 1 l z 1 ). z = k z) with k := k 1 + m 1 + m 1 1. W k1 l such I m 1 < 0, then z = m1 l z 1). However, or m m 1, we have l m z 1) P l, and so, k 1 m Thus, we can write In that case, z 1 = m2 l z) with m 2 := k 1 + m 1 1. z = k z) with k := m 2 + m 1 1. It ollows by decreasing induction on l rom j to 0 that or all z j P j, there is an integer k 1 such that z 0 = k 0 z 0). We will now show that we have a common integer k, valid or all points z j P j.

37 QUADRATIC JULIA SETS WITH POSITIVE AREA. 37 Lemma 13. There is an integer k 0 1 such that or all point z j P j, we have z 0 = k0 0 z 0 ). Proo. We will use the connectivity o P j. For k 1, set O k := {z P j ; 0 k Ψj z) ) is deined } This is an open set. Set X k := { z O k ; k 0 Ψj z) ) = Ψ j j z) ) }. Note that or every component O o O k, either X k O = O, or X k is discrete in O, in particular countable. Indeed, X k is the set o zeroes o the holomorphic unction 0 k Ψ j Ψ j j : O k C. Since P j = X k k 1 there is a smallest integer k 0 1 such that X k0 is not countable. Then, there is a component O o O k0 such that on O, we have k0 0 Ψ j = Ψ j j. Since O is a component o O k0, we have It ollows that O P j C \ O k0. 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 i O P j = since in any neighborhood o a point z C \ O k0, there are uncountably many points in C \ O k0. As a consequence, O = P j, which concludes the proo o the lemma. We must now show that k 0 = q j. Let L j P j be the curve deined by Set L j := jl j ), i.e. the curve L j := { z P j ; Re Φ j z) ) = 1 }. L j := { z P j ; Re Φ j z) ) = 2 }. Those curves both have an end point at z = 0. They both have tangents at z = 0. Since the linear part o j at z = 0 is the rotation o angle α j, the angle between L j and L j at z = 0 is α j. It ollows 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 o k0 0 at z = 0 is the rotation o angle α 0 α 1 α j. It ollows that k 0 = q j. Set D j := V k1 j W k1 j, D j := V j W j, C j := Ψ j D j ) and C j := Ψ j D j). Note that k1 j maps D j to D j. Proposition 15. The map Ψ j k1qj+qj 1) 0 : C j C j. conjugates the map k1 j : D j D j to the map

38 38 XAVIER BUFF AND ARNAUD CHÉRITAT In other words, we have the ollowing commutative diagram: C j Ψ j P j ) k1qj +qj 1) 0 C j Ψ jp j ) Ψ j D j P j k 1 j Ψ j D j P j. Proo. The proo is similar to the one o Prop Neighborhoods o the postcritical set. We can now see that the post-critical set o maps IS α with α Irrat N is ininite. Proposition 16 Inou-Shishikura Cor. 4.2). For all α Irrat N and all IS α, the postcritical set o is ininite. Proo. For j 1, the map k1 j : W k1 j W j is a ramiied covering o degree 2, ramiied above v. Denote by w j the critical point o this ramiied covering. Set w 0 := Ψ j w j ). According to Prop. 15, we can iterate 0 at least k 1 q j + q j 1 times at w 0, w 0 is a critical point o 0 and its critical value is Ψ j v). In particular, Ψ j v) is a point o the postcritical set o 0. Note that v P j. According to Prop. 14, we can iterate 0 at least q j times at Ψ j v). This shows that we can iterate 0 at least q j times at v. Since j 1 is arbitrary, the postcritical set o 0 is ininite. k1qj +qj 1) For every α Irrat N, we are going to deine a sequence U j ) o open sets containing the post-critical set o P α. We still use the notations o the previous paragraph. In particular, or j 1, the j-th renormalization o 0 := P α has a perturbed petal P j, a perturbed Fatou coordinate The set is mapped by k1 j Φ j : P j { w C ; 0 < Rew) < 1/α j R 3 }. to D j := V k1 j W k1 j P j D j := { z P j ; 0 < Re Φ j z) ) < 2 and Im Φ j z) ) > 2 }. There is a map Ψ j, univalent on P j, with values in the dynamical plane o P α, conjugating k1 j : D j D j k1qj +qj 1) to Pα : C j C j with C j := Ψ j D j ) and C j := Ψ jd j ). Deinition 9. For α Irrat N and j 1 we set where l := k 1 R 3 4 N. U j α) := q j+1+lq j k=0 P k α C j ) Figure 19 shows the open set U 1 α) or an α o bounded type. Proposition 17. For all α Irrat N and all j 1, the post-critical set PCP α ) is contained in U j α).

39 QUADRATIC JULIA SETS WITH POSITIVE AREA. 39 Figure 19. I IS α with α Irrat N, the set U 1 ) contains the postcritical set PC). I α is o bounded type, this post-critical set is dense in the boundary o the Siegel disk o. Proo. We will show that or all j 1, there is a point z 0 C j which is a precritical point o P α, and a sequence o positive integers with t 1 < t 2 < t 2 <... such that or all m 1, t m+1 t m < q j+1 + k 1 + R 3 4)q j and Pα tm z 0 ) C j. The proo ollows immediately. Denote by ω j+1 the critical point o j+1. According to Prop. 16 the orbit o ω j+1 under iteration o j+1 is ininite. In particular, or all m 0, j+1 m ω j+1) is in the domain U j+1 o j+1. Remember that the map φ j := Exp Φ j : D j U j+1 is surjective. So, or all m 0, we can ind a point w m D j such that Set φ j w m ) = m j+1ω j+1 ). z m := Ψ j w m ) C j. Then, z 0 is a precritical point o P α and according to lemma 12, there is an increasing sequence t m ) such that z m = Pα tm z 0 ). It is thereore enough to show that or all m 1, t m+1 t m < q j+1 + k 1 + R 3 4)q j. Note that or m 0, w m D j, w m k1 := j w m ) D j. By deinition o the renormalization j+1, we have φ j w m ) = j+1 φj w m ) ) = m+1) j+1 ω j+1 ) = φ j w m ).

40 40 XAVIER BUFF AND ARNAUD CHÉRITAT Thus, Φ j w m+1 ) Φ j w m ) is a positive integer l m. Then, We have w m+1 = lm) j w m ). Re Φ j w m )) 0 and Re Φ j w m+1 ) ) < 1 α j R 3 5. Remember a j+1 = 1/α j. Thus, l m a j+1 R 3 4. Set z m := Ψ jw m ). According to Prop. 14 and 15, we have Thus, z m = P k1qj +qj 1) α z m ) and z m+1 = P lmqj α z m ). 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 ollows immediately rom 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 coeicients o the continued raction are bounded). We will use the additional hypothesis that α has bounded type in order to obtain the ollowing result. Proposition 18. For all α S N, or all ε > 0, i j is large enough, the set U j α) is contained in the ε-neighborhood o the Siegel disk α. Proo. Consider the renormalization tower associated to 0 := P α and let us keep the notations we have introduced so ar. Set Deine Note that D j := aj+1+l) j D j ). N j := a j+1 R 3 1 < 1 α j R 3. D j = { z C ; N j 3 < Re Φ j z) ) < N j 1 and Imw) > 2 }. In particular, D j P j. Set According to Prop. 14 and 15, C j C j := Ψ jd j ). = P qj+1+lqj) α C j ). Lemma 14. There exists M such that or all j 1, the disk D 0, v e 2πM) is contained in the Siegel disk o j. Proo. Let Bα j ) be the Brjuno sum deined by Yoccoz as Bα j ) := + k=0 α j α j+k 1 log 1 α j+k. Since α is o bounded type, there is a constant B such that or all j 1, Bα j ) B. The map j has a univalent inverse branch g j : D 0, v ) C ixing 0 with derivative e 2iπαj. According to a theorem o Yoccoz [Y], there is a constant C,

41 QUADRATIC JULIA SETS WITH POSITIVE AREA. 41 which does not depend on j, such that the Siegel disk o g j centered at 0 with radius v e 2πBαj)+C) v e 2πB+C). The lemma is proved with M := B + C. contains the disk Let us now show that or any ε > 0, or j large enough, C j is contained in the ε-neighborhood o α. Denote by D j the set o points in D j which are mapped by φ j = Exp Φ j in D 0, v e 2πM) and set D j := D j \ D j. In addition, set C j := Ψj D j ) and C j := Ψj D ) j. Points in D 0, v e 2πM) have an ininite orbit under iteration o j+1. It ollows that points in D j have an ininite orbit under iteration o j. Thus, the orbit o 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 o P α, and since it contains 0 in its boundary, C j is contained in the Siegel disk o P α. So, in order to show that C j is contained in the ε-neighborhood o α, it is enough to show that C j is contained in the ε-neighborhood o α. Note that D j is the image o the rectangle { w C ; Nj 3 < Rew) < N j 1 and 2 < Imw) M } by the map Φ 1 j which is univalent on the strip { w C ; 0 < Rew) < 1/αj R 3 }. Since 1 < N j 3 < N j < 1/α j R 3, the modulus o the annulus P j \ D j is bounded rom below independently o j. It ollows rom Koebe s distortion lemma that there is a constant K such that diamc j ) K dzj, z j) where z j := Ψ j Φ 1 j N j 3) and z j := Ψ j Φ 1 j N j 2). According to Prop. 14, z j = Pα Nj 3)qj ω α ) and z j = P qj z j). The boundary o P α is a Jordan curve, and P α : α α is conjugate to the rotation o angle α on R/Z. It ollows that diamc j ) K max Pα qj z) z. z α Without loss o generality, we may assume that M 2. I z U j α), then there is a k q j+1 + lq j such that Pα k z) C j. Then, either Pα k z) C j in which case z α, or Pα k z) C j in which case z belongs to the connected component Oj k o P k ) intersecting α. α C j α

42 42 XAVIER BUFF AND ARNAUD CHÉRITAT In the second case, Oj k contains two points zj k and z j k which are in the boundary o α and which are respectively mapped to z j and z j by P α k. We have z j k = P qj α zj k ). Note that since α is o 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 or all j 1 and all k q j+1 + lq j diamoj k ) K z k j z k j K max P q j α z) z. z α So, we see that sup dz, α ) maxk, K ) max z U jα) This completes the proo o Prop. 18. P q j α z α Assume α R \ Q is o bounded type. I z α, we set r j z) = P q j α z) z. z) z j + 0. Lemma 15. For all α R \ Q o bounded type, all A 1 and all K 1, there exists a K such that the ollowing holds. I j 1, i k A q j, i z 0 α, i z k = Pα k z 0 ) and i O is the connected component o Pα k Dzk, K r j z k )) ) containing z 0, then diamo) K r j z 0 ). Proo. The constants M 1, M 2 and m which will be introduced in the proo 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 zk, 2K r j z k ) ). Since α is a quasicircle and since P α : α α is conjugate to the rotation o angle α on R/Z, the number o critical values o Pα k in D is bounded by a constant M 1 which only depends on α, A and K. k Let O respectively Ô) be the connected component o Pα D) respectively Pα k D)) containing z 0. The degree o Pα k : Ô D is bounded by 2 M1. On the one hand, it easily ollows rom the Grötzsch inequality that the modulus o the annulus Ô \O is bounded rom below by log 2/2π2M ) see or example [ShT] lemma 2.1). On the other hand, it ollows rom Schwarz s lemma that the hyperbolic distance in Ô between z 0 and Pα qj z 0 ) is greater than the hyperbolic distance in D between z k and Pα qj z j ), i.e. a constant m which only depends on α, A and K. Lemma 15 now ollows easily rom the Koebe distortion lemma. Note that or each ixed j, the set U j α) depends continuously on α as long as the irst j + 1 approximants remain unchanged. Hence, given α S N and δ > 0, i α Irrat N is suiciently close to α in particular, the irst j entries o the continued ractions o α and α coincide), then U j α ) is contained in the δ-neighborhood o U j α). This completes the proo o Prop. 11.

43 QUADRATIC JULIA SETS WITH POSITIVE AREA Lebesgue density near the boundary o a Siegel disk. Deinition 10. I α is a Brjuno number and i δ > 0, we denote by the Siegel disk o P α and by Kδ) the set o points whose orbit under iteration o P α remains at distance less than δ o. Our proo will be based on the ollowing theorem o 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 o Kδ). Figure 20. I α = 5 1)/2, the critical point o P α is a Lebesgue density point o the set o points whose orbit remain in D0, 1). Let: the set o points whose orbit remains in D0, 1). Right: a zoom near the critical point. Corollary 5. Assume α is a bounded type irrational and δ > 0. Then d := dz, ) 0 with z = dens Dz,d) C \ Kδ) ) 0. Proo. We proceed by contradiction. Assume we can ind a sequence z j ) such that d j := dz j, ) 0 and ρ j := dens Dzj,d j) C \ Kδ) ) 0. Extracting a subsequence i 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 or i 1, set X i := { w r 1/i) densdw,r) C \ Kδ) ) η }. The sets X i are closed. By McMullen s Theo. 4, X i =. By Baire category, one o these sets X i contains an open subset W o. Then, or all sequence o points w j W and all sequence o real number r j converging to 0, we have 4) lim sup dens Dwj,r j) j + ) ρ C \ Kδ) η = 5. We claim that there is a map g deined and univalent in a neighborhood U o z 0, such that gz 0 ) = w 0 W,

44 44 XAVIER BUFF AND ARNAUD CHÉRITAT g Kδ) U ) = Kδ) gu) and g U) = gu). Indeed, i z 0 is not precritical, we can ind an integer k 0 such that k z 0 ) W and we let g be the restriction o k to a suiciently small neighborhood o z 0. I z 0 is precritical, we can ind a point w 0 W and an integer k 0 such that k w 0 ) = z 0 and we let g coincide the restriction o the branch o k sending z 0 to w 0, to a suiciently small neighborhood o z 0. Let z j be such that z j z j = d j. Then, z j z 0. Let j be suiciently j + large so that z j U and set w j := gz j ). On the one hand, w j w 0. Thus, j + w j W or j large enough. On the other hand, ) 1 dens Dz j,2d j) C \ Kδ) 4 dens ) Dz j,d j) C \ Kδ) and so lim in dens ) ρ Dz j + j,2dj) C \ Kδ) 4. Since g is holomorphic at z 0, lim in j + dens Dw j,r j) This contradicts 4). ) ρ C \ Kδ) 4 with r j := g w 0 ) 2d j j The proo. We will now prove Prop. 3. We let N be suiciently large so that the conclusions o Prop. 11 and Cor. 4 apply. Assume α S N and choose a sequence A n ) such that qn An + and qn log An n + 1. n + Set α n := [a 0, a 1,..., a n, A n, N, N, N,...]. Note that since α is o bounded type, the Julia set J α has zero Lebesgue measure see [P]). Prop. 6 then easily implies that lim in areak αn ) 1 2 areak α). Everything relies on our ability to promote the coeicient 1/2 to a coeicient 1. Denote by K resp. K n ) the illed-in Julia set o P α resp. P αn ) and by resp. n ) its Siegel disk. For δ > 0, set V δ) := { z C dz, ) < δ }, Kδ) := { z V δ) k 0) P k α z) V δ)} and K n δ) := { z V δ) k 0) P k α n z) V δ) }. Deine ρ n : ]0, + [ [0, 1] by ρ n δ) := dens C \ Kn 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 nδ ) + c n The coeicient 3 4 could have been replaced by any λ > 1 2

45 QUADRATIC JULIA SETS WITH POSITIVE AREA. 45 This lemma enables us to complete the proo o Prop. 3 as ollows. We set ρδ) := lim sup ρ n δ) 1). n + Then, or all δ > 0, there is a δ > 0 such that ρδ) 3 4 ρδ ). Since ρ is bounded rom above by 1, this implies that ρ identically vanishes. In other words 5) δ > 0) dens Kn δ) ) n + 1. Since K n δ) K αn, we deduce that dens K αn ) 1. We know that n + P αn converges locally uniormly to P α, the orbit under iteration o P α o any point in K α \ J α eventually lands in and P 1 α n K αn ) = K αn. It ollows that dens Kα\J α K αn ) 1. Since the Julia set J α has Lebesgue n + measure zero, this implies that lim in areak αn ) areak α ). This completes the proo o Prop. 3 up to Lemma 16. Proo o 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 in dens U Kn δ) ) 1. This is an n + 2 immediate consequence o Prop. 6 in section 1.4. B) For all δ > 0, i n is suiciently large, the post-critical set o P αn is contained in V δ). This is just a restatement o Cor. 4 in section 1.5. C) For all η > 0 and all δ > 0, there exists δ 0 ) > 0 such that i δ < δ 0 and i z V 8δ )\V 2δ ), then dens Dz,δ ) C\Kδ) < η. This is an easy consequence o Cor. 5 in section 1.6. Step 1. By Koebe distortion theorem, there exists a constant κ such that or all map φ : D := Da, r) C which extends univalently to Da, 3r/2), we have We choose η > 0 such that sup φ κ in D D φ. 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) i dz, ) < 2δ, then d P α z), ) < 8δ and iii) i z V 8δ ) \ V 2δ ), then dens Dz,δ ) C \ Kδ) ) < η. Indeed, it is well-known that or α R, P α < 4 on Kα. As a consequence, i δ > 0 is suiciently small, then P α < 4 on V 2δ ). It ollows that ii) holds or δ > 0 suiciently small. Claim iii) ollows rom the aorementioned 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 ). 13 Those requirements will be used in step 9.

46 46 XAVIER BUFF AND ARNAUD CHÉRITAT Step 3. Set Y l := { z Kδ) P l α z) }. The set o points in Kδ) whose orbits do not intersect, is contained in the Julia set o P α. This set has zero Lebesgue measure. Thus, Kδ) and Y l coincide up to a set o zero Lebesgue measure. The sequence Y l ) l 0 is increasing. From now on, we assume that l is suiciently large so that ) w W dens Dw,δ )C \ Y l ) < η. Step 4. Assume φ is univalent on Dw, 3δ /2) with w W, r is the radius o the largest disk centered at φw) and contained in φ Dw, δ ) ) and Q is a square contained in φ Dw, δ ) ) with side length at least r/ 8. Set D := Dw, δ ). Then, r in D φ δ and thus, In addition, sup φ κ in D D φ and so, dens Q C \ φy l ) ) area φd \ Y l ) ) As a consequence, areaq) in D φ 2 δ ) 2 8. areaq) dens Q φy l ) ) > 3 4. sup φ 2 πδ ) 2 η D in D φ 2 δ ) 2 /8 1 8πκ2 η < 4. Step 5. I X C is a measurable set, we use the notation m X or the Lebesgue measure on X, extended by 0 outside X. I U C is an open set, X n ) is a sequence o measurable subsets o C and λ [0, 1], we say that lim in n + m X n λ m U i or all non empty open set U relatively compact in U, we have lim in dens U X n) λ. 14 n + Assume : V U is a holomorphic map, nowhere locally constant, and n : V n C) is a sequence o holomorphic maps such that every compact subset o V is eventually contained in V n and the sequence n ) converges uniormly to on every compact subset o V. Then, lim in m X n λ m U = lim in m n + n + n 1 X λ m n) V. Step 6. Set Y l n := { z V δ) j l) P j α n z) V δ) and P l α n z) }. On the one hand, i z Y l n and P l α n z) K n δ), then z K n δ). On the other hand, every compact subset o Y l is eventually contained in Y l n and the sequence 14 Equivalently, or all non empty open set U C with inite area, lim in n + dens U Xn) λ dens U U).

47 QUADRATIC JULIA SETS WITH POSITIVE AREA. 47 P l α n ) converges uniormly to P l α on every compact subset o Y l. By the aorementioned point A), we have So, according to step 5, lim in n + m K nδ) 1 2 m. lim in n + m K nδ) 1 2 m Y l. Step 7. Assume φ n is univalent on Dw n, 3δ /2) with w n W, r n is the radius o the largest disk centered at φ n w n ) and contained in φ n Dwn, δ ) ) and Q n is a square contained in φ n Dwn, δ ) ) with side length at least r n / 8. Then, lim in n + dens Q n φn Kn δ) )) 3 8. Indeed, assume λ is a limit value o the sequence dens Qn φn Kn δ) )). Post-composing the maps φ n with aine maps and extracting a subsequence i necessary, we may assume that w n ) converges to w W, φ n ) converges locally uniormly to φ : Dw, 3δ /2) C, r n converges to the radius r o the largest disk centered at φw) and contained in φ Dw, δ ) ) and Q n converges to a square Q with side length at least r/ 8. According to steps 5 and 6, According to step 4, it ollows that lim in n + m φ nk nδ)) 1 2 m φy l ). λ 1 2 dens Q φy l ) ) 3 8. Step 8. From now on, we assume that n is suiciently large, so that: i) \ K n δ) X n \ K n δ ) with X n := { z k) P k α n z) W } this is possible by step 2); ii) s n < δ with s n := sup d z, K n δ ) ) z this is possible since s n 0 in order or the aorementioned point A) to n + hold); iii) the post-critical set o P αn is contained in V δ /2) this is possible by the aorementioned point B)); iv) i φ is univalent on Dw, 3δ /2) with w W, i r is the radius o the largest disk centered at φw) and contained in φ Dw, δ ) ) and i Q is a square contained in φ Dw, δ ) ) with side length at least r/ 8, then dens Q φ Kn δ) )) 1 4 this is easily ollows rom step 7 by contradiction).

48 48 XAVIER BUFF AND ARNAUD CHÉRITAT Step 9. Assume z 0 X n. Then, we have P αn z 0 X n z1 V 2δ ) Pαn Pαn z k 1 V 2δ ) Pαn z k W or some integer k > 0. Since the post-critical set o P αn is contained in V δ /2), or j k there exists a univalent map φ j : D := Dz k, δ ) C such that φ j is the inverse branch o Pα k j n which maps z k to z j and φ j extends univalently to Dz k, 3δ /2). In particular, sup φ j κ in D D φ j. Let Dz j, r j ) be the largest disk centered at z j and contained in φ j D) and Dz 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, or j k 1, Dz j, r j ) φ j D) C \ K n δ ). On the one hand, dz j, ) < 2δ and on the other hand, every point o is at distance at most s n rom a point o K n δ ). It ollows that R j κr j κ s n + 2δ ). I w 0 φ 0 D) and w j := P j α n w 0 ), then or j k 1, dw j, ) dw j, z j ) + dz j, ) κ s n + 2δ ) + 2δ < 2 + 3κ) δ < δ and or j = k, dw k, ) dw k, z k ) + dz k, ) 9δ < δ. In other words, w 0, w 1,..., w k all belong to V δ). As a consequence, φ 0 Kn δ) ) K n δ). Step 10. Continuing with the notations o step 9, we denote by Q z0 the largest douadic square containing z 0 and contained in Dz 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 Dz 0, r 0 ) V s n ) \ K n δ ). On the other hand, Q z0 has an edge o length greater than r 0 /2 2 and so, according to step 8 point iv), As a consequence dens Qz0 Kn δ) ) > 1 4. dens Qz0 C \ Kn δ) ) < 3 4. Given two douadic squares Q and Q, either Q Q =, or Q Q or Q Q. It ollows that area \ K n δ) ) ) 3 4 area z X n Q z 3 4 area V s n ) \ K n δ ) ) 3 4 area \ K n δ ) ) area V s n ) \ ) = 3 4 area \ K n δ ) ) + c n area )

49 QUADRATIC JULIA SETS WITH POSITIVE AREA. 49 with c n := 3 area V s n ) \ ). 4 area ) Step 11. Since s n 0 and since the boundary o has zero Lebesgue measure, area V s n ) \ ) 0. n + Thus, dens C \ Kn δ) ) < 3 4 dens C \ Kn δ ) ) + c n with c n n + 0. This completes the proo o Lemma The linearizable case In order to ind a quadratic polynomial with a linearizable ixed point and a Julia set o positive area, we need to modiy the argument. Deinition 11. I α is a Brjuno number, we denote by α the Siegel disk o P α and by r α its conormal radius. For ρ r α, we denote by α ρ) the invariant sub-disk with conormal radius ρ and by L α ρ) the set o points in K α whose orbits do not intersect α ρ). Figure 21. Two sets L α ρ) and L α ρ), with α a well-chosen perturbation o α as in Prop. 19. This proposition asserts that i α and α are chosen careully enough, the loss o measure rom L α ρ) to L α ρ) is small. We colored white the basin o ininity, the invariant subdisks α ρ) and α ρ) and their preimages; we colored light grey the remaining parts o 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 o bounded type irrationals such that or 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) ).

50 50 XAVIER BUFF AND ARNAUD CHÉRITAT Proo. We let N be suiciently large so that the conclusions o 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 qn lim An = r α n + 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 o P α. Let us use the notations V δ), Kδ) and K n δ) introduced in section 1.7. With an abuse o notations, set ρ) := α ρ) and n ρ) := αn ρ). Set ρ) := Pα 1 ) ρ) \ ρ). Then, ρ) and ρ) are symmetric with respect to the critical point o P α. The orbit under iteration o P α o a point z / ρ) lands in ρ) i and only i it passes through ρ). We have a similar property or 1 n ρ) := Pα n n ρ) ) \ n ρ). We have proved see equation 5) that δ > 0) dens Kn δ) ) n + 1. The sequence o compact sets n ρ) ) converges to ρ) or the Hausdor topology on compact subsets o C, because lim r αn > ρ. It immediately ollows that or all δ > 0, dens \ ρ) Kn δ) \ n ρ) ) 1. n + Choose δ suiciently small so that V δ) does not intersect ρ). Then, or n large enough V δ) does not intersect nρ). In that case, the orbit under iteration o P αn o a point in K n δ) \ n ρ) cannot pass through n ρ) and so, Thus, K n δ) \ n ρ) L αn ρ). dens \ ρ) Lαn ρ) ) 1. n + The points o L α ρ) whose orbits do not intersect \ ρ) are contained in the union o the Julia set J α and the countably many preimages o ρ). Thus, they orm a set o zero Lebesgue measure. It ollows that area L αn ρ) ) area L α ρ) ). n + Proo o Theo. 2. We start with α 0 S and set ρ 0 := r α0. We then choose ρ ]0, ρ 0 [ and two sequences o real numbers ε n in 0, 1) and ρ n in 0, ρ 0 ) such that 1 εn ) > 0 and ρ n ρ > 0. We can construct inductively a Cauchy sequence α n S) such that or all n 1, r αn ρ n, ρ n 1 ) and area L αn ρ) ) 1 ε n )area L αn 1 ρ) ).

51 QUADRATIC JULIA SETS WITH POSITIVE AREA. 51 Let α be the limit o the sequence α n ). The conormal radius o a ixed Siegel disk depends upper semi-continuously on the polynomial a limit o linearizations linearizes the limit). So, r α lim r αn = ρ. Also, by choosing α n suiciently 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 o pointed domains αn ρ), 0 ) converges or the Carathéodory topology to α, 0). In particular, every compact subset o α is contained in αn ρ) or n large enough. Similarly, every compact subset o C \ K α is contained in C \ K αn or n large enough. It ollows that lim sup L αn ρ) := L αn ρ) L α ρ). m n m Since r α = ρ, α ρ) = α and L α ρ) = J α. Thus, lim sup L αn ρ) J α and areaj α ) arealim sup L αn ρ)) area L α0 ρ) ) 1 ε n ) > The ininitely renormalizable case In order to ind an ininitely renormalizable quadratic polynomial with a Julia set o positive area, we need a modiication based on Sørensen s construction o an ininitely renormalizable quadratic polynomial with a non-locally connected Julia set. Proposition 20. There exists a set S o bounded type irrationals such that or 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 areak α ) 1 ε)areak α ). Figure 22. Two illed-in Julia sets K α and K α, with α a wellchosen perturbation o α as in Prop. 20. This proposition asserts that i α and α are chosen careully enough, P α has a periodic Siegel disk and the loss o measure rom K α to K α is small. Let: we hatched the ixed Siegel disk. Right: we hatched the cycle o Siegel disks.