We prove the existence of quadratic polynomials having a Julia set with positive Lebesgue measure. We find such examples with a Cremer fixed point, with a Siegel disk, or with infinitely many satellite renormalizations.

IntroductionAssume that U is an open subset of C and f: U-+C is a holomorphic map which satisfies f(0)=0 and f'(0)=e 2i~, aER/Z. We say that f is linearizable at 0 if it is topologically conjugate to the rotation R~: z~-~e2i~C~z in a neighborhood of 0. If f: U-+C is linearizable, there is a largest f-invariant domain AcU containing 0 on which f is conjugate to the rotation R~. This domain is simply-connected and is called the Siegel disk of f.A basic but remarkable fact is that the conjugacy can be taken holomorphic.In this article, we are mainly concerned with the dynamics of the quadratic polynomials P~: Z~--~e2i~rC~z~-z2, with c~ER\Q. They have z=0 as an indifferent fixed point.For every aER\Q, there exists a unique formal power series such that r = z+b2z2+baza+... r = p~orWe denote by r~ ~>0 the radius of convergence of the series r It is known (see [Y1], for example) that r~>0 for Lebesgue almost every c~ER. More precisely, r~>0 if and only if c~ satisfies the Bruno condition (see Definition 2 below).From now on, we assume that r~>0. In that case, the map r r~)--+C is univalent, and it is well known that its image As coincides with the Siegel disk of P~ associated to the point 0. The number r~ is called the conformal radius of the Siegel disk. The Siegel disk is also the connected component of C\J(P~) which contains 0, where J(P~) is the Julia set of P~, i.e., the closure of the set of repelling periodic points. Figure 1 shows the Julia sets of the quadratic polynomials P~, for c~ = v~ and c~--v/~. Both polynomials have a Siegel disk colored grey.In this article, we investigate the structure of the boundary of the Siegel disk. It is known since Fatou that this boundary is contained in the closure of the forward orbit A. AVILA, X. BUFF AND A. CHI~RITAT Fig. 1. Left: the Julia set of the polynomial z~-~.e2i'V'~z+z 2. Right: the Julia set of the polynomial z~-~e2in'fV6z+z 2. In both cases, there is a Siegel disk. 1 2~.~ (for example, see [Mi, Theorem 11.17] or [Mi, Corolof the critical point w~= -h e lary 14.4]). By plotting a large number of points in the forward orbit of w~, we should therefore get a good idea of what those boundaries look like. In practice, that works only when c~ is sufficiently well-behaved, the number of iterations needed being otherwise enormous.In 1983, Herman [Hell proved that when a satisfies the Herman condition, the critical point actually belongs to the boundary of the Siegel disk. (Recall that Herman's condition is the optimal arithmetical condition to ensure that every analytic circle diffeomorphism with rotation number a is analytically linearizable near the circle. We will not give a precise description here. See [Y2] for more details.) Using a construction due to Ghys, Herman [He2] also proved the existence of quadratic polynomials P~ for which the boundary of the Siegel disk is a quasicircle which does not contain the critical point. Later, following an idea of Douady [D] and using work of Swi~tek [Sw] (see also [Pt]), he proved that when a is Diophantine of exponent 2, the b...

If α is an irrational number, Yoccoz defined the Brjuno function Φ bywhere α 0 is the fractional part of α and α n+1 is the fractional part of 1/α n .The numbers α such that Φ(α) < +∞ are called the Brjuno numbers. The quadratic polynomial P α : z → e 2iπα z + z 2 has an indifferent fixed point at the origin. If P α is linearizable, we let r(α) be the conformal radius of the Siegel disk and we set r(α) = 0 otherwise.Yoccoz [Y] proved that Φ(α) = +∞ if and only if r(α) = 0 and that the restriction of α → Φ(α) + log r(α) to the set of Brjuno numbers is bounded from below by a universal constant. In [BC2], we proved that it is also bounded from above by a universal constant. In fact, Marmi, Moussa and Yoccoz [MMY] conjecture that this function extends to R as a Hölder function of exponent 1/2. In this article, we prove that there is a continuous extension to R.

Abstract. In the family of quadratic polynomials with an irrationally indifferent fixed point, we show the existence of Siegel disks with a fine control on the degree of regularity of the linearizing map on their boundary. A general theorem is stated and proved. As a particular case, we show that in the quadratic family, there are Siegel disks whose boundaries are C n but not C n+1 Jordan curves.

Abstract. If α is an irrational number, we let {p n /q n } n≥0 , be the approximants given by its continued fraction expansion. The Bruno series B(α) is defined asqn .The quadratic polynomial Pα : z → e 2iπα z + z 2 has an indifferent fixed point at the origin. If Pα is linearizable, we let r(α) be the conformal radius of the Siegel disk and we set r(α) = 0 otherwise. Yoccoz proved that if B(α) = ∞, then r(α) = 0 and Pα is not linearizable. In this article, we present a different proof and we show that there exists a constant C such that for all irrational number α with B(α) < ∞, we haveTogether with former results of Yoccoz (see [Y]), this proves the conjectured boundedness of B(α) + log r(α).