We show that there exist polynomial endomorphisms of C 2 , possessing a wandering Fatou component. These mappings are polynomial skewproducts, and can be chosen to extend holomorphically of P 2 (C). We also find real examples with wandering domains in R 2 . The proof relies on parabolic implosion techniques, and is based on an original idea of M. Lyubich.

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.

Let f : P 1 → P 1 be a rational map with finite postcritical set P f . Thurston showed that f induces a holomorphic map σ f : Teich(P 1 , P f ) → Teich(P 1 , P f ) of the Teichmüller space to itself.

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Given a complex number λ of modulus 1, we show that the bifurcation locus of the one parameter family {f b (z) = λz + bz 2 + z 3 } b∈C contains quasi-conformal copies of the quadratic Julia set J (λz + z 2 ). As a corollary, we show that when the Julia set J (λz + z 2 ) is not locally connected (for example when z → λz + z 2 has a Cremer point at 0), the bifurcation locus is not locally connected. To our knowledge, this is the first example of complex analytic parameter space of dimension 1, with connected but non-locally connected bifurcation locus. We also show that the set of complex numbers λ of modulus 1, for which at least one of the parameter rays has a non-trivial accumulation set, contains a dense G δ subset of S 1 .

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.

This note will study a family of cubic rational maps which carry antipodal points of the Riemann sphere to antipodal points. It focuses particularly on the fjords, which are part of the central hyperbolic component but stretch out to infinity. These serve to decompose the parameter plane into subsets, each of which is characterized by a corresponding rotation number.