Bounded type Siegel disks of finite type maps with few singular values

Let U be an open subset of the Riemann sphere \(\hat {\mathbb{C}}\). We give sufficient conditions for which a finite type map f: U → \(\hat {\mathbb{C}}\) with at most three singular values has a Siegel disk compactly contained in U and whose boundary is a quasicircle containing a unique critical point. The main tool is quasiconformal surgery à la Douady-Ghys-Herman-?wi?tek. We also give sufficient conditions for which, instead, Δ has not compact closure in U. The main tool is the Schwarzian derivative and area inequalities à la Graczyk-?wi?tek.     

How regular can the boundary of a quadratic Siegel disk be?

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 but not Jordan curves.

     

Nouvelle preuve d'un théorème de Yoccoz

If θ is an irrational real number such that ∑(lnq_n+1)/q_n=+∞ where p_n/q_n are the convergents of θ, then the quadratic polynomial $\text{P}{}_{\mathrm{\theta }}\text{(z)=}\text{e}{}^{\mathrm{<mtext>i</mtext><mtext>2\pi \theta </mtext>}}\text{z+z}{}^2$ is not linearizable at 0. This theorem has been proved in 1988 by J.C. Yoccoz, who first constructs a nonlinearizable germ by inverting a renormalisation procedure, and then proves universality of the quadratic family for that question. We give an alternative proof, based on the study of the explosion of parabolic cycles. To cite this article: A. Chéritat, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Ensembles de Julia quadratiques de mesure de Lebesgue strictement positive

In this Note, we prove a variant of a conjecture stated in the thesis of Chéritat. The proof is based on results announced by Inou and Shishikura, and on earlier results of McMullen and of Chéritat. According to Chéritat's thesis, this allows us to complete a plan initiated by Douady and to show that there exist quadratic polynomials having a Julia set of positive Lebesgue measure. To cite this article: X. Buff, A. Chéritat, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Relatively compact Siegel disks with non-locally connected boundaries

We construct holomorphic maps f with a Siegel disk whose boundary is not locally connected (and is an indecomposable continuum), yet compactly contained in the domain of definition of the map. Our examples are injective and defined on a subset of \mathbb C{\mathbb C}.