Siegel disks with smooth boundaries (Q557394)

Let \(P_\alpha\) be the quadratic polynomial \(P_\alpha(z) = e^{2i\pi\alpha}z + z^2\). Recall that \(P_\alpha\) is conjugated to the rotation \(R_\alpha(z) = e^{2i\pi\alpha}z\) in a neighbourhood \(U\) of \(0\) if and only if \(\alpha\) is a Bruno number. By definition, the Siegel disk \(D_\alpha\) associated to \(P_\alpha\) is the largest invariant domain \(U\) on which \(P_\alpha\) is linearizable. If the conjugation map \(\phi_\alpha : D(0,r_\alpha) \to D_\alpha\) is normalized by \(\phi(0)=0\) and \(\phi'(0) = 1\), we call \(r_\alpha\) the conformal radius. In this article, the authors show that there exist many polynomials \(P_\alpha\) whose Siegel disk has a smooth boundary. More precisely, the set of such \(\alpha\) is dense in \(\mathbb{R}\) and has uncountable intersection with any open subset of \(\mathbb{R}\). Note that in a preceding work, \textit{R. Pérez-Marco} [Acta Math. 179, 243--294 (1997; Zbl 0914.58027)] proved an analog for the class of univalent maps in the unit disk. The theorem of the article in review is a consequence of the following fundamental lemma: for any Bruno number \(\alpha \) and \(0 < r_1 < r_\alpha\), there exists a sequence of Bruno numbers \((\alpha_n)_n\) such that \(\lim \alpha_n = \alpha\) and \(\lim r_{\alpha_n} = r_1\). The article contains two proofs of this lemma. The first one (due to X. Buff and A. Chéritat) consists in proving lower and upper semicontinuity properties for the conformal radius. The techniques used are the Douady-Ghys renormalization and the parabolic explosion. In the second proof (due to A. Avila), the parabolic technique is replaced by the Yoccoz's theorem on the optimality of the Bruno condition for the linearization problem in the quadratic family.