Siegel disks with critical points in their boundaries
Let \(f\) be a holomorphic function in a domain \(V \subset \mathbb{C}\) with the neutral fixed-point \(0 \in V\), and let \(S \subset V\) be a maximal domain on which \(f\) is linearizable. This means that there is a conformal map \(h\) from the unit disk \(\mathbb{D}\) onto \(S\) such that \(h \circ R_\sigma = f \circ h\), where \(R_\sigma(z) := e^{2\pi i\sigma}z\) and \(f'(0)=e^{2\pi i\sigma}\). Assume that \(S\) is compactly contained in \(V\). Then \(S\) is called a proper Siegel disk for \(f\) with rotation number \(\sigma\). The existence of Siegel disks was proved by C. Siegel in 1942 [Ann. Math. (2) 43, 613--616 (1942; Zbl 0138.31402)] provided that \(\sigma\) satisfies the following Diophantine condition: There exist constants \(\alpha \geq 0\) and \(K>0\) such that \(| q\sigma-p| \geq Kq^{-1-\alpha}\) for all positive coprime integers \(p\) and \(q\). A number \(\sigma\) is called of bounded type if this condition holds with \(\alpha=0\). \textit{P.~Fatou} [C. R. 168, 501--502 (1919; JFM 47.0298.02)] already proved in 1919 that for a rational function \(f\), the boundary of any Siegel disk is contained in the closure of the forward orbit of the set of critical points. A point \(c \in \mathbb{C}\) is a critical point of \(f\) if \(f'(c)=0\). It is clear that no critical point can be contained in \(S\). A.~Douady in 1980 and D. Sullivan in 1981 raised the question whether the boundary of a Siegel disk contains a critical point. Several authors contributed to this question, for example \textit{A. Douady} [Astérisque 152/153, 151--172 (1987; Zbl 0638.58023)], \textit{É. Ghys} [C. R. Acad. Sci. Paris, Sér. I 298, 385--388 (1984; Zbl 0573.58021)], and \textit{M. R. Herman} [Commun. Math. Phys. 99, 593--612 (1985; Zbl 0587.30040)]. Here, the authors prove that, if \(S\) is a Siegel disk of \(f\) with a rotation number of bounded type such that \(S\) is compactly contained in the domain of holomorphy of \(f\), then \(\partial S\) contains a critical point of \(f\). In particular, this is true if \(f\) is an entire function and \(S\) is a bounded domain. The main idea of the proof is to avoid the standard technique of quasiconformal deformations. Instead, the authors estimate a normalized Schwarzian derivative and apply the Ahlfors-Weill theorem.
scientific article; zbMATH DE number 1991042