All bounded type Siegel disks of rational maps are quasi-disks (Q636830)

The author proves that every bounded type rotation number Siegel disk of a rational map must be a quasi-disk with at least one critical point on its boundary. This extends a result of Shishikura (unpublished) from the setting of polynomials to that of rational maps. It was conjectured by Douady and Sullivan that the boundary of any Siegel disk of a rational map is a Jordan curve [\textit{A. Douady}, ``Systèmes dynamiques holomorphes'', Sémin. Bourbaki, 35e année, 1980/81, Exp. 599, Astérisque 105--106, 39--63 (1983; Zbl 0532.30019)], so the above result verifies this conjecture in the case of bounded type rotation number. This result is deduced from the following main theorem of the present article: Let \(d \geq 2\) be an integer and \(0 < \theta < 1\) be an irrational number of bounded type. Then there exists a constant \(1 < K(d, \theta ) < \infty\) depending only on \(d\) and \(\theta\) such that for any rational map \(f\) of degree \(d\), if \(f\) has a fixed Siegel disk with rotation number \(\theta\), then the boundary of the Siegel disk is a \(K(d, \theta )\)-quasi-circle which passes through at least one critical point of \(f\). The idea of the proof is to show that the invariant curves of such a Siegel disk are uniform quasicircles, with bound depending only on the degree and the rotation number. To establish this, the author extends a result of Herman by means of the following result (Theorem B). To set up the statement, suppose \(d \geq 2\) is an integer, \(m=2d-1\), and \(\theta \in \mathbb{R}\) is of bounded type. ``Let \(\mathbf{B}_\theta^m\) denote the class of all the Blaschke products \[ B(z) = \lambda\prod^d_{i=1}\frac { z - p_i} {1 - \overline{p}_i z}\prod^{d -1}_{j=1}\frac {z - q_j}{1 - \overline{q}_j z}\tag{1} \] such that 1. \(|p_i | < 1\) and \(|q_j | > 1\) for all \(1 \leq i \leq d\) and \(1 \leq j \leq d - 1\), 2. \(|\lambda| = 1\), 3. \(B|_{\mathbb{T}} : \mathbb{T} \rightarrow\mathbb{T}\) is a circle homeomorphism of rotation number \(\theta\). The family \(\mathbf{B}_\theta^m\) contains a distinguished subset consisting of centered Blaschke products -- those for which the unique invariant nonatomic probability measure on the circle has conformal barycenter (in the sense of \textit{A. Douady} and \textit{C. J. Earle} [``Conformally natural extension of homeomorphisms of the circle'', Acta Math. 157, 23--48 (1986; Zbl 0615.30005)]) located at the origin. Theorem B. Let \(m \geq 3\) be an odd integer and \(\theta = [a_1 , \dots, a_n , \dots]\) be a bounded type irrational number. Then there is a constant \(1 < M(m, \theta ) < \infty\) depending only on \(m\) and \(\theta\) such that for any centered Blaschke product \(B\) in \(\mathbf B^m_\theta\), the map \[ h_B : \mathbb{T} \rightarrow\mathbb{T} \] is an \(M(m, \theta )\)-quasisymmetric homeomorphism, where \(h_B : \mathbb{T} \rightarrow \mathbb{T}\) is the circle homeomorphism such that \(B|_{\mathbb{T}} = h^{-1}_B \circ R_\theta \circ h_B\) and \(h_B (1) = 1\). The paper is very clearly written.