The high type quadratic Siegel disks are Jordan domains

Abstract

Let $\alpha$ be an irrational number of sufficiently high type and suppose $P_{\alpha }(z)=e^{2\pi i\alpha }z+z^2$ has a Siegel disk ${\Delta }_{\alpha }$ centered at the origin. We prove that the boundary of ${\Delta }_{\alpha }$ is a Jordan curve, and that it contains the critical point $-e^{2\pi i\alpha }/2$ if and only if $\alpha$ is a Herman number.

arXiv comment50 pages, 8 figures; to appear in J. Eur. Math. Soc

References in corpus (5)

• A Proof of the Marmi-Moussa-Yoccoz conjecture for rotation numbers of high type
• Satellite renormalization of quadratic polynomials
• An answer to a question of Herman, Baker and Rippon concerning Siegel disks
• Herman's condition and critical points on the boundary of Siegel disks of polynomials with two critical values
• Polynomial Siegel disks are typically Jordan domains

Cited by in corpus (3)

• Area of Julia sets of non-renormalizable cubic polynomials
• Siegel disks of the tangent family
• Area and Hausdorff dimension of Sierpiński carpet Julia sets