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\textup{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\textup{i}\alpha}/2 if and only if \alpha is a Herman number.