The high type quadratic Siegel disks are Jordan domains

Abstract

Let $\alpha$ be an irrational number of sufficiently high type and suppose $P_{\alpha }(z)=e^{2\pi \text{i}\alpha }z+z^2$ 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 $-e^{2\pi \text{i}\alpha }/2$ if and only if $\alpha$ is a Herman number.