On David type Siegel Disks of the Sine family

Abstract: In 2008 Petersen posed a list of questions on the application of trans-quasiconformal Siegel surgery developed by Zakeri and himself. In this paper we extend Petersen-Zakeri's idea so that the surgery can be applied to all the premodels which have no "free critical points". We explain how the idea is used in solving three of the questions posed by Petersen. To present the details of the idea, we focus on the solution of one of them: we prove that for typical rotation numbers $0< \theta < 1$, the boundary of the Siegel disk of $f_{\theta}(z) = e^{2 \pi i \theta} \sin (z)$ is a Jordan curve which passes through exactly two critical points $\pi/2$ and $-\pi/2$.