Consider an analytic function f which has a Siegel disk properly contained in the domain of holomorphy. We prove that if the rotation number is of bounded type, then f has a critical point in the boundary of the Siegel disk.
Jacek Graczyk. Grzegorz Świątek. "Siegel disks with critical points in their boundaries." Duke Math. J. 119 (1) 189 - 196, 15 July 2003. https://doi.org/10.1215/S0012-7094-03-11916-X