The golden mean Siegel disk | its conjugate
This is a non matable example. Moreover, since we took a polynomial | its conjugate, the equator remains straight in the process.
The folder CritPtCentered shows it centered on both critical points. It's like a zoom on the asymptotically selft similar critical point of each Julia sets.
The folder IndFixPtCentered shows it centered on the indifferent fixed point. The animation shows only one hemisphere, flattened to a disk by a Möbius map. Now the Siegel disks tend to fill all available space.