File:InfoldingSiegelDisk.gif

This animation shows a series of Siegel disks. These are the Siegel disks of maps \(P_\theta(z) = e^{i2\pi\theta} z +z^2\) with \(\theta\) taking the following successive values:

• \(\theta=[0;1,2,4,4,\tau]\)
• \(\theta=[0;1,2,4,10,\tau]\)
• \(\theta=[0;1,2,4,10^2,\tau]\)
• \(\theta=[0;1,2,4,10^3,\tau]\)
• \(\theta=[0;1,2,4,10^4,\tau]\)
• \(\theta=[0;1,2,4,10^5,\tau]\)
• \(\theta=[0;1,2,4,10^6,\tau]\)
• \(\theta=[0;1,2,4,10^7,\tau]\)

here, \(\tau\) is some fixed bounded type number, arbitrarily chosen to be \(\tau=[4;4,1,2,4,4,4,4,1,1,1,1,\ldots]\). The last frame shows a superposition of all the pictures. The Siegel disks have been translated in the plane so that the critical point is always at the same point on the screen (near the top).

Notation\[[a_0;a_1,a_2,\ldots] = a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\ddots}}\].

http://i.creativecommons.org/l/by-sa/3.0/88x31.png This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. Attribution: Arnaud Chéritat.