File:GoldenMeanExponentialSiegelDisk 4.png

The golden mean Siegel disk for the exponential map z -> exp(z)+kappa (with adequate value of kappa), plus a few invariant circles. The fixed point is marked as a red dot. The strands that seem to grow from the Siegel disk belong to it. They have non-zero width, since the Siegel disk is an open set, but are extremely thin. There are infinitely many of them, and they are dense in Siegel disk. It is conjectured that all of them tend to the right to infinity. Contrarily to what the picture may suggest, they do not reconnect, because the Siegel disk is a simply connected set: in other words, the connected components of its complement (the white components) are all unbounded.

The drawing method was adapted in order to try to respect the following convention with respect to sets: a pixel is lit if it lies not too far away from the set. For this, a special effort was paid to desing an algorithm that would draw the strands. For a picture without the strands, see this image.

So, what is seen is rather a thickened neighborhood of the set. Indeed, if we had respected the following convention:a pixel is lit proprotionally to the area of the set passing through it, then the strands would be invisible.

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.