  We study the post-critical set of a class of holomorphic systems with an
irrationally indifferent fixed point. We prove a trichotomy for the topology of
the post-critical set based on the arithmetic of the rotation number at the
fixed point. The only options are Jordan curves, a one-sided hairy Jordan
curves, and Cantor bouquet. This explains the degeneration of the closed
invariant curves inside the Siegel disks, as one varies the rotation number.
