We study the attractors of a class of holomorphic systems with an irrationally indifferent fixed point. We prove a trichotomy for the topology of the attractor based on the arithmetic of the rotation number at the fixed point. That is, the attractor is either a Jordan curve, a one-sided hairy circle, or a Cantor bouquet. This has a number of remarkable corollaries on a conjecture of M. Herman about the optimal arithmetic condition for the existence of a critical point on the boundary of the Siegel disk, and a conjecture of A. Douady on the topology of the boundary of Siegel disks. Combined with earlier results on the topic, this completes the topological description of the behaviors of typical orbits near such fixed points, when the rotation number is of high type.