If a holomrphic function can be locally linearized near an irrational indifferent fixed point, then we call the maximal linearization domain containing this fixed point a Siegel disk. Since Siegel proved the exitence of the Siegel disks in 1942, much study has been done on the topological and geometric properties of the Siegel disks. In this talk we will give a survey of the results on the Siegel disks, including three conjectures raised by Douady, Sullivan and Herman in 1980s, and some recent developments on these conjectures.

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