Fixed Fatou component classification

A periodic Fatou component of a rational map is an immediate attracting basin, an immediate parabolic basin, a Siegel disc, or a Herman ring. Attracting and parabolic basins capture critical orbits, while the boundaries of rotation domains lie in the closure of the postcritical set.

Every immediate basin of an attracting fixed point of a rational map contains a critical point. Otherwise the map on the basin would be an unbranched covering; lifting it to the unit disc would give an automorphism fixing a point, contradicting strict contraction at the attracting fixed point.

The postcritical set is the closure of the union of the forward orbits of all critical values.

A parabolic basin consists of points whose iterates converge to a parabolic cycle along a specified attracting direction. Each connected basin component lies in the Fatou set, and its boundary lies in the Julia set.

Every Fatou component of a rational map of degree at least two is eventually periodic. Thus the periodic-component classification accounts for the entire Fatou set.