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Dynamics in One Complex Variable. (AM-160), 3rd Edition by John Milnor

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§16. The Sullivan Classification of Fatou Components

The results in this section are due in part to Fatou and Julia, but with major contributions by Sullivan.

A Fatou component for a nonlinear rational map f will mean any connected component of the Fatou set image \ J(f). Evidently f carries each Fatou component U onto some Fatou component U′ by a proper holomorphic map. First consider the special case U = U′.

Theorem 16.1. If f maps the Fatou component U onto itself, then there are just four possibilities, as follows: Either U is the immediate basin for an attracting fixed point or for one petal of a parabolic fixed point which has multiplier λ = ...

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