Classification of Fatou components

In mathematics, Fatou components are components of the Fatou set.

If f is a rational function

defined in the extended complex plane, and if it is a nonlinear function ( degree > 1 )

then for a periodic component ${\displaystyle U}$ of the Fatou set, exactly one of the following holds:

A Siegel disk is a simply connected Fatou component on which f(z) is analytically conjugate to a Euclidean rotation of the unit disc onto itself by an irrational rotation angle. A Herman ring is a double connected Fatou component (an annulus) on which f(z) is analytically conjugate to a Euclidean rotation of a round annulus, again by an irrational rotation angle.

Julia set (white) and Fatou set (dark red/green/blue) for ${\displaystyle f:z\mapsto z-{\frac {g}{g'}}(z)}$ with ${\displaystyle g:z\mapsto z^{3}-1}$ in the complex plane.

...
Wikipedia