In mathematics, Fatou components are components of the Fatou set. They were named after Pierre Fatou.

## Rational case

If f is a rational function

${\displaystyle f={\frac {P(z)}{Q(z)}}}$

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

${\displaystyle d(f)=\max(\deg(P),\,\deg(Q))\geq 2,}$

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

1. ${\displaystyle U}$ contains an attracting periodic point
2. ${\displaystyle U}$ is parabolic [1]
3. ${\displaystyle U}$ is a Siegel disc: 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.
4. ${\displaystyle U}$ is a Herman ring: 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.

### Attracting periodic point

The components of the map ${\displaystyle f(z)=z-(z^{3}-1)/3z^{2}}$ contain the attracting points that are the solutions to ${\displaystyle z^{3}=1}$. This is because the map is the one to use for finding solutions to the equation ${\displaystyle z^{3}=1}$ by Newton–Raphson formula. The solutions must naturally be attracting fixed points.

### Herman ring

The map

${\displaystyle f(z)=e^{2\pi it}z^{2}(z-4)/(1-4z)}$

and t = 0.6151732... will produce a Herman ring. [2] It is shown by Shishikura that the degree of such map must be at least 3, as in this example.

### More than one type of component

If degree d is greater than 2 then there is more than one critical point and then can be more than one type of component

## Transcendental case

### Baker domain

In case of transcendental functions there is another type of periodic Fatou components, called Baker domain: these are " domains on which the iterates tend to an essential singularity (not possible for polynomials and rational functions)" [3] [4] one example of such a function is: [5]

${\displaystyle f(z)=z-1+(1-2z)e^{z}}$

### Wandering domain

Transcendental maps may have wandering domains: these are Fatou components that are not eventually periodic.