Classification of Fatou components
components of the Fatou set
In mathematics, Fatou components are components of the Fatou set. They were named after Pierre Fatou.
Rational case
If f is a rational function
f
=
P
(
z
)
Q
(
z
)
{\displaystyle f={\frac {P(z)}{Q(z)}}}
defined in the extended complex plane, and if it is a nonlinear function (degree > 1)
d
(
f
)
=
max
(
deg
(
P
)
,
deg
(
Q
)
)
≥
2
,
{\displaystyle d(f)=\max(\deg(P),\,\deg(Q))\geq 2,}
then for a periodic component
U
{\displaystyle U}
of the Fatou set, exactly one of the following holds:
U
{\displaystyle U}
contains an attracting periodic point
U
{\displaystyle U}
is parabolic
U
{\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.
U
{\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
f
(
z
)
=
z
−
(
z
3
−
1
)
/
3
z
2
{\displaystyle f(z)=z-(z^{3}-1)/3z^{2}}
contain the attracting points that are the solutions to
z
3
=
1
{\displaystyle z^{3}=1}
. This is because the map is the one to use for finding solutions to the equation
z
3
=
1
{\displaystyle z^{3}=1}
by Newton–Raphson formula. The solutions must naturally be attracting fixed points.
Herman ring
The map
f
(
z
)
=
e
2
π
i
t
z
2
(
z
−
4
)
/
(
1
−
4
z
)
{\displaystyle f(z)=e^{2\pi it}z^{2}(z-4)/(1-4z)}
and t = 0.6151732... will produce a Herman ring. It is shown by Shishikura that the degree of such map must be at least 3, as in this example.
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