Critical points and Fatou theorem

An orbit starting at z = 0 which converges to the attracting fixed
point z_{*} = f(z_{*}) is shown to the left.
In the applets below you see small circles around the attracting fixed points.
For small e
f_{c}(z_{*}+e )
= z_{*} + l e +
O(e^{ 2}),
l < 1
therefore f maps every disk with radius R into the next
smaller one with radius lR (really the
"circles" are distorted a little
by the O(e^{ 2}) term).

Reverse function f^{ 1}(z) maps the disks vice versa.
One can extend the map analitically [1], while
f^{ 1}(z) is a smooth nonsingular function with finite
derivative ^{d}/_{dz} f^{ 1}(z) =
f^{ 1}(z)'. Differentiating f^{ 1}(f(z)) = z we
get f^{ 1}(t)'_{t = f(z)}
= 1 / f '(z). Therefore the map is singular if f '(z) = 0.
Points z_{c} for which f '(z_{c}) = 0 are
called critical points of a map f. E.g. quadratic map
f_{c}(z) = z^{2} + c with derivative
f_{c}(z)' = 2z has the only critical point z_{c} = 0.
Reverse function f_{c}^{1}(z) = +(z  c)^{1/2}
is singular at z = c. We can continue it up to the outside border of
the yellow region. The border contains the point z = c and is mapped to
the figure eight curve with the critical point z_{c} in the
center. Therefore iterations f_{c}^{on}(z_{c})
converge to z_{*} for large n (the orbit is called the
critical orbit). This is the subject of the Fatou theorem.
Fatou theorem: every attracting
cycle for a polynomial or rational function attracts at least one critical
point.
As since quadratic maps have the only critical point z_{c} = 0
then quadratic Js may have the only finite attractive cycle!
(There is one more critical point at infinity which attracts diverging orbits.)
Thus, testing the critical point shows if there is any finite attractive cycle.
[1] John W. Milnor "Dynamics in One Complex Variable" § 8.5
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updated 14 November 2002