Critical points and Fatou theorem
Reverse function f -1(z) maps the disks vice versa.
One can extend the map analitically , 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 zc for which f '(zc) = 0 are
called critical points of a map f. E.g. quadratic map
fc(z) = z2 + c with derivative
fc(z)' = 2z has the only critical point zc = 0.
Reverse function fc-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 zc in the
center. Therefore iterations fcon(zc)
converge to z* for large n (the orbit is called the
critical orbit). This is the subject of the 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 ε
= z* + λ ε +
|λ| < 1
therefore f maps every disk with radius R into the next
smaller one with radius |λ|R (really the
"circles" are distorted a little
by the O(ε 2) term).
Fatou theorem: every attracting
cycle for a polynomial or rational function attracts at least one critical
As since quadratic maps have the only critical point zc = 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.
 John W. Milnor "Dynamics in One Complex Variable" § 8.5
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updated 14 November 2002