Critical points and Fatou theorem
The orbit starting at zo = 0 which converges to the attracting fixed
point z∗ = f(z∗ ) is shown to the left.
For small ε
fc(z∗ + 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) terms).
You see these small circles around the attracting fixed points below
(here |λ| = 0.832). One of two branches of
the inverse function f -1(z) maps the disks vice versa.
We can extend the map analytically, while
f -1(z) is a smooth nonsingular function with finite
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 and inverse function
fc-1(z) = ±(z - c)½
is singular at z = c.
We can continue fc-1 up to the outside border of
the yellow region. The border contains the point z = c and is mapped on
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.
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 J 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 11 Sep 2013