# Critical points and Fatou theorem

The orbit starting at *z*_{o} = 0 which converges to the attracting fixed
point *z*_{∗} = f(z_{∗ }) is shown to the left.
For small *ε*

*f*_{c}(z_{∗} + ε )
= z_{∗} + λ ε +
O(ε^{ 2}),
|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
derivative [1].

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 and inverse function
*f*_{c}^{-1}(z) = ±(z - c)^{½}
is singular at *z = c*.
We can continue *f*_{c}^{-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 *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 **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.

[1] *John W. Milnor* "Dynamics in One Complex Variable" § 8.5

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*updated* 11 Sep 2013