Fractals/Iterations in the complex plane/Koenigs coordinate

Koenigs coordinate are used in the basin of attraction of finite attracting (not superattracting) point (cycle),

=Definition=
 * a rational map f of degree at least two $$ f(z)$$
 * a fixed point $$z_1 = 0$$
 * multiplier of the fixed point is $$\lambda $$
 * fixed point is attracting but not superattracting $$0 < \lambda < 1$$
 * $$\mathcal{A}$$ = the attracting basin of the fixed point zero under function $$ f$$. In other words interior of component containing fixed point =  the open set consisting of all points whose orbits under f converge to 0.

$$\phi_{\lambda}(z) : \mathcal{A} \to \mathbb{C}$$

It is approximated by normalized iterates :

$$\phi_n(z)= \frac{f^n(z)}{\lambda^n}$$

It can be defined by the formula :

$$\phi_{\lambda}(z)= \lim_{n \to \infty} \frac{f^n(z)}{\lambda^n}$$

Function f is locally conjugate to the model linear map $$ z \to \lambda z$$

=Examples=

Dynamics for quadratic 1D polynomials fc(z)=z²+c

=Key words=
 * Koenigs function
 * Kœnigs Linearization of Geometrically Attracting basins

=References=