# Descriptio0n

Koenigs[1] coordinate are used in the basin of attraction of finite attracting (not superattracting) point (cycle),[2][3]

# 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 aproximated 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[4] $z \to \lambda z$

# Key words

• Koenigs function [5]
• Kœnigs Linearization of Geometrically Attracting basins

# References

1. Gabriel Koenigs biographie at The MacTutor History of Mathematics archive
2. G. Koenigs, Recherches sur les intégrales de certaines équations fonctionnelles, Annales École Normale Supérieure, 1(3) (1884), Supplément, 3-41.
3. Inigo Quilez images and tutuorial
4. Classification and Structure of Periodic Fatou Components. Senior Honors Thesis in Mathematics, Harvard College By Benjamin Dozier. Adviser: Sarah Koch 3/19/2012
5. Koenigs function in wikipedia