Description

Animated periodic cycle

Koenigs[1] coordinate[2] are used in the basin of attraction of finite attracting (not superattracting) point (cycle),[3][4]

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[5] $z \to \lambda z$

Key words

• Koenigs function [6][7]
• Kœnigs Linearization of Geometrically Attracting basins

References

1. Gabriel Koenigs biographie at The MacTutor History of Mathematics archive