Fractals/Iterations in the complex plane/Koenigs coordinate

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Animated periodic cycle

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


  • 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[edit]

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


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