Fractals/Iterations in the complex plane/Koenigs coordinate

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Descriptio0n[edit]

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

Definition[edit]

  • 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[edit]

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

References[edit]

  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