Fractals/Iterations in the complex plane/boettcher

Intro

Superattracting fixed points

For complex quadratic polynomial there are many superattracting fixed point ( with multiplier = 0 ):

• infinity ( It is always is superattracting fixed point for polynomials )
• $z_s = 0 \,$ is finite superattracting fixed point for map $f_0\,$
• $z_s = 0 \,$ and $z_s = -1 \,$ are two finite superattracting fixed points for map $f_{-1}\,$

Description

Near super attracting fixed point (for example infinity) the behaviour of discrete dynamical system :

$z_{n+1} = f_c(z_n) = z_n^2 + c \,$

based on complex quadratic polynomial $f_c(z) = z^2 + c\,$ is similar to

$w_{n+1} = f_0(w_n) = w_n^2 \,$

based on $f_0(w) = w^2\,$

It can be treated as one dynamical system viewed in two coordinate systems :

• easy ( w )
• hard to analyse( z )

In other words map $f_c\,$ is conjugate to map $f_0\,$ near infinity.

History

In 1904 LE Boettcher solved Schröder equation in case of supperattracting fixed point

Names

• $w\,$ is Boettcher coordinate
• $\Phi_c \,$ is Boettcher function
• Boettcher Functional Equation : $\Phi_c(f_c(z)) = \Phi_c(z)^2 \,$

where :

$w = \Phi_c(z)\,$