# 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[1] 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 )

[2]

In other words map $f_c\,$ is conjugate [3] to map $f_0\,$ near infinity. [4]

## History

In 1904 LE Boettcher solved Schröder equation[5][6] in case of supperattracting fixed point[7]

## Names

• $w\,$ is Boettcher coordinate
• $\Phi_c \,$ is Boettcher function
• Boettcher Functional Equation [8][9]: $\Phi_c(f_c(z)) = \Phi_c(z)^2 \,$

where :

$w = \Phi_c(z)\,$

# References

1. Neighbourhood in wikipedia
2. The work of George Szekeres on functional equations by Keith Briggs
3. Topological conjugacy in wikipedia
4. How to draw external rays by Wolf Jung
5. Schröder equation in wikipedia
6. Lucjan Emil Böttcher and his mathematical legacy by Stanislaw Domoradzki, Malgorzata Stawiska
7. L. E. Boettcher, The principal laws of convergence of iterates and their aplication to analysis (Russian), Izv. Kazan. fiz.-Mat. Obshch. 14) (1904), 155-234.
8. Böttcher equation at Hyperoperations Wiki
9. wikipedia : Böttcher's equation