Fractals/Iterations in the complex plane/boettcher
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[edit] Intro
[edit] Superattracting fixed points
For complex quadratic polynomial there are many superattracting fixed point ( with multiplier = 0 ):
- infinity ( It is allways is superattracting fixed point for polynomials )
is finite superattracting fixed point for map 
and
are two finite superattracting fixed points for map 
[edit] Description
Near infinity the behaviour of discrete dynamical system :

based on complex quadratic polynomial
is similar to

based on 
It can be treated as one dynamical system viewed in two coordinate systems :
- easy ( w )
- hard to analyse( z )
In other words map
is conjugate to map
near infinity. [2]
[edit] History
In 1904 LE Boettcher solved Schröder equation in case of supperattracting fixed point[3]
[edit] Names
is Boettcher coordinate
is Boettcher function- Boettcher Functional Equation :

where :
[edit] References
- ↑ The work of George Szekeres on functional equations by Keith Briggs
- ↑ How to draw external rays by Wolf Jung
- ↑ 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.
