Fractals/Iterations in the complex plane/boettcher
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Intro [edit]
Superattracting fixed points [edit]
For complex quadratic polynomial there are many superattracting fixed point ( with multiplier = 0 ):
- infinity ( It is always is superattracting fixed point for polynomials )
is finite superattracting fixed point for map 
and
are two finite superattracting fixed points for map 
Description [edit]
Near[1] super attracting fixed point (for example 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 [3] to map
near infinity. [4]
History [edit]
In 1904 LE Boettcher solved Schröder equation[5][6] in case of supperattracting fixed point[7]
Names [edit]
where :
References [edit]
- ↑ Neighbourhood in wikipedia
- ↑ The work of George Szekeres on functional equations by Keith Briggs
- ↑ Topological conjugacy in wikipedia
- ↑ How to draw external rays by Wolf Jung
- ↑ Schröder equation in wikipedia
- ↑ Lucjan Emil Böttcher and his mathematical legacy by Stanislaw Domoradzki, Malgorzata Stawiska
- ↑ 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.
- ↑ Böttcher equation at Hyperoperations Wiki
- ↑ wikipedia : Böttcher's equation
is finite superattracting fixed point for map
are two finite superattracting fixed points for map 
is Boettcher coordinate
is Boettcher function
