Fractals/Iterations in the complex plane/boettcher

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Contents

[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 )
  • 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}^2\,

[edit] Description

Near 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 )

[1]

In other words map f_c\, is conjugate to map f_0\, near infinity. [2]

[edit] History

In 1904 LE Boettcher solved Schröder equation in case of supperattracting fixed point[3]

[edit] 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)\,

[edit] References

  1. ↑ The work of George Szekeres on functional equations by Keith Briggs
  2. ↑ How to draw external rays by Wolf Jung
  3. ↑ 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.
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