Boettcher coordinate ( complex number) = complex potential

It creates polar coordinate system of complement of M-set.

- radial part of Boettcher coordinate = real potential

G_{M}(c) = log |phi(c)| - angle part of Boettcher coordinate = external argument Arg
_{M}

Arg_{M}(c) = Arg ( Phi_{M}( c) )

Log|Phi| made with program by Wolf Jung

Arg(Phi) made with program by Wolf Jung

Image made with ultrafractal and MMF3 colouring 3.8 (Colourings for Ultrafractal 3+)formula (Field line) by Dave Makin (Makin' Magic)

compare it with: Inigo Quilez images , especially Phase of phi(c)

or The Mandelbrot Function 2 and mb1 by John J. G. Savard

Jungreis algorithm gives inverse of Boettcher function.

Point at infinity is fixed point in the parameter plane

It is a superatractive fixed point of mandelbrot function

Complement of M-set is a set of points c in the parameter plane for which iteration of Z0=0 tends to infinity in the dynamical plane.

c --> Phi

C-M --> C-D

it is a conformal mapping ( isomorphism )

Phi

Phi

phi

= c * ( 1 + c/c

= c * Product (( 1 + c/ F

= c * Product (( 1 + c/ Z

where F

It goes like this :

phi

phi

phi

R(t) = { Phi

= { Phi

- Wikibooks
- Superattractive Fixed Points in C^n by John Hubbard, Peter Papadopol,
- First : thx for Inigo Quilez for help and many informations. (:-))
- L. E. Böttcher, The principal laws of convergence of iterates and their aplication to analysis (Russian), Izv. Kazan. fiz.-Mat. Obshch. 14) (1904), 155-234.
- Holomorphic dynamics by Morosawa, Nishimura , Taniguchi and Ueda
- Transcendental ... by Becker and Bergweiler
- Bottcher’s Theorem ... by Shigehiro Ushiki
- Douady-Hubbard Potential at sci.fractals
- Conjugation Theorems via Neumann Series by T.W. Gamelin
- Connectivity of Julia and Mandelbrot sets by David Ben-zvi
- Connectedness Proof by Robert P. Munafo
- Algebraic Dynamics and Transcendental Numbers by Michel Waldschmidt
- Peitgen, Richter : The beauty of fractals. Springer-Verlag 1986 ; Fig. 16 on page 16 ; page 68
- Hypertranscendency of Conjugacies in Complex Dynamics by Paul-Georg Becker
- COMMUTING POLYNOMIALS AND POLYNOMIALS WITH SAME JULIA SET Pau Atela Jun Hu
- THE MANDELBROT SET Math by O. Knill
- ONTINUITY OF LYAPUNOV EXPONENTS FOR POLYNOMIAL AUTOMORPHISMS OF C 2 ROMAIN DUJARDIN
- search google for :Böttcher Function of the Quadratic Map
- Ravings of a Mathematician by Jamie Sawyer
- Böttcher Function
- B¨OTTCHER COORDINATES by XAVIER BUFF, ADAM L. EPSTEIN, AND SARAH KOCH

Autor: Adam Majewski

adammaj1-at-o2-dot-pl

Feel free to e-mail me. (:-))

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