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The Dynamic of the Mandelbrot-set

First we show that $ Q_c(z)$ is a holomorphic function, and therefore $ Q_c(z)$ is a angle-conform map. This means a rectengular grid of horizontzal and vertical lines in a family of paraboli and circles are mapped in circles (see Figure 6)

Figure 6: The upper left picture shows a rectengular grid of the relevant piece of the complex plane. The upper right shows the mapping of the grid with respect to $ Q_0(z)$. The lower pictures show the same process for a polar grid
\scalebox {0.3}{\includegraphics{Pics/cartesian_grid.ps}} \scalebox {0.3}{\includegraphics{Pics/cartesian_map.ps}}
\scalebox {0.3}{\includegraphics{Pics/polar_grid.ps}} \scalebox {0.3}{\includegraphics{Pics/polar_map.ps}}

Theorem 9   The map $ Q_c(z)$ is a holomorphic map.

Proof. You can separate the real and imaginary part of $ Q_c=(z_1+iz_2)^{2}+c_1+ic_2$ as

$\displaystyle \begin{eqnarray}\Re\ Q_c(z) &=& z_1^{2}-z_2^{2} +c_1 \\ \Im \ Q_c...
...eft( \begin{array}{cc} 2x & -2y \\  -2y & 2x \end{array} \right) \end{equation}$ (1)

This yields that the Cauchy-Riemann differential equations are true and therefore $ Q_c(z)$ is holomorph. $ \qedsymbol$

Theorem 10 (The cardiode is part of the set)   The filled cardiodic curve
$\displaystyle c(\varphi)=1/2 e^{i\varphi}-1/4e^{i\varphi}$ (c)

is part of the mandelbrot set. The interior of this curve contains Lyapunov-stable fixed points and the border contains neutral fixed points of the map $ Q_{c(\varphi)}(z)$ for fixed $ \varphi$

Proof.
% latex2html id marker 2567
$\displaystyle To obtain the fixed points, we must s...
...in the closed cardiode and $z_{f2}$\ is a instable fixed point for all $c$\par $ (4a)

$ \qedsymbol$


next up previous
Next: The Algorithm Up: The Mandelbrot set Previous: Some properties of the
Tino Kluge
2000-12-05