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

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

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

$$c(\varphi)=1/2 e^{i\varphi}-1/4e^{i\varphi}$$ (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.

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

$\qedsymbol$