Julia Set

Consider the quadratic map $P_c(z)=z^2+c$ where $z=x+yi$ and $c=a+bi$ are complex numbers $i=\sqrt{-1}$

Example Iterations

$$\begin{array}{rl}{P}_c(z)& =(x+yi{)}^2+a+bi=\underset{real}{\underbrace{(x^2-y^2+a)}}+\underset{imaginary}{\underbrace{(2xyi)}}\\ z_0& =x_0+y_{0}i\\ z_1& =(x_0^2-y_0^2+a)+(2x_0{y}_0+b)i\\ x_{n+1}& =x_n^2-y_n^2+a\\ y_{n+1}& =2x_n{y}_n+b\end{array}$$

If $c=0$, then the map is:

$$P_0(z)=z^2$$

has an Attracting fixed point at origin. The basin is $\{z:|z|<1\}$ ( interior of the unit disk )

Points of the disc have their angle doubled but stay on the disc. Points outside the disc