f 1(x) = f (x) f 2(x) = f ( f (x) ) f 3(x) = f ( f ( f (x) ) ) . . .
The white points are initially inside a disc of radius 2 centered at the origin and 3 iterations are then performed. Some of the orbits tend to remain bounded while others tend to escape towards infinity.
Now let X= C ∪ {∞} be the extended complex plane and consider rational functions f (z) = p(z) | q(z) where p, q are polynomials without any common factors. The filled Julia set of the dynamical system ( C ∪ {∞}, p(z) | q(z) ) consists of all points whose orbits are bounded sets in the complex plane C. The Julia set of the dynamical system ( C ∪ {∞}, p(z) | q(z) ) is the boundary of the filled Julia set. A filled Julia set may be a connected set in the complex plane (aka Fatou region) or a totally disconnected set in the complex plane (aka Cantor dust).
We first consider the case f (z) = z 2 + c where c is a constant. It is a remarkable fact that the corresponding filled Julia sets are connected (i.e. form Fatou regions) if and only if c belongs to the Mandelbrot set. In fact, this can be taken as a definition of the Mandelbrot set, shown below in Figure 1.
The Mandelbrot set (white) shown on [-1.5,0.5] × [-1,1]
Prof. Ashay Dharwadker