3. The Mandelbrot Set and Julia Sets

3. B. Julia Sets

First we define the filled-in Julia set, Kc, for each complex number c.

For each point z0 of the plane, generate a sequence z1, z2, z3, ... by the basic iteration rule

zn+1 = zn2 + c

If the sequence does not run away to infinity, then the point z0 belongs to Kc;

if the sequence does run away to infinity, then z0 does not belong to Kc.

First we note three computational aspects of the definition of Kc:

Finite resolution Run Away to Infinity Accuracy vs. Time

Using the escape criterion, we now describe the coloring schemes for filled-in Julia sets.

Here are some examples of filled-in Julia sets.

Now for the definition of the Julia set.

The definition of the Mandlebrot set is based on the Dichotomy Theorem, that there are only two types of Julia sets. Here is an illustration of the difference.

Return to the Mandelbrot set and Julia sets.