3. The Mandelbrot Set and Julia Sets

3. B. Julia Sets

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

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

z_n+1 = z_n² + c

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

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

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.

Return to the Mandelbrot set and Julia sets.