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: |
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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. |