First we define the filledin Julia set,
K_{c}, for each complex number c. 
For each point
z_{0} of the plane, generate a sequence z_{1}, z_{2},
z_{3}, ... by the basic iteration rule 
z_{n+1} = z_{n}^{2} + c 
If the sequence does not run away to infinity, then the point
z_{0} belongs to K_{c}; 
if the sequence does run away to infinity, then
z_{0} does not belong to K_{c}. 
First we note three computational aspects of the definition of K_{c}: 

Using the escape criterion, we now
describe the coloring schemes for filledin Julia sets. 
Here are some examples of filledin
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. 