Principle

Fractal images as shown here are done
generating a number sequence for a complex number
with a generation function.

Dependent on the properties of the sequence a color is assigned to the number, for which the sequence has been generated.
Simple properties of a sequence are getting bigger and bigger, not getting bigger and bigger and the repetition of values.

Mapping of a complex number to a pixel of an image

A complex number c is just a pair of real numbers, the real part (re) and the imaginary part (im): c = (re,im).
This pair (re,im) can be shown graphically in an image, where you map (re,im) to a pixel of the image:

Complex number c = (re,im) --> pixel (i,j)

Generating a sequence for c

You generate a sequence z0, z1, z2, z3, z4,...
by evaluating a generation function f(z,c):

z0 = s ----- start value s is assigned to z0
z1 = f(z0,c)
z2 = f(z1,c)
z3 = f(z2,c)

Depending on the properties of the sequence you assign a color to your pixel.
Do this for all pixels of your image and you get a fractal image.

The second part of the math basics you can find in: Banach fixed point theorem and dynamical systems

The third part of the math basics you can find in: What about the real world?

Mathematical Rendering

My own tries to render this sequence you can find on the mathematical render page..