# The Mandelbrot, Julia and Fatou sets

The *Mandelbrot set* (**M**) is the set of all points *c* on
complex plane (*parameter space*) such that iterations
*z*_{n+1} = z_{n}^{2} + c do not go to infinity
(the starting point *z*_{o} = 0 will be discussed later).

Points on complex *z* plane (*dynamical* or *variable space*)
which under iterations *f*_{c} for fixed *c* go to an
attractor (attracting fixed point, periodic orbit or infinity) form the *Fatou
set*. The *Julia set* (**J**) is its complement. Therefore the Julia
set includes all repelling fixed points, periodic orbits and their
preimages.

The Mandelbrot set is the black region on this image.
Points outside the **M**-set are colored according to how many
iterations *n* were completed before *|z*_{n}| > 2
(see also the Distance Estimator algorithm).
Each point *c* in the Mandelbrot set specifies the geometric
structure of the corresponding Julia set. If *c* is in the **M**-set,
the **J**-set is connected. If *c* is not in the **M**-set, the
**J**-set is a Cantor dust.

Connected **J**-set for *c1 = -0.71 + 0.1i* in the **M**-set
is shown below. The Julia set is the boundary between colored and black regions
on the left image. You see too the orbit starting at *z*_{o} = 0
which converges to the attracting fixed point *z*_{1} .
"Filled" **J**-set is shown on the right picture
(it is marked by "f"). Colors inside connected **J**-sets show how fast
a point goes to attractor. You can see attractor and infinite sequence of its
preimages *f*^{ -on}(z_{1}).

This is the famous "Douady's rabbit".
The "white" triangle shows the *orbit star* of attracting period-3
orbit *f: z*_{1} → z_{2} →
z_{3} → z_{1 }.
This cycle lies in *n = 3* components of
the interior of the J-set. Moreover, these *n* components are joined
together at one point. The attracting cycle hops among these *n*
components as *f*_{c} is iterated.
The left image below is Cantor dust and the right image is connected pure
"dendrite".

Now we are ready to a The Julia set trip
CPU-based animation.

Contents
Previous: Bifurcation diagram for quadratic maps
Next: Iterations of inverse maps

*updated* 8 Sep 2013