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
zn+1 = zn2 + c do not go to infinity
(the starting point zo = 0 will be discussed later).
Points on complex z plane (dynamical or variable space)
which under iterations fc 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
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 |zn| > 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 zo = 0
which converges to the attracting fixed point z1 .
"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(z1).
This is the famous "Douady's rabbit".
The "white" triangle shows the orbit star of attracting period-3
orbit f: z1 → z2 →
z3 → z1 .
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 fc is iterated.
The left image below is Cantor dust and the right image is connected pure
Now we are ready to a The Julia set trip
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updated 8 Sep 2013