2.3: The Mandelbrot, Julia and Fatou sets
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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 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 |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).
Figure \(\PageIndex{2}\)
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 "dendrite".
Now we are ready to a The Julia set trip CPU-based animation.