Julia set
Examples

For f (z) = z 2 {\displaystyle f(z)=z^{2}} the Julia set is the unit circle and on this the iteration is given by doubling of angles (an operation that is chaotic on the points whose argument is not a rational fraction of 2 π {\displaystyle 2\pi }). There are two Fatou domains: the interior and the exterior of the circle, with iteration towards 0 and ∞, respectively.

For g (z) = z 2 − 2 {\displaystyle g(z)=z^{2}-2} the Julia set is the line segment between −2 and 2. There is one Fatou domain: the points not on the line segment iterate towards ∞. (Apart from a shift and scaling of the domain, this iteration is equivalent to x → 4 (x − 1 2) 2 {\displaystyle x\to 4(x-{\tfrac {1}{2}})^{2}} on the unit interval, which is commonly used as an example of chaotic system.)

The functions f and g are of the form z 2 + c {\displaystyle z^{2}+c}, where c is a complex number. For such an iteration the Julia set is not in general a simple curve, but is a fractal, and for some values of c it can take surprising shapes. See the pictures below.

For some functions f (z) we can say beforehand that the Julia set is a fractal and not a simple curve. This is because of the following result on the iterations of a rational function:

Theorem — Each of the Fatou domains has the same boundary, which consequently is the Julia set.

This means that each point of the Julia set is a point of accumulation for each of the Fatou domains. Therefore, if there are more than two Fatou domains, each point of the Julia set must have points of more than two different open sets infinitely close, and this means that the Julia set cannot be a simple curve. This phenomenon happens, for instance, when f (z) is the Newton iteration for solving the equation P (z):= z n − 1 = 0 : n > 2 {\displaystyle \;P(z):=z^{n}-1=0~:~n>2\;}:

The image on the right shows the case n = 3.
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