The Julia set of a function f(z) in the complex plane has all points z that remain finite upon iterations of the function. In the last posts I have used expanding functions to get fractal images from iteration, as discussed in “self-similar fractals with rotational symmetry“. The Julia set of these functions just has the exact zeros of the iterated functions. Thus the Julia set has only a countable infinite number of points, such as the set of natural numbers. Its area vanishes.

We now use smaller values for the constant c of the function

With c<1 it becomes contracting and a region around the zeros of the iterated function will converge to zero upon iteration. Its Julia set thus has a finite area. Its boundary is a fractal shape.

Similarly as before, we look at the number of iterations needed to get an absolute value smaller than some critical radius R. The pixel at z has then one of two colors depending whether this number is even or odd. The resulting images are quite different from those for expanding functions. An example with n=6, c=0.95 and R=0.2:

You can use the code of “self-similar fractals … – the code” to look at smaller parts of the image and see its self-similarity. Try out other functions to get related fractal images.

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