Mandelbrot and Julia sets

Let fc(z) = z2 + c , and (fc)n = fc o fc o ... o fc.

The Mandelbrot set is the set M of complex numbers c=x+iy for which the series { fcn(0) }n is bounded.

For a given c, the filled-in Julia set is the set Kc of complex numbers z for which the series { fcn(z) }n is bounded.

Look carefully at the difference of the last two lines, especially the different role of c, to see the relationship between the two sets! Notice also that there is one Mandelbrot set, but many Julia sets.

For a given c, depending on the starting value of z, the series { fcn(z) }n may:

For example, if c=0, K0 is the closed Unity Disk; the only finite attractor is zero, the Julia set J0 is the Unity Circle.

This is the only trivial case; here are two other examples:

  • c=0.6i
  • this Julia set still resembles to a deformed circle.
    Every Julia set is self-similar: this means that any zoom-in always reproduced exactly the same structure; the Mandelbrot set is only quasi-self-similar!
  • c=0.7i
  • this Julia set is no longer in one piece (connected).
    Such a figure is called Fatou dust.

    From a theorem proved in 1919 independently by Julia and by Fatou, it follows that:

    Jc is connected <=> c is inside the Mandelbrot set.

    Because of this, the Mandelbrot set is sometimes called the map of all the Julia sets.

    Short history of the Mandelbrot / Julia sets:


    Credits: much of these facts, I learned from the book "The Beauty of Fractals" by Peitgen & Richter.

    1996-12-04
    Your comments are welcome! Mail to Patrick Hahn (phahn@vo.lu)
    Homepage: http://www2.vo.lu/homepages/phahn