Next, for the natural interpretation of the Mandelbrot set, recall the theorem of Fatou and Julia, that Julia sets Jcare either connected or Cantor sets. | ||
Moreover, they showed Jc is connected if and only if the iterates of z0 = 0 does not run away to infinity. | ||
(Specifically, Julia and Fatou showed the iterates of the critical point of the function iterated, z2 + c in this case, determine whether the Julia set is connected.) | ||
Because the Mandelbrot set consists of those c for which the iterates of z0 = 0 do not run away to infinity, we see the Mandelbrot set is exactly those c for which Jc is connected. | ||
Consequently, the Mandelbrot set is a very natural object to study. | ||
To emphasize the relation between the Mandelbrot set and the Julia sets, this movie traces a path of points in the Mandelbrot set (right panel) and shows the corresponding Julia set in the left panel. | ||
Note that when the points go outside the Mandelbrot set, the Julia set disintegrates into a dust of points. | ||
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Here are three more applets from Bob Devaney's dynamical systems website at BU. | ||
With the Julia set computer we can choose a point on the Mandelbrot set and view the corresponding Julia set and its magnifications. | ||
With the Mandelbrot/Julia set applet we can view the Mandelbrot set, a Julia set, and any orbit in or near the Julia set. | ||
With the Mandelbrot movie maker we can define a path in and around the Mandelbrot set and animate the corresponding Julia sets as we move along the path. |
Return to the Mandelbrot set.