Exploration #4

Question: What happens to Julia sets as c leaves the Mandelbrot set?

Answer: The filled Julia sets shatter as c crosses the boundary of the Mandelbrot set.

Click on an arrowhead (not stem) to see an animation of the behavior of the filled Julia sets as c travels along the path shown. You must have a QuickTime player to view the animation. Click here to download a QuickTime player for Macs or PCs. Have patience while these animations are being loaded. They average around 650K per animation.

More explanation: It is a fact that the filled Julia sets for x² + c assume one of only two possible shapes. Either the filled Julia set is

connected, which means it consists of just one piece, or
totally disconneccted, which means that it consists of infinitely many pieces and each piece is a point. Technically, these filled Julia sets are examples of "Cantor sets," very complicated collections of points that appear often in the study of iteration.

In these animations, it appears that the black filled Julia sets suddenly disappear and are replaced by images that contain no black. Actually, this is not the case. There are infinitely many points that should be colored black. Unfortunately, we never have enough accuracy to find these points exactly; roundoff errors force us off the filled Julia set and onto a point whose orbit eventually escapes. That's why our coloring algorithm helps "see" the filled Julia set. As your eye moves from red to orange to violet, you are zeroing in on a point in the filled Julia set.

Further Exploration:

The Java applet below allows you to make your own movie. Simply click on a point in or near the Mandelbrot set and click Add current E. This gives you your starting c-value. Then click on a second c-value in the Mandelbrot set and again click Add current E. This determines a path along which c will move. Click Animate to compute the resulting animation. Then click Animate again to view the movie.

Click here to open the Movie Maker applet.

Please note that your browser must be Java-enabled in order to use this applet. All recent versions of Netsapce or Internet Explorer are Java-enabled.

For Further Information:

The Fractal Geometry of the Mandelbrot Set. I: Periods of the Bulbs. Robert L. Devaney. Especially Section 5. Accessible to high school students with a solid background in complex numbers.
The Mandelbrot and Julia Sets: A Toolkit of Dynamics Activities. Robert L. Devaney. Pages 93-95. Key Curriculum Press. Accessible to advanced high school students.

Devaney home page

Dynamical Systems and Technology Project Home Page