Question: What is the relationship between the Mandelbrot set and filled Julia sets?
Answer: Remember that the Mandelbrot set is a picture of the complex c-plane. Each point in the Mandelbrot set represents a c-value for which the orbit of 0 does not escape under iteration of x2 + c. Now we may also draw the corresponding filled Julia set for that c-value. This filled Julia set assumes one of only two possible "shapes" depending upon whether c is chosen in the Mandelbrot set or outside it. If c lies in the Mandelbrot set, then the corresponding filled Julia set is connected, meaning it is just one piece. If c lies outside the Mandelbrot set, then the filled Julia set shatters into infinitely many pieces (what is known technically as a "Cantor set" or, more popularly, "fractal dust").
Further Exploration: To choose other c-values, you may use the Mandelbrot/Julia Set Applet. This tool will take a few seconds to compute various Julia sets, since, unlike the above images, you really compute each different image.
For Further Information:
Devaney home page
Dynamical Systems and Technology Project