Question: What is a filled Julia set?
Question 2A: Which of the orbits A-G above lie in the filled Julia set? What do the colors mean in this image? Answer
Question 2B: Describe the behavior of the orbits whose seeds lie in the filled Julia set for x2 - 1. Answer
Question 2C: The Julia set of x2 - 1 is defined to be the boundary of the filled Julia set. That is, the Julia set is the set of complex seeds whose orbits do not escape to infinity, but have neighboring seeds whose orbits do escape to infinity. Which of the seeds above seem to lie in the Julia set? What can you say about the behavior of orbits in the Julia set? Answer
Question 2D: Use algebra to find the fixed points F and G of x2 - 1 explicitly. Answer
Further Exploration. Now let's compute filled Julia sets for other values of the parameter c. In order to do this, you need to specify the particular c-value. What more natural way to do this than to choose c from the c-plane, otherwise known as the Mandelbrot set. Open the Mandelbrot/Julia set applet to accomplish this. In this applet you will be able to click on a c-value in or around the Mandelbrot set and then display the corresponding filled Julia set as well as the fate of orbits.
For Further Information:
Devaney home page
Dynamical Systems and Technology Project