Exploration #2

Question: What is a filled Julia set?

Answer: Suppose we are given a particular complex c-value. The filled Julia set for x2 + c is the collection of all seeds whose orbit does not escape to infinity under iteration of x2 + c. Thus there is a different filled Julia set for each c-value.

Example x2 - 1. Below is an image of the filled Julia set of x2 - 1. We have indicated the location of 8 seeds on this image. Click on each seed (the small white circle) to see the corresponding orbit and determine if that seed is in the filled Julia set for c = -1.

ImageMap - turn on 

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 Home Page