Julia Sets

A Julia set can be drawn as a fractal in the complex plane by iterating a function f(z) at each point z and checking if the iterated value eventually exceeds some fixed bound. The set of points for which the iterated value remains bounded is called the filled-in Julia set. For example, consider the function f(z) = z². If we iterate the function f starting at z = 0.5, we get the sequence

which tends toward 0. Thus, 0.5 is in the filled-in Julia set. On the other hand, if we start with z = 2, we get the sequence

which tends toward infinity. Thus, 2 is not in the filled-in Julia set.

Exercise: What is the shape of the filled-in Julia set for f(z) = z²?

In the interactive applet below, we look at functions of the form f(z) = zⁿ + c. You can use the slider to change the exponent n, and by hovering your mouse/finger over the applet you can change the value of the constant c. The black region is the filled-in Julia set for the function f(z) = zⁿ + c. The colored region consists of the points whose iterated value tends toward infinity. The points colored red or orange are those whose iterated value approached infinity quickly; the points colored blue or purple had their iterated values approach infinity more slowly.

Exercise: For which values of c does the filled-in Julia set (the black region) of the quadratic function f(z) = z² + c consist of one connected component?