A few words are in order to explain how to interpret these pictures.
We are studying the dynamical systems given by iterating the functions
f(z) = z2 + c where
z is a complex variable and
c is a complex parameter. The
Mandelbrot set is the set of values of
c such that the orbit of 0 does not escape to infinity. It is colored black in these pictures. For values of
c that are not in the Mandelbrot set the orbit of 0 does escape to infinity. The color of a pixel corresponding to the number
c is the first integer
n such that |
fn(0)| >
2. We know that the orbit of
0 escapes to infinity if and only if some iterate has absolute value greater than 2. (In fact we only check 256 iterates, so some
c values will be colored black even though they lie slightly outside the Mandelbrot set.)
For a fixed value of
c, the
filled Julia set FJc is the set of
z values such that the orbit of
z does not escape to infinity.
In our pictures of the filled Julia set the color of a pixel is black if the orbit of the corresponding
z value does not escape from the circle of radius 2 after 256 iterates. Otherwise the color represents the number of iterates before it does escape from the circle of radius 2 (and therefore escapes to infinity.) The
Julia set Jc is the boundary of the filled Julia set, and is represented by the black pixels in the black-and-white pictures.
We studied the theorem that the Julia set
Jc is connected if
c lies in the Mandelbrot set, and totally disconnected otherwise. As
c moves from 0 to a point outside the Mandelbrot set the Julia set changes shape, breaking up into "fractal dust" as
c crosses the boundary of the Mandelbrot set.