Complex Dynamics:Julia Sets and the Mandelbrot Set
Some examples of filled-in Julia sets. For a complex number c, the filled-in Julia set is the set of those points in ? whose orbits under the function z 2 + c do not approach infinity. The upper left image is the filled-in Julia set where c = −0.123 + 0.745 i, known as Douady’s rabbit; in this case there is an attracting 3-cycle. The upper right corresponds to c = 0.32 + 0.043 i, which has an attracting 11-cycle. At the lower left is the Julia set for a different function, a bifurcation of the quadratic map rz(1−z) at r = 3, discussed in Chapter 7. The image at lower right is the Mandelbrot set, which encodes the collection of c for which the Julia set of z 2 + c is connected.