4.2 • JULIA AND MANDELBROT SETS 139Definition 4.8: n-cycleAn n-cycle for a function l is a set { zo, z 1 , •. • , z,. _ 1} of n complex numbers
such that Zk = l (zk- 1 ), for 1 ::; k::; n - 1 and f (zn-1) = zo.
Definit ion 4.9: Attracting n-cycle
An n-cycle { zo, zi,... , Zn-l} for a function l is said to be attracting if the
condition lg~ (zo)I < 1 holds, where 9n is the composition off with itself n times.
For example, if n = 2, then g2(z) = (! o !) (z) = f (! (z)).
- EXAMPLE 4.12 Example 4.10 shows that {-1 + i, - i} is a 2-cycle for the
function ft. It is not an attracting 2-cycle because g 2 (z) = z^4 + 2iz^2 + i - 1 and
gHz) = 4z^3 + 4iz. Hence lgH-1+i)I= 14 + 4il, so lgH-1+i)I>1.
In the exercises, we ask you to show that if { zo, z 1 , ... , Zn-d is an attracting
n-cycle for a function f, then not only does zo satisfy jg~ (zo)I < 1, but also
lg~ (zk)I < 1, for k = 1, 2, ... , n - 1.
It turns out that the large disk to the left of the cardioid in color plate 6
consists of those points c for which le (z) has a 2-cycle. The large disks above
and below the main cardioid disk are the points c for which le (z) has a 3-cycle.
Continuing with this scheme, we see that the idea of n-cycles explains the
appearance of the "buds" that you see on color plate 6. It does not, however,
begin to do justice to the enormous complexity of the entire set. Even color
plates 7 and 8 are mere glimpses into its awesome beauty. On our website, we
suggest several references for projects t hat you could pursue for a more detailed
study of topics relating to those covered in this section.-------~EXERCISES FOR SECTION 4.21. Consider the function f(z) =z^2 +1, where N(z) = z-f,<(2> = •:;^1 = ~ (z-~).
(a) Show that iflm(zo) > 0, the sequence {zk} formed by successive iter-
ations of zo via N (z) lies entirely within the upper half-plane.
(b) Show that a similar result holds if Im (zo) < 0.