1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

136 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES

fc- Example 4.9 shows that Ko is the closed unit disk D 1 (O). The boundary of
Kc is known as the Julia set for the function le· Thus, the Julia set for /o is
the unit circle C1 (0). It turns out that K c is a nice simple set only when c = 0
or c = -2; otherwise, Kc is a fractal. Color plate 4 shows I<-1.25. The variation
in colors indicate the length of time it takes for points to become "sufficiently
unbounded" according to the following algorithm, which uses the same notation
as our algorithm for iterations via Newton's method.

1. Compute /e (Zi j ). Continue computing successive iterates of this initial point
   until the absolute value of one of the iterations exceeds a certain bound (say,
   L), or until the number of iterations has exceeded a preassigned maximum.


2. If Step 1 leaves us with an iteration whose absolute value exceeds L, we color
   the entire rectangle R;; with a color indicating the number of iterations
   needed before this value was attained (the more iterations required, the
   darker the color). Otherwise, we assume that the orbit of the initial point
   Z;j do not diverge to infinity, and we color the entire rectangle black.

Note, again, that this algorithm doesn't prove anything. It merely guides

the direction of our efforts to do rigorous mathematics.

Color plate 5 shows the Julia set for the function /c, where c = -0.11- 0.67i.

The boundary of this set is different from the boundaries of the other sets we

have seen, in that it is disconnected. Julia and Fatou independently discovered

a simple criterion that can be used to tell when the Julia set for / e is connected

or disconnected. We state their result, but omit the proof, as it is beyond the

scope of this text.

• EXAMPLE 4.10 Show that the Julia set for fi is connected.
  Solution We apply Theorem 4.9 and compute the orbit of 0 for f; (z) = z^2 +i.
  We have/; (0) = i, /di)= -1 +i, f; { - 1 + i) = -i, and/; (-i) = -1 +i. Thus,
  the orbit of 0 is the sequence {O, -1 + i, -i, -1 + i, - i, -1 + i, -i, ... } , which
  is clearly a bounded sequence. Thus, by Theorem 4.9, the Julia set for / ; is
  connected.

In 1980, the Polish-born mathematician Benoit Mandelbrot used computer

graphics to study the set

M = {c: the Julia set for le is connected}

= { c : the orbit of 0 determined by f c is a bounded set}.