4.2 • JULIA AND MANDELBROT SETS 135front cover of this book illustrate the results of applying our algorithm to various
functions. Color plate 1 shows the results for the cubic polynomial J ( z) = z^3 + 1.
The points in the blue, red, and green regions are those "initial guesses" that
will converge to the roots - 1, ~ + 'f.!'i, and ~ - :/fi, respectively. (The roots
themselves are located in the middle of the three largest colored regions.) The
complexity of this picture becomes apparent when you observe that, wherever
two colors appear to meet, the third color emerges between them. But then, a
closer inspection of the area where this third color meets one of the other colors
reveals again a different color between them. This process continues with an
infinite complexity.
There appear to be no yellow regions with any area in color plate 1, in-
dicating that at least most initial guesses zo at a solution to z3 + 1 = 0 will
produce a sequence { zk} that converges to one of the three roots. Color plate 2
demonstrates that this outcome does not always occur. It shows the results of ap-
plying the preceding algorithm to the polynomial J (z) = z^3 + (- 0.26 + 0.02i) z+
(- 0.74 + 0.02i). The yellow area shown is often referred to as the rabbit. It con-
sists of a main body and two ears. Upon closer inspection (color plate 3) you can
see that each of the ears consists of a main body and two ears. Color plate 2 is
an example of a fractal image. Mathematicians use the term fractal to indicate
an object that has this kirid of recursive structure.
In 1918, the French mathematicians Gaston Julia and Pierre Fatou noticed
this fractal phenomenon when exploring iterations of functions not necessarily
connected with Newton's method. Beginning with a function J (z) and a point
zo, they computed the iterates z 1 = J (zo), Z2 = J (z1), ... , Zk+i = J (zk),. .. ,
and investigated properties of the sequence { zk}. Their findings did not receive
a great deal of attention, in part because computer graphics were not available
at that time. With the recent proliferation of computers, it is not surprising
that these investigations were revived in the 1980s. Detailed studies of Newton's
method and the more general topic of iteration were undertaken by a host of
mathematicians including Curry, Devaney, Douady, Garnett, Hubbard, Mandel-
brot, Milnor, and Sullivan. We now t urn our attention to some of their results
by focusing on the iterations produced by quadratics of the form Jc (z) = z^2 + c.
You will be surprised at the startling pictures that graphical iterates of such
simple functions produce.
• EXAMPLE 4.9 For Jc (z) = z^2 + c, analyze all possible iterations when
c = 0, that is, for the function Jo defined by Jo (z) = z^2 + 0.Solution We leave as an exercise the claim that if lzol < 1, the sequence will
converge to O; if lzol > 1, the sequence will be unbounded; and if lzol = 1, the
sequence will either oscillate around the unit circle or converge to 1.
For the function Jc. defined by le (z) = z^2 + c, and an initial seed zo, the
set of iterates given by z 1 =Jc (zo), Z2 =Jc (z 1 ), ... is also called the orbit of
zo generated by Jc. We let Kc denote the set of points with a bounded orbit for