4.2 • JULIA ANO MANDELBROT SETS 187The set M has come to be known as the Mandelbrot set. Color plate 6
shows its intricate nature. The Mandelbrot set is not self-similar, although it
may look that way. There are subtle variations in its infinite complexity. Color
plate 7 shows a zoom over the upper portion of the set shown in color plate- Likewise, color plate 8 zooms in on the upper portion of color plate 7. In
color plate 8 you can see the emergence of another structure very similar to the
Mandelbrot set that we began with. Although it isn't an exact replica, if you
zoomed in on this set at almost any spot, you would eventually see yet another
"Mandelbrot clone" and so on ad infinitum! In the remainder of this section we
look at some of the properties of this amazing set.
- EXAMPLE 4.11 Show that { c: lei ::; !} ~ M.
Solution Let {an}::"=o be the orbit of 0 generated by le (z) = z^2 + c, where
!cl ::; ~. Then
ao = o,
a1 =le (ao) = ~ + c = c,
% = le (ai) = ai + c, and in general,
iln+ I = le (an) =a~+ c.
We show that {an} is bounded, a.nd, in particular, we show that lanl::; 4 for all
n by mathematical induction. Clearly lanl ::; 4 if n = 0 or l. We assume that
Jani ::;! for some value of n ~ 1 (our goal is to show lan+d ::; 4). Now,
la~ +cl
::; ~a~ I + le! (by the triangle inequality)
::; 4 + l = 4 (by our induction assumption and the fact that lcl::; ~).In the exercises, we ask you to show that if !cl > 2, then c ¢ M. Thus,
the Mandelbrot set depicted in color plate 6 contains the disk Di (0) and is
contained in the disk D2 (0).
We can use other methods to determine which points belong to M. To do
so, we need some additional vocabulary.
Definition 4 .6: Fixed p oint