Graphics Reference
In-Depth Information
For example, the “point” • is always an attractive fixed point of the map f
c
defined
by equation (22.2), so that we always have a basin of attraction in this case which we
shall denote by A(f
c
,•). It turns out that for some values of
c
(for example, for
c
with
|
c
| much smaller than 1) there is a second attractive fixed point and that for other values
of
c
(for example, for
c
with |
c
| much larger than 1) • is the
only
attractive fixed point.
With regard to the case
c
= 0.31 + 0.04
i
and Figure 22.2, there is one finite attractive
fixed point and its basin of attraction is the interior of the black region in the figure.
One can show that A(f,
p
) is invariant under f. Some basic questions in the theory
of dynamical systems are:
(1) What does the boundary of A(f,
p
) look like?
(2) How “nice” a set is this boundary?
(3) What is its dimension?
Except for certain special cases such as the
Julia set
J
c
=∂A(f
c
,•), few answers are
known in general. Usually the only way to get some information is via approximation.
Let
A
= A(f,
p
). No point of ∂
A
can converge to
p
, but points arbitrarily close to ∂
A
will. Furthermore, roughly speaking, the closer a point is to ∂
A
, the longer it will take
to converge to
p
. Thus what one can do is to take small disks
D
around
p
and ask
which points will end up in
D
after k iterations. Let us call this set
A
k
. It is an approx-
imation for
A
. The larger k, the better the approximation. The “level sets”
L
k
=-
-
AA
1
k
k
are approximations to ∂
A
.
Finally,
Definition.
The set
{
}
k
Mc
=
f
()
Æ•
0
as k
Æ•
c
is called the
Mandelbrot set
.
See Figure 22.3. Today there are probably very few people, if any, who work with
computers who have not seen some of the beautiful computer-generated images of
Figure 22.3.
The Mandelbrot set.
