In these notes I will use mathematical notation similar to the syntax of Mathematica.

5. The classical Mandelbrot set , February 2.

The classical Mandelbrot set **M**,
possibly the most famous of all fractals, is defined in the same way as
the logistic Mandelbrot set

As before this leads to a discrete dynamical system with orbits, fixed
points, cycles of various periods, and a bifurcation diagram. In this case
** g'[z] = 2z**, so the critical orbit is that of

Figure 5.3. The classical Mandelbrot set ** M**.

The picture above was obtained with Fractint using ** Fractals/Fractal_Formula/mandel**;
it is also the default image in Fractint. The interior color was changed
from blue to black by typing

** M** is the black region,which consists of those

By definition the Mandelbrot set ** M** consists the numbers

**Theorem 5.4 (Escape criterion). ***If an
orbit of the form (5.2) contains a number *`z
`with

Now each pixel on the screen correspond to a value of c. Fractint can
compute the first few numbers in the associated orbit and see if any of
them have absolute value exceeding 2. If it does, then we know that orbit
is unbounded, so that point is not in M. It gets colored according to how
many iterations were required to reach a sufficiently large number. In
Figure 5.3 above, the outer blue region consists
of those numbers ** c** with |

This is fine for recognizing unbounded orbits, but how do we know a
bounded orbit when we see one? We cannot look at all the points in an orbit,
so we have to compromise. Fractint gives up after 150 iterations; if none
of the first 150 numbers in an orbit are too big, it assumes the orbit
is bounded and colors the pixel black. The number 150 can be changed by
typing ** x** and then editing the maximum number of iterations.
Raising it has no visible effect on Figure 5.3,
but it can make a difference in close-ups of

**Julia sets.**

Another fractal associated with (5.1) is the (filled
in) Julia set ** J(c)**. For a fixed value of

`z, z^2 + c, (z^2 + c)^2 + c, ...`

The escape criterion of 5.4 applies and the image
can be computed in a similar way. ** J(c)** itself is shown
in black, and the points near it are colored by escape time as in Figure
5.3. Here are two examples of Julia sets.

Figure 5.5. The Julia set for c = -1.

Figure 5.6. The Julia set for c = -1 + .35I.

The point ** c **used in Figure 5.5
is well within the Mandelbrot set

How do we know there are any points at all in Figure 5.6 if we cannot see them in the picture? For a given value of c, consider the fixed point equation

`z^2 + c = z.`

It will have two solutions, unless ** c = 1/4**, in which
case it will have only one,

`z^2 + c = z0,`

and let ** z1** be a solution Then its orbit is

**The theorem of Julia and Fatou: the significance of
the critical orbit.**

Further experiments of this nature will lead one to the following conclusions about Julia sets:

*When *`c`* is a point in the Mandelbrot set *`M`*,
then the Julia set *`J(c)`* is connected, i.e., all
in one piece.*

*When*` c`* is not in *`M`*,
then the Julia set *`J(c)`* is totally disconnected,
i.e., it consists of infinitely many isolated points.*

Such computer experiments were not possible before 1980, but the above
facts were proved by Julia and Fatou in papers published in 1918-1920.
They considered a polynomial function of degree ** d**,

`p[z] = a[0]*z^d + a[1]*z^(d-1) + ... + a[d]`

where ** a[0],...,a[d]** are constants. One can consider
orbits obtained by iterating this function and define

`p'[z] = d*a[0]*z^(d-1) + ... + a[1]`

is a polynomial of degree ** d -1** , and there are at most

**Theorem 5.7 (Julia and Fatou, 1918-1920). **For
a polynomial function `p `*as above, the set *`J[p]
`*is connected if and only if each of the critical orbits of
*`p`* is bounded.*

In the case ** p[z] = g[z] = z^2 + c**, there is a just
one critical point,

**The main and period 2 bodies.**

Now we will describe the two largest regions in **M**,
the cardioid shaped main body, and the large circular region to its left.
This will be a calculation similar to the one we made for the logistic
Mandelbrot set L in Lecture
4.

First we want to find the values of ** c** for which

** u = g'[z] = 2z. **(5.8)

Then we need to determine when ** |u| < 1** if

** g[z] = z**. (5.9)

Now (5.8) implies that ** z = u/2**, and substituting this
into (5.9) gives

`(u^2)/4 + c = u/2`

`c = u/2 - (u^2)/4.`

For ** |u| < 1**, this defines the cardioid shaped region
r main body of

Next we do a similar computation with the function

`g2[z] = g[g[z]] = z^4 + 2c*z^2 + c^2 + c.`

The derivative is

** u = g2'[z] = 4z^3 + 4c*z = 4z*(z^2 + c) **(5.10)

The fixed point equation for ** g2** is

`0 = g2[z] - z = (g[z]- z)(z^2 + z + c + 1).`

Note that ** g2[z] - z **must be divisible by

`g[g[z]] = g[z] = z.`

We have already considered the case ** g[z] = z**, so we
want to assume instead that

** z^2 + z + c + 1 = 0. **(5.11)

If we multiply this by ** z** and use (5.10) we get

** 0 = z^3 + z^2 + c*z + z = u/4 + z^2 + z**.

Combining this with (5.11) gives

`u/4 = c + 1.`

For ** |u| < 1**, this describes a circular region with
radius 1/4 centered at -1.

These calculations can be done with Mathematica using the following code. For the main body,

`g[z_] := z^2 + c; Solve[{g[z]==z, u==g'[z]}, c, z],`

which gives

`{{c -> ((2 - u)*u)/4}}.`

For the period 2 body, use

`g2[z_] := g[g[z]]; Solve [{g2[z]==z, u==g2'[z]}, c, z]
,`

which gives

`{{c -> (-4 + u)/4}}.`

Mathematica cannot handle a similar computation for the function

`g3[z_] := g[g[g[z]]]`

to describe the period 3 body. However, as in the case of the logistic
Mandelbrot set, we can solve for those values of ** c** for
which the critical point is fixed by

`NSolve[g3[0]/g[0]==0,c]`

gives

`{c -> -1.75488}, {c -> -0.122561 - 0.744862 I},
{c -> -0.122561 + 0.744862 I}}.`

Figure** **5.12` `is a picture of the
area of` M `near

` `

Figure 5.12. The Mandelbrot set near ** c
= -1.75488**, magnified 20 times.

Figure 5.13. The Mandelbrot set near ** c
= -.122561 + .744862I**, magnified 3 times.

*This page was last revised on February 3, 1998.*