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

10. External angles in the Mandelbrot set: the work of Douady and Hubbard.

For more information on material in this section
see Chapter 5 and Adrien Douady's article Julia
sets and the Mandelbrot set in the book *The
Beauty of Fractals* by H. O. Peitgen and P. H. Richter.

Suppose the Mandelbrot set was an electrically
charged metal object that created an electric field around itself. The
following picture gives us an idea of what this filed would look like by
showing contours of equal electric potential (the curves surrounding ** M**)
and lines of force (the curves perpendicular to the countours).

Figure 10.1. The force field created by an electrically charged Mandelbrot set. For details of how this picture was created, see below.

**The results of Douady and Hubbard.**

In the early 1980s Douady and Hubbard proved some remarkable facts about the force field picture.

- The electric potential is
- Each line of force or
*external ray*leads inward to a pointon the boundary of`c0`, and outward to a strait line pointing away from 0 and making a certain angle with the positive real axis, called an`M`*external angle*for. Actually the first statement was proved only in cases when the external angle is a`c0`*rational*multiple of. It is conjectured that all rays, even those with irrational angles, lead inward to points on the boundary of`2Pi`. The is the main unsolved mathematical problem associated with the Mandelbrot set. We will say more about external angles below.`M` - Each point outside of
is uniquely determined by its escape time to infinity and its external angle. One can say the same of the force field associated with a connected Julia set, two of which are shown below.`M`

** V[c_] := Limit[Log[Abs[G[n, c, 0]]]/2^n, n ->
Infinity]. **(10.2)

To see that this limit exists, recall that by definition

`G[n + 1, c, 0] = G[n, c, 0]^2 + c.`

For large n, the c on the right is insignificant and ** G[n + 1,
c, 0] **is approximately

Figure 10.3. The force fields for the Julia sets
** J(0)** (the unit disk) and

- It is
*connected*, i.e. all in one piece. - It is
*simply connected*, i.e., it has "no holes". This also means that its compliment is connected.

The force field picture for the unit disk J(0) is very simple. The contours
are concentric circles around the origin and the rays are straight lines.Using
the escape time to infinity and the external angle of each point as "polar
coordinates", we can identify each point outside of ** M**
with the corresponding point outside of the unit disk. This has deep implications
for the structure of

**Computing external angles.**

Douady and Hubbard found a simple method for computing external angles
for values of ** c** outside of

`{Arg[c], Arg[c^2 +c], Arg[(c^2 + c)^2 + c], ...}.`

We replace ** Arg[z]** by 0 of

`c = -.75 +.0001*I; z = 0; `

`Do[z = z^2 + c; Print[Abs[Floor[Arg[z]/Pi]]], {n, 1, 10}]`

This produces the sequence ** {0, 1, 0, 1, 0, 1, 0, ...}**,
while

**Theorem 10.4 (Douady and Hubbard)**. Suppose
a bud or cardioid (other than the main one) has period `n`*.
Then the two rays leading to its root have angles which are whole multiples
of *`2Pi/(2^n - 1)`*. (The angle for the single ray
leading to the cusp of the main cardioid (*`c = 1/4`*)
is zero.) Conversely, every such whole multiple between 0 and *`2Pi
`*is the external angle of such a root or cusp.*

In particular the two rays leading to ** -3/4 **must have
angles which are multiples of

In many cases one can use Theorem 10.4 to determine external angles.
Consider the period 3 bud above the main cardioid. The angles of its two
rays must be multiples of ** 2Pi/7**. These rays must lie between
the for the main cusp (with angle 0) and the upper ray at

Simlarly one can deduce the following external angles for the points of period 4:

- The period 4 bud above the cusp of the main cardioid has
values of`Ray`and`1/15`.`2/15` - The period 4 cardioid above the period 3 bud has
values of`Ray`and`1/5`.`4/15` - The secondary bud of period 4 has
values of`Ray`and`2/5`.`3/5` - The period 4 cardioid on the real axis has
values of`Ray`and`7/15`.`8/15`

Before proceeding further we need some simple facts about binary arithmetic.

- The binary expansion for a fraction with denominator
is a repeating decimal with period`2^n - 1`.`n` - The binary expansion for a fraction with odd denominator
is a repeating decimal with period`q`, the smallest integer such that`n`is a multiple of`2^n - 1`; such an n always exists and is never more that`q`.`q - 1` - The binary expansion for a fraction with denominator
with`q*2^k`odd consists of`q`digits followed by a repeating decimal with period`k`as above.`n`

Now consider the *doubling
map* ** D **under which the value of a fraction is
doubled and digits to the left of the decimal point are ignored. It follows
from the facts above that fractions with odd denominators are periodic
under

A preperiodic point ** c0** in

- For
,`c0 = -2`is the line segment from -2 to 2 and its its own spine. An easy calculation shows that the ray leading to -2 has`J(c0)`value`Ray`. The external rays for are hyperbolas, and the equipotential lines are ellipses.The`1/2`values for the nearest period n cardioid are`Ray`and`(2^(n-1)-1)/(2^n - 1)`.`2^(n-1)/(2^n - 1)`

Figure 10.5. Force field for ** J(-2)**.

- For
,`c0 = I`is a dendrite and the right fixed point is`J(c0)`, which is also the rightmost point of the Julia set. The orbit of a point near`1.30024 - .624811*I`is near`I`, yielding the sequence`{I, -1 + I, -I, -1 + I, ...}`which makes the`{0, 0, 1, 0, 1,...}`value`Ray`.`1/6`

- For
, there are three external angle to compute. It turns out that their`c0 = -.101096 + 0.965287*I`values are`Ray`,`9/56`and`11/56`. The critical orbit is`15/56`

`{0, -0.101096+0.956287I, -1.00536+0.762932 I, 0.327586-0.577756I,
-0.327586+0.577756 I, -0.327586+0.577756 I, ...}`

Figure 10.6. The Julia sets ** J(c0)**
for

A theorem of Douady and Hubbard says that the external angles of a bud
root or cardioid cusp can be found by applying the method above to the
Julia set for the *center* of the corresponding bud or cardioid. Here
are some examples.

- The center of the period 3 cardioid a is
. Its critical oribit is`c0 = -1.75488` - The center of the period 4 cardioid above the period 3 bud is
. Its critical oribit is`c0 = -0.15652+1.03225 I`

** {0, -1.75488, 1.32472, 0, ...}**.

** {0, -0.15652+1.03225 I, -1.19756+0.709112 I, 0.774779 -0.666155
I,, 0, ...}**.

Figure 10.7. The Julia sets ** J(c0)**
for

**Devaney's method for finding external angles of primary
buds.**

In Devaney's paper The Mandebrot set and the Farey
tree, another method for computing external angles of primary bud is
given. Let ** I** denote the either of the half-open unit intervals,

`I0- = (0, 1 - p/q], I1- = (1 - p/q, 1]`

and

`I0+ = [0, 1 - p/q), I1+ = [1 - p/q, 1).`

We will illustrate with the case ** p/q = 2/5**. In

Figure 10.1 was made with the program ** GrowMandelbrot**
in drc95.frm. Here is the code for it.

`GrowMandelbrot(XAXIS) {;Douglas C. Ravenel January 21, 1995 `

` ;
Floating point must be used with `

` ;
this to get good results. `

` ;
Try this for various (real) values of p1 `

` ;
and explain what you observe. `

` ;
Animate it by altering the value of p1 `

`c=pixel, m = |p1|: `

`z=z*z+c, `

`|z*z-z+c| <= m } `

Using the z-option, the escape criterion ** p1** was set
to 1000 and the magnification to .5. Using the x-option, the

The escape parameter p1 (set to 1000) determines the test Fractint uses
to decide if an orbit is unbounded; instead of checking to see if ** |z|
< 4**, it looks for |

The color in the picture is determined by the value of the angle ** Arg[z]**
when the escape criterion is met. Since

Figure 10.3 was made in a similar way using the
program ** GrowJulia** in drc95.frm,
in which the parameter

`GrowJulia(Origin) {;Douglas C. Ravenel January 21, 1995 `

` ;
Floating point must be used with `

` ;
this to get good results. `

` ;
Try this for various (real) values of p1 `

` ;
and explain what you observe. `

` ;
Animate it by altering the value of p1 or c `

`c=p2, z=pixel, m = |p1|: `

`z=z*z+c, |z*z-z+c| <= m } `

.

*This page was last revised on March 4, 1998.*