Fractal Geometry of the Mandelbrot Set (Cover Page)

3 Periods of the Bulbs (Previous Section)

BU Math Home Page

There is a second, more dynamic way to calculate the periods of these
primary bulbs in ** M**. To explain this, we have to introduce the
notion of a filled Julia set.

The filled Julia set for **x ^{2} + c** is subtly different
from the Mandelbrot set.
For

For the filled Julia sets, we fix a **c**-value and then consider
the fate of
all possible seeds for that fixed value of **c**. Those seeds whose
orbits do not escape form the * filled Julia set * of ** x ^{2}
+ c**.

Orbits
that do escape do not lie in the filled Julia set. Thus we get a different
filled Julia set of each different choice of **c**. That is, the
filled Julia set is a
picture in the dynamical plane, not the parameter plane. We denote the
filled Julia set for **x ^{2} + c** by

** c = -1.037 + 0.17i** in a period 2 bulb

**c = -0.52 + 0.57i** in a period 5 bulb

** c = 0.295 + 0.55i** in a period 4 bulb

** c = -0.624 + 0.435i ** in a period 7 bulb.

** Figure 7. The filled Julia sets for several c-values**

where **t** is the polar angle and **r** is the magnitude of
**z _{0}**
But then the
orbit of

**
**

_{1} = r^{2} exp (i2t)

_{2} = r^{4} exp (i4t)

**
z**

and so forth. The magnitude of **z _{n}** is

It follows that any seed on or inside the circle of radius 1
centered at the origin has an orbit that does not escape to infinity.
It therefore follows
that **J _{0}** consists of all those seeds whose orbits lie on or insider
the unit circle centered at the origin.

Incidentally, the fate of orbits of ** x ^{2}** that lie on the unit circle
is quite an interesting story. These are precisely the orbits that
behave in a chaotic fashion. See [3]
for more details.

Other filled Julia sets are much more difficult to compute. To see them, we
must use a computer. The algorithm is, of course, a direct consequence
of the definition of **J _{c}**. We simply consider a grid of points
centered at the origin and compute the orbit of each of these points
under

The ** Mandelbrot set:**

- is a picture in parameter space
- records the fate of the orbit of
**0**

The ** filled Julia set**

- is a picture in dynamical plane
- records the fate of all orbits

Fractal Geometry of the Mandelbrot Set (Cover Page)

3 Periods of the Bulbs (Previous Section)

BU Math Home Page

Prof. Robert L. Devaney (Boston University)