Exploration #7

Question: What is the rotation number associated to the "primary bulbs"
and how can you compute this number?

Answer: First of all, recall that the "primary bulbs" are the large, disk-like decorations hanging directly off the main cardioid in the Mandelbrot set. If you choose a c-value from inside this bulb, then the corresponding quadratic function has an attracting cycle with some period, say q. You saw three different ways to "read off" this period in the previous Exploration.

Now we are going to associate a fraction p/q to each primary bulb. This fraction is called the "rotation number" of the bulb. You will see in the next exploration that these rotation numbers are ordered in a rather interesting way around the boundary of the main cardioid.

So the question is: What are the numbers p and q that form the numerator and denominator of the rotation number? Well, the denominator q is just the period of the bulb, as described in the previous exploration. The numerator p is found as in these examples:

To the right, you see a period 4 bulb in the Mandelbrot set, and, next to it, a filled Julia set whose c-value was chosen from inside this bulb. Note that the cycle of period 4 in this filled Julia set moves in the counterclockwise direction, jumping essentially 1/4 of a turn at each iteration. That is, the orbit hops from one piece of the filled Julia set to the next in the counterclockwise motion about a particular junction point. For this reason, this bulb has rotation number 1/4.
A period 4 bulb

Filled Julia Set

Here is another example. This time we show a period 7 bulb and a corresponding filled Julia set. Now notice that the attracting cycle jumps three "ears" in the filled Julia set in the counterclockwise direction at each iteration. Therefore this bulb has rotation number 3/7.
A period 7 bulb

Filled Julia Set

Exploration: In the Mandelbrot set below, select one of the white dots. These correspond to specific c-values in one of the primary bulbs. You will then see the corresponding filled Julia set together with the fate of the orbit of 0 as well as a magnification of the primary bulb from which the particular c-value was selected. Note that the orbit of 0 tends to a cycle of some period, and that the cycle has a definite rotation number.

You can see the rotation number of the bulb by simply watching how this cycle rotates around the junction point in the counterclockwise direction. However, as in the previous exploration, there are two other ways that you can read off this rotation number, one involving the geometry of the filled Julia set, the other involving the geometry of the Mandelbrot set. Your job is to figure out how to read off these rotation numbers just from the geometric information. You should be able to look at a filled Julia set drawn from a primary bulb and read off the rotation number of that bulb. You should also be able to look at a primary bulb in the Mandelbrot set and also read off the rotation number.

Warning: These two other methods are not quite exact: sometimes your eyes can fool you. The method tends to work best on the "western" side of the main cardioid. On the eastern side, there is some distortion. However, it is possible to use some advanced mathematics to compensate for this distortion and then get a rigorous geometric method for reading off the rotation numbers.

Further Exploration: To choose other c-values, you may use the Mandelbrot/Julia Set Applet. This tool will take some time to compute various Julia sets, since, unlike the above images, you really compute each different image.

Now answer the following questions:

Question 7A:

Here is a filled Julia set
that comes from a c-value
inside a primary bulb
in the Mandelbrot set.
What is the rotation number
corresponding to this bulb?


Question 7B:

Here is a primary bulb
in the Mandelbrot set.
What is the rotation number
associated to this bulb?


For Further Information: See the papers:

Devaney home page

Dynamical Systems and Technology Project Home Page