4 Rotation Numbers

We will now associate a rational number p/q to each primary bulb. The denominator q will simply be the period of the bulb, so we need only describe how to find p. There are two ways to do this.

Given c in a primary bulb of period q, with q > 1, first compute J_c and then superimpose the attracting cycle of period q on J_c. This cycle will lie in q components of the interior of J_c. Moreover, it is a fact that these q components are joined together at one of the fixed points of F_c. We denote this fixed point by w_c. The attracting cycle hops among these q components as F_c is iterated. If we observe this motion, the cycle jumps exactly p components in the counterclockwise direction at each iteration. Another way to say this is the cycle rotates about w_c by a p/q revolution in the counterclockwise direction under iteration. Thus p/q gives us a type of rotation number. In Figure 5a we display the 2/5 bulb. Click on the legend to see an animation of the period 5 cycle in a Julia set corresponding to a -value from this bulb. animated some movies of the motion in a Julia set dra

The 2/5 bulb Click here to see an animation of a cycle in the 2/5 bulb.

Figure 5. The 3/7 bulb. Click here to see an animation of a cycle in the 3/7 bulb.

There is a second method to determine p/q, which is not quite well defined but which, in practice, is easier to use. Again compute J_c and the attracting cycle. As before, this cycle determines q regions in the interior of J_c which are joined at w_c. Now 0 lies in one of these regions. Note that this region is the largest of the q regions. Now find the "smallest" of the remaining regions. This region is located exactly p/q revolutions about w_c in the counterclockwise direction, again yielding the rotation number.

The reason why this region is smallest is that it contains F_c(0), the image of the critical point of F_c. Since 0 is the only critical point for F_c, the region containing 0 suffers the most amount of contraction as we apply F_c, making its image the smallest.

The reason for the imprecision here is in the word "smallest." How do we measure the size of these regions? It is in general impossible to determine the areas of these regions explicitly. Thus in practice we merely "eyeball" the various regions to see which is smallest. Also, when determining p in this fashion, it is best to choose c near the center of the bulb (as near as possible to the c-value for which 0 lies on the attracting cycle). Of course, when q is large, it is essentially impossible to distinguish the smallest region, so we must resort to the previous method in these cases.

There is a third way to determine p that does not involve computing J_c but rather involves only looking at the antenna attached to the bulb in M. The disadvantage of this method is that we sometimes need "hyperbolic eyeglasses" to view the antenna to read off p. That is, this method does not work for all bulbs. To find p, note that there are q spokes emanating from the junction point in the main antenna attached to the bulb. Locate the shortest of these. For most bulbs, this shortest spoke is located p/q revolutions in the counterclockwise direction from the main spoke.

Now this is not always true if we measure the length using the usual Euclidean distance. Rather, we should use a distance that assigns a shorter length to the spokes closer to the main spoke. Without being precise, this is what we mean by "hyberbolic eyeglasses." In Figure 6 we have displayed the p/q bulbs for various choices of p and q. Click on the appropriate figure to view.

Figure 6a. The 1/3 bulb.

Figure 6b. The 2/5 bulb.

Figure 6c. The 3/7 bulb.

Figure 6d. The 5/7 bulb.

To test your ability to read off the rotation numbers, click here for a "clickable" version of Figure 6a-d which is in turn linked to enlargements and movies.