3 Periods of the Bulbs

As we showed in [7], the fact that the orbit of 0 "finds" the attracting cycle of Fc (if it exists) allows us to assign a number to the bulbs or decorations in M. For example, consider the main cardioid in M. This region is the set of c-values for which c has an attracting fixed point. Indeed, Fc has an attracting fixed point if

x2 + c = x and |F'c(x)| = |2x| < 1.

The boundary of this region consists of c values for which the magnitude of this derivative is 1, i.e., for c-values for which

2x = exp (it).

Solving the equations above using this fact yields

c = 0.5 exp (it) - 0.25 exp (2it)

which parametrizes a cardioid as t runs from 0 to 2 pi. Similarly, the large disk to the left of the main cardioid consists of c-values for which Fc has an attracting cycle of period 2. One may easily check that this region is bounded by the circle of radius 1/4 centered at c = -1.

Figure 2. Periods of the bulbs in M

Experimentally, one may determine the periods of the other bulbs in M. See Figure 2. To do this, one simply computes the orbit of 0 and checks whether it tends to an attracting cycle of period n. If this is the case for some c-value in the interior of a given decoration, then it is a fact that all c-values in the interior of this decoration have this property. That is, we may assign an integer n to each decoration as the period of the bulb or decoration.

The primary bulbs in M are those decorations that are directly attached to the main cardioid. In the sequel we will be mainly concerned with these decorations.

Recall from [7] that we may also read off the period of the primary bulbs by simply counting the spokes of the antenna attached to this decoration. Note that each bulb features a large antenna that contains a junction point from which a number of spokes emanate. It is a fact that the number of these spokes is exactly the period of the bulb. When counting these spokes, it is important to count the main spoke (the spoke that is attached directly to the bulb). For example, in Figure 3 we have displayed several of the primary decorations and their periods.

Period 3

Period 9

Period 19

Figure 3.

There is another way to read off the period of the primary bulbs. Choose any c in the interior of such a bulb and compute Jc. Of course, Jc is a connected set. However, there are infinitely many points in Jc that have the property that Jc becomes disconnected whenever any one of these points is removed. In fact, when such a point p is removed, Jc always separates into n disjoint pieces where n is the period of the primary bulb containing c. The word primary is important here. See Figure 4.

Figure 4

It can be shown that these special points in Jc consist of one of the fixed points of Fc together with all of the preimages of this point under Fck for each k. Note that there are no such points when c lies in the main cardioid (the fixed point bulb).

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