Complex iteration

Since the discovery of the Mandelbrot set the preferred formula for complex iteration has been

w = z² + c,

wherein the quadratic polynomial has been reduced to canonical form by scaling to a unit leading coefficient and suppressing the linear term. According to this interpretation the fixed points are

$$z = \frac{1 \pm \sqrt{1-4c}}{2}.$$

Much insight into the process can be obtained by choosing c=0, for which iteration is the simple process of squaring.

The fixed points are then 0 and 1, although $\infty$ should be considered as well, given its natural role in complex variable theory. The stable fixed points are 0 and $\infty,$ while 1 is unstable. Working with absolute values these conclusions seem to be reasonable; the square of a small number is even smaller, the square of a large number is larger still; only the absolute value 1 will be conserved.

The complex numbers occupying the unit circle exemplify one of the differences between real iteration and complex iteration; the unit circle is a fixed set even though the only fixed point which it contains is 1. While all the rest of the complex plane is drawn towards one of the stable fixed points, the unit circle will wrap around itself; the modulus of each point gets doubled.

Since a second degree polynomial has two roots, quadratic iteration will usually map two distinct points into a common image; in our case

$$z=\pm\sqrt{w-c}$$

shows that +1 and -1 both map into 1, both +i and -i map into -1, both $\frac{1+i}{\sqrt{2}}$ and $\frac{-1-i}{\sqrt{2}}$ map into +i, $\frac{1-i}{\sqrt{2}}$ and $\frac{-1+i}{\sqrt{2}}$ map into -i, and so on. There is an infinite number of counterimages to be arranged into a binary tree, the set of whose nodes is representative of the unit circle. Similar trees exist for other values of c; only for c=0 do the nodes lie on the unit circle.

The Julia curves of Figures 2 and 3 were constructed by this process; the $\pm$ sign in the formula for counterimages explains their symmetry by 180 degree rotation. There are other sets which sample the Julia curve; even if not fixed themselves, they can still outline the curve.

The binary tree constructed from the sign sequence through which the counterimages are obtained gives an invariant representation of the Julia curve, valid even when $c \ne 0.$ Its nodes can be expressed as binary numbers on the range $0 \leq x \leq 1$ by writing a decimal point followed by zeroes or ones according to whether the plus sign or the minus sign was used to form each succeeding counterimage.

For example, .0 represents the positive square root of the unstable fixed point 1, which is 1 itself, while .1 represents -1, its negative square root. Continuing, .01 would represent i, .11 -i, and .001 $\frac{1+i}{\sqrt{2}}.$ Writing the digit representing the most recent sign change on the left, just after the decimal point, ensures that the order of the counterimages along the Julia curve is the same as the order of the angles in the exponential representation of the unit circle. This is because the positive square root tends to halve the angle of the point, while placing the negative square root in the diametrically opposite position. The notation works out well because an arbitrary string of zeroes still represents 1, or the fixed point when $c \ne 0.$

With respect to this binary notation, iteration is nothing but multiplication by 2 modulo 2; the formation of counterimages is division by 2, followed by the adjunction of a ``sign'' bit to distinguish the two possible counterimages. To the extent that the entire Julia curve is accurately approximated by the counterimage tree, the remarkable amount of self symmetry of the Julia curves is summarized by this correspondence.

Take any small arc, say the portion for which the first k binary digits are fixed. Iteration magnifies the arc, doubling the angle which it subtends from the origin, then translates it by c, which does not change its shape. But this is just the arc that would have been constructed with one less binary digit. Inspection of Figures 2 and 3 shows this effect quite clearly, especially if the radial peaks or valleys in the curves are used as reference points.

 

Figure 6: counterimages approaching a Julia curve

Figure 6 shows how the Julia curve can be approached as a series of counterimages of the fixed point in its interior. It was constructed by drawing a small circle around the fixed point, then calculating four levels of counterimages for the points on the circle; a number which fits nicely within the diagram without crowding.

The unstable fixed point sits on the point of the figure at the far right. It is its own counterimage, together with the diametrically opposite point. As successive counterimages are adjoined to the figure, rough squares, octagons, and so on, may be perceived. Evidently the vertices of these polygons move more rapidly toward the fixed point than the edges do, with the vertices of higher order polygons being more reluctant than those of lower order.

It is not surprising that a similar differentiation is to be found in the interior of the Julia curve, accounting for the doubling of the number of protuberances on the counterimage ``circles'' at each stage, producing curves with recognizable even harmonics in their polar coordinate Fourier transform. With increased deviation from the circular form the curves begin to pinch together and even to break up into disconnected sections.

 

Figure 7: ellipse with two of its successive images

Another viewpoint, which is shown in Figure 7, is to take a fairly large curve which still fits within the Julia curve, and calculate its images according to the mapping. In principle they converge to the fixed point; nevertheless the form of successive iterates gives some idea of the speed of convergence and reveals other details of the process. If the starting curve is inadvertently made too big, so that it intersects the Julia curve, it is possible to see a sampling of three types of behavior.

The portion inside converges to the fixed point, the portion outside diverges to infinity, and the points of intersection with the curve itself remain upon the curve. This detail can be seen in Figure 7, where the ellipse touches the curve; the point of tangency has rotated in successive images because the angle of its polar decomposition has doubled each time.