Scaling

The representation of a Julia curve as a binary tree of counterimages of the unstable fixed point accounts for its fractal nature, because the arc at any stage of the construction resembles a doubled image of the arc at the next stage. A polygonal approximation to the Julia curve results from connecting successive points around its circumference by straight lines. If the Julia curve were a circle the lengths of each chord would be the same, but of course they are not.

 

Figure 8: angle and radius in invariant coordinates

As the number of points in the approximating polygon is doubled, the length of the chords would be halved; since the curve is not a circle, the statistics of the fluctuations in the ratio should reflect some of the structure of the curve. Figure 8 shows the absolute value and complex angle of each of the vertices of the Julia curve, plotted as a function of the invariant binary representation of the points.

Superimposed on the figure is the Julia curve itself, for purposes of comparison. The complex argument runs out to 180 degrees in both directions, causing the discontinuity at middiagram in the graph representing the angle. Fractal structure is especially evident in the curve representing the radius.

 

Figure 9: ratio of chord lengths for two successive generations

Figure 9 in turn shows a plot of the ratios of successive chord lengths. The chords are not those forming the 2ⁿ-gon approximating the Julia curve, but rather the ratio of the differences of two successive iterates. Consequently they are taken from a varying assortment of locations. If $\omega$ is a root of unity, then for c=0 the ratio is

$$\frac{\omega^4-\omega^2}{\omega^2-\omega}$$

which simplifies to $2\vert{\rm cos}\frac{t}{2}\vert$, where the t in $\omega=e^{2\pi it}$ is the invariant coordinate. To facilitate comparison, Figure 9 actually shows a family of values of c, together with the associated Julia curves.