Parameter space sets

Points in the space of parameters can conveniently be labelled by a complex number c, where the imaginary direction is taken to increase to the right. The pattern corresponding to each point is the limit of Nest[Flatten[1 + {c #, Conjugate[c] #}]&, {1}, n] when n ∞. Such a limiting pattern exists only within the unit circle Abs[c] < 1. It then turns out that the limiting pattern is either completely connected or completely disconnected; which it is depends on whether it contains any points on the vertical axis Im[c] 0. Every point in the pattern must correspond to some list of left and right branchings, represented by 0's and 1's respectively; in terms of this list the position of the point is given by Fold[1 + {c, Conjugate[c]}〚1 + #2〛 #1&, 1, Reverse[list]]. Patterns are disconnected if there is a gap between the parts obtained from lists starting with 0 and with 1. The magnitude of this gap turns out to be given by

With[{d = Conjugate[c], r = 1 - Abs[c]^{2}}, Which[Im[c] < 0, d, Im[c] 0, 0, Re[c] > 0, With[{n = Ceiling[π/2/Arg[c]]}, Im[c(1 - d^{n})/(1 - d)] + Im[c d^{n}(1 + d)]/r], Arg[c] > 3π/4, Im[c + c^{2}]/r, True, Im[c] + Im[c^{2} + c^{3}]/r]]

The picture below shows the region for which the gap is positive, corresponding to trees which are not connected. (This region was found by Michael Barnsley and others in the late 1980s.) The overall maximum gap occurs at c = 1/2 Sqrt[5 - √17]. The bottom boundary of the region lies along Re[c] = -1/2; the extremal point on the edge of the gap in this case corresponds to {0, 0, 1, 0, 1, 0, 1, …} where the last two elements repeat forever. The rest of the boundary consists of a sequence of algebraic curves, with almost imperceptible changes in slope in between; the first corresponds to {0, 0, 0, 1, 0, 1, 0, 1, …}, while subsequent ones correspond to {0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, …}, {0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, …}, etc.

In the pictures in the main text, the black region is connected wherever it does not protrude into the shaded region, which corresponds to disconnected patterns, in the pictures above. And in general it turns out that near any particular value of c the sets shown in black in the main text always look at sufficient magnification like the pattern that would be obtained for that value of c. The reason for this is that if c changes only slightly, then the pattern to a first approximation deforms only slightly, so that the part seen through the peephole just shifts, and in a small region of c values the peephole in effect simply scans over different parts of the pattern.

A simple way to approximate the pictures in the main text would be to generate patterns by iterating the substitution system a fixed number of times. In practice, however, it is essential to prune the tree of points at each stage. And at least for Abs[c] not too close to 1, this can be done by discarding points that are so far away from the peephole that their descendents could not possibly return to it.

The parameter space sets discussed here are somewhat analogous to the Mandelbrot set discussed on page 934, though in many ways easier to understand.

(See also the discussion of universal objects on page 1127.)