Pictures of Julia and Mandelbrot Sets/History

Newton
Already in the antiquity it was known, that if the fraction $$x_0$$ is near &#8730;a, then the fraction $$x_1 = (x_0 + a/x_0)/2$$ is a better approximation to &#8730;a. For if $$x_0$$ is an approximation to &#8730;a that is smaller than &#8730;a, then $$a/x_0$$ is an approximation to &#8730;a that is larger than &#8730;a, and conversely, therefore $$x_1 = (x_0 + a/x_0)/2$$ is a better approximation to &#8730;a than both $$x_0$$ and $$a/x_0$$. We can, of course, repeat the procedure, and it is very effective: you get an approximation to &#8730;a with ten correct digits, if you start with $$x_0$$ = 1.5 and apply the procedure three times.

To find the square root of $$a$$, is to solve the equation $$x^2 - a = 0$$, and we have solved this equation by the iteration $$x$$ → $$(x + a/x)/2$$. And it is this method we must apply for solving an equation, if we do not have a formula, or if this - after the invention of the computer - seems too laborious to use.

It was Newton who described the general procedure: if $$f(x)$$ is a (continuous) differentiable function which intersects the x-axis near $$x = x_0$$, and we apply the iteration $$x$$ → $$x - f(x)/f'(x)$$ a number of times starting in $$x_0$$, then we get a good approximation to a solution x* of the equation $$f(x)$$ = 0 (because we have assumed that $$f(x)$$ intersects the x-axis, $$f'(x)$$ <> 0 near the point x* where $$f(x*)$$ = 0).

The method can be generalized, so that it works for $$f'(x*)$$ = 0, but if $$f'(x)$$ converges to ∞ for x converging to x*, as it is the case for $$f(x) = x^{1/3}$$ and x* = 0, then the procedure leads away from the solution.

There are situations where a solution cannot be found by this method, and there can be points whose sequence of iteration either converges towards a cycle of points (containing more than one point) or does not converge. This question ought to be studied more closely, advertised the English mathematician Arthur Cayley in 1879 in a paper of only one page. He was aware, that the Newton procedure could as well be applied for solving a complex equation $$f(z)$$ = 0, and he proposed that we "look away from the realities" and examine what can happen when the iteration does not lead to a solution. But this proposal was his only contribution to the theory.

Julia and Fatou
Thirty years passed before anyone took up this question. But then the two french mathematicians Gaston Julia (1893-1978) and Pierre Fatou (1878-1929) published papers where they examined iteration of general complex rational functions, and they proved all the facts about "Julia" sets and "Fatou" domains we have used here. Julia and Fatou were of course not able to produce detailed pictures showing the Julia set for Newton iteration for solving the equation $$z^3$$ = 1, for instance, but they knew that the Julia set in this case is not a curve. Cayley certainly imagined that the plane would be divided up by geometrical curves, and if he did so, it was not unforgivable, for this is namely the case for the so-called weak Newton procedure: this leads more slowly but more safe to a solution.

We have intimated, that our mean results regarding Julia sets, were known to Julia and Fatou. This is not quite true, for the fact that the sequences of iteration in the neutral case can go into annular shaped revolving movements, was first discovered in 1942 by the German mathematician C.L. Siegel. We have said that these revolving movements can be either polygon or annular shaped, that they are lying concentrically and that there is a finite cycle which is centre for the movements. And possibly the reader has wondered: this centre cycle, does it belongs to the Fatou domain or to the Julia set? We will now answer this question. In the section about field lines we have defined a complex number $$\alpha$$ which in the attracting case has norm smaller than 1, but which in the neutral case has norm 1 and therefore corresponds to an angle (its argument). When this angle is rational (with respect to $$2\pi$$), the terminal movements are finite (polygonal, the parabolic case), and the centre cycle belongs to the Julia set, when the angle is irrational, the terminal movements are infinite (annular, the Siegel-disc case), and the centre cycle belongs to the Fatou domain.

Mandelbrot


But apart for such few results, not very much happened in this theory before Benoit B. Mandelbrot (1924-2010) in the late 1970s began his serious study of Julia sets by using computer. It is Mandelbrot who has coined the word fractal geometry. He had had Julia as teacher (at École Polytechnique), and around 1964 he began his "varied forays into unfashionable and lonely corners of the Unknown". But the fractal patterns he first studied were self-similar in the strict sense of this word, namely invariant under linear transformation. It was first in 1978-79 he began his study of Julia sets for rational complex functions. He made some print-outs and he studied families of rational complex functions, formed by multiplying the function by a complex parameter $$\lambda$$. His intention was to let the computer draw a set M of parameters $$\lambda$$ for which the Julia set is not (totally) disconnected (a "fractal dust"). Such a program could be made very simple: he found two critical points for the function, and plotted the points $$\lambda$$ for which the two critical points did not iterate towards the same cycle. For then there would be at least two Fatou domains, and therefore the Julia set would not be a dust cloud. He chose functions for which he knew a real parameter value $$\lambda$$ such that the iteration behaved chaotically on some real interval, for then this $$\lambda$$ might belong to his set M.

He began (in 1979) with the family $$\lambda (1 + z^2)^2/(z(z^2 - 1))$$ (having four real and two imaginary critical points). For $$\lambda$$ = 1/4 it behaves chaotically on an interval, and he "felt that in order to achieve a set having a rich structure, it was best to pick a complicated map (every beginner I have since then watched operate has taken the same tack)". The picture where $$\lambda$$ varied over the complex plane, showed a highly structured but very fuzzy "shadow" of the set. A very blotchy version, but "it sufficed to show that the topic was worth pursuing, but had better be persued in an easier context".

Then (in 1980) he studied the family $$\lambda z(1 - z)$$ (having critical points 1/2 and ∞), which for $$\lambda$$ = -2 and 4 behaves chaotically on the interval [-2, 2]. He saw two discs of radius 1 and centres in 0 and 2: "Two lines of algebra confirmed that these discs were to be expected here, and that the method was working. We also saw, on the real line to the right and left of the above discs, the crude outlines of round blobs which I call "atoms" today. They appeared to be bisected by intervals known in the Myrberg theory, which encoraged us to go on to increasing bold computations. For a while, every investment in computation yielded increasing sharply focussed pictures. Helped by imagination, I saw the atoms fall into a hierarchy, each carrying smaller atoms attached to it".

"After that, however, our luck seemed to break; our pictures, instead of becoming increasingly sharp, seemed to become increasingly messy. Was this the fault of the faltering Textronix [cathode ray tube ("worn out and very faint")]?". Mandelbrot ran the program on another computer: "The mess had failed the vanish! In fact, as you can check, it showed signs of being systematic. We promptly took a much closer look. Many specks of dirt duly vanished after we zoomed in. But some specks failed to vanish; in fact, they proved to solve into complex structures endowed with "sprouts" very similar to those of the whole set M. Peter Moldave and I could not contain our excitement. Some reasons made us redo the whole computation using the equivalent map z → $$z^2 - c$$, and here the main continent of the set M proved to be shaped like each of the islands! Next, we focussed on the sprouts corresponding to different orders of bifurcation, and we compared the corresponding off-shore islands. They proved to lie on the intersection of stellate patterns of logarithmic spirals! (...) We continued to flip in this fashion between the set M and selected Julia sets J, and made an exciting discovery. I saw that the set M goes beyond being a numerical record of numbers of points in limit cycles. It also has uncanny "hieroglyphical" character: including within itself a whole deformed collection of reduced-size versions of all the Julia sets".

References

Mandelbrot, B.B.: Fractals and the Rebirth of Iteration Theory. In: Peitgen & Richter: The Beauty of Fractals (1986), pp 151-160 (in this article you can see Mandelbrots first two print-outs of the last picture).

Mandelbrot, B.B.: Fractal aspects of the iteration of z → $$\lambda z(1 - z)$$ for complex $$\lambda$$ and z. In: Nonlinear Dynamics, Annals New York Acad. Sciences 357 (1980), pp 249-259.