Mandelbrot Indexing Julia Sets

(An index may have a content of its own.)

Given a function f(x) and a starting value x₀ one can construct a new value x₁ = f(x₀). With some persistence, the next value x₂ is obtained by another application of f: x₂ = f(x₁₎) = f(f(x₀)). This is an iterative process that, generally speaking, generates a sequence x₀, x₁, ..., x_k, ..., where x_k is the kth iterate obtained by applying the function f to x₀ k times. The sequence is known as an orbit of its starting point x₀. Gaston Julia (1893-1978) and Pierre Fatou (1878-1929) made a fundamental contribution to the study of iterative processes. Their contribution (Ref [3]), although regarded as a masterpiece, was largely ignored by the mathematical community until a revival in the late 1970s spawned by the discovery of fractals by Benoit Mandelbrot.

For a given function f, behavior of an orbit very much depends on the selection of the starting point x₀. Following is a rough classification of possible behaviors:

Convergence: the sequence of points {x_k} converges to a limit
Periodic cycle: for some p > 0, x₀ = x_p so that the sequence repeats itself
Chaos: none of the above. The points {x_k} go from one place to another in apparently chaotic manner

The set of points with chaotic orbits is called the Julia set for a given function f. Until quite recently the study of iterations and Julia sets has been in a prolonged limbo. B. Mandelbrot has the following to say on the development of the theory,

The resulting revival makes the properties of iterations essential for the theory of fractals. The fact that the Fatou-Julia findings did not develop to become the source of this theory suggests that even classical analysis needs intuition to develop, and can be helped by the computer.

B.Mandelbrot has discovered a way to index Julia sets for parametric families of functions. The applet below illustrates this concept for a simple function f_c(z) = z² + c where z and c are complex. For every c there exists a Julia set; and a related picture appears in the right part of the area. Values for c can be picked from the picture on the left that, loosely speaking, depicts the Mandelbrot set for the family f_c(z). To obtain the Mandelbrot set, run iterations z_k+1 = f_c(z_k) with z₀ = 0 and c varying in some bounded area (below a rectangle with opposite corners (-2.2, -1.4) and (0.8, 1.4)). It's known that once |z_k| becomes greater than 2: |z_k| > 2, the iterations will eventually escape to infinity. For every c, mark the iteration k_c at which this condition first becomes true. Associate with the point c a color number k_c from a given palette of colors. This will produce a picture on the left. The Mandelbrot set is the set of c's for which the iterations starting with x₀ = 0 are bounded. This is the set that consists of the enterior cardioid-like shape with a circle attached on its left. Each of the two has smaller warts attached which have some more, adding to the ugliness (or is it the beauty?) of the curve.

As one can see, the algorithms for obtaining the Mandelbrot and Julia sets are virtually the same. For a given (fixed) c, in order to visualize the Julia set, run iterations z_k+1 = f_c(z_k) starting with various z₀ ranging in a rectangular area. Associate different colors with different starting points depending on how fast (or slow) iterations converge (or diverge).

A few words on the applet itself. Once the Mandelbrot set is drawn, you can select a value of c by clicking anywhere (any time) inside the left portion of the display. The corresponding Julia set will be getting drawn in the right portion of the display.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

What if applet does not run?

References

J. Gleick, Chaos, Viking, 1987
D. R. Hofstadter, Metamagical Themas, Basic Books, Inc., 1985, Chapter 16.
B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Co., NY, 1977.

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