# Mandelbrot Indexing Julia Sets

(An index may have a content of its own.)Given a function f(x) and a starting value x_{0} one can construct a new value _{1} = f(x_{0})._{2} is obtained by another application of _{2} = f(x_{1)}) = f(f(x_{0})).*iterative* process that, generally speaking, generates a sequence x_{0}, x_{1}, ..., x_{k}, ..., where x_{k} is the *k*th iterate obtained by applying the function f to x_{0} k times. The sequence is known as an *orbit* of its starting point x_{0}. 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 x_{0}. 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, x so that the sequence repeats itself_{0}= x_{p} - 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 _{c}(z) = z^{2} + c*Mandelbrot* set for the family f_{c}(z).
To obtain the Mandelbrot set, run iterations _{k+1} = f_{c}(z_{k})_{0} = 0_{k}| becomes greater than _{k}| > 2,_{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 x_{0} = 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 _{k+1} = f_{c}(z_{k})_{0} 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.

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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