The Mandelbrot set and the associated Julia sets are objects in the complex plane. This page will not attempt to give an introduction to the complex numbers, so the reader is assumed to be familiar with them.
Consider the function f(z)=z*z where z is a complex variable. This can be thought of as a formula for moving points in the complex plane. That is, each complex number z is moved to z*z. Thus 1 does not move, i goes to -1, -1 goes to 1, 1+i goes to 2i and so forth. We refer to this motion as saying that f acts on the complex plane.
There are two obvious "fixed points" of f. These are 1 and 0 which f does not move. They are the two solutions to the equation f(z)=z or z*z=z. We also think of infinity as a fixed point since complex numbers near infinity (far from 0) stay near infinity (far from 0).
Now consider the composition f(f(z)). This will take i to 1 in two steps. It takes 1+i to -4 in two steps. If we consider f(f(f(z))), then this takes 1+i to 16 in three steps.
We can ask for the behavior of higher and higher compositions of f with itself. The properties of the long term behavior of higher iterates of a function such as f(z)=z*z make up the "dynamics" of f. The study of the dynamics of functions is called "dynamical systems." The Mandelbrot set and its associated Julia sets are objects that give information about the dynamics of functions such as f(z)=z*z.
Under iterates of f(z)=z*z, each complex number z follows a "path" called the forward orbit of z. This consists of the sequence z, f(z), f(f(z)), f(f(f(z))), f(f(f(f(z)))), etc. We would like to understand the orbits to whatever extent possible. This turns out to be easy to describe for most complex numbers, and more complicated to describe for the rest. For all z inside the unit circle centered at 0, the orbits all tend to 0. For all z outside the unit circle, the orbits all tend to infinity. On the unit circle, the orbits are more complicated.
Some of the orbits on the unit circle end at 1. These are the orbits of fractional powers of 1 with the denominator of the fraction an integral power of 2. Some of the orbits are finite and cyclically repeating. We such an orbit a "periodic orbit" and the number of points in it the "period" of the orbit. The two non-real cube roots of 1 form a finite cyclic orbit of two points. Some orbits end in a periodic orbit, and lastly, some orbits never repeat, never stop and never reach a limit. These orbits are orbits of points on the unit circle that are an irrational multiple of 2*pi from 1. Each of these last orbits gets arbitrarily close to any point on the unit circle.
Thus we have three regions associated with f(z)=z*z. Two regions tend to two of the fixed points of f, namely 0 and infinity. We define a fixed point to be an "attracting fixed point" if all points near the fixed point have orbits that have that fixed point as a limit. Thus 0 and infinity are attracting fixed points, and the regions that tend to them are their "basins of attraction." The fixed point 1 is not attracting since it has points nearby whose orbits do not tend to 1. The third region is the unit circle. No point there has an orbit that has an attracting fixed point as a limit.
One can also have "attracting periodic orbits." For example, consider f(z)=z*z+c where c is a complex constant. Now f(z)=z or z*z-z+c=0 also has two solutions giving two finite fixed points, and the usual infinite fixed point. The equation f(f(z))=z or f(f(z))-z=0 is a quartic with 4 finite solutions. Two are the solutions to f(z)=z since a fixed point for f is a fixed point for f(f). Thus z*z-z+c is a factor of f(f(z))-z and long division gives z*z+z+(c+1) as the other factor. The equation z*z+z+(c+1)=0 has one root if c=-3/4. For other values of c, we get a periodic orbit of period two. It turns out that the orbit will be attracting if the derivative of f(f(z)) at one of the points in the orbit is inside the unit circle. This is easy to calculate since the derivative of f(f(z)) is 4*z*f(z) when f(z)=z*z+c. Now if z is in a periodic orbit of period 2, then f(z) is the other point in the orbit. Thus the derivative of f(f(z)) at such a point is 4 times the product of the two points or 4 times the product of the two roots of z*z+z+(c+1)=0. This calculation eventually leads to the conclusion that 4(c+1) must lie inside the unit circle. Equivalently, c must lie inside the circle of radius 1/4 centered at -1. The points inside this circle will show up later.
We now concentrate on two regions for a given f. The first is the set of points whose orbits approach an attracting periodic orbit or attracting fixed point. The second is what is left over. What is left over is called the "Julia set of f." Thus the Julia set of f(z)=z*z is the unit circle.
We can now define the Mandelbrot set. This is associated with the familiy of functions f(z)=z*z+c. Each of these functions has a Julia set, and it turns out that either the Julia set is connected (as is the case for f(z)=z*z), or breaks into an infinite number of pieces. This gives a rough categorization into two types of behavior of the functions of the form f(z)=z*z+c. The behavior depends on c. The Mandelbrot set is the set of those c for which the function f(z)=z*z+c has a connected Julia set. At the moment we know that c=0 is in the Mandelbrot set.
It turns out that the Mandelbrot set is quite complicated. Here is a small picture of the Mandelbrot set with a small cross at the point c=0. Click on it for a larger image. The larger image covers real and imaginary values from -2 to 2.
THere are two large, obvious features in the image. One is a large cardioid that contains zero and that has most of the area of the set. Its cusp is at 1/4+0i the point opposite the cusp is at -3/4+0i. The next largest feature is a disk tangent to the cardioid at -3/4+0i. This disk has center at -1+0i and radius 1/4. Recall that the interior of this disk is the set of points c for which f(z)=z*z+c has an attracting periodic orbit of period 2. It is not hard to show that the cardioid is the set of points c for which f(z) has an attracting fixed point. (Those c values where z*z-z+c=0 has a root z for which the derivative of f at z is inside the unit circle.)
The coloring of the image needs some explanation. It turns out that if the orbit of z=0 under f(z)=z*z+c tends to infinity, then the Julia set is not connected and c is not in the Mandelbrot set. It further turns out that if the orbit of z=0 ever gets outside the circle of radius 2 about 0 then the orbit tends to infinity. This gives a partial test for not being in the Mandelbrot set. Simply follow the orbit of 0 and see if it ever gets outside the circle of radius 2. Of course one can only follow an orbit for a finite number of steps. This is used to get the picture shown as follows.
A maximum number of steps is chosen. Then orbits of points are followed until either the orbit leaves the circle of radius two around 0, or until the maximum is reached. The number of steps is recorded. The image is created by coloring those points that took the same number of steps the same color. It can be shown that the Mandelbrot set lies inside the circle of radius 2 about 0, so the scope of testing is limited. The bands of one color are the "approximating bands" to the Mandelbrot set. The set inside all of these bands (shown yellow in the large image and white in the small image) is the best approximation the Mandelbrot set that can be obtained with that maximum number of steps. The maximum number of colors used in these images is 240. The image on this page uses exactly one color per band. Images on later pages may combine or split bands to conserve on colors or to enhance the image.
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