This page is an extraction of a small part of Adrien Douady's excellent article Julia sets and the Mandelbrot set in The beauty of fractals: images of complex dynamical systems by H.-O. Peitgen and P. H. Richter, Springer-Verlag publ., 1986, pp. 161-173.
The family of functions f(z)=z*z+c as c varies over the complex plane has its dynamic behavior crudely classified by the Mandelbrot set: the function f(z)=z*z+c has a connected Julia set if and only if c is in the Mandelbrot set. The Mandelbrot set gives a bit more information since the shape of the Mandelbrot set near c gives hints as to the appearance of the Julia set for f(z)=z*z+c.
The family of functions f(z)=z*z+c has infinitely many members, but it is hardly an enormous family. It is a subset of the quadratic polynomials. What about other families of functions from the complex plane to itself? Are there sets like the Mandelbrot set that give information about the dynamics of those families? It turns out that the answer is yes in a strong way. For many other families of functions, sets that classify their dynamic properties incorporate almost identical copies of the Mandelbrot set.
The reason for this is that the family of functions f(z)=z*z+c captures the typical behavior of many high iterates of functions near places where the derivative is zero. That is, in a small region of the complex plane, after rescaling and shifting, a high iterate of a family of functions will resemble the collection f(z)=z*z+c strongly enough to have the dynamics there imitate that of the family f(z)=z*z+c. In such a region, a (somewhat distorted) copy of the Mandelbrot set will show up. In fact, this happens infinitely often to high iterates of the family f(z)=z*z+c itself, and accounts for the infinitely many small copies of the mandelbug inside the Mandelbrot set. Thus the universality of the family f(z)=z*z+c explains some of the features of its own Mandelbrot set.
To what extent are the copies of the Mandelbrot set like the Mandelbrot set? This requires having a description of the Mandelbrot set, so that the copies can be compared to the description. There is a method, Hubbard trees, for describing all the branching information that is known about the Mandelbrot set: what are the braching indexes of the filaments of all the buds on all of the mandelbugs in the Mandelbrot set. Hubbard trees explain all the known images of the Mandelbrot set. It is known that the information given by the Hubbard trees is accurate, but it is not known if it is complete. That is, all the structural information predicted about the Mandelbrot set by the Hubbard trees is correct, but it is not known if Hubbard trees predict all of the structure.
It turns out that the information in the Hubbard trees will be complete if it can be shown that the Mandelbrot set is locally connected at every point. While the Mandelbrot set is known to be locally connected an many points, there are still infinitely many points where local connectivity is not known. For example, it is not known whether the Mandelbrot set is locally connected at the Myrberg-Feigenbaum point.
Given that the Hubbard trees describe what is known about the Mandelbrot set, we can now say to what extent the Mandelbrot set is universal. For many families of functions, the sets that categorize their dynamics incorporate Mandelbrot sets that agree with all the structure predicted by the Hubbard trees. A certain amount of bending and stretching may be evident, but not enough to destroy the basic structure of the set.
Forward to A catalog of images of the
Mandelbrot set (23 embedded color images).
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Mandelbrot set - Page VI (9 embedded images).
Back to The Mandelbrot Set and Julia Sets.
