Julia Sets: Fractal Basin Boundaries

Abstract

The goal of this chapter is to demonstrate how genuine mathematical research experiments open a door to a seemingly inexhaustible new reservoir of fantastic shapes and images. Their aesthetic appeal stems from structures which are beyond imagination and yet, at the same time, look strangely familiar. The ideas we present here are part of a world wide interest in so called complex dynamical systems. They deal with chaos and order, both in competition and coexistence. They show the transition from one condition to the other and how magnificently complex the transitional region generally is. One of the things many dynamical systems have in common is the competition of several centers for the domination of the plane. A single boundary between territories is seldom the result of this contest. Usually, an unending filigree entanglement and unceasing bargaining for even the smallest areas results. We studied the quadratic iterator in chapters 1,10 and 11 and learned that it is the most prominent and important paradigm for chaos in deterministic dynamical systems. Now we will see that it is also a source of fantastic fractals. In fact the most exciting discovery in recent experimental mathematics, i.e., the Man-delbrot set, is an offspring of these studies. Now, about 10 years after Adrien Douady and John Hamal Hubbard started their research on the Mandelbrot set, many beautiful truths have been gained about this ‘most complex object mathematics has ever seen’. Almost all of this progress stems from their work.

I must say that in 1980, whenever I told my friends that I was just starting with J. H. Hubbard a study of polynomials of degree 2 in one complex variable (and more specifically those of the form z → z² + c), they would all stare at me and ask: Do you expect to find anything new? It is, however, this simple family of polynomials which is responsible for producing these objects which are so complicated — not chaotic, but on the contrary, rigorously organized according to sophisticated combinatorial laws.¹

Adrien Douady