ABSTRACT
You may have already seen beautiful and facinating pictures of the Julia set and the Mandelbrot set. These pictures become even more intriguing if we zoom in at finer scales. Images of Julia sets and Mandelbrot sets can be found on posters, book covers, T-shirts, carpets, screen savers, and web pages. Moreover, these images have captured the imagination of mathematicians and the public at large alike. The mathematics behind the beauty of these pictures is part of a branch of mathematics called complex dynamics (see for example [15] for a readable account). Remarkable progress on iterating complex functions, not just real functions, was made by Gaston Julia [52] and Pierre Fatou [38]. This is indeed a remarkable feat considering that all this was done before computer graphics were available to them. The subject stayed moribund, however, until Benoit Mandelbrot made it
popular after the appearance of his seminal book “The Fractal Geometry of Nature” in 1982. Taking advantage of the availability of computer graphics, Mandelbrot has created some of the most facinating pictures ever produced mathematically. More importantly, he was able to lay down the foundation of his new objects “Mandelbrot sets” and to give a great impetus to an otherwise dormant subject.
