We have seen that the introduction of complex numbers into linear polynomials creates interesting new dynamical phenomena involving rotations and spirals. With the elaborate structure of the orbits of quadratic polynomials and the bifurcation diagram described in the previous chapter, we should truly start salivating at the prospect of infusing complex arithmetic into the dynamics of quadratic and higher degree polynomials. Indeed, theoretical studies of iteration of complex polynomials have fascinated pure mathematicians since the early twentieth century. However, in a certain sense the structure was too rich to take in at that time. After a flurry of theorems of great depth, the subject of iteration of complex polynomials lay largely dormant until 1980.
At that time along came the master Synthesist, Mandelbrot. It is true
that even before Mandelbrot's revolutionary article
[Man80] serious pure mathematicians had recognized that
there were extraordinary phenomena lurking in the dynamics of even as
simple a polynomial as 
, where c is an arbitrary complex
constant. The question of whether or not the sequence of iterates
diverges to infinity arose in a study of discrete groups by
Brooks and Matielski, and those authors made a picture of what came to
be known as the Mandelbrot set. Their plot consisted of a set of
asterisks made on a terminal screen (80 by 25 pixels; state-of-the-art
for most mathematicians at the time!). It surely suggested interesting
structure, but nothing of the fractal enormity unearthed by
Mandelbrot's harnessing of the world's finest graphics at the Thomas
J. Watson Research Center. Mandelbrot's attention to detail in
physically describing this object is what caused the world's
mathematicians to begin to pay attention to this subject.
In this chapter, we hope to present further details of this amazing story as it has evolved since then. Unfortunately, our term is at an end and I will have to suspend for the time being production of these notes. Fortunately, this topic has already had a vast number of skilled expositors labor on it, and for the time being (until Spring 1997?) we can link the interested reader directly to Robert Devaney's wonderful web pages listed below:
The master research work at this time in this area is [McM94]. The master (with a few fine pictures) may be found at
Curt McMullen's Home Page