ABSTRACT
This chapter presents some of the mathematics behind one of the most intricate and interesting shapes in dynamics, the Mandelbrot set. This set, first viewed in 1980 by the mathematician Benoit Mandelbrot, has been the subject of intense research ever since. The Fundamental Dichotomy indicates that there are only two basic types of filled Julia sets for Qc, those that are connected and those that consist of infinitely many disjoint components. Moreover, it is the orbit of 0 that determines which of these two cases hold. In the Mandelbrot set, the period-2 bulb is the large circular region visible just to the left of the main cardioid. In the early 1980's, Adrien Douady and John Hubbard joined forces to study the structure of the Mandelbrot set. Using tools from complex analysis and dynamical systems, they managed to prove that, despite appearances, this set is connected.
