Bifurcation in complex quadratic polynomial and some fold theorems involving the geometry of bulbs of the Mandelbrot set

The Mandelbrot set M is a subset of the parameter plane for iteration of the complex quadratic polynomial Q_c( z) = z2 + c. M consists of those c values for which the orbit of 0 is bounded. This set features a basic cardioid shape from which hang numerous ‘bulbs’ or ‘decorations’. Each of these bulbs is a large disk that is directly attached to the main cardioid together with numerous other smaller bulbs and a prominent ‘antenna’. In this thesis we study the geometry of bulbs and some ‘folk theorems’ about the geometry of bulbs involving spokes of the antenna.