Properties of the Julia Set of a Rational Map

Abstract

The theory of iteration of rational maps is a relatively new area in mathematics, which has enjoyed a bit of a renaissance the last three decades thanks to computer images that reveal the beauty of the Mandelbrot set and various Julia sets. I first stumbled upon images of some Julia sets and the Mandelbrot set while searching for images of the Cantor set and the Sierpinski triangle. Certainly, like many, I was first attracted to this area of mathematics because of the obvious complexity of these sets that these images revealed. Though what ultimately hooked me in was the complex analysis involved. I had never heard of pointwise convergence or considered the notion of a sequence of functions before my study of iterating rational functions began. I would have had too much fun in analysis with so many different types of epsilon arguments to choose from. Indeed I am quite pleased with the mathematics behind the theory of iteration of rational maps. To this end, we seek out some basic results, where for the most part the theme involves either a family of maps or the image of a set under a rational map.