Introduction

The existence of functions such as Nest and NestList, together with extensive visualization tools, ensure that Mathematica is a natural system for investigating the iteration of mappings. One favourite is the logistic map that was considered in Chapter 5. Complex numbers play a very natural role, from two quite different points of view. First, we wish to understand why maps such as the logistic map behave the way they do, without relying merely on numerical simulation. Second, we wish to extend the use of numerical simulation to mappings of the complex plane.

You explored the first point in Chapter 5, and the second in Chapters 4 and 6, for simple polynomial maps. Now you will consider some other mappings of the complex plane into itself. These mappings are non-holomorphic (in the sense that will be defined properly in Chapter 10 – for now it suffices to realize that this means the functions involve complex conjugation in an essential way). Normally, when considering the theory of complex numbers, one's interest is quite rightly focused on the analytical properties of holomorphic or meromorphic functions. I hope the constructions described here will suggest that there is much that is both beautiful and interesting in complex structures that are not holomorphic. My approach is based on that of Field and Golubitsky (1992), and the use of Mathematica on this topic was first given by the author (Shaw, 1995).