Julia Sets: An Introduction to Chaos in the Plane

Abstract

In this paper we introduce the reader to a family of functions which give rise to Julia sets. Making their home in the complex plane, these sets exhibit many interesting properties, including chaotic dynamics . The aesthetic quality of Julia sets help us to appreciate more fully the properties which these sets exhibit . We encourage the reader to flip through the graphs presented in Appendix B, evaluating their artistic value now and their scientific value later. The study of these sets began during World War I. The French mathematicians Gaston Julia and Pierre Fatou wrote extensive papers, Julia in 1918 and Fatou in 1919 and 1920, which established the fundamental definitions and theorems of this paper. Largely due to the increased computational power and graphical ability of modern computers, the subject has recently experienced a flood of attention. Among the current mathematicians studying Julia sets, Sullivan has classified the dynamics in the stable set (defined in this paper) while Douady, Hubbard, and Mandelbrot have analyzed the dynamics of quadratic polynomials. This work merely introduces the reader to the subject of Julia sets, which has been explained in much greater depth by more capable mathematicians. In addition to omitting much of the established work of the mathematicians mentioned above , we exclude a great deal of current work on the subject. The references should provide the interested reader with some starting points to pursue the subject in greater detail. [From Introduction]