## ABSTRACT

This chapter introduces very different types of complex functions. The study of cubic polynomials is quite similar to the study of quadratics, with one major exception. The difference is that the typical cubic polynomial has two critical points, not just one. For example, a cubic polynomial may have two distinct attracting fixed or periodic orbits. The chapter focuses on a very different and more complicated class of complex functions, namely, rational maps. What makes these maps more complicated is the fact that rational maps often have many more critical points than polynomials. An amazing theorem in complex dynamics says that, if the critical orbits do not meet the Julia set and the Julia set is connected, then this set must also be locally connected. Another theorem from complex analysis (Montel’s Theorem) says that, if the Julia set ever contains an open set (no matter how small), then, in fact, the Julia set must be the entire Riemann sphere.