ABSTRACT
This chapter examines how the added assumption of analyticity introduces a new wrinkle into a dynamical system. A complex analytic map always decomposes the plane into two disjoint subsets, the stable set, on which the dynamics are relatively tame, and the Julia set, on which the map is chaotic. The chapter describes this chaotic behavior in detail and to sample the types of stable behavior that can occur. It then discusses the family of quadratic maps. There are basically three types of periodic points which may occur for a complex analytic map, attracting, repelling, and indifferent or neutral periodic points. The chapter also derives the basic properties of the Julia set of a polynomial map of the complex plane and presents variety of examples of Julia sets of polynomials and rational functions. It discusses the difficult problem of the behavior of an analytic function in the neighborhood of a neutral or indifferent periodic point.
