Parameter Planes for Complex Analytic Maps

Abstract

In this paper we describe the structure of the parameter planes for certain families of complex analytic functions. These families include the quadratic polynomials z ² + c, the exponentials λ exp(z), and the family of rational maps $z^{n} +\lambda /z^{n}$.These are, in a sense, the simplest polynomial, transcendental, and rational families, as each has essentially one critical orbit.

In this paper we give a brief overview of the structure of the parameter plane for three different families of complex analytic maps, namely quadratic polynomials (the Mandelbrot set), singularly perturbed rational maps, and the exponential family. The goal is to show how these objects allow us to understand almost completely the different dynamical behaviors that arise in these families as well as the accompanying bifurcations.