Critical points of iterates of complex quadratic functions

Abstract

We look at a familiar one-parameter family of quadratic functions on the complex plane. After restricting the parameter to be real, we explore when the critical points of the functions and their iterates are real and when they are not real. We prove that when the parameter is greater than or equal to 2, all critical points are real. When the parameter is between 0 and 2, critical points for the original function are real but there is an iterate with nonreal critical points. When the parameter is equal to 0, all critical points are 0. When the parameter is less than 0, the critical points are all 0 or are nonreal. Finally, we compare the locations of these critical points to the contour plots of the real parts of the functions for different values of the parameter and for different iterates of the function.