The Quadratic Family and the Mandelbrot Set

Abstract

We return once again to the study of the dynamics of quadratic functions. In this chapter, we consider the quadratic family q _c (z)=z²+c. We demonstrated in Exercise 9.5 that all real quadratic functions are topologically conjugate to a real polynomial of the form q _c (x)=x²+c for some c. This fact extends to the complex quadratic polynomials; all complex quadratic polynomials are topologically conjugate to a polynomial of the form q _c (z)=z²+c. (The reader is asked to show this in Exercise 15.1.) We will take direction for our study of the quadratic family from our previous work with the logistic map h _r (x)=rx(1 − x).