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 10.4 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. 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).