Chapter 39 - A generalized Mandelbrot set and the role of critical points

Abstract

Since the appearance of high-resolution images of the Mandelbrot set and Julia sets, a considerable amount of computer time has been devoted to exploring the detailed structure of these sets. The usual Mandelbrot and Julia sets are generated by iterating the function Fc(z) = z2 + c, for the Mandelbrot set always starting the orbit at z = 0. Taking z = 0 for a starting point is the natural choice because work of Fatou and Julia shows the iteration sequences generated from the critical points (points at which the derivative of the function is zero) dominates the dynamical behavior, and for Fc(z) the only critical point is z = 0. This chapter examines the Julia sets for certain cubic and quadratic polynomials and observes the structure of the Julia set as determined by the orbits of the critical points. In examples where the orbits of some critical points diverge and others converge, or where orbits converge to different cycles, it has been noted that the local structure of the Julia set is dominated by the behavior of the nearest critical point. This information is encoded in a generalized Mandelbrot set reflecting the behavior of all the critical points.