138 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER S ERIES
Definition 4.7: Attracting point
The point ZQ is an attracting po int for the function f if If' (zo)I < 1.
T heorem 4.10 explains the significance of these terms.In 1905, Fatou showed that if the function fc defined by fe (z) = z^2 + c has
attracting fixed points, then the orbit of 0 determined by f c mu&t converge to
one of them. Because a convergent sequence is bounded, this condition implies
that c must belong to M. In the exercises we ask you to show that the main
cardioid-shaped body of M in color plate 6 is composed of those points c for
which fe has attracting fixed points. You will find Theorem 4.11 to be a useful
characterization of those points.