III.52 The Mandelbrot Set

Suppose we have a complex polynomial f defined by a formula f(z) = z² + C for some complex number C. Then for any choice of complex number z₀ we can form a sequence z₀, z₁, z₂,. . . by iterating, that is, repeatedly applying, the function f. So we let z₁ = f(z₀), z₂ = f(z₁), and so on. Sometimes the resulting sequence will tend to infinity, but sometimes it remains bounded—that is, it stays within a fixed distance from 0. For example, if we take C = 2 and start with z₀ = 1, then the sequence goes 1, 3, 11, 123, 15 131, . . . and clearly tends to infinity, whereas if we start with z₀ equal to (1 - i), then we find that so the sequence is bounded since all its terms are equal to z₀. The Julia set associated with the constant C is the set of all z₀ for which the sequence remains bounded. Julia sets often have a fractal shape (see [IV.14 §2.5]).

To define a Julia set, one fixes C and considers different possibilities for z₀. What happens if one fixes z₀ and considers different possibilities for C? The result is the Mandelbrot set. The precise definition is that it is the set of all C such that the sequence is bounded if you take z₀ = 0. (One could consider other values of z₀, but the resulting sets are not interestingly different because they are related to each other by a simple change of variables.)

The Mandelbrot set also has an intricate fractal shape—one that has captured the popular imagination. The detailed geometry of the Mandelbrot set is not yet fully understood; some of the resulting open problems are of major importance because they encode very general information about dynamical systems. See DYNAMICS [IV.14 §2.8] for more details.