1 Basic Definitions

The Mandelbrot set is the parameter space for the iteration of the quadratic family of maps F_C(x) = x² + c. Here both x and c are complex numbers. Given a seed x₀, the orbit of x₀ is the sequence x₀, x₁, x₂,...> where

Of particular interest is the orbit of 0, the critical orbit. The Mandelbrot set consists of all c-values for which the critical orbit is bounded. We denote the Mandelbrot set by M. See Figure 1. For example, c = i lies in the Mandelbrot set since the critical orbit for F_i is

which eventually cycles with period 2. On the other hand, c = 2i does not lie in M since the critical orbit for F_2i is

and it is easy to check that this orbit tends to infinity.

Figure 1. The Mandelbrot set.

A closely related object is the filled Julia set of x² + c. This set, denoted J_c , consists of all seeds whose orbits remain bounded under iteration of x² + c. For example, J₀ is the closed unit disk centered at the origin in the plane. Indeed, if x₀ = r exp(it), then

so the orbit of x₀ escapes if and only if r>1.

In general, J_c is a much more complicated set; its boundary is a fractal (unless c = 0 or c = -2). To compute J_c, we make use of the fact that

provided that

Thus any orbit that eventually leaves the both the circle of radius 2 and of radius |c| must escape to infinity. This fact is true because of some elementary inequalities:

by the Triangle Inequality. Therefore

This last fact is true since |x_n| - 1 > 1.