1 Basic Definitions

The Mandelbrot set is the parameter space for the iteration of the quadratic family of maps FC(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

xn+1 = Fc(xn)

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 Fi is

0, i, -1 + i, -i, -1 + i, -i,...

which eventually cycles with period 2. On the other hand, c = 2i does not lie in M since the critical orbit for F2i is

0, 2i, -4 + 2i, 12 - 14i, -52 -334i,...

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 Jc , 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

xn = r{2^n} exp(i 2n t)

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

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

|xn+1| > |xn|

provided that

|xn| > 2 and |xn| > |c|

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:

|xn+1| > |xn|2 - |c| > |xn|2 - |xn|

by the Triangle Inequality. Therefore

|xn+1| > |xn|(|xn| - 1) > |xn|.

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

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