Figure 1. The Mandelbrot set.
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.