The Mandelbrot set is the parameter space for the iteration of the
quadratic family of maps ** F _{C}(x) = x^{2} + c**.
Here both

**x _{n+1} = F_{c}(x_{n})**

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 **
x ^{2} + c**. This
set, denoted

so the orbit of ** x _{0}**
escapes if and only if

In general, ** J _{c}** is a much more complicated set; its boundary is a
fractal (unless

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**.

Fractal Geometry of the Mandelbrot Set (Cover Page)

Fractal Geometry of the Mandelbrot Set (Previous Section)