The Mandelbrot Set and Julia Sets

Julia Sets - Dichotomy Theorem

Fatou and Julia proved that for any polynomial f,

if all the critical points of f belong to Kf, then Jf is connected

and

if none of the critical points of f belong to Kf, then Jf is a Cantor set.

The function f(z) = z2 + c has one critical point, z = 0, and so

if the iterates of z0 = 0 do not escape to infinity, then Jc is connected,

and

if the iterates of z0 = 0 do escape to infinity, then Jc is a Cantor set.

For polynomials with several critical points, the situation can become much more interesting. For example, suppose a function has two critical points, one escaping to infinity, the other not. How will the Julia set look? Can it be something other than connected or a Cantor set? Here is an example.

Return to the Dichotomy Theorem.