Let x1(c) = 1/2 + (1/4 - c)1/2 is unstable
fixed point. In Fig 1. we take c = -2.5
and denote A = -x1 and D = x1 .
For real maps iterations of points |x| > x1 diverge to
infinity, therefore Julia set J(-2.5) Ì
[A,D]. Open interval (B,C) is mapped outside [A,D],
thus we throw away these points too. As since intervals
[A,B] and [C,D] are mapped onto [A,D], therefore we can
continue this process ad infinitum. Thus intervals double every iteration
and turn into separate points when number of iterations go to infinity.
You see these Cantor-like set on the (x, c) plane (the first picture) and on the complex z plane below.
Animated Cantor trip.
Here you can see yellow outline of the Cantor set in the Mandelbrot
and Julia midgets (the Cantor-like J(-2.15) set is shown for