We see that after one application of fc , there are no
points in [-x2, c). The segment (c2+c,
x2] is stretched every iteration. Points leave it and never
return back. Thus eventually all points from Ic come into
[c, c2+c] attractor, bounded by the g1(c) =
c and g2(c) = c2+c curves.
You see below intervals Ic (the left picture) and chaotic attractors (the right one) for different c values.
From x = 2 cos(2p y) one gets
|dx| = 4p |sin(2p y)| dy = 2p (4 - x2)1/2 dy .
For the sawtooth map with the uniform density relative number of points of a chaotic orbit in a small interval dy is r(y) dy = dy. As since all these points are mapped in interval dx, therefore the number is equal to r(x)|dx| for our quadratic map and corresponding invariant density is (there is 2 below because y is mapped in x twice)
r(x) = 2 dy / |dx| = 1/p (4 - x2)-1/2 .
The density is shown to the left in blue-green-red colors.
The average expansion along a chaotic orbit for c = -2 is
ò |2x| r(x) dx = 8/p = 2.54648 .
You see below more complicated chaotic orbits and invariant densities
For quadratic maps it is proven that the set of c values for which attractor is chaotic has positive Lebesque measure and attracting periodic orbits are dence in the set, i.e. between any two chaotic parameter values there is always a periodic interval . We will see these amazing structures on the next page.
 Mikhail Lyubich The Quadratic Family as a Qualitatively Solvable Model of Chaos Notices of the AMS, 47, 1042-1052 (2000)