Fig.1 shows, how the classic Cantor set emerges in dynamics
of the tent map tc(x)
We will get a general Cantor set, if we cut i.g. the central 1/3, then 1/9, then 1/27, etc., of each piece. The resulting set is topologically equivalent, but the holes decrease in size sufficiently fast so that the "fat" Cantor set has positive Lebesque measure and fractal dimension 1 .
You can find more about Cantor sets in the Math World and Math Academy Online.
It is well known (see e.g. ), that for c = -1.7542 value
(corresponding to the regular dynamics of fc )
almost all initial points in the real interval [-a, a] are attracted
to the period-3 orbit (i.e. they belong to a circle). That is all the rest
points have the zero Lebesgue measure and make a classic Cantor set.
On the other hand, in the real interval -2 < c <1/4 , regions with chaotic dynamics have nonzero Lebesgue measure and make a "fat" Cantor set (see below).
But there are many respects in which nonlinear dynamical systems can behave in complicated, unpredictable ways. J. Doyne Farmer noted , that two qualitatively different types of dynamical behavior (regular and chaotic regions in the previous picture) is so tightly interwoven that it becomes impossible to predict when a small change in parameter will cause a change in qualitative properties. Thus quadratic maps have sensitive dependence on parameters.
Moreover even regular maps (with an attracting periodic orbit) are
unpredictable in some way. Consider again the map with period-3
orbit for c = -1.7542 . The map fco3
has 3 attracting fixed points. In the picture below J(0) midgets
attracted to the same fixed points are colored in the same colors.
There is a funny "traffic lights" rule: in any interval, between two
biggest midgets with different colors one more biggest circle has the third
color (red, blue and green in the picture). Therefore all colors are dense
in the [-a, a] interval. You see, that basins of attraction of
different fixed points are tightly interwoven again.
The regular map has sensitive dependence on initial conditions as since you can find two arbitrary close points which diverge under iterations (asymptotically they go to the same cycle but with different "phases").
Close orbits diverge exponentially at every unstable fixed point or periodic orbits. But it seems that something interesting takes place after the period doubling cascade at the Myrberg-Feigenbaum point, as since for c < -1.401155 unstable periodic orbits become very dense.
 J. Doyne Farmer Sensitive dependence on parameters in nonlinear
Phys.Rev.Lett. 55, 351 (1985)