Cantor-like sets

Cantor sets and the Tent map dynamics

Tent map Fig.1 shows, how the classic Cantor set emerges in dynamics of the tent map tc(x)
    tc(x) = cx,
c(1 - x),
x £ 1/2
x > 1/2
for c = 3 . Really, as since xo = 0 is an unstable fixed point, therefore iterations for x outside the [0, 1] interval diverge to infinity. t(x) maps open interval (1/3,2/3) beyond [0,1], thus we can throw away these points too. As since intervals [0,1/3] and [2/3,1] are mapped linearly onto [0,1], therefore we can continue this process ad infinitum. In every iteration we cut the central one third of an interval. They turn into separate points when number of iterations go to infinity (if you believe that a point is a very small circle with radius r->0 :) The limit Cantor set is nowhere dense as since it has holes in any small interval.
The lenght of all removed intervals is
    1/3 + 2/9 + 4/27 + ... = 1/3 Sn=0,¥ 2n / 3n = 1/3 [1 / (1-2/3)] = 1 ,
therefore Lebesque measure (length) of the classic Cantor set is zero. Its fractal dimension is log2 / log3 . You can test easy, that a Cantor set with the zero length will appear again, if the central 1/4 or 1/a (a > 1) piece is removed in each iteration.

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.

Classic or "fat"?

We meet complicated Cantor-like structures made of infinite sequences of tiny midgets in the Mandelbrot and Julia sets filaments.
It is well known (see e.g. [1]), 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).

As we know (see Misiurewicz points and the M-set self-similarity), near a Misiurewicz point the Mandelbrot midgets shrink by a factor of l2 but the Julia midgets miniaturize only as l . Therefore topologically similar M and J sets have different metric properties.

Sensitive dependences

Chaotic dynamics is explained by exponential divergence of close orbits and sensitivity to initial conditions (see Chaotic sawtooth map).

But there are many respects in which nonlinear dynamical systems can behave in complicated, unpredictable ways. J. Doyne Farmer noted [1], 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.

[1] J. Doyne Farmer Sensitive dependence on parameters in nonlinear dynamics
Phys.Rev.Lett. 55, 351 (1985)

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updated 28 May 2003