Middle thirds set

Cantor s middle thirds set. We denote it by the symbol C. It has recently attracted much attention in connection with chaotic scattering and decay processes (see Sections 1.1 above and 2.3 below, Chapter 8 and Chapter 9). Cantor s middle thirds set is also an example of a fractal, a concept very important in chaos theory (see Section 2.3 for more details). 

Fig. 2.2. Sketch of the first three stages in the construction of Cantor s middle thirds set C.

Let us now study the action of T when it is applied to the points in A. In order to make the connection with atomic physics, we interpret the unit interval A as an atom and the action of the mapping T as its dynamics, induced, e.g., by a periodically applied external field. The unit interval A, the atom at rest, is sketched in Fig. 2.8(a). After the first application of T, all points in the interval Aq = [0,1/3] are mapped onto A. The same holds for the points in the interval Ai = [2/3, Ij. The points in the middle thirds set (1/3,2/3) are mapped outside A. In subsequent applications of T they are mapped to —oo, and are lost. Keeping with our atomic physics analogy we say that the points of the middle thirds set are ionized after the first application of T. In order to represent this situation graphically, we delete the middle thirds set from the unit interval. The result is shown in Fig. 2.8(b). 

According to (2.3.2), there will be no probability left in the unit interval in the limit of n 00. Therefore, we say that a point xq in A ionizes with probabiUty 1 according to the ionization process defined by the tent map T. But is there really nothing left in A after the application of n -i- 00 mappings The successive steps in the ionization process defined by repeated application of T (see Fig. 2.8) remind us strongly of the construction scheme of Cantor s middle thirds set C (Fig. 2.2) which was introduced and briefiy discussed in Section 2.1. And indeed, there is a whole infinity of points left in A, even in the limit of an infinite number of applications of T. What kind of infinity We can easily answer this question with the tools developed in Section 2.1. Let us introduce a notation for the set of points in A that never ionize. We call this set A" ". For the tent map T we have A" " = C. 

Figure 5 Construction process of the structure considered in section 4 a one-dimensional line with a distribution of reactive centres localized on a Cantor middle-third set. The two graphs labelled (A ) are copies of the structure at iteration n, while the graph labelled (0) is a one-dimensional chain at iteration n with no reactive centres.

To give a simple example, let us take Q equal to the one-dimensional line (this example corresponds to the classical slab-like model for a cylindrical pore), and let Qr be a Cantor middle-third set, da = log 2/log 3 = 0.631. Figure 5 shows the construction process of the structure applied in the renormalization process. The graph of the structure at... 

Figure 6 Effectiveness factor vs the Thiele modulus for a distribution of reacting centres on the one-dimensional line localized on a Cantor middle-third set. Curve d) is the theoretical behavior, eqs. (18),(23).

Since is mapped onto A after the first application of T, the middle third of the set is ionized after the second application of T. The same holds for Ai- What remains are the sets j4oo, Aqi, A o and An, as shown in Fig. 2.8(c). 

The Cantor-layered chain discussed here is obtained through a substitution rule based on the usual middle-third Cantor set generation algorithm, e.g. ... 

We start with the closed interval =[0.1] and remove its open middle third, i.e., we delete the interval, ) and leave the endpoints behind. This produces the pair of closed intervals shown as 5,. Then we remove the open middle thirds of those two intervals to produce, and so on. The limiting set C = S is the Cantor set. It is difficult to visualize, but Figure 11.2.1 suggests that it consists of an infinite number of infinitesimal pieces, separated by gaps of various sizes. 

S is totally disconnected. This means that 5 contains no connected subsets (other than single points). In this sense, all points in S are separated from each other. For the middle-thirds Cantor set and other subsets of the real line, this condition simply says that S contains no intervals. 

The paradoxical aspects of Cantor sets arise because the first property says that points in S are spread apart, whereas the second property says they re packed together In Exercise 11.3.6, you re asked to check that the middle-thirds Cantor set satisfies both properties. 

We conclude by mentioning a recent development, although we cannot go into details. In the logistic attractor of Example 11.5.2, the scaling varies from place to place, unlike in the middle-thirds Cantor set, where there is a uniform scaling by everywhere. Thus we cannot completely characterize the logistic attractor by its dimension, or any other single number-—we need some kind of distribution function that tells us how the dimension varies across the attractor. Sets of this type are called multifractals. 

Middle-halves Cantor set) Construct a new kind of Cantor set by removing the middle half of each sub-interval, rather than the middle third. 

Show that the middle-thirds Cantor set contain.s no intervals. But also show that no point in the set is isolated. 

Fat fractal) A fat fractal is a fractal with a nonzero measure. Here s a simple example start with the unit interval [0,1] and delete the open middle 1/2, 1/4, 1/8, etc., of each remaining sub-interval. (Thus a smaller and smaller fraction is removed at each stage, in contrast to the middle-thirds Cantor set, where we always remove 1/3 of what s left.)... 

The Cartesian product of the middle-thirds Cantor set with itself. 

A system of three sets of protons, each set separated by a large chemical shift, can be designated AaMmXx. If two sets are separated from each other by a small chemical shift, and the third set is widely separated from the other two, we use an AaBbXx designation. If all shift positions are close, the system is AaBbCc. Both end sets are coupled to the middle set with different coupling constants, whereas the end sets may or may not be coupled to one another. The AMX systems are first-order ABX systems can be approximated by using first-order rules, but ABC systems cannot be analyzed by inspection. These more complex patterns are treated in Section 4.12. 

Protons in the same environment are called chemically equivalent protons. For example, 1-bromopropane has three different sets of chemically equivalent protons. The toee methyl protons are chemically equivalent because of rotation about the C—C bond. The two methylene protons on the middle carbon are chemically equivalent, and the two methylene protons on the carbon bonded to the bromine atom make up the third set of chemically equivalent protons. 

A few years later. Cantor [8] gave the concept of dimension another serious jolt and created in the process the first of what, after Henri Poincare, came to be called the mathematical monsters . Known as the Cantor set, it is also commonly termed the middle third , the ternary or the triadic Cantor set. It is constructed by a sequence... 

Figure 2.1 Initial steps in the construction of the Cantor set hy repeated removal of the middle third of intervals.

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