Cantor fractal

Although it is not the only such measure, the fractal dimension docs quantify the intuitive belief that the Cantor set is somewhere in-between a point and a line. We will consider generalizations of fractal needed in later chapters,... 

Fig. 2.1 First six steps of the construction of the triadic Cantor Set. The infinite-time limit point set has fractal dimension Dfractai = In 2/In 3.

What does the orbit look like for Ooo It is an infinite (and therefore aperiodic) self-similar point set with fractal dimensionality, Dfractai 0.5388 [grass86c]. Figure 4.5 shows the first six stages in the Cantor-set like construction. 

Fig. 4.5 Sets of stable attractors for the first six critical values of a. Note the self-similarity between the boxed subpattern for oe and the entire pattern for 04 appearing two lines above. A Cantor-set-like fractal pattern appears in the limit an-+oo-...

Fig. 14a-c. Mathematical models of dendritic/fractal systems a) Bifurcation of a line segment with connectivity paths to define Cantor dust, b) Bifurcation of a triangle (two-dimensions) to define Koch snowflakes, c) Fitting of a sphere (three-dimensional) to define a series of infinite nesting spheres... 

Figure 50. Schematic picture of mg(r) dependence, f,- are the time moments of the interaction that construct in time a fractal Cantor set with dimension df = In 2/ In 3 = 0.63. (Reproduced with permission from Ref. 2. Copyright 2002, Elsevier Science B.V.)...

It has been said that the three great developments in twentieth century science are relativity, quantum mechanics, and chaos. That strikes me the same as saying that the three great developments in twentieth century engineering are the airplane, the computer, and the pop-top aluminum can. Chaos and fractals are not even twentieth century ideas chaos was first observed by Poincar6 and fractals were familiar to Cantor a century ago, although neither man had the computer at his disposal to show the rest of the world the beauty he was seeing. 

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). 

This shows that the dimension of the Cantor set is in fact fractal . It is certainly noninteger. For the following reasons it is very satisfactory that the dimension of the Cantor set turns out to be between 0 and 1. The Cantor set has measure zero. Therefore, it is hard to associate it with a dimension-1 object. On the other hand, it contains an uncountable number of points which is too much to associate the Cantor set with a dimension-0 object. Therefore, even intuitively, its dimension has to be somewhere in between 0 and 1. 

The Cantor set C, as well as the fractal shown in Fig. 2.10, are selfsimilar fractals with simple construction rules. Many fractals encountered in physical and mathematical apphcations are not at all that simple. In order to compute their dimensions, one has to use numerical methods. The following are two frequently employed numerical methods for computing fractal dimensions. 

The standard Cantor set is a self-similar fractal with fractal dimension d = log 2/log 3. The fractal dimension d of a self-similar set is given by the dependence on of the number v e) of finite intervals (or squares or cubes, etc.) of length needed to cover the set in the limit - 0,... 

There is a Cantor set of trapped trajectories which show up in the deflection functions or in a phase space portrait of the scattering time delays or survival times. This is indicated in Fig. 8 showing the phase space structure at a held strength = 2, where there are no islands of stability. The initial conditions of those trajectories with exactly two zeros are marked black, the white regions in between correspond to trajectories with three or more zeros. A break-down of these regions according to the number of zeros reveals self-similar fractal structure [62]. 

Cantor strings (or Cantor harps) are one-dimensional drums with a Cantor set as a fractal boundary. They have been studied extensively and an important relationship to the Riemann conjecture was shown [6]. Cantor chains are their discrete counterpart, introduced by the author in [7],... 

R. Etienne, The Spectrum of Monoatomic Cantor Chains, in Fractal 2006 Booklet of Poster Abstracts (Emergentis Ltd. 2006). 

Thus, df is a fractional number and defines the Cantor set dimension. Thus, the function of the set measure, Mf = N(ln)lf represents the main characteristics of the fractal. 

Different relaxation functions are derived assuming that the actual (physical) ensemble of relaxation times is confined between the upper and lower limits of self-similarity. In this respect, the temporal fractal differs from a geometrical fractal (e.g., Cantor dust) for which only an upper limit (i.e., the initial segment before its subdivision) is assumed to exist. It is predicted that at times, shorter than the relaxation time at the lowest (primitive) self-similarity level the... 

Now we turn to another of Cantor s creations, a fractal known as the Cantor set. It is simple and therefore pedagogically useful, but it is also much more than that— as we ll see in Chapter 12, the Cantor set is intimately related to the geometry of strange attractors. 

The Cantor set C has several properties that are typical of fractals more generally ... 

Warning The strict self-similarity of the Cantor set is found only in the simplest fractals. More general fractals are only approximately selfsimilar. 

Two other properties of the Cantor set are worth noting, although they are not fractal properties as such C has measure zero and it consists of uncountably many points. These properties are clarified in the examples below. 

Other self-similar fractals can be generated by changing the recursive procedure. For instance, to obtain a new kind of Cantor set, divide an interval into five equal pieces, delete the second and fourth subintervals, and then repeat this process indefinitely (Figure 11.3.5). 

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.)... 

Now try to picture the limiting set. It consists of infinitely many smooth layers, separated by gaps of various sizes. In fact, a vertical cross section through the middle of 5 would resemble a Cantor set. Thus is (locally) the product of a smooth curve with a Cantor set. The fractal structure of the attractor is a consequence of the stretching and folding that created 5 in the first place. 

Consider the following Cantor construction. A unit section is divided into three equal parts and the middle segment removed. Next, the remaining sections are divided into three equal parts and their middle segments removed again. The procedure, repeated an infinite number of times, yields the fractal Cantor set, see Fig. 91, in which the first few steps of the construction are shown. 

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