Cantor set

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  

C is self-similar. It contains smaller copies of itself at all scales. For instance, if we take the left part of C (the part contained in the interval [0,7]) and enlarge it by a factor of three, we get C back again. Similarly, the parts of C in each of the four intervals of Sj are geometrically similar to C, except scaled down by a factor of nine. 

If you re having trouble seeing the self-similarity, it may help to think about the sets S rather than the mind-boggling set S . Focus on the left half of Sj—it looks just like S, except three times smaller. Similarly, the left half of S3 is Sj, reduced by a factor of three. In general, the left half of looks like all of S , scaled down by three. Now set n = 00. The conclusion is that the left half of looks like, scaled down by three, just as we claimed earlier. 

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

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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.

Thus, whenever the set A has a manifest self-similarity, so that, like the Cantor set, it can be defined by a recursive geometric construction, Dfractal oan be easily calculated from this relation. The Koch Curve, for example, the first three steps in the construction of which are shown in figure 2.2, has a length L which scales as... 

If we give E the discrete topology and T the product topology, then T becomes a compact metric space homeomorphic to the Cantor set under the metric... 

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-...

As should have been made plausible by the above discussion, symbolic dynamics provides an intuitive conceptual bridge between continuous and discrete dynamical systems. On the one hand, except for the fact that the coarse-grained discrete dynamics of symbolic systems is typically nonlocal in character (see the following section), symbolic dynamical systems are essentially CA-in-disguise. On the other hand, by exploiting the fundamental CA property of continuity in the Cantor-set topology under the metric defined by equation 4.49, real-valued representations of CA dynamics may be readily obtained. We discuss these two alternative points of... 

A partial analogy between the dynamics of CA and the behaviors of continuous dynamical systems may be obtained by exploiting a fundamental property of CA systems namely, continuity in the Cantor-set Topology. We recall from section 2.2.1 that the collection of all one-dimensional configurations, or the CA phase space, r = where E = 0,1,..., fc 9 cr and Z is the set of integers by which each site of the lattice is indexed, is a compact metric space homeomorphic to the Cantor set under the metric... 

This Cantor set may be explicitly visualized in the n-dimensional euclidean space 7 " by defining a mapping xp F —> 7 " ([grass83] and [packl]). The coordinate of the resulting vector xpid) is given by ... 

The formalism for computing Lyajmnov exponents for continuous dynamical systems that was introduced in the last section can also be used, with only minor modifications, for determining exponents for CA as well. The major modification involves replacing the Euclidean norm, V t) - used for measuring the divergence of two nearby trajectories (see equation 4.60) - by the Cantor-set metric, d t) ... 

The (finite time) limit set Df may be considered to be a Cantor Set, with dimension S given by (see section 4.5.2) ... 

At the risk of oversimplifying, there are essentially three different dynamical regimes of the one-dimensional circle map (we have not yet formed our CML) (I) j A < 1 - for which we find mode-locking within the so-called AmoW Tongues (see section 4.1.5) and the w is irrational (11) k = 1 - for which the non mode-locked w intervals form a self-similar Cantor set of measure zero (111) k > 1 - for which the map becomes noninvertible and the system is, in principle, ripened for chaotic behavior (the real behavior is a bit more complicated since, in this regime, chaotic and nonchaotic behavior is actually densely interwoven in A - w space). 

ThenWs(gfe( j) and Wu 7e(t)) intersect transversely at(xh(rio)+0(e),9 j and consequently (from, theS male-Birkhoff homoclinic theorem) for the map P there exists an integer n > 1 that P has an invariant Cantor set on which it is topologically conjugate to a full shift of N symbols. 

Holmes 1983) states that when the above transversal homoclinic intersection occurs, that there is a structurally stable invariant Cantor set like the one for the Horseshoe map. It has also been shown by Holmes (1982) that this invariant set contains a countable, dense set of saddles of all periods, an uncountable set of non-periodic trajectories and a dense orbit. If nothing else is clear from the above, it is at least certain that homoclinic bifurcations for maps are accompanied by some very unusual phase portraits. Even if homoclinic bifurcations are not necessarily accompanied by the formation of stable chaotic attractors, they lend themselves to extremely long chaotic like transients before settling down to a periodic motion. Because there are large numbers of saddles present, their stable manifolds divide up the phase plane into tiny stability regions and extreme sensitivity to perturbations is expected. 

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.)...

The mapping that generates the Cantor set is known as the tent map. It is defined by... 

Let us now turn to the Cantor set C and calculate its dimension using (2.3.3). Magnifying the Cantor set by a factor 3 gives us two identical copies of the original Cantor set. Denoting the magnified set by C, we obtain ... 

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. 

Let us apply this formula to the Cantor set. The Cantor set consists of two pieces, each of which can be magnified by a factor m = 3 and yields a complete copy of the original set. Therefore, we have k = 2 and m = 3 and d — ln(2)/ln(3) according to (2.3.8). This result agrees with what we computed using the dimension formula (2.3.3). 

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. 

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