Chapter 2: Introduction to Fractals. Topics

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1 ME597B/Math597G/Phy597C Spring 2015 Chapter 2: Introduction to Fractals Topics Fundamentals of Fractals Hausdorff Dimension Box Dimension Various Measures on Fractals Advanced Concepts of Fractals 1

2 C 0 C 1 C 2 C 3 Figure 1: One-third Cantor set Figure 2: Cantor-like sets construction in R 2 (only the first three iterations have been shown) 1 Introduction Fractals are complex mathematical objects that have no natural scale. Even when magnified to great levels, the details in the fractal remain crisp. Inherently, fractals also have a degree of self similarity. This means that a small part of a fractal object may resemble the entire fractal object. For fractals that display a recursive hierarchical structure, information on the recursive algorithm is sufficient to generate the fractal to any arbitrary precision. Figure 1 shows a well-known fractal construction, called the one-third Canter set. It is created by repeatedly deleting the open middle thirds of a set of line segments (e.g., the closed interval [0, 1] on the real line). More general constructions may be done along the lines of the Cantor set. A few examples of these constructions are depicted in Fig. 2. In most cases occurring in nature, fractals only exhibit statistical similarity across scale. These fractals do not look well ordered at first and the apparent random nature may falsely confirm their self-similar character. Analysis of the socalled random fractals is usually a much harder problem. These random fractals are frequently encountered. For example, economic records like the Dow Jones index, physiological data such as ECG records, texture of images, and variations of traffic flow belong to the class of random fractals. 2

3 2 Fundamentals of Fractals This section discusses fundamental concepts such as Hausdorff dimension, box dimension, and pointwise dimension. 2.1 Hausdorff Dimension Let Z be a set in a separable space R n, i.e., Z R n. The definition of the Hausdorff dimension of Z requires several preliminary concepts. A metric d(x,y) is defined on R n. However, an additional constraint is imposed. The topological space as induced by the metric has to be separable. A countable collection of open sets U = {U i } that covers the set Z, i.e., Z i U i (1) Such a collection is called a countable open cover, referred to as a cover, in the sequel. The diameter of a set U i,denoted as (diam U i ), is defined as diam U i = sup{d(x,y) x,y U i } (2) and the diameter of the cover is obtained as diam U = sup diam U i (3) U i U Fixing ε > 0, if diam U ε, i.e., if every open set in U has a diameter less that ε, then U is said to be an ε-cover. For a fixed α > 0, a function m(z,α,ε) is defined as { } m(z,α,ε) inf (diamu i ) α U is an ε-cover of Z i The function m depends on the set Z and also on the parameters α and ε. Let ε 1 > ε 2 > 0; then, it follows that any ε 2 -cover of Z is also an (4) 3

4 ε 1 -cover of Z. Thus, the set of covers over which the infimum is taken shrinks as the value of ε decreases. This implies that m(z,α,ε 2 ) m(z,α,ε 1 ) ε 1 > ε 2 > 0 (5) As a consequence of the above, m(z,α,ε) is a monotonicallyincreasing function of ε. Hence the limit exists. However, the limit may be equal to. m(z,α) = lim m(z,α,ε) (6) ε 0 + The set function m(,α) : Z m(z,α) has the following properties 1. Normalization: m(,α) = 0 α > 0, where is the empty set. 2. Monotonicity: m(z 1,α) m(z 2,α) Z 1 Z 2 3. Countable subadditivity: given any at most countable (i.e., either finite of countably infinite) collection of subsets {Z i } of Z, it follows that m( i Z i,α i m(z i,α)) (7) Proof: The proofs of the above expressions are provided by Pesin & Clemenhaga [3] (see pp ). Let α > 0 be chosen such that m(z,α) <. Then, for every β > α, it follows that m(z,β,ε) = inf U (diamu i ) β (8) i ε β α inf U (diamu i ) α (9) i = ε β α m(z,α,ε) (10) Since ε β α 0 as ε 0, it implies that m(z,β,ε) 0. Thus, m(z,β) = 0 for β > α. Similarly, if β < α, it follows that m(z,β) =. The resulting graph of m(z,α) vs. α is shown in Fig. 3. Here α C is the critical value of α, where m(z,α) changes from to 0, and α C is called the Hausdroff dimension of the set Z, denoted by dim H Z. Formally, dim H Z is defined as 4

5 m( Z,!)! C! Figure 3: Graph of m(z, ) dim H Z = inf{α [0, ) : m(z,α) = 0} (11) or dim H Z = sup{α [0, ) : m(z,α) = } (12) For sets, where dim H is an integer, the value of m(z,dim H Z) is the Lebesgue outer measure. Setting α = 1, m(z,1) yields the length of the set Z. Similarly, m(z,2) is the area of the set Z; in general, for k N, m(z,k) is the k-volume of the set Z. A set that has a Hausdorff dimension equal to 1.5 is infinite 1-volume (i.e., length) and zero 2-volume (i.e., area). Example of such a set is the two-dimensional Brownian motion [1]. Hausdorff dimension is an effective tool for analyzing the nature of fractals [2] for especially those with fractional dimensions. The contents of this section are summarized as:. 1. dim H = 0 2. dim H Z 1 dim H Z 2 Z 1 Z 2 3. dim H ( i Z i ) = sup i dim H Z i, where {Z i } is a countable collection of subsets of Z. 2.2 Box Dimension The hausdroff dimension of a set is often difficult to obtain as it requires finding an infimum over all possible collections of ε-covers. This problem can be circumvented by introducing a competing notion of dimension, namely, the box dimension. The definition of box dimension is similar to that of Hausdorff dimension except for the single condition stated below. For the definition of box dimension, all sets in an open cover must have the same diameter. In this setting, the function r(z, α, ε) is introduced as; 5

6 r( Z,!) m( Z,!) r( Z,!)! Figure 4: Graph of m(z, ),r(z, ) and r(z, ) { } r(z,α,ε) = inf (diamu i ) α : U is an ε cover and diamu i = ε i i (13) Note that r(z,α,ε) is no longer a monotonically increasing function of ε. Therefore, the lower and upper limits of r(z,α,ε) are evaluated as: r(z,α) = liminfr(z,α,ε) and r(z,α) = limsupr(z,α,ε) (14) ε 0 + ε 0 + Accordingly, the upper and lower box dimensions are defined as: dim B Z = sup{α > 0 r(z,α) = } (15) dim B Z = sup{α > 0 r(z,α) = } (16) From the above definition and also from Fig. 4, it is evident that dim H Z dim B Z dim B Z (17) It is shown by Falconer [2] that the above three dimensions are equal for fractals obtained by a Cantor-like construction. An equivalent definition of the box dimension is derived by having the set covered by open balls of diameter ǫ. dim B Z = limsup ε + 0 logn(ε) log 1/ε and dim B Z = liminf ε + 0 logn(ε) log 1/ε (18) where N(ε) is the minimum number of balls of diam ε required to cover the set Z. In other words, N(ε) ε dim BZ. 6

7 C 0 C 1 C 2 C Figure 5: Symbolic space representation of the Cantor set. The symbol set is chosen as {0,1} 3 Extended Concepts of Fractals The notion of an extended concept of fractals is first explained via a simple example. Any basic interval in the n th iterate of the Cantor construction (i.e., the set C n in the cantor set) and let us denote the closed intervals by a sequence of ones and zeroes, i.e., [σ 1,σ 2,...,σ n ], where σ i {0,1} as seen in Fig. 5. Let two positive real p 0 and p 1 be chosen such that p 1 +p 2 = 1. The Bernoulli measure of an interval I is given by µ(i) = where, j is the number of zeros in I. n p σi (19) i=1 = p j 0 pn j 1 (20) The construction of Bernoulli measure is extended to other fractals, such as the Sierpinski s triangle and Cantor dust [2]. In general, for a fractal with k self-similar components, k real positive parameters p 0,p 1,...p k 1 are chosen such that i p i = 1. The measure of an interval is then obtained as a product of the appropriate parameters. A broader class of measures, called the Markov measures, are often used [2]. 3.1 Point-wise Dimension Both Hausdorff dimension and box dimension represent a global property of fractals. However, for certain applications, it is beneficial to study the local behaviors of fractals. To this end, the notion of point-wise dimension is introduced. The point-wise dimension at a point x 0 is defined as log(µ(b(x 0,r)) d µ (x 0 ) = lim r 0 + log(r) provided that the limit exists (21) 7

8 where B(x 0,r) is an open ball of radius r and center x 0. The above equation implies that µ(b(x 0,r)) r d µ(x 0 ) (22) Remark 1. Equation (22) is consistent with the notion of integral dimension (i.e., k-volume for k N). Let C be the one-third cantor set and a Bernoulli measure µ with parameters p 0 and p 1 be induced on C. Let x C be denoted by an infinite sequence of ones and zeros (σ 1,σ 2,..). Then, the point-wise dimension at x is given by log(µ([σ 1,σ 2,...,σ n ]) d µ (x) = lim n log((1/3) n ) n i=1 = lim log(p σ i ) n nlog(1/3) = lim n j n log(p 0)+ ( 1 j n) log(p1 ) log(1/3) (23) (24) (25) where j is the number of zeros in the sequence. The (1/3) n term in the denominator in Eq. (23) is the diameter of the interval [σ 1,σ 2,...,σ n ]. If this limit does not exist, one may find the upper and lower point-wise dimensions. 3.2 Singularity Spectrum It follows from Eq. (23) that the point-wise dimension is different across the points in the Cantor set C. For example, if p 0 < p 1, then the point with the lowest point-wise dimension is the sequence x min (0,0,0,...)and the point with the highest dimension is the sequence x max (1,1,1,...). d µ (x min ) = logp 0 log(1/3) and d µ(x max ) = logp 1 log(1/3) (26) Remark 2. If p 0 = p 1 = 1/2, then all the points in the Cantor set C have the same dimension d µ = log(1/2) log(1/3) = log2 log3 Definition 1. (Singularity Spectrum) Let S α C be defined as S α = {x C : d µ (x) = α}. (27) Singularity spectrum D(α) of C is equal to the Hausdorff dimension of S α. 8

9 In this context, similar to the earlier case of N(ε) ε dim BZ, the number N α (ε) of open balls of diameter ε required to cover the set S α is given as N α (ε) ε D(α) (28) Remark 3.1. The singularity spectrum describes the statistical distribution of the exponent. The definition of singularity spectrum also allows us to differentiate between Homogeneous measure and Multi-fractal measures. Homogeneous measures are characterized by a uniform point-wise dimension, thus the spectrum is supported by a single point. In contrast, multi-fractals involve singularities of various strengths. Generally, the spectrum extends across a finite interval [α min,α max ] References [1] B.B.Mandelbrot. The Fractal Geometry of Nature. W.H. Freeman and Company, New York, NY, USA, Second edition, [2] K. Falconer. Fractal Geometry: Mathematical Foundations and Applications. Wiley, New York, NY, USA, Second edition, [3] Y. Pesin and V. Clemenhaga. Lectures on Fractal Geometry and Dynamical Systems. American Mathematical Society (AMS), Philadelphia, PA, USA,