Chapter 5: Fractals. Topics. Lecture Notes. Fundamentals of Fractals Hausdorff Dimension Box Dimension Various Measures on fractals Multi Fractals

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1 ME/MATH/PHYS 597 Spring 2019 Chapter 5: Fractals Topics Lecture Notes Fundamentals of Fractals Hausdorff Dimension Box Dimension Various Measures on fractals Multi 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 Measure and sdimension Let W be a set in the separable space R n, i.e., W R n. The definitions of Hausdorff measure and dimension of W require 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. Note that a space S is called separable if there exists a countable subset Σ S such that Σ is dense in the space S. Let U = {U i } be a countable collection of open sets, which covers the set W, i.e., W i U i (1 Such a collectionu is called a countable open cover of the set W, 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 than or equal to ε, then U is said to be an ε-cover. For a fixed α > 0, a function m(w,α,ε is defined as { } m(w,α,ε inf (diam U i α U i U that is an ε cover of W i (4 3

4 The function m depends on the set W as well as on the parameters α and ε. If ε 1 > ε 2 > 0, then any ε 2 -cover is also an ε 1 -cover. Thus, the set of covers over which the infimum is taken shrinks as the value of ε is decreased. This implies that m(w,α,ε 2 m(w,α,ε 1 if ε 2 < ǫ 1 (5 As a consequence of the above, m(w,α,ε increases as ε is decreased, i.e., m(z, α, ε is a monotonically decreasing function of ε. Hence, the following limit m(w,α = limm(w,α,ε (6 ε 0 exists. It is noted that 0 m(w,α. Referring to Falconer [3], the set function m(,α : Σ W [0, ], where Σ W is a σ-algebra of W, has the following properties: 1. m(,α = 0 α > 0 2. m(w 1,α m(w 2,α W 1 W 2 3. i m(w i,α m( i Z i,α if {W i } is any countable collection of subsets of W, where the equality exists if W i W j = i j. The function m(w, α leads to the concept of α-dimensional Hausdorff measure of the set W, as seen below. Let U be an ε-cover of W and let α > 0 be chosen such that m(w,α <. Then, for every β > α, it follows that m(w,β,ε = inf (diam U i β (7 U i ε β α inf (diam U i α (8 U i = ε β α m(w,α,ε (9 The above inequality results from the fact that diam U i ε < U i U. Since(β α > 0, it follows that ε β α 0 as ε 0 +, which implies that m(w,β,ε 0 as ε 0. Thus, m(w,β = 0 for β > α. Similarly, if β < α, it follows that m(w,β = because ε β α as ε 0 +. The resulting graph of m(w,α vs. α is shown in Fig. 3. Here α C is the critical value of α, where m(w,α changes from to 0, and α C is called 4

5 m( Z,!! C! Figure 3: Graph of m(w, the Hausdroff dimension of the set W, denoted by dim H W. Formally, the Hausdorff dimension dim H W is defined as: dim H W = inf{α [0, m(w,α = 0} (10 Remark 2.1. Hausdorff dimension can be alternatively defined as: dim H W = sup{α [0, m(w,α = } For sets, where dim H is a (non-negative integer, m(w,dim H W coincides with the Lebesgue outer measure of W. For example, if α = 1, then m(w,α yields the length of the set W. Similarly, m(w,2 is the area of the set W; in general, for k N, m(w,k is the k-volume of the set W. 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 W with dim H W = 1.5 is the two-dimensional Brownian motion [1]. In essence, Hausdorff dimension is an effective tool for analyzing the nature of fractals [3] for especially those with fractional dimensions. The results are summarized as follows. 1. dim H = 0 2. dim H W 1 dim H W 2 W 1 W 2 3. dim H ( i W i = sup i dim H W i, where {W i } is a countable collection of monotonically decreasing (or increasing sets. 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, 5

6 r( Z,! m( Z,! r( Z,!! Figure 4: Graph of m(w,,r(w, and r(w, 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(w,α,ε is introduced as { } r(w,α,ε = inf (diamu i α U is an ε cover and diam U i = ε i i (11 It is noted that r(w,α,ε may no longer be a monotonically decreasing function of ε. Therefore, lower and upper limits of r(w,α,ε are evaluated as r(w,α = liminfr(w,α,ε and r(w,α = limsupr(w,α,ε (12 ε 0 + ε 0 + Accordingly, the upper and lower box dimensions are defined as dim B W = sup{α > 0 r(w,α = } (13 dim B W = sup{α > 0 r(w,α = } (14 From the above definition and also from Fig. 4, it is evident that dim H W dim B W dim B W (15 It is shown by Falconer [3] 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 ε. If dim B W = dim B W, if the limit exists, then logn(ε dim B W = lim (16 ε 0 log(1/ε 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} where N(ε is the minimum number of balls of diam ε required to cover the set W. In other words, N(ε ε dim BW (17 3 Multi-Fractals Any basic interval in the n th iterate of the Cantor construction (i.e., the set C n in the cantor set is denoted 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 non-negative real numbers p 0 and p 1 be chosen such that p 0 +p 1 = 1, signifying that p 0 is the probability of occurrence of a 0 and p 1 the probability of occurrence of a 1. Then, Bernoulli measure of an interval I, containing n points, is given by µ(i = n i=1 where j is the number of zeros in I. p σi = p j 0 pn j 1 (18 The construction of Bernoulli measure is extended to other fractals, such as the Sierpiński s triangle and Cantor dust [3]. 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 [3]. 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 pointwise dimension is 7

8 introduced. The point-wise dimension at a point x 0 is defined as log(µ(b(x 0,ε d µ (x 0 = lim ε 0 + log(ε provided that the limit exists (19 where µ is a pointwise measure of the set under consideration, and B(x 0,ε is an open ball of radius ε with its center at x 0. The above equation implies that µ(b(x 0,ε ε d µ(x 0 (20 Remark 1. Equation (20 is consistent with the notion of integer dimension (i.e., k-volume for k N. Let C be the one-third cantor set and let 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 zeros and ones as {σ i }. Then, the point-wise dimension locally at a point x C 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 (21 (22 (23 where j is the number of zeros in the sequence. The (1/3 n term in the denominator in Eq. (21 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 is clear from Eq. (21 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 lowestpoint-wise dimensionis 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 (24 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 8

9 Definition 1. (Singularity Spectrum Let α [0,1] and let S α C such that S α {x C : d µ (x = α}. (25 Then, singularity spectrum f(α of C is defined to be the Hausdorff dimension of S α. In this context, similar to Eq. (17, 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 f(α (26 where N α is the number of open balls of diameter ε to cover the set S α. Remark 3.1. The singularity spectrum describes the statistical distribution of the exponent d µ (x. 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 ]. 3.3 Renyi Dimension Let the bounded phase space W of a dynamical system be partitioned into identical hypercubes/boxes having the length parameter ε > 0. The number of boxes is approximately given as R ε dim BW. The measure of the i th box, i = 1,2,...,R, is expressed as: p i = µ(w box i ε d µ(x i, where x i is the center of box i. Then, the Renyi dimension (that is an intensive parameter is given in terms of the Renyi entropy H(β,p 1 1 β log i pβ i as: D(β,p = lim ε 0 H(β,p log(1/ε = lim ε log(1/ε 1 β log i (p i β (27 It is noted that logz(β,p (β 1D(β,p, where Z(β,p i (p i β is the partition function. Let us suprress the second argument p in D(β,p to imply that a specific probability vector p is given. Then, D(0 is the same as the box dimension. It has been shown earlier that, for β 1, Renyi entropy approaches the Shannon entropy; therefore, D(1 is called the information dimension. For 9

10 β = 2, the correlation dimension D(2 can be computed by numerical experimentation. The inequalities for the Renyi dimension are identical to those derived for Renyi entropy, as seen below. 1. D(β D( β D(β if β > β. 3. ( β 1D( β (β 1D(β if β > β. 4. D( β ( β 1 β β 1 ( β 1 β ( β 1 β D(β if β > β and ββ > 0. A special case: β + and β yielding the following results: ( D(β β D(+ for β > 1. D(β D( for β < 0. As discussed in earlier chapters, the free energy Ψ(β is defined as the negative of the logarithm of the partition function Z(β ( i (p i β. ( ( Ψ(β = log (p i β = log ε α iβ (28 i where α i is the point-wise dimension d µ (x i and x i is the center of box i. This sum is evaluated over all boxes. Denoting V = logε, it follows that V + as ε 0 +. Then, as V enters into the canonical distribution of a dynamical system as aparameter, it becomes a thermodynamic analog of the the volume in the canonical distribution of thermodynamics. Therefore, V + is considered as the so-called thermodynamiclimit. Therefore, ithe limitlim ε 0 +α(ε,x is denoted as α(x as a thermodynamic parameter. Alternatively, for small ε, Ψ(β is approximately evaluated as an integral over the range of α. ( αmax Ψ(β log dα N α (εε αβ (29 α min where N α (ε (defined earlier is the number of boxes of size ε with pointwise dimension equal to α. Furthermore, N α (ε ε f(α, where f(α is the i 10

11 singularity spectrum. Ψ(β = lim ε 0 log = lim ε 0 log ( αmax dα ε αβ f(α α min ( αmax α min ( dα exp (αβ f(αlog(1/ε (30 The expression in Eq. (30 is simplified by saddle point approximation [4] as Ψ(β = lim ε 0 log(1/εmin α (αβ f(α (31 Explanation of saddle point approximation: Let g be a smooth (at least C 2 function having a single minimal point g(x 0 at x 0 [x 1,x 2 ]; and let V be a large-magnitude parameter involved in the following integral: I = By Taylor series expansion, x2 x 1 dx exp[ g(x V] g(x g(x 0 +(x x 0 g (x (x x 0 2 g (x 0 At the minimal point x 0, we have g (x 0 = 0 and g (x 0 > 0. Therefore, the integral reduces to x2 I exp[ g(x 0 V] dx exp [ (x x 0 2 ] x 1 2 g (x 0 V 1 Since the above integral has a Gaussian structure with a very small variance (because V is large, we approximate the integral as [ ] 1/2 2π I exp[ g(x 0 V] g (x 0 V This is equivalent to logi g(x 0 V +O(logV. RecallingthattheHelmholtzfreeenergyF = U TS or F T = U T S, letusnow examine the significance of the thermodynamic free energy Ψ = βm S, where M αv and V = log(1/ε in the present setting of dynamical systems. Then, f(α can be regarded as the entropy density S/Vof the 11

12 probability distrdistribution in thermodynamic limit, i.e., as V. f(α = lim V S V = lim ε 0 + S log(1/ε A necessary condition for achieving the minimum in Eq. (31 is f α = β. Let us define a β-dependent parameter as It follows from Eqs. (27 and (32 that τ(β (β 1D(β (32 Ψ(β τ(β = lim V V ( Ψ(β = lim ε 0 log(1/ε (33 The parameter τ(β in Eq. (33 is an intensive parameter that is physically interpreted as a free energy density. By combining Eqs. (31 and (33, it follows that τ(β = min(αβ f(α (34 α The following identities are obtained at At an extremal point, f(α = α β τ(β τ β = α and f α = β (35 α=α where α takes on the value of α that minimizes the free energy (αβ f(α (see Eq. (34, and f(α is computed by the Legendre transform [2] as explained below. Havingcomputed τ(β, α is obtainedas a function of β by using Eq. (35and then f(α is computed. Thus, the spectra of scaling exponents, generated by f(α, are obtained from the parameter τ(β. Remark 3.2. Since the parameters f(α and τ(β are mutually related by Legendre transformation, they contain similar information on the system. Next we present certain properties of the singularity spectrum f(α by using the relationship in Eq. (32 between τ(β and D(β in terms of Eq. (35 in the following form. α (β = D(β+(β 1 dd dβ f(α (β = D(β+β(β 1 dd dβ 12 (36

13 For example, the special cases β = 0 and β = 1 yield the following results. f(α (0 = D(0 = α (0+ dd dβ β=0 f(α (1 = D(1 = α (1 (37 which impliesthatthe maximumof f(αis attainedat at df dα = 0 when β = 0; hence, Renyi dimension is D(0. The information dimension D(1 is obtained as the value of f(α such that df dα = 1. Proposition 3.1. The lowerlimit of the range of the scalingindex α is related to the Renyi dimension as: α min = lim β D(β (38 Proof. Let p max be the largest element in the probability vector, i.e., p max max i p i. Therefore, the smallest scaling index α min corresponds to p max because p max exp ( α min log(1/ε = εe α min. For a large positive β, the partition function Z(β = exp( Ψ(β is dominated by (p max β. Hence, ( 1 D(β = lim ε 0 (1 βlog(1/ε ( 1 lim ε 0 (1 βlog(1/ε ( β = α min β 1 Therefore, α min = lim β D(β. ( log (p i β i ( log (p max β Corollary 3.1. The upper limit of the range of the scaling index α is related to the Renyi dimension as: α max = lim D(β (39 β Proof. The proof follows the same argument as in Proposition 3.1 with the exception that we will deal with the smallest (positive element in the probability vector P, i.e., p min min i p i. Therefore, the largest scaling index α max corresponds to p min because p min exp ( α max log(1/ε = εe α max. Note that, for a large negative β, the partition function W(β = exp( Ψ(β is dominated by (p min β = (p min β, which leads to the expression: ( β D(β α max for large negative values of β β 1 Therefore, α max = lim β D(β. 13

14 3.4 Kolmogorov Sinai Entropy Symbolic dynamics is a viable tool for analysis of chaotic dynamical systems. While a d-dimensional phase space can be partitioned in various ways, a conventionalwayistouseuniformpartitioning;inthiscase, allboxesareofequal size (e.g., d-cubes of side length ε. However, it may become advantageous to use boxes of variable sizes (i.e., variable side lengths ε of the d-cubes. Let W be a finite subset of a phase space, which is partitioned by finitely many boxes of (possibly variable size, A 1,,A R such that W = R i=1 A i and A i A j = i j. Let us identify each A i by a unique symbol s i such that the alphabet of symbols is Σ = {s 1,s 2,,s R }. Now let X {x k : k = 0,1,,N 1} be a sequence of vectors in W, which is symbolized as {s ik : k = 0,1,,N 1}, where each s ik Σ; it is noted that since the cardinality Σ = R is much less the length N of the symbol string, each symbol is expected to be repeated many times. This situation can be viewed as a map f : X Σ. The map f represents a dynamical system whose probability measure µ can be attributed to the N-cylinder J(s 0,,s N 1 such that the probability p(s 0,,s N 1 = dµ(θ = dθ ρ(θ θ J(s 0,,s N 1 θ J(s 0,,s N 1 where an N-cylinder J(s 0,,s N 1 is the set of all initial conditions x 0 that will produce the symbol string {s 0,,s N 1 } under given partitioning A 1,,A R of the dynamical system. Given a map f, we re interested in the time evolution of ensemble of trajectories corresponding to the initial values x 0 of the dynamical system. Let µ k be the probability measure after k iterations of the map f. That is, µ k (A is the probabilityof finding an iterate x k in the subset A and the corresponding density is denoted by ρ k as µ k (A = dx ρ k (x x A If the probability measure µ is conserved, then the following condition is satisfied for arbitrary subsets A as: µ k+1 (A = µ k (f 1 (A, where f 1 (A is theinverseimageofaunder f (i.e.,the setofallpointsthataremapped onto A by one iteration. In other words, the relative frequency of iterates x k+1 in the subset A must be equal to that of iterates x k in the subset f 1 (A. We are specifically interested in invariant distributions (i.e., distributions that 14

15 do not change under action of the map f, i.e., µ k+1 (A = µ k (A. Thus, the measure µ and the corresponding density ρ do not change with time. The Shannon entropy is then given by: H(p = p(s 0,,s N 1 log(p(s 0,,s N 1 (s 0,,s N 1 which is a measure of the expected uncertainties of the symbol string. For a given map f, the entropy H is dependent on the following quantities: 1. µ: probability measure of the initial condition x {A}: partition of the phase space 3. N: Length of the symbol string. Since H(µ,{A},N is likely to diverge as N for a dynamical system, we will examine its rate, i.e., h(µ,{a} lim N = lim N H(µ,{A},N N p(s 0,,s N 1 log(p(s 0,,s N 1 1 N (s 0,,s N 1 which depends on both µ and {A}. By taking supremum over all partitions {A}, Kolmogorov Sinai entropy (KS entropy for a given map is obtained as: h(µ = sup {A} h(µ,{a} If µ is taken as the natural invariant measure of the map of the dynamical system, then we simply denote KS entropy by h. Furthermore, if A gen is a generating partition, then h(µ = h(µ,{a gen } as N. However, an infinitely long symbol string mat not have a generating partition; even if it has a generating partition, it may not be known. Remark 3.3. The K-S entropy is very useful in the the analysis chaotic systems because of the following reasons: Invariance under a change of coordinates of the dynamical system. Quantification of chaos. For example, a dynamical system is said to be chaotic if its K-S entropy is strictly positive.in general, we can measure the degree of chaoticity based on the fact that the larger the KS entropy, the stronger are the chaotic properties of the dynamical system. 15

16 To summarize, the K-S entropy measures the average production (respectively, average loss of information per iteration step, as explained below. The Shannon entropy H(µ,{A}, N is the missing information needed to locate a certain event on a symbol trajectory of length N. That is, H N H(µ,{A},N +1 H(µ,{A},N is the missing information that is necessary to predict the symbol at iterate N +1. To summarize, the K-S entropy measures the average production (respectively, average loss of information per iteration step, as explained below. The Shannon entropy H(µ,{A}, N is the missing information needed to locate a certain event on a symbol trajectory of length N. That is, H N H(µ,{A},N +1 H(µ,{A},N is the missing information(or loss of information that is necessary to predict the symbol at iterate N +1. The average loss of information is obtained as: 1 H = lim M M M N=1 H N = lim M 1 = lim M M provided that H(µ,{A},1 is finite. 1 M M H(µ,{A},N +1 H(µ,{A},N N=1 ( H(µ,{A},M +1 H(µ,{A},1 = h(µ,a 3.5 Renyi entropies We have seen in earlier chapters that Renyi information is a generalization of Shannon information in the sense that I(β,p 1 β 1 log R i=1 (p i β converges to S(p R i=1 p ilog(p i as β 1. The generalizations of K-S entropy, which is for β = 1, are known as Renyi entropies for different values of β. References [1] B.B.Mandelbrot. The Fractal Geometry of Nature. W.H. Freeman and Company, New York, NY, USA, [2] C. Beck and F. Schlögl. Thermodynamics of Chaotic Systems. Cambridge University Press, Cambridge, UK, First edition,

17 [3] K. Falconer. Fractal Geometry: Mathematical Foundations and Applications. Wiley, New York, NY, USA, Second edition, [4] K. Huang. Statistical Mechanics. Wiley, New York, NY, USA, Second edition,