A new fractal dimension for curves based on fractal structures

JID:TOPOL

AID:5719 /FLA

[m3L; v1.172; Prn:19/01/2016; 15:00] P.1 (1-17) Topology and its Applications ••• (••••) •••–•••

Contents lists available at ScienceDirect

Topology and its Applications www.elsevier.com/locate/topol

A new fractal dimension for curves based on fractal structures M. Fernández-Martínez a,1 , M.A. Sánchez-Granero b,∗,2 a

University Centre of Defence at the Spanish Air Force Academy, MDE-UPCT, 30720 Santiago de la Ribera, Murcia, Spain b Department of Mathematics, Universidad de Almería, 04120 Almería, Spain

a r t i c l e

i n f o

Article history: Received 8 November 2014 Accepted 30 March 2015 Available online xxxx MSC: primary 37F35 secondary 28A78, 28A80, 54E99 Keywords: Fractal Box-counting dimension Hausdorff dimension Space-filling curve Hilbert’s curve Hurst exponent

a b s t r a c t In this paper, we introduce a new theoretical model to calculate the fractal dimension especially appropriate for curves. This is based on the novel concept of induced fractal structure on the image set of any curve. Some theoretical properties of this new definition of fractal dimension are provided as well as a result which allows to construct space-filling curves. We explore and analyze the behavior of this new fractal dimension compared to classical models for fractal dimension, namely, both the Hausdorff dimension and the box-counting dimension. This analytical study is illustrated through some examples of space-filling curves, including the classical Hilbert’s curve. Finally, we contribute some results linking this fractal dimension approach with the self-similarity exponent for random processes. © 2016 Elsevier B.V. All rights reserved.

1. Introduction The word fractal, which derives from the Latin term frangere (that means “to break”), provided a novel concept in mathematics since Benoît Mandelbrot first introduced it in the early eighties [22]. Since then, both the study and the identification of fractal patterns have become more and more important due to the large number of applications to diverse scientific fields where fractals have been found, including computation, physics, economics and statistics among them (see [12,14,18,19]). Moreover, there has also been a particular interest in the application of fractals to social sciences (see for example [10] and its

* Corresponding author. E-mail addresses: [email protected] (M. Fernández-Martínez), [email protected] (M.A. Sánchez-Granero). The first author, who specially acknowledges the valuable support provided by Centro Universitario de la Defensa en la Academia General del Aire de San Javier (Murcia, Spain), has been partially supported by Fundación Séneca de la Región de Murcia, Grant No. 19219/PI/14. 2 The second author acknowledges the support of the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01. 1

http://dx.doi.org/10.1016/j.topol.2015.12.080 0166-8641/© 2016 Elsevier B.V. All rights reserved.

JID:TOPOL 2

AID:5719 /FLA

[m3L; v1.172; Prn:19/01/2016; 15:00] P.2 (1-17)

M. Fernández-Martínez, M.A. Sánchez-Granero / Topology and its Applications ••• (••••) •••–•••

references). Nevertheless, some interesting non-standard objects had appeared previously as mathematical monsters, due to their novel and counter-intuitive analytical properties. Thus, they were frequently shown as counterexamples or exceptional objects provided by some remarkable mathematicians including Peano and Hilbert space-filling curves [21,24]. Fractals objects have been studied from different points of view, and the main tool that has been applied to study them is the fractal dimension, since it is their main invariant which shows some useful information about their complexity and irregularities. In particular, topology allows the study of this class of non-linear objects by means of fractal structures. They were first sketched in [6] and then formally defined and applied in [1] to characterize non-Archimedeanly quasi-metrizable spaces. This concept has allowed to formalize some topics on fractal theory from both theoretical and applied points of view. A fractal structure is just a countable collection of coverings of a given subset which provides better approximations to the whole space as we explore deeper stages, called levels. Thus, if we analyze the definition of the box-counting dimension, then we can observe that fractal structures provide a suitable context where new models of fractal dimension can be developed. On the other hand, given any patch of the plane, a plane-filling curve is a continuous curve which meets every point in that patch. Thus, though the Peano plane-filling curve appeared in 1890, the later Hilbert’s curve results also quite interesting, since it has no self-intersections nor touching points at any stage of its construction (that will be explain in detail later by means of fractal structures). In this way, a wide variety of space-filling curves were studied after that, though the example proposed by Hilbert still remains one of the most famous, since he provided one of the first graphical visualizations of a fractal in his original 2-page paper Über die stetige Abbildung einer Linie auf ein Flächenstück (1891) [21]. This curve was first sketched during a mathematical annual meeting in Bremen (Germany), where Hilbert and G. Cantor (1845–1918) were working on the foundation of the German mathematical society. Our main purpose is to introduce a new theoretical model of fractal dimension for any fractal structure that becomes especially appropriate to analyze fractal patterns on curves. Additionally, we will explore some interesting connections between that fractal dimension approach and the self-similarity exponent of random processes. The organization of this paper is as follows. In Section 2, we recall some preliminary definitions, notations and results including box-counting dimension, Hausdorff dimension, fractal structures and fractal dimension for a fractal structure. In Section 3, we provide a new theoretical model of fractal dimension especially appropriate to explore both the complexity and the fractal pattern of curves which is based on a novel concept of an induced fractal structure. This new procedure presents some advantages with respect to the classical models that may be applied for the same purpose, since it takes also into account the underlying structure of the curve. In addition to that, this is calculated on the image set of the curve, in contrast to the Hausdorff dimension and the box-counting dimension which are both calculated for its graph. We also show some theoretical properties of this fractal dimension for curves. In Subsection 3.1, we provide a theorem which allows us to generate space-filling curves among other applications. In Subsection 3.2, we explain in detail how to iteratively approach both the classical Hilbert’s curve and a modified Hilbert’s curve using fractal structures. We also calculate, compare and explain the values obtained for their classical fractal dimensions as well as for their new fractal dimensions. These examples show that the new model of fractal dimension we introduce in this paper results more accurate to distinguish and classify space-filling curves generated through different ways of construction. A curve which fills the whole Sierpiński’s gasket is also explored from the point of view of fractal dimension (see Subsection 3.3). Finally, in Section 4, we show some theoretical results connecting the fractal dimension with the self-similarity exponent of random processes.

JID:TOPOL

AID:5719 /FLA

[m3L; v1.172; Prn:19/01/2016; 15:00] P.3 (1-17)

M. Fernández-Martínez, M.A. Sánchez-Granero / Topology and its Applications ••• (••••) •••–•••

3

2. Preliminaries The main goal of this section is to recall some definitions, results and notations that will be useful in this paper. In this way, we will focus on fractal structures, box-counting dimension, Hausdorff dimension and a new theoretical model to calculate the fractal dimension for any fractal structure especially appropriate for curves. 2.1. Fractal structures The concept of fractal structure first appeared in [1] to characterize non-Archimedeanly quasi-metrizable spaces though a more natural use of it is to study fractals. For example, in [5] it was applied to study attractors of iterated function systems. Fractal structures provide a powerful tool to introduce new models valid to calculate fractal dimension, since they constitute a natural context in which the concept of fractal dimension may be revisited. In addition to that, they allow to calculate fractal dimension in new spaces and situations.  Recall that a family Γ of subsets of a space X is called a covering if X = {A : A ∈ Γ}. Let Γ be a   covering of X. Then we will denote St (x, Γ) = {A ∈ Γ : x ∈ A} and UxΓ = X \ {A ∈ Γ : x ∈ / A}. Moreover, if Γ = {Γn : n ∈ N} is a countable family of coverings of X, then we will denote Uxn = UxΓn , UxΓ = {Uxn : n ∈ N}, and St(x, Γ) = {St(x, Γn ) : n ∈ N}. Let us describe a first approach to define a fractal structure on a set X. Indeed, let Γ1 and Γ2 be two coverings of X. Thus, we will write Γ1 ≺ Γ2 to denote that Γ1 is a refinement of Γ2 , namely, for all A ∈ Γ1 there exists B ∈ Γ2 such that A ⊆ B. In addition to that, Γ1 ≺≺ Γ2 means that Γ1 ≺ Γ2 , and for all B ∈ Γ2  it is satisfied that B = {A ∈ Γ1 : A ⊆ B}. A fractal structure on a set X can be defined as a countable family of coverings of X, Γ = {Γn : n ∈ N}, such that Γn+1 ≺≺ Γn for all n ∈ N. Next, we present the definition of a fractal structure on a topological space as it was introduced in [1, Definition 3.1]. Definition 2.1. Let X be a topological space. Then (1) a pre-fractal structure on X is a countable family of coverings, Γ = {Γn : n ∈ N}, such that UxΓ is an open neighborhood base of x, for each x ∈ X. (2) Moreover, if Γn+1 is a refinement of Γn such that for all x ∈ A with A ∈ Γn there exists B ∈ Γn+1 with x ∈ B ⊆ A, then we will say that Γ is a fractal structure on X. (3) If Γ is a (pre-)fractal structure on X, then we will say that (X, Γ) is a generalized (pre-)fractal space, or simply a (pre-)GF-space. If there is no doubt about the fractal structure Γ, then we will say that X is a (pre-)GF-space. The covering Γn is called level n of the fractal structure Γ. Remark 2.2. To simplify the theory, the levels of any fractal structure Γ will not be coverings in the usual sense. Instead of this, we are going to allow that a set can appear more than once in any level of Γ. For instance, Γ1 = {[0, 1/2], [1/2, 1], [0, 1/2]} may be the first level of a fractal structure defined on the closed unit interval [0, 1]. Recall also that if Γ is a pre-fractal structure, then any of its levels is a closure-preserving closed covering (see [3, Proposition 2.4]). If Γ is a fractal structure on X and St (x, Γ) is a neighborhood base of x for all x ∈ X, then we will call Γ a starbase fractal structure. Starbase fractal structures are connected to metrizability (see [2,3]). A fractal structure Γ is said to be finite if all levels Γn are finite coverings. A fractal structure Γ is said to be locally

JID:TOPOL 4

AID:5719 /FLA

[m3L; v1.172; Prn:19/01/2016; 15:00] P.4 (1-17)

M. Fernández-Martínez, M.A. Sánchez-Granero / Topology and its Applications ••• (••••) •••–•••

finite if for each level Γn of Γ, we have that any point x ∈ X belongs to a finite number of elements A ∈ Γn . Further, a fractal structure Γ is said to be Γ-Cantor-complete (see [4, Definition 3.1]) if for each decreasing sequence {An : n ∈ N} (namely, An+1 ⊆ An for all n ∈ N) of subsets of X with An ∈ Γn , then it holds that ∩n∈N An = ∅. In general, if Γn has the property P for all n ∈ N and Γ = {Γn : n ∈ N} is a fractal structure on X, then we will say that Γ is a fractal structure with the property P , and that (X, Γ) is a GF-space with the property P . We also recall the definition of the natural fractal structure that may be defined on any Euclidean space Rd that will be useful for our purposes. Definition 2.3. ([16, Definition 3.1]) The natural fractal structure on any Euclidean space Rd is defined as the countable family of coverings Γ = {Γn : n ∈ N}, whose levels are given as  Γn =

      k2 k2 + 1 kd kd + 1 k1 k1 + 1 × n, × ... × n, : k1 , k2 , . . . , kd ∈ Z . , 2n 2n 2 2n 2 2n

2.2. Box-counting dimension Fractal dimension is one of the main tools applied to study fractals, since it is a single quantity which provides useful information about their complexity as they are examined with enough level of detail. This becomes the main invariant of any fractal set and is usually understood as the classical box-counting dimension, which is also known as Kolmogorov entropy, entropy dimension, metric dimension or logarithmic density, among others (see [12]). It seems that the origins of the box-counting dimension go back to the twenties, when they were first considered by the pioneers of Hausdorff measure and dimension. Nevertheless, it was rejected at a first glance for being less appropriate than the Hausdorff dimension from a theoretical point of view. However, Bouligand adapted the Minkowski content to non-integral dimensions [9], and the classical definition of the box-counting dimension was provided by Pontrjagin and Schnirelman in [25]. Though the Hausdorff dimension can be considered a fractal dimension too, in practical applications it is always used the box-counting dimension, since it is the only one that can be calculated or computationally estimated when working with a finite range of scales, which is the case of empirical applications. Popularity of the box-counting dimension is mainly due to the possibility of its effective calculation in Euclidean contexts. Indeed, in practical applications, the box-counting dimension can be estimated as the slope of the regression line of a log–log graph plotted for a suitable discrete collection of scales. The basic theory about box-counting dimension can be found in [12]. Next, we recall the definition of the standard box-counting dimension. Definition 2.4. The (lower/upper) box-counting dimension of a subset F ⊆ Rd is given by the following (lower/upper) limit: log Nδ (F ) , δ→0 − log δ

dim B (F ) = lim

(1)

where δ is the scale and Nδ (F ) can be calculated, in an equivalent way, as one of the following quantities (see [12, Equivalent Definitions 3.1]): (1) the number of δ-cubes that meet F , where a δ-cube in Rd is a set of the form [k1 δ, (k1 +1) δ] ×[k2 δ, (k2 + 1) δ] × . . . × [kd δ, (kd + 1) δ] where k1 , . . . , kd ∈ Z. (2) the number of δ = 1/2n -cubes that intersect F , with n ∈ N. (3) the smallest number of sets of diameter at most δ that cover F . (4) the largest number of disjoint balls of radius δ with centers in F .

JID:TOPOL

AID:5719 /FLA

[m3L; v1.172; Prn:19/01/2016; 15:00] P.5 (1-17)

M. Fernández-Martínez, M.A. Sánchez-Granero / Topology and its Applications ••• (••••) •••–•••

5

Note also that the limit in Eq. (1) can be discretized by taking, for example, δ = 1/2n , which is formalized in the next remark. Remark 2.5. To calculate the (lower/upper) box-counting dimension of any subset F of an Euclidean space Rd , it suffices with taking limits as δ −→ 0 through any decreasing sequence {δn : n ∈ N} verifying that δn+1 ≥ c δn for all n ∈ N, where c ∈ (0, 1) is a suitable constant. In particular, it holds for δn = 1/2n . 2.3. Hausdorff dimension In this subsection, we include a short sketch about the construction of both Hausdorff measure and dimension. Their definitions and properties can be found in [12, Chapter 2]. The first to define a measure by means of coverings of sets was Carathéodory in [11]. Later (1919), Hausdorff used this method to define the measures that now bear his name [20]. A detailed study about the analytical properties of both Hausdorff measure and dimension was contributed by Besicovitch and his pupils in a series of papers (see for instance, [7,8]). The Hausdorff dimension, which is the oldest definition of fractal dimension, presents the best analytical properties. Indeed, note that this fractal dimension can be defined for any subset of an Euclidean (resp. metrizable) space and its definition is based on a measure which makes it very convenient from a mathematical point of view. Nevertheless, it presents some disadvantages, particularly from the point of view of applications, since it can be hard to calculate or to estimate. Thus, while this fractal dimension is “better” from a theoretical approach, the box-counting dimension becomes more appropriate for a wide range of applications. Next, we recall the analytical construction of the Hausdorff dimension. Let (X, ρ) be a metric space and let δ be a positive real number. Let us denote the diameter of a subset A ⊆ X by diam (A) = sup{ρ(x, y) : x, y ∈ A}, as usual. For any subset F of X, recall that a δ-cover of F is just a countable family of subsets  {Ui : i ∈ I} such that F ⊆ i∈I Ui , with diam (Ui ) ≤ δ for all i ∈ I. Further, let us denote by Cδ (F ) to the collection of all δ-covers of F . The underlying idea to define the Hausdorff measure consists of minimizing the sum of the s-powers of the diameters of all the subsets for any δ-cover, where s is going to be the fractal dimension. In this way, the following quantity is considered:  Hδs (F )

= inf



 diam (Ui ) : {Ui }i∈I ∈ Cδ (F ) . s

(2)

i∈I

Note that when δ decreases, then the class Cδ (F ) of all δ-covers of F is reduced, so the measure of F increases. Accordingly, the next limit always exists: s HH (F ) = lim Hδs (F ), δ→0

(3)

which is called the s-dimensional Hausdorff measure of F . Recall that the Hausdorff measure generalizes the classical Lebesgue measure for Euclidean subspaces. Hence, note that the Hausdorff dimension may be described as follows: s s dim H (F ) = inf{s : HH (F ) = 0} = sup{s : HH (F ) = ∞},

(4)

or equivalently,  s HH (F )

=

∞ 0

if s < dim H (F ) if s > dim H (F ).

(5)

JID:TOPOL 6

AID:5719 /FLA

[m3L; v1.172; Prn:19/01/2016; 15:00] P.6 (1-17)

M. Fernández-Martínez, M.A. Sánchez-Granero / Topology and its Applications ••• (••••) •••–•••

s Accordingly, the Hausdorff dimension of F is the point s where HH (F ) “jumps” from ∞ to 0. In particular, s s if s = dim H (F ), then HH (F ) can be equal to 0, ∞, and even it can be possible that HH (F ) ∈ (0, ∞).

2.4. Fractal dimension for fractal structures Additionally, in [15, Definition 4.2] it was provided a new model of fractal dimension for any fractal structure following the analytical construction of the Hausdorff dimension. This became a first discrete attempt to approach Hausdorff dimension in the context of fractal structures. Next, we recall the definition of the so-called fractal dimension III, which verifies some of the analytical properties of Hausdorff dimension, but presenting the advantage that it may be easily calculated, just like box-counting dimension. Definition 2.6. Let Γ be a fractal structure on a metric space (X, ρ) and let F be a subset of X. Let us suppose that δ(F, Γn ) −→ 0, where δ(F, Γn ) = sup{diam (A) : A ∈ Γn , A ∩ F = ∅}, and let us consider the following expression:  s Hn,3 (F )

= inf



 diam (A) : B ∈ An,3 (F ) , s

A∈B

where An,3 (F ) =



{Am (F )},

(6)

m≥n s and Am (F ) = {A ∈ Γm : A ∩ F = ∅} for all m ∈ N. Take also H3s (F ) = limn→∞ Hn,3 (F ). Then the fractal dimension III of F is defined as

dim 3Γ (F ) = inf{s : H3s (F ) = 0} = sup{s : H3s (F ) = ∞}. s Thus, since the set function Hn,3 provides a monotonic sequence in n ∈ N, then we have that the fractal dimension III of any subset F of X always exists (see [15, Remark 4.4]). In particular, in [15, Theorem 4.20] we proved how this model allows to calculate the fractal dimension of any attractor equipped with its natural fractal structure. In this case, the similarities of the corresponding iterated function system are not required to verify the so-called open set condition hypothesis. Some additional theoretical models to calculate the fractal dimension of a subset with respect to any fractal structure have been previously explored (see [15–17,19]). However, in this paper there will be no doubt about the fractal dimension model we are calculating in each case, so let us denote fractal dimension III as dim Γ to simplify notation. Moreover, we will refer to it simply as fractal dimension, herein. The next result is about the possibility of calculating fractal dimension from an easier expression (see [15, Theorem 4.7]).

Theorem 2.7. Let Γ be a fractal structure on a metric space (X, ρ) and let F be a subset of X. Let us suppose that Hs (F ) = limn→∞ Hns (F ) exists, where Hns (F ) =



{diam (A)s : A ∈ An (F )},

(7)

where An (F ) is given as in Definition 2.6. Then the fractal dimension of F can be calculated as follows: dim Γ (F ) = inf{s : Hs (F ) = 0} = sup{s : Hs (F ) = ∞}.

JID:TOPOL

AID:5719 /FLA

[m3L; v1.172; Prn:19/01/2016; 15:00] P.7 (1-17)

M. Fernández-Martínez, M.A. Sánchez-Granero / Topology and its Applications ••• (••••) •••–•••

7

3. The fractal dimension of a curve Both the Hausdorff dimension and the box-counting dimension constitute the classical fractal dimension models to explore the complexity of fractal patterns. These notions can be defined on any metric space, and while the former is “better” from a theoretical approach since its definition is based on a measure, the latter is “better” from the point of view of empirical applications since it is easier to calculate or to computationally estimate. Overall, almost all empirical applications of fractal dimension have been carried out on Euclidean spaces through the box-counting dimension. One of the main goals of this paper is to study the fractal dimension of a curve. Thus, note that the only choice available for the classical fractal dimension models is to calculate the fractal dimension for the graph of a given curve. On the other hand, fractal dimension (introduced in Definition 2.6 and calculated as in Theorem 2.7) may still be applied for this purpose since it provides a novel alternative to the classical definitions of fractal dimension. Indeed, the effective calculation of this fractal dimension model for a fractal structure also takes into account the structure of the curve as well as the complexity of the procedure used to generate it. To show how to do this, first we define an appropriate fractal structure on (the parametrization of) any curve from a fractal structure on the closed unit interval. We will denote I = [0, 1], herein. Definition 3.1. Let α : I −→ X be a parametrization of a curve and let X be a metric space. Let also Γ be a fractal structure on I. Then the fractal structure induced by Γ on the image set α(I) ⊆ X is defined as the countable family of coverings Δ = {Δn : n ∈ N}, where its levels are given by Δn = α(Γn ) = {α(A) : A ∈ Γn } for all n ∈ N. Fig. 1 illustrates how Definition 3.1 works. In this case, we provide the first two levels of the fractal structure induced on the image set of a given Brownian motion. In the bottom of the figures, the elements (marked by horizontal bars) of both level 1 (in the first image) and level 2 of the fractal structure Γ can be found. The images of these elements by that Brownian motion provide the elements (vertical bars on the left) of both level 1 (first image) and level 2 of the induced fractal structure Δ. Next, we pay attention to a necessary condition to properly define the fractal dimension of a curve with respect to its induced fractal structure. Remark 3.2. To calculate the fractal dimension for any subset F of a metric space X through either Definition 2.6 or Theorem 2.7, it becomes necessary to consider a fractal structure Γ such that δ(F, Γn ) −→ 0 (recall [15, Remark 4.1]). With this in mind, it results quite easy to check (by using uniform continuity of α) that, if the curve is continuous, then δ(F, Γn ) −→ 0 implies that δ(F, Δn ) −→ 0. This fact allows to calculate the fractal dimension of a curve with respect to its corresponding induced fractal structure Δ. The fractal dimension of the parametrization of a given curve can be defined from the induced fractal structure as follows [27]. Definition 3.3. Let ρ be a distance (resp. a metric, semimetric, quasi-metric, . . . , etc.) on X and let α : I −→ X be a parametrization of a curve. Let also Γ be a fractal structure on I and let Δ be the fractal structure induced by Γ on the image set α(I) ⊆ X. Then the fractal dimension of the (parametrization of the) curve α is defined by dim Γ (α) = dim Δ (α(I)). Moreover, if no additional information about the starting fractal structure Γ is provided, then we will assume that Γ is the natural fractal structure on I (as given in Definition 2.3). In that case, the fractal dimension of that curve will be denoted merely by dim (α) = dim Δ (α(I)).

JID:TOPOL 8

AID:5719 /FLA

[m3L; v1.172; Prn:19/01/2016; 15:00] P.8 (1-17)

M. Fernández-Martínez, M.A. Sánchez-Granero / Topology and its Applications ••• (••••) •••–•••

Fig. 1. The two figures above show the elements of the first two levels of the induced fractal structure Δ by the (natural) fractal structure Γ on the image set of a given Brownian motion α, namely Δ1 and Δ2 . The horizontal lines at the bottom of each figure represent the elements of the natural fractal structure on I, namely, Γ1 and Γ2 . In addition to that, note that the image of each dotted line border box by α gives the elements of Δ (vertical bars on the left).

Observe that the fractal dimension model allows us to provide an appropriate definition for the fractal dimension of (the parametrization of) a curve. This is possible since fractal dimension really takes into account all the possible overlappings of the elements in a given level of a fractal structure. This fact may seem a disadvantage of Definition 3.3 at a first glance, though in some cases (including this context) it becomes especially suitable since the fractal dimension of a curve may be calculated depending on each parametrization we select for it. On the other hand, note that the graph of the curve has to be considered instead in order to study the behavior of real curves through the classical fractal dimension models since it has no sense to calculate them for the image set of the curve. So overall, our new model allows a deeper study since it may be applied for different parametrizations of the same curve. Additionally, note that to apply Definition 3.3, the curve α does not need to be continuous. Also, it can be a time series or a financial time series, for instance. Indeed, in Section 4, we will use the previous definitions to obtain some theoretical results that allow us to connect the fractal dimension of a random process with its self-similarity exponent. The following technical lemma is likely well-known. However, by a lack of reference, next we provide a proof for sake of completeness. Lemma 3.4. Let a, b, c be three non-negative real numbers such that a ≤ b + c, and let us consider s ∈ (0, 1). Then as ≤ bs + cs . Proof. Firstly, note that if b > a, then the result is immediate, since bs > as , and this implies that bs +cs ≥ as +cs ≥ as . So, let us focus on the case b < a. Then there exists r ∈ (0, 1) such that b = a r. Hence, bs +cs ≥ as rs +as (1 −r)s = as (rs +(1 −r)s ), since c ≥ a −b = a (1 −r). Thus, the result holds, since xs ≥ x for all s, x ∈ (0, 1). Indeed, this implies that (1 − r)s ≥ 1 − r and rs ≥ r, so (1 − r)s + rs ≥ 1 − r + r = 1. 2

JID:TOPOL

AID:5719 /FLA

[m3L; v1.172; Prn:19/01/2016; 15:00] P.9 (1-17)

M. Fernández-Martínez, M.A. Sánchez-Granero / Topology and its Applications ••• (••••) •••–•••

9

Some analytical properties regarding the new fractal dimension for curves we introduced in Definition 3.3 are presented next. Theorem 3.5. Let α : I −→ X be a parametrization of a curve and let X be a metric space. Let also Γ be the natural fractal structure on I and let Δ be the fractal structure induced by Γ on the image set α(I) ⊆ X. Then the following statements hold. (1) if α is a constant curve, then dim (α) = 0. (2) if α is not a constant curve, then dim (α) ≥ 1. (3) if α is a Lipschitz map, then dim (α) ≤ 1. In particular, this is satisfied for any differentiable map with bounded differential. (4) if α is a differentiable map that is not constant, then dim (α) = 1. Proof. (1) Note that for all B ∈ Δn , there exists A ∈ Γn such that B = α(A) = {α(t) : t ∈ A} = {p}, that is a single point since the curve α is constant. Thus, diam (B) = diam (α(A)) = 0 for all B ∈ Δn . Hence, Hns (α(I)) = {diam (B)s : B ∈ Δn , B ∩ α(I) = ∅} = Card (Δn ) diam (B)s = 2n 0s = 0, so Hs (α(I)) = 0 for all s > 0. Accordingly, since dim (α) = inf{s : Hs (α(I)) = 0}, then we have that dim (α) ≤ s for all s > 0, namely, dim (α) = 0. (2) Firstly, recall that Lemma 3.4 states that if a ≤ b + c, then as ≤ bs + cs for all s ∈ (0, 1). In particular, for all A ∈ Δn , there exists B, C ∈ Δn+1 such that A = B ∪ C. Hence, it is satisfied that diam (A)s ≤ diam (B)s + diam (C)s s for all s ∈ (0, 1). Thus, we have that Hns (α(I)) ≤ Hn+1 (α(I)) for all n ∈ N, and since α is not a constant s s map, then H1 (α(I)) > 0. Consequently, H (α(I)) > 0 for all s < 1, so dim (α) ≥ s for all s < 1. This implies that dim (α) ≥ 1. (3) Note with A ∈ Γn . So we can affirm that B =

 that for all B ∈ Δn , we have that B = α(A) k k+1 α 2kn , k+1 α(b)], where a, b ∈ , with k ∈ {0, 1, . . . , 2n − 1}. Then diam (B)s = = [α(a), , 2n 2n 2n

(α(b) − α(a))s ≤ (L (b − a))s ≤ ( 2Ln )s , where L is the Lipschitz constant associated with α. Thus, Hns (α(I)) = {diam (B)s : B ∈ Δn , B ∩ α(I) = ∅} ≤ Ls 2n(1−s) . Here, we used that Card (Δn ) = 2n , for all n ∈ N. Therefore, Hs (α(I)) = 0 for all s > 1. Hence, dim (α) ≤ s for all s > 1, namely, dim (α) ≤ 1. (4) Just apply both Theorem 3.5 (2) and Theorem 3.5 (3) to reach the result. 2 The interpretation of the values that may be reached by the fractal dimension of a curve (as introduced in Definition 3.3) could be stated as follows. Any non-constant continuous curve has dimension d ∈ [1, ∞). A bigger fractal dimension means that the oscillations of the curve increase at any scale. On the other hand, smaller values of the fractal dimension for a curve imply a greater smoothness in its the graph. In particular, if α is a smooth curve, then dim (α) = 1, and if α is a Brownian motion, then dim (α) = 2. 3.1. Using fractal structures to construct space-filling curves In this subsection, we contribute some applications of the new fractal dimension for curves we introduced in Definition 3.3 to analyze the complexity of space-filling curves. Since G. Peano first described a plane-filling curve in 1890 [24], some curves of this kind have appeared in mathematical literature, being the Hilbert’s curve (whose description was first provided in [21]) one of

JID:TOPOL 10

AID:5719 /FLA

[m3L; v1.172; Prn:19/01/2016; 15:00] P.10 (1-17)

M. Fernández-Martínez, M.A. Sánchez-Granero / Topology and its Applications ••• (••••) •••–•••

the most famous. Our next goal is to show that fractal structures allow the description of this kind of filling curves from a rigorous point of view. The next result (that can be found in [26]) is the key to construct continuous maps between two topological spaces. This will be applied to show how to formalize the construction of space-filling curves in the forthcoming examples. Theorem 3.6. Let Γ = {Γn : n ∈ N} be a starbase fractal structure on a metric space X, and let Δ = {Δn : n ∈ N} be a Δ-Cantor-complete starbase fractal structure on a complete metric space Y . Let also fn : Γn −→ Δn be a family of maps verifying the two following conditions: • if A ∩ B = ∅ with A, B ∈ Γn for some n ∈ N, then fn (A) ∩ fn (B) = ∅, and • if A ⊆ B with A ∈ Γn+1 and B ∈ Γn for some n ∈ N, then fn+1 (A) ⊆ fn (B). Then there exists a unique continuous map f : X −→ Y such that f (A) ⊆ fn (A) for all A ∈ Γn and all n ∈ N. Moreover, if we assume that Γ is a Γ-Cantor-complete fractal structure and that fn also verifies that • fn is onto, as well as  • fn (A) = {fn+1 (B) : B ∈ Γn+1 , B ⊆ A} for each A ∈ Γn , then f is an onto map and f (A) = fn (A) for all A ∈ Γn and all n ∈ N. Proof. First of all, let us define the map f : X −→ Y . For any x ∈ X there exists a sequence {An : n ∈ N} such that An ∈ Γn , An+1 ⊆ An for each n ∈ N, and x ∈ ∩n∈N An . Then {fn (An ) : n ∈ N} is a decreasing sequence with fn (An ) ∈ Δn . Moreover, note also that ∩n∈N fn (An ) is exactly a point since Δ is Δ-Cantor complete and starbase. So let us define it as the image of x, namely, {f (x)} = ∩n∈N fn (An ). The following statements hold. • f is well-defined. Indeed, let x ∈ X and let us consider sequences {An : n ∈ N} and {An : n ∈ N} such that An , An ∈ Γn , An+1 ⊆ An and An+1 ⊆ An for each n ∈ N, with x ∈ ∩n∈N An and x ∈ ∩n∈N An . Let {y} = ∩n∈N fn (An ) and {z} = ∩n∈N fn (An ). If y = z, then there exists n ∈ N such that fn (An ) ∩fn (An ) = ∅ since Δ is starbase. However, note that since x ∈ An ∩An , then fn (An ) ∩fn (An ) = ∅ taking into account the properties of fn , which leads to a contradiction. Hence, y = z.  • f (A) ⊆ fn (A) for each A ∈ n∈N Γn . This becomes clear from the definition of f . • f is continuous. To show this, let n ∈ N and x ∈ X. If y ∈ St (x, Γn ), then there exists A ∈ Γn with x, y ∈ A. Thus, f (x), f (y) ∈ f (A) ⊆ fn (A) and fn (A) ∈ Δn . Hence, f (y) ∈ St (f (x), Δn ).  • Uniqueness of f . Let g : X −→ Y be a continuous map such that g(A) ⊆ fn (A) for all A ∈ n∈N Γn . Therefore, given x ∈ X there exists a sequence {An : n ∈ N} such that An ∈ Γn , An+1 ⊆ An for all n ∈ N with x ∈ ∩n∈N An . Hence, g(x) ∈ ∩n∈N fn (An ) = {f (x)}, so f (x) = g(x). Accordingly, f ≡ g. • Let us suppose that Γ is Γ-Cantor complete and that fn verifies the two conditions that follow: – fn (Γn ) = Δn , and  – fn (A) = {fn+1 (B) : B ∈ Γn+1 , B ⊆ A} for each A ∈ Γn .  Finally, let us show that f is an onto map and that f (A) = fn (A) for all A ∈ n∈N Γn . Note that  we only have to prove that fn (A) ⊆ f (A) for all A ∈ n∈N Γn (since this implies that f is onto). Indeed, let n ∈ N, A ∈ Γn and y ∈ fn (A). Let also Bn = fn (A) and An = A. By hypothesis, there exists An+1 ∈ Γn+1 with An+1 ⊆ An such that y ∈ fn+1 (An+1 ) ⊆ fn (An ). So let Bn+1 = fn+1 (An+1 ). Hence, note that we can construct recursively sequences {Bk : k ≥ n} and {Ak : k ≥ n} with Ak ∈ Γk , fk (Ak ) = Bk ∈ Δk , y ∈ Bk for k ≥ n, and An = A and Bn = fn (A). Since Γ is Γ-Cantor complete

JID:TOPOL

AID:5719 /FLA

[m3L; v1.172; Prn:19/01/2016; 15:00] P.11 (1-17)

M. Fernández-Martínez, M.A. Sánchez-Granero / Topology and its Applications ••• (••••) •••–•••

11

Fig. 2. First two levels (α1 and α2 ) of the Hilbert’s curve.

and starbase, we have that ∩k≥n Ak = {x} for some x ∈ X, and by construction of these sequences, it becomes clear that f (x) = y and that x ∈ A. Thus, y ∈ f (A). 2 Theorem 3.6 becomes the key to define functions or curves through fractal structures. Note that it may be understood as follows: we define the image of the first level of the fractal structure as a first approach to the definition of the function. Then, we refine the definition to the second level, and so on. If this refining process verifies some natural conditions (just for the coherence of the definition), then there really exists a map defined in the space which agrees with the approaches in each level. Moreover, since the proof of Theorem 3.6 is constructive, by choosing different fractal structures in a space or by choosing different chains in the construction, we can get different filling curves. This yields a great flexibility in the construction of filling curves that is worth in applications. In the forthcoming subsections, we explore different situations to illustrate this fact. 3.2. Applying fractal dimension to study the complexity of space-filling curves In this subsection, we show how fractal structures may be used in order to describe the classical Hilbert’s curve as well as a modified Hilbert’s curve. In addition to that, we apply Definition 3.3 to calculate their fractal dimensions which will be compared with their classical fractal dimensions. In Example 2, we can see that the fractal dimension we introduce in this paper provides a natural value for a space-filling curve, whereas in Example 3, we show that this new approach allows a deeper study of curves than the classical models for fractal dimension. Example 1 (The classical Hilbert’s curve (1891)). ([21]) The elegant iterative construction of the classical Hilbert’s plane-filling curve may be easily performed by means of fractal structures as follows. Let Γ be a fractal structure defined on the closed unit interval I, whose levels are given by Γn = {[ 2k2n , k+1 22n ] : k ∈ {0, 1, . . . , 22n −1}} for all n ∈ N. Let us also consider the fractal structure Δ = {Δn : n ∈ N} (that is just the k2 k2 +1 one induced by Γ on the image set α(I) ⊆ R), whose levels are given by Δn = {[ 2kn1 , k12+1 n ] × [ 2n , 2n ] : ki ∈ n {0, 1, . . . , 2 − 1}, i ∈ {1, 2}} for all natural number n. The definition of the classical Hilbert’s plane-filling curve by means of fractal structures may be done as follows. Let us describe the curve α : I −→ X by defining the image of each level of the fractal structure Γ by the map α. To do this, let us consider the sequence of maps {αn : n ∈ N}, where the definition of the maps αn : Γn −→ Δn is illustrated in Fig. 2. Indeed, Fig. 2 can be understood as follows: α([0, 14 ]) = [0, 12 ]2 , α([ 14 , 12 ]) = [ 12 , 1] ×[0, 12 ], α([ 12 , 34 ]) = [ 12 , 1]2 and α([ 34 , 1]) = [0, 12 ] × [ 12 , 1]. This allows us to completely define the set α(Γ1 ) = {α(A) : A ∈ Γ1 }, and now we continue in a similar way with the next levels of Γ. Note that the polygonal line in that image shows how the whole plane is filled by α in each level of the fractal structure Δ. Moreover, this recursive approach allows to refine the definition of αn in each stage of its construction, since more information about

JID:TOPOL 12

AID:5719 /FLA

[m3L; v1.172; Prn:19/01/2016; 15:00] P.12 (1-17)

M. Fernández-Martínez, M.A. Sánchez-Granero / Topology and its Applications ••• (••••) •••–•••

the Hilbert’s curve is provided as we explore deeper levels. Accordingly, if A ∈ Γn is sent to B ∈ Δn by   means of αn , then in the next level, A = {C ∈ Γn+1 : C ⊆ A} and B = {D ∈ Γn+1 : D ⊆ B}, and we have also that each C is sent to any D by αn+1 . Thus, α can be considered as the limit of the sequence of maps {αn : n ∈ N}. Next, we calculate the fractal dimension of the Hilbert’s curve and then we compare it with both its Hausdorff dimension and its box-counting dimension. Example 2. Let α be the Hilbert’s curve whose construction by means of fractal structures was defined in Example 1. Then dim (α) = dim B (α(I)) = dim H (α(I)) = 2. Proof. Let us calculate the fractal dimension of the Hilbert’s curve by counting the number of elements of the induced fractal structure Δ which meet α(I) on each level of that fractal structure. Thus, it suffices with taking into account that Hns (α(I)) =



√ {diam (B)s : B ∈ Δn , B ∩ α(I) = ∅} = ( 2)s 2n(2−s) ,

since there are 22n elements on each level Δn that meet α(I), and all of them have a diameter equal to √ n 2/2 for all natural number n. Hence,  H (α(I)) = s

∞ if s < 2 0 if s > 2,

which leads to dim (α) = 2. Note that the value obtained for the fractal dimension of the Hilbert’s curve becomes natural since this curve fills the whole unit square and accordingly, it must be equal to both its Hausdorff dimension and its box-counting dimension. 2 The next example presents a curve which also fills the whole unit square though its fractal dimension does not agree with its box-counting dimension nor its Hausdorff dimension. Thus, it shows that Definition 3.3 of fractal dimension results more accurate than the classical models of fractal dimension, since it also takes into account the way used to construct it. Example 3 (A modified Hilbert’s curve). Let us consider a modified Hilbert’s curve β which crosses twice some elements of each level of the induced fractal structure Δ and let us calculate its fractal dimension. Thus, we obtain that its fractal dimension does not agree with its box-counting dimension nor its Hausdorff dimensions (calculated for the image set β(I)). Proof. Indeed, let Γ = {Γn : n ∈ N} be a fractal structure on the closed unit interval I, whose levels are n given by Γn = {[ 5kn , k+1 5n ] : k ∈ {0, 1, . . . , 5 − 1}} for all n ∈ N. Let us also consider the curve β : I −→ X that we will define using Theorem 3.6, where X = [0, 1]2 is equipped with the Euclidean distance, and let also Δ be the fractal structure given in Example 1 and Δ the fractal structure induced by Γ on β(I). The definition of the modified Hilbert’s curve is given by the sequence of maps {βn : n ∈ N}, where the description of βn : Γn −→ Δn is illustrated in Fig. 3 for its first two levels. Recall that the polygonal line shows the way to fill the whole unit square in each stage. For instance, note that the elements of the first level Δ1 of the induced fractal structure are given as β([0, 15 ]) = β([ 15 , 25 ]) = [0, 12 ]2 , β([ 25 , 35 ]) = [ 12 , 1] × [0, 12 ], β([ 35 , 45 ]) = [ 12 , 1]2 and β([ 45 , 1]) = [0, 12 ] × [ 12 , 1]. The next levels can be obtained in a similar way. Note also that dim B (β(I)) = dim H (β(I)) = 2 since the modified Hilbert’s curve fills the whole square.

JID:TOPOL

AID:5719 /FLA

[m3L; v1.172; Prn:19/01/2016; 15:00] P.13 (1-17)

M. Fernández-Martínez, M.A. Sánchez-Granero / Topology and its Applications ••• (••••) •••–•••

13

Fig. 3. First two levels (β1 and β2 ) of the modified Hilbert’s curve.

√ On the other hand, we obtain that Hns (β(I)) = ( 2)s ( 25s )n for all natural number n, since there are √ 5n subsquares (of diameter equal to 2/2n each one) which meet β(I) on level n of the induced fractal structure Δ . Thus, ⎧ ⎪ if s < ⎨∞ √ 5 if s = Hs (β(I)) = ⎪ ⎩0 if s > which leads to dim (γ) = log 5/ log 2.

log 5 log 2 log 5 log 2 log 5 log 2 .

2

Accordingly, the obtained value for the fractal dimension of the modified Hilbert’s curve provides more accurate information about its fractal pattern since it also takes into account the underlying structure of the curve. Observe that it allows a deeper study of curves than the classical models for fractal dimension since it can be used with different parametrizations of the same curve. 3.3. Applying fractal dimension to study the complexity of a curve which fills a whole self-similar set As well as in Subsection 3.2 we described some plane-filling curves by means of fractal structures and calculated their fractal dimensions (by means of Definition 3.3), here we explore the possibility of defining curves which fill a whole self-similar set like the Sierpiński’s gasket [28]. The results obtained in the next example become quite natural. Example 4. We describe a curve γ which fills the whole Sierpiński’s gasket whose fractal dimension agrees with both the Hausdorff dimension and the box-counting dimension of the whole Sierpiński’s gasket as a self-similar set. Proof. Let Γ be a fractal structure on the closed unit interval I, where its levels are given by Γn = {[ 3kn , k+1 3n ] : k ∈ {0, 1, . . . , 3n − 1}} for all n ∈ N. Let us consider also the curve γ : I −→ X that we will define using Theorem 3.6, where X is the Sierpiński’s triangle contained in the equilateral triangle whose set of vertices is {(0, 0), ( 12 , 1), (1, 0)}, equipped with the Euclidean distance. Let also Δ be the fractal structure of X as a self similar set (see [5]). Note that in this case Δ will also be the fractal structure induced by Γ on the image set γ(I). Then the definition of the Sierpiński’s gasket filling curve is given by means of the sequence of maps γn : Γn −→ Δn whose definition is illustrated in Fig. 4 for its first levels. Note that the line shows the followed path in order to fill this self-similar set in each stage. Observe that γ([0, 13 ]) is the equilateral triangle whose set of vertices is {(0, 0), ( 41 , 12 ), ( 12 , 0)}. Similarly, γ([ 13 , 23 ]) is the equilateral triangle described by the set of vertices {( 12 , 0), ( 34 , 12 ), (1, 0)}, and γ([ 23 , 1]) is just the equilateral triangle given by {( 41 , 12 ), ( 12 , 1), ( 34 , 12 )}. Note that since all these triangles are in the

JID:TOPOL 14

AID:5719 /FLA

[m3L; v1.172; Prn:19/01/2016; 15:00] P.14 (1-17)

M. Fernández-Martínez, M.A. Sánchez-Granero / Topology and its Applications ••• (••••) •••–•••

Fig. 4. First two levels (γ1 and γ2 ) of a curve defined on the Sierpiński’s gasket which fills all this self-similar set.

Sierpiński’s gasket, they are not fully filled. Hence, Δ1 = γ(Γ1 ) which is the first level of the induced fractal structure by Γ on the image set γ(I). This method can be applied in a similar way for the next levels of Δ. Additionally, some calculations lead to  Hns (γ(I))

=

3 2s

n ,

for all n ∈ N, since there are 3n equilateral triangles (with sides equal to 1/2n each) on level n of the fractal structure Δ. Moreover, note that ⎧ ⎪ ⎨ ∞ if s < s H (γ(I)) = 1 if s = ⎪ ⎩ 0 if s >

log 3 log 2 log 3 log 2 log 3 log 2 .

Hence, dim (γ) = log 3/ log 2, since this is the value of s where it jumps from ∞ to 0. In this case, the fractal dimension of this curve agrees with both the box-counting dimension and the Hausdorff dimension of the whole Sierpiński’s gasket, which results quite natural, since the image of this curve fills the whole set. 2 4. Applying fractal dimension to study random processes In this section, we show how the new fractal dimension model introduced in Definition 3.3 for any fractal structure could be used to study fractal patterns of random processes. In particular, we establish a connection between the fractal dimension and the self-similarity exponent associated with a sample function of a random process. A wide range of random processes, including (fractional) Brownian motions (FBMs for short) and (fractional) Lévy stable motions (FLSMs) have been applied in scientific literature to model financial and other kinds of time series. Accordingly, the theoretical results of this section could be applied to study long-memory of this kind of processes. For the sake of completeness of the study of the fractal dimension of a curve in this paper, we have gathered in this section two key results from [27]. 4.1. Random functions and their increments. Self-affinity properties The definitions, properties and results that we recall next come from both the theory of probability and stochastic processes. In this way, some useful references are [12,23]. Let (X, A, P ) be a probability space and let t ∈ [0, ∞) denote time. We say that X = {X(t, ω) : t ≥ 0} is a random process or a random function from [0, ∞) × Ω to R, if X(t, ω) is a random variable for all t ≥ 0 and all ω ∈ Ω (where ω belongs to a sample space Ω). So we could think of X as defining a sample function

JID:TOPOL

AID:5719 /FLA

[m3L; v1.172; Prn:19/01/2016; 15:00] P.15 (1-17)

M. Fernández-Martínez, M.A. Sánchez-Granero / Topology and its Applications ••• (••••) •••–•••

15

t → X(t, ω) for all ω ∈ Ω. Hence, the points of Ω parametrize functions X : [0, ∞) × Ω → R and P is a probability measure on this class of functions. The definitions of Brownian motion (BM for short), fractional Brownian motion (FBM), Lévy stable motion (LSM) and fractional Lévy stable motion (FLSM) can be consulted in [12,13]. Let X(t, ω) and Y (t, ω) be two random functions. The notation X(t, ω) ∼ Y (t, ω) means that the two preceding random functions have the same finite joint distribution functions. Recall also that (1) A random process X = {X(t, ω) : t ≥ 0} is said to be H-self-similar if for some H > 0, it is satisfied that X(at, ω) ∼ aH X(t, ω), for all a > 0 and t ≥ 0. The parameter H is called the self-similarity index or exponent of the random process X. (2) The increments of a random function X(t, ω) are said to be: (a) stationary, if for each a > 0 and t ≥ 0, X(a + t, ω) − X(a, ω) ∼ X(t, ω) − X(0, ω). (b) self-affine with parameter H ≥ 0, if for any h > 0 and any t0 ≥ 0, X(t0 + τ, ω) − X(t0 , ω) ∼

1 {X(t0 + hτ, ω) − X(t0 , ω)}. hH

(8)

For instance, note that any FBM with exponent H has stationary and self-affine increments with parameter H (see [23, Theorem 3.3]). On the other hand, note that by [23, Corollary 3.6], we have that if a random function X(t, ω) has self-affine increments with parameter H, then a T H -law as the following one is satisfied: M (T, ω) ∼ T H M (1, ω),

(9)

where its cumulative range is given by  M (t, T, ω) =

sup

   X(s, ω) − X(t, ω) − inf s∈[t,t+T ] X(s, ω) − X(t, ω) ,

(10)

s∈[t,t+T ]

and moreover, M (T, ω) = M (0, T, ω). In particular, any FBM with parameter H satisfies a T H -law as that one contained in Eq. (9). Further, if X is a FBM, then it is possible to replace sup and inf by max and min respectively, in Eq. (10) by [23, Proposition 4.1]. Remark 4.1. Let α : I −→ R be a sample function of a random process X with stationary increments. Let also Γ be the natural fractal structure on I and let Δ be the fractal structure induced by Γ on α(I). Then the collection {diam (A) : A ∈ Δn } becomes a sample of the random variable M ( 21n , ω) for all n ∈ N. 4.2. Connecting fractal dimension with Hurst exponent The new model to calculate the fractal dimension allows to define the fractal dimension for the parametrization of a curve. This is possible since it takes into account the overlappings of the elements in a given level of the fractal structure, and though it may seem a disadvantage of the definition at a first glance, it follows that in some cases and in particular for this application, it becomes very appropriate and allows to calculate the fractal dimension of a curve depending on its parametrization. Recall also that the curve under study does not need to be continuous, so it may be a time series and in particular, a financial time series.

JID:TOPOL

AID:5719 /FLA

[m3L; v1.172; Prn:19/01/2016; 15:00] P.16 (1-17)

M. Fernández-Martínez, M.A. Sánchez-Granero / Topology and its Applications ••• (••••) •••–•••

16

Next, we present the main theoretical results of this section. Recall that the sth moment of a random variable X is defined by ms (X) = E(X s ). Firstly, we have a key theorem that allows to connect both the fractal dimension and the self-similarity exponent. Theorem 4.2. (Compare [27, Th. 1]) Let α : I −→ R be a sample function of a random process X with stationary and self-affine increments with parameter H. Let Γ be the natural fractal structure on I and let Δ the fractal structure induced by Γ on α(I). Then the two following hold. 1 (1) M ( 21n , ω) ∼ 2H M ( 2n+1 , ω), and 1 (2) If the H -moment of the cumulative range M ( 21n , ω) is finite then dim (α) =

1 H.

Proof. (1) By hypothesis, X has stationary and self-affine increments with parameter H, so Eq. (9) implies that

1 1 H ,ω ∼ n M (1, ω), 2n 2

(11)

1

1 H , ω ∼ n+1 M (1, ω). 2n+1 2

(12)

M and equivalently, M

Hence, Theorem 4.2 (1) becomes now immediate from both Eqs. (11) and (12). 1 1 1 (2) Firstly, it is clear that Theorem 4.2 (1) leads to M ( 21n , ω) H ∼ 2 M ( 2n+1 , ω) H . In addition to that, note that by Remark 4.1, {diam (A) : A ∈ Δn } becomes a sample of the random variable M ( 21n , ω), so 1 1 {diam (A) H : A ∈ Δn } is a sample of the random variable M ( 21n , ω) H . 1 1 If the H -moment of M ( 21n , ω) is finite, then the mean of the random variable M ( 21n , ω) H will be finite. 1 1 1 1 Thus, since M ( 21n , ω) H ∼ 2 M ( 2n+1 , ω) H , then the mean of any sample of M ( 21n , ω) H must be equal 1 1 to twice the mean of any sample of M ( 2n+1 , ω) H , and hence, it follows that 1 1 {diam (A) H : A ∈ Δn } {diam (B) H : B ∈ Δn+1 } =2 , 2n 2n+1 1

1

1

H which implies that HnH (α(I)) = Hn+1 (α(I)) for all n ∈ N. Thus, there exists H H (α(I)) ∈ (0, ∞), and 1 then dim (α) = H . 2

Theorem 4.2 is quite general. For instance, FBMs verifies the hypothesis of such result. Corollary 4.3. Let α : I −→ R be a sample function of a FBM X with parameter H. Let also Γ be the 1 natural fractal structure on I and let Δ be the fractal structure induced by Γ on α(I). Then dim (α) = H . Proof. The increments of any FBM are stationary and self-affine with parameter H by [23, Theorem 3.3], so it suffices with applying Theorem 4.2 (2) to get the result. 2 Next, we present the second main result of this section which allows to calculate the fractal dimension for a wide range of curves. Theorem 4.4. ([27, Th. 3]) Let α : I −→ R be a sample function of a random process X. Let Γ be the natural fractal structure on I and let Δ be the fractal structure induced by Γ on α(I). Let Xn = M ( 21n , ω) be the

JID:TOPOL

AID:5719 /FLA

[m3L; v1.172; Prn:19/01/2016; 15:00] P.17 (1-17)

M. Fernández-Martínez, M.A. Sánchez-Granero / Topology and its Applications ••• (••••) •••–•••

17

random variable which provides the statistical distribution of the collection of diameters {diam (A) : A ∈ Δn }. Let us suppose also that there exists a positive real number s that verifies the next two conditions: (1) the s-moment of Xn , ms (Xn ), is finite for all n ∈ N, and (2) ms (Xn ) = 2 ms (Xn+1 ) for all n ∈ N. Then dim (α) = s. In [27, Section 6], the authors described in detail how Theorems 4.2 and 4.4 allow to test for long-range properties on S&P500 stocks. This was done by calculating their fractal dimension regarding the evolution of each stock through time as a discrete curve. References [1] F.G. Arenas, M.A. Sánchez-Granero, A characterization of non-archimedeanly quasimetrizable spaces, Rend. Istit. Mat. Univ. Trieste Suppl. XXX (1999) 21–30. [2] F.G. Arenas, M.A. Sánchez-Granero, A new approach to metrization, Topol. Appl. 123 (1) (2002) 15–26. [3] F.G. Arenas, M.A. Sánchez-Granero, A new metrization theorem, Boll. Unione Mat. Ital. (8) 5-B (2002) 109–122. [4] F.G. Arenas, M.A. Sánchez-Granero, Completeness in metric spaces, Indian J. Pure Appl. Math. 33 (8) (2002) 1197–1208. [5] F.G. Arenas, M.A. Sánchez-Granero, A characterization of self-similar symbolic spaces, Mediterr. J. Math. 9 (4) (2012) 709–728. [6] C. Bandt, T. Retta, Topological spaces admitting a unique fractal structure, Fundam. Math. 141 (1992) 257–268. [7] A.S. Besicovitch, Sets of fractional dimensions IV: on rational approximation to real numbers, J. Lond. Math. Soc. 9 (1934) 126–131. [8] A.S. Besicovitch, H.D. Ursell, Sets of fractional dimensions V: on dimensional numbers of some continuous curves, J. Lond. Math. Soc. 12 (1937) 18–25. [9] G. Bouligand, Ensembles impropres et nombre dimensionnel, Bull. Sci. Math. II-52 (1928) 320–334, 361–376. [10] C. Brown, L. Liebovitch, Fractal Analysis, first ed., Quantitative Applications in the Social Sciences, vol. 165, SAGE Publications Inc., New York, 2010. [11] C. Carathéodory, Über das lineare mass von punktmengen-eine verallgemeinerung das längenbegriffs, Nachr. Ges. Wiss. Gött. (1914) 406–426. [12] K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, John Wiley & Sons, 1990. [13] M. Jeanblanc, M. Yor, M. Chesney, Mathematical Methods for Financial Markets, Springer Finance, London, 2009. [14] J. Feder, Fractals, Plenum Press, New York, 1988. [15] M. Fernández-Martínez, M.A. Sánchez-Granero, Fractal dimension for fractal structures: a Hausdorff approach, Topol. Appl. 159 (2012) 1825–1837. [16] M. Fernández-Martínez, M.A. Sánchez-Granero, Fractal dimension for fractal structures, Topol. Appl. 163 (2014) 93–111. [17] M. Fernández-Martínez, M.A. Sánchez-Granero, Fractal dimension for fractal structures: a Hausdorff approach revisited, J. Math. Anal. Appl. 409 (2014) 321–330. [18] M. Fernández-Martínez, M.A. Sánchez-Granero, J.E. Trinidad Segovia, Fractal dimension for fractal structures: applications to the domain of words, Appl. Math. Comput. 219 (2012) 1193–1199. [19] M. Fernández-Martínez, M.A. Sánchez-Granero, J.E. Trinidad Segovia, Fractal Dimensions for Fractal Structures and Their Applications to Financial Markets, Aracne, Roma, 2013. [20] F. Hausdorff, Dimension und äusseres mass, Math. Ann. 79 (1919) 157–179. [21] D. Hilbert, Über die stetige Abbildung einer Linie auf ein Flächenstück, Math. Ann. 38 (1891) 459–460. [22] B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman & Company, New York, 1982. [23] B. Mandelbrot, Gaussian Self-Affinity and Fractals, Springer-Verlag, New York, 2002. [24] G. Peano, Sur une courbe qui remplit toute une aire plane, Math. Ann. 36 (1890) 157–160. [25] L. Pontrjagin, L. Schnirelman, Sur une proprieté métrique de la dimension, Ann. Math. 33 (1) (1932) 156–162. [26] M.A. Sánchez-Granero, Fractal structures, in: Asymmetric Topology and Its Applications, in: Quaderni di Matematica, vol. 26, Aracne, 2012, pp. 211–245. [27] M.A. Sánchez-Granero, M. Fernández-Martínez, J.E. Trinidad Segovia, Introducing fractal dimension algorithms to calculate the Hurst exponent of financial time series, Eur. Phys. J. B 85 (2012) 86, http://dx.doi.org/10.1140/epjb/e201220803-2. [28] W. Sierpiński, Sur une courbe dont tout point est un point de ramification, C. R. Math. Acad. Sci. Paris 160 (1915) 302–305.