fractal dimension for fractal structures

$: Fractal dimension for fractal structures$
Topology and its Applications 163 (2014) 93–111

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Topology and its Applications

www.elsevier.com/locate/topol

Fractal dimension for fractal structures

M. Fernández-Martínez ∗,1, M.A. Sánchez-Granero 2

Area of Geometry and Topology, Department of Mathematics, Universidad de Almería, 04120 Almería,Spain

a r t i c l e i n f o a b s t r a c t

MSC:primary 37F35secondary 28A80, 54E99

Keywords:FractalFractal structureGeneralized-fractal spaceFractal dimensionBox-counting dimensionSelf-similar set

The main goal of this paper is to provide a generalized definition of fractal dimensionfor any space equipped with a fractal structure. This novel theory generalizes theclassical box-counting dimension theory on the more general context of GF-spaces.In this way, if we select the so-called natural fractal structure on any Euclideanspace, then the box-counting dimension becomes just a particular case. This ideaallows to consider a wide range of fractal structures to calculate the effective fractaldimension for any subset of this space. Unlike it happens with the classical theoryof fractal dimension, the new definitions we provide may be calculated in contextswhere the box-counting one can have no sense or cannot be calculated. Nevertheless,the new models can be computed for any space admitting a fractal structure, justas easy as the box-counting dimension in empirical applications.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Since the concept of fractal was first introduced by Benoît Mandelbrot in the early eighties [12], both thestudy and analysis of this kind of non-linear objects have become more and more important. In this way,fractals have been applied to a diverse spectrum of fields in science. The main tool used in these areas is thefractal dimension, understood as the box-counting dimension, since it is a single quantity which providessome information about the complexity of a given set.

The main goal of this paper is to introduce a new concept of fractal dimension for a space equippedwith a fractal structure. A fractal structure is just a countable collection of coverings which provides betterapproximations of the whole space as we go through deeper stages. If we analyze the definition of thebox-counting dimension, then we can observe that fractal structures provide a perfect place to define theconcept of fractal dimension.

The first notion of fractal dimension we propose depends only on the fractal structure. Note that somevalues of this fractal dimension may seem counterintuitive at a first glance. For example, the Cantor sethas dimension one with respect to some fractal structure but this fact is a consequence of depending only

* Corresponding author.E-mail addresses: [email protected], [email protected] (M. Fernández-Martínez), [email protected] (M.A. Sánchez-Granero).

1 The first author appreciates a FPU grant of the Spanish Ministry of Education.2 The second author acknowledges the support of the Spanish Ministry of Science and Innovation, grant MTM2009-12872-C02-01.

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94 M. Fernández-Martínez, M.A. Sánchez-Granero / Topology and its Applications 163 (2014) 93–111

on the fractal structure and not on any metric we can consider on the space. We explain this situation inRemark 4.10.

An alternative notion of fractal dimension we provide depends not only on the fractal structure but alsoon a metric (resp. distance) function. In this way, this second version of fractal dimension agrees with thefirst one when taking into account the semimetric associated with the fractal structure. Thus, the secondnotion of fractal dimension may be applied if it is necessary to consider the size of the elements of each levelof the fractal structure.

One of the main properties of these fractal dimensions is that they are easy to calculate, just as easyas the box-counting dimension of a Euclidean subspace. But these dimensions can also be applied for anyspace with a fractal structure. This allows to use the concept of fractal dimension in new contexts wherethe box-counting dimension can have no sense or may not be applied. And, what kind of spaces admitsa fractal structure? Exactly those spaces which admit a non-Archimedean quasi-metric. This includes anymetrizable space, but also other non-metrizable spaces which can also be interesting. In this way, in [8] theauthors provided an example of how to calculate the fractal dimension with respect to the fractal structureinduced by a (non-metrizable) non-Archimedean quasi-metric.

Note that the box-counting dimension can be defined in a metric space, but the definition that allowsan easy calculation is only available for Euclidean subspaces. This means that if we want to use the fractaldimension in an empirical application on a given (non-Euclidean) metric space, then we cannot use thebox-counting dimension, but we can only use the fractal dimension models introduced in this paper.

On the other hand, note that the box-counting dimension becomes just a particular case of the newfractal dimension definitions since they coincide for the natural fractal structure on any Euclidean spacewhich we formally introduce in Section 3.

The structure of the paper is as follows. In Section 2, we recall some concepts and notations whichbecome necessary to develop our theory of fractal dimension for fractal structures. This section is focusedon fractal structures, iterated function systems, quasi-pseudometrics and box-counting dimension topics. InSections 3 and 4, we introduce the two new definitions of fractal dimension for a set with respect to anyfractal structure. Then, we examine some properties of these notions and relate them with the box-countingdimension in those contexts where the latter can be defined, by providing some conditions on the elements ofthe fractal structure. In this way, we also study the new fractal dimensions for attractors of iterated functionsystems which constitute a kind of fractals which always present a fractal structure in a natural way.

2. Preliminaries

Let us start with some preliminary topics.

2.1. Fractal structures

The main purpose of this subsection is to introduce some notations and basic notions that will be usefulin this paper.

First, we begin with the concept of fractal structure. Recall that it was introduced in [2] to characterizenon-Archimedeanly quasi-metrizable spaces, but it can also be used to study fractals. For example, in [5],it was used to study attractors of iterated function systems.

The use of fractal structures provides a powerful tool to introduce new models for a definition of fractaldimension, since a fractal structure is a natural context in which the concept of fractal dimension can beperformed. Moreover, it will allow to calculate the fractal dimension in new spaces and situations.

Let Γ be a covering of X. We will denote St(x, Γ ) =⋃{A ∈ Γ : x ∈ A} and UxΓ = X \

⋃{A ∈ Γ : x /∈ A}.

Furthermore, if Γ = {Γn: n ∈ N} is a countable family of coverings of X, then we will denote Uxn = UxΓn,

UΓx = {Uxn: n ∈ N} and St(x,Γ ) = {St(x, Γn): n ∈ N}.

M. Fernández-Martínez, M.A. Sánchez-Granero / Topology and its Applications 163 (2014) 93–111 95

Definition 2.1. ([2, Definition 3.1]) Let X be a topological space. A pre-fractal structure on X is a countablefamily of coverings called levels, Γ = {Γn: n ∈ N}, such that UΓ

x is an open neighborhood base of x foreach x ∈ X.

Moreover, if Γn+1 is a refinement of Γn, such that for all x ∈ A with A ∈ Γn, there exists B ∈ Γn+1 suchthat x ∈ B ⊆ A, then we will say that Γ is a fractal structure on X.

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 Γ , then we will say that X is a (pre-)GF-space.

Remark 2.2. To simplify the theory, the levels of any fractal structure Γ will not be coverings in the usualsense. Instead, 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.

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 callΓ a starbase fractal structure. 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 finite if for each level Γn of the fractal structure Γ , we have thatevery point x ∈ X belongs to a finite number of elements A ∈ Γn. In general, if Γn has the property P forall n ∈ N, and Γ = {Γn: n ∈ N} is a fractal structure on X, we will say that Γ is a fractal structure withthe property P , and that (X,Γ ) is a GF-space with the property P .

2.2. Iterated function systems

Next we recall the definition of attractor of an iterated function system, introduced by Hutchinson(see [11]).

Definition 2.3. Let I = {1, . . . ,m} be a finite index set and let {fi: i ∈ I} be a family of contractive mappingsdefined from a complete metric space X into itself. (X, {fi: i ∈ I}) is called an iterated function system(IFS for short). Then, there exists a unique non-empty compact subset K of X such that K =

⋃i∈I fi(K).

K is called the attractor of the IFS. Further, if the mappings fi are similarities, then K is called a strictself-similar set.

Next, we provide an interesting example which describes analytically the so-called Sierpiński gasket, firstdefined in [15], which is a typical example of a strict self-similar set.

Example 1. Let I = {1, 2, 3} be a finite index set and let {fi: i ∈ I} be a finite set of similarities definedfrom the Euclidean plane into itself as follows:

fi(x, y) =

⎧⎨⎩

(x2 ,y2 ) if i = 1

f1(x, y) + (12 , 0) if i = 2

f1(x, y) + (14 ,

12 ) if i = 3

for all (x, y) ∈ R2. Thus, the Sierpiński gasket is fully determined as the unique non-empty compact subsetverifying the next Hutchinson’s equation: K =

⋃i∈I fi(K). In this way, note that each component fi(K) is

a self-similar copy of the whole Sierpiński gasket.

To consult a procedure in order to approach self-similar sets, see [1]. Note that attractors of IFS constitutean interesting kind of fractals that always have a natural fractal structure which was first sketched in [6].Note that this paper is the origin of the term fractal structure. Next, we recall the description of such fractalstructure as it was provided in [5, Definition 4.4].


Definition 2.4. Let I = {1, . . . ,m} be a finite index set, (X, {fi: i ∈ I}) an IFS and K the attractor of thatIFS. The natural fractal structure on K can be defined as the countable family of coverings Γ = {Γn: n ∈ N},where Γn = {fω(K): ω ∈ In}, for each n ∈ N. Here, for n ∈ N and each word ω = ω1 ω2 . . . ωn ∈ In, wedenote fω = fω1 ◦ · · · ◦ fωn

.

Remark 2.5. Another description of the levels of that fractal structure may be performed as follows: Γ1 ={fi(K): i ∈ I}, and Γn+1 = {fi(A): A ∈ Γn, i ∈ I} for all n ∈ N.

In Example 1 we described analytically the IFS whose associated attractor is the Sierpiński gasket. Next,we describe the natural fractal structure defined on this strict self-similar set.

Example 2. The natural fractal structure defined on the Sierpiński gasket can be given as the countablefamily of coverings Γ = {Γn: n ∈ N}, where Γ1 is the union of three equilateral “triangles” with sides equalto 1/2, Γ2 consists of the union of 32 equilateral “triangles” with sides equal to 1/22, and in general, Γn is theunion of 3n equilateral “triangles” whose sides are equal to 1/2n, for each natural number n. Furthermore,this is a finite starbase fractal structure.

2.3. Quasi-pseudometrics

Recall that a quasi-pseudometric on a set X is a non-negative real-valued function d defined on X ×X

such that for all x, y, z ∈ X, the following conditions are verified: (i) d(x, x) = 0, and (ii) d(x, y) �d(x, z) + d(z, y). If in addition d satisfies also the next one: (iii) d(x, y) = d(y, x) = 0 iff x = y, thend is called a quasi-metric. In particular, a non-Archimedean quasi-pseudometric is a quasi-pseudometricwhich also verifies that d(x, y) � max{d(x, z), d(z, y)} for all x, y, z ∈ X. Moreover, we have that eachquasi-pseudometric d on X generates a quasi-uniformity Ud on X which has as a base the family of sets ofthe form {(x, y) ∈ X ×X: d(x, y) < 1/2n}, n ∈ N. The topology τ(Ud) induced by the quasi-uniformity Ud

will be denoted simply by τ(d). Therefore, a topological space (X, τ) is said to be (non-Archimedeanly)quasi-pseudometrizable if there exists a (non-Archimedean) quasi-pseudometric d on X such that τ = τ(d).The theory of quasi-uniform spaces is covered in detail in [9].

2.4. The box-counting dimension

Fractal dimension is one of the main tools applied to study fractals, since it is a single quantity whichprovides information about their complexity when being examined with enough level of detail. In thisway, fractal dimension is usually understood as the classical box-counting dimension, which is also knownas information dimension, Kolmogorov entropy, capacity dimension, entropy dimension, metric dimension,logarithmic density, . . . , etc. (see [7]).

Though the Hausdorff dimension can be considered as a fractal dimension too, in practical applicationsit is always used the box-counting dimension, since it is the only one that can be calculated when workingwith a finite range of scales, which is the case of empirical applications. Popularity of the box-countingdimension is mainly due to the possibility of its effective calculation and empirical estimation in Euclideancontexts. Indeed, in practical applications, the box-counting dimension may be estimated as the slope ofthe regression line of a log–log graph plotted for a suitable discrete collection of scales δ.

The basic theory of box-counting dimension can be found in [7]. Next, we recall the definition of thestandard box-counting dimension.

Definition 2.6. The (lower/upper) box-counting dimension of a subset F ⊆ Rd is given by the following(lower/upper) limit:

dimB(F ) = lim logNδ(F ) (1)
δ→0 − log δ


where δ is the scale and Nδ(F ) can be calculated, in an equivalent way, as one of the following quantities(see [7, 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 [k1δ, (k1 +1)δ]× [k2δ, (k2 +1)δ] × · · · × [kdδ, (kd + 1)δ] where ki ∈ Z for all i ∈ {1, . . . , d};

(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 .

Note also that the limit in Eq. (1) can be discretized by taking, for example, δ = 1/2n, which is formalizedin the next remark.

Remark 2.7. To calculate the (lower/upper) box-counting dimension of any subset F of a Euclidean space Rd,it suffices to take limits as δ → 0 through any decreasing sequence {δn: n ∈ N} verifying that δn+1 � c · δnfor all n ∈ N, where c ∈ (0, 1) is a suitable constant. In particular, it holds for δn = 1/2n.

3. Generalized definition of fractal dimension for a subset with respect to any fractal structure: fractaldimension I

The starting point of the fractal dimension theory for fractal structures we develop in this paper considersNδ(F ) as the number of 1/2n-cubes that meet F with n being a natural number (see equivalent definition (2)in Definition 2.6) and it also applies Remark 2.7. Thus, we define a fractal structure on any Euclideanspace Rd in a natural way which verifies some desirable properties: it is a locally finite starbase fractalstructure.

Definition 3.1. The natural fractal structure on the Euclidean space Rd is defined as the countable familyof coverings Γ = {Γn: n ∈ N}, whose levels are given by

Γn ={[

k1

2n ,k1 + 1

2n

]×

[k2

2n ,k2 + 1

2n

]× · · · ×

[kd2n ,

kd + 12n

]: k1, . . . , kd ∈ Z

}

for each n ∈ N.

Remark 3.2. In particular, it is also possible to consider a natural fractal structure induced on real subsetsfrom the previous one. For instance, the natural fractal structure induced on the interval [0, 1] ⊂ R may bedefined as the family of coverings Γ , whose levels are Γn = {[ k

2n ,k+12n ]: k ∈ {0, 1, . . . , 2n−1}} for all natural

numbers n.

Note that the natural fractal structure on the Euclidean space Rd is just the tiling consisting of 1/2n-cubeson Rd. Then both definition (2) of Nδ(F ) and Definition 3.1 imply that N 1

2n(F ) is just the number of

elements of Γn which meet F . This motivates the introduction of our first model of fractal dimension forany fractal structure.

Definition 3.3. Let Γ be a fractal structure on X and let F be a subset of X. Let also Nn(F ) be thenumber of elements of Γn which meet F . Then the (lower/upper) fractal dimension I of F is defined as the(lower/upper) limit:

dim1Γ (F ) = lim

n→∞logNn(F )n log 2


Remark 3.4. In order to estimate the fractal dimension I of a set F in an empirical application, it may becalculated the slope of the regression line of log2(Nn(F )) vs. n, just like with the box-counting dimensionestimation.

Theorem 3.5. Let Γ be the natural fractal structure on the Euclidean space Rd and let F be a subset of Rd.Then the (lower/upper) box-counting dimension and the (lower/upper) fractal dimension I of F are equal:

dimB(F ) = dim1Γ (F )

Proof. The arguments presented above to motivate Definition 3.3 lead to the result. �The Hausdorff dimension constitutes the main theoretical model for any definition of fractal dimension.

In this way, we can check some of its analytical properties (see [7, Chapter 3]) for fractal dimension I.Indeed, we have the following result.

Proposition 3.6. Let (X,Γ ) be a GF-space. Then,

(1) both lower and upper fractal dimensions I are monotonic;(2) upper fractal dimension I is finitely stable;(3) neither lower nor upper fractal dimension I are countably stable;(4) there exist a countable set F ⊆ X and a fractal structure Γ defined on X such that dim1

Γ (F ) �= 0;(5) there exists a locally finite starbase fractal structure Γ on a suitable space X such that dim1

Γ (F ) �=dim1

Γ (F ) for a given subset F ⊆ X.

Proof.

(1) Let E,F be two subsets of X and let us suppose that E ⊆ F . Then it is clear that Nn(E) � Nn(F ) forall n ∈ N, since any element A ∈ Γn such that A ∩ E �= ∅ verifies also that A ∩ F �= ∅. Accordingly, wehave that

logNn(E)n log 2 � logNn(F )

n log 2

which leads to dim1Γ (E) � dim1

Γ (F ), and the same arguments remain true for upper fractal dimension I.(2) Firstly, recall that a dimension function dim verifies the finitely stable property if it satisfies the next

expression:

dim(E ∪ F ) = max{dim(E),dim(F )

}Let E,F be two subsets of X. Since the fractal dimension I function is monotonic, then we have thatmax{dim1

Γ (E),dim1Γ (F )} � dim1

Γ (E∪F ). On the other hand, note that Nn(E∪F ) � Nn(E)+Nn(F ) forall natural numbers n, and let ε be a positive real number. Let d1 = dim1

Γ (E). Then there exists n1 ∈ Nsuch that Nn(E) � 2n(d1+ε) for all n � n1. Equivalently, if d2 = dim1

Γ (F ), then Nn(F ) � 2n(d2+ε) forall n � n2. Let us suppose without loss of generality that d1 � d2 and let m = max{n1, n2}. Thus, wehave that Nn(E) + Nn(F ) � 2n(d1+ε)+1 for all n � m. Accordingly,

dim1Γ (E ∪ F ) � limn→∞

log(Nn(E) + Nn(F ))n log 2 � limn→∞

n(d1 + ε) + 1n

= d1 + ε

for all ε > 0. Hence, dim1Γ (E ∪ F ) � max{dim1

Γ (E), dim1Γ (F )} which provides the opposite inequality.


(4) The countable stability property for a dimension function dim means that

dim(⋃

i∈I

Fi

)= sup

i∈Idim(Fi)

where {Fi: i ∈ I} is a countable family of subsets of X. Therefore, consider X = [0, 1] with F = Q∩ [0, 1],and let Γ be the natural fractal structure on the real line induced on [0, 1], whose levels can be describedby Γn = {[ k

2n ,k+12n ]: k ∈ {0, 1, . . . , 2n − 1}} for all n ∈ N. Then it is clear that Nn(F ) = 2n, so that

dim1Γ (F ) = 1. Furthermore, note that (4) implies (3).

(5) Consider the fractal structure Γ = {Γn: n ∈ N} whose levels are given by Γn = {[ k2n ,

k+12n ] × {0}: k ∈

{0, 1, . . . , 2n−1}}∪{{ 12m }× [ k

2n ,k+12n ]: k ∈ {0, 1, . . . , 2n−1}, m ∈ N} for all natural numbers n, on the

space X = ([0, 1]×{0})∪ {{ 12n }× [0, 1]: n ∈ N}. Take also F =

⋃k∈N

( 12k+1 ,

12k )×{0} as a subset of X.

It is clear that F = [0, 1] × {0}, so that Nn(F ) = 2n and Nn(F ) = ∞ for each n ∈ N, which impliesthat dim1

Γ (F ) = 1 and dim1Γ (F ) = ∞. �

In Theorem 3.5 we proved that the box-counting dimension and the fractal dimension I are equal forEuclidean subspaces if the fractal dimension I is calculated with respect to the natural fractal structureon Rd. Next, we study how the box-counting dimension of a Euclidean subspace can be related with thefractal dimension I of the subspace with respect to any fractal structure.

Let (X, ρ) be a (quasi-)metric space. Then we will denote the diameter of a set A ⊆ X, as usual, bydiam(A) = sup{ρ(x, y): x, y ∈ A}. In addition to that, we will use the expression Bd(x, ε) to denote theball centered in x ∈ X with respect to the metric (resp. quasi-metric) d, whose radius is ε > 0, namely,Bd(x, ε) = {y ∈ X: d(x, y) < ε}.

First, we define the diameter of each level of a fractal structure as well as the diameter of a subset in agiven level of a fractal structure as follows.

Definition 3.7. Let Γ be a fractal structure on a distance space (X, ρ) and let F be a subset of X. Then,the diameter of each level Γn of the fractal structure Γ is defined as

δ(Γn) = sup{diam(A): A ∈ Γn

}and the diameter of F in any level Γn of the fractal structure Γ , by

δ(F, Γn) = sup{diam(A): A ∈ Γn, A ∩ F �= ∅

}Notice that starbase fractal structures provide GF-spaces with some desirable topological properties.

Taking it as well as Definition 3.7 into account, we found a natural and easy condition about the sequenceof diameters {δ(Γn): n ∈ N} of the fractal structure Γ in order to obtain this kind of fractal structures.

Proposition 3.8. Let Γ be a fractal structure on a compatible metric (resp. quasi-metric) space (X, ρ) andlet us suppose that δ(Γn) → 0. Then Γ is starbase.

Proof. Indeed, let us prove that St(x,Γ ) is a neighborhood base of x, for all x ∈ X. First, it is clear thatx ∈ Uxn ⊂ St(x, Γn), for each x ∈ X and all n ∈ N, where Uxn is defined as in Section 2.1. On the otherhand, let x ∈ X be a fixed point and let ε > 0. Since δ(Γn) → 0, then there exists a natural number n0 ∈ Nsuch that δ(Γn) < ε for all n � n0. Hence, consider St(x, Γm) with m � n0. Then for all y ∈ St(x, Γm)there exists A ∈ Γm with x, y ∈ A. Thus, since δ(Γm) < ε, we have that ρ(x, y) < ε, so that y ∈ Bρ(x, ε).Therefore, there exists m ∈ N such that St(x, Γm) ⊂ Bρ(x, ε) which implies that Γ is starbase. �


Another natural condition that may be verified by a fractal structure is that the sequence of diameters ofeach level of that fractal structure decreases on a geometric way. Indeed, it is the main idea in the followingresult which provides a first approach to the box-counting dimension by means of the fractal dimension Imodel.

Theorem 3.9. Let Γ be a fractal structure on a metric space (X, ρ), with F being a subset of X, and let ussuppose that there exists a suitable constant c ∈ (0, 1) such that the next condition holds:

δ(F, Γn+1) � c · δ(F, Γn)

for all natural numbers n. Then

(1) dimB(F ) � γc · dim1Γ (F ).

(2) dimB(F ) � γc · dim1Γ (F ).

(3) Moreover, if there exist both the box-counting dimension and the fractal dimension I of F , then

dimB(F ) � γc · dim1Γ (F )

where γc is a constant which depends on c.

Proof. To calculate the box-counting dimension of F , let Nδ(F ) be as in equivalent definition (3) of Defi-nition 2.6.

(1) The geometric decrease of the sequence of diameters of each level of the fractal structure implies thatthere exists c ∈ (0, 1) such that

δ(F, Γn) � cn−1 · δ(F, Γ1)

for all n ∈ N. Let us denote δn = cn−1 · δ(F, Γ1) as the general term of a decreasing sequence whichconverges to 0. Thus,

dimB(F ) = limn→∞logNδn(F )− log δn

� limn→∞logNn(F )−n log c = − log 2

log c · limn→∞logNn(F )n log 2

where Remark 2.7 is used in the first equality. Notice that it suffices to take γc = − log 2/ log c.(2) Consider the decreasing sequence {δn: n ∈ N}, where δn = cn−1 · δ(F, Γ1).

Hence, since

limδ→0logNδ(F )− log δ � limn→∞

logNδn(F )− log δn

a similar argument to the previous one can be applied. �Since the sequence of diameters {δ(Γn): n ∈ N} always decreases in a geometric way when working with

the attractor of an IFS equipped with its natural fractal structure as a self-similar set (by its construction),then we can estimate the box-counting dimension of a self-similar set by means of its fractal dimension Ivalue, which becomes easier to calculate. In this way, we present the next result which is an immediateconsequence of Theorem 3.9.

Corollary 3.10. Let K be the attractor of the IFS (X, {fi: i ∈ I}), where X is a complete metric space andlet Γ be the natural fractal structure on K as a self-similar set (see Definition 2.4). Then the inequalities


contained in Theorem 3.9 are verified, with c being the maximum of the contraction factors {ci: i ∈ I}associated with the contraction mappings {fi: i ∈ I}.

The fractal dimension I model studied in this section generalizes the box-counting one in the context ofEuclidean spaces, and on the other hand, under certain conditions on the size of the elements on each levelof the fractal structure, an upper bound for the box-counting dimension has been found in terms of thefractal dimension I. Note that the fractal dimension I depends on the fractal structure that is used in thespace. We show this in the next remark.

Remark 3.11. Let X be a subspace of a Euclidean space. Then it is possible to obtain different values forits fractal dimension I depending on the fractal structure we select in order to calculate it.

Proof. First of all, let Γ 1 = {Γ1,n: n ∈ N} be the natural fractal structure on the real line induced on themiddle third Cantor set X. Thus, by Theorem 3.5, we have that dim1

Γ 1(X) = dimB(X) = log 2/ log 3, where

the second equality is due to [7, Example 3.3]. On the other hand, let Γ 2 be the natural fractal structureon X as a self-similar set (recall Definition 2.4). Then an easy calculation leads to dim1

Γ 2(X) = 1, since on

each level Γ2,n of the fractal structure Γ 2 there are 2n “subintervals” of length equal to 1/3n. �Remark 3.12. Note that a fractal structure is a kind of uniform structure. If we do not have any metric, theonly way to “measure” subsets is by checking if they belong to one level of the fractal structure or anotherone. So it is natural that the fractal dimension I depends on the fractal structure as well as the box-countingdimension depends on the metric.

4. Fractal dimension for fractal structures: a second model

The fractal dimension I introduced in Definition 3.3 regards all elements on each level of the fractalstructure as having the same “size”, namely, 1/2n. So that, the natural fractal structure we use on Euclideanspaces in order to determine the box-counting dimension of a given subset can be extended to other kind oftilings such as triangulations on the plane. Note that it results quite interesting because, for instance, anycompact surface has a triangulation.

If, in addition to the fractal structure, we have any kind of metric in the space, then we can use thatmetric to “measure” the size of the elements of the fractal structure at the same time that we use thefractal structure. For example, in any Euclidean space we can use both the natural fractal structure andthe Euclidean metric.

In this section, we introduce the so-called fractal dimension II, which uses a fractal structure as well as ametric or a distance function in its definition. In this way, the more general concept of a distance functionwill be used.

Definition 4.1. ([14]) A positive real-valued mapping ρ defined on a set X × X is said to be a distancefunction, or merely a distance on X, if it verifies that ρ(x, x) = 0 for all x ∈ X.

Diameters of subsets, coverings, . . . , etc., with respect to a distance function are defined in the same wayas that for a metric. Next, the definition of the fractal dimension II for any subset of a GF-space is provided.

Definition 4.2. Let (X,Γ ) be a GF-space and let F be a subset of X. Let also Nn(F ) be the number of ele-ments of Γn which meet F . The (lower/upper) fractal dimension II of F is defined as the (lower/upper) limit:

dim2Γ (F ) = lim

n→∞logNn(F )

− log δ(F, Γn)

where δ(F, Γn) is given as in Definition 3.7.


In general, the levels of a fractal structure do not have to be finite coverings, but if they are (that is, thefractal structure is finite) then the calculation of Nn(F ) becomes easier. In the next remark, we point outwhat kind of spaces admits a finite fractal structure. This idea is developed in the next example.

Remark 4.3. Recall that a metrizable space is second-countable if and only if it is separable (see [4, Theo-rem 5.7]). On the other hand, it is known that any second-countable T0-topological space has a compatiblefinite fractal structure (see [3, Theorem 4.3]).

Notice that the definition of fractal dimension II for any subset F ⊆ X uses the distance function in thequantity δ(F, Γn) to “measure” the size of the elements of each level of the fractal structure. Note that itmay be possible to consider δ(Γn) instead of δ(F, Γn) in Definition 4.2, but it implies certain disadvantages.For instance, let Γ be a finite fractal structure defined on a distance space (X, ρ) where ρ is a not-boundeddistance function. Thus, δ(Γn) = ∞ for all n ∈ N which does not allow to calculate the fractal dimension IIof any subset F of X.

Example 3. Let Γ be a finite fractal structure whose levels are given by Γn = {[ k12n ,

k1+12n ]× [ k2

2n ,k2+12n ]×· · ·×

[ kd

2n ,kd+12n ]: ki ∈ {−n2n, . . . , n2n − 1}, i ∈ {1, . . . , d}} ∪ {Rd \ (−n, n)d} for all natural numbers n, defined

on the Euclidean space Rd. Then for any bounded subset F of Rd, we have that δ(Γn) = ∞ for all n ∈ N(calculated with respect to the Euclidean distance), but nevertheless, there exists a natural number n0 suchthat δ(F, Γn) < ∞ for all n � n0.

Taking into account both fractal dimension Definitions 3.3 and 4.2, it becomes clear that fractal dimen-sions I and II are going to agree if we select any fractal structure Γ such that δ(F, Γn) = 1/2n for all n ∈ N.Next, we provide some conditions to obtain the equivalence of both fractal dimensions. To do that, we willdefine the semimetric associated with a fractal structure.

Definition 4.4. ([10, Definition 9.5]) A semimetric on a topological space X is a non-negative real-valuedmapping d defined on X ×X such that it verifies the following conditions:

(i) d(x, y) = 0 if and only if x = y,(ii) the mapping d is symmetric, and(iii) the family {Bd(x, ε): ε > 0} is a neighborhood base for all x ∈ X, namely, the topology induced by

the semimetric d agrees with the starting topology.

Next, we also recall the definition of the semimetric associated with a starbase fractal structure.

Definition 4.5. ([5, Theorem 6.5]) Let (X,Γ ) be a starbase GF-space. The semimetric associated with thefractal structure Γ is defined as the mapping ρ : X ×X → R+ given by:

ρ(x, y) =

⎧⎪⎨⎪⎩

0 if x = y12n if y ∈ St(x, Γn) \ St(x, Γn+1)1 if y /∈ St(x, Γ1)

(2)

It follows from Eq. (2) that Bρ(x, 1/2n) = St(x, Γn+1) for all n ∈ N and all x ∈ X, and since Γ isa starbase fractal structure, we conclude that the topology induced by the semimetric ρ agrees with thetopology induced by the fractal structure.

Theorem 4.6. Let (X,Γ ) be a starbase GF-space and let us consider in X the semimetric associated with thefractal structure Γ . Let also F be a subset of X and let us suppose that for each n ∈ N there exists x ∈ F

such that St(x, Γn) �= St(x, Γn+1). Then


(1) dim1Γ (F ) = dim2

Γ (F ).(2) dim1

Γ (F ) = dim2Γ (F ).

(3) If there exists any of the fractal dimensions I or II, then

dim1Γ (F ) = dim2

Γ (F )

Proof.

(1) By hypothesis, given n ∈ N there exists x ∈ F such that St(x, Γn) �= St(x, Γn+1). This implies thatthere exists A ∈ Γn with x ∈ A and such that A � St(x, Γn+1). Thus, diam(A) = 1/2n, and thenδ(F, Γn) = 1/2n. Therefore, the following expression holds:

dim2Γ (F ) = limn→∞

logNn(F )− log δ(F, Γn) = limn→∞

logNn(F )n log 2 = dim1

Γ (F )

(2) The case for lower limits may be dealt with in the same way. �Fractal dimension II also generalizes both fractal dimension I and the box-counting dimension on any Eu-

clidean space equipped with its natural fractal structure. Thus, we get for fractal dimension II an analogousresult to Theorem 3.5.

Theorem 4.7. Let Γ be the natural fractal structure on the Euclidean space Rd and let F be a subset of Rd.Then the (lower/upper) box-counting dimension is equal to the (lower/upper) fractal dimensions I and II :

dim1Γ (F ) = dim2

Γ (F ) = dimB(F )

Proof. First of all, by Theorem 3.5, we have that dim1Γ (F ) = dimB(F ). Note also that diam(A) =

√d/2n

for all A ∈ Γn, which implies that

limn→∞

logNn(F )− log δ(F, Γn) = lim

n→∞logNn(F )n log 2

that is, dim1Γ (F ) = dim2

Γ (F ). �Note that this theorem allows the calculation of the box-counting dimension of a subset of the plane

by counting, for example, triangles instead of squares, since we can define a fractal structure on the planeconsisting of triangulations of it by means of triangles of diameter of order 1/2n.

Remark 4.8. Similarly to Proposition 3.6 for fractal dimension I, we can also study some analytical propertiesfor the fractal dimension II definition. In this way, let Γ be a fractal structure on a distance space (X, ρ).First of all, it is clear that both dim2

Γ and dim2Γ are monotonic. Furthermore, since the fractal dimension II

generalizes fractal dimension I (in the sense of Theorem 4.6), then any counterexample for fractal dimension Ialso remains valid for fractal dimension II. In particular, we can use those given for statements (3), (4) and (5)in Proposition 3.6 for fractal dimension II. Nevertheless, unlike fractal dimension I, the fractal dimension IImodel does not verify the finite stability, as shown in the next example.

Example 4. Let Γ 1 be the natural fractal structure on C1 as a self-similar set, where C1 is the middlethird Cantor set on [0, 1]. Let also Γ 2 be a fractal structure on C2 = [2, 3] given by Γ 2 = {Γ2,n: n ∈ N},with Γ2,n = {[ k

22n ,k+122n ]: k ∈ {22n+1, 22n+1 + 1, . . . , 3 · 22n − 1}} for all natural numbers n. Consider also

Γ = {Γn: n ∈ N} as a fractal structure on C = C1 ∪ C2, where Γn = Γ1,n ∪ Γ2,n for all n ∈ N. A simplecalculation leads to dim2

Γ (C1) = log 2/ log 3 and dim2Γ (C2) = 1, while dim2

Γ (C) = log 4/ log 3 > 1.


Remark 4.9. Recall that in Remark 3.11 we showed that the fractal dimension I and the box-countingdimension of the middle third Cantor set C were not equal: indeed, we got that dimB(C) = log 2/ log 3,while dim1

Γ (C) = 1. Note that these dimensions were calculated with respect to different fractal structures:we used the natural fractal structure on the real line induced on C to calculate the box-counting dimension,and on the other hand, we selected the natural fractal structure on C as a self-similar set to obtain the fractaldimension I of such a space. However, the fractal dimension II agrees with the box-counting dimension of C.In this way, consider again the natural fractal structure on C as a self-similar set. Then it holds that

dim2Γ (C) = lim

n→∞log 2n

− log 3−n= log 2

log 3 = dimB(C)

since on each level Γn of the fractal structure there are 2n elements with diameters equal to 1/3n.

Nevertheless, though the value obtained for the fractal dimension I of C may seem counterintuitive atthe first time (see Remark 3.11), it is possible to explain it by means of its fractal dimension II. Indeed, thereason is that fractal dimension I only depends on the fractal structure we select to calculate it. We studythis fact in the next remark.

Remark 4.10. Fractal dimension I only depends on the fractal structure, while fractal dimension II alsodepends on the diameter of the elements of each level of the fractal structure. We show this difference byconstructing a family of spaces which from the fractal structure point of view are the same.

Proof. Indeed, let us consider slight modifications on the construction of the middle third Cantor set, whichwe are going to denote by Ci, such that their associated contraction factors are ci ∈ [1/3, 1/2) for the twosimilarities that give Ci. Thus, it is clear that δ(Ci, Γn) = cni for all natural numbers n. Therefore, considerthe natural fractal structure Γ i on each space Ci as a self-similar set. Then an easy calculation yields (orapply Theorem 4.19)

dimB(Ci) = dim2Γ (Ci) = log 2

− log ci−→ 1 = dim1

Γ (C)

when ci → 1/2. �It is also possible to find an upper bound for the box-counting dimension of any subset F ⊆ X in terms of

its fractal dimension II under the natural hypothesis that δ(F, Γn) → 0. This bound becomes more accuratethan that one obtained with the fractal dimension I in Theorem 3.9. Moreover, we found a connectionbetween the Hausdorff dimension and the fractal dimension II which is contained in the following theorem.

Theorem 4.11. Let Γ be a fractal structure on a metric space (X, ρ), with F being a subset of X and let ussuppose that δ(F, Γn) → 0. Then

(1) dimH(F ) � dimB(F ) � dim2Γ (F );

(2) if there exist both the box-counting dimension and the fractal dimension II of F , then

dimB(F ) � dim2Γ (F )

(3) suppose also that there exists an appropriate constant c > 0 such that

δ(F, Γn) � c · δ(F, Γn+1)


for all natural numbers n. Then


Proof. To calculate the box-counting dimension of F , let Nδ(F ) be as in equivalent definition (3) of Defini-tion 2.6. Note also that F ⊆

⋃{A ∈ Γn: A∩F �= ∅}. Furthermore, diam(A) � δ(F, Γn) =: δn for all A ∈ Γn

with A ∩ F �= ∅. Accordingly, F can be covered by Nn(F ) sets with diameters at most δn. Hence:

(1) It suffices to apply [7, Proposition 4.1] to get

dimH(F ) � dimB(F ) � limn→∞logNn(F )− log δn

= dim2Γ (F )

(2) This is a consequence of item (1).(3) Let c ∈ (0, 1] be such that δ(F, Γn) � c · δ(F, Γn+1) for all n ∈ N. Thus δ(F, Γn) � 0, which becomes

a contradiction, so that c > 1. Hence, there exists d ∈ (0, 1) such that δn+1 � d · δn for all n ∈ N.Therefore, by applying Remark 2.7, we get that

dimB(F ) = limn→∞logNδn(F )− log δn

� limn→∞logNn(F )− log δn

= dim2Γ (F ) �

As a consequence of Theorem 4.11, it becomes possible to get an approximation to the box-countingdimension of the attractor of an IFS in terms of its fractal dimension II. Indeed, the following corollaryholds.

Corollary 4.12. Let K be the attractor of the IFS (Rd, {fi: i ∈ I}) and let Γ be the natural fractal structureon K as provided in Definition 2.4. Let also F be a subset of K. Then

(1) dimB(F ) � dim2Γ (F );

(2) if there exist both the box-counting dimension and the fractal dimension II of F then


(3) suppose that there exists i ∈ I such that fi is a bilipschitz contractive mapping. Then


In particular, this inequality is verified for any strict self-similar set.

Proof.

(1) Since K is the attractor of the IFS (Rd, {fi : i ∈ I}), it is clear that δ(F, Γn) → 0 for all F ⊆ K, soTheorem 4.11(1) leads to the result.

(2) This is by the first item.(3) Let fi be a bilipschitz contractive mapping. Thus, there exist constants Li and ci with 0 < Li < ci < 1,

such that

Li · d(x, y) � d(fi(x), fi(y)

)� ci · d(x, y)


for all x, y ∈ K, where d denotes the Euclidean distance. Now, let A ∈ Γn with A ∩ F �= ∅ andsuch that diam(A) = δ(F, Γn). By definition of supremum, we have that for all ε > 0, there existx, y ∈ A such that d(x, y) > diam(A) − ε. Let B = fi(A) ∈ Γn+1. Thus, we have that d(fi(x), fi(y)) �Li ·d(x, y) > Li ·(diam(A)−ε), which implies that diam(B) = diam(fi(A)) � Li ·diam(A). Accordingly,δ(F, Γn+1) � diam(B) � Li · δ(F, Γn), so it suffices to apply Theorem 4.11(3). �

In the next theorem, we look for properties on the natural fractal structure of a Euclidean space in orderto generalize Theorem 4.7. Recall that given a scale δ > 0 it is verified that any subset F ⊆ Rd withdiam(F ) � δ meets at most 3d δ-cubes. Thus, a similar property to the latter in a more general contextwould allow to find a suitable connection between fractal dimension II and box-counting dimension.

Theorem 4.13. Let Γ be a fractal structure on a metric space (X, ρ), with F being a subset of X andlet us suppose that there exists a natural number k such that for all n ∈ N, every subset A of X withdiam(A) � δ(F, Γn) meets at most k elements of the level Γn of the fractal structure Γ . Suppose also thatδ(F, Γn) → 0. Then

(1) dimB(F ) � dim2Γ (F ) � dim2

Γ (F ) � dimB(F ). Moreover, if there exists dimB(F ), then

dimB(F ) = dim2Γ (F )

(2) if there exists c ∈ (0, 1) such that

δ(F, Γn+1) � c · δ(F, Γn)

then dimB(F ) = dim2Γ (F ) and dimB(F ) = dim2

Γ (F ).

Proof. To calculate the box-counting dimension of F , let Nδ(F ) be as in equivalent definition (3) of Defi-nition 2.6, and let us define δn := δ(F, Γn).

(1) First, note that by Theorem 4.11(1), we have that dimB(F ) � dim2Γ (F ). On the other hand, the main

hypothesis implies that

Nn(F ) � k ·Nδn(F )

for all n ∈ N, so it is clear that

dim2Γ (F ) = limn→∞

logNn(F )− log δn

� limn→∞logNδn(F )− log δn

= dimB(F ) (3)

(2) Note that, by Remark 2.7, dimB(F ) = limn→∞log Nδn (F )− log δn

, which implies that dimB(F ) � dim2Γ (F ) since

Nδn(F ) � Nn(F ) for all n ∈ N. In addition to that, an application of Eq. (3) leads to the oppositeinequality. The case for lower limits may be dealt with in the same way. �

The condition used in Theorem 4.13 in order to get the equality between the fractal dimension II andthe box-counting dimension becomes necessary as the next remark shows.

Remark 4.14. There exists an attractor K of an IFS on the Euclidean plane equipped with its natural fractalstructure Γ (as it was provided in Definition 2.4), for which dimB(K) �= dim2

Γ (K).


Proof. Let I = {1, . . . , 8} be a finite index set and let (R2, {fi: i ∈ I}) be an IFS whose attractor is theunit square on the Euclidean plane, K = [0, 1]2. Consider the contractive mappings fi : R2 → R2 given asfollows:

fi(x, y) ={

(x2 ,y4 ) + (0, i−1

4 ) if i = 1, 2, 3, 4(x2 ,

y4 ) + (1

2 ,i−54 ) if i = 5, 6, 7, 8

Let Γ be the natural fractal structure on K as the attractor of that IFS. First, note that the mappings fiare not similarities but affinities, and that all mappings have the same contraction factor ci = 1/2 forall i ∈ I. It is also immediate that dimB(K) = 2. On the other hand, note that there are 8n rectangleson each level Γn of the fractal structure Γ , whose dimensions are 1/2n × 1/22n. Thus, it is verified thatdiam(A) = δ(K,Γn) =

√1+22n

24n for all A ∈ Γn. Hence, we can calculate the fractal dimension II of K asfollows:

dim2Γ (K) = lim

n→∞logNn(K)

− log δ(K,Γn) = limn→∞

3n log 2−1

2 log(1+22n

24n

) = limn→∞

3n log 2n log 2 = 3

On the other hand, we can find the connection between δ(K,Γn) and each side of the 1/2n×1/22n-rectangles.

Thus, we get that√

1+22n24n1

22n=

√1 + 22n > 2n, as well as

√1+22n24n1

2n=

√1 + 1

22n � 12n for all n ∈ N. Therefore,

each subset A ⊆ K with diameter at most√

22n+124n is going to meet at most 3 · 2n+1 elements A ∈ Γn for

all n ∈ N. Accordingly, since this quantity depends on each natural number n, then the main hypothesis inTheorem 4.13 is not verified in this counterexample. �

Let f, g : N → R+ be two sequences of positive real numbers. We say that O(f) = O(g) iff they verifythat

limn→∞

f(n)g(n) ∈ (0,∞)

To show that fractal dimensions I and II agree on a GF-space, we will need that all elements of thecovering Γn of the fractal structure have a diameter of order 1/2n as the next theorem establishes.

Theorem 4.15. Let Γ be a fractal structure on a metric space (X, ρ), let F be a subset of X and let ussuppose that diam(A) = δ(F, Γn) for all A ∈ Γn with A ∩ F �= ∅. Suppose that O(δ(F, Γn)) = O( 1

2n ) for alln ∈ N. Then the (lower/upper) fractal dimension I is equal to the (lower/upper) fractal dimension II,

dim1Γ (F ) = dim2

Γ (F )

Proof. Let ε > 0 be a fixed but arbitrarily chosen real number. Consider α = dim1Γ (F ), so we affirm that

there exist n0 ∈ N and a strictly increasing mapping σ : N → N such that∣∣∣∣ logNσ(n)(F )

σ(n) log 2 − α

∣∣∣∣ � ε (4)

for all σ(n) � n0. Moreover it is clear by induction that σ(n) � n for all n ∈ N. Now, from Eq. (4), we havethe following chain of inequalities:

2σ(n)(α−ε) � Nσ(n)(F ) � 2σ(n)(α+ε) (5)


for all σ(n) � n0. Observe that for two given sequences of positive real numbers f, g such that O(f) = O(g),it holds that for any ε1 > 0 there exists n1 ∈ N such that∣∣∣∣f(n)

g(n) − k

∣∣∣∣ � ε1

for all n � n1. This implies that

log f(n) � log g(n) + log(k + ε1) (6)

for all n � n1. Thus, since O(δ(F, Γn)) = O(1/2n) by hypothesis, then it suffices to take f(n) = δ(F, Γn)and g(n) = 1/2n for all n ∈ N in Eq. (6). Then we obtain what follows:

1− log δ(F, Γn) � 1

n log 2 − log(k + ε1)(7)

for all n � n1. On the other hand, taking into account the fact that k − f(n)g(n) � ε for all n � n1, and using

an analogous argument to the previous one, it is clear that

1− log δ(F, Γn) � 1

n log 2 − log(k − ε1)

for all n � n1. Hence, using both Eqs. (5) and (7), we have that

logNσ(n)(F )− log δ(F, Γσ(n))

� σ(n)(α + ε) log 2σ(n) log 2 − log(k + ε1)

(8)

for all σ(n) � N := max{n0, n1}. A similar argument leads to


� σ(n)(α− ε) log 2σ(n) log 2 − log(k − ε1)

(9)

for all σ(n) � N . Hence, it is clear from both Eqs. (8) and (9), that for any σ(n) � N it is verified that

α− ε � limn→∞

supσ(n)�n


� α + ε

so the arbitrariness of ε > 0 implies that limn→∞log Nn(F )

− log δ(F,Γn) = α, namely, dim2Γ (F ) = dim1

Γ (F ), as desired.The case of lower limits may be dealt with in the same way. �

Recall that we have been able to calculate an upper bound for the box-counting dimension of the attractorof any IFS in terms of its fractal dimension II (see Corollary 4.12). However, it is possible to reach the equalitybetween these two quantities under certain conditions on the IFS. In particular, we will get the result ifthe elements on each level of the fractal structure do not overlap too much, and because of the shape thatthe elements of the natural fractal structure on the attractor have, this restriction is going to be associatedwith the contractions fi of the corresponding IFS. Indeed, this property is the so-called open set conditionwhich we recall next.

Definition 4.16. ([7, Section 9.2], [13, Definition 1.1]) Let K be the attractor of the IFS (X, {fi: i ∈ I}).The contractions fi satisfy the open set condition iff there exists a non-empty bounded open subset V of Xsuch that

⋃i∈I fi(V ) ⊂ V , with fi(V ) ∩ fj(V ) = ∅ for all i �= j. In addition to that, if V ∩K �= ∅, then the

contractions fi are said to verify the strong open set condition.


The next remark is quite interesting.

Remark 4.17. ([13, Theorem 2.2]) Both the open set condition and the strong open set condition areequivalent in any Euclidean space.

The following result is a technical lemma which is necessary in order to reach the equality between thefractal dimension II and the box-counting dimension of any self-similar set (in a Euclidean space) equippedwith its natural fractal structure.

Lemma 4.18. Let K be the attractor of the IFS (Rd, {fi: i ∈ I}). Suppose that the mappings fi are injectivecontractions verifying the open set condition for all i ∈ I. Then there exist ε > 0 and x ∈ K such that forall natural numbers and all ω, u ∈ In with ω �= u, the following expression holds:

fω(B(x, ε)

)∩ fu

(B(x, ε)

)= ∅

Proof. First, since K ⊆ Rd is a self-similar set, then we have that the open set condition is equivalent to thestrong open set condition (see Remark 4.17). Thus, there exists a non-empty bounded open subset O ⊆ Rd

such that⋃

i∈I fi(O) ⊂ O, with fi(O) ∩ fj(O) = ∅ for all i, j ∈ I such that i �= j. Moreover, O ∩K �= ∅, sowe can take x ∈ O ∩K ⊂ O. Since O is an open set, then there exists ε > 0 such that B := B(x, ε) ⊂ O.It is also verified that fi(B) ⊂ O for all i ∈ I. The result will be shown by induction on the length ofthe words on In. Indeed, let ωn = inin−1 . . . i1 ∈ In, and let us denote fωn

(F ) = fin ◦ fin−1 ◦ · · · ◦ fi1(F )for all F ⊆ Rd. Let n = 1. Then we can select ω1 = i1 and u1 = j1 ∈ I. Accordingly, we have thatfω1(B) ∩ fu1(B) ⊂ fω1(O) ∩ fu1(O) = ∅, since the contractions fi verify the open set condition for alli ∈ I. Suppose that the induction hypothesis is verified for n, that means that fωn

(B) ∩ fun(B) = ∅, with

ωn, un ∈ In such that ωn �= un. Thus, let us show it for n + 1. Indeed, let ωn+1, un+1 ∈ In+1. Then we candistinguish the two following cases:

(1) Suppose that in+1 = jn+1, so that ωn+1 = in+1 in . . . i1 and un+1 = in+1 jn . . . j1, with ωn+1, un+1 ∈In+1. Thus, using the injectivity of fin+1 and the induction hypothesis, we conclude that

fωn+1(B) ∩ fun+1(B) = fin+1

(fωn

(B))∩ fin+1

(fun

(B))

= ∅

(2) Suppose that ωn+1 �= un+1, so that ωn+1 = in+1 in . . . i1 and un+1 = jn+1 jn . . . j1, with in+1 �= jn+1.Then the following chain of inclusions is clear:

fwn+1(B) ∩ fun+1(B) = fin+1

(fωn

(B))∩ fjn+1

(fun

(B))⊂ fin+1(O) ∩ fjn+1(O) = ∅

Note that fωn(B) ⊂ O for all ωn ∈ In. Indeed, it is clear for words of length equal to 1, since it is

obvious that fi1(B) ⊂ O. Suppose now that fωn(B) ⊂ O for all ωn ∈ In. Then it is verified that

fωn+1(B) = fin+1(fωn(B)) ⊂ fin+1(O) ⊂ O. �

In [7, Theorem 9.3] it was provided an interesting result which allows the calculation of the box-countingdimension of a certain class of self-similar sets on the Euclidean space Rd as the solution of a simpleequation which only involves the contraction factors associated with each mapping fi of the correspondingIFS. Indeed, under the open set condition hypothesis, the box-counting dimension agrees with the Hausdorffdimension of such self-similar sets, and moreover, this value can be easily calculated from the mentionedexpression. In this way, it would be an interesting result to reach the equality between the box-countingdimension and the fractal dimension II of a self-similar set whose similarities fi also verify the open setcondition. Moreover, the calculation of such quantity would be immediate from the number of contractivemappings of the IFS and their common contraction factor.


Theorem 4.19. Let I = {1, . . . ,m} be a finite index set and let (Rd, {fi: i ∈ I}) be an IFS whose associatedattractor is K. Suppose that the similarities fi verify the open set condition and have equal similarity factors,c ∈ (0, 1). Let Γ be the natural fractal structure on K as a self-similar set as provided in Definition 2.4.Then

dimB(K) = dim2Γ (K) = − logm

log c (10)

Proof. To calculate the box-counting dimension of K, let Nδ(K) be as in equivalent definition (4) of Defi-nition 2.6, and let us define δn := δ(K,Γn). Observe that δn = cn ·diam(K) for all natural numbers n, sinceK is a strict self-similar set. By Lemma 4.18, there are so many disjoint balls with radius εn = cn · ε withε > 0 and centered in K as the number of elements of In. Now, since Nεn(K) is the largest number of suchballs, it is obvious that the number of elements of In is at most Nεn(K), namely,

Nn(K) � Nεn(K)

for all n ∈ N. On the other hand, there exists k > 0 such that δ(K,Γn) = k · εn. Indeed, that is true fork = diam(K)/ε. Therefore, it results clearly that

dim2Γ (K) � limn→∞

logNεn(K)− log(k εn) = limn→∞

logNεn(K)− log εn

= dimB(K)

Hence, we get the next chain of inequalities:

dimB(K) � dim2Γ (K) � dim2

Γ (K) � dimB(K) (11)

where the first inequality is by Corollary 4.12(1), and the last one is due to Eq. (11). Now, the existence ofthe box-counting dimension of K implies the existence of the fractal dimension II of K, and the expectedequality: dimB(K) = dim2

Γ (K). Furthermore, apply [7, Theorem 9.3] to get the last equality in Eq. (10).Indeed, note that the expression

∑i∈I c

si = m · cs = 1 leads to s = − logm/ log c. �

The hypothesis based on the equality of the contraction factors in Theorem 4.19 is necessary. By thecounterexample provided in Remark 4.14, the contractions have to be similarities, while by the next coun-terexample, all contraction factors must be the same.

Remark 4.20. There exists a strict self-similar set K whose similarities fi verify the open set conditionand have different contraction factors ci, such that dimB(K) < dim2

Γ (K), where Γ is the natural fractalstructure on K as provided in Definition 2.4.

Proof. Let (R, {fi: i = 1, 2}) be the IFS whose associated attractor is K, where the similarities f1, f2 :R → R are given by f1 : x → x

2 , and f2 : x → x+34 . It is clear that their associated contraction factors are

c1 = 1/2 and c2 = 1/4 respectively, and it is also obvious that K is a strict self-similar set. We can alsojustify that the similarities fi satisfy the open set condition: just take V = (0, 1) as an appropriate openset. Thus, by [7, Theorem 9.3], we can calculate the box-counting dimension of K as the solution of theequation 1

2s + 14s = 1. Hence, we have that dimB(K) = log(1+

√5

2 )/ log 2. On the other hand, it is clear thatthere are 2n subintervals of [0, 1] on each level Γn of the fractal structure, where the largest of them has adiameter equal to 1/2n for all n ∈ N. This implies that dim2

Γ (K) = 1 > dimB(K). �References

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