A survey on fractal dimension for fractal structures

A quasi-pseudometric on a set X is a non-negative real-valued function ρ defined on X × X such that for all x,y,z ∈ X, the two following conditions are satisfied:

ρ(x,x) = 0.

Moreover, if ρ satisfies also the next one:

3. ρ(x,y) = ρ(y,x) = 0 if and only if x = y,

then ρ is called a quasi-metric. In particular, a non-Archimedean quasi-pseudometric is a quasi-pseudometric which also satisfies that ρ(x,y) ≤ max{ρ(x,z),ρ(z,y)} for all x,y,z ∈ X. Each quasi-pseudometric ρ on X generates a quasi-uniformity 𝒰_ρ on X which has as a base the family of sets of the form {(x,y) ∈ X × X : ρ(x,y) < 1/2ⁿ} : n ∈ ℕ. The topology τ(𝒰_ρ) induced by the quasi-uniformity 𝒰_ρ will be denoted τ(ρ), merely. Therefore, a topological space (X,τ) is said to be (non-Archimedeanly) quasi-pseudometrizable if there exists a (non-Archimedean) quasi-pseudometric ρ on X such that τ = τ(ρ). The theory of quasi-uniform spaces is covered in detail in [29].

Let (X,ρ) be a (quasi-)metric space. Then we shall denote the diameter of a subset A ⊆ X by diam(A) = sup{ρ(x,y) : x,y ∈ A}, as usual. In addition, B_ρ(x,ε) will denote the ball centred in x ∈ X with radius ε > 0 with respect to the metric (resp. quasi-metric) ρ, namely, B_ρ(x,ε) = {y ∈ X : ρ(x,y) < ε}.

The concept of fractal structure was first introduced in [23] to characterize non-Archimedeanly quasi-metrizable spaces though it can be also used to deal with fractals. For instance, in [25] it was used to study attractors of iterated function systems.

Fractal structures constitute a powerful tool to introduce new models for a fractal dimension definition since it is a natural context where the concept of fractal dimension can be developed. Further, they will allow to calculate the fractal dimension in new spaces and situations.

A family Γ of subsets of a given space X is called a covering if X = ∪[{A : A ∈ Γ}. Let Γ be a covering of X. Then we shall denote St(x,Γ) = ∪[{A ∈ Γ : x ∈ A} and U_xΓ = X \ ∪[{A ∈ Γ : x ∉ A}. Furthermore, if Γ = {Γ_n}_n∈ℕ is a countable family of coverings of X, then we shall denote U_xn = U_xΓn, UxΓ={Uxn}n∈ℕ$\begin{array}{} \displaystyle {\mathscr U}_x^{\rm{\Gamma }} = {\left\{ {{U_{xn}}} \right\}_{n \in N}} \end{array}$, and St(x,Γ) = {St(x,Γ_n)}_n∈ℕ.

Next, we provide a first approach to define a fractal structure on a set X. Indeed, let Γ₁ and Γ₂ be two coverings of X. Thus, we shall denote Γ₁ ≺ Γ₂ that Γ₁ is a refinement of Γ₂, namely, for all A ∈ Γ₁ there exists B ∈ Γ₂ such that A ⊆ B. In addition to that, the notation Γ₁ ≺≺ Γ₂ means that Γ₁ is a strong refinement of Γ₂, namely, Γ₁ ≺ Γ₂, and in addition, for all B ∈ Γ₂ we can write B = ∪[{A ∈ Γ₁ : A ⊆ B}. Thus, a fractal structure on a set X can be defined as a countable family of coverings of X, Γ = {Γ_n}_n∈ℕ, such that Γ_n+1 ≺≺ Γ_n for all n ∈ ℕ. Next, we provide the definition of a fractal structure on a topological space.

If Γ is a pre-fractal structure, then any of its levels is a closure-preserving closed covering (c.f. [30, Proposition 2.4]). Let Γ be a fractal structure on X such that St(x,Γ) is a neighborhood base for all x ∈ X. Then Γ is called a starbase fractal structure. Interestingly, starbase fractal structures are connected with metrizability (c.f. [30, 31]). A fractal structure Γ is finite provided that all its levels are finite coverings. A fractal structure Γ is said to be locally finite if for each level Γ_n of the fractal structure Γ, it holds that any point x ∈ X belongs to a finite number of elements A ∈ Γ_n. Additionally, a fractal structure Γ is said to be Γ−Cantor-complete if for each decreasing sequence {A_n}_n∈ℕ (namely, A_n+1 ⊆ A_n for all n ∈ ℕ) of subsets of X with A_n ∈ Γ_n, we have ∩_n∈ℕA_n ≠ ∅. In general, if Γ_n has the property 𝒫 for all n ∈ ℕ and Γ = {Γ_n}_n∈ℕ is a fractal structure on X, then we will say that Γ is a fractal structure with the property 𝒫 and also that (X,Γ) is a GF-space under such a property 𝒫.

Self-similar sets are a kind of fractals that can be always endowed with a fractal structure in a natural way. Along this section, we recall a standard approach to construct attractors of iterated function systems. In addition, we shall also describe their natural fractal structure as self-similar sets. The results and properties described below are essential to understand some results appeared in upcoming sections.

Firstly, let f : X → X be a mapping defined on a metric space (X,ρ). Recall that f is said to be Lipschitz if it satisfies the following condition: ρ(f(x), f(y)) ≤ c · ρ(x,y) for all x,y ∈ X, where c > 0 is the Lipschitz constant associated with f . In particular, if c < 1, then f is said to be a contraction and c is called the contraction ratio associated with f. In addition, if it is reached the equality in the previous expression, namely, ρ(f(x), f(y)) = c · ρ(x,y) for all x,y ∈ X, then we have a similarity and its Lipschitz constant is called its similarity ratio.

Next, let us recall the standard construction of self-similar sets provided by Hutchinson (c.f. [32]). Assume that (X,ρ) is a complete metric space and let {fi}i=1m$\begin{array}{} \displaystyle \{ {f_i}\} _{i = 1}^m \end{array}$ be a finite set of contractions defined on X. The scheme (X,{fi}i=1m)$\begin{array}{} \displaystyle (X,\{ {f_i}\} _{i = 1}^m) \end{array}$ is called an iterated function system (IFS). Thus, let us consider that IFS in order to define the map 𝒲 : ℍ → ℍ by W(A)=∪i=1mfi(A)$\begin{array}{} \displaystyle {\mathscr W}(A) = \cup _{i = 1}^m{f_i}(A) \end{array}$ for all A ∈ ℍ, where ℍ denotes the hyperspace of X, namely, the set consisting of all non-empty compact subsets of X.

It can be proved that 𝒲 is a contraction with respect to the Hausdorff metric d_ℍ on H (c.f. [2, Section 9.1]), with associated contraction ratio c < 1. In addition, since (X,ρ) is a complete metric space, then (ℍ,d_H) also is (due to Zenor-Morita’s Theorem). Hence, since we have a contraction on a complete metric space, then the Banach fixed-point Theorem guarantees that there exists a unique non-empty compact subset 𝒦 ⊆ X such that 𝒦 = 𝒲 (𝒦 ). 𝒦 is said to be the attractor of the corresponding IFS. A strict self-similar set is an attractor of an IFS such that all its contractions are similarities. Along the sequel, by an IFS-attractor, we shall understand the attractor of an IFS whose contractions are similarities.

The next example describes analytically the so-called Sierpiński gasket, first defined in [33].

As it was mentioned previously, attractors of IFSs can be always endowed with a natural fractal structure, first sketched in [22] and formally defined later in [25]. It is worth pointing out that the latter provides the definition of a fractal structure as a mathematical concept, whereas the former deals with the (natural) fractal structure of a self-similar set. Next, we recall the description of such a fractal structure as provided in [25, Definition 4.4].

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

Fractal dimension is one of the main tools applied to study fractals since it is a single quantity which throws useful information regarding their complexity when being examined with enough level of detail. It is worth noting that fractal dimension is usually understood as the classical box-counting dimension (box dimension along the sequel, for short) which is also known as information dimension, Kolmogorov entropy, capacity dimension, entropy dimension, metric dimension, logarithmic density, . . . , etc. (c.f. [2, Section 3.1]).

Though the Hausdorff dimension can be also considered a fractal dimension, in practical applications it is always used the box dimension since it can be easily calculated for a finite range of scales, which is the case of empirical applications.Popularity of the box dimension is mainly due to the possibility of its effective calculation and empirical estimation in Euclidean contexts. It is worth pointing out that, concerning applications of fractal dimension, the box dimension can be estimated as the slope of the regression line of a log−log graph plotted for a suitable discrete range of scales.

The basic theory on box dimension is covered in detail in [2]. Next, we recall the definition of the standard box dimension.

Notice also that the limit in Eq. (1) can be discretized by taking, for instance, δ = 1/2ⁿ. This is formalized in the next remark.

The main purpose of this section is to include a sketch about the construction of both Hausdorff measure and dimension whose definitions and properties can be found out in [2, Chapter 2].

The first to define a measure by means of coverings of sets was Carathéodory in [14]. Later (1919), Hausdorff used this method to define the measures that now bear his name, and showed that the middle third Cantor set has positive and finite measure of dimension equal to log2/log3 [15]. A detailed study regarding the analytical properties of both Hausdorff measure and dimension was mainly developed by Besicovitch and his pupils.

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 a Euclidean (resp. metrizable) space and its definition is based on a measure which makes it quite appropriate from a mathematical viewpoint. Nevertheless, it presents some disadvantages, especially from the viewpoint 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 is “better” for a wide range of applications. Next, let us recall the analytical construction of the Hausdorff dimension. Let (X,ρ) be a metric space and δ be a positive real number. For any subset F of X, a δ−cover of F is a countable family of subsets {U_i}_i∈I such that F ⊆ ∪_i∈IU_i with diam(U_i) ≤ δ for all i ∈ I. Let 𝒞_δ(ℱ) denote 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 the fractal dimension to be calculated. In this way, the following quantity can be defined:

Hδs(F)=inf{∑i∈Idiam(Ui)s:{Ui}i∈I∈Cδ(F)}.$$\begin{array}{} \displaystyle {\mathscr H}_\delta ^s(F) = \inf \left\{ {\mathop \sum \limits_{i \in I} {\rm{diam}}{{({U_i})}^s}:{{\left\{ {{U_i}} \right\}}_{i \in I}} \in {{\mathscr C}_\delta }(F)} \right\}. \end{array}$$(2)

Note that when δ decreases, then the class 𝒞_δ(F) of all δ−covers of F is reduced and hence, the measure of F increases. Accordingly, the next limit always exists:

HHs(F)=limδ→0Hδs(F)$$\begin{array}{} \displaystyle {\mathscr H}_{\rm{H}}^s\left( F \right) = \mathop {\lim }\limits_{\delta \to 0} {\mathscr H}_\delta ^s(F) \end{array}$$(3)

which is called the s−dimensional Hausdorff measure of F.

It is worth mentioning that Hausdorff measure generalizes the classical Lebesgue measure for Euclidean subspaces. Indeed, if F is a Borel subset of ℝ^d, then we have HHd(F)=cd⋅vold(F)$\begin{array}{} \displaystyle {\mathscr H}_{\rm{H}}^d(F) = {c_d} \cdot {\rm{vo}}{{\rm{l}}^d}(F) \end{array}$, where the constant cd=πd2/(2d⋅(d2)!)$\begin{array}{} \displaystyle {c_d} = {\pi ^{\frac{d}{2}}}/({2^d} \cdot (\frac{d}{2})!) \end{array}$ is the volume of a d−dimensional ball of diameter equal to 1. In particular, HH0(F)=cd(F)$\begin{array}{} \displaystyle {\mathscr H}_{\rm{H}}^0(F) = {\rm{cd}}(F) \end{array}$, namely, the number of points in F; HH1(F)$\begin{array}{} \displaystyle {\mathscr H}_{\rm{H}}^1(F) \end{array}$ is the length of a smooth curve F; HH2(F)=π4⋅area(F)$\begin{array}{} \displaystyle {\mathscr H}_{\rm{H}}^2(F) = \frac{\pi }{4} \cdot {\rm{area}}(F) \end{array}$, if F is a smooth surface; HH3(F)=43π⋅vol(F)$\begin{array}{} \displaystyle {\mathscr H}_{\rm{H}}^3(F) = \frac{4}{3}\pi \cdot {\rm{vol}}(F) \end{array}$; and in general, HHm(F)=cm⋅volm(F)$\begin{array}{} \displaystyle {\mathscr H}_{\rm{H}}^m(F) = {c_m} \cdot {\rm{vo}}{{\rm{l}}^m}(F) \end{array}$, if F is a smooth m−dimensional submanifold of ℝ^d, namely, an m−dimensional surface in the classical sense.

From Eq. (2), it becomes clear that for any set F and any scale δ ∈ (0,1), the quantity Hδs(F)$\begin{array}{} \displaystyle {\mathscr H}_\delta ^s(F) \end{array}$ is non-increasing with s. Accordingly, HHs(F)$\begin{array}{} \displaystyle {\mathscr H}_{\rm{H}}^s(F) \end{array}$ is also non-increasing with s (c.f. Eq. (3)). Let t > s and let {U_i}_i∈I be a δ -cover of F. Then Σ_i∈I diam(U_i)^t ≤ δ^t−s · Σ_i∈I diam(U_i)^s, and taking infima in the last expression on the class 𝒞_δ(F), it holds that Hδt(F)≤δt−s⋅Hδs(F)$\begin{array}{} \displaystyle {\mathscr H}_\delta ^t(F) \le {\delta ^{t - s}} \cdot {\mathscr H}_\delta ^s(F) \end{array}$. Letting δ → 0, we have HHt(F)≤δt−s⋅HHs(F)$\begin{array}{} \displaystyle {\mathscr H}_{\rm{H}}^t(F) \le {\delta ^{t - s}} \cdot {\mathscr H}_{\rm{H}}^s(F) \end{array}$. If HHs(F)<∞$\begin{array}{} \displaystyle {\mathscr H}_{\rm{H}}^s(F) \lt \infty \end{array}$ when t > s, then HHt(F)=0$\begin{array}{} \displaystyle {\mathscr H}_{\rm{H}}^t(F) = 0 \end{array}$. Thus, the point s where HHs(F)$\begin{array}{} \displaystyle {\mathscr H}_{\rm{H}}^s(F) \end{array}$ “jumps” from ∞ to 0 is called the Hausdorff dimension of F (also called the Hausdorff-Besicovitch dimension). In fact, the Hausdorff dimension of F can be described in the following terms:

dimH(F)=inf{s≥0:HHs(F)=0}=sup{s≥0:HHs(F)=∞},$$\begin{array}{} \displaystyle {\rm{di}}{{\rm{m}}_{\rm{H}}}(F) = \inf \left\{ {s \ge 0:{\mathscr H}_{\rm{H}}^s(F) = 0} \right\} = \sup \left\{ {s \ge 0:{\mathscr H}_{\rm{H}}^s(F) = \infty } \right\}, \end{array}$$

or equivalently,

HHs(F)={∞ifs<dimH(F)0ifs>dimH(F).$$\begin{array}{} \displaystyle {\mathscr H}_{\rm{H}}^s(F) = \left\{ \begin{array}{*{20}{l}} \infty &{{\rm{if }}s \lt {{\dim }_H}(F)}\\ 0&{{\rm{if }}s \gt {{\dim }_H}(F).} \end{array}\right. \end{array}$$(4)

In particular, if s = dim_H(F), then HHs(F)$\begin{array}{} \displaystyle {\mathscr H}_{\rm{H}}^s(F) \end{array}$ can be equal to 0,∞, and even it can happen that HHs(F)∈(0,∞)$\begin{array}{} \displaystyle {\mathscr H}_{\rm{H}}^s(F) \in \left( {0,\infty } \right) \end{array}$.

Next theorem collects several properties that are satisfied by Hausdorff dimension as a dimension function. Such a result will be referred to afterwards to compare these properties for the different models of fractal dimension for fractal structures we shall introduce along upcoming sections.

It is also worth mentioning that the finite stability property consists of dim(E∪F) = max{dim(E),dim(F)}. Clearly, if countable stability implies finite stability. In addition, we say that a dimension function satisfies the closure dimension property if there exists a subset F of X such that dim(F)≠dim(F¯)$\begin{array}{} \displaystyle \dim (F) \ne \dim (\bar F) \end{array}$, where F¯$\begin{array}{} \displaystyle \bar F \end{array}$ denotes the closure of F, namely, the smallest closed subset of ℝ^d containing F.

The countable stability property satisfied by the Hausdorff dimension (c.f. Eq. (5)) is the key for the next result.

Let F be a subset of ℝ^d and 𝒩_δ(F) be the number of 1/2ⁿ−cubes that intersect F (c.f. Definition 3 (2)). Consider also Remark 3. First, we shall define a fractal structure which every Euclidean set can be always endowed with. Such a fractal structure is locally finite and starbase.

Next, we highlight that natural fractal structures can be induced on Euclidean subsets.

Let us introduce our first generalized box dimension type model of fractal dimension.

Recall that Hausdorff dimension constitutes the main theoretical model of fractal dimension that we should be mirrored in when providing new definitions of fractal dimension. In this way, our next goal is to explore some theoretical properties from those listed in both Theorem 3 and Remark 4 for fractal dimension I.

Next step is to explore how box dimension and fractal dimension I are theoretically connected for any generalized-fractal space. To deal with, first we shall define the diameter of any level of a fractal structure and the diameter of any subset in a level of a fractal structure. Recall that a distance is a non-negative map ρ : X × X → ℝ such that ρ(x,x) = 0 for all x ∈ X.

A feasible condition to be satisfied by a fractal structure consists of a geometric decrease regarding the sequence of diameters {δ(F,Γ_n)}_n∈ℕ. That assumption leads to an upper bound to the box dimension of F in terms of its fractal dimension I (up to a constant).

It is worth noting that Theorem 6 can be extended to deal with IFS-attractors. In fact, this is due to the fact that the sequence of diameters {δ(Γ_n)}_n∈ℕ for this kind of sets decrease geometrically. With this aim, we shall select their natural fractal structures as self-similar sets.

Next, we highlight how fractal dimension I depends on a selected fractal structure.

In this subsection, we shall explore several conditions on the elements of each level of a fractal structure to reach the equality between fractal dimensions I and II. To tackle with, first we shall define the concepts of a semimetric on a topological space (c.f. [34, Definition 9.5]) and a semimetric associated with a starbase fractal structure.

It is worth pointing out that Eq. (6) implies B_ρ(x,1/2ⁿ) = St(x,Γ_n+1) for all n ∈ ℕ and each x ∈ X. Moreover, since Γ is starbase, we can affirm that the topology induced by the semimetric ρ matches with the topology induced by the fractal structure. Next, we contribute a condition concerning the levels of a (starbase) fractal structure to reach the equality between fractal dimensions I and II.

In this section, we theoretically explore the behavior of fractal dimension II as a dimension function similarly to Proposition 5 for fractal dimension I.

In Remark 7, we highlighted the dependence of fractal dimension I on the selected fractal structure. Next, we point out the additional dependence of fractal dimension II on a metric. In particular, we shall justify why the fractal dimension II of the standard middle third Cantor set (endowed with its natural fractal structure as a self-similar set) equals its box dimension.

In this subsection, we provide an upper bound for both the Hausdorff and the box dimensions of any subset F in terms of its fractal dimension II.

Theorem 11 also allows to achieve an upper bound for the box dimension of IFS-attractors throughout their fractal dimension II values. That result is stated next.

Our next goal is to explore which properties underlying the natural fractal structure on any Euclidean space (c.f. Definition 4) could allow to generalize Theorem 9. With this aim, observe that given a scale δ > 0, it holds that any Euclidean subspace F of ℝ^d such that diam(F) ≤ δ intersects at most to 3^dδ−cubes. In this way, a similar property in the broader context of fractal structures would lead to an additional connection between fractal dimension II and box dimension.

In addition, let Γ be the natural fractal structure on 𝒦 as a self-similar set. First, notice that the self-maps f_i are not similarities but affinities and all of them have the same contraction ratio, namely, c_i = 1/2. It is also clear that dim_B(𝒦) = 2.

On the other hand, there are 8ⁿ rectangles in level n of Γ whose dimensions are 12n×122n$\begin{array}{} \displaystyle \frac{1}{{{2^n}}} \times \frac{1}{{{2^{2n}}}} \end{array}$. Hence,

diam(A)=δ(K,Γn)=1+22n24n$$\begin{array}{} \displaystyle {\rm{diam}}(A) = \delta ({\mathscr K},{\Gamma _n}) = \sqrt {\frac{{1 + {2^{2n}}}}{{{2^{4n}}}}} \end{array}$$

for all A ∈ Γ_n. Next, we calculate the fractal dimension II of 𝒦.

dimΓ2(K)=limn→∞logNn(K)−logδ(K,Γn)=limn→∞3nlog2−12log1+22n24n=limn→∞3nlog2nlog2=3.$$\begin{array}{} \displaystyle {\rm{dim}}_{\bf{\Gamma }}^2({\mathscr K}) = \mathop {\lim }\limits_{n \to \infty } \frac{{\log {N_n}({\mathscr K})}}{{ - \log \delta ({\mathscr K},{{\bf{\Gamma }}_n})}} = \mathop {\lim }\limits_{n \to \infty } \frac{{3n\log 2}}{{ - \frac{1}{2}\log \frac{{1 + {2^{2n}}}}{{{2^{4n}}}}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{3n\log 2}}{{n\log 2}} = 3. \end{array}$$

We also provide lower bounds for the ratios between ∈(𝒦, Γ_n) and the sides of each 12n×122n$\begin{array}{} \displaystyle \frac{1}{{{2^n}}} \times \frac{1}{{{2^{2n}}}} \end{array}$−rectangle:

1+22n24n122n=1+22n>2n,1+22n24n12n=1+122n≥12n.$$\begin{array}{} \displaystyle \frac{{\sqrt {\frac{{1 + {2^{2n}}}}{{{2^{4n}}}}} }}{{\frac{1}{{{2^{2n}}}}}} = \sqrt {1 + {2^{2n}}} \gt {2^n},\qquad \frac{{\sqrt {\frac{{1 + {2^{2n}}}}{{{2^{4n}}}}} }}{{\frac{1}{{{2^n}}}}} = \sqrt {1 + \frac{1}{{{2^{2n}}}}} \ge \frac{1}{{{2^n}}}. \end{array}$$

Accordingly, each subset A ⊂ 𝒦 whose diameter is at most 1+22n24n$\begin{array}{} \displaystyle \sqrt {\frac{{1 + {2^{2n}}}}{{{2^{4n}}}}} \end{array}$ intersects at most to 3 · 2ⁿ+1 elements in level n of Γ. Since that quantity depends on each n ∈ ℕ, then the κ−condition is not satisfied.

Let us recall when two sequences of positive real numbers are said to be of the same order.

Fractal dimension II provides an upper bound concerning the box dimension of any Euclidean IFS-attractor (c.f. Corollary 12). Going beyond, it is even possible to reach that equality under certain conditions on the corresponding IFS. More specifically, this kind of result stands provided that the elements in each level of the fractal structure do not overlap “too much”. Hence, due to the shape of the elements in the natural fractal structure which any IFS-attractor can be endowed with, this restriction will rely on the similarities of the IFS. In this context, the so-called open set condition (OSC) plays a key role.

Schief proved that both the OSC and the SOSC are equivalent for Euclidean IFSs (c.f. [36, Theorem 2.2]). In 1946, P.A.P. Moran contributed a strong result allowing the calculation of both the box and the Hausdorff dimensions for a certain class of Euclidean IFS-attractors throughout the (unique) solution of an equation only involving the similarity ratios associated with each similarity of the IFS (c.f. [37, Theorem III] and [2, Theorem 9.3]). That classical result is stated next.

Accordingly, under the OSC, the box dimension of any IFS-attractor equals its Hausdorff dimension, and that common value can be easily calculated from Eq. (7). The next result we provide guarantees the equality between the box dimension and the fractal dimension II of IFS-attractors lying under the OSC. Indeed, the calculation of these dimensions follows immediately from the number of similarities in the IFS and their common similarity ratio, as in [37, Theorem II].

We would like to point out that the hypothesis consisting of equal similarity ratios in Theorem 17 is necessary. Recall that Counterexample 14 implies that all the contractions involved in Theorem 17 must be similarities. Further, the following counterexample justifies why all the similarity ratios must be equal.

It is clear that their associated contraction ratios are c₁ = 1/2 and c₂ = 1/4, respectively. Moreover, it holds that 𝒦 is a strict self-similar set. It is also possible to justify that ℱ is under the OSC. In fact, let 𝒱 = (0,1) ⊂ ℝ. Thus, Moran’s Theorem allows to affirm that the box dimension of 𝒦 equals the solution of the equation 12s+14s=1$\begin{array}{} \displaystyle \frac{1}{{{2^s}}} + \frac{1}{{{4^s}}} = 1 \end{array}$. Hence, dimB(K)=log(1+52)/log2$\begin{array}{} \displaystyle {\rm{di}}{{\rm{m}}_{\rm{B}}}({\mathscr K}) = \log (\frac{{1 + \sqrt 5 }}{2})/\log 2 \end{array}$. Finally, observe that there are 2ⁿ “subintervals” of [0,1] in level n of the fractal structure Γ, where the diameter of the largest of them equals 1/2ⁿ. Accordingly, dimΓ2(K)=1>dimB(K)$\begin{array}{} \displaystyle {\rm{dim}}_{\bf{\Gamma }}^2\left( {\mathscr K} \right) = 1 \gt {\rm{di}}{{\rm{m}}_{\rm{B}}}\left( {\mathscr K} \right) \end{array}$.

Let Γ be a fractal structure on a metric space (X,ρ), F be a subset of X, and s ≥ 0. We shall “measure” the size of each element in any collection 𝒜_n(F) by its diameter. Notice that the sum of the s−powers of the diameters of all the elements in each family 𝒜_n(F) allows to “measure” the level of irregularity and complexity of F provided that it is explored by a whole range of scales. In this way, we shall define the following expression for each natural number n:

Hns(F)=∑{diam(A)s:A∈An(F)},$$\begin{array}{} \displaystyle {\mathscr H}_n^s\left( F \right) = \sum \{ {\rm{diam}}{(A)^s}:A \in {{\mathscr A}_n}\left( F \right)\} , \end{array}$$(8)

as well as its asymptotic behavior,

Hs(F)=limn→∞Hns(F).$$\begin{array}{} \displaystyle {{\mathscr H}^s}\left( F \right) = \mathop {{\rm{lim}}}\limits_{n \to \infty } {\mathscr H}_n^s\left( F \right). \end{array}$$(9)

Recall that Hausdorff dimension is fully determined throughout the (unique) value of s satisfying the equality sup{s:HHs(F)=∞}=inf{s:HHs(F)=0}$\begin{array}{} \displaystyle {\rm{sup}}\{ s:{\mathscr H}_{\rm{H}}^s\left( F \right) = \infty \} = {\rm{inf}}\{ s:{\mathscr H}_{\rm{H}}^s\left( F \right) = 0\} \end{array}$, where HHs(F)$\begin{array}{} \displaystyle {\mathscr H}_{\rm{H}}^s\left( F \right) \end{array}$ denotes the s−dimensional Hausdorff measure. Following the above, our next goal is to verify a property of this kind for the set function ℋ^s(F) just defined in Eq. (9) with respect to a fractal structure. To deal with, let t be another non-negative real number and consider Eq. (8). Then

∑A∈An(F)diam(A)t≤δ(F,Γn)t−s⋅∑A∈An(F)diam(A)s.$$\begin{array}{} \displaystyle \mathop \sum \limits_{A \in {{\mathscr A}_n}\left( F \right)} {\rm{diam}}{(A)^t} \le \,\delta {\left( {F,{{\rm{\Gamma }}_n}} \right)^{t - s}}\, \cdot \mathop \sum \limits_{A \in {{\mathscr A}_n}\left( F \right)} {\rm{diam }}{(A)^s}. \end{array}$$(10)

Observe that Eq. (10) is equivalent to

Hnt(F)≤Hns(F)⋅δ(F,Γn)t−s.$$\begin{array}{} \displaystyle {\mathscr H}_n^t(F) \le \,{\mathscr H}_n^s(F) \cdot \delta {\left( {F,{{\rm{\Gamma }}_n}} \right)^{t - s}}. \end{array}$$

Letting n → ∞, we have

Ht(F)≤Hs(F)⋅limn→∞δ(F,Γn)t−s.$$\begin{array}{} \displaystyle {{\mathscr H}^t}\left( F \right) \le {{\mathscr H}^s}\left( F \right) \cdot \mathop {{\rm{lim}}}\limits_{n \to \infty } {\rm{ }}\delta {\left( {F,{{\rm{\Gamma }}_n}} \right)^{t - s}}. \end{array}$$

Hence, if ℋ^s(F) < ∞ and δ(F,Γ_n) → 0 for all t > s, then it holds that ℋ^t(F) = 0. Accordingly, under the natural hypothesis consisting of the sequence of diameters {δ(F,Γ_n)_n∈ℕ goes to 0 (such a condition regarding the elements in each level of Γ makes the fractal structure being starbase, c.f. [26, Proposition 3.8]), our new theoretical method to calculate the fractal dimension of a subset with respect to a fractal structure establishes that this value is exactly the unique critical point where ℋ^s(F) “jumps” from ∞ to 0. Formally, the new fractal dimension for a fractal structure can be described as follows:

sup{s≥0:Hs(F)=∞}=inf{s≥0:Hs(F)=0},$$\begin{array}{} \displaystyle \sup \left\{ {s \ge 0:{{\mathscr H}^s}\left( F \right) = \infty } \right\} = \inf \left\{ {s \ge 0:{{\mathscr H}^s}\left( F \right) = 0} \right\}, \end{array}$$

provided that δ(F,Γ_n) → 0. Going beyond, that hypothesis, which is only a natural constraint concerning the size of the elements in each level of the involved fractal structure, becomes necessary as the next counterexample highlights.

Next, we provide the key definition of fractal dimension for a fractal structure we shall explore along this section.

From Definition 13 of fractal dimension, it follows that the quantity H3s(F)$\begin{array}{} \displaystyle {\mathscr H}_3^s\left( F \right) \end{array}$ can be described similarly to HHs(F)$\begin{array}{} \displaystyle {\mathscr H}_H^s\left( F \right) \end{array}$ in terms of dimΓ3(F)$\begin{array}{} \displaystyle {\rm{dim}}_{\rm{\Gamma }}^3\left( F \right) \end{array}$, instead (c.f. Eq. (4)):

H3s(F)={∞if s<dimΓ3(F)0if s>dimΓ3(F),$$\begin{array}{} \displaystyle {\mathscr H}_3^s(F) = \left\{ \begin{array}{*{20}{l}} \infty &{{\rm{if }}s \lt {\rm{dim}}_{\bf{\Gamma }}^3(F)}\\ 0&{{\rm{if }}s \gt {\rm{dim}}_{\bf{\Gamma }}^3(F),} \end{array}\right. \end{array}$$(13)

provided that δ(F,Γ_n) → 0.

Additionally, the next remark becomes quite useful for fractal dimension III calculation purposes, since it highlights that it is no longer necessary to consider lower/upper limits to define H3s(F)$\begin{array}{} \displaystyle {\mathscr H}_3^s\left( F \right) \end{array}$.

Along this subsection, we contribute several results to theoretically connect fractal dimension III with the classical definitions of fractal dimension, namely, both Hausdorff and box dimensions, as well as with fractal dimension II, previously explored in Section 4 for fractal structures.

For a given subset F ⊆ X, the calculation of each term in the sequence Hns(F)$\begin{array}{} \displaystyle {\mathscr H}_n^s\left( F \right) \end{array}$ (c.f. Eq. (8)) seems to be easier to be calculated than the corresponding in Hn,3s(F)$\begin{array}{} \displaystyle {\mathscr H}_{n,3}^s\left( F \right) \end{array}$ (as described in Eq. (12)). In addition, as Remark 11 points out, fractal dimension III always exists provided that the set function Hn,3s$\begin{array}{} \displaystyle {\mathscr H}_{n,3}^s \end{array}$ is considered to deal with its effective calculation. Following the above, the next theoretical result we provide allows the calculation of fractal dimension III from easier Eqs. (8) and (9).

As well as the Hausdorff dimension definition is based on the s−dimensional Hausdorff measure HHs$\begin{array}{} \displaystyle {\mathscr H}_H^s \end{array}$, next we shall explore some measure properties regarding the set functions Hn,3s,H3s$\begin{array}{} \displaystyle {\mathscr H}_{n,3}^s,{\mathscr H}_3^s \end{array}$, and ℋ^s that allow the calculation of the fractal dimension III for any subset F ⊆ X.

To deal with, let 𝒫(X) denote the class of all subsets of a given space X. Recall that an outer measure is a set function µ : 𝒫(X) → [0,¥] satisfying the three following conditions (c.f., e.g., [38, Section 5.2]):

It assigns the value 0 to the empty set, namely, µ(⌀) = 0.

for all sequence {A_n}_n∈ℕ ⊆ 𝒫(X).

Notice that Hns$\begin{array}{} \displaystyle {\mathscr H}_n^s \end{array}$ is an outer measure for all natural number n. Moreover, the following result concerning the set function Hn,3s$\begin{array}{} \displaystyle {\mathscr H}_{n,3}^s \end{array}$ can be stated.

Another issue naturally arising consists of determining some reasonable conditions on the elements in each level of a fractal structure to guarantee the equality among fractal dimension III and fractal dimensions I and II. In this way, the following result we provide allows the calculation of fractal dimension III from the fractal dimension I formula provided that fractal structures having and appropriate size (of 1/2ⁿ−order) are selected to deal with the calculations.

Next step is to find out appropriate conditions regarding the size of the elements in each level of a fractal structure to reach the equality between fractal dimensions II and III.

Next, we collect several theoretical properties for fractal dimension III similarly to Proposition 5 for fractal dimension I and Proposition 10 for fractal dimension II.

Recall that in Theorem 13, some properties regarding the elements in each level of a fractal structure were provided to reach the equality between fractal dimension II and box dimension. It is worth pointing out that box dimension may be also defined for metrizable spaces. The next result we provide has been carried out in the spirit of Theorem 13 and generalizes Theorem 28.

As it was stated previously in Subsection 4.6, the issue concerning the calculation of the fractal dimension for IFS-attractors via algebraic expressions involving only a finite number of known quantities arises naturally for each new definition of fractal dimension. It is worth pointing out that this kind of theoretical results are inspired on classical Moran’s Theorem (c.f. Theorem 16) and usually assume that the similarities that give rise to the IFS-attractor are under the OSC hypothesis (c.f. Definition 12). In fact, recall that this constitutes the main constraint required to an IFS to reach the equality between the Hausdorff and the box dimensions of its strict self-similar set (c.f. Theorem 16).

The OSC is a strong hypothesis requested to the self-similar copies of the whole IFS-attractor (sometimes called pre-fractals) to guarantee that they do not overlap “too much”. In this way, Theorem 17 stands under the OSC for fractal dimension II. Interestingly, the fractal dimension III model allows the calculation of the fractal dimension of strict self-similar sets via a Moran’s type equation (c.f. Eq. (7)) even if the similarities of the IFS do not lie under the OSC. This allows to generalize Moran’s Theorem in the context of fractal structures. To prove such a theoretical result, both the natural fractal structure which any IFS-attractor can be always endowed with (c.f. Definition 2 or Remark 2) and Equivalent definition (2) in Remark 10 for Hn,3s$\begin{array}{} \displaystyle {\mathscr H}_{n,3}^s \end{array}$ do play a relevant role herein.