A φ-Contractivity and Associated Fractal Dimensions

Abstract

In this paper, we extend the concept of dimension of sets to some general frameworks relative to a gauge function $\phi$, where two simultaneous dimensions are introduced. Unlike the classical cases where one dimension function is introduced based on the diameter power relative to the associated measure power, and where the gauge is a set-valued function or a measure in the majority of cases, we no longer assume this hypothesis. The introduced variant generalizes many existing cases, such as Haudorff, packing, Carathéodory, and Billingsley original variants. Many characteristics of the dimensions are investigated, such as bijectivity, convexity, monotony, asymptotic behavior, and fixed points.

1. Introduction and Motivations

The present manuscript explores the extension of the concept of fractal dimensions based on gauge functions and attempts to introduce new fractal metrics and dimensional variants. In ref. [1], the author introduced an interesting generalization of self-similar sets associated to the concept of $\phi$-contractivity and fixed point theory, provided with some variants of fractal measures and dimensions. This was a first motivation to develop the present work showing that the concept introduced there may be extended more to a general multifractal form. It is already questionable in fractal analysis/geometry to decide about the fractality of a set without investigating its dimension (due to some variants of dimension). In the classical definition, a set (a measure) is fractal if its Hausdorff and packing dimensions coincide. Next, with the scientific progress and the emergence of technology, the concept of dimension began to develop a clear scientific definition, little by little. The dimension was defined in mathematics/physics as the lower number of variables (known as coordinates) necessary to determine the position of an object in its universe or space [2,3,4]. With the development and advancement of scientific research, many cases of sets (measures) appeared that are not well-described via the original fractal measures and dimensions. This was a second motivation leading researchers to explore more variants.

The concept of dimension has been known since the early existence of humans. Since the beginning of existence in the universe, humanity has used this concept in daily life, in building houses, tools, and machines for daily use. Some of the activities show the existence of the dimension concept, even in the mind of the human, appearing in drawings with lines and bodies in caves and on buildings, stones, and so on, without the creators noticing that they were applying the concept of dimension. In ref. [5], the concept of dimension is investigated via the well-known quaternions. The authors developed a reduced biquaternion windowed linear canonical transform by combining the reduced biquaternion signal and the windowed linear canonical transform. The method was proven to be powerful in processing multidimensional data and enhancing the quality and efficiency of signal and image processing. Recall that quaternions are not multiplicatively commutative, which justifies the use of biquaternions in the technique proposed. The authors claimed that their theoretical findings led to fast algorithms in image processing. In ref. [6], the authors introduced a new invertibility concept for matrices over special algebraic structures, especially antirings. The concept states that a matrix is invertible if its product with its “inverse” lies on the line generated by the identity matrix and does not necessarily equal it.

The fractal dimensions were, since their first introduction, useful tools for the roughness of sets and the exploration of the hidden properties of sets. As these types of sets are met everywhere in nature, being natural or artificial, the concept of fractal dimension has known great interest in the scientific community and has been applied in physics, biology, signal/image processing, computer vision, and so on. Therefore, this concept was branched out and extended to other fields and areas. It has been applied in the field of modeling, for example, to represent the number of variables needed to express a phenomenon mathematically through equations, functions, or other modelers [2,4,7]. To model the dynamics, the concept of dimension is explained as a measure of the complexity of invariant sets. This definition or sense was the first step to invent the fractal dimensions, which overcome the existing dimensions by their ability to have better insights into the dynamics hidden in the phenomenon [2,3,8].

The essential difficulty arises already and always in the direct computation of the fractal dimension from the original definition of the fractal measure associated to it. This led mathematicians to introduce many variants, based on gauge functions, for example, to facilitate the task. One of the known cases is self-similarity, which was extended to many variants, such as ref. [1], and based generally on a fixed point theory associated to suitable contractions. See also [9,10,11,12].

Fractal dimensions have been known since the beginning of the 1900s. Many variants have been introduced, such as Hausdorff, packing, Billingsley, Minkowski, Bouligand, and so on [13,14]. Furthermore, fractal geometry and analysis came to add to the concept of dimension a broad framework for both the theory and applications, and, thus, fractal dimensions represented a new vision of things, which contributed to understanding many phenomena and issues. A link to dynamical systems, thermodynamics, images, and signals resulted in many fascinating models [11,15,16,17].

These dimensions were living in our minds since the early age of humanity. Lao-Tzu, a Chinese philosopher (6th century B.C.), said that “the key to growth is the introduction of higher dimensions of consciousness into our awareness”. This early comprehension of the dimension concept confirms that the way of growth is based on the dimension’s comprehension. We have to quantify the growth via a dimension function.

Let us criticize and/or review the mathematical framework. Fractal analysis provides a powerful tool to analyze complex structures through irregularities or singularities. It operates via the so-called fractal spectrum, which describes the scaling behavior of the analyzed object (data, signal, function, measure, time series, etc.) [18,19,20,21,22]. The easiest way used to this aim is based on the Legendre transform of a specific function, defined via the $L^q$-spectrum of a specific measure $\mu$, supported on the set under investigation. Let

$${\displaystyle {\tau }_{\mu }\left(q\right)=\underset{r\to 0}{lim\; sup}\frac{{\displaystyle logsup\sum _i\mu {\left(B(x_i,r)\right)}^q}}{logr},q\in \mathbb{R},}$$

(1)

where ${\left(B(x_i,r)\right)}_i$ is a countable covering of the set with disjoint closed balls with radius r. The spectrum that gives the dimension of the set states that

$$\dim E_{\mu }\left(\alpha \right)={\tau }_{\mu }^{*}\left(\alpha \right)=\underset{q\in \mathbb{R}}{inf}\{\alpha q+{\tau }_{\eta }\left(q\right)\},\alpha \in [0,\infty ),$$

(2)

where $E_{\mu }\left(\alpha \right)$ is the so-called $\alpha$-level set associated to the regularity of the measure $\mu$:

$${\displaystyle E_{\mu }\left(\alpha \right)=\left\{x;\underset{r\downarrow 0}{lim}\frac{log\left(\mu \right(B(x,r)\left)\right)}{logr}=\alpha \right\}.}$$

(3)

By looking deeply at Equation (1), we deduce that it means in some sense that

$$\sum _i\mu {\left(B(x_i,r)\right)}^q\sim r^{{\tau }_{\mu }\left(q\right)},r\to 0.$$

(4)

Equation (4) was the crucial point to consider in the early extension of Olsen [23], which considers a $(q,t)$- form for the Hausdorff and packing measures by considering the gauge function

$${\phi }_{q,t}\left(r\right)=\mu {\left(B(x,r)\right)}^q{r}^t,r>0.$$

On one hand, this formulation compares again the measure of the ball with the diameter powers. On the other hand, this formulation involves changing the measure into the definition of the Hausdorff measure to allow simultaneously taking into consideration the behavior of the measure and the geometry of the set, in particular, when the level sets are supported by fractal measures of the Hausdorff type. Recall that the sets investigated in fractal analysis/geometry are of the zero-Lebesgue measure. It is already mentioned in ref. [24] that the measure $\mu$ is nonatomic, since otherwise there may be no covering at all. Equation (2) indicates an affine behavior of the function $\dim E_{\mu }\left(\alpha \right)$. In many situations of $\alpha$ and $\mu$, like doubling, Hölderian, self-similar, and Gibbs, the computation of the dimension is already known. Olsen, for example [23], used the large deviation formalism and Billingsley theorem to deduce the dimension of $E_{\mu }\left(\alpha \right)$. In ref. [25], the authors considered an extension of [23] and involved a gauge function of the form

$${\phi }_{q,t}\left(r\right)=\mu {\left(B(x,r)\right)}^q{r}^t{e}^{{\phi }_0\left(r\right)},r>0,$$

for a specific function ${\phi }_0$, such that ${\phi }_0\left(r\right)=o(logr)$ as $r\to 0$. Note here that we come back again to a control with the powers of the diameter. In ref. [26], the authors considered the introduction of the gauge function into the level sets and obtained an analog of Equation (3) as

$${\displaystyle E_{\mu ,\phi }\left(\alpha \right)=\left\{x;\underset{r\downarrow 0}{lim}\frac{log\left(\mu \right(B(x,r)\left)\right)}{\phi \left(r\right)}=\alpha \right\}.}$$

(5)

In ref. [27], the author compared the original measure $\mu$ to another appropriate measure $\nu$, and considered two-parameter level sets:

$${\displaystyle E_{\mu ,\nu }(\alpha ,\beta )=\left\{x;\underset{r\downarrow 0}{lim}\frac{log\left(\mu \right(B(x,r)\left)\right)}{logr}=\alpha and\underset{r\downarrow 0}{lim}\frac{log\left(\mu \right(B(x,r)\left)\right)}{log\left(\nu \right(B(x,r)\left)\right)}=\beta \right\}.}$$

These works join the early definitions of the well-known Carathéodory dimension, which links the theory of dimensions to the theory of dynamical systems [7]. The common drawback of all these situations is the use of measures or set-valued functions compared with the diameters, which may not be suitable to handle appropriately many situations, like discrete countable sets. The measures there cannot be evidently compared to power-law diameters. See for instance [28].

Another common point in these existing works reposes on the concavity/convexity of the function ${\tau }_{\mu }$, which is critical in many situations. For example, in ref. [29], an application dealt with the heart health/failure signals via the spectrum of the signal, and showed the possible concavity of it. In ref. [30], the authors suggested that non-concavities may relate to physiological factors but not to the theory of fractal spectrum computation. In financial time series, the task gives a link between fractal dimensions and Hurst exponent [31].

In another context, Graf et al. [32] considered a different method for fractal measures and dimensions. It is similar to the present work in that it considered, instead of gauges based on measures or set-valued functions, gauges depending only on the radius/diameter. They considered the iterated logarithms gauge function

$$\phi \left(r\right)={\left({\displaystyle \frac{1}{log\frac{1}{r}}}\right)}^{{\beta }_1}{\left({\displaystyle \frac{1}{loglog\frac{1}{r}}}\right)}^{{\beta }_2}\dots {\left({\displaystyle \frac{1}{log\dots loglog\frac{1}{r}}}\right)}^{{\beta }_k}$$

for some constants ${\beta }_1,\dots ,{\beta }_k$, and the associated Hausdorff-type measure

$${\mathcal{H}}^{\alpha }\left(E\right)=\underset{\eta \searrow 0}{lim}inf\left\{{\displaystyle \sum _i\phi \left(r_i\right)r_i^{\alpha };x_i\in E\subset \bigcup _{i}B(x_i,r_i),r_i<\eta }\right\}.$$

The authors successfully estimated the Hausdorff-type measure above for special cases of self-similar sets and the special values of the exponents, such as $\alpha +{\beta }_1=1$ and ${\beta }_i=0,i\ge 2$. Precisely, in ref. [32], the authors constructed suitable sets $E_{\mu }\omega$ depending on a parameter $\omega$, for which $\alpha$ is almost the Hausdorff dimension of $E_{\mu }\omega$. Taking ${\beta }_1=1-\alpha$, they obtained $0<{\mathcal{H}}^{\alpha }\left(E\right)<\infty$. General constructions have been also investigated intensively with the so-called iterative logarithms in [33,34,35]. This case will be reexamined later.

In ref. [36], a construction of a special self-similar set E allowed discussion of the question of the choice of the gauge function $\phi$ to obtain a finite, infinite, or zero value for the Hausdorff- or packing-type measures. In ref. [37], the author developed one case yielding an infinite value and one zero-measure case. The case of finite positive values remains not completely resolved. See also ref. [26], where an answer to some cases is provided.

In the present work, we aim to introduce a generalized form of fractal dimensions of sets based on gauge functions. The introduced dimensions include (with the specific choice of the gauges) the original definitions of the fractal dimensions due to Hausdorff, packing, Billingsley, Minkowski, and Bouligand. Moreover, we show that the introduced variants answer many questions that are left open in the fractal analysis/geometry theory, such as the invertibility of the dimensions’ functions and the possibility of the existence of fixed points. From the applied point of view, such as signal/image processing, time series, and other fields, a crucial point remains, of course: the suitable choice of the gauge function to obtain adequate description and comprehension of the case under investigation. Many techniques have been applied in this direction, such as fractal functions interpolation. Obtaining a close model to the data permits the application of the suitable measures by the next.

During this paper, we will adopt the following notations.

• $N\ge 1$ is an integer designating the dimension of the Euclidean real space ${\mathbb{R}}^N$.
• $\parallel ·\parallel$ stands for the usual Euclidean norm on ${\mathbb{R}}^N$.
• $d(·,·)$ stands for the Euclidean distance associated to the norm $\parallel ·\parallel$.
• For $x\in {\mathbb{R}}^N$ and $r>0$, $B(x,r)$ is the open ball of center x and radius r.
• $\overline{\mathbb{R}}=[-\infty ,\infty ]$.

Section 2 is devoted to a review of the original definitions of the Hausdorff and packing measures and dimensions. Section 3 is concerned with the main results of our paper. Section 4 is concerned with the development of some illustrative examples. Section 5 is concerned with some final remarks related to the applicability, limitations and eventual relationship with original definitions in the field. Section 6 is the conclusion of the paper.

2. Old Fractal Measures and Dimensions Revisited

Given a subset $E\subseteq {\mathbb{R}}^N$ and $\eta >0$, we call an $\eta$-covering of E any countable set ${\left(U_i\right)}_i$ composed of non-empty subsets $U_i\subseteq {\mathbb{R}}^N$ satisfying

$${\displaystyle E\subseteq \bigcup _i{U}_i\mathrm{and}\left|U_i\right|=\mathrm{diam}\left(U_i\right)\le \eta ,}$$

(6)

where for any subset $U\subseteq {\mathbb{R}}^N$, $\left|U\right|=\mathrm{diam}\left(U\right)$ is the diameter defined by

$${\displaystyle \left|U\right|=\mathrm{diam}\left(U\right)=\underset{x,y\in U}{sup}\parallel x-y\parallel .}$$

Notice that for ${\eta }_1<{\eta }_2$, any ${\eta }_1$-covering of E is obviously an ${\eta }_2$-covering of it. Therefore, the quantity

$${\displaystyle {\mathcal{H}}_{\eta }^s\left(E\right)=inf\left\{\sum _i{\left|U_i\right|}^s;\left(U_i\right)\mathrm{satisfying}\mathrm{Equation}\left(6\right)\right\}}$$

is a nonincreasing function of the variable $\eta$. Its limit

$${\mathcal{H}}^s\left(E\right)=\underset{\eta \downarrow 0}{lim}{\mathcal{H}}_{\eta }^s\left(E\right)$$

defines the so-called s-dimensional Hausdorff measure of E. It holds that, for any set $E\subseteq {\mathbb{R}}^N$, there exists a critical value $s_E$ in the sense that

$${\mathcal{H}}^s\left(E\right)=0,\forall s<s_E\mathrm{and}{\mathcal{H}}^s\left(E\right)=\infty ,\forall s>s_E,$$

or, otherwise,

$${\displaystyle s_E=sup\left\{s>0;{\mathcal{H}}^s\left(E\right)=0\right\}=inf\left\{s>0;{\mathcal{H}}^s\left(E\right)=\infty \right\}.}$$

Such a value is called the Hausdorff dimension of the set E and is usually denoted by ${\dim }_{H}E$ or simply $\dim E$. When $U_i=B(x_i,r_i)$ is a ball centered at $x_i\in E$ and with diameter $r_i<\eta$, the covering ${\left(B(x_i,r_i)\right)}_i$ is called an $\eta$-centered covering of E. However, surprisingly, the quantity ${\mathcal{H}}^s$ restricted only on centered coverings does not define a measure. To obtain a good measure with centered coverings, one should do more. Denote

$${\displaystyle {\overline{\mathcal{C}}}_{\eta }^s\left(E\right)=inf\{\sum _i{\left(2r_i\right)}^s;{\left(B(x_i,r_i)\right)}_i\mathrm{an}\eta -\mathrm{centered}\mathrm{covering}\mathrm{of}E\},}$$

and similarly as above,

$${\overline{\mathcal{C}}}^s\left(E\right)=\underset{\eta \downarrow 0}{lim}{\overline{\mathcal{C}}}_{\eta }^s\left(E\right).$$

As stated previously, this is not a good measure. To obtain a good candidate, we set for $E\subseteq {\mathbb{R}}^N$,

$${\displaystyle {\mathcal{C}}^s\left(E\right)=\underset{F\subseteq E}{sup}{\overline{\mathcal{C}}}^s\left(F\right).}$$

It is called the centered Hausdorff s-dimensional measure of E. But, although a fascinating relation to the Hausdorff measure exists, stating that

$$2^{-s}{\mathcal{C}}^s\left(E\right)\le {\mathcal{H}}^s\left(E\right)\le {\mathcal{C}}^s\left(E\right),\forall E\subseteq {\mathbb{R}}^N,$$

(7)

indeed, let $F\subseteq E$ be subsets of ${\mathbb{R}}^N$. It follows, from the definition of ${\mathcal{H}}^s$ and ${\overline{\mathcal{C}}}^s$, that

$${\mathcal{H}}^s\left(F\right)\le {\overline{\mathcal{C}}}^s\left(F\right).$$

As this holds for any subset F of E, from the fact that ${\mathcal{H}}^s$ is an outer metric measure on ${\mathbb{R}}^N$ and the definition of ${\mathcal{C}}^s$, it results that

$${\mathcal{H}}^s\left(E\right)\le {\mathcal{C}}^s\left(E\right).$$

Next, let ${\left\{U_j\right\}}_j$ be an $\eta$-covering of F and $r_j=\mathrm{diam}\left(U_j\right)$. For each i fixed, consider a point $x_i\in U_i\cap F$. This results in a centered $\eta$-covering ${\left\{B(x_i,r_i)\right\}}_i$ of F. Consequently,

$${\displaystyle {\overline{\mathcal{C}}}_{\eta }^s\left(F\right)\le \sum _i{\left(2r_i\right)}^s=2^s\sum _i{\left(\mathrm{diam}\left(U_i\right)\right)}^s.}$$

Hence,

$${\overline{\mathcal{C}}}_{\eta }^s\left(F\right)\le 2^s{\mathcal{H}}_{\eta }^s\left(F\right).$$

As $\eta \downarrow 0$, we obtain

$${\overline{\mathcal{C}}}^s\left(F\right)\le 2^s{\mathcal{H}}^s\left(F\right),\forall F\subseteq E,$$

which guaranties that

$${\mathcal{C}}^s\left(E\right)\le 2^s{\mathcal{H}}^s\left(E\right).$$

It holds that ${\mathcal{C}}^s$ gives rise to a critical value in the sense that, for any set $E\subseteq {\mathbb{R}}^N$, there exists a critical value $c_E$ for which

$${\mathcal{C}}^s\left(E\right)=0,\forall s<c_E\mathrm{and}{\mathcal{C}}^s\left(E\right)=\infty ,\forall s>c_E.$$

Otherwise,

$${\displaystyle c_E=sup\{s>0;{\mathcal{C}}^s\left(E\right)=0\}=inf\{s>0;{\mathcal{C}}^s\left(E\right)=\infty \}.}$$

Using Equation (7) above, it holds that $s_E=c_E$, which means that the critical value of ${\mathcal{C}}^s$ coincides with the Hausdorff dimension of the set E introduced previously.

Similarly, we call a centered $\eta$-packing of $E\subseteq {\mathbb{R}}^N$ any countable set ${\left(B(x_i,r_i)\right)}_i$ of disjoint balls centered at points $x_i\in E$ and with diameters $r_i<\eta$. The packing measure and dimension are defined as follows:

$${\displaystyle {\overline{\mathcal{P}}}^s\left(E\right)=\underset{\eta \downarrow 0}{lim}\left(sup\left\{\sum _i{\left(2r_i\right)}^s;{\left(B(x_i,r_i)\right)}_i\eta -\mathrm{packing}\mathrm{of}E\right\}\right),}$$

$${\displaystyle {\mathcal{P}}^s\left(E\right)=inf\left\{\sum _i{\overline{\mathcal{P}}}^s\left(E_i\right);E\subseteq {\cup }_i{E}_i\right\}.}$$

It holds, as for the Hausdorff measure, that there exists critical values ${\Delta }_E$ and $p_E$ satisfying, respectively,

$${\overline{\mathcal{P}}}^s\left(E\right)=\infty \mathrm{for}s<\Delta \left(E\right)\mathrm{and}{\overline{\mathcal{P}}}^s\left(E\right)=0\mathrm{for}\alpha >\Delta \left(E\right)$$

and

$${\mathcal{P}}^s\left(E\right)=\infty \mathrm{for}s<p_E\mathrm{and}{\mathcal{P}}^s\left(E\right)=0\mathrm{for}s>p_E.$$

The critical value $\Delta \left(E\right)$ is called the logarithmic index of E and $p_E$ is called the packing dimension of E, denoted ${\mathrm{Dim}}_{P}E$ or simply $\mathrm{Dim}E$. These quantities may be evaluated as

$$\Delta \left(E\right)=sup\left\{s;{\overline{\mathcal{P}}}^s\left(E\right)=0\right\}=inf\left\{s;{\overline{\mathcal{P}}}^s\left(E\right)=\infty \right\}$$

and

$$\mathrm{Dim}E=sup\left\{s;{\mathcal{P}}^s\left(E\right)=0\right\}=inf\left\{s;{\mathcal{P}}^s\left(E\right)=\infty \right\}.$$

Additionally, we have the inequality

$$\dim E\le \mathrm{Dim}E\le \Delta \left(E\right),\forall E\subseteq {\mathbb{R}}^N.$$

The set $E\subset {\mathbb{R}}^N$ is said to be fractal in the sense of Taylor if $\dim E=\mathrm{Dim}E$ [13,38,39].

In the literature, many generalizations of these measures and dimensions have been investigated. However, the major drawback in most of them is the involvement of, again, a measure or many measures in the expression of $\mathcal{H}$, $\overline{\mathcal{P}}$, and $\mathcal{P}$. Such involved measures are controlled by the radius [12,40,41,42,43,44,45,46].

In the present work, we will not involve a measure, but just a function. We will show that many functions are admissible, in the sense that they permit the computation of the fractal dimension of sets without being related to the set, as in the existing literature. The readers may refer to [7,8,9,10,11,13,14,15,16,17,23].

3. Main Results

Definition 1.

A function $\phi :[0,\infty )⟶[0,\infty )$ is said to be admissible if

• it is nondecreasing,
• it is continuous on the right,
• there exists a constant $c>0$ such that $\phi \left(x\right)\le cx$ on $[0,\infty )$.

3.1. The Hausdorff-Type Measure and Dimension

The $\phi$-generalized Carathédory–Billingsley dimension of sets is introduced via the so-called $\phi$-Carathéodory–Billingsley measure of sets, which will be a general form of the classical cases of the Hausdorff measure, as we will see. Original definitions and backgrounds on such a measure and dimension may be found in [7,8,47].

For a subset $E\subset {\mathbb{R}}^N$, and $q,t\in \mathbb{R}$, write

$$\overline{{\mathcal{H}}_{\phi }^{q,t}}\left(E\right)=\underset{\eta \to 0}{lim}\left(inf\left\{{\displaystyle \sum _i{\left(\phi \left(r_i\right)\right)}^q{r}_I^t,E\subset \underset{i}{\cup }B(x_i,r_i),x_i\in E,\phi \left(r_i\right)\le \eta ,\forall i}\right\}\right).$$

For $\eta >0$, the collection ${\left(B(x_i,r_i)\right)}_i$ is called a $\phi$-$\eta$-covering of E. We write, finally,

$${\displaystyle {\mathcal{H}}_{\phi }^{q,t}\left(E\right)=\underset{F\subseteq E}{sup}\overline{{\mathcal{H}}_{\phi }^{q,t}}\left(F\right).}$$

The following lemma gives a general variant of the well-known Caratéodory, Billingsley, and Hausdorff dimensions of sets.

Lemma 1.

For any admissible function φ, and any subset $E\subset {\mathbb{R}}^N$, there exists a unique cut-off value $(q_t\left(E\right),t_q\left(E\right))\in {\overline{\mathbb{R}}}^2$, such that

$$\left\{\begin{array}{c}{\mathcal{H}}_{\phi }^{q,t}\left(E\right)=0,\forall q>q_t\left(E\right)\mathrm{and}{\mathcal{H}}_{\phi }^{q,t}\left(E\right)=\infty ,\forall q<q_t\left(E\right),\hfill \\ {\mathcal{H}}_{\phi }^{q,t}\left(E\right)=0,\forall t>t_q\left(E\right)\mathrm{and}{\mathcal{H}}_{\phi }^{q,t}\left(E\right)=\infty ,\forall t<t_q\left(E\right).\hfill \end{array}\right.$$

Proof. 

Let $t\in \mathbb{R}$ be such that ${\mathcal{H}}_{\phi }^{q,t}\left(E\right)<\infty$ and $\eta >0$. Let also ${\left(B(x_i,r_i)\right)}_i$ be a $\phi$-$\eta$-covering of E. It holds, for all $s>t$, that

$$\begin{array}{c}{\displaystyle {\mathcal{H}}_{\phi ,\eta }^{q,s}\left(E\right)\le \sum _i{\left(\phi \left(r_i\right)\right)}^q{r}_i^s\le {\eta }^{s-t}\sum _i{\left(\phi \left(r_i\right)\right)}^q{r}_i^t.}\hfill \end{array}$$

Therefore,

$${\mathcal{H}}_{\phi ,\eta }^{q,s}\left(E\right)\le {\eta }^{s-t}{\mathcal{H}}_{\phi ,\eta }^{q,s}\left(E\right).$$

Hence,

$${\mathcal{H}}_{\phi }^{q,s}\left(E\right)=0,\forall s>t.$$

We thus write

$$t_q\left(E\right)=inf\left\{t\in \mathbb{R};{\mathcal{H}}_{\phi }^{q,t}\left(E\right)=0\right\}.$$

Similarly, we perform this for $q_t\left(E\right)$. □

Definition 2.

For any admissible function φ, and any subset $E\subset {\mathbb{R}}^N$, we write

$$\left\{\begin{array}{c}{h}_{\phi }(·,E):q⟼t_q\left(E\right),\hfill \\ H_{\phi }(·,E):t⟼q_t\left(E\right).\hfill \end{array}\right.$$

The function $(q,t)⟼(h_{\phi }(q,E),H_{\phi }(t,E))$ is called the φ-generalized Carathéodory–Billingsley–Hausdorff dimension of the set E and will be symbolized as ${\mathit{gcbh}}_{\phi }\left(E\right)$.

Remark 1.

Notice that

• for $q=0$, we come back to the classical (original) definition of the Hausdorff measure and dimension.
• for $t=0$, we come back to a simple form of the Carathéodory and Billingsley cases.

The following theorem gives some properties of ${\mathit{gcbh}}_{\phi }$, such as the monotony.

Theorem 1.

For any admissible function φ, the dimension functions $h_{\phi }(·,E)$ and $H_{\phi }(·,E)$ are nonincreasing.

Proof. 

Let $q_1<q_2$. We immediately obtain

$${\mathcal{H}}_{\phi ,\eta }^{q_2,t}\left(E\right)\le {\left(\phi \left(\eta \right)\right)}^{q_2-q_1}{\mathcal{H}}_{\phi ,\eta }^{q_1,t}\left(E\right).$$

Consequently,

$${\mathcal{H}}_{\phi ,\eta }^{q_2,t}\left(E\right)=0,\forall t>h_{\phi }\left(q_1\right).$$

Therefore,

$$h_{\phi }\left(q_2\right)\le t,\forall t>h_{\phi }\left(q_1\right),$$

which reads that

$$h_{\phi }\left(q_2\right)\le h_{\phi }\left(q_1\right).$$

□

3.2. The Packing Type Measure and Dimension

In the same way as above, we introduce a generalization for the pre-packing and packing measures and dimensions by setting

$$\widehat{{\mathcal{P}}_{\phi }^{q,t}}\left(E\right)=\underset{\eta \to 0}{lim}\left(sup\left\{{\displaystyle \sum _i{\left(\phi \left(r_i\right)\right)}^q{r}_I^t,B(x_i,r_i)\mathrm{disjoint},x_i\in E,\phi \left(r_i\right)\le \eta ,\forall i}\right\}\right).$$

This is, obviously, not a measure as it lacks the property of sub-additivity. We thus consider

$${\mathcal{P}}_{\phi }^{q,t}\left(E\right)=inf\left\{{\displaystyle \sum _i\widehat{{\mathcal{P}}_{\phi }^{q,t}}\left(E_i\right);E\subset \underset{i}{\cup }{E}_i}\right\},$$

which is an outer metric measure on the usual Euclidean space. The collection ${\left(B(x_i,r_i)\right)}_i$ is called a $\phi$-$\eta$-packing of E. The following lemma is the analog of Lemma 1 and provides the analogs of the $\phi$-generalized Caratéodory–Billingsley–Hausdorff dimensions of sets.

Lemma 2.

For any admissible function φ, and any subset $E\subset {\mathbb{R}}^N$, there exists unique cut-off values $(\widehat{q_t}\left(E\right),\widehat{t_q}\left(E\right))\in {\overline{\mathbb{R}}}^2$, and $(q_t\left(E\right),t_q\left(E\right))\in {\overline{\mathbb{R}}}^2$, such that

$$\left\{\begin{array}{c}\widehat{{\mathcal{P}}_{\phi }^{q,t}}\left(E\right)=0,\forall q>\widehat{q_t}\left(E\right)\mathrm{and}\widehat{{\mathcal{P}}_{\phi }^{q,t}}\left(E\right)=\infty ,\forall q<\widehat{q_t}\left(E\right),\hfill \\ \widehat{{\mathcal{P}}_{\phi }^{q,t}}\left(E\right)=0,\forall t>\widehat{t_q}\left(E\right)\mathrm{and}\widehat{{\mathcal{P}}_{\phi }^{q,t}}\left(E\right)=\infty ,\forall t<\widehat{t_q}\left(E\right),\hfill \end{array}\right.$$

and

$$\left\{\begin{array}{c}{\mathcal{P}}_{\phi }^{q,t}\left(E\right)=0,\forall q>q_t\left(E\right)\mathrm{and}{\mathcal{P}}_{\phi }^{q,t}\left(E\right)=\infty ,\forall q<q_t\left(E\right),\hfill \\ {\mathcal{P}}_{\phi }^{q,t}\left(E\right)=0,\forall t>t_q\left(E\right)\mathrm{and}{\mathcal{P}}_{\phi }^{q,t}\left(E\right)=\infty ,\forall t<t_q\left(E\right).\hfill \end{array}\right.$$

Proof. 

It follows similar arguments and techniques as Lemma 1. □

Definition 3.

For any admissible function φ, and any subset $E\subset {\mathbb{R}}^N$, we write

$$\left\{\begin{array}{c}{\delta }_{\phi }(·,E):q⟼t_q\left(E\right),\hfill \\ {\Delta }_{\phi }(·,E):t⟼q_t\left(E\right).\hfill \end{array}\right.$$

The function $(q,t)⟼({\delta }_{\phi }(q,E),{\Delta }_{\phi }(t,E))$ is called the φ-generalized Carathéodory–Billingsley–Log-Index of the set E and will be symbolized as ${\widehat{\mathit{gcbp}}}_{\phi }\left(E\right)$. Similarly, we write

$$\left\{\begin{array}{c}{p}_{\phi }(·,E):q⟼t_q\left(E\right),\hfill \\ P_{\phi }(·,E):t⟼q_t\left(E\right).\hfill \end{array}\right.$$

The function $(q,t)⟼(p_{\phi }(q,E),P_{\phi }(t,E))$ is called the φ-generalized Carathéodory–Billingsley–packing dimension of the set E and will be symbolized as ${\mathit{gcbp}}_{\phi }\left(E\right)$.

Remark 2.

Notice that

• for $q=0$, we come back to the classical (original) definition of the logarithmic index and the pre-packing measure for $\widehat{{\mathcal{P}}_{\phi }^{q,t}}$.
• for $q=0$, we come back to the classical (original) definition of the packing measure and dimension for ${\mathcal{P}}_{\phi }^{q,t}$.
• for $t=0$, we come back to a simple form of the pre-packing and packing variants corresponding to the original Carathéodory and Billingsley cases.

The following theorem summarizes some properties of the dimensions ${\widehat{\mathrm{gcbp}}}_{\phi }$ and ${\mathrm{gcbp}}_{\phi }$, such as monotony and convexity.

Theorem 2.

For any admissible function φ, the dimension functions ${\delta }_{\phi }(·,E)$, ${\Delta }_{\phi }(·,E)$, $p_{\phi }(·,E)$, and $P_{\phi }(·,E)$ are convex and nonincreasing.

Proof. 

The convexity of ${\delta }_{\phi }(·,E)$ and ${\Delta }_{\phi }(·,E)$ follows from the Hölder-type inequality

$${\displaystyle \sum _i{a}_i^{\alpha }{b}_i^{1-\alpha }\le {(\sum _i{a}_i)}^{\alpha }{(\sum _i{b}_i)}^{1-\alpha },}$$

for any sequences ${\left(a_i\right)}_i$ and ${\left(b_i\right)}_i$ of positive real numbers, and any real number $\alpha \in [0,1]$. The monotony of ${\delta }_{\phi }(·,E)$ and ${\Delta }_{\phi }(·,E)$ follows from similar arguments and techniques as in Theorem 1. We will show the result for $p_{\phi }(·,E)$ and $P_{\phi }(·,E)$. Let $q_1,q_2\in \mathbb{R}$, $\alpha \in [0,1]$ and $\epsilon >0$, and write

$$t_1=p_{\phi }(q_1,E)\mathrm{and}{t}_2=p_{\phi }(q_2,E).$$

We obtain

$${\mathcal{P}}_{\phi }^{q_1,t_1+\epsilon }\left(E\right)={\mathcal{P}}_{\phi }^{q_2,t_2+\epsilon }\left(E\right)=0.$$

Therefore, there exists sequences ${\left(H_i\right)}_i$ and ${\left(K_i\right)}_i$ of sets, both covering E and satisfying

$${\displaystyle \sum _i\widehat{{\mathcal{P}}_{\phi }^{q_1,t_1+\epsilon }}\left(H_i\right)<\infty \mathrm{and}\sum _i\widehat{{\mathcal{P}}_{\phi }^{q_2,t_2+\epsilon }}\left(K_i\right)<\infty .}$$

Write, for $n\in \mathbb{N}$, ${\displaystyle E_n=\bigcup _{1\le i,j\le n}(H_i\cap K_j)}$. We deduce immediately that

$$\begin{array}{ccc}{\mathcal{P}}_{\phi }^{\alpha q_1+(1-\alpha )q_2,\alpha t_1+(1-\alpha )t_2+\epsilon }\left(E_n\right)\hfill & \le \hfill & {\displaystyle \sum _{i,j=1}^n{\mathcal{P}}_{\phi }^{\alpha q_1+(1-\alpha )q_2,\alpha t_1+(1-\alpha )t_2+\epsilon }(H_i\cap K_j)}\hfill \\ & \le \hfill & {\displaystyle \sum _{i,j=1}^n\widehat{{\mathcal{P}}_{\phi }^{\alpha q_1+(1-\alpha )q_2,\alpha t_1+(1-\alpha )t_2+\epsilon }}(H_i\cap K_j)}\hfill \\ & \le \hfill & {\displaystyle {\left(\sum _{i,j=1}^n\widehat{{\mathcal{P}}_{\phi }^{q_1,t_1+\epsilon }}(H_i\cap K_j)\right)}^{\alpha }{\left(\sum _{i,j=1}^n\widehat{{\mathcal{P}}_{\phi }^{q_2,t_2+\epsilon }}(H_i\cap K_j)\right)}^{1-\alpha }.}\hfill \end{array}$$

As a consequence,

$${\mathcal{P}}_{\phi }^{\alpha q_1+(1-\alpha )q_2,\alpha t_1+(1-\alpha )t_2+\epsilon }\left(E_n\right)<\infty ,\forall n,$$

which reads that

$$p_{\phi }(\alpha q_1+(1-\alpha )q_2,E_n)\le \alpha t_1+(1-\alpha )t_2+\epsilon ,\forall \epsilon >0\mathrm{and}\forall n.$$

Hence,

$$p_{\phi }(\alpha q_1+(1-\alpha )q_2,E)\le \alpha p_{\phi }(q_1,E)+(1-\alpha )p_{\phi }(q_2,E).$$

So, as the convexity of $p_{\phi }(·,E)$. The case of $P_{\phi }(·,E)$ is similar. We now investigate the monotony of $p_{\phi }(·,E)$ and $P_{\phi }(·,E)$. For $q_1\ge q_2$ and $t\in \mathbb{R}$, notice from the hypothesis on the function $\phi$ that

$$\widehat{{\mathcal{P}}_{\phi }^{q_1,t}}\left(E\right)\le \widehat{{\mathcal{P}}_{\phi }^{q_2,t}}\left(E\right).$$

Therefore,

$${\mathcal{P}}_{\phi }^{q_1,t}\left(E\right)\le {\mathcal{P}}_{\phi }^{q_2,t}\left(E\right).$$

Hence,

$${\mathcal{P}}_{\phi }^{q_1,t}\left(E\right)=0,\forall t>p_{\phi }(q_2,E),$$

which reads that

$$p_{\phi }(q_1,E)\le t,\forall t>p_{\phi }(q_2,E),$$

which in turn implies that

$$p_{\phi }(q_1,E)\le p_{\phi }(q_2,E).$$

□

The next result deals with a comparison of the fractal dimensions introduced above and is stated as follows.

Theorem 3.

For any admissible function φ, and any subset set $E\subset {\mathbb{R}}^N$, we have

$$\left\{\begin{array}{c}{h}_{\phi }\left(E\right)\le p_{\phi }\left(E\right)\le {\delta }_{\phi }\left(E\right),\hfill \\ H_{\phi }\left(E\right)\le P_{\phi }\left(E\right)\le {\Delta }_{\phi }\left(E\right).\hfill \end{array}\right.$$

Proof. 

Using the well-known Besicovitch Covering Theorem, there exists a fixed integer $N_B$, and ${\mathcal{B}}_i$, $i=1,2,\dots ,N_B$ countable collections of disjoint balls ${\left(B(x_j,r_j)\right)}_j$, $x_j\in E$ satisfying ${\displaystyle E\subseteq \bigcup _{i=1}^{N_B}\bigcup _{B\in {\mathcal{B}}_i}B}$. Therefore, by assuming that ${\left(B(x_i,r_i)\right)}_i$ is a $\phi$-$\eta$-covering of E, we obtain

$$\begin{array}{c}{\displaystyle {\mathcal{H}}_{\phi ,\eta }^{q,t}\left(E\right)\le \sum _{i=1}^{N_B}\sum _j{\left(\phi \left(r_j\right)\right)}^q{r}_j^t\le \sum _{i=1}^{N_B}\widehat{{\mathcal{P}}_{\phi ,\eta }^{q,t}}\left(E\right)=N_B\widehat{{\mathcal{P}}_{\phi ,\eta }^{q,t}}\left(E\right).}\hfill \end{array}$$

As a result,

$${\mathcal{H}}_{\phi }^{q,t}\left(E\right)\le N_B\widehat{{\mathcal{P}}_{\phi }^{q,t}}\left(E\right).$$

Therefore, for any covering ${\left(E_i\right)}_i$ of E, we obtain

$$\begin{array}{c}{\displaystyle {\mathcal{H}}_{\phi }^{q,t}\left(E\right)\le \sum _i{\mathcal{H}}_{\phi }^{q,t}\left(E_i\right)\le N_B\sum _i\widehat{{\mathcal{P}}_{\phi }^{q,t}}\left(E_i\right).}\hfill \end{array}$$

Taking next the inf on all ${\left(E_i\right)}_i$, we obtain

$${\mathcal{H}}_{\phi }^{q,t}\left(E\right)\le N_B{\mathcal{P}}_{\phi }^{q,t}\left(E\right).$$

On the other hand, as E is a covering of itself, we obtain

$${\mathcal{P}}_{\phi }^{q,t}\left(E\right)\le \widehat{{\mathcal{P}}_{\phi }^{q,t}}\left(E\right).$$

As a consequence, we obtain

$${\mathcal{H}}_{\phi }^{q,t}\left(E\right)\le N_B{\mathcal{P}}_{\phi }^{q,t}\left(E\right)\le N_B\widehat{{\mathcal{P}}_{\phi }^{q,t}}\left(E\right).$$

This leads to the desired result. □

3.3. Invertibility and Asymptotic Behavior of the New Dimension Functions

Among the interesting and at the same time difficult questions in fractal analysis, we cite the asymptotic behavior and the bijectivity of the functions defining the multifractal dimensions. One of the classical results is known as Folklore’s theorem and states that in some cases, the multifractal dimension function is increasing, convex, smooth, and asymptotically affine at infinity. Moreover, the affine asymptotic behavior permits the obtaining of a multifractal formalism via a Legendre transform of the dimension function [23].

In the present work, we propose to study the behavior of the new functional dimensions, as well as their possible bijectivity. Our result is stated as follows.

Theorem 4.

For any admissible function φ, and any subset set $E\subset {\mathbb{R}}^N$, the functions $h_{\phi }(·,E)$, $H_{\phi }(·,E)$, $p_{\phi }(·,E)$, $P_{\phi }(·,E)$, ${\delta }_{\phi }(·,E)$, and ${\Delta }_{\phi }(·,E)$ are bijections. Moreover,

$$h_{\phi }^{-1}(·,E)=H_{\phi }(·,E),p_{\phi }^{-1}(·,E)=P_{\phi }(·,E),\mathrm{and}{\delta }_{\phi }^{-1}(·,E)={\Delta }_{\phi }(·,E),$$

and $p_{\phi }(·,E)$, $P_{\phi }(·,E)$, ${\delta }_{\phi }(·,E)$, and ${\Delta }_{\phi }(·,E)$ have affine lower bounds.

Proof. 

Let us show the result for the functions $h_{\phi }(·,E)$ and $H_{\phi }(·,E)$. The remaining cases may be shown by similar techniques. Indeed, for $q\in \mathbb{R}$, we obtain

$${\mathcal{H}}_{\phi }^{q,t}\left(E\right)=0,\forall t>h_{\phi }(q,E).$$

Hence,

$$q\ge H_{\phi }(t,E),\forall t>h_{\phi }(q,E).$$

Therefore,

$$q\ge H_{\phi }(h_{\phi }(q,E),E).$$

Similarly, we show that

$$q\le H_{\phi }(h_{\phi }(q,E),E).$$

We thus conclude that

$$H_{\phi }(h_{\phi }(q,E),E)=q,$$

or, equivalently,

$$h_{\phi }(H_{\phi }(t,E),E)=t.$$

The remaining part of the theorem stating that $p_{\phi }(·,E)$, $P_{\phi }(·,E)$, ${\delta }_{\phi }(·,E)$, and ${\Delta }_{\phi }(·,E)$ are lower bound affine is a consequence of the following classical lemma. □

Lemma 3.

(Affine lower bound) Let $f:I\to \mathbb{R}$ be a convex function on an interval I and $x_0$ an interior point in I. Then, there exists $a,b\in \mathbb{R}$, such that

$$f\left(x_0\right)=a{x}_0+b\mathrm{and}f\left(x\right)\ge ax+b,\forall x\in I.$$

4. Some Illustrative Examples

In this section, in addition to the computation of the new dimensions, we propose to confirm the last result in Theorem 4, which in fact concerns an important question in fractal analysis about the existence of fixed points for the dimension functions introduced and their affine asymptotic behavior.

4.1. Example 1

Take $\phi \left(x\right)=1-e^{-x}$. As we are concerned, in fact, with coverings and packings with radii decreasing to zero, we may notice that $\phi$ satisfies a local inequality:

$$c_{1}x\le \phi \left(x\right)\le c_{2}x,x\in (0,1).$$

In the special present case, we may take $c_1=1-e^{-1}$ and $c_2=1$. Standard computations lead to

$$h_{\phi }\left(q\right)=\dim E-q\mathrm{and}{H}_{\phi }\left(t\right)=\dim E-t,$$

$$p_{\phi }\left(q\right)=\mathrm{Dim}E-q\mathrm{and}{P}_{\phi }\left(t\right)=\mathrm{Dim}E-t,$$

and

$${\delta }_{\phi }\left(q\right)=\Delta E-q\mathrm{and}{\Delta }_{\phi }\left(t\right)=\Delta E-t,$$

where $\dim E$, $\mathrm{Dim}E$, and $\Delta E$ are, respectively, the Hausdorff dimension, the packing dimension, and the logarithmic index of E. More generally, if we take the function $\phi \left(x\right)=1-e^{-x^{\alpha }}$, for some $\alpha >0$, we obtain in this case

$${\displaystyle h_{\phi }\left(q\right)=\dim E-\alpha q\mathrm{and}{H}_{\phi }\left(t\right)=\frac{1}{\alpha }(\dim E-t),}$$

$${\displaystyle p_{\phi }\left(q\right)=\mathrm{Dim}E-\alpha q\mathrm{and}{P}_{\phi }\left(t\right)=\frac{1}{\alpha }(\mathrm{Dim}E-t),}$$

and

$${\displaystyle {\delta }_{\phi }\left(q\right)=\Delta E-\alpha q\mathrm{and}{\Delta }_{\phi }\left(t\right)=\frac{1}{\alpha }(\Delta E-t).}$$

Notice now that this simple example answers the interesting question already raised in fractal analysis about the existence of fixed points of the fractal dimensions’ functions. We notice here that

$$h_{\phi }\left({\displaystyle \frac{\dim E}{1+\alpha }}\right)=H_{\phi }\left({\displaystyle \frac{\dim E}{1+\alpha }}\right)={\displaystyle \frac{\dim E}{1+\alpha }},$$

$$p_{\phi }\left({\displaystyle \frac{\mathrm{Dim}E}{1+\alpha }}\right)=P_{\phi }\left({\displaystyle \frac{\mathrm{Dim}E}{1+\alpha }}\right)={\displaystyle \frac{\mathrm{Dim}E}{1+\alpha }},$$

and

$${\delta }_{\phi }\left({\displaystyle \frac{\Delta E}{1+\alpha }}\right)={\Delta }_{\phi }\left({\displaystyle \frac{\Delta E}{1+\alpha }}\right)={\displaystyle \frac{\Delta E}{1+\alpha }}.$$

4.2. Example 2

Take the particular gauge function $\phi$ inspired from [32,33,34] and defined by

$$\phi \left(r\right)=r^{\alpha }{(log|logr\left|\right)}^{\beta },$$

where $\alpha ,\beta >0$. We obtain the Hausdorff-type measure

$${\mathcal{H}}_{\phi }^{q,t}\left(E\right)=\underset{\eta \searrow 0}{lim}inf\left\{{\displaystyle \sum _i{(log|logr\left|\right)}^{\beta q}{r}_i^{\alpha q+t};x_i\in E\subset \bigcup _{i}B(x_i,r_i),r_i<\eta }\right\}.$$

In ref. [32,33,34,35], the authors constructed specific measure leading to a self-similar-type set, which supports the Hausdorff/packing-type measure. However, the computation of the fractal dimension remains ambiguous. Moreover, the only case that guarantees the support of these measures by the constructed self-similar-type set is the assumption that $\alpha +{\beta }_1=1$. This is due to the difficulty in computing the cut-off values of the fractal measures. In the present case, standard computations yield the result easily.

4.3. Example 3

In this example, we come back to a fascinating case investigated in [26]. Consider a measure $\mu$ supported on $[0,1]$, and with density

$$f_{\mu }\left(t\right)=-{\displaystyle \frac{\gamma {(logt)}^{\gamma -1}}{t}}{e}^{-{\left(\right|logt\left|\right)}^{\delta }}{X}_{]0,1[}\left(t\right)$$

where $\gamma >0$. It is shown that there is no possibility here to obtain level sets as in the existing cases. It will be, however, necessary to add an extra function or parameter or to involve a function in the Hausdorff-type measure that controls well the measure and, thus, obtain a context where the computation of the dimension is possible. A choice of $\phi \left(r\right)={\left(\right|logr\left|\right)}^{\gamma -1}$ may reduce the problem. In this example, the level sets considered in [23] did not allow for obtaining Gibbs or self-similar-type measures.

5. Some Final Remarks

In the present paper, a generalization of fractal dimensions is introduced based on gauge functions, extending classical concepts into a $\phi$-contractivity framework. The main idea is to remove the restriction that gauges must be set-valued, which could open new lines of research in fractal geometry and analysis. We thus develop new functional dimension measures, explore their properties, such as monotonicity, convexity, bijectivity, asymptotics, and give illustrative examples demonstrating fixed-point existence. The paper attempts to unify several strands of dimension theory.

By referring to Taylor’s definition of fractal sets, it holds that, for many situations, the spectrum due to the Hausdorff-type measure and the packing-type measure did not coincide (the Legendre transform of $b_{\mu }$ is different from that of $B_{\mu }$). This opens a question about the correct or general sense of being fractal, as this difference implies that the dimension functions are naturally different.

Examples provided here produce exact affine dimension functions for a specific choice of the gauges and thus open the question of when these dimensions are completely affine. For the moment, we guess that for gauges controlled by a diameter power, the dimensions could be exactly affine. Moreover, the iterated logarithms did not affect the dimension. In addition, from the applicability point of view, the framework’s flexibility is a limitation of the existing works as well as the present one, as the properties of the new dimensions depend heavily on the choice of the admissible gauge function. We intend that in the future a classification of signals, for example, could take place based on a same choice of the gauge. Developing numerical algorithms to estimate these dimensions would be a crucial next step. Fractal interpolations could be a good starting step to deduce the closest model to the data.

6. Conclusions

In this paper, we introduced new variants of the fractal measures and dimensions based on gauge functions. The variants introduced generalization of the classical cases due to Hausdorff and packing constructions and differ from the original case of Carathéodory-Billingsley by the fact that, unlike existing variants where the gauge is a set-valued function or a measure in the majority of cases, this restriction on gauges is removed, which may open new lines of research in fractal geometry and analysis. Under suitable hypotheses, we have shown that our variants answer some questions left open in the literature, such as the existence of fixed points of the dimension functions and their asymptotics. We intend for the results to contribute to the investigation of many complicated situations in fractal analysis/geometry where the estimation of fractal dimensions still need suitable techniques such as gauge functions to be estimated. Recall in this context that in the majority of cases, the existing works serve as self-similar-type objects such as Cantor-type sets to estimate the dimension. This is a major drawback as sets or functions may not be of this characteristic. Functions and sets may be related to natural sources or phenomena such as images, signals, or financial series, which are always characterized by hidden structures, and thus need more investigation. From the applied point of view, the crucial point remains how to chose the suitable gauge function to obtain an adequate description and comprehension of the data. Many techniques have been applied in this direction, such as interpolation with fractal functions to obtain a close model to the data, permitting the application of the suitable measures by the next. Finally, the paper gives the contribution within classical fractal dimension theory and may provide a useful tool for both theoretical mathematics and applied areas.