On the vectorial multifractal analysis in a metric space

: Multifractal analysis is typically used to describe objects possessing some type of scale invariance. During the last few decades, multifractal analysis has shown results of outstanding signiﬁcance in theory and applications. In particular, it is widely used to characterize the geometry of the singularity of a measure µ or to study the time series, which has become an important tool for the study of several natural phenomena. In this paper, we investigate a more general level set studied in multifractal analysis. We use functions deﬁned on balls in a metric space and that are Banach valued which is more general than measures used in the classical multifractal analysis. This is done by investigating Peyri`ere’s multifractal Hausdor ﬀ and packing measures to study a relative vectorial multifractal formalism. This leads to results on the simultaneous behavior of possibly many branching random walks or many local H¨older exponents. As an application, we study the relative multifractal binomial measure in symbolic space ∂ A .


Introduction
The concept of multifractal analysis was developed around 1980, following the work of B. Mandelbrot, when he studied the multiplicative cascades for energy dissipation in the context of turbulence [24,25]. Since then, it has been developed rapidly and discussed by several authors, emphasizing its importance in the study of local properties of functions and measures. In particular, the multifractal spectrum provides a characterization in terms of the geometric properties of the singularities of a distribution. More precisely, let X : R d −→ R be a signal; the multifractal analysis is a processing method that allows the examination of X by using the characteristics of its pointwise regularity, which are measured by α X (x), i.e., the exponent of pointwise regularity. This is done by using the multifractal spectrum, which is the Hausdorff dimension of the set of locations where the function α X (x) is distributed, to characterize the set of x such that α X (x) = α. Specifically, consider the set which gives a geometric and global account of the variations in X's regularity along x. Usually, we use the Hurst exponent H as a quantification of the degree of self-similarity of the time series which is directly correlated with the fractal dimension D and describes the complexity of the signals. A higher value of D indicates a higher irregularity of the signals: D = 2 − H [11,18]. In the last few decades, multifractal analysis has become a powerful tool to study the time series which has become an important tool for the study of several natural phenomena. In fact, such series present complex statistical fluctuations that are associated with long-range correlations between the dynamical variables present in these series, and which obey the behavior usually described by the decay of the fractal power law. This theory in time series was first introduced by B. Mandelbrot in [21][22][23] including early approaches by Hurst and colleagues [18,19]. Since then, fractal and multifractal scaling behavior has been reported in many natural time series generated by complex systems, including medical and physiological time series especially recordings of the heartbeat, respiration, blood pressure wind speed, seismic events, etc.
Recall the set E(α) given in (1.1) and consider, for n ≥ 1, the dyadic interval I n (k) = [(k−1)2 −n , k2 −n ] with 1 ≤ k ≤ 2 n and with length |I n (k)| = 2 −n . In fact, there are various definitions of the exponent α: α = lim n→∞ log A X (I n (k)) log |I n (k)| , where A X (I n (k)) may be chosen to be the wavelet-leaders L X (I n (k)) or the oscillation Osc X (I n (k)) of X over the interval I n (k) [20]. Therefore, it is interesting to introduce the local dimension of a probability measure µ at a point x: as well as the set E µ (α) = x ∈ R d ; dim loc (x, µ) = α , where B(x, r) stands for the closed ball of center x and radius r and α ≥ 0. In the beginning, the multifractal formalism used "boxes", or in other terms took place in a totally disconnected metric space. To get rid of these boxes and have a formalism meaningful in geometric measure theory, Olsen [27] introduced a formalism which is now commonly used. Especially, we compute the Hausdorff multifractal spectrum function f µ defined as where dim denotes the Hausdorff dimension. To this end, multifractal analysis can be considered as another way to describe the local properties of time series. Since then, numerous writers have looked at these measures, stressing their significance for the study of local fractal properties and fractal products [5-7, 13-16, 26]. Moreover, the developments of this field showed that getting a valid variant of the multifractal formalism does not require the application of radius power-laws equivalent measures. This leads one to think about a general framework wherein the restriction of the vector-valued function on balls may be any vector-valued function ξ(B(x, r)) which is not equivalent to power-laws r α and develops a general multifractal analysis. In particular, and in another context, to overcome the problem of being a nondoubling, non-Hölderian measure, Cole, in [10] proposed to control the analyzed measure µ by another suitable measure ν via a relative multifractal analysis of the relative singularity sets. More specifically, he calculated, for α ≥ 0, the size of the set where supp µ is the topological support of the measure µ. For this, he introduced a generalized Hausdorff and packing measures denoted by H q,s µ,ν and P q,s µ,ν respectively. One can emphasize the duality by replacing R d by a general metric space (X, d) and then replacing the diameter by a more general function defined on balls in X and analyzing functions defined on balls which are more general than measures. More precisely, let E be a separable real Banach space, whose dual is denoted by E and the form of the duality , . We denote by B(X) the set of closed balls on X. We consider the functions such that, for all x ∈ X, one has that lim r→0 ξ(B(x, r)) = +∞. For α ∈ E , we consider the set where χ = (κ, ξ). The set X χ (α) may be thought of as the set of points x such that κ(x,r) ξ(x,r) tends to α in the sense of topology σ(E, E ) when r tends to 0. In [28], Peyrière introduced vectorial Hausdorff and packing measures denoted by H q,t χ and P q,t χ respectively. He defined, in a natural way, the Hausdorff and packing dimensions denoted respectively as dim q χ and Dim q χ . In particular, if κ = 0 then dim q χ will be denoted by dim ξ and Dim q χ will be denoted by Dim ξ . In fact, such measures are appropriate for the study of a general formalism by relating dim ξ X χ (α) and Dim ξ X χ (α) to the Legendre transform of the multifractal Hausdorff and packing functions denoted respectively by b χ and B χ (see Section 2 for the definition).
The purpose of this paper is to study the Hausdorff and packing dimensions of the set X χ (α). In fact, it is difficult to compute these dimensions in general, but we can compute a lower bound of the Hausdorff and packing dimensions of this level set. Indeed, we can decompose the set X χ (α) and calculate the size of the subset of X χ (α) whose points satisfy that lim r→0 ξ(x, r) − log r = β. Inspired by [4,10,29], we define α ∈ E and β ≥ 0 ; then the set is given as This article is organized as follows. The next section is devoted to recalling the definitions of the various multifractal dimensions and measures investigated in the paper. In Section 3, we will state and prove our main results concerning the study of Hausdorff and packing dimensions of the set X χ (α, β). In general settings, we have that dim X χ (α, β) Dim X χ (α, β); for this, we give in Section 4 a sufficient condition so that we have the equality. In this case, we say that the relative multifractal formalism holds. As an application, we study the validity of the relative multifractal formalism for the binomial measure in symbolic space ∂A.

Vectorial multifractal measures and dimensions
In this section, we recall the multifractal Hausdorff and packing measures introduced in [28]. We assume throughout this paper that X is a separable metric space verifying the Besicovitch covering property [8,9]. We define B(x, r) := y ∈ X; d(x, y) ≤ r , i.e., the closed ball with center x ∈ X and radius r > 0. We denote by B(X) the set of closed balls on X.
Let ξ : B(X) −→ R be an application such that, for all x ∈ X, one has that lim r→0 ξ(B(x, r)) = +∞. Such a function will be called a valuation on X and we will write that ξ(x, r) = ξ(B(x, r)), for simplicity. When such a valuation is given, one sets We consider the function κ : X × R + −→ E . We denote by , the duality bracket between E and E . Let A ⊆ X, t ∈ R, q ∈ E, χ = (κ, ξ) and δ > 0; we write where the infimum is taken over all families Now H q,t χ is a metric outer measure. In addition, the function t −→ H q,t χ (A) is non-decreasing; nevertheless, it is so if A is a subset of one of the X n . This is why one more step is needed in the construction. We write H q,t Similarly, multifractal packing measures are defined as where the supremum is taken over all families The functions P q,t χ and P q,t χ are metric outer measures. Furthermore, we may prove using the well known Besicovitch covering theorem that there exists an integer θ ∈ N such that The measures H q,t χ and P q,t χ assign in the usual way a multifractal dimension to each subset A of X. They are respectively denoted by dim q χ (A) and Dim q χ (A). More precisely, we have One also defines ∆ q χ , which generalizes the Minkowski-Bouligand dimension; for a bounded set A, one sets As a direct consequence of the definition, the dimensions defined above satisfy that dim q . Moreover, for κ = 0, the functions H q,t χ , P q,t χ and P q,t χ will be denoted respectively by H t ξ , P t ξ and P t ξ ; then, we will write Remark 1. In the special case where κ = 0 and ξ(x, r) = − log r, we come back to the classical definitions of the Hausdorff and packing measures and dimensions in their original forms [27]. In particular, we get Moreover, it is well known [28] that Λ χ and B χ are convex and b χ ≤ B χ ≤ Λ χ .

Example: Homogeneous tree
Let b ≥ 2 and consider the set A * = k≥0 A k as a free monoid consisting of words on A = The empty word ε is the identity element and it is convenient to set A 0 = { }.
The concatenation of two words u and v will be simply denoted by a juxtaposition, that is the word. The length of the word u is denoted by |u|. Moreover, we may define an order " ≺ " on A * : if a word v is a prefix of the word u, we write v ≺ u. The set of infinite sequences of elements of A will be denoted by ∂A. We identify u ∈ A * with the cylinder [u] := {x ∈ ∂A, u ≺ x}. We define an ultrametric distance on ∂A by where u ∧ v stands for their largest common prefix. In this example, we consider X to be the space ∂A and χ = (κ, ξ) defined in (1.2) such that χ constitutes functions defined on the cylinder. Let δ > 0; A is a bounded subset of X. We set where the supremum is taken over by the collection of δ-packings Lemma 1. Let q ∈ E, t ∈ R and k ≥ 1. If ξ is normal, then we have the following as required.
(2) Since, for all n ≥ 1, P * q, t . Now, suppose that ∆ q χ (A)> 0. Let t and be two positive numbers such that 0 < t − < t < ∆ q χ (E). Therefore, P q,t χ (A) = +∞. We define recursively a sequence {η m } m≥0 . First, η 0 = b −k 0 ρ, where ρ given by the normality of ξ and k 0 is chosen so that η 0 ≤ 1/n. Suppose that η m has been defined. Then, there exists an (η m /b)-packing of A ∩ X n with There exists a positive integer k ≥ 1 such that Then we set η m+1 = b −k η m . It follows that Proposition 1. Let q ∈ E, t ∈ R and k ≥ 1. If the valuation ξ is normal, then we have It follows that and, then P * q, t χ < ∞. This implies that Λ χ (q) ≤ f (q). On the other hand, assume that t < f (q); then, there exists a sequence (k m ) m≥1 such that It follows that and then P * q, t χ > 0. This implies that Λ χ (q) ≥ f (q) as required.

Main results
Multifractal analysis is typically used to describe objects possessing some type of scale invariance. The investigation has focused on structures produced by one mechanism which were analyzed with respect to the ordinary volume or metric. The most imported examples include branching random walk and self-similar measures [1,2,27]. In particular, the multifractal spectrum provides a characterization of the singularities of a distribution in terms of the geometrical properties. Unfortunately, we may obtain identical spectra despite having strikingly different measures. For this, we will study a more general level set. More precisely, let (X, d) be a separable metric space verifying the Besicovitch covering property; E is the dual of a separable real Banach space E and χ = (κ, ξ) such that κ and ξ satisfy (1.2). For α ∈ E and β ≥ 0, we recall the set In this section, we will state our main results concerning the estimation of the Hausdorff and packing dimensions of the set X χ (α) by using the Legendre transform of the multifractal Hausdorff and packing functions, where the Legendre transform of a real valued function f : More precisely, we have the following results.
The most common example in this context is considered when we study the multifractal measure µ with respect to arbitrary measure ν. More precisely, take κ(x, r) = − log µ(B(x, r)) and ξ(x, r) = − log ν(B(x, r)), where µ and ν are two Borel measures defined in the metric space X. The major interest of this is to use a partition of the space in sets of equal ν measures instead of equal size (when considering the diameter). In [10] the author formalizes the idea of performing multifractal analysis with respect to an arbitrary reference measure by developing a formalism for the multifractal analysis of one measure with respect to another. This formalism is based on the ideas of the 'multifractal formalism' as first introduced by Halsey et al. [17], and closely parallels Olsen's formal treatment of this formalism in [27]. The Hausdorff and packing dimensions of X χ (α) are fully carried by some subset X χ (α, β).
The following corollary provides us with a sufficient condition that gives the lower bound for the Hausdorff and packing dimensions of X χ (α).

Corollary B.
(1) Assume that, at some point q, the multifractal function b χ is convex and differentiable. Set α = −b χ (q) and (2) Assume that, at some point q, the multifractal function B χ is differentiable. Set α = −B χ (q) and Remark 3. It is not difficult to observe that the second assertion of the preview corollary remains true when we consider Λ χ instead of B χ . In particular, let α = −Λ χ (q) and Then, provided that Λ * χ (α) ≥ 0, we have that dim X χ (α) = Dim X χ (α) ≥ sup β∈ I β Λ * χ (α). In the following example, we will consider a special case when the function Λ χ is differentiable. This fact will be used in Section 4. Example 1. In this example, we will use the same notation as in Section 2.2. Let X = ∂A, E be the Euclidean space R N and (p i, j ) 0≤ j<b 1≤i≤N be a family of positive numbers. Define the recurrence p i,u for given i and u ∈ A * : p i, = 1 and p i,u j = p i,u p i, j .
It follows that the sequence is geometric; then, using Remark 2, which is clearly differentiable.

Upper bound of Hausdorff and packing dimensions
Let A ⊆ E, α ∈ E and β ≥ 0; we define The sets X χ (α, β; E) and X χ (α, β; E) will simply be denoted by X χ (α, β) and X χ (α, β) respectively. We will be interested in the set Theorem 1. For α ∈ E and β ≥ 0, we have the following: Proof. This theorem follows immediately from the following lemma.
(2) If q, α + B χ (q) ≥ 0, then Proof. It is clear that we only have to consider the case when the set A = {q}. Let n and m be two positive integers such that m ≥ n, q ∈ E, t ∈ R and ε 1 and ε 2 are two positive numbers such that ε 1 ≤ q, α + t and ε 2 ≤ β q, α + t − ε 1 .

Lower bound of Hausdorff and packing dimensions
Let v, q ∈ E and assume that B ξ,κ (q) < ∞. We define t .
We will denote by B χ (q) (as an element of E ) the derivative of B χ at q when it exists. When B χ has a partial derivative at point q along the direction v, one has that Assume that the function v −→ ∂ v B χ (q) is lower semi-continuous; then, from [28, Proposition 10] and (2.1), one gets that P (2) Set α = −B χ (q); then, for each Borel set E ⊆ X χ (α, β) ∩ X n , we have Proof.
(1) For m ≥ n, we consider the set Given n and a subset F of A m , let (B(x i , r i )) i a centered δ-covering of F with 0 < δ < min{1/n, 1/m}. We have that Then, for δ ≤ min{1/n, 1/m}, we have that H which gives that tH (2) For m ≥ n, consider Given n and a subset F of A m , 0 < δ < 1 m and let (B(x i , r i )) i be a δ-packing of F. Then, we have that Putting these together we see that (F). Now, let (A i ) i be a covering of A m . Therefore, we have It follows that We can deduce now that P As mentioned above, in the last decay, there has been a great interest in the validity and non-validity of the multifractal formalism. Many positive results have been written in various situations. What follows, we state a sufficient condition so that we obtain the validity of the multifractal formalism. This result will be used to study the binomial measure in symbolic space ∂A.

Application
In this section, we will consider a special case when κ and ξ are two functions defined by using binomial measures. In this situation, we are able to construct an auxiliary measure µ q so that we obtain the validity of the relative multifractal formalism, that is dim X χ (α, β) = Dim X χ (α, β) .
Observe that, for all k ≥ 1, we have We define, for each q ∈ R, the measure µ q on ∂A by µ q ([ ]) = ∅ and µ q ([u]) = p q u ω τ(q) u (4.2) for all u ∈ A * .

Use of AI tools declaration
The authors declare that they have not used artificial intelligence tools in the creation of this article.