The mutual singularity of the relative multifractal measures


 M. Das proved that the relative multifractal measures are mutually singular for the self-similar measures satisfying the significantly weaker open set condition. The aim of this paper is to show that these measures are mutually singular in a more general framework. As examples, we apply our main results to quasi-Bernoulli measures.


Introduction and statements of results
In [4], Billingsley applies methods from ergodic theory to calculate the size of a level sets of the local dimension of µ with respect to another measure ν. Cajar [5] also has studies these sets in the code space. Anyone familiar with multifractal analysis will recognize this as a form of multifractal analysis. In several recent papers on multifractal analysis this type of multifractal analysis has re-emerged as mathematicians and physicists have begun to discuss the idea of performing multifractal analysis with respect to an arbitrary reference measure. Cole [6] has formalised these ideas by introducing a relative formalism for the multifractal analysis of one measure with respect to another. This formalism is based on the ideas of the "multifractal formalism" as clari ed by Olsen [15]. Later, in [1,3,10,21], Selmi et al. justi ed the relative multifractal formalism under less restrictive hypotheses. Other studies have been developed in the same direction such as [2,9,19,20,22,24]. The purpose of this paper is to show that the relative multifractal Hausdor and packing measures are mutually singular. The purpose of this paper is to study the multifractal structure of measures using the formalism introduced in [6].
We rst, let us recall the multifractal formalism introduced by Cole in [6]. Let µ and ν be two probability measures on a metric space R n . For q, t ∈ R, E ⊆ R n and δ > , write where the supremum is taken over all centered δ-packing of E. Moreover we can set P q,t µ,ν,δ (∅) = . Also, we de ne where the in nimum is taken over all centered δ-covering of E. Moreover we can set H q,t µ,ν,δ (∅) = . Especially, we have the conventions q = ∞ for q ≤ and q = for q > .
The functions H q,t µ,ν and P q,t µ,ν are metric outer measures and thus measures on the Borel family of subsets of R n . It is easy to see that P q,t µ,ν ≤ P q,t µ,ν . Moreover, by using Besicovitch's theorem, there exists an integer ξ ∈ N, such that H q,t µ,ν ≤ ξ P q,t µ,ν (see [15]). The measure H q,t µ,ν is a multifractal generalization of the centered Hausdor measure, whereas P q,t µ,ν is a multifractal generalization of the packing measure. In fact, in the case when t ≥ and ν is the Lebesgue measure L n on R n , H ,t µ,ν = H t and P ,t µ,ν = P t , where H t denotes the t-dimensional centered Hausdor measure and P t denotes the t-dimensional packing measure.
The measures H q,t µ,ν and P q,t µ,ν and the pre-measure P q,t µ,ν assign in the usual way a multifractal dimension to each subset E of R n . They are respectively denoted by b q µ,ν (E), B q µ,ν (E) and Λ q µ,ν (E) and satisfy The number b q µ,ν (E) and B q µ,ν (E) are obvious multifractal analogues of the ν-Hausdor dimension dimν(E) and the ν-packing dimension Dimν(E) of E respectively. In fact, it follows immediately from the de nitions that dimν(E) = b µ,ν (E) and Dimν(E) = B µ,ν (E).
Relative multifractal analysis is a natural framework to nely describe geometrically the heterogeneity in the distribution at small scales of the elements of compactly supported Borel positive and nite measures on R n . Speci cally, this heterogeneity can be described via the lower and upper local dimensions of a measure µ with respect to an arbitrary probability measure ν, namely If α µ,ν (x) = αµ,ν(x), we refer to the common value as the local dimension of µ with respect to ν at x, and we denote it by αµ,ν(x). For α ≥ , let us introduce the fractal sets which are also very natural, and the most studied in the literature, Inspired by the observations made by physicists of turbulence and statistical mechanics, mathematicians derived, and in many situations justi ed the heuristic claiming that for a measure possessing a self-conformal like property, its Hausdor spectrum should be obtained as the Legendre transform of a kind of free energy function called L q -spectrum. This gave birth to the abundant literature on the so-called relative multifractal formalisms, which aim at linking the asymptotic statistical properties of a given measure with its ne geometric properties. One of the main importance of the relative multifractal measures H q,t µ,ν and P q,t µ,ν , and the corresponding dimension functions bµ,ν , Bµ,ν, and Λµ,ν is due to the fact that the ν-multifractal spectra functions dimν and Dimν are bounded above by the Legendre transforms of bµ,ν and Bµ,ν, respectively, i.e., These inequalities may be viewed as rigorous versions of the multifractal formalism. Furthermore, for many natural families of measures we have The interest of mathematicians in singularly continuous measures and probability distributions were fairly weak, which can be explained, on the one hand, by the absence of adequate analytic apparatus for speci cation and investigation of these measures, and, on the other hand, by a widespread opinion about the absence of applications of these measures. Due to the fractal explosion and a deep connection between the theory of fractals and singular measures, the situation has radically changed in the last years. It was proved that singular distributions of probabilities are dominant for many classes of random variables. Possible applications in the spectral theory of self-adjoint operators serve as an additional stimulus for a further investigation of singularly continuous measures. The authors in [11,12,15,23] provided some examples of the mutual singularity of multifractal Hausdor and packing measures for graph directed self-similar measures in R n with totally disconnected support, cookie-cutter measures [15], for some homogeneous Moran measures [11,12], in the spacial case where ν is the Lebesgue measure L n . Also, in [8,9], M. Das proved that the relative multifractal Hausdor and packing measures are mutually singular for the self-similar measures satisfying the signi cantly weaker open set condition. The aim of this article is to show that the relative multifractal Hausdor and packing measures are mutually singular in a more general setting. The results in this paper generalize many known results and in particular provides a positive answer to Olsen's questions. Our main results apply to quasi-Bernoulli measures.
These more general results are stated as follows:  ,ν(q)), i.e., the existence of a measure νq on supp µ ∩ supp ν and constants C, C > with C = C − and δ > such that for every x ∈ supp µ ∩ supp ν and every < r < δ, Proof. Let us prove the rst inequality. The proof of the second statement is identical to the proof of the statement in the rst inequality and is therefore omitted. Write a = D+bµ,ν(q). Fix ε, η > and let Then it is su cient to prove that H q,bµ,ν(q) µ,ν (E) = . It follows from the convexity of bµ,ν(q) that there exists h > such that bµ,ν(q + h) − bµ,ν(q) h < a + ε and thus bµ,ν(q + h) < bµ,ν(q) + h(a + ε).
(2.1) Also observe that for each x ∈ E, there exists rx > such that

µ(B(x, r)) q ν(B(x, r)) bµ,ν(q) ≤ η −h µ(B(x, r)) q+h ν(B(x, r)) bµ,ν(q)+h(a+ε)
2. We present some tools, as well as lemmas, which will be used in the proof of our main result. Lemma 2. For any q ∈ R, we have Proof. Fix δ > and let B (x i , r i ) i∈N be a centered δ-covering of K. One gets When δ tend to 0, the set B(F, δ) decreases to F. Then for all ε > , we can choose δ > satisfying Fix δ > and let B(x i , r i ) i be a centered δ-packing of F. Then, one has Which leads to As ε tend to , we can conclude that which proves the desired result.
Now, let us prove Assertion (2). By using Lemmas 2 and 3, we have It results that Since νq is the Gibbs measure for (µ, ν) at (q, Bµ,ν(q)) and if we assume that Bµ,ν is di erentiable, by similar technics in [15,16,21], we have This implies that νq E(−B ′ µ,ν (q)) = . We therefore infer that if p, q ∈ R with B ′ µ,ν (p) ≠ B ′ µ,ν (q), then This completes the proof of Assertion (2). 3. The proof of Assertion (3) is identical to the proof of the statement in the second assertion and is therefore omitted.

An example
Let F = ∪ n≥ Fn such that Fn stands for a sequence of the 5-adic intervals. If x belongs to [ , [, In(x) stands for the interval Fn which contains x. Now, considering I = Iε ε ···εn and J = I ε ′ ε ′ ···ε ′ p , we set A probability measure on [ , [ is said to be quasi-Bernoulli if there exists C > such that, for any I, J ∈ F , one has We say that the quasi-Bernoulli measure µ has a strong separation condition if Throughout this section, we assume that both µ and ν are two quasi-Bernoulli measures that have the above strong separation condition. For any q, t ∈ R, one de nes where the star means that the terms for which µ(I j ) = and ν(I j ) = are removed, and let τµ,ν(q) = sup t ∈ R; Kµ,ν(q, t) = +∞ .
In the next lemma we investigate the relationship between the multifractal functions τµ,ν(q) and Λµ,ν(q).
Let δ > , t > τµ,ν(q) and B(x j , r j ) j be a δ-packing of K. Fix j, since x j ∈ K, there exists n j ∈ N * such that n j ≤ r j < n j − , which implies that Also, each B(x j , r j ) is covered by at most three 5-adic intervals I j , I n j − (x j ), I j . Moreover, the strong separation condition ensures that B(x j , r j ) ⊆ I n j − (x j ).
From the construction of measures µ and ν that, there exists C, C ′ > such that µ(B(x j , r j )) ≤ C µ(In j (x j )) and ν(B(x j , r j )) ≤ C ′ ν(In j (x j )).
Now we will prove the other inequality. For δ > , let (I j ) j be a δ-packing of [ , [ and t > Λµ,ν(q). For any j there exists n j ∈ N * such that I j ∈ Fn j , x j ∈ I j ∩ K, I j ⊆ B(x j , −n j ), µ(I j ) = µ(B(x j , −n j )) and ν(I j ) = ν(B(x j , −n j )).
The strong separation condition implies that B(x j , −n j ) j is a δ-packing of K. Then, one has and we therefore deduce that sup j µ(I j ) q ν(I j ) t ≤ P q,t µ,ν,δ (K) and Kµ,ν(q, t) ≤ P q,t µ,ν (K) < +∞.
2. The interesting case is, of course, the case where the measure ν is di erent from the normalized Lebesgue measure L n on an open and bounded set containing the support of µ. If ν is the normalized Lebesgue measure L n then our main results follow immediately from the (substantially more general) theorems in [11,12,15,23] (provided that certain conditions are satis ed). 3. All the above results hold if we replace the centered δ−coverings (δ−packings) by the centered ν − δ−coverings (ν − δ−packings) and we suppose that the measure ν satis es the following condition For any < λ < given, there exists δ > , such that if ν (B(x, r)) ≤ δ for every x ∈ supp ν, then r ≤ λ.
Note that this assumption is not restrictive, as it encompasses a fairly broad class of measures, namely: quasi-Bernoulli measures, inhomogeneous Bernoulli measures and homogeneous Moran measures, etc. The reader is referred to [14] for a systematic discussion of these measures. 4. Let (X, d) be a metric space and B stand for the set of balls of X and F for the set of maps from B to [ , +∞). The set of µ ∈ F such that µ(B) = implies µ(B ′ ) = for all B ′ ⊆ B will be denoted by F * . For such a µ, one de nes its support supp µ to be the complement of the set {B ∈ B µ(B) = }. Then, all the above results hold for any µ ∈ F * . 5. Our main results in Theorem 1 also hold for the vectorial multifractal measures introduced by Peyrière in [18], the φ-Mixed multifractal measures introduced in [13] and the relative multifractal Hausdor measure and the multifractal packing measure in a probability space [7].
Data Availability Statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.