ON THE MUTUAL SINGULARITY OF MULTIFRACTAL MEASURES

. The aim of this article is to show that the multifractal Hausdorﬀ and packing measures are mutually singular, which in particular provides an answer to Olsen’s questions.


Introduction
The notion of singularity exponents or spectrum and generalized dimensions are the major components of the multifractal analysis. They were introduced with a view of characterizing the geometry of measure and to be linked with the multifractal spectrum which is the map which affects the Hausdorff or packing dimension of the iso-Hölder set  for a given α ≥ 0 and supp µ is the topological support of probability measure µ on R n , B(x, r) is the closed ball of center x and radius r. It unifies the multifractal spectra to the multifractal Hausdorff (packing) function b µ (q) (B µ (q)) via the Legendre transform [3,6,7], i.e., f µ (α) := dim H E(α) = inf q∈R qα + b µ (q) or F µ (α) := dim P E(α) = inf q∈R qα + B µ (q) .
In the last decay, there has been a great interest in understanding the fractal dimensions of the iso-Hölder sets and measures. In the following we aim to introduce the general tools that will be applied next. We will review in brief the notion of multifractal Hausdorff and packing measures already introduced in [6]. The key ideas behind the fine multifractal formalism in [6] are certain measures of Hausdorff-packing type which are tailored to see only the multifractal decomposition sets E(α). These measures are natural multifractal generalizations of the centered Hausdorff measure and the packing measure and are motivated by the τ µ -function which appears in the multifractal formalism. We first recall the definition of the multifractal Hausdorff measure and the multifractal packing measure. We start by introducing the multifractal Hausdorff and packing measures. Let µ be a compactly supported probability measure on R n . For q, t ∈ R, E ⊆ R n and δ > 0, we define where the supremum is taken over all centered δ-packing of E. Moreover we can set P q,t µ,δ (∅) = 0. The packing pre-measure is then given by In a similar way, we define where the infinimum is taken over all centered δ-covering of E. Moreover we can set H q,t µ,δ (∅) = 0. The Hausdorff pre-measure is defined by Especially, we have the conventions 0 q = ∞ for q ≤ 0 and 0 q = 0 for q > 0. H q,t µ is σ-subadditive but not increasing and P q,t µ is increasing but not σsubadditive. That's why Olsen introduced the following modifications on the multifractal Hausdorff and packing measures H q,t µ and P q,t µ , In follows that H q,t µ and P q,t µ are metric outer measures and thus measures on the Borel family of subsets of R n . An important feature of the Hausdorff and packing measures is that P q,t µ ≤ P q,t µ . Moreover, there exists an integer ξ ∈ N, such that H q,t µ ≤ ξP q,t µ . The measure H q,t µ is a multifractal generalization of the centered Hausdorff measure, whereas P q,t µ is a multifractal generalization of the packing measure. In fact, it is easily seen that if t ≥ 0, then H 0,t µ = H t and P 0,t µ = P t , where H t denotes the t-dimensional centered Hausdorff measure and P t denotes the t-dimensional packing measure.
The measures H q,t µ and P q,t µ and the pre-measure P q,t µ assign in the usual way a multifractal dimension to each subset E of R n . They are respectively denoted by b q µ (E), B q µ (E) and Λ q µ (E) and satisfy is an obvious multifractal analogue of the Hausdorff dimension dim H (E) of E whereas B q µ (E) and Λ q µ (E) are obvious multifractal analogues of the packing dimension dim P (E) and the pre-packing dimension ∆(E) of E respectively. In fact, it follows immediately from the definitions that dim H (E) = b 0 µ (E), dim P (E) = B 0 µ (E) and ∆(E) = Λ 0 µ (E). Next, for q ∈ R, we define the separator functions b µ , B µ and Λ µ by b µ (q) = b q µ supp µ , B µ (q) = B q µ supp µ and Λ µ (q) = Λ q µ supp µ .
It is well known that the functions b µ , B µ and Λ µ are decreasing and B µ , Λ µ are convex and satisfying The multifractal formalism based on the measures H q,t µ and P q,t µ and the dimension functions b µ , B µ and Λ µ provides a natural, unifying and very general multifractal theory which includes all the hitherto introduced multifractal parameters, i.e., the multifractal spectra functions f µ and F µ , the multifractal box dimensions. The dimension functions b µ and B µ are intimately related to the spectra functions f µ and F µ , whereas the dimension function Λ µ is closely related to the upper box spectrum (more precisely, to the upper multifractal box dimension function C µ , see [6, Propositions 2.19 and 2.22]).
It should be noted that 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 specification 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. The multifractal and the fractal analysis allows one to perform a certain classification of these measures. Therefore, Olsen in [6, Questions 7.1 and 7.2], posed the following two questions: Let p, q ∈ R.
(1) Assume that b µ is differentiable at p and q with b µ (p) = b µ (q). Then, the following problem remains open: (2) Assume that B µ is differentiable at p and q with B µ (p) = B µ (q). Then, the following problem remains open: The aim of this paper is to focus on the above questions relying on these multifractal measures and functions. More precisely, we study the mutual singularity of multifractal Hausdorff and packing measures on the homogeneous Moran sets and this result completely differ to Olsen's main theorems [6, Theorems 5.1 and 6.1] which are based on graph directed self-similar measures in R n with totally disconnected support, cookie-cutter measures and self-similar measures satisfying the significantly weaker open set condition [4,5].

Main result
Before we set our main result, let us recall the class of homogeneous Moran sets. We denote by {n k } k≥1 a sequence of positive integers and {c k } k≥1 a sequence of positive numbers satisfying n k ≥ 2, 0 < c k < 1, n k c k ≤ 1 for k ≥ 1. Let D 0 = ∅, and for any k ≥ 1, set ,m , we denote σ * τ = (σ 1 , . . . , σ k , τ 1 , . . . , τ m ) . (b) For all k ≥ 0 and σ ∈ D k , J σ * 1 , J σ * 2 , . . . , J σ * n k+1 are subintervals of J σ , and satisfy that J Let Let A = {a, b} be a two-letter alphabet, and A * the free monoid generated by A. Let F be the homomorphism on A * , defined by F (a) = ab and F (b) = a. It is easy to see that F n (a) = F n−1 (a)F n−2 (a). We denote by |F n (a)| the length of the word F n (a), thus F n (a) = s 1 s 2 · · · s |F n (a)| , s i ∈ A.
Let µ be a mass distribution on E, such that for any σ ∈ D k , Now we define an auxiliary function β(q) as follows: For each q ∈ R and k ≥ 1, there is a unique number β k (q) such that By a simple calculation, we get Clearly, for any k 1 we have β k (1) = 0. Thus β k (q) < 0 for all q and β k (q) is a strictly decreasing function. Our auxiliary function is where η 2 + η = 1. The function β is strictly decreasing and differentiable at q, lim q→∓∞ β(q) = ±∞ and β(1) = 0. Note that in [7,Theorem B] it is shown that the dimension of the level sets of the local Hölder exponent E(−β (q)) is given by dim H E(−β (q)) = dim P E(−β (q)) = −qβ (q) + β(q).
Definition 2.2. Let µ, ν be two Borel probability measures on R n . µ and ν are said to be mutually singular and we write µ ⊥ ν if there exists a set A ⊂ R n , such that µ(A) = 0 = ν(R n \ A).
In the following we show that the Olsen's multifractal Hausdorff and packing are mutually singular, which in particular provides an answer to Olsen's questions [6, Questions 7.1 and 7.2]. Theorem 2.3. Suppose that E is a homogeneous Moran set satisfying (SSC) and µ is the Moran measure on E. Then, for all p, q ∈ R where β (p) = β (q) we have Remark 2. The results of Theorem 2.3 hold if we replace the multifractal Hausdorff and packing measures by the multifractal Hewitt-Stromberg measures (see [1,2] for the precise definitions).

Proof of the main result
In this section, we give a proof of the main theorem. Given q ∈ R, it follows from [7, Proposition 3.1] that there exists a probability measure ν q supported by E such that for any k ≥ 1 and σ 0 ∈ D k , However, in [7] it is shown that which implies that ν q E − β (q) = 1. We therefore infer that if p, q ∈ R with β (p) = β (q), then We now prove the following three claims.
Proof of Claim 1. By a simple calculation, we can get β(q) − β k (q) = O( 1 k ). Then, The proof of the lim inf is identical to the proof of the statement in the first part and is therefore omitted.
Then we have If β(q) 0, then If β(q) < 0, then thus we deduce that And this gives us Which leads to the following inequality where k 1 is a suitable constant. If q < 0, it follows from (2) that Since the measure µ satisfies the doubling condition (see [7,Proposition 3.2]) then for all q ≥ 0, there exists a constant A > 0 such that It follows from (5) and (6) that there exists a constant C 2 such that Now combining (3), (4) and (7) shows that Finally, this yields Claim 3. There exists a constant K > 0 such that for any q ∈ R P q,β(q) µ (E) ≤ Kν q (E).
Let B(x i , r i ) i∈N be a centered δ-packing of F . For each i choose σ(i) ∈ D n , for any n 1 such that x i ∈ J σ(i) . For each i ∈ N choose k i , i ∈ N such that J σ(i)|ki+1 r i < J σ(i)|ki and ∆ J σ(i)| i+1 r i < ∆ J σ(i)| i .
This complete the proof of Claim 3.
Proof of Theorem 2.3. It follows from Claim 2 and Claim 3 and since µ satisfies the doubling condition that K ν q ≤ H q,β(q)