A multifractal formalism for Hewitt-Stromberg measures
aa r X i v : . [ m a t h . D S ] O c t A MULTIFRACTAL FORMALISM FOR HEWITT-STROMBERG MEASURES
NAJMEDDINE ATTIA, BILEL SELMIA
BSTRACT . In the present work, we give a new multifractal formalism for which the classical multifractal formalismdoes not hold. We precisely introduce and study a multifractal formalism based on the Hewitt-Stromberg measures andthat this formalism is completely parallel to Olsen’s multifractal formalism which based on the Hausdorff and packingmeasures.
1. I
NTRODUCTION
In certain circumstances, a measure µ gives rise to sets of points where µ has local density of exponent α . Thedimensions of these sets indicate the distribution of the singularities of the measure. To be more precise, for a finitemeasure µ on R n , the pointwise dimension at x is defined as follows α µ ( x ) = lim r → log µ ( B ( x, r ))log r , whenever this limit exists. For α ≥ , define E ( α ) = n x ∈ supp µ (cid:12)(cid:12) α µ ( x ) = α o where B ( x, r ) is the closed ball with center x and radius r . The set E ( α ) may be thought of as the set where the local dimension of µ equals α or as a multifractal component of supp µ . The main problem in multifractal analysisis to estimate the size of E ( α ) . This is done by calculating the functions f µ ( α ) = dim H ( E ( α )) and F µ ( α ) = dim P ( E ( α )) for α ≥ . These functions are generally known as the multifractal spectrum of µ or the singularity spectrum of the measure µ . One of the main problems in multifractal analysis is to understand the multifractal spectrum and the R´enyidimensions and their relationship with each other. During the past 25 years there has been an enormous interest incomputing the multifractal spectra of measures in the mathematical literature. Particularly, the multifractal spectraof various classes of measures in Euclidean space R n exhibiting some degree of self-similarity have been computedrigorously. The reader can be referred to the paper [42], the textbooks [25, 54] and the references therein. Someheuristic arguments using techniques of Statistical Mechanics (see [32]) show that the singularity spectrum shouldbe finite on a compact interval, noted by Dom ( µ ) , and is expected to be the Legendre transform conjugate of the L q -spectrum, given by τ µ ( q ) = lim r → log sup (X i µ ( B ( x i , r )) q )! − log r where the supremum is taken over all centered packing (cid:0) B ( x i , r ) (cid:1) i of supp µ . That is, for all α ∈ Dom ( µ ) , f µ ( α ) = inf q ∈ R n αq + τ µ ( q ) o = τ ∗ µ ( α ) . (1.1)The multifractal formalism (1.1) has been proved rigorously for random and non-random self-similar measures[1, 16, 42, 43, 51], for self-conformal measures [26, 27, 28, 29, 37, 52], for self-affine measures [6, 7, 8, 9, 23, 24,36, 45] and for Moran measures [61, 62, 63, 64]. We note that the proofs of the multifractal formalism (1.1) inthe above-mentioned references [1, 10, 12, 13, 14, 36, 37, 42, 43, 45, 52] are all based on the same key idea. Theupper bound for f µ ( α ) is obtained by a standard covering argument (involving Besicovitch’s Covering Theorem Mathematics Subject Classification.
Key words and phrases.
Multifractal analysis, multifractal formalism, multifractal Hausdorff measure, multifractal packing measure,Hewitt-Stromberg measures, Hausdorff dimension, packing dimension, doubling measures, inhomogeneous multinomial measures, Moranmeasures. or Vitali’s Covering Theorem). However, its lower bound is usually much harder to prove and is related to theexistence of an auxiliary measure (Gibbs measures) which is supported by the set to be analysed. In an attempt todevelop a general theoretical framework for studying the multifractal structure of arbitrary measures, Olsen [42],Pesin [53] and Peyri`ere [55] suggested various ways of defining an auxiliary measure in a very general setting. Thisformalism was motivated by Olsen’s wish to provide a general mathematical setting for the ideas presented by thephysicists Halsey et al. in their seminal paper [32]. In fact, they have been interested in the concept of multifractalspectrum, that is an interesting geometric characteristic for discrete and continuous models of statistical physics.An important thing which should be noted is that there are many measures for which the multifractal formalismdoes not hold (some examples could be found in [11, 13, 42, 64]). An imported question, in which several theoristsare interested, is: can we find a necessary and sufficient condition for the multifractal formalism to hold? Anotherone, asked by Olsen in [42] is: which functions give more information about a multifractal measure, the dimensionfunctions b µ and B µ or the spectra functions f µ and F µ ? Olsen gives examples of measures where the dimensionfunctions can be used to split measures which have the same spectrum. In doing this, he implicitly suggests thata return to the physicists’ original idea of calculating the moments of multifractal measures may be the best wayto characterize them. It always needs some extra conditions to obtain a minoration for the dimensions of the levelsets E ( α ) . Olsen proved the following statement. Theorem 1. [42]
Let µ be a Borel probability measure on R n . Define α = sup q − b µ ( q ) q .Then, dim H ( E ( α )) ≤ b ∗ µ ( α ) and dim P ( E ( α )) ≤ B ∗ µ ( α ) for all α ∈ ( α, α ) . In general, such a minoration is related to the existence of an auxiliary measure which is supported by the set tobe analyzed. Olsen also gives a result in such a way and supposes the existence of a Gibbs’ measure (see [42]) ata state q for the measure µ , i.e., the existence of a measure ν q on supp µ and constants C > , δ > such that forevery x ∈ supp µ and every < r < δ , C µ ( B ( x, r )) q (2 r ) B µ ( q ) ≤ ν q ( B ( x, r )) ≤ C µ ( B ( x, r )) q (2 r ) B µ ( q ) to conclude that dim H ( E ( α )) = dim P ( E ( α )) = b ∗ µ ( α ) = B ∗ µ ( α ) , where α = − B ′ µ ( q ) . In general, one needs some degree of similarity to prove the existence of Gibbs measures. For example, in dynamiccontexts, the existence of such measures are often natural. For this reason, Ben Nasr et al. in [10, 11, 12, 13]improved Olsen’s result and proposed a new sufficient condition that gives the lower bound. For more details andbackgrounds on multifractal analysis as well as their applications the readers may be referred also to the followingessential references [4, 5, 15, 17, 18, 19, 39, 41, 44, 47, 56, 57, 58, 59, 60, 61, 62, 63, 64].In [11, 13, 42, 56, 64], the authors provided some examples for which the classical multifractal formalism doesnot hold. Indeed, for such examples, the functions b µ and B µ differ and dim H ( E ( α )) and dim P ( E ( α )) are givenrespectively by the Legendre transform of b µ and B µ . Motivated by the above papers, the authors in [3] introducednew metric outer measures (multifractal analogues of the Hewitt-Stromberg measure) H q,tµ and P q,tµ lying betweenthe multifractal Hausdorff measure H q,tµ and the multifractal packing measure P q,tµ , and they used the multifractaldensity theorems to prove the decomposition theorem for the regularities of these measures. In the present paper,we give a new multifractal formalism for which the functions b µ and B µ differ. Actually, the main aim of this workis to introduce and study a multifractal formalism based on the Hewitt-Stromberg measures. However, we pointout that this formalism is completely parallel to Olsen’s multifractal formalism introduced in [42] which basedon the Hausdorff and packing measures. Then, we prove that the lower and upper multifractal Hewitt-Strombergfunctions b µ and B µ are intimately related to the spectra functions. More precisely, we have f µ ( α ) := dim MB ( E ( α )) ≤ b ∗ µ ( α ) and F µ ( α ) := dim MB ( E ( α )) ≤ B ∗ µ ( α ) for some α ≥ . Here dim MB and dim MB denote, respectively, the lower and the upper Hewitt-Stromberg dimension (the lowerand the upper modified box-counting dimension), see Section 2.2 for precise definitions of this. One of our pur-poses of this paper is to show the following result: if H q, b µ ( q ) µ ( E ( − b µ ′ ( q ))) > , then dim MB (cid:16) E (cid:0) − b ′ µ ( q ) (cid:1)(cid:17) = b ∗ µ (cid:0) − b ′ µ ( q ) (cid:1) MULTIFRACTAL FORMALISM FOR HEWITT-STROMBERG MEASURES 3 and, if P q, B µ ( q ) µ ( E ( − B µ ′ ( q ))) > , then dim MB (cid:16) E (cid:0) − B ′ µ ( q ) (cid:1)(cid:17) = B ∗ µ (cid:0) − B ′ µ ( q ) (cid:1) . Moreover, we describe a sufficient condition leading to the equalities f µ ( α ) = F µ ( α ) = F µ ( α ) for some α ≥ . Specifically, if we assume that H q, B µ ( q ) µ (supp µ ) > , then dim MB (cid:16) E (cid:0) − B ′ µ ( q ) (cid:1)(cid:17) = dim MB (cid:16) E (cid:0) − B ′ µ ( q ) (cid:1)(cid:17) = b ∗ µ ( − B ′ µ ( q )) = B ∗ µ ( − B ′ µ ( q )) . We also observe that this sufficient condition is very close to being a necessary and sufficient one, see Theorem 7.In particular, we deal with the case where the lower and upper multifractal Hewitt-Stromberg functions b µ and B µ do not necessarily coincide, see Theorem 8.We will now give a brief description of the organization of the paper. In the next section we recall the definitionsof the various fractal and multifractal dimensions and measures investigated in the paper. The definitions of theHausdorff and packing measures and the Hausdorff and packing dimensions are recalled in Section 2.1, and thedefinitions of the Hewitt-Stromberg measures are recalled in Section 2.2, while the definitions of the Hausdorffand packing measures are well-known, we have, nevertheless, decided to include these-there are two main reasonsfor this: firstly, to make it easier for the reader to compare and contrast the Hausdorff and packing measures withthe less well-known Hewitt-Stromberg measures, and secondly, to provide a motivation for the Hewitt-Strombergmeasures. Section 2.3 recalls the multifractal formalism introduced in [42]. In Section 2.4 we recall the definitionsof the multifractal Hewitt-Stromberg measures and separator functions, and study their properties. Section 2.5recalls earlier results on the values of the multifractal Hausdorff measure, the multifractal packing measure, themultifractal Hewitt-Stromberg measures and separator functions; this discussion is included in order to motivateour main results presented in Section 3. Section 4 contains concrete examples related to these concepts. The paperis concluded with Section 5 that, lists some open problems.2. P RELIMINARIES AND STATEMENTS OF RESULTS
Hausdorff measure, packing measure and dimensions.
While the definitions of the Hausdorff and packingmeasures and the Hausdorff and packing dimensions are well-known, we have, nevertheless, decided to brieflyrecall the definitions below. There are several reasons for this: firstly, since we are working in general metricspaces, the different definitions that appear in the literature may not all agree and for this reason it is useful to stateprecisely the definitions that we are using; secondly, and perhaps more importantly, the less well-known Hewitt-Stromberg measures (see Section 2.2) play an important part in this paper and to make it easier for the reader tocompare and contrast the definitions of the Hewitt-Stromberg measures and the definitions of the Hausdorff andpacking measures it is useful to recall the definitions of the latter measures; and thirdly, in order to provide amotivation for the Hewitt-Stromberg measures.Let X be a metric space, E ⊆ X and t > . The Hausdorff measure is defined, for ε > , as follows H tε ( E ) = inf (X i (cid:16) diam ( E i ) (cid:17) t (cid:12)(cid:12)(cid:12) E ⊆ [ i E i , diam ( E i ) < ε ) . This allows to define first the t -dimensional Hausdorff measure H t ( E ) of E by H t ( E ) = sup ε> H tε ( E ) . Finally, the Hausdorff dimension dim H ( E ) is defined by dim H ( E ) = sup n t ≥ (cid:12)(cid:12) H t ( E ) = + ∞ o . The packing measure is defined, for ε > , as follows P tε ( E ) = sup (X i (cid:16) r i (cid:17) t ) , N. ATTIA, B. SELMI where the supremum is taken over all closed balls (cid:16) B ( x i , r i ) (cid:17) i such that r i ≤ ε and with x i ∈ E and d ( x i , x j ) ≥ r i + r j for i = j . The t -dimensional packing pre-measure P t ( E ) of E is now defined by P t ( E ) = sup ε> P tε ( E ) . This makes us able to define the t -dimensional packing measure P t ( E ) of E as P t ( E ) = inf (X i P t ( E i ) (cid:12)(cid:12)(cid:12) E ⊆ [ i E i ) , and the packing dimension dim P ( E ) is defined by dim P ( E ) = sup n t ≥ (cid:12)(cid:12) P t ( E ) = + ∞ o . Hewitt-Stromberg measures and dimensions.
Hewitt-Stromberg measures were introduced in [33, Exer-cise (10.51)]. Since then, they have been investigated by several authors, highlighting their importance in the studyof local properties of fractals and products of fractals. One can cite, for example [30, 31, 35, 50, 65]. In particu-lar, Edgar’s textbook [20, pp. 32-36] provides an excellent and systematic introduction to these measures. Suchmeasures appear also appears explicitly, for example, in Pesin’s monograph [54, 5.3] and implicitly in Mattila’stext [38]. One of the purposes of this paper is to define and study a class of natural multifractal generalizations ofthe Hewitt-Stromberg measures. While Hausdorff and packing measures are defined using coverings and packingsby families of sets with diameters less than a given positive number ε , say, the Hewitt-Stromberg measures aredefined using packings of balls with a fixed diameter ε . For t > , the Hewitt-Stromberg pre-measures are definedas follows, H t ( E ) = lim inf r → N r ( E ) (2 r ) t and P t ( E ) = lim sup r → M r ( E ) (2 r ) t , where the covering number N r ( E ) of E and the packing number M r ( E ) of E are given by N r ( E ) = inf ( ♯ { I } (cid:12)(cid:12)(cid:12) (cid:16) B ( x i , r ) (cid:17) i ∈ I is a family of closed balls with x i ∈ E and E ⊆ [ i B ( x i , r ) ) and M r ( E ) = sup (cid:26) ♯ { I } (cid:12)(cid:12)(cid:12) (cid:16) B ( x i , r i ) (cid:17) i ∈ I is a family of closed balls with x i ∈ E and d ( x i , x j ) ≥ r for i = j (cid:27) . Now, we define the lower and upper t -dimensional Hewitt-Stromberg measures, which we denote respectivelyby H t ( E ) and P t ( E ) , as follows H t ( E ) = inf (X i H t ( E i ) (cid:12)(cid:12)(cid:12) E ⊆ [ i E i ) and P t ( E ) = inf (X i P t ( E i ) (cid:12)(cid:12)(cid:12) E ⊆ [ i E i ) . We recall some basic inequalities satisfied by the Hewitt-Stromberg, the Hausdorff and the packing measure(see [35, 50, Proposition 2.1]) H t ( E ) ≤ P t ( E ) ≤ P t ( E ) and H t ( E ) ≤ H t ( E ) ≤ P t ( E ) ≤ P t ( E ) . The lower and upper Hewitt-Stromberg dimension dim MB ( E ) and dim MB ( E ) are defined by dim MB ( E ) = inf n t ≥ (cid:12)(cid:12)(cid:12) H t ( E ) = 0 o = sup n t ≥ (cid:12)(cid:12)(cid:12) H t ( E ) = + ∞ o and dim MB ( E ) = inf n t ≥ (cid:12)(cid:12) P t ( E ) = 0 o = sup n t ≥ (cid:12)(cid:12) P t ( E ) = + ∞ o . MULTIFRACTAL FORMALISM FOR HEWITT-STROMBERG MEASURES 5
The lower and upper box dimensions, denoted by dim B ( E ) and dim B ( E ) , respectively, are now defined by dim B ( E ) = lim inf r → log N r ( E ) − log r = lim inf r → log M r ( E ) − log r and dim B ( E ) = lim sup r → log N r ( E ) − log r = lim sup r → log M r ( E ) − log r . These dimensions satisfy the following inequalities, dim H ( E ) ≤ dim MB ( E ) ≤ dim MB ( E ) ≤ dim P ( E ) , dim H ( E ) ≤ dim P ( E ) ≤ dim B ( E ) and dim H ( E ) ≤ dim B ( E ) ≤ dim B ( E ) . The reader is referred to [22] for an excellent discussion of the Hausdorff dimension, the packing dimension, lowerand upper Hewitt-Stromberg dimension and the box dimensions. In particular, we have (see [22, 40]) dim MB ( E ) = inf ( sup i dim B ( E i ) (cid:12)(cid:12)(cid:12) E ⊆ [ i E i , E i are bounded in X ) and dim MB ( E ) = inf ( sup i dim B ( E i ) (cid:12)(cid:12)(cid:12) E ⊆ [ i E i , E i are bounded in X ) . Multifractal Hausdorff measure and packing measure.
We start by introducing the generalized centeredHausdorff measure H q,tµ and the generalized packing measure P q,tµ . We fix an integer n ≥ and denote by P ( R n ) the family of compactly supported Borel probability measures on R n . Let µ ∈ P ( R n ) , q, t ∈ R , E ⊆ R n and δ > . We define the generalized packing pre-measure by P q,tµ ( E ) = inf δ> sup (X i µ (cid:0) B ( x i , r i ) (cid:1) q (2 r i ) t (cid:12)(cid:12)(cid:12) (cid:16) B ( x i , r i ) (cid:17) i is a centered δ -packing of E ) . In a similar way, we define the generalized Hausdorff pre-measure by H q,tµ ( E ) = sup δ> inf (X i µ (cid:0) B ( x i , r i ) (cid:1) q (2 r i ) t (cid:12)(cid:12)(cid:12) (cid:16) B ( x i , r i ) (cid:17) i is a centered δ -covering of E ) , with the conventions q = ∞ for q ≤ and q = 0 for q > .The function H q,tµ is σ -subadditive but not increasing and the function P q,tµ is increasing but not σ -subadditive.That is the reason for which Olsen introduced the following modifications of the generalized Hausdorff and packingmeasures H q,tµ and P q,tµ : H q,tµ ( E ) = sup F ⊆ E H q,tµ ( F ) and P q,tµ ( E ) = inf E ⊆ S i E i X i P q,tµ ( E i ) . The functions H q,tµ and P q,tµ are metric outer measures and thus measures on the Borel family of subsets of R n .Moreover, there exists an integer ξ ∈ N , such that H q,tµ ≤ ξ P q,tµ . The measure H q,tµ is of course a multifractalgeneralization of the centered Hausdorff measure, whereas P q,tµ is a multifractal generalization of the packingmeasure. In fact, it is easily seen that, for t ≥ , one has − t H ,tµ ≤ H t ≤ H ,tµ and P ,tµ = P t , where H t and P t denote respectively the t -dimensional Hausdorff and t -dimensional packing measures.We now define the family of doubling measures. For µ ∈ P ( R n ) and a > , we write P a ( µ ) = lim sup r ց sup x ∈ supp µ µ (cid:0) B ( x, ar ) (cid:1) µ (cid:0) B ( x, r ) (cid:1) ! . N. ATTIA, B. SELMI
We say that the measure µ satisfies the doubling condition if there exists a > such that P a ( µ ) < ∞ . It is easilyseen that the exact value of the parameter a is unimportant: P a ( µ ) < ∞ , for some a > if and only if P a ( µ ) < ∞ , for all a > . Also, we denote by P D ( R n ) the family of Borel probability measures on R n which satisfy the doubling condition.We can cite as classical examples of doubling measures, the self-similar measures and the self-conformal ones[42]. In particular, if µ ∈ P D ( R n ) then H q,tµ ≤ P q,tµ . The measures H q,tµ and P q,tµ and the pre-measure P q,tµ assign in a usual way a multifractal dimension to eachsubset E of R n . They are respectively denoted by dim qµ ( E ) , Dim qµ ( E ) and ∆ qµ ( E ) (see [42]) and satisfy dim qµ ( E ) = inf n t ∈ R (cid:12)(cid:12) H q,tµ ( E ) = 0 o = sup n t ∈ R (cid:12)(cid:12) H q,tµ ( E ) = + ∞ o , Dim qµ ( E ) = inf n t ∈ R (cid:12)(cid:12) P q,tµ ( E ) = 0 o = sup n t ∈ R (cid:12)(cid:12) P q,tµ ( E ) = + ∞ o , ∆ qµ ( E ) = inf n t ∈ R (cid:12)(cid:12) P q,tµ ( E ) = 0 o = sup n t ∈ R (cid:12)(cid:12) P q,tµ ( E ) = + ∞ o . The number dim qµ ( E ) is an obvious multifractal analogue of the Hausdorff dimension dim H ( E ) of E whereas Dim qµ ( E ) and ∆ qµ ( E ) are obvious multifractal analogues of the packing dimension dim P ( E ) and the pre-packingdimension ∆( E ) of E respectively. In fact, it follows immediately from the definitions that dim H ( E ) = dim µ ( E ) , dim P ( E ) = Dim µ ( E ) and ∆( E ) = ∆ µ ( E ) . We define the functions b µ ( q ) = dim qµ (supp µ ) and B µ ( q ) = Dim qµ (supp µ ) . It is well known that the functions b µ and B µ are decreasing and B µ is convex and satisfying b µ ≤ B µ . Multifractal Hewitt-Stromberg measures and separator functions.
In the following, we will set up, for q, t ∈ R and µ ∈ P ( R n ) , the lower and upper multifractal Hewitt-Stromberg measures H q,tµ and P q,tµ .For E ⊆ supp µ , the pre-measure of E is defined by C q,tµ ( E ) = lim sup r → M qµ,r ( E )(2 r ) t , where M qµ,r ( E ) = sup (X i µ ( B ( x i , r )) q (cid:12)(cid:12)(cid:12) (cid:16) B ( x i , r ) (cid:17) i is a centered packing of E ) . It’s clear that C q,tµ is increasing and C q,tµ ( ∅ ) = 0 . However it’s not σ -additive. For this, we introduce the P q,tµ -measure defined by P q,tµ ( E ) = inf (X i C q,tµ ( E i ) (cid:12)(cid:12)(cid:12) E ⊆ [ i E i and the E ′ i s are bounded ) . In a similar way we define L q,tµ ( E ) = lim inf r → N qµ,r ( E )(2 r ) t , where N qµ,r ( E ) = inf (X i µ ( B ( x i , r )) q (cid:12)(cid:12)(cid:12) (cid:16) B ( x i , r ) (cid:17) i is a centered covering of E ) . Since L q,tµ is not increasing and not countably subadditive, one needs a standard modification to get an outermeasure. Hence, we modify the definition as follows H q,tµ ( E ) = inf (X i L q,tµ ( E i ) (cid:12)(cid:12)(cid:12) E ⊆ [ i E i and the E ′ i s are bounded ) and H q,tµ ( E ) = sup F ⊆ E H q,tµ ( F ) . MULTIFRACTAL FORMALISM FOR HEWITT-STROMBERG MEASURES 7
The measure H q,tµ is of course a multifractal generalization of the lower t -dimensional Hewitt-Stromberg mea-sure H t , whereas P q,tµ is a multifractal generalization of the upper t -dimensional Hewitt-Stromberg measures P t .In fact, it is easily seen that, for t > , one has H ,tµ = H t and P ,tµ = P t . The following result describes some of the basic properties of the multifractal Hewitt-Stromberg measuresincluding the fact that H q,tµ and P q,tµ are Borel metric outer measures and summarises the basic inequalities satisfiedby the multifractal Hewitt-Stromberg measures, the generalized Hausdorff measure and the generalized packingmeasure. Theorem 2. [3]
Let q, t ∈ R and µ ∈ P ( R n ) . Then for every set E ⊆ R n we have (1) the set functions H q,tµ and P q,tµ are metric outer measures and thus they are measures on the Borel algebra. (2) There exists an integer ξ ∈ N , such that H q,tµ ( E ) ≤ H q,tµ ( E ) ≤ ξ P q,tµ ( E ) ≤ ξ P q,tµ ( E ) . (3) When q ≤ or q > and µ ∈ P D ( R n ) , we have H q,tµ ( E ) ≤ H q,tµ ( E ) ≤ P q,tµ ( E ) ≤ P q,tµ ( E ) . The measures H q,tµ and P q,tµ and the pre-measure C q,tµ assign in the usual way a multifractal dimension to eachsubset E of R n . They are respectively denoted by b qµ ( E ) , B qµ ( E ) and ∆ qµ ( E ) , Proposition 1.
Let q ∈ R , µ ∈ P ( R n ) and E ⊆ R n . Then (1) there exists a unique number b qµ ( E ) ∈ [ −∞ , + ∞ ] such that H q,tµ ( E ) = ∞ if t < b qµ ( E ) , if b qµ ( E ) < t, (2) there exists a unique number B qµ ( E ) ∈ [ −∞ , + ∞ ] such that P q,tµ ( E ) = ∞ if t < B qµ ( E ) , if B qµ ( E ) < t, (3) there exists a unique number ∆ qµ ( E ) ∈ [ −∞ , + ∞ ] such that C q,tµ ( E ) = ∞ if t < ∆ qµ ( E ) , if ∆ qµ ( E ) < t. In addition, we have b qµ ( E ) ≤ B qµ ( E ) ≤ ∆ qµ ( E ) . The number b qµ ( E ) is an obvious multifractal analogue of the lower Hewitt-Stromberg dimension dim MB ( E ) of E whereas B qµ ( E ) is an obvious multifractal analogues of the upper Hewitt-Stromberg dimension dim MB ( E ) of E . In fact, it follows immediately from the definitions that b µ ( E ) = dim MB ( E ) and B µ ( E ) = dim MB ( E ) . Remark 1.
It follows from Theorem 2 that dim qµ ( E ) ≤ b qµ ( E ) ≤ B qµ ( E ) ≤ Dim qµ ( E ) ≤ ∆ qµ ( E ) . The definition of these dimension functions makes it clear that they are counterparts of the τ µ -function whichappears in the multifractal formalism . This being the case, it is important that they have the properties describedby the physicists. The next theorem shows that these functions do indeed have some of these properties. Theorem 3.
Let q ∈ R and E ⊆ R n . (1) The functions q H q,tµ ( E ) , P q,tµ ( E ) , C q,tµ ( E ) are decreasing. (2) The functions t H q,tµ ( E ) , P q,tµ ( E ) , C q,tµ ( E ) are decreasing. (3) The functions q b qµ ( E ) , B qµ ( E ) , ∆ qµ ( E ) are decreasing. N. ATTIA, B. SELMI (4)
The functions q B qµ ( E ) , ∆ qµ ( E ) are convex.Proof. Let q ∈ R and E ⊆ R n .The first and second part of Theorem 3 follows since x a x is decreasing for all a ∈ ]0 , .Observe that part (3) of Theorem 3 follows immediately from (1).We will now prove the part (4). Let α ∈ [0 , and p, s, t ∈ R . Suppose that we have shown that C αp +(1 − α ) q,αt +(1 − α ) sµ ( E ) ≤ (cid:16) C p,tµ ( E ) (cid:17) α (cid:16) C q,sµ ( E ) (cid:17) − α . (2.1)Then, for all ǫ > , we have C αp +(1 − α ) q,α ∆ pµ ( E )+(1 − α ) ∆ qµ ( E )+ ǫµ ( E ) ≤ (cid:16) C p, ∆ pµ ( E )+ ǫµ ( E ) (cid:17) α (cid:16) C q, ∆ qµ ( E )+ ǫµ ( E ) (cid:17) − α = 0 . We therefore conclude that ∆ αp +(1 − α ) qµ ( E ) ≤ α ∆ pµ ( E ) + (1 − α ) ∆ qµ ( E ) + ǫ. Finally, letting ǫ tend to , then the convexity of q ∆ qµ ( E ) follows.We now turn towards the proof of (2.1). Put r > and (cid:16) B ( x i , r ) (cid:17) i be a centered packing of E . It follows fromH¨older inequality that X i µ ( B ( x i , r )) αp +(1 − α ) q = X i (cid:16) µ ( B ( x i , r ) p (cid:17) α (cid:16) µ ( B ( x i , r ) q (cid:17) − α ≤ X i µ ( B ( x i , r ) p ! α X i µ ( B ( x i , r ) q ! − α ≤ (cid:16) M pµ,r ( E ) (cid:17) α (cid:16) M qµ,r ( E ) (cid:17) − α . This shows that M αp +(1 − α ) qµ,r (2 r ) αt +(1 − α ) s ≤ (cid:16) M pµ,r (2 r ) t (cid:17) α (cid:16) M qµ,r (2 r ) s (cid:17) − α . Letting r tend to we get the result.We must now show the convexity of q B qµ ( E ) . Let η > and put t = B pµ ( E ) and s = B qµ ( E ) . Since P q,s + ηµ ( E ) = P p,t + ηµ ( E ) = 0 , we can choose bounded coverings ( H i ) i and ( K i ) i of E such that X i C q,t + ηµ ( H i ) ≤ and X i C q,s + ηµ ( K i ) ≤ . MULTIFRACTAL FORMALISM FOR HEWITT-STROMBERG MEASURES 9
Next, for n ∈ N ∗ , let E n = n [ i,j =1 ( H i ∩ K j ) , we clearly have P αp +(1 − α ) q,αt +(1 − α ) s + ηµ ( E n ) ≤ n X i,j =1 P αp +(1 − α ) q,αt +(1 − α ) s + ηµ ( H i ∩ K j ) ≤ n X i,j =1 C αp +(1 − α ) q,αt +(1 − α ) s + ηµ ( H i ∩ K j ) (2.1) ≤ n X i,j =1 (cid:16) C p,t + ηµ ( H i ∩ K j ) (cid:17) α (cid:16) C q,s + ηµ ( H i ∩ K j ) (cid:17) − α H¨older ≤ n X i,j =1 C p,t + ηµ ( H i ∩ K j ) α n X i,j =1 C q,s + ηµ ( H i ∩ K j ) − α ≤ n X i,j =1 C p,t + ηµ ( H i ) α n X i,j =1 C q,s + ηµ ( K j ) − α ≤ n n X i =1 C p,t + ηµ ( H i ) ! α n n X j =1 C q,s + ηµ ( K j ) − α ≤ n α n − α = n < ∞ . We now obtain, for all n ∈ N ∗ , B αp +(1 − α ) qµ ( E n ) ≤ αt + (1 − α ) s + η. Since clearly E ⊆ S n E n , we therefore conclude that B αp +(1 − α ) qµ ( E ) ≤ B αp +(1 − α ) qµ [ n E n ! ≤ sup n B αp +(1 − α ) qµ ( E n ) ≤ α B pµ ( E ) + (1 − α ) B qµ ( E ) + η. Letting η tend to now yields the desired result. This completes the proof of Theorem 3. (cid:3) Next we define the multifractal separator functions b µ , B µ and Λ µ : R → [ −∞ , + ∞ ] by b µ : q → b qµ (supp µ ) , B µ : q → B qµ (supp µ ) and Λ µ : q → ∆ qµ (supp µ ) . We also obtain the following corollary providing information about the lower and upper multifractal Hewitt-Stromberg functions.
Corollary 1.
Let q ∈ R . We have (1) for q < , b µ ( q ) ≥ . (2) For q = 1 , b µ ( q ) = Λ µ ( q ) = 0 . (3) For q > , Λ µ ( q ) ≤ .Proof. This follow immediately from the above theorem and definitions. (cid:3)
Some characterizations of b µ ( q ) and B µ ( q ) . In this section, we investigate the relation between the lowerand upper multifractal Hewitt-Stromberg functions b µ and B µ and the multifractal box dimension, the multifractalpacking dimension and the multifractal pre-packing dimension. We first note that there exists a unique number Θ qµ ( E ) ∈ [ −∞ , + ∞ ] such that L q,tµ ( E ) = ∞ if t < Θ qµ ( E ) , if Θ qµ ( E ) < t. Proposition 2.
Let q ∈ R and µ be a compact supported Borel probability measure on R n . Then for every E ⊆ supp µ we have Θ qµ ( E ) = lim inf r → log N qµ,r ( E ) − log r and ∆ qµ ( E ) = lim sup r → log M qµ,r ( E ) − log r . Proof.
We will prove the first equality, the second one is similar. Suppose that lim inf r → log N qµ,r ( E ) − log r > Θ qµ ( E ) + ǫ for some ǫ > . Then we can find δ > such that for any r ≤ δ , N qµ,r ( E ) r Θ qµ ( E )+ ǫ > and then L q, Θ qµ ( E )+ ǫµ ≥ Θ qµ ( E )+ ǫ which is a contradiction. We therefore infer lim inf r → log N qµ,r ( E ) − log r ≤ Θ qµ ( E ) + ǫ for any ǫ > . The proof of the following statement lim inf r → log N qµ,r ( E ) − log r ≥ Θ qµ ( E ) − ǫ for any ǫ > is identical to the proof of the above statement and is therefore omitted. (cid:3) Remark 2.
Here we follow the approach of Olsen in [42, 45, 48, 49] . (1) The multifractal dimensions Θ qµ ( E ) and ∆ qµ ( E ) of E represent the upper and lower multifractal box-dimension. In particular, we have Θ µ ( E ) = dim B ( E ) and ∆ µ ( E ) = dim B ( E ) . (2) Let us introduce the multifrcatal generalization of the q -dimensions called also relative R´enyi q -dimensionsbased on integral representations. Let µ be a probability measure on R n . For q ∈ R \ { } , we write D qµ = lim inf r → q log r log Z µ ( B ( x, r )) q dµ ( x ) , and D qµ = lim sup r → q log r log Z µ ( B ( x, r )) q dµ ( x ) . Now we define the generalized entropies due to R´enyi by, h qr ( µ ) = 1 q − M qµ,r (supp µ ) for q = 1 and h r ( µ ) = inf ( − X i µ ( E i ) log µ ( E i ) (cid:12)(cid:12)(cid:12) ( E i ) i is a partition of supp µ ) . We define the upper and lower R´enyi q -dimensions T qµ and T qµ of µ by T qµ = lim sup r → log h qr ( µ )log r and T qµ = lim inf r → log h qr ( µ )log r . If D qµ = D qµ (respectively T qµ = T qµ ) we refer to the common value as the relative R´enyi q -dimension of µ and denote it D qµ (respectively T qµ ). Finally define D µ ( q ) , D µ ( q ) , T µ ( q ) and T µ ( q ) : R → [ −∞ , + ∞ ] by D µ ( q ) = (1 − q ) D q − µ , D µ ( q ) = (1 − q ) D q − µ and T µ ( q ) = (1 − q ) T qµ , T µ ( q ) = (1 − q ) T qµ . Let q ∈ R and µ ∈ P D ( R n ) . Then the following holds ∆ µ ( q ) = D µ ( q ) ∨ D µ ( q ) = T µ ( q ) ∨ T µ ( q ) . MULTIFRACTAL FORMALISM FOR HEWITT-STROMBERG MEASURES 11 (3)
We define the multifractal Minkowski volume as follows. Let E be a subset of R n and r > . We denoteby B ( E, r ) the open r neighbourhood of E , i.e. B ( E, r ) = n x ∈ R n (cid:12)(cid:12)(cid:12) dist ( x, E ) < r o . For a real number q and a Borel measure µ on R n , we define the multifractal Minkowski volume V qµ,r ( E ) of E with respect to the measure µ by V qµ,r ( E ) = 1 r n Z B ( E,r ) µ ( B ( x, r )) q d L n ( x ) . Here L n denotes the n -dimensional Lebesgue measure in R n . The importance of the R´enyi dimensionsin multifractal analysis together with the formal resemblance between the multifractal Minkowski volume V qµ,r ( E ) and the moments R E µ ( B ( x, r )) q − dµ ( x ) used in the definition the R´enyi dimensions may beseen as a justification for calling the quantity V qµ,r ( E ) for the multifractal Minkowski volume. Using themultifractal Minkowski volume we can define multifractal Monkowski dimensions. For a real number q and a Borel measure µ on R n , we define the lower and upper multifractal Minkowski dimension of E , by dim qM,µ ( E ) = lim inf r → log V qµ,r ( E ) − log r and dim qM,µ ( E ) = lim sup r → log V qµ,r ( E ) − log r . We note the close similarity between the multifractal Minkowski dimensions and ∆ qµ . Indeed, the equality (2.2) shows that this similarity is not merely a formal resemblance. In fact, for q ≥ , the multifractalMinkowski dimensions and ∆ qµ coincide, i.e. for q ≥ and µ ∈ P D ( R n ) , we have ∆ µ ( q ) = dim q − M,µ (supp µ ) ∨ dim q − M,µ (supp µ ) . (2.2) Proposition 3.
Let q ∈ R and µ be a compact supported Borel probability measure on R n . Then for every E ⊆ supp µ we have b qµ ( E ) = sup F ⊆ E ( inf ( sup i Θ qµ ( F i ) (cid:12)(cid:12)(cid:12) F ⊆ [ i F i , F i are bounded in R n )) and B qµ ( E ) = inf ( sup i ∆ qµ ( E i ) (cid:12)(cid:12)(cid:12) E ⊆ [ i E i , E i are bounded in R n ) . Proof.
Denote β = sup F ⊂ E ( inf ( sup i Θ qµ ( F i ) (cid:12)(cid:12)(cid:12) F ⊆ [ i F i , F i are bounded in R n )) . Assume that β < b qµ ( E ) and take α ∈ ( β, b qµ ( E )) . Then, for all F ⊆ E , there exists { F i } of bounded subset of F such that F ⊆ ∪ i F i , and sup i Θ qµ ( F i ) < α . Now observe that L q,αµ ( F i ) = 0 which implies that H q,αµ ( F ) = 0 . Thisimplies that H q,αµ ( E ) = 0 . It is a contradiction. Now suppose that b qµ ( E ) < β , then, for α ∈ ( b qµ ( E ) , β ) , we have H q,αµ ( E ) = 0 . It follows from this that H q,αµ ( F ) = 0 for all F ⊆ E . Thus, there exists { F i } of bounded subset of F such that F ⊆ ∪ i F i , and sup i L q,αµ ( F i ) < ∞ . We conclude that, sup i Θ qµ ( F i ) ≤ α . It is also a contradiction.The proof of the second statement is identical to the proof of the statement in the first part and is thereforeomitted. (cid:3) Proposition 4. If q ∈ R and µ ∈ P D ( R n ) , then for any subset E of supp µ , we have B qµ ( E ) = B qµ ( E ) . Proof.
This follows easily from Propositions 2 and 3, Propositions 2.19 and 2.22 in [42] and Lemma 4.1 in [46]. (cid:3)
Remark 3.
The results developed by Falconer in [22] are obtained as a special case of the multifractal results bysetting q = 0 .
3. A
MULTIFRACTAL FORMALISM FOR H EWITT -S TROMBERG MEASURES
Multifractal analysis was proved to be a very useful technique in the analysis of measures, both in theory andapplications. The upper and lower local dimensions of a measure µ on R n at a point x are respectively given by : α µ ( x ) = lim sup r → log µ ( B ( x, r ))log r and α µ ( x ) = lim inf r → log µ ( B ( x, r ))log r , where B ( x, r ) denote the closed ball of center x and radius r . We refer to the common value as the local dimensionof µ at x , and denote it by α µ ( x ) .The level set of the local dimension of µ contains a crucial information on the geometrical properties of µ . Theaim of the multifractal analysis of a measure is to relate the Hausdorff and packing dimensions of these levels setsto the Legendre transform of some concave (convex) function (see for example [2, 10, 11, 12, 13, 17, 41, 42]). For α ≥ , we define the fractal sets, E α = n x ∈ supp µ (cid:12)(cid:12) α µ ( x ) ≤ α o ; E α = n x ∈ supp µ (cid:12)(cid:12) α µ ( x ) ≥ α o and E α = n x ∈ supp µ (cid:12)(cid:12) α µ ( x ) ≤ α o ; E α = n x ∈ supp µ (cid:12)(cid:12) α µ ( x ) ≥ α o . Also, let E ( α ) = E α ∩ E α = n x ∈ supp µ (cid:12)(cid:12) α µ ( x ) = α o . Theorem 4 allows us to consider the relationship between the lower and upper multifractal Hewitt-Strombergfunctions b µ and B µ and the multifractal spectra. We start by giving an upper bound theorem. For µ ∈ P ( R n ) , set α min = sup q − b µ ( q ) q , β min = sup
q − B µ ( q ) q . Before stating this formally, we remind the reader that if ϕ : R → R is a real valued function, then the Legendretransform ϕ ∗ : R → [ −∞ , + ∞ ] of ϕ is defined by ϕ ∗ ( x ) = inf y (cid:16) xy + ϕ ( y ) (cid:17) . Now, we can state our multifractal formalism . Theorem 4.
Let α ≥ , then the following hold (1) α min ≤ inf α µ ( x ) ≤ sup α µ ( x ) ≤ β max and β min ≤ inf α µ ( x ) ≤ sup α µ ( x ) ≤ α max . (2) dim MB E ( α ) ≤ b ∗ µ ( α ) if α ∈ ( α min , α max )= 0 if α / ∈ ( α min , α max ) . (3) dim MB E ( α ) ≤ B ∗ µ ( α ) if α ∈ ( α min , α max )= 0 if α / ∈ ( α min , α max ) . Proof.
This theorem follows immediately from the following lemmas. (cid:3)
Lemma 1. If µ ∈ P ( R n ) and α ≥ , then (1) E α = ∅ for α < β min and E α = ∅ for α > β max . MULTIFRACTAL FORMALISM FOR HEWITT-STROMBERG MEASURES 13 (2) E α = ∅ for α > α max and E α = ∅ for α < α min .Proof. (1) Let x ∈ E α and α < β min . There exists ǫ > and q > such that − q ( α + ǫ ) > B µ ( q ) . Since x ∈ E α , we can thus choose ( r n ) n such that r n → and log µ ( B ( x, r n ))log r n < α + ǫ. For brevity write t = − q ( α + ǫ ) , then we obtain µ ( B ( x, r n )) q (2 r n ) t > t . Hence, for all n ∈ N , M q ,tµ,r n (cid:0) { x } (cid:1) (2 r n ) t ≥ µ ( B ( x, r n )) q (2 r n ) t > t > . It follows that P q ,tµ (cid:0) { x } (cid:1) > . We therefore conclude that t ≤ B q µ (cid:0) { x } (cid:1) ≤ B µ ( q ) which contradicts the fact that − q ( α + ǫ ) > B µ ( q ) .The proof of the second statement is identical to the proof of the first statement in Part (1) and istherefore omitted.(2) Let x ∈ E α and α > α max . Then, we can find ǫ > and q < such that − q ( α − ǫ ) > b µ ( q ) . Since x ∈ E α , we can choose r such that for < r < r we have log µ ( B ( x, r ))log r > α − ǫ. Then, for t = − q ( α − ǫ ) and r < r , we have µ ( B ( x, r )) q (2 r ) t > t . Therefore, it follows that, for all r < r , N q ,tµ,r (cid:0) { x } (cid:1) (2 r ) t = µ ( B ( x, r )) q (2 r ) t > t > . which implies that H q ,tµ (cid:0) { x } (cid:1) > . It now follows from this that t ≤ b q µ (cid:0) { x } (cid:1) ≤ b µ ( q ) which contradicts the fact that − q ( α − ǫ ) > b µ ( q ) .The proof of the second statement in part (2) is very similar to the proof of the first statement and istherefore omitted. (cid:3) Lemma 2.
Let µ ∈ P ( R n ) , α ≥ , q, t ∈ R and δ > such that δ ≤ αq + t. Then the following hold (1) (a) H αq + t + δ ( E α ) ≤ αq + δ H q,tµ ( E α ) for ≤ q. (b) H αq + t + δ ( E α ) ≤ αq + δ H q,tµ ( E α ) for ≥ q. (c) If ≤ αq + b µ ( q ) then dim MB ( E α ) ≤ inf q ≥ αq + b µ ( q ) and dim MB ( E α ) ≤ inf q ≤ αq + b µ ( q ) . In particular dim MB ( E α ) ≤ α . (2) (a) P αq + t + δ ( E α ) ≤ αq + δ P q,tµ ( E α ) for ≤ q. (b) P αq + t + δ ( E α ) ≤ αq + δ P q,tµ ( E α ) for ≥ q. (c) If ≤ αq + b µ ( q ) then dim MB ( E α ) ≤ inf q ≥ αq + B µ ( q ) and dim MB ( E α ) ≤ inf q ≤ αq + B µ ( q ) . In particular dim MB ( E α ) ≤ α .Proof. An exhaustive proof of this lemma would require considerable repetition. To avoid this we prove (1)-(a)and (2)-(a). (1) (a) Clearly, the statement is true for q = 0 . For m ∈ N , write E m = (cid:26) x ∈ E α (cid:12)(cid:12)(cid:12) log µ ( B ( x, r ))log r ≤ α + δq for < r < m (cid:27) . Fix m ∈ N and r > such that < r < m . Let (cid:16) B ( x i , r ) (cid:17) i be a centered covering of E m . Next,we observe that log µ ( B ( x i , r ))log r ≤ α + δq = ⇒ µ ( B ( x i , r )) q ≥ r qα + δ = ⇒ X i µ ( B ( x i , r )) q ≥ N r ( E m ) r qα + δ = ⇒ N qµ,r ( E m ) ≥ N r ( E m ) r qα + δ = ⇒ L q,tµ ( E m ) ≥ − αq − δ H qα + δ + t ( E m )= ⇒ H q,tµ ( E m ) ≥ H q,tµ ( E m ) ≥ − αq − δ H qα + δ + t ( E m ) . Now from this and since E m ր E α we can deduce that H q,tµ ( E α ) ≥ − αq − δ H qα + δ + t ( E α ) . (2) (a) Once again for q = 0 the statement is well known. For m ∈ N , put E m = (cid:26) x ∈ E α (cid:12)(cid:12)(cid:12) log µ ( B ( x, r ))log r ≤ α + δq for < r < m (cid:27) . We therefore fix m ∈ N and r > such that < r < m . Let (cid:16) B ( x i , r ) (cid:17) i ∈{ ,...,M r ( E m ) } be apacking of E m . Then we have log µ ( B ( x i , r ))log r ≤ α + δq = ⇒ µ ( B ( x i , r )) q ≥ r qα + δ = ⇒ X i µ ( B ( x i , r )) q ≥ M r ( E m ) r qα + δ = ⇒ M qµ,r ( E m ) ≥ M r ( E m ) r qα + δ = ⇒ C q,tµ ( E m ) ≥ − αq − δ P qα + δ + t ( E m )= ⇒ P q,tµ ( E m ) ≥ − αq − δ P qα + δ + t ( E m ) . However, since E m ր E α , we conclude that P q,tµ ( E α ) ≥ − αq − δ P qα + δ + t ( E α ) which yields the desired result. (cid:3) Our purpose of the following theorems is to propose a sufficient condition that gives the lower bound.
Theorem 5.
Let q, t ∈ R , and α > such that αq + t ≥ . Let A ⊆ E ( α ) is a Borel set. (1) If H q,tµ ( A ) > , then dim MB E ( α ) ≥ αq + t. In particular, if the multifractal function b µ is differentiable at q, then, provided that b ∗ µ (cid:16) − b ′ µ ( q ) (cid:17) ≥ and H q, b µ ( q ) µ (cid:16) E ( − b µ ′ ( q )) (cid:17) > , we have dim MB E (cid:16) − b ′ µ ( q ) (cid:17) = b ∗ µ (cid:16) − b ′ µ ( q ) (cid:17) . MULTIFRACTAL FORMALISM FOR HEWITT-STROMBERG MEASURES 15 (2) If P q,tµ ( A ) > , then dim MB E ( α ) ≥ αq + t. In particular, if the multifractal function B µ is differentiable at q, then, provided that B ∗ µ (cid:16) − B ′ µ ( q ) (cid:17) ≥ and P q, B µ ( q ) µ (cid:16) E ( − B µ ′ ( q )) (cid:17) > , we have dim MB E (cid:16) − B ′ µ ( q ) (cid:17) = B ∗ µ (cid:16) − B ′ µ ( q ) (cid:17) . Proof.
This follows easily from Theorem 4 and the following lemma. (cid:3)
Lemma 3.
Let µ ∈ P ( R n ) , α ≥ , q, t ∈ R and δ > such that δ ≤ αq + t. Then we have the following (1) (a) If A ⊆ E α , is Borel then P αq + t − δ ( A ) ≥ αq − δ P q,tµ ( A ) for ≥ q. (b) If A ⊆ E α , is Borel then P αq + t − δ ( A ) ≥ αq − δ P q,tµ ( A ) for ≤ q. In particular, if µ ( A ) > then dim MB ( A ) ≥ α . (2) (a) If A ⊆ E α , is Borel then H αq + t − δ ( A ) ≥ αq − δ H q,tµ ( A ) for ≥ q. (b) If A ⊆ E α , is Borel then H αq + t − δ ( A ) ≥ αq − δ H q,tµ ( A ) for ≤ q. In particular, if µ ( A ) > then dim MB ( A ) ≥ α .Proof. An exhaustive proof of this theorem would require considerable repetition. For this we only prove (1)-(a)and (2)-(a), the other assertions are similar.(1) (a) Clearly the statement is true for q = 0 . For m ∈ N , write E m = (cid:26) x ∈ A (cid:12)(cid:12)(cid:12) log µ ( B ( x, r ))log r ≤ α − δq for < r < m (cid:27) . Let m ∈ N and r > such that < r < m . Let (cid:16) B ( x i , r ) (cid:17) i be a centred packing of E m . We have log µ ( B ( x i , r ))log r ≤ α − δq = ⇒ µ ( B ( x i , r )) q ≤ r qα − δ = ⇒ X i µ ( B ( x i , r )) q ≤ M r ( E m ) r qα − δ = ⇒ M qµ,r ( E m ) ≤ M r ( E m ) r qα − δ = ⇒ C µ ( E m ) q,t ≤ − αq + δ P qα − δ + t ( E m )= ⇒ P q,tµ ( E m ) ≤ − αq + δ P qα − δ + t ( E m ) . Finally, since E m ր A we conclude that P q,tµ ( A ) ≤ − αq + δ P qα − δ + t ( A ) . (2) (a) It is well known that the statement is true for q = 0 . For m ∈ N , we define the set E m by E m = (cid:26) x ∈ A (cid:12)(cid:12)(cid:12) log µ ( B ( x, r ))log r ≤ α − δq for < r < m (cid:27) . Next, fix m ∈ N and r > such that < r < m . Let (cid:16) B ( x i , r ) (cid:17) i ∈{ ,...,N r ( F ) } be a centred coveringof F ⊂ E m . We get log µ ( B ( x i , r ))log r ≤ α − δq = ⇒ µ ( B ( x i , r )) q ≤ r qα − δ = ⇒ X i µ ( B ( x i , r )) q ≤ N r ( F ) r qα − δ = ⇒ N qµ,r ( F ) ≤ N r ( F ) r qα − δ = ⇒ L q,tµ ( F ) ≤ t − αq + δ H qα − δ + t ( F )= ⇒ H q,tµ ( F ) ≤ − αq + δ H qα − δ + t ( E m ) . Putting these together we have that H q,tµ ( A ) ≤ − αq + δ H qα − δ + t ( A ) . This proves the lemma. (cid:3)
Theorem 6.
Let q ∈ R and suppose that H q, Λ µ ( q ) µ (supp µ ) > . Then, dim MB (cid:16) E − Λ ′ µ + ( q ) ∩ E − Λ ′ µ − ( q ) (cid:17) ≥ − Λ ′ µ − ( q ) q + Λ µ ( q ) , for q ≤ , − Λ ′ µ + ( q ) q + Λ µ ( q ) , for q ≥ .Proof. It is well known from Lemma 3 that for all δ > and t ∈ R , H − Λ ′ µ + ( q ) q + t − δ ( E ( q )) ≥ − Λ ′ µ + ( q ) q − δ H q,tµ ( E ( q )) , for q ≥ , H − Λ ′ µ − ( q ) q + t − δ ( E ( q )) ≥ − Λ ′ µ − ( q ) q − δ H q,tµ ( E ( q )) , for q ≤ where the set E ( q ) is defined by E ( q ) = E − Λ ′ µ + ( q ) ∩ E − Λ ′ µ − ( q ) . Theorem 6 is then an easy consequence of the following lemma. (cid:3)
Lemma 4.
One has H q, Λ µ ( q ) µ (cid:16) supp µ \ E ( q ) (cid:17) = 0 . Proof.
Let us introduce, for α and β in R X α = supp µ \ E α and Y β = supp µ \ E β . It clearly suffices to prove that H q, Λ µ ( q ) µ (cid:0) X α (cid:1) = 0 , for all α < − Λ ′ µ + ( q ) (3.1)and H q, Λ µ ( q ) µ (cid:0) Y β (cid:1) = 0 , for all β > − Λ ′ µ − ( q ) . (3.2)Indeed, it is clear that ≤ H q, Λ µ ( q ) µ (cid:16) supp µ \ (cid:0) E − Λ ′ µ + ( q ) ∩ E − Λ ′ µ − ( q ) (cid:1)(cid:17) ≤ H q, Λ µ ( q ) µ (cid:16) supp µ \ E − Λ ′ µ + ( q ) (cid:17) + H q, Λ µ ( q ) µ (cid:16) supp µ \ E − Λ ′ µ − ( q ) (cid:17) ≤ H q, Λ µ ( q ) µ [ α< − Λ ′ µ + ( q ) E α + H q, Λ µ ( q ) µ [ β> − Λ ′ µ − ( q ) E β ≤ X α< − Λ ′ µ + ( q ) H q, Λ µ ( q ) µ (cid:0) X α (cid:1) + X β> − Λ ′ µ − ( q ) H q, Λ µ ( q ) µ (cid:0) Y β (cid:1) = 0 . MULTIFRACTAL FORMALISM FOR HEWITT-STROMBERG MEASURES 17
We only have to prove that (3.1). The proof of (3.2) is identical to the proof of (3.1) and is therefore omitted.Let α < − Λ ′ µ + ( q ) and t > , such that Λ µ ( q + t ) < Λ µ ( q ) − αt , we have C q + t, Λ µ ( q ) − αtµ (cid:0) supp µ (cid:1) = 0 . For x ∈ X α and δ > , we can find λ x ≥ and δλ x < r x < δ, such that µ ( B ( x, r x )) > r αx . The family (cid:16) B ( x, r x ) (cid:17) x ∈ X α is a centered δ -covering of X α . Then, we can choose a finite subset J of N such thatthe family (cid:16) B ( x i , r x i ) (cid:17) i ∈ J is a centered δ -covering of X α . Take λ = sup i ∈ J λ x i , then for all i ∈ J , we have µ ( B ( x i , δ )) ≥ µ ( B ( x i , r x i )) > r αx i ≥ (cid:18) δλ (cid:19) α . Since (cid:16) B ( x i , δ ) (cid:17) i ∈ J is a centered covering of X α . Then, using Besicovitch’s covering theorem, we can con-struct ξ finite sub-families (cid:16) B ( x j , δ ) (cid:17) j , . . . , (cid:16) B ( x ξj , δ ) (cid:17) j , such that each X α ⊆ ξ [ i =1 [ j B ( x ij , δ ) and (cid:16) B ( x ij , δ ) (cid:17) j is a packing of X α . We clearly have µ ( B ( x ij , δ ) q δ Λ µ ( q ) ≤ λ αt µ ( B ( x ij , δ ) q + t δ Λ µ ( q ) − αt . It therefore follows that N qµ ( X α ) δ Λ µ ( q ) ≤ λ αt ξ M q + tµ ( X α ) δ Λ µ ( q ) − αt . Letting δ → now yields H q, Λ µ ( q ) µ ( X α ) ≤ L q, Λ µ ( q ) µ ( X α ) ≤ αt λ αt ξ C q + t, Λ µ ( q ) − αtµ ( X α ) ≤ αt λ αt ξ C q + t, Λ µ ( q ) − αtµ (supp µ ) = 0 . Remark that, in the last inequality, we can replace X α by any arbitrary subset of X α . Then, we can finally concludethat H q, Λ µ ( q ) µ ( X α ) ≤ αt λ αt ξ C q + t, Λ µ ( q ) − αtµ (supp µ ) = 0 . This completes the proof of (3.1). (cid:3)
The following result proves that the condition H q, Λ µ ( q ) µ (supp µ ) > is very close to being a necessary andsufficient condition for the validity of our multifractal formalism . Theorem 7.
Let q ∈ R and µ be a compact supported Borel probability measure on R n . Suppose that one of thefollowing hypotheses is satisfied, (1) dim MB (cid:16) E − Λ ′ µ + ( q ) ∩ E − Λ ′ µ − ( q ) (cid:17) ≥ − Λ ′ µ + ( q ) q + Λ µ ( q ) , for q ≤ . (2) dim MB (cid:16) E − Λ ′ µ + ( q ) ∩ E − Λ ′ µ − ( q ) (cid:17) ≥ − Λ ′ µ − ( q ) q + Λ µ ( q ) , for q ≥ . Then, b µ ( q ) = B µ ( q ) = Λ µ ( q ) . In other words, H q,tµ (supp µ ) > for all t < Λ µ ( q ) . Proof.
We have, for q ≥ E − Λ ′ µ + ( q ) ∩ E − Λ ′ µ − ( q ) ⊆ E − Λ ′ µ − ( q ) , it follows immediately that − Λ ′ µ − ( q ) q + Λ µ ( q ) ≤ dim MB (cid:16) E − Λ ′ µ + ( q ) ∩ E − Λ ′ µ − ( q ) (cid:17) ≤ dim MB (cid:16) E − Λ ′ µ − ( q ) (cid:17) . Now, suppose that α = − Λ ′ µ − ( q ) . We only prove the case where q ≥ . The other one is very similar and istherefore omitted. We have dim MB (cid:16) E α (cid:17) ≥ αq + Λ µ ( q ) . Since b µ ( q ) ≤ B µ ( q ) ≤ Λ µ ( q ) , we only have to prove that b µ ( q ) ≥ Λ µ ( q ) . Let t < Λ µ ( q ) and choose β, such that β < α . Then, βq + t < αq + Λ µ ( q ) . For p ∈ N , we consider the set F p = (cid:26) x ∈ E α (cid:12)(cid:12)(cid:12) µ ( B ( x, r )) ≥ r β , < r < p (cid:27) . It is clear that F p ր E α as p → ∞ . It follows that, there exists p > , such that dim MB ( F p ) > βq + t ⇒ H βq + t ( F p ) > . Let < r < p and (cid:16) B ( x i , r ) (cid:17) i be a centered covering of F p . Then, X i µ ( B ( x i , r )) q r t ≥ X i r βq + t ≥ N r ( F p ) r βq + t . We conclude that N qµ,r ( F p ) (2 r ) t ≥ t N r ( F p ) r βq + t and then L q,tµ ( F p ) ≥ − βq H βq + t ( F p ) . This implies that H q,tµ (supp µ ) ≥ H q,tµ ( E α ) ≥ H q,tµ ( F p ) ≥ − βq H βq + t ( F p ) > . It therefore follows that t ≤ b µ ( q ) . Finally, we get b µ ( q ) = B µ ( q ) = Λ µ ( q ) . (cid:3) Corollary 2.
Assume that H q, Λ µ ( q ) µ (supp µ ) > hold for all q ∈ R and that Λ µ is differentiable at q . Let α = − Λ ′ µ ( q ) , there holds dim MB (cid:0) E ( α ) (cid:1) = dim MB (cid:0) E ( α ) (cid:1) = b ∗ µ ( α ) = B ∗ µ ( α ) = Λ ∗ µ ( α ) . Remark 4.
The results of Theorems 6, 7 and Corollary 2 hold if we replace the multifractal function Λ µ by thefunction B µ . Now, we deal with the case where the lower and upper multifractal Hewitt-Stromberg functions b µ and B µ donot necessarily coincide. Theorem 8.
Let q ∈ R and µ be a compact supported Borel probability measure on R n . (1) If the multifractal function b µ is differentiable at q, then, provided that b ∗ µ (cid:16) − b ′ µ ( q ) (cid:17) ≥ and H q, b µ ( q ) µ (cid:16) E (cid:16) − b ′ µ ( q ) (cid:17) (cid:17) > , we have dim H E (cid:16) − b ′ µ ( q ) (cid:17) = dim MB E (cid:16) − b ′ µ ( q ) (cid:17) = b ∗ µ (cid:16) − b ′ µ ( q ) (cid:17) = b ∗ µ (cid:16) − b ′ µ ( q ) (cid:17) . (2) If the multifractal function B µ is differentiable at q, then, provided that B ∗ µ (cid:16) − B ′ µ ( q ) (cid:17) ≥ and P q,B µ ( q ) µ (cid:16) E (cid:16) − B ′ µ ( q ) (cid:17) (cid:17) > , we have dim P E (cid:16) − B ′ µ ( q ) (cid:17) = dim MB E (cid:16) − B ′ µ ( q ) (cid:17) = B ∗ µ (cid:16) − B ′ µ ( q ) (cid:17) = B ∗ µ (cid:16) − B ′ µ ( q ) (cid:17) . Proof.
The proof is similar to the one of Theorem 5. (cid:3)
4. E
XAMPLES
In this section, more motivations and examples related to these concepts, will be discussed.
MULTIFRACTAL FORMALISM FOR HEWITT-STROMBERG MEASURES 19
Example 1.
The classical multifractal formalism has been proved rigorously for random and non-randomself-similar measures [42, 43], for self-affine measures [10, 45], for quasi self-similar measures [41], for quasi-Bernoulli measures [10], for graph directed self-conformal measures [42] and for some Moran measures [61, 62].Specifically, we have b µ ( q ) = b µ ( q ) = B µ ( q ) = B µ ( q ) and for some α ≥ , we get f µ ( α ) = f µ ( α ) = F µ ( α ) = F µ ( α ) = b ∗ µ ( α ) = b ∗ µ ( α ) = B ∗ µ ( α ) = B ∗ µ ( α ) . Example 2 : Multifractal formalism of homogeneous Moran measures.
We will start by defining thehomogeneous Moran sets. Let { n k } k and { Φ k } k ≥ be respectively two sequences of positive integers and positivevectors such that Φ k = (cid:16) c k , c k , . . . , c k nk (cid:17) , n k X j =1 c kj ≤ , k ∈ N . For any m, k ∈ N , such that m ≤ k , let D m,k = n ( i m , i m +1 , . . . , i k ) (cid:12)(cid:12) ≤ i j ≤ n j , m ≤ j ≤ k o and D k = D ,k = n ( i , i , . . . , i k ) (cid:12)(cid:12) ≤ i j ≤ n j , ≤ j ≤ k o . We also set D = ∅ and D = ∪ k ≥ D k , Considering σ = ( i , i , . . . , i k ) ∈ D k , τ = ( j , j , . . . , j m ) ∈ D k +1 ,m , we set σ ∗ τ = ( i , i , . . . , i k , j , j , . . . , j m ) . Definition 1.
Let X be a complete metric space and I ⊆ X a compact set with no empty interior (for convenience,we assume that the diameter of I is 1). The collection F = { I σ , σ ∈ D } of subsets of I is said to have ahomogeneous Moran structure, if it satisfies the following conditions (MSC): a: I ∅ = I . b: For all k ≥ , ( i , i , . . . , i k − ) ∈ D k − , I i i ...i k (cid:0) i k ∈ { , , . . . , n k } (cid:1) are subsets of I i i ...i k − and I ◦ i i ...i k − ,i k ∩ I ◦ i i ...i k − ,i ′ k = ∅ , ≤ i k < i ′ k ≤ n k , where I ◦ denotes the interior of I . c: For all k ≥ and ≤ j ≤ n k , taking ( i , i , . . . , i k − , j ) ∈ D k , we have < c kj = c i i ...i k − j = | I i i ...i k − j || I i i ...i k − | < , k ≥ , where | I | denotes the diameter of I . Suppose that F is a collection of subsets of I having a homogeneous Moran structure. We call E = \ k ≥ [ σ ∈ D k I σ , a homogeneous Moran set determined by F , and call F k = n σ (cid:12)(cid:12) σ ∈ D k o the k -order fundamental sets of E . I is called the original set of E . We assume lim k →∞ max σ ∈ D k | I σ | = 0 . Then, for all i ∈ D , the set \ n ≥ I i i ...i n is asingle point. We use the abbreviation w | k for the first k elements of the sequence w = ( i , i , . . . , i k , . . . ) ∈ D, I k ( w ) = I w | k = I i i ...i k . Here, we consider a class of homogeneous Moran sets E witch satisfy a special property called the strongseparation condition ( SSC ), i.e., take I σ ∈ F . Let I σ ∗ , I σ ∗ , . . . , I σ ∗ n k +1 be the n k +1 basic intervals of order k + 1 contained in I σ arranged from the left to the right, Then we assume that for all ≤ i ≤ n k +1 − ,dist ( I σ ∗ i , I σ ∗ ( i +1) ) ≥ δ k | I σ | , for all i = j, where ( δ k ) k is a sequence of positive real numbers, such that < δ = inf k δ k . We now define a Moran measure. Let n p i,j o n i j =1 be the probability vectors, i.e. p i,j > and P n i j =1 p i,j = 1 ( i = 1 , , , .... ), suppose that p = inf { p i,j } > . Let µ be a mass distribution on E , such that for any I σ ( σ ∈ D k ) µ ( I σ ) = p ,σ p ,σ . . . p ,σ k and µ X σ ∈ D k I σ ! = 1 , we call µ be Moran measure.Finally we define an auxiliary function β k ( q ) as follows: for all k ≥ and q ∈ R , there is a unique number β k ( q ) satisfying X σ ∈ D k p qσ | I σ | β k ( q ) = 1 . Set β ( q ) = lim inf k → + ∞ β k ( q ) and β ( q ) = lim sup k → + ∞ β k ( q ) Theorem 9.
Suppose that E is a homogeneous Moran set satisfying ( SSC ) and µ is the Moran measure on E , (1) then for all q ∈ R , b µ ( q ) = b µ ( q ) = Θ qµ (supp µ ) = β ( q ) and B µ ( q ) = B µ ( q ) = Λ qµ (supp µ ) = β ( q ) . (2) Suppose that β ′ ( q ) exists and for this real number q (a) there is k ∈ N such that β ( q ) ≤ β k ( q ) for all k ≥ k , or (b) there is some c > and n ∈ N such that β k i ( q ) − β ( q ) ≤ ck i for all k i ≥ n with β k i ( q ) < β ( q ) . (c) β ( q ) is smooth.Then there exist numbers ≤ α ≤ α such that f µ ( α ) = f µ ( α ) = b ∗ µ ( α ) = b ∗ µ ( α ) = β ∗ ( α ) , if α ∈ ( α, α )0 , if α / ∈ ( α, α ) . (3) Suppose that β ′ ( q ) exists and for this real number q (a) there is k ∈ N such that β ( q ) ≥ β k ( q ) for all k ≥ k , or (b) there is some c > and n ∈ N such that β ( q ) − β k i ( q ) ≤ ck i for all k i ≥ n with β k i ( q ) < β ( q ) . (c) β ( q ) is smooth.Then there exist numbers ≤ γ ≤ γ such that F µ ( α ) = F µ ( α ) = B ∗ µ ( α ) = B ∗ µ ( α ) = β ∗ ( α ) , if α ∈ ( γ, γ )0 , if α / ∈ ( γ, γ ) . (4) If the limit lim inf k → + ∞ β k ( q ) = β ( q ) exists, and for all k ≥ , k ( β ( q ) − β k ( q )) < + ∞ , suppose that α = − β ′ ( q ) exists, then f µ ( α ) = f µ ( α ) = F µ ( α ) = F µ ( α ) = b ∗ µ ( α ) = b ∗ µ ( α ) = B ∗ µ ( α ) = B ∗ µ ( α ) . Proof.
All of the ideas needed to prove Theorem 9 can be found in [63, 64], Propositions 2, 3 and 4 and Theorem8. (cid:3)
MULTIFRACTAL FORMALISM FOR HEWITT-STROMBERG MEASURES 21
Moran measures for which the classical multifractal formalism is valid . Let n k = , k is odd number, , k is even number. c k = r , k is odd number, r , k is even number,where < r < and < r < . Put p k,j = p ,j , k is odd number, ≤ j ≤ ,p ,j , k is even number, ≤ j ≤ , where X j =1 p ,j = 1 and X j =1 p ,j = 1 . We therefore conclude that β k ( q ) = − log P j =1 p q ,j − k − k +1 log P j =1 p q ,j log r + k − k +1 log r , k is odd number, − log P j =1 p q ,j − log P j =1 p q ,j log r + log r , k is even number,and β ( q ) = lim k → + ∞ β k ( q ) = − log P j =1 p q ,j − log P j =1 p q ,j log r + log r . This clearly implies that k ( β ( q ) − β k ( q )) < + ∞ and β ′ ( q ) exists. Now, it follows immediately from Theorem 9that f µ ( α ) = f µ ( α ) = F µ ( α ) = F µ ( α ) = b ∗ µ ( α ) = b ∗ µ ( α ) = B ∗ µ ( α ) = B ∗ µ ( α ) . Moran measures for which the classical multifractal formalism does not hold . Let { T k } k ≥ be a se-quence of integers such that T = 1 , T k < T k +1 and lim k → + ∞ T k +1 T k = + ∞ We define the family n i = , if T k − ≤ i < T k , , if T k ≤ i < T k +1 .c i = r , if T k − ≤ i < T k ,r , if T k ≤ i < T k +1 , where < r < and < r < . Put p i,j = p ,j , if T k − ≤ i < T k , ≤ j ≤ ,p ,j , if T k ≤ i < T k +1 , ≤ j ≤ , where X j =1 p ,j = 1 and X j =1 p ,j = 1 . We therefore conclude from this β k ( q ) = log P σ ∈ D k µ ( I σ ) q − log c . . . c k . Finally, if N k is the number of integers i ≤ k such that p i,j = p ,j , we have β k ( q ) = − N k k log( p q , + p q , ) + (1 − N k k ) log( p q , + p q , + p q , ) N k k log r + (1 − N k k ) log r . Observing that lim inf k → + ∞ N k k = 0 and lim sup k → + ∞ N k k = 1 . We can then conclude that lim inf k → + ∞ β k ( q ) = inf (cid:26) log( p q , + p q , ) − log r , log( p q , + p q , + p q , ) − log r (cid:27) and lim sup k → + ∞ β k ( q ) = sup (cid:26) log( p q , + p q , ) − log r , log( p q , + p q , + p q , ) − log r (cid:27) . It results that for < q < , we have b µ ( q ) = b µ ( q ) = β ( q ) = lim inf k → + ∞ β k ( q ) = log( p q , + p q , ) − log r , b µ ( q ) = b µ ( q ) = β ( q ) = lim inf k → + ∞ β k ( q ) = log( p q , + p q , + p q , ) − log r
In the following, we give an example of a measure for which the lower and upper multifractalHewitt-Stromberg functions are different and the Hausdorff and packing dimensions of the level sets of the localH¨older exponent E ( α ) are given by the Legendre transform respectively of lower and upper multifractal Hewitt-Stromberg functions. Take < p < ˆ p ≤ / and a sequence of integers t < t < . . . < t n < . . . , such that lim n → + ∞ t n +1 t n = + ∞ . The measure µ assigned to the diadic interval of the n-th generation I ε ε ...ε n is µ (cid:0) I ε ε ...ε n (cid:1) = n Y j =1 ̟ j , where if t k − ≤ j < t k for some k, ̟ j = p if ε j = 0 , ̟ j = 1 − p otherwise , if t k ≤ j < t k +1 for some k, ̟ j = ˆ p if ε j = 0 , ̟ j = 1 − ˆ p otherwise.Now, for q ∈ R we define, τ ( q ) = log (cid:0) p q + (1 − p ) q (cid:1) and ˆ τ ( q ) = log (cid:0) ˆ p q + (1 − ˆ p ) q (cid:1) . It results from [11, 13] that b µ ( q ) = b µ ( q ) = τ ( q ) < ˆ τ ( q ) = B µ ( q ) = B µ ( q ) , for < q < ,b µ ( q ) = b µ ( q ) = ˆ τ ( q ) < τ ( q ) = B µ ( q ) = B µ ( q ) , for q < or q > . Then we have the following result,
Theorem 10.
Let α ≥ . MULTIFRACTAL FORMALISM FOR HEWITT-STROMBERG MEASURES 23 (1)
For α ∈ (cid:16) − log (1 − ˆ p ) , − log (ˆ p ) (cid:17) , then we have f µ ( α ) = f µ ( α ) = b ∗ µ ( α ) = b ∗ µ ( α ) . (2) For α ∈ (cid:16) − log (1 − ˆ p ) , − log (ˆ p ) (cid:17) \ (cid:16)h − B ′ µ + (0) , − B ′ µ − (0) i S h − B ′ µ + (1) , − B ′ µ − (1) i(cid:17) , we have F µ ( α ) = F µ ( α ) = B ∗ µ ( α ) = B ∗ µ ( α ) . Proof.
All of the ideas needed to prove this theorem can be found in [11, Proposition 9], Propositions 2, 3 and 4and Theorem 8. (cid:3)
Example 4.
Given a class of exact dimensional measures (inhomogeneous multinomial measures) whosesupport is the whole interval [0 , , the multifractal functions b µ , b µ , B µ and B µ are real analytic and agree at twopoints only and (for more details, see [56]). These measures satisfy our multifractal formalism in the sensethat, for α in some interval, the Hausdorff dimension of the level sets E ( α ) is given by the Legendre transform oflower multifractal Hewitt-Stromberg function and their packing dimension by the Legendre transform of the uppermultifractal Hewitt-Stromberg function. More specifically, b µ ( q ) = b µ ( q ) < B µ ( q ) = B µ ( q ) for all q / ∈ { , } ,f µ ( α ) = f µ ( α ) = b ∗ µ ( α ) = b ∗ µ ( α ) and F µ ( α ) = F µ ( α ) = B ∗ µ ( α ) = B ∗ µ ( α ) for some α.
5. O
PEN PROBLEMS
Motivated by some results and examples developed in [42, 43, 44, 45, 47, 65], we therefore ask the followingquestions.(1) Let µ ∈ P D ( R n ) , E ⊆ supp µ , p, q ∈ R and α ∈ [0 , . Then, the following problem remains open: b αp +(1 − α ) qµ ( E ) ≤ α B pµ ( E ) + (1 − α ) b qµ ( E )? (2) Let q ∈ R and assume that B µ ( q ) = b µ ( q ) . Are the measures H q,tµ x supp µ and P q,tµ x supp µ proportional, i.e.does there exists a constant c q > such that P q,tµ x supp µ = c q H q,tµ x supp µ ? Even though it seems rather unlikely that the lower and upper multifractal Hewitt-Stromberg measures areproportional in general, the ratio of the measures H q,tµ x supp µ and P q,tµ x supp µ might still be bounded. Wetherefore ask the following question: Does there exists a number < c q < + ∞ such that H q,tµ x supp µ ≤ P q,tµ x supp µ ≤ c q H q,tµ x supp µ ? (3) Let q ∈ R and assume that b µ ( q ) = b µ ( q ) . Are the measures H q,tµ x supp µ and H q,tµ x supp µ proportional, i.e.does there exists a constant C q > such that H q,tµ x supp µ = C q H q,tµ x supp µ ? Even though it seems rather unlikely that the multifractal Hausdorff measure and the lower multifractalHewitt-Stromberg measure are proportional in general, the ratio of the measures H q,tµ x supp µ and H q,tµ x supp µ might still be bounded. We therefore ask the following question: Does there exists a number < C q < + ∞ such that H q,tµ x supp µ ≤ H q,tµ x supp µ ≤ C q H q,tµ x supp µ ? (4) Let p, q ∈ R and assume that b µ ( q ) is differentiable at p and q with b ′ µ ( p ) = b ′ µ ( q ) . Then, the followingproblem remains open: H p, b µ ( p ) µ x supp µ ⊥ H q, b µ ( q ) µ x supp µ ? (5) Let p, q ∈ R and assume that B µ ( q ) is differentiable at p and q with B ′ µ ( p ) = B ′ µ ( q ) . Then, the followingproblem remains open: P p, B µ ( p ) µ x supp µ ⊥ P q, B µ ( q ) µ x supp µ ? (6) Is it true that the weaker condition b µ ( q ) = b µ ( q ) is sufficient to obtain the conclusion of Theorem 8?(7) Let µ ∈ P D ( R n ) , ν ∈ P D ( R m ) and q, s, t ∈ R . Assume that c > , E ⊆ R n , F ⊆ R m , H ⊆ R n + m and H ( y ) = (cid:8) x ; ( x, y ) ∈ H (cid:9) . Then, the following problem remains open: Z H q,sµ ( H ( y )) d H q,tν ( y ) ≤ c H q,s + tµ × ν ( H ) H q,s + tµ × ν ( E × F ) ≤ c H q,sµ ( E ) P q,tν ( F ) Z H q,sµ ( H ( y )) d P q,tν ( y ) ≤ c P q,s + tµ × ν ( H ) P q,s + tµ × ν ( E × F ) ≤ c P q,sµ ( E ) P q,tν ( F ) and b qµ ( E ) + b qν ( F ) ≤ b qµ × ν ( E × F ) ≤ b qµ ( E ) + B qν ( F ) ≤ B qµ × ν ( E × F ) ≤ B qµ ( E ) + B qν ( F )? (8) The multifractal Hausdorff dimension function b µ and the lower multifractal Hewitt-Stromberg function b µ do not necessarily coincide. Motivated by the results developed in [40], we conjecture that there existBorel probability measures µ on R n such that b µ = b µ = B µ = B µ or b µ < b µ < B µ = B µ < Λ µ for all q = 1 . In particular, this will imply that f µ ( α ) = b ∗ µ ( α ) < f µ ( α ) = F µ ( α ) = F µ ( α ) = b ∗ µ ( α ) = B ∗ µ ( α ) = B ∗ µ ( α ) for some α ≥ . A CKNOWLEDGMENTS
The authors would like to thank Professor
Lars Olsen for his first reading of this work, the interest he gave to itand his valuable comment which improves the presentation of the paper. And they thank the anonymous refereesfor their valuable comments and suggestions that led to the improvement of the manuscript.R
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