� EXTRACTA MATHEMATICAE Volumen 33, Número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura EXTRACTA MATHEMATICAE Vol. 35, Num. 2 (2020), 229 – 252 doi:10.17398/2605-5686.35.2.229 Available online September 16, 2020 Multifractal formalism of an inhomogeneous multinomial measure with various parameters A. Samti Analysis, Probability & Fractals Laboratory LR18ES17 University of Monastir, Faculty of Sciences of Monastir Department of Mathematics, 5019-Monastir, Tunisia amal samti@yahoo.fr Received May 18, 2020 Presented by Mostafa Mbekhta Accepted July 7, 2020 Abstract: In this paper, we study the refined multifractal formalism in a product symbolic space and we estimate the spectrum of a class of inhomogeneous multinomial measures constructed on the product symbolic space. Key words: Hausdorff dimension, packing dimension, fractal, multifractal. AMS Subject Class. (2010): 28A80, 28A78, 28A12, 11K55. 1. Introduction The multifractal formalism of a measure µ aims to establish a relationship between the dimension of level set of the local Hölder exponent of µ to the Legendre transform of what is called the ”free energy” function. A problem initially raised and studied for physical motivations [8, 9, 11, 12, 10]. It will be convenient to give a brief description of the multifractal formalism. Let X be a metric space. The local Hölder exponent αµ(x) at the point x ∈ X is defined to be αµ(x) = lim r→0 log µ(B(x,r)) log r where B(x,r) stands for the ball of radius r centered at x. The measure µ is said to satisfy the multifractal formalism at α ≥ 0, if the Hausdorff dimension (dim) and the packing dimension (Dim) of the level set E(α) which is defined by E(α) = {x ∈ supp(µ) : αµ(x) = α} , are equal respectively to the value of the Legendre transform at α of a scale function τµ associated to the measure µ, i.e., dim E(α) = Dim E(α) = τ∗µ(α), ISSN: 0213-8743 (print), 2605-5686 (online) c©The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0) https://doi.org/10.17398/2605-5686.35.2.229 mailto:amal_samti@yahoo.fr https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 230 a. samti where f∗(x) = inf y (xy + f(y)) is the Legendre transform of a function f and supp(µ) is the topological support of µ. The upper bound for dim E(α) (respectively Dim E(α)) is obtained by a standard covering argument as Besicovisch’s covering Theorem and Vitali’s Lemma [13]. However, the lower bound is usually much harder to prove, it is related to the existence of an auxiliary measure such as a Gibbs measure [13] or a Frostman measure [3] which is supported by the set to be analyzed. For this reason, F. Ben Nasr et al. [4] improved the Olsen’s result in de- scribing a class of measures satisfying the multifractal formalism and proposed a new sufficient condition that gives the lower bound. In such a situation, they concluded that Bµ(q) = bµ(q), where bµ and Bµ are Olsen’s functions. Besides, they constructed inhomogeneous Bernoulli products, such measures whose both multifractal dimension functions bµ and Bµ agree at one or two points only. Which implies a valid refined multifractal formalism no more than two points. In [5], Ben Nasr and Peyrière constructed an example of a “bad” measure on the interval {0, 1}N for which the Olsen’s functions bµ and Bµ differ and the Hausdorff dimensions of the sets E(α) are given by the Legendre transform of bµ, and their packing dimensions by the Legendre transform of Bµ, i.e., bµ(q) < Bµ(q) for all q ∈{0, 1} and dim E(α) = b∗µ(α) and Dim E(α) = B ∗ µ(α), for some α ≥ 0. Shen [14] and Wu et al. [17, 18, 19] revisited this example such that the functions Bµ and bµ can be real analytic. Motivated by these examples N. Attia and B. Selmi [1, 2] introduced and studied a new multifractal formalism based on the Hewitt-Stromberg measures and showed that this formalism is completely parallel to Olsen’s multifractal formalism based on the Hausdorff and packing measures. In the present work, let 2 ≤ r1 < r2 be two integers, we consider a class of measures defined on a product symbolic space A1 × A2 endowed with the distance product where Ai = {0, . . . ,ri − 1} for i = 1, 2, and constructed on the rectangles that flatten as their diameters tend to zero. However, these rectangles do not allow the calculation of the Hausdorff dimension, hence the difficulty of the problem. The aim of this paper is to study the validity of the refined multifractal formalism of this class of measures. The paper is organized as follows. In Section 2, we give some notations and definitions which will be useful. In the third section we consider a sequence of finite partitions of a product symbolic space made of rectangles and we show through an example that the almost squares allow the calculation of the multifractal formalism of an inhomogeneous measure 231 Hausdorff and packing dimensions. In Section 4, we consider a variant of the refined multifractal formalism as already introduced by Ben Nasr and Peyrière [5] which we adapt it to almost squares and estimate the dimensions of the level sets E(α). Finally, we apply our results to a class of inhomogeneous measures defined on the product symbolic space. 2. Notations and definitions In this section, we will recall the Hausdorff and packing measures and their dimensions. Let (X,d) be a separable metric space. The diameter of a non-empty set E ⊆ X is given by diam E = sup{d(x,y) : x,y ∈ E} , with the convention that diam(∅) = 0. We define the closed ball with center x ∈ X and radius r > 0 as B(x,r) = {y ∈ X : d(x,y) ≤ r} . A finite or countable collection of subsets {Ui}i of X is called a δ-cover of E ⊆ X, if for each i we have diam Ui ≤ δ and E ⊂ ⋃ i Ui. Suppose that E is a subset of X and s is a non-negative number. For any δ > 0 we define Hsδ(E) = inf {∑ i diam(Ui) s : {Ui}i is a δ-cover of E } . As δ decreases, the class of δ-covers of E is reduced. Therefore, this infimum increases and approaches a limit as δ ↘ 0. Thus we define Hs(E) = lim δ→0 Hsδ(E). We term Hs(E) the s-dimensional Hausdorff measure of E. Then we define the Hausdorff dimension of E as dim(E) = sup{s ≥ 0 : Hs(E) = ∞} = inf {s ≥ 0 : Hs(E) = 0} . Remark 1. Notice that the covering of E with centered balls in E allow the calculation of the Hausdorff dimension of E, for more details see [7]. 232 a. samti We will now define the packing measure. First, let define a δ-packing of E ⊂ X to be a finite or countable collection of disjoint balls {B(xi,ri)}i of diameter at most δ and with centers in E. For s ≥ 0 and δ > 0, let Psδ(E) = sup {∑ i (2ri) s : {B(xi,ri)}i is a δ-packing of E } . From this the s-dimensional pre-packing measure Ps of E is defined by Ps(E) = lim δ→0 Psδ(E). Finally, we define the s-dimensional packing measure Ps(E) of E by Ps(E) = inf {∑ i Ps(Ei) : E ⊂ ∞⋃ i=1 Ei } . The packing dimension of E, denoted by Dim(E), is defined in the same way as Hausdorff dimension, that means Dim(E) = sup{s ≥ 0 : Ps(E) = ∞} = inf {s ≥ 0 : Ps(E) = 0} . For more details about the Hausdorff, packing measures and their dimen- sions see [15, 16, 7]. 3. Calculation of the Hausdorff and packing dimensions on the product symbolic space on different basis For practical reasons, we shall need basic notions about the set of words on an alphabet. Let 2 ≤ r1 < r2 be two integers. For i ∈ {1, 2}, given Ai = {0, . . . ,ri − 1} a finite alphabet. For all n ∈ N∗, each element in Ani is denoted by a string of n letters or digits in Ai that we call a word; by convention A0i is reduced to the empty word ∅. Let A ∗ i = ⋃ n≥0 A n i be the set of finite words built over Ai and Ai = AN ∗ i the symbolic space over Ai. The set A∗i ∪ Ai is endowed with the concatenation operation: If ω ∈ A ∗ i and ω′ ∈ A∗i ∪ Ai, we denote by ω.ω ′ the word obtained by juxtaposition of the two words ω and ω′. For each finite word ω ∈ A∗i , [ω] is the cylinder ω ·Ai = {ω ·ω ′ : ω′ ∈ Ai}. Furthermore, if ω = ω1 · · ·ωk · · · ∈ Ai and n ∈ N then ω|n stands for the prefix ω1 · · ·ωn of ω for n ≥ 1 and the empty word otherwise. Each set Ai multifractal formalism of an inhomogeneous measure 233 is endowed with the ultrametric distance di : (z,z ′) ∈ A2i 7−→ r −|z∧z′| i , where z∧z′ is defined to be the longest prefix common to both z and z′ and |z| the length of a word z ∈ A∗i ∪ Ai. Then the product symbolic space A1 × A2 is endowed with the ultrametric distance. d((x,y), (x′,y′)) = max(d1(x,x ′),d2(y,y ′)). In the next, if ω ∈ Ak1 and ω ′ ∈ Ak ′ 2 , we call R(ω,ω ′) the rectangle obtained as the product of the cylinders [ω] and [ω′]. We denote by ∣∣R(ω,ω′)∣∣ M = sup ( 1 rk1 , 1 rk ′ 2 ) , and ∣∣R(ω,ω′)∣∣ m = inf ( 1 rk1 , 1 rk ′ 2 ) . We say that a sequence {ξn}n≥1 of finite partitions of A1 × A2 made of rectangles satisfies condition (1) if lim n→∞ sup R∈ξn diam(R) = 0 and ξn+1 is a refinement of ξn. (1) In all over this work, we will consider a sequence {ξn}n≥1 of finite partitions of A1 × A2 made of rectangles verifying (1) and we put ξ = ⋃ n≥1 ξn. If R belongs to ξn+1, we define by p(R) the element of ξn that contains it. Let E be a nonempty subset of A1 × A2 and s a strictly positive real number. For all ε > 0, a finite or countable collection of rectangles {Rj}j is called an ε-covering of E if diam(Rj) ≤ ε for all j and E ⊂ ⋃ j Rj. Let Hsξ,ε(E) = inf  ∑ j diam(Rj) s : Rj ∈ ξ,{Rj}j is an ε-covering of E   and Hsξ(E) = lim ε→0 Hsξ,ε(E). Finally, the dimension dimξ(E) is given by dimξ(E) = inf { s > 0 : Hsξ(E) = 0 } = sup { s > 0 : Hsξ(E) = ∞ } . 234 a. samti Here, we define an ε-packing of E ⊂ A1 × A2 to be a finite or countable collection of disjoint rectangles {Rj}j of diameter not exceeding ε and with Rj ∩E 6= ∅. For ε > 0, we define Psξ,ε(E) = sup  ∑ j diam(Rj) s : Rj ∈ ξ,{Rj}j is an ε-packing of E   . Then Psξ,ε(E) decreases as ε increases, so we may take the limit Psξ(E) = lim ε→0 Psξ,ε(E). Unfortunately, Psξ(E) is not an outer measure, to overcome this difficulty we define Psξ (E) = inf  ∑ j Psξ(Ej) : E ⊆ ⋃ j Ej   . The definition of packing dimension parallels that of Hausdorff dimension. So, let Dimξ(E) defined such that Dimξ(E) = inf { s > 0 : Psξ (E) = 0 } = sup { s > 0 : Psξ (E) = ∞ } . In the following proposition we will give some conditions on a family ξ of rectangles of the symbolic space A1×A2 such that for every part E of A1×A2, we have dim(E) = dimξ(E) and Dim(E) = Dimξ(E). Proposition 3.1. Suppose that (i) lim n→∞ sup R∈ξn log |R |m / log |R |M = 1, (ii) lim n→∞ sup R∈ξn log |R |M / log |p(R)|M = 1. Then for any part E of A1 ×A2, we have Dimξ(E) = Dim(E), (2) dimξ(E) = dim(E). (3) multifractal formalism of an inhomogeneous measure 235 Proof. In order to prove the equality (2), we start by proving that Dimξ(E) ≤ Dim(E). Let t > Dim(E) and η > 0 such that t 1+η > Dim(E). It follows from assumption (i) that there exists an integer n0 such that for all n ≥ n0 and for all R ∈ ξn, we have |R|1+ηM ≤ |R|m . Take {Ej}j a cover of E and choose {Rk}k an ε-packing of Ej with ε ≤ inf R∈ξn0 diam(R). For j ∈ N, fix xk ∈ Rk ∩ Ej, we denote by Bk = B(xk, |Rk|m). It is clear that {Bk}k is an ε-packing of Ej. As ε ≤ inf R∈ξn0 diam(R) we get for all integer k, |Rk| 1+η M ≤ |Rk|m (4) and ∑ k diam(Rk) t ≤ ∑ k diam(Bk) t 1+η . Then, Ptξ,ε(Ej) ≤P t 1+η ε (Ej) and as ε goes to 0, yields Ptξ(Ej) ≤P t 1+η (Ej). Therefore, Ptξ(E) ≤P t 1+η (E) < +∞ consequently, Dimξ(E) < t, for all t > Dim(E), which implies that Dimξ(E) ≤ Dim(E). In order to obtain the other inequality, fix t > Dimξ(E) and η > 0 such that t 1+η > Dimξ(E). Using assumption (ii) there exists an integer n0 such that for all n ≥ n0 and for all R ∈ ξn, we have |P(R)|1+ηM ≤ |R |M . (5) 236 a. samti Let {Ej}j be a cover of E and {Bk = B(xk,rk)}k an ε-packing of Ej with ε ≤ inf R∈ξn0 diam(R). If Rk is a rectangle such that Rk ⊂ B(xk,rk) and P(Rk) * B(xk,rk), (6) then {Rk}k is an ε-packing of Ej. Since ε ≤ inf R∈ξn0 diam(R), we have for all k ∈ N, |P(Rk)| 1+η M ≤ |Rk|M . (7) Taking into account relations (6) and (7), we have∑ k diam(Bk) t ≤ ∑ k diam(P(Rk)) t ≤ ∑ k diam(Rk) t 1+η . So, Ptε(Ej) ≤P t 1+η ξ,ε (Ej). As ε goes to zero, Pt(Ej) ≤P t 1+η ξ (Ej). Then, we obtain Pt(E) ≤P t 1+η ξ (E) < +∞. Hence, Dim(E) ≤ Dimξ(E) which achieves the proof of equality (2). Now, we will be interested in proving the equality (3). It is easy to see that Ht(E) ≤Htξ(E) then dim(E) ≤ dimξ(E). Let’s prove that dimξ(E) ≤ dim(E). Fix t > dim(E) and η > 0 such that t 1+η + (2−2(1 +η)3) > dim(E). Let ε be a positive number such that ε ≤ inf R∈ξn0 diam(R). Pick an ε-covering {Rj}j of E and set Bj = B(xj, |Rj|M ) such that Rj ⊆ Bj. For all j ∈ N, there exists a family of disjoint rectangles {Rjk}k∈Lj such that ⋃ k∈Lj Rjk ⊂ Bj , P(Rjk) * Bj and Bj ⊆ ⋃ k∈Lj P(Rjk). multifractal formalism of an inhomogeneous measure 237 In a first step, we will calculate the number of P(Rjk) that cover Bj. We denote by λ the Lebesgue measure on A1 × A2. Using relations (4) and (5), we have λ(P(Rjk)) (1+η)2 ≤ λ(Rjk) and diam(Bj) 2(1+η)3 ≤ λ(Rjk). (8) Let s and s′ be two positive integers such that r −(s+1) 1 < |Rjk|M ≤ r −s 1 and r −(s′+1) 2 < |Rjk|M ≤ r −s′ 2 . We have ∑ k∈Lj λ(Rjk) ≤ λ(Bj) ≤ r−s1 r −s′ 2 ≤ (r1r2) diam(Bj) 2. (9) It follows from inequalities (8) and (9) that∑ k∈Lj diam(Bj) 2(1+η)3 ≤ r1r2 diam(Bj)2. Hence, card(Lj) ≤ r1r2 diam(Bj)2−2(1+η) 3 . In a second step, we have |P(Rjk)| 1+η M ≤ |Rjk|M ≤ diam(Bj) and ∑ k∈Lj |P(Rjk)| t M ≤ ∑ k∈Lj diam(Bj) t 1+η . So, ∑ j diam(Rjk) t ≤ ∑ j ∑ k∈Lj |P(Rjk)|tM ≤ ∑ j (r1r2) diam(Bj) 2−2(1+η)3 diam(Bj) t 1+η and Htξ,ε(E) ≤ (r1r2)H t 1+η +(2−2(1+η)3) ε (E). 238 a. samti Letting ε tend to 0 implies Htξ(E) ≤ (r1r2)H t 1+η +2−(1+η)3 (E). Finally, we obtain dimξ(E) ≤ t. And the result yields. Next, we set a generalization of the Billingsley Theorem [6] in our case. For this purpose, we introduce the following notations. If E is a non empty subset of A1 × A2 and x = (x1,x2) ∈ E, let ξ = ⋃ n≥1 ξn be a family of rectangles satisfying assumptions (i) and (ii) of Proposition 3.1 and Rn(x) be the rectangle of ξn containing x. In the sequel, we define by P(A1×A2) the set of Borel probability measures on A1 ×A2. For all µ ∈P(A1 ×A2) and ε > 0, and E ∈ A1 ×A2, we define µ]ε(E) = inf  ∑ j µ(Rj) : Rj ∈ ξ,{Rj}j an ε-covering of E   , µ](E) = lim ε→0 µ]ε(E) and ess sup x∈E,µ] A(x) = inf { t ∈ R : µ]({x ∈ E : A(x) > t}) = 0 } . Proposition 3.2. Let E be a subset of A1 × A2 and µ ∈ P(A1 × A2), we have (a) dimξ(E) ≤ sup x∈E lim inf n→∞ log µ(Rn(x)) log(diam(Rn(x))) ; (b) Dimξ(E) ≤ sup x∈E lim sup n→∞ log µ(Rn(x)) log(diam(Rn(x))) . If µ](E) > 0, then we have (c) dimξ(E) ≥ ess sup x∈E,µ] lim inf n→∞ log µ(Rn(x)) log(diam(Rn(x))) ; (d) Dimξ(E) ≥ ess sup x∈E,µ] lim sup n→∞ log µ(Rn(x)) log(diam(Rn(x))) . multifractal formalism of an inhomogeneous measure 239 Proof. Let us prove assumption (a). Take δ> sup x∈E lim inf n→∞ log µ(Rn(x)) log(diam(Rn(x))) , then for all x ∈ E, there exists k ≥ n such that µ(Rk(x)) ≥ diam(Rk(x))δ. Let ε be a positive number, there exists {Rj}j a family of pairwise disjoint rectangles such that E ⊂ ⋃ j Rj with µ(Rj) ≥ diam(Rj)δ and diam(Rj) ≤ ε. We have ∑ j diam(Rj) δ ≤ ∑ j µ(Rj) < ∞. Therefore, Hδξ,ε(E) < ∞. Finally, when ε → 0, we get dimξ(E) ≤ δ and the result easily follows. To prove the assumption (b), take δ > sup x∈E lim sup n→∞ log µ(Rn(x)) log(diam(Rn(x))) . For all x ∈ E, there exists n ∈ N such that, for all k ≥ n one has µ(Rk(x)) ≥ diam(Rk(x))δ. Consider the set E(n) = { x ∈ E : for each k ≥ n, µ(Rk(x)) ≥ diam(Rk(x))δ } . Let {Ek}k be a cover of E and {Rj}j be an ε-packing of E(n) ∩ Ek with ε < infR∈ξn0 diam(R). One has∑ j diam(Rj) δ ≤ ∑ j µ(Rj) < ∞. From which Pδξ,ε(E(n) ∩ Ek) < ∞. Then we get P δ ξ(E(n) ∩ Ek) < ∞ when ε → 0. Since E = ⋃ n E(n), we obtain Dimξ(E) ≤ δ. Hence (b). Let us prove assumption (c). Take δ < ess sup x∈E,µ] lim inf n→∞ log µ(Rn(x)) log(diam(Rn(x))) and set Eδ = { x ∈ E : lim inf n→∞ log µ(Rn(x)) log(diam(Rn(x))) > δ } . 240 a. samti Let En = { x ∈ Eδ : for each k ≥ n, µ(Rk(x)) ≤ diam(Rk(x))δ } . It is clear that Eδ = ⋃ n En. As we have µ ](Eδ) > 0, there exists n ∈ N such that µ](En) > 0. Then, for any ε-covering {Rj}j of En, one has µ]ε(En) ≤ ∑ j µ(Rj) ≤ ∑ j diam(Rj) δ. Therefore, µ]ε(En) ≤H δ ξ,ε(En). So, 0 < µ](En) ≤Hδξ(En), which implies dimξ(E) ≥ dimξ(Eδ) ≥ dimξ(En) ≥ δ and assumption (c) yields. In order to prove assumption (d), let δ < ess sup x∈E,µ] lim sup n→∞ log µ(Rn(x)) log(diam(Rn(x))) , and put Eδ = { x ∈ E : lim sup n→∞ log(µ(Rn(x))) log(diam(Rn(x))) > δ } . We have µ](Eδ) > 0, so there exists a subset F of Eδ such that µ ](F) > 0. If x ∈ F , then for all n ∈ N there exists k ≥ n such that µ(Rk(x)) ≤ diam(Rk(x))δ (10) Let ε > 0 and {Rj}j an ε-packing of F satisfying (10). So, µ]ε(F) ≤ ∑ j µ(Rj) ≤ ∑ j diam(Rj) δ. Then µ]ε(F) ≤P δ ξ,ε(F). This implies 0 < µ](F) ≤Pδξ(F). multifractal formalism of an inhomogeneous measure 241 Hence, if F = ⋃ j Fj, one has 0 < µ](F) < ∑ j µ](Fj) ≤ ∑ j Pδξ(Fj). Thus, Pδξ (F) > 0. Therefore, Dimξ(Eδ) ≥ δ, from which the result follows and we achieve the proof of Proposition 3.2. As a consequence of Proposition 3.2, we obtain the following corollary. We adopt the following convention log 0 log ρ = +∞, for each ρ > 0. Corollary 1. Let γ ∈ R. If µ is a probability Borel measure on A1 ×A2 such that µ(E) > 0, we consider a family ξ of rectangles verifying the assump- tions of Proposition 3.1 and E ⊂ { x ∈ A1 ×A2 : lim n→∞ log µ(Rn(x)) log(diam(Rn(x))) = γ } , we have dimξ(E) = Dimξ(E) = γ. Next, we will be interested in adding an example of application of Corollary 1. Example. Let {ξn}n≥1 be a sequence of finite partitions of A1 ×A2 made of rectangles in the form [ω]×[ω′], for all (ω,ω′) ∈ Aq(n)1 ×A n 2 and ξ = ⋃ n≥1 ξn, where the integer q(n) is defined such that, for n ∈ N∗ n log(r2) log(r1) ≤ q(n) < n log(r2) log(r1) + 1. It is clear that the family ξ satisfies the assumptions of Proposition 3.1. For α ≥ 0, we consider the set Eα = { x ∈ A1 ×A2 : lim n→∞ N ω,ω′ n n (x) = αω,ω′ for all (ω,ω ′) ∈ A1 ×A2 } 242 a. samti where for (ω,ω′) ∈ A1 ×A2, N ω,ω′ n (x) stands for the number of appearances of the couple (ω,ω′) in the product word x|n×y|n and α = (αω,ω′)(ω,ω′)∈A1×A2 is a family of positive numbers such that∑ (ω,ω′)∈A1×A2 αω,ω′ = 1. We propose to calculate the Hausdorff dimension of the set Eα. For this purpose, we consider the Bernoulli measure µ in A1 ×A2 defined by µ([ω1 · · ·ωn] × [ω′1 · · ·ω ′ n]) = n∏ k=1 αωk,ω′k for each n ∈ N∗. We have µ([ω1 · · ·ωq(n)] × [ω ′ 1 · · ·ω ′ n]) = n∏ k=1 αωk,ω′k q(n)∏ k=n+1 λωk with λωk = ∑ ω′ k αωk,ω′k . It is clear that Eα ⊂ { x ∈ A1 ×A2 : lim n→∞ log µ(Rn(x)) log(diam(Rn(x))) = γ } , where γ = − ∑ ω,ω′ αω,ω′ log αω,ω′ log r2 + ( 1 log r2 − 1 log r1 )∑ ω λω log λω. So, according to the strong law of large numbers we have µ(Eα) = 1. By using Corollary 1 we have, dimξ(Eα) = Dimξ(Eα) = γ, which implies from Proposition 3.1 that dim(Eα) = Dim(Eα) = γ. Thus, any Borel set of Eα with dimension inferior to γ is of measure µ-zero. multifractal formalism of an inhomogeneous measure 243 4. A variant of the refined multifractal formalism in the product space A1 ×A2 4.1. Problematic. In this section, we will consider a sequence {ξn}n≥1 of finite partitions of A1 ×A2 made of rectangles satisfying condition (1) and we put ξ = ⋃ n≥1 ξn. In the following, we consider a Borel probability measure µ on A1 × A2 and one defines its support supp(µ) to be the complement of the set⋃ {R ∈ ξ : µ(R) = 0} . Then, we intend to underestimate the dimensions of the fractal sets Eµ(γ) for some values of γ, where Eµ(γ) = { x ∈ supp(µ) : lim n→∞ log µ(Rn(x)) log(diam(Rn(x))) = γ } . Notice that the natural coverings of these iso-Hölder sets are made of rectan- gles which become thinner and thinner as their diameter tends to zero which doesn’t allow the calculation of the Hausdorff and packing dimensions. For this purpose, we will consider a variant of the refined multifractal formalism as already introduced by F. Ben Nasr and J. Peyrière [5], adapted to rectangles. Let us consider an auxiliary Borel probability measure ν on A1 ×A2. If E is a nonempty subset of A1 ×A2 then for q,t ∈ R and ε > 0, we introduce the following quantities: Hq,tµ,ν,ε(E) = inf {∑ j µ(Rj) q diam(Rj) tν(Rj) : Rj ∈ ξ,{Rj}j an ε-covering of E } , Hq,tµ,ν(E) = lim ε→0 Hq,tµ,ν,ε(E), and P q,t µ,ν,ε(E) = sup {∑ j µ(Rj) q diam(Rj) tν(Rj) : Rj ∈ ξ, {Rj}j an ε-packing of E } , P q,t µ,ν(E) = lim ε→0 P q,t µ,ν,ε(E). 244 a. samti The function P q,t µ,ν is called the packing pre-measure. In order to deal with an outer measure, one defines Pq,tµ,ν(E) = inf  ∑ j P q,t µ,ν(Ej) : E ⊂ ⋃ j Ej   . Let ϕ be the following function ϕ(q) = inf { t ∈ R : Pq,tµ,ν(supp(µ)) = 0 } . (11) 4.2. Main results. Let µ be a Borel probability measure on A1 × A2. For α,β ∈ R, one sets Eµ(α,β) = Eµ(α) ∩Eµ(β), where Eµ(α) = { x ∈ supp(µ) : lim inf n→∞ log µ(Rn(x)) log(diam(Rn(x))) ≥ α } and Eµ(β) = { x ∈ supp(µ) : lim sup n→∞ log µ(Rn(x)) log(diam(Rn(x))) ≤ β } . Theorem 4.1. Assume that ϕ(0) = 0 and ν](supp(µ)) > 0. Then one has dimξ Eµ ( −ϕ′r(0),−ϕ ′ l(0) ) ≥ inf { lim inf n→∞ log ν(Rn(x)) log(diam(Rn(x))) : x ∈ Eµ(−ϕ′r(0),−ϕ ′ l(0)) } and Dimξ Eµ ( −ϕ′r(0),−ϕ ′ l(0) ) ≥ inf { lim sup n→∞ log ν(Rn(x)) log(diam(Rn(x))) : x ∈ Eµ(−ϕ′r(0),−ϕ ′ l(0)) } , where ϕ′r,ϕ ′ l are respectively the left-hand and right-hand derivatives of ϕ. multifractal formalism of an inhomogeneous measure 245 Remark 2. The same result holds with ψ(q) = inf { t ∈ R : Pq,tµ,ν(supp(µ)) = 0 } . The proof of Theorem 4.1 is an immediate consequence of the following proposition. Proposition 4.1. Assume that ϕ(0) = 0 and ν](supp(µ)) > 0. Then one has ν](Eµ(−ϕ′r(0),−ϕ ′ l(0)) c) = 0. Proof. Take δ > −ϕ′l(0), there exist two positive reals t and δ ′ such that δ > δ′ > −ϕ′l(0) and δ ′t > ϕ(−t) which implies P−t,δ ′t µ,ν (supp(µ)) = 0. So, there exists a partition {Ej}j of supp(µ) such that∑ j P −t,δ′t µ,ν (Ej) ≤ 1. It results that P −t,δt µ,ν (Ej) = 0 for all j. Now, consider the set Eδ = { x ∈ supp(µ) : lim sup n→∞ log µ(Rn(x)) log(diam(Rn(x))) > δ } . If x ∈ Eδ, for all n ∈ N there exists k ≥ n such that µ(Rk(x)) ≤ diam(Rk(x))δ. Let E be a subset of Eδ and set Fj = E ∩Ej. For 0 < ε ≤ inf R∈ξn diam(R) and for all j, one can find an ε-packing {Rjk}k of Fj such that µ(Rjk) ≤ diam(Rjk) δ. So, we have ν]ε(Fj) ≤ ∑ j ν(Rj) ≤ ∑ j ∑ k ν(Rjk) ≤ ∑ j ∑ k µ(Rjk) −t diam(Rjk) δtν(Rjk) ≤ ∑ j P −t,δt µ,ν,ε (Fj) = 0. 246 a. samti Then ν](Eδ) = 0. We conclude that ν] ({ x ∈ supp(µ) : lim sup n→∞ log µ(Rn(x)) log(diam(Rn(x))) > −ϕ′l(0) }) = 0. In the same way, one proves that ν] ({ x ∈ supp(µ) : lim inf n→∞ log µ(Rn(x)) log(diam(Rn(x))) < −ϕ′r(0) }) = 0. Proof of Theorem 4.1. Assume that ϕ(0) = 0 and ν](supp(µ)) > 0. Then we have according to Proposition 4.1 ν](Eµ(−ϕ′r(0),−ϕ ′ l(0))) > 0. So, it is easy to see from Proposition 3.2 that dimξEµ(−ϕ′r(0),−ϕ ′ l(0)) ≥ ess sup x∈Eµ(−ϕ′r(0),−ϕ′l(0)),ν ] lim inf n→∞ log ν(Rn(x)) log(diam(Rn(x))) , and Dimξ Eµ(−ϕ′r(0),−ϕ ′ l(0)) ≥ ess sup x∈Eµ(−ϕ′r(0),−ϕ′l(0)),ν ] lim sup n→∞ log ν(Rn(x)) log(diam(Rn(x))) . However, as a property of ess sup, we know that if ν](Eµ(−ϕ′r(0),−ϕ′l(0))) > 0, then inf x∈Eµ(−ϕ′r(0),−ϕ′l(0)) { lim inf n→0 log ν(R(x)) log(diam(Rn(x))) } ≤ ess sup x∈Eµ(−ϕ′r(0),−ϕ′l(0)),ν ] lim inf n→∞ log ν(R(x)) log(diam(Rn(x))) and the proof of the theorem follows. multifractal formalism of an inhomogeneous measure 247 5. An example In this section we give a large class of measures satisfying the result of Theorem 4.1. Let {ξn}n≥1 be the sequence of finite partitions of A1 × A2 made of rectangles of the form [ω] × [ω′], for all (ω,ω′) ∈ Aq(n)1 × A n 2 and ξ = ⋃ n≥1 ξn, where the integer q(n) is defined such that, for n ∈ N∗ n log(r2) log(r1) ≤ q(n) < n log(r2) log(r1) + 1. For (i,j) ∈ A1×A2, take (pi,j)i,j and (qi,j)i,j two sequences of non negative numbers such that∑ i,j pi,j = ∑ i,j qi,j = 1 and λi = ∑ j pi,j = ∑ j qi,j. Let (Tn)n≥1 be a sequence of integers defined by T1 = 1 , Tn < Tn+1 and lim n→∞ Tn Tn+1 = 0. Consider the family of parameters αik,jk αik,jk = { pik,jk if T2n−1 ≤ k < T2n, qik,jk if T2n ≤ k < T2n+1. We define the measure µ on A1 ×A2 as follows µ ( [i1 · · ·in] × [j1 · · ·jn] ) = n∏ k=1 αik,jk. It is easy to see that µ ( [i1 · · ·iq(n)] × [j1 · · ·jn] ) = µ ( [i1 · · ·in] × [j1 · · ·jn] ) ·λin+1 · · ·λiq(n). In the sequel we will impose those monotony hypotheses p0,0 < p0,1 < · · · < p0,r2−1 < p1,0 < · · · < p1,r2−1 < · · · · · · < pr1−1,0 < · · · < pr1−1,r2−1, q0,0 < q0,1 < · · · < q0,r2−1 < q1,0 < · · · < q1,r2−1 < · · · · · · < qr1−1,0 < · · · < qr1−1,r2−1, p0,0 < q0,0 and pr1−1,r2−1 > qr1−1,r2−1, 248 a. samti which prove the existence of a real x0 such that T(x0) = W(x0), where T(x) = ∑ i,j pxi,j∑ i,j pxi,j logr2 pi,j + ( 1 log r1 − 1 log r2 )∑ i,j pxi,j∑ i,j pxi,j log λi and W(x) = ∑ i,j qxi,j∑ i,j qxi,j logr2 qi,j + ( 1 log r1 − 1 log r2 )∑ i,j qxi,j∑ i,j qxi,j log λi. For this real x0, we denote by p̃i,j = px0i,j∑ i,j px0i,j and q̃i,j = qx0i,j∑ i,j qx0i,j . Our aim is to estimate the dimensions of the sets Eµ(γ) for certain values of γ. To be done, we consider an auxiliary measure ν on A1 × A2 defined as µ with the parameters p̃i,j and q̃i,j instead of pi,j and qi,j by ν([i1 · · ·in] × [j1 · · ·jn]) = n∏ k=1 α̃ik,jk where α̃ik,jk = { p̃ik,jk if T2n−1 ≤ k < T2n, q̃ik,jk if T2n ≤ k < T2n+1. Let λ̃i = ∑ j p̃i,j = ∑ j q̃i,j. Then, we have the following result. Theorem 5.1. For every γ ∈ ( − logr2 ( qr1−1,r2−1λ log(r2) log(r1) −1 r1−1 ) ,− logr2 ( q0,0λ log(r2) log(r1) −1 0 )) we have dim Eµ(γ) ≥ min{h(p̃),h(q̃)} and Dim Eµ(γ) ≥ max{h(p̃),h(q̃)} , multifractal formalism of an inhomogeneous measure 249 where h(p̃) = − ∑ i,j p̃i,j logr2 p̃i,j + ( 1 log r2 − 1 log r1 )∑ i λ̃i log λ̃i and h(q̃) = − ∑ i,j q̃i,j logr2 q̃i,j + ( 1 log r2 − 1 log r1 )∑ i λ̃i log λ̃i. In order to prove this theorem we will calculate the function ϕ defined in equation (11). For that, we need to use the following lemma. Lemma 5.1. For t ∈ R, one has ϕ(t) = lim sup n→∞ 1 n log r2 log ∑ Rn∩supp(µ)6=∅ µ(Rn) tν(Rn). Proof. For t ∈ R, we denote by Φ(t) = lim sup n→∞ 1 n log r2 log ∑ Rn∩supp(µ)6=∅ µ(Rn) tν(Rn). We will prove that ϕ(t) = Φ(t). Let’s begin by proving that ϕ(t) ≤ Φ(t). For α > 0 satisfying Φ(t) ≤ α, there exists n0 ∈ N such that for all n ≥ n0, 1 n log r2 log ∑ Rn∩supp(µ)6=∅ µ(Rn) tν(Rn) ≤ α. So, ∑ Rn∩supp(µ) 6=∅ µ(Rn) tν(Rn)r −nα 2 ≤ 1, for each n ≥ n0. Then P t,α µ,ν(supp(µ)) ≤ 1, and α ≥ ϕ(t), which gives that Φ(t) ≥ ϕ(t). 250 a. samti Next, we prove that ϕ(t) ≥ Φ(t). Let α > ϕ(t), then P t,α µ,ν(supp(µ)) = 0. For ε > 0, there exists an ε-packing {Rn}n of supp(µ) such that∑ Rn∩supp(µ)6=∅ µ(Rn) tν(Rn)r −nα 2 ≤ 1. Thus 1 n log r2 log ∑ Rn∩supp(µ)6=∅ µ(Rn) tν(Rn) ≤ α. So, lim sup n→∞ 1 n log r2 log ∑ Rn∩supp(µ)6=∅ µ(Rn) tν(Rn) ≤ α and Φ(t) ≤ α, which prove Lemma 5.1. Now, we are able to prove Theorem 5.1. It is easy to see that ϕ(t) = sup  logr2 ∑ i,j pti,jp̃i,j, logr2 ∑ i,j qti,jq̃i,j   + ( 1 log r1 − 1 log r2 ) log ∑ i λtiλ̃i and ϕ(0) = 0. By the way, using the definitions of the sequences (p̃i,j) and (q̃i,j) and a simple computation of the derivative of ϕ at 0 we obtain ϕ′(0) = ∑ i,j p̃i,j logr2 pi,j + ( 1 log r1 − 1 log r2 )∑ i λ̃i log λi. Let γ = −ϕ′(0), it is clear that γ ∈ ( − logr2 ( qr1−1,r2−1λ log(r2) log(r1) −1 r1−1 ) ,− logr2 ( q0,0λ log(r2) log(r1) −1 0 )) . multifractal formalism of an inhomogeneous measure 251 Besides, using the strong law of large numbers we can see that lim inf n→∞ logr2 ν(Rn(x)) −n = min{h(p̃),h(q̃)} and lim sup n→∞ logr2 ν(Rn(x)) −n = max{h(p̃),h(q̃)} , for ν-almost every x. Then, it follows from Theorem 4.1 and Proposition 3.2 that dim Eµ(γ) ≥ min{h(p̃),h(q̃)} and Dim Eµ(γ) ≥ max{h(p̃),h(q̃)} which achieve the proof of Theorem 5.1. References [1] N. Attia, B. Selmi, Regularities of multifractal Hewitt-Stromberg measures, Commun. Korean Math. Soc. 34 (2019), 213 – 230. [2] N. Attia, B. Selmi, A multifractal formalism for Hewitt-Stromberg mea- sures, J. Geom. Anal. (2019) https://doi.org/10.1007/s12220-019-00302-3. [3] F. Ben Nasr, I. Bhouri, Spectre multifractal de mesures boréliennes sur Rd, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), 253 – 256. [4] F. Ben Nasr, I. Bhouri, Y. Heurteaux, The validity of the multifractal formalism: results and examples, Adv. Math. 165 (2002), 264 – 284. [5] F. Ben Nasr, J. Peyrière. Revisiting the multifractal analysis of measures, Rev. Mat. Iberoam. 29 (1) (2013), 315 – 328. [6] P. Billingsley, “Ergodic Theory and Information”, John Wiley & Sons, Inc., New York-London-Sydney, 1965. [7] K. Falconer, “Fractal Geometry. Mathematical Foundations and Applica- tions”, John Wiley & Sons, Ltd., Chichester, 1990. [8] U. Frisch, G. Parisi, Fully developped turbulence and intermittency in tur- bulence, and predictability in geophysical fluid dynamics and climate dynam- ics, in “ International School of Physics Enrico Fermi, Course 88 ”, (edited by M. Ghil), North Holland, 1985, 84 – 88. [9] T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia, B.J. Shraiman, Fractal measures and their singularities: the characterization of strange sets, Phys. Rev. A (3) 33 (1986), 1141 – 1151. [10] H.G. Hentschel, I. Procaccia, The infinite number of generalized di- mensions of fractals and strange attractors, Phys. D 8 (1983), 435 – 444. https://doi.org/10.1007/s12220-019-00302-3 252 a. samti [11] B.B. Mandelbrot, Multifractal measures, especially for geophysicist, An- nual Reviews of Materials Sciences. 19 (1989), 514 – 516. [12] B.B. Mandelbrot, A class of multifractal measures with negative (latent) value for the dimension f(α), in “ Fractals: Physical Origin and Properties ”, in Proceedings of the Special Seminar on Fractals, Erice, 1988 (edited by K. Ford and D. Campbell), American Institute of Physics, 1990, 3 – 29. [13] L. Olsen, A multifractal formalism, Adv. Math. 116 (1995), 82 – 196. [14] S. Shen, Multifractal analysis of some inhomogeneous multinomial measures with distinct analytic Olsen’s b and B functions, J. Stat. Phys. 159 (5) (2015), 1216 – 1235. [15] C. Tricot, Jr., Rarefaction indices, Mathematika 27 (1980), 46 – 57. [16] S.J. Taylor, C. Tricot, Packing measure, and its evaluation for a Brow- nian path, Trans. Amer. Math. Soc. 288 (1985), 679 – 699. [17] M. Wu, The singularity spectrum f(α) of some Moran fractals, Monatsh. Math. 144 (2005), 141 – 55. [18] M. Wu, J. Xiao, The singularity spectrum of some non-regularity Moran fractals, Chaos Solitons Fractals 44 (2011), 548 – 557. [19] J. Xiao, M. Wu, The multifractal dimension functions of homogeneous Moran measure, Fractals 16 (2008), 175 – 185. Introduction Notations and definitions Calculation of the Hausdorff and packing dimensions on the product symbolic space on different basis A variant of the refined multifractal formalism in the product space A1A2 Problematic. Main results. An example