Rev. Mat. Iberoam. 29 (2013), no. 1, 315–328 c European Mathematical Society doi 10.4171/rmi/721
Revisiting the multifractal analysis of measures
Fathi Ben Nasr and Jacques Peyri`ere
Abstract. New proofs of theorems on the multifractal formalism are given. They yield results even at points q for which Olsen’s functions b(q) and B(q) differ. Indeed, we provide an example of a measure for which the functions b and B differ and for which the Hausdorff dimensions of the sets Xα (the level sets of the local H¨older exponent) are given by the Legendre transform of b and their packing dimensions by the Legendre transform of B.
1. Introduction
The multifractal formalism aims at expressing the dimension of the level sets of the local H¨older exponent of some set function μ in terms of the Legendre transform of some “free energy” function (see [7], [5], and [6] for early works). If such a formula holds, one says that μ satisfies the multifractal formalism. At first, the formalism used “boxes”, or in other terms took place in a totally disconnected metric space. In this context, the closeness to large deviation theory is patent. To get rid of these boxes and have a formalism meaningful in geometric measure theory, Olsen [8] introduced a formalism which is now commonly used. See also Pesin’s monograph [9] on multifractality and dynamical systems. At this stage of the theory, whether it dealt with boxes or not, the formalism was proven to hold when there exists an auxiliary measure, a so-called Gibbs measure. Later, it was shown that this formalism holds under the condition that Olsen’s Hausdorff-like multifractal measure be positive (see [2] in the totally disconnected case, [3]in general). So, the situation when b(q)=B(q) (in Olsen’s notation) is fairly well understood. Here, we elaborate on the previous proofs. There is a vector version of Olsen’s constructions [10], and, in particular, of the functions b and B. However, in this setting b and B are functions of several variables. In this work, we show that the restriction of these functions to a suitable affine subspace can be used to estimate the Hausdorff and Tricot dimensions of some level sets. In particular, this gives
Mathematics Subject Classification (2010): 28A80, 28A78, 28A12, 11K55. Keywords: Hausdorff dimension, packing dimension, fractal, multifractal. 316 F. Ben Nasr and J. Peyriere` some results even in the case when b = B. Although notation is inherently compli- cated, we provide a simple proof of already known results, and we obtain some new estimates. In particular, we provide an example of a measure on the interval [0, 1] for which the functions b and B differ and for which the Hausdorff dimensions of the sets Xα (the level sets of the local H¨older exponent) are given by the Legendre transform of b, and their packing dimensions by the Legendre transform of B.
2. Notations and definitions
We deal with a metric space (X, d)havingtheBesicovitch property: There exists an integer constant CB such that one can extract CB countable fam- {B } B ilies j,k k ≤j≤C from any collection of balls so that 1 B 1. j,k Bj,k contains the centers of the elements of B, 2. for any j and k = k , Bj,k ∩ Bj,k = ∅.
Notations
B(x, r) stands for the open ball B(x, r)={y ∈ X ; d(x, y)
m m For μ =(μ1,...,μm) ∈ F , E ⊂ X, q =(q1,...,qm) ∈ R , t ∈ R,andδ>0, one sets ∗ m q,t t qk Pμ,δ(E)=sup rj μk(Bj ) ; {Bj} a δ-packing of E , k=1 Revisiting the multifractal analysis of measures 317 where ∗ means that one only sums the terms for which k μk(Bj ) =0, q,t q,t Pμ (E) = lim Pμ,δ(E), δ0 q,t q,t Pμ (E) = inf Pμ (Ej ); E ⊂ Ej , and ∗ m q,t t qk H μ,δ(E) = inf rj μk(Bj ) ; {Bj} a centered δ-cover of E , k=1 q,t q,t H μ (E) = lim H μ,δ(E), δ 0 q,t q,t Hμ (E)=supH μ (F ); F ⊂ E ,
q,t q,t q,t It is known that H μ is σ-subadditive, and that Pμ and Hμ are outer q,t q,t measures. When d is an ultrametric, then Hμ = H μ . When m = 1, these measures have been defined by Olsen [8]. When μ is identically 1 these quantities do not depend on q. They will be simply denoted t t t t t t by Pδ(E), P (E), P (E), H δ(E), H (E), and H (E), respectively. They are the classical packing pre-measures and measures introduced by Tricot [12], and the Hausdorff centered pre-measures and measures [11]. The centered Hausdorff measures also define the Hausdorff dimension. It will prove convenient to use the following notations, when m =1: , , 1 0 1 0 1,0 μδ = H μ,δ, μ = H μ , and μ = Hμ . Also, as usual, one considers the following functions: q,t q,t τμ,E (q) = inf{t ∈ R ; Pμ (E)=0} =sup{t ∈ R ; Pμ (E)=∞} q,t q,t Bμ,E (q) = inf{t ∈ R ; Pμ (E)=0} =sup{t ∈ R ; Pμ (E)=∞}, q,t q,t bμ,E (q) = inf{t ∈ R ; Hμ (E)=0} =sup{t ∈ R ; Hμ (E)=∞}.
It is well known [8], [10]thatτ and B are convex and that b ≤ B ≤ τ.LetJτ , JB, and Jb stand for the interiors of the sets where respectively τ, B,andb are finite.
When μ is identically 1 we will denote these quantities by dimBE,dimP E, and dimH E. The first one is the Minkowski–Bouligand dimension (or upper box- dimension), the second is the Tricot (packing) dimension [12], and the last the Hausdorff dimension. Here is an alternate definition of τμ,E .Fixλ<1 and define ∗ m q,t t qk Pμ,δ(E)=sup rj μk(Bj ) ; {Bj} a packing of E with λδ < rj ≤ δ , k=1 q,t q,t Pμ (E)=lim P (E), δ μ,δ 0 q,t τ μ,E (q)=sup t ∈ R ; Pμ (E)=+∞ . 318 F. Ben Nasr and J. Peyriere`
Lemma 2.1. One has τ μ,E = τμ,E .
q,t q,t Proof. Obviously Pμ (E) ≤ Pμ (E), so τ μ,E ≤ τμ,E . To prove the converse inequality, one only has to consider the case τμ,E (q) > −∞. Choose γ<τμ,E (q)andε>0 such that γ + ε<τμ,E (q). There exists n0 such n that, for all n>n0, there exists a λ -packing {Bj } of E such that m γ+ε qk rj μk(Bj ) > 1. k=1 As m m γ+ε qk γ+ε qk rj μk(Bj ) = rj μk(Bj ) , −(n+i) k=1 i≥0 λ q,γ and Pμ (E)=+∞. 2 Corollary 2.2. For any λ<1, one has − ∗ m 1 qk τμ,E (q)=lim log sup μk(Bj ) ; δ0 log δ k=1 {Bj } a packing of E with λδ < rj ≤ δ . Level sets of local H¨older exponents Let μ be an element of F ∗.Forα, β ∈ R,onesets log μ B(x, r) Xμ(α)= x ∈ Sμ ; lim ≤ α , r0 log r log μ B(x, r) Xμ(α)= x ∈ Sμ ; lim ≥ α , r0 log r Xμ(α, β)=Xμ(α) ∩ Xμ(β), and Xμ(α)=Xμ(α) ∩ Xμ(α). Revisiting the multifractal analysis of measures 319 3. Results First, we revisit the Billingsley and Tricot lemmas [4], [12]. Lemma 3.1. Let E be a subset of X and ν an element of F. a) If Bν,E(1) ≤ 0,then log ν B(x, r) (3.1) dimH E ≤ sup lim , x∈E r log r 0 log ν B(x, r) (3.2) dimP E ≤ sup lim . x∈E r0 log r b) If ν(E) > 0,then log ν B(x, r) (3.3) dimH E ≥ ess sup lim , x∈E,ν r log r 0 log ν B(x, r) (3.4) dimP E ≥ ess sup lim , x∈E,ν r0 log r where ess sup χ(x) = inf t ∈ R; ν E ∩{χ>t} =0 . x∈E,ν Proof. Take log ν B(x, r) γ>sup lim x∈E r log r 0 and η>0. Since Bν,E(1) ≤ 0 there exists a partition E = Ej such that 1,η/2 1,η Pν (Ej )<1. Therefore we have that Pν (Ej )=0. Let F be a subset of Ek and let δ be a positive number. For all x ∈ F ,there exists r ≤ δ such that ν B(x, r) ≥ rγ . By the Besicovitch property, there exists a centered δ-cover {Bj } of F , which can be decomposed into CB packings, such that γ ν(Bj ) ≥ rj .Wethenhave γ+η η 1,η rj ≤ rj ν(Bj ) ≤ CBPν,δ (Ek). γ η + γ+η γ+η Therefore we have H (F )=0,H (Ek) = 0, and finally H (E)=0. Then (3.1) easily follows. To prove (3.2), take log ν B(x, r) γ>sup lim x∈E r0 log r 1,η and η>0. As previously, there exists a partition E = Ej such that Pν (Ej )=0. For all x ∈ E,thereexistsδ>0 such that, for all r ≤ δ, one has ν B(x, r) ≥ rγ . Consider the set E(n)= x ∈ E ; ∀r ≤ 1/n, ν B(x, r) ≥ rγ . 320 F. Ben Nasr and J. Peyriere` Let {Bj } be a δ-packing of Ek ∩ E(n), with δ ≤ 1/n. One has γ+η η 1,η rj ≤ rj ν(Bj ) ≤ Pν,δ (Ek), j γ η P + E ∩ E n from which k () = 0 follows. Pγ+η E n E E n E ≤ γ η So we have ( ) =0.Since = n≥1 ( ), one has dimP + , and hence (3.2). To prove (3.3), take log ν B(x, r) γ Therefore, γ γ γ νδ(G) ≤ H δ (G), and 0 < ν(G) ≤ H (G) ≤ H (G), which implies dimH E ≥ dimH G ≥ γ. To prove (3.4), take log ν B(x, r) γ Lemma 3.2. Let μ and ν be elements of F ∗ and F respectively. Set ϕ(t)= B t, ϕ ν S > (μ,ν),Sμ ( 1) and assume that (0) = 0 and ( μ) 0. Then one has C ν Xμ −ϕr(0), −ϕl(0) =0, where ϕl and ϕr are the left-hand and right-hand derivatives of ϕ. ϕ t τ t, The same result holds with ( )= (μ,ν),Sμ ( 1). Proof. Take γ>−ϕl(0), and choose γ and t>0 such that γ>γ > −ϕl(0) ϕ −t <γt P(−t,1),γ t S and ( ) .Then (μ,ν) ( μ) = 0, so there exists a countable partition Sμ = Ej of Sμ such that −t, ,γ t P( 1) E ≤ , (μ,ν) ( j ) 1 j −t, ,γt P( 1) E j and therefore (μ,ν) ( j ) = 0 for all . Consider the set log μ B(x, r) E(γ)= x ∈ Sμ ; lim >γ . r0 log r If x ∈ E(γ), for all δ>0, there exists r ≤ δ such that μ B(x, r) ≤ rγ .LetF be a subset of E(γ). Set Fj = F ∩ Ej . For δ>0, for all j, one can find a Besicovitch δ-cover {Bj,k} of Fj such that γ μ(Bj,k) ≤ rj,k. We have, −t t −t γt νδ(Fj ) ≤ ν(Bj,k)= μ(Bj,k) μ(Bj,k) ν(Bj,k) ≤ μ(Bj,k) rj,kν(Bj,k), k k k which, together with the Besicovitch property, implies −t, ,γt ν F ≤ C P( 1) E . δ( j ) B (μ,ν),δ ( j ) so −t, ,γt ν F ≤ C P( 1) E . ( j ) B (μ,ν) ( j )=0 This implies ν(F ) = 0, and ν(E(γ)) = 0. 322 F. Ben Nasr and J. Peyriere` We conclude that log μ B(x, r) ν x ∈ Sμ ; lim > −ϕl(0) =0. r0 log r In the same way, one proves that log μ B(x, r) ν x ∈ Sμ ; lim < −ϕr(0) =0. r0 log r 2 Corollary 3.3. With the same notations and hypotheses as in Lemma 3.2,one has log ν B(x, r) dimH Xμ −ϕr(0), −ϕl(0) ≥ inf lim ; x ∈ Xμ −ϕr(0), −ϕl(0) r0 log r and log ν B(x, r) dimP Xμ −ϕr(0), −ϕl(0) ≥ inf lim ; x ∈ Xμ −ϕr(0), −ϕl(0) . r0 log r Note that statements in Corollary 3.3 are weaker than what can be deduced from Lemma 3.2 and Lemma 3.1-b. The previous lemmas contain the now classical results on multifractal analy- sis [8], [3], [10]. Indeed, let μ be a element of F ∗. Until the end of this section, we will write b, τ,andB instead of bμ,Sμ , τμ,Sμ ,andBμ,Sμ .Forq ≥ 0, take ν B μ B qrB(q) ϕ B t, ( )= ( ) . Then the corresponding of Lemma 3.2 is (μ,ν),Sμ ( 1) = B(q + t) − B(q) and, for x ∈ Xμ(α), one has log ν B(x, r) log μ B(x, r) lim = q lim + B(q) ≤ qα + B(q). r0 log r r0 log r So, by (3.2) of Lemma 3.1,onegets dimP Xμ(α) ≤ inf qα + B(q). q≥0 Inthesameway,weget dimP Xμ(α) ≤ inf qα + B(q). q≤0 q,B(q) If moreover we assume that Hμ (Sμ) > 0, we have ν (Sμ) > 0, and there- fore, by Lemma 3.2, ν Xμ −Br(q), −Bl(q) > 0. Therefore, by (3.3) of Lemma 3.1,wehave −qB q B q q ≥ , r( )+ ( )if 0 dimH Xμ −B (q), −B (q) ≥ r l −qBl(q)+B(q)ifq ≤ 0. Revisiting the multifractal analysis of measures 323 Recall that the Legendre transform of a function χ is defined to be χ∗(α)= infq∈R qα + χ(q). All this gives a new proof of the following theorem (see [2] in the totally dis- connected case, [3] in general). q,B(q) Theorem 3.4. If B has a derivative at some point q ∈ JB and if Hμ (Sμ) > 0, then ∗ dimH Xμ −B (q) = B −B (q) . The same statement holds with τ instead of B. ∗ In [3]itisshownthatifB (q) exists and if dimH Xμ −B (q) = B −B (q) , then b(q)=B(q). We now deal with the case when b(q) = B(q). The following notation will prove convenient: for a real function ψ,weset ψ(q − t) − ψ(q) ψ(q + t) − ψ(q) ψl (q)=lim and ψr(q)=lim . t0 −t t0 t Lemma 3.5. Let μ and ν be elements of F ∗ and F respectively. Set ϕ(t)= b t, ϕ ν S > (μ,ν),Sμ ( 1) and assume that (0) = 0 and ( μ) 0. Then one has log μ B(x, r) ν x ∈ Sμ ; lim > −ϕl (0) =0 r0 log r and log μ B(x, r) ν x ∈ Sμ ; lim < −ϕr(0) =0. r0 log r γ>−ϕ ϕ(−t) t> γt > ϕ −t Proof. Take l (0) = limt0 t and choose 0 such that ( ). H (−t,1),γt S We have (μ,ν) ( μ)=0. Consider the set log μ B(x, r) E = x ∈ Sμ ; lim >γ . r0 log r γ For all x ∈ E,thereexistsδ>0 such that, for all r<δ, one has μ B(x, r) Therefore −t, ,γt ν F ≤ H ( 1) F . δ( ) (μ,ν),δ ( ) Then we have −t, ,γt ν F ≤ H ( 1) F ≤ H (−t,1),γt S . ( ) (μ,ν) ( ) (μ,ν) ( μ)=0 This implies ν (En)=0andν (E) = 0. This proves the first assertion. The second one is proved in the same way. 2 324 F. Ben Nasr and J. Peyriere` Proposition 3.6. Let μ be an element of F. Suppose that, for some q ∈ Jb, q,b(q) Hμ (Sμ) > 0, and consider the set log μ (B(x, r)) log μ (B(x, r)) E = x ∈ Sμ ; lim ≤−bl (q) and lim ≥−br(q) . r r0 log r 0 log r Then we have b q − qb q , q ≥ , E ≥ ( ) r( ) if 0 dimP b(q) − qbl (q), if q ≤ 0. In particular, if b(q) exists one has log μ (B(x, r)) log μ (B(x, r)) dimP x ∈ Sμ ; lim ≤−b (q) ≤ lim ≥ b(q)−qb(q). r r0 log r 0 log r Proof. This results from Lemma 3.5 and (3.4) of Lemma 3.1. 2 4. An example Now, we can deal with the example given in [3] (Theorem 2.6). We take for X ∗ the space {0, 1}N endowed with the ultrametric which assigns diameter 2−n to cylinders of order n. We are given two numbers p andp ˜ such that 0 -ift2k−1 ≤ j -ift2k ≤ j We wish to prove the following result: Proposition 4.1. α ∈ − − p , − p 1) For log2(1 ˜) log2 ˜ , we have dimH Xμ(α) = inf b(q)+αq. q∈R α ∈ − − p , − p \ −B , −B ∪ −B , −B 2) For log2(1 ˜) log2 ˜ [ r(0) l(0)] [ r(1) l(1)] ,we have dimP Xμ(α) = inf B(q)+αq. q∈R Proof. We consider the measure ν constructed as μ with parameters r andr ˜ instead of p andp ˜. We impose the condition (4.1) r log p +(1− r)log(1− p)=˜r logp ˜ +(1− r˜) log(1 − p˜). As both r andr ˜ should belong to the interval (0, 1), we must have 1 − p 1 − p 1 − p (4.2) log (4.4) dimH Xμ(α) ≥ min{h(r), h(˜r)} and (4.5) dimP Xμ(α) ≥ max{h(r), h(˜r)}, where r,˜r,andα are linked by (4.1)and(4.3). 326 F. Ben Nasr and J. Peyriere` If α is defined by (4.3), we have log 1−r log 1−r˜ α −θ q q r α −θ˜ q q r˜ . (4.6) = ( )if = 1−p and = (˜)if˜= 1−p˜ log p log p˜ Now fix q andq ˜ as above in (4.6). One can check that, for these values of q andq ˜, one has (4.7) θ(q) − qθ(q)=h(r)andθ˜(˜q) − q˜θ˜(˜q)=h(˜r). In order to have θ(q)=b(q), we must have 0 1 1 (4.8) log <α In order to have θ˜(˜q)=b(˜q), we must haveq< ˜ 0or˜q>1, which means α> 1 (4.9) log2 p˜(1 − p˜) or 1 (4.10) α One can check that at least one of the conditions (4.8), (4.9)and(4.10) is fulfilled. But for any q such that b(q) exists, we have (see [8]or[1]) that (4.11) dimH Xμ −b (q) ≤ b(q) − qb(q). The first assertion then results from (4.4), (4.7), and (4.11). In order to have θ(q)=B(q), we must have q<0orq>1, which means 1 1 α>log = −Bl(0) or α In order to have θ˜(˜q)=B(˜q), we must have 0 < q<˜ 1, which means 1 1 −Bl(1) = log <α Then assertion (2) follows as before. 2 Remark 4.2. Proposition 4.1 also holds for the measure considered in [3]. Indeed, using the Gray code before projecting on [0, 1] yields doubling measures. Revisiting the multifractal analysis of measures 327 5. The vector case As in [10] one may consider expressions of the form exp − q, κ(B) instead of μ(B)q , where κ takes its values in the dual E of a separable Banach space E and q ∈ E. Let ν be an element of F.ForE ⊂ X, q ∈ E, t ∈ R,andδ>0, one sets q,t t − q,κ(Bj ) Pδ (E)=sup rj e ν(Bj );{Bj } a δ-packing of E , q,t q,t P (E) = lim Pδ (E), δ0 q,t q,t P (E) = inf P (Ej ); E ⊂ Ej , and q,t t − q,κ(Bj ) H δ (E) = inf rj e ν(Bj ); {Bj } a centered δ-cover of E , q,t q,t H (E) = lim H δ (E), δ 0 q,t H q,t(E)=supH (F ); F ⊂ E , For a function χ from E to R,andforv ∈ E of norm 1, one defines χ(tv) − χ(0) ∗ χ(tv) − χ(0) ∂vχ(0) = lim and ∂v χ(0) = lim − . t0 t t0 t With these notations we have the following analogues of Lemmas 3.2 and 3.5: Lemma 5.1. Let ϕ(q) be one of the following functions: q,t inf t ; P (X)=0 or inf t ; Pq,t(X)=0 . Assume that ϕ(0) = 0 and that ∂vϕ(0) at 0 is a lower semi-continuous function of v. Then one has v, κ B(x, r) ν x ; lim < −∂vϕ(0) for some v ∈ E =0. r0 − ln r Lemma 5.2. Set ϕ(q) = inf {t ; H q,t(X)=0} and assume that ϕ(0) = 0 and ∗ that ∂v χ(0) is a lower semi-continuous function of v. Then one has v, κ B(x, r) ∗ ν x ; lim < −∂v ϕ(0) for some v ∈ E =0. r0 − ln r The proofs follow the same lines as those above and as the proofs in [10]. As a corollary we get the following result (with the notations of [10]): q,t Theorem 5.3. Let B(q)=inf{t∈R ; Hκ (X)=0}. Assume that, at some point q, q,B(q) the function B is differentiable with derivative B (q) and that Hκ (X) > 0. Then one has v, κ B(x, r) dimH x ; ∀v ∈ E, lim = −B (q)v = B(q) − B (q)q. r0 log r 328 F. Ben Nasr and J. Peyriere` References [1] Barral, J., Fan, A. H. and Peyriere,` J.: Mesures engendr´ees par multiplica- tions. In Quelques interactions entre analyse, probabilit´es et fractals, 57–189. Panor. Syntheses 32, Soc. Math. France, Paris, 2010. [2] Ben Nasr, F.: Analyse multifractale de mesures. C. R. Acad. Sci. Paris S´er. I Math. 319 (1994), 807–810. [3] Ben Nasr, F., Bhouri, I. and Heurteaux, Y.: The validity of the multifractal formalism: results and examples. Adv. Math. 165 (2002), 264–284. [4] Billingsley, P.: Ergodic theory and information. John Wiley, New York-London- Sydney, 1965. [5] Frisch,U.andParisi,U.:On the singularity structure of fully developed turbu- lence (appendix to “Fully developed turbulence and intermittency”, by U. Frisch). In Proceedings of the International School of Physics “Enrico Fermi”, 84–88. North- Holland, 1985. [6] Halsey, T. C., Jensen, M. H., Kadanoff, L. P., Procaccia, I. and Shraiman, B. I.: Fractal measures and their singularities: the characterisation of strange sets. Phys. Rev. A (3) 33 (1986), 1141–1151. [7] Hentschel, H. G. and Procaccia, I.: The infinite number of generalized dimen- sions of fractals and strange attractors. Phys. D 8 (1983), 435–444. [8] Olsen, L.: A multifractal formalism. Adv. Math. 116 (1995), 82–196. [9] Pesin, Y.: Dimension theory in dynamical systems. Contemporary views and applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. [10] Peyriere,` J.: A vectorial multifractal formalism. In Fractal geometry and applica- tions: a jubilee of Benoˆıt Mandelbrot, Part 2, 217–230. Proc. Sympos. Pure Math. 72, Part 2, Amer. Math. Soc., Providence, RI, 2004. [11] Saint Raymond, X. and Tricot, C.: Packing regularity of sets in n-space. Math. Proc. Cambridge Philos. Soc. 103 (1988), 133–145. [12] Tricot, C.: Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc. 91 (1982), 57–74. Received May 14, 2011. F. Ben Nasr:D´epartement de Math´ematiques, Facult´e des Sciences de Monastir, Monastir 5000, Tunisie. E-mail: fathi [email protected] J. Peyriere` :Universit´e Paris-Sud, Math´ematique bˆat. 425, CNRS UMR 8628, 91405 Orsay Cedex, France; and School of Mathematics and Systems Science, Beihang Uni- versity, Beijing 100191, P. R. China. E-mail: [email protected] J. Peyri`ere gratefully acknowledges the support of Project 111, P. R. China.