Sums of Dirac masses and conditioned ubiquity Sommes de masses de Dirac et ubiquit´e conditionnelle

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Sums of Dirac masses and conditioned ubiquity Sommes de masses de Dirac et ubiquit´e conditionnelle

Julien Barral St´ ephane Seuret

^a

aEquipe Complex, INRIA Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France

Abstract

Multifractal formalisms hold for certain classes of atomless measuresµ obtained as limits of multiplicative processes. This naturally leads to ask whether non trivial discontinuous measures obey such formalisms. This is the case for a new kind of measures, whose construction combines additive and multiplicative chaos. This class is defined byνγ,σ=X

j≥1

b^−jγ j²

b^j−1

X

k=0

µ([kb^−j,(k+ 1)b^−j))^σδ_kb−j (supp(µ) = [0,1],binteger≥2,γ≥0,σ≥1).Under suitable assumptions on the initial measureµ,νγ,σobeys some multifractal formalisms. Its Hausdorff multifractal spectrumh7→dν_γ,σ(h) is composed of a linear part forhsmaller than a critical valuehγ,σ, and then of a concave part whenh≥hγ,σ. The same properties hold for the Hausdorff spectrum of some function seriesfγ,σconstructed according to a scheme similar to the one ofνγ,σ. These phenomena are the consequences of new results relating ubiquitous systems to the distribution of the mass ofµ.

R´esum´e

Les formalismes multifractals sont v´erifi´es par certaines classes de mesures diffusesµlimites de processus multiplicatifs. Cela pose naturellement la question de savoir s’ils le sont encore pour des mesures non diffuses non triviales.

C’est effectivement le cas pour des mesures d’un type nouveau, qui mˆelent chaos additifs et multiplicatifs. Cette classe de mesures est d´efinie parνγ,σ =X

j≥1

b^−jγ j²

b^j−1

X

k=0

µ([kb^−j,(k+ 1)b^−j))^σδ_kb−j (supp(µ) = [0,1],bentier ≥2, γ ≥ 0,σ ≥ 1).Sous certaines hypothèses sur µ, plusieurs formalismes multifractals sont en effet satisfaits par νγ,σ. De plus, son spectre multifractal de Hausdorffh7→dν_γ,σ(h) se compose alors d’une partie linéaire pourh plus petit qu’une valeur critiquehγ,σ, puis d’une partie concave pourh≥hγ,σ. Cette propriété est partagée par le spectre de Hausdorff de séries de fonctionsfγ,σ construites de fa¸con analogue àνγ,σ. L’analyse des singularités de ces objets fait appel à de nouveaux résultats combinant la notion d’ubiquité avec les propriétés d’auto-similarité de la mesureµ.

Email addresses:Julien.Barral@inria.fr(Julien Barral),Stephane.Seuret@inria.fr(St´ephane Seuret).

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Version fran¸caise abr´eg´ee

Soitµune mesure borélienne positive sur [0,1]. Soitbun entier≥2,γ≥0 etσ≥1. Considérons la mesure constituée d’atomesνγ,σ =P

j≥1j⁻²Pb^j−1

k=0 b⁻^jγµ([kb⁻^j,(k+ 1)b⁻^j))^σδ_kb−j. Nous nous intéressons au spectre de Hausdorff de νγ,σ, c’est-à-dire à la fonction dν_γ,σ qui à un exposant h ≥ 0 associe la dimension de Hausdorff dimE_h^ν^γ,σ de l’ensembleE_h^ν^γ,σ ={x: lim inf_r_→₀+

logν_γ,σ(B(x,r)) logr =h}.

A une mesure de Borel positive` νsur [0,1], nous associons sa fonction d’´echelleτν(q) d´efinie relativement

`

a la grilleb-adique comme dans [6], et la transform´ee de Legendreτ_ν^∗(h) = infq∈R(qh−τν(q)). On dit que le formalisme multifractal est valide pourν enhsi dimE_h^ν=τ_ν^∗(h).

On a alors le théorème suivant, qui dépend de deux conditions sur µ : C1 demande que le support de µ soit [0,1] et que sup_x_∈_[0,1],r_∈_(0,1)^log^µ(B(x,r))_log_r < ∞. Etant donné un exposant h ≥ 0, C2(h) est vérifiée lorsque la famille{(kb⁻^j,2b⁻^j)}j≥1,k=0,..,b^j forme un système hétérogène d’ubiquité par rapport

`

a µ, h, τ_µ^∗(h)

au sens de [4].

Théorème 0.1 Supposons que C1 soit vérifiée par µ. Soit qγ,σ = inf{q ∈ R : τµ(σq) +γq = 0} et hγ,σ=στ_µ⁰(σq_γ,σ⁻ ) +γ. On a qγ,σ ∈(0,1]et0≤hγ,σ ≤q_γ,σ⁻¹.

(i) Si hγ,σ > 0, alors pour tout h ∈ [0, hγ,σ], dν_γ,σ(h) ≤ qγ,σh. De plus, si C2 ^h^γ,σ_σ⁻^γ

est v´erifi´ee, alors pour touth∈[0, hγ,σ],dν_γ,σ(h) =qγ,σh, et le formalisme multifractal est valide en h.

(ii) Sih≥hγ,σ, alors dν_γ,σ(h)≤τ_µ^∗ ^h⁻_σ^γ

. De plus, siC2 ^h^γ,σ_σ⁻^γ

est v´erifi´ee, dν_γ,σ(h) =τ_µ^∗ ^h⁻_σ^γ , et le formalisme multifractal est valide enh.

Lorsqu’elle est v´erifi´ee, C2 h

permet de calculer la dimension de Hausdorff d’ensembles limsup Sµ(α, δ,ε) é etroitement liés à la fois à la distribution de la masse deµet à la famille des pointsb-adiques {(kb⁻^j,2b⁻^j)}j≥1,k=0,..,b^j. En effet, on a :

Théorème 0.2 Soith >0. Étant donnésδ≥1 et une suite positive eε={ε_j}j, on définit l’ensemble S_µ(h, δ,ε) =e \

N≥1

[

n≥N:b^−j(h+^εj⁾≤µ [kb^−j,(k+1)b^−j)

≤b^−j(h−εj⁾

[kb⁻^j−b⁻^jδ, kb⁻^j+b⁻^jδ].

Supposons queC2 h

soit vérifiée parµ. Il existe une suite positiveεeconvergeant vers 0 telle que pour chaqueδ≥1, il existe une mesure de Borel positivem_δ sur[0,1]telle quem_δ(S_µ(h, δ,ε))e >0, et pour tout borélienE⊂[0,1]tel quedimE < τ_µ^∗(h)/δ,m_δ(E) = 0. En particulier, on adimS_µ(h, δ,ε)e ≥τ_µ^∗(h)/δ.

La connaissance de la dimension de Hausdorff de ces ensemblesS_µ(h, δ,ε) permet le calcul dee d_ν_γ,σ.

1. Introduction

Among the measures whose multifractal analysis can be performed, two families can be distinguished by the shape of their Hausdorff spectrum. On one side, measures built on an additive scheme, such as the Lebesgue-Stieljes measure associated with L´evy subordinators [10] for instance, classically exhibit linear increasing spectrum. On the other side, atomless measures with a construction involving a multiplicative scheme usually have a strictly concave spectrum, including a decreasing part when h is larger than a typical exponentht. Multinomial measures, Mandelbrot multiplicative cascades and their extensions, are classical examples of such measures, with the well-knownT

-shaped spectrum.

We consider a distinct construction scheme which mixes both additive and multiplicative structures.

Definition1.1 If µ is a positive Borel measure on [0,1] andb is an integer greater than 2, let νγ,σ be the measure defined with the help of two parameters γ≥0 andσ≥1 by

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νγ,σ =X

j≥1

j⁻² X

k=0,...,b^j−1

b⁻^jγµ([kb⁻^j,(k+ 1)b⁻^j))^σδ_kb−j. (1)

The multifractal behavior of these measures combines the two typical multifractal behaviors described above. A part of their multifractal spectrum is closely related to the non-emptiness of a new kind of limsup-setsSµ(α, δ, M) defined by (4). The computation of the Hausdorff dimension of these sets requires new ubiquity results (see Section 4 and [4]) which extend to the multifractal frame the classical ubiquity results only valid for a monofractal measureµ[7,11].

2. Multifractal spectrum of the new class of measures ν_γ,σ

The local regularity of a measureµat a pointxis hereafter described by the H¨older exponenthµ(x) = lim inf_r_→₀+ logµ(B(x,r))

logr . The multifractal analysis ofµconsists in computing the Hausdorff dimensions of the level sets of this H¨older exponentE_h^µ ={x:h_µ(x) =h},h≥0. Then one tries to find the multifractal spectrum ofµ,h7→d_µ(h) = dim(E_h^µ),where dim(E) stands for the Hausdorff dimension of the setE.

In order to fully state our result, the notion of multifractal formalism for measures is required. A multifractal formalism is a formula which relates, via a Legendre transform, the multifractal spectrum d_µ to a scaling function associated withµ [6,15]. Here, we adopt the following definition for the scaling function [6]: ∀q ∈ R, τ_µ(q) = lim inf_j_→₊_∞−¹_jlog_bP

0≤k≤b^jµ [kb⁻^j,(k+ 1)b⁻^j)^q

, with the convention 0^q = 0,∀q. The multifractal formalism is said to hold forµat exponenthwhen the multifractal spectrum ofµathis equal to the Legendre transformτ_µ^∗ ofτ_µ ath, i.e. whend_µ(h) =τ_µ^∗(h) = inf_q_∈R(qh−τ_µ(q)).

Our result invokes two technical conditions. C1simply requires that the support of the measureµis [0,1] and that one has the control sup_x_∈_[0,1],r_∈_(0,1)^log^µ(B(x,r))_log_r <∞.C2(h)is said to hold for the measure µ and the exponent h when the family of points {(kb⁻^j,2b⁻^j)}j≥1,k=0,..,b^j−1 forms an heterogeneous ubiquitous system with respect to µ, h, τ_µ^∗(h)

(see Section 4).

Theorem 2.1 Letµbe a positive Borel measure supported by[0,1], and assume thatC1holds forµ. Let γ≥0 andσ≥1, and consider the measureνγ,σ defined in (1). Let qγ,σ = inf{q∈R:τµ(σq) +γq= 0}, andhγ,σ=στ_µ⁰(σq_γ,σ⁻ ) +γ. One has qγ,σ∈(0,1]and0≤hγ,σ≤q⁻_γ,σ¹.

(i) Ifhγ,σ >0, for everyh∈[0, hγ,σ],dν_γ,σ(h)≤qγ,σh. If moreover C2 ^h^γ,σ_σ⁻^γ

holds, then for every h∈[0, hγ,σ] dν_γ,σ(h) =qγ,σh, and the multifractal formalism holds ath.

(ii) If h≥h_γ,σ, thend_ν_γ,σ(h)≤τ_µ^∗ ^h⁻_σ^γ

. If moreoverC2 ^h^γ,σ_σ⁻^γ

holds, d_ν_γ,σ(h) =τ_µ^∗ ^h⁻_σ^γ

, and the multifractal formalism holds at h.

Theorem 2.1 applies to several classes of atomless statistically self-similar measures µ (see [2,5]), as well as to the measureν0,1itself: In this case the process can be iterated, the spectrum being unchanged.

It is shown in [2] that multifractal formalisms that focus on level sets such as{x: limr→0

logν_γ,σ(B(x,r))

logr =

h}(defined using a limit instead of a lim inf) do not hold for these measuresνγ,σ athwhen 0< h < hγ,σ. Theorem 2.1 thus pleads for the choice of the setsE_h^ν^γ,σ defined using the H¨older exponenthνγ,σ(x).

Also, the measuresνγ,σ provide new examples of measures that may have a scaling functionτν_γ,σwhose derivative does not exist at some point (hereqγ,σ), while satisfying a multifractal formalism. Such a non differentiability is called a phase transition in the thermodynamical terminology.

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ht d (h)_µ

h o

1

o

1 1

d (h)_µ d (h)

f

h h

h h h

t hc t γ t

ν f wav

d (h) or d (h)

o

Fig.1: Typical multifractal spectrum of a measureµbuilt on an (Left) additive or (Middle) multiplicative scheme, and (Right) ofνγ,σ(Theorem 2.1) andfγ,σ(Theorem 3.1) withγ= 0 andσ= 1; offwav(Theorem 3.2) with arbitraryγ, σ.

Fig.1: Spectre multifractal typique d’une mesureµconstruite sur un schéma (gauche) additif ou (centre) multiplicatif, et (droite) deνγ,σ(Théorème 2.1) oufγ,σ (Théorème 3.1) avecγ= 0,σ= 1; defwav(Théorème 3.2) avecγ, σquelconques.

3. Multifractal spectrum of function seriesf_γ,σ naturally associated with ν_γ,σ

The multifractal analysis of functions constitutes a companion domain of the multifractal analysis of measures. Given a bounded functionf on [0,1],x0∈(0,1) andh≥0,f ∈C^h(x0) if there exist a constant C and a polynomial P of degree ≤hsuch that

f(x)−P(x−x0)

≤C|x−x0|^h forx close enough to x0. The pointwise H¨older exponent of f at x0 [9] is then defined as hf(x0) = sup{h:f ∈ C^h(x0). The multifractal analysis off consists in computing its Hausdorff spectrumd_f :h7→dim{x:h_f(x) =h}.

Inspired by the construction of the measures ν_γ,σ, it is natural to build functions with a structure comparable with the one of ν_γ,σ. Let π be a positive Borel measure supported by [0,1], φ a bounded function onR,γ≥0 andσ≥1. In two situations, we study the function series defined by

f(x) =

+∞

X

j=1

fj(x) wherefj(x) = b⁻^jγ j²

X

k=0,...,b^j−1

π([kb⁻^j,(k+ 1)b⁻^j))^σφ(b^jx−k). (2)

The first choice ofφcreates a discontinuity at eachb-adic number, the second one uses a wavelet.

Theorem 3.1 Under the assumptions of Theorem 2.1, and ifφ(x) = (x−1)11_(x₋₁₎_∈_[0,1) andπ=µ, the corresponding function series (2), here denoted byf_γ,σ, satisfy:

(i) Ifhγ,σ >0 andC2 ^h^γ,σ_σ⁻^γ

holds, for anyh∈

0,^h^γ,σ_σ⁻^γ

,df_γ,σ(h) =qγ,σh.

(ii) If h≥hγ,σ andC2 ^h⁻_σ^γ

holds, thendf_γ,σ(h) =dµ h−γ σ

=τ_µ^∗ ^h⁻_σ^γ .

Theorem 3.2 b = 2. Under the assumptions of Theorem 2.1, if φ is a C^∞ wavelet as constructed for instance in [14] and ifπ=ν0,1, the corresponding function series (2), denoted herefwav, satisfy:

(i) Ifh_0,1>0andC2(h_0,1)holds, for anyh∈[γ, γ+σh_0,1], one hasd_f_wav(h) =q_0,1^h⁻_σ^γ. (ii) If h≥γ+σh0,1 andC2 ^h⁻_σ^γ

holds, thendf_wav(h) =dµ h−γ σ

=τ_µ^∗ ^h⁻_σ^γ .

Theorem 3.1 is established in [3], and Theorem 3.2 is a consequence of a work achieved in [1].

The interest in function series with bounded variations and having a dense set of discontinuities goes back to the note of C. Jordan [12], where the spaceBV was introduced.

4. Ubiquity revisited via multifractality

Along the proof of Theorem 2.1, the non-emptiness of limsup-sets defined asSµ(α, δ, M) in (4) is crucial.

The study of this kind of sets is comparable with classical problems of ubiquity [7], but here the sets

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Sµ(α, δ, M) are conditioned by a property PM which depends on a (statistically self-similar) measureµ.

We thus need a refinement of the ubiquity results in order to be able to combine the approximation rate by some family of points with the multifractal properties of a measureµ. This is achieved by Theorem 4.2.

Letd ≥1 and µa positive Borel measure on [0,1]^d. The analysis of the local structure of µ may be naturally done on ac-adic grid for somec≥2. This is the case for instance for multinomial measures or Mandelbrot cascades. We shall thus need the following notations. For every integerj ≥1 and for every multi-integer k = (k1, . . . , kd) ∈ N^d, let us denote by Ij,k thec-adic box Π^d_i=1[kic⁻^j,(ki+ 1)c⁻^j). For every pointx∈[0,1)^d,∀j≥1, we denotekj,x the uniquek∈N^d such thatx∈Ij,k_j,x.

Definition4.1 Letµbe a positive Borel measure supported by[0,1]^d, andα,β two positive real numbers.

Let{xn}n∈Nbe a sequence in[0,1]^d, and let{λn}n∈Nbe a non-increasing sequence of positive real numbers converging to 0. The family{(xn, λn)}n∈Nis said to form an heterogeneous ubiquitous system with respect to(µ, α, β)if the following conditions(1-4) are fulfilled (|I|stands for the diameter of the setI).

(1)There exist φandψ, two non-decreasing continuous functions defined on R₊ such that (i)ϕ(0) = ψ(0) = 0,r7→r⁻^ϕ(r)andr7→r⁻^ψ(r)are non-increasing near0⁺, andlim_r_→₀+r⁻^ϕ(r)= +∞;(ii)∀ε >0, r7→r^ε⁻^ϕ(r) is non-decreasing near 0;(iii) (2),(3) and(4)hold.

(2)There exist a measure mwith a support equal to[0,1]^d, which satisfies the following properties:

(i)m-almost every y∈[0,1]^d belongs toT

N≥1

S

n≥NB(xn,^λ₂ⁿ).

(ii) For m-almost everyy∈[0,1]^d, there exists an integerj(y)such that

∀j ≥j(y), ∀k such thatkk−kj,yk_∞≤1, m Ij,k

≤ |Ij,k|^β⁻^ϕ(^|^I^j,k^|⁾ (3) (iii) There exists a constantM (depending onc andµ) such that form-almost everyy∈[0,1]^d, one can find an integerj(y)such that∀j≥j(y),∀ ksuch thatkk−k_j,yk_∞≤1,PM(I_j,k)holds, where PM(I)is said to hold for a setI and a constantM when _M¹|I|^α+ψ(^|^I^|⁾≤µ I

≤M|I|^α⁻^ψ(^|^I^|⁾.

(3) (Self-similarity ofm) For every c-adic box I of [0,1]^d, let fI denote the canonical affine mapping fromI onto[0,1]^d. There exists a measurem^I onI, equivalent to the restriction ofmtoI, such that the measurem^I◦f_I⁻¹ satisfies (3), and with the same exponent β.

Let us define the non-decreasing sequence{E_n^I}n≥1 of sets of [0,1]^d

E_n^I =





 y∈I:







∀ j≥n+ log_c |I|⁻¹ ,

∀k such thatkk−k_j,yk∞≤1,

m^I I_j,k

≤ |I_j,k|

|I|

^β−ϕ ^|^Ij,k^|

|I|





 .

By (3),S

n≥1E_n^I is of fullm^I-measure. DefinenI = inf

n≥1 : mÎ(E_nÎ)≥¹₂kmÎk .

(4)(Speed of renewal of level sets and control of the mass kmÎk) There exists Jm such that for every j≥Jm, for everyc-adic box I=Ij,k,nI ≤log_c |I|⁻¹)ϕ(|I|)and |I|^ϕ(^|Î^|⁾≤ kmÎk.

Theorem 4.2 Let us assume that{(x_n, λ_n)}n∈N forms an heterogeneous ubiquitous system with respect to(µ, α, β). For δandM two real numbers greater than 1, let

Sµ(α, δ, M) = \

N≥1

[

n≥N:PM(B(xn,λn))holds

B(xn, λ^δ_n). (4)

There exists a constant M such that for every δ ≥ 1, one can find a positive measure mδ such that mδ(Sµ(α, δ, M)) > 0, and for everyx ∈ Sµ(α, δ, M), lim sup_r_→₀+

m_δ B(x,r)

rβ/δ−(4+d)ϕ(r) < ∞. In particular, dimSµ(α, δ, M)≥β/δ.

Theorem 4.2 is established in [4]. Heuristically, since PM(I) ensures that µ(I) ∼ |I|^α up to a small correction, Theorem 4.2 allows the computation of the Hausdorff dimension of sets of points xthat are infinitely often close at rateδto pointsxn(i.e. such thatkx−xnk_∞≤λ^δ_n) that verifyµ(B(xn, λn))∼λ^α_n.

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Theorem 4.2 is referred to as “measure-conditioned ubiquity” because it involves an ubiquity property of a family of points that must satisfy some property. Here this property is related to the behavior of µ B(xn, λn)

. The “usual” ubiquity theorems [7,11] must be seen as Theorem 4.2 in the particular case where the measureµis the Lebesgue measure, that is the condition is empty sinceµis strictly monofractal.

•Examples of families {(xn, λn)}:- Theb-adic family

(kb⁻^j,2b⁻^j) _k

∈N^d, j∈N^∗ (whereb∈N\{0,1} can differ fromc) and the family of rational numbers

(p/q,2/q^1+1/d) _p

∈N^d, q∈N∗ satisfy (i) of assumption (2)for any measure m, because lim sup_n_→∞B(x_n, λ_n/2) = [0,1]^d in these cases. Ifmis atomless,d= 1 andα∈R\Q, the family{({nα},2/n)}n∈N∗ satisfies (i) of(2)({x}is the fractional part ofx).

- Item (i) of (2) can also be satisfied almost surely for some random families of points. Let {xn} be a sequence of points independently and uniformly distributed in [0,1]^d, and {λn}n be a non-increasing sequence of positive numbers. If lim sup_n_→₊_∞ Pn

p=1λp/2−dlogn

= +∞then, with probability one, lim sup_n_→∞B(x_n, λ_n/2) = [0,1]^d (see [13]).

• Examples of measures µ: Suppose that lim sup_n_→∞B(xn, λn/2) = [0,1]^d. In [5], the properties required by Definition 4.1 are shown to hold for several classes of statistically self-similar measures (multinomial measures, Gibbs measures and their random counterparts) withα >0 andβ =τ_µ^∗(α) as soon as τ_µ^∗(α)>0. We mention that, in order to treat the Mandelbrot multiplicative cascades, a slight stronger version of Theorem 4.2 is required, see [4,5].

5. Another application of Theorem 4.2: Conditioned Diophantine approximation

Consider for the family {(x_n, λ_n)}n the pairs {p/q,1/q²}p,q∈N², p<q. For any x ∈ [0,1], consider the b-adic expansion of x = P∞

m=1x_mb⁻^m, where ∀m, x_m ∈ {0,1, . . . , b−1}. Let φ_i,n(x) be the function x7→φ_i,n(x) =^#^{^m^≤^n:x_n ^m⁼ⁱ^}.Given (ρ₀, ρ₁, . . . , ρ_b₋₁)∈(0,1)^b such thatPb−1

i=0ρ_i= 1, andδ >1, let E_δ^ρ⁰^,ρ¹^,...,ρ^b−1=







x∈[0,1] :

there is an infinite number of integersp_n, q_n such that |x−p_n/q_n| ≤q⁻_n^2δ and∀i∈ {0,1, . . . , b−1}, lim

n→+∞φ_i,[log

bq²_n] p_n/q_n

=ρ_i





 .

Besicovitch and later Eggleston [8] found dimE^ρ₁⁰^,ρ¹^,...,ρ^b−1 =Pb−1

i=0−ρilog_bρi. We address the problem of the computation of the Hausdorff dimension of the setsE_δ^ρ⁰^,ρ¹^,...,ρ^b−1, which is the set of points which are well-approximated by rational numbers that have given frequencies of digits. This problem is not covered by the usual ubiquity results. Applying Theorem 4.2 with the multinomial measureµassociated with the weights (ρ0, ρ1, . . . , ρb−1) and with the family {(xn, λn)}n∈N = {(p/q,1/q²)}p,q∈N×N∗ yields dimE_δ^ρ⁰^,ρ¹^,...,ρ^b−1≥δ⁻¹Pb−1

i=0−ρ_ilog_bρ_i. The opposite inequality follows from standard arguments.

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