Heterogeneous ubiquitous systems in $\mathbb{R}^{d}$ and Hausdorff dimension

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Heterogeneous ubiquitous systems in R ^d and Hausdorff dimension

Julien Barral, Stephane Seuret

To cite this version:

Julien Barral, Stephane Seuret. Heterogeneous ubiquitous systems in R^d and Hausdorff dimension.

2005. �hal-00004525�

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ccsd-00004525, version 1 - 21 Mar 2005

HETEROGENEOUS UBIQUITOUS SYSTEMS IN R AND HAUSDORFF DIMENSION

JULIEN BARRAL AND ST´EPHANE SEURET

Abstract. Let{xn}n∈Nbe a sequence of [0,1]^d,{λn}n∈N a sequence of positive real numbers converging to 0, andδ >1. The classical ubiquity results are concerned with the computation of the Hausdorff dimension of limsup-sets of the formS(δ) =T

N∈N

S

n≥NB(xn, λ^δ_n).

Letµbe a positive Borel measure on [0,1]^d,ρ∈(0,1] andα >0. Consider the finer limsup-set

Sµ(ρ, δ, α) = \

N∈N

[

n≥N:µ(B(xn,λ^ρn))∼λ^ραn

B(xn, λ^δ_n).

We show that, under suitable assumptions on the measureµ, the Hausdorff dimension of the setsSµ(ρ, δ, α) can be computed. Moreover, whenρ <1, a yet unknown saturation phenomenon appears in the computation of the Hausdorff dimension ofSµ(ρ, δ, α). Our results apply to several classes of multifractal measures, andS(δ) corresponds to the special case whereµis a monofractal measure like the Lebesgue measure.

The computation of the dimensions of such sets opens the way to the study of several new objects and phenomena. Applications are given for the Dio- phantine approximation conditioned by (or combined with)b-adic expansion properties, by averages of some Birkhoff sums and branching random walks, as well as by asymptotic behavior of random covering numbers.

1. Introduction

Since the famous result of Jarnik [30] concerning Diophantine approximation and Hausdorff dimension, the following problem has been widely encountered and studied in various mathematical situations.

Let{xn}^n∈^Nbe a sequence in a compact metric spaceEand{λn}^n∈^Na sequence of positive real numbers converging to 0. Let us define the limsup set

S= \

N∈N

[

n≥N

B(xn, λn),

and let D be its Hausdorff dimension. Let δ > 1. What can be said about the Hausdorff dimension of the subsetS(δ) ofS defined by

S(δ) = \

N∈N

[

n≥N

B(xn, λ^δ_n) ?

Intuitively one would expect the Hausdorff dimension ofS(δ) to be lower bounded byD/δ. This has been proved to hold in many cases which can roughly be separated into two classes:

• when the sequence {(xn, λn)}ⁿ forms a sort of “regular system” [3, 15], which ensures a strong uniform repartition of the points{xn}ⁿ.

1

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2 JULIEN BARRAL AND ST´EPHANE SEURET

• when the sequence{(xn, λn)}ⁿforms an ubiquitous system [18, 19, 29] with respect to a monofractal measure carried by the setS.

Let us mention that similar results are obtained in [43] when E is a Julia set.

When dimS(δ) < D, such subsets S(δ) are often referred to as exceptional sets [17]. Another type of exceptional sets arises when considering the level sets of well-chosen functions:

• the function associating with each pointx∈[0,1] the frequency of the digit i∈ {0,1, . . . , b−1}in theb-adic expansion ofx,

• more generally the function associating with each point xthe average of the Birkhoff sums related to some dynamical systems,

• the functionx7→hf(x), whenf is either a function or a measure onR^dand hf(x) is a measure of the local regularity (typically an H¨older exponent) of f aroundx.

It is a natural question to ask whether these two approaches can be combined to obtain finer exceptional sets. Let us take an example to illustrate our purpose.

On one side, it is known since Jarnik’s results [30] that if the sequence{(xn, λn)}ⁿ is made of the rational pairs{(p/q,1/q²)}p,q∈N∗2, p≤q, then for everyδ >1 the subset S(δ) of [0,1] has a Hausdorff dimension equal to 1/δ. In the ubiquity’s setting, this is a consequence of the fact that the family{(p/q,1/q²)}p,q∈N∗2 forms an ubiquitous systems associated with the Lebesgue measure [18, 19].

On the other side, given (π0, π1, . . . , πb−1) ∈ [0,1]^b such that Pb−1

i=0πi = 1, Besicovitch and later Eggleston [20] studied the setsE^π⁰^,π¹^,...,π^b−1 of pointsxsuch that the frequency of the digiti∈ {0,1, . . . , b−1} in theb-adic expansion ofxis equal toπi. More precisely, for anyx∈[0,1], let us consider theb-adic expansion of x=P∞

m=1xmb^−m, where∀m,xm∈ {0,1, . . . , b−1}. Let φi,n(x) be the mapping (1) x7→φi,n(x) = #{m≤n:xm=i}

n .

Then E^π⁰^,π¹^,...,π^b−1 = {x : ∀i ∈ {0,1, . . . , b−1},limn→+∞φi,n(x) = πi}. They found that dimE^π⁰^,π¹^,...,π^b−1 =Pb−1

i=0−πilog_bπi.

We address the problem of the computation of the Hausdorff dimension of the subsetsE_δ^π⁰^,π¹^,...,π^b−1 of [0,1] defined by

E_δ^π⁰^,π¹^,...,π^b−1=





x:







∃(pn, qn)n ∈(N^∗2)^Nsuch thatqn→+∞,

|x−pn/qn| ≤1/q_n^2δ and∀i∈ {0, . . . , b−1}, limn→+∞φi,[log_b(q²_n)](pn/qn) =πi





 ([x] denotes the integer part of x). In other words, we seek in this example for the Hausdorff dimension of the set of points of [0,1] which are well-approximated by rational numbers fulfilling a given Besicovitch condition (i.e. having given digit frequencies in their b-adic expansion). This problem is not covered by the works mentioned above. The main reason is the heterogeneity of the repartition of the rational numbers satisfying the Besicovitch conditions. As a consequence of Theorems 2.2 and 2.7 of this paper, one obtains

(2) dimE_δ^π⁰^,π¹^,...,π^b−1= Pb−1

i=0−πilog_bπi

δ .

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The key point to achieve this work is to see the Besicovitch condition as a scaling property derived from a multinomial measure. More precisely, the computation of the Hausdorff dimensions of the setsE_δ^π⁰^,π¹^,...,π^b−1 proves to be a particular case of the following problem: Letµ be a positive Borel measure on the compact metric spaceEconsidered above. Givenα >0 andδ≥1, what is the Hausdorff dimension of the set of points xof E that are well-approximated by points of{(xn, λn)}ⁿ at rateδ, i.e. such that for an infinite number of integersn,|x−xn| ≤λ^δ_n, conditionally to the fact that the corresponding sequence of couples (xn, λn) satisfies

(3) lim

n→∞

logµ B(xn, λn) log(λn) =α?

In other words, ifε = (εn)n≥1 is a sequence of positive numbers converging to 0, what is the Hausdorff dimension of

(4) Sµ(δ, α, ε) = \

N≥0

[

n≥N:λ^α+n ^εn≤µ(B(xn,λn))≤λ^α−εnn

B(xn, λ^δ_n) ?

We study the problem inR^d(d≥1). An upper bound for the Hausdorff dimension ofSµ(δ, α, ε) is given by Theorem 2.2 forweakly redundant systems{(xn, λn)}ⁿ (see Definition 2.1). Its proof uses ideas coming from multifractal formalism for measures [14, 39].

Theorem 2.7 (caseρ= 1) gives a precise lower bound of the Hausdorff dimension ofSµ(δ, α, ε) when the family{(xn, λn)}ⁿforms a 1-heterogeneous ubiquitous system with respect to the measure µ(see Definition 2.3 for this notion, which generalizes the notion of ubiquitous system mentioned above). It can specifically be applied to measuresµthat possess some statistical self-similarity property, and to any family {(xn, λn)}ⁿ as soon as the support ofµis covered by lim sup_n→∞B(xn, λn).

To fix ideas, let us state a corollary of Theorems 2.2 and 2.7. This result uses the Legendre transformτ_µ^∗of the “dimension” functionτµconsidered in the multifractal formalism studied in [14] (see Section 2.2 and Definition 8).

Theorem 1.1. Let µ be a multinomial measure on [0,1]^d. Suppose that the family{(xn, λn)}ⁿ forms a weakly redundant 1-heterogeneous ubiquitous system with respect to µ, α, τ_µ^∗(α)

.

There is a positive sequence ε= (εn)n≥1 converging to 0 at∞ such that

∀δ≥1, dimSµ(δ, α, ε) =τ_µ^∗(α)/δ.

Examples of remarkable families {(xn, λn)}ⁿ are discussed in Section 6, as well as examples of suitable statistically self-similar measuresµ. There, the measures µare chosen so that the property (3) has a relevant interpretation (for instance in terms of theb-adic expansion of the pointsxn).

The formula (4) defining the set Sµ(δ, α, ε) naturally leads to the question of conditioned ubiquity into the following more general form: Letρ∈(0,1]. What is the Hausdorff dimension of

(5) Sµ(ρ, δ, α, ε) = \

N≥0

[

n≥N:λ^ρ(α+εn)n ≤µ(B(xn,λ^ρn))≤λ^ρ(α−εn)n

B(xn, λ^δ_n) ?

Remark that, in (4) and (5), if µ equals the Lebesgue measure and ifα=d, the conditions on B(xn, λ^ρ_n) are empty, since they are independent of xn, λn and ρ

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(this remains true for a strictly monofractal measureµof indexα, that is such that

∃C >0,∃r0such that∀x∈supp(µ),∀ 0< r≤r0, C⁻¹r^α≤µ(B(x, r))≤Cr^α).

Again, an upper bound for the Hausdorff dimension ofSµ(ρ, δ, α, ε) is found in Theorem 2.2 for weakly redundant systems.

Theorem 2.7 (case ρ < 1) yields a lower bound of the Hausdorff dimension of Sµ(ρ, δ, α, ε) whenρ <1, as soon as the family{(xn, λn)}ⁿforms aρ-heterogeneous ubiquitous system with respect toµin the sense of Definition 2.5. The introduction of this dilation parameterρsubstantially modifies Definition 2.3 and the proofs of the results in the initial caseρ= 1.

As a consequence of Theorem 2.7, a new saturation phenomenon occurs for systems that are both weakly redundant andρ-heterogeneous ubiquitous systems whenρ <1. This points out the heterogeneity introduced when considering ubiquity conditioned by measures that are not monofractal. The following result is also a corollary of Theorems 2.2 and 2.7.

Theorem 1.2. Let µ be a multinomial measure on[0,1]^d. Let ρ∈(0,1). Suppose that {(xn, λn)}ⁿ forms a weakly redundantρ-heterogeneous ubiquitous system with respect to µ, α, τ_µ^∗(α)

.

There is a positive sequence ε= (εn)n≥1 converging to 0 at∞ such that

∀ δ≥1, dimSµ(ρ, δ, α, ε) = mind(1−ρ) +ρτ_µ^∗(α)

δ , τ_µ^∗(α) .

Under the assumptions of Theorem 1.2, if τ_µ^∗(α) < d, although δ starts to in- crease from 1, dimSµ(ρ, δ, α, ε) remains constant untilδ reaches the critical value

d(1−ρ)+ρτ_µ^∗(α)

τ_µ^∗(α) >1. When δ becomes larger than ^{d(1−ρ)+ρτ}_τ∗ ^µ^∗^(α)

µ(α) , the dimension decreases. This is what we call a saturation phenomenon.

It turns out that conditioned ubiquity as defined in this paper is closely related to the local regularity properties of some new classes of functions and measures having dense sets of discontinuities. In particular, Theorem 2.7 is a determinant tool to analyze measures constructed as the measuresνρ,γ,σ

νρ,γ,σ =X

n≥0

λ^γ_nµ B(xn, λ^ρ_n)σ

δxn,

whereδxn is the probability Dirac mass atxn,ρ∈(0,1], andγ, σare real numbers which make the series converge. Conditioned ubiquity is also essential to perform the multifractal analysis of L´evy processes in multifractal time. These objects have multifractal properties that were unknown until now. Their study is achieved in other works [9, 10].

The definitions of weakly redundant and ρ-heterogeneous ubiquitous systems are given in Section 2. The statements of the main results (Theorems 2.2 and 2.7) then follow. The proofs of Theorem 2.2, Theorem 2.7 (case ρ= 1) and Theorem 2.7 (case ρ < 1) are respectively achieved in Sections 3, 4 and 5. Finally, our results apply to suitable examples of systems{(xn, λn)}ⁿ and measuresµthat are discussed in Section 6.

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2. Definitions and statement of results

It is convenient to endow R^d with the supremum norm k · k^∞ and with the associated distance (x, y)∈R^d×R^d7→ kx−yk^∞= max1≤i≤d(|xi−yi|). All along the paper, for a setS,|S|denotes then the diameter ofS.

We briefly recall the definition of the generalized Hausdorff measures and Haus- dorff dimension inR^d. Letξbe agaugefunction, i.e. a non-negative non-decreasing function onR₊ such that limx→0⁺ξ(x) = 0. LetS be a subset of R^d. For η >0, let us define

H^ξη(S) = inf

{Ci}_i∈I:S⊂S

i∈ICi

X

i∈I

ξ(|Ci|), (the family{Ci}^i∈I coversS) where the infimum is taken over all countable families{Ci}^i∈I such that ∀i ∈ I,

|Ci| ≤η. Asηdecreases to 0,Hη^ξ(S) is non-decreasing, andH^ξ(S) = limη→0H^ξη(S) defines a Borel measure onR^d, called Hausdorffξ-measure.

Defining the family ξα(x) = |x|^α (α ≥ 0), there exists a unique real number 0 ≤ D ≤ d, called the Hausdorff dimension of S and denoted dimS, such that D = sup

α≥0 :H^ξ^α(S) = +∞ = inf

α:H^ξ^α(S) = 0 (with the convention sup∅= 0). We refer the reader to [36, 22] for instance for more details on Hausdorff dimensions.

Let µ be a positive Borel measure with a support contained in [0,1]^d. The analysis of the local structure of the measure µ in [0,1]^d may be naturally done using a c-adic grid (c ≥ 2). This is the case for instance for the examples of measures of Section 6. We shall thus need the following definitions.

Letcbe an integer≥2. For everyj≥0,∀k= (k1, . . . , kd)∈ {0,1, . . . , c^j−1}^d, I_j,^c_k denotes thec-adic box [k1c^−j,(k1+ 1)c^−j)×. . .×[kdc^−j,(kd+ 1)c^−j). ∀x∈ [0,1)^d,I_j^c(x) stands for the uniquec-adic box of generationj that containsx, and k^c_j,x is the unique (multi-)integer such thatI_j^c(x) =I_j,^c_k^c

j,x. Ifk= (k1, . . . , kd) and k^′ = (k₁^′, . . . , k^′_d) both belong toN^d,kk−k^′k^∞= maxi|ki−k_i^′|. The set of c-adic boxes included in [0,1)^d is denoted byI.

Finally, the lower Hausdorff dimension of µ, dim(µ), is classically defined as inf

dimE:E∈ B([0,1]^d), µ(E)>0 .

2.1. Weakly redundant systems. Let {xn}^n∈^N be a family of points of [0,1]^d and {λn}^n∈^N a non-increasing sequence of positive real numbers converging to 0.

For everyj≥0, let

(6) Tj =n

n: 2^−(j+1)< λn≤2^−jo .

The following definition introduces a natural property from which an upper bound for the Hausdorff dimension of limsup-sets (4) and (5) can be derived. Weak re- dundancyis slightly more general thansparsityof [23].

Definition 2.1. The family {(xn, λn)}^n∈^N is said to form a weakly redundant system if there exists a sequence of integers (Nj)j≥0 such that

(i) limj→∞logNj/j= 0.

(ii) for everyj≥1,Tjcan be decomposed intoNjpairwise disjoint subsets (denoted Tj,1, . . . , Tj,Nj) such that for each 1≤i≤Nj, the family

B(xn, λn) : n∈Tj,i is composed of disjoint balls.

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One hasSNj

i=1Tj,i=Tj. Since theTj,iare pairwise disjoint, any pointx∈[0,1]^d is covered by at most Nj balls B(xn, λn), n ∈ Tj. Moreover, for every i and j, the number of balls of Tj,i is bounded by Cd2^dj, whereCd is a positive constant depending only ond. Indeed, if two integersn6=n^′are such thatλnandλn^′ belong toTj,i, then kxn−xn^′k^∞≥2^−j.

2.2. Upper bounds for Hausdorff dimensions of conditioned limsup sets.

Letµbe a finite positive Borel measure on [0,1]^d.

We let the reader verify that if suppµ= [0,1]^d, then the concave function (7) τµ,c: q7→lim inf

j→∞−j⁻¹log_c X

k∈{0,...,c^j−1}^d

µ(I_j,^c_k)^q

does not depend on the integerc≥2, and is consequently simply denotedτµ. This function is considered in the multifractal formalism for measures of [14]. Then, the Legendre transform ofτµ at α∈R₊, denoted by τ_µ^∗, is defined by

(8) τ_µ^∗: α7→inf

q∈R αq−τµ(q)

∈R∪ {−∞}.

Theorem 2.2. Let {xn}^n∈^N be a family of points of [0,1]^d and {λn}^n∈^N a non- increasing sequence of positive real numbers converging to 0. Let µ be a positive finite Borel measure with a support equal to [0,1]^d. Let {εn}^n∈^N be a positive sequence converging to 0, ρ∈(0,1],δ≥1 andα≥0. Let us define

Sµ(ρ, δ, α, ε) = \

N≥1

[

n≥N:λ^ρ(α+εn)n ≤µ(B(xn,λ^ρn))≤λ^ρ(α−εn)n

B(xn, λ^δ_n).

Suppose that{(xn, λn)}^n∈^Nforms a weakly redundant system. Then (9) dimSµ(ρ, δ, α, ε)≤min d(1−ρ) +ρτ_µ^∗(α)

δ , τ_µ^∗(α) . Moreover,Sµ(ρ, δ, α, ε) =∅if τ_µ^∗(α)<0.

The result does not depend on the precise value of the sequence{εn}ⁿ, as soon as limn→+∞εn= 0. The proof of Theorem 2.2 is given in Section 3.

2.3. Heterogeneous ubiquitous systems. Letα >0 andβ ∈(0, d] be two real numbers. They play the role respectively of the H¨older exponent of µ and of the lower Hausdorff dimension of an auxiliary measurem.

The upper bound obtained by Theorem 2.2 is rather natural. Here we seek for conditions that make the inequality (9) become an equality. The following Definitions 2.3 and 2.5 provide properties guarantying this equality.

The notion ofheterogeneous ubiquitous systemgeneralizes the notion of ubiquitous systeminR^d considered in [18].

Definition 2.3. The system{(xn, λn)}^n∈^Nis said to form a 1-heterogeneous ubiquitous system with respect to (µ, α, β) if conditions(1-4)are fulfilled.

(1)There exist two non-decreasing continuous functionsφandψdefined onR₊ with the following properties:

-ϕ(0) =ψ(0) = 0,r7→r^−ϕ(r)andr7→r^−ψ(r) are non-increasing near 0⁺, - limr→0⁺r^−ϕ(r)= +∞, and∀ε >0,r7→r^ε−ϕ(r) is non-decreasing near 0, -ϕandψverify(2), (3)and(4).

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(2)There exist a measurem with a support equal to [0,1]^d with the following properties:

• m-almost everyy∈[0,1]^dbelongs toT

N≥1

S

n≥NB(xn, λn/2), i.e.

m \

N≥1

[

n≥N

B xn, λn/2

=kmk. (10)

• One has:

(11)

(Form-almost everyy∈[0,1]^d,∃ j(y), ∀j≥j(y),

∀ksuch thatkk−k^c_j,yk^∞≤1, P1¹(I_j,^c_k) holds,

wherePM¹ (I) is said to hold for the set Iand for the real numberM ≥1 when (12) M⁻¹|I|^α+ψ(|I|)≤µ I

≤M|I|^α−ψ(|I|).

• One has:

(13)

(Form-almost everyy∈[0,1]^d,∃ j(y), ∀j≥j(y),

∀ksuch thatkk−k^c_j,yk^∞≤1, D1^m(I_j,^c_k) holds,

whereD^mM(I) is said to hold for the set Iand for the real number M >0 when

(14) m(I)≤M|I|^β−ϕ(|I|).

(3) (Self-similarity of m) For every c-adic box L of [0,1)^d, let fL denote the canonical affine mapping from L onto [0,1)^d . There exists a measure m^L on L, equivalent to the restriction m_|L of m to L (in the sense that m_|L and m^L are absolutely continuous with respect to one another), such that property (13) holds for the measurem^L◦f_L⁻¹ instead of the measurem.

For everyn≥1, let us then introduce the sets E_n^L=



x∈L:

∀j ≥n+ log_c |L|⁻¹

, ∀ ksuch thatkk−k^c_j,xk^∞≤1, m^L I_j,^c_k

≤_|I^c

j,k|

|L|

β−ϕ ^|Ic_|L|^j,^k^|



. The sets E_n^L form a non-decreasing sequence in L, and by (13) and property (3), S

n≥1E_n^L is of fullm^L-measure. One can thus consider the integer nL= inf

n≥1 : m^L(E_n^L)≥ km^Lk/2 . Ifx∈(0,1)^d andj≥1, let us define the set of balls

B^j(x) =n

B(xn, λn) :x∈B xn, λn/2

andλn ∈(c^−(j+1), c^−j]o .

Notice that this set may be empty. Then, ifδ >1 andB(xn, λn)∈ B^j(x), consider B(xn, λ^δ_n). This ball contains an infinite number ofc-adic boxes. Among them, let B^δ_n be the set ofc-adic boxes of maximal diameter. Then define

B_j^δ(x) = [

B(xn,λn)∈Bj(x)

B^δ_n.

(4)(Control of the growth speednLand of the masskm^Lk) There exists a subsetD of (1,∞) such that for everyδ∈ D, form-almost everyx∈lim sup_n→∞B xn, λn/2 ,

(9)

there exists an infinite number of integersj for which there existsL∈ B^δj(x) such that

nL≤log_c |L|⁻¹)ϕ(|L|) and |L|^ϕ(|L|)≤ km^Lk. (15)

Remark 2.4. 1. (1)is a technical assumption. In(2), (13) provides a lower bound for the lower Hausdorff dimension of the analyzing measurem. (11) yields a control of the local behavior ofµ,m-almost everywhere. Then (10) is the natural condition on m to analyze ubiquitous properties of {(xn, λn)}ⁿ conditioned byµ. (3) is a kind of self-similarity needed for the measure m, and(4)imposes a control of the growth speed in the level sets for the “copies” m^L◦f_L⁻¹ of m. The combination of assumptions(3)and (4)supplies the monofractality property used in classical ubiquity results.

2. Ifµ is a strictly monofractal measure of exponentd(typically the Lebesgue measure), then(1-4)are always fulfilled withα=β =dandµ=mas soon as (10) holds. In fact, in this case,(1-4)imply the conditions required to be an ubiquitous system in the sense of [18, 19].

3. For some well-chosen measuresm, property(4)automatically holds for any system {(xn, λn)}^n∈^N and D = (1,∞). This is due to the fact that a stronger property holds: (4’)There exists Jm such that for everyj≥Jm, for everyc-adic boxL=I_j,^c_k, (15) holds. The first two classes described in Section 6.2 verify(4’) (see [12]).

The use of the weakened property(4)is needed for the last two examples devel- oped in Section 6.2 and for other measures constructed similarly (see [12]). Indeed, for these kinds of random measures, it was impossible for us to prove the stronger uniform property(4’), and we are only able to derive (see [12]) that, with probability 1,(4)holds with a dense countable setD.

4. Property (4) can be weakened without affecting the conclusions of Theo- rem 2.7 below as follows: weak (4) There exists a subset Dof (1,∞) such that for everyδ ∈ D, for m-almost everyx∈lim sup_n→∞B xn, λn/2

, there exists an increasing sequencejk(x) such that for everyk, there existsB(xn_k, λn_k)∈ B^jk(x)(x) as well as ac-adic boxLkincluded inB(xnk, λ^δn_k) such that (15) holds withL=Lk; moreover limk→∞log|L_k|

logλ_nk =δ. This weakening, necessary in [10], slightly compli- cates the proof and we decided to only discuss this point in this remark.

In order to treat the case of the limsup-sets (5) defined with a dilation parameter ρ <1, conditions(2)and(4)are modified as follows.

Definition 2.5. Let ρ < 1. The system {(xn, λn)}^n∈^N is said to form a ρ- heterogeneous ubiquitous system with respect to (µ, α, β) if the following conditions are fulfilled.

(1)and(3)are the same as in Definition 2.3.

(2(ρ))There exists a measuremwith a support equal to [0,1]^d such that:

• There exists a non-decreasing continuous functionχdefined onR₊ such that χ(0) = 0, r 7→ r^−χ(r) is non-increasing near 0⁺, lim_r→0⁺r^−χ(r) = +∞, and

∀ε, θ, γ >0,r7→rε−θϕ(r)−γχ(r)is non-decreasing near 0.

Moreover, form-almost every pointy, there exists an infinite number of integers {ji(y)}^i∈^N with the following property: the ball B(y, c^−ρjⁱ^(y)) contains at least

(10)

c^jⁱ(y)(d(1−ρ)−χ(c^−ji^(y)))pointsxnsuch that the associated couples (xn, λn) all satisfy λn∈[c^−jⁱ^(y)+1, c^−jⁱ^(y)(1−χ(c^−ji^(y)⁾⁾],

for everyn^′6=n, B(x^′_n, λ^′_n)T

B(xn, λn) =∅. (16)

•(11) and (13) in assumption(2)are also supposed here.

(4’)There existsJm such that for everyj≥Jm, for everyc-adic boxL=I_j,k^c , (15) holds. In particular,(4)holds withD= (1,+∞).

Remark 2.6. 1. Heuristically, condition (16) ensures that form-almost everyy, for infinite many numbersj, approximativelyc^jd(1−ρ)“disjoint” couples (xn, λn) such thatλn ∼c^−j can be found in the neighborhoodB(y, c^−ρj) ofy. This property is much stronger than (10).

2. Again, the uniform property (4’)(the same as in item 3. of Remark 2.4) could be weakened into: (4(ρ)) There exists a subset D of (1,∞) such that for every δ ∈ D, for m-almost every y, the sequence ji(y) of(2(ρ)) can be chosen so that for every B(xn, λn) invoked in (16), among the c-adic boxes of maximal diameterLincluded in B(xn, λ^δ_n), at least one satisfies (15).

Nevertheless, we kept (4’) because we do not know any example of system {xn, λn}^n∈^N and of measure m such that (2(ρ)) and the weak form of (4’) hold but such that (2(ρ)) and(4’)do not.

Before stating the results, a last property has to be introduced. Letρ <1. For every setI, for every constantM >1,PM^ρ (I) is said to hold if

(17) M⁻¹|I|α+ψ(|I|))+2αχ(|I|))

≤µ I

≤M|I|^α−ψ(|I|)−2αχ(|I|). The dependence inρofPM^ρ (I) is hidden in the functionχ(see (16)).

It is convenient for aρ-heterogeneous ubiquitous system{(xn, λn)} (ρ∈(0,1]) with respect to (µ, α, β) to introduce the sequences ε^ρ_M = (ε^ρ_M,n)n≥1 defined for a constantM ≥1 by ε^ρ_M,n= max(ε^ρ,−_M,n, ε^ρ,+_M,n), where

(18) λ^α±ε

ρ,±

n M,n =M^∓(2λn)^α±ψ(2λⁿ^)±2αχ(2λⁿ⁾(by conventionχ≡0 if ρ= 1).

2.4. Lower bounds for Hausdorff dimensions of conditioned limsup-sets.

The triplets (µ, α, β), together with the auxiliary measurem, have the properties required to study the exceptional sets we introduced before.

Letbδ= (δn)n≥1∈[1,∞)^N^∗,eε= (εn)n≥1∈(0,∞)^N^∗,ρ∈(0,1],M ≥1, and (19) Sbµ(ρ,bδ, α,eε) = \

N≥1

[

n≥N:Q(xn,λn,ρ,α,εn)holds

B(xn, λ^δ_nⁿ), whereQ(xn, λn, ρ, α, εn) holds whenλn^ρ(α+εⁿ⁾≤µ B(xn, λ^ρ_n)

≤λ^ρ(α−εn ⁿ⁾. So, ifbδ is a constant sequence equal to someδ≥1, the setSbµ(ρ,bδ, α,eε) coincides with the setSµ(ρ, δ, α,ε) defined in (4) and considered in Theorem 2.2.e

Theorem 2.7. Let µ be a finite positive Borel measure whose support is [0,1]^d, ρ∈(0,1]andα, β > 0. Let {xn}^n∈^N be a sequence in [0,1]^d and {λn}^n∈^N a non- increasing sequence of positive real numbers converging to 0.

Suppose that{(xn, λn)}^n∈^N forms a ρ-heterogeneous ubiquitous system with respect to (µ, α, β). Let Db be the set of points δ of R which are limits of a non- decreasing element of {1} ∪ D^N^∗

(in the case ofρ <1,D= (1,+∞)).

(11)

10 JULIEN BARRAL AND ST ´EPHANE SEURET

There exists a constantM ≥1 such that for every δ∈Db, one can find a non- decreasing sequence bδ converging to δ and a positive measure mρ,δ which satisfy mρ,δ

Sbµ(ρ,δ, α, εb ^ρ_M)

>0, and such that for everyx∈Sbµ(ρ,δ, α, εb ^ρ_M), (recall that χ≡0 if ρ= 1 and the definition ofε^ρ_M (18))

lim sup

r→0⁺

mρ,δ B(x, r) r^{D(β,ρ,δ)−ξ}^ρ,δ^(r) <∞, (20)

where





∀ρ∈(0,1], D(β, ρ, δ) = mind(1−ρ) +ρβ

δ , β

;

∀r >0, ξρ,δ(r) = (4 +d)ϕ(r) +χ(r).

bδcan be taken equal to the constant sequence(δ)n≥1 ifδ∈ {1} ∪ D.

For the two first classes of measures of Section 6.2 (Gibbs measures and products of multinomial measures), (4’) holds instead of (4) and D = (1,+∞), and thus Theorem 2.7 applies with anyρ∈(0,1]. To the contrary, as soon asρ <1, Theorem 2.7 does not apply to the last two classes of Section 6.2 (independent multiplicative cascades and compound Poisson cascades).

Corollary 2.8. Under the assumptions of Theorem 2.7, there existsM ≥1 such that for every δ∈Db, there exists a non-decreasing sequenceδbconverging toδsuch that H^ξ^ρ,δ(Sbµ(ρ,δ, α, εb ^ρ_M))>0. Moreover, bδ= (δ)n≥1 if δ∈ {1} ∪ D.

In particular,dimSbµ(ρ,bδ, α, ε^ρ_M)≥D(β, ρ, δ).

When ρ < 1, D(β, ρ, δ) remains constant and equal to β when δ ranges in [1,^{d(1−ρ)+ρβ}_β ]. This is what we call a saturation phenomenon. Then, as soon as

d(1−ρ)+ρβ

β < δ, we are back to a “normal” situation where D(β, ρ, δ) decreases as 1/δwhenδincreases.

Whenρ= 1,D(β, ρ, δ) =β/δ, thus there is no saturation phenomenon.

Corollary 2.9. Fix eε = (εn)n≥1 a sequence converging to 0 at ∞. Under the assumptions of Theorem 2.2 and Theorem 2.7, if the family {(xn, λn)}^n∈^N both forms a weakly redundant and a ρ-heterogeneous ubiquitous system with respect to (µ, α, τ_µ^∗(α)), then there exists a constant M ≥ 1 such that for every δ ∈ [^{d(1−ρ)+ρτ}_τ∗ ^µ^∗^(α)

µ(α) ,+∞)∩Db, there exists a non-decreasing sequence δb converging to δsuch that

dim Sbµ(ρ,δ, α, εb ^ρ_M)

= dim Sbµ(ρ,δ, α, εb ^ρ_M)\ [

δ^′>δ

Sµ(ρ, δ^′, α,eε)

= D(τ_µ^∗(α), ρ, δ).

Moreover,δbcan be taken equal to (δ)n≥1 if δ∈ {1} ∪ D.

Remark 2.10. 1. Corollary 2.8 is an immediate consequence of Theorem 2.7.

2. In order to prove Corollary 2.9, let us first observe that if δ > 1 and δbis a non-decreasing sequence converging to δ when n tend to ∞, Sbµ(ρ,bδ, α, ε^ρ_M) ⊂ Sµ(ρ, δ^′, α, ε^ρ_M) for all δ^′ < δ. Theorem 2.2 gives the optimal upper bound for dim Sbµ(ρ,bδ, α, ε^ρ_M)

. Again by Theorem 2.2, ifδ≥ ^{d(1−ρ)+ρτ}τ_µ^∗(α)^µ^∗^(α), forδ^′ > δ, the sets Sµ(ρ, δ^′, α, ε^ρ_M) form a non-increasing family of sets of Hausdorff dimension

< D(τ_µ^∗(α), ρ, δ). This implies H^ξ^ρ,δ S

δ^′>δSµ(ρ, δ^′, α,ε)e

= 0. Then the lower

(12)

bound for dim Sbµ(ρ,bδ, α, ε^ρ_M)\S

δ^′>δSµ(ρ, δ^′, α,ε)e

is given by Corollary 2.8. This holds for any sequenceεeconverging to zero.

Whenδ= 1, one necessarily hasρ= 1 andbδ= (1)n≥1. The arguments are then similar to those used forδ >1.

3. The previous statements are still valid if property(4’)is replaced by property (4(ρ)) of Remark 2.6, and in Section 6.2, the measures considered are such that eitherD= (1,∞) orDis dense in (1,∞).

3. Upper bound for the Hausdorff dimension of conditioned limsup-sets: Proof of Theorem 2.2

The sequence{(xn, λn)}ⁿ is fixed, and is supposed to form a weakly redundant system (Definition 2.1). We shall need the functions∀j≥1

τµ,ρ,j(q) =−j⁻¹log₂ X

n∈Tj

µ B(xn, λ^ρ_n)q

and τµ,ρ(q) = lim inf

j→∞ τµ,ρ,j(q), with the convention that the empty sum equals 0 and log(0) =−∞.

In the sequel, the Besicovitch’s covering theorem is used repeatedly

Theorem 3.1. (Theorem 2.7 of [36]) Let d be an integer greater than 1. There is a constant Q(d) depending only on d with the following properties. Let A be a bounded subset ofR^d andF a family of closed balls such that each point ofAis the center of some ball of F.

There are families F¹, ...,F^Q(d) ⊂ F covering A such that each Fⁱ is disjoint, i.e.

A⊂

Q(d)[

i=1

[

F∈Fi

F and ∀F, F^′ ∈ Fⁱ with F=6 F^′, F∩F^′=∅.

Let (Nj)j≥1be a sequence as in Definition 2.1, and let us consider for everyj≥1 the associated partition{Tj,1, . . . , Tj,Nj}ofTj. For every subsetS ofTj, for every 1≤i≤Nj, Theorem 3.1 can be used to extract from

B(xn, λ^ρ_n) : n∈Tj,i∩S Q(d) disjoint families of balls denoted byTj,i,k(S), 1≤k≤Q(d), such that

(21) [

n∈Tj,i∩S

B(xn, λ^ρ_n)⊂

Q(d)[

k=1

[

n∈Tj,i,k(S)

B(xn, λ^ρ_n).

Let us then introduce the functions b

τµ,ρ,j(q) = −j⁻¹ log₂ sup

S⊂Tj

X

n∈SNj i=1

SQ(d) k=1 Tj,i,k(S)

µ B(xn, λ^ρ_n)q

(j≥1)

and bτµ,ρ(q) = lim infj→∞bτµ,ρ,j(q). Recall thatτµ is defined in (7).

Lemma 3.2. Under the assumptions of Theorem 2.2, one has (22) τµ,ρ≥d(1−ρ) +ρτµ and bτµ,ρ≥ρτµ. Proof. •Let us show the first inequality of (22).

First suppose that q ≥ 0. Fix j ≥ 1 and 1 ≤ i ≤ Nj. For every n ∈ Tj,i, B(xn, λ^ρ_n)∩[0,1]^d is contained in the union of at most 3^d distinct dyadic boxes of

(13)

generationjρ:= [jρ]−1 denotedB1(n), . . . , B3^d(n). Hence µ B(xn, λ^ρ_n)q

≤X³^d

i=1

µ Bi(n)^q

≤3^dq

3^d

X

i=1

µ Bi(n)q

.

Moreover, since the ballsB(xn, λn) (n∈Tj,i) are pairwise disjoint and of diameter larger than 2^−(j+1), there exists a universal constantCd depending only ond such that each dyadic box of generationjρ meets less thanCd2^d(1−ρ)jof these balls B(xn, λ^ρ_n). Hence when summing over n ∈ Tj,i the masses µ B(xn, λ^ρ_n)q

, each dyadic box of generationjρ appears at mostCd2^d(1−ρ)j times. This implies that

X

n∈Tj,i

µ B(xn, λ^ρ_n)q

≤ 3^dqCd2^d(1−ρ)j X

k∈{0,...,2^jρ−1}^d

µ(Ij,k)^q (23)

and X

n∈Tj

µ B(xn, λ^ρ_n)q

≤ 3^dqCdNj2^d(1−ρ)j X

k∈{0,...,2^jρ−1}^d

µ(I_j,k)^q. (24)

Since logNj=o(j), one getsτµ,ρ(q)≥d(1−ρ) +ρτµ(q).

Now suppose thatq <0. Let us fixj ≥1 and 1≤i≤Nj. For every n∈Tj,i, B(xn, λ^ρ_n) contains a dyadic boxB(n) of generation [jρ] + 1, andµ B(xn, λ^ρ_n)q

≤ µ B(n)q

. The same arguments as above also yieldτµ,ρ(q)≥d(1−ρ) +ρτµ(q).

•We now prove the second inequality of (22).

Suppose that q ≥ 0. Fix j ≥1 and S a subset of Tj, as well as 1 ≤ i ≤ Nj

and 1 ≤ k ≤ Q(d). We use the decomposition (21). Since the balls B(xn, λ^ρ_n) (n ∈ Tj,i,k(S)) are pairwise disjoint and of diameter larger than 2^−(j+1)ρ, there exists a universal constantC_d^′, depending only ond, such that each dyadic box of generationjρ meets less thanC_d^′ of these balls. Consequently, the arguments used to get (23) yield here

X

n∈Tj,i,k(S)

µ B(xn, λ^ρ_n)q

≤ 3^dqC_d^′ X

k∈{0,...,2^jρ−1}^d

µ(Ij,k)^q

and X

n∈SNj i=1

SQ(d) k=1 Tj,i,k(S)

µ B(xn, λ^ρ_n)q

≤ 3^dqC_d^′Q(d)Nj

X

k∈{0,...,2^jρ−1}^d

µ(I_j,k)^q.

The right hand side in the previous inequality does not depend onS, hence sup

S⊂Tj

X

n∈SNj i=1

SQ(d) k=1 Tj,i,k(S)

µ B(xn, λ^ρ_n)q

≤3^dqC_d^′Q(d)Nj

X

k∈{0,...,2^jρ−1}^d

µ(I_j,k)^q,

and the conclusion follows. The caseq <0 is left to the reader.

Proof of Theorem 2.2. •First case: α≤τ_µ^′(0⁻). Henceτ_µ^∗(α) = infq≥0(αq−τµ(q)).

Let us first prove that dimSµ(ρ, δ, α)≤ ^{d(1−ρ)+ρτ}δ ^µ^∗^(α).

Fix η > 0 and N so that εn < η for n ≥ N. Let us introduce the set Sµ(N, η, ρ, δ, α) =S

n≥N:λ^ρ(α+η)_n ≤µ B(xn,λ_n^ρ)B(xn, λ^δ_n). This set is also written Sµ(N, η, ρ, δ, α) = [

j≥inf_n≥Nlog₂(λ⁻¹_n )

[

n∈Tj:λ^ρ(α+η)_n ≤µ B(xn,λ^ρ_n)B(x n, λ^δ_n).

(14)

Let us fix D≥0. Remark thatSµ(ρ, δ, α,ε)e ⊂Sµ(N, η, ρ, δ, α). We use this set as covering ofSµ(ρ, δ, α,eε) in order to estimate theD-dimensional Hausdorff measure ofSµ(ρ, δ, α,eε).

Fix q ≥ 0 such that τµ(q) > −∞. Let jq be an integer large enough so that j ≥ jq implies τµ,ρ,j(q)≥ τµ,ρ(q)−η. For jN = max jq,infn≥Nlog₂(λ⁻¹_n )

, one gets that for some constantCdepending onD, δ, α, η, ρandq,

H^ξ_2·2^D⁻jN δ Sµ(ρ, δ, α,ε)e

≤ X

j≥jN

X

n∈Tj:λ^ρ(α+η)n ≤µ B(xn,λ^ρn)

B(x_n, λ^δ_n)^D

≤ X

j≥jN

X

n∈Tj

|B(xn, λ^δ_n)^Dλ^{−qρ(α+η)}_n µ B(xn, λ^ρ_n)q

≤ X

j≥jN

(22^−jδ)^D2(j+1)qρ(α+η)2^−jτ^µ,ρ,j^(q)

≤ C X

j≥jN

2−j(Dδ−qρ(α+η)+τµ,ρ(q)−η). Therefore, ifD > ^{ρ(α+η)−τ}_δ^µ,ρ^(q)+η,H_2·2^ξ^D⁻jN δ Sµ(ρ, δ, α,ε)e

converges to 0 asN →

∞, and dimSµ(ρ, δ, α,eε)≤D. This yields dimSµ(ρ, δ, α,ε)e ≤ ^{qρ(α+η)−τ}δ^µ,ρ^(q)+η, which is less than d(1−ρ)+ρ(αq−τµ(q))+(qρ+1)η

δ by Lemma 3.2. This holds for every η > 0 and for every q ≥ 0 such that τµ(q) > −∞. Finally, dim Sµ(ρ, δ, α,ε)e ≤

d(1−ρ)+ρinfq≥0αq−τµ(q)

δ =^{d(1−ρ)+ρτ}_δ ^µ^∗^(α).

Let us now show that dimSµ(ρ, δ, α,eε)≤τ_µ^∗(α). This time, forj≥1 we define Sj ={n∈Tj: λ^ρ(α+η)n ≤µ B(xn, λ^ρ_n)

}. By (21), we remark that Sµ(ρ, δ, α,ε)e ⊂ [

j≥jN

Nj

[

i=1 Q(d)[

k=1

[

n∈Tj,i,k(Sj)

B(xn, λ^ρ_n).

By definition ofbτµ,ρ(q), a computation mimicking the previous one yields H^ξ_2·2^D⁻ρjN Sµ(ρ, δ, α,ε)e

≤ C X

j≥jN

2−j(Dρ−qρ(α+η)+bτµ,ρ(q)−η).

Hence dimSµ(ρ, δ, α,ε)e ≤ ^{qρ(α+η)−}ρ^τ^b^µ,ρ^(q)+η, for every η >0 and everyq≥0 such thatτµ(q)>−∞. The conclusion follows from Lemma 3.2.

Finally, if τ_µ^∗(α) < 0 and Sµ(ρ, δ, α,eε) 6= ∅, the previous estimates show that H_2·2^ξ^D−ρjN(Sµ(ρ, δ, α,ε)) is bounded fore D∈(τ_µ^∗(α),0) (one can formally extend the definition ofH^ξ^D to the caseD <0). This is a contradiction.

•The proof whenα≥τ_µ^′(0⁻) follows similar lines.

4. Conditioned ubiquity. Proof of Theorem 2.7 (case ρ= 1) We assume that a 1-heterogeneous ubiquitous system is fixed. With each couple (xn, λn) is associated the ball In = B(xn, λn). For everyδ ≥1, In^(δ) denotes the contracted ballB(xn, λ^δ_n). The following property is useful in the sequel. Because of the assumption(1)onϕandψ, one has

(25) ∃C >1, ∀0< r≤s≤1, s^−ϕ(s)≤Cr^−ϕ(r) ands^−ψ(s)≤Cr^−ψ(r). We begin with a simple technical lemma

(15)

Lemma 4.1. Let y ∈[0,1]^d, and let us assume that there exists an integer j(y) such that for some integerc≥2, (11) and (13) hold for y and everyj≥j(y).

There exists a constant M independent of y with the following property: for every n such that y ∈ B(xn, λn/2) andlog_cλ⁻¹_n ≥j(y) + 4 ,D^mM(B(y,2λn))and PM¹ (B(xn, λn))hold.

Proof. Let us assume that y ∈ B(xn, λn/2) with λn ≤ c^−j(y)−4. Let j0 be the smallest integer j such that c^−j ≤ λn/2, and j1 the largest integer j such that c^−j≥2λn. One hasj0≥ −log_cλn≥j1≥j(y). One thus ensured by construction thatj0−4≤ −log_cλn≤j1+ 4.

Let us recall that Ij(y) is the unique c-adic box of scale j which contains y, and that kj,y is the unique k ∈ N^d such that y ∈ I_j,k^c = Ij(y). One has I_j^c₀(y)⊂B(xn, λn)⊂S

kk−k^c_j₁_,yk∞≤1I_j^c₁_,k, which yieldsµ(I_j^c₀(y))≤µ(B(xn, λn))≤ X

kk−k^cj1,yk_∞≤1

µ(I_j^c₁_,k). Applying (11) and (12) yields

|c^−j⁰|^α+ψ(|c^−j⁰^|)≤µ(B(xn, λn))≤3^d|c^−j¹|^α−ψ(|c^−j¹^|).

Combining the fact that j0−4≤ −log_cλn≤j1+ 4 with (25) and (18) gives λ^α+ε

1,+

n M,n=M⁻¹|2λn|^α+ψ(2λⁿ⁾≤µ(B(xn, λn))≤M|2λn|^α−ψ(2λⁿ⁾=λ^α−ε

1,−

n M,n

for some constantM that does not depend ony.

Similarly, one gets from (13) and (14) thatD^mM B(y,2λn)

holds for some con-

stantM >0 that does not depend ony.

of Theorem 2.7 in the caseρ= 1. All along the proof,Cdenotes a constant which depends only onc,α,β,δ,ϕandψ.

The caseδ= 1 follows immediately from the assumptions (herem1=m).

Now letM ≥1 be the constant given by Lemma 4.1. Letδ∈D ∩b (1,+∞), and let{dn}^n≥1 be a non-decreasing sequence in D converging toδ (if δ ∈ D, dn =δ for everyn). For everyk≥1,j≥1 andy∈[0,1]^d, let

(26) n^(d_j,y^k⁾= inf (

n:λn ≤c^−j, ∃j^′≥j:

(B(xn, λn)∈ B^j^′(y)

∃ L∈B^d_n^k, (15) holds )

.

We shall find a sequence bδ = (δj)j≥1, converging toδ, to construct a generalized Cantor set Kδ in Sbµ(1,δ, α, εb ¹_M) and simultaneously the measure mδ onKδ. The successive generations of c-adic boxes involved in the construction of Kδ, namely Gn, are obtained by induction.

- First step: The first generation of boxes defining Kδ is taken as follows.

LetL0 = [0,1]^d. Consider the first elementd1 ofD of the sequence converging toδ. We first impose that δj :=d1, for everyj≥1.

Due to assumptions(2),(3)and(4), there existE^L⁰ ⊂E_n^L_L⁰

0 such thatm(E^L⁰)≥ kmk/4 and an integern^′_L₀≥nL0 such that for ally∈E^L⁰:

-y∈T

N≥1

S

n≥NB(xn, λn/2),

- for everyj≥n^′_L₀, both (11) and (13) hold,

- there are infinitely many integersj such that (15) holds for someL∈ Bj^d¹(y).

Heterogeneous ubiquitous systems in $\mathbb{R}^{d}$ and Hausdorff dimension

HAL Id: hal-00004525

https://hal.archives-ouvertes.fr/hal-00004525

Heterogeneous ubiquitous systems in R d and Hausdorff dimension

Julien Barral, Stephane Seuret

To cite this version:

ccsd-00004525, version 1 - 21 Mar 2005

Heterogeneous ubiquitous systems in R ^d and Hausdorff dimension