The Multifractal Formalism for Measures,Review and Extension to Mixed Cases

MohamedMenceur,AnouarBenMabroukandKamelBetina

1Algerba and Number Theory Laboratory,Faculty of Mathematics,University of Sciences and Technology Houari Boumediene,BP 32 EL Alia 16111 Bab Ezzouar Algiers Algeria.

2Computational Mathematics Laboratory,Department of Mathematics,Faculty of Sciences,5019 Monastir,Tunisia.

3Department of Mathematics,HIgher Institute of Applied Mathematics and Informatics,Street of Assad Ibn Alfourat,Kairouan University,3100 Kairouan Tunisia.

The Multifractal Formalism for Measures,Review and Extension to Mixed Cases

MohamedMenceur1,AnouarBenMabrouk2,3,?andKamelBetina1

1Algerba and Number Theory Laboratory,Faculty of Mathematics,University of Sciences and Technology Houari Boumediene,BP 32 EL Alia 16111 Bab Ezzouar Algiers Algeria.

2Computational Mathematics Laboratory,Department of Mathematics,Faculty of Sciences,5019 Monastir,Tunisia.

3Department of Mathematics,HIgher Institute of Applied Mathematics and Informatics,Street of Assad Ibn Alfourat,Kairouan University,3100 Kairouan Tunisia.

.The multifractal formalism for single measure is reviewed.Next,a mixed generalized multifractal formalism is introduced which extends the multifractal formalism of a single measure based on generalizations of the Hausdorff and packing measures to a vector of simultaneously many measures.Borel-Cantelli and Large deviations Theorems are extended to higher orders and thus applied for the validity of the new variant of the multifractal formalism for some special cases of multi-doubling type measures.

Hausdorff measures,packing measures,Hausdorff dimension,packing dimension, renyi dimension,multifractal formalism,vector valued measures,mixed cases,Holderian measures,doubling measures,Borel-Cantelli,large deviations.

AMS SubjectClassif i cations:28A78,28A80

1 Introduction

In the present work,we are concerned with the whole topic of multifractal analysis of measures and the validity of multifractal formalisms.We aim to consider some cases of simultaneous behaviors of measures instead of a single measure as in the classic or original multifractal analysis of measures.We call such a study mixed multifractal analysis.Such a mixed analysis has been generating a great attention recently and thusproved to be powerful in describing the local behavior of measures especially fractal ones(see[1,2,9–14]).

In this paper,multi purposes will be done.Firstly we review the classical multifractal analysis of measures and recall all basics about fractal measures as well as fractal dimensions.We review Hausdorff measures,Packing measures,Hausdorff dimensions,Packing dimensions as well as Renyi dimensions and we recall the eventual relations linking these notions.A second aim is to develop a type of multifractal analysis,multifractal spectra,multifractal formalism which permit to study simultaneously a higher number of measures.As it is noticed from the literature on multifractal analysis of measures, this latter always considered a single measure and studies its scaling behavior as well as the multifractal formalism associated.Recently,many works have been focused on the study of simultaneous behaviors of f i nitely many measures.In[9],a mixed multifractal analysis is developed dealing with a generalization of R ′enyi dimensions for f i nitely many self similar measures.This was one of the motivations leading to our present paper.Secondly,we intend to combine the generalized Hausdorff and packing measures and dimensions recalled after with Olsen’s results in[14]to def i ne and develop a more general multifractal analysis for f i nitely many measures by studying their simultaneous regularity,spectrum and to def i ne a mixed multifractal formalism which may describe better the geometry of the singularities’s sets of these measures.We apply the techniques of L.Olsen especially in[9]and[14]with the necessary modif i cations to give a detailed study of computing general mixed multifractal dimensions of simultaneously many f inite number of measures and try to project our results for the case of a single measure to show the generecity of our’s.

The f i rst point to check in multifractal analysis of a measure is its singularity on its spectrum.Given a measure μ eventually Borel and f i nite,for x ∈ supp(μ),the singularity of μ is estimated via μ(B(x,r))as r → 0.If μ(B(x,r)) ～ rα,the measure μ is said to be α-H ¨older at x.The local lower dimension and the local upper dimension of μ at the point x are respectively def i ned by

Whenthesequantitiesare equal wecall theircommonvalue thelocal dimension,denoted by αμ(x)of μ at x.Next,the α-singularity set is X(α)={x ∈ supp(μ); αμ(x)= α}and fi nally,the spectrum of singularities is the mapping de fi ned by d(α)=dimX(α)where dim stands for the Hausdorff dimension.

The computation of such a spectrum is the delicate point and the most principal aim in the whole multifractal study of the measure.Its computation needs more efforts and special techniques based on the characteristics of the measure,such that self similarity, scalings.In multifractal analysis,it is related to multifractal dimensions and in some cases it is computed by means of the Legendre transform of such dimensions.This fact constitutes the so-called multifractal formalism for measures.

The present work will be organized as follows.The next section concerns a review of Hausdorff and packing measures and dimensions.Section 3 is concerned to Multifractal generalizations of Hausdorff and packing measures as well as the associated dimensions. In Section 4,the mixed multifractal generalizations of Hausdorff and packing measures and dimensions are introduced.Section 5 is devoted to the mixed multifractal generalization of Bouligand-Minkowsky or R ′enyi dimension inspired from Olsen in[14].In Section 6,a mixed multifractal formalism associated to the mixed multifractal generalizations of Hausdorff and packing measures and dimensions is proved in some case based on a generalization of the well known large deviation formalism.

2 Hausdorff and packing measures and dimensions

Given a subset E ? R,and ∈ ＞ 0,we call an ∈-covering of E,any countable set(Ui)iof non-empty subsets Ui? R satisfying

where for any subset U ? R,|U|=diam(U)is the diameter def i ned by

Remark here that for ∈1＜ ∈2,any ∈1-covering of E is obviously an ∈2-covering of E.This implies that the quantity

is a non increasing function in ∈.Its limit

def i nes the so-called s-dimensional Hausdorff measure of E.It holds that for any set E ? R there exists a critical value sEin the sense that

or otherwise,

Such a value is called the Hausdorff dimension of the set E and is usually denoted by dimHE or simply dimE.When Ui=B(xi,ri)is a ball centered at xi∈ E and with diameter ri＜∈,thecovering(B(xi,ri))iis called an ∈-centered covering of E.However,surprisingly,the quantity Hsrestricted only on centered coverings does not def i ne a measure.To obtain a good measure with centered coverings one should do more.Denote

and similarly as above,

As stated previously,this is not a good measure.So,to obtain a good candidate,we set for E ? R,

It is called the centered Hausdorff s-dimensional measure of E.But,although a fascinating relation to the Hausdorff measure exists.It holds that

Indeed,let F ? E be subsets of Rd.It follows from the def i nition of Hsandthat Hs(F) ≤(F).Next,from the fact that Hsis an outer metric measure on Rd,and the def i nition of Cs,il results that Hs(E)≤ Cs(E).Next,let{Uj}jbe an ∈-covering of F and rj=diam(Uj).For each i f i xed,consider a point xi∈ Ui∩ F.This results in a centered∈-covering{B(xi,ri)}iof F.Consequently,

Hence,

Next,as ∈↓ 0,we obtain

which guaranties that

It holds that these measures give rise to some critical values in the sense that,for any set E ? R there exists a critical value hEand cEfor which

and similarly

But using Eq.(2.2)above,it proved that hE=cEand otherwise,

Such a value is called the Hausdorff dimension of the set E and is usually denoted by dimHE or simply dimE.

Similarly,we call a centered ∈-packing of E ? Rd,any countable set(B(xi,ri))iof disjoint balls centered at points xi∈ E and with diameters ri＜ ∈.The packing measure and dimension are def i ned as follows

It holds as for the Hausdorff measure that there exists critical values ?Eand pEsatisfying respectively

and respectively

The critical value ?(E)is called the logarithmic index of E and pEis called the packing dimension of E denote by DimP(E)or simply Dim(E).These quantities may be shown as

and respectively

Usually,we have the inequality

Def i nition 2.1.A set E ?Rdis said to be fractal in the senseof Tayloriff dim(E)=Dim(E).

3 Multifractal generalizations of Hausdorff and packing measures

Letμ beaBorelprobability measureonRd,anonemptyset E?Rdand ∈＞0.Letalso q,t be real numbers.We will recall hereafterthe stepsleading to the multifractal generalizations of the Hausdorff and packing measures due to L.Olsen in[9].Denote

where the inf is taken over the set of all centered ∈-coverings of E,and for the empty set,=0.As for the preceding cases of Hausdorff and packing measures,it consists of a non increasing quantity as a function of ε.We then consider its limit

and f i nally,the multifractal generalization of the s-dimensional Huasdorrf measure

Similarly,we def i ne the multifractal generalization of the packing measure as follows

where the sup is taken over the set of all centered ∈-packings of E.For the empty set,we set as usual=0.Next,

and f i nally,

In[9],it has been proved that the measuresand the pre-measureassign in a usual way a dimension to every set E ? Rdas resumed in the following proposition.

Proposition 3.1(see[9]).Given a subset E ? Rd,

1.There exists a unique number(E)∈ [? ∞ ,+ ∞ ]such that

2.There exists a unique number(E)∈ [? ∞ ,+ ∞ ]such that

3.There exists a unique number(E) ∈ [?∞,+∞]such that

The characteristics of these functions have been studied completely by L.Olsen.He proved among author results thatandare monotones and σ-stables.Furthermore,if E=supp(μ)is the support of the measure μ,one obtains

4 Mixed multifractal generalizations of Hausdorff and packing measures and dimensions

The purpose of this section is to present our ideas about mixed multifractal generalizations of Hausdorff and packing measures and dimensions.Let μ =(μ1,μ2,···,μk)a vector valued measure composed of probability measures on Rd.We aim to study the simultaneous scaling behavior of μ,which we denote

Let E ? Rdbe a nonempty set and ∈＞ 0.Let also q=(q1,q2,···,qk) ∈ Rkand t∈ R.The mixed generalized multifractal Hausdorff measure is def i ned as follows.Denote

and the product

Denote next,

where the inf is taken over the set of all centered ∈-coverings of E,and for the empty set,=0.As for the single case,of Hausdorff measure,it consists of a non increasing function of the variable ε.So that,its limit as ∈↓ 0 exists.Let

Let f i nally

Lemma 4.1.is an outer metric measure on Rd.

The proof of this lemma is technic and follows carefully analogous steps as the single case.

Def i nition 4.1.The restriction ofon Borel sets is called the mixed generalized Hausdorff measure on Rd.

Now,we def i ne the mixed generalized multifractal packing measure.We use already the same notations as previously.Let

where the sup is taken over the set of all centered ∈-packings of E.For the empty set,we set as usual=0.Next,we consider the limit as ∈ ↓ 0,

and f i nally,

Lemma 4.2.is an outer metric measure on Rd.

Theproofofthislemma ismore specif i cthanLemma4.1and usesthefollowing result.

Indeed,let

and(B(xi,ri))ibe a centered ∈-packing of the union A∪B.It can be divided into two parts I and J,

where

Therefore,(B(xi,ri))i∈Iis a centered ∈-packing of A and(B(xi,ri))i∈Jis a centered ∈-packing of the union B.Hence,

Consequently,

and thus the limit for ∈ ↓ 0 gives

The converse is more easier and it states thatand nextare sub-additive.Letbe a centered ∈-packing of A andbe a centered ∈-packing of B.The union???is a centered ∈-packing of A ∪ B.So that

Taking the sup on(B(xi,ri))ias a centered ∈-packing of A and next the sup on(B(yi,ri))ias a centered ∈-packing of B,we obtain

and thus the limit for ∈ ↓ 0 gives

Def i nition4.2.TherestrictionofonBorelsetsis called themixed generalizedpacking measure on Rd.

It holds as for the case of the multifractal analysis of a single measure that the measures Hq,tμ,Pq,tμand the pre-measureassign a dimension to every set E ? Rd.

Proposition 4.1.Given a subset E ? Rd,

1.There exists a unique number(E) ∈ [? ∞ ,+ ∞ ]such that

2.There exists a unique number Dimqμ(E)∈ [? ∞ ,+ ∞ ]such that

3.There exists a unique number ?qμ(E) ∈ [?∞,+∞]such that

Def i nition 4.3.The quantities(E),anddef i ne the so-called mixed multifractal generalizations of the Hausdorff dimension,the packing dimension and the logarithmic index of the set E.

Remark that if we denote Qi=(0,0,···,qi,0,···,0)the vector with zero coordinates except the ith one which equals qi,we obtain the multifractal generalizations of the Hausdorff dimension,the packing dimension and the logarithmic index of the set E for the single measure μi,

Similarly,for the null vector of Rk,we obtain

Proof of Proposition 4.1.We will sketch only the proof of the f i rst point.The rest is analogous.

First,we claim that ?t∈ R such that(E)＜ ∞ it holds that(E)=0 for anyIndeed,let ∈＞ 0,F ? E and(B(xi,ri))ibe a centered ∈-covering of F.We have

Consequently,

Hence,

One can proceed otherwise by claiming that ?t∈ R such that(E) ＞ 0 it holds that(E)=+∞ for anyIndeed,proceeding as previously,we obtain for ∈＞ 0,

Hence,

Next,we aim to study the characteristics of the mixed multifractal generalizations of dimensions.To do this we will adapt the following notations.For q=(q1,···,qk)∈ Rk,

When E=supp(μ)is the support of the measure μ,we will omit the indexation with E and denote simply

Thus,we complete the proof.

The following propositions resume the characteristics of these functions and extends the results of L.Olsen[9]for our case.

Proposition 4.2.(a)andare non decreasing with respect to the inclusion property in Rd.

Proof.(a)Let E ? F be subsets of Rd.We have

(b)Let(An)nbe a countable set of subsets An? Rdand denote A=It holds from the monotony ofthat

Hence,

Consequently,from the sub-additivity property ofit holds that

Which means that

Hence,

Similar arguments permit to prove the properties of Bμ,A(q).

Next,we continue to study the characteristics of the mixed generalized multifractal dimensions.The following result is obtained.

Proposition 4.3.(a)The functionsBμ(q)and qΛμ(q)are convex.

(b)For i=1,2,···,k,the functionsbμ(q),Bμ(q)andΛμ(q),(= (q1,···,qi?1,qi+1,···,qk)f i xed),are non increasing.

Proof.(a)We start by proving that Λμ,Eis convex.Let p,q ∈ Rk,α ∈ [0,1],s ＞ Λμ,E(p)and t＞ Λμ,E(q).Consider next a centered ∈-packing(Bi=B(xi,ri))iof E.Applying H ¨older’s inequality,it holds that

Hence,

The limit on ∈↓ 0 gives

Consequently,

It results that

We now prove the convexity of Bμ,E.We set in this case t=Bμ,E(q)and s=Bμ,E(p).We have

Therefore,there exists(Hi)iand(Ki)icoverings of the set E for which

Consequently,

Hence,

Hence,

When ∈ ↓0,we obtain

Therefore,

This induces the fact that

Consequently

Hence,

The remaining part to prove the monotony Λμ,Eand Bμ,Eis analogous.

Proposition 4.4.(a)0 ≤ bμ(q)≤ Bμ(q)≤ Λμ(q),whenever qi＜ 1 for all i=1,2,···,k.

(b)bμ(?i)=Bμ(?i)= Λμ(?i)=0,where ?i=(0,0,···,1,0,···,0).

(c)bμ(q)≤ Bμ(q)≤ Λμ(q)≤ 0 whenever qi＞ 1 for all i=1,2,···,k.

The proof of this results reposes on the following intermediate ones.

Lemma 4.3.There exists a constant ξ∈ [0,+∞]satisfying for any E ? Rd,

More precisely,ξ is the number related to the Besicovitch covering theorem.

Theorem 4.1(Besicovitch Covering Theorem).There exists a constant ξ∈ N satisfying:For any E ∈ Rdand(rx)x∈Ea bounded set of positive real numbers,there exists ξ sets B1,B2,···,Bξ, that are f i nite or countable composed of balls B(x,rx),x ∈ E such that

? each Biis composed of disjoint balls.

Proof of Lemma 4.3.It suf fi ces to prove the fi rst inequality.Th?e seco?nd is always true for all ξ＞ 0.Let F ? Rd,∈ ＞ 0 and V={B(x,∈/2);x ∈ F}.Let next(Bij)j1≤i≤ξbe the ξ sets of V obtained by the Besicovitch covering theorem.So that,(Bij)i,jis a centered ∈-covering of the set F and for each i,(Bij)jis a centered ∈-packing of F.Therefore,

Hence,

So as Lemma 4.3.

Proof of Proposition 4.4.It follows from Propositions 4.2,4.3 and Lemma 4.3.

5 Mixed multifractal generalization of Bouligand-Minkowski’s dimension

In this section,we propose to develop mixed multifractal generalization of Bouligand-Minkowski’s dimension.Such a dimension is sometimes called the box-dimension or the Renyi dimension.Some mixed generalizations are already introduced in[15].We will see hereafter that the mixed generalizations to be provided resemble to those in[15].We will prove that in the mixed case,these dimensions remain strongly related to the mixed multifractal generalizations of the Hausdorff and packing dimensions.In the case of a single measure μ,the Bouligand-Minkowski dimensions are introduced as follows.For E ? supp(μ),δ＞ 0 and q ∈ R,let

where the inf is over the set of all centered δ-coveringsof the set E.The Bouligand-Minkowski dimensions are

for the upper one and

for the lower.In the case of equality,the common value is denoted(E)and is called the Bouligand-Minkowski dimension of the set E.We can equivalently def i ne these dimensions via the δ-packings as follows.For δ＞ 0 and q ∈ R,we set

where the sup is taken over all the centered δ-packingsof the set E.The upper dimension is

and the lower is

and similarly,when these are equal,the common value will be denotedand it def i nes the dimension of E.We now introduce the mixed multifractal generalization of the Bouligand-Minkowski dimensions.As we have noticed,our idea here is quite thesame as the one in[15].Let μ =(μ1,μ2,···,μk)be a vector valued measure composed of probability measures on Rd.Denote as previously

and for q=(q1,q2,···,qk)∈ Rk,

Next,for a nonempty subset E ? Rdand δ＞ 0,we will use the same notations forandbut without forgetting that we use the new product for the measure μ. Similarly for(E),(E)and(E).

Def i nition 5.1.For E ? supp(μ)and q=(q1,q2,···,qk)∈ Rk,we will call

Remark 5.1.We stress the fact that each quantity def i nes in fact a mixed generalization that can be different from the other.That is,we did not mean thatandare the same(equal)and similarly for the lower ones.We will prove in the contrary that as for the single case,they can be different.

Theorem 5.1.For

1).For all q ∈ Rk,we have

Proof.1).Using Besicovitch covering theorem we get

with some constant C f i xed.So as 1)is proved.

2).We f i rstly prove that

Which means that

Using the assertion 1),we obtain the equalities

This means that for each n ∈ N,there exists a centered δn-coveringof E such that fore,

There balls may be considered to be intersecting the set F.Next,for each i,choose an element yi∈ B(xni,δn)∩ F.This results on a centered 2δn-covering??of F.There-

Hence,

So that,

Consequently,

The remaining part can be proved by following similar techniques.

Next we need to introduce the following quantities which will be useful later.Let μ =(μ1,μ2,···,μk)be a vector valued measure composed of probability measures on Rd. For j=1,2,···,k,a ＞ 1 and E ? supp(μ),denote

Theorem 5.2.For

1).For μ ∈ P0(Rd)and q∈,there holds that

2).For μ ∈ P1(Rd)and q ∈there holds that

Proof.1).The vector valued measure μ ∈ P0(Rd)yields that

where

Therefore,there exists a centered δn-covering(B(xni,δn))iof F satisfying

Let next yni∈ B(xni,δn).Then,(B(xni,2δn))iis a centered 2δn-covering of F.Hence,

where|q|=q1+q2+···+qk.Thus,

Which means that

Consequently,

Using the σ-stability of bμ,·(q)(see Proposition 4.2),it results that

Assertion 2 is left to the reader.

Wenowre-introducethemixedmultifractal generalizationofthe Lq-dimensionscalled also Renyi dimensions based on integral representations.See[15]for more details and other results.For q ∈,μ =(μ1,μ2,···,μk)and δ＞ 0,we set

where,in this case,

and

The mixed multifractal generalizations of the Renyi dimensions are

Proposition 5.1.The following results hold:

Proof.We only prove a).The remaining proofs of points b),c)and d)follow the same ideas.

For δ＞ 0,let(B(xi,δ))ibe a centered δ-covering of supp(μ)and let next(B(xij,δ))j, 1 ≤ i≤ ξ the ξ sets def i ned in Besicovitch covering theorem.It holds that

As a results,

Which implies that

Thus,we complete the proof.

6 A mixed multifractal formalism for vector valued measures

Let μ =(μ1,μ2,···,μk)be a vector valued probability measure on Rd.For x ∈ Rdand j=1,2,···,k,we denote

respectively the local lower dimension and the local upper dimension of μjat the point x and as usually the local dimension αμj(x)of μjat x will be the common value when these are equal.Next for α =(α1,α2,···,αk) ∈let

and

The mixed multifractal spectrum of the vector valued measure μ is def i ned by

where dim stands for the Hausdorff dimension.

In this section,we propose to compute such a spectrum for some cases of measures that resemble to the situation raised by Olsen in[9]but in the mixed case.This will permittodescribebetterthesimultaneousbehavioroff i nitelymany measures.Weintend preciselyto computethemixedspectrumbased onthemixed multifractal generalizations of the Haudorff and packing dimensions bμ,Bμand Λμ.We start with the following technic results.

Lemma 6.1.For

Proof.1).For i),We prove the f i rst part.For m ∈ N?,consider the set

Let next 0 ＜ η ＜ 1/m and(B(xi,ri))ia centered η -covering ofIt holds that

Consequently,

Hence,?η ＞ 0,there holds that

Which means that

Consequently,?η ＞ 0,

Let next,(Ei)ibe a covering ofThus,

Hence,?m,

Consequently,

2).i).and ii).follow similar arguments and techniques as previously.

Proposition 6.1.Let α ∈and q ∈ Rk.The following assertions hold:

a).Whenever 〈α,q〉+bμ(q)≥ 0,we have

b).Whenever 〈α,q〉+Bμ(q)≥ 0,we have

Proof.a).i).It follows from Lemma 6.1,assertion 1)i),

Consequently,

Hence,

a).ii).It follows from Lemma 6.1,assertion 2)i),as previously,that

Hence,

and f i nally,

b).i).Observing Lemma 6.1,assertion 1)ii),we obtain

Consequently,

Hence,

b).ii).observing Lemma 6.1,assertion 2)ii),we obtain

Hence,

and f i nally,

Thus,we complete the proof.

Lemma 6.2.?q ∈ Rksuch that

we have

Proof.It is based on

Hence,

Choosing t= 〈(εI?α),q〉,this induces that({x})＞ 2tand consequently,

Letting ε↓ 0,it results that bμ(q) ≥ ?〈α,q〉which is impossible.So as the f i rst part of 1.

The remaining part as well as 2 can be checked by similar techniques.

Theorem 6.1.Let μ =(μ1,μ2,···,μk)be a vector-valued Borel probability measure on Rdand q ∈ Rkf i xed.Let further tq∈ R,rq＞ 0,＞ 0, νqa Borel probability measure supported by supp(μ), ?q:R+→ R be such that ?q(r)=o(logr),as r → 0.Let f i nally(rq,n)n? [0,1]↓ 0 and satisfying

Assume next the following assumptions:

A1). ?x ∈supp(μ)andr∈ [0,rq],

A2).Cq(p)=exists and fi nite for all p ∈ R ,where

Then,the following assertions hold.

i).

ii).Whenever Cqis differentiable at 0,we have

Theorem 6.2.Assume that the hypotheses of Theorem 6.1 are satisf i ed for all q ∈ Rk.Then,the following assertions hold:

i). αμ= ? Bμ,νq,a.s.,whenever Bμis differentiable at q.

ii).Dom(B)? αμ(supp(μ))and fμ=on Dom(B).

The proof of this result is based on the application of a large deviation formalism. This will permit to obtain a measure ν supported byTo do this,we re-formulate a mixed large deviation formalism to be adapted to the mixed multifractal formalism raised in our work.

Theorem 6.3(The Mixed Large Deviation Formalism).Consider a sequence of vector-valued random variables(Wn=(Wn,1,Wn,2,···,Wn,k))non a probability space(?,A,P)and(an)n?[0,+∞]with limn→+∞an=+∞.Let next the function

Assume that

A1).Cn(t)is f i nite for all n and t.

A2).C(t)=limn→+∞Cn(t)exists and is f i nite for all t.

There holds that

i).The function C is convex.

ii).If ??C(t)≤ ?+C(t)＜ α,for some t∈ Rk,then

iii).If∑ne?εan＜ ∞ for all ε＞ 0,then

iv).If α ＜ ??C(t)≤ ?+C(t),for some t∈ Rk,then

v).If∑ne?εanis f i nite for all ε＞ 0,then

Proof.i).It follows from Holder’s inequality.

ii).Let h ∈ R?,k+be such that C(t)+ 〈α,h〉? C(t+h)＞ 0.We have

Next,by taking the limsup as n ?→ +∞,the result follows immediately. iii).Denote for n,m ∈ N,

By choosing t=0 and α=?+C(0)+1/m in item ii),and observing that C(0)=0,we obtain

which means that limsupnlogP(Tn,m) ＜ 0.Consequently,for some ε＞ 0 and n large enough,there holds that limsupnlogP(Tn,m)＜ ?ε.Thus,P(Tn,m)＜ e?εanwhich implies the convergence of the series ∑nP(Tn,m).Hence,using Borel-Cantelli theorem,we obtain

for all m.Therefore,

and f i nally,

Thus,we complete the proof.

Proof of Theorem 6.1.For simplicity we denote t=tq,K=Kq, ? = ?q,ν = νqand rn=rq,n.Next,for x ∈ supp(μ),let

i).Using the hypothesis A1).and Lemma 4.3 we obtain bμ(q)=Bμ(q)= Λμ(q)=t.Next, it is straightforward that the set

Finally,applying the famous Billingsley’s Theorem[7],we obtain

ii).Remark that if C is differentiable at 0,item i).states that

In the other hand,since the set M is not empty,Lemma 6.2 implies that ??C(0)q+t≥ 0. Hence,Proposition 6.1 yields that dimM ≤ ??C(0)q+t for any q ∈ Rk.Thus,taking the inf on q,we obtain

iii).We f i rstly claim that,there exists β ＞ 0 such that,for all x ∈ supp(μ)and 0 ＜ r? 1,we have

So let(B(xij,rn))1≤iξ,jthe ξ sets relatively to Besicovitch theorem extracted from the set (B(xi,rn))i.A careful computation yields that

where

Theorem 5.1 and Proposition 5.1 guarantee that

Consequently,Cq(p)= Λμ(p+q)? Λμ(p).So,if Λμis differentiable at q,Cqwill be too at 0 and ?Cq(0)= ?Λμ(q).Thus,using the mixed large deviation formalism,we obtain

hence,f i nally,αμ(x)= ??Λμ(q).

iv).Let q be such that ?Λμ(q)exists.Then ?Cq(0)exists too.So,item ii).states that

Which completes the proof.

Proof of Theorem 6.2.i).Using the same notations as previously,we obtain Cqdifferentiable at 0,Bμdifferentiable at q,and ?Cq(0)= ?Bμ(q).In the other hand,we obtain also

ii).Follows immediately from i).and Theorem 6.1.

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Received 2 December 2013;Accepted(in revised version)15 September 2015

?Corresponding author.Email addresses:m.m m@live.fr(M.Menceur),anouar.benmabrouk@issatso.rnu. tn(A.Mabrouk),kamelbetina@gmail.com(K.Betina)