On the projections of generalized upper Lq-spectrum

Chaos, Solitons and Fractals 42 (2009) 1451–1462

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On the projections of generalized upper Lq-spectrum Imen Bhouri Faculté des Sciences de Monastir, 5019 Monastir, Tunisia

a r t i c l e

i n f o

a b s t r a c t In this paper we intend to generalize the Lq-spectrum relatively to two measures and to study its behavior under orthogonal projections. A uniform result is obtained in the case of ‘‘Frostman like measures”. Ó 2009 Elsevier Ltd. All rights reserved.

Article history: Accepted 11 March 2009

Communicated by: Prof. G. Iovane

1. Introduction The notion of singularity exponents or spectrum and generalized dimensions are the major components of the multifractal analysis. They were introduced with a view of characterizing the geometry of a measure l (see [1–7]) and to be linked with the multifractal spectrum which is the map which affects the Hausdorff or packing dimension of the iso-Hölder set

Ea ¼

  logðlðBðx; rÞÞ x 2 suppðlÞ : lim ¼a ; r!0 log r

for a given a P 0. Spectrum and generalized dimensions are very helpful when comparing multifractals appearing in nature with treatable measures. For q 2 R, the Lq-spectrum of l is defined as

sl ðqÞ ¼ limþ

log sup

P

i

lðBðxi ; rÞÞq

log r

r!0

 ;

where the supremum is taken over all the centered packing of the support of l by balls of radius r. It is easy to check that sl ðqÞ is a concave function of q over R and for q > 1 it has an integrand expression. For q P 1, we have

sl ðqÞ ¼ lim r!0

1 log log r

Z

lðBðx; rÞÞq1 dlðxÞ:

suppðlÞ

For a given measure it is usually very hard or impossible to calculate the corresponding multifractal spectrum directly. The multifractal analysis was first introduced according to heuristic arguments in [8,4,3] and have been computed rigorously for various classes of measures in Rn exhibiting some degree of self-similarity [9–11,5,12–15]. It unifies the multifractal spectra to the Lq-spectrum sl ðqÞ via the Legendre transform, i.e.

  dimK El ðaÞ ¼ s ðaÞ :¼ inf aq  sl ðqÞ : q 2 R ;

where K 2 fH; pg, here dimH and dimp denote, respectively, the Hausdorff and packing dimension (see [16,17] for the definitions). Differing notions of singularity exponents and dimensions distributions have been developed in various fields such a measure theory (see [3–6,18,7,19,20]). In 1983, Hentschel and Procaccia [3], Grassberger and Procaccia [21] and Grassberger [22] intoduced a one-parameter family of numbers ðDq Þ based on some generalized entropies due to Rényi [1,2].

E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.03.056

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I. Bhouri / Chaos, Solitons and Fractals 42 (2009) 1451–1462

Dq ¼ limþ r!0

P q log sup i lðBðxi ; rÞÞ ; ð1  qÞ log r

where the supremum is taken over all the centered packing of the support of l by balls of radius r. A parallel development of q Rényi dimensions using integrals was also suggested in [3, Formula (3.14)]. In this paper we provide a generalization of the Lq-spectrum relatively to two compactly supported Borel probability measures on Rn . Let l and m be two compactly supported Borel probability measures on Rn such that suppðlÞ ¼ suppðmÞ. For q 2 R, one defines

T l;m ðqÞ ¼ lim inf r!0

T l;m ðqÞ ¼ lim sup r!0

1 log log r

Z

1 log log r

lðBðx; rÞÞq dmðxÞ;

suppðlÞ

Z

lðBðx; rÞÞq dmðxÞ;

suppðlÞ

respectively, the generalized lower and upper Lq-spectrum of l relatively to m. If T l;m ðqÞ ¼ T l;m ðqÞ, their common value is denoted by T l;m ðqÞ at the value q and we call it the generalized Lq-spectrum of l relatively to m. Notice that these quantities are strictly related to the relative multifractal analysis and the multifractal variation measure developed by Olsen and Cole [23,24]. Other works were carried in this sense in probability and symbolic spaces [25–27]. We note that some scholar such as El Naschie [28–32], Ord et al. [33,34] and Nottale [35] have achieved many valuable results on the same subject and application. We are interested in the study of the behavior of the generalized upper Lq-spectrum relatively to two measures on Rn under orthogonal projections onto a lower dimensional linear subspace. Recently the projectional behavior of dimensions and multifractal spectra of measures have generated an interest in the mathematical literature [36–41]. This is connected to the question of the relationship between the Hausdorff and packing dimensions of a subset E of Rn or a Borel probability measure l and that of its orthogonal projections onto an m-dimensional subspace V. The first results in this direction were by Marstrand [42] who showed that for a Borel set, E in the plane dimH ðpV ðEÞÞ ¼ minðdimH ðEÞ; 1Þ for almost every line V (here pV denotes orthogonal projection onto V). Later, this result has been extended to higher dimensions by Kaufman [43] and Mattila [44]. It is easy to deduce from this that a similar result holds for the Hausdorff dimensions of a measure l on Rn (see [45]). Unlike the Hausdorff dimension, the packing dimension is not preserved under orthogonal projections. Nevertheless, the packing dimensions of the projections cannot be too small: in [38], it has been proved that if dimH l 6 m then

dimp lð1  dimH l=nÞ 6 dimp lV ; 1 þ ð1=m  1=nÞdimp l  dimH =m for almost all m-dimensional subspaces V, where lV is the image of l under pV . Later, Falconer and Howroyd have proved that dimp lV is the same for almost all m-dimensional subspaces V [37]. For this purpose they have characterized the packing dimension in terms of a ptentional obtained by convolving l with a certain kernel. In this paper we pursue those kinds of studies, we begin by investigating a new definition of potentialist type of the generalized upper Lq-spectrum relatively to two measures when q > 0. Then we show that if n and m are integers with 0 < m < n, l and m are two compactly supported Borel probability measures on Rn with suppðlÞ ¼ suppðmÞ, then m

 For 0 < q 6 1, we have T lV ;mV ðqÞ ¼ dimq ðl; mÞ. m  For q > 1, we have T lV ;mV ðqÞ ¼ minðmq; dimq ðl; mÞÞ. m

For almost every linear m-dimensional subspace V where dimq ðl; mÞ is a quantity obtained by convolving the measure m with a certain kernel. Moreover, it can be strictly less then the generalized upper Lq-spectrum relatively to l and m. For ‘‘Frostman like measures”, we can improve substantially this result showing a uniform result: For almost all linear m-dimensional subspace V, for all q > 0. Notice that in [46] we have introduced the generalized lower Lq-spectrum relatively to two measures for q > 0 and we have studied the behavior of these quantities under orthogonal projections. We have generalized the result of Hunt and Kaloshin [39, Theorem 3.1] and we have treated an unsolved case by Hunt and Kaloshin which is q > 1 and the result that we have obtained is optimal. This paper is organized as follows. In the next section we introduce some basic notions and properties. Section 3 contains intermediate results which are used to prove our main results in Section 4. Then we state our general results in Section 4. Moreover, we give an example illustrating that the generalized upper Lq-spectrum cannot be preserved under orthogonal projections. The uniform result for ‘‘Frostman like measures” is stated in Section 5. 2. Preliminaries Casually, we briefly recall some basic definitions and facts which will be repeatedly used in subsequent developments. Let m be an integer with 0 < m < n and Gn;m stand for the Grassmann manifold of all m-dimensional linear subspaces of Rn . For V 2 Gn;m , let pV : Rn ! V stands for the orthogonal projection onto V. Then fpV ; V 2 Gn;m g is compact in the space of all linear maps from Rn to Rm , and the identification of V with pV induces a compact topology for Gn;m . Fixing V 0 2 Gn;m , we can define an orthogonally invariant radon probability measure cn;m on Gn;m by

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I. Bhouri / Chaos, Solitons and Fractals 42 (2009) 1451–1462

cn;m ðAÞ ¼ v n fg 2 OðnÞ; gðV 0 Þ 2 Ag for A  Gn;m ; where v n denotes the unique Haar measure on the orthogonal group OðnÞ of Rn normalized so that v n ðOðnÞÞ ¼ 1. The uniqueness implies that cn;m is independent of V 0 . In other words,

cn;m ¼ fV 0  v n with f V 0 ðgÞ ¼ gðV 0 Þ for g 2 OðnÞ; where fV 0  v n denotes the image of the measure v n under the map fV 0 . For more details, see [47]. For a Borel probability measure l on Rn , supported on the compact set suppðlÞ, and for V 2 Gn;m , we define jection of l onto V by

lV , the pro-

lV ðEÞ ¼ lðp1 8E # V: V ðEÞÞ Thus if f : V ! R is continuous then

Z V

f ðuÞ dlV ðuÞ ¼

Z Rn

f ðpV ðxÞÞ dlðxÞ:

Observe that since l has compact support, suppðlV Þ ¼ pV ðsuppðlÞÞ for all V 2 Gn;m . From now on, 0 < m < n are two integers, l and m denote two compactly supported Borel probability measures on Rn with suppðlÞ ¼ suppðmÞ ¼ K. In [46], we have studied the behavior of the generalized lower Lq-spectrum relatively to l and m under orthogonal projections and we have proved the following result. Theorem 2.1. Let p > 0 we have 1. If p 2 ð0; 1 and T l;m ðpÞ 6 pm then T lV ;mV ðpÞ ¼ T l;m ðpÞ for cn;m -almost every V. 2. If p > 1 and T l;m ðpÞ 6 m then T lV ;mV ðpÞ ¼ T l;m ðpÞ for cn;m -almost every V. Let us notice that assertion (1) is a generalization of the result of Hunt and Kaloshin [39] and due to Example 5.2 in [39], assertion (2) is optimal and the result is the best possible one. 3. Intermediate results In this section we require an alternative characterization of the generalized upper Lq-spectrum relatively to l and m in terms of the convolutions. For this purpose let us introduce some notations. For all k 2 N n f0g and real p > 0, we define

n o n o k dimp ðl; mÞ ¼ sup s P 0 : Lks;p ðl; mÞ < 1 ¼ inf s P 0 : Lks;p ðl; mÞ ¼ 1 ; where for all s P 0,

Lks;p ðl; mÞ ¼ lim inf rs

Z Z

r!0

p n o min 1; rk jx  yjk dlðyÞ dmðxÞ:

Proposition 3.1. For all p > 0 n

T l;m ðpÞ ¼ dimp ðl; mÞ: In order to prove the previous proposition we need the following results. Lemma 3.1. For all q > 0, k > 1 and M P 1 one has

mðfx 2 Rn : lðBðx; kqÞÞ > MmðBðx; qÞÞgÞ 6 4n kn M1 : Proof.  Denote A ¼ fx 2 K : lðBðx; kqÞÞ > MmðBðx; qÞÞg. B x; q2  Bðy; qÞ and Bðy; kqÞ  Bðx; 2kqÞ. So,



Choose

x

such

that

 A \ B x; q2 – ;.

If

 y 2 A \ B x; q2 ,

q 6 M 1 lðBðx; 2kqÞÞ: 2

m A \ B x;

ð3:1Þ

Writing Ln for the Lebesgue measure in Rn and using Fubini theorem we have

mðAÞ ¼

Z A

aðnÞaðnÞ1 dmðxÞ ¼

Z n q A

2

then

Z

q dmðxÞ ¼ aðnÞ1 2n qn 2

aðnÞ1 Ln B x;

where aðnÞ stands for the Lebesgue measure of the unit ball in Rn .



q dLn ðxÞ; 2

m A \ B x;

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I. Bhouri / Chaos, Solitons and Fractals 42 (2009) 1451–1462

Due to (3.1) and Fubini Theorem we have

Z

mðAÞ 6 aðnÞ1 2n qn M1

Z

6 aðnÞ1 2n qn M 1 which achieves the proof.

lðBðx; 2kqÞÞdLn ðxÞ 6 aðnÞ1 2n qn M1

Z

Ln ðBðx; 2kqÞÞ dlðxÞ

ð2kqÞn aðnÞ dlðxÞ 6 M 1 4n kn ;

h

Lemma 3.2. Let 0 < a < 1 and e > 0. There exists a number C 0 > 0 depending only on e and n such that for every compactly supported probability measures l and m on Rn with suppðlÞ ¼ suppðmÞ and for all q0 6 12, we have

(

)!  nð1þeÞ 1 4r x 2 R : there are q and r with 0 < q < q0 and q 6 r 6 1 such that lðBðx; rÞÞ > mðBðx; qÞÞ 2 q n

m

a

neð1aÞ

6 2C 0 q0

: ð3:2Þ

Proof. Let r > q > 1. Using Lemma 3.1, we have

(

)!  nð1þeÞ  ne 1 4r r x 2 R : lðBðx; rÞÞ P mðBðx; qÞÞ 6 24n : 2 q q n

m

In particular for r ¼ 2p and q ¼ 2q where p abd q are two integers such that p < q we have

  nð1þeÞ  1 x 2 Rn : lðBðx; 2p ÞÞ P 2qp mðBðx; 2q ÞÞ 6 24n 2qp ne : 2

m

ð3:3Þ

It is easy to see that



1  qp nð1þeÞ mðBðx; 2q ÞÞ for some p and q with 0 6 p 6 aq and q P q0 2 2  ½aq  1 [ [ nð1þeÞ 1  x 2 Rn : lðBðx; 2p ÞÞ P 2qp mðBðx; 2q ÞÞ : 2 q¼q p¼0

A¼

x 2 Rn : lðBðx; 2p ÞÞ P



0

So,

 ½aq  1 X X 1 m x 2 Rn : lðBðx; 2p ÞÞ P 2qp nð1þeÞ mðBðx; 2q ÞÞ : 2 q¼q p¼0

mðAÞ 6

0

By relation (3.3) we have

mðAÞ 6 4n ð2ne  aÞ1 ð1  2neða1Þ Þ1 ð2q0 Þneð1aÞ 6 C 1 ð2q0 Þneð1aÞ ; where C 1 ¼ 4n ð2ne  aÞ1 ð1  2neða1Þ Þ1 . Let q0 ¼ 21q0 . Then for 0 < q < q0 and qa 6 r 6 1, there exist integers p and q such that 2p1 < r 6 2p and 2q 6 q < 2qþ1 . Thus Bðx; rÞ  Bðx; 2p Þ and Bðx; 2q Þ  Bðx; qÞ. Hence,

( M¼

)  nð1þeÞ 1 4r x 2 R : there are q and r with 0 < q < q0 and q 6 r 6 1 such that lðBðx; rÞÞ > mðBðx; qÞÞ  A; 2 q n

a

neð1aÞ

which implies that mðMÞ 6 2C 0 q0

, where C 0 ¼ C 1 2neð1aÞ and achieves the proof of Lemma 3.2. h

Remark 3.1. Let 0 < a < 1 and 0 < e < 1. Consider x 2 Rn and q > 0 such that

lðBðx; rÞÞ 6

 nð1þeÞ 4r

q

mðBðx; qÞÞ and lðBðx; rÞÞ 6

 nð1þeÞ 4r

q

lðBðx; qÞÞ;

ð3:4Þ

for all r with qa 6 r 6 1. An integration by parts gives that

Zq ðxÞ :¼

Z

  min 1; qn jx  yjn dlðyÞ ¼ nqn

Z

1

un1 lðBðx; uÞÞ du

q

¼ nqn

Z

qa

un1 lðBðx; uÞÞdu þ

Z

1

un1 lðBðx; uÞÞdu þ

qa

q a

nð1þeÞ 1

6 lðBðx; q ÞÞ þ 4

Z 1

ne

e lðBðx; qÞÞq

n

n

þ lðR Þq ;

!

1

un1 lðBðx; uÞÞdu

I. Bhouri / Chaos, Solitons and Fractals 42 (2009) 1451–1462

1455

and due to (3.4) we have

Zq ðxÞ 6 C



mðBðx; qÞÞqða1Þnð1þeÞ þ mðBðx; qÞÞqne þ qn ;



ð3:5Þ



lðBðx; qÞÞqða1Þnð1þeÞ þ lðBðx; qÞÞqne þ qn ;



ð3:6Þ

and

Zq ðxÞ 6 C nð1þeÞ

for C ¼ 4

e

. nð1þeÞ

Definition 3.1. Let 0 < a < 1 and 0 < e < 1. Let q > 0 and C ¼ 4 1. A point x 2 K is called ðq; CÞ-stable relatively to

Zq ðxÞ 6 C

e

.

l and m if



mðBðx; qÞÞqða1Þnð1þeÞ þ mðBðx; qÞÞqne þ qn ;





lðBðx; qÞÞqða1Þnð1þeÞ þ lðBðx; qÞÞqne þ qn :

and

Zq ðxÞ 6 C



2. We say that x 2 K is ðq; CÞ-unstable relatively to

Zq ðxÞ > C



mðBðx; qÞÞq

ða1Þnð1þeÞ

l and m if

ne

þ mðBðx; qÞÞq

þ qn :

Lemma 3.3. Let 0 < a < 1 and 0 < e < 1. There exists a real C 0 > 0 depending on a, e and n, such that for every two compactly supported Borel probability measures on Rn l and m such that suppðlÞ ¼ suppðmÞ and for all q0 6 12 we have



m

x 2 suppðlÞ : x is



q;

 C -unstable relatively to 2



l and m

neð1aÞ

6 2C 0 q0

;

for all q with 0 < q < q0 .  Proof. Let C 0 be as in Lemma 3.2. If x 2 suppðlÞ is q; C2 -unstable relatively to

lðBðx; rÞÞ >



1 4r 2 q

nð1þeÞ

l and m then

mðBðx; qÞÞ;

for some qa 6 r 6 1. The result follows from Lemma 3.2.

h

Lemma 3.4. Let 0 < a < 1, 0 < e < 1 and p > 0. Let l and m be two compactly supported Borel probability measures such that suppðlÞ ¼ suppðmÞ ¼ K. There exist q1 > 0 and C 1 > 0 such that for all q with 0 < q < q1 , one has

Z

 p Z 1 Zq ðxÞp dmðxÞ 6 C 1 log Zq ðxÞp dmðxÞ;

q

Fq

Eq

where Eq ¼ fx 2 K : x is ðq; CÞ-stable relatively to

l and mg and F q ¼ K n Eq .

Proof. The main idea of this proof is that points x 2 K for which lðBðx; qÞÞ is ‘‘big” relatively to mðBðx; qÞÞ are stable and a point y 2 K can be unstable only if there are points x 2 K such that lðBðx; qÞÞ > mðBðy; qÞÞ. We will divide K into parts such that we can control points which cause instability for a given point y 2 K. The proof splits into three parts. neð1aÞ 6 12, where C 0 is as in Lemma 3.2. For 0 < q < q0 , define First step. Firstly, choose 0 6 q0 6 12 such that 2C 0 q0 h i Sq ¼ supx2K lðBðx; qÞÞ. For each x 2 K we will estimate hq ðxÞ as follows. Denote Lq ¼ diamðKÞ þ 1. The set K can be covered by q balls B of radius q and with centers at the distance jq from x, where j ¼ 1; . . . ; Lq , then each ball B can be covered with balls of radius q and centers in B \ K. Using Bescovich’s covering theorem [47] and estimating the integrand in each of these disjoint balls, we have

Zq ðxÞ 6 C 0

Lq  n1  n X jq q j¼1

q

jq

1 Sq 6 C 00 Sq log ;

ð3:7Þ

q

where C0 and C00 are constants depending only on n and diamK. Second step. Define,

E1q ¼



x 2 K : x is ðq; CÞ-stable relatively to

l and m; lðBðx; qÞÞ P

 Sq Sq and mðBðx; qÞÞ P ; 2 2

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I. Bhouri / Chaos, Solitons and Fractals 42 (2009) 1451–1462

and

( F 1q1 ¼

x 2 K : x is ðq; CÞ-unstable relatively to

l and m and Zq ðxÞ 6 2

For all k P 2, we define inductively the sets F 1qk by

F 1qk

8 > > < ¼ x 2 K : x is ðq; CÞ-unstable relatively to > > :

l and m

Z E1q

) minf1; qn jx  yjn g dlðyÞ :

9 > > = n n   : and Zq ðxÞ 6 2 q jx  yj g d l ðyÞ minf1; kS 1 > > E1q [ F 1q ; i Z

i¼1

Finally, we define 1 [

F 1q ¼

F 1qk ;

k¼1

and

A1q ¼ E1q [ F 1q : Notice that these sets are Borel sets since x#lðBðx; qÞÞ is upper semicontinuous and x#Zq ðxÞ is continuous. Now we will show the following inequality

Z F 1q

 p Z 1 Zq ðxÞp dmðxÞ 6 2C 00 log Zq ðxÞp dmðxÞ:

q

We may assume that

Zq ðxÞ 6 2

Z 1

ð3:8Þ

E1q

lðA1q Þ > 0. Let x 2 F 1q , then

  min 1; qn jx  yjn dlðyÞ;

Aq

which implies that x is ðq; CÞ-unstable relatively to lðA1q Þ1 ljA1q and mðA1q Þ1 mjA1q , where ljA1q (respectively mjA1q ) is the restriction of l to A1q (respectively m to A1q ). By Lemma 3.3, we deduce that Zq ðxÞ P lðBðx; qÞÞ P

Z F 1q

Sq 2

mðF 1q Þ 6 mðE1q Þ. Inequality (3.7) and the fact that

1

for all x 2 Eq give that

 p  p Z 1 1 Zq ðxÞp dmðxÞ 6 C 00 Sq log mðF 1q Þ 6 2C 00 log Zq ðxÞp dmðxÞ:

q

q

E1q

Thus inequality (3.8) holds. Third step. Further, for all l P 2, we define

n Elq ¼ x 2 K : x is ðe; CÞ-stable relatively to

o

l and m; 2l Sq 6 lðBðx; qÞÞ and 2l Sq 6 mðBðx; qÞÞ < 2lþ1 Sq ;

and

F lq1

8 > > l1 < [ ¼ x2K Aiq : x is ðq; CÞ-unstable relatively to > > i¼1 :

l and m

9 > > =   minf1; qn jx  yjn g dlðyÞ : and Zq ðxÞ 6 2 l1 S > > Elq [ Aiq ; i¼1

Then for all k P 2, we define

( l

F qk ¼

-

x2K

l1 [

Aiq : x is ðq; CÞ-unstable relatively to

i¼1

9 > > Z =  n  n     62 min 1; q jx  yj dlðyÞ : l1 kS 1 S > > Elq [ Aiq [ F lq ; j i¼1

Denote by

F lq ¼

1 [

F lqk ;

k¼1

and

Alq ¼ Elq [ F lq :

j¼1

Z

l and m and Zq ðxÞ

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I. Bhouri / Chaos, Solitons and Fractals 42 (2009) 1451–1462

In the following we will prove that

8l P 1; 8x 2 K n [li¼1 Aiq ;

lðBðx; qÞÞ < 2l Sq and mðBðx; qÞÞ < 2l Sq :

ð3:9Þ

Let us proceed by induction on l. We will choose in the sequel, q smaller if necessary, such that 4C 00 log if x 2 K and mðBðx; qÞÞ P

Sq 2

then (3.7) yields that x is ðq; CÞ-stable relatively to

for l ¼ 1. We now assume that

 1

q

6 Cr ne . For l ¼ 1,

l and m. Thus x 2 E1q  A1q and so (3.9) is true

i n lðBðx; qÞÞ < 2lþ1 Sq and mðBðx; qÞÞ < 2lþ1 Sq for all x 2 K n [l1 such that i¼1 Aq . Let x 2 R

S

lðBðx; qÞÞ P 2 Sq and mðBðx; qÞÞ P 2l Sq , we will prove that x 2 ðRn n KÞ ð[li¼1 Aiq Þ. For this purpose we may assume that i l l x 2 K n [l1 i¼1 Aq . If x is ðq; CÞ-stable relatively to l and m then x 2 Eq  Aq . If x is ðq; CÞ-unstable relatively to l and m, since l

we may assume that x R F lq then

Zq ðxÞ 6 2

Z Kn[l1 Ai i¼1 q

  min 1; qn jx  yjn dlðyÞ:

i lþ1 Further, by induction hypothesis K n [l1 Sq ; mðBðx; qÞÞ < 2lþ1 Sq g and the same calculai¼1 Aq  A :¼ fy 2 K : lðBðx; qÞÞ < 2 tions as in (3.7) give that

Zq ðxÞ 6 2

Z

  min 1; qn jx  yjn dlðyÞ 6 C qne mðBðx; qÞÞ;

A

S l which is a contradiction since x is ðq; CÞ-unstable relatively to l and m. Hence (3.9) holds. So, F q ¼ 1 l¼1 F q , indeed, if S1 l x 2 F q n l¼1 F q , then (3.9) implies that lðBðx; qÞÞ ¼ mðBðx; qÞÞ ¼ 0, which is a contradiction with the fact that x 2 K. Lemma 3.3 holds if we prove that for all l P 1,

Z F lq

!  p X Z l 1 p 00 pðklÞ Zq ðxÞ dmðxÞ 6 4C log 2 Zq ðxÞ dmðxÞ : p

q

S1

l l¼1 Eq

In fact, since Eq ¼

Z

ð3:10Þ

Elq

k¼1

and Elq are disjoint Borel sets then

! !   p X p Z 1 1 Z X 1 1 p 00 pk Zq ðxÞ dmðxÞ 6 4C log 2 Zq ðxÞ dmðxÞ 6 C 1 log Zq ðxÞp dmðxÞ: p

q

Fq

k¼0

l¼1

q

Elq

Eq

Let us prove (3.10). If l ¼ 1 then the inequality (3.10) follows from (3.8). Let l P 2, we may assume that x 2 F lq , on one hand for all k P 1 we have

Zq ðxÞ P 2

Z

l1

Eq [

   l2 kS 1 S i Aq

[

i¼1

j¼1

 min 1; qn jx  yjn dlðyÞ P 2 F l1 qj

Z

[l1 Ai i¼1 q

lð[li¼1 Aiq Þ > 0. Let

  min 1; qn jx  yjn dlðyÞ:

So,

Zq ðxÞ 6 2

Z l1 Ai Knð[i¼1 qÞ

  min 1; qn jx  yjn dlðyÞ:

On the other hand, (3.9) and with same calculations as in (3.7) imply that for all x 2 F lq

1 Zq ðxÞ 6 4C 00 2l Sq log :

ð3:11Þ

q

Moreover, if x 2 [lj¼1 F jq , then

Zq ðxÞ 6 2

Z

[li¼1 Aiq

  min 1; qn jx  yjn dlðyÞ;

which implies that x is ðq; CÞ-unstable relatively to Hence, Lemma 3.3 gives that

1 2

m1 ð[lj¼1 F jq Þ 6 : Thus

m1

l [ j¼1

! Ejq

¼ m1

l [ j¼1

! Ajq

 m1

l [ j¼1

! F jq

P

1 : 2

l1 ¼ lð[li¼1 Aiq Þ1 lj[l

i¼1

Aiq

and

m1 ¼ mð[li¼1 Aiq Þ1 mj[l

i¼1

Aiq

.

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I. Bhouri / Chaos, Solitons and Fractals 42 (2009) 1451–1462

So, mðF lq Þ 6 mð[lj¼1 Ejq Þ. It follows from (3.11) and Sq 6 2lzq ðxÞ for all x 2 Ejq and 1 6 j 6 l that

Z F lq

 p Z 1 Zq ðxÞp dmðxÞ 6 4C 00 2l log

q

[lj¼1 Ekq

 p l Z 1 X Spq dmðxÞ 6 4C 00 log 2pðjlÞ Zq ðxÞp dmðxÞ;

q

j¼1

Ekq

which achieves the proof of (3.10). h Let us now prove Proposition 3.1. We first show that if s > T l;m ðpÞ then Lns;p ðl; mÞ is infinite. Choose t 2 R such that R s > t > T l;m ðpÞ then lðBðx; rÞÞp dmðxÞ > rt for all small r > 0. So,

Lns;p ðl; mÞ P lim inf r s r!0

Z

lðBðx; rÞÞp dmðxÞ P lim inf rts ¼ 1: r!0

Lns;p ð

Next we show that l; mÞ is finite for s < T l;m ðpÞ. Let T l;m ðpÞ > t > s, then there exists a sequence ðrk Þ tending to zero such R that lðBðx; rk ÞÞp dmðxÞ < r tk for all integer k. Choose 0 < e < 1, 0 < a < 1 and 0 < d < 1 such that t  s þ pða  1Þnð1 þ eÞ > pd and t  s  pne > pd. Let k be large enough such that rk < q1 and log r1 6 C 1 r d k where q1 and C 1 are as in Lemma 3.4. Rek p call that for every a; b P 0, ða þ bÞp 6 supð1; 2p1 Þðap þ b Þ. Using Lemma 3.4 and the inequality (3.6) we have

Z Z

  min 1; r n jx  yjn dlðyÞ

p

 p Z 1 dmðxÞ 6 2C 1 log zrk ðxÞp dmðxÞ rk Er k  p Z  1 e lðBðx; rÞÞp rkpða1Þnð1þeÞ þ lðBðx; rÞÞp rpn þ r pn 6 M log dmðxÞ k k rk

 tþpðða1Þnð1þeÞdÞ tpðneþdÞ pðndÞ þ rk þ rk ; 6 M rk

where M is a constant depending only on p and diamK. Hence, The choice of e, a and d imply that Lns;p ðl; mÞ ¼ 0. 4. Projection results In this section we consider for q > 0, the behavior of the generalized upper Lq-spectrum relatively to two measures on Rn under orthogonal projections.Theorem 4.1Let p > 0 then, 1. For 0 < p 6 1, we have m

T lV ;mV ðpÞ ¼ dimp ðl; mÞ for cn;m -almost every V 2 Gn;m : 2. For p > 1, we have m

T lV ;mV ðpÞ ¼ minðdimp ðl; mÞ; pmÞ for cn;m -almost every V 2 Gn;m :

m Proof. Fix V 2 Gn;m . For s P 0, it is obvious that Lm then by Proposition 3.1 we have s;p ðl; mÞ 6 Ls;p ðlV ; mV Þ m m T lV ;mV ðpÞ ¼ dimp ðlV ; mV Þ 6 dimp ðl; mÞ. m (1) In order to prove the other inequality in the case 0 < p 6 1 let s < dimp ðl; mÞ and r > 0. Jensen’s inequality, Fubini’s Theorm and Lemma 3.11 in [47] imply that

Z Z

lV ðBðpV ðxÞ; rÞÞp dcn;m ðVÞ dmðxÞ 6 ¼

Z Z Z Z

6c

Z Z

lfy 2 Rn : jpV ðxÞ  pV ðyÞj 6 rgdcn;m ðVÞ 

cn;m V 2 Gn;m

p

dmðxÞ

p  : jpV ðxÞ  pV ðyÞj 6 r dlðyÞ dmðxÞ

p   min 1; rm jx  yjm dlðyÞ dmðxÞ;

where c is a constant independent of r. Therefore by Fubini’s Theorem

lim inf r s r!0

Z Z

lV ðBðpV ðxÞ; rÞÞp dmðxÞdcn;m ðVÞ < 1;

R and Fatou’s Lemma implies that lim inf r!0 rs lV ðBðpV ðx; rÞÞp dmðxÞÞ < 1 for cn;m -almost every V 2 Gn;m . Thus T lV ;mV ðpÞ P s which achieves the proof of assertion ð1Þ. (2) Let us prove assertion ð2Þ. For all p > 1, we have m

dimp ðlV ; mV Þ 6 mp for cn;m -almost every V 2 Gn;m : Indeed, it is easy to show that there exists a constant C n;m > 0 such that for r > 0,

ð4:1Þ

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I. Bhouri / Chaos, Solitons and Fractals 42 (2009) 1451–1462

Z Z Z

p p Z Z     min 1; r m jx  yjm dlV ðyÞ dmV ðxÞdcn;m ðVÞ P min 1; r m jx  yjm dlðyÞ dmðxÞdcn;m ðVÞ P C n;m

So,

Z Z Z

  min 1; r m jx  yjm dlðyÞ

p

dmðxÞdcn;m ðVÞ P C n;m P C n;m

Z Z

Z Z Z Z

lV ðBðpV ðxÞ; rÞÞdcn;m ðVÞ

p

dmðxÞ: p

lfy 2 Rn : jpV ðxÞ  pV ðyÞj 6 rgdcn;m ðVÞ 



cn;m V 2 Gn;m : jpV ðxÞ  pV ðyÞj 6 r dlðyÞ

dmðxÞ

p

dmðxÞ:

Lemma 2.7 in [48] gives that

Z Z Z

  min 1; r m jx  yjm dlðyÞ

p

dmðxÞdcn;m ðVÞ P C n;m bðn  1Þp aðmÞp bðn  m  1Þp r mp 

Z Z

un;m ðrÞjx  yjm dlðyÞ

p

dmðxÞ;

n1

n1

fy2S :jyj6rg n1 where bðn  1Þ denotes the ðn  1Þ-area of the unit sphere Sn1 in Rn and for r > 0, un;m ðrÞ ¼ HaðmÞbðnm1Þr ðAÞ dem , with H notes the ðn  1Þ-dimensional Hausdorff measure of a borelean A 2 Rn . Let us remark that un;m is bounded and R jx  yjm dlðyÞ > 0. Thus, let e > 0, for r smaller enough

rmpe

Z Z Z

  min 1; rm jx  yjm dlV ðyÞ

p

dmV ðxÞdcn;m ðVÞ P C 0 re :

So by Fatou Lemma we have,

lim inf Lm for cn;m -almost every V 2 Gn;m : mpe;p ðlV ; mV Þ ¼ 1 r!0

Hence, m

mp þ e > dimp ðlV ; mV Þ for cn;m -almost every V 2 Gn;m : m

The arbitrary on e implies that dimp ðlV ; mV Þ 6 mp for cn;m -almost every V 2 Gn;m . m m Let s > dimp ðlV ; mV Þ we may choose s > mp. Using Proposition 3.1, it is sufficient to prove that s > dimp ðl; mÞ. Similar techniques as to prove (4.1) show that

lim inf Lm s;p ðl; mÞ ¼ 1 for cn;m -almost every V 2 Gn;m : r!0

Hence, m

s > dimp ðl; mÞ; which achieves the proof of Theorem 4.1. h Comments. The following example is constructed in a similar way as in [38, Example 5.1]. It shows that the behavior of the generalized upper Lq-spectrum relatively to l and m under orthogonal projections is different from the generalized lower Lq-spectrum relatively to l and m that is for q > 0, the generalized upper Lq-spectrum relatively to l and m unlike the lower one [46,39], is not preserved under projections.  < n and d < m. Then there exists a Borel probability compactly supported measure l on Rn such Example 4.1. Let 0 < d < d that the following properties hold: 1. There is a positive constant c such that 

crd 6 lðBðx; rÞÞ 6

rd ; c

for all x 2 suppðlÞ, and 0 < r 6 1, 2. there exist

sequences   ðr k Þ and ðRk Þ of positive real numbers tending to zero such that  d r k and l B x; R2k ¼ Rdk for all x 2 suppðlÞ, and 3.

dimB ðpV ðsuppðlÞÞÞ 6

  d=nÞ dð1   d=m ; 1 þ ð1=m  1=nÞd

for all V 2 Gn;m where the upper box-counting dimension is denoted by dimB (see [16]).

  pffiffi

l B x; ðnÞrk



¼

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I. Bhouri / Chaos, Solitons and Fractals 42 (2009) 1451–1462

Let m be a Borel probability compactly supported measure on Rn with suppðlÞ ¼ suppðmÞ such that lðBðx; rÞÞ 6 mðBðx; rÞÞ for all x 2 suppðlÞ and r > 0.  Moreover, by (3) we have Let p > 0, due to assertions (1) and (2) it is easy to show that T l;m ðpÞ ¼ pd and T l;m ðpÞ ¼ pd.

 T ðpÞ T l;m ðpÞ 1  l;nm T lV ;mV ðpÞ 6 T lV ;lV ðpÞ 6 pdimB ðpV ðsuppðlÞÞÞ 6 < T l;m ðpÞ:  T ðpÞ 1 þ m1  1n T l;m ðpÞ  l;mm

5. Uniform results for projections of ‘‘Frostman like measures In tis section we will improve Theorem 4.1 in the case of ‘‘Frostman like measures” by providing uniform results. We introduce technical property (P) on the measure l, which is satisfied by ‘‘Frostman measures” [12,49] and in particular by Gibbs measures on conformal repellers [50] and self-conformal Gibbs measures [51]. Let l be a measure like in Section 2 and define

p 2 R#sl;m ðpÞ ¼ lim sup

nP

log sup

o

i

lðBðxi ; rÞÞp1 mðBðxi ; rÞÞ

;

log r

r!0þ

where the supremum is taken over all the centered packing of suppðlÞ by balls of radius r. Proposition 5.1. Let l and have T l;m ðp  1Þ ¼ sl;m ðpÞ.

m be two compactly supported Borel measures on Rn such that suppl ¼ suppm ¼ K. For all p P 1 we

Proof. Let d > 0 and fBðxi ; dÞg be a family of disjoint balls centered on K. One has,

Z

lðBðx; 2dÞÞp1 dmðxÞ P

Z

K

lðBðx; 2dÞÞp1 dmðxÞ P [i Bðxi ;dÞ

X

lðBðxi ; dÞÞp1 mðBðxi ; dÞÞ:

i

Thus, T l;m ðp  1Þ 6 sl;m ðpÞ. On the other hand, for every d > 0 we can apply Bescovich’s covering theorem to fBðx; 2dÞgx2K , to get a positive integer j depending on n only, as well as F1 ¼ fBðx1;j ; 2dÞg; . . . ; Fj ¼ fBðxj;j ; 2dÞg, j families of disjoint balls of radius 2d, such that S S K  ji¼1 j Bðxi;j ; 2dÞ and

Z

lðBðx; dÞÞp1 dmðxÞ 6

K

j X Z X i¼1

j

lðBðx; dÞÞp1 dmðxÞ 6

Bðxi;j ;2dÞ

j X X i¼1

lðBðxi;j ; 2dÞÞp1 mðBðxi;j ; 2dÞÞ;

j

which implies that T l;m ðp  1Þ P sl;m ðpÞ. h Definition 5.1. If q 2 R we say that property MðqÞ holds if there exists a ‘‘Frostman like measure”: a positive Borel measure lq , a non-negative function uq defined over Rþ such for all x 2 K and r > 0,

lq ðBðx; rÞÞ 6 expðuq ðrÞÞlðBðx; rÞÞq rT l;l ðq1Þ ;

ð5:1Þ

with uq ðrÞ ¼ oðj log rjÞ near 0. Remark 5.1. If Eq. (5.1) holds for all q 2 R then T l;lq ðpÞ P sl;l ðp þ qÞ  sl;l ðqÞ for all p 2 R with equality if measure that is it satisfies for all x 2 K and r > 0,

lq is a Gibbs

expðuq ðrÞÞlðBðx; rÞÞq r T l;l ðq1Þ 6 lq ðBðx; rÞÞ 6 expðuq ðrÞÞlðBðx; rÞÞq r T l;l ðq1Þ : Definition 5.2. We say that l satisfies property (P) if (P): For all q > 0 MðqÞ holds there exists a non-negative continuous function f on ð0; 1Þ2 , null on fðq; qÞ : q > 0g such that for all x 2 K, r 2 ð0; 1Þ and ðq; q0 Þ 2 ð0; 1Þ2



1 lq ðBðx; rÞÞ

log

6 f ðq; q0 Þ:

logðrÞ lq0 ðBðx; rÞÞ

ð5:2Þ

Let l be a compactly supported Borel probability measure satisfying the property (P). We may alleviate the use of cumbersome notations by simply writing m ¼ lq , q > 0. We have the following uniform result. Theorem 5.1. For cn;m -almost every V 2 Gn;m , for all p P 0, m

T lV ;mV ðpÞ ¼ dimp ðl; mÞ:

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I. Bhouri / Chaos, Solitons and Fractals 42 (2009) 1451–1462

Proof. Let S ¼ fq : s0l;lq ð0þ Þ 6 mg. We learn from Proposition 5.1 and Theorem 4.1 that if q 2 S then for cn;m -almost every V, m for all p P 1 we have slV ;mV ðpÞ ¼ T lV ;mV ðp  1Þ ¼ dimp1 ðl; mÞ. Consequently, for cn;m -almost every V, for all q 2 S\ m Q and p P 1 we have slV ;mV ðpÞ ¼ T lV ;mV ðp  1Þ ¼ dimp1 ðl; mÞ. Therefore, if we show that for a fixed p P 1 the mapping q#sl;lq;V ðpÞ is continuous in q, the conclusion will follow. S If V 2 Gn;m , let fBðxi ; rÞg be a centered packing of pV ðKÞ. For each i we write K \ p1 y2K\p1 ðfxi gÞ Bðy; rÞ. By V ðBðxi ; rÞÞ ¼ V Besicovich’s covering theorem, there exists a positive integer j depending on n only such that for each i, we can extract from the family fBðy; rÞgy2K\p1 ðfxi gÞ , j families of pairwise disjoint balls F1 ¼ fBðx1;j ; rÞg; . . . ; Fj ¼ fBðyj;j ; rÞg such that V

K \ p1 V ðBðx; rÞÞ 

j [ [ i¼1

Bðxi;j ; rÞ and 8 i;

j

[

Bðxi;j ; rÞ  p1 V ðBðx; 2rÞÞ:

j

Now, we use (5.2). Let q; q0 2 ð0; 1Þ2 , we have





lq p1 V ðBðx; rÞÞ 6

j X X i¼1

j

0

lq ðBðxi;j ; rÞÞ 6 rf ðq;q Þ

j X X i¼1

0





lq0 ðBðxi;j ; rÞÞ 6 jrf ðq;q Þ lq0 p1 V ðBðx; 2rÞÞ :

j

Consequenly, we get sl;lq;V ðpÞ P sl;lq0 ;V ðpÞ  f ðq; q0Þ for all p P 1. By interchanging the roles of q and q0 we have jsl;lq;V ðpÞ  sl;lq0 ;V ðpÞj 6 f ðq; q0Þ and the result yields. h References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46]

Rényi A. Some findamental questions of information theory. Magyar Tud Akad Math Fiz Ost Közl 1960;10:251–80. Rényi A. On measures of entropy and formation. In: Proceedings of the 4th Berkeley symposium on mathematical statistics and probability; 1961. Hentschel H, Procaccia I. The infinite number of generalized dimensions of fractals and strange attractors. Physica D 1983;8:435–44. Halsey TC, Jensen MH, Kadanoff LP, Procaccia I, Shraiman BI. Fractal measures and their singularities: the characterisation of strange sets. Phys Rev A 1986;33:1141–51. Pesin Y, Weiss H. The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples. Chaos 1997;7:89–106. Riedi R. An improved multifractal formalism and self-affine measures. J Math Anal Appl 1995;189:462–90. Olsen L. A multifractal formalism. Adv Math 1995;116:82–196. Frisch U, Parisi G. Fully developped turbulence and intermittency in turbulence, and predictability in geophysical fluid dynamics and climate dymnamics. In: Ghil M, editor. International school of Physics ‘‘Enrico Fermi course, 88, vol. 84. Amsterdam: North-Holland; 1985. Arbeiter M, Patszchke N. Random self-similar multifractals. Math Nachr 1996;181:5–42. Brown G, Michon G, Peyrière J. On the multifractal analysis of measures. J Stat Phys 1992;66:775–90. Pesin Y. Dimension theory in dynamical systems: contemporary views and applications. Chicago lectures in mathematics, The University of Chicago Press; 1997. Ben Nasr F. Analyse multifractale de mesures. CR Acad Sci Paris Ser I 1994;319:807–10. Falconer K. The multifractal spectrum of statistically self-similar measures. J Theor Probab 1994;7:681–702. Ben Nasr F, Bhouri I, Heurteaux Y. The validity of the multifractal formalism: results and examples. Adv Math 2002;165:264–84. Barral J. Continuity of the multifractal spectrum of a random statistically self-similar measure. J Theor Probab 2000;13:1027–60. Falconer K. Fractal geometry: mathematical foundations and applications. New York: Wiley; 1990. Tricot C. Two definitions of fractional dimension. Math Proc Cambridge Philos Soc 1982;91(1):57–74. Ngai SM. A dimension result arising from the Lq-spectrum of a measure. Proc Am Math Soc 1997;125:2943–51. Heurteaux Y. Estimations de la dimension inférieure et de la dimension supérieure des measures. Ann Inst Henri Poincaré Stat 1998;34:30938. Peyrière J. A vectorial multifractal formalism. In: Lapidus ML, van Frankenhuijsen M, editors. Fractal geometry and applications. Proc. Symp. Pure Math., Proc. Symp. in Pures Math AMS, Providence, RI 2004;72(2):217–30. Grassberger P, Procaccia I. Characterization of strange attractors. Phys Rev Lett 1983;50:346–9. Grassberger P. Generalized dimensions of strange attractors. Phys Lett A 1983;97:227–30. Cole J. Relative multifractal analysis. Chaos, Solitons & Fractals 2000;11:223350. Cole J, Olsen L. Multifractal variation measures and multifractal density theorems. Real Anal Exchange 2002/2003;28(2):501–14. Li Yueling, Dai Chaoshou. A multifractal formalism in a probability space. Chaos, Solitons & Fractals 2006;27:5773. Li Yueling, Dai Chaoshou. Multifractal dimension inequalities in a probability space. Chaos, Solitons & Fractals 2006;27:5773. Barreira L, Saussol B. Variational principles and mixed multifractal spectra. Trans Am Math Soc 2001;353:3919–44. El Naschie MS. The Fractal dimension of spacetime-remarks on theoretical derivation and experimental verification. Chaos, Solitons & Fractals 1998:9. El Naschie MS. Remarks on superstrings, fractal gravity, Nagasawa’s diffusion and Cantorian spacetime. Chaos, Solitons & Fractals 1997;8(11):187386. El Naschie MS. The Bethe lattice and the dimension of micro-spacetime. Chaos, Solitons & Fractals 1997;8:18879. El Naschie MS. Silver mean Hausdorff dimension and Cantor sets. Chaos, Solitons & Fractals 1994;4:186170. El Naschie MS. Average symmetry stability and ergodicity of multi-dimensional Cantor sets. II Nuo Cim 1994;109(2):14957. Ord GN, Nottale L, El Naschie MS. The impact of nonlinear dynamics and fractals on quantum physics and relativity. Chaos, Solitons & Fractals 1998;9(7):100910. Ord GN. Fractal spacetime, a geometric analog of relativistic quantum mechanics. J Phys A 1983;16:186984. Nottale L. Fractal spacetime and microphysics. Singapore: World Scientific; 1993. Falconer K, Howroyd J. Projection theorems for box and packing dimensions. Math Proc Cambridge Philos Soc 1996;119:287–95. Falconer K, Howroyd J. Packing dimensions of projections and dimension profiles. Math Proc Cambridge Philos Soc 1997;121:269–86. Falconer K, Mattila P. The packing dimensions of projections and sections of measures. Math Proc Cambridge Philos Soc 1996;119:695–713. Hunt BR, Kaloshin VY. How projections affect the dimension spectrum of fractal measures? Nonlinearity 1997;10:1031–46. O’Neil T. The multifractal spectrum of projected measures in Euclidean spaces. Chaos, Solitons & Fractals 2000;11:901–21. Falconer K, O’Neil T. Convolutions and the geometry of multifractal measures. Math Nachr 1999;204:61–82. Marstrand JM. Some fundamental geometrical properties of plane sets of fractional dimensions. Proc Lond Math Soc 1954;4(3):257–302. Kaufman R. On Hausdorff dimension of projections. Mathematika 1968;10:153–5. Mattila P. Hausdorff dimension, orthogonal projections and intersections with planes. Ann Acad Sci Fenn Ser A I Math 1975;1(2):227–44. Hu X, Taylor S. Fractal properties of products and projections of measures in Rd . Math Proc Cambridge Philos Soc 1994;115:527–44. Bahroun F, Bhouri I. Multifractals and projections. Extr Math 2006;21:83–91.

1462

I. Bhouri / Chaos, Solitons and Fractals 42 (2009) 1451–1462

[47] Mattila P. Geometry of sets, and measures in euclidean spaces fractals and rectifiability. Cambridges studies in advanced mathematics, vol. 44. Cambridge: Cambridge University Press; 1995. [48] Mattila P. Orthogonal projections, Riez capacities and minkowski contents. Ind Univ Math J 1990;39(1):185–97. [49] Bhouri I, Tlili H. On the multifractal formalism for bernoulli products of invertible matrices. Dis Cont Dynam Syst 2009;24(4). [50] Pesin Y, Weiss H. A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions. J Stat Phys 1997;86:233–75. [51] Feng D-J. Gibbs properties of self-conformal measures and the multifractal formalism. Ergod Th Dynam Syst 2007;27:787–812.