On the fast Khintchine spectrum in continued fractions

(1)

HAL Id: hal-00723315

https://hal.archives-ouvertes.fr/hal-00723315

Submitted on 9 Aug 2012

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

On the fast Khintchine spectrum in continued fractions

Fan Ai-Hua, Lingmin Liao, Bao-Wei Wang, Jun Wu

To cite this version:

Fan Ai-Hua, Lingmin Liao, Bao-Wei Wang, Jun Wu. On the fast Khintchine spectrum in continued fractions. Monatshefte für Mathematik, Springer Verlag, 2013, 171, pp.329–340. �hal-00723315�

(2)

ON THE FAST KHINTCHINE SPECTRUM IN CONTINUED FRACTIONS

AIHUA FAN, LINGMIN LIAO, BAOWEI WANG^†, AND JUN WU

Abstract. Forx∈[0,1), letx=[a1(x),a₂(x),· · ·] be its continued fraction expansion with partial quotients{an(x),n≥1}. Letψ:N→Nbe a function withψ(n)/n→ ∞asn → ∞. In this note, the fast Khintchine spectrum, i.e., the Hausdorffdimension of the set

E(ψ) :=n

x∈[0,1) : lim

n→∞

1 ψ(n)

n

X

j=1

loga_j(x)=1o is completely determined without any extra condition onψ.

1. Introduction

Continued fraction expansions are induced by the Gauss transformation T : [0,1)→[0,1) given by

T(0) := 0, T(x)= 1

x (mod 1), forx∈(0,1).

Leta₁(x)= ⌊x⁻¹⌋(⌊·⌋stands for the integral part) anda_n(x)=a₁(Tⁿ⁻¹(x)) for n ≥ 2. Each irrational numberx∈[0,1) admits a unique infinite continued fraction expansion of the form

x= 1

a1(x)+ 1 a2(x)+ 1

a₃(x)+ . ..

. (1.1)

Sometimes, (1.1) is written as x = [a1,a2,· · ·]. The integersan are called the partial quotients of x. The n-th convergent pn(x)/q_n(x) of xis given by

p_n(x)/q_n(x)=[a₁,· · · ,a_n].

The continued fraction is tightly connected with the classic Diophantine approximation. For example, for anyv≥ 2, the well-known Jarn´ık set

nx:|x−p/q|<q^−v, for infinitely many (p,q)∈Z²o

is equal to Jv−2, where for any β > 0, the set J_β is defined by continued fractions as

J_β:= n

x:a_n+1(x)≥q_n(x)^β, for infinitely many n∈No .

2000Mathematics Subject Classification. 11K50, 28A80.

Key words and phrases. Continued fractions, Fast Khintchine spectrum, Hausdorff dimension.

†Corresponding author.

1

hal-00723315, version 1 - 9 Aug 2012

(3)

The Gauss transformation is identified with an infinite symbolic dynamical system if we consider the partial quotients as symbols. The appearence of infinite symbols brings us new phenomena in relative to the case of finite symbols. For example, consider the set

(

x∈[0,1) :An 1

n♯{1≤ j≤ n:aj(x)= 1}o

n≥1 =[0,1]

)

where A(E) denotes the set of the accumulation points of a set E. The Hausdorffdimension of this set is 1/2 (see [9]), while inb-adic expansion a similar set is of Haudorffdimension 0 (see [12]). Another example is that the multifractal spectrum of the level sets of the Khintchine constant

nx∈[0,1) : lim

n→∞

1 n

n

X

j=1

loga_j(x)= ξo

is neither concave nor convex [3]. Because of the difference from the finite symbolic dynamical systems and of the observed new phenomena, continued fractions attracted much attention. One can find rich properties of the continued fraction dynamical system in [2, 3, 5, 6, 7, 10, 11, 14] and related works therein.

Letψ:N→N. Define E(ψ)=

(

x∈[0,1) : lim

n→∞

loga1(x)+· · ·+logan(x)

ψ(n) =1

) .

When ψ(n) = λn for someλ > 0, the set E(ψ) is a level set of the classic Khintchine constant. Besides a detailed spectrum analysis of the classic Khintchine constant in [3], the authors also studied the fast Khintchine spectrum, i.e. the Hausdorffdimension ofE(ψ) whenψ(n)/n → ∞asn → ∞.

But the result for the latter case is incomplete. Only under the conditions that limn→∞ ψ(n+1)

ψ(n) = b and limn→∞(ψ(n)−ψ(n −1)) = ∞, the dimension of E(ψ) was given [3]. In this note, we show that these extra conditions are unnecessary for determining the dimension of E(ψ) in the case of fast Khintchine spectrum.

Two functionsψand ˜ψdefined onNare said to beequivalentif ^ψ(n)_ψ(n)_˜ → 1 asn→ ∞.

Theorem 1.1. Let ψ : N → N with ψ(n)/n → ∞ as n → ∞. If ψ is equivalent to an increasing function, then E(ψ), ∅and

dimHE(ψ)= 1

1+b, with b =lim sup

n→∞

ψ(n+1) ψ(n) . Otherwise, E(ψ)= ∅.

Remark 1. The method used in [3] does not apply to general ψ. This is explained in Section 3 below.

hal-00723315, version 1 - 9 Aug 2012

(4)

FAST KHINTCHINE SPECTRUM 3

Remark 2. The upper bound of dim_HE(ψ) is the difficult part of the proof of Theorem 1.1. As a byproduct of the proof, we get that for anyβ > 0, the Hausdorffdimension of the set

J^∗_β :=n

x∈ J_β: lim

n→∞

logq_n(x) n = ∞o

is 1/(2+β), i.e. one half of the dimension of the Jarn´ık set Jβ. A detailed explanation is given at the end of this paper.

2. Preliminary

This section is devoted to fixing some notation, recalling some elemen- tary properties enjoyed by continued fractions and citing some technical lemmas in dimension estimation.

Throughout this paper, we use ⌊·⌋ to denote the integral part of a real number, |A|the diameter of a set A ⊂ R, H^s the s-dimensional Hausdorff measure, and dim_Hthe Hausdorffdimension of a subset of [0,1).

Recall that for any irrational number x ∈ [0,1), p_n(x) andq_n(x) are the numerator and denominator of the n-th convergent of x. It is known that p_n = p_n(x) and q_n = q_n(x) can be obtained recursively by the following relations.

p_n =a_n(x)p_n−1+ p_n−2, q_n= a_n(x)q_n−1+q_n−2 (2.1) with the conventions p0= q−1 =0 and p−1 =q0 =1. For eachn≥1,

p_n−1q_n− p_nq_n−1 =(−1)ⁿ. (2.2) For anyn≥1 and (a₁,a₂,· · ·,a_n)∈Nⁿ, define

I_n(a₁,a₂,· · ·,a_n)=

x∈[0,1) : a₁(x)= a₁,· · ·,a_n(x)= a_n ,

which is the set of points beginning with (a₁,· · · ,a_n) in their continued fraction expansions, and is called acylinderof ordern.

Note thatp_nandq_nare determined by the firstnpartial quotients ofx. So all points inIn(a1,· · · ,an) determine the samepnandqn. Hence sometimes, we write pn = pn(a₁,· · · ,an) and qn = qn(a₁,· · ·,an) to denote pn(x) and q_n(x) forx∈I_n(a₁,· · ·,a_n).

Proposition 2.1([8]). For any n≥ 1and(a₁,· · · ,a_n)∈Nⁿ, let q_nbe given recursively by (2.1). The cylinder I_n(a₁,· · ·,a_n)is an interval with the end- points p_n/q_nand(p_n+ pn−1)/(q_n+qn−1). Then

1 2q²_n ≤

I_n(a₁,· · ·,a_n)

= 1

q_n(q_n+q_n−1) ≤ 1

q²_n. (2.3) For each n ≥1, q_n(a1,· · · ,a_n)≥2^(n−1)/2and

n

Y

k=1

a_k ≤q_n(a₁,· · ·,a_n)≤2ⁿ

n

Y

k=1

a_k. (2.4)

hal-00723315, version 1 - 9 Aug 2012

(5)

Now we mention some known results concerning the dimension of sets in continued fractions. Let {s_n}_n≥1 be a sequence of integers and ℓ ≥ 2 be some fixed integer. Set

F({s_n}^∞_n=1;ℓ)=

x∈[0,1) : s_n ≤a_n(x)< ℓs_n, for alln≥1 . Lemma 2.2 ([3]). Under the assumption that ¹_nPn

k=1s_k → ∞ as n → ∞, one has

dim_HF({s_n}^∞_n=₁;ℓ)=lim inf

n→∞

log(s1s2· · ·s_n) 2 log(s₁s₂· · ·s_n)+logs_n+1. Lemma 2.3([3]).

dim_H (

x∈[0,1) : lim sup

n→∞

logq_n(x)

n = ∞

)

= 1 2.

3. Proof ofTheorem1.1

Notice that E(ψ) = E( ˜ψ) ifψand ˜ψare equivalent. We can assume that ψis increasing because of the following simple lemma.

Lemma 3.1. The set E(ψ) ,∅if and only ifψis equivalent to an increasing function.

Proof. IfE(ψ) is nonempty, take anx0 ∈E(ψ). Then put ψ(n)˜ = j

loga₁(x₀)+· · ·+loga_n(x₀)k

for alln≥1. Clearly ˜ψis increasing. The functionsψand ˜ψare equivalent.

On the other hand, if ψis increasing, we have a pointx∈E(ψ) such that for eachn≥ 1

a_n(x)= ⌊eψ(n)−ψ(n−1)+1⌋.

Now we can proceed the proof of Theorem 1.1 with the assumption that ψis increasing.

•Lower bound. Apply Lemma 2.2 to s_n= ⌊eψ(n)−ψ(n−1)⌋andℓ = 2. Let F =n

x∈[0,1) :j

eψ(n)−ψ(n−1)k

≤a_n(x)<2j

eψ(n)−ψ(n−1)k

, for alln≥1o which is subset ofE(ψ). We get immediately that

dim_HE(ψ)≥ 1 1+b.

•Upper bound. This is the main part of the proof.

Let us first recall the method used in [3] under the extra condition that limn→∞ ψ(n+1)

ψ(n) = b ≥ 1. Especially when b > 1, we constructed a set con- tainingE(ψ) by posing precise restrictions on each partial quotients, namely

nx∈[0,1) :e^Lⁿ ≤a_n(x)≤ e^Mⁿ,whenn≫ 1o

, (3.1)

hal-00723315, version 1 - 9 Aug 2012

(6)

where (with a smallǫ >0) L_n= ψ(n)

1+ǫ − ψ(n−1)

1−ǫ and M_n= ψ(n)

1−ǫ − ψ(n−1) 1+ǫ .

By a standard covering argument, together with lim_n→∞^ψ(n+1)_ψ(n) = b, we get the exact upper bound of the dimension of E(ψ). But as far as a general functionψis concerned, the above argument fails. For example, take

ψ(n)= (k+2)!, whenk!≤n< (k+1)!.

Then the set in (3.1) reads as (

x∈[0,1) :

( e^c¹^(k+2)! ≤an(x) ≤e^c²^(k+2)!, whenn=k!;

1≤ a_n(x)≤ e^c³^(k+2)!, whenk!<n< (k+1)!.

)

for suitably chosen constants c₁,c₂,c₃. According to Lemma 2.2, this set has Hausdorffdimension≥ 1/2. However, the dimension ofE(ψ) is equal to zero by Theorem 1.1.

Now we are going to prove the upper bound of dim_HE(ψ) for a general function ψ. Since ψis increasing, we always have b ≥ 1. We distinguish two cases: b=1 andb> 1.

Case 1. b = 1. Lemma 2.3 serves for this case. According to the estimation (2.4), sinceψ(n)/n→ ∞asn→ ∞, we have

n→∞lim

logq_n(x)

ψ(n) = lim

n→∞

loga₁(x)+· · ·+loga_n(x)

ψ(n) .

Thus Lemma 2.3 gives us

dim_HE(ψ)≤ 1 2 = 1

1+b.

Case 2. b> 1. Fix anǫ >0. Choose a sequence of integers{n_k}^∞_k=1 ⊂ N withn₁large enough and for eachk ≥1 one has

ψ(n_k+1)≥ ψ(n_k)b(1−ǫ), n_k ≤ǫψ(n_k). (3.2) For eachN ≥ 1, let

E_N(ψ)=n

x∈[0,1) : (1−ǫ)< 1 ψ(n)

n

X

j=1

loga_j(x) <(1+ǫ), ∀n≥ No . Then

E(ψ)⊂[

N≥1

EN(ψ).

To estimate the dimension ofE_N(ψ) forN ≥1, we proceed in three steps.

Step i. Find a cover ofEN(ψ). For anyn≥N, set D_n(ǫ)=n

(a1,· · · ,a_n)∈Nⁿ : (1−ǫ)< 1 ψ(n)

n

X

j=1

loga_j <(1+ǫ)o

. (3.3) For every (a₁,· · · ,a_n)∈D_n(ǫ), we define

D_n+1

ǫ; (a₁,· · · ,a_n)

= n

a_n+1 ∈N: (a₁,· · ·,a_n,a_n+1)∈D_n+1(ǫ)o .

hal-00723315, version 1 - 9 Aug 2012

(7)

Clearly, by the definition of Dn(ǫ), we have E_N(ψ)⊂

∞

\

n=N

D_n(ǫ), withD_n(ǫ)= [

(a1,···,an)∈Dn(ǫ)

I_n(a1,· · · ,a_n). (3.4) Now instead of considering the intersections in (3.4) from n = N until n= ∞, we only consider the intersection of two consecutive terms. Namely, for anyn≥ N,

EN(ψ)⊂

D_n(ǫ)∩D_n+1(ǫ)

= [

(a1,···,an)∈D_n(ǫ)

Jn(a1,· · ·,an), where

J_n(a₁,· · ·,a_n)= [

a_n+1∈D_n+1(ǫ;(a1,···,an))

I_n+1(a₁,· · · ,a_n,a_n+1).

Hence, for eachn≥ N, we get a cover ofEN(ψ):

nJ_n(a₁,· · · ,a_n) : (a₁,· · ·,a_n)∈D_n(ǫ)o

. (3.5)

Thus the s-dimensional Hausdorffmeasure ofE_N(ψ) can be estimated as H^s(E_N(ψ))≤ lim inf

n→∞

X

(a1,···,an)∈Dn(ǫ)

J_n(a1,· · · ,a_n)

s. (3.6)

As we shall see, J_n(a₁,· · ·,a_n) is a union of cylinders of order (n+1), say I_n+1(a₁,· · ·,a_n,a_n+1) with a taking large values (Lemma 3.3). Using this fact, the length ofJ_n(a₁,· · · ,a_n) will be well estimated.

Step ii.Lengths ofJ_n(a₁,· · ·,a_n). We begin with a fact onD_n+1(ǫ;a₁,· · ·,a_n).

Lemma 3.2. For each(a₁,· · ·,a_n)∈D_n(ǫ), D_n+1

ǫ; (a₁,· · · ,an) ,∅.

Proof. This follows from the following simple constructions.

(a) IfPn

j=1logaj >(1−ǫ)ψ(n+1), we choosea_n+1 =1.

(b) IfPn

j=1loga_j ≤(1−ǫ)ψ(n+1), we can choose a_n+1= j e^ψ(n+1)

a₁· · ·a_n k.

Recall that the sequence of integers{n_k}_k≥1is given in (3.2).

Lemma 3.3. For any(a₁,· · · ,a_n_k)∈D_n_k(ǫ)and a_n_k₊₁∈D_n_k₊₁ ǫ,(a₁,· · · ,a_n_k) , we have

loga_n_k₊₁ ≥(1−ǫ)b(1−ǫ)² 1+ǫ −1

logq_n_k =:βlogq_n_k, (3.7)

hal-00723315, version 1 - 9 Aug 2012

(8)

Proof. By the definitions of Dn(ǫ) and the first inequality in (3.2), for any (a₁,· · ·,a_n_k)∈D_n_k(ǫ) anda_n_k₊₁ ∈D_n_k₊₁ ǫ,(a₁,· · ·,a_n_k)

, one has

nk+1

X

j=1

loga_j ≥ ψ(n_k+1)(1−ǫ)≥ψ(n_k)b(1−ǫ)²

≥ b(1−ǫ)² 1+ǫ

nk

X

j=1

loga_j. (3.8) On the other hand, by (2.4) and the second inequality in (3.2), we get

q_n_k(a₁,· · · ,a_n_k)≤2ⁿ^k

nk

Y

j=1

a_j ≤







nk

Y

j=1

a_j







1 1−ǫ

. (3.9)

Combining (3.8) and (3.9), we obtain the desired result.

Now return back to the cover of EN(ψ) given in (3.5) especially when n = nk. We estimate the length of Jnk(a₁,· · ·,ank) for every (a₁,· · · ,ank) ∈ D_n_k(ǫ). Forn=n_k, by (3.7) and Proposition 2.1, we have

J_n(a₁,· · · ,a_n) ≤ X

a:a≥q^β_n

a· p_n+pn−1

a·q_n+q_n−1 − (a+1)p_n+ pn−1

(a+1)q_n+q_n−1 .

By (2.2), for alla∈N, the differences appearing in the series have the same sign depending only the parity of n. Thus the series is telescopic. Since

(a+1)pn+pn−1

(a+1)qn+qn−1 tends to p_n/q_nasa→ ∞, we get

Jn(a1,· · ·,an) ≤

q^β_npn+pn−1

q^β_nq_n+qn−1

− pn

q_n

= 1

(q^β_nq_n+qn−1)q_n ≤ 1 q²_n^+β. Consider the liminf in (3.6) along the subsequence{n_k}_k≥1, then we obtain

H^s(E_N(ψ))≤lim inf

k→∞

X

(a1,···,a_nk)∈D_nk(ǫ)

1 q_n_k

!s(2+β)

. (3.10)

The last step is devoted to estimating the summation in (3.10) under a suitable choice ofs.

Step iii. Bernoulli measures. A family of measuresµ_t defined on cylinders is constructed firstly. For each t > 1 and for any (a₁,· · ·,a_n) ∈ Nⁿ, set

µ_t(I_n(a₁,· · ·,a_n))=e^−nP(t)−t^Pⁿ^j=1^log^a^j, (3.11) wheree^P(t) = ζ(t) =

∞

P

k=1

k^−t. By Kolmogorov’s consistency theorem, µ_t can be extended into a probability measure on [0,1).

Fix ǫ > 0. By the assumption that limn→∞ψ(n)/n = ∞, one can choose some integerN(ǫ)∈Nsuch that for alln≥ N(ǫ),

nP 1+ ǫ 2

≤ ǫ

2(1−ǫ)ψ(n). (3.12)

hal-00723315, version 1 - 9 Aug 2012

(9)

We claim that for eachn≥N(ǫ) and (a₁,· · ·,an)∈Dn,

q_n^−(1+ǫ) ≤ µ_(1+ǫ/2) I_n(a₁,· · ·,a_n). (3.13) More precisely, for any (a₁,· · ·,a_n)∈ D_n, by (3.3) and (3.12), we have

ǫ 2

n

X

j=1

logaj ≥ nP(1+ ǫ

2). (3.14)

Thus by (2.4) and then (3.14), we get

qn−(1+ǫ) ≤e^−(1+ǫ)^Pⁿ^j=1^log^a^j ≤e^−nP(1+^ǫ²⁾⁻⁽¹⁺^ǫ²⁾^Pⁿ^j=1^log^a^j. Choose s= ^1+ǫ_2+β in (3.10). By (3.13), we have

H^1+ǫ^2+β(E_N(ψ))≤lim inf

k→∞

X

(a1,···,a_nk)∈D_nk(ǫ)

µ_(1+ǫ/2) I_n_k(a₁,· · ·,a_n_k)≤ 1.

Hence

dim_HE(ψ)≤sup

N≥1

ndim_HE_N(ψ)o

≤ 1+ǫ 2+β.

Then the desired result follows by lettingǫ → 0.

Final remark: Now we give a remark on the dimension ofJ_β^∗and that ofJ_β. Recall that J_β^∗ and J_β are defined in Section 1. For any (a₁,· · · ,a_n) ∈ Nⁿ, we define

J˜_n(a₁,· · ·,a_n)= [

a_n+1≥q^β_n

I_n+1(a₁,· · ·,a_n,a_n+1).

Then it is clear that J_β =

∞

\

N=1

∞

[

n=N

[J˜_n(a₂,· · ·,a_n),

where the last union is taken over all (a₁,· · ·,a_n)∈Nⁿ.While J_β^∗⊂

∞

\

N=1

∞

[

n=N

[J˜n(a2,· · ·,an),

where the last union is taken over all (a₁,· · ·,a_n) ∈ Nⁿ with ^log_n^qⁿ being sufficiently large. As a result,

H^s(J_β)≤lim inf

N→∞

∞

X

n=N

X

(a1,···,an)∈Nn

1 q_n

!s(2+β)

,

while

H^s(J_β^∗)≤ lim inf

N→∞

∞

X

n=N

X

(a1,···,an)∈Nⁿ,(logqn)/nlarge

1 qn

!s(2+β)

. By (2.3), we know that

1≤ X

a1,···,an∈Nⁿ

q_n⁻² ≤2. (3.15)

hal-00723315, version 1 - 9 Aug 2012

(10)

While, by (3.13), we get

X

a1,···,an:logqn/nlarge

qn−(1+ǫ)≤ 1. (3.16) Comparing of (3.15) and (3.16) reveals that

dim_HJ_β ≤ 2

2+β, dim_HJ_β^∗ ≤ 1 2+β.

Actually we have proven that dim_HJ_β^∗ = _2+β¹ since E(ψ) can serve as a subset of J_β^∗.

Acknowledgement: This work was partially supported by PICS program No. 5727, RFDP20090141120007, NSFC 10901066 and NSFC 11171124.

The authors thank the Morningside Center of Mathematics, Beijing for its hospitality.

References

[1] Billingsley, P., Henningsen, I.: Hausdorffdimension of some continued-fraction sets.

Z. Wahrscheinlichkeitstheorie verw. Geb. 31, 163-173 (1975)

[2] Fan, A.-H., Liao, L.-M., Ma, J.-H.: On the frequency of partial quotients of regular continued fractions. Math. Proc. Camb. Phil. Soc. 148, 179-192 (2010)

[3] Fan, A.-H., Liao, L.-M., Wang, B. W., Wu, J.: On Kintchine exponents and Lyapunov exponents of continued fractions. Ergod. Th. Dynam. Sys. 29, 73-109 (2009) [4] Jarn´ık, I.: Zur metrischen Theorie der diopahantischen Approximationen. Proc. Mat.

Fyz. 36, 91-106 (1928)

[5] Jaerisch, J., Kesseb¨ohmer, M.: The arithmetic-geometric scaling spectrum for continued fractions. Arkiv f¨or Matematik 48 (2), 335-360 (2010)

[6] Kesseb¨ohmer, M., Stratmann, S.: A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates. J. Reine Angew. Math. 605, 133- 163 (2007)

[7] Kifer, Y., Peres, Y., Weiss, B.: A dimension gap for continued fractions with inde- pendent digits. Israel J. Math. 124(1), 61-76 (2001)

[8] Khintchine, A. Ya.: Continued Fractions. P. Noordhoff, Groningen, The Netherlands (1963)

[9] Liao, L.-M., Ma, J.-H., Wang, B.-W.: Dimension of some non-normal continued fraction sets. Math. Proc. Cambridge Philos. Soc. 145 (1), 215-225 (2008)

[10] Mauldin, R. D., Urba´nski, M.: Conformal iterated function systems with applications to the geometry of continued fractions. Trans. Amer. Math. Soc. 351 (12), 4995-5025 (1999)

[11] Mayer, D.: On the thermodynamics formalism for the Gauss map. Comm. Math.

Phys. 130, 311-333 (1990)

[12] Olsen, L.: Extremely non-normal numbers. Math. Proc. Cambridge Philos. Soc. 137 (1), 43-53 (2004)

[13] Pesin, Y.: Dimension Theory in Dynamical Systems: Contemporary Views and Applications. Chicago Lectures in Mathematics, The University of Chicago Press, Chicago (1998)

[14] Pollicott, M., Weiss, H.: Multifractal analysis of Lyapunov exponent for continued fraction and Manneville-Pomeau transformations and applications to Diophantine approximation. Comm. Math. Phys. 207 (1), 145-171 (1999)

hal-00723315, version 1 - 9 Aug 2012

(11)

Lamfa, Umr7352 (Ex6140), CNRS, Universite de´ PicardieJulesVerne, 33, RueSaint Leu, 80039 AmiensCedex1, France

E-mail address:[email protected]

Lama, Umr8050, CNRS, Universite´Paris-EstCreteil´ Val deMarne, 61,avenue du G´eneral de´ Gaulle94010 Cr´eteilCedexFrance

School ofMathematics andStatistics, HuazhongUniversity ofScience andTechnol- ogy, 430074 Wuhan, China

E-mail address:bwei [email protected]

School ofMathematics andStatistics, HuazhongUniversity ofScience andTechnol- ogy, 430074 Wuhan, China

hal-00723315, version 1 - 9 Aug 2012