Research Article Differentiability Properties of the Pre-Image Pressure

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Discrete Dynamics in Nature and Society Volume 2012, Article ID 951691,14pages doi:10.1155/2012/951691

Research Article

Differentiability Properties of

the Pre-Image Pressure

Kesong Yan,

1, 2

_{Fanping Zeng,}

2, 3

_{and Gengrong Zhang}

3

1_{Department of Mathematics, University of Science and Technology of China, Hefei,}

Anhui 230026, China

2_{Department of Mathematics and Computer Science, Liuzhou Teachers College, Liuzhou,}

Guangxi 545004, China

3_{Institute of Mathematics, Guangxi University, Nanning, Guangxi 530004, China}

Correspondence should be addressed to Kesong Yan,[email protected]

Received 8 March 2012; Accepted 26 March 2012 Academic Editor: M. De la Sen

Copyrightq2012 Kesong Yan et al. This is an open access article distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the diﬀerentiability properties of the pre-image pressure. For a TDSX, Twith finite

topological pre-image entropy, we prove the pre-image pressure functionPpreT,• is Gateaux

diﬀerentiable atf ∈ CX,Rif and only ifPpreT,•has a unique tangent functional atf. Also,

we obtain some equivalent conditions forPpreT,•to be Fr´echet diﬀerentiable atf.

1. Introduction

By a topological dynamical systemfor short TDS, we mean a pairX, TwhereXis a com-pact metric space and T : X → X is a continuous surjection from X to itself. Entropies are fundamental to our current understanding of dynamical systems. The classical measure-theoretic entropy for an invariant measure and the topological entropy were introduced in1,

2, respectively, and the classical variational principle was completed in 3,4. Topological

entropy measures the maximal exponential growth rate of orbits for arbitrary topological dynamical systems, and measure-theoretic entropy measures the maximal loss of information for the iteration of finite partitions in a measure-preserving transformation.

Topological pressure is a generalization of topological entropy for a dynamical system. The notion was first introduced by Ruelle5in 1973 for an expansive dynamical system and later by Walters6for the general case. The theory related to the topological pressure, varia-tional principle, and equilibrium states plays a fundamental role in statistical mechanics, ergodic theory, and dynamical systemssee, e.g., the books7–12. Since the works of Bowen

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related to dynamical systems. One of the basic questions of physical interest in that of dif-ferentiability of the pressure. The diﬀerentiability of the pressure was considered by many peoplesee, e.g.,15–18.

Recently, the pre-image structure of maps has become deeply characterized via en-tropies. In several papers see 19–24, some important pre-image entropy invariants of

dynamical systems have been introduced and their relationships with topological entropy have been established. In a certain sense, these new invariants give a quantitative estimate of how “noninvertible” a system is. In25, we defined the topological pre-image pressure of

topological dynamical systems, which is a generalization of the Cheng-Newhouse pre-image entropy see 19, and proved a variational principle for it. We gave some applications

of the pre-image pressure to equilibrium statessee 25,26. Under the assumption that

hpreT <∞and the metric pre-image entropy functionh{pre,•}Tis upper semicontinuous,

we obtained a way to describe a kind of continuous dependence of equilibrium states. Also, we proved that the set of all continuous functions with unique equilibrium states is a denseGδ-set ofCX,R, and for any finite collection of ergodic measures, we can find some

continuous function such that its set of equilibrium states contains the given setsee26.

The purpose of this paper is to study the diﬀerentiability properties of the pre-image pressure of the TDS X, Twith finite topological pre-image entropy. InSection 2, we con-centrate on reviewing some basic definitions and give some basic properties of tangent functionals to the pre-image pressure.

InSection 3, the Gateaux diﬀerentiability of the pre-image pressure is discussed. We

show that the pre-image pressure functionPpreT,•is Gateaux diﬀerentiable atf ∈CX,R

if and only if it has a unique tangent functional atf.

In Section 4, we discuss the Fr´echet diﬀerentiability of the pre-image pressure. We

obtain some equivalent conditions forPpreT,• to be Fr´echet diﬀerentiable atf. Also, we

show that the pre-image function PpreT,•is Fr´echet diﬀerentiable if and only if X, Tis

uniquely ergodic, and hence the pre-pressure is linear.

2. Preliminaries

Throughout the paper, letX, Tbe a TDS with finite topological pre-image entropyhpreT

see19for definition. In this section, we will recall some basic definitions and give some

useful properties.

LetX, Tbe a TDS and letBXbe the collection of all Borel subsets ofX. Recall that a cover of X is a family of Borel subsets of X whose union isX. An open cover is one that consists of open sets. A partition ofXis a cover ofXconsisting of pairwise disjoint sets. We denote the set of finite covers, finite open covers, and finite partition ofXbyCX,Co_X, andPX, respectively. Given two coversU,V,Uis said to be finer thanVdenoted byU Vif each element ofUis contained in some element ofV. We setU ∨ V{U∩V :U∈ U, V ∈ V}.

Denote by CX,R the Banach space of all continuous, real-valued functions on X endowed with the supremum norm. Forf∈CX, Randn∈N, we denoten−1_i0 fTi_x_by

Snfx.

2.1. Topological Pre-Image Pressure

In an early paper with Zeng et al. 25, following the idea of topological pressure see

Chapter 9,12, we defined a new notion of pre-image pressure, which extends the

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T is a mapPpreT,•:CX,R → Rwhich is convex, Lipschitz continuous, increasing, with

PpreT,0 hpreT. More precisely, letU ∈ C_Xo. Forx∈Xandk∈N, we put

Pn

T, f,U, T−kx:inf

V B∈V

sup

y∈Be

Snfy_,

2.1 where the infimum is taken over all finite subcoversVofn−1_i0T−iUwith respect toT−kx. We define the pre-image pressure ofT related toUatfas

Ppre

T, f,U: lim

n→ ∞

1

nlogPpre,n

T, f,U, 2.2

wherePpre,nT, f,U : sup_{x∈X, k}_nPnT, f,U, T−kx. The pre-image pressure of T atf is

de-fined by

Ppre

T, f: sup

U∈Co

X

Ppre

T, f,U. _2.3

It is clear thatPpreT, f PT, f topological pressure, see12and PpreT,0 hpreT.

PpreT, ffifTis a homeomorphism.

2.2. Measure-Theoretic Pre-Image Entropy

Denote byMX,MX, T, andMe_{X, T}_{the set of all Borel probability measures,}_T-invariant

Borel probability measures andT-invariant ergodic measures, onX, respectively. Note that

MX, T ⊇ Me_{X, T}_/_∅_{, and both} _M_X _and _M_{X, T} _{are convex compact metric spaces}

when endowed with the weak∗-topology; Me_{X, T} _{is a} _G

δ subset of MX, T see 12,

Chapter 6. Beside the weak∗_{-topology on}_M_{X, T, we also have the norm topology arising}

from the metric:

μ−ν :sup

gdμ−

gdν:g∈CX,R, g 1

. 2.4

Note thatμ−ν2 ifμ /ν∈ Me_{X, T}_.

Givenα∈ PX,μ∈ MXand a sub-σ-algebraA ⊆ BX, define Hμα| A:

A∈α

X

−E1A| AlogE1A| Adμ, 2.5

whereE1A | Ais the conditional expectation of 1A with respect toA. It is a standard fact

thatHμα| Aincreases with respect toαand decreases with respect toA. Note thatβ∈ PX

naturally generates a sub-σ-algebraFβofBX; where there is no ambiguity, we writeFβ asβ. It is easy to check, forα, β∈ PX, thatHμα|β Hμα∨β−Hμβ. More generally, for

a sub-σ-algebraA ⊆BX, we have Hμ

α∨β| AHμ

β| AHμ

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Whenμ∈ MX, TandA ⊆ BXis aT-invariant sub-σ-algebra, that is,T−1AAup toμ-null sets, it is not hard to see thatan Hμ

_n−1

i0T−iα| Ais a nonnegative subadditive

sequence for a givenα∈ PX, that is,anm≤anamfor all positive integersnandm. It is well

known that

lim

n→ ∞

an

n infn≥1

an

n. 2.7 The conditional entropy ofαwith respect toAis then defined by

hμT, α| A: lim

n→ ∞

1 nHμ

_n−1

i0

T−iα| A

inf

n≥1

1 nHμ

_n−1

i0

T−iα| A

. 2.8

Moreover, the metric conditional entropy of X, Twith respect toAis defined by hμT, X| A sup

α∈PXhμT, α| A. 2.9

Note that ifN{∅, X}is a trivial sub-σ-algebra, we recover the metric entropy, and we write hμT, α| NandhμT, X| Nsimple byhμT, αandhμT.

Particularly, setB−∞_n0T−nBX, thenB−is aT-invariant sub-σalgebra. We callB− the infinite pastσ-algebra related to BX. We define the measure-theoretic (or metric) pre-image

entropy ofαwith respect toX, Tby

hpre,μT, α:hμ

T, α| B− lim

n→ ∞

1 nHμ

_n−1

i0

T−iα| B−

. 2.10

Moreover, we define the metric pre-image entropy of X, Tby

hpre,μT:sup

α∈PXhpre,μT, α. 2.11

2.3. A Variational Principle for Pre-Image Pressure

The following variational relationship for topological pre-image pressure and measure-theo-retic pre-image entropy is established in25.

Theorem 2.1. LetX, Tbe a TDS andf∈CX,R. Then,

Ppre

T, f sup

μ∈MX,T

hpre,μT

X

fdμ

. _2.12

We also havesee, e.g.,26the following proposition.

Proposition 2.2. Let X, T be a TDS,α ∈ PX and μ ∈ MX, T. Then,μ → hpre,μT, αand

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μ_Me_X,Tθdλθ, then

hpre,μT, α

Me_X,T

hpre,θT, αdλθ, hpre,μT

Me_X,T

hpre,θTdλθ. 2.13

2.4. Equilibrium States and Tangent Functionals to Pre-Image Pressure

Givenf∈CX,R. A finite signed Borel measureμonX,BXis called a tangent functional toPpreT,•atfif

Ppre

T, fg−Ppre

T, f

gdμ, ∀g∈CX,R. 2.14

LetTfX, Tdenote the collection of all tangent functionals toPpreT,•atf. An application

of the Hahn-Banach theorem givesTfX, T/∅. It is easy to see thatμ∈ TfX, Tif and only if

Ppre

T, f−

fdμinf

Ppre

T, g−

gdμ:g ∈CX,R

. 2.15

Also, we haveTfX, T⊆ MX, T see25for details.

Theorem 2.3. The following holds.

1Forf∈CX,R,

TfX, T ∞

n1

μ∈ MX, T:hpre,μT

fdμ > Ppre

T, f− 1 n

. 2.16

2Iff1, f2∈CX,Randμ∈ Tf1X, T∩ Tf2X, T, then

Ppre

T, pf1

1−pf2

Ppre

T, f1

1−p f1−f1

dμ, ∀p∈0,1, 2.17

andTpf11−pf2X, T⊆ Tf1X, T∩ Tf1X, T.

Proof. 1 Let μ ∈ If ≡

_∞

n1{μ∈ MX, T:hpre,μT

fdμ > PpreT, f−1/n} . By

Theorem 2.1, there is μn ∈ MX, T with μn → μand hpre,μnT

fdμn → PpreT, f.

Hence, for eachg∈CX,R,

Ppre

T, fg−Ppre

T, f lim

n→ ∞

hpre,μnT fg

dμn−Ppre

T, f

gdμ, 2.18

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which follows thatμ∈ TfX, T. Now suppose there isμ0∈ TfX, T\If. SinceIf is convex,

the standard separation theorem27, page 417follows that there existsg∈CX,Rwith

gdμ0 >sup

gdμ:μ∈If

. 2.19

ByTheorem 2.1, we can chooseμn∈ MX, Tsuch that

hpre,μnT f

g n

dμn> Ppre

T, fg n

− 1

n2. 2.20

Without loss of generality, we can assumeμn → μ∗. Then,

gdμ0n

g

ndμ0n

Ppre

T, f g n

−Ppre

T, f < n

hpre,μnT f

g n

dμn

1

n2 −hpre,μnT−

fdμn

gdμn

1 n →

gdμ∗.

2.21

However,μ∗∈If follows from the fact:

hpre,μnT

fdμn> Ppre

T, fg n

−

g n dμn−

1 n2

by2.20

> Ppre

T, f−2· g n −

1 n2,

by25, Lemma 4.13,

2.22

which is a contradiction. 2If 0< p <1, then

pPpre

T, f1

1−pPpre

T, f2

Ppre

T, pf1

1−pf2 by25, Lemma 4.13

Ppre

T, f1

1−pf2−f1

Ppre

T, f1

1−p f2−f1

dμ sinceμ∈ Tf1X, T.

2.23

Hence,

Ppre

T, f2

−

f2dμPpre

T, f1

−

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By symmetry,PpreT, f2−

f2dμPpreT, f1−

f1dμ, which implies

Ppre

T, f1

1−p f2−f1

dμpPpre

T, f1

1−pPpre

T, f2

Ppre

T, pf1

1−pf2

. 2.25 A memberμ∈ MX, Tis called an equilibrium state forPpreT,•atfif

Ppre

T, fhpre,μT

fdμ. 2.26

LetMfX, Tdenote the collection of all equilibrium states forPpreT,•atf.

The setTfX, Tis convex and compact in the weak∗-topology. The setMfX, Tis convex but it may be not closed in the weak∗-topology. Note thatMfX, T ⊆ TfX, T ⊆

MX, T and MfX, T could be empty see Example 5.1, 25. We also have MfX, T TfX, Tif and only if the metric pre-image entropy maphpre,•Tis upper semicontinuous at

every element ofTfX, T, Theorem 5.225. The extreme points ofMfX, Tare precisely the ergodic members ofMfX, Tand ifμ∈ MfX, T, then almost every measure in the ergodic decomposition ofμis a member ofMfX, T see Proposition 2.1,26. When the metric

pre-image entropy maph_pre,•Tis upper semicontinuous onMX, T, then_f_∈CX,_RMfX, Tis dense inMX, Tin the norm topology, and given any finite collection of ergodic measures

{μ1, . . . , μn}, there is somef∈CX,Rsuch that{μ1, . . . , μn} ⊆ MfX, T 26, Theorem 4.2.

The following theorem shows when tangent functionals to pre-image pressure are not equilibrium states.

Theorem 2.4. LetX, Tbe a TDS andf ∈CX,R. The following statements are mutually

equiva-lent:

1μ∈ TfX, T\ MfX, T;

2hpre,μT

fdμ < PpreT, f and there exist{μn}_n1∞ ⊆ MX, Twithμn → μand

hpre,μnT

fdμn → PpreT, f;

3μ∈ TfX, Tandhpre,•Tis not upper semicontinuous atμ.

Proof. 1⇒2Follows from the variational principle andTheorem 2.31.

2⇒3ByTheorem 2.31,μ∈ TfX, T. Ifhpre,•Tis upper semicontinuous atμ, then

Ppre

T, f lim

n→ ∞

hpre,μnT

fdμn

hpre,μT

fdμ. 2.27

Hence,μ∈ MfX, Tby the variational principle.

3⇒1If 3 holds, then there areμn ∈ MX, Twith μn → μand limn→ ∞hpre,μnT >

hpre,μT. Hence,

Ppre

T, f lim

n→ ∞

hpre,μnT

fdμn

> hpre,μT

fdμ. 2.28

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3. Gateaux Differentiability of the Pre-Image Pressure

In26, we studied the uniqueness of the equilibrium state for the pre-image pressure. We

showed that when the metric pre-image entropy maph_pre,•Tis upper semicontinuous on

MX, T, then the set of all functions with unique equilibrium state is dense in CX,R. Without the upper semicontinuity assumption, one can show that all functions with unique tangent functional are dense inCX,R can see27, page 450or11, Appendix A.3.6. In

this section, we will show a continuous function with unique tangent functional to pre-image pressure if and only if it is Gateaux diﬀerentiable.

Given f, g ∈ CX,R. Since PpreT,• is convex, the map t → PpreT, f tg −

PpreT, f/tis increasing and hence

dPpre

T, fg lim

t→0

Ppre

T, ftg−Ppre

T, f t , d−Ppre

T, fg lim

t→0−

Ppre

T, ftg−Ppre

T, f t

3.1

exist. Note that dPpreT, fg −d−PpreT, f−g. The pre-image pressure function

PpreT,•is said to be Gateaux diﬀerentiable atfif, for allg∈CX,R,

lim

t→0

Ppre

T, ftg−Ppre

T, f

t 3.2 exist. It is easy to check that PpreT,• is Gateaux diﬀerentiable at f if and only if g →

dPpreT, fgis linear.

Lemma 3.1. LetX, Tbe a TDS andf, g∈CX,R. Then,

dPpre

T, fgsup

gdμ:μ∈ TfX, T

. 3.3

Proof. Ifμ∈ TfX, T, then, forg ∈CX,R,

gdμPpre

T, ftg−Ppre

T, f

t , ∀t >0. 3.4 Hence,

sup

gdμ:μ∈ TfX, T

dPpre

T, fg. 3.5

Next, we prove the converse inequality. Seta dPpreT, fg. Define a continuous linear

functionalγ:{tg:t∈R} → Rby

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The convexity ofPpreT,•implies

γtgt·dPpre

T, fgPpre

T, ftg−Ppre

T, f, ∀t∈R. 3.7 By the Hahn-Banach theorem,γcan be extended to a continuous linear functional onCX,R such that

γhPpre

T, fh−Ppre

T, f, ∀h∈CX,R. 3.8 By the Riesz representation theorem, there isμ∈ MXwith

γh

hdμ, ∀g ∈CX,R. 3.9

Combining3.6,3.8, and3.9, we haveμ∈ TfX, T, and

gdμγgadPpre

T, fg. 3.10

The lemma is proved.

Theorem 3.2Uniqueness of tangent functional and Gateaux diﬀerentiability. The following

statements are mutually equivalent:

1the pre-image pressure functionPpreT,•is Gateaux diﬀerentiable atf∈CX,R;

2the unique tangent functional toPpreT,•atfisμ;

3for eachg ∈CX,R,

lim

t→0

Ppre

T, ftg−Ppre

T, f t

gdμ. 3.11

Proof. 1⇒2If the pre-image pressure functionPpreT,•is Gateaux diﬀerentiable at f,

then the functiong→dPpreT, fgis linear. ByLemma 3.1,

sup

g dμ:μ∈ TfX, T

dPpre

T, fg −dPpre

T, f−g −sup

−g dμ:μ∈ TfX, T

inf g dμ:μ∈ TfX, T

3.12

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2⇒3It directly follows fromLemma 3.1. 3⇒1It follows from the definition.

4. Fr ´echet Differentiability of the Pre-Image Pressure

In this section, we will study the Fréchet differentiability of image pressure. The pre-image pressure functionPpreT,•is said to be Fréchet differentiable atfif there isγ∈CX,R∗

such that

lim

g→0

_P_pre

T, fg−Ppre

T, f−γg

_g 0. 4.1

The pre-image pressure functionPpreT,• is said to be Fréchet differentiable if it is Fréchet

diﬀerentiable at eachf∈CX,R.

Note that ifPpreT,•is Fréchet differentiable atf, then it is Gateaux differentiable at

fandγg gdμ, whereμis the unique tangent functional toPpreT,•atf.

Theorem 4.1. The following conditions are mutually equivalent:

1PpreT,•if Fr´echet diﬀerentiable atf;

2Tf X, T {μf} and μn −μf → 0 for each {μn} ⊆ MX, T with hpre,μnT

fdμn → PpreT, f;

3PpreT,•is locally aﬃne atf;

4TfX, T {μf}and

lim

g→0sup

μ−μf :μ∈ TfgX, T

0. _4.2

Proof. 1⇒2SupposePpreT,•is Fr´echet diﬀerentiable atf. Then,fhas a unique tangent

functionalμf toPpreT,•at f. Let{μn}∞n1 ⊆ MX, Twithhpre,μnT

fdμn →

PpreT, f. Given >0, there areN∈Nandδ >0 such that

Ppre

T, fhpre,μnT

fdμnδ, ∀nN, 4.3

0Ppre

T, fg−Ppre

T, f−

gdμf · g whenever g < δ. 4.4

Hence, ifnNandg< δ, then

gdμn−

gdμPpre

T, f

gdμn−Ppre

T, f−

gdμf

hpre,μnT

fgdμnδ−Ppre

T, f−

g dμf

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Ppre

T, fg−Ppre

T, f−

g dμf δ

by Theorem 2.1 · g δ2δ by 4.4.

4.5 Note that4.5is also true when−ginstead ofg. So,

g dμn−

g dμ2δ whenever g < δ, nN. 4.6

Therefore,

_μ_n₋_μ_f _sup _g _dμ_n₋ _g _dμ_f_: _g _<₁

1 δsup

g dμn−

g dμf

: g < δ

2 by 4.6.

4.7

By arbitrary of,μn−μf → 0.

2⇒3By the variational principle of pre-image pressure, we can pick ergodic measures μn such thathpre,μnT

fdμn → PpreT, f, Thenμn−μf → 0. Note that two

distinct ergodic measures have norm-distance 2. So there isNsuch thatμnμf for

allnN. Hence,μf ∈ MeX, Tand

:Ppre

T, f−sup

hpre,μT

f dμ:μ∈ MeX, T, μ /μf

>0. 4.8

This implies for eachg∈CX,R, withf−g< /2, that sup

hpre,μT

g dμ:μ∈ MeX, T, μ /μf

Ppre

T, f− f−g Ppre

T, g−2 f−g < Ppre

T, g.

4.9

By the variational principle of pre-image pressure again, we have

Ppre

T, ghpre,μfT

gdμf whenever f−g <

2. 4.10 Hence,PpreT,•is aﬃne on the neighborhoodBf, /2 {g∈CX,R:f−g< /2}off.

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3⇒4Is obvious.

4⇒1Letg∈CX,Randμ∈ TfgX, T. By definition, we have

Ppre

T, f−Ppre

T, fg−

gdμ. 4.11

Hence,

0Ppre

T, fg−Ppre

T, f−

g dμf

g dμ−

g dμf

by4.11 g · μ−μf .

4.12

Therefore,

0 Ppre

T, fg−Ppre

T, f−gdμf

g sup μ−μf :μ∈ TfgX, T

−→0 4.13

asg → 0. That isPpreT,•is Fr´echet diﬀerentiable atf.

Corollary 4.2. LetX, Tbe a TDS with finite pre-image entropy. Then,PpreT,•is Fr´echet diﬀ

e-rentiable if and only ifTis uniquely ergodic.

Proof. UsingTheorem 4.1,PpreT,•is locally aﬃne wheneverPpreT,•is Fr´echet

diﬀeren-tiable. Hence, the mapf ∈ CX,R → μf ∈ MX, Tis locally constant, whereTfX, T {μf}for eachf ∈CX,R. SinceCX,Ris connected, the map is constant. Soμf μ0for all

f∈CX,R. Ifμ∈ MX, T\ {μ0}, then we can choosef∈CX,Rsuch thatfdμ >fdμ0.

Then for suﬃciently largek, we have Ppre

T, kfhpre,μT

kfdμ > hpre,μ0T

kfdμ0Ppre

T, kf, 4.14

which is a contradiction. Therefore,MX, T {μ0}.

Remark 4.3. In the situation ofCorollary 4.2, there is only one invariant measure, and the

pre-image pressure is the expectation with respect to this measure, hence, it is linear.

Acknowledgments

The authors would like to thank the referee for many valuable and constructive comments and suggestions for improving this paper. The first and second authors are supported by the National Natural Science Foundation of China no. 11161029 and the Department of

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Education Research Foundation of Guangxi Province nos. 200911MS298, 200103YB157; the third author is supported by the National Natural Science Foundation of China no. 11126342 and the Natural Science Foundation of Guangxi Province nos. 0897012, 2010-GXNSFA013109.

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