For example, taking d=1, B ={0,1}, h(x0,x1)=(x1?x0)2/2+K cos 2πx0, we have the classical FK model[1].In this paper we always take B =Br0={k ∈Zd|‖k‖≤r}, where r >0 is an integer and ‖·‖ denotes a norm on lattice Zddefined as in Section 2.

Like the classical FK model, we will study minimizers for the local potential h (see the definition in Section 2).Unlike the classical FK model,for which all minimizers are Birkhoff[1,3], minimizers for the high-dimensional FK model or even for monotone recurrence relations may not be Birkhoff; see [20] for an example.A configuration x is said to be Birkhoffif the translation{τk,lx|k ∈Zd,l ∈Z} of x is totally ordered(see Section 2 for the definition).Each Birkhoffconfiguration on Zdhas a rotation vector which turns out to be the rotation number for d=1.

Let Mωdenote the set of all Birkhoffminimizers with rotation vector ω ∈Rd.Then the Aubry-Mather theory for the high-dimensional FK model tells us that Mω/=?(see,for example,[5, 6, 18]) if the local potential h satisfies the hypotheses (H1)–(H3) specified in Section 2.

According to Moser [17](the case of partial differential equations),Mωis called a minimal foliation if p0(Mω) = R, where p0is the projection on 0 - site, i.e., p0(x) = x0; otherwise, it is called a minimal lamination.For the classical FK model or monotone twist maps, minimal foliations correspond to invariant circles which play an important role in the discussion of monotone twist maps.

For the high-dimensional FK model, the rotation vector is not enough to distinguish different minimizers.Therefore we introduce secondary invariants of Birkhoffminimizers with the same rotation vector [16] as Bangert used in [4].

If we apply an external driving force F to each particle of the chain of the classical FK model,then the equilibrium states may disappear and the chain may begin to slide if the driving force increases beyond a critical value, referred to as the depinning force Fd(ω), depending on the mean spacing ω of particles.It was proven in[23]that the depinning force can be used as a criterion for the existence of invariant circles for the one-dimensional nearest neighbor coupled FK model.That is,the depinning force Fd(ω)equals zero if and only if there exists an invariant circle with rotation number ω.We extended these results to monotone recurrence relations with finite range interactions [26].

In [26], we considered the tilted FK model

We will consider the tilted FK model with i ∈Z in the above equation replaced by i ∈Zd,and define the depinning force via the average velocity, just as the authors Qin and Wang did in [23].

Using the depinning force Fd(ω), which depends upon the rotation vector ω, we obtain the following conclusions (note that ˉa1=‖ˉω‖?1ˉω and ˉω =(?ω,1)∈Rd+1):

Theorem A For ω ∈Rd, Fd(ω)=0 implies that p0(M(ˉa1))=R; that is, that M(ˉa1) is a foliation.

2 Preliminaries

We denote the support of a configuration v = (vj) by supp(v) = {j ∈Zd|vj/= 0}.Let VA={v|supp(v)?int(A)}.

Definition 2.3 A configuration x is called a minimizer for the local potential h (or for the Lagrangian W) if, for every finite subset A ?Zdand every v ∈VA,

3 Invariant Strictly Ordered Circles

In this section, we shall introduce strictly ordered circles and construct a strictly ordered circle with all arbitrary rotation vector which is invariant for the gradient flow and translations.

Definition 3.1Let g :R →RZdbe continuous and satisfy the following:

(1) Periodicity: g(s+1)=g(s)+1.

(2) Monotonicity: if s1

Then the image ?=g(R)is called an ordered circle and g is called a parameterization of ?.If g satisfies (1) and strict monotonicity i.e., g(s1)?g(s2) for s1

Then ?=g(R) is a strictly ordered circle with a rotation vector ω.Moreover,? is invariant for translations.

3.1 Strictly Ordered Circle with Rational Rotation Vector

where ?(·) represents the number of elements in the set.

Lemma 3.3For each ? ∈?, there is a unique standard parametrization.

ProofLet g be a parametrization of ?.We set

Then F is strictly increasing and satisfies that F(s+1) = F(s)+1.Let F?1be the inverse function of F.It is easy to see that F?1(s+1)=F?1(s)+1.It follows that ˉη(s)=g(F?1(s))is a standard parametrization of ?.

ProofLet ?g1,?g2∈ ?H, and λ ∈[0,1].There exist g1,g2∈H such that p(gi) = ?gi,i = 1,2.Let gλ(s) = λg1(s)+(1 ?λ)g2(s).It is easy to check, from Remark 3.4, that gλis the standard parametrization of some ordered circle in BP,qand that gλis still invariant for translations.Then, gλ∈H and p(gλ)=λp(g1)+(1 ?λ)p(g2)=λ?g1+(1 ?λ)?g2∈ ?H.Thus,?H is convex.

3.2 Strictly Ordered Circle with Irrational Rotation Vector

In this section, we shall construct a strictly ordered circle with rotation vector ω ∈RdQd,which is invariant for translations and gradient flow.To do this, some notations need to be introduced.

Let (X,d) be a metric space and let H be the set of all nonempty bounded closed sets in X.Define a metric in H: DH(A,B) = inf{ε|A ?U(B,ε) and B ?U(A,ε)}, where U(A,ε)denotes the ε neighborhood of set A.It is easy to check that (H,DH) is a metric space; it is called a Hausdorffspace and is determined by (X,d).The following lemma is a well known conclusion which can be found in [9]:

Lemma 3.12If (X,d) is compact, then (H,DH) is also compact.

Next, we define a metric in X/〈1〉 as follows:

Definition 3.14A ?RZdis said to be saturated ifx∈A implies thatx+k1∈A for k ∈Z.

We remark that the subset A of RZdis saturated if and only if A=j?1j(A).

Lemma 3.15Assume that A is a saturated and closed set in X.Then j(A) is a closed set in X/〈1〉.

ProofSince A is saturated, A = j?1j(A).Meanwhile, A is a closed set in X.Thus j(A) is a closed set in X/〈1〉.?

Lemma 3.16 Assume that K ?Rdis compact and that J is a closed and Lipschitz mapping from BKinto itself satisfying that J(x+1)=J(x)+1.Then, for A,B ∈SK,

where L1is a Lipschitz constant of J.

Proof Since J(x+1)=J(x)+1, J(A),J(B)∈SK.We define ?J from BK/〈1〉into itself by ?J([x])=[J(x)].It is easy to check that ?J is well defined.For [x],[y]∈BK/〈1〉, there exists k0such that ?d([x],[y])=‖x ?y+k01‖X.Then,

Theorem 3.20Assume that ω ∈Rd,F ∈R andx∈Bω.The average velocity of φtF(x)exists and does not depend onx.

Let v(ω,F)be the average velocity of φtFon Bω.It is easy to check that v(ω,F)=0 if and only if there is an equilibrium point of φtFin ? (?Bω).

Theorem 3.21v(ω,F) is continuous with respect to ω and F.

Lemma 3.22Assume that t ∈R and F10 such that φtF1(x)+σ1≤φtF2(x).

By this lemma, it is easy to obtain the following theorem:

Theorem 3.23Let ω ∈Rd.Then v(ω,F) is a monotonically increasing function with respect to parameter F.

We remark that the proofs for these conclusions are similar to those for d=1 [23], so they are omitted here.

Let Mωbe the set of all Birkhoffminimizers of rotation vector ω.

Proposition 3.24Assume that ω ∈Rd.Then Mω/=?.

is called the depinning force.

Next, we shall show that the depinning force plays an important role in the existence of foliation in the FK model on Zdlattices with d>1.

Definition 3.27A nonempty set F consisting of stationary points is said to be a foliation if

(1) F is a closed set under the product topology;

(2) F is strictly ordered (or totally ordered);

(3) F is invariant for translations;

(4) p0(F)=R.

To show that M(ˉa1) is a foliation, we need to introduce a lemma.Let

Then, for each n ∈N, ωn∈Qdis a solution of (3.2) and ωn→ω as n →∞.?

Lemma 3.29Let ? ?Bωbe a strictly ordered circle consisted of stationary configurations.Then ? ?Mω.

ProofIt suffices to show that eachx∈? is a minimizer.Let A ?Zd.From the coercive condition (H2), there exists a segmentx?: ˉA →R such thatx?=x|?Aand WA(x?) ≤WA(x?+v) for allv∈VA.We set

This is a contradiction.Thus, ???Bωis a foliation.

Moreover, we will prove that ???M(ˉa1).By Lemma 3.29, we only need to prove that, if x ∈??and (k,l)∈Iω, then τk,lx=x.

For ω ∈Qd, this is obvious.Let ω ∈RdQd.If there exist (k,l)∈Iωand x ∈??such that τk,lx/=x, then there exist b ∈N,a1,a2,···and ar∈Z such that

where {(ki, li)} is a basis of Iω.

By Lemma 3.28, there exists a sequence {ωn}?Qdsuch that b(k,l)∈Iωnfor each n ∈N.By the construction of ?ωn, we derive that, for each y ∈?ωn, τbk,bly = y.By Corollary 3.11 and Theorem 3.17, for each n, there exists xn∈?ωnsuch that xn→x as n →∞.Hence,τbk,blxn=xnand τbk,blx=x, which is a contradiction.

Since M(ˉa1)is closed,totally ordered,and invariant for translations[16],then ??=M(ˉa1).Hence, M(ˉa1) is a foliation.?

Here we just get a sufficient condition for the existence of foliations.In the one-dimensional nearest neighbor-coupled case, the depinning force equals zero if and only if there exists a corresponding foliation, i.e., Fd(ω) = 0 ?p0(M(ˉa1)) = R [23].We also hope that there is a similar conclusion for the d-dimensional case.

4 Hull Functions

In this section, we shall introduce hull functions (for example, see [10]), by which we prove Theorem B.

Definition 4.1Letx∈Bωand ω ∈RdQd.We define the lower hull function f?x:R →R and the upper hull function f+x:R →R ofxby

The following propositions can be found in [10]:

Proposition 4.2Let ω ∈Rdandx∈Bω.If (k,l)·(?ω,1) > 0, then τk,lx?x, and if(k,l)·(?ω,1)<0, then τk,lx?x.

Proposition 4.3Letx∈Bωand ω ∈RdQd.The upper and lower hull functions satisfy the following properties:

Proposition 4.4Letx∈Bωand ω ∈RdQd.Then Γ(x) is totally ordered, translationinvariant, minimal, and closed under pointwise convergence.

Definition 4.5A Birkhoffconfigurationx∈Bω(ω ∈RdQd) is said to be recurrent if it can be approximated by its translations from below or above, i.e.,