Calibrated configurations for Frenkel-Kontorova type models in almost-periodic environments
aa r X i v : . [ m a t h . D S ] S e p Calibrated configurations for Frenkel-Kontorova typemodels in almost-periodic environments
Eduardo Garibaldi ∗ Departamento de Matem´aticaUniversidade Estadual de Campinas13083-859 Campinas - SP, Brasil [email protected]
Samuel Petite † LAMFA, CNRS, UMR 7352Universit´e de Picardie Jules Verne80000 Amiens, France [email protected]
Philippe Thieullen ‡ Institut de Math´ematiques, CNRS, UMR 5251Universit´e Bordeaux 1F-33405 Talence, France
August 17, 2018
Abstract
The Frenkel-Kontorova model describes how an infinite chain of atoms min-imizes the total energy of the system when the energy takes into account theinteraction of nearest neighbors as well as the interaction with an exteriorenvironment. An almost-periodic environment leads to consider a family ofinteraction energies which is stationary with respect to a minimal topologi-cal dynamical system. We introduce, in this context, the notion of calibratedconfiguration (stronger than the standard minimizing condition) and, for con-tinuous superlinear interaction energies, we show the existence of these config-urations for some environment of the dynamical system. Furthermore, in onedimension, we give sufficient conditions on the family of interaction energies toensure, for any environment, the existence of calibrated configurations whenthe underlying dynamics is uniquely ergodic. The main mathematical toolsfor this study are developed in the frameworks of discrete weak KAM theory,Aubry-Mather theory and spaces of Delone sets.
Keywords: almost-periodic environment, Aubry-Mather theory, calibratedconfiguration, Delone set, Frenkel-Kontorova model, Ma˜n´e potential, Matherset, minimizing holonomic probability, weak KAM theory
Mathematical subject classification: ∗ supported by FAPESP 2009/17075-8, CAPES-COFECUB 661/10 and Brazilian-French Net-work in Mathematics † supported by FAPESP 2009/17075-8, CAPES-COFECUB 661/10 and ANR SUBTILE 0879. ‡ supported by FAPESP 2009/17075-8 Garibaldi, Petite and Thieullen A minimizing configuration { x k } k ∈ Z for an interaction energy E : R d × R d → R is a chain of points in R d arranged so that the energy of each finite segment( x m , x m +1 , . . . , x n ) cannot be lowered by changing the configuration inside the seg-ment while fixing the two boundary points. Define E ( x m , x m +1 , . . . , x n ) := n − X k = m E ( x k , x k +1 ) . Then { x k } k ∈ Z is said to be minimizing if, for all m < n , for all y m , y m +1 , . . . , y n ∈ R d satisfying y m = x m and y n = x n , one has E ( x m , x m +1 , . . . , x n ) ≤ E ( y m , y m +1 , . . . , y n ) . (1)If the interaction energy is C , coercive and translation periodic ,lim R → + ∞ inf k y − x k≥ R E ( x, y ) = + ∞ , (2) ∀ t ∈ Z d , ∀ x, y ∈ R d , E ( x + t, y + t ) = E ( x, y ) , (3)it is easy to show (see [14]) that minimizing configurations do exist. If d = 1 and E is a smooth strongly twist translation periodic interaction energy, ∂ E∂x∂y ≤ − α < , (4)a minimizing configuration admits in addition a rotation number (see Aubry andLe Daeron [2]). The interaction energy E is supposed to model the interactionbetween two successive points as well as the interaction between the chain and theenvironment.For environments which are aperiodic, namely, with trivial translation group,few results are known (see, for instance, [8, 12, 26]). If d = 1 and E is a twistinteraction energy describing a quasicrystal environment, Gambaudo, Guiraud andPetite [12] showed that minimizing configurations do exist, they all have a rotationnumber and any prescribed real number is the rotation number of a minimizingconfiguration.We shall make slightly more general assumptions on the properties of E . Wesay that E is translation bounded if ∀ R > , sup k y − x k≤ R E ( x, y ) < + ∞ , (5) translation uniformly continuous if ∀ R > , E ( x, y ) is uniformly continuous in k y − x k ≤ R, (6)and superlinear if lim R → + ∞ inf k y − x k≥ R E ( x, y ) k y − x k = + ∞ . (7) alibrated configurations in almost-periodic environments E . It is not clear that there exist bi-infinite minimizingconfigurations in this general context.We call ground energy the lowest energy per site for all configurations¯ E := lim n → + ∞ inf x ,...,x n n E ( x , . . . , x n ) . (8)A configuration { x n } n ∈ Z is calibrated at the level ¯ E if, for every k < l , E ( x k , . . . , x l ) − ( l − k ) ¯ E ≤ inf n ≥ inf y = x k ,...,y n = x l (cid:2) E ( y , . . . , y n ) − n ¯ E (cid:3) . (9)Notice that the number of sites on the right hand side is arbitrary. A calibratedconfiguration is obviously minimizing; the converse is false in general. More gener-ally, a configuration which is calibrated at some level c (replace ¯ E by c in (9)) isalso minimizing.If d ≥ E is C , coercive and translation periodic (conditions (2) and (3)),an argument using the notion of weak KAM solutions as in [16, 10, 14] shows thatthere exist calibrated configurations at the level ¯ E . Conversely, if d = 1 and E is twist translation periodic, every minimizing configuration is calibrated for somemodified energy E λ ( x, y ) = E ( x, y ) − λ ( y − x ), λ ∈ R , with ground energy ¯ E ( λ ).If d = 1 and E is arbitrary (at least translation bounded, translation uniformlycontinuous and superlinear but not translation periodic), it is not known in generalthat a calibrated configuration does exist.In order to give a positive answer to the question of the existence of calibratedconfigurations, we will consider in this paper an interaction energy which has almostperiodic behavior. This leads to look at a family of interaction energies parame-terized by a minimal topological dynamical system. Such an approach is similar tostudies for the Hamilton-Jacobi equation (see, for instance, [5, 6, 7, 20]), where astationary ergodic setting has been taken into account.Concretely, we will assume there exists a family of interaction energies { E ω } ω depending on an environment ω . Let Ω denote the collection of all possible envi-ronments. We assume that a chain { x k + t } k ∈ Z translated in the direction t ∈ R d and interacting with the environment ω has the same local energy that { x k } k ∈ Z interacting with the shifted environment τ t ( ω ), where { τ t : Ω → Ω } t ∈ R d is supposedto be a group of bijective maps. More precisely, each environment ω defines aninteraction E ω ( x, y ) which is assumed to be topologically stationary in the followingsense ∀ ω ∈ Ω , ∀ t ∈ R d , ∀ x, y ∈ R d , E ω ( x + t, y + t ) = E τ t ( ω ) ( x, y ) . (10)In order to ensure the topological stationarity, the interaction energy will besupposed to have a Lagrangian form . Formally, we will use the following definition.
Definition 1.
Let Ω be a compact metric space. Garibaldi, Petite and Thieullen
1. An almost periodic environment is a couple (cid:0) Ω , { τ t } t ∈ R d (cid:1) , where { τ t } t ∈ R d is aminimal R d -action, that is, a family of homeomorphisms τ t : Ω → Ω satisfying– τ s ◦ τ t = τ s + t for all s, t ∈ R d (the cocycle property),– τ t ( ω ) is jointly continuous with respect to ( t, ω ) ,– ∀ ω ∈ Ω , { τ t ( ω ) } t ∈ R d is dense in Ω .2. A family of interaction energies { E ω } ω ∈ Ω is said to derive from a Lagrangianif there exists a continuous function L : Ω × R d → R such that ∀ ω ∈ Ω , ∀ x, y ∈ R d , E ω ( x, y ) := L ( τ x ( ω ) , y − x ) . (11)
3. An almost periodic interaction model is the set of data (Ω , { τ t } t ∈ R d , L ) where (Ω , { τ t } t ∈ R d ) is an almost period environment and L is a continuous functionon Ω × R d . Notice that the expression “almost periodic” shall not be understood in thesense of H. Bohr. The almost periodicity according to Bohr is canonically reliedto the uniform convergence. See [3] for a discussion on the different concepts ofalmost periodicity in conformity with the uniform topology or with the compactopen topology.Because of the particular form (11) of E ω ( x, y ), these energies are translationbounded and translation continuous uniformly in ω and in k y − x k ≤ R . We makeprecise the two notions of coerciveness and superlinearity in the Lagrangian form. Definition 2.
Let (Ω , { τ t } t ∈ R d , L ) be an almost periodic interaction model. TheLagrangian L is said to be coercive if lim R → + ∞ inf ω ∈ Ω inf k t k≥ R L ( ω, t ) = + ∞ .L is said to be superlinear if lim R → + ∞ inf ω ∈ Ω inf k t k≥ R L ( ω, t ) k t k = + ∞ .L is said to be ferromagnetic if, for every ω ∈ Ω , E ω is of class C ( R d × R d ) and,for every ω ∈ Ω and x, y ∈ R d , x ∈ R d ∂E ω ∂y ( x, y ) ∈ R d and y ∈ R d ∂E ω ∂x ( x, y ) ∈ R d are homeomorphisms. Note that if there is a constant α > P di,j =1 ∂ E ω ∂x∂y v i v j ≤ − α P di =1 v i for all ω ∈ Ω , x, y ∈ R d , then L is ferromagnetic and superlinear.Let us illustrate our abstract notions by three typical examples. alibrated configurations in almost-periodic environments Example 3.
The classical periodic one-dimensional Frenkel-Kontorova model [11]takes into account the family of interaction energies E ω ( x, y ) = W ( y − x ) + V ω ( x ) ,with ω ∈ S , written in Lagrangian form as L ( ω, t ) = W ( t ) + V ( ω ) = 12 | t − λ | + K (2 π ) (cid:0) − cos 2 πω (cid:1) , (12) where λ , K are constants. Here Ω = S and τ t : S → S is given by τ t ( ω ) = ω + t .We observe that { τ t } t is clearly minimal. The following example comes from [12].
Example 4.
Consider, for an irrational α ∈ (0 , \ Q , the set ω ( α ) := { n ∈ Z : ⌊ nα ⌋ − ⌊ ( n − α ⌋ = 1 } , where ⌊·⌋ denotes the integer part. Notice that the distance between two consecutiveelements of ω ( α ) is ⌊ α ⌋ or ⌊ α ⌋ + 1 . Now let U and U be two real valued smoothfunctions with supports respectively in (0 , ⌊ α ⌋ ) and (0 , ⌊ α ⌋ + 1) . Let V ω ( α ) be thefunction defined by V ω ( α ) ( x ) = U ω n +1 − ω n −⌊ α ⌋ ( x − ω n ) , where ω n < ω n +1 are thetwo consecutive elements of the set ω ( α ) such that ω n ≤ x < ω n +1 . The associatedinteraction energy is the function E ω ( α ) ( x, y ) = 12 | x − y − λ | + V ω ( α ) ( x ) . (13) Given any relatively dense set ω ′ of the real line such that the distance between twoconsecutive points is in {⌊ α ⌋ , ⌊ α ⌋ +1 } , we can directly extend the previous definitionand introduce the function V ω ′ . Let Ω ′ be the collection of all such sets ω ′ . Then, forany x, t ∈ R , we have the relation V ω ′ ( x + t ) = V ω ′ − t ( x ) , where ω ′ − t denotes the setof elements of ω ′ ∈ Ω ′ translated by − t . In section 3, we explain how to associatea compact metric space Ω ⊂ Ω ′ , where the group of translations acts minimally, aswell as a Lagrangian from which the family { E ω } ω ∈ Ω derives. As we shall see in section 3, the construction given in example 4 extends to any quasiscrystal ω of R , namely, to any set ω ⊂ R which is relatively dense and uni-formly discrete such that the difference set ω − ω is discrete and any finite patternrepeats with a positive frequency. We will later focus on a particular class of in-teraction models, called almost crystalline, which will include all quasicrystals. Anexample of almost periodic interaction model on R which is not almost crystallinecan be constructed in the following way. Example 5.
The underlying minimal flow is the irrational flow τ t ( ω ) = ω + t (1 , √ acting on Ω = T . The family of interaction energies E ω derives from the La-grangian L ( ω, t ) := 12 | t − λ | + K (2 π ) (cid:0) − cos 2 πω (cid:1) + K (2 π ) (cid:0) − cos 2 πω (cid:1) , (14) where ω = ( ω , ω ) ∈ T . Garibaldi, Petite and Thieullen
For an almost periodic interaction model, the notion of ground energy is givenby the following definition.
Definition 6.
We call ground energy of a family of interactions { E ω } ω ∈ Ω of La-grangian form L : Ω × R d → R the quantity ¯ E := lim n → + ∞ inf ω ∈ Ω inf x ,...,x n ∈ R d n E ω ( x , . . . , x n ) . The above limit is actually a supremum by superadditivity and is finite as soonas L is assumed to be coercive. Besides, we clearly have a priori boundsinf ω ∈ Ω inf x,y ∈ R d E ω ( x, y ) ≤ ¯ E ≤ inf ω ∈ Ω inf x ∈ R d E ω ( x, x ) . (15)The constant ¯ E plays the role of a drift. It is natural to modify the previousnotion of minimizing configurations by saying that { x n } n ∈ Z is calibrated at the level ¯ E if P n − k = m [ E ( x k , x k +1 ) − ¯ E ] realizes the smallest signed distance between x m and x n for every m < n . Hence, we consider the following key notions borrowed fromthe weak KAM theory (see, for instance, [9]). Definition 7.
We call Ma˜n´e potential in the environment ω the function on R d × R d given by S ω ( x, y ) := inf n ≥ inf x = x ,...,x n = y (cid:2) E ω ( x , . . . , x n ) − n ¯ E (cid:3) . We say that a configuration { x k } k ∈ Z is calibrated for E ω (at the level ¯ E ) if ∀ m < n, S ω ( x m , x n ) = E ω ( x m , x m +1 , . . . , x n ) − ( n − m ) ¯ E. As discussed in section 2, the Ma˜n´e potential for any almost periodic environ-ment is always finite and shares the same properties as a pseudometric. In thiscontext, the calibrated configurations may be seen as geodesics. An important factin the framework of almost periodic interaction models is that calibrated configu-rations always exist for some environments ω . This is given below by the statementof the first main result of this paper. In section 2, we introduce minimizing holo-nomic probabilities, which correspond in our discrete setting to Mather measures,and we define the Mather set as the subset of Ω × R d formed by the union of theirsupports (see definition 11). Denoted Mather( L ), we show that its projection by pr : Ω × R d → Ω is contained into the set of environments for which there exists acalibrated configuration passing through the origin of R d . Thus, the next theoremextends Aubry-Mather theory of the classical periodic model. Theorem 8.
Let (Ω , { τ t } t ∈ R d , L ) be an almost periodic interaction model, with L a C superlinear function. Then, for all ω ∈ pr ( Mather ( L )) , there exists a calibratedconfiguration { x k } k ∈ Z for E ω such that x = 0 and sup k ∈ Z k x k +1 − x k k < + ∞ . This theorem states that, in the almost periodic case, there exist at least oneenvironment and one calibrated configuration for that environment (and thus forany environment in its orbit). It may happen that the projected Mather set does notmeet every orbit of the system. Indeed, in the almost periodic Frenkel-Kontorova alibrated configurations in almost-periodic environments λ = 0, we have ¯ E = 0 which is attained bytaking x n = 0 for every n ∈ Z . In addition, it is easy to check that the Mather setis reduced to the point (0 T , R ) and in particular the projected Mather set { T } meets a unique orbit. This phenomenon is related to the fact that the minimum of aLagrangian may not be reached for several orbits of the flow τ on Ω. A similar caseoccurs when there is no exact corrector for the homogenization of Hamilton-Jacobiequations in the stationary ergodic setting [20, 5]. At the difference with theseworks, in our context, we leave open the question whether for any environmentthere are “approximated” calibrated configurations (in a sense to define).However, we shall see (theorem 10) that, for a certain class of one-dimensionalmodels with transversally constant Lagrangians , this symptom disappears and theprojected Mather set meets any orbit of the system. Such a family of Lagrangiansincludes the ones of examples 3 and 4. Notice that, for both examples, given anyfinite configuration, the interaction energy keeps the same value for infinitely manytranslated configurations. Indeed, take any translation by a multiple of the periodin the example 3, and, for example 4, take a relatively dense set of translationsprovided by the collection of return times to the origin of the irrational rotation ofangle α on the circle (recall that this dynamical system is minimal). A transversallyconstant Lagrangian is defined in order to share the same property. We postponeto section 3, more precisely, to definition 34 the details of the technical notion of atransversally constant Lagrangian to be adopted in this article.Let us also precise that we work on a class of environments more general thanthe usual one for one-dimensional quasicrystals, in particular, more general thanthe one considered in [12]. Furthermore, we slightly extend the strongly twistproperty (4), which is the main assumption in Aubry-Mather theory ([2, 22]). Theweakly twist property will allow us to use, for example, | t − λ | instead of | t − λ | in example 5, which would be impossible with the strongly twist property (4). Weformalize all these extensions in the next definition. Definition 9.
Let (Ω , { τ t } t ∈ R , L ) be a one-dimensional almost periodic interactionmodel.– L is said to be weakly twist if, for every ω ∈ Ω , E ω ( x, y ) is C , and ∀ x, y ∈ R , ω ∈ Ω , ∂ E ω ∂x∂y ( x, · ) < and ∂ E ω ∂x∂y ( · , y ) < a.e.– The interaction model (Ω , { τ t } t ∈ R , L ) is said to be almost crystalline if1. { τ t } t ∈ R is uniquely ergodic (with unique invariant probability measure λ ),2. L is superlinear and weakly twist,3. L is locally transversally constant (as in definition 34). We now state the second main result of this paper, which says that, in the caseof almost crystalline interaction models, for any environment, there always existsa calibrated configuration passing close to the origin. This result may be seen asa consequence of the proof of theorem 8, since the strategy to obtain it consists
Garibaldi, Petite and Thieullen in arguing, mainly through a Kakutani-Rohlin tower description of the system(Ω , { τ t } t ∈ R ), that the corresponding projected Mather set intersects all orbits. Asimpler version of the following theorem is given in corollary 37. Theorem 10.
Let (Ω , { τ t } t ∈ R , L ) be an almost crystalline interaction model. Thenthe projected Mather set meets uniformly any orbit of the flow τ t . In particular,for every ω ∈ Ω , there exists a calibrated configuration for E ω with bounded jumpsand at a bounded distance from the origin uniformly in ω , that is, a configuration { x k,ω } k ∈ Z satisfying1. ∀ m < n, S ω ( x m,ω , x n,ω ) = n − X k = m E ω ( x k,ω , x k +1 ,ω ) − ( n − m ) ¯ E ,2. sup ω ∈ Ω sup k ∈ Z | x k +1 ,ω − x k,ω | < + ∞ , sup ω ∈ Ω | x ,ω | < + ∞ . The paper is organized as follows. In section 2.1, we properly introduce theMather set and we show that there exist calibrated configurations for almost pe-riodic interaction models by giving the proof of theorem 8. Besides, we gather,in section 2.2, ordering properties of one-dimensional calibrated configurations inthe presence of the twist hypothesis that will be useful for the demonstration oftheorem 10. In section 3.1, we recall basic definitions and properties concerning De-lone sets and specially quasicrystals. In particular, strongly equivariant functionsassociated with a quasicrystal will serve as a prototype to our notion of locallytransversally constant Lagrangian. Section 3.2 concerns the basic properties offlow boxes and locally transversally constant Lagrangians. Finally, section 4 isdevoted to the proof of theorem 10.
We show here that the Mather set describes the set of environments for which thereexist calibrated configurations. The Mather set is defined in terms of minimizingholonomic measures. Let ω ∈ Ω be fixed. The ground energy (in the environ-ment ω ) measures the mean energy per site of a configuration { x n } n ≥ which dis-tributes in R d so that n E ω ( x , . . . , x n ) → ¯ E . Notice that the previous mean canbe understood as an expectation of L ( ω, t ) with respect to a probability measure µ n,ω := n P n − k =0 δ ( τ xk ( ω ) , x k +1 − x k ) :1 n E ω ( x , . . . , x n ) = Z L ( ω, t ) µ n,ω ( dω, dt ) . (16)Notice also that µ n,ω satisfies the following property of pseudoinvariance Z f ( ω ) µ n,ω ( dω, dt ) − Z f ( τ t ( ω )) µ n,ω ( dω, dt ) = 1 n (cid:16) f ◦ τ x n ( ω ) − f ◦ τ x ( ω ) (cid:17) . (17)This suggests to consider the set of all weak ∗ limits of µ n,ω as n → + ∞ . Follow-ing [21], we call these limit measures holonomic probabilities . alibrated configurations in almost-periodic environments Definition 11.
A probability measure µ on Ω × R d is said to be holonomic if ∀ f ∈ C (Ω) , Z f ( ω ) µ ( dω, dt ) = Z f ( τ t ( ω )) µ ( dω, dt ) . Let M hol denote the set of all holonomic probability measures. The set M hol is certainly not empty since it contains any δ ( ω, , ω ∈ Ω. It is thennatural to look for holonomic measures that minimize L . We show that minimizingholonomic measures do exist and that the lowest mean value of L is the groundenergy. Proposition 12 ( The ergodic formula). If L is C coercive, then ¯ E = inf n Z L dµ : µ ∈ M hol o , and the infimun is attained by some holonomic probability measure. Definition 13.
We say that an holonomic measure µ is minimizing if ¯ E = R L dµ .We denote by M min ( L ) the set of minimizing measures. We call Mather set of L the set Mather( L ) := ∪ µ ∈ M min ( L ) supp( µ ) ⊆ Ω × R d . The projected Mather set is simply pr (Mather( L )) , where pr : Ω × R d → Ω is thefirst projection. Proposition 14.
1. If L is C coercive, then ∃ µ ∈ M min ( L ) with Mather( L ) = supp( µ ) . In particular,
Mather( L ) is closed.2. If L is C superlinear, then Mather( L ) is compact. The set of holonomic measures may be seen as a dual object to the set ofcoboundaries { u − u ◦ τ t : u ∈ C (Ω) , t ∈ R d } . Proposition 12 admits thus a dualversion that will be first proved. Proposition 15 ( The sup-inf formula). If L is C coercive, then ¯ E = sup u ∈ C (Ω) inf ω ∈ Ω , t ∈ R d (cid:2) L ( ω, t ) + u ( ω ) − u ◦ τ t ( ω ) (cid:3) . We do not know whether the above supremum is achieved for general almostperiodic interaction models. There is finally a third way to compute the groundenergy, which says that the exact choice of the environment ω is irrelevant. Proposition 16. If L is C coercive, then ∀ ω ∈ Ω , ¯ E = lim n → + ∞ inf x ,...,x n ∈ R d n E ω ( x , . . . , x n ) . Garibaldi, Petite and Thieullen
Before proving propositions 12, 15 and 16, we note temporarily¯ E ω = lim n → + ∞ inf x ,...,x n ∈ R d n E ω ( x , . . . , x n ) , ¯ L := inf n Z L dµ : µ ∈ M hol o , and ¯ K := sup u ∈ C (Ω) inf ω ∈ Ω , t ∈ R d (cid:2) L ( ω, t ) + u ( ω ) − u ◦ τ t ( ω ) (cid:3) . We first show that the infimum is attained in proposition 12.
Proof of proposition 12.
We shall prove later that ¯ L = ¯ E . We prove now that theinfimum is attained in ¯ L := inf { R L dµ : µ ∈ M hol } . Let C := sup ω ∈ Ω L ( ω, ≥ ¯ L and M hol,C := n µ ∈ M hol : Z L dµ ≤ C o . We equip the set of probability measures on Ω × R d with the weak topology (con-vergence of sequence of measures by integration against compactly supported con-tinuous test functions). By coerciveness, for every ǫ > M > inf L such that ǫ > ( C − inf L ) / ( M − inf L ), there exists R ( ǫ ) > ω ∈ Ω , k t k≥ R ( ǫ ) L ( ω, t ) ≥ M .By integrating L − inf L , we get ∀ µ ∈ M hol,C , µ (cid:0) Ω × { t : k t k ≥ R ( ǫ ) } (cid:1) ≤ Z L − inf LM − inf L dµ ≤ C − inf LM − inf L < ǫ.
We have just proved that the set M hol,C is tight. Let ( µ n ) n ≥ ⊂ M hol,C be a sequenceof holonomic measures such that R L dµ n → ¯ L . By tightness, we may assume that µ n → µ ∞ with respect to the strong topology (convergence of sequence of measuresby integration against bounded continuous test functions). In particular, µ ∞ isholonomic. Moreover, for every φ ∈ C (Ω , [0 , ≤ Z ( L − ¯ L ) φ dµ ∞ = lim n → + ∞ Z ( L − ¯ L ) φ dµ n ≤ lim inf n → + ∞ Z ( L − ¯ L ) dµ n = 0 . Therefore, µ ∞ is minimizing.We next show that there is no need to take the closure in the definition of theMather set. We will show later that it is compact. Proof of proposition 14 – Item 1.
We show that Mather( L ) = supp( µ ) for someminimizing measure µ . Let { V i } i ∈ N be a countable basis of the topology of Ω × R d and let I := { i ∈ N : V i ∩ supp( ν ) = ∅ for some ν ∈ M min ( L ) } . We reindex I = { i , i , . . . } and choose for every k ≥ µ k sothat V i k ∩ supp( µ k ) = ∅ or equivalently µ k ( V i k ) >
0. Let µ := P k ≥ k µ k . Then µ is minimizing. Suppose some V i is disjoint from the support of µ . Then µ ( V i ) = 0and, for every k ≥ µ k ( V i ) = 0. Suppose by contradiction that V i ∩ supp( ν ) = ∅ forsome ν ∈ M min ( L ), then i = i k for some k ≥ µ k , µ k ( V i ) > V i is disjoint from the Mather set and we havejust proved Mather( L ) ⊆ supp( µ ) or Mather( L ) = supp( µ ). alibrated configurations in almost-periodic environments L − ¯ L on the Mather set, that will be proved in lemma 24.The two formulas given in propositions 12 and 15 are two different ways tocompute ¯ E . It is not an easy task to show that the two values are equal. It is thepurpose of lemma 17 to give a direct proof of this fact. We also give a second proofusing the minimax formula. Lemma 17. If L is C coercive, then ¯ L = ¯ K and there exists µ ∈ M hol such that ¯ L = R L dµ .First proof of lemma 17. Part 1.
We show that ¯ L ≥ ¯ K . Indeed, for any holonomicmeasure µ and any function u ∈ C (Ω), Z L dµ = Z [ L ( ω, t ) + u ( ω ) − u ◦ τ t ( ω )] µ ( dω, dt ) ≥ inf ω ∈ Ω , t ∈ R d (cid:2) L ( ω, t ) + u ( ω ) − u ◦ τ t ( ω ) (cid:3) . We conclude by taking the supremum on u and the infimum on µ . Part 2.
We show that ¯ K ≥ ¯ L . Let X := C b (Ω × R d ) be the vector space ofbounded continuous functions equipped with the uniform norm. A coboundary is afunction f of the form f = u ◦ τ − u or f ( ω, t ) = u ◦ τ t ( ω ) − u ( ω ) for some u ∈ C (Ω).Let A := { ( f, s ) ∈ X × R : f is a coboundary and s ≥ ¯ K } and B := { ( f, s ) ∈ X × R : inf ω ∈ Ω , t ∈ R d ( L − f )( ω, t ) > s } . Then A and B are nonempty convex subsets of X × R . They are disjoint by thedefinition of ¯ K and B is open because L is coercive. By Hahn-Banach theorem,there exists a nonzero continuous linear form Λ on X × R which separates A and B . The linear form Λ is given by λ ⊗ α , where λ is a continuous linear form on X and α ∈ R . The linear form λ is, in particular, continuous on C (Ω × R d ) and, byRiesz-Markov theorem, ∀ f ∈ C (Ω × R d ) , λ ( f ) = Z f dµ, for some signed measure µ . By separation, we have λ ( f ) + αs ≤ λ ( u − u ◦ τ ) + αs ′ , for u ∈ C (Ω), f ∈ X and s, s ′ ∈ R such that inf Ω × R d ( L − f ) > s and s ′ ≥ ¯ K . Bymultiplying u by an arbitrary constant, one obtains ∀ u ∈ C (Ω) , λ ( u − u ◦ τ ) = 0 . The case α = 0 is not admissible, since otherwise λ ( f ) ≤ f ∈ X and λ would be the null form, which is not possible. The case α < f = 0 and2 Garibaldi, Petite and Thieullen s → −∞ . By dividing by α > λ/α to λ (as well as µ/α to µ ), oneobtains ∀ f ∈ X, λ ( f ) + inf Ω × R d ( L − f ) ≤ ¯ K. By taking f = c , one obtains c ( λ ( ) − ≤ ¯ K − inf Ω × R d L for every c ∈ R , andthus λ ( ) = 1. By taking − f instead of f , one obtains λ ( f ) ≥ inf Ω × R d L − ¯ K forevery f ≥
0, which (again arguing by contradiction) yields λ ( f ) ≥
0. In particular, µ is a probability measure. We claim that ∀ u ∈ C (Ω) , Z ( u − u ◦ τ ) dµ = 0 . Indeed, given
R >
0, consider a continuous function 0 ≤ φ R ≤
1, with compactsupport on Ω × B R +1 (0), such that φ R ≡ × B R (0). Then u − u ◦ τ ≥ ( u − u ◦ τ ) φ R + min Ω × R d ( u − u ◦ τ )(1 − φ R ) . Since λ and µ coincide on C (Ω × R d ) + R , one obtains0 = λ ( u − u ◦ τ ) ≥ Z ( u − u ◦ τ ) φ R dµ + min Ω × R d ( u − u ◦ τ ) Z (1 − φ R ) dµ. By letting R → + ∞ , it follows that R ( u − u ◦ τ ) dµ ≤ u to − u . In particular, µ is holonomic. We claim that ∀ f ∈ X, Z f dµ + inf Ω × R d ( L − f ) ≤ ¯ K. Indeed, we first notice that the left hand side does not change by adding a constantto f . Moreover, if f ≥ ≤ f R ≤ f is any continuous function with compactsupport on Ω × B R +1 (0) which is identical to f on Ω × B R (0), the claim follows byletting R → + ∞ in Z f R dµ + inf Ω × R d ( L − f ) ≤ λ ( f R ) + inf Ω × R d ( L − f R ) ≤ ¯ K. We finally prove the opposite inequality ¯ L ≤ ¯ K . Given R >
0, denote L R =min( L, R ). Since L is coercive, L R ∈ X . Then L − L R ≥ R L R dµ ≤ ¯ K . Byletting R → + ∞ , one obtains R L dµ ≤ ¯ K for some holonomic measure µ .We give a second proof of lemma 17. We will use basic properties of theKantorovich-Rubinstein topology on the set of probabilities measures on a Pol-ish space ( X, d ) and a version of the Topological Minimax Theorem which is ageneralization of Sion’s classical result [24]. For a recent review on the last topic,see [25]. We state a particular case of theorem 5.7 there.
Theorem 18 ( Topological Minimax Theorem [25]).
Let X and Y be Haus-dorff topological spaces. Let F ( x, y ) : X × Y → R be a real-valued function. Define η := sup y ∈ Y inf x ∈ X F ( x, y ) and assume there exists a real number α ∗ > η such that alibrated configurations in almost-periodic environments ∀ α ∈ ( η, α ∗ ) , for every finite set ∅ 6 = H ⊂ Y , ∩ y ∈ H { x ∈ X : F ( x, y ) ≤ α } iseither empty or connected;2. ∀ α ∈ ( η, α ∗ ) , for every set K ⊂ X , ∩ x ∈ K { y ∈ Y : F ( x, y ) > α } is eitherempty or connected;3. for any y ∈ Y and x ∈ X , F ( x, y ) is lower semi-continuous in x and uppersemi-continuous in y ;4. there exists a finite set M ⊂ Y such that ∩ y ∈ M { x ∈ X : F ( x, y ) ≤ α ∗ } iscompact and non-empty.Then, inf x ∈ X sup y ∈ Y F ( x, y ) = sup y ∈ Y inf x ∈ X F ( x, y ) . We recall basic facts on the Kantorovich-Rubinstein topology (see [23] or [1]).Given a Polish space Z and a point z ∈ Z , let us consider the set of probabilitymeasures on the Borel sets of Z that admit a finite first moment, i.e. , P ( Z ) = (cid:8) µ : Z Z d ( z , z ) dµ ( z ) < + ∞ (cid:9) . Notice that this set does not depend on the choice of the point z . The Wassersteindistance or Kantorovitch-Rubinstein distance on P ( Z ) is a distance between two µ, ν ∈ P ( Z ) defined by W ( µ, ν ) := inf (cid:8) Z Z × Z d ( x, y ) dγ ( x, y ) : γ ∈ Γ( µ, ν ) (cid:9) , where Γ( µ, ν ) denotes the set of all the probability measures γ on Z × Z withmarginals µ and ν on the first and second factors, respectively.Recall that a continuous function L : Z → R is said to be superlinear on a Polishspace Z if the map defined by z ∈ Z L ( z ) / (cid:0) d ( z, z ) (cid:1) ∈ R is proper. Noticethat this definition is also independent of the choice of z and, by considering thedistance ˆ d := min( d,
1) on Z , any proper function is superlinear for ˆ d . The followinglemma is easy to prove and gives us a sufficient condition for relative compactnessin P ( Z ) (see theorem 6.9 in [23] or [1] for a more detailed discussion). Lemma 19.
Let Z be a Polish space, L : Z → R be a continuous function, and X := { µ ∈ P ( Z ) : R L dµ < + ∞} be equipped with the Kantorovich-Rubinsteindistance. Then1. the map µ ∈ X R L dµ is lower semi-continuous;2. if L is a superlinear, then, for every α ∈ R , the set { µ ∈ X : R L dµ ≤ α } iscompact (the map µ ∈ X R L dµ is proper).Second proof of lemma 17.
Lemma 19 applied to the C superlinear Lagrangian L : Ω × R d → R guarantees the existence of a minimizing probability for L . Thisminimizing measure is holonomic since the set of holonomic measures is a closed4 Garibaldi, Petite and Thieullen subset of P (Ω × R d ) for the Kantorovich-Rubinstein distance. Notice that for every u ∈ C (Ω),inf ω ∈ Ω , t ∈ R d ( L + u − u ◦ τ )( ω, t ) = inf ω ∈ Ω , t ∈ R d Z ( L + u − u ◦ τ ) dδ ( ω,t ) ≥ inf µ ∈ P (Ω × R d ) Z ( L + u − u ◦ τ ) dµ ≥ inf ω ∈ Ω , t ∈ R d ( L + u − u ◦ τ )( ω, t ) . Let X := { µ ∈ P (Ω × R d ) : R L dµ < + ∞} and Y := C (Ω). Then¯ K = sup u ∈ Y inf µ ∈ X Z ( L + u − u ◦ τ ) dµ ≤ min ω ∈ Ω L ( ω, . Define α ∗ := min ω ∈ Ω L ( ω,
0) + 1 > ¯ K and F : ( µ, u ) ∈ X × Y Z ( L + u − u ◦ τ ) dµ. Since F is affine in both variables, it satisfies items 1 and 2 of theorem 18. Item 3is also satisfied since F ( µ, u ) is lower semi-continuous in µ and continuous in u .By taking M = { } , the singleton set reduced to the null function in Y , the set ∩ u ∈ M { µ ∈ X : F ( µ, u ) ≤ α ∗ } is compact and non-empty, so that item 4 is satisfied.The Topological Minimax Theorem therefore implies¯ K = inf µ ∈ X sup u ∈ Y Z ( L + u − u ◦ τ ) dµ. (18)We show that every µ ∈ X such that sup u ∈ Y R ( L + u − u ◦ τ ) dµ < + ∞ is holonomic.If not, there would exist a function u ∈ C (Ω) such that R ( u − u ◦ τ ) dµ > u − u ◦ τ ) by a positive scalar λ and letting λ → + ∞ would lead to acontradiction. Thus, the infimum in (18) may be taken over holonomic probabiltymeasures with respect to which L is integrable. We finally conclude that¯ K = inf µ ∈ X sup u ∈ Y Z ( L + u − u ◦ τ ) dµ = inf µ ∈ M hol Z L dµ = ¯ L. The holonomic condition shall not be confused with invariance in the usual senseof dynamical systems. We may nevertheless introduce a larger space than Ω × R d and a suitable dynamics on such a space. We will apply Birkhoff ergodic theoremwith respect to that dynamical system to prove that ¯ L ≥ ¯ E . Notation 20.
Consider ˆΩ := Ω × ( R d ) N equipped with the product topology and theBorel sigma-algebra. In particular, ˆΩ becomes a complete separable metric space.Any probability measure µ on Ω × R d admits a unique disintegration along the firstprojection pr : Ω × R d → Ω , µ ( dω, dt ) := pr ∗ ( µ )( dω ) P ( ω, dt ) , alibrated configurations in almost-periodic environments where { P ( ω, dt ) } ω ∈ Ω is a measurable family of probability measures on R d . Let ˆ µ be the Markov measure with initial distribution pr ∗ ( µ ) and transition probabilities P ( ω, dt ) . For Borel bounded functions of the form f ( ω, t , . . . , t n ) , we have ˆ µ ( dω, dt ) = pr ∗ ( dω ) P ( ω, dt ) P ( τ t ( ω ) , dt ) · · · P ( τ t + ··· + t n − ( ω ) , dt n ) . If µ is holonomic, then ˆ µ is invariant with respect to the shift map ˆ τ : ( ω, t , t , . . . ) ( τ t ( ω ) , t , t , . . . ) . We will call ˆ µ the Markov extension of µ . Conversely, the projection of any ˆ τ -invariant probability measure ˜ µ on Ω × R d is holonomic. This gives a fourth wayto compute ¯ E ¯ E = inf n Z ˆ L d ˜ µ : ˜ µ is a ˆ τ -invariant probability measure on ˆΩ o , where ˆ L ( ω, t , t , . . . ) := L ( ω, t ) is the natural extension of L on ˆΩ .End of proof of propositions 12, 15 and 16. – Part 1: We know that ¯ K = ¯ L by lemma 17.– Part 2: We claim that ¯ E ω = ¯ E for all ω ∈ Ω. By the topological stationar-ity (10) of E ω and by the minimality of τ t , for any n ∈ N , we have thatinf x ,...,x n ∈ R d E ω ( x , . . . , x n ) = inf x ,...,x n ∈ R d inf t ∈ R d E ω ( x + t, . . . , x n + t )= inf x ,...,x n ∈ R d inf t ∈ R d E τ t ( ω ) ( x , . . . , x n )= inf x ,...,x n ∈ R d inf ω ∈ Ω E ω ( x , . . . , x n ) , which clearly yields ¯ E ω = ¯ E for every ω ∈ Ω.– Part 3: We claim that ¯ E ≥ ¯ K . Indeed, given c < ¯ K , let u ∈ C ( R d ) besuch that, for every ω ∈ Ω and any t ∈ R d , u ( τ t ( ω )) − u ( ω ) ≤ L ( ω, t ) − c . Define u ω ( x ) = u ( τ x ( ω )). Then, ∀ x, y ∈ R d , u ω ( y ) − u ω ( x ) ≤ E ω ( x, y ) − c, which implies ¯ E ≥ c for every c < ¯ K , and therefore ¯ E ≥ ¯ K .– Part 4: We claim that ¯ L ≥ ¯ E . Let µ be a minimizing holonomic probabilitymeasure with Markov extension ˆ µ (see notation 20). If ( ω, t ) ∈ ˆΩ, then n − X k =0 ˆ L ◦ ˆ τ k ( ω, t ) = E ω ( x , . . . , x n ) with x = 0 and x k = t + · · · + t k − , and, by Birkhoff ergodic theorem,¯ E ≤ Z lim n → + ∞ n n − X k =0 ˆ L ◦ ˆ τ k d ˆ µ = Z L dµ = ¯ L. Garibaldi, Petite and Thieullen
A backward calibrated sub-action u as given by the Lax-Oleinik operator inthe periodic context (for details, see [14]) is not available in general for an almostperiodic interaction model. A calibrated sub-action u in this setting would be a C (Ω) function such that, if E ω,u is defined by E ω,u ( x, y ) := E ω ( x, y ) − (cid:2) u ◦ τ y ( ω ) − u ◦ τ x ( ω ) (cid:3) − ¯ E, then (cid:26) ∀ ω ∈ Ω , ∀ x, y ∈ R d , E ω,u ( x, y ) ≥ , ∀ ω ∈ Ω , ∀ y ∈ R d , ∃ x ∈ R d , E ω,u ( x, y ) = 0 . We do not know whether such a function exists. We will weaken this notion by in-troducing a notion of measurable subadditive cocycle. Notice first that the function U ( ω, t ) := u ◦ τ t ( ω ) − u ( ω ) is a cocycle, namely, it satisfies ∀ ω ∈ Ω , ∀ s, t ∈ R d , U ( ω, s + t ) = U ( ω, s ) + U ( τ s ( ω ) , t ) . (19)A natural candidate to be a subadditive function is given by the Ma˜n´e potentialin the periodic context. For almost periodic interaction models, we introduce thefollowing definition. Definition 21.
Let L be a coercive Lagrangian. We call Ma˜n´e subadditive cocycleassociated with L the function defined on Ω × R d by Φ( ω, t ) := inf n ≥ inf x ,x ,...,x n = t n − X k =0 (cid:2) L ( τ x k ( ω ) , x k +1 − x k ) − ¯ E (cid:3) . We call Ma˜n´e potential in the environment ω the function on R d × R d given by S ω ( x, y ) := Φ( τ x ( ω ) , y − x ) = inf n ≥ inf x = x ,...,x n = y (cid:2) E ω ( x , . . . , x n ) − n ¯ E (cid:3) . The very definitions of Φ and ¯ E show that Φ takes finite values and is a subad-ditive cocycle, ∀ ω ∈ Ω , ∀ s, t ∈ R , Φ( ω, s + t ) ≤ Φ( ω, s ) + Φ( τ s ( ω ) , t ) , (20) ∀ ω ∈ Ω , ∀ t ∈ R d , Φ( ω, t ) ≤ L ( ω, t ) − ¯ E, (21) ∀ ω ∈ Ω , Φ( ω, ≥ , (22) ∀ ω ∈ Ω , ∀ t ∈ R d , Φ( ω, t ) ≥ ¯ E − L ( τ t ( ω ) , − t ) . (23)Inequality (22) is proved using the fact that, for a fixed ω , the sequence¯ E n ( ω,
0) := inf x ,...,x n − E ω (0 , x , . . . , x n − , n and ¯ E ≤ lim n →∞ n ¯ E n ( ω,
0) = inf n ≥ n ¯ E n ( ω, alibrated configurations in almost-periodic environments Definition 22.
A measurable function U : Ω × R d → [ −∞ , + ∞ [ is called a Mather-calibrated subadditive cocycle if the following properties are satisfied:– ∀ ω ∈ Ω , ∀ s, t ∈ R d , U ( ω, s + t ) ≤ U ( ω, s ) + U ( τ s ( ω ) , t ) ,– ∀ ω ∈ Ω , ∀ s, t ∈ R d , U ( ω, t ) ≤ L ( ω, t ) − ¯ L and U ( ω, ≥ ,– ∀ µ ∈ M hol , if R L dµ < + ∞ , then R U ( ω, P n − k =0 t k ) ˆ µ ( dω, dt ) ≥ , ∀ n ≥ ,– where ˆ µ is the Markov extension of µ . Notice that, provided we know in advance that U is finite, U ( ω, ≥ s = t = 0 in the subadditive cocycle inequality. Lemma 23.
A Mather-calibrated subadditive cocycle U satisfies in addition– U ( ω, t ) is finite everywhere,– sup ω ∈ Ω ,t ∈ R d | U ( ω, t ) | / (1 + k t k ) < + ∞ ,– ∀ µ ∈ M min ( L ) , ∀ n ≥ , U ( ω, P n − k =0 t k ) = P n − k =0 [ ˆ L − ¯ L ] ◦ ˆ τ k ( ω, t ) ˆ µ a.e.Proof. Part 1. We show that U is sublinear. Let K := sup ω ∈ Ω , k t k≤ [ L ( ω, t ) − ¯ L ].Fix t ∈ R d and choose the unique integer n such that n − ≤ k t k < n . Let t k = kn t for k = 0 , . . . , n −
1. Then the subadditive cocycle property implies, on the onehand, ∀ ω ∈ Ω , ∀ t ∈ R d , U ( ω, t ) ≤ n − X k =0 U ( τ t k ( ω ) , t k +1 − t k ) ≤ nK ≤ (1 + k t k ) K. On the other hand, thanks to the hypothesis U ( ω, ≥
0, we get the oppositeinequality ∀ ω ∈ Ω , ∀ t ∈ R d , U ( ω, t ) ≥ U ( ω, − U ( τ t ( ω ) , − t ) ≥ − (1 + k t k ) K. We also have shown that U is finite everywhere. Part 2.
Suppose µ is minimizing. Since ∀ ω ∈ Ω , ∀ t , . . . , t n − ∈ R d , n − X k =0 (cid:2) ˆ L − ¯ L (cid:3) ◦ ˆ τ k ( ω, t ) ≥ U (cid:16) ω, n − X k =0 t k (cid:17) , by integrating with respect to ˆ µ , the left hand side has a null integral whereasthe right hand side has a nonnegative integral. The previous inequality is thus anequality that holds almost everywhere. Lemma 24. If L is C coercive, then the Ma˜n´e subadditive cocycle Φ is uppersemi-continuous and Mather-calibrated. In particular, Φ = L − ¯ L on Mather( L ) ,or more precisely, for every µ ∈ M min ( L ) , being ˆ µ its Markov extension, ∀ ( ω, t ) ∈ supp(ˆ µ ) , ∀ i < j, Φ (cid:16) τ P i − k =0 t k ( ω ) , j − X k = i t k (cid:17) = j − X k = i (cid:2) L − ¯ L (cid:3) ◦ ˆ τ k ( ω, t ) . In an equivalent manner, if ( ω, t ) ∈ supp(ˆ µ ) , x = 0 and x k +1 = x k + t k , ∀ k ≥ ,the semi-infinite configuration { x k } k ≥ is calibrated for E ω as in definition 7: ∀ i < j, S ω ( x i , x j ) = E ω ( x i , x i +1 , . . . , x j ) − ( j − i ) ¯ E. Garibaldi, Petite and Thieullen
Proof. Part 1.
We first show the existence of a particular measurable Mather-calibrated subadditive cocycle U ( ω, t ). From the sup-inf formula (proposition 15),for every p ≥
1, there exists u p ∈ C (Ω) such that ∀ ω ∈ Ω , ∀ t ∈ R d , u p ◦ τ t ( ω ) − u p ( ω ) ≤ L ( ω, t ) − ¯ L + 1 /p. Let U p ( ω, t ) := u p ◦ τ t ( ω ) − u p ( ω ) and U := lim sup p → + ∞ U p . Then U is clearly asubadditive cocycle and satisfies U ( ω,
0) = 0. Besides, U is finite everywhere, since0 = U ( ω, ≤ U ( ω, t ) + U ( τ t ( ω ) , − t ) and U ( ω, t ) ≤ L ( ω, t ) − ¯ L . We just verify thelast property in definition 22. Let µ ∈ M hol be such that R L dµ < + ∞ . For n ≥ S n,p ( ω, t ) := n − X k =0 h ˆ L − ¯ L + 1 p i ◦ ˆ τ k ( ω, t ) − U p (cid:16) ω, n − X k =0 t k (cid:17) ≥ . Since U p (cid:16) ω, n − X k =0 t k (cid:17) = n − X k =0 ˆ U p ◦ ˆ τ k ( ω, t ) , ˆ U p ( ω, t ) := U p ( ω, t ) , by integrating with respect to ˆ µ , we obtain0 ≤ Z inf p ≥ q ˆ S n,p d ˆ µ ≤ inf p ≥ q Z ˆ S n,p ( ω, t ) d ˆ µ ≤ n Z h L − ¯ L + 1 q i dµ. By Lebesgue’s monotone convergence theorem, as q → + ∞ , we have Z h n ( ˆ L − ¯ L ) − U (cid:16) ω, n − X k =0 t k (cid:17)i d ˆ µ ≤ Z n [ L − ¯ L ] dµ and Z U (cid:16) ω, n − X k =0 t k (cid:17) ˆ µ ( dω, dt ) ≥ . Part 2.
We next show that Φ is Mather-calibrated. We have already noticedthat Φ satisfies the subadditive cocycle property, Φ ≤ L − ¯ L , Φ( ω, ≥
0, andΦ( ω, t ) is finite everywhere. Moreover, Φ( ω, t ) ≥ U ( ω, t ) and the third property ofdefinition 22 follows from part 1. Part 3.
We show that Φ is upper semi-continuous. For n ≥
1, let S n ( ω, t ) := inf { E ω ( x , . . . , x n ) : x = 0 , x n = t } . Then Φ = inf n ≥ ( S n − n ¯ E ) is upper semi-continuous if we prove that S n ( ω, t )is continuous on every bounded set with ω ∈ Ω and k t k ≤ D . Denote c :=inf ω,x,y E ω ( x, y ) and K := sup ω ∈ Ω , k t k≤ D E ω (0 , . . . , , t ). By coerciveness, thereexists R > ∀ x, y ∈ R d , k y − x k > R ⇒ ∀ ω ∈ Ω , E ω ( x, y ) > K − ( n − c . Suppose ω, x , . . . , x n are such that E ω ( x , . . . , x n ) ≤ K . Suppose by contradictionthat k x k +1 − x k k > R for some k ≥
0. Then K ≥ E ω ( x , . . . , x n ) ≥ ( n − c + E ω ( x k , x k +1 ) > K, alibrated configurations in almost-periodic environments S n ( ω, t ),for every ω ∈ Ω and k t k ≤ D , can be realized by some points k x k k ≤ kR . By theuniform continuity of E ω ( x , . . . , x n ) on the product space Ω × Π k {k x k k ≤ kR } , weobtain that S n is continuous on Ω × {k t k ≤ D } . Part 4.
Let µ be a minimizing measure with Markov extension ˆ µ . We show thatevery ( ω, t ) in the support of ˆ µ is calibrated. LetˆΣ := n ( ω, t ) ∈ Ω × ( R d ) N : ∀ n ≥ , Φ (cid:16) ω, n − X k =0 t k (cid:17) ≥ n − X k =0 (cid:2) L − ¯ L (cid:3) ◦ ˆ τ k ( ω, t ) o . The set ˆΣ is closed, since Φ is upper semi-continuous. By lemma 23, ˆΣ has fullˆ µ -measure and therefore contains supp(ˆ µ ). Thanks to the subadditive cocycle prop-erty of Φ and the ˆ τ -invariance of supp(ˆ µ ), we obtain the calibration property ∀ ( ω, t ) ∈ ˆΣ , ∀ ≤ i < j, Φ (cid:16) τ x i ( ω ) , j − X k = i t k (cid:17) = j − X k = i (cid:2) L − ¯ L (cid:3) ◦ ˆ τ k ( ω, t ) . Proof of proposition 14 – Item 2.
We now assume that L is superlinear. Fromlemma 23, the Ma˜n´e subadditive cocycle is at most linear. There exists R > ∀ ω ∈ Ω , ∀ t ∈ R d , | Φ( ω, t ) | ≤ R (1 + k t k ) . By superlinearity, there exists
B > ∀ ω ∈ Ω , ∀ t ∈ R d , L ( ω, t ) ≥ R k t k − B. Let µ be a minimizing measure. Since Φ = L − ¯ L µ a.e. (lemma 23), we obtain k t k ≤ ( R + B + | ¯ L | ) /R, µ ( dω, dt ) a.e.We have proved that the support of every minimizing measure is compact. Inparticular, the Mather set is compact. Proof of theorem 8.
We show that, for every environment ω in the projected Matherset, there exists a calibrated configuration for E ω passing through the origin. Let µ be a minimizing measure such that supp( µ ) = Mather( L ). Let ˆ µ denote its Markovextension. For n ≥
1, considerˆΩ n := n ( ω, t ) ∈ Ω × ( R d ) N : Φ (cid:16) ω, n − X k =0 t k (cid:17) ≥ n − X k =0 (cid:2) L − ¯ L (cid:3) ◦ ˆ τ k ( ω, t ) o . From lemma 24, supp(ˆ µ ) ⊆ ˆΩ n . From the upper semi-continuity of Φ, ˆΩ n is closed.To simplify the notations, for every t , we define a configuration ( x , x , . . . ) by x = 0 , x k +1 = x k + t k so that ˆ τ k ( ω, t ) = ( τ x k ( ω ) , ( t k , t k +1 , . . . )) . Garibaldi, Petite and Thieullen
Notice that, if ( ω, t ) ∈ ˆΩ n , thanks to the subadditive cocycle property of Φ andthe fact that Φ ≤ L − ¯ L , the finite configuration ( x , . . . , x n ) is calibrated in theenvironment ω , that is, ∀ ≤ i < j ≤ n, Φ (cid:16) τ x i ( ω ) , j − X k = i t k (cid:17) = j − X k = i (cid:2) L − ¯ L (cid:3) ◦ ˆ τ k ( ω, t ) , or written using the family of interaction energies E ω , ∀ ≤ i < j ≤ n, S ω ( x i , x j ) = E ω ( x i , . . . , x j ) − ( j − i ) ¯ E. Thanks to the sublinearity of S ω , there exists a constant R > ω ∈ Ω and x, y ∈ R d , we have | S ω ( x, y ) | ≤ R (1 + k y − x k ). Besides, thanks tothe superlinearity of E ω , there exists a constant B > E ω ( x, y ) ≥ R k y − x k − B . Since S ω ( x k , x k +1 ) = E ω ( x k , x k +1 ) − ¯ E , we thus obtain a uniformupper bound D := ( R + B + | ¯ E | ) /R on the jumps of calibrated configurations: ∀ ( ω, t ) ∈ ˆΩ n , ∀ ≤ k < n, k x k +1 − x k k ≤ D. Let ˆΩ ′ n = ˆ τ n ( ˆΩ n ). Thanks to the uniform bounds on the jumps, ˆΩ ′ n is again closed.Since ˆ µ ( ˆΩ n ) = 1, ˆ µ ( ˆΩ ′ n ) = 1 by invariance of ˆ τ . Let ν := pr ∗ ( µ ) be the projectedmeasure on Ω. Then supp( ν ) = pr (Mather( L )). By the definition of ˆΩ ′ n , we haveˆ pr ( ˆΩ ′ n ) = { ω ∈ Ω : ∃ ( x − n , . . . , x n ) ∈ R d s.t. x = 0 and S ω ( x − n , x n ) ≥ E ω ( x − n , . . . , x n ) − n ¯ E } . Again by compactness of the jumps, ˆ pr ( ˆΩ ′ n ) is closed and has full ν -measure. Thus,ˆ pr ( ˆΩ ′ n ) ⊇ pr (Mather( L )). By a diagonal extraction procedure, we obtain, for every ω ∈ Mather( L ), a bi-infinite calibrated configuration with uniformly bounded jumpspassing through the origin. Perhaps the most powerful assumption made in one-dimensional Aubry theory [2]is the twist property. It will not be used here in the infinitesimal form. Supposingthat (Ω , { τ t } t ∈ R , L ) is weakly twist (definition 9), we discuss in this section keyproperties on the ordering of minimizing configurations and therefore of calibratedconfigurations. The fundamental Aubry crossing property is explained in lemma 25.We collect in lemmas 26 and 28 intermediate results, that are consequences ofthe weakly twist property, about the order of the points composing a minimizingconfiguration. Such results will be applied in the proof of theorem 10 in section 4.The following lemma is an easy consequence of the definition. It shows that theenergy of a configuration can be lower by exchanging the positions. Lemma 25 ( Aubry crossing lemma). If L is weakly twist, then, for every ω ∈ Ω ,for every x , x , y , y ∈ R satisfying ( y − x )( y − x ) < , (cid:2) E ω ( x , x ) + E ω ( y , y ) (cid:3) − (cid:2) E ω ( x , y ) + E ω ( y , x ) (cid:3) = α ( y − x )( y − x ) > , with α = y − x )( y − x ) R y x R y x ∂ E ω ∂x∂y ( x, y ) dydx < . alibrated configurations in almost-periodic environments Proof.
The inequality is obtained by integrating the function ∂ ∂x∂y E ω on the domain[min( x , y ) , max( x , y )] × [min( x , y ) , max( x , y )].The first consequence of Aubry crossing lemma is that minimizing configurationsshall be strictly ordered. We begin by an intermediate lemma. Lemma 26.
Let L be a weakly twist Lagrangian, ω ∈ Ω , n ≥ , and x , . . . , x n ∈ R be a nonmonotone sequence (that is, a sequence which does not satisfy x ≤ . . . ≤ x n nor x ≥ . . . ≥ x n ).– If x = x n , then E ω ( x , . . . , x n ) > P n − i =0 E ω ( x i , x i ) .– If x = x n , then there exists a subset { i , i , . . . , i r } of { , . . . , n } , with i = 0 and i r = n , such that ( x i , x i , . . . , x i r ) is strictly monotone and E ω ( x , . . . , x n ) > E ω ( x i , . . . , x i r ) + X i i ,...,i r } E ω ( x i , x i ) . (Notice that it may happen that x i = x j for i
6∈ { i , . . . , i r } and j ∈ { i , . . . , i r } .)Proof. We prove the lemma by induction.Let x , x , x ∈ R be a nonmonotone sequence. If x = x , then E ω ( x , x , x ) >E ( x , x ) + E ω ( x , x ). If x = x then x , x , x are three distinct points. Thus, x < x implies x < x and x < x implies x < x . In both cases, lemma 25tells us that E ω ( x , x ) + E ω ( x , x ) > E ω ( x , x ) + E ω ( x , x ) . Let ( x , . . . , x n +1 ) be a nonmonotone sequence. We have two cases: either x ≤ x n or x ≥ x n . We shall only give the proof for the case x ≤ x n . Case x = x n . Then ( x , . . . , x n ) is nonmonotone and by induction E ω ( x , . . . , x n +1 ) > E ω ( x n , x n +1 ) + n − X i =0 E ω ( x i , x i )= E ω ( x , x n +1 ) + n X i =1 E ω ( x i , x i ) . The conclusion holds whether x n +1 = x or not. Case x < x n . Whether ( x , . . . , x n ) is monotone or not, we may choose asubset of indices { i , . . . , i r } such that i = 0, i r = n , x i < x i < . . . < x i r and E ω ( x , . . . , x n +1 ) ≥ (cid:16) E ω ( x i , . . . , x i r ) + X i i ,...,i r } E ω ( x i , x i ) (cid:17) + E ω ( x n , x n +1 ) . If x n ≤ x n +1 , then ( x , . . . , x n ) is necessarily nonmonotone and the previousinequality is strict. If x n = x n +1 , the lemma is proved by modifying i r = n + 1. If x n < x n +1 , the lemma is proved by choosing r + 1 indices and i r +1 = n + 1.If x n +1 < x n = x i r , by applying lemma 25, one obtains E ω ( x i r − , x i r ) + E ω ( x n , x n +1 ) > E ω ( x n , x i r ) + E ω ( x i r − , x n +1 ) ,E ω ( x , . . . , x n +1 ) > E ω ( x i , . . . , x i r − , x n +1 ) + (cid:2) X i i ,...,i r } E ω ( x i , x i ) (cid:3) + E ω ( x n , x n ) . Garibaldi, Petite and Thieullen If x i r − < x n +1 , the lemma is proved by changing i r = n to i r = n + 1. If x i r − = x n +1 , the lemma is proved by choosing r − i r − = n + 1. If x n +1 < x i r − , we apply again lemma 25 until there exists a largest s ∈ { , . . . , r } such that x s < x n +1 or x n +1 ≤ x . In the former case, the lemma is proved bychoosing s + 1 indices and by modifying i s +1 = n + 1. In the latter case, namely,when x n +1 ≤ x < x n , we have E ω ( x , . . . , x n +1 ) > E ω ( x , x n +1 ) + n X i =1 E ω ( x i , x i )and the lemma is proved whether x n +1 = x or x n +1 < x .The Ma˜n´e subadditive cocycle Φ( ω, t ) (definition 21) is obtained by minimizinga normalized energy E ω ( x , . . . , x n ) − n ¯ E on all the configurations satisfying x = 0and x n = t . The following lemma shows that it is enough to minimize on strictlymonotone configurations (unless t = 0). Corollary 27. If L is weakly twist, then, for every ω ∈ Ω , the Ma˜n´e subadditivecocycle Φ( ω, t ) satisfies:– if t = 0 , Φ( ω,
0) = E ω (0 , − ¯ E ,– if t > , Φ( ω, t ) = inf n ≥ inf x
Suppose that L is weakly twist. Then for every ω ∈ Ω , if ( x , . . . , x n ) is a minimizing configuration for E ω , with x = x n , such that x i is strictly between x and x n for every < i < n − , then ( x , . . . , x n ) is strictly monotone.Proof. Let ( x , . . . , x n ) be such a minimizing sequence. We show, in part 1, it ismonotone, and, in part 2, it is strictly monotone. alibrated configurations in almost-periodic environments Part 1.
Assume by contradiction that ( x , . . . , x n ) is not monotone. Accordingto lemma 26, one can find a subset of indices { i , . . . , i r } of { , . . . , n } , with i = 0and i r = n , such that ( x i , . . . , x i r ) is strictly monotone and E ω ( x , . . . , x n ) > E ω ( x i , . . . , x i r ) + X i i ,...,i r } E ω ( x i , x i ) . We choose the largest integer r with the above property. Since ( x , . . . , x n ) is notmonotone, we have necessarily r < n . Since ( x , . . . , x n ) is minimizing, one canfind i
6∈ { i , . . . , i r } such that x i
6∈ { x i , . . . , x i r } . Let s be one of the indices of { , . . . , r } such that x i is between x i s and x i s +1 . Then, by lemma 25, E ω ( x i s , x i s +1 ) + E ω ( x i , x i ) > E ω ( x i s , x i ) + E ω ( x i , x i s +1 ) . We have just contradicted the maximality of r . Therefore, ( x , . . . , x n ) must bemonotone. Part 2.
Assume by contradiction that ( x , . . . , x n ) is not strictly monotone.Then ( x , . . . , x n ) contains a subsequence of the form ( x i − , x i , . . . , x i + r , x i + r +1 )with r ≥ x i − = x i = . . . = x i + r = x i + r +1 . To simplify the proof, we assume x i − < x i + r +1 . We want built a configuration ( x ′ i − , x ′ i , . . . , x ′ i + r , x ′ i + r +1 ) so that x ′ i − = x i − , x ′ i + r +1 = x i + r +1 and E ω ( x i − , x i , . . . , x i + r , x i + r +1 ) > E ω ( x ′ i − , x ′ i , . . . , x ′ i + r , x ′ i + r +1 ) . Indeed, since ( x i − , . . . , x i + r +1 ) is minimizing, we have E ω ( x i − , . . . , x i + r +1 ) = E ω ( x i − , x i + ǫ, x i +1 − ǫ, . . . , x i + r − ǫ, x i + r +1 ) + o ( ǫ ) . Let α = 1 x i − x i − Z x i x i − ∂ E ω ∂x∂y ( x, x i ) dx < ,β = 1 x i + r +1 − x i + r Z x i + r +1 x i + r ∂ E ω ∂x∂y ( x i + r , y ) dy < . By Aubry crossing lemma, E ω ( x i − , x i + ǫ ) + E ω ( x i + ǫ, x i +1 − ǫ )= E ω ( x i − , x i +1 − ǫ ) + E ω ( x i + ǫ, x i + ǫ ) − ǫ ( x i − x i − ) α + o ( ǫ ) . Since x i = x i + r , obviously E ω ( x i + ǫ, x i + ǫ ) = E ω ( x i + r + ǫ, x i + r + ǫ ). Again byAubry crossing lemma, E ω ( x i + r + ǫ, x i + r + ǫ ) + E ω ( x i + r − ǫ, x i + r +1 )= E ω ( x i + r − ǫ, x i + r + ǫ ) + E ω ( x i + r + ǫ, x i + r +1 ) − ǫ ( x i + r +1 − x i + r ) β + o ( ǫ ) . Then, for ǫ small enough, we have E ω ( x i − , . . . , x i + r +1 ) > E ω ( x i − , x i − ǫ, . . . , x i − r − − ǫ, x i + r + ǫ, x i + r +1 ) , which contradicts that ( x i − , . . . , x i + r +1 ) is minimizing. We have thus proved that( x , . . . , x n ) is strictly monotone.4 Garibaldi, Petite and Thieullen
Our purpose in this section is to provide a rich variety of examples of almostcrystalline interaction models. We first recall the basic definitions and propertiesconcerning quasicrystals. More details on such a motivating concept can be found,for instance, in [4, 18, 19]. Associated with quasicrystals, we will consider stronglyequivariant functions (an inspiration to our concept of locally transversally constantLagrangian to be introduced in section 3.2). We recall their main properties hereand we refer the reader to [12, 17] for the proofs.
Definition of a quasicrystal.
For a discrete set ω ⊂ R , a ρ - patch , or a pattern for short, is a finite set P of the form ω ∩ B ρ ( x ) for some x ∈ ω and some constant ρ >
0, where B ρ ( x ) denotes the open ball of radius ρ centered in x . We say that y ∈ ω is an occurrence of P if ω ∩ B ρ ( y ) is equal to P up to a translation. A quasicrystal is a discrete set ω ⊂ R satisfying– finite local complexity : for any ρ > ω has just a finite number of ρ -patchesup to translations;– repetitivity : for all ρ >
0, there exists M ( ρ ) > M ( ρ ) contains at least one occurrence of every ρ -patch of ω ;– uniform pattern distribution : for any pattern P of ω , uniformly in x ∈ R , thefollowing positive limit existslim r → + ∞ { y ∈ R : y is an occurrence of P } ∩ B r ( x ))Leb( B r ( x )) = ν ( P ) > . Notice that the finite local complexity is equivalent to the fact that the inter-section of the difference set ω − ω with any bounded set is finite. Basic examples ofone-dimensional quasicrystals are the lattice Z and the Beatty sequences defined by ω ( α ) = { n ∈ Z : ⌊ nα ⌋ − ⌊ ( n − α ⌋} for α ∈ (0 , α is irrationalas in example 4, the set ω ( α ) provides a non periodic quasicrystal for which therepetitively and the uniform pattern distribution are due to the minimality and theunique ergodicity of an irrational rotation on the circle. For details, we refer to [19].Note that, from the definition, when ω is a quasicrystal, then the discrete set ω + t , obtained by translating any point of ω by t ∈ R , is also a quasicrystal. Aquasicrystal is said to be aperiodic if ω + t = ω implies t = 0, and periodic otherwise.For Beatty sequences, it is simple to check that the quasicrystal ω ( α ) is aperiodicif, and only if, α is irrational. Hull of a quasicrystal.
Given a quasicrystal ω ∗ ⊂ R , we will equip the set ω ∗ + R of all the translations of ω ∗ with a topology that reflects its combinatorialproperties: the Gromov-Hausdorff topology. Roughly speaking, two quasicrystals alibrated configurations in almost-periodic environments ω and ω two translations of ω ∗ , their distance is D ( ω, ω ) := inf (cid:8) r + 1 : ∃ | t | , | t | < r s.t. ( ω + t ) ∩ B r (0) = ( ω + t ) ∩ B r (0) (cid:9) . The continuous hull Ω( ω ∗ ) of the quasicrystal ω ∗ is the completion of this metricspace. The finite local complexity hypothesis implies that Ω( ω ∗ ) is a compactmetric space and that any element ω ∈ Ω( ω ∗ ) is a quasicrystal which has the samepatterns as ω ∗ up to translations (see [18, 4]). Moreover, Ω( ω ∗ ) is equipped with acontinuous R -action given by the homeomorphisms τ t : ω ω − t for ω ∈ Ω( ω ∗ ) . The dynamical system (Ω( ω ∗ ) , { τ t } t ∈ R ) has a dense orbit, namely, the orbitof ω ∗ . Actually, the repetitivity hypothesis is equivalent to the minimality of theaction, and so any orbit is dense. The uniform pattern distribution is equivalentto the unique ergodicity : the R -action has a unique invariant probability measure.For details on these properties, we refer the reader to [18, 4]. We summarize thesefacts in the following proposition. Proposition 29 ([18, 4]) . Let ω ∗ be a quasicrystal of R . Then the dynamicalsystem (Ω( ω ∗ ) , { τ t } t ∈ R ) is minimal and uniquely ergodic. Flow boxes.
The canonical transversal Ξ ( ω ∗ ) of the hull Ω( ω ∗ ) of a quasicrystalis the set of quasicrystals ω in Ω( ω ∗ ) such that the origin 0 belongs to ω . A basis ofthe topology on Ξ ( ω ∗ ) is given by cylinder sets Ξ ω,ρ with ω ∈ Ξ ( ω ∗ ) and ρ >
0. Ingeneral, that is, for every ω ∈ Ω( ω ∗ ) and ρ > ω ∩ B ρ (0) = ∅ , a cylinderset Ξ ω,ρ is defined byΞ ω,ρ := { ω ∈ Ω( ω ∗ ) : ω ∩ B ρ (0) = ω ∩ B ρ (0) } . If ω ∈ Ξ ( ω ∗ ), then Ξ ω,ρ ⊂ Ξ ( ω ∗ ).The designation of transversal comes from the obvious fact that the set Ξ ( ω ∗ )is transverse to the action: for any real t small enough, we have τ t ( ω ) Ξ ( ω ∗ ) forany ω ∈ Ξ ( ω ∗ ). This gives a Poincar´e section. Proposition 30 ([18]) . The canonical transversal Ξ ( ω ∗ ) and the cylinder sets Ξ ω,ρ associated with an aperiodic quasicrystal ω ∗ are Cantor sets. If ω ∗ is a periodicquasicrystal, these sets are finite. This allows us to give a more dynamical description of the hull in one dimensionby considering the return time function Θ : Ξ ( ω ∗ ) → R + defined byΘ( ω ) := inf { t > τ t ( ω ) ∈ Ξ ( ω ∗ ) } , ∀ ω ∈ Ξ ( ω ∗ ) . Garibaldi, Petite and Thieullen
The finite local complexity implies that this function is locally constant. The firstreturn map T : Ξ ( ω ∗ ) → Ξ ( ω ∗ ) is then given by T ( ω ) := τ Θ( ω ) ( ω ) , ∀ ω ∈ Ξ ( ω ∗ ) . Remark that the unique invariant probability measure on Ω( ω ∗ ) induces a finitemeasure on Ξ ( ω ∗ ) that is T -invariant (see [12]).It is straightforward to check that the dynamical system (Ω( ω ∗ ) , { τ t } t ∈ R ) isconjugate to the suspension of the map T on the set Ξ ( ω ∗ ) with the time mapgiven by the function Θ. Thus, when ω ∗ is periodic, the hull Ω( ω ∗ ) is homeomorphicto a circle. Otherwise, Ω( ω ∗ ) has a laminated structure: it is locally the Cartesianproduct of a Cantor set by an interval.To be more precise, in the aperiodic case, for every ω ∈ Ω( ω ∗ ) and r >
0, if ρ islarge enough, the set U ω,ρ,r := { ω − t : t ∈ B r (0) , ω ∈ Ξ ω,ρ } is open and homeomorphic to B r (0) × Ξ ω,ρ by the map ( t, ω ) → τ t ( ω ) = ω − t .Their collection forms a base for the topology of Ω( ω ∗ ). In this case, U ω,ρ,r is called a flow box of the cylinder set Ξ ω,ρ .The next lemma improves the fact that the return time is locally constant. Lemma 31 ([4]) . Let ω ∗ be an aperiodic quasicrystal. Let U i := U ω i ,ρ i ,r i , i = 1 , ,be two flow boxes such that U ∩ U = ∅ . Then there exists a real number a ∈ R such that, for every ω i ∈ Ξ ω i ,ρ i , | t i | < r i , i = 1 , , ω − t = ω − t = ⇒ t = t − a. Strongly equivariant function.
Associated with a quasicrystal ω ∗ of R , wewill consider strongly ω ∗ -equivariant functions, as introduced in [17]. A potential V ω ∗ : R → R is said to be strongly ω ∗ - equivariant (with range R ) if there exists aconstant R > V ω ∗ ( x ) = V ω ∗ ( y ) , ∀ x, y ∈ R with ( B R ( x ) ∩ ω ∗ ) − x = ( B R ( y ) ∩ ω ∗ ) − y. Of course any periodic potential is strongly equivariant with respect to a discretelattice of periods. In example 4, the function V ω ( α ) is strongly ω ( α )-equivariant withrange R = ⌊ α ⌋ + 1. Let us mention another example from [17], which holds for anyquasicrystal ω ∗ . Let δ := P x ∈ ω ∗ δ x be the Dirac comb supported on the points ofa quasicrystal ω ∗ and let g : R → R be a smooth function with compact support.Then, one may check that the convolution product δ ∗ g is a smooth strongly ω ∗ -equivariant function. Actually, any strongly ω -equivariant function can be definedby a similar procedure [17].A strongly equivariant potential factorizes through a continuous function on thehull Ω( ω ∗ ). More precisely, the following lemma shows that strongly ω ∗ -equivariantfunctions arise from functions on the space Ω( ω ∗ ) that are constant on the cylindersets. alibrated configurations in almost-periodic environments Lemma 32 ([12, 17]) . Given a quasicrystal ω ∗ of R , let V ω ∗ : R → R be a contin-uous strongly ω ∗ -equivariant function with range R . Then, there exists a uniquecontinuous function V : Ω( ω ∗ ) → R such that V ω ∗ ( x ) = V ◦ τ x ( ω ∗ ) , ∀ x ∈ R . Moreover, V is constant on any cylinder set Ξ ω,R + S , with ω ∈ Ω( ω ∗ ) and S ≥ . Inaddition, if V ω ∗ is C , then V is C along the flow (that is, for all ω , the function x ∈ R V ( τ x ( ω )) is C ). Note that, for every
S >
0, the function V : Ω( ω ∗ ) → R is transversally constant on each flow box U ω,R + S,S , that is, V ( τ x ( ω )) = V (cid:0) τ x ( ω ′ ) (cid:1) , ∀ | x | < S, ∀ ω, ω ′ ∈ Ξ ω,R + S . This comes from the fact that τ x ( ω ′ ) ∈ Ξ τ x ( ω ) ,R whenever ω, ω ′ ∈ Ξ ω,R + S and | x | < S , since V is constant on such cylinder sets. In order to complete the definition of almost crystalline interaction models (defini-tion 9), we introduce here the technical concept of a locally transversally constantLagrangian that we adopt in this paper. By doing this, we focus on a class of mod-els whose typical examples are provided by suspensions of minimal and uniquelyergodic homeomorphisms on a Cantor set, with locally constant ceiling functions.Such a modeling approach enables us to consider general R -actions, as, for instance,equicontinuous, distal or expansive ones, whereas, in the aperiodic quasicrystal case,one deals always with expansive actions. We also show that strongly equivariantfunctions associated with a quasicrystal provide locally transversally constant La-grangians.In topological dynamics, the study of minimal homeomorphisms on a Cantorset has been enriched by an invaluable combinatorial description of the system viaKakutani-Rohlin towers (see, for instance, [15]). Using a similar strategy in ourcontext, we describe, in a second part, the transverse measures associated with theprobability measures on the space Ω invariant by the flow τ . Characterized by theaverage frequency of return times to a particular transverse section of the flow,these measures are key ingredients in the proof of theorem 10. Precise notion of locally constant Lagrangians.
Our definition of a locallytransversally constant Lagrangian is based on (topological) flow boxes, transversesections, and flow box decompositions. Even if we consider only the one-dimensionalcase, these concepts can be introduced in any dimension.
Definition 33.
Let (Ω , { τ t } t ∈ R ) be an almost periodic environment.– An open set U ⊂ Ω is said to be a flow box of size R > if there exists a compactsubset Ξ ⊂ Ω , called transverse section, such that: (cid:5) the induced topology on Ξ admits a basis of closed and open subsets, calledclopen subsets, Garibaldi, Petite and Thieullen (cid:5) τ ( t, ω ) = τ t ( ω ) , ( t, ω ) ∈ R × Ξ , is a homeomorphism from B R (0) × Ξ onto U .We shall later write B R = B R (0) and τ − i ) = τ − | U i : U i → B R × Ξ for a flow box U i .– Two flow boxes U i = τ [ B R i × Ξ i ] and U j = τ [ B R j × Ξ j ] are said to be admissibleif, whenever U i ∩ U j = ∅ , there exists a i,j ∈ R such that τ − j ) ◦ τ ( t, ω ) = ( t − a i,j , τ a i,j ( ω )) , ∀ ( t, ω ) ∈ τ − i ) ( U i ∩ U j ) . – A flow box decomposition { U i } i ∈ I is a cover of Ω by admissible flow boxes. Lemma 31 implies that the hull of a quasicrystal admits a flow box decompo-sition given by flow boxes of cylinder sets [4]. Standard examplifications of thestructures formalized in definition 33 are provided by the suspensions of minimalhomeomorphisms on Cantor sets, with locally constant ceiling functions. This con-text includes expansive flows (as in the case of one-dimensional quasicrystals) andequicontinuous ones. But, in general, a minimal flow does not possess a cover offlow boxes.An interaction model does not have a canonical notion of vertical section. Such anotion occurs naturally whenever the model admits a flow box decomposition. Moreimportantly, in this situation, we give and exploit a definition of locally transversallyconstant Lagrangian.
Definition 34.
Let (Ω , { τ t } t ∈ R , L ) be an almost periodic interaction model admit-ting a flow box decomposition.– A flow box τ [ B R × Ξ] is said to be compatible with respect to a flow box decom-position { U i } i ∈ I , where U i = τ [ B R i × Ξ i ] , if for every | t | < R , there exist i ∈ I , | t i | < R i and a clopen subset ˜Ξ i of Ξ i such that τ t (Ξ) = τ t i (˜Ξ i ) .– L is said to be locally transversally constant with respect to a flow box decompo-sition { U i } i ∈ I if, for every flow box τ [ B R × Ξ] compatible with respect to { U i } i ∈ I , ∀ ω, ω ′ ∈ Ξ , ∀ | x | , | y | < R, E ω ′ ( x, y ) = E ω ( x, y ) . As in examples 3 and 4, interaction models with weakly twist and locallytransversally constant Lagrangians can be easily built when the interaction energyhas the form E ω ( x, y ) = W ( y − x ) + V ( τ x ( ω )) + V ( τ y ( ω )), where W is superlinearweakly convex (namely, W is C , W ′′ > | W ′ ( t ) | → + ∞ as | t | → + ∞ ),and V and V are locally transversally constant, in the sense described below. Definition 35.
Let (Ω , { τ t } t ∈ R , L ) be an almost periodic interaction model. Afunction V : Ω → R is said to be locally transversally constant with respect to aflow box decomposition { U i } i ∈ I , where U i = τ ( B R i × Ξ i ) , if ∀ i ∈ I, ∀ ω, ω ′ ∈ Ξ i , ∀ | x | < R i , V ( τ x ( ω )) = V ( τ x ( ω ′ )) . Notice that, in example 5, the locally transversally constant property does nothold. We check in the next lemma that locally transversally constant functions V , V : Ω → R indeed enable to construct a transversally constant Lagrangian. Lemma 36.
Let (Ω , { τ t } t ∈ R , L ) be an almost periodic interaction model admittinga flow box decomposition. Let V , V : Ω → R be two locally transversally constantfunctions on the same flow box decomposition, and W = R → R be any function.Define L ( ω, t ) = W ( t ) + V ( ω ) + V ( τ t ( ω )) . Then L is locally transversally constant. alibrated configurations in almost-periodic environments Proof.
Assume V and V are locally transversally constant on a flow box decom-position { U i } i ∈ I . Let τ [ B R × Ξ] be a flow box which is compatible with respect to { U i } i ∈ I . If | x | , | y | < R and ω, ω ′ ∈ Ξ, then E ω ( x, y ) = W ( y − x ) + V ,ω ( x ) + V ,ω ( y ) . There exist i ∈ I , | t i | < R i and ˜Ξ i a clopen subset of Ξ i such that τ x (Ξ) = τ t i (˜Ξ i ).Then τ x ( ω ) = τ t i ( ω i ) and τ x ( ω ′ ) = τ t i ( ω ′ i ) for some ω i , ω ′ i ∈ ˜Ξ i . We have V ,ω ( x ) = V ,ω i ( t i ) = V ,ω ′ i ( t i ) = V ,ω ′ ( x ) . Similarly V ,ω ( y ) = V ,ω ′ ( y ). We have thus proved E ω ′ ( x, y ) = E ω ( x, y ).To give a concrete example of a family of locally transversally constant La-grangians for which the conclusions of Theorem 10 hold, let us recall that a con-tinuous function V : Ω → R is C along the flow if, for each ω ∈ Ω, the function x ∈ R V ( τ x ( ω )) is C . Corollary 37.
Let (Ω , { τ t } t ∈ R ) be an almost periodic environment admitting a flowbox decomposition. Let V , V : Ω → R be C locally transversally constant functions(on the same flow box decomposition) that are C along the flow. Let W : R → R be a C superlinear weakly convex function. Define L ( ω, t ) = W ( t ) + V ( ω ) + V ( τ t ( ω )) . Then L is C , superlinear, weakly twist and locally transversally constant. If more-over (Ω , { τ t } t ∈ R ) is uniquely ergodic, then (Ω , { τ t } t ∈ R , L ) is an almost crystallineinteraction model and all conclusions of theorem 10 apply. Kakutani-Rohlin tower description of the system.
Flow boxes are opensets obtained by taking the union of every orbits of size R starting from any pointbelonging to a closed transverse Poincar´e section. The restricted topology on atransverse section must be special: it must admit a basis of clopen sets. We recallin lemma 40 how to construct a suspension with locally constant return mapscalled Kakutani-Rohlin tower. When the flow is uniquely ergodic, we describe inthe lemmas 41 and 42 how this Kakutani-Rohlin tower enables to characterize theunique transverse measure associated with each transverse section.We begin with some basic properties of systems with a flow box decomposition.Since the proof of the next lemma is standard, we leave it to the reader. Lemma 38.
Let (Ω , { τ t } t ∈ R ) be an almost periodic environment. Assume that theaction is not periodic ( t ∈ R τ t ( ω ) ∈ Ω is injective for every ω ∈ Ω ). Then1. If τ [ B R × Ξ] is a flow box, then there exists R ′ such that Ω = τ [ B R ′ × Ξ] = { τ t ( ω ) : | t | < R ′ and ω ∈ Ξ } .
2. If τ [ B R × Ξ] is a flow box, then τ : R × Ξ → Ω is open and τ [ B R × Ξ ′ ] is againa flow box for every clopen subset Ξ ′ ⊂ Ξ . Garibaldi, Petite and Thieullen
3. If τ [ B R × Ξ] is a flow box, then, for every R ′ > and ω ∈ Ξ , there exists aclopen set Ξ ′ ⊂ Ξ containing ω such that τ [ B R ′ × Ξ ′ ] is again a flow box.4. If U = τ [ B R × Ξ] and U ′ = τ [ B R ′ × Ξ ′ ] are two admissible flow boxes, if τ [ B R +2 R ′ × Ξ] and τ [ B R +2 R ′ × Ξ ′ ] are also flow boxes, then U ∩ U ′ = τ ( ˜ B × ˜Ξ) = τ ( ˜ B ′ × ˜Ξ ′ ) for some clopen sets ˜Ξ , ˜Ξ ′ and some open convex subsets ˜ B ⊂ B R , ˜ B ′ ⊂ B R ′ .5. If { U i } i ∈ I is a flow box decomposition, then, for every ω ∈ Ω and R > , thereexits a flow box τ [ B R × Ξ] , with a transverse section Ξ containing ω , that iscompatible with respect to { U i } i ∈ I . The existence of a flow box decomposition enables us to build a global transversesection of the flow with locally constant return times. We extend for an almostperiodic interaction model what has been done for quasicrystals in [12]. We firstdefine the notion of Kakutani-Rohlin tower and show that an interaction modelpossessing a flow box decomposition admits a Kakutani-Rohlin tower.
Definition 39.
Let (Ω , { τ t } t ∈ R ) be a one-dimensional almost periodic environmentpossessing a flow box decomposition { U i } i ∈ I . We call Kakutani-Rohlin tower apartition { F α } α ∈ A of Ω of the form F α = τ (cid:0) [0 , H α ) × Σ α (cid:1) = ∪ ≤ t
Let (Ω , { τ t } t ∈ R ) be a one-dimensional almost periodic environmentpossessing a flow box decomposition { U i } i ∈ I . Then there exists a Kakutani-Rohlintower { F α } α ∈ A which is compatible with respect to { U i } i ∈ I .Proof. Let { U i } ni =1 be a flow box decomposition, where U i = τ [ B R i × Ξ i ]. Bydefinition, U i is an open set of Ω. We denote V i := τ (cid:0) [ − R i , R i ) × Ξ i (cid:1) . We shallbuild by induction on i = 1 , . . . , n a collection of flow boxes { τ (cid:0) (0 , H i,j ) × Σ i,j (cid:1) } j such that– the sets F i,j := τ (cid:0) [0 , H i,j ) × Σ i,j (cid:1) are pairwise disjoint,– V i \ ∪ k H l ∗ at each step of the construction.2 Garibaldi, Petite and Thieullen
We now assume that the flow (Ω , { τ t } t ∈ R ) is uniquely ergodic. Let λ be theunique ergodic invariant probability measure. The average frequency of returntimes to a transverse section of a flow box measures the thickness of the section.The next lemma gives a precise definition of a family of transverse measures { ν Ξ } Ξ parameterized by every transverse section Ξ. Lemma 41.
Let (Ω , { τ t } t ∈ R ) be an almost periodic and uniquely ergodic environ-ment. Given Ξ a transverse section, let R Ξ ( ω ) be the set of return times to Ξ , R Ξ ( ω ) := { t ∈ R : τ t ( ω ) ∈ Ξ } , ∀ ω ∈ Ω . Then, for every nonempty clopen set Ξ ′ ⊂ Ξ , the following limit exists uniformlywith respect to ω ∈ Ω and is positive: ν Ξ (Ξ ′ ) := lim T → + ∞ R Ξ ′ ( ω ) ∩ B T (0))Leb( B T (0)) > . Moreover, ν Ξ extends to a finite and nonnegative measure on Ξ , called transversemeasure to Ξ , and, for every flow box U = τ [ B R × Ξ] , λ ( τ ( B ′ × Ξ ′ )) = Leb( B ′ ) ν Ξ (Ξ ′ ) , ∀ B ′ ⊂ B R (0) , ∀ Ξ ′ ⊂ Ξ (
Borel sets ) . Proof.
Let U = τ [ B R × Ξ] be a flow box. Let t = t be two return times of R Ξ ( ω ).Since τ is injective on B R (0) × Ξ, it is straightforward that B R ( t ) ∩ B R ( t ) = ∅ .For ω ∈ Ω and
T >
0, consider µ T,ω ( U ′ ) = 1Leb( B T (0)) Z B T (0) U ′ ( τ s ( ω )) ds, ∀ U ′ ⊂ Ω (Borel set) . The unique ergodicity of the action implies that, for all φ ∈ C (Ω), µ T,ω ( φ ) con-verges uniformly in ω to λ ( φ ) as T → + ∞ . Let B ′ ⊂ B R (0) be a Borel set andΞ ′ ⊂ Ξ be a nonempty clopen set. For U ′ = τ ( B ′ × Ξ ′ ), notice then that { s ∈ R : τ s ( ω ) ∈ U ′ } = [ t ∈ R Ξ ′ ( ω ) t + B ′ , µ T,ω ( U ′ ) = X t ∈ R Ξ ′ ( ω ) Leb( B T (0) ∩ ( t + B ′ ))Leb( B T (0)) , and, whenever T > R ,Leb( B ′ ) B T − R (0) ∩ R Ξ ′ ( ω ))Leb( B T (0)) ≤ µ T,ω ( U ′ ) ≤ Leb( B ′ ) B T + R (0) ∩ R Ξ ′ ( ω ))Leb( B T (0)) . Moreover, clearly B T (0) ∩ R Ξ ′ ( ω )) ≤ Leb( B T + R (0))Leb( B R (0)) and lim T → + ∞ Leb( B T + R (0))Leb( B T (0)) = 1.Thus, if B ′ is open in B R (0), then U ′ is open in Ω and λ ( U ′ ) ≤ lim inf T → + ∞ µ T,ω ( U ′ ) ≤ Leb( B ′ )Leb( B R (0)) . In particular, if B ′ is negligible, thanks to the regularity of Leb, λ ( U ′ ) = 0. If B ′ is open, B ′ ⊂ B R (0) and ∂B ′ is negligible, then, for every ǫ >
0, there existnonnegative continuous functions φ ≤ ψ such that φ ≤ τ ( B ′ × Ξ) ≤ τ ( B ′ × Ξ) ≤ ψ and λ ( ψ − φ ) < ǫ. alibrated configurations in almost-periodic environments µ T,ω ( τ ( B ′ × Ξ ′ )) converges uniformly in ω to λ ( τ ( B ′ × Ξ)) as T → + ∞ .On the one hand, for all clopen set Ξ ′ ⊂ Ξ, τ ( B R (0) × Ξ ′ ) is a flow box andlim T → + ∞ B T (0) ∩ R Ξ ′ ( ω ))Leb( B T (0)) := ν Ξ (Ξ ′ ) (exists uniformly in ω ) . On the other hand, for every B ′ = B R ′ ( s ′ ), s ′ ∈ B R (0), k s ′ k + R ′ < R , λ ( τ ( B ′ × Ξ ′ )) = lim T → + ∞ µ T,ω ( τ ( B ′ × Ξ ′ )) = Leb( B ′ ) ν Ξ (Ξ ′ ) . Hence, ν Ξ extends to a measure on the Borel sets of Ξ and by the monotone classtheorem λ ( τ ( B ′ × Ξ ′ )) = Leb( B ′ ) ν Ξ (Ξ ′ ) for every Borel sets B ′ ⊂ B R (0) and Ξ ′ ⊂ Ξ.We finally remark that ν Ξ (Ξ ′ ) > ′ ⊂ Ξ, sinceotherwise there would exist an open set of Ω of λ -measure zero.We come back to Kakutani-Rohlin towers of flows. Let { F lα } α ∈ A l be such a towerof order l and { F l +1 β } β ∈ A l +1 be the subsequent tower as introduced in (24). We recallthe definition of the homology matrix as explained in lemma 2.7 of [12]. For every α ∈ A l and β ∈ A l +1 , β = ( α , . . . , α p ), α = α p , α i = α for i = 1 , . . . , p −
1, wedenote M lα,β := { ≤ k ≤ p − α k = α } . A flow box of order l + 1, τ (cid:0) [0 , H l +1 β ) × Σ l +1 β (cid:1) , is obtained as a disjoint union of flowboxes of order l of the type τ (cid:0) [ t i , t i + H lα i ) × Σ lα i (cid:1) . The integer M lα,β counts thenumber of times a flow box of order l + 1 indexed by β cuts a flow box of order l indexed by α . The main result that we shall need is given by the following lemma. Lemma 42.
Let (Ω , { τ t } t ∈ R ) be a one-dimensional almost periodic and uniquelyergodic environment. Let { F lα } α ∈ A l be a sequence of Kakutani-Rohlin towers builtas in (24). Let ν l be the transverse measure associated with the transverse section ∪ α ∈ A l Σ lα . If ν lα := ν l (Σ lα ) , then ν lα = X β ∈ A l +1 M lα,β ν l +1 β . Proof.
Let Ξ = ∪ β ∈ A l +1 Σ l +1 β . For ω ∈ Ξ, let 0 = t , t , t , . . . be its successivereturn times to Ξ. We introduce as in lemma 41 the set of return times to thetransverse section Σ lα , say, R lα ( ω ) := { t ∈ R : τ t ( ω ) ∈ Σ lα } . The set R l +1 β ( ω ) isdefined similarly. Since (cid:0) R lα ( ω ) ∩ [0 , t n ) (cid:1) = X β ∈ A l +1 M lα,β (cid:0) R l +1 β ( ω ) ∩ [0 , t n ) (cid:1) , we divide by t n and apply lemma 41 to conclude.4 Garibaldi, Petite and Thieullen
This section is devoted to the proof of the second main result of this paper: the-orem 10. We consider an almost periodic environnement (Ω , { τ t } t ∈ R ) admitting aflow box decomposition with respect to which L : Ω × R → R is locally transversallyconstant, and we suppose the Lagrangian L is also weakly twist. We shall studythe properties of the associated minimizing configurations.If E ω ( x, x ) = ¯ E for some ω ∈ Ω and x ∈ R , then δ ( τ x ( ω ) , ∈ M min ( L ), τ x ( ω )belongs to the projected Mather set, and the configuration x k,ω = x fulfills items 1and 2 of theorem 10. We thus suppose later E ω ( x, x ) > ¯ E for every ω and x .Our first nontrivial result is stated in proposition 44: a finite configuration( x n , . . . , x nn ) which realizes the minimum of the energy among all configurationsof the same length must be strictly monotone, and must have uniformly boundedjumps | x nk − x nk − | ≤ R . Our second key result, proposition 47, shows actually thatlim inf n → + ∞ n | x nn − x n | >
0: the frequency of points x nk in a flow box of sufficientlylarge size is positive. We finally conclude this section with the proof of theorem 10. Lemma 43.
Given a weakly twist and transversally constant Lagrangian L , thereexists R > such that, if ω ∈ Ω is any environment, if ( x , . . . , x n ) ∈ R is mini-mizing for E ω and | x n − x | ≥ R , then ( x , . . . , x n ) is strictly monotone.Proof. Let { U i = τ [ B R i × Ξ i ] } i ∈ I be a flow box decomposition with respect to which L is transversally constant. Since { U i } i ∈ I is a finite cover, we may choose R largeenough so that every orbit of size R meets every box entirely: for every ω , for every | y − x | ≥ R , for every i ∈ I , there exists t i ∈ R such that ( t i − R i , t i + R i ) ⊂ [ x, y ]and τ t i ( ω ) ∈ Ξ i .We first show that there cannot exist r ≥ < k < n − r such that x k < x k − , x k = . . . = x k + r and x k < x k + r +1 . Otherwise, Aubry crossing lemma implies that E ω ( x k − , x k ) + E ω ( x k , x k + r +1 ) > E ω ( x k − , x k + r +1 ) + E ω ( x k , x k ) . We rewrite the configuration ( x , . . . , x k − , x k + r +1 , . . . , x n ) as ( y , . . . , y n − r − ). Let U i be a flow box containing τ x k ( ω ). There exists | s | < R i and ω ′ ∈ Ξ i such that τ x k ( ω ) = τ s ( ω ′ ). By the choice of R , there exists t such that ( t − R i , t + R i ) ⊂ [ x , x n ]and τ t ( ω ) ∈ Ξ i . Let z = . . . = z r := t + s and 1 ≤ l ≤ n − r − y l − < z ≤ y l . Using the fact that L is transversally constant on U i , we have E ω ( x k , x k ) = E ω ′ ( s, s ) = E τ t ( ω ) ( s, s ) = E ω ( z , z ) . By applying again Aubry crossing lemma, we obtain E ω ( y l − , y l ) + E ω ( z , z ) ≥ E ω ( y l − , z ) + E ω ( z , y l ) , (possibly with a strict inequality if z < y l ). We have just obtained a new con-figuration ( y , . . . , y l − , z , . . . , z r , y l , . . . , y n − r − ) of n points with a strictly lowerenergy, which contradicts the fact that ( x , . . . , x n ) is minimizing. alibrated configurations in almost-periodic environments r ≥ < k < n − r such that x k > x k − , x k = . . . = x k + r and x k > x k + r +1 . There cannot exist either a sub-configuration ( x k − , x k , . . . , x k + r , x k + r +1 ), r ≥ x k − = x k + r +1 and x k = . . . = x k + r strictly between x k − and x k + r +1 thanks to lemma 28. We are thus left to a configuration of the form x = . . . = x r < . . . < x n − r ′ = . . . = x n or x = . . . = x r > . . . > x n − r ′ = . . . = x n for some r, r ′ ≥
0. Assume by contradiction that x = x (the case x n − = x n is done similarly). Exactly as before, there exist U i containing τ x ( ω ), | s | < R i and ω ′ ∈ Ξ i such that τ x ( ω ) = τ s ( ω ′ ), as well as there exists t ∈ R such that( t − R i , t + R i ) ⊂ [min { x , x n } , max { x , x n } ] and τ t ( ω ) ∈ Ξ i . One can show in ananalogous way that, whenever z := t + s belongs to (min { x l − , x l } , max { x l − , x l } ] for2 ≤ l ≤ n , E ( x , x , . . . , x n ) ≥ E ( x , . . . , x l − , z, x l , . . . , x n ), with strict inequality if z < max { x l − , x l } . Since ( x , x , . . . , x n ) is a minimizing configuration, this impliesthat z = max { x l − , x l } 6∈ { x , x n } , and ( x , . . . , x l − , z, x l , . . . , x n ) is a minimizingconfiguration. The first part of this proof shows that this cannot happen.The proof that ( x , . . . , x n ) is strictly monotone is complete. Proposition 44.
Given a weakly twist and transversally constant Lagrangian L ,there exists R > such that, if ω ∈ Ω is any environment and ( x , . . . , x n ) , n ≥ ,satisfies E ( x , . . . , x n ) = min ( y ,...,y n ) E ω ( y , . . . , y n ) and max ≤ k 2, and ( x , . . . , x n ) realizing the minimum of the energyamong all configurations of length n in the environment ω . Part 1. We show there exists R ′ > ω and n ) such that | x − x | ≤ R ′ and | x − x | ≤ R ′ . Indeed, we have E ω ( x , x ) ≤ E ω ( x , x ) and E ω ( x , x , x ) ≤ E ω ( x , x , x ) , which implies E ω ( x , x ) ≤ sup x ∈ R E ω ( x, x ) and E ω ( x , x ) ≤ x ∈ R E ω ( x, x ) − inf x,y ∈ R E ω ( x, y ) . The existence of R ′ follows then from the coerciveness of L , which is uniform withrespect to ω . Similarly, we have | x n − − x n − | ≤ R ′ and | x n − x n − | ≤ R ′ . Part 2. We show there exists R ′′ > x , . . . , x m ) is strictlymonotone, then | x i − x i − | ≤ R ′′ for every 1 ≤ i ≤ m . It is clear from thedefinition that, if L is transversally constant with respect to a particular flow boxdecomposition { τ [ B r i × Ξ i ] } , then L is transversally constant for any flow boxdecomposition such that its flow boxes are compatible with respect to { τ [ B r i × Ξ i ] } .Therefore, let { U i = τ [ B R ′ × Ξ ′ i ] } be a finite cover of Ω by flow boxes such that τ [ B R ′ × Ξ ′ i ] is again a flow box and L is transversally constant with respect to { τ [ B R ′ × Ξ ′ i ] } . We choose R ′′ > R ′′ meets entirely each τ [ B R ′ × Ξ ′ i ]. Let U i be a flow box containing τ x ( ω ): there6 Garibaldi, Petite and Thieullen exist | s | < R ′ and ω ′ ∈ Ξ ′ i such that τ x ( ω ) = τ s ( ω ′ ). From part 1, we deducethat τ [ B R ′ × Ξ ′ i ] contains { τ x ( ω ) , τ x ( ω ) , τ x ( ω ) } . Denote s := s + x − x and s := s + x − x , so that | s | , | s | < R ′ , τ x ( ω ) = τ s ( ω ′ ) and τ x ( ω ) = τ s ( ω ′ ).Assume by contradiction | x i − x i − | > R ′′ . Then, there exists t ∈ R such that( t − R ′ , t + 2 R ′ ) ⊂ [min { x i − , x i } , max { x i − , x i } ] and τ t ( ω ) ∈ Ξ ′ i . Let z = t + s , z = t + s and z = t + s . Notice that ( x i − , x i ) and ( z , z , z ) are ordered in thesame way. As L is transversally constant on τ [ B R ′ × Ξ ′ i ], we obtain E ω ( x , x , x ) = E ω ′ ( s , s , s ) = E τ t ( ω ) ( s , s , s ) = E ω ( z , z , z ) . Aubry crossing lemma applied twice gives E ω ( x i − , x i ) + E ω ( z , z , z ) > E ω ( x i − , z ) + E ω ( z , x i ) + E ω ( z , z ) ,> E ω ( x i − , z , x i ) + E ω ( z , z ) . As L is transversally constant, E ω ( z , z ) = E ω ( x , x ) as above and we obtain E ω ( x i − , x i ) + E ω ( x , x , x ) > E ω ( x i − , z , x i ) + E ω ( x , x ) . The configuration ( x , x , . . . , x i − , z , x i , . . . , x m ) has a strictly lower energy, whichcontradicts the fact that ( x , . . . , x m ) is minimizing. We obtain similarly that, if( x m , . . . , x n ) is strictly monotone, then | x i − − x i | ≤ R ′′ for every m + 1 ≤ i ≤ n . Part 3. Let R ′′′ be the constant given by lemma 43. Take R > R ′′ + 4 R ′′′ . If | x n − x | > R ′′′ , then ( x , . . . , x n ) is strictly monotone by lemma 43 and the jumps | x i − x i − | are uniformly bounded by R ′′ . The proof is finished.Assume by contradiction that | x n − x | ≤ R ′′′ . Let a = min ≤ k ≤ n x k and b = max ≤ k ≤ n x k . Since diam( { x k : 0 ≤ k ≤ n } ) ≥ R , one of the two inequalities | a − x | > R/ | b − x | > R/ | b − x | > R/ | a − x | > R/ b = x m for some 0 < m < n .Since ( x , . . . , x m ) and ( x m , . . . , x n ) are minimizing and satisfy | x m − x | > R ′′′ and | x m − x n | > R ′′′ , these two configurations are strictly monotone. Then, part 2tells us that the jumps | x i − x i − | are uniformly bounded by R ′′ . In particular, | x m +1 − x m | ≤ R ′′ . The configuration ( x , . . . , x m +1 ) is minimizing and, since | x m − x | > R ′′ + 2 R ′′′ , it satisfies | x m +1 − x | > R ′′′ . By lemma 43, it must bestrictly monotone, which is in contradiction with the maximum x m .Thus, | x n − x | > R ′′′ , ( x , . . . , x n ) is strictly monotone and | x i − x i − | ≤ R ′′ .The proof of the fact that | x k − x k − | is uniformly bounded uses the same ideasas in lemma 3.1 of [12]. The fact that L is transversally constant enables us totranslate subconfigurations without modifying the total energy. For a minimizingand strictly monotone configuration, by minimality of the energy, two consecutivepoints cannot enclose a translated subconfiguration of three points. More precisely,we have the following lemma that extends lemma 3.2 of [12]. Lemma 45. Let L be a weakly twist Lagrangian which is transversally constant fora flow box decomposition { U i } i ∈ I . Suppose that the flow box τ [ B R × Ξ] is compatiblewith respect to { U i } i ∈ I . Let ( x , . . . , x n ) be a strictly monotone minimizing config-uration for some environment ω ∈ Ω . Let ( a − R, a + R ) and ( b − R, b + R ) be two alibrated configurations in almost-periodic environments disjoint intervals such that τ a ( ω ) ∈ Ξ and τ b ( ω ) ∈ Ξ . Assume that ( a − R, a + R ) is a subset of [ x , x n ] . Let A be the number of sites ≤ k ≤ n such that x k belongsto ( a − R, a + R ) and let B be defined similarly. Then B ≤ A + 2 . In particular, if ( b − R, b + R ) ⊂ [ x , x n ] , then | A − B | ≤ .Proof. To simplify we assume that ( x , . . . , x n ) is strictly increasing. The proof isdone by contradiction by assuming B ≥ A + 3. Denote { y , . . . , y A } := { x , . . . , x n } ∩ ( a − R, a + R ) and { y ′ , . . . , y ′ B } := { x , . . . , x n } ∩ ( b − R, b + R ) . Let y be the greatest x k ≤ a − R and y A +1 be the smallest x k ≥ a + R . We write s k := y ′ k − b and z k := a + s k for k = 1 , . . . , B . The partition into A + 1 disjointintervals ∪ A +1 k =1 ( y k − , y k ] must contain A +3 distinct points { z , . . . , z A +3 } . We havetherefore to consider two cases. Case 1. Either some interval ( y k − , y k ], 2 ≤ k ≤ A , contains three points( z i − , z i , z i +1 ). By Aubry crossing lemma, E ω ( y k − , y k ) + E ω ( z i − , z i ) > E ω ( y k − , z i ) + E ω ( z i − , y k ) ,E ω ( z i − , y k ) + E ω ( z i , z i +1 ) ≥ E ω ( z i − , z i +1 ) + E ω ( z i , y k ) . Since L is transversally constant on τ [ B R × Ξ], we obtain E ω ( y ′ i − , y ′ i , y ′ i +1 ) + E ω ( y k − , y k ) = E ω ( z i − , z i , z i +1 ) + E ω ( y k − , y k ) > E ω ( z i − , z i +1 ) + E ω ( y k − , z i , y k )= E ω ( y ′ i − , y ′ i +1 ) + E ω ( y k − , z i , y k ) . We have obtained a configuration (if, for instance, b < a ) of the form( x , . . . , y ′ i − , y ′ i +1 , . . . , y ′ B , . . . , y , . . . , y k − , z i , y k , . . . , x n )with strictly lower energy, which contradicts the fact that ( x , . . . , x n ) is minimizing. Case 2. Or there exist two distinct intervals ( y k − , y k ] and ( y l − , y l ], with 2 ≤ k < l ≤ A , that contain each two points ( z i − , z i ) and ( z j − , z j ), respectively. Noticethat we may have y k = y l − , but we must have z i < z j − , z i +1 ∈ ( a − R, a + R ),and possibly z i +1 = z j − . We want to obtain a contradiction by showing that onecan decrease the sum of energies E ω ( y ′ i − , . . . , y ′ j ) + E ω ( y k − , . . . , y l ) while fixingthe four boundary points.In the case z i = y k , we perturb the point z i slightly by a small quantity ǫ andallow an increase of the energy of order ǫ . Since ( z i − , z i , z i +1 ) is minimizing, wehave E ω ( z i − , z i , z i +1 ) = E ω ( z i − , z i − ǫ, z i +1 ) + o ( ǫ ) . By Aubry crossing lemma, either z i < y k , and the reminder in lemma 25 takes theform reminder := ( z i − − y k − )( z i − y k ) α > , where α = 1( z i − − y k − )( z i − y k ) Z z i − y k − Z z i y k ∂ E ω ∂x∂y ( x, y ) dydx < , Garibaldi, Petite and Thieullen (in that case, we define ǫ := 0), or z i = y k , and the reminder becomesreminder := − ǫ ( z i − − y k − ) α + o ( ǫ ) > o ( ǫ ) , where α = 1 z i − − y k − Z z i − y k − ∂ E ω ∂x∂y ( x, y k ) dx < . In both cases, E ω ( y k − , y k ) + E ω ( z i − , z i − ǫ ) = E ω ( y k − , z i − ǫ ) + E ω ( z i − , y k ) + reminder ,E ω ( y k − , y k ) + E ω ( z i − , z i , z i +1 ) > E ω ( y k − , z i − ǫ, z i +1 ) + E ω ( z i − , y k ) . Again by Aubry crossing lemma, E ω ( y l − , y l ) + E ω ( z j − , z j ) ≥ E ω ( y l − , z j ) + E ω ( z j − , y l ) , with possibly equality if z j = y l . Since L is transversally constant, we obtain E ω ( y ′ i − , . . . , y ′ j ) + E ω ( y k − , . . . , y l )= E ω ( z i − , . . . , z j ) + E ω ( y k − , . . . , y l ) > E ω ( z i − , y k , . . . , y l − , z j ) + E ω ( y k − , z i − ǫ, z i +1 , . . . , z j − , y l )= E ω ( y ′ i − , w k , . . . , w l − , y ′ j ) + E ω ( y k − , z i − ǫ, z i +1 , . . . , z j − , y l ) , with t k := y k − a , w k := b + t k ,. . . , t l − := y l − − a , w l − := b + t l − . Hence, we have aconfiguration ( . . . , y ′ i − , w k , . . . , w l − , y ′ j , . . . , y k − , z i − ǫ, z i +1 , . . . , z j − , y l , . . . ) withstrictly lower energy, which contradicts the fact that ( x , . . . , x n ) is minimizing.It may happen that E ω ( x, x ) = ¯ E for some ω ∈ Ω and x ∈ R . Let x n = x for every n . Then ( x n ) n ∈ Z is a calibrated configuration in the environment ω and δ ( τ x ( ω ) , is a minimizing measure. If L is transversally constant on a flow box τ [ B R × Ξ] such that τ x ( ω ) ∈ Ξ, then δ ( ω ′ , is a minimizing measure for every ω ′ ∈ Ξ. The projected Mather set contains Ξ and theorem 10 is proved. We arethus left to understand the case inf ω ∈ Ω , x ∈ R E ω ( x, x ) > ¯ E . Lemma 46. Let L be a weakly twist Lagrangian for which inf ω ∈ Ω , x ∈ R E ω ( x, x ) > ¯ E. For every ω ∈ Ω , n ≥ , if ( x n , . . . , x nn ) is a configuration realizing the minimum E ω ( x n , . . . , x nn ) = min x ,...,x n ∈ R E ω ( x , . . . , x n ) , then lim n → + ∞ | x nn − x n | = + ∞ .Proof. The proof is done by contradiction. Let ω ∈ Ω and R > 0. Assume thereexist infinitely many n ’s for which every configuration ( x n , . . . , x nn ) realizing theminimum of E ω ( x , . . . , x n ) satisfies | x nn − x n | ≤ R . If ( x n , . . . , x nn ) is not monotone,thanks to lemma 26, we can find distinct indices { i , . . . , i r } of { , . . . , n } such that i = 0, i r = n , ( x ni , . . . , x ni r ) is monotone (possibly not strictly monotone) and E ω ( x n , . . . , x nn ) ≥ E ω ( x ni , . . . , x ni r ) + X i i ,...,i r } E ω ( x ni , x ni ) . alibrated configurations in almost-periodic environments ǫ > E ω ( x, y ) ≥ ¯ E + ǫ for every | y − x | ≤ ǫ . Thus, if θ n denotes the number of indices 1 ≤ k ≤ r such that | x ni k − x ni k − | > ǫ , it is clear that θ n ≤ R/ǫ . Since n ¯ E ≥ E ω ( x n , . . . , x nn ) ≥ ( n − θ n )( ¯ E + ǫ ) + θ n inf x,y ∈ R E ω ( x, y ) , we obtain a contradiction by letting n → + ∞ .We now assume that (Ω , { τ t } t ∈ R , L ) is an almost crystalline interaction model.We show in the following proposition that a sequence of configurations ( x n , · · · , x nn )realizing the minimum of the energy E ω ( x , . . . , x n ) among all configurations oflength n admits a weak rotation number in the sense thatlim inf n → + ∞ | x nn − x n | n > . (25)The existence of a rotation number for an infinite minimizing configuration ( x k ) k ∈ Z has been established in [12]. The following proposition extends partially this resultin two directions: the interaction model is more general; we compute the rotationnumber of a sequence of configurations of increasing length and not the rotationnumber of a unique infinite configuration. Proposition 47. Let (Ω , { τ t } t ∈ R , L ) be an almost crystalline interaction model.Assume that inf ω ∈ Ω , x ∈ R E ω ( x, x ) > ¯ E. For every ω ∈ Ω and n ≥ , let ( x n , . . . , x nn ) be a configuration realizing the mini-mum of the energy among all configurations of length n : E ω ( x n , · · · , x nn ) = min x ,...,x n E ω ( x , . . . , x n ) . Then,– ¯ E = lim n → + ∞ n E ω ( x n , · · · , x nn ) = sup n ≥ n E ω ( x n , · · · , x nn ) ,– for n sufficiently large, ( x n , · · · , x nn ) is strictly monotone,– there is R > (independent of ω ) such that sup n ≥ sup ≤ k ≤ n | x nk − x nk − | ≤ R ,– lim inf n → + ∞ n | x nn − x n | > .Proof. To avoid trivialities, we assume that the flow (Ω , { τ t } t ∈ R ) is not periodic. Step 1. The first item has been proved in proposition 16; the limit can beobtained as a supremum because of superadditivity. Moreover, from lemma 46, | x nn − x n | → + ∞ . From proposition 44, the configuration ( x n , . . . , x nn ) must bestrictly monotone and have uniformly bounded jumps R . We are left to prove thelast item of the proposition. Step 2. By definition of an almost crystalline interaction model, L is transver-sally constant with respect to some flow box decomposition { U i } i ∈ I (definitions 33and 34). Let { F α } α ∈ A be a Kakutani-Rohlin tower that is compatible with respectto { U i } i ∈ I (definition 39) and let Σ = ∪ α ∈ A Σ α be its basis. We may assume thatmin α ∈ A H α is as large as we want and, in particular, larger than R (see the construc-tion (24)). We also assume that n is sufficiently large so that every tower F α of basis0 Garibaldi, Petite and Thieullen Σ α is completely cut by the trajectory τ t ( ω ) for t ∈ (min { x n , x nn } , max { x n , x nn } ).We consider ν the transverse measure to Σ (as defined in lemma 41) and we denote ν α := ν (Σ α ). Step 3. Let S n < T n be the two return times to Σ (namely, τ S n ( ω ) ∈ Σ and τ T n ( ω ) ∈ Σ) that are chosen so that [ S n , T n ) is the smallest interval containing thesequence ( x nk ) nk =0 . From the definition of a Kakutani-Rohlin tower, [ S n , T n ) can bewritten as a disjoint union of intervals of type I α,i := [ t α,i , t α,i + H α ), where the list { t α,i } i , i = 1 , . . . , C nα , denotes the successive return times to Σ α between S n and T n .We distinguish two exceptional intervals among this list: the two intervals whichcontain x n and x nn . If x n < x nn , then N nα,i denotes the number of points ( x nk ) nk =1 belonging to I α,i and N nα denotes the maximum of N nα,i . If x nn < x n , then N nα,i and N nα are defined similarly by considering in this case ( x nk ) n − k =0 . From lemma 45, weobtain N nα − ≤ N nα,i ≤ N nα for every nonexceptional interval I α,i . We show thatsup n ≥ N nα < + ∞ for every α ∈ A . The proof is done by contradiction.Let E nα,i be the energy of the configuration localized in I α,i . More precisely,assume first x n < x nn ; index the part of ( x nk ) nk =1 in I α,i by ( x nk,α,i ) Nk =1 with N = N nα,i ;denote by x n ,α,i the nearest point strictly smaller than x n ,α,i and define the partialenergy E nα,i := E ω ( x n ,α,i , . . . , x nN,α,i ). If x nn < x n , the part of ( x nk ) n − k =0 in I α,i isindexed by ( x nk,α,i ) N − k =0 with N = N nα,i ; denote by x nN,α,i the nearest point strictlylarger than x nN − ,α,i and define E nα,i similarly.Thanks to the hypothesis inf x ∈ R E ω ( x, x ) > ¯ E , one can choose ǫ > E ω ( x, y ) ≥ ¯ E + ǫ as soon as | y − x | ≤ ǫ . Let ¯ H := max α ∈ A H α . Then, if θ nα,i denotesthe number of consecutive points x nk,α,i in I α,i satisfying | x nk,α,i − x nk − ,α,i | > ǫ ,obviously θ nα,i ≤ ¯ H/ǫ . Thus, since n = P α ∈ A P ≤ i ≤ C nα N nα,i , we have that n ¯ E ≥ E ω ( x n , . . . , x nn ) = X α ∈ A X ≤ i ≤ C nα E nα,i ≥ X α ∈ A X ≤ i ≤ C nα h θ nα,i inf x,y ∈ R E ω ( x, y ) + (cid:0) N nα,i − θ nα,i (cid:1) ( ¯ E + ǫ ) i = n ( ¯ E + ǫ ) + X α ∈ A X ≤ i ≤ C nα θ nα,i E ≥ n ( ¯ E + ǫ ) + X α ∈ A C nα ¯ Hǫ E, (26)where E := (inf x,y ∈ R E ω ( x, y ) − ¯ E − ǫ ) < 0. For α fixed, among the intervals ( I α,i ) i , i = 1 , . . . , C nα , at most two of them are exceptional and the other intervals satisfy N nα,i ≥ N nα − 2. We thus get n ≥ P α ∈ A ( C nα − N nα − n sufficiently large,we have C nα T n − S n ≤ (1 + ǫ ) ν α , C nα − T n − S n ≥ (1 − ǫ ) ν α and1 n X α ∈ A C nα ≤ (1 + ǫ ) P α ∈ A ν α (1 − ǫ ) P α ∈ A ν α ( N nα − . If N nα → + ∞ for some α and a subsequence n → + ∞ , then n P α ∈ A C nα → alibrated configurations in almost-periodic environments Step 4. For every α , I α,i ⊂ [ x n , x nn ] except maybe for at most two of them. Then | x nn − x n | n ≥ P α ∈ A ( C nα − H α P α ∈ A C nα N nα . Denote ¯ N α := lim sup n → + ∞ N nα . From step 3 we know that ¯ N α < + ∞ . By dividingby ( T n − S n ) and by letting n → + ∞ , we obtainlim inf n → + ∞ | x nn − x n | n ≥ P α ∈ A ν α H α P α ∈ A ν α ¯ N α = 1 P α ∈ A ν α ¯ N α > . Now we are able to prove theorem 10. Proof of theorem 10. Let (Ω , { τ t } t ∈ R , L ) be an almost crystalline interaction model.We discuss two cases. Case 1. Either inf ω ∈ Ω inf x ∈ R E ω ( x, x ) = ¯ E . Then E ω ∗ ( x ∗ , x ∗ ) = ¯ E for some ω ∗ and x ∗ . By hypothesis, L is transversally constant with respect to a flow boxdecomposition { U i = τ [ B R i × Ξ i ] } i ∈ I . Let i ∈ I be such that τ x ∗ ( ω ∗ ) ∈ U i . Let | t i | < R i and ω i ∈ Ξ i be such that τ x ∗ ( ω ∗ ) = τ t i ( ω i ). Then¯ E = E ω ∗ ( x ∗ , x ∗ ) = E ω i ( t i , t i ) = E ω ( t i , t i ) , ∀ ω ∈ Ξ i . We have just proved that δ ( τ ti ( ω ) , is a minimizing measure for every ω ∈ Ξ i .The projected Mather set contains τ t i (Ξ i ). By minimality of the flow, we haveΩ = τ [ B R × Ξ i ], for some R > 0, thanks to item 1 of lemma 38. The projectedMather set thus meets every sufficiently long orbit of the flow. Case 2. Or inf ω ∈ Ω inf x ∈ R E ω ( x, x ) > ¯ E . Proposition 47 shows that, if ω ∗ ∈ Ωhas been fixed, if for every n ≥ x nk ) ≤ k Gradient flows in metric spaces and in the space ofprobability measures , Lectures in Mathematics ETH Z¨urich, Birkh¨auser Verlag, Basel, 2005.[2] S. Aubry and P.Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions: I.Exact results for the ground states, Physica D (1983), 381–422.[3] L. Auslander and F. Hahn, Real functions coming from flows on compact spaces and conceptsof almost periodicity, Transactions of the American Mathematical Society (1963), 415–426.[4] J. Bellissard, R. Benedetti and J. M. Gambaudo, Spaces of tilings, finite telescopic approxi-mations and gap-labeling, Communications in Mathematical Physics (2006), 1–41.[5] A. Davini and A. Siconolfi, Exact and approximate correctors for stochastic Hamiltonians:the 1-dimensional case, Mathematische Annalen (2009), 749–782.[6] A. Davini and A. Siconolfi, Metric techniques for convex stationary ergodic Hamiltonians, Calculus of Variations and Partial Differential Equations (2011), 391–421.[7] A. Davini and A. Siconolfi, Weak KAM theory topics in the stationary ergodic setting, Cal-culus of Variations and Partial Differential Equations (2012), 319–350.[8] R. de la Llave and X. Su, KAM theory for quasi-periodic equilibria in one-dimensional quasi-periodic media. SIAM Journal of Mathematical Analysis (2012), 3901–3927.[9] A. Fathi, Solutions KAM faibles conjugu´ees et barri`eres de Peierls, Comptes Rendus desS´eances de l’Acad´emie des Sciences, S´erie I, Math´ematique (1997), 649–652.[10] A. Fathi, The weak KAM theorem in Lagrangian dynamics , book to appear, Cambridge Uni-versity Press.[11] Ya. I. Frenkel and T. A. Kontorova, On the theory of plastic deformation and twinning I,II, III, Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki (1938) 89–95 (I), 1340–1349 (II),1349–1359 (III).[12] J. M. Gambaudo, P. Guiraud and S. Petite, Minimal configurations for the Frenkel-Kontorovamodel on a quasicrystal, Communications in Mathematical Physics (2006), 165–188.[13] E. Garibaldi, S. Petite and Ph. Thieullen, Discrete Lax-Oleinik operators for non periodicinteractions, preprint , 2015.[14] E. Garibaldi and Ph. Thieullen, Minimizing orbits in the discrete Aubry-Mather model, Non-linearity (2011), 563–611.[15] T. Giordano, I. F. Putnam and C. F. Skau, Topological orbit equivalence and C ⋆ -crossedproducts, Journal f¨ur die reine und angewandte Mathematik (1995), 51–111.[16] D. A. Gomes, Viscosity solution methods and the discrete Aunbry-Mather problem, Discreteand Continuous Dynamical Systems, Series A (2005), 103–116.[17] J. Kellendonk, Pattern-equivariant functions and cohomology, Journal of Physics A: Mathe-matical and General (2003), 5765–5772. alibrated configurations in almost-periodic environments [18] J. Kellendonk and I. F. Putnam, Tilings, C ∗ -algebras and K -theory, In: M. Baake and R. V.Moody (eds.), Directions in mathematical quasicrystals , CRM Monograph Series , AMS,Providence, RI, 2000, pp. 177–206.[19] J. C. Lagarias and P. A. B. Pleasants, Repetitive Delone sets and quasicrystals, ErgodicTheory and Dynamical Systems (2003), 831–867.[20] P. L. Lions and P. E. Souganidis, Correctors for the homogenization of Hamilton-Jacobi equa-tions in the stationary ergodic setting, Communications on Pure and Applied Mathematics (2003), 1501–1524.[21] R. Ma˜n´e, Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity (1996), 273–310.[22] J. N. Mather, Existence of quasiperiodic orbits for twist homeomorphisms of the annulus, Topology (1982), 457–467.[23] C. Vilani, Optimal transport: old and new , Grundlehren der mathematischen Wissenschaften , Springer-Verlag, 2008.[24] M. Sion, On general minimax theorems,