Eduardo Garibaldi
email: garibaldi@ime.unicamp.br

Philippe Thieullen
email: philippe.thieullen@math.u-bordeaux1.fr

Minimizing orbits in the discrete Aubry-Mather model

We consider a generalization of the Frenkel-Kontorova model in higher dimension leading to a new theory of configurations with minimal energy, as in Aubry's theory or in Mather's twist approach in the periodic case. We consider a one dimensional chain of particles and their minimizing configurations and we allow the state of each particle to possess many degrees of freedom. We assume that the Hamiltonian of the system satisfies some twist condition. The usual "total ordering" of minimizing configurations does not exist any more and new tools need to be developed. The main mathematical tool is to cast the study the minimizing configurations into the framework of discrete Lagrangian theory. We introduce forward and backward Lax-Oleinik problems and interpret their solutions as discrete viscosity solutions as in Hamilton-Jacobi methods. We give a fairly complete description of a particular class of minimizing configurations: the calibrated class. These configurations may be thought of as "ground states" obtained in the thermodynamic limit at temperature zero. We obtain, in particular, Mather's graph property or the non-crossing property of two calibrated configurations and the existence of a rotation number for most of the calibrated configurations.

Introduction

The original Frenkel-Kontorova model, see [START_REF] Frenkel | On the theory of plastic deformation and twinning I, II, III[END_REF], describes a one dimensional chain of (classical) particles coupled to their nearest neighbors and subjected to a periodic environment generated by a one dimensional cristal. The chain is supposed to be located on a line; the local interaction energy L : R 2 → R takes into account an elastic potential between two successive particles in the chain, as well as an external periodic potential modeling the influence of a periodic structure:

L(x k , x k+1 ) = 1 2 (x k+1 -x k -λ) 2 + K (2π) 2 (1 -cos 2πx k ),
where λ is the unperturbed mean interatomic distance and K is the strength of the external periodic environment. The state space in this model is represented by a sequence of positions of each particle constrained to stay on a line. A central problem in this theory is to describe the set of minimizing configurations, that is, configurations which minimize the total energy of any finite sub-chains with fixed boundary states. Among the main results, one may cite the existence of a well defined "rotation number" for any minimizing configuration, the "total ordering property" of the set of minimizing configurations with a fixed rotation number, and the existence of a unique "hull function" up to a phase shift which parametrizes all minimizing configurations with a fixed rotation number. These main results have been obtained by several authors, first by Aubry and Le Daeron [START_REF] Aubry | The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states[END_REF], Aubry, Axel and Vallet [START_REF] Aubry | Devil's staircases and Manhattan profile in an exact model for an incommensurate structure in an electric field[END_REF], Griffiths, Schellnhuber and Urbschat [START_REF] Griffiths | Exact ground state of the Frenkel-Kontorova model with repeated parabolic potential. I. Basic results[END_REF], independently by Mather [START_REF] Mather | Existence of quasi-periodic orbits for twist homeomorphims of the annulus[END_REF][START_REF] Mather | Action minimizing invariant measures for positive definite Lagragian systems[END_REF], more rencently by Gomes and Oberman, Rorro and Falcone [START_REF] Gomes | Viscosity solution method and the discrete Aubry-Mather problem, Discrete and Continuous Dynamical Systems[END_REF][START_REF] Gomes | Computing the effective hamiltonian using a variational approach[END_REF][START_REF] Rorro | An approximation scheme for the effective Hamiltonian and application[END_REF][START_REF] Falcone | On a Variational Approximation of the Effective Hamiltonian[END_REF], Baesens and MacKay [START_REF] Baesens | Discommensuration theory and shadowing in Frenkel-Kontorova models[END_REF], and in a similar context of ergodic optimization by Bousch, Brémont, Jenkinson and Morris, Bressaud and Quas, Leplaideur [START_REF] Th | Un lemme de Mañé bilatéral[END_REF][START_REF] Th | Le lemme de Mañé-Conze-Guivarc'h pour les systèmes amphidynamiques rectifiables[END_REF][START_REF] Brémont | Finite flowers and maximizing measures for generic Lipschitz functions on the circle[END_REF][START_REF] Jenkinson | Ergodic optimization[END_REF][START_REF] Jenkinson | Lyapunov optimizing measures for C 1 expanding maps of the circle[END_REF][START_REF] Bressaud | Rate of approximation of minimizing measures[END_REF][START_REF] Leplaideur | A dynamical proof for the convergence of Gibbs measures at temperature zero[END_REF]. The minimizing configurations previously described correspond to equilibrium fixed points of the steady-state Frenkel-Kontorova equation. Recent progress has been made in the space-time Frenkel-Kontorova model. In this model, each particle evolves in time according to a Newtonian second order differential equation. Without being exhaustive, we just quote recent articles by Forcadel, Imbert and Monneau [START_REF] Forcadel | Homogenization of fully overdamped Frenkel-Kontorova models[END_REF][START_REF] Forcadel | Homogenization of accelerated Frenkel-Kontorova models with n types of particles[END_REF] which use viscosity technics related to our methods and by Baesens and MacKay [START_REF] Baesens | A novel preserved partial order for cooperative networks of units with overdamped second order dynamics, and application to tilted Frenkel-Kontorova chains[END_REF].

Several general monographies may help the reading of the present article: for instance, the notes of G. Forni and J. N. Mather [START_REF] Forni | Action minimizing orbits in Hamiltonian systems[END_REF] for a global description of Aubry-Mather theory, and two textbooks written by G. Contreras and R. Iturriaga [START_REF] Contreras | Global minimizers of autonomous Lagrangians[END_REF] and by A. Fathi [START_REF] Fathi | The weak KAM theorem in Lagrangian dynamics[END_REF] for the Lagrangian point of view. The weak-KAM approach developed by Fathi for Lagrangian flows is an especially important concept we are going to apply in a discrete setting. For more details on the Frenkel-Kontorova model, we refer the reader to [START_REF] Braun | The Frenkel-Kontorova model: concepts, methods, and applications[END_REF][START_REF] Floría | The Frenkel-Kontorova model[END_REF].

It is crucial to point out that all previously mentioned results (except those in [START_REF] Gomes | Viscosity solution method and the discrete Aubry-Mather problem, Discrete and Continuous Dynamical Systems[END_REF][START_REF] Gomes | Computing the effective hamiltonian using a variational approach[END_REF]) depend strongly on the fact that the state x k of the k th particle of the chain is one dimensional. In other words, the order of the real line plays a crucial role; the fact that two configurations can be compared globally is fundamental. Our focus here is on an extension of Frenkel-Kontorova theory to the multidimensional case: the state of each particle will belong to R d . The case where the state space is finite S = {1, 2, . . . , d} and the interaction energy map L(i, j) is a function on S × S is a relevant situation worked out in ergodic optimization theory and in Gibbs fields theory. Although we do not consider specific examples nor numerical experiments in this article, the following interaction energy satisfies the hypotheses of all subsequent theorems:

L(x k , x k+1 ) = 1 2 d i=1 |x i k+1 -x i k -λ i | 2 + K (2π) 2 1 -cos(2π d i ν i x i k ) ,
where (ν 1 , . . . , ν d ) are integers and (x 1 k , . . . , x d k ) are the components of x k in R d . A large part of the theory of minimizing configurations can be done under very weak assumptions on the local interaction energy L(x, y). We first assume that L(x, y) : R d × R d → R is continuous and is doubly periodic L(x, y) = L(x + s, y + s), ∀ s ∈ Z d .

We next assume that L(x, y) is coercive lim R→+∞ inf x-y ≥R L(x, y) = +∞.

The formal total energy of a chain of particles {x k } k∈Z in R d is given by

L tot ({x k } k∈Z ) = k∈Z L(x k , x k+1 ),
which may a priori diverge. A minimizing configuration is a configuration that can only increase its total energy whenever a finite number of particles is moved, that is, a configuration {x k } k∈Z in R d satisfying L(x m , x m+1 , . . . , x n ) := n-1 k=m L(x k , x k+1 ) ≤ L(y m , y m+1 , . . . , y n ), for every m < n and every configuration {y k } k∈Z in R d with y m = x m and y n = x n . If L is in addition C 1 , a minimizing configuration {x k } is critical (or at equilibrium) in the sense that:

∂L ∂y (x k-1 , x k ) + ∂L ∂x (x k , x k+1 ) = 0, ∀ k ∈ Z.
The main objective of this paper is to extend partially the theory of Frenkel-Kontorova model to the case where the state x k of each particle belongs to a ddimensional vector space R d . The natural tool we are going to apply in the multidimensional setting comes from either the notion of effective potential of Chou and Griffiths [START_REF] Chou | Ground states of one-dimensional systems using effective potentials[END_REF] or the notion of viscosity solution of Fathi's weak KAM theory [START_REF] Fathi | The weak KAM theorem in Lagrangian dynamics[END_REF]. This approach allows only the study of a special class of minimizing configurations, called ground states by Chou and Griffiths or calibrated by Fathi. Notice that Aubry-Le Daeron [START_REF] Aubry | The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states[END_REF] used the expression "ground state" to denote a recurrent minimizing configuration. It happens that, in the one dimensional case, recurrent minimizing configurations and calibrated configurations are strongly related through the notion of rotation number; we do not know whether that fact persists in the multidimensional setting. Ground states are physical quantities as shown by N. Anantharaman in [START_REF] Anantharaman | On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics[END_REF]: she proved that Gibbs measures at positive temperatures converge when the system is frozen (in a weak sense along any converging subsequences) to measures whose support contains only ground states. We will show that calibrated configurations exist assuming L(x, y) to be only C 0 and coercive. We will also show that most of the calibrated configurations admit a rotation vector assuming in addition L(x, y) to be superlinear:

lim R→+∞ inf y-x ≥R L(x, y) x -y = +∞.
Uniqueness of a calibrated configuration passing through a given point uses an even stronger assumption called twist condition or ferromagnetic condition (definition 2.4), which is implied, for instance, when L(x, y) is C 2 and uniformly strictly convex (definition 2.7). The definition of calibrated configurations will be explained soon.

Generalizations of Frenkel-Kontorova model have been investigated in several different directions. The state space is still one dimensional as in Aubry Mather theory.

One can consider, for example, a multidimensional setting where the particles are indexed by the sites of a multidimensional lattice Z d . The topology of interactions plays a fundamental role. The interaction can be either local with each particle interacting only with its nearest neighbors, or global as in mean field theory. The state x k ∈ R of the particles is nevertheless one dimensional as in Aubry theory, but indexed by Z d for instance. Notions of minimizing configurations and rotation vectors can be defined similarly. In the context of the multidimensional Frenkel-Kontorova model, one still obtains minimizing configurations having a prescribed rotation vector. For precise definitions and statements, we refer the reader to the work of R. de la Llave and E. Valdinoci [START_REF] De La Llave | Ground states and critical points for generalized Frenkel-Kontorova models in Z d[END_REF]. One should also consult the paper of H. Koch, R. de la Llave and C. Radin [START_REF] Koch | Aubry-Mather theory for functions on lattices[END_REF] for situations where the variables range over a more complicated lattice.

In another direction, the potential could be assumed quasiperiodic instead of periodic, as it is done in the work of J. M. Gambaudo, P. Guiraud and S. Petite [START_REF] Gambaudo | Minimal configurations for the Frenkel-Kontorova model on a quasicrystal[END_REF]. Once again, the state space is one dimensional, all minimizing configurations admit a rotation number and all rotation number is obtained by a minimizing configuration.

Mather [START_REF] Mather | Action minimizing invariant measures for positive definite Lagragian systems[END_REF] has developed a theory in any dimension, closely related to the present article, of minimizing measures for a Tonelli's Lagrangian, that is, 1-periodic superlinear strictly convex Lagrangian L : T M × T 1 → R. In dimension 1, Moser [START_REF] Moser | Monotone twist mappings and the calculus of variations[END_REF] has proved the equivalence between two approaches, monotone twist map versus Tonelli's Lagrangian, showing that such a monotone twist map can always be seen as the time one map of some 1-periodic smooth Hamiltonian H : T * M ×T 1 → R. As noticed by Herman [START_REF] Herman | Inégalités a priori pour des tores lagrangiens invariants par des difféomorphismes symplectiques[END_REF], there are interesting connections between configurations with minimal energy and Lagrangian tori invariant under symplectic diffeormorphisms of the cotangent bundle of the d-dimensional torus. Massart has generalized several results of the Aubry-Mather theory in [START_REF] Massart | Stable norms of surfaces: local structure of the unit ball at rational directions[END_REF][START_REF] Massart | On Aubry sets and Mather's action functional[END_REF][START_REF] Massart | Aubry sets vs Mather sets in two degrees of freedom[END_REF].

We now return to the main results of the present article. We first define the ground energy per particle or the minimizing holonomic value L. The word holonomic is used as in Gomes [START_REF] Gomes | Viscosity solution method and the discrete Aubry-Mather problem, Discrete and Continuous Dynamical Systems[END_REF] or Mañé [START_REF] Mañé | Generic properties and problems of minimizing measures of Lagrangian systems[END_REF] to point out that no dynamical system can be introduced in a natural way. More explanations on this subject will be given in section 3. The minimizing holonomic value may be seen as the lowest mean energy per particle, formally it is defined as L = inf lim inf n→+∞ 1 n L(x 0 , x 1 , . . . , x n ) : {x k } k∈Z is a configuration .

We will describe a special subset of minimizing configurations called calibrated, which is closely related to Aubry's notion of recurrent minimizing configuration or to Chou and Griffiths' concept of ground state. We first define the Aubry set A(L), a similar notion introduced by Fathi [START_REF] Fathi | The weak KAM theorem in Lagrangian dynamics[END_REF]. Informally, A(L) is the set of configurations {x k } k∈Z such that all couple (x k , x k+1 ) is the initial segment of a periodic configurations with minimal energy. More precisely, {x k } k∈Z ∈ A(L) if, and only if, for every k ∈ Z and ǫ > 0, there exist a periodic configuration (x ǫ 0 , x ǫ 1 , . . . , x ǫ q(ǫ) ) and integers q(ǫ) ≥ 1 and p(ǫ) such that x ǫ q(ǫ) = x ǫ 0 + p(ǫ), We first introduce an important notion called effectif potential by Chou and Griffiths, weak KAM solution by Fathi or viscosity solution in the context of Hamilton-Jacobi equation. We prefer to use the term calibrated sub-action. We call sub-action any continuous

|L(x ǫ 0 , x ǫ 1 , . . . , x ǫ q(ǫ) ) -q(ǫ) L| < ǫ and (x k , x k+1 ) = (x ǫ 0 , x ǫ 1 ).
Z d -periodic function u : R d → R such that u(y) -u(x) ≤ L(x, y) -L, ∀ x, y ∈ R d .
We call backward or forward calibrated sub-action a sub-action u satisfying in addition the forward case: u(x) -L = max y∈R d u(y) -L(x, y) ,

∀ x ∈ R d , the backward case: u(y) + L = min x∈R d u(x) + L(x, y) , ∀ y ∈ R d .
Calibrated sub-actions u are solutions of the backward or forward Lax-Oleinik operator. These operators may be seen as a min-plus (or tropical) version of the Ruelle operator used in the thermodynamical formalism. The two sub-actions may be seen as left and right eigenvectors of a min-plus eigenvalue problem. We not only guarantee that calibrated sub-actions do exist in the C 0 coercive case, but we also discuss how the regularity of the local interaction energy affects the regularity of a calibrated sub-action. We show (see proposition 4.7) that, if L is locally Lipschitz or C 2 , then any forward (resp. backward) calibrated sub-action is Lipschitz or semiconvex (resp. semiconcave). See also Gomes [START_REF] Gomes | Viscosity solution method and the discrete Aubry-Mather problem, Discrete and Continuous Dynamical Systems[END_REF] for similar results. We call calibrated configuration (or more precisely u-calibrated if needed) a configuration {x k } k∈Z such that, for some sub-action u,

u(x k+1 ) -u(x k ) = L(x k , x k+1 ) -L, ∀ k ∈ Z.
Let N(L, u) be the set of u-calibrated configurations. It is then obvious that a calibrated configuration is minimizing in the strong sense. It is also easy to see that configurations in the Aubry set are calibrated for any sub-action and therefore minimizing in the strong sense:

A(L) ⊂ N(L, u) ⊂ E(L), ∀ u calibrated.
We show in section 5 that calibrated configurations do exist and have bounded jumps sup k x k+1x k < +∞. We also show in section 10 that the Aubry set can be characterized using the property of calibrated sub-actions, more precisely

A(L) = x = {x k } k∈Z ∈ (R d ) Z :
x is calibrated for any sub-action .

The proof requires a particular class of sub-actions, called separating sub-action. The fact that the class of separating sub-actions is generic in the C 0 topology may be seen as a discrete counterpart of Fathi and Siconolfi's critical subsolutions in the framework of Hamilton-Jacobi equation (see [START_REF] Fathi | Existence of C 1 critical subsolutions of the Hamilton-Jacobi equation[END_REF]).

In the C 1 ferromagnetic and coercive case, we can say more on the Aubry set. For any x 0 ∈ R d , there exists at most one calibrated configuration passing through x 0 . The two nearest neighbors are given by the equation

Du(x 0 ) = ∂L ∂y (x -1 , x 0 ) = - ∂L ∂x (x 0 , x 1 ),
where u is any sub-action and Du is continuous on the projected Aubry set. This property is called the Aubry-Mather graph property. The graph is compact and invariant with respect to the corresponding twist or discrete Euler-Lagange map. Sections 6 and 7 are devoted to more precise properties of the Aubry set in the general C 0 coercive case. We show that the Aubry set is not empty by showing that it contains the Mather set M(L) which is not empty. The Mather set is built using minimizing holonomic measures, a notion introduced by Mather (see [START_REF] Mather | Minimal measures[END_REF] and [START_REF] Mather | Action minimizing invariant measures for positive definite Lagragian systems[END_REF]) but in the context of twist maps where the invariance property is used instead of the holonomic property. In the context of configurations space, we call minimizing holonomic measure a transshipment π(dx, dy) (definition 3.3), that is, a sigmafinite measure π(dx, dy) invariant by the integer translations We show that the minimum is attained by such transshipments with bounded offdiagonal support. We call Mather set M(L) the set of configurations {x k } k∈Z such that, for all k ∈ Z, there exists a minimizing holonomic measure π whose support contains (x k , x k+1 ). We then show that

x ∈ R d → x + k ∈ R d , k ∈ Z d ,
M(L) ⊂ A(L) and L = T d ×R d L(x, y) π(dx, dy), ∀ π minimizing.
The Aubry set may contain "heteroclinic" configurations joining two minimizing periodic orbits and may therefore be stricly larger than the Mather set. Aubry [START_REF] Aubry | The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states[END_REF] uses the term advanced or delayed discommensuration for these minimizing configurations. The Aubry set admits a description in terms of a potential barrier. We call Mañé potential the map

S(x, y) = inf L(x 0 , . . . , x n ) -n L : n ≥ 1, x 0 = x, x n = y + p, x k ∈ R d , p ∈ Z d .
Chou and Griffiths call excitation energy of a configuration {x k } k∈Z , relative to the ground state level, the nonnegative limit

lim k→-∞, l→+∞ L(x k , x k+1 , . . . , x l ) -(l -k) L -S(x k , x l ).
A calibrated configuration has obviously zero excitation energy. We show that the Aubry set can be characterized by the Mañé potential and coincides with the set of pure ground states introduced by Chou and Griffiths, namely,

A(L) = {x k } k∈Z : L(x k , x k+1 ) -L = S(x k , x k+1 ) = -S(x k+1 , x k ), ∀ k ∈ Z = {x k } k∈Z : L(x k , . . . , x l ) -(l -k) L = S(x k , x l ) = -S(x l , x k ), ∀ k < l .
Assuming only L to be C 0 and coercive, we prove in section 8 that S(x, y) is continuous, doubly periodic and satisfies u(y)u(x) ≤ S(x, y), for all x, y ∈ R d and any sub-action u. Let A 0 (L) be the projected Aubry set,

A 0 (L) = x 0 ∈ R d : {x k } k∈Z ∈ A(L) .
We show that (see theorem 8.10), for any x ∈ A 0 (L), S(x, •) is backward calibrated, -S(•, x) is forward calibrated and a characterization in terms of Mañé potential:

A 0 (L) = x 0 ∈ R d : S(x 0 , x 0 ) = 0 .
We finally show (see theorem 9.1) that calibrated sub-actions are completely known as soon as they are known on the projected Aubry set:

the forward case: u(x) = max y∈A 0 (L) u(y) -S(x, y) , ∀ x ∈ R d , the backward case: u(y) = min x∈A 0 (L) u(x) + S(x, y) ,

∀ y ∈ R d .
This result should be compared to the one obtained by G. Contreras [START_REF] Contreras | Action potential and weak KAM solutions[END_REF] for weak KAM solutions.

When a system depends on several parameters, it is important to introduce an order parameter which labels the different phases in a bifurcation diagram called phase diagram. In the one dimensional case, the order parameter is the rotation number of ground states. It is generally believed that, in the (λ, K) phase diagram in the standard Frenkel-Kontorova, apart a set of zero Lebesgue measure, the Aubry set is reduced to a unique periodic orbit: the system is locked at rational rotation number. We introduce in section 11 the basis of a theory of rotation vectors in the multidimensional case.

We call rotation vector ω of a minimizing configuration {x k } k∈Z the following limit when it exists

ω {x k } k∈Z = lim n-m→+∞ x n -x m n -m .
As in Mather theory, we relate this notion to the notion of rotation vector of a minimizing measure

ω[π] = T d ×R d (y -x) π(dx, dy).
The main important observation is that a configuration is minimizing for the interaction energy L(x, y) if, and only if, it is minimizing for the interaction energy L(x, y)λ, (yx) for any parameter λ ∈ R d . In the one dimensional case, a recurrent minimizing configuration admits a rotation number ω and is calibrated for some Lλ. The two numbers ω and λ are related by Legendre transform. If ω is irrational, λ is unique. In the multidimensional case, we follow Gomes [START_REF] Gomes | Viscosity solution method and the discrete Aubry-Mather problem, Discrete and Continuous Dynamical Systems[END_REF] to show that, whenever the minimizing holonomic value of Lλ, denoted L(λ), is differentiable at λ, any calibrated configuration for Lλ admits a rotation vector given by

ω = - ∂ L ∂λ .
In particular, there exists minimizing configuration with rotation vector of arbitrarily large norm.

We have chosen to translate the Frenkel-Kontorova model into the framework of discrete Lagrangian Aubry-Mather theory mainly to have a simple notion of minimizing holonomic measures instead of transshipments. The main object we are interested in is thus a C 0 coercive Lagrangian L(x, v) defined on T d × R d . A family of local interaction energies can be naturally defined by

L τ (x, y) = τ L x (mod Z d ), y -x τ , ∀ x, y ∈ R d ,
where τ > 0 is a parameter that we keep in order to understand later the homogenization theory of Frenkel-Kontorova model as in [START_REF] Forcadel | Homogenization of fully overdamped Frenkel-Kontorova models[END_REF] and [START_REF] Forcadel | Homogenization of accelerated Frenkel-Kontorova models with n types of particles[END_REF]. Notice that L τ is invariant under the diagonal action of Z d ,

L τ (x + k, y + k) = L τ (x, y), ∀ k ∈ Z d .
The two approches are complementary. While the Lagrangian formulation will be more adapted in the description of the support of minimizing measures, the Frenkel-Kontorova setting will be used in the construction of calibrated sub-actions, as well as in the definition of two major notions of action potential: the Mañé potential and the Peierls barrier. We intend later to better understand the limit when the step τ tends to zero and the thermodynamic ground state limit of the system when the temperature goes to zero.

A discrete Lagrangian dynamics

We fix from now on a C 0 coercive Lagrangian L(x, v) : R d × R d → R, Z d -periodic in x, and its associated local interaction energy map

L τ (x, y) = τ L x, y -x τ
defined on R d ×R d and invariant by the diagonal action of Z d . We begin by recalling some well known notions of divergence type at infinity. Coerciveness is our basic assumption, superlinerarity will be used when homology will play a role.

Definition 2.1. Let L(x, v) : T d × R d → R be a C 0 -Lagrangian. i. L(x, v) is said to be coercive if lim R→+∞ inf v ≥R inf x∈T d L(x, v) = +∞. ii. L(x, v) is said to be superlinear if lim R→+∞ inf v ≥R inf x∈T d L(x, v) v = +∞.
We call configuration any sequence {x k } k∈Z of points in R d . Let Σ = (R d ) Z be the set of configurations. We also consider the set of configurations modulo the diagonal action of Z d , that is, the quotient of Σ by the equivalence relation: {x k } k∈Z ∼ {y k } k∈Z if, and only if, there exists s ∈ Z d such that

y k = x k + s for all k ∈ Z. So Σ = (R d ) Z and Σ/ ∼ = (R d ) Z / ∼ .
Let us notice that, for any fundamental domain D of the action of

Z d on R d , the set (R d ) Z * -× D × (R d ) Z *
+ is a fundamental domain for the diagonal action of Z d on Σ. Let σ : Σ → Σ be the left shift given by σ({x k }) = {y k } where y k = x k+1 . Notice that σ commutes with the diagonal action. Definition 2.2. We call minimizing configuration any sequence {x k } k∈Z which minimizes the local interaction energy, namely,

L τ (x n , x n+1 , . . . , x n+m ) := n+m k=n L τ (x k , x k+1 ) ≤ L τ (y n , y n+1 , . . . , y n+m ),
for any finite configuration {y k } n+m k=n with identical boundary conditions x n = y n and x n+m = y n+m .

Although one of our aim is to extend as much as we can the discrete Aubry-Mather theory to just C 0 coercive Lagrangian and to describe precisely the set of minimizing configurations in this general setting, we show in this section that, under a stronger hypothesis on the Lagrangian (C 2 -smoothness and twist condition), we can recover the original theory, where the set of minimizing configurations can be understood through the help of a dynamical system similar to the usual standard map. Let us first recall the notion of critical configuration.

Definition 2.3. Let L(x, v) : T d × R d → R be a C 1 -Lagrangian. We call critical triple a configuration (x -1 , x 0 , x 1 ) of three points in R d satisfying ∂L τ ∂y (x -1 , x 0 ) + ∂L τ ∂x (x 0 , x 1 ) = 0.
We call critical configuration any configuration {x k } k∈Z of points in R d consisting of critical triples (x k-1 , x k , x k+1 ):

∂L τ ∂y (x k-1 , x k ) + ∂L τ ∂x (x k , x k+1 ) = 0, ∀ k ∈ Z.
Let Γ τ (L) ⊂ Σ be the set of critical configurations. We notice that Γ τ (L) is invariant by both the diagonal action of Z d and the shift σ. Let Γ τ (L)/ ∼ be the quotient of Γ τ (L) by the diagonal action of Z d .

The equations defining Γ τ (L) may be seen as a discrete version of the Euler-Lagrange equation. These equations show that, under some stronger hypothesis of twist condition, the knowledge of (x 0 , x 1 ) implies the existence of a unique critical configuration with such initial conditions. More precisely, prefering the use of the ferromagnetic terminology instead of the twist condition as it is done in statistical mechanics, we introduce the following notion. Definition 2.4. A C 1 -Lagrangian L(x, v) is said to be ferromagnetic if, for any sufficiently small τ > 0, the two maps in (I) or equivalently in (II), where

(I)          R d → R d x → ∂L τ ∂y (x, y) y → ∂L τ ∂x (x, y)
and (II)

         R d → R d v → ∂L ∂v (y -τ v, v) v → τ ∂L ∂x (x, v) - ∂L ∂v (x, v)
, are homeomorphisms for all (x, y).

Similarly a discrete version of the Euler-Lagrange flow may be introduced.

Definition 2.5. Let L(x, v) be a C 1 ferromagnetic Lagrangian. For sufficiently small τ > 0, we call discrete Euler-Lagrange map (or standard map), the map

Φ τ = T d × R d → T d × R d (x, v) → (y, w)
where y = x+τ v and w is the unique solution of one of the two equivalent equations

∂L τ ∂y (x, y) + ∂L τ ∂x (y, y + τ w) = 0 or ∂L ∂v (x, v) + τ ∂L ∂x (y, w) - ∂L ∂v (y, w) = 0.
Notice that Φ τ is a homeomorphism on T d × R d . In most part of the article, the dynamical sytem (T d × R d , Φ τ ) will not be used, except, for instance, in section 6, where we prove that minimizing measures are supported on a graph. The main advantage of the standard map approach is that the space of critical configurations modulo the diagonal action is conjugate to a 2d degrees of freedom dynamical system.

Remark 2.6. Let Π τ : R d × R d → T d × R d be the projection given by Π τ (x 0 , x 1 ) = x 0 mod Z d , (x 1 -x 0 )/τ .
We extend Π τ to Σ by writing Π τ ({x k } k∈Z ) = Π τ (x 0 , x 1 ) and notice that the projection

Π τ : Σ/ ∼ → T d × R d is also well defined. If L(x, v) is ferromagnetic, then (Γ τ (L)/ ∼ , σ) is conjugated to (T d ×R d , Φ τ ), that is, the following diagram commutes Γ τ (L)/ ∼ Πτ ----→ T d × R d   σ   Φτ Γ τ (L)/ ∼ Πτ ----→ T d × R d .
A critical configuration is thus completely determined by the 2d data (x, v) and a general configuration plays the role of a virtual deformation as in Mechanics. In order to check that a Lagrangian satisfies the ferromagnetic condition, an easier but stronger assumption may be used instead.

Definition 2.7. Let L(x, v) : T d × R d → R be a C 2 -Lagrangian. We say that L is uniformly strictly convex (with respect to v) if ∂ 2 L
∂v 2 is uniformly positive definite, that is, if there exists α > 0 such that

∂ 2 L ∂v 2 (x, v) • w, w ≥ α w 2 , ∀ x ∈ T d , ∀ v, w ∈ R d .
The following proposition shows that a strictly convex Lagrangian with bounded second derivative is ferromagnetic. The proof is left to the reader.

Proposition 2.8. Let L(x, v) : T d × R d → R be a C 2 strictly convex Lagrangian. Then L(x, v) is superlinear. If L(x, v) satisfies in addition the uniform condition ∂ 2 L ∂x∂v T d ×R d ≤ β for some β > 0, then L(x, v
) is ferromagnetic. Moreover, for any x, y ∈ T d , for any sufficiently small τ > 0, the two maps

v ∈ R d → τ ∂L ∂x (x, v) - ∂L ∂v (x, v) ∈ R d and v ∈ R d → ∂L ∂v (y -τ v, v) ∈ R d ,
or equivalently the two maps

y ∈ R d → ∂L τ ∂x (x, y) ∈ R d and x ∈ R d → ∂L τ ∂y (x, y) ∈ R d ,
are C 1 -diffeomorphisms. In particular, the discrete Euler-Lagrange map Φ τ is a C 1 -diffeomorphism.

Notice that the ferromagnetic condition can be obtained under weaker hypotheses. For instance, assume d = 1. Let f : R → R be an increasing homemorphism with f (0) = 0 and g : T 1 → R be a C 1 -function. Then L(x, v) = g(x) + v 0 f (w) dw is superlinear, ferromagnetic and C 1 . We also notice that ∂L ∂x is a coboundary under the dynamics (

T d × R d , Φ τ ): τ ∂L ∂x (y, w) = ∂L ∂v (y, w) - ∂L ∂v • Φ -1 τ (y, w).

Minimizing holonomic measures

In this section, we would like to make clear the usefulness of the notions of holonomic measure (definition 3.1) and minimizing holonomic measure (definition 3.5). We begin by briefly recalling Mather's approach for the theory of minimizing orbits: L(x, v) is usually assumed to be C 2 , periodic in x (namely, x ∈ T d ), strictly convex in v ∈ R d and, for the purposes of this article, time independent. In Mather's approach, we are interested in finding minimizing absolutely continuous trajectories, that is, trajectories t ∈ R → x(t) such that, for any t 0 < t 1 and any other trajectory t ∈ [t 0 , t 1 ] → y(t) satisfying the boundary conditions x(t 0 ) = y(t 0 ) and x(t 1 ) = y(t 1 ), the local action of x(t) on [t 0 , t 1 ] is bounded from above by the local action of y(t),

t 1 t 0 L(x(t), ẋ(t))dt ≤ t 1 t 0 L(y(t), ẏ(t))dt.
These mimimizing trajectories may be looked for among the trajectories with a prescribed rotation vector

ω ∈ R d , lim t→+∞ x(t) t = lim t→+∞ 1 t t 0 ẋ(t)dt = ω.
Notice that a minimizing trajectory must satisfy the Euler-Lagrange equation

d dt ∂L ∂v (x, ẋ) = ∂L ∂x (x, ẋ)
and is therefore governed by the Euler-Lagrange flow Φ τ (x 0 , ẋ0 ) = (x t , ẋt ). Suppose in addition that (x t , ẋt ) is recurrent or more precisely is regular in the sense of Birkhoff's ergodic theorem for some Φ τ -invariant ergodic probability measure µ on

T d × R d , then T d ×R d L(x, v)dµ(x, v) ≤ T d ×R d L(x, v)dν(x, v)
for any other invariant probability measure ν on

T d × R d .
It is therefore natural to look for minimizing trajectories as regular orbits of the Euler-Lagrange flow located in the support of minimizing measures. Mather's approach can thus be translated into a linear optimization problem

           µ = argmin T d ×R d L(x, v)dµ(x, v) µ is a Φ τ -invariant probability measure T d ×R d v dµ(x, v) = ω.
Following R. Mañé [START_REF] Mañé | Generic properties and problems of minimizing measures of Lagrangian systems[END_REF] and D. A. Gomes [START_REF] Gomes | Viscosity solution method and the discrete Aubry-Mather problem, Discrete and Continuous Dynamical Systems[END_REF], one can weaken this optimization problem by asking µ to be only holonomic, that is, satisfying

T d φ • Φ τ (x, v) dµ(x, v) = T d φ(x) dµ(x, v)
for any bounded Borel (periodic) function φ : T d → R. (Notice that the previous condition does not tell that µ is invariant by Φ τ since the the test function f depends only on x and not on (x, v), see definition 3.1).

We come back to our discrete Aubry-Mather theory and, as in the weak Mather's approach, we try to look for minimizing configurations located in the support of minimizing invariant measures or more precisely in the support of minimizing holonomic measures since the discrete Euler-Lagrange map may not exist. We denote by P(T d × R d ) the convex set of probability measures over the Borel sets of T d × R d . Definition 3.1. We call holonomic measure a probability measure µ ∈ P(T d × R d ) satisfying

T d ×R d φ(x + τ v)dµ(x, v) = T d ×R d φ(x)dµ(x, v)
for any bounded Borel function φ : T d → R. The set of holonomic measures is denoted by

P τ (T d × R d ).
Notice that, in the ferromagnetic case, Φ τ -invariant probability measures are holonomic. Nevertheless, the holonomic class is larger. For example, any finite configuration (x 0 , x 1 , . . . , x n-1 ) gives a holonomic probability µ =

1 n n-1 i=0 δ (x i ,v i ) ,
where v i = x i+1 -x i τ and x n = x 0 . Notice also that the set of holonomic measures is closed under the narrow topology.

We want to show that, although the notion of holonomic measures seems to be unrelated to a dynamical system, the set of these measures is nevertheless in one-to-one correspondence with the set of normalized invariant Markov chain of (Σ, σ). Definition 3.2. We call normalized invariant Markov chain on (Σ, σ) a sigmafinite Markov chain (ν(dx), p(x, dy)), with initial distribution ν(dx) (a sigma-finite measure defined on the Borel sets of R d ) and transition kernel p(x, dy) (a measurable family of probability measures defined on the Borel sets of R d ), satisfying the following properties: i. ν(dx) is invariant under the action of Z d and has mass one on any fundamental domain,

ii. p(x, dy) is invariant under the action of Z d in the following sense

R d ψ(y + s)p(x, dy) = R d ψ(y)p(x + s, dy),
for any bounded Borel function ψ, for any s ∈ Z d , iii. ν(dx) is Markov-stationary in the following sense

R d ×R d ψ(y)p(x, dy)ν(dx) = R d ψ(y)ν(dy)
for any bounded Borel function ψ.

The sigma-finite Markov chain μ on Σ is given as usual as

Σ ψ(x)dμ(x) = R d • • • R d ψ(x 0 , x 1 , . . . , x n )ν(dx 0 )p(x 0 , dx 1 ) • • • p(x n-1 dx n ),
for any bounded Borel function ψ(x) = ψ(x 0 , x 1 , . . . , x n ), for any n ≥ 0. Then μ is both invariant with respect to the Z d and the shift σ action.

The correspondence between the notions of normalized invariant Markov chains and of holonomic measures may be explained through another notion called normalized invariant transshipment measure as in Evans and Gomes [START_REF] Evans | Linear programming interpretations of Mather's variational principle[END_REF]. Definition 3.3. We call normalized invariant transshipment measure π a sigmafinite measure defined on the Borel sets of R d ×R d verifying the following properties: i. π is invariant under the diagonal action of Z d and has mass one on any fundamental domain,

ii. if

pr 1 : R d × R d → R d and pr 2 : R d × R d → R d denote the two canonical projections, then pr 1 * (π) = pr 2 * (π).
The following proposition shows the equivalence between the three notions: holonomic measures, normalized invariant transshipment measures and normalized invariant Markov chains. The proof is left to the reader and uses mainly the notion of disintegration of measures.

Proposition 3.4. The three sets of measures, holonomic measures µ, normalized invariant transshipment measures π and normalized invariant Markov chains (ν(dx), p(x, dy)) are in one-to-one correspondence. The correspondence is given by:

R d ×R d ψ(x, y)π(dx, dy) := R d ×R d ψ(x, x + τ v)µ(dx, dv), R d φ(x) R d ψ(x, y)p(x, dy) ν(dx) := R d ×R d φ(x)ψ(x, y)π(dx, dy), R d ×R d φ(x, v)µ(dx, dv) := R d ×R d φ x, y -x τ p(x, dy)ν(dx),
where µ(dx, dv) has been extended to R d × R d by invariance under the action of Z d on the first factor.

As in the weak Mather's approach, we are interested in finding particular minimizing configurations which are located in the support of minimizing holonomic measures. We thus introduce a similar concept equivalent to Mañé's definition of critical value.

Definition 3.5. Let L(x, v) : T d × R d → R be a continuous coercive Lagrangian.
We call minimizing holonomic value of L the quantity

L(τ ) := inf µ T d ×R d L(x, v) µ(dx, dv),
where the infimum is taken over the set of holonomic measures. A measure µ attaining the infimum is called a minimizing holonomic measure.

Remark 3.6. The three equivalent definitions given in proposition 3.4 show that any holonomic measure µ seen on T d × R d can be lifted to a shift-invariant probability measure μ on Σ/ ∼ obtained from the normalized invariant Markov chain (ν(dx), p(x, dy)). Conversely, the projection µ = (Π τ ) * (μ) of any shift-invariant probability measure μ on Σ/ ∼ is holonomic:

T d ×R d φ(x + τ v)µ(dx, dv) = Σ/∼ φ(x 1 )μ(dx) = = Σ/∼ φ(x 0 )μ(dx) = T d ×R d φ(x)µ(dx, dv).
From proposition 3.4, the minimizing holonomic value of L may be computed using two different ways

L(τ ) = inf π R d ×R d /∼ L x, y -x τ π(dx, dy) = inf μ Σ/∼ L x 0 , x 1 -x 0 τ μ(dx),
where the infimums are taken, respectively, over the set of normalized invariant transshipment measures π and over the set of shift-invariant probability measures on Σ/ ∼ .

Since T d × R d is not compact, the existence of a minimizing holonomic measure is not guarantee at first sight. Nevertheless, periodicity in x and coerciveness in yx implies the existence of such minimizing measures.

Proposition 3.7. Let L(x, v) : T d × R d → R be a continuous coercive Lagrangian.
Then there exists a minimizing holonomic measure having a compact support.

Before going into the proof of this result, we will make use of a special piecewise continuous map

F τ : T d × R d → T d × R d which enables us to replace R d by a compact ball. Let v ∞ = max i |v i | be the maximum norm. Definition 3.8. Suppose L(x, v) is a coercive Lagrangian, then there exists a real number R τ > 1/τ such that inf v ∞≥Rτ inf x∈T d L(x, v) > sup v ∞≤1/τ sup x∈T d L(x, v).
Let ⌊v⌋ ∈ Z d denotes the vector whose coordinates are the greatest integers less or equal than the respective coordinates of v ∈ R d . Lemma 3.9. Let L(x, v) be a C 0 coercive Lagrangian and F τ :

T d × R d → T d × R d defined by F τ (x, v) = x, v -1 τ ⌊τ v⌋ if v ∞ ≥ R τ (x, v) if v ∞ < R τ . Then F τ satisfies i. the image F τ (T d × R d ) is a bounded set; ii. L(x, v) ≥ L • F τ (x, v) ∀ (x, v) ∈ T d × R d ; iii. µ ∈ P τ (T d × R d ) ⇒ (F τ ) * µ ∈ P τ (T d × R d ).
Proof. The first item is obviously satisfied. The second one is just a consequence of the choice of R τ . Finally, since ψ(x

+ τ v -⌊τ v⌋) = ψ(x + τ v) for every ψ ∈ C 0 (T d ),
the third item follows without difficulty.

We can now prove the existence of minimizing holonomic measures for C 0 coercive Lagrangians.

Proof of proposition 3.7. Consider a sequence

{µ n } ⊂ P τ (T d × R d ) of holonomic measures satisfying lim n L(x, v) dµ n (x, v) = L(τ ).
Items ii and iii of lemma 3.9 assure that the sequence {ν n = (F τ ) * µ n } verifies the same properties. Furthermore, by item i of the same lemma, all probability measures ν n are supported on a common compact set. Therefore, any accumulation point

ν ∈ P τ (T d × R d ) of {ν n } for the narrow topology satisfies L(x, v) dν(x, v) = L(τ ).

Lax-Oleinik operators

The Lax-Oleinik semigroup is well known in partial differential equations and in calculus of variations. It was used by A. Fathi (see [START_REF] Fathi | Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens[END_REF]) for obtaining the so-called weak KAM theorem in the framework of continuous-time, autonomous, strictly convex and superlinear C 3 -Lagrangians on a compact manifold.

Several results in this section are comparable to those in D. Gomes [START_REF] Gomes | Viscosity solution method and the discrete Aubry-Mather problem, Discrete and Continuous Dynamical Systems[END_REF]. There are some differences: in the computation of the "effective" L(τ ), we do not use the abstract Fenchel-Rockafellar duality theorem to prove the sup-inf principle (proposition 4.5 in this section or theorem 2.3 in [START_REF] Gomes | Viscosity solution method and the discrete Aubry-Mather problem, Discrete and Continuous Dynamical Systems[END_REF]); we do not use a penalized Lax-Oleinik version as in [START_REF] Gomes | Viscosity solution method and the discrete Aubry-Mather problem, Discrete and Continuous Dynamical Systems[END_REF] which corresponds to a more numerical approach (theorem 4.3 in this section or theorem 3.3 in [START_REF] Gomes | Viscosity solution method and the discrete Aubry-Mather problem, Discrete and Continuous Dynamical Systems[END_REF]). More important, we are carefull about the degree of minimum regularity we need in each step; we give in particular two estimates of regularity which are very comparable to those in D. Gomes but with different hypothesis (proposition 4.7 and lemma 6.8 in this paper or propositions 5.1 and 5.2 in [START_REF] Gomes | Viscosity solution method and the discrete Aubry-Mather problem, Discrete and Continuous Dynamical Systems[END_REF]).

In our context, we are interested in studying operators with similar properties to the Lax-Oleinik semigroup. We recall that

L τ (x, y) = τ L(x, y-x τ ). Definition 4.1. Given a C 0 coercive Lagrangian L = L(x, v) : T d × R d →
R and a constant τ > 0, we call forward and backward Lax-Oleinik operators, respectively, the maps T + and T -defined by

T + u(x) = sup v∈R d [u(x + τ v) -τ L(x, v)] = sup y∈R d [u(y) -L τ (x, y)], T -u(y) = inf v∈R d [u(y -τ v) + τ L(y -τ v, v)] = inf x∈R d [u(y) + L τ (x, y)],
for every Z d -periodic function u ∈ C 0 (R d ) that we identify with u ∈ C 0 (T d ).

Because of the choice of R τ in definition 3.8 and the fact that the minimization of L can be made on the ball v ∞ ≤ R τ as explained in lemma 3.9, T ± are well defined and have the following more restricted definition

T + u(x) = max v ∞≤Rτ [u(x + τ v) -τ L(x, v)] = max y-x ∞≤τ Rτ [u(y) -L τ (x, y)], T -u(y) = min v ∞≤Rτ [u(y -τ v) + τ L(y -τ v, v)] = min y-x ∞≤τ Rτ [u(y) + L τ (x, y)].
Such identities are immediate consequences of the explicit construction of the application

F τ : T d × R d → T d × R d
whose properties are described in lemma 3.9. Indeed, writing

φ + (x, v) = u(x + τ v) -τ L(x, v) and φ -(x, v) = u(x -τ v) + τ L(x -τ v, v), we get φ + • F τ ≥ φ + and φ -• F τ ≤ φ -. So we have max v∈R d φ + (x, v) = max v∈R d φ + • F τ (x, v) = max v ∞≤Rτ φ + (x, v),
and similar equalities for φ -as well.

Let osc(f, D) denote the oscillation of a function f on a subset D of its domain.

Lemma 4.2. Let L(x, v) be a C 0 coercive Lagrangian. Then the Lax-Oleinik operators verify the following properties.

i. For all u ∈ C 0 (T d ), for all x, y ∈ T d ,

|T + u(x) -T + u(y)| ≤ max v * ,w * τ L(x, v * ) -L(y, w * ) and |T -u(x) -T -u(y)| ≤ max v * ,w * τ L(x -τ v * , v * ) -L(y -τ w * , w * )
, where the maxima are taken over

v * ∞ , w * ∞ ≤ 2R τ and v * -w * ∞ ≤ x -y ∞ τ .
ii. The two operators T + and T -map C 0 (T d ) into itself.

iii Proof. On the one hand, for any point

x in R d , there exists z * ∈ R d , such that x -z * ∞ ≤ τ R τ and T + u(x) = u(z * ) -L τ (x, z * ).
On the other hand, for any y ∈ R d , T + u(y) ≥ u(z * ) -L τ (y, z * ). Combining these two estimates, we obtain

T + u(y) -T + u(x) ≥ L τ (x, z * ) -L τ (y, z * ) = τ L(x, v * ) -L(y, w * )
where w * = v * + x-y τ and z * = x + τ v * . A similar estimate holds by permuting x and y which proves the first property for T + . An analogous argument can be used to demonstrate the inequality concerning the backward Lax-Oleinik operator T -. Since L(x, v) is uniformly continuous on T d × B 2Rτ , the two sets T ± (C 0 (T d )) are equicontinuous and the lemma is proved.

We recall that the minimizing holonomic value L(τ ) has been introduced in definition 3.5. So the main theorem of this section can be stated as follows.

Theorem 4.3. If L(x, v) is a C 0 coercive Lagrangian, then there exist continuous periodic solutions of the Lax-Oleinik equation, u + , u -∈ C 0 (T d ), satisfying

T + u + = u + -τ L(τ ) and T -u -= u -+ τ L(τ ).
Moreover u ± satisfies the a priori estimate:

u + 0 , u -0 ≤ osc(τ L, T d × B 2Rτ ).
Proof. If we equip C 0 (T d ) with the topology of the uniform convergence, it is easy to show that T + : C 0 (T d ) → C 0 (T d ) is 1-Lipschitz. So the Lipschitz regularity is also respected by the application T+ :

C 0 (T d ) → C 0 (T d ) defined by T+ u = T + u -max(T + u).
Obviously T+ u ≤ 0 everywhere on C 0 (T d ). Conversely, it follows from lemma 4.2 that T+ u ≥ -osc(τ L,

T d × B 2Rτ ), ∀ u ∈ C 0 (T d ).
Let B ⊂ C 0 (T d ) denote the closed convex hull of the closure of T+ (C 0 (T d )). Since the image T+ (C 0 (T d )) is bounded, B is a compact convex set. As T+ (B) ⊂ B, by the Schauder-Tychonoff fixed point theorem, there exists a function u + ∈ C 0 (T d ) such that

T + u + = u + + max(T + u + ).
Obviously, u + 0 = T+ (u + ) 0 ≤ osc(τ L, T d × B 2Rτ ). It remains to show that max(T + u + ) = -τ L(τ ). On the one hand

τ L(x, v) + u + (x) -u + (x + τ v) ≥ -max(T + u + ) everywhere on T d × R d . For any holonomic probability measure µ ∈ P τ (T d × R d ),
by integrating the previous inequality, we obtain

τ L(x, v) dµ = τ L(x, v) + u + (x) -u + (x + τ v) dµ ≥ -max(T + u + )
and therefore max(T + u + ) ≥ -τ L(τ ).

On the other hand, given x 0 ∈ T d , there exists a vector

v 0 ∈ R d such that v 0 ∞ ≤ R τ and τ L(x 0 , v 0 ) + u + (x 0 ) -u + (x 0 + τ v 0 ) = -max(T + u + ). For every k ≥ 1, if x k = x k-1 + τ v k-1 ∈ T d , we consider inductively v k ∈ R d such that v k ≤ R τ and τ L(x k , v k ) + u + (x k ) -u + (x k + τ v k ) = -max(T + u + ).
Let {µ n } be the probability measure defined by

µ n = 1 n n-1 k=0 δ (x k ,v k ) .
Since their supports are contained in the compact set T d × B Rτ , such a sequence is relatively compact for the narrow topology. Let µ ∈ P τ (T d ×R d ) be some convergent subsequence limit. Note that the equality

[τ L(x, v) + u + (x) -u + (x + τ v)] dµ n (x, v) = -max T + u + ,
goes through the limit µ. Hence, we obtain max T + u + ≤ -τ L(τ ) if we prove that µ is a holonomic probability measure. Indeed, for any function

ψ ∈ C 0 (T d ), [ψ(x + τ v) -ψ(x)] dµ n (x, v) = 1 n n-1 k=0 [ψ(x k + τ v k ) -ψ(x k )] = 1 n |ψ(x n ) -ψ(x 0 )| ≤ 2 n ψ 0 .
Letting n go to infinity, we immediately obtain that µ ∈ P τ (T d ×R d ). The existence of a function u -∈ C 0 (T d ) is obtained in an analogous way.

The following result is an immediate consequence of the previous proof.

Corollary 4.4. Let L(x, v) be a C 0 coercive Lagrangian. If u ∈ C 0 (T d ) satisfies either T + u = u -c or T -u = u + c for some constant c ∈ R, then c = τ L(τ ).
The previous theorem 4.3 may be seen as an important theorical tool: it gives a way to renormalize the initial Lagrangian by a coboundary

L norm (x, v) = L(x, v) -L(τ ) - 1 τ u(x + τ v) -u(x) ≥ 0, ∀ (x, v) ∈ T d × R d .
The existence of a solution of the Lax-Oleinik operator also gives other characterizations of the minimizing holonomic value, either as a max-min optimal value or as an ergodic average asymptotic value.

Proposition 4.5. Let L(x, v) be a C 0 coercive Lagrangian. Then we have

τ L(τ ) = sup ψ∈C 0 (T d ) inf (x,v)∈T d ×R d τ L(x, v) + ψ(x) -ψ(x + τ v) or τ L(τ ) = inf {x k }∈(R d ) Z + lim inf n→∞ 1 n n-1 k=0 L τ (x k , x k+1 ). Proof. Let µ ∈ P τ (T d × R d ) be any minimizing holonomic measure. Then τ L(τ ) = τ L(x, v) dµ(x, v) = [τ L(x, v) + ψ(x) -ψ(x + τ v)] dµ(x, v) ≥ inf (x,v)∈T d ×R d [τ L(x, v) + ψ(x) -ψ(x + τ v)]
for every ψ ∈ C 0 (T d ). By taking the supremum over all ψ ∈ C 0 (T d ), one obtain a lower bound of τ L(τ ). Conversely, theorem 4.3 establishes there is

u + ∈ C 0 (T d ) such that τ L(τ ) = inf (x,v)∈T d ×R d [τ L(x, v) + u + (x) -u + (x + τ v)] .
The first identity is proved. Consider now an arbitrary sequence

{x k } ∈ R d Z + . Then for any n > 0 n L(τ ) ≤ n-1 k=0 L τ (x k , x k+1 ) + u + (x 0 ) -u + (x n ) ≤ n-1 k=0 L τ (x k , x k+1 ) + 2 u + 0 .
Dividing by n and letting n go to infinity, we obtain the above upper bound for τ L(τ ). Conversely, choose any optimal sequence

{x * k } k≥0 in R d such that τ L(τ ) = L τ (x * k , x * k+1 ) + u + (x * k ) -u + (x * k+1 ), ∀ k ≥ 0.
Dividing again by n and letting n go to infinity, we then obtain the above lower bound for τ L(τ ) and the second identity is proved.

Thanks to theorem 4.3, we know that the solutions u ± of the Lax-Oleinik operator are continuous. If in addition L is locally α-Hölder continuous, the same estimate of part i in lemma 4.2 shows that u ± is also α-Hölder continuous. In fact, these solutions possess a stronger regularity if L is supposed to be semiconcave. Definition 4.6. A function F : T d × R d → R is called semiconcave if, for every R > 0, there exists a nondecreasing upper semicontinuous function θ R : R + → R + satisfying lim ρ→0 + θ R (ρ) = 0 and

tF (ξ) + (1 -t)F (η) -F (tξ + (1 -t)η) ≤ t(1 -t) ξ -η θ R ( ξ -η ) for all ξ = (x, v), η = (y, w) in R d × R d with v , w ≤ R and for any t ∈ [0, 1]. We call {θ R } R>0 a family of local modulus of semiconcavity for F . A function G : T d × R d → R is called semiconvex if -G is semiconcave.
Notice that in the case the function F (x) depends only in x ∈ T d , semiconcavity is defined using a unique modulus θ instead of a family {θ R } R>0 . Any

C 2 -Lagrangian L(x, v) is an example of a semiconcave function with modulus of semiconcavity θ R (ρ) = C R ρ and C R := 1 2 max x∈T d max v ≤R Hess(L)(x, v) ∞ . The proof is standard.
For more details on semiconcave functions, we refer the reader to the book of P. Cannarsa and C. Sinestrari (see [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF]). Let us examine how the forward Lax-Oleinik operator T + deals with semiconcavity.

Proposition 4.7. Let L(x, v) be a semiconcave C 0 coercive Lagrangian. Then any solution u ∈ C 0 (T d ) of the forward Lax-Oleinik equation, T + u = u -τ L(τ ), is semiconvex. Proof. Given x, y ∈ R d and t ∈ [0, 1], set z = tx + (1 -t)y. Then there exists an optimal z * ∈ R d such that u(z) = u(z * ) -L τ (z, z * ) + τ L(τ ) with z -z * ∞ ≤ τ R τ . Moreover u(x) ≤ u(z * ) -L τ (x, z * ) + τ L(τ ) and u(y) ≤ u(z * ) -L τ (y, z * ) + τ L(τ ).
Combining these two inequalities and the previous identity, we obtain

tu(x) + (1 -t)u(y) -u(z) ≥ -tL τ (x, z * ) + (1 -t)L τ (y, z * ) -L τ (z, z * ) .
Let v * and w * be defined by z * = x + τ v * and z * = y + τ w * . Hence

z * = z + τ (tv * + (1 -t)w * ) and v * ∞ , w * ∞ ≤ R τ + x -y τ ∞ ≤ 2R τ .
Then

tu(x) + (1 -t)u(y) -u(z) ≥ ≥ -τ tL(x, v * ) + (1 -t)L(y, w * ) -L(z, tv * + (1 -t)w * ) ≥ ≥ -τ t(1 -t) 2 τ x -y ∞ θ 2Rτ 2 τ x -y ∞ using the fact that v * -w * ∞ = 1 τ x-y ∞ .
We have shown that 1 τ u is semiconvex with a modulus of convexity θ(ρ) = -2 τ θ 2Rτ 2 τ ρ . Similarly, any solution u of the backward Lax-Oleinik equation T -u = u+τ L(τ ) is semiconcave as soon as L(x, v) is a semiconcave C 0 coercive Lagrangian.

Calibrated and minimizing configurations

For conciseness, for any given configuration {x k } k∈Z of points in R d , for any m < n, we call normalized interaction energy of a finite configuration the quantity

Lτ (x m , x m+1 , . . . , x n ) := n-1 k=m L τ (x k , x k+1 ) -τ L(τ ) .
Let us recall from the introduction the fundamental definition of mimimizing configurations.

Definition 5.1. Consider a bounded below C 0 -Lagrangian L(x, v). We say that a configuration {x k } k∈Z of points of R d is a minimizing configuration if, for every pair m < n, Lτ (x m , x m+1 , . . . , x n ) ≤ Lτ (y m , y m+1 , . . . , y n ) whenever {y k } k∈Z satisfies y m = x m and y n = x n . A configuration {x k } k∈Z is called strongly minimizing configuration if, for any two pairs m < n, m ′ < n ′ and any configuration {y k } k∈Z satisfying y m ′ = x m and y n ′ = x n (mod Z d ), we have

Lτ (x m , x m+1 , . . . , x n ) ≤ Lτ (y m ′ , y m ′ +1 , . . . , y n ′ ).
For a coercive Lagrangian, notice that definition 3.8 implies consecutive jumps x k+1x k are uniformly bounded for strongly minimizing configurations {x k } k∈Z , namely, sup

k∈Z x k+1 -x k ∞ < τ R τ .
In order to introduce an important class of minimizing configurations, we will need to consider the following notions. Definition 5.2. Let L(x, v) be a C 0 coercive Lagrangian. A function u : R d → R is a called sub-action (or τ -sub-action if necessary) with respect to L if u(x) is Z d -periodic, continuous and satisfies

τ L(τ ) ≤ τ L(x, v) + u(x) -u(x + τ v), ∀ (x, v) ∈ T d × R d .
More restrictively, u is called forward calibrated if it is a sub-action and

u(x) + τ L(τ ) = sup v∈R d u(x + τ v) -τ L(x, v) , ∀ x ∈ T d ,
and similarly u is called backward calibrated if it is a sub-action and

u(x) + τ L(τ ) = inf v∈R d u(x -τ v) + τ L(x -τ v, v) , ∀ x ∈ T d .
Notice that the C 0 periodic functions u + , u -are forward or backward calibrated sub-actions if, and only if, they are solutions of the forward or backward Lax-Oleinik equations given in theorem 4.3. The existence of sub-actions has been proved under the sole hypothesis of coerciveness. Proposition 5.3. Let L(x, v) be a C 0 coercive Lagrangian. If L is locally α-Hölder continuous, then any forward or backward calibrated sub-action u ∈ C 0 (T d ) is α-Hölder continuous too. Besides, if L is semiconcave, then all forward calibrated sub-actions are semiconvex and all backward calibrated sub-actions are semiconcave.

Proof. The Hölder case is an immediate consequence of part i of lemma 4.2. The semiconvex property is just a reinterpretation of proposition 4.7 as well as its analogue for backward calibrated sub-actions mentioned after its demonstration.

Observe that, in terms of the associated local interaction energy L τ (x, y), a sub-action u satisfies

L τ (x, y) ≥ u(y) -u(x) + τ L(τ ) everywhere on R d × R d . Definition 5.4. Consider a C 0 (T d ) sub-action u for a C 0 coercive Lagrangian L(x, v). A configuration {x k } k∈Z in R d is called u-calibrated if, for every k ∈ Z, we have L τ (x k , x k+1 ) = u(x k+1 ) -u(x k ) + τ L(τ ).
It is easy to see that calibrated configurations are minimizing and even strongly minimizing. We show in the following lemma that the coerciveness assumption implies the existence of calibrated configurations and therefore the existence of minimizing configurations.

Lemma 5.5. Suppose L(x, v) is a C 0 coercive Lagrangian. If u ∈ C 0 (T d
) is either a forward or a backward calibrated sub-action, then there exists a u-calibrated configuration {x k } k∈Z in R d passing through some point x 0 ∈ [0, 1) d and satisfying

x k+1 -x k ∞ ≤ τ R τ , where R τ > 1
τ has been defined in 3.8.

Lemma 5.6. Let L(x, v) be a C 0 coercive Lagrangian. If u ∈ C 0 (T d ) is an arbitrary sub-action, then any u-calibrated configuration {x k } k∈Z in R d is a strongly minimizing configuration satisfying x k+1x k ∞ ≤ τ R τ for every k ∈ Z.

Proof of lemma 5.5. Let u ∈ C 0 (T d ) be a forward calibrated sub-action. Thanks to coerciveness,

u verifies τ L(τ ) = min v ∞≤Rτ [τ L(x, v) + u(x) -u(x + τ v)] or τ L(τ ) = min y : y-x ∞≤τ Rτ [L τ (x, y) + u(x) -u(y)] , ∀ x ∈ R d .
Hence, for every positive integer n, consider a configuration

{x n k } k≥-n in R d such that x n k -x n k+1 ∞ ≤ τ R τ and τ L(τ ) = L τ (x n k , x n k+1 ) + u(x n k ) -u(x n k+1 ) for all k ≥ -n. Since L τ (x + s, y + s) = L τ (x, y) for s ∈ Z d , we may assume that x n 0 ∈ [0, 1) d for every n > 0.
In particular, we get that

x n k ∞ ≤ τ R τ |k| + 1 for all k ≥ -n. By a diagonal procedure, we extract a configuration {x k } k∈Z satisfying τ L(τ ) = L τ (x k , x k+1 ) + u(x k ) -u(x k+1
) for any integer k. A similar proof can be developed for C 0 (T d ) backward calibrated sub-actions.

Proof of lemma 5.6. Let {x k } k∈Z be a u-calibrated configuration. Thanks to definition 3.8, if

x k+1 -x k ∞ > τ R τ , then u(x k+1 ) -u(x k ) + τ L(τ ) = L τ (x k , x k+1 ) > > L τ (x k , x k+1 -⌊x k+1 -x k ⌋) ≥ u(x k+1 -⌊x k+1 -x k ⌋) -u(x k ) + τ L(τ ).
The periodicity of u implies that the strict inequality cannot happen. Therefore, x k+1x k ∞ ≤ τ R τ for all k ∈ Z. Moreover, for any m < n, for any configuration

{y k } k∈Z in R d satisfying y m ′ = x m and y n ′ = x n (mod Z d ), since Lτ (x m , x m+1 , . . . , x n ) = u(x n ) -u(x m ) = u(y n ′ ) -u(y m ′ ) ≤ Lτ (y m ′ , y m ′ +1 , . . . , y n ′ ),
we obtain that {x k } k∈Z is a strongly minimizing configuration.

Notice that the existence of a u-calibrated configuration of a sub-action gives an equivalent definition of the holomic minimizing value L(τ ) as defined in 3.5

τ L(τ ) = inf {x k }∈(R d ) Z lim inf n-m→∞ 1 n -m n-1 k=m L τ (x k , x k+1 ).
Furthermore, u-calibrated configurations are examples of critical configurations without assuming any ferromagnetic condition.

Lemma 5.7. Let L(x, v) be a C 1 coercive Lagrangian. Any u-calibrated configuration of some C 0 periodic sub-action u is critical.

Proof. Let {x k } k∈Z be a u-calibrated configuration. Lemma 5.6 implies {x k } k∈Z is minimizing and in particular satisfies

L τ (x k-1 , x k , x k+1 ) ≤ L τ (x k-1 , x, x k+1 ), ∀ x ∈ R d . Therefore ∂Lτ ∂y (x k-1 , x k )+ ∂Lτ ∂x (x k , x k+1 ) = 0 for all k ∈ Z and {x k } k∈Z ∈ Γ τ (L).
We have seen in remark 3.6 that any holonomic measure can be lifted to a shift-invariant probability measure in Σ/ ∼ and that

τ L(τ ) = min μ σ-invariant Σ/∼ L τ (x 0 , x 1 ) dμ(x).
We show in the following proposition how to lift some minimizing holonomic probability measures to (Γ τ (L), σ) or equivalently to (T d × R d , Φ τ ).

Proposition 5.8. Let L(x, v) be a C 1 ferromagnetic coercive Lagrangian, then the minimizing holonomic value of L is given by

L(τ ) = min L(x, v) dµ(x, v) : µ ∈ P τ (T d × R d ), µ Φ τ -invariant .
Proof. We already remarked that any Φ τ -invariant probability is holonomic, then

L(τ ) ≤ min L(x, v) dµ(x, v) : µ ∈ P τ (T d × R d ), µ Φ τ -invariant .
If {x k } k∈Z is a u-calibrated configuration for some C 0 periodic sub-action u, then {x k } k∈Z ∈ Γ τ (L) by lemma 5.7. Therefore, thanks to the conjugation between (Γ τ (L), σ) and (

T d × R d , Φ τ ), if v k := x k+1 -x k τ , then (x k , v k ) = Φ k τ (x 0 , v 0 ) for all
k ∈ Z and v k is uniformly bounded by lemma 5.6. Let µ ∈ P τ (T d × R d ) be a weak limit of some convergent subsequence

µ n l = 1 n l n l -1 k=0 δ Φ k τ (x 0 ,v 0 ) .
Then µ is Φ τ -invariant and we have

L(x, v) dµ(x, v) = lim l→∞ 1 n l n l -1 k=0 L • Φ k τ (x 0 , v 0 ) = lim l→∞ 1 n l u(x n l ) -u(x 0 ) τ + n l L(τ ) = L(τ ).
We will see in the next section that, for ferromagnetic Lagrangians, all minimizing holonomic measures are actually Φ τ -invariant.

Graph property and Mather set

In the setting of continuous-time periodic, strictly convex, superlinear and complete C 2 -Lagrangians on a compact and connected C ∞ manifold, J. N. Mather showed (see [START_REF] Mather | Action minimizing invariant measures for positive definite Lagragian systems[END_REF]) that measures invariant under the Euler-Lagrange flow which are action minimizing can be seen as Lipschitz sections of the tangent bundle. Our main goal in this section (see theorem 6.10) is to obtain a similar graph property. Definition 6.1. Let L(x, v) be a C 0 coercive Lagrangian. We call Mather set the set

M τ (L) = closure supp(µ) : µ ∈ P τ (T d × R d ) and µ is minimizing ,
where supp(µ) denotes the support of the probability µ.

Proposition 3.7 implies that minimizing holonomic measures do exist, which shows that the Mather set is nonempty. Definition 6.2. Let L(x, v) be a C 0 coercive Lagrangian and u be a C 0 periodic sub-action for L. We call nil locus of u the set

N τ (L, u) = (x, v) ∈ T d × R d : τ L(x, v) = u(x + τ v) -u(x) + τ L(τ ) .
We observe that coerciveness guarantees all nil loci are nonempty. The following proposition shows that N τ (L, u) actually contains the support of any minimizing holonomic measure. Proposition 6.3. Let L(x, v) be a C 0 coercive Lagrangian. Then, for any subaction u ∈ C 0 (T d ) with respect to L, we have M τ (L) ⊂ N τ (L, u).

Proof. Consider a minimizing holonomic measure µ

∈ P τ (T d × R d ). Since one has both τ L(x, v) + u(x) -u(x + τ v) -τ L(τ ) ≥ 0 and τ L(x, v) + u(x) -u(x + τ v) -τ L(τ ) dµ(x, v) = 0, τ L(x, v) = u(x + τ v) -u(x) + τ L(τ )
holds everywhere on the support of µ.

As in the proof of lemma 5.6, the coerciveness assumption of L implies that any nil locus is compact. More precisely, we have Corollary 6.4. Let L(x, v) be a C 0 coercive Lagrangian and u ∈ C 0 (T d ) be a subaction with respect to L.

If (x, v) ∈ N τ (L, u), then v ∞ ≤ R τ .
In particular, the support of any minimizing holonomic measure is compact.

Proof. If v ∞ > R τ , then L(x, v) > L(x, v -1 τ ⌊τ v⌋). Assume (x, v) ∈ N τ (L, u), then u(x + τ v) -u(x) = τ L(x, v) -τ L(τ ) > τ L x, v - 1 τ ⌊τ v⌋ -τ L(τ ) ≥ u x + τ v - 1 τ ⌊τ v⌋ -u(x) = u(x + τ v) -u(x).
We obtain a contradiction, therefore v ∞ ≤ R τ .

We assume from now on in this section that L is C 1 and coercive. We prove that any sub-action is continuously differentiable on the projected Mather set pr 1 (M τ (L)), where pr 1 : T d × R d → T d denotes the first canonical projection. When L is in addition ferromagnetic, we prove that M τ (L) is a graph over its projection into T d . Recall that Π τ has been introduced in definition 2.6. Lemma 6.5. Let µ be a holonomic measure with compact support, then for any x ∈ pr 1 (supp(µ)), there exists a configuration

x := {x k } k∈Z in R d such that x 0 = x and Π τ • σ k (x) = x k , x k+1 -x k τ ∈ supp(µ) for all k ∈ Z.
Proof. From proposition 3.4, we naturally associate to µ a normalized invariant transshipment π in R d × R d . Let pr 1,2 : R d × R d → R d be the two canonical projections. Since µ has compact support, the support of π has compact horizontal and vertical slices. Then S 1,2 := pr 1,2 (supp(π)) are closed sets. We always have S 1,2 ⊆ supp(pr 1,2 (π)). Since S 1,2 are closed, necessarily S 1,2 = supp(pr 1,2 * (π)). Since π is a transshipment, pr 1 * (π) = pr 2 * (π) and S 1 = S 2 . Let x 0 ∈ S 1 , then there exists x 1 such that (x 0 , x 1 ) ∈ supp(π). Since x 1 ∈ S 2 = S 1 , there exists x 2 such that (x 1 , x 2 ) ∈ supp(π), and so on. We thus obtain a forward and backward orbit {x k } k∈Z of points (x k , x k+1 ) in the support of π or equivalently an orbit {(x k , v k = (x k+1x k )/τ )} k∈Z of points in the support of µ.

In order to prove the differentiability of any sub-action on the projected Mather set, we introduce two intermediate notions of calibration. Definition 6.6. Let u ∈ C 0 (R d ) be a Z d -periodic sub-action for L. A couple

(x 0 , x 1 ) ∈ R d × R d is called u-calibrated if L τ (x 0 , x 1 ) = u(x 1 ) -u(x 0 ) + τ L(τ ).
A triple (x -1 , x 0 , x 1 ) is called u-calibrated if both (x -1 , x 0 ) and (x 0 , x 1 ) are ucalibrated. Lemma 6.7. Let L(x, v) be a C 1 coercive Lagrangian and u ∈ C 0 (T d ) be a subaction. Then, for any u-calibrated couple (x 0 , x 1 ) ∈ R d × R d , we have

lim sup h ∞→0 1 h ∞ u(x 1 + h) -u(x 1 ) - ∂L τ ∂y (x 0 , x 1 ), h ≤ 0 and lim inf h ∞→0 1 h ∞ u(x 0 + h) -u(x 0 ) -- ∂L τ ∂x (x 0 , x 1 ), h ≥ 0.
Proof. Indeed, since u is a sub-action, we have on the one hand

u(x 1 + h) ≤ u(x 0 ) + L τ (x 0 , x 1 + h) -τ L(τ ) and u(x 1 ) ≤ u(x 0 + h) + L τ (x 0 + h, x 1 ) -τ L(τ ), ∀ h ∈ R d .
On the other hand, u(x 1 ) = u(x 0 ) + L τ (x 0 , x 1 )τ L(τ ), which implies

u(x 1 + h) -u(x 1 ) ≤ L τ (x 0 , x 1 + h) -L τ (x 0 , x 1 ) and u(x 0 + h) -u(x 0 ) ≥ L τ (x 0 , x 1 ) -L τ (x 0 + h, x 1 ) .
The lemma follows from the differentiability of L.

Although we could use the theory of subdifferentiability to derive the next lemma, for the sake of completeness, we prefer to give a direct proof. Lemma 6.8. Let L(x, v) be a C 1 coercive Lagrangian and u ∈ C 0 (T d ) be a subaction. Let K τ (L, u) denote the set of mid-points x 0 of all u-calibrated triples (x -1 , x 0 , x 1 ).

i. If (x -1 , x 0 , x 1 ) is u-calibrated, then u is differentiable at x 0 and

Du(x 0 ) = ∂L τ ∂y (x -1 , x 0 ) = - ∂L τ ∂x (x 0 , x 1 ) = ∂L ∂v x -1 , x 0 -x -1 τ = ∂L ∂v x 0 , x 1 -x 0 τ -τ ∂L ∂x x 0 , x 1 -x 0 τ .
ii. The map Du :

K τ (L, u) → R d is uniformly continuous independently of u.
iii. If L is in addition ferromagnetic, then there exists at most one u-calibrated configuration passing through any x 0 ∈ R d .

Proof. Item i . On the one hand, a u-calibrated triple is critical as in definition 2.3. Let ∇ be the common derivative

∇ := ∂L τ ∂y (x -1 , x 0 ) = - ∂L τ ∂x (x 0 , x 1 ).
On the other hand, lemma 6.7 implies that lim sup

h ∞→0 u(x 0 + h) -u(x 0 ) -∇, h h ∞ ≤ 0 ≤ lim inf h ∞→0 u(x 0 + h) -u(x 0 ) -∇, h h ∞ ,
which shows that Du(x 0 ) = ∇.

Item ii . We begin by showing that there exists a positive function C τ (h) defined for all h ∈ R d , depending only on τ and L, such that C τ (h) → 0 when h → 0 and

|u(x 0 + h) -u(x 0 ) -Du(x 0 ), h | ≤ C τ (h) h ∞ ,
for all h ∈ R d , all x 0 ∈ K τ (L, u) and all sub-action u. Let (x -1 , x 0 , x 1 ) be a u-calibrated triple. On the one hand,

u(x 1 ) -u(x 0 ) -L τ (x 0 , x 1 ) = τ L(τ ) ≥ u(x 1 ) -u(x 0 + h) -L τ (x 0 + h, x 1 ),
and by eliminating u(x 1 ) one obtain

u(x 0 + h) -u(x 0 ) -Du(x 0 ), h ≥ -L τ (x 0 + h, x 1 ) -L τ (x 0 , x 1 ) - ∂L τ ∂x (x 0 , x 1 ).h .
On the other hand,

u(x 0 ) -u(x -1 ) -L τ (x -1 , x 0 ) = -τ L(τ ) ≥ u(x 0 + h) -u(x -1 ) -L τ (x -1 , x 0 + h),
and by eliminating u(x -1 ) one obtain

u(x 0 +h)-u(x 0 )-Du(x 0 ), h ≤ L τ (x -1 , x 0 +h)-L τ (x -1 , x 0 )- ∂L τ ∂y (x -1 , x 0 ), h . Notice that x 0 -x -1 ∞ , x 1 -x 0 ∞ ≤ τ R(τ ) whenever (x -1 , x 0 , x 1 ) ∈ K τ (L, u)
and that L τ (x, y) is invariant by the diagonal Z d -translation. Define

C ′ τ (h) := max x 1 -x 0 ∞≤τ R(τ ) max s∈[0,1] ∂L τ ∂x (x 0 + sh, x 1 ) - ∂L τ ∂x (x 0 , x 1 ) ∞ C ′′ τ (h) := max x 0 -x -1 ∞≤τ R(τ ) max s∈[0,1] ∂L τ ∂y (x -1 , x 0 + sh) - ∂L τ ∂y (x -1 , x 0 ) ∞ . Then C τ (h) = max(C ′ τ (h), C ′′ τ (h))
is the desired function. We now show that Du(x 0 ) is uniformly continuous on K τ (L, u). Notice that

|u(x 0 ) -u(x 0 -h) -Du(x 0 ), h | ≤ C τ (-h) h ∞ , ∀ h ∈ R d .
Let x 0 and x ′ 0 be two distinct mid-points of K τ (L, u). Then, for any h,

|u(x 0 + h) -u(x 0 ) -Du(x 0 ), h | ≤ C τ (h) h ∞ , |u(x ′ 0 ) -u(x 0 + h) -Du(x ′ 0 ), x ′ 0 -x 0 -h | ≤ C τ (x 0 + h -x ′ 0 ) x 0 + h -x ′ 0 ∞ , |u(x 0 ) -u(x ′ 0 ) -Du(x ′ 0 ), x 0 -x ′ 0 | ≤ C τ (x 0 -x ′ 0 ) x 0 -x ′ 0 ∞ .

By adding the three inequalities and by taking any h

∞ = x 0 -x ′ 0 ∞ , we obtain | Du(x ′ 0 ) -Du(x 0 ), h | ≤ C τ (x 0 , x ′ 0 ) h ∞ , where C τ (x 0 , x ′ 0 ) := sup h ∞= x 0 -x ′ 0 ∞ [C τ (h) + 2C τ (x 0 + h -x ′ 0 ) + C τ (x 0 -x ′ 0 )]. Therefore Du(x 0 ) -Du(x ′ 0 ) ∞ ≤ C τ (x 0 , x ′ 0 )
and Du(x 0 ) is uniformly continuous. Item iii . If {x k } k∈Z is a u-calibrated configuration, then Du(x k ) exists for all k and the two equations Du(x k ) = ∂Lτ ∂y (x k-1 , x k ) and Du(x k ) = -∂Lτ ∂x (x k , x k+1 ) shows that x k-1 and x k+1 are known as soon as x k is known and L is ferromagnetic.

The following proposition is now a direct consequence of proposition 6.3 and lemmas 6.5 and 6.8. Proposition 6.9. Let L(x, v) be a C 1 coercive Lagrangian. Then any sub-action u ∈ C 0 (T d ) with respect to L is continuously differentiable on the projected Mather set pr 1 (M τ (L)), where pr 1 :

T d × R d → T d denotes the first canonical projection. If L is in addition C 1,1 , then Du : pr 1 (M τ (L)) → R is Lipschitz uniformly in u.
Proof. Recall from lemma 6.8 that K τ (L, u) denotes the set of mid-points of ucalibrated triples. From lemmas 6.5 and proposition 6.3, we deduce that

pr 1 (M τ (L)) ⊆ K τ (L, u).
From 6.8, we obtain that Du :

T d → R d is continuous.
Mather graph property is then an easy consequence of the previous study in the case of ferromagnetic Lagrangians. Theorem 6.10. Let L(x, v) be a C 1 ferromagnetic coercive Lagrangian. Then there exists a continuous map

V τ : pr 1 (M τ (L)) → R d , V τ ∞ ≤ R τ ,
such that M τ (L) is a graph over its projection, that is,

M τ (L) = graph(V τ ) = {(x, V τ (x)) | x ∈ pr 1 (M τ (L)}.
Moreover, M τ (L) is compact and Φ τ -invariant, any minimizing holonomic probability measure µ is Φ τ -invariant and, for any sub-action u ∈ C 0 (T d ), one has

Du(x) = ∂L ∂v (x, V τ (x)) -τ ∂L ∂x (x, V τ (x)), ∀ x ∈ pr 1 (M τ (L)).
Proof. Let u ∈ C 0 (T d ) be any sub-action. From lemma 6.8, we know that Du(x) exists and is continuous for all x ∈ K τ (L, u). Since L is ferromagnetic, we define uniquely V τ (x) by the following implicit equation

Du(x) = ∂L ∂v (x, V τ (x)) -τ ∂L ∂x (x, V τ (x)), ∀ x ∈ K τ (L, u).
Then V τ becomes continuous on K τ (L, u).

Assume now that x ∈ pr 1 (M τ (L)). Consider a point x -1 ∈ T d with (x -1 , x) u-calibrated. Take v ∈ R d such that (x, v) ∈ M τ (L) and define x 1 = x + τ v. Then (x -1 , x, x 1 ) is u-calibrated and, thanks to lemma 6.8, we have

Du(x) = ∂L ∂v (x, v) -τ ∂L ∂x (x, v). Necessarily v = V τ (x), V τ (x) ∞ ≤ R τ and V τ (x)
is independent of the choice of u. From lemma 6.5, we know there exist u-calibrated triples passing through x consisting of points of pr 1 (M τ (L)). From the ferromagnetic property, we deduce that this triple is unique. Thus

x 1 ∈ pr 1 (M τ (L)) ⊆ K τ (L, u) and Du(x 1 ) = ∂L ∂v (x, V τ (x)) = ∂L ∂v (x 1 , V τ (x 1 )) -τ ∂L ∂x (x 1 , V τ (x 1 )).
From the definition of Φ τ (see definition 2.5), we obtain

Φ τ (x, v) = (x + τ v, V τ (x + τ v)) , ∀ (x, v) ∈ M τ (L).
In particular, Φ τ preserves the Mather set (the reverse inclusion is proved similarly)

Φ τ (M τ (L)) = M τ (L).
Let µ ∈ P τ (T d × R d ) be a minimizing holonomic probability measure. For any bounded Borel function ϕ : T d × R d → R, from the previous identity, we have

ϕ • Φ τ (x, v) dµ(x, v) = Mτ (L) ϕ (x + τ v, V τ (x + τ v)) dµ(x, v) = Mτ (L) ϕ (x, V τ (x)) dµ(x, v) = ϕ (x, v) dµ(x, v),
which means the Φ τ -invariance of the measure µ.

The last statement of theorem 6.10 is similar to a known result in the case of Lagrangian theory. For a continuous-time periodic, strictly convex, superlinear and complete C ∞ -Lagrangian on a closed Riemannian manifold, R. Mañé showed (see proposition 1.3 of [START_REF] Mañé | Generic properties and problems of minimizing measures of Lagrangian systems[END_REF]) that any minimizing holonomic measure is invariant under the Euler-Lagrange equations.

The Aubry set

In Aubry-Mather theory for continuous-time Lagrangian dynamics, there are generally two strategies for introducing the Aubry set: A. Fathi's formulation (see [START_REF] Fathi | The weak KAM theorem in Lagrangian dynamics[END_REF]) using the notion of conjugate weak KAM solutions and G. Contreras and R. Iturriaga construction (see [START_REF] Contreras | Global minimizers of autonomous Lagrangians[END_REF]) using the notion of static curves. Both approaches request intrinsically a differentiable Lagrangian.

We have chosen a different approach which is closer to a more usual definition in ergodic optimization theory and which has the main advantage of requiring only C 0 smoothness. The Aubry set will be introduced using the following concept. Definition 7.1. We call periodic configuration of type (q, p) a finite configuration (x 0 , x 1 , • • • , x q ) of points of R d such that x q = x 0 + p, q ≥ 1 and p ∈ Z d . Such a finite configuration determines uniquely a bi-infinite configuration {x k } k∈Z satisfying x q+k = x k + p for all k ∈ Z.

The notion of Aubry point below is similar to the one of non-wandering point with respect a potential used in ergodic optimization (see, for instance, [START_REF] Contreras | Lyapunov minimizing measures for expanding maps of the circle[END_REF][START_REF] Garibaldi | On the Aubry-Mather theory for symbolic dynamics[END_REF][START_REF] Garibaldi | On calibrated and separating sub-actions[END_REF]). A similar projected Aubry set in [START_REF] Gomes | The Mather measure and a large deviation principle for the entropy penalized method[END_REF] has also been used in the discrete Aubry-Mather problem. Recall that Lτ (

x 0 , x 1 , • • • , x q ) = q-1 k=0 [L τ (x k , x k+1 ) -τ L(τ )]. Definition 7.2. Let L(x, v) : T d × R d → R be a C 0 coercive Lagrangian. A point (x, v) ∈ T d ×R d
is said to be an Aubry point if, for any ǫ > 0, there exists a periodic configuration of type (q, p),

(x 0 , x 1 , • • • , x q ), such that x -x 0 ∞ < ǫ, x + τ v -x 1 ∞ < ǫ and | Lτ (x 0 , x 1 , • • • , x q )| ≤ ǫ.
The Aubry set A τ (L) is by definiton the set of all Aubry points.

Notice that the Aubry set depends on L modulo any coboundary, that is, for all function ψ ∈ C 0 (T d ) and

any constant c ∈ R, A τ (L) = A τ (L -∆ τ ψ -c), where ∆ τ ψ(x, v) := ψ(x + τ v) -ψ(x). Notice also that Lτ (x 0 , x 1 , • • • , x q ) ≥ 0 for any periodic configuration of type (q, p), since Lτ (x 0 , x 1 , • • • , x q ) is unchanged if, instead of L, we use L -1 τ ∆ τ u -L(τ )
≥ 0 for some sub-action u. It is easy to see that the Aubry set is a closed subset of T d × R d . The fact that it is a non empty set is proved in the following proposition. Proof. We begin by proving the second inclusion. Define the associated normalized Lagrangian

E(x, v) := L(x, v) -1 τ [u(x + τ v) -u(x)] -L(τ )
and the corresponding interaction energy E τ (x, y). Then E(x, v) ≥ 0 for all (x, v) ∈ T d × R d and Ē(τ ) = 0. For any ǫ > 0, there exists a periodic configuration (

x 0 , x 1 , • • • , x q ) such that 0 ≤ Ēτ (x 0 , x 1 ) ≤ Ēτ (x 0 , x 1 , • • • , x q ) = Lτ (x 0 , x 1 , • • • , x q ) ≤ ǫ.
Letting ǫ go to 0, we obtain

E τ (x, x + τ v) = 0 or (x, v) ∈ N τ (L, u).
We now prove the first inclusion. The proof of this part is non trivial and requires the use of Atkinson's theorem which we recall in 7.4. Let µ be a minimizing holonomic probability measure and (x, v) ∈ supp(µ). By proposition 3.4, the measure µ can be lifted to a normalized shift-invariant Markov chain μ on Σ/ ∼ . Remark 3.6 tells us that μ is a minimizing shift-invariant probability in following sense

τ L(τ ) = Σ/∼ L τ (x 0 , x 1 ) dμ(x) = inf ν σ-invariant Σ/∼ L τ (x 0 , x 1 ) dν(x).
Take ǫ > 0. Let B ǫ denote an open ball of radius η(ǫ) ∈ (0, ǫ) around the point (x, v) such that the oscilation of L τ (x 0 , x 1 ) on Bǫ := Π -1 τ (B ǫ ) is less than ǫ. Then μ( Bǫ ) = µ(B ǫ ) > 0 and, by the ergodic decomposition theorem (see, for instance, chapter 7 of [START_REF] Gray | Probability, random processes, and ergodic properties[END_REF]), there exists an ergodic minimizing shift-invariant probability ν which satisfies ν( Bǫ ) > 0 and τ L(τ ) = Σ/∼ L τ (x 0 , x 1 ) dν(x).

Atkinson's theorem implies that there exist a point x = {x k } k∈Z ∈ Bǫ and infinitely many positive integers q such that σ q (x) ∈ Bǫ and

| Lτ (x 0 , x 1 , • • • , x q )| < ǫ.
By definition of Bǫ , we may assume x 0 and x 1 close to x and x + τ v within η(ǫ). Moreover, x q is close to x 0 + p within η(ǫ) for some p ∈ Z d . We have obtained a periodic configuration (x qp, x 1 , • • • , x q ) beginning close to (x, x + τ v) and satisfying

| Lτ (x q -p, x 1 , • • • , x q )| < 2ǫ. We have shown that (x, v) ∈ A τ (L).
Atkinson's theorem is well known. We recall the statement for completeness.

Theorem 7.4. (Atkinson's theorem [START_REF] Atkinson | Recurrence of co-cycles and random walks[END_REF]) Let (Z, C, λ) be a probability space, T : Z → Z an ergodic measure preserving map, f : Z → R an integrable function, f ∈ L 1 (λ), and D ∈ C a measurable set of positive measure, λ(D) > 0. Denote

Ξ(f, D) := z ∈ D : ∀ ǫ > 0, ∃ n ≥ 1 with T n (z) ∈ D and n-1 k=0 f • T k (z) -n Z f dλ < ǫ . Then λ (Ξ(f, D)) = λ(D).
We want now to prove that any sub-action is continuously differentiable on the Aubry set. We first show that a finite configuration with bounded interaction energy has bounded jumps independently of the length of the configuration. Lemma 7.5. Let L(x, v) be a C 0 coercive Lagrangian. Then for any E > 0 there exists R E > 0 such that, for any n ≥ 1 and any finite configuration (x 0 , x 1 , • • • , x n ) of length n with interaction energy bounded from above by E,

Lτ (x 0 , x 1 , • • • , x n ) ≤ E =⇒ x k -x k-1 ∞ ≤ R E , ∀ k = 1, • • • , n.
Proof. Let u be a fixed C 0 (T d ) sub-action. By coerciveness of L, for every E > 0 there exists

R E > 0 such that | Lτ (x, y)| ≤ E + 4 u 0 ⇒ y -x ∞ < R E . Then 0 ≤ Lτ (x k-1 , x k ) + u(x k-1 ) -u(x k ) ≤ ≤ Lτ (x 0 , x 1 , • • • , x n ) + u(x 0 ) -u(x n ) ≤ E + 2 u 0 , | Lτ (x k-1 , x k )| ≤ E + 4 u 0 and x k -x k-1 < R E .
We can now extend the conclusion of proposition 6.9.

Proposition 7.6. Let u ∈ C 0 (T d ) be a sub-action with respect to a C 1 coercive Lagrangian L(x, v). Then u is continuously differentiable on the projected Aubry set pr

1 (A τ (L)). If L is in addition C 1,1 then Du : pr 1 (A τ (L)) → R is Lipschitz uniformly in u.

Mañé potential and Peierls barrier

We introduce in this section two new definitions: the Mañé potential and the Peierls barrier. We prove that these notions give an equivalent characterization of the Aubry set and that they give a different way to construct calibrated sub-actions. They will play a fundamental role in the next section to classify all calibrated sub-actions.

Definition 8.1. Let L(x, v) : T d × R d → R be a C 0 coercive Lagrangian. We call Mañé potential the function S τ : R d × R d → R defined by S τ (x, y) = inf n≥1 inf p∈Z d inf x 0 =x xn=y+p Lτ (x 0 , . . . , x n ) = inf n≥1 inf p∈Z d inf x 0 =x+p xn=y Lτ (x 0 , . . . , x n ).
Notice that S τ (x, y) is periodic in both variables x and y.

We first give obvious properties of the Mañé potential. Remark 8.2. For any x, y, z in R d , we have i.

S τ (x, y) ≤ inf p∈Z d L τ (x, y + p) -τ L(τ ) ≤ L τ (x, y) -τ L(τ ) = Lτ (x, y), ii. u(y) -u(x) ≤ S τ (x, y), for any sub-action u ∈ C 0 (T d ), iii. S τ (x, y) ≤ S τ (x, z) + S τ (z, y), iv. S τ (x, x) ≥ 0.
We just have seen that coerciveness implies the Mañé potential is a finite function. We show in the next propostion that S τ (x, y) is continuous with respect to both x and y. Proof. We prove the first assertion. Fix ǫ > 0. Take arbitrary points (x, y), (x ′ , y ′ ) in R d × R d . Then there exist two configurations (x 0 , . . . , x m ), (y 0 , . . . , y n ) and two vectors with integer coordinates r, s ∈ Z d such that ,y) can be bounded from above using the estimates

Lτ (x 0 , . . . , x m-1 , x m + r) ≤ S τ (x, y ′ ) + ǫ/2, x 0 = x, x m = y ′ , Lτ (y 0 + s, y 1 , . . . , y n ) ≤ S τ (x, y) + ǫ/2, y 0 = x, y n = y. Then S τ (x ′ , y ′ )-S τ (x, y) = S τ (x ′ , y ′ )-S τ (x, y ′ )+S τ (x, y ′ )-S τ (x
S τ (x ′ , y ′ ) -S τ (x, y ′ ) ≤ ≤ Lτ (x ′ , x 1 , . . . , x m-1 , y ′ + r) -Lτ (x, x 1 , . . . , x m-1 , y ′ + r) + ǫ/2 ≤ Lτ (x ′ , x 1 ) -Lτ (x, x 1 ) + ǫ/2, S τ (x, y ′ ) -S τ (x, y) ≤ ≤ Lτ (x + s, y 1 , . . . , y n-1 , y ′ ) -Lτ (x + s, y 1 , . . . , y n-1 , y) + ǫ/2 ≤ Lτ (y n-1 , y ′ ) -Lτ (y n-1 , y) + ǫ/2.
Since S τ (x, y) is uniformly bounded from above by periodicity and item ii of remark 8.2, lemma 7.5 guarantees that the points x 1 and y n-1 which depend on ǫ and x, y, x ′ and y ′ are uniformly bounded. So the estimation above shows S τ is a continuous map.

The second assertion is an immediate corollary of item iii of remark 8.2

S τ (x, z) -S τ (x, y) ≤ S τ (y, z) ≤ L τ (y, z) -τ L, ∀ y, z ∈ R d , or in terms of the Lagranigan L S τ (x, y + τ v) -S τ (x, y) ≤ τ L(x, v) -τ L, ∀ (y, v) ∈ T d × R d .
We just have proved that S τ (x, •) is a sub-action. Similarly -S τ (•, y) is a subaction.

For a C 0 coercive Lagrangian, we clearly deduce S τ (x, x) ≥ 0 from item ii of remark 8.2. We show in the following proposition that S τ (x, x) = 0 characterizes the Aubry set.

Proposition 8.4. Suppose L(x, v) is a C 0 coercive Lagrangian. Then S τ (x, x) = 0 if, and only if, x (mod Z d ) ∈ pr 1 (A τ (L)).
Proof. Let us first show that (x, v) ∈ A τ (L) implies S τ (x, x) = 0. One can find a sequence of periodic configurations (x l 0 , x l 1 , . . . , x l q(l) ), x l q(l) = x l 0 + p l , such that x l 0 → x, x l 1 → x + τ v and Lτ (x l 0 , x l 1 , . . . , x l q(l) ) → 0.

Since S τ (x l 0 , x l 0 ) = S τ (x l 0 , x l q(l) ) ≤ Lτ (x l 0 , x l 1 , . . . , x l q(l) ), thanks to the continuity of S τ and item iv of remark 8.2, we obtain S τ (x, x) = 0.

Conversely, assume S τ (x, x) = 0. Then there exists a sequence of periodic configurations (x l 0 , . . . , x l q(l) ) such that x l 0 = x = x l q(l) -p l and Lτ (x l 0 , x l 1 , . . . , x l q(l) ) → 0. Thanks to lemma 7.5, x l 1x l 0 remains uniformly bounded. So one can find a subsequence of l's such that {(x l 1x l 0 )/τ } l converges to some v ∈ R d . By definition of the Aubry set, (x, v) ∈ A τ (L).

Mañé potential enables us to construct continuous sub-actions without using the Lax-Oleinik operator. These sub-actions may not be calibrated. We introduce in the following definition a barrier which is similar to Mañé potential, being continuous, periodic with respect to both variables and in addition defining calibrated sub-actions (see theorem 8.10).

Definition 8.5. Let L(x, v) : T d × R d → R be a C 0 coercive Lagrangian. We call Peierls barrier the function h τ : R d × R d → R ∪ {+∞} defined by h τ (x, y) = lim inf n→+∞ inf p∈Z d inf x 0 =x xn=y+p Lτ (x 0 , . . . , x n ) = lim inf n→+∞ inf p∈Z d inf x 0 =x+p xn=y Lτ (x 0 , . . . , x n ).
Again notice that h τ (x, y) is periodic with respect to both variables x and y.

We first gather simple properties of the Peierls barrier. Remark 8.6. For any x, y, z in

R d i. S τ (x, y) ≤ h τ (x, y), ii. h τ (x, y) ≤ S τ (x, z) + h τ (z, y), iii. h τ (x, y) ≤ h τ (x, z) + S τ (z, y).
We will prove in a moment that h τ (x, y) satisfies additional properties: h τ (x, y) takes finite values (proposition 8.7), h τ (x, •) and -h τ (•, y) are continuous, periodic calibrated sub-actions for all x, y ∈ R d (theorem 8.10).

We first prove that S τ and h τ coincide on the projected Aubry set. Proposition 8.7. Let L(x, v) be a C 0 coercive Lagrangian. Then for any points x, y (mod

Z d ) ∈ pr 1 (A τ (L)), S τ (x, •) = h τ (x, •) and S τ (•, y) = h τ (•, y). In particu- lar h τ (x, y) is finite for all x, y ∈ R d .
Proof. We only prove the first identity. Let x (mod Z d ) ∈ pr 1 (A τ (L)) and y ∈ R d . For every ǫ > 0, there exists a configuration (x, y 1 , . . . , y m-1 , y

+ s) in R d , with m ≥ 1 and s ∈ Z d , such that Lτ (x, y 1 , . . . , y m-1 , y + s) < S τ (x, y) + ǫ/2.
As S τ (x, x) = 0, for every positive integer l, one can also find a finite configuration (x, x 1 , . . . , x n-1 , x + r), with n ≥ 1 and r ∈ Z d , such that Lτ (x, x 1 , . . . , x n-1 , x + r) < ǫ/2l.

Notice that the configuration (x, x 1 , . . . , x n-1 , x + r, x 1 + r, . . . , x n-1 + r, x + 2r, x 1 + 2r, . . . , x n-1 + 2r, . . .

x + (l -1)r, . . . , x n-1 + (l -1)r, x + rl, y 1 + lr, . . . , y m-1 + lr, y + lr + s) is of the form (z 0 , z 1 , . . . , z nl+m ) satisfying z 0 = x, z nl+m = y + lr + s and Lτ (z 0 , . . . , z nl+m ) ≤ l Lτ (x, x 1 , . . . , x n-1 , x) + Lτ (y, y 1 , . . . , y m-1 , y) ≤ S τ (x, y) + ǫ.

Since l can be chosen arbitrarily large, we deduce that h τ (x, y) ≤ S τ (x, y)+ǫ, which immediately yields h τ (x, y) ≤ S τ (x, y).

The fact that h τ (x, y) is finite comes from the inequality

h τ (x, y) ≤ S τ (x, z) + h τ (z, y) = S τ (x, z) + S τ (z, y),
where z ∈ pr 1 (A τ (L)) is arbitrarily chosen.

Thanks to proposition 8.3, we conclude that h τ (x, •) and -h τ (•, y) are continuous, Z d -periodic sub-actions with respect to L as soon as x, y ∈ pr 1 (A τ (L)). As a matter of fact, they are also calibrated sub-actions on the projected Aubry set (which is a first step in the proof of theorem 8.10). Proposition 8.8. Let L(x, v) be a C 0 coercive Lagrangian. i. For any x (mod Z d ) ∈ pr 1 (A τ (L)), S τ (x, •) is backward calibrated.

ii. For any y (mod Z d ) ∈ pr 1 (A τ (L)), -S τ (•, y) is forward calibrated.

Proof. We need to show that, for every y ′ ∈ R d , there exists y ∈ R d satisfying

h τ (x, y ′ ) = h τ (x, y) + Lτ (y, y ′ ). Since h τ (x, •) is a sub-action, we already know that h τ (x, y ′ ) ≤ h τ (x, y) + Lτ (y, y ′ ). Conversely, one can find a sequence of configurations in R d , (x k 0 , x k 1 , . . . , x k n(k) ), such that x k 0 = x + p k for some p k ∈ Z d , x k n(k) = y ′ , n(k) → +∞ and Lτ (x k 0 , x k 1 , . . . , x k n(k) ) → h τ (x, y ′ ).
Thanks to lemma 7.5, a subsequence of {x k n(k)-1 } k converges to some y ∈ R d . Then

h τ (x, x k n(k)-1 ) + Lτ (x k n(k)-1 , y ′ ) ≤ S τ (x, x k n(k)-1 ) + Lτ (x k n(k)-1 , y ′ ) ≤ Lτ (x k 0 , x k 1 , . . . , x k n(k) ).
Letting k go to +∞, we obtain h τ (x, y) + Lτ (y, y ′ ) ≤ h τ (x, y ′ ). In an analogous way, we can prove that -h τ (•, x) = -S τ (•, x) is a forward calibrated sub-action.

We have seen that S τ (x, •) and -S τ (•, y) are continuous, periodic sub-actions for any x, y ∈ R d . The following proposition shows that the Peierls barrier can be defined using Mañé potential. (That fact will be used in the proof of theorem 8.10.) Proposition 8.9. Assume L(x, v) is a C 0 coercive Lagrangian. Then

h τ (x, y) = min z∈pr 1 (Aτ (L)) S τ (x, z) + S τ (z, y) , ∀ x, y ∈ T d .
Proof. Proposition 8.7 tells us h τ (•, y) = S τ (•, y) and h τ (x, •) = S τ (x, •) whenever x, y ∈ pr 1 (A τ (L)). Hence, from item ii of remark 8.6, we immediately get

h τ (x, y) ≤ min z∈pr 1 (Aτ (L)) S τ (x, z) + S τ (z, y) . So it suffices to find z ∈ pr 1 (A τ (L)) satisfying S τ (x, z) + S τ (z, y) ≤ h τ (x, y). Let u be a C 0 (T d ) sub-action. By taking L(x, v) -1 τ [u(x + τ v) -u(x)] -L(τ ), we may assume L ≥ 0 and L(τ ) = 0. Let Lτ (x, y) = τ L(x, 1 τ (y -x)) for x, y ∈ R d . By definition of h τ (x, y), there exists a sequence of configurations (x k 0 , . . . , x k n(k) ) in R d of length n(k) and a sequence p k ∈ Z d such that x k 0 = x, x k n(k) = y + p k , n(k) → +∞ and lim k→∞ Lτ (x k 0 , . . . , x k n(k) ) = h τ (x, y).
Since h τ (x, y) < ∞ and Lτ ≥ 0, for k large enough, one can find m(k) and

m ′ (k) in {0, . . . , n(k) -1} such that m ′ (k) -m(k) = ⌊ √ n k ⌋ and 0 ≤ Lτ (x k m(k) , . . . , x k m ′ (k) ) < h τ (x, y) + 1 ⌊ √ n k ⌋ -1 .
Otherwise, we would reach a contradiction, because for arbitrarily large k

Lτ (x k 0 , . . . , x k n k ) ≥ ⌊ √ n k ⌋-2 i=0 Lτ (x k i⌊ √ n k ⌋ , . . . , x k (i+1)⌊ √ n k ⌋ ) ≥ (⌊ √ n k ⌋ -1) h τ (x, y) + 1 ⌊ √ n k ⌋ -1 = h τ (x, y) + 1.
Thanks to the invariance of L τ by the diagonal action of Z d , L τ (x + s, y + s) = L τ (x, y) for all s ∈ Z d , we may assume x k m(k) ∈ [0, 1) d . Using a diagonal procedure, lemma 7.5 allows us to find a subsequence {k j } of integers and a forward infinite

configuration {z l } l≥0 of R d such that lim j→∞ x k j m(k j ) = z 0 ∈ [0, 1) d and lim j→∞ x k j m(k j )+l = z l ∈ R d , ∀ l ≥ 1.
From the construction of the sequence {m k }, it follows that L τ (z l , z l+1 ) = 0 for any nonnegative integer l, which clearly yields S τ (z l , z l+1 ) = 0. From item iii of remark 8.2, we get

S τ (z l , z l ′ ) = 0 whenever l ′ > l ≥ 0. Therefore, if z ∞ ∈ T d is an arbitrary accumulation point of {z l (mod Z d )} l≥0 , then S τ (z ∞ , z ∞ ) = 0 or z ∞ ∈ pr 1 (A τ (L)). Observe that, for any l ≥ 0, Lτ (x k j 0 , . . . , x k j n(k j ) ) = Lτ (x k j 0 , . . . , x k j m(k j )+l ) + Lτ (x k j m(k j )+l , . . . , x k j n(k j ) ) ≥ S τ (x, x k j m(k j )+l ) + S τ (x k j m(k j )+l , y).
Passing to the limit when j → ∞, we obtain h τ (x, y) ≥ S τ (x, z l ) + S τ (z l , y). Taking then a suitable subsequence of {z l }, we get h τ (x, y) ≥ S τ (x, z ∞ ) + S τ (z ∞ , y). Theorem 8.10. Let L(x, v) be a C 0 coercive Lagrangian. Then the Peierls barrier

h τ : R d × R d → R is continuous, Z d × Z d periodic. Moreover, h τ (x, •) : R d → R
is a forward calibrated sub-action and -h τ (•, y) : R d → R is a backward calibrated sub-action for any x, y ∈ R d .

Proof. Consider arbitrary points (x, y), (x ′ , y ′ ) ∈ T d × T d . Thanks to proposition 8.9, there exists z x,y ∈ pr 1 (A τ (L)) satisfying h τ (x, y) = S τ (x, z x,y )+S τ (z x,y , y). Then

h τ (x ′ , y ′ ) -h τ (x, y) ≤ S τ (x ′ , z x,y ) -S τ (x, z x,y ) + S τ (z x,y , y ′ ) -S τ (z x,y , y) .
Since S τ is uniformly continuous on T d × T d , the estimation above assures that h τ is a continuous map.

We already know that h τ (x, •) and -h τ (•, x) are T d periodic and continuous. Take (y, y ′ ) ∈ R d × R d . Thanks to proposition 8.9, there exists z ∈ pr 1 (A τ (L)) such that h τ (x, y) = S τ (x, z) + S τ (z, y). Then, using the fact that S τ (z, •) is a sub-action

h τ (x, y ′ ) -h τ (x, y) ≤ S τ (z, y ′ ) -S τ (z, y) ≤ L(y, y ′ ).
We have proved that h τ (x, •) is a sub-action. Since S τ (z, •) is also backward calibrated, one can find y ′′ ∈ R d such that S τ (z, y) = S τ (z, y ′′ ) + L(y ′′ , y). Then

h τ (x, y) = S τ (x, z) + S τ (z, y ′′ ) + L(y ′′ , y) ≥ h τ (x, y ′′ ) + L(y ′′ , y).
We have proved that h τ (x, •) is calibrated. Analogously, one can show that -h τ (•, y) is a calibrated sub-action too.

Representation formulas for calibrated sub-actions

In the continuous-time Aubry-Mather theory, Contreras has characterized in [START_REF] Contreras | Action potential and weak KAM solutions[END_REF] the set of weak KAM solutions of the Hamilton-Jacobi equation in terms of their values at each static class. We remark that weak KAM solutions are similar to calibrated sub-actions and the set of static classes is similar to the projected Aubry set. We show in the following theorem how a calibrated sub-action is completely known when it is known on the Aubry set. In the context of ergodic optimization for symbolic dynamics, a similar result has been proved in [START_REF] Garibaldi | On the Aubry-Mather theory for symbolic dynamics[END_REF].

Theorem 9.1. Let L(x, v) be a C 0 coercive Lagrangian. If u + ∈ C 0 (T d ) is a forward calibrated sub-action or u -∈ C 0 (T d ) is a backward calibrated sub-action, then, for every x, y ∈ R d , u + (x) = max y∈pr 1 (Aτ (L)) [u + (y) -S τ (x, y)] = max y∈pr 1 (Aτ (L)) [u + (y) -h τ (x, y)], u -(y) = min x∈pr 1 (Aτ (L)) [u -(x) + S τ (x, y)] = min x∈pr 1 (Aτ (L)) [u -(x) + h τ (x, y)].
Proof. Thanks to proposition 8.7, we just need to prove the two first equalities.

From item ii of remark 8.2, we verify without difficulty that

u + (x) ≥ max y∈pr 1 (Aτ (L)) [u + (y) -S τ (x, y)].
As u + is a forward calibrated sub-action, one can find a forward configuration {x k } k≥0 of R d such that x 0 = x and u + (x k ) = u + (x k+1 )-Lτ (x k , x k+1 ) for every k ≥ 0. From u + (x l )-u + (x k ) ≤ S τ (x k , x l ) ≤ Lτ (x k , . . . , x l ) = u + (x l )-u + (x k ) whenever l > k ≥ 0, we conclude that S τ (x k , x l ) = u + (x l )u + (x k ). Therefore, if y ∈ T d is an arbitrary accumulation point of {x k (mod Z d )}, it follows that S τ (y, y) = 0, namely, y ∈ pr 1 (A τ (L)). Furthermore, by taking a suitable subsequence, u + (x) = u + (x k ) -S τ (x, x k ) tends to u + (x) = u + (y) -S τ (x, y). Analogously, one can demonstrate the existence of a point x ∈ pr 1 (A τ (L)) achieving u -(y) = u -(x) + S τ (x, y). Corollary 9.2. Suppose u, u ′ ∈ C 0 (T d ) are both either forward or backward calibrated sub-actions with respect to a C 0 coercive Lagrangian L(x, v). [ψ(y) -S τ (x, y)] = sup y∈pr 1 (Aτ (L))

i. If u| pr 1 (Aτ (L)) ≤ u ′ | pr 1 (Aτ (L)) , then u ≤ u ′ everywhere on R d . ii. If u| pr 1 (Aτ (L)) = u ′ | pr 1 (Aτ (L)) , then u = u ′ everywhere on R d .
[ψ(y)h τ (x, y)].

ii. If ψ is bounded below, the following u is a continuous backward calibrated sub-action u(y) := inf

x∈pr 1 (Aτ (L))
[ψ(x) + S τ (x, y)] = inf

x∈pr 1 (Aτ (L)) [ψ(x) + h τ (x, y)].
iii. If ψ(y)ψ(x) ≤ S τ (x, y) for all x, y ∈ pr 1 (A τ (L)), then u| pr 1 (Aτ ) = ψ.

Proof. In any case, u : R d → R is clearly a well defined periodic function. Since both constructions are similar, we will discuss just the second one. So let us show that u ∈ C 0 (T d ). Fix ǫ > 0 and consider y, y ′ ∈ R d . Take x (mod Z d ) ∈ pr 1 (A τ (L)) such that ψ(x) + S τ (x, y) < u(y) + ǫ. Thus u(y ′ )u(y) ≤ S τ (x, y ′ ) -S τ (x, y) + ǫ. Since S τ is uniformly continuous on R d × R d and ǫ > 0 is arbitrary, it is easy to deduce that |u(y ′ )u(y)| ≤ max

x∈pr 1 (Aτ (L)) |S τ (x, y ′ ) -S τ (x, y)|,
which guarantees the continuity of u.

We now show that u is backward calibrated. Given y ∈ R d and ǫ > 0, choose x ∈ pr 1 (A τ (L)) satisfying ψ(x) + S τ (x, y) < u(y) + ǫ. Thanks to proposition 8.3, S τ (x, •) is a sub-action and

u(y + τ w) -u(y) -ǫ < S τ (x, y + τ w) -S τ (x, y) ≤ τ L(y, w) -τ L(τ ), ∀ w ∈ R d .
Letting ǫ go to 0, we obtain that u is a sub-action. To prove that u is a calibrated sub-action, we use the fact that the sub-actions {S τ (x, •)} x∈pr 1 (Aτ (L)) are calibrated (see proposition 8.8). Let y ∈ R d . It suffices to show there exists v ∈ R d such that u(y) ≥ u(y-τ v)+τ L(y-τ v, v)-τ L(τ ). By definiton of u(y), there exists a sequence of points x k ∈ pr 1 (A τ (L)) such that ψ(x k ) + S τ (x k , y) < u(y) + 1 k . Moreover, there exists a sequence of v k ∈ R d such that

S τ (x k , y) = S τ (x k , y -τ v k ) + τ L(y -τ v k , v k ) -τ L(τ ).
Remember we can assume v k ∞ ≤ R τ (see lemma 5.5) for some constant R τ > 1 τ (see definition 3.8). Let v ∈ R d be an accumulation point of the sequence {v k } k≥0 . Since u(y

-τ v k ) ≤ ψ(x k ) + S τ (x k , y -τ v k ), we obtain u(y -τ v k ) + τ L(y -τ v k , v k ) -τ L(τ ) < u(y) + 1 k .
Taking a suitable subsequence, we get u(y

-τ v) + τ L(y -τ v, v) -τ L(τ ) ≤ u(y).
Suppose ψ(y)ψ(x) ≤ S τ (x, y) for all x, y ∈ pr 1 (A τ (L)) and u is defined as in item ii . Let y (mod Z d ) ∈ pr 1 (A τ (L)). On the one hand, u(y) ≤ ψ(y) by taking x = y in the definition of u and noticing that S τ (y, y) = 0. On the other hand, for any x ∈ pr 1 (A τ (L)), S τ (x, y) + ψ(x) ≥ ψ(y) by hypothesis on ψ. By taking the infimum on x we obtain u(y) ≥ ψ(y). We have proved that u| pr 1 (Aτ (L)) = ψ.

Thanks to item ii of remark 8.2, an immediate but important consequence of theorem 9.3 is the fact that the restriction of any sub-action to the projected Aubry set behaves as a forward or backward calibrated sub-action. Theorems 9.1 and 9.3 together provide an interesting description of the calibrated sub-actions. In order to present it, we decided to adopt a slightly different point of view. Definition 9.7. Let L(x, v) be a C 0 coercive Lagrangian. We call positive-time Mañé-Peierls transform the application F + defined on C 0 (pr 1 (A τ (L))) by

F + (ψ)(x) = max y∈pr 1 (Aτ (L))
[ψ(y) -S τ (x, y)] = max

y∈pr 1 (Aτ (L)) [ψ(y) -h τ (x, y)], ∀ x ∈ R d .
In the same way, we call negative-time Mañé-Peierls transform the application F - defined on C 0 (pr 1 (A τ (L))) by

F -(ψ)(y) = min x∈pr 1 (Aτ (L))
[ψ(x) + S τ (x, y)] = min

x∈pr 1 (Aτ (L)) [ψ(x) + h τ (x, y)], ∀ y ∈ R d .
We summarize then all the main properties of the Mañé-Peierls transforms.

Theorem 9.8. Let L(x, v) be a C 0 coercive Lagrangian. Consider arbitrary functions ψ, ψ ′ ∈ C 0 (pr viii. F + is a bijective and isometric correspondence between the set of the functions ψ ∈ C 0 (pr 1 (A τ (L))) satisfying, for x, y ∈ pr 1 (A τ (L)), ψ(y)ψ(x) ≤ S τ (x, y) and the set of continuous forward calibrated sub-actions;

1 (A τ (L))). Then i. F -(ψ) ≤ ψ ≤ F + (ψ) everywhere on pr 1 (A τ (L)); ii. ψ ≤ ψ ′ implies F + (ψ) ≤ F + (ψ ′ ) and F -(ψ) ≤ F -(ψ ′ );
ix. F -is a bijective and isometric correspondence between the set of the functions ψ ∈ C 0 (π(A τ (L))) satisfying, for x, y ∈ pr 1 (A τ (L)), ψ(y)ψ(x) ≤ S τ (x, y) and the set of continuous backward calibrated sub-actions.

Proof. Items i and ii follow immediately from the respective definitions of the Mañé-Peierls transforms. In truth, items iii , iv and v can be seen as theorem 9.3 rewritten. Besides, items vi and vii result from theorems 9.1 and 9.3 without difficulty.

Since items viii and ix are very similar, we will discuss just the first one. As F + (ψ) = ψ everywhere on pr 1 (A τ (L)), F + is injective. Moreover, when u ∈ C 0 (T d ) is a forward calibrated sub-action, the identity F + (u| π(Aτ (L)) ) = u guarantees that F + is surjective. In fact, this correspondence is an isometry. Indeed, fixing x ∈ R d , there exists a point y ∈ pr 1 (A τ (L)) such that F + (ψ)(x) = ψ(y) -S τ (x, y). Hence, one has

F + (ψ)(x) -F + (ψ ′ )(x) ≤ ψ(y) -ψ ′ (y) ≤ ψ -ψ ′ 0 .
Since x ∈ R d is arbitrary and since we can interchange the roles of ψ and ψ ′ , we get

F + (ψ) -F + (ψ ′ ) 0 ≤ ψ -ψ ′ 0 . On the other hand, F + (ψ)| π(Aτ (L)) = ψ and F + (ψ ′ )| π(Aτ (L)) = ψ ′ imply F + (ψ) -F + (ψ ′ ) 0 ≥ ψ -ψ ′ 0 .
10 Separating sub-actions

If u ∈ C 0 (T d
) is a sub-action for a C 0 coercive Lagrangian, proposition 7.3 establishes that A τ (L) ⊂ N τ (L, u). So it is natural to ask if there exists a sub-action whose nill locus is the smallest possible one, namely, it is equal to the Aubry set. We introduce then the following concept.

Definition 10.1. Let L(x, v) be a C 0 coercive Lagrangian. We say that a sub-

action u ∈ C 0 (T d ) is separating if N τ (L, u) = A τ (L).
In weak KAM theory, global critical subsolutions of the Hamilton-Jacobi equation are analogous notions to separating sub-actions. Working with continuoustime, autonomous, strictly convex and superlinear C 2 -Lagrangians on a smooth manifold without boundary, A. Fathi and A. Siconolfi (see [START_REF] Fathi | Existence of C 1 critical subsolutions of the Hamilton-Jacobi equation[END_REF]) proved the existence of C 1 critical subsolutions. Keeping the hypotheses on the Lagrangians but focusing on compact manifolds, P. Bernard showed in [START_REF] Bernard | Existence of C 1,1 critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds[END_REF] not only the existence of C 1,1 critical subsolutions but also their density in the set of C 0 subsolutions for the uniform topology.

In a similar way, we prove in theorem 10.2 that separating sub-actions are quite typical in the discrete Aubry-Mather context. During the preparation of this paper, we become aware of a related study by M. Zavidovique [START_REF] Zavidovique | Strict sub-solutions and Mañé potential in a discrete weak KAM theory[END_REF] on separating subactions (or strict sub-solutions) in a general discrete setting given by cost functions defined on certain length spaces. We mention yet that the genericity of separating sub-actions has been proved in [START_REF] Garibaldi | On calibrated and separating sub-actions[END_REF] in the context of ergodic optimization for symbolic dynamics.

Theorem 10.2. Let L(x, v) be a C 0 coercive Lagrangian. Then, in the uniform topology, the subset of the continuous separating sub-actions is generic among all continuous sub-actions.

We will need some preliminary results.

By definition of S τ (x, y), one gets S τ (x, y) ≥ u(y)u(x) + η for any y ∈ R d or equivalently u(x) ≥ F+ (u)(x) + η.

We now prove the main induction step: F+ (u)(x) = u(x) and Lτ (x, y) = F+ (u)(y) -F+ (u)(x) =⇒ F+ (u)(y) = u(y).

Indeed, u(y)u(x) ≤ S τ (x, y) ≤ Lτ (x, y) = F+ (u)(y) -F+ (u)(x), which implies first u(y) ≤ F+ (u)(y) and therefore u(y) = F+ (u)(y). We conclude the proof. If (x, y) is u-calibrated for any sub-action u, on the one hand, F+ (u)(x) = u(x), on the other hand, since (x, y) is also F+ (u)-calibrated, F+ (u)(y) = u(y). We have proved that y

∈ ∩ u { F+ (u) = u} = pr 1 (∩ u N τ (L, u))
thanks to lemma 10.3.

The following proposition gives another equivalent definition of the Aubry set.

Proposition 10.5. Let L(x, v) be a C 0 coercive Lagrangian. Then Lemma 10.4 shows there exists a configuration x = {x k } k≥0 such that Π τ (x) = (x, v) and (x k , x k+1 ) is u-calibrated for any subaction u. Let us first show that l m := Lτ (x 0 , . . . , x m ) + S τ (x m , x 0 ) → 0 when m → +∞.

A τ (L) = u is a sub-action N τ (L, u). Proof. Let (x, v) ∈ ∩ u N τ (L, u).
Since { Lτ (x 0 , . . . , x m ) + S τ (x m , x 0 )} m≥0 is uniformly bounded, one can choose a converging subsequence of {l m } and assume in addition that {x m (mod Z d )} converges to a point x ∞ ∈ T d . Define u(x) := S τ (x ∞ , x), for all x ∈ R d . Proposition 8.3 shows that u is a sub-action. By hypothesis of calibration on {x k }, we have

Lτ (x k , x k+1 ) = S τ (x k , x k+1 ) = u(x k+1 ) -u(x k ), ∀ k ≥ 0. More generally, Lτ (x k , x k+1 , . . . , x m ) = S τ (x k , x m ) = u(x m ) -u(x k ), ∀ m > k ≥ 0.
By taking a subsequence of {x m }, on obtains first S τ (x k , x ∞ ) = u(x ∞ )u(x k ), for all k ≥ 0. By taking a subsequence of {k}, one obtains next

u(x ∞ ) = S τ (x ∞ , x ∞ ) = 0 and S τ (x k , x ∞ ) + S τ (x ∞ , x k ) = 0, ∀ k ≥ 0.
Notice that x ∞ necessarily belongs to pr 1 (A τ (L)). Moreover,

l m = Lτ (x 0 , . . . , x m ) + S τ (x m , x 0 ) = S τ (x 0 , x m ) + S τ (x m , x 0 ), ∀ m ≥ 0.
Letting m → +∞, one gets l m → 0 along a subsequence. We thus have shown that any accumulation point of {l m } is necessarily 0.

Let us prove now that (x, v) ∈ A τ (L). By definition of S τ , there exist finite configurations (x ǫ m , x ǫ m+1 , . . . , x ǫ n(m,ǫ) ) such that x ǫ m = x m , x ǫ n(m,ǫ) = x 0 + p ǫ m for some p ǫ m ∈ Z d and, for any m fixed, Lτ (x ǫ m , x ǫ m+1 , . . . , x ǫ n(m,ǫ) ) → S τ (x m , x 0 ) when ǫ → 0.

twist maps of the annulus (the one dimensional smooth ferromagnetic Frenkel-Kontorova model) has been completely solved by Aubry [START_REF] Aubry | The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states[END_REF], Bangert [START_REF] Bangert | Mather sets for twist maps and geodesics on tori[END_REF] and Mather [START_REF] Mather | Minimal measures[END_REF], [START_REF] Mather | Differentiability of the minimal average action as a function of the rotation number[END_REF] and [START_REF] Mather | Action minimizing invariant measures for positive definite Lagragian systems[END_REF]. One knows that a recurrent minimizing configuration admits a rotation number ω and belongs to the support of a minimizing measure of a Lagrangian of the form L(x, v)λ, v , with λ and ω related by the Legendre relation ω = -∂ L ∂λ , where L(λ) denotes the minimizing holonomic value of Lλ. In particular, they proved that L(λ) is concave, decreasing and continuously differentiable with respect to λ. Moreover, L(λ) does not admit any open interval with irrational constant slope.

Our purpose in this section is to partially extend these results to the multidimensional case and to prove the existence of recurrent minimizing configurations with rotation vector as large in norm as we want.

Definition 11.1. We call rotation vector of a configuration {x k } k∈Z in R d the limit (when it exists)

ω[{x k } k∈Z ] := 1 τ lim n-m→+∞ x n -x m n -m .
We call rotation vector of a holonomic measure µ ∈ P τ (T d × R d ) with bounded support the vector

ω[µ] := T d ×R d v dµ(x, v).
Following Mather's terminology we introduce the so-called α and β functions.

Definition 11.2. Let L(x, v) be a C 0 superlinear Lagrangian. We call Mather's alpha function the opposite of the minimizing holonomic value of L

(x, v) -λ, v , λ ∈ R d , -α L (τ, λ) := L(τ, λ) = min L(x, v) -λ, v dµ(x, v) : µ ∈ P τ (T d × R d ) .
We call Mather's beta function the function

β L (τ, ω) := inf L(x, v) dµ(x, v) : µ ∈ P τ (T d × R d ) has bounded support
and rotation vector v dµ(x, v) = ω .

(We recall that P τ (T d × R d ) denotes the set of holonomic measures.)

We notice that, because of the superlinearity of L, Lλ is again coercive (actually superlinear) and that L(τ, λ) is indeed a minimum and not an infimum. We also point out that, in the definition of β L , we prefer to restrict µ to have bounded support so that v dµ is well defined. We will show in a moment that the set where this infimum is taken is not empty and that the infimum is actually attained. Using standard convex analysis, we obtain the following proposition.

We now show that -L(τ, •) is the Legendre transform of β L (τ, •). Thanks to corollary 6.4, we have

-L(τ, λ) = sup [ λ, v -L(x, v)] dµ(x, v) : µ ∈ P τ (T d × R d ),
and supp(µ) is bounded .

Therefore, one can write We are now able to prove the infimum is attained in the definition of β L (τ, ω).

-L(τ, λ) = sup ω∈R d sup λ, ω -L(x, v) dµ(x, v) : µ ∈ P τ (T d × R d ), supp (
Proposition 11.6. Let L(x, v) be a C 0 superlinear Lagrangian. For every ω ∈ R d , there exists a holonomic measure µ ∈ P τ (T d × R d ) with bounded support such that

T d ×R d v dµ(x, v) = ω and β L (τ, ω) = T d ×R d L(x, v) dµ(x, v).
Proof. We follow Mather's idea which says that the superlinearity of L implies that, given a constant C ∈ R, the set of Borel measures v µ(dx, dv) : µ ∈ P τ (T d × R d ), and L(x, v) dµ(x, v) ≤ C is tight. Let χ R (x, v) be a test function taking its values in [0, 1] and satisfying χ(x, v) = 1 for all v ≤ R -1 and χ R (x, v) = 0 for all v ≥ R. Let {µ n } n≥0 be a sequence of Borel probability measures for which { L(x, v) dµ n (x, v)} is uniformly bounded. So notice that, for every ǫ > 0 and R sufficiently large, we have the inequality

v (1 -χ R ) ≤ ǫ(L(x, v) -inf L), which clearly yields lim R→+∞ lim sup n→+∞ v (1 -χ R ) dµ n (x, v) = 0. Suppose in addition that µ n is holonomic, lim n→+∞ L(x, v) dµ n (x, v) = β L (τ, ω) and v dµ n (x, v) = ω.
We first extract a subsequence, that we again call {µ n } n≥0 , converging to a Borel measure µ in the sense that

f dµ = lim n→+∞ f dµ n , ∀ f ∈ C 0 compact (T d × R d ), f dµ ≤ lim inf n→+∞ f dµ n , ∀ f ∈ C 0 bounded (T d × R d ), f ≥ 0.
The tighness property actually implies for any f

∈ C 0 bounded (T d × R d ) f dµ = lim n→+∞ f dµ n , f v dµ = lim n→+∞ f v dµ n .
In particular, µ is a holonomic measure, it possesses a rotation vector ω and L(x, v) dµ(x, v) ≤ β L (τ, ω). However, as β L (τ, •) is the Legendre transform of -L(τ, •), there exists λ ω ∈ R d such that β L (τ, ω) = L(τ, λ ω ) + λ ω , ω . We obtain

L(x, v) -λ ω , v dµ(x, v) = L(x, v) dµ(x, v) -λ ω , ω ≤ L(τ, λ ω ),
which implies that µ is minimizing for L λω and, by corollary 6.4, has bounded support.

Let us recall the definition of subdifferentials of a convex function. We restrict ourself to superlinear convex functions in order to avoid to pay attention to domains of definition. Definition 11.7. Let α(λ) be a superlinear convex function with full domain. We call subdifferential of α at λ * the set

∂α(λ * ) := ω ∈ R d : α(λ) ≥ α(λ * ) + ω, λ -λ * , ∀ λ ∈ R d .
A simple analysis of the definition of subdifferentials implies the following lemma. In particular α(λ) is affine of slope ω on int(∂β(ω)) (if not empty).

We summarize in the next proposition the properties of Mather's alpha and beta functions that are immediate consequences of convexity arguments. Proposition 11.9. Let L(x, v) be a C 0 superlinear Lagrangian. In particular, if L(τ, .) is differentiable at λ, then all minimizing measures of Lλ have rotation vector ω

[µ] = v dµ(x, v) = -∂ L ∂λ (τ, λ). iii. Let ω ∈ R d be such that int(∂β L (τ, ω)) is not empty. Then L(τ, .
) is affine of slope ω on int(∂β L (τ, ω)) and the Mather set of Lλ is independent of λ ∈ int(∂β L (τ, ω)).

Proof. Part i. Let λ be a subdifferential of β L (τ, .) at ω and µ be a holonomic measure of rotation vector ω. If µ realizes the minimum of β L (τ, ω), then

L(x, v) -λ, v dµ(x, v) = β L (τ, ω) -λ, ω = -α L (τ, λ) = L(τ, λ),
and µ is minimizing for Lλ. Conversely, if µ is minimizing for Lλ, then

L(x, v) dµ(x, v) = L(x, v) -λ, v dµ(x, v) + λ, ω = L(τ, λ) + λ, ω = -α L (τ, λ) + λ, ω = β L (τ, ω),
and µ realizes the minimum of β L (τ, ω).

Part ii. Let ω be the rotation vector of a holonomic measure µ which is minimizing for Lλ. Then

v dµ(x, v) = ω and L(x, v) -λ, v dµ(x, v) = L(τ, λ).
On the one hand, α L (τ, λ) + β L (τ, ω) ≥ λ, ω since α L (τ, .) is the Legendre transform of β L (τ, .). On the other hand,

α L (τ, λ) + β L (τ, ω) ≤ -L(τ, λ) + L(x, v) dµ(x, v) = λ, v dµ(x, v) = λ, ω .
We just have proved that α L (τ, λ) + β L (τ, ω) = λ, ω , that is, ω ∈ ∂α L (τ, λ).

Conversely, suppose ω ∈ ∂α L (τ, λ). Then λ is a subdifferential of β L (τ, .) at ω. Let µ be a holonomic probability measure of rotation ω realizing the minimum of β L (τ, ω). From part i, we know that µ is minimizing for Lλ.

Part iii. Let µ be a holonomic minimizing measure with respect to Lλ for some λ ∈ int(∂β L (τ, ω)). Then L(τ, .) is differentiable at λ and µ has rotation vector ω. From part i, µ realizes the minimum in β L (τ, ω) and therefore is minimizing for Lλ * for any λ * ∈ ∂β L (τ, ω).

An important issue in the multidimensional setting is to know whether or not any rotation vector is obtained by some minimizing configuration. In the ferromagnetic case, a simple application of ergodic theory shows that any ω = -∂ L ∂λ , when λ is a point of differentiability of L(τ, .), is obtained by generic recurrent calibrated minimizing configurations for Lλ. Proposition 11.13 gives a sligth extension of this fact. We follow Gomes' approach (see [START_REF] Gomes | Viscosity solution method and the discrete Aubry-Mather problem, Discrete and Continuous Dynamical Systems[END_REF]) to show that actually all calibrated configurations of Lλ have a rotation vector whenever L(λ) is differentiable at λ. Contrary to Gomes, we do not assume L(x, v) to be ferromagnetic. We also prove the existence of minimizing configurations satisfying a stronger notion of rotation vector: for any λ where L(τ, •) is differentiable in a neighborhood of λ and has constant differential, any minimizing configuration of Lλ stays at a bounded distance from a particular straight line.

By standard convexity argument, the following directional differentials exist for all λ, h ∈ R d ii. If L(τ, •) is differentiable at λ, then any u λ -calibrated configuration {x k } k∈Z has a rotation vector given by

ω {x k } k∈Z = - ∂ L ∂λ (τ, λ) = v dµ(x, v), ∀ µ ∈ M τ (L λ ).
There exist minimizing configurations for L with rotation vector ω of arbitrarily large norm which are calibrated for Lλ, where ω = -∂ L ∂λ .

iii. Let ω ∈ R d be given and Λ ω := {λ ∈ R d : -∂ L ∂λ (τ, λ) exists and is equal to ω}. If the interior of Λ ω is not empty, for any λ ∈ int(Λ ω ), for any u λ -calibrated configuration {x k } k∈Z of Lλ, one has sup k,n∈Z

x k+nx knτ ω ∞ < +∞.

We recall that a configuration is minimizing for L if, and only if, it is minimizing for Lλ for any λ. The previous theorem shows that the rotation vector of a minimizing configuration is in duality with the parameter λ which let the configuration be calibrated by Lλ. In Mather theory, λ is seen as a cohomology and ω as an homology. The parameter λ in the standard one dimensional Frenkel-Kontorova model plays thus a special role. We just have proved ω λ := ω {x k } k∈Z = -∂ L ∂λ (τ, λ) exists. Notice yet that ω λ satisfies the relation λ, ω λ = β L (τ, ω λ ) -L(τ, λ). So if λ → +∞ among the set of points of differentiability of -L(τ, •), the superlinearity of -L(τ, •) implies ω λ → +∞. Part iii. Let λ 0 ∈ int(Λ ω ) and {x k } k∈Z be u 0 -calibrated for some continuous subaction u 0 with respect to Lλ 0 . Then, for any k ∈ Z and n ∈ Z + , L τ (x k , x k+1 , . . . , x k+n ) = u 0 (x k+n )u 0 (x k ) + λ 0 , x k+nx k + nτ L(τ, λ 0 ). Since L(τ, •) is affine on Λ ω , we have L(τ, λ) -L(τ, λ 0 ) =ω, λλ 0 , for any λ ∈ Λ ω . By taking any sub-action u λ of Lλ, one gets L τ (x k , . . . , x k+n )λ, x k+nx knτ L(τ, λ)u λ (x k+n ) + u λ (x k ) = = λ 0λ, x k+nx knτ ω + (u 0u λ )(x k+n ) -(u 0u λ )(x k ).

Since the left hand side of the previous equality is nonnegative, one finally gets λλ 0 , x k+nx knτ ω ≤ 4 sup λ-λ 0 ≤ǫ u λ 0 and x k+nx knτ ω ≤ 4 ǫ sup λ-λ 0 ≤ǫ u λ 0 .

We present an example where a calibrated configuration may not have a well defined rotation vector. However, L does not satisfy a twist condition: there are points (x 0 , y 0 ) ∈ R 2 such that ∂ 2 L ∂x∂y (x 0 , y 0 ) = 0. Indeed, since ∂L ∂y (0, 0) = 0 = ∂L ∂y (1, 0), Rolle's theorem states that ∂ 2 L ∂x∂y (x 0 , 0) = 0 for some x 0 ∈ (0, 1). Let then L : T × R → R denote the corresponding C ∞ superlinear Lagrangian. We will exhibit a configuration {x k } u-calibrated for any sub-action u ∈ C 0 (T) but without a well defined rotation vector. To that end, notice we have (x, v) ∈ A 1 (L) ⇔ L(x, x + v) = 0 ⇔ x = 0 (mod Z) and v ∈ {-1, 0, 1}. So consider any sequence of positive integers {r i } i≥1 such that 1 n n i=1 r i has at least two distinct accumulation points: 1/ω 1 and 1/ω 2 . We define a configuration {x k } by Notice that (x k (mod Z), x k-1x k ) ∈ {(0, -1), (0, 0), (0, 1)} = A 1 (L). Therefore, proposition 7.3 guarantees {x k } is u-calibrated for any sub-action u ∈ C 0 (T). Nevertheless, the fact that

n n i=1 r i ≤ x k k < n n-1 i=1 r i whenever n-1 i=1 r i < k ≤ n i=1 r i
and the choice of the sequence {r i } imply that, when k → +∞, x k /k has ω 1 and ω 2 as accumulation points, which shows the configuration {x k } does not admit a rotation vector.

From now on we assume L to be C 1 , superlinear and ferromagnetic. We first notice that the set of critical configurations Γ τ (Lλ) is independent from λ, that is, Γ τ (L) = Γ τ (Lλ). A configuration calibrated for Lλ is therefore critical for L. We also notice that Lλ is ferromagnetic if, and only if, L is ferromagnetic, and that the definition of the discrete-time Lagrangian dynamics (T d × R d , Φ τ ) is independent of λ too.

According to theorem 7.7, Aubry sets are nonempty compact Φ τ -invariant sets. Hence, as a consequence of theorem 11.10, the next result gives a sufficient condition for the existence of disjoint invariant sets with respect to the discrete-time Lagrangian dynamics. We have seen in theorem 11.10 that, if λ is a point of differentiability of L(τ, •), ω = -∂ L ∂λ (τ, λ) is the rotation vector of all configuration of A τ (Lλ). In the one dimensional case, L(λ) is differentiable everywhere and therefore all rotation numbers come from a minimizing configuration. We do not know whether this result persists in the multidimensional case. The following proposition is a step in that direction. We show that a vector ω which is extremal in ∂α L (τ, λ) for some λ is the rotation vector of some minimizing configuration. Proposition 11.13. Let L(x, v) be a C 1 ferromagnetic superlinear Lagrangian. If ω is an extremal vector of ∂α L (τ, λ) for some λ ∈ R d , then it is the rotation vector of some minimizing configuration {x k } k∈Z calibrated for Lλ.

Proof. Let ω be an extremal point of ∂α L (τ, λ). By hypothesis, there exists a holonomic measure µ such that v dµ(x, v) = ω and [L(x, v)λ, v ] dµ(x, v) = L(τ, λ).

Theorem 6.10 guarantees that µ is Φ τ -invariant. Furthermore, thanks to the extremal conditions on ω and on L(τ, λ), we may assume that µ is Φ τ -ergodic. By the ergodicity of µ, for almost all (x, v) ∈ T d × R d , if x 0 is a representant of x and x n = x 0 + τ n-1 k=0 pr 2 • Φ τ (x, v), then Moreover, {x k } k∈Z is u λ -calibrated for any sub-action u λ of Lλ since

x k ( mod Z d ), x k+1 -x k τ = Φ k τ (x, v) ∈ M τ (L -λ) ⊂ N τ (L -λ, u λ ).

  finite with mass one on any fundamental domain, say T d ×R d , with identical marginals on the two factors R d , pr 1 * (π) = pr 2 * (π), which in addition minimizes π = argmin T d ×R d L(x, y) π(dx, dy).

  . The two sets T + (C 0 (T d )) and T -(C 0 (T d )) are equicontinuous. In particular, osc(T + u, T d ) and osc(T -u, T d ) are bounded by the oscillation of τ L on T d × B 2Rτ , where B 2Rτ denotes the closed ball of center 0 and radius 2R τ .

Proposition 7 . 3 .

 73 Let L(x, v) be a C 0 coercive Lagrangian and u ∈ C 0 (T d ) be a sub-action with respect to L. Then M τ (L) ⊂ A τ (L) ⊂ N τ (L, u).

Proposition 8 . 3 .

 83 Let L(x, v) be a C 0 coercive Lagrangian. Then i. S τ (x, y) : R d × R d → R is continuous and periodic in x and y, ii. For every x, y ∈ R d , S τ (x, •) and -S τ (•, y) are C 0 (T d ) sub-actions.

Theorem 9 .

 9 1 admits a reciprocal. Theorem 9.3. Let L(x, v) be a C 0 coercive Lagrangian and ψ be any function defined on the projected Aubry set pr 1 (A τ (L)).

  i. If ψ is bounded above, the following u is a continuous forward calibrated subaction u(x) := sup y∈pr 1 (Aτ (L))

  iii. F + (ψ) is a continuous forward calibrated sub-action; iv. F -(ψ) is a continuous backward calibrated sub-action; v. if ψ(y)ψ(x) ≤ S τ (x, y) for all x, y ∈ pr 1 (A τ (L)), then F + and F -act as extension operators:F + (ψ)| pr 1 (Aτ (L)) = ψ = F -(ψ)| pr 1 (Aτ (L)) ; vi. if u ∈ C 0 (T d) is a forward calibrated sub-action, thenF + (u| pr 1 (Aτ (L)) ) = u = F + (F -(u| pr 1 (Aτ (L)) )| pr 1 (Aτ (L)) ) everywhere on R d and F -(u| π(Aτ (L))) is the unique sub-action conjugated to u;vii. if u ∈ C 0 (T d ) is a backward calibrated sub-action, then F -(u| pr 1 (Aτ (L)) ) = u = F -(F + (u| pr 1 (Aτ (L)) )| pr 1 (Aτ (L))) everywhere on R d and F + (u| pr 1 (Aτ (L)) ) is the unique sub-action conjugated to u;

  µ) is bounded, and v dµ(x, v) = ω , namely, -L(τ, λ) = sup ω∈R d [ λ, ωβ L (ω)]. Proposition 11.3 follows then from lemma 11.5.

Lemma 11 . 8 .

 118 Let α(λ), β(ω) : R d → R be convex functions with full domain and obtained by Legendre transform one from the other. Then λ ∈ ∂β(ω) ⇐⇒ ω ∈ ∂α(λ) ⇐⇒ α(λ) + β(ω) = λ, ω .

  i. Let ω, λ ∈ R d be related by Legendre relation λ ∈ ∂β L (τ, ω). Let µ be a holonomic measure of rotation vector ω. Then µ realizes the minimum in the definition of β L (τ, ω) if, and only if, µ is minimizing for Lλ.ii. Let λ ∈ R d . Then ∂α L (τ, λ) = ω[µ] : µ is holonomic and minimizing for Lλ .

  ∂ + h L(τ, λ) := lim ρ→0 + L(τ, λ + ρh) -L(τ, λ) ρ , ∂ - h L(τ, λ) := lim ρ→0 + L(τ, λ) -L(τ, λρh) ρ .Theorem 11.10. Suppose L(x, v) is C 0 and superlinear. Let u λ ∈ C 0 (T d ) be an arbitrary sub-action for Lλ. Then i. For any h ∈ R d , for any u λ -calibrated configuration {x k } k∈Z , one has-τ ∂ - h L(τ, λ) ≤ lim inf n-m→∞ h, x nx m nm ≤ lim sup n-m→∞ h, x nx m nm ≤ -τ ∂ + h L(τ, λ).

Proof of theorem 11 . 10 .

 1110 Without loss of generality, we can suppose that L(τ ) = 0. Part i. Since u λ calibrates {x k } k∈Z , one hasu λ (x n ) = u λ (x m ) + L τ (x m , . . . , x n )λ, x nx m -(nm)τ L(τ, λ), ∀ m < n. Let ρ > 0, h ∈ R d and λ h := λρh. If u λ h is a continuous sub-action for Lλ h , u λ h (x n ) ≤ u λ h (x m ) + L τ (x m , . . . , x n )λ h , x nx m -(nm)τ L(τ, λ h ).Therefore, by substracting the first equality to the above inequality, one obtainsτ L(τ, λρh) -L(τ, λ) ρ -2 ρ u λu λ h 0 nm ≤ h, x nx m nm ,from which we immediately deduce-τ ∂ - h L(τ, λ) = τ lim ρ→0 + L(τ, λρh) -L(τ, λ) -L(τ, λ + ρh) ρ = -τ ∂ + h L(τ, λ). Part ii. If L(τ, •) is differentiable at λ ∈ R d , the previous inequalities become lim n-m→∞ x nx m nm + τ ∂ L ∂λ (τ, λ), h = 0, ∀ h ∈ R d .

Example 11 . 11 .

 1111 Assume d = 1 and τ = 1. Let ℓ : R×R → [0, 1] be a C ∞ function such that ℓ -1 (1) = {(0, 0), (0, 1/2), (1/2, 0)} and ℓ -1 (0) ⊃ R 2 -(-1/4, 3/4) 2 . Define then L 1 : R × R → R + by L 1 (x, y) = 1 -s∈Z ℓ(x + s, y + s), ∀ x, y ∈ R. Clearly, L 1 is a C ∞ function invariant by the diagonal action of Z, L 1 ≥ 0, L -1 1 (0) = S := s∈Z {(s, s), (s, s + 1/2), (s + 1/2, s)} and L 1 > 0 everywhere on R 2 -S. If L 2 (x, y) = |x -y| 2 |xy + 1| 2 |xy -1| 2 , letus consider a nonnegative local interaction energy map given byL(x, y) = L 1 (x, y)L 1 (x -1, y)L 1 (x, y -1) + L 2 (x, y), ∀ x, y ∈ R.Notice that L is C ∞ , superlinear, invariant by the diagonal action of Z, L ≥ 0 and L -1 (0) = s∈Z {(s, s), (s, s + 1), (s + 1, s)}.

x 0

 0 = 0 and x k = n if

Proposition 11 . 12 .

 1112 Let L(x, v) be a C 1 ferromagnetic superlinear Lagrangian. Suppose λ, λ * ∈ R d are points of differentiability of L(τ, •). Then∂ L ∂λ (τ, λ) = ∂ L ∂λ (τ, λ * ) =⇒ A τ (Lλ) ∩ A τ (Lλ * ) = ∅.Proof. Suppose on the contrary (x 0 , v 0 ) ∈ A τ (L-λ)∩A τ (L-λ * ). By the invariance of Aubry sets, we have(x k , v k ) := Φ k τ (x 0 , v 0 ) ∈ A τ (Lλ) ∩ A τ (Lλ * ), ∀ k ∈ Z.Define then y 0 = x 0 ∈ [0, 1) d and recursivelyy k+1 = y k + τ v k ∈ R d for k ≥ 0 and y k-1 = y kτ v k-1 ∈ R d for k ≤ 0.Let u λ ∈ C 0 (T d ) be a sub-action for Lλ and let u λ * ∈ C 0 (T d ) be a sub-action for Lλ * . By proposition 7.3, the configuration {y k } k∈Z is simultaneously u λcalibrated and u λ * -calibrated. Hence, theorem 11.10 implies-∂ L ∂λ (τ, λ) = ω {y k } k∈Z = -∂ L ∂λ (τ, λ * ),which is a contradiction. Thus A τ (L-λ) and A τ (L-λ * ) are necessarily disjoint.

2 •

 2 Φ τ (x, v) = v dµ(x, v) = ω.

  It is not difficult but not immediate to show that configurations in the Aubry set are minimizing in a stronger sense. More precisely, they are minimizing with respect to L * (x, y) := inf s,t∈Z d L(x+s, y+t). Notice that the two minimizing holonomic values are identical: L = L * . Let E(L) be the set of energy minimizing configurations in the strong sense. Configurations in E(L) are strongly related to a special class of configurations called calibrated.

* supported by CNPq posdoc scholarship † partially supported by ANR BLANC07-3 187245, Hamilton-Jacobi and Weak KAM Theory Eduardo Garibaldi and Philippe Thieullen

Proof. As in the proof of proposition 6.9, we just need to prove that pr 1 (A τ (L) ⊆ K τ (L, u).

We normalize again by defining

Let (x, v) ∈ A τ (L), x 0 = x and x 1 = x+τ v. Then there exist a sequence of periodic configurations (x l 0 , x l 1 , • • • , x l q(l) ) and a sequence of integers p l ∈ Z d such that

x l 0 → x 0 , x l 1 → x 1 , x l q(l) = x l 0 + p l and Ēτ (x l 0 , x l 1 , • • • , x l q(l) ) → 0.

From lemma 7.5 we obtain that {x l q(l)x l q(l)-1 } l is uniformly bounded. One can extract a converging subsequence of {x l q(l)-1p l } l to some x -1 ∈ R d . Since 0 ≤ Ēτ (x l q(l)-1p l , x l q(l)p l ) ≤ Ēτ (x l 0 , x l 1 , • • • , x l q(l) ), Ēτ (x -1 , x 0 ) = 0 and (x -1 , x 0 , x 1 ) is a u-calibrated triple: x ∈ K τ (L, u).

We can now improve theorem 6.10 in the ferromagnetic case. As in definition 2.5, we recall that (T d × R d , Φ τ ) denotes the discrete Euler Lagrange map.

Theorem 7.7. Let L(x, v) be a C 1 ferromagnetic coercive Lagrangian. Then A τ (L) is compact, Φ τ -invariant and equal to the graph of some continuous map

Proof. The proof is similar to the proof of theorem 6.10 thanks to the fact that pr 1 (A τ (L)) ⊆ K τ (L, u) for any continuous and periodic sub-action u and to the fact that any x ∈ pr 1 (A τ (L)) is the projection of a configuration x = {x k } k∈Z satisfying Π τ (σ k (x)) ∈ A τ (L) for all k ∈ Z. This is similar to lemma 6.5. The proof of this fact is given in the following lemma 7.8.

Proof. We begin by normalizing L by assuming L(x, v) ≥ 0 and L = 0. Let (x, v) ∈ A τ (L), x 0 = x and x 1 = x 0 + τ v. Then there exists a sequence of periodic configurations

From lemma 7.5 we know there exists R > 0 such that all jumps are bounded uniformly, x l kx l k+1 ∞ < R for all k ∈ Z. By a diagonal procedure of extraction, there exists a subsequence of {x l } l , that we call again {x l } l , such that, for all k ≥ 0, when l → ∞ one has x l k → x k and x l q(l)-kp l → x -k for some configuration {x k } k∈Z . By definition of the Aubry set, each Π τ (x k , x k+1 ) belongs to A τ (L). A special care should be given in the previous argument when the length l remains bounded.

Corollary 9.4. Let u ∈ C 0 (T d ) be an arbitrary sub-action for a C 0 coercive Lagrangian L(x, v). Then, for every point x ∈ pr 1 (A τ (L)), we have

Theorem 9.3 motivates the introduction of the following notion. Definition 9.5. Let L(x, v) be a C 0 coercive Lagrangian. Suppose that u + is a C 0 (T d ) forward calibrated sub-action and that u -is a C 0 (T d ) backward calibrated sub-action. We say that u + and u -are conjugated sub-actions, and we use the

Notice that coerciveness is a sufficient condition for the existence calibrated sub-actions. Moreover, corollary 9.2 implies that, given a forward calibrated subaction u + , there exists at most one backward calibrated u -conjugated to u + and vice versa. Finally, theorem 9.3 shows that such a backward calibrated sub-action does exist. More precisely, if u -is given, the conjugated u + takes necessarily the form u + (x) := max

and conversely if u + is given, the conjugated u -has the form

Fathi points out in [START_REF] Fathi | Solutions KAM faibles conjuguées et barrières de Peierls[END_REF] that, for a continuous-time, autonomous, strictly convex, superlinear C 3 -Lagrangian on a compact C ∞ manifold without boundary, the Peierls barrier admits a characterization in terms of conjugated sub-actions. We obtain in the following theorem a similar result in the discrete Aubry-Mather theory.

Proposition 9.6. Let L(x, v) be a C 0 coercive Lagrangian. Then

Proof. For any z ∈ pr 1 (A τ (L)) and a pair of sub-actions u + and u -, we have

Thanks to proposition 8.9, we get u -(y)u + (x) ≤ h τ (x, y), which obviously yields

Fix x, y ∈ R d . Consider then the forward calibrated sub-action u + = -h τ (•, y) and define a backward calibrated sub-action u -∈ C 0 (T d ) by

By construction, u + and u -are conjugated sub-actions. Furthermore, u -(y) = 0. Thus u -(y)u + (x) = h τ (x, y).

Lemma 10.3. Let L(x, v) be a C 0 coercive Lagrangian. Then

In other words, if x ∈ R d and for any sub-action u there exists y ∈ R d such that (x, y) is u-calibrated, then there exists y ∈ R d such that (x, y) is u-calibrated for any sub-action u.

Proof. The inclusion pr

. We want to show there exists a sub-action u ∈ C 0 (T d ) such that x / ∈ pr 1 (N τ (L, u)). Let u 0 ∈ C 0 (T d ) be a fixed sub-action. We know from corollary 6.4 that one can choose a constant R τ > 0 such that (x, v) / ∈ N τ (L, u 0 ) whenever v ∞ > R τ . By hypothesis, for any v ∞ ≤ R τ , there exist a subaction u v ∈ C 0 (T d ) and a constant η v > 0 satisfying (x, w) / ∈ N τ (L, u v ) whenever vw < η v . By extracting a finite subcover, one can find a finite collection of sub-actions {u 1 , . . . ,

Since the set of sub-actions is convex, u turns out to be a sub-action. Besides, from

In other words, if (x, y) ∈ R d × R d is u-calibrated for any sub-action u, then there exists z ∈ R d such that (y, z) is u-calibrated for any sub-action u.

Proof. Let us introduce a similar transform as in definition 9.7 by considering F+ (u)(x) := max

where u ∈ C 0 (T d ) is any sub-action. It is easy to see that F+ (u) ∈ C 0 (T d ) is again a sub-action satisfying F+ (u) ≤ u, with equality everywhere whenever u behaves as a forward calibrated sub-action (see corollary 9.4). We begin by proving

which implies F+ (u)(x) ≥ u(y) -S τ (x, y) = u(x) and therefore F+ (u)(x) = u(x).

Conversely if x / ∈ pr 1 (N τ (L, u)) then, by coerciveness of L and periodicity of u, there exists η > 0 such that Lτ (x, y) ≥ u(y)u(x) + η for any y ∈ R d . For any finite configuration (x 0 , x 1 , . . . , x n ) satisfying x 0 = x, one has Lτ (x 0 , x 1 , . . . , x n ) = Lτ (x 0 , x 1 ) + Lτ (x 1 , . . . , x n )

We conclude that Lτ (x 0 , . . . , x m-1 , x ǫ m , x ǫ m+1 , . . . , x ǫ n(m,ǫ) ) = = l m + Lτ (x ǫ m , x ǫ m+1 , . . . , x ǫ n(m,ǫ) ) -S τ (x m , x 0 ) tends to 0 when m is first chosen large enough and then ǫ is chosen close enough to 0. Thus (x, v) ∈ A τ (L).

We now prove that separating sub-actions are generic among sub-actions.

Proof of theorem 10.2. Let {O n } n be a countable family of open neighborhoods of the Aubry set such that

Since the subset of C 0 (T d ) separating sub-actions is equal to ∩ n U n , the statement of the theorem will be obtained if we show that, for the uniform topology, every U n is open and dense in the set of C 0 (T d ) sub-actions. Suppose on the contrary that U n is not open. So there exists a sequence of C 0 (T d ) sub-actions {u k } k≥0 converging to some u ∈ U n and a sequence of points

From corollary 6.4, we know there exists a positive constant R τ such that v k ≤ R τ for all k. By considering a suitable subsequence, we obtain a point (

Let us prove now that U n is dense. We first notice that, if t ∈ (0, 1), u ∈ U n and u ′ ∈ C 0 (T d ) is any arbitrary sub-action, then

and therefore tu + (1t)u ′ ∈ U n . In particular, in order to prove that U n is dense, it suffices to argue that U n is nonempty. Corollary 6.4 assures that (x, v) / ∈ N τ (L, u) for any v > R τ and any subaction u. Let B τ denote the closed ball of center 0 ∈ R d and radius R τ . Thanks to proposition 10.5, for every point (x, v) ∈ (T d × B τ ) -O n , one can find a sub-action u (x,v) ∈ C 0 (T d ) and an open set

Hence, thanks to the compactness of (T d × B τ ) -O n , there exist a finite cover by open sets {V 1 , . . . , V m } of (T d × B τ ) -O n and a finite collection of sub-actions {u 1 , . . . , u m } ⊂ C 0 (T d ), where

Some results on rotation vectors

We are interested in this section whether a minimizing configuration {x n } n∈Z in R d is distributed according to a periodic pattern, that is, whether 1 n (x nx 0 ) → τ ω, where ω is some fixed vector in R d called rotation vector. The case of monotone Proposition 11.3. Let L(x, v) be a C 0 superlinear Lagrangian. Then the two functions

In particular, both suprema are maxima.

The next lemma shows that Mather's beta function is well defined.

Lemma 11.4. For every ω ∈ R d , there exists a holonomic measure µ such that

where B ω ∞ denotes the closed ball of center 0 and radius ω ∞ .

Proof. If ω = p/q, with q ∈ Z * + and p ∈ Z d , then clearly

is a holonomic measure satisfying the statement of the lemma. For a general ω ∈ R d , consider a sequence {p n /q n }, with q n ∈ Z * + and p n ∈ Z d , such that lim n→∞ p n /q n = ω and p n /q n ∞ ≤ ω ∞ . Let {µ pn/qn } be the corresponding sequence of holonomic measures defined as above. Then this sequence is relatively compact for the narrow topology and any accumulation point µ ω is holonomic, µ ω ∈ P τ (T d × R d ), and admits ω as a rotation vector.

We first recall a standard fact in convex analysis (see [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF] for complete references).

Lemma 11.5. Let f, g : R d → R be convex functions with full domain. Suppose that f is the Legendre transform of g, namely,

Then f and g are superlinear and g is the Legendre transform of f . Moreover, for every λ fixed (respectively ω fixed), the equation f (λ)+g(ω) = λ, ω admits at least one solution in ω (respectively in λ).