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This definition of the Peierls barrier is due to Mather, see [Mat93,

§7, page 1372].

Definition 5.3.1 (Peierls Barrier). For t > 0 fixed, let us define the function ht: M × M → R by

ht(x, y) = inf

γ

Z t

0

L(γ(s), ˙γ(s)) ds,

where the infimum is taken over all the (continuous) piecewise C1 curves γ : [0, t] → M with γ(0) = x and γ(t) = y. The Peierls barrier is the function h : M → R defined by

h(x, y) = lim inf

t→+∞ht(x, y) + c[0]t.

It is not completely clear that h is finite nor that it is contin-uous. We start by showing these two points.

Lemma 5.3.2 (Properties of ht). The properties of htare (1) for each x, y, z ∈ M and each t, t> 0, we have

ht(x, y) + ht(y, z) ≥ ht+t(x, z);

(2) if u ≺ L + c, we have ht(x, y) + ct ≥ u(y) − u(x);

(3) for each t > 0 and each x ∈ M , we have ht(x, x) + c[0]t ≥ 0;

(4) for each t0 > 0 and each u ∈ S, there exists a constant Ct0,u such that

∀t ≥ t0, ∀x, y ∈ M, −2kuk≤ ht(x, y)+c[0]t ≤ 2kuk+Ct0,u; (5) for each t > 0 and each x, y ∈ M , there exists an

ex-tremal curve γ : [0, t] → M with γ(0) = x, γ(t) = y and ht(x, y) = Rt

0L(γ(s), ˙γ(s)) ds. Moreover, an extremal curve γ : [0, t] → M is minimizing if and only if ht(γ(0), γ(t)) = Rt

0L(γ(s), ˙γ(s)) ds;

175 (6) for each t0 > 0, there exists a constant Kt0 ∈ [0, +∞[ such that, for each t ≥ t0 the function ht : M × M → R is Lipschitzian with a Lipschitz constant ≤ Kt0.

Proof. Properties (1) and (2) are immediate, and property (3) results from (2) taking for u a function in S.

To prove property (4), we first remark that the inequality

−2kuk≤ ht(x, y) + c[0]t also results from (2). By compactness of M , we can find a constant Ct0 such that for each x, z ∈ M , there exists a C1 curve γx,z : [0, t0] → M with γx,z(0) = x, γx,z(t0) = z, and Rt0

0 L(γx,z(s), ˙γx,z(s)) ds ≤ Ct0. By the properties of u, we can find an extremal curve γy :] − ∞, 0] → M , with γy(0) = y, and

∀t ≥ 0, u(y) − uy(−t)) = Z 0

−t

L(γy(s), ˙γy(s)) ds + c[0]t.

If t ≥ t0, we can define a (continuous) piecewise C1 curve γ : [0, t]

by γ(s) = γx,γy

(t0−t)(s), for s ∈ [0, t0], and γ(s) = γy(s − t), for s ∈ [t0, t]. This curve γ joins x with y, and we have

Z t 0

L(γ(s), ˙γ(s)) ds + c[0]t ≤ Ct0+ c[0]t0+ u(y) − uy(t0− t)).

It is then enough to set Ct0,u= Ct0+ c[0]t0 to finish the proof of (4).

The first part of the property (5) results from Tonelli’s Theo-rem 3.3.1. The second part is immediate starting from the defini-tions.

To prove property (6), suppose that γ : [0, t] → M is an extremal curve such that γ(0) = x, γ(t) = y, and ht(x, y) = Rt

0 L(γ(s), ˙γ(s)) ds. Since t ≥ t0, we know by the Compactness Lemma that there exists a compact subset K of T M with (γ(s), ˙γ(s)) ∈ K for each x, y ∈ M , each t ≥ t0 and each s ∈ [0, t]. It is then enough to adapt the ideas which made it possible to show that the family {Ttu | t ≥ t0, u ∈ C0(M, R)} is equi-Lipschitzian.

Corollary 5.3.3 (Properties of h). The values of the map h are finite. Moreover, the following properties hold

(1) the h map is Lipschitzian;

(2) if u ≺ L + c[0], we have h(x, y) ≥ u(y) − u(x);

(3) for each x ∈ M , we have h(x, x) ≥ 0;

(4) h(x, y) + h(y, z) ≥ h(x, z);

(5) h(x, y) + h(y, x) ≥ 0;

(6) for x ∈ M0, we have h(x, x) = 0;

(7) for each x, y ∈ M , there exists a sequence of minimizing extremal curves γn : [0, tn] → M with tn → ∞, γn(0) = x, γn(tn) = y and

h(x, y) = lim

n→+∞

Z tn

0

L(γn(s), ˙γn(s)) ds + c[0]tn;

(8) if γn: [0, tn] → M is a sequence of (continuous) piecewise C1 curves with tn → ∞, γn(0) → x, and γn(tn) → y, then we have

h(x, y) ≤ lim inf

n→+∞

Z tn

0

L(γn(s), ˙γn(s)) ds + c[0]tn.

Proof. Properties (1) to (5) are easy consequences of the lemma giving the properties of ht5.3.2. Let us show the property (6). By the continuity of h, it is enough to show that if µ is a Borel proba-bility measure on T M , invariant by φt and such thatR

T M L dµ =

−c[0], then, for each (x, v) ∈ supp(µ), the support of µ, we have h(x, x) = 0. By Poincar´e’s Recurrence Theorem, the recurrent points for φt contained in supp(µ) form a dense set in supp(µ).

By continuity of h, we can thus assume that (x, v) is a recurrent point for φt. Let us fix u∈ S. We have

u(πφt(x, v)) − u(x) = Z t

0

L(φs(x, v)) ds + c[0]t

Since there exists a sequence tn→ ∞ with φtn(x, v) → (x, v), it is not difficult, for each ǫ > 0 and each t ≥ 0, to find a (continuous) piecewise C1 curve γ : [0, t] → M , with t ≥ t, γ(0) = γ(t) = x, and such that Rt

0 L(γ(s), ˙γ(s)) ds + c[0]t ≤ ǫ. Consequently, we

177 obtain h(x, x) ≤ 0. The inequality h(x, x) ≥ 0 is true for each x ∈ M .

Property (7) results from part (5) of the lemma giving the properties of ht, since there is a sequence tn → +∞ such that h(x, y) = limtn→+∞htn(x, y) + c[0]tn.

Let us show property (8). By the previous lemma, there is a constant K1such that

∀t ≥ 1, ∀x, x, y, y ∈ M, |ht(x, y)−ht(x, y)| ≤ K1(d(x, x)+d(y, y)).

In addition, we also have htnn(0), γn(tn)) ≤

Z tn

0

L(γn(s), ˙γn(s)) ds + c[0]tn. For n large, it follows that

htn(x, y) + c[0]tn≤ htnn(0), γn(tn)) + c[0]tn+ K1(d(x, γn(0)) + d(y, γn(tn)))

≤ c[0]tn+ K1(d(x, γn(0)) + d(y, γn(tn))) + Z tn

0

L(γn(s), ˙γn(s)) ds + c[0]tn. Since d(x, γn(0)) + d(y, γn(tn)) → 0, we obtain the sought

inequal-ity.

The following lemma is useful.

Lemma 5.3.4. Let V be an open neighborhood of ˜M0 in T M . There exists t(V ) > 0 with the following property:

If γ : [0, t] → M is a minimizing extremal curve, with t ≥ t(V ), then, we can find s ∈ [0, t] with (γ(s), ˙γ(s)) ∈ V .

Proof. If the lemma were not true, we could find a sequence of extremal minimizing curves γi : [0, ti] → M , with ti → ∞, and such that (γi(s), ˙γi(s)) /∈ V , for each s ∈ [0, ti]. Since ti → +∞, by corollary 4.3.2, there exists a compact subset K ⊂ T M with (γi(s), ˙γi(s)) ∈ K, for each s ∈ [0, ti] and each i ≥ 0. We then consider the sequence of probability measures µn on T M defined by

Z

T M

θ dµn= 1 tn

Z tn

0

θ(γn(s), ˙γn(s)) ds,

for θ : T M → R continuous. All the supports of these mea-sures are contained in the compact subset K of T M . Extract-ing a subsequence, we can assume that µn converge weakly to a probability measure µ. Since (γn(s), ˙γn(s)), s ∈ [0, tn] are pieces of orbits of the flow φt, and since tn → +∞, the measure µ is invariant by φt. Moreover, its support supp(µ) is contained in T M \V , because this is the case for all supp(µn) = {(γn(s), ˙γn(s)) | s ∈ [0, tn}. Since the γn are minimizing extremals, we have R L dµn = htnn(0), γ(tn))/tn. By the lemma giving the prop-erties of ht 5.3.2, if u ∈ S, we can find a constant C1 such that

∀t ≥ 1, ∀x, y ∈ M, −2kuk0≤ ht(x, y) + c[0]t ≤ 2kuk0+ C1. It follows that limn→+∞R

L dµn= −c[0]. HenceR

T ML dµ = −c[0]

and the support of µ is included in Mather’s set ˜M0. This is a contradiction, since we have already observed that supp(µ) is disjoint from the open set V which contains ˜M0.

Corollary 5.3.5. For each pair u ∈ S, u+ ∈ S+ of conjugate functions, we have

∀x, y ∈ M, u(y) − u+(x) ≤ h(x, y).

Proof. We pick a sequence of extremals γn: [0, tn] → M joining x to y, and such that

h(x, y) = lim

n→∞

Z tn

0

L(γn(s), ˙γn(s)) ds + c[0]tn.

By the previous lemma 5.3.4, extracting a subsequence if neces-sary, we can find a sequence tn ∈ [0, tn] such that γn(tn) → z ∈ M0. If u∈ S, and u+∈ S+, we have

u+n(tn)) − u+(x) ≤ Z tn

0

L(γn(s), ˙γn(s)) ds + c[0]tn, u(y) − un(tn)) ≤

Z tn

tn

L(γn(s), ˙γn(s)) ds + c[0](tn− tn).

If we add these inequalities, and we let n go to +∞, we find u(y) − u(z) + u+(z) − u+(x) ≤ h(x, y).

179

Proof. We first show that the function hx is dominated by L+c[0].

If γ : [0, t] → M is a (continuous) piecewise C1 curve, we have ht(γ(0), γ(t)) ≤ Rt

0L(γ(s), ˙γ(s)) ds and thus, by part (1) of the lemma giving the properties of ht 5.3.2, we obtain

ht+t(x, γ(t)) ≤ ht(x, γ(0)) +

Since tn → ∞ and the γn are all minimizing extremal curves, by extracting a subsequence if necessary, we can suppose that the sequence of extremal curves γn : [−tn, 0] → M, t 7→ γn(tn+ t) γ(t), by part (8) of the corollary giving the properties of h, we obtain h(x, γ(t)) ≤ lim infn→∞Rtn+t

0 L(γn(s), ˙γn(s)) ds+c[0](tn+ t). By taking the liminf in the equality (∗), we do indeed find

h(x, y) ≥ h(x, γ(t)) + Z 0

t

L(γ(s), ˙γ(s)) ds − c[0]t.

It remains to be seen that ux+∈ S+, the conjugate function of hx, vanishes at x. For that, we define

a(t) = −c[0]t + sup limt→∞a(t). For each t > 0, we can choose an extremal curve γt: [0, t] → M , with γt(0) = x and to a point of M which we will call y. By continuity of h we have

181 h(x, y) = limn→+∞h(x, γtn(tn)). Moreover, by part (8) of the corollary giving the properties of h, we have

h(x, y) ≤ lim inf

n→+∞

Z t 0

L(γt(s), ˙γt(s)) ds + c[0]t.

It follows that ux+(x) = lim a(tn) ≤ 0. Since we already showed the inequality h(x, y) ≥ u(y) − u+(x), for any pair of conjugate functions u ∈ S, u+ ∈ S+, we have h(x, x) ≥ hx(x) − ux+(x).

However h(x, x) = hx(x), which gives ux+(x) ≥ 0.

Corollary 5.3.7. For each x, y ∈ M , we have the equality h(x, y) = sup

(u,u+)

u(y) − u+(x),

the supremum being taken on pairs (u, u+) of conjugate functions u∈ S, u+∈ S+.

We can also give the following characterization for the Aubry set A0.

Proposition 5.3.8. If x ∈ M , the following conditions are equiv-alent

(1) x ∈ A0;

(2) the Peierls barrier h(x, x) vanishes;

(3) there exists a sequence γn : [0, tn] → M of (continuous) piecewise C1 curves such that

–for each n, we have γn(0) = γn(tn) = x;

–the sequence tn tends to +∞, when n → ∞;

–for n → ∞, we have Rtn

0 L(γn(s), ˙γn(s)) ds + c[0]tn→ 0;

(4) there exists a sequence γn : [0, tn] → M minimizing ex-tremal curves such that

–for each n, we have γn(0) = γn(tn) = x;

–the sequence tn tends to +∞, when n → ∞;

–for n → ∞, we have Rtn

0 L(γn(s), ˙γn(s)) ds + c[0]tn→ 0.

Proof. Equivalence of conditions (1) and (2) results from the pre-vious corollary. Equivalence of (2), (3) and (4) results from the definition of h.