A dynamical condition for differentiability of Mather's average action

We prove the differentiability of $\beta $ of Mather function on all homology classes corresponding to rotation vectors of measures whose supports are contained in a Lipschitz Lagrangian absorbing graph, invariant by Tonelli Hamiltonians. We also show the relationship between local differentiability of $\beta $ and local integrability of the Hamiltonian flow.


Introduction
Given a Tonneli Lagrangian L, Mather introduced the β-function of L, which is a convex and superlinear function. Many interesting properties of the Euler-Lagrange flow can be derived from the study of the behaviour of the β-function. Understanding whether or not this function is differentiable and what are the implications of its regularity to the dynamics of the system is an interesting problem. This type of problem was developed by D. Massart in several works as, for example, [14] and [15].
Even in this context, D. Massart and A. Sorrentino get in the work Differentiability of Mather's average action and integrability on closed surfaces (see [16]) the relation, on closed surfaces, between the differentiability of β-function and the integrability of the system. However there are examples of systems which are not integrable but have invariant Lipschitz Lagrangian graphs, i.e. invariant graphs of the form G η,u = graph (η + du) where η is a closed one-form and u is a function of class C 1 with Lipschitz differential.
Motivated by these problems, in this work we study the differentiability of β at homologies h whose the measures with vector rotation h are supported on an invariant Lipschitz Lagrangian graph. We obtain differentiability of β in these homologies if the invariant graph is an absorbing graph, i.e. a graph which not contain ω-limit of minimizing curves out of it 1 . More precisely, we prove the following theorem: Theorem 1 Let G η,u be an invariant Lipschitz Lagrangian graph. Then G η,u is absorbing if and only if β is differentiable at h for all h ∈ ∂α ([η]) and A * [η] = G η,u .
One can derive some consequences of this result. For instance, if the system is locally Lipschitz integrable on an invariant Lipschitz Lagrangian graph G η,u ⊂ T * M, i.e.
there exists a neighborhood V of G η,u in T * M foliated by disjoint invariant Lipschitz Lagrangian graphs, of course that the graphs contained in V are absorbing, so the following result is a local version of a result of D. Massart and A. Sorrentino (See [16], Lemma 5).
Theorem 2 Let G η,u be an invariant Lipschitz Lagrangian graph. If H is locally Lipschitz integrable on G η,u , then there exists a neighborhood U 0 ⊂ H 1 (M; R) of [η] such that β is differentiable at any point of V = c∈U 0 ∂α (c) .
We prove the converse of Theorem 2 in the case M equals torus T 2 (see Theorem 15). In this case, the set V = c∈U 0 ∂α (c) , obtained in the above statement, is open in H 1 (T 2 ; R). Then we geralize ( [16], Theorem 3) to local case.
We also give a particular attention to existence of neighborhood contained in the tiered Mañé, introduced by M-C. Arnaud in [1], and its relation with the local integrability of system and therefore with the local differentiability of β. Indeed, we prove a local version of a result of M-C. Arnaud (See [2], Theorem 1), in the Section 5, Corollary 13.

Preliminaries
Let M be a compact connected manifold and T M its tangent bundle. A Tonelli's Lagrangian is a function L : T M → R of class at least C 2 which is convex and superlinear. Let us recall the main concepts introduced by Mather in [17]. The minimal action value, which also depends only on the cohomology class c = [η], is denoted by −α(c), that is: Mather proved that the function c → α (c) , so-called α of Mather function, is convex and superlinear. It is known that α(c) is the energy level that contains the Mather set for the cohomology class c: where the union is taken over the set of Borel probability measures µ ∈ M (L) called The set M c is a compact invariant set which is a graph over a compact subset M c of M, the projected Mather set (see [17]). M c is laminated by curves, which are global (or time independent) minimizers.
Given a probability measure µ ∈ M (L), its homology or its rotation vector is defined as the unique ρ (µ) ∈ H 1 (M; R) such that for all closed 1-forms ω on M. By convexity, we can consider the dual Fenchel of α, called β of Mather function, as Mather also proved that the β function is convex and superlinear. We say that a the union of supports of probability measures h-minimizing.
In general, the maps α and β are neither strictly convex, nor differentiable. The projection on domain of regions of graph where either map is affine are called flats.
Actually, if the map is strictly convex at a point, the flat is this only point and if the map is not strictly convex, the flat is non-trivial. By duality we have the inequality called Fenchel inequality. Given c ∈ H 1 (M; R) (resp. h ∈ H 1 (M; R)), the homology class h ∈ H 1 (M; R) (resp. c ∈ H 1 (M; R)) achieving equality in the Fenchel inequality is called subderivative of α in c (resp. subderivative of β in h). The set composed by subderivatives of α in c (resp. subderivatives of β in h) is called Legendre transform of c (resp. h), and denoted ∂α (c) (resp. ∂β (h)). Therefore, ∂α (c) is a flat of β and ∂β (h) is a flat of α. By convexity, the sets ∂α (c) and ∂β (h) are non-empty. Many interesting properties of the Euler-Lagrange flow can be derived from the study of the behaviour of the β-function. For instance, if h is an extremal point of β-function, i.e. h is not convex combination of two elements in a same flat of β, then there exist ergodic measures with homology h (see [13]).
Let us recall that we can associate to such a Tonelli's Lagrangian L the Hamil- our assumption, is a diffeomorphism of class at least C 1 , defined in coordinates by Actually, H is the dual Fenchel of L and also is convex and superlinear. Given a cohomology class c and a closed 1-form η c with [η c ] = c, we consider the Hamilton-Jacobi equation A Lipschitz function u : M → R is called a subsolution of Hamilton-Jacobi for the at almost every point. Note that this definition is equivalent to the notion of viscosity subsolutions (see [9]). We denote by C 1,1 the set of differentiable functions with Lipschitz differential. Observe that a C 1,1 function u is solution of (HJ) if and only if the graph of η c + du, denoted by G ηc,u , is invariant under Hamiltonian flow.
We now recall the definition of calibrated curves (see [9]). If u : M → R is a subsolution of Hamilton-Jacobi for L − c, we say that the curve γ : I → M is (u, L − c, α (c))-calibrated if, for the representative η c of the cohomology class c given in (1), we have the equality for all t ′ , t ∈ I. The subset I c (u) of T M is defined by The set I c (u) is invariant and the curves contained in it are called curves c-minimizing.
Using the sets I c (u), one can give (see [10]) the following characterization of the Mañé set and of the Aubry set: where SS c is the set of subsolution of (HJ) for L − c. These invariant sets contain the Mather set and have interesting dynamical properties, for instance A c also is graph whose projection is laminated by global minimizers and it is chain recurrent. The Mañé set N c is connected and chain transitive (see for instance [7]).
Using the duality between Lagrangian and Hamiltonian, via Legendre transform, we define the sets of Mather, Aubry and Mañé in the cotangent bundle, respectively One useful way to produce invariant Lipschitz Lagrangian graphs is to show that π (A * c ) = M. If this is the case, the Theorem 2.5 of [10] says that there exists an unique solution u of (HJ) for the Lagrangian L − c which is C 1,1 and such that A * c is the graph of η c + du, for some η c representative of cohomology class c.

Absorbing sets
If u : M → R is a subsolution of (HJ) for L − c, we denote by I + c (u) the subset of T M defined as where SS c is the set of critical subsolutions for the Lagrangian L − c. We define the N + c .

Definition 3 We say that an invariant set Λ ⊂ T M is an absorbing set if for all
Definition 4 Let G ⊂T * M be an invariant Lipschitz Lagrangian graph. We say that G is an absorbing graph if L −1 (G) is an absorbing set.
Proof: In fact, let (x, v) ∈ N + c and γ : R → M the projection of Euler-Lagrange solution γ (t) = π • ϕ L t (x, v) curve such that γ| [0,+∞) is (u, L − c, α (c))-calibrated for some u ∈ SS c . This means that there exists a closed 1-form η c with [η c ] = c such that We can consider V a C ∞ (M) function such that η c = ξ c + dV. Thus, if s > 0 and (t k ) and (t m ) are two subsequences of (t n ) such that t k − s > t m + s and t k , t m > s, we have The opposite inequality holds because u is a Hamilton-Jacobi subsolution (see [9], Examples of absorbing graphs are the so-called Schwartzman strictly ergodic graphs (see [11]), i.e. invariant graphs Λ which support an invariant measure with full support. In fact, let µ the invariant measure supported in Λ with ρ (µ) = h. If Actually, the same argument can be used to prove that an invariant graph Λ such that all invariant probability measures with support contained in Λ have the same rotation vector h and the union of their supports equals Λ, also it is absorbing.
, then Λ projects onto the whole manifold M.
Proof: Let us consider a closed 1-form η c representative of the cohomology class c. It follows from ( [9], Theorem 4.9.3) that there exists a weak KAM of positive type u + for the Lagrangian L − η c . Then u + is subsolution of (HJ) and given x ∈ M we can find a Therefore (γ x ,γ x ) ∈ I + c (u + ) ⊂ N + c and, by Lemma 5, we have that the ω-limit set of (γ x ,γ x ) is contained in Aubry set A c . Since Λ is an absorbing set that contains A c , we obtain (x,γ x (0)) ∈ Λ.
where η c is a representative of the cohomology class c. This show that c ∈ ∂β (h) .
Conversely, let µ minimizing measure with rotation vector h. If c ∈ ∂β (h) , then This proves that µ is c-minimizing.

Proof of Theorems 1 and 2
In order to prove Theorems 1 and 2 we begin by proving the following lemma: . By Lemma 5, the ω-limit set of points in Mañé set is contained in the Aubry set. Thus

This shows that the intersection
is non-empty. Then, by a result of D. Massart (see [14], Proposition 6), α has a flat F containing [ω] and [η]. Let c be a cohomology class belonging to the relative interior of F. It follows from ( [14], . Let us consider a closed 1-form η c , with [η c ] = c. It follows from ( [9], Theorem 4.9.3) that there exists a weak KAM of positive type u + for the Lagrangian L−η c . Then u + is subsolution of (HJ) and given x ∈ M, we can find a C 1 curve γ Therefore (γ x ,γ x ) ∈ N + c and, by Lemma 5, we have that the ω-limit set of (γ x ,γ x ) is contained in Aubry set A c . Recall that by invariance of G η,u we have G η,u = L I η (u) and the Aubry set Moreover, it follows from G η,u being an absorbing graph, that the Hamiltonian orbit L (γ x ,γ x ) is entirely contained in G η,u . As a consequence, given T > 0, we have

Now let us consider another weak KAM
Similarly, we conclude that Moreover, the points L (γ x (0) ,γ x (0)) and L δ x (0) ,δ x (0) belong to the graph Since this equality holds for all x ∈ M and M is connected, we have that u + differ of v + by a constant. By arbitrariness of the two weak KAM, we conclude that any two weak KAM for the Lagrangian L − η c differ by a constant. It follows from ([10], Proposition On the other hand, since u + is differentiable everywhere in M, by ( [9], Lemma 4.13.1), we have This means that the graph G ηc,u + is invariant, so G ηc,u + ⊂ N * c = A * c . By the graph property, we have A * c = N * c = G ηc,u + . As a consequence of , we obtain We also have that the projected Aubry set A [ω] is the whole manifold M, so there exists We can now prove the main result stated in Introduction.
Proof: (of Theorem 1) Suppose that G η,u is an absorbing graph. Thus This implies that A * c ∩ A * [η] = ∅ and, by ( [14], Proposition 6), there exists a flat F of α such that c and [η] belong to F. As a consequence, there exists h ∈ H 1 (M; R) Dividing by t, and letting t → 0 + , we get Now we can to prove the Theorem 2: Proof: (of Theorem 2) Let V be a neighborhood of G η,u foliated by invariant Lipschitz Lagrangian graphs. Note that an invariant Lipschitz Lagrangian graph G contained in V is absorbing. Since G η,u ⊂ V, by Theorem 1, We . We study the relation of this hypothesis with the existence of absorbing graphs contained in the neighborhood V.
In particular, (γ n ,γ n ) is a periodic orbit contained in some Mañé set N [λn] for some closed 1-form λ n . Therefore (γ n ,γ n ) supports a measure µ n which is [λ n ]-minimizing. Now we will show that A [η] ⊂ A c . Indeed, let us consider y ∈ A [η] and q ∈ T * y M such that (y, q) ∈ A * Theorem, we know that for each T n , there exists a Tonelli minimizing curve Γ n : [0, T n ] → M with Γ n (0) = Γ n (T n ) = y for the Lagrangian L − η which is homologous to γ n . Since T n > T 0 (η) , we have d L y,Γ n (0) , (y, q) < ǫ which implies L y,Γ n (0) ∈ B ǫ (y, q) ⊂ V.
As a consequence, Γ n ,Γ n is a periodic orbit which supports a measure ν n which is λ n -minimizing for some closed 1-form λ n . However, we have Thus, Or, Therefore, This shows that y ∈ A c and we obtain A such that N * c ⊂ V for all c ∈ U 0 . In particular, for all c ∈ U 0 we have A * c ⊂ V and, by the Lemma 10, we conclude that A c = A [η] . Moreover, since G η,u = A * [η] , we have A c = M. Therefore, there exists a closed 1-form η c with [η c ] = c and a function u c ∈ C 1,1 such that A * c = G ηc,uc . It remains to show that G ηc,uc is absorbing. In fact, if (x, p) ∈ L N T + (L) and ω (x, p) ⊂ G ηc,uc = A * c , then the curve γ (t) = π • ϕ H t (x, p) , the projection of the Hamiltonian flow ϕ H t , intersects V for some time τ > 0. In particular ϕ H τ (x, p) belongs to some Mañé set N * c . This implies that ω (x, p) ⊂ A * c , so A * c ∩ A * c = ∅. The Lemma 10 implies that A c = M and, as a consequence, there exist v ∈ C 1,1 and η c a representative of the cohomology class c such that A * c = G η c ,v . By Proposition 6 of [14], c and c belong to the same flat F of α. Moreover, if It follows from Lemma 10 that A c 0 = M and, as a consequence, there exist a closed 1-form η c 0 with [η c 0 ] = c 0 and a function u 0 ∈ C 1,1 such that A * c 0 = G ηc 0 ,u 0 . This shows that and, by the graph property, N * c = G η c ,v = G ηc,u . This shows that ϕ H t (x, p) belongs to G ηc,u for all t ∈ R. In particular, (x, p) belongs to G ηc,u and we conclude that G ηc,u is an absorbing graph.
The conclusion that β is differentiable at any point of V = c∈U 0 ∂α (c) follows from Theorem 1. Proof: Let us define the map where the closed 1-form η c is a representative of c and u c ∈ C 1,1 such that G ηc,uc is an invariant absorbing graph. In particular, by Theorem 1, we have A * c = G ηc,uc . This map is injective because the absorbing graphs are disjoint. Moreover, F is also continuous.
In fact, let (x n , c n ) → (x 0 , c) and consider its associated graphs A * cn = G * ηc n ,uc n . Let us consider the sequence of Lipschtz functions λ n = η cn + du cn . Note that, by the graph property, we have A * cn = N * cn . Moreover, by upper semicontinuity of the Mañé set (see [1], Proposition 13), we conclude that for n sufficiently large, the sequence (x n , λ n (x n )) ∈ N * cn remains in a compact set. We can conclude that -up to selecting a subsequence -(x n , λ n (x n )) converges to some point that belongs to N * c . Moreover, since N * cn = A * cn and x n → x 0 , by the graph property, we conclude that The result follows from Invariance Domain Theorem for manifolds. In fact, and F is continuous and injective. Therefore that is, the Hamiltonian is locally Lipschitz integrable on G η,u .
As an immediate consequence of Corollary 11 and Theorem 12, we present a local version of Arnaud's Theorem ( [2], Theorem 1).
and V is contained in the dual tiered Mañé N T * (L) . Then the Hamiltonian is locally Lipschitz integrable on G η,u .
6 The Case M = T 2 We say that a homology h ∈ H 1 (T 2 ; R) is rational if there exists λ > 0 such that λh ∈ i * H 1 (T 2 ; Z) , where i * : H 1 (T 2 ; Z) → H 1 (T 2 ; R) is the natural map. Since the manifold treated in this section is the torus T 2 , we can say that a homology h ∈ H 1 (T 2 ; R) is irrational if it is not rational. For general manifolds, there is the concept of k-irrationality (see for instance [14]).
Note that if two cohomology classes lie in the relative interior of a flat of α, by Proof: Let c ∈ U 0 be a cohomology class and let h be a rational homology class belonging to ∂α (c) . Suppose by contradiction that ∂α (c) = {h} . Recall that, in case M = T 2 , all flats of β flats are radial (see [5]) . Therefore, exchanging, if necessary, h and a nonzero extremal point of ∂α (c) , which exists because ∂α (c) has two extremal points which also are rational, we can assume for some λ < 1. Let us consider a sequence t n > 1 such that t n → 1. We will show that, for n sufficiently large, β is differentiable at t n h. In fact, there exists a sequence c n ∈ H 1 (T 2 ; R) such that c n ∈ ∂β (t n h) .Thus The sequence c n has subsequence bounded. Otherwise, let us consider a subsequence of cn cn convergent, i.e.
where . denotes a norm on H 1 (T 2 ; R) (see for instance [9], Section 4.10). So we have c n k → ∞ and which contradicts the superlinearity of α. Then we can assume, extracting a subsequence if necessary, c n → c. Thence This shows that c ∈ ∂β (h) . Since β is differentiable at h, we conclude that c = c and c n ∈ U 0 for n sufficiently large. Therefore β is differentiable at t n h.
Observe that t n h is non-singular for all n ∈ N. Otherwise, there exists a fixed point, which comprise the support of a minimizing measure µ 0 , in M (∂β (t n h)) = M cn .
Since ρ (µ 0 ) = 0, the set [0, t n h] is contained in a flat of β and the maximal flat [λh, h] may be extended because t n > 1.
Therefore we can apply ( [16], Corollary 1) to deduce that T 2 is foliated by periodic orbits and M cn = A cn = T 2 . By ( [8], Proposition 2.1) we can consider a point (x, v) ∈ T T 2 such that the orbit ϕ t (x, v) is periodic with period T and which comprise the support of a minimizing measure with rotation vector λh. If (x, v) is a fixed point, take T = +∞. Since x ∈ A cn , there exists a only v n such that (x, v n ) ∈ A cn is a periodic point with period T n . By semicontinuity of the Aubry set, (x, v n ) converges to some point (x, w) of A c . By the graph property, w = v.
We now prove that for all δ > 0, there exists n 0 ∈ N such that T n 0 ≥ T − δ. In fact, if some δ > 0 is such that 0 < T n < T − δ for all n ∈ N, extracting a subsequence if necessary, T n → S < T and ϕ Tn (x, v n ) → ϕ S (x, v) = (x, v) which contradicts the minimality of period of the orbit of (x, v). Let us consider h 0 ∈ H 1 (T 2 ; Z) such that the probability measure carried by the orbit ϕ t (x, v) have homology 1 T h 0 = λh and the probability measure carried by the orbit ϕ t (x, v n ) have homology 1 Tn h 0 = t n h. Given δ = T (1 − λ) , there exists n 0 ∈ N such that T n 0 ≥ T − T (1 − λ) = T λ. Then, since h 0 = T λh and h 0 = T n 0 t n 0 h, we have T λ Tn 0 = t n 0 ≤ 1 which is a contradiction. Now suppose that h be an irrational homology class belongs to ∂α (c) . Therefore any probability measure h-minimizing is supported on a lamination of the torus without closed leaves. Moreover, this measure is uniquely ergodic. In particular, h is not contained in any non-trivial flat of β.
As a consequence of the above Proposition, we present a local version of ( [14],

Theorem 3).
Theorem 15 Suppose that M = T 2 and let [η] be a cohomology class. Then the following statements are equivalent: (1) There exists u ∈ C 1,1 (M) and a neighborhood V of G η,u in T * M such that A * [η] = G η,u and V ⊂ N T (L) .   [14], Lemma 7) is dense in H 1 (T 2 ; R). Now note that zero is the only (possibly) singular class belongs to ∂α (c) such that c ∈ U 0 , because if a non-zero class h is singular belongs to ∂α (c) such that c ∈ U 0 , then there is a fixed point in the Mather set of c. Thus ∂α (c) contains the homology of the Dirac measure on the fixed point. This contradicts the Proposition 14, which says that α is It follows from ( [16], Corollary 1) that A c h = T 2 is foliated by periodic orbits. By semicontinuity of Aubry set, we have that A c = T 2 for all c ∈ U 0 and that there exist η c a representative of class c and u c ∈ C 1,1 (T 2 ) such that A * c = G ηc,uc . In particular, A * [η] = G η,u for some u ∈ C 1,1 . Moreover, these graphs are absorbing. Indeed, this follows from assumption that β is differentiable at any point V = c∈U 0 ∂α (c) and from converse of Theorem 1. Therefore, since dim H 1 (T 2 ; R) = dim R 2 , by the Theorem 12, we obtain a neighborhood of G η,u foliated by invariant Lipschitz Lagrangian graphs.
(3) ⇒ (1) All point of V belongs to some invariant Lipschitz Lagrangian graphs. In particular, belongs to some Mañé set, so V ⊂ N T (L) . Since graphs of a foliation are absorbing, by Theorem 1 we have A * [η] = G η,u .

An Example: vertical exact magnetic Lagrangian
In this section we present a Lagrangian on the two torus T 2 such that the β- The Euler-Lagrange flow associated with this Lagrangian is generated by the vector field: where J is the 2 × 2 canonical sympletic matrix. Since the energy function for L is E (x, y, v) = 1 2 v 2 , for each energy level E > 0, we can consider the angle ϕ (with horizontal line) of trajectories of the Euler-Lagrange flow. This means that ϕ is the It is easy to see that H (x, ϕ) = cos (2πx) + √ 2E sin ϕ is a first integral. The critical points of H are 0, π 2 -maximum, 0, − π 2 -saddle, 1  the two graphs of ϕ 1 and ϕ 2 with ϕ 2 = π − ϕ 1 , given implicitly by equation are invariant for E > 1 2 and |F | < √ 2E − 1.
The figure 1 describes the projection these graphs in the section xϕ, in the energy level E > 1 2 . Let us consider the closed 1-forms dependent of E and F. Since L (x, v) = (x, (v 1 , v 2 + cos (2πx)) , · ) , we have G η i ,0 = L x, √ 2E cos ϕ i , √ 2E sin ϕ i . These graphs compose a local foliation in T * T 2 , then are absorbing graphs. It follows from Theorem 1 that β is differentiable at any point for all i = 1, 2, E > 1 2 and F with |F | < √ 2E − 1. Moreover, by Proposition 14, we conclude that α function is differentiable at [η i ] .