A Generic Property of Exact Magnetic Lagrangians

We prove that for the set of Exact Magnetic Lagrangians the property"There exist finitely many static classes for every cohomology class"is generic. We also prove some dynamical consequences of this property.


Introduction
Let M be a closed manifold equipped with an Riemannian metric g = ., . . A Lagrangian L : T M → R is called Exact Magnetic Lagrangian if for some non-closed 1-form η.
This type of Lagrangian fits into Mather's theory, as developed by R. Mañé and A. Fathi, about Tonelli Lagrangians, namely, it is fiberwise convex and superlinear.
We refer the reader to the references Fathi in [6], Contreras and Iturriaga in [4] for expositions of this theory. M c is a compact invariant set which is a graph over a compact subset M c of M, the projected Mather set (see [11]). M c is laminated by curves, which are global (or time independent) minimizers. Mather also proved that the function c → α(c) is convex and superlinear.
In general, M c is contained in another compact invariant set, which also a graph whose projection is laminated by global minimizers: the Aubry set for the cohomology class c, denoted by A c . Mañé proved that A c is chain recurrent and it is a challenging question to describe the dynamics of the Euler-Lagrange flow restricted to A c . The definition of Aubry set and some its properties are given in Section 3.
Of course this question only makes sense if it is posed for generic Lagrangians, since many pathological examples can be constructed. The notion of genericity in the context of Lagrangian systems is provided by Mañé in [9]. The idea is to make special perturbations by adding a potential: L(x, v) + Ψ(x), for Ψ ∈ C ∞ (M).
A property is generic in the sense of Mañé if it is valid for all Lagrangians L(x, v)+ Φ(x) with Φ contained in a residual subset O.
In this setting, G. Contreras and P. Bernard proved in the work A Generic Property of Families of Lagrangian Systems (see [1]) that generically, in the sense of Mañé, for all cohomology class c there is only a finite number of minimizing measures. This theorem is a consequence of an abstract result which is useful in different situations.
In general, when we are dealing with an specific class of Lagrangians, perturbations by adding a potential are not allowed. However, due to the abstract nature of Bernard-Contreras proof it may be addapted to the specific case like the one treated here.
The objective of this paper is to prove the genericity of finitely many minimizing measures for Exact Magnetic Lagrangians and apply it to the dynamics of the Aubry set.
Let us consider Γ 1 (M) the set of smooth 1-forms in M endowed with the metric denoting by ω k the C k -norm of the 1-form ω. With this metric Γ 1 (M) is a Frechet space, it means that Γ 1 (M) is a locally convex topological vector space whose topology is defined by a translation-invariant metric, and that Γ 1 (M) is complete for this metric.
The main result of this paper is the following: Theorem 1 Let A be a finite dimensional convex family of Exact Magnetic Lagrangians.
Then there exists a residual subset O of Γ 1 (M) such that, Hence there exist at most 1 + dim A ergodic minimizing measures of L + ω. The last part of this work is dedicated to prove some consequences about the dynamics. For instance, using the work of Contreras and Paternain, [5] we obtain connecting orbits between the elements of the Aubry set that contain the support of minimizing measures (the so called "static classes").
2 Adapting the abstract setting of Bernard and

Contreras
As it was pointed out previously, the proof of Theorem 1 is an application of the work of Contreras and Bernard. Here we state their result.
Assume that we are given (i) Three topological vector spaces E, F, G.
(iv) Two metrizable convex compact subsets H ⊂ F and K ⊂ G such that π (H) ⊂ K.
Suppose that 1. The restriction of the map given by (iii), , | E×K is continuous.
2. The compact K is separated by E. This means that, if µ and ν are two different points of K, then there exists a point ω in E such that ω, µ − ν = 0.
3. E is a Frechet space.
Note then that E has the Baire property, that is any residual subset of E is dense.
We shall denote by H * the set of affine and continuous functions defined on H. GivenL In order to apply this theorem, we need to define the above objects in an adequate setting as follows: Let C be the set of continuous functions f : T M → R with linear growth, that is endowed with the norm . ℓin .
• F = C * is the vector space of continuous linear functionals µ : C → R provided with the weak-⋆ topology: • G is the vector space of continuous linear functionals µ : is the space of continuous 1-forms on M. Note that the Riemannian metric g = ., .
allows us to represent any continuous 1-form as X, . , for some C 0 vector field X. We endow G with the weak-⋆ topology: • The continuous linear π : F → G is given by • For a given natural number N, let Then, by following classical theorem of Analysis, it is enough show that Γ 0 (M) is a separable vector space.
Theorem 4 Let E a Banach's space. Then E is separable if, and only if, the unit ball B E * ⊂ E * in the weak-⋆ topology is metrizable.
The separability of Γ 0 (M) follows from the lemma below and of the duality between 1-forms and vector fields provided by the Riemannian metric.

Lemma 5
The space X 0 (M) of continuous vector fields in a compact manifold M is separable.
Proof: By compactness of M, we can consider a number finite local trivializationŝ Let us consider (X n ) a dense sequence in X 0 (M) and ω n = X n , · ∈ Γ 0 (M) . Let ω = X, · ∈ Γ 0 (M) and U ε be a ball in Γ 0 (M) centered at ω., of radius ε > 0. Then This shows that ω n ∈ U ε and Γ 0 (M) is separable, so K N is metrizable. This finishes the proof of the Claim 1.
Observe that K N is compact and convex since K N = π (M 1 N ) , π is a continous map and M 1 N is a compact subset of probability measures in T M. • The bilinear mapping , : E × G → R is given by integration: Note that here we apply the Hahn-Banach Theorem for extends the functional µ and that the above integral does not depend on the extension of µ to a signed measure on and d (ω n , ω) → 0 implies that given ǫ > 0, there exists n 0 ∈ N such that ∀n ≥ n 0 , ω n − ω ℓin < ǫ (N + 1) .
When n → ∞, property A is a graph on π (A) and we can write where π −1 is Lipschitz on the projected Mather set. Let and consider f n : M → [0, 1] sequence of smooth bump functions where B n (π (A)) is a neighborhood of the compact π (A) : Let us consider X a continuous extension of X| π(A) on M. Then the vector field X n = f n X ∈ X 0 (M) , converges pointwise to X (x) and
Then we have Then by previous Lemma, dim M (L + ω) ≤ dim A for all L ∈ A and all ω ∈ O (A) .
This finishes the proof.

Some Dynamical Consequences
As it was pointed out in the Introduction, the Mather set M c associated to a cohomology class c is contained in another compact invariant set called the Aubry set A c . It is also a graph over a compact subset of the manifold M and it is contained in the same energy level α(c) as M c . Moreover, A c is chain recurrent set. All these properties are proven in [4], see also [6].
In order to state the dynamical consequences of our Theorem 1, we need to introduce the Aubry set and the concept of static classes for a general Tonelli Lagrangian.
Let us consider the action on a curve γ : [0, T ] → M defined by where k is a real number and η is a representative of the class c. The energy level α(c), Recall that, for a given real number k the action potential Φ L−c+k : M × M → R is defined by infimum taken over the curves γ joining x the y.

Mañé proved that
where η is a representative of the class c and that α(c) is the smallest number such that the action potential is finite, in other words, if k < α(c), then Φ L−c+k (x, y) = −∞ and for k ≥ α(c), Φ L−c+k (x, y) ∈ R.
Observe that by Tonelli's Therorem (See for example in [4]), for fixed t > 0, there always exists a minimizing extremal curve connecting x to y in time t. The potential calculates the global (or time independent) infimum of the action. This value may not be realized by a curve.
The potential Φ L−c+α(c) is not symmetric in general but is a pseudo-metric. A curve γ : R → M is called semistatic if minimizes action between any of its points: and γ is called static if is semistatic and δ M (γ (a) , γ (b)) = 0 for all a, b ∈ R.
For example, the orbits contained in the Mather set M c project onto static curves.
The Aubry set A c is the set of the points (x, v) ∈ T M such that the projection Proof: It suffices to show that each static class supports at least one ergodic minimizing measure. In fact, let Λ be a static class for L + ω − c and (p, v) ∈ A c with p ∈ Λ. For T > 0 we define a Borel probability measure µ T on T M by All these probability measures have their supports contained in A c that is a compact subset, consequently, we can extract a sequence µ Tn weakly convergent to µ: which is a ergodic minimizing measure whose support is contained in Λ (See [6] for details). Contreras and Paternain prove in [5] that between two static classes there exists a chain of static classes connected by heteroclinic semistatic orbits. More precisely they show Theorem 9 Suppose that the number of static classes is finite. Then given two static classes Λ k and Λ l , there exist classes Λ 1 = Λ k , Λ 2 , ..., Λ n = Λ l and θ 1 , θ 2 , ..., θ n−1 ∈ T M such that for all i = 1, ..., n − 1 we have that γ i (t) = π • ϕ t (θ i ) are semistatic curves, Another important property, demonstrated by P. Bernard in [2], is the semicontinuity of the Aubry set when A M is finite. In order to be more precise he showed the following Theorem Theorem 10 Let L k be a sequence of Tonelli Lagrangians converging to L. Then given a neighborhood U of A 0 in T M, there exists k 0 such that A 0 (L k ) ⊂ U for each k ≥ k 0 , where A 0 (L k ) is the Aubry set for the Lagrangian L k .
In fact Bernard showed that this Theorem is true with a weaker hypothesis than A M be finite, namely coincidence hypothesis (See [2]).

Example
In this section we present an example of a Exact Magnetic Lagrangian on flat torus T 2 whose quotient Aubry set A M is a Cantor set, therefore not every Exact Magnetic Lagrangian has finitely many static classes.
Let L : T T 2 → R be a Exact Magnetic Lagrangian defined by where f is a C 2 nonpositive and periodic function whose set of minimum points Γ min is a Cantor set and f | Γ min is a negative constant.
In this case the system of Euler-Lagrange is given by where J is the canonical sympletic matrix.
Moreover, the closed curves γ a defined by γ a (t) = (a, −f (a) t) , are static curves.
Proof: Given any curve β (t) = (x (t) , y (t)) on T 2 , we have Then and we obtain α(0) ≤ f (a) 2 2 . Observe that if 0 < k ≤ f (a) 2 2 , the closed curve given by γ k (t) = a, √ 2kt , where a ∈ Γ min , is Euler-Lagrange solution and its energy is This shows that α(0) = f (a) 2 2 and the curve γ a is semistatic, i.e., realizes the action potential. Since γ a is a semistatic closed curve, it is static curve.
To complete the example, it suffices to show that the application Ψ : Γ min → A M , given by Ψ (a) = [(a, 0)] , where [(a, 0)] is a representative of the static class conteining the curve γ a , is a Lipschitz bijection. In fact, since the action potential Φ L+α(0) is Lipschitz, the distance δ M on quotient Aubry set A M also is Lipschitz.
In order to show the surjectivity of Ψ it is enough to show that the projected Aubry set A 0 is exactly the union of the closed curves γ a with a ∈ Γ min . Suppose that there exists p ∈ A 0 such that π (p) / ∈ Γ min , where π is the canonical projection of T 2 on R/Z. Then there exists a neighborhood V p of p such that f (x) > f (a) for all a ∈ Γ min and x ∈ V p . Let γ be a piece, contained in V p , of the static curve passing through p.
This is a contradiction.
By total disconnectedness of Γ min , there exists q ∈ π ([(a, 0)]) − Γ min . The contradiction follows of the inequality 3 by same argument above.