A GENERIC PROPERTY OF FAMILIES OF LAGRANGIAN SYSTEMS

PATRICK BERNARD AND GONZALO CONTRERAS

Abstract. We prove that a generic lagrangian has finitely many minimizing measures for every cohomology class.

1. Introduction

Let M be a compact boundaryless smooth manifold.

Let 𝕋 be either the group (ℝ/ℤ, +) or the trivial group ({0}, +).

A Tonelli Lagrangian is a C^{2} function L : 𝕋 × T M → ℝ such that

• The restriction to each fiber of 𝕋 × T M → 𝕋 × M is a convex function.

• It is fiberwise superlinear:

lim_{|θ|→+∞} L(t, θ)/|θ| = +∞, (t, θ) ∈ 𝕋 × T M.

• The Euler-Lagrange equation

\frac{d}{dt}L_{v} = L_{x}

defines a complete flow φ : ℝ × (𝕋 × T M) ⟶ 𝕋 × T M.

We say that a Tonelli Lagrangian L is strong Tonelli if L + u is a Tonelli Lagrangian for each u ∈ C^{∞}(𝕋 × M, ℝ). When 𝕋 = {0} we say that the lagrangian is autonomous.

Let P(L) be the set of Borel probability measures on 𝕋 × T M which are invariant under the Euler-Lagrange flow φ. The action functional A_{L} : P(L) → ℝ ∪ {+∞} is defined as

A_{L}(μ) := ⟨L, μ⟩ := ∫_{𝕋×T M} L dμ.

The functional A_{L} is lower semi-continuous and the minimizers of A_{L} on P(L) are called minimizing measures. The ergodic components of a minimizing measure are also minimizing, and they are mutually singular, so that the set 𝔐(L) of minimizing measures is a simplex whose extremal points are the ergodic minimizing measures.

In general, the simplex 𝔐(L) may be of infinite dimension. The goal of the present paper is to prove that this is a very exceptional phenomenon. The first results in that direction were obtained by Mañé in [4]. His paper has been very influential to our work.

We say that a property is generic in the sense of Mañé if, for each strong Tonelli Lagrangian L, there exists a residual subset O ⊂ C^{∞}(𝕋 × M, ℝ) such that the property holds for all the Lagrangians L − u, u ∈ O. A set is called residual if it is a countable intersection of open and dense sets. We recall which topology is used on C^{∞}(𝕋 × M, ℝ). Denoting by ∥u∥_{k} the C^{k}-norm of a function u : 𝕋 × M ⟶ ℝ, define

Gonzalo Contreras was partially supported by CONACYT-México grant # 46467-F.

∥u∥_{∞} := ∑_{k∈ℕ} \frac{arctan(∥u∥_{k})}{2^{k}}.

Note that ∥.∥_{∞} is not a norm. Endow the space C^{∞}(𝕋 × M, ℝ) with the translationinvariant metric ∥u − v∥_{∞}. This metric is complete, hence the Baire property holds: any residual subset of C^{∞}(𝕋 × M, ℝ) is dense.

Theorem 1. Let A be a finite dimensional convex family of strong Tonelli Lagrangians. Then there exists a residual subset O of C^{∞}(𝕋 × M, ℝ) such that,

u ∈ O, L ∈ A ⟹ dim 𝔐(L − u) ⩽ dim A.

In other words, there exist at most 1 + dim A ergodic minimizing measures of L − u.

The main result of Mañé in [4] is that having a unique minimizing measure is a generic property. This corresponds to the case where A is a point in our statement. Our generalization of Mañé’s result is motivated by the following construction due to John Mather:

We can view a 1-form on M as a function on T M which is linear on the fibers. If λ is closed, the Euler-Lagrange equation of the Lagrangian L − λ is the same as that of L. However, the minimizing measures of L − λ, are not the same as the minimizing measures of L. Mather proves in [5] that the set 𝔐(L − λ) of minimizing measures of the lagrangian L − λ depends only on the cohomology class c of λ. If c ∈ H^{1}(M, ℝ) we write 𝔐(L − c) := 𝔐(L − λ), where λ is a closed form of cohomology c.

It turns out that important applications of Mather theory, such as the existence of orbits wandering in phase space, require understanding not only of the set 𝔐(L) of minimizing measures for a fixed or generic cohomology classes but of the set of all Mather minimizing measures for every c ∈ H^{1}(M, L). The following corollaries are crucial for these applications.

Corollary 2. The following property is generic in the sense of Mañé:

For all c ∈ H^{1}(M, ℝ), there are at most 1 + dim H^{1}(M, ℝ) ergodic minimizing measures of L − c.

We say that a property is of infinite codimension if, for each finite dimensional convex family A of strong Tonelli Lagrangians, there exists a residual subset O in C^{∞}(𝕋 × M, ℝ) such that none of the Lagrangians L − u, L ∈ A, u ∈ O satisfy the property.

Corollary 3. The following property is of infinite codimension:

There exists c ∈ H^{1}(M, ℝ), such that L − c has infinitely many ergodic minimizing measures.

Another important issue concerning variational methods for Arnold diffusion questions is the total disconnectedness of the quotient Aubry set. John Mather proves in [7, § 3] that the quotient Aubry set \overline{A} of any Tonelli lagrangian on 𝕋 × T M with 𝕋 = ℝ/ℤ and dim M ⩽ 2 (or with 𝕋 = {0} and dim M ⩽ 3) is totally disconnected. See [7] for its definition.

The elements of the quotient Aubry set are called static classes. They are disjoint subsets of 𝕋 × T M and each static class supports at least one ergodic minimizing measure. We then get

Corollary 4. The following property is generic in the sense of Mañé:

For all c ∈ H^{1}(M, ℝ) the quotient Aubry set \overline{A}_{c} of L − c has at most 1 + dim H^{1}(M, ℝ) elements.

2. Abstract Results

Assume that we are given

• Three topological vector spaces E, F, G.

• A continuous linear map π : F → G.

• A bilinear pairing ⟨u, ν⟩ : E × G → ℝ.

• Two metrizable convex compact subsets H ⊂ F and K ⊂ G such that π(H) ⊂ K.

Suppose that

(i) The map

E × K ∋ (u, ν) ⟼ ⟨u, ν⟩

is continuous.

We will also denote ⟨u, π(μ)⟩ by ⟨u, μ⟩ when μ ∈ H. Observe that each element u ∈ E gives rise to a linear functional on F

F ∋ μ ⟼ ⟨u, μ⟩

which is continuous on H. We shall denote by H^{∗} the set of affine and continuous functions on H and use the same symbol u for an element of E and for the element u ⟼ ⟨u, μ⟩ of H^{∗} which is associated to it.

(ii) The compact K is separated by E. This means that, if η and ν are two different points of K, then there exists a point u in E such that ⟨u, η − ν⟩ ≠ 0.

Note that the topology on K is then the weak topology associated to E. A sequence η_{n} of elements of K converges to η if and only if we have ⟨u, η_{n}⟩ ⟶ ⟨u, η⟩ for each u ∈ E. We shall, for notational conveniences, fix once and for all a metric d on K.

(iii) E is a Frechet space. It means that E is a topological vector space whose topology is defined by a translation-invariant metric, and that E is complete for this metric.

Note then that E has the Baire property. We say that a subset is residual if it is a countable intersection of open and dense sets. The Baire property says that any residual subset of E is dense.

Given L ∈ H^{∗} denote by

M_{H}(L) := arg min_{H} L

the set of points μ ∈ H which minimize L|_{H}, and by M_{K}(L) the image π(M_{H}(L)). These are compact convex subsets of H and K.

Our main abstract result is:

Theorem 5. For every finite dimensional affine subspace A of H^{∗}, there exists a residual subset O(A) ⊂ E such that, for all u ∈ O(A) and all L ∈ A, we have

(1)  dim M_{K}(L − u) ≤ dim A.

Proof: We define the ε-neighborhood V_{ε} of a subset V of K as the union of all the open balls in K which have radius ε and are centered in V. Given a subset D ⊂ A, a positive number ε, and a positive integer k, denote by O(D, ε, k) ⊂ E the set of points u ∈ E such that, for each L ∈ D, the convex set M_{K}(L − u) is contained in the ε-neighborhood of some k-dimensional convex subset of K.

We shall prove that the theorem holds with

O(A) = ⋂_{ε>0} O(A, ε, dim A).

If u belongs to O(A), then (1) holds for every L ∈ A. Otherwise, for some L ∈ A, the convex set M_{K}(L − u) would contain a ball of dimension dim A + 1, and, if ε is small enough, such a ball is not contained in the ε-neighborhood of any convex set of dimension dim A.

So we have to prove that O(A) is residual. In view of the Baire property, it is enough to check that, for any compact subset D ⊂ A and any positive ε, the set O(D, ε, dim A) is open and dense. We shall prove in 2.1 that it is open, and in 2.2 that it is dense.

□

2.1. Open.

We prove that, for any k ∈ ℤ^{+}, ε > 0 and any compact D ⊂ A, the set O(D, ε, k) ⊂ E is open. We need a Lemma.

Lemma 6. The set-valued map (L, u) ⟼ M_{H}(L − u) is upper semi-continuous on A × E. This means that for any open subset U of H, the set

{(L, u) ∈ A × E : M_{H}(L − u) ⊂ U} ⊂ A × E

is open in A × E. Consequently, the set-valued map (L, u) ⟼ M_{K}(L − u) is also upper semi-continuous.

Proof: This is a standard consequence of the continuity of the map

A × E × H ∋ (L, u, μ) ⟼ (L − u)(μ) = L(μ) − ⟨u, μ⟩.
□

Now let u_{0} be a point of O(D, ε, k). For each L ∈ D, there exists a k-dimensional convex set V ⊂ K such that M_{K}(L − u_{0}) ⊂ V_{ε}. In other words, the open sets of the form

{(L, u) ∈ D × E : M_{H}(L − u) ⊂ V_{ε}} ⊂ D × E,

where V is some k-dimensional convex subset of K, cover the compact set D × {u_{0}}. So there exists a finite subcovering of D × {u_{0}} by open sets of the form Ω_{i} × U_{i}, where Ω_{i} is an open set in A and U_{i} ⊂ O(Ω_{i}, ε, k) is an open set in E containing u_{0}. We conclude that the open set ∩U_{i} is contained in O(D, ε, k), and contains u_{0}. This ends the proof.

□

2.2. Dense.

We prove the density of O(A, ε, dim A) in E for ε > 0. Let w be a point in E. We want to prove that w is in the closure of O(A, ε, dim A).

Lemma 7. There exists an integer m and a continuous map

T_{m} = (w_{1}, …, w_{m}) : K ⟶ ℝ^{m},

with w_{i} ∈ E such that

(2)  ∀ x ∈ ℝ^{m}  diam T_{m}^{−1}(x) < ε,

where the diameter is taken for the distance d on K.

Proof: In K × K, to each element w ∈ E we associate the open set

U_{w} = { (η, μ) ∈ K × K : ⟨w, η − μ⟩ ≠ 0 }.

Since E separates K, the open sets U_{w}, w ∈ E cover the complement of the diagonal in K ×K. Since this complement is open in the separable metrizable set K ×K, we can extract a countable subcovering from this covering. So we have a sequence U_{w_{k}}, with w_{k} ∈ E, which covers the complement of the diagonal in K × K. This amounts to say that the sequence w_{k} separates K. Defining T_{m} = (w_{1}, …, w_{m}), we have to prove that (2) holds for m large enough. Otherwise, we would have two sequences η_{m} and μ_{m} in K such that

T_{m}(μ_{m}) = T_{m}(η_{m}) and d(μ_{m}, η_{m}) ⩾ ε.

By extracting a subsequence, we can assume that the sequences μ_{m} and η_{m} have different limits μ and η, which satisfy d(η, μ) ⩾ ε. Take m large enough, so that T_{m}(η) ≠ T_{m}(μ). Such a value of m exists because the linear forms w_{k} separate K. We have that

T_{m}(μ_{k}) = T_{m}(η_{k}) for k ≥ m.

Hence at the limit T_{m}(η) = T_{m}(μ). This is a contradiction.

□

Define the function F_{m} : A × ℝ^{m} → ℝ ∪ {+∞} as

F_{m}(L, x) := min_{{μ∈H \atop T_{m}∘π(μ)=x}}(L − w)(μ),

when x ∈ T_{m}(π(H)) and F_{m}(L, x) = +∞ if x ∈ ℝ^{m}\T_{m}(π(H)). For y = (y_{1}, …, y_{m}) ∈ ℝ^{m}, let

M_{m}(L, y) := arg min_{x∈ℝ^{m}}[F_{m}(L, x) − y · x] ⊂ ℝ^{m}

be the set of points which minimize the function x ⟼ F_{m}(L, x) − y · x. We have that

M_{K}(L − w − y_{1}w_{1} − ⋯ − y_{m}w_{m}) ⊂ T_{m}^{−1}(M_{m}(L, y)).

Let

O_{m}(A, dim A) := { y ∈ ℝ^{m} | ∀ L ∈ A : dim M_{m}(L, y) ⩽ dim A }.

From Lemma 7 it follows that

y ∈ O_{m}(A, dim A) ⟹ w + y_{1}w_{1} + ⋯ + y_{m}w_{m} ∈ O(A, ε, dim A).

Therefore, in order to prove that w is in the closure of O(A, ε, dim A), it is enough to prove that 0 is in the closure of O_{m}(A, dim A), which follows from the next proposition.

Proposition 8. The set O_{m}(A, dim A) is dense in ℝ^{m}.

Proof: Consider the Legendre transform of F_{m} with respect to the second variable,

G_{m}(L, y) = max_{x∈ℝ^{m}} y · x − F_{m}(L, x)

= max_{μ∈H}⟨w + y_{1}w_{1} + ⋯ + y_{m}w_{m}, μ⟩ − L(μ).

It follows from this second expression that the function G_{m} is convex and finite-valued, hence continuous on A × ℝ^{m}.

Consider the set Σ̃ ⊂ A × ℝ^{m} of points (L, y) such that dim ∂G_{m}(L, y) ≥ dim A + 1, where ∂G_{m} is the subdifferential of G_{m}. It is known, see the appendix, that this set has Hausdorff dimension at most

(m + dim A) − (dim A + 1) = m − 1.

Consequently, the projection Σ of the set Σ̃ on the second factor ℝ^{m} also has Hausdorff dimension at most m − 1. Therefore, the complement of Σ is dense in ℝ^{m}. So it is enough to prove that

y ∉ Σ ⟹ ∀ L ∈ A : dim M_{m}(L, y) ≤ dim A.

Since we know by definition of Σ that dim ∂G_{m}(L, y) ≤ dim A, it is enough to observe that

dim M_{m}(L, y) ≤ dim ∂G_{m}(L, y).

The last inequality follows from the fact that the set M_{m}(L, y) is the subdifferential of the convex function

ℝ^{m} ∋ z ⟼ G_{m}(L, z)

at the point y.

□

3. Application to Lagrangian dynamics

Let C be the set of continuous functions f : 𝕋 × T M → ℝ with linear growth, i.e.

∥f∥_{ℓ} := sup_{(t,θ)∈𝕋×T M} \frac{|f(t, θ)|}{1 + |θ|} < +∞,

endowed with the norm ∥·∥_{ℓ}.

We apply Theorem 5 to the following setting:

• F = C^{∗} is the vector space of continuous linear functionals μ : C → ℝ provided with the weak-⋆ topology. Recall that

lim_{n} μ_{n} = μ ⟺ lim_{n} μ_{n}(f) = μ(f), ∀ f ∈ C.

• E = C^{∞}(𝕋 × M, ℝ) provided with the C^{∞} topology.

• G is the vector space of finite Borel signed measures on T M, or equivalently the set of continuous linear forms on C^{0}(𝕋 × M, ℝ), provided with the weak-⋆ topology.

• The pairing E × G → ℝ is given by integration:

⟨u, ν⟩ = ∫_{𝕋×M} u dν.

• The continuous linear map π : F ⟶ G is induced by the projection 𝕋 × T M ⟶ 𝕋 × M.

• The compact K ⊂ G is the set of Borel probability measures on 𝕋 × M, provided with the weak-⋆ topology. Observe that K is separated by E.

• The compact H_{n} ⊂ F is the set of holonomic probability measures which are sup-

ported on

B_{n} := {(t, θ) ∈ 𝕋 × T M | |θ| ⩽ n }.

Holonomic probabilities are defined as follows: Given a C^{1} curve γ : ℝ → M of period T ∈ ℕ define the element μ_{γ} of F by

⟨f, μ_{γ}⟩ = \frac{1}{T} ∫_{0}^{T} f(s, γ(s), γ̇(s)) ds

for each f ∈ C. Let

Γ := { μ_{γ} | γ ∈ C^{1}(ℝ, M) is periodic of integral period } ⊂ F.

The set H of holonomic probabilities is the closure of Γ in F. One can see that H is convex (cf. Mañé [4, prop. 1.1(a)]). The elements μ of H satisfy ⟨1, μ⟩ = 1 therefore we have π(H) ⊂ K.

Note that each Tonelli Lagrangian L gives rise to an element of H_{n}^{∗}.

Let 𝔐(L) be the set of minimizing measures for L and let supp 𝔐(L) be the union of their supports. Recalling that we have defined M_{H_{n}}(L) as the set of measures μ ∈ H_{n} which minimize the action ∫ L dμ on H_{n}, we have:

Lemma 9. If L is a Tonelli lagrangian then there exists n ∈ ℕ such that

dim π(M_{H_{n}}(L)) = dim 𝔐(L).

Proof:

Birkhoff theorem implies that 𝔐(L) ⊂ H (cf. Mañé [4, prop. 1.1.(b)]). In [5, Prop. 4, p. 185] Mather proves that supp 𝔐(L) is compact, therefore 𝔐(L) ⊂ H_{n} for some n ∈ ℕ.

In [4, §1] Mañé proves that minimizing measures are also all the minimizers of the action functional A_{L}(μ) = ∫ L dμ on the set of holonomic measures, therefore 𝔐(L) = M_{H_{n}}(L) for some n ∈ ℕ.

In [5, Th. 2, p. 186] Mather proves that the restriction supp 𝔐(L) → M of the projection T M → M is injective. Therefore the linear map π : 𝔐(L) → G is injective, so that dim π(M_{H_{n}}(L)) = dim π(𝔐(L)) = dim 𝔐(L). □

Proof of Theorem 1.

Given n ∈ ℕ apply Theorem 5 and obtain a residual subset O_{n}(A) ⊂ E such that

L ∈ A, u ∈ O_{n}(A) ⟹ dim π(M_{H_{n}}(L − u)) ⩽ dim A.

Let O(A) = ∩_{n}O_{n}(A). By the Baire property O(A) is residual. We have that

L ∈ A, u ∈ O(A), n ∈ ℕ ⟹ dim π(M_{H_{n}}(L − u)) ⩽ dim A.

Then by Lemma 9, dim 𝔐(L − u) ⩽ dim A for all L ∈ A and all u ∈ O(A). □

Appendix A. Convex Functions

Given a convex function f : ℝ^{n} → ℝ and x ∈ ℝ^{n}, define its subdifferential as

∂f(x) := { ℓ : ℝ^{n} → ℝ linear | f(y) ⩾ f(x) + ℓ(y − x) , ∀y ∈ ℝ^{n} }.

Then the sets ∂f(x) ⊂ ℝ^{n} are convex. If k ∈ ℕ, let

Σ_{k}(f) := { x ∈ ℝ^{n} | dim ∂f(x) ⩾ k }.

The following result is standard.

Proposition 10. If f : ℝ^{n} → ℝ is a convex function then for all 0 ⩽ k ⩽ n the Hausdorff dimension HD(Σ_{k}(f)) ⩽ n − k.

We recall here an elegant proof due to Ambrosio and Alberti, see [1]. Note that much more can be said on the structure of Σ_{k}, see [2, 9] for example.

By adding |x|^{2} if necessary (which does not change Σ_{k}) we can assume that f is superlinear and that

(3)  f(y) ⩾ f(x) + ℓ(y − x) + \frac{1}{2} |y − x|^{2} ∀x, y ∈ ℝ^{n}, ∀ℓ ∈ ∂f(x).

Lemma 11. ℓ ∈ ∂f(x), ℓ′ ∈ ∂f(x′) ⟹ |x − x′| ⩽ ∥ℓ − ℓ′∥.

Proof: From inequality (3) we have that

f(x′) ⩾ f(x) + ℓ(x′ − x) + \frac{1}{2} |x′ − x|^{2} ,
f(x) ⩾ f(x′) + ℓ′(x − x′) + \frac{1}{2} |x − x′|^{2} .

Then

0 ⩾ (ℓ′ − ℓ)(x − x′) + |x − x′|^{2} (4)

∥ℓ − ℓ′∥ |x − x′| ⩾ (ℓ − ℓ′)(x − x′) ⩾ |x − x′|^{2} . (5)

Therefore ∥ℓ − ℓ′∥ ⩾ |x − x′|. □

Since f is superlinear, the subdifferential ∂f is surjective and we have:

Corollary 12. There exists a Lipschitz function F : ℝ^{n} → ℝ^{n} such that

ℓ ∈ ∂f(x) ⟹ x = F (ℓ).

Proof of Proposition 10:

Let A_{k} be a set with HD(A_{k}) = n − k such that A_{k} intersects any convex subset of dimension k. For example

A_{k} = {x ∈ ℝ^{n} | x has at least k rational coordinates }.

Observe that

x ∈ Σ_{k} ⟹ ∂f(x) intersects A_{k} ⟹ x ∈ F (A_{k}).

Therefore Σ_{k} ⊂ F (A_{k}). Since F is Lipschitz, we have that HD(Σ_{k}) ⩽ HD(A_{k}) = n−k. □

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CEREMADE, Université de Paris Dauphine, Pl. du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France,

E-mail address: patrick.bernard@ceremade.dauphine.fr

CIMAT, P.O. Box 402, 36.000 Guanajuato, GTO, México.

E-mail address: gonzalo@cimat.mx