LAGRANGIAN FLOWS: THE DYNAMICS OF GLOBALLY MINIMIZING ORBITS - II

GONZALO CONTRERAS, JORGE DELGADO, AND RENATO ITURRIAGA

To the memory of Ricardo Mañé.

Introduction.

In this work we prove most of the theorems of Mañé’s unfinished work “Lagrangian Flows the dynamics of Globally Minimizing Orbits”, [7]. Exceptions are theorem III, whose proof is divided in [6] and [3] and theorem IV which was proved in [7]. Also, we provide proofs for slightly different statements of theorems VII, XI and XIV. We would like to emphasize that all the theorems in this paper are due to Mañé and all the responsibility of the proofs is ours.

We encourage the reader to use Mañé’s original paper [7] as the introduction of this work. In section 1 we prove theorems I and II, in section 2 we prove theorem V, in section 3 we prove theorems VI, VII, VIII and IX, in section 4 we prove theorems X and XI, and in section 5 we prove theorems XII, XIII and XIV.

The first and second authors want to thank the hospitality of CIMAT.

We want to use this space to say how much we admired Ricardo’s clearness and brightness and how grateful we are to his enormous generosity. This paper is to his memory.

1. Basic properties of the critical value.

Let M be a smooth closed manifold. We say that a smooth function L : T M → ℝ is a Lagrangian if it satisfies the following conditions:

(a) Convexity: For all x ∈ M, v ∈ T_{x}M, the Hessian matrix \frac{∂^{2}L}{∂v_{i}∂v_{j}} (x, v) (calculated

with respect to linear coordinates on T_{x}M) is positive definite.

(b) Superlinearity: lim_{∥v∥→+∞} \frac{L(x,v)}{∥v∥} = +∞.

G. Contreras was partially supported by CNPq-Brazil.

R. Iturriaga was partially supported by Conacyt-Mexico, grant 3398E9307.

Given an absolutely continuous curve x : [0, T] → M define its L-action by

S_{L}(x) := ∫_{a}^{b} L (x(t), ẋ(t)) dt .

Fixing p, q ∈ M and T > 0, the critical points of the action functional on the set

AC(p, q, T) := { x : [0, T] → M | x(0) = p, x(T) = q, x absolutely continuous }.

are solutions of the Euler-Lagrange equation, which in local coordinates is given by

\frac{d}{dt}L_{v} = L_{x} .  (E-L)

Because of the convexity of the Lagrangian this equation can be thought as a first order differential equation on T M. The Lagrangian flow f_{t} on T M is defined by f_{t}(x, v) = (γ(t), γ̇(t)), where γ is the solution of (E-L) with γ(0) = x and γ̇(0) = v. Define the energy function E : T M → ℝ as

E(x, v) := \frac{∂L}{∂v}(x, v) · v − L .

It can be seen that the value of E(x, v) is constant along the orbits of f_{t}. The superlinearity condition implies that the level sets of the energy function have bounded velocities and hence that are compact. This, in turn, implies that the solutions of (E-L) are defined for all values t ∈ ℝ, i.e., that the flow f_{t} is complete.

Finally, for p, q ∈ M, let

AC(p, q) := ⋃_{T >0} AC(p, q, T) ,

and define the action potential as

Φ_{k}(p_{1}, p_{2}) := inf { S_{L+k}(x) | x ∈ AC(p_{1}, p_{2}) } , k ∈ ℝ .

Theorem I. There exists c(L) ∈ ℝ such that

(a) k < c(L) ⟹ Φ_{k}(p_{1}, p_{2}) = −∞ , ∀p_{1} , p_{2} ∈ M.

(b) k ≥ c(L) ⟹ Φ_{k}(p_{1}, p_{2}) > −∞ , ∀p_{1} , p_{2} ∈ M and Φ_{k} is Lipschitz.

(c) k ≥ c(L) ⟹ Φ_{k}(p_{1}, p_{3}) ≤ Φ_{k}(p_{1}, p_{2}) + Φ_{k}(p_{2}, p_{3}) ∀p_{1} , p_{2} , p_{3} ∈ M

Φ_{k}(p_{1}, p_{2}) + Φ_{k}(p_{2}, p_{1}) ≥ 0 ∀p_{1} , p_{2} ∈ M

(d) k > c(L) ⟹ Φ_{k}(p_{1}, p_{2}) + Φ_{k}(p_{2}, p_{1}) > 0 ∀p_{1} ≠ p_{2} .

1.1. Remark. This theorem, with the same proof, holds for coverings π : M̂ → M of a compact manifold M, with the lifted Lagrangian L̂ = L ∘ π.

Proof:

We first prove that if for some p_{1}, p_{2} ∈ M, Φ_{k}(p_{1}, p_{2}) = −∞, then Φ_{k}(q_{1}, q_{2}) = −∞ for all q_{1}, q_{2} ∈ M. Let γ, η: [0, 1] → M, γ ∈ AC(q_{1}, p_{1}), η ∈ AC(p_{2}, q_{2}). Let x_{n} ∈ AC(p_{1}, p_{2}) be such that lim_{n→∞} S_{L+k}(x_{n}) = −∞. Then

lim_{n→∞} S_{L+k}(γ ∗ x_{n} ∗ η) = S_{L+k}(γ) + S_{L+k}(η) + lim_{n→∞} S_{L+k}(x_{n}) = −∞ .

Thus the number

c(L) := inf { k ∈ ℝ | Φ_{k}(p, q) > −∞ }

does not depend on (p, q). We have to see that −∞ < c(L) < +∞. Since the function k ↦ Φ_{k}(p, q) is nondecreasing, it is enough to see that there exist k_{1}, k_{2} ∈ ℝ such that Φ_{k_{1}}(p, q) = −∞ and Φ_{k_{2}}(p, q) > −∞. We first prove the existence of k_{1}. Since L is bounded on {(x, v) ∈ T M | v ∈ T_{x}M , |v| ≤ 2 }, there exists B > 0 such that

|L(x, v)| < B if |v| < 2 .  (1)

Let x_{n} : [0, n] → M, x_{n} ∈ AC(p, q) be such that |ẋ_{n}| ≤ 2. Then S_{L}(x_{n}) ≤ B n and hence, for k_{1} = −B − 1, we have that

Φ_{k_{1}}(p, q) ≤ lim inf_{n} S_{L−B−1}(x_{n}) ≤ lim inf_{n} (∫_{0}^{n} L(x_{n}, ẋ_{n}) dt − (B + 1)n)

≤ lim inf_{n}(Bn − (B + 1)n) = −∞.

Now we prove the existence of k_{2}. The superlinearity hypothesis implies that L is bounded below. Let A be a lower bound for L on T M. We claim that it is enough to take k_{2} > −A + 1. Indeed

S_{L+k_{2}}(x) ≥ ∫_{0}^{T} (A + k_{2}) dt ≥ 0 for all x ∈ AC(p, q).

hence Φ_{k_{2}}(p, q) ≥ 0.

It remains to prove that Φ_{c}(p, q) > −∞ for all p, q ∈ M, where c = c(L). Suppose not. Take p ∈ M, then Φ_{c}(p, p) = −∞. Let γ ∈ AC(p, p) be such that S_{L+c}(γ) < −a < 0. Then there exists ε > 0 such that S_{L+c+ε}(γ) < −\frac{1}{2}a < 0. Let δ_{N} := γ ∗ …^{N} ∗ γ, then

Φ_{c+ε}(p, p) ≤ lim_{N} S_{L+c+ε}(δ_{N}) ≤ lim_{N} −\frac{1}{2} a N = −∞ .

This contradicts the definition of c(L). In particular, we have also proven that Φ_{c}(p, p) ≥ 0 for all p ∈ M. By taking γ ∈ AC(p, p) with bounded velocities and arbitrarily small parameter intervals, we have that

Φ_{c}(p, p) = 0 for all p ∈ M .  (2)

Similarly

Φ_{k}(p, p) = 0 for all p ∈ M for all k ≥ c(L).  (3)

We now prove (c). Let k ≥ c(L) and p_{1}, p_{2}, p_{3} ∈ M. Let x_{n}^{ij} : [0, T_{n}] → M, x_{n}^{ij} ∈ AC(p_{i}, p_{j}), be such that

lim_{n} S_{L+k}(x_{n}^{ij}) = Φ_{k}(p_{i}, p_{j}) i, j ∈ {1, 2, 3}.

Then x_{n}^{12} ∗ x_{n}^{23} ∈ AC(p_{1}, p_{3}) and

Φ_{k}(p_{1}, p_{3}) ≤ S_{L+k}(x_{n}^{12} ∗ x_{n}^{23}) = S_{L+k}(x_{n}^{12}) + S_{L+k}(x_{n}^{23}).

Taking the limit when n → ∞ we get that

Φ_{k}(p_{1}, p_{3}) ≤ Φ_{k}(p_{1}, p_{2}) + Φ_{k}(p_{2}, p_{3}) .  (4)

Finally using (3) and (4) we obtain that

0 = Φ_{k}(p_{1}, p_{1}) ≤ Φ_{k}(p_{1}, p_{2}) + Φ_{k}(p_{2}, p_{1})  (5)

when k ≥ c(L).

We prove that Φ_{k} is Lipschitz when k ≥ c(L). Let γ : [0, d(p, q)] → M be a geodesic joining p and q, using (1) we obtain

Φ_{k}(p, q) ≤ ∫_{0}^{d(p,q)} [L (γ(t), γ̇(t)) + k] dt
Φ_{k}(p, q) ≤ (B + k) d(p, q) for k ≥ c(L) .  (6)

Therefore if k ≥ c(L),

Φ_{k}(p_{1}, p_{2}) − Φ_{k}(q_{1}, q_{2}) ≤ Φ_{k}(p_{1}, q_{1}) + Φ_{k}(q_{1}, p_{2}) − Φ_{k}(q_{1}, q_{2})
≤ Φ_{k}(p_{1}, q_{1}) + Φ_{k}(q_{1}, q_{2}) + Φ_{k}(q_{2}, p_{2}) − Φ_{k}(q_{1}, q_{2})
≤ Φ_{k}(p_{1}, q_{1}) + Φ_{k}(q_{2}, p_{2})
≤ (B + k)(d(p_{1}, q_{1}) + d(p_{2}, q_{2})).

Changing the roles of p_{i} and q_{i}, i = 1, 2 we get that

|Φ_{k}(p_{1}, p_{2}) − Φ_{k}(q_{1}, q_{2})| ≤ (B + k) (d(p_{1}, q_{1}) + d(p_{2}, q_{2})) .

We now prove that if k ≥ c(L) and p ≠ q, then the function k ↦ Φ_{k}(p, q) is strictly increasing. By (5) this implies (d). Let p ≠ q and ℓ > k ≥ c(L). Let x_{n} : [0, T_{n}] → M, x_{n} ∈ AC(p, q) be such that lim_{n} S_{L+ℓ}(x_{n}) = Φ_{ℓ}(p, q). We have that

S_{L+ℓ}(x_{n}) = S_{L+k}(x_{n}) + (ℓ − k)T_{n}
Φ_{ℓ}(p, q) ≥ Φ_{k}(p, q) + (ℓ − k) lim inf_{n} T_{n}

It is enough to prove that lim inf_{n} T_{n} > 0, because then Φ_{ℓ}(p, q) > Φ_{k}(p, q). Suppose that lim inf_{n} T_{n} = 0. By the superlinearity of L, for all B > 0 there exists A > 0 such that

|L(x, v)| ≥ B |v| − A .

Then

Φ_{ℓ}(p, q) = lim_{n} S_{L+ℓ}(x_{n}) ≥ lim inf_{n} [∫_{0}^{T_{n}} B \left|{ \atop ẋ_{n}}\right| dt + (k − A) T_{n} ]
≥ B d(p, q) + 0 ,

for all B > 0. Therefore Φ_{L+ℓ}(p, q) = +∞, which contradicts (6).

Through the rest of the paper we shall neeed the following results:

1.2. Theorem. (Mather [8])

For all C > 0 there exists A_{1} = A_{1}(C) such that if T > 0, p, q ∈ M and x ∈ AC(p, q, T) satisfy

(a) S_{L}(x) = min{ S_{L}(y) | y ∈ AC(p, q, T) }.

(b) S_{L}(x) ≤ C T.

Then

(c) ∥ẋ(t)∥ < A_{1} for all t ∈ [0, T].
(d) x|_{[0,T]} is a solution of (E-L).

1.3. Corollary. There exists A > 0 such that if T > 1, p, q ∈ M and x ∈ AC(p, q, T) satisfy

S_{L}(x) = min{ S_{L}(y) | y ∈ AC(p, q, T) } ,

then ∥ẋ(t)∥ < A for all t ∈ [0, T].

Proof: Let

C := sup{ |L(x, v)| | ∥v∥ < diam (M) } .

6 G. CONTRERAS, J. DELGADO, AND R. ITURRIAGA

There exists a geodesic γ ∈ AC(p, q, T) with

\left|{ \atop γ̇}\right| ≡ \frac{d(p, q)}{T} < diam(M) and S_{L}(γ) < C T .

Then the corollary follows from theorem 1.2. 1.4. Corollary. There exists A > 0 such that if p, q ∈ M and x ∈ AC(p, q, T) satisfy

(a) S_{L}(x) = min{ S_{L}(y) | y ∈ AC(p, q, T) }.

(b) S_{L}(x) < Φ_{c}(p, q) + d_{M}(p, q).

Then

(c) T > \frac{1}{A} d_{M}(p, q) .
(d) ∥ẋ(t)∥ < A for all t ∈ [0, T].

Proof:

Let B be from (1) and let γ : [0, d_{M}(p, q)] → M be a minimal geodesic with \left|{ \atop γ̇}\right| ≡ 1 joining p and q. By the superlinearity of L, for D = 2 B + 1 > 0 there exists E > 0 such that

|L(x, v)| > D |v| − E for all (x, v) ∈ T M.

We have that

B d_{M}(p, q) > S_{L}(γ) ≥ Φ_{c}(p, q) ≥ S_{L}(x) − d_{M}(p, q)
> ∫_{0}^{T} (D |ẋ(t)| − E) dt − d_{M}(p, q)
> D d_{M}(p, q) − E T − d_{M}(p, q) .

Hence

T > \frac{(D − B − 1)}{E} d_{M}(p, q) = (\frac{B}{E} ) d_{M}(p, q) .  (7)

Now let

C := max\{ |L(x, v)| | |v| ≤ \frac{E}{B} \} .

Let η ∈ AC(p, q, T) be a minimal geodesic. Then, by (7),

\left|{ \atop η̇}\right| ≡ \frac{d_{M}(p, q)}{T} < \frac{E}{B}

and hence S_{L}(η) ≤ C T. Let A_{1} = A_{1}(C) be from theorem 1.2, then it is enough to use A = max \{ A_{1}(C), \frac{E}{B} \} .

Let M(L) be the set of invariant borel probability measures for the Lagrangian flow.

Theorem II.

c(L) = − min {∫ L dμ | μ ∈ M(L) } .

Proof:

Let μ ∈ M(L) be ergodic. Let (p, v) ∈ T M be such that

lim_{T →∞} \frac{1}{T} ∫_{0}^{T} L(f_{t}(p, v)) dt = ∫ L dμ .

Let B > 0 be such that

|L(x, v)| < B if |v| ≤ 2 .

For N > 0 let q_{N} := π(f_{N}(p, v)) and let γ_{N} : [0, d(p, q_{N})] → M be a geodesic joining q_{N} to p. Let x_{N} : [0, N] → M be defined by x_{N}(t) = π(f_{t}(p, v)). Then f_{t}(p, v) = (x_{N}(t), ẋ_{N}(t)). For k ∈ ℝ, we have that

S_{L+k}(γ_{N}) = ∫_{0}^{d(p,q_{N})} L(γ_{N}(t), γ̇_{N}(t)) dt ≤ (B + k) diam(M) .
lim_{N} \frac{1}{N} S_{L+k}(x_{N} ∗ γ_{N}) = lim_{N} \frac{1}{N} S_{L+k}(x_{N}) + 0
= S_{L+k}(μ) = S_{L}(μ) + k .

If k < −S_{L}(μ), then

Φ_{k}(p, p) ≤ lim_{N} S_{L+k}(x_{N} ∗ γ_{N}) = −∞ .

Hence k ≤ c(L). Therefore

c(L) ≥ sup {−S_{L}(μ) | μ ∈ M_{erg}(L) }
≥ − min {S_{L}(μ) | μ ∈ M(L) } .

Now let k < c(L) and p, q ∈ M. Then Φ_{k}(p, q) = −∞ and there exists a sequence of paths x_{n} : [0, T_{n}] → M, x_{n} ∈ AC(p, q) such that

lim_{n} S_{L+k}(x_{n}) = −∞ .  (8)

Since L is bounded below, we have that

lim_{n} T_{n} = +∞ .  (9)

Let y_{n} : [0, T_{n}] → M, y_{n} ∈ AC(p, q) be a minimizer of the action among the curves in AC(p, q, T_{n}). Then (y_{n}(t), ẏ_{n}(t)) is an orbit segment of f_{t} and by 1.3, |ẏ| is bounded. Let ν_{n} be the probability measure defined by

∫ h dν_{n} = \frac{1}{T_{n}} ∫_{0}^{T_{n}} h(y_{n}(t), ẏ_{n}(t)) dt

= \frac{1}{T_{n}} ∫_{0}^{T_{n}} h(f_{t}(y_{n}(0), ẏ_{n}(0))) dt ,

for all h : T M → ℝ continuous. There exists a subsequence ν_{n_{i}} which converges weakly* to a probability measure μ. By (9), μ is invariant under the flow f_{t}. Since the velocities are bounded we have that lim_{n_{i}} \frac{1}{T_{n_{i}}} S_{L+k}(y_{n_{i}}) = S_{L+k}(μ) = S_{L}(μ) + k. Since

lim_{n} S_{L+k}(y_{n}) = Φ_{k}(p, q) = −∞ and T_{n} > 0 for all n, then S_{L}(μ) + k ≤ 0. For any k < c(L) we found an invariant measure μ such that k ≤ −S_{L}(μ). Therefore

c(L) ≤ sup {−S_{L}(μ) | μ ∈ M(L) }
≤ − min { S_{L}(μ) | μ ∈ M(L) } .

We state now theorem III. The proof of theorem III is split in Mañé [6] and [3]. We say that a property holds for a generic Lagrangian if given any Lagrangian L, there exists a residual set O ⊆ C^{∞}(M, ℝ) such that the property holds for all the Lagrangians L + ψ with ψ ∈ O. We say that an invariant measure is uniquely ergodic if it is the only invariant measure on its support. A periodic orbit for the Lagrangian flow is said hyperbolic if it is a hyperbolic periodic orbit of the flow restricted to its energy level.

Definition. We say that μ ∈ M(L) is a minimizing measure if

∫ L dμ = −c(L).

Denote by M̂(L) the set of minimizing measures in M(L).

Theorem III.

(a) For generic L, M̂(L) contains a single measure and this measure is uniquely ergodic.

(b) When this measure is supported on a periodic orbit, this orbit is hyperbolic.

Item (a) is proved in Mañé [6] and item (b) is proved in [3].

Conjecture. (Mañé)

For a generic L, M̂(L) consists on a single measure supported in a periodic orbit.

2. Recurrence properties.

The prerequisite of the following definition is this remark: since d_{k} ≥ 0 for k ≥ c(L), then for any absolutely continuous curve x : [a, b] → M and k ≥ c(L) we have that

S_{L+k}(x) ≥ Φ_{k}(x(a), x(b)) ≥ −Φ_{k}(x(b), x(a)) .  (10)

Definition. Set c = c(L). We say that x : [a, b] → M is a semistatic curve if it is absolutely continuous and:

S_{L+c} (x|_{[t_{0},t_{1}]}) = Φ_{c} (x(t_{0}), x(t_{1})) ,  (11)

for all a < t_{0} ≤ t_{1} < b; and that it is a static curve if

S_{L+c} (x|_{[t_{0},t_{1}]}) = −Φ_{c}(x(t_{1}), x(t_{0}))  (12)

for all a < t_{0} ≤ t_{1} < b.

By (10), equality (12) implies (11). Hence static curves are semistatic. Semistatic curves are solutions of (E-L) because they minimize the action of L + c in AC(x(t_{0}), x(t_{1}), t_{1} − t_{0}). If p, q ∈ M are on a static curve then d_{c}(p, q) = 0.

If w ∈ T M, denote by x_{w} : ℝ → M the solution of (E-L) with ẋ_{w}(0) = w.

Definition.

Σ(L) := { w ∈ T M | x_{w} : ℝ → M is semistatic } ,
Σ̂(L) := { w ∈ T M | x_{w} : ℝ → M is static } ,
Σ^{+}(L) := { w ∈ M | x_{w}|_{[0,∞[} is semistatic } .

Remark. Replacing c by any other real number in the definition of semistatic solution, the set Σ^{+}(L) (and then Σ̂(L) ⊂ Σ(L) ⊂ Σ^{+}(L)) becomes empty.

For k > c(L) this remark follows from the following estimates:

+∞ > max_{p,q∈M} Φ_{k}(p, q) ≥ S_{L+k}(x|_{[0,T]}) = Φ_{k}(x(0), x(T))

≥ Φ_{c}(x(0), x(T)) + (k − c) T

≥ max_{p,q∈M} Φ_{c}(p, q) + (k − c) T.

The following theorem is proven in Mañé [7].

Theorem IV. (Characterization of minimizing measures)

A measure μ ∈ M(L) is minimizing if and only if supp(μ) ⊆ Σ̂(L).

We include a proof below, using theorem V(c).

Given an f_{t} invariant subset Λ ⊆ T M, and ε > 0, T > 0, an (ε, T)-chain joining ξ_{a} ∈ Λ and ξ_{b} ∈ Λ is a finite sequence {(ζ_{i}, t_{i})}_{i=1}^{N} ⊂ Λ×ℝ such that ζ_{1} = ξ_{a}, ζ_{N+1} = ξ_{b}, t_{i} > T and d(f_{t_{i}}(ζ_{i}), ζ_{i+1}) < ε for i = 1, …, N. We say that a set Λ ⊂ T M is chain transitive if for all ξ_{a}, ξ_{b} ∈ Λ, and all ε > 0, T > 0 there exists and (ε, T)-chain in Λ joining ξ_{a} and ξ_{b}. When this condition holds only for all ξ_{a} = ξ_{b} ∈ Λ we say that Λ is chain recurrent.

Theorem V. (Recurrence Properties)

a) Σ(L) is chain transitive.

b) Σ̂(L) is chain recurrent.

c) The α and ω-limit set of a semistatic orbit is contained in Σ̂(L).

Proof:

Lets first prove (c). Let w ∈ Σ and let u ∈ ω(w). We prove that ω(w) ⊆ Σ̂(L), the proof that α(w) ⊆ Σ̂(L) is similar. It is enough to prove that for all T > 0, x_{u}|_{[0,T]} is static. Let p := x_{u}(0), q := x_{u}(T). Let p_{n} = x_{w}(s_{n}), q_{n} = x_{w}(t_{n}) be sequences of points

in M with s_{n} < s_{n}+T = t_{n} < s_{n+1}, s_{n}, t_{n} →_{n} +∞ and ẋ_{w}(s_{n}) →_{n} ẋ(0), ẋ_{w}(t_{n}) →_{n} ẋ_{w}(T).

We have that

S_{L+c} (x_{w}|_{[s_{n},t_{n}]}) = Φ_{c}(p_{n}, q_{n}) ,

x_{w}|_{[s_{n},t_{n}]} ⟶^{C^{1}} x_{u}|_{[0,T]} ,

S_{L+c} (x_{u}|_{[0,T]}) = lim_{n} S_{L+c} (x_{w}|_{[s_{n},t_{n}]})

= lim_{n} Φ_{c}(p_{n}, q_{n})

= Φ_{c}(p, q) .

Moreover, by the continuity of Φ_{c}, we have that

Φ_{c}(p, q) + Φ_{c}(q, p) = lim_{n} Φ_{c}(p_{n+1}, q_{n+1}) + Φ_{c}(q_{n}, p_{n+1})

= lim_{n} {S_{L+c} (x_{w}|_{[t_{n},s_{n+1}]}) + S_{L+c} (x_{w}|_{[s_{n+1},t_{n+1}]})}

= lim_{n} {S_{L+c} (x_{w}|_{[t_{n},t_{n+1}]})}

= lim_{n} Φ_{c}(q_{n}, q_{n+1})

= Φ_{c}(q, q) = 0 .

Proof of theorem IV:

Suppose that μ ∈ M(L) and supp(μ) ∈ Σ̂(L). Let θ be a generic poinmt for μ and L + c. Then θ is static and if x_{θ}(t) = π ∘ φ_{t}(θ), we have

∫ (L + c) dμ = lim_{T →+∞} \frac{1}{T} ∫_{0}^{T} [L(x_{θ}, ẋ_{θ}) + c] dt

= lim_{T →+∞} \frac{1}{T} Φ_{c}(x_{θ}(0), x_{θ}(T))

≤ lim_{T →+∞} \frac{1}{T} sup_{p,q∈M} Φ_{c}(p, q) = 0.

where the second equality is because θ is (semi)static. Hence μ is minimizing.

Now suppose that μ ∈ M(L) is minimizing. Applying lemma II.2 in Mañé [6], we get that for μ-almost every θ there is a sequence T_{j} = T_{j}(θ) → +∞ such that

lim_{j→+∞} d_{T M}(θ, φ_{T_{j}}(θ)) = 0 ,  (13)
lim_{j→+∞} ∫_{0}^{T_{j}}[L(x_{θ}, ẋ_{θ}) + c] dt = 0 .  (14)

By theorem V(c), it is enough to prove that supp(μ) ⊆ Σ(L). Since Φ_{c} is continuous, it is enough to prove that μ-almost every θ is semistatic. Now let θ satisfy (13), (14) and define

δ_{θ}(T) := S_{L+c} (x_{θ}|_{[0,T]}) − Φ_{c}(x_{θ}(0), x_{θ}(T)) .

Then δ_{θ}(t) is non-derecasing and δ_{θ}(t) ≥ 0.

By the continuity of Φ_{c} and (13) we have that

lim_{j→+∞} Φ_{c}(x_{θ}(0), x_{θ}(T_{j})) = 0 .  (15)

From (15) and (14) we have that lim_{j→+∞} δ_{θ}(T_{j}) = 0. Since δ_{θ}(t) is non-decerasing, then δ_{θ}(t) ≡ 0 for all t ≥ 0. Hence θ is semistatic.

Proof of theorem V(a):

Given ε > 0, S > 0, u, v ∈ Σ(L), we have to find an (ε, S)-chain in Σ(L) joining u to v. It is easy to see that such (ε, S)-chain exists if ω(u) ∩ α(v) ≠ ø. Let

A := α(v) , Ω := ω(u)  (16)

and suppose that A ∩ Ω = ø. Let p ∈ π(Ω), q ∈ π(A). Let η_{n} := [0, T_{n}] → M be such that

p := η_{n}(0) ∈ π(Ω) , q := η_{n}(T_{n}) ∈ π(A)
S_{L+c}(η_{n}) ≤ Φ_{c}(p, q) + \frac{1}{n}  (17)

By corollary 1.4, we can assume that η_{n} is a solution of (E-L) and satisfies

|η̇_{n}(t)| < A for all 0 ≤ t ≤ T_{n}  (18)

Given \frac{ε}{2} > δ_{0} > 0 there exists 0 < δ < δ_{0} such that if |v| , |w| < A and d_{T M}(v, w) < δ then

d_{T M}(f_{t}(v), f_{t}(w)) < δ_{0} for all |t| ≤ S.  (19)

Let 𝕄 be the union of A, Ω and the set of accumulation points of the tangent vectors of the η_{n}’s:

𝕄 := A ∪ Ω ∪ \{v ∈ T M | \begin{matrix}∃ ⟨n_{k}⟩ ⊆ ℕ & 0 ≤ t_{n_{k}} ≤ T_{n_{k}} \\ ∃ ⟨t_{n_{k}}⟩ ⊆ ℝ & v = lim_{k} η̇_{n_{k}}(t_{n_{k}}) \end{matrix} \} .

Then

𝕄 ⊆ { v ∈ T M | ∥v∥ ≤ A } .  (20)

We shall need the following lemmas

2.1. Lemma.

(a) 𝕄 ⊆ Σ(L).

(b) 𝕄 is invariant.

Let 𝕂 be the set of vectors which are on the ω-limit of vectors of 𝕄.

𝕂 := ⋃_{v∈𝕄}[ω(v)] .

Since 𝕄 is closed and foward invariant, then the closure \overline{𝕂} ⊆ 𝕄 ⊆ Σ(L). Moreover, the vectors in 𝕂 are chain recurrent. By (20) \overline{𝕂} is compact. Given δ > 0 let

𝕂_{δ} := \{ v ∈ T M | d_{T M}(v, \overline{𝕂}) < δ \} .

Since \overline{𝕂} is compact, the number of connected components of 𝕂_{δ} is finite. Let Λ_{i} = Λ_{i}(δ), i = 0, 1, …, N, Ω ⊆ Λ_{0}, be the connected components of 𝕂_{δ}.

2.2. Lemma. Each component Λ_{i} is (2δ, T)-chain transitive for any T > 0.

If A ⊂ Λ_{0} the proposition is proved. Suppose that A∩Λ_{0} = ø. Consider the oriented graph Γ with vertices Λ_{i}, i = 0, 1, …, N and an edge Λ_{i} → Λ_{j} if there exists v ∈ 𝕄 and t_{i} < t_{j} such that f_{t_{i}}(v) ∈ Λ_{i} and f_{t_{j}}(v) ∈ Λ_{j}.

2.3. Lemma. There exists a path in the graph joining Λ_{0} ⊇ Ω to Λ_{j} ⊇ A.

We need that the connecting orbits between the Λ_{j}’s have time intervals greater than S. Let Λ_{i} → Λ_{j} be an edge of the graph. Take a connecting orbit f_{t}(v), t_{i} < t_{j}, v ∈ 𝕄, from Λ_{i}(δ) to Λ_{j}(δ). Since f_{t_{j}}(v) is in a δ-neighbourhood of \overline{𝕂} ∩ Λ_{j}(δ) and \overline{𝕂} is invariant, (19) implies that f_{t_{j}+S}(v) is in a δ_{0}-neighbourhood of \overline{𝕂} ∩ Λ_{i}(δ).

Now we can construct an (ε, S)-chain joining u ∈ A to v ∈ Ω by using the connecting orbits between the Λ_{i}’s and joining them by (ε, S)-chains inside each Λ_{i}. This completes the proof of theorem V(a).

Now we prove the lemmas

Proof of lemma 2.1(a):

Equation (17) implies that

S_{L+c}(η_{n}|_{[α,β]}) ≤ Φ_{c}(η_{n}(α), η_{m}(β)) + \frac{1}{n}

for all 0 < α < β < T_{n}. If 0 < a_{n_{k}} < T_{n_{k}}, η̇_{n_{k}}(a_{n_{k}}) →^{k} v, b_{n_{k}} − a_{n_{k}} →^{k} τ, then

(η_{n_{k}}(t), η̇_{n_{k}}) → (x_{v}(t), ẋ_{v}(t)) = f_{t}(v) for 0 ≤ t < b_{n} − a_{n}

and by the continuity of Φ_{c} we have that S_{L+c}(x_{v}|_{[0,T]}) ≤ Φ_{c}(x_{v}(0), x_{v}(T)) .

Hence x_{v}|_{[0,T]} is semistatic.

Proof of 2.1(b):

Since the set A ∪ Ω ⊆ 𝕄 is invariant, we only have to prove that 𝕄 \ (A ∪ Ω) is invariant. Let ω ∈ 𝕄 \ (A ∪ Ω) and ⟨k⟩ ⊆ ℕ, ⟨t_{n_{k}}⟩ ⊆ ℝ^{+} be such that η̇_{n_{k}}(t_{n_{k}}) →_{k} w. It is enough to show that

lim inf_{k} t_{n_{k}} = +∞ ,  (21)
lim inf_{k} |T_{n_{k}} − t_{n_{k}}| = +∞ .  (22)

We need the folowing claim

Claim. The limit points of { η̇_{n}(0) | n ∈ N } and { η̇(T_{n}) | n ∈ ℕ } are in Ω and A respectively.

Proof:

We prove it only for limit points η̇_{n}(0) → u ∈ T M. Since η_{n}(0) ∈ π(Ω), then π(u) ∈ π(Ω). Since Ω ⊆ Σ̂(L) and 𝕄 ⊆ Σ^{+}(L), then the claim follows from part (b) of theorem VI.

We only prove (21). Suppose that

lim inf_{k} t_{n_{k}} = a < +∞ .

Then there exists a convergent subsequence ⟨η_{n_{ℓ}}|_{[0,a]}⟩ in the C^{1} topology, to a semistatic solution γ such that

γ̇(a) = lim_{ℓ} η̇_{n_{ℓ}}(a) = w .

Then by the claim we have that

γ̇(0) = f_{−a}(w) = lim_{ℓ} η̇_{n_{ℓ}}(0) ∈ Ω .

and hence w ∈ Ω. This contradicts our assumption that w ∈ 𝕄 \ (A ∪ Ω).

Proof of lemma 2.2:

Let T > 0. Since Λ_{i} = Λ_{i}(δ) is open and connected, it is pathwise connected. Let ξ, ζ ∈ Λ_{i} and let Γ : [0, S] be a continuous path such that Γ(0) = ξ, Γ(s) = ζ. Let 0 = s_{0} < s_{1} < ⋯ < s_{n} = S be such that d_{T M}(Γ(s_{i}), Γ(s_{i+1})) < \frac{δ}{2}. For each s_{i} let v_{i} ∈ Λ_{i}(δ) ∩ \overline{𝕂} be such that d(Γ(s_{i}), v_{i}) < δ. Since the orbits of \overline{𝕂} are recurrent, there exist τ_{i} > T such that d(f_{τ_{i}}(v_{i}), v_{i}) < \frac{δ}{2}. We have that

d(f_{τ_{i}}(v_{i}), v_{i+1}) < d(f_{τ_{i}}(v_{i}), Γ(s_{i})) + d(Γ(s_{i}), Γ(s_{i+1})) + d(Γ(s_{i+1}), v_{i+1}) < 2 δ.

The orbit segments { f_{t}(v_{i}) | 0 ≤ t ≤ τ_{i}, i = 0, …, n − 1 } give the required (2 δ, T)- chain.

In order to prove lemma 2.3 we need first

2.4. Lemma. For all δ > 0 there exists N = N(δ) > 0 and S = S(δ) > 0 such that if n > N, then for all c ∈ [0, T − n] we have that

𝕂_{δ} ∩ {(η_{n}(t), η̇_{n}(t)) | t ∈ [c − S, c + S] ∩ [0, T_{n}] } ≠ ø .

Proof: Suppose it is not true. Then there exists δ > 0 and sequences n_{k} → ∞ and c_{k} ∈ [0, T_{n_{k}}], such that 0 < c_{k} − k < c_{k} + k < T_{n_{k}} and

𝕂_{δ} ∩ { (η_{n_{k}}(t), η̇_{n_{k}}(t)) | t ∈ [c_{k} − k, c_{k} + k] } = ø .

Let { v ∈ T M | ∥v∥ ≤ A, v ∉ 𝕂_{δ} } = B, then (η_{n_{k}}(t), η̇_{n_{k}}(t)) ∈ B for all k ∈ ℕ and t ∈ [c_{k} − k, c_{k} + k]. Consider the measures μ_{k} defined by

∫_{B} ϕ dμ_{k} := \frac{1}{2k} ∫_{c_{k}−k}^{c_{k}+k} ϕ(η_{n_{k}}(t), η̇_{n_{k}}(t)) dt .

Since the η_{n_{k}}’s are solutions of (E-L), B is compact and k → +∞, then there exists a convergent subsequence ν_{k} → ν in the weak* topology to an invariant measure ν of the Lagrangian flow, with supp(ν) ⊂ B. Moreover, supp(ν) ⊂ 𝕄. Let v ∈ supp(ν). Since B is compact, we have that the ω-limit ø ≠ ω(v) ⊂ 𝕂 and

ø ≠ ω(v) ⊆ supp(ν) ⊂ B ∩ 𝕂 ⊆ B ∩ 𝕂_{δ} = ø .

This is a contradiction.

Proof of lemma 2.3:

Suppose that A ∩ Λ_{0} = ø, otherwise there is nothing to prove. For each n ∈ ℕ let

a_{n} := sup{ t ∈ [0, T_{n}] | (η_{n}(t), η̇_{n}(t)) ∈ Λ_{0} } ,
a_{n} + s_{n} := inf{ t ∈]a_{n}, T_{n}] | (η_{n}(t), η̇_{n}(t)) ∈ 𝕂_{δ} } .

By lemma 2.4, for n > N(δ) we have that 0 < s_{n} < S(δ). Choose a sequence η_{n}^{1} such that (η_{n}^{1}(a_{n}), η̇_{n}^{1}(a_{n})) converges. Renumbering the Λ_{i}’s if necessary, we can assume that s_{n} → s^{1} ∈ [0, S(δ)] and (η_{n}^{1}(a_{n} + s^{1}), η̇_{1}^{n}(a_{n} + s^{1})) ∈ \overline{Λ_{1}}. By lemma 2.1(a), the sequence (η_{n}^{1}, η̇_{n}^{1})|_{[a_{n},a_{n}+s^{1}]}(s−a_{n}) converges to a semistatic solution of (E-L) (γ^{1}, γ̇^{1})|_{[0,s^{1}]}, whose points are in 𝕄 and such that γ^{1}(0) ∈ Λ_{0}, γ^{1}(s^{1}) ∈ Λ_{1} ≠ Λ_{0}. Hence there is and edge Λ_{0} ⟶ Λ_{1} in the graph Γ.

Suppose that Λ_{1} ∩ A = ø, otherwise the lemma is proved. Let

b_{n} := sup{ t ∈ [0, T_{n}] | (η_{n}^{1}(t), η̇_{n}^{1}(t)) ∈ Λ_{0} ∩ Λ_{1} } .

The same arguments show that there exists 0 < s^{2} ≤ S(δ) and a subsequence η_{n}^{2}|_{[b_{n},b_{n}+s^{2}]} such that η_{n}^{2}(b_{n} + s^{2}) ∈ Λ_{2}, Λ_{2} ≠ Λ_{i}, i = 0, 1, and an edge Λ_{1} ⟶ Λ_{2} or Λ_{0} ⟶ Λ_{2}. We can repeat this argument each time that the final Λ_{j} does not contain A. Since the number of the Λ_{j}’s is finite, we obtain a path in the graph Γ from Λ_{0} to Λ_{j} with A ⊆ Λ_{j}.

Proof of theorem V(b):

Let w ∈ Σ̂(L) and let A = α(w), Ω = ω(w). It is easy to see that if A ∩ Ω ≠ ø then w is chain recurrent in Σ̂(L). Suppose that A ∩ Ω = ø. For all s < t and all ε > 0 there exist T = T (γ) > 0 and γ = γ_{ε,s,t} : [0, T] → M such that γ(0) = x_{w}(t), γ(T) = x_{w}(s) and

S_{L+c}(γ) ≤ −Φ_{c}(x_{w}(t), x_{w}(s)) + ε

Let

η_{n} := γ_{\frac{1}{n} ,−n,n}  ,  T_{n} := T (γ_{\frac{1}{n} ,−n,n})  (23)

We can assume that |η̇(t)| < A for all 0 ≤ t ≤ T_{n}.

The rest of the proof is similar to item (a), but now the corresponding 𝕄 ⊆ Σ̂(L).

3. Graph, covering and coboundary properties.

Theorem VI. (Graph Properties)

(a) If γ(t), t ≥ 0 is an orbit in Σ^{+}(L), then, denoting π : T M → M the canonical projection, the map π|_{{γ̇|t>0}} is injective with Lipschitz inverse.

(b) Denoting Σ_{0}(L) ⊂ M, the projection of Σ̂(L), for every p ∈ Σ_{0}(L) there exists a unique ξ(p) ∈ T_{p}M such that

(p, ξ(p)) ∈ Σ^{+}(L) .

Moreover

(p, ξ(p)) ∈ Σ̂(L) ,

and the vector field ξ is Lipschitz. Obviously

Σ̂(L) = graph(ξ) .

For a proof of the following lemma see Mather [8] or Mañé [4].

3.1. Lemma. (Mather) Given A > 0 there exists K > 0 ε_{1} > 0 and δ > 0 with the following property: if |v_{i}| < A, (p_{i}, v_{i}) ∈ T M, i = 1, 2 satisfy d(p_{1}, p_{2}) < δ and d((p_{1}, v_{1}), (p_{2}, v_{2})) ≥ K^{−1}d(p_{1}, p_{2}) then, if a ∈ ℝ and x_{i} : ℝ → M, i = 1, 2, are the solutions of L with x_{i}(a) = p_{i}, ẋ_{i}(p_{i}) = v_{i}, there exist solutions γ_{i} : [a − ε, a + ε] → M of L with 0 < ε < ε_{1}, satisfying

γ_{1}(a − ε) =x_{1}(a − ε) , γ_{1}(a + ε) = x_{2}(a + ε) ,
γ_{2}(a − ε) =x_{2}(a − ε) , γ_{2}(a + ε) = x_{1}(a + ε) ,
S_{L}(x_{1}|_{[a−ε,a+ε]}) + S_{L}(x_{2}|_{[a−ε,a+ε]}) > S_{L}(γ_{1}) + S_{L}(γ_{2})

Proof of theorem VI:

(a) Since the curve γ is semistatic, corollary 1.4 implies that there exists A > 0 such that |γ̇(t)| < A. Let K > 0, ε_{1} = 1, δ > 0 be from lemma 3.1. We prove that if (p, v), (q, w) ∈ { γ̇(t) | t > 0 } and d_{M}(p, q) < δ, then

d_{T M}((p, v), (q, w)) < K d_{M}(p, q) .

This implies item(a). Suppose it is false. Then there exist 0 < t_{1} < t_{2} such that

d_{M}(γ(t_{1}), γ(t_{2})) < δ
d_{T M}(γ̇(t_{1}), γ̇(t_{2})) > K d_{M}(γ(t_{1}), γ(t_{2}))

By lemma 3.1 there exist 0 < ε < t_{1}, and solutions α, β : [−ε, ε] → M, of (E-L) such that

α(−ε) = γ(t_{1} − ε) =: p  α(ε) = γ(t_{2} + ε) =: q
β(−ε) = γ(t_{2} − ε) =: s  β(ε) = γ(t_{1} + ε) =: r
S_{L}(α) + S_{L}(β) < S_{L}(γ|_{[t_{1}−ε , t_{1}+ε]}) + S_{L}(γ|_{[t_{2}−ε , t_{2}+ε]})

18 G. CONTRERAS, J. DELGADO, AND R. ITURRIAGA

Since γ|_{]0,+∞[} is semistatic, we have that

Φ_{c}(p, q) + Φ_{c}(s, r) ≤ S_{L+c}(α) + S_{L+c}(β)
< S_{L+c}(γ|_{[t_{1}−ε , t_{1}+ε]}) + S_{L+c}(γ|_{[t_{2}−ε , t_{2}+ε]})
< Φ_{c}(p, r) + Φ_{c}(s, q)

Since 0 ≤ Φ_{c}(s, r) + Φ_{c}(r, s),

Φ_{c}(p, q) < Φ_{c}(p, r) − Φ_{c}(s, r) + Φ_{c}(s, q)
Φ_{c}(p, q) < Φ_{c}(p, r) + Φ_{c}(r, s) + Φ_{c}(s, q)  (24)

Since γ|_{]0,+∞[} is semistatic, we have that

Φ_{c}(p, q) = S_{L+c}(γ|_{[t_{1}−ε , t_{1}+ε]}) + S_{L+c}(γ|_{[t_{1}+ε , t_{2}−ε]}) + S_{L+c}(γ|_{[t_{2}−ε , t_{2}+ε]}) ,
Φ_{c}(p, q) = Φ_{c}(p, r) + Φ_{c}(r, s) + Φ_{c}(s, q) .

This contradicts (24).

(b) Now we prove item (b). We prove that if p, q ∈ Σ_{0}(L), (p, v) ∈ Σ̂(L), (q, w) ∈ Σ^{+}(L), and d(p, q) < δ, then

d_{T M}((p, v), (q, w)) < K d_{M}(p, q) .

Observe that this implies item (b). Suppose it is false. Then by lemma 3.1 there exist α, β : [−ε, ε] → M such that

α(−ε) = f_{−ε}(q, w) =: q_{−ε}  α(ε) = f_{ε}(p, v) =: p_{−ε}
β(−ε) = f_{−ε}(p, v) =: p_{ε}  β(ε) = f_{ε}(q, w) =: q_{ε}

and

S_{L}(α) + S_{L}(β) < S_{L}(x_{w}|_{[−ε,ε]}) + S_{L}(x_{v}|_{[−ε,ε]})

So

Φ_{c}(q_{−ε}, p_{ε}) + Φ_{c}(p_{−ε}, q_{ε}) < Φ_{c}(q_{−ε}, q_{ε}) + Φ_{c}(p_{−ε}, p_{ε})
= Φ_{c}(q_{−ε}, q_{ε}) − Φ_{c}(p_{ε}, p_{−ε})

theorem vi b.

Thus

Φ_{c}(q_{−ε}, q_{ε}) ≤ Φ_{c}(q_{−ε}, p_{ε}) + Φ_{c}(p_{ε}, p_{−ε}) + Φ_{c}(p_{−ε}, q_{ε}) < Φ_{c}(q_{−ε}, q_{ε})

which is a contradiction.

Denote by L the set of equivalence classes of the equivalence relation d_{c}(a, b) = 0 lifted to Σ̂(L) by the vectorfield ξ of theorem VI. Denote by ω(v) the ω-limit of v ∈ T M by the flow f_{t}. If Γ ∈ L set

Γ^{+} := { w ∈ Σ^{+}(L) | ω(w) ∈ Γ } .

Clearly Γ^{+} is foward invariant. Let

Γ_{0}^{+} := ⋃_{t>0} π f_{t} Γ^{+} .

Theorem VII. (Covering Property)

(a) πΣ^{+}(L) = M.

(b) For all p ∈ Γ_{0}^{+}, there exists a unique ξ_{Γ}(p) ∈ T_{p}M such that

(p, ξ_{Γ}(p)) ∈ Γ^{+} .

Moreover, ξ_{Γ} is Lipschitz.

Observe that π : Σ^{+}(L) → M is not necessarily injective. We recall that Mañé stated item (a) in a stronger form: πΓ = M for every equivalence class Γ. This may not be true as the following example shows. Let M = S^{1} = ℝ/ℤ be the unit circle and L = \frac{1}{2}v^{2}−cos 6πx. Then the three maximums of the potential A = 0 = 2π, B = \frac{2π}{3} and C = \frac{4π}{3} are singular (hyperbolic saddle) points of the Lagrangian flow. For mechanical Lagrangians L = \frac{1}{2}v^{2} − ϕ(x), we have that

c(L) = e_{0} = − min_{x∈M} L(x, 0) ,

and the static points are the critical points (x, 0) of the Lagrangian flow such that L(x, 0) = −e_{0}. In this example c(L) = 1 and Σ̂(L) = {(A, 0), (B, 0), (C, 0)}. We shall prove below that d_{1}(A, B) = Φ_{1}(A, B) + Φ_{1}(B, A) > 0. Hence (A, 0) and (B, 0) are not in the same equivalence class. By (30), Σ^{+}(L) ⊆ E^{−1}{c(L)}, hence if Γ is the equivalence class of (A, 0), then π Γ^{+} ⊆ [0, \frac{2π}{3} [∪]\frac{2π}{3} , 2π] ≠ S^{1}. This example can easily generalized to higher dimensions.

We show now that in the example above d_{1}(A, B) > 0. By theorem V, the set Σ(L) is chain transitive and contains Σ̂(L) = {(A, 0), (B, 0), (C, 0)}. Moreover, by (30), Σ^{+}(L) ⊆ E^{−1}{c(L)}. In our example this is only possible if Σ(L) contains complete components of the saddle connections (so that πΣ^{+}(L) = M). From the symmetry of this Lagrangian we get that Σ(L) contains all the saddle connections. Hence Σ(L) = E^{−1}{c(L)}. Let (x, ẋ) be the orbit on E^{−1}{c(L)} with α-limit (A, 0) and ω-limit (B, 0). Then it is semistatic and hence

L(x, ẋ) + c(L) = ẋ L_{v}(x, ẋ) = \left|{ \atop ẋ}\right|^{2} ,

Φ_{1}(A, B) = lim_{T →+∞} S_{L+c}(x|_{[−T,+T]}) = ∫_{−∞}^{+∞} |ẋ(t)|^{2} dt > 0 .

The same argument gives that Φ_{1}(B, A) > 0 and thus d_{1}(A, B) > 0.

Proof of theorem VII:

We prove (a). We may assume that πΣ̂(L) ≠ M otherwise the proof is trivial. Let p ∈ M \ πΣ̂(L) and q ∈ πΣ̂(L) ≠ ø. For each n ∈ ℕ let γ_{n} : [0, T_{n}] → M be such that

1. γ_{n}(0) = p, γ_{n}(T_{n}) = q.

2. S_{L+c}(γ_{n}) ≤ Φ_{c}(p, q) + \frac{1}{n}.

3. γ_{n} minimizes L + c in AC(p, q, T_{n}).

Then

(i) γ_{n} is a solution of (E-L). (ii) \left|{ \atop γ̇_{n}}\right| < A for all t ∈ [0, T_{n}].

Since Σ̂(L) is invariant under the Lagrangian flow, item (b) of theorem VI implies that

(iii) T_{n} → +∞.

Let η(t) := πf_{t}(p, v). By (i) and (iii), for any fixed T > 0 we have that γ_{n}|_{[0,T]} → η|_{[0,T]} in the C^{1} topology, and hence

S_{L+c}(η|_{[0,T]}) = lim S_{L+c}(γ_{n}|_{[0,T]}) .

Clearly η(0) = p. It is enough to show that η|_{[0,+∞[} is semistatic. For, we have that

S_{L+c}(γ_{n}) = S_{L+c}(γ_{n}|_{[0,T]}) + S_{L+c}(γ_{n}|_{[T,T_{n}]})
≤ Φ_{c}(p, q) + \frac{1}{n}
≤ Φ_{c}(p, γ_{n}(T)) + Φ_{c}(γ_{n}(T), q) + \frac{1}{n}
≤ Φ_{c}(p, γ_{n}(T)) + S_{L+c}(γ_{n}|_{[T,T_{n}]}) + \frac{1}{n} .

Hence

S_{L+c}(γ_{n}|_{[0,T]}) ≤ Φ_{c}(p, γ_{n}(T)) + \frac{1}{n} .

Taking the limit when n → +∞ we obtain that η is semistatic.

We now prove item (b). Let A > 0 be from corollary 1.4 and K = K(A) > 0, ε_{1} = ε_{1}(A) > 0, δ = δ(A) > 0 be from Mather’s lemma 3.1. It is enough to prove that if p_{1}, p_{2} ∈ Γ_{0}^{+}, v_{1}, v_{2} ∈ Γ^{+}, π(v_{i}) = p_{i}, i = 1, 2 and d_{M}(p_{1}, p_{2}) < δ, then d_{T M}(v_{1}, v_{2}) ≤ K d_{M}(p_{1}, p_{2}).

Suppose it is false. Then there exists p_{i} ∈ Γ_{0}^{+}, v_{i} ∈ Γ^{+}, π(v_{i}) = p_{i} such that d_{M}(p_{1}, p_{2}) < δ and d_{T M}(v_{1}, v_{2}) > K d_{M}(p_{1}, p_{2}). Since p_{i} ∈ Γ_{0}^{+}, there exists 0 < ε < ε_{1} such that f_{t}(v_{i}) ∈ Γ^{+} for t > −ε. Let q_{i} := πf_{−ε}(v_{i}), r_{i} := f_{+ε}(v_{i}). By Mather’s lemma 3.1, there exist η_{i} : [−ε, ε] → M such that η_{i}(−ε) = q_{i}, η_{i}(+ε) = r_{i} and

S_{L+c}(η_{1}) + S_{L+c}(η_{2}) < S_{L+c}(x_{v_{1}}|_{[−ε,+ε]}) + S_{L+c}(x_{v_{2}}|_{[−ε,+ε]}).

Thus

Φ_{c}(q_{1}, r_{2}) + Φ_{c}(q_{2}, r_{1}) <  Φ_{c}(q_{1}, r_{1}) +  Φ_{c}(q_{2}, r_{2}).  (25)

Let u_{i} ∈ ω(v_{i}) and z_{i} := π(u_{i}). If T_{n}^{i} ⟶^{n} +∞ is such that f_{T_{n}^{i}}(v_{i}) ⟶^{n} u_{i}, we have that

Φ_{c}(q_{i}, z_{i}) = lim_{n} Φ_{c}(q_{i}, x_{v_{i}}(T_{n}^{i})) = lim_{n} S_{L+c}(x_{v_{i}}|_{[−ε,T_{n}^{i}]})

= S_{L+c}(x_{v_{i}}|_{[−ε,+ε]}) + lim_{n} S_{L+c}(x_{v_{i}}|_{[+ε,T_{n}^{i}]})

≥ Φ_{c}(q_{i}, r_{i}) + Φ_{c}(r_{i}, z_{i}) .

By the triangle inequality for Φ_{c} we get that

Φ_{c}(q_{i}, z_{i}) = Φ_{c}(q_{i}, r_{i}) + Φ_{c}(r_{i}, z_{i})  (26)

From (26) and (25) we have that

Φ_{c}(q_{1}, z_{1}) + Φ_{c}(q_{2}, z_{2}) = Φ_{c}(q_{1}, r_{1}) + Φ_{c}(r_{1}, z_{1}) + Φ_{c}(q_{2}, r_{2}) + Φ_{c}(r_{2}, z_{2})
> Φ_{c}(q_{1}, r_{2}) + Φ_{c}(q_{2}, r_{1}) + Φ_{c}(r_{1}, z_{1}) + Φ_{c}(r_{2}, z_{2})
≥ Φ_{c}(q_{1}, z_{2}) + Φ_{c}(q_{2}, z_{1}) .  (27)

Since u_{i} ∈ Γ, then d_{c}(z_{1}, z_{2}) = Φ_{c}(z_{1}, z_{2}) + Φ_{c}(z_{2}, z_{1}) = 0. Adding d_{c}(z_{1}, z_{2}) = 0 to the right of inequality (27), we obtain that

Φ_{c}(q_{1}, z_{1}) + Φ_{c}(q_{2}, z_{2}) > Φ_{c}(q_{1}, z_{2}) + Φ_{c}(z_{2}, z_{1}) + Φ_{c}(z_{1}, z_{2}) + Φ_{c}(q_{2}, z_{1})
≥ Φ_{c}(q_{1}, z_{1}) + Φ_{c}(q_{2}, z_{2}) .

This is a contradiction.

Theorem VIII. (Generic Structure of Σ̂(L))

For a generic Lagrangian L, Σ̂(L) is a uniquely ergodic set. If it is a periodic orbit then it is a hyperbolic periodic orbit.

Proof:

This should be thought as a corollary of theorems III and IV. Take the Generic set given by theorem III of Lagrangians L that satisfy #M̂(L) = 1 and call this unique minimizing measure μ(L). Then if μ is an invariant measure of L and it is supported in Σ̂(L) then by theorem IV it is minimizing. Thus μ = μ(L). This proves the theorem.

Theorem IX. (Coboundary Property)

If c = c(L), then (L + c)|_{Σ̂(L)} is a Lipschitz coboundary. More precisely, taking any

p ∈ M and defining G : Σ̂(L) → ℝ by

G(w) = Φ_{c}(p, π(w))

Then

(L + c)|_{Σ̂(L)} = \frac{dG}{df} ,

where

\frac{dG}{df}(w) := lim_{h→0} \frac{1}{h}[ G(f_{h}(w)) − G(w) ] .

LAGRANGIAN FLOWS 23

Proof:

Let w ∈ Σ̂(L) and define F_{w}(v) := Φ_{c}(π(w), π(v)). We have that

\frac{dF_{w}}{df} |_{w} = lim_{h→0} \frac{1}{h} [ F_{w}(f_{h} w) − F_{w}(w) ]

= lim_{h→0} \frac{1}{h} [ Φ_{c}(πw, πf_{h} w) − Φ_{c}(πw, πw) ]

= lim_{h→0} \frac{1}{h} [ S_{L+c} (x_{w}|_{[0,h]}) ]

= lim_{h→0} \frac{1}{h} ∫_{0}^{h}[L(x_{w}(s), ẋ_{w}(s)) + c] ds

= L(w) + c

We claim that for any p ∈ M and any w ∈ Σ̂(L), h ∈ ℝ,

G(f_{h} w) = Φ_{c}(p, π(f_{h} w)) = Φ_{c}(p, π(w)) + Φ_{c}(π(w), π(f_{h} w))
G(f_{h} w) = Φ_{c}(p, π(w)) + F_{w}(f_{h}(w)) .  (28)

This is enough to prove the theorem because then

\frac{dG}{df}|_{w} = \frac{d}{dh}F_{h}(f_{h} w)|_{h=0} = \frac{F_{w}}{df} |_{w} = L(w) + c,

and G is Lipschitz by theorem I.

We now prove (28). Let q := π(w), x := π(f_{h} w). We have to prove that

Φ_{c}(p, x) = Φ_{c}(p, q) + Φ_{c}(q, x) .  (29)

Since the points q and x can be joined by the static curve x_{w}|_{[0,h]}, then

Φ_{c}(x, q) = −Φ_{c}(q, x) .

Using twice the triangle inequality for Φ_{c} we get that

Φ_{c}(p, q) ≤ Φ_{c}(p, x) + Φ_{c}(x, q) = Φ_{c}(p, x) − Φ_{c}(q, x) ≤ rΦ_{c}(p, q) .

This implies (29).

4. Connecting orbits inside fixed energy levels.

We quote a paragraph from Mañé [7]:

“Exploiting that the energy, E : T M → ℝ, defined as usual by E(x, v) = \frac{∂L}{∂v} v − L, is a first integral of the flow generated by L, leads to information on the position of Σ^{+}(L). First observe that it is easy to check that a semistatic curve x : [a, b] → M satisfies:

E(x(t), ẋ(t)) = c(L) .  (30)

This follows from calculating the derivative at λ = 1 of the function F : ℝ → ℝ given by:

F (λ) = ∫a^{\frac{b}{λ}} (L + c)(x_{λ}(t), ẋ_{λ}(t)) dt ,

where x_{λ} : [a, \frac{b}{λ}] → M is given by x_{λ}(t) = x(λt). From (30) follows that:

Σ^{+}(L) ⊂ E^{−1}(c) ,

that together with π Σ^{+}(L) = M implies:

π E^{−1}(c) = M .

Hence,

c ≥ max_{q} E(q, 0) .

Moreover Σ̂(L) ⊂ E^{−1}(c) implies:”

4.1. Corollary. μ ∈ M(L) is minimizing if and only if

∫ (\frac{∂L}{∂v}) v dμ = 0 ,

supp(μ) ⊂ E^{−1}(c(L)) .

Theorem X.

If k > c(L) then for all p, q ∈ M, p ≠ q there exists a solution x(t) of (E-L) such that x ∈ AC(p, q) and

Φ_{k}(p, q) = S_{L+k}(x)

Moreover the solution (x(t)ẋ(t)) is contained in the energy level E^{−1}{k}.

4.2. Remark. If p = q then the infimum

Φ_{k}(p, p) = inf{S_{L+k}(γ) | γ ∈ AC(p, q) } = 0

can not be realized by a path defined on an interval with nonzero length. For otherwise if γ : [0, T] → M, T > 0 is a minimum, then

Φ_{c}(p, p) ≤ S_{L+c}(γ) = Φ_{k}(p, p) − (k − c(L)) T < Φ_{k}(p, p) = 0 .

Contradicting theorem I.

4.3. Remark. This theorem, with the same proof, holds for coverings π : M̂ → M of a compact manifold M, with the lifted Lagrangian L̂ = L ∘ π.

Proof:

Suppose that p, q ∈ M and p ≠ q. For each T > 0 there exists a minimizer x_{T} of S_{L} on AC(p, q, T). Then

Φ_{k}(p, q) = inf_{T >0} S_{L}(x_{T}) + k T .

Observe that

lim_{T →+∞}(S_{L}(x_{T}) + k T ) = lim_{T →+∞}(S_{L+c}(x_{T}) + (k − c) T )

≥ lim_{T →+∞}(Φ_{c}(p, q) + (k − c) T )

= +∞ .

Choose a sequence {T_{i}} such that lim_{i} S_{L}(x_{T_{i}} + k T_{i}) = Φ_{k}(p, q). Then {T_{i}} must be bounded and we may assume that it has a limit T_{0}. Since \left∥{ \atop ẋ_{T_{i}}}\right∥ < A and p ≠ q, then T_{0} > 0. We shall prove that we actually have a minimal solution with time interval [0, T_{0}]. We proceed as in the proof of Tonelli’s theorem (see Mather [8] or Mañé [4], lemma 6.1). Observe that by corollary 1.3 the sequence {x_{T_{i}}} is absolutely

equicontinuous because {S_{L}(x_{T_{i}})} is bounded. Since T_{i} ⟶^{i} T_{0} there is a subsequence of {x_{T_{i}}} which converges to a curve x_{T_{0}} ∈ AC(p, q, T_{0}). It remains to prove that

S_{L+k}(x_{T_{0}}) = Φ_{k}(p, q) .  (31)

The following lemma uses the superlinearity of the Lagrangian. A proof can be found in Maǹé [4]. We may assume that M = ℝ^{n}.

4.4. Lemma. Given C > 0 and ε > 0 and a compact subset S ⊂ M = ℝ^{n} there exist δ > 0 such that if (x, v), (y, w) ∈ T M = ℝ^{n} ×ℝ^{n} satisfy x ∈ S, d(x, y) < δ and |v| < C then

L(y, w) ≥ L(x, v) + \frac{∂L}{∂v}(x, v)(w − v) − ε

Let Δ := Φ_{k}(p , q) − k T_{0}. Given C > 0 define

S(C) := { t ∈ [0, T_{0}] | |ẋ(t)| ≤ C } .

26 G. CONTRERAS, J. DELGADO, AND R. ITURRIAGA

Since L is bounded below, we can assume that L is positive. We will prove that for all h > 0 and C > 0,

∫_{S(C)} L(x(t), ẋ(t))dt ≤ Δ + h .  (32)

Taking the limits when h → 0 and C → ∞ we get that S_{L}(x) ≤ Δ, completing the proof of (31). If i is big enough

Δ + h > ∫_{S(C)} L(x_{T_{i}}(t), ẋ_{T_{i}}(t)) dt

≥ ∫_{S(C)} L(x(t), ẋ(t)) dt − ε T_{0} + ∫_{S(C)} \frac{∂L}{∂v}(ẋ_{T_{i}}(t) − ẋ(t)) dt .

Then

∫_{S(C)} L(x(t), ẋ(t)) dt ≤ Δ + h + ε T − lim inf_{i} ∫_{S(C)} \frac{∂L}{∂v}(ẋ_{T_{i}}(t) − ẋ(t)) dt . (33)

We claim that

lim inf ∫_{S(C)} \frac{∂L}{∂v}(ẋ_{T_{i}}(t) − ẋ(t)) dt = 0.  (34)

This follows from the fact that x is absolutely continuous, \frac{∂L}{∂v} is bounded in S(C) and that for any interval (a, b) ⊂ [0, T_{0}]

lim_{i→∞} ∫_{a}^{b}(ẋ_{T_{i}}(t) − ẋ(t)) dt = lim_{i→∞}(x_{T_{i}}(b) − x(b)) − (x_{T_{i}}(a) − x(a)) = 0.

Since ε > 0 is arbitrary (34) together with (33) prove (32).

We now prove that minimizers of S_{L+k} are in the energy level E = k. Suppose that x ∈ AC(p, q, T) is such that

S_{L+k}(x) = min_{y∈AC(p,q)} S_{L+k}(y) .

Define

F (λ) = ∫_{0}^{λT} (L + k)(x_{λ}, ẋ_{λ})

where x_{λ}(t) : [0, λT] → M is defined as x_{λ}(t) = x(\frac{t}{λ}). By the minimizing condition F′(1) = 0. On the other hand

F′(λ) = T L(x_{λ}(λT), ẋ_{λ}(λT)) + ∫_{0}^{λT} \frac{∂L}{∂λ} dt + kT

now

\frac{∂L}{∂λ} = \frac{∂L}{∂x} \frac{∂x_{λ}}{∂λ} + \frac{∂L}{∂v} \frac{∂ẋ_{λ}}{∂λ}

= −\frac{∂L}{∂x}ẋ_{λ} (\frac{t}{λ}) \frac{t}{λ^{2}} − \frac{∂L}{∂v} (\frac{1}{λ^{3}} ẍ_{λ} (\frac{t}{λ}) t + \frac{ẋ_{λ}(\frac{t}{λ})}{λ^{2}})

So

0 = T L(x(T), ẋ(T)) + T k − ∫_{0}^{T} (\frac{∂L}{∂x}ẋ + \frac{∂L}{∂v}ẍ) t dt − ∫_{0}^{T} \frac{∂L}{∂v}ẋ dt

= T L(x(T), ẋ(T)) + T k − ∫_{0}^{T} \frac{∂L}{∂v}ẋ dt − ∫_{0}^{T} \frac{d}{dt}(L)t dt

= T L(x(T), ẋ(T)) + T k − ∫_{0}^{T} \frac{∂L}{∂v}ẋ dt + ∫_{0}^{T} L dt − L t|_{0}^{T}

= T k − ∫_{0}^{T} E dt

= T (k − E).

This proves that the energy level of the solution x is k.

Observe that

L + k = (∂L/∂v)v on E^{−1}{k} .  (35)

4.5. Corollary.

(a) If k > c(L) and a, b ∈ M, there exists a solution x(t) of (E-L) such that x(0) = a, x(T) = b for some T > 0, E(x(t), ẋ(t)) = k for all t ∈ ℝ, and:

∫_{0}^{T} \frac{∂L}{∂v}(x, ẋ) ẋ dt = min ∫_{0}^{T_{1}} \frac{∂L}{∂v}(y, ẏ) ẏ dt ,  (36)

where the minimum is taken over all the absolutely continuous y : [0, T_{1}] → M, T_{1} ≥ 0, with y(0) = a, y(T_{1}) = b and E(y(t), ẏ(t)) = k for a.e. t ∈ [0, T_{1}].

(b) Conversely, if given k > c(L) and a, b ∈ M, there exists an absolutely continuous x : [0, T] → M with x(0) = a, x(T) = b, E(x(t), ẋ(t)) = k for a.e. t ∈ [0, T] and satisfying the minimization property (36), then x(t) is a solution of (E-L).

If p, q ∈ πE^{−1}{k}, define

ACE(p, q; k) := {x ∈ AC(p, q) | E(x, ẋ) = k a.e. } .

Item (a) of corollary 4.5 follows from (35) and item (b) follows from the fact that since minimizers of S_{L+k} have energy k, then minimizing S_{L+k} on AC(p, q) is equivalent to minimize it on ACE(p, q; k).

Given μ ∈ M(L) define its homology or asymptotic cycle (cf. Schwartzman [10]), by ρ(μ) ∈ H_{1}(M, ℝ) = H^{1}(M, ℝ)^{∗} such that

ρ(μ) · [θ] = ∫_{T M} θ dμ , ∀ [θ] ∈ H^{1}(M, ℝ) ,

where θ is a closed 1-form and [θ] its homology class. Define the Mather’s betha function β : H_{1}(M, ℝ) → ℝ, by

β(h) := min{ S_{L}(μ) | μ ∈ M(L), ρ(μ) = h } .

Since, for any h ∈ H_{1}(M, ℝ) the set K(h) := { μ ∈ M(L) | ρ(μ) = h } is convex, it follows that β is a convex function. Let β^{∗} be its Legendre transform: β^{∗} : H^{1}(M, ℝ) → ℝ,

β^{∗}(w) := max_{h∈H_{1}(M,ℝ)}{ w(h) − β(h) } .

The reader can check that β^{∗}([θ]) = −c(L − θ), ∀ [θ] ∈ H^{1}(M, ℝ) .

Define the strict critical value c_{0}(L) by

c_{0}(L) = min_{θ} c(L − θ) .

Then

c_{0}(L) = −β^{∗}(0) = − min{ S_{L}(μ) | μ ∈ M(L), ρ(μ) = 0 } .

Observe that

\frac{d}{dt} E(x, tv)|_{t=1} = v · L_{vv}(x, v) · v > 0 ,  (37)

Therefore if (x, v) is a critical point of the energy function E then v = 0 and \frac{∂L}{∂x}(x, 0) = 0. Let

e_{0} := max_{p∈M} E(p, 0).

By (37), we have that

E(p, 0) = min_{v∈T_{p}M} E(p, v) .  (38)

In particular

e_{0} = min{ k ∈ ℝ | π(E^{−1}{k}) = M } .

Let θ_{0} be a closed 1-form such that c_{0}(L) = c(L − θ_{0}). Then the energy function and the Euler-Lagrange equations for L − θ_{0} and L are the same. Theorem X implies that π(E^{−1}{k}) = M for all k > c(L − θ_{0}) = c_{0}(L). Hence

e_{0} ≤ c_{0}(L) .

As observed in Mañé [7], for mechanical Lagrangians L(x, v) = \frac{1}{2}⟨v, v⟩_{x} − ϕ(x), with ⟨ , ⟩_{x} a riemannian metric, we have that d_{0} = e_{0} = c_{0}(L) = c(L). There is an example in [7] of a Lagrangian L with e_{0} < c_{0}(L). It uses the following corollary to the proof of theorem X:

4.6. Corollary. If k > c_{0}(L), for every free homotopy class H ≠ 0 of M, there exists a periodic orbit in E^{−1}{k} such that its projection on M belongs to that free homotopy class.

Proof:

Fix k > c_{0}(L). By adding a closed 1-form we can assume that c(L) = c_{0}(L). Let AC(H) be the set of absolutely continuous closed curves in M with free homotopy class H. Let x_{n} ∈ AC(H) with x_{n} : [0, T_{n}] → M and

lim_{n→∞} S_{L+k}(x_{n}) = inf_{x∈AC(H)} S_{L+k}(x).

Let x̃_{n} be a lift of x_{n} to the universal cover M̃ of M. We can assume that x̃_{n} is a minimizer of _{L̃+k} on AC(x̃_{n}(0), x̃_{n}(T_{n}); T_{n}) and in particular, that it is a solution of (E-L). Then the same arguments as in the proof theorem X yield that {T_{n}} must be bounded and \left∥{ \atop ẋ_{n}}\right∥ < A. We can assume thet lim_{n} T_{n} = T_{0}. Moreover

T_{0} > \frac{1}{A} min\{ length(γ) | γ ∈ AC(H) \} > 0.

The same arguments as in theorem X give a closed curve γ : [0, T_{0}] → M wich is a uniform limit of a subsequence of x_{n} and hence γ ∈ AC(H). Moreover

S_{L+k}(γ) = min_{x∈AC(H)} S_{L+k}(x) ,  (39)

and (γ, γ̇) is in the energy level E^{−1}{k}.

It remains to prove that γ̇(0) = γ̇(T_{n}). Suppose that γ̇(0) ≠ γ̇(T_{n}). Let M̃ be the universal cover of M and L̃ the lift of L. Let γ̃ be a lift of γ. Consider the path η|_{[−ε,ε]} = γ̃|_{[T_{0}−ε,T_{0}]} ∗ γ̃|_{[0,ε]}. We have that η is not C^{1} and hence it is not a solution of (E-L). Since k > c_{0}(L) ≥ e_{0}, then k is a regular value of the energy Ẽ of L̃. By the Maupertius principle (see theorem 3.8.5 of Abraham & Marsden [1]), η|_{[−ε,ε]} is not a minimizer of the (L̃ + k)-action on Ẽ^{−1}{k}. Then there exists ξ ∈ (η(−ε), η(ε)), with energy Ẽ(ξ, ξ̇) ≡ k and S_{L̃+k}(ξ) < S_{L+k}(η). Moreover, since M̃ is simply connected, the paths ξ and η are homotopic by a homotopy which fixes their endpoints. Hence π(ξ ∗ γ̃|_{[−ε,T_{0}−ε]}) and γ̇|_{[0,T_{0}]} are in the same free homotopy class of M. We have that

S_{L+k} (π(ξ ∗ γ̃|_{[ε,T_{0}−ε]})) = S_{L̃+k}(ξ ∗ γ̃|_{[ε,T_{0}−ε]})
= S_{L̃+k}(ξ) + S_{L+k}(γ̃|_{[ε,T_{0}−ε]})
< S_{L+k}(γ) .

This contradicts the minimizing property (39) of γ.

“An interesting characterization of the critical value c(L), in terms of an analogous to Tonelli’s theorem (Mather [8]) in a prescribed energy level is given by the following result:” — Mañé [7]

Theorem XI.

Assume that k is a regular value of E and dim M > 1. Suppose that k has the following property: for all a, b in πE^{−1}(k) there exists an absolutely continuous curve x : [0, T] → M such that:

(i) x(0) = a and x(T) = b. (ii) E(x(t), ẋ(t)) = k a.e. in [0, T].

(iii) ∫_{0}^{T} \frac{∂L}{∂v}(x(t), ẋ(t)) ẋ(t) dt = min ∫_{0}^{T_{1}} \frac{∂L}{∂v}(y(t), ẏ(t)) ẏ(t) dt,

where the minimum is taken over all absolutely continuous y : [0, T_{1}] → M, with y(0) = a, y(T_{1}) = b and E(x(t), ẋ(t)) = k a.e. in [0, T_{1}].

Then k > c(L) and x(t) is a solution of the Euler Lagrange equation.

The hypothesis dim M > 1 is necessary as the example of a simple pendulum shows. Indeed, for L(x, v) = \frac{1}{2} |v|^{2} − cos x and any regular value k < e_{0} = − min_{p∈S^{1}} L(p, 0) = c(L) = 1 there are such minimizers. This is because a non-empty energy level E^{−1}(k) with k < e_{0} is a topological circle and given a, b ∈ π(E^{−1}(k)) there are two injective paths on E^{−1}(k) from π^{−1}{a} to π^{−1}{b}. One of them must minimize the (L + k)- action because L + k = v · L_{v} = \frac{1}{2} |v|^{2} ≥ 0 on E^{−1}(k). In fact, the minimizer is the one whose projection on S^{1} is injective.

Another example in M = S^{1} is 𝕃 = L + θ where θ_{x}(v) is a (closed) 1-form such that c(𝕃) = c(L + θ) > c(L) = e_{0}. The energy functions and the Lagrangian flows for 𝕃 and L are the same. For e_{0} < k < c(𝕃) the energy levels E^{−1}(k) support two periodic orbits. Lemma 4.8 below shows that these orbits have negative (L + k)-action, and hence there are no minimizers on these levels. On the other hand, a regular energy level k < e_{0} consists of one periodic orbit (γ, γ̇). By the symmetry of E^{−1}(k), we have that ∫_{γ} v 𝕃_{v} = ∫_{γ} \frac{1}{2} |v|^{2} + 0 > 0. Then the same arguments as for L show that there exists minimizers for k < c(L).

We comment now the hypothesis of the regularity of the energy value k. Recall that all the critical points of E are in the zero section of T M. The following lemma shows that if (p_{0}, 0) is a critical point of E, then it is a singularity of the Lagrangian flow and that the point p_{0} can not be joined to other points by a path with energy k which is differentiable at p_{0}. Thus completing the picture of theorem XI: If we require the differentiability of the minimizers x(t) at the endpoints, then the same statement holds for critical values k of E if E^{−1}{k} consists of more than one point, and if #E^{−1}{k} = 1 the (unique) minimizing curve is a (singular) solution of (E-L). Conversely, if (q_{0}, 0) ∈ E^{−1}{k} is not a critical point, then it is necessarily reached on finite time by curves on E^{−1}{k}.

Define

d_{0} := min_{p∈M} E(p, 0) = − max_{p∈M} L(p, 0)

If k < d_{0} then E^{−1}(k) = ø. Given d_{0} ≤ k ≤ e_{0} define

θ(k) := { p ∈ M | E(p, 0) = k } ,

then θ(k) ≠ ø. The proof of the following lemma is delayed to the end of the section.

4.7. Lemma.

(a) If p_{0} ∈ θ(k) and D_{p}E(p_{0}, 0) = 0, then the only curve p : [0, δ] → M with E(p(t), ṗ(t)) = k and p(0) = p_{0} is the constant curve p(t) = p_{0}.

(b) If p : [0, t_{0}[→ M is such that E(p, ṗ) ≡ k, lim_{t→t_{0}^{−}} p(t) = q_{0} ∈ θ(k) and D_{p}E(q_{0}, 0) ≠ 0, then t_{0} is finite.

Proof of theorem XI:

For k ≤ c(L) we have to show that there are points in E^{−1}{k} which can not be joined by a curve which minimizes the action of S_{L+k} on ACE(p, q; k).

Observe that d_{0} is necessarily a critical value of E. Suppose first that d_{0} < k < e_{0}. Since k is a regular value of E, by the Maupertius principle (see theorem 3.8.5 of Abraham & Marsden [1]), the critical points of the functional S_{L+k} on ACE(p, q, k) are solutions of (E-L). Let p ∈ θ(k) ≠ ø. Then {(p, 0)} = T_{p}M ∩ E^{−1}{k}. Hence there is only one solution x of (E-L) with E(x, ẋ) ≡ k and x(0) = p. The set { x(t) | ẋ = 0 } = { x(t) | x(t) ∈ θ(k) } has at most countably many points. But since k is a regular value of E then θ(k) is a submanifold of M of codimension 1 and dim θ(k) > 0. Hence there are points in θ(k) = ∂πE^{−1}(k) which can not be joined to p by (L+k)-minimizers with E(x, ẋ) ≡ k.

Suppose now that k = c(L) =: c. The arguments of equation (30) show that minimizers of S_{L+c} in AC(p, q) are in the energy level E = c(L). Thus minimizers of S_{L+c} on ACE(p, q; c(L)) are also minimizers on AC(p, q) and hence semistatic curves. Let (q, ξ(q)) ∈ Σ̂ ⊆ E^{−1}{k}. Let p ∉ π{f_{t}(q, ξ(q)) | t ∈ ℝ }. Suppose that there exists a semistatic curve x ∈ AC(p, q, T). Then by theorem VI (b), we have that ẋ(T) = ξ(q). This contradicts the choice of p.

Now suppose that e_{0} < k < c(L). We shall prove in lemma 4.8 that there exists a closed curve γ(t) such that E(γ, γ̇) ≡ k and

∫_{γ}(L + k) < 0 .

By making a large number of loops along γ one can produce a curve with arbitrarily negative action. By adding two connecting segments this implies that for any p, q ∈ M there are not minimizers of S_{L+k} on ACE(p, q; k).

This completes the proof of theorem XI.

Given an absolutely continuous closed curve Γ : [0, T] → M, define the probability measure μ_{Γ} on T M by

∫_{T M} φ dμ_{Γ} := \frac{1}{T} ∫_{0}^{T} φ(Γ(t), Γ̇(t)) dt .

Let C(k) be the set of measures μ_{Γ} supported on the energy level E^{−1}(k) and let \overline{C(k)} be its closure in the weak* topology. Define

γ(k) := min_{μ∈\overline{C}(k)} ∫ v · L_{v} dμ .

Measures realizing this minimum may not be invariant. For example if L(x, v) = \frac{1}{2} |v|^{2} − ϕ(x) is a mechanical lagrangian and k < e_{0} is a regular energy level, then any such measure will be supported on ∂E^{−1}\{k\}, where v = 0 and vL_{v} = \frac{1}{2} |v|^{2} = 0. Nevertheless, ∂E^{−1}{k} has no invariant subsets.

In the following lemma we use ideas of Dias Carneiro [2]. 4.8. Lemma. For all e_{0} < k < c(L) we have that γ(k) < 0.

Proof:

From (38) we get that given any k > e_{0} and (p, v) ∈ T M, there exists a unique λ > 0 such that E(p, λ v) = k. Moreover, λ = λ(p, v, k) : T M×]e_{0}, +∞[→]0, +∞[ is a smooth function.

Let μ be an invariant measure μ ∈ M(L) such that

∫ (L + c(L)) dμ = 0 .  (40)

By theorem 4 and the fact that static curves have energy level c(L) we have that supp(μ) ⊆ E^{−1}(c(L)), and by the Poincaré recurrence theorem we have that μ ∈ \overline{C(c(L))}.

For k > e_{0} define the measure ν_{k} on E^{−1}(k) by

∫_{E^{−1}(k)} f dν_{k} := A(k) ∫_{E^{−1}(k)} \frac{f(p, λ(p, v, k)v)}{λ(p, v, k)} dμ(p, v)  (41)

for any continuous function f : E^{−1}(k) → ℝ, where

A(k) := (∫ \frac{1}{λ(p, v, k)} dμ(p, v))^{−1} .

Then ν_{k} ∈ \overline{C(k)} and ν_{c(L)} = μ. This measure ν_{k} is just the (probability) measure obtained by reparametrizing the solutions of (E-L) on π(supp(μ)) so as to have energy k. This process is reversible, i.e. we can recover μ by reparametrizing ν_{k}. Let

g(k) := ∫ v · L_{v} dν_{k} = k + ∫ L dν_{k} .

Then g is a differentiable function with derivative

g′(k) = 1 + A(k) ∫ \frac{∂}{∂k} (\frac{L(p, λv)}{λ}) dμ + A′(k) ∫ \frac{L(p, λv)}{λ} dμ .

34 G. CONTRERAS, J. DELGADO, AND R. ITURRIAGA

If we change the reference energy level c(L) to k_{1} > e_{0}, we can use ν_{k_{1}} instead of μ on formula (41). The function g(k) does not change but now λ(k_{1}) ≡ 1, A(k_{1}) = 1 and

g′(k_{1}) = 1 + ∫ \frac{∂}{∂k} (\frac{𝕃}{λ}) dν_{k_{1}} + A′(k_{1}) ∫ L dν_{k_{1}} ,

where 𝕃(p, v) := L(p, λ v). We compute this derivative:

\frac{∂}{∂k} (\frac{𝕃}{λ}) |_{k=k_{1}} = \frac{𝕃_{k} λ − 𝕃 λ_{k}}{λ^{2}} = 𝕃_{k} − λ_{k} 𝕃|_{k=k_{1}} ,
𝕃_{k} = v · L_{v} λ_{k} .

Since E(p, λ v) = k, we have that

\frac{∂E}{∂k} = E_{v} · (λ_{k} v) = (v · L_{vv} · v) λ_{k} = 1 ,

λ_{k} = \frac{1}{v · L_{vv} · v} and 𝕃_{k} = \frac{v · L_{v}}{v · L_{vv} · v} .

Moreover

\frac{∂}{∂k} (\frac{1}{λ}) |_{k=k_{1}} = −\frac{λ_{k}}{λ^{2}} |_{k=k_{1}} = −\frac{1}{v · L_{vv} · v}
A′(k_{1}) = −\frac{1}{A(k_{1})} ∫ \frac{∂}{∂k} (\frac{1}{λ}) dμ = ∫ \frac{1}{v · L_{vv} · v} dν_{k_{1}} .

Therefore

\begin{aligned}g′(k) & = 1 + ∫ \frac{v · L_{v}}{v · L_{vv} · v} dν_{k} − ∫ \frac{L}{v · L_{vv} · v} dν_{k} + (∫ \frac{1}{v · L_{vv} · v} dν_{k}) (∫ L dν_{k}) \\ g′(k) & = 1 + ∫ \frac{k}{v · L_{vv} · v} dν_{k} + (∫ \frac{1}{v · L_{vv} · v} dν_{k}) (∫ L dν_{k}) \\ g′(k) & = (∫ (L + k) dν_{k}) (∫ \frac{1}{v · L_{vv} · v} dν_{k}) + 1 \end{aligned}

g′(k) = b(k) g(k) + 1 ,  (42)

where b(k) := ∫ (v · L_{vv} · v)^{−1} dν_{k} > 0. Let

B(k) := ∫_{e_{0}}^{k} b(s) ds

and h(k) := e^{−B(k)} g(k). From (42) we get that h′(k) = e^{−B(k)} > 0 .

By (40), h(c(L)) = 0, therefore h(k) < 0 for all e_{0} < k < c(L). And thus

γ(k) ≤ g(k) = e^{B(k)} h(k) < 0

for all e_{0} < k < c(L).

Proof of lemma 4.7:

We first prove (a). Using local coordinates we can assume that M = ℝ^{n} and T M = ℝ^{n} × ℝ^{n}. For v small and p near p_{0} such that E(p, v) = k, we have that

E = v · L_{v} − L ,
k = (v · L_{v}(p, 0) + v · L_{vv}(p, 0) · v + O(v^{3}))

− (L(p, 0) + v · L_{v}(p, 0) + \frac{1}{2} v · L_{vv}(p, 0) · v + O(v^{3})) ,

\frac{1}{2} v · L_{vv}(p, 0) · v + O(v^{3}) = k + L(p, 0) .

Since k = −L(p_{0}, 0), there exists a function F (p, v) defined on a neighbourhood of (p_{0}, 0) on the energy level E(p, v) = k by

F (p, v) |v|^{2} := k + L(p, 0)
= L_{p}(p_{0}, 0) Δp + \frac{1}{2} Δp L_{pp}(p_{0}, 0) Δp + O(Δp^{3}) ,  (43)

where Δp = p − p_{0}, F (p, v) is smooth and F (p_{0}, v) > 0. We have that

D_{p}E(p_{0}, 0) = −L_{p}(p_{0}, 0) = 0 .  (44)

Since the left hand side of (43) is positive, using (44) we have that Δp L_{pp}(p_{0}, 0) Δp > 0. Then there exists a function G(p, v) > A > 0 such that for v small and p near p_{0} such that E(p, v) = k, we have

|v|^{2} = G(p, v)^{2} |Δp|^{2} ,
|v| ≥ A |Δp| .  (45)

Suppose that there exists a differentiable curve p : [0, δ] → M such that p(0) = p_{0} and E(p(t), ṗ(t)) = k. For simplicity suppose that δ = 2. Let x(t) := |p(t) − p_{0}|. Writing v(t) = ṗ(t), we have that

\frac{d}{dt}x(t)^{2} = 2 x ẋ = 2 ⟨p(t) − p_{0}, v(t)⟩ ,
ẋ(t) = ⟨\frac{p(t) − p_{0}}{|p(t) − p_{0}|}, v(t)⟩ ≥ −B |v(t)| .

for some B > 0, because p(t) is differentiable at t = 0. From (45), we have that

ẋ(t) ≥ −B |v(t)| ≥ −AB x(t) ,
x(t) ≥ x(1) exp(−AB (t − 1)) .

In particular x(0) ≠ 0. This contradicts the choice of p(t).

Now we prove (b). Since D_{p}E(q_{0}, 0) = L_{p}(q_{0}, 0) ≠ 0, then θ(k) is locally a codimension 1 submanifold near q_{0}. Let q(t) ∈ θ(k) be given by the condition

L_{p}(q(t), 0) (p(t) − q(t)) = 0 .

Using formula (43) with Δp = p(t) − q(t), we have that

F (p, v) |v|^{2} = L_{p}(q(t), 0) Δp + O (|Δp|^{2})

and then

|v| ≥ A \sqrt{|Δp|} ,

for some A > 0. Let y(t) := |p(t) − q(t)|. Computing \frac{d}{dt} y(t)^{2}, we obtain

ẏ(t) = ⟨\frac{p(t) − q(t)}{|p(t) − q(t)|} , v(t)⟩ ≤ −B |v(t)| ≤ −A B \sqrt{y(t)} ,

for some B > 0. Therefore

\frac{ẏ}{2 y^{\frac{1}{2}}} ≤ −\frac{A B}{2} and y(t) ≤ \frac{A^{2} B^{2}}{4} (t − t_{0})^{2} .

For future reference, we also note that

|ẏ(t)| ≥ b |v(t)| ≥ b_{1} \sqrt{y(t)} ,  (46)
y(t) ≥ b_{2} (t − t_{0})^{2} ,  (47)

for some b, b_{1}, b_{2} > 0.

5. Properties of weaker global minimizers.

Definition. We say that a solution x(t) of (E-L) is a minimizer (resp. forward minimizer) if

S_{L}(x|_{[t_{0},t_{1}]}) ≤ S_{L}(y)

for every t_{0} ≤ t_{1} (resp. 0 < t_{0} ≤ t_{1}) and every absolutely continuous y : [t_{0}, t_{1}] → M, with y(t_{i}) = x(t_{i}), i = 1, 2.

Denote by Λ(L) (resp. Λ^{+}(L)) the set of (p, v) ∈ T M such that the solution x(t) of (E-L) with initial condition (x(0), ẋ(0)) = (p, v) is a minimizer (resp. forward minimizer).

Let w ∈ Λ^{+}(L) for 0 ≤ s ≤ t define δ_{w}(s, t) by

S_{L+c}(x_{w}|_{[s,t]}) = Φ_{c}(x_{w}(s), x_{w}(t)) + δ_{w}(s, t).  (48)

It is clear that

δ_{w}(s, t) ≥ 0.  (49)

The triangle inequality for Φ implies that

δ_{w}(s, t) + δ_{w}(t, r) ≤ δ_{w}(s, r)  (50)

for any w ∈ Λ^{+}(L) and 0 ≤ s ≤ t ≤ r.

Claim. δ_{w}(s, t) is uniformly bounded on w ∈ Λ^{+}(L), and 0 ≤ s ≤ t < ∞. We shall prove first theorems XIV, XII and XIII and then the claim.

Theorem XIV.

(a) There exists C > 0 such that setting c = c(L)

|S_{L+c}(x_{w}|_{[s,t]})| ≤ C

for every w ∈ Λ^{+}(L) and all 0 ≤ s ≤ t.

(b) If w ∈ Λ^{+}(L) and p ∈ M is such that p = lim_{n→∞} x_{w}(t_{n}) for some sequence

t_{n} → ∞ then the limit

lim_{n→∞} S_{L+c}(x|_{[0,t_{n}]})

exists and does not depend in the sequence {t_{n}}.

Proof:

Item (a) is straightforward consequence of the claim and the definition of δ_{w}. From equation (50) it follows that the function t → δ_{w}(0, t) is increasing. Then the claim implies that D := lim_{t→+∞} δ_{w}(0, t) exists. Thus

lim_{n→∞} S_{L+c}(x_{w}|_{[0,t_{n}]}) = lim_{n→∞} Φ_{c}(x_{w}(0), x_{w}(t_{n})) + lim_{n→∞} δ_{w}(0, t_{n})
= Φ_{c}(x_{w}(0), p) + D .

Theorem XII.

(a) The ω-limit set of an orbit in Λ^{+}(L) is contained in Σ̂(L).

38 G. CONTRERAS, J. DELGADO, AND R. ITURRIAGA

(b) The α and ω-limit sets of an orbit in Λ(L) are contained in Σ̂(L). Proof:

We only prove (a). Let w ∈ Λ^{+}(L). Let (p, v) ∈ ω(w), T > 0 and let ⟨n_{k}⟩ be a sequence in ℝ^{+} such that

n_{k+1} > n_{k} + T ,

(p, v) = lim_{k}(x_{w}(n_{k}), ẋ_{w}(n_{k})) .

Let

p_{k} := x_{w}(n_{k}) , q_{k} := x_{w}(n_{k} + T) ,

(q, u) := (x_{v}(T), ẋ_{v}(T)) = lim_{k} (q_{k} , ẋ_{w}(n_{k} + T)) .

we have that

S_{L+c}(x_{w}|_{[n_{k},n_{k}+T]}) = Φ_{c}(p_{k}, q_{k}) + δ_{w}(n_{k}, n_{k} + T) ,
S_{L+c}(x_{w}|_{[n_{k}+T,n_{k+1}]}) = Φ_{c}(q_{k}, p_{k}) + δ_{w}(n_{k} + T, n_{k+1}) ,
S_{L+c}(x_{w}|_{[n_{k},n_{k+1}]}) = Φ_{c}(p_{k}, p_{k+1}) + δ_{w}(n_{k}, n_{k+1}) .

By the claim and equation (50) there is a constant Q such that

∑_{k=1}^{∞}(δ_{w}(n_{k}, n_{k} + T) + δ_{w}(n_{k} + T, n_{k+1})) ≤ ∑_{k=1}^{∞} δ_{w}(n_{k}, n_{k+1})

≤ lim_{t→∞} δ_{w}(0, t) ≤ Q .

Therefore

lim_{k→∞} δ_{w}(n_{k}, n_{k+1}) = lim_{k→∞} δ_{w}(n_{k}, n_{k} + T) = lim_{k→∞} δ_{w}(n_{k} + T, n_{k+1}) = 0 . (51)

Hence

S_{L+c}(x_{w}|_{[0,T]}) = lim_{k} S_{L+c}(x_{w}|_{[n_{k},n_{k}+T]})

= lim_{k} Φ_{c}(p_{k}, q_{k}) = Φ_{c}(p, q) .

LAGRANGIAN FLOWS 39

and from (51),

Φ_{c}(p, q) + Φ_{c}(q, p) = lim_{k} Φ_{c}(p_{k}, q_{k}) + lim_{k} Φ_{c}(q_{k}, p_{k})

= lim_{k} [ S_{L+c}(x_{w}|_{[n_{k},n_{k}+T]}) + S_{L+c}(x_{w}|_{[n_{k}+T,n_{k+1}]}) ]

= lim_{k} [ Φ_{c}(p_{k}, p_{k}) + δ_{w}(n_{k}, n_{k+1}) ]

= Φ_{c}(p, p) = 0 .

This implies that ω(w) ⊆ Σ̂(L). Theorem XIII. f_{t}|_{Λ(L)} is chain transitive. Proof: Let v, w ∈ Λ(L) and ε > 0. It is enough to prove that there exists an ε-chain in Λ(L) joining ω(w) to α(v). By theorem XIII, α(v)∪ω(w) ⊆ Σ̂(L) ⊆ Σ(L). By theorem V there exists such ε-chain contained in Σ(L) ⊆ Λ(L). This completes the proof. Proof of the claim:

Let

A := 2 max { L(p, v) + c(L) | ∥v∥ ≤ diamM + 1 }
B := max { |Φ_{c}(p, q)| | p, q ∈ M }
Q := 3 max { A, B } + 2 .

Suppose that there exist ω ∈ Λ^{+}(L), 0 ≤ a ≤ b such that δ_{w}(a, b) > Q.

Let γ : [a, T_{b}] → M be in AC(x_{w}(a), x_{w}(b)) and such that

S_{L+c}(γ) < Φ_{c}(x_{w}(a), x_{w}(b)) + 1 .

We have that

S_{L+c}(γ) < S_{L+c}(x_{w}|_{[a,b]}) − δ_{w}(a, b) + 1 ,
S_{L+c}(γ) < S_{L+c}(x_{w}|_{[a,b]}) − Q + 1 ,  (52)
S_{L+c}(γ) < S_{L+c}(x_{w}|_{[a,b]}) − (A + B) .  (53)

Suppose that T_{b} ≥ b − a. Let η : [0, 1] → M be a geodesic on M such that

η(0) = x_{w}(b) , η(1) = x_{w}(T_{b} + 1) , \left∥{ \atop η̇}\right∥ ≤ diam(M) .

40 G. CONTRERAS, J. DELGADO, AND R. ITURRIAGA

Then

S_{L+c}(η) < A .

Since

−B ≤ Φ_{c}(x_{w}(b), x_{w}(T_{b} + 1)) ≤ S_{L+c}(x_{w}|_{[b,T_{b}+1]}) ,

using (53) (or (54)), we have that

S_{L+c}(γ ∗ η) < S_{L+c}(x_{w}|_{[a,b]}) − (A + B) + A
< S_{L+c}(x_{w}|_{[a,b]}) + S_{L+c}(x_{w}|_{[b,T_{b}+1]})
S_{L+c}(γ ∗ η) < S_{L+c}(x_{w}|_{[a,T_{b}+1]}) .

This contradicts the hypothesis w ∈ Λ^{+}(L).

Now suppose that T_{b} < b − a. Let u ∈ Σ̂(L) and let λ : [0, 1] → M be a geodesic on M such that

λ(0) = x_{w}(b) = γ(T_{b}) , λ(1) = π u , ∥λ̇∥ ≤ diam M ,

and let \overline{λ} : [0, 1] → M be \overline{λ}(t) := λ(1 − t). Then

S_{L+c}(λ) + S_{L+c}(\overline{λ}) ≤ \frac{A}{2} + \frac{A}{2} = A .

Let τ > (b − a) − T_{b} and let σ : [0, T_{σ}] → M be a curve in AC(x_{u}(τ), π u) such that

S_{L+c}(σ) < −Φ_{c}(π u, x_{u}(τ)) + 1 .

Let \overline{γ} := γ ∗ λ ∗ x_{u}|_{[0,τ]} ∗ σ ∗ \overline{λ}. This curve is in AC(x_{w}(a), x_{w}(b)), it is defined on a time interval of length

T_{b} + 1 + τ + T_{σ} + 1 > T_{b} + τ > b − a ,

and has (L + c)-action

S_{L+c}(\overline{γ}) ≤ S_{L+c}(γ) + (S_{L+c}(λ) + S_{L+c}(\overline{λ})) + (S_{L+c}(x_{u}|_{[0,τ]}) + S_{L+c}(σ))
≤ S_{L+c}(γ) + A + (Φ_{c}(π u, x_{u}(τ)) − Φ_{c}(π u, x_{u}(τ)) + 1)
< (S_{L+c}(x_{w}|_{[a,b]}) − Q + 1) + A + 1 ,
S_{L+c}(\overline{γ}) < S_{L+c}(x_{w}|_{[a,b]}) − (A + B) .  (54)

Now the same argument as in the case T_{b} ≥ b, using \overline{γ} instead of γ, gives a contradiction.

References

[1] R. Abraham & J. E. Marsden. Foundations of Mechanics. Benjamin: London, 1978.

[2] M.J. Dias Carneiro. On Minimizing Measures of the Actions of Autonomous Lagrangians. Nonlinearity, 8 (1995) no. 6, 1077-1085.

[3] G. Contreras, R. Iturriaga. Convex Hamiltonians without conjugate points. Preprint.

[4] R. Mañé. Global Variational Methods in Conservative Dynamics. 18\underline{o} Coloquio Bras. de Mat. IMPA. Rio de Janeiro, 1991.

[5] R. Mañé. On the minimizing measures of Lagrangian dynamical systems. Nonlinearity, 5, (1992), no. 3, 623-638.

[6] R. Mañé. Generic properties and problems of minimizing measures of lagrangian systems. Nonlinearity, 9, (1996), no.2, 273-310.

[7] R. Mañé. Lagrangian Flows: The Dynamics of Globally Minimizing Orbits. To appear. Proc. Int. Congress in Dyn. Sys., Montevideo 1995, Pitman Research Notes in Math.

[8] J. Mather. Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207, (1991), no. 2, 169-207.

[9] M. Morse. Calculus of variations in the large. Amer. Math. Soc. Colloquium Publications vol. XVIII, 1934.

[10] S. Schwartzman. Asymptotic cycles. Ann. of Math. (2) 66 (1957) 270, 284.

Depto. de Matemática. PUC-Rio, R. Marquês de São Vicente, 225, 22453-900 Rio de Janeiro, Brasil.

  E-mail address: gonzalo@mat.puc-rio.br

Depto. de Matemática. PUC-Rio, R. Marquês de São Vicente, 225, 22453-900 Rio de Janeiro, Brasil.

  E-mail address: jdelgado@mat.puc-rio.br

CIMAT, A.P. 402, 3600, Guanajuato. Gto., México.

  E-mail address: renato@fractal.cimat.mx