A MINIMAX SELECTOR FOR A CLASS OF HAMILTONIANS ON COTANGENT BUNDLES

RENATO ITURRIAGA AND HÉCTOR SÁNCHEZ-MORGADO

Abstract. We construct a minimax selector for eventually quadratic hamiltonians on cotangent bundles. We use it to give a relation between Hofer’s energy and Mather’s action minimizing function. We also study the local flatness of the set of twist maps.

1. Introduction

One purpose of this paper is to have some understanding of the relation between two approaches to the study of hamiltonian systems, namely the dynamical point of view of Mather [7] and the geometric point of view of Hofer and Zehnder [4].

The main result concerning this purpose is an extension to arbitrary cotangent bundles of a result given by Siburg [10] for the cotangent bundle of an n - torus. See Theorem 3 below.

To do so we follow the suggestion of Bialy and Polterovich [1] and construct a minimax selector for a certain class of Hamiltonians. They conjectured that a selector could be constructed for symplectic manifolds admiting a nice Floer homology. This program has been carried out by Schwarz [8] for symplectic compact manifolds using Floer homology.

Since our interest is mainly on convex superlinear Hamiltonians defined on cotangent bundles, we found very appealing to use the methods developed by Golé [3] to find periodic orbits for Hamiltonians that are not necessarily convex but quadratic outside of a neighbourhood of the zero section.

More precisely, let M be a compact connected riemannian manifold and let π : T^{∗}M → M be its cotangent bundle endowed with its natural symplectic structure Ω = −dλ. Fix R > 0 smaller than the injectivity radius for the given metric of M and let

U_{r} = {(q, p) : |p| ≤ r}.

Date: Jan-2000.

Both authors were partially supported by CONACYT-México grant # 28489-E.

2000 Mathematics Subject Classification 37J10, 37J50.

2 R. ITURRIAGA AND H. SÁNCHEZ M.

Let H_{0}(U_{r}) be the set of smooth functions on T^{∗}M ×𝕊^{1} with support contained in U_{r}×𝕊^{1}. Let H be the set of smooth functions on T^{∗}M ×𝕊^{1} such that H = \frac{R}{2}(|p|^{2} − 1) for |p| ≥ 1. For H ∈ H_{0} or H ∈ H, let φ_{t}(x) = φ_{t}^{H}(x) be the solution to the Hamiltonian system ẋ = X_{H}(x, t) with φ_{0}(x) = x and define φ_{s}^{t}(x) = φ_{t} ∘ φ_{s}^{−1}(x).

Letting

D_{0} = {φ_{1}^{H} : H ∈ H_{0}(U_{1})}, D = {φ_{1}^{H} : H ∈ H},

for any ψ ∈ D we have D = {ψ ∘ ϕ : ϕ ∈ D_{0}}.

For H ∈ H_{0} or H ∈ H, let

E^{+}(H) = − ∫_{0}^{1} min_{x∈U_{1}} H(x, t)dt, E^{−}(H) = − ∫_{0}^{1} max_{x∈U_{1}} H(x, t)dt,

and

∥H_{t}∥ = max_{x∈U_{1}} H(x, t) − min_{x∈U_{1}} H(x, t).

If H ∈ H_{0} or H ∈ H, then H = 0 on ∂U_{1} and so E^{+}(H) ≥ 0 ≥ E^{−}(H).

For H ∈ H, define the contractible action spectrum

σ_{c}(H) = {∫_{Γ} λ − Hdt : Γ ⊂ U_{1} is a contractible 1-periodic orbit of φ_{t}^{H}}

Theorem 1. For any nonzero v ∈ H^{∗}(M, ℝ) and H ∈ H there is c_{v}(H) ∈ ℝ such that

1. c_{v}(H) ∈ σ_{c}(H)

2. For K ∈ H, E^{−}(K − H) ≤ c_{v}(K) − c_{v}(H) ≤ E^{+}(K − H).

In particular, for H ∈ H define the selectors

γ_{−}(H) = min σ_{c}(H), γ_{+}(H) = max σ_{c}(H).

These selectors have the following key properties:

Theorem 2. Let H, K ∈ H. Then 1. If φ_{1}^{H} = φ_{1}^{K} then γ_{±}(H) = γ_{±}(K).

2. If H < 0 in the interior of U_{1} then γ_{−}(H) > 0.

3. E^{−}(H) ≤ γ_{−}(H) ≤ γ_{+}(H) ≤ E^{+}(H).

4. If H ≤ 0 then γ_{−}(H) ≥ 0.

If φ is generated by H ∈ H i.e. φ_{1}^{H} = φ, define γ_{±}(φ) = γ_{±}(H). For φ ∈ D define its energy as

E(φ) = inf {∫_{0}^{1} ∥H_{t}∥ dt : H ∈ H, φ_{1}^{H} = φ} .

For ϕ ∈ D_{0} let

∥ϕ∥ = inf{∫_{0}^{1} ∥K_{t}∥ dt : K ∈ H_{0}(U_{1}), φ_{1}^{K} = ϕ}

If φ, ψ ∈ D define d(φ, ψ) = ∥ψ^{−1}φ∥.

As proved by Lalonde and Mc Duff in [6], ∥·∥ is a nondegenerate norm and then d is a metric in D.

For L : T M ×𝕊^{1} → ℝ a convex superlinear Lagrangian with complete Euler-Lagrange flow, let β = β_{L} : H_{1}(M, ℝ) → ℝ be the Mather’s beta function. The following theorem gives a conection between the points of view of Mather and Hofer-Zehnder.

Theorem 3. If φ is generated by a convex Hamiltonian H ∈ H, then

β_{L}(0) ≤ E(φ)

where L is the Legendre transform of H.

A twist map F is a diffeomorphism of a neighborhood U of the zero section in T^{∗}M onto itself satisfying the following:

• If F (q, p) = (Q, P), then the map Φ(q, p) = (q, Q) is an embedding of U in M × M

• F is exact symplectic, that is, F^{∗}λ − λ = d(S ∘ Φ) for some real

function S on Φ(U). The function S is called a generating function for F.

Let T be the set of twist maps φ|U_{1} with φ ∈ D. Any map in T has a generating function satisfying

• \frac{∂^{2}S}{∂q∂Q} is negative definite.

• S(q, Q) = \frac{d(q, Q)^{2}}{2R} for d(q, Q) ≥ R.

The following proposition gives the local flatness of Hofer’s metric for twist maps and is a generalization of Theorem 1 in Siburg’s paper [9].

Proposition 1. For any φ ∈ T there is a C^{1} neighborhood O of φ such that if φ_{0}, φ_{1} ∈ O and S_{0}, S_{1} are their generating functions as above, then

d(φ_{0}, φ_{1}) = ∥S_{0} − S_{1}∥ .

The plan of the article is as follows, in section 2 we introduce the generating families whose critical values belong to the contractible action spectrum. In section 3 we prove theorems 1 and 2. In section 4 we recall the definition of Mather’s beta function and prove theorem 3. In section 5 we apply the results of Lalonde and Mc Duff [5] describing

4 R. ITURRIAGA AND H. SÁNCHEZ M.

the geodesics of Hofer’s metric for certain classes of symplectic manifolds that include cotangent bundles of compact manifolds and prove proposition 1.

We thank K.F. Siburg for his useful comments on a first draft of this paper.

2. Generating families

Let g_{t} be the geodesic flow on T^{∗}M and d be the distance in M defined by the metric. The map g_{R}|U_{1} is twist since G_{R}(q, p) = (q, exp_{q}(Rp)) is an embedding of U_{1} in M × M and S(q, Q) = d(q, Q)^{2}/2R is a generating function. Write g_{−R} = (Q_{g}, P_{g}).

Since the C^{1} distance from φ_{s}^{t} to the identity on U_{1} is arbitrarly small when s − t is suficiently small, there is N such that

Φ_{s}^{t}(q, p) = (q, Q_{g} ∘ φ_{s}^{t}(q, p))

is an embedding of U_{1} in M × M for |t − s| ≤ 1/N. For 0 ≤ j < N, let s_{j} = j/N,

φ^{j} = φ_{s_{j}}^{s_{j+1}}, F_{2j} = g_{−R} ∘ φ^{j}, F_{2j+1} = g_{R}, Φ_{2j} = Φ_{s_{j}}^{s_{j+1}}, Φ_{2j+1} = G_{R}

Let Γ_{x}^{j}(t) = φ_{s_{j}}^{t}(x), t ∈ [s_{j}, s_{j+1}], and define the action A_{j} : U_{1} → ℝ by

A_{j}(x) = ∫_{Γ_{x}^{j}} λ − Hdt, (1)

then dA_{j} = (φ^{j})^{∗}λ − λ. Thus, defining

S_{2j} = (A_{j} − S ∘ G_{R} ∘ F_{2j}) ∘ Φ_{2j}^{−1},

we have

\begin{aligned}d S_{2j} ∘ Φ_{2j} & = dA_{j} − F_{2j}^{∗} d S ∘ G_{R} \\ & = (φ^{j})^{∗}λ − λ − (φ^{j})^{∗}g_{−R}^{∗}(g_{R}^{∗}λ − λ) \\ & = F_{2j}^{∗}λ − λ. \end{aligned} (2)

Therefore F_{2j} is a twist map with generating function S_{2j}. Let S_{2j+1} = S and

M_{N} = {(q_{0}, q_{1}, …, q_{2N}) : (q_{j}, q_{j+1}) ∈ Φ_{j}(U_{1}), q_{2N} = q_{0}}.

The function W_{N} : M_{N} → ℝ given by

W_{N} : q = (q_{0}, q_{1}, …, q_{2N}) ↦ ∑_{j=0}^{2N−1} S_{j}(q_{j}, q_{j+1}).

is called a generating family. Letting (q_{j}, p_{j}) = Φ_{j}^{−1}(q_{j}, q_{j+1}), p_{2N} = p_{0}, (q_{j+1}, P_{j+1}) = F_{j}(q_{j}, p_{j}), P_{2N} = P_{0}, we have by (2)

dW_{N} = ∑_{j=0}^{2N−1}(P_{j} − p_{j})dq_{j}. (3)

Therefore q is a critical point of W_{N} if and only if (q_{j+1}, p_{j+1}) = F_{j}(q_{j}, p_{j}) for 0 ≤ j < 2N, so (q_{2j}, p_{2j}) = φ_{j/N} = (q_{0}, p_{0}), (q_{0}, p_{0}) is a fixed point of φ_{1}, and Γ(t) = φ_{t}(q_{0}, p_{0}) is a 1-periodic orbit in U_{1}. In such a case

W_{N}(q) = ∑_{j=0}^{N−1} A_{j}(q_{j}, p_{j}) = ∫_{Γ} λ − Hdt.

Proposition 2. Given ε > 0 there is N(ε) such that for N ≥ N(ε) we have

∥\frac{∂P_{2j+1}}{∂q_{2j+1}} − \frac{∂p_{2j+1}}{∂q_{2j+1}} ∥ < ε (4)

for any j such that F_{2j} ∘ Φ_{2j}^{−1}(q_{2j}, q_{2j+1}) = G_{R}^{−1}(q_{2j+1}, q_{2j+2}), and

∥\frac{∂P_{2j}}{∂q_{2j}} − \frac{∂p_{2j}}{∂q_{2j}} ∥ < ε (5)

for any j such that g_{R} ∘ G_{R}^{−1}(q_{2j−1}, q_{2j}) = Φ_{2j}^{−1}(q_{2j}, q_{2j+1}).

Proof A. From (q_{2j+1}, P_{2j+1}) = F_{2j} ∘Φ_{2j}^{−1}(q_{2j}, q_{2j+1}), F_{2j} = g_{−R} ∘φ^{j} one gets

\frac{∂P_{2j+1}}{∂q_{2j+1}} = \frac{∂(P_{g} ∘ φ^{j})}{∂p} ∘ Φ_{2j}^{−1} (\frac{∂(Q_{g} ∘ φ^{j})}{∂p} ∘ Φ_{2j}^{−1})^{−1} .

B. From (q_{2j+1}, p_{2j+1}) = G_{R}^{−1}(q_{2j+1}, q_{2j+2}) one gets

\frac{∂p_{2j+1}}{∂q_{2j+1}} = \frac{∂P_{g}}{∂p} ∘ g_{R} ∘ G_{R}^{−1} (\frac{∂Q_{g}}{∂p} ∘ g_{R} ∘ G_{R}^{−1})^{−1} .

C. From q_{2j+1} = Q_{g} ∘ φ^{j}(q_{2j}, p_{2j}) one gets

\frac{∂p_{2j}}{∂q_{2j}} = − (\frac{∂(Q_{g} ∘ φ^{j})}{∂p} ∘ Φ_{2j}^{−1})^{−1} \frac{∂(Q_{g} ∘ φ^{j})}{∂q} ∘ Φ_{2j}^{−1}.

D. From q_{2j−1} = Q_{g}(q_{2j}, P_{2j}) one gets

\frac{∂P_{2j}}{∂q_{2j}} = − (\frac{∂Q_{g}}{∂p} ∘ g_{R} ∘ G_{R}^{−1})^{−1} \frac{∂Q_{g}}{∂q} ∘ g_{R} ∘ G_{R}^{−1}.

6 R. ITURRIAGA AND H. SÁNCHEZ M.

Suppose that F_{2j} ∘ Φ_{2j}^{−1}(q_{2j}, q_{2j+1}) = G_{R}^{−1}(q_{2j+1}, q_{2j+2}) so that

φ^{j} ∘ Φ_{2j}^{−1}(q_{2j}, q_{2j+1}) = g_{R} ∘ G_{R}^{−1}(q_{2j+1}, q_{2j+2}),

then

D(g_{−R} ∘ φ^{j})(Φ_{2j}^{−1}(q_{2j}, q_{2j+1})) =

Dg_{−R}(g_{R} ∘ G_{R}^{−1}(q_{2j+1}, q_{2j+2}))Dφ^{j}(Φ_{2j}^{−1}(q_{2j}, q_{2j+1}))

Suppose g_{R} ∘ G_{R}^{−1}(q_{2j−1}, q_{2j}) = Φ_{2j}^{−1}(q_{2j}, q_{2j+1}), then

D(g_{−R} ∘ φ^{j})(Φ_{2j}^{−1}(q_{2j}, q_{2j+1})) = D(g_{−R} ∘ φ^{j})(g_{R} ∘ G_{R}^{−1}(q_{2j−1}, q_{2j})).

Taking N sufficiently large, one makes any φ^{j}|U_{1} as C^{1}- close to the identity as one wants. Thus, (4) follows from items A, B, and (5) follows from items C,D.

As observed by Golé [3] one can define a path σ_{k}(q, Q) between q and Q ∈ Φ_{k}(U_{1} ∩ T_{q}^{∗}M) that for k odd concides with the unique geodesic between these points. Therefore to each q ∈ M_{N} we associate a polygonal loop c(q). We can work with the component M_{N}^{e} of M_{N} consisting of points q such that c(q) is homotopically trivial. Thus, σ_{c}(H) is the set of critical values of W_{N} : M_{N}^{e} → R.

Golé proved that the gradient flow of W_{N} has the index pair (B_{N}, B_{N}^{−}),

B_{N} = {q ∈ M_{N}^{e} : d(q_{j}, q_{j+1}) ≤ |a_{j}|R},
B_{N}^{−} = {q ∈ B_{N} : d(q_{j}, q_{j+1}) = |a_{j}|R for some even j},

where

a_{j} = \begin{cases}1 & j odd \\ \frac{1}{N} − 1 & j even \end{cases}

Lemma 1. Floer [2].

Let ζ^{τ} be a one parameter family of flows on a manifold M. Suppose that Σ^{0} is a compact submanifold invariant under ζ^{0}. Assume moreover that Σ^{0} is normally hyperbolic i. e. there is a decomposition

T_{Σ^{0}}M = T Σ^{0} ⊕ E^{+} ⊕ E^{−}

invariant under the covariant linearization of the vector field V_{0} generating ζ^{0} with respect to some metric ⟨, ⟩, so that for some m > 0:

⟨ξ, DV_{0}ξ⟩ ≤ −m⟨ξ, ξ⟩, ξ ∈ E^{−}
⟨ξ, DV_{0}ξ⟩ ≥ m⟨ξ, ξ⟩, ξ ∈ E^{+}

Suppose that there is a retraction α : M → Σ^{0} and that there is an index pair (B, B^{−}) for all ζ^{τ}. Then • There is u ∈ H^{∗}(B, B^{−}) of dimension dim E^{+} so that the map

T : H^{∗}(Σ^{0}) → H^{∗}(B, B^{−}), T (v) = (α|_{B})^{∗}v ∪ u

is an isomorphism.

• If Σ^{τ} denotes the maximal invariant for the flow ζ^{τ}, the homomorphism in Cech cohomology (α|Σ^{τ})^{∗} : H^{∗}(Σ^{0}) → H^{∗}(Σ^{τ}) is injective.

3. Critical values of the action

Let H ∈ H, and define H_{0} = \frac{R}{2}(|p|^{2} − 1). Let H_{τ} = H_{0} + τ(H − H_{0}) and φ_{t}^{τ} be the corresponding flow. Let N be sufficiently large to have a decomposition of all φ_{1}^{τ} in 2N twist maps. Let W_{N}^{τ} be the corresponding generating family and ζ^{τ} its gradient flow.

Lemma 2. Golé [3]. Let Σ^{0} = {q ∈ M_{N} : q_{k} = q_{0}, ∀k}. Then Σ^{0} is a normally hyperbolic invariant set for ζ^{0} and it is a retract of M_{N}^{e}

By Lemma 1, T_{N} : H^{∗}(M) → H^{∗+k}(B_{N}, B_{N}^{−}) is an isomorphism. Let B_{N}^{a} = {q ∈ B_{N} : W_{N}(q) ≤ a}, j_{a} : B_{N}^{a} ↪ B_{N} and consider the

induced map j_{a}^{∗} : H^{∗}(B_{N}, B_{N}^{−}) → H^{∗}(B_{N}^{a}, B_{N}^{−}).

For N of the form N = 2^{m}, we follow Viterbo [11] and define for any nonzero v ∈ H^{∗}(M) = H^{∗}(Σ^{0})

c_{v}(H, m) = inf{a : j_{a}^{∗}T_{N}(v) ≠ 0}.

so that c_{v}(H, m) is a critical point of W_{N}.

Proposition 3. There is m such that for any v ∈ H^{∗}(M) and m ≥ m, c_{v}(H, m) = c_{v}(H, m).

Proof. Represent the points of M_{2N} in the form

q = (η_{0}, ξ_{0}, ξ_{1}, η_{1}, …, η_{2N−2}, ξ_{2N−2}, ξ_{2N−2}, η_{2N−1}, η_{2N} = η_{0})

so that

W_{2N}(q) = W_{2N}(η, ξ) =

∑_{j=0}^{N−1} S_{4j}(η_{2j}, ξ_{2j}) + S(ξ_{2j}, ξ_{2j+1}) + S_{4j+2}(ξ_{2j+1}, η_{2j+1}) + S(η_{2j+1}, η_{2j+2}).

Let h_{j} = G_{R} ∘ g_{−R} ∘ φ_{j/N}^{2j+1/2N} ∘ (Φ_{j/N}^{j+1/N})^{−1} and define h : M_{N} → M^{2N} by h(η) = (h_{0}(η_{0}, η_{1}), …, h_{N−1}(η_{2N−2}, η_{2N−1})). Then W_{2N}(η, h(η)) = W_{N}(η) and

\frac{∂W_{2N}}{∂ξ_{j}} (η, h(η)) = 0.

Define f : M_{2N} → ℝ by f(η, ξ) = W_{2N}(η, ξ) − W_{N}(η), then

(6)  \frac{∂f}{∂ξ_{2j+i}} = \frac{∂W_{2N}}{∂ξ_{2j+i}} = P_{4j+1+i} − p_{4j+1+i}, i = 0, 1
(7)  \frac{∂f}{∂η} = \frac{∂W_{2N}}{∂η} − \frac{∂W_{2N}}{∂η} (η, h(η)) − \frac{∂W_{2N}}{∂ξ} Dh.

Thus Df|graph h = 0 and

\frac{∂^{2}f}{∂(ξ_{2j}, ξ_{2j+1})} = [\begin{matrix}\frac{∂(P_{4j+1} − p_{4j+1})}{∂ξ_{2j}} & \frac{∂^{2}S}{∂q∂Q}(ξ_{2j}, ξ_{2j+1}) \\ \frac{∂^{2}S}{∂q∂Q}(ξ_{2j}, ξ_{2j+1}) & \frac{∂(P_{4j+2} − p_{4j+2})}{∂ξ_{2j+1}} \end{matrix}] .

Since \frac{∂^{2}S}{∂q∂Q} is negative definite on G_{R}(U_{1}), we have that

\frac{∂^{2}f}{∂(ξ_{2j}, ξ_{2j+1})}|graph h is invertible if \frac{∂(P_{4j+1+i} − p_{4j+1+i})}{∂ξ_{2j+i}}|graph h, i =

0, 1 are sufficiently small.

We know from Proposition 2 that this holds if N = 2^{m} is sufficiently large. In such a case graph h is a nondegenerate critical manifold of f. By the generalized Morse’s lemma there is a tubular neighborhood ψ : E → M_{2N}, with E = E^{+} ⊕ E^{−} a vector bundle over M_{N}, such that f(ψ(η, ζ)) = |ζ_{−}|^{2} − |ζ_{+}|^{2} and so

W_{2N}(ψ(η, ζ)) = W_{N}(η) + |ζ_{−}|^{2} − |ζ_{+}|^{2}. (8)

Consider the commutative diagram

\begin{aligned}& T_{2N} \\ & ↗ \\ H^{∗}(M) \\ & ↘ \\ & T_{N} \end{aligned} \begin{matrix}H^{∗+k+l}(B_{2N}, B_{2N}^{−}) & ⟶^{j_{a}^{∗}} & H^{∗+k+l}(B_{2N}^{a}, B_{2N}^{−}) \\ ↓^{ψ^{∗}} & & ↓^{ψ^{∗}} \\ H^{∗+k+l}(ψ^{−1}(B_{2N}, B_{2N}^{−})) & ⟶^{J_{a}^{∗}} & H^{∗+k+l}(ψ^{−1}(B_{2N}^{a}, B_{2N}^{−})) \\ ↑^{Thom} & & ↑≅ \\ H^{∗+k}(B_{N}, B_{N}^{−}) & ⟶^{j_{a}^{∗}} & H^{∗+k}(B_{N}^{a}, B_{N}^{−}) \end{matrix}

where the upwards isomorphisms are as in [11].

The maximal invariant set Σ_{a} for the gradient flow of W_{2N} in B_{2N}^{a} − B_{2N}^{−} consists of critical points and heteroclinic orbits. The critical points belong to graph h by (6) and so do the heteroclinic orbits by (8). Thus (B_{2N}^{a}, B_{2N}^{−}) and (B_{2N}^{a}, B_{2N}^{−}) ∩ ψ(E) are index pairs for Σ_{a} and then

ψ^{∗} : H^{∗}(B_{2N}^{a}, B_{2N}^{−}) ≅ H^{∗}(ψ^{−1}(B_{2N}^{a}, B_{2N}^{−})).

Therefore j_{a}^{∗}T_{2N}(v) = 0 if and only if j_{a}^{∗}T_{N}(v) = 0 and so

c(v, m) = c(v, m + 1).

We consider in particular

γ_{−}(H) = min_{q∈Σ^{1}} W (q) = min σ_{c}(H),
γ_{+}(H) = max_{q∈Σ^{1}} W (q) = max σ_{c}(H).

The following Proposition exhibits γ_{−}(H) as a minimax selector.

Proposition 4.

γ_{−}(H) = sup_{t<0} inf{W (q) : q ∈ ζ_{t}(B)}

Proof. For every t,

inf{W (q) : q ∈ ζ_{t}(B)} ≤ W (x_{0}) = γ(H),

so

sup_{t<0} inf{W (q) : q ∈ ζ_{t}(B)} ≤ γ(H).

On the other hand for every q ∈ B either ζ_{s}(q) is outside of B for some s < 0 and hence W (ζ_{t}(q)) ≥ W (x_{0}) for some t < 0 or ζ_{t}(q) → z for some z ∈ I. In any case, given ε > 0 there is t > 0 such that

inf{W (q) : q ∈ ζ_{t}(B)} ≥ W (x_{0}) − ε

Lemma 3. Viterbo [11]. Let S_{τ} be a smooth family of smooth functions. Let c(τ) = S_{τ}(x_{τ}) be a critical value obtained by minimax as

above. Assume that dS_{τ}(x) = 0 implies \frac{∂S_{τ}(x)}{∂τ} ≥ 0 (≤ 0). Then c(τ) is increasing (decreasing).

The following Lemma completes the proof of Theorem 1

Lemma 4. For v ∈ H^{∗}(M), H, K ∈ H

E^{−}(K − H) ≤ c_{v}(K) − c_{v}(H) ≤ E^{+}(K − H).

Proof. For τ ∈ [0, 1], define H_{τ} = H + τ(K − H),

Let S_{τ} = W_{τ} − E^{−}(K − H)τ then c_{v}(H_{τ}) − E^{−}(K − H)τ is a critical value of S_{τ} and

c_{v}(H_{τ}) = W_{τ}(x_{τ}) = ∫_{S^{1}} ψ_{τ}^{∗}(λ − H_{τ}dt)

where ψ_{τ}(t) = φ_{t}^{τ}(x_{τ}). Then

\frac{∂W_{τ}}{∂τ} (x_{τ}) = ∫_{S^{1}} dψ_{τ}^{∗}λ(\frac{∂ψ_{τ}}{∂τ}) + ψ_{τ}^{∗}i(\frac{∂ψ_{τ}}{∂τ})dλ

− ∫_{0}^{1} (\frac{∂H_{τ}}{∂τ} (ψ_{τ}(t), t) + dH_{τ}(ψ_{τ}(t), t)\frac{∂ψ_{τ}}{∂τ} ) dt

= ∫_{0}^{1}(H − K)(φ_{t}^{τ}(x_{τ}), t)dt ≥ E^{−}(K − H).

By Lemma, 3 c_{v}(H_{τ}) − E^{−}(K − H)τ is increasing. Similarly c_{v}(H_{τ}) − E^{+}(K − H)τ is decreasing. Then

c_{v}(K) − E^{+}(K − H) ≤ c_{v}(H) ≤ c_{v}(K) − E^{−}(K − H)

Corollary 1. If H ≤ K then c_{v}(H) ≥ c_{v}(K) for any nonzero v ∈ H^{∗}(M).

Proof of Theorem 2.

Item 1 is a consequence of the following Lemma.

Lemma 5. If H, K ∈ H are such that the corresponding time one maps φ_{1} and ψ_{1} are equal then σ_{c}(H) = σ_{c}(K).

Proof. Let x_{0} in the border of U_{1}. Let x_{1} in Fix(φ_{1}) = Fix(ψ_{1}). Then φ_{t}(x_{0}) = ψ_{t}(x_{0}) for all t. Let β be a curve on T^{∗}M such that β(0) = x_{0} and β(1) = x_{1}. Define σ_{1}, σ_{2} : [0, 1]^{2} → T^{∗}M × ℝ by

σ_{1}(s, t) = (φ_{t}(β(s)), t)
σ_{2}(s, t) = (ψ_{t}(β(s)), t).

Then

0 = ∫_{σ_{1}} d(λ − Hdt) = ∫_{φ_{t}(x_{1})} λ − Hdt − ∫_{φ_{t}(x_{0})} λ − Hdt − ∫_{φ_{1}(β)} λ + ∫_{β} λ

and

0 = ∫_{σ_{2}} d(λ − Kdt) = ∫_{ψ_{t}(x_{1})} λ − Kdt − ∫_{ψ_{t}(x_{0})} λ − Kdt − ∫_{ψ_{1}(β)} λ + ∫_{β} λ

So

∫_{φ_{t}(x_{1})} λ − Hdt = ∫_{ψ_{t}(x_{1})} λ − Kdt,

It remains to prove that φ_{t}(x_{1}) is contractible if and only if ψ_{t}(x_{1}) is. To see that, define the path

h_{t}(x) = \begin{cases}φ_{t}(x) & if t ∈ [0, 1] \\ ψ_{2−t}(x) & if t ∈ [1, 2] \end{cases} .

Let Ω(M) be the free loop space of M, and define the continuous function χ : M → Ω(M) by χ(x) = (h_{t}(x))_{t∈[0,2]}.

Since M is connected all the loops in χ(M) are homotopic to χ(x_{0}) which is homotopically trivial. Thus φ_{t}(x_{1}) is contractible if and only if ψ_{t}(x_{1}) is and then

σ_{c}(H) = σ_{c}(K).

For any ε > 0 let K_{ε} be the set of functions f(|p|) in H such that f is convex, f(s) = −ε for s ≤ 1 − 2ε, and f′(s) ≤ 1 for s ≤ 1.

We claim that γ_{±}(K) = ε for any K ∈ K_{ε} . In fact, the only 1periodic orbits on U_{1} for K are the constants and these have action ε.

Item 2. Since H is negative for |p| ≤ 1 and equals \frac{R}{2}(|p|^{2} − 1) for |p| ≥ 1, we can find ε > 0 small enough and K ∈ K_{ε} such that K ≥ H and then by Corollary 1 we have γ_{−}(H) ≥ γ_{−}(K) > 0.

Item 3. By Lemma 4, for K_{ε} ∈ K_{ε} we have

E^{−}(H − K_{ε}) ≤ γ_{±}(H) − γ_{±}(K_{ε}) ≤ E^{+}(H − K_{ε}).

Now let ε → 0.

Item 4. Since H ≤ 0, E^{−}(H) = 0. By item 3, γ_{±}(H) ≥ 0.

The following Corollary is an immediate consequence of item 3 of Theorem 2.

Corollary 2. If φ ∈ D

γ_{−}(φ) ≤ γ_{+}(φ) ≤ E(φ), γ_{+}(φ) − γ_{−}(φ) ≤ E(φ).

4. Proof of Theorem 3

Let us recall the main concepts introduced by Mather in [7]. Let L : T M × 𝕊^{1} → ℝ be a convex superlinear Lagrangian with complete Euler-Lagrange flow. Let M(L) be the set of probabilities on the Borel σ-algebra of T M that have compact support and are invariant under the Euler-Lagrange flow ϕ_{t}. Let H_{1}(M, ℝ) be the first real homology group of M. Given a closed one-form ω on M and ρ ∈ H_{1}(M, ℝ), let < ω, ρ > denote the integral of ω on any closed curve in the homology class ρ. If μ ∈ M(L), its rotation vector is defined as the unique ρ(μ) ∈ H_{1}(M, ℝ) such that

< ω, ρ(μ) >= ∫ ω dμ,

for all closed one-forms on M. The integral on the right-hand side is with respect to μ with ω considered as a function ω : T M → ℝ. The function ρ : M(L) → H_{1}(M, ℝ) is surjective [7]. The action of μ ∈ M(L) is defined by

A_{L}(μ) = ∫ L dμ.

Finally we define the function β : H_{1}(M, ℝ) → ℝ by

β(γ) = inf{A_{L}(μ) : ρ(μ) = γ}.

The function β is convex and superlinear and the infimum can be shown to be a minimum [7] and the measures at which the minimum is attained are called minimizing measures. In other words, μ ∈ M(L) is a minimizing measure iff

β(ρ(μ)) = A_{L}(μ).

Let us recall how the convex dual α : H^{1}(M, ℝ) → ℝ of β is defined. Since β is convex and superlinear we can set

α([ω]) = max{< ω, γ > −β(γ) : γ ∈ H_{1}(M, ℝ)},

where ω is any closed one-form whose cohomology class is [ω]. The function α is also convex and superlinear. It is not difficult to see that [7]

α([ω]) = − min {∫ (L − ω) dμ : μ ∈ M(L)} . (9)

Each contractible 1-periodic orbit is the support of an invariant probability measure with zero homology and the same action. So

β(0) ≤ γ_{−}(H).

By Corollary 2, γ_{−}(φ) ≤ E(φ), and this concludes the proof of Theorem 3.

5. A Hofer’s distance and twist maps

For ϕ ∈ D_{0} let

∥ϕ∥ = inf{∫_{0}^{1} ∥L_{t}∥ dt : L ∈ H_{0}(U_{1}), φ_{1}^{L} = ϕ}.

If φ, ψ ∈ D define d(φ, ψ) = E(ψ^{−1}φ).

If H, K ∈ H generate φ, ψ respectively, then L(x, t) = (H−K)(ψ_{t}(x), t) belongs to H_{0}(U_{1}) and generates ψ^{−1}φ. Reciprocally, suppose that K ∈ H generates ψ. If L ∈ H_{0}(U_{1}) generates ψ^{−1}φ then H(x, t) = K(x, t) + L(ψ_{t}^{−1}(x), t) belongs to H and generates φ.

Thus, for any ψ, φ ∈ D and K ∈ D generating ψ we have

{∥(H − J)_{t}∥ : H, J ∈ H, φ = φ_{1}^{H}, ψ = φ_{1}^{J}}

⊂ {∥L_{t}∥ : L ∈ H_{0}, ψ^{−1}φ = φ_{1}^{L}}
⊂ {∥(H − K)_{t}∥ dt : H ∈ H, φ = φ_{1}^{H}} .

Therefore

d(φ, ψ) = inf {∫_{0}^{1} ∥(H − J)_{t}∥ dt : H, J ∈ H : φ = φ_{1}^{H}, ψ = φ_{1}^{J}}
= inf {∫_{0}^{1} ∥(H − K)_{t}∥ dt : H ∈ H : φ = φ_{1}^{H}} .

Corollary 3. Let ϕ ∈ D_{0} and define

E^{+}(ϕ) = inf{E^{+}(L) : L ∈ H_{0}, φ_{1}^{L} = ϕ}
E^{−}(ϕ) = sup{E^{−}(L) : L ∈ H_{0}, φ_{1}^{L} = ϕ}.

Let ψ ∈ D, then

E^{−}(ϕ) ≤ γ^{±}(ψ ∘ ϕ) − γ^{±}(ψ) ≤ E^{+}(ϕ).

Proof Let L ∈ H_{0}, K ∈ H with φ_{1}^{L} = ϕ and φ_{1}^{K} = ψ. As above H(x, t) = K(x, t) + L(ψ_{t}^{−1}(x), t) belongs to H and generates ψ ∘ ϕ. By item 2 in Theorem 1

E^{−}(L) ≤ γ^{±}(H) − γ^{±}(K) ≤ E^{+}(L),

from which Corollary 3 follows.

Definition 1. A function H ∈ H_{0} is called quasi-autonomous if there are x_{−}, x_{+} ∈ U_{1} such that

H(x_{−}, t) = min H_{t}, H(x_{+}, t) = max H_{t}

for all t ∈ 𝕊^{1}.

Recall that G_{R}(q, p) = (q, exp_{q}(Rp)). Let G be the set of functions S : G_{R}(U_{1+δ}) → ℝ such that

• \frac{∂^{2}S}{∂q∂Q} is negative definite.

• S(q, Q) = \frac{d(q, Q)^{2}}{2R} for d(q, Q) ≥ R.

14 R. ITURRIAGA AND H. SÁNCHEZ M.

Proposition 5. Let S_{t}, t ∈ [a, b] be a path in G. Then the corresponding path φ_{t} in T is generated by a Hamiltonian H ∈ H_{0}(U_{1}) satisfying the Hamilton-Jacobi equation

\frac{∂S_{t}}{∂t} (q, Q) + H(Q, \frac{∂S_{t}}{∂Q} (q, Q), t) = 0. (10)

Moreover, H is quasi-autonomous if and only if \frac{∂S_{t}}{∂t} is.

The proof of equation 10 is standard and the proof of the last statement is the same as the given by Bialy and Polterovich [1].

Consider the length function L on the fundamental group of D_{0}

L([φ_{t}]) = inf_{ψ_{t}∈[φ_{t}]} length(ψ_{t}).

We claim that the image of L is {0}. In fact, given any loop φ_{t} = (Q_{t}, P_{t}) in D_{0} with generating Hamiltonian H ∈ H_{0}(U_{1}), we can define the loop ψ : 𝕊^{1} → D_{0} by ψ_{t}(q, p) = (Q_{t}(q, p/ε), εP_{t}(q, p/ε)) with generating Hamiltonian F (q, p, s) = εH(q, p/ε, s). Therefore

length(ψ) = ∫_{0}^{1} ∥F_{t}∥ dt = ε ∫_{0}^{1} ∥H_{t}∥ dt.

Thus, for any class [φ] ∈ π_{1}(D_{0}) we have L([φ]) = 0.

Proposition 6. There is a C^{2} neighborhood U of zero in H_{0}(U_{1}) such that

∥φ_{1}^{H} ∥ = ∫_{0}^{1} ∥H_{t}∥ dt

for any quasi-autonomous H ∈ U .

Proof. By Remark 3.3 and item (ii) of Theorem 1.3 in [5] II, there is a C^{2} neighborhood U of zero in H_{0}(U_{1}) such that for any quasiautonomous H ∈ U the path φ_{t∈[0,1]}^{H} is length-minimizing amongst all paths homotopic (with fixed end points) to φ_{t∈[0,1]}^{H}.

Suppose that ϕ is other path in D_{0} with the same end points and length(ϕ) < length(φ^{H}). Choose a loop ψ ∈ [−ϕ ∗ φ^{H}] such that

length(ψ) < length(φ^{H}) − length(ϕ).

the path ϕ ∗ ψ is homotopic to φ^{H} and shorter: a contradiction.

Proof of Proposition 1. For φ_{0}, φ_{1} ∈ T , let S_{0}, S_{1} ∈ G be their generating functions. For t ∈ [0, 1], S_{t} = (1 − t)S_{0} + tS_{1} defines a map φ_{t} ∈ T . Consider the path φ_{t}φ_{0}^{−1} in D_{0} and its generating Hamiltonian H. As in [1], one easily shows that \frac{∂S_{t}}{∂t} = S_{1} −S_{0} is quasi-autonomous. By Proposition 5, H is quasi-autonomous and

∥H_{t}∥ = ∥S_{0} − S_{1}∥ .

By Proposition 6, there is a C^{1} neighborhood O of φ such that if φ_{0}, φ_{1} ∈ O, the length of the path φ_{t}φ_{0}^{−1} equals d(φ_{0}, φ_{1}).

References

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[4] H. Hofer, E. Zehnder. Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser. Basel 1994.

[5] F. Lalonde, D. Mc Duff. Hofer’s L^{∞} geometry: energy and stability of hamiltonian flows I, II. Invent. Math. 122 (1995) 1-69.

[6] F. Lalonde, D. Mc Duff. The geometry of symplectic energy. Ann. Math. 141(1995) 349-371.

[7] J. Mather. Action minimizing measures for positive definite Lagrangian systems. Math. Z. 207 (1991) 169–207.

[8] M. Schwarz. On the action spectrum for closed symplectically aspheric manifolds. Preprint 1998.

[9] K.F. Siburg. Bi-invariant metrics for simplectic twist mappings in T^{∗}𝕋^{n} and an application in Aubry-Mather theory. CRM Proc. Lec. Notes 15 (1998) 137-148.

[10] K.F. Siburg. Action mininizing measures and the geometry of the hamiltonian diffeomorphism group. Duke Math. J. 92 No.2 (1998) 295-318.

[11] C. Viterbo. Symplectic Topology as the geometry of generating functions. Math. Ann. 292 (1992) 685-710.

CIMAT, A.P. 402, 36000, Guanajuato. Gto., México.

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

Instituto de Matemáticas. UNAM, Ciudad Universitaria C. P. 04510, México, DF, México.

E-mail address: hector@matem.unam.mx