CONVEX HAMILTONIANS WITHOUT CONJUGATE POINTS

GONZALO CONTRERAS AND RENATO ITURRIAGA

Abstract. We construct the Green bundles for an energy level without conjugate points of a convex Hamiltonian. In this case we give a formula for the metric entropy of the Liouville measure and prove that the exponencial map is a local diffeomorphism. We prove that the Hamiltonian flow is Anosov if and only if the Green bundles are transversal. Using the Clebsch transformation of the index form we prove that if the unique minimizing measure of a generic Lagrangian is supported on a periodic orbit, then it is a hyperbolic periodic orbit.

We also show some examples of differences with the behaviour of a geodesic flow without conjugate points, namely: (non-contact) flows and periodic orbits without invariant transversal bundles, segments without conjugate points but with crossing solutions and non-surjective exponential maps.

Introduction.

Let M be a closed connected riemannian manifold and T^{∗}M its cotangent bundle. By a convex Hamiltonian on T^{∗}M we shall mean a C^{3} function H : T^{∗}M → ℝ satisfying the following conditions:

(a) Convexity: For all q ∈ M, p ∈ T_{q}^{∗}M, the Hessian matrix \frac{∂^{2}H}{∂p_{i} ∂p_{j}} (q, p) (calculated

with respect to linear coordinates on T_{q}^{∗}M) is positive definite.

(b) Superlinearity:

lim_{|p|→∞} \frac{H(q, p)}{|p|} = +∞ .

The Hamiltonian equation for H is defined as

\begin{aligned}q′ & = H_{p} , \\ p′ & = −H_{q} \end{aligned} (1)

where H_{q} and H_{p} are the partial derivatives with respect to q and p. Observe that the Hamiltonian function H is constant along the solutions of (1). Its level sets Σ = H^{−1}{e} are called energy levels of H. Then the compactness of M and the superlinearity hypothesis imply that the energy levels are compact. Since the

G. Contreras was partially supported by CNPq-Brazil and PUC-Rio. R. Iturriaga was partially supported by Conacyt-Mexico, grant 0324P-E.

2 G. CONTRERAS AND R. ITURRIAGA

Hamiltonian vectorfield (1) is Lipschitz, the solutions of (1) are defined on all ℝ. Denote by ψ_{t} the corresponding Hamiltonian flow on T^{∗}M.

Our main interest in this paper are disconjugate orbits of the Hamiltonian flow, i.e. orbits without conjugate points. Let π : T^{∗}M → M be the canonical projection and define the vertical subspace on θ ∈ T^{∗}M by V (θ) = ker(dπ). Two points θ_{1}, θ_{2} ∈ T^{∗}M are said conjugate if θ_{2} = ψ_{τ}(θ_{1}) for some τ ≠ 0 and dψ_{τ}(V (θ_{1}))∩V (θ_{2}) ≠ {0}.

In this work we generalize some results of the theory of geodesics without conjugate points. A first step is to construct the Green subspaces for disconjugate orbits:

Proposition A.

Suppose that the orbit of θ ∈ T^{∗}M does not contain conjugate points and H(θ) = e is a regular value of H. Then there exist two φ-invariant Lagrangian subbundles 𝔼, 𝔽 ⊂ T (T^{∗}M) along the orbit of θ given by

𝔼(θ) = lim_{t→+∞} dψ_{−t}(V (ψ_{t}(θ))) ,
𝔽(θ) = lim_{t→+∞} dψ_{t}(V (ψ_{−t}(θ))) .

Moreover, 𝔼(θ)∪𝔽(θ) ⊂ T_{θ}Σ, 𝔼(θ)∩V (θ) = 𝔽(θ)∩V (θ) = {0}, ⟨X(θ)⟩ ⊆ 𝔼(θ)∩𝔽(θ) and dim 𝔼(θ) = dim 𝔽(θ) = dim M, where X(θ) = (H_{p}, −H_{q}) is the Hamiltonian vectorfield and Σ = H^{−1}{e}.

These bundles where constructed for disconjugate geodesics of riemannian metrics by Green [19] and of Finsler metrics by Foulon [18]. In the Finsler case this is done by defining a new connection ˜∇ such that the solutions γ of the Euler-Lagrange equation satisfy ˜∇γ̇(γ̇) ≡ 0. This approach leads to extensions of many important theorems in Riemannian geometry to Finsler geometry. See Chern [6] and the references therein for a survey and an introduction to this field.

If the energy level Σ does not have conjugate points, the Green bundles 𝔼 and 𝔽 can be defined on all θ ∈ Σ. Nevertheless, in general they are neither continuous (cf. [4]) nor transversal in T Σ (i.e. dim 𝔼 ∩ 𝔽 = 1).

An example of the relationship between the transversality of the Green subspaces and hyperbolicity appears in the following

Proposition B.

Let Γ be a periodic orbit of ψ_{t} without conjugate points. Then Γ is hyperbolic (on its energy level) if and only if 𝔼(θ) ∩ 𝔽(θ) = ⟨X(θ)⟩ for some θ ∈ Γ, where ⟨X(θ)⟩ is the 1-dimensional subspace generated by the Hamiltonian vectorfield X(θ). In this case 𝔼 and 𝔽 are its stable and unstable subspaces.

We can compare proposition A with the following theorem by G. Paternain and M. Paternain [32].

0.1. Theorem. [Paternain] [32]

Suppose that e is a regular value of H, and that ψ_{t}|_{Σ} admits a continuous invariant Lagrangian subbundle E. Then,

(a) E(θ) ∩ V (θ) = {0} ∀θ ∈ Σ.

(b) π(Σ) = M.

(c) Σ contains no conjugate points.

It is not difficult to see (e.g. [24]) that when the flow ψ_{t}|_{Σ} is Anosov, the stable and unstable subbundles of T Σ are Lagrangian. Paternain’s theorem then implies that Anosov Lagrangian flows do not have conjugate points. This result was proven for geodesic flows by Klingenberg [23]. An extension of Klingenberg result appears in Mañé [24].

Define

B(θ) = {ξ ∈ T_{θ}Σ | sup_{t∈ℝ}|dψ_{t}(θ) ξ| < +∞ } .

An important property (cf. proposition 1.11) in the proof of proposition B is that if the orbit of θ is disconjugate then B(θ) ⊆ 𝔼(θ) ∩ 𝔽(θ). This in turn implies that ⟨X(θ)⟩ ⊆ 𝔼(θ) ∩ 𝔽(θ) and 𝔼(θ) ∪ 𝔽(θ) ⊆ T_{θ}Σ (cf. corollary 1.12). The global version of proposition B is given by

Theorem C.

Let Σ = H^{−1}{e} be a regular energy level without conjugate points.

Then the following statements are equivalent:

(a) ψ_{t}|_{Σ} is Anosov.

(b) For all θ ∈ Σ, 𝔼(θ) and 𝔽(θ) are transversal in T_{θ}Σ.

(c) For all θ ∈ Σ, 𝔼(θ) ∩ 𝔽(θ) = ⟨X(θ)⟩.

(d) If θ ∈ Σ, v ∈ T_{θ}Σ, v ∉ ⟨X(θ)⟩ then sup_{t∈ℝ}|dψ_{t}(θ) · v| = +∞ .

Where ⟨X(θ)⟩ ⊂ T_{θ}Σ is the 1-dimensional subspace generated by the vectorfield X(θ) of ψ.

This theorem was proven by Eberlein (cf. [13]) for geodesic flows. For further applications of this theorem see [9] and [10]. An interesting ingredient on its proof is the characterization of recurrent quasi-hyperbolic ℝ-actions of vector bundles. Given a vector bundle π : E → B consider a continuous ℝ-action Λ : ℝ → Isom(E, E) where Λ_{t} : E → E is a bundle map which is a linear isomorphism on each fiber

4 G. CONTRERAS AND R. ITURRIAGA

and Λ_{s+t} = Λ_{s} ∘ Λ_{t}. The action Λ induces a continuous flow ϕ_{t} : B ↩ such that ϕ_{t} ∘ π = π ∘ Λ_{t}. We say that Λ is quasi-hyperbolic if

sup_{t∈ℝ}|Λ_{t}(ξ)| = +∞ for all ξ ∈ E, ξ ≠ 0 ,

and we say that it is hyperbolic if there exists a continuous splitting E = E^{s} ⊕ E^{u} and C > 0, λ > 0 such that

(i) |Λ_{t}(ξ)| ≤ C e^{−λt} for all t > 0, ξ ∈ E^{s}. (ii) |Λ_{−t}(ξ)| ≤ C e^{−λt} for all t > 0, ξ ∈ E^{u}.

It is easy to see that hyperbolic actions are quasi-hyperbolic. The converse is false by counterexamples in Robinson [35] and Franks & Robinson [14]. But it is true with an additional recurrency hypothesis, (that in our case is always true because the Lagrangian flow preserves the Liouville measure).

Define the non-wandering set Ω(ϕ) of ϕ as the set of points b ∈ B such that for every neighbourhood U of b there exists T > 0 such that ϕ_{T} (U) ∩ U ≠ 0. The following theorem is a slight modification of a theorem in Freire [15], written there for quasi-Anosov flows (i.e when Λ_{t} = dϕ_{t} and E is a continuous invariant subbundle of T B transversal to the vectorfield X, see theorem 3.1).

0.2. Theorem. Let π : E → B be a continuous vector bundle and Λ : ℝ → Isom(E, E) a continuous ℝ-action with induced flow ϕ_{t}: π ∘ Λ_{t} = ϕ_{t} ∘ π. If Λ is quasi-hyperbolic and Ω(ϕ|_{B}) = B, then Λ is hyperbolic.

For completeness of the exposition we present a proof of this following theorem in section §3. The reason to use this version of Freire’s result is that our Hamiltonian flows are not of contact type and a priori they may not preserve any continuous bundle transversal to the vectorfield X(θ). Indeed, in appendix I we show an example of a convex Hamiltonian without conjugate points where this phenomenon occurs. The solution is to study the behaviour of an (orbitwise) reparametrization of the flow which preserves a continuous transversal bundle E ⊂ T (T^{∗}M). This reparametrization can not be made global but instead we use the quasi-hyperbolicity of the action Λ_{t}(θ) = P (ψ_{t}(θ)) ∘ dψ_{t}(θ) ∘ P (θ) where P (θ) : T_{θ}T^{∗}M → E(θ) is the projection along the direction of the vectorfield X(θ).

Generic Lagrangians.

A C^{4} function L : T M → ℝ is called a (convex autonomous) Lagrangian if it satisfies the following properties:

(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|} = +∞ ,

uniformly on x ∈ ℝ. Observe that by the compactness of M, the Euler-Lagrange equation, that in local

coordinates is:

\frac{d}{dt} \frac{∂L}{∂v}(x, ẋ) = \frac{∂L}{∂x}(x, ẋ) , (2)

generates a complete flow φ : T M × ℝ → T M defined by

φ_{t}(x_{0}, v_{0}) = (x(t), ẋ(t)),

where x : ℝ → M is the maximal solution of (2) with initial conditions x(0) = x_{0} and ẋ(0) = v_{0}.

A Lagrangian flow is conjugated to a Hamiltonian flow by the Legendre transform F_{L} : T M → T^{∗}M defined by

F_{L}(x, v) = (x, \frac{∂L}{∂v}(x, v)) .

The corresponding Hamiltonian is given by H(q, p) = E(F_{L}^{−1}(q, p)), where E : T M → ℝ is the energy function defined by

E(x, v) := \frac{∂L}{∂v}|_{(x,v)} · v − L(x, v) . (3)

Conversely, given a convex Hamiltonian H on T^{∗}M, the corresponding convex Lagrangian is given by the Legendre transform of the Hamiltonian F_{H} : T^{∗}M → T^{∗∗}M ≈ T M which can be seen as

L(x, v) = max_{p} { p(v) − H(x, p) } = (p H_{p} − H) ∘ (F_{H})^{−1} .

Observe that item (b) in theorem 0.1 implies that the energy e of an Anosov level for the Lagrangian flow satisfies

e > e_{0} := − max_{p∈M} L(p, 0) ,

because e_{0} is the minimal energy level e for which π(Σ) = M. Define

c(L) := − min {∫_{T M} L dμ | μ is a φ − invariant probability} .

It is proven in [28], [8], that c(L) ≥ e_{0}.

We say that a φ-invariant probability μ is minimizing if

c(L) = − ∫ L dμ ,

6 G. CONTRERAS AND R. ITURRIAGA

and we say that it is uniquely ergodic if it is the only invariant probability measure supported inside supp(μ).

Action minimizing curves do not contain conjugate points. This fact is known since Morse earlier works (cf. [30]). We present a simple proof of this fact in section §4 and we use proposition B to prove the following theorem, which was stated as theorem III in Mañé’s unfinished work [28].

Theorem D.

There exists a generic set O ⊆ C^{∞}(M, ℝ) such that for all ϕ ∈ O the Lagrangian L + ϕ has a unique minimizing measure and this measure is uniquely ergodic. When this measure is supported on a periodic orbit or a fixed point, this orbit (point) Γ is hyperbolic and its stable and unstable manifolds intersect transversally W^{s}(Γ) ⋔ W^{u}(Γ).

On Mañé [28], it is conjectured that there exists a generic set O such that this unique minimizing measure is supported on a periodic orbit or an equilibrium point.

The first statement of theorem D was proved by Mañé in [26]. Here we prove the generic hyperbolicity of the minimizing periodic orbit and the transversality property W^{s}(Γ) ⋔ W^{u}(Γ). The hyperbolicity of Γ was announced in Mañé [28] and the transversality W^{s}(Γ) ⋔ W^{u}(Γ) was conjectured in Mañé [27]. We use the Clebsch transformation [7] to derive formulas for the index form on an orbit without conjugate points. Then we use similar arguments to the Rauch comparison theorem to obtain the transversality of the Green bundles of the periodic orbit. Finally, proposition B implies the hyperbolicity of the periodic orbit.

The metric entropy.

The machinery developed for the proof of theorem C can be used to obtain formulas for the metric entropy of a Hamiltonian flow without conjugate points. Oseledec’s theory [31] gives a splitting

T_{θ}Σ = E^{s}(θ) ⊕ E^{c}(θ) ⊕ E^{u}(θ)

for a set of points θ ∈ Λ ⊆ Σ of total measure, such that

lim_{t→±∞} \frac{1}{t} log |dψ_{t}(θ) ξ| < 0 for ξ ∈ E^{s}(θ) ,
lim_{t→±∞} \frac{1}{t} log |dψ_{t}(θ) ξ| = 0 for ξ ∈ E^{c}(θ) ,
lim_{t→±∞} \frac{1}{t} log |dψ_{t}(θ) ξ| > 0 for ξ ∈ E^{u}(θ) .

Following Freire and Mañé [16], we see in proposition 1.14 that

E^{u}(θ) ⊆ 𝔽(θ) ⊆ E^{u}(θ) ⊕ E^{c}(θ) ,
E^{s}(θ) ⊆ 𝔼(θ) ⊆ E^{s}(θ) ⊕ E^{c}(θ) .

HAMILTONIANS WITHOUT CONJUGATE POINTS 7

Given a riemannian metric on M, define the horizontal subspace as the kernel of the connection map. Then there exists symmetric linear isomorphisms 𝕌 : H(θ) → V (θ) such that 𝔽(θ) = graph (𝕌(θ)) for all θ ∈ Σ. It satisfies the Ricatti equation (7)

𝕌̇ + 𝕌 H_{pp} 𝕌 + 𝕌 H_{pq} + H_{qp} 𝕌 + H_{qq} = 0,

where H_{pp}, H_{qp}, H_{pq} and H_{qq} are linear maps which coincide with the second derivatives of H in local coordinates, and 𝕌̇ is the covariant derivative of 𝕌 defined as

𝕌̇(θ) v = lim_{h→0} \frac{1}{h}[τ_{h} 𝕌(ψ_{h}(θ)) v − 𝕌(θ) v] ∈ V (θ) ≈ T_{θ}^{∗}M,

where τ_{h} : T_{πψ_{h}(θ)}^{∗}M → T_{π(θ)}^{∗}M is the parallel transport.

For θ ∈ Λ define the unstable Jacobian by

χ(θ) := lim_{T →+∞} \frac{1}{T} log |det dψ_{T} |_{E^{u}(θ)}| .

We prove in proposition 1.14 that

χ(θ) = lim_{T →+∞} \frac{1}{T} ∫_{0}^{T} tr [H_{pq} + H_{pp} 𝕌] (ψ_{t}(θ)) dt ,

where tr ( ) is trace. From Ruelle’s inequality [36], for any invariant measure μ for ψ_{t}|_{Σ}, we obtain that the metric entropy satisfies

h_{μ}(ψ|_{Σ}) ≤ ∫_{Σ} χ dμ ≤ ∫_{Σ} tr[H_{pq} + H_{pp} 𝕌] dμ .

Pesin’s formula states that for a smooth invariant measure μ of a C^{1+α} diffeomorphism f : N ↩, we have that

h_{μ}(f) = ∫_{N} χ dμ .

In our case we obtain Theorem E.

Let m be the Liouville measure on an energy level Σ without conjugate points. Then

h_{m}(ψ|_{Σ}) = ∫_{Σ} tr[H_{pq} + H_{pp} 𝕌] dm .

This theorem was proved for geodesic flows by Freire and Mañé [16], Finsler metrics by Foulon [18] and mechanical Lagrangians by Innami [22].

The exponential map.

The quasi-hyperbolicity in the case 𝔼(θ) ⋔ 𝔽(θ) is obtained from proposition 1.10 which states the horizontal growth of vertical vectors, i.e.

lim_{t→+∞} ∥dπ ∘ dψ_{t}|_{V (θ)}∥ = +∞.

The bounds obtained in proposition 1.10 are not uniform, by an example in Ballmann, Brin & Burns [4]. In section §6 we give a uniform lower bound for ∥dπ ∘ dψ_{t}|_{V (θ)}∥, θ ∈ Σ, |t| ≥ 1. This bound was given by Freire and Mañé [16] in the case of geodesic flows.

Define the exponential map of an energy level Σ as exp_{q} : T_{q}^{∗}M → M, exp_{q}(tθ) = π(ψ_{t}(θ)), where θ ∈ Σ ∩ T_{q}^{∗}M and q is a regular value of π : Σ → M. This map is not differentiable at 0 ∈ T_{q}^{∗}M, but its Lagrangian version is differentiable with d exp_{q}(0) = I. An important question is whether the lift of this map to the universal cover, exp_{q} : T_{q}^{∗}M → M̃, is a diffeomorphism in an energy level without conjugate points and, in particular, if the universal cover M̃ is homeomorphic to ℝ^{n}. We show in corollaries 1.18 and 1.13 that if the energy level Σ has no conjugate points then exp_{q} is a local diffeomorphism and π(Σ) = M.

Let Θ = p dq be the Liouville’s 1-form (4) in T^{∗}M. Given a regular energy level Σ and q ∈ π(Σ), let Σ_{q} := (π|_{Σ})^{−1}{q}. Since M̃ is simply connected, in order to obtain a global diffeomorphism exp_{q} : T_{q}^{∗}M → M̃ it is sufficient to show that exp_{q} is a covering map. For this it is enough the condition Θ(X) > 0. Indeed, in section §6 we prove the following

Theorem F.

Let H : T^{∗}M → ℝ be convex and Σ = H^{−1}{e} a regular energy level. Suppose that q ∈ π(Σ) satisfies

(a) The positive orbit of all θ ∈ Σ_{q} has no conjugate points.

(b) inf{ |Θ(X(ϑ))| | ϑ ∈ ψ_{ℝ^{+}}(Σ_{q}) } > 0. Then the (Lagrangian) exponential map exp_{q} : T_{q}^{∗}M̃ → M̃ associated to (ψ_{t}, Σ) is a diffeomorphism.

We give an example (cf. ex. I.4) in appendix I in which exp_{q} is not surjective and where all the orbits in Σ of ψ_{t} starting at π^{−1}{q} have no conjugate points. This shows that condition (a) is not sufficient in theorem F. By theorem X in Mañé [28] (see also [8]), this can not happen for high energy levels. Example 1 in appendix I also shows that the norm of the derivative ∥d_{θ}(exp_{q})∥ may not be bounded below on an energy level without conjugate points. We don’t know if the exponential map is a diffeomorphism when the whole energy level has no conjugate points.

HAMILTONIANS WITHOUT CONJUGATE POINTS 9

The condition Θ(X) = p H_{p} > 0 is written Θ(X) = v L_{v} > 0 in Lagrangian form. It is satisfied by Finsler Lagrangians, L(x, v) = \frac{1}{2} ∥v∥_{x}^{2}, mechanical Lagrangians L(x, v) = \frac{1}{2}⟨v, v⟩_{x} − ϕ(x) (with Θ(X) = 1, H(q, p) = \frac{1}{2}⟨p, p⟩_{q} + ϕ(q)) and all convex Hamiltonians on sufficiently high energy levels. It fails for magnetic Lagrangians L(x, v) = \frac{1}{2}⟨v, v⟩_{x} − ϕ(x) + η_{x}(v) (with dη ≠ 0) on lower energy levels. If one adds a closed 1-form to a Lagrangian, the Euler-Lagrange equation and hence the Lagrangian flow does not change. This may help to make a given Lagrangian flow satisfy condition (b). See remark 6.5 and [10] for a characterization of the energy levels in which this condition holds for a convex Lagrangian. See [11] for other proofs of theorem F and its converse.

The obstruction to obtain a bound on ∥d_{ψ_{t}(θ)}(exp_{q})∥ is the angle ∢(X(ψ_{t}(θ)), dψ_{t}(V (θ) ∩ T_{θ}Σ)). If the orbit of θ is disconjugate this angle is always nonzero by proposition 1.16. On Example 3 in appendix I this angle reaches zero before the first conjugate point appear. Moreover, there are crossings of solutions on Σ starting at π(θ) before the first conjugate point.

On section §1 we prove proposition A, theorem E and we show the proof of item (c) in theorem 0.1. In section §2 we study the transversal behaviour of the Hamiltonian flow and prove proposition B. On section §3 we show the proof of theorem 0.2 and we prove theorem C. On section §4 we give a characterization of the index form and on section §5 we prove theorem D. On section §6 we give a uniform lower bound for dπ ∘ dψ_{t}|_{V (θ)} and prove theorem F. Finally, we add some examples in the appendix.

We want to thank G. Paternain, M. Paternain and R.O. Ruggiero for useful conversations and J. Sotomayor for pointing us reference [20]. We also want to thank the hospitality of CMAT, Uruguay; IMPA, Brazil; CIMAT, Mexico and PUC-Rio, Brazil.

1. The Green Bundles.

For proofs of the basic results of this section see Abraham & Marsden [1]. There is a canonical symplectic structure on T^{∗}M given by a closed non-degenerate 2-form ω = dΘ, where Θ is the Liouville’s 1-form, defined by

Θ_{θ}(ξ) = θ(dπ(θ) ξ) , (4)

where π : T^{∗}M → M is the canonical projection. The Hamiltonian vectorfield X of a function H : T^{∗}M → ℝ is defined by ω(X, ·) = −dH. The Hamiltonian flow ψ_{t} preserves the function H and the 2-form ω. The level sets of H are called energy levels. Given a local chart q : U ⊆ M → ℝ^{n} it induces a natural chart (q, p) : T^{∗}U → ℝ^{n} × ℝ^{n} of T^{∗}M. In natural charts the Liouville’s 1-form is written

10 G. CONTRERAS AND R. ITURRIAGA

as Θ = p dq. Hence in natural charts ω = dp ∧ dq and the Hamiltonian equations are given by

q̇ = H_{p} , ṗ = −H_{q} . (5)

In general, a set of local coordinates (q, p) of T^{∗}M is called symplectic if the canonical 2-form is written as ω = dp ∧ dq. A subspace E ⊂ T (T^{∗}M) is said Lagrangian if dim E = n = dim M and ω(x, y) = 0 for all x, y ∈ E.

Fix a riemannian metric on M and the corresponding induced metric on T^{∗}M. Then T_{θ}T^{∗}M splits as a direct sum of two Lagrangian subspaces: the vertical subspace V (θ) = ker (dπ(θ)) and the horizontal subspace H(θ) given by the kernel of the connection map. Using the isomorphism K : T_{θ}T^{∗}M → T_{π(θ)}M × T_{π(θ)}^{∗}M, ξ ↦ (dπ(θ) ξ , ∇_{θ}(π ξ)), we can identify H(θ) ≈ T_{π(θ)}M ×{0} and V (θ) ≈ {0}×T_{π(θ)}^{∗}M ≈

T_{π(θ)}M. If we choose local coordinates along t ↦ π ψ_{t}(θ) such that t ↦ \frac{∂}{∂q_{i}} (π ψ_{t}(θ)) are parallel vectorfields, then this identification becomes ξ ↔ (dq(ξ), dp(ξ)). Let E ⊂ T_{θ}T^{∗}M be an n-dimensional subspace such that E ∩ V (θ) = {0}. Then E is a graph of some linear map S : H(θ) → V (θ). It can be checked that E is Lagrangian if and only if in symplectic coordinates S is symmetric.

Proof of Proposition A.

Take θ ∈ T^{∗}M and ξ = (h, v) ∈ T_{θ}T^{∗}M = H(θ) ⊕ V (θ) ≈ T_{π(θ)}M ⊕ T_{π(θ)}M. Consider a variation

α_{s}(t) = (q_{s}(t), p_{s}(t))

such that for each s ∈]−ε, ε[, α_{s} is a solution of the Hamiltonian H such that α_{0}(0) = θ and \frac{d}{ds}α_{s}(0)|_{s=0} = ξ. Writing dψ_{t}(ξ) = (h(t), v(t)), we obtain the Hamiltonian Jacobi equations

ḣ = H_{pq} h + H_{pp} v ,
(6)  v̇ = −H_{qq} h − H_{qp} v ,

where the covariant derivatives are evaluated along π(α_{o}(t)), and H_{qq}, H_{qp}, H_{pp} and H_{qq} are linear operators on T_{π(θ)}M, that in local coordinates coincide with the

matrices of partial derivatives (\frac{∂^{2}H}{∂q_{i}∂q_{j}} ), (\frac{∂^{2}H}{∂q_{i} ∂p_{j}} ), (\frac{∂^{2}H}{∂p_{i} ∂p_{j}} ) and (\frac{∂^{2}H}{∂p_{i} ∂q_{j}} ). Moreover,

since the Hamiltonian H is convex, then H_{pp} is positive definite.

We derive now the Ricatti equation. Let E be a Lagrangian subspace of T_{θ}T^{∗}M. Suppose that for t in some interval ] − ε, ε[ we have that dψ_{t}(E) ∩ V (ψ_{t}(θ)) = {0}. Then we can write dψ_{t}(E) = graph S(t), where S(t) : H(ψ_{t} θ) → V (ψ_{t} θ) is a symmetric map. That is, if ξ ∈ E then

dψ_{t}(ξ) = (h(t), S(t) h(t)) .

HAMILTONIANS WITHOUT CONJUGATE POINTS 11

Using equation (6) we have that

Ṡh + S(H_{pq}h + H_{pp}Sh) = −H_{qq}h − H_{qp}Sh .

Since this holds for all h ∈ H(ψ_{t}(θ)) we obtain the Ricatti equation: Ṡ + SH_{pp}S + SH_{pq} + H_{qp}S + H_{qq} = 0 . (7)

Take the symmetric map S(t) for which dψ_{t}(V (θ)) = graph(S(t)), t > 0. Let Y (t) : T_{π(θ)}M ≈ V (θ) → H(ψ_{t}θ) ≈ T_{πψ_{t} θ}M, t ∈ ℝ, be the family of isomorphisms Y (t)w = dπ·dψ_{t}(θ) (0, w), where (0, w) ∈ H(θ)⊕V (θ). Then for t > 0, Y (t) satisfies

(8) Ẏ = (H_{pq} + H_{pp} S) Y
Y (0) = 0 , Ẏ (0) = H_{pp} ,

where Ẏ (0) is calculated from (8) using that

lim_{t→0} S(t) Y (t) = I .

Given u ∈ T_{π(θ)}M, define

h_{1}(t) = Y (t) u ,
v_{1}(t) = S(t) Y (t) u , t > 0 .

Then (h_{1}, v_{1}) is a solution of equation (6) with h_{1}(0) = 0, v_{1}(0) = u. Let ξ ∈ T_{θ}T^{∗}M and let (h(t), v(t)) = dψ_{t}(ξ) ∈ H ⊕ V. Since dψ_{t} preserves the symplectic form, we have that

⟨h(t), v_{1}(t)⟩ − ⟨v(t), h_{1}(t)⟩ = ⟨h(0), u⟩

Hence

⟨Y^{∗}(t) S(t) h(t) , u⟩ − ⟨Y^{∗}(t) v(t) , u⟩ = ⟨h(0) , u⟩

for all u ∈ T_{π(θ)}M. Therefore v(t) = S(t) h(t) − (Y^{∗}(t))^{−1} h(0) . (9) Since

ḣ = H_{pq} h + H_{pp} v

we get ḣ = (H_{pq} + H_{pp}S)h − H_{pp}(Y^{∗})^{−1}h(0) (10) Differentiating Y^{−1} h and using equation (8), we obtain

h(t) = Y (t) Y (c)^{−1} h(c) + Y (t) ∫_{t}^{c} Y^{−1}(s) H_{pp} (Y^{∗}(s))^{−1} h(0) ds , for 0 < t < c. (11)

12 G. CONTRERAS AND R. ITURRIAGA

Now consider the map dψ_{t} restricted to the subspace dψ_{−c}(V (ψ_{c} θ)), c > 0. Its projection Z_{c}(t) to the horizontal subspace is given by the solutions of (6) with h(c) = 0. Then Z_{c}(t) is the family of isomorphisms given by

Z_{c}(t) = Y (t) ∫_{t}^{c} Y^{−1}(s) H_{pp} (Y^{∗}(s))^{−1} ds , 0 < t < c , (12)

because Z_{c}(t) satisfies (6) for 0 < t < c, Z_{c}(c) = 0 and

Z_{c}(0) = lim_{t→0^{+}} Z_{c}(t) = I .

Observe that from its definition, the linear map Z_{c}(t) is defined for all t ∈ ℝ and the no-conjugate points condition implies that Z_{c}(t) is an isomorphism for all t ≠ c. Moreover, since Z_{c}(t) is a matrix solution of the Jacobi equation (6), then Ż_{c}(0) exists, and taking the limit in (12), we have that the linear isomorphism

(13)  Ż_{d}(0) − Ż_{c}(0) = H_{pp}(θ) ∫_{c}^{d} Y (s)^{−1} H_{pp} (Y^{∗}(s))^{−1} ds

is symmetric and the second factor on the right is positive definite for all 0 < c < d. In particular

(14)  Z_{d}(t) − Z_{c}(t) = Y (t) H_{pp}(θ)^{−1} [Ż_{d}(0) − Ż_{c}(0)] , 0 < c < d .

For ε > 0 define Z_{−ε}(t) := Y (t) N_{c} + Z_{c}(t) , t ∈ ℝ , (15)

where N_{c} := −Y (−ε)^{−1} Z_{c}(−ε). We have that Z_{−ε}(t) is the solution of the Jacobi equation (6) satisfying Z_{−ε}(−ε) = 0, Z_{−ε}(0) = I and Ż_{−ε}(0) − Ż_{c}(0) = H_{pp}(θ) N_{c} . (16)

1.1. Claim. Y^{−1}(t) Z_{c}(t) is symmetric for all t ≠ 0. In particular N_{c} is symmetric for all c > 0.

Proof: Using (10), we have that

Ż_{c} = A Z_{c} + H_{pp} (Y^{∗})^{−1} ,
Ẏ = A Y +  0  , t ≠ 0 ,

where A = H_{pq} + H_{pp} S. And then

Ż_{−ε}= A Z_{−ε} + H_{pp}(Y^{∗})^{−1} .

Using (9) we have that the functions

\begin{aligned}h_{2}(t) & = Z_{−ε}(t) u , \\ v_{2}(t) & = S(t) h_{2}(t) − (Y^{∗}(t))^{−1} u , \end{aligned} (17)

are solutions of (6) with h_{2}(−ε) = 0, h_{2}(0) = u. Similarly, the functions

\begin{aligned}h_{3}(t) & = Z_{c}(t) w , \\ v_{3}(t) & = S(t) h_{3}(t) − (Y^{∗}(t))^{−1} w , \end{aligned} (18)

are solutions of (6) with h_{3}(0) = w, h_{3}(c) = 0.

Since the flow preserves the symplectic form, we have that the following expressions do not depend on t for all u, w ∈ T_{π(θ)}M:

⟨ h_{2}(t), v_{3}(t) ⟩ − ⟨ v_{2}(t), h_{3}(t) ⟩
⟨ Z_{−ε} u , (SZ_{c} − (Y^{∗})^{−1}) w ⟩ − ⟨ (SZ_{−ε} − (Y^{∗})^{−1}) u , Z_{c} w ⟩
Z_{c}^{∗}SZ_{−ε} − Y^{−1}Z_{−ε} − Z_{c}^{∗}SZ_{−ε} + Z_{c}^{∗}(Y^{∗})^{−1} = Z_{c}^{∗}(Y^{∗})^{−1} − Y^{−1}Z_{−ε} .

Using formula (15), this is equal to (Y^{−1}Z_{c})^{∗} − Y^{−1}Z_{c} − N_{c} = −N_{c}^{∗} , (19)

where the right hand side corresponds to its value when t = −ε. Since Z_{c}(c) = 0, evaluating the equation at t = c, we get that N_{c} is symmetric for all c > 0. Using (19) again, we obtain that Y^{−1}Z_{c} is symmetric for all t ≠ 0.

1.2. Claim. N_{c} is positive definite for all c > 0.

Proof: It is enough to show it for N_{c}^{−1}. We have that

\frac{d}{dt} − Z_{c}(t)^{−1} Y (t)|_{t=0} = Z_{c}^{−1} Ż_{c} Z_{c}^{−1} Y − Z_{c}^{−1} Ẏ |_{t=0} = −H_{pp} .

Therefore −Z_{c}(t)^{−1}Y (t) is positive definite for small t < 0. By claim 1.1, this matrix is symmetric for all t < 0. Since for t < 0 its determinant never vanishes, it is positive definite for all t < 0, in particular for t = −ε, N_{c}^{−1} is positive definite.

Define a partial order on the symmetric isomorphisms of T_{π(θ)}M by writing A ⊳ B if H_{pp}(θ)^{−1} (A − B) is positive definite. From (13) we have that the family F = {Ż_{c}(0) − Ż_{1}(0) | c > 1 } is monotone increasing. Moreover, by (16) and claim 1.2, we have that

Ż_{c}(0) − Ż_{1}(0) ⊲ H_{pp} N_{c} + (Ż_{c}(0) − Ż_{1}(0))
⊲ Ż_{−ε}(0) − Ż_{1}(0) .

So that the family F is bounded above. It can be seen that this implies the existence of a lower upper bound which is the limit of the family. We obtain the first part of

1.3. Claim.

(a) lim_{c→+∞} Ż_{c}(0) − Ż_{1}(0) = Q exists and is symmetric.

(b) lim_{c→+∞} Z_{c}(t) = D(t) exists (uniformly for bounded t intervals).

Moreover,

h(t) = D(t) ,
v(t) = S(t) D(t) + (Y^{∗}(t))^{−1} , t ≠ 0

is a matrix solution of (6) such that D(0) = I, Ḋ(0) = Q + Ż_{1}(0) and det D(t) ≠ 0 for all t ∈ ℝ.

Part (b) is a consequence of (14) and the continuous dependence of the solutions of equation (6) on the initial data.

We have that the subspace

𝔼_{θ}(0) := { (u, v(0) u) | u ∈ T_{π(θ)}M } ⊂ H(θ) ⊕ V (θ)

is the limit of the subspaces

E_{c}(θ) := dψ_{−c}(V (ψ_{c}(θ))) = { (h_{3}(θ), v_{3}(θ)) | u ∈ T_{π(θ)}M }

where h_{3}(t) and v_{3}(t) are given by (18). Moreover

𝔼_{θ}(t) := dψ_{t}(𝔼_{θ}(0)) = { (h(t) u, v(t) u) | u ∈ T_{π(θ)}M }

satisfies 𝔼_{θ}(t) ∩ V (ψ_{t}θ) = {0} and

𝔼_{θ}(t) = lim_{c→+∞} dψ_{t−c}(V (ψ_{c}(θ))) = lim_{d→+∞} dψ_{−d}(V (ψ_{d}(ψ_{t}θ))) = 𝔼_{ψ_{t}θ}(0) .

Since the vertical subspace is Lagrangian, then E_{c}(θ) is Lagrangian. By the continuity of the symplectic form, the subspaces 𝔼_{θ}(t) are Lagrangian.

To obtain the “unstable” Green bundle 𝔽(θ), observe that the flow of the Hamiltonian \overline{H}(θ) = −H(θ) is the flow ψ_{t} of H with the time reversed. In this case \overline{H}_{pp} is negative definite, but similar arguments apply to obtain the subbundle 𝔽. Applying the lemma to \overline{H}, we obtain the subbundle 𝔽. Moreover, in this case the family G = { Ż_{−c}(0) − Ż_{−1}(0) | c > 1 } is monotone decreasing and bounded below by Ż_{ε}(0)−Ż_{−1}(0). This finishes the proof of the first part of proposition A, we defer the proof of the second statement to corollary 1.12.

Let

\begin{aligned}𝕊(t) : & = v(t) h(t)^{−1} \\ & = S(t) + Y^{∗}(t)^{−1} D(t)^{−1}. \end{aligned} (20)

Then 𝕊(t) is a symmetric solution of the Ricatti equation (7), which is defined on all t ∈ ℝ. The symmetry of 𝕊(t) can be checked either because 𝔼(t) = graph (𝕊(t)) is a Lagrangian subspace or because

Y (t)^{−1} D(t) = lim_{c→∞} Y (t)^{−1} Z_{c}(t)

is symmetric and this implies that (Y^{∗})^{−1} D^{−1} is symmetric.

Let 𝕊(θ), 𝕌(θ) be the the symmetric solutions of the Ricatti equation (7) corresponding to the green bundles 𝔼(θ) = graph(𝕊(θ)) and 𝔽(θ) = graph(𝕌(θ)). Let K_{c}(θ) : H(θ) → V (θ) be the symmetric linear map such that graph(K_{c}(θ)) = dψ_{−c}(V (ψ_{c}(θ))). Define a partial order on the symmetric isomorphisms of T_{π(θ)}M by writing A ≻ B if A − B is positive definite.

1.4. Proposition. For all ε > 0,

(a) If d > c > 0 then K_{−ε} ≻ K_{d} ≻ K_{c}.

(b) If d < c < 0 then K_{ε} ≺ K_{d} ≺ K_{c}.

(c) lim_{d→+∞} K_{d} = 𝕊, lim_{d→−∞} K_{d} = 𝕌.

(d) 𝕊 ≼ 𝕌.

Proof:

Let (Z_{c}(t), V_{c}(t)) be the matrix solution of the Jacobi equation (6) with Z_{c}(0) = I, Z_{c}(c) = 0, c > 0. From (9) we have that V_{c}(t) = S(t) Z_{c}(t) − (Y^{∗}(t))^{−1}. (21)

Hence, using (14) and (13) we have that

(22)  V_{d}(t) − V_{c}(t) = S(t) [Z_{d}(t) − Z_{c}(t)]

= S(t) Y (t) ∫_{c}^{d} Y^{−1} H_{pp} (Y^{∗})^{−1} ds. (23)

When t → 0 we have that S(t) Y (t) → I. Therefore

V_{d}(0) − V_{c}(0) = ∫_{c}^{d}(Y^{−1} H_{pp} (Y^{∗})^{−1}) ds. (24)

Let K_{c}(t) = V_{c}(t) Z_{c}(t)^{−1} be the corresponding solution of the Ricatti equation (7). Since Z_{c}(0) = I, we have that K_{c}(0) = V_{c}(0). By (24), K_{d}(0) − K_{c}(0) is positive definite if d > c > 0, hence the sequence K_{d}(0), d > 0 is strictly increasing.

From (17) we have that equation (22) also holds for c = −ε. Using (15), we have that

V_{d}(t) − V_{−ε}(t) = S(t) (Z_{d}(t) − Z_{−ε}(t))
= S(t) (−Y (t) N_{d}) .

Hence V_{d}(0) − V_{−ε}(0) = −N_{d} is negative definite. Since Z_{−ε}(0) = I, then K_{−ε}(0) = V_{−ε}(0). Therefore K_{−ε} ≻ K_{d} for all d > 0.

This completes the proof of item (a). Item (c) has already been proven on claim 1.3 above. Item (b) is proven similarly using the concave Hamiltonian \overline{H} = −H and item (d) is a corollary of items (a), (b) and (c).

1.5. Remark. If a bounded open segment { ψ_{t}(θ) | a^{−} < t < a^{+} } has no conjugate points, the arguments above apply to obtain limit solutions lim_{c→a^{±}} Z_{c}(t) = D^{±}(t), D^{±}(0) = I, det D(t) ≠ 0, Lagrangian subspaces E^{±}(θ) = lim_{t→a^{±}} dψ_{−t}(V (ψ_{t} θ)) = dψ_{a^{±}}(V (ψ_{a^{±}}(θ))) and the monotonicity properties of proposition 1.4. Nevertheless, in general E^{±}(θ) ⊄ T_{θ}Σ, see remark 1.17

1.6. Remark. The existence of the Green bundles, the solutions h, v, 𝕌, 𝕊 and the monotonicity properties in proposition 1.4 do not need the uniform boundedness of the operators H_{pp}, H_{pq}, H_{qq}. But in the unbounded case we can not guarantee neither that X(θ) ∈ 𝔼(θ) ∩ 𝔽(θ) nor 𝔼(θ) ∪ 𝔽(θ) ⊂ T_{θ}Σ.

1.7. Proposition.

(a) Any symmetric solution S_{θ}(t) : H(ψ_{t}(θ)) → V (ψ_{t}(θ)) of the Ricatti equation (7) which is defined on t > 0 is uniformly bounded on t ≥ 1, i.e. there exists A > 0 such that

∥S_{θ}(t)∥ < A for all θ ∈ Σ, t > 1.

(b) Any measurable symmetric solution of the Ricatti equation defined on the energy level Σ is uniformly bounded.

An immediate consequence is the following

1.8. Corollary.

(a) The solution S_{θ}(t) of equation (8) given by Image(Y_{θ}(t) , S_{θ}(t) Y_{θ}(t)) = dψ_{t} V (θ) is uniformly bounded for |t| > 1.

(b) The solutions 𝕊_{θ}(t), 𝕌_{θ}(t) corresponding to the Green bundles are uniformly bounded on Σ.

We shall need the following lemma, whose proof is elementary:

1.9. Lemma. The function w(r) = R coth (R t − d), R > 0, satisfies

(i) ẇ + w^{2} − R^{2} = 0. (ii) ẇ < 0, w (−t +\frac{d}{R}) = −w (t +\frac{d}{R}). (iii) lim_{t→+∞} w(t) = R, lim_{t→−∞} w(t) = −R. (iv) lim_{t→(\frac{d}{R})^{+}} w(t) = +∞, lim_{t→(\frac{d}{R})^{−}} w(t) = −∞.

HAMILTONIANS WITHOUT CONJUGATE POINTS 17

Proof of proposition 1.7:

We first prove (a). By the spectral theorem, it is enough to prove that x^{∗} S_{θ}(t) x is uniformly bounded for ∥x∥ = 1 and t > 1. Since H_{pq}^{∗} = H_{qp} and S^{∗} = S, we have that

Ṡ + S^{∗} H_{pp} S + H_{pq}^{∗} S + S^{∗} H_{pq} + H_{qq} = 0.

Let C := (H_{pp})^{−1} H_{pq}, D := H_{qq} − C^{∗}H_{pp}C − Ċ and V_{θ}(t) := S_{θ}(t) + C(ψ_{t}(θ)). Then

V̇ + V^{∗} H_{pp} V + D = 0 .

Since C(θ) and D(θ) are continuous, then they are uniformly bounded on the energy level. Hence it is enough to prove that x^{∗}V_{θ}(t) x is uniformly bounded on t > 1 and θ ∈ Σ. Observe that V_{θ}(t) is not necessarily symmetric. Let M > 0 be such that

y^{∗}H_{pp}(θ) y ≥ \frac{1}{M} ∥y∥^{2} for all θ ∈ Σ. (25)

Let R > 0 be such that | y^{∗}D(θ) y | < M R^{2} for all θ ∈ Σ, ∥y∥ = 1. (26)

We claim that

| y^{∗}V_{θ}(t) y | ≤ M R coth(R) for all ∥y∥ = 1, t > 1, θ ∈ Σ.

For, let θ_{0} ∈ Σ, ∥x∥ = 1. Write V (t) := V_{θ}(t), H_{pp}(t) := H_{pp}(ψ_{t} θ). Suppose that there exists t_{o} ∈ ℝ such that

x^{∗}V (t_{0}) x =: M α > M R coth(R).

There exists d_{0} ∈ ℝ such that R coth (R t_{0} − d_{0}) = α. Observe that d_{0} > 0 and t_{0} > \frac{d_{0}}{R} > 0. Write w(t) := R coth (R t − d_{0}). Then w(t) is a solution of ẇ + w^{2} − R^{2} = 0 for t > \frac{d_{0}}{R}. In particular,

M ẇ + M w^{2} − M R^{2} = 0.

Let f(t) := x^{∗} V (t) x − M w(t). Then f(t_{0}) = 0 and

f′(t) + (x^{∗} V (t)^{∗}H_{pp}(t) V (t) x − M w(t)^{2}) + (x^{∗}D(t) x + M R^{2}) = 0 . (27)

Using Schwartz inequality and (25), we have that

M w(t_{0})^{2} = M α^{2} = \frac{1}{M} (M α)^{2} = \frac{1}{M} ⟨x, V (t_{0}) x⟩^{2}

≤ \frac{1}{M} |V (t_{0}) x|^{2} ≤ (V (t_{0}) x)^{∗}H_{pp}(t_{0}) (V (t_{0}) x)

M w(t_{0})^{2} ≤ x^{∗} V (t_{0})^{∗}H_{pp}(t_{0}) V (t_{0}) x (28)

18 G. CONTRERAS AND R. ITURRIAGA

Then (28), (26) and (27) imply that f′(t_{0}) < 0. The same argument can be applied each time that f(t) = 0. Therefore x^{∗} V (t) x ≤ M w(t) for all t > t_{0} and

x^{∗} V (t) x ≥ M w(t) for all \frac{d_{0}}{R} < t < t_{0}.

Then

lim_{t→(\frac{d_{0}}{R})^{+}} x^{∗} V (t) x ≥ lim_{t→(\frac{d_{0}}{R})^{+}} M w(t) = +∞ .

Since \frac{d_{o}}{R} > 0, this contradicts the existence of V (t) for t > 0. Hence such t_{0} does not exists and

x^{∗} V (t) x ≤ M R coth(R) for all t > 0.

Now suppose that there exists t_{1} ∈ ℝ, ∥z∥ = 1 such that

x^{∗} V (t_{1}) x < −M R .

We compare v(t) = x^{∗} V (t) x with M w_{1}(t), where w_{1}(t) = R coth (R t − c_{0}) is such that M w_{1}(t_{1}) = v(t_{1}). Observe that w_{1}(t) is defined for t < \frac{c_{0}}{R} and in this case c_{0} > 0 and \frac{c_{0}}{R} > t_{1} > 0. The same argument as above shows that v(t) ≤ M w_{1}(t) for t_{1} ≤ t < \frac{c_{0}}{R}. Since lim_{t→(\frac{c_{0}}{R})^{−}} w_{1}(t) = −∞, this contradicts the fact that V (t) is

defined for all t > 0.

The proof of (b) is similar to that of (a). In this case we can change the MR coth(R) bound by MR. We allow t_{0} ∈ ℝ and d_{0} ∈ ℝ to be nonpositive if necessary. The second inequality has also the same proof, but here we allow t_{1} ∈ ℝ and c_{0} ∈ ℝ.

The following proposition states that lim_{t→±∞} ∥dπ ∘ dψ_{t}(θ)|_{V (θ)}∥ = +∞ for all θ ∈ Σ. This limit is not uniform in θ ∈ Σ. A uniform lower bound for this norm is given in section §6.

1.10. Proposition. For all R > 0 there exists T > T (R, θ) > 0 such that |Y_{θ}(t) v| > R |v| for all |t| > T and all v ∈ T_{θ}M \ {0}.

Proof:

Let

M(t) := ∫_{t}^{∞} Y^{−1}(s) H_{pp}(s) (Y^{∗}(s))^{−1} ds .

From (12), using that D(t) = lim_{c→∞} Z_{c}(t), we have that D(s) = Y (s) M(s) for s > 0 . (29)

Consider the solutions S(t), 𝕊(t) of the Ricatti equation given on (8) and (20). From (20) and (29) we have that

𝕊(t) − S(t) = Y^{∗}(t)^{−1} M(t)^{−1} Y (t)^{−1} for t > 0.

HAMILTONIANS WITHOUT CONJUGATE POINTS 19

Since the solutions S(t), 𝕊(t) are defined on all t > 0, by proposition 1.7, there exist t_{0} > 0 and k > 0 such that ∥S(t)∥ < k and ∥𝕊(t)∥ < k for all t > t_{0}. Then for |x| = 1 and t > t_{0},

|⟨ M(t)^{−1} Y (t)^{−1} x , Y (t)^{−1} x ⟩| = |⟨ S(t) x , x ⟩| + |⟨ 𝕊(t) x , x ⟩| ≤ 2 k .

Let λ(t) be the largest eigenvalue of M(t), t > 0. Since M(t) is positive definite then λ(t) > 0 and ∥M(t)∥ = λ(t). Moreover,

2 k ≥ |⟨ M(t)^{−1} Y (t)^{−1} x , Y (t)^{−1} x ⟩| ≥ \frac{1}{λ(t)} | Y (t)^{−1} x |^{2} ,

| Y (t)^{−1} x | ≤ \sqrt{2 k} ∥M(t)∥^{\frac{1}{2}} for all |x| = 1, t > t_{0} .

Then if |v| = 1, we have that

|Y (t) v| ≥ \frac{1}{∥Y (t)^{−1}∥} ≥ \frac{1}{\sqrt{2 k} ∥M(t)∥^{\frac{1}{2}}} for t > t_{0} .

Since M(t) → 0 when t → +∞, given R > 0 there exists T > t_{0} such that (2 k ∥M(t)∥)^{\frac{1}{2}} < \frac{1}{R} for all t > T and hence |Y (t) v | > R for all t > T and |v| = 1.

Using the Hamiltonian \overline{H} := −H, we get the result for t < −T.

For θ ∈ T M, define

B(θ) := {ξ ∈ T_{θ}T^{∗}M | sup_{t∈ℝ}|dψ_{t}(θ) · ξ| < +∞ }.

1.11. Proposition. If the ψ-orbit of θ ∈ T M does not contain conjugate points, then B(θ) ⊆ 𝔼(θ) ∩ 𝔽(θ).

An example showing that in general 𝔼(θ) ∩ 𝔽(θ) ⊄ B(θ) is given in Ballmann, Brin & Burns [4].

Proof: Let ξ ∈ B(θ) and let ζ_{Γ} ∈ E_{T} (θ) := dψ_{−T} (V (ψ_{T} (θ))) be such that dπ(θ) · ζ_{T} = dπ(θ) · ξ, in particular ξ − ζ_{T} ∈ V (θ). Since lim_{T →+∞} E_{T} (θ) = 𝔼(θ) and 𝔼(θ) ⋔ V (θ) then there exists ζ := lim_{T →+∞} ζ_{T} ∈ 𝔼(θ). Moreover, dπ∘dψ_{T} (ξ − ζ_{T}) = dπ∘dψ_{T} (ξ). Since dπ∘dψ_{T} (ξ) is bounded on T > 0, by proposition 1.10, lim_{T →+∞}(ξ − ζ_{T}) = 0. Since ζ_{T} → ζ ∈ 𝔼(θ), then ξ ∈ 𝔼(θ).

Similarly if η_{T} ∈ F_{T} (θ) := dψ_{T} (V (ψ_{−T} (θ))) is such that dπ(θ) · η_{T} = dπ(θ) · ξ. Then lim_{T →+∞} η_{T} ∈ 𝔽(θ) and lim_{T →+∞}(ξ − η_{T}) = 0.

1.12. Corollary. If the orbit of θ has no conjugate points then

(a) ⟨X(θ)⟩ ⊆ 𝔼(θ) ∩ 𝔽(θ).

(b) 𝔼(θ) ∪ 𝔽(θ) ⊆ T_{θ}Σ.

20 G. CONTRERAS AND R. ITURRIAGA

Proof:

The property ⟨X(θ)⟩ ⊆ 𝔼(θ) ∩ 𝔽(θ) is a direct consequence of proposition 1.11. It remains to show that 𝔼(θ) ∪ 𝔽(θ) ⊂ T_{θ}Σ. Observe that since −dH = ω(X, ·), then

T_{θ}Σ = { ξ ∈ T_{θ}T^{∗}M | ω(X(θ), ξ) = 0 } .

Since 𝔽(θ) is Lagrangian and X(θ) ∈ 𝔽(θ), then ω(X(θ), ξ) = 0 for all ξ ∈ 𝔽(θ). Hence 𝔽(θ) ⊆ T_{θ}Σ. Similarly 𝔼(θ) ⊆ T_{θ}Σ.

1.13. Corollary. If M is connected and Σ is a regular energy level without conjugate points, then π(Σ) = M.

Proof: For θ ∈ Σ, dπ(θ) : 𝔽(θ) → T_{π(θ)}M is an isomorphism. By corollary 1.12, 𝔽(θ) ⊂ T_{θ}Σ. Hence π : Σ → M is an open map. Since M is connected then π(Σ) = M.

The metric entropy.

Let Λ be the set of total measure having the Oseledec’s splitting T_{θ}T^{∗}M = E^{s}(θ)⊕ E^{c}(θ) ⊕ E^{u}(θ).

1.14. Proposition. For all θ ∈ Λ (a) E^{s}(θ) ⊆ 𝔼(θ) ⊆ E^{s}(θ) ⊕ E^{c}(θ).

(b) E^{u}(θ) ⊆ 𝔽(θ) ⊆ E^{u}(θ) ⊕ E^{c}(θ).

(c) χ(θ) = lim_{T →+∞} \frac{1}{T} ∫_{0}^{T} tr [H_{pq} + H_{pp} 𝕌](ψ_{t}(θ)) dt.

Proof:

We only prove E^{u}(θ) ⊆ 𝔽(θ) ⊆ E^{u}(θ) ⊕ E^{c}(θ) , (30)

which shall be used for theorem E, the other inclusions are similar.

We first prove the second inclusion. Observe that for \overline{v} = (0, v) ∈ V (θ) we have that dψ_{t}(θ) \overline{v} = (Y_{θ}(t) v, ∗). Then, by proposition 1.10, we have that V (θ) ∩ E^{s}(θ) = {0}. For every θ ∈ Λ there exists a subspace R(θ) ⊂ E^{u}(θ) ⊕ E^{c}(θ) and a linear map L(θ) : R(θ) → E^{s}(θ) such that V (θ) = graph L(θ).

Let μ be an ergodic invariant measure for ψ_{t}. Given ε > 0 let K ⊂ Λ be a compact subset such that μ(K) > 1 − ε and

(a) sup { ∥L(θ)∥ | θ ∈ K } < +∞.

(b) There exists T > 0 and 0 < η < 1 < λ < η^{−1} such that for t > T and θ ∈ K

∥dψ_{t}|_{E^{s}(θ)}∥ < η^{t} and ∥dψ_{t}^{−1}|_{E^{u}(θ)⊕E^{c}(θ)}∥ < λ^{t}.

Let V_{−s}(θ) := dψ_{s}(V (ψ_{−s}(θ))). Then

V_{−s}(θ) = dψ_{s}(graph L(ψ_{−s}(θ)))
= graph [dψ_{s} ∘ L(ψ_{−s}(θ)) ∘ (dψ_{s}|_{R(ψ_{−s}(θ)})^{−1}] .

For θ ∈ K take a sequence s_{n} → +∞ such that ψ_{−s_{n}}(θ) ∈ K. Since dψ_{s_{n}}(R(ψ_{−s_{n}}(θ))) ⊆ E^{u}(θ) ⊕ E^{c}(θ), by properties (a) and (b), we have that

∥dψ_{s_{n}} ∘ L(ψ_{−s_{n}}(θ)) ∘ (dψ_{−s_{n}}|_{dψ_{sn}R(ψ_{−sn}(θ))})∥ ≤ η^{s_{n}} λ^{s_{n}} sup_{θ∈K} ∥L(θ)∥ .

Since η^{s_{n}}λ^{s_{n}} → 0 when n → +∞, we have that

𝔽(θ) = lim_{n→+∞} V_{−s_{n}}(θ) = lim_{n→+∞} dψ_{s_{n}}(R(ψ_{−s_{n}}(θ))) ⊆ E^{u}(θ) ⊕ E^{c}(θ) .

Therefore the second inclusion in (30) is satisfied for θ ∈ K. Since μ(K) ≥ 1 − ε and ε and μ are arbitrary, this inclusion holds for a set of total measure in Λ.

We prove now the first inclusion. There exist (cf. [1] theorem 3.1.19) a continuous riemannian metric ⟨ , ⟩ on T^{∗}M and a continuous family of linear isomorphisms J(θ) : T_{θ}T^{∗}M ↩ such that J(θ)^{2} = −I and the symplectic forms is written as ω(x, y) = ⟨x, J(θ)y⟩ for all x, y ∈ T_{θ}T^{∗}M and θ ∈ T^{∗}M. From now we use this metric. Let {e_{1}(θ), …, e_{n}(θ)} be an orthonormal basis of 𝔽(θ) and define e_{n+i}(θ) = Je_{i}(θ) for i = 1, …, n. Since the subspace 𝔽(θ) is Lagrangian, the subspace J𝔽(θ) = span {e_{n+1}(θ), …, e_{2n}(θ)} is the orthogonal complement of 𝔽(θ) with respect to ⟨ , ⟩. The matrices of ω and dψ_{t}(θ) with respect to the family of basis {e_{1}(θ), …, e_{2n}(θ)} are

J = [\begin{matrix}& I \\ −I \end{matrix}] and dψ_{t}(θ) = \left[{A_{t}(θ) C_{t}(θ) \atop B_{t}(θ)}\right]

Since dψ_{t} preserves the symplectic form we have that (dψ_{t})^{∗} J (dψ_{t}) = J. Hence B_{t}^{∗}(θ) = A_{t}(θ)^{−1} and B_{t}^{∗}(θ) C_{t}(θ) is symmetric.

Since 𝔽(θ) ⊆ E^{u}(θ) ⊕ E^{c}(θ) then

lim_{t→±∞} \frac{1}{t} log ∥B_{t}^{∗}^{−1}(θ) w∥ = lim_{t→±∞} \frac{1}{t} log ∥A_{t}(θ) w∥ ≥ 0

for all w ∈ ℝ^{n}, θ ∈ Λ. This implies that

lim_{t→±∞} \frac{1}{t} log ∥B_{t}^{−1}(θ) w∥ ≥ 0 . (31)

Suppose that E^{u}(θ) ⊄ 𝔽(θ). Take v ∈ E^{u}(θ) \ 𝔽(θ). Then v = w + z with z ∈ 𝔽(θ) and 0 ≠ w ∈ J𝔽(θ). We have that dψ_{−t} v = u_{−t} + (∗, B_{t}^{−1}(θ) w) with u_{−t} ∈ 𝔽(ψ_{−t}(θ)) ⊆ E^{u}(ψ_{−t}(θ)) ⊕ E^{c}(ψ_{−t}(θ)). Since B_{t}^{−1}(θ) w ∈ J𝔽(ψ_{−t}(θ)) and 𝔽

22 G. CONTRERAS AND R. ITURRIAGA

and J𝔽 are orthogonal, we have that ∥dψ_{−t}(v)∥ ≥ ∥B_{t}^{−1}(θ) w∥. Hence from (31) we obtain

lim_{t→+∞} \frac{1}{t} log ∥dψ_{−t}(θ) v∥ ≥ 0 .

This contradicts the choice v ∈ E^{u}(θ).

Finally we prove (c). Let π_{θ} : 𝔽(θ) → H(θ) be the restriction of dπ. Then π_{θ}^{−1}(v) = (v, 𝕌(θ) v) and by corollary 1.8

sup_{θ∈Σ} ∥π_{θ}^{−1}∥ < +∞ . (32)

Fix θ ∈ Λ, let Z_{θ}(t) : V (θ) → H(ψ_{t}(θ)) be defined by Z_{θ}(t) v = π_{ψ_{t}(θ)} dψ_{t}(θ) (v, 𝕌(θ)v). Then

dψ_{t}(θ)|_{𝔽(θ)} = π_{ψ_{t}(θ)}^{−1} ∘ Z_{θ}(t) ∘ π_{θ}

By (30) and (32), we have that

χ(θ) = lim_{T →+∞} \frac{1}{T} log |det dψ_{t}(θ)|_{𝔽(θ)}|

= lim_{T →+∞} \frac{1}{T} log |det Z_{θ}(t)| .

Since the isomorphisms h(t) = Z_{θ}(t), v(t) = 𝕌(ψ_{t}(θ)) Z_{θ}(t) satisfy the Jacobi equation (6), we have that

Ż_{θ} = (H_{pq} + H_{pp} 𝕌) Z_{θ} .

Since det Z_{θ}(t) ≠ 0 for all t ∈ ℝ, then det Z_{θ}(t) has constant sign and

\frac{d}{dt} |det Z_{θ}(t)| = \frac{d}{dt} det Z_{θ}(t) = tr (H_{pq} + H_{pp} 𝕌) |det Z_{θ}(t)| ,
\frac{d}{dt} log |det Z_{θ}(t)| = tr [H_{pq} + H_{pp} 𝕌] ,
lim_{T →+∞} \frac{1}{T} log |det dψ_{t}(θ)|_{𝔽(θ)}| = lim_{T →+∞} \frac{1}{T} ∫_{0}^{T} tr [H_{pq} + H_{pp} 𝕌] dt .

The exponential map.

The following proposition proves item (c) in theorem 0.1. We present a modern proof that shall be needed in the sequel.

1.15. Proposition. [Hartman][20] Let H : T^{∗}M → ℝ be a convex Hamiltonian. Then

(a) A half open segment { ψ_{t}(θ) | t ∈ [0, a[ }, a ∈]0, +∞[∪{+∞} has no conjugate points if and only if there exists a Lagrangian subspace E ⊂ T_{θ}T^{∗}M such that dψ_{t}(E) ∩ V (ψ_{t}(θ)) = {0} for all t ∈]0, a[.

(b) A closed segment { ψ_{t}(θ) | t ∈ [0, a] } has no conjugate points if and only if there exists a Lagrangian subspace E ⊂ T_{θ}T^{∗}M such that dψ_{t}(E) ∩ V (ψ_{t}(θ)) = {0} for all t ∈ [0, a].

(c) If a closed segment { ψ_{t}(θ) | t ∈ [0, a] } has no conjugate points, then there exists δ > 0 such that the segment { ψ_{t}(θ) | t ∈ [−δ, a + δ] } has no conjugate points.

Proof:

(a) If the half open segment is disconjugate, then the Lagrangian subspace E = V (θ) satisfies dψ_{t}(V (θ)) ∩ V (ψ_{t}(θ)) = {0}.

Conversely, suppose that such Lagrangian subspace E is given. Let E(t) := dψ_{t}(E). Then the projection dπ : E(t) → H(ψ_{t}(θ)) is an isomorphism. Let S(t) : H(ψ_{t}(θ)) → V (ψ_{t}(θ)) be the linear isomorphism such that E(t) = graph (S(t)) for t ∈]0, a[. Since E(t) is Lagrangian, then S(t) is symmetric and it satisfies the Ricatti equation (7). Let Z(t) : E(0) → H(ψ_{t}(θ)) be given by Z(t) w = dπ dψ_{t}(θ) w. Then Ż = (H_{pq} + H_{pp} S) Z , Z(0) = dπ|_{E}. (33)

Let w ∈ E and (h_{1}(t), v_{1}(t)) = dψ_{t}(w) ∈ E(t), then h_{1}(t) = Z(t) w, v_{1}(t) = S(t) Z(t) w. Let ξ ∈ T_{θ}T^{∗}M and (h(t), v(t)) = dψ_{t}(ξ). Suppose that h(c) = 0, c ∈]0, a[. Using these h_{1}(t), v_{1}(t), the same arguments as in (9), (10), (11) give that

h(t) = Z(t) Z(c)^{−1} h(c) + Z(t) ∫_{c}^{t} Z^{−1}(s) H_{pp} (Z^{∗}(s))^{−1}(Z(c)^{∗}v(c)) ds . (34)

Since h(c) = 0, then

⟨Z(t)^{−1} h(t) , Z(c)^{∗}v(c)⟩ = ∫_{c}^{t} ⟨H_{pp} (Z(s)^{∗})^{−1} Z(c)^{∗}v(c) , (Z(s)^{∗})^{−1} Z(c)^{∗}v(c)⟩ ds . (35)

In particular h(t) ≠ 0 for all t ∈]0, a[\{c}. Hence the segment ]0, a[ does not have conjugate points.

It remains to prove the case c = 0. Let 0 < b < a. We prove that θ is not conjugate to ψ_{b}(θ). We already know that ψ_{b}(θ) has no conjugate points in { ψ_{t}(θ) | 0 < t < a }. For 0 < d < b let E_{d} = dψ_{b−d}(V (ψ_{d}(θ))) = graph (K_{d}), where K_{d} : H(ψ_{b}(θ)) →

24 G. CONTRERAS AND R. ITURRIAGA

V (ψ_{b}(θ)) is the corresponding solution of the Ricatti equation (7). Let ε > 0 be such that b < b + ε < a and let E_{b+ε} = ψ_{−ε}(V (ψ_{b+ε}(θ))) = graph (K_{b+ε}). By proposition 1.4(b) and remark 1.5, the family K_{d} is monotone decreasing on d and has the (lower) bound K_{b+ε}. Hence there exists 𝔼 := lim_{d→0^{+}} E_{d} = dψ_{b}(V (θ)) and 𝔼 ∩ V (ψ_{b}(θ)) = {0}.

(c) If the closed segment [0, a] has no conjugate points, then dψ_{t}(V (θ))∩V (ψ_{t}(θ)) = {0} for 0 < t ≤ a. Then dψ_{t}(V (θ)) ∩ V (ψ_{t}(θ)) = {0} for 0 < t ≤ a + δ for some δ > 0. By item (a), using E(t) = dψ_{t}(V (θ)), the segment [0, a + δ] has no conjugate points. The same argument using F (t) = dψ_{a+δ−t}(V (ψ_{a+δ}(θ))) shows that there exists δ_{1} > 0 such that the segment [−δ_{1}, a + δ] has no conjugate points.

(b) Suppose that the segment [0, a] is disconjugate. By item (c) there exists δ > 0 such that the segment [−δ, a + δ] is disconjugate. Now apply item (a) to the segment [−\frac{δ}{2}, a + δ[. Conversely, if there is a Lagrangian subspace E ⊂ T_{θ}T^{∗}M with dψ_{t}(E) ∩ V (ψ_{t}(θ)) = {0} for all 0 ≤ t ≤ a, then the argument of equation (35) applies for all c ∈ [0, a].

It may be impossible to find such Lagrangian subspace E of theorem 1.15 (a),(b) satisfying E ⊂ T_{θ}Σ as explained in remark 1.17. Compare this with theorem 0.1. If there exists a continuous invariant Lagrangian bundle E(θ), one can always suppose that E(θ) ⊂ T_{θ}Σ by taking (E(θ) ∩ T_{θ}Σ) ⊕ ⟨X(θ)⟩. Then it must satisfy the transversality condition 0.1(a).

1.16. Proposition.

For θ ∈ Σ define W (θ) := (V (θ) ∩ T_{θ}Σ) ⊕ ⟨X(θ)⟩. Suppose that the orbit of θ has no conjugate points. Then for all t ≠ 0, dψ_{t}(W (θ)) ∩ V (ψ_{t}(θ)) = {0}.

Proof:

Suppose that there exists b > 0 such that (0, w_{b}) ∈ dψ_{b}(W (θ)) ∩ V (ψ_{b}(θ)) ≠ {0}. Let (k_{0}, w_{0}) := dψ_{−b}(0, w_{b}) ∈ W (θ). Since the segment { ψ(θ) | t ∈ [0, +∞[ } has no conjugate points, then k_{0} ≠ 0. Write H_{p}(t) := H_{p}(ψ_{t}(θ)), H_{q}(t) := H_{q}(ψ_{t}(θ)). Since W (θ) = (V (θ) ∩ T_{θ}Σ) ⊕ ⟨X(θ)⟩ and X(θ) = (H_{p}(0), −H_{q}(0)), then k_{0} = α H_{p}(0) for some α ≠ 0. Multiplying w_{b} by \frac{1}{α}, we can assume that α = 1. Let

(h(t), v(t)) := dψ_{t}(X(θ)) − (k_{0}, w_{0}) ∈ dψ_{t}(W (θ)) ⊆ T_{ψ_{t}(θ)}Σ .

Then

h(0) = H_{p}(0) − k_{0} = 0 , v(0) ≠ 0.
(36)  h(b) = dπ [X(ψ_{t}(θ)) − (0, w_{b})] = H_{p}(b) .

Since (h(0), v(0)) ∈ T_{θ}Σ, then 0 = dH(h(0), v(0)) = H_{q} h(0) + H_{p} v(0) = H_{p}(0) · v(0) . (37)

HAMILTONIANS WITHOUT CONJUGATE POINTS 25

Let 𝔼(t) = 𝔼(ψ_{t}(θ)) be the stable Green bundle of ψ_{ℝ}(θ), let 𝕊(t) be from (20) and let Z(t) := h(t) : H(ψ_{t}(θ)) → V (ψ_{t}(θ)) be from claim 1.3. So that Z(t) w = dπ dψ_{t}(w, 𝕊(0) w) and Z(t) satisfies (33). Using formula (34) for c = 0, (with Z(t) = h(t), Z(0) = I and h(0) = 0) we obtain that

h(t) = Z(t) ∫_{0}^{t} Z^{−1}(s) H_{pp}(s) (Z^{∗}(s))^{−1}v(0) ds .

Hence, since v(0) ≠ 0, we have that

⟨Z(b)^{−1}h(b), v(0)⟩ = ∫_{0}^{b}⟨H_{pp} (Z^{∗}(s)^{−1}v(0)), (Z^{∗}(s)^{−1}v(0))⟩ ds > 0 . (38)

Since X(θ) ∈ 𝔼(θ), then

(H_{p}(b), −H_{q}(b)) = X(ψ_{b}(θ)) = dψ_{b}(X(θ)) = dψ_{b}(H_{p}(0), 𝕊(0) H_{p}(0))
= (Z(b) H_{p}(0), 𝕊(b) · Z(b) H_{p}(0)) .

From the first component we get that Z(b)^{−1}H_{p}(b) = H_{p}(0). Using (36) we have that Z(b)^{−1}h(b) = H_{p}(0). Replacing this equation on (38), we obtain that

⟨H_{p}(0), v(0)⟩ > 0 .

This contradicts equation (37).

1.17. Remark.

The same proof applies for the following statement: If H : T^{∗}M → ℝ is convex and there is a Lagrangian subspace E ⊂ T_{θ}Σ such that dψ_{t}(E) ∩ V (ψ_{t}(θ)) = {0} for t ∈ [0, a], then dψ_{t}(W (θ)) ∩ V (ψ_{t}(θ)) = {0} for all t ∈]0, a].

In particular this hypothesis holds if { ψ_{t}(θ) | t ≥ 0 } or { ψ_{t}(θ) | t ≤ 0 } have no conjugate points. Just take the corresponding Green bundles.

The example I.2 does not satisfy the conclusion of proposition 1.16 on t ∈ [0, π], nevertheless the segment { ψ_{t}(θ) | t ∈ [0, π] } has no conjugate points. This implies that the Lagrangian subspace given by proposition 1.15(b) may not satisfy E ⊂ T_{θ}Σ.

1.18. Corollary. Let H : T^{∗}M → ℝ be convex, Σ = H^{−1}{e}, q ∈ π(Σ) and suppose that the orbits { ψ_{t}(θ) | t ≥ 0 } have no conjugate points for all θ ∈ π^{−1}{q}∩Σ. Then the exponential map exp_{q} : T_{q}^{∗}M → M is an immersion.

Proof:

We have that exp_{q}(tθ) = π∘ψ_{t}(θ) for θ ∈ Σ_{q} = π^{−1}{q} ∩ Σ. Consider the splitting

T_{tθ}(T_{q}^{∗}M) ≈ T_{q}^{∗}M = T_{θ}(Σ_{q}) ⊕ ⟨θ⟩ = (V (θ) ∩ T_{θ}Σ) ⊕ ⟨θ⟩ .

26 G. CONTRERAS AND R. ITURRIAGA

The derivative d exp_{q}(tθ) : T_{q}^{∗}M → T_{π(ψ_{t} θ)}M is given by

(39)  d exp_{q}(tθ)|_{V (θ)∩T_{θ}Σ} = dπ ∘ dψ_{t}(θ)|_{V (θ)∩T_{θ}Σ} ,
(40)  d exp_{q}(tθ)(θ) = dπ(X(ψ_{t}(θ))) = dπ ∘ dψ_{t}(X(θ)) .

Therefore

d exp_{q}(tθ)(T_{q}^{∗}M) = dπ ∘ dψ_{t}((V (θ) ∩ T_{θ}Σ) ⊕ ⟨X(θ)⟩)

= dπ ∘ dψ_{t}(W (θ)) . (41)

Since dim W (θ) = n, then dim dψ_{t}(W (θ)) = n. By proposition 1.16 dψ_{t}(W (θ)) ∩ V (ψ_{t}(θ)) = {0}. Hence dπ : dψ_{t}(W (θ)) → T_{π(ψ_{t} θ)}M is a linear isomorphism. From (41) d exp_{q}(tθ) is a linear isomorphism for all t > 0, θ ∈ Σ_{q}.

2. The transversal behaviour.

Given θ ∈ Σ, consider H_{p}(θ) = dπ X(θ) ≠ 0. Let N(θ) ⊆ T_{θ}Σ be defined by

N(θ) := { ξ ∈ T_{θ}Σ | ⟨ dπ ξ, dπ X(θ) ⟩_{π(θ)} = 0 } . (42)

Then N(θ)⊕⟨X(θ)⟩ = T_{θ}Σ. Fix θ_{0} ∈ Σ and a smooth coordinate system (q_{1}, …, q_{n}, t) of M × ℝ along the projection (π(ψ_{t}(θ_{0})), t) of the orbit of θ_{0} such that

(2.a) \frac{∂}{∂q_{1}} |_{(π(ψ_{t}θ_{0}),t)} = dπ X(ψ_{t}(θ_{0})).

(2.b) \frac{∂}{∂q_{2}} , …, \frac{∂}{∂q_{n}} is an orthonormal basis for dπ N(ψ_{t}(θ_{0})) = ⟨\frac{∂}{∂q_{1}} ⟩^{⊥} along π(ψ_{t}(θ_{0}), t).

Write p_{i} = dq_{i}|_{T_{q}M}, i = 1, 2, …, n. From (2.a) we have that

1 = \frac{d}{dt} q_{1}|_{(π(ψ_{t}θ_{0}), t)} = H_{p_{1}}(ψ_{t}(θ_{0})) ≠ 0 .

Then the equation H(q_{1}, q_{2}, …, q_{n}; p_{1}, p_{2}, …, p_{n}) = h can be solved locally for p_{1}:

p_{1} = −K(Q, P, T; h) ,

where P = (p_{2}, …, p_{n}), Q = (q_{2}, …, q_{n}), T = q_{1}. This solution can be extended to a simply connected neighbourhood W of the orbit { (ψ_{t}(θ_{0}), t) | t ∈ ℝ } ⊆ T^{∗}M × ℝ.

From now on we omit the ℝ-coordinate in W ⊆ T^{∗}M × ℝ. Let ϕ_{T} be the reparametrization of the flow ψ_{t} on W such that it preserves the foliation q_{1} = T = constant. In particular dϕ_{T} (θ_{0}) preserves the transversal bundle N(ϕ_{T} (θ_{0})) along the orbit of θ_{0}. Define τ(T, θ) by ϕ_{T} (θ) = ψ_{τ(T,θ)}(θ). Then τ(T, θ_{0}) = T for all T ∈ ℝ. The following reduction appears in Arnold [2].

2.1. Lemma. The orbits ϕ_{T} (θ) = (T = q_{1}, Q(T), p_{1}(T), P (T); τ(T, θ)) ∈ W satisfy the equations

\frac{dQ}{dT} = \frac{∂K}{∂P}  ,  \frac{dP}{dT} = −\frac{∂K}{∂Q} ,

where Q = (q_{2}, …, q_{n}), P = (p_{2}, …, p_{n}), T = q_{1} and K(Q, P; T) is defined by H(T, q_{2}, …, q_{n}; −K, p_{2}, …, p_{n}) = h.

Proof: Consider the canonical 1-form Θ = p dq on T^{∗}M and the symplectic form ω = dΘ = dp ∧ dq. On T Σ we have that ω(X(θ), ·) = −dH = 0. Let \frac{d}{dT} ϕ_{T} (θ) = (Y (ϕ_{T} (θ)), \frac{d}{dT} τ(T, θ)) be the vectorfield of ϕ_{T}. Then Y (ϕ_{T} (θ)) is a multiple of the vectorfield X(ϕ_{T} (θ)) of ψ_{t}, and hence ω(Y (ϕ_{T} (θ)), · ) = 0 on T Σ. We have

Θ = p dq = P dQ − K dT,

ω|_{T Σ} = dΘ|_{T Σ} = ∑_{i=2}^{n} [dP_{i} ∧ dQ_{i} − K_{Q_{i}} dQ_{i} ∧ dT − K_{P_{i}} dP_{i} ∧ dT] .

The matrix of ω|_{T Σ} is given by

[\begin{matrix}0 & −I & −K_{Q}^{∗} \\ I & 0 & −K_{P}^{∗} \\ K_{Q} & K_{P} & 0 \end{matrix}] \begin{matrix}\} Q \\ \} P \\ \} T =q_{1} \end{matrix}

Then the vector (K_{P} , −K_{Q}, 1 = \frac{dq_{1}}{dT} ) =: Z(ϕ_{T} (θ)) satisfies ω(Z(ϕ_{T} (θ)), ·) = 0 on Σ ≈ { (T, Q, p_{1}, P) | p_{1} = −K(Q, P, T) }. Since the form ω has maximal rank on Σ, we have that Z(ϕ_{T} (θ)) = Y (ϕ_{T} (θ)) for θ ∈ W. This proves the lemma.

Since for ϑ ∈ W, ϕ_{T} (ϑ) = ψ_{τ(T,ϑ)}(ϑ), then

dϕ_{T} (ϑ) · ξ = dψ_{τ(T,ϑ)} · ξ + ( \frac{∂τ}{∂ϑ}|_{(T,ϑ)} · ξ) \frac{d}{dt}ψ_{t}(ϑ)|_{τ(T,ϑ)}

= dψ_{τ(T,ϑ)} · ξ + α(T, ξ) X(ϕ_{T} (ϑ)) , (43)

where α(T, ξ) = \frac{∂τ}{∂ϑ} |_{(T,ϑ)} · ξ.

Let Λ(ψ_{t}(θ_{0})) : T_{ψ_{t}(θ_{0})}Σ → N(ψ_{t}(θ_{0})) be the projection along the direction of the vectorfield, i.e.

Λ ξ = ξ + β(ξ) X(ψ_{t}(θ_{0})) with β(ξ) ∈ ℝ such that Λ ξ ∈ N(ψ_{t}(θ_{0})) .

Through the proof of lemma 2.2 all the quantities will be understood in local coordinates T = q_{1}, Q = (q_{2}, …, q_{n}), P = (p_{2}, …, p_{n}) of W ∩ Σ. In particular

T_{ϕ_{T} (θ_{0})}Σ = ℝ × ℝ^{n−1} × ℝ^{n−1} for all T ∈ ℝ, with coordinates (\frac{∂}{∂T} , \frac{∂}{∂Q}, \frac{∂}{∂P} ). Since along the orbit of θ_{0} we have that

q̇ = (Ṫ , Q̇) = (1, 0_{n−1}) = (H_{p_{1}}, H_{P}) along ϕ_{T} (θ_{0}) = (T, 0_{n−1}, 0_{n−1}),

then the subspace N(ϕ_{T} (θ_{0})) in (42) is written as N := {0} × ℝ^{n−1} × ℝ^{n−1} in coordinates. Define ℙ : T_{ϑ}Σ = ℝ × ℝ^{n−1} × ℝ^{n−1} → N = {0} × ℝ^{n−1} × ℝ^{n−1} as the projection

ℙ(h_{1}, H, V) := (0, H, V).

Using β = −h_{1} on (43) and H_{p} = (1, 0), the projection Λ is written as

Λ(h_{1}, H, V) = (0, H, V + h_{1} · H_{Q}) ,

where H_{Q} := (H_{q_{2}}, …, H_{q_{n}}). Define the matrices

ℍ(T) = [\begin{matrix}H_{pq} & H_{pP} \\ −H_{Qq} & −H_{QP} \end{matrix} ]_{(2n−1)×(2n−1)}

𝕂(T) = [\begin{matrix}0_{1×n} & 0_{1×(n−1)} \\ K_{P q} & K_{P P} \\ −K_{Qq} & −K_{QP} \end{matrix}]_{(2n−1)×(2n−1)}

along the orbit of θ_{0}.

2.2. Lemma. Along the orbit θ_{T} := ϕ_{T} (θ_{0}) of θ_{0}, we have that

(a) The matrix K_{P P} (θ_{T}) is positive definite.

(b) For ξ ∈ N(θ_{T}), 𝕂(θ_{T}) ξ = Λ(θ_{T}) ℍ(θ_{T}) ξ.

(c) The operator 𝕂(θ_{T})|_{N(θ_{T})} is uniformly bounded on T ∈ ℝ.

Proof:

(b) Write θ_{T} = ϕ_{T} (θ_{0}) = ψ_{T} (θ_{0}) = (T, 0, 0). From lemma 2.1, the Jacobi equation for ϕ_{T} is written as

\frac{d}{dT} [ℙ dϕ_{T} ξ] = 𝕂 [ℙ dϕ_{T} ξ] ,

where dϕ_{T} ξ = (h_{1}(T), H(T), V (T)) ∈ ℝ × ℝ^{n−1} × ℝ^{n−1}. for any ξ ∈ T_{θ_{0}}Σ = ℝ × ℝ^{n−1} × ℝ^{n−1}. Also, from (6),

\frac{d}{dt} [dψ_{t} ξ] = ℍ [dψ_{t} ξ] .

Since dϕ_{T} (θ_{0}) preserves the foliation q_{1} = constant, then it preserves the subspace N. Therefore

ℙ dϕ_{T} ξ = dϕ_{T} ξ = Λ dϕ_{T} ξ for ξ ∈ N .

HAMILTONIANS WITHOUT CONJUGATE POINTS 29

In particular, the first coordinate of dϕ_{T} ξ is h_{1}(T) ≡ 0 and also \frac{d}{dT} h_{1} ≡ 0. Hence if ξ ∈ N, then

\frac{d}{dT} [ℙ dϕ_{T} ξ] = \frac{d}{dT} [dϕ_{T} ξ]
= \frac{d}{dT} [ dψ_{T} ξ + α(T) X(θ_{T}) ]
= ℍ(dψ_{T} ξ) + α̇(T) X(θ_{T}) + α(T) \frac{d}{dT} X(θ_{T})
= ℍ[dϕ_{T} ξ − α(T) X(θ_{T})] + α̇(T) X(θ_{T}) + α(T) \frac{d}{dT} X(θ_{T}) ,

where α(T) = α(T, ξ) is from (43).

Since \frac{d}{dT} X(θ_{T}) = ℍ X(θ_{T}), then

\frac{d}{dT} [ℙ dϕ_{T} ξ] = ℍ [dϕ_{T} ξ] + α̇(T) X(θ_{T}) .

Hence 𝕂 dϕ_{T} ξ = ℍ [dϕ_{T} ξ] + α̇(T) X(θ_{T}) for ξ ∈ N . (44)

Observe that this equation must hold in all the 2n − 1 coordinates of T_{ϑ}Σ. Since Image(𝕂) ⊆ N, then Λ 𝕂 = 𝕂. So

𝕂(dϕ_{T} ζ) = Λ 𝕂(dϕ_{T} ζ) = Λ [ ℍ dϕ_{T} ζ + α̇(T, ζ) X(θ_{T}) ]
= Λ ℍ(dϕ_{T} ζ) for ζ ∈ N .

Since dϕ_{T} : N(θ_{0}) = N → N(θ_{T}) = N is surjective, this proves item (b). (a) Let V ∈ ℝ^{n−1}, then

V^{∗} 𝕂_{P P} V = [(0, V^{∗}), 0_{n−1}] 𝕂 [\begin{matrix}0 & n \\ V \end{matrix}] .

Since ξ = (0_{n}, V) ∈ N, we can use equation (44) to obtain

V^{∗} K_{P P} V = [(0, V^{∗}), 0_{n−1}] [ℍ [\begin{matrix}0 & n \\ V \end{matrix}] + α̇(T, dϕ_{−T} ξ) [\begin{matrix}H_{p}^{∗} \\ −H_{Q}^{∗} \end{matrix}]]
= 0 + V^{∗} H_{P P} V + α̇ ((0, V^{∗}) · H_{p}^{∗})
= V^{∗} H_{P P} V + α̇ ⟨(1, 0_{n−1}), (0, V^{∗})⟩
= V^{∗} H_{pp} V > 0 .

We now prove (c). Since the operator ℍ(ϑ) is uniformly bounded on ϑ ∈ Σ, from (b) it is enough to prove that the projection Λ is bounded on the orbit of θ_{0}. For, it is enough to see that the angle ∢(X(θ_{T}), N(θ_{T})) is uniformly bounded away from 0. Given a vector

ξ = (h, v) = ∑_{i=1}^{n} (h_{i} \frac{∂}{∂q_{i}} + v_{i} \frac{∂}{∂p_{i}} ) ∈ N(θ_{T}) ,

30 G. CONTRERAS AND R. ITURRIAGA

written in our coordinates (2.a), (2.b), define the norm |ξ| := (∑_{i=1}^{n} h_{i}^{2} + v_{i}^{2})^{\frac{1}{2}} . Suppose that |ξ| = 1. Since ξ ∈ N(θ_{T}), then H_{p} · h = 0. Moreover, H_{p} = \frac{∂}{∂q_{1}} and hence

\frac{|X(θ_{T}) · ξ|}{|X(θ_{T})|} = \frac{|H_{p} · h − H_{q} · v|}{(|H_{p}|^{2} + |H_{q}|^{2})^{\frac{1}{2}}} = \begin{matrix}|H_{q}| & |v| \\ √ & \overline{1 + |H_{q}|^{2}} \end{matrix}

≤ 1 · max_{0≤x≤A} \frac{x}{\sqrt{1 + x^{2}}} = \frac{A}{\sqrt{1 + A^{2}}} < 1 ,

where A := sup_{T ∈ℝ} |H_{q}(θ_{T})| < +∞ with the norm in our coordinates \frac{∂}{∂p_{i}}.

2.3. Corollary. Along the orbit θ_{T} = ϕ_{T} (θ_{0}), we have

(a) The orbit of θ_{0} under ϕ_{T} has no conjugate points.

(b) Existence of the Green bundles for dϕ_{s}|_{N(θ_{T})}:

𝔼^{⊤}(θ_{T}) = lim_{s→+∞} dϕ_{−s} (V (θ_{T +s}) ∩ N(θ_{T +s}))
𝔽^{⊤}(θ_{T}) = lim_{s→−∞} dϕ_{−s} (V (θ_{T +s}) ∩ N(θ_{T +s}))

Moreover, 𝔼^{⊤}(θ) = 𝔼(θ) ∩ N(θ) and 𝔽^{⊤}(θ) = 𝔽(θ) ∩ N(θ) for all θ ∈ Σ.

(c) Horizontal growth of iterates of vertical vectors: For all R > 0 there exists S = S(R, θ_{T}) > 0 such that for all |s| > S(R, θ_{T}) and all ξ ∈ N(θ_{T}) ∩ V (θ_{T}) we have that |dπ(dϕ_{s}(θ_{T}) · ξ)| > R |ξ|.

(d) Define

B^{⊤}(θ_{T}) = { ξ ∈ N(θ_{T}) | sup_{s∈ℝ}|dπ · dψ_{s} · ξ| < +∞ } .

Then B^{⊤}(θ_{T}) ⊆ 𝔼^{⊤}(θ_{T}) ∩ 𝔽^{⊤}(θ_{T}).

(e) Let Ψ_{s} := dϕ_{s}|_{N} = Λ ∘ dψ_{s}|_{N}, then

dϕ_{s}|_{N(θ_{T})} = Λ(θ_{T +s}) ∘ dΨ_{s} ∘ Λ(θ_{T}) = Ψ_{s} ,
Ψ_{s+t} = Ψ_{s} ∘ Ψ_{t} .

Proof:

Let U(θ_{T}) := V (θ_{T}) ∩ N(θ_{T}) = V (θ_{T}) ∩ T_{θ_{T}} Σ. From (43) we have that

dϕ_{T} (θ_{0}) · ξ = dψ_{T} (θ_{0}) · [ξ + α(T, ξ) X(θ_{0})] .

Hence dϕ_{T} (U(θ_{0})) ⊆ dψ_{T} (W (θ_{0})), where W (θ) := (V (θ) ∩ T_{θ}Σ) ⊕ ⟨X(θ)⟩. From proposition 1.16,

U(θ_{T}) ∩ dϕ_{T} (U(θ_{0})) ⊆ V (θ_{T}) ∩ dψ_{T} (W (θ_{0})) = {0} for all T ≠ 0 . This implies item (a).

HAMILTONIANS WITHOUT CONJUGATE POINTS 31

From lemma 2.2, the first part of item (b) and items (c) and (d) have the same proofs as propositions A, 1.10 and proposition 1.11 respectively.

Since dϕ_{s} and dψ_{s} satisfy (43) and dϕ_{s} preserves the bundle N(θ_{T}), we have that

dϕ_{s}|_{N(θ_{T})} = Λ(θ_{T +s}) ∘ dψ_{s}|_{N(θ_{T})} = Λ(θ_{T +s}) ∘ dψ_{s} ∘ Λ(θ_{T}) = Ψ_{s} .

Moreover,

Ψ_{s+t} = dϕ_{s+t}|_{N(θ_{T})} = dϕ_{s}|_{N(θ_{T +t})} ∘ dϕ_{t}|_{N(θ_{T})} = Ψ_{s} ∘ Ψ_{t} .

This proves item (e).

Finally, we have

𝔼^{⊤}(θ_{0}) = lim_{T →+∞} [dϕ_{−T} (V (θ_{T}) ∩ N(θ_{T}))]

= Λ(θ_{0}) lim_{T →+∞} [dψ_{−T} (V (θ_{T}) ∩ T_{θ_{T}} Σ)]

⊆ Λ(θ_{0}) · 𝔼(θ_{0}) ⊆ 𝔼(θ_{0}) .

Because X(θ_{0}) ∈ 𝔼(θ_{0}). Thus 𝔼^{⊤}(θ_{0}) ⊆ 𝔼(θ_{0})∩N(θ_{0}). Since dim 𝔼^{⊤}(θ_{0}) = dim U = n − 1 = dim(𝔼(θ_{0}) ∩ N(θ_{0})), we have that 𝔼^{⊤}(θ_{0}) = 𝔼(θ_{0}) ∩ N(θ_{0}). The equation 𝔽^{⊤}(θ_{0}) = 𝔽(θ_{0}) ∩ N(θ_{0}) is proven similarly. This completes the proof of item (b).

Proof of proposition B.

Let θ_{0} ∈ Γ and let τ > 0 be the period of θ_{0}. Suppose that Γ is hyperbolic, then by proposition 1.14, 𝔼(θ_{0}) ⊆ E^{s}(θ_{0}) ⊕ ⟨X(θ_{0})⟩ and 𝔽(θ_{0}) ⊆ E^{u}(θ_{0}) ⊕ ⟨X(θ_{0})⟩. Hence 𝔼(θ_{0}) ∩ 𝔽(θ_{0}) = ⟨X(θ_{0})⟩.

Now observe that if we construct the coordinates (q_{1}, …, q_{n}) in (2.a), (2.b) so that they are periodic with period τ, then dϕ_{τ}(θ_{0}) : N(θ_{0}) ↩ is the derivative of the Poincaré map of ψ_{t} defined on the cross-section { ϑ | q_{1}(ϑ) = 0 }.

Suppose that 𝔼(θ_{0}) ∩ 𝔽(θ_{0}) = ⟨X(θ_{0})⟩. By corollary 2.3 (a) and (b) we have that B^{⊤}(θ_{0}) ⊆ 𝔼^{⊤}(θ_{0}) ∩ 𝔽^{⊤}(θ_{0}) = ⟨X(θ_{0})⟩ ∩ N(θ_{0}) = {0}. This implies that Γ is hyperbolic.

3. Quasi-hyperbolic actions.

Let B be a compact metric space and π : E → B a vector bundle provided with a continuous norm | |_{p} on each fibre π^{−1}{p}. Let Ψ be an ℝ-action Ψ : ℝ → Isom(E), i.e.

(i) There exists a continuous flow ψ_{t} on B such that π ∘ Ψ_{t} = ψ_{t} ∘ π. (ii) Ψ_{t} : E(p) → E(ψ_{t}(p)) is a linear isomorphism, where E(p) := (π|_{E})^{−1}{p},

p ∈ B.

(iii) Ψ_{s+t} = Ψ_{s} ∘ Ψ_{t}. We say that the action Ψ_{t} is quasi-hyperbolic if

sup_{t∈ℝ}|Ψ_{t}(ξ)| = +∞  for all  ξ ∈ E, ξ ≠ 0.

We say that Ψ is hyperbolic if there exists an invariant continuous splitting E(p) = E^{u}(p) ⊕ E^{s}(p), p ∈ B and C > 0, λ > 0 such that

(3.a) Ψ_{t}(E^{u}(p)) = E^{u}(ψ_{t}(p)), Ψ_{t}(E^{s}(p)) = E^{s}(ψ_{t}(p)), for all x ∈ B, t ∈ ℝ. (3.b) |Ψ_{t}(ξ)| ≤ C e^{−λt} |ξ| for all t > 0, ξ ∈ E^{u}(p), p ∈ B. (3.c) |Ψ_{−t}(ξ)| ≤ C e^{−λt} |ξ| for all t > 0, ξ ∈ E^{u}(p), p ∈ B.

The aim of this section is to prove the following

0.2. Theorem. If the action Ψ is quasi-hyperbolic and the non-wandering set Ω(ψ|_{B}) = B, ψ ∘ π = π ∘ Ψ, then Ψ|_{E} is hyperbolic.

Applying this theorem to the case in which B = M is a compact manifold, ψ_{t} is a differentiable flow on M, Ψ_{t} = dψ_{t} and E is a dψ_{t}-invariant continuous bundle such that E ⊕ ⟨\frac{d}{dt}ψ_{t}⟩ = T M, we get

3.1. Theorem. [Freire] [15]

If M is a closed manifold and ψ is a quasi-Anosov flow on M such that Ω(ψ) = M, then ψ is Anosov.

The proof of theorem 0.2 is similar to that of theorem 3.1. We include it here for completeness.

Suppose that the action of Ψ is quasi-hyperbolic. For p ∈ B define

E^{s}(p) := { v ∈ E(p) | sup_{t>0} |Ψ_{t}(p)(v)| < +∞ } ,
E^{u}(p) := { v ∈ E(p) | sup_{t<0} |Ψ_{t}(p)(v)| < +∞ } .

Observe that by the quasi-hyperbolicity we have that

E^{s}(p) ∩ E^{u}(p) = {0} for all p ∈ B.

3.2. Lemma. There exists τ > 0 such that for all p ∈ B,

∥Ψ_{τ}|_{E^{s}(p)}∥ < \frac{1}{2} , ∥Ψ_{−τ}|_{E^{u}(p)}∥ < \frac{1}{2}

for all p ∈ M.

Proof:

Suppose that the lemma is false for E^{s}. The proof for E^{u} is similar. Then there exist sequences x_{n} ∈ B, v_{n} ∈ E^{s}(x_{n}), |v_{n}| = 1 such that |Ψ_{n}(v_{n})| ≥ \frac{1}{2} for all n ∈ ℕ. We claim that there exists C such that

sup_{t≥0} ∥Ψ_{t}(p)|_{E^{s}(p)}∥ < C < +∞ .

for all p ∈ B.

Suppose the claim is true. Let w_{n} := Ψ_{n}(v_{n}) and y_{n} := ψ_{n}(x_{n}). We have that \frac{1}{2} ≤ |w_{n}| ≤ C for all n and

|Ψ_{t}(w_{n})| = |Ψ_{t+n}(v_{n})| ≤ C for all t ≥ −n .

Since B is compact there exists a convergent subsequence of (y_{n}, w_{n}). If y_{n} →_{n} y and

w_{n} →_{n} w, we have that y ∈ B, w ∈ E(y), |w| ≥ \frac{1}{2} and

|Ψ_{t}(w)| ≤ C for all t ∈ ℝ .

This is a contradiction.

Now we prove the claim. Suppose it is false. Then there exists x_{n} ∈ B, t_{n} ≥ 0, v_{n} ∈ E^{s}(x_{n}), |v_{n}| = 1, such that

sup_{n} |Ψ_{t_{n}}(v_{n})| = +∞ . (45)

Let s_{n} > 0 be such that

|Ψ_{s_{n}}(v_{n})| > \frac{1}{2} sup_{s≥0} |Ψ_{s}(v_{n})| ≥ \frac{1}{2} |Ψ_{t_{n}}(v_{n})| .

By (45) we have that s_{n} →_{n} +∞. Let y_{n} := ψ_{s_{n}}(x_{n}) and

w_{n} := \frac{Ψ_{s_{n}}(v_{n})}{|Ψ_{s_{n}}(v_{n})|} .

Then |w_{n}| = 1 and if t > −s_{n} we have

|Ψ_{t}(w_{n})| = \frac{|Ψ_{t+s_{n}}(v_{n})|}{|Ψ_{s_{n}}(v_{n})|} ≤ \frac{2 |Ψ_{t+s_{n}}(v_{n})|}{sup_{s≥0} |Ψ_{s}(v_{n})|} ≤ 2 .

Since |w_{n}| = 1 and y_{n} ∈ M, there exists a convergent subsequence (y_{n}, w_{n}) → (y, w). We would have that y ∈ B, w ∈ E(y), |w| = 1 and

|Ψ_{t}(w)| ≤ 2 for all t ∈ ℝ .

This is a contradiction.

3.3. Lemma. There exists K > 0 such that for all x ∈ B, v ∈ E(x)

|Ψ_{t}(v)| ≤ K ( |v| + |Ψ_{s}(v)| ) for all 0 ≤ t ≤ s.

Proof:

Suppose it is false. Then there exist x_{n} ∈ B, 0 ≠ v_{n} ∈ E(x_{n}) and 0 ≤ t_{n} ≤ s_{n}, such that

|Ψ_{t_{n}}(v_{n})| ≥ n( |v_{n}| + |Ψ_{s_{n}}(v_{n})| ) .

Then t_{n} → +∞ and s_{n} − t_{n} → +∞ when n → +∞. We can assume that

|Ψ_{t_{n}}(v_{n})| = sup_{0≤t≤s_{n}} |Ψ_{t}(v_{n})|. Let

w_{n} := \frac{Ψ_{t_{n}}(v_{n})}{|Ψ_{t_{n}}(v_{n})|} ·

For −t_{n} < t < s_{n} − t_{n}, we have that

|Ψ_{t}(w_{n})| = \frac{|Ψ_{t+t_{n}}(v_{n})|}{|Ψ_{t_{n}}(v_{n})|} ≤ \frac{1}{|Ψ_{t_{n}}(v_{n})|} sup_{0≤t≤s_{n}} |Ψ_{t}(v_{n})| = 1.

Taking a subsequence, if necessary, we can assume that x_{n} → x and w_{n} → w ∈ E(x). Then |Ψ_{t}(w)| ≤ 1 for all t ∈ ℝ, with |w| = 1. This is a contradiction.

Proof of theorem 0.2:

Standard methods (cf. Hirsch-Pugh-Shub [21]) show that the continuity of the strong stable and unstable bundles is redundant in the definition of a hyperbolic action. So it remains to prove that for all x ∈ B, we have that E^{s}(x)⊕E^{u}(x) = E(x). Given x ∈ B, since x ∈ Ω(ψ|_{B}), there exist x_{n} → x, s_{n} → +∞ such that ψ_{s_{n}}x_{n} → x. Let E_{n} be a subspace of E(x_{n}) such that

dim E_{n} = dim E(x) − dim E^{s}(x) ,

E^{s}(x) ⊕ lim_{n} E_{n} = E(x) .

We claim that there exists C > 0 such that

∥Ψ_{−t}| Ψ_{s_{n}}(E_{n})∥ ≤ C for all 0 ≤ t ≤ s_{n}.

Suppose the claim is true. Taking a subsequence if necessary, we can assume that the limit lim_{n} Ψ_{s_{n}}(E_{n}) exists. Then

∥Ψ_{−t}| lim_{n} Ψ_{s_{n}}(E_{n})∥ ≤ C for all t > 0.

Then lim_{n} Ψ_{s_{n}}(E_{n}) ⊂ E^{u}(x) and dim E^{u}(x) + dim E^{s}(x) ≥ dim E_{n} + dim E^{s}(x) = dim E(x). Since E^{s}(x) ∩ E^{u}(x) = {0}, this completes the proof.

Now we prove the claim. Suppose it is false, then there exist v_{n} ∈ Ψ_{s_{n}}(E_{n}), |v_{n}| = 1 and 0 < t_{n} < s_{n} such that |Ψ_{−t_{n}}(v_{n})| ≥ n. By lemma 3.3, we have that

n ≤ |Ψ_{−t_{n}}(v_{n})| ≤ K (|Ψ_{−s_{n}}(v_{n})| + |v_{n}|) .

Hence |Ψ_{−s_{n}}(v_{n})| ≥ \frac{n−K}{K}. From lemma 3.3 we also have that

\frac{|Ψ_{t}(Ψ_{−s_{n}}(v_{n}))|}{|Ψ_{−s_{n}}(v_{n})|} ≤ K + \frac{K}{|Ψ_{−s_{n}}(v_{n})|} for 0 < t < s_{n} .

Let w_{n} := \frac{Ψ_{−s_{n}}(v_{n})}{|Ψ_{−s_{n}}(v_{n})|}. The estimates above give that

|Ψ_{t}(w_{n})| ≤ K + \frac{K^{2}}{n − K} for 0 < t < s_{n} .

If w_{n} → w, then |w| = 1 and w ∈ lim E_{n}, thus w ∉ E^{s}(x). But |Ψ_{t}(w)| ≤ K for all t ≥ 0. This is a contradiction.

Proof of theorem C.

For θ ∈ Σ, let

N(θ) := { ξ ∈ T_{θ}Σ | ⟨ dπ · ξ , dπ X(θ) ⟩_{π(θ)} = 0 },

and let Λ(θ) : T_{θ}Σ → N(θ) be the projection along the direction of X(θ):

Λ(θ) · ξ = ξ + β(ξ) X(θ) such that Λ(θ) · ξ ∈ N(θ) .

Let Ψ_{s}(θ) = Λ(ψ_{s}(θ)) ∘ dψ_{s}|_{N(θ)}. By corollary 2.3 (e), Ψ_{s} defines an ℝ-action on the vector bundle π : N → Σ.

Suppose that the Green bundles satisfy 𝔼∩𝔽 = ⟨X⟩ on Σ. Then by corollary 2.3(b) and 2.3(d),

B^{⊤}(θ) ⊆ 𝔼^{⊤}(θ) ∩ 𝔽^{⊤}(θ) = 𝔼(θ) ∩ 𝔽(θ) ∩ N(θ) = {0} .

Therefore the action Ψ_{s} is quasi-hyperbolic. Since ψ_{s} preserves the Liouville measure, which is positive on open sets, then Ω(ψ|_{Σ}) = Σ. Hence Ψ_{s} is a hyperbolic action.

This implies that ϕ_{t} is Anosov. By the methods in Hirsch-Pugh-Shub [21], we get that ψ_{t} is Anosov. We sketch the proof below.

For θ ∈ Σ let E^{u}(θ) ⊆ N(θ) be the subspace given by item (3.b) in the definition of hyperbolic action. Then E^{u} is a continuous subbundle of T Σ. Define

F := { L : E^{u} → ℝ continuous | L(θ) : E^{u}(θ) → ℝ is linear ∀θ ∈ Σ },

with ∥L∥ = sup_{ξ∈E^{u}\\{0\}} \frac{|L(ξ)|}{|ξ|}.

36 G. CONTRERAS AND R. ITURRIAGA

To each functional in F associate a subbundle W_{L} of T Σ by

W_{L}(θ) := graph(L(θ)) = { ξ + (L(θ) · ξ) X(θ) | ξ ∈ E^{u}(θ) }.

Let τ > 0 be such that C e^{−λτ} < e^{−λ} < 1, where C and λ are from (3.b). Consider the following “graph transformation” T : F → F, corresponding to W_{T L} = dψ_{τ}(W_{L}) and defined by

dψ_{τ}(ξ + L(ξ) X(θ)) = Ψ_{τ}(ξ) + [T (L) (Ψ_{τ}(ξ))] X(ψ_{τ}(θ)), θ ∈ Σ, ξ ∈ E^{u}(θ).

We claim that T is a contraction. Indeed

|\frac{T (L_{1} − L_{2}) (Ψ_{τ}ξ)}{∥Ψ_{τ} ξ∥}| ∥X(ψ_{τ}θ)∥ ≤ \frac{∥ dψ_{τ}[(L_{1}(ξ) − L_{2}(ξ)) X(θ)]∥}{e^{λ} ∥ξ ∥}

≤ \frac{|(L_{1} − L_{2})(ξ)|}{e^{λ} ∥ξ∥} ∥X(ψ_{τ}θ)∥ .

Hence ∥T (L_{1}) − T (L_{2})∥ ≤ e^{−λ} ∥L_{1} − L_{2}∥.

The fixed point L^{∗} of T gives the (continuous) strong unstable bundle of ψ_{t}. Indeed, if E^{u} = W_{L^{∗}}, then E^{u} is clearly dψ_{t}-invariant. Moreover, if ζ ∈ E^{u}(θ), then ζ = ξ + (L^{∗}ξ) X(θ) with ξ = Λζ ∈ E^{u}(θ). Since L^{∗} is continuous then there exists Q_{1} > 0 such that |ζ| ≤ Q |ξ| = Q |Λζ| for all ζ ∈ E^{u}. Since N(θ) is transversal to ⟨X(θ)⟩ and both are continuous subbundles of T Σ, (Σ compact), then the angle ∢(X(θ), N(θ)) is bounded below and hence there exists Q_{2} > 0 such that \frac{1}{Q_{2}} |Λζ| = \frac{1}{Q_{2}} |ξ| < |ζ|. Then

|dψ_{t}(ζ)| = | Ψ_{t}(ξ) + (T (L^{∗}) · ξ) X(ψ_{t}θ) |

≥ \frac{1}{Q_{2}} |Ψ_{t}(ξ)| ≥ \frac{C^{−1} e^{λ t}}{Q_{2}} |ξ| ≥ \frac{e^{λt}}{C Q_{1} Q_{2}} |ζ| .

Finally,

dim E^{u} = dim graph(L^{∗}) = dim E^{u} = n − 1 .

The existence of a continuous strong stable subbundle of dimension n − 1 is proven similarly.

4. The Index Form.

Let L = p H_{p} − H be the Lagrangian associated to H. Consider \frac{∂L}{∂v}(x, v) : T_{(x,v)}(T_{x}M) ≈ T_{x}M → ℝ as an element L_{v} ∈ T^{∗}M. The Legendre transform F_{L}(x, v) = (x, L_{v}) identifies q = x, p = L_{v}. Similarly, using F_{L}^{−1} = F_{H}, we get that v = H_{p} under the same identification. Also

H = v L_{v} − L = p H_{p} − L(q, H_{p}) .

HAMILTONIANS WITHOUT CONJUGATE POINTS 37

Hence H_{q} = −L_{q} = −L_{x}. With these identifications the Hamiltonian equations (1) become

\frac{d}{dt} x = v  ⟷  \frac{d}{dt} q = H_{p} ,
(46)  \frac{d}{dt} L_{v} = L_{x}  ⟷  \frac{d}{dt} p = −H_{q} ,

Hence, identifying L_{x} = −H_{q} ∈ T_{(p,q)}(T^{∗}M), the Euler-Lagrange equation (46) is understood as a first order differential equation on T^{∗}M, where \frac{d}{dt}L_{v} is the tangent vector to the path t ↦ L_{v}(γ(t), γ̇(t)) in T^{∗}M.

We derive the Jacobi equation in this Lagrangian setting. Let γ(t) be a solution of the Euler-Lagrange equation (2). Considering a variation f(s, t) of γ(t) = f(0, t) made of solutions t ↦ f(s, t) of the Euler-Lagrange equation (2) and taking the covariant derivative \frac{D}{ds}, we obtain the Jacobi equation

\frac{D}{dt} (L_{vx} k + L_{vv} k̇) = L_{xx} k + L_{xv} k̇ , (47)

where k = \frac{∂f}{∂s} (0, t), k̇ = \frac{D}{dt} \frac{∂f}{∂s} and the derivatives of L are evaluated on γ(t) = f(0, t). Here we have used that \frac{D}{ds} \frac{∂F}{∂t} = \frac{D}{dt} \frac{∂F}{∂s} for the variation map F (s, t) = L_{v}(f(s, t), \frac{∂f}{∂t} (s, t)) ∈ T^{∗}M, where \frac{D}{ds} and \frac{D}{dt} are the covariant derivatives on the riemannian manifold T^{∗}M. The linear operators L_{xx}, L_{xv}, L_{vv} coincide with the corresponding matrices of partial derivatives in local coordinates. The solutions of (47) satisfy

Dφ_{t}(k(0), k̇(0)) = (k(t), k̇(t)) ∈ T_{γ(t)}(T M) ,

where φ_{t} is the Lagrangian flow on T M. A solution of (47) is called a Jacobi field.

Let Ω_{T} be the set of continuous piecewise C^{2} vectorfields ξ along γ_{[0,T]}. Define the index form on Ω_{T} by

I(ξ, η) = ∫_{0}^{T} (ξ̇ L_{vv} η̇ + ξ̇ L_{vx} η + ξ L_{xv} η̇ + ξ L_{xx} η) dt , (48)

which is the second variation of the action functional for variations f(s, t) with \frac{∂f}{∂s} ∈ Ω_{T}. For general results on this form see Duistermaat [12].

The following transformation of the index form is taken from Hartman [20] and originally due to Clebsch [7]. Let θ ∈ T^{∗}M and suppose that the orbit of θ, ψ_{t}(θ), 0 ≤ t ≤ T does not have conjugate points. Let E ⊂ T_{θ}T^{∗}M be a Lagrangian subspace such that dψ_{t}(E) ∩ V (ψ_{t}(θ)) = {0} for all 0 ≤ t ≤ T. Such subspace E always exists by proposition 1.15(b). Let E(t) := dψ_{t}(E) and let H(t), V (t) be a matrix solution of the Hamiltonian Jacobi equation (6) such that det H(t) ≠ 0 and E(t) = Image (H(t), V (t)) ⊂ T_{ψ_{t}(θ)}(T^{∗}M) is a Lagrangian subspace. In particular

38 G. CONTRERAS AND R. ITURRIAGA

H(t) satisfies the Lagrangian Jacobi equation (47). We have that

[\begin{matrix}H(t) \\ V (t) \end{matrix}] = DF_{L} (φ_{t}(θ)) · [\begin{matrix}H(t) \\ H′(t) \end{matrix}] = [\begin{matrix}I & 0 \\ L_{vx} & L_{vv} \end{matrix}] [\begin{matrix}H \\ H′ \end{matrix}] (49)

From (49) and (47), we have that

\begin{aligned}V & = L_{vv} H′ + L_{vx} H \\ V′ & = L_{xv} H′ + L_{xx} H \end{aligned} (50)

Moreover, since E(t) is a Lagrangian subspace, then the Hamiltonian solution of the Ricatti equation (7) S(t) = V (t) H(t)^{−1} is symmetric, i.e. H(t)^{∗} V (t) = V (t)^{∗} H(t). (51)

Let η ∈ Ω_{T} and define ζ ∈ Ω_{T} by η(t) = H(t) ζ(t). Then the integrand of I(η, η) in (48) is

(L_{vv} H ζ′ + V ζ) · (H′ ζ + H ζ′) + (V′ ζ + L_{xv} H ζ′) · Hζ .

Since (L_{vv} Hζ′) · (H′ζ) = (Hζ′) · (L_{vv} H′ζ) and (L_{xv} Hζ′) · (Hζ) = (Hζ′) · (L_{vx} Hζ), the integrand can be written as

L_{vv} Hζ′ · Hζ′ + Hζ′ · V ζ + V ζ · H′ζ + V ζ · Hζ′ + V′ζ · Hζ .

Using (51), we have that V^{∗}Hζ′ · ζ = H^{∗}V ζ′ · ζ = V ζ′ · Hζ. hence the integrand in (48) is L_{vv} Hζ′ · Hζ′ + (V ζ · Hζ)′. Since η and ζ are continuous, we have that I(η, η) = ∫_{0}^{T} (L_{vv} Hζ′ · Hζ′) dt + Hζ · V ζ|_{0}^{T} if η = Hζ ∈ Ω_{T} . (52)

Let F_{H}(q, p) = (q, H_{p}) be the Legendre transform of the Hamiltonian. Then F_{L} and F_{H} are inverse maps, in particular

(DF_{H})^{−1} = [\begin{matrix}I & 0 \\ H_{pq} & H_{pp} \end{matrix}]^{−1} = [\begin{matrix}I & 0 \\ L_{vx} & L_{vv} \end{matrix}] = DF_{L} ,

and hence L_{vv} = (H_{pp})^{−1}. We obtain that I(ξ, η) = ∫_{0}^{T} (Hζ′)^{∗} (H_{pp})^{−1} (Hρ′) dt + (Hζ)^{∗}(V ρ)|_{0}^{T} . (53) for ξ = Hζ ∈ Ω_{T}, η = Hρ ∈ Ω_{T}. This formula can also be written as I(ξ, η) = ∫_{0}^{T} (Hζ′)^{∗} (H_{pp})^{−1} (Hρ′) dt + ξ^{∗} S η|_{0}^{T}, (54) where S(t) = V (t) H(t)^{−1} is the corresponding solution to the Ricatti equation (7).

HAMILTONIANS WITHOUT CONJUGATE POINTS 39

4.1. Corollary. If θ ∈ T^{∗}M and the segment { ψ_{t}(θ) | t ∈ [0, T] } has no conjugate points then the index form is positive definite on

Γ_{T} = { ξ : [0, T] → T M | ξ(t) ∈ T_{πψ_{t}θ}M, ξ is piecewise C^{2}, ξ(0) = 0, ξ(T) = 0 } .

Moreover, I_{T} (η, η) = 0 if and only if η is a Jacobi field on Γ_{T}.

Proof:

Let ξ ∈ Γ_{T}, ξ ≠ 0. Write ξ(t) = H(t)ζ(t). Since det H(t) ≠ 0, ζ(0) = 0, ζ(T) = 0 and ζ(t)≢ 0, then ζ′≢ 0. Now use formula (53).

We now extend formula (53) to the case in which H(t) may be singular (or ζ′(t), η′(t) may be discontinuous) at a finite set of points. Let {t_{1}, …, t_{N}} be the points in [0, T] such that det H(t_{i}) = 0. Then

I(ξ, η) = ∫_{0}^{T} (Hζ′)^{∗} (H_{pp})^{−1} (Hρ′) dt + (Hζ)^{∗}(V ρ)|_{0}^{T} − ∑_{i=1}^{N}(Hζ)^{∗}(V ρ)|_{t_{i}^{−}}^{t_{i}^{+}} . (55)

where ξ = Hζ, η = Hρ. But now ζ, ρ are piecewise C^{2} but may be discontinuous at t_{1}, …, t_{N}. Alternatively, we can use a sum of formulas (53) or (55) using different Lagrangian subspaces dψ_{t}(E_{i}) and corresponding solutions (H_{i}, V_{i}) on disjoint subintervals ]t_{i}, t_{i+1}[⊂ [0, T].

Observe that for any convex Hamiltonian on T^{∗}M and any θ ∈ M there exists ε > 0 such that the segment {ψ_{t}(θ) | |t| < ε} has no conjugate points. Indeed, let Y_{θ}(t) v = dπ dψ_{t}(θ) (0, v), where (0, v) ∈ V (θ). Then from the Jacobi equation (6) we have that

\frac{D}{dt} Y_{θ}(t)|_{t=0} = H_{pp}(θ).

Since Y_{θ}(0) = 0 and H_{pp}(θ) is nonsingular, then Y_{θ}(t) is an isomorphism for |t| < ε, some ε > 0. In particular dψ_{t}(θ) V (θ) ∩ V (θ) = {0} for |t| < ε.

We say that a curve γ(t) ∈ M, t ∈ [0, T] is minimizing if it minimizes the action functional

∫_{0}^{T} L(δ(t), δ′(t)) dt

over all absolutely continuous curves δ(t) ∈ M, 0 ≤ t ≤ T, such that δ(0) = γ(0) and δ(T) = γ(T).

40 G. CONTRERAS AND R. ITURRIAGA

4.2. Corollary. If θ ∈ T^{∗}M and the segment { πψ_{t}(θ) | t ∈ [0, S[ } is minimizing, then it has no conjugate points.

Proof:

Suppose it is false. Let ψ_{T} (θ), T < S be the first conjugate point in { πψ_{t}(θ) | t ∈ [0, S[ }. Then there exists ξ = (0, v) ∈ V (θ) such that dψ_{T} (θ) ξ ∈ V (ψ_{T} (θ)). Let η(t) = dπ dψ_{t}(θ) ξ. Using the limit of (53) on the interval [0, T [ we have that I(η, η)|_{0}^{T} = 0.

Let ε > 0 be such that (cf. prop. 1.15) the segment {ψ_{t}(θ) | T −ε ≤ t ≤ T +ε} has no conjugate points. Let ζ(t) = Y_{ψ_{T +ε}(θ)}(t − T − ε) w = dπ dψ_{t−T −ε}(ψ_{T +ε}(θ)) (0, w), T − ε ≤ t ≤ T + ε, where w is such that ζ(T − ε) = η(T − ε). Let η̂ = η(t) if T − ε ≤ t ≤ T and η̂(t) = 0 if T ≤ t ≤ T + ε. Then using H(t) = Y_{ψ_{T +ε}(θ)}(t − T − ε) on (53), we have that

I(η̂, η̂)|_{T −ε}^{T +ε} > I(ζ, ζ)|_{T −ε}^{T +ε}.

Extend ζ and η̂ by ζ(t) = η̂(t) = η(t) for 0 ≤ t ≤ T − ε. Then

I(ζ, ζ)|_{0}^{T +ε} < I(η̂, η̂)|_{0}^{T +ε} = 0.

Now consider the variation f(s, t) = exp_{πψ_{t}(θ)} s ζ(t). We have that f(s, 0) = π(θ), f(s, T + ε) = π(ψ_{T +ε}(θ)), \frac{∂f}{∂s} (0, t) = ζ(t) and if

A(s) = ∫_{0}^{T +ε} L (f(s, t), \frac{∂f}{∂t} (s, t)) dt

is the action of f(s, t), then A′(0) = 0 because f(0, t) = π ψ_{t}(θ) satisfies the Euler-Lagrange equation and

A′′(0) = I(ζ, ζ) < 0 .

Therefore the segment { π ψ(θ) | 0 ≤ t ≤ T + ε } is not minimizing.

4.3. Corollary. If Γ_{T} is as in corollary 4.1, then

index I_{T} |_{Γ_{T}} = ∑_{0<t≤T} dim [ dψ_{t}(V (θ)) ∩ V (ψ_{t}(θ)) ] ,

where the sum is over the discrete set of points conjugate to θ.

A proof of this corollary can be obtained from the following remarks:

1. By corollary 4.1, if 0 < t ≤ T then I_{t}(η, η) = 0 only if η is a Jacobi field on Γ_{t}. Hence the index of I_{t} can only change with t when ψ_{t}(θ) is conjugate to θ.

2. If I_{t}(ζ, ζ) < 0 and 0 < t ≤ T, then the extension ζ̂(s) = ζ(s) on 0 ≤ s ≤ t and ζ̂(s) = 0 on 0 ≤ s ≤ T satisfies I_{T} (ζ̂, ζ̂) = I_{t}(ζ, ζ) < 0. Hence the function t ↦ index(I_{t}|_{Γ_{t}}) is non-decreasing.

HAMILTONIANS WITHOUT CONJUGATE POINTS 41

3. The proof of corollary 4.2 constructs a linearly independent vectorfield ζ ∈ T_{T} with I_{T} (ζ, ζ) < 0 for each linearly independent vector ξ ∈ V (θ)∩dψ_{−t}(V (ψ_{t}(θ)), 0 < t ≤ T.

5. Proof of theorem D.

We begin by quoting the following theorem by R. Mañé [26]. Given a (periodic) Lagrangian L : T M × S^{1} → ℝ and ω ∈ H^{1}(M, ℝ), let M^{ω}(L) be the set of minimizing measures of L + \overline{ω}, where \overline{ω} is any 1-form in the class ω.

5.1. Theorem. (Mañé [26])

(a) For every ω ∈ H^{1}(M, ℝ) there exists a residual subset O(ω) ⊂ C^{∞}(M × S^{1}) such that ψ ∈ O(ω) implies #M^{ω}(L + ψ) = 1.

(b) There exist residual subsets O ⊂ C^{∞}(M × S^{1}) and H ⊂ H^{1}(M, ℝ) such that ψ ∈ O and ω ∈ H imply #M^{ω}(L + ψ) = 1.

Now take an autonomous Lagrangian L and ω = 0 ∈ H^{1}(M, ℝ). Then the item (a) implies the first part of theorem D. Let O be the residual subset given by this theorem. Let A be the subset of O of potentials ψ for which the measure on M(L + ψ) is supported on a periodic orbit. Let B := O \ A and let A_{1} be the subset of A on which the minimizing periodic orbit is hyperbolic. We prove that A_{1} is relatively open on A. For, let ψ ∈ A_{1} and

M(L + ψ) = {μ_{γ}}

where μ_{γ} is the invariant probability measure supported on the hyperbolic periodic orbit γ for the flow of L + ψ. We claim that if ϕ_{k} ∈ A, ϕ_{k} → ψ and M(L + ϕ_{k}) = {μ_{η_{k}}}, then η_{k} → γ. Indeed, since L is superlinear, the velocities in the support of the minimizing measures μ_{k} := μ_{η_{k}} are bounded (cf. [29], [25]), and hence there exists a subsequence μ_{k} → ν converging weakly* to a some invariant measure ν for L + ψ. Then if ν ≠ μ_{γ},

lim_{k} S_{L+ϕ_{k}}(μ_{k}) = S_{L+ψ}(ν) > S_{L+ψ}(μ_{γ}) . (56)

Thus if δ_{k} is the analytic continuation of the hyperbolic periodic orbit γ to the flow of L + ϕ_{k} in the original energy level c(L + ψ), since lim_{k} S_{L+ψ_{k}}(μ_{δ_{k}}) = S_{L+ψ}(μ_{γ}), for k large we have that,

S_{L+ϕ_{k}}(μ_{δ_{k}}) < S_{L+ϕ_{k}}(μ_{η_{k}}) ,

which contradicts the choice of η_{k}. Therefore ν = μ_{γ}. For energy levels h near to c(L+ψ) and potentials ϕ near to ψ, there exist hyperbolic periodic orbits γ_{ϕ,h} which are the continuation of γ. Now, on a small neighbourhood of a hyperbolic orbit there exists a unique invariant measure supported on it, and it is in fact supported in the periodic orbit. Thus, since η_{k} → γ, then η_{k} is hyperbolic. Hence ϕ_{k} ∈ A_{1} and A_{1} contains a neighbourhood of ϕ in A.

Let U be an open subset of C^{∞}(M, ℝ) such that A_{1} = U ∩ A. We shall prove below that A_{1} is dense in A. This implies that A_{1} ∪ B is generic. For, let V := int (C^{∞}(M, ℝ)\U), then U ∪V is open and dense in C^{∞}(M, ℝ). Moreover, V ∩A = ø because A ⊆ \overline{A_{1}} ⊆ \overline{U} and V ∩ A ⊆ A \ \overline{U} = ø. Since O = A ∪ B is generic and

(U ∪ V) ∩ (A ∪ B) = (U ∩ A) ∪ ((U ∪ V) ∩ B)
⊆ A_{1} ∪ B ,

then A_{1} ∪ B is generic.

Note: The perturbation to achieve hyperbolicity in the case of a periodic orbit and a singularity follow the same spirit. However to prove it in the case of a singular point is much easier because the Jacobi equation (the linear part of the flow) is autonomous. We suggest to read it first.

We have to prove that A_{1} is dense in A, i.e. given that a Lagrangian L has a unique minimizing measure supported on a periodic orbit γ, then there exists a perturbation by an arbitrarily small C^{∞}-potential ϕ, such that the new Lagrangian L + ϕ has a unique minimizing measure supported on a hyperbolic periodic orbit.

Fix ρ ∈ A and let Γ be the periodic orbit in M(L+ρ), we can assume that ρ = 0. By the graph property (cf. [29], [25]), the projection π|_{Γ} : Γ → M, π(x, v) = x is injective. In particular π(Γ) ⊆ M is a simple closed curve. Choose coordinates \overline{x} = (x_{1}, …, x_{n}) : U → S^{1} × ℝ^{n−1} on a tubular neighbourhood of π(Γ) such that

\overline{x}(Γ) = S^{1} × \{0\} and \{ \frac{∂}{∂x_{1}} , \frac{∂}{∂x_{2}} , …, \frac{∂}{∂x_{n}} \} is an orthonormal frame over the points

of π(Γ). In particular \frac{∂}{∂x_{1}} is parallel to π(Γ). Define L̃ = L+ϕ with ϕ(x_{1}, …, x_{n}) = \frac{1}{2} f(\overline{x}) ε (x_{2}^{2} + x_{3}^{2} + ⋯ + x_{n}^{2}), where f(\overline{x}) is a C^{∞} nonnegative bump function with support in U which is 1 on a small neighbourhood of π(Γ). Clearly ϕ can be made C^{∞} arbitrarily small. On π(Γ) we have that L̃_{vv} = L_{vv}, L̃_{vx} = L_{vx} and L̃_{xx} = L_{xx} + ε [ \begin{matrix}0 & 0 \\ 0 & I \end{matrix}]. Observe that since L̃ ≥ L then M(L + ϕ) = \{μ_{Γ}\}.

Let H and H̃ = H − ϕ be the Hamiltonians associated to L and L̃ respectively. Let Γ be the corresponding periodic orbit for H and H̃ on T^{∗}M. We have to prove that the periodic orbit Γ is hyperbolic for the flow of H̃. Since it is minimizing, by corollary 4.2 the orbit Γ has no conjugate points. By proposition B it is enough to prove that the Green bundles 𝔼̃, 𝔽̃ satisfy 𝔼̃(θ) ∩ 𝔽̃(θ) = ⟨X̃(θ)⟩ on a point θ ∈ Γ, where X̃ is the Hamiltonian vectorfield of H̃.

Fix θ ∈ Γ and let Z̃_{T} (t) = dπ∘dψ̃_{t}∘[dπ|_{dψ̃_{−T} (V (ψ̃_{T} (θ)))}]^{−1}. Let (Z̃_{T} , Ṽ_{T}) and K̃_{T} (t) =

Ṽ_{T} (t) (Z̃_{T} (t))^{−1}, 0 ≤ t < T, be the corresponding solutions of the Jacobi equation

HAMILTONIANS WITHOUT CONJUGATE POINTS 43

(6) and the Ricatti equation (7) respectively: graph(K̃_{T} (t)) = dψ̃_{t}(dψ̃_{−T} (V (ψ̃_{T} θ))), Z̃_{T} (T) = 0, Z̃_{T} (0) = I.

Define N(θ) := { w ∈ T_{π(θ)}M | ⟨w, γ̇⟩ = 0 }. Then N(θ) is the subspace of T_{π(θ)}M generated by the vectors \frac{∂}{∂x_{2}} ,…, \frac{∂}{∂x_{n}}. Let v_{o} ∈ N(θ), |v_{o}| = 1 and let ξ^{T} (t) := Z̃_{T} (t) v_{o}. Denote by Ĩ_{T} and I_{T} be the index forms on [0, T] for L̃ and L respectively. Using the solution (Z̃_{T} , Ṽ_{T}) on formula (53), we obtain that Ĩ_{T} (ξ^{T} , ξ^{T}) = −(Z̃_{T} (0) v_{o})^{∗} (Ṽ_{T} (0) v_{o}) = −v_{o}^{∗} K̃_{T} (0) v_{o} . (57)

Moreover, in the coordinates (x_{1}, …, x_{n}; \frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{n}}) on T U we have that

Ĩ_{T} (ξ^{T} , ξ^{T}) = ∫_{0}^{T} (ξ̇^{T} L̃_{vv} ξ̇^{T} + 2 ξ̇^{T} L̃_{xv} ξ^{T} + ξ^{T} L̃_{xx} ξ^{T} ) dt

= ∫_{0}^{T} (ξ̇^{T} L_{vv} ξ̇^{T} + 2 ξ̇^{T} L_{xv} ξ^{T} + ξ^{T} L_{xx} ξ^{T} ) dt + ∫_{0}^{T} ε ∑_{i=2}^{n} |ξ_{i}^{T} |^{2} dt . (58)

We have that Z̃_{T} (0) = I and for all t > 0, lim_{t→∞} Z̃_{T} (t) = h̃(t) with h̃(t) given on claim 1.3 for H̃. Writing π_{N}(ξ) = (ξ_{2}, ξ_{3}, … ξ_{n}) then |π_{N} h̃(0) v_{0}| = |v_{0}| = 1 because v_{0} ∈ N(θ). Hence there exists λ > 0 and T_{0} > 0 such that |π_{N}ξ^{T} (t)| = |π_{N} Z̃_{T} (t) v_{0}| > \frac{1}{2} for all 0 ≤ t ≤ λ and T > T_{0}. Therefore

Ĩ_{T} (ξ^{T} , ξ^{T}) ≥ I_{T} (ξ^{T} , ξ^{T}) + \frac{ελ}{4} . (59)

Let (h(t), v(t)) = dψ_{t} ∘ (dπ|_{𝔼(θ)})^{−1} be the solution of the Jacobi equation for H given by claim 1.3 and let 𝕊(ψ_{t}(θ)) = v(t) h(t)^{−1} be the corresponding solution of the Ricatti equation, with graph[𝕊(ψ_{t}(θ)] = 𝔼(ψ_{t}(θ)). Using formula (53), and writing ξ^{T} (t) = h(t) ζ(t), we have that

I_{T} (ξ^{T} , ξ^{T}) = ∫_{0}^{T} (h ζ̇)^{∗} H_{pp}^{−1} (h ζ̇) dt + 0 − (h(0) ζ(0))^{∗}(v(0) ζ(0))
(60)  I_{T} (ξ^{T} , ξ^{T}) ≥ −v_{o}^{∗} 𝕊(θ) v_{o} .

From (57), (59) and (60), we get that

v_{o}^{∗} 𝕊(θ) v_{o} ≥ v_{o}^{∗} K̃_{T} v_{o} + \frac{ελ}{4} .

From proposition 1.4, we have that lim_{T →+∞} K̃_{T} (0) = 𝕊̃(θ), where graph(𝕊̃(θ)) = 𝔼̃(θ), the stable Green bundle for H̃. Therefore

v_{o}^{∗} 𝕊(θ) v_{o} ≥ v_{0}^{∗} 𝕊̃(θ) v_{o} + \frac{ελ}{4} . (61)

44 G. CONTRERAS AND R. ITURRIAGA

Similarly, for the unstable Green bundles we obtain that

v_{o}^{∗} 𝕌(θ) v_{o} + λ_{2} ≤ v_{o}^{∗} 𝕌̃(θ) v_{o} for v_{o} ∈ N(θ), |v_{o}| = 1. (62)

for some λ_{2} > 0 independent of v_{o}.

From proposition 1.4 we have that 𝕌(θ) ≽ 𝕊(θ). From (61) and (62) we get that 𝕌̃|_{N} ≻ 𝕌|_{N} ≽ 𝕊|_{N} ≻ 𝕊̃|_{N}. Since 𝔼̃(θ) = graph(𝕊̃(θ)) and 𝔽̃(θ) = graph(𝕌̃(θ)), we get that 𝔼̃(θ) ∩ 𝔽̃(θ) ⊆ ⟨X̃(θ)⟩. Then proposition B shows that Γ is a hyperbolic periodic orbit for L + ϕ.

This proves that A_{1} is dense in A. Let A_{2} be the subset of A_{1} of potentials ψ for which the minimizing hyperbolic periodic orbit Γ has transversal intersections W^{s}(Γ) ⋔ W^{u}(Γ). Then A_{2} is relatively open on A_{1}. Indeed, if ϕ_{n} ∈ A_{1} and ϕ_{n} → ψ ∈ A_{2} and M(L + ϕ_{n}) = {μ_{γ_{n}}}, M(L + ψ) = {μ_{Γ}}, we have seen that the γ_{n}’s are continuations of the hyperbolic orbit Γ for nearby flows (on nearby energy levels): γ_{n} = Γ_{ϕ_{n},h_{n}}. Since compact subsets (fundamental domains) of the invariant manifolds W^{s}(Γ_{ϕ,h}) depend continuously in the C^{1} topology on (ϕ, h), then the transversality property W^{s}(Γ_{ϕ,h}) ⋔ W^{u}(Γ_{ϕ,h}) persists on a neighbourhood of (ψ, h_{0}). In particular it holds for γ_{n} = Γ_{ϕ_{n},h_{n}} for (ϕ_{n}, h_{n}) sufficiently close to (ψ, h_{0}).

By the same arguments as above, it is enough to prove that A_{2} is dense in A. For it we need the following lemma,

5.2. Lemma. Let Γ be a hyperbolic periodic orbit without conjugate points of a convex Hamiltonian. Then for all θ ∈ W^{s}(Γ) there exists S = S(θ) > 0 such that the segment { ψ_{t}(θ) | t ≥ S } has no conjugate points.

Proof:

Let 𝔼 be the stable Green subspace of Γ. Then 𝔼(ϑ) ∩ V (ϑ) = {0} for all ϑ ∈ Γ. Since the weak stable manifold W^{s}(Γ) is tangent to 𝔼; then for all θ ∈ W^{s}(Γ) there exists S = S(θ) > 0 such that E(t) := T_{ψ_{t}(θ)}W^{s}(Γ) satisfies E(t) ∩ V (ψ_{t}(θ)) = {0}. Moreover, dim E(t) = n and for all u ∈ E(t), lim_{s→+∞} dψ_{s}(u) ∈ ⟨X(θ)⟩. Since ψ_{t} preserves the symplectic form ω, we have that ω(u, v) = lim_{s→+∞} ω(dψ_{s}(u), dψ_{s}(v)) = 0 for all u, v ∈ E(t). Hence E(t) is Lagrangian and E(t) ∩ V (ψ_{t}(θ)) = {0} for all t > S. Then the lemma follows from proposition 1.15.

Now we prove that A_{2} is dense in A. Let ρ ∈ A \ A_{2} and make the perturbation L̃ = (L + ρ) + ϕ explained above so that ρ + ϕ ∈ A_{1}. We can assume that ρ + ϕ = 0 and also that ρ + ϕ ∉ A_{2}, otherwise there is nothing to prove. Write M(L) = {μ_{Γ}} and let θ ∈ W^{s}(Γ) ∩ W^{u}(Γ) be such that dim(T_{θ}W^{s}(Γ) ∩ T_{θ}W^{u}(Γ)) > 1. Observe that the α and ω limit sets of θ are Γ. In particular the orbit of θ has

HAMILTONIANS WITHOUT CONJUGATE POINTS 45

no autoaccumulation points. Let S = S(θ) > 0 be from lemma 5.2 and ϵ > 0 small. Then there exists τ > S(θ) and a neighbourhood W ⊂ M of π(ψ_{t}(θ)) such that { t ∈ ℝ | πψ_{t}(θ) ∈ W } =]τ, τ + 3ϵ[ and W ∩ π(Γ) = ø. Let U ⊂ M × ℝ be a tubular neighbourhood of \{ (πψ_{t}(θ); t) | t > S(θ) \}. Choose coordinates \overline{y} = (y_{1}, …, y_{n}; t) : U → ℝ^{n} × ℝ such that \overline{y}(πψ_{t}(θ), t) = (α(t), 0; t) ∈ ℝ × ℝ^{n−1} × ℝ, and \{\frac{∂}{∂y_{1}} , \frac{∂}{∂y_{2}} , ⋯, \frac{∂}{∂y_{n}} \} is an orthonormal frame over the points πψ_{t}(θ), t > S(θ). Let φ(y_{1}, …, y_{n}) = \frac{1}{2} f(\overline{y}) δ (y_{2}^{2}+y_{3}^{2} ⋯+y_{n}^{2}), where f(\overline{y}) is a C^{∞} bump function with support in W which is 1 on a small neighbourhood of { πψ_{t}(θ) | t ∈ [τ + ϵ, τ + 2ϵ] }. By choosing a small δ > 0 the function φ can be made C^{∞} arbitrarily small. Choose δ small enough such that the orbit Γ remains hyperbolic. Then we still have that M(L+ϕ) = {μ_{Γ}} and the orbit of θ is the same for both flows. Write L̃ = L+ρ+ϕ, L̂ = L̃ + ψ and Ĩ_{T}, Î_{T} the corresponding index forms on { ψ_{t}(θ) | t ∈ [τ, T] }. Since { ψ_{t}(θ) | t > S(θ) } has no conjugate points, the stable Green subspace 𝔼̂(ψ_{t}(θ)), t > S(θ) and the stable solution of the Ricatti equation 𝕊̂(ψ_{t}(θ)) exist. The same arguments as in equations (58)–(61) give for ξ^{T} = Ẑ_{T} (t) v_{o}, v_{o} ∈ N(ψ_{τ}(θ)), |v_{o}| = 1, Ẑ_{T} (t) = dπ ∘ dψ̂_{t−T} ∘ [dπ|_{dψ_{−T} (V (ψ̂_{T +τ} (θ)))}]^{−1}, that

Î_{T} (ξ^{T} , ξ^{T}) = Ĩ_{T} (ξ^{T} , ξ^{T}) + ∫_{τ+ϵ}^{τ+2ϵ} δ ∑_{i=2}^{n} |ξ_{i}^{T} |^{2} dt ,

v_{0}^{∗} 𝕊(ψ_{τ}(θ)) v_{o} ≥ v_{o} 𝕊̂(ψ_{τ}(θ)) v_{o} + \frac{δ λ̂}{4} ,

for some λ̂ > 0. In particular 𝔼̂(ψ_{τ}(θ)) ∩ 𝔼̃(ψ_{τ}(θ)) = ⟨X(ψ_{τ}θ)⟩ . (63)

By the same arguments as in proposition 1.14, we have that 𝔼̂(ψ_{τ}(θ)) =

= T_{ψ_{τ} (θ)}(W u ; L̃(ψ_{τ}θ)). Observe that T_{ψ_{τ} (θ)}(W u ; L̃(ψ_{τ}θ)) = T_{ψ_{τ} (θ)}(W u ; L̃(ψ_{τ}θ)) didn’t change.

Write T_{ψ_{τ} (θ)}(W u ; L̃(ψ_{τ}θ)) = 𝕍 ⊕ 𝕎, with 𝕍 ⊆ 𝔼̃(ψ_{τ}θ) and 𝕎 ∩ 𝔼̃(ψ_{τ}θ) = {0}. By

diminishing δ if necessary we can get (63) and 𝔼̂(ψ_{τ}θ) ∩ 𝕎 = {0}. Thus

T_{ψ̂_{τ} (θ)}W s ; L̂(Γ) ∩ T_{ψ̂_{τ} (θ)}W u ; L̂(Γ) = 𝔼̂(ψ_{τ}θ) ∩ T_{ψ_{τ} (θ)}W^{u}(Γ) = ⟨X̂(ψ_{τ}θ)⟩ .

Finally, by a finite number of these perturbations on a fundamental domain of W^{s}(Γ) one can remove all tangencies of W^{s}(Γ) and W^{u}(Γ) and obtain a potential in A_{2} arbitrarily C^{∞}-near ρ ∈ A.

46 G. CONTRERAS AND R. ITURRIAGA

The case of a singularity.

Suppose that the minimizing measure μ is supported on a singularity (x_{0}, 0) of the Lagrangian flow. From the Euler-Lagrange equation (2) we get that L_{x}(x_{0}, 0) = 0. Differentiating the energy function (3) we see that (x_{0}, 0) is a singularity of the energy level c(L). Moreover, the minimizing property of μ implies that x_{0} is a minimum of the function x ↦ L_{xx}(x, 0). In particular, L_{xx}(x_{0}, 0) is positive semidefinite in linear coordinates in T_{x_{0}}M.

Choose coordinates \overline{y} = \{y_{1}, …, y_{n}\} on a neighborhood of x_{0} such that \overline{y}(x_{0}) = 0 and L_{vv}(x_{0}, 0) is the identity on the basis \{ \frac{∂}{∂y_{1}} |_{x_{0}}, ⋯, \frac{∂}{∂y_{n}} |_{x_{0}} \}. Use the coordinates ( \{y_{1}, …, y_{n}\} , \{ \frac{∂}{∂y_{1}} , ⋯, \frac{∂}{∂y_{n}} \}) on a neighbourhood of x_{0}. On the orbit φ_{t}(x_{0}, 0) ≡ (x_{0}, 0) the matrices L_{xx}, L_{xv}, L_{vv} are constant with respect to the time t. The Jacobi equation (47) at (x_{0}, 0) becomes

k̈ = L_{xx}(x_{0}, 0) k ,

where L_{vv}(x_{0}, 0) = I. Let f be a nonnegative C^{∞} function on ℝ^{n} with support on |y| < ε < 1 and f ≡ 1 on a neighbourhood of 0. Let ϕ(y) = \frac{1}{2} δ f(y) (y_{1}^{2} + ⋯ + y_{n}^{2}). Adjusting δ > 0 we can make ϕ arbitrarily C^{∞} small. Let L̃ := L + ϕ. Then the atomic measure μ supported on (x_{0}, 0) is still the unique minimizing measure for L̃ and the Jacobi equation for the orbit φ̃_{t}(x_{0}, 0) ≡ (x_{0}, 0) is k̈ = (L_{xx}(x_{0}, 0) + δ I) k̇ . (64)

Since L_{xx}(x_{0}, 0) is positive semidefinite, then A := (L_{xx}(x_{0}, 0) + δ I) is positive definite. Equation (64) is linear

[\begin{matrix}k′ \\ k′′ \end{matrix}] = [\begin{matrix}0 & I \\ A & 0 \end{matrix}] [\begin{matrix}k \\ k′ \end{matrix}] (65)

with constant coefficients where k ≡ 0 is a hyperbolic singularity.

To obtain the transversality property W^{s} ⋔ W^{u} at homoclinic orbits we can use the same arguments as above if we have a replacement for lemma 5.2. The arguments of lemma 5.2 apply if we show that the stable subspace of (x_{0}, 0) is transversal to the vertical subspace. For, it is enough to prove that the eigenvectors (x, y) of the (nonsingular) matrix in (65) have coordinate x ≠ 0. These eigenvectors satisfy y = λx, Ax = λ^{2}x, (λ ∈ ℝ). Hence if x = 0, then (x, y) = (0, 0) is not an eigenvector.

6. The Exponential map.

The following lemma was proven for geodesic flows by Freire and Mañé [16] and has intrinsic interest (compare with proposition 1.10).

6.1. Lemma. Let H be a convex Hamiltonian. Choose a riemannian metric ∥ ∥ on M. If ψ_{t} has no conjugate points on Σ = H^{−1}{e}, then there exists B > 0 such that

∥Y_{θ}(t) · v∥ > B ∥v∥

for all θ ∈ Σ, |t| > 1 and v ∈ T_{θ}M. Where Y_{θ}(t) · v = dπ ∘ dψ_{t}(θ) · (0, v), (0, v) ∈ H(θ) ⊕ V (θ).

We sketch the proof of theorem F. In the case in which Θ(X) > 0 on Σ one can define the vectorfield Y = \frac{1}{|Θ(X)|} X. This vectorfield preserves the 1-form Θ on T Σ. Since Σ has no conjugate points, then π(Σ) = M, (cf. corollary 1.13) and we can define a Hamiltonian on T^{∗}M by ℍ(t θ) = t^{2}, where θ ∈ Σ. This new Hamiltonian ℍ is convex and has no conjugate points in Σ. The Hamiltonian vectorfield 𝕐 of ℍ is called the (convex) symplectification of the contact vectorfield Y. The transversal bundle N(θ) = ker Θ|_{T_{θ}Σ} is invariant under the flow of 𝕐. Let E_{q} : T_{q}M̃ → M̃ be the exponential map for (𝕐, Σ). The derivative dE_{q}(tθ) is given in formulas (39) and (40). The norm of the directional derivative in formula (40) is clearly bounded below. By lemma 6.1 the norm of the derivative dE_{q}|_{V (θ)∩T_{θ}Σ} in formula (39) is uniformly bounded away from zero. Since V (θ) ∩ T_{θ}Σ ⊂ N(θ), and the angle ∢(N(θ), X(θ)) is bounded below, we obtain that ∥dE_{q}∥ is bounded away from zero. This implies that E_{q} : T_{q}M̃ → M̃ is a covering map and hence a diffeomorphism. In particular, the universal cover M̃ of M is homeomorphic to ℝ^{n}. By corollary 1.18, the exponential map exp_{q} for X is a local diffeomorphism. Once we know that E_{q} is a diffeomorphism, it is easy to show that exp_{q} is bijective and hence a diffeomorphism.

Proof of Lemma 6.1:

Since Y_{θ}(t) satisfies (8) (with H_{qp} and H_{pp} uniformly bounded on Σ) and by corollary 1.8, S_{θ}(t) is uniformly bounded for |t| > 1, then there exists D_{1} > 1 such that

∥Ẏ_{θ}(t) · w∥ ≤ D_{1} ∥Y_{θ}(t) · w∥ for all |t| > 1, w ∈ V (θ), θ ∈ Σ.

Let

A := (1 + D_{1})^{2} sup_{θ∈Σ} ∥H_{pp}(θ)^{−1}∥ .

Let H_{θ}(t) := Y_{ψ_{−2}(θ)}(t+2)∘(Y_{ψ_{−2}(θ)}(2))^{−1}, then det H_{θ}(t) ≠ 0 for t > −2, H_{θ}(0) = I and there exists a linear homomorphism V_{θ}(t) : H(θ) → V (ψ_{t}(θ)) such that (H_{θ}, V_{θ}) is a matrix solution of the Jacobi equation (6) and Image(H_{θ}(t), V_{θ}(t)) = dψ_{t+2}(V (ψ_{−2}(θ))) is a Lagrangian subspace. Let

D_{2} := max_{\begin{matrix}−2≤t≤2 \\ θ∈Σ \end{matrix}} ∥H_{θ}(t)^{∗} (H_{pp})^{−1} H_{θ}(t)∥ < +∞.

Let ρ : ℝ → [0, 1] be a C^{∞} function such that ρ(t) = 0 for |t| ≥ 1 and ρ(0) = 1. Let

C := D_{2} ∫_{−1}^{1} |ρ′(t)|^{2} dt .

Let

0 < B < (A C)^{\frac{1}{2}} . (66)

Suppose that ∥Y_{θ}(T) v_{o}∥ < B ∥v_{o}∥ for some θ ∈ Σ, v_{o} ∈ T_{π(θ)}M, ∥v_{o}∥ = 1 and |T | > 1. Assume that T > 1, the case T < −1 is proven similarly. Let J(t) ∈ H(ψ_{t}(θ)) = T_{πψ_{t}(θ)}M be defined by J(t) = 0 for −1 ≤ t ≤ 0, J(t) = Y_{θ}(t) v_{o} for 0 ≤ t ≤ T and J(t) = (T + 1 − t) τ_{t} J(T) for T ≤ t ≤ T + 1, where τ_{t} : T_{πψ_{T} (θ)}M → T_{πψ_{t}(θ)}M is the parallel transport along the path t ↦ πψ_{t}(θ). Then J(t) is continuous and piecewise C^{2}.

Let v(t) ∈ V (θ) ≈ T_{π(θ)}M be defined by J(t) = Y_{θ}(t) v(t). Then v(t) = 0 for −1 ≤ t ≤ 0 and v(t) = v_{o} for 0 ≤ t ≤ T. Let {e_{1}, …, e_{n}} be a basis of T_{π(θ)}M and write v(t) = ∑_{i=1}^{n} v_{i}(t) e_{i}, Y_{i}(t) = Y (t) e_{i}. Then the covariant derivative along πψ_{t}(θ) of J(t) = Y (t)v(t) is

\frac{D}{dt} (Y v) = \frac{D}{dt} (∑_{i=1}^{n} v_{i}(t) Y_{i}(t)) = ∑_{i} (\frac{DY_{i}}{dt} ) v_{i} + ∑_{i} (\frac{dv_{i}}{dt} ) Y_{i} ,

Y v′ = \frac{D}{dt} (Y v) − \frac{DY}{dt} v .

For T ≤ t ≤ T + 1, since J(t) = Y_{θ}(t) v(t) = (T + 1 − t) τ_{t} J(T), we have that

∥Y v′∥ ≤ ∥\frac{D}{dt} (Y v)∥ + ∥\frac{DY}{dt} v∥
≤ ∥\frac{D}{dt} J(t)∥ + D_{1} ∥J(t)∥
≤ ∥J(T)∥ + D_{1} ∥J(T)∥
≤ (1 + D_{1}) B .

Using Y_{θ}(t) on formula (55) we have that

I(J, J) = ∫_{T}^{T +1}(Y v′)^{∗} (H_{pp})^{−1} (Y v′) dt
≤ (1 + D_{1})^{2} B^{2} ∥H_{pp}^{−1}∥ = A B^{2}.

Let Z(t) ∈ H(ψ_{t}(θ)) = T_{πψ_{t}(θ)}M be defined by Z(t) = ρ(t) H_{θ}(t) v_{o}. Then Z(t) = Y_{θ}(t) ξ(t) for t ≠ 0, with ξ(t) = 0 for |t| > 1. Since dψ_{t}(θ) · V (θ) = Image (Y_{θ}(t) , S_{θ}(t) Y_{θ}(t)), we have that S_{θ}(t) Y_{θ}(t) → I when t → 0. Using Y_{θ}(t) on formula (55), we have that the only non zero term is

I(Z, J) = −Z(0^{+}) · v_{o} + Z(0^{−}) · 0 = − ∥v_{o}∥^{2} = −1.

Write Z(t) = H(t) ζ(t), ζ(t) = ρ(t) v_{o}. Using H(t) and V (t) on formula (53), we have that

I(Z, Z) = ∫_{−1}^{1}(Hζ′)^{∗} (H_{pp})^{−1} (Hζ′) dt
= ∫_{−1}^{1}(ρ′(t))^{2} (H(t) v_{o})^{∗} (H_{pp}^{−1}) (H(t) v_{o}) dt
≤ D_{2} ∫_{−1}^{1} |ρ′(t)|^{2} dt = C.

Since by (48) the index form is symmetric, for λ ∈ ℝ we have that

I(Z − λ J , Z − λ J) ≤ C + 2 λ + λ^{2} A B^{2}.

Since by (66) 4 − 4 A B^{2} C > 0, this polynomial is λ has two real roots. Therefore I(Z − λJ , Z − λJ) < 0 for some value of λ. Hence there must be conjugate points in the orbit segment { ψ_{t}(θ) | t ∈ [−1, T + 1] }.

6.2. Proposition. Let e be a regular value of H. If Θ(X) > 0 on the energy level Σ = H^{−1}{e} and the flow of X has no conjugate points, then there exists a Hamiltonian ℍ on T^{∗}M with vectorfield Z such that

(i) ℍ is convex and ℍ^{−1}{e} = Σ. (ii) Z = \frac{1}{θ(X)} · X on Σ. (iii) The flow ζ_{t} of Z has no conjugate points on Σ. (iv) ζ_{t} preserves the 1-form Θ on T Σ.

(v) If N(θ) := \frac{d}{ds}sθ|_{s=1}, then for θ ∈ Σ we have

dζ_{t}(θ) · N(θ) = N(ζ_{t}(θ)) + t Z(ζ_{t}(θ)) .

50 G. CONTRERAS AND R. ITURRIAGA

6.3. Remark. This proposition holds if Θ(X) ≠ 0 on a connected compact forward invariant set K ⊂ Σ. In this case we obtain an open connected neighbourhood U of K such that inf_{ϑ∈U} |Θ(X(ϑ))| > 0 and a convex Hamiltonian ℍ without conjugate points, defined on a neighbourhood of U in T^{∗}M. To apply this on theorem F, we can take K = closure of ψ_{ℝ^{+}}((π|_{Σ})^{−1}{q}) ⊆ Σ.

6.4. Remark. Given q ∈ πΣ, let S(q) := T_{q}^{∗}M ∩ Σ be the energy sphere at q. Then it is impossible to have Θ(X(p)) < 0 on all p ∈ S(q) and the condition Θ(X(q, p)) > 0 is equivalent to H(q, 0) < e, i.e. the zero section M ×0 lies inside the (vertical convex hull of the) energy level σ. Indeed, S(q) is convex and if H(q, 0) < e then, since for p ∈ S(q) the vector H_{p}(q, p) is outwards normal to S(q), then Θ(X(q, p)) = p · H_{p}(q, p) > 0. Otherwise, if H(q, 0) ≥ e, then there exists a supporting hyperplane E for S(q) in T_{q}^{∗}M containing (q, 0). Let p ∈ E ∩S(q). Then Θ(X(q, p)) = p · H_{p}(q, p) = 0 because the vector p is tangent to S(q) at (q, p).

6.5. Remark. By remark 6.4, the condition Θ(X) ≠ 0 is equivalent to H(q, 0) < e, ∀q ∈ M. We can extend the result of theorem F to the case H(q, d_{q}f) < e, ∀q ∈ M for some differentiable function f : M → ℝ because the flow of the new Hamiltonian H_{df}(q, p) := H(q, p + d_{q}f) is conjugate to the flow of H on H = e and satisfies H_{df}(q, 0) < e, ∀q ∈ M. See theorem A in [10] for a characterization of the energy levels where the last condition holds. See [11] for other proofs of theorem F and its converse.

Proof of Proposition 6.2:

Recall that ω(X, ·) = −dH. Let Θ = p dq be the canonical 1-form on T^{∗}M. Suppose that Θ(X) > 0 on Σ. Let Y be the vectorfield on Σ given by Y = \frac{1}{Θ(X)} · X. Then Y preserves the 1-form Θ on T Σ because the Lie derivative

L_{Y} Θ = d i_{Y} Θ + i_{Y} dΘ = d[Θ(Y)] + ω(Y, ·)
= d(1) − \frac{1}{Θ(X)} dH = 0 on T Σ.

Let ϕ_{t}(q, p) be the flow of Y on Σ. Define the symplectification ϕ̃_{t} of ϕ_{t} (cf. Arnold [3], [2]) on the cotangent bundle without its zero section T^{∗}M \ M by ϕ̃_{t}(q, λp) = λ ϕ_{t}(q, p), (q, p) ∈ Σ. Observe that ϕ̃_{t} preserves the canonical 1-form Θ. Indeed, we have that π ∘ ϕ_{t} = π ∘ ϕ̃_{t} and if ϕ_{t}(q, p) = (q_{t}, p_{t}),

Θ(q_{t}, λ p_{t}) · (dϕ̃_{t} w) = (λ p_{t}) · dπ (dϕ̃_{t} w)
= λ [p_{t} · (dπ dϕ_{t} w)]
= λ [p · (dπ w)] = Θ(q, λ p) · w .

Let Ỹ be the vectorfield of ϕ̃_{t}. We have that

LỸΘ = d[Θ(Ỹ)] + ω(Ỹ, ·) = 0 .

Therefore the function H̃ = Θ(Ỹ) is a Hamiltonian for Ỹ. But Ỹ has conjugate points on Σ and H̃ is not convex. In fact H̃ is homogeneous of degree 1 on the fibres of T^{∗}M.

Let ℍ = \frac{1}{2} H̃^{2} and let Z be the Hamiltonian vectorfield of ℍ. We have that

Z = H̃ · Ỹ. Since H̃|_{Σ} ≡ 1, then Z is also an extension of Y. Let ζ_{t} be the flow of Z. Observe that

ζ_{t}(q, sp) = s ϕ̃_{tH̃(q,sp)}(q, p) = s ϕ̃_{t s}(q, p) , for (q, p) ∈ Σ .

Taking the derivative \frac{d}{ds} |_{s=1}, we obtain

dζ_{t}(θ) · N(θ) = N(ζ_{t}(θ)) + t Y (ζ_{t}(θ)) for θ ∈ Σ.

Write ζ(t, θ) = ψ(s(t, θ), θ) with s(t, θ) > 0, θ ∈ Σ. Then

D ζ_{t} = D ψ_{s} + X(ψ_{s}(θ)) \frac{∂s}{∂θ} on T Σ . (67)

Let 𝔼 ⊂ T Σ be the stable Green bundle for X on Σ. Since X(θ) ∈ 𝔼(θ) and 𝔼 is ψ_{t}-invariant, then equation (67) shows that the flow ζ_{t} of Z preserves the Lagrangian bundle 𝔼 ⊂ T Σ, which satisfies 𝔼(θ)∩V (θ) = {0} for all θ ∈ Σ. By proposition 1.15, ζ_{t} has no conjugate points on Σ.

The following lemma completes the proof of proposition 6.2

6.6. Lemma. The Hamiltonian ℍ is convex, i.e. ℍ_{pp} is positive definite.

Proof:

Let q ∈ π(Σ) and S(q) := T_{q}^{∗}M ∩Σ. Since ℍ is homogeneous of degree 2, then ℍ_{pp} is homogeneous of degree 0 and it is enough to prove that ℍ_{pp}(q, p) is positive definite for p ∈ S(q). Let p_{0} ∈ S(q). Let h : T_{p_{0}}S(q) × ℝ → T_{q}^{∗}M be h(x, λ) := x + λ p_{0}.

Let G = ℍ ∘ h and K = H̃ ∘ h. Then G = \frac{1}{2}K^{2}. Since D^{2}h = 0, then D^{2}ℍ(q_{0}) is positive definite if D^{2}G(0, 1) is positive definite. Since K(0, λ) = λ, we have that

G_{λλ}(0, 1) = [K_{λ}^{2} + K · K_{λλ}] |_{(0,1)} = 1 .

Since in the coordinates (x, λ) the vectors \frac{∂}{∂x_{i}} |_{(0,1)}, i = 1, …, n − 1 are tangent at

p_{0} to S(q) and H̃ ≡ 1 on S(q), then K_{x}(0, 1) = 0. Hence

G_{xx}(0, 1)(v, v) = |K_{x} · v|^{2} + K K_{xx}(v, v)
= K_{xx}(v, v) .

Observe that K is homogeneous of degree 1. Therefore K_{x}(0, λ) = K_{x}(0, 1) for all λ > 0. Hence K_{xλ}(0, 1) = 0, so that

G_{xλ}(0, 1) · (v, μ) = μ [K_{λ} (K_{x} · v) + K (K_{xλ} · v)] = 0 .

And then

D^{2}G(0, 1)(v, μ)^{(2)} = μ^{2} G_{λλ} + 2μ(G_{λx} · v) + G_{xx}(v, v)
= μ^{2} + K_{xx}(0, 1) · (v, v) .

So it is enough to see that K_{xx}(0, 1) is positive definite.

Let v ∈ T_{p_{0}}S(q) be a unit vector and define f : ℝ → ℝ by f(s, t) = H(t (p_{0} +sv)). Observe that H_{p}(p_{0})·v = 0 and \frac{∂f}{∂t} |_{(0,1)} = p_{0}·H_{p}(p_{0}) = Θ(X) > 0. Let t :]−ε, ε[→ ℝ be a local C^{2} function such that f(s, t(s)) = e and t(0) = 1. Differentiating this equation with respect to s we obtain H_{p} [t′(s) (p_{0} + sv) + t(s) v] = 0, (68)

H_{pp} [t′(s) (p_{0} + sv) + t(s) v]^{(2)} + H_{p} [t′′(s) (p_{0} + sv) + 2 t′(s) v] = 0. (69)

Evaluating (68) at s = 0, t(0) = 1, we have that t′(0)(H_{p} · p_{0}) = 0. Since H_{p} · p_{0} = Θ(X) > 0, then t′(0) = 0. Evaluating (69), we get that

t′′(0) = \frac{−v^{∗}H_{pp} v}{Θ(X(p_{o}))} < −δ

for some δ > 0 uniform for all (q, p_{0}) ∈ Σ, v ∈ T_{p_{0}}S(q), |v| = 1.

Since K(sv, 1) = H̃(p_{0} + sv) = \frac{1}{t(s)} and t′(0) = 0, t(0) = 1, then

v^{∗} K_{xx} v = −t′′(0) > δ

for all v ∈ T Σ, |v| = 1.

Proof of Theorem F.

Through the proof of theorem F we work with the lifted Hamiltonian to the cotangent bundle of the universal cover M̃ of M. Let ζ_{t} be the flow of the Hamiltonian ℍ of proposition 6.2. Let q ∈ M̃ and let E_{q} : T_{q}^{∗}M → M̃ be the exponential map associated to the flow ζ_{t} and the energy level Σ. We first prove that E_{q} is

HAMILTONIANS WITHOUT CONJUGATE POINTS 53

a diffeomorphism. The derivative dE_{q}(tθ) is non-singular by corollary 1.18 and is given by formulas (39) and (40). We claim that the norm ∥dE_{q}(tθ)∥ is uniformly bounded below. By lemma 6.1 the norm ∥dE_{q}(tθ)|_{V (θ)∩T_{q}Σ}∥ is uniformly bounded below for t > ϵ. From equation (40) ∥dE_{q}(tθ)|_{⟨θ⟩}∥ is also uniformly bounded below. It remains to see that the angle ∢[dE_{q}(tθ)(V (θ) ∩ T_{q}Σ) , dπ(X(ζ_{t} θ))] is uniformly bounded below.

Since ζ_{t} preserves the 1-form Θ, the bundle K(θ) = ker Θ|_{T_{θ}Σ} is invariant under dζ_{t}. Since V (θ) ∩ T_{q}Σ ⊂ K(θ), then dE_{q}(tθ) (V (θ) ∩ T_{q}Σ) ⊆ dπ dζ_{t}(θ) (K(θ)) ⊆ dπ(K(ζ_{t}(θ))) . (70) For θ = (q, p) ∈ Σ write H_{p}(q, p) = α p+ k, with ⟨k, p⟩ = 0. Since Θ(X) = p·H_{p} ≠ 0 on Σ there exists 0 ≤ λ < 1 such that | k| ≤ λ |H_{p}| on Σ. Let h ∈ dπ(K(ζ_{t} θ)) with |h| = 1, i.e. ⟨p, h⟩ = 0, |h| = 1. Then

⟨h, H_{p}⟩ = α ⟨p, h⟩ + ⟨k, h⟩ = ⟨k, h⟩ .
|⟨h, H_{p}⟩| ≤ 1 · λ |H_{p}| .

Hence cos ∢(h, H_{p}) ≤ λ < 1. Therefore the angle ∢[dπ(K(θ)), dπ(X(θ))] is uniformly bounded below on θ ∈ Σ. By (70) ∢[dE_{q}(tθ)(V (θ) ∩ T_{θ}Σ), dπ(X(ζ_{t} θ))] > ∢[dπ(K(ζ_{t} θ)), dπ(X(ζ_{t} θ))] is uniformly bounded below. This proves that ∥dE_{q}(tθ)∥ is uniformly bounded below on t ∈ ℝ^{+}, θ ∈ Σ. Hence the map E_{q} : T_{q}M̃ → M̃ is a differentiable covering map. Since M̃ is simply connected, it must be a diffeomorphism.

Now let exp_{q} : T_{q}M̃ → M̃ be the exponential map of (ψ_{t}, Σ). Since ζ_{t} is a reparametrization of ψ_{t} for t > 0, θ ∈ Σ there exists τ(t, θ) > 0 such that ψ_{t}(θ) = ζ_{τ(t,θ)}(θ). By corollary 1.18 exp_{q} is a local diffeomorphism. It remains to prove that it is bijective. Since |Θ(X)| = \frac{|X|}{|Y |} is bounded then t ↦ τ(t, θ) is a (bijective) homeomorphism of ℝ^{+}. This implies that the map t θ ↦ τ(t, θ) θ, with θ ∈ T_{q}^{∗}M ∩Σ, is a homeomorphism of T_{q}^{∗}M. Since E_{q} is bijective and exp_{q}(tθ) = E_{q}(τ(t, θ) θ), then exp_{q} is bijective.

APENDIX I.

I. Examples.

The following formalism for twisted Hamiltonians on surfaces was derived by G. Paternain and M. Paternain in [33]. Let M be a closed orientable riemannian surface with riemannian metric ⟨ , ⟩_{x}. Let K : T T M → T M be the connection map Kξ = ∇ẋv, where ξ = \frac{d}{dt}(x(t), v(t)). Let π : T M → M be the canonical projection. Let ω_{0} be the symplectic form in T M obtained by pulling back the canonical symplectic form via the Legendre transform associated to the riemannian metric, i.e.

ω_{0}(ξ, ζ) = ⟨dπ ξ , Kζ⟩ − ⟨dπ ζ , Kξ⟩ .

Denote by Ω the area 2-form on M. Given a smooth function F : M → ℝ, define a new symplectic form ω_{F} on T M by

ω_{F} = ω_{0} + (F ∘π) (π^{∗}Ω) .

This is called a twisted symplectic structure on T M (cf. [33]). Let H : T M → ℝ be the Hamiltonian

H(x, v) = \frac{1}{2}⟨v, v⟩_{x} .

Consider the Hamiltonian vectorfield X_{F} corresponding to (H, ω_{F}), i.e.

ω_{F} (X_{F} (θ), ·) = dH . (71)

Define Y : T M → T M as the bundle map such that

Ω_{x}(u, v) = ⟨Y (u), v⟩_{x} . (72)

Then Y (u) = iu is the rotation of angle +\frac{π}{2}. From (71) and (72) we have that

dH_{θ}(ξ) = ω_{0}(X_{F} (θ), ξ) + F (π(θ)) ⟨Y (dπ X_{F} (θ)), dπξ⟩_{π(θ)} . (73)

Hence if ξ = (ξ_{1}, ξ_{2}) = (dπ ξ, Kξ) ∈ H(θ)⊕V (θ) and X_{F} = (X_{1}, X_{2}) ∈ H(θ)⊕V (θ), equation (73) becomes

⟨θ, ξ_{2}⟩ = ⟨X_{1}, ξ_{2}⟩ − ⟨X_{2}, ξ_{1}⟩ + F ⟨Y (X_{1}), ξ_{1}⟩ ,

X_{F} (θ) = (θ, F Y (θ)) . (74)

In particular, the orbits of the Hamiltonian flow are of the form (γ(t), γ̇(t)) for γ : ℝ → M, with

\frac{D}{dt} γ̇ = F (γ) Y (γ̇) . (75)

HAMILTONIANS WITHOUT CONJUGATE POINTS 55

If ∫_{M} F Ω = 0 this flow can be seen as a Lagrangian flow as follows. The condition ∫_{M} F Ω = 0 implies that the cohomology class of F Ω is zero. Hence there exists a 1-form η on M such that dη = F Ω. Consider the Lagrangian

L(x, v) = \frac{1}{2}⟨v, v⟩_{x} + η_{x}(v) .

The corresponding Euler-Lagrange equation is

\frac{D}{dt} ⟨γ̇, · ⟩ = dη_{x}(γ̇) = ⟨F Y (γ̇), · ⟩ .

which gives the same differential equation (75). This flow corresponds to a magnetic field with Lorentz force F (x) Y (v).

Observe that the theory developed in the previous sections also applies to the case in which ∫_{M} F Ω ≠ 0. Indeed, let q : U ⊆ M → ℝ^{2} be a local chart defined on a simply connected domain U ⊂ M. Then the 2-form F Ω is exact on U and there exists a 1-form η|_{U} such that dη = F Ω on U. The Hamiltonian flow on T U ⊂ T M is the Lagrangian flow of L(x, v) = \frac{1}{2}⟨v, v⟩_{x} + η_{x}(v), x ∈ U. Let (q, p) be the natural chart on T^{∗}U ⊆ T^{∗}M corresponding to the chart q and let F^{−1} be the inverse of the Legendre transform corresponding to L. Let (q, w) = F^{−1} ∘ (q, p). Since F^{∗}(dΘ) = ω_{F}, then (q, w) is a symplectic chart for ω_{F} and sends the vertical bundle of T_{q(U)}ℝ^{n} to the vertical bundle of T_{U}M.

We derive the Jacobi equation. Let (c(t), ċ(t)) be an orbit of the flow. Consider a variation γ_{s}(t) = f(s, t), where f(0, t) = c(t) and the paths t ↦ f(s, t) are solutions of (75):

\frac{D}{dt} γ̇_{s} = F Y (γ̇_{s}) . (76)

Denote the variational field of f by J(t) = \frac{∂f}{∂s} (0, t). Using the formula

\frac{D}{ds} \frac{D}{dt} \frac{∂f}{∂t} = \frac{D}{dt} \frac{D}{ds} \frac{∂f}{∂t} + R (\frac{∂f}{∂t} , \frac{∂f}{∂s} ) \frac{∂f}{∂t} .

and taking the covariant derivative of equation (76) in the direction J(t), we obtain

J̈ + R(ċ, J) ċ = \frac{D}{ds}[F Y (γ̇_{s})] .

Since Y (v) is linear on the fibers of T M and J̇ = \frac{D}{ds} \frac{∂f}{∂t} = \frac{D}{dt} \frac{∂f}{∂s}, we have

\frac{D}{ds} F Y (γ̇_{s}) = [∇_{J}(F Y)](γ̇_{s}) + F Y (J̇) ,
= (∇_{J}F) Y (ċ) + F Y (J̇) .

Hence we obtain the Jacobi equation J̈ + R(ċ, J) ċ − (∇_{J}F) Y (ċ) − F Y (J̇) = 0 . (77)

56 G. CONTRERAS AND R. ITURRIAGA

Consider the orthonormal basis { ċ, Y (ċ) = i ċ } of T_{c(t)}M. We have that

\frac{D}{dt} ċ = F Y (ċ) ,
\frac{D}{dt} Y (ċ) = (∇ċY)(ċ) + Y (\frac{D}{dt} ċ)
= −F ċ .

Write

J = x ċ + y Y (ċ) .

Then

J̇ = (ẋ − F y) ċ + (ẏ + F x) Y (ċ) ,
J̈ = (ẍ − 2 F ẏ − Ḟ y − F^{2} x) ċ + (ÿ + 2 F ẋ + Ḟ x − F^{2} y) Y (ċ) ,
F Y (J̇) = −F (ẏ + F x) ċ + F (ẋ − F y) Y (ċ) .

Replacing these formulas on equation (77) we obtain ẍ − (F y)′ = 0 , (78) ÿ + K y + F ẋ = y (∇_{Y (ċ)}F) , (79) where K = ⟨R(ċ, Y (ċ)) ċ , Y (ċ)⟩ is the sectional curvature along c(t). Fix the energy level Σ = H^{−1}\{\frac{1}{2}\} = SM. If (J, J̇) ∈ T_{ċ(t)}Σ ⊂ H(θ)⊕V (θ), we have that dH(J, J̇) = ⟨ċ, J̇⟩ = 0. Hence ẋ − F y = 0 . (80) This implies equation (78), and from (79) and (80), we get ÿ + (K + F^{2} − ∇_{Y (ċ)}F ) y = 0 . (81) Hence the Jacobi equations on T_{ċ(t)}Σ are given by (80) and (81).

Example 1. A convex Hamiltonian without conjugate points and no continuous invariant transversal bundle.

Take K ≡ −1, F ≡ 1. Then equations (80) and (81) become

ẋ = y ,
(82) ÿ = 0 .

The solutions have the form

x(t) = \frac{1}{2} a t^{2} + b t + c ,
(83)  y(t) = a t + b ,
ẏ(t) = a .

The derivative of the Hamiltonian flow ψ_{t} on (c(t), ċ(t)) is given by

dψ_{t}(J(0), J̇(0)) = (J(t), J̇(t)) ∈ H(ψ_{t}θ) ⊕ V (ψ_{t}θ)
= (x ċ + y Y (ċ) ; 0 ċ + (ẏ + x) Y (ċ))

= (x(t) ċ + y(t) Y (ċ) ; z(t) Y (ċ)) , (84)

where z(t) = ẏ(t) + x(t) and θ = (c(0), ċ(0)).

Every orbit (c, ċ) has no conjugate points because the only solution of (83) with x(0) = y(0) = 0 and y(T) = 0 = x(T) , T ≠ 0 is x(t) = y(t) ≡ 0. The Green bundles coincide and are generated by X(θ) = (ċ; −Y (ċ)) and (Y (ċ); 0). The last vector can be found by solving x(0) = 0, y(0) = 1, x(T) = y(T) = 0 and letting T → ±∞.

Now suppose that there exists an invariant continuous subbundle N(θ) ⊆ T_{θ}Σ which is transversal to the vectorfield X(θ). Let θ ∈ Σ be a recurrent point and let t_{n} →_{n} +∞ be such that ψ_{t_{n}}(θ) →_{n} θ. Let ξ ∈ N(θ) \ {0}. If dψ_{t}(ξ) =

(x(t), y(t); z(t)) = (J(t); J̇(t)), then

lim_{n→±∞} \frac{1}{|x(t_{n})|} dψ_{t_{n}}(ξ) = lim_{n} \frac{1}{x(t_{n})} (x(t_{n}), y(t_{n}); z(t_{n}))

= (1, 0; 1) = X(θ) . (85)

By the continuity of N we have

X(θ) = lim_{n} \frac{1}{x(t_{n})}dψ_{t}(ξ) ∈ N(θ) .

This contradicts the transversality hypothesis.

The same argument as in equation (85) shows that for every θ ∈ Σ, the angle

∢(dψ_{t}(V (θ) ∩ T_{θ}Σ), X(ψ_{t}θ)) ⟶_{t→+∞} 0.

In this case the trajectories of the Hamiltonian flow are the unit tangent vectors of the horospheres of the riemannian metric, i.e.

ψ_{t}(θ) = −Y (h_{t}(Y (θ))) , (86)

where h_{t} is the horocycle flow. In particular, the exponential map exp_{q} : T_{q}M̃ → M̃ is a diffeomorphism for all q ∈ M. This flow has no periodic orbits because the horocycles W^{uu} ⊂ Σ = T^{1}M̃/_{π_{1}(M)} are homeomorphic to ℝ.

example I.1

We prove (86). Let θ ∈ T_{q}M̃, ∥θ∥ = 1. Let ϕ_{t} : T M̃ ↩ be the geodesic flow of M̃. Consider a sphere

S_{T} (θ) = { q ∈ M̃ | dM̃(q, πϕ_{T} (θ)) = T }

and a parametrization by arc length γ_{T} (s) of S_{T} (θ) with γ_{T} (0) = π(θ). Then \frac{D}{ds} γ̇_{T} (s) is orthogonal to γ̇_{T} (s). Moreover, it has constant norm, because it is the image of \frac{D}{ds} γ̇_{T} (0) by the isometry given by a rotation about the center πϕ_{T} (θ) of S_{T} (θ). Therefore γ_{T} satisfies

\frac{D}{ds} γ̇_{T} = λ_{T} Y (γ̇_{T})

for some constant λ_{T} > 0. Passing to the limit when T → +∞ we obtain that the reparametrizations by arc length γ of the horospheres satisfy \frac{D}{ds} γ̇ = μ Y (γ̇) , (87)

with μ = lim_{T →+∞} λ_{T}, (in fact T → λ_{T} is strictly decreasing). We claim that μ = 1. Indeed, the solutions of (87) of are closed circles if μ is big enough. The infimum μ_{0} of such μ’s corresponds to the horospheres . For μ > μ_{0} the flow has conjugate points corresponding to the period (length) of these closed circles (see example 3). From the Jacobi equation (81) with F = μ, we see that Σ = H^{−1}\{\frac{1}{2}\} has conjugate points for μ > 1 and the flow is Anosov for μ < 1. Hence μ_{0} = 1.

Indeed, for μ < 1 the solution of (80) and (81) are y = Ae^{λt} + B e^{−λt}, x = μ(\frac{A}{λ} e^{λt} − \frac{B}{λ} e^{−λt} + C), with λ = \sqrt{1 − μ^{2}}. Writing ξ = (x(0), y(0); ẋ(0), ẏ(0)) we have that dψ_{t}(ξ) = (x(t), y(t); ẋ(t), ẏ(t)). For A ≠ 0 the function t ↦ ∥dψ_{t}(ξ)∥ has exponential growth and for B ≠ 0, t ↦ ∥dψ_{−t}(ξ)∥ has exponential growth. As in section §3, this implies that the flow is Anosov for μ < 1. For μ > 1, the Jacobi equations (80), (81) exhibit conjugate points, by the same arguments as for equations (89) and (90).

Example 2. A periodic orbit without conjugate points and no transversal invariant subspace.

We construct a convex Lagrangian which has the Jacobi equations (82) on a periodic orbit. Let θ ∈ Σ be such a periodic point with period τ > 0. Changing the coordinates (x, y; z) on T_{θ}Σ of equation (84) to (x, y; ẏ) = (x, y; z − x), the matrix of the derivative dψ_{τ}(θ) is

dψ_{τ}(θ) = [\begin{matrix}1 & t & \frac{1}{2} t^{2} \\ & 1 & t \\ & & 1 \end{matrix}] \begin{matrix}\} x \\ \} y \\ \} ẏ \end{matrix} ≈  [\begin{matrix}1 & 1 \\ & 1 & 1 \\ & & 1 \end{matrix}] in Jordan form.

Let {e_{1}, e_{2}, e_{3}} be the basis of T_{θ}Σ in which dψ_{τ}(θ) is written in canonical form. The transformations dψ_{τ} and

dψ_{τ} − I = [\begin{matrix}0 & 1 \\ & 0 & 1 \\ & & 0 \end{matrix}]

have the same invariant subspaces. The vectorfield corresponds to the unique eigenspace ⟨e_{1}⟩ and the only invariant subspaces are ⟨e_{1}⟩ = ⟨X(θ)⟩, ⟨e_{1}, e_{2}⟩ = 𝔼(θ) = 𝔽(θ) and ⟨e_{1}, e_{2}, e_{3}⟩ = T_{θ}Σ. None of them is transversal to X(θ) = e_{1}.

Now we make the construction. Consider the torus 𝕋^{2} = ℝ^{2}/_{5ℤ^{2}} with the flat metric and polar coordinates (r, θ) centered at (0, 0) ∈ ℝ^{2}/_{5ℤ^{2}} defined on r ≤ 2. Let F : 𝕋^{2} → ℝ be a C^{∞} function such that F (r, θ) = F (r), F (r = 1) = 1, \frac{∂F}{∂r} |_{r=1} = −1, and ∫_{𝕋^{2}} F Ω = 0. Let η be a 1-form in 𝕋^{2} such that dη = F Ω. Define L(x, v) = \frac{1}{2} ⟨v, v⟩_{x} + η_{x}(v). At r = 1, the Euler-Lagrange equation (75) is

\frac{D}{dt} γ̇ = F (γ) Y (γ̇) = Y (γ̇) at r = 1 .

Hence r = 1, θ̇ = −1 is a periodic solution. Moreover, at r = 1 we have that F = 1, ∇_{Y (ċ)}F = −\frac{∂F}{∂r} = +1. Hence the Jacobi equations (80), (81) are the same as (82).

Example 3. A segment without conjugate points with crossing solutions.

Let W = { (x, y) ∈ ℝ^{2} | x^{2} + y^{2} ≤ 4 } and consider the Lagrangian

L(x, v) = \frac{1}{2} ⟨v, v⟩ + η_{x}(v) ,

where ⟨ , ⟩ is the euclidean metric and dη = −Ω. This Lagrangian corresponds to F |_{W} = −1 in the formalism above and can be embedded into a convex Lagrangian on a closed surface. The Euler-Lagrange equation (75) is v̇ = −i v . (88)

60 G. CONTRERAS AND R. ITURRIAGA

The solutions of (88) on the energy level |v| = 1, are all the circles contained in W with constant angular velocity θ̇ = −1. Fix C: R(t) = 1, θ(t) = π − t in polar coordinates in W with center (0, 0). Let ϑ = (0, 1)_{(−1,0)} be the initial vector of this solution. Then the circles which are rotations of C fixing the initial point p = (−1, 0) are also solutions of (88) (see figure I.3) with the same energy E = \frac{1}{2} |v| = \frac{1}{2}. The solution C has conjugate points, but the conjugate points appear at t = 2π. In particular the segment \{ ψ_{t}(ϑ) | 0 ≤ t ≤ \frac{3π}{2} \} has no conjugate points. But the intersections with other solutions starting at p = (−1, 0) appear at t < \frac{3}{2}π and at t = π there is a vector ξ ∈ V (ϑ) ∩ T_{ϑ}Σ such that dπ (dψ_{t} ξ) = dπ X(ψ_{t}ϑ) = (0, −1). Proposition 1.16 says that these phenomena can not occur if all the positive (negative) orbit of ϑ has no conjugate points. The claims stated above can be proved using the Jacobi equations (80) and (81):

ẋ + y = 0 ,  ÿ + y = 0 , (89)

whose solutions are

x(t) = a cos t + b sin t + c ,
(90)  y(t) = a sin t − b cos t .

Then if x(0) = y(0) = 0 and ẏ(0) ≠ 0 then (x(t), y(t)) ≠ 0 for all 0 < t < 2π. Moreover, if a = −1, b = 0, c = +1, then x(0) = y(0) = 0, y(π) = 0, x(π) = +2.

example I.3  example I.4

Example 4. A non surjective exponential map without conjugate points.

Consider the flat torus 𝕋^{2} = ℝ^{2}/_{6ℤ} = [−3, 3]^{2}/_{6ℤ} and polar coordinates (r, θ) on { (x, y) ∈ 𝕋^{2} | x^{2} + y^{2} < 3 } centered at 0 = (0, 0). Define the Lagrangian

L(x, v) = \frac{1}{4} ⟨v, v⟩_{x} + η_{x}(v) + ϕ(x) ,

where ϕ is a C^{∞} function on 𝕋^{2} such that ϕ(r, θ) = \frac{1}{4}r^{2} + cos πr on r ≤ 2 and ϕ(r, θ) ≥ 1 outside r ≤ 2; and η(r, θ) = −\frac{1}{2} r^{2} f(r) dθ with f(r) a C^{∞} function with support on r ≤ \frac{5}{2} such that f(r) = 1 on r ≤ 2 and f(r) ≤ 1.

Observe that the path C : r(t) ≡ 1, θ(t) = t is static (cf. Mañé [28]). Indeed, the Lagrangian

L(r, θ; ṙ, θ̇) = \frac{1}{4} (ṙ^{2} + r^{2} θ̇^{2}) − \frac{1}{2} r^{2} θ̇ + \frac{1}{4} r^{2} + cos πr on r ≤ 2 ,

is minimized on ṙ = 0, θ̇ = 1, r = 1, with L = −1; and on r ≥ 2, we have that ϕ(r) ≥ 1 and f ≤ 1, so that

\frac{1}{4} r^{2} θ̇^{2} + η(v) ≥ r^{2} (\frac{1}{4} θ̇^{2} − \frac{1}{2} θ̇) ≥ −\frac{1}{4} r^{2} ≥ −1

Hence L ≥ ϕ(r) ≥ 1 on r ≥ 2.

By theorem VII in Mañé [28] (see also [8]), there exists a semistatic (hence forward minimizing) solution γ(t) such that γ(0) = (0, 0) and the ω-limit of (γ, γ̇) is the (hyperbolic) closed orbit C. By the rotational symmetry of the Lagrangian, for every 1-dimensional subspace L ⊂ T_{(0,0)}𝕋^{2} we obtain a semistatic curve γ, which is tangent to L, with γ_{L}(0) = (0, 0) and ω-limit(γ_{L}, γ̇_{L}) = C. Since the semistatic solutions are minimizing, none of them γ_{L}|_{[0,+∞[} have conjugate points. Moreover, by theorem X in Mañé [28] (also [8]), C and all the (γ_{L}, γ̇_{L})’s are in the same energy level E = c(L). By the graph property (theorem VII, Mañé [28] and also [8]), the γ_{L}|_{]0,+∞[}’s don’t intersect π(C) neither intersect each other.

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