THE ASYMPTOTIC MASLOV INDEX AND ITS APPLICATIONS

GONZALO CONTRERAS, JEAN-MARC GAMBAUDO, RENATO ITURRIAGA, AND GABRIEL P. PATERNAIN

Abstract. Let N be a 2n-dimensional manifold equipped with a symplectic structure ω and Λ(N) be the Lagrangian Grassmann bundle over N. Consider a flow ϕ^{t} on N that preserves the symplectic structure and a ϕ^{t}-invariant connected submanifold Σ. When there exists a continuous section Σ → Λ(N), we can associate to any finite, ϕ^{t}-invariant measure with support in Σ, a quantity: The Asymptotic Maslov Index, that describes the way Lagrangian planes are asymptotically wrapped in average around the Lagrangian Grassmann bundle. A particular attention is paid to the case when the flow is derived from an optical Hamiltonian and when the invariant measure is the Liouville measure on compact energy levels. The situation when the energy levels are not compact is discussed in an appendix.

1. Introduction

1.1. The Maslov Cocycle.

In his book Théorie des perturbations et méthodes asymptotiques [15], V. P. Maslov introduced an index of curves relevant in quantum mechanics. V. Arnold [2], in an Appendix to Maslov’s book, settled down the main geometric features of this index introducing a characteristic class. This introductive section is very much guided by Arnold’s Appendix.

Consider the standard 2n-dimensional vector space ℝ^{2n} and its canonical decomposition ℝ^{2n} = ℝ^{n} ⊕ ℝ^{n}, and denote by x = (p, q) a point in ℝ^{2n}, where p = (p_{1}, …, p_{n}) and q = (q_{1}, …, q_{n}). The space ℝ^{2n} can be endowed with three structures:

• An Euclidean Structure i.e. a positive definite quadratic form on ℝ^{2n}:

⟨x, x⟩ = ∑_{i=1}^{n}(p_{i}^{2} + q_{i}^{2}).

• A complex structure i.e. an endomorphism J on ℝ^{2n}

satisfying J^{2} = −Id:

J(p, q) = (−q, p).

• A symplectic structure i.e. an antisymmetric non degenerate bilinear form:

ω(x, y) = ⟨J(x), y⟩ = ∑_{i=1}^{n} dp_{i} ∧ dq_{i}.

G. Contreras was partially supported by CNPq-Brazil. G. Contreras and R. Iturriaga were partially supported by Conacyt-México, grant 28489-E.

The group of automomorphisms of ℝ^{2n} that preserve these structures^{1} is called the unitary group and is denoted by U(n). This group is isomorphic to the group of linear isometries of ℂ^{n} = ℝ^{n} ⊕ iℝ^{n} ≈ (ℝ^{n} × {0}) ⊕ J(ℝ^{n} × {0}) = ℝ^{2n}.

A subspace W in ℝ^{n} is said isotropic if the symplectic form ω vanishes on W. A Lagrangian plane is an isotropic subspace of maximal dimension n. The subspaces p = 0, q = 0 and p = q are examples of Lagrangian planes.

The elements of the unitary group map Lagrangian planes onto Lagrangian planes. Actually the unitary group acts transitively on the set of Lagrangian planes and its stationary group is isomorphic to the orthogonal group O(n). This provides the set of Lagrangian planes Λ(n) the structure of a compact manifold: Λ(n) = U(n)/O(n), called the Lagrangian Grassmann manifold.

This last identification can be seen as follows. Given an orthonormal basis {e_{1}, …, e_{n}} of the Lagrangian plane 𝔼_{0} := {q = 0}, consider another Lagrangian plane λ and an orthonormal basis {ℓ_{1}, …, ℓ_{n}} for λ. The automorphism of ℝ^{2n} that maps, for i = 1, …, n, each vector e_{i} on ℓ_{i} and each vector Je_{i} on Jℓ_{i} is clearly an unitary automorphism. Any unitary automorphism that leaves the Lagrangian plane λ invariant, transforms the orthogonal basis {e_{1}, …, e_{n}} into another orthogonal basis {e′_{1}, …, e′_{n}} of 𝔼_{0} and the corresponding orthogonal basis {Je_{1}, …, Je_{n}} of J𝔼_{0} into the orthogonal basis {Je′_{1}, …, Je′_{n}} of J𝔼_{0}. Consequently an unitary isomorphism U that fixes 𝔼_{0} fixes also J𝔼_{0}. Furthermore the matrix of U restricted to 𝔼_{0} written in the basis {e_{1}, …, e_{n}} coincides with the matrix of U restricted to J𝔼_{0} written in the basis {Je_{1}, …, Je_{n}} and is an element of the orthogonal group O(n). Thus, given two Lagrangian planes 𝔼_{0} and λ, there exists a unitary automorphism u(λ) mapping the plane 𝔼_{0} onto the plane λ. This automorphism is unique up to orthogonal self transformations of the plane 𝔼_{0}, hence its complex determinant is a complex number with modulus one which is unique up to multiplication by −1. This defines a map:

det^{2} : Λ(n) → 𝕊^{1},

which associates to each Lagrangian plane, the square of the determinant of the automorphism u(λ).

The Maslov cocycle 𝔐 (see [2]) is the element of the first cohomology group H^{1}(Λ(n), Z) which associates to any oriented closed curve γ in Λ(n) the degree of the map:

𝕊^{1} ⟶^{γ} Λ(n) ⟶^{det^{2}} 𝕊^{1}.

The Maslov cycle of the Lagrangian plane 𝔼_{0} (also called the train of the Lagrangian plane) is the subset Λ^{𝔼_{0}}(n) of Λ(n), which consists of all the Lagrangian planes whose intersection with 𝔼_{0} is non-trivial. V. Arnold proved [2] that the Maslov cycle of a Lagrangian plane defines a codimension 1, co-oriented cycle in the Lagrangian Grassmann manifold Λ(n)^{2}. Writing Λ(n) = U(n)/O(n), the transversal orientation of Λ^{𝔼_{0}}(n) coincides with the orientation of θ ↦ e^{iθ} 𝔼_{0}.

1In fact an automorphism that preserves two of these structures, preserves necessarily the third one.

Using the homotopy exact sequence of the fibration

O(n) ⟶ U(n) ⟶ Λ(n) ,

we get that π_{1}(Λ(n)) = ℤ. Moreover, the following commutative diagram of fiber bundles

[figure: SO(n); O(n); 𝕊^{0}; SU(n); U(n); 𝕊^{1}; SΛ(n); Λ(n); 𝕊^{1}; det; det; z^{2}; det^{2}]

where SΛ(n) = {λ ∈ Λ(n)|det^{2}λ = 1}, implies that the generator of π_{1}(Λ(n)) is sent by det^{2} to the generator of π_{1}(𝕊^{1}). This implies that H^{1}(Λ(n), ℤ) ≈ H_{1}(Λ(n), ℤ) ≈ π_{1}(Λ(n)) ≈ ℤ and that the Maslov cocycle is a generator of H^{1}(Λ(n), ℤ) (cf. [2]).

The Maslov index 𝔐(h) ∈ ℤ of a homology class h ∈ H_{1}(Λ(n), ℤ) is the value the cocycle 𝔐 takes on the cycle h. The Maslov cycle Λ^{𝔼_{0}}(n) is the Poincaré dual of 𝔐, so that 𝔐(h) is also equal to the oriented intersection number of h and Λ^{𝔼_{0}}(n).

1.1. Remark. The Maslov cocycle 𝔐 in H^{1}(Λ(n), ℤ) is induced by a 1-form η in H^{1}(Λ(n), ℝ) which is defined for every λ in Λ(n) by η_{λ} = d_{λ}u, where u : Λ(n) → ℝ is determined by det^{2}(λ) = exp(2iπu(λ)). Thus, for any piecewise differentiable oriented closed curve γ in Λ(n) which is in the homology class h, we have:

𝔐(h) = ∫_{0}^{1} η_{γ(t)}(γ̇(t))dt.

In the particular case when the dimension n is 1, Lagrangian planes are lines in ℝ^{2} and the Lagrangian Grassmann manifold Λ(1) is the projective line ℝℙ(1) = 𝕊^{1}. The unitary group U(1) is the group 𝕊^{1} of complex numbers with modulus 1. The orthogonal group O(1) is the group with 2 elements ±Id and the identification Λ(1) = U(1)/O(1) reflects the fact that each line in ℝ^{2} is image of a given line by a multiplication by a complex number with modulus 1, exp 2iπθ,where θ is defined up to translation by \frac{1}{2} in ℝ/ℤ. The Maslov cocycle is simply the morphism that associates to any curve in Λ(1) its degree and the Maslov cycle of a Lagrangian plane (i.e. a line) is reduced to the line itself. Thus it is straightforward to notice that any two lines (i.e. Lagrangian planes) are joined by an arc in Λ(1) whose intersection number with (the Maslov cycle of) any given line is bounded in norm by 1.

When the dimension n is greater than 1, even if more complex, the situation keeps part of this rigidity. More precisely, consider two Lagrangian planes λ_{0} and λ_{1}, a basis {e_{1}, …, e_{n}} for λ_{0} and a basis {e′_{1}, …, e′_{n}} for λ_{1} such that:

2In fact, for k ≥ 1, the set Λ^{k}(n) = {λ ∈ Λ| dim λ∩𝔼_{0} = k} is an open submanifold of Λ(n) with codimension \frac{1}{2}k(k + 1), (see Arnold [2]). In particular Λ^{𝔼_{0}}(n) = \overline{Λ^{1}(n)} = ∪_{k≥1}Λ^{k}(n) is a cycle in Λ(n) of codimension 1 (actually it is an algebraic variety with singular set \overline{Λ^{2}(n)} = ∪_{k≥2}Λ^{k}(n) of codimension ≥ 3 and thus its boundary chain is null).

ω = ∑_{i=1}^{n} e_{i}^{∗} ∧ e′_{i}^{∗}

where {e_{i}^{∗}, e′_{i}^{∗}} is the dual basis of {e_{i}, e′_{i}}. Any Lagrangian plane λ whose intersection with and λ_{1} is reduced to 0 can be seen as a graph of a map from λ_{0} to λ_{1}. Actually, when written in coordinates with the basis {e_{1}, …, e_{n}} for λ_{0} and {e′_{1}, …, e′_{n}} for λ_{1}, a straightforward calculation shows that this graph is given by a n × n-symmetric matrix A:

y = A x.

Furthermore if the intersection of λ_{0} and λ is reduced to 0, the matrix A is invertible. The path Γ_{λ_{0},λ_{1},λ} defined for every t ∈ [0, 1] by

Γ_{λ_{0},λ_{1},λ}(t) = {(x, y) ∈ ℝ^{n} × ℝ^{n} | y = tA x}

is a path from [0, 1] to Λ(n) joining Γ_{λ_{0},λ_{1},λ}(0) = λ to Γ_{λ_{0},λ_{1},λ}(1) = λ_{0}.

The following lemma has is roots deep in the history of symplectic geometry and we refer to [3] for a nice comment on its origins. However for the sake of completeness and since it will be useful in the following, we include a simple proof of it.

1.2. Lemma. Let γ a be Lagrangian plane whose intersection with λ_{0} and λ_{1} is reduced to 0 and 𝔼_{0} be a Lagrangian plane whose intersection with λ_{1} is reduced to 0, then the intersection number of the path Γ_{λ_{0},λ_{1},λ} with the Maslov cycle of the Lagrangian plane 𝔼_{0} is bounded in norm by n.

Proof: Written in coordinates in the basis {e_{1}, …, e_{n}} for λ_{0} and {e′_{1}, …, e′_{n}} for λ_{1} the equation of 𝔼_{0} reads:

y = B x.

where B is a n × n-symmetric matrix. Assume that for some t = t_{1} the two planes 𝔼_{0} and Γ_{λ_{0},λ_{1},λ}(t_{1}) have a non trivial intersection. This means that the kernel of t_{1}A − B that we denote by W_{t_{1}}, is a subspace of ℝ^{n} not reduced to 0. Let t_{1}, …, t_{k} be a strictly increasing sequence of values of t such that W_{t_{i}} is not reduced to 0 for i = 1, …, k. Since A is invertible, all these subspaces are linearly independent and thus:

∑_{i=1}^{k} dim W_{t_{i}} ≤ n.

It follows that for ϵ > 0 small, the path Γ_{λ_{0},λ_{1},λ}|_{t∈[t_{i}−ϵ,t_{i}+ϵ]} is homotopic, with fixed endpoints, to a path h which intersects 𝔼_{0} at most dim W_{t_{i}}-times and such that, for each t ∈ [0, 1], dim h(t) ∩ 𝔼_{0} ≤ 1.

1.3. Remark. Consider a transformation T of ℝ^{2n} that preserves the symplectic structure. It is clear that the path Γ_{T λ_{0},T λ_{1},T λ} constructed with the bases {T e_{1}, …, T e_{n}} for T λ_{0} and {T e′_{1}, …, T e′_{n}} for T λ_{1} is the image under T of the path Γ_{λ_{0},λ_{1},λ} and is again given for every t ∈ [0, 1] by the equation:

Γ_{T λ_{0},T λ_{1},T λ}(t) = T ∘ Γ_{λ_{0},λ_{1},λ}(t) = {(x, y) ∈ ℝ^{n} × ℝ^{n} | y = tA x}.

1.2. Symplectic manifolds and the Lagrangian Grassmann bundle.

Let N be a 2n-dimensional manifold equipped with a symplectic structure, i.e. a non degenerated closed 2-form ω.

Given x in N, a subspace W in the tangent space T_{x}N is isotropic if the 2-form ω vanishes on W. Isotropic subspaces with maximal dimension are called Lagrangian planes and have dimension n. The Lagrangian Grassmann bundle Λ(N) is the bundle over N whose fibers consist of all the Lagrangian planes. We denote by

Π : Λ(N) → N ,

the standard projection.

Given x in N, an almost complex structure in the tangent space T_{x}N is an endomorphism J_{x} of T_{x}N, such that J_{x}^{2} = −Id. We say that an almost complex structure J_{x} and a symplectic structure ω_{x} are compatible if u ↦ ω_{x}(J_{x}u, u) is a positive quadratic form on T_{x}N. This quadratic form induces an Euclidean structure associated to the compatible pair of symplectic and almost complex structures.

Let J (N) be the bundle over N whose fiber over any point x in N consists of all compatible almost complex structures in T N_{x}. Since the fibers of this bundle are contractible, there exists a continuous section

J : N → J (N),

which is unique up to homotopy. Consider a connected submanifold Σ of N (neither necessarily compact nor with finite first homology group) and the corresponding bundle Λ(Σ) = Π^{−1}(Σ). Assume that there exists a continuous section:

𝔼 : Σ → Λ(Σ).

This section induces, over each point x in Σ, a continuous splitting of the tangent space:

T N_{x} = 𝔼(x) ⊕ J_{x}𝔼(x).

Since the unitary group acts transitively on the set of Lagrangian planes of T N_{x} and its stationary group is isomorphic to the orthogonal group O(n) (see §1.1), this section yields a trivialization of the bundle Λ(Σ) :

I_{𝔼} : Λ(Σ) → Σ × Λ(n) = Σ × U(n)/O(n).

The Maslov cocycle of the section 𝔼, 𝔐_{𝔼}, is the element in H^{1}(Λ(Σ), ℤ) which associates to any oriented closed curve γ in Λ(Σ) the degree of the map:

𝕊^{1} ⟶^{γ} Λ(Σ) ≈ Σ × Λ(n) ⟶^{τ} Λ(n) →^{det^{2}} 𝕊^{1},

where τ : Σ × Λ(n) → Λ(n) stands for the projection on the second factor.

The Maslov cycle of the section 𝔼 is the sub-bundle Λ^{𝔼}(Σ) of Λ(Σ), whose fiber F_{x} over any point x in Σ consists in all the Lagrangian planes in T N_{x} whose intersection with 𝔼(x) is non-trivial, and is given by: Λ_{x}^{𝔼} = I_{𝔼}^{−1}(Λ^{{q=0}}(n)). This sub-bundle defines a codimension 1, co-oriented cycle in the bundle Λ(Σ), the co-orientation of this cycle being induced by the co-orientation of Λ^{𝔼_{0}}(n).

The Maslov index of a homology class h ∈ H_{1}(Λ(Σ), ℤ) is the value that the cocycle 𝔐_{𝔼} takes on the cycle h. The Maslov cycle Λ^{𝔼}(Σ) is the Poincaré dual of 𝔐_{𝔼}, so that 𝔐_{𝔼}(h) is also equal to the oriented intersection number of h and Λ^{𝔼}(Σ).

If 𝔼 is a C^{1} section and γ is a piecewise differentiable closed oriented curve in Λ(Σ) with homology class h, Remark 1.1 gives an integral version of the Maslov index of h:

𝔐_{𝔼}(h) = ∫_{0}^{1} η_{τ∘I_{𝔼}∘γ(t)}(\frac{d}{dt}τ ∘ I_{𝔼}γ(t)) dt = ∫_{0}^{1} η_{𝔼}_{γ(t)}(\frac{d}{dt}γ(t)) dt,

where η_{𝔼} is the pullback by τ ∘I_{𝔼} of the form η. When 𝔼 is only a continuous section we can approximate 𝔼 by a C^{∞} section 𝔼̂ such that 𝔐_{𝔼}(h) = 𝔐𝔼̂(h).

From the very definition, the Maslov cocycle depends on the choice of the section 𝔼, however this dependence can be made completely explicit. Consider another continuous section:

𝔽 : Σ → Λ(Σ)

and the trivialization obtained from this section

I_{𝔽} : Λ(Σ) → Σ × Λ(n) = Σ × U(n)/O(n).

Let us estimate the difference cocycle 𝔐_{𝔼} − 𝔐_{𝔽} evaluated on a homology class h in H_{1}(Λ(Σ), ℤ). For this purpose consider a closed curve γ : 𝕊^{1} → Λ(Σ) whose homology class is h. The curve γ induces, through both trivializations, two closed curves γ_{𝔼}, γ_{𝔽} : 𝕊^{1} → U(n)/O(n). On the other hand, the curve γ induces a third curve γ_{𝔽,𝔼} : 𝕊^{1} → U(n)/O(n) which is given by the section 𝔽 above the projected curve Π ∘ γ : 𝕊^{1} → Σ, seen through the trivialization I_{𝔼}. For each θ ∈ 𝕊^{1}, these three curves are related as follows:

det^{2}(γ_{𝔼}(θ)) = det^{2}(γ_{𝔽}(θ)) . det^{2}(γ_{𝔽,𝔼}(θ)).

Consequently

𝔐_{𝔼}(h) − 𝔐_{𝔽}(h) = 𝔐_{𝔼}(𝔽_{⋆} ∘ Π_{⋆}(h)),

where 𝔽_{⋆} and Π_{⋆} are respectively the maps induced by 𝔽 and Π on the first homology groups. In terms of 1-forms we have:

η_{𝔼} − η_{𝔽} = Π^{⋆} ∘ 𝔽^{⋆}(η_{𝔼}). (1)

It is worth noticing that the difference of the two 1-forms η_{𝔼} − η_{𝔽} contains in its kernel the tangent space to the fibers of Λ(Σ).

This work is organized as follows:

• In section 2, we consider a flow ϕ^{t} defined on a neighborhood of a submanifold Σ of N, which leaves Σ invariant and preserves the symplectic 2-form. When there exists a continuous section Σ → Λ(N), we associate to any finite, ϕ^{t}-invariant measure with support in Σ, a quantity: The Asymptotic Maslov Index. The dependence of this asymptotic Maslov index on the section and the invariant measure is discussed. In particular we prove that there is no dependence on the section if the Schwartzman asymptotic cycle of the measure vanishes;

• In section 3 we focus on Hamiltonian flows. We show that the asymptotic cycle of the Liouville measure for a compact energy level is zero if the form ω^{n−1} is exact which is always true for Hamiltonian flows on the cotangent bundle of a manifold equipped with its canonical symplectic structure.

• In section 4 we prove that for Hamiltonians that are optical with respect to a given section (see the definition below or [5]) the asymptotic Maslov index of the Liouville measure with respect to this section is always non-negative and it is strictly positive if and only if there are conjugate points. As an application we conclude that an optical Hamiltonian on a compact energy level which possesses a ϕ^{t}-invariant Lagrangian section (this is the case in particular when the flow on the energy level is Anosov ) does not have conjugate points; a result already proved by W. Klingenberg [12] for geodesic flows on compact manifolds; by R. Mañé [13] for geodesic flows with dense non-wandering set; and by G. Paternain and M. Paternain [16] for convex Hamiltonians;

• The cotangent bundle of a manifold is equipped with a canonical symplectic form and a canonical section (the vertical section). Convex Hamiltonian flows -which are optical with respect to the vertical section- are studied in Appendix A;

• Finally, in Appendix B, we give an example of a convex Hamiltonian with an invariant Lagrangian section and conjugate points on a non compact regular energy level with finite volume.

2. The Asymptotic Maslov Index

Consider a flow ϕ^{t} defined on a neighborhood of the submanifold Σ of N, which leaves Σ invariant and preserves the symplectic 2-form:

ϕ^{t}^{∗}ω = ω, (2)

and denote by X the vector field induced by the flow ϕ^{t}.

We denote by n_{𝔼}(Γ) the algebraic intersection number of an oriented curve Γ in Λ(Σ) with Λ^{𝔼}(Σ). To a Lagrangian plane λ_{x} in the fiber Π^{−1}(x) of Λ(Σ) we associate the path: Γ_{λ_{x},T} : [0, T] → Λ(Σ) defined for any t ∈ [0, T] by:

Γ_{λ_{x},T} (t) = dϕ^{t}_{x}(λ_{x}).

The following lemma (or some very similar version) is, like Lemma 1.2, a well known fact in symplectic dynamics and again for the sake of completeness we give here a very simple proof.

2.1. Lemma. Let x be a point in N and λ_{x} and λ′_{x} two Lagrangian planes in the fiber Π^{−1}(x) of Λ(Σ). Then, for any T ≥ 0:

|n_{𝔼}(Γ_{λ_{x},T}) − n_{𝔼}(Γ_{λ′_{x},T})| ≤ 4n.

Proof: Let λ_{0,x} and λ_{1,x} be two Lagrangian planes in Π^{−1}(x) such that:

• λ_{0,x} ∩ λ_{1,x} = 0;

• λ_{0,x} ∩ λ_{x} = λ_{0,x} ∩ λ′_{x} = 0;

• λ_{1,x} ∩ λ_{x} = λ_{1,x} ∩ λ′_{x} = 0;

• λ_{1,x} ∩ 𝔼(x) = 0.

Consider the two paths Γ_{λ_{0,x},λ_{1,x},λ_{x}} and Γ_{λ_{0,x},λ_{1,x},λ′_{x}} : [0, 1] → Π^{−1}(x) respectively joining λ_{0,x} to λ_{x} and λ_{0,x} to λ′_{x}. Let us denote by Γ_{x}(λ_{x}, λ′_{x}), the path connecting λ to λ′ obtained by concatenating Γ_{λ_{0,x},λ_{1,x},λ_{x}}^{−1} and Γ_{λ_{0,x},λ_{1,x},λ′_{x}}. The path Γ_{x}(λ_{x}, λ′_{x}) possesses the following two properties:

i) from Lemma 1.2, its intersection number with the Maslov cycle Λ^{𝔼}(Σ) of the section 𝔼 is bounded in norm by 2n;

ii) from Remark 1.3, it is transported by the flow ϕ^{t}. More precisely

dϕ_{x}^{t}(Γ_{x}(λ_{x}, λ′_{x})) = Γ_{ϕ^{t}(x)}(dϕ_{x}^{t}λ_{x}, dϕ_{x}^{t}λ′_{x}),

and consequently the intersection number of dϕ_{x}^{t}(Γ_{x}(λ_{x}, λ′_{x})) with the Maslov cycle Λ^{𝔼}(Σ) is also bounded in norm by 2n.

The 2-chain A : [0, 1] × [0, T] → Λ(Σ) defined by:

A(s, t) = dϕ_{x}^{t}(Γ_{x}(λ_{x}, λ′_{x})(s))

has a boundary:

Γ_{λ_{x},T} + dϕ_{x}^{T} (Γ_{x}(λ_{x}, λ′_{x})) − Γ_{λ′_{x},T} − Γ_{x}(λ_{x}, λ′_{x})

which is null homologous. Thus:

|n(Γ_{λ_{x},T}) − n(Γ_{λ′_{x},T})| ≤ |n(dϕ_{x}^{T} (Γ_{x}(λ_{x}, λ′_{x}))| + |n(Γ_{x}(λ_{x}, λ′_{x}))| ≤ 4n.

Notice that the bound in Lemma 2.1 can be improved with more sophisticated arguments. For instance a very elegant formulation is given in [4].

The asymptotic Maslov index of point x in Σ is the limit when it exists:

𝔐_{𝔼}(x) := lim_{T →+∞} \frac{1}{T} n_{𝔼}(Γ_{λ_{x},T}). (3)

It is clear from Lemma 2.1 that this limit is independent of the choice of the Lagrangian plane λ_{x} in the fiber Π^{−1}(x).

The flow ϕ^{t} induces a flow Φ^{t} in the Lagrangian Grassmann bundle defined for all x in Σ and for all λ_{x} in Π^{−1}(x) by:

Φ^{t}(x, λ_{x}) := (ϕ^{t}(x), dϕ^{t}_{x}(λ_{x})).

We denote by 𝕏 the vector field corresponding to the flow Φ^{t} and we say that the data (𝔼, 𝕏) satisfies the bounding condition if the map:

\begin{matrix}Λ(Σ) & → & ℝ \\ (x, λ_{x}) & ↦ & η_{𝔼}_{(x,λ_{x})}(𝕏(x, λ_{x})) \end{matrix}

is uniformly bounded on Λ(Σ).

From the path Γ_{λ_{x},T}, we construct a piecewise smooth loop Γ′_{λ_{x},T} = Γ_{0}∪Γ_{1}∪Γ_{2}∪Γ_{3} in Λ(Σ) as follows:

• Γ_{0} = Γ_{λ_{x},T};

• Γ_{1} is a path (as in Lemma 2.1) from Φ^{T} (x, λ_{x}) to (ϕ^{T} (x), J_{ϕ^{T} (x)}𝔼_{ϕ}^{T} (x)) in Π^{−1}(ϕ^{T} (x)) whose intersection number with the Maslov cycle Λ^{𝔼}(Σ) is bounded in norm by 2n;

• Γ_{2} is the path t ↦ J_{ϕ^{t}(x)}𝔼_{ϕ^{t}(x)} for t going from T to 0;

• Γ_{3} is a path (as in Lemma 2.1) from (x, J_{x}𝔼_{x}) to (x, λ_{x}) in Π^{−1}(x) whose intersection number with the Maslov cycle Λ^{𝔼}(Σ) is bounded in norm by 2n.

It is clear from the construction that the limit 𝔐_{𝔼}(x) when it exists, satisfies:

𝔐_{𝔼}(x) = lim_{T →+∞} \frac{1}{T} 𝔐_{𝔼}([Γ′_{λ_{x},T}]),

where [Γ′_{λ_{x},T}] stands for the homology class of the path Γ′_{λ_{x},T}. Using the integral version of the Maslov index, we have:

𝔐_{𝔼}([Γ′_{λ_{x},T}]) = ∫_{Γ_{0}} η_{𝔼} + ∫_{Γ_{1}} η_{𝔼} + ∫_{Γ_{2}} η_{𝔼} + ∫_{Γ_{3}} η_{𝔼}.

It is plain that ∫_{Γ_{2}} η_{𝔼} = 0 and that ∫_{Γ_{2}} η_{𝔼} and ∫_{Γ_{2}} η_{𝔼} are bounded in norm by 2n. It follows that the limit 𝔐_{𝔼}(x) when it exists, is also given by the average:

𝔐_{𝔼}(x) = lim_{T →+∞} \frac{1}{T} ∫_{0}^{T} η_{𝔼}(𝕏(Φ^{t}(x, λ_{x}))) dt.

2.2. Proposition. Let ν be an invariant probability measure of the flow ϕ^{t} with support in Σ and assume that the data (𝔼, 𝕏) satisfies the bounding condition. Then:

i) for ν-almost every x in Σ, the limit 𝔐_{𝔼}(x) exists; ii) the map x → 𝔐_{𝔼}(x) is integrable.

We denote by 𝔐_{𝔼}(ν) the integral ∫_{Σ} 𝔐_{𝔼}(x)dν(x) and call it the asymptotic Maslov index of the measure ν.

The proof of this proposition clearly rely on an ergodic theorem. The peculiarity here is that the probability measures we consider are invariant measures for the flow ϕ^{t} on Σ and the quantities we average are computed for the flow Φ^{t} on Λ(Σ). There are actually two ways to prove Proposition 2.2. One way consists in forgetting a part of the dynamics of the flow Φ^{t} by showing that the quantity n_{𝔼}(Γ_{λ_{x},T} ) up to a bounded error is independent of the Lagrangian plane λ_{x} and is quasi-additive. This allows us to define a bounded cocycle over Σ. The proof of Proposition 2.2 is then a direct application of the sub-additive ergodic theorem and can be found for instance in the work of Ruelle [17] and Barge and Ghys [4] (in a slightly different context). The second way, that we shall follow in a moment, consists in lifting the invariant measures for the flow ϕ^{t} to invariant measures for the flow Φ^{t} and then use the standard Birkhoff ergodic theorem. We shall see in the sequel that the advantage of this second approach is double:

• it gives a simple proof of the continuity of the asymptotic Maslov index with respect to the invariant measure for the weak^{∗} topology;

• it gives a better understanding of the dependence of the asymptotic Maslov index with respect to the section 𝔼.

Before proceeding to the proof and in order to understand better the way invariant measures of the flow ϕ^{t} can be lifted to invariant measures of the flow Φ^{t}, let us recall some basic definitions and results in ergodic theory.

Consider a metric space S, a family M of Borel probability measures on S is tight if for each ε > 0, there is a compact set K_{ε} ⊂ S for which P (K_{ε}) > 1 − ε for all P ∈ M. Tight families are characterized by the fact that their closures are compact in the weak* topology (see for instance [7]). This characterization will be used extensively in the proofs of Propositions 2.2 and 2.4.

Proof of proposition 2.2:

Given x ∈ Σ and T > 0, let ν_{x,T} be the equidistributed probability measure on the arc of ϕ^{t}-orbit joining x to ϕ^{T} (x). For an ergodic ϕ-invariant probability ν on Σ, regular points for ν are these points for which the measures ν_{x,T} converge weakly* to ν. If x is a regular point, the family M := {ν_{x,N}|n ∈ ℕ} is tight on Σ because its closure \overline{M} = M ∪ \{ν\} is compact. Choose λ_{x} ∈ Π^{−1}(x) and consider the probability measure μ_{λ_{x},T} which is equidistributed on the arc of Φ^{t}-orbit joining (x, λ_{x}) to Φ^{T} (x, λ_{x}). The pushforward by the projection Π of the measure μ_{λ_{x},T} coincides with the measures ν_{x,T}:

Π_{∗}μ_{λ_{x},T} = ν_{x,T} .

The compactness of the fibers of Π yields the tightness of the family {μ_{λ,N}|N ∈ ℕ}. Thus, there exists a convergent subsequence μ_{λ,N_{k}} →_{k} μ. It is plain to check that μ is invariant and Π_{∗}μ = ν. Since the data (𝔼, 𝕏) satisfies the bounding condition, we can apply the Birkhoff ergodic theorem to the dynamical system (Λ(Σ), μ, Φ^{t}) and the observable (x, λ_{x}) ↦ η_{𝔼_{x}}(𝕏(x, λ_{x})). This yields that for μ-a.e. (x, λ_{x}) in Λ(Σ) (i.e. for ν-a.e. x in Σ) the quantity \frac{1}{T} ∫_{0}^{T} η_{𝔼}(𝕏(Φ^{t}(x, λ_{x})))dt converges, when T goes to +∞ to a limit which does not depend on λ_{x} in Π^{−1}(x) and is a ν-integrable function of x in Σ.

2.3. Remark. The hypothesis that the data (𝔼, 𝕏) satisfies the bounding condition is actually too strong and the proof of Proposition 2.2 works if we require only the map (x, λ_{x}) ↦ η_{𝔼_{x}}(𝕏(x, λ_{x})) to be μ-integrable where μ is an invariant measure of Φ^{t} which is a lift of ν. However in the sequel we will see that it is convenient to keep a hypothesis which does not depend on the invariant measure we consider.

2.4. Proposition. The asymptotic Maslov index of a probability measure depends continuously on the invariant probability measure for the weak^{∗}-topology.

Proof: Suppose that ν_{N} is a sequence of ϕ^{t}-invariant probabilities on Σ with lim_{N} ν_{N} = ν. We have seen in the proof of Proposition 2.2 that any ϕ^{t}-invariant probability measure ν_{N} on Σ can be lifted to a Φ^{t}-invariant probability measure μ_{N} on Λ(Σ): Π_{∗}μ_{N} = ν_{N}. Since the fibers of Π are compact, the family {μ_{N}} is tight. Thus, given any subsequence ⟨N_{k}⟩, there exists a convergent subsequence ⟨N_{k_{ℓ}}⟩ such that the measures μ_{N_{k_{ℓ}}} converge when l goes to +∞ to a Φ^{t}-invariant measure μ, which satisfies Π_{∗}μ = ν. Since the data (𝔼, 𝕏) satisfies the bounding condition:

𝔐_{𝔼}(ν_{N_{k_{ℓ}}}) = ∫ η_{𝔼}(𝕏)dμ_{N_{k_{ℓ}}} →^{ℓ} ∫ η_{𝔼}(𝕏)dμ = 𝔐_{𝔼}(ν).

The subsequence ⟨N_{k}⟩ has been chosen arbitrarily and the limit 𝔐_{𝔼}(ν) does not depend on this subsequence, thus lim_{N→+∞} 𝔐_{𝔼}(ν_{N}) = 𝔐_{𝔼}(ν).

2.1. Change of reference section. Let us describe now how the asymptotic Maslov index behaves under a change of the section 𝔼. Let 𝔽 be another continuous section of Π : Λ(Σ) → Σ and assume that the data (𝔽, 𝕏) satisfies also the bounding condition. From (1), we get that the map:

\begin{matrix}Σ & → & ℝ \\ x & ↦ & 𝔽^{⋆}η_{𝔼}_{x}(𝕏(x)) \end{matrix}

is uniformly bounded on Σ. From the Birkhoff ergodic theorem, we immediately get:

𝔐_{𝔼}(ν) − 𝔐_{𝔽}(ν) = ∫ (𝔽^{∗}η_{𝔼})(X) dν . (4)

Notice that Equation (4) can be made more explicit when the Schwartzman asymptotic cycle of a ϕ^{t}-invariant measure ν [18, 19] can be computed. This is the case when the following two assertions^{3} are satisfied:

• the first homology group H_{1}(Σ, ℝ) has a finite dimension;

• the measure ν is X-tame i.e. the first de Rham cohomology group H^{1}(Σ, ℝ) is generated by a finite number of 1-forms ω_{i} such that for each i the map:

\begin{matrix}Σ & → & ℝ \\ x & ↦ & ω_{i}_{x}(X(x)). \end{matrix}

is ν-integrable.

3Notice that both assertions are satisfied when Σ is compact.

In this context, the asymptotic cycle of the measure ν is the unique element S(ν) in H_{1}(Σ, ℝ) = H^{1}(Σ, ℝ)^{∗}, which satisfies for any closed 1-form ζ:

⟨[ζ], S(ν)⟩ = ∫ ζ(X) dν ,

where [ζ] is the cohomology class ζ and the brackets ⟨ , ⟩ mean evaluation. The section 𝔽 read in the trivialization I_{𝔼} induces a map denoted I_{𝔼}−I_{𝔽} and defined by:

\begin{matrix}Σ & → & 𝕊^{1} \\ x & ↦ & det^{2}I_{𝔼}(x, 𝔽_{x}). \end{matrix}

Thus, the homotopy classes of trivializations are parametrized by the homotopy classes of maps from Σ to a circle: namely the cohomology group H^{1}(Σ, ℤ). We denote by [I_{𝔼} − I_{𝔽}] the cohomology class of this map. With these notations Equation (4) reads^{4}:

𝔐_{𝔼}(ν) − 𝔐_{𝔽}(ν) = ⟨[I_{𝔼} − I_{𝔽}], S(ν)⟩. (5)

If some Φ^{t}-invariant measure μ which is a lift of ν is 𝕏-tame, the asymptotic cycle S(μ) in H_{1}(Λ(Σ), ℝ) is well defined. The asymptotic cycle S(ν) in H_{1}(Σ, ℝ) and the Maslov index 𝔐_{𝔽}(ν) are also well defined (see Remark 2.3). When projecting onto Σ, we have:

Π_{⋆}S(μ) = S(ν).

When projecting onto Λ(n) through the trivialization I_{𝔼}, we get:

τ_{⋆} ∘ I_{𝔼}_{⋆}(S(μ)) = 𝔐_{𝔼}(ν).

Notice that the asymptotic Maslov index of a ϕ^{t}-invariant measure ν does not depend on the section in the following two cases:

• if H_{1}(Σ, ℝ) = 0;

• or if the Schwartzman asymptotic cycle of the measure ν is zero.

2.5. Remark. In the case of an odd dimensional manifold M (that we choose compact to make things simpler) equiped with a contact structure, there exists an analogous definition the asymptotic Maslov index. More precisely a contact structure is the data of a field of 1-form η such that the plane field corresponding to the kernels is nowhere integrable, i.e. for every x in M:

η_{x} ∧ dη_{x} ≠ 0.

It is clear that the 2-form dη restricted to ker η is symplectic. To a contact structure can be associated a vector field X whose flow ϕ^{t} is called the Reeb flow and which is defined by:

• i_{X}η = 1;

• i_{X}dη = 0.

4See [11] for a similar dependence on the trivialization in the case of non singular vector fields in a 3-dimensional manifold.

This flow preserves the planes ker η, the symplectic form dη on ker η and the volume form η ∧ dη. Assume there exists a continuous section which associates to any point x in M a Lagrangian plane 𝔼(x) in ker η_{x}. Then, it is possible to associate to each ϕ^{t}-invariant probability measure ν, its asymptotic Maslov index, in the same way we did for the even dimensional case. In particular, this can be done for the measure induced by the volume form η ∧ dη.

3. Hamiltonian flows.

Let N be a 2n-dimensional complete connected Riemannian manifold. A Hamiltonian on N is a C^{∞} function H : N → ℝ. The Hamiltonian vector field X of H is defined by

ω(X, ·) = −dH. (6)

The level sets of the Hamiltonian Σ = H^{−1}{e} are called the energy levels of H and are invariant under the flow of X. In the sequel of this paper we focus our attention on the dynamics on regular energy levels Σ, i.e. the energy levels which correspond to a regular value of H. Thus, Σ is a codimension 1 submanifold of N and X is a non singular vector field on Σ. Furthermore we shall always assume the following hypothesis:

(a) Completeness: The Hamiltonian vector field X gives rise to a complete flow ϕ : Σ × ℝ → Σ on the energy level Σ.

If the energy level Σ is compact, this last hypothesis is satisfied because the Hamiltonian vector field is Lipschitz.

3.1. The Asymptotic Maslov Index for the Liouville’s Measure.

The Hamiltonian flow on the energy level Σ inherits a canonical smooth invariant measure m̄, called the Liouville’s measure which correspond to the volume form m = i^{∗}σ, where σ is a form such that ω^{n} = dH ∧ σ and i : Σ ↪ T^{∗}M is the inclusion map [1].

3.1. Proposition.

If the energy level Σ is compact and the form ω^{n−1} is exact^{5}, then the asymptotic cycle S(m̄) of the Liouville’s measure m̄ is zero.

Proof: Let Y be a vector field Y ∈ T_{Σ}N such that ω(X, Y) ≡ 1. The sequence of equalities:

dH ∧ i_{Y} ω^{n} = +i_{Y} (dH ∧ ω^{n}) − (i_{Y} dH) ∧ ω^{n}
= 0 + (i_{Y} i_{X} ω) ω^{n}
= ω^{n},

5We shall discuss later on the strength of the hypothesis on the exactness of ω^{n−1}.

yields the explicit formulation:

m = i^{∗}(i_{Y} ω^{n}).

In order to prove that the asymptotic cycle is zero, it is enough to check that for any closed 1-form η on Σ such that x ↦ η_{x}(X(x)) is m̄-integrable:

⟨η, S(m̄)⟩ = ∫ η(X) dm̄ = 0 . (7)

We have:

η(X) m = i_{X}(η ∧ m) + η ∧ (i_{X}m) = η ∧ (i_{X}m) .

On the other hand:

i_{X}m = i_{X}(i^{∗}i_{Y} ω^{n}) = i^{∗}(n ω^{n−1}).

Since ω^{n−1} is exact, there exists a (2n − 1)-form, τ such that dτ = i^{∗}(n ω^{n−1}). Consequently

η(X) m = η ∧ dτ = d(η ∧ τ).

A direct application of Stoke’s theorem gives S(m̄) = 0.

3.2. Corollary.

If a regular energy level Σ is compact and ω^{n−1} is exact, then:

(i) for any two continuous Lagrangian sections 𝔼, 𝔽:

𝔐_{𝔼}(m̄) = 𝔐_{𝔽}(m̄);

(ii) the asymptotic Maslov index of the Liouville measure is invariant under symplectic conjugacies of the Hamiltonian flow.

Proof: (i) is a direct consequence of equation (5). In order to prove (ii), suppose that F : (Σ, ω) → (Σ′, ω′) is a symplectic diffeomorphism i.e. F^{⋆}ω′ = ω and let 𝔼 be a continuous Lagrangian section of Σ. A direct computation yields F_{⋆}m̄ = m̄′ where m̄ and m̄′ stand respectively for the Liouville measures on Σ and Σ′. The asymptotic Maslov index of the Liouville measure m̄ on the energy level Σ can be computed (thanks to (i)) by using any Lagrangian section, 𝔼 for instance.

It follows that 𝔐_{𝔼}(m̄) = 𝔐_{F_{⋆}𝔼}(F_{⋆}m̄) = 𝔐_{F_{⋆}𝔼}(m̄′) which is the asymptotic Maslov index of the Liouville measure m̄′ on the energy level Σ′ computed by using the Lagrangian section F_{⋆}𝔼.

4. Optical Hamiltonians.

Now we restrict our attention to Hamiltonian flows whose lift to Λ(Σ) cuts the Maslov cycle transversely with positive orientation. In the next two subsections, we relate the work of M. Bialy and L. Polterovich [5] with the asymptotic Maslov index.

4.1. Positive tangent vectors. Consider again the Lagrangian manifold Λ(n) introduced in Section 1, let us make more explicit here the tranverse orientation of the train of a given Lagrangian plane. For t in [−1, 1], let t ↦ λ(t) ∈ Λ(n) be a curve in Λ(n) passing through the Lagrangian plane λ(0) = λ. There exists a curve of symplectic automorphisms of ℝ^{2n}, t ↦ Sp(t) such that λ(t) = Sp(t)λ, Sp(0) = 𝕀. To the pair (λ, λ′(0)) (where λ′(0) stands for the tangent vector λ′(0) = \frac{dλ}{dt}|_{t=0} ∈ T_{λ}Λ(n)), we can associate the bilinear form β_{λ,λ′(0)} on λ:

(ξ, η) ⟼ ω(ξ, Sp′(0)η) for ξ, η ∈ λ . (8)

Notice that:

• this form is well defined: another choice of curve of symplectic transformations t ↦ S̃p(t) such that λ(t) = S̃p(t)λ satisfies S̃p(t) = Sp(t)Q(t), where Q(t) preserves λ and Q(0) = 𝕀, hence \frac{d}{dt}ω(ξ, Q(t)η) = ω(ξ, Q′(0)η), a quantity that vanishes since λ is Lagrangian;

• this form is symmetric since Sp(t) preserves the symplectic form ω;

• the map λ′(0) ↦ β_{λ,λ′(0)} is linear since it is the derivative of the function F (ℓ) = ω(ξ, Sp(ℓ)η).

Recall that the train Λ^{𝔼_{0}}(n) of a given Lagrangian plane 𝔼_{0} is the union for k ≥ 1 of the collection of open submanifolds Λ^{k} = { λ ∈ Λ(n) | dim(λ ∩ 𝔼_{0}) = k }. There exists a morphism from the normal bundle to Λ^{k}, T_{λ}Λ(n)/T_{λ}Λ^{k}, to the space S^{2}(λ ∩ 𝔼_{0}) of symmetric bilinear forms on λ ∩ 𝔼_{0} which is defined as follows: to each vector u + T_{λ}Λ^{k} we associate the restriction of the bilinear form β_{λ,u} to the subspace λ ∩ 𝔼_{0}. In order to check that this map is well defined it is enough to show that β_{λ,u}|_{λ∩𝔼_{0}} = 0 for u ∈ T_{α}Λ^{k} (remind that the map u ↦ β_{λ,u} is linear), which can be proved by using an arc of symplectic linear maps t ↦ Sp(t) such that Sp(0)λ = λ, Sp′(0)λ = u and Sp(t)(λ ∩ 𝔼_{0}) = λ(t) ∩ 𝔼_{0} for all t in [−1, 1]. Actually, this morphism is an isomorphism: it is injective since β_{λ,u}|_{λ∩𝔼_{0}} ≠ 0 when u is not in the symplectic orthogonal of λ ∩ 𝔼_{0} which contains λ and 𝔼_{0} and has dimension 2n − k, it is onto because the source and the target have same dimension.

We say that a vector u ∈ T_{z}Λ is positive if β_{λ,u}|_{λ∩𝔼_{0}} is positive definite whenever λ ∩ 𝔼_{0} ≠ {0}. A curve λ(t) ∈ Λ(n) is positive if its tangent vectors are positive. For a positive curve λ(t), the set of t′s for which λ(t) belongs to the train of 𝔼 is discrete; Indeed, if λ ∈ Λ^{k}, the fact that β_{λ,u}|_{λ∩𝔼_{0}} is positive definite implies that λ′(t) is not in the tangent space over λ of the closure of Λ^{p} for all 1 ≤ p ≤ k.

In particular, if λ ∈ Λ^{1} i.e. dim(λ ∩ 𝔼_{0}) = 1, S^{2}(λ ∩ 𝔼_{0}) ≈ T_{λ}Λ/T_{λ}Λ^{1} ≈ ℝ. A vector u in T_{λ}Λ(n) is transversal to Λ^{1} if and only if β_{λ,u}|_{λ∩𝔼} ≠ 0. This gives a transversal orientation on the cycle Λ^{𝔼_{0}}. Notice that for λ ∈ Λ^{1}, the curves t ↦ e^{it} 𝕀 λ are positive. Thus this orientation fits with on one given in Section 1.

It is fairly clear how to extend the above definitions to the case of a manifold N equipped with a symplectic structure, a connected submanifold Σ of N and a continuous section 𝔼 from Σ to the Grassman Lagrangian bundle.

4.2. Optical Hamiltonians. A Hamiltonian H : N → ℝ is optical or positively twisted with respect to a differentiable Lagrangian section 𝔼 on a regular energy level Σ if the flow lines of the lifted Hamiltonian flow Φ^{t} on Λ(Σ) are positive curves. That is, for any λ ∈ Λ^{𝔼}(Σ), the form

(ξ, η) ⟼^{β} ω(ξ, \frac{d}{dt}(Φ^{t}η)) ξ, η ∈ λ ∩ 𝔼,

restricted to λ ∩ 𝔼, is positive definite. Examples

1) A convex Hamiltonian in the cotangent bundle of a manifold equipped with is canonical symplectic structures is optical with respect to the vertical section (see Appendix A);

2) A Riemannian metric with a twisted symplectic structure is optical with respect to the vertical section (see Remark B.2);

3) A Riemannian metric with positive curvature is optical with respect to the section given by the kernel of the Riemannian connection i.e. the horizontal bundle;

4) C^{2}-perturbations of the above examples remain optical with respect to the perturbed sections.

From now on, we consider a regular energy level Σ of an optical Hamiltonian system H, on which the flow ϕ^{t} induced by the Hamiltonian is complete^{6}. In this case, it is still possible to define the asymptotic Maslov index 𝔐_{𝔼}(ν) for any ϕ^{t}- invariant probability measure ν on Σ. The Maslov cocycle 𝔐_{𝔼} measures the oriented intersection with the Maslov cycle Λ^{𝔼}. In order to evaluate the Maslov cocycle on a cycle in Λ(σ) we can use a 1-form ζ_{𝔼} (instead of η_{𝔼}) which is supported on a small neighborhood of Λ^{𝔼} and is cohomologous to η_{𝔼}. If (H, Σ, 𝔼) is optical, the form ζ_{𝔼} can be chosen such that ζ_{𝔼}(𝕏) ≥ 0 and ζ_{𝔼}(𝕏(x, λ_{x})) > 0 for all (x, λ_{x}) ∈ Λ^{𝔼}. Since this function is positive and ν-measurable, it is integrable and we can define

𝔐_{𝔼}(ν) := ∫ ζ_{𝔼}(𝕏) dμ ∈ [0, +∞] , (9)

for any invariant probability ν on Σ and any invariant lift μ of ν to Λ(Σ).

4.1. Remark. Notice that when the data (𝔼, 𝕏) satisfies the bounding condition (which occurs if the second derivatives of the Hamiltonian are uniformly bounded on Σ) this new definition fits with the previous one and the integral is finite.

We say that a curve ℓ(t) on Λ(Σ) is quasi-invariant if whenever ℓ(t_{0}) ∩ 𝔼 ≠ {0}, there exists δ > 0 such that ℓ(t_{0} + s) = Φ^{s}(ℓ(t_{0})) for |s| < δ. To a Lagrangian plane λ_{x} in the fiber Π^{−1}(x) of Λ(Σ) we associate the path: Γ_{λ_{x},T} : [0, T] → Λ(Σ) defined for any t ∈ [0, T] by:

Γ_{λ_{x},T} (t) = dϕ^{t}_{x}(λ_{x}).

6Notice that we do not require neither the vector field induced by ϕ^{t} nor its derivatives to be uniformly bounded and consequently the data (𝔼, 𝕏) does not necessarily satisfy the bounding condition.

Since H is optical, the set of intersection points of Γ_{λ_{x},T} with the Maslov cycle Λ^{𝔼} is discrete. If dim(Γ_{λ_{x},T} (t_{0})∩𝔼) = k ≥ 1, there exists ϵ > 0 and a quasi-invariant curve ℓ : [t_{0} − ϵ, t_{0} + ϵ] → Λ(Σ) homotopic with fixed endpoints, to the arc Γ_{λ_{x},T} |_{[t_{0}−ϵ,t_{0}+ϵ]} and such that dim(ℓ(t) ∩ 𝔼) ≤ 1 for all t. It follows that

n_{𝔼}(Γ_{λ_{x},T}) = ∑_{Γ_{λx,T} (t)∩𝔼≠{0}}dim(Γ_{λ_{x},T} (t) ∩ 𝔼). (10)

A probability measure μ is ergodic for a flow ψ^{t} if any measurable function invariant under ψ^{t} is μ-a.e constant. Ergodic measures play an central role in ergodic theory, in particular because of the ergodic decomposition theorem (see for instance [14]), that we briefly recall now. Let us consider for every point x in the ambient space, the probability measure μ(x, T) which is equidistributed along the arc of orbit joining x to ψ^{T} (x). It turns out that for μ-a.e. x, the probabilities μ(x, T) converges in the weak^{⋆} topology, to an ergodic invariant probability measure μ_{x}. Furthermore, we get a decomposition of the measure μ in the measures μ_{x} in the following sense: for any measurable function f bounded on the ambient space:

∫ fdμ = ∫ (∫ fdμ_{x}) dμ. (11)

The measure μ_{x} is the ergodic component measure of μ associated to x. Notice that when f is a measurable positive function, by approximating f from below by positive measurable bounded functions, 0 ≤ f_{n} ↑ f, Equation (11) remains true for f (even if both terms of the equation may be infinite).

Let us come back now to our particular situation and consider a Φ^{t}-invariant probability measure μ on Λ(Σ) which is a lift of a ϕ^{t}-invariant probability measure ν on Σ. It is clear that the pushforward by the projection Π of any ergodic component measure μ_{(x,λ_{x})} of μ, associated to a point (x, λ_{x}), is the ergodic component measure of ν associated to the point x. Applying Equation (11) to the map: (x, λ_{x}) ↦ (ζ_{𝔼})_{x}(𝕏(x, λ_{x})) we get for every ν probability measure on Σ:

𝔐_{𝔼}(ν) = ∫ 𝔐_{𝔼}(ν_{x}) dν. (12)

4.2. Lemma. If the invariant measure ν is ergodic, then for ν almost every point x in Σ and for every λ_{x} in Π^{−1}(x), we have:

lim_{T →+∞} \frac{1}{T} ∑_{Γ_{λx,T} (t)∩𝔼≠\{0\}}dim(Γ_{λ_{x},T} (t) ∩ 𝔼) = 𝔐_{𝔼}(ν).

Proof: When 𝔐_{𝔼}(ν) < +∞ the lemma is a direct application of the Birkhoff ergodic theorem (notice that any Φ^{t}-invariant measure μ which is a lift of an ergodic measure is also ergodic). When the integral in (9) is infinite the Birkhoff’s theorem still holds by approximating ζ_{𝔼}(𝕏) from below by positive functions with finite integral: 0 ≤ f_{n} ↑ ζ_{𝔼}(𝕏). For ν-a.e. x in Σ and for every λ_{x} in Π^{−1}(x), we have:

∫ f_{n} dμ = lim_{T →+∞} \frac{1}{T} ∫_{0}^{T} f_{n}(Ψ_{t}(x, λ_{x}))dt

≤ lim inf_{T →+∞} \frac{1}{T} ∫_{0}^{T} ζ_{𝔼}(𝕏(Ψ_{t}(x, λ_{x}))dt

≤ lim inf_{T →+∞} \frac{1}{T} ∑_{Γ_{λx,T} (t)∩𝔼≠\{0\}}dim(Γ_{λ_{x},T} (t) ∩ 𝔼)

Letting n → +∞ the left hand side tends to +∞ and we get:

lim_{T →+∞} \frac{1}{T} ∑_{Γ_{λx,T} (t)∩𝔼≠\{0\}}dim(Γ_{λ_{x},T} (t) ∩ 𝔼) = 𝔐_{𝔼}(ν) = +∞.

4.3. Conjugate points. Two points x_{1}, x_{2} ∈ T^{∗}M are said 𝔼-conjugate if x_{2} = ϕ^{τ}(x_{1}) and Φ^{τ}(𝔼(x_{1}))∩𝔼(x_{2}) ≠ {0} for some τ ≠ 0. The following theorem provides a simple criterium for the existence of conjugate points.

4.3. Theorem.

Let ν be an invariant probability measure on a regular energy level of an optical Hamiltonian (H, Σ, 𝔼) which satisfies the completeness hypothesis (a). Then 𝔐_{𝔼}(ν) > 0 if and only if there are 𝔼-conjugate points in the support of ν.

Proof: If the orbit of any point x in Σ has no conjugate points then for all T > 0: n_{𝔼}(Γ_{𝔼_{x},T}) = 0, and consequently by using Lemma 2.1: 0 ≤ n_{𝔼}(Γ_{λ_{x},T}) ≤ 4n for any Lagrangian plane λ_{x} in Π^{−1}(x). Hence:

lim_{T →+∞} \frac{1}{T} ∑_{Γ_{λx,T} (t)∩𝔼≠\{0\}}dim(Γ_{λ_{x},T} (t) ∩ 𝔼) = 0.

From Lemma 4.2 we get 𝔐_{𝔼}(ν_{x}) = 0, where ν_{x} is the ergodic component measure of μ associated to the point x. The decomposition (12) gives 𝔐(ν) = 0. Conversely, suppose that there exists x_{1} in the support of ν such that x_{2} = ϕ^{τ}(x_{1}) is conjugate to x_{1}. Since H is optical, for any small enough flow box U_{1} containing x_{1} and for every y_{1} in U_{1}, there exists a conjugate point to y_{1} on the arc of orbit issuing from y_{1} before the next return to U_{1}. Choose now an ergodic component measure ν_{x} of ν. By using the Birkhoff ergodic theorem, we know that for ν_{x}-a.e. y in Σ:

lim_{T →+∞} \frac{1}{T} θ(y, T, U_{1}) = ν_{x}(U_{1}),

where θ(y, T, U_{1}) is the time spent in U_{1} by the arc of orbit with length T issuing from y. By calling α the minimal time length of a connected component of an arc of orbit crossing U_{1}, we get:

lim inf_{T →+∞} \frac{1}{T} n(y, T, U_{1}) ≥ \frac{ν_{x}(U_{1})}{α},

where n(y, T, U_{1}) is the number of times the arc of orbit issuing from y of length T, visits U_{1}. Thus, for ν_{x}-a.e. y in Σ and for every λ_{y} in Π^{−1}(y):

lim inf_{T →+∞} \frac{1}{T} ∑_{Γ_{λy,T} (t)∩𝔼≠\{0\}}dim(Γ_{λ_{y},T} (t) ∩ 𝔼) ≥ \frac{ν_{x}(U_{1})}{α}.

Hence using Lemma 4.2:

𝔐_{𝔼}(ν_{x}) ≥ \frac{ν_{x}(U_{1})}{α}.

By integrating against ν, Equation (12) yields:

𝔐_{𝔼}(ν) ≥ \frac{ν(U_{1})}{α} > 0.

Observe that when the energy level is compact and ω^{n−1} is exact, by Corollary 3.2.(i), the Maslov index of the Liouville’s measure does not depend on the choice of Lagrangian section. Thus we get

4.4. Corollary. Let (H, Σ, 𝔼) be an optical Hamiltonian with Σ compact and ω^{n−1} an exact form. Then the asymptotic Maslov index of the Liouville’s measure is nonzero (𝔐(m̄) > 0) if and only if Σ has conjugate points with respect to 𝔼.

In fact, the same result holds for any invariant probability ν with supp(ν) = Σ.

We now give the first application of the average Maslov index.

4.5. Proposition. Let (M, g) and (N, h) be two closed Riemannian manifolds. Suppose that there exists a contactomorphism between the unit sphere bundle of M and the unit sphere bundle of N. Then 𝔐_{M}(m̄) = 𝔐_{N}(m̄).

Proof: It is well known that the geodesic flow of a Riemannian manifold is optical with respect to the vertical section which is given by the kernel of the differential of the projection map from the tangent bundle to the manifold (see Section 5).By Corollary 3.2 and Remark 2.5 if there exists a contactomorphism between the unit sphere bundle of M and the unit sphere bundle of N we have 𝔐_{M}(m̄) = 𝔐_{N}(m̄).

4.6. Remark. In particular M has conjugate points if and only if N does. This was proved by C. Croke and B. Kleiner [9] under the much weaker assumption of the existence of a C^{0}-conjugacy between geodesc flows. This naturally raises the question: is it true that the asymptotic Maslov index is an invariant of C^{0}- conjugacy?

We now give our main application of the asymptotic Maslov index.

4.7. Theorem. Let (H, Σ, 𝔼) be an optical Hamiltonian level with Σ compact and ω^{n−1} exact. Assume that e : Σ → Λ(Σ) is a continuous semi-conjugacy between the Hamiltonian flow ϕ^{t} on Σ and its lift Φ^{t} to Λ(Σ). Then (a) Σ has no conjugate points.

(b) e(z) ∩ 𝔼(Π e(z)) = {0} for all z ∈ Σ.

Mañé [13] gave a proof of this theorem in the particular case of recurrent geodesic flows and G. Paternain and M. Paternain [16] proved it for convex Hamiltonians.

Proof:

(a) Since e is a semi-conjugacy, then measure e_{∗}m̄ is a Φ^{t}-invariant lift of the Liouville measure m̄. Using (9), we have that

𝔐_{𝔼}(m̄) = ∫ ζ_{𝔼}(𝕏) d(e_{∗}m̄) = ∫ ζ_{𝔼}(𝕏 ∘ e) dm̄ = ⟨e^{∗}i^{∗}𝔐_{𝔼} , S(m̄)⟩ = 0,

where e^{∗}i^{∗}𝔐_{𝔼} is the cohomology class obtained from the maps induced by Σ →^{e} Λ(Σ) ↪^{i} Λ(Σ) in H^{1}(Σ, ℝ) and the last equality follows from Proposition 3.1. Corollary 4.4 yields the proof of (a).

(b) Suppose that e(z) ∩ 𝔼(z) ≠ {0} for some z ∈ Σ. Then there is a neighborhood U of z in Σ such that ζ_{𝔼}(𝕏 ∘ e) > 0 on U. Since m̄(U) > 0 and ζ_{𝔼}(𝕏 ∘ e) ≥ 0, then

𝔐_{𝕍}(m̄) ≥ ∫_{U} ζ_{𝕍}(𝕏 ∘ e) dm̄ > 0.

This contradicts (a).

When the energy level is non-compact the conclusion of Theorem 4.7 does not hold. An example is given by the geodesic flow of the paraboloid of revolution (cf. G. Paternain & M. Paternain [16]). In this case e = ⟨X⟩ ⊕ ⟨Y ⟩, is an invariant Lagrangian section, where X = \frac{∂}{∂t}ψ_{t} is the vector field of the geodesic flow and Y = \frac{∂}{∂t} dR_{t}, where R_{t} is the flow on the paraboloid given by rotation isometries. However this example has infinite Liouville measure. In Appendix B, we give a more elaborate example that shows that the compactness hypothesis in Theorem 4.7 cannot be replaced just by finite Liouville measure.

4.8. Remark. M. Bialy and L. Polterovich proved in [6] that if H : N → ℝ is proper,^{7} bounded from below and optical then H^{2n−2}(N , ℝ) = 0 for n ≥ 3 and therefore any closed 2n − 2 form is exact. This implies in particular the exactness of the form ω^{n−1}.

4.4. Anosov Flows. A complete flow ψ^{t} on a manifold V is Anosov if there exists a continuous splitting of the tangent space over each point x in V:

T V_{x} = E_{x}^{s} ⊕ E_{x}^{u} ⊕ Y_{x},

(where Y stands for the direction of the vector field induced by ψ^{t}) such that, for some Riemannian metric on V, ψ^{t}_{⋆}|_{E^{s}} (resp. ψ^{t}_{⋆}|_{E^{u}}) is uniformly exponentially contracting as t goes to +∞ (resp. −∞).

7In particular with compact energy levels.

Anosov flows possess a very rich and well understood dynamics and consequently, it is important to know whether the flow of an optical Hamiltonian which is complete on a regular energy level can be Anosov.

4.9. Lemma. Let (Σ, H, 𝔼) be an optical regular energy level with complete Hamiltonian flow ϕ^{t} and finite Liouville’s measure such that the flow (Σ, ϕ^{t}) is Anosov. Then, for all x in Σ, the spaces E_{x}^{cs} := E_{x}^{s} ⊕ Y_{x} and E^{cu} := E_{x}^{u} ⊕ Y_{x} are Lagrangian planes.

The proof is already standard in slightly different contexts. We give it here for sake of completeness and to make it fit with our hypotheses.

Proof: Since the sum of the dimensions of E^{cs} and E^{cu} is 2n, it is enough to prove that both spaces are isotropic. Actually, from the definition of the Hamiltonian vector field (6), we only have to prove that E_{x}^{s} and E_{x}^{u} are isotropic. Choose a neighborhood U of the point x in Σ whose closure Ū is compact. Recall that since the Liouville measure is finite m̄-a.e. point is recurrent. Thus, it is possible to find a recurrent point z in U. We consider a sequence of times (t_{n}) such that t_{n} goes to +∞ with n and ϕ^{t_{n}}(z) is in U. Choose now two vectors v_{1} and v_{2} in E_{z}^{s}. The quantity ω_{ϕ^{tn}(z)}(dϕ^{t_{n}}_{z}(v_{1}), dϕ^{t_{n}}_{z}(v_{2})) decreases in norm at least exponentially at n goes to +∞ (since ω is bounded on Ū). On the other hand the same quantity is independent on n since ϕ^{t} preserves ω. Thus ω_{z}(v_{1}, v_{2}) = 0. A similar argument works for E_{z}^{u}. Since recurrent points are dense we get that E_{x}^{s} and E_{x}^{u} are isotropic for all x in Σ.

Lemma 4.9 shows that a prerequisite to find a flow ϕ^{t} of an optical Hamiltonian, complete and Anosov on a regular energy level Σ, is to find a continuous dϕ^{t}-invariant Lagrangian section e. Theorem 4.7 shows some clear obstructions to the existence of such sections.

4.10. Corollary.

Let (Σ, H, 𝔼) be an optical Hamiltonian with Σ compact and ω^{n−1} exact and such that (Σ, ϕ^{t}) is Anosov. Then E^{s} and E^{u} are transversal to 𝔼 and Σ has no conjugate points.

Corollary 4.10 was proved by W. Klingenberg [12] for geodesic flows on compact manifolds; by R. Mañé [13] for geodesic flows with dense non-wandering set; and by G. Paternain and M. Paternain [16] for convex Hamiltonians.

Appendix A.

Convex Hamiltonians.

An important special class of optical Hamiltonians is given by convex Hamiltonians on the cotangent bundle T^{∗}M.

Let M be a n-dimensional, connected manifold without boundary, N = T^{⋆}M its cotangent bundle and π : N → M the standard projection. The cotangent bundle is equipped with a canonical 1-form, called the Liouville’s 1-form, defined by:

Θ_{p}(ζ) := p(dπ(ζ)),

for all p ∈ T^{∗}M and ζ ∈ T_{p}T^{∗}M. The 2-form ω = dΘ is a symplectic form on N^{8}. The Lagrangian Grassmann bundle Λ(N) is also equipped with a canonical smooth section 𝕍 = ker dπ, called the vertical section.

A Hamiltonian H : T^{∗}M → ℝ is convex if 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. Convex Hamiltonian systems play a central role in physics and have been extensively studied.

A.1. Lemma ([5]).

A convex Hamiltonian is optical with respect to the vertical section 𝕍.

Proof: Locally (T^{∗}M, ω, 𝕍) can be identified with (ℝ^{2n}, dp∧dq, q = 0). We consider a Lagrangian plane in the train of the plane q = 0 and in order to fix the notations, we assume that dim(λ ∩ 𝕍) = k (notice that it is not restrictive to assume also that λ ∩ [p = 0] = {0}). As we already saw in Section 1, λ, written in coordinates is a graph: λ = { q = Ap | p ∈ ℝ^{n} }, where A is a (symmetric) linear map, A : ℝ^{n} → ℝ^{n}. Then λ ∩ 𝕍 = { (q, p) | q = 0, p ∈ ker A }. Consider now a curve t ↦ λ(t) passing through λ at t = 0 and write λ(t) = S(t)λ with

S(t) = [\begin{matrix}a(t) & b(t) \\ c(t) & d(t) \end{matrix}] .

The symmetric two-form β_{λ,λ′(0)}|_{λ∩𝕍} reads:

β_{λ,λ′(0)}|_{λ∩𝕍}(ξ, ξ) = p ∗ (ḃ(0)p),

where p ∈ KerA is the unique vector in ℝ^{n} such that ξ = (0, p) and p∗ stands for the dual form associated to p.

The Hamiltonian vector field (6) is given by X = (H_{p}, −H_{q}) = (q̇, ṗ). From the equation \frac{d}{dt} dψ_{t} = DX · dψ_{t}, we get that the orbit of the Lagrangian plane λ under the lifted Hamiltonian flow is a curve t ↦ S(t)λ passing through λ at t = 0 such that:

S′(t) = DX = [\begin{matrix}H_{pq} & H_{pp} \\ −H_{qq} & −H_{qp} \end{matrix}] .

In particular ḃ(0) = H_{pp} is positive definite.

A.2. Remark. Notice that our definition of 𝕍-conjugate points coincides with the usual definition of conjugate points for convex Hamiltonians and also with the more standard definition that uses Jacobi fields.

8In this situation, ω^{n−1} is exact.

In the case of convex Hamiltonians, Theorem 4.7 can be improved as follows:

A.3. Corollary. Assume that ϕ^{t} is a convex Hamiltonian flow on a regular energy level Σ ⊂ T^{∗}M with Σ compact. If there is a continuous semiconjugacy e : Σ → Λ(Σ) between ϕ^{t} and its lift Φ^{t} = dϕ^{t} to Λ(Σ), then

(a) Σ has no conjugate points.

(b) e(z) ∩ 𝕍(Π e(z)) = {0} for all z ∈ Σ.

(c) π(Σ) = M.

The fact that item c) is a supplementary condition to get an Anosov flow on energy levels has been first observed in [16].

Proof: We only have to prove item (c). We have to show that the map π : Σ → M is a submersion since in such a case, π(Σ) is open and closed on M (which is connected) and then equal to M. Note that the existence of a continuous dϕ^{t}- invariant Lagrangian section 𝔼 implies the existence of a continuous dϕ^{t}-invariant Lagrangian section 𝔽 such that for all x in Σ, 𝔽_{x} ⊂ T Σ_{x}. Indeed, this section 𝔽 is defined by 𝔽_{x} = 𝔼_{x} ∩ T Σ_{x} ⊕ Y (x) where Y (x) is the direction in T Σ_{x} tangent to the flow ϕ^{t}. Let x be a point in Σ where π is not a submersion. Then 𝕍_{x_{0}} ⊂ T Σ_{x_{0}} and consequently:

dim 𝕍_{x_{0}} ∩ 𝔽_{x_{0}} ≥ 1,

a contradiction with (b).

Appendix B.

The aim of this appendix is to prove the following result:

B.1. Theorem. There exists a convex Hamiltonian on a surface with a regular energy level that satisfies the following properties:

(a) the flow is complete on this energy level;

(b) the Liouville measure of the energy level is finite;

(c) there exists a smooth invariant Lagrangian section;

(d) every orbit in the energy level has conjugate points.

We know from Theorem 4.7 that if a compact energy level of a convex Hamiltonian admits a continuous invariant Lagrangian section, then there are no conjugate points in the level. On the other hand, it is also known ( see [16]) that this result extends to a special class of Hamiltonians (namely the symmetric Hamiltonians) when the energy level is not compact but has a finite Liouville measure. Theorem B.1 shows that the symmetry in the Hamiltonian cannot be removed.

B.1. Magnetic flows in general. Let M^{n} be a closed n-dimensional manifold endowed with a C^{∞} Riemannian metric g, and let π : T M → M be the canonical projection. The symplectic form on T M obtained by pulling back the canonical symplectic form of T^{∗}M via the Riemannian metric is denoted by ω_{0}. Consider a closed 2-form Ω of M and the new symplectic form ω_{1} defined by:

ω_{1} =^{def} ω_{0} + π^{∗}Ω.

The 2-form ω_{1} is a symplectic form. We say that it defines a twisted symplectic structure.

Let E : T M → ℝ be given by

E(x, v) = \frac{1}{2} g_{x}(v, v).

The Hamiltonian flow of E with respect to ω_{1} models the motion of a particle of unit mass and charge under the effect of a magnetic field, whose Lorentz force Y : T M → T M is the bundle map determined by:

Ω_{x}(u, v) = g_{x}(Y_{x}(u), v),

for all x ∈ M and all u and v in T_{x}M. In other words, the curve

t ↦ (γ(t), γ̇(t)) ∈ T M

is an orbit of the Hamiltonian flow iff

\frac{Dγ̇}{dt} = Y_{γ}(γ̇), (13)

where D stands for the covariant derivative of g. The Hamiltonian flow of E with respect to ω_{1} leaves the unit sphere bundle SM = E^{−1}(1/2) invariant and therefore it defines a flow

ϕ^{t} : SM → SM,

that we call the magnetic flow of the pair (g, Ω). The magnetic flow of the pair (g, 0) is the geodesic flow of the Riemannian metric g. A curve γ that satisfies (13) will be called a magnetic geodesic.

B.2. Remark. The magnetic flow is locally conjugate to the Lagrangian flow of the Lagrangian L(x, v) = \frac{1}{2} ⟨v, v⟩_{x}+η_{x}(v), where η_{x}(v) is a local 1-form such that dη = Ω. The conjugacy preserves the vertical section. This Lagrangian flow, in turn, is conjugate to the Hamiltonian flow of the convex Hamiltonian H(x, p) = \frac{1}{2} ∥p − η_{x}∥_{x}^{2}, in T^{∗}M with the canonical symplectic structure. In particular the twisted geodesic flow is optical with respect to the vertical section. A computation also shows that the Liouville measure on a regular energy level of the twisted geodesic flow coincides with the Liouville measure on a regular energy level of the geodesic flow itself.

B.2. Magnetic flows for constant fields on rotationally symmetric surfaces. Let M be an oriented surface. Given (x, v) ∈ SM, let iv be the unique vector in T_{x}M such that {v, iv} is a positively oriented orthonormal basis of T_{x}M. The area form Ω is given by

Ω_{x}(u, v) = g_{x}(iu, v),

hence the Lorentz force Y is given by

Y_{x}(v) = iv.

Define the λ-magnetic flow as the magnetic flow of (g, λ Ω). It follows from equation (13) that t ↦ γ(t) is a λ-magnetic geodesic iff:

\frac{Dγ̇}{dt} = λ iγ̇.

In other words, γ is a λ-magnetic geodesic iff γ has constant geodesic curvature λ.

Suppose now that M = ℝ×S^{1} is a rotationally symmetric surface, i.e. if (s, φ) are the obvious coordinates in M, then the Riemannian metric of M in these coordinates has the expression:

g = ds^{2} + r(s)^{2}dφ^{2},

where r : ℝ → (0, ∞) is a smooth function.

We orient S^{1} counterclockwise and we give M the product orientation. This means that

\{\frac{∂}{∂s}, \frac{∂}{∂φ}\}

is a positively oriented basis of M and hence the area form Ω is given by

Ω = r ds ∧ dφ.

Define

R(s) := ∫_{0}^{s} r(u) du.

B.3. Lemma. A curve t ↦ (s(t), φ(t)) is a λ-magnetic geodesic iff

s̈ = rφ̇ (r′φ̇ − λ)

\frac{d}{dt} (r^{2}φ̇ − λ R) = 0,

where dot indicates derivative with respect to t and prime indicates derivative with respect to the s-parameter. In particular the quantity r(s)^{2}φ̇ − λ R(s) is a first integral of the flow called the Clairaut integral.

Proof: Let us consider the Lagrangian:

L(s, φ, ṡ, φ̇) = \frac{1}{2} (ṡ^{2} + r(s)^{2}φ̇^{2}) − λ R(s) φ̇.

Note that

d(R dφ) = r ds ∧ dφ = Ω.

Hence the extremals of L are the λ-magnetic geodesics. The lemma follows from a simple computation derived from the Euler-Lagrange equations:

\frac{d}{dt} (\frac{∂L}{∂ṡ}) − \frac{∂L}{∂s} = 0

\frac{d}{dt} (\frac{∂L}{∂φ̇}) − \frac{∂L}{∂φ} = 0.

B.4. Lemma.

The parallel t ↦ (s_{0}, at) (a ≠ 0) is a unit speed λ-magnetic geodesic iff

|a^{−1}| = r(s_{0})

\frac{r′(s_{0})}{r(s_{0})} = ±λ,

where the positive sign holds if a > 0 and the negative signs holds if a < 0.

Proof: The parallel has unit speed iff r^{2}(s_{0}) a^{2} = 1. Using Lemma B.3 we see that the parallel is a λ-magnetic geodesic iff r′(s_{0}) a = λ.

Let θ ∈ [0, π] be the angle that a unit speed λ-magnetic geodesic makes with the parallel t ↦ (s, t/r(s)) at the point (s, φ). We have:

r cos θ = r^{2} φ̇,

and therefore the Clairaut integral from Lemma B.3 reads:

r cos θ − λ R = c, (14)

where c is a constant that depends only on the λ-magnetic geodesic.

Define

f_{+,λ}(s) := r(s) − λ R(s),
f_{−,λ}(s) := −r(s) − λ R(s),

B.5. Lemma. Suppose that there exists a λ-magnetic geodesic for which θ = 0 at values r(s_{1}) and r(s_{2}) with s_{1} < s_{2}. Then there exists s_{0} ∈ (s_{1}, s_{2}) on which f_{+,λ} has a local maximum and such that the parallel t ↦ (s_{0}, t/r(s_{0})) is a λ-magnetic geodesic.

Similarly, suppose that there exists a λ-magnetic geodesic for which θ = π at values r(s_{1}) and r(s_{2}) with s_{1} < s_{2}. Then there exists s_{0} ∈ (s_{1}, s_{2}) on which f_{−,λ} has a local minimum and such that the parallel t ↦ (s_{0}, −t/r(s_{0})) is a λ-magnetic geodesic.

Proof: We only give here a proof of the first statement (the proof of the second one is completely analogous). If for all s ∈ (s_{1}, s_{2}) the λ-magnetic geodesic is always tangent to the parallel at s the parallel is an integral curve and the lemma is proved. If not, there exists s_{3} ∈ (s_{1}, s_{2}) such that the angle θ_{3} at which the λ-magnetic geodesic crosses the parallel at s_{3} is not zero and then cos θ_{3} < 1. Since r(s_{3}) > 0, by using the Clairaut integral (Lemma B.3) we get:

f_{+,λ}(s_{3}) > f_{+,λ}(s_{1}) = f_{+,λ}(s_{2}).

Consequently there exists s_{0} ∈ (s_{1}, s_{2}) such that f_{+,λ} presents a local maximum at s_{0}. But

f′_{+,λ}(s) = r′(s) − λ r(s) = r(s) (\frac{r′(s)}{r(s)} − λ) .

Hence f′_{+,λ}(s_{0}) = 0 iff \frac{r′(s_{0})}{r(s_{0})} = λ. By Lemma B.4 this implies that the parallel t ↦ (s_{0}, t/r(s_{0})) is a λ-magnetic geodesic.

B.3. Proof of Theorem B.1. The example that achieves the properties described in Theorem B.1, is choosen among λ-magnetic flows associated to rotationally symmetric surfaces. Let us first construct the rotationally symmetric surface. Consider a smooth function u : ℝ → ℝ with the following properties:

1. u is odd i.e. u(s) = −u(−s) for all s ∈ ℝ;

2. For all s ∈ ℝ, −1 < u(s) < 1.

3. For all s > 3, u(s) = −2/s.

Now let r(s) : ℝ → (0, ∞) be defined by

\frac{r′(s)}{r(s)} = u(s),

r(0) = r_{0} > 0.

In other words,

r(s) = r_{0} exp (∫_{0}^{s} u(t) dt) .

Observe that r(s) is an even function of s and that for s > 3 we have

r(s) = r(3) 9/s^{2}. (15)

Let M = ℝ × S^{1} be the rotationally symmetric surface determined by such a function s ↦ r(s). The estimate (15) implies that the area of M is finite:

∫_{M} Ω = 2π ∫_{−∞}^{+∞} r(s) ds = 4π R(+∞).

Consequently (using Remark B.2) the Liouville measure of the λ-magnetic flow on the unit tangent bundle SM is also finite. This ensures item (b) of Theorem B.1.

We consider now the λ-magnetic flow on SM which corresponds to λ = 1 and prove items (a), (c) and (d) for this particular flow. Since the vector field generated by the circle action is tangent to the energy levels, above each point in SM the vector space generated by the magnetic vector field and the vector field generated by the circle action is isotropic. Since u(s) ∈ (−1, 1), Lemma B.4 ensures that no parallel is a magnetic geodesic. In other words, no magnetic geodesic is an orbit of the circle action and consequently we have constructed an invariant Lagrangian section E spanned by the magnetic vector field and the lift of the circle action that generates the symmetry. This ensures item (c) of Theorem B.1. Observe that the Clairaut integral C : SM → ℝ satisfies

E(x, v) = ker d_{(x,v)}C

for all points (x, v) in SM and consequently C does not have critical points. Notice that

f_{−,1}(s) ≤ C(s, φ, ṡ, φ̇) ≤ f_{+,1}(s),

for all s ∈ ℝ and that f_{−,1} and f_{+,1} are strictly decreasing functions which have the same finite limit when s → +∞ (resp. when s → −∞). Hence the projection to M of any level set of C has to be contained in a compact set. Therefore this level set is compact and hence its connected components are finitely many tori. In conclusion the energy level SM is foliated by tori on which the Clairaut integral is constant. This implies in particular that the magnetic vector field is complete: the item (a) of Theorem B.1.

Actually it is possible to give a better visualization of the dynamical behavior of the magnetic flow: The connected components of the level sets of the Clairaut integral C are 2-tori that project on M onto strips bounded by two parallels. The magnetic geodesics on these tori oscillate between these two parallels. The difference with the case of the geodesic flow of a surface of revolution comes from the fact that in our example a parallel cannot be a magnetic geodesic. It follows from Lemma B.5 that a magnetic geodesic makes an angle θ = 0 with the bottom parallel and an angle θ = π with the top one. In other words it “curls” like in Figure B.3 below.

Figure 1. The figure on the left hand side shows a typical geodesic while the figure on the right hand side shows a typical magnetic geodesic that “curls”.

Let us prove now item (d) of Theorem B.1. More precisely let us show that every magnetic geodesic has conjugate points. Consider a magnetic geodesic. The invariant section E (spanned by the magnetic vector field and the vector field induced by the circle action) intersects non trivially the vertical fibers above the points of the magnetic geodesic at which the latter is tangent to the parallels. Consequently, by Lemma 2.1 the magnetic geodesic has conjugate points.

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