Mélanie Bertelson

Mihai Damian

Samuel Lisi

Michael Hutchings

Jungsoo Kang

Yiming Long

Maksim Maydanskiy

Vinicius Ramos

Keywords:

.. .. .. .

Prolegomenon

As the title indicates, periodic Reeb orbits are at the centre of this work. The general framework is contact geometry. A contact structure on a manifold M of dimension 2n -1 is a hyperplane field ξ which is maximally non integrable; i.e if we write locally ξ = ker α where α is a differential 1-form, then α ∧ (dα) n-1 = 0 everywhere. If α is globally defined, we say that ξ is coorientable and we call α a contact form. In this thesis we consider only coorientable contact structures. The 1-form α is not unique; for any function f : M → R, the 1-form e f α defines the same contact structure. The Reeb vector field R α associated to a contact form α is the unique vector field on M characterized by: ι(R α )dα = 0 and α(R α ) = 1. Since this vector field does not vanish anywhere, there are no fixed points of its flow. Periodic orbits are thus the most noticeable objects in the flow. Poincaré pointed out their interest in his "traité de la mécanique céleste": "Ce qui nous rend ces solutions périodiques si précieuses, c'est qu'elles sont, pour ainsi dire, la seule brèche par où nous puissions essayer de pénétrer dans une place jusqu'ici réputée inabordable." Does there exist a periodic Reeb orbit for any contact form on any manifold?

The answer to this question is negative as shown by the following example. Let R 2n-1 be endowed with the "standard" contact form α = dz -n-1 i=1

x i dy i .

The Reeb vector field is given by R α = ∂ z , and there are no periodic Reeb orbits.

The same question for a compact oriented manifold endowed with a contact 1-form is still open. It is called the Weinstein conjecture. Some partial results are known. Taubes answered positively this question for manifolds in dimension 3. Theorem 0.0.1 ( [START_REF] Henry | The Seiberg-Witten equations and the Weinstein conjecture[END_REF]) If Y is a closed oriented three-manifold with a contact form, then the associated Reeb vector field has a closed orbit.

In higher dimensions, only partial results have been obtained. Let us mention a first result due to Viterbo:

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Theorem 0.0.2 ( [START_REF] Viterbo | A proof of Weinstein's conjecture in R 2n[END_REF]) If Σ ⊂ (R 2n , ω 0 ) is a compact hypersurface of "contact type", then Σ has at least one periodic Reeb orbit.

More recently Albers and Hofer proved:

Theorem 0.0.3 ( [START_REF] Albers | On the Weinstein conjecture in higher dimensions[END_REF]) Let (M, ξ) be a closed "PS-overtwisted1 " contact manifold. Then the Reeb vector field associated to any contact form α inducing ξ has a contractible periodic orbit.

One could also quote many particular results related for example to Hamiltonian dynamics.

Another natural question is to ask how many periodic Reeb orbits can exist. Since any periodic orbit can be iterated any number of times, it is more reasonable to ask how many geometrically distinct periodic orbits can exist. Consider the sphere S 2n-1 , naturally embedded in R 2n , endowed with the standard contact structure,

α = α std S 2n-1 = 1 2 n i=1 (x i dy i -y i dx i )) S 2n-1 .
The Reeb vector field is R α = n i=1 2(x i ∂ y iy i ∂ x i ) and all Reeb orbits are periodic. On the other hand, if we look on the same sphere at the "deformed" contact form

α ′ := 1 2 n i=1 a i (x i dy i -y i dx i ) S 2n-1 ,
where all the a i are rationally independent, the Reeb vector field is R α ′ = n i=1 2 a i (x i ∂ y iy i ∂ x i ). There are only n distinct periodic Reeb orbits, one in each "coordinate plane". Now the contact forms α and α ′ on the sphere are isotopic (i.e. there exists a smooth path of contact forms on the sphere joining them). Gray's stability theorem asserts that there exists a diffeomorphism of the sphere, which maps the contact structure ξ ′ = ker α ′ to the contact structure ξ = ker α. Hence, depending on the contact form defining the standard contact structure on the sphere, one can get different answers concerning the number of distinct periodic orbits. In view of this fact, a natural question is:

If (M, ξ) is a compact contact manifold, can one say something about the minimal number of geometrically distinct periodic Reeb orbits for any contact form α (eventually in a subclass) defining the contact structure ξ?

In particular what is the answer for the sphere? Some results are known in this case; in particular, it was solved in dimension 3 by Hofer, Wysocki and Zehnder for the class of "dynamically convex" contact forms.

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2. H * (W 0 , M ; Q) = 0 for all even degree * ≤ 2n -4 where W 0 is the relatively compact domain of W bounded by M . Then there are at least two periodic Reeb orbits contractible in W for any nondegenerate contact form α on (M, ξ) such that αλ | M is exact.

In this thesis I present new results on the minimal number of periodic Reeb orbits on some contact type hypersurfaces in negative line bundles (Proposition 0.0.15 and Theorem 0.0.16).

Strategy followed in this thesis

The problem of finding periodic Reeb orbits on a contact manifold which is embedded in a symplectic manifold can often be translated into the problem of finding periodic orbits of a Hamiltonian vector field on a prescribed energy level. For instance, if C is a starshaped domain in R 2n such that 0 ∈ Int C, finding periodic orbits of the Reeb vector field on the boundary of C (for the standard contact form α std C ) amounts to finding periodic orbits of the Hamiltonian vector field defined by a power of the gauge function, on the boundary of C which is a level set of this Hamiltonian. Indeed, the gauge function of C, j C : R 2n → [0, ∞) is defined by j C (x) := min λ| x λ ∈ C and the Hamiltonian vector field associated to

H β = j C (x) β is X H β = β 2 R α std C
. The first idea to tackle the question of the minimal number of periodic Reeb orbits was to use a homological invariant of the contact structure, constructed from periodic Reeb orbits. To build such an invariant is the aim of contact homology. At the time of this writing, contact homology is still in development and encounters "transversality" problems. Instead we consider positive S 1 -equivariant symplectic homology which is built from periodic orbits of Hamiltonian vector fields in a symplectic manifold whose boundary is the given contact manifold. In this spirit, Bourgeois and Oancea, in [START_REF] Bourgeois | S 1 -equivariant symplectic homology and linearized contact homology[END_REF], relate, in the case where it could be defined, the linearised contact homology of the boundary to the positive S 1 -equivariant symplectic homology of the symplectic manifold. The positive S 1 -equivariant symplectic homology is one of the main objects considered in this thesis.

Our first aim is to analyse the relation between the symplectic homologies of an exact compact symplectic manifold with contact type boundary (also called Liouville domain) and the periodic Reeb orbits on the boundary. The next point is to prove some properties of these homologies. For a Liouville domain embedded into another one, we construct a morphism between their homologies. We study the invariance of these homologies with respect to the choice of the contact form on the boundary. Finally, we use the positive S 1 -equivariant symplectic homology to give a new proof of Theorem 0.0.7 and see how it can extend to the framework of hypersurfaces in negative line bundles.

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Another approach to solve the question of the minimal number of periodic Reeb orbits on hypersurfaces in R 2n , developed by Long, uses variational methods and a thorough study of their Conley-Zehnder index. With this in mind, we study the generalisation of the Conley-Zehnder index defined for any path of symplectic matrices. This led us to analyse in details normal forms of symplectic matrices. Those results could be useful to study degenerate orbits.

Content of the thesis and statements of the results

The first four chapters develop the approach using positive S 1 -equivariant symplectic homology.

In Chapter 1, we recall the definition of positive S 1 -equivariant symplectic homology, first describing Floer homology (Section 1.1), symplectic homology (Section 1.2.2), positive symplectic homology (Section 1.2.4), then recalling two equivalent definitions of S 1 -equivariant symplectic homology in Sections 1.3.2 and 1.4.

The link between the generators of positive S 1 -equivariant symplectic homology and periodic Reeb orbits is explained in Chapter 2. The explicit computation gives the following result:

Theorem 0.0.10 Let (W, λ) be a Liouville domain. Assume there exists a contact form α on the boundary ∂W such that the Conley-Zehnder index of all periodic Reeb orbits have the same parity. Then

SH S 1 ,+ (W, Q) = γ∈P(Rα)
Q γ where P(R α ) denotes the set of periodic Reeb orbits on ∂W .

In Chapter 3, we show that positive S 1 -equivariant symplectic homology has good functorial properties. In the first section, we construct a "transfer morphism" between all the above mentioned variants of symplectic homology in the case of two Liouville domains embedded one into the other. This construction generalises a construction given by Viterbo [START_REF] Viterbo | Functors and computations in Floer homology with applications[END_REF]). We prove that this morphism has nice composition properties:

Theorem 0.0.11 Let (V 1 , λ V 1 ) ⊆ (V 2 , λ V 2 ) ⊆ (V 3 , λ V 3
) be Liouville domains with Liouville embeddings. Then the following diagram commutes:

SH † (V 3 , λ V 3 ) φ † V 3 ,V 2 / / φ † V 3 ,V 1 6 6 SH † (V 2 , λ V 2 ) φ † V 2 ,V 1 / / SH † (V 1 , λ V 1 )
where † can be any of the following symbol: ∅, +, S 1 , (S 1 , +).

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where SH denotes the symplectic homology, SH + , the positive symplectic homology, SH S 1 the S 1 -equivariant symplectic homology and SH S 1 ,+ the positive S 1 -equivariant symplectic homology.

The second section of Chapter 3 is dedicated to the invariance of the different variants of symplectic homology. In particular, we prove Theorem 0.0.12 Let (W 0 , λ 0 ) and (W 1 , λ 1 ) be two Liouville manifolds 6 of finite type such that there exists a symplectomorphism f : (W 0 , λ 0 ) → (W 1 , λ 1 ). Then SH † (W 0 , λ 0 ) ∼ = SH † (W 1 , λ 1 ).

Theorem 0.0.13 Let (M 0 , ξ 0 ) and (M 1 , ξ 1 ) be two contact manifolds that are exactly fillable; i.e. there exist Liouville domains (W 0 , λ 0 ) and (W 1 , λ 1 ) such that ∂W 0 = M 0 , ξ 0 = ker(λ 0| M 0 ), ∂W 1 = M 1 and ξ 1 = ker(λ 1| M 1 ). Assume there exists a contactomorphism ϕ : (M 0 , ξ 0 ) → (M 1 , ξ 1 ). Assume moreover that ξ 0 admits a contact form α 0 such that all periodic Reeb orbits are nondegenerate and their Conley-Zehnder indices have all the same parity. Then SH S 1 ,+ (W 0 , λ 0 ) ∼ = SH S 1 ,+ (W 1 , λ 1 ).

This Theorem, together with Theorem 0.0.10 reproves Ustilovsky's result on the existence of non diffeomorphic contact structures on the spheres S 4m+1 . The original proof depends on a theory of cylindrical contact homology, which is not yet rigorously established due to transversality problems.

Theorem 0.0.14 ( [START_REF] Ustilovsky | Infinitely many contact structures on S 4m+1[END_REF]) For each natural number m, there exist infinitely many pairwise non isomorphic contact structures on S 4m+1 .

In Chapter 4 we use positive S 1 -equivariant symplectic homology to give a new proof of Theorem 0.0.7, about the minimal number of periodic Reeb orbits on some hypersurfaces in R 2n , when all periodic Reeb orbits on Σ are non degenerate. It appears as Theorem 4.1.1 in the following.

We extend the definitions of positive symplectic homology and positive S 1 -equivariant symplectic homology in Sections 4.2.1 and 4.2.2 to the non exact case. It allows us to extend the techniques developed for the proof of Theorem 0.0.7 to start the study of hypersurfaces in negative line bundles. This framework is a natural generalisation of hypersurfaces in C n . Indeed the sphere is the boundary of the ball in C n ≃ R 2n but also the boundary of the ball blown up7 at the origin. The blown up ball, at the origin, in C n is

B2n := z, [t] ∈ C n × CP n-1 | z ∈ [t] .

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It is canonically isomorphic to the canonical disk bundle over CP n-1 which is a subbundle of the tautological complex (negative) line bundle over CP n-1 O(-1) -→ CP n-1 . This generalisation gives: Proposition 0.0.15 Let Σ be a contact type hypersurface in a negative line bundle over a closed symplectic manifold L → B such that the intersection of Σ with each fiber is a circle. The contact form is the restriction of r 2 θ ∇ where θ ∇ is the transgression form on L and r is the radial coordinate on the fiber. Then Σ carries at least 2n i=0 β i geometrically distinct periodic Reeb orbits (the β i are the Betti numbers of B).

Theorem 0.0.16 Let Σ be a contact type hypersurface in a negative line bundle L, over a symplectic manifold B. Suppose that there exists a Liouville domain W ′ (such that its first Chern class vanishes on all tori) whose boundary coincides with the circle bundle of radius R 1 in L, denoted S R 2 1 . Suppose there exists a Morse function f : B → R such that all critical points of f have a Morse index of the same parity. Let α be the contact form on Σ induced by r 2 θ ∇ on L (θ ∇ is the transgression form on L and r is the radial coordinate on the fiber). Assume that Σ is "pinched" between two circle bundles S R 2 1 and S R 2 2 of radii R 1 and R 2 such that 0 < R 1 < R 2 and R 2 R 1 < √ 2. Assume that the minimal period of any periodic Reeb orbit on Σ is bounded below by R 2 1 . Then Σ carries at least 2n i=0 β i geometrically distinct periodic Reeb orbits, where the β i denote the Betti numbers of B.

In this Theorem, the assumption on the existence of a Morse function all of whose critical points have Morse indices of the same parity is of a technical nature. Its purpose is to bring the situation within the scope of Theorem 0.0.10, which is our tool for computing the positive S 1 -equivariant symplectic homology. The lower bound on the period of any periodic Reeb orbit is semi-technical; it is now the only way we have to distinguish the images of the orbits. The "pinching" assumption is more conceptual, its main implication is that the "n first generators" of the positive S 1 -equivariant symplectic homology are simple orbits.

The techniques developed in the thesis should prove, extending the homologies to the setup of monotone compact symplectic manifolds using coefficients in the Novikov ring, the following: Conjecture 0.0.17 Let Σ be a contact type hypersurface in a negative line bundle, L, over a closed monotone symplectic manifold B. The bundle is endowed with a hermitian structure and a connection. Suppose there exists a Morse function f : B → R such that all critical points of f have a Morse index of the same parity. Let α be the contact form on Σ induced by r 2 θ ∇ on L. Assume that Σ is "pinched" between two circle bundles S R 1 and S R 2 of radii R 1 and R 2 respectively such that 0 < R 1 < R 2 and R 2 R 1 < √ 2. Assume that the Contents minimal action of any periodic Reeb orbit on Σ is bounded below by R 2 1 . Then Σ carries at least 2n i=0 β i geometrically distinct periodic Reeb orbits.

The last two chapters are a contribution to a detailed study of indices. I hope it can be used in the future to tackle the study of degenerate periodic orbits. The Conley-Zehnder index is an integer associated to a path of symplectic matrices. It relies on a precise description of symplectic matrices. Chapter 5, which will appear as a paper in Portugaliae Mathematica, gives new normal forms for symplectic matrices. Let us present here the normal form on the generalised eigenspace of eigenvalue ±1.

Theorem 0.0.18 Any symplectic endomorphism A of a finite dimensional symplectic vector space (V, Ω) is the direct sum of its restrictions A |V [λ] to the real A-invariant symplectic subspace V [λ] whose complexification is the direct sum of the generalized eigenspaces of eigenvalues λ, 1 λ , λ and 1 λ :

V C [λ] := E λ ⊕ E 1 λ ⊕ E λ ⊕ E 1 λ .
If λ ∈ {±1}, there exists a symplectic basis of V [λ] in which the matrix representing the restriction of A to V [λ] is a symplectic direct sum of matrices of the form J(λ, r j ) -1 C(r j , s j , λ) 0 J(λ, r j ) τ

where C(r j , s j , λ) := J(λ, r j ) -1 diag 0, . . . , 0, s j with s j ∈ {0, 1, -1} and where J(λ, r j ) is the elementary Jordan matrix of dimension r j associated to λ. If s j = 0, then r j is odd. The dimension of the eigenspace of the eigenvalue λ is given by 2 Card{j | s j = 0} + Card{j | s j = 0}. The number of s j equal to +1 (resp. -1) arising in blocks of dimension 2k (i.e. with corresponding r j = k) is equal to the number of positive (resp. negative) eigenvalues of the symmetric 2-form

Qλ 2k : Ker (A -λ Id) 2k × Ker (A -λ Id) 2k → R (v, w) → λ Ω (A -λ Id) k v, (A -λ Id) k-1 w .
The decomposition is unique up to a permutation of the blocks and is determined by λ, by the dimension dim Ker(Aλ Id) r for each r ≥ 1, and by the rank and the signature of the symmetric bilinear 2-form Qλ 2k for each k ≥ 1.

The last Chapter, which will appear as a paper in Annales de la faculté des Sciences de Toulouse, is devoted to the study of a generalised version of the Conley-Zehnder index defined by Robbin and Salamon in [START_REF] Robbin | The Maslov index for paths[END_REF]. We start by giving a new formula for the "classical" Conley-Zehnder index.

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Theorem 0.0.19 Let ψ : [0, 1] → Sp(R 2n , Ω 0 ) be a continuous path of matrices linking the matrix Id to a matrix which does not admit 1 as an eigenvalue. Let ψ : [0, 2] → Sp(R 2n , Ω 0 ) be an extension such that ψ coincides with ψ on the interval [0, 1] , such that ψ(s) does not admit 1 as an eigenvalue for all s 1 and such that the path ends either at ψ(2) = W + := -Id either at ψ(2) = W -:= diag(2, -1, . . . , -1, 1 2 , -1, . . . , -1). The Conley-Zehnder index of ψ is equal to the integer given by the degree of the map ρ2 • ψ : [0, 2] → S 1 :

µ CZ (ψ) = deg(ρ 2 • ψ) (1)
for ANY continuous map ρ : Sp(R 2n , Ω 0 ) → S 1 which coincides with the (complex) determinant det C on U (n) = O R 2n ∩ Sp R 2n , Ω 0 ; so that ρ(W -) ∈ {±1} and so that

deg (ρ 2 • ψ 2-) = n -1 for ψ 2-: t ∈ [0, 1] → exp tπJ 0   0 0 -log 2 π 0 0 Id n-1 0 0 -log 2 π 0 0 0 0 0 0 Id n-1   .
In particular, two alternative ways to compute the Conley-Zehnder index are :

• Using the polar decomposition of a matrix, µ CZ (ψ) = deg(det C 2 • U • ψ) where U : Sp(R 2n , Ω 0 ) → U (n) : A → AP -1 with P the unique symmetric positive definite matrix such that P 2 = A τ A.

• Using the normalized determinant of the C-linear part of a symplectic matrix,

µ CZ (ψ) = deg(ρ 2 • ψ) where ρ : Sp(R 2n , Ω 0 ) → S 1 : A → ρ(A) = det C( 1 2 (A-J 0 AJ 0 )) |detC( 1 2 (A-J 0 AJ 0 ))|
with J 0 = 0 -Id Id 0 the standard complex structure on R 2n .

We give a characterisation of the generalised the Conley-Zehnder index defined by Robbin and Salamon.

Theorem 0.0.20 The Robbin-Salamon index for a continuous path of symplectic matrices is characterized by the following properties:

• (Homotopy) it is invariant under homotopies with fixed end points;

• (Catenation) it is additive under catenation of paths;

• (Zero) it vanishes on any path ψ : [a, b] → Sp(R 2n , Ω) of matrices such that dim Ker ψ(t)-

Id = k is constant on [a, b];
• (Normalization) if S = S τ ∈ R 2n×2n is a symmetric matrix with all eigenvalues of absolute value < 2π and if ψ(t) = exp(J 0 St) for t ∈ [0, 1] , then µ RS (ψ) = 1 2 Sign S where Sign S is the signature of S.

We give a new way to compute this index.

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Theorem 0.0.21 Let ψ : [0, 1] → Sp(R 2n , Ω 0 ) be a path of symplectic matrices. Decompose ψ(0) = ψ ⋆ (0) ⊕ ψ (1) (0) and ψ(1) = ψ ⋆ (1) ⊕ ψ (1) (1) where ψ ⋆ (•) does not admit 1 as eigenvalue and ψ (1) (•) is the restriction of ψ(•) to its generalized eigenspace of eigenvalue 1. Consider a continuous extension Ψ : [-1, 2] → Sp(R 2n , Ω 0 ) of ψ such that • Ψ(t) = ψ(t) for t ∈ [0, 1];

• Ψ -1 2 = ψ ⋆ (0) ⊕ e -1 Id 0 0 e Id and Ψ(t) = ψ ⋆ (0) ⊕ φ 0 (t) where φ 0 (t) has only real positive eigenvalues for t ∈ -1 2 , 0 ;

• Ψ 3 2 = ψ ⋆ (1) ⊕ e -1 Id 0 0 e Id and Ψ(t) = ψ ⋆ (1) ⊕ φ 1 (t) where φ 1 (t) has only real positive eigenvalues for t ∈ 1, 3 2 ;

• Ψ(-1) = W ± , Ψ(2) = W ± and Ψ(t) does not admit 1 as an eigenvalue for t ∈ -1, -1 2 and for t ∈ 3 2 , 2 .

Then the Robbin Salamon index is given by

µ RS (ψ) = deg(ρ 2 • Ψ) + 1 2 k≥1 Sign Q(ψ(0)) k - 1 2 k≥1 Sign Q(ψ(1))
k with ρ as in theorem 0.0.19, and with

QA k : Ker (A -Id) 2k × Ker (A -Id) 2k → R (v, w) → Ω (A -Id) k v, (A -Id) k-1 w .
The advantage of this new formula is that we can compute the index of any path without perturbing the path. The drawback is that we have to extend the initial path.

Prolégomènes

Les orbites périodiques de Reeb sont au centre de ce travail. Le cadre général est la géométrie de contact. Une structure de contact sur une variété M de dimension 2n -1 est un champ d'hyperplans ξ maximalement non intégrable; i.e. si on écrit, localement, ξ = ker α ou α est une 1-forme différentielle, alors α ∧ (dα) n-1 est partout non nulle. Si α est globalement définie, on dit que ξ est coorientée et on appelle α une forme de contact. Dans cette thèse, nous ne considérons que des structures de contact coorientées. La 1forme α n'est pas unique; pour toute fonction f : M → R, la 1-forme e f α définit la même structure de contact. Le champ de vecteurs de Reeb R α associé à une forme de contact α est l'unique champ de vecteurs sur M caractérisé par: ι(R α )dα = 0 et α(R α ) = 1. Ce champ de vecteurs ne s'annulant nulle part, son flot n'a pas de point fixe. Les orbites périodiques sont donc les objets les plus remarquables de ce flot. Poincaré en a montré l'intérêt dans son "Traité de la mécanique céleste": "Ce qui nous rend ces solutions périodiques si précieuses, c'est qu'elles sont, pour ainsi dire, la seule brèche par où nous puissions essayer de pénétrer dans une place jusqu'ici réputée inabordable."

Existe t-il une orbite de Reeb périodique pour toute forme de contact sur n'importe quelle variété?

La réponse à cette question est négative comme illustré par l'exemple suivant. Soit R 2n-1 muni de la forme de contact "standard"

α = dz - n-1 i=1 x i dy i .
Le champ de vecteurs de Reeb est donné par R α = ∂ z , et il n'y a pas d'orbites de Reeb périodiques.

La même question pour des variétés compactes orientées munies d'une 1-forme de contact est toujours ouverte; c'est la conjecture de Weinstein. Quelques résultats partiels sont connus. Taubes a répondu de manière affirmative à cette question pour les variétés de dimension 3.

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Theorem 0.0.22 ( [START_REF] Henry | The Seiberg-Witten equations and the Weinstein conjecture[END_REF]) Si Y est une variété fermée, orientée, de dimension 3, munie d'une forme de contact, alors le champ de vecteurs de Reeb associé possède une orbite périodique.

En plus grande dimension, seuls des résultats partiels sont connus. Nous commençons par mentionner un résultat dû à Viterbo.

Theorem 0.0.23 ( [START_REF] Viterbo | A proof of Weinstein's conjecture in R 2n[END_REF]) Si Σ ⊂ (R 2n , ω 0 ) est une hypersurface de "type contact", alors Σ possède au moins une orbite de Reeb périodique.

Plus récemment, Albers et Hofer ont prouvé:

Theorem 0.0.24 ([AH09]) Soit (M, ξ) une variété de contact fermée et "PS-vrillée8 ". Alors le champ de vecteurs de Reeb associé à n'importe quelle forme de contact α déterminant ξ, possède au moins une orbite de Reeb périodique et contractible.

Nous pourrions citer beaucoup d'autres résultats, par exemple liés à la dynamique Hamiltonienne.

Une autre question naturelle est de demander combien d'orbites de Reeb périodiques peuvent exister. Comme toute orbite de Reeb périodique peut être itérée un nombre arbitraire de fois, il est plus raisonnable de demander combien d'orbites de Reeb périodiques géométriquement distinctes peuvent exister. Considérons la sphère S 2n-1 naturellement plongée dans R 2n , munie de la structure de contact standard,

α = α std S 2n-1 = 1 2 n i=1 (x i dy i -y i dx i )) S 2n-1 .
Le champ de Reeb est R α = n i=1 2(x i ∂ y iy i ∂ x i ) et toutes les orbites de Reeb sont périodiques. D'un autre côté, si nous regardons la même sphère munie de la forme de contact "déformée"

α ′ := 1 2 n i=1 a i (x i dy i -y i dx i ) S 2n-1 ,
où tous les a i sont rationnellement indépendants, le champ de Reeb est

R α ′ = n i=1 2 a i (x i ∂ y i - y i ∂ x i ).
Il y a seulement n orbites de Reeb géométriquement distinctes, une dans chaque "plan de coordonnées". Les formes de contact α et α ′ sur la sphère sont isotopes (i.e. il existe un chemin lisse de formes de contact sur la sphère les reliant). Le théorème de stabilité de Gray assure l'existence d'un difféomorphisme de la sphère envoyant la structure de contact ξ ′ = ker α ′ sur la structure de contact ξ = ker α. Donc, en fonction de la forme de contact définissant la structure de contact standard sur la sphère, nous pouvons avoir différentes réponses concernant le nombre d'orbites périodiques distinctes. Theorem 0.0.28 ([EL80, BLMR85]) Soit Σ une hypersurface de type contact dans R 2n . Soit ξ = ker α la structure de contact induite par la forme de contact standard sur R 2n . Supposons qu'il existe un point x 0 ∈ R 2n et des nombres 0 < r ≤ R tels que:

∀x ∈ Σ, r ≤ x -x 0 ≤ R avec R r < √ 2.
Supposons également que ∀x ∈ Σ, ν Σ (x), x > r où ν Σ (x) est le vecteur unitaire normal extérieur à Σ en x. Alors Σ possède au moins n orbites de Reeb périodiques géométriquement distinctes.

Un premier résultat de cette thèse est une preuve alternative (géométrique) de ce résultat quand toutes les orbites de Reeb périodiques sur Σ sont non dégénérées10 . Une conjecture raisonnable est que toute hypersurface étoilée dans R 2n possède au moins n orbites de Reeb périodiques géométriquement distinctes. Pour d'autres variétés que la sphère, fort peu est connu; en dimension 3, Cristofaro-Gardiner et Hutchings ont prouvé:

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Theorem 0.0.29 ([CGH12]) Toute forme de contact (possiblement dégénérée11 ) sur une variété de dimension 3 possède au moins 2 orbites de Reeb périodiques plongées.

En plus grande dimension, un résultat récent est le suivant: Theorem 0.0.30 ( [START_REF] Kang | Equivariant symplectic homology and multiple closed reeb orbits[END_REF]) Supposons que la variété de contact fermée (M, ξ) de dimension 2n -1 admet un plongement de contact exact et déplaçable dans une variété symplectique (W, ω = dλ) convexe à l'infini12 et satisfaisant c 1 (W ), π 2 (W ) = 0. Supposons qu'au moins une des conditions suivantes est satisfaite

1. H * (W 0 , M ; Q) = 0 pour un * ∈ 2N -1 2. H * (W 0 , M ; Q) = 0 pour tout degré pair * ≤ 2n -4
où W 0 est le domaine relativement compact de W bordé par M . Alors il y a au moins 2 orbites de Reeb périodiques contractibles dans W pour toute forme de contact non dégénérée α sur (M, ξ) telle que αλ | M est exacte.

Dans cette thèse, je présente de nouveaux résultats sur le nombre minimal d'orbites de Reeb périodiques géométriquement distinctes sur certaines hypersurface dans des fibrés en droites négatifs (Proposition 0.0.36 and Théorème 0.0.37).

Stratégie suivie dans cette thèse

Le problème de trouver des orbites de Reeb périodiques sur une variété de contact plongée dans une variété symplectique peut souvent être traduit en le problème de trouver des orbites périodiques d'un champ de vecteurs Hamiltonien sur un niveau d'énergie fixé. Par exemple, si C est un domaine étoilé dans R 2n tel que 0 ∈ Int C, trouver les orbites de Reeb périodiques sur le bord de C (pour la forme de contact standard α std C ) revient à trouver les orbites périodiques du champ de vecteurs Hamiltonien défini comme une puissance de la fonction de jauge, sur le bord de C qui est un niveau de ce Hamiltonien. En effet, la fonction de jauge de C, j C : R 2n → [0, ∞), est définie par

j C (x) := min λ| x λ ∈ C et le champ de vecteurs Hamiltonien associé à H β = j C (x) β est X H β = β 2 R α std C
. La première idée pour aborder la question du nombre minimal d'orbites de Reeb périodiques était d'utiliser un invariant homologique de la structure de contact construit à partir des orbites de Reeb périodiques. La construction d'un tel invariant est le but de l'homologie de contact. Actuellement, l'homologie de contact est toujours en développement Contents et rencontre des problèmes de "transversalité". Nous considérons, à la place, l'homologie symplectique S 1 -équivariante positive construite à partir d'orbites périodiques de champs de vecteurs Hamiltoniens sur une variété symplectique dont le bord est la variété de contact considérée. Dans cet esprit, Bourgeois et Oancea [START_REF] Bourgeois | S 1 -equivariant symplectic homology and linearized contact homology[END_REF] ont lié, l'homologie de contact linéarisée du bord (dans le cas où elle peut être définie) avec l'homologie S 1 -équivariante positive de la variété symplectique. L'homologie symplectique S 1 -équivariante positive est un des objets principaux considérés dans les quatre premiers chapitres de cette thèse.

Notre premier but est d'analyser les relations entre les homologies symplectiques d'une variété symplectique exacte avec un bord de type contact (également appelé domaine de Liouville) et les orbites de Reeb périodiques sur le bord. Le point suivant est de prouver quelques propriétés de ces homologies. Pour un domaine de Liouville plongé dans un autre, nous construisons un morphisme entre leurs homologies. Nous étudions l'invariance de ces homologies par rapport au choix d'une forme de contact sur le bord. Finalement, nous utilisons l'homologie S 1 -équivariante positive pour donner une nouvelle preuve du Théorème 0.0.28 et regardons comment elle peut s'étendre au cadre d'hypersurfaces dans des fibrés en droites négatifs.

Une autre approche à la question du nombre minimal d'orbites de Reeb périodiques sur des hypersurfaces dans R 2n , développée par Long, utilise des méthodes variationnelles et une étude détaillée de l'indice de Conley-Zehnder. Dans cette optique, nous étudions une généralisation de l'indice de Conley-Zehnder définie pour tout chemin de matrices symplectiques. Ceci nous a mené à une analyse détaillée de formes normales de matrices symplectiques. Ces résultats peuvent être utiles pour une étude d'orbites dégénérées.

Contenu de la thèse et énoncés de résultats

Les quatre premiers chapitres développent l'approche utilisant l'homologie symplectique S 1 -équivariante positive.

Dans le Chapitre 1, nous rappelons la définition de l'homologie symplectique S 1équivariante positive. Nous présentons l'homologie de Floer (Section 1.1), l'homologie symplectique (Section 1.2.2), l'homologie symplectique positive (Section 1.2.4). Ensuite nous exposons deux définitions équivalentes de l'homologie symplectique S 1 -équivariante positive dans les Sections 1.3.2 et 1.4.

Le lien entre les générateurs de l'homologie symplectique S 1 -équivariante positive et les orbites de Reeb périodiques est expliqué dans le Chapitre 2. Un calcul explicite donne: Theorem 0.0.31 Soit (W, λ) un domaine de Liouville. Supposons qu'il existe une forme de contact α sur le bord ∂W telle que les indices de Conley-Zehnder de toutes les orbites de Reeb périodiques ont la même parité. Alors

SH S 1 ,+ (W, Q) = γ∈P(Rα)
Q γ où P(R α ) est l'ensemble des orbites de Reeb périodiques sur ∂W .

Contents

Dans le Chapitre 3, nous montrons que l'homologie symplectique S 1 -équivariante positive a de bonnes propriétés de fonctorialité. Dans la première Section, nous construisons un "morphisme de transfert" pour toutes les variantes précitées d'homologie symplectique dans le cas de deux domaines de Liouville emboîtés. Cette construction généralise une construction de Viterbo ([Vit99]). Nous prouvons que ce morphisme possède de bonnes propriétés de composition:

Theorem 0.0.32 Soit (V 1 , λ V 1 ) ⊆ (V 2 , λ V 2 ) ⊆ (V 3 , λ V 3
) des domaines de Liouville avec des plongements de Liouville. Alors le diagramme suivant commute:

SH † (V 3 , λ V 3 ) φ † V 3 ,V 2 / / φ † V 3 ,V 1 6 6 SH † (V 2 , λ V 2 ) φ † V 2 ,V 1 / / SH † (V 1 , λ V 1 )
où † est l'un des symboles suivants: ∅, +, S 1 , (S 1 , +). où SH dénote l'homologie symplectique, SH + , l'homologie symplectique positive, SH S 1 l'homologie symplectique S 1 -équivariante et SH S 1 ,+ l'homologie symplectique S 1 -équivariante positive.

La seconde Section du Chapitre 3 est consacrée à l'invariance des différentes variantes d'homologie symplectique. En particulier, nous prouvons Theorem 0.0.33 Soit (W 0 , λ 0 ) et (W 1 , λ 1 ) deux variétés de Liouville13 de type fini tels que il existe un symplectomorphisme f :

(W 0 , λ 0 ) → (W 1 , λ 1 ). Alors SH † (W 0 , λ 0 ) ∼ = SH † (W 1 , λ 1 ).
Theorem 0.0.34 Soit (M 0 , ξ 0 ) and (M 1 , ξ 1 ) deux variétés de contact exactement remplissables; i.e. il existe des domaines de Liouville (W 0 , λ 0 ) et (W 1 , λ 1 ) tels que ∂W 0 = M 0 , ξ 0 = ker(λ 0| M 0 ), ∂W 1 = M 1 and ξ 1 = ker(λ 1| M 1 ). Supposons qu'il existe un contactomorphisme ϕ : (M 0 , ξ 0 ) → (M 1 , ξ 1 ). Supposons de plus que ξ 0 admet une forme de contact α 0 telle que toute les orbites de Reeb périodiques sont non dégénérées et leurs indices de Conley-Zehnder ont tous la même parité. Alors

SH S 1 ,+ (W 0 , λ 0 ) ∼ = SH S 1 ,+ (W 1 , λ 1 ).
Ce Théorème, couplé au Théorème 0.0.31 donne une preuve du résultat d'Ustilovsky sur l'existence de structures de contact non difféomorphes sur les sphères S 4m+1 . La preuve originelle repose sur une théorie d'homologie de contact cylindrique qui n'est pas encore établie rigoureusement.

:= z, [t] ∈ C n × CP n-1 | z ∈ [t] .
Elle est canoniquement isomorphe au fibré en disques canonique au-dessus de CP n-1 qui est un sous-fibré du fibré en droites complexes négatif tautologique au-dessus de

CP n-1 O(-1) -→ CP n-1 .

Cette généralisation donne:

Proposition 0.0.36 Soit Σ une hypersurface de type contact dans un fibré en droites négatif au-dessus d'une variété symplectique fermée L → B tel que l'intersection de Σ avec chaque fibre est un cercle. Alors Σ possède au moins 2n i=0 β i orbites de Reeb périodiques et géométriquement distinctes (les β i étant les nombres de Betti de B).

Theorem 0.0.37 Soit Σ une hypersurface de type contact dans un fibré en droites négatif L au-dessus d'une variété symplectique B. Supposons qu'il existe un domaine de Liouville W ′ (tel que sa première classe de Chern s'annule sur tous les tores) dont le bord coincide avec le fibré en cercles de rayon R 1 dans L, denoté S R 2 1 . Supposons qu'il existe une fonction de Morse f : B → R telle que tous les points critiques de f ont un indice de même parité. Soit ξ = ker α la structure de contact sur Σ induite par r 2 θ ∇ sur L (θ ∇ étant la forme de transgression sur L et r est la coordonnée radiale dans la fibre). Supposons que Σ est "pincée" entre deux fibrés en cercles

S R 2 1 et S R 2 2 de rayon R 1 et R 2 tels que R 2 R 1 < √ 2.
Supposons que la période minimale de toute orbite de Reeb périodique sur Σ est bornée inférieurement par R 2 1 . Alors Σ possède au moins 2n i=0 β i orbites de Reeb périodiques et géométriquement distinctes, où les β i sont les nombres de Betti de B.

Contents

Dans ce Théorème, l'hypothèse de l'existence d'une fonction de Morse dont tous les points critiques ont un indice de même parité est de nature technique. Son but est de nous amener dans les hypothèse du Théorème 0.0.31, qui est notre outil pour calculer l'homologie symplectique S 1 -équivariante positive. La borne inférieure sur la période de toute orbite de Reeb périodique est semi-technique; c'est le seul moyen actuellement de distinguer les différentes images des orbites. L'hypothèse de "pincement" est plus conceptuelle, son implication majeure est que les "n premiers générateurs" de l'homologie symplectique S 1équivariante positive sont des orbites simples.

Les techniques développées dans cette thèse devraient prouver, en étendant les homologies au cadre des variétés symplectiques compactes et monotones, et en introduisant l'anneau de Novikov comme anneau de coefficients, le résultat suivant: Conjecture 0.0.38 Soit Σ une hypersurface de type contact dans un fibré en droites négatif L au-dessus d'une variété symplectique fermée et monotone B. Le fibré est muni d'une structure hermitienne et d'une connexion. Supposons qu'il existe une fonction de Morse f : B → R telle que tous les points critiques de f ont un indice de même parité. Soit ξ = ker α la structure de contact sur Σ induite par r 2 θ ∇ sur L (θ ∇ étant la forme de transgression sur L et r est la coordonnée radiale dans la fibre). Supposons que Σ est "pincée" entre deux fibrés en cercles

S R 2 1 et S R 2 2 de rayon R 1 et R 2 tels que R 2 R 1 < √ 2.
Supposons que la période minimale de toute orbite de Reeb périodique sur Σ est bornée inférieurement par R 2 1 . Alors Σ possède au moins 2n i=0 β i orbites de Reeb périodiques et géométriquement distinctes, où les β i sont les nombres de Betti de B.

Les deux derniers chapitres sont une contribution à une étude d'indices de type Conley-Zehnder. J'espère que cela pourra être utilisé pour l'étude des orbites périodiques dégénérées. L'indice de Conley-Zehnder généralisé est un entier associé à un chemin de matrices symplectiques. Notre étude s'appuie sur une description détaillée de matrices symplectiques. Le Chapitre 5, qui apparaîtra comme article dans Portugaliae Mathematica, donne de nouvelles formes normales des matrices symplectiques. Nous présentons ici la forme normale sur l'espace propre généralisé de valeur propre ±1.

Theorem 0.0.39 Tout endomorphisme symplectique A d'un espace vectoriel symplectique de dimension finie (V, Ω) est la somme directe de ses restrictions A |V [λ] au sous-espace symplectique réel A-invariant V [λ] dont la complexification est la somme directe de ses espaces propres généralisés de valeur propre λ, 1 λ , λ et 1 λ :

V C [λ] := E λ ⊕ E 1 λ ⊕ E λ ⊕ E 1 λ .
Si λ ∈ {±1}, il existe une base symplectique de V [λ] dans laquelle la matrice représentant la restriction de A à V [λ] est une somme directe symplectique de matrices de la forme:

J(λ, r j ) -1 C(r j , s j , λ) 0 J(λ, r j ) τ
Contents où C(r j , s j , λ) := J(λ, r j ) -1 diag 0, . . . , 0, s j avec s j ∈ {0, 1, -1} et où J(λ, r j ) est la matrice de Jordan élémentaire de dimension r j associée à λ. Si s j = 0, alors r j est impair. La dimension de l'espace propre de valeur propre λ est donnée par 2 Card{j | s j = 0} + Card{j | s j = 0}. Le nombre de s j égaux à +1 (reps. -1) apparaissant dans des blocs de dimension 2k (i.e. avec r j = k) est égal au nombre de valeurs propres positives (resp. négatives) de la 2-forme symétrique

Qλ 2k : Ker (A -λ Id) 2k × Ker (A -λ Id) 2k → R (v, w) → λ Ω (A -λ Id) k v, (A -λ Id) k-1 w .
(2) = W + := -Id, soit en ψ(2) = W -:= diag(2, -1, . . . , -1, 1 2 , -1, . . . , -1)
. L'indice de Conley-Zehnder de ψ est égal à l'entier donné par le degré de l'application ρ2

• ψ : [0, 2] → S 1 : µ CZ (ψ) = deg(ρ 2 • ψ) (2) pour TOUTE application continue ρ : Sp(R 2n , Ω 0 ) → S 1 coïncidant avec le déterminant complexe det C sur U (n) = O R 2n ∩ Sp R 2n , Ω 0 ; telle que ρ(W -) ∈ {±1} et telle que deg (ρ 2 • ψ 2-) = n -1 pour ψ 2-: t ∈ [0, 1] → exp tπJ 0   0 0 -log 2 π 0 0 Id n-1 0 0 -log 2 π 0 0 0 0 0 0 Id n-1   . En
particulier, deux manières alternatives de calculer l'indice de Conley-Zehnder sont:

• En utilisant la décomposition polaire des matrices,

µ CZ (ψ) = deg(det C 2 • U • ψ) où U : Sp(R 2n , Ω 0 ) → U (n) : A → AP -1 avec P l'unique matrice symétrique déifie positive telle que P 2 = A τ A.
• En utilisant le déterminant normalisé de la partie C-linéaire d'une matrice sympplectique,

Contents µ CZ (ψ) = deg(ρ 2 • ψ) où ρ : Sp(R 2n , Ω 0 ) → S 1 : A → ρ(A) = det C( 1 2 (A-J 0 AJ 0 )) |detC( 1 2 (A-J 0 AJ 0 ))| avec J 0 = 0 -Id Id 0
la structure complexe standard sur R 2n .

Nous donnons une caractérisation de l'indice de Conley-Zehnder généralisé défini par Robbin et Salamon.

Theorem 0.0.41 L'indice de Robbin-Salamon pour un chemin continu de matrices symplectiques est caractérisé par les propriétés suivantes:

• (Homotopie) il est invariant par homologies à extrémités fixées;

• (Caténation) il est additif sous caténation de chemins;

• (Zéro) il s'annule sur tout chemin de matrices ψ : [a, b] → Sp(R 2n , Ω) tel que dim Ker ψ(t) -Id = k est constant sur [a, b];
• (Normalisation) si S = S τ ∈ R 2n×2n est une matrice symétrique dont toutes les valeurs propres sont en valeur absolue < 2π et si ψ(t) = exp(J 0 St) pour t ∈ [0, 1], alors µ RS (ψ) = 1 2 Sign S où Sign S est la signature de S.

Nous donnons une nouvelle façon de calculer cet indice.

Theorem 0.0.42 Soit ψ : [0, 1] → Sp(R 2n , Ω 0 ) un chemin de matrices symplectiques. Décomposons ψ(0) = ψ ⋆ (0) ⊕ ψ (1) (0) et ψ(1) = ψ ⋆ (1) ⊕ ψ (1) (1) où ψ ⋆ (•) n'admet pas 1 comme valeur propre et ψ (1) (•)
est la restriction de ψ(•) à son espace propre généralisé de valeur propre 1. Considérons une extension continue Ψ : [-1, 2] → Sp(R 2n , Ω 0 ) de ψ telle que

• Ψ(t) = ψ(t) pour t ∈ [0, 1];

• Ψ -1 2 = ψ ⋆ (0) ⊕ e -1 Id 0 0 e Id et Ψ(t) = ψ ⋆ (0) ⊕ φ 0 (t) où φ 0 (t) n'a que des valeurs propres réelles pour t ∈ -1 2 , 0 ;

• Ψ 3 2 = ψ ⋆ (1) ⊕ e -1 Id 0 0 e Id et Ψ(t) = ψ ⋆ (1) ⊕ φ 1 (t) où φ 1 (t)
n'a que des valeurs propres réelles pour t ∈ 1, 3 2 ;

• Ψ(-1) = W ± , Ψ(2) = W ± et Ψ(t) n'admet pas 1 comme valeur propre pour t ∈ -1, -1 2 and for t ∈ 3 2 , 2 .
Alors, l'indice de Robbin-Salamon est donné par

µ RS (ψ) = deg(ρ 2 • Ψ) + 1 2 k≥1 Sign Q(ψ(0)) k - 1 2 k≥1 Sign Q(ψ(1))
k Contents avec ρ comme dans le Théorème 0.0.40, et avec

QA k : Ker (A -Id) 2k × Ker (A -Id) 2k → R (v, w) → Ω (A -Id) k v, (A -Id) k-1 w .
L'avantage de cette nouvelle formule est de pouvoir calculer l'indice de tout chemin sans avoir à la perturber. L'inconvénient est que nous devons étendre le chemin initial.

1 Background on symplectic homology

Floer Homology

Let (W, ω) be a compact symplectic manifold. For simplicity of the presentation, we assume Assumption 1.1.1 that W is symplectically aspherical, i.e the symplectic form vanishes on the second fundamental group ω, π 2 (W ) = 0, Assumption 1.1.2 and that the first Chern class of the manifold (i.e. the first Chern class of its tangent bundle, endowed with a compatible almost complex structure) vanishes on the second fundamental group c 1 (W ), π 2 (W ) = 0.

Assumption 1.1.1 will ensure that the action of a contractible loop is well-defined. To deal with other free homotopy classes of loops, one has to assume a stronger version of atoroidality. Assumption 1.1.2 implies that the Conley-Zehnder index of a 1-periodic orbit of a Hamiltonian is well-defined on Z. One can get rid of this assumption by looking at the homology with coefficients in the Novikov ring. Both assumptions will ensure that there are no holomorphic spheres, which is a necessary requirement for the moduli spaces of Floer trajectories to be nice manifolds with boundaries. Floer homology for W is a kind of Morse homology on the loop space of W , L(W ). It has been developed in the late eighties [START_REF] Floer | Symplectic fixed points and holomorphic spheres[END_REF][START_REF] Floer | Transversality in elliptic Morse theory for the symplectic action[END_REF]; a detailed account with proofs can be found in the book [START_REF] Audin | Théorie de Morse et homologie de Floer[END_REF]. This homology is based on functionals defined on the space of contractible loops L contr (W ); this is the connected component of the loop space containing the constant loops. A functional A H on L contr (W ) is associated to a time dependent Hamiltonian on W , H : R × W → R such that H(t + 1, x) = H(t, x) for all t ∈ R. Since the Hamiltonian is periodic in the R variable, we will see H as a function S 1 × W → R and denote by θ the variable in S 1 ∼ = R/ Z .

Background on symplectic homology

Definition 1.1.3 The Hamiltonian action functional A H : L contr (W ) → R is defined by

A H (γ) := - D u ⋆ ω - 1 0 H θ, γ(θ) dθ
where D denotes the disk, D = {z ∈ C| |z| ≤ 1}, and where u is an extension of γ to the disk : u : D → W with u(e 2πiθ ) = γ(θ).

This functional is well-defined (independent of the choice of u) thanks to assumption 1.1.1. Lemma 1.1.4 A loop γ is a critical point of A H if and only if it is induced by a 1-periodic solution γ : R → W of the Hamiltonian system

γ(t) = X t H γ(t)
where the vector field X t H is the Hamiltonian vector field corresponding to the function H(t, •), i.e. ι(X t H )ω = dH(t, •). Such a 1-periodic solution will be called a 1-periodic orbit of X H and we shall denote by P(H) the set of contractible 1-periodic orbits of X H . If γ is a 1-periodic orbit of X H with γ(0) = x and if ϕ X H denotes the flow of the time dependent vector field X H , then x is a fixed point of the flow after time 1, ϕ X H 1 (x) = x, and the differential of ϕ X H 1 at x yields an endomorphism of T x W which preserves ω x ; it is called the Poincaré return map.

Definition 1.1.5 A 1-periodic orbit γ of X H is non degenerate if 1 is not an eigenvalue of the Poincaré return map.

We look only at Hamiltonians whose 1-periodic orbits are all non degenerate; this implies that the 1-periodic orbits are isolated. Such a Hamiltonian is called non degenerate. We associate to each 1-periodic orbit its Conley-Zehnder index which is an integer, defined as follows. We choose as above an extension u of the orbit γ to the disk, u : D → W ; we choose a symplectic trivialization of the bundle on the disk defined by the pullback by u of the tangent bundle to W ; since the differential of the flow of the Hamiltonian vector field ϕ X H t at x = γ(0) yields a symplectic endomorphism from T γ(0) W to T γ(t) W , it is represented in the trivialization by a symplectic matrix. We associate in this way a path ψ : [0, 1] → Sp(R 2n , Ω 0 ) of symplectic matrices, starting from the identity and such that 1 is not an eigenvalue of ψ(1). The Conley-Zehnder index of the orbit γ is defined to be the Conley-Zehnder index of the path ψ as defined in section 6.2.

This index does not depend on the chosen trivialisation of u * T W , and is invariant under a continuous deformation of the extension u. If two extensions u and u ′ differ, up to homotopy, by an element S in π 2 (W ), the difference between the Conley-Zehnder indices of γ is equal to twice the evaluation of the first Chern class of W on this element S (see, for instance, [START_REF] Mcduff | Introduction to symplectic topology[END_REF]). The assumption 1.1.2 ensures that the Conley-Zehnder index does not depend on the choice of the extension u.

Floer Homology

The Floer complex is the Z-vector space generated by the contractible 1-periodic orbits of X H , graded by minus their Conley-Zehnder index.

In Morse homology, to define the boundary operator, one has to count negative gradient trajectories between critical points. In our setting, to define negative gradient trajectories of A H , one needs a metric on the loop space. One chooses a smooth loop J : S 1 → End(T W ) : θ → J θ of almost complex structures on W which are compatible with ω, i.e ω(J θ X, J θ Y ) = ω(X, Y ) for all X, Y ∈ T W and ω(X, J θ X) > 0 for all 0 = X ∈ T W . The resulting inner product on the tangent space to the loop space at the loop γ, T γ L(W ) = Γ ∞ (S 1 , γ * T W ), is defined by

ξ, η := S 1 ω γ(θ) ξ(θ), J θ (γ(θ))η(θ) dθ .
Lemma 1.1.6 Negative gradient trajectories of A H correspond to maps u : R × S 1 → W satisfying perturbed nonlinear Cauchy-Riemann equations called Floer equations:

∂u ∂s (s, θ) + J θ u(s, θ) ∂u ∂θ (s, θ) -X θ H u(s, θ) = 0 (1.1)
with ι(X θ H )ω = dH(θ, •). Such maps are called Floer trajectories. As in Morse theory, we want to "count" the number of negative gradient trajectories between some pairs of critical points. A first important issue is to know whether a Floer trajectory converges to 1-periodic orbits. We define the energy of a Floer trajectory as

E(u) := 1 2 S 1 R ∂u ∂s 2 + ∂u ∂θ -X θ H • u 2 dsdθ = S 1 R ∂u ∂s 2 dsdθ.
Proposition 1.1.7 ( [START_REF] Floer | Symplectic fixed points and holomorphic spheres[END_REF], see also [START_REF] Audin | Théorie de Morse et homologie de Floer[END_REF] theorem 6.5.6) A Floer trajectory with finite energy converges at ±∞ to 1-periodic orbits of X H , assuming that all the 1-periodic orbits of X H are non degenerate.

Let us consider the space M of contractible smooth Floer trajectories with finite energy. By contractible, we mean that u(s, •) : S 1 → W is contractible for one, and hence all, s ∈ R. This contractibility assumption is considered only when studying contractible 1-periodic orbits.

M := {u : R × S 1 → W | u is a contractible solution of (1.1), C ∞ and E(u) < ∞}.

1. Background on symplectic homology Theorem 1.1.8 ( [START_REF] Floer | Symplectic fixed points and holomorphic spheres[END_REF], see also [START_REF] Audin | Théorie de Morse et homologie de Floer[END_REF], theorem 6.5.4 ) M is compact in C ∞ loc (R× S 1 , W ), where C ∞ loc is the space of smooth maps endowed with the topology of uniform convergence on compact subsets.

Let γ -, γ + be two 1-periodic orbits of X H and let M(γ -, γ + , H, J) denote the space of u ∈ M such that lim

s→-∞ u(s, •) = γ -and lim s→+∞ u(s, •) = γ + .
A naive but crucial remark is that one can "count" the points of a 0-dimensional compact set. One shows that the space of Floer trajectories between two 1-periodic orbits of the Hamiltonian vector field X H is a manifold, provided one has perturbed a little -if neededthe Hamiltonian. The way to prove that is to describe M(γ -, γ + , H, J) as the zero set of a smooth Fredholm map between two Banach manifolds. The perturbation of H is introduced so that the differential of the map is surjective at all solutions. Indeed recall :

Proposition 1.1.9 Let E (resp. F) be a connected Banach manifold, locally modelled on the Banach space E (resp. F ). Let D : E → F be a smooth map which is Fredholm, i.e. such that the differential at each point x ∈ E, T x D : E → F is a Fredholm operator. Let y be an element in F. If, for any u ∈ D -1 (y), the differential -also called linearization-T u D : E → F is surjective, then D -1 (y) is a submanifold; its dimension is the index of T u D (which is independent of u) and its tangent space at u is the kernel of T u D.

For D -1 (y) to be a manifold, it is enough, (since the set of Fredholm operators is open in the set of bounded linear maps), to show that T u D is Fredholm and surjective at any point u ∈ D -1 (y), and to prove that the index of T u D is constant on D -1 (y).

To do this in the Floer homology context, one considers the Floer operator ∂ defined, given a Hamiltonian H and a loop of almost complex structures J, by

∂ : C ∞ loc (R × S 1 , W ) → C ∞ loc (R × S 1 , W ) u → ∂u ∂s + J θ • u ∂u ∂θ -X θ H • u
and one extends it to suitable Banach spaces.

Theorem 1.1.10 ( [START_REF] Floer | Symplectic fixed points and holomorphic spheres[END_REF]; see also [START_REF] Audin | Théorie de Morse et homologie de Floer[END_REF] theorem 8.1.5) The linearization D u of the Floer operator at the point u is a Fredholm operator whose index is equal to the difference of the Conley-Zehnder indices of the limiting 1-periodic orbits

index(D u ) = µ CZ (γ + ) -µ CZ (γ -).
To apply Proposition 1.1.9, one shows the existence of a pair (H, J) so that the corresponding linearisation D u is surjective for all u ∈ M(γ -, γ + , H, J) and for all γ -, γ + 1-periodic 1.1. Floer Homology orbits of X H . One first shows that the set of all Floer trajectories X (γ -, γ + , J) defined for a fixed J and for a Hamiltonian varying in a class H 0 + h for H 0 fixed, non degenerate, and h in a Banach space B (so that the 1-periodic orbits remain unchanged -and non degenerate) is a Banach manifold. For this, one considers X (γ -, γ + , J) as the set of zeroes of the section σ defined by Floer equations

σ(u, h) = ∂u ∂s + J θ • u ∂u ∂θ -X θ H 0 +h • u
and one uses the following proposition.

Proposition 1.1.11 Let Z π → P be a Banach vector bundle over a Banach manifold and let σ : P → Z be a smooth section. Then the intersection of σ and the 0-section is a Banach manifold, whenever the section σ is transversal to the 0-section at every point of the intersection.

One then considers the natural projection π(σ(u, h)) = h from this Banach manifold X (γ -, γ + , J) on the Banach manifold B and one uses Sard-Smale theorem.

Proposition 1.1.12 (Sard-Smale) Let π : X → B be a smooth map between Banach manifolds, whose differential is a Fredholm operator. Then the set of regular values of π is of second Baire category, i.e. is the intersection of a numerable set of open dense subsets.

Let us observe that the regular values of π are exactly the elements h ∈ B such that, for any u ∈ M(γ -, γ + , H, J), the linearized map D u is surjective. Let us denote by (HJ ) reg the space of all pairs (H, J) such that the 1-periodic orbits of X H are all nondegenerate and the operator D u is surjective for all u ∈ M(γ -, γ + , H, J) and all 1-periodic orbits γ -, γ + . One gets Theorem 1.1.13 ([FHS95]) Let (W, ω) be a compact manifold. Let J ∈ J (W, ω) and H 0 : S 1 × W → R be a Hamiltonian such that the 1-periodic orbits of X H 0 are all nondegenerate. Denote by C ∞ (H 0 ) the set of Hamiltonians that coincide with H 0 up to the second order on the 1-periodic orbits of X H 0 . Then the set

H reg = {H ∈ C ∞ (H 0 ) | (H, J) ∈ (HJ ) reg } is dense (of second Baire category) in C ∞ (H 0 ).

This implies :

Theorem 1.1.14 ( [START_REF] Floer | Symplectic fixed points and holomorphic spheres[END_REF][START_REF] Floer | Transversality in elliptic Morse theory for the symplectic action[END_REF]; see also [START_REF] Audin | Théorie de Morse et homologie de Floer[END_REF] Theorem 8.1.2) For a generic choice of the Hamiltonian H (by this we mean a H in H reg ) and every γ -, γ + , 1-periodic orbits of X H , each M(γ -, γ + , H, J) is a smooth compact manifold of dimension µ CZ (γ + )µ CZ (γ -).

Background on symplectic homology

We assume in what follows that the Hamiltonian H is generic in the above sense. There is an R-action on M(γ -, γ + , H, J) defined by reparametrization of u in the s coordinate. Thus, if µ CZ (γ + ) = µ CZ (γ -) + 1, the quotient M(γ -, γ + , H, J)/ R of M(γ -, γ + , H, J) under this R-action is 0-dimensional. In order to get compact spaces, one studies the limit of a sequence of Floer trajectories.

Theorem 1.1.15 [START_REF] Floer | Symplectic fixed points and holomorphic spheres[END_REF] or for instance [AD10] Theorem 9.1.6) Let (u n ) be a sequence of elements in M(γ 1 , γ 2 , H, J). There exists 1. a subsequence of (u n ); 2. a finite number of critical points

x 0 = γ 1 , x 1 , . . . , x l+1 = γ 2 of A H ; 3. sequences of real numbers (s i n ) for 0 ≤ i ≤ l; 4. elements u i ∈ M(x i , x i+1 , H, J) for 0 ≤ i ≤ l ; such that, for 0 ≤ i ≤ l, lim n→∞ u n • s i n = u i in C ∞ loc with (u • s)(s ′ , t) := u(s + s ′ , t).
This means that, up to the R-action, the limit of a sequence of Floer trajectories can be a broken Floer trajectory. We define the closure of M(γ -, γ + , H, J)/ R as the union of M(γ -, γ + , H, J)/ R and all those broken trajectories (up to reparametrization) so that those closures are compact. In particular, when µ CZ (γ -) = µ CZ (γ + ) -1 the space M(γ -, γ + , H, J)/ R is equal to its closure and is thus compact. This means it consists of a finite number of points. Each of these points comes with a sign induced by the choice of a system of coherent orientations; for simplicity, we postpone their description to section 1.1.1. Without those orientations, everything is well defined for a Z/ 2Z valued Floer homology.

Definition 1.1.16 The Floer complex is the Z-vector space generated by the 1-periodic orbits of X H , graded by minus their Conley-Zehnder index

F C(H, J) := γ∈P(H) Z γ . The Floer differential ∂ : F C * (H, J) → F C * -1 (H, J) is defined by ∂(γ -) := γ + ∈P(H) -µ CZ (γ -)=-µ CZ (γ + )+1 #M(γ -, γ + , H, J)/ R γ +
where # is a count of points with signs. (Those signs are defined in section 1.1.1).

Floer Homology

The fact that ∂ 2 = 0 follows from the study of the boundary of M(γ -, γ + , H, J)/ R when the difference of the Conley-Zehnder indices is 2. When -µ CZ (γ -) = -µ CZ (γ + ) + 2, the space M(γ -, γ + , H, J)/ R is of dimension 1. Its boundary ∂M/ R consists of an even number of points and is given by

{x | -µ CZ (x)=-µ CZ (γ + )+1} M(γ -, x, H, J)/ R × M(x, γ + , H, J)/ R . The coefficient of γ + in ∂ 2 (γ -) is given by {x | -µ CZ (x)=-µ CZ (γ + )+1} #M(γ -, x, H, J)/ R #M(x, γ + , H, J)/ R = #∂M/ R = 0.
(see the end of section 1.1.1).

Definition 1.1.17 Floer homology is defined as the homology of the Floer complex with the Floer differential, for a pair (H, J) ∈ (HJ ) reg ; it is denoted F H * (H, J) :

F H * (H, J) := H * F C * (H, J), ∂ .
Theorem 1.1.18 ( [START_REF] Floer | Symplectic fixed points and holomorphic spheres[END_REF], see also [START_REF] Audin | Théorie de Morse et homologie de Floer[END_REF] chapter 11) Floer homology does not depend on the choice of the regular pair (H, J). It is isomorphic to the singular homology of W .

Corollary 1.1.19 ( [START_REF] Floer | Symplectic fixed points and holomorphic spheres[END_REF]) The number of 1-periodic orbits of a non degenerate time dependent periodic Hamiltonian on a compact symplectic manifold W is bounded below by the sum of the Betti numbers of W .

Remark 1.1.20 One can replace assumptions 1.1.1 and 1.1.2 by a monotonicity condition namely there exist k ≥ 0 such that [ω] = k[c 1 ]. Then the action functional is defined on a cover of the loop space and the Conley-Zehnder index depends on the choice of the trivialisation disk. The way to deal with this is to look at Floer homology with coefficients in the Novikov ring. We shall not use this generalisation here.

Parenthesis : A glimpse on signs

We indicate in this section how to define the signs attached to Floer trajectories, and mention the steps which prove that #∂M/ R = 0 ( and hence, by the above, that ∂ 2 = 0).

Background on symplectic homology

Operator gluing lemma

We look at operators

D : W 1,p (R × S 1 , R 2n ) → L p (R × S 1 , R 2n ),
with p > 2, which are of the form

D = ∂ s + J∂ t + S(s, t)
with S(s, •) → S ± (•) for s → ±∞, where S ± belongs to the following space S of loops of symmetric matrices. Given a loop S of symmetric matrices, one considers the corresponding path ψ of symplectic matrices defined by the differential equations ψ = JSψ. The loop S belongs to the space S if and only if 1 is not an eigenvalue of ψ(1), i.e. det ψ(1) -Id = 0. We denote by O(R × S 1 ; S -, S + ) the space of such operators.

For R >> 0 big enough and S ± , S ∈ S, one defines a gluing operation

O(R × S 1 ; S -, S) × O(R × S 1 ; S, S + ) → O(R × S 1 ; S -, S + ) : (D 1 , D 2 ) → D 1 # R D 2
in the following way. Fix a smooth function β : R → [0, 1] such that β(s) = 0 for s ≤ 0, β(s) = 1 for s ≥ 1.Define,

D R i := ∂ s + J∂ t + S(t) + β(-s + R) S i (s, t) -S(t) for i = 1, 2
The glued operator, D 1 # R D 2 is defined by

D 1 # R D 2 := D R 1 (s + R) if s ≤ 0 D R 2 (s -R) if s ≥ 0 .
Theorem 1.1.21 (Operator gluing lemma) Assume D 1 and D 2 are surjective. Then D 1 # R D 2 is surjective for R >> 0 and has uniformly bounded right inverse.

Proof: Choose Q 1 , Q 2 , right inverse for D 1 , D 2 . We first construct an approximate right inverse T R for D R = D 1 # R D 2 i.e an operator T R : L p (R × S 1 , R 2n ) → W 1,p (R × S 1 , R 2n ) such that D R T R -Id < 1 2 and T R is uniformly bounded. Then Q R := T R (D R T R ) -1
is a genuine right inverse for D R and is uniformly bounded.

Floer Homology

We construct T R according to the following diagram:

L p (R × S 1 , R 2n ) × L p (R × S 1 , R 2n ) Q 1 ×Q 2 / / W 1,p (R × S 1 , R 2n ) × W 1,p (R × S 1 , R 2n ) G R L p (R × S 1 , R 2n ) S R O O T R / / W 1,p (R × S 1 , R 2n ) We define T R := G R • Q 1 × Q 2 • S R
where G R is a "gluing map" and S R is a "splitting map". The splitting map is defined as

S R (ζ) = (ζ 1 , ζ 2 ) with ζ 1 (s, •) = 1 -β(s -R) ζ(s -R, •) ζ 2 (s, •) = β(s + R)ζ(s + R, •) .
Given L > 0, we define β L (s) := β s L and we assume that β is such that β L (s) = 0 for s ∈ [0, 1] if L is big. We define the gluing map to be

G R (ξ 1 , ξ 2 ) = 1 -β R 2 (s) ξ 1 (s + R) + 1 -β R 2 (-s) ξ 2 (s -R).
Note that since G R and S R are uniformly bounded, so is T R . To conclude the theorem, one can show that

D R T R -Id → 0 as R → ∞.

Coherent orientations

Let D : X → Y be a Fredholm operator. Its determinant is the 1-dimensional vector space det(D) := Λ max ker(D) ⊗ Λ max coker(D)

∨ .

An orientation of D is an orientation of this vector space. One considers the real line bundle det → F(X, Y ) over the space of Fredholm operators, whose fiber above D is det(D).

Lemma 1.1.22 Assume D 1 ∈ O(R × S 1 ; S -, S) and D 2 ∈ O(R × S 1 ; S, S + ) are surjective. For R big enough, if D R := D 1 # R D 2 and Q R is its right inverse, there is a canonical isomorphism φ R : ker(D 1 ) ⊕ ker(D 2 ) → ker(D R ) defined by φ R := (Id -Q R D R ) • G R
where G R is the gluing map defined above.

Background on symplectic homology

A consequence of this Lemma is that there is a canonical isomorphism

det(D 1 ) ⊗ det(D 2 ) → det(D R ) for R ≫ 0. (1.2)
Definition 1.1.23 A system of coherent orientations on the space of operators {O(R × S 1 ; S -, S + )|S -, S + as above } is an orientation of the determinant line bundle over each O(R × S 1 ; S -, S + ), which is compatible with the gluing operation via the canonical isomorphism (1.2):

det(D 1 # R D 2 ) ≃ det(D 1 ) ⊗ det(D 2 ).
This can be done because O(R × S 1 ; S -, S + ) is contractible.

Theorem 1.1.24 ([FH94, BM04, BO09b]) There exists a system of coherent orientations.

Definition 1.1.25 Assume a system of coherent orientations is given. We shall define the sign attached in Floer coboundary operator to a Floer trajectory between two 1-periodic orbits with a difference of Conley-Zehnder index equal to 1. The space of trajectories is of dimension 1, its quotient by the action of R, M(γ ) which are the two ends of a 1-dimensional moduli space M(x, y). Then

-, γ + ), is of dimension 0. Given [u] ∈ M(γ -, γ + ),
ǫ [u] • ǫ [v] + ǫ [u ′ ] • ǫ [v ′ ] = 0.
This shows that #∂M(x, y) = 0.

Symplectic Homology

Symplectic homology is defined for a compact symplectic manifold W with boundary of contact type. It is defined as a direct limit of Floer homologies of the symplectic completion of W , using some special Hamiltonians. This homology was developed by Viterbo in [START_REF] Viterbo | Functors and computations in Floer homology with applications[END_REF], using works of Cieliebak, Floer, Hofer [START_REF] Floer | Symplectic homology. I. Open sets in[END_REF][START_REF] Cieliebak | Symplectic homology. II. A general construction[END_REF]. The class of admissible Hamiltonians was extended by Oancea in his PhD thesis [START_REF] Oancea | Fibered symplectic cohomology and the Leray-Serre spectral sequence[END_REF]. The case of autonomous Hamiltonians is treated by Cieliebak-Floer-Hofer-Wysocki, Hermann, Bourgeois-Oancea in [START_REF] Cieliebak | Applications of symplectic homology. II. Stability of the action spectrum[END_REF][START_REF] Hermann | Homologie symplectique et épaisseur de Gromov d'un ouvert de type contact restreint[END_REF][START_REF] Bourgeois | Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces[END_REF].

1.2. Symplectic Homology

Setup

Let (W, ω) be a compact symplectic manifold with contact type boundary M := ∂W . This means that there exists a Liouville vector field X (i.e. a vector field X such that L X ω = ω) defined on a neighbourhood of the boundary M , and transverse to M . In the sequel, we shall assume that the Liouville vector field has been chosen and we shall denote by (W, ω, X) such a manifold. We denote by λ the 1-form defined in a neighbourhood of M by λ := ι(X)ω

and by α the contact 1-form on M which is the restriction of λ to M :

α := ι(X)ω | M .
We denote by ξ the contact structure defined by α, i.e ξ := ker α. The Reeb vector field R α is the vector field on M defined by :

ι(R α )dα = 0 α(R α ) = 1.
The action spectrum of (M, α) is the set of all periods of the Reeb vector field

Spec(M, α) := {T ∈ R + | ∃γ periodic orbit of R α of period T }.
The symplectic completion of (W, ω, λ) is the symplectic manifold defined by

W := W G (M × R + ) := W ⊔ (M × [-δ, +∞]) / ∼ G
with the symplectic form

ω := ω on W d(e ρ α) on M × [-δ, +∞]
.

The equivalence ∼ G , between a neighbourhood U of M in W and M × [-δ, 0], is defined by the diffeomorphism G : M × [-δ, 0] → U : (p, ρ) → ϕ X ρ (p)
where ϕ X is the flow of the Liouville vector field X. This is always possible since

G ⋆ ω = e ρ (dα + dρ ∧ α) = d(e ρ α).

Background on symplectic homology

Observe indeed that G ⋆(y,ρ)

∂ ∂ρ = X G(y,ρ) , G ⋆(y,ρ) (Y y ) = ϕ X ρ ⋆y Y y . Since λ(X) = (ι(X)ω)(X) = 0, we also have L X λ = λ and ϕ X ρ ⋆ λ = e ρ λ. Hence, ∀ Y y , Z y ∈ T y M : (G ⋆ ω) (y,ρ) ∂ ∂ρ , Y y = ω ϕ X ρ y X G(y,ρ) , ϕ X ρ ⋆y Y y = ϕ X ρ ⋆ ω y (X y , Y y ) = e ρ dλ y (X y , Y y ) = e ρ (L X λ) y (Y y ) = e ρ (dα + dρ ∧ α) ∂ ∂ρ , Y y (G ⋆ ω) (y,ρ) (Y y , Z y ) = ω ϕ X ρ y ϕ X ρ ⋆y Y y , ϕ X ρ ⋆y Y z = ϕ X ρ ⋆ ω y (Y y , Z y ) = e ρ dλ y (X y , Y y ) = e ρ dα y (X y , Y y ) = e ρ (dα + dρ ∧ α) (Y y , Z y ) .
We still assume throughout that ω is symplectically aspherical and that the first Chern class vanishes on the second fundamental group.

Symplectic homology

Given a time-dependent Hamiltonian

H : S 1 × W → R, we define for each θ ∈ S 1 the Hamiltonian vector field X θ H by ω(X θ H , .) = dH(θ, •), θ ∈ S 1 . Definition 1.2.1
The class H std of admissible Hamiltonians consists of smooth functions H : S 1 × W → R satisfying the following conditions:

1. H is negative and C 2 -small on S 1 × W ;

2. there exists

ρ 0 ≥ 0 such that H(θ, p, ρ) = βe ρ + β ′ for ρ ≥ ρ 0 , with 0 < β / ∈ Spec(M, α) and β ′ ∈ R; 3. H(θ, p, ρ) is C 2 -close to h(e ρ ) on S 1 × M × [0, ρ 0 ], for h a convex increasing function.
We say furthermore that it is non degenerate if all 1-periodic orbits of X H are nondegenerate, i.e the Poincaré return map has no eigenvalue equal to 1.

The complex SC(H, J) considered is the Floer complex, i.e. the complex generated by 1-periodic orbits of the Hamiltonian vector field X H with boundary δ defined as before through Floer trajectories.

Symplectic Homology

Remark 1.2.2 Condition 1 implies that the only 1-periodic orbits of X H in W are constants; they correspond to critical points of H. On S 1 × M × [0, +∞[, if a Hamiltonian is the pullback of a function on [0, +∞[, H 1 (θ, p, ρ) = h 1 (e ρ ), then the corresponding Hamiltonian vector field is proportional to the Reeb vector field, X θ H 1 (p, ρ) = -h ′ 1 (e ρ )R α (p). Hence, for such a Hamitonian H 1 , with h 1 increasing, the image of a 1-periodic orbit of X H 1 is the image of a periodic orbit of the Reeb vector field -R α of period T := h ′ 1 (e ρ ) located at level M × {ρ}. In particular, condition 2 implies that there is no 1-periodic orbit of X H in M × [ρ 0 , +∞[ for a Hamiltonian H in H std ; indeed; h(e ρ ) = βe ρ + β ′ so h ′ (e ρ ) = β which is assumed to be different from the period of any closed orbit of the Reeb vector field. Condition 3 ensures that for any non constant 1-periodic orbit γ H of X H for a Hamiltonian H in H std , there exists a closed orbit of the Reeb vector field R α of period T < β (with β the slope of H "at ∞"), such that γ H is close to this closed orbit of (minus) the Reeb vector field located in M × {ρ} with T = h ′ (e ρ ).

Remark 1.2.3 We can consider a larger class of admissible Hamiltonians, removing conditions 1 and 3. It will not change the definition of the symplectic homology. However condition 1 allows to identify 1-periodic orbits of small action with critical points of H in W . It will be important in the definition of positive symplectic homology.

We denote again by P(H) the set of 1-periodic orbits of X H . Definition 1.2.4 The class J of admissible J : S 1 → End(T W ) : θ → J θ consists of smooth loops of compatible almost complex structures J θ on W , such that, at infinity (i.e. for ρ large enough) J is autonomous (i.e. independent of θ), invariant under translations in the ρ variable, and satisfies

J θ ξ = ξ J θ (∂ ρ ) = R α .
The space W is not compact; it is proven in [FH94, CFH95, FHS95, Vit99, Oan08] that the Floer homology SH(H, J) := HSC((H, J), ∂) is well-defined for a pair (H, J) in a set of so-called regular pairs (HJ ) reg which is of second Baire category in the set of pairs of admissible non degenerate Hamitonians and admissible loops of almost complex structures. One can even fix a non degenerate admissible H and the set of H-regular J's, i.e. admissible J so that (H, J) ∈ (HJ ) reg , is of second Baire category in the set of admissible loops of almost complex structures. Furthermore, SH(H, J) is independent of the choice of the H-regular J [START_REF] Floer | Symplectic fixed points and holomorphic spheres[END_REF]. It is however dependent of the choice of H. An important point in the proofs is that the Floer trajectories with finite energy are confined due to a maximum principle.

The symplectic homology is defined as a direct limit of SH(H, J) over H non degenerate in H std . To define the direct limit one needs a partial order ≤ on H std and morphisms SH(H 1 , J 1 ) → SH(H 2 , J 2 ) whenever H 1 ≤ H 2 are non degenerate, and these morphisms should have nice composition rules.

Background on symplectic homology

• The partial order on H std is given by

H 1 ≤ H 2 if H 1 (θ, x) ≤ H 2 (θ, x) for all (θ, x) ∈ S 1 × W (for more general Hamiltonians, it is enough to have H 1 (θ, x) ≤ H 2 (θ, x
) for all (θ, x) outside a compact domain).

• Let (H 1 , J 1 ) and (H 2 , J 2 ) be two regular pairs for ( W , ω) with H 1 ≤ H 2 , and consider a smooth increasing homotopy of regular pairs between them, (H s , J s ) where s ∈ R and (H s , J s ) is constant for |s| large. By increasing, we mean ∂ s H s ≥ 0 (again for more general H it is enough to consider ∂ s H s ≥ 0 outside a compact subset to be able to define as below a continuation map). The morphism SH(H 1 , J 1 ) → SH(H 2 , J 2 ) is the continuation map induced by this increasing homotopy, when it is regular, as described below.

Consider the Floer equation

∂u ∂s + J θ s • u ∂u ∂θ -X θ Hs • u = 0 (1.3) defined on the set of maps u : R × S 1 → W of class C ∞ loc . If u : R × S 1 → W is a solution of (1.
3), we define its energy to be

E(u) := +∞ -∞ S 1
∂u ∂s 2 dθds.

(1.4)

The fact that the homotopy is increasing insures that Floer trajectories with finite energy are confined. Let γ 1 ∈ P(H 1 ) and γ 2 ∈ P(H 2 ), we denote by

M(γ 1 , γ 2 , H s , J s ) the space of solutions u : R×S 1 → W of (1.3) with finite energy, E(u) < ∞ such that lim s→-∞ u(s, •) = γ 1 and lim s→+∞ u(s, •) = γ 2 .
Remark that in this case there is no R-action on the space M(γ 1 , γ 2 , H s , J s ).

Theorem 1.2.5 ([Oan08, FHS95]) For a generic choice of the homotopy (H s , J s ), the spaces M(γ 1 , γ 2 , H s , J s ), are manifolds of dimension (-µ CZ (γ 1 )) -(-µ CZ (γ 2 )) for all γ 1 ∈ P(H 1 ) and

γ 2 ∈ P(H 2 ). Moreover, if µ CZ (γ 1 ) = µ CZ (γ 2 ), then M(γ 1 , γ 2 , H s , J s ) is compact.
We will call such a homotopy, regular.

Definition 1.2.6 Let (H 1 , J 1 ) and (H 2 , J 2 ) be two regular pairs for ( W , ω) with H 1 ≤ H 2 , and consider a smooth regular increasing homotopy (H s , J s ) between them. Such a regular homotopy induces a morphism, called a continuation morphism

φ (Hs,Js) : SC * (H 1 , J 1 ) → SC * (H 2 , J 2 ) γ → γ ′ ∈P(H 2 ) -µ CZ (γ ′ )=-µ CZ (γ) #M(γ, γ ′ , H s , J s )γ ′
where, again, # is a count of points in a compact 0-dimensional space, with signs.

Symplectic Homology

The fact that it is a chain map relies on the study of the boundary of the 1-dimensional manifold M(γ 1 , γ 2 , H s , J s ) with -µ CZ (γ 1 ) = -µ CZ (γ 2 ) + 1.

Theorem 1.2.7 (for instance [START_REF] Audin | Théorie de Morse et homologie de Floer[END_REF] theorem 11.3.10) Let (u n ) be a sequence of elements in M(γ 1 , γ 2 , H s , J s ). There exists 1. a subsequence of (u n ),

2. x 0 = γ 1 , x 1 , . . . , x k critical points of A H 1 , 3. y 0 , . . . , y l-1 , y l = γ 2 critical points of A H 2 , 4. sequences of real numbers (s i n ) for 0 ≤ i ≤ k -1 and (s ′j n ) for 0 ≤ j ≤ l -1 such that s i n → n→∞ -∞ and s ′j n → n→∞ +∞, 5. elements u i ∈ M(x i , x i+1 , H 1 , J 1 ) for 0 ≤ i ≤ k -1 and, for 0 ≤ j ≤ l -1, elements v j ∈ M(y j , y j+1 , H 2 , J 2 ), 6. an element w ∈ M(x k , y 0 , H s , J s ) such that, for 0 ≤ i ≤ k -1 and for 0 ≤ j ≤ l -1, lim n→+∞ u n • s i n = u i , lim n→+∞ u n • s ′j n = v j in C ∞ loc with (u • s)(s ′ , t) = u(s + s ′ , t
), and such that

lim n→+∞ u n = w in C ∞ loc .
Theorem 1.2.8 (for instance [START_REF] Audin | Théorie de Morse et homologie de Floer[END_REF] proposition 11.2.8) At the homological level, the induced morphism, φ (Hs,Js) is independent of the choice of the regular homotopy between (H 1 , J 1 ) and (H 2 , J 2 ).

The theorem above is one of the versions of the so-called "homotopy of homotopies" theorem; we give a proof in section 1.2.3. The fact that the continuation morphisms SH(H 1 , J 1 ) → SH(H 2 , J 2 ) for H 1 ≤ H 2 compose nicely results from the following theorem.

Theorem 1.2.9 (for instance [START_REF] Audin | Théorie de Morse et homologie de Floer[END_REF] proposition 11.2.9) Consider three regular pairs (H 1 , J 1 ), (H 2 , J 2 ) and (H 3 , J 3 ) for ( W , ω). Let (H s , J s ) and (H ′ s , J ′ s ) be two regular homotopies between (H 1 , J 1 ) and (H 2 , J 2 ) and (H 2 , J 2 ) and (H 3 , J 3 ) respectively. Then there exists a regular homotopy

(H ′′ s , J ′′ s ) between (H 1 , J 1 ) and (H 3 , J 3 ) such that φ (Hs,Js) • φ (H ′ s ,J ′ s )
and φ (H ′′ s ,J ′′ s ) induces the same homomorphism in homology.

This theorem is based on the gluing lemma as explained in subsection 1.1.1 (see also Theorem 3.1.13).

1. Background on symplectic homology Definition 1.2.10 The symplectic homology of (W, ω) is defined as the direct limit

SH * (W, ω, X) := lim -→ H∈H std SH * (H, J)
where, for each H, J : S 1 → End (Γ(T W ) is chosen so that (H, J) is a regular pair.

Example: the ball B 2n

We consider the ball B 2n with the symplectic form which is the restriction of the standard symplectic 2-form ω std = i 2 dz∧dz on C n and with the Liouville radial vector field defined by

X rad = 1 2 (z∂ z + z∂ z ).
The completion is given by B 2n = C n with the standard symplectic form ω std = i 2 dz ∧ dz. We look at Hamiltonians

H C : C n → R : z → C z 2
such that C π / ∈ Z. These Hamiltonians are not in H std but form an admissible cofinite family by Remark 1.2.3. For each C, the Hamiltonian vector field is

X H C = -2iC(z∂ z -z∂ z ).
The integral trajectories are of the form z(t) = e -2iCt z 0 ; therefore, the only 1-periodic orbit of X H is the critical point z = 0. The Floer chain groups are thus

SC * (H C ) = Z if * = -µ CZ (0) 0 otherwise
and, since the differential is 0, the homology groups are the same as the chain groups. The Conley-Zehnder index of the constant orbit at z = 0 depends on C and is given by

-µ CZ (0) = 2n C π + n. Let C k := kπ + ǫ where k ∈ N ≥0 and ǫ > 0. The continuation maps ϕ k : SC * (H C k ) → SC * (H C k+1
) are all identically zero, thus the symplectic homology is also 0

SH * (B 2n , ω std , X rad ) = lim -→ k SH * (H C k ) = 0.

Parenthesis on the homotpy of homotopies Theorem

This classical material can be found, for instance, in [AD10, Sal99, Rit].

Definition 1.2.11 Let H 1 and H 2 be two Hamiltonians. We say that an increasing homotopy H s between H 1 and H 2 is regular if for all 1-periodic orbits γ 1 ∈ P(H 1 ) and

γ 2 ∈ P(H 2 ), M(γ 1 , γ 2 , H s , J s ) is a manifold of dimension -µ CZ (γ 1 ) + µ CZ (γ 2 ).

Symplectic Homology

Theorem 1.2.12 The morphism

φ : SH(H 1 , J 1 ) → SH(H 2 , J 2 )
is independent of the choice of the regular homotopy between H 1 and H 2 .

Proof: Consider two regular homotopies K 0 and K 1 joining H 1 and H 2 . We are going to construct an homotopy between φ K 0 and φ K 1 in other word a

S : SC * (H 1 , J 1 ) → SC * +1 (H 2 , J 2 ) satisfying the relation φ K 1 -φ K 0 = S • ∂ H 1 + ∂ H 2 • S.
Consider a homotopy of homotopies K η , η ∈ [0, 1] such that in a neighbourhood of 0,

K η ≡ K 0 and in a neighbourhood of 1, K η ≡ K 1 . For γ 1 ∈ P(H 1 ), γ 2 ∈ P(H 2 ) and η fixed, we denote by M(γ 1 , γ 2 , K η ) the space of Floer trajectories u : R × S 1 → R ∂ s u + J η (∂ θ u -X Kη ) = 0
and define the parametrized moduli space

M K (γ 1 , γ 2 ) := η∈[0,1] M(γ 1 , γ 2 , K η ).
We now use the following two theorems:

Theorem 1.2.13 If µ CZ (γ 1 ) -µ CZ (γ 2 ) + 1 = 0, then M K (γ 1 , γ 2 ) is a compact manifold of dimension 0.
Theorem 1.2.14 ([AD10]) Let us define

Π K (γ 1 , γ 2 ) :=      γ ′ 1 ∈P(H 1 ) µ CZ (γ 1 )-µ CZ (γ ′ 1 )=1 M(γ 1 , γ ′ 1 , H 1 , J 1 )/ R × M K (γ 1 , γ ′ 1 )           γ ′ 2 ∈P(H 2 ) µ CZ (γ ′ 2 )-µ CZ (γ 2 )=1 M K (γ ′ 2 , γ ′ 2 ) × M(γ ′ 2 , γ 2 , H 2 , J 2 )/ R      . If µ CZ (γ 1 ) = µ CZ (γ 2 ), then M K (γ 1 , γ 2 ) ∪ Π K (γ 1 , γ 2 ) is a compact manifold of dimension 1 with boundary equal to Π K (γ 1 , γ 2 ) ∪ {0} × M(γ 1 , γ 2 , K 0 ) with opposite orientation ∪ {1} × M(γ 1 , γ 2 , K 1 ) .

Background on symplectic homology

We are now ready to proceed with the proof of Theorem 1.2.12. We define the homotopy

S : SC * (H 1 , J 1 ) → SC * +1 (H 2 , J 2 ) as follows: if γ 1 ∈ P(H 1 ) such that µ CZ (γ 1 ) = k then S k (γ 1 ) = γ 2 ∈P(H 2 ) µ CZ (γ 2 )=k+1 #M K (γ 1 , γ 2 )γ 2 .
We have, for

γ 1 ∈ P(H 1 ) such that µ CZ (γ 1 ) = k, S • ∂ H 1 (γ 1 ) + ∂ H 2 • S(γ 1 ) = S k-1 γ ′ 1 ∈P(H 1 ) µ CZ (γ ′ 1 )=k-1 #M(γ 1 , γ ′ 1 , H 1 , J 1 )/ R γ ′ 1 + ∂ H 2 γ ′ 2 ∈P(H 2 ) µ CZ (γ ′ 2 )=k+1 #M K (γ 1 , γ ′ 2 )γ ′ 2 = γ 2 ∈P(H 2 ) µ CZ (γ 2 )=k γ ′ 1 ∈P(H 1 ) µ CZ (γ ′ 1 )=k-1 #M(γ 1 , γ ′ 1 , H 1 , J 1 )/ R #M K (γ 1 , γ 2 )γ 2 + γ 2 ∈P(H 2 ) µ CZ (γ 2 )=k γ ′ 2 ∈P(H 2 ) µ CZ (γ ′ 2 )=k+1 #M K (γ 1 , γ ′ 2 ) #M(γ ′ 2 , γ 2 , H 2 , J 2 )/ R γ 2 = γ 2 ∈P(H 2 ) µ CZ (γ 2 )=k #Π K (γ 1 , γ 2 )γ 2 .
On the other side

φ K 1 -φ K 0 = - γ 2 ∈P(H 2 ) µ CZ (γ 2 )=k #M(γ 1 , γ 2 , K 0 ) γ 2 + γ 2 ∈P(H 2 ) µ CZ (γ 2 )=k #M(γ 1 , γ 2 , K 1 ) γ 2 .
Therefore we reach the conclusion using theorem 1.2.14.

Positive symplectic homology

Let (W, ω, X) be a compact symplectic manifold with contact type boundary, satisfying assumptions 1.1.1 and 1.1.2 and let ( W , ω) be its symplectic completion. The idea of positive symplectic homology is to "remove" the data of constant 1-periodic orbits from symplectic homology. We assume in this section that (W, ω, X) is an exact symplectic manifold , i.e. there exists a globally defined 1-form λ such that dλ = ω. We need this assumption in order to identify the set of critical points of a Hamiltonian with its 1-periodic orbits of small action.

Symplectic Homology Let

H : S 1 × W → R be a Hamiltonian in H std (cf. Definition 1.2.

1). Recall that the Hamiltonian action functional

A H : C ∞ contr (S 1 , W ) → R is defined as A H (γ) := - D 2 σ ⋆ ω - S 1 H θ, γ(θ) dθ
where σ : D 2 → W is an extension of γ to the disc D 2 . When the symplectic form is exact, ω = dλ, the action becomes

A H (γ) := - S 1 γ ⋆ λ - S 1 H θ, γ(θ) dθ.
The 1-periodic orbits of H ∈ H std fall into two classes (see Remark 1.2.2)

1. critical points in W ; whose action is strictly less than some small positive constant ǫ; indeed, if (θ, x) is a critical point of H, the action of the constant orbit is equal to -H(θ, x);

2. non-constant periodic orbits lying in W \ W whose action is strictly greater than ǫ; indeed, the action of such an orbit is close, for a given ρ in [0, ρ 0 ] with T = h ′ (e ρ ) in Spec(M, α), to the action of the orbit of the vector field -h ′ (e ρ )R α located in M ×{ρ}; this is given by

-S 1 e ρ α(-h ′ (e ρ )R α )dθ-S 1 h(e ρ )dθ = e ρ h ′ (e ρ )-h(e ρ ) = e ρ T -h(e ρ ); it is positive since h is convex.
The ǫ above is chosen (for instance) as half the minimal value of the periods of closed orbits of the Reeb vector field on M = ∂W . Functions H are chosen so that the value of |H| in S 1 × W is less than ǫ, so that h(e ρ ) is less than 1 2 ǫ (hence e ρ Th(e ρ ) is greater than 3 2 ǫ) and the C 2 -closeness to an autonomous function is such that the actions differ at most by

1 2 ǫ.
Lemma 1.2.15 The action decreases along Floer trajectories, i.e. if u : R × S 1 → W is a solution of equation (1.1) such that

lim s→-∞ u(s, •) = γ -(•) and lim s→+∞ u(s, •) = γ + (•) then A(γ -) ≥ A(γ + ).
For a proof we refer to the more general case of lemma 1.3.18. Let SC ≤ǫ (H, J) be the complex generated by the 1-periodic orbits of action no greater than ǫ. It is built out of critical points of H and it is a subcomplex of SC(H, J). It has been proven by Viterbo in [Vit99, Proposition 1.3] that

H * (SC ≤ǫ (H, J), ∂) ∼ = H * +n (W, ∂W ).
1. Background on symplectic homology Definition 1.2.16 The positive Floer complex is defined as the quotient of the total complex by the subcomplex of critical points;

SC + (H, J) := SC(H, J)/ SC ≤ǫ (H, J)
Remark that the differential induces a differential on the quotient which we still denote ∂.

Positive symplectic homology is defined as a direct limit over non degenerate H ∈ H std of the homology of SC + (H, J). The continuation morphisms defined in Definition 1.2.6 descend to the quotient since the action decreases along a solution of (1.3) cfr lemma 1.3.18 (when the homotopy is increasing everywhere).

Definition 1.2.17 The positive symplectic homology of (W, ω) is defined as

SH + (W, ω, X) := lim -→ H∈H std H * SC + * (H, J), ∂ Remark 1.2.18 The short exact sequence 0 → SC ≤ǫ (H, J) → SC(H, J) → SC + (H, J) → 0 induces a long exact sequence in homology H * +n (W, ∂W ) / / SH * (W, ω, X) v v SH + * (W, ω, X) [-1] h h . (1.5)
Positive symplectic homology can be defined in a wider context. We refer to section 4.2.1 for an explicit construction on compact symplectic manifold with contact type boundary satisfying assumptions 1.1.1 and 1.1.2.

Example: the ball B 2n

We use the long exact sequence :

H * +n (B 2n , ∂B 2n = S 2n-1 ) / / SH * (B 2n , ω std , X rad ) t t SH + * (B 2n , ω std , X rad ) [-1] j j
and the fact that SH * (B 2n ) = 0 to deduce that

SH + * (B 2n , ω std , X rad ) ∼ = H * +n-1 (B 2n , S 2n-1 ).
To compute the relative homology H * (B 2n , S 2n-1 ), we use the exact sequence

H * (S 2n-1 ) / / H * (B 2n ) w w H * (B 2n , S 2n-1 ) [-1] h h .
Thus we have

H * (B 2n , S 2n-1 ) = Z if * = 2n 0 otherwise
and this implies

SH + * (B 2n , ω std , X rad ) = Z if * = n + 1, 0 otherwise.
In particular the positive symplectic homology of the ball has only one generator. This shows that this homology invariant cannot detect all distinct periodic Reeb orbits on the sphere (with a contact structure which is non degenerate).

1.3 S 1 -equivariant symplectic homology

1.3.1 S 1 -equivariant homology
Let X be a topological space endowed with an S 1 -action. If the S 1 -action is free, X/ S 1 is a topological space. The aim of S 1 -equivariant homology is to build on the space X a homology which coincides, when the action is free, with the singular homology of the quotient. One considers the universal principal S 1 -bundle ES 1 → BS 1 .The diagonal action on X × ES 1 is free and one denotes by X × S 1 ES 1 the quotient (X × ES 1 )/ S 1 .

Definition 1.3.1 (Borel) Let X be a topological space endowed with an S 1 -action. The S 1 -equivariant homology of X with Z-coefficients is

H S 1 * (X) := H * (X × S 1 ES 1 , Z).
An axiomatic definition of equivariant homology was stated later by Basu, [Bas], based on the following Proposition:

Proposition 1.3.2
The S 1 -equivariant homology with Z-coefficients is a functor H S 1 * from the category of S 1 -spaces and S 1 -maps to the category of abelian groups and homomorphisms. Let X be a topological space endowed with a S 1 -action, H S 1 * associates to X a sequence of abelian groups:

H S 1 i (X, Z), i ≥ 0. Let f : X → Y be an S 1 -equivariant map between topological spaces endowed with an S 1 -action. It induces homomorphisms f S 1 i : H S 1 i (X, Z) → H S 1 i (Y, Z).
The functor H S 1 * satisfy the two following conditions:

1. Background on symplectic homology

1. If the S 1 -action on X is free, then H S 1 * (X, Z) = H * (X/ S 1 , Z) (the singular homology of X/ S 1 ). 2. If f : X → Y induces an isomorphism f * : H * (X, Z) → H * (Y, Z), then it also induces an isomorphism f S 1 * : H S 1 * (X, Z) → H S 1 * (Y, Z).
Any functor satisfying the two conditions of Proposition 1.3.2 is given by Definition 1.3.1. Indeed, the projection pr 1 : X × ES 1 → X : (x, e) → x is an S 1 -equivariant map which induces an isomorphism

pr 1 * : H * (X × ES 1 , Z) → H * (X, Z)
since ES 1 is contractible. By 2, pr 1 * induces an isomorphism

pr 1 S 1 * : H S 1 * (X × ES 1 , Z) → H S 1 * (X, Z).
Condition 1 then implies

H S 1 * (X, Z) ∼ = H * (X × S 1 ES 1 , Z).

S 1 -equivariant symplectic homology

The setup is the same as for symplectic homology (cf. section 1.2.1). The S 1 -equivariant symplectic homology is defined for any compact symplectic manifold with contact type boundary (W, ω, X). The S 1 -action we are referring to in this section is the reparametrization action on the loop space,

ϕ • γ(θ) = γ(θ -ϕ)
not an action on W . This homology was first introduced by Viterbo in [START_REF] Viterbo | Functors and computations in Floer homology with applications[END_REF]; a different approach, which will be presented in section 1.4 was sketched by Seidel in [START_REF] Seidel | A biased view of symplectic cohomology[END_REF] and a detailed study by Bourgeois and Oancea appears in [BO12, [START_REF] Bourgeois | Fredholm theory and transversality for the parametrized and for the S 1 -invariant symplectic action[END_REF][START_REF] Bourgeois | The index of Floer moduli problems for parametrized action functionals[END_REF].

The functional

Viterbo's idea is to adapt Borel's construction for Morse theory to the space of contractible loops in W with the S 1 -action. We consider the model of ES 1 given as a limit of spheres S 2N +1 for N going to ∞ with the Hopf S 1 -action. To provide S 1 -invariant functionals, we use S 1 -invariant Hamiltonians :

H : S 1 × W × S 2N +1 → R. The S 1 -invariance condition reads, H(θ + ϕ, x, ϕz) = H(θ, x, z), ∀θ, ϕ ∈ S 1 , ∀z ∈ S 2N +1 .
The action functional A : C ∞ (S 1 , W ) × S 2N +1 → R, called the parametrised action functional, is defined as

A(γ, z) := - D 2 σ ⋆ ω - S 1 H θ, γ(θ), z dθ. (1.6) where σ : D 2 → W is an extension of γ to the disc D 2 . It is invariant under the diagonal S 1 -action on C ∞ (S 1 , W ) × S 2N +1 .
The critical points of the parametrised action functional are pairs (γ, z) such that γ ∈ P(H z ) and

S 1 ∂H z ∂z θ, γ(θ) dθ = 0, (1.7)
where H z is the function on S 1 × W defined by H z (θ, x) := H(θ, x, z) and where P(H z ) denote, as before, the set of 1-periodic orbits of X Hz . The set of critical points of A, denoted by

P S 1 (H), is S 1 -invariant. If q = (γ, z) ∈ P S 1 (H), we denote by S q the S 1 -orbit of q S q := {ϕ • q := ϕ • (γ, z) = (ϕ • γ, ϕz) | ϕ ∈ S 1 }.
Such an S q is called nondegenerate if the Hessian d 2 A(γ, z) has a 1-dimensional kernel for some (and hence any) (γ, z) ∈ S q .

Definition 1.3.3 An S 1 -invariant Hamiltonian H is called admissible if H z is in H std (as in Definition 1.2.1) with constant slope independent of z for all z ∈ S 2N +1 and if for any critical point q ∈ P S 1 (H), the S 1 -orbit S q is non degenerate. Let H S 1 ,N be the family of such hamiltonians.

Again, one can consider more general Hamiltonian in H std ; the main point is that it coincides with a linear function with constant slope outside a compact set.

Proposition 1.3.4 ([BO10], Proposition 5.1) The set H S 1 ,N is of second Baire category in the space of S 1 -invariants Hamiltonians H such that H z is in H std with constant slope independent of z for all z ∈ S 2N +1 .

The chain complex

The chain complex of our homology will be generated by the set of S 1 -orbits of critical points of A.

Definition 1.3.5 ( [START_REF] Bourgeois | The index of Floer moduli problems for parametrized action functionals[END_REF]) The parametrized index of a non degenerate circle of critical points S q , with q = (γ, z), is defined as follows. The Hamiltonian H :

S 1 × W × S 2N +1 → R is extended to a function H : S 1 × W × T * S 2N +1 → R by pullback H θ, x, (z, ζ) = H(θ, x, z) = H z (θ, x).
The cotangent bundle T * S 2N +1 is endowed with its canonical symplectic structure dz ∧ dζ and one considers the Hamiltonian vector field

X H = X Hz -∂H ∂z ∂ ζ . The 1-periodic orbits of X H are of the form γ := γ(•), z, ζ(•) with γ a 1-periodic orbit of H z and ζ(θ) = ζ(0)- θ 0 ∂H ∂z θ ′ , γ(θ ′ ), z dθ ′ with (γ, z) a critical

Background on symplectic homology

The parametrized index of the circle of critical points S (γ,z) , µ param S (γ,z) , is defined as the Robbin-Salamon index (see section 6.4) of the path of symplectic matrices defined by the differential of the flow of X H along γ using a trivialisation of T W × T * S 2N +1 over a disk bounded by γ.

The grading in the chain complex of the element S q is equal to -µ param (S q ) + N .

Floer trajectories

To define negative gradient trajectories of the parametrized action functional A, one needs a metric on the loop space and on the sphere. One chooses again a parametrized smooth loop J : S 1 ×S 2N +1 → End(T W ) of almost complex structures on W which are compatible with ω, and a metric g on S 2n+1 ; since the functional is S 1 invariant, we want the following S 1 invariance.

Definition 1.3.6 A parametrized loop of almost complex structures

J : S 1 × S 2N +1 → End(T W ), (θ, z) → J θ z is called S 1 -invariant if J θ+ϕ ϕz = J θ z , ∀θ, ϕ ∈ S 1 , ∀z ∈ S 2N +1
and is called admissible if for all z in S 2N +1 , the loop of almost complex structures J z is in J as defined in Definition 1.2.4. We denote by J S 1 ,N the set of pairs (J, g) consisting of an admissible S 1 -invariant parametrised loop of almost complex structures and an S 1 -invariant Riemannian metric g on S 2N +1 . Definition 1.3.7 Given H ∈ H S 1 ,N , (J, g) ∈ J S 1 ,N and q -, q + ∈ P S 1 (H), we denote by M(S q -, S q + ; H, J, g) the set of S 1 -equivariant Floer trajectories, consisting of pairs (u, z) with u : R × S 1 → W and z : R → S 2N +1 such that

     ∂ s u + J θ z(s) • u ∂ θ u -X θ H z(s) • u = 0 ż(s) - S 1 ∇ z H θ, u(s, θ), z(s) dθ = 0 (1.8) with the asymptotic conditions lim s→-∞ u(s, •), z(s) ∈ S q -, lim s→+∞ u(s, •), z(s) ∈ S q + .
By ∇ z H θ, x, z , we mean the gradient at the point z with respect to the metric g, of the function on S 2n+1 defined by H(θ, x, •).

1.3. S 1 -equivariant symplectic homology Proposition 1.3.8 ([BO10], Proposition 5.2) Assume S q -, S q + ⊂ P S 1 (H) are nondegenerate. Then for any (u, z) ∈ M(S q -, S q + ; H, J, g), the linearisation D u,z of the equation (1.8), extended to suitable Banach spaces is Fredholm of index -µ param S q -+ µ param S q + + 1.

Transversality results

Bourgeois and Oancea have proven a transversality result involving the two following classes of Hamiltonians (section 7 of [START_REF] Bourgeois | Fredholm theory and transversality for the parametrized and for the S 1 -invariant symplectic action[END_REF]):

1. Generic Hamiltonians. The set, denoted H gen , of Hamiltonians H ∈ H S 1 ,N such that:

a) For all (γ, z) ∈ P S 1 (H), γ is a simple embedded curve ;

b) For all distinct elements (γ 1 , z 1 ), (γ 2 , z 2 ) ∈ P S 1 (H), we have γ 1 = γ 2 .

2. Split Hamiltonians. The set, denoted H split , of Hamiltonians H ∈ H S 1 ,N of the form K(x) + f (z), with K C 2 -small on W such that K has either constant and non degenerate 1-periodic orbits, or non constant and transversally non degenerate ones and f is an S 1 -invariant function.

We denote by H * the union of those two sets:

H * = H gen ∪ H split . Definition 1.3.9 ([BO12]) An admissible Hamiltonian H ∈ H S 1 ,N is called strongly admissible if
1. For every (γ, z) ∈ P S 1 (H) such that γ is not constant, we have

X θ Hz γ(θ) = 0, ∀θ ∈ S 1 .
2. For every (γ, z)

∈ P S 1 (H) such that γ is constant (equal to x ∈ W ), there exists a neighbourhood U of {x}×(S 1 •z) in W ×S 2N +1 such that H(θ, x ′ , z ′ ) = K(x ′ )+f (z ′ )
for all θ ∈ S 1 and for all (x ′ , z ′ ) ∈ U . Moreover, x is an isolated critical point of K.

Remark that any Hamiltonian in H * is strongly admissible.

Definition 1.3.10 ([BO12]) Given a strongly admissible Hamiltonian H, a pair (J, g) ∈ J S 1 ,N is called adapted to H if the following hold :

1. For every (γ, z 0 ) ∈ P S 1 (H), we have

J θ z X θ Hz , X θ Hz γ(θ) / ∈ Span J θ z X θ Hz , X θ Hz , ∀θ ∈ S 1 , z ∈ S 1 • z 0 .
1. Background on symplectic homology 2. for every (γ, z 0 ) ∈ P S 1 (H) such that γ is constant (equal to x ∈ W ), there exists a neighbourhood U of {x} × (S 1 • z) in W × S 2N +1 such that J θ z is independent of θ and z on U , i.e. J θ z (x ′ ) = J(x ′ ) for all (x ′ , z) ∈ U and θ ∈ S 1 .

We denote by HJ ′ the set of triples (H, J, g) such that H is a strongly admissible Hamiltonian and (J, g) is adapted to H and we denote by H * J ′ the subset of HJ ′ corresponding to elements H ∈ H * , asking, furthermore, if H ∈ H split , that J is independent of θ, of ρ and of z for ρ ≥ 1.

Given H ∈ H S 1 ,N , we say that a pair (J, g) ∈ J S 1 ,N is regular for H if the linearisation D u,z of the equation (1.8), extended to suitable Banach spaces, is surjective for any q -, q + ∈ P S 1 (H) and any (u, z) ∈ M(S q -, S q + ; H, J, g). We denote the set of those regular pairs by

J S 1 ,N reg (H).
Theorem 1.3.11 ([BO10], Theorem 7.4) There exists an open subset HJ ′ reg ⊂ HJ ′ which is dense in a neighbourhood of H * J ′ ⊂ HJ ′ and consisting of triples (H, J, g) such that H ∈ H S 1 ,N and (J, g) ∈ J S 1 ,N reg (H).

For (H, J, g) ∈ HJ ′ reg , the moduli space M(S q -, S q + ; H, J, g) is a manifold whose dimension is -µ param S q -+ µ param S q + + 1 ; it carries an action of R (by reparametrization in the s-variable) and an action by S 1 coming from the S 1 -invariance of the action A and of the almost complex structure (J, g). We denote by M S 1 (S q -, S q + ; H, J, g) the moduli space quotiented by those two actions. It is a smooth manifold of dimension -µ param S q -+ µ param S q + -1. Definition 1.3.12 The S 1 -equivariant Floer complex SC S 1 ,N * (H, J, g) is the following chain complex:

SC S 1 ,N * (H) := SC S 1 ,N * (H, J, g) := Sq∈P S 1 (H) Z S q
where the grading is defined by |S q | = -µ param S q + N . The S 1 -equivariant differential is defined as

∂ S 1 : SC S 1 ,N * (H, J, g) -→ SC S 1 ,N * -1 (H, J, g) ∂ S 1 (S q -) := S q + ⊂P S 1 (H) |S q -|-|S q + |=1 #M S 1 (S q -, S q + ; H, J, g)S q +
where # is a count with signs defined in [START_REF] Bourgeois | S 1 -equivariant symplectic homology and linearized contact homology[END_REF], obtained by comparing the coherent orientations on M S 1 (S q -, S q + ; H, J, g) with the orientation induced by the infinitesimal generator of the action.

The fact that it is indeed a differential follows from:

1.3. S 1 -equivariant symplectic homology Proposition 1.3.13 ([BO13a], Proposition 2.2) The map ∂ S 1 satisfies ∂ S 1 • ∂ S 1 = 0.
Definition 1.3.14 The S 1 -equivariant Floer homology groups are defined as

SH S 1 ,N * (H, J, g) := H * SC S 1 ,N * (H), ∂ S 1
The S 1 -equivariant Floer homology groups are independent of the choice of the regular pair (J, g).

Proposition 1.3.15 ([BO13a]) Given a Hamiltonian in H S 1 ,N , and two regular pairs for H, (J 1 , g 1 ) and (J 2 , g 2 ) in J S 1 ,N reg , there exists a canonical isomorphism

SH S 1 ,N * (H, J 1 , g 1 ) ≃ SH S 1 ,N * (H, J 2 , g 2 ).
In the sequel we shall denote

SH S 1 ,N * (H, J, g) by SH S 1 ,N * (H). Let H 1 , H 2 ∈ H S 1 ,N
reg be two S 1 -invariant Hamiltonians and let H s be an increasing homotopy between them (i.e ∂ s H s ≥ 0 and there exists s 0 such that for |s| ≥ s 0 , we have

H -|s| = H 1 and H |s| = H 2 ). We consider the solutions of      ∂ s u + J θ s,z(s) ∂ θ u -X H s,z(s) (u) = 0 ż(s) - S 1 ∇ z H s, θ, u(s, θ), z(s) dθ = 0 (1.9)
with the asymptotic conditions

lim s→-∞ u(s, •), z(s) ∈ S q 1 , lim s→+∞ u(s, •), z(s) ∈ S q 2
where q 1 ∈ P S 1 (H 1 ) and q 2 ∈ P S 1 (H 2 ). Let M(S q 1 , S q 2 , H s , J s , g s ) be the space of solutions of (1.9) ; it carries an S 1 -action (but no R-action) and we denote the quotient by M S 1 (S q 1 , S q 2 , H s , J s , g s ).

Proposition 1.3.16 ([BO12], Proposition 2.1) For all q 1 ∈ P S 1 (H 1 ) and q 2 ∈ P S 1 (H 2 ), the M S 1 (S q 1 , S q 2 , H s , J s , g s ) are smooth manifolds of dimension -µ param (S q 1 )+µ param (S q 2 ).

We can define the continuation morphism :

φ : SC S 1 ,N * (H 1 , J 1 , g 1 ) → SC S 1 ,N * (H 2 , J 2 , g 2 ) by φ(S q 1 ) := Sq 2 ⊂P S 1 (H 2 ) |Sq 1 |-|Sq 2 |=1 #M S 1 (S q 1 , S q 2 ; H s , J s , g s ) S q 2 (1.10)
where q 1 ∈ P S 1 (H 1 ) and q 2 ∈ P S 1 (H 2 ). These morphisms are chain maps and thus pass to the quotient where we still denote them by φ. As previously, we define the S 1 -equivariant homology groups of W to be the direct limit over continuation maps of the S 1 -equivariant Floer homology groups

SH S 1 ,N * (W, ω, X) := lim -→ H∈H S 1 ,N SH S 1 ,N * (H).
(1.11)

We then take the direct limit over N with respect to the S 1 -equivariant embeddings

S 2N +1 ֒→ S 2N +3 which induce maps SH S 1 ,N * (W, ω, X) → SH S 1 ,N +1 * (W, ω, X) for each N .
Definition 1.3.17 The S 1 -equivariant symplectic homology of W is

SH S 1 * (W, ω, X) := lim -→ N SH S 1 ,N * (W, ω, X).

Example: the ball B 2n

The idea is the same as in section 1.2.2. We take a Hamiltonian

H C : C n × S 2N +1 → R : (z, p) → C z 2 + f (p)
where f is the S 1 -invariant lift of a perfect Morse function on CP n-1 . The critical points of A H are (0, p 0 ), . . . , (0, p N -1 ) where

{p i | i = 0 . . . N -1} = Crit( f ). The index of (0, p i ) is -µ param (0, p i ) + N = 2n C π + n + 2i. Therefore SC S 1 ,N * (H C ) = Z if * = 2n C π + n + 2i, i ∈ {0, . . . N -1} 0 otherwise
and, the differential is 0 since the complex is lacunary, thus

SH S 1 ,N * (H C ) ≃ SC S 1 ,N * (H C ). If we let C k = kπ(1 + 2N ) + ǫ for ǫ > 0, the continuation maps ϕ k : SC S 1 ,N * (H C k ) → SC S 1 ,N * (H C k+1 ) are identically 0 and thus SH S 1 ,N * (B 2n , ω std , X rad ) = lim -→ N lim -→ k SH S 1 ,N * (H C k ) = 0. 1.3. S 1 -equivariant symplectic homology 1.3.3 Positive S 1 -

equivariant symplectic homology

As in section 1.2.4, we assume that (W, ω) is an exact compact symplectic manifold with contact type boundary in order to identify the complex generated by 1-periodic orbits of X H of action ≤ ǫ with the complex generated by the critical points of H. To see that it is a subcomplex, we study the action along the Floer trajectories.

Proposition 1.3.18 The action decreases along a parametrized Floer trajectory; i.e a solution of (1.9).

Proof: Since the action for the Hamiltonian H s along the pair u(s, •), z(s) is given by

A H s,z(s) u(s, •), z(s) = - D 2 σ ⋆ s ω - S 1 H s, θ, u(s, θ), z(s)
where

σ s : D 2 → W is an extension of γ s = u(s, •) to the disc D 2 . By the asphericity condition, D 2 σ ⋆ s ω = D 2 σ ⋆ s 0 ω + S 1 ×[s 0 ,s] u ⋆ ω so that ∂ ∂s A H s,z(s) u(s, •), z(s) = - S 1 ω(∂ s u, ∂ θ u)dθ - S 1 ∂ ∂u H s, θ, u(s, θ), z(s) ∂ ∂s u(s, θ)dθ - S 1 ∇ z H s, θ, u(s, θ), z(s) • ż(s)dθ - S 1 ∂ ∂s H s, θ, u(s, θ), z(s) dθ = - S 1 ω(∂ s u, ∂ θ u)dθ - S 1 dH(∂ s u)dθ - S 1 ∇ z H s, θ, u(s, θ), z(s) dθ • S 1 ∇ z H s, θ, u(s, θ), z(s) dθ - S 1 ∂ ∂s H s, θ, u(s, θ), z(s) dθ
The last two terms -S 1 ∇ z H s, θ, u(s, θ), z(s) dθ

- S 1 ω(∂ s u, ∂ θ u)dθ - S 1 ω(X H , ∂ s u)dθ = - S 1 ω(∂ s u, ∂ θ u -X H z(s) )dθ = - S 1 ω(∂ s u, J θ z(s) ∂ s u)dθ = -∂ s u 2 gJ θ z(s) ≤ 0.
Corollary 1.3.19 The action decreases along a solution of (1.8).

Background on symplectic homology

Let H ∈ H S 1 ,N . By the above Corollary, the complex generated by 1-periodic orbits of X H of action ≤ ǫ (denoted SC S 1 ,N,≤ǫ (H, J, g)) is a subcomplex of SC S 1 ,N (H, J, g). It is built out of critical points of H. As in Viterbo [Vit99, Proposition 1.3] we have

H * (SC S 1 ,N,≤ǫ (H, J, g), ∂ S 1 ) ∼ = H S 1 * +n (W, ∂W ).
where the S 1 -action on the pair (W, ∂W ) is the trivial one.

Definition 1.3.20 Let H ∈ H S 1 ,N reg be a Hamiltonian. The positive S 1 -equivariant complex is defined as

SC S 1 ,N,+ (H, J, g) := SC S 1 ,N (H, J, g)/ SC S 1 ,N,≤ǫ (H, J, g) .
Remark that the differential induces a differential on the quotient which we still denote ∂ S 1 . The continuation morphisms defined in equation (1.10) descend to the quotient since the action decreases along a solution of (1.9) cfr Proposition 1.3.18. The positive S 1 -equivariant Floer groups are defined as

SH S 1 ,N,+ (H) := H SC S 1 ,N,+ (H), ∂ S 1 .
Definition 1.3.21 The positive S 1 -equivariant symplectic homology is defined as

SH S 1 ,+ * (W, ω, X) := lim -→ N lim -→ H∈H S 1 ,N SH S 1 ,N,+ * (H).

Example: the ball B 2n

The short exact sequence 0 → SC S 1 ,N,≤ǫ (H, J, g) → SC S 1 ,N (H, J, g) → SC S 1 ,N,+ (H, J, g) → 0 induces a long exact sequence in homology

H S 1 * +n (B 2n , ∂B 2n = S 2n-1 ) / / SH S 1 * (B 2n , ω std , X rad ) t t SH S 1 ,+ * (B 2n , ω std , X rad ) [-1] j j
.

(1.12) The fact that SH S 1 ,N * (B 2n , ω std , X rad ) = 0 implies

SH S 1 ,+ * (B 2n , ω std , X rad ) ∼ = H S 1
* +n-1 (B 2n , S 2n-1 ).

1.4. An alternative presentation of the S 1 -equivariant symplectic homology

The S 1 -action on the pair (B 2n , S 2n-1 ) is trivial ; therefore

H S 1 * (B 2n , S 2n-1 ) = H * (B 2n , S 2n-1 ) ⊗ H * (BS 1 ).
We have as in 1.2.4 that

H * (B 2n , S 2n-1 ) = Z if * = 2n 0 otherwise.
Thus,

SH S 1 ,+ * (B 2n , ω std , X rad ) = Z if * = n + 1 + 2i, i ∈ N, 0 otherwise.

An alternative presentation of the S 1 -equivariant symplectic homology

The definition of S 1 equivariant symplectic homology that we have presented fits nicely in the general picture of symplectic homologies but is hard to tackle. An alternative homology has been suggested by Seidel in [START_REF] Seidel | A biased view of symplectic cohomology[END_REF] and developed by Bourgeois and Oancea in [START_REF] Bourgeois | S 1 -equivariant symplectic homology and linearized contact homology[END_REF].

It has the advantage to use a special class of Hamiltonians and simplified equations for Floer trajectories, so that properties and computations are often feasible. The important point is that this homology coincides with the S 1 equivariant symplectic homology defined above.

We shall look at a subfamily of the S 1 equivariant Hamiltonians H S 1 ,N as defined in Definition 1.3.3. We shall define the generators of our complex and the equations for the Floer trajectories considered to define a differential.

The data

We consider a compact symplectic manifold with compact type boundary (W, ω, X). We choose a perfect Morse function f : CP N → R and a Riemannian metric ḡ on CP N for which the gradient flow of f has the Morse-Smale property (and this will be for each N ). Let f : S 2N +1 → R be the S 1 -invariant lift of f . Let g be the lifted S 1 invariant metric on the sphere S 2N +1 . We denote by Crit( f ) the critical set of f ; it is a union of circles. We choose a point z 0 on each critical circle and we fix a local slice transverse in S 2N +1 to the circle in Crit( f ) at z 0 , considering the hypersurface T z 0 spanned by the stable and the unstable manifold at z 0 (with respect to the gradient ∇ f of f with respect to g). Let U be a neighbourhood of Crit( f ) and let ρ : S 2N +1 → R be a cut-off function on U which is equal to 1 in a neighbourhood U ′ ⊂ U of Crit( f ) and 0 outside U . We define

ǫ := min z∈S 2N +1 \U ′ ∇ f (z) > 0.
1. Background on symplectic homology

Class of admissible Hamiltonians

We look at the subfamily

H S 1 ,N (f ) ⊂ H S 1 ,N consisting of Hamiltonians of the form H + f with H : S 1 × W × S 2N +1 → R in H S 1 ,N (cf. Definition 1.3.3) such that
1. Each critical point (γ, z) of the parametrized action functional A H+ f defined by H + f lies over a z which is a critical point of f ; 2. For every z ∈ Crit( f ), H(•, •, z) has non degenerate periodic orbits;

3. H + f has nondegenerate S 1 -orbits;

4. ∇ z H θ, x, z < ǫ, for all z ∈ S 2N +1 \ U ′ ; 5. For all z ∈ U ′ , ∇ z H • ∇ f (z) = 0.
Remark 1.4.1 Condition 3 can be replaced by the following : near every critical orbit of f , we have H(θ, x, z) = H ′ (θφ z , x), where φ z ∈ S 1 is the unique element such that the action of its inverse brings z into T z 0 , i.e. φ -1 z • z ∈ T z 0 and H ′ ∈ H std . In fact, we shall consider elements H which are built from an H ′ : S 1 × W → R in H std as in Definition 1.2.1, close to an autonomous Hamiltonian; we shall develop this in the next chapter.

The chain complex

Given an admissible H + f , the set P S 1 (H + f ) of critical points (γ, z) of the parametrized action functional A H+ f arise in circles and each one of those circle gives a generator of the chain complex. So the complex is generated by the set

S (γ,z) := S 1 • γ, z .
The index of the generator S (γ,z) is defined to be µ(S γ,z ) := -µ CZ (γ) + µ M orse (z; -f ).

The differential operator

Let (J θ z ) be an S 1 -invariant family of almost complex structures independent of z along each local slice. Let p -= (γ -, z -) and p + = (γ + , z + ) be two critical points of A H+ f . We denote by M(S p -, S p + ; H, f, J θ z , g) the space of solutions (u, z), u : R × S 1 → W , z : R → S 2N +1 to the system of equations

∂ s u + J θ z(s) • u ∂ θ u -X H z(s) • u = 0 ż -∇ f (z) = 0
1.4. An alternative presentation of the S 1 -equivariant symplectic homology with the conditions

lim s→-∞ u(s, •), z(s) ∈ S p - lim s→∞ u(s, •), z(s) ∈ S p + . (1.13)
This system is to be compared to the system (1.9). If S p -= S p + , we denote by M(S p -, S p + ; H, f, J θ z , g) the quotient of M(S p -, S p + ; H, f, J θ z , g) by the reparametrization R-action. M(S p -, S p + ; H, f, J θ z , g) carries a free S 1 -action and we denote by M S 1 (S p -, S p + ; H, f, J θ z , g) the quotient of M(S p -, S p + ; H, f, J θ z , g) by this S 1 -action.

Proposition 1.4.2 ([BO12]) For generically chosen J θ z and g, the spaces

M S 1 (S p -, S p + ; H, f, J θ z , g) are smooth manifolds of dimension -µ(S p -) + µ(S p + ) -1.
The chain complex is defined as:

S C S 1 ,N * (H, f ) := Sp⊂P S 1 (H+ f ) Z S p . The differential ∂ S 1 : S C S 1 ,N * (H, f ) → S C S 1 ,N * -1 (H, f ) is defined by ∂ S 1 (S p -) := S + p ⊂P S 1 (H+ f ) µ(S p -)-µ(S p + )=1 #M S 1 (S p -, S p + ; H, f, J θ z , g)S p +
with signs defined in [START_REF] Bourgeois | S 1 -equivariant symplectic homology and linearized contact homology[END_REF]. The fact that the homology defined above is the same as the one defined in section 1.3.2 is proven in:

Theorem 1.4.3 ([BO12], Proposition 2.7) For any non degenerate K ∈ H S 1 ,N that coincides with H + f outside a compact set and for any pair (J ′ , g ′ ) ∈ J S 1 ,N which is regular for K and coincides with (J, g) outside a compact set, there is a canonical isomorphism

SH S 1 ,N * (K, J ′ , g ′ ) ≃ H * S C S 1 ,N * (H, f ), ∂ S 1 .
Continuation maps are defined as usual, using the space of solutions (u, s) of

∂ s u + J θ s,z(s) • u ∂ θ u -X H s,z(s) • u = 0 ż -∇ f (z) = 0 (1.14)
with H s + f an increasing homotopy between H 0 + f and H 1 + f . The isomorphisms of Theorem 1.4.3 commute with continuation maps when H 0 ≤ H 1 and when N → ∞.

Hence we have an alternative definition:

1. Background on symplectic homology Definition 1.4.4 The S 1 -equivariant Floer homology groups are defined as

SH S 1 ,N * (H, f, J, g) := H * S C S 1 ,N * (H, f ), ∂ S 1
The S 1 -equivariant symplectic homology groups of W are defined as

SH S 1 * (W, ω, X) := lim -→ N lim -→ H∈H S 1 ,N (f ) SH S 1 ,N * (H, f, J, g)
We show here below that the action decreases along these new trajectories. This allows to define SH S 1 ,+ in the context of exact compact symplectic manifolds with contact type boundary.

Proposition 1.4.5 Let H 0 + f and H 1 + f be Hamiltonians in H S 1 ,N (f ) and let Hs := H s + f be an increasing homotopy between H 0 + f and H 1 + f . If (u, z), u : R × S 1 → W and z : R → S 2N +1 is a solution of

∂ s u + J θ s,z(s) • u ∂ θ u -X H s,z(s) • u = 0 ż -∇ f (z) = 0 with lim s→-∞ u(s, •), z(s) = γ -(•), z - and lim s→+∞ u(s, •), z(s) = γ + (•), z + then A(γ -, z -) ≥ A(γ + , z + ).
Proof: The proof proceeds as in proposition 1.3.18. The parametrized action for the Hamiltonian H s + f on the pair (u(s, •), z(s)) is given by

- D 2 σ ⋆ s ω - S 1 (H s + f ) (θ, u(s, θ), z(s)) dθ
where σ s : D 2 → W is an extension of γ s = u(s, •) to the disc D 2 . By the asphericity condition,

D 2 σ ⋆ s ω = D 2 σ ⋆ s 0 ω + S 1 ×[s 0 ,s] u ⋆ ω so that ∂ ∂s A Hs+ f u(s, •), z(s) = - S 1 ω(∂ s u, ∂ θ u)dθ - S 1 ∂ ∂u H s θ, u(s, θ), z(s) ∂ ∂s u(s, θ)dθ - S 1 ∇ z (H s + f ) θ, u(s, θ), z(s) • ż(s)dθ - S 1 ∂ ∂s (H s + f ) θ, u(s, θ), z(s) dθ = - S 1 ω(∂ s u, ∂ θ u)dθ - S 1 dH s,z(s) (∂ s u)dθ - S 1 ∇ z (H s + f ) s, θ, u(s, θ), z(s) • ∇f (z)dθ - S 1 ∂ ∂s (H s + f ) θ, u(s, θ), z(s) dθ.
The first line can be rewritten as

- S 1 ω(∂ s u, ∂ θ u)dθ - S 1 ω(X H s,z(s) , ∂ s u)dθ = - S 1 ω(∂ s u, ∂ θ u -X H s,z(s) )dθ = - S 1 ω(∂ s u, J θ z(s) ∂ s u)dθ = -∂ s u 2 gJ θ z(s) ≤ 0.
The last two terms rewrite as

- S 1 ∂ ∂s (H s + f ) θ, u(s, θ), z(s) dθ - S 1 ∇ z H s, θ, u(s, θ), z(s) • ∇f (z)dθ - S 1 ∇f (z) 2 dθ ≤ 0 .
Condition 4 and 5 conclude.

Remark 1.4.6 With the assumptions of Proposition, 1.4.5, it appears in the proof above that

∂ s u 2 gJ θ z(s) dsdθ ≤ A(γ -, z -) -A(γ + , z + ). 57
1. Background on symplectic homology Definition 1.4.7 Let H ∈ H S 1 ,N (f ) be a Hamiltonian. The positive S 1 -equivariant complex is defined as

S C S 1 ,N,+ (H, f ) := S C S 1 ,N (H, f )/ S C S 1 ,N,≤ǫ (H, f )
where S C S 1 ,N,≤ǫ (H, f ) is the set of critical points of A H+ f of action less than ǫ. The differential passes to the quotient where we still denote it ∂ S 1 and the positive S 1 -equivariant Floer groups are defined as

SH S 1 ,N,+ (H, f ) := H S C S 1 ,N,+ (H, f ), ∂ S 1 .
Observe that the f should more precisely read f N in all the above construction.

Definition 1.4.8 The positive S 1 -equivariant symplectic homology is defined by

SH S 1 ,+ * (W, ω, X) := lim -→ N lim -→ H∈H S 1 ,N (f N ) SH S 1 ,N,+ * (H, f N ).
We assume (W, ω, X) to be exact and we assume the function f to be small in order to a identify 1-periodic orbits of small action with a pair (p, z), p a critical points of H.

2 SH S 1 ,+ and periodic Reeb orbits

The goal of this section is to relate the positive S 1 -equivariant homology of an exact compact symplectic manifold (W, dλ), which is a so-called Liouville domain, to the Reeb orbits on (M = ∂W, α = λ | M ); this relation is expressed in Theorem 2.2.2. We use the alternative description of S 1 -equivariant symplectic homology ([Sei08] , [START_REF] Bourgeois | S 1 -equivariant symplectic homology and linearized contact homology[END_REF]) presented in section 1.4, for a nice subclass of Hamiltonians.

Liouville domains

Definition 2.0.9 A Liouville domain (also called compact symplectic manifold with restricted contact type boundary) is a compact manifold W with boundary ∂W = M , together with a 1-form λ such that ω := dλ is symplectic and the Liouville vector field X defined by ι(X)ω = λ points strictly outwards along ∂W . The Liouville domain will be denoted (W, λ).

Let us observe that the asphericity condition is automatically satisfied. We still assume that c 1 (T W ), π 2 (W ) = 0. We have defined the (symplectic) completion of a compact symplectic manifold with contact type boundary in section 1.2.1. We consider the completion

W = W ∪ (∂W × R + )
of a Liouville domain (W, λ), built from the flow of the Liouville vector field X. We denote by λ the 1-form on W defined by λ on W and by e ρ α on ∂W × R + with α := λ | ∂V . The completion will be denoted ( W , λ).

The multicomplex defining positive S 1 -equivariant homology

The "nice subclass" of Hamiltonians that we use was introduced in [START_REF] Bourgeois | S 1 -equivariant symplectic homology and linearized contact homology[END_REF]. The Hamiltonians are constructed using elements in H std which are small perturbations of autonomous Hamiltonians as we shall now indicate.

2. SH S 1 ,+ and periodic Reeb orbits 2.1.1 Construction of admissible Hamiltonians from elements in H std .

As in section 1.4, we fix a sequence of perfect Morse functions f N : CP N → R, which are C 2 -small, together with a Riemannian metric ḡN for which the gradient flow of f N has the Morse-Smale property. For instance, we can take

f N [w 0 : . . . : w N ] = C N j=0 a j w j 2 N j=0 |w j | 2 with a i+1 > a i ∈ R and C < 0 ∈ R
and the standard metric. We denote by fN : S 2N +1 → R their S 1 -invariant lift, and by Crit( fN ) the set of critical points of fN (which is a union of circles).

We choose a point z j on the critical circle which projects on the critical point of -f N of index 2j. In our example, the point z j ∈ S 2N +1 can be taken as the point (w 0 , . . . , w N ) with w i = δ i j . We fix a local slice transverse in S 2N +1 to the circle in Crit( fN ) at z j . This local slice is the hypersurface T z j spanned by the stable and the unstable manifold at z j (with respect to the gradient ∇ fN of fN with respect to g N ). In our example,

T z j = {(w 0 . . . , w N ) ∈ S 2N +1 | w j ∈ R + }.
We consider U N a neighbourhood of Crit( fN ) and ρN :

S 2N +1 → R a S 1 invariant cut-off function on U N which is equal to 1 in a neighbourhood U ′ N ⊂ U N of Crit( fN ) and 0 outside U N . We set ǫ N := min z∈S 2N +1 \U ′ N ∇ fN (z) > 0. If H ′ ∈ H std , we can create an S 1 -invariant Hamiltonian H : S 1 × W × S 2N +1 → R from H ′ and f . Define H : S 1 × W × U N → R by H(θ, x, z) := H ′ (θ -φ z , x) where φ z ∈ S 1
is the unique element such that φ -1 z • z ∈ T z j when z is close to the critical circle including z j . We extend H to H :

S 1 × W × S 2N +1 → R, by H(θ, x, z) := ρN (z) H(θ, x, z) + 1 -ρN (z) β(x)H ′ (θ, x)
(2.1) using the cutoff function ρN on S 2N +1 and a function β : W → R which is 0 where H ′ is time-dependent and equal to 1 outside a compact set. The element H is admissible, i.e. is in H S 1 ,N (f ) as defined in section 1.4, when assumption 4 is satisfied; this will be the case when we restrict ourselves to a subclass of H ′ ∈ H std , consisting of small perturbations of some autonomous functions. This will be developed in section 2.1.3.

The complex for a subclass of special Hamiltonians

Let H ′ : S 1 × W → R in H std be fixed, with non degenerate 1-periodic orbits. We consider a sequence

H N ∈ H S 1 ,N , N ≥ 1 such that H N (θ, x, z) = H ′ (θ -φ z , x)
for every z ∈ Crit( fN ) (see for instance construction 2.1) and a sequence J N ∈ J S 1 ,N such that J N is regular for H N .

2.1. The multicomplex defining positive S 1 -equivariant homology Let i 0 : CP N ֒→ CP N +1 : [w 0 : . . . : w N -1 ] → [w 0 : . . . : w N -1 : 0] and i 1 : CP N ֒→ CP N +1 : [w 0 : . . . : w N -1 ] → [0 : w 0 : . . . : w N -1 ] and denote by ĩ0 : S 2N +1 → S 2N +3 : z → (z, 0) and ĩ1 : S 2N +1 → S 2N +3 : z → (0, z) their lifts. We assume:

1. Im(i 0 ) and Im(i 1 ) are invariant under the gradient flow of f N +1 ;

2. f N = f N +1 • i 0 = f N +1 • i 1 + cst and i ⋆ 1 ḡN+1 = i ⋆ 0 ḡN+1
= ḡN (this will be true for our example when a i+1 = a i + 1 for all i);

3. H N +1 •, •, ĩ1 (z) = H N +1 •, •, ĩ0 (z) = H N (•, •, z); 4. J N +1, ĩ1 (z) = J N +1, ĩ0 (z) = J N,z .
The critical points of A H N + fN are pairs (γ z , z) where z is a critical point of fN and where γ z is a φ z -translation of a 1-periodic orbit γ of H ′ in W (i.e γ z (θ) = γ(θφ z ) which writes γ z = φ z • γ). We have thus a natural identification (with gradings)

S C S 1 ,N * (H N , f N ) ≃ Z[u]/ u N +1 ⊗ Z SC * (H ′ , J) S 1 • (γ z j , z j ) → u j ⊗ γ =: u j γ
where z j is the chosen critical point of -fN of index 2j and u is a formal variable of degree 2.

The differential, under this identification of complexes, writes

∂ S 1 (u l ⊗ γ) = l j=0 u l-j ⊗ ϕ j (γ). (2.2)
for maps ϕ j : SC * (H ′ ) → SC * +2j-1 (H ′ ), j = 0, . . . N defined by counting elements of M S 1 (S (γ - z j ,z j ) , S (γ + z 0 ,z 0 ) ; H N , f N , J N , g N ) which is the quotient by the R and the S 1 -action of the space of solutions of

∂ s u + J θ z(s) • u ∂ θ u -X H N,z(s) • u = 0 ż -∇ f (z) = 0 going from S 1 • (γ - z j , z j ) to S 1 • (γ + z 0 , z 0 )
. It follows from the assumptions (1), (2), (3) and (4) that for a fixed j, the maps ϕ j obtained for varying values of N ≥ j coincide. Therefore we can encode the limit as

N → ∞ of all the S C S 1 ,N * (H N , f N ) into a complex denoted S C S 1 * (H ′ ) := Z[u] ⊗ Z SC * (H ′ ) 61
2. SH S 1 ,+ and periodic Reeb orbits with differential induced by (2.2) that we can formally write as

∂ S 1 = ϕ 0 + u -1 ϕ 1 + u -2 ϕ 2 + . . .
As before, there are well-defined continuation maps induced by increasing homotopies of Hamiltonians and we have Proposition 2.1.1 [BO12] The S 1 equivariant homology of W is given by:

SH S 1 * (W ) := lim -→ H∈H std H S C S 1 * (H ′ ), ∂ S 1 .

Perturbation of Morse-Bott Hamiltonians

We show that we have a good control on the generators of the complex SC * (H ′ ) defining symplectic homology when we choose the admissible Hamiltonian H ′ in H std to be close to an autonomous Hamiltonian. We shall use techniques taken from [START_REF] Bourgeois | Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces[END_REF] where symplectic homology is computed directly from autonomous Hamiltonians. This we do in the context of a compact symplectic manifold with contact type boundary (W, ω, X). We denote by M the boundary ∂W . As before, we denote by ( W , ω) the completion, by ρ the second coordinate on M × R + , and by α the contact form on M defined by α = ι(X)ω | M . We denote by R α the corresponding Reeb vector field. Note that the 1-periodic orbits of X H in W , for H ∈ H MB , are constant and non degenerate by assumption 1. For (p, ρ) ∈ M × R + and H ∈ H MB , we have

X H (p, ρ) = -e ρ h ′ (ρ)R α .
(2.3)

Thus the 1-periodic orbit of X H are either critical points of H in W or non constant 1periodic orbits, located on levels M × {ρ}, ρ ∈ (0, ρ 0 ), which are in correspondence with periodic -R α -orbits of period e ρ h ′ (ρ). Since H is autonomous, every 1-periodic orbit, γ H of X H , corresponding to the periodic Reeb orbit γ, gives birth to a S 1 family of 1-periodic orbits of X H which is denoted by S γ .

We shall modify an element H ∈ H MB , as in [START_REF] Cieliebak | Applications of symplectic homology. II. Stability of the action spectrum[END_REF], to deform this autonomous Hamiltonian into a time-dependent Hamiltonian H δ with only non degenerate 1-periodic orbits. The Hamiltonian H δ (θ, p) will coincide with H(p) outside a neighbourhood of the image of the non-constant 1-periodic orbits of X H . We proceed as follows:

We choose a perfect Morse function on the circle, f : S 1 → R.

For each 1-periodic orbit γ H of X H , we consider the integer l γ H so that γ H is a l γ H -fold cover of a simple periodic orbit:

l γ H := max k ∈ N | γ H (θ + 1 k ) = γ H (θ) ∀θ ∈ S 1 .
This number l γ H is constant on the S 1 -family of 1-periodic orbits of X H corresponding to the periodic Reeb orbit γ. We set l γ = l γ H = 1 T where T is the period of γ. We choose a symplectic trivialization ψ := (ψ 1 , ψ 2 ) :

U γ → V ⊂ S 1 × R 2n-1 between open neighborhoods U γ ⊂ ∂W × R + ⊂ W of the image of γ H and V of S 1 × {0} such that ψ 1 γ H (θ) = l γ θ.
Here S 1 × R 2n-1 is endowed with the standard symplectic form. Let ǧ : S 1 × R 2n-1 → [0, 1] be a smooth cutoff function supported in a small neighborhood of S 1 × {0} such that ǧ| S 1 ×{0} ≡ 1. We denote by fγ the function defined on S γ by f • ψ 1| Sγ .

For δ > 0 and (θ, p, ρ) ∈ S 1 × U γ , we define

H δ (θ, p, ρ) := h(ρ) + δǧ ψ(p, ρ) f ψ 1 (p, ρ) -l γ θ .
(2.4)

The Hamiltonian H δ coincides with H outside the open sets S 1 × U γ .

Lemma 2.1.3 ([CFHW96, BO09b]) The 1-periodic obits of H δ , for δ small enough, are either constant orbits (the same as those of H) or nonconstant orbits which are non degenerate and form pairs (γ min , γ Max ) which coincide with the orbits in S γ starting at the minimum and the maximum of fγ respectively, for each Reeb orbit γ such that S γ appears in the 1-periodic orbits of H. Their Conley-Zehnder index is given by µ CZ (γ min ) = µ CZ (γ)-1 and µ CZ (γ Max ) = µ CZ (γ).

Computing SH S 1 ,+

We consider now the symplectic homologies with coefficients in Q, denoted SH † (W, Q) on a Liouville domain (W, λ). The goal of this section is to show that if the Conley-Zehnder indices of all periodic Reeb orbits on M = ∂W have the same parity, then the positive S 1 equivariant symplectic homology is generated by those periodic Reeb orbits. Let f N : CP N → R be as before a sequence of perfect Morse functions, which we assume here to be C 2 -small.

The class of Hamiltonians

We consider a Hamiltonian denoted H δ,N which is a S 1 -equivariant lift, as in construction 2.1, of a Hamiltonian H δ which is a perturbation, as in section 2.1.3, of a Hamiltonian H in H MB (cf definition 2.1.2) such that the slope a is big and ρ 0 is small.

2. SH S 1 ,+ and periodic Reeb orbits

As mentioned before, the non constant critical points of A H δN + fN are pairs (γ z , z) where z is a critical point of fN and where γ z is a φ z -translation of a non constant 1-periodic orbit γ ′ of H δ in W . Such a γ ′ is of the form γ min or γ Max , located on a level M × {ρ}, ρ ∈ (0, ρ 0 ) corresponding to a periodic orbit of -R α of period T = e ρ h ′ (ρ).

Remark 2.2.1 The action of this critical point is given by

- S 1 γ ⋆ z λ - S 1 (H N + fN )(θ, γ z (θ), z) dθ.
With our assumptions (f small, ρ 0 small), the second term is close to zero. The first term is equal to -S 1 (γ ′ ) ⋆ λ and this is given by e ρ T . Recall that T is called the contact action of the Reeb orbit γ of period T . Hence the action of this critical point is close to T .

We shall take f and H so that the difference between the action of a non constant critical orbit and the period T of the corresponding Reeb orbit is, for any critical orbit, smaller than a quarter of the smallest period, the smallest spectral gap and the smallest distance between two geometrically distinct periodic Reeb orbits.

Theorem 2.2.2 Let (W, λ) be a Liouville domain. Assume there exists a contact form α on ∂W such that the Conley-Zehnder index of all periodic Reeb orbits have the same parity.

Then SH S 1 ,+ (W, Q) = γ∈P(Rα)
Q γ

where P(R α ) denotes the set of periodic Reeb orbits on ∂W .

Proof: Let H be a Hamiltonian in H MB such that the action is distinct for S 1 -families of orbits corresponding to Reeb orbits of different period. This is possible by Remark 2.2.1. We consider, as mentionned above, the S 1 -equivariant functions H δ,N which are lifts of a perturbation H δ of H. We use the natural identification, described in section 2.1.2:

S C S 1 ,N,+ (H δ,N , f N ) ≃ Z[u]/ u N +1 ⊗ SC + (H δ )
and the description of SC + (H δ ) given by Lemma 2.1.3. The complex S C S 1 ,N,+ (H δ,N , f N ) is filtered by the action thanks to Proposition 1.4.5. We take the filtration F p S C S 1 ,N,+ (H δ,N , f N ), p ∈ Z such that for every p ∈ Z, the quotient

F p+1 S C S 1 ,N,+ (H δ,N , f N )/ F p S C S 1 ,N,+ (H δ,N , f N ) is a union of sets {1 ⊗ γ Max , . . . , u N ⊗ γ Max , 1 ⊗ γ min , . . . , u N ⊗ γ min }
corresponding to underlying Reeb orbits γ of the same period T .

We consider the zero page of the associated spectral sequence.

E 0,N p,q := F p+1 S C S 1 ,N,+ p+q (H δ,N , f N )/ F p S C S 1 ,N,+ p+q (H δ,N , f N ) 2.2. Computing SH S 1 ,+
We have "twin towers of generators", one tower corresponding to each periodic Reeb orbit of period T on ∂W ,

u N ⊗ γ Max u -1 ϕ 1 & & u N ⊗ γ min ϕ 0 o o . . . . . . u 2 ⊗ γ Max u -1 ϕ 1 ' ' u 2 ⊗ γ min ϕ 0 o o u ⊗ γ Max u -1 ϕ 1 ' ' u ⊗ γ min ϕ 0 o o 1 ⊗ γ Max 1 ⊗ γ min ϕ 0 o o
with induced differential as in the above diagram with the notation of section 2.1.2. The differential between two elements in distinct towers vanishes. Indeed the corresponding Reeb orbits are geometrically distinct in ∂W so any Floer trajectory linking elements in distinct towers should satisfy

∂ s u 2 gJ θ z(s) dsdθ ≥ dist(γ -, γ + ).
On the other hand, by Remark 1.4.6,

∂ s u 2 gJ θ z(s) dsdθ ≤ A(γ -, z -) -A(γ + , z + )
≤ T γ --T γ + + 1 2 min{distance between two distinct Reeb orbits}. So there cannot be a Floer trajectory between elements in different towers.

To study any given tower, we use the explicit description of ϕ 0 and ϕ 1 .

1. [BO09b, Lemma 4.28] Let γ min , γ Max and H δ be as above. For δ small enough, the moduli space M(γ min , γ Max ; H δ , J)/ R consists of two elements; they have opposite signs, due to the choice of a system of coherent orientations, if and only of the underlying Reeb orbit γ is good. This implies that in our case,

ϕ 0 = 0. 2. [BO12, Lemma 3.3] The map ϕ 1 : SC + * (H δ ) → SC + * +1 (H δ ) acts by ϕ 1 (γ Max ) = k γ γ min if γ is good, 0 if γ is bad.
where k γ is the multiplicity of the underlying Reeb orbit γ i.e. γ is a k γ -fold cover of a simple periodic Reeb orbit. A Reeb orbit is called bad if its Conley-Zehnder index is not of the same parity as the Conley-Zehnder index of the simple Reeb orbit with same image. The factor k γ comes from the fact that γ min (θ) = γ Max (θ + τ 0 ) for a real number τ 0 and that if the underlying Reeb orbit is of multiplicity k γ , γ min (θ) = γ min θ + 1 kγ . So that γ min (θ) = γ Max (θ + τ 0 + m kγ ) for any integer 0 ≤ m ≤ k γ . We have here k γ trajectories and they all appear with the same sign.

To compute the first page E 1;N p,q of the spectral sequence we have the complex E 0;N p,q :

Q 0 / / Q (×kγ ) / / . . . (×kγ ) / / Q 0 / / Q (×kγ ) / / Q 0 / / Q
and thus, in the homology E 1;N p,q , we are left with one copy of Q in degree -µ CZ (γ) and one copy of Q in degree -µ CZ (γ) + 2N . The first page is given by

E 1;N = γ∈P(H δ ) Q γ Max ⊕ Q u N ⊗ γ min .
Due to the assumption, the differential on the first page of the spectral sequence vanishes (because of the same parity of the Conley-Zehnder indices) therefore, for N large enough, it gives the homology

SH S 1 ,N,+ (H δ,N ) = γ∈P(H δ ) Q γ Max ⊕ Q u N ⊗ γ min .
The morphism induced by a regular homotopy between two such Hamiltonians (built from standard Hamiltonians close to Morse Bott Hamiltonians) respects the filtration, thanks to proposition 1.4.5. We can therefore take the direct limit on the pages over those Hamiltonians which form a cofinal family. The inclusion S 2N +1 ֒→ S 2N +3 induces a map

E 1;N = γ∈P(Rα) Q γ Max ⊕ Q u N ⊗ γ min → E 1;N +1 .
which is the identity on the first factor and zero on the second factor. Taking the direct limit over the inclusion S 2N +1 ֒→ S 2N +3 we have

SH S 1 ,+ (W ; Q) = lim -→ N E 1;N = γ∈P(Rα) Q γ .
Remark 2.2.3 Stricto sensu, in the above Theorem, we have only proven that

SH S 1 ,+ (W, Q) = γ∈ P(Rα)
Q γ where P(R α ) are the periodic Reeb orbits contractible in the Liouville domain W . Nonetheless Theorem 2.2.2 is true after extending the definition of SH S 1 ,+ (H) to all 1-periodic orbits of H. This is done in section 2.2.2.

An orbit γ H is bad if the underlying Reeb orbit is bad.

Proposition 2.2.4 There are no bad orbits in the generators of the S 1 -equivariant symplectic homology.

Proof: In the spectral sequence, as above, the twin tower over a bad orbit is as follows:

u N ⊗ γ Max 0 & & u N ⊗ γ min ×(±2) o o . . . . . . u 2 ⊗ γ Max 0 ' ' u 2 ⊗ γ min ×(±2) o o u ⊗ γ Max 0 ' ' u ⊗ γ min ×(±2) o o 1 ⊗ γ Max 1 ⊗ γ min ×(±2) o o
So the complex is:

Q ×(±2) / / Q 0 / / . . . 0 / / Q ×(±2) / / Q 0 / / Q ×(±2) / / Q .
Therefore, on the first page of the spectral sequence, E 1;N p,q = 0.

Corollary 2.2.5 The only generators that may appear in the positive S 1 -equivariant homology are of the form u 0 ⊗ γ Max with γ Max a good orbit.

Corollary 2.2.6 The number of good periodic Reeb orbits of periods ≤ T is bounded below by the rank of the positive S 1 -equivariant symplectic homology of action ≤ T .

2. SH S 1 ,+ and periodic Reeb orbits Theorem 2.2.2 establishes a link between periodic orbits of the Reeb vector field of a contact form on M and the positive S 1 -equivariant symplectic homology of an exact symplectic filling W of M (i.e. a compact symplectic manifold (M, ω = dλ) with contact type boundary such that ∂W = M , α = λ | M )). We shall study, in chapter 3, the invariance of positive S 1 -equivariant symplectic homology. The idea is to use this homology to get information on Reeb orbits on some contact manifolds.

The example of Brieskorn spheres

The Brieskorn manifold Σ(a 0 , . . . , a n ), with all a i ≥ 2 positive integers is defined as the intersection of the singular hypersurface

{(z 0 , . . . , z n ) ∈ C n+1 | z a 0 0 + • • • + z an n = 0} in C n+1 with the unit sphere S 2n+1 ⊂ C n+1 . It is a smooth 2n -1-dimensional manifold which admits a contact form α = i 8 n j=0
a j (z j dz jz j dz j )

with corresponding Reeb vector field

R α = 4i a 0 z 0 , . . . , 4i a n z n .
For any odd number n = 2m + 1 and any p ≡ ±1 mod 8, the Brieskorn manifold Σ(p, 2, . . . , 2) is diffeomorphic to the standard sphere S 4m+1 [START_REF] Brieskorn | Beispiele zur Differentialtopologie von Singularitäten[END_REF]. One defines the contact structures ξ p on S 4m+1 defined as the kernel of the contact form α p with

α p := ip 8 (z 0 dz 0 -z 0 dz 0 ) + i 4 n j=1
(z j dz jz j dz j ).

The fact that the Brieskorn sheres are exactly fillable can be found, for instance, in the book of Geiges [START_REF] Geiges | Contact geometry[END_REF].

Proposition 2.2.7 For p 1 = p 2 , the positive S 1 equivariant homologies of symplectic fillings of the Brieskorn spheres are different.

Proof: We consider the description of the chain complex for those homologies in terms of good periodic orbits of the Reeb vector field, graded by minus their Conley indices. We shall show that all Conley-Zehnder indices are even. To compute them, the first thing to do is to build an explicit perturbation of the contact form so that all periodic Reeb orbits are non degenerate. We proceed as in [START_REF] Ustilovsky | Infinitely many contact structures on S 4m+1[END_REF]. For that one makes the change of coordinate

w 0 = z 0 , w 1 = z 1 w 2j w 2j+1 = 1 √ 2 1 i 1 -i z 2j z 2j+1
, forj ≥ 1.

In these coordinates

Σ(p, 2, . . . 2) =    w ∈ C n+1 w p 0 + w 2 1 + 2 m j=1 w 2j w 2j+1 = 0, |w| 2 = 1    .
Consider the real positive function f : Σ(p, 2, . . . 2) → R given by

f (w) = |w| 2 + m j=1 ǫ j |w 2j | 2 -|w 2j+1 | 2 , where 0 < ǫ j < 1.
The contact form f α defines the same contact structure on Σ(p, 2, . . . 2) as α and its associated Reeb vector field is given by

R f α (w) = 4i p w 0 , 2iw 1 , 2i(1 + ǫ 1 )w 2 , 2i(1 -ǫ 1 )w 3 , . . . , 2i(1 + ǫ m )w n-1 , 2i(1 -ǫ m )w n .
If all the ǫ j are irrational and linearly independent over Q, the only periodic orbits are

γ 0 (t) = re 4it p , ir
p 2 e 2it , 0, . . . , 0 , r > 0, r p + r 2 = 1, 0 ≤ t ≤ pπ; γ + j (t) = 0, . . . , 0, e 2it(1+ǫ j ) 2j , 0, . . . , 0 , 0

≤ t ≤ π 1 + ǫ j , j = 1, . . . , m; γ - j (t) = 0, . . . , 0, e 2it(1-ǫ j ) 2j+1 , 0, . . . , 0 , 0 ≤ t ≤ π 1 -ǫ j , j = 1, . . . , m
and all their iterates, γ N 0 , γ + j N , γ - j N , for all N ≥ 1. Their Conley-Zehnder index is given by

µ CZ γ N 0 = 2N p(n -2) + 4N ; µ CZ γ ± j N = 2 2N p(1 ± ǫ j ) + 2 N 1 ± ǫ j + 2 m k=1 k =j N (1 + ǫ k ) 1 ± ǫ j + N (1 -ǫ k ) 1 ± ǫ j + n -1.
All indices have the same parity, thus applying Theorem 2.2.2, theS 1 -equivariant positive symplectic homologies are generated by the periodic orbits of the Reeb vector field graded by their Conley indices. If p 1 = p 2 , those positive S 1 -equivariant symplectic homologies are different as proven in [Ust99].

Homology with non contractible orbits

To deal with non contractible orbits, one chooses for any free homotopy class of loops a, a representative l a and one chooses a trivialisation of the tangent space along that curve.

2. SH S 1 ,+ and periodic Reeb orbits

For the free homotopy class of contractible loop, l 0 is chosen to be constant loop with constant trivialisation. One ask moreover that l a -1 is l a in the reverse order and with the corresponding trivialisation. One replaces assumption 1.1.1 of asphericity by an assumption of atoroidality namely for any v :

T 2 → W T 2 v ⋆ ω = 0.
We also replace assumption 1.1.2, asking that the first Chern class of the tangent bundle vanishes on all toruses inside W . The action functional induced by a Hamiltonian H becomes A(γ) := -

[0,1]×S 1 u ⋆ ω - S 1 H(θ, γ(θ))dθ
where u : [0, 1] × S 1 → W is a homotopy from l a to γ.

For any loop γ belonging to the free homotopy class a, one chooses a homotopy u : [0, 1] × S 1 → W from l a to γ and one considers the trivialisation of T W on γ induced by u and by the choice of the trivialisation along l a . Let us observe that any Floer trajectory can only link two orbits in the same free homotopy class and as before, the action decreases along Floer trajectories.

As before, the Floer complex is generated by the 1-periodic orbits of H graded by minus their Conley-Zehnder index. The differential "counts" Floer trajectories between two orbits whose difference of grading is 1.

The positive version of symplectic homology is defined as before since the set of critical points of H is still a subcomplex : Floer trajectories can only link a critical point to a contractible orbit.

All the results stated above extend to this framework.

3 Structural properties of symplectic homology

Transfer morphism

In this section, we prove that symplectic homology, positive symplectic homology, S 1equivariant symplectic homology and positive S 1 -equivariant symplectic homology are functors (reversing the arrows) defined on the category where objects are Liouville domains, and morphisms are embeddings. Precisely, we construct a morphism between the (S 1 -equivariant positive) symplectic homologies when one Liouville domain is embedded in another one, and we show that those morphisms compose nicely. Such a morphism, called a transfer morphism, has been studied by Viterbo [Vit99] in the case of the symplectic homology. We adapt his construction to extend it to all the variants of the symplectic homology considered above. Recall definition 2.0.9: A Liouville domain (W, λ) is a compact manifold W with boundary ∂W = M , together with a 1-form λ such that ω := dλ is symplectic and such that the Liouville vector field X defined by ι(X)ω = λ points strictly outwards along ∂W . We still assume that c 1 (T W ), π 2 (W ) = 0. We have defined the (symplectic) completion of a compact symplectic manifold with contact type boundary in section 1.2.1. We consider the completion

W = W ∪ (∂W × R + )
of a Liouville domain (W, λ), built from the flow of the Liouville vector field X. We denote by λ the 1-form on W defined by λ on W and by e ρ α on ∂W × R + with α := λ | ∂V . We shall denote by SH † (W, λ) the symplectic homology SH † (W, dλ, X). Definition 3.1.1 Let (V, λ V ) and (W, λ W ) be two Liouville domains. A Liouville embedding j : (V, λ V ) → (W, λ W ) is a symplectic embedding j : V → W with V and W of codimension 0 such that j ⋆ λ W = λ V . (One can consider, more generally, a symplectic embedding j of codimension 0 such that λ W coincides in a neighbourhood of j(∂V ) in W with λ V + df .)

Structural properties of symplectic homology

The goal of this section is to adapt Viterbo's definition of transfer morphisms between symplectic homology [START_REF] Viterbo | Functors and computations in Floer homology with applications[END_REF] so that it extends to

SH † (W, λ W ) → SH † (V, λ V ) with † = +, S 1 or (S 1 , +).
We first present the construction for symplectic and positive symplectic homology. The idea, as in [START_REF] Viterbo | Functors and computations in Floer homology with applications[END_REF], is to use increasing homotopies between

H 1 : S 1 × W → R ∈ H std (W ) and an H 2 : S 1 × W → R in a special class H stair (V, W ). Let U be a neighbourhood of ∂V in W \ V so that (U, ω W ) is symplectomorphic to ∂V × [0, δ], d(e ρ α V ) . Definition 3.1.2 A Hamiltonian H 2 : S 1 × W → R is in H stair (V, W ) if and only if • on S 1 × V , H 2 is negative and C 2 -small ; • on S 1 × U ∼ = S 1 × ∂V × [0, δ], with ρ the last coordinate, H 2 is of the following form -there exists 0 < ρ 0 ≪ δ such that H 2 (θ, p, ρ) = βe ρ + β ′ for ρ 0 ≤ ρ ≤ δ -ρ 0 , with 0 < β / ∈ Spec(∂V, α) ∪ Spec(∂W, α) and β ′ ∈ R; -H 2 (θ, p, ρ) is C 2 -close on S 1 × ∂V × [0, ρ 0 ]
to a convex increasing function of e ρ which is independent of θ and p;

-H 2 (θ, p, ρ) is C 2 -close on S 1 × ∂V × [δ -ρ 0 ,
δ] to a concave increasing function of e ρ which is independent of θ and p;

• on S 1 × W \ (V ∪ U ), H 2 is C 2 -close to a constant ; • on S 1 × ∂W × [0, +∞[, with ρ ′ the R + coordinate on ∂W × R + , H 2 is of the following form -there exists ρ ′ 1 > 0 such that H 2 (θ, p, ρ ′ ) = µe ρ ′ + µ ′ for ρ ′ ≥ ρ ′ 1 , with 0 < µ / ∈ Spec(∂V, α) ∪ Spec(∂W, α), µ ≤ β, µ ′ ∈ R; -H 2 (θ, p, ρ ′ ) is C 2 -close on S 1 × ∂W ×]0, ρ ′
1 ] to a concave increasing function of e ρ ′ which is independent of θ and p;

• all 1-periodic orbits of X θ H 2 are non-degenerate, i.e the Poincaré return map has no eigenvalue equal to 1.

A representation of H 2 is given in Figure 3.1.

The 1-periodic orbits of H 2 lie either in the interior V (which we call region I), either in

∂V × [0, ρ 0 ] (region II), either in ∂V × [δ -ρ 0 , δ] (region III), either in W \ (V ∪ U ) (region IV) or in ∂W × [0, ρ 1 ] (region V).
We consider their action using the following obvious lemma: Proof:

Transfer morphism

|A H (γ) -A H ( γ)| = S 1 γ ⋆ λ + S 1 H θ, γ(θ) - S 1 γ ⋆ λ - S 1 H θ, γ(θ) ≤ S 1 λ γ(θ) ˙ γ(θ) -λ γ(θ) γ(θ) dθ + S 1
H θ, γ(θ) -H θ, γ(θ) dθ ≤ ǫ I In region I, there are only critical points so the action of the critical point q is non negative and small (< ǫ).

II In region II, H 2 is C 2 -close to a convex function H = h(r) (with r = e ρ ); since dH = h ′ (r)dr = ι(X H )ω W = ι(X H )d(rα V ) = ι(X H )(dr ∧ α V + rdα V ),
we have X H = -h ′ (r)R α V where R α V is the Reeb vector field on ∂V associated to the contact form α V = λ V | ∂V . So an orbit of X H lies on a constant level for r and 3. Structural properties of symplectic homology its action is given by:

A(γ) = - S 1 γ ⋆ (rα V ) - S 1 H γ(θ) dθ = - S 1 (rα V ) γ(θ) γ(θ) - S 1 h(r)dθ = - S 1 rα V -h ′ (r)R α V -h(r) = h ′ (r)r -h(r).
Since ρ 0 is small we have e ρ 0 ∼ 1 and h(e ρ 0 ) ∼ 0, so the actions of 1-periodic orbits of H 2 in this region are close to the periods of closed orbits of the Reeb vector field on the boundary of V of periods T < β and they are greater than ǫ.

III In region III, the computation is similar to the case of region II:

A(γ H 2 ) is close to h ′ (r)r -h(r) which is close to e δ (T -β) < 0
hence the actions of 1-periodic orbits of H 2 in this region are negative.

IV In region IV, there are only critical points so the action of the critical point q is given by -H 2 (q) which is close to -e δ β.

V In region V, the computation of the action is similar to the case of region II:

A(γ) is close to h ′ (r)r -h(r) with r = e ρ ′ .
Observe that here the 1-periodic orbits are close to 1-periodic orbits of -h ′ (r)R α W where now R α W is the Reeb vector field on ∂W . The action of any 1-periodic orbit of H 2 in this region is close to e ρ ′ T ′h(e ρ ′ ) where T ′ is the period of a closed orbit of the Reeb vector field on the boundary of W with T ′ < µ < β and where h(e ρ ′ ) > e δ β.

So, for nice parameters (for instance ρ ′ 1 < δ), we have

A(IV ) < A(III), A(V ) < 0 < A(I) < ǫ < A(II).
We denote by C IV,III,V,I (H 2 , J) the subcomplex of the Floer complex for H 2 generated by critical orbits lying in regions IV, III, V, and I and by C IV,III,V (H 2 , J) the subcomplex of the Floer complex for H 2 generated by critical orbits lying in regions IV, III and V. Observe that C IV,III,V,I (H 2 , J) coincides with the subcomplex generated by 1-periodic orbits of action ≤ ǫ and C IV,III,V (H 2 , J) coincides with subcomplex generated by 1-periodic orbits of action ≤ -η (for a well chosen small positive η). With similar notations, we have the identifications:

C I,II (H 2 , J) = C IV,III,V,I,II (H 2 , J)/ C IV,III,V (H 2 , J) = SC(H 2 , J), ∂)/ SC ≤-η (H 2 , J), ∂) , 3.1. Transfer morphism C II (H 2 , J) = C IV,III,V,I,II (H 2 , J)/ C IV,III,V,I (H 2 , J) = SC(H 2 , J), ∂)/ SC ≤ǫ (H 2 , J), ∂) .
Since the action decreases along Floer trajectories, the Floer differential passes to the quotient where we still denote it ∂. Remark that the function H 2 is not in H std (V ). We want to relate the homology of C I,II (H 2 , J), ∂ to the homology of a function in H std (V ).

Definition 3.1.4 Let H 2 ∈ H stair (V, W ); we denote by β the slope of the linear part close to ∂V , as in Definition 3.1.2. The associated function

H = ι V (H 2 ) ∈ H std (V ), defined on S 1 × V , is the function which coincides with H 2 on V ∪ (∂V × [0, δ -ρ 0 ]
) and which is linear with slope β "further" in the completion: H(θ, e ρ ) = βρ + β ′ for all ρ ≥ δρ 0 .

Proposition 3.1.5 Let H 2 be an function in H stair and let H = ι V H 2 be the associated function in H std (V ) as defined above. We assume furthermore that the Hamiltonians are generic in the sense that the homologies are well-defined for a good choice of J's. Then

H C I,II (H 2 , J), ∂ = H SC(H, J) and H C II (H 2 , J), ∂ = H SC + (H, J) .
Proof: We need to check that there is no Floer trajectory u : R × S 1 → W going from an orbit in C I,II (resp. C II ) to an orbit in C I,II (resp. C II ) with points in W \ (U ∪ V ). We prove it by contradiction, as a direct application of Abouzaid maximum principle which we prove below as theorem 3.1.6. Assume that u : R × S 1 → W is a Floer trajectory whose image intersects W \ (U ∪ V ). We consider the intersection of the image with a slice ∂V × {ρ} for any ρ 0 < ρ < δρ 0 and we choose a regular value

ρ 0 + ǫ of ρ • u. The manifold W ′ := W \ (V ∪ (∂V × [0, ρ 0 + ǫ[))
is symplectic with contact type with boundary ∂V × {ρ 0 + ǫ} and Liouville vector field pointing inwards. Let S be the inverse image of W ′ under the map u; it is a compact Riemann surface with boundary ; the complex structure j is the restriction to S of the complex structure j on the cylinder defined by j(∂ s ) = ∂ θ .

We define β to be the restriction of dθ to S. The fact that u is a Floer trajectory is equivalent to (du -X H ⊗ β) 0,1 := 1 2 ((du -X H ⊗ β) + J(du -X H ⊗ β)j) = 0, where du is the differential of the map u viewed as a section of T * S ⊗ u * T W ′ . Then theorem 3.1.6, which is slight generalisation of a theorem of Abouzaid, concludes. Theorem 3.1.6 (Abouzaid, [START_REF] Ritter | Topological quantum field theory structure on symplectic cohomology[END_REF]) Let (W ′ , ω ′ = dλ ′ ) be an exact symplectic manifold with contact type boundary ∂W ′ , such that the Liouville vector field points inwards. Let ρ be the coordinate near ∂W ′ defined by the flow of the Liouville vector field starting from the boundary and let r := e ρ ; near the boundary the symplectic form writes ω ′ = d(rα) with α the contact form on ∂W ′ given by the restriction of λ ′ . Let J be a compatible almost complex structure such that J * λ ′ = dr on the boundary. Let H : W ′ → R be non negative, and such that H = h(r) where h is a convex increasing function near the boundary. Let S be a compact Riemann surface with boundary and let β be a 1-form such that dβ ≥ 0.

Structural properties of symplectic homology

Then any solution u : S → W ′ of (du -X H ⊗ β) 0,1 = 0 with u(∂S) ⊂ ∂W ′ is entirely contained in ∂W ′ .

Proof: The energy of a map u : S → W ′ is defined as

E(u) := 1 2 S du -X H ⊗ β 2 vol S
where du is viewed as a section of T * S ⊗ u * T W ′ . If s + it is a local holomorphic coordinate on S, so that j(∂ s ) = ∂ t and vol S = ds ∧ dt we have

1 2 du -X H ⊗ β 2 vol S = ω ′ (∂ s u -X H β(∂ s ), ∂ t u -X H β(∂ t )) ds ∧ dt = ω ′ (∂ s u, ∂ t u) -dH(∂ t u)β(∂ s ) + dH(∂ s u)β(∂ t ) ds ∧ dt = u * ω ′ + u * (dH) ∧ β.
It is obviously non negative for any path. Since

d(u ⋆ Hβ) = u ⋆ (dH) ∧ β + u ⋆ Hdβ ≥0
, we have

E(u) = S u ⋆ dλ ′ + u ⋆ (dH) ∧ β ≤ S d(u ⋆ λ ′ ) + d(u ⋆ Hβ) ≤ ∂S u ⋆ λ ′ -λ ′ (X H )β using Stokes's theorem and H = h(r) ≤ rh ′ (r) = rα h ′ (r)R α = -λ ′ (X H ) on u(∂S) ⊂ ∂V = ∂S λ ′ (du -X H ⊗ β) = ∂S -λ ′ J(du -X H ⊗ β)j since (du -X H ⊗ β) 0,1 = 0 = ∂S -dr(du -X H ⊗ β)j since J * λ ′ = dr along u(∂S) ⊂ ∂W ′ = ∂S
-dr du j since dr vanishes on X H on u(∂S) ⊂ ∂W ′

Let ν be the outward normal direction along ∂S. Then (ν, jν) is an oriented frame, so ∂S is oriented by jν. Now dr(du)j(jν) = d(r • u)(-ν) ≥ 0 since in the inward direction, -ν, r • u can only increase because r is minimum on ∂W ′ . So E(u) ≤ 0 hence E(u) = 0. This implies that du -X H ⊗ β = 0 which shows that the image of du is in the span of X H which is the span of R α ∈ T ∂W ′ on ∂W ′ . Hence the image of u is entirely in contained in ∂W ′ .

For any element H 1 ∈ H std (W ), one can consider an element in H 2 ∈ H stair (V, W ) such that H 1 and H 2 coincide "far in the completion", i.e. on ∂W ×

[ρ ′ 2 , +∞[⊂ W . Let H = ι V (H 2 ) ∈ H std (V ).
We want to build a morphism from the homology defined by H 1 to the homology defined by H. We shall first construct a morphism in the homology defined by H 2 . With H 1 ∈ H std (W ) and H 2 ∈ H stair (V, W ) as above, we can consider an increasing homotopy H s , s ∈ R, between H 1 and H 2 , i.e d ds H s ≥ 0, with the property that 3.1. Transfer morphism there exists s 0 such that H s ≡ H 1 for s ≤ -s 0 and H s ≡ H 2 for s ≥ s 0 . We define a morphism SC(H 1 , J 1 ) → SC(H 2 , J 2 ) by counting Floer trajectories for the homotopy. Denote by M(γ 1 , γ 2 , H s , J s ) the space of Floer trajectories from γ 1 to γ 2 i.e maps u : R × S 1 → W such that: ∂Js,Hs (u

) := ∂ s u + J θ s • u(∂ θ u -X θ Hs • u) = 0 (3.1) lim s→-∞ u(s, •) = γ 1 (•) and lim s→∞ u(s, •) = γ 2 (•).
Again it is proven in [START_REF] Oancea | Fibered symplectic cohomology and the Leray-Serre spectral sequence[END_REF][START_REF] Floer | Transversality in elliptic Morse theory for the symplectic action[END_REF] that for a generic choice of the pair (H s , J s ), the spaces M(γ 1 , γ 2 , H s , J s ) are manifolds of dimension µ CZ (γ 2 )µ CZ (γ 1 ) for any γ 1 in P(H 1 ) and γ 2 in P(H 2 ). Let us observe that there is no general R-action on this space. The homotopy H s gives rise to a morphism

φ Hs : SC(H 1 , J 1 ) → SC(H 2 , J 2 ) γ 1 → γ 2 ∈P(H 2 ) µ CZ (γ 2 )=µ CZ (γ 1 ) #M(γ 1 , γ 2 , H s , J s )γ 2
where the count involves, as always, signs. Proposition 3.1.7 The morphism φ Hs is a chain map.

Proof: As before this follows from the study of the boundary of a space of Floer trajectories. Let γ 1 ∈ P(H 1 ) and γ 2 ∈ P(H 2 ) be such that µ CZ (γ 1 ) = µ CZ (γ 2 ) + 1. The 1-dimensional manifold M(γ 1 , γ 2 , H s , J s ) has the following boundary

∪ γ∈P(H 1 ) M(γ 1 , γ, H 1 , J 1 )×M(γ, γ 2 , H s , J s ) ∪ γ∈P(H 2 ) M(γ 1 , γ, H s , J s )×M( γ, γ 2 , H 2 , J 2 ).
The first part yields the coefficient of γ 2 in φ Hs • ∂ H 1 (γ 1 ) and the second part corresponds to ∂ H 2 • φ Hs (γ 1 ).

So φ Hs induces a morphism in homology, still denoted by φ Hs φ Hs : SH(H 1 , J) → SH(H 2 , J).

The fact that φ Hs is independent of the choice of the homotopy is a consequence of the homotopy of homotopies theorem (section 1.2.3). We denote it by φ H 1 ,H 2 . Definition 3.1.8 Given an element H 1 in H std (W ), consider an element H 2 ∈ H stair (V, W ) such that H 1 and H 2 coincide "far in the completion", and let H = ι V (H 2 ) ∈ H std (V ). We define the transfer morphism

SH(H 1 , J) → SH(H, J ′ ) = SH(H 2 , J), ∂)/ SH ≤-η (H 2 , J), ∂) = H C I,II (H 2 , J), ∂
which is the composition of φ H 1 ,H 2 followed by the natural projection.The action decreases along Floer trajectories, so this maps SH ≤ǫ (H 1 , J) to SH ≤ǫ (H 2 , J), ∂)/ SH ≤-η (H 2 , J), ∂) and induces a transfer morphism for the positive homology

SH + (H 1 , J) → SH + (H, J ′ ) = SH ≤ǫ (H 2 , J), ∂)/ SH ≤-η (H 2 , J), ∂) = H C II (H 2 , J), ∂ .
With our identification, the map is obtained by counting solutions of

∂u ∂s + J s • u ∂u ∂θ -X Hs • u = 0 going from a 1-periodic orbit of X H 1 to a 1-
: SH(H 2 ) → SH(H ′ 2 ) is a continuation for H 2 then φ H ′ 1 ,H ′ 2 • ρ 1 = ρ 2 • φ H 1 ,H 2 .
Proposition 3.1.9 The transfer map φ H H 1 : SH(H 1 , J) → SH(H, J ′ ) commutes with continuations.

Proof: To show this, we still have to show that a continuation map built in W from SH(H 2 , J) to SH(H ′ 2 , J ′ ), defined by an increasing homotopy H s :

S 1 × W → R, induces a continuation map in V from SH(H = ι V (H 2 ), J) to SH(H ′ = ι V (H ′
2 ), J ′ ). For this, it is enough to check that there is no Floer trajectory corresponding to the homotopy, i.e. u : R × S 1 → W solution of

∂u ∂s + J s • u ∂u ∂θ -X Hs • u = 0,
going from an orbit in C I,II (H 2 , J) (resp. C II (H 2 , J)) to an orbit in C I,II (H ′ 2 , J ′ ) (resp. C II (H ′ 2 , J ′ )) with points in W \ (U ∪ V ). We prove it by contradiction, proceeding as in the proof of Proposition 3.1.5, using a generalized Abouzaid maximum principle which we prove below as proposition 3.1.10. Assume that u : R × S 1 → W is a Floer trajectory whose image intersects W \ (U ∪ V ). We consider the intersection of the image with a slice ∂V × {ρ} for any ρ 0 < ρ < δρ 0 and we choose a regular value

ρ 0 + ǫ of ρ • u. The manifold W ′ := W \ (V ∪ (∂V × [0, ρ 0 + ǫ[))
is symplectic with contact type with boundary ∂V × {ρ 0 + ǫ} and the Liouville vector field pointing inwards. Let S be the inverse image of W ′ under the map u; it is a compact Riemann surface embedded in the cylinder with boundary ; the complex structure j is the restriction to S of the complex structure j on the cylinder defined by j(∂ s ) = ∂ θ . The fact that u is a Floer trajectory is equivalent to (du -X Hs ⊗ dθ) 0,1 := 1 2 ((du -X Hs ⊗ dθ) + J(du -X Hs ⊗ dθ)j) = 0, where du is the differential of the map u viewed as a section of T * S ⊗ u * T W ′ . Then Proposition 3.1.10 concludes.

Proposition 3.1.10 Let (W ′ , ω ′ = dλ ′ ) be an exact symplectic manifold with contact type boundary ∂W ′ , such that the Liouville vector field points inwards. Let ρ be the coordinate near ∂W ′ defined by the flow of the Liouville vector field starting from the boundary and let r := e ρ . Let J be a compatible almost complex structure such that J * λ ′ = dr on the boundary. Let H : R × S 1 × W ′ → R be an increasing homotopy, such that H(s, θ, p, ρ) = H θ s (p, ρ) = h s (r) where h s are convex increasing functions near the boundary. Let S be a compact Riemann surface with boundary embedded in the cylinder. Then any solution u : S → W ′ of (du -X Hs ⊗ dθ) 0,1 = 0 with u(∂S) ⊂ ∂W ′ is entirely contained in ∂W ′ .

Proof: The proof starts as in Theorem 3.1.6. The energy of u is non negative and given by and the proof proceeds as in Theorem 3.1.6.

E(u) := 1 2 S du -X Hs ⊗ dθ 2 vol S = S u * ω ′ + u * (dH θ s ) ∧ dθ. We have u ⋆ (dH θ s ) ∧ dθ = d(u ′ ⋆ H) ∧ dθ -u ⋆ ∂ s H θ s ds ∧ dθ ≥0 , for u ′ : S → R × S 1 × W ′ which maps an element (θ, s) ∈ S to the element (s, θ, u ′ (θ, s)). Hence E(u) = S u ⋆ dλ ′ + u ⋆ (dH) ∧ dθ ≤ S d(u ⋆ λ ′ ) + d(u ′ ⋆ Hdθ) ≤ ∂S u ⋆ λ ′ -λ ′ (X
Corollary 3.1.11 The transfer maps { φ H H 1 }induce a transfer map:

φ W,V : SH(W, λ W ) → SH(V, λ V ).
and, on the quotient, the morphism

φ + = φ + W,V : SH + (W, λ W ) → SH + (V, λ V ).

Structural properties of symplectic homology

Theorem 3.1.12 (Composition)

Let (V 1 , λ V 1 ) ⊆ (V 2 , λ V 2 ) ⊆ (V 3 , λ V 3
) be Liouville domains with Liouville embeddings. Then the following diagram commutes:

SH + (V 3 , λ V 3 ) φ + V 3 ,V 2 / / φ + V 3 ,V 1 5 5 SH + (V 2 , λ V 2 ) φ + V 2 ,V 1 / / SH + (V 1 , λ V 1 ) (3.2)
Proof: The proof results from the comparison of a count of Floer trajectories. On one hand, one counts Floer trajectories corresponding to an increasing homotopy H 13 , going from a 1-periodic orbit of X H 1 for an admissible Hamiltonian H 1 on S 1 × V 3 to the C II,I part of a stair Hamiltonian H 3 with two "steps". On the other hand, one counts trajectories relative to the composition of two increasing homotopies, H 12 going from H 1 to H 2 (a stait hamiltonian with one step) and H 23 going from H 2 to H 3 . The property is a consequence of the composition of homotopies that we now present.

Let H 1 , H 2 , H 3 be three Hamiltonians on the completion of a symplectic manifold (W, ω) with contact type boundary such that there exist two increasing homotopies H 12 from H 1 to H 2 and H 23 from H 2 to H 3 . We assume, as always here, that there exists s 0 such that H 12 ≡ H 1 for s ≤ -s 0 and H 12 ≡ H 2 for s ≥ s 0 and similarly for H 23 . Denote, for R ∈ R, R ≥ s 0 >> 0, by H 12 # R H 23 the gluing of the two homotopies;

H 12 # R H 23 = H 12 (s + R, •, •) s ≤ 0 H 23 (s -R, •, •) s ≥ 0
The almost complex structures J 12 and J 23 are glued similarly. We choose J 12 and J 23 such that the operators D 12 and D 23 , which linearize the equations (3.1) on the Banach tangent space to a Floer trajectory, are surjective at any solution. By the operator gluing lemma (cfr section 1.1.1), the operator D 12 # R D 23 , corresponding to the linearisation of the Floer equation for H 12 # R H 23 , is surjective and the index of D 12 # R D 23 , is the sum of the indices of D 12 and D 23 . Hence the space of Floer trajectories M(γ 1 , γ 3 , H 12 # R H 23 , J 12 # R J 23 ), for γ 1 ∈ P(H 1 ) and γ 3 ∈ P(H 3 ), is a smooth manifold of dimension µ CZ (γ 1 )µ CZ (γ 3 ).

Theorem 3.1.13 Let γ 1 ∈ P(H 1 ) and γ 3 ∈ P(H 3 ) such that µ CZ (γ 1 ) = µ CZ (γ 3 ). Then for R ′ large enough,

M(γ 1 , γ 3 , H 12 # R ′ H 23 , J 12 # R ′ J 23 ) = γ 2 ∈P(H 2 ) M(γ 1 , γ 2 , H 12 , J 12 ) × M(γ 2 , γ 3 , H 23 , J 23 )
Proof: Let us consider the 1-dimensional manifold defined as the union of 0-dimensional manifolds:

R≥R ′ ≥s 0 M(γ 1 , γ 3 , H 12 # R H 23 , J 12 # R J 23 ).
Its boundary consists of M(γ 1 , γ 3 , H 12 # R ′ H 23 , J 12 # R ′ J 23 ) and of broken trajectories. It can not include

γ∈P(H 1 ) M(γ 1 , γ, H 1 , J 1 ) × M(γ, γ 3 , H 12 # R H 23 , J 12 # R J 23 ).
since the elements of M(γ 1 , γ, H 1 , J 1 ) have index at least equal to 1 and the elements in M(γ, γ 3 , H 12 # R H 23 , J 12 # R J 23 ) are of index at least 0. Thus this boundary is just

M(γ 1 , γ 3 , H 12 # R ′ H 23 , J 12 # R ′ J 23 ) M(γ 1 , γ 2 , H 12 , J 12 ) × M(γ 2 , γ 3 , H 23 , J 23 ).
The conclusion follows.

Transfer morphism for S 1 -equivariant symplectic homology

We extend the definition of the transfer morphisms of the previous section to S 1 -equivariant and positive S 1 -equivariant symplectic homology. We consider two embedded Liouville domains (V, λ V ) ⊂ (W, λ W ) and we want to define a morphism SH S 1 (W, λ W ) → SH S 1 (V, λ V ). We use the alternative definition of the S 1 equivariant symplectic homology, considering the cofinal family of Hamiltonians which are built as in section 2.2 : starting with autonomous Hamiltonians H in H std , we do small Morse Bott type deformations H δ as presented in section 2.1.3 and then lift those to S 1equivariant functions H N δ as in section 2.1.1. In this setting, the S 1 -equivariant symplectic homology can be computed by a simplified complex as described in proposition 2.1.1:

S C S 1 * (H δ ) := Z[u] ⊗ Z SC * (H δ ) with differential ∂ S 1 = ϕ 0 + u -1 ϕ 1 + u -2 ϕ 2 + . . .
where the maps ϕ j counts Floer trajectories for parametrized Hamiltonians

   ∂ s u + J θ z(s) • u ∂ θ u -X H N δ,z(s) • u = 0 ż -∇ f (z) = 0 going from S 1 • (γ -, z j ) to S 1 • (γ + , z 0 ) with z j the critical point of f of index -2j.
We have seen in section 2.2 that the action of the element represented by u k ⊗ γ is very close to the action of γ. To define transfer morphisms, we start with an autonomous Hamiltonian H 1 in H std (W ) and an autonomous H 2 in H stair (W ), and we do small Morse Bott type deformations H 1δ et H 2δ . We define as in the previous section the subcomplex Z[u] ⊗ Z (C III,IV,V (H 2δ )) corresponding to points with negative action and we identify the quotient

Z[u] ⊗ Z SC * (H 2δ )/ Z[u] ⊗ Z (C III,IV,V (H 2δ )) to Z[u] ⊗ Z C I,II (H 2δ ). We consider the Hamiltonian ι V H 2δ in H std (W ).

Structural properties of symplectic homology

Proposition 3.1.14 For δ small enough, the S 1 equivariant homology of the quotients coincide with the S 1 equivariant homology of the small domain:

H Z[u] ⊗ Z C I,II (H 2δ ), ∂ = H SC(Z[u] ⊗ Z SC(ι V H 2δ )) H Z[u] ⊗ Z C II (H 2δ ), ∂ = H SC + (Z[u] ⊗ Z SC(ι V H 2δ ))
Proof: What remains to be checked is again there is no parametrized Floer trajectory u : R × S 1 → W going from an orbit in C I,II (H 2δ ) to an orbit in C I,II (H 2δ ) with points in W \ (U ∪ V ). This is proven by contradiction. If there was a parametrized trajectory going from an orbit in C I,II (H 2δ ) to an orbit in C I,II (H 2δ ) with points in W \ (U ∪ V ) for all δ's, then, by a theorem of Bourgeois and Oancea [BO09b, Proposition 4.7], there would be such a broken trajectory for the autonomous Hamiltonian and we have proven in Proposition 3.1.5 that this can not exist.

To get a transfer map, we use an autonomous increasing homotopy between H 1 and H 2 and we deform it into an increasing homotopy between H 1δ and H 2δ ; this induces a map

Z[u] ⊗ Z SC * (H 1δ ) → Z[u] ⊗ Z SC * (H 2δ ).
This map decreases the action (which is defined on the second factor) and commutes with the differential so it induces a map going to the quotient

H (Z[u] ⊗ Z SC * (H 1δ , ∂ ) → H Z[u] ⊗ Z C II (H 2δ ), ∂ .
This maps commutes with continuation maps. Proposition 3.1.15 For δ small enough, a continuation map in the homology defined from an H 2δ induces a continuation continuation map in the homology defined from ι V H 2δ .

Proof: One checks again that there is no parametrized Floer trajectory, corresponding to a homotopy, going from an orbit in C I,II (H 2δ ) to an orbit in C I,II (H ′ 2δ ) with points in W \ (U ∪ V ). This is done as in the former proposition, using the fact that the existence of such a trajectory for all δ's would imply the existence of such a broken trajectory for the autonomous Hamiltonian and we have proven in Proposition 3.1.9 that this can not exist.

We thus get a transfer morphism

φ S 1 W,V : SH S 1 (W, λ W ) → SH S 1 (V, λ V )
. and, on the quotient, the morphism

φ S 1 ,+ = φ S 1 ,+ W,V : SH S 1 ,+ (W, λ W ) → SH S 1 ,+ (V, λ V )
. By the same arguments as before, those morphisms compose nicely.

{ ϕ ⋆ H ′ } of Hamiltonians is cofinal and thus we reach the conclusion.

We now restrict our attention to Liouville domains, also called symplectic manifolds with restricted contact type boundary. Recall that a Liouville domain is a compact manifold W with boundary, together with a one form λ such that ω := dλ is symplectic and the Liouville vector field X defined by ι(X)ω = λ points strictly outwards along ∂W .

The asphericity condition being satisfied, we assume that c 1 (T W ), π 2 (W ) = 0. Recall that the completion of a Liouville domain (W, λ) is W = W ∪ (∂W × R + ) with symplectic form given by ω = ω on W and ω = de ρ λ | ∂W on ∂W × R + . We refer to section 1.2.1 for more details. We denote, as in the previous chapter, by SH † (W, λ) the symplectic homology SH † (W, dλ, X).

Lemma 3.2.3 Let (W, λ) be a Liouville domain. Then for all R ∈ R + , we have

SH † (W, λ) ∼ = SH † W ∪ (∂W × [0, R]), λ ′
where the 1-form λ ′ on ∂W × [0, R] is the restriction of the 1-form λ, thus (e ρ α) with α := λ | ∂W .

Proof: Denote by ϕ X t the flow of X; since L X λ = λ we have ϕ X t ⋆ λ = e t λ. This gives a symplectomorphism

ϕ X R : (W, e R ω) → W ∪ (∂W × [0, R]), ω ′
mapping the boundary ∂W to the boundary {R} × ∂W and such that ϕ X R * λ = e R λ.

One concludes by the two lemmas above. Explicitely, the diffeomorphism ϕ X R : W → W maps Hamiltonian vector fields as follows : (ϕ X R ) * (X H ′ ) = X H when H ′ = e -R (ϕ X R ) * H; hence ϕ X R gives a bijection between 1-periodic orbits of X H ′ and 1-periodic orbits of X H , and, with suitable choices of J's, a bijection between Floer trajectories between 1-periodic orbits of X H ′ and Floer trajectories between 1-periodic orbits of X H . Hence it yields an isomorphism

SH † (W, e -R (ϕ X R ) * H) ∼ = SH † W ∪ (∂W × [0, R]
), H . Furthermore, the diffeomorphism ϕ X R intertwines a continuation morphism defined by a homotopy H ′ s to the corresponding continuation morphism defined by H s when again H ′ s = e -R (ϕ X R ) * H s . This yields the isomorphism mentionned above.

Lemma 3.2.4 The transfer morphism

SH † W ∪ (∂W × [0, R]), λ ′ → SH † (W, λ)
is an isomorphism which coincides with the natural identification of Lemma 3.2.3.

Proof: Let H be an admissible Hamiltonian for W ∪(∂W ×[0, R]). Consider the homotopy

H 1 s := e -f (s) ϕ X f (s)
⋆ H with f : R → [0, R] a smooth function so that H 1 s = H for large negative s and H 1 s = H := e -R (ϕ X R ) * H for large positive s. The set of 1 periodic orbits for H 1 s is constant (since, as in the Lemma above, the diffeomorphism ϕ X f (s) of the completion is a bijection between 1-periodic orbits of X H 1 s and 1-periodic orbits of X H ). This homotopy defines the "transfer morphism"

φ : SH W ∪ (∂W × [0, R]), H → SH(W, H).

Let {H η

s } η∈[0,1] be a family of homotopies (with non fixed endpoint) such that H 0 s is the constant homotopy H 0 s = H for all s, and such that all H η s are of the form e -f ′ (s,η) ϕ X f ′ (s,η)

⋆ H with f ′ (., η) : R → [0, ηR] and f ′ (., 1) = f . We have

H η +∞ = e -ηR ϕ X ηR ⋆ H = H 1 f -1 (ηR)
. The set of 1-periodic orbits of H η s is in bijection with the set of orbits of H. We consider, for a given η, the space of Floer trajectories

M(H η s , J η s ) := (γ η -,γ η + )∈P(H η -∞ )×P(H η +∞ ) µ CZ (γ η -)=µ CZ (γ η + ) M(γ η -, γ η + , H η s , J η s )
and the parametrized moduli space

M({H η s , J η s }) := η∈[0,1] M(H η s , J η s )
which could have boundaries for some η = 0, 1. It defines a cobordism between M(H 0 s , J 0 s ) and M(H 1 s , J 1 s ). Now M(H 0 s , J 0 s ) = M(H, J) is the space of constant trajectories { u(s, •) = γ 0 (•) | γ 0 ∈ P(H)}. Thus for small η's, say η ≤ η 0 , the cobordism is a bijection, M(H η s , J η s ) consists of exactly one Floer trajectory starting from each orbit in P(H) and arriving at the corresponding orbit in P(H η +∞ ). The morphism induced by H η 0 s is thus the natural identification of periodic orbits. Hence the transfer

φ : SH W ∪ (∂W × [0, R]), H → SH W ∪ (∂W × [0, R -ǫ]), e ǫ ϕ X ǫ ⋆ H
is the natural identification for ǫ = η 0 R. Now we use the flow of the Liouville vector field, ϕ X ǫ , to carry all this construction further and we get the natural identification as the transfer morphism

φ : SH W ∪ (∂W × [0, R -ǫ]), e ǫ ϕ ǫ X ⋆ H → SH W ∪ (∂W × [0, R -2ǫ]), e 2ǫ ϕ 2ǫ X ⋆ H .
By induction and functoriality, we get the result.

Structural properties of symplectic homology

Lemma 3.2.5 Let W be a compact symplectic manifold with contact type boundary. Let λ t , t ∈ [0, 1] be an isotopy of Liouville forms on W such that in a neighbourhood U of the boundary, λ t = λ 0 . Then SH † (W, λ 0 ) ∼ = SH † (W, λ 1 ).

Proof: Remark that we do not require the dλ t to be equal. Let X t be the time dependent vector field defined by

ι(X s )(dλ s ) = -d dt λ(t) |s
and let ϕ t be its flow. In the neighbourhood U , the vector field vanishes, X s = 0, and so

ϕ ⋆ 1 λ 1 = λ 1 = λ 0 on U . Furthermore ϕ ⋆ 1 dλ 1 = dλ 0 because d dt ϕ ⋆ t λ t s = ϕ ⋆ s dλ t dt s + ϕ ⋆ s L Xs λ s = ϕ ⋆ s dλ t dt s + ϕ ⋆ s ι(X s )dλ s + dι(X s )λ s = d ϕ ⋆ s λ s (X s ) .
This implies that the completions for λ 0 and ϕ ⋆ 1 λ 1 are the same, therefore, by lemma 3.2.2,

SH † (W, λ 1 ) = SH † (W, ϕ ⋆ 1 λ 1 ) = SH † (W, λ 0 ).
Theorem 3.2.6 Let W be a compact symplectic manifold with contact type boundary. Let λ t , t ∈ [0, 1] be a homotopy of Liouville forms on W . Then SH † (W, λ 0 ) ∼ = SH † (W, λ 1 ).

To prove this Proposition, we use the following Proposition from Cieliebak and Eliashberg:

Proposition 3.2.7 ([CE12], Proposition 11.8) Let W be a compact symplectic manifold with contact type boundary. Let λ t , t ∈ [0, 1] be a homotopy of Liouville forms on W . Then there exists a diffeomorphism of the completions f : W 0 → W 1 such that f ⋆ λ 1λ 0 = dg where g is a compactly supported function.

Proof of Theorem 3.2.6: There exists a positive real ρ 0 such that supp(g)

⊂ W ∪ (∂W × [0, ρ 0 ]). We choose positive real numbers ρ 1 , ρ ′ 0 and ρ ′ 1 such that f -1 (W ∪ (∂W × [0, ρ 1 ]) contains W ∪ (∂W × [0, ρ 0 ]), f -1 (W ∪ (∂W × [0, ρ 1 ])) ⊂ W ∪ (∂W × [0, ρ ′ 0 ]) and W ∪ (∂W × [0, ρ ′ 0 ]) ⊂ f -1 (W ∪ (∂W × [0, ρ ′ 1 ])). The situation is represented in Figure 3.2 supp(g) ∂W × {0} ∂W × {ρ 0 } ∂W × {ρ ′ 0 } f -1 (∂W 1 × {ρ 1 }) f -1 (∂W 1 × {ρ ′ 1 }) Figure 3.2: The choice of ρ 0 , ρ 1 , ρ ′ 0 and ρ ′ 1
The diffeomorphism f and the flow of X 1 on W 1 give

f -1 (W ∪ (∂W × [0, ρ 1 ])), f ⋆ λ 1 ∼ = W ∪ (∂W × [0, ρ 1 ], λ 1 ∼ = (W, e ρ 1 λ 1 ). The completion of f -1 (W ∪ (∂W × [0, ρ 1 ])), f ⋆ λ 1 coincides with ( W 0 , λ 0 ) since close to the boundary f ⋆ X 0 = X 1 . SH(W, λ 1 ) ∼ = SH W ∪ (∂W × [0, ρ 1 ]), λ 1 by Lemma 3.2.3 ∼ = SH f -1 (W ∪ (∂W × [0, ρ 1 ])), f ⋆ λ 1 by Lemma 3.2.2 ∼ = SH f -1 (W ∪ (∂W × [0, ρ 1 ])), λ 0 + dg by Proposition 3.2.7 ∼ = SH f -1 (W ∪ (∂W × [0, ρ 1 ])) =:W 1 , λ 0 by Lemma 3.2.5.
Denoting by ϕ X 0 t the flow of X 0 and by W 0 the manifold W ∪ (∂W × [0, ρ 0 ]), we have

ϕ X 0 ρ ′ 1 -ρ 1 (W 1 ) = f -1 (W ∪ (∂W × [0, ρ ′ 1 ])) and ϕ X 0 ρ ′ 0 -ρ 0 (W 0 ) = W ∪ (∂W × [0, ρ ′ 0 ]
) Using the functoriality of the transfer morphism,

SH(ϕ X 0 ρ ′ 1 -ρ 1 (W 1 ), λ 0 ) / / ∼ = 4 4 SH(ϕ X 0 ρ ′ 0 -ρ 0 (W 0 ), λ 0 ) ∼ = ) ) / / SH(W 1 , λ 0 ) / / SH(W 0 , λ 0 ); therefore SH(W, λ 1 ) ∼ = SH(W 1 , λ 0 ) ∼ = SH(W 0 , λ 0 ) ∼ = SH(W, λ 0 ).
Seidel in [START_REF] Seidel | A biased view of symplectic cohomology[END_REF] has extended the definition of symplectic homology (and all its variants) to Liouville manifolds. Definition 3.2.8 (see for instance [START_REF] Cieliebak | From Stein to Weinstein and back[END_REF]) A Liouville manifold is an exact symplectic manifold (W, ω, X), where the vector field X is an expanding Liouville vector field, i.e L X ω = ω and ϕ X t ω = e t ω such that

• the vector field X is complete and

• the manifold is convex in the sense that there exists an exhaustion W = ∪ ∞ k=1 W k by compact domains W k ⊂ W with smooth boundaries along which X is outward pointing.

In the following we will denote a Liouville manifold either by (W, ω, X) or by (W, λ := ι(X)ω).

The set

Skel(V, ω, X) := ∞ k=1 t>0 ϕ X -t (W k )
is called the skeleton of the Liouville manifold (W, ω, X). It is independent of the choice of the exhausting sequence of compact sets W k . A Liouville manifold (W, ω, X) is said to be of finite type if its skeleton is compact. Every finite type Liouville manifold is the completion of a Liouville domain1 .

Definition 3.2.9 ([Sei08]) Let (W, ω, X) be a Liouville manifold non necessarily of finite type and let W k be an exhaustion by compact domains W k ⊂ W with smooth boundaries along which X is outward pointing such that W k ⊂ W k+1 . The symplectic homology (and its variants) of (W, λ) is defined as the inverse limit of the symplectic homologies of

(W k , λ | W k ) SH † (W, λ) := lim ←- SH † (W k , λ | W k ).
The morphisms appearing in this inverse limit are the transfer morphisms.

This definition is independent of the chosen exhaustion. Remark that in the case of finite type Liouville manifolds, this definition coincides with the previous one.

Invariance of the homology of contact fillings

In this section we shall prove:

Theorem 3.2.13 Let (M 0 , ξ 0 ) and (M 1 , ξ 1 ) be two contact manifolds that are exactly fillable; i.e. there exist Liouville domains (W 0 , λ 0 ) and (W 1 , λ 1 ) such that ∂W 0 = M 0 , ξ 0 = ker(λ 0| M 0 ), ∂W 1 = M 1 and ξ 1 = ker(λ 1| M 1 ). Assume there exists a contactomorphism ϕ : (M 0 , ξ 0 ) → (M 1 , ξ 1 ) which is "oriented" in the sense that ϕ ⋆ λ 1 M 1 = e f λ 0 M 0 . Assume moreover that there exists a contact form α0 on M 0 such that all periodic Reeb orbits are nondegenerate and their Conley-Zehnder index have all the same parity. Then

SH S 1 ,+ (W 0 , λ 0 ) ∼ = SH S 1 ,+ (W 1 , λ 1 ).
Lemma 3.2.14 ( [START_REF] Cieliebak | Handle attaching in symplectic homology and the chord conjecture[END_REF]) Let (α t ) t∈[0,1] be a smooth family of contact forms on a closed manifold M of dimension 2n -1. Then there exists a R > 0 and a non-decreasing function

f : [0, R] → [0, 1] such that f ≡ 0 close to ρ = 0 and f ≡ 1 close to ρ = R and d e ρ α f (ρ) is symplectic on M × [0, R] Proof:
The proof is a computation: Proof: Since α 1 = e g α 0 , for a smooth function g on M , we consider the smooth family of contact forms α t = e tg α 0 , t ∈ [0, 1]. We define on W ∪ M × [0, R] ⊂ W the 1-form λ:

d e ρ α f (ρ) = e ρ dρ ∧ α f (ρ) + e ρ dα f (ρ) + e ρ f ′ (ρ)dρ ∧ αf(ρ) . d e ρ α f (ρ) n = ne nρ dρ ∧ α f (ρ) + f ′ (ρ) αf(ρ) ∧ dα f (ρ) n-1 and thus d e ρ α f (ρ) is symplectic if and only if α f (ρ) + f ′ (ρ) αf(ρ) R α f (ρ) > 0. This is true if f ′ is small.
λ = λ 0 on M e ρ α f (ρ) on M × [0, R]
with f as in Lemma 3.2.14, so that d λ is symplectic. The flow of the vector field X 0 , where ι(X 0 )dλ 0 = λ 0 , ϕ X 0 -r induces a diffeomorphism from W ∪ M × [0, r] to W . The pull-back by this flow of e -r λ gives the desired λ f (r) .

Combining with Theorem 3.2.6 and Theorem 2.2.2, this yields Lemma 3.2.16 Let (M 0 , ξ 0 ) be a contact manifold that is exactly fillable by the Liouville domains (W 0 , λ 0 ). Assume that there exists a (oriented) contact form α0 on M 0 such that all periodic Reeb orbits are nondegenerate and their Conley-Zehnder index have all the same parity. Then

SH S 1 ,+ (W 0 , λ 0 ) = γ∈P(R α0 )
Q γ

where P(R α0 ) denotes the set of periodic Reeb orbits on (M 0 , α0 ).

Proof of Theorem 3.2.13: Given the contactomorphism ϕ : (M 0 , ξ 0 ) → (M 1 , ξ 1 ) and the contact form α0 , we define the form α1 := (ϕ -1 ) ⋆ α0 ; it is a contact form on M 1 and its periodic orbits are non degenerate, in bijection with those of α0 with the same Conley-Zehnder index. We apply twice Lemma 3.2.16; once for (W 0 , λ 0 , α0 ) and for (W 1 , λ 1 , α1 ).

This gives a tool to prove Ustilovsky's Theorem.

Corollary 3.2.17 (Ustilovsky, [Ust99]) For each natural number m, there exist infinitely many pairwise non isomorphic contact structures on S 4m+1 .

Proof: We see that one can build contact structures on S 4m+1 , which are exactly fillable, but which do not yield isomorphic SH S 1 ,+ homologies of the filling. The result then follows from Theorem 3.2.13. The contact structures in question are those defined by the Brieskorn sheres; see section 2.2.1. The fact that the homologies are different follows from proposition 2.2.7.

On the minimal number of periodic Reeb orbits

We shall now use the properties of positive S 1 -equivariant symplectic homology to get results on the minimal number of geometrically distinct periodic Reeb orbits on some contact manifolds. We first give an alternative proof of a result of Ekeland and Lasry on the minimal number of distinct periodic Reeb orbits on a hypersurface in R 2n . We also obtain information on the minimal number of simple periodic Reeb orbits on some hypersurfaces in some negative complex bundles over a compact symplectic manifold. We extend our machinery to some non exact symplectic manifolds with contact type boundary.

Minimal number of periodic Reeb orbits on a hypersurface in R 2n

We show how to use the transfer morphism to give an alternative proof of a result by Ekeland and Lasry concerning the number of simple periodic Reeb orbits on a hypersurface in R 2n , pinched between two spheres, endowed with the restriction of the standard contact form on R 2n .

Theorem 4.1.1 (Ekeland, Lasry, [EL80, Eke90]) Let Σ be a contact type hypersurface in R 2n . Let ξ = ker α be the contact structure induced by the standard contact form on R 2n . Assume there exists a point x 0 ∈ R 2n and numbers 0 < R 1 ≤ R 2 such that:

∀x ∈ Σ, R 1 ≤ x -x 0 ≤ R 2 with R 2 R 1 < √ 2 Assume also that ∀x ∈ Σ, T x Σ ∩ B R 1 (x 0 ) = ∅.
Assume moreover that all periodic Reeb orbits are non degenerate. Then Σ carries at least n geometrically distinct periodic Reeb orbits.

Remark 4.1.2

The assumption ∀x ∈ Σ,

T x Σ ∩ B R 1 (x 0 ) = ∅ (which is weaker than convexity) can be stated as ν Σ (z), z > R 1 , ∀z ∈ Σ (4.1)
where ν Σ (z) is the exterior normal vector of Σ at point z and •, • denotes the Euclidean scalar product on R 2n .

Proof: We consider ellipsoids, very close to the spheres, defined by

S ′ R 1 = n i=1 a -1 i (x i ) 2 + (y i ) 2 = R 2 1
with a 1 < • • • < a n real numbers arbitrarily close to 1 and rationally independent, and

S ′ R 2 = { n i=1 a -1 i (x i ) 2 + (y i ) 2 = R 2
2 }, and we denote by S ′ R 1 , Σ and S ′ R 2 the compact regions in R 2n bounded respectively by S ′ R 1 , Σ and S ′ R 2 , endowed with the restriction of the standard symplectic form ω on R 2n . We take the parameters a i sufficiently close to 1 so that we have the inclusion

S ′ R 1 ⊂ Σ ⊂ S ′ R 2
of Liouville domains. The contact form on the boundaries is the one induced by ι(X rad )ω, where X rad is the radial vector field X rad = 1 2

x i ∂ x i + y i ∂ y i .
The completion of those Liouville domain is (R 2n , ω). By Theorem 3.1.16, the transfer morphisms yields the following commutative diagram:

SH S 1 ,+ ( S ′ R 2 , ω) φ / / ∼ = 5 5 SH S 1 ,+ ( Σ, ω) / / SH S 1 ,+ ( S ′ R 1 , ω). (4.2)
We can consider the positive S 1 -equivariant symplectic homology truncated by the action at level ≤ T , SH S 1 ,+,T . Since all Floer trajectories inducing the morphisms lower the action, we still have the commutative diagram for the truncated positive invariant symplectic homology:

SH S 1 ,+,T ( S ′ R 2 , ω) φ / / ∼ = 4 4 SH S 1 ,+,T ( Σ, ω) / / SH S 1 ,+,T ( S ′ R 1 , ω). (4.3)
where we have chosen a number T such that

πa n R 2 2 < T < 2πa 1 R 2 1 . (4.4)
This is possible thanks to the "pinching" hypothesis

R 2 R 1 < √ 2. By Theorem 2.2.2, SH S 1 ,+,T ( S ′ R 2 , ω) is generated by n elements u 0 ⊗γ 1 Max , . . . u 0 ⊗γ n Max cor- responding to n simple periodic Reeb orbits on S ′ R 2 , γ 1 , . . . , γ n of action πa 1 R 2 2 , . . . , πa n R 2 2 . The analogous is true for SH S 1 ,+,T ( S ′ R 1 , ω) with actions πa 1 R 2 1 , . . . , πa n R 2 1
. By (4.3), SH S 1 ,+,T ( Σ, ω) is thus of rank at least n. All applications in the above diagrams decrease the action thus the action of each of those n generators in

SH S 1 ,+,T ( Σ, ω) ∩ Im(φ) is pinched between πa 1 R 2 1 and πa n R 2 2 < 2πa 1 R 2 1 .
4.1. Minimal number of periodic Reeb orbits on a hypersurface in R 2n

By Corollary 2.2.5, the only generators that may appear in SH S 1 ,+,T ( Σ, ω) are elements of the form u 0 ⊗ γ Max with γ a good Reeb orbit on Σ.

It remains to prove that the n elements in the image of φ are geometrically distinct. By the pinching condition on their action, we know that they are not iterate one from another but we still need to prove that two of them can not be the iterates of a same orbit of smaller action. This we do by proving that the smallest possible action for any periodic Reeb orbit on Σ is greater than πa 1 R 2 1 . Let γ : [0, T ] → Σ be a simple periodic Reeb orbit. We have :

2T = T 0 α γ(t) γ(t) dt = T 0 γ(t), Jγ(t) dt since α x (X x ) = 1 2 X x , Jx = T 0 γ(t), J γ(t) dt with γ(t) := γ(t) - T 0 γ(t)dt ≤ γ L 2 γ L 2 ≤ γ 2 L 2 T 2π
via the Wirtinger's inequality

= T 2π T 0 γ(t) 2 dt = T 2π T 0 (R α ) γ(t) 2 dt (4.5)
For any point x in Σ, the norm of the Reeb vector field is bounded by (

R α ) x ≤ 2 R 1 . Indeed, R α is proportional to Jν Σ since ι(Jν Σ )dα = 0 because ι(Jν Σ )dα(Y ) = ω(Jν Σ , Y ) = -ν Σ , Y = 0 for all Y ∈ T Σ. Thus R α = cJν Σ with |c| = R α . But α x (R αx ) = 1 = 1 2 c x Jν Σ (x), Jx = cx 2 ν Σ (x),
x . Therefore, by assumption (4.1),

c x = 2 ν Σ (x),x ≤ 2 R 1 . And thus (4.5) ≤ 4 R 2 1 T T 2π . Then 2T ≤ 2T T πR 2 1
and we reach the conclusion

T ≥ πR 2 1 .
Hence the conclusion of the Theorem.

The original proof of Theorem 4.1.1 uses variational methods that work only in R 2n . This new proof may extend to other cases such as hypersurfaces in negative line bundles. For this, we have to extend our machinery to some non exact compact symplectic manifolds with contact type boundaries.

4. On the minimal number of periodic Reeb orbits

Extension of the definitions

Let (W, ω, X) be a compact symplectic manifold with contact type boundary, satisfying assumptions 1.1.1 and 1.1.2 and let ( W , ω) be its symplectic completion. We want to define the positive symplectic homology and the positive S 1 -equivariant symplectic homology of (W, ω). In this general situation, the action does not distinguish between constant and nonconstant 1-periodic orbits since we have no control on the disk bounded by a 1-periodic orbit.

Positive symplectic homology

This section is joint results with Strom Borman. Definition 4.2.1 Let H be a Hamiltonian in H std . The submodule of the Floer complex generated by the critical points of H is denoted by SC - * ( H, J).

We restrict ourselves to the cofinal family of perturbation of Morse-Bott Hamiltonians as defined in section 2.1.3. We use the same notations as in section 2.1.3.

Theorem 4.2.2 Let H be a Hamiltonian in H MB . There exists a real number δ 0 > 0 such that for all δ < δ 0 , the submodule SC - * (H δ , J) is a subcomplex of the Floer complex SC * (H δ , J).

Proof: We prove this by contradiction, assuming that no such δ 0 exists. Then there is a decreasing sequence δ n converging to 0 such that SC - * (H δn , J) is not a subcomplex of SC * (H δn , J). This implies that for all n, there exists a Floer trajectory for H δn going from a fixed critical point of H δn to a nonconstant 1-periodic orbit of H δn . We can assume (considering, if needed, a subsequence) that the 1-periodic orbit is the same for all n. This sequence of Floer trajectories converges to a Floer trajectory for the autonomous Hamiltonian by the following theorem due to Bourgeois and Oancea:

Theorem 4.2.3 ([BO09b], Proposition 4.7) Let v n ∈ M(γ -, γ + , H δn , J) be a sequence of Floer trajectories with δ n → 0 as n → ∞. Then there exists a Broken Floer trajectory with gradient fragments u and a subsequence (still denoted by v n ) such that v n → u.

To conclude the proof of theorem 4.2.2, we shall now prove that it is impossible for an autonomous Hamiltonian H to have such a Floer trajectory. Precisely, we shall show that, for any critical point p of H and any nonconstant 1-periodic orbit γ of H, the moduli space M(p, γ; H, J) of solutions u : R × S 1 → W of the Floer equation

   ∂ s u + J • u ∂ θ u -X H • u = 0 u(-∞, •) = p (critical point) u(+∞, •) = γ(•) ⊂ ∂W × {ρ γ } (4.6)
is empty. (We have considered the image of γ sitting in ∂W ×{ρ γ }.) We choose a compatible almost complex structure J i.e a J ∈ End T (∂W × R + ) such that dα(•, J•) > 0 on ξ ⊂ ∂W × R + , such that Jξ = ξ and such that J∂ ρ = R α near the Hamiltonian orbits. We assume u is a Floer trajectory, solution of (4.6), and we consider a real number s 0 ∈ R such that u(s, •) stays in a neighbourhood of γ for all s ≥ s 0 . We can decompose u: u(s, θ) := f (s, θ), a(s, θ) ∈ ∂W × R + with a(s, θ) close to ρ γ for s ≥ s 0 . We distinguish two cases:

Case 1: there exist s and θ such that a(s, θ) > ρ γ ;

Case 2: a(s, θ) ≤ ρ γ for all s and θ.

We shall see that neither case can arise, and this will conclude our proof.

Case 1

We prove by contradiction that it can not happen, as a direct application of Abouzaid maximum principle (proven above as theorem 3.1.6). Assume that u : R × S 1 → W is a Floer trajectory whose image intersects W \ W ∪ (∂W × [0, ρ γ ]) . There exists δ > 0 such that the intersection of the image of u with a slice ∂W × {ρ} is non-empty for any ρ γ < ρ < ρ γ + δ and we choose between them a regular value

ρ γ + ǫ of ρ • u. The manifold W ′ := W \(W ∪(∂W ×[0, ρ γ +ǫ[))
is symplectic with contact type boundary ∂W ×{ρ 0 +ǫ} and Liouville vector field pointing inwards. Let S be the inverse image of W ′ under the map u; it is a compact Riemann surface with boundary embedded in R × S 1 ; the complex structure j is the restriction to S of the complex structure j on the cylinder defined by j(∂ s ) = ∂ θ . We define β to be the restriction of dθ to S. The fact that u is a Floer trajectory is equivalent to (du -X H ⊗ β) 0,1 := 1 2 ((du -X H ⊗ β) + J(du -X H ⊗ β)j) = 0, where du is the differential of the map u viewed as a section of T * S ⊗u * T W ′ . Then theorem 3.1.6 concludes.

Case 2

To prove that this situation can not happen, we use an argument taken from Bourgeois and Oancea [START_REF] Bourgeois | An exact sequence for contactand symplectic homology[END_REF]. We restrict to s ∈ [s 0 , ∞) and we define ā(s) := S 1 a(s, θ)dθ. Equation (4.6) decomposes as:

   ∂ s a -α(∂ θ f ) -e -a h ′ (a) = 0 α(∂ s f ) + ∂ θ a = 0 π ξ (∂ s f ) + π ξ (J∂ θ f ) = 0 (4.7)
where

π ξ : T (∂W × R) → ξ is the projection on ξ along R ∂ ρ ⊕ R R α . The first equation of (4.7) implies ∂ s ā(s) = S 1 α(∂ θ f )dθ + S 1 h ′ (a)e -a dθ ≤ S 1 α(∂ θ f )dθ + T (4.8)
where T = h ′ (ρ γ )e ργ . The inequality is true since h ′ (b)e -b is increasing in b.

For s 0 ≤ s ≤ s ′ , we have, by Stokes,

S 1 α ∂ θ f (s ′ , θ) dθ - S 1 α ∂ θ f (s, θ) dθ = [s,s ′ ]×S 1 f ⋆ dα (4.9)
and

[s,s ′ ]×S 1 f ⋆ dα > 0 (4.10)
because dα(•, J•) > 0 on the contact distribution and dα "kills" ∂ ρ and R α ; we assume of course that f is non constant. Hence the map s → S 1 α ∂ θ f (s, θ) dθ is an increasing function. Since we know that lim s→∞ f (s, θ) = γ ′ (-T θ), where γ ′ is the corresponding Reeb orbit, we have

S 1 α ∂ θ f (+∞, θ) dθ = -T. (4.11) Hence ∂ s ā(s) ≤ S 1 α ∂ θ f (s, θ) dθ + T ≤ 0. (4.12) Now ā(+∞) = ρ γ since a(+∞, θ), f (+∞, θ) = ρ γ , γ ′ (-T θ)
. Since a ≤ ρ γ by assumption and since ā(s) is non increasing by the above, we have

a(s, θ) ≡ ρ γ . (4.13) So (4.7) becomes:    α(∂ θ f ) -h ′ (a)e -a = 0 α(∂ s f ) = 0 π ξ (df • j) -Jπ ξ (df ) = 0 (4.14)
Thus α(∂ θ f ) = h ′ (a)e -a = h ′ (ρ γ )e ργ = constant. Using (4.9), we get

[s 0 ,∞)×S 1 f ⋆ dα = 0 (4.15)
which implies that f is constant, which yields a contradiction.

We denote by H ′ std the subfamily of Hamiltonians in H std such that the conclusion of Theorem 4.2.2 holds. Definition 4.2.4 Let H δ be a Hamiltonian in H ′ std , we define the positive Floer complex to be the quotient

SC + * (H δ , J) := SC * (H δ , J)/ SC - * (H δ , J) .
We would like to define as in section 1.2.4 the direct limit over Hamiltonians of the positive homologies defined by those quotients.

Theorem 4.2.5 Let H 0δ and H 1δ be two Hamiltonian in H ′ std such that H 0δ ≤ H 1δ . Let H s be a increasing homotopy between them. The continuation map maps SC - * (H 0 , J 0 ) to SC - * (H 1 , J 1 ).

Proof: Let H 1 be the Hamiltonian in H MB from which H 1δ is the perturbation. It suffices to show, as before, that for any nonconstant orbit γ of H 1 the moduli space M(p, γ; H s , J s ) of solutions u : R × S 1 → W to the Floer continuation equation

   ∂ s u + J s • u ∂ θ u -X Hs • u = 0 u(-∞, θ) = p (critical point) u(+∞, θ) = t γ , γ(-T θ) (4.16)
is empty. We consider again a real number s 0 ∈ R such that u s, θ) stays in a neighbourhood of the image of γ for all s ≥ s 0 and such that H s is constant (in s) for s ≥ s 0 . We can again decompose u in

u(s, θ) = f (s, θ), a(s, θ) ∈ M × R + .
Again,we distinguish two cases:

1. there exist s and θ such that a(s, θ) > ρ γ ; 2. a(s, θ) ≤ ρ γ ∀s, θ.

The proof to rule out the second case is the same as in Theorem 4.2.2. For the first case, the argument is the same and the conclusion comes from the generalised version of "Abouzaid maximum principle" stated in proposition 3.1.10.

We are now ready to define the positive symplectic homology in the framework of a compact symplectic manifold with contact type boundary (W, ω) which is aspherical and such that its first Chern class vanishes on the second homotopy group : Definition 4.2.6 The positive symplectic homology of (W, ω) is defined as

SH + (W, ω) := lim -----→ H∈H ′ std H * (SC + * (H, J), d).
4. On the minimal number of periodic Reeb orbits

Positive S 1 -equivariant symplectic homology

We shall use the special complex in the alternative definition of S 1 -equivariant symplectic homology presented in section 2.1.2. One considers an autonomous Hamiltonian H in H MB (cf definition 2.1.2) such that the slope a is big and ρ 0 is small; one denotes by H δ a small perturbation, as in section 2.1.3. One defines the positive S 1 -equivariant Floer complex of

H δ by S C S 1 ,+ * (H δ ) := Z[u] ⊗ Z SC + * (H δ )
with differential induced by the differential given by (2.2). The fact that SC S 1 ,- *

(H δ ) is a subcomplex of SC S 1
* (H δ ) for δ small enough is a consequence of Theorem 4.2.3, extended to parametrized Floer trajectories in chapter 5.2 of [START_REF] Bourgeois | The Gysin exact sequence for S 1 -equivariant symplectic homology[END_REF]; one reduces the problem to the situation of Theorems 4.2.5 and 4.2.2. Similarly, there are well-defined continuation maps induced by increasing homotopies of Hamiltonians and we have

SH S 1 ,+ * (W ) := lim -→ H δ ∈H ′ std SH S 1 ,+ * (H δ ).

Reeb orbits on hypersurfaces in negative line bundles

We sketch a procedure to detect a minimal number of geometrically distinct periodic Reeb orbits for hypersurfaces in some negative line bundles.

Symplectic structure on the complement of the zero section

Our framework here is a complex line bundle L π → B 2n over a closed symplectic manifold (B 2n , ω B ), endowed with a Hermitian structure h and a connection ∇. We assume L to be negative i.e.

c 1 (L) = -k[ω B ]
for a real number k > 0. The transgression 1-form,

θ ∇ ∈ Ω 1 L \ O L , R is defined by θ ∇ u (u) = 0, θ ∇ u (iu) = 1 2π u ∈ L \ O L θ ∇ | H ∇ ≡ 0 where H ∇ is the horizontal distribution.
(4.17)

We have

dθ ∇ = kπ ⋆ ω B .
We denote by r the radial function on the fiber, i.e. r :

L → R : u → h π(u) (u, u) 1 2 =: |u|. Observe that d(r 2 θ ∇
) is symplectic except on the zero section O L . We want to have information about the minimal number of periodic orbits of the Reeb vector field on a hypersurface in L \ O L endowed with the contact form defined by the restriction of (r 2 θ ∇ ) to Σ. 100 4.3. Reeb orbits on hypersurfaces in negative line bundles

Reeb orbits on the circle bundle with varying radius

Let f : B → R be a smooth function. Define the contact hypersurface

S e f = {u ∈ L | |u| = e 1 2 f (π(u)) }, α := (r 2 θ ∇ ) | S e f
The Reeb vector field on S e f is given by:

R α = e -f (π(u)) 2π∂ θ + Xf (4.18)
where ∂ θ is the infinitesimal rotation in the fiber (∂ θ at the point u identifies with iu), where X f is the Hamiltonian vector field on B corresponding to the function f (i.e. ι(X f )ω B = df ) and where X denotes the horizontal lift of a vector X ∈ T B. The periodic Reeb orbits correspond to the critical points of f . The contact action (i.e. the period) of a simple orbit γ which lies above a critical point p is

A(γ) = e f (p) . (4.19)
The Conley-Zehnder index of an orbit which is a k iterate of a simple orbit over the critical point p is given by

µ CZ (γ) = 2k - 1 2 Sign(Hess p f ) (4.20)
It is given by the Conley-Zehnder index of the path of symplectic matrices,

φ : 0, e f (p) → Sp(R 2n , Ω 0 ) φ(t) := ϕ Rα t ⋆q ξq
given by the expression of the differential of the flow in a symplectic trivialisation of the contact structure ξ along γ.

We have thus proven the following, using Morse's inequalities:

Proposition 4.3.1 Let Σ be a contact type hypersurface in L such that the intersection of Σ with each fiber is a circle. The contact form is the restriction of r 2 θ ∇ . Then Σ carries at least 2n i=0 β i geometrically distinct periodic Reeb orbits, where β i denote the Betti numbers of B.

We do not make here any assumptions on B except that B is closed.

Symplectic structure on a negative line bundle Definition 4.3.2 Let ρ : [0, ∞[→ [0, 1] be a smooth decreasing function such that on a neighbourhood of 0, ρ is equal to 1 and ρ vanishes outside a compact set. Let ǫ > 0 be such that 2r + ǫρ ′ (r) > 0. The two form defined by

ω ρ,ǫ := d(r 2 θ ∇ ) + ǫd(ρ(r)θ ∇ ) (4.21)
is smooth and well defined on L; indeed, it is obviously well defined (and exact) on L \ O L . On the zero section, it coincides with the sum of ǫπ * (ω B ) and the standard form on the fiber. It is symplectic, non exact. For a fixed ǫ those forms are cohomologous. For two different choices of ǫ and ρ, one can interpolate between ω ρ 0 ,ǫ 0 and ω ρ 1 ,ǫ 1 staying in the class of elements of the form ω ρ,ǫ .

When the circle bundle is the boundary of a Liouville domain Theorem 4.3.3 Let Σ be a contact type hypersurface in L, negative line bundle over a symplectic manifold. Suppose that there exists a Liouville domain W ′ (such that its first Chern class vanishes on all tori) whose boundary coincides with the circle bundle S R 2 1 . Suppose there exists a Morse function f : B → R such that all critical points of f have a Morse index of the same parity. Let α be the contact form on Σ induced by r 2 θ ∇ on L.

Assume that Σ is "pinched" between two circle bundles S R 2 1 and S R 2 2 of radii R 1 and R 2 such that 0 < R 1 < R 2 and R 2 R 1 < √ 2.
Assume that the minimal action of any periodic Reeb orbit on Σ is bounded below by R 2 1 . Then Σ carries at least 2n i=0 β i geometrically distinct periodic Reeb orbits, where the β i denote the Betti numbers of B.

In this Theorem, the assumption on the existence of a Morse function all of whose critical points have Morse indices of the same parity is of a technical nature. Its purpose is to bring the situation within the scope of Theorem 2.2.2, which is our tool for computing the positive S 1 -equivariant symplectic homology. The lower bound on the period of any periodic Reeb orbit is semi-technical; it is now the only way we have to distinguish the images of the orbits. The "pinching" assumption is more conceptual, its main implication is that the "n first generators" of the positive S 1 -equivariant symplectic homology are simple orbits. Proof: The proof is the same as for Theorem 4.1.1 using transfer morphisms for Liouville domains. We see the hypersurfaces as lying in the completion of the Liouville domain W ′ which we assumed to exist. We find a small ǫ so that the convex domain Σ bounded by the hypersurface Σ is such that

S R 2 1 e ǫf ⊂ Σ ⊂ S R 2 2 e ǫf ⊂ W ′
where S f is the domain bounded by S f . We can compute the positive S 1 -equivariant symplectic homology, which is spanned by periodic orbits of the Reeb vector field by Theorem 2.2.2. This is possible by the pinching condition. One uses then the transfer morphisms with truncated action. We have seen that there are 2n i=0 β i simple periodic orbits on S R 1 e ǫf whose actions are very close to R 2 1 and the same number of simple periodic orbits on S R 2 e ǫf whose actions are very close to R 2 2 . The transfer morphism imply the existence of at least 2n i=0 β i periodic orbits on Σ with action between R 2 1 et R 2 2 . Since we have assumed here that the minimal action of any periodic Reeb orbit on Σ is bounded below by R 2 1 , those orbits are geometrically distinct.

Symplectic homology for a hypersurface in a negative line bundle

Let W ⊂ L be a compact symplectic manifold of codimension 0 in L with contact type boundary such that O L ⊂ W ; the symplectic form on W is ω ρ,ǫ | W . We choose the function ρ such that ρ vanishes in a neighbourhood of the boundary of W . For the symplectic homology to be well-defined, we assume that the closed symplectic manifold B is atoroidal and that the first Chern class of its tangent bundle vanishes on the tori. We first observe that symplectic homology does not depend on the choice of such a function ρ. The completions (L) are all symplectomorphic outside a neighbourhood of O L .

Lemma 4.3.4 The (positive, S 1 -equivariant, positive S 1 -equivariant) symplectic homology of W is independent of ρ and ǫ.

The proof is the same as that of Lemma 3.2.5, using the fact that for a fixed ǫ the difference of two of those symplectic forms is exact. We even have a stronger result:

Lemma 4.3.5 Let β be a positive real number and consider Hamiltonians

H β : S 1 × W → R and H β : S 1 × W × S 2N +1
→ R which are of the form βr + β ′ for r big. Then SH(H β ) and SH S 1 ( H β ) are independent of ρ and ǫ.

Proof: We consider a curve ω t , t ∈ [0, 1] of symplectic structures in our class and two 1-periodic orbits γ -, γ + of H β . Remark the those orbits are the same for all t since the symplectic forms coincide on ∂W × R + in the completion. We consider the moduli spaces M t (γ -, γ + , H ωt β , J t , ) built from Floer trajectories which are solutions of

∂u ∂s + J θ t • u ∂u ∂θ -X t H β • u = 0 (4.22)
going from one of these orbits to the other, where X t H is the Hamiltnian symplectic vector field corresponding to H for the symplectic structure ω t . Observe that this equation coincides with the classical Floer equation (with no t dependance) outside a compact set. So it behaves in the same way as the continuation equation 1.3 for a homotopy with compact support. Hence it defines a map which intertwines the differential. This gives an isomorphism in homology, since an inverse is defined by following the path of symplectic matrices in the reverse way.

One does a similar reasoning to see that this isomorphism commutes with continuation maps. Hence we get 4. On the minimal number of periodic Reeb orbits Lemma 4.3.6 The symplectic homology SH(W ) of W is independent of ρ and ǫ.

Remark that in this case of negative line bundle, the first term in the action i.e. D 2 σ ⋆ ω ρ,ǫ where σ : D 2 → W with σ | ∂D 2 = γ is given by

D 2 σ ⋆ ω ρ,ǫ = D 2 σ ⋆ d(r 2 θ ∇ ) + ǫd(ρ(r)θ ∇ ) = D 2 σ ⋆ d(r 2 θ ∇ ) + D 2 σ ⋆ ǫd(ρ(r)θ ∇ ) = S 1 γ ⋆ d(r 2 θ ∇ ) + ǫ D 2 σ ⋆ d(ρ(r)θ ∇ ).
For a chosen Hamiltonian H β (or an S 1 equivariant lift H β ), one chooses ǫ sufficiently small so that the actions are close to

A H β (γ) = - S 1 γ ⋆ d(r 2 θ ∇ ) - S 1 H β θ, γ(θ) A H β (γ, z) = - S 1 γ ⋆ d(r 2 θ ∇ ) - S 1 H β θ, γ(θ), z .
and by close we mean that the difference with the action is smaller than the smallest value of the action and smaller than the difference of two actions (this is possible since, for a given β, we only have a finite number of 1-periodic orbits). We proceed equivalently for non contractible orbits for pullbacks on tori linking the orbit to a given loop in the same homotopy class defined on the manifold B. This allows, when looking at a "stair Hamiltonian" to define a transfer morphism, to use the action to distinguish the different subcomplexes. The transfer is then defined as in section 3.1. All reasonings are the same, since the exactness of the symplectic form was only used to distinguish different classes of orbits by the value of their action. To see that the transfer does not depend on ǫ, one proceeds as in Lemma 4.3.5.

At this point, extending the above results to the positive S 1 equivariant homology, we would have a theorem for hypersurfaces in negative line bundles over atoroidal symplectic closed manifolds endowed with a Morse function such that all critical points of f have a Morse index of the same parity. This is asking too much. So we should use the wider context of symplectic manifolds which are monotone and work in the homology with coefficients in the Novikov ring. Everything should extend to this new setup and should lead to Conjecture 4.3.7 Let Σ be a contact type hypersurface in L, negative line bundle over a closed monotone symplectic manifold. The bundle is endowed with a hermitian structure and a connection. Suppose there exists a Morse function f : B → R such that all critical points of f have a Morse index of the same parity. Let α be the contact form on Σ induced by r 2 θ ∇ on L. Assume that Σ is "pinched" between two circle bundles S R 1 and S R 2 of radii 5 Normal forms for symplectic matrices This chapter is to appear as an homonymous paper in Portugaliae Mathematica [Gutb].

We give here a self contained and elementary description of normal forms for symplectic matrices, based on geometrical considerations. The normal forms in question are expressed in terms of elementary Jordan matrices and integers with values in {-1, 0, 1} related to signatures of quadratic forms naturally associated to the symplectic matrix.

Let V be a real vector space of dimension 2n with a non degenerate skewsymmetric bilinear form Ω. The symplectic group Sp(V, Ω) is the set of linear transformations of V which preserve Ω:

Sp(V, Ω) = { A : V → V | A linear and Ω(Au, Av) = Ω(u, v) for all u, v ∈ V } .
A symplectic basis of the symplectic vector space (V, Ω) is a basis {e 1 , . . . , e 2n } in which the matrix representing the symplectic form is Ω 0 = 0 Id -Id 0 . In a symplectic basis, the matrix A ′ representing an element A ∈ Sp(V, Ω) belongs to

Sp(2n, R) = A ′ ∈ M at(2n × 2n, R) | A ′ τ Ω 0 A ′ = Ω 0
where (•) τ denotes the transpose of a matrix. Given an element A in the symplectic group Sp(V, Ω), we want to find a symplectic basis of V in which the matrix A ′ representing A has a distinguished form; to give a normal form for matrices in Sp(2n, R) means to describe a distinguished representative in each conjugacy class. In general, one cannot find a symplectic basis of the complexified vector space for which the matrix representing A has Jordan normal form. The normal forms considered here are expressed in terms of elementary Jordan matrices and matrices depending on an integer s ∈ {-1, 0, 1}. They are closely related to the forms given by Long in [START_REF] Long | Normal forms of symplectic matrices[END_REF][START_REF] Long | Index theory for symplectic paths with applications[END_REF] ; the main difference is that, in those references, some indeterminacy was left in the choice of matrices in each conjugacy class, in particular when the matrix admits 1 as an eigenvalue. We speak in this case of quasi-normal forms. Other constructions can be found in [Wim91, LM74, LMX99, Spe72, MT99] but they are either quasi-normal or far from Jordan normal forms. Closely related are the constructions of normal forms for real matrices that are selfadjoint, skewadjoint or unitary with respect to an indefinite inner product where sign characteristics are introduced; they have been 5. Normal forms for symplectic matrices studied in many sources; for instance -mainly for selfadjoint and skewadjoint matricesin the monograph of I. Gohberg, P. Lancaster and L. Rodman [START_REF] Gohberg | Indefinite linear algebra and applications[END_REF], and for unitary matrices in the papers [START_REF] Au-Yeung | H-unitary and Lorentz matrices: a review[END_REF][START_REF] Gohberg | On H-unitary and block-Toeplitz Hnormal operators[END_REF][START_REF] Mehl | On classification of normal matrices in indefinite inner product spaces[END_REF][START_REF] Rodman | Similarity vs unitary similarity and perturbation analysis of sign characteristics: complex and real indefinite inner products[END_REF]. Normal forms for symplectic matrices have been given by C. Mehl in [START_REF] Mehl | Essential decomposition of polynomially normal matrices in real indefinite inner product spaces[END_REF] and by V. Sergeichuk in [START_REF] Sergeȋchuk | Classification problems for systems of forms and linear mappings[END_REF] ; in those descriptions, the basis producing the normal form is not required to be symplectic.

We construct here normal forms using elementary geometrical methods. The choice of representatives for normal (or quasi normal) forms of matrices depends on the application one has in view. Quasi normal forms were used by Long to get precise formulas for indices of iterates of Hamiltonian orbits in [START_REF] Long | Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics[END_REF]. The forms obtained here were useful for us to give new characterisations of Conley-Zehnder indices of general paths of symplectic matrices [Guta]. We have chosen to give a normal form in a symplectic basis. The main interest of our description is the natural interpretation of the signs appearing in the decomposition, and the description of the decomposition for matrices with 1 as an eigenvalue. It also yields an easy natural characterization of the conjugacy class of an element in Sp(2n, R). We hope it can be useful in other situations.

Assume that V decomposes as a direct sum V = V 1 ⊕ V 2 where V 1 and V 2 are Ωorthogonal A-invariant subspaces. Suppose that {e 1 , . . . , e 2k } is a symplectic basis of V 1 in which the matrix representing

A| V 1 is A ′ = A ′ 1 A ′ 2 A ′ 3 A ′ 4
. Suppose also that {f 1 , . . . , f 2l } is a symplectic basis of V 2 in which the matrix representing

A| V 2 is A ′′ = A ′′ 1 A ′′ 2 A ′′ 3 A ′′ 4
. Then {e 1 , . . . , e k , f 1 , . . . , f l , e k+1 , . . . , e 2k , f l+1 , . . . , f 2l } is a symplectic basis of V and the matrix representing A in this basis is

    A ′ 1 0 A ′ 2 0 0 A ′′ 1 0 A ′′ 2 A ′ 3 0 A ′ 4 0 0 A ′′ 3 0 A ′′ 4     .
The notation A ′ ⊕A ′′ is used in Long [START_REF] Long | Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics[END_REF] for this matrix. It is "a direct sum of matrices with obvious identifications". We call it the symplectic direct sum of the matrices A ′ and A ′′ . We C-linearly extend Ω to the complexified vector space V C and we C-linearly extend any

A ∈ Sp(V, Ω) to V C . If v λ denotes an eigenvector of A in V C of the eigenvalue λ, then Ω(Av λ , Av µ ) = Ω(λv λ , µv µ ) = λµΩ(v λ , v µ ), thus Ω(v λ , v µ ) = 0 unless µ = 1 λ .
Hence the eigenvalues of A arise in "quadruples"

[λ] := λ, 1 λ , λ, 1 λ .
(5.1)

• If λ is a real eigenvalue of A (λ / ∈ S 1 so λ = ±1), there exists a symplectic basis of V [λ] in which the matrix representing the restriction of A to V [λ] is a symplectic direct sum of k matrices of the form J(λ, q j ) -1 0 0 J(λ, q j ) τ with q = q 1 ≥ q 2 ≥ • • • ≥ q k and J(λ, m) is the elementary m × m Jordan matrix associated to λ

J(λ, m) =      λ 1 λ 1 0 λ 1 . . . . . . 0 λ 1 λ 1 λ      .
This decomposition is unique, when λ has been chosen in {λ, λ -1 }. It is determined by the chosen λ and by the dimension dim Ker(Aλ Id) r for each r > 0.

• If λ = re iφ / ∈ (S 1 ∪ R) is a complex eigenvalue of A, there exists a symplectic basis of V [λ] in which the matrix representing the restriction of A to V [λ] is a symplectic direct sum of k matrices of the form

J R re -iφ , 2q j -1 0 0 J R re -iφ , 2q j τ with q = q 1 ≥ q 2 ≥ • • • ≥ q k and J R (re iφ , 2m
) is the 2m × 2m block upper triangular matrix defined by

J R (re iφ , 2m) :=         R(re iφ ) Id R(re iφ ) Id 0 R(re iφ ) Id . . . . . . 0 R(re iφ ) Id R(re iφ ) Id R(re iφ )        
with R(re iφ ) = r cos φ -r sin φ r sin φ r cos φ . This decomposition is unique, when λ has been chosen in {λ, λ -1 , λ, λ -1 }. It is is determined by the chosen λ and by the dimension dim Ker(A-λ Id) r for each r > 0.

Normal form for A |V [λ] for λ = ±1 : Let λ = ±1 be an eigenvalue of A. There exists a symplectic basis of V [λ] in which the matrix representing the restriction of A to V [λ] is a symplectic direct sum of matrices of the form J(λ, r j ) -1 C(r j , s j , λ) 0 J(λ, r j ) τ

where C(r j , s j , λ) := J(λ, r j ) -1 diag 0, . . . , 0, s j with s j ∈ {0, 1, -1}. If s j = 0, then r j is odd. The dimension of the eigenspace of the eigenvalue λ is given by 2 Card{j | s j = 0} + Card{j | s j = 0}. The number of s j equal to +1 (resp. -1) arising in blocks of dimension 2k (i.e. with corresponding r j = k) is equal to the number of positive (resp. negative) eigenvalues of the symmetric 2-form

Qλ 2k : Ker (A -λ Id) 2k × Ker (A -λ Id) 2k → R (v, w) → λ Ω (A -λ Id) k v, (A -λ Id) k-1 w .
The decomposition is unique up to a permutation of the blocks and is determined by λ, by the dimension dim Ker(Aλ Id) r for each r ≥ 1, and by the rank and the signature of the symmetric bilinear 2-form Qλ 2k for each k ≥ 1. Normal form for A |V [λ] for λ ∈ S 1 \ {±1} : Let λ ∈ S 1 , λ = ±1 be an eigenvalue of A. There exists a symplectic basis of V [λ] in which the matrix representing the restriction of A to V [λ] is a symplectic direct sum of 4k j × 4k j matrices (k j ≥ 1) of the form where J R (e iφ , 2k) is defined as above, where

   J R (λ,2k j ) -1 0 . . . 0 ••• ••• 0 . . . 0 s j V 1 k j (φ) s j V 2 k j (φ) ) 0 J R (λ,2k j ) τ    (5.2) and (4k j + 2) × (4k j + 2) matrices (k j ≥ 0) of the form           J R (λ,2k j ) -1 s j U 2 k j (φ) 0 . . . 0 ••• ••• 0 . . . 0 s j 2 V 2 k j (φ) -s j 2 V 1 k j (φ) U 1 k j (
V 1 k j (φ) V 2 k j (φ) is the 2k j × 2 matrix defined by V 1 k j (φ) V 2 k j (φ) =    (-1) k j -1 R(e ik j φ ) . . . R(e iφ )    (5.4)
with R(e iφ ) = cos φ -sin φ sin φ cos φ , where

U 1 k j (φ) U 2 k j (φ) = V 1 k j (φ) V 2 k j (φ) R(e iφ ) (5.5)
and where s j = ±1. The complex dimension of the eigenspace of the eigenvalue λ in V C is given by the number of such matrices. The number of s j equal to +1 (resp. -1) arising in blocks of dimension 2m in the normal decomposition given above is equal to the number of positive (resp. negative) eigenvalues of the Hermitian 2-form Qλ m defined on Ker (Aλ Id) m by:

Qλ m : Ker (A -λ Id) m × Ker (A -λ Id) m → C (v, w) → 1 λ Ω (A -λ Id) k v, (A -λ Id) k-1 w if m = 2k (v, w) → i Ω (A -λ Id) k v, (A -λ Id) k w if m = 2k + 1.
This decomposition is unique up to a permutation of the blocks, when λ has been chosen in {λ, λ}. It is determined by the chosen λ, by the dimension dim Ker(Aλ Id) r for each r ≥ 1 and by the rank and the signature of the Hermitian bilinear 2-form Qλ m for each m ≥ 1.

The normal form for A |V [λ] is given in Theorem 5.2.1 for λ / ∈ S 1 , in Theorem 6.5.1 for λ = ±1, and in Theorem 5.4.2 for λ ∈ S 1 \ {±1}. The characterisation of the signs is given in Proposition 5.3.3 for λ = ±1 and in Proposition 5.4.4 for λ ∈ S 1 \ {±1}.

A direct consequence of Theorem 5.0.8 is the following characterization of the conjugacy class of a matrix in the symplectic group.

Theorem 5.0.9 The conjugacy class of a matrix A ∈ Sp(2n, R) is determined by the following data:

• the eigenvalues of A which arise in quadruples [λ] = {λ, λ -1 , λ, λ -1 };

• the dimension dim Ker(Aλ Id) r for each r ≥ 1 for one eigenvalue in each class [λ];

• for λ = ±1, the rank and the signature of the symmetric form Qλ 2k for each k ≥ 1 and for an eigenvalue λ in S 1 \ {±1} chosen in each [λ], the rank and the signature of the Hermitian form Qλ m for each m ≥ 1, with

Qλ m : Ker (A -λ Id) m × Ker (A -λ Id) m → C (v, w) → 1 λ Ω (A -λ Id) k v, (A -λ Id) k-1 w if m = 2k (v, w) → i Ω (A -λ Id) k v, (A -λ Id) k w if m = 2k + 1.

Preliminaries

Lemma 5.1.1 Consider A ∈ Sp(V, Ω) and let 0 = λ ∈ C. Then Ker(Aλ Id) j in V C is the symplectic orthogonal complement of Im(A -1 λ Id) j .

Preliminaries

Proof:

Ω (Aλ Id)u, Av = Ω(Au, Av) -λΩ(u, Av) = Ω(u, v) -λΩ(u, Av)

= -λΩ u, A -1 λ Id v
and by induction Ω (Aλ Id) j u, A j v = (-λ) j Ω u, A -1 λ Id j v .

(5.6)

The result follows from the fact that A is invertible.

Corollary 5.1.2 If E λ denotes the generalized eigenspace of eigenvalue λ, i.e E λ := v ∈ V C | (Aλ Id) j v = 0 for an integer j > 0 , we have

Ω(E λ , E µ ) = 0 when λµ = 1.
Indeed the symplectic orthogonal complement of E λ = ∪ j Ker(Aλ Id) j is the intersection of the Im(A-1 λ Id) j . By Jordan normal form, this intersection is the sum of the generalized eigenspaces corresponding to the eigenvalues which are not 1 λ .

If v = u+iu ′ is in Ker(A-λ Id) j with u and u ′ in V then v = u-iu ′ is in Ker(A-λ Id) j so that E λ ⊕ E λ is the complexification of a real subspace of V . From this remark and corollary 5.1.2 the space

W [λ] := E λ ⊕ E 1 λ ⊕ E λ ⊕ E 1 λ (5.7)
is the complexification of a real and symplectic A-invariant subspace V [λ] and

V = V [λ 1 ] ⊕ V [λ 2 ] ⊕ . . . ⊕ V [λ K ] (5.8)
where we denote by [λ] the set {λ, λ, 1 λ , 1 λ } and by [λ 1 ] , . . . , [λ K ] the distinct such sets exhausting the eigenvalues of A. We denote by A [λ i ] the restriction of A to V [λ i ] . It is clearly enough to obtain normal forms for each A [λ i ] since A will be a symplectic direct sum of those.

We shall construct a symplectic basis of W [λ] (and of V [λ] ) adapted to A for a given eigenvalue λ of A. We assume that (A-λ Id) p+1 = 0 and (A-λ Id) p = 0 on the generalized eigenspace E λ . Since A is real, this integer p is the same for λ. By lemma 5.1.1, Ker(Aλ Id) j is the symplectic orthogonal complement of Im A-1 λ Id j for all j, thus dim Ker(Aλ Id) j = dim Ker A -1 λ Id j ; hence the integer p is the same for λ and 1 λ . We decompose W [λ] (and V [λ] ) into a direct sum of A-invariant symplectic subspaces. Given a symplectic subspace Z of V [λ] which is A-invariant , its orthogonal complement (with respect to the symplectic 2-form) V ′ := Z ⊥ Ω is again symplectic and A-invariant. so that Ω (Aλ Id) p v, w = (-λ 2 ) p Ω v, A -1 λ Id p w (5.13)

and

Ω (A -λ Id) k v, A -1 λ Id j w = 0 if k + j > p.
(5.14)

Proof: We have:

Ω (A -λ Id) i v, A -1 λ Id j w = - 1 λ Ω A -λ Id -A (A -λ Id) i v, A -1 λ Id j w = - 1 λ Ω (A -λ Id) i+1 v, A -1 λ Id j w + 1 λ Ω A(A -λ Id) i v, A -1 λ Id A -1 λ Id j-1 w = - 1 λ Ω (A -λ Id) i+1 v, A -1 λ Id j w + 1 λ Ω (A -λ Id) i v, A -1 λ Id j-1 w - 1 λ 2 Ω A(A -λ Id) i v, A -1 λ Id j-1 w
and formula (5.11) follows.

For any integers k, j ≥ 0 and any v such that (Aλ Id) p v = 0, we have, by (5.6), (-λ) j Ω (Aλ Id) p+k+1-j v, A -1 λ Id j w = Ω (Aλ Id) p+k+1 v, A j w = 0.

Hence, applying formula (5.11) with a decreasing induction on j, we get formula (5.12). The other formulas follow readily.

Definition 5.1.5 For λ ∈ S 1 an eigenvalue of A and v ∈ E λ a generalized eigenvector, we define

T i,j (v) := 1 λ i λ j Ω (A -λ Id) i v, (A -λ) j v .
(5.15)

We have, by equation (5.11) :

T i,j (v) = -T i+1,j (v) -T i+1,j-1 (v), (5.16)
and also, T i,j (v) = -T j,i (v).

(5.17) Lemma 5.1.6 Let λ ∈ S 1 be an eigenvalue of A and v ∈ E λ be a generalised eigenvector such that the largest integer p so that (Aλ Id) p v = 0 is odd, say, p = 2k -1. Then, in the A-invariant subspace E v λ of E λ generated by v, there exists a vector v ′ generating the same A-invariant subspace E v ′ λ = E v λ , so that (Aλ Id) p v ′ = 0 and so that

T i,j (v ′ ) = 0 for all i, j ≤ k -1.
If λ is real (i.e. ±1), and if v is a real vector (i.e. in V ), the vector v ′ can be chosen to be real as well.

Proof: Observe that

T k,k-1 (v) = -T k,k (v) -T k-1,k (v) by (5.11) = -T k-1,k (v) by (5.14) = T k,k-1 (v) by (5.17)
is real and can be put to d = ±1 by rescaling the vector. We use formulas (5.11) and (5.17) and we proceed by decreasing induction on i + j as follows:

• if T k-1,k-1 (v) = α 1 , this α 1 is purely imaginary, we replace v by v ′ := v - α 1 2λd (A -λ Id)v; clearly E v ′ λ = E v λ and T i,j (v ′ ) = T i,j (v) for i + j ≥ 2k -1 but now T k-1,k-1 (v ′ ) = α 1 - α 1 2d T k,k-1 (v) - α 1 2d T k-1,k (v) = 0;
so we can now assume T k-1,k-1 (v) = 0; observe that if λ is real and v is in V , then α 1 = 0 and v ′ = v;

• if T k-2,k-1 (v) = α 2 = -T k-1,k-2 (v), this α 2 is real and we replace v by v - α 2 2λ 2 d (A -λ Id) 2 v;
the space E v λ does not change and the quantities T i,j (v) do not vary for i + j ≥ 2k -2; now

T k-2,k-1 (v ′ ) = α 2 - α 2 2d T k,k-1 (v) - α 2 2d T k-2,k+1 (v) = 0, hence also T k-1,k-2 (v ′ ) = 0; observe that if λ is real and v is in V , then v ′ is in V .
• we now assume by induction to have a J > 0 so that T i,j (v) = 0 for all 0 ≤ i, j ≤ k -1 so that i + j > 2k -1 -J;

• if T k-J,k-1 (v) = α J , then T k-J,k-1 (v) = (-1) J-1 T k-1,k-J (v)
so that α J is real when J is even and is imaginary when J is odd; we replace v by

v - α J 2λ J d (A -λ Id) J v;
the space E v λ does not change and the quantities T i,j (v) do not vary for i+j ≥ 2k -J; but now

T k-J,k-1 (v ′ ) = α J - α J 2d T k,k-1 (v) - α J 2d T k-J,k+J-1 (v) = α J - α J 2 -(-1) J α J 2 = 0.
Hence also T k-J+1,k-2 (v ′ ) = 0, . . . T k-1,k-J+1 (v ′ ) = 0; so the induction proceeds.

Observe that if λ is real and v is in V then v ′ is in V .

We shall use repeatedly that a n × n block triangular symplectic matrix is of the form

A ′ = B C 0 D ∈ Sp(2n, R) ⇔ B = (D τ ) -1 C = (D τ ) -1 S with S symmetric.
(5.18)

Normal forms for A |V

[λ] when λ / ∈ S 1 .
As before, p denotes the largest integer such that (Aλ Id) p does not vanish identically on the generalized eigenspace E λ . Let us choose an element v ∈ E λ and an element w

∈ E 1 λ such that Q p [v], [w] = Ω (A -λ Id) p v, w = 0.
Let us consider the smallest A-invariant subspace E v λ of E λ containing v; it is of dimension p + 1 and a basis is given by a 0 := v, . . . , a i := (Aλ Id) i v, . . . , a p := (Aλ Id) p v .

Observe that Aa i = (Aλ Id)a i + λa i so that Aa i = λa i + a i+1 for i < p and Aa p = a p .

Similarly, we consider the smallest A-invariant subspace E w We choose v and w such that Ω A -1 λ Id p w, v = 1. We define inductively on j b

′ p := 1 Ω(bp,a 0 ) b p = b p ; b ′ p-j =
1 Ω(b p-j ,a j ) b p-j -k<j Ω(b p-j , a k )b ′ p-k , so that any b ′ j is a linear combination of the b r with r ≥ j.

In the symplectic basis b ′ p , . . . , b ′ 0 , a 0 , . . . , a p the matrix representing A is

B 0 0 J(λ, p + 1) τ where J(λ, m) =      λ 1 λ 1 0 λ 1 . . . . . . 0 λ 1 λ 1 λ      (5.19)
is the elementary m × m Jordan matrix associated to λ. Since the matrix is symplectic, B is the transpose of the inverse of J(λ, p + 1) τ by (5.18), so B = J(λ, p + 1) -1 . This is the normal form for A restricted to

E v λ ⊕ E w 1 λ . If λ = re iφ / ∈ R
} of E v λ ⊕ E w 1 λ . Writing b ′ j = 1 √
2 (u j + iv j ) and a j = 1 √ 2 (w jix j ) for all 0 ≤ j ≤ p with the vectors u j , v j , w j , x j in the real vector space V, we get a symplectic basis {u p , v p . . . , u 0 , v 0 , w 0 , x 0 . . . , w p , x p } of the real subspace of V whose complexification is

E v λ ⊕ E w 1 λ ⊕ E v λ ⊕ E w 1 λ
. In this basis, the matrix

representing A is J R λ, 2(p + 1) -1 0 0 J R λ, 2(p + 1) τ
where J R (re iφ , 2m) is the 2m × 2m matrix written in terms of 2 × 2 matrices as

J R (re iφ , 2m) :=         R(re iφ ) Id R(re iφ ) Id 0 R(re iφ ) Id . . . . . . 0 R(re iφ ) Id R(re iφ ) Id R(re iφ )        
(5.20) with R(re iφ ) = r cos φ -r sin φ r sin φ r cos φ . By induction, we get Theorem 5.2.1 (Normal form for A |V [λ] for λ / ∈ S 1 .) Let λ / ∈ S 1 be an eigenvalue of A. Denote k := dim C Ker(A-λ Id) (on V C ) and p the smallest integer so that (A-λ Id) p+1 is identically zero on the generalized eigenspace E λ .

• If λ = ±1 is a real eigenvalue of A, there exists a symplectic basis of V [λ] in which the matrix representing the restriction of A to V [λ] is a symplectic direct sum of k matrices of the form J(λ, p j + 1) -1 0 0 J(λ, p j + 1) τ with p = p 1 ≥ p 2 ≥ • • • ≥ p k and J(λ, k) defined by (5.19). To eliminate the ambiguity in the choice of λ in [λ] = {λ, λ -1 } we can consider the real eigenvalue such that λ > 1. The size of the blocks is determined knowing the dimension dim (Ker(Aλ Id) r ) for each r ≥ 1.

• If λ = re iφ / ∈ (S 1 ∪ R) is a complex eigenvalue of A, there exists a symplectic basis of V [λ] in which the matrix representing the restriction of A to V [λ] is a symplectic direct 5. Normal forms for symplectic matrices sum of k matrices of the form

J R re -iφ , 2(p j + 1) -1 0 0 J R re -iφ , 2(p j + 1) τ with p = p 1 ≥ p 2 ≥ • • • ≥ p k and J R (re iφ ,
k) defined by (5.20). To eliminate the ambiguity in the choice of λ in [λ] = {λ, λ -1 , λ, λ -1 } we can choose the eigenvalue λ with a positive imaginary part and a modulus greater than 1. The size of the blocks is determined, knowing the dimension dim C (Ker(Aλ Id) r ) for each r ≥ 1.

This normal form is unique, when a choice of λ in the set [λ] is fixed.

Normal forms for A |V

[λ] when λ = ±1.
In this situation [λ] = {λ} and V [λ] is the generalized real eigenspace of eigenvalue λ, still denoted -with a slight abuse of notation-E λ . Again, p denotes the largest integer such that (Aλ Id) p does not vanish identically on E λ . We consider

Q p : E λ/Ker(A -λ Id) p × E λ/Ker(A -λ Id) p → R the non degenerate form defined by Q p [v], [w] = Ω (A - λ Id) p v, w .
We see directly from equation (5.13) that Q p is symmetric if p is odd and antisymmetric if p is even.

If

p = 2k -1 is odd we choose v ∈ E λ such that Q [v], [v] = Ω (A -λ Id) p v, v = 0
and consider the smallest A-invariant subspace E v λ of E λ containing v; it is spanned by a p := (Aλ Id) p v, . . . , a i := (Aλ Id) i v, . . . , a 0 := v .

We have Ω(a i , a j ) = 0 if i + j ≥ p + 1(= 2k) by equation (5.14); Ω(a i , a p-i ) = 0; by equation(5.12) and by the choice of v.

Hence E v λ is a symplectic subspace because, in the basis defined by the e i 's, Ω has the triangular form We can choose v in E λ ⊂ V so that Ω (Aλ Id) k v, (Aλ Id) k-1 v = λs with s = ±1 by rescaling the vector and one may further assume, by lemma 5.1.6, that

T i,j (v) = 1 λ i 1 λ j Ω (A -λ Id) i v, (A -λ Id) j v = 0 for all 0 ≤ i, j ≤ k -1.
We now construct a symplectic basis a ′ p , . . . , a ′ k , a 0 , . . . , a k-1 of E v λ , extending {a 0 , . . . , a k-1 }, by a Gram-Schmidt procedure, having chosen v as above. We define inductively on 0 ≤ j ≤ k -1

a ′ p := 1 Ω(ap,a 0 ) a p ; a ′ p-j =
1 Ω(a p-j ,a j ) a p-j -k<j Ω(a p-j , a k )a ′ p-k , so that any a ′ j is a linear combination of the a r 's with r ≥ j and in particular a

′ k = 1 sλ a k + k-1 j=1 c j a k+j .
In the symplectic basis a ′ p , . . . , a ′ k , a 0 , . . . , a k-1 the matrix representing A is

A ′ = B C 0 J(λ, k) τ
with J(λ, m) defined by (5.19) and with C identically zero except for the last column, and the coefficient C k k = sλ. Since the matrix is symplectic, B is the transpose of the inverse of J(λ, p + 1) τ by (5.18), so B = J(λ, k) -1 and J(λ, k)C is symmetric with zeroes except in the last column, hence diagonal of the form diag 0, . . . , 0, s . Thus

J(λ, k) -1 J(λ, k) -1 diag 0, . . . , 0, s 0 J(λ, k) τ , with s = ±1, is the normal form of A restricted to E v λ . Recall that s = λ -1 Ω (A -λ Id) k v, (A -λ Id) k-1 v .

If p = 2k is even

we choose v and w in E λ such that

Q [v], [w] = Ω (A -λ Id) p v, w = λ p = 1 and we consider the smallest A-invariant subspace E v λ ⊕ E w λ of E λ containing v and w. It is of dimension 4k + 2. Remark that Ω (A -λ Id) p v, v = 0. We can choose v so that T r,s (v) = 1 λ r+s Ω (A -λ Id) r v, (A -λ Id) s v = 0 for all r, s. Indeed, by formula (5.11) we have T i,j (v) = -T i+1,j (v)-T i+1,j-1 (v). Observe that T i,j (v) = -T j,i (v) so that T i,i (v) = 0 and T j,i (v) = -T j,i+1 (v)-T j-1,i+1 (
v). We proceed by induction, as in lemma 5.1.6 :

• T p,0 (v) = 0 implies T p-r,r (v) = 0 for all 0 ≤ r ≤ p by equation (5.12).

• We assume by decreasing induction on J, starting from J = p, that we have T i,j (v) = 0 for all i + j ≥ J. Then we have T J-1-s,s (v) = -T J-1-s,s+1 (v) -T J-2-s,s+1 (v); the first term on the righthand side vanishes by the induction hypothesis, so T J-1,0 (v) = (-1) s T J-1-s,s (v) = (-1) J-1 T 0,J-1 (v) = (-1) J T J-1,0 .

If T J-1,0 (v) = α = 0, J must be even and we replace v by

v ′ = v + α 2λ p-J+1 (A -λ Id) p-J+1 w. Then v ′ ∈ E v λ ⊕ E w λ , E v λ ⊕ E w λ = E v ′ λ ⊕ E w λ , Ω (A -λ Id) p v ′ , w = λ p and T i,j (v ′ ) = T i,j (v) = 0 for all i + j ≥ J but now T J-1,0 (v ′ ) = T J-1,0 (v) + α 2λ p Ω (A -λ Id) p w, v + α 2λ p Ω (A -λ Id) J-1 v, (A -λ Id) p-J+1 w + α 2 4λ p Ω (A -λ Id) p w, (A -λ Id) p-J+1 w = α - α 2 - α 2 = 0
so that T i,j (v ′ ) = 0 for all i + j ≥ J -1 and the induction proceeds.

We assume from now on that we have chosen v and w in E λ so that Ω (Aλ Id) p v, w = 1 and Ω (Aλ Id) r v, (A -1 λ Id) s v = 0 for all r, s. We can proceed similarly with w so we can thus furthermore assume that Ω (Aλ Id) j w, Aλ Id k w = 0 for all j, k.

A basis of E v λ ⊕ E w λ is given by a p = (Aλ Id) p v, . . . , a 0 = v, b 0 = w, . . . , b p = (Aλ Id) p w .

We have Ω(a i , a j ) = 0 and Ω(b i , b j ) = 0 by the choice of v and w; Ω(a i , b j ) = 0 if i + j > p by equation (5.14) ; Ω(a i , b p-i ) = 0 by equation (5.12) and the choice of of v, w.

The matrix representing Ω has the form

      0 * 0 . . . * * * * . . . 0 * 0       hence is non singular and the subspace E v λ ⊕E w λ is symplectic. We now construct a symplectic basis a ′ p , . . . , a ′ 0 , b 0 , . . . , b p of E v λ ⊕E w 1 λ
, extending {b 0 , . . . , b p }, using a Gram-Schmidt procedure on the a i 's. We define inductively on j Proof: On the intersection of Ker (Aλ Id) 2k with one of the symplectically orthogonal subspaces E v λ constructed above for an odd p = 2k -1, the form Qλ 2k vanishes identically. On the intersection of Ker (Aλ Id) 2k with a subspace E v λ for a v so that p = 2k -1 and Ω

(A -λ Id) k v, (A -λ Id) k-1 v = λs the only non vanishing component is Qλ 2k (v, v) = s. Indeed, Ker (A -λ Id) 2k ∩ E v λ is spanned by {(A -λ Id) r v ; r ≥ 0 and r + 2k > p },
and Ω (Aλ Id) k+r v, (Aλ Id) k-1+r ′ v = 0 when 2k + r + r ′ -1 > p so the only non vanishing cases arise when r = r ′ = 0 and p = 2k -1. Similarly, the 2 form Qλ 2k vanishes on the intersection of Ker (Aλ Id) 2k with a subspace E v λ ⊕ E w λ constructed above for an even p.

The numbers s j appearing in the decomposition of A are thus invariant of the matrix.

Corollary 5.3.4 The normal decomposition described in theorem 6.5.1 is determined by the eigenvalue λ, by the dimension dim Ker(Aλ Id) r for each r ≥ 1, and by the rank and the signature of the symmetric bilinear 2-forms Qλ 2k for each k ≥ 1. It is unique up to a permutation of the blocks.

Normal forms for A |V

[λ] when λ = e iφ ∈ S 1 \ {±1}.
We denote again by p the largest integer such that (Aλ Id) p does not vanish identically on E λ and we consider the non degenerate sesquilinear form

Q : E λ/Ker(A -λ Id) p × E λ/Ker(A -λ Id) p → C Q [v], [w] = λ p Ω (A -λ Id) p v, w . Since Q is non degenerate, we can choose v ∈ E λ such that Q([v], [v]) = 0 thus (A - λ Id) p v =
0 and we consider the smallest A-invariant subspace, stable by complex conjugaison, and containing v :

E v λ ⊕ E v λ ⊂ E λ ⊕ E λ . A basis is given by a i := (A -λ Id) i v, b j := (A -λ Id) j v 0 ≤ i, j ≤ p .
We have a i = b i and

• Ω(a i , a j ) = 0, Ω(b i , b j ) = 0 because Ω(E λ , E λ ) = 0; • Ω(a i , b k ) = 0 if i + k ≥ p + 1 by equation (5.14);
• Ω(a i , b k ) = 0 if p = i + k by equation (5.12) and by the choice of v.

We conclude that E v λ ⊕ E v λ is a symplectic subspace.

If

p = 2k -1 is odd observe that T k,k-1 (v) := 1 λ Ω (A -λ Id) k v, (A -λ Id) k-1 v = s
is real and can be put to ±1 by rescaling the vector (we could even put it to 1 exchanging if needed λ and its conjugate). One may further assume, by lemma 5.1.6 that

T i,j (v) = 1 λ i 1 λ j Ω (A -λ Id) i v, (A -λ Id) j v = 0 for all 0 ≤ i, j ≤ k -1.
We consider the basis {a 2k-1 , . . . , a k , b p , . . . , b k , b 0 , . . . b k-1 , a 0 , . . . a k-1 } for such a vector v with T k,k-1 (v) = s = ±1 and T i,j (v) = 0 for all 0 ≤ i, j ≤ k -1; the matrix representing Ω has the form

                 0 * 0 . . . * * 0 0 * 0 . . . * * * * . . . 0 * 0 0 * * . . . 0 * 0                 
and we transform it by a Gram-Schmidt method into a symplectic basis composed of pairs of conjugate vectors, extending {b 0 , . . . , b k-1 , a 0 , . . . , a k-1 } on which Ω identically vanishes. We define

a ′ 2k-1 = 1 Ω(a 2k-1 , b 0 ) a 2k-1 , b ′ 2k-1 = 1 Ω(b 2k-1 , a 0 ) b 2k-1 = a ′ 2k-1
and, inductively on increasing j with 1

< j ≤ k a ′ 2k-j = 1 Ω(a 2k-j , b j-1 ) a 2k-j - j-1 r=1 Ω(a 2k-j , b r-1 ) a ′ 2k-r , b ′ 2k-j = a ′ 2k-j .
Any a ′ 2k-j is a linear combination of the a 2k-i for 1 ≤ i ≤ j; reciprocally any a 2k-j can be written as a linear combination of the a ′ 2k-i for 1 ≤ i ≤ j, and the coefficient of a ′ 2k-j is equal to Ω(a 2k-j , b j-1 ).

The basis {a

′ 2k-1 , . . . , a ′ k , b ′ 2k-1 , . . . , b ′ k , b 0 , . . . , b k-1 , a 0 , . . . , a k-1 } is symplectic,
   * 0 0 C * C 0 J(λ, k) τ 0 0 J(λ, k) τ    
where C is a k × k matrix such that the only non vanishing terms are on the last column (C i j = 0 when j < k) and C k k = Ω(a k , b k-1 ) = sλ. The fact that the matrix is symplectic implies that S := J(λ, k)C is hermitean; since S i j = 0 when j = k, we have,

C = J(λ, k) -1 0 ... 0 0 . . . . . . . . . . . . 0 ... 0 0 0 ... 0 s = C(k, s, λ)
and the matrix of the restriction of A to the subspace

E v λ ⊕ E v λ has the block triangular normal form     J(λ, k) -1 0 0 C(k, s, λ) J(λ, k) -1 C(k, s, λ) 0 J(λ, k) τ 0 0 J(λ, k) τ     . (5.23) Writing a ′ 2k-j = 1 √ 2 (e 2j-1 -ie 2j ), b ′ 2k-j = a ′ 2k-j = 1 √ 2 (e 2j-1 + ie 2j ), as well as a j-1 = 1 √ 2 (f 2j-1 -if 2j ) and b j-1 = a j-1 = 1 √ 2 (f 2j-1 + if 2j ) for 1 ≤ j ≤ k, the vectors e i , f j all belong to the real subspace denoted V v [λ] of V whose complexification is E v λ ⊕ E v λ and
J R (λ, 2k) -1 C R (k, s, λ) 0 J R (λ, 2k) τ (5.24)
where J R (e iφ , 2k) is defined as in (5.20) and where C R (k, s, e iφ ) is the (p + 1) × (p + 1) matrix written in terms of two by two matrices as

C R (k, s, e iφ ) τ = s   0 ... 0 0 . . . . . . . . . 0 ... 0 0 (-1) k-1 R(e ikφ ) ... -R(e i2φ ) R(e iφ )  
(5.25)

with R(e iφ ) = cos φsin φ sin φ cos φ as before and s = ±1. This is the normal form of A

restricted to V v [λ] ; recall that s = λ -1 Ω (A -λ Id) k v, (A -λ Id) k-1 v .

Normal forms for symplectic matrices

Remark 5.4.1 For such a v, all T i,j (v) are determined inductively and we have

T i,j (v) = 0 if i + j ≥ 2k + 1 and for all 0 ≤ i, j ≤ k -1 T k-r,k+r (v) = (-1) r+1 si for all 0 ≤ r ≤ k T k-r,k+m (v) = (-1) r+1 si 2 (r + m)(r -1)! m!(r -m)! for all 0 ≤ m ≤ r ≤ k, r > 1 T i,j (v) = T j,i (v)
for all i, j.

With the notation a

i = (A -λ Id) i v, b i = (A -λ Id) i v,
                      0 0 0 * 0 . . . * * 0
0 -si * ••• * 0 0                       .
We transform (by a Gram-Schmidt method) the basis above into a symplectic basis, composed of pairs of conjugate vectors (up to a factor) and extending b 0 , . . . , b k-1 , a 0 , . . . , a k-1

Normal forms for A |V

[λ] when λ = e iφ ∈ S 1 \ {±1}.
on which Ω identically vanishes. We define inductively, for increasing j with 1 ≤ j ≤ k -1 This implies

a ′ 2k : = 1 Ω (A -λ Id) 2k v, v (A -λ Id) 2k v = 1 Ω(a 2k , b 0 ) a 2k b ′ 2k : = 1 Ω (A -λ Id) 2k , v, v (A -λ Id) 2k v = 1 Ω(b 2k , a 0 ) b 2k = a ′ 2k a ′ 2k-j = 1 Ω(a 2k-j , b j ) a 2k-j - j-1 r=0 Ω(a 2k-j , b r )a ′ 2k-r b ′ 2k-j = 1 Ω(b 2k-j , a j ) b 2k-j - j-1 r=0 Ω(b 2k-j , a r )b ′ 2k-r = a ′ 2k-j a ′ k = a k - k-1 r=0 Ω(a k , b r )a ′ 2k-r b ′ k = 1 Ω(b k , a k ) b k - k-1 r=0 Ω(b k , a r )b ′ 2k-r = 1 is a ′ k . Each a ′ 2k-j is a linear combination of the (A -λ Id) 2k-r v for 0 ≤ r ≤ j. The basis {a ′ 2k , . . . , a ′ k+1 , b ′ 2k , . . . , b ′ k+1 , b ′ k ; b 0 , . . . , b k-1 , a 0 , . . . , a k-1 , a ′ k } is now symplectic. Since A(a r ) = λa r + a r+1 for all r < 2k, and A(a 2k ) = λa 2k , the matrix representing A in that basis is of the form                A 1 0 0    0 c 2k d 2k . . . . . . c k+1 d k+1    0 A 2     0 e 2k . . . e k+1 e k     0 0 0 J(λ,k) τ 0 0 0 0 J(λ,k+1) τ                with A(b k-1 ) = λb k-1 + k j=0 e k+j b ′ k+j , A(a k-1 ) = λa k-1 + a ′ k + k j=1 c k+j a ′ k+j and A(a ′ k ) = λa ′ k + k j=1 d k+j a ′ k+j . Since a matrix A ′ E 0 D is symplectic if and only if A ′ = (D τ ) -1 and D τ E is symmetric, we have A 1 = J(λ, k) -1 A 2 = J(λ, k +
J(λ, k)   c 2k d 2k . . . . . . c k+2 d k+2 c k+1 d k+1   = 0 0 . . . . . . 0 0 s 1 s 2 J(λ, k + 1)    e 2k
. . .

e k+2 e k+1 e k    =   0 . . . 0 s 1 s 2   so that s 1 = λc k+1 and s 2 = λd k+1 . Now A(a ′ k ) = A a k + j≥1 F j k a k+j = λa ′ k + a k+1 + j≥1 F j k a k+j+1 = λa ′ k + a ′ k+1 Ω(a k+1 , b k-1 ) + j≥1 F ′ j k a ′ k+j+1 so that d k+1 = Ω(a k+1 , b k-1 ) = λ 2 is and s 2 = λis. We also have A(a k-1 ) = λa k-1 + a k = λa k-1 + a ′ k + Ω(a k , b k-1 )a ′ k+1 + j≥2 G j a ′ k+j so that c k+1 = Ω(a k , b k-1 ) = λ 1 2 is and s 1 = 1 2 is.
We have thus shown that the matrix representing A in the chosen basis has the block upper-triangular normal form

    J(λ, k) -1 0 0 J(λ, k) -1 S J(λ, k + 1) -1 J(λ, k + 1) -1 S τ 0 J(λ, k) τ 0 0 J(λ, k + 1) τ     (5.27)
where S is the k × (k + 1) matrix defined by

S = S(k, d, λ) :=      0 . . . 0 0 0 . . . . . . . . . . . . 0 . . . 0 0 0 0 . . . 0 1 2 is λis      . (5.28) We write a ′ 2k+1-j = 1 √ 2 (e 2j-1 -ie 2j ), b ′ 2k+1-j = a ′ 2k+1-j = 1 √ 2 (e 2j-1 + ie 2j ), as well as a j-1 = 1 √ 2 (f 2j-1 -if 2j ) and b j-1 = a j-1 = 1 √ 2 (f 2j-1 + if 2j ) for 1 ≤ j ≤ k, and a ′ k = 1 √ 2 (e 2k+1 + id f 2k+1 ), b ′ k = -ida ′ k = 1 √ 2 (-f 2k+1 -id e 2k+1 ).
The vectors e i , f j all belong to the real subspace

V v [λ] of V whose complexification is E v λ ⊕ E v λ and we get a symplectic basis {e 1 , . . . , e 2k+1 , f 1 , . . . , f 2k+1 } of V v [λ]
. In this basis, the matrix representing A is :

          J R (λ,2k) -1 s U 2 (φ) 0 . . . 0 ••• ••• 0 . . . 0 s 2 V 2 (φ) -s 2 V 1 (φ) U 1 (φ) 0 cos φ 0 ... 0 1 0 s sin φ 0 0 . . . 0 J R (λ,2k) τ 0 . . . 0 0 -s sin φ 0 ... 0 0 -s cos φ           where s = ±1, U 1 (φ), U 2 (φ), V 1 (φ) and V 2 (φ) are real 2k × 1 column matrices such that V 1 (φ) V 2 (φ) =    (-1) k-1 R(e ikφ ) . . . R(e iφ )    U 1 (φ) U 2 (φ) =    (-1) k-1 R(e i(k+1)φ ) . . . R(e i2φ )    = V 1 (φ) V 2 (φ) R(e iφ ) .

This is the normal form of

A restricted to V v [λ] . Recall that s = iΩ (A -λ Id) k v, (A -λ Id) k v .
Theorem 5.4.2 (Normal form for A |V [λ] for λ ∈ S 1 \ {±1}.) Let λ ∈ S 1 \ {±1} be an eigenvalue of A. There exists a symplectic basis of V [λ] in which the matrix representing the restriction of

A to V [λ] is a symplectic direct sum of 4k j × 4k j matrices (k j ≥ 1) of the form    J R (λ,2k j ) -1 0 . . . 0 ••• ••• 0 . . . 0 s j V 1 k j (φ) s j V 2 k j (φ) ) 0 J R (λ,2k j ) τ    (5.29)
5. Normal forms for symplectic matrices and (4k j + 2) × (4k j + 2) matrices (k j ≥ 0) of the form 

         J R (λ,2k j ) -1 s j U 2 k j (φ) 0 . . . 0 ••• ••• 0 . . . 0 s j 2 V 2 k j (φ) -s j 2 V 1 k j (φ) U 1 k j (φ) 0 cos φ 0 ... 0 1 0 s j sin φ 0 0 . . . 0 J R (λ,2k j ) τ 0 . . . 0 0 -s j sin φ 0 ... 0 0 -s j cos φ          
(5.30)

where J R (e iφ , 2k) is defined as in (5.20), where

V 1 k j (φ) V 2 k j (φ) is the 2k j ×2 matrix defined by V 1 k j (φ) V 2 k j (φ) =    (-1) k j -1 R(e ik j φ ) . . . R(e iφ )    (5.31)
with R(e iφ ) = cos φsin φ sin φ cos φ , where

U 1 k j (φ) U 2 k j (φ) = V 1 k j (φ) V 2 k j (φ) R(e iφ ) (5.32)
and where s j = ±1. The complex dimension of the eigenspace of eigenvalue λ in V C is given by the number of such matrices.

Definition 5.4.3 Given λ ∈ S 1 \ {±1}, we define, for any integer m ≥ 1, a Hermitian form Qλ m on Ker ((Aλ Id) m ) by:

Qλ m : Ker (A -λ Id) m × Ker (A -λ Id) m → C (v, w) → 1 λ Ω (A -λ Id) k v, (A -λ Id) k-1 w if m = 2k (v, w) → i Ω (A -λ Id) k v, (A -λ Id) k w if m = 2k + 1.
Proposition 5.4.4 For λ ∈ S 1 \ {±1}, the number of positive (resp. negative) eigenvalues of the Hermitian 2-form Qλ m is equal to the number of s j equal to +1 (resp. -1) arising in blocks of dimension 2m in the normal decomposition of A on V [λ] given in theorem 5.4.2.

Proof: On the intersection of Ker (Aλ Id) m with one of the symplectically orthogonal subspaces

E v λ ⊕ E v λ constructed above from a v such that (A -λ Id) p v = 0 and (A -λ Id) p+1 v = 0, the form Qλ m vanishes identically, except if p = m -1 and the only non vanishing component is Qλ m (v, v) = s. Indeed, Ker (A -λ Id) m ∩ E v λ is spanned by {(A -λ Id) r v ; r ≥ 0 and r + m > p },
and Qλ m (Aλ Id) r v, (Aλ Id) r ′ v = 0 when m + r + r ′ -1 > p so the only non vanishing cases arise when r = r ′ = 0 and m = p + 1 so for Qλ m (v, v). This is equal to

1 λ Ω (A -λ Id) k v, (A -λ Id) k-1 v = 1 λ λs = s if m = 2k, and to i Ω (A -λ Id) k v, (A - λ Id) k v = i(-is) = s if m = 2k + 1.
The numbers s j appearing in the decomposition are thus invariant of the matrix.

Corollary 5.4.5 The normal decomposition described in theorem 5.4.2 is unique up to a permutation of the blocks when the eigenvalue λ has been chosen in {λ, λ}, for instance by specifyng that its imaginary part is positive. It is completely determined by this chosen λ, by the dimension dim C Ker(Aλ Id) r for each r ≥ 1 and by the rank and the signature of the Hermitian bilinear 2-forms Qλ m for each m ≥ 1.

Generalised Conley-Zehnder index

This chapter will appear as an homonymous paper in Annales de la Faculté des Sciences de Toulouse, [Guta]. The Conley-Zehnder index associates an integer to any continuous path of symplectic matrices starting from the identity and ending at a matrix which does not admit 1 as an eigenvalue. We give new ways to compute this index. Robbin and Salamon define a generalization of the Conley-Zehnder index for any continuous path of symplectic matrices; this generalization is half integer valued. It is based on a Maslov-type index that they define for a continuous path of Lagrangians in a symplectic vector space (W, Ω), having chosen a given reference Lagrangian V . Paths of symplectic endomorphisms of (R 2n , Ω 0 ) are viewed as paths of Lagrangians defined by their graphs in (W = R 2n ⊕ R 2n , Ω = Ω 0 ⊕ -Ω 0 ) and the reference Lagrangian is the diagonal. Robbin and Salamon give properties of this generalized Conley-Zehnder index and an explicit formula when the path has only regular crossings. We give here an axiomatic characterization of this generalized Conley-Zehnder index. We also give an explicit way to compute it for any continuous path of symplectic matrices.

Introduction

The Conley-Zehnder index associates an integer to any continuous path ψ defined on the interval [0, 1] with values in the group Sp R 2n , Ω 0 = 0 Id -Id 0 of 2n × 2n symplectic matrices, starting from the identity and ending at a matrix which does not admit 1 as an eigenvalue. This index is used in the definition of the grading of Floer homology theories. If the path ψ were a loop with values in the unitary group, one could define an integer by looking at the degree of the loop in the circle defined by the (complex) determinant -or an integer power of it. The construction [SZ92, Sal99, AD10] of the Conley-Zehnder index is based on this idea. One uses a continuous map ρ from the sympletic group Sp(R 2n , Ω 0 ) into S 1 and an "admissible" extension of ψ to a path ψ : [0, 2] → Sp(R 2n , Ω 0 ) in such a way that ρ 2 • ψ : [0, 2] → S 1 is a loop. The Conley-Zehnder index of ψ is defined as the degree of this loop µ CZ (ψ) := deg(ρ 2 • ψ).

6.1. Introduction for a path having only regular crossings. A crossing for Λ is a number t ∈ [a, b] for which dim Λ t ∩ L = 0, and a crossing t is regular if the crossing form Γ(Λ, L, t) is nondegenerate. We recall the precise definitions in section 6.3. Robbin and Salamon define the index of a continuous path of symplectic matrices ψ : [0, 1] → Sp(R 2n , Ω 0 ) : t → ψ t as the index of the corresponding path of Lagrangians in (W := R 2n × R 2n , Ω = -Ω 0 × Ω 0 ) defined by their graphs, Λ = Gr ψ : [0, 1] → L (W,Ω) : t → Gr ψ t = {(x, ψ t x)|x ∈ R 2n }.

The reference Lagrangian is the diagonal ∆ = {(x, x)|x ∈ R 2n }. They prove that this index coincide with the Conley Zehnder index on continuous paths of symplectic matrices which start from the identity and end at a matrix which does not admit 1 as an eigenvalue. To be complete, we include this in section 6.4. They also prove that this index vanishes on a path of symplectic matrices with constant dimensional 1-eigenspace. Robbin and Salamon present also another way to associate an index to a continuous path ψ of symplectic matrices. One chooses a Lagrangian L in L (R 2n ,Ω 0 ) and one considers the index of the path of Lagrangians t → ψ t L, with L as the reference Lagrangian. We show in section 6.4.2 that those two indices do not coincide in general.

We use the normal form of the restriction of a symplectic endomorphism to the generalized eigenspace of eigenvalue 1 obtained in [Gutb] to construct special paths of symplectic endomorphisms with a constant dimension of the eigenspace of eigenvalue 1. This leads in section 6.5 to a characterization of the generalized half-integer valued Conley Zehnder index defined by Robbin and Salamon : Theorem 6.1.2 The Robbin-Salamon index for a continuous path of symplectic matrices is characterized by the following properties:

• (Homotopy) it is invariant under homotopies with fixed end points;

• (Catenation) it is additive under catenation of paths;

• (Zero) it vanishes on any path ψ : [a, b] → Sp(R 2n , Ω) of matrices such that dim Ker ψ(t)-Id = k is constant on [a, b];

• (Normalization) if S = S τ ∈ R 2n×2n is a symmetric matrix with all eigenvalues of absolute value < 2π and if ψ(t) = exp(J 0 St) for t ∈ [0, 1] , then µ RS (ψ) = 1 2 Sign S where Sign S is the signature of S.

The same techniques lead in section 6.6 to a new formula for this index : Theorem 6.1.3 Let ψ : [0, 1] → Sp(R 2n , Ω 0 ) be a path of symplectic matrices. Decompose ψ(0) = ψ ⋆ (0) ⊕ ψ (1) (0) and ψ(1) = ψ ⋆ (1) ⊕ ψ (1) (1) where ψ ⋆ (•) does not admit 1 as eigenvalue and ψ (1) (•) is the restriction of ψ(•) to its generalized eigenspace of eigenvalue 1. Consider a continuous extension Ψ : [-1, 2] → Sp(R 2n , Ω 0 ) of ψ such that Proof: Assume µ ′ : SP(n) → Z is a map satisfying those properties. Let ψ : [0, 1] → Sp(R 2n , Ω 0 ) be an element of SP(n); Since ψ is in the same component of SP(n) as its prolongation ψ : [0, 2] → Sp(R 2n , Ω 0 ) we have µ ′ (ψ) = µ ′ ( ψ).

Observe that W + = exp π(J 0 S + ) with S + = Id and W -= exp π(J 0 S -) with

S -=   0 0 -log 2 π 0 0 Id n-1 0 0 -log 2 π 0 0 0 0 0 0 Id n-1   .
The catenation of ψ and ψ - 2 (the path ψ 2 in the reverse order, i.e followed from end to beginning) when ψ 2 : [0, 1] → Sp(R 2n , Ω 0 ) t → exp tπJ 0 S ± is a loop φ. Hence ψ is homotopic to the catenation of φ and ψ 2 , which is homotopic to the product φψ 2 (see, for instance, [Gutb]).

Thus we have µ ′ (ψ) = µ ′ (φψ 2 ). By the loop condition µ ′ (φψ 2 ) = µ ′ (ψ 2 ) + 2µ(φ) and by the signature condition µ ′ (ψ 2 ) = 1 2 Sign(S ± ). Thus µ ′ (ψ) = 2µ(φ) + 1 2 Sign(S ± ).

Since the same is true for µ CZ (ψ), this proves uniqueness.

Remark that we have only used the signature property to know the value of the Conley-Zehnder index on the paths ψ 2± : t ∈ [0, 1] → exp tπJ 0 S ± . Hence we have : (6.9)

Proof: Remark that for any A ∈ Sp(R 2n , Ω 0 ) the element C A := 1 2 (A -J 0 AJ 0 ), which clearly defines a complex linear endomorphism of C n since it commutes with J 0 , is always invertible. Indeed for any non-zero v ∈ V 4Ω 0 (C A v, J 0 C A v) = 2Ω 0 (v, J 0 v) + Ω 0 (Av, J 0 Av) + Ω 0 (AJ 0 v, J 0 AJ 0 v) > 0.

If A ∈ U (n), then C A = A so that ρ(A) = det C (A) hence ρ is a continuous map which coincide with det C on U (n). Furthermore 1 2 2 t 0 0 2 -t -J 0 2 t 0 0 2 -t J 0 = 1 2 2 t +2 -t 0 0 2 t +2 -t hence its complex determinant is equal to 1 2 (2 t +2 -t ) and its normalized complex determinant is equal to 1 so that ρ(exp tπJ 0 S -) = e πi(n-1)t and deg(ρ 2 • ψ 2-) = n -1.

The Robbin-Salamon index for a path of Lagrangians

A Lagrangian in a symplectic vector space (V, Ω) of dimension 2n is a subspace L of V of dimension n such that Ω| L×L = 0. Given any Lagrangian L in V, there exists a Lagrangian M (not unique!) such that L ⊕ M = V. With the choice of such a supplementary M any Lagrangian L ′ in a neighborhood of L (any Lagrangian supplementary to M ) can be identified to a linear map α : L → M through L ′ = {v + α(v) | v ∈ L}, with α such that Ω α(v), w + Ω v, α(w) = 0 ∀v, w ∈ L. Hence it can be identified to a symmetric bilinear form α : L × L → R : (v, v ′ ) → Ω v, α(v ′ ) . In particular the tangent space at a point L to the space L (V,Ω) of Lagrangians in (V, Ω) can be identified to the space of symmetric bilinear forms on L.

If Λ : [a, b] → L n := L (R 2n ,Ω 0 ) : t → Λ t is a smooth curve of Lagrangian subspaces in (R 2n , Ω 0 ) , we define Q(Λ t 0 , Λt 0 ) to be the symmetric bilinear form on Λ t 0 defined by

Q(Λ t 0 , Λt 0 )(v, v ′ ) = d dt α t (v, v ′ ) t 0 = d dt Ω v, α t (v ′ )
t 0 (6.10) where α t : Λ t 0 → M is the map corresponding to Λ t for a decomposition R 2n = Λ t 0 ⊕ M with M Lagrangian. Then [RS93] :

• the symmetric bilinear form Q(Λ t 0 , Λt 0 ) : Λ t 0 × Λ t 0 → R is independent of the choice of the supplementary Lagrangian M to Λ t 0 ; Proof: One has

Λ ′ t := Gr ψ ′ t = {(x, φ t ψ t φ -1 t x) | x ∈ R 2n } = {(φ t y, φ t ψ t y) | y ∈ R 2n } = (φ t × φ t ) Gr ψ t = (φ t × φ t )Λ t
and (φ t × φ t )∆ = ∆. Furthermore (φ t × φ t ) ∈ Sp(R 2n × R 2n , Ω). Hence t ∈ [0, 1] is a crossing for the path of Lagrangians Λ ′ = Gr ψ ′ if and only if dim Gr ψ ′ t ∩ ∆ = 0 if and only if dim(φ t × φ t )(Gr ψ t ∩ ∆) = 0 if and only if t is a crossing for the path of Lagrangian Λ = Gr ψ.

By homotopy with fixed endpoints, we can assume that Λ has only regular crossings and φ is locally constant around each crossing t so that d dt (φψφ -1 )(t) = φ t ψt φ -1 t .

Then at each crossing Γ(Gr ψ ′ , ∆, t) = Q(Λ ′ t , Λ′ t )| Gr ψ ′ t ∩∆ = Q((φ t × φ t )Λ t , (φ t × φ t ) Λt )| (φt×φt) Gr ψt∩∆ = Q(Λ t , Λt )| Gr ψt∩∆ • (φ -1 t × φ -1 t ) ⊗ (φ -1 t × φ -1 t ) in view of (6.11), so that Sign Γ(Gr ψ ′ , ∆, t) = Sign Γ(Gr ψ, ∆, t). Definition 6.4.4 For any smooth path ψ of symplectic matrices, define a path of symmetric matrices S through ψt = J 0 S t ψ t .

This is indeed possible since ψ t ∈ Sp(R 2n , Ω 0 ) ∀t, thus ψ -1 t ψt is in the Lie algebra sp(R 2n , Ω 0 ) and every element of this Lie algebra may be written in the form J 0 S with S symmetric.

6. Generalised Conley-Zehnder index 0, ȧ(1) = 0 and a(t) > 1 for 0 < t < 1; c(0) = c(1) = 0, c(t)d(t) ≥ 0 ∀t and ċ(t) = 0 (resp.= 0) when d(t) = 0 (resp.= 0) for t = 0 or 1. The only crossings are t = 0 and t = 1 since the trace of ψ(t) is > 2 for 0 < t < 1. Now, at those points ( t = 0 and t = 1) ψt = ȧ(t) ḋ(t) ċ(t) -ȧ(t)+d(t) ċ(t) so that S t = -J 0 ψt ψ -1 t = ċ(t)

-ȧ(t)

-ȧ(t) ȧ(t)d(t)-ḋ(t) .

Clearly, at the crossings, we have Ker ψ t = R 2 iff d(t) = 0 and Ker ψ t is spanned by the first basis element iff d(t) = 0, so that from definition 6.4.5 Γ(ψ, t) = ċ(t) when • it is invariant under homotopies with fixed endpoints and two paths with the same endpoints are homotopic with fixed endpoints if and only if they have the same µ ′ RS index;

• it is additive under catenation of paths;

• it has the product property µ RS (ψ ′ ⊕ ψ ′′ ) = µ RS (ψ ′ ) + µ RS (ψ ′′ );

• it vanishes on a path whose image lies in

{A ∈ Sp(R 2n , Ω 0 ) | dim AL ∩ L = k}
for a given k ∈ {0, . . . , n};

• µ ′ RS (ψ) = 1 2 Sign B(0) -1 2 Sign B(1) when ψ t = Id B(t) 0 Id . Robbin and Salamon [START_REF] Robbin | The Maslov index for paths[END_REF] prove that those properties characterize this index. Proof: We have seen in the previous section that the index µ RS defined by Robbin and Salamon satisfies all the above properties. To see that those properties characterize this index, it is enough to show (since the group Sp(R 2n , Ω 0 ) is connected and since we have the catenation property) that those properties determine the index of any path starting from the identity. Since it must be a generalization of the Conley-Zehnder index and must be additive for catenations of paths, it is enough to show that any symplectic matrix A which admits 1 as an eigenvalue can be linked to a matrix B which does not admit 1 as an eigenvalue by a continuous path whose index is determined by the properties stated. From proposition 6.5.1, there is a basis of R 2n such that A is the symplectic direct sum of a matrix which does not admit 1 as eigenvalue and matrices of the form

A (1)
r j ,d j := J(1,r j ) -1 J(1,r j ) -1 diag(0,...,0,d j ) 0 J(1,r j ) τ

; with d j equal to 0, 1 or -1. The dimension of the eigenspace of eigenvalue 1 for A

(1) r j ,d j is equal to 1 if d j = 0 and is equal to 2 if d j = 0. In view of the naturality and the product property of the index, we can consider a symplectic direct sum of paths with the constant path on the symplectic subspace where 1 is not an eigenvalue and we just have to build a path in Sp(R 2r j , Ω 0 ) from A

(1) r j ,d j to a matrix which does not admit 1 as eigenvalue and whose index is determined by the properties given in the statement. This we do by the catenation of three paths : we first build the path ψ 1 : [0, 1] → Sp(R 2r j , Ω 0 ) defined by ψ 1 (t) := D(t,r j ) -1 D(t,r j ) -1 diag c(t),0,...,0,d(t) and this last matrix does not admit 1 as eigenvalue.

Clearly dim ker ψ 1 (t) -Id = 2 for all t ∈ [0, 1] when d j = 0; we now prove that dim ker(ψ 1 (t) -Id) = 1 for all t ∈ [0, 1] when d j = 0. Hence the index of ψ 1 must always be zero by the zero property. To prove that dim ker(ψ 1 (t)-Id) = 1 we have to show the non vanishing of the determinant of the 2r j -1 × 2r j -1 matrix

                  E t
                 
where E t := D(t, r j ) -1 is upper triangular. This determinant is equal to (-1) r j +1 c(t)(e -t -1) r j -1 (e t -1) r j -1 + (-1) r j -1 d(t)(1t) r j -1 det E ′ (t)

where E ′ (t) is obtained by deleting the first column and the last line in E(t) -Id so given by the (r j -1) × (r j -1) matrix

          
(t-1)e -t (t-1) 2 e -2t ... ... (t-1) r j -1 e -(r j -1)t e -t -1 (t-1)e -2t (t-1) 2 e -3t ... (t-1) r j -2 e -(r j -1)t 0 e -t -1 (t-1)e -2t

. . .

(t-1) r j -3 e -(r j -2)t .

and we have det F m (t) = ((t-1)e -2t -(e -t -1)(t-1)e -t ) det F m-1 (t) = (t-1)e -t det F m-1 (t) so that, by induction on m, det F m (t) = (t -1) m e -(m+1)t hence the determinant we have to study is (-1) r j -1 c(t)(2e te -t ) r j -1 + d(t)(t -1) r j det F r j -2 (t) which is equal to (-1) r j -1 c(t)(2e te -t ) r j -1 + d(t)(t -1) r j (t -1) r j -2 e -(r j -1)t hence to c(t)(e t + e -t -2) r j -1 + d(t)(1t) 2r j -2 e -(r j -1)t which never vanishes if c(t) = td j and d(t) = (1t)d j since e t + e -t -2 and (1t) are ≥ 0.

We then construct a path ψ 2 : [0, 1] → Sp(R 2r j , Ω 0 ) which is constant on the symplectic subspace where 1 is not an eigenvalue and which is a symplectic shear on the first two dimensional symplectic vector space, i.e. ψ 2 (t) := 1 (1-t)d j 0 1 ⊕ e -1 Id r j -1 0 0 e Id r j -1

; then the index of ψ 2 is equal to 1 2 Sign d j . Observe that ψ 2 is constant if d j = 0; then the index of ψ 2 is zero. In all cases ψ 2 (1) = Id 2 ⊕ e -1 Id r j -1 0 0 e Id r j -1 . We then build ψ 3 : [0, 1] → Sp(R 2r j , Ω 0 ) given by ψ 3 (t) := e -t 0 0 e t ⊕ e -1 Id r j -1 0 0 e Id r j -1

  point of the parametrized action functional and where ζ(0) can be chosen arbitrarily in T * z S 2N +1 .

2

  and -S 1 ∂ ∂s H s, θ, u(s, θ), z(s) dθ are clearly non positive. The first line can be rewritten as

  Definition 2.1.2 Let H MB be the set of Hamiltonians H :W → R such that 1. H | W is a negative C 2 -small Morse function;2. H(p, ρ) = h(ρ) outside W , where h is a strictly increasing function, which coincides with h(ρ) = ae ρ + b for ρ > ρ 0 , a, b ∈ R and a / ∈ Spec(M, α), and we assume that h ′′h ′ > 0 on [0, ρ 0 ).

Figure 3 . 1 :

 31 Figure 3.1: Example of H 2 on W

  Hs )dθ using Stokes's theorem and H = h s (r) ≤ rα h ′ s (r)R α = -λ ′ (X Hs ) on u(∂S) ⊂ ∂V = ∂S λ ′ (du -X Hs ⊗ dθ)

  Lemma 3.2.15 If (M, ξ) is a compact contact manifold which is exactly fillable by a Liouville domain (W, λ 0 ) (i.e. ∂W = M and ξ = ker α 0 where α 0 = λ 0 M ) then, for any contact form α 1 such that ξ = ker α 1 (and α 1 defines the same orientation on M ), there exists a homotopy of Liouville form λ s , s ∈ [0, 1] on W such that λ 1 M = α 1 .

1 λ of E 1 λ

 1 containing w; it is also of dimension p + 1 and a basis is given byb 0 := w, . . . , b j := A -1 λ Id j w, . . . b p := A -1 λ Id p w . One has Ω(a i , a j ) = 0 and Ω(b i , b j ) = 0 because Ω(E λ , E µ ) = 0 if λµ = 1;5. Normal forms for symplectic matricesΩ(a i , b j ) = 0 if i + j > p by equation (5.14) ; Ω(a i , b p-i ) = -1 λ 2p-i Ω Aλ Id p v, w by equation (5.12) and is non zero by the choice of v, w.The matrix representing Ω in the basis {b p , . . . , b 0 , a 0 , . . . , a p } is thus of the form 

  Normal forms for A |V [λ] when λ = ±1.

and in that 5 .

 5 Normal forms for symplectic matrices basis, since A(a r ) = λa r + a r+1 and A(b r ) = λb r + b r+1 for all r < 2k -2, the matrix representing A is of the block upper triangular form 

  we consider the basis {a 2k , . . . , a k+1 , b 2k , . . . , b k+1 , b k ; b 0 , . . . , b k-1 , a 0 , . . . , a k-1 , a k } for such a vector v; the matrix representing Ω in this basis has the form

6. 2 .

 2 The Conley-Zehnder index5. (Loop) If φ : [0, 1] → Sp(R 2n , Ω 0 ) is a loop with φ(0) = φ(1) = Id, then µ CZ (φψ) = µ CZ (ψ) + 2µ(φ)where µ(φ) is the Maslov index of the loop φ, i.e. µ(φ) = deg(ρ • φ); 6. (Signature) If S = S τ ∈ R 2n×2n is a symmetric non degenerate matrix with all eigenvalues of absolute value < 2π ( S < 2π) and if ψ(t) = exp(J 0 St) for t ∈ [0, 1] , then µ CZ (ψ) = 1 2 Sign(S) (where Sign(S) is the signature of S).7. (Determinant) (-1) n-µ CZ (ψ) = sign det Id -ψ(1) 8. (Inverse) µ CZ (ψ -1 ) = µ CZ (ψ τ ) = -µ CZ (ψ) Proposition 6.2.6 ([Sal99,AD10]) The properties 2, 5 and 6 of homotopy, loop and signature characterize the Conley-Zehnder index.

Proposition 6.2. 7

 7 Let ψ ∈ SP(n) be a continuous path of matrices in Sp(R 2n , Ω 0 ) linking the matrix Id to a matrix in Sp ⋆ (R 2n , Ω 0 ) and let ψ : [0, 2] → Sp(R 2n , Ω 0 ) be an extension such that ψ coincides with ψ on the interval [0, 1] , such that ψ(s) ∈ Sp ⋆ (R 2n , Ω 0 ) for all 6. Generalised Conley-Zehnder index s 1 and such that the path ends either in ψ(2) = -Id = W + either in ψ(2) = W -:= diag(2, -1, . . . , -1, 1 2 , -1, . . . , -1). The Conley-Zehnder index of ψ is equal to the integer given by the degree of the map ρ2 • ψ : [0, 2] → S 1 :µ CZ (ψ) := deg(ρ 2 • ψ) (6.6)for any continuous map ρ : Sp(R 2n , Ω 0 ) → S 1 which coincide with the (complex) determi-nant det C on U (n) = O R 2n ∩ Sp R 2n , Ω 0 , such that ρ(W -) = ±1, and such that deg (ρ 2 • ψ 2-) = n -1 for ψ 2-: t ∈ [0, 1] → exp tπJ 0 S -.Proof: This is a direct consequence of the fact that the map defined by deg(ρ 2 • ψ) has the homotopy property, the loop property (since any loop is homotopic to a loop of unitary matrices where ρ and det C coincide) and we have added what we need of the signature property to characterize the Conley-Zehnder index. Indeed 1 2 Sign S -= n -1, S + = Id 2n , 1 2 Sign S + = n and exp tπJ 0 S + = exp tπ 0 -Idn Idn 0 = cos πt Idn -sin πt Idn sin πt Idn cos πt Idn is in U (n) so that ρ2 exp tπ 0 -Idn Idn 0 = e 2πint and deg(ρ 2 • ψ 2+ ) = n. Corollary 6.2.8 The Conley-Zehnder index of a path ψ ∈ SP(n) is given by µ CZ (ψ) := deg(det C 2 • U • ψ) (6.7) where U : Sp(R 2n , Ω 0 ) → U (n) is the projection defined by the polar decomposition U (A) = AP -1 with P the unique symmetric positive definite matrix such that P 2 = A τ A. Proof: The map ρ := det C • U satisfies all the properties stated in proposition 6.2.7; it is indeed continuous, coincides obviously with det C on U (n) and we have that exp tπJ t is a positive symmetic matrix so that U (exp tπJ 0 S -πt Id n-1 0 -sin πt Id n-1 πt Id n-1 0 cos πt Id n-1 ; hence det 2 C • U (exp tπJ 0 S -) = e 2πi(n-1)t and deg(det C 2 • U • ψ 2-) = n -1. Formula (6.7) is the definition of the Conley-Zehnder index used in [dG09, HWZ95]. Another formula is obtained using the parametrization of the symplectic group introduced in [RR89]: Corollary 6.2.9 The Conley-Zehnder index of a path ψ ∈ SP(n) is given by µ CZ (ψ) := deg(ρ 2 • ψ) (6.8) 6.3. The Robbin-Salamon index for a path of Lagrangianswhere ρ : Sp(R 2n , Ω 0 ) → S 1 is the normalized complex determinant of the C-linear part of the matrix: -J 0 AJ 0 ) .

6.

  Generalised Conley-Zehnder index Proposition 6.4.2 If ψ : [a, b] → Sp(R 2n , Ω) is a path of matrices such that dim Ker(ψ(t)-Id) = k for all t ∈ [a, b] then µ RS (ψ) = 0. Indeed, Gr ψ t ∩ ∆ = {v ∈ R 2n |ψ t v = v} so dim(Gr ψ t ∩ ∆) = k if and only if dim Ker(ψ(t) -Id) = k.Proposition 6.4.3 (Naturality) Consider two continuous paths of symplectic matrices ψ, φ : [0, 1] → Sp(R 2n , Ω 0 ) and define ψ ′ = φψφ -1 . Then µ RS (ψ ′ ) = µ RS (ψ)

  d(t) = 0 and Γ(ψ, t) = 0 -ȧ(t) -ȧ(t) 0 when d(t) = 0. Hence both crossings are regular and Sign Γ(ψ, t) = Sign ċ(t) when d(t) = 0 and Sign Γ(ψ, t) = 0 when d(t) = 0. Since d(t)c(t) ≥ 0 for all t, we clearly have Sign ċ(0) = Sign d(0) and Sign ċ(1) = -Sign d(1). Proposition 6.4.6 then gives µ RS (ψ) = 1 2 Γ(ψ, 0) + 1 2 Sign Γ(ψ, 1) = 1 2 Sign d(0) -1 2 Sign d(1). Remark 6.4.10 Robbin and Salamon introduce another index µ ′ RS for paths of symplectic matrices built from their index for paths of Lagrangians. Consider the fixed Lagrangian L = {0} × R n in (R 2n , Ω 0 ), observe that AL is Lagrangian for any A ∈ Sp(R 2n , Ω 0 ), and define, for ψ : [0, 1] → Sp(R 2n , Ω 0 ) µ ′ RS (ψ) := µ RS (ψL, L). (6.18) This index has the following properties [RS93] :

  The Robbin-Salamon index is characterized by the following properties:1. (Generalization) it is a correspondence µ RS which associates a half integer to any continuous path ψ : [a, b] → Sp(R 2n , Ω 0 ) of symplectic matrices and it coincides with µ CZ on paths starting from the identity matrix and ending at a matrix for which 1 is not an eigenvalue;2. (Naturality) if φ, ψ : [0, 1] → Sp(R 2n , Ω 0 ), we have µ RS (φψφ -1 ) = µ RS (ψ);3. (Homotopy) it is invariant under homotopies with fixed end points; 4. (Catenation) it is additive under catenation of paths; 5. (Product) it has the product property µ RS (ψ ′ ⊕ ψ ′′ ) = µ RS (ψ ′ ) + µ RS (ψ ′′ ); 6. (Zero) it vanishes on any path ψ : [a, b] → Sp(R 2n , Ω) of matrices such that dim Ker(ψ(t) -Id) = k is constant on [a, b]; 7. (Shear)on a symplectic shear , ψ : [0, 1] → Sp(R 2n , Ω 0 ) of the form ψ t = Id -tB B symmetric, it is equal to µ RS (ψ) = 1 2 Sign B.

  . 0 e t 1-t 0 0 ... ... 0 e t 1-t 0 ... ... ... 0 e t with c(t) = td j , d(t) = (1t)d j . Observe that ψ 1 (0) = A (1)r j ,d j and ψ 1 (1) is the symplectic direct sum of 1 c(1)=d j 0 1 and e -1 Id r j -1 0 0 e Id r j -1

  En vue de quoi, une question naturelle est Contents Si (M, ξ) est une variété de contact compacte, que pouvons-nous dire sur le nombre minimal d'orbites de Reeb périodiques et géométriquement distinctes pour toute forme de contact α (éventuellement dans une sous-classe) définissant la structure de contact ξ? En particulier, quelle est la réponse pour la sphère? Quelques résultats sont connus dans ce cas. Le problème a été résolu en dimension 3 par Hofer, Wysocki et Zehnder pour la classe des formes de contact "dynamiquement convexes". Supposons que la forme de contact α = f • α std sur S 3 est dynamiquement convexe 9 , où f : S 3 → (0, ∞) est une fonction lisse positive. Soit R α le champ de vecteurs de Reeb associé. Alors il y a soit exactement 2 soit une infinité d'orbites périodiques de R α .

	Theorem 0.0.25 ([HWZ98])

En grande dimension, moins de choses sont connues: Theorem 0.0.26 ([LZ02]) Toute hypersurface Σ ⊂ R 2n compacte et strictement convexe possède au moins ⌊ n 2 ⌋ + 1 orbites de Reeb périodiques géométriquement distinctes. Theorem 0.0.27 ([WHL07]) Toute hypersurface Σ ⊂ R 6 compacte et strictement convexe possède au moins 3 orbites de Reeb périodiques géométriquement distinctes.

Contents

  Theorem 0.0.35 ([Ust99]) Pour tout nombre naturel m, il existe une infinité de structures de contact non isomorphes sur S 4m+1 . Dans le Chapitre 4, nous utilisons l'homologie symplectique S 1 -équivariante positive pour donner une nouvelle preuve du Théorème 0.0.28 sur le nombre minimal d'orbites de Reeb périodiques sur certaines hypersurfaces dans R 2n quand toutes les orbites de Reeb sont non dégénérées. Cela apparaît dans la suite comme le Théorème 4.1.1. Nous étendons les définitions d'homologie symplectique positive et d'homologie symplectique S 1 -équivariante positive au cas non exact dans les Sections 4.2.1 et 4.2.2. Ceci nous permet d'étendre les techniques développées pour la preuve du Théorème 0.0.28 pour commencer l'étude d'hypersurfaces dans des fibrés en droites négatifs. Ce cadre est la généralisation naturelle d'hypersurfaces dans C n . En effet la sphère est le bord de la boule dans C n ≃ R 2n mais également de la boule éclatée 14 à l'origine. La boule éclatée en l'origine dans C n est B2n

  La décomposition est unique à permutation des blocs près et est déterminée par λ, par la dimension dim Ker(Aλ Id) r pour tout r ≥ 1 et par le rang et la signature de la 2-forme bilinéaire symétrique Qλ

2k pour tout k ≥ 1. Le dernier Chapitre, qui apparaîtra comme article dans les Annales de la faculté des Sciences de Toulouse, est dévolu à l'étude d'une version généralisée de l'indice de Conley-Zehnder définie par Robbin et Salamon dans [RS93]. Nous commençons par donner une nouvelle formule pour l'indice de Conley-Zehnder "classique". Theorem 0.0.40 Soit ψ : [0, 1] → Sp(R 2n , Ω 0 ) un chemin continu de matrices symplectiques liant la matrice Id à une matrice n'admettant pas 1 comme valeur propre. Soit ψ : [0, 2] → Sp(R 2n , Ω 0 ) une extension telle que ψ coincide avec ψ sur l'intervalle [0, 1], telle que ψ(s) n'admette pas 1 comme valeur propre pour tout s 1 et telle que le chemin se termine soit en ψ

  the dimension of ker(D u ) is equal to 1; this ker(D u ) is spanned by ∂ s u .

	The sign associated to [u], ǫ [u] , is given by
	+1 if orientation of ker(D u ) given by ∂ s u coincides with the coherent orientation
	-1 if it is not the case.
	Proposition 1.1.26 Consider two broken Floer trajectories ([u], [v]) and ([u ′ ], [v ′ ]

  periodic orbit of X H 2 lying in region I or II. The homotopy of homotopies theorem shows that the map does not depend on the choice of stair function H 2 such that ι V H 2 = H and such that H 1 and H 2 coincide far in the completion; we shall denote it φ H H 1 . It also shows that the map φ H 1 ,H 2 commutes with continuation, i.e if ρ 1 : SH(H 1 ) → SH(H ′ 1 ) is a continuation for H 1 and ρ 2

  , . . . , a p }, using a Gram-Schmidt procedure on the b i 's. This gives a normal form for A on E v λ ⊕ E w defined by the conjugate vectors {b ′ p , . . . , b ′ 0 , a 0 , . . . , a p } and this yields a conjugate normal form on E λ ⊕ E 1 λ , hence a normal form on W [λ] and this will induce a real normal form on V [λ] .

		
		   
	0	0
	with non vanishing * . Hence Ω is non degenerate on E v λ ⊕ E w 1 λ A-invariant subspace.	which is thus a symplectic
	We now construct a symplectic basis b ′ p , . . . , b ′ 0 , a 0 , . . . , a p of E v λ ⊕ E w 1 λ {a 0 1 .	, extending
	λ If λ is real, we take v, w in the real generalized eigenspaces E R λ and E R 1 λ symplectic basis of the real A-invariant symplectic vector space , E Rv 1 λ λ ⊕E Rw one considers the basis of E v 1 λ ⊕ E w λ	and we obtain a . If λ is not real,

  we consider the symplectic basis {b ′ p , . . . , b ′ 0 , a 0 , . . . , a p } of E v λ ⊕ E w

	1	as
	λ
	above and the conjugate symplectic basis {b ′ p , . . . , b ′ 0 , a 0 , . . . , a p

  Thus det E ′ (t) = (t -1)(e -t -(e -t -1)) det F r j -2 (t) where

									
		. . . . . .	. . .	. . . . . .	e -t -1	. . . (t-1)e -2t	. . .	. . . (t-1 2 )e -3t	          .
		0	...		0	e -t -1		(t-1)e -2t
			(t-1)e -2t (t-1) 2 e -3t	...	...		(t-1) r j -2 e -(r j -1)t	
	F m (t) :=	         	e -t -1 0 . . . . . .	(t-1)e -2t e -t -1 . . . . . . . . . e -t -1 . . . (t-1)e -2t . . .	... . . . (t-1)e -2t	(t-1) r j -3 e -(r j -2)t . . . . . . (t-1 2 )e -3t	         
			0	...		0	e -t -1	(t-1)e -2t

(t-1) r j -3 e -(r j -2)t

We refer to the cited reference for a precise definition. It will not be used in the sequel

We refer to Definition 3.2.8 for a precise definition of Liouville manifold.

We refer to[START_REF] Mcduff | Introduction to symplectic topology[END_REF] for a detailed definition of blow ups

Nous référons à la référence citée pour une définition précise qui ne sera pas utilisée dans la suite de ce travail.

Nous référons à la référence citée pour une définition précise qui ne sera pas utilisée dans la suite de ce travail.

La définition d'orbites non dégénérée est donnée Définition 1.1.5.

Une forme de contact est non dégénérée si toutes ses orbites de Reeb périodiques sont non dégénérées.

Nous référons à la référence citée pour les définitions précises qui ne seront pas utilisées dans la suite de ce travail.

Nous référons à la Définition 3.2.8 pour une définition précise de variété de Liouville.

Nous référons à [MS98] pour une définition détaillée d'éclatement.

We refer to the book by Cieliebak and Eliashberg for more details,[START_REF] Cieliebak | From Stein to Weinstein and back[END_REF] Chapter 11]

Acknowledgements

I am grateful to the Foundation for Scientific Research (FNRS-FRS) for its support and to the Collège Doctoral Européen for providing me lodging in Strasbourg. I acknowledge partial support from the ERC via the grant StG-259118-STEIN, from the ERC via the grant StG-239781-ContactMath, from an ARC of the Communauté française de Belgique and from the Pôle d'Attraction Interuniversitaire Dygest.

This thesis would not have been fun without all the friends that supported me during these 4 years. Thanks to the persons who shared an office with me (and congratulations for not killing me for my stupid jokes), thanks also to the PhD students of Bruxelles, Strasbourg and Paris

Contents

Theorem 0.0.4 ( [START_REF] Hofer | The dynamics on three-dimensional strictly convex energy surfaces[END_REF]) Assume the contact form α = f • α std on S 3 is dynamically convex 2 , where f : S 3 → (0, ∞) is a smooth, positive function. Let R α be the associated 3.2. Invariance of symplectic homology Theorem 3.1.16 (Composition) Let (V 1 , λ V 1 ) ⊆ (V 2 , λ V 2 ) ⊆ (V 3 , λ V 3 ) be Liouville domains with Liouville embeddings. Then the following diagram commutes:

Invariance of symplectic homology

In this section, we study the invariance of the (S 1 -equivariant) positive symplectic homology with respect to the choice of the Liouville vector field in a neighbourhood of the boundary. This has been studied by Viterbo [Vit99], Cieliebak [START_REF] Cieliebak | Handle attaching in symplectic homology and the chord conjecture[END_REF] and Seidel [START_REF] Seidel | A biased view of symplectic cohomology[END_REF] in the case of the symplectic homology, in the framework of Liouville domains.

Lemma 3.2.1 Let (W, ω, X) be a compact symplectic manifold with contact type boundary and let k be a positive real number. Then SH † (W, ω, X) = SH † (W, kω, X).

Where † denotes any of the variants that we have considered ∅, +, S 1 or (S 1 , +).

Proof: The symplectic completions are ( W , ω) and ( W , k ω); the chain complexes for a pair (H, J) on ( W , ω) and the pair (kH, J) on ( W , k ω) are the same, since the 1 periodic orbits are the same, and the Floer trajectories satisfy the same equations; indeed X ω H = X kω kH . Similarly, continuation maps are equivalent taking as homotopies H s and kH s . The result follows, observing that kH form a cofinal family.

For positive or S 1 -equivariant positive homology, we clearly assume that we are in a framework where it is well-defined. Lemma 3.2.2 Let (W, ω, X) and (W ′ , ω ′ , X ′ ) be two compact symplectic manifolds with contact type boundary. If there exists a symplectomorphism ϕ : W → W ′ such that ϕ(∂W ) = ∂W ′ , and such that ϕ ⋆ (X) = X ′ on a neighbourhood of ∂W then SH † (W, ω, X) ∼ = SH † (W ′ , ω ′ , X ′ ).

Proof: We can extend ϕ to a symplectomorphism ϕ : W → W ′ of the completions. For J ′ an almost complex structure on W ′ , we take the corresponding almost complex structure J on W defined by

and if H ′ is a Hamiltonian on W ′ , we take the Hamiltonian H on W defined by H := ϕ ⋆ H ′ . Then the 1 periodic orbirs are in bijection and so are the Floer trajectories. The subfamily Proposition 3.2.10 Let (W 0 , λ 0 ) and (W 1 , λ 1 ) be two Liouville manifolds not necessarily of finite type. Assume there exists an exact symplectomorphism f : W 0 → W 1 i.e. such that f ⋆ λ 1λ 0 = dg with g a function on W 0 . Then SH † (W 0 , λ 0 ) ∼ = SH † (W 1 , λ 1 ).

Proof: Let W k 0 be an exhaustion for W 0 and W k 1 be an exhaustion for W 1 such that for all k,

where the inclusion at each level means the inclusion in the interior of the next compact space. Let η be a smooth function η : W 0 → [0, 1] such that η = 1 in a neighbourhood of ∪ ∞ k=1 f -1 (∂W k 1 ) and η = 0 in a neighbourhood of ∪ ∞ k=1 ∂W k 0 . We define the 1-form λ on W 0 to be λ := λ 0 + d(ηg).

We have

The functoriality of the transfer morphism implies that the following diagram is commutative:

Therefore,

The above result may be extended thanks to the following Lemma:

Lemma 3.2.11 ( [START_REF] Bourgeois | Effect of legendrian surgery[END_REF], see also [START_REF] Cieliebak | From Stein to Weinstein and back[END_REF], Lemma 11.2) Any symplectomorphism between finite type Liouville manifolds f : (W 0 , λ 0 ) → (W 1 , λ 1 ) is diffeotopic to an exact symplectomorphism.

We have thus

Theorem 3.2.12 Let (W 0 , λ 0 ) and (W 1 , λ 1 ) be two Liouville manifolds of finite type such that there exists a symplectomorphism f :

Assume that the minimal action of any periodic Reeb orbit on Σ is s bounded below by R 2 1 . Then Σ carries at least 2n i=0 β i geometrically distinct periodic Reeb orbits; where the β i are the Betti numbers of B.

We find a symplectic basis of V C so that A is a symplectic direct sum of block-uppertriangular matrices of the form

Here, J(λ, k) is the elementary k × k Jordan matrix corresponding to an eigenvalue λ,

and S(k, s, λ) is the k × (k + 1) matrix defined by

with s an integer in {-1, 0, 1}. Each s ∈ {±1} is called a sign and the collection of such signs appearing in the decomposition of a matrix A is called the sign characteristic of A.

More precisely, on the real vector space V , we shall prove:

Theorem 5.0.8 (Normal forms for symplectic matrices) Any symplectic endomorphism A of a finite dimensional symplectic vector space (V, Ω) is the direct sum of its restrictions A |V [λ] to the real A-invariant symplectic subspace V [λ] whose complexification is the direct sum of the generalized eigenspaces of eigenvalues λ, 1 λ , λ and 1 λ :

We distinguish three cases :

) and q be the smallest integer so that (Aλ Id) q is identically zero on the generalized eigenspace E λ .

Normal forms for symplectic matrices

The generalized eigenspace for A on V ′ C are E ′ µ = V ′ C ∩ E µ , and the smallest integer p ′ for which (Aλ Id) p ′ +1 = 0 on E ′ λ is such that p ′ ≤ p. Hence, to get the decomposition of W [λ] (and V [λ] ) it is enough to build a symplectic subspace of W [λ] which is A-invariant and closed under complex conjugation and to proceed inductively. We shall construct such a subspace, containing a well chosen vector v ∈ E λ so that (Aλ Id) p v = 0.

We shall distinguish three cases; first λ / ∈ S 1 then λ = ±1 and finally λ ∈ S 1 \ {±1}. We first present a few technical lemmas which will be used for this construction.

A few technical lemmas

Let (V, Ω) be a real symplectic vector space. Consider A ∈ Sp(V, Ω) and let λ be an eigenvalue of A in V C . Lemma 5.1.3 For any positive integer j, the bilinear map

is well defined and non degenerate. In the formula, [v] denotes the class containing v in the appropriate quotient.

Proof: The fact that Q j is well defined follows from equation (5.6); indeed, for any integer j, we have

(5.10)

The map is non degenerate because Q j [v], [w] = 0 for all w if and only if (Aλ Id) j v = 0 since Ω is a non degenerate pairing between E λ and E 1 λ , thus if and only if [v] = 0. Similarly,

[w] = 0 for all v if and only if w is Ω-orthogonal to Im(Aλ Id) j , thus if and only if w ∈ Ker A -1 λ Id j hence [w] = 0.

Lemma 5.1.4 For any v, w ∈ V , any λ ∈ C \ {0} and any integers i ≥ 0, j > 0 we have:

In particular, if λ is an eigenvalue of A, if v ∈ E λ is such that p ≥ 0 is the largest integer for which (Aλ Id) p v = 0, we have for any integers k, j ≥ 0:

1 Ω(a p-j ,b j ) a p-j -k<j Ω(a p-j , b k )a ′ p-k , so that any a ′ j is a linear combination of the a ′ k with k ≥ j.

In the symplectic basis a ′ p , . . . , a ′ 0 , b 0 , . . . , b p the matrix representing A is

Hence, the matrix J(λ, p + 1) -1 0 0 J(λ, p + 1) τ is a normal form for A restricted to E v λ ⊕ E w λ . Thus we have:

Theorem 5.3.1 (Normal form for A |V [λ] for λ = ±1.) Let λ = ±1 be an eigenvalue of A. There exists a symplectic basis of V [λ] in which the matrix representing the restriction of A to V [λ] is a symplectic direct sum of matrices of the form

where C(r j , s j , λ) := J(λ, r j ) -1 diag 0, . . . , 0, s j with s j ∈ {0, 1, -1}. If s j = 0, then r j is odd. The dimension of the eigenspace of eigenvalue 1 is given by 2 Card{j | s j = 0} + Card{j | s j = 0}.

Definition 5.3.2 Given λ ∈ {±1}, we define, for any integer k ≥ 1, a bilinear form Qλ 2k on Ker (Aλ Id) 2k :

It is symmetric.

Proposition 5.3.3 Given λ ∈ {±1}, the number of positive (resp. negative) eigenvalues of the symmetric 2-form Qλ 2k is equal to the number of s j equal to +1 (resp. -1) arising in blocks of dimension 2k (i.e. with corresponding r j = k) in the normal decomposition of A on V [λ] given in theorem 6.5.1. On V [λ] , we have:

If p = 2k is even

we observe that Ω (Aλ Id) k v, (Aλ Id) k v is purely imaginary and we choose v so that it is Ω (Aλ Id) k v, (Aλ Id) k v = si where s = ±1 (remark that the sign changes if one permutes λ and λ). We can further choose the vector v so that :

Indeed, as before, by (5.11), we have T i,j (v) = -T i+1,j (v) -T i+1,j-1 (v) and T i,j (v) = -T j,i (v) and we proceed as in lemma 5.1.6 by decreasing induction on i + j:

the imaginary part of α 1 is equal to 1 2 si and we replace v by v -α 1 2λsi (Aλ Id)v; it generates the same A-invariant subspace and the quantities T i,j (v) do not vary for i+j ≥ 2k but now

this α 2 is purely imaginary and we replace v by v -α 2 2λ 2 si (Aλ Id) 2 v; it generates the same A-invariant subspace and the quantities T i,j (v) do not vary for i

We may thus assume this property to hold for v.

, this α 3 is real and we replace v by v -α 3 2λ 3 si (Aλ Id) 3 v; it generates and the the same A-invariant subspace and the quantities T i,j (v) do not vary for i + j ≥ 2k -2; now

• we now assume by induction to have a J > 1 so that T i,j (v) = 0 for all 0 ≤ i, j ≤ k -1 so that i + j > 2k -1 -J;

so the induction step is proven.

Generalised Conley-Zehnder index

We recall this construction in section 6.2 with the precise definition of the map ρ. The value of ρ(A) involves the algebraic multiplicities of the real negative eigenvalues of A and the signature of natural symmetric 2-forms defined on the generalised eigenspaces of A for the non real eigenvalues lying on S 1 . We give alternative ways to compute this index : Theorem 6.1.1 Let ψ : [0, 1] → Sp(R 2n , Ω 0 ) be a continuous path of matrices linking the matrix Id to a matrix which does not admit 1 as an eigenvalue. Let ψ : [0, 2] → Sp(R 2n , Ω 0 ) be an extension such that ψ coincides with ψ on the interval [0, 1] , such that ψ(s) does not admit 1 as an eigenvalue for all s 1 and such that the path ends either at ψ(2)

). The Conley-Zehnder index of ψ is equal to the integer given by the degree of the map ρ2

for ANY continuous map ρ : Sp(R 2n , Ω 0 ) → S 1 with the following properties:

1. ρ coincides with the (complex

In particular, two alternative ways to compute the Conley-Zehnder index are :

• Using the polar decomposition of a matrix,

where U : Sp(R 2n , Ω 0 ) → U (n) : A → AP -1 with P the unique symmetric positive definite matrix such that P 2 = A τ A.

• Using the normalized determinant of the C-linear part of a symplectic matrix,

the standard complex structure on R 2n .

In [START_REF] Robbin | The Maslov index for paths[END_REF], Robbin and Salamon define a Maslov-type index for a continuous path Λ from the interval [a, b] to the space L (W,Ω) of Lagrangian subspaces of a symplectic vector space (W, Ω), having chosen a reference Lagrangian L. They give a formula of this index

where φ 0 (t) has only real positive eigenvalues for t ∈ -1 2 , 0 ;

where φ 1 (t) has only real positive eigenvalues for t ∈ 1, 3 2 ; • Ψ(-1) = W ± , Ψ(2) = W ± and Ψ(t) does not admit 1 as an eigenvalue for t ∈ -1, -1 2 and for t ∈ 3 2 , 2 . Then the Robbin Salamon index is given by

k with ρ as in theorem 6.1.1, and with

In the theorem above, we have used the notation A ⊕ B for the symplectic direct sum of two symplectic endomorphisms with the natural identification of Sp(

This writes in symplectic basis as

We recall the definition of the Conley-Zehnder index in section 6.2 and obtain a new way of computing this index in Proposition 6.2.7 and its corrolaries (stated above as Theorem 6.1.1). In sections 6.3 and 6.4, we present known results about the Robbin Salamon index of a path of Lagrangians and the Robbin Salamon index of a path of symplectic matrices, including the fact that it is a generalization of the Conley-Zehnder index; in section 6.4.2, we stress the fact that another index introduced by Robbin and Salamon does not coincide with this generalization of the Conley-Zehnder index. In section 6.5, we give a characterization of the generalization of the Conley-Zehnder index (stated above as Theorem 6.1.2). Section 6.6 gives a new formula to compute this index (stated above as Theorem 6.1.3).

The Conley-Zehnder index

The Conley-Zehnder index is an application which associates a integer to a continuous path of symplectic matrices starting from the identity and ending at a matrix in the set Sp ⋆ (R 2n , Ω 0 ) of symplectic matrices which do not admit 1 as an eigenvalue.

6.2. The Conley-Zehnder index Definition 6.2.1 ([SZ92, Sal99]) We consider the set SP(n) of continuous paths of matrices in Sp(R 2n , Ω 0 ) linking the matrix Id to a matrix in Sp ⋆ (R 2n , Ω 0 ) :

ψ(0) = Id and 1 is not an eigenvalue of ψ(1) .

Definition 6.2.2 ([SZ92, AD10]) Let ρ : Sp(R 2n , Ω 0 ) → S 1 be the continuous map defined as follows. Given A ∈ Sp(R 2n , Ω), we consider its eigenvalues {λ i }. For an eigenvalue λ = e iϕ ∈ S 1 \ {±1}, let m + (λ) be the number of positive eigenvalues of the symmetric non degenerate 2-form Q defined on the generalized eigenspace E λ by

Then ρ(A) := (-1)

where m -is the sum of the algebraic multiplicities m λ = dim C E λ of the real negative eigenvalues.

Proposition 6.2.3 ([SZ92, AD10]) The map ρ : Sp(R 2n , Ω 0 ) → S 1 has the following properties:

1.

[determinant] ρ coincides with det C on the unitary subgroup

[invariance] ρ is invariant under conjugation :

for matrices which have no eigenvalue on the unit circle;

4.

[multiplicativity] ρ behaves multiplicatively with respect to direct sums : if

, Ω 0 ) and ⊕ expressing as before the obvious identification of

The construction [Sal99, AD10] of the Conley-Zehnder index is based on the two following facts 6. Generalised Conley-Zehnder index

, Ω 0 ) has two connected components, one containing the matrix W + := -Id and the other containing

Thus any path ψ :

Observe that ρ(Id)

and the contractibility property shows that its degree does not depend on the extension chosen.

Definition 6.2.4 The Conley-Zehnder index of ψ is defined by:

for an extension ψ of ψ as above.

with the identification of Sp(R 2n ′ , Ω 0 ) × Sp(R 2n ′′ , Ω 0 ) with a subgroup of Sp(R 2n , Ω 0 );

6. Generalised Conley-Zehnder index 1. (Naturality) For ψ ∈ Sp(R 2n , Ω) µ RS (ψΛ, ψL) = µ RS (Λ, L).

(Catenation) For

where A(t) is a path of symmetric matrices, then the index of Λ is given by

are homotopic with fixed endpoints if and only if they have the same index.

(Zero) Every path

6.4 The Robbin-Salamon index for a path of symplectic matrices

Generalized Conley-Zehnder index

Consider the symplectic vector space (R 2n × R 2n , Ω = -Ω 0 × Ω 0 ). Given any linear map

A particular Lagrangian is given by the diagonal

Remark that Gr(-ψ) is a Lagrangian subspace which is always supplementary to Gr ψ for ψ ∈ Sp(R 2n , Ω 0 ). In fact Gr φ and Gr ψ are supplementary if and only if φψ is invertible. Note that this index is defined for any continuous path of symplectic matrices but can have half integer values.

A crossing for a smooth path Gr ψ is a number t ∈ [0, 1] for which 1 is an eigenvalue of ψ t and Gr

is in bijection with Ker(ψ t -Id). The properties of homotopy, catenation and product of theorem 6.3.2 imply that [RS93]

• µ RS is invariant under homotopies with fixed endpoints,

• µ RS is additive under catenation of paths and

The zero property of the Robbin-Salamon index of a path of Lagrangians becomes:

6.4. The Robbin-Salamon index for a path of symplectic matrices

The symmetric bilinear form Q Gr ψ, d dt Gr ψ is given as follows. For any t 0 ∈ [0, 1], write R 2n × R 2n = Gr ψ t 0 ⊕ Gr(-ψ t 0 ). The linear map α t : Gr ψ t 0 → Gr(-ψ t 0 ) corresponding to Gr ψ t is obtained from: (x, ψ t x) = (y, ψ t 0 y) + α t (y, ψ t 0 y) = (y, ψ t 0 y) + ( α t y, -ψ t 0 α t y) if and only if (Id + α t )y = x and ψ t 0 (Idα t )y = ψ t x, hence ψ -1 t 0 ψ t (Id + α t ) = Idα t and

Hence the restriction of Q to Ker(ψ t 0 -Id) is given by , Ω 0 ) be a continuous path of symplectic matrices such that ψ(0) = Id and such that 1 is not an eigenvalue of ψ(1) (i.e. ψ ∈ SP(n)). The Robbin-Salamon index of ψ defined by (6.15) coincides with the Conley-Zehnder index of ψ In particular, for a smooth path ψ ∈ SP(n) having only regular crossings, the Conley-Zehnder index is given by

Sign Γ(ψ, t) (6.17)

with S 0 = -J 0 ψ0 .

Proof: Since the Robbin-Salamon index for paths of Lagrangians is invariant under homotopies with fixed end points, the Robbin-Salamon index for paths of symplectic matrices is also invariant under homotopies with fixed endpoints. Its restriction to SP(n) is actually invariant under homotopies of paths in SP(n) since for any path in SP(n), the starting point ψ 0 = Id is fixed and the endpoint ψ 1 can only move in a connected component of Sp * (R 2n , Ω 0 ) where no matrix has 1 as an eigenvalue.

To show that this index coincides with the Conley-Zehnder index, it is enough, in view of proposition 6.2.6, to show that it satisfies the loop and signature properties.

Let us prove the signature property. Let ψ t = exp(tJ 0 S) with S a symmetric nondegenerate matrix with all eigenvalues of absolute value < 2π, so that Ker(exp(tJ 0 S) -Id) = {0} for all t ∈]0, 1]. Hence the only crossing is at t = 0, where ψ 0 = Id and ψt = J 0 Sψ t so that S t = S for all t and

To prove the loop property, note that µ RS is additive for catenation and invariant under homotopies with fixed endpoints. The path (φψ) is homotopic to the catenation of φ and ψ; it is thus enough to show that the Robbin-Salamon index of a loop is equal to 2 deg(ρ • φ). Since two loops φ and φ ′ are homotopic if and only if deg(ρ

The crossings of φ n arise at t = m n with m an integer between 0 and n. At such a crossing, Ker φ n (t) is R 2 for 0 < t < 1 and is R 2n for t = 0 and t = 1. We have

so that, extending ⊕ to symmetric matrices in the obvious way,

Thus Sign Γ(φ n , t) = 2 for all crossings t = m n , 0 ≤ m ≤ n. From equation (6.16) we get

and the loop property is proved. Thus the Robbin-Salamon index for paths in SP(n) coincides with the Conley-Zehnder index.

The formula for the Conley-Zehnder index of a path ψ ∈ SP(n) having only regular crossings, follows then from (6.16). Indeed, we have Ker(ψ 1 -Id) = {0}, while Ker(ψ 0 -Id) = R 2n and Γ(ψ, 0) = S 0 . Proof: We write B(t) = A(t) τ D(t)A(t) with A(t) ∈ O(R n ) and D(t) a diagonal matrix.

Another index defined by Robbin and Salamon

The matrix

is in Sp(R 2n , Ω 0 ) and

. By proposition 6.4.3 µ RS (ψ) = µ RS (ψ ′ ); by the product property it is enough to show that µ RS (ψ) = 1 2 Sign d(0) -1 2 Sign d(1) for the path

.

Since µ RS is invariant under homotopies with fixed end points, we may assume

with a and c smooth functions such that a(0) = 1, a(1) = 1, ȧ(0) = 6.5. Characterization of the Robbin-Salamon index

The two indices µ RS and µ ′ RS defined on paths of symplectic matrices DO NOT coincide in general. Indeed, consider the path ψ :

and ψ ′ = φψφ -1 , then

. Then

which is in general different from µ ′ RS (ψ). Whereas, by (6.4.3), µ RS (ψ) = µ RS (ψ ′ ). The index µ ′ RS vanishes on a path whose image lies into one of the (n+1) strata defined by {A ∈ Sp(R 2n , Ω 0 ) | dim AL ∩ L = k} for 0 ≤ k ≤ n, whereas the index µ RS vanishes on a path whose image lies into one of the (2n + 1) strata defined by the set of symplectic matrices whose eigenspace of eigenvalue 1 has dimension k (for 0 ≤ k ≤ 2n).

However, the two indices µ RS and µ ′ RS coincide on symplectic shears.

Characterization of the Robbin-Salamon index

In this section, we prove theorem 6.1.2 stated in the introduction. Before proving this theorem, we show that the Robbin-Salamon index is characterized by the fact that it extends Conley-Zehnder index and has all the properties stated in the previous section. This is made explicit in Lemma 6.5.2. We then use the characterization of the Conley-Zehnder index given in Proposition 6.2.6 to give in Lemma 6.5.3 a characterization of the Robbin-Salamon index in terms of six properties. We use explicitly the normal form of the restriction of a symplectic endomorphism to its generalised eigenspace of eigenvalue 1 that we have proven in [Gutb] and that we summarize in the following proposition Proposition 6.5.1 (Normal form for A |V [λ] for λ = ±1.) Let λ = ±1 be an eigenvalue of A ∈ Sp(R 2n , Ω 0 ) and let V [λ] be the generalized eigenspace of eigenvalue λ. There exists a symplectic basis of V [λ] in which the matrix associated to the restriction of A to V [λ] is a symplectic direct sum of matrices of the form

where C(r j , d j , λ) := J(λ, r j ) -1 diag 0, . . . , 0, d j with d j ∈ {0, 1, -1}. If d j = 0, then r j is odd. The dimension of the eigenspace of eigenvalue 1 is given by 2 Card{j

For any integer k ≥ 1, the bilinear form on Ker (Aλ Id) 2k defined by

which is the direct sum of a path whose Conley-Zehnder index is known and a constant path whose index is zero. Clearly 1 is not an eigenvalue of ψ 3 (1).

Combining the above with the characterization of the Conley-Zehnder index, we now prove: Lemma 6.5.3 The Robbin-Salamon index for a path of symplectic matrices is characterized by the following properties:

• (Homotopy) it is invariant under homotopies with fixed end points;

• (Catenation) it is additive under catenation of paths;

is a symmetric non degenerate matrix with all eigenvalues of absolute value < 2π and if ψ(t) = exp(J 0 St) for t ∈ [0, 1] , then µ RS (ψ) = 1 2 Sign S where Sign S is the signature of S;

Proof: Remark first that the invariance by homotopies with fixed end points, the additivity under catenation and the zero property imply the naturality; they also imply the constancy on the components of SP(n). The signature property stated above is the signature property which arose in the characterization of the Conley-Zehnder index given in proposition 6.2.6. To be sure that our index is a generalization of the Conley-Zehnder index, there remains just to prove the loop property. Since the product of a loop φ and a path ψ starting at the identity is homotopic to the catenation of φ and ψ, it is enough to prove that the index of a loop φ with φ(0) = φ(1) = Id is given by 2 deg(ρ • φ). Since two loops φ and φ ′ are homotopic if and only if deg(ρ • φ) = deg(ρ • φ ′ ), it is enough to consider the loops φ n defined by φ n (t) := cos 2πnt -sin 2πnt sin 2πnt cos 2πnt ⊕Id; since φ n (t) = φ 1 (t) n , it is enough to show, using the homotopy, catenation, product and zero properties that the index of the loop given by φ(t) = cos 2πt -sin 2πt sin 2πt cos 2πt

for t ∈ [0, 1] is equal to 2. This is true, using the signature property, writing φ as the catenation of the path ψ 1 (t) := φ( t 2 ) = exp tJ 0 ( π 0 0 π ) for t ∈ [0, 1] whose index is 1 and the path ψ 2 (t) := φ( t 2 ) = exp tJ 0 ( π 0 0 π ) for t ∈ [1, 2]. We introduce the path in the reverse direction ψ - 2 (t) := exp -tJ 0 ( π 0 0 π ) for t ∈ [0, 1] whose index is -1; since the catenation of ψ - 2 and ψ 2 is homotopic to the constant path whose index is zero, the index of φ 1 is given by the index of ψ 1 minus the index of ψ - 2 hence is equal to 2.

Generalised Conley-Zehnder index

We are now ready to prove the characterization of the Robbin-Salamon index stated in the introduction.

Proof of theorem 6.1.2: Observe that any symmetric matrix can be written as the symplectic direct sum of a non degenerate symmetric matrix S and a matrix S ′ of the form 0 0 0 B where B is symmetric and may be degenerate. The index of the path ψ t = exp tJ 0 S ′ is equal to the index of the path ψ ′ t = exp tλJ 0 S ′ for any λ > 0. Hence the signature and shear conditions, in view of the product condition, can be simultaneously written as: if S = S τ ∈ R 2n×2n is a symmetric matrix with all eigenvalues of absolute value < 2π and if ψ(t) = exp(J 0 St) for t ∈ [0, 1] , then µ RS (ψ) = 1 2 Sign S. This is the normalization condition stated in the theorem.

From Lemma 6.5.3, we just have to prove that the product property is a consequence of the other properties. We prove it for paths with values in Sp(R 2n , Ω 0 ) by induction on n, the case n = 1 being obvious. Since ψ ′ ⊕ ψ ′′ is homotopic with fixed endpoints to the catenation of ψ ′ ⊕ ψ ′′ (0) and ψ ′ (1) ⊕ ψ ′′ , it is enough to show that the index of A ⊕ ψ is equal to the index of ψ for any fixed A ∈ Sp(R 2n ′ , Ω 0 ) with n ′ < n and any continuous path ψ : [0, 1] → Sp(R 2n ′′ , Ω 0 ) with n ′′ < n.

Using the proof of lemma 6.5.2, any symplectic matrix A can be linked by a path φ(s) with constant dimension of the 1-eigenspace to a matrix of the form exp(J 0 S ′ ) with S ′ a symmetric n ′ × n ′ matrix with all eigenvalues of absolute value < 2π. The index of A ⊕ ψ is equal to the index of exp(J 0 S ′ ) ⊕ ψ; indeed A ⊕ ψ is homotopic with fixed endpoints to the catenation of the three paths φ s ⊕ ψ(0), exp(J 0 S ′ ) ⊕ ψ and the path φ s ⊕ ψ(1) in the reverse order, and the index of the first and third paths are zero since the dimension of the 1-eigenspace does not vary along those paths.

Hence it is enough to show that the index of exp(J 0 S ′ ) ⊕ ψ is the same as the index of ψ. This is true because the map µ sending a path ψ in Sp(R 2n ′′ , Ω 0 ) (with n ′′ < n) to the index of exp(J 0 S ′ ) ⊕ ψ has the four properties stated in the theorem, and these characterize the Robbin-Salamon index for those paths by induction hypothesis. It is clear that µ is invariant under homotopies, additive for catenation and equal to zero on paths ψ for which the dimension of the 1-eigenspace is constant. Furthermore µ(exp t(J 0 S)) which is the index of exp(J 0 S ′ ) ⊕ exp t(J 0 S) is equal to 1 2 Sign S, because the path exp tJ 0 (S ′ ⊕ S) whose index is 1 2 Sign(S ′ ⊕ S) = 1 2 Sign S ′ + 1 2 Sign S is homotopic with fixed endpoints with the catenation of exp t(J 0 S ′ ) ⊕ Id = exp tJ 0 (S ′ ⊕ 0), whose index is 1 2 Sign S ′ , and the path exp(J 0 S ′ ) ⊕ exp t(J 0 S).

A formula for the Robbin-Salamon index

Let ψ : [0, 1] → Sp(R 2n , Ω 0 ) be a path of symplectic matrices. The symplectic transformation ψ(1) of V = R 2n decomposes as

where ψ ⋆ (1) does not admit 1 as eigenvalue and ψ (1) (1) is the restriction of ψ(1) to the generalized eigenspace of eigenvalue 1

By proposition 6.5.1, there exists a symplectic matrix A such that Aψ (1) (1)A -1 is equal to

with each d

(1)

It is a path from ψ(1) to the matrix defined in 6.21. Clearly, µ RS (ψ I ) = 0 and ρ is constant on ψ I . Let ψ II : [0, 1] → Sp(R 2n , Ω 0 ) be the path from ψ I (1) to

defined as in the proof of lemma 6.5.2 in each block by . Note that µ RS (ψ III ) = 1 2 j d

(1) j by proposition 6.4.9 and ρ is constant on ψ III . Finally, consider ψ IV : [0, 1] → Sp(R 2n , Ω 0 ) from ψ III (1) to ψ ⋆ (1) ⊕ e -1 Id 0 0 e Id which is given by ψ ⋆ (1) ⊕ e -t 0 0 e t ⊕ e -1 Id 0 0 e Id . Note that µ RS (ψ IV ) = 0, ρ is constant on ψ IV and ψ IV (1) is in Sp ⋆ (R 2n , Ω 0 ). Since two paths of matrices with fixed ends are homotopic if and only if their image under ρ are homotopic, the catenation of the paths ψ III and ψ IV is homotopic to any path from ψ I (1) to ψ ⋆ (1) ⊕ e Id 0 0 e -1 Id of the form ψ ⋆ (1)⊕) ⊕ φ 1 (t) where φ 1 (t) has only real positive eigenvalues. We proceed similarly for ψ(0) and we get Theorem 6.6.1 Let ψ : [0, 1] → Sp(R 2n , Ω 0 ) be a path of symplectic matrices. Decompose ψ(0) = ψ ⋆ (0) ⊕ ψ (1) (0) and ψ(1) = ψ ⋆ (1) ⊕ ψ (1) (1) where ψ ⋆ (0) (resp. ψ ⋆ (1)) does not admit 1 as eigenvalue and ψ (1) (0) (resp. ψ (1) (1)) is the restriction of ψ(0) (resp. ψ(1)) to the generalized eigenspace of eigenvalue 1 of ψ(0) (resp. ψ(1)). Consider a prolongation

where φ 0 (t) has only real positive eigenvalues for t ∈ -1 2 , 0 ;

• Ψ 3 2 = ψ ⋆ (1) ⊕ e -1 Id 0 0 e Id and Ψ(t) = ψ ⋆ (1) ⊕ φ 1 (t) where φ 1 (t) has only real positive eigenvalues for t ∈ 1, 3 2 ;

• Ψ(-1) = W ± , Ψ(2) = W ± and Ψ(t) ∈ Sp ⋆ (R 2n , Ω 0 ) for t ∈ -1, -1 2 ∪ 3 2 , 2 . (1) j .

Remark that we can replace in the formula above ρ by ρ as in proposition 6.2.7. By proposition 6.5.1, we have theorem 6.1.3 :

Remark 6.6.2 The advantage of this new formula is that to compute the index of a path whose crossing with the Maslov cycle is non transverse we do not need to perturb the path. The drawback is that we have to extend the initial path.

Summary

This thesis deals with the question of the minimal number of distinct periodic Reeb orbits on a contact manifold which is the boundary of a compact symplectic manifold.

The positive S 1 -equivariant symplectic homology is one of the main tools considered in this thesis. It is built from periodic orbits of Hamiltonian vector fields in a symplectic manifold whose boundary is the given contact manifold.

Our first result describes the relation between the symplectic homologies of an exact compact symplectic manifold with contact type boundary (also called Liouville domain), and the periodic Reeb orbits on the boundary. We then prove some properties of these homologies. For a Liouville domain embedded into another one, we construct a morphism between their homologies. We study the invariance of the homologies with respect to the choice of the contact form on the boundary. We use the positive S 1 -equivariant symplectic homology to give a new proof of a Theorem by Ekeland and Lasry about the minimal number of distinct periodic Reeb orbits on some hypersurfaces in R 2n . We indicate how it extends to some hypersurfaces in some negative line bundles.

We also give a characterisation and a new way to compute the generalized Conley-Zehnder index defined by Robbin and Salamon for any path of symplectic matrices. A tool for this is a new analysis of normal forms for symplectic matrices.

Résumé

Le sujet de cette thèse est la question du nombre minimal d'orbites de Reeb distinctes sur une variété de contact qui est le bord d'une variété symplectique compacte.

L'homologie symplectique S 1 -équivariante positive est un des outils principaux de cette thèse; elle est construite à partir d'orbites périodiques de champs de vecteurs hamiltoniens sur une variété symplectique dont le bord est la variété de contact considérée.

Nous analysons la relation entre les différentes variantes d'homologie symplectique d'une variété symplectique exacte compacte (domaine de Liouville) et les orbites de Reeb de son bord. Nous démontrons certaines propriétés de ces homologies. Pour un domaine de Liouville plongé dans un autre, nous construisons un morphisme entre leurs homologies. Nous étudions ensuite l'invariance de ces homologies par rapport au choix de la forme de contact sur le bord. Nous utilisons l'homologie symplectique S 1 -équivariante positive pour donner une nouvelle preuve d'un théorème de Ekeland et Lasry sur le nombre minimal d'orbites de Reeb distinctes sur certaines hypersurfaces dans R 2n . Nous indiquons comment étendre au cas de certaines hypersurfaces dans certains fibrés en droites complexes négatifs.

Nous donnons une caractérisation et une nouvelle façon de calculer l'indice de Conley-Zehnder généralisé, défini par Robbin et Salamon pour tout chemin de matrices symplectiques. Ceci nous a mené à développer de nouvelles formes normales de matrices symplectiques.