Fast finite-energy planes in symplectizations and applications

We define the notion of fast finite-energy planes in the symplectization of a closed 3-dimensional energy level $M$ of contact type. We use them to construct special open book decompositions of $M$ when the contact structure is tight and induced by a (non-degenerate) dynamically convex contact form. The obtained open books have disk-like pages that are global surfaces of section for the Hamiltonian dynamics. Let $S \subset \R^4$ be the boundary of a smooth, strictly convex, non-degenerate and bounded domain. We show that a necessary and sufficient condition for a closed Hamiltonian orbit $P\subset S$ to be the boundary of a disk-like global surface of section for the Hamiltonian dynamics is that $P$ is unknotted and has self-linking number -1.


Introduction
We intend to give a systematic treatment to the procedure of constructing global surfaces of section for the Hamiltonian dynamics on strictly convex 3-dimensional energy levels. (1) X is transversal to Σ \ ∂Σ and ∂Σ consists of periodic orbits of X.
(2) For every x ∈ M \ ∂Σ one finds sequences t ± n → ±∞ such that φ t ± n (x) ∈ Σ. Here φ t denotes the flow of X.
In [17] Hofer, Wysocki and Zehnder studied the Hamiltonian dynamics on a bounded and strictly convex regular level S = H −1 (1) ⊂ R 4 , where H : R 4 → R is a smooth Hamiltonian. If z = (q 1 , p 1 , q 2 , p 2 ) are coordinates in R 4 equipped with its canonical symplectic form (1) ω 0 = dq 1 ∧ dp 1 + dq 2 ∧ dp 2 , then Hamilton's equations can be rewritten aṡ where X H is the so-called Hamiltonian vector field. It is uniquely determined by i XH ω 0 = dH and its flow preserves level-sets of H. They consider the case when S = ∂K for some bounded, smooth and strictly convex domain K ⊂ R 4 . If G is some other Hamiltonian realizing S as a regular energy level then RX G | S = RX H | S , that is, Hamiltonian dynamics depends only on S and ω 0 , up to time-reparametrization. This can be checked by inspection, or by noting that where (T z S) ω0 := v ∈ R 4 : ω 0 (v, u) = 0, ∀u ∈ T z S is the ω 0 -symplectic orthogonal of T z S. The line bundle (T S) ω0 is called the characteristic line field.
The study of Hamiltonian dynamics on such strictly convex hypersurfaces is now a classical subject. In 1978 P. Rabinowitz [27] and A. Weinstein [33] proved existence of periodic orbits on these energy levels. In [17] Hofer, Wysocki and Zehnder proved the following remarkable result. Theorem 1.2 (Hofer, Wysocki and Zehnder). Let S ⊂ R 4 be a bounded, smooth and strictly convex hypersurface. Then S carries an unknotted periodic Hamiltonian orbit P 0 bounding a disk-like global surface of section D for the Hamiltonian dynamics.
The Poincaré return map toD preserves a smooth area-form, with total area D ω 0 < ∞. Brower's translation theorem provides a second periodic orbit P 1 , geometrically distinct of P 0 . It corresponds to a fixed point of the first return map toD. One can, as described in [17], apply results of J. Franks [6] on periodic points of area-preserving diffeomorphisms of the open annulus to obtain the following important corollary. Corollary 1.3 (Hofer, Wysocki and Zehnder). Hamiltonian dynamics on a bounded, smooth, strictly convex energy level inside R 4 has either 2 or ∞-many periodic orbits. Theorem 1.2 immediately prompts the following question: What are the necessary and sufficient conditions for a periodic Hamiltonian orbit to bound an embedded disk-like global surface of section?
Our main result answers this question when S is non-degenerate, that is, when 1 is not a transversal eigenvalue of the linearized Poincaré return map of every closed orbit. This is a C ∞ -generic condition on S. Our answer is stated in terms of a certain contact-topological invariant, called the self-linking number, which we now describe.
A 1-form λ on a 3-manifold M is a contact form if λ ∧ dλ = 0. The associated contact structure is the hyperplane distribution (2) ξ = ker λ.
A co-oriented contact 3-manifold is a pair (M, ξ) such that (2) holds for some contact form λ. We call λ tight if ξ is a tight contact structure, see Subsection 3.4. The associated Reeb vector field R is defined implicitly by (3) i R dλ = 0 and i R λ = 1.
Definition 1.4 (Self-linking Number). Let L ⊂ M be a knot transversal to ξ, and let Σ ֒→ M be a Seifert surface 1 for L. Since the bundle ξ| Σ carries the symplectic bilinear form dλ, there exists a smooth non-vanishing section Z of ξ| Σ , which can be used to slightly perturb L to another transversal knot L ǫ = {exp x (ǫZ x ) : x ∈ L}.
Here exp is any exponential map. A choice of orientation for Σ induces orientations of L and of L ǫ . The self-linking number is defined as the oriented intersection number where M is oriented by λ∧dλ. It is independent of Σ when c 1 (ξ) ∈ H 2 (M ) vanishes.
Recall that ω 0 has a special primitive λ 0 = 1 2 2 k=1 q k dp k − p k dq k . We assume 0 is in the bounded component K of R 4 \ S, so that λ 0 | S is a contact form. If R is the associated Reeb field on S then RR = RX H . Our main result can be stated as follows.
Theorem 1.5. Let S ⊂ R 4 be a non-degenerate, bounded, smooth and strictly convex hypersurface. A necessary and sufficient condition for a periodic Hamiltonian orbit P ⊂ S to be the boundary of a disk-like global surface of section for the Hamiltonian dynamics is that P is unknotted and has self-linking number −1.
Necessity is an easy computation, see Proposition 2.1. The assumption of S being non-degenerate is rather technical and will be removed in [20]. The reader acquainted with the work of Hofer, Wysocki and Zehnder will notice that it can be removed by arguments found in [17]. In [20] we shall also prove that a periodic orbit associated to any fixed point of Poincaré's first return map to the global surface of section obtained from Theorem 1.2 has self-linking number −1. As a consequence, it also bounds a disk-like global surface of section.
Sufficiency in Theorem 1.5 will follow from a more general result, which we now describe. Motivated by [17], we consider systems of global surfaces of section organized in the form of open book decompositions. Definition 1. 6. An open book decomposition of an oriented 3-manifold M is a pair (L, p) where L is an oriented link in M , and p : M \ L → S 1 is a fibration such that each fiber p −1 (θ) is the interior of an oriented compact embedded surface S θ ֒→ M satisfying ∂S θ = L (including orientations). L is called the binding and the fibers are called pages. It is said to be adapted to the dynamics of a vector field X if L consists of periodic orbits, X orients L positively, the pages are global surfaces of section and the orientation of M together with X induce the orientation of the pages. 1 By a Seifert surface for L we mean an orientable embedded connected compact surface Σ ֒→ M such that L = ∂Σ.
We are interested in constructing open book decompositions adapted to the dynamics of Reeb vector fields. The study of this problem was initiated by Hofer, Wysocki and Zehnder in [15,16]. Their proofs are based on the theory of pseudoholomorphic curves in symplectizations, introduced by Hofer in [9]. In the constructions they use disk-filling methods, bubbling-off analysis and their own perturbation theory. This work is the first of three articles extending their results.
Consider a contact form λ on a closed 3-manifold M . A periodic Reeb orbit will be denoted by P = (x, T ), where x : R → M solvesẋ = R • x and T > 0 is a period. It is called simply covered when T is the minimal positive period of x, and unknotted if it is simply covered and x(R) is the unknot. When Σ is a Seifert surface for the transverse knot x(R) we write sl(P, Σ) instead of sl(x(R), Σ). Since the Reeb flow φ t preserves λ we have well-defined dλ-symplectic maps dφ t (x(t 0 )) : ξ x(t0) → ξ| x(t0+t) . The periodic orbit P is non-degenerate if 1 is not an eigenvalue of dφ T (x(t 0 ))| ξ , ∀t 0 ∈ R. When every P is non-degenerate one says λ is non-degenerate. See Subsection 3.1 for a more precise discussion.
In [18] we find the following axiomatic characterization of the Conley-Zehnder index in the case n = 1. For other discussions on this topic see [28] or [29]. Theorem 1.7. There exists a unique surjective map µ : Σ * → Z characterized by the following axioms: (1) Homotopy: If ϕ s is a homotopy of arcs in Σ * then µ (ϕ s ) is constant.
(2) Maslov index: If ψ : (R/Z, 0) → (Sp(1), I) is a loop and ϕ ∈ Σ * then µ(ψϕ) = 2Maslov(ψ) + µ(ϕ). Let P = (x, T ) be a non-degenerate periodic Reeb orbit and denote by ξ P the bundle x * T ξ → R/Z, where x T (t) = x(T t). Consider the set S P of homotopy classes of smooth dλ-symplectic trivializations of ξ P , and fix β ∈ S P . Any trivialization in class β can be used to represent the linear maps dφ T t : ξ x(0) → ξ x(T t) by some path ϕ ∈ Σ * . We write µ CZ (P, β) = µ(ϕ). By axiom (1) above this is independent of the particular trivialization in class β. When P is contractible we shall say c 1 (ξ) vanishes along P if the 2-sphere obtained by gluing any two disk-maps spanning the map e i2πt → x(T t) lies in the kernel of c 1 (ξ). In this case, there exists a special class β P ∈ S P induced by some, and hence any, such disk-map. We can define µ CZ (P ) := µ CZ (P, β P ), see Subsection 3.1.
In [9] Hofer introduced special almost complex structures on the symplectization R × M and considered solutions of the associated Cauchy-Riemann equations in order to study the dynamics of Reeb vector fields. As a consequence of his groundbreaking work, the three-dimensional Weinstein Conjecture 2 was confirmed in many cases, including all contact forms on S 3 . We need to recall a few concepts from [9] in order to discuss our results.
A complex structure J on ξ is dλ-compatible if dλ(·, J·) is a metric on ξ. We write J (ξ, dλ| ξ ) for the set of such complex structures. Following [9], every J ∈ J (ξ, dλ| ξ ) induces an almost complex structureJ on R × M by where a denotes the R-component. A finite-energy plane is aJ-holomorphic map u : (C, i) → (R × M,J) with positive and finite Hofer energy, see Subsection 3.5 for the precise definitions. The following result is central in [16], see also [15]. Theorem 1.8 (Hofer, Wysocki and Zehnder). Let λ be a tight contact form on a closed 3-manifold M and assume c 1 (ξ) vanishes. Let P 0 = (x 0 , T 0 ) be an unknotted non-degenerate closed Reeb orbit satisfying µ CZ (P 0 ) = 3 and sl(P 0 ) = −1. Suppose every contractible orbit P = (x, T ) with T ≤ T 0 is non-degenerate and satisfies µ CZ (P ) ≥ 3. Then, for generic choice of J ∈ J (ξ, dλ| ξ ), there exists an embedded finite-energyJ-holomorphic planeũ 0 in the sympelctization R × M asymptotic to P 0 at ∞. Its projection onto M does not intersect x 0 (R) and is only one page of an open book decomposition adapted to the Reeb dynamics. The binding is P 0 and the pages are open disks. In particular, M = S 3 and ξ is its unique (up to contactmorphism) positive tight contact structure.
As explained before, Brower's translation theorem and results of J. Franks from [6] provide an important corollary. Corollary 1.9 (Hofer, Wysocki and Zehnder). Under the assumptions of Theorem 1.8 the Reeb dynamics has either 2 or ∞-many periodic orbits.
Here we shall define a class of pseudo-holomorphic curves suitable for this construction. They generalize the curves used in [15], [16] and [17]. Let us briefly describe the invariants wind ∞ and cov, originally introduced in [12]. If we writeũ = (a, u) ∈ R × M and assume λ is non-degenerate then the loops t → u(e 2π(s+it) ) converge to x(T t + c) in C ∞ (R/Z, M ) as s → +∞, where P = (x, T ) is some closed Reeb orbit and c ∈ R. This follows from Theorem 3.9 below and the definition of non-degenerate punctures given in Subsection 3.6, see [11]. In this case one saysũ is asymptotic to P . Condition cov(ũ) = 1 means that P is simply covered. The identity wind ∞ (ũ) = 1 holds if, and only if, u : C → M is an immersion transversal to the Reeb vector field, see Lemma 3.19.
It provides a smooth homotopy from the vector θ → W (ϕ(e i2πθ )) to the vector θ → A(θ) through non-vanishing vectors. This shows P W ǫ · D = 0 and completes the proof that sl(P, D) = −1.

2.2.
Compactness. Since we deal with higher Conley-Zehnder indices, we need new compactness arguments replacing those given by Hofer, Wysocki and Zehnder, see [12], [15] and [16]. Loosely speaking, we shall prove that, under convexity assumptions on λ, breaking of Morse trajectories for the action functional does not occur in families of unparametrized fast planes through a compact H ⊂ R × M . As an example, this will be the case for non-degenerate dynamically convex contact forms on S 3 .
Let λ be a contact form on a closed 3-manifold M , and let P = (x, T ) be a simply covered periodic Reeb orbit. Assume every contractible periodic Reeb orbit P = (x,T ) withT ≤ T is non-degenerate. Consider the set A c of positive periods of contractible periodic Reeb trajectories and define A k c = {τ ∈ A c : τ ≤ k}. Following [18], we define and fix a number Recall the almost complex structureJ (5) associated to some J ∈ J (ξ, dλ| ξ ). We fix a subset H ⊂ R × M and define by requiring thatũ ∈ Θ(H, P, λ, J) if, and only if,ũ is a fast finite-energyJholomorphic plane asymptotic to P ,ũ(0) ∈ H and C\D u * dλ = γ. We define for a given L > 0. When (λ, J) are fixed we write Θ(H, P ), Λ(H, P ), Θ L (H, P ) and Λ L (H, P ) for simplicity. Note that if P is not simply covered then all these sets of fast planes are empty. Our compactness result is as follows.
Theorem 2.2. Let λ be a contact form on a closed 3-manifold M , P = (x, T ) be a periodic Reeb orbit and H ⊂ R × M be compact. Suppose the following properties hold for every contractible periodic Reeb orbitP = (x,T ) withT ≤ T : (i)P is non-degenerate, (ii) c 1 (ξ) vanishes alongP and µ CZ (P ) ≥ 3.
Then the following assertions are true: The above theorem can be rephrased in the terminology of Symplectic Field Theory (SFT) originally introduced by Eliashberg, Givental and Hofer in [4]. A plane in Θ(H, P ) is a stable connected smooth holomorphic curve, in the sense of [2]. The SFT Compactness Theorem [2] describes the compactification of the set of such curves with a priori bounds on energy and genus. It generalizes the notion of Gromov convergence of pseudo-holomorphic curves in closed symplectic manifolds [8] to, for example, the non-compact setting of symplectizations. The notion of a stable curve, adapted in [2] to symplectic cobordisms, was first introduced by Kontsevich in [23]. More general than a stable connected smooth holomorphic curve is a holomorphic building of height 1, which is a finite energy map defined on the components of a (not necessarily stable) nodal Riemann surface, plus compatibility conditions. These are not enough to compactify the set of smooth curves and one needs to introduce higher buildings. In our situation, the possible limiting holomorphic buildings of a sequence of fast planes can be described as a rooted graph, the vertices of which represent stable connected smooth holomorphic curves. An edge represents a closed Reeb orbit which is a common limit of the curves at the corresponding vertices. This is always the case when dealing with curves of genus 0 with one positive puncture. The edges can be oriented as going away from the root, and this divides the graph into levels, these being the levels of the holomorphic building as described in [2]. The proof of Theorem 2.2 consists of showing this graph has exactly one vertex. In order to accomplish this we only need to analyze the root, more precisely, we shall prove that the convexity assumptions on λ will discard outgoing edges.
As a final remark, assertions (1) and (2) of Theorem 2.2 about the sets Λ L (H, P ) and Λ(H, P ) follow independently from Theorem 4 of [34], we explain. The elegant analysis of Wendl can be applied to embedded fast planes. In view of Theorem 2.3, these are examples of "nicely embedded" curves in the sense of [34] which, in the symplectization R × M , are holomorphic curves with embedded projections onto M . With the appropriate asymptotic constraint on the orbit P , a fast plane has a vanishing "constrained" normal first Chern number, so that the limiting holomorphic building of a sequence of embedded fast planes is either smooth or one of its levels contains a plane with µ-index equal to 2. But these do not exist under our crucial additional assumption that λ is dynamically convex. However, the results of [34] do not cover the corresponding statements made in Theorem 2.2 about the sets Θ L (H, P ) and Θ(H, P ), even when λ is dynamically convex.
A proof using the SFT-Compactness Theorem or the results of [34] directly would make the exposition non-elementary and highly not self-contained, forcing the introduction of a large amount of notation, and making this work less accessible to a wider public.
Also, we would like to emphasize that our arguments are independent of any transversality assumptions.
2.3. Fredholm theory. The second analytical tool for proving Theorem 1.11 is a perturbation theory. Embedded fast planes are always regular in a suitably defined index-2 Fredholm theory with exponential weights. Theorem 2.3. Let λ be any contact form on a 3-manifold M and ξ = ker λ be the associated contact structure. Fix any J ∈ J (ξ, dλ| ξ ) and supposeũ = (a, u) is an embedded fast finite-energyJ-holomorphic plane asymptotic to a periodic Reeb orbit is a smooth proper embedding. Moreover, for any l ≥ 1 there exists an open ball B r (0) ⊂ R 2 and a C l embedding f : C × B r (0) → R × M satisfying: (1) f (z, 0) =ũ(z).
In view of definitions 1.10 and 3.11 the map u : C → M provides a capping disk for P 0 , singling out a homotopy class βũ ∈ S P0 defined by the following property: a dλ-symplectic trivialization Ψ of ξ P0 extends to u * ξ if, and only if, it is in class βũ. In the above statement µ(ũ) = µ CZ (P 0 , βũ).
Note that P 0 is not assumed non-degenerate, instead we assume the planes have non-degenerate asymptotic behavior in the sense of Definition 3.11. This allows us to handle arbitrary contact forms on S 3 , see [20]. We also emphasize that no assumptions on λ, like being Morse-Bott, are made. This justifies Theorem 2.3.
A degenerate Fredholm theory as described above was only hinted at in [17]. The above statement does not follow from the results of [13] but, of course, its proof follows their arguments closely. We refer the reader to Section 6 for the proof.

2.4.
Existence of fast finite-energy planes. We shall prove the following existence result of fast planes. It partially generalizes the existence statement in Theorem 1.8 since it deals with higher Conley-Zehnder indices.
Theorem 2.4. Let λ be a tight contact form on a closed 3-manifold M such that the following properties hold for every contractible Reeb orbitP : (i)P is non-degenerate, (ii) c 1 (ξ) vanishes alongP and µ CZ (P ) ≥ 3.
Suppose P is an unknotted periodic Reeb orbit satisfying sl(P, disk) = −1. Then, for a suitable dλ-compatible complex structure J : ξ → ξ, there exists an embedded fast finite-energyJ-holomorphic plane asymptotic to P at ∞.
As before, the integer sl(P, disk) denotes the self-linking number computed with respect to any embedded disk spanning the unknot x(R). It is independent of this disk since c 1 (ξ) vanishes along P .
The general idea of the proof is standard, see [16]. Since P is unknotted and sl(P ) = −1 we can find, using arguments of Giroux [7] and Hofer [9], an embedded disk F ֒→ M spanning P (including orientations) such that its characteristic foliation has exactly one positive elliptic singularity e ∈ F with real eigenvalues. Denote F * = F \ {e}. The surface {0} × F * ֒→ R × M is totally real with respect toJ and there exists a so-called Bishop family emanating from (0, e). It is a one-dimensional family of unparametrized embeddedJ-holomorphic disks with boundary on {0} × F * . This family was discovered by E. Bishop in [1] and used by Hofer in [9] in order to establish the Weinstein Conjecture in S 3 and in many other closed 3-manifolds. It should be noted that disk-filling methods were also used in [8] and in [5] in order to show that symplectically fillable contact structures are tight.
Each connected component of the Bishop family is an open interval. At one end the family converges to the constant (0, e). At the other end bubbling-off occurs and one can prove, using the convexity assumptions on λ, that bubbling-off does not happen before the disks reach the boundary ∂F = P . As a result of this bubbling-off analysis we have, in the language of Symplectic Field Theory (see [2]), a holomorphic building with height k ≥ 2. Each level is a collection of smooth finite-energy surfaces in R × M . The first level is a half trivial cylinder over P and the curves on other levels have no boundary. All this is proved in [16]. Now we need to introduce new arguments. In [16] the authors use the fact that the µ-index of the orbit in question is 3. They heavily rely on the compactness argument explained in [12] to conclude that the second level of the stable curve consists of a single plane, hence there are no more levels. This is in great contrast with our situation since we allow µ CZ (P ) ≥ 3. We overcome this difficulty by slightly perturbing the boundary condition F in order to ensure there are no Reeb tangencies near its boundary (of course the Reeb vector is tangent at the boundary since it is a Reeb orbit). This allows us to reach the same conclusions as in [16]: k = 2 and the second level is a single plane asymptotic to P . Also, this plane is embedded and fast.
One easily checks that Λ k (H, P ) is C ∞ loc -closed, ∀k. Thus, each Λ k (H, P ) is C ∞ loccompact whenever Λ(H, P ) is C ∞ loc -compact. In Section 7 we prove Theorem 2.5. Let λ be a non-degenerate contact form on the closed 3-manifold M and letJ be the almost complex structure on R × M defined by equations (5). Suppose there exists an embedded fastJ-holomorphic finite-energy planeũ 0 asymptotic to P = (x, T ) with µ(ũ 0 ) = k ≥ 3. We also suppose that the set of planes Λ k (H, P ) is C ∞ loc -compact for every compact subset H ⊂ R× M satisfying H ∩(R× x(R)) = ∅. Then for every l ≥ 1 there exists a C l mapũ = (a, u) : S 1 × C → R × M with the following properties: (1)ũ(ϑ, ·) is an embedded fast finite-energy plane asymptotic to P at (the positive puncture) ∞ satisfying µ(ũ(ϑ, ·)) = k, ∀ϑ ∈ S 1 . (2) u(ϑ, C) ∩ x(R) = ∅ ∀ϑ ∈ S 1 and the map u : is a smooth global surface of section for the Reeb dynamics.
The purpose of the above statement is to isolate the compactness properties of fast planes which allow us to construct the desired open book decompositions. Here these compactness properties follow from convexity assumptions on λ, see Theorem 2.2. In [21] we shall prove that the assumptions of Theorem 2.5 hold under much less restrictive assumptions on λ, allowing us to investigate general Reeb flows on the tight 3-sphere.
Let us assume the hypotheses of Theorem 1.11. If P is an unknotted, simply covered, periodic Reeb orbit satisfying sl(P ) = −1 then Theorem 2.4 provides an embedded fast finite-energy planeũ 0 asymptotic to P at ∞. Theorem 2.2 now shows that the hypotheses of Theorem 2.5 hold. Theorem 1.11 follows immediately.
Before proving Theorem 1.5 we briefly outline the proof of Theorem 2.5 for convenience of the reader. Suppose M , λ andũ 0 satisfy the hypotheses of Theorem 2.5. If we writeũ 0 = (a 0 , u 0 ) ∈ R × M then it follows from lemmas 6.22, 6.23 and 6.24 below that u 0 is a proper embedding into M \ x(R). The identity wind ∞ (ũ 0 ) = 1 proves u 0 (C) is transverse to the Reeb vector field. By Theorem 2.3,ũ 0 is only one embedded fast plane in a small 2-parameter family. Let p 0 = u 0 (0) ∈ M and assume, without loss of generality, that a 0 (0) = 0. Denoting by φ t the Reeb flow, we can single out a one-dimensional subfamily {ũ t = (a t , u t )} by requiring The family of embedded planes {u t (C)} inside M \ x(R) provides a smooth foliation of a neighborhood of u 0 (C). Using the compactness assumptions we continue the familyũ t for all values t ∈ R, satisfying the above normalization conditions. By Poincaré recurrence we could have assumed, without loss of generality, that p 0 ∈ ω-limit(p 0 ). Then the trajectory φ t (p 0 ) will eventually come close to p 0 . The completeness statement in Theorem 2.3 can be used to show that the familyũ t can be glued to provide a S 1 -family. It foliates the whole of M \ x(R). This S 1 -family can be made minimal if we require is a diffeomorphism. This provides an open book decomposition with disk-like pages that are transverse to the Reeb field. To prove the pages are global surfaces of section, fix q 0 ∈ M \ x(R). If x(R) ∩ ω-limit(q 0 ) = ∅ then φ t (q 0 ) hits every page in forward time, by an easy compactness argument. If x(R) ∩ ω-limit(q 0 ) = ∅ then the condition µ CZ (P ) ≥ 3 makes the flow wind around x(R) for long enough times, forcing it to hit every page. This is proved in Section 5 of [17], see Lemma 6.9 below. The argument is the same for negative times. This concludes the proof of Theorem 2.5. Theorem 1.5 follows easily from Theorem 1.11 and from the following result from [17]. Theorem 2.6 (Hofer, Wysocki and Zehnder). If S ⊂ R 4 is the boundary of a bounded, smooth, strictly convex domain containing 0 then λ 0 | S is dynamically convex, that is, µ CZ (P ) ≥ 3 for every periodic orbit of the Reeb vector field associated to the contact form λ 0 | S .
The arguments are immediate in view of a famous result of Bennequin asserting that ξ 0 = ker λ 0 | S ⊂ T S is a tight contact structure.

Basic definitions and facts
Unless otherwise stated, M denotes a closed 3-manifold, λ ∈ Ω 1 (M ) is a contact form and ξ = ker λ is the induced contact structure.
3.1. Periodic Reeb orbits and winding numbers. We shall identify a periodic Reeb orbit P = (x, T ) with the class in C ∞ (S 1 , M )/S 1 of the loop Here we let S 1 act on the loop space by rotations on the domain. Hence, we view the collection P of periodic Reeb orbits as a subset of C ∞ (S 1 , M )/S 1 . The geometrical image of P = (x, T ) ∈ P is the set x(R) and We shall agree with the following convention: for every periodic Reeb orbit we select a point in its geometrical image, and it will be implicit from the notation P = (x, T ) that x(0) is the chosen point.
We can define the map f 1 #f 2 : We say that c 1 (ξ) vanishes along P if c 1 ((f 1 #f 2 ) * ξ) = 0 for every pair f 1 , f 2 as above. We denote by P * the set of contractible orbits with this property.
Clearly all contractible periodic Reeb orbits belong to P * if c 1 (ξ) vanishes.  3.3. Given P ∈ P, recall the set S P of homotopy classes of dλ-symplectic trivializations of ξ P . One can identify S P with the set of homotopy classes of nonvanishing sections of ξ P in a straightforward way. In the following we shall always assume this is done. Consequently, if P = (x, T ) ∈ P * then a section Z is in the special class β P discussed in the introduction if, and only if, for some (and hence any) continuous map f : D → M satisfying f (e i2πt ) = x(T t) the section Z extends to a non-vanishing section of f * ξ.
The following lemma, which is a trivial consequence of standard degree theory, will be stated without proof.
Lemma 3.4. Suppose P = (x, T ) ∈ P * and U is a small tubular neighborhood of x(R) in M . Let Z be a non-vanishing section of ξ| U such that x * T Z ∈ β P . If f : D → M is a continuous map such that f (∂D) ⊂ U and t → f (e i2πt ) is homotopic to x T in U then (f | ∂D ) * Z extends to a non-vanishing section of f * ξ.

Dynamical Convexity.
Here we shall modify slightly an important definition from [17]. 3.5. Finite-energy surfaces in symplectizations. In 1985 pseudo-holomorphic curves were introduced in symplectic geometry by M. Gromov [8]. In 1993 they were used by H. Hofer to study Reeb flows on contact manifolds. Let (Σ, j) be a Riemann surface, possibly with non-empty boundary and not necessarily compact, and Γ ⊂ Σ \ ∂Σ be a finite subset. The notion of finite-energy surfaces was introduced by H. Hofer in [9].
Let us writeũ = (a, u) ∈ R × M . The points of Γ are called punctures. Let us fix a puncture z ∈ Γ and let ϕ : (U, 0) → (ϕ(U ), z) be a holomorphic chart of (Σ, j) centered at z. Writeũ(s, t) =ũ • ϕ e −2π(s+it) . It follows easily from E(ũ) < ∞ that the limit (14) m = lim exists. The puncture z is removable if m = 0, positive if m > 0 and negative if m < 0. A removable singularity can actually be removed, meaning thatũ can be smoothly continued across the singularity, see [9]. If Σ is closed then a finite-energy surface must have non-removable punctures because the forms ω φ are exact. Finite-energy surfaces are closely related to periodic Reeb orbits. This is the content of the following fundamental result from [9]. Theorem 3.9 (H. Hofer). In the notation explained above, suppose z is nonremovable and let ǫ = ±1 be the sign of m in (14). Then every sequence s n → +∞ has a subsequence s n k such that the following holds: there exists a real number c and a periodic Reeb orbit P = (x, T ) such that u(s n k , t) → x(ǫT t+c) in C ∞ (S 1 , M ) as k → +∞.
In his seminal work [9] H. Hofer is able to partially solve the three-dimensional Weinstein conjecture using techniques of pseudo-holomorphic curves.
Remark 3.10. R × M carries a R-action given by translating the first coordinate andJ is R-invariant. Ifũ = (a, u) is a finite-energy surface then so is c·ũ := (a+c, u).

3.6.
Asymptotic behavior near the punctures. Let π : T M → ξ denote the projection along the Reeb direction.
(3) If π · du does not vanish identically over S \ Γ then π · du(s, t) = 0 when s is large enough. (4) If we define ζ(s, t) by u(s, t) = exp x(ǫT t+c) ζ(s, t) then ∃b > 0 such that sup t∈S 1 e bs |ζ(s, t)| → 0 as s → +∞. In this case we sayũ is asymptotic to P at z. The puncture z is positive or negative according to the sign ǫ. This definition is independent of ϕ and of the exponential map exp. Definition 3.12. We say thatũ has non-degenerate asymptotic behavior at z if z is a non-degenerate puncture, and thatũ has non-degenerate asymptotics if this holds for every puncture.
The behavior ofũ near a puncture is studied in [11]. Here is a partial result. Theorem 3.13 (Hofer, Wysocki and Zehnder). Letũ, z and ϕ be as in Theorem 3.9. If a Reeb orbit P obtained by Theorem 3.9 is non-degenerate thenũ has non-degenerate asymptotic behavior at z. In particular,ũ is asymptotic to P at z.

3.7.
Algebraic invariants. In [12] a number of algebraic invariants of finiteenergy surfaces were introduced. In the next two definitions we fix a finite-energy planeũ : C = S 2 \ {∞} → R × M with non-degenerate asymptotics and consider its asymptotic limit P = (x, T ) at ∞. Definition 3.14 (Covering Number). We define cov(ũ) := T /T min ∈ Z + where T min > 0 is the minimal period of x.
For the next two definitions we fix a closed Riemann surface (Σ, j), a finite set Γ ⊂ Σ and a finite-energy surfaceũ = (a, u) : (Σ \ Γ, j) → (R × M,J) with nondegenerate asymptotics. We assume that π · du does not vanish identically and that Γ consists of non-removable punctures.
Definition 3.17 (wind ∞ ). Split Γ = Γ + ⊔ Γ − where Γ + is the set of positive punctures and Γ − is the set of negative punctures. The bundle u * ξ is trivializable since Γ = ∅. Let α be a homotopy class of non-vanishing sections of u * ξ. Choose a non-vanishing section σ in class α and writeũ(s, t) around a puncture z ∈ Γ as in Definition 3.11. If ǫ = ±1 is the sign of z then define wind ∞ (ũ, α, z) = lim s→+∞ wind(t → πu s (s, ǫt), t → σ(s, ǫt), J) where σ(s, t) = σ(ϕ(e −2π(s+it) )). The invariant wind ∞ (ũ) is defined in [12] by Each wind ∞ (ũ, α, z) depends on the choice of α but is independent of ϕ. By standard degree theory wind ∞ (ũ) is independent of α. Definition 3.18 (wind π ). The bundle E = Hom C (T (Σ \ Γ), u * ξ) is a complex line bundle and the section π · du satisfies a perturbed Cauchy-Riemann equation, see Remark 3.16. Thus its zeros are isolated and count positively when computing the intersection number with the zero section of E. As a consequence of Definition 3.11 the number of zeros is finite. Following [12] we define (15) wind π (ũ) = algebraic count of zeros of π · du where the zeros are counted with multiplicities. The inequality wind π (ũ) ≥ 0 can be seen as a linearized version of positivity of self-intersections.
The Gauss-Bonet formula proves the following lemma, as shown in [12].

Compactness
This section is devoted to the proof of Theorem 2.2. We fix the riemannian metric g 0 on R × M given by (16) g where J : ξ → ξ is dλ-compatible. All norms of maps or objects in R × M are taken with respect to the metric g 0 .

4.1.
Asymptotic operators and their spectral properties. We endow R 2 with its standard euclidean structure ·, · , inducing a Hilbert space structure on (1) is a smooth path and S := −J 0 ϕ ′ ϕ −1 then S T = S. We identify S 1 = R/Z and consider the unbounded self-adjoint operator L S has compact resolvent and discrete real spectrum σ(L S ) accumulating only at ±∞. Each point of the spectrum is an eigenvalue with the same (finite) algebraic and geometric multiplicities, see [22]. This is so because L S is homotopic to −J 0 ∂ t through compact symmetric perturbations. See [12] for more details. To each non-zero eigenvector e of L S one can consider the winding number wind(e). It is proved in [12] that: (1) If e 1 , e 2 are two non-zero eigenvectors of L S such that L S e j = νe j for j = 1, 2 then wind(e 1 ) = wind(e 2 ). (2) If L S e j = ν j e j for j = 1, 2 and ν 1 ≤ ν 2 then wind(e 1 ) ≤ wind(e 2 ).
Thus one has a well-defined winding wind(ν) associated to an eigenvalue ν of L S satisfying ν 1 ≤ ν 2 ⇒ wind(ν 1 ) ≤ wind(ν 2 ). It is also proved in [12] that for every k ∈ Z there are exactly two eigenvalues (counting multiplicities) with winding equal to k. Following [12], we have a well-defined integer (17) where b δ is the number of eigenvalues ν < δ such that wind(ν) = wind(ν neg δ ), counting multiplicities. Remark 4.2. The above lemma provides an extension of the index to symplectic paths ϕ that are not in Σ * . By the spectral properties of L S , if δ is not an eigenvalue then the term 1 2 1 + (−1) b δ is equal to wind (ν pos δ ) − wind (ν neg δ ).

Definition 4.3 (Asymptotic operators)
. Fix a dλ-compatible complex structure J on ξ and let P = (x, T ) be a periodic Reeb orbit. Then the metric dλ(·, J·) induces a Hilbert space structure on L 2 (ξ| P ). Choose a symmetric connection ∇ on T M . The unbounded self-adjoint operator is independent of ∇ (∇ t denotes the covariant derivative along the curve t → x T (t)). A P is the so-called asymptotic operator at P .
The linear flow generated by Thus A P has all the spectral properties explained before.
Notation 4.4. With respect to a symplectic frame σ = {e 1 , e 2 } for ξ| P the eigenvectors and eigenvalues of A P have well-defined winding numbers. These, of course, depend on the homotopy class β ∈ S P of the section t → e 1 (t) and will be denoted by (ν, β) ∈ Z. They are comparable via the formula (ν, β 1 ) = (ν, β 0 ) + wind(β 0 , β 1 , J). For any δ ∈ R we define where b δ is the number of eigenvalues ν < δ such that (ν, β) = (ν neg δ , β), counting multiplicities. If P is non-degenerate then The following lemma is an easy consequence of the definitions.
is a finite-energy sphere satisfying π · dv ≡ 0. Suppose further that ∞ is its unique positive puncture. There exists a non-constant polynomial p : C → C and a periodic orbitP

4.3.
Bubbling-off points. The basic tool for the bubbling-off analysis is the following lemma. In the statement below norms are taken with respect to g 0 (16) and the euclidean metric on C.
Lemma 4.7. Let Γ ⊂ C be finite and U n ⊂ C \ Γ be an increasing sequence of open sets such that ∪ n U n = C \ Γ. Letũ n = (a n , u n ) : (U n , i) → (R × M,J) be a sequence ofJ-holomorphic maps satisfying sup n E(ũ n ) = C < ∞, and z n ∈ U n be a sequence such that |dũ n (z n )| → +∞. If z n stays bounded away from Γ ⊔ {∞}, or if there exist m and ρ > 0 such that C \ B ρ (0) ⊂ U m and z n stays bounded away from Γ, then the following holds: ∀0 < s < 1 there exist subsequences {ũ nj } and {z nj }, sequences z ′ j ∈ C and r j (s) ∈ R, and a contractible periodic Reeb orbitP = (x,T ) We do not include a proof here since it is standard.
Then lim sup |z n | ≤ 1 and for any 1 < s < γ −1 min{γ 1 , γ 2 } there exist subsequences Proof. Writeũ n = (a n , u n ). The conclusion follows easily from the previous lemma since C u * n dλ = T for all n.
We assume every P ∈ Q C is non-degenerate, so that Q C is finite. We follow [18] and fix a number (19) 0 < e < min{a 1 , a 2 }, where a 1 = min{T :P = (x,T ) ∈ Q C } and Choose also an arbitrary The proof of the following lemma is found all over the literature, however its statement is not. We shall not give the arguments here since they are almost identical to the ones given to prove Lemma 4.9 from [18], see also [19].
Lemma 4.11. Let η > 0 be fixed. Suppose that, in addition to the assumptions made above, every contractibleP = (x,T ) withT ≤ C belongs to Q. Then ∃h > 0 with the following significance.
It is not hard to check that, under the assumptions of Theorem 2.2, Q = P * and C = T satisfy the hypotheses of Lemma 4.11. 4.5. An auxiliary lemma. This subsection is independent of the previous discussion. Our goal is to prove Lemma 4.15 below. Let ..N such thatw is asymptotic to P j at z j , according to Definition 3.11.
Assume P j ∈ P * ∀j. There are distinguished homotopy classes β j = β Pj ∈ S Pj induced by capping disks for the maps x j Tj , as explained in Remark 3.3. We can find, see Remark 3.6, small open neighborhoods U j of x j (R) and non-vanishing sections Z j of ξ| Uj such that x j * Tj Z j ∈ β j . Lemma 4.12. The sections Z j • w (defined only near the punctures z j ) extend to a non-vanishing section B of w * ξ.
The proof follows from standard degree theory, we only sketch it here.
Sketch of proof. We can glue capping disks D j for P j along the punctures z j , for j = 2 . . . , N , with the surface w to obtain a capping disk for P 1 . This follows from the asymptotic behavior described in Definition 3.11. The sections {Z j } j≥2 extend to a section σ of ξ| D1 . We used that Z j | Pj ∈ β Pj ∀j ≥ 2 and that the class β Pj has special properties described in Subsection 3.1. Since Z 1 | P1 ∈ β P1 then σ does not wind with respect to Z 1 near ∂D 1 and, consequently, can be patched with Z 1 .
where δ is the homotopy class of the section B given by Lemma 4.12.
The proof requires the following very non-trivial theorem proved in [11].
Theorem 4.14 (Hofer, Wysocki and Zehnder). Suppose P j is non-degenerate and choose a holomorphic chart ϕ : (V, z j ) → (ϕ(V ), 0) centered at z j . Write w(s, t) = w • ϕ(e −2π(s+it) ) if z j is a positive puncture or w(s, t) = w • ϕ(e 2π(s+it) ) if z j is a negative puncture. By rotating the chart ϕ we can assume that w(s, t) → x j (T j t) as |s| → +∞ in C ∞ . Then either π · dw vanishes identically or where e is an eigenvector of A Pj associated to an eigenvalue ν ≤ ν neg .
where e is an eigenvector of A Pj associated to an eigenvalue ν ≥ ν pos .
In Section 6 we will generalize the above theorem, replacing the assumption that P j is non-degenerate by the assumption thatw has non-degenerate asymptotic behavior at z j .
Proof of Lemma 4.13. Let ϕ be as in the statement of Theorem 4.14. Write w(s, t) = w•ϕ e 2π(s+it) and B(s, t) = B•ϕ e 2π(s+it) , where B is the non-vanishing section given by Lemma 4.12. We compute where ν is a positive eigenvalue of A Pj and c ∈ R. The inequality µ CZ (P j ) ≥ 3 implies (ν, β Pj ) ≥ 2 in view of Lemma 4.5.
Proof. Let us assume that µ CZ (P j ) ≥ 3 ∀j ≥ 2. Using Lemma 4.13 we have the following chain of inequalities proving thatΓ = ∅. This contradiction concludes the argument.
As the proof demonstrates, the above lemma follows essentially from the inequality wind π ≥ 0. This should be seen as some kind of linearized version of positivity of self-intersections, and is violated when the asymptotic winding is ≤ 1 at the positive puncture and ≥ 2 at the negative punctures. One should also note that this very simple argument is independent of any transversality results. 4.6. Bubbling-off analysis. We start with a technical lemma.
We proceed indirectly. Suppose ∃{z n } ⊂ C such that z n → z * = 0 and {d n (z n )} is bounded. Hence the sequenceṽ n (z n ) is compactly contained in R×M . We proved Elliptic estimates provide C ∞ loc -bounds and a subsequence {ṽ nj } converging to aJholomorphic cylinder f : It satisfies E(f ) ≤ T and C\{0} g * dλ ≤ γ. We write f = (h, g) and estimate (20) |z|=ρ Hence E(f ) > 0 and 0 is not a removable puncture.
Consequently 0 is a positive puncture. Let us write g e −2π(s+it) = g(s, t). By Theorem 3.9 there exists a periodic orbit P = (x,T ) and a sequence s k → +∞ such that for some c ∈ R. Thus lim k→+∞ |z|=e −2πs k g * λ = −T < 0 contradicting (20). (1) Definingw n : C → R × M byw n (z) = (a n (z) − a n (2), u n (z)) thenw nj →w Proof. By Corollary 4.10 we find a subsequence {ũ nj } ⊂ {ũ n } and a finite set Γ ⊂ D such that Definew n (z) = (a n (z)−a n (2), u n (z)) and writew n = (b n , w n ). Note that |dw n (z)| = |dũ n (z)|∀z ∈ C since the metric g 0 is R-invariant. This proves thatw nj is C 1 locbounded on C \ Γ sincew nj (2) ⊂ {0} × M . Elliptic estimates provide C ∞ loc -bounds. Thus we can assume, without loss of generality, that we can find a smoothJ- We split the remaining arguments into a few steps. STEP 1:w is not constant, all punctures in Γ are negative and ∞ is the unique positive puncture.
By Corollary 4.8 we assume, without loss of generality, that we can find sequences z nj → z * and r j (s) → 0 + such that lim sup This contradiction showsw is not constant. Assume again Γ = ∅, fix z * ∈ Γ and suppose, by contradiction, that it is a positive puncture. By Theorem 3.9 we find r > 0 small and a periodic Reeb orbit P * = (x * , T * ) such that This contradiction shows that z * is a negative puncture. If ∞ is also a negative puncture then E(w) < 0, which is impossible.
We fix a S 1 -invariant neighborhood W of the (discrete) set of S 1 -orbits
Proof of STEP 2. We can estimate we obtain h > 0 and j 0 such that the loop t → u nj (re i2πt ) is in W whenever j ≥ j 0 and r ≥ 2e h . By the path-connectedness of W 1 these loops can not leave W 1 .

STEP 3:
The curvew has non-degenerate asymptotics and is asymptotic to P at the puncture ∞.
Proof of STEP 3. We only deal with ∞, the other punctures are subject to analogous arguments. Use Theorem 3.9 to find c + ∈ R, r k → +∞ and an orbit Fix k large. Then for j large the loop u nj r k e i2πt is C ∞ close to w r k e i2πt and homotopic to x(T t). It follows that P + is contractible. Clearly T + ≤ E(w) ≤ T . Thus, by the assumptions of Theorem 2.2, P + is non-degenerate and belongs to P * . By Theorem 3.13w has non-degenerate asymptotic behavior at the puncture ∞ and the associated asymptotic Reeb orbit is P + . It follows from STEP 2 that P + = P .
By our assumptions on λ and P , the asymptotic limits at the punctures z ∈ Γ are non-degenerate orbits in P * , with periods ≤ T and indices µ CZ ≥ 3.
Proof of STEP 4. Write z 1 = ∞ and Γ = {z 2 , . . . , z N }. Supposew is asymptotic to P j = (x j , T j ) at the puncture z j . As remarked above, P 1 = P , {P 1 , . . . , P N } ⊂ P * and P j is non-degenerate ∀j. Moreover, max j T j ≤ T and µ CZ (P j ) ≥ 3 ∀j. Let U j be open neighborhoods of x j (R) and Z j be non-vanishing sections of ξ| Uj satisfying Z j | Pj ∈ β Pj . Let B be the non-vanishing section of w * ξ given by Lemma 4.12 and δ be its homotopy class. By Theorem 3.13 we can find R 0 ≫ 2 so that r ≥ R 0 implies π · ∂ r w(re i2πt ) = 0 and w(re i2πt ) ∈ U 1 . Let h > 0 be given by STEP 2 and suppose W 1 is small enough so that c ∈ W 1 ⇒ c(S 1 ) ⊂ U 1 . If we assume R 0 ≥ 2e h then w nj (re i2πt ) ∈ U 1 whenever r ≥ R 0 and j is large. Perhaps after making j larger, we can also assume π · ∂ r w nj (R 0 e i2πt ) = 0 ∀t ∈ S 1 because w nj → w in the sense of C ∞ on the set {|z| = R 0 }. Define l j and l by Then l = wind ∞ (w, δ, ∞). Note that Z 1 • w nj , only defined on {|z| ≥ R 0 }, extends to a non-vanishing section of w * nj ξ by the properties of the class β P1 = β P .
Consequently, ∀j ∃R j ≫ R 0 such that The Gauss-Bonet formula proves It follows that l j ≤ 1. We know that l j → l, proving l ≤ 1. If Γ = ∅ we can apply Lemma 4.15 to obtain a contradiction.
Proof of STEP 5. We use Lemma 4.6. Let p be a polynomial of degree k ≥ 1 satisfying p −1 (0) = Γ and letP = (x,T ) be a periodic orbit such thatw = f (x,T ) •p.
It follows from STEP 3 that (x, T ) = (x, kT ). Thus x =x and k = 1. In fact, if k ≥ 2 then T is not the minimal period of x, contradicting the fact that P is simply covered. Consequently p(z) = Az + D and Γ = {−D/A} ⊂ D for some A ∈ C, A = 0. It follows from Lemma 4.6 and STEP 3 thatw(C \ Γ) ⊂ R × x(R). If | − D/A| < 1 then we obtain the following contradiction Proof of STEP 6. Let {z j } be so that a nj (z j ) = inf C a nj → −∞. Suppose, by contradiction, that Γ = ∅. Then |dũ nj | is bounded on compact subsets of C. We claim that |z j | → +∞. If not we can assume, after selecting a subsequence, that we have an uniform bound |a nj (z j ) − a nj (0)| ≤ c for some c > 0. This proves Then the planes {ṽ n } satisfy the hypotheses of Lemma 4.16 with C = T , e = T − γ, r j = 2|z nj | −1 and N = 0, in view of the properties of the set Θ(H, P ). However, {ṽ n (1)} ⊂ {0} × M and this is in contradiction to Lemma 4.16. Assume Γ = ∅ and suppose, by contradiction, that A > −∞. First we claim that a nj (2) → +∞. If not we can assume, after selecting a subsequence, that ∃N < +∞ satisfying sup j a nj (2) < N . Choose z ∈ Γ. By STEP 1 z is a negative puncture. In view of the asymptotic behavior described in Theorem 3.13 and of Definition 3.11, This contradicts the definition of A, proving a nj (2) → +∞. We now claim that 0 ∈ Γ. If not then we find a smooth curve c : This is a contradiction since {a n (0)} is a bounded sequence (H is assumed compact) and a nj (2) → +∞. Hence 0 ∈ Γ. By STEP 4 we must have π · dw ≡ 0. By STEP 5 we conclude that 0 ∈ S 1 , absurd.

4.7.
End of the proof of Theorem 2.2. Suppose λ and P satisfy the assumptions of Theorem 2.2 and that H ⊂ R × M is compact. Take {ũ n } ⊂ Θ L (H, P ) and set A n = inf C a n . We can assume, after selecting a subsequence, that A n → A ∈ [L, +∞). By Lemma 4.17 we find a subsequence {ũ nj } and someũ . We already know ∃ũ ∈ Θ L (H, P ) and a subsequence {ũ nj } such thatũ nj →ũ in C ∞ loc . We must show thatũ is an embedding. It must be an immersion since wind π (ũ) = wind ∞ (ũ) − 1 = 0. Let ∆ be the diagonal in C × C and consider the set If D has a limit point in C × C \ ∆ then we find, using the similarity principle as in [26], a polynomial p : C → C of degree ≥ 2 and aJ-holomorphic map f : C → R × M such thatũ = f • p. This forces zeros of dũ, a contradiction. Thus D is closed and discrete in C × C \ ∆. By stability and positivity of intersections of pseudo-holomorphic immersions we find self-intersections of the mapsũ nj for large values of j if D = ∅. This is a contradiction since eachũ n is an embedding. We proved D = ∅ and Λ L (H, P ) is C ∞ loc -compact. The same reasoning as above shows The proof of Theorem 2.2 is complete.

Existence of fast planes
In this section we prove Theorem 2.4.

Special boundary conditions for the Bishop Family.
We will need special totally real boundary conditions for our Bishop family ofJ-holomorphic disks described in Subsection 2.4. (1) The goal of this subsection is to prove the above statement. We start with some technical lemmas. Let sp(1) denote the Lie algebra of Sp(1).
The case det T > 0 is similar and the claim is proved.
We continue the proof of the lemma considering the case det T < 0. Then Maslov(N ) = k. Using e tY = T −1 e itγ T we compute The conclusion follows by noting thatỸ := T −1 i(γ + 2πk)T is another logarithm of ψ(1). The case det T > 0 is treated similarly. The eigenvalues ofỸ are ±i( √ det Y + 2πk) ∈ i2πZ, and we still have detỸ > 0.
Before starting with the proof, we fix the notation and make some initial constructions. Let U , D 0 and P be as in Proposition 5.1. Perhaps after making U smaller, we can find a Martinet tube where the smooth function f > 0 satisfies f (θ, 0, 0) ≡ T and df (θ, 0, 0) ≡ 0. Note that T is the minimal positive period of x by assumption. The map t → x(T t) is represented by t → (t, 0, 0). We still denote by R the Reeb vector in the local coordinates (θ, x, y). The proof of Proposition 5.1 is based on the following lemma. Lemma 5.3. Let π 0 : R/Z × R 2 → R 2 be the projection onto the second factor. There exists ǫ > 0 and a smooth embedding h : (1−ǫ, 1]×R/Z → R/Z×B satisfying Proof. We denote the Reeb flow by φ t and assume, for simplicity and without loss of generality, that T = 1. In the local coordinates (θ, x, y) introduced above we have R(θ, 0, 0) = (1, 0, 0), φ t (θ 0 , 0, 0) = (t + θ 0 , 0, 0) and (1)) with respect to the splitting T (R/Z × R 2 ) = R ⊕ R 2 . We used that the linearized Reed flow preserves the splitting T M = RR ⊕ ξ. The formula and prove From now on we work in these new coordinates, obtained by pushing forward with G. We still denoted them by (θ, x, y) without fear of ambiguity. The Reeb vector is still denoted by R and its flow by φ t . The above equations imply DR(θ, The case det Y = 0 is ruled out since P is non-degenerate. Let k = Maslov(M ) and consider the smooth 2π-periodic function (24) g of the real variable ϑ. We split the proof in two cases.

The Bishop
Family. From now on we suppose λ is a contact form on a 3-manifold M and P is a non-degenerate, unknotted and simply covered periodic Reeb orbit. Following [15], we construct a Bishop family ofJ-holomorphic disks in the symplectization R × M . We orient x(R) along the Reeb field and let D 0 ⊂ M be an embedded disk with ∂D 0 = x(R), orientations included. By Proposition 5.1 we obtain another embedded disk D 1 spanning the orbit P with special properties near the boundary. These properties will be crucial for the proof of Theorem 2.4.
If h : D → M is a smooth embedding such that h(D) = D 1 , we will consider the transverse unknot l and the disk F given by where 0 < ǫ ≪ 1. We orient l so that λ| T l > 0, and F accordingly. If ǫ is small enough then sl(l, F ) = sl(P, D 1 ) and ξ| p = T p D 1 , ∀p ∈ D 1 \ F .
Generically these are lines since ξ is maximally non-integrable, except at the socalled singular points, where ξ = T F . Given a smooth function H on a neighborhood of F , having F inside a regular level set, the equations define a vector field V tangent to both F and the contact structure ξ| F . The zero set of V is precisely the singular set of ξ ∩ T F . Clearly V does not vanish over ∂F = l since l is transverse to ξ. All these facts are standard, see [15] and [16]. The integral lines of V define the so-called characteristic singular foliation of F .

5.2.2.
A convenient spanning disk for P . Let dvol be a smooth 2-form on D 1 defining the orientation induced by the Reeb vector along ∂D 1 = x(R). We have two symplectic bundles over F , namely (T F, dvol) and (ξ| F , dλ), and V is a section of both. Perhaps after changing H by −H in (33), we can assume V points out of F at ∂F . One can, as done in [15] and [16], use the topological information of both bundles in order to understand the zero set of V . The singular distribution (32) is said to be Morse-Smale if so is V . One can show, see [9], that F can be C ∞ -perturbed, keeping ∂F fixed, so that its characteristic distribution becomes Morse-Smale. This perturbation can be arbitrarily C ∞ -small. V becomes a non-degenerate section of T F and, consequently, also of ξ| F . We assume this is done and examine a zero p ∈ F of V . Let o and o ′ be the orientations of T F and ξ|F induced by dvol and dλ respectively. The zero p has two associated numbers ǫ and ǫ ′ (both equal to ±1), namely, the intersection numbers of V with the zero sections 0 T F and 0 ξ of T F and ξ| F , respectively. Let a 1 and a 2 be the two eigenvalues of the linearization DV | p ∈ GL(T p F = ξ| p ). The zero p is elliptic if a 1 a 2 > 0, or hyperbolic if a 1 a 2 < 0. An elliptic point is nicely elliptic if the eigenvalues are real. These notions relate to the bundle T F . If p is elliptic then ǫ = 1, if p is hyperbolic then ǫ = −1. Now, following [9], we relate them to the bundle ξ| In view of the formula (4) for the self-linking number, and of standard degree theory (see the proof of Proposition 2.1), one has where the sums are taken over the (non-degenerate) zeros of V . We used the fact that V points in the outward normal direction of the disk F at its boundary and hence pushes it off from F . We state the following proposition which can be extracted from [15].
Proposition 5.4 (Giroux and Hofer). The disk F can be smoothly perturbed, keeping l = ∂F fixed, so that its singular characteristic distribution satisfies the following properties.
(2) All its elliptic points are nicely elliptic. It is crucial for the above statement that ξ is tight, so that no closed leaves arise when perturbing F . The proof that h + = e − = 0 is carried out in section 3 of [16], more precisely, they show that (5) is implied by (1)- (4).
From now on we assume the disk F was perturbed using Proposition 5.4. The compact strip S := (D 1 \ F ) ∪ ∂F has two boundary components, namely x(R) and ∂F . Recall that T p S = ξ| p , for every p ∈ S. Since the perturbation using the above lemma can be arbitrarily C ∞ -small near ∂F , we can obtain a smooth embedded disk, still denoted by D 1 , constructed by joining F with S along ∂F , and smoothing it out near ∂F . This smoothing process clearly has support arbitrarily near ∂F . The singular distribution ξ ∩ T D 1 has the same singular points as F . If sl(P, D 0 ) = −1 then sl(l, F ) = −1, by the invariance of the self-linking number under homotopy through transverse knots. Equations (34) and the above construction imply e + = 1 and h − = 0. This follows from e − = h + = 0. Summarizing, we proved Theorem 5.5. Let M be a closed 3-manifold with a tight contact form λ. Let P = (x, T ) be a non-degenerate, unknotted, simply covered, periodic Reeb orbit. Suppose there exists an embedded disk D 0 ⊂ M satisfying ∂D 0 = x(R) and sl(P, D 0 ) = −1. Then there exists an embedded disk D 1 spanning P , arbitrarily C 0 -close to D 0 , such that the singular characteristic distribution ξ∩T D 1 has precisely one positive, nicely elliptic singular point. In addition, there exists a neighborhood O ⊂ D 1 of ∂D 1 such that R p ∈ T p D 1 for every p ∈ O \ ∂D 1 , where R is the Reeb vector associated to λ.

One last perturbation. Let
Consider the set Then X is closed in the complete metric space C ∞ (D, M ) endowed with the C ∞ topology. Hence it is also a complete metric space. For a fixed non-trivial periodic Reeb trajectory y : R → M we define By the definition of δ and the properties of D 1 we have It is easy to show X c y is open and dense in X if y(R) = x(R). Let us assume every contractible closed Reeb orbit P ′ = (x ′ , T ′ ) is non-degenerate. There are only countably many such P ′ . It follows from Baire's category theorem that is dense in X. Hence, by an arbitrarily small C ∞ -perturbation supported away from ∂D 1 , we may assume that our disk D 1 contains no periodic Reeb orbits other than x(R).

5.2.4.
Filling byJ-holomorphic disks. We now recall a construction done by Hofer, Wysocki and Zehnder in [15] and [16]. Let M , ξ, λ and P satisfy the hypotheses of Theorem 5.5. Let e be the (unique) singular point of the characteristic foliation of D 1 and denote D * 1 = D 1 \{e}. Recall that e is a nicely elliptic singularity. Following [16], consider for each J ∈ J (ξ, dλ| ξ ) the boundary value problem Here J (ξ, dλ) denotes the set of dλ-compatible complex structures on ξ andJ is given by (5).
Theorem 5.6 (Hofer, Wysocki and Zehnder). Let the closed 3-manifold M be equipped with a tight contact form λ. Assume every contractible closed Reeb orbit P is non-degenerate, c 1 (ξ) vanishes alongP and µ CZ (P ) ≥ 3. Let P = (x, T ) be an unknotted, simply covered periodic Reeb orbit satisfying sl(P, disk) = −1. Suppose D 1 is an embedded disk spanning P such that its characteristic foliation has precisely one positive nicely elliptic singular point e, and that D 1 \x(R) contains no periodic orbits. There exists J ∈ J (ξ, dλ| ξ ) for which the following holds. The disk D 1 can be smoothly perturbed on an arbitrarily small neighborhood of e so that e still is the only (nicely elliptic) singularity of the characteristic foliation. Moreover, there exists a smooth 1-parameter family {ũ t = (a t , u t )} t∈(0,1) ⊂ M(J) satisfying: (1)ũ t converges in C ∞ to the constant map (0, e) as t → 0.
(2) There exists η > 0 such that Here the norms are taken with respect to euclidean metric on D and to any R-invariant riemannian metric on R × M .
Here Möb(D) denotes the group of holomorphic diffeomorphisms of D. The arguments for proving Theorem 5.6 are very delicate and we refer to [16] for details.

5.3.
Obtaining the fast plane. Suppose now we are under the assumptions of Theorem 2.4. Let D 1 be the smooth embedded disk spanning the orbit P = (x, T ) obtained by Theorem 5.5. As explained in Subsection 5.2.3 we can assume, in addition and without loss of generality, that D 1 contains no periodic Reeb trajectories other than x(R). Consequently we can apply Theorem 5.6 to obtain a 1-parameter family {ũ t } t∈(0,1) of solutions of (35). Select a sequence t n → 1 + and denotẽ u tn =ũ n = (a n , u n ). Theorem 5.6 tells us that, after selecting a subsequence, we can assume there exists g n ∈ Möb(D) such thatũ n • g n → f P in C ∞ loc (D \ {0}, R × M ). Replacingũ n bỹ u n • g n we assume g n = id ∀n.
In the following we denote f P (z) = (d, w) ∈ R × M and follow [18] closely. The bubbling-off point 0 has a well-defined mass m(0) = lim (37) a n (z n ) = inf D a n then z n → 0. Choose δ n > 0 so that where γ is the constant (6). It follows easily from the definition of m(0) that δ n → 0 + . Now definẽ v n (z) = (b n (z), v n (z)) := (a n (z n + δ n z) − a n (z n + 2δ n ), u n (z n + δ n z)) for z ∈ B δ −1 n (1−|zn|) (0). By standard arguments, we knoŵ Γ := {z ∈ C : ∃ζ n → z such that |dṽ n (ζ n )| → ∞} is finite andΓ ⊂ D. Moreover, there is a subsequence, still denoted by {ṽ n }, with uniform gradient bounds on compact subsets of C \Γ. Thus we have C 1 loc -bounds sinceṽ n (2) ∈ {0}×M . Standard elliptic boot-strapping arguments give C ∞ loc -bounds for the sequenceṽ n on C \Γ. A particular subsequence, again denoted {ṽ n }, must have aJ-holomorphic limit The following important lemma can be extracted from [16]. Note that very similar arguments were used to prove Lemma 4.17 above.
Let π : T M → ξ be the projection along the Reeb direction. If π · dv ≡ 0 then we can apply Lemma 4.6 to find a non-constant polynomial p and a periodic orbitP = (x,T ) such that p −1 (0) =Γ andṽ = f (x,T ) • p. Here f (x,T ) (e 2π(s+it) ) = (T s,x(T t)). Let k := deg p. Sinceṽ is asymptotic to the orbit P at the unique positive puncture ∞, we must have (x, T ) = (x, kT ). Thus x =x and k = 1. In fact, if k ≥ 2 then T is not the minimal period of x, contradicting our assumptions on P . Consequently we must have p(z) = Az + D andΓ = {−D/A} ⊂ D for some A ∈ C * . Thus D = 0 since 0 ∈Γ. We now get the following contradiction Remark 5.8. Further bubbling-off analysis would reveal an entire bubbling-off tree. The first level of this tree has only one vertex representing the sphereṽ. All this is showed in [18] via the so-called "soft-rescaling". We shall not make any use of these facts.
We fix a small tubular neighborhood U of x(R) and a non-vanishing section (40) Z : U → ξ| U satisfying x * T Z ∈ β P . Here β P is the special homotopy class of non-vanishing sections of x * T ξ discussed in Remark 3.3. Theorem 5.6 tells us that the sequence of loops t → u n (e i2πt ) converges in C ∞ to x T . Thus u n (S 1 ) ⊂ U for n ≫ 1. By Lemma 3.4 the sections t → Z • u n (e i2πt ) extend to non-vanishing sections (41) Z n : D → u * n ξ.
As usual, π : T M → ξ denotes the projection along the Reeb direction.
Lemma 5.9. If n is large enough then the sections π · du n do not vanish on D.
Proof. In view of Theorem 5.5 there exists a neighborhood O ⊂ D 1 of ∂D 1 such that R p ∈ T p D 1 for every p ∈ O \ ∂D 1 . There exists n 0 ∈ Z + such that n ≥ n 0 implies u n (S 1 ) ⊂ O. We can, of course, assume O ⊂ U . From now on we consider only n ≥ n 0 . For every z ∈ S 1 the linear map π · du n (z) does not vanish. In fact, π · du n (z) has rank 0 or 2, since it satisfies (42) π · du n (z) · i = J · π · du n (z).
However, by the strong maximum principle, we have λ(u n (e i2πt )) · ∂ θ u n (e i2πt ) > 0. This is a contradiction. We proved The section V of ξ| D1 has a unique simple positive zero inside D 1 . By Lemma 3.4 the section Z| ∂D1 extends to a non-vanishing section of ξ| D1 . It follows from standard degree theory that (45) wind(t → V (x(T t)), t → Z(x(T t)), J) = 1.
Consequently wind t → V • u n e i2πt , t → Z • u n e i2πt , J → 1, proving (46) wind t → V • u n e i2πt , t → Z • u n e i2πt , J = 1 if n ≫ 1, because winding numbers are Z-valued and u n e i2πt converges in C ∞ to x(T t). Now recall the non-vanishing sections Z n of u * n ξ (41) and compute, for n ≫ 1, wind t → π · ∂ x u n e i2πt , t → Z n e i2πt , J = wind t → π · ∂ x u n e i2πt , t → π · ∂ θ u n e i2πt , J The last line follows from (43), (44) and (46). This proves the algebraic count of zeros of the section π · ∂ x u n on D is zero. Since zeros count positively (this follows from (42)) we conclude π · ∂ x u n never vanishes on D, if n ≫ 1.
We writeΓ = {z 2 , . . . , z N } and z 1 = ∞. The mapṽ was obtained by bubbling-off analysis of the disksũ n , following a standard procedure described in [18]. Further bubbling-off analysis would reveal an entire bubbling-off tree. As is well-known, it follows from this procedure that there exist contractible periodic orbits {P j = (x j , T j )} j=1...N such thatṽ is asymptotic to P j at z j as in Definition 3.11. P 1 = P by Lemma 5.7. We used the ongoing assumption that every contractible orbit is non-degenerate. We know P j ∈ P * ∀j, by hypothesis. There are distinguished homotopy classes β j = β Pj ∈ S Pj induced by capping disks for the maps x j Tj , as explained in Remark 3.3. We can find, see Remark 3.6, small open neighborhoods U j of x j (R) and non-vanishing sections Z j of ξ| Uj such that x j * Tj Z j ∈ β j . Here β 1 is the class β P from (40) and we set Z 1 = Z.
By Proof. We can assume t → v(Re i2πt ) converges to t → x(T t) in C ∞ , as R → +∞. Let z = re i2πθ be polar coordinates centered at 0, and z − z n = ρe i2πϕ be polar coordinates centered at z n . Here z n is the sequence defined in (37). There exists R 0 ≫ 1 such that r ≥ R 0 implies v(re i2πθ ) ∈ U , π · dv(re i2πθ ) = 0 and B(re i2πθ ) = Z • v(re i2πθ ). This follows from Lemma 5.7, Theorem 3.13 and from the asymptotic behavior described in Definition 3.11. We compute for R ≫ R 0 wind π · d dρ ρ=Rδn u n z n + ρe iϕ , Z • u n z n + δ n Re iϕ , J .
All windings are taken with respect to angular variables.
Here Z n are the sections (41).
Proof of CLAIM. If n ≫ 1 then the loop ϕ → u n z n + Rδ n e iϕ = v n Re iϕ is arbitrarily C ∞ -close to the loop ϕ → v Re iϕ , which can be made arbitrarily C ∞close to the loop t → x(T t) if R is fixed large enough. By Lemma 3.4 the section ϕ → Z u n z n + Rδ n e iϕ extends to a non-vanishing section of u * n ξ| {|z−zn|≤Rδn} . Consequently it does not wind with respect to ϕ → Z n z n + δ n Re iϕ .
Letṽ be the finite-energy sphere obtained from Lemma 5.7. IfΓ = ∅ then Lemma 5.10 and Lemma 4.15 provide a contradiction to µ CZ (P j ) ≥ 3 ∀j ≥ 2. We showedΓ = ∅. Consequentlyṽ is a finite-energy plane satisfying wind ∞ (ṽ) ≤ 1. By Lemma 3.19 we must have wind ∞ (ṽ) = 1. By Lemma 5.7,ṽ asymptotic to the orbit P at the puncture ∞. We must now show thatṽ is an embedding. To that end we argue as in Subsection 4.7. The mapṽ must be an immersion since wind π (ṽ) = wind ∞ (ṽ) − 1 = 0 implies π · dv does not vanish. Thus, if it is not an embedding, we find self-intersections. Let ∆ be the diagonal in C × C and consider If D has a limit point in C × C \ ∆ then we find, using the similarity principle as in [26], a polynomial p : C → C of degree at least 2 and aJ-holomorphic map f : C → R × M such thatṽ = f • p. This forces zeros of dṽ, a contradiction. Thus D is closed and discrete in C × C \ ∆. By stability and positivity of intersections of pseudo-holomorphic immersions we find self-intersections of the mapsũ n for large values of n if D = ∅. This is a contradiction since eachũ n is an embedding. Theorem 2.4 is proved.

Fredholm theory
Our goal in this section is to prove Theorem 2.3.
6.1. The asymptotic analysis for∂ 0 . In order to understand the asymptotic behavior of finite-energy surfaces near non-removable punctures we need the fundamental analytical tools from [11]. The next three statements can not be explicitly found in the literature, but the proofs are completely contained in [11], see also [14] and the appendix of [2]. We include the arguments in the appendix for completeness.
We can write Y = Dz where D(θ, z) We refer to [11] for more details.
Lemma 6.4. Ifũ satisfies (55) then The proof is exactly the same as that of Lemma 2.4 from [11] and is omitted. We need one more computation: The values of f and its partial derivatives are evaluated at w(s, t) = Ψ • u(s, t).
Lemma 6.5. Supposeũ satisfies (55). Thenũ has non-degenerate asymptotics as in Definition 3.12 if, and only of, ∃b > 0 such that e bs |z(s, t)| is bounded. In this case ∃r > 0 such that The technical proof is postponed to the appendix. The arguments are essentially found in section 4 of [11]. We include them here since Lemma 6.5 is not proved in [11] and is crucial for our results.
Here we denoted by ·, · the standard euclidean inner-product on R 2 . In fact, it follows from (52) that The proof of the following corollary of Theorem 6.1 is found in [11], and is included in the appendix for completeness. Corollary 6.6. If the finite-energy planeũ has non-degenerate asymptotic behavior at ∞ either z(s, t) ≡ 0 or it has the form Proof. The lemma is an easy consequence of (61) and (66) since |∆| = o(|z(s, t)|) as s → +∞.
Given any β ∈ S P we recall the winding numbers (µ, β) ∈ Z associated to eigenvalues of A P . Those were defined in [12] and are discussed in section 4. Lemma 6.8. If the finite-energy planeũ = (a, u) has non-degenerate asymptotics and if β Ψ P = βũ then wind ∞ (ũ) = (µ, β Ψ P ). Here µ < 0 is the negative eigenvalue of A P given by an application of Corollary 6.6 to the function z(s, t).
is purely imaginary and the real part r of u ′ (iu) −1 is equal to −µ > 0. However r is the derivative of the argument of u. It follows that γ ∞ intersects the embedded surface h (R × R) transversely at time t.
As consequence we obtain (transverse) intersections of γ n with the surface h n for n large enough, proving that φ t (q) intersects the surface u(C) for t > t n .
T 0 is the minimal positive period of x 0 sinceũ is fast. We writeũ(s, t) = (a(s, t), u(s, t)) =ũ e 2π(s+it) and assume, without loss of generality, that 6.3.1. Normal coordinates atũ. We work on a Martinet tube Ψ (12) defined on an open neighborhood U of the set x 0 (R). We assume β Ψ P = βũ and use all the notation from Subsection 6.2. Define the complex structureĴ on R × S 1 × B by The basis {e 1 , e 2 } is a dλ-symplectic frame for ξ| U .Ĵ is represented by is the matrix (54). Note that (70) is independent of the first coordinate. For s large enough we have a well-defined map w(s, t) := (id R × Ψ) (ũ(s, t)) = (a(s, t), θ(s, t), z(s, t) = (x(s, t), y(s, t))).
It follows from Lemma 6.5 thatĴ(s, t) :=Ĵ(w(s, t)) satisfies On R × M we have projections π R : R × M → R and π M : R × M → M . The metric g 0 (16) is R-invariant andJ is a pointwise isometry with respect to g 0 . Fix two non-vanishing smooth sections n and m of ξ| U such that m = Jn. We can assume in addition that {n, m} is dλ-symplectic since GL(1, C) is homotopy equivalent to U (1). Thenñ = π * M n andm = π * M m are smooth sections of π * M ξ over R × U satisfyingJñ =m. Define The bundle with fibers E (s,t) = π * M ξ|ũ (s,t) is smooth,J-invariant and transverse to Tũ over [s 0 , +∞) × S 1 for a fixed s 0 > 0 large. Thus we can assume there exists a smoothJ-invariant subbundle L ofũ * T (R × M ) such that (1) L = E over points z = e 2π(s+it) with s ≥ s 0 + 1.
Here Nũ is the normal bundle toũ with respect to the metric g 0 . Nũ is alsõ J-invariant. Standard degree theory shows that there is precisely one homotopy class β N ∈ S P with the following property: the sectionñ extends to a smooth non-vanishing section of L if, and only if, the section t → n(x 0 (T 0 t)) is in class β N . From now on we assume this is the case and thatñ can be extended. We extend m bym =Jñ. The following non-trivial identity is proved in [13] (73) wind(β N , βũ, J) = +1.
where exp is the exponential map associated to g and B ′ is a small open ball around 0 ∈ R 2 . We need to examine the map Φ in more detail. Let us define By the asymptotic behavior ofw,n andm there exists b > 0 such that where F ∞ (s, t, v) = (T 0 s, t, v 1n∞ (t)+v 2m∞ (t)). We used Lemma 6.5. In particular, is independent of s. DefineJ := Φ * J on C × B ′ and write Note that Proof. The lemma follows easily from the formulā and the estimates (76) and (77), whereĴ is the matrix (70). It is only important to note thatĴ is defined on R × S 1 × B and is independent of the first variable.
6.3.2. The non-linear Cauchy-Riemann equations in normal coordinates. We look for smooth maps z → v(z) such that z → Φ(z, v(z)) has aJ-invariant tangent space. This is equivalent to In view of (80) one can write where the prime denotes a derivative with respect to the v-variable. Analogously we write Plugging (83) and (84) into (82) we find Note that C(z) and W (z, v, L) are linear maps R 2 → R 2×2 for fixed (z, v, L), and (85) is an identity on 2 × 2 matrices. LetC(s, t) be the 2 × 2-matrix given byC(s, t)u = [C(e 2π(s+it) )u]∂ t , and write W (s, t, v, L) = W (e 2π(s+it) , v, L). Then W (s, t, 0, 0) ≡ 0 and, by Lemma 6.11, where K is any compact subset of B ′ ×R 2×2 , b > 0 and C ∞ (t) is some smooth matrix loop. Later we will need the following statement which is an easy consequence of the chain rule and the asymptotic behavior (86). The following important lemma is proved in [13].
is the linearized Reeb flow restricted to ξ along t → x 0 (T 0 t) represented in the basis {n, m}. In other words, the differential operator represents the asymptotic operator A P in the frame {n, m}.
H is smooth in the following Banach space set-up defined in [13].
Thus ζ never vanishes or vanishes identically. If there are 3 linearly independent sections in ker DH(0) then a linear combination of them would have to vanish at some point, which is impossible by the above discussion. This proves that DH(0) is surjective. 6.3.6. Consequences of the implicit function theorem. We fix δ < 0 as in (87)  Writing v(τ )(z) = v(τ, z), the maps z → Φ(z, v(τ, z)) are not necessarilyJholomorphic. This is taken care of in the appendix of [13]. Analyzing a suitable Fredholm problem, fixing 0 < ǫ < 2π and possibly making U smaller, it is possible to find a smooth function with the following properties: is a diffeomorphism of C, and if we define then the maps f (·, τ ) areJ-holomorphic. For τ ∈ U small, t ∈ S 1 and s ≫ 1 we can define smooth functions S = S(τ, s, t) ∈ R and T = T (τ, s, t) ∈ S 1 by (92) e 2π(S+iT ) = ψ τ (e 2π(s+it) ).
6.3.8. Further consequences of the asymptotic analysis. Fix τ ∈ U and ζ in the kernel of DH(v(τ )). We can write . As a consequence of the definition of the space C l,α,δ 0 (C, R 2 ) and of (86) we have where W τ (s, t) = W τ e 2π(s+it) , for some d > 0. Since we chose l ≥ 4, these estimates and Lemma 6.13 allow us to apply Theorem 6.1 and find that either v(τ ) vanishes identically or The definition of the space C l,α,δ 0 (C, R 2 ) forces η < δ andη < δ. The maps v(τ ) and ζ satisfy perturbed Cauchy-Riemann equations. By the similarity principle, they have only isolated zeros or vanish identically, see [9]. Moreover, each zero counts positively in the algebraic count of the intersection number with the zero map z → (z, 0). The above asymptotic behavior tells us that if v τ or ζ do not vanish identically then they do not vanish near ∞. Standard degree theory implies Thus v(τ ) and ζ never vanish or vanish identically. This has important consequences. The map is an immersion. In fact, DG(z, 0) is non-singular ∀z ∈ C since non-zero elements ζ ∈ ker DH(0) never vanish. Now fix any τ ∈ U. Applying the implicit function theorem and doing the same analysis centered at a given fast plane with image Π τ := {Φ(z, v(τ )(z)) : z ∈ C} we conclude that DG(z, τ ) is also non-singular ∀z ∈ C. Analogously one shows that G is 1-1. In fact, if τ = 0 then since v(τ ) has no zeros. This shows that v(τ )(z 1 ) = v(0)(z 2 ) for all (z 1 , z 2 ) ∈ C 2 . Doing the same analysis by applying the implicit function theorem centered at a given map fast plane with image Π τ0 we conclude that v(τ 0 )(z 1 ) = v(τ 1 )(z 2 ) for all (z 1 , z 2 ) ∈ C 2 when τ 0 = τ 1 .
We proved that the map f in (91) is an embedding. This concludes the proof of Lemma 6.10. We collect a useful lemma that follows from our arguments so far.
The proof is technical but straightforward, we only sketch it here.
Sketch of Proof. Consider the map Φ in (74). Clearly we can find v n : and v n → 0 in C ∞ loc . It follows from wind ∞ (c n ·ũ) = 1 that v n ∈ C l,α,δ 0 (C, R 2 ). A long and technical argument, again using the very particular fact that c n ·ũ are translations ofũ, shows that v n → 0 in the space C l,α,δ 0 (C, R 2 ). The uniqueness statement in the implicit function theorem concludes the proof.
Proof. It follows from wind π (ũ) = wind ∞ (ũ) − 1 = 0 that π · du does not vanish. Hence u and x only intersect transversely. Let us assume, by contradiction, that u(C) ∩ x(R) = ∅. In view of Definition 3.11, the map u has self-intersections and we find intersections of the planesũ and c ·ũ for c ≫ 1. Here c ·ũ denotes the translation of the planeũ in the R-direction of R × M by c, as explained in Remark 3.10. Consider, for each c > 0, the closed set A c can not accumulate in C×C since otherwise one could argue, using the similarity principle as in [26], thatũ(C) = (c ·ũ)(C). It is not hard to show ∃R 0 ≫ 1 such that This follows essentially from Lemma 6.5 since P is simply covered. It follows from (100) that given any ǫ > 0 one finds a compact K ⊂ C such that A c ⊂ K × K for every c ≥ ǫ. This is so since intersections of the planeũ with any of its translations c ·ũ induce self-intersections of the map u. We can now use the homotopy invariance of intersection numbers together with positivity of intersections of pseudo-holomorphic maps to conclude thatũ intersects c ·ũ for any c > 0. Let f : C × B r (0) → R × M be the embedding (91) obtained by the implicit function theorem. Choose c n → 0 + . By Lemma 6.20 ∃τ n → 0 satisfying τ n = 0 and (c n ·ũ) (C) = f (C, τ n ) for n large enough. This is an absurd because f is 1-1 and c n ·ũ intersectsũ for every n. Proof. It is a straightforward consequence of lemmas 6.23 and 6.24 that either u(C)∩v(C) = ∅ or u(C) = v(C). Suppose u(C) = v(C). One finds a diffeomorphism ϕ : C → C such that v = u•ϕ since both u and v are embeddings of C into M \x(R). Let s + it be a complex parameter on C and compute The condition wind π (ũ) = 0 implies that π · du is a linear isomorphism from T z C to ξ| u(z) for every z ∈ C. Hence so is π · (du • ϕ). We conclude from the above computation that ϕ s + iϕ t = 0, that is, ϕ is a biholomorphism. It must have the form ϕ(z) = Az + B. The Cauchy-Riemann equations We now complete the proof of Theorem 2.3. Let f be the C l -embedding (91) and write f = (h, g) ∈ R × M . Fix τ 0 ∈ B r (0). Letṽ n = (b n , v n ) be a sequence of embedded fast finite-energy planes with µ(ṽ n ) = µ andṽ n → f (·, τ 0 ) in C ∞ loc . We find (B n , τ n ) → (0, τ 0 ) such thatṽ n (0) = f (B n , τ n ). By Lemma 6.25 we find c n ∈ R, A n → 1 such that (c n ·ṽ n )(z) = f (A n z + B n , τ n ) ∀z ∈ C since v n (0) = g(B n , τ n ). Note that c n = h(B n , τ n ) − b n (0) = 0. The conclusion follows.

Open book decompositions
In this section we prove Theorem 2.5. Recall the families Λ(H, P ) in (8) and the C ∞ loc -closed subfamilies Λ k (H, P ) defined in (11). From now on we assume the contact form λ and the periodic orbit P satisfy the hypotheses of Theorem 2.5. Proof. The sets u(C) and v(C) intersect at the point q ∈ M . Lemma 6.25 provides constants A ∈ C * and B ∈ C such thatũ(Az + B) =ṽ(z) for all z ∈ C. The definition of Λ({(r, q)}, P ) implies B = 0 and A ∈ S 1 .
Letx : R → M be a Reeb trajectory. We denote for some t ∈ R and k ≥ 3. In the following I r (a) denotes the open interval (a − r, a + r).
Proof. By the inverse function theorem we can find ρ > 0 and ǫ > 0 small so that is a C 1 -embedding onto an open set of M since wind π (ũ(t 0 , ·)) = 0 and u(t, 0) = x(t). We now argue indirectly. Take sequences t n , t * n → t 0 and z n , z * n ∈ C such that (t n , z n ) = (t * n , z * n ) and u(t n , z n ) = u(t * n , z * n ). By lemmas 6.22 and 6.25 we know that t n = t * n implies z n = z * n . Consequently we must have t n = t * n . Again by Lemma 6.25 we find a sequence {ζ n } ⊂ C \ {0} such that u(t n , ζ n ) =x(t * n ) since u(t n , C) = u(t * n , C) ∀n. The sequence {ζ n } is bounded since the compact sets x(R) andx I ǫ (t 0 ) do not intersect. Suppose lim inf |ζ n | = 0. Then we find a subsequence ζ nj → 0 as j → +∞. For j large enough the points (t nj , ζ nj ) and (t * nj , 0) are distinct points in I ǫ (t 0 ) × B ρ (0) and satisfy u(t nj , ζ nj ) =x(t * nj ) = u(t * nj , 0). This is in contradiction to the injectivity of the map (102). This proves lim inf n |ζ n | > 0. After selecting a subsequence we can assume ζ n → ζ * = 0. Consequently u(t 0 , ζ * ) = u(t 0 , 0) =x(t 0 ). However, u(t 0 , ·) is an embedding by Lemma 6.22. This is a contradiction.
(3) Supposeṽ n are embedded fast finite-energy planes asymptotic to P at ∞, as in definition 3.11. Ifṽ n →ũ 0 in C ∞ loc and µ(ṽ n ) = k ∀n then, for n large enough, one finds {A n } ⊂ C \ {0}, {B n } ⊂ C, {r n } ⊂ R and {t n } ⊂ R such that A n → 1, B n → 0, r n → 0, t n → t 0 and u(t n , z) = (r n ·ṽ n ) (A n z + B n ) ∀z ∈ C.
Proof. We split the proof into two steps.

STEP 2:
The map u is an immersion.
The following statement is an easy consequence of lemmas 7.3 and 7.4. Lemma 7.5. Suppose I is an open real interval and l ≥ 1. Letũ = (a, u) : I × C → R × M be a C l map such thatũ(t, ·) ∈ Λ k t ∀t ∈ I for some k ≥ 3. Then ∀t ′ ∈ I ∃ǫ > 0 such that u : Later we will need the following claim.
Proof. Set U := M \ (u 0 (C) ∪ x(R)). Clearly U is open and connected. It follows from lemmas 6.24 and 6.23 and from the definition of L that u ((0, L) × C) ⊂ U. We claim u ((0, L) × C) is closed in U. In fact, suppose the sequence y n satisfies ∀n∃t n ∈ (0, L) such that y n ∈ u(t n , C) and y n → y for some y ∈ U. One finds a unique sequence B n ∈ C such that u(t n , B n ) = y n .
Let W be a S 1 -invariant neighborhood of the (discrete) set of S 1 -orbits {c ∈ C ∞ (S 1 , M ) : ∃P = (x,T ) ∈ P * such that c ∈P andT ≤ T } We can assume no connected component of W contains loops in distinct classes of C ∞ (S 1 , M )/S 1 . Let W P be the component containing the class P . We can assume, without loss of generality, that c ∈ W P ⇒ c(S 1 ) ∩ {y n } = ∅. Consider the cylinders Z t (s, ϑ) :=ũ t, e 2π(s+iϑ) for t ∈ [0, L] and set s 0 = (2π) −1 log 2. The definition of Λ k t implies that . Applying Lemma 4.11 we find r 0 ≫ 1 such that |z| ≥ r 0 ⇒ũ(t, z) ∈ {y n } ∀t ∈ [0, L]. It follows that sup n |B n | < ∞. We can assume B n → B for some B ∈ C and t n → t * for some t * ∈ [0, L]. If t * = 0 then y n = u(t n , B n ) → u(0, B) contradicting y ∈ u 0 (C). Hence t * = 0. Analogously t * = L and we have t * ∈ (0, L). It follows that y n = u(t n , B n ) → u(t * , B) and (t * , B) ∈ (0, L) × C. We proved y ∈ u ((0, L) × C).  Proof. Suppose (t 0 , z 0 ) = (t 1 , z 1 ) are points of (0, L) × C satisfying u(t 0 , z 0 ) = u(t 1 , z 1 ). It follows from lemmas 6.24 and 6.23 that u(t 0 , C) = u(t 1 , C). In view of Lemma 6.22 every u(t, ·) is 1-1. Thus t 0 = t 1 implies z 0 = z 1 . Consequently we may assume 0 < t 0 < t 1 < L. In view of Lemma 7.5, we can cover [0, L] by finitely many open intervals {I k } such that u : I k × C → M is an embedding for each k. Examining the Lebesgue number of this cover we find η > 0 such that u : F × C → M is an embedding whenever F is a subinterval of [0, L] of length at most η. It follows that t 0 ≤ t 1 − η.
The proof of Lemma 7.7 is complete.

7.2.2.
Reparametrizing the foliation. We now glue the foliatioñ given by Lemma 7.7 near the ends to produce a closed S 1 -family of planes.
We finally defineŨ : By constructionŨ is C l since F (t) = t − L near L. EachŨ (t, ·) is an embedded fast finite-energy plane asymptotic to P at the positive puncture ∞. It follows that U satisfies (1) and (2) of Theorem 2.5.

7.2.3.
Each page is a global surface of section. LetŨ = (a, u) be the map (108). We claim that ∀s ∈ R/LZ and ∀p ∈ M \ x(R) we find sequences t ± n such that t + n → +∞, t − n → −∞ and φ t ± n (p) ∈ u(s, C).
Let p ∈ M \ x(R) be arbitrary, denote by y 0 : R → M be the Reeb trajectory satisfying y 0 (0) = p. We have to prove that the sets C + n = {s ∈ R/LZ : u(s, C) ∩ y 0 ([n, +∞)) = ∅} C − n = {s ∈ R/LZ : u(s, C) ∩ y 0 ((−∞, n]) = ∅} are equal to R/LZ for every n ∈ Z. We only prove C + n = R/LZ ∀n, the arguments for C − n are analogous. Let ω(p) be the ω-limit set of p. If ω(p) ∩ x(R) = ∅ then we find a neighborhood O of x(R) in M such that y 0 ([n, +∞)) ∩ O = ∅. Using Lemma 4.11 exactly as in the proof of Lemma 7.9 we find r > 0 such that |z| > r ⇒ u(s, z) ∈ O ∀s. It follows that y 0 ([n, +∞)) ⊂ u(R/LZ × B r (0)). It follows easily that C + n is closed ∀n. It is also open (each u(s, ·) : C → M \ x(R) is an embedding transversal to the Reeb vector) and clearly non-empty. Thus C + n = R/LZ ∀n ∈ Z in this case. Now suppose ω(p) ∩ x(R) = ∅ and fix s ∈ R/LZ. SinceŨ (s, ·) is an embedded fast finie-energy plane asymptotic the the orbit P at the positive puncture ∞ then we conclude from Lemma 6.9 that s ∈ C + n for every n ∈ Z. The conclusion follows.
It is proved in [17] that the Poincaré return map ψ : u(s, C) → u(s, C) is an areapreserving diffeomorphism with respect to the smooth area form ω = dλ| u(s,C) , for every s. Clearly ω = T . They also show that ψ is conjugated to a diffeomorphism of the open unit diskD preserving the measure T π dx∧dy. The proof of Theorem 2.5 can now be completed as explained in the introduction. : z 0 = 0} has length less than π. This follows easily from the linearity of (109) and is explained in [18]. Thus either I ϕ ∩ 2πZ = ∅ or I ϕ ∩ 2πZ = ∅. Moreover, det[ϕ(1) − I] = 0 if, and only if, 2πZ ∩ ∂I ϕ = ∅. It turns out, see [10], that the µ-index given by Theorem 1.7 satisfies µ(ϕ) =μ(I ϕ ) ∀ϕ ∈ Σ * .
This discussion provides an extension of the µ-index to paths ϕ ∈ Σ * , which coincides with the extension explained in Section 4.
Proof. We follow [11] and proceed in four steps.
This concludes the induction argument.
Proof of STEP 3. Fix γ. We have equations ∂ s D γ X + J 0 ∂ t D γ X = D γ h =: h γ and sup t e ds D β h γ → 0 as s → +∞, for any β. The conclusion follows from STEP 2. Lemma 6.3 follows from STEP 3 by the Sobolev embedding theorem, since we can assume, possibly after making d smaller, that 0 < d < 1 2 .
Appendix C. Proofs of technical lemmas C.1. Proofs of Lemma 6.5 and Corollary 6.6. Suppose e bs |z(s, t)| is bounded for some b > 0. Then we can assume, possibly after making b > 0 smaller, that Equations (132) and (131) prove (62). Obviously ifũ has non-degenerate asymptotics then (129) is true. This completes the proof of Lemma 6.5.
We now turn to the proof of Corollary 6.6. For simplicity we assume σ =d = 0. Define L ∞ (t) = L(kt, 0). The definition of L(s, t) and Lemma 6.
This proves −J 0 ∂ t − Λ ∞ is just a representation of the asymptotic operator A P in a different symplectic frame. Moreover, Λ(θ) T = Λ(θ) ∀θ since L is Sp(1)-valued. The asymptotic behavior of Λ and (130) allow us to apply Theorem 6.1 to ζ(s, t). The conclusion follows immediately from the formula z(s, t) = L −1 (s, t)ζ(s, t) and from the asymptotic behavior of L(s, t).