Symplectic homology of convex domains and Clarke's duality
SSymplectic homology of convex domains andClarke’s duality
Alberto Abbondandolo and Jungsoo Kang
Abstract
We prove that the Floer complex that is associated with a convex Hamiltonianfunction on R n is isomorphic to the Morse complex of Clarke’s dual action functionalthat is associated with the Fenchel-dual Hamiltonian. This isomorphism preservesthe action filtrations. As a corollary, we obtain that the symplectic capacity fromthe symplectic homology of a convex domain with smooth boundary coincides withthe minimal action of closed characteristics on its boundary. Introduction
Let H : T × R n → R be a smooth time-periodic Hamiltonian function on R n , endowedwith its standard symplectic structure ω := n (cid:88) j =1 dp j ∧ dq j . Here T := R / Z denotes the 1-torus. The corresponding time-periodic Hamiltonian vectorfield X H is defined as usual by ı X H ω = − dH, where d denotes differentiation with respect to the spatial variables. The 1-periodic orbitsof X H are precisely the critical points of the action functionalΦ H ( x ) := 12 ˆ T J ˙ x ( t ) · x ( t ) dt − ˆ T H t ( x ( t )) dt, x ∈ C ∞ ( T , R n ) , where J denotes the standard complex structure on R n mapping ( q, p ) into ( − p, q ), andwe are using the notation H t ( x ) := H ( t, x ). Assuming non-degeneracy of all 1-periodicorbits of X H and suitable assumptions on the behaviour of H t ( x ) for | x | large, one canassociate a Floer complex with H . This is a chain complex F ∗ ( H ) of Z -vector spaces thatare generated by the 1-periodic orbits of X H , whose boundary operator ∂ : F ∗ ( H ) → F ∗− ( H )1 a r X i v : . [ m a t h . S G ] J u l s obtained by a suitable count of the spaces of cylinders u : R × T → R n that satisfy theFloer equation ∂ s u + J t ( u )( ∂ t u − X H t ( u )) = 0 , ( s, t ) ∈ R × T , (1)and are asymptotic to pairs of periodic orbits for s → ±∞ . Here, J is a generic time-periodic almost complex structure on R n that is compatible with ω and has a suitablebehaviour at infinity. Actually, one can work with coefficients in an arbitrary abelian groupinstead of Z , but in this paper we stick to Z coefficients, as this simplifies the presentationand the proofs.The Floer complex is graded by the Conley-Zehnder index µ CZ ( x ), an integer thatcounts the half-windings of the differential of the flow of X H along x in the symplecticgroup. The homology of this chain complex, which is known as the Floer homology of H , is a considerably stable object and depends only on the behaviour at infinity of theHamiltonian H . It is denoted by HF ∗ ( H ). See e.g. [Sal99], [AD14], [AS18] and referencestherein for more information on Hamiltonian Floer theory.The Floer equation (1) can be seen as a negative gradient equation for the actionfunctional Φ H , and Floer homology should be thought as a kind of Morse theory for thisfunctional, which does not have a Morse theory in the usual sense because all its criticalpoints have infinite Morse index and co-index.When H is strongly convex (i.e. has an everywhere positive definite second differential)and superlinear in the spatial variable, there is another way of deriving a Morse theoryfor the 1-periodic orbits of X H . Indeed, following Clarke’s [Cla79] one can introduce thefollowing dual action functionalΨ H ∗ ( x ) := − ˆ T J ˙ x ( t ) · x ( t ) dt + ˆ T H ∗ t ( J ˙ x ( t )) dt, where H ∗ denotes the Fenchel conjugate of H in the spatial variable z = ( q, p ), whichis still a strongly convex and superlinear function (see also [CE80, Cla81, CE82]). Thisfunctional is invariant under translations, so it can be seen as a functional on the quotientspace of smooth closed curves in R n modulo translations, that we identify with the spaceof closed curves with zero mean. The crucial observation of Clarke was that there is anatural one-to-one correspondence between the critical points of Φ H and Ψ H ∗ : A closedcurve x is a critical point of Ψ H ∗ if an only if x + v is a critical point of Φ H , for a suitabletranslation vector v ∈ R n . Moreover, the direct action functional and the dual one havethe same value at their corresponding critical points:Ψ H ∗ ( x ) = Φ H ( x + v ) . See Ekeland’s book [Eke90] for a general approach to Clarke’s duality, with special emphasison Hamiltonian systems.Clarke’s dual functional Ψ H ∗ has better analytical properties than the direct actionfunctional Φ H : In Φ H , the indefinite quadratic form12 ˆ T J ˙ x ( t ) · x ( t ) dt, (2)2s the leading part (the part involving derivatives of x , as opposed to the integral of H t ( x ),which does not), whereas in Ψ H ∗ the leading term is the integral of H ∗ t ( J ˙ x ), which definesa convex functional. A consequence of this is that the critical points of Ψ H ∗ have finiteMorse index. Moreover, under suitable assumptions on H (e.g. H subquadratic, so that H ∗ is superquadratic), one can find critical points of Ψ H ∗ just by minimization.Here we are interested in the global properties of Ψ H ∗ and its critical points. Theseproperties can be encoded in the Morse complex of Ψ H ∗ . Constructing such a Morsecomplex presents some analytical difficulties, which we will mention in due time, butit is possible and the outcome is a Morse theory that, unlike Floer’s theory for Φ H , isessentially finite dimensional. It is then a natural question to compare the Floer chaincomplex associated with Φ H to the Morse complex induced by Ψ H ∗ . It is to this questionthat this paper is devoted.Before stating our main results, we need to clarify the class of convex Hamiltonianswe are going to work with. We shall assume that H ∈ C ∞ ( T × R n ) is non degenerate,meaning that all the 1-periodic orbits of X H are non degenerate, and quadratically convex,meaning that h | u | ≤ d H t ( x )[ u, u ] ≤ h | u | ∀ x, u ∈ R n , for suitable positive numbers h and h . Moreover, we shall assume that H is non-resonantat infinity. This means that there are positive numbers (cid:15) and r such that every smoothcurve x : T → R n satisfying (cid:107) ˙ x − X H ( x ) (cid:107) L ( T ) < (cid:15) has L -norm bounded by r . This is a kind of Palais-Smale condition for the action func-tional Φ H and implies in particular that all 1-periodic orbits are contained in a compactset. By the non-degeneracy assumption, X H has then only finitely many 1-periodic orbits.Here is a concrete condition on the behaviour of H at infinity that guarantees that itis non-resonant at infinity: H ( t, z ) = η | z | + ξ for | z | ≥ R, ∀ t ∈ T , where R > ξ ∈ R , and η ∈ (0 , + ∞ ) \ π N . A more general class of non-resonantHamiltonians is described in Lemma 4.1 below.Fix a Hamiltonian H ∈ C ∞ ( T × R n ) that is non-degenerate, quadratically convex andnon-resonant at infinity. The Floer complex F ∗ ( H ) is a finitely generated chain complex. Itis filtered by the action: F
Assume that the Hamiltonian H ∈ C ∞ ( T × R n ) is non-degenerate, quadrati-cally convex and non-resonant at infinity. Then there is a chain complex isomorphism Θ : M ∗− n ( ψ H ∗ ) → F ∗ ( H ) from the Morse complex of the reduced dual functional ψ H ∗ to the Floer complex of thedirect action functional Φ H . This isomorphism preserves the action filtrations. The theorem is proven in Section 10. The main ideas in the construction of the chainisomorphism Θ will be sketched at the end of this introduction.Our next result is a corollary of the above theorem and concerns the symplectic homol-ogy of convex domains in R n and the resulting SH -capacity of such domains.Symplectic homology is an algebraic invariant that is associated with certain symplecticcompact manifolds with boundary. It was introduced by Floer and Hofer in [FH94] andfurther developed in [FHW94], [CFH95] and [FCHW96]. See also [Vit99] for a somehowdifferent and quite fruitful approach, and the surveys [Oan04] and [Sei08].In Section 11, we recall its definition for a smooth starshaped domains W , that is, abounded open subset of R n that is starshaped with respect to a point and has a smoothboundary that is transverse to all the lines through this point. Being a smooth hypersur-face, the boundary of W carries a 1-dimensional foliation, that is called the characteristicfoliation and is tangent to the kernel of the restriction of ω to the tangent spaces of ∂W .The closed leaves of this foliation are called closed characteristics, and their action is de-fined to be the absolute value of the integral of ω over a disk in R n capping them. The setof the actions of all closed characteristics - including their iterations - is called the action4pectrum of ∂W . It is a closed subset of R consisting of positive numbers and having zeromeasure.Given a smooth starshaped domain W , the filtered symplectic homology SH (cid:15) | SH <(cid:15)n ( W ) → SH Let C be a convex bounded open subset of R n with smooth boundary. Then c SH ( C ) coincides with the minimum of the action spectrum of ∂C . The above result has been very recently proven also by Kei Irie, by extending to arbi-trary convex bodies in R n the approach that he had developed in [Iri14] for the cotangentdisk bundle of domains in R n . A paper about this is in preparation.The above corollary adds some more evidence to the conjecture that all symplecticcapacities should coincide on convex bodies. Indeed, it has been known for a long time5hat the above results holds for the Hofer-Zehnder capacity and the Ekeland-Hofer capacity,see [EH89, HZ90]. Moreover, Hermann has shown in [Her04] that the Viterbo capacity andthe symplectic homology capacity c SH coincide on domains with contact type boundary.The above corollary allows us to conclude that these four capacities coincide on the set ofconvex bodies.The proof of this corollary is contained in Section 11. The idea is to perturb C to makeit strongly convex and non-degenerate, and then to see SH <η ∗ ( C ) for η just above theminimum of the spectrum of ∂C as the Floer homology of a suitable Hamiltonian H thatis non-degenerate, quadratically convex and non-resonant at infinity. The correspondingFloer complex is generated by two generators for each closed characteristic of ∂C of minimalaction plus an extra generator z corresponding to the global minimum of H . The generator z defines a cycle in F n ( H ) whose homology class generates the image of the homomorphism SH <(cid:15)n ( C ) → SH <ηn ( C ) . The generator z corresponds to a local minimizer π ( z ) of the reduced dual action functional ψ H ∗ , while the pairs of generators given by the closed characterstics of minimal actioncorrespond to pairs of critical points of ψ H ∗ of Morse index 1 and 2, respectively. Moreover,the function ψ H ∗ is unbounded from below, and this easily implies that π ( z ) is the boundaryof a critical point of index 1 in the Morse complex of ψ H ∗ . Using the isomorphism of ourmain theorem, we deduce that z is a boundary in F ∗ ( H ), and hence vanishes in SH <η ∗ ( C ).This proves that c SH ( C ) does not exceed the minimum of the spectrum of α C . Being anelement in this set, c SH ( C ) must then coincide with the minimum of the spectrum of ∂C .We conclude this introduction by sketching the construction of the isomorphism Θ fromour main theorem. As it is now customary in Floer homological theories, see in particular[AS06], the chain isomorphism Θ : M ∗− n ( ψ H ∗ ) → F ∗ ( H )is defined by counting solutions of a suitable hybrid problem that relates the negativegradient flow lines of ψ H ∗ and the solutions of the Floer equation (1). We now wish todescribe this hybrid problem.The quadratic form (2) is continuous on the Sobolev space H / of closed curves x : T → R n of Sobolev class H / . The corresponding bounded self-adjoint operator on H / is a Fredholm operator having a finite dimensional kernel - the space of constant curves,which we denote by R n - and two infinite dimensional positive and negative eigenspaces H +1 / and H − / . So we have the orthogonal splitting H / = H +1 / ⊕ H − / ⊕ R n . Let x and y be two 1-periodic orbits of X H . By what we have seen above, π ( x ) belongsto M and is a critical point of ψ H ∗ . As before, π is the standard projection on the space ofcurves with zero mean. We denote by W u ( π ( x )) ⊂ M its unstable manifold with respect to6he negative gradient flow of ψ H ∗ . The hybrid problem we are interested in is the following:We look for smooth solutions u : [0 , + ∞ ) × T → R n of the Floer equation (1) that converge to the periodic orbit y for s → + ∞ and satisfy thefollowing boundary condition for s = 0: u (0 , · ) ∈ π − ( W u ( π ( x ))) + H − / . The set appearing on the right-hand side turns out to be a submanifold of the Sobolevspace H / with infinite dimension and infinite codimension. Its tangent space at everypoint is a closed vector subspace that is a compact perturbation of H − / .This is a non-local and somehow non-standard boundary condition for the Floer equa-tion - standard boundary conditions would require u (0 , · ) to take values into a Lagrangiansubmanifold of R n - but conditions of this kind have been considered in [Hec12], [Hec13]and [AS15]. The Fredholm analysis for the linearization of the above hybrid problemultimately relies on the identity ˆ [0 , + ∞ ) × T |∇ u | dsdt = ˆ [0 , + ∞ ) × T | ∂u | dsdt +2 ˆ T u (0 , · ) ∗ λ , ∀ u ∈ C ∞ c ([0 , + ∞ ) × T , R n ) , where ∂ = ∂ s + J ∂ t is the Cauchy-Riemann operator, and on the fact that the secondintegral on the right-hand side is the quadratic form2 ˆ T x ∗ λ = ˆ T J ˙ x ( t ) · x ( t ) dt, which is negative definite on H − / . This analysis is carried out in Section 8.A good functional space for studying the above hybrid problem is the space of maps u : [0 , + ∞ ) × T → R n of Sobolev class H , as the trace at s = 0 of these maps belongs to H / . The usual arguments from Floer theory for showing that solutions of (1) are smoothrequire the solutions to be in W ,p loc for some p > C ∩ H (see [IS99, Section 2.3] or [IS00]), and hence cannot be applied directly here.In Appendix A we show how interior regularity can be obtained also starting from H solutions, while in Section 7 we deal with regularity up to the boundary.The compactness of the spaces of solutions of the hybrid problem relies on the followinginequality relating the direct and the dual action functionalsΦ H ( x + y ) ≤ Ψ H ∗ ( π ( x )) − (cid:107) P − y (cid:107) / , (5)where x : T → R n is of Sobolev class H and y ∈ H − / ⊕ R n . This inequality followsfrom Fenchel duality and is proven in Proposition 5.2. Here, (cid:107) · (cid:107) / denotes the H / -normon H / and P − is the orthogonal projection onto H − / . The equality holds, in particular,when y is a constant loop and x + y is a critical point of Φ H .7nce all these facts have been proven, the homomorphismΘ : M ∗− n ( ψ H ∗ ) → F ∗ ( H )is defined in the usual way by counting the zero-dimensional spaces of solutions of the hybridproblem. The inequality (5) implies that the space of solutions of the hybrid problem withasymptotic Hamiltonian orbits x and y can be non-empty only when Φ H ( x ) ≥ Φ H ( y ).This fact is used in the proofs of the fact that Θ is an isomorphism and of the fact that itpreserves the action filtrations. Outlook The argument behind our main theorem is quite flexible and it should be pos-sible to adapt it to several different situations. A natural direction is to extend our iso-morphism to the S -equivariant setting. This is particularly interesting in view of somerecent results of Gutt and Hutchings, who defined a sequence of symplectic capacities for astarshaped domain with extremely good properties using S -equivariant symplectic homol-ogy, see [GH18]. This sequence of symplectic capacities is reminiscent of another sequenceof symplectic capacities that was defined by Ekeland and Hofer [EH90] using the directfunctional Φ H on the space H / together with the Fadell-Rabinowitz index [FR78]. On theother hand, using Clarke’s duality and the Fadell-Rabinowitz index, one can also obtain asequence of positive numbers belonging to the action spectrum of the boundary of a smoothconvex domain as in [EH87, Eke90], which is monotone with respect to inclusions betweensmooth convex domains. It should be also possible to build an isomorphism between theFloer homology for the Hamiltonian H and the Morse homology for Φ H on H / . Once thecorresponding isomorphisms are upgraded to the S -equivariant setting, this could suggesthow to compare Gutt and Hutchings’ symplectic capacities with the sequence of actionsdefined using Clarke’s duality and with Ekeland and Hofer’s symplectic capacities. Acknowledgments We are very grateful to Urs Fuchs for explaining us how to proveinterior regularity of solutions of the Floer equation of Sobolev class H , which is the con-tent of Appendix A. We would like to thank Kei Irie for discussing with us his proof of theabove corollary and the comparison of the two approaches. The research of A. Abbondan-dolo is supported by the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra andDynamics”, funded by the Deutsche Forschungsgemeinschaft. The research of J. Kang issupported by Samsung Science and Technology Foundation under Project Number SSTF-BA1801-01. Some part of this paper was written during several visits of the second authorto the Ruhr-Universit¨at Bochum and the Universit¨at Heidelberg. He would like to thankA. Abbondandolo, P. Albers, and G. Benedetti for their warm hospitality. Contents Uniform bounds for solutions of the Floer equation 164 The Floer complex of H 205 The dual action functional 236 The Morse complex of the dual action functional 327 The functional setting for the hybrid problem 398 The Fredholm index of the hybrid problem 469 Compactness properties of the hybrid problem 5610 The chain complex isomorphism 6011 Symplectic homology and SH -capacity of convex domains 62A Appendix: Interior regularity of solutions of the Floer equation 66 We equip R n with coordinates ( q , p , . . . , q n , p n ), with the standard Liouville form λ := 12 n (cid:88) j =1 ( p j dq j − q j dp j )and with the standard symplectic form ω := dλ = n (cid:88) j =1 dp j ∧ dq j . Note that λ ( u )[ v ] = 12 ω ( u, v ) ∀ u, v ∈ R n . (1.1)The linear automorphism J : R n → R n , ( q, p ) (cid:55)→ ( − p, q ) , is the standard complex structure on R n , according to the identification R n ∼ = C n givenby ( q, p ) (cid:55)→ q + ip . The symplectic form ω and the complex structure J are related tothe standard Euclidean scalar product on R n by the identity u · v = ω ( J u, v ) ∀ u, v ∈ R n . X H associated with a smooth Hamiltonian H : R n → R is defined by the identity ω ( X H , · ) = − dH, or equivalently by X H = − J ∇ H, where ∇ denotes the Euclidean gradient on R n .We now fix a time-periodic smooth Hamiltonian H : T × R n → R , where T := R / Z denotes the 1-torus, and use the notation H t ( x ) := H ( t, x ). We recall that a 1-periodicorbit x of X H is said to be non-degenerate if 1 is not an eigenvalue of the linearization ofthe Hamiltonian flow along x : 1 / ∈ σ (cid:0) dφ X H ( x (0)) (cid:1) , where φ tX H denotes the (possibly non-autonomous) local flow of X H . When needed, thetime-periodic Hamiltonian H will be assumed to be non-degenerate: Non-degeneracy: The Hamiltonian H ∈ C ∞ ( T × R n ) is said to be non-degenerate if allthe 1-periodic orbits of X H are non-degenerate.The 1-periodic orbits of X H are exactly the critical points of the action functionalΦ H : C ∞ ( T , R n ) → R given byΦ H ( x ) := ˆ T x ∗ λ − ˆ T H t ( x ( t )) dt = 12 ˆ T J ˙ x ( t ) · x ( t ) dt − ˆ T H t ( x ( t )) dt. If the second differential of H in the spatial variable z = ( q, p ) ∈ R n has polynomialgrowth, meaning that there are c > N > | d H t ( x ) | ≤ c (1 + | x | N ) ∀ ( t, x ) ∈ R n , then Φ H is twice continuously differentiable on the Sobolev space H / := H / ( T , R n ) . This space consists of all L curves x : T → R n such that the coefficients (ˆ x k ) k ∈ Z of theFourier decomposition x ( t ) = (cid:88) k ∈ Z e − πkJ t ˆ x k , ˆ x k ∈ R n , (1.2)satisfy (cid:88) k ∈ Z | k | | ˆ x k | < + ∞ . See [HZ94, Section 3.3 and Appendix A.3] for more information on the properties of theaction functional on the Sobolev space H / .10he non-degeneracy of the 1-periodic orbits of X H translates into the fact that Φ H isa Morse functional on H / . The critical points of Φ H have infinite Morse index, but onecan associate with them a finite relative Morse index. Indeed, this is due to the fact thatthe leading part of this functional has the form12 ˆ T J ˙ x ( t ) · x ( t ) dt = 12 (cid:0) (cid:107) P + x (cid:107) / − (cid:107) P − x (cid:107) / (cid:1) , where (cid:107) · (cid:107) / denotes the H / -Hilbert norm (cid:107) x (cid:107) / := | ˆ x | + 2 π (cid:88) k ∈ Z | k | | ˆ x k | , and P + and P − are the orthogonal projectors onto the closed subspaces H +1 / := (cid:8) x ∈ H / | ˆ x k = 0 ∀ k ≤ (cid:9) , H − / := (cid:8) x ∈ H / | ˆ x k = 0 ∀ k ≥ (cid:9) , defined by the Fourier decomposition (1.2). The Hilbert space H / has the orthogonalsplitting H / = H +1 / ⊕ H − / ⊕ R n , where R n denotes the space of constant curves. We also denote by P the orthogonalprojector onto R n . It is convenient to split the space of constant curves R n into twoorthogonal subspaces of dimension n : R n = E + ⊕ E − . The fact that the Hessian of the functional x (cid:55)→ ˆ T H t ( x ( t )) dt is a compact operator on H / implies that the negative eigenspace V − ( ∇ Φ H ( x )) of theHessian ∇ Φ H ( x ) of Φ H at a critical point x is a compact perturbation of the space H − / ⊕ E − , meaning that the difference of the orthogonal projectors onto these subspacesis compact. Then we define the relative Morse index of x as the relative dimension of V − ( ∇ Φ H ( x )) with respect to H − / ⊕ E − , that is, the integerind H − / ⊕ E − ( x ; Φ H ) := dim (cid:0) V − ( ∇ Φ H ( x )) , H − / ⊕ E − ):= dim V − ( ∇ Φ H ( x )) ∩ ( H +1 / ⊕ E + ) − dim (cid:0) V + ( ∇ Φ H ( x )) ⊕ ker ∇ Φ H ( x ) (cid:1) ∩ ( H − / ⊕ E − ) . Here V + ( ∇ Φ H ( x )) denotes the positive eigenspace of ∇ Φ H ( x ) at x . An equivalent defi-nition is the following: any maximal closed subspace W of H / on which the bilinear form11 Φ H ( x ) is negative definite is in Fredholm pair with the subspace H +1 / ⊕ E + , and therelative Morse index of x is the Fredholm index of this pair:ind H − / ⊕ E − ( x ; Φ H ) = ind( W, H +1 / ⊕ E + ) . See [Abb01][Chapter 3] for more details about relative dimensions and relative Morse in-dices. The nullity of x is defined as usual as the dimension of the kernel of the Hessian ofΦ H at x : null( x ; Φ H ) := dim ker ∇ Φ H ( x ) . We shall make use of the following fact, which is proven in [Abb01][Corollary 3.3.1]. Proposition . Assume that x is a 1-periodic orbit of X H . Then the nullity of x coincides with the geometric multiplicity of the eigenvalue 1 of the linearization of the flowalong x : null( x ; Φ H ) = dim ker (cid:0) I − dφ X H ( x (0)) (cid:1) , and the relative Morse index of x coincides with its Conley-Zehnder index: ind H − / ⊕ E − ( x ; Φ H ) = µ CZ ( x ) . The Conley-Zehnder index is an integer assigned to every element in the space SP (2 n ) = (cid:8) Z ∈ C ([0 , , Sp(2 n ) | Z (0) = I and det( I − Z (1)) (cid:54) = 0 (cid:9) (1.3)which we extend to degenerate paths, i.e. det( I − Z (1)) = 0, by lower semi-continuity as in[LZ90] and [Lon02]. In the above proposition µ CZ ( x ) is the Conley-Zehnder index of thesymplectic path t (cid:55)→ dφ tX H ( x (0)), t ∈ [0 , t (cid:55)→ e − (cid:15)J t , t ∈ [0 , , is n for every (cid:15) ∈ (0 , π ). We recall that the characteristic line bundle of a smooth hypersurface Σ ⊂ R n is given bythe kernel of the restriction of ω to the tangent bundle of Σ. Its integral lines are calledcharacteristics. The action of a closed characteristic γ on Σ is defined as the absolute valueof the integral of a primitive of ω over γ . Stokes’ theorem implies that this definitiondoes not depend on the choice of the primitive of ω . The set of all actions of closedcharacteristics of Σ is called action spectrum of Σ, or just spectrum of Σ, and denoted byspec(Σ). Here, iterates of closed characteristics are also considered, so the spectrum of Σis a subset of [0 , + ∞ ) that is invariant under multiplication by positive integers.We now restrict the attention to those hypersurfaces that are obtained as boundariesof starshaped domains. In this paper, by a smooth starshaped domain we mean a bounded12pen subset W which is starshaped with respect to the origin and has a smooth boundarywhich is transverse to all lines through the origin. The restriction of the Liouville 1-form λ to the boundary of W is denoted by α W := λ | ∂W . It is a contact form on ∂W , meaning that the restriction of the differential dα W to thekernel of α W is non-degenerate. The corresponding Reeb vector field on ∂W is denoted by R α W and is defined by dα W ( R α W , · ) = 0 , α W ( R α W ) = 1 . This vector field is a smooth non-vanishing section of the characteristic line bundle of thehypersurface ∂W . Therefore, the orbits of R α W are parametrizations of the characteristiccurves on ∂W . The action of a closed characteristic on ∂W coincides with its period asclosed orbit of R α W : If γ : R /T Z → ∂W is a closed orbit of R α W of (not necessarilyminimal) period T , then ˆ R /T Z γ ∗ λ = ˆ R /T Z γ ∗ α W = T. The spectrum of ∂W is a measure zero nowhere dense closed subset of R consisting ofpositive numbers and invariant under the multiplication by positive integers.We denote by H W : R n → R the positively 2-homogeneous function that takes the value 1 on ∂W . This function is con-tinuously differentiable on R n and smooth on R n \{ } . The restriction of the Hamiltonianvector field X H W to the boundary of W coincides with the Reeb vector field R α W : R α W = X H W | ∂W . Indeed, this follows from the fact that for every x ∈ ∂W the vector X H W ( x ) spansker dα W ( x ) = ker ω | T x ∂W and from the identity α W ( x )[ X H W ( x )] = λ ( x )[ X H W ( x )] = 12 ω ( x, X H W ( x )) = 12 dH W ( x )[ x ] = H W ( x ) = 1 , where we have used the Euler identity for the positively 2-homogeneous function H W .The symplectization of the contact manifold ( ∂W, α W ) is the manifold (0 , + ∞ ) × ∂W equipped with the Liouville form λ W := rα W and the symplectic form ω W := dλ W , where r ∈ (0 , + ∞ ) denotes the variable in the first factor. The symplectization of ( ∂W, α W ) canbe identified with ( R n \ { } , λ ) thanks to the following well known fact: Lemma . The diffeomorphism ϕ : R n \ { } −→ (0 , + ∞ ) × ∂W, ϕ ( x ) = (cid:32) H W ( x ) , x (cid:112) H W ( x ) (cid:33) , satisfies ϕ ∗ λ W = λ . In particular, ϕ is a symplectomorphism from ( R n \ { } , ω ) to ((0 , + ∞ ) × ∂W, ω W ) . roof. We denote by µ W : R n → R , µ W := (cid:112) H W , the Minkowski gauge function of W , which is continuous on R n , smooth on R n \ { } andpositively 1-homogeneous. Let x ∈ R n \ { } . For every v ∈ ker dµ W ( x ) we have( ϕ ∗ λ W )( x )[ v ] = λ W ( ϕ ( x )) (cid:2) dϕ ( x )[ v ] (cid:3) = H W ( x ) α W (cid:18) xµ W ( x ) (cid:19) (cid:20) vµ W ( x ) (cid:21) = H W ( x ) µ W ( x ) λ ( x )[ v ] = λ ( x )[ v ] . There remains to check that the one-forms ( ϕ ∗ λ W )( x ) and λ ( x ) coincide on a vector whichis transverse to ker dµ W ( x ). The vector x has this property, and we compute( ϕ ∗ λ W )( x )[ x ] = λ W ( ϕ ( x )) (cid:2) dϕ ( x )[ x ] (cid:3) = H W ( x ) α W (cid:18) xµ W ( x ) (cid:19) (cid:20) xµ W ( x ) − xµ W ( x ) dµ W ( x )[ x ] (cid:21) = H W ( x ) α W (cid:18) xµ W ( x ) (cid:19) (cid:20) xµ W ( x ) − xµ W ( x ) (cid:21) = 0 = λ ( x )[ x ] , where we have used the Euler identity for the 1-homogeneous function µ W .Any T -periodic Reeb orbit of R α W on ∂W can be seen as a 1-periodic orbit of X T H W ,after time reparametrization. More generally, it can be seen as a 1-periodic orbit of X ϕ ◦ H W ,where ϕ is any smooth function on R such that ϕ (cid:48) (1) = T . In the next proposition westudy how the index and nullity of this orbit change, when it is seen as a critical point ofΦ T H W or Φ ϕ ◦ H W . Proposition . Assume that γ : R /T Z → ∂W is a periodic orbit of R α W . Let ϕ : R → R be a smooth function with ϕ (cid:48) (1) = T . Then x γ ( t ) := γ ( T t ) is a critical point of both Φ T H W and Φ ϕ ◦ H W and ind H − / ⊕ E − ( x γ , Φ ϕ ◦ H W ) = ind H − / ⊕ E − ( x γ , Φ T H W ) if ϕ (cid:48)(cid:48) (1) ≤ H − / ⊕ E − ( x γ , Φ T H W ) + 1 if ϕ (cid:48)(cid:48) (1) > x γ , Φ ϕ ◦ H W ) = (cid:40) null( x γ , Φ T H W ) if ϕ (cid:48)(cid:48) (1) = 0null( x γ , Φ T H W ) − if ϕ (cid:48)(cid:48) (1) (cid:54) = 0 . Proof. From the hypothesis ϕ (cid:48) ( H W ( x γ )) = ϕ (cid:48) (1) = T , we have for every u ∈ H / d Φ ϕ ◦ H W ( x γ )[ u ] = ˆ T (cid:0) J ˙ x γ − ϕ (cid:48) ( H W ( x γ )) ∇ H W ( x γ ) (cid:1) · u dt = d Φ T H W ( x γ )[ u ] . Since x γ satisfies J ˙ x γ = T ∇ H W ( x γ ), the above formula shows that x γ is a critical pointof both Φ T H W and Φ ϕ ◦ H W . 14onsider the following continuous symmetric bilinear forms on H / : a ( u, v ) := ˆ T u · (cid:0) J ˙ v − T ∇ H W ( x γ ) v (cid:1) dt,b ( u, v ) := ˆ T (cid:0) ∇ H W ( x γ ) · u (cid:1) ( ∇ H W ( x γ ) · v (cid:1) dt. Then we have d Φ T H W ( x γ ) = a, d Φ ϕ ◦ H W ( x γ ) = a − ϕ (cid:48)(cid:48) (1) b. (2.1)From the 2-homogeneity of H W we deduce the identity ∇ H W ( z ) z = ∇ H W ( z ) ∀ z ∈ R n \ { } , which implies that x γ belongs to the kernel of a . This reflects the isochronicity property of2-homogeneous Hamiltonians, namely the fact that the Hamiltonian flows on its differentenergy levels are conjugated. The kernel of b is the L -orthogonal complement of the line R ∇ H W ( x γ ) in H / . Since b ( x γ , x γ ) = ˆ T (cid:0) ∇ H W ( x γ ) · x γ (cid:1) dt > , we have H / = ker b ⊕ R x γ . Let s (cid:54) = 0. A vector u = v + λx γ , v ∈ ker b , λ ∈ R , belongs to the kernel of a + sb if andonly if ( a + sb )( u, x γ ) = 0 and ( a + sb )( u, w ) = 0 ∀ w ∈ ker b. The first identity is equivalent to0 = ( a + sb )( v + λx γ , x γ ) = sλ b ( x γ , x γ ) , and hence to λ = 0. The second identity then reads0 = ( a + sb )( v, w ) = a ( v, w ) ∀ w ∈ ker b. Since x γ belongs to the kernel of a , the latter requirement is equivalent to the fact that v belongs to the kernel of a . We conclude that for every real number s (cid:54) = 0 the kernel of a + sb is the following spaceker( a + sb ) = ker a ∩ ker b ∀ s ∈ R \ { } , which is independent of s and has codimension 1 in ker a . The formula for the nullity of d Φ ϕ ◦ H W ( x γ ) immediately follows from this and (2.1).The path s (cid:55)→ a + sb describes a 1-parameter family of continuous symmetric bilinearforms that are rank one perturbations of the Fredholm form a . The fact that the nullity15f a + sb is constant for s (cid:54) = 0 implies that the relative index of a + sb with respect to H − / ⊕ E − is constant for s > s < 0. Since the kernel of a + sb increases by onedimension - and precisely by addition of the line R x γ - when s becomes 0, the inequality dds (cid:12)(cid:12)(cid:12) s =0 ( a + sb )( x γ , x γ ) = b ( x γ , x γ ) > (cid:0) V − ( a + sb ) , H − / ⊕ E − ) = (cid:40) dim (cid:0) V − ( a ) , H − / ⊕ E − ) ∀ s ≥ , dim (cid:0) V − ( a ) , H − / ⊕ E − ) + 1 ∀ s < . The formula for the relative index of x γ as a critical point of Φ ϕ ◦ H W immediately followsfrom this and (2.1). In this section, we wish to prove uniform bounds for solutions u : I × T → R n of the Floer equation ∂ s u + J ( s, t, u ) (cid:0) ∂ t u − X H t ( u ) (cid:1) = 0 . (3.1)Here I is an open unbounded interval, i.e. either R , or ( a, + ∞ ), or ( −∞ , a ) for some a ∈ R . The symbol J denotes a smooth almost complex structure on R n , which is allowedto depend on the variables ( s, t ) ∈ I × T and is supposed to be ω -compatible, meaningthat the formula g J ( u, v ) := ω ( J u, v )defines an ( s, t )-dependent family of Riemannian metrics on R n . The corresponding familyof norms will be denoted by | · | J , and ∇ J will denote the gradient operator with respectto these Riemannian metrics. By our sign conventions we have X H = − J ∇ J H. The Floer equation can be rewritten as ∂ J u = ∇ J H ( u ) , where ∂ J u := ∂ s u + J ( u ) ∂ t u. We shall also assume that J is uniformly bounded as a map valued into the space ofendomorphisms of R n . It follows that the family of metrics g J is uniformly globallyequivalent to the Euclidean metric g J ( u, v ) = u · v .We will consider the following growth assumptions on Hamiltonian functions H ∈ C ∞ ( T × R n ). 16 inear growth of the Hamiltonian vector field: The Hamiltonian vector field X H issaid to have linear growth at infinity if there exists a positive number c such that | X H t ( z ) | ≤ c (1 + | z | ) for every ( t, z ) ∈ T × R n . Non-resonance at infinity: The Hamiltonian H is said to be non-resonant at infinityif there exist positive numbers (cid:15) > r > x : T → R n satisfying (cid:107) ˙ x − X H ( x ) (cid:107) L ( T ) ≤ (cid:15), there holds (cid:107) x (cid:107) L ( T ) ≤ r .The latter requirement is a version of the Palais-Smale condition for the direct actionfunctional Φ H : It is equivalent to saying that every sequence ( x h ) ⊂ C ∞ ( T , R n ) on whichthe L -gradient of Φ H tends to zero in the L -norm is bounded in L ( T , R n ). The abovetwo assumptions imply in particular that 1-periodic orbits of X H are uniformly bounded.At the end of the next section, we shall discuss some sufficient conditions for H to benon-resonant at infinity in the above sense. Here, we shall prove the following a prioribounds. Proposition . Let I ⊂ R be an open unbounded interval, J be a uniformly bounded ω -compatible almost complex structure on R n , smoothly depending on ( s, t ) ∈ I × T , and H ∈ C ∞ ( T × R n ) be a smooth Hamiltonian which is non-resonant at infinity and whoseHamiltonian vector field has linear growth at infinity. Let I (cid:48) be a (possibly unbounded)interval such that I (cid:48) ⊂ I . For every E > there is a positive number M = M ( E ) suchthat every solution u ∈ C ∞ ( I × T , R n ) of (3.1) with energy bound ˆ I × T | ∂ s u | J dsdt ≤ E satisfies sup ( s,t ) ∈ I (cid:48) × T | u ( s, t ) | ≤ M. When I = R , we are allowed to take I (cid:48) = R and we obtain uniform bounds on thewhole cylinder. The above formulation allows us to get uniform bounds also for solutionson half-cylinders, as long as we stay far away from the boundary. Proof. Without loss of generality, we may assume that I is unbounded from below, thecase in which it is unbounded from above being completely analogous. Since the family ofmetrics g J is uniformly globally equivalent to the Euclidean metric, the energy bound on u translates into ˆ I × T | ∂ s u | dsdt ≤ E (cid:48) , (3.2)for a suitable number E (cid:48) = E (cid:48) ( E ). Consider the set S = S ( u ) = { s ∈ R | (cid:107) ∂ s u ( s, · ) (cid:107) L ( T ) < (cid:15)/ (cid:107) J (cid:107) ∞ } , (cid:15) is the positive number appearing in the assumption of non-resonance at infinityand the L ∞ -norm of J is induced by the Euclidean metric on R n . By the Chebichevinequality, the complement of S in I has uniformly bounded measure: | I \ S | ≤ (cid:107) J (cid:107) ∞ (cid:15) ˆ I (cid:107) ∂ s u ( s, · ) (cid:107) L ( T ) ds = (cid:107) J (cid:107) ∞ (cid:15) ˆ I × T | ∂ s u | dsdt < L, (3.3)where we have set L = L ( E ) := (cid:107) J (cid:107) ∞ E (cid:48) (cid:15) . Let I (cid:48)(cid:48) ⊂ I be an interval of length L . By (3.3), I (cid:48)(cid:48) intersects S in at least one point s .For such a point s we have, using the fact that u solves the Floer equation (3.1): (cid:107) ∂ t u ( s , · ) − X H ( u ( s , · )) (cid:107) L ( T ) = (cid:107) J ( s , · , u ( s , · )) ∂ s u ( s , · ) (cid:107) L ( T ) ≤ (cid:107) J (cid:107) ∞ (cid:107) ∂ s u ( s , · ) (cid:107) L ( T ) < (cid:15), and the non-resonance at infinity implies the L -bound (cid:107) u ( s , · ) (cid:107) L ( T ) ≤ r. Together with the energy bound on u , we easily get a uniform L -bound for u on I (cid:48)(cid:48) × T .Indeed, from the identity u ( s, t ) = u ( s , t ) + ˆ ss ∂ s u ( σ, t ) dσ and the bound (3.2), we get the inequality | u ( s, t ) | ≤ | u ( s , t ) | + 2 (cid:12)(cid:12)(cid:12)(cid:12) ˆ ss ∂ s u ( σ, t ) dσ (cid:12)(cid:12)(cid:12)(cid:12) ≤ | u ( s , t ) | + 2 | s − s | (cid:12)(cid:12)(cid:12)(cid:12) ˆ ss | ∂ s u ( σ, t ) | dσ (cid:12)(cid:12)(cid:12)(cid:12) ≤ | u ( s , t ) | + 2 LE (cid:48) . for every s ∈ I (cid:48)(cid:48) . Integration over T yields ˆ T | u ( s, t ) | dt ≤ ˆ T | u ( s , t ) | dt + 2 LE (cid:48) < (cid:15) (cid:107) J (cid:107) ∞ + 2 LE (cid:48) =: C, ∀ s ∈ I (cid:48)(cid:48) , and hence ˆ I (cid:48)(cid:48) × T | u ( s, t ) | dsdt ≤ LC. This inequality holds for every interval I (cid:48)(cid:48) ⊂ I of length L . It follows that ˆ I (cid:48)(cid:48) × T | u ( s, t ) | dsdt ≤ C ( | I (cid:48)(cid:48) | + L ) , (3.4)18or any bounded interval I (cid:48)(cid:48) ⊂ I . Together with the assumption of linear growth at infinityfor X H = − J ∇ J H and the boundedness of J , we get the bound ˆ I (cid:48)(cid:48) × T |∇ J H t ( u ( s, t )) | J dsdt ≤ C (cid:48) ( | I (cid:48)(cid:48) | + L ) , (3.5)for a suitable positive number C (cid:48) . Now let I (cid:48) be the possibly unbounded interval such that I (cid:48) ⊂ I which appears in the statement. Fix a positive number δ such that I (cid:48) + [ − δ, δ ] ⊂ I. In order to prove a uniform bound for | u | on I (cid:48) × T it is enough to prove such a uniformbound on I (cid:48)(cid:48) × T for all intervals I (cid:48)(cid:48) ⊂ I (cid:48) of length 1. Let I (cid:48)(cid:48) be such an interval.By the Calderon-Zygmund estimates in L for the uniformly bounded almost complexstructure J (see e.g. [MS04][Proposition B.4.9]), we have (cid:107)∇ u (cid:107) L (( I (cid:48)(cid:48) +[ − δ,δ ]) × T ) ≤ c (cid:0) (cid:107) ∂ J u (cid:107) L (( I (cid:48)(cid:48) +[ − δ, δ ]) × T ) + (cid:107) u (cid:107) L (( I (cid:48)(cid:48) +[ − δ, δ ]) × T ) (cid:1) = c (cid:0) (cid:107)∇ J H ( u ) (cid:107) L (( I (cid:48)(cid:48) +[ − δ, δ ]) × T ) + (cid:107) u (cid:107) L (( I (cid:48)(cid:48) +[ − δ, δ ]) × T ) (cid:1) . Together with the estimates (3.4) and (3.5) we obtain a bound (cid:107)∇ u (cid:107) L (( I (cid:48)(cid:48) +[ − δ,δ ]) × T ) ≤ C (cid:48)(cid:48)(cid:48) , holding for all intervals I (cid:48)(cid:48) ⊂ I (cid:48) of length 1. Using again (3.4), we deduce that the restrictionof u to ( I (cid:48)(cid:48) + [ − δ, δ ]) × T is uniformly bounded in the Sobolev space H , and hence in allLebesgue spaces L p for p < + ∞ . Using again the linear growth at infinity of X H = − J ∇ J H we deduce that the function ∂ J u = ∇ J H t ( u )has a uniform L p bound on ( I (cid:48)(cid:48) + [ − δ, δ ]) × T . By the Calderon-Zygmund estimate in L p (see again [MS04][Proposition B.4.9]) (cid:107)∇ u (cid:107) L p ( I (cid:48)(cid:48) × T ) ≤ c p (cid:0) (cid:107) ∂ J u (cid:107) L p (( I (cid:48)(cid:48) +[ − δ,δ ]) × T ) + (cid:107) u (cid:107) L p (( I (cid:48)(cid:48) +[ − δ,δ ]) × T ) (cid:1) , we conclude that the restriction of u to I (cid:48)(cid:48) × T has a uniform W ,p bound for every p < ∞ .For p > L ∞ bound on the restriction of u to I (cid:48)(cid:48) × T , where I (cid:48)(cid:48) ⊂ I (cid:48) is an arbitrary interval of length 1. Remark . A straightforward modification of the above argument allows one to extendthe above result to s -dependent Hamiltonians. The precise assumptions are that H ∈ C ∞ ( I × T × R n ) depends on s only for s in a bounded interval, that for s outside of thisinterval H ( s, · ) is non-resonant at infinity, and that the Hamiltonian vector field of H haslinear growth at infinity, uniformly on s ∈ I . The Floer complex of H Assume that the Hamiltonian H ∈ C ∞ ( T × R n ) is non-degenerate, non-resonant at infinityand that the corresponding Hamiltonian vector field X H has linear growth at infinity. Sucha Hamiltonian has a well-defined Floer complex, whose construction we quickly recall here.Let J = J ( t, z ) be a time-periodic smooth almost complex structure on R n , thatis compatible with ω and uniformly bounded. By Proposition 3.1, energy bounds onsolutions of the Floer equation (3.1) on the whole cylinder R × T imply L ∞ -bounds. Onceuniform bounds in L ∞ have been established, the standard bubbling-off argument andan elliptic bootstrap imply that solutions with uniformly bounded energy are compactin C ∞ loc ( R × T , R n ). Actually, the fact that we are working in R n would allow us toprove the above results using only a bootstrap argument involving the Calderon-Zygumndinequalities, as in the last part of the proof of Proposition 3.1, avoiding the bubbling-offargument.Let x and y be two 1-periodic orbits of X H with µ CZ ( x ) − µ CZ ( y ) = 1. Standard indexand transversality arguments imply that, if J is chosen generically, the space of solutionsof the Floer equation (3.1) that for s → −∞ are asymptotic to x and for s → + ∞ to y is finite, after modding out translation in the s -variables. Let n F ( x, y ) ∈ Z denote theparity of this finite set. The Floer complex is then defined as usual by ∂ F : F k ( H ) → F k − ( H ) , ∂ F x := (cid:88) y n F ( x, y ) y, where F k ( H ) denotes the Z -vector space generated by the 1-periodic orbits of X H withConley-Zehnder index k , x is any 1-periodic orbit with µ CZ ( x ) = k , and the sum rangesover all 1-periodic orbits y of Conley-Zehnder index k − 1. A standard cobordism argumentimplies that ∂ F ◦ ∂ F = 0, so { F ∗ ( H ) , ∂ F } is a chain complex of finite dimensional Z -vectorspaces.The fact that n F ( x, y ) = 0 whenever Φ H ( y ) > Φ H ( x ) implies that the Floer complex of H is graded by the Hamiltonian action: If F 0, and hence byrescaling we obtain an η -periodic orbit of R α W = X H W | ∂W on ∂W , contradicting the factthat η is not a period of closed Reeb orbits on ( ∂W, α W ).We claim that the number a := inf x ∈ H / (cid:107) x (cid:107) / =1 (cid:13)(cid:13) d Φ ηH W + ξ ( x ) (cid:13)(cid:13) H ∗ / is positive. If by contradiction a is zero, then there exists a sequence ( x h ) ⊂ H / suchthat (cid:107) x h (cid:107) / = 1 and d Φ ηH W + ξ ( x h ) → H ∗ / . Up to considering a subsequence, we mayassume that ( x h ) converges weakly to some x ∈ H / . The sequence v h := P + ( x h − x ) − P − ( x h − x )is bounded, and hence the real sequence d Φ ηH W + ξ ( x h )[ v h ]is infinitesimal. This sequence has the form d Φ ηH W + ξ ( x h )[ v h ] = ( x h , P + v h ) / − ( x h , P − v h ) / − η ˆ T ∇ H W ( x h ) · v h dt = (cid:107) x h − x (cid:107) / + ( x, x h − x ) / − ( x h , P ( x h − x )) / − η ˆ T ∇ H W ( x h ) · v h dt, P : H / → H / is the orthogonal projector onto the space of constant loops.Since x h converges to x weakly in H / , the term ( x, x h − x ) / is infinitesimal. Moreoversince P has finite rank, the sequence P ( x h − x ) converges to zero strongly and hencethe sequence ( x h , P ( x h − x )) / is also infinitesimal. Since ( v h ) converges to 0 weaklyin H / , it converges to 0 strongly in L ( T ). Since the sequence ∇ H W ( x h ) is bounded in L , we deduce that the last integral defines an infinitesimal sequence. We conclude thatthe sequence (cid:107) x h − x (cid:107) / is also infinitesimal, so the convergence of ( x h ) to x is actuallystrong in H / . But then x is a critical point of Φ ηH W + ξ lying on the unit sphere and thisis impossible, because 0 is the only critical point of Φ ηH W + ξ . This contradiction proves theclaim.The above claim and the homogeneity of Φ ηH W + ξ imply that (cid:13)(cid:13) d Φ ηH W + ξ ( x ) (cid:13)(cid:13) H ∗ / ≥ a (cid:107) x (cid:107) / ∀ x ∈ H / . (4.2)The continuously differentiable function K : T × R n → R , K t ( z ) := ηH W ( z ) + ξ − H t ( z ) , is compactly supported and hence has uniformly bounded first derivatives. From theidentity Φ H ( x ) − Φ ηH W + ξ ( x ) = ˆ T ( ηH W ( x ) + ξ − H t ( x )) dt = ˆ T K t ( x ) dt we deduce that d Φ H ( x )[ u ] − d Φ ηH W + ξ ( x )[ u ] = ˆ T dK t ( x )[ u ] dt ∀ x, u ∈ H / , and hence there is a number b such that (cid:13)(cid:13) d Φ H ( x ) − d Φ ηH C + d ( x ) (cid:13)(cid:13) H ∗ / ≤ b ∀ x ∈ H / . Together with (4.2) we deduce the desired first bound.From the identity d Φ H ( x )[ v ] = ˆ T ( J ˙ x − ∇ H ( x )) · v dt = (cid:0) J ( ˙ x − X H ( x )) , v (cid:1) L ( T ) ∀ x, v ∈ C ∞ ( T , R n ) , we deduce that (cid:107) d Φ H ( x ) (cid:107) H ∗ / = (cid:107) J ( ˙ x − X H ( x )) (cid:107) H − / ( T ) ≤ c (cid:107) ˙ x − X H ( x ) (cid:107) L ( T ) ∀ x ∈ C ∞ ( T , R n ) , and the second bound follows from the first one and from the inequality (cid:107) x (cid:107) L ( T ) ≤ (cid:107) x (cid:107) / .22 emark . The condition on H that appears in the above lemma clearly implies alsothe linear growth condition on X H , so a Hamiltonian of this kind has a well-defined Floercomplex. Another class of Hamiltonians that are non-resonant at infinity and have a Hamil-tonian vector field with linear growth is given by functions of the form H ( t, x ) = 12 (cid:104) A ( t ) x, x (cid:105) + K ( t, x ) , where the function K ∈ C ∞ ( T × R n ) satisfies ∇ K ( t, x ) = o ( (cid:107) x (cid:107) ) for (cid:107) x (cid:107) → ∞ uniformlyin t ∈ T and t (cid:55)→ A ( t ) is a smooth loop of symmetric endomorphisms of R n such that thelinear Hamiltonian system ˙ x ( t ) = − J A ( t ) x ( t ) does not have any non-zero 1-periodic orbit. We now focus our attention on convex Hamiltonians. More precisely, we consider thefollowing convexity assumption. Quadratic convexity: The Hamiltonian H ∈ C ∞ ( T × R n ) is said to be quadraticallyconvex if there are positive numbers h and h such that h I ≤ ∇ H t ( x ) ≤ h I for all x ∈ R n and t ∈ T .Note that the upper bound on the Hessian of H implies that the Hamiltonian vector field X H is globally Lipschitz-continuous in the space variables, | X H t ( y ) − X H t ( x ) | ≤ h | y − x | ∀ t ∈ T , ∀ x, y ∈ R n , (5.1)and in particular X H has linear growth at infinity.Let C ⊂ R n be a bounded convex open set with smooth boundary. Assume moreoverthat all the sectional curvatures of ∂C are positive. This is equivalent to the fact that, aftershifting C so that the origin belongs to its interior, the second differential of the positively2-homogeneous function H C at every point in R n \ { } is positive definite. We shall referto such a set as a smooth strongly convex domain. From the compactness of ∂C and fromthe 2-homogeneity of H C we deduce the bounds c I ≤ ∇ H C ( x ) ≤ c I for all x ∈ R n \ { } , for suitable positive numbers c and c . Therefore, the square H C ofthe Minkowski gauge function of the strongly convex domain C is quadratically convex,except for the lack of differentiability at the origin.23hroughout this section, the Hamiltonian H : T × R n → R is assumed to be smoothand quadratically convex. We denote by H ∗ t : R n → R , H ∗ t ( x ) := max y ∈ R n (cid:0) x · y − H ( t, y ) (cid:1) , the Fenchel conjugate of H t . The properties of Fenchel duality imply that the function H ∗ : T × R n → R is smooth and satisfies h − I ≤ ∇ H ∗ t ( x ) ≤ h − I ∀ x ∈ R n . Clarke’s dual action functional is defined by the formulaΨ H ∗ ( x ) := − ˆ T J ˙ x ( t ) · x ( t ) dt + ˆ T H ∗ t (cid:0) J ˙ x ( t ) (cid:1) dt. The functional Ψ H ∗ is continuously differentiable on the Hilbert space H := H ( T , R n ) / R n , where the action of R n onto the Sobolev space H ( T , R n ) is given by translations. Ratherthan working with equivalence classes of curves modulo translations, it is convenient towork with genuine curves by identifying H with the space of closed curves with zero mean: H = (cid:110) x ∈ H ( T , R n ) | ˆ T x ( t ) dt = 0 (cid:111) . On this space, it is convenient to use the inner product( x, y ) H := ( ˙ x, ˙ y ) L which induces a norm equivalent to the H -norm in H . We denote by π : H ( T , R n ) −→ H , π ( x ) = x − ˆ T x ( t ) dt the quotient projection. The differential of Ψ H ∗ is given by the formula d Ψ H ∗ ( x )[ v ] = − ˆ T J ˙ v ( t ) · x ( t ) dt + ˆ T dH ∗ t (cid:0) J ˙ x ( t ) (cid:1) [ J ˙ v ( t )] dt = − ˆ T J ˙ v ( t ) · x ( t ) dt + ˆ T J ˙ v ( t ) · ∇ H ∗ t (cid:0) J ˙ x ( t ) (cid:1) dt. (5.2)Since ∇ H ∗ t is Lipschitz continuous, so is the differential d Ψ H ∗ .There is a one-to-one correspondence between the critical points of Φ H and Ψ H ∗ . Moreprecisely, we have the following well known fact, of which we include a proof for sake ofcompleteness. 24 emma . If x is a critical point of Φ H , then π ( x ) is a critical point of Ψ H ∗ . Conversely,every critical point x of Ψ H ∗ is smooth and there exists a unique vector v ∈ R n such that x + v is a critical point of Φ H . In this case, we have Φ H ( x + v ) = Ψ H ∗ ( x ) . Proof. Assume that x is a critical point of Φ H . Then x is smooth and is a 1-periodic orbitof X H , that is, J ˙ x ( t ) = ∇ H t ( x ( t )) ∀ t ∈ T . Fenchel duality implies that for every t ∈ T the map ∇ H ∗ t is the inverse of the map ∇ H t .Therefore, the above identity implies ∇ H ∗ t (cid:0) J ˙ x ( t ) (cid:1) = x ( t ) ∀ t ∈ T . Then y := π ( x ) satisfies ∇ H ∗ t (cid:0) J ˙ y ( t ) (cid:1) = y ( t ) + ˆ x ∀ t ∈ T , where ˆ x ∈ R n denotes the average of x . For every v ∈ H we have d Ψ H ∗ ( y )[ v ] = − ˆ T J ˙ v ( t ) · y ( t ) dt + ˆ T J ˙ v ( t ) · ∇ H ∗ t (cid:0) J ˙ y ( t ) (cid:1) dt = ˆ T J ˙ v ( t ) · ˆ x dt = 0 , so π ( x ) is a critical point of Ψ H ∗ .Now assume that x ∈ H is a critical point of Ψ H ∗ . Then ˆ T J ˙ v ( t ) · (cid:16) x ( t ) − ∇ H ∗ t (cid:0) J ˙ x ( t ) (cid:1)(cid:17) dt = 0for every v ∈ H , and hence there exists v ∈ R n such that ∇ H ∗ t (cid:0) J ˙ x ( t ) (cid:1) − x ( t ) = v for a.e. t ∈ T , that is, ∇ H ∗ t (cid:0) J ˙ x ( t ) (cid:1) = x ( t ) + v for a.e. t ∈ T . By applying the map ∇ H t to the above identity we obtain J ˙ x ( t ) = ∇ H t ( x ( t ) + v ) for a.e. t ∈ T . The above identity and a standard bootstrap argument imply that the curve x is smoothand x + v is a 1-periodic orbit of X H , and hence a critical point of Φ H .The fact that x ( t ) + v is the image of J ˙ x ( t ) by ∇ H ∗ t implies that H t ( x ( t ) + v ) + H ∗ t ( J ˙ x ( t )) = J ˙ x ( t ) · ( x ( t ) + v ) ∀ t ∈ T , T we find ˆ T H t ( x ( t ) + v ) dt + ˆ T H ∗ t ( J ˙ x ( t )) dt = ˆ T J ˙ x ( t ) · x ( t ) dt. We conclude thatΨ H ∗ ( x ) − Φ H ( x + v ) = − ˆ T J ˙ x ( t ) · x ( t ) dt + ˆ T H ∗ t ( J ˙ x ( t )) dt + ˆ T H t ( x ( t ) + v ) dt = 0 . The following result plays a fundamental role in the construction of the isomorphismbetween the Morse complex induced by Ψ H ∗ and the Floer complex of Φ H . Proposition . Let x ∈ H ( T , R n ) and y ∈ R n ⊕ H − / . Then we have Φ H ( x + y ) ≤ Ψ H ∗ ( π ( x )) − (cid:107) P − y (cid:107) / , (5.3) with the equality holding if and only if J ˙ x = ∇ H t ( x + y ) almost everywhere. In particular,the equality Φ H ( x + y ) = Ψ H ∗ ( π ( x )) holds if and only if y ∈ R n and x + y is a critical point of Φ H .Proof. Fenchel duality implies that H t ( x + y ) ≥ J ˙ x · ( x + y ) − H ∗ t ( J ˙ x ) a.e. (5.4)with equality if and only if J ˙ x = ∇ H t ( x + y ) a.e. (5.5)By integration we getΦ H ( x + y ) = 12 ˆ T J ( ˙ x + ˙ y ) · ( x + y ) dt − ˆ T H t ( x + y ) dt ≤ ˆ T J ( ˙ x + ˙ y ) · ( x + y ) dt − ˆ T J ˙ x · ( x + y ) dt + ˆ T H ∗ t ( J ˙ x ) dt = − ˆ T J ˙ x · x dt + 12 ˆ T J ˙ y · y + ˆ T H ∗ t ( J ˙ x ) dt = Ψ H ∗ ( x ) − (cid:107) P − y (cid:107) / , where in the last equality we have used the fact that y belongs to H − / ⊕ R n . This showsthat (5.3) holds, with equality if and only if the Fenchel inequality (5.4) is an equality, thatis if and only if (5.5) holds. In particular,Φ H ( x + y ) ≤ Ψ H ∗ ( π ( x )) . with equality if and only if P − y = 0 and (5.5) holds. This is equivalent to the fact that y is a constant loop and the loop x + y is a 1-periodic orbit of X H , or equivalently a criticalpoint of Φ H . 26he functional Ψ H ∗ is in general not twice differentiable, unless the function x (cid:55)→ H t ( x )is quadratic for every t ∈ T , but it is twice Gateaux differentiable, meaning that for every x, v ∈ H the limit ∇ Ψ H ∗ ( x ) v := lim h → h (cid:0) ∇ Ψ H ∗ ( x + hv ) − ∇ Ψ H ∗ ( x ) (cid:1) exists and defines a continuous linear operator ∇ Ψ H ∗ ( x ) on H . The corresponding seconddifferential of Ψ H ∗ at x has the form d Ψ H ∗ ( x )[ u, v ] = − ˆ T J ˙ u · v dt + ˆ T ∇ H ∗ t ( J ˙ x ) J ˙ u · J ˙ v dt. The second integral defines a coercive bilinear form on H . The first integral defines ablinear form which is continuous in H / and, thanks to the compactness of the embedding H (cid:44) → H / , this bilinear form is represented by a compact operator with respect to theinner product of H . It follows that the critical point x of Ψ H ∗ has finite Morse index andfinite nullityind( x ; Ψ H ∗ ) := dim V − (cid:0) ∇ Ψ H ∗ ( x ) (cid:1) = max { dim W | W linear subspace of H , d Ψ H ∗ ( x ) negative definite on W } , null( x ; Ψ H ∗ ) := dim ker ∇ Ψ H ∗ ( x ) . The next result could be deduced from Proposition 1.1 and from the relationship betweenthe Conley-Zehnder index and the Morse index of the dual action functional, see [Bro86,Bro90, Lon02]. Here we give a direct proof. Proposition . Let x be a critical point of Φ H and let π ( x ) be the corresponding criticalpoint of Ψ H ∗ . Then null( x ; Φ H ) = null( π ( x ); Ψ H ∗ ) and ind H − / ⊕ E − ( x ; Φ H ) = ind( π ( x ); Ψ H ∗ ) + n. Proof. We denote by (cid:98) Ψ H ∗ : H ( T , R n ) −→ R , (cid:98) Ψ H ∗ = Ψ H ∗ ◦ π the natural lift of the functional Ψ H ∗ . Notice that x ∈ H ( T , R n ) is a critical point of (cid:98) Ψ H ∗ if and only if π ( x ) is a critical point of Ψ H ∗ . In this case we havenull( x ; (cid:98) Ψ H ∗ ) = null( π ( x ); Ψ H ∗ ) + 2 n, ind( x ; (cid:98) Ψ H ∗ ) = ind( π ( x ); Ψ H ∗ ) . (5.6)Now let x ∈ H / be a critical point of Φ H . Being a 1-periodic orbit of X H , x : T → R n issmooth and in particular in H ( T , R n ). Let S be the smooth loop of positive symmetricendomorphisms of R n defined by S ( t ) := ∇ H t ( x ( t )) . 27y Fenchel duality we have ∇ H ∗ t ◦ ∇ H t = id , and differentiation gives us ∇ H ∗ t ( ∇ H t ( z )) ∇ H t ( z ) = I ∀ z ∈ R n , ∀ t ∈ T . If z = x ( t ) then ∇ H t ( z ) = J ˙ x ( t ), so the above identity yields ∇ H ∗ t ( J ˙ x ( t )) ∇ H t ( x ( t )) = I, ∀ t ∈ T , and hence ∇ H ∗ t ( J ˙ x ( t )) = S ( t ) − , ∀ t ∈ T . If we denote by Ω the symmetric bilinear formΩ : H / × H / → R , Ω[ u, v ] = ˆ T J ˙ u · v dt, the second differentials of Φ H and (cid:98) Ψ H ∗ at x take the form d Φ H ( x )[ u, v ] = Ω[ u, v ] − ˆ T Su · v dt, ∀ u, v ∈ H / d (cid:98) Ψ H ∗ ( x )[ u, v ] = − Ω[ u, v ] + ˆ T S − ( J ˙ u ) · ( J ˙ v ) dt, ∀ u, v ∈ H . (5.7)Since S ( t ) is symmetric and positive definite, the formula( u, v ) S := ˆ T S ( t ) u ( t ) · v ( t ) dt. defines an equivalent inner product on L ( T , R n ). Let T be the symmetric operator whichrepresents the bilinear form Ω with respect to the inner product ( · , · ) S :Ω[ u, v ] = ( T u, v ) S , where T := S − J ddt . Here T is an unbounded operator on L ( T , R n ) with domain H ( T , R n ) and the aboveidentity holds for every u ∈ H ( T , R n ) and v ∈ L ( T , R n ). The operator T is self-adjointand has a compact resolvent. Its spectrum is discrete and consists of real eigenvalues withfinite multiplicities which are unbounded from above and from below. The space L ( T , R n )admits a Hilbert basis { ϕ j } j ∈ Z of eigenvectors of T which is orthonormal with respect tothe inner product ( · , · ) S : T ϕ j = µ j ϕ j ∀ j ∈ Z . Here, the real eigenvalues µ j satisfylim j →−∞ µ j = −∞ , lim j → + ∞ µ j = + ∞ . J ˙ ϕ j = µ j Sϕ j , the eigenvectors ϕ j ’s are smooth loops. There are exactly 2 n eigenvalues µ j ’s with value 0and the corresponding eigenvectors ϕ j are constant loops forming a basis of R n .Let V be the closure in H / of the linear subspacespan { ϕ j | µ j < } . Then V is a maximal subspace of H / on which Ω is negative definite. Therefore, H / = V ⊕ R n ⊕ H +1 / . (5.8)By the first formula in (5.7), the second differential of Φ H at x has the following represen-tation with respect to the inner product ( · , · ) S : d Φ H ( x )[ u, v ] = (( T − I ) u, v ) S . Therefore, ker d Φ H ( x ) = ker( T − I ) = span { ϕ j | µ j = 1 } (5.9)Moreover, the H / -closure of the linear subspacespan { ϕ j | µ j < } is a maximal subspace of H / on which d Φ H ( x ) is negative definite. This space has theform V ⊕ R n ⊕ W, where W is the following finite dimensional subspace W := span { ϕ j | < µ j < } . Together with (5.8), we deduce thatind H − / ⊕ E − ( x ; Φ H ) = dim( V ⊕ R n ⊕ W, H − / ⊕ E − )= dim( V, H − / ) + dim W + dim E + = dim W + n. (5.10)Since ˆ T S − ( J ˙ u ) · ( J ˙ v ) dt = ˆ T ( J ˙ u ) · S − ( J ˙ v ) dt = ˆ T SS − ( J ˙ u ) · S − ( J ˙ v ) dt = ( T u, T v ) S = ( T u, v ) S , the second formula in (5.7) gives us the following representation for d (cid:98) Ψ H ∗ ( x ) with respectto the inner product ( · , · ) S : d (cid:98) Ψ H ∗ ( x )[ u, v ] = (( T − T ) u, v ) S = ( T ( T − I ) u, v ) S . { ϕ j } j ∈ Z is a basis of eigenvectors for T ( T − I ) and the eigenvalue corre-sponding to ϕ j is µ j ( µ j − µ j = 0 or µ j = 1. Therefore,the kernel of d (cid:98) Ψ H ∗ ( x ) is given byker d (cid:98) Ψ H ∗ ( x ) = span { ϕ j | µ j = 0 } ⊕ span { ϕ j | µ j = 1 } = R n ⊕ ker d Φ H ( x ) , where we have used (5.9). Together with the first identity in (5.6), we deduce thatnull( x ; Φ H ) = null( x ; (cid:98) Ψ H ∗ ) − n = null( π ( x ); Ψ H ∗ ) . The eigenvalue µ j ( µ j − 1) is negative if and only if 0 < µ j < 1. Therefore, the finitedimensional space W defined above is a maximal subspace of H ( T , R n ) on which d (cid:98) Ψ H ∗ ( x )is negative definite and henceind ( x ; (cid:98) Ψ H ∗ ) = dim W = ind H − / ⊕ E − ( x ; Φ H ) − n, where we have used (5.10). By the second identity in (5.6), we conclude thatind ( π ( x ); Ψ H ∗ ) = ind H − / ⊕ E − ( x ; Φ H ) − n. Let C be a smooth strongly convex domain, as defined at the beginning of this section.We conclude this section by showing how Clarke’s duality can be used to determine therelative Morse index - and hence by Proposition 1.1 the Conley-Zehnder index - of periodicHamiltonian orbits that correspond to closed characteristics of ∂C having minimal action. Proposition . Let C be a smooth strongly convex domain. If γ : R /T Z → ∂C is aperiodic orbit of R α C = X H C | ∂C with minimal period T among all periodic Reeb orbits on ∂C , then setting x γ ( t ) := γ ( T t ) we have ind H − / ⊕ E − ( x γ , Φ T H C ) = n. Proof. Set H := ϕ ◦ H C , with ϕ ( r ) = 2 Tp r p , for some real number p ∈ (1 , x γ is a criticalpoint of Φ H . Let q > /p + 1 /q = 1. By Lemma 5.1, π ( x γ ) is a critical pointof the dual functional Ψ H ∗ defined on the space W ,q ( T , R n ) of W ,q -loops in R n withzero mean. We use the well-known fact, which we will recall below, that Ψ H ∗ is boundedfrom below and attains a global minimizer that corresponds exactly to a periodic orbit of X H C on ∂C with minimal period. From this, we deduce that π ( x γ ) is a minimizer of Ψ H ∗ and therefore ind ( π ( x γ ); Ψ H ∗ ) = 0 . H − / ⊕ E − ( x γ ; Φ T H C ) = ind H − / ⊕ E − ( x γ ; Φ H ) = ind ( π ( x γ ); Ψ H ∗ ) + n = n. For sake of completeness, we include a proof of the well-known fact mentioned above.Applying the Poincar´e inequality and using the fact that H ∗ is q -homogeneous with q > x of W ,q ( T , R n ):Ψ H ∗ ( x ) = − ˆ T J ˙ x · x dt + ˆ T H ∗ ( J ˙ x ) dt ≥ − (cid:107) ˙ x (cid:107) L (cid:107) x (cid:107) L + c (cid:107) ˙ x (cid:107) qL q ≥ − (cid:107) ˙ x (cid:107) L + c (cid:107) ˙ x (cid:107) qL q ≥ c (cid:107) ˙ x (cid:107) qL q − d for some suitable constants c, d > 0. Therefore, Ψ H ∗ is coercive on W ,q ( T , R n ). On thisspace, the functional Ψ H ∗ is weakly lower semi-continuous, because the quadratic form12 ˆ T J ˙ x · x dt is weakly continuous, being strongly continuous on the space H / where W ,q ( T , R n )embeds compactly into, and the term ˆ T H ∗ ( J ˙ x ) dt is strongly continuous and convex. Being coercive and weakly lower semi-continuous, Ψ H ∗ attains its minimum on the reflexive Banach space W ,q ( T , R n ).Moreover, the minimum of Ψ H ∗ on W ,q ( T , R n ) is negative, since for any non-zeroelement x of W ,q ( T , R n ) and any λ > H ∗ ( λx ) = − λ ˆ T J ˙ x · x dt + λ q ˆ T H ∗ ( − J ˙ x ) dt < , because q > 2. Since the loop z mapping to the origin is a unique constant critical pointof Ψ H ∗ with Ψ H ∗ ( z ) = 0, every minimizer of Ψ H ∗ is non-constant.Next we compute critical values of Ψ H ∗ . If x is a critical point of Ψ H ∗ and y is thecritical point of Φ H with π ( y ) = x , thenΨ H ∗ ( x ) = Φ H ( y ) = 12 ˆ T J ˙ y · y dt − ˆ T H ( y ) dt, = 12 ˆ T ∇ H ( y ) · y dt − H ( y (0)) = (cid:16) p − (cid:17) H ( y (0))where we have used the Euler identity and the fact that H is constant along y . In particularif x is non-constant, Ψ H ∗ ( x ) < 0. A simple computation shows that the curve t (cid:55)→ H ( y (0)) − p y (cid:18) p H ( y (0)) − pp t (cid:19) , t ∈ R 31s a periodic orbit of X H C sitting on ∂C with period p H ( y (0)) p − p = p (cid:18) p − H ∗ ( x ) (cid:19) p − p . The above formula shows the following: There exists a one-to-one correspondence betweenthe closed orbits of X H C on ∂C and the critical points of Ψ H ∗ with negative critical value,and the function that to every negative critical value of Ψ H ∗ associates the period of thecorresponding closed orbit - or orbits - of X H C on ∂C is strictly monotonically increasing.In particular, the global minimizers of Ψ H ∗ correspond to periodic Reeb orbits on ∂C withminimal period, as claimed above. The first aim of this section is to prove the Palais-Smale condition for the dual actionfunctional that is associated with a quadratically convex Hamiltonian that is non-resonantat infinity. We begin with the following lemma: Lemma . Let K : T × R n → R be a smooth function such that | dK t ( x ) | ≤ c (1 + | x | ) ∀ ( t, x ) ∈ T × R n , (6.1) and ∇ K t ( x ) ≥ δI ∀ ( t, x ) ∈ T × R n , (6.2) for some positive number δ . Then the functional Ψ K : H → R , Ψ K ( x ) = − ˆ T J ˙ x ( t ) · x ( t ) dt + ˆ T K t ( J ˙ x ( t )) dt, has the following property: If the sequence ( x h ) ⊂ H converges weakly to x ∈ H and d Ψ K ( x h ) converges to zero strongly in the dual of H , then ( x h ) converges to x strongly in H .Proof. Assumption (6.1) guarantees that Ψ K is continuously differentiable on H . Fromthe assumption on ( d Ψ K ( x h )) and the boundedness of ( x h − x ) in H we deduce that thereal sequence d Ψ K ( x h )[ x h − x ] = − ˆ T J ( ˙ x h − ˙ x ) · x h dt + ˆ T dK t ( J ˙ x h )[ J ( ˙ x h − ˙ x )] dt is infinitesimal. The fact that ( ˙ x h − ˙ x ) converges to zero weakly in L and ( x h ) convergesto x weakly in H and hence strongly in L implies that the first integral in the aboveexpression is infinitesimal. Therefore, the second integral must be infinitesimal too: ˆ T dK t ( J ˙ x h )[ J ( ˙ x h − ˙ x )] dt = o (1) . (6.3)32rom assumption (6.2) we deduce the inequality dK t ( J ˙ x h )[ J ( ˙ x h − ˙ x )] − dK t ( J ˙ x )[ J ( ˙ x h − ˙ x )]= ˆ d K t ( J ˙ x + sJ ( ˙ x h − ˙ x ))[ J ( ˙ x h − ˙ x ) , J ( ˙ x h − ˙ x )] ds ≥ δ | J ( ˙ x h − ˙ x ) | = δ | ˙ x h − ˙ x | a.e.By integrating this inequality over T we get δ ˆ T | ˙ x h − ˙ x | ds ≤ ˆ T dK t ( J ˙ x h )[ J ( ˙ x h − ˙ x )] dt − ˆ T dK t ( J ˙ x )[ J ( ˙ x h − ˙ x )] dt. The first integral on the right-hand side is infinitesimal because of (6.3). The secondintegral is also infinitesimal, because ( ˙ x h − ˙ x ) converges to zero weakly in L and dK t ( J ˙ x )is an L function, thanks to (6.1). We conclude that the L -norm of ˙ x h − ˙ x is infinitesimal,that is, ( x h ) converges to x in H . Proposition . Assume that the Hamiltonian H ∈ C ∞ ( T × R n ) is quadratically convexand non-resonant at infinity. Then the dual action functional Ψ H ∗ : H → R satisfiesthe Palais-Smale condition. More precisely, any sequence ( x h ) ⊂ H such that d Ψ H ∗ ( x h ) converges to zero strongly in the dual of H has a convergent subsequence.Proof. By (5.2), the differential of Ψ H ∗ at x ∈ H has the form d Ψ H ∗ ( x )[ v ] = (cid:0) ˙ v, J ( x − ∇ H ∗ t ( J ˙ x ) (cid:1) L ( T ) . Therefore, endowing H with the inner product given by the L -product of the derivatives,the gradient of Ψ H ∗ has the form ∇ Ψ H ∗ ( x ) = Π (cid:0) J ( x − ∇ H ∗ t ( J ˙ x ) (cid:1) , where Π : L ( T , R n ) → H , (Π v )( t ) = ˆ t v ( s ) ds − ˆ T (cid:18) ˆ t v ( s ) ds (cid:19) dt. (6.4)is the linear operator mapping each v into the primitive of v with zero mean.Let ( x h ) ⊂ H be a sequence such that ( d Ψ H ∗ ( x h )) converges to zero strongly in H ∗ , orequivalently ( ∇ Ψ H ∗ ( x h )) converges to zero strongly in H . ThenΠ( x h − ∇ H ∗ t ( J ˙ x h )) = y h where ( y h ) ⊂ H converges to zero strongly. Differentiation in t yields x h − ∇ H ∗ t ( J ˙ x h ) = ˙ y h . By applying the nonlinear map ∇ H t to the identity ∇ H ∗ t ( J ˙ x h ) = x h − ˙ y h , 33e obtain J ˙ x h = ∇ H t ( x h − ˙ y h ) , or equivalently ˙ x h = X H t ( x h − ˙ y h ) . (6.5)Therefore, using the fact that X H t is globally Lipschitz-continuous (see (5.1)), we find | ˙ x h − X H t ( x h ) | = | X H t ( x h − ˙ y h ) − X H t ( x h ) | ≤ h | ˙ y h | , and integrating over T we obtain the bound (cid:107) ˙ x h − X H ( x h ) (cid:107) L ( T ) ≤ h (cid:107) ˙ y h (cid:107) L ( T ) . Therefore, the sequence ( ˙ x h − X H ( x h )) is infinitesimal in L ( T ), and hence the non-resonance at infinity assumption implies that ( x h ) is uniformly bounded in L . But thenthe identity (6.5) and the linear growth of X H imply that ( ˙ x h ) is uniformly bounded in L ( T ), and hence ( x h ) is uniformly bounded in H .Up to passing to a subsequence, we may assume that ( x h ) converges to some x weaklyin H . By Lemma 6.1 we conclude that this convergence is strong. This proves that Ψ H ∗ satisfies the Palais-Smale condition.If the quadratically convex Hamiltonian H ∈ C ∞ ( T × R n ) is non-degenerate, then thefunctional Ψ H ∗ is Morse, meaning that the (Gateaux) second differential of Ψ H ∗ at eachcritical point is non-degenerate. However, the functional Ψ H ∗ is in general not of class C (it is not even twice differentiable), so some care is needed in order to associate a Morsecomplex with it.One way of doing this would be to show that Ψ H ∗ admits a smooth pseudo-gradientvector field on H with good properties. This has been done in another setting for afunctional whose analytical properties are similar to those of Ψ H ∗ , see [AS09]. Here weprefer to use a different strategy and to use the fact that Ψ H ∗ is smooth when restrictedto a suitable finite dimensional smooth submanifold of H , which contains all the criticalpoints of Ψ H ∗ and is defined by a saddle-point reduction. This approach has also beenused by Viterbo in [Vit89].Given a natural number N ∈ N , consider the splitting H = H N, +1 ⊕ (cid:98) H N, +1 , where H N, +1 := (cid:40) x ∈ H | x ( t ) = N (cid:88) k =1 e − πkJ t ˆ x k , ˆ x k ∈ R n (cid:41) , (cid:98) H N, +1 := (cid:40) x ∈ H | x ( t ) = (cid:88) k ≤− e − πkJ t ˆ x k + (cid:88) k ≥ N +1 e − πkJ t ˆ x k , ˆ x k ∈ R n (cid:41) . This splitting is orthogonal with respect to the H and to the L inner products. Weidentify H with the product space H N, +1 × (cid:98) H N, +1 . The following proposition summarizesthe main properties of the saddle point reduction.34 roposition . Assume the Hamiltonian H ∈ C ∞ ( T × R n ) to be quadratically convexand non-resonant ay infinity. If N ∈ N is large enough, then the following facts hold:(a) For every x ∈ H N, +1 the restriction of Ψ H ∗ to { x } × (cid:98) H N, +1 has a unique critical point ( x, Y ( x )) , which is a non-degenerate global minimizer of this restriction.(b) The map Y : H N, +1 → (cid:98) H N, +1 takes values into C ∞ ( T , R n ) and is smooth with respectto the C k -norm on the target, for any k ∈ N . In particular, its graph M := { ( x, y ) ∈ H N, +1 × (cid:98) H N, +1 | y = Y ( x ) } is a smooth nN -dimensional submanifold of H consisting of smooth loops.(c) The restriction of Ψ H ∗ to M , which we denote by ψ H ∗ : M → R , is smooth.(d) A point z ∈ H is a critical point of Ψ H ∗ if and only if it belongs to M and is a criticalpoint of ψ H ∗ . In this case, the Morse index and the nullity with respect to the twofunctionals coincide: ind( z ; Ψ H ∗ ) = ind( z ; ψ H ∗ ) , null( z ; Ψ H ∗ ) = null( z ; ψ H ∗ ) . (e) If M is endowed with the Riemannian metric induced by the inclusion into H , thenthe functional ψ H ∗ satisfies the Palais-Smale condition. The proof of this proposition is contained at the end of this section. If we furtherassume that the Hamiltonian H is non-degenerate, we obtain that ψ H ∗ is a smooth Morsefunction with finitely many critical points and satisfying the Palais-Smale condition on thefinite-dimensional manifold M . As such, it has a Morse complex, which we denote by { M ∗ ( ψ H ∗ ) , ∂ M } and is uniquely defined up to chain isomorphisms. The space M ∗ ( ψ H ∗ ) is the Z -vectorspace generated by the critical points of ψ H ∗ , graded by the Morse index. The boundaryoperator ∂ M : M ∗ ( ψ H ∗ ) → M ∗− ( ψ H ∗ )is defined by the formula ∂ M x = (cid:88) y n M ( x, y ) y ∀ x ∈ crit ψ H ∗ , where y ranges over all critical points with Morse index equal to the index of x minus 1 and n M ( x, y ) ∈ Z is the parity of the finite set of negative gradient flow lines of ψ H ∗ going from x to y . Here, the negative gradient vector field of ψ H ∗ is induced by a generic Riemannianmetric on M , uniformly equivalent to the standard one and such that the negative gradientflow is Morse-Smale, meaning that stable and unstable manifolds of pairs of critical pointsmeet transversally. Changing the generic metric changes the Morse complex by a chain35somorphism. The homology of the Morse complex of ψ H ∗ is isomorphic to the singularhomology of the pair ( M, { ψ H ∗ < a } ), where a is any number which is smaller than thesmallest critical level of ψ H ∗ : HM k ( ψ H ∗ ) ∼ = H k ( M, { ψ H ∗ < a } ) . (6.6)We conclude this section by proving Proposition 6.3. Proof of Proposition 6.3. By the inequality ∇ H ∗ t ( x ) ≥ h − I , we have for every x, u ∈ H , d Ψ H ∗ ( x )[ u, u ] = − ˆ T J ˙ u · u dt + ˆ T ∇ H ∗ t ( J ˙ x ) J ˙ u · J ˙ u dt ≥ − ˆ T J ˙ u · u dt + 1 h (cid:107) ˙ u (cid:107) L = − π (cid:88) k ∈ Z k | ˆ u k | + 4 π h (cid:88) k ∈ Z k | ˆ u k | We choose N ∈ N so that 2 π ( N + 1) > h . For all x ∈ H and all u ∈ (cid:98) H N, +1 , we have d Ψ H ∗ ( x )[ u, u ] ≥ π h (cid:88) k ≤− k | ˆ u k | + 2 π (cid:88) k ≥ N +1 (cid:18) πkh − (cid:19) k | ˆ u k | ≥ π δ (cid:32) (cid:88) k ≤− k | ˆ u k | + (cid:88) k ≥ N +1 k | ˆ u k | (cid:33) = δ (cid:107) u (cid:107) H (6.7)where δ > x ∈ H N, +1 , thesecond differential of the function (cid:98) H N, +1 → R , y (cid:55)→ Ψ H ∗ ( x + y )is bounded from below by a coercive quadratic form. In particular, this function is strictlyconvex and coercive and hence has a unique non-degenerate critical point Y ( x ) which is aminimizer. This proves (a).From the expression (5.2) for d Ψ H ∗ we deduce that the gradient of Ψ H ∗ with respectto the inner product ( · , · ) H is ∇ Ψ H ∗ ( x ) = Π (cid:0) J x − J ∇ H ∗ t ( J ˙ x ) (cid:1) (6.8)where Π : L ( T , R n ) → H is the inverse of the derivative given by (6.4). The vector y ∈ (cid:98) H N, +1 satisfies y = Y ( x ) for some x ∈ H N, +1 if and only if ∇ Ψ H ∗ ( x + y ) ∈ H N, +1 whichby (6.8) is equivalent to J ( x + y ) − J ∇ H ∗ t ( J ( ˙ x + ˙ y )) = ˙ u u ∈ H N, +1 . The above equality can be reformulated as˙ y = − J ∇ H t ( x + y + J ˙ u ) − ˙ x. (6.9)The fact that x and u are smooth implies that y = Y ( x ) is also smooth.We now deal with the regularity of the map Y and start by showing that Y is Lipschitzcontinuous. We use subscripts 1 and 2 to denote partial derivatives with respect to thesplitting H = H N, +1 × (cid:98) H N, +1 . Then by (a) we have for every x ∈ H N, +1 , ∇ Ψ H ∗ ( x, Y ( x )) = 0 . From the fact that Ψ H ∗ is twice Gateaux-differentiable and from the lower bound (6.7), wededuce for all x ∈ H N, +1 and all u ∈ (cid:98) H N, +1 (cid:107)∇ Ψ H ∗ ( x, Y ( x ) + u ) (cid:107) H = (cid:107)∇ Ψ H ∗ ( x, Y ( x ) + u ) − ∇ Ψ H ∗ ( x, Y ( x )) (cid:107) H = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ˆ ∇ Ψ H ∗ ( x, Y ( x ) + tu ) dt (cid:19) u (cid:13)(cid:13)(cid:13)(cid:13) H ≥ δ (cid:107) u (cid:107) H . (6.10)The above inequality with x replaced by x + h , h ∈ H N, +1 and with u = Y ( x ) − Y ( x + h )gives us δ (cid:107) Y ( x + h ) − Y ( x ) (cid:107) H ≤ (cid:107)∇ Ψ H ∗ ( x + h, Y ( x )) (cid:107) H = (cid:107)∇ Ψ H ∗ ( x + h, Y ( x )) − ∇ Ψ H ∗ ( x, Y ( x )) (cid:107) H ≤ c (cid:107) h (cid:107) H for some constant c > ∇ Ψ H ∗ is Lipschitz continuous. Thisproves that Y is Lipschitz continuous.Let x, h ∈ H N, +1 . By the Gateaux differentiability of ∇ Ψ H ∗ , a first order expansionyields ∇ Ψ H ∗ (cid:0) x + th, Y ( x ) − t ∇ Ψ H ∗ ( x, Y ( x )) − ∇ Ψ H ∗ ( x, Y ( x )) h (cid:1) = ∇ Ψ H ∗ ( x, Y ( x )) + t ∇ Ψ H ∗ ( x, Y ( x )) h − t ∇ Ψ H ∗ ( x, Y ( x )) ∇ Ψ H ∗ ( x, Y ( x )) − ∇ Ψ H ∗ ( x, Y ( x )) h + o ( t )= o ( t ) , where ∇ Ψ( x, Y ( x )) is invertible since it is self adjoint and bounded from below as ob-served in (6.7). On the other hand, the bound (6.10) with x replaced by x + th and with u = − Y ( x + th ) + Y ( x ) − t ∇ Ψ H ∗ ( x, Y ( x )) − ∇ Ψ H ∗ ( x, Y ( x )) h gives δ (cid:107) Y ( x + th ) − Y ( x ) + t ∇ Ψ H ∗ ( x, Y ( x )) − ∇ Ψ H ∗ ( x, Y ( x )) h (cid:107) H ≤ (cid:13)(cid:13) ∇ Ψ H ∗ (cid:0) x + th, Y ( x ) − t ∇ Ψ H ∗ ( x, Y ( x )) − ∇ Ψ H ∗ ( x, Y ( x ) (cid:1) h (cid:13)(cid:13) H = o ( t ) . Y : H N, +1 → (cid:98) H N, +1 is Gateaux-differentiable with Gateaux-gradient ∇ Y ( x ) = −∇ Ψ H ∗ ( x, Y ( x )) − ∇ Ψ H ∗ ( x, Y ( x )) . (6.11)We have already seen that the map Y takes values in C ∞ ( T , R n ). We claim that Y iscontinuous with respect to the C ∞ -topology on the target space. Indeed, we assume that( x n ) ⊂ H N, +1 converges to x in the H -norm. Since H N, +1 is contained in C ∞ ( T , R n ) andis finite dimensional, ( x n ) converges to x in the C k -norm for any k ∈ N . The vector u n = ∇ Ψ H ∗ ( x n , Y ( x n ))converges to u = ∇ Ψ H ∗ ( x, Y ( x )) in the H -norm due to the continuity of Y and ∇ Ψ H ∗ .Being a sequence in H N, +1 , the sequence ( u n ) converges to u in any C k -norm. As seen in(6.9), the vector y n = Y ( x n ) is characterized by˙ y n = − J ∇ H t ( x n , y n , J ˙ u n ) − ˙ x n . This ODE shows that ( y n ) converges to the solution y = y ( x ) of˙ y = − J ∇ H t ( x, y, J ˙ u ) − ˙ x in any C k -norm. This proves the claim.Although Ψ H ∗ is not of class C , the map x (cid:55)→ ∇ Ψ H ∗ ( x ) is easily seen to be continuousfrom the C -topology on H to the operator norm topology on L ( H , H ). Then the identity(6.11) and the regularity property of Y proven above yield that the map ∇ Y : H N, +1 → L ( H N, +1 , (cid:98) H N, +1 )is continuous. This together with the Gateaux-differentiability of Y implies that the map Y : H N, +1 → (cid:98) H N, +1 is of class C by the total differential theorem.Since the restriction of Ψ H ∗ to C k ( T , R n ) is smooth for all k ∈ N , the above argumentcan be bootstrapped and implies that the map Y is smooth with respect to the C k -normfor all k ∈ N on the target. This completes the proof of (b).Statement (c) follows from (b) and the smoothness property of Ψ H ∗ mentioned above.To prove (d) we first observe that all critical points of Ψ H ∗ are contained in M . A point( x, Y ( x )) ∈ M is a critical point of the restriction ψ H ∗ of Ψ H ∗ to M if and only if d Ψ H ∗ ( x, Y ( x )) | T ( x,Y ( x )) M = 0which is equivalent to d Ψ H ∗ ( x, Y ( x )) = 0 since d Ψ H ∗ ( x, Y ( x )) | (cid:98) H N, +1 = 0 and H = T ( x,Y ( x )) M ⊕ (cid:98) H N, +1 . This proves the first statement of (d).The estimate (6.7) that d Ψ H ∗ ( x, Y ( x )) is positive on (cid:98) H N, +1 guarantees that the indexand the nullity does not change when restricting Ψ H ∗ to M . This completes the proof of (d).The statement (e) follows immediately from Proposition 6.2 since any sequence ( z h ) ⊂ M with the property that dψ H ∗ ( z h ) converges to zero with respect to the Riemannianmetric induced from H satisfies also that d Ψ H ∗ ( z h ) strongly converges to zero.38 The functional setting for the hybrid problem Throughout this section, we assume that H ∈ C ∞ ( T × R n ) is non-degenerate, quadraticallyconvex, and non-resonant at infinity.Let x and y be 1-periodic orbits of X H . We shall see π ( x ) ∈ H as a critical pointof Ψ H ∗ , and hence of ψ H ∗ on the finite dimensional manifold M that is introduced inSection 6, and y ∈ H / as a critical point of Φ H . Let J be a family of uniformly bounded ω -compatible almost complex structures on R n parametrized by [0 , + ∞ ) × T such that J = J on [0 , × T and J ( s, t ) is independent of s for s large. We denote by M ( x, y ) = M ( x, y ; H, J )the space of smooth maps u : [0 , + ∞ ) × T → R n which solve the Floer equation ∂ s u + J ( s, t, u )( ∂ t u − X H t ( u )) = 0 on [0 , + ∞ ) × T with the asymptotic conditionlim s → + ∞ u ( s, · ) = y in C ∞ ( T , R n ) , and the boundary condition u (0 , · ) ∈ π − (cid:0) W u ( π ( x ); −∇ ψ H ∗ ) (cid:1) + H − / . Here W u ( π ( x ); −∇ ψ H ∗ ) is the unstable manifold of the negative gradient vector field of ψ H ∗ at π ( x ) in the finite dimensional submanifold M of H , which is used to construct theMorse complex of ψ H ∗ in Section 6. In other words, the trace of u at the boundary of thehalf-cylinder is the sum of a loop in W u ( π ( x ); −∇ ψ H ∗ ), seen as a submanifold of the spaceof loops with zero mean, and a loop in R n ⊕ H − / .The proposition below will be used in the following sections. Proposition . If u ∈ M ( x, y ) , we have Φ H ( x ) − Φ H ( y ) ≥ (cid:107) ∂ s u (cid:107) L ([0 , + ∞ ) × T ) Moreover Φ H ( x ) = Φ H ( y ) if and only if x = y , and in this case M ( x, x ) consists of aunique solution, namely the constant half-cylinder mapping to x .Proof. If u ∈ M ( x, y ), then u (0 , · ) = v + w for some v ∈ W u ( π ( x ); −∇ ψ H ∗ ) and w ∈ R n ⊕ H − / . This yields the estimate ˆ [0 , + ∞ ) × T | ∂ s u | dsdt = Φ H ( u (0 , · )) − Φ H ( y ) ≤ Ψ H ∗ ( v ) − Φ H ( y ) ≤ Ψ H ∗ ( π ( x )) − Φ H ( y ) = Φ H ( x ) − Φ H ( y ) (7.1)39here we have used Lemma 5.1 and Proposition 5.2. If we have Φ H ( x ) = Φ H ( y ), thetwo inequalities in (7.1) become equalities and u ( s, · ) = y for all s ∈ [0 , + ∞ ). The lastinequality is equality if and only if π ( x ) = v . That the first one is equality is equivalent to w ∈ R n and u (0 , · ) = y is a critical point of Φ H by Proposition 5.2. Therefore π ( x )+ w = y and by Lemma 5.1 again, this shows that x = y .In this section, we exhibit the functional setting which allows us to see M ( x, y ) as theset of zeroes of a nonlinear Fredholm map. We consider the following space of R n -valuedmaps on the positive half-cylinder converging to y for s → + ∞ and having the prescribedboundary condition at s = 0: H x,y := (cid:8) u : (0 , + ∞ ) × T → R n | u − y ∈ H ((0 , + ∞ ) × T , R n ) ,u (0 , · ) ∈ π − (cid:0) W u ( π ( x ); −∇ ψ H ∗ ) (cid:1) + H − / (cid:9) . Notice that the tangent space of M at any point has trivial intersection with H − / andtherefore the set π − (cid:0) W u ( π ( x ); −∇ ψ H ∗ ) (cid:1) + H − / is a smooth submanifold of H / having infinite dimension and codimension. The boundarycondition at s = 0 in the definition of H x,y is well posed because the trace of an H mapon the half-cylinder belongs to H / . Therefore, H x,y is a smooth submanifold of the affineHilbert space y + H ((0 , + ∞ ) × T , R n ) . On H x,y we shall consider the topology and the differentiable structure which is inducedby this embedding. This Hilbert manifold is the domain of the map ∂ J,H : H x,y → L ((0 , + ∞ ) × T , R n ) , ∂ J,H u = ∂ J u − ∇ J H t ( u ) , where the Cauchy-Riemann operator ∂ J = ∂ s + J ∂ t is to be understood in the distributionalsense. It is easy to check that this map is well defined, meaning that ∂ J,H u belongs indeedto L ((0 , + ∞ ) × T ). Indeed, if u = y + u , u ∈ H ((0 , + ∞ ) × T , R n ), is an element of H x,y we have ∂ J,H u = ∂ J u + J ( s, t, u )( y (cid:48) − X H t ( y + u )) . Since (cid:107) J (cid:107) ∞ < ∞ , the map ∂ J u belongs to L ((0 , + ∞ ) × T , R n ). The fact that y is a1-periodic orbit of X H implies that y (cid:48) − X H t ( y + u ) belongs to L ((0 , + ∞ ) × T , R n ) aswell. To see this, we compute y (cid:48) − X H t ( y + u ) = y (cid:48) − X H t ( y ) − ˆ ddθ X H t ( y + θu ) dθ = ˆ J ∇ H t ( y + θu ) u dθ, and obtain the pointwise estimate | y (cid:48) ( t ) − X H t ( y ( t ) + u ( s, t )) | ≤ (cid:107)∇ H (cid:107) ∞ | u ( s, t ) | (cid:107) y (cid:48) − X H t ( y + u ) (cid:107) L ((0 , + ∞ ) × T ) ≤ (cid:107)∇ H (cid:107) ∞ (cid:107) u (cid:107) L ((0 , + ∞ ) × T ) < + ∞ . This shows that ∂ J,H u belongs to L ((0 , + ∞ ) × T , R n ). The regularity and growth as-sumptions on H easily imply that the map ∂ J,H is smooth.We claim that the set of zeroes of ∂ J,H coincides with M ( x, y ). In order to prove theinclusion M ( x, y ) ⊂ ∂ − J,H (0), we have just to prove that every u ∈ M ( x, y ) is also in H x,y . This is true because by the non-degeneracy of y the elements u of M ( x, y ) convergeto y for s → + ∞ exponentially fast together with all their derivatives, and in particular u − y ∈ H ((0 , + ∞ ) × T ). The opposite inclusion instead follows from the next regularityresult: Proposition . Let u ∈ H x,y be such that ∂ J,H u = 0 . Then u is smooth on [0 , + ∞ ) × T and u ( s, · ) → y for s → + ∞ in C ∞ ( T , R n ) . The regularity of u on the open half-cylinder (0 , + ∞ ) × T does not follow from thestandard regularity results in Floer theory (see e.g. [MS04, Appendix B.4]), because theserequire the map u to be in W ,p loc for some p > 2, or at least in H ∩ C (see [IS99, Section2.3] or [IS00]). However, a different argument implies that interior regularity holds also forsolutions of the Floer equation that are just H . This argument is explained in AppendixA. The convergence to y in C ∞ ( T , R n ) is due to the non-degeneracy of y , see e.g. [Sal99,Section 2.7]. It remains to prove that u is smooth up to the boundary. The proof of thisfact is based on a bootstrap argument which makes use of the following lemmas. In whatfollows, we omit the subscript in the standard Cauchy-Riemann operator ∂ J = ∂ . Lemma . Let −∞ < a < b < + ∞ and u ∈ H (( a, b ) × T , R n ) . Denote by α and β theboundary traces of u , α ( t ) := u ( a, t ) , β ( t ) := u ( b, t ) , which are almost everywhere well defined functions belonging to H / ( T , R n ) . Then ˆ ( a,b ) × T |∇ u | dsdt = ˆ ( a,b ) × T | ∂u | dsdt − ˆ T β ∗ λ + 2 ˆ T α ∗ λ . and ˆ ( a,b ) × T |∇ u | dsdt = ˆ ( a,b ) × T | ∂ ∗ u | dsdt + 2 ˆ T β ∗ λ − ˆ T α ∗ λ . where ∂ ∗ = − ∂ s + J ∂ t . The simple proof of the lemma is based on Stokes’ theorem, see [AS15, Lemma 1.1]. Lemma . Let u ∈ H ((0 , × T , R n ) ∩ C ∞ ((0 , × T , R n ) be such that ∂u belongs to H ((0 , × T , R n ) and u (0 , · ) = v + w, ith v ∈ C ∞ ( T , R n ) and w ∈ H − / . Then u belongs to H ((0 , × T , R n ) . Moreover, (cid:107)∇ u (cid:107) L ((0 , × T ) ≤ C (cid:0) (cid:107) ∂u (cid:107) H ((0 , × T ) + (cid:107) ∂ t u (1 , · ) (cid:107) / + (cid:107) v (cid:48) (cid:107) / (cid:1) , (7.2) for some C > .Proof. Set ∆ h u ( s, t ) := u ( s, t + h ) − u ( s, t ) h , where h ∈ R . The fact that u is in H implies thatlim h → ∆ h u = ∂ t u in L ((0 , × T , R n ) . Indeed, this follows from the inequality (cid:107) ∆ h u − ∂ t u (cid:107) L ((0 , × T ) = ˆ (0 , × T (cid:12)(cid:12)(cid:12)(cid:12) ˆ (cid:0) ∂ t u ( s, t + θh ) − ∂ t u ( s, t ) (cid:1) dθ (cid:12)(cid:12)(cid:12)(cid:12) dsdt ≤ ˆ (0 , × T (cid:18) ˆ (cid:12)(cid:12) ∂ t u ( s, t + θh ) − ∂ t u ( s, t ) (cid:12)(cid:12) dθ (cid:19) dsdt = ˆ (cid:107) T θh ∂ t u − ∂ t u (cid:107) L ((0 , × T ) dθ, where T h is the translation operator T h u ( s, t ) = u ( s, t + h ), and from the fact that T h ∂ t u → ∂ t u in L ((0 , × T ) for h → ∂ t u belongs to L ((0 , × T ).Analogously, the function f := ∂u ∈ H ((0 , × T , R n ) ∩ C ∞ ((0 , × T , R n )satisfies lim h → ∆ h f = ∂ t f in L ((0 , × T ) . The fact that u is in C ∞ ((0 , × T ) implies that for every (cid:15) > h → ∆ h u = ∂ t u in C ∞ ([ (cid:15), × T ) . In particular, ∇ ∆ h u → ∇ ∂ t u pointwise in (0 , × T , and by the Fatou Lemma (cid:107)∇ ∂ t u (cid:107) L ((0 , × T ) ≤ lim inf h → (cid:107)∇ ∆ h u (cid:107) L ((0 , × T ) . (7.3)We claim that the right-hand side of this inequality is finite. Indeed, by the identity ofLemma 7.3 we have ˆ (0 , × T |∇ ∆ h u | dsdt = ˆ (0 , × T | ∂ ∆ h u | dsdt − ˆ T (∆ h u (1 , · )) ∗ λ + 2 ˆ T (∆ h u (0 , · )) ∗ λ = ˆ (0 , × T | ∆ h f | dsdt − ˆ T (∆ h u (1 , · )) ∗ λ + 2 ˆ T (∆ h v ) ∗ λ + 2 ˆ T (∆ h w ) ∗ λ . L norm of ∂ t f . Thesecond and third one converge to the integral of ∂ t u (1 , · ) ∗ λ and ( v (cid:48) ) ∗ λ over T , because thefunctions u (1 , · ) and v are smooth. The last integral is non-positive, because ∆ h w belongsto H − / . We conclude thatlim sup h → (cid:107)∇ ∆ h u (cid:107) L ((0 , × T ) ≤ ˆ (0 , × T | ∂ t f | dsdt − ˆ T ( ∂ t u (1 , · )) ∗ λ + 2 ˆ T ( v (cid:48) ) ∗ λ < + ∞ , (7.4)and by (7.3) the L norm of ∇ ∂ t u is finite. Equivalently, the functions ∂ s ∂ t u and ∂ t u havefinite L norm on (0 , × T . From the identity ∂ s u = ∂ s ∂u − J ∂ s ∂ t u = ∂ s f − J ∂ s ∂ t u (7.5)and the fact that f is in H ((0 , × T ), we deduce that also the L norm of ∂ s u is finite.We conclude that u is in H ((0 , × T ).From (7.3) and (7.4) we deduce the bound (cid:107) ∂ t u (cid:107) L ((0 , × T ) + (cid:107) ∂ s ∂ t u (cid:107) L ((0 , × T ) = (cid:107)∇ ∂ t u (cid:107) L ((0 , × T ) ≤ (cid:107) ∂ t ∂u (cid:107) L ((0 , × T ) + 2 (cid:107) ∂ t u (1 , · ) (cid:107) / + 2 (cid:107) v (cid:48) (cid:107) / . The bound (7.2) follows from the above inequality together with (7.5).In order to complete the bootstrap argument, we need the following easy consequenceof the chain rule. Lemma . Let H ∈ C ∞ ( T × R n ) and u ∈ C ∞ ((0 , × T , R n ) . Let h ≥ and k ≥ beintegers with h + k ≥ . Then ∂ hs ∂ kt ( ∇ H t ◦ u ) = ∇ H t ( u ) ∂ hs ∂ kt u + p, where p is a R n -valued polynomial mapping of the partial derivatives ∂ is ∂ jt u with ≤ i ≤ h , ≤ j ≤ k , ≤ i + j ≤ h + k − , whose coefficients are of the form A ( t, u ( s, t )) , where A is smooth.Proof. We argue by induction on h + k . If h + k = 1, then either h = 1 and k = 0 or h = 0and k = 1. In the first case we find ∂ s ( ∇ H t ◦ u ) = ∇ H t ( u ) ∂ s u, so the desired conclusion holds with p = 0. In the second case we have ∂ t ( ∇ H t ◦ u ) = ∇ H t ( u ) ∂ t u + ∇ ( ∂ t H t )( u ) , so the desired conclusion holds with p being the polynomial map of degree 0 p = A ( t, z ) := ∇ ( ∂ t H t )( z ) ∀ ( t, z ) ∈ T × R n , h ≥ k ≥ ≤ h + k ≤ (cid:96) ,for a given (cid:96) ≥ 1. Let h (cid:48) ≥ k (cid:48) ≥ h (cid:48) + k (cid:48) = (cid:96) + 1. Our aim is toshow that ∂ h (cid:48) s ∂ k (cid:48) t ( ∇ H t ◦ u ) has the desired form.We first assume that k (cid:48) ≤ (cid:96) , so that h (cid:48) ≥ 1. Then the inductive assumption impliesthat ∂ h (cid:48) − s ∂ k (cid:48) t ( ∇ H t ◦ u ) = ∇ H t ( u ) ∂ h (cid:48) − s ∂ k (cid:48) t u + p, where p is a polynomial map of the partial derivatives ∂ is ∂ jt u with 0 ≤ i ≤ h (cid:48) − 1, 0 ≤ j ≤ k (cid:48) ,1 ≤ i + j ≤ h (cid:48) + k (cid:48) − 2, whose coefficients are of the form A ( t, u ( t )), where A is smooth.By differentiating the above identity with respect to s we obtain ∂ h (cid:48) s ∂ k (cid:48) t ( ∇ H t ◦ u ) = ∇ H t ( u ) ∂ h (cid:48) s ∂ k (cid:48) t u + ∇ H t ( u )[ ∂ s u, ∂ h (cid:48) − s ∂ k (cid:48) t u ] + ∂ s p. The middle term on the right-hand side is a bilinear map in ∂ s u and ∂ h (cid:48) − s ∂ k (cid:48) t u , whichare partial derivatives of u of order not exceeding h (cid:48) + k (cid:48) − 1, with coefficient of the form A ( t, u ( s, t )), where A ( t, z ) := ∇ H t ( z ) ∀ ( t, z ) ∈ T × R n , is smooth. When we differentiate p with respect to s we obtain terms of two kinds. Theterms of the first kind are obtained by differentiating a given coefficient A ( t, u ( s, t )) withrespect to s . This produces a term of the form ∇ A ( t, u ) ∂ s u . This term is a multilinear inthe set of partial derivatives of u of admissible order. Such a term has the required form.The terms of the second kind are obtained by differentiating with respect to s a givenpartial derivative of u . This operation produces a monomial in which a term of the form ∂ is ∂ jt u , with 0 ≤ i ≤ h (cid:48) − 1, 0 ≤ j ≤ k (cid:48) , 1 ≤ i + j ≤ h (cid:48) + k (cid:48) − 2, is replaced by ∂ i +1 s ∂ jt u .Since i + 1 ≤ h (cid:48) , the new monomial satisfies the required conditions. We conclude that ∂ s p is a polynomial map of the partial derivatives ∂ is ∂ jt u with 0 ≤ i ≤ h (cid:48) , 0 ≤ j ≤ k (cid:48) ,1 ≤ i + j ≤ h (cid:48) + k (cid:48) − 1, whose coefficients are of the required form.There remains to consider the case k (cid:48) = (cid:96) + 1, which implies that h (cid:48) = 0. By theinductive assumption we have ∂ (cid:96)t ( ∇ H t ◦ u ) = ∇ H t ( u ) ∂ (cid:96)t u + p, where p is a polynomial map in ∂ t u, ∂ t u, . . . , ∂ (cid:96) − t u , whose coefficients have the requiredform. Differentiation with respect to t gives ∂ (cid:96) +1 t ( ∇ H t ◦ u ) = ∇ H t ( u ) ∂ (cid:96) +1 t u + ∇ H t ( u )[ ∂ t u, ∂ (cid:96)t u ] + ∇ ( ∂ t H t )( u ) ∂ (cid:96)t u + ∂ t p. The maps ( t, z ) (cid:55)→ ∇ H t ( z ) ( t, z ) (cid:55)→ ∇ ( ∂ t H t )( z )are smooth. Moreover, an argument analogous to the previous one shows that ∂ t p is apolynomial map of the partial derivatives ∂ jt u with 1 ≤ j ≤ (cid:96) , whose coefficients are of therequired form.This proves that in both cases ∂ h (cid:48) s ∂ k (cid:48) t ( ∇ H t ◦ u ) has the desired form and concludes theproof of the induction step. 44 roof of Proposition 7.2. We can now conclude the proof of Proposition 7.2 by showingthat u is smooth up to the boundary. By the Sobolev embedding theorems, it is enough toprove that the restriction of u to (0 , × T belongs to H k ((0 , × T , R n ) for every naturalnumber k . It certainly belongs to H ((0 , × T , R n ) by the definition of H x,y . Note thaton (0 , × T , J = J and the equation ∂ J,H u = 0 simplifies to ∂u = ∇ H t ( u ) . (7.6)Since the Hessian of H t is globally bounded, the maps ∂ s ( ∇ H t ( u )) = ∇ H t ( u ) ∂ s u, ∂ t ( ∇ H t ( u )) = ∇ H t ( u ) ∂ t u + ∇ ( ∂ t H t )( u ) , belong to L ((0 , × T , R n ). Therefore, the right-hand side of (7.6) belongs to H ((0 , × T , R n ). Thanks to the boundary conditions satisfied by u , Lemma 7.4 implies that therestriction of u to (0 , × T belongs to H ((0 , × T , R n ). In particular, u extendscontinuously to the closed half-cylinder [0 , + ∞ ) × R and is globally bounded.Arguing by induction, we assume that the restriction of u to (0 , × T belongs to H k ((0 , × T , R n ) for some integer k ≥ H k +1 ((0 , × T , R n ). By differentiating (7.6) k − t we obtain, thanks to Lemma7.5: ∂∂ k − t u = ∇ H t ( u ) ∂ k − t u + p, (7.7)where p is a R n -valued polynomial mapping of the partial derivatives ∂ t u, . . . , ∂ k − t u whosecoefficients are of the form A ( t, u ( t )), where A is smooth. By the inductive assumption,the function ∂ k − t u belongs to H ((0 , × T ) ∩ C ∞ ((0 , × T ). Therefore, its trace at s = 0is in H / and, since u (0 , · ) ∈ π − ( W u ( x ; −∇ ψ H ∗ )) + H − / where the first set consists of smooth loops by Proposition 6.3, it has the form ∂ k − t u (0 , · ) = v + w, where v is a smooth loop and w is an element of H / which is the ( k − H − / . As such, w also belongs to H − / .By differentiating the right-hand side of (7.7) we get ∇ ( ∇ H t ( u ) ∂ k − t u + p ) = ∇ H t ( u ) ∂ k − t ∇ u + q, where q = ∇ ( ∂ t H t )( u ) ∂ k − t u + ∇ H t ( u )[ ∇ u, ∂ k − t u ] + ∇ p is a R n -valued polynomial mapping of the partial derivatives ∂ t u, . . . , ∂ k − t u, ∂ s u, ∂ s ∂ t u, . . . , ∂ s ∂ k − t u whose coefficients are of the form A ( t, u ( t )), where A is smooth. Note that the coefficients A ( t, u ( s, t )) are uniformly bounded since u is bounded as observed above. The function45 H t ( u ) ∂ k − t ∇ u is in L ((0 , × T ) because of the inductive assumption and the bound-edness of ∇ H . The polynomial mapping q has the pointwise estimate | q | ≤ C (cid:0) | ∂ t u | N + · · · + | ∂ k − t u | N + | ∂ s u | N + | ∂ s ∂ t u | N + · · · + | ∂ s ∂ k − t u | N (cid:1) on (0 , × T for a suitable positive number C and a suitable natural number N . Thanks tothe Sobolev embedding of H k ((0 , × T ) into W k − ,N ((0 , × T ), all the partial derivativeswhich appear in the right-hand side of the above estimate are in L N ((0 , × T ) and hence q is in L ((0 , × T ). We conclude that the right-hand side of (7.7) is in H ((0 , × T ).Then we can apply Lemma 7.4 to the function ∂ k − t u and we obtain that this functionbelongs to H ((0 , × T ), which means that the functions ∂ s ∂ k − t u , ∂ s ∂ kt u and ∂ k +1 t u arein L ((0 , × T ).The fact that u solves the equation (7.6) easily implies that all other derivatives oforder k + 1 of u are in L ((0 , × T ). Indeed, by applying the differential operator ∂ s ∂ k − t to (7.6) we obtain, thanks to Lemma 7.5, ∂ s ∂ k − t u = − J ∂ s ∂ k − t u + ∇ H t ( u ) ∂ s ∂ k − t u + r, where r is a R n -valued polynomial mapping of the partial derivatives ∂ t u, . . . , ∂ k − t u, ∂ s u, ∂ s ∂ t u, . . . , ∂ s ∂ k − t u, ∂ s u, ∂ s ∂ t u, . . . , ∂ s ∂ k − t u whose coefficients are of the form A ( t, u ( t )), where A is smooth. Arguing as above, wededuce that ∂ s ∂ k − t u belongs to L ((0 , × T ). Iteratively, we conclude that all derivativesof order k + 1 of u belong to L ((0 , × T ) and hence u is in H k +1 ((0 , × T ), as we wishedto prove. We continue to assume that H ∈ C ∞ ( T × R n ) is non-degenerate, quadratically convex andnon-resonant at infinity. Let J be as before a family of uniformly bounded ω -compatiblealmost complex structures smoothly parametrized by [0 , + ∞ ) × T , such that J ( s, t ) = J for all s ∈ [0 , 1] and J ( s, t ) is independent of s for all s large. The aim of this section is toprove the following result. Proposition . Let x and y be 1-periodic orbits of X H . For every u ∈ M ( x, y ) , thelinear operator D∂ J,H ( u ) : T u H x,y −→ L ((0 , + ∞ ) × T , R n ) is Fredholm of index µ CZ ( x ) − µ CZ ( y ) . Given a closed linear subspace V of H / = H / ( T , R n ) we define H V ([0 , + ∞ ) × T , R n ) := { u ∈ H ([0 , + ∞ ) × T , R n ) | u (0 , · ) ∈ V } . This is a well-defined closed subspace of H ([0 , + ∞ ) × T , R n ) because the trace operator tr : H ([0 , + ∞ ) × T , R n ) → H / , u (cid:55)→ u (0 , · ) , is continuous. The proof of the above result relies on the following proposition.46 roposition . Let A ∈ C ([0 , + ∞ ] × T , L ( R n )) be a continuous map into the space oflinear endomorphisms of R n . Let J ∈ C ([0 , + ∞ ] × T , L ( R n )) be a continuous map suchthat J ( s, t ) is an ω -compatible almost complex structure for every ( s, t ) ∈ [0 , + ∞ ] × T and J ( s, t ) = J for every s ∈ [0 , × T . We assume that − J J (+ ∞ , t ) A (+ ∞ , t ) is symmetricfor every t ∈ T . Denote by Z A : [0 , → Sp(2 n ) be the symplectic path which is defined by Z A (0) = I, Z (cid:48) A ( t ) = − J (+ ∞ , t ) A (+ ∞ , t ) Z A ( t ) ∀ t ∈ [0 , , and assume that Z A is non-degenerate, meaning that 1 is not an eigenvalue of Z A (1) . So Z A ∈ SP (2 n ) and its Conley-Zehnder index is denoted by µ CZ ( Z A ) . Let V be a closedlinear subspace of H / which is a compact perturbation of H − / . Then the linear operator T : H V ([0 , + ∞ ) × T , R n ) → L ([0 , + ∞ ) × T , R n ) , u (cid:55)→ ∂ s u + J ∂ t u − Au, is Fredholm of index ind T = dim( V, H − / ⊕ E − ) − µ CZ ( Z A ) . Proof of Proposition 8.1. Let u ∈ M ( x, y ) and write u (0 , · ) = v + w, where v ∈ W u ( π ( x ); −∇ ψ H ∗ ), seen as a loop with zero mean, and w ∈ R n ⊕ H − / . Thetangent space of H x,y at u is T u H x,y = H V ([0 , + ∞ ) × T , R n ) , where V := T v W u ( π ( x ); −∇ ψ H ∗ ) ⊕ R n ⊕ H − / is a closed linear subspace of H / . This subspace is clearly a compact perturbation of H − / and dim( V, H − / ⊕ E − ) = dim T v W u ( π ( x ); −∇ ψ H ∗ ) + dim( R n ⊕ H − / , H − / ⊕ E − )= ind( π ( x ); Ψ H ∗ ) + n. Together with Propositions 5.3 and 1.1, we finddim( V, H − / ⊕ E − ) = ind H − / ⊕ E − ( x ; Φ H ) − n + n = µ CZ ( x ) . The differential of ∂ J,H at u is an operator of the form considered in Proposition 8.2, thatis D∂ J,H ( u ) : H V ((0 , + ∞ ) × T , R n ) → L ((0 , + ∞ ) × T , R n ) , v (cid:55)→ ∂ s v + J ( s, t, u ) ∂ t v − A ( s, t ) v, where A ( s, t ) v = J ( s, t, u ) ∇ v X H t ( u ( s, t )) − ∇ v J ( s, t, u ) (cid:0) ∂ t u ( s, t ) − X H t ( u ( s, t )) (cid:1) Z A ( t ) = dφ tX H ( y (0)) . Thanks to the above computation of the relative dimension of V with respect to H − / ⊕ E − ,Proposition 8.2 implies that this operator is Fredholm of index µ CZ ( x ) − µ CZ ( y ) . Proof of Proposition 8.2. Let Λ( s, t ) be a family of endomorphisms smoothly dependingon ( s, t ) ∈ [0 , + ∞ ] × T such thatΛ( s, t ) ∗ ω = ω , Λ( s, t ) ∗ J ( s, t ) = J , Λ( s, t ) = I for all ( s, t ) ∈ [0 , × T . Then T conjugates to an operator T Λ by Λ of the form T Λ : H V ([0 , + ∞ ) × T , R n ) → L ([0 , + ∞ ) × T , R n ) , u (cid:55)→ ∂ s u + J ∂ t u − A Λ u where A Λ : [0 , + ∞ ] × T → L ( R n ). To show that T Λ is Fredholm, we choose smooth cut-offfunctions β , β , β : [0 , + ∞ ) → [0 , 1] such thatsupp β ⊂ [0 , , supp β ∈ [1 / , τ − , supp β ⊂ [ τ − , + ∞ ) , β + β + β = 1 . where τ > u ∈ H V ([0 , + ∞ ) × T , R n ), let u = β u , u = β u ,and u = β u .Since u has support in [0 , × T , using the first identity in Lemma 7.3 we estimate (cid:107)∇ u (cid:107) L ([0 , + ∞ ) × T ) = (cid:107) ∂u (cid:107) L ([0 , + ∞ ) × T ) + 2 ˆ T u (0 , · ) ∗ λ ≤ (cid:107) ∂u (cid:107) L ([0 , + ∞ ) × T ) + 2 ˆ T (cid:0) P + u (0 , · ) (cid:1) ∗ λ = (cid:107) ∂u (cid:107) L ([0 , + ∞ ) × T ) + (cid:107) P + tr u (cid:107) / . This implies (cid:107) u (cid:107) H ([0 , + ∞ ) × T ) ≤ c (cid:0) (cid:107) T Λ u (cid:107) L ([0 , + ∞ ) × T ) + (cid:107) u (cid:107) L ([0 , × T ) + (cid:107) P + tr u (cid:107) / (cid:1) (8.1)where c = 1 + (cid:107) A Λ (cid:107) L ∞ .The estimates for u and u below are standard. Applying the Calderon-Zygmundestimate to u , we know that there exists c > (cid:107) u (cid:107) H ([0 , + ∞ ) × T ) ≤ c (cid:0) (cid:107) T Λ u (cid:107) L ([0 , + ∞ ) × T ) + (cid:107) u (cid:107) L ([0 ,τ ] × T ) (cid:1) . Since Z A is non-degenerate, for sufficiently large τ , there exists c > (cid:107) u (cid:107) H ([0 , + ∞ ) × T ) ≤ c (cid:107) T Λ u (cid:107) L ([0 , + ∞ ) × T ) c > (cid:107) u (cid:107) H ([0 , + ∞ ) × T , R n ) ≤ c (cid:0) (cid:107) T Λ u (cid:107) L ([0 , + ∞ ) × T , R n ) + (cid:107) u (cid:107) L ([0 ,τ ] × T , R n ) + (cid:107) P + tr u (cid:107) / (cid:1) . (8.2)Since V is a compact perturbation of H − / , P + | V : V → H / and hence P + | V ◦ tr is a compact operator. A standard arguments using (8.2) shows that T Λ is semi-Fredholm. To conclude that T Λ is Fredholm, we analyze the formal adjointoperator T ∗ Λ which enjoys the property ker T ∗ Λ = coker T Λ . In view of the computation ˆ [0 , + ∞ ) × T v · T Λ u dsdt = ˆ [0 , + ∞ ) × T ( − ∂ s v + J ∂ t v − A ∗ Λ ) · u dsdt − ˆ { }× T v · u dt for a compactly supported smooth map v : [0 , + ∞ ) × T → R n , where we have usedintegration by parts, it is defined as T ∗ Λ : H V ⊥ ([0 , + ∞ ) × T , R n ) → L ([0 , + ∞ ) × T , R n ) , − ∂ s + J ∂ t − A ∗ Λ , where A ∗ Λ denotes the transpose of A Λ . Here V ⊥ is the orthogonal complement of V withrespect to the L -metric which is a compact perturbation of H +1 / . Arguing as above,one can readily see that T ∗ Λ also satisfies an inequality like (8.2) with P + replaced by P − using the second identity in Lemma 7.3. This proves that T ∗ Λ is also semi-Fredholm andconsequently T Λ is Fredholm.In order to compute the Fredholm index of T , we homotope T to another Fredholmoperator in two steps and use the following well-known facts. First, the Fredholm index islocally constant in the space of Fredholm operators and therefore the index does not changealong a continuous homotopy of Fredholm operators. Second, two paths of symplecticmatrices in SP (2 n ) lie in the same connected component if and only if their Conley-Zehnder indices coincide.We choose a continuous path of ω -compatible almost complex structures { J r } r ∈ [0 , such that J ( s, t ) = J and J ( s, t ) = J ( s, t ) for all ( s, t ) ∈ [0 , + ∞ ] × T and J r ( s, t ) = J for all ( r, s, t ) ∈ [0 , × [0 , × T , which exists due to the contractibility of the space of ω -compatible almost complex structures. We set A r ( s, t ) = − J r ( s, t ) J ( s, t ) A ( s, t ) ∈ L ( R n )and consider a continuous family of operators T r : H V ((0 , + ∞ ) × T , R n ) → L ((0 , + ∞ ) × T , R n ) , v (cid:55)→ ∂ s v + J r ( s, t ) ∂ t v − A r ( s, t ) v, such that T = T . Since its asymptotic operator at s = + ∞ being of the form H V ( T , R n ) → L ( T , R n ) , v (cid:55)→ J r (+ ∞ , t )( ∂ t v + J (+ ∞ , t ) A (+ ∞ , t ) v )49s invertible, the arguments as above show that T r is Fredholm for all r ∈ [0 , T = ind T = ind T . Moreover, the symplectic path Z A r ( t ) defined by Z A r (0) = I, Z (cid:48) A r ( t ) = − J r (+ ∞ , t ) A r (+ ∞ , t ) Z A r ( t ) ∀ t ∈ [0 , , is independent of r which yields that the Conley-Zehnder index µ CZ ( Z A r ) does not changein r . It remains to show ind T = dim( V, H − / ⊕ E − ) − µ CZ ( Z A ) . (8.3) Case 1 : The Conley-Zehnder index µ CZ ( Z A ) is odd.We pick any number θ ∈ R \ π Z satisfying µ CZ ( Z A ) = 2 (cid:22) θ π (cid:23) + 1 (8.4)and define a symmetric matrix A odd = (cid:18) θ θ (cid:19) ⊕ (cid:18) − (cid:19) ⊕ n − . The associated non-degenerate path of symplectic matrices Z A odd = e − tJ θ ⊕ (cid:18) cosh t sinh t sinh t cosh t (cid:19) ⊕ n − satisfies µ CZ ( Z A odd ) = µ CZ ( Z A ) since (cid:18) − (cid:19) has zero signature and thus do not con-tribute to the Conley-Zehnder index. Then we choose another continuous family A r : [0 , + ∞ ] × T → L ( R n ) , r ∈ [ − , A ( s, t ) = A ( s, t ), A − ( s, t ) = A odd ( t ) for all ( s, t ) ∈ [0 , + ∞ ] × T ,- A r (+ ∞ , t ) is symmetric for all t ∈ T and r ∈ [ − , Z A r ∈ SP (2 n ) for all r ∈ [ − , T r = ∂ s + J ∂ t − A r : H V ([0 , + ∞ ) × T , R n ) → L ([0 , + ∞ ) × T , R n ) . In particular, ind T = ind T − . Since the Fredholm index, the relative dimension, theConley-Zehnder index are additive under direct sum, it is enough to establish (8.3) for T θ = ∂ s + J ∂ t − θI : H V ([0 , + ∞ ) × T , R ) → L ([0 , + ∞ ) × T , R ) , T Q = ∂ s + J ∂ t − Q : H V ([0 , + ∞ ) × T , R ) → L ([0 , + ∞ ) × T , R )where I denotes the identity map on R , Q = (cid:18) − (cid:19) , and V is now a closed subspaceof H / = H / ( T , R ). Since V is a compact perturbation of H − / , the spaces V ⊥ ∩ H − / and V ∩ ( R ⊕ H +1 / ) are finite dimensional. In other words, there exist (cid:96) ∈ N , a subspace W k of R e − πkJ t for − (cid:96) ≤ k ≤ (cid:96) , and a closed subspace W − of H − / such that H − / = W − ⊕ (cid:77) − (cid:96) ≤ k ≤− R e − πkJ t and V = W − ⊕ (cid:77) − (cid:96) ≤ k ≤ (cid:96) W k . Then we have V ⊥ = W + ⊕ (cid:77) − (cid:96) ≤ k ≤ (cid:96) W ⊥ k where W + is a closed subspace of H +1 / such that H +1 / = W + ⊕ (cid:77) ≤ k ≤ (cid:96) R e − πkJ t and W ⊥ k is the orthogonal complement of W k in R e − πkJ t .We first compute the index of T θ . Let u ∈ H V ((0 , + ∞ ) × T , R n ). We represent u asits Fourier series u ( s, t ) = (cid:88) k ∈ Z e − πkJ t ˆ u k ( s ) , ˆ u k ( s ) ∈ R . Then the condition u ∈ ker T θ implies0 = T θ u = ∂ s u + J ∂ t u − θu = (cid:88) k ∈ Z e − πkJ t (cid:0) ˆ u (cid:48) k ( s ) + (2 πk − θ )ˆ u k ( s ) (cid:1) we deduce ˆ u k ( s ) = e − (2 πk − θ ) s ˆ u k (0) . The condition that u has finite H -norm translates toˆ u k (0) = 0 , ∀ k ≤ θ π . Next we study the boundary condition u (0 , · ) ∈ V . Let us consider the case θ < − π(cid:96) . Inthis case, u ∈ ker T θ if and only if u (0 , · ) ∈ − (cid:96) − (cid:77) k = (cid:100) θ π (cid:101) R e − πkJ t ⊕ (cid:96) (cid:77) k = − (cid:96) W k T θ = − (cid:18)(cid:24) θ π (cid:25) + (cid:96) (cid:19) + (cid:96) (cid:88) k = − (cid:96) dim W k . If θ > − π(cid:96) , u ∈ ker T θ exactly when u (0 , · ) ∈ (cid:96) (cid:77) k = (cid:100) θ π (cid:101) W k and thus, dim ker T θ = (cid:96) (cid:88) k = (cid:100) θ π (cid:101) dim W k . In particular if θ > π(cid:96) , then dim ker T θ = 0.To compute the dimension of the cokernel of T θ , we use its formal adjoint operator T ∗ θ : H V ⊥ ([0 , + ∞ ) × T , R n ) → L ([0 , + ∞ ) × T , R n ) , u (cid:55)→ − ∂ s u + J ∂ t u − θu. Arguing as above we can show that if u ∈ ker T ∗ θ , u ( s, t ) = (cid:88) k ∈ Z e − πkJ t e (2 πk − θ ) s ˆ u k (0) . Since u ∈ H V ⊥ ([0 , + ∞ ) × T , R n ), we have ˆ u k (0) = 0 for all k ≥ θ π and u (0 , · ) ∈ V ⊥ . If θ < π(cid:96) , u ∈ ker T ∗ θ is equivalent to u (0 , · ) ∈ (cid:98) θ π (cid:99) (cid:77) k = − (cid:96) W ⊥ k and this computes dim ker T ∗ θ = (cid:98) θ π (cid:99) (cid:88) k = − (cid:96) dim W ⊥ k . In particular if θ < − π(cid:96) , dim ker T ∗ θ = 0. In the case of θ > π(cid:96) , u ∈ ker T ∗ θ if and only if u (0 , t ) ∈ (cid:98) θ π (cid:99) (cid:77) k = (cid:96) +1 R e − πkJ t ⊕ (cid:96) (cid:77) k = − (cid:96) W ⊥ k and therefore, dim ker T ∗ θ = 2 (cid:18)(cid:22) θ π (cid:23) − (cid:96) (cid:19) + (cid:96) (cid:88) k = − (cid:96) dim W ⊥ k . W ⊥ k = 2 − dim W k and (cid:100) θ π (cid:101) = (cid:98) θ π (cid:99) + 1, we see that in allthe cases ind T θ = dim ker T θ − dim ker T ∗ θ = − (cid:18)(cid:24) θ π (cid:25) + (cid:96) (cid:19) + (cid:96) (cid:88) k = − (cid:96) dim W k . (8.5)On the other hand, we havedim( V, H − / ⊕ E − ) = dim (cid:0) V ∩ ( H +1 / ⊕ E + ) (cid:1) − dim (cid:0) V ⊥ ∩ ( H − / ⊕ E − ) (cid:1) = dim( W ∩ E + ) + (cid:96) (cid:88) k =1 dim W k − dim( W ⊥ ∩ E − ) − (cid:88) k = − (cid:96) dim W ⊥ k = dim W − (cid:96) (cid:88) k =1 dim W k − (cid:88) k = − (cid:96) (2 − dim W k )= (cid:96) (cid:88) k = − (cid:96) dim W k − (cid:96) − . (8.6)Combining (8.4), (8.5), and (8.6), we concludeind T θ = dim( V, H − / ⊕ E − ) − µ CZ ( Z θ ) . To compute the index of T Q , where Q = (cid:18) − (cid:19) , we write as before u ∈ H V ((0 , + ∞ ) × T , R n ) as u ( s, t ) = (cid:88) k ∈ Z e − πkJ t ˆ u k ( s ) , ˆ u k ( s ) = (cid:0) ˆ a k ( s ) , ˆ b k ( s ) (cid:1) ∈ R = E + ⊕ E − . then the condition u ∈ ker T Q translates to (cid:88) k ∈ Z e − πkJ t (cid:0) ˆ a (cid:48) k ( s ) + (2 πk + 1)ˆ a k ( s ) , ˆ b (cid:48) k ( s ) + (2 πk − b k ( s ) (cid:1) = 0Therefore we have ˆ a k ( s ) = e − (2 πk +1) s ˆ a k (0) , ˆ b k ( s ) = e − (2 πk − s ˆ b k (0)Since u has finite H -norm, ˆ a k (0) = 0 , ∀ k ≤ − b k (0) = 0 , ∀ k ≤ . u (0 , · ) ∈ V yields that u ∈ ker T Q if and only ifˆ u (0) ∈ (cid:77) ≤ k ≤ (cid:96) W k ⊕ ( W ∩ E + )and therefore dim ker T Q = (cid:96) (cid:88) k =1 dim W k + dim( W ∩ E + ) . To compute the dimension of the cokernel of T Q , we use its formal adjoint T ∗ Q : H V ⊥ ([0 , + ∞ ) × T , R n ) → L ([0 , + ∞ ) × T , R n ) , u (cid:55)→ − ∂ s u + J ∂ t u − Qu. Arguing as above, we see that u ∈ T ∗ Q if and only if ˆ u k is such thatˆ a k ( s ) = e (2 πk +1) s ˆ a k (0) , ˆ b k ( s ) = e (2 πk − s ˆ b k (0)with ˆ u (0) ∈ (cid:77) − (cid:96) ≤ k ≤− W ⊥ k ⊕ ( W ⊥ ∩ E − ) . Thus, dim coker T Q = dim ker T ∗ Q = − (cid:88) k = − (cid:96) dim W ⊥ k + dim( W ⊥ ∩ E − ) . Finally we haveind T Q = (cid:96) (cid:88) k = − (cid:96) dim W k − dim W − (cid:96) + dim( W ∩ E − ) − dim( W + E − ) ⊥ = (cid:96) (cid:88) k = − (cid:96) dim W k − (cid:96) − 1= dim( V, H − / ⊕ E − ) − µ CZ ( Z Q )where the last equality is again by (8.6) and µ CZ ( Z Q ) = 0. Case 2 : The Conley-Zehnder index µ CZ ( Z A ) is even.Suppose that n ≥ 2. We pick any θ , θ ∈ R \ π Z and define A even = (cid:18) θ θ (cid:19) ⊕ (cid:18) θ θ (cid:19) ⊕ (cid:18) − (cid:19) ⊕ n − . such that µ CZ ( Z A even ) = 2 (cid:22) θ π (cid:23) + 2 (cid:22) θ π (cid:23) + 2 = µ CZ ( A ) . T θ and ind T Q in Case 1, we obtainagain (8.3) in this case.If n = 1, we consider A ⊕ A : [0 , + ∞ ] × T → L ( R ) and the associated Fredholmoperator. We also double V to have a closed subspace V ⊕ V in H / ⊕ H / = H / ( T , R ).Due to the additivity properties of the indices and the relative dimension, the case n = 1follows from the case n = 2, that we have just shown.When x (cid:54) = y , a generic choice of the ω -compatible almost complex structure J andof the Riemannian metric on M makes the operator D∂ J,H ( u ) surjective for every u ∈M ( x, y ). When x = y , M ( x, y ) = M ( x, x ) consists of just the constant half-cylinder map-ping to x , see Proposition 7.1. At such a stationary solution u , changing the almost complexstructure and the metric does not affect the linearized operator D∂ J,H ( u ). Therefore, weneed the following automatic transversality result. Proposition . Let x be 1-periodic orbit of X H . Assume that J satisfies in addition that J ( s, t, x ( t )) = J for all ( s, t ) ∈ [0 , + ∞ ) × T . Let u ∈ M ( x, x ) be the constant half-cylindermapping to x . Then the linear operator D∂ J,H ( u ) : T u H x,x −→ L ((0 , + ∞ ) × T , R n ) is invertible.Proof. Since we have seen in Proposition 8.1 that D∂ J ,H ( u ) is Fredholm of index 0, itis enough to show that the kernel of D∂ J ,H ( u ) is trivial. Let v ∈ ker D∂ J ,H ( u ). Since u ( s, · ) = x is a critical point of Φ H , this translates into ∂ s v + ∇ L Φ H ( x ) v = 0where ∇ L Φ H ( x ) = J ∂ t − ∇ H ( x ) is the L -Hessian of Φ H at x which satisfies (cid:0) ∇ L Φ H ( x ) · , · (cid:1) L ( T ) = d Φ H ( x ) . We define a function ϕ : [0 , + ∞ ) → [0 , + ∞ ) by ϕ ( s ) := (cid:107) v ( s, · ) (cid:107) L ( T ) . Its first and secondderivatives are ϕ (cid:48) ( s ) = − (cid:0) v ( s, · ) , ∇ L Φ H ( x ) v ( s, · ) (cid:1) L ( T ) , ϕ (cid:48)(cid:48) ( s ) = 4 (cid:13)(cid:13) ∇ L Φ H ( x ) v ( s, · ) (cid:13)(cid:13) L ( T ) , where we used the fact that ∇ L Φ H ( x ) is symmetric with respect to the L -inner product.Since v has finite H -norm, ϕ ( s ) converges to 0 as s goes to + ∞ . Unless v = 0, thishappens only if − (cid:0) v (0 , · ) , ∇ L Φ H ( x ) v (0 , · ) (cid:1) L ( T ) = ϕ (cid:48) (0) < ϕ (cid:48)(cid:48) ≥ 0. Using that x is a critical point of Φ H again, we deduceΦ H ( x + (cid:15)v (0 , · )) = Φ H ( x ) + (cid:15) d Φ H ( x )[ v (0 , · ) , v (0 , · )] + O ( (cid:15) ) . (cid:15) ,Φ H ( x + (cid:15)v (0 , · )) > Φ H ( x ) . (8.8)On the other hand, the condition v (0 , · ) ∈ V = T x W u ( π ( x ); −∇ ψ H ∗ ) ⊕ R n ⊕ H − / implies the opposite. To see this, we write v (0 , · ) = v + v + v , v ∈ T x W u ( π ( x ); −∇ ψ H ∗ ) , v ∈ R n , v ∈ H − / . Then d Ψ H ∗ ( π ( x ))[ v , v ] ≤ . Arguing as above, this implies that for small (cid:15) ,Ψ H ∗ (cid:0) π ( x + (cid:15) ( v + v )) (cid:1) = Ψ H ∗ (cid:0) π ( x + (cid:15)v ) (cid:1) ≤ Ψ H ∗ ( π ( x )) . Using this together with Proposition 5.2, we deduceΦ H ( x + (cid:15)v (0 , · )) ≤ Ψ H ∗ (cid:0) π ( x + (cid:15) ( v + v )) (cid:1) − (cid:107) (cid:15)v (cid:107) / ≤ Ψ H ∗ ( π ( x )) = Φ H ( x ) . This contradicts (8.8). This proves v = 0 and completes the proof. In this section, we investigate the compactness properties of the set M ( x, y ). We keep thesame assumptions on the Hamiltonian and the almost complex structure: H ∈ C ∞ ( T × R n )is non-degenerate, quadratically convex and non-resonant at infinity, while J = J ( s, t, x ) is ω -compatible, globally bounded, equal to J if s ∈ [0 , 1] and independent of s for s large.We start by observing that the elements of M ( x, y ) have uniform energy bounds due toProposition 7.1. Then arguments similar to those of Section 4 and building on Proposition3.1 lead to the following result. Proposition . For every σ > the set { u | [ σ, + ∞ ) × R | u ∈ M ( x, y ) } is bounded in L ∞ ([ σ, + ∞ ) × T , R n ) . Moreover, for every S > σ the set { u | [ σ,S ] × R | u ∈ M ( x, y ) } is pre-compact in C ∞ ([ σ, S ] × T , R n ) . In order to find bounds near the boundary for the elements of M ( x, y ) we start withthe following: 56 emma . Let u ∈ M ( x, y ) and write u (0 , · ) = v + w + w , where v ∈ W u ( π ( x ); −∇ ψ H ∗ ) , w ∈ R n and w ∈ H − / . Then:(i) v belongs to a compact subset of C ∞ ( T , R n ) which depends only on x and y ;(ii) (cid:107) w (cid:107) / ≤ H ∗ ( x ) − Φ H ( y )) ;(iii) w belongs to a compact subset of R n which depends only on x and y .Proof. By Proposition 5.2, we haveΦ H ( y ) ≤ Φ H ( u (0 , · ) = Φ H ( v + w + w ) ≤ Ψ H ∗ ( v ) − (cid:107) w (cid:107) / , which immediately implies (ii). The above inequality also implies that v belongs to the set W u ( π ( x ); −∇ ψ H ∗ ) ∩ { Ψ H ∗ ≥ Φ H ( y ) } , which is pre-compact in C ∞ ( T , R n ). So (i) holds.The quadratic convexity assumption on H guarantees that H t ( z ) ≥ a | z | − b ∀ ( t, z ) ∈ T × R n , for suitable positive numbers a, b . Then we find ˆ T H t ( v + w + w ) dt ≥ a ˆ T | v + w + w | dt − b ≥ a ˆ T ( | w | − | v | − | w | ) dt − b ≥ a | w | − a | w | ˆ T ( | v | + | w | ) dt − b The integral ´ T ( | v | + | w | ) dt is uniformly bounded thanks to (i) and (ii), so we get theuniform bound ˆ T H t ( v + w + w ) dt ≥ a | w | − ac | w | − b = a ( | w | − c ) − ac − b, for a suitable positive number c which depends only on x and y . From the above boundwe deduceΦ H ( y ) ≤ Φ H ( v + w + w ) = 12 ˆ T J ( ˙ v + ˙ w ) · ( v + w ) dt − ˆ T H t ( v + w + w ) dt ≤ (cid:107) v + w (cid:107) / − a ( | w | − c ) + ac + b. This inequality, together with the fact that (cid:107) v + w (cid:107) / is uniformly bounded thanks to (i)and (ii), implies that w is uniformly bounded in R n . This concludes the proof of (iii).57e can now prove the compactness properties of the restrictions of elements of M ( x, y )near the boundary. The proof consists in making the argument of the regularity resultProposition 7.2 quantitative. Proposition . For every S > the set { u | [0 ,S ] × T | u ∈ M ( x, y ) } is pre-compact in C ∞ ([0 , S ] × T , R n ) . Moreover, M ( x, y ) is bounded in L ∞ ([0 , + ∞ ) × T , R n ) .Proof. Once the first statement has been proven, the boundedness of M ( x, y ) in the space L ∞ ([0 , + ∞ ) × T , R n ) follows immediately from the first statement of Proposition 9.1.Moreover by the second statement of Proposition 9.1, the restriction of every u ∈ M ( x, y )to [1 , S ] × T is uniformly bounded in C ∞ ([1 , S ] × T , R n ) for every S ≥ 1, so it suffices toprove the first statement for S = 1.The restriction of each u ∈ M ( x, y ) to [0 , × T satisfies the equation ∂u = ∇ H t ( u ) on [0 , × T (9.1)and the boundary condition u (0 , · ) = v + w + w , where v ∈ W u ( π ( x ); −∇ ψ H ∗ ) , w ∈ R n , w ∈ H − / . (9.2)The fact that u has uniformly bounded energy, together with the uniform bound on (cid:107) u (0 , · ) (cid:107) L ( T ) following from Lemma 9.2, implies a uniform bound for the L norm of u on (0 , × T . By the linear growth of ∇ H , we deduce that also ∇ H ( u ) has a uniform L bound on (0 , × T . Since both u (0 , · ) and u (1 , · ) are uniformly bounded in H / , becauseof Lemma 9.2 and the second statement in Proposition 9.1, the first formula of Lemma7.3 and (9.1) give us a uniform L bound for ∇ u on (0 , × T . Therefore, the elements of M ( x, y ) are uniformly bounded in H ((0 , × T , R n ).Now we wish to prove that the elements of M ( x, y ) are uniformly bounded in H k ((0 , × T , R n ) for every natural number k . By the Sobolev embedding theorem, this will give usthe boundedness of { u | [0 , × T | u ∈ M ( x, y ) } in C k ([0 , × T , R n ) for every every natural number k , and by the Ascoli-Arzel`a theoremits pre-compactness in C ∞ ([0 , × T , R n ).The uniform bound in H ((0 , × T , R n ) has just been proven. Then the Floer equation(9.1) and the growth conditions on H imply that ∂u has a uniform bound in H ((0 , × T , R n ). Thanks to the boundary condition (9.2), the bound (7.2) from Lemma 7.4, togetherwith Lemma 9.2 and the fact that all the derivatives of u (1 , · ) are uniformly bounded,implies a uniform bound for u in H ((0 , × T , R n ). In particular, the restriction of u to(0 , × T is uniformly bounded in L ∞ ((0 , × T , R n ).Arguing by induction, we now assume that the elements of M ( x, y ) are uniformlybounded in H k ((0 , × T , R n ) for some integer k ≥ 2, and we wish to prove a uniform58ound in H k +1 ((0 , × T , R n ). By differentiating the Floer equation k times with respectto t , we find, thanks to Lemma 7.5, ∂∂ kt u = ∂ kt ∂u = ∇ H t ( u ) ∂ kt u + p, where p is a R n -valued polynomial map in ∂ t u, ∂ t u, . . . , ∂ k − t u whose coefficients are smoothfunctions of ( t, u ( s, t )). The fact that u has a uniform bound in L ∞ ((0 , × T , R n ) im-plies that these coefficients are also uniformly bounded in this space. In particular, thepolynomial map p has the pointwise estimate | p | ≤ C (1 + | ∂ t u | N + · · · + | ∂ k − t u | N ) , for suitable constants C and N . Then the inductive hypothesis and the continuity of theSobolev embedding of H k into W k − ,N imply a uniform bound of the form (cid:107) ∂∂ kt u (cid:107) L ((0 , × T ) ≤ C ∀ u ∈ M ( x, y ) , for some constant C .By (9.2) we have ∂ kt u (0 , · ) = v ( k ) + w ( k )1 . Here, v ( k ) is uniformly bounded in H / ( T , R n ) thanks to statement (i) in Lemma 9.2.On the other hand, w ( k )1 is an element of H − / , because the time derivative of a loop in H − / ∩ C ∞ ( T , R n ) belongs to H − / . Therefore (cid:12)(cid:12)(cid:12)(cid:12) ˆ T ( v ( k ) ) ∗ λ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) v ( k ) (cid:107) / ≤ C and ˆ T ( w ( k )1 ) ∗ λ = −(cid:107) w ( k )1 (cid:107) / ≤ , for some constant C . By the second statement in Proposition 9.1, we also have (cid:12)(cid:12)(cid:12)(cid:12) ˆ T ( ∂ kt u (1 , · )) ∗ λ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) ∂ kt u (1 , · ) (cid:107) / ≤ C , for some constant C . Then the first formula of Lemma 7.3 applied to ∂ kt u gives ˆ (0 , × T |∇ ∂ kt u | dsdt = ˆ (0 , × T | ∂∂ kt u | dsdt − ˆ T ( ∂ kt u (1 , · )) ∗ λ + 2 ˆ T ( v ( k ) ) ∗ λ + 2 ˆ T ( w ( k )1 ) ∗ λ ≤ C + 2 C + 2 C . This shows that the partial derivatives ∂ s ∂ kt u and ∂ k +1 t u have uniform L -bounds on (0 , × T . The uniform L bounds on all the other partial derivatives of order k + 1 now followeasily from the Floer equation and Lemma 7.5. Indeed, by applying the differential operator ∂ s ∂ k − t to the Floer equation we find ∂ s ∂ k − t u = − J ∂ s ∂ kt u + ∇ H t ( u ) ∂ s ∂ k − t u + q, (9.3)59here q is a R n -valued polynomial map in the variables ∂ t u, ∂ t u, . . . , ∂ k − t u, ∂ s u, ∂ s ∂ t u, . . . , ∂ s ∂ k − t u, (9.4)whose coefficients are uniformly bounded. The first term on the right-hand side of (9.3) isbounded in L ((0 , × T ), as shown above. The middle term is also bounded in L ((0 , × T )by the inductive hypothesis. Being a polynomial in the partial derivatives (9.4) withuniformly bounded coefficients, q has the pointwise estimate | q | ≤ C (cid:0) | ∂ t u | N + · · · + | ∂ k − t u | N + | ∂ s u | N + · · · + | ∂ s ∂ k − t u | N (cid:1) , for a suitable positive number C and a suitable natural number N . Integration over(0 , × T and the observation that the partial derivatives appearing above have order atmost k − (cid:107) q (cid:107) L ((0 , × T ) ≤ C (cid:0) (cid:107) u (cid:107) NW k − ,N ((0 , × T ) (cid:1) . By the continuity of the Sobolev embedding H k ((0 , × T )) (cid:44) → W k − ,N ((0 , × T )and by the inductive hypothesis we deduce that q has a uniform L bound on (0 , × T .Then all terms on the right-hand side of (9.3) have uniform L bounds on (0 , × T andhence the same is true for the term on the left-hand side, that is, ∂ s ∂ k − t u . By iterating thisargument inductively in h , we obtain that ∂ hs ∂ k − h +1 t has a uniform L bound on (0 , × T for every h ∈ { , . . . , k + 1 } . We conclude that the elements of M ( x, y ) are uniformlybounded in H k +1 ((0 , × T ). This proves the induction step and concludes the proof. 10 The chain complex isomorphism Let H ∈ C ∞ ( T × R n ) be non-degenerate, quadratically convex and non-resonant at in-finity. As we have seen in Section 6, the dual action functional Ψ H ∗ restricts to a smoothMorse function ψ H ∗ on a finite dimensional manifold M ⊂ H , whose Morse complex { M ∗ ( ψ H ∗ ) , ∂ M } is well defined. On the other hand, we have the Floer complex { F ∗ ( H ) , ∂ F } of the Hamiltonian H . These two complexes depend on auxiliary data - a generic metric on M and a generic ω -compatible almost complex structure on R n - but choices of differentauxiliary data change them by isomorphisms preserving gradings and actions.The generators of these two chain complexes are in one-to-one correspondence: Thegenerator x of F ∗ ( H ) - a 1-periodic orbit of X H - induces the generator π ( x ) of M ∗ ( ψ H ∗ )and the relationships between grading and actions areind( π ( x ); ψ H ∗ ) = µ CZ ( x ) − n, ψ H ∗ ( π ( x )) = Φ H ( x )by Propositions 1.1, 5.3, 6.3, and Lemma 5.1. In this section, we wish to construct a chaincomplex isomorphism Θ : { M ∗− n ( ψ H ∗ ) , ∂ M } −→ { F ∗ ( H ) , ∂ F } ∂W then SH <(cid:15)n ( W ) ∼ = H n ( W, ∂W ; Z ) = Z . On the other hand, the full symplectic homology of W vanishes, so we can define the SH -capacity of W as the positive number c SH ( W ) := inf { a > (cid:15) | SH <(cid:15)n ( W ) → SH 0) for all s ≤ ϕ (cid:48) (1) = A min ( ∂C ).There are two types of 1-periodic orbits of X ϕ ◦ H C :- the constant curve z : T → R n mapping to the origin with actionΦ ϕ ◦ H C ( z ) = − ϕ ◦ H C ( z ) = − ϕ (0) ∈ (0 , (cid:15)/ y i : T → ∂C , i ∈ { , . . . , m } where y i ( t/A min ( ∂C )) is a closed orbit of R α C with period A min ( ∂C ), with actionΦ ϕ ◦ H C ( y i ) = A min ( ∂C ) − ϕ (1) ∈ (cid:0) A min ( ∂C ) , A min ( ∂C ) + (cid:15)/ (cid:1) , and with index µ CZ ( y i ) = n + 1 , ∀ i ∈ { , . . . , m } by Propositions 1.1, 2.2 and 5.4.Note that ϕ ◦ H C is everywhere smooth, ∇ ( ϕ ◦ H C ) is positive definite on R n \ { } and ∇ ( ϕ ◦ H C )(0) = 0. We choose a C -small function f : R n → R supported in a smallneighborhood of the origin such that the origin is a critical point of f and the Hessian ∇ f is sufficiently small and positive definite near the origin. Then the function ϕ ◦ H C + f has positive definite Hessian, has a unique critical point at the origin, and coincides with ϕ ◦ H C outside a neighborhood of the origin. In particular, ϕ ◦ H C + f is quadraticallyconvex. Since the origin is a minimizer of ϕ ◦ H C + f , the Conley-Zehnder index of z is µ CZ ( z ) = ind (0; ϕ ◦ H C + f ) + n = n, where ind (0; ϕ ◦ H C + f ) denotes the Morse index of ϕ ◦ H C + f at the origin. Addingan additional small perturbation supported in neighborhoods of periodic orbits y i ’s as in[BO09] to ϕ ◦ H C + f , we obtain a smooth function H : T × R n → R such that:- H is quadratically convex;- H = ϕ ◦ H C outside the neighborhoods of the origin and of y i ( T )’s;- H satisfies all the properties in the statement except possibly for (11.1).64n order to establish (11.1), we use Clarke’s duality. By Lemma 5.1 and Proposition6.3, the reduced dual functional ψ H ∗ : M → R has non-degenerate critical points π ( z ) , π ( y − i ) , π ( y + i ) , i ∈ { , . . . , m } with Morse indicesind ( π ( z ) , ψ H ∗ ) = 0 , ind ( π ( y − i ) , ψ H ∗ ) = 1 , ind ( π ( y + i ) , ψ H ∗ ) = 2 , ∀ i ∈ { , . . . , m } . Thanks to the Theorem from the introduction and (6.6), we have HF k + n ( H ) ∼ = HM k ( ψ H ∗ ) ∼ = H k ( M, { ψ H ∗ < a } ) , ∀ k ∈ Z (11.2)for any a < ψ H ∗ ( π ( z )). Since y + i ’s represent non-zero class in F H n +2 ( H ), by (11.2) H ( M, { ψ H ∗ < a } ) (cid:54) = 0 . Since M is diffeomorphic to R nN , this implies that { ψ H ∗ < a } (cid:54) = ∅ , and therefore we have H ( M, { ψ H ∗ < a } ) = 0 . Applying this to (11.2), we conclude that the cycle z vanishes in HF n ( H ). Hence, ∂ F y − j = z for some j ∈ { , . . . , m } . This completes the proof.We can now prove the corollary stated in the introduction. The inequality c SH ( C ) ≥ A min ( ∂C ) follows from the already mentioned fact that c SH ( C ) belongs to the action spec-trum of ∂C . We prove the opposite inequality. Fix some positive numbers (cid:15) < A min ( ∂C ) , η ∈ ( A min ( ∂C ) , A ( ∂C )) , and let H be as in Lemma 11.2. The Hamiltonian H belongs to H ( C ) and HF <(cid:15)n ( H ) isisomorphic to Z and generated by z . Moreover the homomorphism σ : HF <(cid:15)n ( H ) −→ SH <(cid:15)n ( C )in the direct limit defining SH <(cid:15)n ( C ) is an isomorphism. One way to see this is to observethat there is a cofinal subset { H ν } ν ∈ N of H ( C ) such that H = H , and for every ν ∈ N , X H ν has a unique constant orbit z mapping to the origin and all other 1-periodic orbitshave Φ H ν -action larger than (cid:15) .Consider the following commutative diagram: HF <(cid:15)n ( H ) HF
In this appendix we prove the following interior regularity result for solutions of the Floerequation. The argument used in the proof was explained to us by Urs Fuchs. Proposition A.1 . Let U be an open subset of R × T . Let J be a uniformly bounded ω -compatible almost complex structure on R n , smoothly depending on ( s, t ) ∈ U . Let H : T × R n → R be a smooth Hamiltonian function such that | X H t ( z ) | ≤ c (1 + | z | ) ∀ ( t, z ) ∈ T × R n (A.1) for some c > . If u ∈ H ( U, R n ) is a weak solution of the Floer equation ∂ s u + J ( s, t, u ) (cid:0) ∂ t u − X H t ( u ) (cid:1) = 0 , then it is smooth. Since u is a priori only in H , it may not be continuous and hence the map ( s, t ) (cid:55)→ J ( s, t, u ( s, t )) may as well be not continuous. This prevents us from using standard argu-ments, in which one looks at a small neighbourhood of a point in U and sees J ( s, t, u ( s, t ))there as a small perturbation of a constant complex structure on R n . We overcome thisdifficulty by the following result. Lemma A.2 . Let U be an open subset of R , let { J ( s, t ) } ( s,t ) ∈ U be a bounded measurablefamily of ω -compatible almost complex structures, and let f be a map in L p loc ( U, R n ) forsome p > . Then there exists a number q > , depending only on (cid:107) J (cid:107) ∞ and p , such thatevery u ∈ L p loc ( U, R n ) solving the linear equation ∂ s u + J ∂ t u = f (A.2) in the distributional sense belongs to W ,q loc ( U, R n ) . s, t ) (cid:55)→ J ( s, t, u ( s, t )) is measurable and bounded and the map f ( s, t ) = J ( s, t, u ( s, t )) X H t ( u ( s, t ))belongs to L p loc ( U, R n ) for every p ∈ (1 , + ∞ ), thanks to the growth assumption on X H andto the fact that u belongs to L p loc ( U, R n ) for every p ∈ (1 , + ∞ ), by the Sobolev embeddingtheorem. Then this lemma allows us to upgrade the regularity of u to W ,q loc regularity,for some q > 2. Basing on this, standard regularity arguments (see e.g. [MS04, AppendixB.4]) imply that the solution u of (A.1) is smooth.The remaining part of this appendix is devoted to the proof of Lemma A.2. Theargument consists of transforming the linear Floer equation (A.2) for a variable almostcomplex structure J into a Beltrami equation, and then proving regularity for solutions ofthis equation using the standard Calderon-Zygmund estimates. The reader interested inlearning more about the Beltrami equation and its regularity theory might refer to [Vek62]and [Boj10].In order to reduce the linear Floer equation to a Beltrami equation, we shall make useof the following well known facts about complex structures (see e.g. [IS99, Section 1.2]).Here, | · | and · denote the euclidean norm and scalar product on R n and (cid:107) · (cid:107) the inducedoperator norm on the space L ( R n ) of linear endomorphisms of R n . Lemma A.3 . Let J be a complex structure on R n .(i) Assume that there exists α > such that ω ( J u, u ) ≥ α | u | , ∀ u ∈ R n . Then α ≤ , J + J is invertible, and the following inequalities hold (cid:107) ( J + J ) − (cid:107) ≤ α + 1 , (cid:107) ( J + J ) − ( J − J ) (cid:107) ≤ (cid:115) − α ( α + 1) . (ii) Assume that J is ω -compatible, meaning that ( u, v ) (cid:55)→ ω ( J u, v ) is a scalar producton R n . Then ω ( J u, u ) ≥ (cid:107) J (cid:107) | u | ∀ u ∈ R n . Proof. (i) The invertibility of J + J readily follows from( α + 1) | u | ≤ ω ( J u, u ) + ω ( J u, u ) ≤ ω (cid:0) ( J + J ) u, u (cid:1) . Moreover by the Cauchy-Schwarz inequality, we have( α + 1) | u | ≤ ω (cid:0) ( J + J ) u, u (cid:1) = − J ( J + J ) u · u ≤ | ( J + J ) u || u | . u = ( J + J ) − v , we obtain | ( J + J ) − v | ≤ ( α + 1) − | v | , ∀ v ∈ R n . This proves the first inequality in the lemma. From α | J u | ≤ ω ( J J u, J u ) = ω ( J u, u ) ≤ | J u || u | we deduce the inequality (cid:107) J (cid:107) ≤ α − , and from 1 = (cid:107) J (cid:107) ≤ (cid:107) J (cid:107) ≤ α − we obtain α ≤ 1. If we set S := I − J J = − J ( J + J ),where I is the identity map, we find (cid:107) S − (cid:107) = (cid:107) ( J + J ) − J (cid:107) ≤ (cid:107) ( J + J ) − (cid:107)(cid:107) J (cid:107) ≤ ( α + 1) − α − Moreover, if we denote W := ( J + J ) − ( J − J ), a straightforward computation showsthat S ( I − W W T ) S T = 2( J J T − J J ) . From the estimate( J J T − J J ) u · u = − J J u · u = 2 ω ( J J u, J u ) ≥ α | u | , we deduce I − W W T ≥ α (cid:107) S − (cid:107) I ≥ α + 1) − α − I. This in turn implies W W T ≤ (cid:18) − α ( α + 1) (cid:19) I and hence the last inequality in the statement (i) follows.(ii) The assumption, together with the identity ω ( J u, v ) = J u · J v = − J J u · v ∀ u, v ∈ R n , implies that the endomorphism − J J is self-adjoint and positive. Therefore, its spectrum σ ( − J J ) is contained in the positive real axis and we have ω ( J u, u ) ≥ min σ ( − J J ) | u | = 1max σ (( − J J ) − ) | u | = 1max σ ( − J J ) | u | ∀ u ∈ R n . The desired inequality now follows from the identitymax σ ( − J J ) = (cid:107) − J J (cid:107) = (cid:107) J (cid:107) . roof of Lemma A.2. We identify R with C by mapping ( s, t ) into w = s + it , and we set ∂ := ∂ s − J ∂ t , ∂ := ∂ s + J ∂ t . Moreover, we identify R n with C n by identifying J with the multiplication by i , and weconsider the linear convolution operator T : C ∞ c ( C , R n ) → C ∞ ( C , R n ) , ( T v )( z ) := 12 π ˆ C v ( w ) z − w dsdt, where w = s + it . The operator T commutes with partial derivatives and satisfies ∂T v = T ∂v = v, see [HZ94, Appendix A.4]. Moreover by the Calderon-Zygmund inequality, see [MS04,Theorem B.2.7], for every p ∈ (1 , + ∞ ) there exists C p > v ∈ C ∞ c ( C , R n ), (cid:107) ∂T v (cid:107) L p = (cid:107) T ∂v (cid:107) L p ≤ C p (cid:107) v (cid:107) L p . Therefore ∂T extends to a bounded linear operator on L p ( C , R n ). If p = 2, this operatoris an isometry. Indeed, for v ∈ C ∞ c ( C , R n ) we have (cid:107) ∂T v (cid:107) L = (cid:107) ∂ s T v − ∂T v (cid:107) L = 4 (cid:107) ∂ s T v (cid:107) L − ∂ s T v, ∂T v ) L + (cid:107) ∂T v (cid:107) L = 4 (cid:107) ∂ s T v (cid:107) L − ∂ s T v, ∂ s T v + J ∂ t T v ) L + (cid:107) v (cid:107) L = − ∂ s T v, J ∂ t T v ) L + (cid:107) v (cid:107) L = (cid:107) v (cid:107) L , where the last equality follows from partial integration and the skew symmetry of J .Applying the Riesz-Thorin interpolation theorem, we also have (cid:107) ∂T (cid:107) L p ≤ (cid:107) ∂T (cid:107) − θL (cid:107) ∂T (cid:107) θL p (cid:48) = (cid:107) ∂T (cid:107) θL p (cid:48) for p (cid:48) ≥ θ ∈ (0 , p ≥ /p = (1 − θ ) / θ/p (cid:48) . Byfixing p (cid:48) > θ go to 0, or equivalently p go to 2, we obtainlim sup p ↓ (cid:107) ∂T (cid:107) L p ≤ . (A.3)Let u ∈ L p loc ( U, R n ) be a solution of the linear equation (A.2), which we rephrase as (cid:0) J + J (cid:1) ∂u + (cid:0) J − J (cid:1) ∂u = 2 J f. (A.4)Since J is ω -compatible and uniformly bounded, Lemma A.3 (ii) implies that ω ( J ( w ) v, v ) ≥ α | v | ∀ w ∈ U, ∀ v ∈ R n , where α := 1 / (cid:107) J (cid:107) ∞ . In particular, all the complex structures J ( w ) satisfy the assumptionof A.3 (i) with the same constant α . 69ultiplying the both sides of (A.4) by ( J + J ) − , which exists as observed in LemmaA.3 (i), we obtain the following Beltrami equation on U∂u + G∂u = g (A.5)where G := ( J + J ) − ( J − J )and g := 2( J + J ) − J f. The map g belongs to L p loc ( U, R n ). By the second inequality in Lemma A.3 (i),sup ( s,t ) ∈ U | G ( s, t ) | < . Let Ω ⊂ U be an open subset with compact closure contained in U . Let ρ : C → R be asmooth function with compact support contained in U and taking the value 1 on Ω. Wedefine v ∈ L p ( C , R n ) by v := ρu. Since Ω is arbitrary, it is enough to show that v belongs to W ,q ( C , R n ) for some q > v solves the following Beltrami equation on C ∂v + G∂v = h, (A.6)where the map h := ρg + ( ∂ρ ) u + ( ∂ρ ) Gu belongs to L p ( C , R n ), and G is extended to the whole C by setting it equal to 0 outside U , so that we still have sup w ∈ C | G ( w ) | < . (A.7)Having compact support, v and h belong also to L p (cid:48) ( C , R n ) for every p (cid:48) ≤ p . Since T ∂ = I ,equation (A.6) can be rewritten as ( I + G∂T ) ∂v = h. (A.8)Thanks to (A.3) and (A.7), we can find q > (cid:107) G∂T (cid:107) L q < q ∈ [2 , q ]. The above inequality implies that the operator I + G∂T is invertibleon L q ( C , R n ) for every q ∈ [2 , q ]. Let q ∈ (2 , q ] be a number not larger than p . Since h is in L q ( C , R n ), the equation (A.8) can be restated as ∂v = ( I + G∂T ) − h, and shows that ∂v belongs to L q ( C , R n ). 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Ruhr Universit¨at Bochum, Fakult¨at f¨ur Mathematik, Geb¨aude NA 4/33, D-44801 Bochum,Germany E-mail address : [email protected] Seoul National University, Department of Mathematical Sciences, Research Institute inMathematics, Gwanak-Gu, Seoul 08826, South Korea E-mail address : [email protected]@snu.ac.kr