A generalized Poincaré-Birkhoff theorem
aa r X i v : . [ m a t h . S G ] N ov A GENERALIZED POINCAR ´E–BIRKHOFF THEOREM
AGUSTIN MORENO, OTTO VAN KOERT
To H. Poincar´e, who taught us much;To A. Floer, who followed suit;To C. Viterbo, now on his 60th birthday, who took the cue;and to all those who stand on the Shoulders of Giants. A BSTRACT . We prove a generalization of the classical Poincar´e–Birkhoff theorem for Liouville do-mains, in arbitrary even dimensions. This is inspired by the existence of global hypersurfaces of sectionfor the spatial case of the restricted three-body problem [MvK]. C ONTENTS
1. Introduction 12. Motivation and background 53. Preliminaries on symplectic homology 64. Proof of the Generalized Poincar´e–Birkhoff Theorem 11Appendix A. Hamiltonian twist maps: examples and non-examples 16Appendix B. Symplectic homology of surfaces 19Appendix C. On symplectic return maps 20Appendix D. Strong convexity implies strong index-positivity 24Appendix E. Strongly index-definite symplectic paths 25References 271. I
NTRODUCTION
Poincar´e–Birkhoff theorem, and the planar restricted three-body problem.
The problem offinding closed orbits in the planar case of the restricted three-body problem goes back to ground-breaking work in celestial mechanics of Poincar´e [P12, P87], building on work of G.W. Hill on thelunar problem [H78]. The basic scheme for his approach may be reduced to:(1) Finding a global surface of section for the dynamics;(2) Proving a fixed point theorem for the resulting first return map.This is the setting for the celebrated Poincar´e–Birkhoff theorem, proposed and confirmed in specialcases by Poincar´e and later proved in full generality by Birkhoff in [Bi13]. The statement can besummarized as: if τ : A → A is an area-preserving homeomorphism of the annulus A = [ − , × S that satisfies a twist condition at the boundary, then it admits infinitely many periodic points ofarbitrary large period.In [MvK], the authors proved the existence of S -families of global hypersurfaces of section forthe spatial restricted three-body problem (in the low-energy range, i.e. below and slightly above the first critical value, and independent of mass ratio), fully and non-perturbatively generalizingstep (1) in the above approach to the spatial situation. The relevant return map is a Hamiltoniansymplectomorphism τ : ( D ∗ S , ω ) → ( D ∗ S , ω ) of a Liouville domain ( D ∗ S , ω ) , where ω is defor-mation equivalent to the standard symplectic form. This map extends to the boundary [MvK, Thm.B]. Drawing inspiration from this situation, in this paper, we propose a general fixed-point theoremfor Liouville domains, as an attempt to address step (2) for the spatial case. Fixed-point theory of Hamiltonian twist maps.
The periodic points of τ are either boundary pe-riodic points, which give planar orbits, or interior periodic points which are in 1:1 correspondencewith spatial orbits. We are interested in finding interior periodic points. The Hamiltonian twist condition.
We propose a generalization of the twist condition intro-duced by Poincar´e, for the Hamiltonian case and for arbitrary Liouville domains. Let ( W, ω = dλ ) be a n -dimensional Liouville domain, and consider a Hamiltonian symplectomorphism τ . Let ( B, ξ ) = ( ∂W, ker α ) be the contact manifold at the boundary where α = λ | B , and R α the Reebvector field of α . Recall that τ is Hamiltonian if τ = φ H , where φ tH is the isotopy of W defined by φ H = id , ddt φ tH = X H t ◦ φ tH , where we write H t = H ( t, · ) , and X H t is the Hamiltonian vector fieldof H t defined via i X Ht ω = − dH t . The Liouville vector field V λ is defined via i V λ ω = λ . Definition 1.1. ( Hamiltonian twist map ) We say that τ is a Hamiltonian twist map (with respect to α ), if τ is generated by a smooth Hamiltonian H : R × W → R which satisfies X H t | B = h t R α forsome positive and smooth function h : R × B → R + .In particular, H t | B ≡ const on B , and τ ( B ) ⊂ B . We have h t = dH t ( V λ ) | B is the derivative of H t in the Liouville direction V λ along B , which we assume strictly positive. Also, τ | B is the time- map of a positive reparametrization of the Reeb flow on B . But note that, while the latter conditionis only localized at B , the twist condition is of a global nature, as it requires global smoothness ofthe generating Hamiltonian (cf. [MvK, Rk. 1.3]).Here is a simple example illustrating why the smoothness of the Hamiltonian is relevant for thepurposes of fixed points: Example 1.2 (Integrable twist maps) . Let M = S n for n ≥ with the round metric, and H : T ∗ M → R , H ( q, p ) = 2 π | p | ( not smooth at the zero section); φ H extends to all of D ∗ M as the identity. It is apositive reparametrization of the Reeb flow at S ∗ M , a full turn of the geodesic flow, and all orbitsare fixed points with fixed period. If we smoothen H near | p | = 0 to K ( q, p ) = 2 πg ( | p | ) , with g (0) = g ′ (0) = 0 , then τ = φ K : D ∗ M → D ∗ M , τ ( q, p ) = φ πg ′ ( | p | ) H ( q, p ) , is now a Hamiltonian twistmap. If g ′ ( | p | ) = l/k ∈ Q with l, k coprime, then τ has a simple k -periodic orbit; therefore τ hassimple interior orbits of arbitrary large period (cf. [KH95, p. 350], [M86], for the case M = S ).The Hamiltonian twist condition will be used to extend the Hamiltonian to a Hamiltonian thatis admissible for computing symplectic homology. The extended Hamiltonian can have additional -periodic orbits and these, as well as -periodic orbits on the boundary, need be distinguishedfrom the interior periodic points of τ . We impose the following conditions to do so. Index growth.
We consider a suitable index growth condition on the dynamics on the boundary,which is satisfied in the three-body problem whenever the planar dynamics is strictly convex (seeThm. D.1). This assumption will allow us to separate boundary and extension orbits from interiorones via the index.
GENERALIZED POINCAR´E–BIRKHOFF THEOREM 3
We call a strict contact manifold ( Y, ξ = ker α ) strongly index-definite if the contact structure ( ξ, dα ) admits a symplectic trivialization ǫ with the property that • There are constants c > and d ∈ R such that for every Reeb chord γ : [0 , T ] → Y of Reebaction T = R T γ ∗ α we have | µ RS ( γ ; ǫ ) | ≥ cT + d, where µ RS is the Robbin–Salamon index [RS93].Index-positivity is defined similarly, where we drop the absolute value. A variation of this no-tion was explored in Ustilovsky’s thesis [U99]. He imposed the additional condition π ( Y ) = 0 .With this extra assumption, the concept of index positivity becomes independent of the choice oftrivialization, although the exact constants c and d still depend on the trivialization ǫ . The globaltrivialization will be important when considering extensions of our Hamiltonians, as it will allowus to measure the index growth of potential new orbits. Fixed-point theorems.
We propose the following generalization of the Poincar´e–Birkhoff theo-rem:
Theorem A (Generalized Poincar´e–Birkhoff theorem) . Suppose that τ is an exact symplectomorphismof a connected Liouville domain ( W, λ ) , and let α = λ | B . Assume the following: • (Hamiltonian twist map) τ is a Hamiltonian twist map, where the generating Hamiltonian is atleast C ; • (index-definiteness) If dim W ≥ , then assume c ( W ) | π ( W ) = 0 , and ( ∂W, α ) is stronglyindex-definite. In addition, assume all fixed points of τ are isolated; • (Symplectic homology) SH • ( W ) is infinite dimensional.Then τ has simple interior periodic points of arbitrarily large (integer) period. Remark 1.3.
Let us discuss some aspects of the theorem:(1) (Grading)
We impose the assumptions c ( W ) | π ( W ) = 0 (i.e. W is symplectic Calabi-Yau)to have a well-defined integer grading on symplectic homology.(2) (Surfaces) If dim W = 2 , then the condition that SH • ( W ) is infinite dimensional just meansthat W is not D (see App. B); for D we have SH • ( D ) = 0 , and a rotation on D gives anobvious counterexample to the conclusion. In the surface case, the argument simplifies, andone can simply work with homotopy classes of loops rather than the grading on symplectichomology. The Hamiltonian twist condition implies the classical twist condition for W = D ∗ S , due to orientations.(3) (Cotangent bundles) The symplectic homology of the cotangent bundle of a closed man-ifold is well-known to be infinite dimensional, due to a result of Viterbo [V18, V99] (seealso [AS06]), combined e.g. with a theorem of Gromov [G78, Sec. 1.4]. We have c ( T ∗ M ) = 0 whenever M is orientable. As for the existence of a global trivialization of the contact struc-ture ( ξ, dλ can ) , we note the following: • if Σ is an oriented surface, then S ∗ Σ admits such a global symplectic trivialization; • if M is an orientable -manifold, then S ∗ M also admits such a global symplectictrivialization; • In addition, we know that symplectic trivializations of the contact structure on ( S ∗ S , λ can ) are unique up to homotopy, since [ S ∗ S , Sp (2)] = H ( S ∗ S ; Z ) = 0 . AGUSTIN MORENO, OTTO VAN KOERT (4) (Fixed points)
If fixed points are non-isolated, then we vacuously obtain infinitely many ofthem, although we cannot conclude that their periods are unbounded; “generically”, oneexpects finitely many fixed points.(5) (Long orbits) If W is a global hypersurface of section for some Reeb dynamics, with returnmap τ , interior periodic points with long (integer) period for τ translates into spatial Reeborbits with long (real) period; see Lemma C.1.(6) (Katok examples) There are well-known examples due to Katok [K73] of Finsler metricson spheres with only finitely many simple geodesics, which are arbitrarily close to theround metric (we review them in App. A.2); they admit global hypersurfaces of sectionwith Hamiltonian return maps, for which the index-definiteness and the condition on sym-plectic homology hold. It follows that the return map does not satisfy the twist conditionfor any choice of Hamiltonians.(7) (Spatial restricted three-body problem)
From the above discussion and [MvK], we gather:the only standing obstruction for applying the above result to the spatial restricted three-body problem, in case where the planar problem is strictly convex, is the Hamiltonian twistcondition. Here, note that symplectic homology is invariant under deformations of Liou-ville domains; see e.g. [BR] for a paper with detailed proofs. This would give a proof ofexistence of spatial long orbits in the spirit of Conley [C63], which could in principle be col-lision orbits. Since the geodesic flow on S arises as a limit case (i.e. the Kepler problem),it should be clear from the discussion on Katok examples that this is a subtle condition.In [MvK], we have computed a generating Hamiltonian for the integrable case of the rotat-ing Kepler problem; it does not satisfy the twist condition in the spatial case (in the planarcase, a Hamiltonian twist map was essentially found by Poincar´e). This does not mean apriori that there is not another generating Hamiltonian which does, but this seems ratherunlikely.As a particular case of Thm. A, we state the above result for star-shaped domains in cotangentbundles, as of independent interest (cf. [H11]): Theorem B.
Suppose that W is a fiber-wise star-shaped domain in the Liouville manifold ( T ∗ M, λ can ) ,where M is simply connected, orientable and closed, and assume that τ : W → W is a Hamiltonian twistmap. If the Reeb flow on ∂W is strongly index-positive, and if all fixed points of τ are isolated, then τ hassimple interior periodic points of arbitrarily large period. The above also holds for M = S , as explained in Remark 1.3 (2). One difference with [H11] isthat we work with compact domains in cotangent bundles and conclude that periodic points areinterior, at the expense of imposing index-positivity. Sketch of the proof.
The proof is fairly simple: due to the twist condition we can extend themap τ to a Hamiltonian diffeomorphism b τ that is generated by an admissible Hamiltonian. Thisallows us to appeal to symplectic homology. In particular, we will show lim −→ k HF • ( b τ k ) = SH • ( W ) . Using an index filtration (via index-definiteness and the twist condition), we can show that allgenerators contributing to homology are actually fixed points of some τ k , rather than fixed pointsof the extension. The crucial technical input is Lemma 4.6. If the minimal periods of periodic pointsof τ are bounded, then we can show using a spectral sequence, involving local Floer homologygroups, that the rank of the resulting symplectic homology should also be bounded, leading to acontradiction. Alternatively, one could use the methods used for the proof of the Conley conjecture[G10, H11] to finish the proof. GENERALIZED POINCAR´E–BIRKHOFF THEOREM 5
Remarks on the twist condition and generalizations.
If the Liouville domain is a surface, thisdefinition of the Hamiltonian twist condition is not restrictive, and implements the idea sketchedabove in a simple way. In higher dimensions, the Hamiltonian twist condition is much more re-strictive. Some examples illustrating the nature of the twist condition and applications of the abovetheorem will be presented in Section A. Given the above sketch of the proof, there is obviously somefreedom in Def. 1.1 that allows the same methods to work. For example, if the vector field X H t issufficiently C -close to a positive reparametrization of the Reeb vector field, then the methods willstill go through. However, we will not pursue this generalization because its depends on detailsthat make the formulation awkward and difficult to check. We list some other generalizations,whose proofs will not be worked out in detail: • ( Action positivity ) One can impose constraints on the functions h t in the Hamiltonian twistcondition that force the periodic orbits in the extension to have large action under iterates.In the setting of cotangent bundles, one can then use a theorem of Gromov [G78, Sec. 1.4]cited below, to construct infinitely many interior periodic points. • ( Isolated sets ) The assumption that the fixed points are isolated can be replaced by theweaker assumption that the fixed point set consists of a finite union of submanifolds. Thisis based on a slight generalization of local Floer homology, and is useful when studyingintegrable systems and their perturbations. • ( Non-vanishing symplectic homology ) The condition dim SH • ( W ) = ∞ can be replacedby the condition SH • ( W ) = 0 . The key point here is that non-vanishing symplectic ho-mology implies its unit is non-trivial. Then the methods of the proof of the Conley conjec-ture [G10, H11] can be applied to conclude the existence of infinitely many simple periodicpoints. Strong index-definiteness is needed to show that these periodic points do not corre-spond to boundary and extension orbits, and so are interior. Remark 1.4.
Concerning the last generalization, we remark that we don’t know a single exampleof a Liouville domain ( W, λ ) with c ( W ) = 0 , SH • ( W ) = 0 , and dim SH • ( W ) < ∞ . Acknowledgements.
The authors thank Urs Frauenfelder, for suggesting this problem to thefirst author, for his generosity with his ideas and for insightful conversations throughout the project;Lei Zhao, Murat Saglam, Alberto Abbondandolo, and Richard Siefring, for further helpful inputs,interest in the project, and discussions. The first author has also significantly benefited from severalconversations with Kai Cieliebak in Germany and Sweden, as well as with Alejandro Passeggi inMontevideo, Uruguay. This research was started while the first author was affiliated to AugsburgUniversit¨at, Germany. The first author is also indebted to a Research Fellowship funded by theMittag-Leffler Institute in Djursholm, Sweden, where this manuscript was finalized.2. M
OTIVATION AND BACKGROUND
Hypersurfaces of section, return maps, and open books.Definition 2.1.
Suppose that Y is a compact, oriented, smooth manifold with a non-singular au-tonomous flow φ t . We call an oriented, compact hypersurface Σ in Y a global hypersurface of section for φ t if • the set ∂ Σ is an invariant set for the flow φ t (if non-empty); • the flow φ t is positively transverse to the interior of Σ ; • for all x ∈ Y \ ∂ Σ there are t + > and t − < such that φ t + ( x ) ∈ Σ and φ t − ( x ) ∈ Σ . AGUSTIN MORENO, OTTO VAN KOERT
Given a global hypersurface of section we can define a return map τ as follows: for each x ∈ int (Σ) we choose a minimal t + ( x ) > as in the definition above. Then we put τ ( x ) = φ t + ( x ) ( x ) .Periodic points of τ then correspond to closed orbits of φ t . In general, there is no continuousextension to the boundary, although it is unique whenever exists. Although global hypersurfacesof section do not have good stability properties in higher dimensions, we found that they can beconstructed in certain classes of Hamiltonian dynamical systems that admit an involution. Thisclass includes the restricted three-body problem and several variations (e.g. suitable Stark-Zeemansystems [MvK]).This notion is also closely related to the notion of an open book decomposition. This consists ofa fiber bundle π : Y \ B → S , where B ⊂ Y is a codimension submanifold with trivial normalbundle (called the binding ), such that π coincides with the angular coordinate along some choice ofcollar neighbourhood B × D of B . The pages of the open book are the closure of the fibers of π , allhaving B as boundary. Whenever φ t is a Reeb dynamics of a contact form α on Y which is adaptedto the open book (i.e. α | B is also contact, and dα is symplectic on the pages), each page is a globalhypersurface of section, and the return map preserves the symplectic form dα . This is precisely thesituation in [MvK].In App. C, we will collect some standard facts which apply for return maps arising from Reebdynamics, as described here, for which Thm. A may be applied.3. P RELIMINARIES ON SYMPLECTIC HOMOLOGY
Liouville domains and Hamiltonian dynamics.
There are various forms of Hamiltonian Floerhomology for Liouville domains: these are all referred to as symplectic homology . We will review theversion due to Viterbo, [V18, V99]. Roughly speaking, this is a ring with unit that encodes bothtopological and dynamical data; it is the homology of a chain complex that is freely generated by -periodic Hamiltonian orbits.We now fix conventions. Consider a Liouville domain ( W, λ ) , i.e. ( W, dλ ) is a compact symplecticmanifold with boundary, and the vector field X defined by the equation ι X dλ = λ is outwardpointing along each boundary component of W . This vector field is the Liouville vector field . The -form λ is the Liouville form , and its restriction to ∂W , which we denote by α , is a contact form.Given a Liouville domain ( W, λ ) we build its completion to a Liouville manifold by attaching acylindrical end: ( c W , b λ ) := ( W, λ ) ∪ ∂ ([1 , ∞ ) × ∂W, rα ) . Throughout the paper we will consider smooth functions of the form H : W × S → R , a (time-dependent) Hamiltonian on W . Given such a Hamiltonian, we define its Hamiltonian vector field X H via ι X H dλ = − dH. We denote the set of -periodic orbits of X H by P ( H ) . For the purpose of Floer theory on non-compact manifolds we will need a suitable class of Hamiltonians to work with. First, we recall thespectrum of a contact form α . If P ( α ) denotes the set of all periodic Reeb orbits (including coversand without period bound), thenspec ( α ) = { a ∈ R | there is γ ∈ P ( α ) such that a = A ( γ ) } , where the action is defined as A ( γ ) = R γ α . Definition 3.1.
We recall some standard terminology.
GENERALIZED POINCAR´E–BIRKHOFF THEOREM 7 • A -periodic orbit γ ∈ P ( H ) is non-degenerate if dF l X H − id invertible. • The Hamiltonian H is non-degenerate if all γ ∈ P ( H ) are non-degenerate. • A Hamiltonian H on c W is linear at infinity if at the cylindrical end H has the form H ( r, b, t ) = cr + d for some constants c > and d . In this case we write slope ( H ) := c . • A Hamiltonian H that is non-degenerate and linear at infinity with slope ( H ) / ∈ spec ( α ) willbe called admissible .We call an almost complex structure J = J t on a Liouville manifold ( c W , b λ ) SFT-like if • it is compatible with ( T c W , d b λ ) ; and • on the cylindrical end it satisfies L ∂r J = 0 , Jξ = ξ , and Jr∂ r = R α .We denote by J the space of such J .We will also need invariants of Hamiltonian orbits, i.e. the Conley-Zehnder index, or more gen-erally, the Robbin-Salamon index, and the mean index. Assume that x : R → c W is an orbit of X H .Take a symplectic trivialization ǫ : R × R n → x ∗ T c W , ( t, v ) ǫ t ( v ) ∈ T x ( t ) c W . Then we get apath of symplectic matrices associated with x , namely ψ t = ǫ − t ◦ dF l X H t ◦ ǫ . We can then definethe Robbin-Salamon index of x as µ RS ( x | [0 ,T ] , ǫ ) := µ RS ( ψ | [0 ,T ] ) . If ψ T − id is invertible, then theRobbin-Salamon index reduces to the Conley-Zehnder index. The case of Reeb flows is done simi-larly; we simply restrict the linearized Reeb flow to the symplectic vector bundle ( ξ, dα ) . Similarly,we define the mean index of x as ∆( x, ǫ ) := ∆( ψ ) , where ∆( ψ ) is the mean index of the symplecticpath ψ . We have the following properties (see e.g. section 3.1.1 of [GG15]): (1) | µ RS ( x | [0 ,T ] , ǫ ) − ∆( x, ǫ ) | ≤ dim W , for all T ;(2) lim T → + ∞ µ RS ( ψ | [0 ,T ] ,ǫ ) T = ∆( x, ǫ );(3) ∆( x ( k ) , ǫ ) = k ∆( x, ǫ ) , where we interpret the k -fold catenation x ( k ) , a k -periodic orbit of H , as a -periodic orbit of theiterated Hamiltonian H k . Definition 3.2.
We will call a Hamiltonian flow on W strongly index-definite if there is a symplectictrivialization ǫ W : W × R n → T W , and constants c > , d and such that for every orbit of X H , wehave | µ RS ( x | [0 ,T ] , ǫ ) | ≥ cT + d. The notion of strong index-positivity is obtained by dropping the absolute value in the abovedefinition, and similarly for strong index-negativity . As in the Introduction, we can also define it forReeb flows. Here are some examples:
Lemma 3.3.
Suppose that ( M, g ) is a closed Riemannian manifold with positive sectional curvature. As-sume in addition that the contact structure ( ST ∗ M, ( ξ, dα )) admits a global symplectic trivialization. Then ( ST ∗ M, dα ) is strongly index-positive. Other examples are complements of Donaldson hypersurfaces in monotone symplectic mani-folds provided the degree is sufficiently high and symplectically trivial: these manifolds are indexnegative. A description of the mean index can be found on page 1318 of of [SZ].
AGUSTIN MORENO, OTTO VAN KOERT
Hamiltonian Floer homology and symplectic homology.
Given
Floer data ( J, H ) of an SFT-like J and an admissible H , we note the following: • There are no -periodic orbits of X H on the cylindrical end, because of the spectrum as-sumption. • Non-degenerate -periodic orbits of X H are isolated.Then P ( H ) consists of finitely many -periodic orbits. Informally speaking, we think of Floer ho-mology as “Morse homology” of the following action functional: A H : W , ( S = R / Z , c W ) −→ R , γ Z S γ ∗ b λ − Z H ( γ ( t ) , t ) dt. This functional has the property A H k ( x ( k ) ) = k A H ( x ) for iterates. A computation shows that crit A H = P ( H ) , and we define the Floer chain complex as: CF • ( c W , b λ, H, J ) := M γ ∈P ( H ) Z h γ i . We grade this chain complex by the Conley-Zehnder index, so deg γ := µ CZ ( γ, ǫ ) . This grad-ing depends on the trivialization ǫ , but not if c ( W ) | π ( W ) = 0 . If we define an L -metric on W , ( S , x ∗ T c W ) by h X, Y i = Z ω ( X ( t ) , J t ( x ( t )) Y ( t ) ) dt, then the Floer equation is the L -gradient “flow” of the above functional: for a cylinder u : Z = R × S → c W , this is ( du − X H ⊗ dt ) , = 0 , lim s →±∞ u ( s, t ) = x ± ( t ) . (3.1)Solutions to this equation are called Floer trajectories.
Given -periodic orbits x + , x − ∈ P ( H ) , themoduli space of Floer trajectories is M ( x + , x − ) := { u : Z → c W | u satisfies (3.1) } . In general, this space does not need to have a manifold structure. To obtain this extra structure, wefirst interpret Equation (3.1) as a section of a vector bundle, via ¯ ∂ F : P ( x + , x − ) −→ E ( x + , x − ) , u ( du − X H ⊗ dt ) , ∈ L p ( Z, Ω , ( u ∗ T c W ) ) . Here P ( x + , x − ) is a Banach manifold of cylinders of class W ,p that are W ,p -pushoffs of smoothcylinders that exponentially converge to the given asymptotes x + and x − , and E ( x + , x − ) is a Ba-nach bundle over P ( x + , x − ) whose fiber over u ∈ P ( x + , x − ) is L p ( Z, Ω , ( u ∗ T c W ) ) . For details, seeCh. 8 in [AD]. We will denote the linearization of ¯ ∂ F at u ∈ P ( x + , x − ) by D u ¯ ∂ F . Proposition 3.4.
For Floer data ( J, H ) and u ∈ M ( x + , x − ) , D u ¯ ∂ F is a Fredholm operator of index ind D u ¯ ∂ F = µ CZ ( x + , ǫ ) − µ CZ ( x − , ǫ ) . In addition, we can always choose suitable Floer data close to initial Floer data such all modulispace are transversely cut out: The flow is strictly speaking not defined, since it leads to an ill-posed initial value problem.
GENERALIZED POINCAR´E–BIRKHOFF THEOREM 9
Proposition 3.5.
There is a dense set J reg ⊂ J with the property for all J ∈ J reg , the linearized op-erator D u ¯ ∂ F is surjective for all u ∈ M ( x + , x − ) , and so M ( x + , x − ) is a smooth manifold of dimension µ CZ ( x + , ǫ ) − µ CZ ( x − , ǫ ) . Floer data ( J, H ) as in Proposition 3.5 will be called regular Floer data . We now have all the basicingredients in place: choose regular Floer data ( J, H ) , and define the boundary operator for thechain complex CF • ( c W , b λ, H, J ) via ∂x + = X x − ∈P ( H ) , µ CZ ( x − )= µ CZ ( x + ) − Z ( M ( x + , x − ) / R ) · x − . Here we have modded out M ( x + , x − ) by the reparametrization action in the domain, and theresulting quotient spaces can be compactified, so the coefficients in the above sum are actuallyfinite. Lemma 3.6.
This linear map is a differential: ∂ ◦ ∂ = 0 . The Floer homology of ( c W , b λ, J, H ) is then defined as the homology HF • ( c W , b λ, J, H ) := H • ( CF • ( c W , b λ, J, H ) , ∂ ) . Remark 3.7.
In the case of closed symplectic manifolds, Floer homology is independent of thechoice of Floer data. This is not the case for Liouville domains, and this is the next topic we willdeal with.3.3.
Continuation maps and symplectic homology.
Assume that H and H admissible Hamilto-nians on a Liouville manifold c W . We interpolate between them via K : c W × S × R −→ R , ( w, t, s ) K s ( w, t ) , where K s ( w, t ) = ( H ( w, t ) , if s ≫ H ( w, t ) , if s ≪ . We then consider the parametrized Floer equation for u : Z → c W : ( du − X K ⊗ dt ) , = 0 , lim s →∞ u ( s, t ) = x + ( t ) ∈ P ( H ) , lim s →−∞ u ( s, t ) = x − ( t ) ∈ P ( H ) . The results of the Fredholm theory mentioned in the previous section also apply in this setup, andwe can define a continuation map as c : CF • ( c W , b λ, J, H ) −→ CF • ( c W , b λ, J, H ) ,x + X x − ∈P ( H ) , deg( x − )=deg( x + ) Z M ( x + , x − , J, K ) · x − . Lemma 3.8.
The map c is a chain map, and the induced map on homology is independent of J, K . We also write c for the induced map on Floer homology: c : HF • ( c W , b λ, J, H ) −→ HF • ( c W , b λ, J, H ) . Symplectic homology is then defined as the direct limit over a directed system { H i } i of admissibleHamiltonians for whose slopes slope( H i ) increase to ∞ , SH • ( W, λ, J, { H i } i ) := lim −→ c ij , j>i HF • ( c W , b λ, J, H i ) . (3.2) Remark 3.9.
Symplectic homology is independent of J , and the sequence of Hamiltonians { H i } i .We will henceforth write SH • ( W, λ ) , or SH • ( W ) (omitting the dependence on λ for notationalsimplicity), for symplectic homology. We similarly use the notation CF • ( H ) when ( W, λ ) is fixed.3.4. Degenerate Hamiltonians and local Floer homology.
In case there is a -periodic orbit of H that is degenerate, we perturb H to a non-degenerate Hamiltonian e H with the same slope as H ,choose regular Floer data ( e J, e H ) , and define HF • ( c W , b λ, H ) := HF • ( c W , b λ, e J, e H ) . Lemma 3.10.
This is well-defined, i.e. it is independent of the choice of perturbation, and of e J . Instead of choosing explicit perturbed Hamiltonians, we package them in local Floer homology,which we now review. Suppose H is a Hamiltonian and assume that x ∈ P ( H ) is isolated . Weneed the following lemma, which we adapt from [CFHW]: Lemma 3.11.
Suppose that γ is an isolated -periodic orbit of X H with an isolating neighborhood U . Thenfor every neighborhood V of γ with V ⊂ U , there is a C -small perturbation e H of H with the followingproperties: • All -periodic orbits of X e H contained in U are already contained in V ; • For a compatible almost complex structure e J , all Floer trajectories contained in U are already con-tained in V . Take a C -small perturbation e H as in the lemma so that -periodic orbits in U are non-degenerate(via [SZ, Thm. 9.1]). As in [CFHW], we define the local Floer homology HF loc • ( γ, H ) of γ as the ho-mology of the complex CF loc • ( U, e H, e J ) generated by -periodic orbits of e H , with differential count-ing Floer solutions lying in U . This is well-defined and independent of the isolating neighborhood U , and the perturbed Floer data ( e J, e H ) .We have the following (see for formula (3.1) in [GG15]):supp HF loc • ( γ, H ) ⊂ [∆( γ ) − n, ∆( γ ) + n ] , (3.3)where supp HF loc • ( γ, H ) = { i : HF loci ( γ, H ) = 0 } , and n = dim( W )2 .3.5. Spectral sequence.
Suppose now that H is a Hamiltonian that is linear at infinity with slope( H ) / ∈ spec( α ) . We assume furthermore that the -periodic orbits of H are all isolated. Hence there arefinitely many -periodic orbits with finite action spectrum A H ( P ( H )) . We order the actions valuesin a strictly increasing sequence { a i } ki =1 . Choose a strictly increasing function f : N → R such that f ( i ) < a i +1 < f ( i + 1) . Proposition 3.12.
There is a spectral sequence converging to the Floer homology HF • ( W, λ, H ) , whose E -page is given by E pq := M γ ∈P ( H ) f ( p − < A H ( γ ) GENERALIZED POINCAR´E–BIRKHOFF THEOREM 11 Index-definiteness and grading. We shall need the following: Lemma 3.13. Suppose that SH • ( W, λ ) is infinite-dimensional, and assume that λ | ∂W is an index-definitecontact form. Then { i | SH i ( W, λ ) = 0 } = ∞ .Proof. To prove this, choose a family { H N } N of admissible Hamiltonians with increasing slopessuch that H N is independent of N on W , and so that CF • ( H N ) injects into CF • ( H M ) for M > N . Bynon-degeneracy, each CF • ( H N ) is finitely generated, so the chain complexes get more generatorswith increasing N (since dim SH • ( W, λ ) = ∞ ). By the index-definiteness assumption, these newgenerators have a degree whose absolute value is strictly increasing if N increases sufficiently. Thissettles the claim. (cid:3) 4. P ROOF OF THE G ENERALIZED P OINCAR ´ E –B IRKHOFF T HEOREM Let ( W, λ ) be a Liouville domain with completion ( c W , b λ ) , r the coordinate in the cylindricalend, B = ∂W , α = λ | B , and τ a Hamiltonian twist map generated by H = H t . The symplecticform on the cylindrical end is d ( tα ) , so by the Hamiltonian twist condition, we get h t : B → R + such that X H t | B = h t R α . This means that H t | r =1 ≡ C t > , with ∂ r H | r =1 = h . The family ofHamiltonians H t is not necessarily linear at infinity, and might hence be unsuitable to computesymplectic homology. To deal with this we will construct an extension b H to the cylindrical end of c W that is linear at infinity. Expand H near r = 1 as H = H ( b, t ) + ( r − H ( b, t ) + ( r − H ( b, t ) + . . . . We extend this Hamiltonian to the cylindrical end of c W as b H = b H ( r, b, t ) + ( r − b H ( r, b, t ) + ( r − b H ( r, b, t ) + . . . , (4.4)where we use the following procedure to define the family of smooth functions b H j : • Choose δ > δ > and choose a decreasing cutoff function ρ with ρ | [1 , δ ] = 1 and ρ ( r ) = 0 for r > δ ; • put b H j ( r, b, t ) = H j ( b, t ) · ρ ( r ) for j = 2 , , . . . ; • put b H ( r, b, t ) = C ≥ max t ( C t ) , b H ( r, b, t ) = A ≥ max t,b ( h t ( b )) for r ≥ δ ; • and put b H j ( r, b, t ) = H j ( b, t ) · ρ ( r ) + (1 − ρ ( r ) ) b H j (1 + δ , b, t ) , for j = 0 , . Remark 4.1. The above extension procedure is meant for smooth Hamiltonians. If less regularitysuffices for some application, then the reader can simply truncate the above expansion. Keep inmind that we need at least C -Hamiltonians, since we need to work with a controlled linearizedHamiltonian flow.By the above, we see that H = C t and H = h t , so with our choices, we conclude that b H = A ( r − 1) + C for large r . The extension b H is therefore linear at infinity, and by perturbing A we canassume that A / ∈ spec( α ) . The same can be arranged for all iterates b H k . Hence we have provedthe following lemma: Lemma 4.2. The extended Hamiltonian b H is linear at infinity, so we obtain an admissible family b H k suchthat F l X c H k | W = b τ k and for which SH • ( W, λ ) = lim −→ k HF • ( c W , b λ, b H k ) . (cid:3) For later purposes, we need the explicit form of X b H t . This is given by X b H t = (cid:18) ∂ r b H + b H + ( r − ∂ r b H + ( r − b H + ( r − ∂ r b H + . . . (cid:19) R α + r − r (cid:18) X ξ b H + r − X ξ b H + · · · − (cid:18) d b H ( R α ) + r − d b H ( R α ) + . . . (cid:19) Y (cid:19) . (4.5)Here, Y = r∂ r is the Liouville vector field, and X ξh ∈ ξ is the ξ -component of the contact Hamilton-ian vector field X h = hR α + X ξh of a Hamiltonian h : B → R , defined implicitly by the equation dα ( X ξh , · ) = − dh | ξ . Due to our choice of interpolation, the second term will be smaller in C -normif we choose δ smaller. We denote the coefficient of R α by F = ∂ r b H + b H + ( r − ∂ r b H + ( r − b H + ( r − ∂ r b H + . . . . Lemma 4.3. If δ is chosen to be sufficiently small, then F is positive.Proof. To see this, we note that the first three terms are non-negative, and the second term is at least min t,b h t ( b ) > . The later terms come in pairs of the following form, ( r − k − ( k − b H k + ( r − k k ! ∂ r b H k , with k ≥ . The function b H k has a bound independent of δ , and ∂ r b H k is bounded by C k /δ , where C k is independent of δ . Because the terms are multiplied by a factor ( r − k , which is bounded by δ k , the claim follows. (cid:3) As a result we see that X b H is mostly following the positive Reeb direction if we choose δ suffi-ciently small. In the proof of Lemma 4.6 below we will investigate the linearization of X b H , whichideally would require closeness to a reparametrized Reeb flow in C -norm rather than C -norm.However, C -closeness does not hold, but we will perform a finer analysis with additional assump-tions, which will allow us to fix δ .Lemma 4.2 allows us to compute symplectic homology with the extended Hamiltonian, butit does, by itself, not give any control over periodic orbits in the extension. To prove our maintheorem, we want to show that all generators of SH • ( W, λ ) represent periodic points of τ (i.e. liein W ). To do so, we need to show that the additional periodic points of b τ do not contribute to thesymplectic homology. Depending on the situation, we will use a filtration by homotopy classes ora filtration by index. More specifically, for p ∈ Fix ( b τ k ) , consider the loop γ p ( t ) = F l X c Ht t ( p ) . Then: • If dim W = 2 , the free homotopy class of γ p in ˜ π ( W ) can be used to see that the additionalperiodic orbits do not contribute homologically; • if dim W > , the CZ-index and the index-definiteness assumption will be used to arrive atthe same conclusion.4.1. Filtration by homotopy class. Assume dim W = 2 . Let Fix ∂ ( b τ k ) := Fix ( b τ k ) ∩ ([1 , + ∞ ) × B ) .Given p ∈ Fix ∂ ( b τ k ) , let [ γ p ] be the free homotopy class in ˜ π ( ∂W ) ∼ = Z . We denote the absolutevalue of this integer by | [ γ p ] | . Lemma 4.4. Assume the hypothesis of Thm. A, and that dim W = 2 . Then there is A > such that for all p ∈ Fix ∂ ( b τ k ) , we have | [ γ p ] | ≥ Ak . GENERALIZED POINCAR´E–BIRKHOFF THEOREM 13 Proof. On each circle component of B , choose an angular coordinate φ such that R α = ∂ φ . FromEq. (4.5) and the definition of F we see that X b H has component in the ∂ φ -direction that is boundedfrom below by some constant A > . Iterating, we get a bound of the form Ak . Since the chaincomplex of Floer homology is generated by -periodic orbits, the claim holds. (cid:3) Corollary 4.5. Suppose W and τ are as in the assumptions of Thm. A, with dim W = 2 . Then Thm. Aholds.Proof. Fix a positive integer N and let A be as in Lemma 4.4. Let δ denote a free homotopy classin ˜ π ( W ) that is represented by a simple Reeb orbit (a boundary parallel simple loop). For i ∈{ , . . . , N } and the iterate iδ , from Cor. B.3, we have rk SH iδ ( W ) = 2 (here we forget about integergrading, and use the notation from App. B). As SH iδ ( W ) = lim −→ k HF iδ ( b H k ) , for large k we have rk CF iδ ( b H k ) ≥ . From Lemma 4.4, every p ∈ Fix ∂ ( b τ k ) has [ γ p ] = jδ with j ≥ Ak . If we choose k > N/A we see that j > N , so the generators in CF iδ ( b H k ) are represented by fixed points of τ k .This works for all N , so by sending k to infinity we get infinitely many periodic points of τ .To see that these are geometrically distinct, note that if p ∈ Fix ( τ k ) with k its minimal period and a := [ γ p ] = iδ is boundary parallel, then γ ℓp is a generator of CF ℓa ( H ℓk ) , but γ p is not a generatorof CF a ( H ℓk ) . Taking limit in k , we see that new generators in homotopy class a need appearto generate SH a ( W ) . This gives infinitely many geometrically distinct interior periodic points (indifferent boundary parallel homotopy classes). (cid:3) Filtration by index. We now deal with the second case, so we assume now that dim W > , c ( W ) | π ( W ) = 0 , and that the Reeb flow is strongly index-definite. To set up the argument, we firstneed to establish that index-definiteness of the linearized Reeb flow equation at the boundary (inthe sense of Definition E.1 in Appendix E) implies index-definiteness of the linearized Hamiltonianequation along the cylindrical end: Lemma 4.6. Assume that ( ξ | B , dα | B ) is symplectically trivial, and that the linearized Reeb flow equation ˙ ψ = ∇ ψ R α along B = ∂W is strongly index-definite. Then, the linearized Hamiltonian flow equation ˙ ψ = ∇ ψ X H of the extension of H given by Equation (4.4) is also strongly index-definite along the cylindricalend [1 , + ∞ ) × B .Proof. We prove this using a matrix representation. To do this, we need to symplectically trivializethe full tangent bundle on the cylindrical ends. Given a symplectic trivialization of ( ξ | B , dα | B ) ,we only need to trivialize the symplectic complement of ξ . We do this using the trivialization L = h Y = r∂ r , R i , where R = R α /r is the Reeb vector field at the r -slice.We will work with the usual formalism of time-dependent Hamiltonians, and we do not includethis time-dependence in the notation. Exterior and covariant derivatives are computed using thebase manifold only, and do not involve time derivatives. We will also use the following notation: X ξ := X ξ b H + r − X ξ b H + ( r − X ξ b H + . . . ,G := d b H ( R α ) + r − d b H ( R α ) + ( r − d b H ( R α ) + . . . . To compute the linearization, we choose a convenient connection ∇ , namely the Levi-Civita con-nection for the metric /r · dr ⊗ dr + α ⊗ α + dα ( · , J · ) . This connection has the following properties: • ∇ Y = 0 . Keep in mind that Y is the Liouville vector field r∂ r ; • ∇ R α R α = 0 and ∇ Y R α = 0 ; • ∇ X R α ∈ ξ for all X ∈ ξ .With respect to this connection we compute the linearization as ∇ X b H = F ∇ R α + dF ⊗ R α + 1 r dr ⊗ ( X ξ − GY ) + r − r ( ∇ X ξ − dG ⊗ Y ) . (4.6)Before we continue our analysis of the linearization, we first need to discuss the behaviour of theHamiltonians b H j and their derivatives under rescaling the interpolation parameter δ . We willwrite the terms in the expression (4.4) as b H ′ j if we use δ ′ as interpolation parameter. We have thefollowing: • derivatives in the B -direction (denoted ∂ b ) admit a uniform bound, independent of δ , i.e. max [1 , + ∞ ) × B ∂ kb b H ′ j = max [1 , + ∞ ) × B ∂ kb b H j , for all k ≥ • derivatives in the r -direction scale as follows: max [1 , + ∞ ) × B ∂ kr b H ′ j = (cid:18) δ δ ′ (cid:19) k max [1 , + ∞ ) × B ∂ kr b H j , for all k ≥ . Keeping this scaling behaviour in mind, we regroup terms in Equation (4.6) to obtain the followingrepresentation: ∇ X b H = L + L , where L = F ∇ R α + dF ⊗ R α + 1 r dr ⊗ ( X ξ − GY ) − r − r dG ( Y ) dr ⊗ Y + r − r dr ⊗ ∇ Y X ξ and L = r − r (cid:0) ∇ ξ X ξ + α ⊗ ∇ R α X ξ − R α ( G ) α ⊗ Y − d ξ G ⊗ Y (cid:1) . Here, ∇ ξ = P ξ ∇| ξ , where P ξ is the orthogonal projection to ξ , and d ξ = d | ξ . We will explain belowthat the matrices L and L have the following matrix representations: L = F · ∇ ξ R α U V 00 0 W Z a b c , L = r − r ∇ ξ X ξ U ′ V ′ W ′ Z ′ a ′ This is clear for L . We further want to show that L ∈ sp (2 n ) , which will constrain the entriesmore. Since we know that L + L ∈ sp (2 n ) , we will show that L ∈ sp (2 n ) , since the latter containsfewer terms. For this we note the following: • the matrix representation for ∇ ξ X ξ is in sp (2 n ) . This is because these entries come from the ξ -part of a contact Hamiltonian; • the matrix representation for R α ( G ) α ⊗ Y is in sp (2 n ) : the non-trivial entry corresponds tothe element a ′ ; • non-trivial entries in the matrix representation of − d ξ G ⊗ Y appear only on the first row ofthe lower left block. These corresponds to the elements W ′ , Z ′ ; GENERALIZED POINCAR´E–BIRKHOFF THEOREM 15 • non-trivial entries in the matrix representation of α ⊗ ∇ R α X ξ appear as the last column. Wewill show that these correspond to the elements U ′ and V ′ . We claim that h∇ R α X ξ , R α i = 0 .Indeed, since the contact structure is orthogonal to the Reeb vector field with our choice ofmetric, we have R α h X ξ , R α i = h∇ R α X ξ , R α i + h X ξ , ∇ R α R α i = h∇ R α X ξ , R α i . Similarly, we obtain h∇ R α X ξ , Y i = 0 . This means that the L -entries in the matrix represen-tation of α ⊗ ∇ R α X ξ are zero. • we have ( W ′ , Z ′ ) T = J · ( U ′ , V ′ ) T = ( − V ′ , U ′ ) T . This follows since − d ξ G is dual to ∇ R α X ξ ,i.e. dα ( ∇ R α X ξ , · ) = − d ξ G .We conclude that L ∈ sp (2 n ) , and hence L is, too. Observe also that for all ǫ > we can choose δ > such that k L k < ǫ due to the scaling behaviour we discussed earlier: this can be done in away that is compatible with Lemma 4.3, i.e. δ getting smaller as ǫ gets smaller.Since J L is symmetric, we can fix the terms of L . They must necessarily have the followingform: L = F · ∇ ξ R α U V 00 0 V − U a b − a ∈ sp (2 n ) . This matrix has precisely the form that we consider in Appendix E. Moreover, note that strongindex-definiteness is invariant under scaling by a positive (possibly time-dependent) function ofthe generating matrix. Indeed, this scaling has the effect of positively reparametrizing the flow,and so the new flow intersects the Maslov cycle as often as the original one (although the constantsin the definition of strong index-definiteness might change). Therefore, since the ODE ˙ ψ = ∇ ξψ R α isstrongly index-definite by assumption and F > , then so is the ODE ˙ ψ = F · ( ∇ ξψ R α ) . Appendix Enow tells us that the system ˙ ψ = L ψ is strongly index-definite. By choosing δ sufficiently small,we can make the matrix L get arbitrarily C -close to L + L = ∇ X ˆ H . Since the system ˙ ψ = L ψ is strongly index-definite, we can adapt Lemma 2.2.9 from [U99] to see that ˙ ψ = ∇ ψ X b H is stronglyindex-definite, too. This concludes the proof of Lemma 4.6. (cid:3) Proof of Thm. A ( dim W > ). Write τ = φ H for H as in Def. 1.1. Assuming its interior fixed pointsare isolated, we have finitely many isolated interior -periodic orbits of H , say γ , . . . , γ k . Assumeby contradiction that the minimal periods of all interior periodic points of τ are, in increasing order,given by , m , . . . , m ℓ . Take an increasing sequence { p i } ∞ i =1 going to infinity, and such that each p i is indivisible by the m j ’s. We have SH • ( W ) = lim −→ i HF • ( b H p i ) . By Lemma 3.13, for all N > nk , where dim( W ) = 2 n , we find distinct degrees i , . . . , i N such that SH i j ( W ) = 0 , ordered byincreasing absolute value. By Lemma 4.6, we can choose p i sufficiently large such that the followinghold:(1) Each -periodic orbit of b H p i that is contained in c W \ int ( W ) has RS-index whose absolutevalue is larger than | i N | + 2 n ;(2) the Floer homology groups HF i j ( b H p i ) are non-trivial for j = 1 , . . . , N .Now consider the spectral sequence from Proposition 3.12 for b H p i . We deduce from (2) that theremust be non-trivial summands on E pq with p + q = i j for j = 1 , . . . , N . Since the terms of thespectral sequence are made up from local Floer homology groups, and we know from (1) that no -periodic orbit in c W \ int ( W ) can contribute to local Floer homology of degree i j , we conclude thatevery term E pq in the spectral sequence with p + q = i j must come from the local Floer homologyof an orbit γ in int ( W ) .Because we have assumed that the p i ’s are indivisible by the m j ’s we conclude that each suchorbit γ must be an iterate of one of the orbits γ , . . . , γ k . Moreover, by (3.3):supp HF loc • ( γ p i j , H p i ) ⊂ [ p i ∆( γ j ) − n, p i ∆( γ j ) + n ] . This covers at most nk different degrees, leaving some of the degree i j uncovered as we hadchosen N > nk . This is a contradiction. (cid:3) Proof of Thm B.