Orderability, contact non-squeezing, and Rabinowitz Floer homology
OORDERABILITY, CONTACT NON-SQUEEZING, AND RABINOWITZFLOER HOMOLOGY
PETER ALBERS AND WILL J. MERRY
Abstract.
We study Liouville fillable contact manifolds (Σ , ξ ) with non-zero RabinowitzFloer homology and assign spectral numbers to paths of contactomorphisms. As a conse-quence we prove that (cid:94)
Cont (Σ , ξ ) is orderable in the sense of Eliashberg and Polterovich.This provides a new class of orderable contact manifolds. If the contact manifold is in addi-tion periodic or a prequantization space M × S for M a Liouville manifold, then we constructa contact capacity. This can be used to prove a general non-squeezing result, which amongstother examples in particular recovers the beautiful non-squeezing results from [EKP06]. Introduction and Results
Suppose (Σ , ξ ) is a closed coorientable contact manifold. Denote by Cont (Σ , ξ ) the identitycomponent of the group of contactomorphisms, and denote by P Cont (Σ , ξ ) the set of smoothpaths of contactomorphisms starting at the identity. The universal cover (cid:93) Cont (Σ , ξ ) is then P Cont (Σ , ξ ) / ∼ , where ∼ denotes the equivalence relation of being homotopic with fixedendpoints. Suppose α ∈ Ω (Σ) is a contact form defining ξ , and θ t its Reeb flow. To a path ϕ = { ϕ t } ≤ t ≤ ∈ P Cont (Σ , ξ ) we can associate its contact Hamiltonian h t h t ◦ ϕ t := α (cid:18) ddt ϕ t (cid:19) : Σ → R , (1.1)which uniquely determines the path ϕ . In this article we are interested in four classes ofcontact manifolds, labelled (A) , (A) + , (B) , and (C) . See Section 2 for precise definitions ofthe terms involved. Assumption (A): (Σ , ξ ) admits a Liouville filling W such that the Rabinowitz Floerhomology RFH ∗ (Σ , W ) is non-zero. Theorem 1.1.
Suppose (Σ , ξ ) satisfies Assumption (A) . Then for any non-zero class Z ∈ RFH ∗ (Σ , W ) there is a map c ( · , Z ) : P Cont (Σ , ξ ) → R with the following properties. (1) If ϕ ∼ ψ then c ( ϕ, Z ) = c ( ψ, Z ) . Thus c ( · , Z ) descends to define a map (denoted bythe same symbol) c ( · , Z ) : (cid:93) Cont (Σ , ξ ) → R . (2) For any T ∈ R , c ( t (cid:55)→ θ tT , Z ) = − T + c (id Σ , Z ) . (3) The map c is continuous with respect to the C -norm on P Cont (Σ , ξ ) . (4) If ϕ resp. ψ is generated by the contact Hamiltonian h t resp. k t with h t ( x ) ≥ k t ( x ) forall x ∈ Σ and t ∈ [0 , then c ( ϕ, Z ) ≤ c ( ψ, Z ) . Corollary 1.2.
If there exists a constant δ > such that h t ( x ) ≥ δ for all x ∈ Σ and t ∈ [0 , then c ( ϕ, Z ) < c (id Σ , Z ) . Corollary 1.3. If (Σ , ξ ) satisfies Assumption (A) then (cid:93) Cont (Σ , ξ ) is orderable in the senseof Eliashberg–Polterovich [EP00] . a r X i v : . [ m a t h . S G ] J u l PETER ALBERS AND WILL J. MERRY
Proof of Corollary 1.2.
Note that the constant function δ generates the path { t (cid:55)→ θ tδ } thusTheorem 1.1 (2) & (4) implies c ( ϕ, Z ) ≤ c ( t (cid:55)→ θ tδ , Z ) = − δ + c (id Σ , Z ) < c (id Σ , Z ) . (1.2) (cid:3) Proof of Corollary 1.3.
Recall from [EP00, Criterion 1.2.C.] that (cid:93)
Cont (Σ , ξ ) is orderable ifand only if no contractible loop ϕ of contactomorphisms exists whose contact Hamiltonian h t satisfies h t ( x ) > x ∈ Σ and t ∈ [0 , ϕ is sucha loop. Then (1) in Theorem 1.1 implies that c ( ϕ, Z ) = c (id Σ , Z ) since ϕ is contractible. Onthe other hand Corollary 1.2 implies that c ( ϕ, Z ) < c (id Σ , Z ). This contradiction proves theCorollary. (cid:3) Remark 1.4.
Together with its companion article [AFM13] this article is the first to establishRabinowitz Floer homology as a tool for studying orderability and non-squeezing questions incontact geometry. The aim of the article [AFM13] is very different from the present one sinceit is solely concerned with a link between the famous Weinstein conjecture and orderability.In this article we derive obstructions from Rabinowitz Floer homology to non-orderability andto squeezing phenomena. Since Rabinowitz Floer homology is nowadays rather computablethis delivers checkable criteria for orderability and non-squeezing. In particular, we reproducemany of the previously known examples of orderable contact manifolds and similarly for thenon-squeezing results. At the same time our approach gives entirely new classes of orderablecontact manifolds and an abstract non-squeezing results.A precursor to this development is the article by Frauenfelder [AF12] and the first authorin which a rather different version of Rabinowitz Floer homology is used to mimic Givental’sconstruction of the non-linear Maslov index. On unit cotangent bundles this also leads to anobstruction to a (strong form) of non-orderability.Corollary 1.3 has the following rephrasing.
Corollary 1.5.
Let (Σ , ξ ) be a closed contact manifold for which (cid:93) Cont (Σ , ξ ) is not orderable.Then for any Liouville filling W of Σ one has RFH ∗ (Σ , W ) = 0 . (1.3)We illustrate the above at some examples. Example 1.6. • The sphere S n − with its standard contact structure is not orderable by [EKP06,Theorem 1.10]. The equivalent statement of vanishing symplectic homology of anyLiouville filling of the standard contact sphere was proved before by Smith, see [Sei06,Corollary 6.5]. • A new class of orderable contact manifolds is given by links of weighted homogeneoussingularities with positive Milnor number. This includes certain
Brieskorn manifolds ,and in particular non-standard structures on spheres (the Ustilovsky spheres), as wellas contact structures on exotic spheres. This was communicated to us by Otto vanKoert, see Example 1.11 below, which also includes more examples. • Let Σ = S ∗ g B be the unit cotangent bundle of the closed manifold B equipped withits standard contact structure ξ . The Liouville filling by the unit codisk bundle D ∗ g B always has RFH ∗ ( S ∗ g B, D ∗ g B ) (cid:54) = 0 due to Cieliebak-Frauenfelder-Oancea [CFO10], see RDERABILITY, CONTACT NON-SQUEEZING, AND RABINOWITZ FLOER HOMOLOGY 3 also Abbondandolo-Schwarz [AS09]. Thus, (cid:93)
Cont ( S ∗ g B, ξ ) is orderable, which wasproved by Eliashberg-Kim-Polterovich [EKP06] and Chernov-Nemirovski [CN10]. • Since symplectic homology and thus Rabinowitz Floer homology is unchanged undersubcritical handle attachment ([Cie02]) any Liouville fillable contact manifold withnon-zero Rabinowitz Floer homology remains (cid:93)
Cont -orderable under subcritical han-dle attachment.A slight strengthening of Assumption (A) is the following, see the discussion on page 19for details. Assumption (A) + : (Σ , ξ ) admits a Liouville filling ( W, dλ ) such that α := λ | Σ isMorse-Bott. Moreover there exists a non-zero class µ Σ ∈ RFH n (Σ , W ) such that p − ε ( µ Σ ) = j ε ([Σ]). Example 1.7.
If (Σ , ξ ) admits a Liouville filling (
W, dλ ) such that the Reeb vector field of α := λ | Σ has no contractible Reeb orbits (e.g. T with its standard contact structure, which isfilled by D ∗ T ) then Assumption (A) + is trivially satisfied, since in this case RFH ∗ (Σ , W ) ∼ =H ∗ + n − (Σ; Z ).The class µ Σ has the property that c (id Σ , µ Σ ) = 0. Assumption (A) + allows us tostrengthen Statement (4) of Theorem 1.1 to the following statement: Theorem 1.8.
Suppose ϕ ∈ (cid:93) Cont (Σ , ξ ) has contact Hamiltonian h t . Assume h t ≤ andthere exists x ∈ Σ such that h t ( x ) < for all t ∈ [0 , then c ( ϕ, µ Σ ) > . Remark 1.9.
In Section 1.1 below we provide an example to show that the same implicationwith opposite inequalities in the above theorem does not hold. See Remarks 3.13 and 5.13,and Appendix A.It’s worthwhile pointing out that replacing α with − α changes the sign of the contactHamiltonian h t . However in general if (Σ , α ) is Liouville fillable then there is no canonicalLiouville filling of (Σ , − α ), and thus for our purposes the distinction between positive andnegative is not arbitrary. A good non-trivial example to bear in mind is Σ = R P , with ξ thecontact structure induced from from the standard contact structure on S . Then Σ is Liouvillefillable with W = T ∗ S , and RFH ∗ ( R P , T ∗ S ) is infinite dimensional (cf. [AS09, CFO10]).However we can also view R P as a prequantization space over S , which corresponds toreplacing α with − α . Indeed, in this case the completion W is the total space of O (2) → S ,as opposed to T ∗ S which is the total space of O ( − → S .For us the main relevance of Theorem 1.8 is that it implies the contact capacity ¯ c ( · , µ Σ ) wedefine below is non-trivial, see Remark 1.19 below. Definition 1.10.
We call a contact form α periodic if its Reeb flow θ t is a 1-periodic loop: θ = id Σ .Let us now assume that (Σ , ξ ) satisfies the following condition: Assumption (B): (Σ , ξ ) admits a Liouville filling ( W, dλ ) such that the RabinowitzFloer homology RFH ∗ (Σ , W ) is non-zero and such that α := λ | Σ is periodic. Example 1.11 (Communicated to us by Otto van Koert) . An interesting class of exampleswhere our results apply is the following. Assume that (
Q, ω ) is a simply connected symplecticmanifold. Assume in addition that [ ω ] is integral and ( Q, ω ) is monotone, with non-positivemonotonicity constant. Let K ⊂ Q denote a closed connected symplectic submanifold of PETER ALBERS AND WILL J. MERRY codimension 2 such that K is Poincar´e dual to k [ ω ] for some k ∈ N . Such a hypersurface isknown as a Donaldson hypersurface , since Donaldson showed that every symplectic manifoldwith an integral symplectic form admits a symplectic submanifold Poincar´e dual to k [ ω ] for k ∈ N sufficently large [Don96]. Assume in addition that H ( K ; Z ) = 0. Let ν ( K ) denote acollar neighborhood of K in Q . Then the complement Q \ ν ( K ) is the interior of a Liouvilledomain ( W , λ ) with the property that the Reeb flow on Σ := ∂W is periodic (see for instance[Dio12]). Denote by W the completion of W . If we assume that the inclusion Σ (cid:44) → W inducesan injection on π (e.g. if dim Q ≥ ∗ ( W ) is non-zero (seebelow), and hence so is the Rabinowitz Floer homology RFH ∗ (Σ , W ). Thus Assumption (B) is satisfied.There are several ways to see that the symplectic homology of SH ∗ ( W ) is non-zero. Thesimplest one is an index argument, and goes as follows. Since ( Q, ω ) is monotone it followseasily that c ( T W ) is torsion, and hence the Conley-Zehnder index is a well defined integer forcontractible orbits. Next the monotonicity assumption and a suitable choice of Hamiltonianfunctions imply that the index of all contractible Reeb orbit is at most n = dim W . Howeverthere is a well defined map H ∗ + n ( W , Σ) → SH ∗ ( W ), and the image of the fundamentalclass has degree n . This class therefore remains non-zero in SH ∗ ( W ) due to index reasons.Alternatively, one can argue using S -equivariant symplectic homology: the proof of Lemma7.6 in [RDvK14] implies that SH S , + ∗ ( W ) has no generators with large positive degree, sincethe index growth of non constant one periodic orbits is proportional to the (non-positive)monotonicity constant. However if SH ∗ ( W ) = 0 then work of Bourgeois-Oancea [BO12]implies that SH S ∗ ( W ) = 0. The Viterbo long exact sequence (see [BO13, Lemma 4.8]) thenimplies SH S , + ( W ) ∼ = H ∗ + n ( W, Σ) ⊗ H ∗ ( C P ∞ ; Z ), which has generators with arbitrary positivedegree, which is a contradiction.If we assume that π ( Q ) = 0 then it follows from the homotopy exact sequence of thefibration that all the Reeb orbits on Σ are non-contractible. If in addition H ( Q ; Z ) = 0 andH ( K ; Z ) = 0 then the construction described above yields examples satisfying Assumption (A) + and (B) (cf. Example 1.7). Moreover, very explicit examples are the complement of adegree k -curve in C P with k ≥ ( K ; Z ) (cid:54) = 0).Finally, another more general class of examples where our results apply are links of weightedhomogeneous singularities with positive Milnor number. The latter guarantees the existenceof Lagrangian spheres which in turn implies non-vanishing of RFH, see [KvK13]. In particular,this includes certain Brieskorn manifolds , see [KvK13, Theorem 1.2] for a precise statement.We mention here only that these include non-standard contact structures on spheres (theUstilovsky spheres [Ust99]), as well as contact structures on exotic spheres.The advantage of Assumption (B) is the following. As before Z denotes a non-zero classin RFH ∗ (Σ , W ). The basic definitions and results that follow are based on Sandon’s article[San11]. Definition 1.12.
We define for ϕ ∈ (cid:93) Cont (Σ , ξ ) an integer c ( ϕ, Z ) by c ( ϕ, Z ) := (cid:100) c ( ϕ, Z ) (cid:101) . (1.4) Proposition 1.13.
The function c ( · , Z ) : (cid:93) Cont (Σ , ξ ) → Z is conjugation invariant: if ψ ∈ Cont (Σ , ξ ) and ϕ ∈ (cid:93) Cont (Σ , ξ ) then ¯ c ( ψϕψ − , Z ) = ¯ c ( ϕ, Z ) . (1.5) RDERABILITY, CONTACT NON-SQUEEZING, AND RABINOWITZ FLOER HOMOLOGY 5
Remark 1.14.
In contrast to spectral invariants in Hamiltonian Floer homology, Proposition1.13 is a non-trivial result. See Remark 1.20 below.
Definition 1.15.
For an open set U ⊂ Σ we define the contact capacity c ( U, Z ) := sup (cid:110) c ( ϕ, Z ) | ϕ ∈ (cid:93) Cont (Σ , ξ ) , S ( ϕ ) ⊂ U (cid:111) ∈ Z ∪ {±∞} , (1.6)where by convention we declare that sup ∅ = −∞ . Remark 1.16.
The notion of contact capacity was introduced by Sandon in [San11]. Shewas the first to discover a connection between translated points and orderability and othercontact rigidity phenomena.Proposition 1.13 has the following immediate corollary.
Corollary 1.17.
For all ψ ∈ Cont (Σ , ξ ) , one has c ( ψ ( U ) , Z ) = c ( U, Z ) . (1.7)Using the contact capacity we obtain the following abstract non-squeezing results. Theorem 1.18.
Let U ⊂ V ⊂ Σ be open sets and assume that there exists ϕ ∈ Cont (Σ , ξ ) with ϕ ( V ) ⊂ U . Then c ( U, Z ) = c ( V, Z ) . (1.8) In particular, if c ( U, Z ) < c ( V, Z ) then there exists no contact isotopy mapping V into U . Proof . Suppose ϕ is as in the statement of the Theorem. Then trivially we have c ( U, Z ) ≤ c ( V, Z ) and c ( ϕ ( V ) , Z ) ≤ c ( U, Z ). By Proposition 4.7 we also have c ( ϕ ( V ) , Z ) = c ( V, Z ), andhence c ( U, Z ) = c ( V, Z ) as claimed. (cid:3)
Remark 1.19.
If we assume both Assumptions (A) + and (B) then it follows from Theorem1.8 that ¯ c ( U, µ Σ ) ≥ U ⊂ Σ, and hence the capacity ¯ c ( · , µ Σ )is non-trivial. In the next section we provide a class of examples where ¯ c ( · , µ Σ ) is computable,and thus derive applications of Theorem 1.18.1.1. Prequantization spaces.
Fix a Liouville manifold (
M, dγ ) (i.e. the completion of Li-ouville domain, cf. Definition 2.1). The prequantization space of M is the contact manifoldΣ := M × S , equipped with the contact structure ξ := ker α , where α := γ + dτ, (1.9)where τ is the coordinate on S = R / Z . These contact manifolds are the last type we studyin this paper: Assumption (C): (Σ , ξ = ker α ) is a prequantization space Σ = M × S , where ( M, dγ )is a Liouville manifold, and α = γ + dτ .Let P denote a torus with a small disc removed, so that ∂P = S . Equip P with an exactsymplectic form dβ such that β | ∂P = dτ . Let ( P, dβ ) denote the completion of the Liouvilledomain ( P , dβ ), and consider W := M × P, equipped with the symplectic form dλ where λ := γ + β . Even though Σ is periodic, W is not a Liouville filling of Σ, and in fact Σ does not satisfy either Assumptions (A) or (B) - forinstance, Σ is non-compact. Nevertheless, it is still possible to define the Rabinowitz Floerhomology RFH ∗ (Σ , W ), and we prove thatRFH ∗ (Σ , W ) ∼ = HF ∗ ( M ) ⊗ H ∗ ( S ; Z ) . (1.10) PETER ALBERS AND WILL J. MERRY
Here HF ∗ ( M ) ∼ = H n −∗ ( M ; Z ) (1.11)denotes the Hamiltonian Floer homology of M , defined using compactly supported Hamiltoni-ans (see Frauenfelder-Schlenk [FS07]). Moreover the Rabinowitz Floer homology RFH ∗ (Σ , W )constructed in this way satisfies the analogue of Assumption (A) + , that is, there is a suitablenon-zero class µ Σ ∈ RFH n (Σ , W ). Indeed, in this case one simply takes µ Σ to be the imageof the class { pt } × [ S ] ∈ H ( M ; Z ) ⊗ H ( S ; Z ) under the isomorphisms (1.10) and (1.11).Since the Hamiltonian Floer homology is non-zero, one can associate a spectral number c M ( f ) to any f ∈ (cid:93) Ham c ( M, dγ ), the universal cover of the group of compactly supportedHamiltonian diffeomorphisms (see eg. Schwarz [Sch00] or Frauenfelder-Schlenk [FS07]) . Asin the contact case described above, c M can then be used to define a symplectic capacity c M ( O ) for O ⊂ M open, by setting c M ( O ) := sup { c M ( f ) | S ( f ) ⊂ O} . (1.12) Remark 1.20.
In contrast to the contact case (see Proposition 1.13 and Remark 1.14),the proof that c M ( f ) is invariant under conjugation, that is, c M ( hf h − ) = c M ( f ) for f ∈ (cid:93) Ham c ( M, dγ ) and h ∈ Symp c ( M, dγ ) is immediate, since in this case the action spectrum of hf h − is the same as the action spectrum of f (see for instance [HZ94, Chapter 5, Proposition7]). This in turn immediately implies that c M is a symplectic capacity, that is, c M ( f ( O )) = c M ( O ) for any symplectomorphism f and any open set O ⊂ M .Going back to Σ = M × S , let us denote by Cont ,c (Σ , ξ ) those contactomorphisms ϕ withcompact support. There is a natural way to lift an element f ∈ (cid:93) Ham c ( M, dγ ) to obtain anelement ϕ ∈ (cid:93) Cont ,c (Σ , ξ ), as we now explain. The equation f ∗ t γ − γ = da t , a ≡ , (1.13)determines a smooth compactly supported function a t : M → R . Define ϕ t : Σ → Σ by ϕ t ( y, τ ) := (cid:0) f t ( y ) , τ − a t ( y ) (cid:124) (cid:123)(cid:122) (cid:125) mod 1 (cid:1) . (1.14)As explained in Section 5.1 below, one can define for any non-zero class Z the spectralnumbers c ( ϕ, Z ) for ϕ ∈ (cid:93) Cont ,c (Σ , ξ ) in much the same way as before. Similarly one candefine the capacity ¯ c ( U ) for U ⊂ Σ open in the same way as before (if U is not precompactthen one must again only use elements of (cid:93) Cont ,c (Σ , ξ ) when defining ¯ c ( U )). Moreover mostof the results stated thus far in the paper continue to hold (this statement is made moreprecise in Section 5.1). In particular, Parts (1), (3), and (4) of Theorem 1.1 remain true, andso do Proposition 1.13 and Theorem 1.18.It is natural to ask the question: if ϕ ∈ (cid:93) Cont ,c (Σ , ξ ) is the lift of f ∈ (cid:93) Ham c ( M, dγ ), how is c ( ϕ ) := c ( ϕ, µ Σ ) related to c M ( f )? Note that S ( ϕ ) = S ( f ) × S , and hence another questionis how the c M capacity of O ⊂ M is related to the c capacity of O × S . The following resultanswers these questions in the nicest possible way. Theorem 1.21.
Suppose f ∈ (cid:93) Ham c ( M, dγ ) , and let ϕ ∈ (cid:93) Cont ,c (Σ , ξ ) denote the lift of f .Then c M ( f ) = c ( ϕ ) . (1.15) Moreover, if
O ⊂ M is open and has compact closure then c M ( O ) = c ( O × S ) . (1.16) RDERABILITY, CONTACT NON-SQUEEZING, AND RABINOWITZ FLOER HOMOLOGY 7
Theorem 1.21 allows us to prove non-squeezing results on Σ by making use of the knownresults on M . Let l t : M → M denote the flow of the Liouville vector field on M and set ζ t := l log t . We will prove the following general result. Theorem 1.22.
Suppose
O ⊂ M is a non-empty open set with compact closure and finitecapacity: c M ( O ) < ∞ . Suppose there exists a contact isotopy ϕ ∈ (cid:93) Cont ,c (Σ , ξ ) such that ϕ (cid:0) ζ r ( O ) × S (cid:1) ⊂ ζ r ( O ) × S (1.17) for r , r ∈ R . Then (cid:100) r (cid:101) ≤ (cid:100) r (cid:101) . More generally if O ⊂ Q ⊂ M are open sets with theproperty that there exists ϕ ∈ (cid:93) Cont (Σ , ξ ) with ϕ ( Q× S ) ⊂ O× S then (cid:100) c M ( Q ) (cid:101) = (cid:100) c M ( O ) (cid:101) . Proof . Note that for any r ∈ R , c M ( ζ r ( O )) = rc M ( O ) (cid:54) = 0 , (1.18)since c M ( O ) > O is non-empty. Thus without loss of generality we may assume that c M ( O ) = 1. Then c ( ζ r ( O ) × S ) = (cid:100) c M ( ζ r ( O )) (cid:101) = (cid:100) r (cid:101) . (1.19)The result is now an immediate consequence of Theorem 1.18 (which, as remarked above,does indeed remain true in this setting). The last statement follows similarly. (cid:3) Remark 1.23.
Theorem 1.22 recovers the beautiful non-squeezing result of [EKP06, Theorem1.2]. In this case one takes M = R n and U the unit ball. They prove that if (cid:100) r (cid:101) < (cid:100) r (cid:101) then it is not possible to squeeze B ( r ) × S into the cylinder C ( r ) × S . This result wasalso recovered by Sandon [San11] using generating functions.A further applications of Theorem 1.21 is the following. Here we denote by c HZ the Hofer-Zehnder capacity (see Definition 5.18 below or [HZ94]).
Theorem 1.24.
Let ( M, dγ ) denote a Liouville manifold. Equip R m with the standardsymplectic form dλ std , and consider the contact manifold ( (cid:101) Σ , α + λ std ) , where (cid:101) Σ := M × R m × S . Suppose O ⊆ M is open and c HZ ( O , M ) < ∞ . Choose r > such that (cid:6) πr (cid:7) < (cid:100) c HZ ( O , M ) (cid:101) (1.20) and set r := (cid:113) π c HZ ( O , M ) + 1 (1.21) Then there does not exist ϕ ∈ Cont ,c ( (cid:101) Σ , α + λ std ) such that ϕ ( O × B ( r ) × S ) ⊂ O × B ( r ) × S . (1.22)The proof of Theorem 1.24 is given in Section 5.4. See also Corollary 5.21 for an applicationof Theorem 1.24.Finally, following [EKP06, Section 1.7] we investigate a rigidity phenomenon of positivecontractible loops of contactomorphisms. Suppose now that ( M, dγ ) is the completion of aLiouville domain ( M , dγ ). Set S := ∂M and κ := γ | S , so that ( S, κ ) is a contact manifold.Abbreviate M r := (cid:40) M \ ( S × ( r, , < r < ,M ∪ S ( S × [1 , r ] , r ≥ . (1.23) Note that whilst C ( r ) := B ( r ) × R n − does not have compact closure in R n , and thus c ( C ( r ) × S )is not defined, since we only work with compactly supported contactomorphisms we can deduce this from thesecond statement of Theorem 1.22 by taking O = B ( r ) and Q a sufficiently large ellipse contained in C ( r ). PETER ALBERS AND WILL J. MERRY
We can prove the following result, which roughly speaking says that if (cid:93)
Cont ( S, κ ) is non-orderable , so there exists a positive contractible loop χ = { χ t } t ∈ S ⊂ Cont ( S, κ ) of contacto-morphisms, then it is not possible to homotope ζ through positive loops to id S . In [EKP06,Theorem 1.11] this was proved for S = S n − . Theorem 1.25.
Set c := c M ( M ) and assume that c < ∞ . Suppose that χ = { χ t } t ∈ S ⊂ Cont ( S, κ ) a positive contractible loop of contactomorphisms. Let g t : S → (0 , ∞ ) denote thecontact Hamiltonian of χ , and set ε := min ( t,y ) ∈ S × S g t ( y ) > . (1.24) Then if { χ s,t } ≤ s ≤ is any homotopy of loops of contactomorphisms such that χ ,t = χ t and χ ,t = id S with corresponding contact Hamiltonian g s,t : S → R then there exists ( s, t, y ) ∈ [0 , × S × S such that g s,t ( y ) ≤ − (1 − ε ) c . Proof . This follows directly from Theorem 1.22 and the material from [EKP06, Section2.1]. Indeed, suppose there exists δ > g s,t ( y ) > − (1 − ε )( c − δ ) for all ( s, t, y ) ∈ [0 , × S × S . Set a := min { ε, εc } . Then as proved in [EKP06, Section 2.1] for any r < c − δ it is possible to squeeze M r × S into M r ar × S . Fix 0 < λ < min { a, δ } and take r = c − λ .Then c ( M r × S ) = (cid:100) c M ( M r ) (cid:101) = (cid:100) rc M ( M ) (cid:101) = (cid:108) cc − λ (cid:109) = 2 . (1.25)But c ( M r ar ) × S ) = (cid:108) c M ( M r ar ) (cid:109) = (cid:24) cr ar (cid:25) = (cid:24) cc − λ + a (cid:25) = 1 . (1.26)This contradicts Theorem 1.22. (cid:3) Acknowledgement.
We are very grateful to Otto van Koert for explaining Example 1.11to us, and for many helpful discussions. We are also very grateful to Urs Fuchs, LeonidPolterovich, Daniel Rosen, Sheila Sandon and Frol Zapolsky for their helpful comments anduseful remarks. Finally we thank Irida Altman for her help with Figure 1. PA is supported bythe SFB 878 - Groups, Geometry and Actions, and WM is supported by an ETH PostdoctoralFellowship. 2.
Preliminaries
Introductory definitions.
Suppose (Σ , ξ ) is a connected closed coorientable contactmanifold. We denote by P Cont (Σ , ξ ) the set of all smoothly parametrized paths { ϕ t } ≤ t ≤ with ϕ = id Σ . We introduce an equivalence relation ∼ on P Cont (Σ , ξ ) by saying that twopaths ϕ and ψ are equivalent if ϕ = ψ and we can connect ϕ and ψ via a smooth family ϕ s = { ϕ st } ≤ s,t ≤ of paths such that ϕ = ϕ , ϕ = ψ and such that ϕ s is independent of s . RDERABILITY, CONTACT NON-SQUEEZING, AND RABINOWITZ FLOER HOMOLOGY 9
The universal cover (cid:93)
Cont (Σ , ξ ) of Cont (Σ , ξ ) is then P Cont (Σ , ξ ) / ∼ . We now give theprecise definition of a Liouville manifold, and what it means for Σ to be Liouville fillable. Definition 2.1. A Liouville domain ( W , λ ) is a compact exact symplectic manifold suchthat λ | ∂W is a positive contact form on ∂W . Equivalently the vector field Z on W definedby ι Z λ = dλ should be transverse to ∂W and point outwards. Z is called the Liouvillevector field , and we denote by l t the flow of Z , which is defined for all t ≤
0, and thusinduces an embedding ∂W × (0 , → W defined by ( x, r ) (cid:55)→ l log r ( x ). Thus we can form the completion ( W, dλ ) of ( W , λ ) by attaching ∂W × [1 , ∞ ) onto ∂W : W := W ∪ ∂W ( ∂W × [1 , ∞ )) . We extend λ to a 1-form λ on W by setting λ = rλ | ∂W on ∂W × [1 , ∞ ). Thus dλ is asymplectic form on W . Similarly we extend Z to a vector field Z on W by setting Z = r∂ r on ∂W × [1 , ∞ ). One calls ( W, dλ ) a
Liouville manifold - thus Liouville manifolds are exactnon-compact symplectic manifolds obtained by completing a Liouville domain.We say that a closed connected coorientable contact manifold (Σ , ξ ) is
Liouville fillable ifthere exists a Liouville domain ( W , dλ ) such that Σ = ∂W and such that if α := λ | Σ then α is a positive contact form on Σ with ker α = ξ . By a slight abuse of notation we will generallyrefer to the Liouville manifold ( W, dλ ) obtained from completing ( W , dλ ) as “the” filling ofΣ. The symplectization S Σ of a contact manifold (Σ , ξ = ker α ) is the symplectic manifoldΣ × (0 , ∞ ) equipped with the symplectic form d ( rα ). If Σ is Liouville fillable with filling( W, dλ ) then one can embed S Σ (cid:44) → W by using the flow l t of the Liouville vector field Z of V . Next we recall how to lift a path ϕ = { ϕ t } ≤ t ≤ to a symplectic isotopy Φ = { Φ t } ≤ t ≤ on the symplectization S Σ. Write ϕ ∗ t α = ρ t ϕ t . Then define Φ t : S Σ → S Σ byΦ t ( x, r ) := (cid:18) ϕ t ( x ) , rρ t ( x ) (cid:19) . (2.1)The path Φ t is Hamiltonian (in fact it preserves λ ) with Hamiltonian function H t ( x, r ) := rh t ( x ) : S Σ → R . (2.2)We next define precisely what it means for a contact form α generating ξ to be of Morse-Bott type. Definition 2.2.
A contact 1-form α ∈ Ω (Σ) generating ξ is said to be of Morse-Bott type iffor each
T >
0, the set P T ⊂ Σ of points x ∈ Σ satisfying θ T ( x ) = x is a closed submanifoldof Σ, with the property that rank dα | P T is locally constant and T x P T = ker (cid:0) Dθ T ( x ) − T x Σ (cid:1) for all x ∈ P T . (2.3)A Liouville fillable contact manifold (Σ , ξ ) is said to admit a Morse-Bott Liouville filling ifthere exists a filling (
W, dλ ) such that α := λ | Σ is of Morse-Bott type.Let us now recall the definition of a translated point of a contactomorphism. This notionwas introduced by Sandon in [San12]. Definition 2.3.
Let (Σ , ξ ) denote a closed connected coorientable contact manifold, and fixa contact form α ∈ Ω (Σ) generating ξ . Fix ψ ∈ Cont (Σ , ξ ). We can write ψ ∗ α = ρα for a smooth positive function ρ on Σ. A translated point of ψ is a point x ∈ Σ with the propertythat there exists η ∈ R such that ψ ( x ) = θ η ( x ) , and ρ ( x ) = 1 . (2.4)We call η the time-shift of x . Note that if the leaf { θ t ( x ) } t ∈ R is closed (which is always thecase when α is periodic) then the time-shift is not unique. Indeed, if the leaf { θ t ( x ) } t ∈ R hasperiod T > ψ ( x ) = θ η + νT ( x ) for all ν ∈ Z .Now let us define what it means for a translated point x of an element [ ϕ ] ∈ (cid:93) Cont (Σ , ξ )to be contractible with respect to a Liouville filling ( W, dλ ). Definition 2.4.
Let (
W, dλ ) denote a Liouville filling of (Σ , ξ ), with α = λ | Σ . Suppose[ ϕ ] ∈ (cid:93) Cont (Σ , ξ ) and x is a translated point of ϕ of time-shift η . We say that the pair( x, η ) is a contractible translated point if the continuous loop l : R / Z → Σ obtained fromconcatenating the path { ϕ t ( x ) } ≤ t ≤ with the path { θ − ηt ( x ) } ≤ t ≤ is contractible in W . Itis easy to see that this does not depend on path ϕ = { ϕ t } ≤ t ≤ ∈ P Cont (Σ , ξ ) representing[ ϕ ].For us, the usefulness of translated points stems from the fact that the translated points of ϕ are the generators of the Rabinowitz Floer homology associated to ϕ , when the RabinowitzFloer homology is well defined; see Lemma 2.7 or [AM13] for more information.2.2. The Rabinowitz action functional A ϕ on Λ( S Σ) × R . Write Λ( S Σ) := C ∞ contr ( S , S Σ)for the component of the free loop space containing the contractible loops.
Definition 2.5.
Fix a path ϕ ∈ P Cont (Σ , ξ ) as above, and let H t denote the Hamiltonian(2.2). We define the perturbed Rabinowitz action functional A ϕ : Λ( S Σ) × R → R (2.5)by A ϕ ( u, η ) := (cid:90) u ∗ λ − η (cid:90) β ( t )( r ( t ) − dt − (cid:90) ˙ χ ( t ) H χ ( t ) ( u ( t )) dt, (2.6)where β : R / Z → R is a smooth function with β ( t ) = 0 ∀ t ∈ [ , , and (cid:90) β ( t ) dt = 1 , (2.7)and χ : [0 , → [0 ,
1] is a smooth monotone map with χ ( ) = 0, χ (1) = 1, and r ( t ) is the R -component of the map u : S → S Σ = Σ × R . Denote by Crit( A ϕ ) the set of critical pointsof A ϕ , and denote by Spec( ϕ ) := A ϕ (Crit( A ϕ )). Remark 2.6.
In this paper we define the Rabinowitz action functional only on the con-tractible component of the free loop space, as all the applications we have in mind herepertain only to the contractible component. Nevertheless, it is possible to carry out all ofour constructions on the full loop space without any changes. This is because the symplecticform (on S Σ or on the Liouville filling W ) is exact, and so there are no ambiguities in thedefinition of A ϕ on non-contractible loops.The following lemma explains why the perturbed Rabinowitz action functional is useful indetecting translated points. It is a minor variant of an argument originally due to the firstauthor and Frauenfelder [AF10, Proposition 2.4]. For the convenience of the reader we recallthe proof again here. RDERABILITY, CONTACT NON-SQUEEZING, AND RABINOWITZ FLOER HOMOLOGY 11
Lemma 2.7. [AM13]
A pair ( u, η ) is a critical point of A ϕ only if, writing u ( t ) = ( x ( t ) , r ( t )) ∈ Σ × (0 , ∞ ) , p := x ( ) is a translated point of ϕ , with time-shift − η . Conversely every suchpair ( p, η ) gives rise to a unique critical point of A ϕ . Moreover if ( u, η ) is a critical point of A ϕ then A ϕ ( u, η ) = η. (2.8) If ϕ is an exact path of contactomorphisms then r ( t ) ≡ for every critical point ( u = ( x, r ) , η ) . Proof . Denote by Φ t : S Σ → S Σ the symplectic isotopy (2.1). A pair ( u, η ) with u = ( x, r ) : S → Σ × (0 , ∞ ) belongs to Crit( A ϕ ) if and only if (cid:40) ˙ u ( t ) = ηβ ( t ) R ( x ( t )) + ˙ χ ( t ) X H χ ( t ) ( u ( t )) , (cid:82) β ( t )( r ( t ) − dt = 0 . (2.9)Thus for t ∈ [0 , ], we have r ( t ) = 1 and ˙ x ( t ) = − ηR ( x ( t )), and x (1) = Φ ( x ( )). Sup-pose ( u, η ) ∈ Crit( A ϕ ). Thus u ( ) = ( θ − η ( x (0)) , t ∈ [ ,
1] we have ˙ u ( t ) =˙ χ ( t ) X H χ ( t ) ( u ( t )). In particular, ϕ ( x ( )) = θ − η ( x ( )), and thus x ( ) is a translated point of ϕ . Moreover the time shift of x is given by − η mod 1.Next, we note that λ ( X H ( x, r )) = dH ( x, r ) (cid:18) r ∂∂r (cid:19) = H ( x, r ) , (2.10)and hence A ϕ ( u, η ) = (cid:90) ( rα )( ηβ ( u ) R ( x )) dt + (cid:90) (cid:2) λ ( ˙ χX H χ ( u )) − ˙ χH χ ( u ) (cid:3) dt = η + 0 . (2.11)Finally if ϕ t is exact for each t then the path Φ t of symplectomorphisms defined in (2.1) issimply given by Φ t ( x, r ) = ( ϕ t ( x ) , r ), from which the last statement immediately follows. (cid:3) Remark 2.8.
We emphasize again that if ( u = ( x, r ) , η ) is a critical point of A ϕ then thetime-shift of the translated point x ( ) is the negative of the action value. This explains theReeb flow will turn out to have a negative spectral number (cf. part (1) of Theorem 1.1).We point out that there is a distinguished Morse-Bott component of Crit( A id Σ ) diffeomor-phic to Σ corresponding to critical points (( x, ,
0) for x ∈ Σ.We now define what if means for ϕ to be non-degenerate. In the periodic case we alsointroduce the notion of being non-resonant. Definition 2.9.
A path ϕ is non-degenerate if A ϕ : Λ( S Σ) × R → R is a Morse-Bott function.In the periodic case we say that ϕ is non-resonant if Spec( ϕ ) ∩ Z = ∅ . Remark 2.10.
The identity id Σ is non-degenerate if and only if α is of Morse-Bott type(see [CF09, Appendix B]). In [AM13] we explained why a generic path ϕ is non-degenerate(actually a stronger result is true: for generic ϕ the functional A ϕ is even Morse). It is alsoeasy to see that a generic ϕ is non-resonant. Moreover Spec( ϕ ) is always a nowhere densesubset of R (even in the degenerate case), cf. [Sch00, Lemma 3.8]. Finally we note thatSpec( ϕ ) depends only on the terminal map ϕ . The next lemma explains why we pay particular attention to periodic contact manifolds.It will prove crucial in the construction of the contact capacity (cf. Section 4, in particularProposition 4.3). Fix ϕ ∈ (cid:93) Cont (Σ , ξ ) and fix a contactomorphism ψ ∈ Cont (Σ , ξ ). Lemma 2.11.
Assume α is periodic. If ( u = ( x, r ) , η ) ∈ Crit( A ϕ ) with η ∈ Z then thereexists a critical point ( u = ( x , r ) , η ) of A ψϕψ − with x ( ) = ψ ( x ( )) . In particular, Spec( ϕ ) ∩ Z = ∅ ⇔ Spec( ψϕψ − ) ∩ Z = ∅ . (2.12) Moreover ( u, η ) is non-degenerate if and only if ( u , η ) is non-degenerate. Proof . If ( u, η ) ∈ Crit( A ϕ ) with η ∈ Z then since θ t is 1-periodic, this means that if we write u ( t ) = ( x ( t ) , r ( t )) then x ( ) is a fixed point of ϕ . Thus ψ ( x ( )) is a fixed point of ψϕψ − .Thus by Lemma 2.7 for each ν ∈ Z there exists a critical point ( u ν = ( x ν , r ν ) , ν ) of A ψϕψ − with x ν ( ) = ψ ( x ν ( )). In particular, this is true for ν = η . The final statement follows fromthe fact that the linearised equation is also conjugation invariant. (cid:3) Rabinowitz Floer homology.
Let us now assume that (Σ , ξ ) is Liouville fillable witha Morse-Bott Liouville filling (
W, dλ ). We would like to extend A ϕ to a functional defined onall of Λ( W ) × R , where Λ( W ) := C ∞ contr ( S , W ) as before. In order to do this we must extendthe function ( x, r ) (cid:55)→ r − H t to functions defined on all of W . At thesame time, it is convenient to truncate them. As in [AF10], we proceed as follows. Define m : W → R so that m ( x, r ) := r − × ( , ∞ ) , (2.13) ∂m∂r ( x, r ) ≥ x, r ) ∈ S Σ , (2.14) m | W \ S Σ := − . (2.15)Next, for κ > ε κ ∈ C ∞ ([0 , ∞ ) , [0 , ε κ ( r ) = (cid:40) , r ∈ [ e − κ , e κ ] , , r ∈ [0 , e − κ ] ∪ [ e κ + 1 , ∞ ) , (2.16)and such that 0 ≤ ε (cid:48) κ ( r ) ≤ e κ for r ∈ [ e − κ , e − κ ] , (2.17) − ≤ ε (cid:48) κ ( r ) ≤ r ∈ [ e κ , e κ + 1] . (2.18)Then define H κt : W → R by setting H κt | W \ S Σ = − and H κt ( x, r ) := ε κ ( r ) H t ( x, r ) for ( x, r ) ∈ S Σ . (2.19)We denote by A κϕ : Λ( W ) × R → R the Rabinowitz action functional defined using theHamiltonians m and H κt : A κϕ ( u, η ) := (cid:90) u ∗ λ − η (cid:90) β ( t ) m ( u ( t )) dt − (cid:90) ˙ χ ( t ) H κχ ( t ) ( u ( t )) dt. (2.20) Definition 2.12.
Now let us recall the definition of the oscillation ‘norm’ on (cid:93)
Cont (Σ , ξ ).Firstly, suppose { ϕ t } ≤ t ≤ ∈ P Cont (Σ , ξ ). Let h t : Σ → R denote the contact Hamiltonian.The oscillation norm (cid:107) h (cid:107) osc is defined by (cid:107) h (cid:107) osc := (cid:107) h (cid:107) + + (cid:107) h (cid:107) − , (2.21) RDERABILITY, CONTACT NON-SQUEEZING, AND RABINOWITZ FLOER HOMOLOGY 13 (cid:107) h (cid:107) + := (cid:90) max x ∈ Σ h t ( x ) dt, (cid:107) h (cid:107) − := − (cid:90) min x ∈ Σ h t ( x ) dt. (2.22)We then define the oscillation norm (cid:107) ϕ (cid:107) and its positive and negative parts (cid:107) ϕ (cid:107) ± for ϕ ∈ (cid:93) Cont (Σ , ξ ) by taking the infimum of the oscillation norms (cid:107) h (cid:107) osc [resp. (cid:107) h (cid:107) ± ] over all possiblepaths { ϕ t } ≤ t ≤ representing ϕ (with corresponding contact Hamiltonians h t ). Remark 2.13.
Denote by Φ κ ∈ Ham c ( W, dλ ) the Hamiltonian diffeomorphism generated by H κt . The Hofer norm (cid:107) Φ κ (cid:107) of H κt is related to (cid:107) ϕ (cid:107) by (cid:107) Φ κ (cid:107) ≤ e κ (cid:107) ϕ (cid:107) . (2.23)Here by definition the Hofer norm (cid:107) Φ κ (cid:107) is the infimum of the oscillation norms of all possibleHamiltonians generating Φ κ . One such Hamiltonian is H κt , and it is clear that (cid:107) H κ (cid:107) osc ≤ e κ (cid:107) h (cid:107) osc . Definition 2.14.
Suppose ϕ ∈ P Cont (Σ , ξ ). Let ρ t : Σ → (0 , ∞ ) is defined by ϕ ∗ t α = ρ t α .Define a constant κ ( ϕ ) > κ ( ϕ ) := max t ∈ [0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t max x ∈ Σ ˙ ρ τ ( x ) ρ τ ( x ) dτ (cid:12)(cid:12)(cid:12)(cid:12) (2.24)Note that if ϕ is exact then κ ( ϕ ) = 0.In [AM13, Proposition 2.5] we proved: Lemma 2.15. If κ > κ ( ϕ ) then if ( u, η ) ∈ Crit( A κϕ ) then u ( S ) ⊆ S Σ , and moreover if wewrite u ( t ) = ( x ( t ) , r ( t )) then r ( S ) ⊆ ( e − κ/ , e κ/ ) . If ϕ is non-degenerate in the sense of Definition 2.9, then as explained in [AM13], Lemma2.15 allows to define for a, b ∈ [ −∞ , ∞ ] \ Spec( ϕ ) the Rabinowitz Floer homology
RFH ( a,b ) ∗ ( A ϕ , W ) . (2.25)This is a semi-infinite dimensional Morse theory associated to the functional A κϕ (for some κ > κ ( ϕ )), and we sketch the definition here and refer to e.g. [AF10, AM13] for moreinformation.Let J conv ( S Σ) denote the set of time dependent almost complex structures J = { J t } t ∈ S on S Σ that are d ( rα )-compatible and that are convex . Here we use the sign convention that J is dλ -compatible if d ( rα )( J · , · ) defines a Riemannian metric, and to say that J is convex isto ask that there exists S > dr ◦ J t = d ( rα ) on Σ × [ S , ∞ ) (2.26)(in particular, J is independent of t on Σ × [ S , ∞ )). We denote by J conv ( W ) the set of time-dependent almost complex structures J = { J t } t ∈ S with the property that J | S Σ ∈ J conv ( S Σ).Given J ∈ J conv ( W ) we can define an L -inner product (cid:104)(cid:104)· , ·(cid:105)(cid:105) J on Λ( W ) × R : for ( u, η ) ∈ Λ( W ) × R , ζ, ζ (cid:48) ∈ Γ( u ∗ T W ) and b, b (cid:48) ∈ R , set (cid:10)(cid:10) ( ζ, b ) , ( ζ (cid:48) , b (cid:48) ) (cid:11)(cid:11) J := (cid:90) dλ ( J t ζ, ζ (cid:48) ) dt + bb (cid:48) . (2.27)We denote by ∇ J A κϕ the gradient of A κϕ with respect to (cid:104)(cid:104)· , ·(cid:105)(cid:105) J . Remark 2.16.
In this paper all sign conventions are the same as in [AS09, AM13]. Note that this is not a norm; the same is true of the quantity (cid:107) ϕ (cid:107) κ defined below. Assume that ϕ is non-degenerate and fix κ > κ ( ϕ ) and J ∈ J conv ( W ). By assumption A κϕ is a Morse-Bott function. Pick a Morse function g : Crit( A κϕ ) → R , and choose a Riemannianmetric (cid:37) on Crit( A κϕ ) such that the negative gradient flow of ∇ (cid:37) g is Morse-Smale . Given twocritical points w − , w + ∈ Crit( g ), with w ± = ( u ± , η ± ), we denote by M w − ,w + ( A κϕ , g, J, (cid:37) ) themoduli space of gradient flow lines with cascades of −∇ J A κϕ and −∇ (cid:37) g running from w − to w + . See [Fra04, Appendix A] or [CF09, Appendix A] for the precise definition.Introduce a grading on Crit( g ) by setting µ ( u, η ) := µ CZ ( u ) − dim ( u,η ) Crit( A κϕ ) + ind g ( u, η ) , η > ,µ CZ ( u ) − dim ( u,η ) Crit( A κϕ ) + ind g ( u, η ) + 1 , η < , − n + ind g ( u, η ) , η = 0 . (2.28)Here µ CZ ( u ) denotes the Conley-Zehnder index of the loop t (cid:55)→ u ( t/η ) and dim ( u,η ) Crit( A κϕ )denotes the local dimension of Crit( A κϕ ) at ( u, η ). Actually in most cases of interest in thispaper, we may assume that A ϕ is actually Morse. In this case the Morse function g is takento be identically zero, and (2.28) continues to hold. Remark 2.17.
Our normalization convention for the Conley-Zehnder index is that if H is a C -small Morse function on W and x is a critical point of W then µ CZ ( x ) = n − ind H ( x ) , (2.29)where ind H ( x ) denotes the Morse index of x .Given −∞ < a < b < ∞ denote by RFC ( a,b ) ∗ ( A κϕ , g ) := Crit ( a,b ) ∗ ( g ) ⊗ Z , where Crit ( a,b ) ∗ ( g )denotes the set of critical points w of g with a < A ϕ ( w ) < b . We only do this when a, b / ∈ Spec( ϕ ), even if this is not explicitly stated. Generically the moduli spaces M w − ,w + ( A ϕ , g, J, (cid:37) )carry the structure of finite dimensional smooth manifolds, whose components of dimensionzero are compact. One defines a boundary operator ∂ on RFC ( a,b ) ∗ ( A κϕ , g ) by counting theelements of the zero-dimensional parts of the moduli spaces M w − ,w + ( A ϕ , g, J, (cid:37) ).The homology RFH ( a,b ) ∗ ( A ϕ , W ) := H ∗ (RFC ( a,b ) ∗ ( A κϕ , g ) , ∂ ) does not depend on any of theauxiliary choices we made. We emphasize though that RFH ( a,b ) ∗ ( A ϕ , W ) depends on the choiceof filling ( W, dλ ). Finally we defineRFH b ∗ ( A ϕ , W ) := lim −→ a ↓−∞ RFH ( a,b ) ∗ ( A ϕ , W ) , (2.30)RFH ( a, ∞ ) ∗ ( A ϕ , W ) := lim ←− b ↑∞ RFH ( a,b ) ∗ ( A ϕ , W ) , (2.31)RFH ∗ ( A ϕ , W ) := lim −→ a ↓−∞ lim ←− b ↑∞ RFH ( a,b ) ∗ ( A ϕ , W ) (2.32)(the order of the limits in (2.32) matters). As pointed out by Ritter [Rit13], it followsfrom work of Cieliebak-Frauenfelder-Oancea [CFO10] that the Rabinowitz Floer homologyRFH ∗ (Σ , W ) vanishes if and only if the symplectic homology SH ∗ ( W ) vanishes.We briefly summarize now the key properties that we will need about the Rabinowitz Floerhomology RFH ( a,b ) ∗ ( A ϕ , W ): RDERABILITY, CONTACT NON-SQUEEZING, AND RABINOWITZ FLOER HOMOLOGY 15 (1) The Rabinowitz Floer homology is independent of ϕ in the following strong sense.There is a universal object RFH ∗ (Σ , W ) (which may be thought as corresponding tothe case ϕ = id Σ ) together with canonical isomorphisms ζ ϕ : RFH ∗ (Σ , W ) → RFH ∗ ( A ϕ , W ) . (2.33)Given two paths ϕ and ψ , there is a map ζ ϕ,ψ : RFH ∗ ( A ϕ , W ) → RFH ∗ ( A ψ , W ) withthe property that ζ ψ = ζ ϕ,ψ ◦ ζ ϕ . (2.34)In particular, if Z ∈ RFH ∗ (Σ , W ) is a non-zero class then RFH ∗ ( A ϕ , W ) contains anon-zero class Z ϕ defined by ζ ϕ,ψ (cid:0) Z ϕ (cid:1) = Z ψ and Z id Σ = Z ∈ RFH ∗ (Σ , W ) . (2.35)(2) If a ≤ b ≤ ∞ there is a natural map j a,bϕ : RFH a ∗ ( A ϕ , W ) → RFH b ∗ ( A ϕ , W ) . (2.36)Similarly there is a natural map p a,bϕ : RFH b ∗ ( A ϕ , W ) → RFH ( a,b ) ∗ ( A ϕ , W ) . (2.37)If b = ∞ we abbreviate j a, ∞ ϕ = j aϕ , and we write j a for the map RFH a ∗ (Σ , W ) → RFH ∗ (Σ , W ), with similar conventions for the maps p a,bϕ . If Spec( ϕ ) ∩ [ a, b ] = ∅ then the map j a,bϕ : RFH a ∗ ( A ϕ , W ) → RFH b ∗ ( A ϕ , W ) is an isomorphism and p a,bϕ :RFH b ∗ ( A ϕ , W ) → RFH ( a,b ) ∗ ( A ϕ , W ) is the zero map (as RFH ( a,b ) ∗ ( A ϕ , W ) = 0).(3) Moreover there is a filtered version of (2.34), which gives the existence of a maps ζ aϕ,ψ : RFH a ∗ ( A ϕ , W ) → RFH a + K ( ϕ,ψ ) ∗ ( A ψ , W ) (2.38)for some constant K ( ϕ, ψ ) ≥
0. The maps (2.38) are a special case of [AF10, Lemma2.7]. It will be important however to note that if the paths ϕ, ψ have contact Hamil-tonians h t and k t then then the constant K ( ϕ, ψ ) satisfies K ( ϕ, ψ ) ≤ e max { κ ( ϕ ) ,κ ( ψ ) } max (cid:8) (cid:107) h − k (cid:107) + , (cid:9) , (2.39)where we are using the notation from (2.21)-(2.22). Finally one has for all Z ∈ RFH a ∗ ( A ϕ , W ) that ζ ϕ,ψ (cid:0) j aϕ ( Z ) (cid:1) = j a + K ( ϕ,ψ ) ψ (cid:0) ζ aϕ,ψ ( Z ) (cid:1) . (2.40)(4) We recall from Remark 2.8 that Crit( A id Σ ) contains Σ as a Morse-Bott componentvia the constants. For ε > ( − ε,ε ) ∗ (Σ , W ) ∼ = H ∗ + n − (Σ; Z ) . (2.41)Even though it is more or less standard, the estimate (2.39) is extremely important in allthat follows, and hence we prove it here. To define the continuation homomorphism ζ ϕ,ψ wedenote by H t = rh t and K t = rk t the Hamiltonian functions of ϕ and ψ , respectively, andchoose a linear homotopy L st := ν ( s ) H t + (1 − ν ( s )) K t (2.42) for a smooth function ν : R → [0 ,
1] with ν ( s ) = 1 for s ≤ − ν ( s ) = 0 for s ≥ ν (cid:48) ( s ) ≤
0. We define the ( s -dependent) action functional A s as in (2.6): A s ( u, η ) = (cid:90) u ∗ λ − η (cid:90) β ( t ) m ( u ( t )) dt − (cid:90) ˙ χ ( t ) ε κ ( r ) L sχ ( t ) ( u ( t )) dt. (2.43)where ϕ s has corresponding Hamiltonian function L st . Then counting solutions of ∂ s ( u, η ) + ∇ J A s ( u, η ) = 0 (2.44) RDERABILITY, CONTACT NON-SQUEEZING, AND RABINOWITZ FLOER HOMOLOGY 17 with ( u − , η − ) := (cid:0) u ( −∞ ) , η ( −∞ ) (cid:1) ∈ Crit( A ϕ ) and ( u + , η + ) := (cid:0) u (+ ∞ ) , η (+ ∞ ) (cid:1) ∈ Crit( A ψ )defines the continuation homomorphism. We recall that κ > max { κ ( ϕ ) , κ ( ψ ) } and estimate0 ≤ E J ( u, η ) (2.45)= (cid:90) ∞−∞ (cid:90) | ∂ s ( u, η ) | J dtds (2.46)= − (cid:90) ∞−∞ (cid:90) (cid:104)(cid:104)∇A s ( u, η ) , ∂ s ( u, η ) (cid:105)(cid:105) J dtds (2.47)= − (cid:90) ∞−∞ (cid:90) dds A s ( u, η ) dtds + (cid:90) ∞−∞ (cid:90) ∂ A s ∂s ( u, η ) dtds (2.48)= A ϕ ( u − , η − ) − A ψ ( u + , η + ) − (cid:90) ∞−∞ (cid:90) ˙ χ ( t ) ε κ ( r ( t )) ∂L sχ ( t ) ∂s ( u ( t )) dtds (2.49)= A ϕ ( u − , η − ) − A ψ ( u + , η + ) (2.50) − (cid:90) ∞−∞ (cid:90) ν (cid:48) ( s ) ε κ ( r ( t )) ˙ χ ( t ) (cid:16) H χ ( t ) ( u ( t )) − K χ ( t ) ( u ( t )) (cid:17) dtds (2.51)= A ϕ ( u − , η − ) − A ψ ( u + , η + ) (2.52) − (cid:90) ∞−∞ (cid:90) ν (cid:48) ( s ) (cid:124) (cid:123)(cid:122) (cid:125) ≤ ε κ ( r ( t )) r ( t ) (cid:124) (cid:123)(cid:122) (cid:125) ≤·≤ e κ ˙ χ ( t ) (cid:124)(cid:123)(cid:122)(cid:125) ≥ (cid:16) h χ ( t ) ( u ( t )) − k χ ( t ) ( u ( t )) (cid:17) dtds (2.53) ≤ A ϕ ( u − , η − ) − A ψ ( u + , η + ) (2.54) − (cid:90) ∞−∞ (cid:90) ν (cid:48) ( s ) (cid:124) (cid:123)(cid:122) (cid:125) ≤ ε κ ( r ( t )) r ( t ) (cid:124) (cid:123)(cid:122) (cid:125) ≤·≤ e κ ˙ χ ( t ) (cid:124)(cid:123)(cid:122)(cid:125) ≥ max x ∈ Σ (cid:16) h χ ( t ) ( x ) − k χ ( t ) ( x ) (cid:17) dtds (2.55) ≤ A ϕ ( u − , η − ) − A ψ ( u + , η + ) (2.56) − (cid:90) ∞−∞ (cid:90) ν (cid:48) ( s ) (cid:124) (cid:123)(cid:122) (cid:125) ≤ ε κ ( r ( t )) r ( t ) (cid:124) (cid:123)(cid:122) (cid:125) ≤·≤ e κ ˙ χ ( t ) (cid:124)(cid:123)(cid:122)(cid:125) ≥ max (cid:110) max x ∈ Σ (cid:16) h χ ( t ) ( x ) − k χ ( t ) ( x ) (cid:17) , (cid:111) dtds (2.57) ≤ A ϕ ( u − , η − ) − A ψ ( u + , η + ) (2.58) − e κ (cid:90) ∞−∞ (cid:90) ν (cid:48) ( s ) ˙ χ ( t ) max (cid:110) max x ∈ Σ (cid:16) h χ ( t ) ( x ) − k χ ( t ) ( x ) (cid:17) , (cid:111) dtds (2.59)= A ϕ ( u − , η − ) − A ψ ( u + , η + ) (2.60) − e κ (cid:90) ∞−∞ ν (cid:48) ( s ) ds (cid:124) (cid:123)(cid:122) (cid:125) = − (cid:90) ˙ χ ( t ) max (cid:110) max x ∈ Σ (cid:16) h χ ( t ) ( x ) − k χ ( t ) ( x ) (cid:17) , (cid:111) dt (2.61)= A ϕ ( u − , η − ) − A ψ ( u + , η + ) + e κ (cid:90) max (cid:110) max x ∈ Σ (cid:16) h t ( x ) − k t ( x ) (cid:17) , (cid:111) dt (2.62) ≤ A ϕ ( u − , η − ) − A ψ ( u + , η + ) + e κ max (cid:8) (cid:107) h − k (cid:107) + , (cid:9) . (2.63)This proves estimate (2.39). Spectral invariants and orderability
Throughout this section we require Assumption (A) from the Introduction to hold. Moreprecisely, recall we say that a closed connected coorientable contact manifold (Σ , ξ ) satisfiesAssumption (A) if:
Assumption (A): (Σ , ξ ) admits a Liouville filling ( W, dλ ) such that α := λ | Σ is Morse-Bott and the Rabinowitz Floer homology RFH ∗ (Σ , W ) is non-zero. Definition 3.1.
Fix a non-zero class Z ∈ RFH ∗ (Σ , W ) and let ϕ denote a non-degeneratepath. We define its spectral number by c ( ϕ, Z ) := inf (cid:8) a ∈ R | Z ϕ ∈ j aϕ (RFH a ∗ ( A ϕ , W )) (cid:9) , (3.1)where we use the notation Z ϕ from (2.35).Throughout the rest of the paper, the letter Z denotes a non-zero class in RFH ∗ (Σ , W ). Proposition 3.2.
Let ϕ and ψ be two non-degenerate paths. Then we have the estimate c ( ψ, Z ) ≤ c ( ϕ, Z ) + K ( ϕ, ψ ) (3.2) ≤ c ( ϕ, Z ) + e max { κ ( ϕ ) ,κ ( ψ ) } max (cid:8) (cid:107) h − k (cid:107) + , (cid:9) , (3.3) where h and k are the contact Hamiltonians of ϕ and ψ , respectively. In particular, we have h t ( x ) ≤ k t ( x ) ∀ x ∈ Σ , t ∈ [0 ,
1] = ⇒ c ( ϕ, Z ) ≥ c ( ψ, Z ) . (3.4) Proof . This follows immediately from the definition of the spectral number together with(2.38) and the estimate (2.39). (cid:3)
Lemma 3.3.
For any non-degenerate path ϕ ∈ P Cont (Σ , ξ ) the spectral numbers are allcritical values of A ϕ , i.e. c ( ϕ, Z ) ∈ Spec( ϕ ) .Moreover c ( · , Z ) admits a unique extension to all of P Cont (Σ , ξ ) : given a degenerate path ϕ , set c ( ϕ, Z ) := lim k c ( ϕ k , Z ) , (3.5) where ϕ k → ϕ is any sequence of non-degenerate paths converging to ϕ in C . The extensionstill satisfies c ( ϕ, Z ) ∈ Spec( ϕ ) and the estimates (3.2) and (3.4) . In particular, c ( · , Z ) : P Cont (Σ , ξ ) → R is a continuous function when we equip P Cont (Σ , ξ ) with the C -topology. Proof . The assertion c ( ϕ, Z ) ∈ Spec( ϕ ) follows immediately from the fact that RFH c ∗ ( A ϕ , W )only changes if c ( · , Z ) crosses a critical value of A ϕ , compare the discussion below (2.37).To prove the existence of the extension we are required to prove that the limit exists and isindependent of the choice of approximating sequence ϕ k . We denote by h k the correspondingcontact Hamiltonians. Since we assume that ϕ k converges to ϕ in C it follows that κ ( ϕ k ) → κ ( ϕ ) and h k → h , the contact Hamiltonian of ϕ . From Proposition 3.2 we conclude that( c ( ϕ k , Z )) converges and in the same way independence of the approximating sequence ( ϕ k )is proved. That c ( ϕ, Z ) ∈ Spec( ϕ ) and the estimates (3.2) and (3.4) hold follows from thedefinition of c ( · , Z ) as a limit. (cid:3) Lemma 3.4.
The map c ( · , Z ) : P Cont (Σ , ξ ) → R descends to give a well defined map c ( · , Z ) : (cid:93) Cont (Σ , ξ ) → R . RDERABILITY, CONTACT NON-SQUEEZING, AND RABINOWITZ FLOER HOMOLOGY 19
Proof . We recall from Remark 2.10 that Spec( ϕ ) ⊂ R is nowhere dense and actually onlydepends on the endpoint ϕ of the path ϕ . Moreover, Lemma 3.3 implies that c ( · , Z ) is acontinuous map. If we vary the path ϕ while keeping the endpoints fixed the continuous map c ( · , Z ) takes values in the fixed, nowhere dense set Spec( ϕ ), thus is constant. This provesthe Lemma. (cid:3) Lemma 3.5.
For any T ∈ R one has c ( θ T , Z ) = − T + c (id Σ , Z ) . (3.6) Proof . One has Spec( θ T ) = − T + Spec(id Σ ). Since Spec(id Σ ) is nowhere dense and c ( · , Z )is continuous the result follows. (cid:3) Remark 3.6.
Proposition 3.2 and Lemmata 3.3, 3.4, 3.5 constitute Theorem 1.1 from theintroduction.Given a path ϕ of contactomorphisms, we define the support of ϕ , S ( ϕ ) := (cid:91) ≤ t ≤ supp( ϕ t ) , (3.7)where supp( ϕ t ) := { x ∈ Σ | ϕ t ( x ) (cid:54) = x } . Definition 3.7.
For an open set U ⊂ Σ we set c ( U, Z ) := sup (cid:110) c ( ϕ, Z ) | ϕ ∈ (cid:94) Cont (Σ , ξ ) , S ( ϕ ) ⊂ U (cid:111) ∈ ( −∞ , ∞ ] . (3.8) Example 3.8.
By Lemma 3.5 one has immediately that c (Σ , Z ) = ∞ for any non-zero class Z .Recall from (2.41) that the fundamental class [Σ] defines a class in RFH ( − ε,ε ) n (Σ , W ). Wenow strengthen Assumption (A) as follows: Assumption (A) + : (Σ , ξ ) admits a Liouville filling ( W, dλ ) such that α := λ | Σ isMorse-Bott. Moreover there exists a non-zero class µ Σ ∈ RFH n (Σ , W ) such that p − ε ( µ Σ ) = j ε ([Σ]). See statement (2) on page 15 for the definition of the maps p − ε and j ε . Remark 3.9.
Here are two instances where this assumption is satisfied. Firstly, if Σ is aunit cotangent bundle then such a non-zero class µ Σ exists; this follows from eg. work ofAbbondandolo-Schwarz [AS09]. Secondly, if (Σ , ξ ) admits a Liouville filling ( W, dλ ) such thatthere are no contractible (in W ) Reeb orbits, then again such a class µ Σ trivially exists sincein this case we can take ε = + ∞ in (2.41), ie. RFH ∗ (Σ , W ) = H ∗ + n − (Σ; Z ). Lemma 3.10.
It holds c (id Σ , µ Σ ) = 0 . (3.9) Proof . This follows from the definition of spectral numbers and the fact that p − ε ( µ Σ ) = j ε ([Σ]) (cid:3) Let us show that c ( U ) > U ⊂ Σ. Proposition 3.11.
Given any non-empty open set U ⊂ Σ , there exists ψ ∈ (cid:93) Cont (Σ , ξ ) suchthat S ( ψ ) ⊂ U and c ( ψ, µ Σ ) > . Proof . We prove the proposition in three steps.
Step 1.
We use an idea from Sandon [San13]. Fix a C -small function b : Σ → R . We use b to build a contactomorphism Ψ : T ∗ Σ × R → T ∗ Σ × R , where the 1-jet bundle T ∗ Σ × R isequipped with the standard contact form λ + dτ and λ = pdx in local coordinates. Namely,we set Ψ( x, p, τ ) = ( x, p − db ( x ) , τ + b ( x )) . (3.10)Note that critical points of b are in 1-1 correspondence with Reeb chords between the twoLegendrians Σ × { } and Ψ(Σ × { } ) (where Σ ⊂ T ∗ Σ is the zero section). Since b is assumedto be C -small, Ψ determines a contactomorphism of ψ of (Σ , α ), defined as follows. Firstly,Weinstein’s neighborhood theorem for Legendrian submanifolds (see [AH, Theorem 2.2.4])implies that there is an exact contactomorphismΞ : N × ( − δ, δ ) → Q × ( − ε, ε ) (3.11)between a neighborhood N × ( − δ, δ ) of Σ ×{ } inside T ∗ Σ × R and a neighborhood Q × ( − ε, ε )of ∆ × { } inside Σ × Σ × R , where ∆ is the diagonal in Σ × Σ. Here Σ × Σ × R is equippedwith the contact form e r pr ∗ α − pr ∗ α , where pr j is the projection onto the j th factor. Thecontactomorphism ψ is then defined by looking at the restriction of Ξ ◦ Ψ ◦ Ξ − to ∆ × { } inside Q × ( − ε, ε ); we can writeΞ ◦ Ψ ◦ Ξ − ( x, x,
0) =: ( x, ψ ( x ) , log ρ ( x )) , (3.12)for ψ : Σ → Σ and ρ : Σ → (0 , ∞ ). One readily checks that ψ ∗ α = ρα , and hence ψ is acontactomorphism.Similarly, if we start with an isotopy { b t } ≤ t ≤ with b = 0 then we obtain a path ψ = { ψ t } ≤ t ≤ of contactomorphisms with ψ = id Σ . In this case one can check that the contactHamiltonian of ψ is − b t : α (cid:18) ∂∂t ψ t (cid:19) = − b t ◦ ψ t . (3.13)The key point now is that the translated points x ∈ Σ of ψ with time-shift η ∈ ( − ε, ε ) arein 1-1 correspondence with the critical points of b : if x ∈ Crit( b ) then ψ ( x ) = θ − b ( x ) ( x ) . (3.14)Thus for each x ∈ Crit( b ) there is a critical point ( u x , b ( x )) ∈ Crit( A ψ ), and any criticalpoint ( u, η ) of A ψ not of this form necessarily satisfies | η | > ε . Step 2.
Suppose now that we start with a C -small Morse-Bott function b on Σ. Define b t := tb for t ∈ [0 , ψ = { ψ t } ≤ t ≤ denote the corresponding path of contacto-morphisms. Then if x ∈ Crit( b ) then the critical point ( u x , b ( x )) belongs to a Morse-Bottcomponent of A ψ , and moreover we claim that µ ( u x , b ( x )) = 1 − n + ind b ( x ) , (3.15)where ind b ( x ) denotes the maximal dimension of a subspace on which the Hessian Hess b ( x )of b at x is strictly negative definite.To see this, we consider the Hamiltonian diffeomorphism Φ of T ∗ Σ × R × R obtained bylifting Ψ, which as Ψ is exact, is given simply byΦ( q, p, τ, σ ) = (Ψ( q, p, τ ) , σ ) . (3.16)A translated point x of ψ gives rise to the following path of Lagrangian subspaces: L t := { (ˆ x, − t Hess b ( x )(ˆ x ) , , ˆ σ ) | ˆ x ∈ T x Σ , ˆ σ ∈ R } ⊂ T ( x, , ,σ ) ( T ∗ Σ × R × R ) , (3.17) RDERABILITY, CONTACT NON-SQUEEZING, AND RABINOWITZ FLOER HOMOLOGY 21
The desired index is then given by µ ( u x , b ( x )) = 1 − n + µ RS ( L , L ) , (3.18)which in this case is just 1 − n + ind b ( x ) as claimed; note that the 1 − n summand comesfrom the normalization used in the definition of the Rabinowitz index (2.28) above, and weare using the grading convention from Remark 2.17. Step 3.
We now prove the theorem. Suppose U ⊂ Σ is open and non-empty. Choose afunction b : Σ → [0 , ∞ ) such that supp( b ) ⊂ U and such that b is Morse on the interior ofits support. Moreover we insist that b has a unique maximum x ∈ Σ, with 0 < b ( x max ) < ε ,where ε is as in (3.11). Let ψ be as in Step 2. Since the contact Hamiltonian of ψ is − b wecan estimate K (id Σ , ψ ) ≤ e κ ( ψ ) 12 b ( x max ) . (3.19)From Proposition 3.2 and Lemma 3.10 we obtain c ( ψ, µ Σ ) ≤ c (id Σ , µ Σ ) (cid:124) (cid:123)(cid:122) (cid:125) =0 + K (id Σ , ψ ) (3.20) ≤ e κ ( ψ ) 12 b ( x max ) . (3.21)We now assume in addition that e κ ( ψ ) 12 b ( x max ) < (cid:15) , too. Since the contact Hamiltonian − b of ψ is non-positive, we have from (3.4) that0 ≤ c ( ψ, µ Σ ) < (cid:15). (3.22)We recall from Step 1 that any critical point ( u, η ) of A ψ which is not of the form ( u x , b ( x ))satisfies | η | > ε . Thus c ( ψ, µ Σ ) is necessarily a critical value of b . Since µ Σ has index n , and x max is the only critical point of b of index 2 n − u x max , b ( x max )) has index 1 − n + 2 n − n ), we see that c ( ψ, µ Σ ) = b ( x max ) > . (3.23)The proof is complete. (cid:3) The following Corollary is Theorem 1.8 from the Introduction.
Corollary 3.12.
Suppose ϕ ∈ (cid:93) Cont (Σ , ξ ) has contact Hamiltonian h t . Assume h t ≤ andthere exists x ∈ Σ such that h t ( x ) < for all t ∈ [0 , . Then c ( ϕ, µ Σ ) > . Proof . There exists a function b : Σ → [0 , ∞ ) satisfying all the properties from the proof ofProposition 3.11 and in addition that − tb ( x ) ≥ h t ( x ) ∀ x ∈ Σ , t ∈ [0 , . (3.24)Let ψ = { ψ t } ≤ t ≤ denote the contact isotopy whose contact Hamiltonian is − tb . ThenProposition 3.2 and Proposition 3.11 imply that0 < b ( x max ) = c ( ψ, µ Σ ) ≤ c ( ϕ, µ Σ ) . (3.25) (cid:3) Remark 3.13.
One might wonder whether the analogue of Corollary 3.12 continues to holdif instead we assume that h t is non-negative and not identically zero. In the non-compactsetting discussed in Section 5 we will see that this is false. See Remark 5.13 and Appendix Afor more information. Contact capacities
Let us now assume that (Σ , ξ ) satisfies Assumption (B) from the Introduction:
Assumption (B): (Σ , ξ ) admits a Liouville filling ( W, dλ ) such that the RabinowitzFloer homology RFH ∗ (Σ , W ) is non-zero and such that α := λ | Σ is periodic.As before Z denotes a non-zero class in RFH ∗ (Σ , W ). Definition 4.1.
We define for ϕ ∈ (cid:93) Cont (Σ , ξ ) an integer c ( ϕ, Z ) by c ( ϕ, Z ) := (cid:100) c ( ϕ, Z ) (cid:101) . (4.1)The reason periodicity is helpful is this function c ( · , Z ) is conjugation invariant . We willprove this shortly in Proposition 4.3 below, but to begin with we present the following lemma.Recall from Definition 2.9 that we say ϕ is non-resonant if Spec( ϕ ) ∩ Z = ∅ . Lemma 4.2.
Suppose ϕ is both resonant and degenerate with c ( ϕ, Z ) ∈ Z . Then there exists ϕ ν → ϕ such that ϕ ν is resonant and non-degenerate such that for all ν sufficiently large onehas c ( ϕ ν , Z ) = c ( ϕ, Z ) . Proof . Start with any sequence ( ϕ ν ) of non-degenerate paths such that ϕ ν → ϕ . Since ϕ isresonant, for each ν ∈ N there exists a translated point x ν ∈ Σ for ϕ ν with time-shift η ν suchthat η ν → ν sufficiently large one has c ( ϕ ν , Z ) = c ( ϕ, Z ) + η ν . (4.2)The sequence θ − η ν ◦ ϕ ν still converges to ϕ , and it is easy to check that θ − η ν ◦ ϕ ν is stillnon-degenerate, and for all ν sufficiently large one has that c ( θ − η ν ◦ ϕ ν , Z ) = c ( ϕ ν , Z ) − η ν = c ( ϕ, Z ) (4.3)since Spec( θ T ◦ ϕ ) = T + Spec( ϕ ) (4.4)and c ( · , Z ) is continuous. (cid:3) The following is Proposition 1.13 from the Introduction.
Proposition 4.3.
The function c ( · , Z ) : (cid:93) Cont (Σ , ξ ) → Z is conjugation invariant: if ψ ∈ Cont (Σ , ξ ) and ϕ ∈ (cid:93) Cont (Σ , ξ ) then ¯ c ( ψϕψ − , Z ) = ¯ c ( ϕ, Z ) . (4.5) Proof . Assume firstly that ϕ is non-resonant, that is, Spec( ϕ ) ∩ Z = ∅ (see Definition 2.9).Fix ψ ∈ Cont (Σ , ξ ) and let ψ s ∈ Cont (Σ , ξ ) be a path connecting id Σ to ψ . Then weconsider the map s (cid:55)→ c ( ψ s ϕψ − s , Z ) . (4.6)Proposition 3.2 implies that this map is continuous. Lemma 2.11 implies that Spec( ψ s ϕψ − s ) ∩ Z = ∅ for all s ∈ [0 , (cid:6) c ( ψϕψ − , Z ) (cid:7) = (cid:100) c ( ϕ, Z ) (cid:101) as required.There are now three cases to consider. Suppose that ϕ is resonant but that c ( ϕ, Z ) / ∈ Z .Suppose ψ ∈ Cont (Σ , ξ ). Then for ϕ (cid:48) non-resonant and sufficiently close to ϕ we have c ( ψϕψ − , Z ) = c ( ψϕ (cid:48) ψ − , Z ) = c ( ϕ (cid:48) , Z ) = c ( ϕ, Z ) , (4.7)where the second equality used the step above. The next case is when ϕ is resonant andnon-degenerate, with c ( ϕ, Z ) ∈ Z . As before, given ψ ∈ Cont (Σ , ξ ) we choose a path ψ s connecting id Σ to ψ . The key point now is that for any s ∈ [0 , u s , η s ) is a critical RDERABILITY, CONTACT NON-SQUEEZING, AND RABINOWITZ FLOER HOMOLOGY 23 point of A ψ s ϕψ − s with η s ∈ Z then ( u s , η s ) is automatically non-degenerate by the laststatement of Lemma 2.11. It follows that there exists ε > ψ s ϕϕ − s ) ∩ [ c ( ϕ, Z ) − ε, c ( ϕ, Z ) + ε ] = { c ( ϕ, Z ) } , (4.8)and the result follows as above. The final case is when ϕ is both resonant and degenerateand c ( ϕ, Z ) ∈ Z . In this case we employ Lemma 4.2 to find a sequence ϕ ν → ϕ such that ϕ ν is both resonant, non-degenerate, and such that for all large ν one has c ( ϕ ν , Z ) = c ( ϕ, Z ).The argument above then implies that for any ψ ∈ Cont (Σ , ξ ) and for all ν sufficientlylarge, c ( ψϕ ν ψ − , Z ) = c ( ϕ ν , Z ) is an integer. Since c ( ψϕ ν ψ − , Z ) → c ( ψϕψ − , Z ) the resultfollows. (cid:3) Corollary 4.4.
One has ¯ c ( t (cid:55)→ θ tT , Z ) = (cid:100)− T + c (id Σ , Z ) (cid:101) for any T ∈ R . Proof . Lemma 3.5. (cid:3)
We now define c ( U, Z ) in the same way as c ( U, Z ) was defined in Definition 3.7.
Definition 4.5.
For an open set U ⊂ Σ we define the contact capacity c ( U, Z ) := sup (cid:110) c ( ϕ, Z ) | ϕ ∈ (cid:93) Cont (Σ , ξ ) , S ( ϕ ) ⊂ U (cid:111) ∈ Z ∪ {∞} . (4.9) Remark 4.6.
The notion of contact capacity was introduced by Sandon in [San11]. Shewas the first to discover a connection between translated points and orderability and othercontact rigidity phenomena.The following is Corollary 1.17 from the Introduction.
Proposition 4.7.
For all ψ ∈ Cont (Σ , ξ ) , one has c ( ψ ( U ) , Z ) = c ( U, Z ) . (4.10) Proof . Since S ( ψϕψ − ) = ψ ( S ( ϕ )) , (4.11)we conclude from Proposition 4.3 that c ( U, Z ) = sup (cid:110) c ( ϕ, Z ) | ϕ ∈ (cid:93) Cont (Σ , ξ ) , S ( ϕ ) ⊂ U (cid:111) = sup (cid:110) c ( ϕ, Z ) | ϕ ∈ (cid:93) Cont (Σ , ξ ) , ψ ( S ( ϕ )) ⊂ ψ ( U ) (cid:111) = sup (cid:110) c ( ϕ, Z ) | ϕ ∈ (cid:93) Cont (Σ , ξ ) , S ( ψϕψ − ) ⊂ ψ ( U ) (cid:111) = sup (cid:110) c ( ψϕψ − , Z ) | ϕ ∈ (cid:93) Cont (Σ , ξ ) , S ( ψϕψ − ) ⊂ ψ ( U ) (cid:111) = sup (cid:110) c ( µ, Z ) | µ ∈ (cid:93) Cont (Σ , ξ ) , S ( µ ) ⊂ ψ ( U ) (cid:111) = c ( ψ ( U ) , Z ) . (4.12) (cid:3) For completeness we recall Theorem 1.18 which is proved in the Introduction.
Theorem 4.8.
Let U ⊂ V ⊂ Σ be open sets and assume that there exists ϕ ∈ Cont (Σ , ξ ) with ϕ ( V ) ⊂ U . Then c ( U, Z ) = c ( V, Z ) . (4.13) In particular, if c ( U, Z ) < c ( V, Z ) then there exists no contact isotopy mapping V into U . Remark 4.9.
If we assume that (Σ , ξ ) satisfies both assumption (A) + and (B) then weknow c ( U, µ Σ ) > U ⊂ Σ is a nonempty open subset. Unfortunately in general wedo not know how to prove that ¯ c ( U, Z ) is ever finite. Nevertheless, in certain situations it is possible to prove finiteness of the capacities, for instance when the subset U is displaceable.In particular this is the case in the setting described in the next section, see Corollary 5.16.5. Prequantization spaces
Hamiltonian Floer homology.
Fix a Liouville domain ( M , dγ ). Let S := ∂M and κ := γ | S , so that ( S, κ ) is a contact manifold. Let (
M, dγ ) denote the completion of M , sothat M = M ∪ S ( S × [1 , ∞ )). It is convenient in this section to introduce the notation M σ := (cid:40) M \ (cid:0) S × ( σ, (cid:1) if 0 < σ < ,M ∪ S (cid:0) S × [1 , σ ] (cid:1) if σ ≥ . (5.1)Note here we are using σ to denote the R -coordinate on the end of M - this is so as to avoidconfusion in Section 5.3, when a second Liouville domain will come into play.Denote by Ham c ( M, dγ ) the group of Hamiltonian diffeomorphisms f on M with compactsupport. As before, a path f = { f t } ≤ t ≤ of compactly supported Hamiltonian diffeomor-phisms is assumed to be smoothly parametrized and begin at the identity: f = id M . Givensuch a path f = { f t } ≤ t ≤ , let X f denote the time-dependent vector field on M defined by ∂∂t f t = X f t ◦ f t . (5.2)The equation f ∗ t γ − γ = da t , a ≡ a t : M → R . If we define F t = i X ft γ − (cid:18) ∂∂t a t (cid:19) ◦ f − t , (5.4)then F t generates f t : f t = f tF . We can recover a t from F t via a t = (cid:90) t (cid:16) i X ft γ − F s (cid:17) ◦ f s ds (5.5)(see for instance [MS98, p294]).We briefly explain the construction of the Hamiltonian Floer homology of f in this section.The setting we consider here is a special case of the one considered by Frauenfelder andSchlenk in [FS07], to which we refer to for more details. However it will be convenient forus to use the Morse-Bott framework developed by Frauenfelder [Fra04], in order to make thelink with the Rabinowitz Floer homology of Σ := M × S clearer in the next section.Let us first note that for a given F ∈ C ∞ c ( S × M, R ), the flow f tF has many 1-periodicorbits, since f tF is compactly supported. Of course, constant 1-periodic orbits outside thesupport of f are uninteresting, and hence we introduce the following notation. Denote by σ ( F ) := inf { σ > | S ( f F ) ⊆ M σ } . (5.6)Given a path f = { f t } ≤ t ≤ in Ham c ( M, dγ ), we set σ ( f ) := σ ( F ), where F is given by (5.4).Next, we set P F := (cid:8) y ∈ M σ ( f ) | f F ( y ) = y (cid:9) . (5.7) RDERABILITY, CONTACT NON-SQUEEZING, AND RABINOWITZ FLOER HOMOLOGY 25
Definition 5.1.
Define a subset H mb c ⊆ C ∞ c ( S × M, R ) (here the “mb” stands for Morse-Bott) to consist of those functions F with the property that P F is either a closed submanifoldof M or an open domain whose closure is a compact manifold, and for which T y P F = ker( Df F ( y ) − ) for all y ∈ P F . (5.8)It is well known that the subset H mb c is generic in C ∞ c ( S × M, R ). We say that a path f = { f t } ≤ t ≤ is non-degenerate if the function F defined in (5.4) belongs to H mb c .We denote by R κ the Reeb vector field of κ . Denote by (cid:98) H the set of time-dependent smoothfunctions (cid:98) F on M with the property that there exists C > (cid:98) F t | S × [ C, ∞ ) is of theform (cid:98) F t ( y, σ ) = e ( σ ) for some smooth function e : [ C, ∞ ) → R satisfying0 ≤ e (cid:48) ( σ ) < ℘ ( S, κ ) . (5.9)Here ℘ ( S, κ ) := inf { T > | ∃ a closed Reeb orbit of R κ of period T > } . (5.10)This ensures that if ϕ (cid:98) F denotes the flow of (cid:98) F then ϕ (cid:98) F has no non-constant 1-periodic orbitson S × ( C, ∞ ). Note that if F ∈ C ∞ c ( S × M, R ) then one can find (cid:98) F ∈ (cid:98) H such that (cid:98) F | S × M σ ( F ) = F .As a special case of this construction, consider a function O on M defined by requiringthat O = 0 on the interior M ◦ of M and that O ( y, σ ) = e ( σ ) (5.11)on S × [1 , ∞ ), where e (1) = 0 and e satisfies (5.9). In this case one has P O = M , (5.12)where points in M are thought of as constant loops. In particular, f tO | M ◦ = id M ◦ . Thus O is an extension of the zero function (generating the Hamiltonian diffeomorphism id M to (cid:98) H ). Definition 5.2.
Fix a path f = { f t } ≤ t ≤ , and let F denote the function defined in (5.4),and fix an extension (cid:98) F ∈ (cid:98) H such that (cid:98) F | S × M σ ( F ) = F . Recall that Λ( M ) := C ∞ contr ( S , M ).Define the Hamiltonian action functional A f : Λ( M ) → R by A f ( v ) := (cid:90) v ∗ γ − (cid:98) F t ( v ) dt. (5.13)Denote by Crit ◦ ( A f ) the set of critical points v of A f with v ( S ) ⊆ M σ ( f ) . Then Crit ◦ ( A f )doesn’t depend on the extension (cid:98) F - in factCrit ◦ ( A f ) = P F , (5.14)and hence the assumption (5.8) implies that each component of Crit ◦ ( A f ) is a Morse-Bottcomponent for A f .Fix J ∈ J conv ( M ) (cf. (2.26)). We define an L -inner product (cid:104)(cid:104)· , ·(cid:105)(cid:105) J on Λ( M ) as before(cf. (2.27), only this time there is no bb (cid:48) term). We denote by ∇ J A F the gradient of A F with respect to (cid:104)(cid:104)· , ·(cid:105)(cid:105) J . Pick a Morse function g : Crit ◦ ( A f ) → R and a Riemannian metric (cid:37) on Crit ◦ ( A f ) such that ( g, (cid:37) ) is a Morse-Smale pair. In the case where P F is an opendomain in M whose boundary is a compact manifold, g must be chosen so that (cid:104) dg, n (cid:105) < n is an outward pointing normal. As before we define moduli spaces M v − ,v + ( A f , g, J, (cid:37) ) of gradient flow lines with cascades for critical points v ± ∈ Crit( g ). This time we grade v ∈ Crit( g ) simply by µ ( v ) := µ CZ ( v ) + ind g ( v ), where µ CZ ( v ) is the Conley-Zehnder index. A standard convexity argument gives the necessary compactness needed todefine Floer homology - see Frauenfelder-Schlenk [FS07].Given −∞ < a < b < ∞ denote by CF ( a,b ) ∗ ( A f , g ) := Crit ( a,b ) ∗ ( g ) ⊗ Z , where Crit ( a,b ) ∗ ( g )denotes the set of critical points v of g with a < A f ( v ) < b . As before one defines aboundary operator ∂ on CF ( a,b ) ∗ ( A f , g ) by counting the elements of the zero-dimensionalparts of the moduli spaces M v − ,v + ( A f , g, J, (cid:37) ) for v − (cid:54) = v + . We denote by HF ( a,b ) ∗ ( A f )the associated homology, which as the notation suggests, is independent of the auxiliarydata ( g, J, (cid:37) ). In fact, one can also show it is also independent of the choice of path f .We abbreviate HF a ∗ ( A f ) := HF ( −∞ ,a ) ∗ ( A f ) and HF ∗ ( A f ) := HF ( −∞ , ∞ ) ∗ ( A f ). We denote thenatural maps HF a ∗ ( A f ) → HF ∗ ( A f ) by j af in the same way as before. Under our gradingconvention explained in Remark 2.17, there is a canonical isomorphismHF ∗ ( A f ) ∼ = H n + ∗ ( M , ∂M ; Z ) ∼ = H n −∗ ( M ; Z ) . (5.15)See the proof of Lemma 5.3 below for one way to see this.Next, the Floer homology HF ∗ ( A f ) carries the structure of a unital ring. The unit livesin degree n according to our sign conventions, and under the isomorphism (5.15), the unitcorresponds to the fundamental class [ M ] ∈ H n ( M , ∂M ; Z ); see Lemma 5.3 below. Wedenote the unit by f ∈ HF n ( A f ). Since HF ∗ ( A f ) is necessarily non-zero, as before we candefine the spectral number c M ( f ) := inf (cid:8) a ∈ R | f ∈ j af (HF a ∗ ( A f )) (cid:9) . (5.16)As before, c M is a well defined function c M : (cid:93) Ham c ( M, dγ ) → R . (5.17)We can use c M to define a capacity on open subsets O ⊂ M , c M ( O ) := sup { c M ( f ) | S ( f ) ⊂ O} , (5.18)in the same way as before. We use the subscript c M to differentiate it from the function c associated to Σ := M × S that we will define shortly. Lemma 5.3.
In the case of id = id M the unit = id is simply given by the fundamentalclass [ M ] , and thus c M (id) = 0 . Proof . We define A id using the function O defined in (5.11). Thus Crit ◦ ( A id ) = M , andevery element of Crit ◦ ( A id ) has action value zero. Thus there are no gradient flow lines of A id ,and hence the Floer complex CF ∗ ( A id , g ) reduces to the Morse complex of a Morse function g on M . Such a Morse function g can be chosen so that g > M ◦ and such that g is therestriction of a Morse function (cid:98) g : M → R such that (cid:98) g ( y, σ ) = σ on S × [1 , ∞ ). Thus thisshows that HF ∗ ( A id ) ∼ = HM n + ∗ ( g ) ∼ = H n + ∗ ( M , ∂M ; Z ) , (5.19)which proves (5.15).It is possible to prove directly using Morse-Bott techniques that the isomorphisms in (5.19)are ring maps, and thus the unit in HF ∗ ( A id ) is exactly the unit in Morse homology for g . Thelatter is of course the fundamental class [ M ] under the isomorphism of the Morse homologyof g with the relative homology of ( M , ∂M ). In this situation however, we can simply makea degree argument: if the Morse function g has a unique maximum at a point y max in M ◦ then RDERABILITY, CONTACT NON-SQUEEZING, AND RABINOWITZ FLOER HOMOLOGY 27 one necessarily has that [ y max ] is a cycle in HF n ( A id ), and that fact HF n ( A id ) = Z [ y max ].Since the unit lives in degree n , it must therefore be precisely [ y max ]. (cid:3) The prequantization space
Σ = M × S . The prequantization space of M is thecontact manifold Σ := M × S , equipped with the contact structure ξ := ker α , where α := γ + dτ, (5.20)and τ is the coordinate on S ∼ = R / Z . The last class of contact manifolds we study in thispaper are these prequantization spaces, which for convenience we refer to as Assumption (C) : Assumption (C): (Σ , ξ = ker α ) is a prequantization space Σ = M × S , where ( M, dγ )is a Liouville manifold, and α = γ + dτ .In this case Σ is obviously periodic, but it is not Liouville fillable in the previous sense. Asidefrom anything else, Σ is necessarily non-compact . However Σ does still retain enough of theproperties needed above in order to define a Rabinowitz Floer homology, as will explain inthe next section.Let us denote by Cont ,c (Σ , ξ ) those contactomorphisms ϕ with compact support. Thereis a natural way to obtain a path ϕ = { ϕ t } ≤ t ≤ of compactly supported contactomorphismson Σ from a path f = { f t } ≤ t ≤ of compactly supported Hamiltonian diffeomorphisms on M .Indeed, given such a path f , define ϕ t : Σ → Σ by ϕ t ( y, τ ) := (cid:0) f t ( y ) , τ − a t ( y ) (cid:124) (cid:123)(cid:122) (cid:125) mod 1 (cid:1) , (5.21)where a t was defined in (5.3). One easily checks that ϕ t is an exact contactomorphism. Wesay that the contact isotopy ϕ is the lift of the Hamiltonian isotopy f . In this case the contactHamiltonian h t associated to ϕ t is simply F t : h t ◦ ϕ t = α (cid:18) ∂∂t ϕ t (cid:19) = F t ◦ ϕ t , (5.22)where F t was defined in (5.4). Fix a function (cid:98) F ∈ (cid:98) H such that (cid:98) F = F on S × M σ ( F ) , anddefine (cid:98) H t : S Σ → R by (cid:98) H t := r (cid:98) F t .Consider again the Rabinowitz action functional A ϕ : Λ( S Σ) × R → R defined as in (2.6),using (cid:98) H t . Suppose ( u, η ) ∈ Crit( A ϕ ). Write u ( t ) = ( v ( t ) , τ ( t ) , r ( t )) ∈ M × S × (0 , ∞ ). Thenfrom (2.9) we have ( f ( v (cid:0) (cid:1) , τ (cid:0) (cid:1) − a (cid:0) v (cid:0) (cid:1)(cid:1)(cid:124) (cid:123)(cid:122) (cid:125) mod 1 (cid:1) = ϕ (cid:0) u (cid:0) (cid:1)(cid:1) , = θ − η (cid:0) v (cid:0) (cid:1) , τ (cid:0) (cid:1)(cid:1) = (cid:0) v (cid:0) (cid:1) , τ (cid:0) (cid:1) − η (cid:124) (cid:123)(cid:122) (cid:125) mod 1 (cid:1) (5.23)and hence if y := v ( ) then f ( y ) = y and a ( y ) = η mod 1. Moreover since ϕ t is exactone has r ( t ) ≡ t (cf. the last statement of Lemma 2.7). Since we only consider contractible critical points of A ϕ , we deduce: Lemma 5.4.
There exists a bijective map π : Crit( A ϕ ) → Crit( A f ) (5.24) given by π ( u = ( v, τ, r ) , η ) := (cid:0) t (cid:55)→ f t (cid:0) v (cid:0) (cid:1)(cid:1)(cid:1) . (5.25) Moreover A ϕ ( u, η ) = A f ( π ( u, η )) . (5.26) In particular, every critical point ( u, η ) of A ϕ has u ( S ) ⊆ M σ ( f ) × S × { } . (5.27)Given a contactomorphism ϕ ∈ Cont ,c (Σ , ξ ), we denote by σ ( ϕ ) = inf (cid:8) σ > | S ( ϕ ) ⊆ M σ × S (cid:9) . (5.28)Thus if ϕ is the lift of f then σ ( ϕ ) = σ ( f ) . (5.29)5.3. Rabinowitz Floer homology on Σ . Let P denote a torus with a disc removed, sothat ∂P = S . Equip P with an exact symplectic form dβ such that β | ∂P = dτ . Denoteby ( P, dβ ) the completion of P , so that β = rdτ on ∂P × [1 , ∞ ) . (5.30)Consider W := M × P, (5.31)equipped with the symplectic form dλ where λ := γ + β . Note that W is not a Liouville fillingof Σ. Indeed, firstly W is not compact, and moreover when equipped with the symplecticform dλ , there is no embedding ( S Σ , d ( rα )) (cid:44) → ( W, dλ ) that we can use in order to extendthe Rabinowitz action functional A ϕ to a functional defined on all on Λ( W ) × R . Ideally wewould like to work with a symplectic form ω (cid:48) such that ω (cid:48) | M × ∂P × (0 , ∞ ) = d ( rα ). Sadly nosuch symplectic form exists, and even if one did it would not have the “right” properties atinfinity.To circumvent this problem we will work with a family { ω s = dλ s } s ≥ of symplectic formswhich satisfy: • If ϕ = { ϕ t } ≤ t ≤ is any path of compactly support contactomorphisms on Σ thenthere exists s ( ϕ ) > s > s ( ϕ ) the 1-form λ s agreeswith rα on S ( ϕ ). • For every s ≥ ω s is split-convex at infinity.More precisely, we prove: Lemma 5.5.
There exists a family { λ s } s ≥ ⊂ Ω ( W ) of 1-forms such that for all s ≥ : (1) ω s := dλ s is a symplectic form on W . (2) Define W + s := ( M \ M s − × P ) ∪ ( M × P \ P s − ) , (5.32) W − s := M s × P / (2 s − . (5.33) Then λ s | W + s = (2 s − γ + β, λ s | W − s = 12 s − γ + β. (5.34) Thus ω s is split-convex at infinity, and hence we can achieve compactness, see theproof of Theorem 5.7 below and also [FS07] . Moreover λ = λ everywhere. RDERABILITY, CONTACT NON-SQUEEZING, AND RABINOWITZ FLOER HOMOLOGY 29 (3)
For s > , define V s := M s × S × ( / s , s ) ⊂ S Σ . (5.35) Then for each s > , the natural embedding ι s : V s (cid:44) → W (5.36) satisfies ι ∗ s λ s = rα . Proof . Define a family { f s } s ≥ of smooth functions, see Figure 1 : f s : [0 , ∞ ) × [0 , ∞ ) → (0 , ∞ ) (5.37)such that f s ( σ, r ) = r, ( σ, r ) ∈ [0 , s ) × (1 /s, s ) , s − , ( σ, r ) ∈ [0 , s ) × (0 , s − ) , s − , ( σ, r ) ∈ [0 , s ) × (2 s − , ∞ ) , s − , ( σ, r ) ∈ [ s, ∞ ) × (0 , ∞ ) . (5.38)and finally such that ∂f s ∂σ ( σ, r ) ≥ , for all ( s, σ, r ) ∈ [1 , ∞ ) × [0 , ∞ ) × [0 , ∞ ) . (5.39)The fact that such functions f s exist is clear from Figure 1.0 s s − σ s − s − s sr f s ( σ, r ) = 2 s − f s ( σ, r ) = rf s ( σ, r ) = s − Figure 1.
The function f s ( σ, r )On M \ M × P \ P , where both the σ and r -coordinates are defined, we set λ s := f s γ + β. (5.40)The condition (5.39) guarantees that ω s := dλ s is symplectic where defined, and it is clearthat statements (2) and (3) from the Lemma are satisfied. It remains to extend λ s to all of W . This is done simply by “continuity”: λ s = f s (0 , r ) γ + β, on M × P \ P ,f s ( σ, γ + β, on M \ M × P , s − γ + β, on M × P . (5.41) (cid:3) Definition 5.6.
Given a path ϕ = { ϕ t } ≤ t ≤ of compactly supported contactomorphisms,we define the number s ( ϕ ) ≥ s ( ϕ ) := inf { s ≥ | S ( ϕ ) ⊂ ι s ( V s ) } , (5.42)where ι s is the embedding (5.36).We now prove the following result. Theorem 5.7.
For any non-degenerate path ϕ if s > s ( ϕ ) then it is possible to definethe Rabinowitz Floer homology RFH ∗ ( A ϕ , W, ω s ) (here the notation indicates that we areworking with the symplectic structure ω s on W ). Moreover the Rabinowitz Floer homology isindependent of the choice of s > s ( ϕ ) . Proof . Let J s ( W ) denote the set of time-dependent almost complex structures J = { J t } t ∈ S on W that are ω s compatible, and satisfy:(1) If ι s is the embedding (5.36) then ι ∗ s J ∈ J conv ( V s ⊂ S Σ).(2) The restriction of J to the subset W + s defined in (5.32) is split - that is, there existalmost complex structures J (cid:48) ∈ J conv ( M ) and J (cid:48)(cid:48) ∈ J conv ( P ) such that J = J (cid:48) ⊕ J (cid:48)(cid:48) on this set.Extend A ϕ to a functional A κϕ defined on all of Λ( W ) × R in the same way as before, byreplacing (cid:98) H t with a truncated function (cid:98) H κt as in (2.19). As with the Hamiltonian Floerhomology, we are now only interested in the set Crit ◦ ( A κϕ ) of critical points ( u = ( v, τ, r ) , η )of A κϕ with v ( S ) ⊂ M σ ( ϕ ) . For s large enough, all elements of Crit ◦ ( A κϕ ) are containedin ι ( V s ) and (cid:98) H κt is constant outside on W +2 s − ε for some small ε >
0. Thus if we workwith an almost complex structure J ∈ J s ( W ), the maximum principle prohibits the cylinderpart of flow lines of −∇ J A κϕ from ever entering W +2 s − ε , see for instance [FS07]. Thus theRabinowitz Floer homology is well defined for this s . We point out that, since the cylinderpart of flow lines stay in a compact subset of W , L ∞ -bounds on the Lagrange multiplier arederived as in [CF09, Theorem 3.1].In order to prove independence of s , first note that for s > max { s ( ϕ ) , s ( ψ ) } the contin-uation maps from points (1)-(4) on page 15 show thatRFH ∗ ( A ϕ , W, ω s ) ∼ = RFH ∗ ( A ψ , W, ω s ) . (5.43)Next we note that if id := id M × S is the contactomorphism with contact Hamiltonian O as defined in (5.11) then s (id) = 1. More generally, this is true for any exact path ϕ ofcontactomorphisms, since in this case for any ε >
0, every critical point of A εϕ is containedin Σ × { } - see Lemma 2.15. Thus by (5.43) it suffices to show that RFH ∗ ( A id , W, ω s ) isindependent of s >
1. But this is clear, since every critical point of the Rabinowitz actionfunctional A id has action value zero, as we are only looking at contractible critical pointsand we have filled S with a punctured torus P rather than a disc D . Thus Crit ◦ ( A id ) = M × S × { } , and hence regardless of which symplectic structure we use, as in Lemma 5.3, RDERABILITY, CONTACT NON-SQUEEZING, AND RABINOWITZ FLOER HOMOLOGY 31 the Rabinowitz complex reduces to the Morse complex of a Morse function ˜ g : M × S → R .In particular, it does not depend on s . (cid:3) We denote by RFH ∗ ( A ϕ , W ) the groups RFH ∗ ( A ϕ , W, ω s ) for any s > s ( ϕ ). Theorem 5.8. If ϕ = { ϕ t } ≤ t ≤ is the lift of f = { f t } ≤ t ≤ then there exists a naturalisomorphism RFH ∗ ( A ϕ , W ) ∼ = HF ∗ ( A f ) ⊗ H ∗ ( S ; Z ) . (5.44) Proof . By naturality it suffices to prove the theorem in the case f = id M and ϕ = id :=id M × S . In this case as in the proof of the last part of Theorem 5.7, one hasRFH ∗ ( A id , W ) ∼ = HM ∗ + n (˜ g ) , (5.45)where ˜ g is a Morse function on M × S . We choose ˜ g = ( g, g (cid:48) ), where g is the Morse functionconsidered in the proof of Lemma 5.3, and g (cid:48) : S → R is a Morse function with two criticalpoints τ min and τ max . This givesRFH ∗ ( A id , W ) ∼ = HM n + ∗ (˜ g ) (5.46) ∼ = HM ∗ ( g ) ⊗ HM ∗ ( g (cid:48) ) (5.47) ∼ = HM n + ∗ ( M , ∂M ) ⊗ H ∗ ( S ; Z ) . (5.48)This completes the proof. (cid:3) Remark 5.9.
As pointed out in Remark 3.9 there exists a certain non-zero class µ Σ ∈ RFH(Σ , W ). The image of µ Σ under the isomorphisms from Theorem 5.8 and (5.15) is theclass [ M ] ⊗ [ S ].5.4. Relating the capacities.Definition 5.10.
We define c ( ϕ ) := c ( ϕ, µ Σ ) in the same way as before for ϕ ∈ (cid:93) Cont ,c (Σ , ξ ).As long as we work with compactly supported contactomorphisms Proposition 3.2 remainstrue and its prove is literally the same. Proposition 5.11.
Let ϕ, ψ ∈ (cid:93) Cont ,c (Σ , ξ ) be two non-degenerate paths. Then we have theestimate c ( ψ ) ≤ c ( ϕ ) + K ( ϕ, ψ ) (5.49) ≤ c ( ϕ ) + e max { κ ( ϕ ) ,κ ( ψ ) } (cid:107) h − k (cid:107) + , (5.50) where h and k are the contact Hamiltonians of ϕ and ψ , respectively. In particular, we haveIn particular, we have h t ( x ) ≤ k t ( x ) ∀ x ∈ Σ , t ∈ [0 ,
1] = ⇒ c ( ϕ ) ≥ c ( ψ ) (5.51) and the same implication with nonstrict inequalities. The analogue of Corollary 3.12 remains true, too, again with the same proof.
Corollary 5.12.
Suppose ϕ ∈ (cid:93) Cont ,c (Σ , ξ ) has contact Hamiltonian h t . Assume h t ≤ and h t (cid:54) = 0 for all t ∈ [0 , . Then c ( ϕ ) > . Remark 5.13.
Recall in the closed case we proved that c ( t (cid:55)→ θ tT , Z ) = − T + c (id Σ , Z )for any T ∈ R (cf. Statement (2) of Theorem 1.1. In this setting the Reeb flow θ t is ofcourse not compactly supported, and thus its spectral value is not defined. Nevertheless itis still possible to define a “compactly supported Reeb flow” ϑ t : Σ → Σ which agrees withthe normal Reeb flow on a neighborhood of a given closed Reeb orbit. For small T it is stillpossible to compute the spectral numbers c ( ϑ T ), but it is no longer the case that c ( ϑ T ) = − T .Indeed, whilst for negative T one still has c ( ϑ T ) = − T , for positive T one has c ( ϑ T ) = 0.This shows that Corollary 5.12 fails if one instead assume h t ≥
0. Details are contained inAppendix A.
Definition 5.14.
For an open non-empty set U ⊂ Σ with compact closure we set c ( U ) := sup (cid:110) c ( ϕ ) | ϕ ∈ (cid:94) Cont ,c (Σ , ξ ) , S ( ϕ ) ⊂ U (cid:111) ∈ ( −∞ , ∞ ] . (5.52)and c ( U ) := (cid:100) c ( U ) (cid:101) . (5.53) Theorem 5.15.
Suppose f ∈ (cid:93) Ham c ( M, dγ ) , and let ϕ ∈ (cid:93) Cont ,c (Σ , ξ ) denote the lift of f .Then c M ( f ) = c ( ϕ ) . (5.54) Moreover, if
O ⊂ M is open with compact closure then c M ( O ) = c ( O × S ) . (5.55) Proof . The first statement follows from Lemma 5.4, Theorem 5.8, and Remark 5.9. Thusclearly c M ( O ) ≤ c ( O × S ). In order to complete the proof, we must show that given any ψ ∈ (cid:93) Cont ,c (Σ , ξ ) with S ( ψ ) ⊂ O × S there exists f ∈ (cid:93) Ham c ( M, dγ ) with S ( f ) ⊂ O andsuch that lifted contactomorphism ϕ satisfies c ( ψ ) ≤ c ( ϕ ) . (5.56)This follows from Proposition 5.11: if h t denotes the contact Hamiltonian of ψ we choosefunctions F t : M → R supported inside O satisfying h t ≥ F t . (5.57)The lift ϕ of the corresponding path f of Hamiltonian diffeomorphisms generated by F satisfiesthe required inequality. (cid:3) In this setting, we can use the fact that c M satisfies the triangle inequality to obtain moreinformation on c . In particular, we obtain a criterion for c ( U ) to be finite (cf. Remark 4.9). Corollary 5.16.
Suppose U ⊂ Σ is a non-empty open set with compact closure, and supposethat pr M ( U ) is a Hamiltonian displaceable subset of M . Then c ( U ) < ∞ . Proof . We have c ( U ) ≤ c (pr M ( U ) × S ) = c M (pr M ( U )), and c M (pr M ( U )) < ∞ by Theorem5.19 below. (cid:3) We also have the following result:
Proposition 5.17.
Suppose that ϕ ∈ (cid:93) Cont ,c (Σ , ξ ) has the property that pr M ( S ( ϕ )) is aHamiltonian displaceable subset of M . Then c ( ϕ ) ≥ . RDERABILITY, CONTACT NON-SQUEEZING, AND RABINOWITZ FLOER HOMOLOGY 33
Proof . First assume that ϕ is the lift of an element f ∈ (cid:93) Ham c ( M, dγ ). The fact that c M ( f ) ≥ S ( f ) is Hamiltonian displaceable is well known, but for the convenience of thereader we give the short argument here. Suppose that g ∈ Ham c ( M, dγ ) displaces S ( f ). Let { g t } ≤ t ≤ denote some path connecting g to id M . Choose a path of paths f s = { f st } ≤ s,t ≤ connecting f = f with the constant path f t ≡ id M such that g displaces S ( f s ) for each0 ≤ s ≤
1. Then we claim that c M ( gf ) = c M ( g ) . (5.58)Indeed, the point is that any fixed point of g f lies outside of S ( f ), and hence is necessarilyalso a fixed point of g . The same is true if we replace f with f s for any 0 ≤ s ≤
1, and thusit follows that Spec( A gf s ) is independent of s . Since the function s (cid:55)→ c M ( gf s ) is continuousand Spec( A gf s ) is discrete, it must be constant. This proves (5.58). We then argue as follows: c M ( g ) = c M ( gf − f ) (5.59) ≤ c M ( gf − ) + c M ( f ) (5.60) ≤ c M ( g ) + c M ( f ) (5.61)where (5.60) used the triangle inequality for c M and (5.61) used (5.58) applied to f − . Thisimplies that c M ( f ) ≥
0. Finally to prove the general case where ϕ is not necessarily the lift ofa Hamiltonian path f , we use the same argument from the proof of Theorem 5.15. Namely,we can find a path f of Hamiltonians with support inside pr M ( S ( ϕ )) such that c M ( f ) ≤ c ( ϕ ).Then the argument above shows that c M ( f ) ≥
0, and hence the same is true of c ( ϕ ). (cid:3) Let us quickly recall the definition of the
Hofer-Zehnder capacity . See for instance [HZ94]for an in depth treatment.
Definition 5.18.
Let O be an open subset of M . We define the Hofer-Zehnder capacity c HZ ( O , M ) of O to c HZ ( O , M ) := sup {(cid:107) H (cid:107) | H is admissible } , (5.62)where H ∈ C ∞ c ( O , R ) is admissible if there exists an open set O ⊂ O such that H | O = max H ,and if the flow ϕ tH has no non-constant periodic orbits of period ≤ displacement energy by e ( O , M ) := inf (cid:8) (cid:107) H (cid:107) | ϕ H ( O ) ∩ O = ∅ (cid:9) . (5.63)The following result is due to Frauenfelder and Schlenk [FS07, Corollary 8.3], see also [FGS05,Sch00]. Theorem 5.19. If ( M , γ ) is a Liouville domain then c HZ ( O , M ) ≤ c M ( O ) ≤ e ( O , M ) . (5.64)Denote by B ( r ) the open ball of radius r in R m . Then c HZ ( B ( r ) , R m ) = πr . We cannow prove the following result, which was stated as Theorem 1.24 in the Introduction. Theorem 5.20.
Let ( M, dγ ) denote a Liouville manifold. Equip R m with the standardsymplectic form dλ std , and consider the contact manifold ( (cid:101) Σ , α + λ std ) , where (cid:101) Σ := M × R m × S . Suppose O ⊆ M is open and c HZ ( O , M ) < ∞ . Choose r > such that (cid:6) πr (cid:7) < (cid:100) c HZ ( O , M ) (cid:101) (5.65) and set r := (cid:113) π c HZ ( O , M ) + 1 (5.66) Then there does not exist ϕ ∈ Cont ,c ( (cid:101) Σ , α + λ std ) such that ϕ ( O × B ( r ) × S ) ⊂ O × B ( r ) × S . (5.67) Proof . We first prove that for r > r , c HZ ( O × B ( r ) , M × R m ) ≥ c HZ ( O , M ) . (5.68)Fix ε >
0. We consider a cutoff function β : [0 , ∞ ) → [0 ,
1] such that β ( s ) = 1 for s ∈ [0 , r − − ε ] and β ( s ) = 0 for s > r , and such that − ≤ β (cid:48) ( s ) ≤ s ∈ [0 , ∞ ). Nowsuppose H is any admissible function on O . Define H β : M × R m → R by H β ( x, y ) := β ( | y | ) H ( x ) . (5.69)The symplectic gradient of H β with respect to dγ ⊕ dλ std is X H β ( x, y ) = ( β ( | y | ) X H ( x ) , H ( x ) X β ( y )) . (5.70)Suppose γ : R → M × R m is a non-constant periodic orbit of X H β , with γ ( t + T ) = γ ( t ) forall t ∈ R . We shall show that T >
1, so that H β is admissible. Write γ ( t ) = ( γ x ( t ) , γ y ( t )).Then ˙ γ x = β ( | γ y | ) X H ( γ x ) , ˙ γ y = H ( γ x ) X β ( γ y ) . (5.71)Since | β (cid:48) | ≤ γ x is non-constant then T >
1. But if γ x is constant, say γ x ( t ) = x , then we must have H ( x ) (cid:54) = 0. Since β (cid:48) is non-zero only for | γ y | ∈ ( r − − ε, r )we necessarily have T ≥ H ( x ) π ( r − − ε ) ≥ c HZ ( O , M ) π ( r − − ε ) . (5.72)Thus as long as π ( r − − ε ) > c HZ ( O , M ) , (5.73) H β is indeed admissible. Since clearly max H β = max H , we see that c HZ ( O × B ( r ) , M × R n ) ≥ c HZ ( U, M ) (5.74)provided that (5.73) holds. Since ε was arbitrary we obtain (5.68). Moreover for any r > e ( O × B ( r ) , M × R n ) ≤ πr , (5.75)as can be checked directly. The remainder of the proof is an easy application of Theorem5.15, Theorem 5.19 and Theorem 4.8. Indeed, we have c ( O × B ( r ) × S ) = (cid:6) e ( O × B ( r ) , M × R m ) (cid:7) ≤ (cid:6) πr (cid:7) < (cid:100) c HZ ( O , M ) (cid:101)≤ (cid:6) c HZ ( O × B ( r ) , M × R m ) (cid:7) ≤ c M × R m ( O × B ( r ))= c ( O × B ( r ) × S ) . (5.76) (cid:3) Here is an application of Theorem 5.20, which can be seen as a more quantitive (albeitweaker, and with more hypotheses) version of the infinitesimal result of [EKP06, Theorem1.18].
RDERABILITY, CONTACT NON-SQUEEZING, AND RABINOWITZ FLOER HOMOLOGY 35
Corollary 5.21.
Suppose X is a closed connected oriented Riemannian manifold which ad-mits a circle action S × X → X such that the loop t (cid:55)→ t · p is not contractible for some p ∈ X . Then if O ⊂ T ∗ X is any neighborhood of the zero section then the conclusion ofProposition 5.20 holds. Proof . A result of Kei Irie [Iri11] proves that in this setting the Hofer-Zehnder capacityof the unit disc bundle D ∗ X ⊂ T ∗ X is finite. Thus the same is true of any neighborhood O ⊂ T ∗ X of the zero section, and hence the hypotheses of Theorem 5.20 are satisfied. (cid:3) Appendix A. The “compactly supported Reeb flow”
In this Appendix we continue to work in the setting from the previous section. ThusΣ = M × S is a prequantisation space associated to the completion of a Liouville domain( M , dγ ). Our aim is to construct a “compactly supported Reeb flow” whose support iscontained in a tubular neighborhood of a closed Reeb orbit, and explicitly compute thespectral value. This result has been alluded to in Remarks 3.13 and 5.13. Theorem A.1.
Suppose (Σ = M × S , ξ ) satisfies Assumption (C) . Let x ( t ) = ( y , t ) denotea closed embedded Reeb orbit (for some fixed y ∈ M .) Then there exists ρ > and aneighborhood B of y in M with the following significance: For all ρ ∈ R with | ρ | < ρ , thereexists an exact contactomorphism ϑ ρ ∈ (cid:93) Cont (Σ , ξ ) with S ( ϑ ρ ) ⊂ B × S with the propertythat if x ∈ S ( ϑ ρ ) is a translated point with of ϑ ρ then ϑ ρ ( x ) = θ ρ ( x ) . (A.1) In other words, from the point of view of translated points, ϑ ρ is “the Reeb flow supportedon x ”. Moreover if B (cid:48) ⊂ B is any neigborhood of y then for | ρ | sufficiently small we have S ( ϑ ρ ) ⊂ B (cid:48) × S . The spectral value c ( ϑ ρ ) is given by c ( ϑ ρ ) = (cid:40) , ≤ ρ < ρ , − ρ, − ρ < ρ ≤ . (A.2) Convention:
In this appendix we equip R n \{ } with polar coordinates ( s, φ ) where s ∈ (0 , ∞ ) and φ = ( φ , . . . , φ n − ) with φ j ∈ R / π Z . In these coordinates the standard contactform α std is given by α std = (cid:88) j s dφ j + dτ. (A.3)This has the slightly unfortunate consequence that τ is 1-periodic but the φ j are 2 π -periodic!These conventions are chosen so that c R n ( B ( r )) = πr instead of r . Proof of Theorem A.1.
The argument is local in M , and hence it is sufficient to prove theresult in the special case M = R n . Thus Σ = R n × S and α = α std is given by (A.3). TheReeb vector field R of α is just ∂∂τ , and the Reeb flow θ t is given by θ t ( s, φ, τ ) = ( s, φ, τ + t (cid:124) (cid:123)(cid:122) (cid:125) mod 1 ) . (A.4)Fix ρ ∈ R such that 0 < | ρ | < πr . Let f : [0 , ∞ ) × [0 , ∞ ) → R denote a smooth functionwith the following properties:(1) There exists ε > f ( s ) = ρ for 0 ≤ s ≤ ε and f ( s ) = 0 for r − ε ≤ s ≤ r .(2) If ρ < f (cid:48) ( s ) ≥ s . If ρ > f (cid:48) ( s ) ≤ s . (3) If ρ < πs − f (cid:48) ( s ) > s >
0. If ρ < πs + f (cid:48) ( s ) < s > | ρ | < πr . Indeed, if ρ < πs − f (cid:48) ( s ) > − ρ = (cid:90) r f (cid:48) ( s ) ds < (cid:90) r πsds = πr . (A.5)Conversely it is easy to see that when | ρ | < πr such functions really do exist. Now considerthe contactomorphism ϑ ρ of R n × S whose contact Hamiltonian h t : R n × S is given by h t ( s, ζ, τ ) = f ( r ) . (A.6)The contact vector field X t of h t is defined by the equations α ( X t ) = h t , i X t dα = dh t ( R ) α − dh t . (A.7)This gives X t ( s, φ, τ ) = (cid:88) j f (cid:48) ( s ) s ∂∂φ j + (cid:18) f ( s ) − sf (cid:48) ( s )2 (cid:19) ∂∂τ . (A.8)We can integrate this to obtain ϑ ρt ( s, φ, τ ) = (cid:18) s, φ + f (cid:48) ( s ) s t, . . . , φ n − + f (cid:48) ( s )2 t, τ + (cid:18) f ( s ) − sf (cid:48) ( s )2 (cid:19) t (cid:19) , (A.9)and hence translated points of ϑ ρ are tuples ( s, φ, τ ) with f (cid:48) ( s ) s ∈ π Z , (A.10)and the time-shift is given by η = f ( s ) − sf (cid:48) ( s )2 . (A.11)By assumption one never has f (cid:48) ( s ) / πs ∈ Z unless f (cid:48) ( s ) = 0. In other words, translatedpoints only occur when 0 ≤ s ≤ ε or when r − ε ≤ s ≤ ∞ . In particular, the only translatedpoints of ϑ ρ that lie in the interior of the support of ϑ ρ are the points in B ( ε ) × S . Since ϑ ρ = θ ρ on B ( ε ) × S , this justifies our claim that ‘from the point of view of translated points’, ϑ ρ is the Reeb flow.To complete the proof let us compute the spectral value of ϑ ρ . Note that the contractible action spectrum of A ϑ ρ is just { , − ρ } , and hence we certainly have c ( ϑ ρ ) ∈ { , − ρ } . For ρ <
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E-mail address : [email protected] Will J. Merry, Department of Mathematics, ETH Z¨urich
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