Legendrian persistence modules and dynamics
LLegendrian persistence modules and dynamics M ichael Entov , Leonid Polterovich January 12, 2021
Abstract
We relate the machinery of persistence modules to the Legendriancontact homology theory and to Poisson bracket invariants, and use itto show the existence of connecting trajectories of contact and sym-plectic Hamiltonian flows.
Contents pb + -invariant and Hamiltonian chords 19 Partially supported by the Israel Science Foundation grant 1715/18. Partially supported by the Israel Science Foundation grant 1102/20. a r X i v : . [ m a t h . S G ] J a n Persistence modules 284 Legendrian contact homology 30 pb + via LCH persistence modules . . . . . . 43 ϕ t . . . . . . . . . . . . . 525.2 Existence of chords of h from Λ to Λ . . . . . . . . . . . . . 53 J Q
617 The case of ST ∗ R n In the present paper we discuss a new facet of a method, introduced in[7, 27], of finding orbits of Hamiltonian systems connecting a pair of dis-joint subsets ( X , X ) in the phases space. The method manifests a dynam-ical phenomenon called interlinking , which involves a quadruple of subsets( X , X , Y , Y ) in a symplectic manifold ( M, ω ) satisfying X ∩ X = Y ∩ Y = ∅ . A Hamiltonian H : M × S → R separates Y , Y if∆( H, Y , Y ) := inf Y × S H − sup Y × S H > . (1)Let µ >
0. According to the definition in [27], ( Y , Y ) µ -interlinks (re-spectively, autonomously µ -interlinks ) ( X , X ), if for every Hamiltonian (re-spectively, every autonomous Hamiltonian) H separating Y , Y and generat-ing a flow { φ t } t ∈ R on M , there exist t ∈ R , x ∈ M and a positive T ≤ µ/ ∆,2o that φ t x ∈ X and φ t + T x ∈ X . The piece of the trajectory { φ t x } , t ∈ [ t , t + T ] is called a chord of time-length T connecting X and X .The pair ( Y , Y ) interlinks (respectively, autonomously interlinks ) thepair ( X , X ) if it µ -interlinks (respectively, autonomously µ -interlinks) itfor some µ > IMPORTANT REMARK:
In this paper we will consider only autonomousinterlinking and for brevity will omit the word “autonomous”.
Thus, “in-terlinking” further on in this paper is the same as “autonomousinterlinking” in [27].
We will focus on quadruples of a special form lying in the symplectization S Σ of a contact manifold (Σ , ξ ), and its fillings. Let λ be a contact form on Σ.Recall that S Σ is Σ × R + ( s ) equipped with the symplectic form ω = d ( sλ ).We set Y = { s = s − } and Y = { s = s + } for some 0 < s − ≤ s + , and take X and X to be disjoint Lagrangian cobordisms whose boundaries projectto Legendrian submanifolds in Σ. We mainly concentrate on the case ofcylindrical cobordisms Y i = Λ i × [ s − , s + ] (2)for some disjoint Legendrians Λ , Λ . The main result of the present paperprovides a sufficient condition for interlinking in terms of contact topologyof this pair of Legendrians.A very rough sketch of the method of [7, 27] is as follows. By using Pois-son bracket invariants coming from function theory on symplectic manifolds,one deduces the desired interlinking from obstructions to some special defor-mations of the symplectic form ω constant near Z := X ∪ X ∪ Y ∪ Y . Theconstraints on such deformations are often provided by pseudo-holomorphiccurves with the boundary on Z . The main new idea of the present paperis to extract such curves from filtered Legendrian contact homology of Λ and Λ . Thus our main point is to deduce “dynamical interlinking” from“contact-topological interlinking”.The realization of this idea requires tools from the relative symplecticfield theory (RSFT) [26] combined with the theory of persistence moduleswhich originated in topological data analysis [19, 43] and brought to symplec-tic topology in [39]. RSFT associates to a (non-degenerate) pair formed bya Legendrian submanifold and a contact form on the manifold an algebraicobject, called the Legendrian contact homology, and to an exact Lagrangian3obordism between Legendrian submanifolds a morphism between the corre-sponding Legendrian contact homologies. In the simplest setting (say, wherea contact manifold is the contactization of an exact symplectic manifold) theenhancement of the RSFT using the action filtration gives rise to the filteredrelative symplectic field theory (FRSFT) . Namely, the Legendrian contact ho-mology can be viewed as a persistence module formed by vector spaces (over Z ), while an exact Lagrangian cobordism between Legendrian submanifoldsdefines a morphism between the corresponding persistence modules, see pa-pers [17, 15] by Dimitroglou Rizell and M. Sullivan. In the present paperwe rely on some special instances of this theory only leaving more generalresults for a forthcoming manuscript [29].In fact, theory of persistence modules enables us, in certain situations, todetect chords of contact flows even in the absence of interlinking, providedthe contact Hamiltonians have a sufficiently small oscillation in the uniformnorm. This robustness of the existence mechanism for contact chords withrespect to C -small perturbations of contact Hamiltonians is a feature of ourapproach.Let us mention also that in a special case when the cobordisms are cylin-drical as in (2) and the Hamiltonian H : S Σ → R is R + -homogeneous andpositive, interlinking reduces to existence of Reeb chords of a modified con-tact form λ/H connecting Legendrian submanifolds Λ and Λ . Existence ofsuch a chord with a good upper bound on it time-length would follow froma Legendrian contact homology theory valid for arbitrary contact forms. Atthe moment such a theory is still under construction, though it sounds likelythat Pardon’s work [38] on foundations of absolute contact homologies willbe eventually extended to the relative (i.e. Legendrian) case. Meanwhile,our results yield existence of Reeb chords for arbitrary forms on certain con-tact manifolds. Applications of these results include, in particular, contact-topological methods in non-equilibrium thermodynamics [28]. In the present paper we focus on contact manifolds of the form Σ = P × R ( z ), where ( P n , dϑ ), n ∈ Z > , is an exact symplectic manifold witha symplectic form dϑ and bounded geometry at infinity (see [3], [24] for thedefinition of this notion) and the contact structure on Σ is defined by thecontact form λ = dz + ϑ . We will call such contact manifolds nice : to thebest of our knowledge, this is the largest class of contact manifolds for which4he details of the Legendrian contact homology theory have been worked outrigorously in the published literature. Note that the Reeb vector field R of λ is ∂/∂s and its flow has no periodic orbits.A specific example of a nice contact manifold is the 1-jet space J Q = T ∗ Q ( p, q ) × R ( z ) of a closed manifold Q equipped with the contact form dz ± pdq . The forms corresponding to different choices of the sign are relatedby the involution p → − p . We freely use both forms depending on thecontext, hoping that this will not cause a confusion.Write ST ∗ R n for the space of co-oriented contact elements of R n . Thereexists a contactomorphism( J S n − , dz − pdq ) → ( ST ∗ R n , pdq )(known as “the hodograph map”, see e.g. [2, pp.48-49]) identifying the stan-dard contact forms on both spaces: it sends a point ( p, q, z ) ∈ J S n − to theunit cotangent vector q in the cotangent space of zq + p ∈ R n . (Here p, q, z are local Darboux vector-coordinates on J S n − ).An important construction used on several occasions in this paper is sta-bilization: Given a nice contact manifold Σ = P × R , define another nicecontact manifold by (cid:98) Σ := Σ × R ( r ) × S ( τ ) equipped with the contact form (cid:98) λ := λ + rdτ = ds + ϑ + rdτ . Before discussing our results in a general setting, let us give a few basicdefinitions and present a sample of dynamical applications. In Remark 1.3below we discuss the relation between these applications and the dynamicalresults in our previous work.A (time-dependent) Hamiltonian on a symplectic manifold is called com-plete if its Hamiltonian flow is defined for all times. Similarly, a (time-dependent) contact Hamiltonian (with respect to a contact form) on a con-tact manifold is called complete if its contact flow is defined for all times.We extend the definition of a chord of a symplectic Hamiltonian (see Sec-tion 1.1) to the contact setting as follows: given a (time-dependent) contactHamiltonian h (with respect to a contact form λ ) on a contact manifold Σand two-disjoint subsets Z , Z of Σ, a chord of h from Z to Z of time-length T > (with respect to λ ) is a trajectory of the contact flow of h (with5espect to λ ) that passes through Z at a time t and through Z at the time t + T . If h >
0, such a chord is a Reeb chord of the contact form λ/h of thesame time-length.Consider R n ( p, q ) = R n ( p ) × R n ( q ) with the standard symplectic form dp ∧ dq . We view R n as the symplectization of the contact manifold ST ∗ R n filled by the zero section { p = 0 } . Let | · | denote the Euclidean norm on R n ( q ).Let 0 < s − < s + . Let x , x ∈ R n ( q ), x (cid:54) = x . If n = 1, assume that x < x .Define X , X , Y , Y ⊂ R n as follows.If n >
1, set X := { ( p, x ) ∈ R n | s − ≤ | p | ≤ s + } ,X := { ( p, x ) ∈ R n | s − ≤ | p | ≤ s + } ,Y := { | p | = s − } ,Y := { | p | = s + } . If n = 1, set X := { ( p, x ) ∈ R n | s − ≤ p ≤ s + } ,X := { ( p, x ) ∈ R n | s − ≤ p ≤ s + } ,Y := { ( s − , q ) ∈ R n | x ≤ q ≤ x } ,Y := { ( s + , q ) ∈ R n | x ≤ q ≤ x } . Let H : R n × S → R be a complete Hamiltonian.Set: c min := min X × S H, c max := max X × S H. Theorem 1.1.
A. Assume ∆( H ; X , X ) > and the following conditions are satisfied: H is time-independent , (3) H | X ≥ , (4)6 upp H ∩ { s − ≤ | p | ≤ s + } is compact . (5) Then there exists a chord of H from Y to Y of time-length bounded fromabove by | x − x | ( s + − s − )∆( H ; X , X ) .In the case n = 1 the claim holds even without assuming (3) , (4) , (5) .B. Assume ∆( H ; Y , Y ) =: ∆ > and for some < e < / the followingcondition is satisfied: sup c min − e ∆ ≤ H ≤ c max + e ∆ (cid:12)(cid:12)(cid:12)(cid:12) ∂H∂t (cid:12)(cid:12)(cid:12)(cid:12) < (1 − e ) e ∆ ( s + − s − ) | x − x | . (6) Then there exists a chord of H from X to X of time-length boundedfrom above by | x − x | ( s + − s − )(1 − e )∆( H ; Y , Y ) .In particular, if H is time-independent (and thus (6) holds for all Remark 1.2. In the case of autonomous Hamiltonians, part B of Theo-rem 1.1 implies that ( Y , Y ) µ -interlinks ( X , X ), where µ = | x − x | ( s + − s − ). Part A of Theorem 1.1 does not imply that ( Y , Y ) µ -interlinks ( X , X )but is a somewhat weaker claim.In the proofs of the two parts we use two slightly different Poisson bracketinvariants – the tools used for detecting interlinking in Section 2.2 below.The appearance of the same constant µ = | x − x | ( s + − s − ) in both parts ofTheorem 1.1 is due to the fact that the proofs of both claims are based on thesame lower bound on both Poisson bracket invariants, coming from areas ofcertain pseudo-holomorphic curves used in the theory of Legendrian contacthomology. In fact, one can think of µ as the area of a (pseudo-holomorphic)quadrilateral whose edges lie in X , X , Y , Y . Remark 1.3. In the case n = 1 the claim of Theorem 1.1 follows ratherdirectly from the results in [27] (cf. [7, Thm. 1.20]) – see the proof ofTheorem 1.1 in Section 7. 7n the case n > R n – see [12], [11]).Let us note that the existence of chords as in Theorem 1.1 for similarlydefined sets in the cotangent bundle of the torus and compactly supportedHamiltonians can be proved along the lines of the proof of [7, Thm. 1.13]using symplectic quasi-states, with the upper bound on the time-length of thechord depending on the size of the support of the Hamiltonian. The methodsof this paper do not allow us to treat this case because the foundations ofthe Legendrian contact homology have not been worked out rigorously yetfor the relevant setting. Remark 1.4. For (time-independent) mechanical Hamiltonians H the exis-tence of Hamiltonian chords of H from X to X , as in part B of Theorem 1.1,can be obtained by the classical Maupertius’s least action principle (see e.g.[1, p.247]).Namely, assume that H is a complete mechanical Hamiltonian of theform H ( p, q ) = | p | / U ( q ), where 0 ≤ sup R n | U | < + ∞ . Let x , x ∈ R n ,0 < s − < s + and assume that for some C > sup R n | U | the level set { H = C } intersects the sets X , X .Consider the Riemannian metric (cid:101) g on R n of the form (cid:101) g = (cid:112) C − U ( q ) g ,where g is the Euclidean metric (the metric (cid:101) g is called the Jacobi metric).It is not hard to verify that the metric (cid:101) g is complete and therefore, by theHopf-Rinow theorem, there exists a minimal geodesic of (cid:101) g from x to x . ByMaupertius’s least action principle, the lift of the geodesic to the level set { H = C } of H in T ∗ R n = R n is a Hamiltonian chord of H from X to X . Let us present sample applications to contact dynamics. Let (Σ , ξ ) be a contact manifold equipped with a contact form λ . An or-dered pair (Λ , Λ ) of disjoint Legendrian submanifolds Λ , Λ ⊂ Σ is called µ -interlinked if there exists a constant µ = µ (Λ , Λ , λ ) > h on Σ with h ≥ c > ≤ µ/c starting at Λ and arriving at Λ . Thepair (Λ , Λ ) is called interlinked if it is µ -interlinked for some µ > 0. Thepair (Λ , Λ ) is called robustly interlinked , if every pair (Λ (cid:48) , Λ (cid:48) ) of Legendri-ans from sufficiently small C ∞ -neighborhoods of, respectively, Λ and Λ isinterlinked.Write R t for the Reeb flow of λ . Given two Legendrian sybmanifoldsΛ , Λ ⊂ Σ, a Reeb chord R t x , t ∈ [0 , τ ] with x ∈ Λ and y := R τ x ∈ Λ iscalled non-degenerate if D x R τ ( T x Λ ) ⊕ T y Λ = ξ y , (7)where ξ y stands for the contact hyperplane at y .Let Σ be the jet space J Q = T ∗ Q ( p, q ) × R ( z ) of a closed manifold Q equipped with the contact form dz − pdq . Denote by R the Reeb vector fieldof λ . Let Λ = { p = 0 , z = 0 } be the zero section. Theorem 1.5. (i) Let ψ be a positive function on Q , and let Λ := { z = ψ ( q ) , p = ψ (cid:48) ( q ) } be the graph of its 1-jet. Then the pair (Λ , Λ ) isrobustly interlinked.(ii) Assume that Λ ⊂ Σ = J Q is a Legendrian submanifold with followingproperty: there is unique chord of the Reeb flow R t starting on Λ andending on Λ , and this chord is non-degenerate. Then the pair (Λ , Λ ) is interlinked. The proof is given in Section 5. Let ψ be a positive function on Q , and let Λ := { z = ψ ( q ) , p = ψ (cid:48) ( q ) } be the graph of its 1-jet in T ∗ Q × R . As we have seen in Theorem 1.5(i), thepair (Λ , Λ ) is interlinked. Note that the order matters: the pair (Λ , Λ ) isnot interlinked. Indeed, there is no Reeb chord from Λ to Λ . Assume nowthat Λ (cid:48) is Legendrian isotopic to Λ outside Λ , and that there exist exactlytwo non-degenerate Reeb chords A, a starting on Λ (cid:48) and ending on Λ , andtheir time lengths | A | , | a | satisfy0 < | A | − | a | < | b | , (8)9or every Reeb chord b starting and ending on Λ (cid:116) Λ (cid:48) . The next result statesthat for contact Hamiltonians with a sufficiently small oscillation (i.e., forsmall perturbations of the constant contact Hamiltonian 1), one can establishexistence of a chord even in the absence of interlinking. Theorem 1.6. Let h be any positive bounded contact Hamiltonian on Σ with c := inf Σ h ≤ h ≤ sup Σ h =: C . If Cc < | A || a | , (9) then there is a chord of h starting on Λ (cid:48) and ending on Λ of time length ≤ (cid:107) a | ( | A | − | a | ) / ( | A | c − | a | C ) . The proof is given in Section 5.2. For an example of a Legendrian submanifoldΛ (cid:48) ⊂ T ∗ S × R satisfying the assumption of Theorem 1.6 we refer to Figures1 and 2 describing the front projection of Λ (cid:48) . As we shall see later on, theproof of this result involves the machinery persistence modules (as opposedto the usual spectral invariants), as the finite bars of persistence barcodesenter the play. Figure 1: Isotopy from Λ to Λ (cid:48) Next, we relax the setting of contact interlinking and work with contactHamiltonians which may be time-dependent, unbounded or may change sign.Let l ∈ R , l > 0. Let Λ , Λ be the following Legendrian submanifolds of(Σ , ξ ): if Σ = ST ∗ R n , then Λ is the unit cotangent sphere at some x ∈ R n while(i) either Λ is the image of Λ under the time- l Reeb flow;10igure 2: Λ (cid:48) , zoomed in(ii) or Λ is the unit cotangent sphere at some x ∈ R n , | x − x | = l , inwhich case there exists a unique non-degenerate Reeb chord starting atΛ and ending on Λ .Note that these pairs (Λ , Λ ) are interlinked by Theorem 1.5(i) and (ii), re-spectively. We shall also allow Σ = J Q , where Λ is the zero-section and Λ is its image under the time- l Reeb flow, as in Theorem 1.5(i). We shall dis-cuss applications concerning the existence of chords of contact Hamiltoniansfrom Λ to Λ .Assume h : Σ × S → R is a complete (time-periodic) contact Hamiltonian(with respect to λ ). Write h t := h ( · , t ), t ∈ S . Denote by v t , t ∈ S , the(time-periodic) contact Hamiltonian vector field of h with respect to thecontact form λ . If v t is time-independent, we write just v . Let { ϕ t } be theflow of v t – that is, the contact Hamiltonian flow of h . Definition 1.7. Let us say that h : Σ × S → R is C -cooperative with Λ , Λ for C > h < C on Λ × S and either the set { h ≥ C } = (cid:83) t ∈ S { h t ≥ C } is emptyor dh t ( R ) ≥ { h t ≥ C } for all t ∈ S .(b) h < C on Λ × S and either the set { h ≥ C } = (cid:83) t ∈ S { h t ≥ C } is emptyor dh t ( R ) ≤ { h t ≥ C } for all t ∈ S .We will say that h is cooperative with Λ , Λ if it is C -cooperative with Λ ,Λ for some C > 0. 11ote that conditions (a) and (b) hold, in particular, for a sufficiently large C if sup Σ × S h < + ∞ . Conditions (a) and (b) guarantee that a chord of h from Λ to Λ , if it exists, does not leave the set { h ≤ C } at any time. Theorem 1.8 (cf. Rem. 1.14 in [27]) . Assume that h is cooperative with Λ , Λ and that inf Σ × S h > . Assume also that for some < e < / Σ × S | ∂h/∂t | < (1 − e ) e (cid:0) inf Σ × S h (cid:1) (cid:0) max Λ × S h + e inf Σ × S h (cid:1) l . Then there exists a chord of h from Λ to Λ of time-length bounded fromabove by l (1 − e ) inf Σ × S h .In particular, if h is time-independent, then the time-length can be boundedfrom above by l inf Σ × S h . For the proofs of Theorems 1.12, 1.8 see Section 7.For other results on the existence of Reeb chords between different Legen-drian submanifolds (or equivalently, chords of positive contact Hamiltonians)see the papers of G.Dimitroglou-Rizell and M.Sullivan [16, 17] (for a com-parison of their results with the results in [27] and here, see [17, Sec. 1.3]).Theorem 1.8, together with a basic dynamical assumption, allows to ob-tain the following results concerning the chords of contact Hamiltonians thatare not positive everywhere . Corollary 1.9. Assume that h is cooperative with Λ , Λ and there existsa (possibly non-compact or disconnected) closed codimension-0 submanifold Ξ ⊂ Σ with a (possibly non-compact or disconnected) boundary ∂ Ξ , so that(1) inf Ξ × S h > (but h may be negative outside Ξ × S ).(2) sup ∂ Ξ × S h < + ∞ .(3) For each t ∈ S the vector field v t is transverse to ∂ Ξ (in particular, ∂ Ξ is a convex surface in the sense of contact topology – see [35]) and eitherpoints inside Ξ everywhere on ∂ Ξ or points outside Ξ everywhere on ∂ Ξ .(4) Both Λ and Λ lie in Ξ . ssume also that for some < e < / Ξ × S | ∂h/∂t | < (1 − e ) e (cid:0) inf Ξ × S h (cid:1) (cid:0) max Λ × S h + e inf Ξ × S h (cid:1) l . Then there exists a chord of h from Λ to Λ whose time-length is boundedfrom above by l (1 − e ) inf Ξ × S h .If h is time-independent, then the time-length of the chord is boundedfrom above by l/ inf Ξ h . For the proof of Corollary 1.9 see Section 7. Remark 1.10. Assume h is time-independent and Ξ := { h ≥ c } for some c > 0. Then the conditions (1) and (2) are satisfied automatically, whilecondition (3) is equivalent to ∂ Ξ being transverse to the Reeb vector field R ,because dh ( v ) = hdh ( R ). Example 1.11. An example satisfying the assumptions of Corollary 1.9 canbe constructed as follows.Consider the case where Λ is the zero-section of J Q and Λ is its imageunder the time- l Reeb flow, l > c > , Λ ⊂ Ξ := { z ≥ c } . Thus condition (4) inCorollary 1.9 is satisfied.Consider a time-independent contact Hamiltonian h = az + g on Σ = J Q = T ∗ Q × R , where a > z is the coordinate along the R -factor and g is a smooth bounded function on T ∗ Q . Assume that inf Ξ h > a is sufficiently large compared to || g || L ∞ . Then condition(1) in Corollary 1.9 is satisfied. Condition (2) is satisfied since g is bounded.Finally, condition (3) is satisfied since the Reeb vector field R of the standardcontact form on J Q is ∂/∂z and therefore dh ( v ) = hdh ( R ) = ah > ∂ Ξ = { z = c } . It is also easy to verify that h is C -cooperative with Λ , Λ for a sufficiently large C .We have verified that the objects above – and accordingly their preimagesunder ψ – satisfy the assumptions of Corollary 1.9. Consequently, Corol-lary 1.9 yields the existence of a chord of h from Λ to Λ of time-length ≤ l/ inf Ξ h . 13 .4.4 Contact flows with large conformal factor Our next result illustrates that contact Hamiltonians separating (in asuitable sense) certain pairs of Legendrian submanifolds generate contactflows with an arbitrarily large conformal factor. Here Λ and Λ are as inthe beginning of Section 1.4.3. Theorem 1.12. Assume that h is time-independent, compactly supportedand h | Λ ≥ , h | Λ < . Then the conformal factor of ϕ t takes arbitrarily large values as t variesbetween and + ∞ : inf t ∈ (0 , + ∞ ) ,y ∈ Σ (cid:0) ϕ − t (cid:1) ∗ λ ( ϕ t ( y )) λ ( ϕ t ( y )) = + ∞ . Recall that the conformal fact is an important dynamical characteristic play-ing the role of the contact Lyapunov exponent. For the proof see Section 7. Let us outline a key property of Σ, λ , Λ , Λ above that allows to provethe results in Sections 1.3,1.4 and outline the general scheme of the proofs.Let Λ , Λ ⊂ Σ be disjoint Legendrian (not necessarily connected) com-pact submanifolds without boundary of a nice contact manifold (see Sec-tion 1.2). Let Λ := Λ (cid:116) Λ . Assume that the pair (Λ , λ ) is non-degenerate –that is, there are only finitely many Reeb chords of Λ and they are all non-degenerate (this can be always achieved by a C ∞ -small Legendrian perturba-tion of either of the two Legendrian submanifolds). Then one can associateto the pair (Λ = Λ (cid:116) Λ , λ ) its (full) Chekanov-Eliashberg algebra – a freenon-commutative unital algebra over Z generated by all the Reeb chords ofΛ. The algebra is filtered by the action (the action of a Reeb chord is itstime-length; the action of a monomial, or a product of the Reeb chords, isthe sum of the actions of its factors).We consider a vector subspace of the Chekanov-Eliashberg algebra, whichwe will call the 01 -subspace – it is generated by the monomials a · . . . · a k , k ∈ Z > , where a starts at Λ , a k ends at Λ , and for each m = 1 , . . . , k − a m lies in the same component of Λ as the origin of a m +1 .14ecall that the differential ∂ J on the Chekanov-Eliashberg algebra de-pends on an almost complex structure J on the symplectization of (Σ , ker λ )and is defined as follows: the differential of a generator (that is, a Reeb chord)is defined using the count of J -holomorphic disks in the symplectization withone positive and possibly several negative punctures on the boundary, whoseboundary lies in Λ and that converge near the punctures to cylinders overReeb chords of Λ; the differential is then extended to the whole algebra usingthe Leibniz rule and the condition ∂ J (1) := 0 (see Section 4.2).The differential preserves the 01-subspace and lowers the filtration. Thisallows to view the resulting homology of the 01-subspace – the filtered Legen-drian contact homology of (Λ := Λ (cid:116) Λ , λ ) – as a persistence module definedover ( −∞ , + ∞ ) and apply the theory of persistence modules to its study. Inparticular, one can associate to it its barcode – a collection of intervals, called bars , lying in (0 , + ∞ ).For s ∈ (1 , + ∞ ) let l min,s (Λ , Λ , λ ) denote the smallest left end of a barof multiplicative length greater than s in the barcode. (The multiplicativelength of a bar in (0 , + ∞ ) is the ratio of its right and left ends; note thatit may be infinite). Let l min, ∞ (Λ , Λ , λ ) denote the smallest left end ofan infinite bar. If there are no such bars, set l min,s (Λ , Λ , λ ) := + ∞ or,respectively, l min, ∞ (Λ , Λ , λ ) := + ∞ .If the pair (Λ = Λ (cid:116) Λ , λ ) is degenerate, then Λ = Λ (cid:116) Λ can beapproximated by Legendrian submanifolds Λ (cid:48) = Λ (cid:48) (cid:116) Λ (cid:48) obtained from Λ bya C ∞ -small Legendrian isotopy, so that the pair (Λ (cid:48) , λ ) is non-degenerate.Extend the definition of l min,s (Λ , Λ , λ ), s ∈ (1 , + ∞ ], to all pairs (Λ =Λ (cid:116) Λ , λ ) as follows: l min,s (Λ , Λ , λ ) := lim inf l min,s (Λ (cid:48) , Λ (cid:48) , λ ) , where the lim inf is taken over all such Λ (cid:48) = Λ (cid:48) (cid:116) Λ (cid:48) converging to Λ (inthe C ∞ -topology). One can show that for non-degenerate pairs (Λ = Λ (cid:116) Λ , λ ) this definition and the original one yield the same l min,s (Λ , Λ , λ ) –cf. Remark 4.9.If l min,s (Λ , Λ , λ ) < + ∞ for all s ∈ (1 , + ∞ ), we will say that the pair(Λ (cid:116) Λ , λ ) is weakly homologically bonded . If l min, ∞ (Λ , Λ , λ ) < + ∞ , wewill say that the pair (Λ (cid:116) Λ , λ ) is homologically bonded. If the pair is non-degenerate, these conditions mean that the corresponding barcode containsbars of arbitrarily large multiplicative length, or, respectively, an infinite bar.The key property of the setting in Sections 1.3,1.4 is that the pair (Λ (cid:116) Λ , λ ) is homologically bonded . 15onsider the stabilization of Σ which is the manifold (cid:98) Σ := Σ × R ( r ) × S ( τ ) equipped with the contact form (cid:98) λ := λ + rdτ = ds + ϑ + rdτ . Thiscontact manifold is also nice. For a Legendrian submanifold ∆ ⊂ Σ define aLegendrian submanifold (cid:98) ∆ ⊂ (cid:98) Σ by (cid:98) ∆ := ∆ × { r = 0 } . For each s ∈ (1 , + ∞ ] define (cid:98) l min,s (Λ , Λ , λ ) := l min,s ( (cid:98) Λ , (cid:98) Λ , (cid:98) λ ) . The pair (Λ (cid:116) Λ , λ ) is said to be stably homologically bonded if (cid:98) l min, ∞ (Λ , Λ , λ ) < + ∞ . Remark 1.13. It is likely that homological bondedness implies stable ho-mological bondedness. If Σ is the standard contact R , this follows from aresult of Ekholm-K´alm´an [23, Thm. 1.1], but, to the best of our knowledge,the case of a general (nice) contact manifold has not been worked out so far.For the Legendrian submanifolds Λ = Λ (cid:116) Λ in Section 1.4 the pair(Λ = Λ (cid:116) Λ , λ ) is homologically bonded and stably homologically bonded– see Sections 6, 7.Let us now explain how (stable) homological bondedness is used to provethe results in Sections 1.3,1.4.For a pair (Λ (cid:116) Λ , λ ) in a general nice contact manifold (Σ , λ ), considerthe trivial exact Lagrangian cobordism (Λ (cid:116) Λ ) × [ s − , s + ] in the trivial exactsymplectic cobordism (Σ × [ s − , s + ] , d ( sλ )). For instance, in the setting ofTheorem 1.1 the latter exact symplectic cobordism can be identified withthe manifold { ( p, q ) ∈ R n | s ≤ | p | ≤ s } whose boundary components –the sets Y , Y – are identified, respectively, with Σ × s − and Σ × s − . Theparts Λ × [ s − , s + ], Λ × [ s − , s + ] of the trivial exact Lagrangian cobordism(Λ (cid:116) Λ ) × [ s − , s + ] are then identified, respectively, with the sets X , X .If the pair (Λ (cid:116) Λ , λ ) is non-degenerate, the exact Lagrangian cobordismdefines a cobordism map (in the category of the persistence modules) fromthe persistence module associated to (Λ = Λ (cid:116) Λ , s + λ ) to the one associatedto (Λ = Λ (cid:116) Λ , s − λ ). These persistence modules are multiplicative shifts ofeach other and the cobordism map is the multiplicative shift by s + /s − .16f the pair (Λ (cid:116) Λ , λ ) is weakly homologically bonded (and, in particu-lar, if it is homologically bonded), then the cobordism map is not the zeromorphism between persistence modules and the pseudo-holomorphic curvesused to define the map can be also used to prove that a version of the Poissonbracket invariant of quadruples of sets is positive for the following quadrupleof sets: Λ × [ s − , s + ], Λ × [ s − , s + ], Σ × s − , Σ × s + . This is the key resultin the paper – see Section 2 for the precise definition of the Poisson bracketinvariant (it is a version of the invariant defined previously in [7, 27]) andTheorem 4.10 for the precise statement of the result. If the pair (Λ (cid:116) Λ , λ ) isdegenerate but still weakly homologically bonded, the same result is obtainedusing a semi-continuity property of the Poisson bracket invariant.The existence of chords for time-independent symplectic Hamiltonians asin Theorem 1.1 follows then from the positivity of the Poisson bracket invari-ant by Fathi’s dynamical Urysohn lemma (see Theorem 2.1 for its statement).If the pair (Λ (cid:116) Λ , λ ) is stably homologically bonded, then a similarargument yields the existence of chords as in part B of Theorem 1.1 for time-dependent Hamiltonians.The existence of a chord from Λ to Λ for a contact Hamiltonian coop-erative with Λ , Λ , as in Section 1.4.3, is deduced then from the existenceof the chords of the corresponding homogeneous symplectic Hamiltonian onthe symplectization of Σ. Remark 1.14. The claims of Theorem 1.1 (and its analogues for other ho-mologically bonded pairs) on the existence of Hamiltonian chords remain trueif X , X are perturbed as exact Lagrangian cobordisms (cylindrical near theboundaries) in the trivial exact symplectic cobordism { ( p, q ) ∈ R n | s ≤| p | ≤ s } so that the Legendrian isotopies, induced on the boundaries by theexact Lagrangian isotopies, are sufficiently small – e.g. sufficiently C ∞ -small.(Note that away from the boundary the perturbations may be arbitrarilylarge, as long as the perturbed X , X are disjoint!). The upper bound onthe time-length of the Hamiltonian chords between the perturbed X , X isthen only slightly larger than the one for the original X , X .Similarly, the claims of the results in Section 1.4 remain true if the Leg-endrian submanifolds Λ , Λ are perturbed by sufficiently C ∞ -small Legen-drian isotopies into Legendrian submanifolds Λ (cid:48) , Λ (cid:48) . The upper bound onthe time-length of the chord between Λ (cid:48) , Λ (cid:48) is then only slightly larger thanthe one for the original Λ , Λ and tends to it as the sizes of the Legendrianisotopies tend to zero. 17et us also remark that the scheme of the proof can be extended toa more general setting and, in particular, to non-trivial exact Lagrangiancobordisms; one can also use the linearized Legendrian contact homologyinstead of the full one [29].For more details and a reference to the proofs see more general Re-marks 4.9, 4.14, 5.9. Let us outline the plan of the paper.In Section 2 we define a Poisson bracket invariant of a quadruple of sets(a modified version of the invariant defined previously in [7, 27]) and state arecent theorem of Fathi (a generalization of the result in [30]), which allowsto deduce the existence of a Hamiltonian chord from the positivity of theinvariant.In Section 3 we recall basic facts about persistence modules.In Section 4 we describe the Legendrian contact homology setting that weneed and explain how to associate a persistence module, and a correspondingbarcode, to a (non-degenerate) pair formed by a Legendrian submanifold anda contact form. Then we show how the existence of bars of sufficiently largemultiplicative length in the barcode implies the positivity of the Poissonbracket invariant for appropriate quadruples of sets, which in turn yields theexistence of the wanted chords.In Section 5 we discuss applications of the result proved in Section 4 tocontact dynamics.In Sections 6,7 we explain how the results of Section 4 can be applied tothe Legendrian submanifolds in J Q and ST ∗ R n , which yields the results ofSections 1.3, 1.4. Acknowledgements: We thank F.Bourgeois, B.Chantraine, G.Dimitro-glou Rizell, T.Ekholm, Y.Eliashberg, E.Giroux, M.Sullivan for useful discus-sions. We also thank A.Fathi for communicating to us his unpublished note[31] containing Theorem 2.1 and its proof.18 A modified pb + -invariant and Hamiltonianchords In this section we discuss a Poisson bracket invariant of quadruples of setsin a symplectic manifold. The proof of the result relating the Poisson bracketinvariant to the existence of Hamiltonian chords is based on the theorems ofA.Fathi in general smooth/topological dynamics [30], [31] and is similar tothe proof of the relevant results in [27] (with the only difference that we cannow use the results from [31] that were unavailable when [27] was written).Let us recall these results of Fathi. In this section let M be any smooth manifold (without boundary), v a complete smooth time-independent vector field on M (meaning that itsflow is defined for all times) and X , X ⊂ M disjoint closed subsets of M .Denote by T ( X , X ; v ) the infimum of the time-lengths of the chords (thatis, integral trajectories) of v from X to X . If there is no such chord, set T ( X , X ; v ) := + ∞ .The following theorem (a “dynamical Urysohn lemma”) was proved byA.Fathi in [30] in the case when X , X are compact and in [31] in the casewhen X , X are arbitrary closed sets. Theorem 2.1 (A.Fathi, [30], [31]) . Assume T > T ( X , X ; v ) .Then there exists a smooth function F : M → R such that F | X ≤ , F | X > , and L v F < /T .If X , X are compact, then F can be chosen to be compactly supported. Define S ( X , X ) := { F ∈ C ∞ ( M ) | F | X ≤ , F | X ≥ } , S (cid:48) ( X , X ) := { F ∈ C ∞ ( M ) | Im ( F ) ⊂ [0 , , F | Op ( X ) = 0 , F | Op ( X ) = 1 } , where Op ( · ) denotes some open neighborhood of a set. Define K ( X , X ) := S ( X , X ) ∩ C ∞ c ( M ) , K (cid:48) ( X , X ) := S (cid:48) ( X , X ) ∩ C ∞ c ( M ) , C ∞ c ( M ) is the space of compactly supported smooth functions on M .Define L ( X , X ; v ) := inf F ∈S ( X ,X ) sup M L v F, and if X and X are compact, L c ( X , X ; v ) := inf F ∈K ( X ,X ) sup M L v F. Clearly, if X and X are compact, then L ( X , X ; v ) ≤ L c ( X , X ; v ) . Proposition 2.2. In the definition of L ( X , X ; v ) and L c ( X , X ; v ) one canreplace S ( X , X ) and K ( X , X ) respectively by S (cid:48) ( X , X ) and K (cid:48) ( X , X ) : L ( X , X ; v ) = inf F ∈S (cid:48) ( X ,X ) sup M L v F, and if X and X are compact, L c ( X , X ; v ) = inf F ∈K (cid:48) ( X ,X ) sup M L v F. Proof of Proposition 2.2: Let us prove the claim for L ( X , X ; v ) – thecase of L c ( X , X ; v ) is similar.Clearly, L ( X , X ; v ) ≤ inf F ∈S (cid:48) ( X ,X ) sup M L v F and thus it suffices toprove that inf F ∈S (cid:48) ( X ,X ) sup M L v F ≤ L ( X , X ; v ) . (10)Let δ > 0. Pick a non-decreasing smooth function χ : R → R so thatsup t ∈ R χ (cid:48) ( t ) ≤ δ and for some (cid:15) > χ ( t ) = 0 on ( −∞ , (cid:15) ] and χ ( t ) = 1 on [1 − (cid:15), + ∞ ).Then χ ◦ F ∈ S (cid:48) ( X , X ) for any F ∈ S ( X , X ) and L v ( χ ◦ F ) = ( χ (cid:48) ◦ F ) L v F ≤ (1 + δ ) L v F. Taking the infimum over F ∈ S ( X , X ) in both sides we getinf F ∈S (cid:48) ( X ,X ) sup M L v F ≤ (1 + δ ) L ( X , X ; v ) . Since this is true for any δ > 0, we obtain (10) and this finishes the proof ofthe proposition.The following corollary follows readily from Theorem 2.1.20 orollary 2.3. T ( X , X ; v ) = 1 /L ( X , X ; v ) .Consequently, if L ( X , X ; v ) > , then for any (cid:15) > there exists a chordof v from X to X of time-length ≤ /L ( X , X ; v ) + (cid:15) (if supp v is compact,then one can drop (cid:15) from the bound).If X , X are compact, then L ( X , X ; v ) = L c ( X , X ; v ) and T ( X , X ; v ) =1 /L c ( X , X ; v ) .Consequently, if X and X are compact and L c ( X , X ; v ) > , thenthere exists a chord of v from X to X of time-length ≤ /L c ( X , X ; v ) . Let ( M, ω ) be a (not necessarily compact) connected symplectic manifold,possibly with boundary.We use the following sign conventions in the definitions of a Hamiltonianvector field and the Poisson bracket on M : the Hamiltonian vector field sgrad H of a Hamiltonian H is defined by i sgrad H ω = − dH and the Poisson bracket of two Hamiltonians F , G is given by { F, G } := ω ( sgrad G, sgrad F ) = dF ( sgrad G ) = − dG ( sgrad F ) == L sgrad G F = − L sgrad F G. Assume X , X , Y , Y are closed subsets of M such that X ∩ X = Y ∩ Y = ∅ . Such a collection of sets X , X , Y , Y will be called an admissible quadru-ple .Consider the following conditions on pairs ( F, G ) ∈ C ∞ ( M ) × C ∞ ( M ):(1) sup M { F, G } < + ∞ and the vector field F sgrad G on M is complete.(1’) supp F is compact and supp ( F sgrad G ) lies in the interior of M .(2) F | X ≤ , F | X ≥ G | Y ≤ , G | Y ≥ F | X ≤ , F | X ≥ G | Op ( Y ) ≡ , G | Op ( Y ) ≡ 1. (Here Op ( · ) denotessome open neighborhood of a set).(2”) F | Op ( X ) ≡ , F | Op ( X ) ≡ G | Op ( Y ) ≡ , G | Op ( Y ) ≡ F ⊂ [0 , ⇒ (1), since supp { F, G } ⊂ supp ( F dG ) = supp ( F sgrad G ).Define: F M ( X , X , Y , Y ) := { ( F, G ) ∈ C ∞ ( M ) × C ∞ ( M ) | ( F, G ) satisfies (1) and (2) } , F (cid:48) M ( X , X , Y , Y ) := { ( F, G ) ∈ C ∞ ( M ) × C ∞ ( M ) | ( F, G ) satisfies (1),(2) and (3) } , F (cid:48)(cid:48) M ( X , X , Y , Y ) := { ( F, G ) ∈ C ∞ ( M ) × C ∞ ( M ) | ( F, G ) satisfies (1),(2”) and (3) } , G M ( X , X , Y , Y ) := { ( F, G ) ∈ C ∞ ( M ) × C ∞ ( M ) | ( F, G ) satisfies (1’) and (2) } , G (cid:48) M ( X , X , Y , Y ) := { ( F, G ) ∈ C ∞ ( M ) × C ∞ ( M ) | ( F, G ) satisfies (1’), (2’) } , G (cid:48)(cid:48) M ( X , X , Y , Y ) := { ( F, G ) ∈ C ∞ ( M ) × C ∞ ( M ) | ( F, G ) satisfies (1’), (2”) and (3) } . For brevity we will omit the sets X , X , Y , Y from this notation, whenneeded.Clearly, G (cid:48)(cid:48) M ⊂ G (cid:48) M ⊂ G M ⊂ F M , F (cid:48)(cid:48) M ⊂ F (cid:48) M ⊂ F M , G M ⊂ F M , G (cid:48) M ⊂ F (cid:48) M , G (cid:48)(cid:48) M ⊂ F (cid:48)(cid:48) M . It is also easy to see that the sets G M , G (cid:48) M and G (cid:48)(cid:48) M are non-empty if only if X is compact.Set pb + M ( X , X , Y , Y ) := inf F M sup M { F, G } ,pb + M,comp ( X , X , Y , Y ) := inf G M max M { F, G } . If the set over which the infimum is taken is empty, we set the infimum tobe + ∞ .Clearly, pb + M ( X , X , Y , Y ) ≤ pb + M,comp ( X , X , Y , Y ) . emark 2.4. The quantities pb + M , pb + M,comp are versions of the pb invariant ofquadruples of sets defined originally in [7] (where the C -norm of { F, G } wasused instead of max M { F, G } ) and of the pb +4 invariant defined in [27] (wherethe sets X , X , Y , Y were assumed to be compact and both F and G wereassumed to be compactly supported). See also [34] for a result that allows todefine pb +4 ( X , X , Y , Y ) in terms of the topology of the set X ∪ X ∪ Y ∪ Y .Note that, unlike pb and pb +4 , the invariants pb + M , pb + M,comp are not symmetricwith respect to the permutation ( X , X , Y , Y ) (cid:55)→ ( Y , Y , X , X ).Similarly to Proposition 2.2 (also see [7]), one can prove that the sets F M , G M in the definitions of pb + M , pb + M,comp can be replaced, respectively, by F (cid:48) M , F (cid:48)(cid:48) M and by G (cid:48) M , G (cid:48)(cid:48) M : pb + M ( X , X , Y , Y ) = inf F (cid:48) M sup M { F, G } = inf F (cid:48)(cid:48) M sup M { F, G } ,pb + M,comp ( X , X , Y , Y ) = inf G (cid:48) M max M { F, G } = inf G (cid:48)(cid:48) M max M { F, G } . We will need the following basic properties of pb + M , pb + M,comp . Monotonicity: Proposition 2.5 (cf. [7], [27]) . Assume that M is a codimension-zero sub-manifold (with boundary) of a symplectic manifold N (without boundary),which is closed as a subset of N . Let U ⊂ N be an open set. Assume that X , X , Y , Y is an admissible quadruple lying in M , so that X , X ⊂ U ∩ M , Y ∩ U, Y ∩ U (cid:54) = ∅ and ∂M ⊂ Y ∪ Y .Then pb + N ( X , X , Y , Y ) ≥ pb + M ( X , X , Y , Y ) , (11) pb + N,comp ( X , X , Y , Y ) ≥ pb + M,comp ( X , X , Y , Y ) , (12) pb + M ∩ U,comp ( X , X , Y ∩ U, Y ∩ U ) ≥ pb + M,comp ( X , X , Y , Y ) . (13) Proof. If ( F, G ) ∈ F (cid:48)(cid:48) N ( X , X , Y , Y ), then it follows easily from the defini-tions that ( F | M , G | M ) ∈ F (cid:48)(cid:48) M ( X , X , Y , Y ). This yields (11). The inequality(12) follows similarly.Let us prove (13). Assume ( F, G ) ∈ G (cid:48)(cid:48) M ∩ U ( X , X , Y ∩ U, Y ∩ U ). Inparticular, this means that supp F ⊂ M ∩ U is compact and G is equal to 0and 1 on some open neighborhoods (in U ) of, respectively, Y ∩ U and Y ∩ U .23xtend F by zero outside M ∩ U to a smooth compactly supported function (cid:101) F : M → [0 , 1] and extend G to a smooth function (cid:101) G : M → R so that (cid:101) G is equal to 0 and 1 on some open neighborhoods (in M ) of, respectively, Y and Y . Then ( (cid:101) F , (cid:101) G ) ∈ G (cid:48)(cid:48) M ( X , X , Y , Y ), while { (cid:101) F , (cid:101) G } = { F, G } (becauseoutside U both Poisson brackets vanish, while on U they coincide since (cid:101) F | U = F , (cid:101) G | U = G ). This immediately yields (13).The following property follows from the definitions (cf. [7], [27]). Semi-continuity: Suppose that ( X , X , Y , Y ) is an admissible quadruple in ( M, ω ), X , X are compact, and { X ( j )0 } , { X ( j )1 } , j ∈ N , are sequences of compact subsets of M converging (in the sense of the Hausdorff distance between sets) respec-tively to X , X , so that the quadruples ( X ( j )0 , X ( j )1 , Y , Y ) are admissible forall j ∈ N .Then lim sup j → + ∞ pb + M ( X ( j )0 , X ( j )1 , Y , Y ) ≤ pb + M ( X , X , Y , Y ) , (14)lim sup j → + ∞ pb + M,comp ( X ( j )0 , X ( j )1 , Y , Y ) ≤ pb + M,comp ( X , X , Y , Y ) . (15)The next proposition is proved as in [27] using Corollary 2.3. Proposition 2.6. Assume that M is a codimension-zero submanifold (withboundary) of a symplectic manifold N (without boundary), so that M is closedas a subset of N . Assume that X , X , Y , Y is an admissible quadruple lyingin M , so that ∂M ⊂ Y ∪ Y .Let H : N → R be a complete time-independent Hamiltonian. Then thefollowing claims hold:I. Assume ∆( H ; Y , Y ) =: ∆ > . If pb + M ( X , X , Y , Y ) =: p > , then forany (cid:15) > there exists a chord of H from X to X of time-length ≤ p ∆ + (cid:15) .If X , X are compact and pb + M,comp ( X , X , Y , Y ) =: p comp > , then thereexists a chord of H from X to X of time-length ≤ p comp ∆ .II. Assume that X , X are compact supp H ∩ M is compact, H | X ≥ and H | X ≤ − ∆ for some ∆ > . Assume also that pb + M,comp ( X , X , Y , Y ) =:24 comp > . Then there exists a chord of H from Y to Y of time-length ≤ p comp ∆ . Proof of Proposition 2.6: Let us prove part I. We may assume withoutloss of generality that H | Y ≤ H | Y ≥ H with aH + b for some a, b ∈ R , a (cid:54) = 0). For any F ∈S ( X , X ; sgrad H ) satisfying sup N L sgrad H F = sup N { F, H } < + ∞ we have( F, H ) ∈ F (cid:48) N , and if supp F is compact, then ( F, H ) ∈ G N . Indeed, since thevector field sgrad H is complete, then so is the vector field F sgrad H , since0 ≤ F ≤ 1. Hence, by (11) and (12),sup N L sgrad H F = sup N { F, H } ≥ pb + N ( X , X , Y , Y ) ≥≥ pb + M ( X , X , Y , Y ) , or, if supp F is compact,sup N L sgrad H F = sup N { F, H } ≥≥ pb + N,comp ( X , X , Y , Y ) ≥ pb + M,comp ( X , X , Y , Y ) . Taking the infimum over all such F , we get L ( X , X ; sgrad H ) ≥ pb + M ( X , X , Y , Y ) , and if X , X are compact, L c ( X , X ; sgrad H ) ≥ pb + M,comp ( X , X , Y , Y ) . Now the claims of part I follow from Corollary 2.3.Let us prove part II. We may assume without loss of generality that H | X ≥ H | X ≤ − H with H/ ∆). For any G ∈ S (cid:48) ( Y , Y ; sgrad H ) we have ( − H | M , G | M ) ∈ G (cid:48) M (recallthat here supp H ∩ M is assumed to be compact and G is constant near Y and Y , since G ∈ S (cid:48) ( Y , Y ; sgrad H )). Hence,sup N L sgrad H G ≥ sup M L sgrad H G = sup M {− H, G } ≥ pb + N,comp ( X , X , Y , Y ) . G ∈ S (cid:48) ( Y , Y ; sgrad H ) and using Proposi-tion 2.2, we get L ( Y , Y ; sgrad H ) ≥ pb + N,comp ( X , X , Y , Y ) ≥≥ pb + M,comp ( X , X , Y , Y ) = p comp . Now the claim of part II follows from Corollary 2.3.Let us now discuss an implication of Proposition 2.6 for the existence ofchords of time-dependent Hamiltonians.Let E > 0. Let r ∈ ( − E, E ) and τ ∈ S = R / Z be the coordinates,respectively, on ( − E, E ) and S . Set (cid:102) M E := M × ( − E, E ) × S and equip (cid:102) M E with the product symplectic form ω ⊕ dr ∧ dτ . Let X , X , Y , Y ⊂ M be an admissible quadruple, such that X , X are compact. Set (cid:101) X := X × { r = 0 } , (cid:101) X := X × { r = 0 } , (cid:101) Y ( E ) := Y × ( − E, E ) × S , (cid:101) Y ( E ) := Y × ( − E, E ) × S . Proposition 2.7. Assume that ∂M = ∅ . With an admissible quadruple X , X , Y , Y ⊂ M as above, let H : M × S → R be a complete Hamiltonianand { φ t } t ∈ R its flow. Let E > . Assume that(a) pb + (cid:102) M E ,comp ( (cid:101) X , (cid:101) X , (cid:101) Y ( E ) , (cid:101) Y ( E )) =: (cid:101) p E > .(b) ∆( H ; Y , Y ) =: ∆ > E .(c) sup t ∈ R (cid:0) sup t ∈ [ t ,t + T ] H ( φ t ( x ) , t ) − inf t ∈ [ t ,t + T ] H ( φ t ( x ) , t ) (cid:1) < E for any x ∈ X , where T := 1 (cid:101) p E (∆ − E ) .Then there exists a chord of H from X to X of time-length ≤ T =1 (cid:101) p E (∆ − E ) . roof of Proposition 2.7: In view of (c) we can pick 0 < E < E < E sothat sup t ∈ [ t ,t + T ] H ( φ t ( x ) , t ) − inf t ∈ [ t ,t + T ] H ( φ t ( x ) , t ) < E for all t ∈ R . (16)Pick a smooth cut-off function χ : R → R such that χ ( x ) = 0 if | x | ≥ E and χ ( x ) = 1 if | x | ≤ E .Define a time-independent Hamiltonian (cid:101) H : (cid:102) M E → R as (cid:101) H ( x, r, τ ) := r + H ( x, τ ). One easily verifies that the Hamiltonian χ (cid:101) H is complete and, inview of (b), ∆ (cid:16) χH ; (cid:101) Y ( E ) , (cid:101) Y ( E ) (cid:17) ≥ ∆ − E > . Together with (a), this implies, by part I of Proposition 2.6, that there existsa chord γ of χ (cid:101) H from (cid:101) X to (cid:101) X of time-length ≤ T = 1 (cid:101) p E (∆ − E ) .We claim that γ is, in fact, a chord of (cid:101) H from (cid:101) X to (cid:101) X – this would implythat the projection of γ to M is a chord of H from X to X of time-length ≤ T .Indeed, note that χ (cid:101) H = (cid:101) H = H on M × [ − E , E ] × S ⊂ (cid:102) M E and theprojection to M of each trajectory of the Hamiltonian flow of (cid:101) H on (cid:102) M E is atrajectory of the Hamiltonian flow of H on M . Since the time-independentHamiltonian (cid:101) H is preserved by its own Hamiltonian flow, (16) implies thatfor any t ∈ R any time-[ t , t + T ] trajectory of the Hamiltonian flow of (cid:101) H passing at some moment t ∈ [ t , t + T ] through (cid:101) X stays in M × [ − E , E ] × S for all t ∈ [ t , t + T ]. Therefore for any t ∈ R any time-[ t , t + T ] trajectoryof the Hamiltonian flow of χ (cid:101) H passing at some moment t ∈ [ t , t + T ] through (cid:101) X is, in fact, a trajectory of the Hamiltonian flow of (cid:101) H for all t ∈ [ t , t + T ].In particular, γ is a chord of (cid:101) H from (cid:101) X to (cid:101) X , which proves the claim andfinishes the proof of the proposition. Remark 2.8. Proposition 2.7 admits an analogue for the case when X , X are not necessarily compact – in that case one should replace the quantity pb + (cid:102) M E ,comp ( (cid:101) X , (cid:101) X , (cid:101) Y ( E ) , (cid:101) Y ( E )) in the claim by pb + (cid:102) M E ( (cid:101) X , (cid:101) X , (cid:101) Y ( E ) , (cid:101) Y ( E )) =: pb + (cid:102) M E and then there would exist a chord of H from X to X of time-length ≤ T = 1 pb + (cid:102) M E (∆ − E ) . The proof of this claim virtually repeats the proof ofProposition 2.7. 27 Persistence modules In this section we recall basic facts about persistence modules. For a moredetailed introduction to persistence modules see e.g. [18], [9], [37], or [40].We work over the base field Z .Let I := ( a, + ∞ ), −∞ ≤ a < + ∞ . (In fact, we will be concerned onlywith I = (0 , + ∞ ) and I = ( −∞ , ∞ )). Definition 3.1. A persistence module over I is given by a pair( V = { V t } t ∈ I , π = { π s,t } s,t ∈ I ,s ≤ t ) , where all V t , t ∈ I , are finite-dimensional Z -vector spaces and π s,t : V s → V t are linear maps, so that(i) (Persistence) π t,t = Id , π s,r = π t,r ◦ π s,t , for all s, t, r ∈ I , s ≤ t ≤ r .(ii) (Discrete spectrum and semicontinuity) There exists a (finite or count-able) discrete closed set of points spec ( V ) = { l min ( V ) := t < t < t < . . . < + ∞} ⊂ I , called the spectrum of V , so that • for any r ∈ I \ spec ( V ) there exists a neighborhood U of r in I suchthat π s,t is an isomorphism for all s, t ∈ U , s ≤ t ; • for any r ∈ spec ( V ) there exists (cid:15) > π s,t is an isomorphismof vector spaces for all s, t ∈ ( r − (cid:15), r ] ∩ I .(iii) (Semi-bounded support) For the smallest point l min ( V ) := t of spec ( V )one has l min ( V ) > a and V t = 0 for all t ≤ l min ( V ).In [40] such persistence modules are called persistence modules of finitetype . The zero (or trivial) persistence module (over I ) is a persistence moduleformed by zero vector spaces and trivial maps between them.The notions of a persistence submodule, the direct sum of persistencemodules, a morphism/isomorphism between persistence modules (over I ) aredefined in a straightforward manner.Right-left equivalent morphisms between persistence modules will be calledjust equivalent . 28 xample 3.2. A persistence module V over (0 , + ∞ ) can be extended to apersistence module over ( −∞ , + ∞ ) by setting V s = 0 and π s,t = 0 for all s ∈ ( −∞ , Example 3.3. Let J ⊂ I be either of the form ( a J , b J ] for some 0 < a J
For every persistence module ( V, π ) over I there exists aunique (finite or countable) collection of intervals J j = ( a j , b j ] ⊂ I – wherethe intervals may not be distinct but each interval appears in the collectiononly finitely many times – so that ( V, π ) is isomorphic to ⊕ j Q ( J j ) : ( V, π ) = ⊕ j Q ( J j ) . The collection { J j } of the intervals is called the barcode of V . The intervals J j themselves are called the bars of V . Note that the same bar may appear inthe barcode several (but finitely many) times (this number of times is calledthe multiplicity of the bar) – in other words, a barcode is a multiset of bars.The barcode of the trivial persistence module is empty. Example 3.5. Let I = ( −∞ , + ∞ ) and let V be a persistence module over I . Let c ∈ R .Define a new persistence module V [+ c ] over I by adding c to all indices of V t and π s,t – in particular, V [+ c ] t := V t + c . The barcode of V [+ c ] is the barcode of V shifted by c to the left.29f c ≥ 0, one can also define a morphism Sh V [+ c ] : V → V [+ c ] , called the additive shift of V by c , as follows: Sh V [+ c ] := (cid:110) π t,t + c : V t → V [+ c ] t (cid:111) t ∈ I . Example 3.6. Let I = (0 , + ∞ ) and let V be a persistence module over I .Let c > V = ( V, π ) be a persistence module. Define a new persistence module V [ × c ] over I by multiplying all the indices of V t and π s,t by c – in particular, V [ × c ] t := V ct . The barcode of V [ × c ] is the barcode of V divided by c .If c ≥ 1, one can also define a morphism Sh V [ × c ] : V → V [ × c ] , called the multiplicative shift of V by c , as follows: Sh V [ × c ] := (cid:110) π t,ct : V t → V [ × c ] t (cid:111) t ∈ I . Remark 3.7. One easily sees that additive/multiplicative shifts of isomor-phic persistence modules by the same constant are also isomorphic. Thus,one can speak about additive/multiplicative shifts of isomorphism classes ofpersistence modules. Let ( P n , dϑ ), n ∈ Z ≥ , be an exact symplectic manifold with boundedgeometry at infinity (see [3] for the definition of this class of manifolds; in30articular, this class includes symplectic manifolds that are convex in thesense of [24]). Consider the contactization of ( P, dϑ ): LetΣ := P × R ( z ) , and let ξ = ker λ be the contact structure on Σ defined by the contact form λ := dz + ϑ. Let { ψ t } be the Reeb flow of λ – each ψ t is a shift by t in the coordinate s . Further on we will always assume that P , and consequently Σ ,are connected. Consider a compact (not necessarily connected) Legendrian submanifoldΛ ⊂ (Σ , ξ ) without boundary.If Λ = Λ (cid:116) Λ is a disjoint union of compact (not necessarily connected)Legendrian submanifolds Λ , Λ without boundary we call Λ a two-part Leg-endrian submanifold and Λ , Λ the ( - and -) parts of Λ.Denote the set of all Reeb chords of (Λ , λ ) by R (Λ , λ ).If Λ = Λ (cid:116) Λ is a two-part Legendrian submanifold, we say that a Reebchord of (Λ , λ ) is an ij -chord for i, j = 0 , i and ends on Λ j .Denote the set of all ij -chords of (Λ , λ ) by R ij (Λ , λ ). The ii -chords will becalled pure , while ij -chords for i (cid:54) = j will be called mixed .We say that the pair (Λ , λ ) is non-degenerate if the following conditionsare satisfied: • For each Reeb chord a : [0 , T ] → Σ, a ( t ) = ψ t ( a (0)) (for all t ∈ [0 , T ]), a (0) , a ( T ) ∈ Λ, of Λ with respect to λ , the Legendrian submanifold ψ T (Λ) is transversal to Λ at the point a ( T ) = ψ T ( a (0)). • Each trajectory of the Reeb flow { ψ t ( x ) } , −∞ < t < + ∞ , x ∈ Σ,intersects Λ at most in two points. (Equivalently, the images of distinctReeb chords are disjoint).If (Λ , λ ) is non-degenerate and Λ is compact, then the set R (Λ , λ ) is finite.31 .1 Exact Lagrangian cobordisms The symplectization of (Σ , ξ ) can be identified with (cid:0) Σ × R + ( s ) , d ( sλ ) (cid:1) .Let 0 < s − < s + .Consider the manifold with boundary Σ × [ s − , s + ] ⊂ Σ × R + equipped withthe symplectic form ω := d ( sλ ) – it is a trivial exact symplectic cobordism whose positive and negative boundaries and the restrictions of sλ to them areidentified, respectively, with (Σ , s + λ ) and (Σ , s − λ ).A differential 1-form θ on Σ × [ s − , s + ] will be called a cobordism 1-form if the following conditions are satisfied: • dθ = ω ; • θ coincides with sλ near the boundaries of Σ × [ s − , s + ];In particular, sλ itself is a cobordism 1-form.Let Λ ± ⊂ (Σ , ξ ) be compact Legendrian submanifolds without boundary,viewed respectively as submanifolds of Σ × s ± . A Lagrangian cobordism in(Σ × [ s − , s + ] , sλ ) between Λ ± is a smooth compact cobordism in Σ × [ s − , s + ]between Λ + ⊂ Σ × s + and Λ − ⊂ Σ × s − which is a Lagrangian submanifoldof (Σ × R + , ω ) so that there exist δ ± > L ∩ Σ × [ s + − δ + , s + ] = Λ + × [ s + − δ + , s + ] ,L ∩ Σ × [ s − , s − + δ − ] = Λ − × [ s − , s − + δ − ] . The sets L ∩ Σ × [ s + − δ + , s + ] and L ∩ Σ × [ s − , s − + δ − ] will be called the positive and the negative collars of L . The Legendrian submanifolds Λ − andΛ + will be called, respectively, the negative and the positive boundary of L .We say that L as above is a two-part Lagrangian cobordism if L is adisjoint union of two (not necessarily connected) Lagrangian cobordisms L and L : L = L (cid:116) L , where L is a Lagrangian cobordism between the Legendrian submanifoldsΛ +0 := Λ + ∩ L and Λ − := Λ − ∩ L and L is a Lagrangian cobordism betweenthe Legendrian submanifolds Λ +1 := Λ + ∩ L and Λ − := Λ − ∩ L . In particular,Λ ± = Λ ± (cid:116) Λ ± are two-part Legendrian submanifolds.Note that in our terminology a two-part Lagrangian cobordism includesa numbering of its parts. 32et θ be a cobordism 1-form. We say that a two-part Lagrangian cobor-dism L = L (cid:116) L is θ -exact if θ | L i = df i , i = 0 , 1, for a smooth function f i : L i → R which is zero on the negative collar of L and is identically equalto a constant C i on the positive collar of L i .If a two-part Lagrangian cobordism L is θ -exact for some cobordism 1-form θ , we call it just exact .The constant C := C − C will be called the gap of L with respect to θ .We will also say that L is C -gapped (with respect to θ ).We say that a two-part Lagrangian cobordism L is belted if there existsa null-homologous piecewise-smooth closed path γ in Σ × [ s − , s + ] tracingΣ × s + from Λ +0 = L ∩ (Σ × s + ) to Λ +1 = L ∩ (Σ × s + ), then tracing L from Λ +1 = L ∩ (Σ × s + ) to Λ − = L ∩ (Σ × s − ), then following possiblyseveral arcs in L connecting points in Λ − , then tracing Σ × s − from Λ − = L ∩ (Σ × s − ) to Λ − = L ∩ (Σ × s − ), then following possibly several arcs in L connecting points in Λ − , and finally tracing L from Λ − = L ∩ (Σ × s − )to Λ +0 = L ∩ (Σ × s + ). We will call such a γ a belt path of L .We claim that if L is belted, then the gap of L with respect to a cobordism1-form (with respect to which L is exact) does not depend on the form.Indeed, assume θ and θ (cid:48) are cobordism 1-forms on Σ × [ s − , s + ] so that L isexact with respect to both θ and θ (cid:48) . Then θ − θ (cid:48) is a closed 1-form vanishingnear the positive and the negative boundaries of Σ × [ s − , s + ]. Since a beltpath γ of L is null-homologous, the integral of θ − θ (cid:48) over γ vanishes. Since θ − θ (cid:48) vanishes near the positive and the negative boundaries of Σ × [ s − , s + ],the latter zero integral is the sum of the integrals of θ − θ (cid:48) over the parts of γ lying in L and L , which readily implies the claim. A trivial Lagrangian cobordism in (Σ × [ s − , s + ] , sλ ) is a cobordism Λ × [ s − , s + ] where Λ is a Legendrian submanifold of (Σ , ξ ).A trivial two-part Lagrangian cobordism L = (Λ (cid:116) Λ ) × [ s − , s + ]is belted: to construct a belt path γ for L , take a path Γ in Σ from x ∈ Λ to y ∈ Λ (it exists, since, by our assumption, Σ is connected). Now defined γ as the path tracing Γ × s + from x × s + ∈ Λ × s + to y × s + ∈ Λ × s + , thentracing y × [ s − , s + ] from y × s + to y × s − , then tracing Γ × s − from y × s − to x × s − , and finally tracing x × [ s − , s + ] from x × s − to x × s + . Note alsothat L is ( sλ )-exact with the gap 0.Given a (two-part exact) Lagrangian cobordism L ⊂ Σ × [ s − , s + ] between33 ± , consider the Lagrangian submanifold L ⊂ (cid:0) Σ × R + , d ( sλ ) (cid:1) defined as L := (cid:0) Λ − × (0 , s − ] (cid:1) ∪ L ∪ (cid:0) Λ + × [ s + , + ∞ ) (cid:1) . We call L the completion of L .Let L ⊂ (Σ × [ s − , s + ] , sλ ) be an exact two-part Lagrangian cobordismbetween two-part Legendrian submanifolds Λ ± ⊂ (Σ , ξ ). By an exact La-grangian cobordism isotopy of L we mean a smooth family { L τ , θ τ } ≤ τ ≤ T ,where { L τ } ≤ τ ≤ T is a Lagrangian isotopy of L = L in Σ × [ s − , s + ] and { θ τ } ≤ τ ≤ T is a smooth family of cobordism 1-forms so that(1) Each L τ is a two-part θ τ -exact Lagrangian cobordism between its pos-itive and negative boundaries that will be denoted by Λ ± τ . In particular, { Λ ± τ } ≤ τ ≤ T are Legendrian isotopies in (Σ ± , ξ ± ). We will say that these are the Legendrian isotopies induced by the exact Lagrangian cobordism isotopy { L τ , θ τ } ≤ τ ≤ T .(2) There exist δ ± > τ ∈ [0 , T ] L τ ∩ Σ × [ s + − δ + , s + ] = Λ + τ × [ s + − δ + , s + ] ,L τ ∩ Σ × [ s − , s − + δ − ] = Λ − τ × [ s − , s − + δ − ] . Note that the gaps C ( τ ) of L τ , 0 ≤ τ ≤ T , with respect to θ τ form asmooth function of τ .Clearly, if L is belted, then so are all L τ , 0 ≤ τ ≤ T . Let us recall the definition of the Chekanov-Eliashberg algebra associatedto a non-degenerate pair (Λ , λ ), where Λ ⊂ (Σ , ξ ) is a compact Legendriansubmanifold without boundary. See [21] (cf. [20],[22], [14]) for more detailsand [10], [25], [26] for the original ideas underlying the construction. We willassume that Λ = Λ (cid:116) Λ is a two-part Legendrian submanifold.A sequence of Reeb chords a , . . . , a k ∈ R (Λ , λ ) is called ij -composable for i = 0 and j = 1, or i = 1 and j = 0, if a starts at Λ i , a k ends at Λ j ,and for each m = 1 , . . . , k − a m lies in the same part of Λ as theorigin of a m +1 . Note that an ij -composable sequence of Reeb chords mustcontain at least one chord from R ij (Λ , λ ).34enote by A (Λ , λ ) a free non-commutative unital algebra over Z gener-ated by the elements of Z and by R (Λ , λ ).For each i = 0 , A (Λ i , λ ) the subalgebra of A (Λ , λ ) freelygenerated (over Z ) by Z and by ii -chords of (Λ , λ ).For each i, j = 0 , i (cid:54) = j , denote by A (Λ i , Λ j , λ ) the vector subspaceof A (Λ , λ ) (over Z ) generated by the monomials a · . . . · a k for all ij -composable sequences a , . . . , a k ∈ R (Λ , λ ), k ∈ Z > – we call such mono-mials ij -composable . (Note that the polynomials appearing in A (Λ i , Λ j , λ )have no constant terms!).The algebra A (Λ , λ ) comes with a filtration defined by the action: theaction of a Reeb chord is its time-length and the action of a monomial whichis a product of Reeb chords is the sum of the actions of the factors. Theaction of the constant monomial 1 ∈ Z is set to be zero and the action of 0is defined as −∞ . For r ∈ ( −∞ , + ∞ ) define A r (Λ , λ ) as the vector subspaceof A (Λ , λ ) spanned over Z by the monomials whose action is smaller than r . It is easy to see that for a finite r the vector space A r (Λ , λ ) is finite-dimensional.Let ∂ J be a differential on A (Λ , λ ) defined for an appropriate (a so-called cylindrical) almost complex structure J on Σ × R + using a countof J -holomorphic maps of a disk with one positive and several (possiblyno) negative boundary punctures into Σ × R + . Such a map should send theboundary of the disk to Λ × R + and converge near positive/negative punctureto a positive/negative cylinder over a Reeb chord in R (Λ , λ ).The set J (Λ) of J for which ∂ J is well-defined and ∂ J = 0 is connectedand dense in the space of all cylindrical almost complex structures on Σ × R + [20], [21], see also [22], [14].Let J ∈ J (Λ).It is easy to see that A (Λ i , λ ), A (Λ i , Λ j , λ ) for all i, j = 0 , ∂ J and so are the spaces A r (Λ , λ ) for all r . We call A (Λ , Λ , λ ) the01 -subspace of A (Λ , λ ).For each r ∈ ( −∞ , + ∞ ) define a vector space V r (Λ , Λ , λ ) over Z : V r (Λ , Λ , λ, J ) := Ker ∂ J | A r (Λ , Λ ,λ ) Im ∂ J | A r (Λ , Λ ,λ ) . The inclusion maps A r (Λ , Λ , λ ) → A t (Λ , Λ , λ ), r ≤ t , induce morphismsof vector spaces V r (Λ , Λ , λ, J ) → V t (Λ , Λ , λ, J ).One easily sees that the vector spaces V r (Λ , Λ , λ, J ), r ∈ (0 , + ∞ ), to-gether with the morphisms between them, form a persistence module over350 , + ∞ ). We will denote this persistence module by V (Λ , Λ , λ, J ). Below,whenever needed, we will also view V (Λ , Λ , λ, J ) as a persistence moduleover ( −∞ , + ∞ ) using the trivial extension as in Example 3.2.One can show (see [29]) that for fixed Λ , Λ , λ different choices of J ∈J (Λ) yield isomorphic persistence modules. The isomorphism class of thepersistence module V (Λ , Λ , λ, J ) will be denoted by V (Λ , Λ , λ ). Abusingthe terminology we will call it the LCH persistence module associated to (Λ , λ ), where LCH stands for “Legendrian contact homology”.It is easy to see that if c > V (Λ , Λ , cλ ) = V [ × /c ] (Λ , Λ , λ ) , where V [ × /c ] (Λ , Λ , λ ) is the isomorphism class of persistence modules ob-tained from V (Λ , Λ , λ ) by the multiplicative shift by 1 /c (see Remark 3.7). Let L = L (cid:116) L ⊂ (cid:0) Σ × [ s − , s + ] , ω = d ( sλ ) (cid:1) , s − < s + , be a two-part exact Lagrangian cobordism between two-part Legendrian submanifoldsΛ ± = Λ ± (cid:116) Λ ± ⊂ (Σ , ξ ).Assume the pairs (Λ ± , λ ) are non-degenerate – in this case we will saythat L is non-degenerate .For an appropriate – a so-called adapted (to L and ω ) – almost complexstructure I on Σ × R + define a unital algebra morphismΦ L,I : A (Λ + , s + λ, I + ) → A (Λ − , s − λ, I − )by prescribing its values on the generators:Φ L,I (1) := 1and for any a ∈ R (Λ + , s + λ )Φ L,I ( a ) := (cid:88) dim M L,I ( a ; b ,...,b m )=0 |M L,I ( a ; b , . . . , b m ) | b · . . . · b m , where M L,I ( a ; b , . . . , b m ) is the moduli space of I -holomorphic maps of a diskwith one positive and m ≥ × R + that36end the boundary of the disk to L and converge near the positive puncture toa cylinder over a and the negative punctures to the cylinders over the chords b , . . . , b m ∈ R (Λ − , s − λ ). Here I ± are cylindrical almost complex structureson Σ × R + induced by the restrictions of I at the ends of Σ × R + . See [21],cf. [22], for more details.The set I ( L ) of I for which Φ L,I is a well-defined unital algebra morphismis dense in the space of all adapted almost complex structures on Σ × R + and the map I ( L ) → J (Λ + ) × J (Λ − ) , I (cid:55)→ ( I + , I − ) , is surjective [21], cf. [22].The map Φ L,I is called the cobordism map associated to L, I . Remark 4.1. Assume that I ∈ I ( L ) and the restriction of Φ L,I to the 01-subspace A (Λ +0 , Λ +1 , s + λ ) is not the zero map. Then for some 01-chord a andsome non-empty set of chords b , . . . , b m the moduli space M L,I ( a ; b , . . . , b m )is non-empty. Rescale an I -holomorphic map of a disk D (cid:48) , with boundarypunctures, that defines an element of the moduli space and obtain a map of D (cid:48) into Σ × [ s − , s + ]. Concatenating the image of ∂D (cid:48) under the latter mapwith the chords a, b , . . . , b m we get a belt path for the two-part Lagrangiancobordism L . In other words, we have obtained that, unless L is belted, Φ L,I has to be the zero map.The following two claims are proved in [29] using [22, Lemma 3.14] (achain homotopy result for the cobordism maps, which is a version of [21,Lemma B.15]) – see the proof of Proposition 4.5 below for a similar use ofthe same result. Proposition 4.2 ([29]) . Assume that L is a non-degenerate belted two-partexact Lagrangian cobordism. Let I ∈ I ( L ) and assume that the restriction ofthe cobordism map Φ L,I to A (Λ +0 , Λ +1 , s + λ ) is non-trivial. Let C be the gapof L (since L is belted, the gap is independent of the cobordism 1-form withrespect to which L is exact).Then Φ L,I maps A r (Λ , Λ , s + λ, I + ) into A r − C (Λ , Λ , s − λ, I − ) for each r ∈ ( −∞ , + ∞ ) and, accordingly, defines a morphism of persistence modulesover ( −∞ , + ∞ ) : Φ L,I ∗ : V (Λ , Λ , s + λ, I + ) → V [ − C ] (Λ , Λ , s − λ, I − ) , where V [ − C ] (Λ , Λ , s − λ, I − ) is the isomorphism class of persistence modulesover ( −∞ , + ∞ ) obtained from V (Λ , Λ , s − λ, I − ) by the additive shift by − C (see Remark 3.7). 37f the restriction of the cobordism map Φ L,I to A (Λ +0 , Λ +1 , s + λ ) is the zeromap, we set Φ L,I ∗ to be the zero morphism into the trivial persistence module. Proposition 4.3 ([29]) . The morphism Φ L,I ∗ is independent of I ∈ I ( L ) up to the right-left equivalence in the category of persistence modules over ( −∞ , + ∞ ) . Further on, we will denote the equivalence class of the persistence modulemorphism Φ L,I ∗ by Φ L ∗ .Abusing the terminology, we will not distinguish between an isomorphismclass of a persistence module and a specific persistence module representing it,as well as between an equivalence class of morphisms of persistence modulesand a specific morphism representing it. In particular, we will writeΦ L ∗ : V (Λ , Λ , s + λ ) → V [ − C ] (Λ , Λ , s − λ ) . Note that compositions of equivalence classes of morphisms of persistencemodules are well-defined.The following proposition is proved in [29] using the well-known results(see [21]) about the cobordism map associated to a trivial exact Lagrangiancobordism. Proposition 4.4 ([29]) . Assume L = (Λ (cid:116) Λ ) × [ s − , s + ] is a trivial non-degenerate two-part exact Lagrangian cobordism. Set U := V (Λ , Λ , s + λ ) . Then Φ L ∗ = Sh U [ × s + /s − ] . Having recalled the needed preparational statements, we present now thekey result of this section. Proposition 4.5. With L being a non-degenerate belted two-part exact La-grangian cobordism as above, suppose that { L τ = L τ (cid:116) L τ , θ τ } ≤ τ ≤ T is anexact Lagrangian cobordism isotopy of L with fixed boundary. Assume that Φ L ∗ is non-trivial. et C ( τ ) be the gap of L τ , ≤ τ ≤ T (since L is belted, it is independentof the cobordism 1-form with respect to which L τ is exact). Let U := V (Λ , Λ , s + λ ) ,W := V (Λ , Λ , s − λ ) .C min := min τ ∈ [0 ,T ] C ( τ ) ,C := C (0) − C min , C T := C ( T ) − C min . Then Sh W [ − C (0)] [+ C ] ◦ Φ L = Sh W [ − C ( T )] [+ C T ] ◦ Φ L T . (17) Here the equality is between equivalence classes of morphisms U → W [ − C min ] . Proof of Proposition 4.5: Pick I ∈ I ( L ), I T ∈ I ( L T ). It follows from[22, Lemma 3.14] (which is a version of [21, Lemma B.15]) that there existsan Z -linear map K : A (Λ + , s + λ ) → A (Λ − , s − λ )such that K ◦ ∂ I +0 + ∂ I − T ◦ K = Φ L T ,I T − Φ L ,I . (18)The map K is defined on each monomial a · . . . · a k , a , . . . , a k ∈ R (Λ + , s + λ ),as K ( a · . . . · a k ) := k (cid:88) j =1 Φ L T ,I T ( a · . . . · a j − ) K ( a j )Φ L ,I ( a j +1 · . . . · a k ) , (19)so that for each a ∈ R (Λ + , s + λ ) K ( a ) := (cid:88) b ,...,b m ∈R (Λ − ,s − λ ) n { L T } , { I T } ( a ; b , . . . , b m ) b · . . . · b m , (20)where { I τ } ≤ τ ≤ T is a generic family of adapted almost complex structures on M connecting I and I T , and n { L T } , { I T } ( a ; b , . . . , b m ) is the mod-2 number ofelements (counted using an abstract perturbation scheme, see [21] for details)of a certain moduli space which is non-empty only if ∪ ≤ τ ≤ T M L τ ,I τ ( a ; b , . . . , b m ) (cid:54) = ∅ . M L τ ,I τ ( a ; b , . . . , b m ) is the moduli spaces of pseudo-holomorphic mapsof the disk with boundary punctures into Σ × R + used to define Φ L τ ,I τ .(This moduli space is non-empty only for some finite set of τ ∈ [0 , T ] – thecorresponding pseudo-holomorphic maps have index − M L τ ,I τ ( a ; b , . . . , b m ) (cid:54) = ∅ , then m (cid:88) i =1 s − l ( b i ) < s + l ( a ) − C ( τ ) for a ∈ R (Λ +0 , Λ +1 , s + λ ) , m (cid:88) i =1 s − l ( b i ) < s + l ( a ) + C ( τ ) for a ∈ R (Λ +1 , Λ +0 , s + λ ) , m (cid:88) i =1 s − l ( b i ) < s + l ( a ) for a ∈ R (Λ + i , Λ + i , s + λ ) , i = 0 , , where l ( · ) is the action (time-length) of a Reeb orbit. Using these inequali-ties together with (19), (20) it is not hard to see that K τ maps the subspace A r (Λ +0 , Λ +1 , s + λ ) into A r − C min (Λ − , Λ − , s − λ ) for any r ∈ ( −∞ , + ∞ ). There-fore the chain homotopy formula (18) implies that the restrictions of Φ L T ,I T and Φ L ,I to A r (Λ +0 , Λ +1 , s + λ ) induce the same map on homology – that is,a map V r (Λ +0 , Λ +1 , s + λ, I +0 ) → V r − C min (Λ − , Λ − , s − λ, I − T )for any r ∈ ( −∞ , + ∞ ). This latter map equals, on one hand, the compositionof Φ L r : V r (Λ +0 , Λ +1 , s + λ, I +0 ) → V r − C (0) (Λ − , Λ − , s − λ, I − T )and the shift V r − C (0) (Λ − , Λ − , s − λ, I − T ) → V r − C min (Λ − , Λ − , s − λ, I − T )and, on the other hand, the composition ofΦ L T r : V r (Λ +0 , Λ +1 , s + λ, I +0 ) → V r − C ( T ) (Λ − , Λ − , s − λ, I − T )and the shift V r − C ( T ) (Λ − , Λ − , s − λ, I − T ) → V r − C min (Λ − , Λ − , s − λ, I − T ) . The equality between the two compositions yields (17). This finishes theproof of part B and of the proposition.40 .4 An invariant of two-part Legendrians via LCH per-sistence modules With Σ and λ as above, assume Λ = Λ (cid:116) Λ is a two-part Legendriansubmanifold in (Σ , ξ = Ker λ ). Definition 4.6. If the pair (Λ , λ ) is non-degenerate, then for each s > l min,s (Λ , Λ , λ ) as the smallest left end of the bars of multiplicative lengthgreater than s in the barcode of V (Λ , Λ , λ ). Denote by l min, ∞ (Λ , Λ , λ ) thesmallest left end of the infinite bars in the barcode of V (Λ , Λ , λ ). If there areno such bars, set l min,s (Λ , Λ , λ ) := + ∞ or, respectively, l min, ∞ (Λ , Λ , λ ) :=+ ∞ .For a general, possibly degenerate, pair (Λ , λ ) and s > l min,s (Λ , Λ , λ ) := lim inf l min,s (Λ (cid:48) , Λ (cid:48) , λ ) ,l min, ∞ (Λ , Λ , λ ) := lim inf l min, ∞ (Λ (cid:48) , Λ (cid:48) , λ ) , where each Λ (cid:48) = Λ (cid:48) (cid:116) Λ (cid:48) is a two-part Legendrian submanifold obtainedfrom Λ by a C ∞ -small Legendrian isotopy and such that the pair (Λ (cid:48) , λ ) isnon-degenerate, and the liminf is taken over all such Λ (cid:48) as the C ∞ -size ofthe Legendrian isotopy converges to zero. (One can show – see [29] – thatfor a non-degenerate pair (Λ = Λ (cid:116) Λ , λ ) both definitions yield the same l min,s (Λ , Λ , λ ) and l min, ∞ (Λ , Λ , λ )).The next proposition illustrates this definition. We work in the setting ofTheorem 1.6 above (the notations are changed for the sake of simplicity).Let ψ be a positive function on Q , and let Λ := { z = ψ ( q ) , p = ψ (cid:48) ( q ) } bethe graph of its 1-jet in J Q = T ∗ Q × R . Assume that K is a Legendriansubmanifold of J Q Legendrian isotopic to Λ outside the zero section K , sothat there exist exactly two non-degenerate Reeb chords A, a starting on K and ending on K , and their time-lengths | A | , | a | satisfy0 < | A | − | a | < | b | , (21)for every Reeb chord b starting and ending on K (cid:116) K . Proposition 4.7. For any s < | A | / | a | , l min,s ( K , K , λ ) = | a | . (22)41 roof. Write A ij for the subalgebra of A generated by all ij -composablemonomials. Observe that A = Span( a, A ) , A = 0 , while the homologies satisfy V ∞ ( K , K ) = 0 , V ∞ ( K , K ) (cid:54) = 0 . Let d := ∂ J , for J ∈ J ( K (cid:116) K ), be the differential on A .Note that da = 0 as there is no 01-composable monomial with a smalleraction. In what follows we write | b | for the action of a monomial b .Since there are no 11-chords, we can write dA = ua + vA . Note that v = 0, since otherwise | vA | > | A | while d lowers the action. By assumption(21), | ua | > | A | for a non-scalar u , and hence either u = 0 or u = 1 (recallthat the base field is Z ). Case 1: u = 0, i.e., dA = 0. We claim that in this case a (cid:54) = dx for any x .Indeed, write x = pa + qA , where p, q ∈ A . Then a = ( dp ) a +( dq ) A yielding dp = 1. But then for every closed y ∈ A we have y = d ( yp ), meaning that y is exact, in contradiction to V ∞ ( K , K ) (cid:54) = 0. Case 2: u = 0, i.e. dA = a . Note that by assumption (21), A has theminimal action among all 01-monomials of action > | a | . It follows thatthe barcode of the persistence module V ( K , K , λ ) contains a bar ( | a | , | A | ].Moreover, since | a | is the minimal action among all 01-chords, we get that(22). This completes the proof.Note that l min,s (Λ , Λ , λ ) is a non-decreasing function of s with values in(0 , + ∞ ]. In particular, l min,s (Λ , Λ , λ ) ≤ l min, ∞ (Λ , Λ , λ ) for all s ∈ (1 , + ∞ ) , and therefore if the number l min, ∞ (Λ , Λ , λ ) is finite, then so is l min,s (Λ , Λ , λ ).Recall from Section 1.5 that the stabilization of Σ is the manifold (cid:98) Σ :=Σ × T ∗ S ( r, τ ), τ ∈ S , equipped with the contact form (cid:98) λ := λ + rdτ = dz + ϑ + rdτ . Since (Σ = P × R ( z ) , λ = dz + ϑ ) is nice, so is ( (cid:98) Σ , (cid:98) λ ) (if ( P, dϑ )has bounded geometry at infinity, then so does ( P × T ∗ S , dϑ + dr ∧ dt )). Fora two-part Legendrian submanifold Λ (cid:116) Λ ⊂ Σ and s > (cid:98) Λ i := Λ i × { r = 0 } ⊂ (cid:98) Σ , i = 0 , , l min,s (Λ , Λ , λ ) := l min,s ( (cid:98) Λ , (cid:98) Λ , (cid:98) λ ) , (cid:98) l min, ∞ (Λ , Λ , λ ) := l min, ∞ ( (cid:98) Λ , (cid:98) Λ , (cid:98) λ ) . Definition 4.8. We say that the pair (Λ (cid:116) Λ , λ ) is weakly homologicallybonded , if l min,s (Λ , Λ , λ ) < + ∞ for all s > V (Λ , Λ , λ ).We say that the pair (Λ (cid:116) Λ , λ ) is homologically bonded , if l min, ∞ (Λ , Λ , λ ) < + ∞ – that is, there are infinite bars in the barcode of V (Λ , Λ , λ ).We say that the pair (Λ (cid:116) Λ , λ ) is stably homologically bonded , if (cid:98) l min, ∞ (Λ , Λ , λ ) < + ∞ .Clearly, homological bondedness implies weak homological bondedness.We do not know whether homological bondedness implies stable homologicalbondedness – see Remark 1.13. Remark 4.9. Assume that l min,s (Λ , Λ , λ ) < + ∞ for some s ∈ (1 , + ∞ ]. LetΛ (cid:48) = Λ (cid:48) (cid:116) Λ (cid:48) be a two-part Legendrian submanifold in (Σ , ξ ) obtained fromΛ = Λ (cid:116) Λ by a contact isotopy { φ t } with the conformal factor satisfying || φ ∗ t λ/λ − || < δ for all t . If δ = δ ( s ) is small enough, then l min,s (Λ (cid:48) , Λ (cid:48) , λ )is also a finite number which tends to l min,s (Λ , Λ , λ ) as δ → 0. The proofwill appear in [29]. pb + via LCH persistence modules With Σ and 0 < s − < s + as above, denote for brevity M := Σ × [ s − , s + ] . Recall that ω := d ( sλ ). Assume Λ = Λ (cid:116) Λ is a two-part Legendriansubmanifold in (Σ , ξ ). Let L = Λ × [ s − , s + ] be the corresponding trivialtwo-part exact Lagrangian cobordism in ( M = Σ × [ s − , s + ] , sλ ).Define an admissible quadruple X , X , Y , Y ⊂ M as follows: X := Λ × [ s − , s + ] , X := Λ × [ s − , s + ] , (23) Y := Σ × s − , Y := Σ × s + . (24)43 heorem 4.10. Assume that l min,s + /s − (Λ , Λ , λ ) < + ∞ .Then pb + M ( X , X , Y , Y ) ≥ s + − s − ) l min,s + /s − (Λ , Λ , λ ) > . Proof of Theorem 4.10: Assume first that the pair (Λ , λ ) is non-degenerate.Set V := V (Λ , Λ , λ ) ,U := V [ × /s + ] = V (Λ , Λ , s + λ ) ,W := V [ × /s − ] = U [ × s + /s − ] = V (Λ , Λ , s − λ ) . Let ( F, G ) ∈ F (cid:48)(cid:48) M ( X , X , Y , Y ). We need to show thatsup M { F, G } ≥ s + − s − ) l min,s + /s − (Λ , Λ , λ ) . (25)Following [7], consider the deformation ω τ := ω + τ dF ∧ dG, τ ∈ R , of ω . A direct calculation shows that dF ∧ dG ∧ ω n − = − n { F, G } ω n . and therefore ω nτ = (1 − τ { F, G } ) ω n . Thus ω τ is symplectic for any τ ∈ I := [0 , / sup M { F, G } ). Note that L isLagrangian with respect to ω τ for all τ ∈ I .Following the idea underlying Moser’s method [36], define a (time-depen-dent) vector field v τ , τ ∈ I , on M by F dG = − i v τ ω τ . One can check that v τ = F − τ { F, G } sgrad G. F, G ) ∈ F (cid:48)(cid:48) M ( X , X , Y , Y ), the vector field F sgrad G is complete.Also the function 1 − τ { F, G } is bounded from below by the constant 1 − τ sup M { F, G } which is positive since τ ∈ I . Thus, for each T ∈ I the time-dependent vector field v τ , 0 ≤ τ ≤ T , equals to the product of F sgrad G with a non-negative function bounded from above by a constant dependingon F, G and T . Therefore the time-[0 , T ] flow σ τ : M → M of v τ , 0 ≤ τ ≤ T ,is well-defined. It is identity near the boundary of M , because v τ vanishesthere (since G is constant near the positive and negative boundaries of M ).For each τ ∈ I define L τ := ( σ τ ) − ( L ) . A direct check shows that σ ∗ τ ω τ = ω for all τ ∈ I . Therefore L τ is Lagrangianwith respect to ω for all τ ∈ I .For each τ ∈ I set θ τ := σ ∗ τ ( sλ + τ F dG )Since d ( sλ + τ F dG ) = ω τ and σ ∗ τ ω τ = ω and since G is constant and σ τ is identity near the positive and negative boundaries of M , one readily getsthat each θ τ is a cobordism 1-form.Since the trivial cobordism L is ( sλ )-exact and F is 0 near L and 1near L , we get that ( sλ + τ F dG ) | L i = df i , i = 0 , 1, for a smooth function f i : L i → R equal to 0 near the negative boundary of L i and to some constantnear the positive boundary of L i . Consequently, for each τ ∈ I the two-partLagrangian cobordism L τ is θ τ -exact.Therefore for each T ∈ I the family { L τ , θ τ } ≤ τ ≤ T is an exact Lagrangiancobordism isotopy.Since L = L is a trivial exact Lagrangian cobordism, Proposition 4.4yields Φ L ∗ = Sh U [ × s + /s − ] : U → W. Note that since L τ is belted, the gap C ( τ ) of L τ is independent of the cobor-dism 1-form with respect to which L τ is exact. Using the Stokes theoremand the assumptions on F and G one easily sees that C ( τ ) = τ. Since C = C min = C (0) = 0 and C T = C ( T ) = T , by Proposition 4.5, Sh U [ × s + /s − ] = Sh W [ − T ] [+ T ] ◦ Φ L T . t > U t → U ts + /s − in the persistencemodule U is a composition of a linear map U t → U ( t − T ) s + /s − , given by Φ L T ,and a morphism U ( t − T ) s + /s − → U ts + /s − in the persistence module U . Thus,if for some t > t and ts + /s − lie in a bar J in the barcode of U ,then by restricting the above-mentioned decomposition of U t → U ts + /s − tothe interval persistence submodule Q ( J ) of U t , we get that T has to besufficiently small so that ( t − T ) s + /s − also lies in J . Take t arbitrarily closefrom above to s + l min,s + /s − (Λ , Λ , λ ), which the smallest left end of the barsof multiplicative length greater than s + /s − in the barcode of U , so that t and ts + /s − lie in the same bar. Then T has to satisfy T ≤ ( s + − s − ) l min,s + /s − (Λ , Λ , λ ) . Since this is true for any T ∈ I = [0 , / max M { F, G } ), we obtain (25) asrequired. This finishes the proof of the theorem for the case where the pair(Λ , λ ) is non-degenerate.The general case now follows from the non-degenerate one by the semi-continuity of the Poisson bracket invariant – see (14).As in Section 4.4, let (cid:98) Σ := Σ × R ( r ) × S ( τ ) = Σ × T ∗ S ( r, τ )be the stabilization of Σ equipped with the contact form (cid:98) λ := λ + rdτ = dz + ϑ + rdτ. Set (cid:102) M := M × T ∗ S ( r, τ )and equip (cid:102) M with the symplectic form ω + dr ∧ dτ = dλ + dr ∧ dτ .For each E > (cid:102) M E ⊂ (cid:102) M as (cid:102) M E := M × ( − E, E ) × S ⊂ (cid:102) M . With the admissible quadruple X , X , Y , Y ⊂ M as above, set (cid:101) X := X × { r = 0 } , (cid:101) X := X × { r = 0 } ⊂ (cid:102) M E ⊂ (cid:102) M , (cid:101) Y := Y × T ∗ S , (cid:101) Y := Y × T ∗ S ⊂ (cid:102) M , (cid:101) Y ( E ) := Y × ( − E, E ) × S , (cid:101) Y ( E ) := Y × ( − E, E ) × S ⊂ (cid:102) M E . heorem 4.11. Assume that (cid:98) l min,s + /s − (Λ , Λ , λ ) < + ∞ .Then for each E > pb + (cid:102) M E ,comp (cid:16) (cid:101) X , (cid:101) X , (cid:101) Y ( E ) , (cid:101) Y ( E ) (cid:17) ≥ pb + (cid:102) M,comp ( (cid:101) X , (cid:101) X , (cid:101) Y , (cid:101) Y ) ≥≥ s + − s − ) (cid:98) l min,s + /s − (Λ , Λ , λ ) > . Proof of Theorem 4.11: Consider the exact symplectic cobordism (cid:99) M := (cid:16)(cid:98) Σ × [ s − , s + ] , d ( s (cid:98) λ ) (cid:17) . Define (cid:98) X , (cid:98) X , (cid:98) Y , (cid:98) Y ⊂ (cid:99) M by (cid:98) X := Λ × S × [ s − , s + ] = (cid:98) Λ × [ s − , s + ] , (cid:98) X := Λ × S × [ s − , s + ] = (cid:98) Λ × [ s − , s + ] , (cid:98) Y := Σ × T ∗ S × s − , (cid:98) Y := Σ × T ∗ S × s + . Since (cid:98) l min,s + /s − (Λ , Λ , λ ) < + ∞ , Theorem 4.10, applied to (cid:99) M , (cid:98) X , (cid:98) X , (cid:98) Y , (cid:98) Y , together with the inequality pb + (cid:99) M,comp ( (cid:98) X , (cid:98) X , (cid:98) Y , (cid:98) Y ) ≥ pb + (cid:99) M ( (cid:98) X , (cid:98) X , (cid:98) Y , (cid:98) Y ),yields pb + (cid:99) M,comp ( (cid:98) X , (cid:98) X , (cid:98) Y , (cid:98) Y ) ≥ s + − s − ) (cid:98) l min,s + /s − (Λ , Λ , λ ) > . (26)Consider a map (cid:99) M = (cid:98) Σ × [ s − , s + ] = Σ × T ∗ S ( r, τ ) × [ s − , s + ] →→ (cid:102) M = Σ × [ s − , s + ] × T ∗ S ( u, τ )that sends each ( x, r, τ, s ) ∈ Σ × T ∗ S ( r, τ ) × [ s − , s + ] to ( x, s, u = sr, τ ) ∈ Σ × [ s − , s + ] × T ∗ S . (We use two copies of T ∗ S – one with the coordinates r, τ and one with the coordinates u, τ ). It is a symplectomorphism – it identifiesthe symplectic form d ( s (cid:98) λ ) = d ( s ( λ + rdτ )) = d ( sλ ) + d ( srdτ ) = ω + d ( srdτ )47n (cid:99) M with the symplectic form ω + du ∧ dτ on (cid:102) M . This symplectomorphism maps the sets (cid:98) X , (cid:98) X , (cid:98) Y , (cid:98) Y ⊂ (cid:99) M , respec-tively, to the sets (cid:101) X , (cid:101) X , (cid:101) Y , (cid:101) Y ⊂ (cid:102) M . Therefore (26) yields pb + (cid:99) M,comp ( (cid:98) X , (cid:98) X , (cid:98) Y , (cid:98) Y ) = pb + (cid:102) M,comp ( (cid:101) X , (cid:101) X , (cid:101) Y , (cid:101) Y ) ≥≥ s + − s − ) (cid:98) l min,s + /s − (Λ , Λ , λ ) > . Combining this with the inequality pb + (cid:102) M E ,comp (cid:16) (cid:101) X , (cid:101) X , (cid:101) Y ( E ) , (cid:101) Y ( E ) (cid:17) ≥ pb + (cid:102) M,comp ( (cid:101) X , (cid:101) X , (cid:101) Y , (cid:101) Y )(that follows from (13)) finishes the proof of the theorem.Assume that ( M, d ( sλ )) is a codimension-zero symplectic submanifoldwith boundary of a symplectic manifold ( N, Ω) which is closed as a subsetof N . Consequently, X , X , Y , Y can be viewed as subsets of N . Let H : N × S ( t ) → R be a complete Hamiltonian. Corollary 4.12. Assume that the Hamiltonian H is time-independent and l min,s + /s − (Λ , Λ , λ ) < + ∞ .Then the following claims hold:A. If ∆( H ; Y , Y ) > , then there exists a chord of H from X to X oftime-length ≤ ( s + − s − ) l min,s + /s − (Λ , Λ , λ )∆( H ; Y , Y ) .B. If supp H ∩ M is compact and H | X ≥ and ∆( H ; X , X ) > , thenthere exists a chord of H from Y to Y of time-length bounded from above by ( s + − s − ) l min,s + /s − (Λ , Λ , λ )∆( H ; X , X ) . Proof of Corollary 4.12: Follows directly from (11), Proposition 2.6 andTheorem 4.10. 48et us now consider the case where H is time-dependent. For each t ≤ t denote by φ t ,t : N → N the time-[ t , t ] flow of H . Set∆ := ∆( H ; Y , Y ) , (cid:98) l min,s + /s − := (cid:98) l min,s + /s − (Λ , Λ , λ ) ,c min := min X × S H, c max := max X × S H. Corollary 4.13. Let < e < / . Set E := e ∆ ,T := ( s + − s − ) (cid:98) l min,s + /s − (1 − e )∆ . Assume that(a) (cid:98) l min,s + /s − < + ∞ ,(b) ∆ > ,(c) sup c min − E ≤ H ≤ c max + E | ∂H/∂t | < E/T = e (1 − e )∆ ( s + − s − ) (cid:98) l min,s + /s − .Then there exists a chord of H from X to X of time-length boundedfrom above by T = (cid:98) l min,s + /s − ( s + − s − )(1 − e )∆ . Proof of Corollary 4.13: Define (cid:101) N E := N × ( − E, E ) × S and equip it with the symplectic form Ω+ dr ∧ dτ , where r, τ are, respectively,the coordinates on ( − E, E ) and S . Then (cid:102) M E = M × ( − E, E ) × S is aclosed codimension-zero symplectic submanifold with boundary of (cid:101) N E . With (cid:101) X , (cid:101) X , (cid:101) Y ( E ) , (cid:101) Y ( E ) ⊂ (cid:102) M E ⊂ (cid:101) N E defined as above, we get, by (11),(12) andTheorem 4.11, that (cid:101) p E := pb + (cid:101) N E ,comp (cid:16) (cid:101) X , (cid:101) X , (cid:101) Y ( E ) , (cid:101) Y ( E ) (cid:17) ≥ pb + (cid:102) M E ,comp (cid:16) (cid:101) X , (cid:101) X , (cid:101) Y ( E ) , (cid:101) Y ( E ) (cid:17) ≥ s + − s − ) (cid:98) l min,s + /s − > . Note that for any x ∈ N and t ∈ R ddt H ( φ t ,t ( x ) , t ) = ∂H∂t ( φ t ,t ( x ) , t ) . Therefore condition (c) implies that for any x ∈ X and any t ∈ R , t ∈ [ t , t + T ], φ t ,t ( x ) ∈ { c min − E ≤ H ≤ c max + E } and | ∂H/∂t ( φ t ,t ( x ) , t ) | < E/T. In particular, this means thatsup t ∈ R (cid:0) sup t ∈ [ t ,t + T ] H ( φ t ,t ( x ) , t ) − inf t ∈ [ t ,t + T ] H ( φ t ,t ( x ) , t ) (cid:1) < ET · T = E. This inequality, together with the positivity of (cid:101) p E and of ∆, allows to applyProposition 2.7 to H , N and (cid:101) N E , which yields the existence of the chord of H from X to X of time-length ≤ (cid:101) p E (∆ − E ) = T . Remark 4.14. The claim on the positivity of pb + M ( X , X , Y , Y ) in Theo-rem 4.10 can be generalized to arbitrary two-part exact Lagrangian cobor-disms, with ( s + − s − ) l min,s + /s − (Λ , Λ , λ ) being replaced by a certain invariantassociated to the morphism of the persistence modules defined by the cobor-dism. In particular, the positivity of pb + M ( X , X , Y , Y ) remains true if thesets X , X as in Theorem 4.10 are perturbed as exact Lagrangian cobor-disms in ( M, d ( sλ )) (cylindrical near the boundaries) so that the Legendrianisotopies, induced on the boundaries by the exact Lagrangian isotopies, aresufficiently small (say, C ∞ -small). (Note that away from the boundary theperturbations may be arbitrarily large, as long as the perturbed X , X aredisjoint!). The lower bound on pb + M ( X , X , Y , Y ) for the perturbed X , X is then only slightly larger than the one appearing in Theorem 4.10 – thedifference between the bounds tends to zero as the sizes of the Legendrianisotopies above tend to zero. In the particular case where a trivial Lagrangiancobordism is deformed among trivial Lagrangian cobordisms the robustnessfollows from Theorem 4.10 and Remark 4.9.50or the proofs and details see [29].Consequently, the results of Theorem 4.11 and Corollaries 4.12 and 4.13are also robust with respect to the above perturbations. With Σ and λ as above, assume that Λ = Λ (cid:116) Λ is a two-part Legendriansubmanifold in (Σ , ξ = ker Λ).Let h : Σ × S → R be a complete (time-dependent) contact Hamiltonian(with respect to λ ). Set h t := h ( · , t ) : Σ → R . Let v t , t ∈ S , denote thecontact vector field of h t . If h and v are time-independent we write v insteadof v t . Denote by { ϕ t } the time-[0 , t ] contact flow of h – that is, the time-[0 , t ]flow of v t . The flow { ϕ t } lifts to a Hamiltonian flow { ψ t } on (Σ × R + , d ( sλ ))equivariant with respect to the multiplicative R + -action on Σ × R + andgenerated by the Hamiltonian H : Σ × R + × S → R , H ( y, s, t ) := s · h ( y, t ).The flow { ψ t } has the form ψ t ( y, s ) = (cid:32) ϕ t ( y ) , s (cid:0) ϕ − t (cid:1) ∗ λ ( ϕ t ( y )) λ ( ϕ t ( y )) (cid:33) . (27)Since the contact flow { ϕ t } of h is defined for all times, so is the Hamiltonianflow { ψ t } of H , meaning that H is complete.Let us recall Definition 1.7. Definition 5.1. Let us say that h : Σ × S → R is C -cooperative with Λ , Λ for C > h < C on Λ × S and either the set { h ≥ C } = (cid:83) t ∈ S { h t ≥ C } is emptyor dh t ( R ) ≥ { h t ≥ C } for all t ∈ S .(b) h < C on Λ × S and either the set { h ≥ C } = (cid:83) t ∈ S { h t ≥ C } is emptyor dh t ( R ) ≤ { h t ≥ C } for all t ∈ S .We will say that h is cooperative with Λ , Λ if it is C -cooperative with Λ ,Λ for some C > 0. 51 .1 Largeness of the conformal factor of ϕ t Theorem 5.2. Assume that h is time-independent and compactly supported.Assume also that h | Λ ≥ , h | Λ < . and the pair (Λ (cid:116) Λ , λ ) is weakly homologically bonded.Then the conformal factor of ϕ t takes arbitrarily large values as t variesbetween and + ∞ : inf t ∈ (0 , + ∞ ) ,y ∈ Σ (cid:0) ϕ − t (cid:1) ∗ λ ( ϕ t ( y )) λ ( ϕ t ( y )) = + ∞ . Remark 5.3. It would be interesting to generalize Theorem 5.2 to contactHamiltonians h that are not necessarily compactly supported but rather areconstant outside a compact set K ⊂ Σ, meaning that the contact Hamil-tonian flow of such an h outside K is a reparameterized Reeb flow and theconformal factor of the flow outside K is identically equal to 1, since theReeb flow preserves the contact form. (Recall that, by our assumptions, theReeb flow is defined for all times so such an h would be complete). Such ageneralization would be true if in this particular setting – for Y = Σ × s − , Y = Σ × s + and the vector field sgrad ( sh ) on (Σ × R + , d ( sλ )) – Fathi’sTheorem 2.1 would provide a function G : Σ × R + → R , G ∈ S (cid:48) ( Y , Y ) (seeProposition 2.2), so that the flow of sgrad G on Σ × [ s − , s + ] is complete. Proof of Theorem 5.2: Pick 0 < s − < s + . Let X , X , Y , Y ⊂ Σ × [ s − , s + ]be the admissible quadruple defined for s − , s + as in (23),(24).Clearly, H | X ≥ 0, ∆( H ; X , X ) > H from Y to Y . Indeed, consider acut-off function χ : R + → [0 , 1] which equals 1 on [ s − , s + ] and 0 outside( s − − (cid:15), s + + (cid:15) ) for some 0 < (cid:15) < s − . Since h : Σ → R is compactlysupported, the Hamiltonian χ ( s ) h : Σ × R + → R is also compactly supportedand coincides with H = sh on Σ × [ s − , s + ]. In particular, χ ( s ) H | X ≥ χ ( s ); X , X ) > 0. Together with the assumption that the pair (Λ (cid:116) Λ , λ )is weakly homologically bonded, this allows to apply part B of Corollary 4.12to the symplectic manifold ( N, Ω) = (Σ × R + , d ( sλ )) and the Hamiltonian χ ( s ) h . This yields the existence of a chord of χ ( s ) h from Y to Y in N . An52asy topological argument then yields that there exists a chord of χ ( s ) h from Y to Y in N that lies in Σ × [ s − , s + ] and therefore is a chord H . This provesthe claim.The existence of the chord of H from Y to Y , together with (27), impliesthat (cid:0) ϕ − t (cid:1) ∗ λ ( ϕ t ( y )) /λ ( ϕ t ( y )) ≥ s + /s − . Since s + /s − can be made arbitrarily large, we get thatinf t ∈ (0 , + ∞ ) ,y ∈ Σ (cid:0) ϕ − t (cid:1) ∗ λ ( ϕ t ( y )) λ ( ϕ t ( y )) = + ∞ , which finishes the proof of the theorem. h from Λ to Λ Theorem 5.4 (Cf. Rem. 1.14 in [27]) . Assume < inf Σ × S h ≤ sup Σ × S h < + ∞ , and let s + > sup Σ × S h inf Σ × S h ≥ . Then the following claims hold:A. Assume that h is time-independent and l min,s + (Λ , Λ , λ ) =: l min,s + < + ∞ .Then there exists a chord of h from Λ to Λ of time-length bounded fromabove by ( s + − l min,s + s + inf Σ h − sup Σ h .B. Assume that (cid:98) l min,s + (Λ , Λ , λ ) =: (cid:98) l min,s + < + ∞ . Let ∆ s + := s + inf Σ × S h − sup Σ × S h. Assume also that for some < e < / Σ × S | ∂h/∂t | < (1 − e ) e ∆ s + inf Σ × S h ( s + − s + max Λ × S h + e ∆ s + ) (cid:98) l min,s + . (28)53 hen there exists a chord of h from Λ to Λ of time-length bounded fromabove by ( s + − (cid:98) l min,s + (1 − e )∆ s + . Proof of Theorem 1.6: The theorem immediately follows from Proposi-tion 4.7 and Theorem 5.4.A. Proof of Theorem 5.4: Pick s − := 1 < s + . Let X , X , Y , Y ⊂ Σ × [1 , s + ]be the admissible quadruple defined for s − = 1 , s + as in (23),(24). Clearly,∆( H ; Y , Y ) := ∆ s + = s + inf Σ h − sup Σ h. Since, by the hypothesis of the theorem,0 < inf Σ × S h ≤ sup Σ × S h < + ∞ , we get that ∆( H ; Y , Y ) = ∆ s + > s + > sup Σ × S h inf Σ × S h .Let us now prove part A of the theorem. Its hypothesis allows to applypart A of Corollary 4.12 to ( N, Ω) = (Σ × R + , d ( sλ )) and the Hamiltonian H on it as long as s + > sup Σ × S h inf Σ × S h . This yields the existence of a Hamiltonianchord of H from X to X of time-length bounded from above by( s + − l min,s + (Λ , Λ , λ ) s + inf Σ h − sup Σ h . The projection of this Hamiltonian chord to Σ is a chord of h from Λ to Λ of the same time-length. This finishes the proof of part A the theorem.Let us prove part B of the theorem. Similarly to the setting of Corol-lary 4.13, for a given 0 < e < / E := e ∆ s + ,T := ( s + − (cid:98) l min,s (1 − e )∆ s + ,c min := min X × S H = min Λ × S h.c max := max X × S H = s + max Λ × S h. S := { c min − E ≤ H = sh ≤ c max + E } . We wouldlike to apply Corollary 4.13 and in order to this we need to verify that theupper bound on the restriction of the function | ∂H/∂t | to S , required inCorollary 4.13, does hold in our case.Note that on S s ≤ c max + E inf Σ × S h . Together with the upper bound on the restriction of the function | ∂H/∂t | = s | ∂h/∂t | to S in the hypothesis of part B of the theorem, this yields thefollowing upper bound on the function | ∂H/∂t | = s | ∂h/∂t | on the set S :sup S s | ∂h/∂t | ≤ c max + E inf Σ × S h · sup Σ × S | ∂h/∂t | << c max + E inf Σ × S h · (1 − e ) e ∆ s + inf Σ × S h ( s + − s + max Λ × S h + e ∆ s + ) (cid:98) l min,s + == c max + E inf Σ × S h · (1 − e ) e ∆ s + inf Σ × S h ( s + − c max + E ) (cid:98) l min,s + == (1 − e ) e ∆ s + ( s + − (cid:98) l min,s + = ET , yielding the bound required in Corollary 4.13. Thus, Corollary 4.13 can beapplied to ( N, Ω) = (Σ × R + , d ( sλ )) and the Hamiltonian H on it (since, byour assumptions, (cid:98) l min,s + < + ∞ and ∆ s + > h from Λ to Λ of time-length ≤ T = ( s + − (cid:98) l min,s (1 − e )∆ s + .This finishes the proof of part B of the theorem. Corollary 5.5. Assume that h is time-independent and inf Σ h > .If h is C -cooperative with Λ , Λ for some C > inf Σ h (see Definition 1.7)and the pair (Λ (cid:116) Λ , λ ) is weakly homologically bonded, then there exists achord of h from Λ to Λ of time-length ≤ inf s>C/ inf Σ h ( s − l min,s (Λ , Λ , λ ) s inf Σ h − C .Furthermore, if h is cooperative with Λ , Λ and the pair (Λ (cid:116) Λ , λ ) ishomologically bonded, the time-length of the chord can be also bounded fromabove by µ := l min, ∞ (Λ , Λ , λ ) / inf Σ h . In particular, the pair (Λ , Λ ) is µ -interlinked. roof of Corollary 5.5: Let us assume that condition (a) from Defini-tion 1.7 of C -cooperativeness is satisfied – that is, h < C on Λ and eitherthe set { h ≥ C } is empty or dh ( R ) ≥ { h ≥ C } (the case of condition(b) from the same definition is similar).Consider a smooth increasing function χ : R + → R such that χ ( s ) = s for s ∈ [0 , C ] and χ ( s ) = C + (cid:15) for some (cid:15) > s ∈ R + . Consider the time-independent contact Hamiltonian (cid:101) h := χ ◦ h . Onereadily sees that it is complete and satisfies 0 < inf Σ (cid:101) h ≤ sup Σ (cid:101) h ≤ C + (cid:15) < + ∞ . Since the pair (Λ (cid:116) Λ , λ ) is weakly homologically bonded, part A ofTheorem 5.4, applied to (cid:101) h , shows that there exists a chord γ : [0 , T ] → Σ of (cid:101) h such that γ (0) ∈ Λ and γ ( T ) ∈ Λ for T ≤ T := inf s ( s − l min,s (Λ , Λ , λ ) s inf Σ (cid:101) h − sup Σ (cid:101) h ,where the infimum is taken over all s > sup Σ (cid:101) h inf Σ (cid:101) h .We claim that γ ([0 , T ]) lies in the set { (cid:101) h ≥ C } . Indeed, for all td ( (cid:101) h ◦ γ ) /dt = d (cid:101) h ( R ) · (cid:101) h = ( χ (cid:48) ◦ h ) · dh ( R ) · (cid:101) h. Therefore, if (cid:101) h ( γ ( t )) >C for some t ∈ [0 , T ], then (cid:101) h ( γ ( t )) ≥ C for all t ∈ [ t , T ], in contradiction to (cid:101) h ( γ ( T )) = h ( γ ( T )) < C (the latter holdssince γ ( T ) ∈ Λ and h < C on Λ ). Thus, γ ([0 , T ]) lies in { (cid:101) h ≤ C } where (cid:101) h coincides with h , meaning that γ is, in fact, the chord of h of time-length T bounded from above by T . Since inf Σ h = inf Σ (cid:101) h , sup Σ (cid:101) h ≤ C + (cid:15) and (cid:15) canbe taken arbitrarily small, we get that T ≤ inf s>C/ inf Σ h ( s − l min,s (Λ , Λ , λ ) s inf Σ h − C ,yielding the required upper bound on the time-length of the chord.If the pair (Λ (cid:116) Λ , λ ) is homologically bonded, then we can replace in thebound above l min,s (Λ , Λ , λ ) by l min, ∞ (Λ , Λ , λ ), remove the infimum andlet s go to + ∞ . This shows that the time-length of the chord is boundedfrom above by l min, ∞ (Λ , Λ , λ ) / inf Σ h .This finishes the proof of the corollary. Corollary 5.6. Assume that inf Σ × S h > , h is cooperative with Λ , Λ and the pair (Λ (cid:116) Λ , λ ) is stably homologically bonded. Denote (cid:98) l min, ∞ := (cid:98) l min, ∞ (Λ , Λ , λ ) > . Assume also that for some < e < / Σ × S | ∂h/∂t | < (1 − e ) e (cid:0) inf Σ × S h (cid:1) (cid:0) max Λ × S h + e inf Σ × S h (cid:1)(cid:98) l min, ∞ . hen there exists a chord of h from Λ to Λ of time-length bounded fromabove by (cid:98) l min, ∞ (1 − e ) inf Σ × S h . Proof of Corollary 5.6: Let us assume that h is C -cooperative with Λ , Λ for some C > C -cooperativenessis satisfied (the case of condition (b) from the same definition is similar).Without loss of generality, assume C > inf Σ × S h .Similarly to the proof of Corollary 5.5, consider a smooth increasing func-tion χ : R + → R such that χ ( s ) = s for s ∈ [0 , C ] and χ ( s ) = C + (cid:15) for some (cid:15) > s ∈ R + . Consider the time-dependent contactHamiltonian (cid:101) h := χ ◦ h . One readily sees that it is complete and satisfies0 < inf Σ × S h = inf Σ × S (cid:101) h ≤ sup Σ × S (cid:101) h ≤ C + (cid:15) < + ∞ . Note that for any sufficiently large s + > s + := s + inf Σ × S (cid:101) h − sup Σ × S (cid:101) h > . Also note thatlim s + → + ∞ (1 − e ) e ∆ s + inf Σ × S h ( s + − s + max Λ × S h + e ∆ s + ) = (1 − e ) e (cid:0) inf Σ × S h (cid:1) max Λ × S h + e inf Σ × S h and for all s + > (cid:98) l min,s + (Λ , Λ , λ ) ≤ (cid:98) l min, ∞ (Λ , Λ , λ ) . Therefore the upper bound on sup Σ × S | ∂h/∂t | in the hypothesis of the corol-lary implies that for any sufficiently large s + we can bound sup Σ × S | ∂ (cid:101) h/∂t | from above as in (28):sup Σ × S | ∂ (cid:101) h/∂t | < (1 − e ) e ∆ s + inf Σ × S h ( s + − s + max Λ × S h + e ∆ s + ) (cid:98) l min,s + . Since the pair (Λ (cid:116) Λ , λ ) is stably homologically bonded, part B of Theo-rem 5.4, applied to (cid:101) h , shows that there exists a chord γ s + : [ t s + , t s + + T s + ] → Σof (cid:101) h , for some t s + ∈ R , so that γ s + ( t s + ) ∈ Λ and γ s + ( t s + + T s + ) ∈ Λ for57 s + ≤ ( s + − (cid:98) l min,s + (1 − e )∆ s + . Similarly to the proof of Corollary 5.5, we get that γ s + is in fact a chord of h .Note thatlim s + → + ∞ ( s + − (cid:98) l min,s + (1 − e )∆ s + ≤ lim s + → + ∞ ( s + − (cid:98) l min, ∞ (1 − e )∆ s + = (cid:98) l min, ∞ (1 − e ) inf Σ × S h . Also note that since h is time-periodic with period 1, we can assume that t s + ∈ [0 , 1] for all s + . Now, since Λ is compact, a standard compactnessargument allows to obtain the existence of a chord of h from X to X oftime-length ≤ ( s + − (cid:98) l min, ∞ (1 − e )∆ s + .The following corollary yields the existence of a chord of h in case where h is not necessarily everywhere positive. Corollary 5.7. Assume there exists a (possibly non-compact or disconnected)closed codimension 0 submanifold Ξ ⊂ Σ with a (possibly non-compact or dis-connected) boundary ∂ Ξ , so that(1) inf Ξ × S h > (but h may be negative outside Σ + × S ).(2) sup ∂ Ξ × S h < + ∞ .(3) For each t ∈ S the contact Hamiltonian vector field v t of h is transverseto ∂ Ξ (in particular, ∂ Ξ is a convex surface in the sense of contact topology– see [35]) and either points inside Ξ everywhere on ∂ Ξ or points outside Ξ everywhere on ∂ Ξ .(4) Both Λ and Λ lie in Ξ .Then the following claims hold:(I) Assume the pair (Λ (cid:116) Λ , λ ) is weakly homologically bonded. Assume alsothat h is time-independent and C -cooperative with Λ , Λ for C > inf Ξ h .Then there exists a chord of h from Λ to Λ of time-length bounded fromabove by inf s>C/ inf Ξ h ( s − l min,s (Λ , Λ , λ ) s inf Ξ h − C . f the pair (Λ (cid:116) Λ , λ ) is homologically bonded, then the time-length ofthe chord can be bounded from above by l min, ∞ (Λ , Λ , λ )inf Ξ h .(II) Assume that the pair (Λ (cid:116) Λ , λ ) is stably homologically bonded andset (cid:98) l min, ∞ := (cid:98) l min, ∞ (Λ , Λ , λ ) . Assume also that h is time-dependent andcooperative with Λ , Λ and for some < e < / Ξ × S | ∂h/∂t | < (1 − e ) e (cid:0) inf Ξ × S h (cid:1) (cid:0) max Λ × S h + e inf Ξ × S h (cid:1)(cid:98) l min, ∞ . Then there exists a chord of h from Λ to Λ whose time-length is boundedfrom above by (cid:98) l min, ∞ (1 − e ) inf Ξ × S h . Proof of Corollary 5.7: Let assume that h is C -cooperative with Λ , Λ for C ≥ sup ∂ Ξ × S h (this is possible since, by (2), sup ∂ Ξ × S h < + ∞ ).For any sufficiently small (cid:15) > (cid:101) h (cid:15) : Σ × S → R so that (cid:101) h = h on a neighborhood of Ξ and inf Ξ × S h − (cid:15) ≤ (cid:101) h (cid:15) ≤ sup Ξ × S h on (Σ \ Ξ) × S . Since C ≥ sup ∂ Ξ × S h , the contact Hamiltonian (cid:101) h (cid:15) satisfies the assumptions of Corollary 5.5 (in case (I)), or of Corollary 5.6(in case (II)). Consequently, by these corollaries, there exists a chord γ (cid:15) ( t ) of (cid:101) h (cid:15) , γ (cid:15) ( t (cid:15) ) ∈ Λ , γ (cid:15) ( t (cid:15) + T (cid:15) ) ∈ Λ , so that- In case (I): T (cid:15) ≤ inf s ( s − l min,s (Λ , Λ , λ ) s inf Σ (cid:101) h (cid:15) − C ≤ inf s ( s − l min,s (Λ , Λ , λ ) s (inf Ξ h − (cid:15) ) − C (the infimum is taken over all s > C/ inf Σ (cid:101) h (cid:15) ), and T (cid:15) ≤ l min, ∞ (Λ , Λ , λ )inf Σ (cid:101) h (cid:15) ≤ l min, ∞ (Λ , Λ , λ )inf Ξ h − (cid:15) . - In case (II): T (cid:15) ≤ (cid:98) l min, ∞ (1 − e ) inf Σ × S (cid:101) h (cid:15) ≤ (cid:98) l min, ∞ (1 − e ) inf Ξ × S h − (cid:15) . Σ (cid:101) h (cid:15) ≥ inf Ξ h − (cid:15) for all (sufficientlysmall) (cid:15) > γ (cid:15) cannot cross ∂ Ξ. Indeed, by (3), if it had crossed ∂ Ξ, itwould have had to cross it transversally from Ξ to M \ Ξ. This would meanthat v t (the contact Hamiltonian vector field of (cid:101) h (cid:15) | Ξ = h | Ξ ) points outside Ξeverywhere on ∂ Ξ for all t ∈ S . But then the chord would not have beenable to return to Ξ to reach Λ . Thus the chord lies in the interior of Ξ andis, in fact, a chord of h from Λ to Λ .We have such a chord γ (cid:15) of h from Λ to Λ for any sufficiently small (cid:15) > (cid:15) . We canalso assume that for all (cid:15) > t s (cid:15) = 0 in case (I) (since h is time-independent)and t s (cid:15) ∈ [0 , 1] in case (II) (since h is 1-periodic in time). Now, since Λ iscompact, a standard compactness argument allows to obtain the existence ofthe chord of h from X to X of time-length T , so that- In case (I): T ≤ inf s>C/ inf Ξ h ( s − l min,s (Λ , Λ , λ ) s inf Ξ h − C and if the pair (Λ (cid:116) Λ , λ ) is homologically bonded, then T ≤ l min, ∞ (Λ , Λ , λ )inf Ξ h . - In case (II) T ≤ (cid:98) l min, ∞ (1 − e ) inf Ξ × S h . This finishes the proof of the corollary. Remark 5.8. Assume h is time-independent and Ξ := { h ≥ c } , ∂ Ξ = { h = c } for some c > 0. Then the conditions (1) and (2) are satisfied automatically,while condition (3) is equivalent to ∂ Ξ being transverse to the Reeb vectorfield, because dh ( v ) = hdh ( R ). 60 emark 5.9. The claim of Theorem 5.4 is robust – for a fixed h – withrespect to perturbations of Λ = Λ (cid:116) Λ by Legendrian isotopies, as longas the perturbation is sufficiently C ∞ -small, depending on s + – this followsfrom Remark 4.9.Accordingly, the claims of Corollaries 5.5, 5.6 and 5.7 are robust – fora fixed h – with respect to perturbations of Λ = Λ (cid:116) Λ by Legendrianisotopies, as long as the perturbation is sufficiently C ∞ -small, depending on C inf Σ × S h , where C is the constant such that h is C -cooperative with Λ , Λ .Namely, if the pair (Λ (cid:116) Λ , λ ) is weakly homologically bonded, then l min,s (Λ , Λ , λ ) < + ∞ for any s > C inf Σ × S h . Fix such an s . Then, byRemark 4.9, l min,s (Λ (cid:48) , Λ (cid:48) , λ ) is finite and close to l min,s (Λ , Λ , λ ) for anyΛ (cid:48) = Λ (cid:48) (cid:116) Λ (cid:48) obtained from Λ by a Legendrian isotopy, as long as the isotopyis small (depending on the chosen s ). The proofs of Corollaries 5.5, 5.6 and5.7 then go through for Λ (cid:48) instead of Λ and yield the existence of a chord of h between Λ (cid:48) and Λ (cid:48) . The cases when the pair (Λ (cid:116) Λ , λ ) is homologicallybonded or stably homologically bonded are similar. J Q In this section let Σ = J Q = T ∗ Q × R ( z ) be the 1-jet space of a smoothmanifold Q , together with the standard contact form λ on it. The Reeb flowof λ is the shift in the z -coordinate. This is a nice contact manifold. Let Λ be the zero section. Proposition 6.1. Assume that Λ ⊂ Σ = J Q is a Legendrian submanifoldwith following property: there is unique Reeb chord starting on Λ and endingon Λ , and this chord is non-degenerate in the sense of (7) . Then the pair (Λ (cid:116) Λ , λ ) is homologically bonded.Proof. Denote by a the unique chord connecting Λ and Λ . All non-vanishing01-composable monomials w from the algebra A (Λ , Λ , λ ) have the form ab ab · · · ab k a, (29)where b j are some Reeb chords starting and ending on Λ (cid:116) Λ . Their actionis at least as large as the one of a . Since the LCH-differential decreasesthe action, it follows that da = 0. Applying the Leibniz rule, we see that61he differential of a monomial of the form (29) is a linear combination ofmonomials of the same form. It follows that a (cid:54) = db for any b , and henceit defines an infinite ray in the LCH-persistence module V (Λ , Λ , λ ). Thisyields the statement of the proposition.Assume now that l > is the image of Λ under the time- l Reebflow. Proposition 6.2. The pair (Λ (cid:116) Λ , λ ) is homologically bonded and l min,s (Λ , Λ , λ ) ≤ l for all s ∈ (1 , + ∞ ] . Proof of Proposition 6.2: The pair (Λ (cid:116) Λ , λ ) is degenerate and we haveto perturb, say, Λ to make it non-degenerate.Namely, consider a Morse function f : Q → R which is a C ∞ -small per-turbation of the constant function z + . Its 1-jet is a Legendrian submanifoldΛ (cid:48) ⊂ J Q which is a small perturbation of Λ . It is not hard to check thatthe pair (Λ (cid:116) Λ (cid:48) , λ ) is non-degenerate: the Reeb chords of Λ = Λ (cid:116) Λ (cid:48) are then the Reeb chords from Λ to Λ (cid:48) corresponding to the critical pointsof f and there actions are the critical values of f shifted down by z − . The01-subspace is then the Z -span of the Reeb chords of Λ.One can show that, under the identification between the Reeb chords ofΛ and the critical points of f , the differential in the Chekanov-Eliashbergalgebra is identified with the Morse differential in the Morse chain complexof f (over Z ). More precisely, the differential in the Chekanov-Eliashbergalgebra is identified with the differential in the Lagrangian Floer complexof the projections of Λ and Λ (cid:48) to T ∗ Q (that are embedded Lagrangiansubmanifolds) and the latter is identified with the Morse differential by theoriginal work of Floer [32] – see e.g. the comment preceding Cor. 1.10 in [8].Thus, the persistence module associated to (Λ , Λ (cid:48) , λ ) is the Morse homol-ogy persistence module associated to f . The corresponding barcode containsinfinite bars – their number is equal to the sum of the Betti numbers of Q over Z . Hence, l min,s (Λ , Λ (cid:48) , λ ) ≤ max Q for all s ∈ (1 , + ∞ ]. Letting f convergeuniformly to the constant function l we readily get that l min,s (Λ , Λ , λ ) ≤ l for all s ∈ (1 , + ∞ ], meaning that the pair (Λ (cid:116) Λ , λ ) is homologicallybonded. 62 roof of Theorem 1.5: The Legendrian submanifolds appearing inthe formulation of the theorem are homologically bonded. For item (i) ofthe theorem, this follows from Proposition 6.2 combined with the fact thatthe property of being homologically bonded is invariant under a Legendrianisotopy of a pair. For item (ii) this follows from Proposition 6.1. Therefore,the pair (Λ , Λ ) is interlinked by Corollary 5.5. Proposition 6.3. The pair (Λ (cid:116) Λ , λ ) is stably homologically bonded and (cid:98) l min, ∞ (Λ , Λ , λ ) ≤ l . Proof of Proposition 6.3: There is a natural identification (cid:98) Σ = J Q × T ∗ S = J ( Q × S ) identifying the contact forms, and hence the contactstructures. The Legendrian submanifold (cid:98) Λ is then the zero-section of J ( Q × S ) and (cid:98) Λ is its image under the time- l Reeb flow. Now the result followsfrom Proposition 6.2. ST ∗ R n In this section let Σ := ST ∗ R n , n > . Denote by ξ the standard contact structure on Σ defined by the contact form λ := pdq. Let x , x ∈ R n ( q ), x (cid:54) = x , and letΛ i := { q = x i , | p | = 1 } , i = 0 , , be the Legendrian submanifolds of (Σ , ξ ) defined in Theorem 1.8. SetΛ := Λ (cid:116) Λ . One easily checks that the pair (Λ = Λ (cid:116) Λ , λ ) is non-degenerate.Consider also the manifold (cid:98) Σ := Σ × T ∗ S ( r, τ ) , (cid:98) λ := pdq − rdτ, where r ∈ R , τ ∈ S , are the standard coordinates on T ∗ S = R × S . Let S := { r = 0 } ⊂ T ∗ S be the zero-section. One easily sees that (cid:98) λ is a contactform, defining a contact structure ˆ ξ on (cid:98) Σ and the Reeb vector field of (cid:98) λ canbe described as follows: its projection to the ST ∗ R n factor is the Reeb vectorfield of λ , while its projection to the T ∗ S factor is zero. Denote (cid:98) Λ := Λ × S , (cid:98) Λ := Λ × S . (cid:98) Λ := (cid:98) Λ (cid:116) (cid:98) Λ = Λ × S . For each δ > (cid:98) Λ ,δ := Λ × graph ( δdf ) ⊂ (cid:98) Σ , (cid:98) Λ δ := (cid:98) Λ (cid:116) (cid:98) Λ ,δ , where graph ( δdf ) ⊂ T ∗ S is the graph of d ( δf ) for a Morse function f : S → R that has only two critical points. The set (cid:98) Λ δ is a two-part Legendriansubmanifold of (Σ , ker (cid:98) λ ). Since the pair (Λ , λ ) is non-degenerate, so is thepair ( (cid:98) Λ δ , (cid:98) λ ). Proposition 7.1. The barcode of the persistence module V (Λ , Λ , λ ) con-sists of infinitely many infinite bars, of multiplicity , whose left ends are (2 k − | x − x | , k ∈ Z > .Consequently, the pair (Λ (cid:116) Λ , λ ) is homologically bonded and l min,s (Λ , Λ , λ )= l min, ∞ (Λ , Λ , λ ) = | x − x | for all s ∈ (1 , + ∞ ] . Proof of Proposition 7.1: The set R (Λ , λ ) consists only of two Reebchords: a 01-chord a and a 10-chord b , which are the lifts of the Euclideangeodesics in R n going from x to x and from x to x . Thus, A (Λ , Λ , λ )is spanned over Z by the monomials of the form abab · . . . · ba , where eachmonomial contains k factors a and k − b , for k ∈ Z > . For each k ∈ Z > there is exactly one such monomial of length 2 k − 1. The actions of a and b are both equal to | x − x | and therefore the action of the monomialof length 2 k − k − | x − x | . 64et J ∈ J (Λ). Since the differential ∂ J on A (Λ , Λ , λ ) lowers the ac-tion, we immediately get that ∂ J ( a ) = ∂ J ( b ) = 0 and therefore ∂ J is identi-cally zero on A (Λ , Λ , λ ). Therefore the barcode of the persistence module V (Λ , Λ , λ ) consists of infinitely many infinite bars of multiplicity 1 whoseleft ends are (2 k − | x − x | , k ∈ Z > . This finishes the proof. Proposition 7.2. The barcode of the persistence module V ( (cid:98) Λ , (cid:98) Λ ,δ , (cid:98) λ ) con-sists of infinitely many infinite bars whose left ends are (2 k − | x − x | , k ∈ Z > . The multiplicity of the bar with the left end (2 k − | x − x | is k − .Consequently, the pair (Λ (cid:116) Λ , λ ) is stably homologically bonded and (cid:98) l min,s (Λ , Λ , λ ) = = (cid:98) l min, ∞ (Λ , Λ , λ ) = | x − x | for all s ∈ (1 , + ∞ ] . Proof of Proposition 7.2: Let a, b ∈ R (Λ , λ ) be the 01-chord and the10-chord of Λ as above. The Reeb chords of (cid:98) Λ δ are the direct productsof the Reeb chords of R (Λ , λ ) lying in Σ and the constant paths in T ∗ S corresponding to the 2 critical points of f (that is, the intersections of thegraph of df with the zero-section). Thus, R ( (cid:98) Λ δ , (cid:98) λ ) consists only of exactly 4Reeb chords: 01-chords a , a and 10-chords b , b , where a , a project onto a and b , b project onto b under the projection (cid:98) Σ = Σ × T ∗ S → Σ.Thus, A ( (cid:98) Λ , (cid:98) Λ ,δ , λ ) is spanned over Z by the monomials of the form a j b j · . . . · b j k − a j k − , where each monomial contains k factors a j i , j i = 1 , k − b j i , j i = 1 , 2, for k ∈ Z > . There are 2 k − such monomialsof length 2 k − 1. The actions of all a j and b j , j = 1 , 2, are equal to | x − x | and therefore the action of a monomial as above is (2 k − | x − x | .Let J ∈ J ( (cid:98) Λ δ ). Since the differential ∂ J on A ( (cid:98) Λ , (cid:98) Λ ,δ , (cid:98) λ ) lowers theaction, we immediately get that ∂ J ( a j ) = ∂ J ( b j ) = 0 for all j = 1 , ∂ J is identically zero on A ( (cid:98) Λ , (cid:98) Λ δ , (cid:98) λ ). Therefore the barcode ofthe persistence module V ( (cid:98) Λ , (cid:98) Λ ,δ (cid:98) λ ) consists of infinitely many infinite barswhose left ends are (2 k − | x − x | , k ∈ Z > . The multiplicity of the barwith the left end (2 k − | x − x | is 2 k − .Consequently, for any s ∈ (1 , + ∞ ] and δ > l min,s ( (cid:98) Λ , (cid:98) Λ ,δ , (cid:98) λ ) = | x − x | (cid:98) l min,s (Λ , Λ , λ ) = lim inf δ → l min,s ( (cid:98) Λ , (cid:98) Λ ,δ , (cid:98) λ ) = | x − x | . This finishes the proof. Proof of Theorem 1.1: In the case n = 1 the claim follows from theresults in [27]. Namely, in this case the sets X , X , Y , Y form a Lagrangiantetragon in ( R ( p, q ) , dp ∧ dq ) built from the point x ∈ R for T = x − x (see[27, Sec. 5.1]). In the terminology of [27], this Lagrangian tetragon is stably κ -interlinked, for κ = | x − x | ( s + − s − ) – this follows e.g. from Cor. 5.3and Thm. 5.8 in [27] (see [7, Thm. 1.20, Prop.1.21] for a different approachto the proof). By the definition of a κ -interlinked Lagrangian tetragon (see[27, Sec. 1.2]), this yields the dynamical claims of Theorem 1.1 in the case n = 1.Assume now that n > l min,s + /s − (Λ , Λ , λ ) = (cid:98) l min,s + /s − (Λ , Λ , λ ) = | x − x | . The symplectization Σ × R + ( s ) is identified symplectically with R n ( p, q ) \{ p = 0 } by the map ( p, q, s ) (cid:55)→ ( sp, q ). Thus, (Σ × [ s − , s + ] , d ( sλ )) can beviewed as a codimension-zero submanifold with boundary of ( N = R n , Ω = dp ∧ dq ).Now the claims of the theorem follow from Corollaries 4.12, 4.13 appliedto the Hamiltonian H on ( N, Ω).This finishes the proof of the theorem. Proofs of Theorems 1.12, 1.8 and Corollary 1.9: We need to proveTheorems 1.12, 1.8 and Corollary 1.9 in the following three cases:(I) Λ is the zero-section of J Q and Λ is its image under the time- l Reebflow.(II) Λ is a cotangent unit sphere in ST ∗ R n and Λ is its image under thetime- l Reeb flow.(III) Λ , Λ ⊂ ST ∗ R n are the cotangent unit spheres at x , x ∈ R n , | x − x | = l . 66he needed results in cases (I), (II), (III) follow from the correspondinggeneral results in Theorem 5.2, Corollaries 5.5, 5.6 and Corollary 5.7, as soonas we check that in all the three cases l min, ∞ (Λ , Λ , λ ) ≥ l , (cid:98) l min, ∞ (Λ , Λ , λ ) ≥ l . 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