A local contact systolic inequality in dimension three
aa r X i v : . [ m a t h . S G ] F e b A local contact systolic inequality in dimension three
Gabriele Benedetti and Jungsoo KangFebruary 7, 2019
Abstract
Let α be a contact form on a connected closed three-manifold Σ. The systolic ratio of α is defined as ρ sys ( α ) := α ) T min ( α ) , where T min ( α ) and Vol( α ) denote the minimalperiod of periodic Reeb orbits and the contact volume. The form α is said to be Zoll if itsReeb flow generates a free S -action on Σ. We prove that the set of Zoll contact forms onΣ locally maximises the systolic ratio in the C -topology. More precisely, we show thatevery Zoll form α ∗ admits a C -neighbourhood U in the space of contact forms such that,for every α ∈ U , there holds ρ sys ( α ) ≤ ρ sys ( α ∗ ) with equality if and only if α is Zoll. Contents C -close to the identity . . . . . . . . . . . . . . . . . 314.3 Generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Quasi-autonomous diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 38 Let Σ be a connected closed manifold of dimension 2 n + 1 and let C (Σ) be the set of contactforms on it. Namely, the elements α ∈ C (Σ) are one-forms on Σ such that the (2 n + 1)-form α ∧ (d α ) n is nowhere vanishing. This property implies that there exists a unique vector field R α on Σ determined by the relations d α ( R α , · ) = 0 and α ( R α ) = 1. The vector field R α is called the Reeb vector field and the associated flow Φ α the Reeb flow. Periodic orbits ofΦ α are fundamental objects in contact and symplectic geometry. The Weinstein conjecture,which asserts that every contact form on a closed manifold possesses at least one periodic1rbit [Wei79], has played a prominent role in the field. The conjecture has been establishedin many particular situations, most notably when Σ is three-dimensional [Tau07]. In thesecases a more refined question arises: What can be said about the period of the orbits thatone finds? A natural problem is, namely, to give an explicit upper bound on T min ( α ), theminimal period of periodic orbits of Φ α , in terms of some geometric quantity associated with α . Following [ ´APB14], we use here the contact volume (other choices are possible and canlead to different results, as in [AFM17])Vol( α ) := Z Σ α ∧ (d α ) n > , (1.1)and consider the systolic ratio ρ sys : C (Σ) → (0 , ∞ ] , ρ sys ( α ) := T min ( α ) n +1 Vol( α ) . Inside C (Σ) one can consider the subset C ( ξ ) of contact forms defining a given co-orientedcontact structure ξ on Σ. This means that ξ is a co-oriented hyperplane field such thatker α = ξ for all α ∈ C ( ξ ). Following the breakthrough result in dimension three obtained byAbbondandolo, Bramham, Hryniewicz and Salom˜ao in [ABHS17a], Sa˘glam showed thatsup α ∈C ( ξ ) ρ sys ( α ) = + ∞ , i.e. the systolic ratio does not admit a global upper bound on C ( ξ ), for any contact contactstructure ξ in any dimension [Sa˘g18].Such bound might hold, however, if one takes a special subclass of contact forms in C ( ξ ).For instance, a celebrated theorem of Viterbo [Vit00, Theorem 5.1] (see also [AAMO08])asserts that the systolic ratio is bounded from above on the set of contact forms on S n +1 arising from convex embeddings into R n +1) . Another distinguished subclass is given by thecanonical contact forms on the unit tangent bundle of closed Riemannian or Finsler manifolds.This is the setting where systolic geometry originated and has been hitherto tremendouslystudied (see [Ber03, Chapter 7.2]).In a similar vein, for a general Σ, one is led to study the local behaviour of ρ sys around itscritical set. This direction of inquiry was initiated in [ ´APB14] by ´Alvarez-Paiva and Balacheff,who showed that Crit ρ sys is exactly the set of Zoll contact forms. Definition 1.1.
A contact form α on a manifold Σ is called Zoll of period T ( α ) >
0, if theflow Φ α induces a free R /T ( α ) Z -action (all orbits are periodic and have prime period T ( α )).We write Z (Σ) for the set of all Zoll contact forms on Σ.Once a Zoll form α ∗ is given, it is easy to deform it through a path s α s of Zoll formswith α = α ∗ . We can just set α s := T s Ψ ∗ s α ∗ , where s Ψ s is any isotopy of Σ and s T s a path of positive numbers. By a theorem of Weinstein [Wei74], these represent all possibledeformations of α ∗ through Zoll contact forms.While the local structure of Z (Σ) is well understood, describing when Z (Σ) is non-emptyand investigating its global structure are more subtle issues. In this regard, a classical con-struction by Boothby and Wang represents a useful tool [BW58, Theorem 2 and 3]. FromDefinition 1.1, it follows that if α ∗ is Zoll of period 1, the quotient by the action of the Reebflow yields an oriented S -bundle p : Σ → M , where M is a closed manifold of dimension 2 n S = R / Z . The Zoll contact form α ∗ becomes a connection form for p , while the two-formd α ∗ descends to a symplectic form ω ∗ on M , representing minus the Euler class of p . Viceversa, given a symplectic manifold ( M, ω ∗ ) such that the cohomology class of ω ∗ is integral,one can construct an oriented S -bundle p : Σ → M with a connection form α ∗ satisfyingd α ∗ = p ∗ ω ∗ , so that α ∗ is a Zoll form of period 1 on Σ.The Boothby-Wang construction tells us exactly which connected three-manifolds admita Zoll contact form: they are total spaces of non-trivial oriented S -bundles over connectedoriented closed surfaces. In this case, an easy topological argument shows that the diffeomor-phism type of the quotient M and the Euler number of p depend only on Σ and not on α ∗ . Inparticular, minus the Euler number equals | H tor1 (Σ; Z ) | , namely the cardinality of the torsionsubgroup of the first integral homology of Σ. Hence, as shown in [ ´APB14, Proposition 3.3],we have the identity ρ sys ( α ∗ ) = 1 | H tor1 (Σ; Z ) | . The next result classifies Zoll contact forms on Σ up to diffeomorphisms and up to isotopies.When Σ is SO (3) or S , the diffeomorphism classification was carried out in [ABHS17b,Theorem B.2] and [ABHS18, Proposition 3.9]. Proposition 1.2.
Let Σ be the total space of a non-trivial orientable S -bundle over a con-nected orientable closed surface.1. If α and α ′ are Zoll contact forms on Σ , there is a diffeomorphism Ψ : Σ → Σ and apositive constant T > such that Ψ ∗ α ′ = T α.
2. The space Z (Σ) has exactly two connected components. We provide a proof of the proposition together with a detailed description of the connectedcomponents of Z (Σ) in Section 2.Actually, the results in [ ´APB14] go beyond the characterisation of Crit ρ sys and imply thatif s α s is a smooth deformation of α ∗ ∈ Z (Σ) with α = α ∗ , then s ρ sys ( α s ) attainsa strict maximum at 0, provided the deformation is not tangent in s = 0 to all orders to Z (Σ). On the other hand, by Weinstein’s theorem, if the deformation is contained in Z (Σ),then ρ sys ( α s ) = ρ sys ( α ∗ ) for all s . As communicated to us by the authors, an implicit goal in[ ´APB14] was to answer the following question on a sharp local upper bound for ρ sys . Question 1.3 (Local contact systolic inequality) . Let α ∗ be a Zoll contact form on a connectedclosed manifold Σ of dimension n + 1 and let k ≥ be an integer. Does there exist a C k -neighbourhood U of α ∗ in the set of contact forms on Σ such that ρ sys ( α ) ≤ ρ sys ( α ∗ ) , ∀ α ∈ U and the equality holds if and only if α is a Zoll form? For Zoll Riemannian metrics on a compact rank one symmetric space, an analogous ques-tion was formulated in [Bal06, ´APB14]. This Riemannian question is answered positively for S with k = 2 in [ABHS17b].In their seminal paper [ABHS18], Abbondandolo, Bramham, Hryniewicz, and Salom˜aogive a positive answer to Question 1.3 with k = 3 when Σ is the three-sphere (or more3enerally, by means of a simple covering argument, when the base M is the two-sphere).Moreover, they give a negative answer to the question in dimension three, if one replaces the C k -closeness of contact forms with the C -closeness of the Reeb flows.In the present paper, building on their beautiful result, we answer Question 1.3 affirma-tively with k = 3 for every closed three-manifold admitting a Zoll contact form (so, comparedwith [ABHS18], here the base M can be an arbitrary orientable closed surface), including astatement regarding the diastolic ratio ρ dia ( α ) := T max ( α ) n +1 Vol( α ) , where T max ( α ) is the maximal period of prime periodic orbits of Φ α . If α is Zoll, then thereholds T min ( α ) = T ( α ) = T max ( α ) so that ρ sys ( α ) = ρ dia ( α ). To get a stronger result, forevery free-homotopy class of loops h in Σ, we also define the minimal and maximal period ofprime periodic orbits of Φ α in the class h and we denote them by T min ( α, h ) and T max ( α, h ),respectively. Finally, we write ρ sys ( α, h ) and ρ dia ( α, h ) for the corresponding systolic anddiastolic ratios. Clearly, ρ sys ( α ) ≤ ρ sys ( α, h ) ≤ ρ dia ( α, h ) ≤ ρ dia ( α ). Theorem 1.4.
Let α ∗ be a Zoll contact form on a connected closed three-manifold Σ , andlet h be the free-homotopy class of the prime periodic orbits of Φ α ∗ . There exists a C -neighbourhood U of d α ∗ in the space of exact two-forms on Σ such that, for every contactform α on Σ with d α ∈ U , we have ρ sys ( α, h ) ≤ | H tor1 (Σ; Z ) | ≤ ρ dia ( α, h ) and any of the two equalities holds if and only if α is Zoll. In particular, Zoll contact formsare strict local maximisers of the systolic ratio in the C -topology. Remark 1.5.
This result can be used to prove a systolic inequality for magnetic flows onclosed oriented surfaces, as we discuss in [BK19b].
Sketch of proof of Theorem 1.4.
The strategy of the proof closely follows the one in[ABHS18]. We divide the proof of the theorem into two parts, corresponding to Section 3and 4, respectively.In the first part we start by assuming without loss of generality that all prime orbits of α ∗ have period equal to 1, the form α is C -close to α ∗ and d α is C -close to d α ∗ . Then, weshow that there exists a real number T with 1 < T < P T ( α, h ) of primeperiodic orbits γ of Φ α in the class h with period T ( γ ) ≤ T is not empty (see Proposition3.4). Moreover, given γ ∈ P T ( α, h ), we construct a global surface of section N → Σ for Φ α ,which is diffeomorphic to M with an open disc removed and such that its boundary covers | H tor1 (Σ; Z ) | -times the orbit γ (see Section 3.3). If λ is the restriction of α to N , then d λ is symplectic in the interior ˚ N and vanishes of order one at the boundary ∂N . The first-return time, a priori only defined on ˚ N , extends to a function τ : N → (0 , ∞ ), which is C -close to the constant 1. The first-return map, a priori only defined on ˚ N , extends to adiffeomorphism ϕ : N → N , which is C -close to id N . Moreover, there holds ϕ ∗ λ − λ = d σ ,where σ := τ − T ( γ ) is a C -small function, called the action of ϕ . The volume of α is relatedto the Calabi invariant CAL( ϕ ) := R N σ d λ of the map ϕ through the formulaVol( α ) = Z N τ d λ = Z N (cid:0) σ + T ( γ ) (cid:1) d λ = 2CAL( ϕ ) + | H tor1 (Σ; Z ) | T ( γ ) . q ∈ ˚ N of ϕ yields a periodic orbit γ q ∈ P T ( α, h ) with period T ( γ q ) = σ ( q ) + T ( γ ) . In particular, when α is not Zoll, ϕ = id N . The properties of the return time and thereturn map are collected in Theorem 3.13. As a consequence, in Corollary 3.14 we argue thatTheorem 1.4 is proven if we take γ to have minimal, respectively, maximal period amongorbits in P T ( α, h ), and are able to show that ϕ = id N , CAL( ϕ ) ≤ ⇒ ∃ q − ∈ ˚ N ∩ Fix ( ϕ ) , σ ( q − ) < ,ϕ = id N , CAL( ϕ ) ≥ ⇒ ∃ q + ∈ ˚ N ∩ Fix ( ϕ ) , σ ( q + ) > . (1.2)Indeed, if we take γ ∈ P T ( α, h ) with minimal period and assume ρ sys ( α, h ) ≥ | H tor1 (Σ; Z ) | , thenCAL( ϕ ) ≤
0. But, if α is not Zoll, the first implication in (1.2) yields γ q − ∈ P T ( α, h ) with T ( γ q − ) < T ( γ ). This contradiction proves ρ sys ( α, h ) < | H tor1 (Σ; Z ) | for a contact form α whichis not Zoll. The inequality for ρ dia ( α, h ) follows analogously.In the second part of the proof, we establish implications (1.2). When N is the two-disc,these implications were already shown to hold in [ABHS18, Corollary 5]. The key step there isto find a formula for the Calabi invariant in terms of the generating function of ϕ [ABHS18,Proposition 2.20]. Instead of finding such a formula for general N , we show implications(1.2) by constructing a path t ϕ t of d λ -Hamiltonian diffeomorphisms of N with ϕ = id N , ϕ = ϕ , which is generated by a quasi-autonomous Hamiltonian H : N × [0 , → R (seeProposition 4.15). We recall from [BP94] that a function H is quasi-autonomous if thereexist q min , q max ∈ N such thatmin q ∈ N H ( q, t ) = H ( q min , t ) , max q ∈ N H ( q, t ) = H ( q max , t ) , ∀ t ∈ [0 , . In particular, q min and q max are fixed points of ϕ , if they lie in ˚ N . In order to exhibit such apath, we construct a Weinstein neighbourhood of the diagonal in (cid:0) N × N, ( − d λ ) ⊕ d λ (cid:1) (seeProposition 4.1). This yields a generating function G : N → R for ϕ . Let [0 , ǫ ) × S ⊂ N be acollar neighbourhood of the boundary with radial coordinate R . At this point, crucially usingthat d λ vanishes at ∂N of order one in the radial direction, we can show (see Proposition4.10) that the generating function belongs to G := n G : N → R (cid:12)(cid:12)(cid:12) G = 0 on ∂N, G is C -small on N, R d G is C -small on [0 , ǫ ) × S o . Conversely, every G ∈ G is the generating function of some diffeomorphism ϕ G : N → N ,which is C -close to the identity (see Proposition 4.12). Therefore, since the set G is star-shaped around the zero function, the Hamilton-Jacobi equation (see (4.32)) tells us that, forevery ϕ = ϕ G , the Hamiltonian function H : N × [0 , → R associated with the path t ϕ tG , t ∈ [0 , ϕ and the action of q min (andsimilarly of q max ), provided it lies in ˚ N , asCAL( ϕ ) = Z N × [0 , H d λ ∧ d t, σ ( q min ) = Z H ( q min , t )d t. (1.3)This finishes the second part of the proof and the whole sketch.5 emark 1.6. Relations (1.3) suggest that one could interpret implications (1.2) as a localsystolic (resp. diastolic) inequality for quasi-autonomous Hamiltonian diffeomorphisms. Suchan inequality yields an upper (resp. lower) bound on the minimal (resp. maximal) action ofa contractible fixed point in terms of the Calabi invariant. The bound can be readily provenfor closed symplectic manifolds in arbitrary dimension. On the other hand, Reeb flows andHamiltonian diffeomorphisms are two special incarnations of the characteristic foliation of anodd-symplectic form (also known as a Hamiltonian structure [CM05]) on an oriented circlebundle over a closed symplectic manifold. These observations prompted us to formulate aconjectural systolic inequality for odd-symplectic forms, which we discuss in [BK19a].
Acknowledgements.
This work is part of a project in the Collaborative Research Center
TRR 191 - Symplectic Structures in Geometry, Algebra and Dynamics funded by the DFG.It was initiated when the authors worked together at the University of M¨unster and partiallycarried out while J.K. was affiliated with the Ruhr-University Bochum. We thank Peter Al-bers, Kai Zehmisch, and the University of M¨unster for having provided an inspiring academicenvironment. We are grateful to Alberto Abbondandolo for valuable discussions and sugges-tions. We are indebted to the anonymous referee for the careful reading of the manuscriptand for helpful comments on its first draft. G.B. would like to express his gratitude to Hans-Bert Rademacher and the whole Differential Geometry group at the University of Leipzig.G.B. was supported by the National Science Foundation under Grant No. DMS-1440140 whilein residence at the Mathematical Sciences Research Institute in Berkeley, California, duringthe Fall 2018 semester. J.K. is supported by Samsung Science and Technology Foundation(SSTF-BA1801-01).
This section is devoted to establish Proposition 1.2. For a clear exposition, we divide the proofinto two lemmas. In the first one, we show that all Zoll contact forms on Σ are isomorphic.This was proved in [ABHS17b, Theorem B.2] when Σ = SO (3) and in [ABHS18, Proposition3.9] when Σ = S . Lemma 2.1.
Let Σ be a connected closed three-manifold. Let α and α ′ be two Zoll contactforms on Σ with unit period. There exists a diffeomorphism Ψ : Σ → Σ such that Ψ ∗ α ′ = α. Proof.
The Reeb flows of α and α ′ yield S -actions on Σ and let p : Σ → M and p ′ : Σ → M ′ be the associated oriented S -bundles. We write e and e ′ for minus the real Euler classof p and p ′ . Let us orient M and M ′ through the forms ω and ω ′ , where d α = p ∗ ω andd α ′ = p ′∗ ω ′ . By a standard topological argument, the surfaces M and M ′ have the samegenus and h e, [ M ] i = h e ′ , [ M ′ ] i . As the Euler number h e, [ M ] i is a complete invariant forprincipal S -bundles over oriented surfaces, there exists an S -equivariant diffeomorphismΨ : Σ → Σ such that p ′ ◦ Ψ = ψ ◦ p , for some orientation-preserving diffeomorphism ψ : M → M ′ . As a result, if α := Ψ ∗ α ′ , then there exists a one-form η on M such that α = α + p ∗ η and d α = p ∗ ω , where ω := ψ ∗ ω ′ . We construct now a diffeomorphismΨ : Σ → Σ with the property Ψ ∗ α = α , so that Ψ := Ψ ◦ Ψ is the desired map. Using astability argument, we seek an isotopy Φ u : Σ → Σ generated by a vector field X u such thatΦ ∗ u α u = α, (2.1)6here α u := α + u p ∗ η , for all u ∈ [0 , := Φ . We observe that ω u := (1 − u ) ω + uω is a path of symplectic forms on M , as ψ preserves the orientation.Differentiating (2.1) with respect to u , we see that (2.1) is satisfied once X u is chosen as thevector field in ker α u with the property that d p ( X u ) = ¯ X u , where ¯ X u is the unique vectorfield on M satisfying the relation ι ¯ X u ω u = − η .Recall that Z (Σ) is the space of Zoll contact forms on Σ. Let ξ be an isotopy class ofco-oriented contact structures on Σ, and let Z ( ξ ) be the set of all Zoll forms defining someelement in ξ . We denote by − ξ the isotopy class obtained by reversing the co-orientation ofthe contact structures in ξ . Lemma 2.2.
Let Σ denote the total space of a non-trivial orientable S -bundle over a con-nected closed orientable surface.1. If Σ is either S or RP , then Z (Σ) has exactly two connected components Z ( ξ st ) and Z ( ξ st ) . Here ξ st is the isotopy class of the standard contact structure and ξ st the isotopyclass obtained from ξ st by applying an orientation-reversing diffeomorphism.2. If Σ is neither S nor RP , then Z (Σ) has exactly two connected components Z ( ξ + ) and Z ( ξ − ) . Here ξ + and ξ − are two distinct isotopy classes with ξ − = − ξ + .Proof. Let us fix a Zoll contact form α on Σ with unit period and bundle map p : Σ → M .We consider any other Zoll form α ′ with unit period on Σ and we distinguish two cases. Case 1: M = S . Here Σ is the lens space L ( p,
1) for some p ≥
1. Lemma 2.1 yields adiffeomorphism Ψ : Σ → Σ with the property α ′ = Ψ ∗ α . Suppose that Σ is either S or RP and let ¯Υ : Σ → Σ be a diffeomorphism of Σ reversing the orientation. By Cerf’s Theorem(see [Cer68], and [Bon83, Th´eor`eme 3] or [HR85, Theorem 5.6]), Ψ is either isotopic to theidentity or to ¯Υ, thus showing that α ′ is either homotopic to α or to ¯Υ ∗ α within Z (Σ).Suppose now that Σ is neither S nor RP . Then, a p -fibre is not homotopic to itself withreverse orientation. Therefore, α and − α are not homotopic in Z (Σ). Moreover, by Lemma2.1, there exists a diffeomorphism Υ − : Σ → Σ such that Υ ∗− α = − α . In particular, Υ − changes the orientation of the fibres and is not isotopic to the identity. By [Bon83, Th´eor`eme3] or [HR85, Theorem 5.6] again, the map Ψ is either isotopic to the identity or to Υ − . Hence, α ′ is either homotopic to α or to − α within Z (Σ). Case 2: M = S . The long exact sequence of homotopy groups shows that a p -fibre is nothomotopic to itself. Therefore, α and − α are not homotopic within Z (Σ). Moreover, [Wal67,Satz 5.5] implies that there exists a diffeomorphism Ψ : Σ → Σ isotopic to the identity andsuch that Ψ ∗ α ′ is an S -connection for p or for p with reversed orientation. The stabilityargument contained in the proof of Lemma 2.1 shows that Ψ ∗ α ′ is homotopic to α or − α .We finally observe that if Σ is not S nor RP , then ξ + and ξ − are not isotopic. To thispurpose, we use the last statement of Theorem D in [Mas08]. The fact that Σ = S , RP is equivalent to the fact that h e, [ M ] i > χ ( M ), and implies the hypothesis − b − r < g − b = h e, [ M ] i , r = 0 and 2 g − − χ ( M ). Therefore, one only needsto check that the twisting number t ( ξ ± ) defined in [Mas08, p. 1730] is equal to −
1. If wesuppose that α has period 1, then it is an S -connection for p with d α = p ∗ ω and there existsa positively immersed disc D ֒ → M , whose lift to the universal cover of M is embedded andsuch that R D ω = 1. One readily sees that the horizontal lift of the boundary of D traversedin the negative direction is a ξ ± -Legendrian curve in Σ, which is isotopic to an oriented p -fibreand has twisting number −
1. 7
A global surface of section for contact forms near Zoll ones
Let us start by fixing some notation which will be used below. As before, we set S = R / Z .Let Σ be a connected closed three-manifold and let α ∗ be a Zoll contact form on Σ with unitperiod (see Definition 1.1). Let R ∗ denote the Reeb vector field of α ∗ . Since α ∗ is Zoll, R ∗ induces a free S -action on Σ and yields an oriented S -bundle p : Σ → M , where M is thequotient of Σ by the action and p is the canonical projection. We write h for the free-homotopyclass of the oriented p -fibres. Throughout this section, we fix auxiliary Riemannian metricson Σ and M , in order to compute the distance between points and between diffeomorphisms,and the norm of sections of vector bundles over these manifolds. The space M is a connectedclosed surface having a symplectic form ω ∗ satisfyingd α ∗ = p ∗ ω ∗ . We endow M with the orientation induced by ω ∗ .Let g st and i be the standard scalar product and complex structure on R ∼ = C , respec-tively. If a > B (respectively B ′ ) the closedEuclidean ball in R of radius a (respectively a/ x = ( x , x ) for a point in B and let λ st = π ( x d x − x d x ) be the standard Liouville form (up to a constant) on B . Weconsider the trivial bundle p st : B × S → B and we write φ for the fibre coordinate. We set α st := d φ + p ∗ st λ st , R st := ∂ φ . We now define a finite Darboux covering for M . To this purpose, let Z ⊂ Σ be a finite set ofpoints. We consider S -equivariant embeddings D z : B × S −→ Σ , D z (0 ,
0) = z, ∀ z ∈ Z. This means that there are corresponding embeddings d q : B −→ M, d q (0) = q, ∀ q ∈ p ( Z )such that p ◦ D z = d p ( z ) ◦ p st , ∀ z ∈ Z. We write Σ z := D z ( B × S ), Σ ′ z := D z ( B ′ × S ), and M q := d q ( B ), M ′ q := d q ( B ′ ). Finally,we denote by ( x z , φ z ) ∈ B × S the coordinates given by D z . By the compactness of Σ, wesee that, if a is small enough, the following three properties can be assumed to hold( DF1 ) M = [ q ∈ p ( Z ) M ′ q , ( DF2 ) ∃ d ∗ > , dist( M ′ q , M \ M q ) > d ∗ , ∀ q ∈ p ( Z ) , ( DF3 ) D ∗ z α ∗ = α st , ∀ z ∈ Z. (3.1)In this section, we define a neighbourhood of d α ∗ in the space of exact two-forms on Σ withspecial properties. The elements of the neighbourhood will be exterior differentials of contactforms, whose Reeb flow has a distinguished set of periodic Reeb orbits, which can be used toconstruct a global surface of section for the flow.8 .1 A distinguished class of periodic Reeb orbits For any contact form α on Σ, let R α be its Reeb vector field. Let P ( α ) denote the set ofprime periodic orbits of the Reeb flow Φ α of α . For all T ∈ (0 , ∞ ), we also denote by P T ( α )the subset of P ( α ), whose elements have period less than or equal to T . We write P ( α, h ) forthe subset of P ( α ), whose elements are in the class h , namely they are freely homotopic toan oriented p -fibre. We abbreviate P T ( α, h ) := P ( α, h ) ∩ P T ( α ).If γ ∈ P ( α ), we write T ( γ ) for the period of γ and define the auxiliary one-periodic curves γ rep , ¯ γ : S → Σ , γ rep ( u ) := γ ( uT ( γ )) , ¯ γ ( u ) := Φ α ∗ u ( γ (0)) . We define T min ( α, h ) := inf γ ∈P ( α, h ) T ( γ ) , T max ( α, h ) := sup γ ∈P ( α, h ) T ( γ ) . We now explore how much information of the Reeb dynamics is already encoded in the exteriordifferential of the contact form.
Lemma 3.1.
Let α and α be contact forms such that d α = d α . The forms α ∧ d α and α ∧ d α induce the same orientation on Σ and Vol( α ) = Vol( α ) . Moreover, there isa bijection between P ( α ) and P ( α ) which preserves the oriented support of curves. Thebijection is period-preserving when restricted to P ( α , h ) and P ( α , h ) . If α is Zoll, then α is also Zoll, and T ( α ) = T ( α ) .Proof. Since d α = d α , we have R α = α ( R α ) R α and α = α + η for some closed one-form η . We orient Σ so that α ∧ d α is positive and computeVol( α ) = Z Σ α ∧ d α = Z Σ α ∧ d α + Z Σ η ∧ d α = Z Σ α ∧ d α = Vol( α ) . In particular, the orientations induced by α and by α coincide. Therefore, for all z ∈ Σ,there holds Φ α t ( t ,z ) ( z ) = Φ α t ( z ) , t ( t , z ) := Z t (cid:0) t Φ α t ( z ) (cid:1) ∗ α , so that t t ( t , z ) is strictly increasing. Hence, Φ α and Φ α have the same trajectories,up to an orientation-preserving reparametrisation, and we have a bijective correspondencebetween P ( α ) and P ( α ) preserving the oriented support of periodic orbits. Let γ ∈ P ( α , h )and γ ∈ P ( α , h ) be corresponding periodic orbits. Since the homology class of γ and γ istorsion, the fact that η is closed implies T ( γ ) = Z R /T ( γ ) Z γ ∗ α = Z R /T ( γ ) Z γ ∗ α + Z R /T ( γ ) Z γ ∗ η = T ( γ ) + 0 . Finally, if α is Zoll with period T , then α is also Zoll with period T := t ( T , z ) (indepen-dent of z ∈ Σ), as t t ( t , z ) is monotone increasing. Since every prime periodic orbit ofΦ α has torsion homology class, we conclude as above that T = T .On the space of one-forms α on Σ we consider the norm k · k C − defined by k α k C − := k α k C + k d α k C . C D > α on Σ, 1 C D k α k C − ≤ max z ∈ Z k D ∗ z α k C − ≤ C D k α k C − . (3.2)For every ǫ >
0, we denote the C − -ball with center α ∗ and radius ǫ by B ( ǫ ) := (cid:8) α one-form on Σ (cid:12)(cid:12) k α − α ∗ k C − < ǫ (cid:9) . The next result shows why it is natural to consider the C − -norm for our purposes. Lemma 3.2.
There exists a constant C > such that for all one-forms α ′ on Σ , there is aone-form α on Σ with the property that • d α = d α ′ , • ∀ ǫ > , k d α ′ − d α ∗ k C < ǫ = ⇒ α ∈ B ( C ǫ ) . Proof.
By standard elliptic arguments (see for instance, [Nic07, Chapter 10]), there exists aconstant C ′ > η Ω withd η Ω = Ω , k η Ω k C ≤ C ′ k Ω k C . Setting α := α ∗ + η d α ′ − d α ∗ and applying the above fact to η d α ′ − d α ∗ , we have d α = d α ′ and k α − α ∗ k C ≤ C ′ k d α ′ − d α ∗ k C , k d α − d α ∗ k C = k d α ′ − d α ∗ k C . The statement follows with C := C ′ + 1.We can now proceed to study the Reeb dynamics for one-forms in the sets B ( ǫ ). Lemma 3.3.
There exist ǫ > and C > with the following properties. Every α ∈ B ( ǫ ) is a contact form, and if z ′ ∈ Σ , z ∈ Z and T ∈ (0 , ∞ ) are such that the integral curve t Φ αt ( z ′ ) lies in Σ z for all t ∈ [0 , T ] , then the curve γ z := ( x z ( t ) , φ z ( t )) = D − z (Φ αt ( z )) satisfies k ˙ γ z − R st k C ≤ C k α − α ∗ k C − . (3.3) Thus, if b φ z : [0 , T ] → R with b φ z (0) = 0 is a lift of φ z − φ z (0) , then | x z ( t ) − x z (0) | ≤ C t k α − α ∗ k C − , | b φ z ( t ) − t | ≤ C t k α − α ∗ k C − , ∀ t ∈ [0 , T ] . (3.4) Proof. If α is a one-form on Σ and we set α z := D ∗ z α , then the estimate (3.2) yields k α z − α st k C − ≤ C D k α − α ∗ k C − . (3.5)If ǫ > α ∈ B ( ǫ ) is a contact form and there exists A > k R α z − R st k C ≤ A k α z − α st k C − , ∀ α ∈ B ( ǫ ) . (3.6)Moreover, using (3.5), we have k R α z − R st k C ≤ AC D k α − α ∗ k C − . (3.7)10herefore, we just need to estimate the left-hand side of (3.3) against k R α z − R st k C . We knowthat ˙ γ z = R α z ( γ z ), which yields k ˙ γ z − R st k C ≤ k R α z − R st k C . For the higher derivatives,we just observe that k ˙ γ z k C is uniformly bounded by 1 + AC D anddd t ( ˙ γ z − R st ) = ¨ γ z = d γ z R α z · ˙ γ z = d γ z ( R α z − R st ) · ˙ γ z ;d d t ( ˙ γ z − R st ) = d γ z R α z ( ˙ γ z , ˙ γ z ) + d γ z R α z · ¨ γ z = d γ z ( R α z − R st )( ˙ γ z , ˙ γ z ) + d γ z ( R α z − R st ) dd t ( ˙ γ z − R st ) . This shows (3.3). Finally, integrating ˙ φ z and ˙ x z and using (3.3), we obtain (3.4). Proposition 3.4.
There exist C > , and for all real numbers T in the interval (1 , , aradius ǫ = ǫ ( T ) ∈ (0 , ǫ ] such that for all α ∈ B ( ǫ ) the following properties are true:(i) A periodic orbit γ of Φ α belongs to the set P T ( α ) if and only if for all z ∈ Z such that γ (0) ∈ Σ ′ z , then γ is contained in Σ z and γ rep is homotopic to ¯ γ within Σ z . In this case,if we set γ z := D − z ◦ γ , ¯ γ z := D − z ◦ ¯ γ , there holds (cid:12)(cid:12) T ( γ ) − (cid:12)(cid:12) ≤ C k α − α ∗ k C − , k γ z, rep − ¯ γ z k C ≤ C k α − α ∗ k C − . (ii) The set P T ( α, h ) is compact, non-empty and coincides with P T ( α ) .Proof. We claim that item (i) holds with ǫ := 12 C min n d ∗ , − T, T − o , C := 10 · C . Moreover, if a periodic curve γ is contained in Σ z , then we can write γ = ( x γ , φ γ ) in thecoordinates D z . Moreover, if b φ γ : R → R is the unique lift of φ γ − φ γ (0) such that b φ γ (0) = 0,then b φ γ ( T ( γ )) = 1 if and only if γ rep is homotopic to ¯ γ within Σ z .Let us now assume that α ∈ B ( ǫ ) and that γ ∈ P T ( α ). Let us take z ∈ Z such that γ (0) ∈ Σ ′ z . Since T <
2, inequalities (3.4) and (
DF2 ) imply that γ is contained in Σ z . By(3.3), we see that ˙ φ γ ≥ − | − ˙ φ γ | ≥ − k ˙ γ z − R st k C > − C ǫ ≥ > . Hence, b φ γ (cid:0) T ( γ ) (cid:1) >
0. On the other hand, using (3.4) and the fact that ǫ ≤ − T C , we get b φ γ (cid:0) T ( γ ) (cid:1) < T ( γ ) + C T ǫ ≤ T + 2 C ǫ ≤ . Since b φ γ (cid:0) T ( γ ) (cid:1) is an integer, we conclude that b φ γ (cid:0) T ( γ ) (cid:1) = 1.Conversely, we assume that γ = ( x γ , φ γ ) ⊂ Σ z and that γ rep is homotopic to ¯ γ insideΣ z and prove that γ ∈ P T ( α ). The curve γ is prime since b φ γ (cid:0) T ( γ ) (cid:1) = 1 has no non-trivialinteger divisor. Substituting t = T ( γ ) in the second inequality in (3.4) yields | T ( γ ) − | ≤ C T ( γ ) k α − α ∗ k C − . (3.8)11sing that k α − α ∗ k C − < ǫ , we solve for T ( γ ) and get T ( γ ) < (1 − C ǫ ) − . This impliesthat T ( γ ) ≤ T since 1 − C ǫ ≥ − T − ≥ − T − T = 1 T .
We suppose that γ ∈ P T ( α ) and prove the estimates in item (i). The first inequalitycomes from (3.8) using that T ( γ ) < C ≥ C . For the second inequality, exploiting(3.4) and (3.8) we have | γ z, rep ( s ) − ¯ γ z ( s ) | ≤ | x γ ( sT ( γ )) | + | b φ γ ( sT ( γ )) − s |≤ C k α − α ∗ k C − T ( γ ) + | b φ γ ( sT ( γ )) − sT ( γ ) | + | T ( γ ) − | s ≤ C k α − α ∗ k C − T ( γ ) + C k α − α ∗ k C − T ( γ ) + | T ( γ ) − |≤ C k α − α ∗ k C − . The higher derivatives can be bounded through (3.3) and (3.8): (cid:13)(cid:13)(cid:13) d γ z, rep d s − d¯ γ z d s (cid:13)(cid:13)(cid:13) C ≤ (cid:13)(cid:13) T ( γ )( ˙ x γ , ˙ φ γ ) rep − R st (cid:13)(cid:13) C ≤ T (cid:13)(cid:13) ( ˙ x γ , ˙ φ γ − rep (cid:13)(cid:13) C + | T ( γ ) − |≤ T · T (cid:13)(cid:13) ( ˙ x γ , ˙ φ γ − (cid:13)(cid:13) C + 2 C k α − α ∗ k C − ≤ C k α − α ∗ k C − + 2 C k α − α ∗ k C − . Let us prove (ii). From [Gin87, Section III] or [ ´APB14, Section 3.2], up to shrinking ǫ , forevery α ∈ B ( ǫ ), there exists a differentiable function S α : Σ → R with the following property.The set Crit S α is the union of the supports of the orbits γ ∈ P T ( α ). Therefore, P T ( α )is non-empty as Crit S α is non-empty. The set P T ( α ) is also compact by the Arzel`a-Ascolitheorem, as its elements have uniformly bounded period, and Σ is compact. Finally, by item(i) we have P T ( α ) = P T ( α, h ). In this subsection, we show that if α lies in B ( ǫ ) and γ ∈ P T ( α, h ), we can suppose that γ isa given flow line of R ∗ , up to rescaling α and applying a diffeomorphism of Σ. Lemma 3.5.
There is a constant C > with the following property. For all z , z ∈ Σ ,there exists an S -equivariant diffeomorphism Ψ z ,z : Σ → Σ isotopic to the identity with • Ψ z ,z ( z ) = z , • Ψ ∗ z ,z α ∗ = α ∗ , • k dΨ z ,z k C ≤ C , k d(Ψ − z ,z ) k C ≤ C . Proof.
We start with a local construction. Let K : B → [0 ,
1] be a function which is equalto 1 in a neighbourhood of B ′ and whose support is contained in the interior of B . For every( x ′ , φ ′ ) ∈ B ′ × S , let ˆ φ ′ ∈ [0 ,
1) be a lift of φ ′ . We define K : B → R , K ( x ) := ˆ φ ′ + g st ( x, ix ′ ) . We let K x ′ : B → R be the function K x ′ := K K and Φ Xt the flow on B × S generated bythe unique vector field X such that α st ( X ) = K x ′ ◦ p st , ι X d α st = − d( K x ′ ◦ p st ) . K x ′ ◦ p st is the contact Hamiltonian of Φ Xt according to [Gei08, Section 2.3]. Thevector field X is compactly supported and an application of Moser’s trick shows that(Φ Xt ) ∗ α st = α st , ∀ t ∈ R . (3.9)The flow Φ Xt lifts the Hamiltonian flow of the function K x ′ with respect to ω st on B . Moreover,since the curve t ( tx ′ ,
0) is α st -Legendrian and K x ′ ( tx ′ ) = ˆ φ ′ , we see thatΦ Xt (0 ,
0) = ( tx ′ , t b φ ′ ) ∈ B ′ × S , ∀ t ∈ [0 , . Then, the map Ψ B, ( x ′ ,φ ′ ) := Φ X is a compactly supported diffeomorphism of B × S sending(0 ,
0) to ( x ′ , φ ′ ) and there exists a positive constant C ′ , independent of ( x ′ , φ ′ ), such that k dΨ B, ( x ′ ,φ ′ ) k C ≤ C ′ , k d(Ψ − B, ( x ′ ,φ ′ ) ) k C ≤ C ′ . (3.10)This completes the local construction. For the global argument, we observe that there exists m ∈ N ∗ independent of z , z and a chain of points( z u ) ⊂ Σ , u ∈ U := (cid:8) ju (cid:12)(cid:12) j = 0 , · · · , m (cid:9) , u := 1 /m such that ∀ u ∈ U \ { } , ∃ y u ∈ Z, z u , z u + u ∈ Σ ′ y u . We construct Ψ z ,z as the composition of m maps Ψ z u ,z u + u : Σ → Σ, u ∈ U \ { } . Considerthe trivialisation D y u : B × S → Σ y u and defineΨ z u ,z u + u : Σ → Σ , Ψ z u ,z u + u := D y u ◦ (cid:16) Ψ B, D − yu ( z u + u ) ◦ Ψ − B, D − yu ( z u ) (cid:17) ◦ D − y u . The lemma follows from (3.9) and (DF3) together with (3.10) and the classical estimate onthe C -norm of the differential of a composition of maps. In particular, the constant C thatwe find depends only on Σ and the Darboux family. Definition 3.6.
Let us fix a reference point z ∗ ∈ Z with q ∗ := p ( z ∗ ) and define γ ∗ : S → Σto be the prime periodic orbit of R ∗ passing through z ∗ at time 0. We say that a contactform α is normalised , if γ ∗ ∈ P ( α ). For every ǫ ∈ (0 , ǫ ], we define the set B ∗ ( ǫ ) := (cid:8) α ∈ B ( ǫ ) (cid:12)(cid:12) α is normalised (cid:9) . Definition 3.7.
Let c be a positive number and Ψ : Σ → Σ a diffeomorphism. For everycontact form α on Σ, we write α c, Ψ := c Ψ ∗ α , so that Vol( α ) = c Vol( α c, Ψ ) and we have abijection P ( α ) −→ P ( α c, Ψ ) ,γ γ c, Ψ ( γ c, Ψ ( s ) : = (Ψ − ◦ γ )( cs ) , ∀ s ∈ R ,T ( γ c, Ψ ) = c T ( γ ) . The next result is analogous to [ABHS18, Proposition 3.10].
Proposition 3.8.
Let T be a number in (1 , . For every ǫ ∈ (0 , ǫ ] , there is ǫ ∈ (0 , ǫ ] (depending on ǫ and T ) with the following properties. For all α ∈ B ( ǫ ) and all γ ∈ P T ( α, h ) ,there exists a diffeomorphism Ψ : Σ → Σ isotopic to the identity such that α T ( γ ) , Ψ ∈ B ∗ ( ǫ ) , γ T ( γ ) , Ψ = γ ∗ . Moreover, the bijection P ( α ) → P ( α T ( γ ) , Ψ ) restricts to a bijection P T ( α, h ) → P T ( α T ( γ ) , Ψ , h ) . roof. Let α be an element of B ( ǫ ) for some ǫ ≤ ǫ to be determined later on, and let γ be a periodic orbit in P T ( α, h ). Here the constant ǫ is given by Proposition 3.4. We applyLemma 3.5 with z = z ∗ and z = γ (0) and get a diffeomorphism Ψ := Ψ z ∗ ,γ (0) : Σ → Σ anda constant C satisfying the properties described therein. We abbreviate α := Ψ ∗ α . We getsome C ′ ≥ C such that k α − α ∗ k C − ≤ C ′ k α − α ∗ k C − . (3.11)The periodic curve γ := Ψ − ◦ γ belongs to P T ( α , h ) and has period T ( γ ). As γ (0) = z ∗ ,we have ¯ γ = γ ∗ . If ǫ ≤ C ′′ ǫ , then α ∈ B ( ǫ ) and Proposition 3.4 implies that γ ∈ Σ z ∗ and k ( γ ) z ∗ , rep − ( γ ∗ ) z ∗ k C ≤ C ′′ k α − α ∗ k C − , C ′′ := C C ′ . (3.12)We write ( γ ) z ∗ , rep = ( x , φ ) in the coordinates given by D z ∗ . If ǫ is small enough, from(3.12), we see that k x k C < / φ : S → S is a diffeomorphism of degree 1.In particular, there exists a unique map ∆ φ : S → R , which lifts φ − id S . We define adiffeomorphism Ψ ,z ∗ : B × S → B × S byΨ ,z ∗ ( x, s ) = (cid:16) x + K (cid:0) | x | (cid:1) x ( s ) , s + K (cid:0) | x | (cid:1) ∆ φ ( s ) (cid:17) , ∀ ( x, s ) ∈ B × S , where K : [0 , → [0 ,
1] is a function which is equal to 1 on [0 , /
2] and equal to 0 close to 1.By (3.12), we have k Ψ ,z ∗ − id B × S k C ≤ C ′′ k K k C k α − α ∗ k C − , which also implies k dΨ ,z ∗ k C ≤ C ′′ k K k C C ′ ǫ . (3.13)Since Ψ ,z ∗ is compactly supported in the interior of B × S , we can define Ψ : Σ → Σ asΨ := D z ∗ ◦ Ψ ,z ∗ ◦ D − z ∗ inside Σ z ∗ and as the identity in Σ \ Σ z ∗ . We have Ψ ◦ γ ∗ = γ , rep ,and thanks to (3.13), we see that k dΨ k C is bounded by a constant depending only on theDarboux family and on C ′′ k K k C C ′ ǫ . Therefore, there is also a constant C ′′′ > k Ψ ∗ ( α − α ∗ ) k C − ≤ C ′′′ k α − α ∗ k C − . (3.14)We define Ψ := Ψ ◦ Ψ : Σ → Σ , ǫ ′ := min n ǫ , ǫ , C T − T + 1 o , and prove that α T ( γ ) , Ψ belongs to B ∗ ( ǫ ′ ), provided ǫ is suitably small. We take δ := ǫ ′ ( T + 1) C D , and let δ > k Ψ ,z ∗ − id B × S k C ≤ δ = ⇒ k (Ψ ,z ∗ ) ∗ α st − α st k C − ≤ δ . (3.15)We assume further that ǫ ≤ min n δ C ′′ k K k C , C T − T + 1 o . k Ψ ,z ∗ − id B × S k C ≤ δ and we compute k α T ( γ ) , Ψ − α ∗ k C − ≤ (cid:12)(cid:12)(cid:12)(cid:12) T ( γ ) − (cid:12)(cid:12)(cid:12)(cid:12) k α ∗ k C − + 1 T ( γ ) k Ψ ∗ α − α ∗ k C − . For the first summand of the right-hand side, we first estimate T ( γ ) − ≤ ( T + 1) and then (cid:12)(cid:12)(cid:12)(cid:12) T ( γ ) − (cid:12)(cid:12)(cid:12)(cid:12) k α ∗ k C − ≤ T + 12 C ǫ k α ∗ k C − . For the second summand, we estimate k Ψ ∗ α − α ∗ k C − ≤ k Ψ ∗ ( α − α ∗ ) k C − + k Ψ ∗ α ∗ − α ∗ k C − ≤ C ′′′ k α − α ∗ k C − + C D k (Ψ ,z ∗ ) ∗ α st − α st k C − ≤ C ′ C ′′′ ǫ + C D δ , where we used (3.2), (3.11), (3.14), and (3.15). Using the definition of δ and putting thecomputations together, we find that k α T ( γ ) , Ψ − α ∗ k C − ≤ T + 12 (cid:16) C k α ∗ k C − + C ′ C ′′′ (cid:17) ǫ + ǫ ′ . The quantity on the right is smaller than ǫ ′ ≤ ǫ , if ǫ is small enough. Finally, we compute γ T ( γ ) , Ψ = Ψ − ◦ γ rep = Ψ − ◦ Ψ − ◦ γ rep = Ψ − ◦ γ , rep = γ ∗ . Let us now deal with the second part of the statement. Let e γ e γ T ( γ ) , Ψ be the bijectionbetween P ( α ) and P ( α T ( γ ) , Ψ ) introduced in Definition 3.7. Let us assume that T ( e γ ) ≤ T .Since ǫ ≤ ǫ we can use Proposition 3.4.(i), and from C ǫ ≤ T − T +1 , we see that T ( e γ T ( γ ) , Ψ ) = T ( e γ ) T ( γ ) ≤ C ǫ − C ǫ ≤ T. Assume, conversely, that T ( e γ T ( γ ) , Ψ ) ≤ T . Since ǫ ′ ≤ ǫ , we can use Proposition 3.4.(i) andfind that T ( e γ ) = T ( e γ T ( γ ) , Ψ ) T ( γ ) ≤ (1 + C ǫ ′ )(1 + C ǫ ) ≤ TT + 1 2 TT + 1 = T T ( T + 1) ≤ T. As in the previous subsection, let z ∗ be a reference point on Σ with q ∗ := p ( z ∗ ) and setˇ M := M \ { q ∗ } . Let e ∈ H ( M ) be minus the real Euler class of p and let us adopt the notation t Σ := h e, [ M ] i > , where, as observed in the introduction, h e, [ M ] i = | H tor1 (Σ; Z ) | . We define the annulus A := [0 , a ) × S .
15e consider the inclusion i : ˚ A → A , where ˚ A = (0 , a ) × S , and the map i : ˚ A → ˇ M , i ( r, θ ) = d z ∗ ( re πiθ ) , where we identify the domain of d z ∗ with a subset of the complex plane. We glue together A and ˇ M along the maps i and i to get a smooth compact surface N with the same genus as M and one boundary component denoted by ∂N . Namely, we have the following commutativediagram ˚ A i / / i (cid:15) (cid:15) ˇ M (cid:15) (cid:15) A / / N so that ˇ M is diffeomorphic to the interior ˚ N = N \ ∂N and A to a collar neighbourhoodof ∂N . On ˇ M we have the orientation given by ω ∗ , while on A the one given by d r ∧ d θ .These two orientations glue together to an orientation of N , since i and i are orientationpreserving. Using the usual convention of putting the outward normal first, we see that theorientation induced on ∂N is given by − d θ . As for M and Σ, we fix on N some auxiliaryRiemannian metric to compute norms of sections, and distances between points and betweendiffeomorphisms. In particular, we write the C -distance on the space of diffeomorphismsfrom N to itself as dist C : Diff( N ) × Diff( N ) → R . Consider now the map S A : A → Σ , S A ( r, θ ) = D z ∗ ( re πiθ , − t Σ θ )and observe that, for all θ ∈ S , there holds S A (0 , θ ) = γ ∗ ( − t Σ θ ), so thatd (0 ,θ ) S A · ∂ θ = − t Σ R ∗ . (3.16)The map S A ◦ i − : i (˚ A ) → Σ is a local section of the bundle p with a singularity of order − t Σ at q ∗ . Since − t Σ is the Euler number of p , this section extends to a section on ˇ M and yieldsa map S ˇ M : ˇ M → Σ. By the commutativity of the diagram above, we get a map S : N → Σfitting into the diagram A ~ ~ ⑤⑤⑤⑤⑤⑤⑤⑤ S A ! ! ❈❈❈❈❈❈❈❈ N S / / Σ . ˇ M ` ` ❆❆❆❆❆❆❆❆ S ˇ M > > ⑤⑤⑤⑤⑤⑤⑤⑤ Moreover, S ∗ ˇ M d α ∗ = ( p ◦ S ˇ M ) ∗ ω ∗ = ω ∗ , and S ∗ A d α ∗ = r d r ∧ d θ . In particular, S ∗ d α ∗ is a two-form on N , which is symplectic on the interior of N and vanishes of order 1 at the boundaryof N . The one-form λ ∗ := S ∗ α ∗ is a primitive for S ∗ d α ∗ such that λ ∗ | A = ( D − z ∗ ◦ S A ) ∗ (d φ + p ∗ st λ st ) = (cid:0) − t Σ + r (cid:1) d θ. α is a normalised form, so that R α = R ∗ on p − ( q ∗ ), we set λ := S ∗ α, and by equation (3.16), we have • λ (cid:12)(cid:12) T( ∂N ) = λ ∗ (cid:12)(cid:12) T( ∂N ) , • d λ = 0 at ∂N. (3.17) Proposition 3.9.
For all ǫ > , there exists a number ǫ ∈ (0 , ǫ ] such that, if α ∈ B ∗ ( ǫ ) ,there exist a map ζ : N → N isotopic to the identity and a function b : N → R satisfying thefollowing properties. ( i ) Triviality at the boundary: ζ | ∂N = id ∂N , b | ∂N = 0 , ( ii ) C -smallness: max (cid:8) dist C ( ζ, id N ) , k b k C (cid:9) < ǫ , ( iii ) Uniformisation: ζ ∗ λ − λ ∗ = d b. Proof.
Let α ∈ B ∗ ( ǫ ), for some ǫ ∈ (0 , ǫ ] to be determined. For all u ∈ [0 , λ u := λ ∗ + u ( λ − λ ∗ ). On A , we get λ − λ ∗ = c d r + c d θ, d λ ∗ = r d r ∧ d θ, d λ = f d r ∧ d θ, d λ u = ( r + u ( f − r ))d r ∧ d θ, for some functions c , c , f : A → R . By (3.17), we have c (0 , θ ) = 0 and f (0 , θ ) = 0. Definethe auxiliary function c : A → R , c ( r, θ ) := c ( r, θ ) − Z r ∂ θ c ( r ′ , θ )d r ′ . From the definition of c , c , c and f , we have the chain of identities( ∂ r c )d r ∧ d θ = ( ∂ r c − ∂ θ c )d r ∧ d θ = d( λ − λ ∗ ) = d λ − d λ ∗ = ( f − r )d r ∧ d θ, which implies ∂ r c = f − r. As a result, c (0 , θ ) = 0 , ∂ r c (0 , θ ) = 0 and there exists a function ˆ c : A → R with c = r ˆ c ,ˆ c | ∂N = 0, and a function ˆ f : A → R with f = r ˆ f , defined byˆ c ( r, θ ) := Z ∂ r c ( vr, θ )d v = Z (cid:0) f ( vr, θ ) − vr (cid:1) d v, ˆ f ( r, θ ) := Z ∂ r f ( vr, θ )d v. In particular, d λ u = r (cid:0) u ( ˆ f − (cid:1) d r ∧ d θ (3.18)and we have the estimate max (cid:8) k ˆ c k C , k ˆ f − k C (cid:9) ≤ k f − r k C . (3.19)We now look for paths u ζ u and u b u with ζ = id N and b = 0 such that ζ ∗ u λ u − d b u = λ ∗
17o that, for u = 1, we get a solution to item (iii) in the statement. Let X u denote the vectorfield generating ζ u and set a u := dd u b u . By differentiating the equation above with respect to u , we find that such an equation can be solved for ζ u and b u if and only if( λ − λ ∗ ) + ι X u d λ u + d (cid:0) λ u ( X u ) (cid:1) − d( a u ◦ ζ − u ) = 0 . Introducing an auxiliary function h : N → R , we see that ( X u , a u ) is a solution if and only if ( ι X u d λ u = − ( λ − λ ∗ ) + d h,a u = (cid:0) λ u ( X u ) + h (cid:1) ◦ ζ u . (3.20)We define A ′ := [0 , a/ × S and choose h := h A · K , where K : N → [0 ,
1] is a bump function which is equal to 1 on A ′ and its support is contained in A , and h A : A → R is defined by h A ( r, θ ) := Z r c ( r ′ , θ )d r ′ . This function has the crucial property that λ − λ ∗ − d h = c d r + c d θ − c d r − (cid:16) Z r ∂ θ c ( r ′ , θ )d r ′ (cid:17) d θ = r ˆ c d θ on A ′ , (3.21)which implies that the first equation in (3.20) admits a smooth solution X u . Indeed, on theannulus A ′ , we divide both sides of the equation by r and using (3.18), (3.21), we get X u = − ˆ c u ( ˆ f − ∂ r . On N \ A ′ , X u is uniquely determined by the fact that d λ u | N \ A ′ is symplectic. The vector X u vanishes at ∂N , since ˆ c vanishes there, as observed before. This shows that ζ u is the identityat the boundary. If we choose ǫ small, we see that the C -norm of c , c and f − r are small,and consequently, also the C -norm of h . By (3.19), we conclude that the C -norm of X u issmall, as well. As a consequence, also dist C ( ζ u , id N ) is small. Therefore, by defining ζ := ζ and taking ǫ small enough, we get dist C ( ζ, id N ) < ǫ . We can now define a u through thesecond equation in (3.20). From the estimates on λ, X u , ζ u and h , we see that, if ǫ is small, k a u k C < ǫ and the same is true for b := b . Since h and X u vanish at the boundary, we alsohave a u | ∂N = 0, and as b | ∂N = 0, the function b vanishes at the boundary, as well. Combining the map S with the Reeb flow of α ∗ , we get a rational open book for Σ:Ξ : N × S −→ Σ( q, s ) Φ α ∗ s (cid:0) S ( q )) . If i N : N ֒ → N × S is the canonical embedding i N ( x ) = ( x, S = Ξ ◦ i N . On the collarneighbourhood A × S of ∂ ( N × S ), Ξ has the coordinate expressionΞ A : A × S −→ B × S (cid:0) ( r, θ ) , s (cid:1) (cid:0) re πiθ , s − t Σ θ (cid:1) . (3.22)18he restricted map ˚Ξ : ˚ N × S → p − ( ˇ M ) = Σ \ p − ( q ∗ ) is a diffeomorphism, and ˚Ξ ∗ α ∗ isa contact form on ˚ N × S with Reeb vector field R ˚Ξ ∗ α ∗ = ∂ s , which smoothly extends tothe whole N × S . If we write i ∂N × S : ∂N × S → N × S for the standard embedding ofthe boundary, the map Ξ ◦ i ∂N × S : ∂N × S → p − ( q ∗ ) ⊂ Σ has the coordinate expression( θ, s ) ( s − t Σ θ ). Therefore we haved(Ξ ◦ i ∂N × S ) · ∂ θ = − t Σ R ∗ , d(Ξ ◦ i ∂N × S ) · ∂ s = R ∗ . (3.23)If α is a normalised contact form, we define the pull-back form β := Ξ ∗ α. The next result is the analogue of [ABHS18, Proposition 3.6].
Proposition 3.10. If α is a normalised contact form, then(i) There hold i ∗ ∂N × S β = d s − t Σ d θ, d β | ∂N × S = 0 . By the latter identity we mean that d β z ( ξ ) = 0 for all z ∈ ∂N × S and ξ ∈ T z ( N × S ) .(ii) The Reeb vector field R ˚Ξ ∗ α of ˚Ξ ∗ α on ˚ N × S smoothly extends to a vector field R β onthe whole N × S , so that, at every point in ∂ ( N × S ) , R β is tangent to ∂ ( N × S ) .(iii) If we denote by Φ β the flow of R β , we have β ( R β ) = 1 , ι R β d β = 0 , (Φ βt ) ∗ β = β, ∀ t ∈ R . (iv) For every ǫ > , there exists ǫ ∈ (0 , ǫ ] , independent of α , such that α ∈ B ∗ ( ǫ ) = ⇒ k R β − ∂ s k C < ǫ . Proof.
By (3.23), equation α ( R α ) = 1, and the fact that R α = R ∗ on p − ( q ∗ ) as α is nor-malised, we get the first equality in item (i). Since ∂N × S has co-dimension 1 in N × S , toprove the second equality it is enough to show that for all vectors v ∈ T( ∂N × S ), we have ι v d β = 0. As v is a linear combination of ∂ θ and ∂ s , this follows again from (3.23) and thefact that R α annihilates d α .Now we prove (ii). We set α z ∗ := D ∗ z ∗ α , which is a contact form on B × S with corre-sponding Reeb vector field R z ∗ . Using coordinates ( x, φ ) ∈ B × S , we have the splitting R z ∗ ( x, φ ) = R xz ∗ ( x, φ ) + R φz ∗ ( x, φ ) ∂ φ . Since R α is tangent to p − ( q ∗ ), there holds R xz ∗ (0 , φ ) = 0, and therefore, there exists a matrix-valued function W z ∗ such that R xz ∗ ( x, φ ) = W z ∗ ( x, φ ) · x, k W z ∗ k C ≤ k R xz ∗ k C , by Lemma 4.8. We can then write R xz ∗ in polar coordinates on ( B \ { } ) × S as R xz ∗ ( re πiθ , φ ) = g st (cid:16) W z ∗ ( re πiθ , φ ) · re πiθ , e πiθ (cid:17) ∂ r + g st (cid:16) W z ∗ ( re πiθ , φ ) · re πiθ , ie πiθ r (cid:17) ∂ θ .
19n particular, if we set R rz ∗ ( r, θ, φ ) := g st (cid:16) W z ∗ ( re πiθ , φ ) · re πiθ , e πiθ (cid:17) ,R θz ∗ ( r, θ, φ ) := g st (cid:16) W z ∗ ( re πiθ , φ ) · e πiθ , ie πiθ (cid:17) , then R rz ∗ ◦ Ξ A and R θz ∗ ◦ Ξ A are smooth functions on A × S ⊂ N × S withmax n k R rz ∗ ◦ Ξ A k C , k R θz ∗ ◦ Ξ A k C o ≤ (1 + k dΞ A k C ) k R xz ∗ k C . (3.24)Differentiating formula (3.22), we getdΞ A · ∂ r = ∂ r , dΞ A · ∂ θ = ∂ θ − t Σ ∂ φ , dΞ A · ∂ s = ∂ φ . Thus, we conclude that R β := ( R rz ∗ ◦ Ξ A ) ∂ r + ( R θz ∗ ◦ Ξ A ) ∂ θ + (cid:0) t Σ R θz ∗ ◦ Ξ A + R φz ∗ ◦ Ξ A (cid:1) ∂ s is the desired extension of R ˚Ξ ∗ α in the collar neighbourhood A × S of ∂N × S . As R rz ∗ ◦ Ξ A vanishes at r = 0, the extended vector field is tangent to ∂N × S and (ii) is proven.By the very definition of the Reeb vector field, (˚Ξ ∗ α )( R ˚Ξ ∗ α ) = 1 and ι R ˚Ξ ∗ α d(˚Ξ ∗ α ) = 0.By continuity of the extended vector field R β , the first two relations in item (iii) follow. Thethird one is a consequence of the first two and Cartan’s formula. Point (iii) is established.We assume that α ∈ B ∗ ( ǫ ), for some ǫ to be determined independently of α and we prove(iv) by estimating k R β − ∂ s k C separately on N \ A × S and A × S . Since ( N \ A ) × S iscompact, there exists a constant C ′ > k d˚Ξ k C but not on α such that k ˚Ξ ∗ α − ˚Ξ ∗ α ∗ k C − ≤ C ′ k α − α ∗ k C − . Therefore, as in (3.6), k R β − ∂ s k C is smaller than ǫ on ( N \ A ) × S , if ǫ is small enough.On A × S , there is some C ′′ > k R β − ∂ s k C ≤ C ′′ max n k R rz ∗ ◦ Ξ A k C , k R θz ∗ ◦ Ξ A k C , k R φz ∗ ◦ Ξ A − k C o ≤ C ′′ (1 + k dΞ A k C ) k R z ∗ − R st k C ≤ C ′′ (1 + k dΞ A k C ) A C D k α − α ∗ k C − , where we have used (3.24), the equality R st = ∂ φ and inequality (3.7). This proves that k R β − ∂ s k C is smaller than ǫ on A × S , if ǫ is small enough.We can now show that S is a global surface of section for Φ α with certain properties. Definition 3.11.
Let Φ be a flow on Σ without rest points and N a compact surface. Amap S : N → Σ is a global surface of section for Φ if the following properties hold: • The map S | ˚ N is an embedding and the map S | ∂N is a finite cover onto its image; • The surface S ( ˚ N ) is transverse to the flow Φ and S ( ∂N ) is the support of a finitecollection of periodic orbits of Φ; 20 For each z ∈ Σ \ S ( ∂N ), there are t − < < t + such that Φ t − ( z ), Φ t + ( z ) lie in S ( ˚ N ).Before stating the proposition, we introduce the following notation. Let q ∈ ∂N ∼ = S anddenote by − q ∈ ∂N its antipodal point. By Z q ′ q λ ∗ , q ′ ∈ ∂N \ {− q } , (3.25)we mean the integral of λ ∗ over any path connecting q and q ′ within ∂N \ {− q } . This numberdoes not depend on the choice of such path. Proposition 3.12.
Let T be a real number in the interval (1 , . For all ǫ > , there exists ǫ ∈ (0 , ǫ ] with the following properties. If α ∈ B ∗ ( ǫ ) , then S : N → Σ is a global surface ofsection for Φ α with the first return time admitting an extension to the boundary τ : N → R , τ ( q ) := inf (cid:8) t > (cid:12)(cid:12) Φ βt ( q, ∈ N × { } (cid:9) and the first return map admitting an extension to the boundary P : N → N, ( P ( q ) ,
0) := Φ βτ ( q ) ( q, . Moreover, the following properties hold. ( i ) C -smallness: max (cid:8) dist C ( P, id N ) , k τ − k C (cid:9) < ǫ , ( ii ) Normalisation: τ ( q ) = 1 + Z P ( q ) q λ ∗ , ∀ q ∈ ∂N, ( iii ) Exactness: P ∗ λ = λ + d τ, ( iv ) Volume:
Vol( α ) = Z N τ d λ, ( v ) Fixed points: q ∈ ˚ N = ⇒ h q ∈ Fix ( P ) ⇐⇒ γ q ( t ) := Φ αt (cid:0) S ( q ) (cid:1) ∈ P T ( α, h ) i , ( vi ) Period: q ∈ ˚ N ∩ Fix ( P ) = ⇒ T ( γ q ) = τ ( q ) , ( vii ) Zoll case: if α is Zoll, then P = id N . Proof.
Let ǫ ∈ (0 , ǫ , and let ǫ be thenumber associated with ǫ in Proposition 3.10. If α ∈ B ∗ ( ǫ ), then 1 − ǫ < d s ( R β ) < ǫ .This implies at once that S : N → Σ is a global surface of section. In particular, if q ∈ N ,there exists a smallest positive time τ ( q ) such that Φ βτ ( q ) ( q,
0) belongs to N ×{ } . We estimatethe return time more precisely as (1 + ǫ ) − < τ ( q ) < (1 − ǫ ) − . In particular, if ǫ is smallenough, there holds max τ < T < τ. (3.26)Shrinking ǫ further, if necessary, we also get k τ − k C < ǫ from Proposition 3.10.(iv).We define the return point P ( q ) by the equation ( P ( q ) ,
0) = Φ βτ ( q ) ( q, C ( P, id N ) < ǫ , if ǫ is small enough, so that item (i)is established. Let q ∈ ˚ N and let us prove the statement in square brackets in item (v). If q ∈ Fix ( P ), then γ q is prime, since intersects S ( N ) only once, and has period τ ( q ). By (3.26),21e have τ ( q ) < T . Hence, by Proposition 3.4.(ii) the curve γ q belongs to P T ( α, h ). Supposeconversely that γ q has period T ( γ q ) ≤ T . If q = P ( q ), then we would get the contradiction T ( γ q ) ≥ τ ( q ) + τ ( P ( q )) ≥ τ > T. This establishes item (v) and (vi), at once. Let us assume that α is Zoll. Since γ ∗ ∈ P ( α, h ),then, if q ∈ ˚ N , the orbit γ q belongs to P ( α, h ) and satisfies T ( γ q ) = T ( γ ∗ ) = 1 < T. By item (v), we conclude that q ∈ Fix ( P ). This shows ˚ N ⊂ Fix ( P ), and by continuityFix ( P ) = N . Namely, P = id N and item (vii) holds.Let q ∈ ∂N and denote by δ q : [0 , τ ( q )] → ∂N × S the curve δ q ( t ) = Φ βt ( q, θ, s ) on ∂N × S , we can write δ q ( s ) = ( θ q ( t ) , s q ( t )), so that θ q : [0 , τ ( q )] → ∂N is a path between θ q (0) = q and θ q ( τ ( q )) = P ( q ), and s q (0) = 0 , s q ( τ ( q )) = 1. We compute τ ( q ) = Z τ ( q )0 d t = Z τ ( q )0 δ ∗ q ( i ∗ ∂N × S β ) = Z τ ( q )0 δ ∗ q (cid:0) d s − t Σ d θ (cid:1) = Z τ ( q )0 (cid:0) d s q + θ ∗ q ( − t Σ d θ ) (cid:1) = 1 + Z τ ( q )0 θ ∗ q λ ∗ and the integral of λ ∗ over θ q is equal to R P ( q ) q λ ∗ , as θ q is short if ǫ is small enough. Thisestablishes item (ii). Therefore, we can choose ǫ := ǫ in the statement of the corollary.We prove now item (iii) and (iv) by considering the map Q : [0 , × N → N × S , Q ( t, q ) := Φ βt τ ( q ) ( i N ( q )) , where i N : N ֒ → N × S is the canonical embedding. Its differential is given byd ( t,q ) Q = d ( t,q ) ( t τ ) ⊗ R β ( Q ( t, q )) + d i N ( q ) Φ βt τ ( q ) · d q i N . Hence, using Proposition 3.10.(iii) we compute Q ∗ β = β Q (cid:16) d( t τ ) ⊗ R β ( Q ) + d i N Φ βt τ · d i N (cid:17) = d( t τ ) β ( R β ) + (Φ βt τ ◦ i N ) ∗ β = d( t τ ) + i ∗ N (Φ βt τ ) ∗ β = d( t τ ) + i ∗ N β = d( t τ ) + λ. We define i : N → [0 , × N by i ( q ) = (1 , q ) and observe that Q ◦ i = i N ◦ P . Therefore, P ∗ λ = P ∗ i ∗ N β = ( i N ◦ P ) ∗ β = ( Q ◦ i ) ∗ β = i ∗ Q ∗ β = i ∗ (cid:0) d( t τ ) + λ (cid:1) = 1d τ + λ. This establishes item (iii). We calculate the volume of α pulling back by Ξ ◦ Q :Vol( α ) = Z N × S β ∧ d β = Z [0 , × N (cid:0) d( t τ ) + λ (cid:1) ∧ d λ = Z [0 , × N d( t τ ) ∧ d λ = Z [0 , × N d (cid:0) t τ d λ (cid:1) = Z N τ d λ − Z N τ d λ, which yields item (iv). 22 .5 Reduction to a two-dimensional problem Putting together all the results of this section, we are able to translate the systolic-diastolicinequality into a statement for maps on N . We recall the set-up. Let α ∗ be a Zoll contactform on a closed three-manifold Σ with associated bundle p : Σ → M . Let S : N → Σ be aglobal surface of section for the Reeb flow Φ α of α ∈ B ∗ ( ǫ ) as described at the beginning ofSection 3.3 and in Proposition 3.12. Let λ ∗ = S ∗ α ∗ and remember that t Σ = h e, [ M ] i . Thenext result is the analogous of [ABHS18, Lemma 3.7 & Proposition 3.8] Theorem 3.13.
For any T ∈ (1 , and ǫ > , there is ǫ > such that for all contactforms α ′ with k d α ′ − d α ∗ k C < ǫ , the set P T ( α ′ , h ) is compact and non-empty. Moreoverfor every γ ∈ P T ( α ′ , h ) , there exist a diffeomorphism ϕ : N → N , and a function σ : N → R with the following properties. ( i ) C -smallness: d ( ϕ, id N ) C < ǫ , ( ii ) Normalisation: σ ( q ) = Z ϕ ( q ) q λ ∗ , ∀ q ∈ ∂N. ( iii ) Exactness: ϕ ∗ λ ∗ = λ ∗ + d σ, ( iv ) Volume:
Vol( α ′ ) − t Σ T ( γ ) = T ( γ ) Z N σ d λ ∗ , ( v ) Fixed points: There is a map ˚ N ∩ Fix ( ϕ ) → P T ( α ′ , h ) , q γ q such that T ( γ q ) = T ( γ )(1 + σ ( q )) . ( vi ) Zoll case: if α ′ is Zoll, then ϕ = id N . Proof.
Let C be the constant given by Lemma 3.2 and let ǫ ≤ C ǫ be some positive realnumber, which will be determined in the course of the proof depending on T and ǫ . If α ′ isa contact form with k d α ′ − d α ∗ k C < ǫ , then Lemma 3.2 yields a contact form α ∈ B ( C ǫ )with d α = d α ′ . Since C ǫ ≤ ǫ , by Lemma 3.1, we have a period-preserving bijection P T ( α ′ , h ) → P T ( α, h ) and by Proposition 3.4.(ii) the set P T ( α, h ) is compact and non-empty.We fix henceforth an element γ ∈ P T ( α ′ , h ). If ǫ ∈ (0 , ǫ ] is an auxiliary number, wecan find a corresponding ǫ ∈ (0 , ǫ ] according to Proposition 3.8, so that, if ǫ ≤ ǫ thereexists a diffeomorphism Ψ : Σ → Σ such that α T ( γ ) , Ψ ∈ B ∗ ( ǫ ) and the map e γ e γ T ( γ ) , Ψ ofDefinition 3.7 restricts to a bijection P T ( α, h ) → P T ( α T ( γ ) , Ψ , h ). Thus, we get a bijection P T ( α ′ , h ) −→ P T ( α c, Ψ , h ) ,γ ′ γ ′ T ( γ ) , Ψ T ( γ ′ ) = T ( γ ) T ( γ ′ T ( γ ) , Ψ ) . (3.27)We choose now an auxiliary ǫ > ǫ ∈ (0 , ǫ ] from Proposition 3.9,so that if ǫ ≤ ǫ , then there exist ζ : N → N and b : N → R associated with α T ( γ ) , Ψ ∈ B ∗ ( ǫ )satisfying the properties contained therein. Finally, let ǫ > ǫ ∈ (0 , ǫ ], the number given by Proposition 3.12, so that, if ǫ ≤ ǫ , thestatements contained therein hold for α T ( γ ) , Ψ , the associated return time τ : N → R andreturn map P : N → N .Now we set • ϕ : N → N, ϕ := ζ − ◦ P ◦ ζ, • σ : N → R , σ := τ ◦ ζ − b ◦ ϕ + b − . ǫ and ǫ small enough, we obtain item (i). Then, wehave ϕ | ∂N = P | ∂N and σ | ∂N = τ | ∂N −
1, so that item (ii) follows from Proposition 3.12.(ii).As far as item (iii) is concerned, we compute ϕ ∗ λ ∗ = ζ ∗ P ∗ λ − ϕ ∗ d b = ζ ∗ ( λ + d τ ) − d( b ◦ ϕ ) = λ ∗ + d (cid:0) b + τ ◦ ζ − b ◦ ϕ (cid:1) = λ ∗ + d σ. For item (iv), we recall from Lemma 3.1, Definition 3.7 and Proposition 3.12.(iv) thatVol( α ′ ) = Vol( α ) = T ( γ ) Vol( α T ( γ ) , Ψ ) = T ( γ ) Z N τ d λ, and we will show that Z N τ d λ = Z N σ d λ ∗ + t Σ . (3.28)We can compute the integral of σ d λ ∗ as Z N σ d λ ∗ = Z N ( τ ◦ ζ )d λ ∗ − Z N ( b ◦ ϕ )d λ ∗ + Z N b d λ ∗ − Z N d λ ∗ . We deal with the first summand. The map ζ preserves the orientation on N , as it is isotopicto the identity, and satisfies d λ ∗ = ζ ∗ (d λ ). Hence, Z N ( τ ◦ ζ )d λ ∗ = Z N ( τ ◦ ζ ) ζ ∗ (d λ ) = Z N ζ ∗ ( τ d λ ) = Z N τ d λ. The second and third summand cancel out. Indeed, as ϕ preserves d λ ∗ , we get Z N ( b ◦ ϕ )d λ ∗ = Z N ( b ◦ ϕ ) ϕ ∗ (d λ ∗ ) = Z N ϕ ∗ ( b d λ ∗ ) = Z N b d λ ∗ . We deal with the last summand. By Stokes’ Theorem, the fact that λ ∗ | ∂N = − t Σ d θ and thatthe induced orientation on ∂N is given by − d θ , we get Z N d λ ∗ = Z ∂N λ ∗ = − Z − t Σ d θ = t Σ . Plugging these last three identities in the computation above, we arrive at (3.28).We move to item (v). We take q ∈ ˚ N ∩ Fix ( ϕ ) and observe that ζ ( q ) ∈ ˚ N ∩ Fix ( P ).By Proposition 3.12.(v), there exists a periodic orbit γ ′ q ∈ P ( α T ( γ ) , Ψ , h ) through S ( ζ ( q )) withperiod τ ( ζ ( q )) ≤ T . We denote by γ q ∈ P T ( α ′ , h ) the orbit assigned to γ ′ q by the bijectiongiven in (3.27), so that T ( γ q ) = T ( γ ) τ ( ζ ( q )). Finally, we observe that τ ( ζ ( q )) = 1 + σ ( q ) + b ( ϕ ( q )) − b ( q ) = 1 + σ ( q ) + b ( q ) − b ( q ) = 1 + σ ( q ) . The implication in item (vi) follows at once, since α ′ is Zoll if and only if α T ( γ ) , Ψ is Zollby Lemma 3.1, and moreover, P = id N if and only if ϕ = id N . Corollary 3.14.
Suppose that we can choose ǫ in Theorem so that, with the corre-sponding ǫ > , we have the following implications for a pair ( ϕ, σ ) as above: ϕ = id N , Z N σ d λ ∗ ≤ ⇒ ∃ q − ∈ ˚ N ∩ Fix ( ϕ ) , σ ( q − ) < ,ϕ = id N , Z N σ d λ ∗ ≥ ⇒ ∃ q + ∈ ˚ N ∩ Fix ( ϕ ) , σ ( q + ) > . (3.29) Then, Theorem holds taking U := (cid:8) Ω exact two-form on Σ (cid:12)(cid:12) k Ω − d α ∗ k C < ǫ (cid:9) . roof. Let α ′ be a contact form such that d α ′ ∈ U as defined in the statement. If α ′ is Zoll,the conclusion follows from Proposition 1.2. Thus, we assume that α ′ is not Zoll and we wantto prove that ρ sys ( α ′ , h ) < t Σ < ρ dia ( α ′ , h ). We first prove the inequality for the systolic ratio.Suppose by contradiction that ρ sys ( α ′ , h ) ≥ t Σ and let γ ∈ P T ( α ′ , h ) be such that T ( γ ) = T min ( α ′ , h ) , (3.30)where T min ( α ′ , h ) is the minimal period of prime periodic Φ α ′ -orbits in the class h . The orbit γ exists by Theorem 3.13. Thus, the assumption ρ sys ( α ′ , h ) ≥ t Σ impliesVol( α ′ ) − t Σ T ( γ ) ≤ . (3.31)Theorem 3.13 assigns to γ the pair ( ϕ, σ ) with the properties listed therein. In particular, byTheorem 3.13.(iv) and (3.31) above, we have that Z N σ d λ ∗ ≤ . As α ′ is not Zoll, ϕ = id N and we can use the first implication in (3.29) to produce a point q − ∈ ˚ N ∩ Fix ( ϕ ) with σ ( q − ) <
0. By Theorem 3.13.(v), this yields an element γ q − ∈ P T ( α ′ , h )with T ( γ q − ) < T ( γ ), which contradicts (3.30). This proves ρ sys ( α ′ , h ) < t Σ .The inequality with the diastolic ratio is analogously established. Suppose by contradic-tion that ρ dia ( α ′ , h ) ≤ t Σ . We take this time γ ∈ P T ( α ′ , h ) to satisfy T ( γ ) = T max ( α ′ , h ) . (3.32)If the pair ( ϕ, σ ) is associated with γ , the assumption ρ dia ( α ′ , h ) ≤ t Σ implies Z N σ d λ ∗ ≥ . The second implication in (3.29) and Theorem 3.13.(v) yield an orbit γ q + ∈ P T ( α ′ , h ) with T ( γ q + ) > T ( γ ). This contradicts (3.32) and proves ρ dia ( α ′ , h ) > t Σ .In view of the last result, we only need to prove implications (3.29) above to establishTheorem 1.4. This will be done in the next section. For a >
0, we recall the notation for the annuli A = [0 , a ) × S , A ′ = [0 , a/ × S , where S = R / Z . As before if M is a manifold, we write ˚ M for the interior of M .In this section, N will denote a connected oriented compact surface with one boundarycomponent. We fix a collar neighbourhood of the boundary i A : A → N with positivelyoriented coordinates ( r, θ ) ∈ A , where r = 0 corresponds to ∂N . Hence, we have the identifi-cation S ∼ = ∂N and the orientation induced by N on ∂N is given by the one-form − d θ .On N , we consider a one-form λ such that d λ is a positive symplectic two-form on ˚ N and λ A := i ∗ A λ = (cid:0) − k + r (cid:1) d θ, where, by Stokes’ Theorem, k = R N d λ >
0. In particular, d λ A = r d r ∧ d θ vanishes of order 1at r = 0. The pair ( N, λ ) is an instance of an ideal Liouville domain , a notion due to Giroux(see [Gir17]). 25 .1 A neighbourhood theorem
In this subsection, we will develop a version of the Weinstein neighbourhood theorem for thediagonal ∆ N ⊂ (cid:0) N × N, ( − d λ ) ⊕ d λ (cid:1) . More precisely, we will consider the zero section O N ⊂ (cid:0) T ∗ N, d λ can (cid:1) in the standard cotangent bundle of N and look for an exact symplectic map W : N → T ∗ N from a neighbourhood N of ∆ N in N × N , so that W ◦ i ∆ N = i O N , where i ∆ N : N ֒ → N × N, i ∆ N ( q ) = ( q, q ) , i O N : N ֒ → T ∗ N, i O N ( q ) = ( q, N . We start by giving an explicit construction of W on A × A .Let us endow the product A × A with coordinates ( r, θ, R, Θ), so that the diagonal is∆ A := { r = R, θ = Θ } . We make the identification T ∗ A = A × R and let ( ρ, ϑ, p ρ , p ϑ ) bethe corresponding coordinates on T ∗ A . We consider an open neighbourhood Y of ∆ A definedas Y := (cid:8) ( r, θ, R, Θ) ∈ A × A (cid:12)(cid:12) | θ − Θ | < (cid:9) and define the auxiliary sets Y ′ := Y ∩ ( A ′ × A ′ ) , ∂ Y := Y ∩ ( ∂ A × ∂ A ) . We have a well-defined difference function Y → ( − , ) , ( r, θ, R, Θ) θ − Θ . We consider the map W A : Y → T ∗ A given in coordinates by ρ = R,ϑ = θ,p ρ = R ( θ − Θ) ,p ϑ = ( R − r ) , (4.1)so that W A ◦ i ∆ A = i O A . The restriction W ˚ A := W A | ˚ Y : ˚ Y → W A (˚ Y ) is a diffeomorphism withinverse given by r = p ρ − p ϑ ,θ = ϑ,R = ρ, Θ = θ − p ρ ρ . (4.2)We also consider the restriction W A ′ := W A | Y ′ : Y ′ → T ∗ A ′ . Its image has the followingexpression, which will be useful later on: W A ′ ( Y ′ ) = n ( ρ, ϑ, p ρ , p ϑ ) ∈ T ∗ A ′ (cid:12)(cid:12)(cid:12) p ρ ∈ (cid:0) − ρ, ρ (cid:1) , p ϑ ∈ (cid:0) (cid:0) ρ − a (cid:1) , ρ (cid:3)o . (4.3)26inally, let us define the function K A : Y → R , K A ( r, θ, R, Θ) := ( k − R )( θ − Θ) , (4.4)and set K A ′ := K A | Y ′ : Y ′ → R . There holds K A | ∆ A = 0 and W ∗ A λ can = ( − λ A ) ⊕ λ A − d K A . (4.5)Indeed, we have W ∗ A λ can + λ A ⊕ ( − λ A ) = R ( θ − Θ)d R + ( R − r )d θ + ( − k + r )d θ − ( − k + R )dΘ= ( θ − Θ)d( R ) + R d( θ − Θ) − k d( θ − Θ)= ( θ − Θ)d( − k + R ) + ( − k + R )d( θ − Θ)= − d K A . Since, for all q ∈ A , ( λ can ) ( q, = 0, we also deduced ( q,q ) K A = (cid:0) ( − λ A ) ⊕ λ A (cid:1) ( q,q ) . (4.6)Finally, if i ∂ Y : ∂ Y → Y is the natural inclusion, from (4.5) we conclude that i ∗ ∂ Y (cid:0) ( − λ A ) ⊕ λ A (cid:1) = d (cid:0) K A ◦ i ∂ Y (cid:1) (4.7)Indeed, ( W A ◦ i ∂ Y ) ∗ λ can = 0 from the explicit formula for W A given in (4.1) and the fact thatboth r and R vanish on ∂ Y .We can now state the neighbourhood theorem. The proof will be an adaptation of [MS98,Theorem 3.33] (with different sign convention). Proposition 4.1.
There exist an open neighbourhood
N ⊂ N × N of the diagonal ∆ N , amap W : N → T ∗ N , and a function K : N → R with the following properties.(i) The set N contains Y ′ . If we write T := W ( N ) , then ˚ T ⊂ T ∗ ˚ N is an open neighbour-hood of O ˚ N and the restriction W| ˚ N : ˚ N → ˚ T is a diffeomorphism.(ii) W ∗ λ can = ( − λ ) ⊕ λ − d K .(iii) W ◦ i ∆ N = i O N , W| Y ′ = W A ′ .(iv) K ◦ i ∆ N = 0 , K | Y ′ = K A ′ .(v) If ∂ Y ′ := Y ′ ∩ ( ∂N × ∂N ) and i ∂ Y ′ : ∂ Y ′ → N × N is the inclusion, then d( K ◦ i ∂ Y ′ ) = i ∗ ∂ Y ′ (cid:0) ( − λ ) ⊕ λ (cid:1) . Proof.
Let us denote by ( q, p ) the points in T ∗ A ∼ = A × R , where q = ( ρ, ϑ ) and p = ( p ρ , p ϑ ).Let g A and g T ∗ A be the standard metrics on A and T ∗ A : g A := d ρ + d ϑ , g T ∗ A := d ρ + d ϑ + d p ρ + d p ϑ . Then, the metric g T ∗ A is compatible with the canonical symplectic form d λ can . Namely, g T ∗ A = d λ can ( J T ∗ A · , · ) , J T ∗ A : T(T ∗ A ) → T(T ∗ A ) is the standard complex structure given by J T ∗ A ∂ ρ = ∂ p ρ , J T ∗ A ∂ ϑ = ∂ p ϑ , J T ∗ A ∂ p ρ = − ∂ ρ , J T ∗ A ∂ p ϑ = − ∂ ϑ . If ( q, ∈ O A , then we have the horizontal and vertical embeddingsd q i O A : T q A → T ( q, (T ∗ A ) , T ∗ q A → T ( q, (T ∗ A ) , p p ∗ , so that, if ♯ : T ∗ q A → T q A is the metric duality given by g A , there holds p ∗ = J T ∗ A · d q i O A · p ♯ , ∀ p ∈ T ∗ q A . We now combine this formula with the fact that, for every ( q, p ) ∈ T ∗ A , the ray t ( q, tp ), t ∈ [0 ,
1] is a geodesic for g T ∗ A with initial velocity p ∗ . Thus, if exp T ∗ A denotes the exponentialmap of g T ∗ A , we arrive at ( q, p ) = exp T ∗ A i O A ( q ) (cid:16) J T ∗ A · d q i O A · p ♯ (cid:17) . (4.8)We consider the pulled back objects g ˚ Y := W ∗ ˚ A g T ∗ A and J ˚ Y := W ∗ ˚ A J T ∗ A . In particular, W ˚ A is a local isometry between g ˚ Y and g T ∗ A . Moreover, since W ∗ ˚ A (d λ can ) = (( − d λ A ) ⊕ d λ A ) by(4.5), we see that the structure J ˚ Y is compatible with ( − d λ ) ⊕ d λ , since J T ∗ A is compatiblewith d λ can and λ A = i ∗ A λ . Namely, (cid:0) ( − d λ ) ⊕ d λ (cid:1)(cid:12)(cid:12) ˚ Y = g ˚ Y ( J ˚ Y · , · ) . Furthermore, using (4.8), we can compute the pre-image of a point ( q, p ) ∈ W A (˚ Y ) as W − A ( q, p ) = W − A (cid:16) exp T ∗ A i O A ( q ) (cid:0) J T ∗ A · d q i O A · p ♯ (cid:1)(cid:17) = exp ˚ Y i ∆ A ( q ) (cid:16) d i O A ( q ) W − A · J T ∗ A · d q i O A · p ♯ (cid:17) = exp ˚ Y i ∆ A ( q ) (cid:16) J ˚ Y · d i O A ( q ) W − A · d q i O A · p ♯ ) (cid:17) = exp ˚ Y i ∆ A ( q ) (cid:0) J ˚ Y · d q i ∆ A · p ♯ (cid:1) . (4.9)The space of almost complex structures, which are compatible with the symplectic form(( − d λ ) ⊕ d λ ) | ˚ N × ˚ N , is contractible. Therefore, we can find an almost complex structure J on˚ N × ˚ N , which is compatible with (( − d λ ) ⊕ d λ ) | ˚ N × ˚ N and such that J (cid:12)(cid:12) ˚ Y ′ = J ˚ Y (cid:12)(cid:12) ˚ Y ′ . (4.10)We denote by g the corresponding metric on ˚ N × ˚ N , which satisfies g | ˚ Y ′ = g ˚ Y | ˚ Y ′ . (4.11)We write g N := i ∗ ∆ N g for the restricted metric on N . We observe that g N | ˚ A ′ = g A | ˚ A ′ , andtherefore, we denote the metric duality given by g N also by ♯ : T ∗ N → T N . Let us considerthe set T made by all the points ( q, p ) ∈ T ∗ ˚ N with the property that the g -geodesic startingat time 0 from i ∆ N ( q ) with direction J · d q i ∆ N · p ♯ is defined up to time 1. We claim that T is a fibre-wise star-shaped neighbourhood of O ˚ N and it contains W A (˚ Y ′ ) . (4.12)28he second assertion follows from equations (4.9) and (4.10), (4.11). For the first one, wesee from the homogeneity of the geodesic equation that T contains O ˚ N , and it is fibre-wisestar-shaped around O ˚ N . Finally, since T \ W A (˚ Y ′ ) is bounded away from ∂ (T ∗ N ), the set T is a neighbourhood of O ˚ N . We define the mapΥ : T → N × N, Υ( q, p ) := exp g i ∆ N ( q ) (cid:16) J · d q i ∆ N · p ♯ (cid:17) . It satisfies Υ | W A (˚ Y ′ ) = W − A ′ , Υ ◦ i O N = i ∆ N . (4.13)If q ∈ ˚ N , the differential of Υ at i O N ( q ) in the direction u = p ∗ + d q i O N · v ∈ T i O N ( q ) T ∗ N ,where p ∈ T ∗ q N and v ∈ T q N , is given byd i O N ( q ) Υ · u = J · d q i ∆ N · p ♯ + d q i ∆ N · v. If we abbreviate Ω = ( − d λ ) ⊕ d λ , we claim that (Υ ∗ Ω) i O N ( q ) = (d λ can ) i O N ( q ) , for all q ∈ ˚ N .For u , u ∈ T i O N ( q ) T ∗ N , we computeΥ ∗ Ω( u , u ) = Ω (cid:0) J · d q i ∆ N · p ♯ + d q i ∆ N · v , J · d q i ∆ N · p ♯ + d q i ∆ N · v (cid:1) = Ω (cid:0) J · d q i ∆ N · p ♯ , d q i ∆ N · v (cid:1) − Ω (cid:0) J · d q i ∆ N p ♯ , d q i ∆ N · v (cid:1) = g (cid:0) d q i ∆ N · p ♯ , d q i ∆ N · v (cid:1) − g (cid:0) d q i ∆ N · p ♯ , d q i ∆ N · v (cid:1) = ( i ∗ ∆ N g ) (cid:0) p ♯ , v (cid:1) − ( i ∗ ∆ N g ) (cid:0) p ♯ , v (cid:1) = g N (cid:0) p ♯ , v (cid:1) − g N (cid:0) p ♯ , v (cid:1) = p ( v ) − p ( v )= d λ can ( u , u ) , (4.14)where in the second equality we used the fact that ∆ N is Lagrangian and that J is a symplecticendomorphism.We move now the first steps in constructing the function K : N → R . We abbreviateΛ := (Υ − ) ∗ (( − λ ) ⊕ λ ). This is a one-form on T ⊂ T ∗ ˚ N and satisfies i ∗O N Λ = i ∗ ∆ N (( − λ ) ⊕ λ ) = − λ + λ = 0 . We consider any K : T → N satisfying, for all q ∈ ˚ N , K ◦ i O N ( q ) = 0 , d i O N ( q ) K = Λ i O N ( q ) . (4.15)For example, we can set K ( q, p ) := Z Λ ( q,tp ) ( p ∗ )d t. The first property in (4.15) is immediate and it implies thatd i O N ( q ) K · d q i O N · v = 0 = Λ i O N ( q ) (cid:0) d q i O N · v (cid:1) . Thus, we just need to check the second property on vertical tangent vectors p ∗ ∈ T ( q, (T ∗ N ):d i O N ( q ) K · p ∗ = lim s → K ( q, sp ) − K ( q, s = lim s → s Z Λ ( q,tsp ) ( sp ∗ )d t = lim s → Z Λ ( q,tsp ) ( p ∗ )d t = Λ i O N ( q ) ( p ∗ ) .
29t this point, we transfer the attention on N × N . First, we can shrink T in such a waythat (4.12) still holds and that Υ is a diffeomorphism onto its image Υ( T ). We define theopen neighbourhood N of ∆ N by N := Υ( T ) ∪ Y ′ and the map W : N → T ∗ N obtained by gluing: W | Υ( T ) = Υ − , W | Y ′ = W A ′ . (4.16)Such a map is well-defined because of (4.13) and satisfies W ◦ i ∆ N = i O N . Let χ : N → [0 , Y ′ and equal to 1 on N \ Y .We set K N : N → R , K N := χ · ( K ◦ W ) + (1 − χ ) · K A . We readily see that K N ◦ i ∆ N = 0 , K N | Y ′ = K A | Y ′ . (4.17)Furthermore, for all q ∈ ˚ N , there holdsd i ∆ N ( q ) K N = χ · W ∗ (d i O N ( q ) K ) + (1 − χ ) · d i ∆ N ( q ) K A = χ · W ∗ (Λ i O N ( q ) ) + (1 − χ ) · (cid:0) ( − λ ) ⊕ λ (cid:1) i ∆ N ( q ) = χ · (cid:0) ( − λ ) ⊕ λ (cid:1) i ∆ N ( q ) + (1 − χ ) · (cid:0) ( − λ ) ⊕ λ (cid:1) i ∆ N ( q ) , = (cid:0) ( − λ ) ⊕ λ (cid:1) i ∆ N ( q ) , where we used K ◦ W ◦ i ∆ N ( q ) = 0 = K A ◦ i ∆ N ( q ) in the first equality. while the secondequality followed from (4.6) and (4.15). Since ( λ can ) i O N ( q ) = 0, we deduce( W ∗ λ can ) i ∆ N ( q ) = (cid:0) ( − λ ) ⊕ λ (cid:1) i ∆ N ( q ) − d i ∆ N ( q ) K N . (4.18)The rest of the proof follows Moser’s argument. We setΛ t := t (cid:0) W ∗ λ can + d K N (cid:1) + (1 − t ) (cid:0) ( − λ ) ⊕ λ (cid:1) , t ∈ [0 , . By (4.5), (4.16), and (4.17), we haveΛ t = ( − λ ) ⊕ λ on Y ′ . (4.19)Moreover, for all q ∈ ˚ N , by (4.14) and (4.18), we have(dΛ t ) i ∆ N ( q ) = (cid:0) ( − d λ ) ⊕ d λ (cid:1) i ∆ N ( q ) , (Λ t ) i ∆ N ( q ) = (cid:0) ( − λ ) ⊕ λ (cid:1) i ∆ N ( q ) . (4.20)In particular dΛ t is non-degenerate on ∆ ˚ N . Therefore, up to shrinking the neighbourhoodaway from Y ′ , we can assume that dΛ t is non-degenerate on ˚ N . Let X t be a time-dependentvector field and L t a time-dependent function on ˚ N defined by ι X t dΛ t = − dΛ t d t , L t := − Z t Λ t ′ ( X t ′ ) ◦ Φ t ′ d t ′ , t is the flow of X t . By (4.19), we see that X t and L t vanish on ˚ Y ′ and we can extendthem trivially to the whole N . Relations (4.20) imply that X t and L t vanish on ∆ ˚ N . Inparticular, Φ t is the identity map on ∆ N , and up to shrinking the neighbourhood N awayfrom Y ′ , we can suppose that Φ t is defined up to time 1. For all t ∈ [0 ,
1] we havedd t (cid:16) Φ ∗ t Λ t + d L t (cid:17) = Φ ∗ t (cid:16) ι X t dΛ t + d (cid:0) Λ t ( X t ) (cid:1) + dΛ t d t (cid:17) + d (cid:16) d L t d t (cid:17) = 0 . Together with Φ ∗ Λ + d L = Λ , this implies Φ ∗ Λ + d L = Λ . Hence,Φ ∗ W ∗ λ can = ( − λ ) ⊕ λ − d (cid:0) L + K N ◦ Φ (cid:1) , and properties (i) and (ii) in the statement follow with W := W ◦ Φ , K := L + K N ◦ Φ . Properties (iii) and (iv) hold as well, since they are satisfied by W and K N and we haveshown that Φ | ∆ N = id, Φ | Y ′ = id and L | ∆ N = 0, L | Y ′ = 0. Property (v) follows from (iv)and equation (4.7). C -close to the identity Let E denote the set of all exact diffeomorphisms ϕ : N → N , namely ϕ ∗ λ − λ is an exactone-form. We endow from now on E with the uniform C -topology, whose associated distancefunction we denote by dist C . For ǫ >
0, we consider the open ball around id N of radius ǫ E ( ǫ ) := (cid:8) ϕ ∈ E (cid:12)(cid:12) dist C ( ϕ, id N ) < ǫ (cid:9) . If ϕ ∈ E , we write Γ ϕ : N → N × N for its graph Γ ϕ ( q ) = ( q, ϕ ( q )), and we haveΓ ϕ ( ∂N ) ⊂ ∂N × ∂N . There is ǫ ∗ > ϕ ∈ E ( ǫ ∗ ) enjoy the following properties:(a) Γ ϕ ( N ) ⊂ N .(b) If π N : T ∗ N → N is the foot-point projection, then the map ν ϕ : N → N, ν ϕ := π N ◦ W ◦ Γ ϕ is a diffeomorphism. Indeed, ν ϕ is C -close to id N , if the same is true for ϕ . Henceforth,we write ν instead of ν ϕ when the map ϕ is clear from the context.(c) We have the inclusions ϕ (cid:0) A ′′ (cid:1) ⊂ A ′ , ν − ( A ′′ ) ⊂ A ′ , where A ′′ := [0 , a/ × S .If ϕ ∈ E ( ǫ ∗ ), then we can write its restriction to A ′′ as ϕ ( r, θ ) = (cid:0) R ϕ ( r, θ ) , Θ ϕ ( r, θ ) (cid:1) . By (4.1), the restriction of ν to A ′′ reads ν ( r, θ ) = (cid:0) R ϕ ( r, θ ) , θ (cid:1) , (4.21)31hich implies that ν ϕ | ∂N = id ∂N . Let i ∂N : ∂N → N be the inclusion and observe that Γ ϕ ◦ i ∂N takes values in ∂ Y ′ . Therefore,taking the pull-back by Γ ϕ ◦ i ∂N in Proposition 4.1.(v), we getd (cid:0) K ◦ Γ ϕ ◦ i ∂N (cid:1) = i ∗ ∂N (cid:0) ϕ ∗ λ − λ (cid:1) . With this relation we can single out a special primitive of ϕ ∗ λ − λ called the action of ϕ ∈ E ( ǫ ∗ ).It is the unique C -function σ : N → R such that( i ) ϕ ∗ λ − λ = d σ, ( ii ) σ | ∂N = K ◦ Γ ϕ | ∂N . (4.22) Remark 4.2.
We observe that the normalisation of σ at the boundary coincides with theone considered in (3.25) and Theorem 3.13. Indeed, we have the explicit formulas λ = − k d θ on ∂N and K (0 , θ, , Θ) = k ( θ − Θ) on ∂ Y ′ , and for all θ ∈ S ∼ = ∂N , there holds K ◦ Γ ϕ ( θ ) = − k (cid:0) Θ ϕ ( θ ) − θ (cid:1) = − k Z ϕ ( θ ) θ d θ = Z ϕ ( θ ) θ λ. We describe the tangent space of E ( ǫ ∗ ). To this purpose we introduce a space of functions. Definition 4.3.
We write V for the vector space of all smooth functions f : N → R suchthat both f and d f vanish at ∂N . We endow this space with the pre-Banach norm k · k V defined by k f k V := k f k C + k r d f | A k C , ∀ f ∈ V . Choosing the restriction to a smaller annulus in the second term above yields an equivalentnorm on V . For all δ >
0, we denote by V ( δ ) the open ball of radius δ in ( V , k · k V ).Let ϕ denote some element in E ( ǫ ∗ ) with action σ . First, we take any differentiable path t ϕ t with values in E ( ǫ ∗ ) such that ϕ = ϕ , and write t σ t for the corresponding pathof actions with σ = σ . Let X t be the C -vector field on N uniquely defined byd ϕ t d t = X t ◦ ϕ t . (4.23)The associated Hamiltonian function is defined by H t : N → R , H t := d σ t d t ◦ ϕ − t − λ ( X t ) . (4.24)Differentiating ϕ ∗ t λ = λ + d σ t with respect to t , we get ι X t d λ = d H t . From this last equation and the fact that d λ = R d R ∧ dΘ vanishes at ∂N , we see that d H t vanishes at ∂N . Hence, if we write X t = X Rt ∂ R + X Θ t ∂ Θ on the annulus A , we find X Rt = R ∂ Θ H t , X Θ t = − R ∂ R H t . (4.25)We also observe that H t = 0 at the boundary ∂N since d σ t d t = λ ( X t ) ◦ ϕ t there. Indeed, from(4.22) and Proposition 4.1.(v), we compute at ∂N d σ t d t = d Γ ϕt K · (0 ⊕ X t ) = (cid:0) ( − λ ) ⊕ λ (cid:1) (0 ⊕ X t ) (cid:12)(cid:12) Γ ϕt = λ ( X t ) ◦ ϕ t . H t belongs to V and k H t k V is equivalent to k X t k C .Conversely, let H ∈ V and take any path t H t with values in V and such that H = H .We claim that there is a uniquely defined path t X t of C -vector fields with ι X t d λ = d H t .The vector fields are well defined away from ∂N , since d λ is symplectic there. On A , instead,they are well defined because of (4.25). Let t ϕ t be the path of diffeomorphisms obtainedintegrating X t with the condition ϕ = ϕ . Differentiating with respect to t , we getdd t (cid:0) ϕ ∗ t λ (cid:1) = ϕ ∗ t (cid:0) ι X t d λ + d( λ ( X t )) (cid:1) = d (cid:0) ( H t + λ ( X t )) ◦ ϕ t (cid:1) so that all the maps ϕ t are exact with some action σ t . Relation (4.24) is also satisfied since H t and d σ t d t ◦ ϕ − t − λ ( X t ) have the same differential and both vanish at ∂N . We sum up theprevious discussion in a lemma. Lemma 4.4.
There is an isomorphism between the pre-Banach spaces (cid:0) T ϕ E ( ǫ ∗ ) , k · k C (cid:1) −→ ( V , k · k V ) given by the map X H , where X and H are defined in (4.23) and (4.24) . In this subsection, we describe how to build the correspondence between C -small exactdiffeomorphisms and generating functions in our setting. For a classical treatment, we referthe reader to [MS98, Chapter 9]. Let ϕ be an exact diffeomorphism in E ( ǫ ∗ ). There exists aone-form η : N → T ∗ N such that W ◦ Γ ϕ = η ◦ ν. Since λ can has the tautological property η ∗ λ can = η , we have ν ∗ η = ν ∗ η ∗ λ can = Γ ∗ ϕ W ∗ λ can = Γ ∗ ϕ (cid:0) ( − λ ) ⊕ λ − d K (cid:1) = ϕ ∗ λ − λ − d( K ◦ Γ ϕ )= d( σ − K ◦ Γ ϕ ) . (4.26)If we denote the generating function of ϕ ∈ E ( ǫ ∗ ) by G ϕ : N → R , G ϕ := ( σ − K ◦ Γ ϕ ) ◦ ν − ϕ , (4.27)we have the equality W ◦ Γ ϕ = d G ϕ ◦ ν. (4.28)Henceforth, we will simply write G instead of G ϕ when the map ϕ is clear from the context.We write the restriction of ν − to A ′′ as ν − ( ρ, ϑ ) = (cid:0) r ϕ ( ρ, ϑ ) , ϑ (cid:1) , so that, for every θ = ϑ ,the functions R ϕ ( · , θ ) and r ϕ ( · , ϑ ) are inverse of each other. Moreover, since r ϕ (0 , ϑ ) = 0, byTaylor’s theorem with integral remainder, there exists a function s ϕ : A ′′ → R such that r ϕ = ρ (1 + s ϕ ) . By (4.21), (4.28) and (4.1), we have ( ∂ ρ G ( ρ, ϑ ) = ρ (cid:0) ϑ − Θ ϕ ( r ϕ ( ρ, ϑ ) , ϑ ) (cid:1) ,∂ ϑ G ( ρ, ϑ ) = (cid:0) ρ − r ϕ ( ρ, ϑ ) (cid:1) = − ρ (cid:0) s ϕ ( ρ, ϑ ) + s ϕ ( ρ, ϑ ) (cid:1) . (4.29)33 roposition 4.5. If G : N → R is the generating function of ϕ ∈ E ( ǫ ∗ ) , there holds ˚ N ∩ Fix ( ϕ ) = ˚ N ∩ Crit G. Moreover, if z ∈ ˚ N ∩ Fix ( ϕ ) , then ν ( z ) = z and σ ( z ) = G ( z ) .Proof. Let z be a point in ˚ N . We suppose first that ϕ ( z ) = z . Then, Γ ϕ ( z ) ∈ ∆ N , and by (iii)in Proposition 4.1, we have W (Γ ϕ ( z )) = i O N ( z ), which implies that ν ( z ) = z and d z G = 0.Moreover, by (4.27) and Proposition 4.1.(iv), we have G ( z ) = σ ( ν − ( z )) − K (cid:0) Γ ϕ ( ν − ( z )) (cid:1) = σ ( z ) − K ◦ i ∆ N ( z ) = σ ( z ) . Conversely, suppose that z is a critical point G . Then, by (4.28)( z, z ) = i ∆ N ( z ) = W − (d G ( z )) = Γ ϕ ( ν − ( z )) = ( ν − ( z ) , ϕ ( ν − ( z ))) , which implies ν − ( z ) = z , and hence, ϕ ( z ) = z . Lemma 4.6.
The generating function G belongs to V . Moreover, there holds ∂ ρρ G | ∂N = id ∂N − Θ ϕ ◦ i ∂N . In particular, for every z ∈ ∂N , we have ϕ ( z ) = z ⇐⇒ ∂ ρρ G ( z ) = 0 , ⇐⇒ σ ( z ) = 0 . Proof.
The vanishing of G at the boundary follows from (ii) in (4.22). To prove the vanishingof the differential of G at the boundary, we just substitute ρ = 0 in (4.29). Moreover, dividingthe first equation in (4.29) by ρ and taking the limit for ρ going to 0, we obtain the formulafor ∂ ρρ G , which also implies the first equivalence above. The second one follows from (4.4)and (4.22).By the previous lemma we have a map G : E ( ǫ ∗ ) → V , G ( ϕ ) = G ϕ , whose properties we will study. To this aim, we need a definition and two lemmas aboutfunctions on A . Definition 4.7.
Fix a positive integer m . Let us denote by F the space of all smooth functionsˆ f : A → R m and by k · k F the norm on F defined by k ˆ f k F := k ˆ f k C + k r d ˆ f k C , ∀ ˆ f ∈ F . Let F ⊂ F be the subspace of those functions f : A → R m such that f (0 , θ ) = 0, for all θ ∈ S . In this case, there exists a unique ˆ f ∈ F such that f ( r, θ ) = r ˆ f ( r, θ ) , ∀ ( r, θ ) ∈ A . Lemma 4.8.
The following two statements hold.(i) The map ( F , k · k C ) → ( F , k · k F ) , f ˆ f is an isomorphism of pre-Banach spaces. ii) Let U be an open set of R m , and let A : U → R m be a C -function with k A k C < ∞ .If F U is the set of all functions ˆ f ∈ F such that the image of ˆ f is a relatively compactsubset of U , then the following map is continuous: ( F U , k · k F ) → ( F , k · k F ) , ˆ f A ◦ ˆ f . Proof.
By Taylor’s theorem with integral remainder, the function ˆ f is defined asˆ f ( r, θ ) = Z ∂ r f ( ur, θ ) d u. (4.30)Moreover, differentiating the identity f = r ˆ f , we deduce thatd f = r d ˆ f + ˆ f d r. (4.31)We see from (4.30) that the C -norm of ˆ f is controlled by the C -norm of f . Consequently,from (4.31), we conclude that the C -norm of r d ˆ f = d f − ˆ f d r is also controlled by the C -norm of f . On the other hand, we deduce from (4.31) that the C -norm of d f is controlledby the C -norm of r d ˆ f and ˆ f . As f vanishes at r = 0, the C -norm of f is controlled, as well.Finally, we consider a map A : U → R m as in the statement. Let ˆ f ∈ F U be fixed andˆ f ∈ F U such that ˆ f + r ( ˆ f − ˆ f ) ∈ F U , for all r ∈ [0 , f is C -close toˆ f since the images of ˆ f and ˆ f are relatively compact in U , by assumption. Then, we canestimate with the help of the mean value theorem: k A ◦ ˆ f − A ◦ ˆ f k C ≤ k A k C k ˆ f − ˆ f k C ; (cid:13)(cid:13) r d( A ◦ ˆ f − A ◦ ˆ f ) (cid:13)(cid:13) C = (cid:13)(cid:13) ( r d ˆ f A · d ˆ f − r d ˆ f A · d ˆ f ) + ( r d ˆ f A · d ˆ f − r d ˆ f A · d ˆ f ) (cid:13)(cid:13) C ≤ (cid:13)(cid:13) d ˆ f A · r d( ˆ f − ˆ f ) (cid:13)(cid:13) C + (cid:13)(cid:13) (d ˆ f A − d ˆ f A ) · r d ˆ f (cid:13)(cid:13) C ≤ k A k C k r d( ˆ f − ˆ f ) k C + k A k C k ˆ f − ˆ f k C k r d ˆ f k C , from which the continuity of the map ˆ f A ◦ ˆ f at ˆ f follows. Lemma 4.9.
Let f : A → R be a function such that, for all ϑ ∈ S , we have f (0 , ϑ ) = 0 , d (0 ,ϑ ) f = 0 . Then, there exist functions f ρ , f ϑ : A → R such that ∂ ρ f = ρf ρ , ∂ ϑ f = ρ f ϑ . Moreover, there exists a constant
C > (independent of f ) such that C (cid:13)(cid:13)(cid:13)(cid:13) ρ d f (cid:13)(cid:13)(cid:13)(cid:13) C ≤ k f ρ k C + k f ϑ k F ≤ C (cid:13)(cid:13)(cid:13)(cid:13) ρ d f (cid:13)(cid:13)(cid:13)(cid:13) C . Proof.
By Taylor’s theorem with integral remainder, for all ( ρ, ϑ ) ∈ A , we can write f ( ρ, ϑ ) = ρ ˆˆ f ( ρ, ϑ ) , for a function ˆˆ f : A → R , so that f ρ := 2 ˆˆ f + ρ∂ ρ ˆˆ f , f ϑ := ∂ ϑ ˆˆ f yield the desired functions.In order to prove the equivalence of the norms, we observe that ρ d f = f ρ d ρ + ρf ϑ d ϑ . Thus, ρ d f is C -small if and only if f ρ is C -small and ρf ϑ is C -small. The conclusion now followsfrom Lemma 4.8.(i). 35 roposition 4.10. The map G : E ( ǫ ∗ ) → V is continuous from the C -topology to the topologyinduced by k · k V .Proof. Since we can write d G = W ◦ Γ ϕ ◦ ν − , we readily see that the map G is continuousfrom the C -topology to the topology induced by the C -norm. The lemma follows if wecan establish the continuity from the C -topology to the topology induced by the semi-norm k ρ d( · ) | A ′′ k C . If π S : A ′′ → S is the standard projection, then, using equations (4.29), thisamounts to showing that the map ϕ π S − Θ ϕ ◦ ν − ϕ is continuous from the C -topology to the C -topology, and further employing Lemma 4.8.(i),that the map ϕ
7→ − s ϕ − s ϕ is continuous from the C -topology to the k · k F -topology. The former map is continuous since( f , f ) f ◦ f is continuous from the product C -topology into the C -topology and ϕ Θ ϕ , ϕ ν − ϕ = ( r ϕ , π S )are continuous in the C -topology. The latter map is continuous since(a) the map ϕ s ϕ = ρ ( r ϕ − ρ ) is continuous from the C -topology to the k · k F -topologyby Lemma 4.8.(i);(b) the map ˆ f A ◦ ˆ f with A ( x ) = − x − x is continuous from the k · k F -topology to the k · k F -topology by Lemma 4.8.(ii).Putting everything together, we have shown that G is continuous.It is well known that the map W translates the standard Hamiltonian-Jacobi equationfor exact Lagrangian graphs in T ∗ N to the Hamilton-Jacobi equation for C -small exactdiffeomorphisms. Namely, for every differentiable path t ϕ t ⊂ E ( ǫ ∗ ) with ν t := ν ϕ t andgenerated by some t H t , the corresponding path t G t := G ( ϕ t ) ⊂ V has a smoothpointwise derivative t d G t d t ⊂ V which satisfies the Hamilton-Jacobi equation:d G t d t ◦ ν t = H t ◦ ϕ t . (4.32)By continuity it is enough to show (4.32) on the interior ˚ N where dλ is symplectic. We definethe extended Hamiltonian e H t : ˚ N × ˚ N → R by e H t ( q, Q ) := H t ( Q ). It generates ˜ ϕ t := id × ϕ t on˚ N × ˚ N which is Hamiltonian with respect to ( − d λ ) ⊕ d λ . By definition, Γ ϕ t = ˜ ϕ t ◦ i ∆ ˚ N . Thus,if we write ψ t : T → T ∗ ˚ N for the Hamiltonian diffeomorphisms defined on a neighbourhoodof O ˚ N ⊂ T ∗ ˚ N generated by e H t ◦ W − , we get d G t ◦ ν t = ψ t ◦ i O ˚ N by (4.28). Hence, G t solves the classical Hamilton-Jacobi equation with respect to e H t ◦ W − [Arn89, Section 46D],i.e. d G t d t = e H t ◦ W − (d G t ). From the definition of e H t and identity (4.28), we obtain (4.32). Remark 4.11.
If we endow E ( ǫ ∗ ) with the C -topology (instead of the coarser C -topology),then the map G becomes of class C , and for all ϕ ∈ E ( ǫ ∗ ) and H ∈ V ∼ = T ϕ E ( ǫ ∗ ), we canrephrase the equation in the statement of the proposition asd ϕ G · H = H ◦ ( ϕ ◦ ν − ) . roposition 4.12. There are δ ∗ , ǫ ∗∗ > and a continuous map E : V ( δ ∗ ) → E ( ǫ ∗ ) such that(i) we have the inclusion G ( E ( ǫ ∗∗ )) ⊂ V ( δ ∗ ) ;(ii) the map E is the inverse of G , namely, • G (cid:0) E ( G ) (cid:1) = G, ∀ G ∈ V ( δ ∗ ) , • E ( G ( ϕ )) = ϕ, ∀ ϕ ∈ E ( ǫ ∗∗ ) . Proof.
Let δ ∗ be a positive number. We first show that if G ∈ V ( δ ∗ ), then d G takes valuesinto T = W ( N ), provided δ ∗ is small enough. Since T is a neighbourhood of the zero sectionaway from the boundary of N , we see that d G ( N \ A ′′ ) is contained in T if δ ∗ is small.On the other hand, since T ⊃ W ( Y ′ ) from Proposition 4.1.(i), we just need to show thatd G ( A ′′ ) ⊂ W ( Y ′ ) ∩ (T ∗ A ′′ ). Recall from (4.3) the description W ( Y ′ ) ∩ (T ∗ A ′′ ) = n ( ρ, ϑ, p ρ , p ϑ ) ∈ T ∗ A ′′ (cid:12)(cid:12)(cid:12) p ρ ∈ (cid:0) − ρ, ρ (cid:1) , p ϑ ∈ (cid:0) (cid:0) ρ − a (cid:1) , ρ (cid:3)) , so that the implication 0 ≤ ρ < a = ⇒ (cid:0) ρ − a (cid:1) > − ρ , yields the implication( ρ, ϑ, p ρ , p ϑ ) ∈ W ( Y ′ ) ∩ (T ∗ A ′′ ) = ⇒ p ϑ ∈ ( − ρ , ρ ] . By Lemma 4.9, we have the expressions ∂ ρ G = ρG ρ and ∂ ϑ G = ρ G ϑ . Therefore, in order tohave d G ( A ′′ ) ⊂ W ( Y ′ ) ∩ (T ∗ A ′′ ), we just need k G ρ k C ( A ′′ ) < and k G ϑ k C ( A ′′ ) < , whichare true if δ ∗ is small, thanks to the inequality in Lemma 4.9 and the definition of k · k V .Since d G ( ˚ N ) ⊂ ˚ T , we can consider the map˚ µ : ˚ N → ˚ N , ˚ µ := π ◦ W − ◦ d G | ˚ N , (4.33)where π : N × N → N is the projection on the first factor. On the annulus A ′′ , we consider,furthermore, the map µ A ′′ : A ′′ → A ′ , µ A ′′ ( ρ, ϑ ) = (cid:0) ρ p − G ϑ ( ρ, ϑ ) , ϑ (cid:1) . Thanks to (4.2), ˚ µ and µ A ′′ glue together and yield a map µ G : N → N . We claim that G µ G is continuous from the topology induced by k · k V to the C -topology. We argueseparately for ˚ µ | N \ A ′′ and µ A ′′ . For the former map, the continuity is clear from the expression(4.33) and the fact that k G k C ≤ k G k V . For the latter map, the continuity is clear in thesecond factor, and we only have to deal with the continuity of G ρ √ − G ϑ . By Lemma4.8, this happens if and only if G
7→ √ − G ϑ is continuous from the k · k V -topology tothe k · k F -topology. The latter map is the composition of G G ϑ with f A ◦ f , where A : ( − , + ) → (0 , ∞ ) is defined by A ( x ) = √ − x . The map G G ϕ is continuous fromthe k · k V -topology to the k · k F -topology by Lemma 4.9. The map f A ◦ f is continuousfrom the k · k F -topology to the k · k F -topology by Lemma 4.8.(ii). The claim is established.Thus, taking δ ∗ small enough, we can assume that µ G : N → N is so C -close to theidentity that is a diffeomorphism and we write ν G : N → N for its inverse, which satisfies ν G ( r, θ ) = (cid:0) R G ( r, θ ) , θ (cid:1) , ∀ ( r, θ ) ∈ A ′′ , R G : A ′′ → [0 , a/ G ν G is continuous in the C -topology.We now construct a diffeomorphism ϕ G : N → N . Let π : N × N → N be the projectionon the second factor and set˚ ϕ : ˚ N → ˚ N , ˚ ϕ := π ◦ W − ◦ d G ◦ ν G | ˚ N . (4.34)On the annulus A ′′ , we set ϕ A ′′ : A ′′ → A ′ , ϕ A ′′ ( r, θ ) = (cid:0) R G ( r, θ ) , θ − G ρ ( R G ( r, θ ) , θ ) (cid:1) . Thanks to (4.2), the maps ˚ ϕ and ϕ A ′′ glue together to yield ϕ G : N → N . We claim that ϕ is exact. Indeed, from (4.33) and (4.34), we get W ◦ Γ ˚ ϕ = d G ◦ ˚ ν . Since ν G and ϕ G arecontinuous up to the boundary, we deduce W ◦ Γ ϕ G = d G ◦ ν G . (4.35)Repeating the computation as in (4.26), it follows that ϕ G is exact with action σ ϕ G := G ◦ ν G + K ◦ Γ ϕ G . (4.36)Therefore, we have a map E : V ( δ ∗ ) → E defined by E ( G ) = ϕ G . We claim that this mapis continuous. As before, we argue separately for ˚ ϕ | N \ A ′′ and ϕ A ′′ . For the former map, thecontinuity follows since we have a control on the C -norm of G . For the latter map, it followsfrom the continuity of G R G from the k · k V -topology to the C -topology, which we havealready established, the continuity of G G ρ from the k · k V -topology to the C -topology,which follows from Lemma 4.9, and the continuity of ( f , f ) f ◦ f from the product C -topology into the C -topology. The claim is established. In particular, up to shrinking δ ∗ , we can assume that E ( V ( δ ∗ )) ⊂ E ( ǫ ∗ ). On the other hand, the existence of ǫ ∗∗ > G ( E ( ǫ ∗∗ )) ⊂ V ( δ ∗ ) is a consequence of the continuity of G .Next, we verify that G ( ϕ G ) = G . First, recalling that π N : T ∗ N → N , we see that ν ϕ G (4.28) = π N ◦ ( W ◦ Γ ϕ G ) (4.35) = π N ◦ (d G ◦ ν G ) = ( π N ◦ d G ) ◦ ν G = ν G . Therefore, comparing (4.36) with (4.27), we get G ( ϕ G ) = G ϕ G = G .Finally, let ϕ ∈ E ( ǫ ∗∗ ). We show that ϕ = E ( G ϕ ). First, we get ν − ϕ | ˚ N (4.28) = π ◦ W − ◦ d G ϕ | ˚ N (4.35) = π ◦ Γ ϕ Gϕ ◦ ν − G ϕ | ˚ N = ν − G ϕ | ˚ N . By continuity, this implies ν ϕ = ν G ϕ , and we arrive at ϕ | ˚ N (4.28) = π ◦ W − ◦ d G ϕ ◦ ν ϕ | ˚ N = π ◦ W − ◦ d G ϕ ◦ ν G ϕ | ˚ N (4.35) = π ◦ Γ ϕ Gϕ | ˚ N = ϕ G ϕ | ˚ N . By continuity again, ϕ = ϕ G ϕ = E ( G ϕ ) as required, and the proof is completed. In this subsection, we complete the proof of Theorem 1.4 using arguments inspired by[ABHS18, Remark 2.8]. We begin with the following well-known lemma whose proof canbe found in [MS98, Lemma 10.27] and [ABHS18, Proposition 2.6 & 2.7].38 emma 4.13.
Let ϕ ∈ E ( ǫ ∗ ) be an exact diffeomorphism and let σ : N → R denote its action.Suppose that there exists a differentiable path t ϕ t in E ( ǫ ∗ ) with ϕ = id N and ϕ = ϕ .We write by t H t ∈ V the Hamiltonian associated with the path. There holds σ ( q ) = Z h H t + λ ( X t ) i ( ϕ t ( q )) d t = Z (cid:0) t ϕ t ( q ) (cid:1) ∗ λ + Z H t ( ϕ t ( q )) d t, ∀ q ∈ N. As a consequence, we have Z N σ d λ = 2 Z (cid:16) Z N H t d λ (cid:17) d t. We recall that, according to [BP94], a Hamiltonian path t H t ∈ V , parametrised in someinterval I , is called quasi-autonomous if there exist a minimiser q min ∈ N and a maximiser q max ∈ N independent of time, i.e.min N H t = H t ( q min ) , max N H t = H t ( q max ) , ∀ t ∈ I. A diffeomorphism ϕ ∈ E ( ǫ ∗ ) is called quasi-autonomous, if there exists a differentiable path t ϕ t ∈ E ( ǫ ∗ ) parametrised in [0 ,
1] with ϕ = id N , ϕ = ϕ , whose associated Hamiltonian t H t ∈ V is quasi-autonomous. Lemma 4.14.
Let ϕ ∈ E ( ǫ ∗ ) be quasi-autonomous with associated Hamiltonian t H t . Thefollowing implications hold: ∃ t − ∈ [0 , , H t − ( q min ) < , = ⇒ q min ∈ Fix ( ϕ ) ∩ ˚ N , σ ( q min ) < , ∃ t + ∈ [0 , , H t + ( q max ) < , = ⇒ q max ∈ Fix ( ϕ ) ∩ ˚ N , σ ( q max ) < . Proof.
We show only the first implication. Since H t − ( q min ) < H t − | ∂N = 0, we deducethat q min ∈ ˚ N . Moreover, since q min minimises H t for all t ∈ [0 , q min H t = 0.Since d λ is symplectic on ˚ N , by ι X t d λ = d H t , we conclude that X t ( q min ) = 0, which impliesthat ϕ t ( q min ) = q min . We estimate the action of q min using Lemma 4.13 and rememberingthat, for all t ∈ [0 , H t ( q min ) ≤
0, since H t vanishes on the boundary: σ ( q min ) = Z (cid:2) H t + λ ( X t ) (cid:3) ( ϕ t ( q min ))d t = Z H t ( q min )d t < . Proposition 4.15.
Every ϕ ∈ E ( ǫ ∗∗ ) is quasi-autonomous.Proof. By Proposition 4.12, the generating function G of ϕ belongs to V ( δ ∗ ). Thus, for all t ∈ [0 , tG belongs to V ( δ ∗ ), and again by Proposition 4.12, we can considerthe path t ϕ t := E ( tG ) ∈ E ( ǫ ∗ ). Let t H t be the associated Hamiltonian. By (4.32), wededuce G = dd t ( tG ) = H t ◦ ( ϕ t ◦ ν − t ) , ∀ t ∈ [0 , , (4.37)which implies min H t = min G, max H t = max G, ∀ t ∈ [0 , . (4.38)Let q min and q max be the minimiser and the maximiser of G , respectively. We claim that G ( q min ) = H t ( q min ) , G ( q max ) = H t ( q max ) , ∀ t ∈ [0 , . (4.39)39e give only the argument for q min . If q min ∈ ∂N , we have G ( q min ) = 0 = H t ( q min ), as G and H t belong to V . If q min ∈ ˚ N , then q min ∈ Crit G . We deduce that ϕ t ( q min ) = q min = ν t ( q min ),as ϕ t and ν t act as the identity on ˚ N ∩ Crit ( tG ) ⊃ ˚ N ∩ Crit G by Proposition 4.5. Theequality G ( q min ) = H t ( q min ) follows then from (4.37). Now that the claim is established,relations (4.38) and (4.39) imply that t H t is quasi-autonomous.We are now ready to prove implications (3.29) in Corollary 3.14, which are the last missingpiece to establish the Main Theorem 1.4. Corollary 4.16.
Let ϕ ∈ E ( ǫ ∗∗ ) be an exact diffeomorphism with action σ : N → R . If ϕ = id N , the following implications hold: • Z N σ d λ ≤ ⇒ ∃ q − ∈ Fix ( ϕ ) ∩ ˚ N with σ ( q − ) < , • Z N σ d λ ≥ ⇒ ∃ q + ∈ Fix ( ϕ ) ∩ ˚ N with σ ( q + ) < . Proof.
The implications follow with q − = q min , q + = q max . We show only the former, thelatter being analogous. Suppose that the integral of σ is non-positive. By Proposition 4.15, ϕ is quasi-autonomous, namely, there exists a quasi-autonomous t H t generating t ϕ t with ϕ = id N and ϕ = ϕ . By Lemma 4.14, the corollary is established, if we show thatthere exists t − ∈ [0 ,
1] such that H t − ( q min ) <
0. Indeed, assume by contradiction that H t ( q min ) ≥
0, for all t ∈ [0 , H t ≥
0. Furthermore, as ϕ = id N , thereexists ( s, w ) ∈ [0 , × N with H s ( w ) >
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E-mail address : [email protected] Seoul National University, Department of Mathematical Sciences, Research institute in Math-ematics, Gwanak-Gu, Seoul 08826, South Korea
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